1
On the study of various equations concerning the Isoperimetric Theorems.
Possible mathematical connections with some sectors of Number Theory, String
Theory and some cosmological parameters.
Michele Nardelli
1
, Antonio Nardelli
2
Abstract
In this paper, we analyze various equations concerning the Isoperimetric Theorems.
We describe the new possible mathematical connections with some sectors of
Number Theory, String Theory and cosmological parameters
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
From Wikipedia:
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).
We propose that some equations concerning the “balls”, can be related with various
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.
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.
4
From:
Isoperimetric Theorems, Open Problems and New Results – Francesco Maggi –
ICTP, Trieste, 22 February 2017
We have:
((3-1)^(13/8)) / ((5-1)^(3/2))*sqrt((5+3-1)/(3*5*x*y))*2^(-12) =
c/(R^2)*(3/5)^(9/4)*5^(1/4)
Where u
k
= x ; ω
h
= y ; k = 3 and h = 5
Input
Exact result
Alternate form assuming c, R, x, and y are real
5
Alternate form
Alternate form assuming c, R, x, and y are positive
Real solutions
Solution for the variable y
6
From the following alternate form:
we obtain:
(5 sqrt(35) R^2 sqrt(1/(x y)))/(73728 2^(3/8) 3^(3/4))
Input
Exact result
Real roots
Properties as a function
Domain
Range
7
Parity
Series expansion at x=0
Series expansion at x=∞
Derivative
Indefinite integral
8
Global minimum
Limit
Series representations
9
From the above derivative
we obtain, from the result:
-(5 sqrt(35) R^2 (1/(x y))^(3/2) y)/(147456 2^(3/8) 3^(3/4))
Input
10
Exact result
Real roots
Properties as a function
Domain
Range
Parity
Series expansion at x=0
11
Series expansion at x=∞
Derivative
Indefinite integral
Limit
12
Series representations
From the above derivative:
we obtain, from the result:
Input
Exact result
13
Real roots
Properties as a function
Domain
Range
Parity
Series expansion at x=0
14
Series expansion at x=∞
Derivative
Indefinite integral
Limit
15
Series representations
For:
αs = [−π/2, π/2] ,
|c| ≥ 1/4,
16
From:
(1-1/16)*Pi/6 * 1/2
Input
Result
Decimal approximation
R = 0.245436926….
Property
Alternative representations
17
Series representations
Integral representations
18
We have:
For C = 8 : θ = 1/16 ; R = 0.245436926
From the previous derivative
we obtain, from the result:
(175 sqrt(35) ((5Pi)/64)^2 sqrt(1/(x y)))/(393216 2^(3/8) 3^(3/4) x^4)
Input
Exact result
19
3D plots
Real part (figures that can be related to the D-branes/Instantons)
Imaginary part
20
Contour plots
Real part
Imaginary part
Roots
Properties as a function
Domain
21
Range
Parity
Series expansion at x=∞
Partial derivatives
Indefinite integral
22
Limit
From the above result
For x = -0.4 and y = -4 , we obtain :
(4375 sqrt(35) π^2 sqrt(1/(-0.4* -4)))/(1610612736 2^(3/8) 3^(3/4) *(-0.4)^4)
Input
Result
0.00165689….
23
Series representations
24
Inverting
we obtain:
1/(((4375 sqrt(35) π^2 sqrt(1/(-0.4* -4)))/(1610612736 2^(3/8) 3^(3/4) *(-0.4)^4)))
Input
Result
603.541….
Series representations
25
From the previous alternate form:
we obtain also:
(5 sqrt(35) ((5Pi)/64)^2 sqrt(1/(x y)))/(73728 2^(3/8) 3^(3/4))
Input
26
Exact result
3D plots
Real part (figures that can be related to the D-branes/Instantons)
Imaginary part
27
Contour plots
Real part
Imaginary part
Roots
Properties as a function
Domain
28
Range
Parity
Series expansion at x=0
Series expansion at x=∞
Partial derivatives
29
Indefinite integral
Limit
Series representations
30
From the above result
for x = y = 0.001, we obtain:
(125 sqrt(35) π^2 sqrt(1/(0.001* 0.001)))/(301989888 2^(3/8) 3^(3/4))
Input
Result
0.00817569….
31
Series representations
Inverting, we obtain:
1/((((125 sqrt(35) π^2 sqrt(1/(0.001* 0.001)))/(301989888 2^(3/8) 3^(3/4)))))
Input
32
Result
122.314….
Series representations
33
From the sum between the two previous inverted expressions, we obtain:
1/(((4375 sqrt(35) π^2 sqrt(1/(-0.4* -4)))/(1610612736 2^(3/8) 3^(3/4) *(-0.4)^4))) +
(((1/((((125 sqrt(35) π^2 sqrt(1/(0.001* 0.001)))/(301989888 2^(3/8) 3^(3/4))))))))+Pi
Input
Result
728.996… ≈ 729
Series representations
34
35
From which:
10^3+1/(((4375 sqrt(35) π^2 sqrt(1/(-0.4* -4)))/(1610612736 2^(3/8) 3^(3/4) *(-
0.4)^4))) + (((1/((((125 sqrt(35) π^2 sqrt(1/(0.001* 0.001)))/(301989888 2^(3/8)
3^(3/4))))))))+Pi
Input
36
Result
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)
Series representations
37
38
39
(((10^3+1/(((4375 sqrt(35) π^2 sqrt(1/(-0.4* -4)))/(1610612736 2^(3/8) 3^(3/4) *(-
0.4)^4))) + (((1/((((125 sqrt(35) π^2 sqrt(1/(0.001* 0.001)))/(301989888 2^(3/8)
3^(3/4))))))))+Pi)))^1/15
Input
Result
1.64381497748…. ≈ ζ(2) =
2
6
= 1.644934 (trace of the instanton shape)
From the result of the previous partial derivative:
we obtain:
-(125 sqrt(35) π^2 x (1/(x y))^(3/2))/(603979776 2^(3/8) 3^(3/4))
Input
40
Exact result
3D plots
Real part (figures that can be related to the D-branes/Instantons)
Imaginary part
41
Contour plots
Real part
Imaginary part
Roots
Properties as a function
Domain
42
Range
Parity
Series expansion at x=0
Partial derivatives
Indefinite integral
43
And from:
we obtain:
(125 sqrt(35) π^2 (1/(x y))^(3/2))/(1207959552 2^(3/8) 3^(3/4))
Input
Exact result
3D plots
Real part (figures that can be related to the D-branes/Instantons)
44
Imaginary part
Contour plots
Real part
Imaginary part
45
Roots
Properties as a function
Domain
Range
Parity
Series expansion at x=0
Series expansion at x=∞
46
Partial derivatives
Indefinite integral
Limit
Series representations
47
And again, from:
-(125 sqrt(35) π^2 (1/(x y))^(5/2) y)/(805306368 2^(3/8) 3^(3/4))
Input
Exact result
48
3D plots
Real part (figures that can be related to the D-branes/Instantons)
Imaginary part
Contour plots
Real part
49
Imaginary part
Alternate form assuming x and y are positive
Roots
Properties as a function
Domain
Range
Parity
50
Series expansion at x=0
Series expansion at x=∞
Partial derivatives
Indefinite integral
Limit
51
Series representations
From:
For x = -0.8 and y = -3 , we obtain:
52
-(125 sqrt(35) π^2 (1/(-0.8* -3))^(5/2) *(-3))/(805306368 2^(3/8) 3^(3/4))
Input
Result
1.03074…*10
-6
Series representations
From which:
1/((-(125 sqrt(35) π^2 (1/(-0.8* -3))^(5/2) *(-3))/(805306368 2^(3/8) 3^(3/4))))^20 *
((1/2 (5 e^π + π + log(16) + 3 log(π) + 3 tan^(-1)(π))))
53
Input
Result
0.351599…*10
122
≈ Λ
Q
The observed value of ρ
Λ
or Λ today is precisely the classical dual of its quantum
precursor values ρ
Q
, Λ
Q
in the quantum very early precursor vacuum U
Q
as
determined by our dual equations. With regard the Cosmological constant,
fundamental are the following results: Λ = 2.846 * 10
-122
and Λ
Q
= 0.3516 * 10
122
(New Quantum Structure of the Space-Time - Norma G. SANCHEZ - arXiv:1910.13382v1
[physics.gen-ph] 28 Oct 2019)
Alternative representations
54
Series representations
55
56
Integral representations
57
Continued fraction representations
58
59
60
From:
SHARP STABILITY INEQUALITIES FOR THE PLATEAU PROBLEM - G.
De Philippis & F. Maggi - j. differential geometry 96 (2014) 399-456
We have that:
From:
x* (h*k)/(m-1) * R^(m-1) / ((h-1)^((k-1)/2)) * y
Input
61
Result
Alternate form
Roots
Derivative
62
Indefinite integral
Limit
Series representations
63
For: k = 3; h = 5 ; m = 8
From:
(3*5 x y (5 - 1)^((1 - 3)/2) 2^(8 - 1))/(8 - 1)
Input
Result
3D plot (figure that can be related to a D-brane/Instanton)
64
Contour plot
Geometric figure
Properties as a function
Domain
Range
Parity
65
Partial derivatives
Indefinite integral
Definite integral over a disk of radius R
Definite integral over a square of edge length 2 L
For x = y = 0.5 :
(480*0.5*0.5)/7
Input
Result
66
Repeating decimal
17.142857
We have:
From:
for: k = 3; h = 5 ; m = 8
(5*3)/(8-1) * (2^(8-1))/(((5-1)^(7/2)))
Input
Exact result
Decimal approximation
2.142857142….
67
We have:
For: k = 3; h = 5 ; m = 8
68
From:
Sqrt(2^13((5-1)/(3-1))*(5*3)/(8-1)*1/(3-1)^0.25)
Input
Result
171.82162803….
From the algebraic sum of the three above expressions, after some calculations, we
obtain:
12*((-(((480*0.5*0.5)/7) + ((5*3)/(8-1) * (2^(8-1))/(((5-1)^(7/2)))) - (Sqrt(2^13((5-
1)/(3-1))*(5*3)/(8-1)*1/(3-1)^0.25))))-8)-(2Pi)
Input
Result
1728.15….
69
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)
Series representations
70
(1/27(12*((-(((480*0.5*0.5)/7) + ((5*3)/(8-1) * (2^(8-1))/(((5-1)^(7/2)))) -
(Sqrt(2^13((5-1)/(3-1))*(5*3)/(8-1)*1/(3-1)^0.25))))-8)-(2Pi)))^2-Φ
Input
Result
4096.08…. ≈ 4096 = 64
2
(12*((-(((480*0.5*0.5)/7) + ((5*3)/(8-1) * (2^(8-1))/(((5-1)^(7/2)))) - (Sqrt(2^13((5-
1)/(3-1))*(5*3)/(8-1)*1/(3-1)^0.25))))-8)-(2Pi))^1/15
Input
Result
1.643761200788…. ≈ ζ(2) =
2
6
= 1.644934 (trace of the instanton shape)
71
We have:
From:
(8sqrt3*16)/(sqrt3)
Input
Result
128
From which:
27*1/2*(((8sqrt3*16)/(sqrt3)))+1
Input
72
Exact result
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*1/2*(((8sqrt3*16)/(sqrt3)))+1))^1/15
Input
Result
Decimal approximation
1.6438152287…. ≈ ζ(2) =
2
6
= 1.644934 (trace of the instanton shape)
(1/2*(((8sqrt3*16)/(sqrt3))))^2
Input
73
Exact result
4096 = 64
2
From:
STABILITY INEQUALITIES FOR LAWSON CONES - Zhenhua Liu -
arXiv:1711.06927v6 [math.DG] 22 Aug 2018
We have that
((1/16(u-v)*v^0.25*(27u^2-123uv+98v^2)))/((((1/16*sqrt(v)*(9u^2-
34uv+49v^2)))))^(3/2)
Input
Result
74
3D plots
Real part (figures that can be related to the D-branes/Instantons)
Imaginary part
75
Contour plots
Real part
Imaginary part
76
Expanded forms
Alternate forms assuming u and v are positive
Real roots
77
Roots for the variable u
Series expansion at u=0
Series expansion at u=∞
78
Derivative
Indefinite integral
From:
(4 (-27 u^3 v^1.75 + 2541 u^2 v^2.75 - 8297 u v^3.75 + 5831 v^4.75))/(v (9 u^2 - 34
u v + 49 v^2)^2 sqrt(sqrt(v) (9 u^2 - 34 u v + 49 v^2)))
Input
79
Result
3D plots
Real part (figures that can be related to the D-branes/Instantons)
Imaginary part
80
Contour plots
Real part
Imaginary part
Alternate form assuming u and v are real
Alternate forms
81
Alternate form assuming u and v are positive
Expanded forms
82
Real roots
Roots for the variable u
83
Series expansion at u=0
Series expansion at u=∞
Derivative
84
Indefinite integral
From:
(24 (81 u^4 v^1.75 - 11358 u^3 v^2.75 + 56320 u^2 v^3.75 - 72754 u v^4.75 +
14847 v^5.75))/(v (9 u^2 - 34 u v + 49 v^2)^3 sqrt(sqrt(v) (9 u^2 - 34 u v + 49 v^2)))
Input
85
3D plots
Real part (figures that can be related to the D-branes/Instantons)
Imaginary part
Contour plots
Real part
86
Imaginary part
Alternate form assuming u and v are real
Alternate forms
Alternate form assuming u and v are positive
87
Expanded forms
88
Real roots
Roots for the variable u
89
Series expansion at u=0
Series expansion at u=∞
90
Derivative
Indefinite integral
From:
91
-(24 (2187 u^5 v^1.75 - 407511 u^4 v^2.75 + 2711610 u^3 v^3.75 - 5131410 u^2
v^4.75 + 1600091 u v^5.75 + 1798153 v^6.75))/(v (9 u^2 - 34 u v + 49 v^2)^4
sqrt(sqrt(v) (9 u^2 - 34 u v + 49 v^2)))
Input
3D plots
Real part (figures that can be related to the D-branes/Instantons)
Imaginary part
92
Contour plots
Real part
Imaginary part
Alternate form assuming u and v are real
93
Alternate forms
Alternate form assuming u and v are positive
94
Expanded forms
95
Series expansion at u=0
Series expansion at u=∞
96
Derivative
Indefinite integral
From:
-(24 (2187 u^5 v^0.5 - 407511 u^4 v^1.5 + 2711610 u^3 v^2.5 - 5131410 u^2
v^3.5 + 1600091 u v^4.5 + 1798153 v^5.5))/(9 u^2 - 34 u v + 49 v^2)^(9/2)
97
Input
3D plots
Real part (figures that can be related to the D-branes/Instantons)
Imaginary part
98
Contour plots
Real part
Imaginary part
Alternate form
99
Expanded form
Roots
v = 0.270764 u
Roots for the variable u
100
Series expansion at u=0
Series expansion at u=∞
101
Derivative
Indefinite integral
From:
2
3
2
1
2
3
2
1
22
i
uv
i
vu
pvuvu
For u = -v(1/2+(i*sqrt3)/2) ; v = -u(1/2+(i*sqrt3)/2)
102
-(24 sqrt(-u(1/2+(i*sqrt3)/2)) (2187 (-v(1/2+(i*sqrt3)/2))^5 - 407511 (-
v(1/2+(i*sqrt3)/2))^4 *(-u(1/2+(i*sqrt3)/2))+ 2711610 (-v(1/2+(i*sqrt3)/2))^3 (-
u(1/2+(i*sqrt3)/2))^2 - 5131410 (-v(1/2+(i*sqrt3)/2))^2 (-u(1/2+(i*sqrt3)/2))^3 +
1600091 (-v(1/2+(i*sqrt3)/2)) (-u(1/2+(i*sqrt3)/2))^4 + 1798153 (-
u(1/2+(i*sqrt3)/2))^5))/(9 (-v(1/2+(i*sqrt3)/2))^2 - 34 (-v(1/2+(i*sqrt3)/2))( -
u(1/2+(i*sqrt3)/2)) + 49 (-u(1/2+(i*sqrt3)/2))^2)^(9/2)
Dividing the above long expression:
-(24 sqrt(-u(1/2+(i*sqrt3)/2))
Input
Exact result
Alternate form
For u = -1 :
-24 sqrt(((sqrt(3) i)/2 + 1/2))
Input
103
Result
Decimal approximation
Polar coordinates
24
(2187 (-v(1/2+(i*sqrt3)/2))^5 - 407511 (-v(1/2+(i*sqrt3)/2))^4 *(-
u(1/2+(i*sqrt3)/2))+ 2711610 (-v(1/2+(i*sqrt3)/2))^3 (-u(1/2+(i*sqrt3)/2))^2
Input
Exact result
104
407511 u((sqrt(3) i)/2 + 1/2) v((sqrt(3) i)/2 + 1/2)^4 - 2711610 u((sqrt(3) i)/2 +
1/2)^2 v((sqrt(3) i)/2 + 1/2)^3 - 2187 v((sqrt(3) i)/2 + 1/2)^5
Input
Exact result
Alternate forms
105
407511 *-((sqrt(3) i)/2 + 1/2) ((sqrt(3) i)/2 + 1/2)^4 - 2711610 *-((sqrt(3) i)/2 +
1/2)^2 ((sqrt(3) i)/2 + 1/2)^3 - 2187 ((sqrt(3) i)/2 + 1/2)^5
Input
Result
Decimal approximation
Polar coordinates
2301912
- 5131410 (-v(1/2+(i*sqrt3)/2))^2 (-u(1/2+(i*sqrt3)/2))^3 + 1600091 (-
v(1/2+(i*sqrt3)/2)) (-u(1/2+(i*sqrt3)/2))^4 + 1798153 (-u(1/2+(i*sqrt3)/2))^5
Input
106
Exact result
For u = -1 ; v = 1 :
-1600091 *-((sqrt(3) i)/2 + 1/2)^4 ((sqrt(3) i)/2 + 1/2) + 5131410 *-((sqrt(3) i)/2 +
1/2)^3 ((sqrt(3) i)/2 + 1/2)^2 - 1798153 *-((sqrt(3) i)/2 + 1/2)^5
Input
Result
Decimal approximation
Polar coordinates
1733166
107
(9 (-v(1/2+(i*sqrt3)/2))^2 - 34 (-v(1/2+(i*sqrt3)/2))( -u(1/2+(i*sqrt3)/2)) + 49 (-
u(1/2+(i*sqrt3)/2))^2)^(9/2)
Input
Exact result
(-34 *-((sqrt(3) i)/2 + 1/2) ((sqrt(3) i)/2 + 1/2) + 49*-((sqrt(3) i)/2 + 1/2)^2 + 9
((sqrt(3) i)/2 + 1/2)^2)^(9/2)
Input
Result
Decimal approximation
Polar coordinates
1296√6
108
Polar forms
Approximate form
Alternate forms
Expanded forms
(24(2301912-1733166))/(1296*sqrt6)
Input
109
Result
Decimal approximation
4299.807077928….
Alternate form
From:
For u = 1 ; v = -1 :
((1/32*u^0.25*(u-v)(49u^2-72uv+27v^2)))/((((1/32*sqrt(u)*(49u^2-
10uv+9v^2)))))^(3/2)
Input
Result
110
3D plots
Real part (figures that can be related to the D-branes/Instantons)
Imaginary part
111
Contour plots
Real part
112
Imaginary part
113
Expanded forms
Alternate forms assuming u and v are positive
Real root
114
Roots for the variable u
Series expansion at u=0
Series expansion at u=∞
Derivative
115
Indefinite integral assuming all variables are real
116
From:
(-2.82843 u^6 + 19.799 u^5 v - 25.9754 u^4 v^2 + 9.39354 u^3 v^3 - 0.222646 u^2
v^4 + 0.286259 u v^5)/(u^2.25 (u^2 - 0.204082 u v + 0.183673 v^2)^2 sqrt(sqrt(u)
(49 u^2 - 10 u v + 9 v^2)))
Input interpretation
Result
117
3D plots
Real part (figures that can be related to the D-branes/Instantons)
Imaginary part
118
Contour plots
Real part
Imaginary part
Alternate form assuming u and v are real
119
Alternate forms
120
Expanded forms
121
Alternate forms assuming u and v are positive
122
Derivative
From:
123
For u = 1; v = -1 :
(-2.82843 - 19.799 - 25.9754 - 9.39354 - 0.222646 - 0.286259)/(1^2.25 (1 +
0.204082 + 0.183673)^2 sqrt(sqrt(1) (49 +10 + 9)))
Input interpretation
Result
-3.683960519….
From which:
1+1/(-(-2.82843 - 19.799 - 25.9754 - 9.39354 - 0.222646 - 0.286259)/(1^2.25 (1 +
0.204082 + 0.183673)^2 sqrt(sqrt(1) (49 +10 + 9))))^1/3
Input interpretation
Result
1.6474829612…. ≈ ζ(2) =
2
6
= 1.644934 (trace of the instanton shape)
124
(24(2301912-1733166))/(1296*sqrt6) - (((-2.82843 - 19.799 - 25.9754 - 9.39354 -
0.222646 - 0.286259)/(1^2.25 (1 + 0.204082 + 0.183673)^2 sqrt(sqrt(1) (49 +10 +
9)))))-233+21+5-Pi/6
Input interpretation
Result
4095.9674…. ≈ 4096 = 64
2
Series representations
125
27(((24(2301912-1733166))/(1296*sqrt6) - (((-2.82843 - 19.799 - 25.9754 - 9.39354
- 0.222646 - 0.286259)/(1^2.25 (1 + 0.204082 + 0.183673)^2 sqrt(sqrt(1) (49 +10 +
9)))))-233+21+5-Pi/6))^1/2
Input interpretation
126
Result
1727.99313…. ≈ 1728
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)
Series representations
127
128
((27(((24(2301912-1733166))/(1296*sqrt6) - (((-2.82843 - 19.799 - 25.9754 -
9.39354 - 0.222646 - 0.286259)/(1^2.25 (1 + 0.204082 + 0.183673)^2 sqrt(sqrt(1)
(49 +10 + 9)))))-233+21+5-Pi/6))^1/2))^1/15
Input interpretation
Result
1.643751394…. ≈ ζ(2) =
2
6
= 1.644934 (trace of the instanton shape)
129
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
130
“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
131
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
132
We have, in certain cases, the following connections:
Fig. 1
Fig. 2
133
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
134
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:
135
Series representations:
Integral representations:
136
From which:
(2Pi/(ln(2)))*(1/12 π log(2))
Input:
Exact result:
Decimal approximation:
1.6449340668…. = ζ(2) =
2
6
= 1.644934
137
From:
Modular equations and approximations to - Srinivasa Ramanujan
Quarterly Journal of Mathematics, XLV, 1914, 350 – 372
We have that:
138
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
139
Minimal polynomial
Expanded forms
Series representations
140
Or:
27((4096+276)^1/2-2-1/2(((√(10-2√5) -2))⁄((√5-1))))+φ
141
Input
Result
Decimal approximation
1729.0526944…. as above
Alternate forms
142
Minimal polynomial
Expanded forms
Series representations
143
144
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
145
Minimal polynomial
Expanded forms
All 15th roots of ϕ + 27 (-2 + 2 sqrt(1093) - (sqrt(10 - 2 sqrt(5)) - 2)/(2 (sqrt(5) -
1)))
146
Series representations
147
Integral representation
148
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.
149
For
exp((-Pi*sqrt(18)) we obtain:
Input:
Exact result:
Decimal approximation:
1.6272016… * 10
-6
Property:
Series representations:
150
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...
151
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:
152
Now:
6+
= 0.0066650177536
=
= 0.00666501785…
From:
ln(0.00666501784619)
Input interpretation:
Result:
-5.010882647757…
153
Alternative representations:
Series representations:
Integral representation:
In conclusion:
6 + = 5.010882647757
and for C = 1, we obtain:
154
= 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:
155
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:
156
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:
157
Result:
Solutions:
Alternate forms:
Expanded form:
Alternate form assuming b, M, and V are positive:
Alternate form assuming b, M, and V are real:
158
Derivative:
Implicit derivatives
159
Global minimum:
Global minima:
From:
we obtain
(225.913 (-0.054323 M^2 + 6.58545×10^-10 sqrt(M^4)))/M^2
Input interpretation:
160
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:
161
Expanded form:
Property as a function:
Parity
Series expansion at M = 0:
Series expansion at M = ∞:
Derivative:
162
Indefinite integral:
Global maximum:
Global minimum:
Limit:
Definite integral after subtraction of diverging parts:
163
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:
164
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:
165
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)
166
Input interpretation:
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
167
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
168
Input interpretation:
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:
169
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
170
Planck’s Electric field strength
1.820306 * 10
61
V*m Lorentz-Heaviside value
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
171
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
172
Acknowledgments
M. Nardelli thanks Francesco Maggi, Professor of Mathematics at University of
Texas - Austin, Department of Mathematics, for his availability and kindness towards
him
173
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
STABILITY INEQUALITIES FOR LAWSON CONES - Zhenhua Liu -
arXiv:1711.06927v6 [math.DG] 22 Aug 2018
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