Ph. D. & Dr. Sc. Lev Gelimson's Physical Unit Removal Theory in Fundamental Science of General Problem Transformation

Physical Unit Removal Theory in Fundamental Science of General Problem Transformation

© 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), 58

Introduction

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 [5] 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].

Physical unit removal theory (PURT) in fundamental science of general problem transformation [5] gives methods of removing physical units to transform a general quantitative mathematical problem, or simply a general problem, into a pure number general quantitative mathematical problem, or simply a pure number general problem (including contradictory problems).

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 .

_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 .

_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 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.

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 Proportional Transformation

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.

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.

Physical Unit Removal Theory

Physical unit removal theory (PURT) in fundamental science of general problem transformation gives methods of removing physical units to transform a general quantitative mathematical problem, or simply a general problem, into a pure number general quantitative mathematical problem, or simply a pure number general problem (including contradictory problems).

A trivial algorithm of general problem physical unit removal transformation is as follows:

1) in a general problem, determine all the known and unknown variables, their values, and constants which all have physical units;

2) for each of such known and unknown variables, their values, and constants with physical units, select any suitable physical unit and fix it;

3) separate all these fixed physical units and explicitly introduce all them into the general problem structure itself;

4) explicitly represent each of such known and unknown variables, their values, and constants 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 can be many distinct physical unit removal general problem transformations.

2. In a general problem with physical units, for known and unknown variables, their values, and constants 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 itself, of a solving method or theory, and of their results by distinct physical unit removal general problem transformations.

Namely, to begin with, explicitly divide each of the known and unknown variables, their values, and constants with physical units v' in a general problem with physical units by its selected and fixed physical unit [v'] to obtain pure number ratio (v'). Then designate (v') as a new pure number known or unknown variable, its value, or constant v .

We obviously have

v = (v') = v'/[v'],

v' = (v')[v'] = v[v'],

[v] = [(v')] = 1,

([v']) = 1.

Show the essence of physical unit removal theory (PURT) in the simplest but most typical case of a finite overdetermined quantiset [2-5] of n (n > m; m , n ∈ N⁺ = {1, 2, ...}) linear equations

_q(i)(Σ_i=1^m a'_ijx'_i = c'_j) (j = 1, 2, ... , n) (L'_j)

with their own positive number quantities q(i), m unknown variables x'_i (i = 1, 2, ... , m), and any given factor constants a'_ij and free constants c'_j.

Nota bene: Both m unknown variables x'_i (i = 1, 2, ... , m) and (m + 1)n given constants a'_ij and c'_j(j = 1, 2, ... , n) may and can have their individual physical units

[x'_i] (i = 1, 2, ... , m),

[a'_ij] (i = 1, 2, ... , m ; j = 1, 2, ... , n),

[c'_j] (j = 1, 2, ... , n).

We may and can either keep these given individual physical units "as is" or rationally transform them if necessary and/or useful.

A rational algorithm of selecting a suitable system of physical units for all the known and unknown variables, their values, and constants with physical units in this general problem with physical units can be as follows:

1. For all the given free constants c'_j(j = 1, 2, ... , n), take physical units from a suitable physical unit system common for all them if possible. In the simplest case, if all the given free constants are homogeneous and allow to take a physical unit common for all them, then take namely it.

2. For all the unknown variables x'_i (i = 1, 2, ... , m), take physical units namely from the same physical unit system (for all the given free constants c'_j) common for all them if possible. In the simplest case, if all the unknown variables are homogeneous and allow to take a physical unit common for all them, then take namely it.

3. For each of the given factor constants a'_ij (i = 1, 2, ... , m ; j = 1, 2, ... , n), take physical units from the same physical unit system (for all the given free constants c'_j and all the unknown variables x'_i) common for all them if possible, namely ratios

[a'_ij] = [c'_j]/[x'_i]

with the corresponding indices. In the simplest case, if all the given free constants are homogeneous and allow to take a physical unit common for all them, then take namely it.

4. Divide each jth of all the given linear equations

_q(i)(Σ_i=1^m a'_ijx'_i = c'_j) (j = 1, 2, ... , n) (L'_j)

by the physical unit [c'_j] for free constant c'_j namely in this equation with using product representation

[c'_j] = [a'_ij][x'_i]

and obtain equivalent quantiset of linear equations

_q(i)(Σ_i=1^m a'_ij/[a'_ij] x'_i/[x'_i] = c'_j/[c'_i]) (j = 1, 2, ... , n) (L''_j).

5. Introduce new pure number unknown variables

x_i = (x'_i) = x'_i/[x'_i] (i = 1, 2, ... , m),

new pure number factor constants

a_ij = (a'_ij) = a'_ij/[a'_ij] (i = 1, 2, ... , m ; j = 1, 2, ... , n),

and new pure number free constants

c_j= (c'_j) = c'_j/[c'_i] (j = 1, 2, ... , n)

6. Substitute all these new pure number unknown variables, factor constants, and free constants into the above equivalent quantiset of linear equations and obtain the desired general pure number problem which is equivalent quantiset of linear equations in the Cartesian m-dimensional "space" [1]

_q(i)(Σ_i=1^m a_ijx_i = c_j) (j = 1, 2, ... , n) (L_j).

Due to natural bijection

x'_i ↔ x_i ,

each pseudosolution (in particular, (precise) solution, quasisolution, supersolution, or antisolution)

x'_i (i = 1, 2, ... , m)

to general problem with physical units

_q(i)(Σ_i=1^m a'_ijx'_i = c'_j) (j = 1, 2, ... , n) (L'_j)

can be obtained using the corresponding pseudosolution (in particular, (precise) solution, quasisolution, supersolution, or antisolution, respectively)

x_i (i = 1, 2, ... , m)

via transformation

x'_i = (x'_i)[x'_i] = x_i[x'_i] (i = 1, 2, ... , m).

Physical unit removal theory (PURT) 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