Kaniadakis deformed $\kappa$-mathematics is an area of mathematics that has found relevance in the analysis of complex systems. Specifically, the mathematical framework in the context of a first-order decay $\kappa$-differential equation is investigated, facilitating an in-depth examination of the $\kappa$-mathematical structure. This framework serves as a foundational platform, representing the simplest non-trivial setting for such inquiries which are demonstrated for the first time in the literature. Finally, additional avenues of study are discussed.

We examine phenomenological consequences of a scale-dependent gravity model in which the gravitational coupling runs with the cosmological scale factor as G(a)=G0an. In the companion paper, this running was shown to modify the Friedmann equation through effective starred couplings and to produce corresponding changes in the Wheeler–DeWitt minisuperspace potential. For values of n, we ask whether the same running can survive into observationally tested epochs. Applying three consistency tests—the expansion rate during Big-Bang nucleosynthesis, the growth of linear matter perturbations, and the CMB acoustic scale—we find that BBN and CMB consistency require n to be very small if the power-law form persists to nucleosynthesis or recombination. These tests constrain the epoch over which the pure power-law form can remain valid. The power-law running is best considered as an early-universe scaling regime that must shut off before nucleosynthesis — the question is not whether, but when and how. Any realistic extension of the strong-running regime must cut off, screen, or smoothly transition the gravitational coupling back to standard gravity before the nucleosynthesis era.

We study an effective quantum cosmological model with a scale-dependent gravitational coupling G(a). Starting from a quantum-corrected Newtonian cosmology, we derive a modified Friedmann equation and quantize the corresponding minisuperspace model to obtain a Wheeler–DeWitt equation. For the power-law running G(a)=G0an, the small-scale behavior is controlled by the exponent n. In the flat dust model with a positive quantum correction term Ξ, the values n=3/2 and n=2 mark distinct analytic boundaries: n=3/2 marks the sign change of the quantum-correction coefficient, while n=2 marks the degeneracy of the leading powers of a. A genuine bounce occurs only when a root of H2(a) separates a forbidden small-a region from an allowed expanding branch, so that nonsingular sectors appear without spatial curvature, exotic matter, or nonstandard quantization prescriptions. We obtain Bessel, Airy, numerical, and WKB solutions, and show that the model separates genuine DeWitt suppression from regularity at the origin. The WKB analysis exhibits a forbidden region, a turning point, and an oscillatory semiclassical branch, while the conserved probability current distinguishes expanding and contracting sectors. A scalar-field clock yields wave packets that track the classical trajectory at large scale factor; the WKB Hamilton–Jacobi limit recovers the modified Friedmann dynamics as the leading-order phase of the Wheeler–DeWitt wave function.

The problem of an electrically charged pendulum above a grounded conducting surface is examined using traditional analyses, such as forces, energy, and torque. The system was numerically modelled using a finite difference method and analysed. A number of classroom activities have been suggested.

We study an effective quantum cosmological model with a scale-dependent gravitational coupling $G\left(a\right)$. Starting from a Newtonian cosmological construction inspired by a corrected two-body gravitational potential, we derive an effective Friedmann-type equation and quantize the corresponding minisuperspace model to obtain a Wheeler–DeWitt equation. For the power-law running $G\left(a\right)=G_0{a}^n$, the small-scale behavior is controlled by the exponent n. In the flat dust model under the repulsive effective-potential prescription adopted here, with a quantum correction term $\mathrm{\Xi }>0$, the values $n=3/2$ and $n=2$ mark distinct analytic boundaries: $n=3/2$ marks the sign change of the quantum-correction coefficient, while $n=2$ marks the degeneracy of the leading powers of a. A genuine bounce occurs only when a root of $H^2\left(a\right)$ separates a forbidden small-a region from an allowed expanding branch, so that nonsingular sectors appear without spatial curvature, exotic matter, or nonstandard quantization prescriptions. We obtain Bessel, Airy, numerical, and WKB solutions, and show that the model separates genuine DeWitt suppression from regularity at the origin. The WKB analysis exhibits a forbidden region, a turning point, and an oscillatory semiclassical branch, while the conserved probability current distinguishes expanding and contracting sectors. A scalar field treated as a weakly backreacting internal clock yields wave packets that track the leading-order classical relational trajectory at a large scale factor; the WKB Hamilton–Jacobi limit recovers the modified Friedmann dynamics as the leading-order phase of the Wheeler–DeWitt wave function.