General Problem Transformation 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), 33
Introduction. General Problem
In classical mathematics [1], known equivalent transformations applied to contradictory problems can lead to results with no objective sense [2-5].
General problem transformation theory (GPTT) in fundamental science of general problem transformation 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].
General problem type and setting theory (GPTST) in fundamental science on general problem 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.
In particular, a general problem can be a quantiset
w(λ)Rλ[φ∈Φ fφ[ω∈Ω zω]] (λ∈Λ)
of known relations Rλ over indexed unknown functions (dependent variables), or simply unknowns, fφ of indexed independent known variables zω , all of them belonging to their possibly individual vector spaces, in their initial forms without any further transformations
where
Rλ is a known relation with index λ from an index set Λ ;
fφ is an unknown function (dependent variable) with index φ from an index set Φ ;
zω is a known independent variable with index ω from an index set Ω ;
[ω∈Ω zω]
is a set of indexed elements zω ;
w(λ) is the own, or individual, quantity (former hyperquantity) [2-5] as a weight of the relation with index λ .
Replace all the unknown quantisubsystems, or simply unknown variables, or unknowns (in the above particular case, unknown functions), with their possible ”values” which are known quantisubsystems (in the above particular case, some known functions). 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 P , fφ , etc. 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]).
In quantitative mathematical problems, namely equations and inequations are the most typical relations.
General Pure Equations Problem
Let us define a general quantitative mathematical pure equations problem, or simply a general pure equations problem, to be a quantiset [2-5] (former hyperset) of equations over indexed functions (dependent variables) fφ of indexed independent variables zω , 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
w(λ)(Lλ[φ∈Φ fφ[ω∈Ω zω]] = 0) (λ∈Λ)
where
Lλ is an operator with index λ from an index set Λ ;
fφ is a function (dependent variable) with index φ from an index set Φ ;
zω is an independent variable with index ω from an index set Ω ;
[ω∈Ω zω]
is a set of indexed elements zω ;
w(λ) is the own, or individual, 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 fφ .
Let us use the concept of a nonzero proportional transformation of a quantiset or set of equations with multiplying each equation by a nonzero number individual for this equation.
Classical mathematics [1] considers a nonzero proportional transformation as an equivalent transformation of a set of equations. However, this holds for exact solutions only. Otherwise, namely by contradictory (e.g. overdetermined) problems without precise solutions, this also holds for any pseudosolutions but only by nonzero proportional transformation invariant theories and methods of solving problems and estimating their pseudosolutions [2-5].
Nota bene: The least square method (LSM) [1] by Legendre and Gauss is the only method well-known in classical mathematics [1] and applicable to contradictory (e.g. overdetermined) problems. Overmathematics [2-4] and the system of fundamental sciences on general problems [5] have discovered many principal shortcomings of this method (and all theories and methods based on this method) which is nonzero proportional transformation noninvariant and hence gives results without any objective sense.
General Pure Inequations Problem
Let us define a general quantitative mathematical pure inequations problem, or simply a general pure inequations problem, to be a quantiset [2-5] (former hyperset) of equations over indexed functions (dependent variables) fφ of indexed independent variables zω , all of them belonging to their possibly individual vector spaces. We may gather (in the left-hand sides of the inequations) 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 inequations. The quantiset can be brought to the form
w(λ)(Lλ[φ∈Φ fφ[ω∈Ω zω]] Rλ 0) (λ∈Λ)
where
Lλ is an operator with index λ from an index set Λ ;
Rλ is an inequality relation (e.g., ≈ , ∼ , ≠ , < , > , ≤ , ≥) with index λ from an index set Λ ;
fφ is a function (dependent variable) with index φ from an index set Φ ;
zω is an independent variable with index ω from an index set Ω ;
[ω∈Ω zω]
is a set of indexed elements zω ;
w(λ) is the own, or individual, quantity (former hyperquantity) [2-5] as a weight of the inequation with index λ .
When replacing all the unknowns (unknown functions) with their possible ”values” (some known functions), the above quantiset of inequations is transformed into the corresponding quantiset of formal functional inequalities. To conserve the quantiset form, let us use for these known functions the same designations fφ .
By using unstrict inequality relations such as ≈ , ∼ , ≤ , ≥ , etc. only, a general pure inequations problem clearly further generalizes a general pure equations problem.
Let us use the concept of a nonzero proportional transformation of a quantiset or set of inequations with multiplying each inequation by a nonzero number individual for this equation and, by a negative factor, replacing inequality signs: < with > ; > with < ; ≤ with ≥ ; ≥ with ≤ .
Let us use the concept of a positive proportional transformation of a quantiset or set of inequations with multiplying each inequation by a positive number individual for this equation.
Let us use the concept of a negative proportional transformation of a quantiset or set of inequations with multiplying each inequation by a negative number individual for this equation and replacing inequality signs: < with > ; > with < ; ≤ with ≥ ; ≥ with ≤ .
General Pure Number Problem
A general problem as a quantisystem [2-5] (former hypersystem) P includes both unknown and known quantisubsystems. The expressions both of this quantisystem and of its quantisubsystems can also include some physical units.
A general pure number problem is such a general problem that the expressions both of this quantisystem and of its quantisubsystems explicitly include no physical units.
A trivial algorithm of general problem physical unit removal transformation is as follows:
1) in a general problem, select all the known and unknown variables and values which all have physical units;
2) for each of such known and unknown variables and values with physical units, select any suitable physical unit and fix it;
3) separate all these fixed physical units and explicitly introduce all these fixed physical units into the general problem structure itself;
4) explicitly represent each of such known and unknown variables and values with pure numbers only, i.e. without physical units;
5) explicitly represent the general problem itself with pure numbers only, i.e. without physical units.
Notata bene:
1. For a general problem with physical units, there are many distinct physical unit removal general problem transformations.
2. In a general problem with physical units, for known and unknown variables and values with physical units, it is reasonable to select those suitable physical units namely in correlation.
3. For a general problem with physical units, there are many distinct reasonable physical unit removal general problem transformations.
4. It is reasonable to investigate the invariances of a general problem, of a solving method or theory, and of their results by distinct physical unit removal general problem transformations.
General problem transformation theory (GPTT) 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