The Generalized Uncertainty Principle (GUP) represents a fundamental modification to Heisenberg's uncertainty principle, incorporating quantum gravitational effects through the introduction of both minimal and maximal length scales. In this work, we investigate the cosmological implications of GUP by examining its effects on the solutions of the Friedmann equations across different evolutionary eras of the universe. We employ the Hamiltonian formalism to derive modified Friedmann equations that incorporate GUP corrections. The Hamilton procedure provides a systematic framework for obtaining these quantum-corrected gravitational equations while preserving the consistency of the theoretical structure. To solve these modified equations, we apply the classical perturbation method, treating GUP corrections as small perturbations to the standard cosmological solutions. This approach allows us to obtain analytical expressions for the modified cosmological parameters while maintaining tractability. Our analysis spans multiple epochs of cosmic evolution, including the radiation-dominated, matter-dominated, and dark energy-dominated eras. For each epoch, we derive the corresponding modified solutions and analyze how GUP-induced corrections alter the universe's expansion dynamics. We place particular emphasis on the Cosmic Microwave Background (CMB) era, which provides robust observational constraints on cosmological models. Using our GUP-modified framework, we compute the corrected value of the Hubble constant and perform a detailed comparison with the observationally determined value from CMB measurements. This comparison serves as a critical test for the viability of GUP modifications in cosmology and enables us to constrain the phenomenological parameters associated with minimal and maximal length scales. Our results contribute to understanding how quantum gravitational effects, as encoded in the GUP framework, might influence large-scale cosmological observables and potentially address tensions in current cosmological measurements.

Understanding dynamical transitions between periodic and chaotic regimes in nonlinear fiber lasers is essential for stability control and performance optimization. In this study, we investigate regime transitions in an erbium-doped fiber laser (EDFL) by integrating nonlinear dynamical analysis with Topological Data Analysis (TDA). The governing differential equations are solved numerically to generate time series across varying control parameters, and Lyapunov exponents are computed to identify stability boundaries and chaotic behavior. To capture the global geometric structure of the underlying attractors, we apply persistent homology to the time series and compute persistence diagrams and Betti curves as quantitative topological descriptors. In addition, the Mapper algorithm is used to construct graph representations of the attractors, enabling the extraction of topological and statistical features that characterize structural organization. Our results reveal clear topological transitions between periodic and chaotic regimes, with chaotic dynamics exhibiting significantly higher geometric complexity and distinct structural patterns in the corresponding graphs. Importantly, the evolution of topological invariants shows strong agreement with Lyapunov-based chaos indicators while simultaneously capturing global attractor reorganization that is not accessible through classical local stability measures alone. These findings demonstrate that TDA provides a robust and complementary geometric framework for detecting, classifying, and understanding dynamical regime transitions in nonlinear fiber laser systems, with potential applications in stability monitoring and control of advanced photonic devices.

In this study, we investigate the qualitative behavior of a higher-order nonlinear system of difference equations. Such systems naturally arise in a wide range of applications, including population dynamics, epidemiology, economics, and discrete-time control theory, where time delays and nonlinear interactions significantly influence the long-term evolution of the process under consideration. The presence of higher-order delays increases the mathematical complexity of the model and enriches its dynamical structure.

We begin by establishing the existence and uniqueness of equilibrium points and deriving the necessary conditions for their positivity. The local asymptotic stability of these equilibria is then examined through linearization methods and spectral analysis of the associated Jacobian matrix. Explicit stability criteria are obtained in terms of the system parameters. Furthermore, we provide sufficient conditions that guarantee the boundedness and persistence of positive solutions, ensuring that trajectories remain biologically and physically meaningful over time.

Special attention is devoted to the role of the delay order, a positive constant parameter, and a nonlinear exponent governing the interaction terms. Their combined influence on stability, convergence, and qualitative behavior is carefully analyzed. In addition, we investigate the possible oscillatory nature of solutions and determine parameter regions that may generate oscillations or complex dynamics.

The theoretical results are rigorously proved and supported by several numerical simulations. These simulations confirm the analytical findings and demonstrate the robustness, applicability, and effectiveness of the derived qualitative and stability criteria.

Semilinear parabolic tridiagonal reaction-diffusion systems of order [ m ] model coupled diffusion-reaction processes in physics, biology, and chemistry. These systems take the form\frac{\partial U}{\partial t}-D\Delta U=F\left( U\right) \text{ \ \ \ \ dans \ }\Omega \times \left( 0,+\infty \right) , with Neumann boundary conditions and positive initial data . Here, [ \Delta_m ] denotes the tridiagonal Laplacian matrix. Proving global existence, uniqueness, and positivity of solutions is vital for understanding long-term dynamics, yet remains challenging due to nonlinear reactions. This work establishes these properties for a broad class of reaction terms [ f_i ].

We employ compact semigroup theory generated by the tridiagonal diffusion operator [ A = \operatorname{diag}(d_1,\dots,d_m) \Delta_m ] on [ [C(\overline{\Omega})]^m ]. Key tools include positivity preservation through maximum principles and a priori bounds via comparison principles and fixed-point arguments in suitable Banach spaces, handling general nonlinearities with sublinear growth

Under assumptions of positive initial data and reaction terms satisfying [ f_i(t,x,\xi) \geq 0 ] for [ \xi \geq 0 ] with controlled growth, the system admits a unique global positive solution remaining bounded for all [ t > 0 ].

These results provide a robust framework for tridiagonal reaction-diffusion systems, applicable to multi-species models. The semigroup-[ L^1 ] approach extends to higher-order or non-local interactions, opening avenues for complex pattern formation studies.

In this work, we present a systematic and comprehensive analysis of vector breather dynamics in inhomogeneous optical media, modelled through a variable-coefficient coupled nonlinear Schrödinger (vc-CNLS) equation. This generalized model encompasses the fundamental components of Kerr-type nonlinear processes, including self-phase modulation, cross-phase modulation, and four-wave mixing, with temporally varying dispersion and nonlinear coefficients. Such variability effectively captures realistic physical settings where the optical properties of the medium evolve along the propagation direction or over time.

To explore the nonlinear wave dynamics supported by this framework, we construct explicit analytical breather solutions by employing a combination of similarity transformations and Darboux transformation techniques. This approach enables the generation of both Akhmediev breathers, which are localized in time and periodic in space, and Kuznetsov–Ma breathers, which display spatial localization and temporal periodicity. The resulting solutions reveal rich structures with controllable amplitudes, periods, and localization features. Our analysis demonstrates that the inhomogeneous nature of the medium plays a crucial role in shaping the evolution of optical breathers. In particular, variations in system parameters induce a diverse range of dynamical behaviors, including amplification, partial suppression, splitting, trapping, asymmetric deformation, and merging of oscillatory localized wave packets. These modulation effects highlight how engineered inhomogeneity can be used to tune or stabilize nonlinear excitations in practical optical systems. Overall, the findings contribute to a deeper understanding of localized coherent structures in variable nonlinear environments and offer potential pathways for controlling breather excitations in advanced optical communication, nonlinear photonic devices, and related applications.

In the present study, a detailed parametric investigation is carried out to examine the influence of the horizontal and vertical location parameters d₁ and d₂ on surface deformation caused by a long inclined tensile fault located at an arbitrary position (d₁,d₂) in a homogeneous isotropic elastic half-space, where d₁ denotes the horizontal distance of the upper edge of the fault and d₂ denotes the burial depth of the upper edge of the fault. Closed-form analytical expressions for displacement components from Singh et al., 2016, are utilized to analyze the sensitivity of horizontal and vertical surface displacements with respect to variations in fault position. It is shown that in a uniform elastic half-space, deformation is invariant under horizontal translation, i.e., the horizontal parameter d₁ produces only a translational shift in the deformation field without altering its amplitude. In contrast, deformation is highly sensitive to burial depth, i.e., the burial depth parameter d₂ significantly affects the magnitude, decay rate, and spatial distribution of deformation. The results demonstrate that in physical interpretations, a deeper tensile fault produces weaker surface deformation and shallow tensile faults produce intense surface deformation, while moving the fault sideways only shifts the deformation zone. Further, the effect of dip angle is also considered in the analysis of this study.

Li-Fi (Light Fidelity) uses visible light for wireless data transmission. System performance depends on both the light source's modulation and secondary optics. Spatial light distribution is critical for coverage uniformity and communication stability. This study presents a comparative analysis of two lenses, LiA and LiB, conducted using Techno Team 3D software. The goal was to evaluate how geometric modification affects photometric properties and identify the optimal configuration for a Li-Fi luminaire requiring wider beam angles and uniform light distribution.

Lens LiA Analysis
Simulation results show LiA has a maximum luminous intensity of 4689 cd and a beam angle of approximately 60° at half maximum. Estimated luminous flux is around 2755 lm. LiA produces a concentrated beam with distinct high-intensity zones. While suitable for directional lighting, this pattern limits the stable signal reception area and creates uneven coverage, which is undesirable for indoor optical wireless communication.

Lens LiB Analysis
After geometric modification, LiB shows a maximum intensity of 4304 cd. The beam angle at half maximum increases to about 80°. Simulations indicate wider light diffusion and smoother intensity distribution. Peak values are reduced, and luminous flux is spread more evenly. This is particularly beneficial for Li-Fi luminaires, as a larger illumination area improves channel accessibility.

An additional advantage of LiB is reduced contrast between the center and periphery of the light spot. This homogeneous illumination minimizes "dead zones" and enhances communication reliability. The optical redesign yields parameters better suited for advanced Li-Fi devices. For Li-Fi systems, uniform light distribution is more critical than high directionality. Expanding the emission angle increases coverage and reduces the likelihood of signal gaps. Calculated data confirm that transitioning from LiA to LiB broadens the solid angle and provides more homogeneous illumination, making LiB the preferred choice for integration into Li-Fi lighting systems.

This study develops and analyzes a nonlinear dynamical model describing the coupled evolution of corruption, organized crime, and cybercriminality under the influence of institutional trust, economic stress, and digital transparency. Unlike traditional enforcement-driven frameworks, the proposed model incorporates endogenous feedback between corruption prevalence and institutional trust, allowing corruption to both erode and be amplified by governance quality. Economic stress is introduced as a recruitment amplifier, while digital transparency functions as a structural suppression mechanism. The model is formulated as a system of nonlinear ordinary differential equations and rigorously examined. We establish positivity, boundedness, and the existence of a positively invariant region. Using the next-generation matrix approach, a modified basic reproduction number is derived that explicitly incorporates trust erosion and transparency effects. Local and global stability of the crime–corruption–free equilibrium are proven via Lyapunov techniques. Furthermore, center manifold analysis demonstrates the potential for backward bifurcation, indicating governance tipping thresholds and multi-stability regimes. An optimal control framework is constructed to determine cost-effective strategies combining transparency investment, prosecution intensity, and employment stabilization policies. Numerical investigations calibrated to India-specific governance and crime indicators illustrate how strengthening institutional trust and digital transparency can produce nonlinear reductions in corruption prevalence. The findings provide analytically grounded insights for sustainable governance reform in digitally evolving socio-economic systems.

The Central Limit Theorem (CLT) offers asymptotic results on the convergence of sample means to normality, but its finite sample properties are relatively less clear. This paper reports the findings of a simulation study on the convergence process of CLT for various continuous distributions, such as Normal, Exponential, Uniform, Gamma, Beta, and Cauchy. By conducting Monte Carlo simulations in R and employing the Shapiro–Wilk test for normality, we have found distribution-wise “threshold” and “optimum” sample sizes for which approximate normality is achieved.

One of the most important results of this study is that the convergence process to normality is not monotonic for finite samples. Even after achieving normality at a threshold sample size, statistically significant deviations are often found at intermediate sample sizes. This “post-threshold instability” is found to occur for all studied distributions, including the Normal distribution itself, for which non-negligible failure rates are found for specified ranges of sample size.

We also investigate the possible causes of these anomalies, focusing on the importance of outliers, simulation methods, and the limitations of normality tests. The empirical findings indicate that the behavior of outliers and the properties of the tails of distributions are of prime importance, especially for skewed and heavy-tailed distributions. In the context of the Cauchy distribution, the traditional CLT is invalid because of the lack of finite variance. Nevertheless, the use of the sample median and its asymptotic distribution offers a promising alternative.

Moreover, we illustrate that the number of iterations is not necessarily a factor that improves the accuracy of convergence with certainty, contrary to general beliefs in Monte Carlo simulation. The effect of the size of the iteration is also shown to be dependent on the distribution.

Nonlinear Fredholm integro-differential equations of the second kind play a pivotal role in modeling complex physical systems involving nonlocal interactions, memory effects, and long-range dependencies. However, when posed on extended intervals (τ ≫ 1), these equations present severe numerical challenges due to the loss of operator compactness, spectral degradation, and instability of conventional discretization schemes. Developing robust, structure-preserving algorithms for such large-domain problems remains an open and critical challenge in computational mathematics.

This work introduces a novel Dual Transformation–Linearization–Discretization (DT-L-D) framework specifically designed for nonlinear Fredholm integro-differential equations on extended intervals. The approach first applies a dual transformation: differentiating the original equation and decomposing the global domain into quasi-uniform subintervals, yielding a block-structured system that preserves coupling topology. Nonlinearity is then handled via Newton–Kantorovich iterative linearization in Banach space, followed by Nyström discretization to obtain a finite-dimensional linear algebraic system. Rigorous convergence analysis establishes quadratic convergence under explicit Lipschitz continuity and uniform invertibility conditions.

Numerical experiments on benchmark problems confirm the method's high accuracy, with global errors reaching 10⁻⁹ and residuals down to 10⁻¹⁵. The algorithm demonstrates remarkable stability across intervals up to τ = 500, requiring only 5–13 iterations for convergence. Crucially, the discrete iterates converge to the exact solution as the iteration count increases, independently of the discretization size—a theoretical advantage over classical discretize-then-linearize approaches.

The DT-L-D framework provides a reliable, efficient, and structure-preserving paradigm for solving nonlinear integro-differential equations on large domains. Its dimension-reduction capability (O(nN) complexity) and parallelization potential make it particularly valuable for scientific computing applications requiring high-precision solutions over extended intervals.