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Rational Objects

In number theory, the partition functionp(n) represents the number of possible partitions of a non-negative integer n. Integers can be considered either in themselves or as solutions to equations (Diophantine geometry).

The central problem is to determine when a Diophantine equation has solutions, and if it does, how many. Two examples of an elliptic curve, that is, a curve of genus 1 having at least ***one rational point***. Either graph can be seen as a slice of a torus in four-dimensional space _([Wikipedia](https://en.wikipedia.org/wiki/Number_theory#Diophantine_geometry))_.

Number theory

One of the main reason is that one does not yet have a mathematically complete example of a quantum gauge theory in four-dimensional space-time. It is even a sign that Einstein’s equations on the energy of empty space are somehow incomplete.

Throughout his life, Einstein published hundreds of books and articles. He published more than 300 scientific papers and 150 non-scientific ones. On 5 December 2014, universities and archives announced the release of Einstein's papers, comprising more than 30,000 unique documents _([Wikipedia](https://en.wikipedia.org/wiki/Albert_Einstein#Scientific_career))_.

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Speculation is that the unfinished book of Ramanujan's partition, series of Dyson's solutions and hugh of Einstein's papers tend to solve it.

Dyson introduced the concept in the context of a study of certain congruence properties of the partition function discovered by the mathematician Srinivasa Ramanujan who the one that found the interesting behaviour of the taxicab number 1729.

The concept was introduced by [Freeman Dyson](https://en.wikipedia.org/wiki/Freeman_Dyson)in a paper published in the journal [Eureka](https://en.wikipedia.org/wiki/Eureka_(University_of_Cambridge_magazine)). It was [presented](https://en.wikipedia.org/wiki/Rank_of_a_partition#cite_note-Dyson-1) in the context of a study of certain [congruence](https://en.wikipedia.org/wiki/Congruence_relation) properties of the [partition function](https://en.wikipedia.org/wiki/Partition_function_(number_theory)) discovered by the Indian mathematical genius [Srinivasa Ramanujan](https://en.wikipedia.org/wiki/Srinivasa_Ramanujan). _([Wikipedia](https://e
n.wikipedia.org/wiki/Rank_of_a_partition))_

Rank_of_a_partition

Young tableaux were introduced by Alfred Young, a mathematician at Cambridge University, in 1900. They were then applied to the study of the symmetric group. Their theory was further developed by many mathematicians, including W. V. D. Hodge

In _[number theory](https://gist.github.com/eq19/e9832026b5b78f694e4ad22c3eb6c3ef#number-theory)_ and combinatorics, [rank of a partition](https://en.wikipedia.org/wiki/Rank_of_a_partition) of a positive integer is a certain integer associated with the partition meanwhile the [crank of a partition](https://en.wikipedia.org/wiki/Crank_of_a_partition) of an integer is a certain integer associated with that partition _([Wikipedia](https://en.wikipedia.org/wiki/Freeman_Dyson#Crank_of_a_partition))_.

Supersymmetry

In mathematics, the rank of a partition is the number obtained by subtracting the number of parts in the partition from the largest part in the partition.

On the other hand, one does not yet have a mathematically complete example of a quantum gauge theory in 4D Space vs Time, nor even a precise definition of quantum gauge theory in four dimensions. Will this change in the 21st century? We hope so! _([Clay Institute's - Official problem description](https://claymath.org/sites/default/files/yangmills.pdf))_.

image

25 + 19 + 13 + 7 = 64 = 8 × 8 = 8²

The True Prime Pairs:
(5,7), (11,13), (17,19)

|--------------- 7¤ ---------------|
|-------------- {89} --------------|👈
+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
|  5 |  7 | 11 |{13}| 17 | 19 | 17 |{12}| 11 | 19 |{18}| 18 | 12 |{13}|
+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
        ∆         ∆      |---- {48} ----|---- {48} ----|---- {43} ----|👈
        7        13      |----- 3¤ -----|----- 3¤ -----|----- 3¤ -----|
                         |-------------------- 9¤ --------------------|
                            ∆                               |-- 25 ---|
                           19                                    ∆
                                                               5 x 5
SU(5) fermions of standard model in 5+10 representations. The sterile neutrino singlet's 1 representation is omitted. Neutral bosons are omitted, but would occupy diagonal entries in complex superpositions. X and Y bosons as shown are the opposite of the conventional definition

SO(10)

SU(5)_representation_of_fermions

$True Prime Pairs:
(5,7), (11,13), (17,19)

     |    168    |    618    |
-----+-----+-----+-----+-----+                                             ---
 19¨ |  3¨ |  4¨ |  6¨ |  6¨ | 4¤  ----->  assigned to "id:30"             19¨
-----+-----+-----+-----+-----+                                             ---
 17¨ |  5¨ |  3¨ | ..  |  .. | 4¤ ✔️ --->  assigned to "id:31"              |
     +-----+-----+-----+-----+                                              |
{12¨}|  .. |  .. |  2¤ (M & F)     ----->  assigned to "id:32"              |
     +-----+-----+-----+                                                    |
 11¨ |  .. |  .. |  .. | 3¤ ---->  Np(33)  assigned to "id:33"  ----->  👉 77¨
-----+-----+-----+-----+-----+                                              |
 19¨ |  .. |  .. |  .. |  .. | 4¤  ----->  assigned to "id:34"              |
     +-----+-----+-----+-----+                                              |
{18¨}|  .. |  .. |  .. | 3¤        ----->  assigned to "id:35"              |
     +-----+-----+-----+-----+-----+-----+-----+-----+-----+               ---
 43¨ |  .. |  .. |  .. |  .. |  .. |  .. |  .. |  .. |  .. | 9¤ (C1 & C2)  43¨
-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+               ---
139¨ |  1     2     3  |  4     5     6  |  7     8     9  |
                    Δ                 Δ                 Δ       
Family Number Group +3, +6, +9 being activated by the Aetheron Flux Monopole Emanations, creating Negative Draft Counterspace, Motion and Nested Vortices.) _([RodinAerodynamics](https://rense.com/RodinAerodynamics.htm))_

guest7

This idea was taken as the earliest in 1960s Swinnerton-Dyer by using the University of Cambridge Computer Laboratory to get the number of points modulo p (denoted by Np) for a large number of primes p on elliptic curves whose rank was known.

From these numerical results ***the conjecture predicts that the data should form a line of slope equal to the rank of the curve***, which is 1 in this case drawn in red in red on the graph _([Wikipedia](https://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture#Current_status))_.

Dyson discovered that the eigenvalue of these matrices are spaced apart in exactly the same manner as _[Mo Unfortunately the rotation of this eigenvalues deals with four-dimensional space-time which was already a big issue.

Geometry of 4D rotations

In 1904 the French mathematician Henri Poincaré asked if the three dimensional sphere is characterized as the unique simply connected three manifold. This question, the Poincaré conjecture, was a special case of Thurston's geometrization conjecture.

Perelman's proof tells us that every three manifold is built from a set of standard pieces, each with one of eight well-understood geometries _([ClayMath Institute](https://www.claymath.org/millennium-problems/poincar%C3%A9-conjecture))_.

Poincaré Conjecture

More generally, the central problem is to determine when an equation in n-dimensional space has solutions. However at this point, we finaly found that the prime distribution has something to do with the subclasses of rank and crank partitions.

Ricci Flow

guest5

p r i m e s
1 0 0 0 0 0
2 1 0 0 0 1
3 2 0 1 0 2
4 3 1 1 0 3
5 5 2 1 0 5
6 7 3 1 0 7
7 11 4 1 0 11
8 13 5 1 0 13
9 17 0 1 1 17 ◄--- has a total of 18-7 = 11 composite √
10 19 1 1 11 ◄--- 0th ∆prime ◄--- Fibonacci Index #18
-----
11 23 2 1 12 ◄--- 1st ∆prime ◄--- Fibonacci Index #19
12 29 2 -1 13 ◄--- 2nd ∆prime ◄--- Fibonacci Index #20
13 31 1 -1 14
14 37 1 1 15 ◄--- 3th ∆prime ◄--- Fibonacci Index #21
15 41 2 1 16
16 43 3 1 17 ◄--- 4th ∆prime ◄--- Fibonacci Index #22
17 47 4 1 18
18 53 4 -1 19
19 59 4 1 110
20 61 5 1 111 ◄--- 5th ∆prime ◄--- Fibonacci Index #23
21 67 5 -1 112
22 71 4 -1 113 ◄--- 6th ∆prime ◄--- Fibonacci Index #24
23 73 3 -1 114
24 79 3 1 115
25 83 4 1 116
26 89 4 -1 117 ◄--- 7th ∆prime ◄--- Fibonacci Index #25
27 97 3 -1 118
28 101 2 -1 119 ◄--- 8th ∆prime ◄--- Fibonacci Index #26
29 103 1 -1 120
30 107 0 -1 121
31 109 5 -1 022
32 113 4 -1 023 ◄--- 9th ∆prime ◄--- Fibonacci Index #27
33 127 3 -1 024
34 131 2 -1 025
35 137 2 1 026
36 139 3 1 027
37 149 4 1 028
38 151 5 1 029 ◄--- 10th ∆prime  ◄--- Fibonacci Index #28
39 157 5 -1 030
40 163 5 1 031 ◄--- 11th ∆prime ◄--- Fibonacci Index #29
-----
41 167 0 1 10
42 173 0 -1 11
43 179 0 1 12 ◄--- ∆∆1
44 181 1 1 13 ◄--- ∆∆2 ◄--- 1st ∆∆prime ◄--- Fibonacci Index #30
45 191 2 1 14
46 193 3 1 15 ◄--- ∆∆3 ◄--- 2nd ∆∆prime ◄--- Fibonacci Index #31
47 197 4 1 16
48 199 5 1 17 ◄--- ∆∆4
49 211 5 -1 18
50 223 5 1 19
51 227 0 1 210
52 229 1 1 211 ◄--- ∆∆5 ◄--- 3rd ∆∆prime ◄--- Fibonacci Index #32
53 233 2 1 212
54 239 2 -1 213 ◄--- ∆∆6
55 241 1 -1 214
56 251 0 -1 215
57 257 0 1 216
58 263 0 -1 217 ◄--- ∆∆7 ◄--- 4th ∆∆prime ◄--- Fibonacci Index #33
59 269 0 1 218
60 271 1 1 219 ◄--- ∆∆8
61 277 1 -1 220
62 281 0 -1 221
63 283 5 -1 122
64 293 4 -1 123 ◄--- ∆∆9
65 307 3 -1 124
66 311 2 -1 125
67 313 1 -1 126
68 317 0 -1 127
69 331 5 -1 028
70 337 5 1 029 ◄--- ∆∆10
71 347 0 1 130
72 349 1 1 131 ◄--- ∆∆11 ◄--- 5th ∆∆prime ◄--- Fibonacci Index #34
73 353 2 1 132
74 359 2 -1 133
75 367 1 -1 134
76 373 1 1 135
77 379 1 -1 136
78 383 0 -1 137 ◄--- ∆∆12
79 389 0 1 138
80 397 1 1 139
81 401 2 1 140
82 409 3 1 141 ◄--- ∆∆13 ◄--- 6th ∆∆prime ◄--- Fibonacci Index #35
83 419 4 1 142
84 421 5 1 143 ◄--- ∆∆14
85 431 0 1 244
86 433 1 1 245
87 439 1 -1 246
88 443 0 -1 247 ◄--- ∆∆15
89 449 0 1 248
90 457 1 1 249
91 461 2 1 250
92 463 3 1 251
93 467 4 1 252
94 479 4 -1 253 ◄--- ∆∆16
95 487 3 -1 254
96 491 2 -1 255
97 499 1 -1 256
98 503 0 -1 257
99 509 0 1 258
100 521 0 -1 259 ◄--- ∆∆17 ◄--- 7th ∆∆prime ◄--- Fibonacci Index #36
-----
101 523 5 -1 12 ◄--- ∆∆18 ◄--- 1st ∆∆∆prime ◄--- Fibonacci Index #37 √
102 541 5 1 13 ◄--- ∆∆∆1 ◄--- 1st ÷÷÷composite ◄--- Index #(37+2)=#39 √
103 547 5 -1 14
104 557 4 -1 15 ◄--- ∆∆∆2 ◄---2nd ∆∆∆prime √
105 563 4 1 16
106 569 4 -1 17 ◄--- ∆∆∆3 ◄--- 3rd ∆∆∆prime √
107 571 3 -1 18
108 577 3 1 19
109 587 4 1 110
110 593 4 -1 111 ◄--- ∆∆∆4 ◄--- 2nd ÷÷÷composite ◄--- Index #(37+3)=#40 √
111 599 4 1 112
112 601 5 1 113 ◄--- ∆∆∆5 ◄--- 4th ∆∆∆prime √
113 607 5 -1 114
114 613 5 1 115
115 617 0 1 216
116 619 1 1 217 ◄--- ∆∆∆6 ◄--- 3rd ÷÷÷composite ◄--- Index #(37+5)=#42 √
117 631 1 -1 218
118 641 0 -1 219 ◄--- ∆∆∆7 ◄--- 5th ∆∆∆prime √
119 643 5 -1 120
120 647 4 -1 121
121 653 4 1 122
122 659 4 -1 123 ◄--- ∆∆∆8 ◄--- 4th ÷÷÷composite ◄--- Index #(37+7)=#44 √
123 661 3 -1 124
124 673 3 1 125
125 677 4 1 126
126 683 4 -1 127
127 691 3 -1 128
128 701 2 -1 129 ◄--- ∆∆∆9 ◄--- 5th ÷÷÷composite ◄--- Index #(37+11)=#48 √
129 709 1 -1 130
130 719 0 -1 131 ◄--- ∆∆∆10 ◄--- 6th ÷÷÷composite ◄--- Index #(37+13)=#50 √
131 727 5 -1 032
132 733 5 1 033
133 739 5 -1 034
134 743 4 -1 035
135 751 3 -1 036
136 757 3 1 037 ◄--- ∆∆∆11 ◄--- 6th ∆∆∆prime √
137 761 4 1 038
138 769 5 1 039
139 773 0 1 140
140 787 1 1 141 ◄--- ∆∆∆12 ◄--- 7th ÷÷÷composite ◄--- Index #(37+17)=#54 √
141 797 2 1 142
142 809 2 -1 143 ◄--- ∆∆∆13 ◄--- 7th ∆∆∆prime √
143 811 1 -1 144
144 821 0 -1 145
145 823 5 -1 046
146 827 4 -1 047 ◄--- ∆∆∆14 ◄--- 8th ÷÷÷composite ◄--- Index #(37+19)=#56 √
147 829 3 -1 048
148 839 2 -1 049
149 853 1 -1 050
150 857 0 -1 051
151 859 5 -1 -152
152 863 4 -1 -153 ◄--- ∆∆∆15 ◄--- 9th ÷÷÷composite ◄--- Index #(37+23)=#60 √
153 877 3 -1 -154
154 881 2 -1 -155
155 883 1 -1 -156
156 887 0 -1 -157
157 907 5 -1 -258
158 911 4 -1 -259 ◄--- ∆∆∆16 ◄--- 10th ÷÷÷composite ◄--- Index #(37+29)=#66 √
159 919 3 -1 -260
169 929 2 -1 -261 ◄--- ∆∆∆17 ◄--- 8th ∆∆∆prime √
161 937 1 -1 -262
162 941 0 -1 -263
163 947 0 1 -264
164 953 0 -1 -265
165 967 5 -1 -366
166 971 4 -1 -367 ◄--- ∆∆∆18 ◄--- 11th ÷÷÷composite ◄--- Index #(37+31)=#68 √
167 977 4 1 -368
168 983 4 -1 -369
169 991 3 -1 -370
170 997 3 1 -3271 ◄--- ∆∆∆19 ◄--- 9th ∆∆∆prime √

Scot_Number_Map_Diag

The Ricci flow is a pde for evolving the metric tensor in a Riemannian manifold to make it rounder, in the hope that one may draw topological conclusions from the existence of such “round” metrics.

Poincaré hypothesized that if such a space has the additional property that each loop in the space can be continuously tightened to a point, then it is necessarily a three-dimensional sphere _([Wikipedia](https://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture))_

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The Ricci Flow method has now been developed not only in to geometric but also to the conversion of facial shapes in three (3) dimensions to computer data. A big leap in the field of AI (Artificial intelligence). No wonder now all the science leads to it.

So what we've discussed on this wiki is entirely nothing but an embodiment of this solved Poincare Conjecture. This is the one placed with id: 10 (ten) which stands as the basic algorithm of π(10)=(2,3,5,7).

Many relevant topics, such as trustworthiness, explainability, and ethics are characterized by implicit anthropocentric and anthropomorphistic conceptions and, for instance, the pursuit of human-like intelligence. AI is one of the most debated subjects of today and there seems little common understanding concerning the differences and similarities of human intelligence and artificial intelligence _([Human vs AI](https://www.frontiersin.org/articles/10.3389/frai.2021.622364/full))_.

Poincaré Conjecture

Finite collections of objects are considered 0-dimensional. Objects that are "dragged" versions of zero-dimensional objects are then called one-dimensional. Similarly, objects which are dragged one-dimensional objects are two-dimensional, and so on.

The basic ideas leading up to this result (including the dimension invariance theorem, domain invariance theorem, and Lebesgue covering dimension) were developed by **Poincaré**, Brouwer, Lebesgue, Urysohn, and Menger _([MathWorld](https://mathworld.wolfram.com/Dimension.html))_.

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Spacetime Patterns

toroid_color

In vector calculus, the Jacobian matrix of a vector-valued function of several variables is the matrix of all its first-order partial derivatives.

It's possible to build a _[Hessian matrix](https://en.wikipedia.org/wiki/Hessian_matrix)_ for a _[Newton's method](https://en.wikipedia.org/wiki/Newton%27s_method_in_optimization)_ step using the Jacobian method. ***You would first flatten out its axes into a matrix, and flatten out the gradient into a vector.*** _([Tensorflow](https://www.tensorflow.org/guide/advanced_autodiff#batch_jacobian))_

Tensorflow - Batch Jacobian

When the subclasses of partitions are flatten out into a matrix, you want to take the Jacobian of each of a stack of targets with respect to a stack of sources, where the Jacobians for each target-source pair are independent.

***When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components of its output, its determinant is referred to as the Jacobian determinant***. Both the matrix and (if applicable) the determinant ad  often referred to simply as the Jacobian in literature. _([Wikipedia](https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant))_

Hessian matrix for Newton Method

Double Strands

Here we adopt an analysis of variance called N/P-Integration that was applied to find the best set of environmental variables that describe the density out of distance matrices.

With collaborators, we regularly work on projects where we want to understand the taxonomic and functional diversity of microbial community in the context of metadata often recorded under specific hypotheses. Integrating (***N-/P- integration***; see figure below) these datasets require a fair deal of multivariate statistical analysis for which I have shared the [code](https://userweb.eng.gla.ac.uk/umer.ijaz/bioinformatics/ecological.html) on this website. _([Umer.Ijaz](https://userweb.eng.gla.ac.uk/umer.ijaz/#intro))_

N-/P- integration

It can be used to build parsers/compilers/interpreters for various use cases ranging from simple config files to full fledged programming languages.

With theoretical foundations in [Information Engineering](https://en.wikipedia.org/wiki/Information_engineering) (Discrete Mathematics, Control Theory, System Theory, Information Theory, and Statistics), my research has delivered a suite of systems and products that has allowed me to carve out a niche within an extensive collaborative network (>200 academics). _([Umer.Ijaz](https://userweb.eng.gla.ac.uk/umer.ijaz/#intro))_

information engineering

Since such interactions result in a change in momentum, they can give rise to classical Newtonian forces of rotation and revolution by means of orbital structure.

torus

As you can see on the left sidebar (dekstop mode) a total of 102 items will be reached by the end of Id: 35.

So when they transfered to Id: 36 it will cover 11 x 6 = 66 items thus the total will be 102 + 66 = 168

Clone this wiki locally