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.

Lorentz factor

When you move, your light clock is slower about T times. This is called Lorentz factor.

c is the speed of light.

2a/c=1

This is the light clock, so you always have 1 second.

When you move from A to C, your light clock move slower.

T=2L/c

T is time, and c is the speed of light.

You see Pythagorean theorem.

∴

You see 2a=c from the light clock.

This is called Lorentz factor.

Upper half-plane

H is the upper half-plane which contains the set of complex numbers with positive imaginary part.

This is modular group.

ad − bc = 1

This includes point at infinity.

Modular Group

a,b,c,d are integers.

ad-bc=1

This is modular group which is also isomorphic of PSL(2, Z).

a/b, a/c, c/d and b/d are irreducible.

(a,b,c,d)=(7,5,4,3),(4,11,17,47),(4,7,13,23),(2,5,17,43),(6,19,23,73),(8,13,19,31),(48,13,11,3)

You see 3 prime numbers and 1 even number in it.

Τ=3P+N

(P:prime number, N:even number)
2 is prime number, but it is also even number. This has special meaning in my blog.

This is also irreducible .

You see (7,5,4,3).

7p+5q

4p+3q

(p=2, q=3)

14+15=29

8+9=17

29/17=1.705882353

PSL(V)=SL(V)/SZ(V)

PSL is projective linear group, and V is vector space. Moreover, SZ(V) is the center, and SL(V) is factor group.

This is order.

d is the center.