A. A. Salama

In this article, the main definitions and differentiation concepts of neutrosophic fuzzy environment will be reviewed. This article will introduce an analytical methodology for solving the second-order linear ordinary differential problem with neutrosophic fuzzy boundary values, this analysis will be under generalized Hukuhara differentiability to show the analytical solutions from a different point of view for the uncertain system, some of these solutions may be decreasing in uncertainty or maybe reflecting the behavior of some real-world systems better. Some applications and numeral examples will be solved to show the behavior of the solution.

In this paper, we present an expansion of one of the well-known classical inventory management models, which is the static model of inventory management without a deficit and for a single substance, based on the neutrosophic logic, where we provide through this study a basis for dealing with all data, whether specific or undefined in the field of inventory management, as it provides safe environment to manage inventory without running into deficit , and give us an approximate ideal volume of inventory. Since the ideal size is affected by the rate of demand for inventory, we present in this paper a study of the rate of demand for inventory when it is not precisely refined, and we find that the indeterminate values of the demand rate cannot be ignored, because they actually affect determining the ideal size of inventory and calculating its costs, thus affecting on the efficiency of the facility and achieve great profits for it.

In recent decades, Information and Communication Technology (ICT) has been advanced and widely spread around the globe in addition to ICT revolution and technological advances are considered the major role in the evolution of modern age, which is called "Digital Transformation Age". Therefore, Electronic Technology (E-Technology) has become one of the most prominent approaches such as Electronic Learning (E-Learning), Electronic Training (E-Training), Mobile Learning (M-Learning), Virtual Lab (V-Lab), Virtual University, etc. E-Technology includes some features, for instance anyone, anywhere, anytime and reducing of geographical barriers. ETechnology is a great trend and influences many fields and sectors such as Learning, Training, Military, Navy, Aviation, Medicine, and Digital space. According to nature of usage, E-Technology can be used in positive or negative trends. E-Technology is considered as a valuable tool in providing several opportunities for learning and training processes for individuals and organizations, especially in critical issues. Finally, we give a quick overview of neutrosophic data and some recent applications.

Neutrosophy has been introduced by Smarandache [3, 4] as a new branch of philosophy. Salama et al.[5] in 2012 introduced and studied the operations on neutrosophic sets. The purpose of this paper is to introduce a new type of data called the neutrosophic data. After given the fundamental definitions of neutrosophic set operations due to Salama [5], we obtain several properties, and discussed the relationship between neutrosophic sets and others. Finally, we discuss and derived a formula for correlation coefficient, defined on the domain of neutrosophic sets.

In this paper we introduce definitions of generalized neutrosophic sets. After given the fundamental definitions of generalized neutrosophic set operations, we obtain several properties, and discussed the relationship between generalized neutrosophic sets and others. Finally, we extend the concepts of neutrosophic topological space [9], intuitionistic fuzzy topological space [5, 6], and fuzzy topological space [4] to the case of generalized neutrosophic sets. Possible application to GIS topology rules are touched upon.

Neutrosophy has been introduced by Smarandache [7, 8] as a new branch of philosophy. The purpose of this paper is to construct a new set theory called the neutrosophic set. After given the fundamental definitions of neutrosophic set operations, we obtain several properties, and discussed the relationship between neutrosophic sets and others. Finally, we extend the concepts of fuzzy topological space [4], and intuitionistic fuzzy topological space [5, 6] to the case of neutrosophic sets. Possible application to superstrings and space–time are touched upon.

Fuzzy ideals, fuzzy local function and the notion of compatibility of fuzzy ideals with fuzzy topologies were introduced and studied by D. Sarker (Fuzzy Sets and Systems 87 (1997) 117). The purpose of this paper deals with new sort of fuzzy local function namely fuzzy α-local function. Many of its characterizations, properties and connections between it and other corresponding fuzzy notions are studied. Also we introduce and study two types of fuzzy compatibility namely fuzzy α-compatibility and fuzzy weak α-compatibility of fuzzy topologies via fuzzy ideals. Possible application to superstrings and space-time are touched upon.

The aim of this paper is to introduce and study some new fuzzy pairwise notion in fuzzy bitopological ideals spaces. We also generalize the notion of fuzzy L-open sets due to Abd El-Monsef et al [1]. In addition to generalize the concept of fuzzy L-closed sets, fuzzy L-continuity and L-open functions due to Abd El-Monsef et al [1]. Relationships between the above new fuzzy pairwise notions and there other relevant classes are investigated. Recently, we define and study two different types of fuzzy pairwise functions.

In this paper we introduce the notion of generalized intuitionistic fuzzy ideals which is considered as a generalization of fuzzy intuitionistic ideals studies in [6], the important generalized intuitionistic fuzzy ideals has been given. The concept of generalized intuitionistic fuzzy local function is also introduced for a generalized intuitionistic fuzzy topological space. These concepts are discussed with a view to find new generalized intuitionistic fuzzy topology from the original one in [5, 7]. The basic structure, especially a basis for such generated generalized intuitionistic fuzzy topologies and several relations between different generalized intuitionistic fuzzy ideals and generalized intuitionistic fuzzy topologies are also studied here.

This paper presents an approach for underdetermined blind source separation in the case of additive Gaussian white noise and pink noise. Likewise, the proposed approach is applicable in the case of separating I + 3 sources from I mixtures with additive two kinds of noises. This situation is more challenging and suitable to practical real world problems. Moreover, unlike to some conventional approaches, the sparsity conditions are not imposed. Firstly, the mixing matrix is estimated based on an algorithm that combines short time Fourier transform and rough-fuzzy clustering. Then, the mixed signals are normalized and the source signals are recovered using modified Gradient descent Local Hierarchical Alternating Least Squares Algorithm exploiting the mixing matrix obtained from the previous step as an input and initialized by multiplicative algorithm for matrix factorization based on alpha divergence. The experiments and simulation results show that the proposed approach can separate I + 3 source signals from I mixed signals, and it has superior evaluation performance compared to some conventional approaches.

This paper presents an approach for underdetermined blind source separation that can be applied even if the number of sources is unknown. Moreover, the proposed approach is applicable in the case of separating I+3 sources from I mixtures without additive noise. This situation is more challenging and suitable to practical real world problems. Also, the sparsity conditions are not imposed unlike to those employed by some conventional approaches. Firstly, the number of source signals are estimated followed by the estimation of the mixing matrix based on the use of short time Fourier transform and rough-fuzzy clustering. Then, source signals are normalized and recovered using modified Lin's projected gradient algorithm with modified Armijo rule. The simulation results show that the proposed approach can separate I+3 source signals from I mixed signals, and it has superior evaluation performance compared to conventional approaches.

A Neutrosophic crisp set is a new type of crisp set, and the concept of Neutrosophic crisp set appeared for the first time in 2013, where I. Hanfya and A. Salama studied the Neutrosophic crisp set in a special case in order to study Neutrosophic crisp events and their possibilities [3]. In the context of studying this type of sets, A. Salama and S. Broumi distinguished three types of Neutrosophic crisp set in its special case, in addition to defining the Neutrosophic crisp set in its general case [9]. The aim of this paper is to study Neutrosophic crisp set in its general case to find families of Neutrosophic crisp sets each of which preserves the properties of classical algebra of crisp sets by reformulating some of the basic concepts associated with classical Neutrosophic crisp sets.

Cardiotocography data uncertainty is a critical task for the classification in biomedical field. Constructing good and efficient classifier via machine learning algorithms is necessary to help doctors in diagnosing the state of fetus heart rate. *e proposed neutrosophic diagnostic system is an Interval Neutrosophic Rough Neural Network framework based on the backpropagation algorithm. It benefits from the advantages of neutrosophic set theory not only to improve the performance of rough neural networks but also to achieve a better performance than the other algorithms. *e experimental results visualize the data using the boxplot for better understanding of attribute distribution. *e performance measurement of the confusion matrix for the proposed framework is 95.1, 94.95, 95.2, and 95.1 concerning accuracy rate, precision, recall, and F1-score, respectively. WEKA application is used to analyse cardiotocography data performance measurement of different algorithms, e.g., neural network, decision table, the nearest neighbor, and rough neural network. *e comparison with other algorithms shows that the proposed framework is both feasible and efficient classifier. Additionally, the receiver operation characteristic curve displays the proposed framework classifications of the pathologic, normal, and suspicious states by 0.93, 0.90, and 0.85 areas that are considered high and acceptable under the curve, respectively. Improving the performance measurements of the proposed framework by removing ineffective attributes via feature selection would be suitable advancement in the future. Moreover, the proposed framework can also be used in various real-life problems such as classification of coronavirus, social media, and satellite image.

This paper is devoted for presenting new neutrosophic similarity measures between neutrosophic graphs. We propose two ways to determine the neutrosophic distance between neutrosophic vertex graphs. The two neutrosophic distances are based on the Haussdorff distance, and a robust modified variant of the Haussdorff distance, moreover we show that they both satisfy the metric distance measure axioms. Furthermore, a similarity measure between neutrosophic edge graphs, that is based on a probabilistic variant of Haussdorff distance, is introduced. The aim is to use those measures for the purpose of matching neutrosophic graphs whose structure can be described in the neutrosophic domain.

In this paper we introduce the notion of intuitionistic fuzzy ideals which is considered as a generalization of fuzzy ideals studies in [1, 2, 3, 11], the important intuitionistic fuzzy ideal has been given. The concept of intuitionistic fuzzy local function is also introduced here by utilizing theεneighborhood structure for an intuitionistic fuzzy topological space. These concepts are discussed with a view to find new intuitionistic fuzzy topology from the original one in [10, 12]. The basic structure, especially a basis for such generated intuitionistic fuzzy topologies and several relations between different intuitionistic fuzzy ideals and intuitionistic fuzzy topologies are also studied here. Finally, several properties of all investigated new notions are discussed.

In this paper we introduce a new type of classical set called the neutrosophic classical set. After given the fundamental definitions of neutrosophic classical set operations, we obtain several properties, and discussed the relationship between neutrosophic classical sets and others. Finally, we generalize the classical probability to the notion of neutrosophic probability. This kind of probability is necessary because it provides a better representation than classical probability to uncertain events. Possible applications to computer sciences are touched upon.

: Neutrosophy has been introduced by Smarandache [3, 4] as a new branch of philosophy. Salama et al. [5] in 2012 introduced and studied the operations on neutrosophic sets. The purpose of this paper is to introduce a new type of data called the neutrosophic data. After given the fundamental definitions of neutrosophic set operations due to Salama [5], we obtain several properties, and discussed the relationship between neutrosophic sets and others. Finally, we discuss and derived a fomula for correlation coefficient, defined on the domain of neutrosophic sets

In this paper we introduce definitions of generalized neutrosophic sets. After given the fundamental definitions of generalized neutrosophic set operations, we obtain several properties, and discussed the relationship between generalized neutrosophic sets and others. Finally, we extend the concepts of neutrosophic topological space [9], intuitionistic fuzzy topological space [5, 6], and fuzzy topological space [4] to the case of generalized neutrosophic sets. Possible application to GIS topology rules are touched upon.

Since the world is full of indeterminacy, the neutrosophics found their place into contemporary research. The fundamental concepts of neutrosophic set, introduced by Smarandache in [30, 31, 32] and Salama et al. in [4-29]. In Geographical information systems (GIS) there is a need to model spatial regions with indeterminate boundary and under indeterminacy. In this paper the structure of some classes of neutrosoph-ic crisp nearly open sets are investigated and some applications are given. Finally we generalize the crisp topological and intuitioistic studies to the notion of neutrosophic crisp set. Possible applications to GIS topological rules are touched upon.

Since the world is full of indeterminacy, the Neutrosophics found their place into contemporary research. We now introduce for the first time the notions of Neutrosophic Crisp Sets and Neutrosophic Topology on Crisp Sets. We develop the 2012 notion of Neutrosophic Topological Spaces and give many practical examples. Neutrosophic Science means development and applications of Neutrosophic Logic, Set, Measure, Integral, Probability etc., and their applications in any field. It is possible to define the neutrosophic measure and consequently the neutrosophic integral and neutrosophic probability in many ways, because there are various types of indeterminacies, depending on the problem we need to solve. Indeterminacy is different from randomness. Indeterminacy can be caused by physical space, materials and type of construction, by items involved in the space, or by other factors. In 1965 [51], Zadeh generalized the concept of crisp set by introducing the concept of fuzzy set, corresponding to the situation in which there is no precisely defined set;there are increasing applications in various fields, including probability, artificial intelligence, control systems, biology and economics. Thus, developments in abstract mathematics using the idea of fuzzy sets possess sound footing. In accordance, fuzzy topological spaces were introduced by Chang [12] and Lowen [33]. After the development of fuzzy sets, much attention has been paid to the generalization of basic concepts of classical topology to fuzzy sets and accordingly developing a theory of fuzzy topology [1-58]. In 1983, the intuitionistic fuzzy set was introduced by K. Atanassov [55, 56, 57] as a generalization of the fuzzy set, beyond the degree of membership and the degree of non-membership of each element. In 1999 and 2002, Smarandache [71, 72, 73, 74] defined the notion of Neutrosophic Sets, which is a generalization of Zadeh’s fuzzy set and Atanassov's intuitionistic fuzzy set. Some neutrosophic concepts have been investigated by Salama et al. [61-70]. Forwarding the study of neutrosophic sets, this book consists of seven chapters, targeting to:  generalize the previous studies in [1-59], and[91-94] so to define the neutrosopic crisp set and neutrosophic set concepts;  discuss their main properties; A. A. Salama & Florentin Smarandache  introduce and study some concepts of neutrosophic crisp and neutrosophic topological spaces and deduce their properties;  deduce many types of functions and give the relationships between different neutrosophic topological spaces, which helps to build new properties of neutrosophic topological spaces;  stress once more the importance of Neutrosophic Ideal as a nontrivial extension of neutrosophic set and neutrosophic logic [71, 72, 73, 74];  propose applications on computer sciences by using neutrosophic sets.

The aim of this paper is devoted to introduce and study the concepts of semi-compact (resp. semi-Lindelӧf, locally semi-compact) spaces in a neutrosophic crisp topological space. Several properties, functions properties of neutrosophic crisp semi-compact spaces are studied. In addition to these, we introduce and study the definition of neutrosophic crisp semi-Lindelӧf spaces and neutrosophic crisp locally semi-compact spaces. We show that neutrosophic crisp semi-compact spaces is preserved under neutrosophic crisp irresolute function and neutrosophic crisp pre-semi-closed function with neutrosophic crisp semi-compact point inverses.

Neutrosophy has been introduced by Smarandache [3, 4] as a new branch of philosophy. Salama et al.[5] in 2012 introduced and studied the operations on neutrosophic sets. The purpose of this paper is to introduce a new type of data called the neutrosophic data. After given the fundamental definitions of neutrosophic set operations due to Salama [5], we obtain several properties, and discussed the relationship between neutrosophic sets and others. Finally, we discuss and derived a fomula for correlation coefficient, defined on the domain of neutrosophic sets.

In this paper we introduce definitions of generalized neutrosophic sets. After given the fundamental definitions of generalized neutrosophic set operations, we obtain several properties, and discussed the relationship between generalized neutrosophic sets and others. Finally, we extend the concepts of neutrosophic topological space [9], intuitionistic fuzzy topological space [5, 6], and fuzzy topological space [4] to the case of generalized neutrosophic sets. Possible application to GIS topology rules are touched upon.