Cooperative Game

This is the game theory, but prime numbers are quite similar because of 2^N. This is complicated. Coalition is the maximum profits. Everybody should be happy, but you need to choose your partners.

v is called characteristic function. N is finite players.

This is dual game.

∀S⊆N

S is opportunity cost which is excluded from N. However, this is subset. S=N

Subgames is S≠N.

This is S⊆N and S≠N.

∀T⊆S

S seems to be isolated.

Knotty Problems

Knotty Problems are unsolvable problems computer can't answer. Now we talk about AI, and semiconductors keep expanding although Moore's law seems to be ending. Tech companies have huge profit in this difficult time. Paradox is philosophical matter, but we can see it in topology.

This ring is crossing. You pile rings, so this is three dimensions although this is flat. You need to connect two circles in one. This is tricky.

g is genus, and K is the minimal genus of a surface. K is unknotting, if g(K)=0. K1#K2 is adding together. When K1 and K2 are unknotting, K1+k2 is unknotting.

Laplace transform

It is hard to handle the infinity.

t is time, and this is t function which is quite huge. You transform it to s function. This is called Laplace transform.

s=σ+iω

This is an imaginary space.

This is s function.

Noiseless coding theorem

How is information source shorten?
There are M sources.

S=(a1,a2,・・, aM)

a is each information and p is possibility.

pi=P(ai)

You encode it.

K(ai)=ci

L is the length.

li=｜K(ai)｜

Entropy is I(E), and possibility of E is P(E).

∴

EX.

This is the first dimension.

Monty Hall problem

This is the popular math problem in an American TV show. They are panic because of the strange result. This is probability. We still don't understand in the AI era.

There are 3 doors. When you open the one of three, there is the gold. Two doors are empty.

You think that the probability is 1/3. However, you have the second chance. You can see where the empty door is.

Therefore, you have 50% chance to get the gold.

You are in front of the 3 door. This is zero, but you don't know yet. You have 1/3 probability. You can change your decision.

You got the gold. When you change the door, you increase the probability. This is 2/3 which is about 67%>50%. Each door has 1/3 probability, so (1)+(3)=2/3.

E(X) is expectation. Xi is the gold and Pi is probability.

AI is getting smarter because they have more chance for better solution. AlphaGo is well known.

Keller's conjecture

Keller's conjecture is solved, but you may not understand what it is. This seems to be true in 6 dimensions, but it doesn't work in more than 10 dimensions. We are living in 4 dimensions, so it is hard to capture it. At first, Keller's conjecture is to cover an area with equal-size tiles without any gaps or overlap. The conjecture is that at least two of the tiles will have to share an edge and that this is true for spaces of every dimension.

This is square. I guess that 6 dimensions are the square, and 0 is the dot. You see the space. 6 dimensions work in this conjecture. Calabi–Yau manifold is well known, so I think that 6 dimensions is rolled up like the square. Moreover, each square has the different color and the same size with sharing the edge. It is not overlapping.

We know 4 dimensions, so when we add this 6 dimensions, we are living in 10 dimensions. However, nobody knows it. I think that square is disappeared like strings. We can't see it. This is called SAT which is the problem using a propositional formula—(A or not B) and (B or C), etc. Collatz problem is the same.

Collatz problem

Every positive integer except for zero includes 1. This is apparent 5*1=5. Therefore, every positive integer must be reduced to 1. This is called Collatz problem.

When you have even numbers, you divide them by 2. Then if the integer is the even number, you divide it by 2. However, 10/2 is 5. In this case, you have odd numbers, so you multiply 3 to them and you plus 1 to it. You repeat over and over again, until you reach to 1.

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ex.

18/2=9, 9*3+1=28, 28/2=14, 14/2=7, 7*3+1=22, 22/2=11, 11*3+1=34, 34/2=17, 17*3+1=52, 52/2=26, 26/2=13, 13*3+1=40, 40/2=20, 20/2=10, 10/2=5, 5*3+1=16, 16/2=8, 8/2=4, 4/2=2, 2/2=1

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Euler's constant

I found an interesting tweet. This is expanding, and you find pattern so it must be a good tool for AI.

σ(n) is sum of divisors and n is infinite.

(n> 5040)

If you find it, you can disprove the Riemann Hypothesis.

p is the i-th prime number.

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ex.

30=2*3*5
900=(2^2)*(3^2)*(5^2)=30^2
27000=(2^3)*(3^3)*(5^3)=30^3

・
・

30^α=(2^α)*(3^α)*(5^α)

α=1,2,3,4・・・・

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Therefore, you can see this.

Φ is multiplication of infinite prime numbers.
I define that multiplication of fractal is still fractal.

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n=5050=2*(5^2)*101
σ(5050)=1+2+5+10+25+50+101+202+505+1010+2525+5050=9486=X
5050 ln(ln 5050)≒10823.43=Y
X/Y≒0.8764

Capacitance

You need high capacitance for your electric cars and solar chargers. Electricity may be cheaper, and you have your own energy. Independence is democratic flows, but it depends on technological innovation. You need efficient batteries.

C=εA/d

C is capacitance, ε is dielectric constant. Electron move to A, and d is the distance.

C=Q/V

V is voltage. Q is the ratio of the maximum charge.

σ=Q/A

σ is the surface charge density.

This is nonlinear and a Taylor series.

δ→0 is very small, so you can ignore but electron jump to A for energy efficiency.

d should be narrowed. Electrons flow like water to black pole (+) even in low voltage. This is called tunneling effect. You see solar chargers.

The voltage provided by the battery pushes electrons along the negative direction.

Deep Boltzmann machine

Deep learning is recognized as more sophisticated artificial intelligence such as AlphaGo. Computer has already gone beyond human ability. AI can learn by itself through past experience. This is the difference between machine learning. It is more than gathering data.

You add deep units.

You still need visibility, and the possibility should be developed. It is almost perfect.

Restricted Boltzmann machine

Machine learning is well known as self driving cars and P=NP. This treats huge numbers, so you need to reject infinity. You want practical machines. You may call it efficiency.

This is restricted because V and H are related but V1 and Vn are separated. H1 and Hm are also not connected.

E is the whole energy. ai and bj are bias because of uncertainty. Moreover, Wij=Wji is symmetry. This is the mathematical beauty.

As you know, image recognition is visibility. Therefore, you need only i.

P is possibility, and T is temperature. You need to cool your machine.

Euler–Lagrange equation

This is expansion.

Lagrange(L) is [kinetic energy - potential energy].

This is Newtonian Equation of motion.

The space move, but it is still zero.

neighborhood

In general topology, a neighborhood is in a set which is opened but still closed. This is like the concept of my website.

X is the topological space, and p is the neighborhood. Moreover, U is the open set.

Therefore, V include p.

In a metric space M=(X,d), V is a set and the neighborhood of p which is the center of the open ball.

r is the radius of Br(p).