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

・
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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).