General Problem Synthesis Theory in Fundamental Science of General Problem Synthesis

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 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 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 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 rotations, 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 synthesis theory (GPSyT) in fundamental science of general problem synthesis naturally deals with general problem synthesis.

General Problem Synthesis

By solving contradictory (e.g. overdetermined [1]) problems without precise solutions, it is necessary to find the best pseudosolutions, so-called quasisolutions [2-5]. If such a problem is a sets of equations, then their graphs in a Cartesian coordinate system have no point in common but in many cases determine a certain (limited if possible) point set whose center (in some reasonable sense) could be considered as the desired quasisolution. The straightforward basic idea is as follows. If it is impossible to precisely satisfy all the given equations and each point (pseudosolution) gives deviations (e.g., errors), then it is logical to try to equally (uniformly, homogeneously) distribute them among all the given equations. Such an approach corresponds to intuition and leads to the intuitive concept of the center (in some reasonable sense) of that point set.

In classical mathematics [1], to solve such overdetermined sets of equations, the least square method (LSM) [1] by Legendre and Gauss only usually applies. Overmathematics [2-4] and the system of fundamental sciences on general problems [5] have discovered many principal shortcomings of this method, by methods of finite elements, points, etc. Additionally consider its simplest approach which is typical. Minimizing the sum of the squared differences of the alone preselected coordinates (e.g., ordinates in a two-dimensional problem) of the graph of the desired approximation function and of everyone among the given functions depends on this preselection, ignores the remaining coordinates, and provides no coordinate system rotation invariance and hence no objective sense of the result.

The implicit center criterion of the least square method (LSM) [1] is based, in particular, on the following:

1) determining the componentwise deviation by a separate equation via the absolute error;

2) determining the total deviation by a whole set of equations as the quadratic mean value of the componentwise deviations by all the separate equations.

Overmathematics [2-4] and the system of fundamental sciences on general problems [5] have discovered many principal shortcomings of both the absolute and the relative error. And the second power is often insufficient to find realistic point set centers.

Solving general problems in fundamental science of general problem essence and solving strategy is based, in particular, on the following:

1) general problem analysis theory (GPAT) determining the componentwise deviation by a separate equation via adequate estimators such as distances which are invariant by coordinate system rotations, unierrors and reserves [2-5];

2) general problem synthesis theory (GPSyT) determining the total deviation by a whole set of equations via much more general and adequate functions of the componentwise deviations by all the separate equations.

Let us consider once more 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

_q(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 ;

q(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

_q(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.

For the complete quantiset of the equalities, an estimating fraction can be chosen in the forms

E(m) = ∑_λ∈Λ q(λ)E_λ(m) // ∑_λ∈Λ q(λ)

(the arithmetic mean value) by the linear law,

²E(m) = {∑_λ∈Λ q(λ)[E_λ(m)]² // ∑_λ∈Λ q(λ)}^1/2

(the quadratic mean value) by the quadratic law, and

ⁿE(m) = {∑_λ∈Λ q(λ)[E_λ(m)]ⁿ // ∑_λ∈Λ q(λ)}^1/n

by the law of the nth power (n > 0).

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).

Subproblems Estimations Unification Function

Suppose 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 are nonnegative estimators E_β [2-5]

E_β(P_β) ≥ 0 (β∈Β)

(e.g., distances which are invariant by coordinate system rotations, unierrors, etc.) individual for all these general subproblems. In particular, there can be a nonnegative estimator E [2-5]

E(P_β) ≥ 0 (β∈Β)

common for all these general subproblems, which is typical.

Our present task is to explicitly give some suitable nonnegative subproblems estimations unification functions F of all

E_β(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_β(P_β)] ≥ 0.

Some suitable nonnegative subproblems estimations unification functions follow.

1. The weighted power mean of the (componentwise) subproblems estimations

^tE(P) = {Σ_β∈Β q(β)[E_β(P_β)]^t / Σ_β∈Β q(β)}^1/t

where t is a positive number and

Q(P) = Σ_β∈Β q(β)

is the uniquantity [2-5] of quantisystem

P = {_{β∈Β q(β)}P_β}.

2. The weighted power product mean of the (componentwise) subproblems estimations

^tE(P) = {[(Σ_β∈Β q(β)E_β(P_β))^t - Σ_β∈Β q(β)E_β^t(P_β)] / [(Σ_β∈Β q(β))^t - Σ_β∈Β q(β)]}^1/t

where t > 1.

2. The weighted power difference mean of the (componentwise) subproblems estimations

^{s , t , u}E(P) = {|[Σ_β∈Β q(β)E_β^u/s(P_β)]^s - [Σ_β∈Β q(β)E_β^u/t(P_β)]^t| / |[Σ_β∈Β q(β)]^s - [Σ_β∈Β q(β)]^t|}^1/u

where u and s ≠ t are positive numbers.

3. The weighted geometric mean of the (componentwise) subproblems estimations

E(P) = [Π_β∈Β E_β^q(β)(P_β)]^{1/Σ_β∈Β q(β)} .

4. The weighted power mean of the moduli (absolute values) of all the pairwise differences of the powers of the (componentwise) subproblems estimations

^{t , u}E(P) = {Σ_{β∈Β ,β'∈Β, β'≠β} q(β)q(β')|E_β^u(P_β) - E_β'^u(P_β')|^t / Σ_{β∈Β ,β'∈Β, β'≠β} q(β)q(β')}^1/(tu)

where t and u are positive numbers.

5. The weighted power difference mean of the moduli (absolute values) of all the pairwise differences of the powers of the (componentwise) subproblems estimations

^{s , t , u , v , w}E(P) = {|[Σ_{β∈Β , β'∈Β , β'≠β} q(β)q(β')|E_β^v(P_β) - E_β'^v(P_β')|^u/(vs)]^s - [Σ_{β∈Β , β'∈Β , β'≠β} q(β)q(β')|E_β^w(P_β) - E_β'^w(P_β')|^u/(wt)]^t| /

|[(Σ_{β∈Β , β'∈Β , β'≠β} q(β)q(β')]^s - [(Σ_{β∈Β , β'∈Β , β'≠β} q(β)q(β')]^t|}^1/u

where s ≠ t , u , v , and w are positive numbers.

6. The weighted power mean of all the groupwise products of the moduli (absolute values) of all the pairwise differences of the powers of the (componentwise) subproblems estimations, e.g., the weighted power mean of all the pairwise products of the moduli (absolute values) of all the pairwise differences of the powers of the (componentwise) subproblems estimations

^{t , u}E(P) = {Σ_{β∈Β , β'∈Β , β'≠β ,β''∈Β , β'''∈Β , β'''≠β'' , (β'', β''')≠(β , β')} q(β)q(β')q(β'')q(β''')[|E_β^u(P_β) - E_β'^u(P_β')| |E_β''^u(P_β'') - E_β'''^u(P_β''')|]^t /

Σ_{β∈Β , β'∈Β , β'≠β ,β''∈Β , β'''∈Β , β'''≠β'' , (β'', β''')≠(β , β')} q(β)q(β')q(β'')q(β''')}^1/(2tu)

where t and u are positive numbers and all the different index pairs (β , β') and (β'', β''') are unordered.

Nota bene: Uncountable operations and their results are not considered in classical mathematics [1] at all. In particular, this holds both for addition (and its result, namely a sum) and a set of equations (also with completely ignoring their quantities). On the contrary, overmathematics [2-4] considers any (also uncountable) sets, quantisets, systems, and quantisystems of any objects, operations, and relations. In particular, this holds both for addition (and its result, namely a sum) and a quantiset of equations (also with completely taking their quantities into account).

To provide the best pseudosolutions, minimize the above and other means of distances or unierrors but maximize reserves.

The above nonnegative subproblems estimations unification functions are based on existing a nonnegative estimator E common for all the subproblems. Now consider the much more general case.

By a finite set of n relations with unit own quantities q(β) = 1 and (also individual) estimators values E_j (j = 1, 2, ... , n), the above weighted power mean of the (componentwise) subproblems estimations

^tE(P) = {Σ_β∈Β q(β)[E_β(P_β)]^t / Σ_β∈Β q(β)}^1/t

(where t is a positive number)

simply gives

^tE(P) = (Σ_j=1ⁿ E_j^t / n)^1/t .

By infinitely increasing t we obtain as the limiting case

E(P) = max{_j=1ⁿ E_j}.

Generalizing this for any (possibly infinite) set of relations with using the least upper bound sup gives

E(P) = sup{_β∈Β E_β}.

The last two formulae hold for any nonnegative estimator E , e.g., distances, unierrors, etc. For reserve R opposite to unierror E (R = - E) we have

R(P) = min{_j=1ⁿ R_j},

R(P) = inf{_β∈Β R_β}

with using the greatest lower bound inf.

It is very useful to further generalize the least upper bound sup and the greatest lower bound inf.

Least Upper Quantibound

The least upper quantibound sup M on an ordered quantiset M [2-5] is the quantiset of the least upper bounds on the subsets of M reduced from above. The least upper quantibounds on two quantisets are ordered by ordering the usual least upper bounds on their quantisubsets minimally equally reduced from above to discriminate them.

Example. One pseudosolution to a set of four equations brings for them the unierrors

0, 1, 1, 1,

respectively; another one

1, 0, 0, 0.

Intuitively, the second one is better than the first. But their Cantor sets [1] of the unierrors are both

{0, 1}

and hence provide no discriminating these pseudosolutions by their quality. The quantisets [2-5] of the unierrors are

{₃1, 0}°

and

{1, ₃0}°,

respectively. Again the usual least upper bounds are both 1. Minimally reducing the quantisets from above is subtracting the quantiset {1}° and brings the required discrimination:

sup({₃1, 0}° -° {1}°) = sup{₂1, 0}° = 1 >

sup({1, ₃0}° -° {1}°) = sup{₃0}° = 0

and therefore

sup{₃1, 0} > sup{1, ₃0}°.

Greatest Lower Quantibound

The greatest lower quantibound inf M on an ordered quantiset M [2-5] is the quantiset of the greatest lower bounds on the subsets of M reduced from below. The greatest lower quantibounds on two quantisets are ordered by ordering the usual bounds on their quantisubsets minimally equally reduced from below to discriminate them.

Example. One pseudosolution to a set of four equations brings for them the reserves

0, -1, -1, -1,

respectively; another one

-1, 0, 0, 0.

Intuitively, the second one is better than the first. But their Cantor sets [1] of the reserves are both

{0, -1}

and hence provide no discriminating these pseudosolutions by their quality. The quantisets [2-5] of the reserves are

{₃-1, 0}°

and

{-1, ₃0}°,

respectively. Again the usual greatest lower bounds are both -1. Minimally reducing the quantisets from below is subtracting the quantiset {-1}° and brings the required discrimination:

inf({₃-1, 0}° -° {-1}°) = inf{₂-1, 0}° = -1 <

inf({-1, ₃0}° -° {-1}°) = inf{₃0}° = 0

and therefore

inf{₃-1, 0} < inf{-1, ₃0}°.

Quantibound Estimator

Using the least upper quantibound sup and the greatest lower quantibound inf, we obtain

E(P) = inf{_β∈Β E_β}

for any nonnegative estimator E , e.g., distances, unierrors, etc. and

R(P) = sup{_β∈Β R_β}

for reserve R opposite to unierror E (R = - E).

Difference Modulus Quantibound Estimator

Now consider any quantiset

{_{α∈Α q(α)}a_α}° = {_{α∈Α qα}a_α}°

with positive individual quantity q(α) = q_α of each element a_α with index α from index set Α . Additionally consider any such quantiset

{_{β∈Β q(β)}b_β}° = {_{β∈Β qβ}b_β}°

with positive individual quantity q(β) = q_β of each element b_β with index β from index set Β . If there exist all the pairwise mutual element products a_αb_β , then consider both elementwise and quantity-wise product quantiset

{_{α∈Α q(α)}a_α}° {_{β∈Β q(β)}b_β}° = {_{α∈Α ,β∈Β q(α)q(β)}(a_αb_β)}° = {_{α∈Α ,β∈Β qαqβ}(a_αb_β)}°

of the above two quantisets. In particular, square quantiset

{_{α∈Α q(α)}a_α}°² = {_{α∈Α ,α'∈Α q(α)q(α')}(a_αa_α')}° =

{_{α∈Α q²(α)}a_α²}° + {_{α∈Α ,α'∈Α , α'≠α , (α ,α') ordered q(α)q(α')}(a_αa_α')}° =

{_{α∈Α q²(α)}a_α²}° + {_{α∈Α ,α'∈Α , α'≠α , (α ,α') unordered 2q(α)q(α')}(a_αa_α')}°.

Replacing all the pairwise mutual element products a_αb_β with the moduli (absolute values) |a_α - b_β| of all the pairwise mutual element differences and keeping (which is logical) the above products

q(α)q(β) = q_αq_β

of individual element quantities, we obtain elementwise difference modulus and quantity-wise product quantiset

|{_{α∈Α q(α)}a_α}° _*-° {_{β∈Β q(β)}b_β}°| = {_{α∈Α ,β∈Β q(α)q(β)}|a_α - b_β|}° = {_{α∈Α ,β∈Β qαqβ}|a_α - b_β|}°

along with the definition of the binary elementwise difference modulus and quantity-wise product quantiset operation _*- .

In particular, elementwise difference modulus and quantity-wise square quantiset

|{_{α∈Α q(α)}a_α}° _*- {_{α∈Α q(α)}a_α}°| = {_{α∈Α ,α'∈Α q(α)q(α')}|a_α - a_α'|}° = {_{α∈Α ,α'∈Α qαqα'}|a_α - a_α'|}° =

{_{α∈Α q²(α)}0}° + {_{α∈Α ,α'∈Α , α'≠α , (α ,α') ordered q(α)q(α')}|a_α - a_α'|}° =

{_{α∈Α q²α}0}° + {_{α∈Α ,α'∈Α , α'≠α , (α ,α') ordered qαqα'}|a_α - a_α'|}° =

{_{α∈Α q²(α)}0}° + {_{α∈Α ,α'∈Α , α'≠α , (α ,α') unordered 2q(α)q(α')}|a_α - a_α'|}° =

{_{α∈Α q²α}0}° + {_{α∈Α ,α'∈Α , α'≠α , (α ,α') unordered 2qαqα'}|a_α - a_α'|}°.

These definitions satisfy the conservation law in the sense that if any elements among

{_{α∈Α q(α)}a_α}° = {_{α∈Α qα}a_α}°,

{_{β∈Β q(β)}b_β}° = {_{β∈Β qβ}b_β}°

can coincide, then the above representations hold (but the quantisets could be reduced via adding the individual quantities of coinciding elements).

Now apply the least upper quantibound sup and the greatest lower quantibound inf NOT to subproblemwise estimations quantiset

{_{β∈Β q(β)}E_β}° = {_{β∈Β qβ}E_β}°

itself but to subproblemwise estimations elementwise difference modulus and quantity-wise product quantiset

|{_{β∈Β q(β)}E_β}° _*-° {_{β∈Β q(β)}E_β}°| = {_{β∈Β ,α'β∈Β q(β)q(β')}|E_β - E_β'|}° = {_{β∈Β ,β∈Β qβqβ'}|E_β - E_β'|}° =

{_{β∈Β q²(β)}0}° + {_{β∈Β ,β'∈Β , β'≠β , (β ,β') ordered q(β)q(β')}|E_β - E_β'|}° =

{_{β∈Β q²β}0}° + {_{β∈Β ,β'∈Β , β'≠β , (β ,β') ordered qβqβ'}|E_β - E_β'|}° =

{_{β∈Β q²(β)}0}° + {_{β∈Β ,β'∈Β , β'≠β , (β ,β') unordered 2q(β)q(β')}|E_β - E_β'|}° =

{_{β∈Β q²β}0}° + {_{β∈Β ,β'∈Β , β'≠β , (β ,β') unordered 2qβqβ'}|E_β - E_β'|}°.

Then we obtain

E(P) = inf|{_{β∈Β q(β)}E_β}° _*-° {_{β∈Β q(β)}E_β}°| = inf{_{β∈Β ,α'β∈Β q(β)q(β')}|E_β - E_β'|}° = inf{_{β∈Β ,β∈Β qβqβ'}|E_β - E_β'|}° =

inf{{_{β∈Β q²(β)}0}° + {_{β∈Β ,β'∈Β , β'≠β , (β ,β') ordered q(β)q(β')}|E_β - E_β'|}°} =

inf{{_{β∈Β q²β}0}° + {_{β∈Β ,β'∈Β , β'≠β , (β ,β') ordered qβqβ'}|E_β - E_β'|}°} =

inf{{_{β∈Β q²(β)}0}° + {_{β∈Β ,β'∈Β , β'≠β , (β ,β') unordered 2q(β)q(β')}|E_β - E_β'|}°} =

inf{{_{β∈Β q²β}0}° + {_{β∈Β ,β'∈Β , β'≠β , (β ,β') unordered 2qβqβ'}|E_β - E_β'|}°}.

Apply this estimator E to compare any two pseudosolutions p₁ and p₂ to any general problem

P = {_{β∈Β q(β)}P_β}.

The estimation quantiset for pseudosolution p₁ is

{_{β∈Β q(β)}E_β(p₁)}° = {_{β∈Β qβ}E_β(p₁)}°.

The estimation quantiset for pseudosolution p₂ is

{_{β∈Β q(β)}E_β(p₂)}° = {_{β∈Β qβ}E_β(p₂)}°.

The total estimation for pseudosolution p₁ is

E(p₁ , P) = inf|{_{β∈Β q(β)}E_β(p₁)}° _*-° {_{β∈Β q(β)}E_β(p₁)}°| = inf{_{β∈Β ,α'β∈Β q(β)q(β')}|E_β(p₁) - E_β'(p₁)|}° = inf{_{β∈Β ,β∈Β qβqβ'}|E_β(p₁) - E_β'(p₁)|}° =

inf{{_{β∈Β q²(β)}0}° + {_{β∈Β ,β'∈Β , β'≠β , (β ,β') ordered q(β)q(β')}|E_β(p₁) - E_β'(p₁)|}°} =

inf{{_{β∈Β q²β}0}° + {_{β∈Β ,β'∈Β , β'≠β , (β ,β') ordered qβqβ'}|E_β(p₁) - E_β'(p₁)|}°} =

inf{{_{β∈Β q²(β)}0}° + {_{β∈Β ,β'∈Β , β'≠β , (β ,β') unordered 2q(β)q(β')}|E_β(p₁) - E_β'(p₁)|}°} =

inf{{_{β∈Β q²β}0}° + {_{β∈Β ,β'∈Β , β'≠β , (β ,β') unordered 2qβqβ'}|E_β(p₁) - E_β'(p₁)|}°}.

The total estimation for pseudosolution p₂ is

E(p₂ , P) = inf|{_{β∈Β q(β)}E_β(p₂)}° _*-° {_{β∈Β q(β)}E_β(p₂)}°| = inf{_{β∈Β ,α'β∈Β q(β)q(β')}|E_β(p₂) - E_β'(p₂)|}° = inf{_{β∈Β ,β∈Β qβqβ'}|E_β(p₂) - E_β'(p₂)|}° =

inf{{_{β∈Β q²(β)}0}° + {_{β∈Β ,β'∈Β , β'≠β , (β ,β') ordered q(β)q(β')}|E_β(p₂) - E_β'(p₂)|}°} =

inf{{_{β∈Β q²β}0}° + {_{β∈Β ,β'∈Β , β'≠β , (β ,β') ordered qβqβ'}|E_β(p₂) - E_β'(p₂)|}°} =

inf{{_{β∈Β q²(β)}0}° + {_{β∈Β ,β'∈Β , β'≠β , (β ,β') unordered 2q(β)q(β')}|E_β(p₂) - E_β'(p₂)|}°} =

inf{{_{β∈Β q²β}0}° + {_{β∈Β ,β'∈Β , β'≠β , (β ,β') unordered 2qβqβ'}|E_β(p₂) - E_β'(p₂)|}°}.

Our aim is to determine the sign function

sign[E(p₁ , P) - E(p₂ , P)]

of difference E(p₁ , P) - E(p₂ , P) to establish either equality

E(p₁ , P) = E(p₂ , P)

or one of the inequality relations

E(p₁ , P) > E(p₂ , P),

E(p₁ , P) < E(p₂ , P).

Reduced Difference Modulus Quantibound Estimator

Naturally, we may directly compare the both complete quantisets. But to simplify this comparison, we may preliminarily reduce the both complete quantisets by subtracting quantiset

{_{β∈Β q²(β)}0}° = {_{β∈Β q²β}0}°

they have in common, to use namely unordered pairs (β , β'), and to halve all the remaining individual element quantities. In this case we use preliminarily reduced estimator

e(P) = inf{_{β∈Β ,β'∈Β , β'≠β , (β ,β') unordered q(β)q(β')}|E_β - E_β'|}° =

inf{_{β∈Β ,β'∈Β , β'≠β , (β ,β') unordered qβqβ'}|E_β - E_β'|}°

and compare preliminarily reduced quantisets

{_{β∈Β ,β'∈Β , β'≠β , (β ,β') unordered q(β)q(β')}|E_β(p₁) - E_β'(p₁)|}° =

{_{β∈Β ,β'∈Β , β'≠β , (β ,β') unordered qβqβ'}|E_β(p₁) - E_β'(p₁)|}°

and

{_{β∈Β ,β'∈Β , β'≠β , (β ,β') unordered q(β)q(β')}|E_β(p₂) - E_β'(p₂)|}° =

{_{β∈Β ,β'∈Β , β'≠β , (β ,β') unordered qβqβ'}|E_β(p₂) - E_β'(p₂)|}°

to determine the sign function

sign[e(p₁ , P) - e(p₂ , P)]

of difference e(p₁ , P) - e(p₂ , P) to establish either equality

e(p₁ , P) = e(p₂ , P)

or one of the inequality relations

e(p₁ , P) > e(p₂ , P),

e(p₁ , P) < e(p₂ , P).

Supplemented Difference Modulus Quantibound Estimator

If for any general problem P , there is an incenter and there are excenters, then both the incenter and each of the excenters give as pseudosolutions the same zero estimation because all the radii of any circle coincide and all the radii differences of any circle vanish. Therefore, the above estimators cannot discriminate at all the incenter and each of the excenters. To provide discriminating them, add to those quantisets a quantielement whose individual quantity q is any positive number and whose element is a strictly monotonically increasing function F of all the subproblems estimations. Then, in order to determine the desired relation between greatest lower quantibounds inf, reducing such supplemented quantisets for such pseudosolutions leads to dropping all zeros and then to comparing the values of this function. Now we obtain a strictly less result namely for the incenter, q.e.d. (quod erat demonstrandum). For example, we may simply take q = 1 and

F = Σ_β∈Β E_β .

Then we obtain supplemented complete estimator

E(P) = inf{|{_{β∈Β q(β)}E_β}° _*-° {_{β∈Β q(β)}E_β}°| +° _qF} =

inf{{_{β∈Β ,α'β∈Β q(β)q(β')}|E_β - E_β'|}° +° _qF} =

inf{{_{β∈Β ,β∈Β qβqβ'}|E_β - E_β'|}° +° _qF} =

inf{{_{β∈Β q²(β)}0}° + {_{β∈Β ,β'∈Β , β'≠β , (β ,β') ordered q(β)q(β')}|E_β - E_β'|}° +° _qF} =

inf{{_{β∈Β q²β}0}° + {_{β∈Β ,β'∈Β , β'≠β , (β ,β') ordered qβqβ'}|E_β - E_β'|}° +° _qF} =

inf{{_{β∈Β q²(β)}0}° + {_{β∈Β ,β'∈Β , β'≠β , (β ,β') unordered 2q(β)q(β')}|E_β - E_β'|}° +° _qF} =

inf{{_{β∈Β q²β}0}° + {_{β∈Β ,β'∈Β , β'≠β , (β ,β') unordered 2qβqβ'}|E_β - E_β'|}° +° _qF}.

The total estimation for pseudosolution p₁ is

E(p₁ , P) = inf{|{_{β∈Β q(β)}E_β(p₁)}° _*- {_{β∈Β q(β)}E_β(p₁)}°| +° _qF(p₁)} =

inf{{_{β∈Β ,α'β∈Β q(β)q(β')}|E_β(p₁) - E_β'(p₁)|}° +° _qF(p₁)} =

inf{{_{β∈Β ,β∈Β qβqβ'}|E_β(p₁) - E_β'(p₁)|}° +° _qF(p₁)} =

inf{{_{β∈Β q²(β)}0}° + {_{β∈Β ,β'∈Β , β'≠β , (β ,β') ordered q(β)q(β')}|E_β(p₁) - E_β'(p₁)|}° +° _qF(p₁)} =

inf{{_{β∈Β q²β}0}° + {_{β∈Β ,β'∈Β , β'≠β , (β ,β') ordered qβqβ'}|E_β(p₁) - E_β'(p₁)|}° +° _qF(p₁)} =

inf{{_{β∈Β q²(β)}0}° + {_{β∈Β ,β'∈Β , β'≠β , (β ,β') unordered 2q(β)q(β')}|E_β(p₁) - E_β'(p₁)|}° +° _qF(p₁)} =

inf{{_{β∈Β q²β}0}° + {_{β∈Β ,β'∈Β , β'≠β , (β ,β') unordered 2qβqβ'}|E_β(p₁) - E_β'(p₁)|}° +° _qF(p₁)}.

The total estimation for pseudosolution p₂ is

E(p₂ , P) = inf{|{_{β∈Β q(β)}E_β(p₂)}° _*- {_{β∈Β q(β)}E_β(p₂)}°| +° _qF(p₂)} =

inf{{_{β∈Β ,α'β∈Β q(β)q(β')}|E_β(p₂) - E_β'(p₂)|}° +° _qF(p₂)} =

inf{{_{β∈Β ,β∈Β qβqβ'}|E_β(p₂) - E_β'(p₂)|}° +° _qF(p₂)} =

inf{{_{β∈Β q²(β)}0}° + {_{β∈Β ,β'∈Β , β'≠β , (β ,β') ordered q(β)q(β')}|E_β(p₂) - E_β'(p₂)|}° +° _qF(p₂)} =

inf{{_{β∈Β q²β}0}° + {_{β∈Β ,β'∈Β , β'≠β , (β ,β') ordered qβqβ'}|E_β(p₂) - E_β'(p₂)|}° +° _qF(p₂)} =

inf{{_{β∈Β q²(β)}0}° + {_{β∈Β ,β'∈Β , β'≠β , (β ,β') unordered 2q(β)q(β')}|E_β(p₂) - E_β'(p₂)|}° +° _qF(p₂)} =

inf{{_{β∈Β q²β}0}° + {_{β∈Β ,β'∈Β , β'≠β , (β ,β') unordered 2qβqβ'}|E_β(p₂) - E_β'(p₂)|}° +° _qF(p₂)}.

Supplemented Reduced Difference Modulus Quantibound Estimator

The preliminarily reduced estimator is

e(P) = inf{{_{β∈Β ,β'∈Β , β'≠β , (β ,β') unordered q(β)q(β')}|E_β - E_β'|}° +° _qF} =

inf{{_{β∈Β ,β'∈Β , β'≠β , (β ,β') unordered qβqβ'}|E_β - E_β'|}° +° _qF}.

Compare preliminarily reduced quantisets

{{_{β∈Β ,β'∈Β , β'≠β , (β ,β') unordered q(β)q(β')}|E_β(p₁) - E_β'(p₁)|}° +° _qF(p₁)} =

{{_{β∈Β ,β'∈Β , β'≠β , (β ,β') unordered qβqβ'}|E_β(p₁) - E_β'(p₁)|}° +° _qF(p₁)}

and

{{_{β∈Β ,β'∈Β , β'≠β , (β ,β') unordered q(β)q(β')}|E_β(p₂) - E_β'(p₂)|}° +° _qF(p₂)} =

{{_{β∈Β ,β'∈Β , β'≠β , (β ,β') unordered qβqβ'}|E_β(p₂) - E_β'(p₂)|}° +° _qF(p₂)}.

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)].

Some suitable nonnegative subproblems estimations unification functions follow.

1. The weighted power mean of the (componentwise) subproblems estimations

^tD(p , P) = {Σ_β∈Β q(β)[D(p , P_β)]^t / Σ_β∈Β q(β)}^1/t

where t is a positive number and

Q(P) = Σ_β∈Β q(β)

is the uniquantity [2-5] of quantisystem

P = {_{β∈Β q(β)}P_β}.

2. The weighted power product mean of the (componentwise) subproblems estimations

^tD(P) = {[(Σ_β∈Β q(β)D(P_β))^t - Σ_β∈Β q(β)D^t(P_β)] / [(Σ_β∈Β q(β))^t - Σ_β∈Β q(β)]}^1/t

where t > 1.

2. The weighted power difference mean of the (componentwise) subproblems estimations

^{s , t , u}D(P) = {|[Σ_β∈Β q(β)D^u/s(P_β)]^s - [Σ_β∈Β q(β)D^u/t(P_β)]^t| / |[Σ_β∈Β q(β)]^s - [Σ_β∈Β q(β)]^t|}^1/u

where u and s ≠ t are positive numbers.

3. The weighted geometric mean of the (componentwise) subproblems estimations

D(P) = [Π_β∈Β D^q(β)(P_β)]^{1/Σ_β∈Β q(β)} .

4. The weighted power mean of the moduli (absolute values) of all the pairwise differences of the powers of the (componentwise) subproblems estimations

^{t , u}D(P) = {Σ_{β∈Β ,β'∈Β, β'≠β} q(β)q(β')|D^u(P_β) - D^u(P_β')|^t / Σ_{β∈Β ,β'∈Β, β'≠β} q(β)q(β')}^1/(tu)

where t and u are positive numbers.

5. The weighted power difference mean of the moduli (absolute values) of all the pairwise differences of the powers of the (componentwise) subproblems estimations

^{s , t , u , v , wD}(P) = {|[Σ_{β∈Β , β'∈Β , β'≠β} q(β)q(β')|D^v(P_β) - D^v(P_β')|^u/(vs)]^s - [Σ_{β∈Β , β'∈Β , β'≠β} q(β)q(β')|D^w(P_β) - D^w(P_β')|^u/(wt)]^t| /

|[(Σ_{β∈Β , β'∈Β , β'≠β} q(β)q(β')]^s - [(Σ_{β∈Β , β'∈Β , β'≠β} q(β)q(β')]^t|}^1/u

where s ≠ t , u , v , and w are positive numbers.

6. The weighted power mean of all the groupwise products of the moduli (absolute values) of all the pairwise differences of the powers of the (componentwise) subproblems estimations, e.g., the weighted power mean of all the pairwise products of the moduli (absolute values) of all the pairwise differences of the powers of the (componentwise) subproblems estimations

^{t , u}D(P) = {Σ_{β∈Β , β'∈Β , β'≠β ,β''∈Β , β'''∈Β , β'''≠β'' , (β'', β''')≠(β , β')} q(β)q(β')q(β'')q(β''')[|D^u(P_β) - D^u(P_β')| |D^u(P_β'') - D^u(P_β''')|]^t /

Σ_{β∈Β , β'∈Β , β'≠β ,β''∈Β , β'''∈Β , β'''≠β'' , (β'', β''')≠(β , β')} q(β)q(β')q(β'')q(β''')}^1/(2tu)

where t and u are positive numbers and all the different index pairs (β , β') and (β'', β''') are unordered.

Pseudosolution Distance

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)].

Some suitable nonnegative subproblems estimations unification functions follow.

1. The weighted power mean of the (componentwise) subproblems estimations

^td(p , P) = {Σ_β∈Β q(β)[d(p , P_β)]^t / Σ_β∈Β q(β)}^1/t

where t is a positive number and

Q(P) = Σ_β∈Β q(β)

is the uniquantity [2-5] of quantisystem

P = {_{β∈Β q(β)}P_β}.

2. The weighted power difference mean of the (componentwise) subproblems estimations

^{s , t , u}d(P) = {|[Σ_β∈Β q(β)d^u/s(P_β)]^s - [Σ_β∈Β q(β)d^u/t(P_β)]^t| / |[Σ_β∈Β q(β)]^s - [Σ_β∈Β q(β)]^t|}^1/u

where u and s ≠ t are positive numbers.

3. The weighted geometric mean of the (componentwise) subproblems estimations

d(P) = [Π_β∈Β d^q(β)(P_β)]^{1/Σ_β∈Β q(β)} .

4. The weighted power mean of the moduli (absolute values) of all the pairwise differences of the powers of the (componentwise) subproblems estimations

^{t , u}d(P) = {Σ_{β∈Β ,β'∈Β, β'≠β} q(β)q(β')|d^u(P_β) - d^u(P_β')|^t / Σ_{β∈Β ,β'∈Β, β'≠β} q(β)q(β')}^1/(tu)

where t and u are positive numbers.

5. The weighted power difference mean of the moduli (absolute values) of all the pairwise differences of the powers of the (componentwise) subproblems estimations

^{s , t , u , v , w}d(P) = {|[Σ_{β∈Β , β'∈Β , β'≠β} q(β)q(β')|d^v(P_β) - d^v(P_β')|^u/(vs)]^s - [Σ_{β∈Β , β'∈Β , β'≠β} q(β)q(β')|d^w(P_β) - d^w(P_β')|^u/(wt)]^t| /

|[(Σ_{β∈Β , β'∈Β , β'≠β} q(β)q(β')]^s - [(Σ_{β∈Β , β'∈Β , β'≠β} q(β)q(β')]^t|}^1/u

where s ≠ t , u , v , and w are positive numbers.

6. The weighted power mean of all the groupwise products of the moduli (absolute values) of all the pairwise differences of the powers of the (componentwise) subproblems estimations, e.g., the weighted power mean of all the pairwise products of the moduli (absolute values) of all the pairwise differences of the powers of the (componentwise) subproblems estimations

^{t , u}d(P) = {Σ_{β∈Β , β'∈Β , β'≠β ,β''∈Β , β'''∈Β , β'''≠β'' , (β'', β''')≠(β , β')} q(β)q(β')q(β'')q(β''')[|d^u(P_β) - d^u(P_β')| |d^u(P_β'') - d^u(P_β''')|]^t /

Σ_{β∈Β , β'∈Β , β'≠β ,β''∈Β , β'''∈Β , β'''≠β'' , (β'', β''')≠(β , β')} q(β)q(β')q(β'')q(β''')}^1/(2tu)

where t and u are positive numbers and all the different index pairs (β , β') and (β'', β''') are unordered.

Quantisystem Unierror

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)].

Some suitable nonnegative subproblems estimations unification functions follow.

1. The weighted power mean of the (componentwise) subproblems estimations

^tE(p , P) = {Σ_β∈Β q(β)[E(p , P_β)]^t / Σ_β∈Β q(β)}^1/t

where t is a positive number and

Q(P) = Σ_β∈Β q(β)

is the uniquantity [2-5] of quantisystem

P = {_{β∈Β q(β)}P_β}.

2. The weighted power difference mean of the (componentwise) subproblems estimations

^{s , t , u}E(P) = {|[Σ_β∈Β q(β)E^u/s(P_β)]^s - [Σ_β∈Β q(β)E^u/t(P_β)]^t| / |[Σ_β∈Β q(β)]^s - [Σ_β∈Β q(β)]^t|}^1/u

where u and s ≠ t are positive numbers.

3. The weighted geometric mean of the (componentwise) subproblems estimations

E(P) = [Π_β∈Β E^q(β)(P_β)]^{1/Σ_β∈Β q(β)} .

4. The weighted power mean of the moduli (absolute values) of all the pairwise differences of the powers of the (componentwise) subproblems estimations

^{t , u}E(P) = {Σ_{β∈Β ,β'∈Β, β'≠β} q(β)q(β')|E^u(P_β) - E^u(P_β')|^t / Σ_{β∈Β ,β'∈Β, β'≠β} q(β)q(β')}^1/(tu)

where t and u are positive numbers.

5. The weighted power difference mean of the moduli (absolute values) of all the pairwise differences of the powers of the (componentwise) subproblems estimations

^{s , t , u , v , w}E(P) = {|[Σ_{β∈Β , β'∈Β , β'≠β} q(β)q(β')|E^v(P_β) - E^v(P_β')|^u/(vs)]^s - [Σ_{β∈Β , β'∈Β , β'≠β} q(β)q(β')|E^w(P_β) - E^w(P_β')|^u/(wt)]^t| /

|[(Σ_{β∈Β , β'∈Β , β'≠β} q(β)q(β')]^s - [(Σ_{β∈Β , β'∈Β , β'≠β} q(β)q(β')]^t|}^1/u

where s ≠ t , u , v , and w are positive numbers.

6. The weighted power mean of all the groupwise products of the moduli (absolute values) of all the pairwise differences of the powers of the (componentwise) subproblems estimations, e.g., the weighted power mean of all the pairwise products of the moduli (absolute values) of all the pairwise differences of the powers of the (componentwise) subproblems estimations

^{t , u}E(P) = {Σ_{β∈Β , β'∈Β , β'≠β ,β''∈Β , β'''∈Β , β'''≠β'' , (β'', β''')≠(β , β')} q(β)q(β')q(β'')q(β''')[|E^u(P_β) - E^u(P_β')| |E^u(P_β'') - E^u(P_β''')|]^t /

Σ_{β∈Β , β'∈Β , β'≠β ,β''∈Β , β'''∈Β , β'''≠β'' , (β'', β''')≠(β , β')} q(β)q(β')q(β'')q(β''')}^1/(2tu)

where t and u are positive numbers and all the different index pairs (β , β') and (β'', β''') are unordered.

Pseudosolution Unierror

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)].

Some suitable nonnegative subproblems estimations unification functions follow.

1. The weighted power mean of the (componentwise) subproblems estimations

^te(p , P) = {Σ_β∈Β q(β)[e(p , P_β)]^t / Σ_β∈Β q(β)}^1/t

where t is a positive number and

Q(P) = Σ_β∈Β q(β)

is the uniquantity [2-5] of quantisystem

P = {_{β∈Β q(β)}P_β}.

2. The weighted power difference mean of the (componentwise) subproblems estimations

^{s , t , u}e(P) = {|[Σ_β∈Β q(β)e^u/s(P_β)]^s - [Σ_β∈Β q(β)e^u/t(P_β)]^t| / |[Σ_β∈Β q(β)]^s - [Σ_β∈Β q(β)]^t|}^1/u

where u and s ≠ t are positive numbers.

3. The weighted geometric mean of the (componentwise) subproblems estimations

e(P) = [Π_β∈Β e^q(β)(P_β)]^{1/Σ_β∈Β q(β)} .

4. The weighted power mean of the moduli (absolute values) of all the pairwise differences of the powers of the (componentwise) subproblems estimations

^{t , u}e(P) = {Σ_{β∈Β ,β'∈Β, β'≠β} q(β)q(β')|e^u(P_β) - e^u(P_β')|^t / Σ_{β∈Β ,β'∈Β, β'≠β} q(β)q(β')}^1/(tu)

where t and u are positive numbers.

5. The weighted power difference mean of the moduli (absolute values) of all the pairwise differences of the powers of the (componentwise) subproblems estimations

^{s , t , u , v , w}e(P) = {|[Σ_{β∈Β , β'∈Β , β'≠β} q(β)q(β')|e^v(P_β) - e^v(P_β')|^u/(vs)]^s - [Σ_{β∈Β , β'∈Β , β'≠β} q(β)q(β')|e^w(P_β) - e^w(P_β')|^u/(wt)]^t| /

|[(Σ_{β∈Β , β'∈Β , β'≠β} q(β)q(β')]^s - [(Σ_{β∈Β , β'∈Β , β'≠β} q(β)q(β')]^t|}^1/u

where s ≠ t , u , v , and w are positive numbers.

6. The weighted power mean of all the groupwise products of the moduli (absolute values) of all the pairwise differences of the powers of the (componentwise) subproblems estimations, e.g., the weighted power mean of all the pairwise products of the moduli (absolute values) of all the pairwise differences of the powers of the (componentwise) subproblems estimations

^{t , u}e(P) = {Σ_{β∈Β , β'∈Β , β'≠β ,β''∈Β , β'''∈Β , β'''≠β'' , (β'', β''')≠(β , β')} q(β)q(β')q(β'')q(β''')[|e^u(P_β) - e^u(P_β')| |e^u(P_β'') - e^u(P_β''')|]^t /

Σ_{β∈Β , β'∈Β , β'≠β ,β''∈Β , β'''∈Β , β'''≠β'' , (β'', β''')≠(β , β')} q(β)q(β')q(β'')q(β''')}^1/(2tu)

where t and u are positive numbers and all the different index pairs (β , β') and (β'', β''') are unordered.

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 Distance Center

By solving contradictory (e.g., overdetermined [1]) problems without precise solutions, it is necessary to find the best pseudosolutions, so-called quasisolutions [2-5]. If such a problem is a sets of equations, then their graphs in a Cartesian coordinate system have no point in common but in many cases determine a certain (limited if possible) point set whose center (in some reasonable sense) could be considered as the desired quasisolution. The straightforward basic idea is as follows. If it is impossible to precisely satisfy all the given equations and each point (pseudosolution) gives deviations (e.g., errors), then it is logical to try to equally (uniformly, homogeneously) distribute them among all the given equations. Such an approach corresponds to intuition and leads to the intuitive concept of the center (in some reasonable sense) of that point set.

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 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.

The general center of a general problem depends on a general problem estimator which can be simply a distance. If for the graphs of all the subproblems, there is an inscribed general sphere (usual circumference in the two-dimensional case or usual sphere in the three-dimensional case), then regard its center (so-called incenter [1]) as the proper center (of a general problem) which is naturally the only and is simultaneously the general center of a general problem. Otherwise, define and determine the general center of a general problem via a reasonable and suitable general problem estimator as a criterion. Naturally, there can be many reasonable and suitable general problem estimators and hence many reasonable and suitable general centers of a general problem. To test general problem estimators for their reasonability, adequacy, usability, and suitability, apply them to general problems whose proper centers exist and can be relatively simply defined and determined. In the two-dimensional case, consider (naturally, convex) polygons with existing inscribed circles (so-called incircles [1]) (which is the case by any triangle) and the set of the equations of the straight lines each of which includes a certain side of such a polygon. In the three-dimensional case, there are two possibilities:

1) consider (naturally, convex) polyhedrons with existing incenters (centers of inspheres, i.e., spheres inscribed in all their flat faces) (which is the case by any tetrahedron) and the set of the equations of the planes each of which includes a certain flat face of such a polyhedron;

2) consider (naturally, convex) polyhedrons with existing carcass incenters (centers of carcass inspheres, i.e., spheres inscribed in all their straight edges) (which is the case by any tetrahedron) and the set of the equations of the straight lines each of which includes a certain straight edge of such a polyhedron.

Similarly consider further multidimensional spaces if necessary.

Nota bene: If there is no incenter but there are excenters [1], it is inadmissible to simply replace above the incenter with one of the excenters. The reason is that at an excenter, a general problem estimator takes equal values by all the subproblems but a suitable nonnegative subproblems estimations unification function F can take smaller values at other points than its value at an excenter. For example, consider a circle arc with a relatively small central angle (e.g., π/1800), divide this arc via 9 points into 10 equal parts, and add the both arc endpoints. Build 11 straight lines touching this circle (its tangents) at these 11 points. Compare the power mean distances (by any common power p ≥ 1) both of the circle center (which is here an excenter) and of the arc midpoint (which lies here near the general center of the set of these 11 straight lines) from these 11 straight lines. Now consider the circle radius infinitely increasing. Then the limit of that arc is a straight line segment included in each of these 11 straight lines. The limit of the general center of the set of these 11 straight lines is the midpoint of that segment. The limit of the power mean distance of the general center of the set of these 11 straight lines from these 11 straight lines vanishes. But the power mean distance of the circle center (which is here an excenter) from these 11 straight lines equals the circle radius and infinitely increases together with it. The critical value of the central angle of the arc is namely π . By central angles not greater than π , the excenter keeps this role whereas by central angles greater than π , the excenter becomes the incenter and, naturally, the general centerjumps by precisely half a circle to this incenter (former excenter).

General Unierror Center

The general unierror center of a general problem depends on a general problem unierror as a general problem estimator. If for the graphs of all the subproblems, there is a point at which both all the subproblems unierrors are equal to one another and a general problem unierror as a general problem estimator takes its minimum value, then regard this point (so-called unierror incenter) as the proper unierror center (of a general problem) which is naturally the only and is simultaneously the general unierror center of a general problem. Otherwise, define and determine the general unierror center of a general problem via a reasonable and suitable general problem unierror as a general problem estimator and a criterion. Naturally, there can be many reasonable and suitable general problem unierrors and hence many reasonable and suitable general unierror centers of a general problem. To test general problem unierrors for their reasonability, adequacy, usability, and suitability, apply them to general problems whose proper unierror centers exist and can be relatively simply defined and determined. In the two-dimensional case, consider specially constructed (naturally, convex) polygons with existing unierror incenters (which is the case by any triangle) and the set of the equations of the straight lines each of which includes a certain side of such a polygon. In the three-dimensional case, there are two possibilities:

1) consider (naturally, convex) polyhedrons with existing unierror incenters (with respect to all their flat faces) (which is the case by any tetrahedron) and the set of the equations of the planes each of which includes a certain flat face of such a polyhedron;

2) consider (naturally, convex) polyhedrons with existing carcass unierror incenters (with respect to all their straight edges) (which is the case by any tetrahedron) and the set of the equations of the straight lines each of which includes a certain straight edge of such a polyhedron.

Similarly consider further multidimensional spaces if necessary.

Nota bene: If there is no unierror incenter but there are unierror excenters, it is inadmissible to simply replace above the unierror incenter with one of the unierror excenters. The reason is that at a unierror excenter, a general problem estimator takes equal values by all the subproblems but a suitable nonnegative subproblems estimations unification function F can take smaller values at other points than its value at a unierror excenter.

General problem synthesis theory (GPSyT) in fundamental science of general problem synthesis 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