Search Results (437)

The integration of AI tools like ChatGPT into educational assessments, particularly in the context of Multivariable Calculus, represents a transformative approach to personalized and scalable learning. This study examines the Exams as a Service (EaaS)-Flipped Chatbot Test (FCT) framework, implemented through the AIQuest platform, to explore how chatbots can support assessment processes while addressing risks related to automation and academic integrity. The methodology combines static and dynamic assessment modes within a cloud-based environment that generates, evaluates, and provides feedback on student responses. Quantitative survey data and qualitative written reflections were analyzed using a mixed-methods approach, incorporating Grounded Theory to identify emerging cognitive patterns. The results reveal differences in students’ engagement, performance, and reasoning patterns between AI-assisted and non-AI assessment conditions, highlighting the role of structured AI-generated feedback in supporting reflective and metacognitive processes. Quantitative results indicate higher and more homogeneous performance under the reverse evaluation, while survey responses show generally positive perceptions of feedback usefulness and task appropriateness. This study contributes integrated quantitative and qualitative evidence on the design of AI-assisted evaluation frameworks as formative and diagnostic tools, offering guidance for educators to implement AI-based evaluation systems. Full article

A fractal approach for the Timoshenko beam theory by applying differential vector calculus in a three-dimensional continuum with a fractal metric is developed. First, a summary of the tools needed, mathematical relationships, and background of fractal continuum mechanics is presented. Then, the static and dynamical parts of the Timoshenko beam equation are extended to fractal manifolds. Afterwards, an intrafractal beam constructed as a Cartesian product is suggested and the fractal dimensionalities of the Balankin beam are scrutinized. This allows comparing both intrafractal beams when they have the same Hausdorff dimension but different connectivity. Finally, the effects of fractal attributes on the mechanical properties of the deformable fractal medium are highlighted. Some applications of the developed tools are briefly outlined. Full article

During the last decade, F-contraction has been a widely investigated problem in the fixed point theory. There are various outcomes regarding the extensions and generalizations of F-contraction in different perspectives, along with the findings concerning the application of those ideas, mostly in the area of differential and difference equations, fractional calculus, etc. The present article concludes some existence and uniqueness outcomes on fixed points for $(\phi ,\mathrm{F})$–contractions in the context of a metric space endowed with a local class of transitive binary relations. Some illustrative examples are furnished to justify that our contraction conditions are more general than many others in this area. The findings presented herein are used to obtain a unique solution to certain fractional boundary value problems. Full article

The goal of this paper is to use a Boole-type inequality framework to provide better estimates for differentiable functions. Using majorization theory, fractional integral operators are incorporated into a new auxiliary identity. The method establishes sharp bounds by combining the properties of convex functions with classical inequalities like the Power mean and Hölder inequalities, as well as the Niezgoda–Jensen–Mercer (NJM) inequality for majorized tuples. Additionally, the study presents real-world examples involving special functions and examines pertinent quadrature rules. This work’s primary contribution is the extension and generalization of a number of results that are already known in the current body of mathematical literature. Full article

The first aim of this study is to point out new aspects of approximation theory applied to a few classes of holomorphic functions via Vitali’s theorem. The approximation is made with the aid of the complex moments of the functions involved, which are defined similarly to the moments of a real-valued continuous function. By applying uniform approximation of continuous functions on compact intervals via Korovkin’s theorem, the hard part concerning uniform approximation on compact subsets of the complex plane follows according to Vitali’s theorem. The theorem on the set of zeros of a holomorphic function is also applied. In the end, the existence and uniqueness of the solution for a multidimensional moment problem are characterized in terms of limits of sums of quadratic expressions. This is the application appearing at the end of the title. Consequences resulting from the first part of the paper are pointed out with the aid of functional calculus for self-adjoint operators. Full article

Modernizing public security risk governance demands a paradigm shift from reactive response to proactive, systems-oriented prevention. Prevailing governance models, with their focus on institutions and technology, often neglect the micro-foundational mechanisms of risk generation: the internal psychological processes of individuals. To address this gap, this study develops a novel theoretical model—the F(T, P, C, R) framework—which integrates self-organization theory with a psychological gaming perspective. We conceptualize an individual’s behavioral choice (F_behavior) as an emergent outcome of the dynamic interplay among four constitutive factors: the situational context of Time (T) and Place (P), and the cognitive assessments of perceived Risk Control power (C) and perceived Risk Destructive power (R). Employing automotive driving behavior—specifically decisions regarding safe distance maintenance and the adoption of autonomous driving technologies—as our primary analytical scenario, we derive a dynamic risk-decision matrix. This matrix categorizes behavioral outcomes into four distinct quadrants (Confirm, Tend-to-Confirm, Tend-to-Deny, Deny) based on the subjective calculus between C and R, thereby elucidating the internal logic of risk-related choices. The study’s main contribution is constituted by this novel micro-behavioral analytical framework that integrates cognitive science with systems-based governance principles. It offers theoretical insights for behavioral public policy and provides a structured toolkit for diagnosing and designing targeted interventions, ultimately aiming to enhance proactive risk management and systemic resilience. Full article

This study introduces a nonlinear fractional bi-susceptible model for COVID-19 using the Atangana–Baleanu derivative in Caputo sense (ABC). The fractional framework captures nonlocal effects and temporal decay, offering a realistic presentation of persistent infection cycles and delayed recovery. Within this setting, we investigate multiple transmission modes, determine the major risk factors, and analyze the long-term dynamics of the disease. Analytical results are obtained at equilibrium states, and fundamental properties of the model are validated. Numerical simulations based on the Toufik–Atangana method further endorse the theoretical results and emphasize the effectiveness of the ABC derivative. Bifurcation analysis illustrates that adjusting time-invariant treatment and awareness efforts can accelerate pandemic control. Sensitivity analysis identifies the most significant parameters, which are used to construct an optimal control problem to determine effective disease control strategies. The numerical results reveal that the proposed control interventions minimize both infection levels and associated costs. Overall, this research work demonstrates the modeling strength of the ABC derivative by integrating fractional calculus, bifurcation theory, and optimal control for efficient epidemic management. Full article

This paper introduces a rigorously defined fractional Heun operator constructed through a symmetric composition of left and right Riemann–Liouville fractional derivatives. By deriving a compatible fractional Pearson-type equation, a new weight function and Hilbert space setting are established, ensuring the operator’s self-adjointness under natural fractional boundary conditions. Within this framework, we prove the existence of a real, discrete spectrum and demonstrate that the corresponding eigenfunctions form a complete orthogonal system in $L_{{\omega }_{\alpha }}^2(a,b)$. The central theoretical result shows that the fractional eigenpairs $({\lambda }_n^{\left(\alpha \right)},u_n^{\left(\alpha \right)})$ converge continuously to their classical Heun counterparts $({\lambda }_n^{\left(1\right)},u_n^{\left(1\right)})$ as $\alpha \to 1^{-}$. This provides a rigorous analytic bridge between fractional and classical spectral theories. A numerical study based on the fractional Legendre case confirms the predicted self-adjointness and spectral convergence, illustrating the smooth deformation of the classical eigenfunctions into their fractional counterparts. The results establish the fractional Heun operator as a mathematically consistent generalization capable of generating new families of orthogonal fractional functions. Full article

This paper uses the theory of self-affine fractal functions to model the dynamic flight graphs of starling flocks, integrating the fractional calculus of self-affine fractal functions to quantitatively characterize the intrinsic nonlinear dynamics and memory effects within the system, employing statistical inference methods to find the fractal fit for the images. The changes in box dimensions over time could characterize the phase transition process of the starling flight flocks. By analyzing the rate of change of fractal dimensions, we identify critical points corresponding to phase transitions during collective flight behavior. During the flight of the starling flocks, a real-time phase transition process for evading attacks and effective advancement has been identified. Experimental data confirms the effectiveness of controlling the phase transition. Full article

In this paper, we develop a geometric, structure-preserving semi-discrete formulation of Maxwell’s equations in both three- and two-dimensional settings within the framework of discrete exterior calculus. The proposed approach preserves the intrinsic geometric and topological structures of the continuous theory while providing a consistent spatial discretization. We analyze the essential properties of the proposed semi-discrete model and compare them with those of the classical Maxwell’s equations. As a representative example, the framework is applied to a combinatorial two-dimensional torus, where the semi-discrete Maxwell system reduces to a set of first-order linear ordinary differential equations. An explicit expression for the general solution of this system is also derived. Full article

The present work centers on the significance of q-calculus in geometric function theory and its expanding applications within the domain of Te-univalent functions, especially those associated with special polynomials like the q-Bernoulli polynomials. Motivated by recent interest in these polynomials, our study introduces and analyzes a generalized subclass of Te-univalent functions that intimately relate to q-Bernoulli polynomials. For this new family, we establish explicit bounds for $|d_2|$ and $|d_3|$, and provide estimates for the Fekete–Szegö functional $|d_3-\xi d_2^2|, \xi \in \mathbb{R}$. Our findings contribute new results and demonstrate meaningful connections to prior work involving Te-univalent and subordinate functions, thereby broadening and integrating various strands of the existing literature. Full article

In this study, we introduce and rigorously formalize the notion of (P, m)-superquadratic stochastic processes, representing a novel and far-reaching generalization of classical convex stochastic processes. By exploring their intrinsic structural characteristics, we establish advanced Jensen and Hermite–Hadamard ($\mathbb{H}.\mathbb{H}$)-type inequalities within the mean-square stochastic calculus framework. Furthermore, we extend these inequalities to their fractional counterparts via stochastic Riemann–Liouville ($\mathbb{RL}$) fractional integrals, thereby enriching the analytical machinery available for fractional stochastic analysis. The theoretical findings are comprehensively validated through graphical visualizations and detailed tabular illustrations, constructed from diverse numerical examples to highlight the behavior and accuracy of the proposed results. Beyond their theoretical depth, the developed framework is applied to information theory, where we introduce new classes of stochastic divergence measures. The proposed results significantly refine the approximation of stochastic and fractional stochastic differential equations governed by convex stochastic processes, thereby enhancing the precision, stability, and applicability of existing stochastic models. To ensure reproducibility and computational transparency, all graph-generation commands, numerical procedures, and execution times are provided, offering a complete and verifiable reference for future research in stochastic and fractional inequality theory. Full article

In this paper, we establish some Simpson-type inequalities within the framework of Riemann–Liouville fractional calculus, specifically tailored for differentiable harmonically convex functions. By introducing a novel fractional integral identity for differentiable functions with harmonic arguments, we derive several estimates that generalize and refine existing results in the literature. The theoretical findings are validated through a numerical example supported by graphical illustration, and potential applications in approximation theory and numerical analysis are discussed. Full article

This work explores the issue of identifying sub-synchronous oscillations (SSOs). Regular detection techniques face issues with response timings to variations in viewpoint and adaptability to variations in conditions of the system but our proposed method overcomes them. We have actually come up with a new framework called Tempered Fractional Deterministic Learning (TF-DL) that successfully combines tempered fractional calculus with deterministic learning theory. This method makes a memory-based learner that works best for oscillatory dynamics. This lets SSO identification happen faster through a recursive structure that can run in real time. Theoretical analysis validates exponential convergence in the context of persistent excitation. Simulations show that detection time is 62.7% shorter than gradient descent, with better convergence and better parameters. Full article

Quantum calculus is a powerful extension of classical calculus, providing novel tools for deriving sharper and more efficient analytical results without relying on limits. This study investigates error estimations for Milne formula-type inequalities within the framework of quantum calculus, offering a fresh perspective on numerical integration theory. New variants of Milne’s formula-type inequalities are established for q-differentiable convex functions by first deriving a key quantum integral identity. The primary aim of this work is to obtain sharper and more accurate bounds for Milne’s formula compared to existing results in the literature. The validity of the proposed results is demonstrated through illustrative examples and graphical analysis. Furthermore, applications to special means of real numbers, the Mittag–Leffler function, and numerical integration formulas are presented to emphasize the practical significance of the findings. This study contributes to advancing the theoretical foundations of both classical and quantum calculus and enhances the understanding of integral inequality theory. Full article