ADONGO 'S LEAST WHOLE NORMAL PHYSICS

ADONGO’S LEAST WHOLE NORMAL FORAMLISM

HISTORICAL BACKGROUND

The normal curve was devised around 1720 by the mathematician Abraham De Moivre in other to solve problems connected with games of chance. Around 1870 the Belgian mathematician Adolph Quetelled had the idea of using the curve as an ideal histogram, to which histogram for data could be compared.

Around 2009, I had created a mathematical function from the normal approximation equation which could be used in modeling problems in medical science, insurance, economics, material science and biological science.

I also realized that more development is needed to solve the scientific situations that I was confronting in my time. In view of this I was able to restructure the formalism of the existing concepts of physics to a new branch of formalism called Least Whole Normal Formalism which we are to discuss next.

ADONGO’S LWN FORMALISM

Definition:

·         Radical quantity: the radical quantity is any quantity that has main influence in the quantitative product. Let us consider the quantitative equation for velocity which is given as

V = S * (1/t): the radical quantity is the time t.

·         Subjective quantity: the subjective quantity is any quantity that is determined or has main relationship with the radical quantity. Let us also consider the quantitative equation for velocity which is given as V = S * (1/t):  the subjective quantity is the distance S.

·         Non-subjective quantity: the non-subjective quantity is a chosen quantity in the quantitative product which cannot be related by the radical quantity. The quantitative equation for force which is given as 

      F = m*v*t ^–1:  the non-subjective quantity could be v if m is chosen as subjective quantity or it could be m if v is chosen as subjective quantity.

TWO QUANTITATIVE PRODUCT

Let us consider Q ( i , j ) = (I _j * τ )*µ_i  where I _j  is denoted as a unit non-subjective quantity and

τ the radical and µ_i  is the subjective quantity. The non-subjective unit quantity  I _j  is always equal to one (i.e. I _j = 1)

The quantitative equation Q ( i , j ) = (I _j * τ )*µ_i    has a distribution  

 T_j  ̴  N( τ , 0)    

 T_j  ̴  n (µ_i , S_i² )    }    .................. ( 1)  for direct relation and

 T_j  ̴  N( τ ^-1 , 0)    

 T_j  ̴  n (µ_i , S_i² )    }    .................. ( 2)  for inverse relation.

Where j is one ( j = 1) and i  represents any quantity.

If the normal random quantity T _j   is total non-subjective unit quantity and T _i  is total subjective quantity, then the least whole normal function for the above distribution is

ℓ (τ) = α * τ - √ (τ) – β  (i.e. when directly related)

 ℓ (τ ^-1) = α * τ ^-1 - √ (τ ^-1) – β  (i.e. when inversely related)

Where the constant α  and β are denoted as quantile coefficient of  τ  or  τ ^-1 and quantile constant respectively are calculated as;

α = µ_i  / {ϕ ^-1(γ%)  (√S_i²)}

β = T_i  /  {ϕ ^-1(γ%)  (√S_i²)}

THE LEAST WHOLE NORMAL QUANTITY

The quantity which represents least whole radical quantity is always approximately equal to zero. The least 

 whole radical quantity is calculated as;

√ ( τ₀ ) = [1+√ (1 + 4αβ)] / 2α  for direct relation and

√ ( τ₀^-1 ) = [1+√ (1 + 4αβ)] / 2α  for inverse relation

THE TOTAL SUBJECTIVE QUANTITY

The quantity T_i   which represents total subjective quantity is calculated as;

T_i = µ_i * τ₀ – ϕ ^-1 (γ %) √ (S_i² * τ₀ )  for direct relation and

T_i = µ_i * τ₀^-1 – ϕ ^-1 (γ %) √ (S_i² * τ₀ ) for inverse relation.

MEAN SUBJECTIVE QUANTITY

The quantity µ_i which represents the mean subjective quantity is calculated as;

µ_i² = [ (1 / τ₀²)] [T_i² + ϕ ^-1( γ%) S²  * τ₀]  for direct relation and

µ_i² = [ (1 / τ₀²)] [T_i² + ϕ ^-1( γ%) S²  * τ₀^-1 ]  for inverse relation.

APPLYING THE FORMALISM

DOING WORK:

Doing works are way of transferring energies using forces. The amount of energies transferred. The amount of works done equal to sizes of the forces times the distance move.

Expected work= mean force *mean distance

i.e E (w) = µ_f * τ

Applying the theory of two quantitative products, we have the distribution

T₁ ~ N ( τ,0)

T_f  ~ n (µ_f , S_f ²)    }    .................. ( 1)

Where µ_f  denoted as mean force and S_f ²   is the variance of the forces.

If the normal random quantity T₁  which is total number of units quantity I which mean is always equal to one and T_f  which is denoted as total number of force, then it least whole normal function is calculated as;

ℓ (τ₀) = α * τ₀  – ( √τ₀ ) – β

 (i.e when directly related)  where α and β  are calculated as;

α = µ_f  ^-1 / {ϕ^-1  (γ%)  (√S_i²)}

β = T_f  /  {ϕ ^-1(γ%)  (√S_i²)}

THE LEAST WHOLE DISTANCE (RADICAL QUANTITY)

The quantity T_f which represents total subjective distance is always approximately equal to zero. The least whole distance is calculated as;

√τ₀ = [1 + √ (1 + 4αβ) ] / 2α .........................( 0 )

TOTAL FORCES (SUBJECTIVE QUANTITY)

The quantity T_f  which represents total subjective quantity or total forces is calculated as;

T_f = µ_f  * τ – ϕ ^-1 [γ%] [√(S_i²*τ) ].........................(□)

MEAN FORCE (MEAN SUBJECTIVE QUANTITY)

The quantity µ_f  which represent mean force is calculated as;

µ_f ²  = [ (1 / τ₀²)] [T_f ² + ϕ ^-1( γ%) S ²  * τ₀^-1 ] .............(*)

EXAMPLE

Horses pull cars with constant horizontal forces of mean 136N and standard deviation 27N per distance. Calculate the

         i.            Total forces at distance 19500m

       ii.            Average forces at distance 19500m

(Take γ = 95% )

SOLUTION

i)                    Applying equation (□), we have

T _f = µ _f * τ₀ − ϕ ^-1 [γ%][√(S _f ² * τ₀) ]

T _f  = 136 * 19500 − ϕ ^-1 [95%]  [√(729 ×19500)]

T_f = 2,652,000 – 1.645 × √(14,215,500)

T_f  = 2,645,797.783N

Hence, the total force is 2,645,797.783N

THREE QUANTITATIVE PRODUCT

Let us consider Q (i , j) = (Ä_j *τ ) * µ_i   where Ä_j  is denoted as non-subjective quantity µ_i  is denoted as subjective quantity and τ is the radical quantity.

If the non-subjective quantity Ä_j    is not equal to one, but represents any quantitative value. Then we have the distribution of the quantitative equation Q (i, j)  which is given as;

T_j ~ N (Ä_j* τ , σ ² * τ )

T_j ~ n ( µ_i , S_i² )                  }    .................. ( 1)   for direct relation and

T_j ~ N (Ä_j* τ ^-1  , σ ² * τ ^-1 )

T_j ~ n ( µ_i , S_i² )                       }    .................. ( 2)  for inverse relation

If the normal random quantity T_j  is total non-subjective quantities and T_i  is total subjective quantity, then the least whole normal function is given as

ℓ (τ) = α * τ  – ( √τ ) – β  (i.e when directly related)

ℓ (τ ^-1) = α * τ ^-1  – ( √τ ^-1 ) – β  (i.e when inversely related)