Distribution Theory in Fundamental Science of General Problem Testing

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), 34

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 best (in some reasonable sense) distribution of the total deviation (in some reasonable sense) of a contradictory system (e.g., a contradictory problem).

Nota bene: Such a total deviation (in some reasonable sense) of a contradictory system (e.g., a contradictory problem) is its contradictoriness (contrariness, inconsistency, inconsistence, discrepancy) measure [2-5].

To provide sich equal (uniform, homogeneous) distribution, we need its criteria whose universality is their great advantage. Therefore, to begin with, consider distributing a given positive number S as a sum between addends a and b whose number n = 2 is also given. This seems to be the simplest nontrivial (due to n ≠ 1) distribution problem.

Nota bene: Such a distribution problem has to be consistent.

The same problem also allows other problem settings equivalent to the above problem setting in the sense that their results coincide. But such problem settings can be contradictory (inconsistent).

Consider, e.g., the problem of best approximating any two given points with namely one point. If those two given points coincide, then, naturally, the desired (required) best point coincides with them and obviously provides the best approximation to those two (and any number of) coinciding points. Such a problem is consistent. Otherwise (if those two given points are distinct), by any reasonable best approximation criterion, we have to obtain namely the midpoint of the segment connecting the both given points as the best approximation to them, which corresponds to intuition. Such a problem is contradictory (inconsistent).

Nota bene: Coinciding the midpoint of the segment connecting the both given points with the best approximation to them is a metacriterion of reasonable best approximation criteria. And we may regard the existing unique reasonable best approximation (problem quasisolution) as the precise best approximation (problem quasisolution).

Distribution theory in fundamental science of general problem testing deeply investigates simple distribution problems to find and test both reasonable best approximation criteria and their metacriteria with the aim to further extend and apply them to much more complicated general problems if possible and admissible.

In classical mathematics [1], to solve contradictory (inconsistent) problems, practically the only possibility is to use the least square method (LSM) [1] by Legendre and Gauss applicable to finite overdetermined sets of equations. This method minimizes the sum of the squares of the differences of the both parts of all the equations, or, equivalently, the sum of the squares of the absolute errors [1] of all the equations. Overmathematics [2-4] and the system of fundamental sciences on general problems [5] have discovered many principal shortcomings [2-5] of this method with no objective sense of its result.

Nota bene: The second power is the only providing simplest possible namely pure analytic formulae and and is the most suitable to deal with. But overmathematics [2-4] and the system of fundamental sciences on general problems [5] show such a suitability usually provides no adequacy.

In the problem of best approximating any two given points with namely one point, the least square method (LSM) always gives correct results only if those two given points coincide. Otherwise (if those two given points are distinct), the least square method (LSM) gives either correct or totally inadequate results dependently on mathematically modeling this problem. Namely, the result is not invariant by equivalent transformations of a problem.

Example. Let us approximate two given points 1 and 2 on a number axis with namely one point x . For three equivalent sets of two equations

x = 1,

x = 2;

10x = 10,

x = 2;

x = 1,

10x = 20,

the method returns

x = 3/2,

x = 102/101,

x = 201/101,

respectively. The first result is correct whereas the both remaining results (close to the solution to an equation with greater factors) are totally inadequate.

To always obtain correct results only (which satisfy the midpoint metacriterion) in solving the problem of best approximating any two given points (either coinciding or distinct) with namely one point, we may use, e.g., the following simple reasonable best approximation criteria building their systems.

Average (Arithmetic Mean) Criteria System

A1. Directly taking the point whose coordinates in a Cartesian coordinate system are the half-sums of the same coordinates of the both given points in a Cartesian coordinate system. This trivial criterion holds for two points and a straight line segment connecting them only.

A2. Directly taking the midpoint of the segment connecting the both given points. This trivial criterion generalizes A1 for any curvilinear segment connecting them with an existing length coordinate and allows additional conditions such as using curves lying in a certain surface, etc.

A3. Directly taking the point whose coordinates in a Cartesian coordinate system are the arithmetic mean values of the same coordinates of all the given points in a Cartesian coordinate system. In classical mathematics [1], this trivial criterion generalizes A1 for any at most countable set of points not necessarily lying on one straight line or in one plane. In overmathematics [2-4] and the system of fundamental sciences on general problems [5], this trivial criterion further generalizes A1 for any infinite (also uncountable) set of points not necessarily lying on one straight line or in one plane.

A4. Directly taking the point whose coordinates in a Cartesian coordinate system are the weighted arithmetic mean values of the same coordinates of all the given points in a Cartesian coordinate system. This trivial criterion generalizes A3 for any quantities (as weights) of the given points. In overmathematics [2-4] and the system of fundamental sciences on general problems [5], this trivial criterion further generalizes A3 for any infinite (also uncountable) quantiset of points.

A5. Directly taking the object which is the weighted arithmetic mean values of all the given objects. This trivial criterion generalizes A4 for anyobjects. In overmathematics [2-4] and the system of fundamental sciences on general problems [5], this trivial criterion further generalizes A4 for any infinite (also uncountable) quantiset of objects.

Nota bene: Average (arithmetic mean) criteria are natural and give centers of gravity (length, area, volume, generally uniquantity [2-5]) which are invariant by coordinate system translation and rotation as well as by any coordinate axes units variation. These criteria give results always adequate only if we determine the centers namely of the objects to be approximated with their centers. For example, in a plane, best approximating any three straight lines with a point needs considering its deviations namely from those three straight lines rather than from their intersections (the triangle vertices). But if those three straight lines have a common point, then namely it is the desired best approximation both to those three straight lines and to their coinciding intersections.

Equalization-Minimization Criteria System

E1. Determining all the objects equidistant (with equal distances) from all the given objects and then determining the object with the minimum common distance. Nota bene: Along with the usual straight line square distance, we may also use, e.g., other possibly curvilinear (by additional limitations and other conditions such as using curves lying in a certain surface, etc.) power distances. By point objects and the usual straight line square distance, e.g., we obtain the only quasisolution by two points on a straight line, three points in a plane, or four points in the three-dimensional space. Using distances only makes this criterion invariant by coordinate system translation and rotation.

E2. Determining all the objects with the equal values of a nonnegative binary function (e.g., distance, norm of difference, absolute error [1], relative error [1], unierror [2-5], etc.) of each of these objects and each of all the given objects and then determining the object with the minimum common value of this function. Nota bene: This criterion generalizes E1 for any nonnegative binary function but if it (unlike a distance) is NOT invariant by coordinate system translation and rotation, then the same holds for the criterion results in general.

Power Mean Minimization Criteria System

PoMi1. Determining the object with the minimum weighted sum of the powers of the distances of this object from all the given objects with power exponent t > 1. Nota bene: To obtain the midpoint of a straight line segment, any t > 1 suffices whereas in many typical problems we need sufficiently great power exponents, e.g., 64, 100, etc. Along with the usual straight line square distance, we may also use, e.g., other possibly curvilinear (by additional limitations and other conditions such as using curves lying in a certain surface, etc.) power distances. By point objects and the usual straight line square distance, e.g., we obtain the only quasisolution by two points on a straight line, three points in a plane, or four points in the three-dimensional space. Using distances only makes this criterion invariant by coordinate system translation and rotation.

PoMi2. Determining the object with the minimum weighted sum of the powers of the values of a nonnegative binary function (e.g., distance, norm of difference, absolute error [1], relative error [1], unierror [2-5], etc.) of this object and each of all the given objects with power exponent t > 1. Nota bene: This criterion generalizes PoMi1 for any nonnegative binary function but if it (unlike a distance) is NOT invariant by coordinate system translation and rotation, then the same holds for the criterion results in general.

Power Mean Maximization Criteria System

PoMa1. Determining the object with the maximum weighted sum of the powers of the distances of this object from all the given objects with power exponent 0 < t < 1. Nota bene: To obtain the midpoint of a straight line segment, any t > 1 suffices whereas in many typical problems we need sufficiently great power exponents, e.g., 64, 100, etc. Along with the usual straight line square distance, we may also use, e.g., other possibly curvilinear (by additional limitations and other conditions such as using curves lying in a certain surface, etc.) power distances. By point objects and the usual straight line square distance, e.g., we obtain the only quasisolution by two points on a straight line, three points in a plane, or four points in the three-dimensional space. Using distances only makes this criterion invariant by coordinate system translation and rotation.

PoMa2. Determining the object with the maximum weighted sum of the powers of the values of a nonnegative binary function (e.g., distance, norm of difference, absolute error [1], relative error [1], unierror [2-5], etc.) of this object and each of all the given objects with power exponent 0 < t < 1. Nota bene: This criterion generalizes PoMa1 for any nonnegative binary function but if it (unlike a distance) is NOT invariant by coordinate system translation and rotation, then the same holds for the criterion results in general.

Product Maximization Criteria System

PrMa1. Determining the object with the maximum weighted product (of the powers whose exponents are the given objects quantities as their weights) of the distances of this object from all the given objects. Nota bene: Along with the usual straight line square distance, we may also use, e.g., other possibly curvilinear (by additional limitations and other conditions such as using curves lying in a certain surface, etc.) power distances. By point objects and the usual straight line square distance, e.g., we obtain the only quasisolution by two points on a straight line, three points in a plane, or four points in the three-dimensional space. Using distances only makes this criterion invariant by coordinate system translation and rotation.

PrMa2. Determining the object with the maximum weighted product (of the powers whose exponents are the given objects quantities as their weights) of the values of a nonnegative binary function (e.g., distance, norm of difference, absolute error [1], relative error [1], unierror [2-5], etc.) of this object and each of all the given objects. Nota bene: This criterion generalizes PrMa1 for any nonnegative binary function but if it (unlike a distance) is NOT invariant by coordinate system translation and rotation, then the same holds for the criterion results in general.

Distribution theory (DT) 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