Unimathematical Estimation Fundamental Sciences System (Essential)

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), 28

UDC 501:510

2010 Math. Subj. Classification: primary 00A71; second. 03E10, 03E72, 08B99, 26E30, 28A75.

Keywords: Overmathematics, unimathematical estimation fundamental sciences system, measurement, concession, reserve, reliability, risk, approximation, deviation.

Classical mathematics [1] possibilities in modeling, expressing, measuring, evaluating, and estimating objects are very limited, nonuniversal, and inadequate. This holds for its very fundamentals such as:

the real numbers with gaps;

the Cantor sets, relations, and at most countable only restrictedly reversible operations with ignoring elements quantities, absorption, and contradicting the conservation law of nature;

the cardinality sensitive to finite unions of disjoint finite sets only and giving the same continuum cardinality C for distinct point sets between two parallel lines or planes differently distant from one another;

the measures which are finitely sensitive within a certain dimensionality, give either 0 or +∞ for distinct point sets between two parallel lines or planes differently distant from one another, and cannot discriminate the empty set ∅ and null sets, namely zero-measure sets [1];

the probabilities which cannot discriminate impossible and some differently possible events.

The same holds for classical mathematics estimators:

the absolute error alone is noninvariant and insufficient for quality estimation;

the relative error is uncertain in principle and has a very restricted domain of applicability.

Applied megamathematics [2] based on pure megamathematics [2] and on overmathematics [2] with its uninumbers, quantielements, quantisets, and uniquantities with quantioperations and quantirelations provides efficiently, universally and adequately strategically unimathematically modeling, expressing, measuring, evaluating, and estimating objects, as well as setting and solving general problems in science, engineering, and life. This all creates the basis for many further fundamental sciences systems developing, extending, and applying overmathematics. Among them is, in particular, the unimathematical estimation fundamental sciences system [2] including:

fundamental science of unimathematical measurement which includes general theories and methods of developing and applying overmathematical uniquantity as universal perfectly sensitive quantimeasure of general objects, systems, and their mathematical models with possibly recovering true measurement information using incomplete changed data;

fundamental science of unimathematical estimation including general theories and methods of applying overmathematical uninumbers and also operable quantisets to estimating (generalizing measurement) general objects, systems, and their mathematical models. It is proved that the classical relative error and least square method have many interconnected basic lacks and the narrowest areas of adequacy;

fundamental science of concessions which for the first time regularly applies and develops universal overmathematical theories and methods of measuring and estimating contradictions, infringements, damages, hindrances, obstacles, restrictions, mistakes, distortions, and errors, and also of rationally and optimally controlling them and even of their efficient utilization for developing general objects, systems, and their mathematical models, as well as for solving general problems;

fundamental science of reserves further naturally generalizing fundamental science of concessions and for the first time regularly applying and developing universal overmathematical theories and methods of measuring and estimating not only contradictions, infringements, damages, hindrances, obstacles, restrictions, mistakes, distortions, and errors, but also harmony (consistency), order (regularity), integrity, preference, assistance, open space, correctness, adequacy, accuracy, reserve, resource, and also of rationally and optimally controlling them and even of their efficiently utilization for developing general objects, systems, and their mathematical models, as well as for solving general problems;

fundamental sciences of reliability and risk for the first time regularly applying and developing universal overmathematical theories and methods of quantitatively measuring, evaluating, and estimating the reliabilities and risks of real general objects and systems and their ideal mathematical models with avoiding unjustified artificial randomization in deterministic problems;

fundamental science of approximation which includes universal overmathematical theories and methods of approximating (as a particular case of estimating other than measuring) objects, systems, and their mathematical models;

fundamental science on deviation for the first time regularly applying overmathematics to measuring and estimating deviations of real general objects and systems from their ideal mathematical models, and also of mathematical models from one another. And in a number of other fundamental sciences at rotation invariance of coordinate systems, general (including nonlinear) theories of the moments of inertia establish the existence and uniqueness of the linear model minimizing its square mean deviation from an object whereas least square distance (including nonlinear) theories are more convenient for the linear model determination. And the classical least square method by Legendre and Gauss ("the king of mathematics") is the only known (in classical mathematics) applicable to contradictory (e.g., overdetermined) problems. In the two-dimensional Cartesian coordinate system, this method minimizes the sum of the squares of ordinate differences and ignores a model inclination. This leads not only to the systematic regular error breaking invariance and growing together with this inclination and data variability but also to paradoxically returning rotating the linear model. By coordinate system linear transformation invariance, power (e.g., square) mean (including nonlinear) theories lead to optimum linear models. Theories and methods of measuring and estimating data scatter and trend give corresponding invariant and universal measures and estimations concerning linear and nonlinear models. Group center theories sharply reduce this scatter, raise data scatter and trend, and for the first time also consider their outliers. Overmathematics even allows to divide a point into parts and to refer them to different groups. Coordinate division theories and especially principal bisector (as a model) division theories efficiently form such groups. Note that there are many reasonable deviation arts, e.g., the following:

the value of a nonnegative binary function (e.g., the norm of the difference of the parts of an equation as a subproblem in a problem after substituting a pseudosolution to this problem, distance from the graph of this equation, its absolute error [1], relative error [1], unierror [2], etc.) of this object and each of all the given objects;

the value of a nonnegative function (e.g., the power mean value) of these values for all the equations in a general problem by some positive power exponent.

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.

The unimathematical estimation fundamental sciences system is universal and very efficient.

References

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

[2] Lev Gelimson. Elastic Mathematics. General Strength Theory. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2004, 496 pp.