Quantum-Causality And Properties of a Class of Nonlinear Covariant Systems (of the Type xi yi zi vi ui = dXYZVU)

This article develops “existence” properties and associated proofs for the equations x^2+y^2+z^2+v^2=dXYZV; x^2+y^2+z^2+v^2+u^2=dXYZVU, x^2+y^2+z^2+v^2+u^2+X^2+Y^2+Z^2+V^2+U^2=dXYZVU and x^i+y^i+z^i+v^i =dXYZV (i is a positive integer), and x│X (ie. X is a multiple of x), y│Y, z│Z and v│V and u│U are real numbers; and each of the variables x,y,z,v,u, dXYZV and dXYZVU are multiples of (n-f) in real numbers. The proofs are within the context of Sub-Rings. The solutions derived herein can be extended to other problems wherein (n-f) can take the form of functions and polynomials such as (3a-1), (12b-5), (as-bs), etc.. This article also: i) summarizes the relationships of these equations to Homotopy Theory, PDEs, Mathematical Cryptography and Analysis; ii) explains why (n-f) can serve as a measure of multicollinearity, and iii) introduces several Anomaly-Detection indicators (including (n-f)); and iv) develops simple Java code for solving the equations x^i+y^i+z^i+v^i+u^i+=dXYZVU and x^2+y^2+z^2+v^2+u^2+X^2+Y^2+Z^2+V^2+U^2=dXYZVU (where: x є X, y єY, z є Z, v є V, and u є U) and similar classes of equations and for Integers in the interval: (-10^1677721600000000) < x,y,z,v,u, d, X,Y,Z,V,U< 10^1677721600000000 (and even larger positive-integers and smaller negative integers depending on available computing power)