Ph. D. & Dr. Sc. Lev Gelimson's General Center Theory in Fundamental Science of General Problem Testing

General Center Theory in Fundamental Science of General Problem Testing

© 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), 35

Introduction

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 set 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 the absolute error. And the second power is often insufficient to find realistic point set centers.

General center theory (GCT) in fundamental science of general problem testing is based, in particular, on the following:

1) 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) 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.

General Problem Estimator

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.

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(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.

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).

General Distance Center

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 center jumps 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 an 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 center theory (GCT) in fundamental science of general problem testing is very efficient by solving many urgent (including contradictory) problems.

Acknowledgements to Anatolij Gelimson for our constructive discussions on coordinate system transformation invariances and his very useful remarks.

References

[1] Encyclopaedia of Mathematics. Ed. M. Hazewinkel. Volumes 1 to 10. Kluwer Academic Publ., Dordrecht, 1988-1994

[2] Lev Gelimson. Elastic Mathematics. General Strength Theory. The ”Collegium” International Academy of Sciences Publishers, Munich, 2004

[3] Lev Gelimson. Providing Helicopter Fatigue Strength: Flight Conditions. In: Structural Integrity of Advanced Aircraft and Life Extension for Current Fleets – Lessons Learned in 50 Years After the Comet Accidents, Proceedings of the 23rd ICAF Symposium, Dalle Donne, C. (Ed.), 2005, Hamburg, Vol. II, 405-416

[4] Lev Gelimson. Overmathematics: Fundamental Principles, Theories, Methods, and Laws of Science. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010

[5] Lev Gelimson. General Problem Fundamental Sciences System. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2011