Power Theory in Fundamental Science of General Problem Estimation

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), 36

In classical mathematics [1], there is no sufficiently general concept of a quantitative mathematical problem. There is the concept of a finite or countable set of equations only with completely ignoring their quantities like any Cantor set [1]. They are very important by contradictory (e.g. overdetermined) problems without precise solutions. Besides that, without equations quantities, by subjoining an equation coinciding with one of the already given equations of such a set, this subjoined equation is simply ignored whereas any (even infinitely small) changing this subjoined equation alone at once makes this subjoining essential and changes the given set of equations. Therefore, the concept of a finite or countable set of equations is ill-defined [1]. Uncountable sets of equations (also with completely ignoring their quantities) are not considered in classical mathematics [1] at all.

The least square method (LSM) [1] by Legendre and Gauss is the only known method applicable to finite overdetermined sets of equations. This method minimizes the sum of the squares of the differences of the both parts of all the equations, or, equivalently, the sum of the squares of the absolute errors [1] of all the equations. Overmathematics [2-4] and the system of fundamental sciences on general problems [5] have discovered many principal shortcomings [2-5] of this method with no objective sense of its result.

Nota bene: The second power is the only providing simplest possible namely pure analytic formulae and and is the most suitable to deal with. But overmathematics [2-4] and the system of fundamental sciences on general problems [5] show such a suitability usually provides no adequacy.

Power theory in fundamental science of general problem estimation corrects and generalizes the least square method (LSM) [1] in the following directions:

applicability to any general problem which can be represented as any (e.g., infinite and even uncountable due to uninumbers and uniquantities [2-5]) quantiset [2-5] of general subproblems with their own quantities rather than to finite overdetermined sets of equations only;

using any nonnegative (e.g., individual) estimators of subproblems rather than the differences of the both parts of all the equations, or, equivalently, the absolute errors [1] of all the equations only;

using any power (exponent) t > 1 rather than the second power only.

Suppose that a general problem P consists of separate general subproblems (e.g., relations) P_β with their own positive quantities q(β)

P = {_{β∈Β q(β)}P_β}

(where index β belongs to index set Β)

and there are nonnegative estimators E_β [2-5]

E_β(P_β) ≥ 0 (β∈Β)

(e.g., absolute errors [1], relative errors [1], distances (which are invariant by coordinate system rotations), unierrors [2-5], etc.) possibly individual for all these general subproblems. In particular, there can be a nonnegative estimator E [2-5]

E(P_β) ≥ 0 (β∈Β)

common for all these general subproblems (naturally, with its possibly individual values as the general subproblems estimations), which is typical.

Our present task is to explicitly give some suitable nonnegative subproblems estimations unification functions F of all

E_β(P_β) ≥ 0 (β∈Β)

with the same own quantities q(β). Each of such functions has to provide applying nonnegative estimator E to the whole general problem P with building its nonnegative total estimation

E(P) = F[_{β∈Β q(β)}E_β(P_β)] ≥ 0.

Power theory in fundamental science of general problem estimation gives the following suitable nonnegative subproblems estimations unification function:

The weighted power mean of the (componentwise) subproblems estimations

^tE(P) = {Σ_β∈Β q(β)[E_β(P_β)]^t / Σ_β∈Β q(β)}^1/t

where t is a positive number and

Q(P) = Σ_β∈Β q(β)

is the uniquantity [2-5] of quantisystem

P = {_{β∈Β q(β)}P_β}.

Naturally, it is admissible to directly minimize this weighted power mean of the (componentwise) subproblems estimations. But it is simpler and much more suitable to directly minimize the weighted sum of powers

^tS(P) = Σ_β∈Β q(β)[E_β(P_β)]^t .

Example. Consider a general pure equations problem as a quantiset [2-5] (former hyperset) of equations over indexed functions (dependent variables) f_φ of indexed independent variables z_ω , all of them belonging to their possibly individual vector spaces. We may gather (in the left-hand sides of the equations) all the functions available in the initial forms without any further transformations. The unique natural exception is changing the signs of the functions by moving them to the other sides of the same equations. The quantiset can be brought to the form

_q(λ)(L_λ[_φ∈Φ f_φ[_ω∈Ω z_ω]] = 0) (λ∈Λ)

where

L_λ is an operator with index λ from an index set Λ ;

f_φ is a function (dependent variable) with index φ from an index set Φ ;

z_ω is an independent variable with index ω from an index set Ω ;

[_ω∈Ω z_ω]

is a set of indexed elements z_ω ;

q(λ) is the own, or individual, positive quantity (former hyperquantity) [2-5] as a weight of the equation with index λ .

When replacing all the unknowns (unknown functions) with their possible ”values” (some known functions), the above quantiset of equations is transformed into the corresponding quantiset of formal functional equalities. To conserve the quantiset form, let us use for these known functions the same designations f_φ .

If we simply take the absolute error [1] as a common estimator of all the given equations, then power theory minimizes the weighted sum of the tth powers of the moduli (norms or simply absolute values) of the differences of the both parts of every equation with its own quantity as a weight:

^tS{_q(λ)(L_λ[_φ∈Φ f_φ[_ω∈Ω z_ω]] = 0) (λ∈Λ)} = Σ_λ∈Λ q(λ)||L_λ[_φ∈Φ f_φ[_ω∈Ω z_ω]]||^t.

Power theory in fundamental science of general problem estimation 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