In this paper, we express the generating function of a combination of cranks of partitions that belong to congruence classes 3, 4, 5, 6, and 7 modulo 15, in terms of the Eisenstein series. Our results rely on the observation that the generating function of the combination we consider is a weight 1 modular form on Γ0(225) with the non-trivial modulo 3 primitive Dirichlet character. We obtain the modularity result by analyzing the transformation properties of related Klein forms. The main results are proven using the fact that the corresponding space is generated by Eisenstein series.
In prey-predator interactions, the introduction of the fear effect is crucial. It captures non-lethal predator impacts on prey behavior, making the interaction more realistic and accurate. The introduction of the fear effect enriches the understanding of population dynamics by providing a comprehensive insight into the interaction. In this study, the dynamics of an updated prey-predator model augmented by the fear effect are analyzed. We have identified the conditions under which the system’s fixed points switch from stable to unstable behavior, as verified by graphs. Through Neimark-Sacker and period-doubling bifurcations, we have highlighted the complex dynamics that fluctuate between stable, periodic, and chaotic regimes. We have used mathematical, graphical, and dynamical results to validate the complex dynamics. We have shown that chaos exists in the model using Marotto’s criterion. To control the model’s bifurcation, we employed a hybrid control technique. We have proved the local and global stability of the system in a stochastic environment. In particular, we have drawn two-parameter plots to unfold the rich dynamics. We have observed four distinct dynamic regimes in the plots: periodic, quasiperiodic, bifurcation, and chaotic. The Arnold tongues are observed as a sign of periodicity in the model. To identify different periods of Arnold’s tongues, we have drawn iso-periodic plots, which show that Arnold’s tongues produce period-adding sequences. From the numerical results, we observe that increased fear leads to greater vigilance, which in turn stabilizes the system. In contrast, reduced fear leads to reduced vigilance, resulting in a bifurcation in the system. Furthermore, increased noise causes instability in the population, while controlled noise has a less significant impact on the dynamics of the model. Through this graphical abstract, the dynamics of a novel model of the environmental population are presented. The dynamics are divided into two environments: stochastic and deterministic. In the stochastic environment, the model is proven to be locally and globally stochastically stable. In a deterministic environment, the population model exhibits rich dynamics. Initially, the topological types of the three calculated fixed points are classified. Later, two types of bifurcations were identified: the period-doubling bifurcation and the Neimark-Sacker bifurcation. The periodicity of the deterministic model, arising from the Neimark-Sacker bifurcation, is demonstrated via Arnold tongues, and the associated periods are identified using isoperiodic plots. The existence of chaos in the model is mathematically proven, and a control technique is proposed to mitigate the chaos. Finally, it is concluded that the increased fear effect causes increased vigilance, which in turn causes stability in the system, while the decreased fear effect results in less vigilance, which in turn causes bifurcation in the system.
Why have some communist regimes persisted into the twenty-first century while others have collapsed? This paper examines regime resilience by analyzing leadership succession in two of the five remaining communist states: China and Vietnam. Drawing on comparative historical analysis, it introduces the concept of “revolutionary regency,” a mechanism of generational transition that enables the gradual and voluntary withdrawal of revolutionary elders from formal authority while allowing their strategic intervention during moments of crisis. This process simultaneously reinforces leadership norms and ensures regime stability. Theoretically, the theory-building exercise contributes to the study of authoritarian durability by highlighting how revolutionary regency serves as a key mechanism underpinning the resilience of post-revolutionary communist regimes.
Mining activities significantly alter environmental conditions by increasing atmospheric aerosol loads and degrading water quality. This study employs Google Earth Engine (GEE) to analyze spatiotemporal anomalies in Aerosol Optical Depth (AOD), PM₂.₅ concentrations, chlorophyll-a levels, and turbidity in Tonk, Rajasthan, between 2016 and 2022. Data from Copernicus Atmosphere Monitoring Service (CAMS) and Landsat 8/9 Surface Reflectance (SR) were used to compute anomalies based on deviations from the 2000–2015 baseline. Results indicate a strong correlation between mining density and AOD (r = 0.82), PM₂.₅ (r = 0.69), chlorophyll-a (r = 0.76), and turbidity (r = 0.88), highlighting significant environmental degradation. Increased AOD and PM₂.₅ values were observed near active mining sites, confirming mining-induced dust emissions. Water quality analysis revealed elevated chlorophyll and turbidity anomalies, likely due to mining runoff, sedimentation, and nutrient enrichment leading to eutrophication. The findings emphasize the need for sustainable mining practices, stricter regulatory measures, and enhanced pollution monitoring to mitigate environmental degradation in the region.
This paper critically engages with Mohammad Ayish’s (2025) meta-analysis of 1,226 publications and 88 curricula in Arab media studies to argue that the field’s digital turn necessitates not only methodological innovation but also an epistemological reconceptualization grounded in postcolonial and decolonial thought. While affirming Ayish’s findings of a paradigm shift toward digital topics (54.41
