As an extension of the classical Poisson model, this paper presents the Neutrosophic Poisson Distribution (NPD), which incorporates an indeterminacy parameter to manage uncertainty in practical situations. We obtain analytically the basic characteristics of the NPD, such as reliability measures, skewness, kurtosis, variance, mean, and moments about the origin. Important generating functions like the characteristic function and moment generating function (MGF) are also developed. The effect of uncertainty is demonstrated by numerical analyses of mean, variance, and higher-order moments for various parameter values and degrees of indeterminacy. Simulation findings show that the NPD is more flexible and efficient than traditional models, as evidenced by bias, mean squared error (MSE), mean relative error (MRE), and entropy metrics. The usefulness of the NPD in quantifying uncertainty and making decisions is validated by real-world applications, such as simulating customer arrivals and equipment breakdowns.
This study investigates the space-like and time-like osculating curves in four-dimensional Minkowski space, [Formula: see text]. We establish necessary and sufficient conditions for a space-like or time-like curve to be congruent to an osculating curve by deriving relationships between their curvature functions. As a special case, we obtain a curvature relation for a space-like curve with a space-like principal normal vector. Furthermore, we examine cases, where two curvatures are constant and derive additional relationships between the curvatures and the various components of space-like and time-like osculating curves.
We compare a radial-basis-function support vector machine (SVM) with a Gaussian naive Bayes (NB) classifier on a reproducible synthetic binary task designed for controlled evaluation. The data matrix contains 1,000 samples and 20 features, of which 15 are informative and 5 are redundant linear combinations; class labels include a controlled flip rate of 0.1. Models were fitted in scikit-learn 1.3.0 (SVM: RBF kernel, C = 1.0, gamma = 'scale'; NB: GaussianNB with default variance smoothing) under an 80/20 stratified hold-out and 5-fold stratified cross-validation. On the test partition, NB attained accuracy 89.95%, precision 91.35%, and F1-score 90.48%, while SVM attained 89.45%, 90.48%, and 90.05%, respectively; recall was identical at 89.62%. McNemar's test (p = 0.8231) and Cohen's kappa (NB 0.7984; SVM 0.7882; Δκ = 0.0102) indicate that the two predictors are statistically equivalent in discriminative accuracy. The practical distinction is computational: NB trained 22.5× faster, predicted 12× faster, used 2.9× less memory, and produced a 12× smaller model file. Under resource limits, therefore, NB is the preferred choice even though predictive accuracy does not differ significantly from SVM.
Fermatean fuzzy set, a significant extension of intuitionistic fuzzy set offer enhanced flexibility to address imprecise information than intuitionistic fuzzy set. In this, study, the parameters of the transportation problem are considered as Fermatean fuzzy numbers and aim to develop a solution framework based on this advanced fuzzy set theory. In order to do this, firstly a ranking function for Fermatean fuzzy number is proposed based on the concept of information reliability. Secondly, three types of transportation problems are considered in which the parameters are Fermatean fuzzy numbers. Moreover, the parameters of the transportation problem are converted in to their equivalent crisp values using the proposed ranking function and then solved by the standard algorithmic approaches using LINGO software. Finally, the significant of the proposed ranking function is discussed and validated through the results of the provided numerical examples.
In this article, we explore the intricate concepts of [Formula: see text]-statistical convergence, which generalizes the notions of [Formula: see text] -statistical convergence and [Formula: see text]-summability, extending the classical framework of [Formula: see text]-summability. In Section 3, we introduce foundational definitions, including the notions of [Formula: see text]-statistical limit points and [Formula: see text] -statistical cluster points, all within the context of neutrosophic [Formula: see text] -normed linear spaces. Section 6 presents significant results on [Formula: see text]-statistical convergence, laying a robust foundation for deeper analysis. The deep interplay between [Formula: see text]-statistically convergent sequences and [Formula: see text] -summable sequences, specific to neutrosophic [Formula: see text]-normed spaces, is revealed in Section 4. In Section 5, we establish an intricate connection between [Formula: see text]-statistically convergent sequences and [Formula: see text]-statistically convergent sequences, emphasizing the role of [Formula: see text] in shaping convergence behavior relative to the neutrosophic [Formula: see text]-norm. Finally, Section 7 delves into the properties of [Formula: see text] -statistical limit points and cluster points, uncovering surprising insights and broadening the understanding of sequence behavior in neutrosophic [Formula: see text] -normed spaces.
In this study, a mathematical model is proposed which analyze the dynamics of co-abuse of cigarette smoking and drugs. Cigarette smoking and illicit drug use aggregate into a massive public health concern, particularly when both substances are abused simultaneously. Numerous studies are present where single-substance dynamics are studied, but the mathematical investigation of co-abuse is limited. The main objective of this study is to construct a novel mathematical model to study the interplay of cigarettes and drugs and their combined dynamics. This co-epidemiological model contains a system of non-linear equations which segregates the population into six compartments: susceptible, cigarette users, drug users, co-abusers, under treatment and recovered. Positivity and boundedness are analyzed to explore whether the system is well-posed. The reproduction number is calculated using the next-generation method, and stability analysis is conducted for the proposed model. Sensitivity analysis is conducted for the identification of dominant parameters of the proposed model. It is determined that the parameters representing the peer influence rate of smoking and progression of smoking to co-abuse are the two most sensitive parameters. Optimal control analysis is performed by yielding three controls on the proposed model using Pontryagin’s Maximum Principle to prevent strategies of co-abuse. It reveals that early awareness and fast recovery treatment break the chain of transmission. This model helps to understand the initiation and progression of addiction and the treatment of co-abuse. MATLAB is used for numerical simulation and visualization of graphs. The model analysis revealed that early prevention and better rehabilitation are helpful in smoking and drug addiction control.
An uncertain set assigns to each element a generalized uncertainty value and thus provides a unified formal language encompassing fuzzy, intuitionistic fuzzy, neutrosophic, plithogenic, and other related degree-based models. In most existing frameworks, a single universe of discourse is fixed in advance, and the associated structures and properties are studied within that single universe. In this paper, we introduce the notion of a multi-universe set, in which multiple universes of discourse are considered simultaneously. Within this framework, we define multi-universe fuzzy, neutrosophic, soft, and rough sets, and examine their fundamental properties.
In this work, the Boubaker wavelet collocation method (BWCM) is used to numerically investigate a fractional-order fowlpox disease model. The intricate dynamics of transmission between birds and mosquitoes are captured by the analysis of the fowlpox model, which was developed using fractional-order derivatives. The model's solutions are approximated using the Boubaker wavelet technique, which is well-known for its effectiveness and precision in resolving fractional ordinary differential equations (FODEs). By using the Boubaker wavelets, we built the integrated operational matrices. To effectively solve fractional-order systems, the BWCM is employed. Numerical results produced using the Runge-Kutta and NDSolve methods are compared to show how successful the proposed method is at solving fractional-order disease models. In contrast to traditional numerical methods, the BWCM offers better accuracy and efficiency while consuming minimal computer power. To determine how fractional-order parameters and other important factors affect the disease's dynamics, sensitivity studies have been carried out. The study reveals new insight on the dynamics of fowlpox transmission and emphasizes the value of fractional-order models and sophisticated numerical methods in epidemiological studies.
In this paper, we investigate the local dynamics, bifurcation phenomena, multistability, chaotic behavior, and bi-parameter space analysis of a discrete population model. Specifically, we identify three equilibrium points of the model - the trivial equilibrium, the boundary equilibrium (which exists for all parameter values), and the interior equilibrium, which is derived under specific parameter condition. Using stability theory, we analyze the local dynamic behavior around these equilibria and classify their stability properties. Based on these classifications, we determine the one-parameter bifurcation sets and carry out a detailed analysis of the resulting bifurcation phenomena. To address the emergence of complex dynamics, particularly those associated with Neimark-Sacker and flip bifurcations, we explore chaotic behavior and implement hybrid as well as OGY control strategies for chaos suppression. Furthermore, we examine multistability and conduct a comprehensive bi-parameter space analysis to illustrate transitions between different dynamical regimes. Finally, all theoretical findings are substantiated through numerical simulations, which confirm the validity and accuracy of the analytical results.
This paper introduces the concept of SR-[Formula: see text][Formula: see text] and [Formula: see text]-ideals in [Formula: see text]-algebras. These algebras generalize Boolean algebras and have applications in computer science, artificial intelligence, and control theory. The proposed SR-[Formula: see text][Formula: see text] and [Formula: see text]-ideals incorporate a square root operation, which provides greater flexibility and improved representation of uncertainty. This study examines their fundamental properties and explores the relationships among them within [Formula: see text]-algebras. Several examples are included to illustrate the theoretical results. The paper concludes with possible directions for future research, contributing to the development of fuzzy set theory in algebraic structures.
Decision-making tasks in practical environments frequently involve incomplete or ambiguous information. Intuitionistic fuzzy soft sets offer a flexible way to represent such data through membership and non-membership degrees. Nevertheless, many existing intuitionistic fuzzy soft set-based multi-criteria decision-making methods mainly depend on aggregation mechanisms and often ignore the relational interactions among alternatives. To overcome this limitation, this study introduces near intuitionistic fuzzy soft (NIFS) sets within the setting of nearness approximation spaces and develops a relational structure called the near intuitionistic fuzzy soft relation (NIFSR). Based on this framework, a relation-oriented decision-making method is constructed in which relational composition is employed to incorporate both direct evaluations and interdependence among alternatives. The proposed approach is evaluated using three benchmark problems related to agricultural land assessment, sustainable supplier selection and dengue fever diagnosis. Experimental results demonstrate that the method generates consistent and interpretable rankings under different parameter values and produces outcomes comparable with several established decision-making techniques.
This paper presents an analytical approach for solving the fuzzy Riccati differential equation using the fuzzy differential transformation method (FDTM) within the framework of generalized Hukuhara differentiability. By employing an extended definition of the fuzzy derivative, the method accommodates a wider class of differentiable fuzzy functions and facilitates the formulation of fuzzy initial value problems. The FDTM constructs the solution as a finite Taylor series through a recursive procedure, avoiding the need for symbolic differentiation and reducing computational complexity. This approach offers a mathematically rigorous and efficient framework for fuzzy differential equations, ensuring the continuity and consistency of the solution in the fuzzy context. The proposed method demonstrates the strength of FDTM in addressing uncertainty in dynamical systems and contributes to the theoretical advancement of fuzzy calculus and fuzzy system modeling.
This paper is devoted to the study of the approximate controllability for a class of Sobolev-type fractional differential systems of order sigma is an element of (0, 1) in a separable Hilbert space. The proposed control system is governed by the Hilfer fractional derivative, which provides a unified framework that interpolates between the Riemann-Liouville and Caputo fractional derivatives. By employing resolvent operator theory and semigroup techniques combined with suitable fixed-point arguments, sufficient conditions for the existence of mild solutions and approximate controllability are established under non-local initial conditions. Unlike many existing results, this analysis does not rely on the compactness assumption of the associated semigroup, thereby extending the applicability of the controllability criteria to a broader class of systems with implicit dynamics. The obtained results are further extended to Sobolev-type fractional integro-differential systems. An illustrative example is presented to demonstrate the effectiveness and applicability of the theoretical findings.
In this paper, we develop a unified categorical framework for structures associated with F-transforms and fuzzy pretopological spaces. Unlike existing approaches based on function-induced morphisms, we introduce categories whose morphisms are defined as pairs of L-valued fuzzy relations, allowing a more general and flexible representation of transformations. Specifically, we construct the categories of spaces with L-valued fuzzy partitions, L-valued fuzzy lower transformation systems, L-valued fuzzy pretopological spaces, and & Ccaron;ech L-valued fuzzy interior spaces, and establish isomorphisms and functorial relationships among them. We further prove the existence of adjoint functors between these categories. A central contribution of this work is to demonstrate that these categories naturally embed into the category Qua, thereby providing a common relational framework that captures their structural similarities. This unification not only generalizes existing categorical constructions but also reveals deeper connections between fuzzy relational systems and transformation-based models.
In this paper, we investigate the convergence behavior of lower and upper approximation spaces and boundary regions in dynamic rough set theory under two frameworks. The first is a set-theoretic framework, where convergence is defined through the eventual stability of sets. In this setting, we show that the convergence of the lower and upper approximations always leads to the convergence of the boundary regions, and that the convergence of the boundary regions also ensure the convergence of the lower-upper approximation spaces. The second framework adopts a metric-based approach, where convergence is determined by a distance between sets. In general, this relationship between the lower approximations, upper approximations and the boundary regions do not hold. However, when the distance is defined by the symmetric difference between sets, the convergence relationships among the lower, upper and boundary regions are preserved. These results provide a unified understanding of convergence in evolving rough set systems.
In this paper, we propose a modified discrete-time Bazykin–Berezovskaya prey–predator model integrating the strong Allee effect and fear induced in prey to reflect important ecological details. The qualitative behavior of the developed model is investigated by identifying equilibrium points, analyzing their stability, and studying period-doubling (PD) and Neimark–Sacker (NS) bifurcations by using center manifold and bifurcation theory. The study examines the influence of the Allee effect and fear-induced parameters in prey on the model’s dynamic behavior, highlighting transitions from stable states to chaotic regimes when both the parameters increase. Numerical simulations support the theoretical results by visualizing phase portraits, 2D and 3D diagrams of the maximum Lyapunov exponent (MLE), bifurcation diagrams, and analyses of local stability. The influence of the complex network interaction, driven by the coupling strength parameter, is also examined to assess the system’s dynamic behavior. We apply the 0–1 indicator of chaos to detect chaotic characteristics and conduct parameter sensitivity analysis to evaluate how the model’s output responds to variations in particular parameters. The system’s bifurcations and oscillations can be controlled using Ott, Grebogi, and Yorke (OGY) control techniques.
In this paper, we introduce the homotopy category [Formula: see text] by identifying soft continuous mappings that are related by soft homotopy. We consider the soft fundamental group as a functor on the category [Formula: see text] of pointed soft topological spaces and show that it assigns the same group homomorphism to homotopic soft continuous mappings. Consequently, the soft fundamental group functor naturally factors through the homotopy category, yielding a well-defined functor from [Formula: see text] to the category of groups. This establishes a homotopy-invariant categorical framework for soft topology and provides a natural categorical setting for studying the functorial properties of soft fundamental groups.
Spectral clustering is a powerful paradigm for analyzing non-convex data, yet standard spectral clustering is oblivious to sensitive attributes, often producing clusters that are systematically biased with respect to protected groups. Existing Fairness SC methods face a critical dilemma: analytical approaches lack flexibility for complex constraints, while relaxation-based iterative methods compromise the core orthogonality constraint, thereby deviating from the graph’s intrinsic structure. To bridge this gap, we propose a novel optimization framework that directly solves the original non-convex fair SC problem on the Stiefel manifold. By integrating the Augmented Lagrangian Multiplier method to enforce linear fairness constraint and employing Projected Gradient Descent on the manifold, our approach rigorously guarantees orthogonality without convex relaxation. Extensive experiments on synthetic and real-world datasets demonstrate that our method efficiently optimizes the Normalized Cut objective, achieving numerical precision comparable to analytical solutions. Furthermore, it significantly outperforms state-of-the-art baselines in balancing clustering utility and fairness, exhibiting superior generalization and flexibility in parameter tuning.
Best proximity point theory provides an effective framework for treating non-self mappings in situations where fixed points cannot arise. This work develops such a framework within the setting of intuitionistic fuzzy metric spaces in the sense of Park. Several new proximal structures are introduced, including the intuitionistic fuzzy [Formula: see text]-property for pairs of subsets, intuitionistic fuzzy proximal compatibility, and two generalized continuity notions termed intuitionistic fuzzy proximally reciprocal continuity and proximally weak reciprocal continuity. In addition, two classes of intuitionistic fuzzy [Formula: see text]-proximally weak reciprocal commuting mappings, referred to as Type I and Type II, are formulated and studied. Under suitable completeness and proximity assumptions, an existence and uniqueness theorem for common best proximity points of two non-self mappings [Formula: see text] is obtained. The results extend and unify several known theorems in classical metric, fuzzy metric, and intuitionistic fuzzy metric settings, and examples are provided to demonstrate the applicability and non-emptiness of the new concepts. When [Formula: see text], the theory reduces to common fixed point results in intuitionistic fuzzy metric spaces, thereby highlighting the generality of the developed approach.
A subset of Euclidean space is called a fractal if its Hausdorff dimension strictly exceeds its topological dimension. In this paper, we construct a set [Formula: see text] through an iterative process inspired by QR code structures. We prove that [Formula: see text] is uncountable, nowhere dense, and has Lebesgue measure zero. Further, we show that [Formula: see text] is the attractor of an iterated function system on [Formula: see text]. We then compute its Hausdorff dimension directly from the geometric structure of the underlying QR code pattern and establish that its Hausdorff dimension strictly exceeds its topological dimension. Consequently, [Formula: see text] is a fractal.
