Lorentzian manifold

Lorentzian manifold is based on general relativity. When you board the airplane, it is necessary to avoid delaying your flight. All passengers need to have their seats before their flight. If some of them stopped other passengers to board, they would be delayed to take them off.

When a passenger had baggage, he would stop near his seat. His move must be slow. This is not efficient. You can't stop the flow.

γ(t) is causal sets. t∈[a,t]

Moreover, L(t) is arc length.

p and q are two points such as a seat (green) and a passenger (red).

p,q∈M

C(p,q)

M×M→Ｒ∪∞

This is called strong causality condition.

All passengers can't stop the flow, so you need to choose passengers who have baggage first. They move slowly to have their seats. Other passengers wait for having their seats. Therefore, nobody needs to stop before having a seat.

Bertrand Arthur William Russell

The real world is often different from logical thinking. We perceive the reality through our five senses like eyes and nose. Then, we describe the logic.

I am a Japanese, and we know that there is no Darth Vader in democratic society.

He was the last shogunate.

I mention it mathematically.

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Every shogunate is bald.

(α) There is at least a shogunate in Japan even now.
(β) There is at most a shogunate in Japan.
(γ) Every shogunate is bald in Japan.

x and y are variable. B(x) means the baldness.

∴

This is completely arbitrary. You see the baldness.

PageRank

Google search keep expanding. You may fear for stealing your personal data, although you still click on the web. Web is quite democratized from your shopping to election. Your opportunity is concentrated on your small screen in these days. You can find whatever you want in it recently. Value is abstract, and it may depend on your country. AI is getting smarter and smarter. This is social evolution. Your life is convenient. You keep searching. This is PageRank. It is hard to escape from it.

X is your website. You may call it your virtual shop. Someone links your website, such as I1,I2, and In.

You also link your favorite pages, such as O1, O2, and Om.

Therefore, if your page is valuable, P must be bigger.

Feit–Thompson theorem

It is quite mysterious that every finite group of odd order is solvable. You see that odd factorial order is even number. Prime number is included in this theory.

3!/2=3

5!/2=60

7!/2=2520

9!/2=181440

11!/2=19958400

13!/2=3113510400

15!/2=653837184000

17!/2=177843714048000

19!/2=60822550204416000

21!/2=25545471085854720000

3,5,7,11,13,17,19 are prime numbers.

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G is a group, and P and Q are odd primes. (P < Q)


(1+KQ) is the subgroups of G, so it is a divisor of ｜G｜ . K is an integer.
Therefore, P^2 is divided by (1+KQ).


∴

(1+KQ)=1,P or P^2

Twistor theory

PN is the space of real null rays in Minkowski space.

PT is the projective twistor space.

Z is a twistor, and Z2=Z3=0

You see Penrose tiling.

Minkowski space

Minkowski space is based on Lorentz transformation.

w = i ct

This is imaginary time.

You can't exceed the speed of light, so you must be inside of red lines.

Future ←

Past →

Lorentz transformation

You move from S(x,y,z) to S'(x',y',z') by uniform linear motion (v) on x.

This is called Galilean transformation.

c is the speed of light.

This is S'.

You see that the speed of light should be stable, so S' is the same.

You know Galilean transformation.

∴

Then you see S.

∴

A=E

∴

You know Lorentz factor.