General Subproblem Estimation 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), 38

Introduction

In classical mathematics, such problem solution estimators as the absolute and relative errors [1] are well-known.

Overmathematics [2-4] and the system of fundamental sciences on general problems [5] have discovered many principal shortcomings of these estimators with no objective sense of their results.

General subproblem estimation theory (GSPET) in fundamental science of general problem estimation gives general subproblem estimators via applying overmathematics [2-4] and the system of fundamental sciences on general problems [5] to estimating general subproblem pseudosolution. In particular, this theory applies:

uninumbers, quantisets, quantisystems, and uniquantities [2-5] to general problem analysis and synthesis;

invariant distances and new universal and adequate unierrors and reserves [2-5] to general subproblem pseudosolution test (check) and estimation.

General problem type and setting theory (GPTST) in fundamental science of general problem essence defines a general quantitative mathematical problem, or simply a general problem, to be a quantisystem [2-5] (former hypersystem)

_w(λ)R_λ[_φ∈Φ f_φ[_ω∈Ω z_ω]] (λ∈Λ)

of known relations R_λ over indexed unknown functions (dependent variables), or simply unknowns, f_φ of indexed independent known variables z_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 z_ω]

is a set of indexed elements z_ω ;

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

When replacing all the unknowns (unknown functions) with their possible ”values” (some known functions), or pseudosolutions [2-5], the above quantisystem of relations is transformed into the corresponding quantisystem of formal functional relations without any unknowns. To conserve the quantisystem form, let us use the same designations f_φ for these known functions, too. This known quantisystem can be further estimated both 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 quantisystem of pseudosolutions is a quantisystem of solutions to this general problem.

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

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) 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

_w(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 ;

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 f_φ .

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

where

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

R_λ is an inequality relation (e.g., ≈ , ∼ , ≠ , < , > , ≤ , ≥) 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 ;

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 f_φ .

By using unstrict inequality relations such as ≈ , ∼ , ≤ , ≥ , etc. only, a general pure inequations problem clearly further generalizes a general pure equations problem.

To provide general problem analysis, suppose (which is typical) 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).

To estimate a pseudosolution to any subproblem P_β (β∈Β), we need a nonnegative estimator E_β [2-5]

E_β(P_β) ≥ 0 (β∈Β)

either individual E_β (for separate subproblem P_β) or common E (for all subproblems P_β , β∈Β).

The least square method (LSM) [1] determines the componentwise deviation by a separate subproblem via the absolute error with many principal shortcomings [2-5]. Overmathematics [2-4] and the system of fundamental sciences on general problems [5] propose much more adequate nonnegative estimators such as distances which are invariant by coordinate system rotations, unierrors and reserves.

Quantisystem Distance

For any general problem P , a pseudosolution p to it transforms this general problem to a known general quantisystem for which it is possible to determine whether it is true (then this pseudosolution is a solution to this general problem) or false (then this pseudosolution is no solution to this general problem).

Let us introduce a quantisystem distance D of a pseudosolution p to general problem P from this general problem P as follows.

If a pseudosolution p is a solution to general problem P , then

D(p , P) = 0.

Naturally generalize this approach for the case when a pseudosolution p to general problem P can be either a solution or no solution to this general problem. Consider a set

{_α∈Α P'_α}

(where index α belongs to index set Α)

of such true known general quantisystems P'_α which are similar to general problem P(p) after substituting this pseudosolution p that it is possible to introduce a nonnegative distance

d[P(p), P'_α]

of a pseudosolution p with respect to general quantisystem P'_α .

Then, using the greatest lower bound inf, simply take

D(p , P) = inf{_α∈Α d[P(p), P'_α]}.

If (which is typical) a general problem P consists of separate 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),

then consider each relation P_β of them as a separate general subproblem in a general problem P , define and determine (componentwise) quantisystem relation distance

D(p , P_β) = inf{_α∈Α d[P_β(p), P'_βα]}.

Finally, define and determine (total) quantisystem distance as a suitable nonnegative function F of all D(p , P_β) (β∈Β) with the same own quantities q(β):

D(p , P) = F[_{β∈Β q(β)}D(p , P_β)].

Pseudosolution Distance

For any general problem P , a pseudosolution p to it transforms this general problem to a known general quantisystem for which it is possible to determine whether it is true (then this pseudosolution is a solution to this general problem) or false (then this pseudosolution is no solution to this general problem).

Let us introduce a pseudosolution distance d of a pseudosolution p to general problem P from this general problem P as follows.

If a pseudosolution p is a solution to general problem P , then

d(p , P) = 0.

Naturally generalize this approach for the case when a pseudosolution p to general problem P can be either a solution or no solution to this general problem. If general problem P is exactly solvable (i.e., has exact solutions), then consider nonempty set

{_α∈Α s_α}

(where index α belongs to index set Α)

of all the solutions s_α to general problem P , and introduce a nonnegative distance

d(p , s_α)

between pseudosolution p and each solution s_α and (using the greatest lower bound inf) simply take

d(p , P) = inf{_α∈Α d(p , s_α)}.

If (which is typical) a general problem P consists of separate exactly solvable relations P_β with their own positive quantities q(β)

P = {_{β∈Β q(β)}P_β}

and their relations solutions sets

{_α∈Α s_βα} (β∈Β)

(where index β belongs to index set Β),

then consider each relation P_β of them as a separate general subproblem in a general problem P , define and determine (componentwise) pseudosolution relation distance

d(p , P_β) = inf{_α∈Α d(p , s_βα)} (β∈Β).

Finally, define and determine the (total) pseudosolution distance as a suitable nonnegative function f of all d(p , P_β) (β∈Β) with the same own quantities q(β):

d(p , P) = f[_{β∈Β q(β)}d(p , P_β)].

Quantisystem Unierror

For any general problem P , a pseudosolution p to it transforms this general problem to a known general quantisystem for which it is possible to determine whether it is true (then this pseudosolution is a solution to this general problem) or false (then this pseudosolution is no solution to this general problem).

Let us introduce a quantisystem unierror E of a pseudosolution p to general problem P as follows.

If a pseudosolution p is a solution to general problem P , then

E(p , P) = 0.

Naturally generalize this approach for the case when a pseudosolution p to general problem P can be either a solution or no solution to this general problem. Consider a set

{_α∈Α P'_α}

(where index α belongs to index set Α)

of such true known general quantisystems P'_α which are similar to general problem P(p) after substituting this pseudosolution p that it is possible to introduce a nonnegative unierror

E[P(p), P'_α]

of a pseudosolution p with respect to general quantisystem P'_α .

Then, using the greatest lower bound inf, simply take

E(p , P) = inf{_α∈Α E[P(p), P'_α]}.

If (which is typical) a general problem P consists of separate 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),

then consider each relation P_β of them as a separate general subproblem in a general problem P , define and determine (componentwise) quantisystem relation unierror

E(p , P_β) = inf{_α∈Α E[P'_βα , P_β(p)]}.

Finally, define and determine the (total) quantisystem unierror as a suitable nonnegative function F of all E(p , P_β) (β∈Β) with the same own quantities q(β):

E(p , P) = F[_{β∈Β q(β)}E(p , P_β)].

Pseudosolution Unierror

For any general problem P , a pseudosolution p to it transforms this general problem to a known general quantisystem for which it is possible to determine whether it is true (then this pseudosolution is a solution to this general problem) or false (then this pseudosolution is no solution to this general problem).

Let us introduce a pseudosolution unierror e of a pseudosolution p to general problem P as follows.

If a pseudosolution p is a solution to general problem P , then

e(p , P) = 0.

Naturally generalize this approach for the case when a pseudosolution p to general problem P can be either a solution or no solution to this general problem. If general problem P is exactly solvable (i.e., has exact solutions), then consider nonempty set

{_α∈Α s_α}

(where index α belongs to index set Α)

of all the solutions s_α to general problem P , and introduce a nonnegative unierror

e(p , s_α)

of pseudosolution p with respect to solution s_α and (using the greatest lower bound inf) simply take

e(p , P) = inf{_α∈Α e(p , s_α)}.

If (which is typical) a general problem P consists of separate exactly solvable relations P_β with their own positive quantities q(β)

P = {_{β∈Β q(β)}P_β}

and their relations solutions sets

{_α∈Α s_βα} (β∈Β)

(where index β belongs to index set Β),

then consider each relation P_β of them as a separate general subproblem in a general problem P , define and determine (componentwise) pseudosolution relation unierror

e(p , P_β) = inf{_α∈Α e(p , s_βα)} (β∈Β).

Finally, define and determine the (total) pseudosolution unierror as a suitable nonnegative function f of all d(p , P_β) (β∈Β) with the same own quantities q(β):

e(p , P) = f[_{β∈Β q(β)}e(p , P_β)].

Quantisystem Reserve

For any general problem P , a pseudosolution p to it transforms this general problem to a known general quantisystem for which it is possible to determine whether it is true (then this pseudosolution is a solution to this general problem) or false (then this pseudosolution is no solution to this general problem).

If general problem P includes equations which are no identities, then quantisystem reserve R of a pseudosolution p to such general problem P is opposite to quantisystem unierror E of the same pseudosolution p to the same general problem P :

R(p , P) = - E(p , P).

If a pseudosolution p is a solution to such general problem P , then

R(p , P) = 0.

Nota bene: For such general problem P , quantisystem reserve R is nonpositive because a unierror is nonnegative.

To obtain a quasisolution to such general problem P via optimizing a pseudosolution p to this general problem, either minimize the nonnegative quantisystem unierror E(p , P) of this pseudosolution p to this general problem or maximize the nonpositive quantisystem reserve R(p , P) of this pseudosolution p to this general problem because these both approaches are equivalent.

If general problem P consists of equations which are identities and of inequations which all are consistent and has a set of precise solutions which includes internal points, then quantisystem unierror E(p , P) identically vanishes and its minimization brings nothing whereas maximizing the nonnegative quantisystem reserve R(p , P) of a pseudosolution p to such general problem P provides obtaining a supersolution S to such general problem P .

Pseudosolution Reserve

For any general problem P , a pseudosolution p to it transforms this general problem to a known general quantisystem for which it is possible to determine whether it is true (then this pseudosolution is a solution to this general problem) or false (then this pseudosolution is no solution to this general problem).

If general problem P includes equations which are no identities, then pseudosolution reserve r of a pseudosolution p to such general problem P is opposite to pseudosolution unierror e of the same pseudosolution p to the same general problem P :

r(p , P) = - e(p , P).

If a pseudosolution p is a solution to such general problem P , then

r(p , P) = 0.

Nota bene: For such general problem P , pseudosolution reserve r is nonpositive because a unierror is nonnegative.

To obtain a quasisolution to such general problem P via optimizing a pseudosolution p to this general problem, either minimize the nonnegative pseudosolution unierror e(p , P) of this pseudosolution p to this general problem or maximize the nonpositive pseudosolution reserve R(p , P) of this pseudosolution p to this general problem because these both approaches are equivalent.

If general problem P consists of equations which are identities and of inequations which all are consistent and has a set of precisesolutions which includes internal points, then pseudosolution unierror e(p , P) identically vanishes and its minimization brings nothing whereas maximizing the nonnegative pseudosolution reserve r(p , P) of a pseudosolution p to such general problem P provides obtaining a supersolution S to such general problem P .

General subproblem estimation theory (GSPET) 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] . General Problem Fundamental Sciences System. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2011