General Problem 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), 29

UDC 501:510

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

Keywords: Overmathematics, general problem fundamental sciences system, pseudosolution, quantiset, subproblem, strategy, invariance, quantibound, estimation, trend, bisector, iteration.

In classical mathematics [1], there is no sufficiently general concept of a quantitative mathematical problem. The concept of a finite or countable set of equations ignores 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.

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 general problem fundamental sciences system [2]. It defines a general problem as a quantisystem [2]

_q(λ)R_λ[_φ∈Φ f_φ[_ω∈Ω z_ω]] (λ∈Λ)

of known relations R_λ over indexed unknown functions (dependent variables), or simply unknowns, f_φ of indexed independent known variables z_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 z_ω] is a set of indexed elements z_ω ; q(λ) is the own, or individual, quantity [2] as a weight of the relation with index λ .

The general problem fundamental sciences system includes:

fundamental science on general problem essence including general problem type and setting theory, general problem quantiobject theory, general problem quantisystem theory, general problem quantioperation theory, general problem uniquantity theory, general quantirelation problem theory, general quantiequation problem theory, and general quantiinequation problem theory;

fundamental science of general problem pseudosolution including general problem pseudosolution theory, general problem quasisolution theory, general problem supersolution theory, and general problem antisolution theory;

fundamental science on general problem solving strategy including finite solving possibility theory, finite solving suitability theory, infinite solving possibility theory, infinite solving suitability theory, and general problem solving strategy theory;

fundamental science of general problem transformation including general problem transformation theory, distance and unit unknown factor power normalization theories families, linear and quadratic unierrors power normalization theories families, unknown factor power normalization theories family, general problem structuring theory, general problem restructuring theory, and general problem partitioning theory;

fundamental science of general problem analysis including general problem analysis theory, general subproblem theory, and general subproblem criterion theory;

fundamental science of general problem synthesis including general problem synthesis theory, general problem symmetry theory, and general problem criterion theory;

fundamental science on general problem invariance including general problem homogeneous coordinate system theory, general problem nonhomogeneous coordinate system theory, general problem invariance theory, general problem data invariance theory, general problem method invariance theory, general problem pseudosolution invariance theory, general problem quasisolution, supersolution, and antisolution invariance theories;

fundamental science of general subproblem estimation including general subproblem estimation theory, difference norm estimation theory, deviation estimation theory, distance estimation theory, linear unierror estimation theory, quadratic unierror estimation theory, reserve estimation theory, reliability estimation theory, and risk estimation theory;

fundamental science of general problem estimation including general problem estimation theory, power estimation theories family, product estimation theories family, power difference estimation theories family, and quantibound estimation theories family;

fundamental science on general problem solving criteria including distance minimization theory, linear unierror minimization theory, square unierror minimization theory, reserve maximization theory, distance equalization theory, linear unierror equalization theory, square unierror equalization theory, reserve equalization theory, distance quantiinfimum theory, linear and quadratic unierrors quantiinfimum theories, and reserve quantisupremum theory;

fundamental science on general problem solving methods including subproblem subjoining theory, linear combination theory, exhaustive solution theory, unit unknown factor power theories family, distance function theories family, linear unierror function theories family, square unierror function theories family, power increase theory, distance product theories family, linear and square unierrors product theories families, distance power difference theories family, linear and square unierroros power difference theories families, distance quantibound theory, linear and square unierrors quantibound theories, reserve quantibound theory, trial pseudosolution and direct solving theories families;

fundamental science on general problem iteration including single-source iteration theory, multiple-sources iteration theory, intelligent iteration theory, general trend multistep theory, trend multistep distance function theories family, trend multistep linear and square unierrors function theories families, and iteration acceleration theory;

fundamental science on general problem bisectors including general center and bisector theory, distance, linear and square unierrors bisector theories, recurrent bisector theories family, incenter theories family, triangles incenters theories family, equidistance theories family, linear unierror equalizing theories family, square unierror equalizing theories family, internal bisectors intersections center theories family, sides pairs bisectors and equidistance theories families, adjacent sides bisectors theories family, adjacent corners bisectors theories family, opposite sides bisectors theories family, and opposite corners bisectors theories family;

fundamental science of general problem testing including directed test system theory, distribution theory, general center theory, triangle, tangential polygon, and quadrilateral theories;

fundamental science of general problem application including overmathematics development theory, pure megamathematics development theory, applied megamathematics development theory, computational fundamental megascience development theory, fundamental mechanical, strength, and physical sciences systems development theories.

The general problem 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.