General Problem Analysis Theory in Fundamental Science of General Problem Analysis

by

© Ph. D. & Dr. Sc. Lev Gelimson

Academic Institute for Creating Fundamental Sciences (Munich, Germany)

Mathematical Monograph

The “Collegium” All World Academy of Sciences Publishers

Munich (Germany), 2011

Introduction

In classical mathematics [1], there is no sufficiently general concept of a quantitative mathematical problem. There is the concept of a finite or countable set of equations only with completely ignoring 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 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 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 Problem Analysis

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)

and there is a nonnegative estimator E [2-5]

E(P_β) ≥ 0 (β∈Β)

(e.g., distance which is invariant by coordinate system rotationces , unierror, etc.) common for all these general subproblems.

To provide general problem synthesis, explicitly give some suitable nonnegative subproblems estimations unification functions F of all

E(P_β) ≥ 0 (β∈Β)

with the same own quantities q(β). Each of such functions has to provide applying nonnegative estimator E to the whole general problem P with building its nonnegative total estimation

E(P) = F[_{β∈Β q(β)}E(P_β)] ≥ 0.

General problem analysis theory (GPAT) in fundamental science of general problem analysis naturally deals with general problem analysis.

Absolute Error

In classical mathematics [1], for estimating any inexact data, e.g., for discriminating two real numbers a and b , the absolute error Δ , e.g.,

Δ = |a - b|,

and the relative error δ , e.g.,

δ₁ = |a - b| / |a|

or

δ₂ = |a - b| / |b|,

are commonly used.

But the absolute error [1] alone offers no sufficient quality estimation giving, for example, the same result

Δ = 1

for the acceptable formal (correct or not) equality

1000 =? 999

and for the inadmissible one

1 =? 0.

Here and further we use the general designation (introduced by the author [2-5]) of any formal relation by placing the question sign after the relation sign. Such a relation may be true or false in any sense or even have no sense at all. In the spirit of constructive philosophy and overmathematics [2-5], there can be no necessity to consider the problem of the sense of any formal relation by its designation.

Further the absolute error is not invariant by equivalent transformations of a problem because, for instance, when multiplying a formal equality by a nonzero number, the absolute error is multiplied by the modulus (absolute value) of that number.

Relative Error

The relative error δ [1] should play a role supplement to that of the absolute error. But even in the case of the simplest formal equality

a =? b

with two numbers, there are at once two propositions, namely to use either

δ₁ = |a - b|/|a|

or

δ₂ = |a - b|/|b|

as an estimating fraction. It is a generally inadmissible uncertainty that could be acceptable only if ratio |a|/|b| is close to 1. Further the relative error is so intended that it should always belong to segment [0, 1]. But for

1 =? 0

by choosing 0 as the denominator, the result is +∞ ,

for

1 =? -1

by each denominator choice the result is 2.

Hence, the relative error has a restricted range of applicability amounting to the equalities of two elements whose ratio can be considered close to 1. By more complicated formal equalities with at least three elements, e.g., by

100 - 99 =? 0

or

1 - 2 + 3 - 4 =? -1,

the choice of a denominator seems to be vague at all. This is why the relative error is practically used only in the above simplest case and very seldom for variables and functions.

Unierror

Overmathematics and fundamental science of errors propose a unierror [2-5] irreproachably correcting the relative error and generalizing it possibly for any conceivable range of applicability.

For the same simplest formal equality

a =? b

of two numbers, a unierror can be represented by linear estimating fraction

E_{a =? b} = |a - b|/(|a| + |b|)

in the case

|a| + |b| > 0,

which should simply vanish by

a = b = 0.

We can get a shorter and more explicit notation by introducing extended division a//b :

a//b = a/b

by

a ≠ 0;

a//b = 0

by

a = 0

independently of the existence and value of b .

The above linear estimating fraction is then explicitly expressed:

E_{a =? b} = |a - b|//(|a| + |b|).

To generalize the above linear estimating fraction via the general linear estimating fraction, it is possible to add a positive uninumber (former hypernumber) [2-5] p suitable for a specific problem to the denominator and/or to replace the origin 0 with a uninumber h :

E_{a =? b} (p, h) = |a - b|//(|a - h| + |b - h| + p).

Another possibility is using quadratic estimating fraction

²E_{a =? b} = |a - b|//[2(a² + b²)]^1/2

instead of the above linear estimating fraction.

The outputs (return values) of such unierrors always belong to segment [0, 1], which should hold for the relative error.

Examples:

E_{0 =? 0} = 0;

E_{1 =? 0} = 1;

E_{100 =? 99} = 1/199;

²E_{0 =? 0} = 0;

²E_{1 =? 0} = 1/2^1/2;

²E_{1 =? -1} = 1.

By the principle of tolerable simplicity [2-5], it is reasonable to use the linear estimating fraction alone if it suffices.

For a formal vector equality

∑_ω∈Ω z_ω =? 0,

a unierror can be represented by linear estimating fraction

E(∑_ω∈Ω z_ω =? 0) = ||∑_ω∈Ω z_ω||//∑_ω∈Ω ||z_ω||

whose denominator contains all elements that have been initially in the equality, i.e., before any transformations. If all the vectors are replaced with numbers, the norms can be replaced with the moduli (absolute values).

The quadratic estimating fraction is

²E(∑_ω∈Ω z_ω =? 0) = ||∑_ω∈Ω z_ω||//(Q(Ω)∑_ω∈Ω ||z_ω||²)^1/2.

Examples.

E_{100 - 99 =? 0} = 1/199 = E_{100 =? 99} ;

E_{1 - 2 + 3 - 4 =? -1}= |1 - 2 + 3 - 4 + 1|/

(1 + 2 + 3 + 4 + 1) = 1/11.

For a formal functional equality

g[_ω∈Ω z_ω] =? 0

with a definition domain Z and a range of values in a normed vector space, a natural generalization idea is to define a unierror by linear estimating fraction

E_{g =? 0} = med_Z||g[_ω∈Ω z_ω]||//sup_Z||g[_ω∈Ω z_ω]||.

Its numerator is a mean value of the norm of the function in its definition domain Z . The denominator is the least upper bound on the norm of this function in the same domain of definition. If the range of the function contains numbers only, the norms can be replaced with the moduli (absolute values).

For a formal equality

g[_ω∈Ω z_ω] =? h[_ω∈Ω z_ω]

of two functions with a common definition domain Z and with ranges of values in a common normed vector space, the following holds. A further natural generalization idea is to define a unierror by linear estimating fraction

E_{g =? h} = sup_Z||g[_ω∈Ω z_ω] - h[_ω∈Ω z_ω]||//

sup_Z(||g[_ω∈Ω z_ω]|| + ||h[_ω∈Ω z_ω]||).

Its numerator is the least upper bound on the norm of the difference of these functions in their definition domain Z . The denominator is the least upper bound on the sum of the norms of these functions in the same domain of definition. If the ranges of values of the functions consist of numbers only, the norms can be replaced with the moduli (absolute values).

If a formal functional equality of algebraic sums in its initial form contains more than two functions, then the numerator of the estimating fraction is the least upper bound on the norm of the difference of the sides of the equality. The denominator is then the least upper bound on the sum of the norms of all the functions available in the initial form. For example, in the case of a formal equality

g[_ω∈Ω z_ω] - h[_ω∈Ω z_ω] =? k[_ω∈Ω z_ω],

the linear estimating fraction is

E_{g - h =? k} = sup_Z||g[_ω∈Ω z_ω] - h[_ω∈Ω z_ω] - k[_ω∈Ω z_ω]||//

sup_Z(||g[_ω∈Ω z_ω]|| + ||h[_ω∈Ω z_ω]|| + ||k[_ω∈Ω z_ω]||).

Let us consider 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), this quantiset of equations is transformed to the corresponding quantiset of formal functional equalities. To conserve the quantiset form, let us use for these known functions the same designations f_φ . For the equality with index λ

_w(λ)(L_λ[_φ∈Φ f_φ[_ω∈Ω z_ω]] =? 0) (λ∈Λ),

an estimating fraction may be

E_λ(m(λ)) = {lim [V(z_λ’)]^-1∫(||L_λ[_φ∈Φ f_φ[_ω∈Ω z_ω]]||_λ//

sup||L_λ[_φ∈Φ f_φ’[_ω∈Ω z_ω]]||_λ)^m(λ) dV(z_λ’)}^1/m(λ)

(z_λ’ → z_λ)

where

m(λ) is a positive number, we shall take 1;

in the denominator, a direct (not composite) function of independent variables is used and by determining the least upper bound, all different isometric transformations (conserving the norms)

||f_φ’[_ω∈Ω z_ω]||_φ = ||f_φ [_ω∈Ω z_ω]||_φ

of even equal elements are considered.

Reserve

The absolute error, the relative error, and the unierror of any exact object or model always vanish. It is often reasonable to additionally discriminate exact objects or models by the confidence in their exactness reliability. For example, both

x₁ = 1 + 10^-10

and

x₂ = 1 + 10¹⁰

are exact solutions to the inequation

x > 1,

x₁ practically unreliable and x₂ guaranteed. Their discrimination is especially important by any inexact data. Classical mathematics [1] cannot provide this at all.

For this purpose, overmathematics and fundamental science of reserves [2-5] propose the basic concept of reserve which is quite new in mathematics and extends the unierror in the following sense. The values of an unierror E belong to the segment [0, 1], those of a reserve R to [-1, 1]. For each inexact object I ,

E(I) > 0

and we can take

R(I) = - E(I).

For each exact (precise) object P,

E(P) = 0

and

R(P) ≥ 0.

A proposition to determine the reserve of an inexact object as its unierror with the opposite sign is at once evident. For an exact object, it seems to be reasonable to first define a suitable mapping of the object with respect to its exactness boundary and to further take the unierror of the mapped object. The last is exact if and only if the object itself precisely lies on its exactness boundary where the reserve vanishes. Otherwise, the mapped object is inexact and the object itself has a positive reserve.

For inequalities, such a mapping can be replaced with negating inequality relations and conserving equality ones. In the above example, we have (using the linear estimating fraction)

R_{x > 1}(x₁) =

R_{x > 1}(1 + 10^-10) =

E_{x <? 1}(1 + 10^-10) =

10^-10/(2 + 10^-10)

very small and

R_{x > 1}(x₂) =

R_{x > 1}(1 + 10¹⁰) =

E_{x <? 1}(1 + 10¹⁰) =

10¹⁰/(2 + 10¹⁰)

very near to 1, both in accordance with intuition.

Example. Let us determine the supersolution to the compound inequation

1 < x < 2

as the set of two simple inequations. The simplest way is to solve the equation

R_{x > 1}(x) = R_{x < 2}(x)

on the interval defined by the compound inequation itself. We receive (using the linear estimating fraction):

E_{x <? 1} = E_{x >? 2} ,

|x - 1|/(|x| + |1|) = |x - 2|/(|x| + |2|),

(x - 1)/(x + 1) = - (x - 2)/(x + 2),

x = 2^1/2

as the only element of the supersolution.

Example. Let us further determine the supersolution to the generalized compound inequation

a < x < b

in the case a < b also by solving the equation

R_{x > a}(x) = R_{x < b}(x)

on the interval defined by this compound inequation. We receive (using the linear estimating fraction):

E_{x <? a} = E_{x >? b} ,

|x - a|//(|x| + |a|) = |x - b|//(|x| + |b|),

by a > 0

(x - a)/(x + a) = - (x - b)/(x + b),

x = (ab)^1/2,

by b < 0

(x - a)/(- x - a) = - (x - b)/(- x - b),

x = - (ab)^1/2,

by a < 0 < b

x = 0

as the only element of the supersolution;

by ab = 0 the supersolution is the empty set ∅ .

For a ≥ b, the same is the quasisolution only.

Similarly, in the overdetermined set of equations

x = a ,

x = b ,

where variable x is an unknown real number, a and b are known real numbers, for the quasisolution holds the following:

if a and b have the same sign, then the only element of the quasisolution is their geometric mean value with the same sign;

if a and b have distinct signs, then the only element of the quasisolution is 0;

if

ab = 0

then the quasisolution is ∅.

If in this case we are dissatisfied by

a < 0 < b

and

ab = 0,

it is also possible to use the above general linear estimating fraction where normally one parameter only (either p > 0 or h) suffices. Let us first use p :

|x - a|/(|x| + |a| + p) = |x - b|/(|x| + |b| + p),

by a ≥ 0

(x - a)/(x + a + p) = - (x - b)/(x + b + p),

x = - p/2 + [(p/2 + a)(p/2 + b)]^1/2,

by b ≤ 0

(x - a)/(- x - a + p) = - (x - b)/(- x - b + p),

x = p/2 - [(p/2 - a)( p/2 - b)]^1/2.

By

p = 1,

for the compound inequation

0 < x < 1,

we receive the only element of the supersolution

x = (3^1/2 - 1)/2 ≈ 0.36603

and for -1 < x < 0

x = - (3^1/2 - 1)/2 ≈ - 0.36603.

Let us now use the parameter h and introduce

X = x - h ,

A = a - h ,

B = b - h .

We have

|X - A|//(|X| + |A|) = |X - B|//(|X| + |B|)

and, for example, by h < a

(X - A)/(X + A) = - (X - B)/(X + B),

X = (AB)^1/2,

x = [(a - h)( b - h)]^1/2 + h .

Let us take h = -1 for the compound inequation

0 < x < 1.

We then receive the only element of the supersolution

x = 2^1/2 - 1 ≈ 0.41421.

By

h → -∞

in the general case of the compound inequation

a < x < b ,

we receive the only element of the supersolution

x = (a + b)/2

by

a < b ,

which is the quasisolution only by

a ≥ b ,

corresponding to the use of the absolute errors or the least square method.

Each of the above problems has one parameter only. But typically, a general problem to be solved has a certain quantiset (or even quantisystem, which is the most general) of its initial parameters and a certain quantiset (or even quantisystem) of its target parameters.

The main idea [20, 21] to realistically determine the general reserve of a system under consideration is separately taking the reserves of its initial parameters into account, each of these reserves being expressed via a common additional number. It is obtained from the condition that, by the worst realizable combination of the values of these parameters arbitrarily modified within the bounds determined by the corresponding reserves, the state of at least one element of the system becomes limiting, no element of it being in overlimiting state [2-5].

This is a further generalization of the universalization method for limiting criteria [2] in fundamental mechanical and strength sciences [2].

In general, in overmathematics and fundamental science of reserves [2-5], for any function of an arbitrary set of variables

z = f[_α∈Α z_α],

Z = f[_α∈Α Z_α],

z_(α)∈Z_(α)

where (α) means that index α∈Α is optional,

the genuine values of the independent variables, z_α, and of the dependent one, z, usually deviate from their calculated values and, if the problem has certain limitations like strength criteria in strength problems [2], should belong to their admissible sets (domains), [Z_(α)]. If

f[_α∈Α [Z_α]] ⊆ [Z],

the problem is already solved. Otherwise, it is necessary to determine such a combination of the restrictions, Z_α , of the admissible sets, [Z_α], that the inclusion

f[_α∈Α Z_α] ⊆ [Z]

is true. For the existence of the numeric measures of those restrictions, or the reserves of the independent variables, it is sufficient that, for any α∈Α , [Z_(α)] is included into a certain Hilbert space [1], L_(α) , with the norm,

||z_(α)||_(α) ,

of each element, z_(α) , and the scalar product,

(z_(α), z’_(α))_(α) ,

of each pair of elements, z_(α) and z’_(α) .

Along with reserves R with the closed interval [-1, 1] as a range, let us introduce reserve factors n > 1. This bound corresponds to their factor nature even by the below additive approach. Please do not confuse such a real number n as a local variable only holding here with the above natural number n of points as a global variable holding not here only.

The additive approach to obtaining reserve factors develops, generalizes, and extends the relative error in a certain sense and naturally determines the neighborhood,

Z_(α)(δ_(α), z_0(α)),

of set Z_(α) with respect to element z_0(α)∈L_(α) with error δ_(α) ≥ 0 as the set of all z’_(α)∈L_(α) with

||z’_(α) - z_(α)||_(α) ≤ δ_(α)||z_(α) - z_0(α)||_(α).

The additive reserve of set Z_(α) by set [Z_(α)] with respect to element z_0(α) is defined as

n_a(α) = 1 + sup{δ_(α) ≥ 0: Z_(α)(δ_(α), z_0(α)) ⊆ [Z_(α)]}.

The multiplicative approach to obtaining reserve factors develops, generalizes, and extends the reserve factor in a certain sense and gives the neighborhood,

Z_(α)(n_(α)exp(iφ_(α)), z_0(α)),

of set Z_(α) with respect to element z_0(α)∈L_(α)

where

n_(α) ≥ 1 is a multiplicative reserve,

0 ≤ φ_(α) ≤ π ,

i² = -1,

as the set of all z’_(α)∈L_(α) with

n_(α)^-1|| z_(α) - z_0(α)||_(α) ≤ || z’_(α) - z_0(α)||_(α) ≤ n_(α)|| z_(α) - z_0(α)||_(α)

and

arccos[(z’_(α) - z_0(α), z_(α) - z_0(α))_(α)/(||z’_(α) - z_0(α)||_(α) ||z_(α) - z_0(α)||_(α))] ≤ φ_(α)

with two independent parameters n_(α) and φ_(α), if the dimensionality of space L_(α) is at least two, possibly with their relation, φ_(α)(n_(α)). If the space L_(α) is one-dimensional, φ_(α) = 0.

The multiplicative reserve of set Z_(α) by set [Z_(α)] with respect to element z_0(α) is defined as

n_m(α) = sup{n_(α) ≥ 1: Z_(α)(n_(α)exp(iφ_(α)), z_0(α)) ⊆ [Z_(α)]}.

By any of the both approaches, reserves n_α can be expressed via different nondecreasing functions of an additive reserve, n_fa , or a multiplicative one, n_fm , respectively, the both being common for reserves n_α and determined by the condition that there is an element z ∈ Z in a limiting state by the worst realizable combination of all z_α :

n_fa = sup{n ≥ 1: f[_α∈ΑZ_α(n_α(n), z_0α)] ⊆ [Z]},

n_fm = sup{n ≥ 1: f[_α∈ΑZ_α(n_α(n)exp(iφ_α(n_α(n))), z_0α) ⊆ [Z]}.

Reserve factor estimates are especially important in strength of materials to authentically determine the domain of the safe combinations of the loads. It is not sufficient to determine a usual safety factor, n_l , as the limiting stress, σ_l , divided by the equivalent stress, σ_e . This safety factor concept has been initially proposed for uniaxial stress states of solids simply (proportionally) loaded. Otherwise, it normalizes no loading, geometrical, and rheologic parameters themselves (really determining the stress states of solids) but their result, σ_e , as their composite function via the principal stresses,

σ₁ ≥ σ₂ ≥ σ₃ ,

by a limiting criterion. Moreover, experiments give only a limiting surface

F(σ₁, σ₂, σ₃) = σ_l

or, equally,

F^γ(σ₁, σ₂, σ₃)/σ_l^γ-1 = σ_l

where γ - any nonzero number.

So, instead of

σ_e = F(σ₁, σ₂, σ₃),

it is possible to consider

σ_eγ = F^γ(σ₁, σ₂, σ₃)/σ_l^γ-1

also for unlimiting states (n_l ≠ 1). The usual safety factor,

n_lγ = σ_l/σ_eγ = σ_l^γ/F^γ(σ₁, σ₂, σ₃) = n_l^γ,

can then take on any positive values when choosing suitable values of γ.

Even if namely γ = 1 is chosen by the principle of tolerable simplicity [2-5], then n_l does not show by what maximum number each of the loads can be multiplied or divided independently of one another so that, by the most dangerous realizable combination of such modified values of the loads, the stress state at the most dangerous point of the solid becomes limiting.

Thus (example 1), by

σ₁ = 250 MPa,

σ₂ = 240 MPa,

σ₃ = 210 MPa,

σ_l = 235 MPa

the Tresca strength theory [2] gives the usual safety factor

n_l = 5.9,

but

n_lσ₁ - σ₃/n_l = 1439 MPa >> σ_l .

If (example 2) a bar with strength

σ_t = 100 MPa

in tension and

σ_c = 800 MPa

in compression is contracted and stretched by two pairs of forces independent from each other and causing the stresses

σ = σ^- + σ⁺ = -500 MPa + 400 MPa = -100 MPa

then, by the usual safety factor determination method,

n_l = σ_c/|σ| = 8,

but

n_lσ^- + σ⁺/n_l = -3950 MPa << -σ_c ,

σ^-/n_l + n_lσ⁺ = 3137.5 MPa >> σ_t .

For simply (proportionally) loading, the multiplicative reserve factor is obtained from the condition

F(n_fmσ₁, n_fmσ₂, n_fmσ₃) = σ_l .

In the simplest case of the equal reserves of all z_α , we obtain the additive and multiplicative reserve factors

n_fa = 1.423,

n_fm = 1.5

in example 1 and the multiplicative reserve factors

n_fmt = 1.25,

n_fmc = 2

in example 2 in tension and compression, respectively. These realistic reserve factors are significantly less than the usual ones unwarrantedly optimistic.

The method of determining the above generalized reserve factors provides rational control of the authentic reserves of the strength of materials and structures.

Determining the relation between a reserve and a reserve factor is based on the definitions of their particular cases.

Example. For a typical inequality

0 < p ≤ P

where p is a parameter and P is its limiting value, the above multiplicative reserve factor

n = P/p

and the above reserve

R_{p ≤ P} = δ_{p > P} =

|p - P|//(|p| + |P|) =

(P - p)/(P + p) =

(P/p - 1)/(P/p + 1) =

(n - 1)/(n + 1).

Then, using simply R rather than R_{p ≤ P} and reversing the above formulae, we obtain

n = (1 + R)/(1 - R).

General problem analysis theory (GPAT) in fundamental science of general problem analysis is very efficient by solving many urgent general (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