Deformations

This is the last conjecture of Stephen Hawking. You need the singularity to exceed his theory. This is the the holographic no-boundary wave function.

(hij, φ) is the wave function, and S is the usual surface terms. Z is the Euclidean AdS/CFT.

The black birds move to the infinite on the surface without directions. Moreover, this is like a cylinder. Euclidean AdS/CFT generalized to complex relevant deformations implies an approximate realisation.

This is the mass deformed free model partition function, and m^2 is supergravity theories which typically contain scalars of mass.

This is the action of the free O(N) model, and O(N) is the partition function of supersymmetry breaking deformations.

You plot two one-dimensional slices of the distribution for two different values of ~m^2.

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①

~m^2=0

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②

~m^2=0.05i

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This conjecture strengthens the intuition that holographic cosmology implies a significant reduction of the multiverse to a much more limited set of possible universes.

1+2+3+4+・・・

This is irrational because this must increase infinitely basically.

This is like a black hole.

Srinivasa Aiyangar Ramanujan solved it.

2F=1

∴

F=1/2

∴

A=1/4

∴

S=-1/12

This is tricky in infinity because he moves the series.

You can also use the Zeta Function.

This is A=1-2+3-4+・・=1/4

This is the Abel's summability method.

This is F=1-1+1-1+・・・=1/2

∴

ζ(-1)=1+2+3+4+・・・=-1/12

Whole Genome Shotgun Sequencing

This is the Hamiltonian path problem, which is closed.

V(G)={a,b,c,d,e,f,g,h}

This is the graph of each dot.

E(G)={ab,bc,cd,de,ef,fg,gh,ha}

This is the graph of the Hamiltonian path.

Then you see the Whole Genome Shotgun Sequencing.

This isn't closed, but there is the Hamiltonian path.

S4→S2→S1→S3→S5

You cut the chain of your DNA, and you reorganize it.

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S1→ACCTG

S2→CGACC

S3→CTGAG

S4→CGTCG

S5→AGTAC

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∴

S4→CGTCG

S2→XXXCGACC

S1→XXXXXACCTG

S3→XXXXXXXCTGAG

S5→XXXXXXXXXXAGTAC

You connect the pattern.

CGTCGACCTGAGTAC

Eulerian trail

You need to go back the start point in Eulerian trail.

You can go through the path once.

This is closed.

Each edge has even links.

Perfect number

A perfect number is a number that is half the sum of all of its positive divisors, and it is including itself.

σ(N)=2N

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28 is the perfect number.

28=1 + 2 + 4 + 7 + 14

σ(28)=1 + 2 + 4 + 7 + 14 + 28 = 2*28 = 56

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Euler proved that all even perfect numbers were in the Euclid–Euler theorem.

The sum of divisors of a number must include the number itself, not just the proper divisors.

σ(2^2(2^3-1))=σ(4)σ(7)=(1+2+4)(1+7)=7*8=56=σ(28)

This is known as Mersenne primes.

2(2^2-1)=6

2^2(2^3-1)=28

2^4(2^5-1)=496

These are perfect numbers.

3,7,31 are primes, so you can find perfect numbers infinitely because of fractal. However, odd perfect numbers are unknown.

Amicable numbers

The smallest pair of amicable numbers is (220, 284). You expand it infinitely.

M and N are amicable. σ(M) is the sum of proper divisors. σ(220)=1+2+4+5+10+11+20+22+44+55+110+220=504.
Moreover, σ(N)=σ(284)=1+2+4+71+142+284=504.

σ(M)-M=N and σ(N)-N=M

In this case, M=220 and N=284.
σ(220)-220=504-220=284

σ(284)-284=504-284=220

∴

σ(M)=M+N=σ(N)

This is crossing over.

M=apq, N=ar

pqr are different prime numbers, and a is the prime relatively.

Leonhard Euler found it.

σ(p)=p+1, p is the prime.
σ(pq)=σ(p)σ(q)=1+p+q+pq=(1+p)+q(1+p)=(1+p)(1+q), p and q are prime numbers.

Therefore,

σ(M)=σ(N), σ(apq)=σ(ar)

σ(a)σ(p)σ(q)=σ(a)σ(r)

σ(p)σ(q)=σ(r)

∴

(p+1)(q+1)=r+1

This is based on prime numbers. Then, you put x=p+1, y=q+1.

xy=r+1, r=xy-1

σ(M)=M+N=apq+ar=a(pq+r)

σ(M)=σ(a)σ(p)σ(q)=σ(a)(p+1)(q+1)=a(pq+r)

σ(a)xy=a[(x-1)(y-1)+(xy-1)]

ax=[2ax-σ(a)x-a]y

y=ax/[2ax-σ(a)x-a]

Then, you put a/[2a-σ(a)]=b/c.

2a-σ(a)=ac/b, σ(a)=2a-(ac/b)

∴
y=ax/[(ac/b)x-a]=bx/(cx-b)

cy-b=c[bx/cx-b]-b=(b^2)/cx-b

b^2=(cx-b)(cy-b)

This square includes prime numbers infinitely.

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For example, a=4.

4/[8-σ(4)]=4/(8-7)=4/1=b/c

σ(4)=1+2+4=7

∴
b=4, c=1

16=(x-4)(y-4)

x-4	y-4	x	y	p=x-1	q=y-1	r=xy-1
16	1	20	5	19	4 (not prime)
8	2	12	6	11	5	71

Therefore

M=apq=4*11*5=220
N=ar=4*71=284

M and N are amicable, and you see prime numbers in it.

(5020,5564) is also amicable.

5020=2^2*5*251
5564=2^2*13*107

This is prime factorization.

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You need to calculate all amicable numbers, but they must exist infinitely because of fractal.

Euler's formula

Euler's formula is based on Maclaurin's expansion.

The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full period centered at the origin.

x is 0,
so f(x)=(sin x,cos x,-sin x,-cos x,・・・)=(0,1,0,-1,・・・)=sin x
and f(x)=(cos x,-sin x,-cos x,sin x,・・・)=(1,0,-1,0,・・・)=cos x

Euler's identity is well known as the beauty of mathematics.