1
On the study of various equations concerning the Isoperimetric Theorems.
Possible mathematical connections with some sectors of Number Theory, String
Theory and Inflationary Cosmology. II
Michele Nardelli
1
, Antonio Nardelli
2
Abstract
In this paper (part II), we analyze various equations concerning the Isoperimetric
Theorems. We describe the new possible mathematical connections with some sectors
of Number Theory, String Theory and Inflationary Cosmology
1
M.Nardelli studied at Dipartimento di Scienze della Terra Università degli Studi di Napoli Federico II,
Largo S. Marcellino, 10 - 80138 Napoli, Dipartimento di Matematica ed Applicazioni “R. Caccioppoli” -
Università degli Studi di Napoli “Federico II” – Polo delle Scienze e delle Tecnologie Monte S. Angelo, Via
Cintia (Fuorigrotta), 80126 Napoli, Italy
2
A. Nardelli studies at the Università degli Studi di Napoli Federico II - Dipartimento di Studi Umanistici –
Sezione Filosofia - scholar of Theoretical Philosophy
3
Introduction
In 1983, it was shown that inflation could be eternal, leading to a multiverse in which
space is broken up into bubbles or patches whose properties differ from patch to
patch spanning all physical possibilities.
When the false vacuum decays, the lower-energy true vacuum forms through a
process known as bubble nucleation. In this process, instanton effects cause a
bubble containing the true vacuum to appear. The walls of the bubble (or domain
walls) have a positive surface tension, as energy is expended as the fields roll over
the potential barrier to the true vacuum.
In mathematics, a ball is the space bounded by a sphere. It may be a closed ball
(including the boundary points that constitute the sphere) or an open ball (excluding
them). (From Wikipedia)
We propose that some equations concerning the “balls”, thus various sectors and
theorems of Geometric Measure Theory, can be related with several parameters of
some cosmological models as the “Multiverse” and the “Eternal Inflation” linked to
it, which provides that space is divided into bubbles or patches whose properties
differ from patch to patch and spanning all physical possibilities.
4
From:
Isoperimetric Theorems, Open Problems and New Results – Francesco Maggi –
ICTP, Trieste, 22 February 2017
We have:
From:
((3-1)^(13/8) / (5-1)^(3/2)) * sqrt((5+3-1)/(5*3*ω
k
ω
h
)) * 2^(-12)
Input
5
Exact result
For ω
h
= x ; ω
k
= y
(sqrt(7/15) sqrt(1/(x*y)))/(8192 2^(3/8))
Input
Exact result
3D plots
Real part (figures that can be related to the D-branes/Instantons)
6
Imaginary part
Contour plots
Real part
Imaginary part
7
Roots
Properties as a function
Domain
Range
Parity
Series expansion at x=0
Series expansion at x=∞
8
Partial derivatives
Indefinite integral
Limit
9
Series representations
From:
(sqrt(7) x sqrt(1/(x y)))/(4096 2^(3/8) sqrt(15))
Input
10
Exact result
3D plots
Real part (figures that can be related to the D-branes/Instantons)
Imaginary part
11
Contour plots
Real part
Imaginary part
12
Alternate form assuming x and y are positive
Roots
Properties as a function
Domain
Range
Parity
Series expansion at x=0
13
Series expansion at x=∞
Partial derivatives
Indefinite integral
Limit
14
Series representations
15
From:
SHARP STABILITY INEQUALITIES FOR THE PLATEAU PROBLEM
G. De Philippis & F. Maggi - J. Differential Geometry 96 (2014) 399-456
We have that:
((2^12*sqrt(ω
k
ω
h
))) / (((3-1)^(1/8))) * sqrt((5*3)/(5+3-1)) * ((5-1)/(3-1))^1.5
Input
16
Result
15551.5 sqrt(x*y)
Input interpretation
Result
3D plots
Real part (figures that can be related to the D-branes/Instantons)
17
Imaginary part
Contour plots
Real part
18
Imaginary part
Roots
Root for the variable y
Series expansion at x=0
Series expansion at x=∞
19
Partial derivatives
Indefinite integral
Global minima
Series representations
20
From:
and
(see page 5)
(((15551.5 sqrt(x y)))) * ((((sqrt(7/15) sqrt(1/(x y)))/(8192 2^(3/8)))))
Input interpretation
Result
21
3D plot (figure that can be related to a D-brane/Instanton)
Contour plot
Alternate form assuming x and y are positive
Roots
Properties as a function
Domain
22
Range
Parity
Series expansion at x=0
Series expansion at x=∞
Partial derivatives
23
Indefinite integral
Global maximum
Global minimum
Limit
24
Definite integral over the region where the integrand is negative
From the right-hand side:
we obtain:
(sqrt(7) *0.002* sqrt(1/(0.002*0.002)))/(4096 2^(3/8) sqrt(15))
Input
Result
0.0001286048046….
From:
25
10367.7 *0.5 sqrt(0.5*0.5)
Input interpretation
Result
2591.925
Inverting
we obtain:
1/((((sqrt(7) *0.002* sqrt(1/(0.002*0.002)))*1/(4096 2^(3/8) sqrt(15)))))
Input
Result
7775.76….
From the ratio between the above expression and
we obtain:
26
(((1/((((sqrt(7) *0.002* sqrt(1/(0.002*0.002)))*1/(4096 2^(3/8) sqrt(15)))))))) 1/
(((10367.7 *0.5 sqrt(0.5*0.5))))
Input interpretation
Result
2.99999…. ≈ 3
From which:
64((((((1/((((sqrt(7) *0.002* sqrt(1/(0.002*0.002)))*1/(4096 2^(3/8) sqrt(15)))))))) 1/
(((10367.7 *0.5 sqrt(0.5*0.5)))))))^3+1
Input interpretation
Result
1728.99…. ≈ 1729
This result is very near to the mass of candidate glueball f
0
(1710) scalar meson.
Furthermore, 1728 occurs in the algebraic formula for the j-invariant of an elliptic
curve. (1728 = 8
2
* 3
3
) The number 1728 is one less than the Hardy–Ramanujan
number 1729 (taxicab number)
27
((64((((((1/((((sqrt(7) *0.002* sqrt(1/(0.002*0.002)))*1/(4096 2^(3/8) sqrt(15))))))))
1/ (((10367.7 *0.5 sqrt(0.5*0.5)))))))^3+1))^1/15
Input interpretation
Result
1.64381…. ≈ ζ(2) =
2
6
= 1.644934 (trace of the instanton shape)
From:
(4.9)
For k = 3 , h = 5
1/512*((3-1)/(5-1))^2 (3-1)^(3/4)
Input
28
Result
Decimal approximation
0.0008211879055….
Alternate form
From: (Isoperimetric Theorems, Open Problems and New Results – Francesco Maggi – ICTP,
Trieste, 22 February 2017)
We have:
For x = ((2^(3/4)/(2048)))*1/64*(3/5)^(9/4)*5^(1/4)
((1/512*((3-1)/(5-1))^2 (3-1)^(3/4)))*1/64*(3/5)^(9/4)*5^(1/4)
Input
29
Result
Decimal approximation
6.079185349….*10
-6
Alternate form
From which, after some calculations, we obtain:
5sqrt[1/(((((1/512*((3-1)/(5-1))^2 (3-1)^(3/4)))*1/64*(3/5)^(9/4)*5^(1/4))))]-
(23*(23+3)/2)
Input
Result
30
Decimal approximation
1728.90362656….. ≈ 1729
This result is very near to the mass of candidate glueball f
0
(1710) scalar meson.
Furthermore, 1728 occurs in the algebraic formula for the j-invariant of an elliptic
curve. (1728 = 8
2
* 3
3
) The number 1728 is one less than the Hardy–Ramanujan
number 1729 (taxicab number)
Alternate forms
((5sqrt[1/(((((1/512*((3-1)/(5-1))^2 (3-1)^(3/4)))*1/64*(3/5)^(9/4)*5^(1/4))))]-
(23*(23+3)/2)))^1/15
Input
31
Exact result
Decimal approximation
1.6438091202…. ≈ ζ(2) =
2
6
= 1.644934 (trace of the instanton shape)
Alternate forms
32
We have:
From:
C(A) ((integrate(f'(0) + x f''(0) + 1/2 f^(3)(0) x^2 + 1/6 f^(4)(0) x^3 + 1/24 f^(5)(0)
x^4 + O(x^5)))-Id)^2 = (A Δk)^2
Input
33
Result
((d^2 f(x))/(dx^2))
Derivative
Series expansion at x=0
E^2 k^2 ((f''(0) + f^(3) x + 1/2 f^(4) x^2 + 1/6 f^(5) x^3 + 1/24 f^(6) x^4 + (x^5)))^2
Input
Alternate forms
34
Expanded form
Derivative
From:
1/576 e^2 3^2 (x^4 + 4 x^3 + 12 x^2 + 24 x + 24 x^5)^2
Input
35
Result
Plots (figures that can be related to the open strings)
Alternate forms
36
Expanded form
Real root
Complex roots
Polynomial discriminant
Properties as a real function
Domain
Range
37
Derivative
Indefinite integral
Global minimum
From:
1/8 e^2 x (24 + 12 x + 4 x^2 + x^3 + 24 x^4) (6 + 6 x + 3 x^2 + x^3 + 30 x^4)
Input
38
Plots (figures that can be related to the open strings)
Alternate forms
Real root
Polynomial discriminant
39
Properties as a real function
Domain
Range
Bijectivity
Derivative
Indefinite integral
40
90 e^2 0.7^9 + (27 e^2 0.7^8)/4 + 193 e^2 1/8 0.7^7 + 511 e^2 1/8 0.7^6 + (447 e^2
0.7^5)/4 + (45 e^2 0.7^4)/4 + 21 e^2 0.7^3 + 27 e^2 0.7^2 + 18 e^2 0.7
Input
Result
502.74046747….
Alternative representation
Series representations
41
From which:
Pi((90 e^2 0.7^9 + (27 e^2 0.7^8)/4 + 193 e^2 1/8 0.7^7 + 511 e^2 1/8 0.7^6 + (447
e^2 0.7^5)/4 + (45 e^2 0.7^4)/4 + 21 e^2 0.7^3 + 27 e^2 0.7^2 + 18 e^2
0.7))+144+5+Φ
Input
Result
1729.02….
This result is very near to the mass of candidate glueball f
0
(1710) scalar meson.
Furthermore, 1728 occurs in the algebraic formula for the j-invariant of an elliptic
curve. (1728 = 8
2
* 3
3
) The number 1728 is one less than the Hardy–Ramanujan
number 1729 (taxicab number)
42
(1/27(((Pi((90 e^2 0.7^9 + (27 e^2 0.7^8)/4 + 193 e^2 1/8 0.7^7 + 511 e^2 1/8 0.7^6
+ (447 e^2 0.7^5)/4 + (45 e^2 0.7^4)/4 + 21 e^2 0.7^3 + 27 e^2 0.7^2 + 18 e^2
0.7))+144+5+Φ))-1))^2
Input
Result
4096.11… ≈ 4096 = 64
2
((Pi((90 e^2 0.7^9 + (27 e^2 0.7^8)/4 + 193 e^2 1/8 0.7^7 + 511 e^2 1/8 0.7^6 + (447
e^2 0.7^5)/4 + (45 e^2 0.7^4)/4 + 21 e^2 0.7^3 + 27 e^2 0.7^2 + 18 e^2
0.7))+144+5+Φ))^1/15
Input
Result
1.6438167368…. ≈ ζ(2) =
2
6
= 1.644934 (trace of the instanton shape)
43
From:
A. FIGALLI - F. MAGGI - A. PRATELLI - A MASS TRANSPORTATION APPROACH TO
QUANTITATIVE ISOPERIMETRIC INEQUALITIES - Inserted: 12 nov 2007 - Last
Updated: 16 feb 2015 - Journal: Invent. Math. - Year: 2010
We have:
An isoperimetric inequality relates the perimeter to the area of a closed curve in the
plane. If P is the perimeter of the curve and A is the area of the region enclosed by
the curve, the inequality has the form
4πA P
2
In the case of a circle of radius r we have A = πr
2
and P = 2πr, and replacing these
in the inequality we see that the circle, among all the curves of fixed perimeter,
maximizes the area. In fact, the circle is the only curve that maximizes the area.
4Pi*Pi*r
2
(2Pi*r)
2
44
For : δ(E) = 0.25 ; K = E = (2πr)^2 ; n = 3
From:
(Sqrt56 * 9 * (2πr)^2 * sqrt0.25)
Input
Result
1329.43 r
2
Plot (figure that can be related to an open string)
Geometric figure
Alternate form assuming r is real
45
Root
Polynomial discriminant
Property as a function
Parity
Derivative
Indefinite integral
Global minimum
Definite integral after subtraction of diverging parts
From:
i.e.
46
((8*3^2*(2πr)^2))
Input
Result
2842.446067 r
2
Plot (figure that can be related to an open string)
Geometric figure
Root
Polynomial discriminant
Property as a function
Parity
Derivative
47
Indefinite integral
Global minimum
From:
we obtain:
576 π^2 r
Input
Plot
Geometric figure
Property as a function
Parity
48
Derivative
Indefinite integral
From:
Sqrt(((7*3^2*2842.446067 * r^2 * ((2πr)^2) * 0.25)))
Input interpretation
Result
49
Plots (figures that can be related to the open strings)
Alternate forms assuming r is real
1329.43 r
2
Root
Property as a function
Parity
50
Series expansion at r=0
Series expansion at r=∞
Derivative
Indefinite integral
Global minimum
Series representations
51
Definite integral after subtraction of diverging parts
From:
we obtain:
(2658.86 r^3)/sqrt(r^4)
Input interpretation
Result
52
Plots
Alternate form assuming r is real
2658.86 r
Expanded form
Roots
Property as a function
Parity
53
Series expansion at r=0
Series expansion at r=∞
Derivative
Indefinite integral
Series representations
54
From:
and
we obtain, after some easy calculations:
(((e)*((2658.86 r^3)/sqrt(r^4))*1/(576 π^2 r)))^2
Input interpretation
Result
1.6163501476…. result that is a very good approximation to the value of the golden
ratio 1.618033988749...
Series representations
55
Integral representation
For:
k(n) = 0.16 ; δ(E) = 0.08
From:
56
(3/(0.16))*0.08
Input
Result
δ(G) = 1.5
From:
for E = (2πr)^2 :
1/2* ((2πr)^2)
Input
Result
Input interpretation
19.739208802 r
2
57
Plot (figure that can be related to an open string)
Geometric figure
Root
Polynomial discriminant
Property as a function
Parity
Derivative
Indefinite integral
58
Global minimum
From:
and
for k(n) = 0.16 ; δ(E) = 0.08 , we obtain:
((4sqrt2)/(0.16))*(sqrt(0.08))
Input
Result
10
For: δ(G) = 1.5 ; k(n) = 0.16 ; δ(E) = 0.25 ; K = (2πr)^2 ; n = 3
From:
59
((20*3^3)/(0.16))*(x/y)^4*sqrt(1.5)
Input
Result
3D plot (figure that can be related to a D-brane/Instanton)
Contour plot
60
Alternate form assuming x and y are real
Root
Properties as a function
Domain
Range
Parity
Partial derivatives
61
Indefinite integral
Global minimum
Limit
Series representations
62
From:
for x = y = 0.5 :
(4133.51 0.5^4)/0.5^4
Input interpretation
Result
4133.51
From:
for x = y = 0.5 , we obtain:
(16534.1 * 0.5^3)/(0.5^4)
Input interpretation
Result
33068.2
63
From (see ref. pag.27):
for: δ(G) = 1.5 ; k(n) = 0.16 ; δ(E) = 0.25 ; K = (2πr)^2 ; n = 3
((181*3^3))/((2-2^(1/x)))^1.5 * sqrt(0.25)
Input
Result
Plots (figures that can be related to the open strings)
64
Alternate form
Roots
Properties as a real function
Domain
Range
Series expansion at x=1
Series expansion at x=∞
65
Derivative
Limit
Series representations
66
From:
for x = 8 :
2443.5/(2 - 2^(1/8))^1.5
Input interpretation
Result
2817.18….
From:
For x = 8, we obtain:
-(2540.56 2^(1/8))/((2 - 2^(1/8))^(5/2) 8^2)
Input interpretation
Result
-54.8758…
67
From the two previous highlighted expressions, after some calculations, we obtain:
(((4133.51 0.5^4)/0.5^4))*((2443.5/(2 - 2^(1/8))^1.5))* ((1/2 (-1 + π + 5 π^2)))
where
Input interpretation
Result
2.99794…*10
8
≈ 299792458 = c = speed of light
Alternative representations
68
Series representations
Integral representations
69
From:
And
we obtain:
(((16534.1 * 0.5^3)/(0.5^4))) 1/ ((-(2540.56 2^(1/8))/((2 - 2^(1/8))^(5/2) 8^2)))
Input interpretation
70
Result
-602.60100485….
From which:
-3(((16534.1 * 0.5^3)/(0.5^4))) 1/ ((-(2540.56 2^(1/8))/((2 - 2^(1/8))^(5/2) 8^2)))-64-
16+2Φ
Input interpretation
Result
1729.04….
This result is very near to the mass of candidate glueball f
0
(1710) scalar meson.
Furthermore, 1728 occurs in the algebraic formula for the j-invariant of an elliptic
curve. (1728 = 8
2
* 3
3
) The number 1728 is one less than the Hardy–Ramanujan
number 1729 (taxicab number)
(((-3(((16534.1 * 0.5^3)/(0.5^4))) 1/ ((-(2540.56 2^(1/8))/((2 - 2^(1/8))^(5/2) 8^2)))-
64-16+2Φ)))^1/15
Input interpretation
71
Result
1.64382…. ≈ ζ(2) =
2
6
= 1.644934 (trace of the instanton shape)
(1/27(((((-3(((16534.1 * 0.5^3)/(0.5^4))) 1/ ((-(2540.56 2^(1/8))/((2 - 2^(1/8))^(5/2)
8^2)))-64-16+2Φ)))-1)))^2
Input interpretation
Result
4096.19… ≈ 4096 = 64
2
72
From:
Isoperimetry and Stability Properties of Balls with Respect to Nonlocal Energies
A. Figalli, N. Fusco, F. Maggi, V. Millot, M. Morini - Commun. Math. Phys Digital
Object Identifier (DOI) 10.1007/s00220-014-2244-1
We have that:
73
Now, we analyze the eqs. (7.2), (7.3), (7.4), (7.5) and (7.6)
For n = 3, k = 2, s = α = 1/2
2(2+3-2) = 6
74
From:
(2^(1-0.5)*Pi)/(1+1/2)* gamma(1/2*(1-
1/2))/gamma(1/2*(3+1/2))*(((((gamma(2+1/2(3+1/2)))/gamma(2+1/2(3-2-1/2))-
gamma(1/2(3+1/2))/gamma(1/2(3-2-1/2)))))
Input
Result
42.6517…
Alternative representations
75
Series representations
76
Integral representations
77
For n = 3, k = 2, α = 1/2
From:
78
(2^(1+0.5)*Pi)/(1-1/2)* gamma(1/2*(1+1/2))/gamma(1/2*(3-
1/2))*(((((gamma(2+1/2(3-1/2)))/gamma(2+1/2(3-2+1/2))-gamma(1/2(3-
1/2))/gamma(1/2(3-2+1/2)))))
Input
Result
20.3103…
Alternative representations
79
Series representations
80
Integral representations
81
For n = 3, k = 2, α = 2
From:
(2^2*Pi)*gamma(1/2*(2-1))/gamma(1/2*(3-2))*(((((gamma(1/2(3-2)))/gamma(1/2(3-
2+2))-gamma(2+1/2(3-2))/gamma(2+1/2(3-2+2)))))
Input
82
Exact result
Decimal approximation
20.106192982….
Property
Alternative representations
83
Series representations
Integral representations
84
For n = 3
From:
(2^2*Pi)/gamma(1/2(3-1))*(((((digamma(2+1)))/gamma(2+1/2(3-1))-
digamma(1/2(3-1))/gamma(1/2(3-1)))))
Input
Exact result
Decimal approximation
13.051530945…
85
Alternate forms
Expanded form
Alternative representations
86
Series representations
Integral representations
87
From the sum of the previous results/expressions, we obtain:
6+42.6517+20.3103+((32π)/5)+[((((4(1/2(3/2-0.5772156649)+0.5772156649)π))))]
Input interpretation
Result
102.120….
Alternative representations
88
Series representations
Integral representations
89
From which:
17((6+42.6517+20.3103+((32π)/5)+[((((4(1/2(3/2-
0.5772156649)+0.5772156649)π))))]))-e*Pi+φ
Input interpretation
Result
1729.11….
This result is very near to the mass of candidate glueball f
0
(1710) scalar meson.
Furthermore, 1728 occurs in the algebraic formula for the j-invariant of an elliptic
curve. (1728 = 8
2
* 3
3
) The number 1728 is one less than the Hardy–Ramanujan
number 1729 (taxicab number)
Alternative representations
90
Series representations
91
Integral representations
(1/27((17((6+42.6517+20.3103+((32π)/5)+[((((4(1/2(3/2-euler-mascheroni
constant)+euler-mascheroni constant)π))))]))-e*Pi+Φ)))^2-euler-mascheroni constant
Input interpretation
Result
4095.96… ≈ 4096 = 64
2
92
((17((6+42.6517+20.3103+((32π)/5)+[((((4(1/2(3/2-euler-mascheroni
constant)+euler-mascheroni constant)π))))]))-7))^1/15
Input interpretation
Result
1.643817466…. ≈ ζ(2) =
2
6
= 1.644934 (trace of the instanton shape)
From:
Inflation after Planck and BICEP – Andrei Linde - “Quantum Gravity and All of
That” - https://qgholqi.inpcs.net/ - 13.01.2022 -
We have:
For R = 5 ; = 2.25647383035 ; α = 1/6 ; φ = 2 ; m = 1 * 10^-5 ;
σ = 0.5 ; V = -453875.19999….
93
5/2-(((2.25647383035)^2))/((2(1-(2.25647383035^2))^2))-((0.5)^2)/((2(1-
(0.5^2)^2)+453875.19999
Input interpretation
Result
453877.325703….
From which:
(2+0.5683000031)(((5/2-(((2.25647383035)^2))/((2(1-(2.25647383035^2))^2))-
((0.5)^2)/((2(1-(0.5^2))^2))+453875.19999)))^1/2-1-((((√(10-2√5) -2))⁄((√5-1))))
where ((√(10-2√5) -2))⁄((√5-1)) = κ = 8πG ; G = 0.011303146014
Input interpretation
Result
1728.9903753…. ≈ 1729
This result is very near to the mass of candidate glueball f
0
(1710) scalar meson.
Furthermore, 1728 occurs in the algebraic formula for the j-invariant of an elliptic
curve. (1728 = 8
2
* 3
3
) The number 1728 is one less than the Hardy–Ramanujan
number 1729 (taxicab number)
94
(1/27((2+0.5683000031)(((5/2-(((2.25647383035)^2))/((2(1-
(2.25647383035^2))^2))-((0.5)^2)/((2(1-(0.5^2))^2))+453875.19999)))^1/2-2-
((((√(10-2√5) -2))⁄((√5-1))))))^2
Input interpretation
Result
4095.954372…. ≈ 4096 = 64
2
(((2+0.5683000031)(((5/2-(((2.25647383035)^2))/((2(1-(2.25647383035^2))^2))-
((0.5)^2)/((2(1-(0.5^2))^2))+453875.19999)))^1/2-1-((((√(10-2√5) -2))⁄((√5-
1))))))^1/15
Input interpretation
95
Result
1.64381461871…. ≈ ζ(2) =
2
6
= 1.644934 (trace of the instanton shape)
From:
-453875.19999(tanh(2))
Input interpretation
Result
-437548.21070….
Alternative representations
96
Series representations
Integral representation
-453875.19999(tanh(0.5))
Input interpretation
Result
-2.09744…*10
5
Alternative representations
97
Series representations
Integral representation
98
From the two previous expressions, after some calculations, we obtain:
[-(((-453875.19(tanh(2))))+((453875.19(tanh(0.5)))))]-
(14258+11468+1010+728+1729+172+138+135+791)-8^3+6^2-8*2
Input interpretation
Result
196883.688518504….
Alternative representations
99
Series representations
Integral representation
100
And again, we obtain:
[-(((-453875.19(tanh(2))))+((453875.19(tanh(0.5)))))]-
(14258+11468+1010+728+1729+172+138+135+791)-8^3+6^2-8*2-Φ
We note that 14258, 11468, 1010, 728, 1729, 172, 138, 135 and 791 are third
roots of Ramanujan cubes
Input interpretation
Result
196883.070484516….
196884/196883 is a fundamental number of the following j-invariant
(In mathematics, Felix Klein's j-invariant or j function, regarded as a function of
a complex variable τ, is a modular function of weight zero for SL(2, Z) defined on
the upper half plane of complex numbers. Several remarkable properties of j have to
do with its q expansion (Fourier series expansion), written as a Laurent series in
terms of q = e
2πiτ
(the square of the nome), which begins:
101
Note that j has a simple pole at the cusp, so its q-expansion has no terms below q
−1
.
All the Fourier coefficients are integers, which results in several almost integers,
notably Ramanujan's constant:
The asymptotic formula for the coefficient of q
n
is given by
as can be proved by the Hardy–Littlewood circle method)
Furthermore, 196884 is the coefficient of q of the partition function Z
1
(q) that is the
number of quantum states of the minimal black hole for the value of k equal to 1.
[-(((-453875.19(tanh(2))))+((453875.19(tanh(0.5)))))]^1/25
Input interpretation
Result
1.63795703751…. result very near to the mean between ζ(2) =
2
6
= 1.644934 , the
value of golden ratio 1.61803398… and the 14th root of the Ramanujan’s class
invariant =
505
/
101/5
3
= 1164.2696 i.e. 1.65578..., i.e. 1.63958266
102
From:
For R = 5 ; α = 1/6
1/6 = 1+1/(6x)
Input
Plot (figure that can be related to an open string)
Alternate form assuming x is real
103
Alternate form
Solution
ξ = -1/5
Thus, from:
we obtain:
Sqrt(-1) (5/2-3/4*1/6(((partial derivative(1.009))/(1.009)))^2-x*(1.009))
Input interpretation
Result
Plot
104
Alternate forms
Expanded form
Alternate form assuming x is real
Real root
x = U = 2.4777
Complex root
Properties as a real function
Domain
Range
105
Derivative
Indefinite integral
From:
we obtain:
Sqrt(-1) (5/2-3/4*1/6(((partial derivative(1.009))/(1.009)))^2-2.4777*(1.009))
Input interpretation
Result
Polar coordinates
7*10
-7
106
From:
1/4*(-1/5)* (((partial derivative(x))^2/(1.009)))
Input interpretation
Result
-0.049554013….
Rational approximation
(1.009)^2*2.4777(1.009)
Input interpretation
Result
2.5452017873433
107
From the three above expressions, after some calculations, we obtain:
(-(1/2(-1/(((Sqrt(-1) (5/2-3/4*1/6(((partial derivative(1.009))/(1.009)))^2-
2.4777*1/(1.009)))*(1/4*(-1/5)* (((partial
derivative(x))^2/(1.009))))*((1.009)^2*2.4777(1.009))))+55i-5i))^2)-34-φ^2
Input interpretation
Result
4096.04…. ≈ 4096 = 64
2
27sqrt((-(1/2(-1/(((Sqrt(-1)(5/2-3/4*1/6(((partial derivative(1.009))/(1.009)))^2-
2.4777*1/(1.009)))(1/4*(-1/5)(((partial
derivative(x))^2/(1.009))))*((1.009)^2*2.4777(1.009))))+55i-5i))^2)-34-φ^2)+1
Input interpretation
108
Result
1729.01….
This result is very near to the mass of candidate glueball f
0
(1710) scalar meson.
Furthermore, 1728 occurs in the algebraic formula for the j-invariant of an elliptic
curve. (1728 = 8
2
* 3
3
) The number 1728 is one less than the Hardy–Ramanujan
number 1729 (taxicab number)
(27sqrt((-(1/2(-1/(((Sqrt(-1)(5/2-3/4*1/6(((derivative(1.009))/(1.009)))^2-
2.4777*1/(1.009)))(1/4*(-
1/5)(((derivative(x))^2/(1.009))))*((1.009)^2*2.4777(1.009))))+55i-5i))^2)-34-
φ^2)+1)^1/15
Input interpretation
Result
1.64381574…. ≈ ζ(2) =
2
6
= 1.644934 (trace of the instanton shape)
109
For R = 5 ; = 2.25647383035 ;
From:
(1/2*1.009*2.25647383035*5-1/2*(-50/1009)* 2.25647383035*2*2.25647383035-
2.5452017873433)
Input interpretation
Result
3.399066341….
From which:
1+1/[(1/napier number(((1/2*1.009*2.25647383035*5-1/2*(-50/1009)*
2.25647383035*2*2.25647383035-2.5452017873433))))^2]
Input interpretation
110
Result
1.63954…. result very near to the mean between ζ(2) =
2
6
= 1.644934 , the value
of golden ratio 1.61803398… and the 14th root of the Ramanujan’s class invariant
=
505
/
101/5
3
= 1164.2696 i.e. 1.65578..., i.e. 1.63958266
Alternative representation
Series representations
111
[((((1/2*1.009*2.25647383035*5-1/2*(-50/1009)*
2.25647383035*2*2.25647383035-2.5452017873433))))^6+128+64-8+e
Input interpretation
Result
1728.98…. ≈ 1729
This result is very near to the mass of candidate glueball f
0
(1710) scalar meson.
Furthermore, 1728 occurs in the algebraic formula for the j-invariant of an elliptic
curve. (1728 = 8
2
* 3
3
) The number 1728 is one less than the Hardy–Ramanujan
number 1729 (taxicab number)
112
Alternative representation
Series representations
113
(1/27([((((1/2*1.009*2.25647383035*5-1/2*(-50/1009)*
2.25647383035*2*2.25647383035-2.5452017873433))))^6+128+64-8+e]-1))^2
Input interpretation
Result
4095.9… ≈ 4096 = 64
2
114
[((((1/2*1.009*2.25647383035*5-1/2*(-50/1009)*
2.25647383035*2*2.25647383035-2.5452017873433))))^6+128+64-8+e]^1/15
Input interpretation
Result
1.64381390868…. ≈ ζ(2) =
2
6
= 1.644934 (trace of the instanton shape)
115
Observations
We note that, from the number 8, we obtain as follows:
We notice how from the numbers 8 and 2 we get 64, 1024, 4096 and 8192, and that 8
is the fundamental number. In fact 8
2
= 64, 8
3
= 512, 8
4
= 4096. We define it
"fundamental number", since 8 is a Fibonacci number, which by rule, divided by the
previous one, which is 5, gives 1.6 , a value that tends to the golden ratio, as for all
numbers in the Fibonacci sequence
116
“Golden” Range
Finally we note how 8
2
= 64, multiplied by 27, to which we add 1, is equal to 1729,
the so-called "Hardy-Ramanujan number". Then taking the 15th root of 1729, we
obtain a value close to ζ(2) that 1.6438 ..., which, in turn, is included in the range of
what we call "golden numbers"
Furthermore for all the results very near to 1728 or 1729, adding 64 = 8
2
, one obtain
values about equal to 1792 or 1793. These are values almost equal to the Planck
multipole spectrum frequency 1792.35 and to the hypothetical Gluino mass
117
Appendix
From: A. Sagnotti – AstronomiAmo, 23.04.2020
In the above figure, it is said that: “why a given shape of the extra dimensions?
Crucial, it determines the predictions for α”.
We propose that whatever shape the compactified dimensions are, their geometry
must be based on the values of the golden ratio and ζ(2), (the latter connected to 1728
or 1729, whose fifteenth root provides an excellent approximation to the above
mentioned value) which are recurrent as solutions of the equations that we are going
to develop. It is important to specify that the initial conditions are always values
belonging to a fundamental chapter of the work of S. Ramanujan "Modular equations
and Appoximations to Pi" (see references). These values are some multiples of 8 (64
and 4096), 276, which added to 4096, is equal to 4372, and finally e
π√22
118
We have, in certain cases, the following connections:
Fig. 1
Fig. 2
119
Fig. 3
Stringscape - a small part of the string-theory landscape showing the new de Sitter solution as a local
minimum of the energy (vertical axis). The global minimum occurs at the infinite size of the extra
dimensions on the extreme right of the figure.
Fig. 4
120
With regard the Fig. 4 the points of arrival and departure on the right-hand side of the
picture are equally spaced and given by the following equation:
we obtain:
2Pi/(ln(2))
Input:
Exact result:
Decimal approximation:
9.06472028365….
Alternative representations:
121
Series representations:
Integral representations:
122
From which:
(2Pi/(ln(2)))*(1/12 π log(2))
Input:
Exact result:
Decimal approximation:
1.6449340668…. = ζ(2) =
2
6
= 1.644934
123
From:
Modular equations and approximations to - Srinivasa Ramanujan
Quarterly Journal of Mathematics, XLV, 1914, 350 – 372
We have that:
124
We note that, with regard 4372, we can to obtain the following results:
27((4372)^1/2-2-1/2(((√(10-2√5) -2))⁄((√5-1))))+φ
Input
Result
Decimal approximation
1729.0526944….
This result is very near to the mass of candidate glueball f
0
(1710) scalar meson.
Furthermore, 1728 occurs in the algebraic formula for the j-invariant of an elliptic
curve. (1728 = 8
2
* 3
3
) The number 1728 is one less than the Hardy–Ramanujan
number 1729 (taxicab number)
Alternate forms
125
Minimal polynomial
Expanded forms
Series representations
126
Or:
27((4096+276)^1/2-2-1/2(((√(10-2√5) -2))⁄((√5-1))))+φ
127
Input
Result
Decimal approximation
1729.0526944…. as above
Alternate forms
128
Minimal polynomial
Expanded forms
Series representations
129
130
From which:
(27((4372)^1/2-2-1/2(((√(10-2√5) -2))⁄((√5-1))))+φ)^1/15
Input
Exact result
Decimal approximation
1.64381856858…. ≈ ζ(2) =
2
6
= 1.644934
Alternate forms
131
Minimal polynomial
Expanded forms
All 15th roots of ϕ + 27 (-2 + 2 sqrt(1093) - (sqrt(10 - 2 sqrt(5)) - 2)/(2 (sqrt(5) -
1)))
132
Series representations
133
Integral representation
134
From:
An Update on Brane Supersymmetry Breaking
J. Mourad and A. Sagnotti - arXiv:1711.11494v1 [hep-th] 30 Nov 2017
From the following vacuum equations:
we have obtained, from the results almost equals of the equations, putting
instead of
a new possible mathematical connection between the two exponentials. Thence, also
the values concerning p, C, β
E
and correspond to the exponents of e (i.e. of exp).
Thence we obtain for p = 5 and β
E
= 1/2:
6+
= 4096
18
Therefore, with respect to the exponentials of the vacuum equations, the Ramanujan’s
exponential has a coefficient of 4096 which is equal to 64
2
, while -6C+ is equal to -
18. From this it follows that it is possible to establish mathematically, the dilaton
value.
135
For
exp((-Pi*sqrt(18)) we obtain:
Input:
Exact result:
Decimal approximation:
1.6272016… * 10
-6
Property:
Series representations:
136
Now, we have the following calculations:
6+
= 4096
18
18
= 1.6272016… * 10^-6
from which:
1
4096
6+
= 1.6272016… * 10^-6
0.000244140625
6+
=
18
= 1.6272016… * 10^-6
Now:
ln
18
= 13.328648814475 =
18
And:
(1.6272016* 10^-6) *1/ (0.000244140625)
Input interpretation:
Result:
0.006665017...
137
Thence:
0.000244140625
6+
=
18
Dividing both sides by 0.000244140625, we obtain:
0.000244140625
0.000244140625
6+
=
1
0.000244140625
18
6+
= 0.0066650177536
((((exp((-Pi*sqrt(18)))))))*1/0.000244140625
Input interpretation:
Result:
0.00666501785…
Series representations:
138
Now:
6+
= 0.0066650177536
=
= 0.00666501785…
From:
ln(0.00666501784619)
Input interpretation:
Result:
-5.010882647757…
139
Alternative representations:
Series representations:
Integral representation:
In conclusion:
6 + = 5.010882647757
and for C = 1, we obtain:
140
= 5.010882647757 + 6 = . =
Note that the values of n
s
(spectral index) 0.965, of the average of the Omega mesons
Regge slope 0.987428571 and of the dilaton 0.989117352243, are also connected to
the following two Rogers-Ramanujan continued fractions:
(http://www.bitman.name/math/article/102/109/)
Also performing the 512
th
root of the inverse value of the Pion meson rest mass
139.57, we obtain:
((1/(139.57)))^1/512
Input interpretation:
141
Result:
0.99040073.... result very near to the dilaton value . = and to
the value of the following Rogers-Ramanujan continued fraction:
From
Properties of Nilpotent Supergravity
E. Dudas, S. Ferrara, A. Kehagias and A. Sagnotti - arXiv:1507.07842v2 [hep-th] 14
Sep 2015
We have that:
We analyzing the following equation:
142
We have:
(M^2)/3*[1-(b/euler number * k/sqrt6) * (φ- sqrt6/k) * exp(-(k/sqrt6)(φ- sqrt6/k))]^2
i.e.
V = (M^2)/3*[1-(b/euler number * k/sqrt6) * (φ- sqrt6/k) * exp(-(k/sqrt6)(φ-
sqrt6/k))]^2
For k = 2 and φ = 0.9991104684, that is the value of the scalar field that is equal to
the value of the following Rogers-Ramanujan continued fraction:
we obtain:
V = (M^2)/3*[1-(b/euler number * 2/sqrt6) * (0.9991104684- sqrt6/2) * exp(-
(2/sqrt6)(0.9991104684- sqrt6/2))]^2
Input interpretation:
143
Result:
Solutions:
Alternate forms:
Expanded form:
Alternate form assuming b, M, and V are positive:
Alternate form assuming b, M, and V are real:
144
Derivative:
Implicit derivatives
145
Global minimum:
Global minima:
From:
we obtain
(225.913 (-0.054323 M^2 + 6.58545×10^-10 sqrt(M^4)))/M^2
Input interpretation:
146
Result:
Plots:
Alternate form assuming M is real:
-12.2723 result very near to the black hole entropy value 12.1904 = ln(196884)
Alternate forms:
147
Expanded form:
Property as a function:
Parity
Series expansion at M = 0:
Series expansion at M = ∞:
Derivative:
148
Indefinite integral:
Global maximum:
Global minimum:
Limit:
149
Definite integral after subtraction of diverging parts:
From b that is equal to
From:
we obtain:
1/3 (0.0814845 ((225.913 (-0.054323 M^2 + 6.58545×10^-10 sqrt(M^4)))/M^2 ) +
1)^2 M^2
Input interpretation:
Result:
150
Plots: (possible mathematical connection with an open string)
M = -0.5; M = 0.2
(possible mathematical connection with an open string)
M = 2 ; M = 3
Root:
Property as a function:
Parity
Series expansion at M = 0:
151
Series expansion at M = ∞:
Definite integral after subtraction of diverging parts:
For M = - 0.5 , we obtain:
1/3 (0.0814845 ((225.913 (-0.054323 (-0.5)^2 + 6.58545×10^-10 sqrt((-0.5)^4)))/(-
0.5)^2 ) + 1)^2 * (-0.5^2)
Input interpretation:
152
Result:
-4.38851344947*10
-16
For M = 0.2:
1/3 (0.0814845 ((225.913 (-0.054323 0.2^2 + 6.58545×10^-10 sqrt(0.2^4)))/0.2^2 ) +
1)^2 0.2^2
Input interpretation:
Result:
7.021621519159*10
-17
153
For M = 3:
1/3 (0.0814845 ((225.913 (-0.054323 3^2 + 6.58545×10^-10 sqrt(3^4)))/3^2 ) + 1)^2
3^2
Input interpretation:
Result:
1.57986484181*10
-14
For M = 2:
1/3 (0.0814845 ((225.913 (-0.054323 2^2 + 6.58545×10^-10 sqrt(2^4)))/2^2 ) + 1)^2
2^2
Input interpretation:
154
Result:
7.021621519*10
-15
From the four results
7.021621519*10^-15 ; 1.57986484181*10^-14 ; 7.021621519159*10^-17 ;
-4.38851344947*10^-16
we obtain, after some calculations:
sqrt[1/(2Pi)(7.021621519*10^-15 + 1.57986484181*10^-14 +7.021621519*10^-17 -
4.38851344947*10^-16)]
Input interpretation:
Result:
5.9776991059*10
-8
result very near to the Planck's electric flow 5.975498 × 10
−8
that
is equal to the following formula:
155
We note that:
1/55*(([(((1/[(7.021621519*10^-15 + 1.57986484181*10^-14 +7.021621519*10^-17
-4.38851344947*10^-16)])))^1/7]-((log^(5/8)(2))/(2 2^(1/8) 3^(1/4) e log^(3/2)(3)))))
Input interpretation:
Result:
1.6181818182… result that is a very good approximation to the value of the golden
ratio 1.618033988749...
From the Planck units:
Planck Length
5.729475 * 10
-35
Lorentz-Heaviside value
Planck’s Electric field strength
1.820306 * 10
61
V*m Lorentz-Heaviside value
156
Planck’s Electric flux
5.975498*10
-8
V*m Lorentz-Heaviside value
Planck’s Electric potential
1.042940*10
27
V Lorentz-Heaviside value
157
Relationship between Planck’s Electric Flux and Planck’s Electric Potential
E
P
* l
P
= (1.820306 * 10
61
) * 5.729475 * 10
-35
Input interpretation:
Result:
Scientific notation:
1.042939771935*10
27
≈ 1.042940*10
27
Or:
E
P
* l
P
2
/ l
P
= (5.975498*10
-8
)*1/(5.729475 * 10
-35
)
Input interpretation:
Result:
1.042939885417*10
27
≈ 1.042940*10
27
158
Acknowledgments
We would like to thank Professor Augusto Sagnotti theoretical physicist at Scuola
Normale Superiore (Pisa – Italy) for his very useful explanations and his availability
159
References
Isoperimetric Theorems, Open Problems and New Results – Francesco Maggi –
ICTP, Trieste, 22 February 2017
SHARP STABILITY INEQUALITIES FOR THE PLATEAU PROBLEM
G. De Philippis & F. Maggi - J. Differential Geometry 96 (2014) 399-456
A. FIGALLI - F. MAGGI - A. PRATELLI - A MASS TRANSPORTATION APPROACH TO
QUANTITATIVE ISOPERIMETRIC INEQUALITIES - Inserted: 12 nov 2007 - Last
Updated: 16 feb 2015 - Journal: Invent. Math. - Year: 2010
Isoperimetry and Stability Properties of Balls with Respect to Nonlocal Energies
A. Figalli, N. Fusco, F. Maggi, V. Millot, M. Morini - Commun. Math. Phys Digital
Object Identifier (DOI) 10.1007/s00220-014-2244-1
Inflation after Planck and BICEP – Andrei Linde - “Quantum Gravity and All of
That” - https://qgholqi.inpcs.net/ - 13.01.2022 -
Modular equations and approximations to - Srinivasa Ramanujan
Quarterly Journal of Mathematics, XLV, 1914, 350 – 372
An Update on Brane Supersymmetry Breaking
J. Mourad and A. Sagnotti - arXiv:1711.11494v1 [hep-th] 30 Nov 2017
Properties of Nilpotent Supergravity
E. Dudas, S. Ferrara, A. Kehagias and A. Sagnotti - arXiv:1507.07842v2 [hep-th] 14
Sep 2015