The solutions to the radial Schrödinger equation for a pseudoharmonic potential and Kratzer potential have been studied separately in the past. Despite different reports on the Kratzer potential, the fundamental theoretical quantities such as Fisher information have not been reported. In this study, we obtain the solution to the radial Schrödinger equation for the combination of the pseudoharmonic and Kratzer potentials in the presence of a constant-dependent potential, utilizing the concepts and formalism of the supersymmetric and shape invariance approach. The position expectation value and momentum expectation value are calculated employing the Hellmann–Feynman Theory. These expectation values are then used to calculate the Fisher information for both position and momentum spaces in both the absence and presence of the constant-dependent potential. The results obtained revealed that the presence of the constant-dependent potential leads to an increase in the energy eigenvalue, as well as in the position and momentum expectation values. Additionally, the constant-dependent potential increases the Fisher information for both position and momentum spaces. Furthermore, the product of the position expectation value and the momentum expectation value, along with the product of the Fisher information, satisfies both Fisher’s inequality and Cramer–Rao’s inequality.

The abundance of ions within the ionosphere makes it an important region for both long range and satellite communication systems. However, characterizing the complexity in the ionosphere within the equatorial region of Abuja, with geographic coordinates of 8.99° N and 7.39° E and a geomagnetic latitude of −1.60, and Lagos, with geographic coordinates of 3.27° E and 6.48° N and a dip latitude of −1.72°, is a challenging and daunting task due to the intrinsic and external forces involved. In this study, chaos theory was applied on data from both an HF Doppler sounding system and the Global Navigation Satellite System (GNSS) for the characterization of the ionosphere over these two tropical locations during 2020–2021 with respect to the quality of high-frequency radio signals between the two locations. Our results suggest that the ionosphere at the two locations is chaotic, with its largest Lyapunov exponent values being greater than 0 ($0.011\le \lambda \le 0.041$) and its correlation dimension being in the range of $1.388\le D_2\le 1.775$. Furthermore, it was revealed that there exists a negative correlation between the state of the ionosphere and signal quality at the two locations. Using transfer entropy, it was confirmed that the ionosphere interfered more with signals during 2020, a year of lower solar activity (sunspot number, 8.8) compared to 2021 (sunspot number, 29.6). On a monthly scale, the influence of the ionosphere on signal quality was found to be complicated. The results obtained in this study will be useful in communication systems design, modelling, and prediction.

The solutions of the radial Schrödinger equation for a pseudoharmonic potential and Kratzer potentials has been studied separately in the past. The Kratzer potential despite different reports, the basic theoretic quantity such as Fisher information has not been reported. In this study, the solution of the radial Schrödinger equation for the combination of the pseudoharmonic and Kratzer potentials in the presence of a constant-dependent potential is obtained using the concepts and formalism of the supersymmetric and shape invariance approach. The position expectation value and momentum expectation value are calculated employing the Hellmann-Feynman Theory. The expectation values are used to calculate the Fisher information for both the position and momentum spaces in both the absence and presence of the constant-dependent potential. The obtained results revealed that the presence of the constant-dependent potential leads to an increase in the energy eigenvalue, as well as in the position and momentum expectation values. The presence of the constant-dependent potential also increases the Fisher information for both position and momentum spaces. Furthermore, the product of the position expectation value and the momentum expectation value along with the product of the Fisher information, satisfies both the Fisher’s inequality and the Cramer-Rao’s inequality.

Rising and volatile prices for conventional protein ingredients constrain smallholder livestock production across Kenya, creating strong incentives for locally produced feed alternatives. This study assesses the feasibility of integrating duckweed (Lemnaceae) cultivation with anaerobic digestion, nutrient recovery and livestock-feed production within a smallholder cattle system at Ngobit, Laikipia County. Four candidate configurations were evaluated: duckweed-tilapia aquaponics, duckweed-integrated swine wastewater treatment, duckweed in poultry waste management and feed production, and duckweed in bioenergy, biofertiliser and feed production (DBBF). A structured synthesis of 110 academic and institutional sources was combined with site observation and a SWOT analysis, then converted into a transparent weighted decision matrix across seven criteria derived from the study objectives and site constraints. DBBF scored highest (4.60 of 5.00) and weakly dominates all alternatives across the criterion set, so its first-ranked position is invariant to the weighting applied. A first-order water and nitrogen balance, developed from published pilot parameters, indicates that the herd's own manure saturates a pond of 500 to 750 m ² , yielding 8 to 12g of crude protein per head per day. Nitrogen rather than water is therefore limiting at this scale, and the configuration functions as a marginal protein supplement within a joint energy, nutrient and feed product rather than as a substitute for purchased concentrate. A staged implementation framework with explicit decision gates is proposed, and the parameters requiring measurement before deployment are identified. The weighted score is a structured decision-support indicator, not an experimentally measured performance index.

This study obtained the energy levels and examined the partition function (Z) of a quantum system described by an Eckart-like potential model. By adopting the Greene–Aldrich approximation scheme for the centrifugal term, the radial Schrödinger equation (SE) is solved and the analytic expression of the energy eigenvalues is obtained. The ro-vibrational Z is computed by explicitly incorporating the rotational quantum number, a feature often neglected or misapplied in many studies. This result is used to evaluate the key thermodynamic properties (TP), including the Gibbs free energy (G), entropy (S), and enthalpy (H). Numerical analysis reveals that the Z increases monotonically with temperature, while the G decreases in accordance with statistical thermodynamics. The S exhibits saturation-like behaviour at higher temperatures, while the H displays convex growth with increasing thermal energy. Parametric studies demonstrate that the Eckart-like potential allows for the controlled tuning of TP, with variations in the potential parameters, including the screening parameter, having distinct effects. The results generalise existing models, reproduce the Hulthén potential under specific conditions, show the effect of the rotational quantum number of TP, and provide new insights into the ro-vibrational statistical mechanics of exponential-type potentials.