Electrodynamics (maps)

It is shown that a considerable simplification can be attained in writing down matrix elements for complex processes in electrodynamics.

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This section is referring to wiki page-22 of gist section-18 that is inherited from the gist section-113 by prime spin-31 and span- with the partitions as below.

All matrix elements are now finite, with the exception of those relating to problems of vacuum polarization. The more conventional Hamiltonian point of view is discussed.

Basic Transformation

The first appearance of e in a printed publication was in Euler's Mechanica (1736). It is unknown why Euler chose the letter e.

Note

Leonhard Euler started to use the letter e for the constant in 1727 or 1728, in an unpublished paper on explosive forces in cannons, and in a letter to Christian Goldbach on 25 November 1731. (Wikipedia)

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It turns out that the basic idea of QED can be communicated while assuming that the square of the total of the probability amplitudes mentioned above (P(A to B), E(C to D) and j) acts just like our everyday probability (a simplification made in Feynman’s book). Later on, this will be corrected to include specifically quantum-style mathematics, following Feynman.

The basic rules of probability amplitudes that will be used are:

If an event can occur via a number of indistinguishable alternative processes (a.k.a. “virtual” processes), then its probability amplitude is the sum of the probability amplitudes of the alternatives.
If a virtual process involves a number of independent or concomitant sub-processes, then the probability amplitude of the total (compound) process is the product of the probability amplitudes of the sub-processes.

The indistinguishability criterion in (a) is very important: it means that there is no observable feature present in the given system that in any way “reveals” which alternative is taken. In such a case, one cannot observe which alternative actually takes place without changing the experimental setup in some way (e.g. by introducing a new apparatus into the system). (Wikipedia)

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It should be remembered that the expression hides a lot of complexity. We have summed over all possible timeorderings and summed over all polarization states of the virtual photon. If we are then presented with a new Feynman diagram we don’t want to go through the full calculation again. Fortunately this isn’t necessary – can just write down matrix element using a set of simple rules Basic Feynman Rules: e+ g m+ Propagator factor for each internal line (i. e. each internal virtual particle) Dirac Spinor for each external line e–

Mapping Scheme

Within a cycle this scheme would generate the prime platform which is performing the rank of 10 shapes starting with the primes 2,3,5,7.

Via the 11 partitions as the lexer and 13 frames as the parser we do a recombination to build the grammar in 6 periods.

When recombination is occur then the prime 13 is forced to → 12 where the impact (Δ1) goes to 18+13+12=43 on the last 7th row. This sequence is simulated by a flowchart having 12 arrows flowing on 10 (ten) shapes of prime 31 up to 71 (40 nodes).

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With theoretical foundations in 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)

Mapping the quantum way within a huge of primes objects (5 to 19) by lexering (11) the ungrammared feed (7) and parsering (13) across syntax (17) (₠Quantum).

By this project the above would be deployed as default layout. It is set to be avalaible throughout the whole platform via a single page within a parser repository which is acting as prime 13. Their interface will be in json and xml format.

Here is for the sample:

{
  "title":"Mapping System",
  "content":"<p>Hello, <strong>world</strong>.\nI am here.</p>\n",
  "links": [
    {"title":"Introduction","url":"https://www.eq19.com/intro/"},
    {"title":"Go tour on Mapping System ","url":"https://www.eq19.com/maps/"},
    {"title":"A backed pretty display for markdown","url":"https://www.eq19.com/gistio/"},
    {"title":"Gist.io for programmers","url":"https://gist.io/@eq19/d2336e28e79702acf38edd182003d5e0"}
  ]
}
  

Using a kind of interface such as docker then it could be developed cross platform. Evenso. Let assume that all alpabethic letter in the sequence is representing a local disk so you may build your own pattern in your PC such as shown below:

The tensor product of a triplet with an octet reducing to a deciquintuplet, an anti-sextet, and a triplet appears diagrammatically as a total of 24 states.

Using the same procedure, any direct product representation is easily reduced.

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Prime Identity

We are going to assign prime identity as a standard model that attempts to stimulate a quantum field model called eQuantum for the four (4) known fundamental forces.

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This section is referring to wiki page-22 of gist section-18 that is inherited from the gist section-113 by prime spin-31 and span- with the partitions as below.

This presentation was inspired by theoretical works from Hideki Yukawa who in 1935 had predicted the existence of mesons as the carrier particles of strong nuclear force.

Addition Zones

Here we would like to recompile the way we take on getting the arithmetic expresion of an individual unit expression (identity) such as a taxicab number below.

Note

It is a taxicab number, and is variously known as Ramanujan’s number and the Ramanujan-Hardy number, after an anecdote of the British mathematician GH Hardy when he visited Indian mathematician Srinivasa Ramanujan in hospital (Wikipedia).

These three (3) number are twin primes. We called the pairs as True Prime Pairs. Our scenario is mapping the distribution out of these pairs by taking the symmetrical behaviour of 36 as the smallest power (greater than 1) which is not a prime power.

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The smallest square number expressible as the sum of four (4) consecutive primes in two ways (5 + 7 + 11 + 13 and 17 + 19) which are also two (2) couples of prime twins! (Prime Curios!).

$True Prime Pairs:
 (5,7), (11,13), (17,19)
 
 layer|  i  |   f
 -----+-----+---------
      |  1  | 5
   1  +-----+
      |  2  | 7
 -----+-----+---  } 36 » 6®
      |  3  | 11
   2  +-----+
      |  4  | 13
 -----+-----+---------
      |  5  | 17
   3  +-----+     } 36 » 6®
      |  6  | 19
 -----+-----+---------
  

Thus in short this is all about the method that we called as the 19 vs 18 Scenario of mapping the quantum way within a huge of primes objects (5 to 19) by lexering (11) the ungrammared feed (7) and parsering (13) across syntax (17).

Φ(1,2,3) = Φ(6,12,18) = Φ(13,37,61)

$True Prime Pairs:
(5,7), (11,13), (17,19)
 
layer | node | sub |  i  |  f
------+------+-----+----------
      |      |     |  1  | 
      |      |  1  +-----+          
      |  1   |     |  2  | (5)
      |      |-----+-----+
      |      |     |  3  |
  1   +------+  2  +-----+----
      |      |     |  4  |
      |      +-----+-----+
      |  2   |     |  5  | (7)
      |      |  3  +-----+
      |      |     |  6  |
------+------+-----+-----+------      } (36)
      |      |     |  7  |
      |      |  4  +-----+
      |  3   |     |  8  | (11)
      |      +-----+-----+
      |      |     |  9  |
  2   +------|  5  +-----+-----
      |      |     |  10 |
      |      |-----+-----+
      |  4   |     |  11 | (13)
      |      |  6  +-----+
      |      |     |  12 |
------+------+-----+-----+------------------
      |      |     |  13 |
      |      |  7  +-----+
      |  5   |     |  14 | (17)
      |      |-----+-----+
      |      |     |  15 |
  3   +------+  8  +-----+-----       } (36)
      |      |     |  16 |
      |      |-----+-----+
      |  6   |     |  17 | (19)
      |      |  9  +-----+
      |      |     |  18 |
------|------|-----+-----+------
  

The main background is that, as you may aware, the prime number theorem describes the asymptotic distribution of prime numbers which is still a major problem in mathematic.

Multiplication Zones

Instead of a proved formula we came to a unique expression called zeta function. This expression first appeared in a paper in 1737 entitled Variae observationes circa series infinitas.

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This expression states that the sum of the zeta function is equal to the product of the reciprocal of one minus the reciprocal of primes to the powers. But what has this got to do with the primes? The answer is in the following product taken over the primes p (discovered by Leonhard Euler):

zeta function

This issue is actually come from Riemann hypothesis, a conjecture about the distribution of complex zeros of the Riemann zeta function that is considered to be the most important of unsolved problems in pure mathematics.

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In addition to the trivial roots, there also exist complex roots for real t. We find that the he first ten (10) non-trivial roots of the Riemann zeta function is occured when the values of t below 50. A plot of the values of ζ(1/2 + it) for t ranging from –50 to +50 is shown below. The roots occur each time the locus passes through the origin. (mathpages).

Meanwhile obtaining the non complex numbers it is easier to look at a graph like the one below which shows Li(x) (blue), R(x) (black), π(x) (red) and x/ln x (green); and then proclaim "R(x) is the best estimate of π(x)." Indeed it is for that range, but as we mentioned above, Li(x)-π(x) changes sign infinitely often, and near where it does, Li(x) would be the best value.

And we can see in the same way that the function Li(x)-(1/2)Li(x1/2) is ‘on the average' a better approximation than Li(x) to π(x); but no importance can be attached to the latter terms in Riemann's formula even by repeated averaging.

Exponentiation Zones

The problem is that the contributions from the non-trivial zeros at times swamps that of any but the main terms in these expansions.

Warning

A. E. Ingham says it this way: Considerable importance was attached formerly to a function suggested by Riemann as an approximation to π(x)… This function represents π(x) with astonishing accuracy for all values of x for which π(x) has been calculated, but we now see that its superiority over Li(x) is illusory… and for special values of x (as large as we please) the one approximation will deviate as widely as the other from the true value (primes.utm.edu).

Moreover in it was verified numerically, in a rigorous way using interval arithmetic, that The Riemann hypothesis is true up to 3 · 10^12. That is, all zeroes β+iγ of the Riemann zeta-function with 0<γ≤3⋅1012 have β=1/2.

Danger

We have Λ ≤ 0.2. The next entry is conditional on taking H a little higher than 10*13, which of course, is not achieved by Theorem 1. This would enable one to prove Λ < 0.19. Given that our value of H falls between the entries in this table, it is possible that some extra decimals could be wrought out of the calculation. We have not pursued this (arXiv:2004.09765).

This Euler formula represents the distribution of a group of numbers that are positioned at regular intervals on a straight line to each other. Riemann later extended the definition of zeta(s) to all complex numbers (except the simple pole at s=1 with residue one). Euler's product still holds if the real part of s is greater than one. Riemann derived the functional equation of zeta function.

Danger

The Riemann zeta function has the trivial zeros at -2, -4, -6, … (the poles of gamma(s/2)). Using the Euler product (with the functional equation) it is easy to show that all the other zeros are in the critical strip of non-real complex numbers with 0 < Re(s) < 1, and that they are symmetric about the critical line Re(s)=1/2. The unproved Riemann hypothesis is that all of the nontrivial zeros are actually on the critical line (primes.utm.edu).

If both of the above statements are true then mathematically this Riemann Hypothesis is proven to be incorrect because it only applies to certain cases or limitations. So first of all the basis of the Riemann Hypothesis has to be considered.

Warning

The solution is not only to prove Re(z)= 1/2 but also to calculate ways for the imaginary part of the complex root of ζ(z)=0 and also to solve the functional equations. (Riemann Zeta - pdf)

On the other hand, the possibility of obtaining the function of the distribution of prime numbers shall go backwards since it needs significant studies to be traced.

Or may be start again from the Euleur Function.

Identition Zones

Freeman Dyson discovered an intriguing connection between quantum physics and Montgomery's pair correlation conjecture about the zeros of the zeta function which dealts with the distribution of primes.

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The Mathematical Elementary Cell 30 (MEC30) standard unites the mathematical and physical results of 1972 by the mathematician Hugh Montgomery and the physicist Freeman Dyson and thus reproduces energy distribution in systems as a path plan more accurately than a measurement. (Google Patent DE102011101032A9)

The path plan assume that a symmetric distribution of prime numbers with equal axial lengths from a middle zero axis = 15 is able to determine the distribution of primes in a given number space. This assumption finally bring us to the equation of Euler's identity.

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Euler’s identity is considered to be an exemplar of deep mathematical beauty as it shows a profound connection between the most fundamental numbers. Three (3) of the basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation (Wikipedia).

The finiteness position of Euler's identity by the said MEC30 opens up the possibility of accurately representing the self-similarity based on the distribution of True Prime Pairs so that all number would belongs together with their own identitities.

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Another fascinating feature of this array is that any even number of–not necessarily contiguous–factors drawn from any one of the 32 angles in this modulo 120 configuration distribute products to 1(mod 120) or 49 (mod 120), along with the squares.

We see from the graphic above that the digital roots of the Fibonacci numbers indexed to our domain (Numbers ≌ to {1,7,11,13,17,19,23,29} modulo 30) repeat palindromically every 32 digits (or 4 thirts) consisting of 16 pairs of bilateral 9 sums.

Nothing is going to be easly about the nature of prime numbers but they demonstrably congruent to something organized. Let's discuss starting with the addition zones.

Reference: