The most important applications of an algebra like BCK-Algebra. As a generalization of ring, we study γ- semi-ring and γ-ring in invarianent neutrosophic set. Neutrosophic concepts are widely used in the field of mathematics and other sciences, especially in studying the Algebra. In this paper, we present the concept of neutrosophic γ-BCK-Algebras as an example of this generalization. We also present neutrosophic sub-algebra, neutrosophic ideal and some other type structure algebraic. We proved that if f : AI → N I is a homomorphism of neutrosophic γ-BCK-algebras AI and NI, then f is injective if and only if neutrosophic ker(f ) = {0I}. Also, we presented, if NI be a normal neutrosophic subalgebra of neutrosophic γ-BCK- algebra AI, then ” ∼ N I ” is a congruence relation.
Read MoreDoi: https://doi.org/10.54216/IJNS.270102
Vol. 27 Issue. 1 PP. 10-18, (2026)
The present work explores the features of new kinds of neutrosophic continuous mappings, including neutrosophic irresolute β^*−continuous mapping (NIβ^*CM) and neutrosophic continuous mappings, including neutrosophic contra β^*−continuous mapping (NCOβ^*CM) and investigates some properties related them. Moreover, we study the relationships between these two concepts with the concept of irresolute α^* and contra α^*−continuous mapping. Finally, we introduced m-polar neutrosophic irresolute β^*−continuous mapping (MPNIβ^*CM) and neutrosophic continuous mappings, including m-polar neutrosophic contra β^*−continuous mapping (MPNCOβ^*CM) with investigates some properties related them.
Read MoreDoi: https://doi.org/10.54216/IJNS.270101
Vol. 27 Issue. 1 PP. 01-09, (2026)
Integral equations, including Abel’s integral equation and linear Volterra integral equations of both the first and second kinds and neutrosophic Abel’s integral equation and linear Volterra integral equations of both the first and second kinds, regularly appear in advanced problems across biology, chemistry, physics, and engineering, often modeling systems with memory effects or time-dependent interactions. This study explores the GALM transform as a powerful and unified method for solving these equations. The exact solution of Abel’s integral equation and its neutrosophic version is derived, demonstrating the transform’s simplicity and efficiency through practical applications. Additionally, the GALM transform is employed to solve linear Volterra integral equations of the first and second kinds with their neutrosophic generalizations, with illustrative examples provided to validate its effectiveness. By addressing a wide range of problems, this research establishes the GALM transform as an accurate, reliable, and versatile tool, offering significant advantages over traditional methods in solving complex scientific and engineering equations.
Read MoreDoi: https://doi.org/10.54216/IJNS.270103
Vol. 27 Issue. 1 PP. 19-35, (2026)
In this paper, we present some results about the neutrosophic-generalized version of finite-difference method, where we prove its essential properties, and we apply it to many different examples to clarify the validity of our work. In addition, some numerical tables related to the results will be clarified and presented.
Read MoreDoi: https://doi.org/10.54216/IJNS.270104
Vol. 27 Issue. 1 PP. 36-42, (2026)
Algebraic operations, which include addition, subtraction, division, scalar multiplication, and exponentiation, are the fundamental mathematical operations utilised in decision-making analysis. When performing on numbers, the algebraic operations are commonly referred to as arithmetic operations. Another alternative for algebraic operations, known as Einstein operations, has gained recognition for its smooth approximation and utilisation of Archimedean norms. However, it is crucial to note that Einstein operations are not designed to effectively address issues of indeterminacy, uncertainty, and lower-upper approximation. Thus, this paper defines some rough neutrosophic-based Einstein operations known as RNS Einstein addition, RNS Einstein multiplication, RNS Einstein scalar multiplication, and RNS Einstein exponentiation. By adopting rough neutrosophic sets (RNS), which incorporate neutrosophic lower and upper approximations, the proposed RNS Einstein operations offer a practical approach for handling uncertain situations. Some examples are provided to demonstrate the applicability of the RNS Einstein operations. Several desirable properties related to the defined RNS Einstein operations are investigated. Finally, the proposed RNS Einstein operations are applied in solving multi-criteria decision-making problems within a rough neutrosophic environment.
Read MoreDoi: https://doi.org/10.54216/IJNS.270105
Vol. 27 Issue. 1 PP. 43-58, (2026)
In this paper, we will use the family of regular open sets in a topological space (Z, τ ) to define an operator ΦR : 2Z → 2Z by ΦR(F) = {s ∈ Z : ∃ D ∈ RO(Z, s) with (D − F )c /∈ P} in frame of primal topological spaces. Then we introduce the notion of topology δ-compatible for a primal in a primal topological space and study some of its properties. Finally, we use the concept of δ-semi-open sets to provide additional properties for the operators (⋄ R) and ΦR(F ), and we add many illustrative examples that help clarify the relationships between the concepts that are presented.
Read MoreDoi: https://doi.org/10.54216/IJNS.270106
Vol. 27 Issue. 1 PP. 59-72, (2026)
Predicting future energy consumption plays a vital role in maximizing resource utilization, reducing costs, and enhancing sustainability. Researchers employ advanced statistical and machine learning models to improve the accuracy of time series forecasting. Real-world energy consumption data is analyzed using State-Space Models (SSMs), Vector Auto Regression (VAR), Structural VAR (SVAR), Generalized Additive Models for Location, Scale, and Shape (GAMLSS), and Bayesian Structural Time Series (BSTS). An evaluation of Long Short-Term Memory (LSTM) networks and the Prophet model is conducted alongside a comparison with the aforementioned models. The proposed method integrates neutrosophic statistical models for feature extraction and residual analysis, generating outputs suitable for machine learning processing. The results indicate that incorporating judgment-based neutrosophic statistical approaches with AI-driven neutrosophic prediction models yields superior forecasts of power consumption, contributing to more comprehensive and effective energy usage prediction methodologies.
Read MoreDoi: https://doi.org/10.54216/IJNS.270107
Vol. 27 Issue. 1 PP. 73-84, (2026)
Group theory is one of the significant parts of mathematical algebra. This theory is characterized by its ability to address various applications, including the classification of the symmetry of crystals, atoms, molecules, and polyhedral structures. In this work, we study a newly introduced concept, namely BIVFSs, which is an extension of previous concepts discussed in the previous studies section of this work. In this work, we establish and apply basic algebraic concepts applicable to this concept. We combine this concept with group theory, which has important properties and applications, generating important results, which are explained in the third section of this work. An important result of this work is BIVF-level set, support, BIVF-kernel and bipolar BIVF- characteristic function, and BCF point. Then, we interpret the BIVF-subgroup. Furthermore, we present the associated examples and theorems and prove these associated theorems.
Read MoreDoi: https://doi.org/10.54216/IJNS.270108
Vol. 27 Issue. 1 PP. 85-92, (2026)
In this paper, separation axioms are discussed in neutrosophic crisp topological spaces from a new perspective. This is generally useless because any neutrosophic set does not necessarily have a union of its neutrosophic points under any union and for any kind of points. Hence, the separation properties are studied concerning stable neutrosophic crisp topological spaces, which are determined by two special types of complement. Moreover, various examples are illustrated in these cases.
Read MoreDoi: https://doi.org/10.54216/IJNS.270109
Vol. 27 Issue. 1 PP. 93-99, (2026)
The probability distribution holds considerable importance within the realm of probability theory, a concept that permeates nearly all scientific disciplines. Nevertheless, the principal aim of the present research endeavor is to introduce a novel distribution referred to as the neutrosophic Alpha logarithm Exponential, abbreviated as NALE. Various mathematical attributes that elucidate life survival and associated characteristics such as hazard rates, moment’s functions, moment-generating functions, and additional metrics including mean and variance are also examined. Two methods were used to estimate the parameters; the Monte Carlo simulation has been employed to evaluate the efficacy of the NALE distribution estimation and to compare the two estimation methods. Therefore, the outcomes from the simulation executed in this research imply that a satisfactory level of precision in estimation is feasible only when the sample size is notably large. The real data has been utilized to demonstrate the specific manner in which the proposed NALE distribution has been recommended for application. Based on the analyses presented in the preceding sections, it can be inferred that the NALE distribution possesses a broad applicability since it is capable of accommodating of neutrosophic data; it does not differentiate between certainty, probabilities of uncertainty, ambiguities, or imprecisions.
Read MoreDoi: https://doi.org/10.54216/IJNS.270110
Vol. 27 Issue. 1 PP. 100-110, (2026)
The most efficient device for modelling uncertainty in decision-making issues is the neutrosophic set (NS) and its add-ons, such as NS of complex, interval, and interval complex. An efficient device for establishing uncertainty in decision-making by inserting three grades of truth, indeterminacy, and falsehood of an established statement. Recently, financial globalization has significantly expanded various methods for enhancing service quality using advanced resources. The practical application of the blockchain (BC) model enables stakeholders concerned about the hazard and return prediction models of economic products. To explore the application of deep learning (DL) in processing financial trading data, a neural network (NN) and DL data are utilized. Absolute stock indices and financial data are utilized for analyzing the efficiency of these models in financial prediction and analysis. This paper presents an Enhanced Risk Prediction Framework for Financial Transactions System Using Interval Neutrosophic Covering Rough Sets (ERPFFTS-INCRS) model. The aim is to develop an effective risk prediction model that enhances the reliability and security of BC financial transactions under uncertain conditions, utilizing neutrosophic logic. Initially, the z-score standardization method is used to clean, transform, and organize raw data into a structured and meaningful format. Furthermore, the ERPFFTS-INCRS method implements the INCRS method for the financial classification process. Finally, the hyperparameter selection for the INCRS model is performed by implementing the Elephant Herding Optimisation (EHO) algorithm. The experimental evaluation of the ERPFFTS-INCRS approach is examined under the metaverse financial transactions (MFT) dataset. The comparison analysis of the ERPFFTS-INCRS approach revealed a superior accuracy value of 98.77% compared to existing methods.
Read MoreDoi: https://doi.org/10.54216/IJNS.270111
Vol. 27 Issue. 1 PP. 111-124, (2026)
This paper proposes a novel hybrid framework that integrates Neutrosophic Logic with Artificial Intelligence (AI) for robust spatiotemporal modeling of urban land parcel transactions. The approach captures the uncertainty, inconsistency, and incompleteness often found in public land auction data through the application of neutrosophic triplets, defined by degrees of truth, indeterminacy, and falsity. Using longitudinal transaction records from Tashkent, the model transforms raw data into neutrosophic representations and feeds them into a Long Short-Term Memory (LSTM) network for forecasting. The enriched feature space enhances interpretability and prediction accuracy across administrative zones. Experimental evaluations demonstrate the superiority of the proposed Neutrosophic-AI model over conventional methods in terms of forecasting precision and uncertainty handling. This study offers a foundational contribution to neutrosophic-based urban analytics and supports transparent digital governance frameworks.
Read MoreDoi: https://doi.org/10.54216/IJNS.270112
Vol. 27 Issue. 1 PP. 125-138, (2026)
Facility location problems assigned for determining the location of different types of facilities as factories, warehouses, hospitals,…, etc. It also helps to find the quantity of products and goods delivered to customers from the assigned facilities. As in other fields, uncertainty occurs in facility location problems, when the cost, time and other information seem in deterministic and unknown. The uncertainty in facility location problems promoted scientists to apply robust optimization such as stochastic techniques for solving complex locations problems. However, in stochastic problems some uncertain parameters need highly approaches such as neutrosophic sets, which is an extension of fuzzy sets to tackle the stochastic parameters. In this paper, a neutrosophic approaches based on neutrosophic sets applied for solving capacitated and uncapacitated stochastic facility location problems. The normal and neutrosophic models designed and some applications illustrated for testing the neutrosophic stochastic facility location problems in two cases capacitated and uncapacitated facilities. The result various for the two different situations and shows that decision maker therefore offers flexibility of various solutions when applying the neutrosophic case under different situations.
Read MoreDoi: https://doi.org/10.54216/IJNS.270113
Vol. 27 Issue. 1 PP. 139-146, (2026)
Personalized learning pathways in digital education platforms have become essential for addressing the unique needs and behaviors of individual learners. However, traditional adaptive systems often fail to account for the uncertainty, ambiguity, and inconsistency inherent in educational data. This paper proposes a novel neutrosophic decision-support framework that models learner profiles using truth (T), indeterminacy (I), and falsity (F) scores derived from student interaction and performance data. Utilizing the Open University Learning Analytics Dataset (OULAD), we compute neutrosophic learner vectors based on assessment outcomes, engagement patterns, and virtual learning environment (VLE) activity. A rule-based decision engine then recommends adaptive learning pathways—ranging from remedial to advanced—by interpreting the T/I/F distributions through a neutrosophic logic framework. Experimental results demonstrate that the proposed model enhances pathway assignment accuracy and provides better support for learners with incomplete or uncertain data compared to traditional fuzzy and crisp models. The neutrosophic approach also ensures interpretability and flexibility, making it well-suited for real-world educational platforms aiming to achieve adaptive learning at scale.
Read MoreDoi: https://doi.org/10.54216/IJNS.270114
Vol. 27 Issue. 1 PP. 147-165, (2026)
The fuzzy reliability estimate for the Benktander distribution, a model appropriate for heavy-tailed data, is investigated in this work. By adding membership functions and α-cuts, we extend the Benktander distribution to a fuzzy framework and compute its probability density function and reliability function. The fuzzy reliability is estimated using two methods: maximum likelihood and Bayesian approaches. The Bayesian method uses special loss functions, gamma priors, and squared error. The effectiveness of these estimators is examined in a simulated study using varying sample sizes and parameter values. The findings show that, especially for smaller samples, Bayesian techniques—in particular, the cautious Bayes estimator—perform better in terms of accuracy and stability than maximum likelihood estimation. The results emphasize how crucial it is to choose suitable prior distributions and loss functions while doing reliability analysis.
Read MoreDoi: https://doi.org/10.54216/IJNS.270115
Vol. 27 Issue. 1 PP. 147-158, (2026)