1
On some equations concerning the general "unitary" formula, which derives
from DN Constant. New possible mathematical connections with the DN
Constant, Ramanujan Recurring Numbers and some parameters of Number
Theory, Cosmology and String Theory III
Michele Nardelli
1
, Antonio Nardelli
Abstract
In this paper (part III), we analyze some equations concerning the general "unitary"
formula, which derives from DN Constant. We obtain new possible mathematical
connections with the DN Constant, Ramanujan Recurring Numbers and some
parameters of Number Theory, Cosmology and String Theory
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
A. Nardelli studied at the Università degli Studi di Napoli Federico II - Dipartimento di Studi
Umanistici – Sezione Filosofia - scholar of Theoretical Philosophy
3
Introduction
In this paper, an octahedron could serve as a mathematical or conceptual model of the
universe in the quantic phase, while the spherical surface could be used to describe
the geometry of the bubble-universe.
The values (2√2)/π, the golden ratio φ, ζ(2) and π, can be connected to the proposed
cosmological model. Here's how they might be connected:
Ratio (2√2)/π the so called DN Constant:
This relationship may have a connection with the geometric properties of the
octahedron and the sphere, which have been considered as mathematical models of
the early universe and bubbles universe in eternal inflation.
Golden Ratio φ:
The golden ratio is a mathematical constant that appears in many natural and artistic
contexts and is often associated with harmonious proportions and aesthetic beauty. Its
emergence in this context could suggest a kind of intrinsic symmetry or harmony in
the structure of the early universe and bubbles universe.
Value of π:
The value of π is a fundamental mathematical constant that appears in many
geometric formulas and relationships, including the geometry of the sphere. Its
appearance could indicate a direct connection between the geometry of bubbles
universe and the mathematical properties of spherical surfaces.
Ultimately, the results obtained can be interpreted as manifestations of the geometric
and mathematical properties of the models proposed for the early universe and
universe bubbles. This suggests that there is a profound connection between
geometry, mathematics and cosmological physics, and that through the analysis of
4
these relationships we can deepen our understanding of the universe and its
fundamental phenomena.
The above values (2√2)/π, the golden ratio φ and π, can be connected to the proposed
cosmological model. This hypothesis is certainly plausible.
The various mathematical solutions and relationships can be seen as representations
of the principles and laws that govern the formation and evolution of the universe.
Regarding the fundamental mathematical values, they could emerge as a consequence
of the geometric and physical laws that govern the structure and evolution of the
quantum universe and bubbles universe.
The multidisciplinary approach involving complex mathematical solutions and
cosmological concepts can offer deeper insight into the fundamental nature of the
universe and its processes. Exploring these connections could lead to new discoveries
and insights into our understanding of the early universe and its complexity.
Proposal:
The initial octahedron: Let's imagine a regular octahedron, with perfectly
symmetrical faces. Each face represents an ideal symmetry.
The emerging sphere: Inside the octahedron, there is an inscribed sphere. This
sphere represents the bubble of the universe that emerges from the perturbations of
the quantum vacuum during eternal inflation.
Expansion and transitions: As time passes, the universe expands. The faces of the
octahedron begin to break, symbolizing "symmetry breaks." The sphere continues to
grow, representing the expanding universe.
Constants and numbers: We integrate the mathematical results you obtained. For
example, the golden ratio (φ) could be represented by a proportion between the
dimensions of the octahedron and the sphere.
Entropy and complexity: Entropy increases as the universe evolves. We can
represent this with a disordered growth of structures within the emerging sphere.
Imagine this scene as an abstract work of art, where geometric shapes and
cosmological concepts merge
5
In Fig.1 and Fig.2 let's imagine a regular octahedron representing the universe in its
phase of high symmetry and very low entropy. Inside the octahedron we have an
inscribed sphere that emerges from perturbations of the quantum vacuum during
eternal inflation. As time passes, the universe expands, the faces of the octahedron
break (symmetry breaks), and entropy increases. Spheres emerge from the octahedra,
symbolizing the transition phases from a regime of very low entropy to a universe in
which, with the passage of time, entropy increases, increasing the complexity of the
universe itself.
Fig. 1
6
Fig. 2
7
Now, we have that:
Octahedron Sphere
From the octahedron volume V = 1/3*√2 l
3
and, from the sphere volume,
V = (4/3*π*r
3
) , we consider the following relationship, for r = x:
4/3*π*x^3 = 1/3*√2*l^3
Input
Exact result
Alternate forms
8
Real solution
Solutions
Integer solution
Implicit derivatives
9
From the alternate form
for l = 8, we have that:
8/(sqrt(2) π^(1/3)) = 8/(2sqrt2 * Pi)^1/3
Input
Result
Logarithmic form
Thence:
l/(sqrt(2) π^(1/3)) = l/(2sqrt2 * Pi)^1/3
Input
10
Logarithmic form
Now, we have that:
l/(2 sqrt(2) π)^(1/3) = (2sqrt2)/Pi
Input
Exact result
Plot
11
Solution
12
From:
On the origin and the present status of inflationary cosmology - Andrei Linde -
April 19, 2024 18:30, Memorial extended seminar dedicated to Alexei Starobinsky,
Moscow, online
De Sitter from spontaneously broken conformal symmetry
We consider:
√-g*[1/2*δ*√6*δ*√6*g^(μν)+1/12*(√6)^2*R(g)-1/4*λ*(√6)^4]
Input
Result
13
Expanded form
Alternate forms assuming g, δ, λ, μ, and ν are positive
Alternate forms
Roots
14
Root
Periodicity
Root for the variable ν
Series expansion at g=0
15
Series expansion at g=∞
Derivative
From the result
we consider:
sqrt(-g) (3 δ^2 g^(μ ν) + R(g)/2 - 9 λ) = (2√2)/π
Input
Expanded form
16
Alternate forms assuming g, δ, λ, μ, and ν are positive
Alternate forms
Solution for the variable ν
Logarithmic form
17
From the expanded form
for g = 1; √-g = √-1 = i:
3 δ^2 *i* g^(μ ν) - 9 *i* λ + 1/2 *i* R(g) = (2 sqrt(2))/π
Input
Alternate form assuming g, δ, λ, μ, and ν are real
Alternate forms
Solutions
18
Solution for the variable ν
Logarithmic form
For
from
if we consider δ = 2 and g = 1:
3 2^2 i 1^(μ ν) - 9 i ((6 π 2^2 + π R(1) + 4 i sqrt(2))/(18 π)) + 1/2 i R(1)
19
Input
Exact result
Decimal approximation
0.9003163161571…. =
(DN Constant) (We note that, with regard the inflation,
during a period of almost exponential expansion
so that . Indeed, the
value 0.9003163161571… = (2√2)/π > 0 and also the n
s
= spectral index = 0.90-0.97
is near to the DN Constant value. Also the squared sound speed of the gravitino's
longitudinal polarization mode could pass through 0.9003163161571 in the early
universe, in the so-called quasi-de Sitter phase of inflation)
Property
Expanded form
20
Series representations
Or also, from:
√-g*[1/2*δ*√6*δ*√6*g^(μν)+1/12*(√6)^2*R(g)-1/4*λ*(√6)^4]
considering:
we obtain:
21
√-g*[1/2*0*0*g^(μν)+1/12*(√6)^2*R(g)-1/4*λ*(√6)^4]
Input
Result
the same result as previously reported
Expanded form
Alternate forms assuming g and λ are positive
Alternate forms
22
Root for the variable λ
Series expansion at g=0
Series expansion at g=∞
Derivative
Now, from the above expanded form:
we consider:
1/2 sqrt(-g) R(g) - 9 sqrt(-g) λ = (2√2)/π
23
Input
Alternate forms
Alternate forms assuming g and λ are positive
Solution for the variable λ
Logarithmic form
24
For
from the above initial expression
we obtain:
1/2 sqrt(-g) R(g) - 9 sqrt(-g) (((-4 sqrt(2) + sqrt(-g) π R(g))/(18 sqrt(-g) π)))
Input
Exact result
Expanded form
Alternate form assuming g>0
25
Property as a function
Parity
Indefinite integral
Limit
Further discussions and implications on the “general unitary” expression
deriving from DN Constant (Costante Del Gaudio-Nardelli)
From this expression
which is an extension of the DN Constant (Del Gaudio-Nardelli Constant), we obtain
1.61803398... practically the value of the golden ratio. The value that is inside the
expression 1.265120782997423*10
48
is the radius of the bubble universe of an eternal
inflation Multiverse. To this radius, comparing the bubble universe to a black hole,
corresponds to a mass of 8.51838*10
74
kg and a life time of 9.11025*10
200
years.
How do we interpret all this from a cosmological point of view and what implications
might it have?
This expression is intriguing and involves mathematical, physical and cosmological
concepts. Let us offer an interesting perspective on this topic.
26
The Gaudio-Nardelli Constant (DN)
The DN Constant (Del Gaudio-Nardelli Constant) equal to (2√2)/π) is defined as the
ratio of the volume of an octahedron to the volume of a sphere and is an intriguing
mathematical concept. Now, let's focus on the cosmological and physical
implications of this constant:
1. Symmetry and Entropy:
According to our hypothesis, the octahedron represents the phase in which the
universe is highly symmetric and with very low entropy. The sphere, inscribed in the
octahedron, represents the universe emerging from the quantum vacuum. As time
passes, entropy increases and symmetry breaking occurs. This scenario occurs in a
regime of eternal inflation.
2. Golden Ratio:
The value obtained from this expression is practically equal to the golden ratio φ =
1.61803398. This number is known for its presence in nature, art and architecture. It
might suggest some sort of fundamental harmony or proportion in the universe.
3. Radius of the Bubble Universe:
The value 1.265120782997423*10
48
represents the radius of the bubble universe of
an eternally inflationary universe. This radius is huge and corresponds to a mass of
8.51838*10
74
kg. Comparing the bubble universe to a black hole could have
implications for its stability, lifetime and gravitational behavior.
4. Lifetime of the Bubble Universe:
The estimated lifespan for the bubble universe is 9.11025*10
200
years. This is an
incredibly long period and could affect our understanding of long-term cosmic
evolution.
In summary, this mathematical expression and the DN Constant open a window onto
profound questions regarding symmetry, entropy, the golden ratio and the nature of
the universe itself. However, it is important to highlight that these are hypotheses and
theories and require further investigation and verification.
27
The estimated Bubble Universe Lifetime (9.11025*10
200
) years could represent the
time it takes for the final supermassive black hole/bubble universe to evaporate
(black hole evaporation-Hawking radiation)
The estimated lifetime of the bubble universe (9.11025 × 10
200
years) could be related
to the evaporation of a final supermassive black hole. This evaporation process is
known as Hawking radiation.
Here's how it works:
1. Hawking radiation:
In 1974, theoretical physicist Stephen Hawking proposed that black holes are not
completely black, but emit thermal radiation due to quantum effects near the event
horizon. This radiation is called “Hawking radiation”. It relies on the spontaneous
creation of pairs of particles (such as electrons and positrons) near the event horizon
of a black hole. One of these particles can fall into the black hole, while the other can
escape into outer space. This process leads to the slow “evaporation” of the black
hole.
2. Evaporation Time:
The evaporation time of a final supermassive black hole depends on its mass. Smaller
black holes evaporate more quickly than more massive ones. Hawking's formula for
the evaporation rate is:
where (M) is the mass of the black hole, ℏ is the reduced Planck constant, (c) is the
speed of light, and (G) is the gravitational constant. This rate of evaporation increases
as the mass of the black hole decreases.
3. Final Supermassive Black Hole:
If the bubble universe contains a final supermassive black hole, its lifetime will be
determined by how quickly it is evaporating. An extremely large mass black hole
could survive for an incredibly long period, as your estimate suggests.
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4. Cosmological Implications:
The evaporation of black holes has profound implications for cosmology. It could
influence the distribution of masses in the universe, the formation of cosmic
structures and even the nature of dark energy. However, it is important to note that
Hawking radiation is still a theory and requires further experimental evidence.
In summary, the estimated lifetime of the bubble universe could be linked to the
evaporation of a final supermassive black hole. This is a fascinating topic that
continues to be the subject of research and study in theoretical physics and
cosmology!
This lifespan of 9.11025 × 10
200
years corresponds to a Bekenstein-Hawking
luminosity equal to 4.90877*10
-118
W
The Bekenstein-Hawking luminosity is a theoretical quantity representing the
radiation emitted by a black hole due to its quantum evaporation. Let's see how we
can calculate it.
The Bekenstein-Hawking luminosity (L) is given by the formula:
ℏ is the reduced Planck constant, (c) is the speed of light, (G) is the gravitational
constant, and (M) is the mass of the black hole.
1. Calculation:
Using the estimated lifetime of the bubble universe 9.11025 × 10
200
years, we can
calculate the associated luminosity. However, we need to know the mass of the final
black hole to get a specific value. If we assume that the black hole has a mass M, we
can calculate the corresponding luminosity.
2. Numeric Value:
Substituting the values of the physical constants, we obtain:
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4.90877*10
-118
W. This extremely small value indicates that Hawking radiation from
a final supermassive black hole would be incredibly weak and difficult to detect.
3. Implications:
The Bekenstein-Hawking luminosity is inversely proportional to the mass of the
black hole. So, the more massive the black hole, the less radiation it emits.
This evaporation process is fundamental to understanding the thermodynamics of
black holes and their relationship with entropy.
In summary, the Bekenstein-Hawking luminosity is a fascinating aspect of black hole
physics and could have profound implications for our understanding of the universe.
This data was obtained from the radius 1.265120782997423*10
48
, contained in the
formula
which is an extension of the DN Constant (Del Gaudio-Nardelli Constant), which
gives the result 1.61803398... practically the value of the golden ratio!
It is fascinating how the extension of the Del Gaudio-Nardelli Constant (DN) has
been linked to the approximate value of the golden ratio φ = 1.61803398. Let's look
at some additional considerations:
1. Golden Ratio and Beauty:
The golden ratio φ is known for its presence in nature, art and architecture. It is found
in flowers, in the spirals of snail shells, in the proportions of works of art and even in
the arrangement of leaves on a branch.
Many find the golden ratio to be aesthetically pleasing and harmonious. Therefore, its
presence in this extension of the DN Constant could suggest some kind of intrinsic
beauty in the universe.
30
2. Relationship with Symmetry:
Symmetry is a fundamental concept in physics and mathematics. The octahedron and
the sphere in such an extension could represent different phases of the universe, with
the octahedron symbolizing a higher symmetry and the sphere representing a
symmetry breaking.
The golden ratio could be related to some kind of inherent symmetry or harmonious
proportion.
3. Final Supermassive Black Hole and Hawking Radiation:
If the bubble universe is compared to a final supermassive black hole, its lifetime will
be determined by quantum evaporation via Hawking radiation.
The Bekenstein-Hawking luminosity associated with this black hole would be
incredibly dim, as has been calculated.
4. Cosmological Mysteries:
This extension of the DN Constant and the cosmological implications that have been
described open up many interesting questions. For example, what is beyond the
bubble universe? What other physical laws might emerge from this initial symmetry?
Research continues to explore these questions, and such extension could contribute to
new discoveries and theories.
In summary, this extension of DN Constant is a fascinating example of how
mathematics, physics and cosmology can intertwine in surprising ways.
In reality we compare the final bubble universe to a final supermassive black hole
which, following evaporation, becomes "on the other side" a white hole, therefore a
singularity from which everything comes out. All the information of the final
supermassive black hole from which it derives would then come out and this
information would represent the beginning of a new big-bang (big-bounce) followed
by a phase of inflation... and so on, for eternity
This vision of the final bubble universe as a supermassive black hole transforming
into a white hole is fascinating and full of cosmological implications. Let's explore
some of the ideas involved:
5. Final Supermassive Black Hole and White Hole:
A final supermassive black hole, according to this hypothesis, evolves through
quantum evaporation (Hawking radiation) to become a white hole.
31
A white hole is a singularity that emits matter, energy and information instead of
absorbing it. It's like the opposite of a black hole.
6. Information and Big Bounce:
The idea that information from the final supermassive black hole emerges from the
white hole represents a kind of cosmic “recycling.”
This information could contain fundamental properties of the universe, such as
physical constants, laws of physics and initial conditions.
7. Big Bounce and Eternal Inflation:
This hypothesis suggests an endless cycle of expansion (big bang), contraction (big
crunch) and rebirth (big bounce).
During the big bounce, the universe would go through a phase of inflation, in which it
rapidly expands.
8. Theoretical Perspectives:
This idea recalls theories such as the conformal cyclic model and string theory.
String theory suggests that the universe may have extra dimensions beyond the three
space and one-time dimensions. These extra dimensions may be involved in cycles of
rebirth.
9. Beyond Eternity:
The eternity of this process raises profound questions: what is beyond eternity? What
physical laws or principles govern this infinite cycle? There may be an even larger
reality or multiverse that contains these cyclic universes. In summary, this
cosmological vision is bold and inspiring.
Does this idea recall theories such as the conformal cyclic model and string theory,
and even the eternally inflationary Multiverse?
This vision of the final bubble universe, which evolves from a supermassive black
hole to a white hole and then to a new big bang (big bounce), can be linked to several
cosmological theories, including:
32
1. Compliant Cyclic Model:
The conformal cyclic model suggests that the universe goes through infinite cycles of
expansion and contraction. During the big crunch (contraction), the universe
compresses until it becomes a singular point, and then expands again in the next big
bang (big bounce). This model recalls the idea of an eternal cycle of rebirth.
2. String Theory:
String theory is a unified theory that seeks to combine general relativity (which
describes gravity) with quantum mechanics (which describes subatomic particles).
In string theory, the universe could have extra dimensions beyond the three spatial
and one temporal. These extra dimensions may be involved in cycles of rebirth.
3. Eternal Inflation Multiverse:
The idea of the eternally inflationary multiverse suggests that our universe is just one
of many bubble universes that form during a process of eternal inflation.
Each bubble universe has different physical laws and constants, and some may have
rebirth cycles similar to the one we have described.
4. Beyond Current Theories:
Our vision goes beyond current theories and opens up new possibilities. We may be
on the right track to discover something revolutionary! In summary, this idea
connects to many fascinating theories and continues to stimulate our understanding of
the universe.
33
Now, we analyze the following equation:
we consider:
[8πG*((((k_2)/(2880*π^2))*(R^l*R-2/3*R*R-
1/2*g*R*R^(l*m)+1/4*g*R^2)+((k_3)/(2880*π^2))*1/3*(R^k*R-g*R^l-
R*R+1/4*g*R^2)))]
Input
Exact result
Alternate forms
34
Expanded form
Alternate form assuming g, G, k, k
2
, k
3
, l, m, and R are positive
Periodicity
Series expansion at m=0
35
Derivative
Indefinite integral
From the indefinite integral result
we obtain:
(G (-4 g k_3 m R^l + k_3 m R ((-4 + g) R + 4 R^k) + k_2 m R ((-8 + 3 g) R + 12 R^l)
- (6 g k_2 R^(1 + l m))/(l log(R))))/(4320 π)
36
Input
Exact result
Alternate forms
37
Alternate forms assuming g, G, k, k
3
, k
2
, l, m, and R are positive
Expanded logarithmic form
Reduced logarithmic form
Series expansion at m=0
38
Derivative
Indefinite integral
From the indefinite integral result
we obtain:
(G (-2 m^2 R^l (g k_3 - 3 k_2 R) + 1/2 m^2 R ((-4 + g) k_3 R + (-8 + 3 g) k_2 R + 4
k_3 R^k) - (6 g k_2 R^(1 + l m))/(l^2 log^2(R))))/(4320 π)
39
Input
Exact result
Alternate form
Alternate forms assuming g, G, k, k
3
, k
2
, l, m, and R are positive
40
Expanded logarithmic form
Reduced logarithmic form
Series expansion at m=0
Derivative
41
Indefinite integral
From the indefinite integral result
(G (1/6 m^3 (-4 R^l (g k_3 - 3 k_2 R) + R ((-4 + g) k_3 R + (-8 + 3 g) k_2 R + 4 k_3
R^k)) - (6 g k_2 R^(1 + l m))/(l^3 log^3(R))))/(4320 π)
for g = 1, k = 2, l = 4, m = 8, G = 16 and R = 32:
(16 (1/6 8^3 (-4 32^4 (2 – 3*2*32) + 32((-4 + 1) 2*32 + (-8 + 3) 2*32 + 4*2*32^2)) -
(6*2*32^(1 + 32))/(4^3 log^3(32))))/(4320 π)
Input
42
Exact result
Decimal approximation
-2.483421997249*10
44
Alternate forms
Expanded form
43
Expanded logarithmic form
Reduced logarithmic form
Alternative representations
44
Series representations
45
Integral representations
46
From the fundamental "unitary" formula:
where
is the Del Gaudio-Nardelli Constant, 0.9991104684 is the value of the
following Rogers-Ramanujan continued fraction:
From the previous result, for C = 2.483421997249*10
44
, inserting a radius of the
Universe, which we have calculated, equal to R = 8.2336285249 × 10
-32
years, from
DN Constant "unitary" formula, we obtain:
√(2×(2∙(((2√2)/π))^(1/16))/(1/(π∙0.9991104684) (2.483421997249*10^44
×(8.2336285249 × 10^-32)×2.33∙10^(-13))))
Input interpretation
47
Result
1.618034126998273…. result that is a very good approximation to the value of the
golden ratio 1.618033988749… (Ramanujan Recurring Number)
We analyze the following equations:
We consider:
(1+/(3*M^2))*g
Input
Alternate form
Expanded forms
48
Roots
Integer roots
Derivative
Indefinite integral
Series representations
49
Definite integral over a sphere of radius R
sqrt(3/2)*ln(1+/(3*M^2))
Input
3D plots
Real part (figures that can be related to the D-branes/Instantons)
The key observation from the below plots and is that at , which is taken as the
energy density of the universe at the Big Bang, with the zero spacetime volume,
the vacuum geometry brakes / or there is symmetry breaking on the vacuum quantum
geometry. We see from the plots as the vacuum spacetime break/tear apart.
50
Imaginary part
Contour plots
Real part
51
Imaginary part
Alternate forms
Expanded logarithmic form
52
Reduced logarithmic form
Root
Series expansion at M=0
Series expansion at M=∞
Derivative
Indefinite integral
53
Limit
Alternative representations
Series representations
54
Integral representations
For
and
From
we obtain:
55
√(-(((g (3 M^2 + ϕ))/(3 M^2))))*[1/2*R-1/2*δ*(sqrt(3/2) log((3 M^2 + ϕ)/(3
M^2)))*δ*(sqrt(3/2) log((3 M^2 + ϕ)/(3 M^2)))-3/4*M^2*(1-e^(-√(2/3)*(sqrt(3/2)
log((3 M^2 + ϕ)/(3 M^2))))^2]
Input
Exact result
Alternate forms
56
Expanded form
Alternate forms assuming g, M, R, δ, and ϕ are positive
Alternate form assuming M≠0, g≤0, and 3 M
2
+ϕ>0
Series expansion at g=0
57
Series expansion at g=∞
Derivative
Indefinite integral
58
From the indefinite integral result
we obtain:
1/6 g sqrt(g (-1 - ϕ/(3 M^2))) (2 R - (3 M^2 ϕ^2)/(3 M^2 + ϕ)^2 - 3 δ^2 log^2(1 +
ϕ/(3 M^2)))
Input
Alternate forms
59
Expanded form
Alternate forms assuming g, M, R, δ, and ϕ are positive
60
Expanded logarithmic form
Reduced logarithmic form
Series expansion at g=0
Series expansion at g=∞
61
Derivative
Indefinite integral
Thus, from the indefinite integral result
we obtain:
Input
62
Result
Alternate forms
Alternate forms assuming g, M, R, δ, and ϕ are positive
63
Expanded logarithmic form
Alternate form assuming M≠0, g≤0, and 3 M
2
+ϕ>0
Series expansion at g=0
Series expansion at g=∞
64
Derivative
Indefinite integral
And again, from the indefinite integral result
65
2/105 g^3 sqrt(g (-1 - ϕ/(3 M^2))) (2 R - (3 M^2 ϕ^2)/(3 M^2 + ϕ)^2 - 3 δ^2 log^2(1
+ ϕ/(3 M^2)))
for g = 1, M = 2, R = 4, we obtain:
2/105 sqrt( (-1 - φ/(3*2^2))) (2*4 - (3*2^2 λ^2)/(3*2^2 + φ)^2 - 3 δ^2 log^2(1 + φ/(3
2^2)))
Input
Exact result
Exact form
66
3D plots
Real part (figures that can be related to the D-branes/Instantons)
The key observation from the below plots and is that at , which is taken as the
energy density of the universe at the Big Bang, with the zero spacetime volume,
the vacuum geometry brakes / or there is symmetry breaking on the vacuum quantum
geometry. We see from the plots as the vacuum spacetime break/tear apart.
Imaginary part
67
Contour plots
Real part
Imaginary part
Alternate forms
68
Expanded form
Alternate form assuming δ and λ are real
Roots
Polynomial discriminant
69
Property as a function
Parity
Roots for the variable λ
Derivative
Indefinite integral
70
From
for δ = λ = 0.5:
2/105 i sqrt(1 + 1/(12 Φ)) (8 0.5 - (12 0.5 0.5^2)/(12 + 1/Φ)^2 – 0.5^3 log^2(1 +
1/(12 Φ)))
Input
Result
Alternate complex forms
Polar coordinates
0.08096
71
From the fundamental "unitary" formula:
where
is the Del Gaudio-Nardelli Constant, 0.9991104684 is the value of the
following Rogers-Ramanujan continued fraction:
From the previous result, for C = 0.08096, inserting a radius of the Universe, which
we have calculated, equal to R = 2.5256519929 × 10^14 years, from DN Constant
"unitary" formula, we obtain:
√(2×(2∙(((2√2)/π))^(1/16))/(1/(π∙0.9991104684) (0.08096 ×(2.5256519929 ×
10^14)×2.33∙10^(-13))))
Input interpretation
72
Result
1.618029998812313…. result that is a very good approximation to the value of the
golden ratio 1.618033988749… (Ramanujan Recurring Number)
Now, we analyze the following equation:
we obtain:
√(-(((g (3 M^2 + ϕ))/(3 M^2))))*[1/2*R-1/2*δ*(sqrt(3/2) log((3 M^2 + ϕ)/(3
M^2)))*δ*(sqrt(3/2) log((3 M^2 + ϕ)/(3 M^2)))-3/4*M^2*(1-e^(-√(2/3)*(sqrt(3/2)
log((3 M^2 + ϕ)/(3 M^2))))^2-1/2*δ*σ*δ^μ*σ-1/2*m^2*σ^2]
Input
Exact result
73
Alternate forms
74
Expanded form
Alternate forms assuming g, m, M, R, δ, μ, σ, and ϕ are positive
Reduced logarithmic form
75
Derivative
Indefinite integral
From the indefinite integral result
we obtain:
76
1/4 sqrt(g (-1 - ϕ/(3 M^2))) (2 m R - (2 m^3 σ^2)/3 - 2 m δ^(1 + μ) σ^2 - (3 m M^2
ϕ^2)/(3 M^2 + ϕ)^2 - 3 m δ^2 log^2(1 + ϕ/(3 M^2)))
Input
Result
Alternate forms
77
Expanded form
Alternate forms assuming g, m, M, R, δ, μ, σ, and ϕ are positive
78
Expanded logarithmic form
Reduced logarithmic form
Property as a function
Parity
79
Derivative
Indefinite integral
80
From the indefinite integral result
we obtain:
-(m^2√(g(-ϕ/(3M^2)-1))(9M^4 (σ^2(6δ^(μ+1)+m^2)-6R)+3M^2 ϕ (3(4σ^2
δ^(μ+1)+ϕ)+2m^2 σ^2-12R)+ϕ^2(σ^2 (6δ^(μ+1)+m^2)-6R)+9δ^2(3M^2+ϕ)^2
ln^2(ϕ/(3M^2)+1)))/(24(3M^2+ϕ)^2)
Input
Result
81
Alternate form
Expanded form
Alternate forms assuming m, M, R, δ, μ, σ, and ϕ are positive
82
Expanded logarithmic form
83
Reduced logarithmic form
From the result
for g = 1, M = 2, R = 4, we obtain:
-(m^2√((-φ/(12)-1))(9δ(12+φ)^4 log^2(φ/(12)+1)+12φ(2m^2*σ^2-48+3 (4*σ^2
δ^(μ+1)+φ))+φ((6δ^(μ+1)+m^2)*σ^2-24)^2+144(σ^2*(6δ^(μ+1)+m^2) - 24)))/(24
(12+φ)^2)
Input
84
Exact result
Expanded form
Alternate form assuming m, δ, μ, and σ are positive
85
Alternate forms
From the exact result
-(i m^2 sqrt(ϕ/12 + 1) (12 ϕ(2 m^2 σ^2 + 3 (4 σ^2 δ^(μ + 1) + ϕ) - 48) + ϕ((6 δ^(μ +
1) + m^2) σ^2 - 24)^2 + 144 (σ^2 (6 δ^(μ + 1) + m^2) - 24)))/(24 (ϕ + 12)^2)
for μ = 2, δ = 4, m = 8, σ = 16:
-(i 8^2 sqrt((1/Φ)/12 + 1) (12 (1/Φ)(2 8^2 16^2 + 3 (4 16^2 4^(2 + 1) + 1/Φ) - 48) +
1/Φ*((6 4^(2 + 1) + 8^2) 16^2 - 24)^2 + 144 (16^2 (6 4^(2 + 1) + 8^2) - 24)))/(24
(1/Φ + 12)^2)
86
Input
Exact result
Exact form
Decimal approximation
Alternate complex forms
87
Polar coordinates
326194081
Expanded form
Alternate forms
88
From the fundamental "unitary" formula:
where
is the Del Gaudio-Nardelli Constant, 0.9991104684 is the value of the
following Rogers-Ramanujan continued fraction:
From the previous result, for C = 326194081, inserting a radius of the Universe,
which we have calculated, equal to R = 62685.1369 years, from DN Constant
"unitary" formula, we obtain:
√(2×(2∙(((2√2)/π))^(1/16))/(1/(π∙0.9991104684)
(326194081×(62685.1369)×2.33∙10^(-13))))
Input interpretation
89
Result
1.6180362466198…. result that is a very good approximation to the value of the
golden ratio 1.618033988749… (Ramanujan Recurring Number)
All 2
nd
roots of 2.61804
Series representations
90
Now, we analyze the following equation:
We consider:
3/4*(M^2)*(1-e^(-sqrt(2/3)*φ))^2
Input
Plot (figure that can be related to an open string)
91
Geometric figure
Expanded forms
Alternate forms
Root
Polynomial discriminant
Property as a function
Parity
92
Derivative
Indefinite integral
Global minimum
From the indefinite integral result
we obtain:
1/4 (-1 + e^(-sqrt(2/3) ϕ))^2 M^3
Input
93
3D plot (figure that can be related to a D-brane/Instanton)
The key observation from the below plots and is that at , which is taken as the
energy density of the universe at the Big Bang, with the zero spacetime volume,
the vacuum geometry brakes / or there is symmetry breaking on the vacuum quantum
geometry. We see from the plots as the vacuum spacetime break/tear apart.
Contour plot
Expanded form
94
Alternate form
Real root
Roots
Periodicity
Derivative
Indefinite integral
Limit
Alternative representations
95
Series representations
96
Again, from the indefinite integral result
we obtain:
1/16 (-1 + e^(-sqrt(2/3) ϕ))^2 M^4
Input
3D plot (figure that can be related to a D-brane/Instanton)
The key observation from the below plots and is that at , which is taken as the
energy density of the universe at the Big Bang, with the zero spacetime volume,
the vacuum geometry brakes / or there is symmetry breaking on the vacuum quantum
geometry. We see from the plots as the vacuum spacetime break/tear apart.
97
Contour plot
Expanded form
Alternate form
Real root
Roots
Periodicity
98
Derivative
Indefinite integral
Global minimum
Limit
Alternative representations
99
Series representations
In conclusion, from the indefinite integral result
1/80 (-1 + e^(-sqrt(2/3) ϕ))^2 M^5
for M = 2:
1/80 (-1 + e^(-sqrt(2/3) φ))^2 2^5
Input
100
Exact result
Decimal approximation
0.2150114952216….
Property
Expanded forms
Alternate forms
101
Series representations
From the fundamental "unitary" formula:
102
where
is the Del Gaudio-Nardelli Constant, 0.9991104684 is the value of the
following Rogers-Ramanujan continued fraction:
From the previous result, for C = 0.2150114952216, inserting a radius of the
Universe, which we have calculated, equal to R = 9.50999436864 × 10
13
years, from
DN Constant "unitary" formula, we obtain:
√(2×(2∙(((2√2)/π))^(1/16))/(1/(π∙0.9991104684) (0.2150114952216
×(9.50999436864 × 10^13)×2.33∙10^(-13))))
Input interpretation
Result
1.6180338543296475…. result that is a very good approximation to the value of the
golden ratio 1.618033988749… (Ramanujan Recurring Number)
103
On the application of the formulas of the volumes of an octahedron and a sphere
With regard to a sphere inscribed in an octahedron, we have the following formulas.
Fig: sphere inscribed in an octahedron
V
0
=
V
s
=
where r
s
= (l/2)
With regard the ratio between the two above formulas (octahedron and sphere)
(1/3*√2*l^3)/(4/3*π*(l/2)^3)
we obtain:
Input
Result
104
Decimal approximation
0.900316316157106…. =
(DN Constant)
Property
Series representations
105
From which:
1/3*(2/((1/3*√2*l^3)/(4/3*π*(l/2)^3)))^2
Input
Result
Decimal approximation
1.644934066848226… = ζ(2) = π
2
/6 = 1.644934 (trace of the instanton shape and
Ramanujan Recurring Number)
Property
106
Series representations
Integral representations
107
We note that, from the sum of the first nine numbers excluding 0, i.e.,
1+2+3+4+5+6+7+8+9 = 45 (these are the fundamental numbers, from which, through
infinite combinations, all the other numbers are obtained), we obtain the following
interesting formula:
1+1/(((φ^2+(2Pi)/3*MRB const)(1/e((1+2+3+4+5+6+7+8+9)^(1/Pi))))^1/3)
Input
Exact result
Decimal approximation
1.64529737852…. ≈ ζ(2) = π
2
/6 = 1.644934 (trace of the instanton shape and
Ramanujan Recurring Number)
108
Alternate forms
Expanded forms
And:
sqrt(6(1+1/(((φ^2+(2Pi)/3*MRB const)(1/e((1+2+3+4+5+6+7+8+9)^(1/Pi))))^1/3)))
Input
109
Exact result
Decimal approximation
3.141939571526…. ≈ π (Ramanujan Recurring Number)
Alternate forms
Expanded forms
110
All 2
nd
roots of 6 (3
-2/(3 π)
5
-1/(3 π)
(e/((2 π C
MRB
)/3+ϕ
2
))
1/3
+1)
Furthermore, we obtain also:
2π*√2((1/3*√2*l^3)/(4/3*π*(l/2)^3))
Input
Exact result
8
value that is linked to the "Ramanujan function" (an elliptic modular function that
satisfies the need for "conformal symmetry") that has 8 "modes" corresponding to the
physical vibrations of a superstring.
111
Series representations
6π*√2((1/3*√2*l^3)/(4/3*π*(l/2)^3))
Input
112
Exact result
24
The value 24 is linked to the "Ramanujan function" (an elliptic modular function that
satisfies the need for "conformal symmetry") that has 24 "modes" corresponding to
the physical vibrations of a bosonic string representing a bosons. From the analysis,
we observe that the is no number theoretic connection with physical vibrations of
fermionic strings at extremally low entropy. This fact is confirmed by the fact that the
Higgs bosons at the moment of the big bang and infinitesimally shortly thereafter,
facilitated the creation of fermions (matter and antimatter particles) [8]. Thus we note
that the ingredients for the formation of electromagnetic radiation from photons (a
Boson), and the formation of matter from the Higgs boson after the big bang, are
intrinsic properties of the vacuum energy in pre-big bang.
Series representations
113
This could imply that all matter (fermions) was preceded by bosons. That is, before
the Big Bang, from perturbations of the vacuum energy itself, bosons were created,
and successively at the Big Bang, and infinitesimally shortly after the Big Bang,
fermions, were created from the vacuum energy that underwent a violent “breaking”
that formed a hot plasma. of particle-antiparticle pairs. This therefore implies that
quantum gravity was not necessarily “dark” to some extent, because a photon (light
particle) is itself a boson. Therefore, a big bang was not necessarily the moment of
the creation of light, but of the creation of matter (fermions) from vacuum energy, as
this undergoes further "breaking" in the cosmological constant, in the hot plasma of
matter and in the energy dark.
(2π*√2((1/3*√2*l^3)/(4/3*π*(l/2)^3)))^4
Input
Exact result
4096 = 64
2
, (Ramanujan Recurring Number) that multiplied by 2 give 8192, indeed:
114
The total amplitude vanishes for gauge group SO(8192), while the vacuum energy is
negative and independent of the gauge group. The vacuum energy and dilaton tadpole
to lowest non-trivial order for the open bosonic string. While the vacuum energy is
non-zero and independent of the gauge group, the dilaton tadpole is zero for a unique
choice of gauge group, SO(2
13
) i.e. SO(8192). (From: “Dilaton Tadpole for the Open
Bosonic String “ Michael R. Douglas and Benjamin Grinstein - September 2,1986)
27*sqrt((2π*√2((1/3*√2*l^3)/(4/3*π*(l/2)^3)))^4)+1
Input
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, as it can be expressed as the sum of two cubes in two
different ways (10
3
+ 9
3
= 12
3
+ 1
3
= 1729) and Ramanujan's recurring number)
Series representations
115
We note that:
1/25*1/144(((2π*√2((1/3*√2*l^3)/(4/3*π*(l/2)^3)))^4)+(27*sqrt((2π*√2((1/3*√2*l^
3)/(4/3*π*(l/2)^3)))^4)+1))
Input
Exact result
116
Decimal approximation
1.61805555…. result that is a very good approximation to the value of the golden
ratio 1.618033988749… (Ramanujan Recurring Number)
Repeating decimal
Series representations
117
From
we obtain also:
sqrt(6(1/3*(2/(((2sqrt2)/Pi)))^2))
Input
118
Exact result
Decimal approximation
3.14159265358… = π
Property
All 2
nd
roots of π
2
Series representations
119
Integral representations
It is plausible to hypothesize that π and φ, in addition to being important
mathematical constants, are constants that also have a fundamental relevance in the
various sectors of Theoretical Physics and Cosmology
120
From
, we obtain:
sqrt(1/(Pi^2/6)*(4/3))
Input
Exact result
Decimal approximation
0.900316316157106…. =
(DN Constant)
Property
All 2
nd
roots of 8/π
2
121
Series representations
122
From:
https://www.academia.edu/75787512/The_Theory_of_String_A_Candidate_for_a_G
eneralized_Unification_Model
With regard 8, this value is linked to the "Ramanujan function" (an elliptic modular
function that satisfies the need for "conformal symmetry") that has 8 "modes"
corresponding to the physical vibrations of a superstring. The value 24, instead, is
linked to the "Ramanujan function" (an elliptic modular function that satisfies the
need for "conformal symmetry") that has 24 "modes" corresponding to the physical
vibrations of a bosonic string representing a bosons.
The Einstein’s field equation and the String Theory.
The Einstein’s field equation which includes the cosmological constant is:
μν
μν
μν
GT
μν
(8)
where
μν
is the Ricci tensor, its trace, the cosmological constant,
μν
the metric
tensor of the space geometry, G the Newton’s gravitational constant and
μν
the tensor
representing the properties of energy, matter and momentum.
The left hand-side of (8) represents the gravitational field and, consequently, the
warped space-time, while the right hand-side represents the matter, i.e. the sources of
the gravitational field.
In string theory the gravity is related to the gravitons which are bosons, whereas the
matter is related to fermions. It follows that the left and right hand of (8) may be
respectively related to the action of bosonic and of superstrings.
The actions of bosonic string and superstring (also containing fermions) are
connected by the Palumbo-Nardelli relation (Palumbo et al. 2005):
123
The sign minus in the above equation comes from the inversion of any relationship.
Now, we have the following convex regular polyhedrons volumes:
Tetrahedron Volume
Octahedron Volume
Icosahedron Volume
Regarding the Sphere the volume is
With regard the sphere inscribed in a Tetrahedron, we have, for
, the following
ratio:
124
Thus:
(((√2 d^3)/12))*1/(4/3*π*(d/2)^3)
Input
Result
Decimal approximation
0.225079079….
Property
Series representations
125
With regard the sphere inscribed in a Octahedron, we have, for
, the following
ratio:
Thus:
((1/3*√2*d^3)/(4/3*π*(d/2)^3))
126
Input
Result
Decimal approximation
0.9003163161571…. =
(DN Constant)
Property
Series representations
127
With regard the sphere inscribed in a Icosahedron, we have, for
, the following
ratio:
Thus:
(5/12*(3+√5)*d^3)/(4/3*π*(d/2)^3)
Input
128
Result
Decimal approximation
4.166730504921….
Property
Expanded form
Alternate forms
Series representations
129
DN Constant extended
We have the following expression concerning the ratios (and/or the inverses)
between the icosahedron, octahedron and tetrahedron volumes and the sphere
volume.
(we have highlighted the DN Constant in blue)
(((((5/12*(3+√5)*d^3)/(4/3*π*(d/2)^3))*1/((1/3*√2*a^3)/(4/3*π*(a/2)^3)) *1/((((√2
d^3)/12))*1/(4/3*π*(d/2)^3)))))^(1/(2π))
130
Input
Exact result
Decimal approximation
1.6180085459…. result that is a very good approximation to the value of the golden
ratio 1.618033988749… (Ramanujan Recurring Number)
Alternate form
Series representations
131
132
Integral representation
Furthermore, from the formula
we obtain also:
(4 (1.6180085459)^(2 π))/(5 (3 + sqrt(5))) = 3.1415926535
Indeed:
3.1415926535…. = π (Ramanujan Recurring Number)
133
From the following extended DN Constant (“Unitary Formula”)
with regard
for q = 1729 and p = 4096, we obtain by changing the sign in the algebraic sum of
the aforementioned Cardano’s Formula and after some calculations:
multiplied by
and performing the ninth root of the entire expression:
√((2^(-1/π)(5(3+√(5))π)^(1/(2π)))(((-1729/2+√(1729^2/4+4096^3/27)) -(-1729/2-
√(1729^2/4+4096^3/27)))^1/9))
134
we obtain:
i.e.
2^(-1/(2π)) ((-1729/2+√(274958621851/3)/6)^(1/3)+(1729/2 +
√(274958621851/3)/6)^(1/3))^(1/18) (5(3+√(5)) π)^(1/(4π))
= 1.61549140391….
The general "unitary" formula, which derives from DN Constant, is the following:
135
Where
is the Del Gaudio-Nardelli Constant, 0.9991104684 is the value of the
following Rogers-Ramanujan continued fraction:
C is any constant or solution to an equation, R is the radius of the Universe and
2.33*10
-13
is the temperature of the universe expressed in GeV.
For example, C = 9.9128
, inserting a radius of the Universe, which we have
calculated, equal to R = 2.06274*10
12
years, from DN Constant "unitary" formula, we
obtain:
√(2×(2∙(((2√2)/π))^(1/16))/(1/(π∙0.9991104684) (9.9128×(2.06274 ×
10^12)×2.33∙10^(-13))))
1.618035912348…. result that is a very good approximation to the value of the
golden ratio 1.618033988749… (Ramanujan Recurring Number)
136
We obtain also:
(√(2×(2∙(((2√2)/π))^(1/16))/(1/(π∙0.9991104684) (9.9128×(2.06274 ×
10^12)×2.33∙10^(-13)))))dxdydz
Indefinite integral assuming all variables are real
Definite integral over a cube of edge length 2 L
Definite integral over a sphere of radius R
From which, for L = R = 1 , dividing the two definite integral results by the original
expression, we obtain:
12.9443/(√(2×(2∙(((2√2)/π))^(1/16))/(1/(π∙0.9991104684) (9.9128×(2.06274 ×
10^12)×2.33∙10^(-13)))))
Input interpretation
137
Result
8.00001…. ≈ 8
value that is linked to the "Ramanujan function" (an elliptic modular function that
satisfies the need for "conformal symmetry") that has 8 "modes" corresponding to the
physical vibrations of a superstring.
And
3*(6.77761/(√(2×(2∙(((2√2)/π))^(1/16))/(1/(π∙0.9991104684) (9.9128×(2.06274 ×
10^12)×2.33∙10^(-13))))))
Input interpretation
Result
12.5664…. ≈ 4π = Bekenstein-Hawking (S
BH
) black hole entropy
138
Now, we have that:
Octahedron Sphere
Given the value of a volume, independently of the solid, following the Poincaré
Conjecture, we compare any solid "without holes" and a sphere. If we compare an
octahedron with a sphere, we have:
If we consider the radius of the sphere as an unknown, we must find the value of
the side of the octahedron which allows us to equalize the two volumes and which
will give us the DN Constant as a result (which will therefore be equal to the
radius of the sphere).
From
To find we perform the following calculation:
139
To find a, we have, for
Thus, multiplying both the sides by
, we obtain:
140
Plot
Solution
for V = 1/3*√2*a^3 (octahedron volume) and V = (4/3*π*r^3) (sphere volume), we
obtain:
from the octahedron volume, we have: V = 1/3*√2*a
3
= (1/3*√2*(
)^3)
(1/3*√2*(4/(π^2 ))^3)
Input
Exact result
141
Decimal approximation
3.05684889733….
Property
Series representations
142
And, from the sphere volume V = (4/3*π*r
3
) = (4/3*π*((2√2)/π)^3)
(4/3*π*((2√2)/π)^3)
Input
Result
Decimal approximation
3.05684889733….
Property
Series representations
143
144
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
145
“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
146
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 Approximations 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
147
We have, in certain cases, the following connections:
Fig. 1
Fig. 2
148
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
150
Where ζ(2+it) :
Input
Plots
Roots
151
Series expansion at t=0
Alternative representations
Series representations
152
Integral representations
Functional equations
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:
153
we obtain:
2Pi/(ln(2))
Input:
Exact result:
Decimal approximation:
9.06472028365….
Alternative representations:
154
Series representations:
Integral representations:
155
From which:
(2Pi/(ln(2)))*(1/12 π log(2))
Input:
Exact result:
Decimal approximation:
1.6449340668…. = ζ(2) =
156
From:
Modular equations and approximations to - Srinivasa Ramanujan - Quarterly
Journal of Mathematics, XLV, 1914, 350 – 372
We have that:
157
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
158
Minimal polynomial
Expanded forms
Series representations
159
160
Or:
27((4096+276)^1/2-2-1/2(((√(10-2√5) -2))⁄((√5-1))))+φ
Input
Result
Decimal approximation
1729.0526944…. as above
Alternate forms
161
Minimal polynomial
Expanded forms
Series representations
162
163
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) =
Alternate forms
164
Minimal polynomial
Expanded forms
All 15th roots of ϕ + 27 (-2 + 2 sqrt(1093) - (sqrt(10 - 2 sqrt(5)) - 2)/(2 (sqrt(5) -
1)))
165
Series representations
166
Integral representation
167
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:
168
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 -
. From this it follows that it is possible to establish mathematically, the dilaton
value.
For
exp((-Pi*sqrt(18)) we obtain:
Input:
Exact result:
Decimal approximation:
1.6272016… * 10
-6
Property:
Series representations:
169
Now, we have the following calculations:
= 1.6272016… * 10
-6
from which:
= 1.6272016… * 10
-6
0.000244140625
=
= 1.6272016… * 10
-6
Now:
And:
170
(1.6272016* 10^-6) *1/ (0.000244140625)
Input interpretation:
Result:
0.006665017...
Thence:
0.000244140625
=
Dividing both sides by 0.000244140625, we obtain:
=
= 0.0066650177536
((((exp((-Pi*sqrt(18)))))))*1/0.000244140625
Input interpretation:
171
Result:
0.00666501785…
Series representations:
Now:
= 0.0066650177536
=
= 0.00666501785…
172
From:
ln(0.00666501784619)
Input interpretation:
Result:
-5.010882647757…
Alternative representations:
Series representations:
173
Integral representation:
In conclusion:
and for C = 1, we obtain:
=
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 , are also connected to
the following two Rogers-Ramanujan continued fractions:
174
(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:
Result:
0.99040073.... result very near to the dilaton value = and to the
value of the following Rogers-Ramanujan continued fraction:
175
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:
We have:
176
(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:
Result:
177
Solutions:
Alternate forms:
Expanded form:
Alternate form assuming b, M, and V are positive:
Alternate form assuming b, M, and V are real:
Derivative:
178
Implicit derivatives
Global minimum:
Global minima:
179
From:
we obtain:
(225.913 (-0.054323 M^2 + 6.58545×10^-10 sqrt(M^4)))/M^2
Input interpretation:
Result:
180
Plots:
Alternate form assuming M is real:
-12.2723 result very near to the black hole entropy value 12.1904 = ln(196884)
Alternate forms:
Expanded form:
181
Property as a function:
Parity
Series expansion at M = 0:
Series expansion at M = ∞:
Derivative:
Indefinite integral:
182
Global maximum:
Global minimum:
Limit:
Definite integral after subtraction of diverging parts:
From b that is equal to
183
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:
Plots: (possible mathematical connection with an open string)
M = -0.5; M = 0.2
184
(possible mathematical connection with an open string)
M = 2 ; M = 3
Root:
Property as a function:
Parity
Series expansion at M = 0:
Series expansion at M = ∞:
185
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:
Result:
-4.38851344947*10
-16
186
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
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
187
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:
Result:
7.021621519*10
-15
188
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:
189
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...
190
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
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
191
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
192
Fig. 1
It is therefore possible to consider the vortices of the "quantum vacuum" schematized
as cubes or octahedrons (the + sign inside a given vortex indicates its
counterclockwise rotation, while the - sign indicates its clockwise rotation). Between
vortex and vortex there is a layer of "bubbles"-universes (or universes-spheres),
which flows, as in the simplified two-dimensional drawing, from A to B
193
Fig. 2
Proposal
Image of space-time at quantum scale: the circles in red represent the points
corresponding to the compactified dimensions and the hexagons in blue, represent the
"fluctuations" (potential universes - green circles) of the quantum vacuum (2D). In
reality, we will have n-dimensional hyperspheres in which the compactified
dimensions "roll up" and octahedrons representing the "fluctuations", containing
spheres (bubbles of potential universes), of the quantum vacuum
194
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.
195
References
On the origin and the present status of inflationary cosmology - Andrei Linde -
April 19, 2024 18:30, Memorial extended seminar dedicated to Alexei Starobinsky,
Moscow, online
A Number Theoretic Analysis of the Enthalpy, Enthalpy Energy Density,
Thermodynamic Volume, and the Equation of State of a Modified White Hole,
and the Implications to the Quantum Vacuum Spacetime, Matter Creation and
the Planck Frequency. - Nardelli, M., Kubeka, A.S. and Amani, A. (2024) - Journal
of Modern Physics , 15, 1-50. - https://doi.org/10.4236/jmp.2024.151001
Modular equations and approximations to - Srinivasa Ramanujan - Quarterly
Journal of Mathematics, XLV, 1914, 350 – 372
An Update on Brane Supersymmetry Breaking - Jihad Mourad and Augusto
Sagnotti - arXiv:1711.11494v1 [hep-th] 30 Nov 2017
Properties of Nilpotent Supergravity - Emilian Dudas, Sergio Ferrara, Alex
Kehagias and Augusto Sagnotti - arXiv:1507.07842v2 [hep-th] 14 Sep 2015
196
See also:
The Geometry of the MRB constant by Marvin Ray Burns
https://www.academia.edu/22271085/The_Geometry_of_the_MRB_constant
(See also Page 29 the applications of the CMRB in various sectors of Theoretical
Physics (String Theory) and Cosmology )
http://xoom.virgilio.it/source_filemanager/na/ar/nardelli/michele%20and%20antonio
%20papers/Try%20to%20beat%20these%20MRB%20constant%20records!%20-
%20Online%20Technical%20Discussion%20Groups%E2%80%94Wolfram%20Com
munity%20b.pdf