Paul Bird's Scientific Research

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) ...

Existence of Solutions to Systems of Polynomial Inequalities I was wondering whether there is a general method to deciding if a system of polynomial inequalities has a solution. If there was such an algorithm this could be used to solve questions such as the sphere packing question. (The kissing number). Simply find all the inequalities for each sphere and determine if a solution can be found. To skip to the end it turns out that all systems of polynomial inequalities can be put in the following form: {A nm x n x m +B n x n +C  ≥ 0} or using the notation X ={x1,x2,x3,...,1} {Q nm X n X m   ≥ 0} Thus all conditions are built up from matrices, A , vectors B and constants C. We must deal with all the special cases where various components are zero. For the sphere packing problem we can set all the { B =0} which makes it much simpler. Simplest Case The simplest case is a single linear equation A x +B ≥ 0 This has no solution when A=0 & B<0 Star...

N=8 Supergravity coupled to E 8 xE 8 N=4 Super Yang-Mills unified in 19 Dimensions? Pure N=8 supergravity can be derived from a compactification of an 11 dimensional action. The massless(?) sector of Heterotic String Theory contains 256 + 248x2x16 = 2 13 particles. This suggests that it could be the compactification of a higher dimension action in 19 dimensions . It should be related to a 16 dimensional lattice. Only up to 11D has group structure . fermion = (D-3)x2 (D-3)/2 graviton = (D-2)(D-1)/2-1 antisymmetric n-tensor (D-2)(D-3)(D-4)..(D-n-1)/n! 11D: graviton ( 44 ) + antisymmetric-3-tensor( 84 ) + fermion ( 128 )  = 2 8 components. (Root vectors of E8 240 roots, 16 less) 19D: graviton ( 152 ) +  232  x vector( 17 )+ fermion ( 4096 ) = 2 13 components. (136, 680, 2380, 6188) (Root vectors of 16D lattice...? E 8 xE 8 ? D 16 ? (too small) Coexter? ~ 8160 roots? 32 less? 4320 densest in 16 dimensions?) 27D:  196608 components. ...

Conjugate of Lie Algebra A little article noting the similarities between momentum conservation (continuous) and charge conservation (discrete). We all know that the conjugate coordinate to position is momentum . Thus a term such as  ∫O(-k)O(-q)O(k+q)dk 4 dq 4 = ∫O'(x)O'(x)O'(x)dx 4 But a lie group given by it's root vectors (charges) also has a similar form which looks a lot like momentum: O {-A} O {-B} O {A+B} + (A n )O {A} O {-A} O n So we should be able to find conjugate variables to the root vectors. So we can write the above as: O' C O' C O' C The O' C  being linear combinations of the Os. In the case of E 8 , the {A} are 8 dimensional unit vectors. One difficulty are the neutral particles, O n  , which seemingly have no equivalent in the momentum picture. Such a term would look like:  ∫O(k)O(-k)kμAμ(0)dk 4 =  ∫O(x)Bμ∂μO(x)dx 4   which is not Lorenz invariant.... [ unless O(x) were spinors and Bμ were Dirac matrices?? B...

Why should N=8 Supergravity be finite? I would like to attempt to explain here why adding N=8 supersymmetry to General Relativity should have the chance of making the theory finite. Firstly, it is known that the only renormalizable theories are those of spin 0, spin 1/2 and spin 1 particles.  General Relativity, however, is quantized by spin 2 particles - gravitons . (Adding simple N=1 supersymmetry to GR also adds spin 3/2 particles - gravitinos ). If we add N=8 supersymmetry we have a single super-particle whose components compose of spin 2, 3/2, 1, 1/2, 0 , -1/2, -3/2, -2 particles. We can think of this as some kind of super-vector. There are (1,8,28,56,72,56,28,8,1) of each of these types of particles respectively. What we would like to do is get rid of these spin 2 and spin 3/2 components (18 of them) because they are non-renormalizable. So, we should like to be able to rotate this super-vector in such a way that these components become zero. We also, need to rotate th...

Three-point Green's function in position space In this post I'm trying to calculate the 3-point interaction function for quantum field theory in position space. It sounds simple enough. It is just calculating the 4 dimensional integral over w of:   Δ(x-w) Δ(y-w) Δ(z-w)   which should produce a symmetric function in three variables: A(|x-y|,|y-z|,|z-x|) however it turns out to be extremely complicated to get a nice formula in terms of symmetric variables. See here for my notes on it so far It would be nice to relate this function to geometric properties such as the area of the triangle but that doesn't seem to be the case. Only the sums of the powers of the lengths of the sides which are symmetric forms. I have a series solution for the case when m is not zero but can't find a solution for when m=0. If anyone can help let me know!

Concise Rules of the Universe In this post I was taking the challenge of instilling all the rules of the Universe that we know of into one side of A4. More technically: taking an embedding of the Standard Model into the low energy sector of a Superstring Theory and try to give all the rules for this as concisely as possible. It is a useful exercise because, as people say, if you can't explain something simply, you probably don't really understand it yourself. It turns out that it is not so simple to do. Particularly describing the particle content of N=8 Supergravity is trivial but stating which interactions are allowed is more complicated. Unless someone knows a simple description? Concise Rules of the Universe Have a go yourself!