This paper presents an adaptive block time-stepping algorithm designed for the numerical integration of first-order stiff oscillatory problems, which typically involve multiple time scales and stringent stability conditions. The proposed method integrates a block formulation enabling the simultaneous evaluation of several solution points with an adaptive step size mechanism to effectively regulate local truncation errors. A predictor-corrector strategy is employed, where an explicit predictor generates initial estimates, and an implicit block corrector enhances stability for stiff systems. The resulting nonlinear equations are resolved using Newton's iterative method. The scheme is proven to... More >

Generative artificial intelligence (GenAI) is increasingly embedded in enterprise workflows, yet the models that support these workflows are updated at discrete and sometimes frequent intervals. Each update can improve technical effectiveness while simultaneously invalidating prompts, interfaces, controls, and employee routines. This study develops an impulsive dynamical system to analyze the resulting tension between technical improvement and organizational disruption. The continuous subsystem tracks enterprise process stability, organizational adaptation, and effective model capability between updates. The impulsive subsystem represents the instantaneous effects of a model release on workf... More >

This study develops and analyzes a nonlinear mathematical model to describe the dynamics of marriage, extramarital affairs, marital breakdown, and divorce. The population is divided into five compartments: single $S(t)$, married $M(t)$, individuals in extramarital affairs $A(t)$, broken marriages $B(t)$, and divorced $D(t)$. The model is formulated as a system of nonlinear ordinary differential equations capturing transitions between these states through parameters representing marriage formation, affair initiation, reconciliation, separation, and divorce. Basic qualitative properties, including positivity and boundedness of solutions, are established to ensure the model is well posed. Equil... More >

This paper develops a deliberately expanded analytic program around the three-dimensional incompressible Navier--Stokes equations. The exposition is intentionally heavy in algebra, index notation, multilinear expansions, differentiated hierarchies, and page-filling calculations. The central idea is to organize the continuation problem around a mixed pressure--vorticity--anisotropic functional, while keeping visible every major nonlinear channel: energy, Hessian growth, high-order commutators, vorticity stretching, elliptic pressure reconstruction, dyadic localization, paradifferential splitting, kernel representation, cylindrical-coordinate geometry, and dense symbolic closure systems. The r... More >