The empirical method of Maurey

It is hard to define boundary mathematically. Zeta function is well known. ζ(s)=0

You need to fill all spaces for integration.

A⊂R^d

x is the convex hull of A.

When λn, n>d+1. Then, n-1 go to 0.

x2-x1=0.00000000000000000009. μ is the vector.

x1	x2	・・・・	xn
λ1(0.9)	λ11(0.00000000009)
λ2(0.09)	λ12(0.000000000009)
λ3(0.009)	λ13(0.0000000000009)
λ4(0.0009)	λ14(0.00000000000009)
λ5(0.00009)	λ15(0.000000000000009)
λ6(0.000009)	λ16(0.0000000000000009)
λ7(0.0000009)	λ17(0.00000000000000009)
λ8(0.00000009)	λ18(0.000000000000000009)
λ9(0.000000009)	λ19(0.0000000000000000009)
λ10(0.0000000009)	λ20(0.00000000000000000009)

λ10 is λ1 because of x1 which is the same group. You use Σ in your Excel. μ10 is also μ1. This is 0.0000000001. Then you keep expanding, but you close to Zero. However, you never reach Zero.

This is known as power law.

Van Aubel's theorem

Each square is connected, and the center is lined.

PR and SQ is vertical, and they are symmetry.

AB=2a, BC=2b, CD=2c, DA=2d, and they are complex numbers.

2a+2b+2c+2d=0

∴

a+b+c+d=0

You see the complex number, so A=0. Then, AP=p is z.

∴

p=a+ia=(1+i)a

You see Euler's formula.
π/2=90°

Then, you turn around squares.

SQ=A and PR=B

B=iA

iB=i^2A

∴

A+iB=0

Squares are curved.

Monte Carlo method

The sum of probability in high dimensions is chaotic. You can divide your places infinitely. You try n times, and you find viruses m times.

m/n

n is the dimension.

N=1

I=Σf(x)

This is zeta function.

n=1/σ

If you have r dimensions, this is too large.

n=(1/σ)^r

Petri net

Infection is graphical mathematically.

N=(P,T,F)

N is a finite ordered list of elements. P and T are disjoint finite sets of places 〇 and transitions □, respectively.

F⊆(P×T)∪(T×P)

This is quite chaotic, so viruses pass through places.
You are closed, so dead or alive is uncertain. We still use masks to decrease possibility blindly.

Automatic differentiation

The chain rule connect different functions for calculus. Our neural network is based on it, but it includes infinity, so adding much data cause some troubles.

 We need to calculate faster. Symbolic differentiation faces the difficulty of converting a computer program into a single mathematical expression, and Numerical differentiation can introduce round-off errors in the discretization process and cancellation. Therefore, optimization process is needed.

●Forward accumulation is the chain rule from inside to outside.

●Reverse accumulation has the traversal from outside to inside.

This is NP-complete.

Analytic geometry

I found this article on twitter. AI search it according to my interest, although privacy on your Cookie cause political difficulty.

r=(α-β)/2π

As long as the parallel curves are closed, you can find r. The problem is that convex and concave curves are chaotic

I think that this is Knotty Problems.

g(α#β)=g(α)+g(β)

g(X)=0, so you can decide α and β alternatively, and you still define r=ρ.

Lie Algebra

This is n×n complex matrices.

[A,B]=AB-BA

Lie Algebra is the standard commutator on M.

T-commutator is nonzero matrix.

A,B∈M

[A,A]T=0

This is a bilinear map.

M^T

M^1=M

It satisfies Jacobi identity.

When T is invertible, M is isomorphic.

ad-bc≠0

When T isn't invertible, M isn't isomorphic.

Moreover, M^T and M^S are Lie isomorphic, if T and S aren't invertible, although T and S are the same rank.

When T is all trace, [A,B]T=0