[2] **viXra:1910.0245 [pdf]**
*submitted on 2019-10-15 11:22:08*

**Authors:** Teo Banica

**Comments:** 200 Pages.

A complex Hadamard matrix is a square matrix $H\in M_N(\mathbb C)$ whose entries are on the unit circle, $|H_{ij}|=1$, and whose rows and pairwise orthogonal. The main example is the Fourier matrix, $F_N=(w^{ij})$ with $w=e^{2\pi i/N}$. We discuss here the basic theory of such matrices, with emphasis on geometric and analytic aspects.

**Category:** Functions and Analysis

[1] **viXra:1910.0064 [pdf]**
*replaced on 2019-10-15 10:26:40*

**Authors:** Robert Jackson

**Comments:** 15 Pages.

The current gold standard for solving nonlinear partial differential equations, or PDEs, is the simplest equation method, or SEM. As a matter of fact, another prior technique for solving such equations, the G'/G-expansion method, appears to branch from the simplest equation method (SEM). This study will discuss a new method for solving PDEs called the generating function technique (GFT) which may establish a new precedence with respect to SEM. First, this study shows how GFT relates to SEM and the G'/G-expansion method. Next, the paper describes a new theorem that incorporates GFT, Ring and Knot theory in the finding of solutions to PDEs. Then the novel technique is applied in the derivation of new solutions to the Benjamin-Ono, QFT and Good Boussinesq equations. Finally, the study concludes via a discourse on the reasons why the technique is likely better than SEM and G'/G-expansion method, the scope and range of what GFT could ultimately accomplish, and the elucidation of a putative new branch of calculus, called "diversification".

**Category:** Functions and Analysis