Functional programming is frequently taught in isolation from its mathematical roots, particularly category theory, leading to a fragmented understanding for students. Simultaneously, category theory is often perceived as too abstract and difficult to grasp, despite its foundational role in programming. This gap between theory and practice creates barriers for students, preventing them from fully appreciating the deep connections between functional programming and its underlying mathematical structures. Although there are resources aimed at bridging this divide, such as works by Milewski, MacLane, and Leinster, they often either lack practical examples or fail to delve deeply into the mathematical rigor required for a comprehensive understanding of category theory. This paper presents a novel pedagogical approach that integrates category theory with functional programming in a unified and accessible framework. By leveraging monadic programming, particularly through the list and Maybe monads, we offer concrete examples of how abstract mathematical concepts can address real-world programming challenges, such as handling missing data. Our approach builds on and generalizes Dayou Jiang's method of using programming to teach partially ordered relations. In doing so, we concurrently teach functional programming and category theory, making the abstract more tangible and applicable. This interdisciplinary method not only enhances comprehension of both fields but also aligns with contemporary educational reforms that prioritize integrated learning across mathematical and computational domains.

The spreading of misinformation in Online Social Networks (OSNs) is quite similar to the spreading of infection in biological diseases. As the biological virus spreads and makes one infected as well as those who come into contact with the infected person, the nature and behavior of the spread of misinformation in an online social network is similar. So in order to understand the functioning of misinformation and to control over epidemic outbreak of misinformation in OSNs, epidemic models can be quite handy. The introduction of Bifurcation theory in the epidemic model explains the qualitative behavior of the system with changes in parameters. In this paper, we have introduced the Susceptible, Infectious, and Protected (SIP) model with Bifurcation for the propagation of misinformation in OSNs. Here Bifurcation is due to the limited number of users who are ready to adopt the Security and Privacy Policies of Social Network Sites. We define the threshold number for the system of equations and explain the stability of an Infection Free Equilibrium (IFE), representing the absence of misinformation. We have discussed about the endemic equilibrium point and bifurcation conditions along with its nature and stability at those points. Also, we have shown global stability at the endemic Equilibrium of the system. Finally, numerical simulation has been used to show the existence of bifurcation in the system with a change in the value of parameters.

Pseudo Random Number Generators (PRNGs) are deterministic and periodic in nature. Hybrid Pseudo Random Number Generators (HPRNGs) address some limitations by using time-based seeding with a modified Linear Congruential Generator (LCG). While HPRNGs improve upon the deterministic nature by using dynamic time-based seeds, they still suffer from periodicity and potential seed-related issues. This study addresses the deterministic nature further as well as the periodicity of PRNGs by proposing an enhanced HPRNG, making it more suitable for high-security applications.

Effective path planning is essential for autonomous mobile robots navigating unknown environments. Fuzzy Logic Controllers (FLCs) are well-suited for this task due to their robustness in handling uncertainty, vagueness, and nonlinearities. Among the core elements that influence FLC behaviour are membership functions (MFs), which define how sensory inputs are translated into fuzzy linguistic terms. Despite their importance, specific impact of different MF shapes on navigation performance remains underexplored. This study investigates the effect of three widely used MF types i.e., triangular, trapezoidal, and Gaussian on the traversal efficiency of a nonholonomic wheeled mobile robot operating in a static, obstacle filled environment. A series of simulations were conducted using MATLAB and CoppeliaSim, with traversal time serving as the primary performance metric. One-way ANOVA results (F = 342.33, p < 0.001) revealed statistically significant differences across MF types, with triangular MFs yielding the shortest average traversal time of 177.95 s, followed by trapezoidal with 179.08 s and Gaussian at 181.05 s. These findings highlight that MF shape significantly influences control responsiveness and path efficiency. By isolating MF type within a consistent rule base and simulation setup, this work provides baseline guidance for MF selection and sets the stage for future research involving hybrid MFs, real-world validation, and adaptive fuzzy systems. 

This paper presents the development of fuzzy multiplicative metric spaces, an extended framework that combines the principles of multiplicative metric spaces with fuzzy logic to better address uncertainty and imprecision inherent in many real-world problems. By replacing additive distance measures with multiplicative ones, this approach proves particularly effective in contexts where relative variations or proportional relationships are more meaningful than absolute differences. Within this generalized setting, we establish a series of fixed-point theorems of Banach, Kannan, and Chatterjee types, along with related corollaries, each supported by concrete examples. The theoretical results are further validated through their application to the solution of nonlinear integral equations, demonstrating the versatility and applicability of the proposed framework across mathematical analysis and applied disciplines.