The Laplacian equation above is equivalent to

U( X ,y z , t+ dt ) = B . U ( X,y ,z ,t )

Where the transition matrix B is subject to,

D(N)= B + B^2 + , + .+ .+B^N

And,

É = D(N) for N tends to infinity..

Where E is given by.,

E = 1 / ( I - B)@

E = D + I¹

The transition matrix B is subject to,

D(N)=B + B^2 . . . . +B^N

And,

D(N) tends to E as N tends to ♾️.

Where E is given by

E= 1/( I - B )

It is clear that E ,= D + I

And D(N)=B+B^2 +.+..+B^N

Also,

D(N)= E as N tends to infinity.

Where E is given by,

E=1/(I-B) and E=D+I

AAlso,

Note that E=1/(I-B) and.

E=D+i

The classical Laplacian operator in the classical 3D + t space generates nothing, unlike the generalized Laplacian in the 4D unit space.

The reason is that in 4D unit space, the feald and its source are generalized.

Just as the loop and divergence differential operators have two forms differential and integral, the theorem of Laplacian operators in 4D unit space also does the same.

The differential form of Laplacian's theorem is provided in the first answer while the integral form of Laplacian's theorem follows:

∫∫ U dS = ∫∫∫ S(U) dV.

To be continued.

Finally, here is a simple understanding of spaces:

The Laplacian theorem lives and functions in the 4D-unit space but not in the classical 3D+t space.

The theory of relativity and the speed of light c lives and functions in 4D unit space but not in classical 3D+t space.

. . etc.

To our knowledge the diffusivity α of energy density U(x,y,z,t) has never been properly defined.

The naive definition of thermal diffusivity as α = k/ρ s in normal conventions is also ugly. The need for a common definition for all types of energy density is obvious.

We start by asserting that for a volume of sample space V surrounded by a closed surface A, there exist two types of diffusivity α:

i-microscopic diffusivity with n free nodes inside V given by,

α = dU(x,y,z,t)/dt)partial/ U(x,y,z,t) . . . . . (11)

And,

ii-macroscopic diffusivity at the boundary surface surrounding V given by,

α = Wave C. Limit / Unodes

A striking fact is that the macroscopic diffusivity is exactly equal to the microscopic diffusivity.

This equivalence is a form of Laplacian's theorem. Another form of Laplacian's theorem is given by equation 11*.

U(x,y,z,t+dt) = B . U(x,y,z,t) . . . . . . (11*)

Furthermore, the solution of the heat diffusion equation in 4D unit space via B-matrix string technique predicts that the speed of the signal which is the speed of light in vacuum c as [9,10,11,12]

c=T1/2 *log 2 * L^2/ (thermal diffusivity α) . . . . (12)

α (Al) = 1.18 E-5 MKS units

α (Iron) = 2.5 E-5 MKS units

And,

T1/2 for aluminum cube of 10 cm side length = 45 sec.

T1/2 for iron cube of 10 cm side length = 100 sec.

Substituting the above values for α and T1/2 in equation 12 we arrive at a value of c close to Cemw = 2.9 E3 in both cases.

Believe it or not,

We can assume that the Laplacian theoem alone encompasses more than half of current mathematics.

The Laplacian theoem has many equivalent forms and many capabilities [1].

i-It can numerically calculate single, double and triple integration.

finite numerical integration.

ii-deduce Gauss's statistical law.

iii-define and describe the properties of the vacuum such as the energy diffusivity α.

iii- Solve the theory of the Schrödinger wave equation or its square.

iv-explain the formation and explosion of the Big Bang.

v-Solve the time-dependent PDE in its most general form.

..etc..etc.

1-Useless mathematics, ResearchGate, IJISRT journal,

September 2024

Unfortunately, during the 20th century, tons of useless and misleading mathematics came in and stabilized this science.

Laplacian theorem and statistical theory of Cairo techniques.  are an advanced and complete format of the energy continuity equation, and can rectify and create new mathematics.

It is now clear that the term useless mathematics refers to incomplete mathematical spaces and rules of a classical 3D+t variety that have entered and stabilized in mathematical science, even though they are misleading and inadequate for generating definitions and well-defined hypotheses.

The same goes for all kinds of time-dependent partial differential equations. In conclusion, mathematics today constitutes two main groups, useful mathematics and useless mathematics. We predict that the former will sooner or later replace the latter.

The Laplacian theorem itself is a particular product of the Cairo theory of techniques.

We briefly present and explain the so-called Laplacian theorem in the 4D x-t unit space and highlight its importance and how it can generate new mathematics.

The Laplacian partial differential equation (not to be confused with Laplace's PDE ∇^2. Ves=0) which interests us is the one describing the energy density field U(x,y,z,t) in an isolated sample spatial control volume (V) surrounded by a closed surface (A) and subjected to Dirichlet boundary conditions. The Laplacian partial differential equation has the format,

dU/dt)partial = α ∇^2 (U) + S. . . . . (1)

in normal conventions and subject to Dirichlet boundary conditions.

So-called Laplacian systems are those described by PDE 1 and subject to their own particular rules.

This very special case of equation 1 is still treated mathematically in a classical way via 3D + external time which is lazy and misleading.

The effective treatment of the Laplacian PDE with Dirichlet boundary conditions is rather a physico-mathematical treatment using its natural physical properties described by the Laplacian theorem.

Let us recall the considerable success of the Schrödinger partial differential equation (which is in a way the SQRT of the Laplacian PDE) and of quantum theory as a whole where it also relies on an advanced and exhaustive form of the equation of energy continuity which is analogous to that of the energy continuity equation proposed in Laplacian mathematics.

The Laplacian process or the Laplacian system itself can be defined by PDE 1.[if and only if].

The spatio-temporal evolution of the Laplacian process is described by the recurrence relation,

U(x,y,z,t+dt) = B. U(x,y,z,t) . . . . . . (2)

Where B is the well-defined statistical transition matrix.

There exists a transfer matrix D(N) given by,

D(N)=B+B^2+B^3+. . +B^N. . . . (3)

For N sufficiently large, we arrive at the steady-state time-independent solution given by,

D(N)= [1/(I-B)] – i . . . . . (4)

Obviously,

B^0 =I,

and B^N tends to zero as N tends to infinity.

Finally the Laplacian process can also be defined as that having the spatio-temporal solution,

U(x,y,z,t)= D(N) . (b+S) + IC. B^N. . . . . (5)

Where b is the vector of Dirichlet boundary conditions arranged in the correct order.

Equation 5 shows that there is some sort of inherent classical energy entanglement between free nodes inside the control volume V and walls.

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(PDF) Useless Mathematics – The Complex and Untold Story. Available at: https://www.researchgate.net, IJISRT journal, September 2024.