### Mathematical Definition

A BPC can be characterized by adding two extra constraints and to (47) where and are the Branch Point test functions. The complete BVP defining a BPC point using the minimal extended system is

|  |  | (84)

where

is defined by requiring

| (85)

Here and are functions, and are scalars and

| (86)

where the bordering operators , function , vector and scalars and are chosen so that is nonsingular [15][16].