General Problem Estimation Theory in Fundamental Science of General Problem Estimation
by
© Ph. D. & Dr. Sc. Lev Gelimson
Academic Institute for Creating Fundamental Sciences (Munich, Germany)
Mathematical Journal
of the "Collegium" All World Academy of Sciences
Munich (Germany)
11(2011), 39
Introduction. General Problem
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 setting theory (GPST) in fundamental science of general problem essence defines a general quantitative mathematical problem, or simply a general problem, to be a quantisystem [2-5] (former hypersystem)
w(λ)Rλ[φ∈Φ fφ[ω∈Ω zω]] (λ∈Λ)
of known relations Rλ over indexed unknown functions (dependent variables), or simply unknowns, fφ of indexed independent known variables zω , all of them belonging to their possibly individual vector spaces, in their initial forms without any further transformations
where
Rλ is a known relation with index λ from an index set Λ ;
fφ is an unknown function (dependent variable) with index φ from an index set Φ ;
zω is a known independent variable with index ω from an index set Ω ;
[ω∈Ω zω]
is a set of indexed elements zω ;
w(λ) is the own, or individual, quantity (former hyperquantity) [2-5] as a weight of the relation with index λ .
When replacing all the unknowns (unknown functions) with their possible ”values” (some known functions), or pseudosolutions [2-5], the above quantisystem of relations is transformed into the corresponding quantisystem of formal functional relations without any unknowns. To conserve the quantisystem form, let us use the same designations fφ for these known functions, too. This known quantisystem can be further estimated both qualitatively and (or) quantitatively discretely (e.g., either true or false) or continuously (e.g., via absolute and relative errors [1], unierrors, reserves, reliabilities, and risks [2-5]). If a quantisystem of pseudosolutions to a general problem transforms this problem into a known true quantisystem, then this quantisystem of pseudosolutions is a quantisystem of solutions to this general problem.
In quantitative mathematical problems, namely equations and inequations are the most typical relations.
Let us define a general quantitative mathematical pure equations problem, or simply a general pure equations problem, to be a quantiset [2-5] (former hyperset) of equations over indexed functions (dependent variables) fφ of indexed independent variables zω , 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)(Lλ[φ∈Φ fφ[ω∈Ω zω]] = 0) (λ∈Λ)
where
Lλ is an operator with index λ from an index set Λ ;
fφ is a function (dependent variable) with index φ from an index set Φ ;
zω is an independent variable with index ω from an index set Ω ;
[ω∈Ω zω]
is a set of indexed elements zω ;
w(λ) is the own, or individual, quantity (former hyperquantity) [2-5] as a weight of the equation with index λ .
When replacing all the unknowns (unknown functions) with their possible ”values” (some known functions), the above quantiset of equations is transformed into the corresponding quantiset of formal functional equalities. To conserve the quantiset form, let us use for these known functions the same designations fφ .
Let us define a general quantitative mathematical pure inequations problem, or simply a general pure inequations problem, to be a quantiset [2-5] (former hyperset) of equations over indexed functions (dependent variables) fφ of indexed independent variables zω , all of them belonging to their possibly individual vector spaces. We may gather (in the left-hand sides of the inequations) all the functions available in the initial forms without any further transformations. The unique natural exception is changing the signs of the functions by moving them to the other sides of the same inequations. The quantiset can be brought to the form
q(λ)(Lλ[φ∈Φ fφ[ω∈Ω zω]] Rλ 0) (λ∈Λ)
where
Lλ is an operator with index λ from an index set Λ ;
Rλ is an inequality relation (e.g., ≈ , ∼ , ≠ , < , > , ≤ , ≥) with index λ from an index set Λ ;
fφ is a function (dependent variable) with index φ from an index set Φ ;
zω is an independent variable with index ω from an index set Ω ;
[ω∈Ω zω]
is a set of indexed elements zω ;
q(λ) is the own, or individual, quantity (former hyperquantity) [2-5] as a weight of the inequation with index λ .
When replacing all the unknowns (unknown functions) with their possible ”values” (some known functions), the above quantiset of inequations is transformed into the corresponding quantiset of formal functional inequalities. To conserve the quantiset form, let us use for these known functions the same designations fφ .
By using unstrict inequality relations such as ≈ , ∼ , ≤ , ≥ , etc. only, a general pure inequations problem clearly further generalizes a general pure equations 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 ryy’(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 Rx(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 estimation theory (GPET) in fundamental science of general problem estimation naturally deals with building suitable nonnegative subproblems estimations unification functions to estimate pseudosolutions to a general problem.
Subproblems Estimations Unification Function
Suppose that a general problem P consists of separate general subproblems (e.g., relations) Pβ with their own positive quantities q(β)
P = {β∈Β q(β)Pβ}
(where index β belongs to index set Β)
and there are nonnegative estimators Eβ [2-5]
Eβ(Pβ) ≥ 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(Pβ) ≥ 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
Eβ(Pβ) ≥ 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)Eβ(Pβ)] ≥ 0.
Some suitable nonnegative subproblems estimations unification functions follow.
1. The weighted power mean of the (componentwise) subproblems estimations
tE(P) = {Σβ∈Β q(β)[Eβ(Pβ)]t / Σβ∈Β q(β)}1/t
where t is a positive number and
Q(P) = Σβ∈Β q(β)
is the uniquantity [2-5] of quantisystem
P = {β∈Β q(β)Pβ}.
2. The weighted power product mean of the (componentwise) subproblems estimations
tE(P) = {[(Σβ∈Β q(β)Eβ(Pβ))t - Σβ∈Β q(β)Eβt(Pβ)] / [(Σβ∈Β q(β))t - Σβ∈Β q(β)]}1/t
where t > 1.
3. The weighted power difference mean of the (componentwise) subproblems estimations
s , t , uE(P) = {|[Σβ∈Β q(β)Eβu/s(Pβ)]s - [Σβ∈Β q(β)Eβu/t(Pβ)]t| / |[Σβ∈Β q(β)]s - [Σβ∈Β q(β)]t|}1/u
where u and s ≠ t are positive numbers.
4. The weighted geometric mean of the (componentwise) subproblems estimations
E(P) = [Πβ∈Β Eβq(β)(Pβ)]1/Σβ∈Β q(β) .
5. The weighted power mean of the moduli (absolute values) of all the pairwise differences of the powers of the (componentwise) subproblems estimations
t , uE(P) = {Σβ∈Β ,β'∈Β, β'≠β q(β)q(β')|Eβu(Pβ) - Eβ'u(Pβ')|t / Σβ∈Β ,β'∈Β, β'≠β q(β)q(β')}1/(tu)
where t and u are positive numbers.
6. 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 , wE(P) = {|[Σβ∈Β , β'∈Β , β'≠β q(β)q(β')|Eβv(Pβ) - Eβ'v(Pβ')|u/(vs)]s - [Σβ∈Β , β'∈Β , β'≠β q(β)q(β')|Eβw(Pβ) - Eβ'w(Pβ')|u/(wt)]t| /
|[(Σβ∈Β , β'∈Β , β'≠β q(β)q(β')]s - [(Σβ∈Β , β'∈Β , β'≠β q(β)q(β')]t|}1/u
where s ≠ t , u , v , and w are positive numbers.
7. 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 , uE(P) = {Σβ∈Β , β'∈Β , β'≠β ,β''∈Β , β'''∈Β , β'''≠β'' , (β'', β''')≠(β , β') q(β)q(β')q(β'')q(β''')[|Eβu(Pβ) - Eβ'u(Pβ')| |Eβ''u(Pβ'') - Eβ'''u(Pβ''')|]t /
Σβ∈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 Ej (j = 1, 2, ... , n), the above weighted power mean of the (componentwise) subproblems estimations
tE(P) = {Σβ∈Β q(β)[Eβ(Pβ)]t / Σβ∈Β q(β)}1/t
(where t is a positive number)
simply gives
tE(P) = (Σj=1n Ejt / n)1/t .
By infinitely increasing t we obtain as the limiting case
E(P) = max{j=1n Ej}.
Generalizing this for any (possibly infinite) set of relations with using the least upper bound sup gives
E(P) = sup{β∈Β Eβ}.
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 Rj},
R(P) = inf{β∈Β Rβ}
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 Eβ}
for any nonnegative estimator E , e.g., distances, unierrors, etc. and
R(P) = sup{β∈Β Rβ}
for reserve R opposite to unierror E (R = - E).
Difference Modulus Quantibound Estimator
Now consider any quantiset
{α∈Α q(α)aα}° = {α∈Α qαaα}°
with positive individual quantity q(α) = qα of each element aα with index α from index set Α . Additionally consider any such quantiset
{β∈Β q(β)bβ}° = {β∈Β qβbβ}°
with positive individual quantity q(β) = qβ of each element bβ with index β from index set Β . If there exist all the pairwise mutual element products aαbβ , then consider both elementwise and quantity-wise product quantiset
{α∈Α q(α)aα}° {β∈Β q(β)bβ}° = {α∈Α ,β∈Β q(α)q(β)(aαbβ)}° = {α∈Α ,β∈Β qαqβ(aαbβ)}°
of the above two quantisets. In particular, square quantiset
{α∈Α q(α)aα}°2 = {α∈Α ,α'∈Α q(α)q(α')(aαaα')}° =
{α∈Α q2(α)aα2}° + {α∈Α ,α'∈Α , α'≠α , (α ,α') ordered q(α)q(α')(aαaα')}° =
{α∈Α q2(α)aα2}° + {α∈Α ,α'∈Α , α'≠α , (α ,α') unordered 2q(α)q(α')(aαaα')}°.
Replacing all the pairwise mutual element products aαbβ with the moduli (absolute values) |aα - bβ| of all the pairwise mutual element differences and keeping (which is logical) the above products
q(α)q(β) = qαqβ
of individual element quantities, we obtain elementwise difference modulus and quantity-wise product quantiset
|{α∈Α q(α)aα}° *-° {β∈Β q(β)bβ}°| = {α∈Α ,β∈Β q(α)q(β)|aα - bβ|}° = {α∈Α ,β∈Β qαqβ|aα - bβ|}°
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)aα}° *- {α∈Α q(α)aα}°| = {α∈Α ,α'∈Α q(α)q(α')|aα - aα'|}° = {α∈Α ,α'∈Α qαqα'|aα - aα'|}° =
{α∈Α q2(α)0}° + {α∈Α ,α'∈Α , α'≠α , (α ,α') ordered q(α)q(α')|aα - aα'|}° =
{α∈Α q2α0}° + {α∈Α ,α'∈Α , α'≠α , (α ,α') ordered qαqα'|aα - aα'|}° =
{α∈Α q2(α)0}° + {α∈Α ,α'∈Α , α'≠α , (α ,α') unordered 2q(α)q(α')|aα - aα'|}° =
{α∈Α q2α0}° + {α∈Α ,α'∈Α , α'≠α , (α ,α') unordered 2qαqα'|aα - aα'|}°.
These definitions satisfy the conservation law in the sense that if any elements among
{α∈Α q(α)aα}° = {α∈Α qαaα}°,
{β∈Β q(β)bβ}° = {β∈Β qβbβ}°
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)Eβ}° = {β∈Β qβEβ}°
itself but to subproblemwise estimations elementwise difference modulus and quantity-wise product quantiset
|{β∈Β q(β)Eβ}° *-° {β∈Β q(β)Eβ}°| = {β∈Β ,α'β∈Β q(β)q(β')|Eβ - Eβ'|}° = {β∈Β ,β∈Β qβqβ'|Eβ - Eβ'|}° =
{β∈Β q2(β)0}° + {β∈Β ,β'∈Β , β'≠β , (β ,β') ordered q(β)q(β')|Eβ - Eβ'|}° =
{β∈Β q2β0}° + {β∈Β ,β'∈Β , β'≠β , (β ,β') ordered qβqβ'|Eβ - Eβ'|}° =
{β∈Β q2(β)0}° + {β∈Β ,β'∈Β , β'≠β , (β ,β') unordered 2q(β)q(β')|Eβ - Eβ'|}° =
{β∈Β q2β0}° + {β∈Β ,β'∈Β , β'≠β , (β ,β') unordered 2qβqβ'|Eβ - Eβ'|}°.
Then we obtain
E(P) = inf|{β∈Β q(β)Eβ}° *-° {β∈Β q(β)Eβ}°| = inf{β∈Β ,α'β∈Β q(β)q(β')|Eβ - Eβ'|}° = inf{β∈Β ,β∈Β qβqβ'|Eβ - Eβ'|}° =
inf{{β∈Β q2(β)0}° + {β∈Β ,β'∈Β , β'≠β , (β ,β') ordered q(β)q(β')|Eβ - Eβ'|}°} =
inf{{β∈Β q2β0}° + {β∈Β ,β'∈Β , β'≠β , (β ,β') ordered qβqβ'|Eβ - Eβ'|}°} =
inf{{β∈Β q2(β)0}° + {β∈Β ,β'∈Β , β'≠β , (β ,β') unordered 2q(β)q(β')|Eβ - Eβ'|}°} =
inf{{β∈Β q2β0}° + {β∈Β ,β'∈Β , β'≠β , (β ,β') unordered 2qβqβ'|Eβ - Eβ'|}°}.
Apply this estimator E to compare any two pseudosolutions p1 and p2 to any general problem
P = {β∈Β q(β)Pβ}.
The estimation quantiset for pseudosolution p1 is
{β∈Β q(β)Eβ(p1)}° = {β∈Β qβEβ(p1)}°.
The estimation quantiset for pseudosolution p2 is
{β∈Β q(β)Eβ(p2)}° = {β∈Β qβEβ(p2)}°.
The total estimation for pseudosolution p1 is
E(p1 , P) = inf|{β∈Β q(β)Eβ(p1)}° *-° {β∈Β q(β)Eβ(p1)}°| = inf{β∈Β ,α'β∈Β q(β)q(β')|Eβ(p1) - Eβ'(p1)|}° = inf{β∈Β ,β∈Β qβqβ'|Eβ(p1) - Eβ'(p1)|}° =
inf{{β∈Β q2(β)0}° + {β∈Β ,β'∈Β , β'≠β , (β ,β') ordered q(β)q(β')|Eβ(p1) - Eβ'(p1)|}°} =
inf{{β∈Β q2β0}° + {β∈Β ,β'∈Β , β'≠β , (β ,β') ordered qβqβ'|Eβ(p1) - Eβ'(p1)|}°} =
inf{{β∈Β q2(β)0}° + {β∈Β ,β'∈Β , β'≠β , (β ,β') unordered 2q(β)q(β')|Eβ(p1) - Eβ'(p1)|}°} =
inf{{β∈Β q2β0}° + {β∈Β ,β'∈Β , β'≠β , (β ,β') unordered 2qβqβ'|Eβ(p1) - Eβ'(p1)|}°}.
The total estimation for pseudosolution p2 is
E(p2 , P) = inf|{β∈Β q(β)Eβ(p2)}° *-° {β∈Β q(β)Eβ(p2)}°| = inf{β∈Β ,α'β∈Β q(β)q(β')|Eβ(p2) - Eβ'(p2)|}° = inf{β∈Β ,β∈Β qβqβ'|Eβ(p2) - Eβ'(p2)|}° =
inf{{β∈Β q2(β)0}° + {β∈Β ,β'∈Β , β'≠β , (β ,β') ordered q(β)q(β')|Eβ(p2) - Eβ'(p2)|}°} =
inf{{β∈Β q2β0}° + {β∈Β ,β'∈Β , β'≠β , (β ,β') ordered qβqβ'|Eβ(p2) - Eβ'(p2)|}°} =
inf{{β∈Β q2(β)0}° + {β∈Β ,β'∈Β , β'≠β , (β ,β') unordered 2q(β)q(β')|Eβ(p2) - Eβ'(p2)|}°} =
inf{{β∈Β q2β0}° + {β∈Β ,β'∈Β , β'≠β , (β ,β') unordered 2qβqβ'|Eβ(p2) - Eβ'(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 q2β0}°
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')|Eβ - Eβ'|}° =
inf{β∈Β ,β'∈Β , β'≠β , (β ,β') unordered qβqβ'|Eβ - Eβ'|}°
and compare preliminarily reduced quantisets
{β∈Β ,β'∈Β , β'≠β , (β ,β') unordered q(β)q(β')|Eβ(p1) - Eβ'(p1)|}° =
{β∈Β ,β'∈Β , β'≠β , (β ,β') unordered qβqβ'|Eβ(p1) - Eβ'(p1)|}°
and
{β∈Β ,β'∈Β , β'≠β , (β ,β') unordered q(β)q(β')|Eβ(p2) - Eβ'(p2)|}° =
{β∈Β ,β'∈Β , β'≠β , (β ,β') unordered qβqβ'|Eβ(p2) - Eβ'(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 = Σβ∈Β Eβ .
Then we obtain supplemented complete estimator
E(P) = inf{|{β∈Β q(β)Eβ}° *-° {β∈Β q(β)Eβ}°| +° qF} =
inf{{β∈Β ,α'β∈Β q(β)q(β')|Eβ - Eβ'|}° +° qF} =
inf{{β∈Β ,β∈Β qβqβ'|Eβ - Eβ'|}° +° qF} =
inf{{β∈Β q2(β)0}° + {β∈Β ,β'∈Β , β'≠β , (β ,β') ordered q(β)q(β')|Eβ - Eβ'|}° +° qF} =
inf{{β∈Β q2β0}° + {β∈Β ,β'∈Β , β'≠β , (β ,β') ordered qβqβ'|Eβ - Eβ'|}° +° qF} =
inf{{β∈Β q2(β)0}° + {β∈Β ,β'∈Β , β'≠β , (β ,β') unordered 2q(β)q(β')|Eβ - Eβ'|}° +° qF} =
inf{{β∈Β q2β0}° + {β∈Β ,β'∈Β , β'≠β , (β ,β') unordered 2qβqβ'|Eβ - Eβ'|}° +° qF}.
The total estimation for pseudosolution p1 is
E(p1 , P) = inf{|{β∈Β q(β)Eβ(p1)}° *- {β∈Β q(β)Eβ(p1)}°| +° qF(p1)} =
inf{{β∈Β ,α'β∈Β q(β)q(β')|Eβ(p1) - Eβ'(p1)|}° +° qF(p1)} =
inf{{β∈Β ,β∈Β qβqβ'|Eβ(p1) - Eβ'(p1)|}° +° qF(p1)} =
inf{{β∈Β q2(β)0}° + {β∈Β ,β'∈Β , β'≠β , (β ,β') ordered q(β)q(β')|Eβ(p1) - Eβ'(p1)|}° +° qF(p1)} =
inf{{β∈Β q2β0}° + {β∈Β ,β'∈Β , β'≠β , (β ,β') ordered qβqβ'|Eβ(p1) - Eβ'(p1)|}° +° qF(p1)} =
inf{{β∈Β q2(β)0}° + {β∈Β ,β'∈Β , β'≠β , (β ,β') unordered 2q(β)q(β')|Eβ(p1) - Eβ'(p1)|}° +° qF(p1)} =
inf{{β∈Β q2β0}° + {β∈Β ,β'∈Β , β'≠β , (β ,β') unordered 2qβqβ'|Eβ(p1) - Eβ'(p1)|}° +° qF(p1)}.
The total estimation for pseudosolution p2 is
E(p2 , P) = inf{|{β∈Β q(β)Eβ(p2)}° *- {β∈Β q(β)Eβ(p2)}°| +° qF(p2)} =
inf{{β∈Β ,α'β∈Β q(β)q(β')|Eβ(p2) - Eβ'(p2)|}° +° qF(p2)} =
inf{{β∈Β ,β∈Β qβqβ'|Eβ(p2) - Eβ'(p2)|}° +° qF(p2)} =
inf{{β∈Β q2(β)0}° + {β∈Β ,β'∈Β , β'≠β , (β ,β') ordered q(β)q(β')|Eβ(p2) - Eβ'(p2)|}° +° qF(p2)} =
inf{{β∈Β q2β0}° + {β∈Β ,β'∈Β , β'≠β , (β ,β') ordered qβqβ'|Eβ(p2) - Eβ'(p2)|}° +° qF(p2)} =
inf{{β∈Β q2(β)0}° + {β∈Β ,β'∈Β , β'≠β , (β ,β') unordered 2q(β)q(β')|Eβ(p2) - Eβ'(p2)|}° +° qF(p2)} =
inf{{β∈Β q2β0}° + {β∈Β ,β'∈Β , β'≠β , (β ,β') unordered 2qβqβ'|Eβ(p2) - Eβ'(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')|Eβ - Eβ'|}° +° qF} =
inf{{β∈Β ,β'∈Β , β'≠β , (β ,β') unordered qβqβ'|Eβ - Eβ'|}° +° qF}.
Compare preliminarily reduced quantisets
{{β∈Β ,β'∈Β , β'≠β , (β ,β') unordered q(β)q(β')|Eβ(p1) - Eβ'(p1)|}° +° qF(p1)} =
{{β∈Β ,β'∈Β , β'≠β , (β ,β') unordered qβqβ'|Eβ(p1) - Eβ'(p1)|}° +° qF(p1)}
and
{{β∈Β ,β'∈Β , β'≠β , (β ,β') unordered q(β)q(β')|Eβ(p2) - Eβ'(p2)|}° +° qF(p2)} =
{{β∈Β ,β'∈Β , β'≠β , (β ,β') unordered qβqβ'|Eβ(p2) - Eβ'(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 Pβ with their own positive quantities q(β)
P = {β∈Β q(β)Pβ}
(where index β belongs to index set Β),
then consider each relation Pβ of them as a separate general subproblem in a general problem P , define and determine (componentwise) quantisystem relation distance
D(p , Pβ) = inf{α∈Α d[Pβ(p), P'βα]}.
Finally, define and determine (total) quantisystem distance as a suitable nonnegative function F of all D(p , Pβ) (β∈Β) with the same own quantities q(β):
D(p , P) = F[β∈Β q(β)D(p , Pβ)].
Some suitable nonnegative subproblems estimations unification functions follow.
1. The weighted power mean of the (componentwise) subproblems estimations
tD(p , P) = {Σβ∈Β q(β)[D(p , Pβ)]t / Σβ∈Β q(β)}1/t
where t is a positive number and
Q(P) = Σβ∈Β q(β)
is the uniquantity [2-5] of quantisystem
P = {β∈Β q(β)Pβ}.
2. The weighted power product mean of the (componentwise) subproblems estimations
tD(P) = {[(Σβ∈Β q(β)D(Pβ))t - Σβ∈Β q(β)Dt(Pβ)] / [(Σβ∈Β q(β))t - Σβ∈Β q(β)]}1/t
where t > 1.
2. The weighted power difference mean of the (componentwise) subproblems estimations
s , t , uD(P) = {|[Σβ∈Β q(β)Du/s(Pβ)]s - [Σβ∈Β q(β)Du/t(Pβ)]t| / |[Σβ∈Β q(β)]s - [Σβ∈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) = [Πβ∈Β Dq(β)(Pβ)]1/Σβ∈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 , uD(P) = {Σβ∈Β ,β'∈Β, β'≠β q(β)q(β')|Du(Pβ) - Du(Pβ')|t / Σβ∈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) = {|[Σβ∈Β , β'∈Β , β'≠β q(β)q(β')|Dv(Pβ) - Dv(Pβ')|u/(vs)]s - [Σβ∈Β , β'∈Β , β'≠β q(β)q(β')|Dw(Pβ) - Dw(Pβ')|u/(wt)]t| /
|[(Σβ∈Β , β'∈Β , β'≠β q(β)q(β')]s - [(Σβ∈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 , uD(P) = {Σβ∈Β , β'∈Β , β'≠β ,β''∈Β , β'''∈Β , β'''≠β'' , (β'', β''')≠(β , β') q(β)q(β')q(β'')q(β''')[|Du(Pβ) - Du(Pβ')| |Du(Pβ'') - Du(Pβ''')|]t /
Σβ∈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 sα}
(where index α belongs to index set Α)
of all the solutions sα to general problem P , and introduce a nonnegative distance
d(p , sα)
between pseudosolution p and each solution sα and (using the greatest lower bound inf) simply take
d(p , P) = inf{α∈Α d(p , sα)}.
If (which is typical) a general problem P consists of separate exactly solvable relations Pβ with their own positive quantities q(β)
P = {β∈Β q(β)Pβ}
and their relations solutions sets
{α∈Α sβα} (β∈Β)
(where index β belongs to index set Β),
then consider each relation Pβ of them as a separate general subproblem in a general problem P , define and determine (componentwise) pseudosolution relation distance
d(p , Pβ) = inf{α∈Α d(p , sβα)} (β∈Β).
Finally, define and determine the (total) pseudosolution distance as a suitable nonnegative function f of all d(p , Pβ) (β∈Β) with the same own quantities q(β):
d(p , P) = f[β∈Β q(β)d(p , Pβ)].
Some suitable nonnegative subproblems estimations unification functions follow.
1. The weighted power mean of the (componentwise) subproblems estimations
td(p , P) = {Σβ∈Β q(β)[d(p , Pβ)]t / Σβ∈Β q(β)}1/t
where t is a positive number and
Q(P) = Σβ∈Β q(β)
is the uniquantity [2-5] of quantisystem
P = {β∈Β q(β)Pβ}.
2. The weighted power difference mean of the (componentwise) subproblems estimations
s , t , ud(P) = {|[Σβ∈Β q(β)du/s(Pβ)]s - [Σβ∈Β q(β)du/t(Pβ)]t| / |[Σβ∈Β q(β)]s - [Σβ∈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) = [Πβ∈Β dq(β)(Pβ)]1/Σβ∈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 , ud(P) = {Σβ∈Β ,β'∈Β, β'≠β q(β)q(β')|du(Pβ) - du(Pβ')|t / Σβ∈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) = {|[Σβ∈Β , β'∈Β , β'≠β q(β)q(β')|dv(Pβ) - dv(Pβ')|u/(vs)]s - [Σβ∈Β , β'∈Β , β'≠β q(β)q(β')|dw(Pβ) - dw(Pβ')|u/(wt)]t| /
|[(Σβ∈Β , β'∈Β , β'≠β q(β)q(β')]s - [(Σβ∈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 , ud(P) = {Σβ∈Β , β'∈Β , β'≠β ,β''∈Β , β'''∈Β , β'''≠β'' , (β'', β''')≠(β , β') q(β)q(β')q(β'')q(β''')[|du(Pβ) - du(Pβ')| |du(Pβ'') - du(Pβ''')|]t /
Σβ∈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 Pβ with their own positive quantities q(β)
P = {β∈Β q(β)Pβ}
(where index β belongs to index set Β),
then consider each relation Pβ of them as a separate general subproblem in a general problem P , define and determine (componentwise) quantisystem relation unierror
E(p , Pβ) = inf{α∈Α E[P'βα , Pβ(p)]}.
Finally, define and determine the (total) quantisystem unierror as a suitable nonnegative function F of all E(p , Pβ) (β∈Β) with the same own quantities q(β):
E(p , P) = F[β∈Β q(β)E(p , Pβ)].
Some suitable nonnegative subproblems estimations unification functions follow.
1. The weighted power mean of the (componentwise) subproblems estimations
tE(p , P) = {Σβ∈Β q(β)[E(p , Pβ)]t / Σβ∈Β q(β)}1/t
where t is a positive number and
Q(P) = Σβ∈Β q(β)
is the uniquantity [2-5] of quantisystem
P = {β∈Β q(β)Pβ}.
2. The weighted power difference mean of the (componentwise) subproblems estimations
s , t , uE(P) = {|[Σβ∈Β q(β)Eu/s(Pβ)]s - [Σβ∈Β q(β)Eu/t(Pβ)]t| / |[Σβ∈Β q(β)]s - [Σβ∈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) = [Πβ∈Β Eq(β)(Pβ)]1/Σβ∈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 , uE(P) = {Σβ∈Β ,β'∈Β, β'≠β q(β)q(β')|Eu(Pβ) - Eu(Pβ')|t / Σβ∈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 , wE(P) = {|[Σβ∈Β , β'∈Β , β'≠β q(β)q(β')|Ev(Pβ) - Ev(Pβ')|u/(vs)]s - [Σβ∈Β , β'∈Β , β'≠β q(β)q(β')|Ew(Pβ) - Ew(Pβ')|u/(wt)]t| /
|[(Σβ∈Β , β'∈Β , β'≠β q(β)q(β')]s - [(Σβ∈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 , uE(P) = {Σβ∈Β , β'∈Β , β'≠β ,β''∈Β , β'''∈Β , β'''≠β'' , (β'', β''')≠(β , β') q(β)q(β')q(β'')q(β''')[|Eu(Pβ) - Eu(Pβ')| |Eu(Pβ'') - Eu(Pβ''')|]t /
Σβ∈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 sα}
(where index α belongs to index set Α)
of all the solutions sα to general problem P , and introduce a nonnegative unierror
e(p , sα)
of pseudosolution p with respect to solution sα and (using the greatest lower bound inf) simply take
e(p , P) = inf{α∈Α e(p , sα)}.
If (which is typical) a general problem P consists of separate exactly solvable relations Pβ with their own positive quantities q(β)
P = {β∈Β q(β)Pβ}
and their relations solutions sets
{α∈Α sβα} (β∈Β)
(where index β belongs to index set Β),
then consider each relation Pβ of them as a separate general subproblem in a general problem P , define and determine (componentwise) pseudosolution relation unierror
e(p , Pβ) = inf{α∈Α e(p , sβα)} (β∈Β).
Finally, define and determine the (total) pseudosolution unierror as a suitable nonnegative function f of all d(p , Pβ) (β∈Β) with the same own quantities q(β):
e(p , P) = f[β∈Β q(β)e(p , Pβ)].
Some suitable nonnegative subproblems estimations unification functions follow.
1. The weighted power mean of the (componentwise) subproblems estimations
te(p , P) = {Σβ∈Β q(β)[e(p , Pβ)]t / Σβ∈Β q(β)}1/t
where t is a positive number and
Q(P) = Σβ∈Β q(β)
is the uniquantity [2-5] of quantisystem
P = {β∈Β q(β)Pβ}.
2. The weighted power difference mean of the (componentwise) subproblems estimations
s , t , ue(P) = {|[Σβ∈Β q(β)eu/s(Pβ)]s - [Σβ∈Β q(β)eu/t(Pβ)]t| / |[Σβ∈Β q(β)]s - [Σβ∈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) = [Πβ∈Β eq(β)(Pβ)]1/Σβ∈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 , ue(P) = {Σβ∈Β ,β'∈Β, β'≠β q(β)q(β')|eu(Pβ) - eu(Pβ')|t / Σβ∈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 , we(P) = {|[Σβ∈Β , β'∈Β , β'≠β q(β)q(β')|ev(Pβ) - ev(Pβ')|u/(vs)]s - [Σβ∈Β , β'∈Β , β'≠β q(β)q(β')|ew(Pβ) - ew(Pβ')|u/(wt)]t| /
|[(Σβ∈Β , β'∈Β , β'≠β q(β)q(β')]s - [(Σβ∈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 , ue(P) = {Σβ∈Β , β'∈Β , β'≠β ,β''∈Β , β'''∈Β , β'''≠β'' , (β'', β''')≠(β , β') q(β)q(β')q(β'')q(β''')[|eu(Pβ) - eu(Pβ')| |eu(Pβ'') - eu(Pβ''')|]t /
Σβ∈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 problem estimation theory (GPET) in fundamental science of general problem estimation 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