Mathematical insights into neuroendocrine transdifferentiation of human prostate cancer cells

Prostate cancer represents the second most common cancer diagnosed in men and the fifth most common cause of death from cancer worldwide. In this paper, we consider a nonlinear mathematical model exploring the role of neuroendocrine transdifferentiation in human prostate cancer cell dynamics. Sufficient conditions are given for both the biological relevance of the model’s solutions and for the existence of its equilibria. By means of a suitable Liapunov functional the global asymptotic stability of the tumour-free equilibrium is proven, and through the use of sensitivity and bifurcation analyses we identify the parameters responsible for the occurrence of Hopf and saddle-node bifurcations. Numerical simulations are provided highlighting the behaviour discovered, and the results are discussed together with possible improvements to the model

Rigorous computer-assisted bounds on the period doubling renormalisation fixed point and eigenfunctions in maps with critical point of degree 4

We gain tight rigorous bounds on the renormalisation fixed point for period doubling in families of unimodal maps with degree $4$ critical point. We use a contraction mapping argument to bound essential eigenfunctions and eigenvalues for the linearisation of the operator and for the operator controlling the scaling of added noise. Multi-precision arithmetic with rigorous directed rounding is used to bound operations in a space of analytic functions yielding tight bounds on power series coefficients and universal constants to over $320$ significant figures.Comment: 15 pages, 8 figure

A graph-theoretic condition for irreducibility of a set of cone preserving matrices

Given a closed, convex and pointed cone K in R^n , we present a result which infers K-irreducibility of sets of K-quasipositive matrices from strong connectedness of certain bipartite digraphs. The matrix-sets are defined via products, and the main result is relevant to applications in biology and chemistry. Several examples are presented