Some qualitative results are provided on the cylindrical version of the Lane-Emden equation. Although the explicit form of the solution is not exactly known, we establish some properties on the critical points of the solution in some specific cases and we propose a technique to evaluate the distance between a pair of solutions in small intervals.

The process of revitalising quiescent cells in order for them to proliferate plays a pivotal role in the repair of warn out tissues as well as for tissue homeostasis. This process is also crucial in the growth, development and well-being of higher multi-cellular organisms such as mammals. Deregulation of quiescent-proliferation transition is related to many diseases, in particular cancer. Recent studies have revealed that this process is regulated tightly by the Rb−E2F bistable switch mechanism. Based on experimental observations, in this study, we formulate a mathematical model to examine the effect of the growth factor concentration on the proliferation-quiescence transition in human cells. Working with a non-dimensionalised model, we prove positivity, boundedness and uniqueness of solutions. To understand model solution behaviour close to bifurcation points, we carry out bifurcation analysis, which is further verified by use of numerical bifurcation analysis, sensitivity analysis and numerical simulations. Indeed, bifurcation and numerical analysis of the model predicted a transition between stable, bistable and stable states, which are dependent on the growth factor concentration parameter (GF). The derived predictions confirm experimental observations.

This study demonstrates a global, non-parametric, non-iterative optimization of time-mean value of a kind of index vibrated by time-varying forcing. It is based on the fact that the (steady) forced vibration of non-autonomous ordinary differential equation systems is well approximated by an analytical solution when the amplitude of forcing is sufficiently small and its base state without forcing is linearly stable and steady. It is applied to optimize a time-averaged heat-transfer rate on a two-dimensional thermal convection field in a square cavity with horizontal temperature difference, and the globally optimal way of vibrational forcing, i.e. the globally optimal, spatial distribution of vibrational heat and vorticity sources, is first obtained. The maximized vibrational thermal convection corresponds well to the state of internal gravity wave resonance. In contrast, the minimized thermal convection is weak, keeping the boundary layers on both sidewalls thick.