This study develops a treatment-modulated discrete framework for breast-tumor dynamics, emphasizing dimensional consistency, treatment dependence, and the distinction between scalar tumor burden and spatial tumor-density fields. Starting from the dimensional logistic growth law, treatment is introduced through a cycle-dependent survival factor. Normalization by the continuous carrying capacity $K_c$ yields the exact Euler update $y_{n+1}=S_n[y_n+g\Delta t_n y_n(1-y_n)]$, whereas the canonical non-autonomous map $x_{n+1}=r_nx_n(1-x_n)$, with $r_n=(1+g\Delta t_n)S_n$, is retained as a reduced nonlinear benchmark. This distinction prevents discretization-induced period-doubling and chaos from b... 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 >

This article develops a constructive approach to the Painlevé equivalence problem based on explicit normalization and residual obstruction analysis. Instead of starting from the full invariant complexity of the general point-equivalence problem, we use target-adapted reductions that convert equivalence into a finite algebraic comparison problem inside canonical normalized families. For derivative-free polynomial classes, the method yields exact results. We obtain a constructive criterion for equivalence to the first Painlevé equation in the quadratic case and an exact criterion for equivalence to the second Painlevé equation in the cubic case. In both settings, the method provides explici... More >

This study presents a mathematical and computational analysis of blood flow in the abdominal aorta under healthy, aneurysmal, and rupture-related conditions. The proposed framework combines a Newtonian fluid approximation with a Lattice Boltzmann formulation to simulate the hemodynamic behavior of the aorta in three-dimensional geometry. The governing equations were implemented in Python, allowing the generation of numerical simulations and corresponding three-dimensional visualizations of aneurysm progression. The analysis considered clinically inspired stages ranging from the healthy vessel to advanced aneurysmal dilation and rupture. The results showed that aneurysm growth significantly a... More >