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Big data analytics is one of the most promising areas of new research and development in computer science, enterprises, e-commerce, and defense. For many organizations, big data is regarded as one of their most important strategic assets. This explosive growth has made it necessary to develop effective techniques for examining and analyzing big data from a mathematical perspective. Among various methods of analyzing big data, topological data analysis (TDA) is now considered one of the useful tools. However, there is no fundamental concept related to topological structure in big data. In this paper, we introduce some fundamental ideas related to the neighborhood structure of words in data searching, which can be extended to form important topological structures of big data in the future. Additionally, we introduce big data primal in big data searching and discuss the application of neighborhood structures in detecting anomalies in data searching using the Jaccard similarity coefficient.

In 1999, Molodtsov \cite{1} developed the idea of soft set theory, proving it to be a flexible mathematical tool for dealing with uncertainty. Several researchers have extended the framework by combining it with other theories of uncertainty, such as fuzzy set theory, intuitionistic fuzzy soft set theory, rough soft set theory, and so on. These enhancements aim to increase the applicability and expressiveness of soft set theory, making it a more robust tool for dealing with complex, real-world problems characterized by uncertainty and vagueness. The notion of fuzzy soft sets and their associated operations were introduced by Maji et al. \cite{7}. However, Molodtsov \cite{3} identified numerous incorrect results and notions of soft set theory that were introduced in the paper \cite{7}. Therefore, the derived concept of fuzzy soft sets is equally incorrect since the basic idea of soft sets in \cite{7} is flawed. Consequently, it is essential to address these incorrect notions and provide an exact and formal definition of the idea of fuzzy soft sets. This reevaluation is important to guarantee fuzzy soft set theory's theoretical stability and practical application across a range of domains. In this paper, we propose fuzzy soft set theory based on Molodtsov's correct notion of soft set theory and demonstrate a fuzzy soft set in matrix form. Additionally, we derive several significant findings on fuzzy soft sets.

In 1998, Fields medalist Stephen Smale [S. Smale, Mathematical problems for the next century, The mathematical Intelligencer, 20(2) (1998), 7-15] proposed his famous eighteen problems to the mathematicians of this century. The statement of his eighteenth problem is very simple but very important. He asked "What are the limits of intelligence, both artificial and human?". In this paper, we prove that human intelligence is limitless. Moreover, we provide justifications to state that artificial intelligence has limitations. Thus, human intelligence will always remain superior to artificial intelligence. Moreover, we provide justifications to conclude the limitations of artificial intelligence.

In ecology, foraging requires animals to expend energy in order to obtain resources. The cost of foraging can be reduced through kleptoparasitism, the theft of a resource that another individual has expended effort to acquire. Thus, kleptoparasitism is one of the most significant feeding techniques in ecology. In this study, we investigate a two predator one prey paradigm in which one predator acts as a kleptoparasite and the other as a host. This research considers the post-kleptoparasitism scenario, which has received little attention in the literature. Parametric requirements for the existence as well as local and global stability of biologically viable equilibria have been proposed. The occurrences of various one parametric bifurcations, such as saddle-node bifurcation, transcritical bifurcation, and Hopf bifurcation, as well as two parametric bifurcations, such as Bautin bifurcation, are explored in depth. Relatively low growth rate of first predator induces a subcritical Hopf bifurcation although a supercritical Hopf bifurcation occurs at relatively high growth rate of first predator making coexistence of all three species possible. Some numerical simulations have been provided for the purpose of verifying our theoretical conclusions.

The main purpose of this paper is to introduce and study the primal-proximity spaces. Also, we define two new operators via primal proximity spaces and investigate some of their fundamental properties. In addition, we obtain a new topology, which is weaker than old one, via these new operators. Moreover, we not only discuss some of their properties but also enrich with some examples., Comment: Submitted to journal on 23.02.2023

The predator-prey dynamic appertaining to two species is explored, wherein the predator species is structured into different stages. As evidenced from natural documentation, the immature predators possess the potential to predate albeit not as competently as the adults. Nevertheless, this potentiality is not acquired immediately after their incipience of life, hence, the immature stage is branched off into the infant stage, the stage with extensive reliance on the adults, and the juvenile stage, the stage with the potential to predate but not to procreate. In this paper, this inaugural concept is coupled with injuries in the juvenile stage as the repercussion of their incompetency in predating, thereby ensuing a delay in their maturation. With the incentive to investigate the ascendancy of these refinements over the whole system, stability analyses along with various bifurcation analyses around the equilibrium points of the system are corroborated. In addition to Hopf, transcritical, and saddle node bifurcations, the existence of Bogdanov-Takens point, cusp point, Bautin bifurcation point, bloom phenomenon, twice occurring Hopf bifurcation, and bi-stability phenomenon make the paper appreciably more rich and efficacious.

Homotopy deals with the intuitive idea of continuous deformation of a continuous map between two topological spaces. In this paper, we introduce homotopical selection principles in bitopological dynamical systems. Here, we define $BTDS-$homotopy, $BTDS-$iteration homotopy and $BTDS-$path homotopy, and then using these notions we define various selection properties viz. H-Rothberger property, H-Menger property, PH-Rothberger property, PH-Menger property and their weaker versions. We also discuss several results connecting these concepts in bitopological dynamical systems.

Recently, Acharjee et al. [S. Acharjee, M. \"Ozko\c{c} and F. Y. Issaka, Primal topological spaces, arXiv:2209.12676[math.GM]] introduced a new structure in topology named primal. Primal is the duel structure of grill. The main purpose of this paper is to introduce an operator using primal and obtain some of its fundamental properties. Also, we define the notion of topology suitable for a primal. We not only obtain some characterizations of this new notion, but also investigate many properties.

The purpose of this paper is to introduce a new structure `primal'. Primal is dual to grill. Like ideal, dual of filter, this new structure also generates a new topology named `primal topology'. We introduce a new operator using primal, which satisfies Kuratowski's closure axioms. Mainly, we prove that primal topology is finer than the topology of a primal topological space. We provide structure of base of primal topology and prove other fundamental results related to this new structure.

Recently, Zielinski and Sokal [ Zielinski, P., Sokal, P. (2016). Full spinal cord regeneration after total transection is not possible due to entropy change. Medical Hypotheses, 94, 63-65] proposed one hypothesis by considering three assumptions against the spinal cord regeneration after total transection. Thus, their claims are concluding that head transplant is not possible. But, using theoretical justifications, we show that head transplant is possible without any information loss. We design a spinal cord logic circuit bridge (SCLCB), which is a reversible logic circuit and thus, we show that there is no information loss.

After the paper of Molodtsov ( D. Molodtsov, Soft set theory first results. Computers and Mathematics with Applications, 37(4-5) (1999), 19-31), soft set theory grew at a breakneck pace. Several authors have introduced various operations, relations, results, as well as other aspects in soft set theory and hybrid structures incorrectly, despite their widespread use in mathematics and allied areas. In his paper (D. A. Molodtsov, Equivalence and correct operations for soft sets. International Robotics and Automation Journal, 4(1) (2018), 18-21), Molodtsov, the father of soft set theory, pointed out the wrong results and notions. Molodtsov (D. A. Molodtsov, Structure of soft sets. Nechetkie Sistemy i Myagkie, 12(1) (2017), 5-18) also stated that the concept of soft set had not been fully understood and used everywhere. As a result, it seemed reasonable to revisit the quirks of those conceptions and provide a formal account of the notion of soft set equivalency. Molodtsov already explored the correct operations on soft sets. We use some notions and results of Molodtsov (Molodtsov D.A. (2017) Structure of soft sets. Nechetkie Sistemy i Myagkie, 12(1) (2017), 5-18) to create matrix representations as well as certain associated operations of soft sets, and to quantify the similarity between two soft sets.

We present a communication which deals with some new methods to solve multiple attribute decision-making (MADM) problems based on Fermatean neutrosophic normal number (FNNN). Fermatean neutrosophic sets based on further generalization of neutrosophic and Pythagorean neutrosophic sets. To develop some Fermatean neutrosophic normal aggregation operators. The notion of FNNN holds for commutative and associative laws. There are many aggregation operators that have been defined up to date, but we concentration of this article is to introduce a new concept of Fermatean neutrosophic normal weighted averaging (FNNWA), Fermatean neutrosophic normal weighted geometric(FNNWG), generalized Fermatean neutrosophic normal weighted averaging(GFNNWA) and generalized Fermatean neutrosophic normal weighted geometric(GFNNWG). Also, we obtain an algorithm that deals with the MADM problems based on these operators. We interact applicability of the euclidean and hamming distance measures which are further extended in real life example. Finally, some important properties of these sets under algebraic operations are to be elaborated in this communication.

In this paper we introduce the notion of Hurewicz fibrations in the class of embedding maps of orbifold charts by giving the concept of E-fibration embedding. We study the fundamental properties of this concept such as the restriction, product and its relationship with Hurewicz fibration, etc. Furthermore, we introduce the notion of lifting functions of E-fibration embedding and study preserving projection property of these lifting functions.

Dynamic exploration for a predator-prey bio-system of two species with ratio-dependent functional response is carried out, where the capability to predate in both the stages of the predator, the juvenile and the matured, is taken into account. But, only the matured predators are inferred to be efficient in killing the prey without any negative repercussions. The mortality risks for the juvenile predators are attributable to the inefficiency rate of juveniles coupled with habitat complexity which is either in the form of anti-predator behavior of the prey taken with the aid of their habitat or in the form of a territorial generalist mesopredator. So as to avoid extinction of either of the species and to preserve the food chain of the ecological system, the results pertaining to the existence and stability of all the equilibrium points of the bio-system along with permanence, transcritical and Hopf bifurcation has been thoroughly studied. Corroboration of the results along with the dependence of the biosystem on some crucial parameters is done through numerical simulation. It is found that juvenile predators' inefficiency relative to the resistance confronted, plays a crucial role to control each species density of the ecosystem, as an intriguing limit cycle between the trivial and axial equilibriums of the proposed system along with the co-existing periodic point, because of some ineffeciency parametric value of the juvenile predator has been witnessed.

In this paper, theorem 3.2 and theorem 4.1 of \"{O}z\c{c}a\u{g} and Eysen [S. \"{O}z\c{c}a\u{g} and A.E. Eysen, Almost Menger property in bitopological spaces, Ukrainian Math. J., {\bf 68}, No 6, 950-958 (2016)] are proven to be incorrect. An example is provided to disprove them and thus, correct versions of the theorems are restated.

In this paper, we study trust levels in social networks from the perspective of Dunbar's number., Comment: This is the first version of our work on trust level in social networks. We welcome constructive suggestions, criticism, etc

In this article we first correct a recent misconception about a topology that was suggested by Zeeman as a possible alternative to his Fine topology. This misconception appeared while trying to establish the causality in the ambient boundary-ambient space cosmological model. We then show that this topology is actually the intersection topology (in the sense of G.M. Reed) between the Euclidean topology on $\mathbb{R}^4$ and the order topology whose order, namely horismos, is defined on the light cone. Last, but not least, we show that the order topology from horismos belongs to the class of Zeeman topologies. These results accelerate the need for a deeper and more systematic study of the global topological properties of spacetime manifolds., Comment: arXiv admin note: text overlap with arXiv:1706.07488