This is my personal website, which holds links to my research notes, and any related projects I happen to be working on, that mostly tend to be in the area of mathematics and computer science.

Currently my main areas of interest are: formal proof construction, group theory, algebraic graph theory and networks, coding and information theory, and the application of symmetry and invariance in developing models to speed up machine learning tasks. I am also interested in using ML techniques to automate data engineering problems.

Jamie Gabriel

I mostly use JupyterLab running through Docker to manage most things related to research. Full details and set up instructions here. Occasionally I like to work with the HTM Machine Intelligence algorithms which can be tricky to get going so I have included that in these notes. This set up is really nice across lots of different math research and works well across different operating systems.

If I am not using JupyterLab, it's usually RStudio & R these days and I tend to build anything web related in Shiny with Django as the backend.

If I am not using JupyterLab, it's usually RStudio & R these days and I tend to build anything web related in Shiny with Django as the backend.

I tend not to use Sage these days but here is a really nice way to install and config the Sage JupyterLab environment running on Docker. For full details of the Docker Image see https://hub.docker.com/r/cemulate/sagemath-jupyterlab/.

*16th January 22:*Ascertaining the criteria needed for graph populations to be bounded or unbounded (ES2)*15th January 22:*Exploring the fundamental properties of graphs and graph populations(ES1_2)*10th January 22:*A new series of notes on Exceptional Structures in mathematics(ES1_1)*15th December 21:*Ascertaining the uniqueness of algebraic solutions to the geometric quintic equation (SPE26)*19th December 21:*Deriving an algebraic solution of the general quintic polynomial equation (SPE25)*3rd December 21:*An overview of the fundamental structures of Algebraic Calculus and area on a parabola (AC1_2)*1st December 21:*Outlining the foundations of Affine geometry to underpin the construction of dCB curves(DCB3)*13th November 21:*Using De Casteljau's algorithm and splitting dCB curves (DCB2)*2nd November 21:*Transforming the general quintic to geometric form (SPE24)*27th October 21:*Incorporating the patterns found in signs of coefficient to generate a general solution to polynomial equations (SPE23)*23rd October 21:*Introduction to DeCastelau Bezier Curves and the DCB Parametrization theorem (DCB1)*10th October 21:*Continuing to explore the bridge between BiTri numbers and polygonal subdivisions (SPE22)*6th October 21:*Exploring ternary operations on Fuss polygonal subdivisions and thier associated ternary trees (SPE21)*2nd October 21:*Exploring the link between polygon subdivisions and polynomial equations (SPE20)*22nd September 21:*Understanding the formal object that is a sum of all polygons with all of their subdivisions into triangles (SPE18)*20th September 21:*Understanding how Catalan and Fuss numbers relate to binary and ternary trees and how these trees can be created from n-gons (SPE16)*2nd September 21:*Exploring the structure of the quartic equation in relation to BiTriQuad numbers (SPE15)*15th August 21:*Exploring the relationship between solutions to polynomial equations, sides of polygons and BiTri numbers (SPE13_14)*10th August 21:*Continuing to explore patterns of matrices created by the solutions of polynomial equations, referencing Knuth, Stanley and Dickau (SPE11)*4th August 21:*Exploring different pattterns in the matrices created by the solutions of polynomial equations (SPE10)*29th July 21:*Extending the conjecture beyond Catalan numbers using matrices and examining resulting patterns using OEIS to find Catalan and Fuss numbers (SPE9)*26th July 21:*Comparing the conjecture for a solution to the general cubic using Catalan Numbers to traditional formulas (SPE8)*15th July 21:*Testing a conjecture for a solution to the general cubic using Catalan Numbers (SPE7)*10th July 21:*Extending the power series approach to the quartic case (SPE6)*28th June 21:*Unpacking the structure of coefficients in the solutions of general polynomials (SPE5)*20th June 21:*Finding patterns in solutions of general polynomials (SPE4)*18th June 21:*Extending the power series approach to the cubic case (SPE3)*7th June 21:*Discovering Catalan numbers in the solutions of general quadratics when using power series subsitution (SPE2)*2nd June 21:*The inherent difficulties that arise when solving higher degree polynomials using radicals (SPE1)