General Problem Type and Setting Theory in Fundamental Science on General Problem Essence

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), 30

Introduction

In classical mathematics [1], there is no sufficiently general concept of a quantitative mathematical problem. The concept of a finite or countable set of equations only completely ignores 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.

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).

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.

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

Further general problem type and setting theory (GPTST) in fundamental science on general problem essence [5] naturally defines a general pure equation problem and a general pure inequation problem.

General Pure Equation Problem

General problem type and setting theory (GPTST) in fundamental science on general problem essence [5] defines a general quantitative mathematical pure equation problem, or simply a general pure equation problem, to be a general problem that can be represented in a form in which all relations are namely equality relations.

In the left-hand sides of all the equations in a general pure equation problem, gather all the expressions available namely in the initial forms of these equations without any further transformations. The unique natural exception is changing the signs of the expressions by moving them to the other sides of the same equations. Then a general pure equation problem can be represented, in particular, as a quantiset

_q(λ){L_λ{_{φ∈Φ r(φ)}f_φ[_{ω∈Ω s(ω)}z_ω]} = 0} (λ∈Λ)

of indexed known quantiequations (with their own, or individual, quantities q(λ)) [2-5] in a form of vanishing operators L_λ 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

L_λ is a known operator 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 equation L_λ = 0 with index λ in a quantiset

_q(λ){L_λ{_{φ∈Φ r(φ)}f_φ[_{ω∈Ω s(ω)}z_ω]} = 0} (λ∈Λ).

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

General Pure Inequation Problem

General problem type and setting theory (GPTST) in fundamental science on general problem essence [5] defines a general quantitative mathematical pure inequation problem, or simply a general pure inequation problem, to be a general problem that can be represented in a form in which all relations are namely inequality relations.

In the left-hand sides of all the inequations in a general pure inequation problem, gather all the expressions available namely in the initial forms of these inequations without any further transformations. The unique natural exception is changing the signs of the expressions by moving them to the other sides of the same inequations. Then a general pure inequation problem can be represented, in particular, as a quantiset

_q(λ){L_λ{_{φ∈Φ r(φ)}f_φ[_{ω∈Ω s(ω)}z_ω]} R_λ 0} (λ∈Λ)

of indexed known inequality quantirelations (with their own, or individual, quantities q(λ)) [2-5] in a form of the comparison with zero of the values of operators L_λ 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

L_λ is a known 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 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 inequation L_λ R_λ 0 with index λ in a quantiset

_q(λ){L_λ{_{φ∈Φ r(φ)}f_φ[_{ω∈Ω s(ω)}z_ω]} R_λ 0} (λ∈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.

General Approximation Problem

Let us now consider a general approximation problem.

Let

Z ⊆ X × Y

be any given subset of the direct product of two sets X and Y and have a projection Z/X on X consisting of all x ∈ X really represented in Z , i.e., of all such x that for each of them there is a y ∈ Y such that

(x, y) ∈ Z .

Let further

{ y = F(x) }

where

x ∈ X

y ∈ Y

be a certain class of functions defined on X with range in Y .

Then the graph of such a function is a curve in X × Y .

The problem consists in finding (in class { y = F(x) }) functions with graphs nearest to Z in a certain reasonable sense.

To exactly fit this with a specific function

y = F(x),

the set Z ⊆ X × Y has to be included in the graph of this function:

Z ⊆ { (x, F(x)) | x ∈ X },

or, equivalently,

F(x) = y

for each

x ∈ Z/X .

But this inclusion (or equality) does not necessarily hold in the general case. Then it seems to be reasonable to estimate the error

E( F(x) =? y | x ∈ Z/X )

of the formal equality (true or false)

F(x) =? y

on this set Z/X via a certain error function E defined at least on Z/X .

To suitably construct such an error function, it seems to be reasonable to first consider two stages of its building:

1) defining local error functions to estimate errors at separate points x ;

2) defining global error functions using the values of local error functions to estimate errors on the whole set Z/X .

Possibly the simplest and most straightforward approach includes the following steps:

1) defining on Y × Y certain nonnegative functions r_yy’(y, y’) generally individual for different y , y’ and, e.g., similar to a distance [1] between any two elements y, y’ of Y (but not necessarily with holding the distance axioms [1]),

2) defining certain nonnegative functions R_x(r(F(x), y)) generally individual for different x ,

3) summing (possibly including integrating) their values on Z/X , and

4) using this sum (possibly including integrals) as a nearness measure.

General Problem Settings

Nota bene: The essence of a general problem includes, in particular, its origin (source) which can give very different settings (and hence both mathematical models and results) of a general problem even if graphical interpretations seem to be very similar or almost identical. For example, in the two-dimensional case, the same graphical interpretation with a triangle corresponds to many very different general problem settings and, moreover, to many very different general problems and even their systems (sets, families, etc.). Among them are, e.g., the following with determining:

1) the point nearest to the to the set or to the quantiset (with own quantities, which is very important by coinciding straight lines) of the three straight lines including the three sides, respectively, of the given triangle by different nearness criteria;

2) the point nearest to the triangle boundary, i.e. either to the set or to the quantiset (with counting the vertices twice) of all the points of the three sides of the given triangle by different nearness criteria;

3) the incenter and/or all the three excenters [1] of the given triangle;

4) the circumference (circle containing all the three vertices) of the given triangle;

5) the gravity (mass, length, uniquantity [2-5]) center of the triangle boundary, e.g., either of the set or of the quantiset (with counting the vertices twice) of all the points of the three sides of the given triangle;

6) the gravity (mass, area, uniquantity [2-5]) center of the triangle area including its interior and either including or not including its boundary, e.g., either of the set or of the quantiset (with counting the vertices twice) of all the points of the three sides of the given triangle.

The similar holds for a tetrahedron in the three-dimensional case with natural additional possibilities (the incenter/excenters for its flat faces along with the carcass incenter/excenters for its straight edges etc.).

By curvilinearity, the usual distance from any selected point to a certain point which lies on the curve or in the curvilinear surface is not the only. It is also possible to consider the distance from the selected point to the tangent (straight line or plane, respectively, if it exists) to the curve or curvilinear surface at that certain point if this tangent is the only. Otherwise, consider a certain suitable nonnegative function of the distances from the selected point to all the tangents. Additionally, if the selected point lies on the same curve or in the same curvilinear surface, then the usual straight line distance is not the only. It is also possible to consider the curvilinear distance as the greatest lower bound of the lengths of the curves lying on that curve or in that curvilinear surface and connecting those both points (simply the length of the shortest curve lying on that curve or in that curvilinear surface and connecting the both points if it exists). The similar can hold for polygons and polyhedra. Naturally, it is also possible to consider other conditions and limitations.

General problem type and setting theory (GPTST) in fundamental science on general problem essence 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