arXiv subjects

Santanu Acharjee

Publications and source records attributed to Santanu Acharjee.

At least 19 recordsLinked to original sources

arXiv2026

Roughness and entropy measures of a soft set

Soft set theory is an important and emerging area within soft computing, owing to its attribute-oriented mathematical framework and its wide applicability in diverse domains, including science and social sciences. The theoretical constraints associated with the selection of subsets of the sets of attributes in soft set theory have further motivated the development of hybrid and extended theoretical models. In this paper, we introduce two distinct roughness measures and six entropy measures for soft sets and systematically investigate their properties using both theoretical analysis and computational techniques. The proposed roughness measures are defined within two distinct conceptual frameworks. Throughout the development of these measures and the corresponding results, the foundational principles of soft set theory, as established by Molodtsov, are strictly preserved. Furthermore, the proposed framework is shown to be novel with respect to roughness characterization, and a comparative analysis with classical rough set theory is presented to highlight the theoretical distinctions and contributions of this work.

math.GM↗

arXiv2025

Information retrieval in big data using cognitive approaches

Due to the exponential growth of big data in this digital era, an advanced method for effective information retrieval becomes essential. The basic objective of this paper is to propose a topology-based method for cognitive information retrieval (CIR) in big data environments. By using concepts such as cognitive similarity distances, metric spaces, retrieval topologies, etc., this paper aims to propose the semantic alignment between user queries and document repositories. The paper also extends this approach to incorporate logical connectives in cognitive information retrieval.

math.GM↗

arXiv2025

Orbitwise expansive maps

This study defines an orbitwise expansive point (OE) as a point, such as $x$ in a metric space $(X,\rho)$, if there is a number $d>0$ such that the orbits of a few points inside an arbitrary open sphere will maintain a distance greater than $d$ from the corresponding points of the orbit of $x$ at least once. The point $x$ is referred to as the relatively orbitwise expansive point (ROE) in the previously described scenario if $d$ is replaced with the radius of the open sphere whose orbit is investigated and whose centre is $x$. %The function generating the orbit is considered to be continuous. We also define OE (ROE) set. We prove that arbitrary union of OE (ROE) set is again OE (ROE) set and every limit point of an OE set is an OE point. We show that, rather than the other way around, Utz's expansive map or Kato's CW-expansive map implies OE (ROE) map. We utilise the concept of OE(ROE) to analyse a time-varying dynamical system and investigate its relevance to certain traits associated with expansiveness.

math.DS↗

arXiv2024

Logical connectives of fuzzy soft set theory

Soft set theory, introduced by Molodtsov [Molodtsov, D. (1999). Soft set theory-first results. Comput. Math. Appl., 37(4-5), 19-31], provides a flexible framework for managing uncertainty and vagueness, addressing limitations in traditional approaches such as fuzzy set theory, rough set theory, and probability theory. Over time, fuzzy soft set theory has emerged as a significant extension, blending the principles of fuzzy set theory and soft set theory to support applications in various decision-making processes. This study revisits fuzzy soft set theory, addressing conceptual errors and inaccuracies in the definitions of t-norm, t-conorm, strong negation, and implication that deviated from Molodtsov's foundational principles. Corrected definitions-fuzzy soft t-norm, fuzzy soft t-conorm, fuzzy soft negation, and fuzzy soft implication-are proposed to ensure theoretical rigor. The paper rectifies conceptual errors in prior work by Ali and Shabir [Ali, M. I., Shabir, M. (2013). Logic connectives for soft sets and fuzzy soft sets. IEEE Transactions on Fuzzy Systems, 22(6), 1431-1442] and introduces refined results to strengthen the logical framework, providing a consistent foundation for future research and hybrid model development in this domain.

math.GM↗

arXiv2024

Big data searching using words

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 considered 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 mathematical perspectives. 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 the topological structure in big data. In this paper, we present fundamental concepts related to the neighborhood structures of words in big data search, laying the groundwork for developing topological frameworks for big data in the future. We also introduce the notion of big data primal within the context of big data search and explore how neighborhood structures, combined with the Jaccard similarity coefficient, can be utilized to detect anomalies in search behavior.

cs.IR↗

arXiv2024

The correct structures in fuzzy soft set theory

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.

math.GM↗

arXiv2023

The limit of human intelligence

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.

math.GM↗

arXiv2023

Two predators one prey model that integrates the effect of supplementary food resources due to one predator's kleptoparasitism under the possibility of retribution by the other predator

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.

q-bio.PE↗

arXiv2023

Primal-Proximity Spaces

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.

math.GN↗

arXiv2023

Predator-prey dynamics pertaining to structuralizing predator species into three stages coupled with maturation delay owing to juvenile hunting

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.

math.DS↗

arXiv2022

Homotopical selection principles in bitopological dynamical systems

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.

math.DS↗

arXiv2022

A New Operator of Primal Topological Spaces

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.

math.GN↗

arXiv2022

Primal Topological Spaces

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.

math.GM↗

arXiv2022

Theoretical possibilities of head transplant

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.

q-bio.TO↗

arXiv2022

MADM Approach For Fermatean Neutrosophic Normal Aggregation Operator

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.

math.GM↗

arXiv2022

On correct structures and similarity measures of soft sets along with historic comments of Prof. D. A. Molodtsov

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.

math.GM↗

arXiv2022

Fiibrations property of embedding maps of orbifold charts

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.

math.GT↗

arXiv2022

Erratum to " Almost Menger property in bitopological spaces"

In this paper, theorem 3.2 and theorem 4.1 of Özçağ and Eysen [S. Özçağ 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.

math.GN↗