Ph. D. & Dr. Sc. Lev Gelimson's Unit Factor Theory in Fundamental Science of General Problem Transformation

Unit Factor Theory in Fundamental Science of General Problem Transformation

© Ph. D. & Dr. Sc. Lev Gelimson

Academic Institute for Creating Fundamental Sciences (Munich, Germany)

Mathematical Journal

of the "Collegium" All World Academy of Sciences

Munich (Germany)

11 (2011), 57

In classical mathematics [1], known equivalent transformations applied to contradictory problems can lead to results with no objective sense [2-5].

Unit factor theory (UFT) in fundamental science of general problem transformation [5] gives methods of invariantly transforming a general problem to efficiently and adequately solve it with applying overmathematics [2-4] and the system of fundamental sciences on general problems [5].

Show the essence of this theory in the simplest but most typical case of a finite overdetermined quantiset [2-5] of n (n > m; m , n ∈ N⁺ = {1, 2, ...}) linear equations

_q(i)(Σ_i=1^m a_ijx_i + c_j= 0) (j = 1, 2, ... , n) (L_j)

with their own positive number quantities q(i), m unknown variables x_i (i = 1, 2, ... , m), and any given real numbers a_ij and c_jin the Cartesian m-dimensional "space" [1].

Unit factor theory (UFPT) [5] is based on the following algorithm for a quantiset of linear equations:

Step 1) for every preselected unknown variable separately, select all the equations with initial nonzero factors at this preselected unknown variable and transform each of these equations via its division by the initial nonzero factor at this unknown variable in this equation so that the transformed factor becomes 1 (unit);

Step 2) for this preselected unknown variable and for each of the remaining equations (with initial zero factor at this preselected unknown variable in this equation) separately,

determine all the nonzero factor unknown variables with initial nonzero factors in this equation;

Step 3) for this preselected unknown variable, for each of the remaining equations (with initial zero factor at this preselected unknown variable in this equation), and for each of these nonzero factor unknown variables separately,

determine the ratio of a mean (e.g., a power mean) of the moduli (absolute values) of the factors at this nonzero factor unknown variable in all the already transformed equations with initial nonzero factors at this preselected unknown variable to the factor at this nonzero factor unknown variable in this equation;

Step 4) for this preselected unknown variable, for each of the remaining equations (with initial zero factor at this preselected unknown variable in this equation), and for all the nonzero factor unknown variables separately,

determine a mean value (e.g., a geometric mean) of the moduli (absolute values) of the above ratios for each of all these nonzero factor unknown variables separately, give this mean value a sign which is the product of the signs of the above ratios, and multiply this equation with this signed mean value.

Separately consider every preselected unknown variable x_h (h = 1, 2, ... , m) and along with it all the linear equations (L_j) (j = 1, 2, ... , n).

If factor a_hj at this preselected unknown variable x_h in linear equation (L_j) is nonzero, then divide linear equation (L_j) by a_hj :

_q(i)(Σ_i=1^m a_ij/a_hj x_i + c_j/a_hj = 0) (j = 1, 2, ... , n) (L_j).

In particular, by m = 2, replacing x₁ with x , x₂ with y , a_1j with a_j , and a_2j with b_j , we have

_q(i)(a_jx + b_jy + c_j= 0) (j = 1, 2, ... , n) (L_j).

If all

a_j (j = 1, 2, ... , n)

are nonzero, then divide linear equation (L_j) by a_j and obtain set of n linear equations

_q(i)(x + b_j/a_j y + c_j/a_j = 0) (j = 1, 2, ... , n) (L_j)

for preselected unknown variable x .

If all

b_j (j = 1, 2, ... , n)

are nonzero, then divide linear equation (L_j) by b_j and obtain set of n linear equations

_q(i)(a_j/b_j x + y + c_j/b_j = 0) (j = 1, 2, ... , n) (L_j)

for preselected unknown variable y .

Zero factors naturally lead to complications to consider in particular problems.

In particular, consider isosceles triangles in two-dimensional coordinate systems with using the symmetry axis of an isosceles triangle as one of the two-dimensional coordinate system axes, e.g., x-axis. Namely, regard two test triangles:

1) an origin-top isosceles triangle TOP with vertices (and their coordinates x , y) T(a , -b), O(0, 0), P(a , b) where a , b are any positive numbers. Then isosceles triangle TOP sides (edges) equations are

OT: bx + ay = 0,

OP: bx - ay = 0,

TP: x - a = 0;

2) an origin-base isosceles triangle BAS with vertices (and their coordinates x , y) B(0, -b), A(a , 0), S(0 , b) where a , b are any positive numbers. Then isosceles triangle BAS sides (edges) equations are

AB: bx + ay - ab = 0,

AS: bx - ay - ab = 0,

BS: x = 0.

Forst apply this algorithm to

OT: bx + ay = 0,

OP: bx - ay = 0,

TP: x - a = 0.

For preselected unknown variable x separately,

in all the equations, the initial factors at this preselected unknown variable x are nonzero. Hence simply omit steps I.2, I.3, and I.4.

Step 1) Simply select all the equations (with initial nonzero factors at this preselected unknown variable x).

Transform each of these equations via its division by the initial nonzero factor at this unknown variable in this equation so that the transformed factor becomes 1 (unit).

Compose the initial separate set of linear equations for preselected unknown variable x :

x + a/b y = 0,

x - a/b y = 0,

x - a = 0.

Hence tansforming the initial separate set of linear equations for this preselected unknown variable x is completed.

For preselected unknown variable y separately,

in the equations

bx + ay = 0,

bx - ay = 0,

the initial factors at this preselected unknown variable y are nonzero. Hence simply omit steps 2, 3, and 4.

Step 1) Simply select all the equations (with initial nonzero factors at this preselected unknown variable y).

Transform each of these equations via its division by the initial nonzero factor at this unknown variable in this equation so that the transformed factor becomes 1 (unit).

Compose the nonzero factor subset

b/a x + y = 0,

- b/a x + y = 0

of the initial separate set of linear equations for preselected unknown variable y .

The only remaining equation (with initial zero factor at this preselected unknown variable y in this equation) is

x - a = 0.

Step 2) for this preselected unknown variable y and for each of the remaining equations (with initial zero factor at this preselected unknown variable y in this equation) separately,

determine all the nonzero factor unknown variableswith initial nonzero factors in this equation:

Here we determine nonzero factor unknown variablex only.

Step 3) for this preselected unknown variable y , for each of the remaining equations (with initial zero factor at this preselected unknown variable y in this equation)

x - a = 0,

and for each of these nonzero factor unknown variables x separately,

determine the ratio b/a of a mean (e.g., a power mean) b/a of the moduli (absolute values) of the factors b/a and - b/a at this nonzero factor unknown variable x in all the already transformed equations with initial nonzero factors at this preselected unknown variable y

b/a x + y = 0,

- b/a x + y = 0

to the factor 1 at this nonzero factor unknown variable x in this equation

x - a = 0.

Here the moduli (absolute values) of the factors b/a and - b/a are both b/a (coinciding).

Hence any reasonable mean of them has to equal b/a , too.

Step 4) for this preselected unknown variable y , for each of the remaining equations (with initial zero factor at this preselected unknown variable in this equation)

x - a = 0,

and for all the nonzero factor unknown variables separately,

determine a mean value b/a (e.g., a geometric mean) of the moduli (absolute values) of the above ratios b/a for each of all these nonzero factor unknown variables x separately, give this mean value b/a the sign + 1 which is the product of the signs + 1 of the above ratios b/a , and multiply this equation

x - a = 0

with this signed mean value b/a :

b/a x - b = 0.

This only equation forms the zero factor subset of the initial separate set

b/a x + y = 0,

- b/a x + y = 0,

b/a x - b = 0

of linear equations for preselected unknown variable y .

Hence transforming the initial separate set of linear equations for this preselected unknown variable y is completed.

Now apply a modified algorithm of unit factor theory (UFT) [5] to

AB: bx + ay - ab = 0,

AS: bx - ay - ab = 0,

BS: x = 0.

For preselected unknown variable x separately,

in all the equations, the initial factors at this preselected unknown variable x are nonzero. Hence simply omit steps 2, 3, and 4.

Step 1) Simply select all the equations (with initial nonzero factors at this preselected unknown variable x).

Transform each of these equations via its division by the initial nonzero factor at this unknown variable in this equation so that the transformed factor becomes 1 (unit).

Compose the initial separate set of linear equations for preselected unknown variable x :

x + a/b y - a = 0,

x - a/b y - a = 0,

x = 0.

Hence transforming the initial separate set of linear equations for this preselected unknown variable x is completed.

For preselected unknown variable y separately,

in the equations

bx + ay - ab = 0,

bx - ay - ab = 0,

the initial factors at this preselected unknown variable y are nonzero. Hence simply omit steps 2, 3, and 4.

Step 1) Simply select all the equations (with initial nonzero factors at this preselected unknown variable y).

Transform each of these equations via its division by the initial nonzero factor at this unknown variable in this equation so that the transformed factor becomes 1 (unit).

Compose the nonzero factor subset

b/a x + y - b = 0,

- b/a x + y + b = 0

of the initial separate set of linear equations for preselected unknown variable y .

The only remaining equation (with initial zero factor at this preselected unknown variable y in this equation) is

x = 0.

Now consider two modifications of unit factor quadrat theory (UFQT) [5] which have additional subalgorithms regarding the remaining equations (with initial zero factors at variables) and show their essences in our case of an origin-base isosceles triangle BAS with vertices (and their coordinates x , y)

B(0, -b), A(a , 0), S(0 , b)

where a , b are any positive numbers. Then isosceles triangle BAS sides (edges) equations are

AB: bx + ay - ab = 0,

AS: bx - ay - ab = 0,

BS: x = 0.

The only remaining equation (with initial zero factor at preselected unknown variable y in this equation) is

x = 0.

Its above transformation for preselected unknown variable y gave

b/a x = 0.

Our goal is to artificially introduce preselected unknown variable y into this equation in which this preselected unknown variable y is absent.

Due to overmathematics and other fundamental mathematical sciences systems [2-5], consider this equation as a quantielement

₁(b/a x = 0)

with naturally implicit quantity 1 and element

b/a x = 0.

Represent this quantielement as a quantisum (symbol ° shows quantioperations and their quantiresults)

₁(b/a x = 0) =° _1/2(b/a x = 0) +° _1/2(b/a x = 0)

of two equal half-quantielements

_1/2(b/a x = 0),

_1/2(b/a x = 0).

Now artificially introduce a positive number c and symmetrically replace these two equal half-quantielements with the following two quantielements:

_1/2(b/a x - b/c y = 0),

_1/2(b/a x + b/c y = 0).

As the result, replace quantielement

₁(b/a x = 0)

with the quantisum

_1/2(b/a x + b/c y = 0) +° _1/2(b/a x - b/c y = 0).

Now consider two natural approaches:

1) take these two half-equations "as is" with keeping "true" factors b/a (at x) obtained before (and independently of) artificially introducing c ;

2) provide namely unit factors at preselected unknown variable y in these two half-equations like the both already obtained (basic, nonzero-factor) equations

b/a x + y - b = 0,

- b/a x + y + b = 0

via dividing these two half-equations by nonzero numbers - b/c and b/c , respectively.

First apply approach

1) take these two half-equations "as is" with keeping "true" factors b/a (at x) obtained before (and independently of) artificially introducing c .

We have the initial separate quantiset

b/a x + y - b = 0,

- b/a x + y + b = 0,

_1/2(b/a x + b/c y = 0),

_1/2(b/a x - b/c y = 0)

of linear equations for preselected unknown variable y .

Hence transforming the initial separate set of linear equations for this preselected unknown variable y is completed.

Now apply approach

2) provide namely unit factors at preselected unknown variable y in these two half-equations like the both already obtained (basic, nonzero-factor) equations

b/a x + y - b = 0,

- b/a x + y + b = 0

via dividing these two half-equations by nonzero numbers - b/c and b/c , respectively.

We have the initial separate quantiset

b/a x + y - b = 0,

- b/a x + y + b = 0,

_1/2(c/a x + y = 0),

_1/2(- c/a x + y = 0)

of linear equations for preselected unknown variable y .

Hence transforming the initial separate set of linear equations for this preselected unknown variable y is completed.

Unit factor theory (UFT) in fundamental science of general problem transformation [5] is very efficient by solving many urgent (including contradictory) problems.

Acknowledgements to Anatolij Gelimson for our constructive discussions on coordinate system transformation invariances and his very useful remarks.

References

[1] Encyclopaedia of Mathematics. Ed. M. Hazewinkel. Volumes 1 to 10. Kluwer Academic Publ., Dordrecht, 1988-1994

[2] Lev Gelimson. Elastic Mathematics. General Strength Theory. The ”Collegium” International Academy of Sciences Publishers, Munich, 2004

[3] Lev Gelimson. Providing Helicopter Fatigue Strength: Flight Conditions. In: Structural Integrity of Advanced Aircraft and Life Extension for Current Fleets – Lessons Learned in 50 Years After the Comet Accidents, Proceedings of the 23rd ICAF Symposium, Dalle Donne, C. (Ed.), 2005, Hamburg, Vol. II, 405-416

[4] Lev Gelimson. Overmathematics: Fundamental Principles, Theories, Methods, and Laws of Science. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010

[5] Lev Gelimson. General Problem Fundamental Sciences System. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2011