Unknown Balance Theory in Fundamental Science of General Problem Transformation

by

© 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), 61

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

General problem type and setting theory (GPTST) in fundamental science on general problem [5] essence defines a general quantitative mathematical problem, or simply a general problem, to be a quantisystem [2-5] (former hypersystem) P which includes unknown quantisubsystems and possibly includes its general subproblems.

General problem transformation theory (GPTT) 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].

Physical unit removal theory (PURT) in fundamental science of general problem transformation [5] gives methods of removing physical units to transform a general quantitative mathematical problem, or simply a general problem, into a pure number general quantitative mathematical problem, or simply a pure number general problem (including contradictory problems).

But even in a pure number general problem, distinct unknown variables can have very different relevant value ranges, which makes it difficult, both to adequately and to suitably graphically interpret pseudosolutions (in particular, (precise) solutions, quasisolutions, supersolutions, and antisolutions) to this problem.

Unknown balance theory (UBT) in fundamental science of general problem transformation [5] gives methods of balancing distinct unknown variables with initially possibly very different relevant value ranges to adequately and to suitably graphically interpret pseudosolutions (in particular, (precise) solutions, quasisolutions, supersolutions, and antisolutions) to a general quantitative mathematical problem, or simply a general problem (including contradictory problems).

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) (j = 1, 2, ... , n) (L'_j)

with their own positive number quantities q(i), m pure number unknown variables x'_i (i = 1, 2, ... , m), and any given real numbers a'_ij and c'_j in the Cartesian m-dimensional "space" [1].

For a quantiset of linear equations, unknown balance theory (UBT) is based on the following algorithm:

For every preselected unknown variable separately, select the subset (with unit own quantities) of 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).

Notata bene:

1. This subset is nonempty (otherwise, a preselected unknown variable would be absent in a given quantiset of linear equations and could be ignored at all).

2. For this preselected unknown variable, at this step, keep each of the remaining equations (with initial zero factor at this preselected unknown variable in this equation) without any transformation.

Hence 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) (j = 1, 2, ... , n) (L'_j).

In particular, by m = 2, replacing x'₁ with x' , x'₂ with y' , a'_1j with a'_j , and a'_2j with b'_j , we have

_q(i)(a'_jx + b'_jy = c'_j) (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) (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) (j = 1, 2, ... , n) (L'_j)

for preselected unknown variable y .

Unknown balance theory (UBT) allows avoiding complications with zero factors due to the following natural approach.

If all

c'_j = 0 (j = 1, 2, ... , n),

then the quantiset of linear equations (L'_j) would have a trivial precise zero solution

x'_i = 0 (i = 1, 2, ... , m).

Otherwise, i.e. if there is a nonzero

c'_j = 0 (j = 1, 2, ... , n),

then take any power exponent t > 0. Determine the weighted (with the own quantities of separate equations) power mean value

[Σ_j=1ⁿ_a'(hj)≠0 q(i)|c'_j/a'_hj|^t / Σ_j=1ⁿ_a'(hj)≠0 q(i)]^1/t (h = 1, 2, ... , m)

of the moduli (absolute values) of the ratios c'_j /a'_hj in all the equations with initial nonzero factors at this preselected unknown variable

x'_h (h = 1, 2, ... , m).

Nota bene: Attaching a condition to an operation designation means that only operands satisfying this condition are selected and considered whereas all the remaining operands (i.e., not satisfying this condition) are ignored in such a conditional operation. In our case, this holds for addition and a sum as its result with condition

a'(hj) ≠ 0,

or, equivalently and using the above designations,

a'_hj ≠ 0.

Now divide

x'_h (h = 1, 2, ... , m)

by this weighted (with the own quantities of separate equations) power mean value of the moduli (absolute values) of the ratios c'_j /a'_hj in all the equations with initial nonzero factors at this preselected unknown variable and introduce new preselected unknown variable

x_h = x'_h / [Σ_j=1ⁿ_a'(hj)≠0 q(j)|c'_j/a'_hj|^t / Σ_j=1ⁿ_a'(hj)≠0 q(j)]^1/t (h = 1, 2, ... , m).

We may also replace here h with i :

x_i = x'_i / [Σ_j=1ⁿ_a'(ij)≠0 q(j)|c'_j/a'_ij|^t / Σ_j=1ⁿ_a'(ij)≠0 q(j)]^1/t (i = 1, 2, ... , m).

Then we have

x'_i = [Σ_j=1ⁿ_a'(ij)≠0 q(j)|c'_j/a'_ij|^t / Σ_j=1ⁿ_a'(ij)≠0 q(j)]^1/tx_i (i = 1, 2, ... , m).

Substitute these m formulae into the initial quantiset of n linear equations

_q(i)(Σ_i=1^m a'_ijx'_i = c'_j) (j = 1, 2, ... , n) (L'_j)

and obtain

_q(i){Σ_i=1^m a'_ij[Σ_j=1ⁿ_a'(ij)≠0 q(j)|c'_j/a'_ij|^t / Σ_j=1ⁿ_a'(ij)≠0 q(j)]^1/tx_i = c'_j} (j = 1, 2, ... , n) (L'_j)

Now naturally introduce new designations

a''_ij = a'_ij[Σ_j=1ⁿ_a'(ij)≠0 q(j)|c'_j/a'_ij|^t / Σ_j=1ⁿ_a'(ij)≠0 q(j)]^1/tx

and obtain

_q(i){Σ_i=1^m a''_ijx_i = c'_j} (j = 1, 2, ... , n) (L'_j).

Finally, apply distance transformation theory (DTT) in fundamental science of general problem transformation [5] to the last quantiset of n linear equations

Unknown balance theory (UBT) in fundamental science of general problem transformation 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