P^2+Σ2P: 10月 2020

2020年10月30日金曜日

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.

2020年10月25日日曜日

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.

2020年10月15日木曜日

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.

2020年10月13日火曜日

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.

2020年10月12日月曜日

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.

ex.