Search This Blog

Paul Bird's Scientific Research

This is a blog containing some scientific research I have done in various fields. My scientific interests are in the areas of physics, mathematics, and the brain. They mostly consist of theories I come up with now and again. Some of them are a bit unconventional or speculative for traditional media.

Set Theory in Javascript

Get link
Facebook
X
Pinterest
Email
Other Apps

May 27, 2020

Get link
Facebook
X
Pinterest
Email
Other Apps

Comments

Vortex Solution to Navier Stokes Equations

August 01, 2011

This is a solution I found to the Navier Stokes equations . It uses the flow lines of the Hopf Fibration (which is basically just a way of covering the entire 3D space in circles). The flow lines can be seen in this image below: You can see my paper here. Solution to Navier-Stokes equation . Unfortunately I won't get a million pounds for winning the Clay Millennium prize because this is a compressible not incompressible solution. :( Oh, well. Nice try! I think it's pretty nice anyway. :) Also it uses the nice identity: which paramaterizes the Pythagorean quadruples!

Solving two polynomials in two variables

April 10, 2014

Solving two polynomials in two variables I had an algorithm for solving two polynomials in 2 variables by converting it into an equation in 1 variable. I'll see if I can remember it. Let us start with 2 polynomials. (solution: x=2 y=3). We try to eliminate x. (1) x 2 +4xy+2y 2 +3x+4y - 64 = 0 (2) 3x 2 -xy + y 2 +5x-y -22 = 0 First eliminate x 2 (3) =- (2) +3 (1) : 13xy +5y 2 +4x +13y -170 = 0 or (13y+4)x + (5y 2 +13y -170) = 0 Next use this equation to eliminate x 2 again from (1) . (1) .(13y+4) : (13y+4)x 2 + (52y 2 +55y+12)x + (13y+4)(2y 2 +4y - 64) = 0 - (3) .x: (13y+4)x 2 + (5y 2 +13y -170)x = 0 =(4) : (47y 2 +42y+182)x + (13y+4)(2y 2 +4y - 64) = 0 Now use these two linear equations to eliminate x: (13y+4) (4) - (47y 2 +42y+182) (3) : (13y+4)(13y+4)(2y 2 +4y - 64) - (47y 2 +42y+182)(5y 2 +13y -170) =0 or (y-3) ...

Tarski-Seidenberg theorem

April 21, 2014

Tarski-Seidenberg theorem States that a set of equalities and inequalities f(x1,x2,x3,....xn)>=0 can be reduced down to a set of equalities and inequalities f(x1,x2,x3,....xn-1)>=0 with one less variable. Thus all equations can be reduced to a set of equalities and inequalities on their coefficients. See also: cylindrical algebraic decomposition See: Elimination These algorithms aren't very simple.