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.

Discrepancy theory

There is a finite set of elements {1,2,・・n}.

S1,...,Sm ⊆ {1,...,n}

There are two colors.

S={{い,ろ,は},{い,に},{に,ほ},{は,ほ}}

∴

This is the discrepancy.

Then you see the binary distribution.

You see -1={{い,ろ,は}{い,に}{に,ほ}} and 1={は,ほ}. This is at most 50%.

B-tree

Every node has at most m children, and every node has at least m/2 children. This is known as binary.

This is the maximum number of potential search keys for each node in a B-tree.

4 is the maximum, so you have the root.

This is expansion. Therefore, your computer need to optimize your huge data.

This is back and forth.

Stochastic block model

Stochastic block model is random graphs which contain communities, subsets of nodes. There are n vertices and disjoint subsets C1・・・Cr. C is a community. P is a symmetric matrix with edge probabilities.

G(n,M)

Thi is the Erdős–Rényi random graph model which have n nodes and M edges.

G(3,2)

G(n,p)

You have more vertices. It is almost Zero.

Tschirnhaus transformation

Tschirnhaus transformation is a polynomial equation of degree (n≧2) with some nonzero intermediate coefficients.

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Ex.
n=3

a’1=0 and a’2=0

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K is a field, and P(t) is polynomial over K.

K(t)/P(t)=L

L=K(α)

α is t modulo P.

β=F(α), α=G(β)

F and G over K are polynomials. Moreover, Q is the minimal polynomial for β over K. This is the Tschirnhaus transformation of P.

L is a Galois extension of K.

Huffman coding

In your fractal data, you can compress the huge chain. It depends on frequency of occurrence, so if you use more, you can shorten the code.

You have BACE. This is 001000010100 which is 4*3=12 bits. You put A is 0 and B is 1. BACE is 10010100. This is 8 bits which is about 67% compression. MP3 and JPEG are well known.

This is the information content h (in bits) of each symbol ai with non-null probability, which are binary.

The entropy H (in bits) is the weighted sum, across all symbols ai with non-zero probability wi, of the information content of each symbol.