Montgomery's pair correlation conjecture

Hybrid orbitals are the explanation of molecular geometry and atomic bonding.

This is Montgomery's pair correlation conjecture, which is the pair correlation between pairs of zeros of the Riemann zeta function.

γ and　γ' are imaginary.

This is like piling prime numbers.

Methane and hydrogen are gasses.

C=C is carbon bonding which is in surface. Gasses move like ↑ ↑ ↓. This is called π bond.

Hybrid orbital

P^2+Σ2P is the pattern of prime numbers which move like atom.

This is Carbon.

There are 6 electrons.

Σ is hybrid, and atomic orbitals are mixtures. Then, you see sp^3.

You move one electron in 2S to 2P. This is called promotion. 2S lose one electron.

There are 4 hydrogen atoms.

This is tetrahedron and sp^3. Methane is CH4.

In sp^2 hybridisation, 2S move to 2P.

This is triangle and sp^2.　Ethylen is C2H4 which is the double bond. C=C is connected tightly.

Golden ratio

This is quite popular.

Knots

We are all living things. This is big picture. The boundary of being disappear in Zero, which is called Knotty Problems. Topology is mechanical deformations, although we don't understand completely. We are in black hole, and AI may be enlightening but we still don’t know. We have different languages and cultures. The fusion is not easy.

qi = ±1

±1 is two colors.

N is the crossing number.

This is almost Zero, but the line is expanding. Wr/N is the average writhe.

Tuple

This is how AI think.

a1, a2, ..., an

This is n-tuple.
There are two sets such as (a1, a2, ..., an), (b1, b2, ..., bn), and (a1 = b1) ∧ (a2 = b2) ∧ ... ∧ (an = bn), which is called Cartesian product.

∴
a ∈ A, b ∈ B ⇒ (a, b) ∈ A × B

This is ordered pair. I use Python.

Moreover, this must be expansion.

Fibonacci number

F is Fibonacci number. Fm divide Fn, and m divide n.

F2=F0+F1=0+1=1

F3=F1+F2=1+1=2

F4=F2+F3=1+2=3

n=3m → F(2,8,34,144,610,2584,10946,・・・)

∴ Fm=F3=2

n=4m → F(3,21,144,987,6765,・・・)

∴ Fm=F4=3

n=5m → F(5,55,610,6765,・・・）

∴ Fm=F5=5

This is fractal and expansion.

The Boltzmann fair division for distributive justice

You have a cake, and there are three people. You need to divide it fair. ⅓ and 120° are the good choices. This is an easy one, which is 1/n in n people. Our society is more complicated. The demand of the cake is required by your ability and contribution. This is often greedy and competitive. The Boltzmann distribution is based on entropy maximization and provides the most probable, natural, and unbiased distribution of a physical system.

Ej is the division potential, and j is players.

β is a division constant. (β≧0)

When β is Zero, all players receive an equal amount of cake. When β increases to a large value, only a few players having made the highest cake contributions receive most of the cake.

The player’s need for the cake as the need values Dj satisfy: uj(0) = 0, uj(Dj) = tanh(1)≅ 0.762. This means that if a player receives what they need (Dj), they satisfy 76.2%.

Homogeneous cake cutting is the total number of cake units, Ej is the division potential of player j, and Pj is the Boltzmann probability that a cake unit is allocated to player j.

Heterogeneous cake cutting is the total number of cake units with flavor i which is the weight factor expressing player j’s preference for flavor i, and the Boltzmann probability that a cake unit of flavor i is allocated to player j.

In β≧0.029,equality starts decreasing.