Ph. D. & Dr. Sc. Lev Gelimson's Applied Unimathematics (Mega-Overmathematics) as a System of Revolutions in Applied Mathematics

Applied Unimathematics (Mega-Overmathematics) as a System of Revolutions in Applied Mathematics

© Ph. D. & Dr. Sc. Lev Gelimson

Academic Institute for Creating Fundamental Sciences (Munich, Germany)

Mathematical Monograph

The “Collegium” All World Academy of Sciences Publishers

Munich (Germany)

12th Edition (2012)

11th Edition (2010)

10th Edition (2004)

9th Edition (2003)

8th Edition (2002)

7th Edition (2001)

6th Edition (2000)

5th Edition (1997)

4th Edition (1995)

3nd Edition (1994)

2nd Edition (1993)

1st Edition (1992)

Abstract

2010 Mathematics Subject Classification: primary 00A69; secondary 15A06, 39B72, 65G99, 68W25.

Keywords: applied mathematics, megascience, revolution, megamathematics, unimathematics, mega-overmathematics, unimathematical test fundamental metasciences system, knowledge, philosophy, strategy, tactic, analysis, synthesis, object, operation, relation, criterion, conclusion, evaluation, measurement, estimation, expression, modeling, processing, symmetry, invariance, bound, level, worst case, defect, mistake, error, reserve, reliability, risk, supplement, improvement, modernization, variation, modification, correction, transformation, generalization, replacement.

Mathematics is usually divided into pure, applied, and computational mathematics. Pure mathematics can be further divided into fundamental and advanced mathematics.

Classical mathematics, its concepts, approaches, methods, and theories are based on inflexible axiomatization, intentional search for artificial contradictions, and even their purposeful creation to desist from further research. These and other fundamental defects do not allow us to acceptably and adequately consider, formulate, and solve many classes of typical urgent problems in science, engineering, and life. Mathematicians select either set theory or mereology as if these were incompatible. The real numbers cannot fill the number line because of gaps between them and hence evaluate even not every bounded quantity. The sets, fuzzy sets, multisets, and set operations express and form not all collections. The cardinalities and measures are not sufficiently sensitive to infinite sets and even to intersecting finite sets due to absorption. No conservation law holds beyond the finite. Infinity seems to be a heap of very different infinities the cardinality only can very roughly discriminate and no tool can exactly measure. Known hypernumber systems, starting with nonstandard analysis, demonstrate the possibility of their construction and use to more intuitively prove well-known theorems but cannot namely quantitatively solve many classes of typical urgent problems. Operations are typically considered for natural numbers or countable sets of operands only and cannot model any mixed magnitude. Exponentiation is well-defined for nonnegative bases only. Exponentiation and further hyperoperations are noncommutative. Division by zero is considered when unnecessary, ever brings insolvable problems, and is never efficiently utilized. The probabilities not always existing cannot discriminate impossible and other zero-measure events differently possible. The absolute error is noninvariant and alone insufficient for quality estimation. The relative error applies to the simplest formal equalities of two numbers only and even then is ambiguous and can be infinite. Mathematical statistics and the least square method irreplaceable in overdetermined problems typical for data processing are based on the noninvariant absolute error and on the second degree analytically simplest but usually very insufficient. This method is unreliable and not invariant by equivalent transformations of a problem, makes no sense by noncoinciding physical dimensions (units) in a problem to be solved, and can give predictably inacceptable and even completely paradoxical outputs without any estimation and improvement. Artificial randomization brings unnecessary complications. One-source iteration with a rigid algorithm requires an explicit expression of the next approximation via the previous approximations with transformation contractivity and often leads to analytic difficulties, slow convergence, and even noncomputability. Real number computer modeling brings errors via built-in standard function rounding and finite signed computer infinities and zeroes, which usually excludes calculation exactness, limits research range and deepness, and can prevent executing calculation for which even the slightest inconsistencies are inadmissible, e.g. in accounting. The finite element method gives visually impressive "black box" results not verifiable and often unacceptable and inadequate.

Every new alternative mathematics can be considered as an external revolution in mathematics which becomes megamathematics. In any new alternative mathematics itself, creating its own cardinally new very fundamentals replacing the very fundamentals of classical mathematics can be considered as an internal revolution in alternative mathematics also if classical mathematics itself remains unchanged.

Mega-overmathematics (by the internal entity), or unimathematics (by the external phenomenon), created and developed has the character of a superstructure (with useful creative succession, or inheritance) over conventional mathematics as a basis without refusing any achievement of ordinary mathematics. Moreover, unimathematics even calls for usefully applying ordinary mathematics if possible, permissible, acceptable, and adequate.

In these names, the prefix "mega" means infinitely many distinct overmathematics with including different infinities and overinfinities into the real numbers.

The prefix "uni" is here associated both with the union, or the general system, of these infinitely many distinct overmathematics and with the universality of these union and system.

The prefix "over" here means:

1) the superstructural character of mega-overmathematics, or unimathematics, with respect to conventional mathematics;

2) the addional nature of new possibilities offered by mega-overmathematics besides the usual opportunities of ordinary mathematics;

3) overpossibilities as the qualitatively new features of mega-overmathematics in setting, considering, and solving whole classes of typical urgent problems so that these overpossibilities often have a much higher order of magnitude compared with the possibilities of conventional mathematics. For example, one of such overpossibilities is oversensitivity as perfect unlimited sensitivity with exactly satisfying universal conservation laws and with complete exclusion of any absorption so that infinitely or overinfinitely great magnitudes are exactly separated from one another even by infinitesimal or overinfinitesimal differences.

Unimathematics can be called not only universal and unified but also general, natural, physical, intuitive, nonrigorous, free, flexible, perfectly sensitive, practical, useful, exclusively constructive, creative, inventive, etc.

Mega-overmathematics is a system of infinitely many diverse overmathematics which differ by possible hyper-Archimedean structure-preserving extensions of the real numbers via including both specific subsets of some infinite cardinal numbers as canonic positive infinities and signed zeroes reciprocals as canonic overinfinities, which gives the uninumbers. They provide adequately and efficiently considering, setting, and namely quantitatively solving many typical urgent problems. In created uniarithmetics, quantialgebra, and quantianalysis of the finite, the infinite, and the overinfinite with quantioperations and quantirelations, the uninumbers evaluate, precisely measure, and are interpreted by quantisets algebraically quantioperable with any quantity of each element and with universal, perfectly sensitive, and even uncountably algebraically additive uniquantities so that universal conservation laws hold. Quantification builds quantielements, integer and fractional quantisets, mereologic quantiaggregates (quanticontents), and quantisystems with unifying mereology and set theory. Negativity conserving multiplication, base sign conserving exponentiation, exponentiation hyperefficiency, composite (combined) commutative exponentiation and hyperoperations, root-logarithmic overfunctions, self-root-logarithmic overfunctions, the voiding (emptifying) neutral element (operand), and operations with noninteger and uncountable quantities of operands are also introduced. Division by zero is regarded when necessary and useful only and is efficiently utilized to create overinfinities. Unielements, unisets, mereologic uniaggregates (unicontents), unisystems, unipositional unisets, unimappings, unisuccessions, unisuccessible unisets, uniorders, uniorderable unisets, unistructures, unicorrespondences, and unirelation unisystems are also introduced. The same holds for unitimes, potential uniinfinities, general uniinfinities, subcritical, critical, and supercritical unistates and uniprocesses, as well as quasicritical unirelations. Unidestructurizators, unidiscriminators, unicontrollers, unimeaners, unimean unisystems, unibounders, unibound unisystems, unitruncators, unilevelers, unilevel unisystems, unilimiters, uniseries uniestimators, unimeasurers, unimeasure unisystems, uniintegrators, uniintegral unisystems, uniprobabilers, uniprobability unisystems, and unicentral uniestimators efficiently provide unimeasuring and uniestimating. The universalizing separate similar (proportional) limiting reduction of objects, systems, and their models to their own similar (proportional) limits as units provides the commensurability and comparability of disproportionate and, therefore, not directly commensurable and comparable objects, systems, and their models. The unierror irreproachably corrects and generalizes the relative error. The unireserve, unireliability, and unirisk based on the unierror additionally estimate and discriminate exact objects, models, and solutions by the confidence in their exactness with avoiding unnecessary randomization. All these uniestimators for the first time evaluate and precisely measure both the possible inconsistency of a uniproblem (as a unisystem which includes unknown unisubsystems) and its pseudosolutions including quasisolutions, supersolutions, and antisolutions. Multiple-sources iterativity and especially intelligent iterativity (coherent, or sequential, approximativity) are much more efficient than common single-source iterativity. Intelligent iterability universalization leads to collective coherent reflectivity, definability, modelablity, expressibility, evaluability, determinability, estimability, approximability, comparability, solvability, and decisionability. This holds, in particular, in truly multidimensional and multicriterial systems of the expert definition, modeling, expression, evaluation, determination, estimation, approximation, and comparison of the qualities of objects, systems, and models which are disproportionate and hence incommensurable and not directly comparable, as well as in truly multidimensional and multicriterial decision-making systems. Sufficiently increasing the exponent in power mean theories and methods can bring adequate results. This holds for linear and nonlinear unibisector theories and methods with distance or unierror minimization, unireserve maximization, as well as for distance, unierror, and unireserve equalization, respectively. Unimathematical data coordinate and/or unibisector unipartitioning, unigrouping, unibounding, unileveling, scatter and trend unimeasurement and uniestimation very efficiently provide adequate data processing with efficiently utilizing outliers and even recovering true measurement information using incomplete changed data. Universal (in particular, infinite, overinfinite, infinitesimal, and overinfinitesimal) continualization provides perfect computer modeling of any uninumbers. Perfectioning built-in standard functions brings always feasible and proper computing. Universal transformation and solving algorithms ensure avoiding computer zeroes and infinities with computer intelligence and universal cryptography systems hierarchies. It becomes possible to adequately consider, model, express, measure, evaluate, estimate, overcome, and even efficiently utilize many complications such as contradictions, infringements, damages, hindrances, obstacles, restrictions, mistakes, distortions, errors, information incompleteness, variability, etc. Unimathematics (mega-overmathematics) also includes knowledge universal test and development fundamental metasciences.

Unimathematics as a megasystem of revolutions in mathematics is divided into fundamental, advanced, applied, and computational unimathematics as systems of revolutions in fundamental, advanced, applied, and computational mathematics.

Uniphilosophy (Exclusively Constructive Creative Philosophy) Principles as a System of Revolutions in Philosophy

Fundamental principles of uniphilosophy (exclusively constructive creative philosophy) build a fundamental system of revolutions in philosophy, in particular, the following subsystems.

1. Fundamental Principles of Uniphilosophy as a Fundamental Subsystem of Revolutions in Philosophy

The fundamental subsystem of revolutions in philosophy includes the following fundamental principles of uniphilosophy:

1. Exceptional natural constructivism (with the complete absence of artificial destructiveness).

2. Free efficient creativity (exclusively practically purposeful, verified, and efficient unlimitedly free creativity, intuition, and phantasy flight).

3. Scientific optimism and duty (each urgent problem can and must be solved adequately and efficiently enough).

4. Complication utilization (creating, considering, and efficiently utilizing only necessary and useful also contradictory objects and models, as well as difficulties, problems, and other complications).

5. Symbolic feasibility (at least symbolic existence of all the necessary and useful even contradictory objects and models).

2. Advanced Principles of Uniphilosophy as an Advanced Subsystem of Revolutions in Philosophy

The advanced subsystem of revolutions in philosophy includes the following advanced principles of uniphilosophy:

1. Exclusively efficient intuitive evidence and provability (reasonable fuzziness, intuitive ideas without axiomatic rigor if necessary and useful).

2. Unrestrictedly flexible constructivism (if necessary even creating new knowledge (concepts, approaches, methods, theories, doctrines, and even sciences) to adequately set, consider, and solve urgent problems).

3. Tolerable simplicity (choosing the best in the not evidently unacceptable simplest).

4. Perfect sensitivity, or conservation laws universality (no uncompensated change in a general object conserves its universal measures).

5. Exact discrimination of noncoinciding objects and models (possibly infinitely or overinfinitely large with infinitesimal or overinfinitesimal distinctions and differences).

6. Separate similar (proportional) limiting universalizability (the reduction of objects, systems, and their models to their own similar (proportional) limits as units).

7. Collective coherent reflectivity, definability, modelablity, expressibility, evaluability, determinability, estimability, approximability, comparability, solvability, and decisionability (in particular, in truly multidimensional and multicriterial systems of the expert definition, modeling, expression, evaluation, determination, estimation, approximation, and comparison of objects, systems, and models qualities which are disproportionate and hence incommensurable and not directly comparable, as well as in truly multidimensional and multicriterial decision-making systems).

3. Some Other Principles of Uniphilosophy

Among other principles of uniphilosophy are the following:

1. Truth priority (primacy of practically verified purely scientific truths and criteria prior to commonly accepted dogmas, views, agreements, and authority, with all due respect to them).

2. Peaceful pluralism (with peaceful development of scientific and life diversity).

3. Efficient creative inheritance (efficiently using, analyzing, estimating, and developing already available knowledge and information).

4. Efficient constructive freedom (unrestrictedly free exclusively constructive and useful self-determination and activity, in particular, in knowledge and information research, creation, and development).

5. Fundamentality priority (primacy of conceptual and methodological fundamentals).

6. Knowledge efficiency (only useful quality (acceptability, adequacy, depth, accuracy, etc.) and amount (volume, completeness, etc.) of knowledge, information, data, as well as creation, analysis, synthesis, verification, testing, structuring, systematization, hierarchization, generalization, universalization, modeling, measurement, evaluation, estimation, utilization, improvement, and development of objects, models, knowledge, information, and data along with intelligent management and self-management of activity).

7. Mutual definability and generalizability (relating successive generalization of concepts in definitions with optional linear sequence in knowledge construction).

8. Efficient unificability of opposites only conditionally distinguished (such as real/potential, real/ideal, specific/abstract, exact/inexact, definitively/possibly, pure/applied, theory/experiment/practice, nature/life/science, for example, the generally inaccurate includes the accurate as the limiting particular case with the zero error).

9. Partial laws sufficiency (if there are no known more general laws).

10. Focus on discoveries and inventions (dualistic unity and harmony of academic quality and originality, discovering phenomena of essence, inventive climbing, helpful knowledge bridges, creative multilingualism, scientific art, anti-envy, learnability, teachability, and terminology development).

Principles of Unimathematics as a System of Revolutions in the Principles of Mathematics

The principles of exclusively constructive creative unimathematics (mega-overmathematics) constitute a system of scientific revolutions in the principles of mathematics including the following subsystems.

1. Fundamental Principles of Unimathematics as a Fundamental Subsystem of Revolutions in the Principles of Mathematics

The fundamental subsystem of revolutions in the principles of mathematics includes the following principles of unimathematics:

1. Typical urgent problems priority and exclusiveness (adequately setting and solving and efficiently using urgent problems only with completely avoiding unnecessary considerations is the only criterion of the necessity and usefulness of creating and developing new knowledge including concepts, approaches, methods, theories, doctrines, and sciences).

2. Intuitive conceptual and methodological fundamentality priority (creating and efficiently using unified knowledge foundation due to fundamental general systems including objects, models, and intuitive fuzzy principles, concepts, and methodology).

3. Collective coherent reflectivity, definability, modelablity, expressibility, evaluability, determinability, estimability, approximability, comparability, solvability, and decisionability (in particular, in constructing nonlinear conceptual systems of knowledge and in truly multidimensional and multicriterial systems of the expert definition, modeling, expression, evaluation, determination, estimation, approximation, and comparison of objects, systems, and models qualities which are disproportionate and hence incommensurable and not directly comparable, as well as in truly multidimensional and multicriterial decision-making systems).

4. Reasonable fuzziness with useful rigor only (exclusively practically useful axiomatization, deductivity, and rigorously proving, as well as intuitive ideas without axiomatic strictness if necessary and useful).

5. Unrestrictedly flexible constructivism (even creating new sciences to adequately set, consider, and solve typical urgent problems).

2. Noncontradictoriness Principles of Unimathematics as a Noncontradictoriness Subsystem of Revolutions in the Principles of Mathematics

The noncontradictoriness subsystem of revolutions in principles of mathematics includes the following principles of unimathematics:

1. The unificability of membership, inclusion, and part-whole relations.

2. Necessary and useful creativity exclusiveness (efficiently and intelligently creating and considering exclusively necessary and useful objects and models with completely ignoring any artificial contradictions typical in classical mathematics).

3. The efficient utilizability of contradictoriness and other complications (creating, considering, and efficiently utilizing exclusively necessary and useful contradictory objects and models, as well as difficulties, problems, and other complications).

4. Symbolic feasibility (at least symbolic existence of all the necessary and useful even contradictory objects and models).

5. Decision-making delayability (if necessary and useful, e.g. by estimating existence and sense with a possible further revaluation in the course of review).

3. Universalizability Principles of Unimathematics as a Universalization Subsystem of Revolutions in the Principles of Mathematics

The universalizability subsystem of revolutions in principles of mathematics includes the following principles of unimathematics:

1. Infinite cardinals canonizability (infinite cardinal numbers as canonical positive infinities namely real but not potential).

2. Zeroes reciprocals overinfinities canonizability (signed zeroes reciprocals as canonical overinfinities namely real but not potential).

3. Hyper-Archimedean axiomability (naturally generalizing the Archimedes axiom to the infinite and the overinfinite).

4. Exactness of the infinite and the overinfinite (perfectly sensitive, invariant, and universal infinite and overinfinite, infinitesimal and overinfinitesimal generalization of the numbers by the uninumbers with exact measurement generalizing counting, unlimited (possibly even noninteger and uncountable) manipulation and operability, as well as exact discrimination in the infinite and the overinfinite even by infinitesimal and overinfinitesimal distinctions and differences).

5. General (nonlogical) quantificability (assignment, definition, determination, and measurement of the individual quantity of a element becoming a quantielement and of the individual quantities of elements in a set which becomes a quantiset).

6. Separate similar (proportional) limiting universalizability (the reduction of objects, systems, and their models to their own similar (proportional) limits as units, in particular, of magnitudes to the moduli of their own unidirectional limits with the same signs).

7. Perfect manipulability (perfectly sensitive, invariant, and universal useful modeling, expression, evaluation, counting measurement, estimation, and essential generalization of urgent objects, relations, structures, systems, and their contents extending sets and quantisets).

8. Conservation laws universalizability (in the overinfinitesimal, the infinitesimal, the finite, the infinite, and the overinfinite).

4. Efficiency Principles of Unimathematics as an Efficiency Subsystem of Revolutions in the Principles of Mathematics

The efficiency subsystem of revolutions in principles of mathematics includes the following principles of unimathematics:

1. Uniproblem unisolvability (existence and expressibility of the best quasisolution, solution, and supersolution among possibly inexact meaningful pseudosolutions to any urgent uniproblem with setting as a unisystem with unknown unisubsystems).

2. Tolerable simplicity (selecting the best in the class of not evidently unacceptable simplest meaningful pseudosolutions).

3. Efficient knowledge (efficient quality (acceptability, adequacy, profundity, exactness, structurality, systematization, inheritance, universality, invariance, strength, stability, reliability, flexibility, etc.) and quantity (volume, completeness, etc.) of objects, models, knowledge, information, data, and their perfectly sensitive creation, analysis, synthesis, verification, testing, structuring, systematization, hierarchization, generalization, universalization, modeling, evaluation, measurement, estimation, utilization, improvement, development, and reasonable control).

4. Free intuitive intelligent iterativity (coherent, or sequential, approximativity) (possibly with many sources and directions, unrestrictedly flexible universal algorithms with avoiding computer zeroes and infinities and independent of analytic solvability with providing mapping contractivity).

5. Collective coherent reflectivity, definability, modelablity, expressibility, evaluability, determinability, estimability, approximability, comparability, solvability, and decisionability (in particular, in truly multidimensional and multicriterial systems of the expert definition, modeling, expression, evaluation, determination, estimation, approximation, and comparison of objects, systems, and models qualities which are disproportionate and hence incommensurable and not directly comparable, as well as in truly multidimensional and multicriterial decision-making systems).

6. General noncriticality (subcritical, critical, and supercritical states, processes, and phenomena in a general structured system which are defined and determined by generally noncritical relationships).

7. General nonlimitability (underlimiting, limiting, and overlimiting states, processes, and phenomena in a general structured system which are defined and determined by generally nonlimiting relationships).

The Principles of Applied Unimathematics

The principles of applied unimathematics build the uniproblem system of revolutions in the principles of mathematics and include:

1. Separate limiting universalizability (the reduction of objects, systems, and their models to their own commonly qualitative (equiqualitative) limits as units, in particular, of magnitudes to the moduli of their own commonly directional (unidirectional, equidirectional) and equisigned (with the same sign) limits as units).

2. Unitransformability (the freedom of efficiently transforming unimodels of real objects and systems and their mathematical and physical models with the absolute and full compliance with the universality of the conservation laws in the overinfinite, infinite, finite, infinitesimal, and infinitesimal, including partitionability by unibisectors, as well as unicombinability by unigrouping, for example dividing a point into any separate parts and including them into different unigroups).

3. Uniestimability (perfect sensitivity, invariance, universality, and efficiency of estimating and reasonably controlling urgent uniobjects, unirelations, unistructures, unisystems, and uniaggregates (unicontents) generalizing unisets and sets, as well as their exact or approximate unimodels via unierrors for approximations and via unireserves, unireliabilities, and unirisks for approximations and even exactness confidence).

4. Uniapproximability (of real objects and systems and their mathematical and physical models via their approximate models with the uniqualitative and uniquantitative uniestimability of this uniapproximability).

5. Unisolvability (the definability of the best exact solutions (supersolutions), of approximate quasisolutions, and of the antisolutions (if necessary and useful) to uniproblems as unisystems with unknown unisubsystems).

6. Intelligent uniiterativity (coherent, or sequential, approximativity) (free intuitive reasonable multi-sources and multidirectional uniiterativity with unrestrictedly flexible universal algorithms with avoiding computer zeroes and infinities and with independence of analytical solvability with mapping contractivity).

8. Unisystematic developing testability (of uniobjects, unisystems, and unimodels, in particular, knowledge including concepts, approaches, methods, theories, doctrines, and sciences).

Applied Unimathematics Contents

Applied unimathematics includes:

1. The system of the fundamental sciences of uniestimation.

2. The system of the fundamental sciences of uniapproximation.

3. The system of fundamental uniproblem sciences.

4. The system of the fundamental metasciences of testing and developing knowledge.

The system of the fundamental sciences of uniestimation includes:

1) the fundamental science of universal estimation including general theories and methods of applying unimathematical uninumbers and also operable unisets to uniestimating (which generalizes unimeasuring) universal objects, systems, and their mathematical models. It is proved that the classic absolute and relative errors and and the least square method by Legendre and "king of mathematics" Gauss irreplaceable in standard mathematics have many interrelated fundamental defects and very narrow domains of applicability and, moreover, adequacy;

2) the fundamental science of concessions which first systematically applies and develops theories and methods of unimathematical unimeasuring and uniestimating contradictions, violations, damage, interference, obstacles, limitations, mistakes, distortions, and errors, as well as their reasonable and best control and even their beneficial utilization for developing uniobjects, unisystems, and unimathematical unimodels, and for solving uniproblems;

3) the fundamental science of unireserving which naturally further generalizes the fundamental science of concessions and for the first time systematically applies and develops theories and methods unimathematical unimeasuring and uniestimating not only contradictions, violations, damage, interference, obstacles, limitations, mistakes, distortions, and errors, but also harmony (consistency), order (regularity), integrity, favored assistance, open space, correctness, suitability, accuracy, safety, resource, as well as their reasonable and best control and even their beneficial utilization for developing uniobjects, unisystems, and unimathematical unimodels, and for solving uniproblems;

4) the fundamental sciences of unireliability and unirisk which first systematically apply and develop theories and methods of unimathematical namely quantitative unimeasuring and uniestimating the unireliabilities and unirisks of uniobjects and unisystems, as well as their real and ideal unimathematical unimodels without undue artificial randomization in the deterministic cases;

5) the fundamental science of unideviations which first systematically applies unimathematical unimeasuring and uniestimating to unideviations of real uniobjects and unisystems from their ideal unimathematical unimodels, as well as of some unimathematical unimodels from others. And in a number of other fundamental sciences by the rotational invariance of the general coordinate system theory (including nonlinear) of moments of inertia establish the existence and uniqueness of the linear model which reduces its maximum deviation from the object whereas distance power theories (including nonlinear) are more convenient for model determination. And the classical least-squares method by Gauss and Legendre almost the only one (in usual mathematics) applicable to overdetermined problems minimizes the sum of the squared differences of the ordinates of the equiabscissa points on the object and on the model without regard to the possible, or general, variability of the model slope in the two-dimensional space. This leads to such fundamental defects as systematic errors violating the rotational invariance principle and growing together both with this slope and data scatter, to impermissibly bounding the model slope, and even to the paradoxical approximation (by the abscissa axis) of data symmetric about the ordinate axis and relatively near to the ordinate axis. By the invariance of the linear transformation of the coordinate system, power mean theories (including nonlinear by the model) with exponents which can equal many thousands if required lead to the best linear models. Power mean and multibisector theories and methods of unimeasuring and uniestimating data scatter and trends provide the corresponding invariant and universal measures and estimates of linear and nonlinear models. Unigroup unicenter theories dramatically reduce this scatter, increase data directedness, and efficiently utilize data outliers for the first time. Unimathematics even allows dividing a single point into any parts and include them into different unigroups. In particular, coordinate and (even more useful) unibisector (even nonlinear) partition theories efficiently divide data into suitable unigroups.

The system of the fundamental sciences of uniapproximation (in particular, by parts) includes the fundamental sciences of uniapproximation, unicenters, linear and nonlinear unibisectors, multi-sources, multidirectional, and intelligent uniiteration including unimathematical theories and methods of uniapproximating (as a particular case of uniestimating other than unimeasuring) uniobjects, unisystems, and their unimathematical unimodels, which are based on applying unimathematic to setting and solving uniapproximation uniproblems.

The system of fundamental uniproblem sciences includes:

1) the fundamental science of uniproblem essence including general theories of unisolving (determining not only solutions but also pseudosolutions, quasisolutions, supersolutions, and even antisolutions) uniproblems, e.g. in data processing;

2) the fundamental science of unisolving uniproblems including general theories and methods of uniparametrization, own classes, general (possibly infinite or overinfinite) linear combinations, exhaustive unisolving, uninormalization, unigrouping, unistructuring, unirestructuring, unibisector, multi-sources, multidirectional, and intelligent uniiteration and its acceleration, distance power, increasing exponents even up to many thousands if necessary and useful, minimizing universal power mean deviations and their equalization, maximizing universal power mean reserves and their equalization, as well as moments of inertia, directed unisolving, and purposeful numerical test systems;

3) the fundamental science of the invariance of uniproblem unisolutions by coordinate systems transformations.

The system of fundamental metasciences of testing and developing knowledge (concepts, approaches, methods, theories, doctrines, and sciences) includes:

1) fundamental metascience of the philosophy, methodology, strategy, and tactics of testing knowledge including relevant metatheories;

2) fundamental metascience of reviewing knowledge including metatheories of determining its principles, approaches, methods, and conclusions;

3) fundamental metascience of knowledge analysis including metatheories of analyzing knowledge principles, approaches, methods, and conclusions;

4) fundamental metascience of knowledge synthesis including metatheories of synthezing knowledge principles, approaches, methods, and conclusions;

5) fundamental metascience of knowledge objects, operations, relations, and criteria including relevant metatheories and metacriteria;

6) fundamental metascience of quantitatively expressing, evaluating, measuring, and estimating knowledge including relevant metatheories;

7) fundamental metascience of representing, modeling, and processing knowledge including relevant metatheories;

8) fundamental metascience of the symmetry and invariance of knowledge including relevant metatheories;

9) fundamental metascience of the boundaries and levels of knowledge including relevant metatheories;

10) fundamental metascience of directedly and purposefully testing knowledge including metatheories of the directions and steps of testing;

11) fundamental metascience of analyzing and synthezing the simplest tolerable limiting, critical, and worst cases in knowledge including metatheories of analyzing and synthezing such cases and of constructing counterexamples;

12) fundamental metascience of knowledge testability, verifiability, flaws, mistakes, errors, correctability, inviolability, strength, stability, reserves, reliability, and risk including relevant metatheories;

13) fundamental metascience of knowledge test result determination, expression, evaluation, measurement, estimation, analysis, and synthesis including relevant metatheories;

14) fundamental metascience of knowledge addition, conversion, modernization, reformation, change, correction, improvement, development, generalization, universalization, structuring, systematization, hierarchization, and replacement including relevant metatheories;

15) fundamental metametascience of applying the systems of fundamental metasciences of unimathematically testing knowledge including metatheories of efficiently developing sciences, as well as unimathematical, unimetrological, unimechanical, and unistrength metatheories of developing the systems of mathematical, metrological, mechanical, and strength sciences, respectively.

Introduction

There are many separate scientific achievements of mankind but they often bring rather unsolvable problems than really improving himan life quality. One of the reasons is that the general level of earth science is clearly insufficient to adequately solve and even consider many urgent himan problems. To provide creating and developing applicable and, moreover, adequate methods, theories, and sciences, we need their testing via universal if possible, at least applicable and, moreover, adequate test metamethods, metatheories, and metasciences whose general level has to be high enough. Mathematics as universal quantitative scientific language naturally has to play here a key role.

But classical mathematics [1] with hardened systems of axioms, intentional search for contradictions and even their purposeful creation cannot (and does not want to) regard very many problems in science, engineering, and life. This generally holds when solving valuation, estimation, discrimination, control, and optimization problems as well as in particular by measuring very inhomogeneous objects and rapidly changeable processes. It is discovered [2-10] that classical fundamental mathematical theories, methods, and concepts [1] are insufficient for adequately solving and even considering many typical urgent problems.

Megamathematics including overmathematics [2-10] based on its uninumbers, quantielements, quantisets, and uniquantities with quantioperations and quantirelations provides universally and adequately modeling, expressing, measuring, evaluating, and estimating general objects. This all creates the basis for many further megamathematics fundamental sciences systems developing, extending, and applying overmathematics. Among them are, in particular, science unimathematical test fundamental metasciences systems [11] which are universal.

The present monograph is dedicated to a revolution in applied mathematics [1] dealing with modeling, measuring, and estimating real objects, as well as solving problems.

Applied Science Unimathematical Test Fundamental Metasciences System

Applied science unimathematical test fundamental metasciences system [11] in megamathematics including overmathematics [2-10] is one of such systems and can efficiently, universally and adequately strategically unimathematically test any pure science. This system includes:

fundamental metascience of applied science test philosophy, strategy, and tactic including applied science test philosophy metatheory, applied science test strategy metatheory, and applied science test tactic metatheory;

fundamental metascience of applied science consideration including applied science fundamentals determination metatheory, applied science approaches determination metatheory, applied science methods determination metatheory, and applied science conclusions determination metatheory;

fundamental metascience of applied science analysis including applied subscience analysis metatheory, applied science fundamentals analysis metatheory, applied science approaches analysis metatheory, applied science methods analysis metatheory, and applied science conclusions analysis metatheory;

fundamental metascience of applied science synthesis including applied science fundamentals synthesis metatheory, applied science approaches synthesis metatheory, applied science methods synthesis metatheory, and applied science conclusions synthesis metatheory;

fundamental metascience of applied science objects, operations, relations, and criteria including applied science object metatheory, applied science operation metatheory, applied science relation metatheory, and applied science criterion metatheory;

fundamental metascience of applied science evaluation, measurement, and estimation including applied science evaluation metatheory, applied science measurement metatheory, and applied science estimation metatheory;

fundamental metascience of applied science expression, modeling, and processing including applied science expression metatheory, applied science modeling metatheory, and applied science processing metatheory;

fundamental metascience of applied science symmetry and invariance including applied science symmetry metatheory and applied science invariance metatheory;

fundamental metascience of applied science bounds and levels including applied science bound metatheory and applied science level metatheory;

fundamental metascience of applied science directed test systems including applied science test direction metatheory and applied science test step metatheory;

fundamental metascience of applied science tolerably simplest limiting, critical, and worst cases analysis and synthesis including applied science tolerably simplest limiting cases analysis and synthesis metatheories, applied science tolerably simplest critical cases analysis and synthesis metatheories, applied science tolerably simplest worst cases analysis and synthesis metatheories, and applied science tolerably simplest limiting, critical, and worst cases counterexamples building metatheories;

fundamental metascience of applied science defects, mistakes, errors, reserves, reliability, and risk including applied science defect metatheory, applied science mistake metatheory, applied science error metatheory, applied science reserve metatheory, applied science reliability metatheory, and applied science risk metatheory;

fundamental metascience of applied science test result evaluation, measurement, estimation, and conclusion including applied science test result evaluation metatheory, applied science test result measurement metatheory, applied science test result estimation metatheory, and applied science test result conclusion metatheory;

fundamental metascience of applied science supplement, improvement, modernization, variation, modification, correction, transformation, generalization, and replacement including applied science supplement metatheory, applied science improvement metatheory, applied science modernization metatheory, applied science variation metatheory, applied science modification metatheory, applied science correction metatheory, applied science transformation metatheory, applied science generalization metatheory, and applied science replacement metatheory.

The applied science unimathematical test fundamental metasciences system [11] in megamathematics including overmathematics [2-10] is universal and very efficient.

In particular, apply the applied science unimathematical test fundamental metasciences system to classical applied mathematics [1].

Nota bene: Naturally, all the fundamental defects of classical pure mathematics [1] discovered due to the pure science unimathematical test fundamental metasciences system in megamathematics [2] also hold in classical applied mathematics [1].

Fundamental Defects of Classical Pure Mathematics

Even the very fundamentals of classical pure mathematics [1] have evident cardinal defects of principle.

1. The real numbers R evaluate no unbounded quantity and, because of gaps, not all bounded quantities. The same probability p_n = p of the random sampling of a certain n ∈ N = {0, 1, 2, ...} does not exist in R , since ∑_n∈N p_n is either 0 for p = 0 or +∞ for p > 0. It is urgent to exactly express (in some suitable extension of R) all infinite and infinitesimal quantities, e.g., such a p for any countable or uncountable set, as well as distributions and distribution functions on any sets of infinite measures.

2. The Cantor sets [1] with either unit or zero quantities of their possible elements may contain any object as an element either once or not at all with ignoring its true quantity. The same holds for the Cantor set relations and operations with absorption. That is why those set operations are only restrictedly invertible. In the Cantor sets, the simplest equations X ∪ A = B and X ∩ A = B in X are solvable by A ⊆ B and A ⊇ B only, respectively [uniquely by A = ∅ (the empty set) and A = B = U (a universal set), respectively]. The equations X ∪ A = B and X = B \ A in the Cantor sets are equivalent by A = ∅ only. In a fuzzy set, the membership function of each element may also lie strictly between these ultimate values 1 and 0 in the case of uncertainty only. Element repetitions are taken into account in multisets with any cardinal numbers as multiplicities and in ordered sets (tuples, sequences, vectors, permutations, arrangements, etc.) [1]. They and unordered combinations with repetitions cannot express many typical objects collections (without structure), e.g., that of half an apple and a quarter pear. For any concrete (mixed) physical magnitudes (quantities with measurement units), e.g., "5 L (liter) fuel", there is no suitable mathematical model and no known operation, say between "5 L" and "fuel" (not: "5 L" × "fuel" or "fuel" × "5 L"). Note that multiplication is the evident operation between the number "5" and the measurement unit "L". The Cantor set relations and operations only restrictedly reversible and allowing absorption contradict the conservation law of nature because of ignoring element quantities and hinder constructing any universal degrees of quantity.

3. The cardinality is sensitive to finite unions of disjoint finite sets only but not sufficiently sensitive to infinite sets and even to intersecting finite sets (because of absorption). It gives the same continuum cardinality C for clearly very distinct point sets in a Cartesian coordinate system between two parallel lines or planes differently distant from one another.

4. The measures 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].

5. The probabilities cannot discriminate impossible and some differently possible events.

6. The operations are considered to be at most countable.

7. All existing objects and systems in nature, society, and thinking have complications, e.g., contradictoriness, and hence exist without adequate models in classical mathematics [1]. It intentionally avoids, ignores, and cannot (and possibly hence does not want to) adequately consider, model, express, measure, evaluate, and estimate many complications. Among them are contradictions, infringements, damages, hindrances, obstacles, restrictions, mistakes, distortions, errors, information incompleteness, multivariant approach, etc.

Naturally, there are very many other lacks and shortcomings of classical pure mathematics [1]. For example, a power of a negative number is well-defined for even positive integer exponents only, see counterexamples

(-1)³ = -1 ≠ 1 = [(-1)⁶]^1/2 = (-1)^6/2 ,

(-1)^1/3 = -1 ≠ 1 = [(-1)²]^1/⁶ = (-1)^2/⁶ .

Therefore, the very fundamentals of classical pure mathematics [1] have a lot of obviously deep and even cardinal defects of principle.

Fundamental Defects of Classical Applied Mathematics

In the very fundamentals of classical applied mathematics [1] with its own evident cardinal defects of principle, there were well-known attempts to consider some separate objects and systems with chosen complications, e.g., approximation and finite overdetermined sets of equations. To anyway consider them, classical mathematics only has very limited, nonuniversal, and inadequate concepts and methods such as the absolute error, the relative error, and the least square method (LSM) [1] by Legendre and Gauss ("the king of mathematics") with producing own errors and even dozens of principal mistakes. The same holds for classical mathematics estimators and methods.

8. The absolute error Δ [1] alone is noninvariant and insufficient for quality estimation giving, for example, the same result 1 for acceptable formal (correct or not) equality 1000 =? 999 and for inadmissible formal equality 1 =? 0. Further the absolute error is not invariant by equivalent transformations of a problem because, for instance, when multiplying a formal equality by a nonzero number, the absolute error is multiplied by the norm (modulus, absolute value) of that number.

9. The relative error δ [1] should play a supplement role. But even in the case of the simplest formal equality a =? b with two numbers, there are at once two propositions, to use either δ₁ = |a - b|/|a| or δ₂ = |a - b|/|b| as an estimating fraction. It is a generally inadmissible uncertainty that could be acceptable only if the ratio a/b is close to 1. Further the relative error is so intended that it should always belong to segment [0, 1]. But for 1 =? 0 by choosing 0 as the denominator, the result is +∞ , for 1 =? -1 by each denominator choice, the result is 2. Hence the relative error has a restricted range of applicability amounting to the equalities of two elements whose ratio is close to 1. By more complicated equalities with at least three elements, e.g., by 100 - 99 =? 0 or 1 - 2 + 3 - 4 =? -1, the choice of a denominator seems to be vague at all. This is why the relative error is uncertain in principle, has a very restricted domain of applicability, and is practically used in the simplest case only and very seldom for variables and functions.

10. The least square method [1] can give adequate results in very special cases only. Its deep analysis [2] by the principles of constructive philosophy, overmathematics, and other fundamental mathematical sciences has discovered many fundamental defects both in the essence (as causes) and in the applicability (as effects) of this method that is adequate in some rare special cases only and even in them needs thorough adequacy analysis. The method is based on the absolute error alone not invariant by equivalent transformations of a problem and ignores the possibly noncoinciding physical dimensions (units) of relations in a problem. The method does not correlate the deviations of the objects approximations from the approximated objects with these objects themselves, simply mixes those deviations without their adequately weighing, and considers equal changes of the squares of those deviations with relatively less and greater moduli (absolute values) as equivalent ones. The method foresees no iterating, is based on a fixed algorithm accepting no a priori flexibility, and provides no own a posteriori adapting. The method uses no invariant estimation of approximation, considers no different approximations, foresees no comparing different approximations, and considers no choosing the best approximation among different ones. These defects in the method essence lead to many fundamental shortcomings in its applicability. Among them are applicability sense loss by a set of equations with different physical dimensions (units), no objective sense of the result noninvariant by equivalent transformations of a problem, restricting the class of acceptable equivalent transformations of a problem, no essentially unique correction of applicability sense loss, possibly ignoring subproblems of a problem, paradoxical approximation, no analyzing the deviations of the result, no adequate estimating and evaluating its quality, no refining the results, no choice, and the best quasisolution illusion as the highest truth fully ungrounded and inadequate. Additionally consider the simplest least square method [1] 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 data depends on this preselection, ignores the remaining coordinates, and provides no coordinate system rotation invariance and hence no objective sense of the result. Moreover, the method is correct by constant approximation or no data scatter only and gives systematic errors increasing together with data scatter and the deviation (namely declination) of an approximation from a constant. Therefore, the least square method [1] has many fundamental defects both in the essence (as causes) and in the applicability (as effects), is adequate only in some rare special cases and even in them needs thorough adequacy analysis. Experimental data are inexact, and their amount is always taken greater than that of the parameters in an approximating function often geometrically interpretable by a straight line or curve, plane or surface. That is why this method was possibly the most important one for any data processing and seemed to be irreplaceable.

11. Further 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.

Therefore, the very fundamentals of classical applied mathematics [1] have a lot of obviously deep and even cardinal defects of principle.

Estimation

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:

the 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;

the 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;

the 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;

the fundamental science of reserves further naturally generalizing the fundamental science of concessions ae 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;

the 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;

the 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;

the fundamental science of 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.

Revolution in Applied Mathematics

Applied megamathematics [8-20] based on pure megamathematics [2-7] and on overmathematics [2-7] revolutionarily replaces the inadequate very fundamentals of classical pure mathematics [1] via adequate very fundamentals.

I. Unierror

Applied megamathematics [8-20] proposes an unierror irreproachably correcting the relative error and generalizing it possibly for any conceivable range of applicability. For a =? b, the linear estimating fraction is

δ_{a =?
b} = |a - b|/(|a| + |b|)

by |a| + |b| > 0, which should simply vanish by a = b = 0. Introduce extended division:

a//b = a/b by a ≠ 0 and a//b = 0 by a = 0

independently of the existence and value of b. Then

δ_{a =?
b} = |a - b|//(|a| + |b|).

The quadratic estimating fraction is

²δ_{a
=? b} = |a - b|//[2(a² + b²)]^1/2.

The outputs (return values) of such unierrors always belong to [0, 1]. By the principle of tolerable simplicity [2], it is reasonable to use the linear estimating fraction alone if it suffices.

Examples:

δ_{100 -
99 =? 0} = 1/199 = δ_{100 =? 99};

δ_{1 - 2 +
3 - 4 =? -1}= |1 - 2 + 3 - 4 + 1|/(1 + 2 + 3 + 4 + 1) = 1/11.

II. Reserve

The absolute error, the relative error, and the unierror of any exact object or model always vanish. It is often reasonable to additionally discriminate exact objects or models by the confidence in their exactness reliability. For example, both x₁ = 1 + 10^-10 and x₂ = 1 + 10¹⁰ are exact solutions to the inequation x > 1, x₁ practically unreliable and x₂ guaranteed. Their discrimination is especially important by any inexact data. Classical mathematics [1] cannot provide this at all.

Applied megamathematics [8-20] proposes for this purpose the basic concept of the reserve which is quite new in mathematics and extends the unierror in the following sense. The values of an unierror A belong to the segment [0, 1], those of a reserve R to [-1, 1]. For each inexact object I, A(I) > 0 and we can take R(I) = -A(I). For each exact object E, A(E) = 0 and R(E) ≥ 0. A proposition to determine the reserve of an inexact object as its unierror with the opposite sign is at once evident. For an exact object, it seems to be reasonable, to first define a suitable mapping of the object with respect to its exactness boundary and to further take the unierror of the mapped object. It is exact if and only if the object itself precisely lies on its exactness boundary where the reserve vanishes. Otherwise, the mapped object is inexact and the object itself has a positive reserve. For inequalities, such a mapping can be replaced with negating inequality relations and conserving equality ones. In our example, we have

R_x>1(x₁) = R_x>1(1 + 10^-10) = A_x<?1(1 + 10^-10) = 10^-10/(2 + 10^-10),

R_x>1(x₂) = R_x>1(1 + 10¹⁰) = A_x<?1 (1 + 10¹⁰) = 10¹⁰/(2 + 10¹⁰).

III. Reliability

Classical mathematics [1] brings key concepts that are very often insufficient, have evident lacks, and are not suitable for solving many typical problems. For example, to estimating reliability, a stochastic approach [1] often applies. Also in deterministic problems, their parameters are often artificially randomized. The corresponding distributions are considered known (even if they are really unknown) and most suitable for calculation. But even such a simplification brings complicated formulae and difficulties by analysis.

Applied megamathematics [8-20] based on the principles of constructive philosophy brings many new general sciences, theories, concepts, and methods suitable for research, engineering, and life. Among them is general reliability science.

The reliability of an object (e.g., element, structure, etc.) can be naturally defined as the probability that the object holds in some reasonable sense (e.g., exists and successfully functions during a given time interval under certain conditions such as loading, temperature, etc.). The above object can take so called values as its particular states, realizations, etc. which are any objects, too. A value is called admissible if the object taking this value holds. A value is called inadmissible if the object taking this value does not hold. A value is called limiting if it belongs to the common boundary of the admissible values and inadmissible ones. In other words, in each, arbitrarily small, neighborhood (in a certain reasonable sense) of such a value, there are both admissible and inadmissible values. A limiting value is either admissible or inadmissible.

The actual value of the object can deviate from its nominal value corresponding to available information always incomplete, usually inexact, and sometimes partially unreliable. This can be not only quantitatively, but also qualitatively significant. The last can especially hold for the limiting values and values near to those in some reasonable sense. For example, by such an admissible, limiting, or inadmissible nominal value, the actual one can be admissible, limiting, or inadmissible in any combination of these properties.

The introduced reserve makes it possible for the first time, to also deterministically estimate reliability in many types of problems and to provide relatively simple formulae very suitable for analysis. Let us determine reliability S via reserve R (whose values belong to the closed interval [-1; 1]) as S = (1 + R)/2. Then the range of S is the closed interval [0; 1] like that of probability, which is natural. For admissible values (states etc.) of the object that are infinitely far from their boundary (its limiting values) in certain reasonable sense, S = 1. For the limiting values themselves, S = 1/2. For inadmissible values infinitely far from their boundary (its limiting values) in certain reasonable sense, S = 0. All these values of reliability are natural and correspond to intuition. For a limiting value, due to possible deviations from it in the reality, the value of a parameter with risk can also be either admissible or inadmissible. It is reasonable to consider equal the measures, probabilities, and some similar estimates (e.g., uniquantities [2]) of the both kinds of the deviations. The same estimates of the limiting values only can be regarded as infinitely small in comparison with the previous both because the dimension of the above boundary (the set of the limiting values) is usually less than the dimension of the set of the admissible values and that of the inadmissible ones. In the one-dimensional case, e.g., for a unique real parameter, the boundary typically consists of some discrete values and is zero-dimensional whereas both the set of the admissible values and the set of the inadmissible values are one-dimensional. In our example, we have

S_x>1(x₁) = S_x>1(1 + 10^-10) = [1 + R_x>1(1 + 10^-10)]/2 =

[1 + 10^-10/(2 + 10^-10)]/2 = (1 + 10^-10)/(2 + 10^-10)

a little greater than 1/2 and

S_x>1(x₂) = S_x>1(1 + 10¹⁰) = [1 + R_x>1(1 + 10¹⁰)]/2 =

[1 + 10¹⁰/(2 + 10¹⁰)]/2 = (1 + 10¹⁰)/(2 + 10¹⁰)

a little smaller than 1, both in accordance with intuition.

Introduced reliability brings (also by distributions) adequate estimations of approximation quality, exactness confidence, and risk. General reliability science is very effective by setting, reasonably simulating, and solving many types of problems in research, engineering, and life.

IV. Risk

For many typical problems in science, engineering, and life, there are no concepts and methods adequate and general enough. Even artificially simplifying by randomizing parameters in deterministic problems to apply stochastic approaches in classical mathematics [1] to estimating risk brings complicated formulae and analysis difficulties.

The unierrors and reserves estimating and measuring also exactness, contradictoriness, and distributions. The unierrors correct and generalize the relative error. The reserves new in principle estimate exactness reliability. They both form a basis for general risk science.

The risk of an object (e.g., element, structure, etc.) can be naturally defined as the probability of the event that the object does not hold in some reasonable sense (e.g., cannot exist and/or successfully function during a given time interval under certain conditions such as loading, temperature, etc.). The above object can take so called values as its particular states, realizations, etc. which are any objects, too. A value is called admissible if the object taking this value holds. A value is called inadmissible if the object taking this value does not hold. A value is called limiting if it belongs to the common boundary of the admissible values and inadmissible ones. In other words, in each, arbitrarily small, neighborhood (in a certain reasonable sense) of such a value, there are both admissible and inadmissible values. A limiting value is either admissible or inadmissible.

The reserve [2] makes it possible for the first time, to also deterministically estimate risk in many types of problems in science, engineering, and life and to provide relatively simple formulae very suitable for analysis.

Let us determine risk r via reserve R (whose values belong to the closed interval [-1; 1]) [2] as

r = (1 - R)/2.

Then the range of r is the closed interval [0; 1] like that of the unierror (with certain sense similarity). By no risk we have r = 0. For a limiting value, r = 1/2. For inadmissible values infinitely far from a limiting value, r = 1. All these values of risk are natural and correspond to intuition. For a limiting value, due to possible deviations from it in the reality, the value of a parameter with risk can be either greater or less than the limiting value. It is reasonable to consider equal the probabilities, measures, and some similar estimates (e.g, uniquantities [2]) of the both kinds of the deviations whereas the same estimates of the only limiting value can be regarded as infinitely small in comparison with the previous both. The remaining cases r = 0 and r = 1 are trivial. In our example, we have

r_x>1(x₁) = r_x>1(1 + 10^-10) = [1 - R_x>1(1 + 10^-10)]/2 =

[1 - 10^-10/(2 + 10^-10)]/2 = 1/(2 + 10^-10)

a little smaller than 1/2 and

r_x>1(x₂) = r_x>1(1 + 10¹⁰) = [1 - R_x>1(1 + 10¹⁰)]/2 =

[1 - 10¹⁰/(2 + 10¹⁰)]/2 = 1/(2 + 10¹⁰)

very small, both in accordance with intuition.

Introduced risk brings (also by distributions) adequate estimations of approximation quality, exactness confidence, and reliability. General risk science is very effective by setting, reasonably simulating, and solving many types of problems in research, engineering, and life.

VII. Unimathematical Estimation Fundamental Sciences System. Summary

Applied megamathematics [8-20] based on pure megamathematics [2-7] and on overmathematics [2-7] 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 [16-19] including:

the fundamental science of reserves further naturally generalizing the 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;

the fundamental science of 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.

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

II. Unimathematical Approximation Fundamental Sciences System. Summary

Computational fundamental megascience [21-24] based on applied megamathematics [8-20] and hence on pure megamathematics [2-7] and on overmathematics [2-7] with its uninumbers, quantielements, quantisets, and uniquantities with quantioperations and quantirelations provides efficiently, universally and adequately strategically unimathematically modeling (expressing, representing, etc.) and processing (measuring, evaluating, estimating, approximating, calculating, etc.) data. This all creates the basis for many further fundamental sciences systems developing, extending, and applying overmathematics. Among them is, in particular, the unimathematical approximation fundamental sciences system [22] including:

the fundamental science of general approximation problem essence including general approximation problem type and setting theory, general approximation problem quantiobject theory, general approximation problem quantisystem theory, and general approximation problem quantioperation theory;

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

the fundamental science of general approximation problem solving strategy and tactic including general approximation problem solving strategy theory and general approximation problem solving tactic theory;

the fundamental science of general approximation problem transformation including general approximation problem transformation theory, general approximation problem structuring theory, general approximation problem restructuring theory, and general approximation problem partitioning theory;

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

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

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

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

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

the fundamental science of general approximation 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 unierroro equalization theory, reserve equalization theory, distance quantiinfimum theory, linear and square unierrors quantiinfimum theories, and reserve quantisupremum theory;

the fundamental science of general approximation problem solving methods including approximation subproblem subjoining theory, 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 unierrors power difference theories families, distance quantibound theory, linear and square unierrors quantibound theories, reserve quantibound theory, trial pseudosolution and direct solving theories families;

the fundamental science of general approximation 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;

the fundamental science of general approximation 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;

analytic solving fundamental science including general power solution theory, power analytic macroelement theory, and integral analytic macroelement theory;

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

the fundamental science of general approximation 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 unimathematical approximation fundamental sciences system is universal and very efficient.

VIII. General Problem Fundamental Sciences System. Summary

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 [8-20] based on pure megamathematics [2-7] and on overmathematics [2-7] 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 [20]. It defines a general problem as a quantisystem [2-7]

_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:

the fundamental science of 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;

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

the fundamental science of 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;

the 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;

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

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

the fundamental science of 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;

the 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;

the 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;

the fundamental science of 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;

the fundamental science of 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 unierrors power difference theories families, distance quantibound theory, linear and square unierrors quantibound theories, reserve quantibound theory, trial pseudosolution and direct solving theories families;

the fundamental science of 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;

the fundamental science of 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;

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

the 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.

Science Unimathematical Test Fundamental Metasciences Systems

Megamathematics including overmathematics [2] based on its uninumbers, quantielements, quantisets, and uniquantities with quantioperations and quantirelations provides efficiently, universally and adequately strategically unimathematically modeling, expressing, measuring, evaluating, and estimating general objects. This all creates the basis for many further megamathematics fundamental sciences systems developing, extending, and applying overmathematics. Among them are, in particular, science unimathematical test fundamental metasciences systems [3-17] including:

Science Unimathematical Test General Fundamental Metasciences System

Among science unimathematical test general fundamental metasciences are:

fundamental metascience of knowledge test philosophy, strategy, and tactic including knowledge test philosophy metatheory, knowledge test strategy metatheory, and knowledge test tactic metatheory;

fundamental metascience of knowledge consideration including knowledge fundamentals determination metatheory, knowledge approaches determination metatheory, knowledge methods determination metatheory, and knowledge conclusions determination metatheory;

fundamental metascience of knowledge analysis including subknowledge analysis metatheory, knowledge fundamentals analysis metatheory, knowledge approaches analysis metatheory, knowledge methods analysis metatheory, and knowledge conclusions analysis metatheory;

fundamental metascience of knowledge synthesis including knowledge fundamentals synthesis metatheory, knowledge approaches synthesis metatheory, knowledge methods synthesis metatheory, and knowledge conclusions synthesis metatheory;

fundamental metascience of knowledge objects, operations, relations, and criteria including knowledge object metatheory, knowledge operation metatheory, knowledge relation metatheory, and knowledge criterion metatheory;

fundamental metascience of knowledge evaluation, measurement, and estimation including knowledge evaluation metatheory, knowledge measurement metatheory, and knowledge estimation metatheory;

fundamental metascience of knowledge expression, modeling, and processing including knowledge expression metatheory, knowledge modeling metatheory, and knowledge processing metatheory;

fundamental metascience of knowledge symmetry and invariance including knowledge symmetry metatheory and knowledge invariance metatheory;

fundamental metascience of knowledge bounds and levels including knowledge bound metatheory and knowledge level metatheory;

fundamental metascience of knowledge directed test systems including knowledge test direction metatheory and knowledge test step metatheory;

fundamental metascience of knowledge tolerably simplest limiting, critical, and worst cases analysis and synthesis including knowledge tolerably simplest limiting cases analysis and synthesis metatheories, knowledge tolerably simplest critical cases analysis and synthesis metatheories, knowledge tolerably simplest worst cases analysis and synthesis metatheories, and knowledge tolerably simplest limiting, critical, and worst cases counterexamples building metatheories;

fundamental metascience of knowledge defects, mistakes, errors, reserves, reliability, and risk including knowledge defect metatheory, knowledge mistake metatheory, knowledge error metatheory, knowledge reserve metatheory, knowledge reliability metatheory, and knowledge risk metatheory;

fundamental metascience of knowledge test result evaluation, measurement, estimation, and conclusion including knowledge test result evaluation metatheory, knowledge test result measurement metatheory, knowledge test result estimation metatheory, and knowledge test result conclusion metatheory;

fundamental metascience of knowledge supplement, improvement, modernization, variation, modification, correction, transformation, generalization, and replacement including knowledge supplement metatheory, knowledge improvement metatheory, knowledge modernization metatheory, knowledge variation metatheory, knowledge modification metatheory, knowledge correction metatheory, knowledge transformation metatheory, knowledge generalization metatheory, and knowledge replacement metatheory;

fundamental metametascience of unimathematical test fundamental metasciences systems application including megascience development metatheory, megamathematics and megastrength fundamental mathematical, mechanical, strength, and physical sciences systems development metatheories.

Science Unimathematical Test Special Fundamental Metasciences System

Among science unimathematical test special fundamental metasciences are, e.g.:

Pure Mathematics Unimathematical Test Fundamental Metascience

Pure mathematics unimathematical test fundamental metascience includes:

metatheory of pure mathematics test philosophy, strategy, and tactic;

metatheory of pure mathematics consideration;

metatheory of pure mathematics analysis;

metatheory of pure mathematics synthesis;

metatheory of pure mathematics objects, operations, relations, and criteria;

metatheory of pure mathematics evaluation, measurement, and estimation;

metatheory of pure mathematics expression, modeling, and processing;

metatheory of pure mathematics symmetry and invariance;

metatheory of pure mathematics bounds and levels;

metatheory of pure mathematics directed test systems;

metatheory of pure mathematics tolerably simplest limiting, critical, and worst cases analysis and synthesis;

metatheory of pure mathematics defects, mistakes, errors, reserves, reliability, and risk;

metatheory of pure mathematics test result evaluation, measurement, estimation, and conclusion;

metatheory of pure mathematics fundamentals development, supplement, improvement, modernization, variation, modification, correction, transformation, generalization, and replacement.

Applied Mathematics Unimathematical Test Fundamental Metascience

Applied mathematics unimathematical test fundamental metascience includes:

metatheory of applied mathematics test philosophy, strategy, and tactic;

metatheory of applied mathematics consideration;

metatheory of applied mathematics analysis;

metatheory of applied mathematics synthesis;

metatheory of applied mathematics objects, operations, relations, and criteria;

metatheory of applied mathematics evaluation, measurement, and estimation;

metatheory of applied mathematics expression, modeling, and processing;

metatheory of applied mathematics symmetry and invariance;

metatheory of applied mathematics bounds and levels;

metatheory of applied mathematics directed test systems;

metatheory of applied mathematics tolerably simplest limiting, critical, and worst cases analysis and synthesis;

metatheory of applied mathematics defects, mistakes, errors, reserves, reliability, and risk;

metatheory of applied mathematics test result evaluation, measurement, estimation, and conclusion;

metatheory of applied mathematics fundamentals development, supplement, improvement, modernization, variation, modification, correction, transformation, generalization, and replacement.

Computational Science Unimathematical Test Fundamental Metascience

Computational science unimathematical test fundamental metascience includes:

metatheory of computational science test philosophy, strategy, and tactic;

metatheory of computational science consideration;

metatheory of computational science analysis;

metatheory of computational science synthesis;

metatheory of computational science objects, operations, relations, and criteria;

metatheory of computational science evaluation, measurement, and estimation;

metatheory of computational science expression, modeling, and processing;

metatheory of computational science symmetry and invariance;

metatheory of computational science bounds and levels;

metatheory of computational science directed test systems;

metatheory of computational science tolerably simplest limiting, critical, and worst cases analysis and synthesis;

metatheory of computational science defects, mistakes, errors, reserves, reliability, and risk;

metatheory of computational science test result evaluation, measurement, estimation, and conclusion;

metatheory of computational science fundamentals development, supplement, improvement, modernization, variation, modification, correction, transformation, generalization, and replacement.

Material Strength Science Unimathematical Test Fundamental Metascience

Material strength science unimathematical test fundamental metascience includes:

metatheory of material strength science test philosophy, strategy, and tactic;

metatheory of material strength science consideration;

metatheory of material strength science analysis;

metatheory of material strength science synthesis;

metatheory of material strength science objects, operations, relations, and criteria;

metatheory of material strength science evaluation, measurement, and estimation;

metatheory of material strength science expression, modeling, and processing;

metatheory of material strength science symmetry and invariance;

metatheory of material strength science bounds and levels;

metatheory of material strength science directed test systems;

metatheory of material strength science tolerably simplest limiting, critical, and worst cases analysis and synthesis;

metatheory of material strength science defects, mistakes, errors, reserves, reliability, and risk;

metatheory of material strength science test result evaluation, measurement, estimation, and conclusion;

metatheory of material strength science fundamentals development, supplement, improvement, modernization, variation, modification, correction, transformation, generalization, and replacement.

Object Strength Science Unimathematical Test Fundamental Metascience

Object strength science unimathematical test fundamental metascience includes:

metatheory of object strength science test philosophy, strategy, and tactic;

metatheory of object strength science consideration;

metatheory of object strength science analysis;

metatheory of object strength science synthesis;

metatheory of object strength science objects, operations, relations, and criteria;

metatheory of object strength science evaluation, measurement, and estimation;

metatheory of object strength science expression, modeling, and processing;

metatheory of object strength science symmetry and invariance;

metatheory of object strength science bounds and levels;

metatheory of object strength science directed test systems;

metatheory of object strength science tolerably simplest limiting, critical, and worst cases analysis and synthesis;

metatheory of object strength science defects, mistakes, errors, reserves, reliability, and risk;

metatheory of object strength science test result evaluation, measurement, estimation, and conclusion;

metatheory of object strength science fundamentals development, supplement, improvement, modernization, variation, modification, correction, transformation, generalization, and replacement.

Science unimathematical test fundamental metasciences systems are universal and very efficient.

Overmathematics: Fundamental Principles, Theories, Methods, and Laws of Science

(former Elastic Mathematics: Theoretical Fundamentals)

Monograph

Parts 4, 5, and 6

Chapter 4.2.

General Approximation Theories

4.2.1. Basic Theory

of General Inexactness

Known inexactness in classical mathematics [1] is only qualitative and not general enough.

Definition A general object is exact if it satisfies all the requirements under a given general consideration.

Definition A general object is inexact if it satisfies not all the requirements under a given general consideration.

Definition A general object is generally inexact if it can be either exact or inexact.

Definition A general inexactness estimator, or unierror estimator, is a quantidiscriminator of exact, inexact and generally inexact general objects.

Definition A inexactly equivalent quantitransformator is a quanticorrespondence conserving a general inexactness.

Definition A quanti-inexactificator is a quanticorrespondence whose images are generally inexact.

4.2.2. Basic Theory of General Approximators

Known approximations in classical mathematics [1] are indeterminate and not sufficiently general.

Definition A general object is generally approximate if its general inexactness is acceptable in a given consideration.

Definition A general approximation to a general object is such a general object that their equality is generally approximate.

Definition A general approximator is a quanticorrespondence in which any image is a general approximation of its preimage.

4.2.3. Basic Theory of Unierrors

Absolute errors in classical mathematics [1] express no degree of inexactness.

Relative errors in classical mathematics [1] are indeterminate over the choice of denominators, their domains are bounded, and their ranges are unbounded (e.g., for

0 =? 1,

-1 =? 1

where =? denotes a generally inexact equality), which is contrary to intuition.

Definition An unierror (system) estimator, or (quantisystem) autoerrer, is a (quantisystem) quantiestimator

E : A → E(A) = (E(a) | a ∈ A)

that distinguishes inexact general objects, possibly vanishes for exact general objects, and has a quantirange quantisystem whose generative quantiset is included into [0I1].

Remark A unierror estimator/errer can be flexibly constructed [11, 12, 20, 25, 26] for a given problem according to the principle of tolerable simplicity.

Examples

E(a =? b) = 0 if a = b = 0,

E(a =? b) = ||a - b|| / (||a|| + ||b||) else;

Eα(a =? b) = ||a - b|| / (||a|| + ||b|| + αu)

for any a , b

where

u may be a physical unit of a , b ,

α > 0 maybe provides

supα(sup{Eα(a) | a ∈ A} -

inf{Eα(a) | a ∈ A}).

Remark E0 provides no distinguishing different x ≠ 0 in

x =? 0.

Example

Eα(∑j∈J aj =? 0) =

||∑j∈J aj|| / (α + ∑j∈J ||aj||).

Example

Let us consider the equation

(1)

Lλ[φ∈Φ fφ [ω∈Ω zω]] = 0 (λ ∈ Λ)

where

Lλ is a given operator having an index λ from a set Λ ;

fφ is an unknown desired function having an index φ from a set Φ ;

zω is an independent variable having an index ω from a set Ω ;

[ω∈Ω zω]

is a set of indexed elements.

The local unierror may be defined by the formula

(2)

Eλ[ω∈Ω zω] =

αλ ||Lλ’[ω∈Ω zω]||λ /

(||Lλ’[ω∈Ω zω]||λ + αλ ||Lλ“[ω∈Ω zω]||λ) +

βλ ||Lλ’[ω∈Ω zω]||λ /

(|||Lλ’[ω∈Ω zω]|||λ + βλ |||Lλ“[ω∈Ω zω]|||λ) +

γλ ||Lλ’[ω∈Ω zω]||λ /

(sup||Lλ’[ω∈Ω zω]||λ + γλ sup ||Lλ“[ω∈Ω zω]||λ)

where

αλ , βλ , γλ

are positive uninumbers, their sum be equal to 1;

αλ , βλ , γλ

are positive uninumbers;

Lλ’[ω∈Ω zω]

is the left-hand side of (1) as a direct (not composite) function of the independent variables;

the both sup are taken in the domain of definition zλ of the equation;

|||Lλ’[ω∈Ω zω]|||λ

is the usual least upper bound on the norm of

||Lλ’[ω∈Ω zω]||λ

when all possibly different isometric (conserving the norms) transformations even of equal elements in

Lλ’[ω∈Ω zω]

are considered;

Lλ“[ω∈Ω zω]

is some function that is chosen (along with

αλ , βλ , γλ ,

αλ , βλ , γλ)

by the principle of tolerable simplicity so that the estimation (2) is the most sensitive one over the set of the classes of the functions fφ under consideration, i.e., the difference

sup δ - inf δ

has the greatest possible value.

Example

If in (1), the equality sign is replaced by an inequality sign, let

Eλ[ω∈Ω zω] = 0

if the inequation is true, and let us use the formula (2) otherwise.

Example

For a comparison

a ≡ b (mod d)

(i.e.,

(a - b)/d

is an integer),

where

a , b , d

are complex numbers

(d ≠ 0),

the unierror may be given by the formula

E = min({|a - b| / |d|}, 1 - {|a - b| / |d|})

where {x} is the fractional part of a real number x .

Example

The unierror of any pseudosolution x to the combined inequalities

[α∈Αaα ⇐ x ⇐ bβ β∈Β]

where

a , b , x

are real numbers;

⇐ is one of the two inequality signs

< , ≤ ,

can be defined as 0 if all the inequalities are true; otherwise

E(x , [α∈Αaα ⇐ x ⇐ bβ β∈Β]) =

supα∈Α , β∈Β ((aα - x)/(|x| + 2|aα| + 1),

(x - bβ)/(|x| + 2|bβ| + 1)).

Definition The unierror E(p|P) of a pseudosolution p to a general problem P is the unierror of all its input data P(p) after substituting that pseudosolution.

4.2.4. Basic Theory of Reserves

Even a unierror estimator cannot distinguish any exact general objects whose exactness reserves can differ. Compare, e.g., solutions

x1 = 1 + 10-10

and

x2 = 1 + 1010

to the inequation

x > 1

[11, 12, 21, 26, 27].

Definition A reserver is a general estimator distinguishing exact general objects by their reserves.

Definition An errer-reserver is a quantiestimator distinguishing both inexact and exact general objects by their unierrors-reserves possibly included into [-1I0] and [0I1], respectively.

Remark An errer-reserver is simultaneously, or a generalization of, an autoerrer and a reserver.

Remark A(n) (autoerrer-)reserver can be naturally and flexibly constructed using the corresponding autoerrer by accepting

R(a) ::= -E(a)

if and only if a is inexact.

Example

The reserve of any pseudosolution x to the combined inequalities

[α∈Αaα ⇐ x ⇐ bβ β∈Β]

where

a , b , x

are real numbers;

⇐ is one of the two inequality signs

< , ≤ ,

can be defined as

R(x, [α∈Αaα ⇐ x ⇐ bβ β∈Β]) =

infα∈Α, β∈Β ((x - aα)/(|x| + 2|aα| + 1),

(bβ - x)/(|x| + 2|bβ| + 1)).

Example

For the previous solutions

x1 = 1 + 10-10

and

x2 = 1 + 1010

to inequation

x > 1,

we have

Rx>1(x1) =

Rx>1(1 + 10-10) =

10-10/(2 + 10-10)

and

Rx>1(x2) =

Rx>1(1 + 1010) =

1010/(2 + 1010).

4.2.5. Basic Theory of Reliabilities

Definition The reliability of a certain general object is the probability that the general object holds in certain reasonable sense (e.g., exists and successfully functions during a certain time interval under certain conditions such as loading, temperature, competition, struggle, war, etc.). The above object can take so called general values as its particular states, realizations, etc. which are any general objects, too.

Definition A general value is admissible if the general object taking this general value holds.

Definition A general value is inadmissible if the general object taking this value does not hold.

Definition A general value is called limiting if it belongs to the common boundary of the admissible general values and inadmissible ones. In other words, in each, arbitrarily small, neighborhood (in a certain reasonable sense) of such a general value, there are both admissible and inadmissible general values.

Remark An actual general value of a general object can deviate from its nominal general value corresponding to available information always incomplete, usually inexact, and sometimes partially unreliable. This can be not only quantitatively, but also qualitatively significant. The last can especially hold for limiting general values and general values near to those in some reasonable sense.

Example By such an admissible, limiting, or inadmissible nominal general value, the actual one can be admissible, limiting, or inadmissible in any combination of these properties.

Remark The introduced reserve makes it possible for the first time, to also deterministically define and estimate reliability in many types of problems in reliability theory, economics, etc. and to provide relatively simple formulae very suitable for analyzing.

Example Let us quantitatively define reliability S via reserve R as

S = (1 + R)/2.

Then the range of S is the closed interval [0, 1] like that of probability, whish is natural. For admissible general values (states etc.) of a general object that are infinitely far from their boundary (its limiting general values) in certain reasonable sense,

S = 1.

For the limiting general values themselves,

S = 1/2.

For inadmissible general values infinitely far from their boundary (the limiting general values of a general object) in certain reasonable sense,

S = 0.

Remark All these values of reliability are natural and correspond to intuition. For a limiting general value, due to possible deviations from it in the reality, a general value of a general object with risk can also be either admissible or inadmissible. It is reasonable to consider equal the measures, probabilities, or some similar estimates (e.g., uniquantities) of the both kinds of the deviations. The same estimates of the limiting general values only can be regarded as infinitely small in comparison with the previous both because the dimension of the above boundary (the set of the limiting general values) is usually less than the dimension of the set of the admissible general values and that of the inadmissible ones.

Example In the one-dimensional case, e.g., for a unique real parameter, the boundary typically consists of some discrete values and is zero-dimensional whereas both the set of the admissible values and that of the inadmissible ones are one-dimensional.

Example

For the previous solutions

x1 = 1 + 10-10

and

x2 = 1 + 1010

to inequation

x > 1,

we have

Sx>1(x1) =

Sx>1(1 + 10-10) =

(1 + Rx>1(1 + 10-10))/2 =

(1 + 10-10/(2 + 10-10))/2 =

(1 + 10-10)/(2 + 10-10)

a little greater than 1/2 and

Sx>1(x2) =

Sx>1(1 + 1010) =

(1 + Rx>1(1 + 1010))/2 =

(1 + 1010/(2 + 1010))/2 =

(1 + 1010)/(2 + 1010)

a little smaller than 1, both in accordance with intuition.

4.2.6. Basic Theory of Risks

Definition The risk of a certain general object is the reliability of the object adverse to the given one in certain reasonable sense.

Remark The introduced reserve makes it possible for the first time, to also deterministically estimate risk in many types of problems in risk theory, economics, etc. and to provide relatively simple formulae very suitable for analyzing.

Example Let us define risk r as

r = (1 - R)/2.

Then the range of r is the closed interval [0, 1] like that of the above unierror with certain sense similarity. By no risk, we have

r = 0.

For a limiting general value,

r = 1/2.

For inadmissible general values infinitely far from a general limiting value,

r = 1.

All these values of risk are natural and correspond to intuition. For a limiting general value, due to possible deviations from it in the reality, a general value of a general object with risk can be also nonlimiting. It is reasonable to consider equal the probabilities, measures or some similar characteristics (e.g., uniquantities) of the both kinds of the deviations whereas the same characteristic of the only limiting value can be regarded as infinitely small in comparison with the previous both. The remaining cases r = 0 and r = 1 are trivial.

Example

For the previous solutions

x1 = 1 + 10-10

and

x2 = 1 + 1010

to the inequation

x > 1,

we have

rx>1(x1) =

rx>1(1 + 10-10) =

(1 - Rx>1(1 + 10-10))/2 =

(1 - 10-10/(2 + 10-10))/2 =

1/(2 + 10-10)

a little smaller than 1/2 and

rx>1(x2) =

rx>1(1 + 1010) =

(1 - Rx>1(1 + 1010))/2 =

(1 - 1010/(2 + 1010))/2 =

1/(2 + 1010)

very small, both in accordance with intuition.

4.2.7. Basic Theory of Quantinonlinearity

Known nonlinearity in classical mathematics [1] is qualitative only and not sufficiently general.

Definition A quantinonlinearity estimator is a quantiestimator NL (L is a quantisystem of some linear systems) replacing a quantisystem A by its quantinonlinearity

NL(A) = inf{E(A =? L) | L ∈ L}.

Definition A quantinonlinearity system estimator is a quantisystem estimator NL (L is a quantisystem of some linear systems) replacing a quantisystem A by its quantinonlinearity system

NL(A) = inf{E (A =? L) | L ∈ L}.

Part 5. General Problems

Chapter 5.1. General Concepts

5.1.1. Basic Theory

of General Problems

Definition A general problem is an incompletely known, or given, quantisystem.

Example The quantisystem (see 4.2.3) of quantirelations rλ

(**)

Lλ[φ∈Φ fφ [ω∈Ω zω]] rλ 0 (λ∈Λ)

generalizes many diverse known problems [1, 5-7].

Definition A solution to a general problem is a general object whose substitution into that problem makes it a known quantisystem that is consistent.

Definition A pseudosolution to a general problem is a general object whose substitution into that problem makes it a known quantisystem that has an actual sense.

Definition An error quasisolution to a general problem is its pseudosolution whose unierror is not greater than that of any other pseudosolution.

Definition A reserve quasisolution to a general problem is its pseudosolution whose reserve is not less than that of any other pseudosolution.

Definition An error antisolution to a general problem is its pseudosolution whose unierror is not less than that of any other pseudosolution.

Definition A reserve antisolution to a general problem is its pseudosolution whose reserve is not greater than that of any other pseudosolution.

Definition A supersolution to a general problem is its solution that is its reserve quasisolution.

Definition The general solution to a general problem is the quantiset of all its solutions with their own, or individual, quantities.

Definition The general pseudosolution to a general problem is the quantiset of all its pseudosolutions with their own, or individual, quantities.

Definition The unierror quasisolution to a general problem is the quantiset of all its error quasisolutions with their own, or individual, quantities.

Definition The reserve quasisolution to a general problem is the quantiset of all its reserve quasisolutions with their own, or individual, quantities.

Definition The unierror-reserve quasisolution to a general problem is the quantiset of all its error-reserve quasisolutions with their own, or individual, quantities.

Definition The general supersolution to a general problem is the quantiset of all its supersolutions with their own, or individual, quantities.

Definition The unierror of a general problem is inf of the unierrors in its general pseudosolution.

Definition The reserve of a general problem is sup of the general reserves in its general pseudosolution.

Definition General problems are equivalent if all the following conditions hold:

1) their general solutions coincide,

2) their general pseudosolutions coincide,

3) their unierror quasisolutions coincide,

4) their reserve quasisolutions coincide,

5) their unierror-reserve quasisolutions coincide,

6) their supersolutions coincide.

Definition Equivalent general problems are superequivalent if by any of their common quantiestimators, the unierrors and reserves of any of their (common) pseudosolutions coincide, respectively.

5.1.2. Basic Theory of General Typification

Problems are usually solved [1] as separate ones without their hierarchy.

Definition A basic subtype of some general problems is their subtype whose solvability ensures the solvability of their whole type.

Definition A basic quantisubsystem of some general problems is their quantisubsystem whose solvability ensures the solvability of their whole quantisystem.

Definition A linearly basic subtype of a type of general problems is their basic subtype whose ordinary linear quanticombinations cover that type.

Definition A linearly basic quantisubsystem of a quantisubsystem of general problems is their basic quantisubsystem whose ordinary linear quanticombinations cover that quantisystem.

Chapter 5.2. General Methods

5.2.1. Parameterization Method

Definition A parametric aggregate of quantisystems is their quantiset in which they are distinguished by the quantisystems of general values of quantiparameters.

Definition The parameterization method is a general method of solving general problems in some parametric aggregate of quantisystems.

Example The punctiformization method of solving continual general problems.

5.2.2. Linear-Combination Method

Definition An eigenaggregate for a quantisystem of quanticorrespondences is a quantisystem of their quantidomain subsystems on which any image in any correspondence is an ordinary homogeneous linear quanticombination of a quantilinearly independent quantisystem which can be individual for that correspondence.

Definition The linear-combination method is the parameterization method applied to an eigenaggregate for a considered general problem.

Example For a general problem (see (1) in 4.2.3)

Lλ[φ∈Φ fφ [ω∈Ω zω]] rλ 0 (λ∈Λ),

an eigenaggregate includes not only all the eigenfunctions [1].

5.2.3. Restructurization Method

Definition The restructurization method is a general method of solving a general problem by its preliminary quantirestructurizing.

Example If a general problem is a quantisystem of nonequicomplicated quantirelations then its most complete solvable decision subsystem of relatively simpler quantirelations gives an explicit quasisolution (of that problem) whose unierror or/and reserve is/are estimated over the rest estimation subsystem [11, 12, 20, 25].

5.2.4. Unierror Minimean Method

The method of least squares in classical mathematics [1] is unique known applicable to generally inconsistent problems but

1) it loses any sense if quantidimensions are not common

2) reasonable reductions to a common unit can give diverse results

3) a result depends on a complicated multiplication of a problem

4) equations with relatively small coefficients can be almost ignored

5) the greater an absolute value, the less even an absolute error

6) the second power can be not enough for over three relations

7) a result cannot be specified by this method.

Definition The unierror minimean method is an iterative general method of solving a general problem by generally minimizing a weighted mean modified unierror (quantisystem)

Mn(Eλ , n+1 | λ ∈ Λ)

where Eλ , n+1 is the unierror of the n+1st approximation (n ∈ N) to an unierror quasisolution to that problem by its λth quantirelation with respect to the nth approximation (see [11, 26], 4.2.3).

Example

Eλ , n+1[φ∈Φ fφ [ω∈Ω zω]] =

αλ ||L’λ , n+1[ω∈Ω zω]||λ /

(||L’λn[ω∈Ω zω]||λ + α’λ ||L“λn[ω∈Ω zω]||λ) +

βλ ||L’λ , n+1[ω∈Ω zω]||λ /

(|||L’λn[ω∈Ω zω]|||λ + β’λ |||L“λn[ω∈Ω zω]|||λ) +

γλ ||L’λ , n+1[ω∈Ω zω]||λ /

(sup||L’λn[ω∈Ω zω]||λ + γλ sup ||L“λn[ω∈Ω zω]||λ)

where

αλ , βλ , γλ

are positive uninumbers, their sum be equal to 1;

αλ , βλ , γλ

are positive uninumbers;

Lλ’[ω∈Ω zω]

is the left-hand side of (1) (see 4.2.3) as a direct (not composite) function of the independent variables;

the both sup are taken in the domain of definition zl of the equation;

|||Lλ’[ω∈Ω zω]|||λ

is the usual least upper bound on the norm of

||Lλ’[ω∈Ω zω]||λ

when all possibly different isometric (conserving the norms) transformations even of equal elements in

Lλ’[ω∈Ω zω]

are considered;

Lλ“[ω∈Ω zω]

is some function that is chosen (along with

αλ , βλ , γλ ,

αλ , βλ , γλ)

by the principle of tolerable simplicity so that the above estimation Eλ, n + 1 is the most sensitive one over the set of the classes of the functions fφ under consideration, i.e., the difference

sup δ - inf δ

has the greatest possible value.

Example The method of least normalized squares (n = 2) uses the Gauß formulae [1].

5.2.5. Unierror Minimax Method

Definition The error minimax method is an iterative general method of solving a general problem by generally minimizing

sup(wλEλ , n+1 | λ ∈ Λ).

5.2.6. Unierror Equalizing Method

Definition The unierror equalizing method is an iterative general method of solving a general problem by generally equalizing the most complete subsystem of possibly greater

wλEλ , n+1 (λ ∈ Λ)

[11, 12, 26].

5.2.7. Reserve Maximean Method

Definition The reserve maximean method is an iterative general method of solving a general problem by generally maximizing

Mn(Rλ , n+1 | λ ∈ Λ)

where

n ∈ {1, 3, 5, ...},

Rλ , n+1

is a modified reserve of the n+1st approximation (n ∈ N) to a reserve quasisolution to that problem with respect to its nth approximation.

5.2.8. Reserve Maximin Method

Definition The reserve maximin method is an iterative general method of solving a general problem by generally maximizing

inf(wλEλ , n+1 | λ∈Λ).

5.2.9. Reserve Equalizing Method

Definition The reserve equalizing method is an iterative general method of solving a general problem by generally equalizing the most complete subsystem of possibly less

wλEλ , n+1 (λ ∈ Λ)

[12].

Chapter 5.3. General Theories

5.3.1. Basic Theory

of General Method Choice

The most effective general method can be chosen as follows:

1) If a continual general problem permits any parameterization, the parameterization method can be used.

2) A generally linear general problem can be solved by the linear-combination method.

3) For a quantisystem of nonequicomplicated quantirelations, the restructurization method is usable.

4) If a general problem can be inconsistent and estimating its solutions is optional then the error minimean/minimax/equalizing method can be used.

5) If a general problem can be inconsistent and estimating its solutions is necessary then the reserve maximean/maximin/equalizing method is usable.

Remark Each general method can be more effectively used after the preliminary typification of a general problem to be solved.

Example To the overdetermined set of the combined equations

29 x + 21 y = 50,

50 x -17 y = 33,

x + 2y = 7,

2 x - 3 y = 0

with the solution

x = 1,

y = 1

to the subset of the first two equations and the solution

x = 3,

y = 2

to the subset of the last two equations, the both subsets being determined, the method of least squares [1] gives the final quasisolution

x≈ 1.00234,

y ≈ 1.00750

with the unierror

E ≈ 0.286

almost ignoring the last two equations with relatively small factors.

The method of least normalized squares after the unit 0th approximation

x0 = 1,

y0 = 1

gives

x1 ≈ 1.263,

y1 ≈ 1.051

with the unierror

E1 ≈ 0.250

and its further correction by this method is unessential because the number of the equations is 4 > 3.

The error/reserve equalizing method gives

x3 ≈ 1.599,

y3 ≈ 1.467

with the unierror

E3 ≈ 0.192

further stabilized so that

Eℵ ≈ 0.192

is this problem's error.

5.3.2. Basic Theory

of Best General Approximations

Definition A best general approximation to an explicit general object e in an approximation quantisystem A is such an a ∈ A that for an intuitively justified autoerrer E with

E(a) ::= E(a =? e)

all the following relations hold:

1) E(a=) ≥ E(a) for any a= ∈ A of the same complication as a ,

2) E(a<) >> E(a) for any a< ∈ A of a less complication as a ,

3) E(a>) << E(a) for no a> ∈ A of a greater complication as a .

Definition A best unierror approximation to an implicit general object i that has to be a (possibly inexistent) solution to a (generally inconsistent) general problem P in an approximation quantisystem A is such an a ∈ A that for an intuitively justified autoerrer E with

E(a) ::= E(P(a)),

all the following relations hold:

1) E(a= ) ≥ E(a) for any a= ∈ A of the same complication as a ,

2) E(a<) >> E(a) for any a< ∈ A of a less complication as a ,

3) E(a>) << E(a) for no a> ∈ A of a greater complication as a .

Definition A best reserve approximation to an implicit general object i that has to be a (possibly inexistent) solution to a (generally inconsistent) general problem P in an approximation quantisystem A is such an a ∈ A that for an intuitively justified reserver R with

R(a) ::= R(P(a)),

all the following relations hold:

1) R(a=) ≤ R(a) for any a= ∈ A of the same complication as a ,

2) R(a<) << R(a) for any a< ∈ A of a less complication as a ,

3) R(a>) >> R(a) for no a> ∈ A of a greater complication as a .

Definition The best (error/reserve) general approximation is the quantiset of all the corresponding best approximations.

Example The best linear approximation to the set of 4 points

{(xi, yi) | i ∈ {1, 2, 3, 4}}° =

{(-1, -1), (1, 1), (8, 10), (12, 10)}

y = kx + b

with

k = 1,

b = 0.

The unique known applicable method of least squares [1] gives the final result (see 5.3.1)

k ≈ 0.927,

b ≈ 0.364

with

E ≈ 0.146.

k0 = 1,

b0 = 1,

then the method of least normalized squares gives

k1 ≈ 0.998,

b1 ≈ 0.008

with

E1 ≈ 0.086.

5.3.3. Basic Theory

of Correcting Quasisolving

Definition A best unierror quasisolution to a general problem is its pseudosolution that is a best unierror approximation to some its (implicit, possibly inexistent) solution.

Definition A best reserve quasisolution to a general problem is its pseudosolution that is a best reserve approximation to some its (implicit, possibly inexistent) solution.

Definition The best unierror quasisolution to a general problem is the quantiset of all its best error quasisolutions.

Definition The best reserve quasisolution to a general problem is the quantiset of all its best reserve quasisolutions.

Definition Correctingly quasisolving a generally inexact general problem is determining its best unierror and/or reserve quasisolution.

Remark The simpler, the truer.

Example Correcting quasisolving the Volterra nonhomogeneous integral equation for inverting the Steklov operator [1]

MΔ : p(s) → pΔ(s) = Δ-1∫s-Δ/2s+Δ/2 p(t)dt

is realizable by means of standard transformations [11, 20] such as

MΔ(as + b) = as + b

for a linear function;

MΔ(p(s)) = sh(0.5nΔ)/(0.5nΔ) p(s)

for

p(s) ∈ {exp(ns), sh ns, ch ns};

MΔ(p(s)) = sin(0.5nΔ)/(0.5nΔ) p(s)

for

p(s) ∈ {sin ns, cos ns}.

Part 6. Applied Methods

and General Results

Chapter 6.1. Applications

of the General Methods

6.1.1. Basic Theory of Reasonable

Mathematical Simulation

Definition A reasonable mathematical simulation of a general problem is its best approximation in the approximation quantisystem of all the general problems solvable by the quantisystem both of all the available methods and of all the individual methods whose creation is possible and reasonable in given solving.

6.1.2. Basic Theory

of Solving General Problems

Definition Reasonable solving a general problem is its correcting quasisolving after its preliminary reasonable mathematical simulation by the quantisystem of all the available general and creatable individual methods.

6.1.3. Summation Methods

for Divergent Series

DefinitionA quantisum of a quantiseries

∑i∈N ai

by one of the following summation methods [12] is a general object s generally minimizing the corresponding quasielement of the Cartesian product

{supn∈N , limn∈N´, limn∈N´} ×

{

||s - ∑i=1n ai|| / (p + ||s - ∑i=1n ai||),

(n-1∑i=1n (||s - ∑i=1n ai || /

(p + ||s - ∑i=1n ai ||))q(n))1/q(n)

}

where

p is a positive uninumber,

lim is the upper limit,

q(n) is a positive function of n , which can be, in particular, n itself or a constant q .

6.1.4. Basic Theory of Solving

Linear General Problems

Only particular solutions to many urgent even linear equations ((bi)harmonic etc.) are known in classical mathematics [1].

The basic theory of solving linear general problems is based on the parametrization method and the linear-combination method and includes

1) determination of a reasonable general pseudosolution,

2) quantiparameterization of that quantiset,

3) substitution of that quantiparametric expression, and

4) determination of the general solution to a given general problem.

Example The most general solution to the homogeneous harmonic equation

∇2φ(x , y , z) = 0

over desired function φ(x , y , z) in the Cartesian coordinates, x , y , z , in the class of power functions (power series) (or, equivalently, in a general pseudosolution

φ(x , y , z) = ∑i=0∞∑j=0∞∑k = 0∞aijkxiyjzk)

due to the linear-combination method is [11, 25]

φ(x , y , z) =

∑i=0∞∑j=0∞∑k=0∞ (-1)[i/2] [i/2]!(i!j!k!)-1 ×

∑l=0[i/2] (j + 2l)!(k + 2[i/2] - 2l)! ×

(l! ([i/2] - l)!)-1 ×

ai-2[i/2], j+2l , k+2[i/2]-2l xiyjzk

where

[m] = entier m

is the integral part of a real number m ,

a0jk and a1jk are two arbitrary number sequences, 0 ≤ j , k < ∞ .

Example The most general solution to the homogeneous biharmonic equation

∇2∇2L(r , z) = 0

over desired function L(r , z) in the cylindrical coordinates, r, z, in the class of power functions (power series) (or, equivalently, in a general pseudosolution

L(r , z) = ∑i=0∞∑j=0∞aijr2izj)

due to the linear-combination method is [11, 20, 25]

L(r , z) =

∑i=0∞∑j=0∞ (-1)i+1 i!-2 j!-1 ×

[(2i + j - 2)! i 22-2ia1, 2i+j-2 +

(2i + j)! (i -1) 2-2ia0, 2i+j] r2izj

where conventionally by k < 0 we consider

k! = 1

and

a1k = 0;

a0j , a1j are two arbitrary number sequences, 0 ≤ j < ∞.

6.1.5. Direct Decomposition Method

Definition The direct decomposition method is the special case [11, 20] of the theory of general typification when a general problem of its general type can be explicitly expressed over some general problems in its basic subtype.

Remark A linearly basic subtype of general problems is often used.

6.1.6. Balanced Decomposition Method

Definition The general balance in a general problem is its quantisubsystem of all the available requirements.

Definition The balanced decomposition method is the application [11] of the theory of general typification when a general problem of its general type can be balancedly decomposed, or expressed over some generally balanced general problems in a basic subtype.

6.1.7. Preliminary Correction Method

Definition The preliminary correction method is the application [11] of the theory of general typification when a general problem of its general type can be preliminarily balancedly corrected and then balancedly decomposed.

6.1.8. Basic Theory of Best

Analytic Quasisolutions

Definition A best error analytic quaslsolution to a general problem is its best error quasisolution in a general analytic pseudosolution chosen by the principle of tolerable simplicity.

Definition A best reserve analytic quaslsolution to a general problem is its best reserve quasisolution in a general analytic pseudosolution chosen by the principle of tolerable simplicity.

Chapter 6.2. Analytic Macroelement Fundamental Science

6.2.1. Power Analytic Macroelement Theory

and New Phenomena in General Problems

Unlike the known finite element method, analytlc macroelement fundamental science [11, 20, 28] gives best analytic quasisolutions.

Definition Power analytic macroelement theory is an application of the linear-combination method to a linear general problem and foresees generally subdividing a general domain into its several subdomains and generally minimizing and correcting all the general conjugation residuals (as absolute errors) between the corresponding subquasisolutions on the quantiboundaries of those subdomains.

Example In spatial elastic etc. problems [2], harmonic solutions can have general power representations (see 6.1.4).

Example In axially symmetric elastic etc. problems [2], biharmonic solutions can have general power representations (see 6.1.4). Such biharmonicity is sufficient and even necessary [11].

Remark Applications of this theory [11, 20, 28] permit to discover new phenomena in general problems such as the following:

1) A general problem can limit the degree of a power representation of its general pseudosolution from above.

2) A (whole) general type of natural general problems can be generally overdetermined.

3) The general overdetermination of a general type of natural general problems can be multiple.

6.2.2. Integral Analytic Macroelement Theory

and New Phenomena in General Problems

Definition Integral analytic macroelement theory is an application of the restructurization method to a quantisystem of nonequicomplicated quantirelations.

Example An axially symmetric elastic problem in stresses [2]: One equation of continuity of the second order is obviously more complicated than all the rest (that form the decision subsystem) and forms the estimation subsystem.

Remark Applications of this theory [11, 20, 28-32] permit to discover new phenomena in general problems such as the following:

1) Continuously varying a quantisystem can lead to spasmodically varying its quantistructure.

2) A critical relation for a quantisystem can exist.

3) A critical relation can bifurcate.

4) A critical relation can be invariant when a quantisystem varies.

5) There can exist total critical relations forming the quantiboundary subsystem of the quantisystem of critical relations.

6) The skips of the quantistructure of a quantisystem can be both successed and reversive.

7) There can exist the determining quantiparameter and the equivalent quantiparameter for a quantisystem.

8) A rational control by the determining quantiparameter of a quantisystem can raise its equivalent quantiparameter by an order.

9) A coincidence of a rational control and of a critical relation can depend on that relation.

10) An equivalent quantiparameter can be uniform if a determining quantiparameter is (symmetrically or asymmetrically) nonuniform.

11) A quantistate can be initial out of the quanticenter of a quantisystem.

12) A quantistate of a quanticenter of a quantisystem is initial for some critical relation.

13) The quantimeasure of a changed-structured general subsystem of a quantisystem can be invariant.

14) The total general measure of all the changed-structured general subsystems with variable general measures can be invariant.

15) The well-posedness and the ill-posedness of a quantisystem are relative.

Chapter 6.3. Universal Laws

6.3.1. Basic Theory

of Quantisystem Statics

Definition Quantisystem statics is a general consideration of a quantisystem at a fixed general instant of a general time.

Definition A phenomenon is an unexpected and/or a nontrivial general object.

Definition A law is a general relation between typical general objects.

Definition An inherent law of science is an axiom of constructing a science.

Definition A scientific phenomenon is a nontrivial consequence of laws.

Remark Some laws and phenomena can form hierarchical quantisystems [11, 21], e.g.:

1) Universal Law of a Solid State:

The general state of any loaded solid as a quantisystem is determined by a general relation between some generally reduced pure general state parameters including the principal stresses whose general values are generally reduced to their corresponding uniaxial critical values.

2) Supergeneral Law of a Solid State:

The general state at any point of any loaded solid is subcritical, critical, or supercritical according to a certain quasicritical relation between those reduced general parameters.

3) General Law of a Solid State (for a certain general reduction).

4) Subgeneral Law of a Solid State (for a certain type of loading).

5) Particular Law of a Solid State (for a certain type of anisotropy).

6) Subparticular Law of a Solid State (for a certain orientation relation between loading and a solid anisotropy).

7) Special Law of a Solid State (for a certain type of a solid nonequiresistance).

8) Subspecial Law of a Solid State (for a certain type of a quasicritical state).

9) Concrete Law of a Solid State (for a certain choice among quasicritical states).

6.3.2. Basic Theory

of Quantisystem

Stochastic Dynamics

Definition A general motion of a general object is generally changing its general position and/or positionality in a general time.

Definition A general passage of a general object through a quasielement is a general motion of that object in which one of the available general positions and/or positionalities includes that element.

Definition A general collision of general objects is their mutual general motion if their general intersection (at a certain instant of a general time) is nonempty.

Definition The general passage (and damage) multiplicity of a quantisystem G through a quantisystem-target T is the general passage (and damage) multiplicity of all the general passages of any quasielement g ∈ G through any quasielement t ∈ T.

Definition The general passage (and damage) quantity of a quantisystem G through a quantisystem-target T is the general passage (and damage) quantity of all the general passages of any quasielement g ∈ G through any quasielement t ∈ T.

Definition The general passage (and damage) measure of a quantisystem G through a quantisystem-target T is the general passage (and damage) measure of all the general passages of any quasielement g ∈ G through any quasielement t ∈ T.

Definition The general passage (and damage) integral of a quantisystem G through a quantisystem-target T is the general passage (and damage) integral of all the general passages of any quasielement g ∈ G through any quasielement t ∈ T.

Definition The general passage (and damage) multiplicity system of a quantisystem G through a quantisystem-target T is the general passage (and damage) multiplicity system of all the general passages of any quasielement g ∈ G through any quasielement t ∈ T.

Definition The general passage (and damage) quantity system of a quantisystem G through a quantisystem-target T is the general passage (and damage) quantity system of all the general passages of any quasielement g ∈ G through any quasielement t ∈ T.

Definition The general passage (and damage) measure system of a quantisystem G through a quantisystem-target T is the general passage (and damage) measure system of all the general passages of any quasielement g ∈ G through any quasielement t ∈ T.

Definition The general passage (and damage) integral system of a quantisystem S through a quantisystem-target T is the general passage (and damage) integral system of all the general passages of any quasielement s ∈ S through any quasielement t ∈ T.

Definition A quantisystem or subsystem is generally multiply undamaged if the general passage quantity through it is the zero one.

Definition A quantisystem or subsystem is generally quantitatively undamaged if the general passage quantity through it is the zero one.

Definition A quantisystem or subsystem is generally measurely undamaged if the general passage measure through it is the zero one.

Definition A quantisystem or subsystem is generally integrally undamaged if the general passage integral through it is the zero one.

Definition A quantisystem or subsystem is generally multiply systematically undamaged if the general passage multiplicity system through it is the zero one.

Definition A quantisystem or subsystem is generally quantitatively systematically undamaged if the general passage quantity system through it is the zero one.

Definition A quantisystem or subsystem is generally measurely systematically undamaged if the general passage measure system through it is the zero one.

Definition A quantisystem or subsystem is generally integrally systematically undamaged if the general passage integral system through it is the zero one.

Example

1) The local implantation quantity

m(S , t)

of a quantisystem S of quasielements s that are implanted into a general target T and are passing through its quasielement-point t .

2) m(S , t) for general subdividing S by

Q(s) = β

from their quantiset B with

m(S , t) = Q(βσ(β , t) | β ∈ Β) =

∑β∈Β βQ(β , t)

where

Q(β , t) = Q(s | Q(s) = β ∧ s ∩ t ≠ #).

3) T is a substance (solid etc.),

s is a projectile, or shell, or bullet, or shot, or grit, or particle,

β is an effective cross section general measure (area),

u is such a general measure (area) unit that any β << u ,

Q(S , t) is a uniquantity/dose per unit u .

m(S , t) = ∑β∈Β βσ(β, t)

shows how many times a general point t is generally passed.

4) U is a quantisystem (figure) whose general measure (area) is u and which generally moves so that it always (at any general time instant) includes t and is generally normal in some reasonable sense (of a general scalar product, etc.) to the general motion of any s at the general instant of general passing s ∩ U ,

V(S) is the general event that S generally passes through U for a general subsystem G of general instants of a general time,

P(S) is the general event that S generally passes through t for the same general time subsystem G ,

P(P(S) | V(S)) is the conditional quantiprobability of P(S) if V(S) occurs.

If all those general passages are generally independent then according to the theorem of multiplying probabilities [1]

P(P(S) | V(S)) = ∏β∈Β(1 - β/u)σ(S, t)u.

But such a notation including u in the explicit form is not quasiclassical and convenient, and a reasonable choice of u is nonunique and must give no effect. Besides that,

β/u << 1

and that is why

1 - β/u ≈ e-β/u.

Definition

(1 - β)Q(S, t) = e-βσ(S, t),

m(S , t) = ∑β∈Β βσ(β, t).

Definition The general undamagedness of t by S is

X(S , t) ::= e-βσ(S, t).

Definition The general undamagedness system of T by S is the S(T)-structured quantisystem

X(S , T) ::= (X(S , t) | t ∈ T).

Remark An invariant universal pure generally additive implantation quantity in particle physics shows how many (in the mean) particles pass through a point of a solid (matrix, target) and naturally distinguishes low, medium, and high implantation doses D. If the effective radius of a particle is about 0.1 nm then

m = 0.01, 1, 100

for

D ≈ 3×1013, 3×1015, 3×1017 cm-2,

respectively.

Remark β can be any effective quantiparameter and even the quantisystem of all of them.

Definition The first critical general passage multiplicity, quantity, measure, or integral is that which is unit.

Definition The first critical general passage multiplicity, quantity, measure, or integral system is that which is unit.

Preliminary Damage Law

A new general damage depends directly on the quantisystems of effective quantiparameters for each participating (damaging or damaged) quantisystem and inversely on the available general undamagedness.

Remark The theory of quantisystem stochastic dynamics permits to discover new phenomena (in ion implantation etc.) such as the following:

1) The existence of the nonequiresistance critical specific energy.

2) The abrupt supercritical decrease of specific resistance.

3) The fuzziness of the boundary between the damaged and the strengthened layers of a solid (target).

4) The coincidence and the common displacement of all the depths of the principal implantation maximums for ions having diverse sizes and initial energies.

6.3.3. Basic Theory

of the Superuniverse

Definition The superuniverse is the quantisystem of all the general objects.

Principle of Laws Hierarchy

For any law in the superuniverse there can be a more general law.

Definition A universal law is a law for which there is no essentially more general law.

Examples

Universal Law of General Kinematics

A general state process law for an isolated quantisystem is invariant.

(Nothing gives no change.)

Universal Law of General Statics

The quantisystem of the general changes in all the quasielements of a quantisystem is self-compensating.

(Action gives effect.)

Universal Law of General Dynamics

The specific general damagedness of a general target depends directly on the effective generally damaging quanti(sub)systems and inversely on the effective generally damaged quanti(sub)systems.

(The more work, the less object, the more result.)

Remark See the Newton laws.

Overmathematics and other fundamental mathematical, strength, creation, and organization sciences permit to generalize not only the most important scientific laws but also many law-like thoughts in philosophy, poetry [33], as well as proverbs, etc.

Basic Results and Conclusions

1. Overmathematics as a universal language of the new scientific thought is advanced on the base of their proposed fundamental principles.

2. The unified theories of quasielements, quantielements, sets, and systems satisfying the general conservation law can adequately express many urgent general objects in nature, society, and thinking.

3. The unified theories of general dimensions, infinities, infinitesimals, quasinumbers, and supernumbers complete the universal scale for perfect expressing arbitrary uniquantities.

4. The unified theories of general meaners, bounders, truncators, measurers, integrators, probabilers, errers, and reservers give subtle general estimators, discriminators, and controllers for inexact, approximate, and even exact general objects.

5. The unified theories and general methods of reasonable simulating and effective solving many urgent fundamental and applied general problems determine their best analytic quasisoltitions and even supersoltitions in their general pseudosoltitions.

6. Overmathematics and other fundamental mathematical and strength sciences permit to discover quasicritical general phenomena in diverse quantisystems.

7. Overmathematics and other fundamental mathematical and strength sciences give a hierarchical quantisystem of the new general laws of the superuniverse.

References

1. Encyclopaedia of Mathematics / Managing ed. M. Hazewinkel (1988) Reidel, Dordrecht etc.

2. Encyclopaedia of Physics / Chief Editor S. Flugge (1973 etc.) Springer, Berlin etc.

3. Zadeh, L. A. (1973) The Concept of a Linguistic Variable and Its Application to Approximate Reasoning. Elsevier, N.Y.

4. Dubois, D., and Prade, H. (1980) Fuzzy Sets and Systems: Theory and Applications. Academic Press, N.Y.

5. Bourbaki, N. (1949 etc.) Elements de mathematique. Hermann, Paris.

6. Schwartz, L. (1967) Analyse mathematique. Hermann, Paris.

7. Korn, G. A., and Korn, T. M. (1968) Mathematical Handbook for Scientists and Engineers. McGraw-Hill, N.Y. etc.

8. Cantor, G. (1932) Gesammelte Abhandlungen mathematischen und philosophischen Inhalts. Springer, Berlin.

9. Hausdorff, F. (1914) Grundzüge der Mengenlehre. Veit, Leipzig.

10. Klaua, D. (1964) Allgemeine Mengenlehre: Ein Fundament der Mathematik. Akad.-Verl., Berlin.

11. Lev Gelimson (1993) The Generalization of the Analytic Methods of Solving Strength Problems for Typical Constructive Elements in High-Pressure Engineering / Dr. Sc. Dissertation. Strength Problems Institute, Ukraine National Academy of Sciences, Kiev.

12. Lev Gelimson (1994) General estimation theory. Transactions of the Ukraine Glass Inst., 1, 214-221.

13. Leibniz, G. W. (1846) Leibnizens gesammelte Werke. Hansch, Hannover.

14. Fontenelle, B. (1727) Elements de la geometrie de l'infini. L'Imprimerie Royal, Paris.

15. Euler, L. (1748) Introductio in Analysis infitorum.

16. Robinson, A. (1970) Non-Standard Analysis. North-Holland, Amsterdam, L.

17. Davis, M. (1977) Applied Nonstandard Analysis. Wiley, N.Y. etc.

18. Einstein, A. (1961) Relativity: The Special and the General Theory. Crown Publishers, N.Y.

19. Blekhman, I. I., Myshkis, A. D., and Panovko, Ya. G. (1990) Mechanics and Applied Mathematics [In Russian]. Nauka, Moscow.

20. Lev Gelimson (1992) The Generalization of the Analytic Methods of Solving Strength Problems [In Russian]. Drukar Publishers, Sumy.

21. Lev Gelimson (1993) General Strength Theory. Drukar Publishers, Sumy.

22. Lev Gelimson (1993) Method of generalizing critical state criteria [In Russian]. Proc. of the Int. Conf. “Glass Technology and Quality”, 98-100.

23. Lev Gelimson (1993) Method of the linear correction for critical state criteria [In Russian]. Proc. of the Int. Conf. “Glass Technology and Quality”, 100-102.

24. Lev Gelimson (1994) The generalized structure for critical state criteria. Transactions of the Ukraine Glass Inst., 1, 204-209.

25. Lev Gelimson (1993) The generalized methods of solving combined functional equations [In Russian]. Proc. of the Int. Conf. “Glass Technology and Quality”, 106-108.

26. Lev Gelimson (1994) The method of least normalized powers and the method of equalizing errors to solve functional equations. Transactions of the Ukraine Glass Inst., 1, 209-214.

27. Lev Gelimson (1993) The generalized determination of a reserve factor [In Russian]. Proc. of the Int. Conf. “Glass Technology and Quality”, 102 - 103.

28. Lev Gelimson (1993) The analytic method of macroelements in axially symmetric elastic problems [In Russian]. Proc. of the Int. Conf. “Glass Technology and Quality”, 104-106.

29. Ol’khovik, O. E., Kaminskii, A. A., Gelimson, L. G., et al. Study of the strength of acrylic plastic under a complex stress state. Strength Mater. (USA), 15 (1984), no. 8, 1127-1129.

30. Kaminskii, A. A., Gelimson, L. G., Karintsev, I. B., and Morachkovskii, O. K. Relationship between the strength of glass and the number of cracks at fracture. Strength Mater. (USA), 17 (1986), no. 12, 1691-1693.

31. Amel’yanovich, K. K., Gelimson, L. G., and Karintsev, I. B. Stress-strain state and strength of transparent elements of portholes. Sov. J. Opt. Technol. (USA), 59 (1993), no. 11, 664-667.

32. Amel’yanovich, K. K., Gelimson, L. G., and Karintsev, I. B. Problem of the criterial evaluation of the strength of cylindrical glass elements of illuminators. Strength Mater. (USA), 25 (1994), no. 10, 772-777.

33. Lev Gelimson (1991) My Inmost. Grafika Publishers, Sumy.

Revolutions in Applied Mathematics

The system of revolutions in applied mathematics includes:

the uniestimation subsystem of revolutions including unierrors, unireserves, unireliabilities, and unirisks;

the uniapproximation subsystem of revolutions including a uniapproximation uniproblem, a unicenter, a linear and nonlinear unibisector, multiple-sources, multidirectional, and intelligent uniiteration;

the uniproblem subsystem of revolutions including the setting of a uniproblem, its pseudosolutions, quasisolutions, supersolutions, and antisolutions, freely increasing the exponents in power mean theories and methods, power mean distance minimization, power mean unierror minimization, power mean reserve maximization, distance equalization, unierror equalization, and reserve equalization;

the contradictoriness subsystem of revolutions including contradictoriness legalization, validation, expression, evaluation, measurement, estimation, and efficient utilization;

the metasciences subsystem of revolutions in applied mathematics including unimathematically testing and developing knowledge.

Basic Results and Conclusions

1. Classical mathematics brings key concepts of the absolute and relative errors that are very often insufficient, have evident lacks, and are not suitable for solving many typical problems. The relative errors of distributions, exactness confidence, and problem contradictoriness are not estimated at all. The known least square method uniquely applicable to contradictory problems has a very restricted range of applicability and, all the more, of adequacy. Otherwise, it brings false and misleading results that cannot be corrected and improved by the method itself.

2. Applied unimathematics includes fundamental sciences systems based on pure megamathematics and hence on overmathematics with its uninumbers, quantielements, quantisets, and uniquantities with quantioperations and quantirelations provides universally and adequately unimathematically modeling (expressing, representing, etc.) and processing (measuring, evaluating, estimating, approximating, calculating, etc.) general objects and systems, as well as setting and solving scientific, engineering, and life problems. This all creates the basis for many further fundamental sciences systems developing, extending, and applying overmathematics. Among them are, in particular, the unimathematical modeling fundamental sciences system, unimathematical measurement fundamental sciences system, unimathematical estimation fundamental sciences system, and general problem fundamental sciences system including many universal, adequate, and very efficient theories, methods, and algorithms.

3. The unimathematical modeling fundamental sciences system provides efficiently, universally and adequately strategically unimathematically modeling (expressing, representing, etc.) general objects and systems in science, engineering, and life with mathematical and physical model analysis, synthesis, invariance and symmetry, data unification and grouping, structuring and restructuring, scatter, trend, and outliers.

4. The unimathematical measurement fundamental sciences system provides efficiently, universally and adequately strategically unimathematically measuring general objects and systems and their mathematical and physical models, as well as deviations, distances, and for the first time also exactness, contradictoriness, and distributions in science, engineering, and life with possibly recovering true measurement information using incomplete changed data maybe with scatter, trend, and outliers.

5. The unimathematical estimation fundamental sciences system provides efficiently, universally and adequately strategically unimathematically estimating general objects and systems and their mathematical and physical models, as well as deviations, distances, and for the first time also exactness, contradictoriness, and distributions in science, engineering, and life with possibly recovering true information using incomplete changed data maybe with scatter, trend, and outliers. The unierrors correct and generalize the relative error. The reserves new in principle estimate exactness reliability and risk. General meaners, bounders, truncators, measurers, integrators, probabilers, errers, and reservers give subtle general estimators, discriminators, and controllers for inexact, approximate, and even exact general objects. The fundamental science of reserves further naturally generalizes the fundamental science of concessions and for the first time regularly applies and develops 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. The fundamental sciences of reliability and risk for the first time regularly apply and develop 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.

6. The general problem fundamental sciences system provides efficiently, universally and adequately strategically unimathematically setting, transforming, analyzing, synthezing, and solving general problems in science, engineering, and life, as well as testing and estimating their pseudosolutions, quasisolutions, supersolutions, and antisolutions via their universal invariant unierrors, reserves, reliabilities, and risks. Many proposed general theories and methods are from the beginning exclusively practice-oriented, effective, universal, and successfully applicable to any scientific, engineering, and life problem mathematically simulated. They apply even to contradictory general problems and adequately determine their best quasisoltitions and for the first time the invariant measures of the contradictorinesses of such problems by the unierrors of quasisolutions to them.

7. Applied megamathematics fundamental sciences systems revolutionarily replaces the inadequate very fundamentals of classical applied mathematics via adequate very fundamentals, applies, develops, and tests overmathematics and pure megamathematics.

References

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

[2] Lev Gelimson. General Estimation Theory. Transactions of the Ukrainian Glass Institute 1 (1994), p. 214-221

[3] Lev Gelimson. Basic New Mathematics. Drukar Publishers, Sumy, 1995

[4] Lev Gelimson. Quantianalysis: Uninumbers, Quantioperations, Quantisets, and Multiquantities (now Uniquantities). Abhandlungen der WIGB (Wissenschaftlichen Gesellschaft zu Berlin), 3 (2003), Berlin, 15-21

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

[6] Lev Gelimson. Providing helicopter fatigue strength: Flight conditions [Megamathematics]. 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, Vol. II, Dalle Donne, C. (Ed.), Hamburg, 2005, p. 405-416

[7] Lev Gelimson. Overmathematics: Fundamental Principles, Theories, Methods, and Laws of Science. Mathematical Monograph. The “Collegium” All World Academy of Sciences Publishers, Munich (Germany), 2009

[8] Lev Gelimson. Overmathematics: Principles, Theories, Methods, and Applications. Mathematical Monograph. The “Collegium” All World Academy of Sciences Publishers, Munich (Germany), 2009

[9] Lev Gelimson. Uniarithmetics, Quantialgebra, and Quantianalysis: Uninumbers, Quantielements, Quantisets, and Uniquantities with Quantioperations and Quantirelations. Mathematical Monograph. The “Collegium” All World Academy of Sciences Publishers, Munich (Germany), 2010

[10] Lev Gelimson. Uniarithmetics, Quantianalysis, and Quantialgebra: Uninumbers, Quantielements, Quantisets, and Uniquantities with Quantioperations and Quantirelations (Essential). Mathematical Journal of the “Collegium” All World Academy of Sciences, Munich (Germany), 11 (2011), 26

[11] Lev Gelimson. Science Unimathematical Test Fundamental Metasciences Systems. Monograph. The “Collegium” All World Academy of Sciences, Munich (Germany), 2011

[12] Encyclopaedia of Physics / Chief Editor S. Flugge. Springer, Berlin etc., 1973 etc.

[13] Encyclopaedia of Physical Science and Technology, Ed. R. A. Meyers, Vols. 1 to 18, Academic Press Publ., San Diego, 2002

[14] Weierstraß K. T. W. Mathematische Werke. Berlin, Leipzig, 1894-1927, Bd. 1-7

[15] N. Bourbaki. Elements de mathematique. Hermann, Paris, 1949 etc.

[16] L. Schwartz. Analyse mathematique. Hermann, Paris, 1967

[17] G. A. Korn, T. M. Korn. Mathematical Handbook for Scientists and Engineers. McGraw-Hill, N.Y. etc., 1968

[18] I. N. Bronstein, K. A. Semendjajew. Taschenbuch der Mathematik. Frankfurt/M., 1989

[19] I. I. Blekhman, A. D. Myshkis, and Ya. G. Panovko. Mechanics and Applied Mathematics [In Russian]. Nauka, Moscow, 1990

[20] A. Fetzer, H. Fränkel, Mathematik, Bände 1, 2, Springer-Verlag, Berlin, 1999, 2000

[21] U. Storch, H. Wiebe, Lehrbuch der Mathematik, Spectrum-Verlag, Heidelberg, 2001

[22] G. Cantor. Gesammelte Abhandlungen mathematischen und philosophischen Inhalts. Springer-Verlag, Berlin, 1932

[23] F. Hausdorff F. Grundzüge der Mengenlehre. Leipzig, 1914

[24] A. Fraenkel. Einleitung in die Mengenlehre. Berlin, 1928

[25] F. Hausdorff. Mengenlehre. Berlin, 1935

[26] L. A. Zadeh. Fuzzy sets. Information and Control 8 (1965), 338-353

[27] D. Dubois, H. Prade. Fuzzy Sets and Systems: Theory and Applications. New York, 1980

[28] W. Blizard. The development of multiset theory. Modern Logic 1 (1991), no. 4, 319-352

[29] B. Bolzano. Paradoxien des Unendlichen. Bei C. H. Reclam Sen., Leipzig, 1851

[30] G. W. Leibniz. Sur les monades et le calcul infinitesimal, etc. Letter to Dangicourt, Sept. 11, 1716. In: Leibniz, G. W.: Opera Omnia (ed. by L. Dutens), Vol. 3 (1789), 499-502

[31] G. W. Leibniz. Leibnizens gesammelte Werke. Hansch, Hannover, 1846

[32] B. Fontenelle. Elements de la geometrie de l'infini. L'Imprimerie Royal, Paris, 1727

[33] L. Euler. Introductio in Analysis infitorum, 1748

[34] R. Sikorski. On an ordered algebraic field. Sci. Math. Phys. 41 (1950), 69-96

[35] D. Klaua. Zur Struktur der reellen Ordinalzahlen. Zeitschrift für math. Logik and Grundlagen der Math. 6 (1960), 279-302

[36] A. Robinson. Non-Standard Analysis. North-Holland, Amsterdam, 1966

[37] J. H. Conway. On Numbers and Games. Academic Press, London, 1976

[38] Martin Davis. Applied Nonstandard Analysis. Wiley, New York, London, Sydney, Toronto, 1977

[39] Real Numbers, Generalizations of the Reals, and Theories of Continua (ed. by Ph. Ehrlich). Kluwer Academic Publisher Group, Dordrecht, 1994

[40] Rechnen mit dem Unendlichen. Beiträge zur Entwicklung eines kontroversen Gegenstandes (ed. by D. D. Spalt). Birkhäuser, Basel, 1990

[41] G. P. Zedgenidze, R. Sh. Gogsadze. Mathematical Methods in Measurement Technology [in Russian]. Standards Committee Publishers, Moscow, 1970

[42] Barford N. C. Experimental Measurements: Precision, Error, and Truth. Addison-Wesley, 1967

[43] Taylor J. R. An Introduction to Errors Analysis. University Science Books Mill Valley, California, 1982

[44] Fréchet M. Recherches théoriques modernes sur la théorie des probabilites. Paris, 1937-1938

[45] A. M. Legendre, Nouvelles méthodes pour la détermination the orbites the comètes, Appendice sur la méthode the moindres carrés, Paris, 1806

[46] C. F. Gauß, Theoria motus corporum coelestium, Hamburg, 1809

[47] O. C. Zienkiewicz, Y. K. Cheung (1967) The Finite Elements Method in Structural and Continuum Mechanics. McGraw-Hill, N. Y.

[48] O. C. Zienkiewicz, R. L. Taylor. Finite Element Method. Volumes 1 to 3. Butterworth Heinemann Publ., London, 2000

[49] Encyclopaedia of Optimization, Ed. C. A. Floudas, P. M. Pardalos, Vols. 1 to 6, Kluwer Academic Publ., Dordrecht, 2001

[50] Lev Gelimson. Quantisets Algebra. Abhandlungen der WIGB (Wissenschaftlichen Gesellschaft zu Berlin), 3 (2003), Berlin, 262-263

[51] Lev Gelimson. Elastic Mathematics. Abhandlungen der WIGB (Wissenschaftlichen Gesellschaft zu Berlin), 3 (2003), Berlin, 264-265

[52] Lev Gelimson. Quantisets Algebra. The Second International Science Conference “Contemporary methods of coding in electronic systems”, Sumy, 26-27 October 2004

[53] Lev Gelimson. Quantisets and Their Quantirelations. The Third International Science Conference “Contemporary methods of coding in electronic systems”, Sumy, 24-25 October 2006

[54] Lev Gelimson. Quantiintervals and Semiquantiintervals. The Third International Science Conference “Contemporary methods of coding in electronic systems”, Sumy, 24-25 October 2006

[55] Lev Gelimson. Multiquantities (now Uniquantities). The Third International Science Conference “Contemporary methods of coding in electronic systems”, Sumy, 24-25 October 2006

[56] Lev Gelimson. Sets with Any Quantity of Each Element. Mathematical Monograph. The “Collegium” All World Academy of Sciences Publishers, Munich (Germany), 2006

[57] Lev Gelimson. Unimathematical Modeling Fundamental Sciences System. Monograph. The “Collegium” All World Academy of Sciences, Munich (Germany), 2011

[58] Lev Gelimson. General Implantation Theory in the New Mathematics. Second International Conference "Modification of Properties of Surface Layers of Non-Semiconducting Materials Using Particle Beams" (MPSL'96). Sumy, Ukraine, June 3-7, 1996. Session 3: Modelling of Processes of Ion, Electron Penetration, Profiles of Elastic-Plastic Waves Under Beam Treatment. Theses of Reports

[59] Lev Gelimson. The Stress State and Strength of Transparent Elements in High-Pressure Portholes (Side-Lights) [In Russian]. Ph. D. dissertation. Institute for Strength Problems, Academy of Sciences of Ukraine, Kiev, 1987

[60] Lev Gelimson. Measurement Theory in Elastic Mathematics. Physical Monograph. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2001

[61] Lev Gelimson. Objektorientierte Mathematik in der Messtechnik. Fachgebiet: "Informatik und Mathematik in der Messtechnik". Manuskript der Arbeit zur Teilnahme am Wettbewerb um den Helmholtz-Preis 2001. Physikalisch-Technische Bundesanstalt, Braunschweig

[62] Lev Gelimson. General Theory of Measuring Inhomogeneous Distributions. In: Review of Aeronautical Fatigue Investigations in Germany during the Period May 2007 to April 2009, Ed. Dr. Claudio Dalle Donne, Pascal Vermeer, CTO/IW/MS-2009-076 Technical Report, International Committee on Aeronautical Fatigue, ICAF 2009, EADS Innovation Works Germany, 2009, 60-61

[63] Lev Gelimson. Unimathematical Measurement Fundamental Sciences System. Monograph. The “Collegium” All World Academy of Sciences, Munich (Germany), 2011

[64] Lev Gelimson. General estimations and approximations [in Russian]. International Scientific and Technical Conference “Energy and Resource Saving Technologies in Glass Production”, General Problems, Theses of Reports, 1995, p. 72-74

[65] Lev Gelimson. General Estimation Theory. Mathematical Monograph. 9th Edition. The “Collegium” All World Academy of Sciences Publishers, Munich (Germany), 2010

[66] Lev Gelimson. General Estimation Theory Fundamentals (along with its line by line translation into Japanese). Mathematical Journal of the “Collegium” All World Academy of Sciences, Munich (Germany), 9 (2009), 1

[67] Lev Gelimson. General Estimation Theory (along with its line by line translation into Japanese). Mathematical Monograph. The “Collegium” All World Academy of Sciences Publishers, Munich (Germany), 2011

[68] Lev Gelimson. Unimathematical Estimation Fundamental Sciences System. Monograph. The “Collegium” All World Academy of Sciences, Munich (Germany), 2011

[69] Lev Gelimson. Corrections and Generalizations of the Absolute and Relative Errors. In: Review of Aeronautical Fatigue Investigations in Germany During the Period May 2005 to April 2007, Ed. Dr. Claudio Dalle Donne, Pascal Vermeer, CTO/IW/MS-2007-042 Technical Report, Aeronautical fatigue, ICAF2007, EADS Innovation Works Germany, 2007, 49-50

[70] Lev Gelimson. General Strength Theory. Monograph. Drukar Publishers, Sumy, 1993

[71] Lev Gelimson. Generalized Reserve Determination Methods [In Russian]. International Scientific and Technical Conference "Glass Technology and Quality". Mathematical Methods in Applying Glass Materials (1993), Theses of Reports, p. 102-103

[72] Lev Gelimson. General System Reserve and Methods to Determine It. Mathematical Journal of the “Collegium” All World Academy of Sciences, Munich (Germany), 1 (2001), 2

[73] Lev Gelimson. Basic Reliability Science in Overmathematics. Mathematical Journal of the “Collegium” All World Academy of Sciences, Munich (Germany), 7 (2007), 1

[74] Lev Gelimson. General Reliability Theory in Elastic Mathematics. In: Review of Aeronautical Fatigue Investigations in Germany during the Period May 2007 to April 2009, Ed. Dr. Claudio Dalle Donne, Pascal Vermeer, CTO/IW/MS-2009-076 Technical Report, International Committee on Aeronautical Fatigue, ICAF 2009, EADS Innovation Works Germany, 2009, 31-32

[75] Lev Gelimson. Basic Risk Science in Overmathematics. Mathematical Journal of the “Collegium” All World Academy of Sciences, Munich (Germany), 8 (2008), 2

[76] Lev Gelimson. General Risk Theory in Elastic Mathematics. In: Review of Aeronautical Fatigue Investigations in Germany during the Period May 2007 to April 2009, Ed. Dr. Claudio Dalle Donne, Pascal Vermeer, CTO/IW/MS-2009-076 Technical Report, International Committee on Aeronautical Fatigue, ICAF 2009, EADS Innovation Works Germany, 2009, 32-33

[77] Lev Gelimson. Generalized Methods for Solving Functional Equations and Their Sets [In Russian]. International Scientific and Technical Conference"Glass Technology and Quality". Mathematical Methods in Applying Glass Materials (1993), Theses of Reports, p. 106-108

[78] Lev Gelimson. The method of least normalized powers and the method of equalizing errors to solve functional equations. Transactions of the Ukraine Glass Institute, 1 (1994), 209-214

[79] Lev Gelimson, General problems and methods of solving them, International Scientific and Technical Conference “Energy and Resource Saving Technologies in Glass Production”, General Problems, Theses of Reports, 1995, p. 74 (Russ.)

[80] Lev Gelimson. General Problem Theory. Abhandlungen der WIGB (Wissenschaftlichen Gesellschaft zu Berlin), 3 (2003), Berlin, 26-32

[81] Lev Gelimson. General Analytic Methods. Abhandlungen der WIGB (Wissenschaftlichen Gesellschaft zu Berlin), 3 (2003), Berlin, 260-261

[82] Lev Gelimson. General Problem Theory. The Second International Science Conference “Contemporary methods of coding in electronic systems”, Sumy, 26-27 October 2004

[83] Lev Gelimson. Fundamental Methods of Solving General Problems via Unierrors, Reserves, Reliabilities, and Risks. Mathematical Monograph. The “Collegium” All World Academy of Sciences Publishers, Munich (Germany), 2007

[84] Lev Gelimson. Corrections and Generalizations of the Least Square Method. In: Review of Aeronautical Fatigue Investigations in Germany during the Period May 2007 to April 2009, Ed. Dr. Claudio Dalle Donne, Pascal Vermeer, CTO/IW/MS-2009-076 Technical Report, International Committee on Aeronautical Fatigue, ICAF 2009, EADS Innovation Works Germany, 2009, 59-60

[85] Lev Gelimson. Least Biquadratic Method in Fundamental Sciences of Estimation, Approximation, and Data Processing. In: Review of Aeronautical Fatigue Investigations in Germany during the Period 2009 to 2011, Ed. Dr. Claudio Dalle Donne, Katja Schmidtke, EADS Innovation Works, CTO/IW/MS-2011-055 Technical Report, International Committee on Aeronautical Fatigue, ICAF 2011, 2011, 44-45

[86] Lev Gelimson. Least Squared Distance Theories in Fundamental Sciences of Estimation, Approximation, Data Modeling and Processing. In: Review of Aeronautical Fatigue Investigations in Germany during the Period 2009 to 2011, Ed. Dr. Claudio Dalle Donne, Katja Schmidtke, EADS Innovation Works, CTO/IW/MS-2011-055 Technical Report, International Committee on Aeronautical Fatigue, ICAF 2011, 2011, 45-47

[87] Lev Gelimson. Least Squared Distance Theories in Fundamental Sciences of Solving General Problems. In: Review of Aeronautical Fatigue Investigations in Germany during the Period 2009 to 2011, Ed. Dr. Claudio Dalle Donne, Katja Schmidtke, EADS Innovation Works, CTO/IW/MS-2011-055 Technical Report, International Committee on Aeronautical Fatigue, ICAF 2011, 2011, 47-49

[88] Lev Gelimson. Signed Geometric and Quadratic Mean Theories in Fundamental Sciences of Estimation, Approximation, and Data Processing. In: Review of Aeronautical Fatigue Investigations in Germany during the Period 2009 to 2011, Ed. Dr. Claudio Dalle Donne, Katja Schmidtke, EADS Innovation Works, CTO/IW/MS-2011-055 Technical Report, International Committee on Aeronautical Fatigue, ICAF 2011, 2011, 70-72

[89] Lev Gelimson. General Theories of Moments of Inertia in Fundamental Sciences of Estimation, Approximation, and Data Processing. In: Review of Aeronautical Fatigue Investigations in Germany during the Period 2009 to 2011, Ed. Dr. Claudio Dalle Donne, Katja Schmidtke, EADS Innovation Works, CTO/IW/MS-2011-055 Technical Report, International Committee on Aeronautical Fatigue, ICAF 2011, 2011, 72-73

[90] Lev Gelimson. Group Center Theories in Fundamental Sciences of Estimation, Approximation, Data Modeling and Processing. In: Review of Aeronautical Fatigue Investigations in Germany during the Period 2009 to 2011, Ed. Dr. Claudio Dalle Donne, Katja Schmidtke, EADS Innovation Works, CTO/IW/MS-2011-055 Technical Report, International Committee on Aeronautical Fatigue, ICAF 2011, 2011, 74-75

[91] Lev Gelimson. Coordinate Partition Theories in Fundamental Sciences of Estimation, Approximation, Data Modeling and Processing. In: Review of Aeronautical Fatigue Investigations in Germany during the Period 2009 to 2011, Ed. Dr. Claudio Dalle Donne, Katja Schmidtke, EADS Innovation Works, CTO/IW/MS-2011-055 Technical Report, International Committee on Aeronautical Fatigue, ICAF 2011, 2011, 75-77

[92] Lev Gelimson. Principal Bisector Partition Theories in Fundamental Sciences of Estimation, Approximation, Data Modeling and Processing. In: Review of Aeronautical Fatigue Investigations in Germany during the Period 2009 to 2011, Ed. Dr. Claudio Dalle Donne, Katja Schmidtke, EADS Innovation Works, CTO/IW/MS-2011-055 Technical Report, International Committee on Aeronautical Fatigue, ICAF 2011, 2011, 77-79

[93] Lev Gelimson. General Problem Fundamental Sciences System. Mathematical Monograph. The “Collegium” All World Academy of Sciences Publishers, Munich (Germany), 2011

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L. B. Tsvik. Finite Element Method Application to Static Deformation [In Russian]. Irkutsk State University Publishers, Irkutsk, 1995

L. B. Tsvik. Triaxial Stress and Strength of Single-Layered and Multilayered High Pressure Vessels with Branch Pipes [In Russian]. Dr. Sc. Dissertation, Irkutsk, 2001

L. B. Tsvik. Schwarz's algorithm generalization for areas interfaced without overlapping [In Russian]. Proceedings of the USSR Academy of Sciences, 1975, 224 (2), 309-312

L. B. Tsvik. Sequential continuity principle in solving field theory problems in parts [In Russian]. Proceedings of the USSR Academy of Sciences, 1978, 243 (1), 74-77

L. B. Tsvik, Yu. L. Vaynapel, G. G. Zorina. Locally controlled finite-element modeling of the physical state of two-dimensional domains [In Russian]. Siberian Conference on Industrial and Applied Mathematics. Institute of Mathematics Publishers, Novosibirsk, 1994, 46-48

L. B. Tsvik. On quasi-exact conjugation operators in separating domains and conjugating solutions [In Russian]. Modern machine mechanics problems: Proceedings of International Conference, Ulan-Ude, June 21-25, 2000, VSGTU Publishers, Ulan Ude, 2000, 140-143