This study conducts a bivariate volatility analysis of crude oil prices and Nigeria’s exchange rate using data from 2010 to 2024 (180 observations). It employs descriptive statistics, cross-correlation, Granger causality, Johansen cointegration tests, and both VAR and DBEKK MGARCH frameworks to capture dynamic interactions and volatility persistence. The VAR results show a positive effect of crude oil prices on the exchange rate (0.1574, p = 0.0368) and mean reversion in the exchange rate (−0.158, p = 0.0344), with limited short-run causality at lag 1. DBEKK MGARCH estimates reveal pronounced volatility clustering with asymmetric persistence across series. Crude oil price volatility exhibits strong baseline and ARCH/GARCH dynamics, with A1(1,1) = 0.674 and B1(1,1) = 0.529, while cross-channel instantaneous spillovers are weak (M(1,2) = 0.287, not statistically significant, p = 0.872). The unconditional baseline reflects substantial volatility (M(1,1) = 12.986, p = 0.0216) and a persistent second channel with B1(2,2) = 1.010 and A1(2,2) = 0.286, indicating very strong persistence and long-lasting volatility in that channel. Exchange rate volatility is highly responsive to shocks, with A1(2,2) = 0.286 and B1(2,2) = 1.010, and M(2,2) = -2.474 (p<0.001), suggesting a damping but dominant persistence pattern in the second variance component. The cross-channel influence remains weak under diagonal BEKK, as evidenced by a non-significant M(1,2) and relatively small cross-covariance terms. Overall, a substantial long-run correlation exists between the volatility processes, linking oil-price and currency risks (strong co-movement in volatility). Implications emphasize continuous volatility monitoring and hedging to mitigate persistent risk, and policies to stabilize currency markets and enhance resilience to oil-price shocks through hedging, reserves management, and macroprudential tools.

Ranked Set Sampling is becoming more prevalent since this method proposes more accurate estimates without increasing sample size. It is especially useful when the process of ranking is simple but measurement is costly or difficult. Ranked Set Sampling enhances productivity and decision-making accuracy by reducing variance and Mean Squared Error. As a highly relevant and cost-effective sampling method, it is currently being used extensively for dependability analysis, industrial quality control, medical, agricultural, and environmental investigations. This review paper provides a structured discussion on the Ranked set sampling framework, ranking mechanisms, major modifications, inferential procedures, and practical applications.

Agri-entrepreneurship has emerged as a transformative strategy for strengthening rural livelihoods, enhancing income diversification and building resilience among farming households. In Chhattisgarh, where a significant proportion of the population depends on agriculture and allied activities, fluctuating market prices, climate variability and limited non-farm opportunities often exacerbate rural vulnerability. This study examines the role of agri-entrepreneurial initiatives including value addition, dairy enterprises, fisheries, minor forest produce processing, and agri-input services in improving household income stability and adaptive capacity. Using secondary data from government reports, NABARD publications, and state economic surveys, supplemented with field insights from selected districts, the paper demonstrates that agri-entrepreneurship contributes to income diversification by 25-40% among participating households. The findings suggest that targeted policy support, access to credit, skill development and market linkages can significantly enhance rural resilience and reduce poverty in the state.

Multi-modal biometric authentication systems face a critical trade-off: deep learning models achieve high accuracy but require substantial computational resources and lack interpretability making them unsuitable for edge devices and regulated environments. This paper addresses this gap by conducting the first comprehensive empirical comparison of Adaptive Neuro-Fuzzy Inference System (ANFIS) against five state-of-the-art deep learning architectures (CNN, DNN, RNN/LSTM, Transformer) for real-time multi-modal fusion of face, iris, and palmprint biometrics. Our novel contributions include: (1) a systematic evaluation framework demonstrating ANFIS achieves superior accuracy (98.1%) and lowest EER (1.5%) while requiring 93.7% less training time and 91.1% less inference time than Transformers; (2) statistical validation via 10-fold cross-validation proving ANFIS improvements are statistically significant (p<0.05); and (3) an adaptive weight adjustment mechanism enabling dynamic modality reweighting based on input quality. Results on three benchmark datasets (LFW, CASIA-Iris-V4, PolyU) with 20-user real-world validation demonstrate ANFIS as the optimal solution for resource-constrained, explainable biometric systems addressing critical deployment gaps in mobile, IoT, and privacy-sensitive applications.

Another important and highly investigated category of Partially Balanced Incomplete Block Designs (PBIBDs) with two associate classes is triangular designs. This paper describes a construction procedure of triangular designs with an unreduced Balanced Incomplete Block Design (BIBD). Triangular designs form a special case of PBIBDs, and are therefore also applicable to experimental design, in coding theory, and in finite geometry by their clear associate structure and incidence properties. The suggested construction offers a systematic structure through the parameters and incidence relationships of an unreduced BIBD to generate a triangular scheme of association. The relation between the parameters of the unreduced BIBD and those of the triangular design obtained is determined, and the conditions that are required to be the case are checked. A number of examples are provided to prove the usefulness of the proposed approach. This method underlines a combinatorial association between BIBDs and PBIBDs.

Using bacteriophage predation, bacterial resistance, and Monod-type growth kinetics, this study creates and evaluates a multi-stage mathematical model for the dynamics of Escherichia coli (E. coli) in wastewater treatment systems. The pre-primary, primary, secondary, and tertiary stages of wastewater treatment are represented by a seven-compartment deterministic model. Memory and hereditary effects in bacterial dynamics are captured by extending the traditional integer-order model to a Caputo–Fabrizio fractional-order framework, which includes both constant and variable fractional orders. Fixed-point theory is used to establish the existence and uniqueness of solutions. The three-step Adams-Bashforth approach is used to do numerical simulations that compare formulations of integer order, constant fractional order, and variable fractional order. Results indicate that the variable fractional-order model provides enhanced flexibility and improved representation of transient dynamics relative to classical approaches. The proposed framework offers a mathematically rigorous and adaptable tool for analysing E. coli control strategies in wastewater treatment, with potential relevance to public health and Sustainable Development Goals related to water and sanitation.

Topological indices play vital role in mathematical chemistry. The first entire Zagreb index (FEZI) is one among them. This parameter has proven to be essential in various real-life scenarios, such as networking businesses and traffic management on roads. In this paper, explore the first entire Zagreb index for various class of generalized fuzzy transformation graphs. This research paper is also deals with the bounds for FEZI for generalized fuzzy transformation graphs in terms of elements of a fuzzy graph G.

We investigate the concept of Moore-Penrose inverses by relating annihilator and ideal in Ternary Semiring. We discussed some outcomes and also characterized the annihilator using Moore Penrose inverses.

This study numerically investigates the significance of non-Fourier heat conduction on the magnetohydrodynamic (MHD) flow of a Maxwell fluid over an inclined vertical plate, incorporating the effects of thermal radiation and couple stresses. The Cattaneo-Christov heat flux model is employed to account for finite thermal relaxation time, moving beyond the classical Fourier's law. The governing equations are transformed into a system of ordinary differential equations and solved using the bvp4c solver in MATLAB. The results indicate that the fluid velocity declines with increasing Weissenberg and couple stress parameters, while the temperature profile is enhanced by thermal radiation but diminished by the thermal relaxation parameter. Furthermore, a multiple linear regression analysis yields predictive models, revealing that the skin friction coefficient increases with the magnetic field and couple stress parameters, and the Nusselt number is augmented by the Prandtl number and thermal radiation.