Stable Matching

This is the marriage problem, so it must be better than insects morally. However, this is computer science, so it is quite mechanical. In this algorithm, everybody must be married. Everybody may hate you, but you are included in this circle. Therefore, everybody has the partner. This is idealistic, but it is just gaming.

They are singles. (1,2,3,4) are males. (a,b,c,d) are females. Male(1) prefers female(a) the most, and Male(2) prefers female(c) the most. This is the competition, so you downgrade your preference like (3:a,b,d,c).

You minimize it for stable matching.

min{n,k}

n is the total number of males, and k is the total number of females.

(1,a) (2,c) (3,b) (4,d)

Everybody is happy mathematically, although male(4) likes female(c) and female(a). Patience is necessary sometimes.

This is not stable. (3,d) and (4,d) are conflict. They may fight each other, although d isn't their favorite woman. They may abandon the marriage. Male(4) chooses female(b) for stable matching, but his life may be the nightmare.

Offspring

We often read arguing about the gender gap on the web. Women have the power politically in these days, but you are just like insects in computer science. In developed countries, there are more women. They have many opportunities. They may have children, or they may pursue the business success. Some men enjoy their single lives. However, some sociologists often insist that the increasing population is necessary for the economic stability. It is criticized that women are not the machine for reproduction, but we are still in the animal instinct mathematically.

There is the free choice to have opposite gender partners. This is chaotic. You call it natural greedy heuristic in computer science.

G=(V,E), V={x1,x2,x3・・・}, E= φ

V is the population, and E is the edge. Men1 has three partners, and women6 has two.

Am is the average of opposite gender partners for men.

Aw is the average of opposite gender partners for women.

∴

Am/Aw=｜Vw｜/｜Vm｜>1

The symmetry is broken in the developed country. This is NP hard, but you have offspring like insects.

You just follow the statistics except for Japanese and German.

Robertson Walker Metric

General Relativity and gravity describe the universe with the large scale, and it is homogeneous and isotropic.

Prime numbers also move the same pattern.

This is called Robertson Walker Metric.

X=a(t)x, Y=a(t)y, Z=a(t)z

t is time.

∴　

dS^2=X^2+Y^2+Z^2

Then you describe a 3D space of constant curvature.

The Hoyle-Narlikar theory of gravitation

Pascal’s triangle is symmetric, and prime numbers are fractal.

You need time to find all numbers even in quantum computers.

Then we talk about gravity.

The universe expands infinitely, but it is finite because of masses. They have the same sign.

a and b are particles over the world lines. G is the green function which satisfies the wave equation.

G* must be symmetric.

rst is retarded, and adv is advanced.

It cannot be satisfied for any models of the universe, so it can’t contain an infinite amount of matter.

φ=-κρ (ρ>0)

ρ is the density. φ is infinite and the same sign. Infinity destroys the Newton theory, so it expands the universe but it is also retarded.

Fermat's little theorem

p is a prime number and {a} can't be divided by p.

You use the strong induction to prove Fermat's little theorem.

∴

This is the 7th row of Pascal’s triangle.

Odd numbers are 1, and even numbers are 0.

This is fractal.

i	0	1	2	3	4	5	6	7
p	1	7	21	35	35	21	7	1

---

(p,i)≡0 mod p

(i≠0,7)

---

For example, (7,2)=7!/2!(7-2)!=5040/240=21

p!=p(p-1)!

∴

1^p≡1 mod p

This is apparent.

Therefore

2^p=(1+1)^p=1+(p,1)+(p,2)+・・・+(p,p-1)+1≡1+0+0+0+・・・+1=2 mod p

3^p=(1+2)^p=1+2(p,1)+2^2(p,2)+・・・+2^(p-1)(p,p-1)+2^p≡1+0+0+0+・・・+2=3 mod p

4^p=(1+3)^p=1+3(p,1)+3^2(p,2)+・・・+3^(p-1)(p,p-1)+3^p≡1+0+0+0+・・・+3=4 mod p

You can expand it because of Pascal’s triangle which is the binomial theorem.

(n,i)=n!/i!(n-i)!

Then you define a^(p-1).

5^16≡1 mod 17

This is also clear, so you can prove the strong induction.