Solving a problem of angiogenesis of degree three

TL;DR

This paper introduces a novel approach to modeling angiogenesis of degree three using weighted Fermat-Torricelli trees, deriving new tree structures from generalized Gauss trees and analyzing their residual absorbing rates.

Contribution

It presents a new mathematical framework connecting Fermat-Torricelli and Gauss trees to model angiogenesis of degree three, including the concept of universal absorbing sets.

Findings

Derived a family of limiting tree structures from generalized Gauss trees.

Identified a universal absorbing set governing the residual rates.

Linked the minimal residual rate to the creation of a degree three Gauss tree.

Abstract

An absorbing weighted Fermat-Torricelli tree of degree four is a weighted Fermat-Torricelli tree of degree four which is derived as a limiting tree structure from a generalized Gauss tree of degree three (weighted full Steiner tree) of the same boundary convex quadrilateral in R^2: By letting the four variable positive weights which correspond to the fixed vertices of the quadrilateral and satisfy the dynamic plasticity equations of the weighted quadrilateral, we obtain a family of limiting tree structures of generalized Gauss trees which concentrate to the same weighted Fermat-Torricelli tree of degree four (universal absorbing Fermat-Torricelli tree). The values of the residual absorbing rates for each derived weighted Fermat-Torricelli tree of degree four of the universal Fermat-Torricelli tree form a universal absorbing set. The minimum of the universal absorbing Fermat-Torricelli set is responsible for the creation of a generalized Gauss tree of degree three for a boundary convex quadrilateral derived by a weighted Fermat-Torricelli tree of a boundary triangle (Angiogenesis of degree three). Each value from the universal absorbing set contains an evolutionary process of a generalized Gauss tree of degree three.