The current mathematics in 3D plus real time t as external control is incomplete and misleading.
We should start immediately in replacing this rotten mathematics one century old with the modern 4D unit space math which can do miracles !.
Let focus here on the correct definition of the Laplacian operator in 4D:
Nabla^(N)= B + B^2 . . . + B^N
Note that,
Nabla^2(infinity/ = 1/( I - B ) - I
This correct definition of the Laplacian can:
1-Solve time- dependent PDE in its most general form.
2- Derive and solve Schrodinger PDE quite simp.
3-Find formulas for numerical integration like Sympson.s rules without the need for Lagrange multipliers.
Etc . .
A question arises,
Can the B-mtrix chains predict the speed of sound Vs and the speed of light c in any medium?
Answer:
The B matrix chains in 4D unit space can predict Vs and c as function of two variables :T1/2 and the diffusivity of the medium space .
It predicts Vs in air at NPT ( from W. Sabine measurements ) as 330 m/s and the speed of thermal radiation signal in Aluminium and iron ( from I. Abbas cooling curve measurements ) as c= 3E8 m/s.
I
The B matrix chains in 4D unit space predicts Vs and c as function of only two variables :
The half time period T1/2 and the diffusivity of the medium.
It predicts Vs in air at NPT as 330 m/s.
It predicts thermal radiation signal speed c as 3 E 8 m/s.
On a similar basis, the B matrix chains in 4D unity space also predicts that the speed of thermal radiation signal in vacuum c, equal to 3 E 8 m/s.
A second question arises:
What is the difference between Einstein space of special and general relativity and the proposed B matrix chains space?
Both spaces are unitary 4D x-t but Einstein space is continuous in time whereas B matrix space is discrete with equal time intervals dt..dt is not arbitrary but fixed according to the system configuration such that the time intervals are woven on the system 4 D geometry.
The results is that Einstein space is not probabilistic and eligible only for relativity
eHWd
In conclusion, the dream of theoretical physicists and mathematician is at hand:
Demolish all current mathematics and replace it by a single statistical universal law:
UU UUÚUÚUUUDDFddD
This universal numerical statistical law is expressed as:
U ( x , y , X , t+, dt ) = B. U ( x , y ,z ,t) ,
No more PDE or any other mathematical expression in classical 3 D + t space.
The idea of this Q&A work is to replace current mathematics and theoretical physics that live and operate in an R^4 spatial manifold with those operating in a modern 4D x-t space.
this would make it possible to resolve in a simple manner almost all classical and quantum physical situations and also to introduce new general rules in science.
We first define the discrete 4D unit space from the numerical statistical theory of matrix chains B of the Cairo techniques and compare it to the current 4D unit spaces of Einstein and Markov.
Next, you need to introduce what is called the modern Laplacian theorem in 4D unit space.
Finally, you explain the unexpected and striking relationship between the speed of light c=3E8 and the thermal diffusivity of metals when both live in 4D unit space.
The accuracy and precision of the numerical results show beyond doubt that the proposed 4D unit space is the one in which mother nature operates. This space forms the basis of a unified field theory of all types of energy density, whereas the classical manifold space R^4 is inferior and incomplete.
In conclusion, we recommend the proposed 4D unit space which constitutes a real breakthrough in the search for a new 4D numerical statistical theory to replace the classical incomplete R^4 mathematical space.
Note that throughout this work, the author uses his own double precision algorithm. In other words,
Python or MATLAB library is not required.
The idea of this work is to replace current mathematics and theoretical physics that live and operate in an R^4 spatial manifold with those operating in a modern 4D x-t space.
You should first describe what ugly mathematics is and its dangerous consequences and how to solve it via the use of 4D discrete unit space from the numerical statistical theory of matrix strings B of Cairo techniques.
Next, you should introduce what we call microscopic energy density diffusion and macroscopic energy density diffusion into the modern Laplacian theorem in 4D unit space.
Finally, one has to explain the unexpected and striking results when we replace the current ugly mathematics with the superior B-matrix string techniques.
Top 10 questions and answers on ugly mathematics in classical physics, quantum mechanics and pure mathematics are explained and validate the superiority of 4D xt unit space.
The accuracy and precision of the numerical results show beyond doubt that the proposed 4D unit space is the one in which mother nature operates.
One can state that this space forms the basis of a unified field theory of all types of energy density, while the classical manifold space R^4 is inferior and incomplete.
In conclusion, we recommend the proposed 4D unit space which constitutes a real breakthrough in the search for a new 4D numerical statistical theory to replace the classical incomplete R^4 mathematical space.
Quite surprisingly, it is possible to calculate a finite numerical integration via the statistical mechanics of the B matrix.
It is obvious that here it is not necessary to use FDM, Mat-Lab or any similar numerical calculation method [1].
Single, double or triple finite statistical integration for 1D, 2D, 3D space is not complicated but a bit cautious. [1].
The starting point is to discretize the space considered into n equidistant free nodes then to calculate the transfer matrix Dnxng given by the following expression Dnxn=1 /(I-Bnxn)- I
Where B is the well-defined nxn statistical transition matrix in 1D with RO=0.
The resulting finite integration statistical formula I = ∫fromx1tox7 y dx is the following numerical formula,
I = 6h/77 (6.Y1 +11.Y2 + 14. Y3+15.Y4 +14.Y5 + 11.Y6 + 6.Y7). . ..(1)
for n = 7 nodes,
And the statistical integration formula I, for 11 nodes gives:
I=10 h/ 145*[6.191Y1+ 10.715 Y2 +13.862 Y3+15.921 Y4 +17.056 Y5+17.4344025 Y6+. . +. . . ] . . (2)
Equations 1,2 are the equivalence of Simpson's rule (which does not exist ?) for 7 and 11 nodes respectively.
It is worth mentioning that the numerical results of equations 1,2 are more accurate than those obtained from the trapezoidal rule and those obtained from Sympson's rule [1].
Also note that the artificial method of Lagrange multipliers or any other statistical assumptions are not required. This clearly shows that the numerical statistical method of matrix chains B in a 4D unit space is more complete than the classical 3D+t method.
The current complications of mathematical calculations are weak and tend to be useless in borderline cases.
1- I. Abbas, Useless mathematics, ResearchGate, IJISRT magazine, September 24
- Badges
- Science topic