P^2+Σ2P

This is taxicab geometry. You move from X to Y. Each square is 1. Red:1+2+2+1+2+2=10 Green:5+5=10 They are the same line. This is called Manhattan distance. n is the dimension. Then, this is Euclidean norm. This is 1*1 squares. ∴ There are 25 suares, so K=25. √50≒7.07. 50=5^2+5^2. This is the green.

Higher dimensions are chaotic . However, you close to Zero, although you keep expanding. This is almost 1-1=0. k go to ∞, which is called Euclidean norm. Then you pile squares . They are almost circles, and cubes are also balls. They are closed. However, higher dimensions are opened, so they are chaotic.

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) ...

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.

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

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.

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 .