General Problem Pseudosolution Theory in Fundamental Science of General Problem Pseudosolution

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), 31

Introduction

In classical mathematics [1], there is no sufficiently general concept of a pseudosolution to a quantitative mathematical problem. Even a Cantor set [1] of solutions to a problem is ill-defined [1] because infinitely small changing a problem with multiple solutions to it can finitely change the number of solutions and hence the Cantor set of solutions to this problem. The known concept of a quasisolution [1] to a problem is not general enough, too. A general quantitative mathematical problem is not considered in classical mathematics [1] at all.

General Problem

General problem type and setting theory (GPTST) in fundamental science on general problem essence [5] 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.

In particular, a general problem can be a quantiset

_q(λ)R_λ{_{φ∈Φ r(φ)}f_φ[_{ω∈Ω s(ω)}z_ω]} (λ∈Λ)

of indexed known quantirelations _q(λ)R_λ (with their own, or individual, quantities q(λ)) [2-5] over indexed unknown quantifunctions (dependent variables), or simply unknowns (with their own, or individual, quantities r(φ)), _r(φ)f_φ , of indexed independent known quantivariables _s(ω)z_ω (with their own, or individual, quantities s(ω), all of them belonging to their possibly individual vector spaces, in their initial forms without any further transformations

where

R_λ is a known relation with index λ from an index set Λ ;

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

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

[_{ω∈Ω s(ω)}z_ω]

is a quantiset of indexed quantielements _s(ω)z_ω .

Nota bene: Own, or individual, quantities play the roles of the weights of quantiset elements. In particular, q(λ) is the own, or individual, quantity (former hyperquantity) [2-5] as a weight of relation R_λwith index λ in a quantiset

_q(λ)R_λ{_{φ∈Φ r(φ)}f_φ[_{ω∈Ω s(ω)}z_ω]} (λ∈Λ).

In quantitative mathematical problems, namely equations and inequations are the most typical relations.

General Problem Pseudosolution

General problem pseudosolution theory (GPPST) in fundamental science of general problem pseudosolution defines both a pseudosolution to a general problem and arts (particular cases) of a pseudosolution which are conditional pseudosolutions.

Replace all the unknown quantisubsystems, or simply unknown variables, or unknowns (in the above particular case, unknown quantifunctions), with their possible ”values” which are known quantisubsystems (in the above particular case, some known quantifunctions). Then a general problem which is a quantisystem [2-5] (former hypersystem) P including unknown quantisubsystems becomes the corresponding known quantisystem without any unknowns. To conserve the quantisystem form, let us use the same designations for these known quantisystem and quantisubsystems, too. This known quantisystem can be further estimated qualitatively and (or) quantitatively discretely (e.g., either true or false) or continuously (e.g., via absolute and relative errors [1], unierrors, reserves, reliabilities, and risks [2-5]). If a quantisystem of pseudosolutions to a general problem transforms this problem into a known true quantisystem, then this pseudosolutions quantisystem (or simply a pseudosolution by obviously using the system meta-level) is a solutions quantisystem (or simply a solution by obviously using the system meta-level) to this general problem.

Further we need some useful definitions and agreements [2-5].

A pseudosolution to a general problem is an arbitrary quantisystem of such values of all the corresponding variables that, after replacing each variable with its value, the general problem becomes a determinable (e.g., true or false) known quantisystem. In the above particular case, this is a known quantiset of relations containing known elements only, and each of its relations becomes determinable (e.g., true or false).

A (precise) solution to a general problem is an arbitrary quantisystem of such values of all the corresponding variables that, after replacing each variable with its value, the general problem becomes a true known quantisystem. In the above particular case, this is a known quantiset of relations containing known elements only, and each of its relations becomes true.

A quasisolution to a general problem by a specific realization of a certain method or theory is a pseudosolution (to this general problem) which has the least unierror and/or the greatest reserve (by this realization of this method or theory) among all the pseudosolutions to this general problem.

Nota bene: A quasisolution is not necessarily a solution, which is especially important in contradictory general problems that have no solutions in principle but can possess quasisolutions.

A supersolution to a general problem by a specific realization of a certain method or theory is a solution (to this general problem) which has the greatest reserve (by this realization of this method or theory) among all the solutions to this general problem.

Nota bene: A supersolution a general problem not necessarily coincides with its quasisolution because the set of the solutions is a subset of the set of the pseudosolutions. If the both exist, then the quasisolution (which is not necessarily a solution) has a not less reserve in comparison with the supersolution. If in the last comparison, namely the strict inequality holds, then the quasisolution is certainly no solution.

An antisolution to a general problem by a specific realization of a certain method or theory is a pseudosolution (to this general problem) which has the greatest unierror and/or the least reserve (by this realization of this method or theory) among all the pseudosolutions to this general problem.

Notata bene:

1. Quasisolutions and supersolutions, as well as antisolutions, not necessarily exist because a set of unierrors or reserves not necessarily contains its greatest lower bound and its least upper one, respectively.

2. The concepts of conditional pseudosolutions (in particular, quasisolutions, supersolutions, and antisolutions) are relative depending not only on the corresponding condition, criterion, method, or theory, but also on the precise setting of a general problem. For example, a quasisolution to a contradictory general problem is namely a quasisolution if the precise setting of a general problem is precisely satisfying all the contradictory conditions of a given general problem. But the same quasisolution becomes a precise solution to the same contradictory general problem by its other setting when general problem contradictoriness measure minimization (instead of precisely satisfying all the contradictory conditions of a given general problem) is required (desired). All the more, an antisolution to a contradictory general problem is namely an antisolution if the precise setting of a general problem is precisely satisfying all the contradictory conditions of a given general problem. But the same antisolution becomes a precise solution to the same contradictory general problem by its other setting when general problem contradictoriness measure maximization (instead of precisely satisfying all the contradictory conditions of a given general problem) is required (desired).

General problem pseudosolution theory (GPPST) in fundamental science of general problem pseudosolution 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