Paul Bird's Scientific Research

This is an article I wrote on the intrinsic Riemann Curvature of Implicit Surfaces (Algebraic Varieties) . If you have any thoughts on this please comment below.

We shall define the  Generalised Complex Contour Integral as: \[  \oint_X \Delta[\phi]D\phi \equiv i \int\int \delta[X]\left( \overline{\Delta}[\overline{\phi}]\det\left[\frac{\delta X}{\delta \phi}\right] + \Delta[\phi] \det\left[\frac{\delta X}{\delta \overline{\phi}}\right] \right) D\phi D\overline{\phi} \] The function \[ \Delta[\phi] \]  is for \[ S_0 \] : \[ \Delta[\phi]  = \frac{1}{2\pi i (\phi_1 - \phi_2)}\] For \[S_n\] where n>0 \[ \Delta[\phi]  = \frac{1}{\int \varepsilon_{abc..}\phi^a(x) \partial_1 \phi^b(x) \partial_2 \phi^c(x)...     dx^n}\]  

 Type Theory My understanding of Type theory. Defining natural numbers: zero: Nat succ(n: Nat ): Nat So for example defining the number 2: succ(succ(zero)): Nat same as??? isNat(zero) ForAll(n) :  isNat(n)=>isNat(succ(n)) We can define operations like plus and times and equals //define proofs that  n==n equals(zero,zero): Equals< zero,zero > equals(succ(n: Nat ),succ(n: Nat )): Equals< succ ( n :Nat ) , succ ( n :Nat ) > Proofs and Dependent types Define a dependent type Even <n: Nat > which is a proof that n is even.  {a: Nat, p: Equals <times(2,a),n>}: Even <n: Nat > So that for example the following is a proof that 6 is even: {3, equals(times(2,3),6)}: Even <6> which becomes {3: Nat ,equals(6,6): Equals <6,6>} : Even <6> {a: Nat ,  p: Equals <times(2,a),n>}: Even <n: Nat > isEven(n) = n =>    (a => (isNat(a) & equals(times(2,a),n))) isEven(6) (3)  In C syntax typedef Even < Nat n> { Nat a, Equals <ti

We can define a distance invariant under superspace transformations: \[ R^2 = |x-y|^2 + \theta \gamma_\mu(x-y)^\mu \zeta \] For D space-time dimensions and E Grassman dimensions, does this satisfy a Cayley-Menger type equation? In 2 dimensions we know the volume of a tetrahedron is zero. Or: \[ \det \begin{bmatrix} 0 & R_{12} & R_{13} & R_{14} & 1\\ R_{12} & 0 & R_{23} & R_{24} & 1\\ R_{13} & R_{23} & 0 & R_{34} &1\\ R_{14} & R_{24} & R_{34} & 0 &1 \\ 1&1&1&1&0 \end{bmatrix} =0 \] What about when D=11 and E=32 ? Simplest is D=1, E=2 with (?): \[ R^2 = (x-y)^2 + (x-y)(\theta \zeta + \zeta^* \theta^* ) \] with for example: \[ R^4 = (x-y)^4 + 2 (x-y)^3(\theta \zeta + \zeta^* \theta^* ) + 2(x-y)^2( \theta \theta^* \zeta \zeta^* ) \] \[ R^6 = (x-y)^6 + 3 (x-y)^5(\theta \zeta + \zeta^* \theta^* ) + 6(x-y)^4( \theta \theta^* \zeta \zeta^* ) \] Then the area of the triangle is 0 (or an

A network of belts would be equivalent to a network of directed graphs. A half twist in a belt corresponds to a directed arrow. In particular a sub structure rotating 360 degrees would reverse the arrows going into the substructure:   Can we be more clear on graph equivalences? Therefor such a patch as fermionic structure. But what are fermions? (Perhaps all nodes are fermions.)