Advanced Unimathematics (Mega-Overmathematics) as a System of Revolutions in Advanced Mathematics
by
© 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 00A71; secondary 00A72, 26A12, 28A25, 93A10.
Keywords: advanced 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 Advanced Unimathematics
The principles of advanced unimathematics build the operation system of revolutions in the principles of mathematics and include:
1. Universal perfect operability (the universality, invariance, sensitivity, and usefulness of modeling and unimeasurement generalizing counting, as well as a substantial generalization of urgent objects, elements, relationships, structures, systems, and contents extending sets via uniobjects, unielements, unirelationships, unistructures, unisystems, and unicontents extending unisets as further generalizations of quantiobjects, quantielements, quantirelationships, quantistructures, quantisystems, and quanticontents extending quantisets, respectively).
2. 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).
3. Conservation laws universality (in the overinfinitesimal, the infinitesimal, the finite, the infinite, and the overinfinite).
4. Operations and overoperations utility universality.
5. The commutativity of composite (combined) alternatives to exponentiation and hyperoperations.
Advanced unimathematics includes unialgebra and unianalysis as further generalizations of quantialgebra and quantianalysis, respectively, in particular:
the fundamental science of universalizing unification further extending general (nonlogical) quantification;
the system of fundamental sciences on unielements, unisets, unioperations, unirelations, uniaggregates, unistructures, unisystems, unistates, uniprocesses, and unilaws as further generalizations of quantielements, quantisets, quantioperations, quantirelations, quantiaggregates, quantistructures, quantisystems, quantistates, quantiprocesses, and quantilaws, respectively;
the system of fundamental sciences on the hierarchy of commutative composite (combined) overoperations;
the system of fundamental unimodeling sciences;
the system of fundamental unimeasurement and unicomparison sciences.
The system of fundamental sciences on the hierarchy of commutative composite (combined) overoperations includes:
1) the fundamental science of nondistributive rings and fields which includes arithmetics and algebra with alternative multiplication preserving the negativity (and the absolute value of the conventional product) so that the negativity-preserving product of nonzero factors is positive if and only if all these factors are positive and is negative if and only if at least one of these factors is negative. Such alternative multiplication is although unusual but no less natural than usual multiplication when the product of an even number of negative factors is positive, which has nothing to do with intuition and follows only from a desire to provide the distributivity of multiplication over addition in rings and fields. But usual multiplication leads to the inacceptable restrictions of the domains of power and exponential functions to the cases of nonnegative bases only. Alternative negativity-preserving multiplication naturally leads to alternative raising to the power with preserving the base sign and the absolute value of the ordinary power, which removes all the restrictions for raising arbitrary negative bases to any power and for the domains of power and exponential functions. In many classes of typical urgent problems, this advantage is necessary for their successful solving, and the nondistributivity of alternative negativity-preserving multiplication over addition does not create any difficulties. It should be also noted that in mathematical logic and set algebra, the both distributivity laws are true, namely for multiplication over addition and for addition with respect to multiplication. In mathematical logic, disjunction plays the role of addition whereas conjunction plays the role of multiplication. In set algebra, the union operation plays the role of addition whereas the intersection operation plays the role of multiplication. At the same time, in arithmetics and algebra of numbers, only one of these two distributivity laws is valid, namely, multiplication is distributive over addition while the another distributivity law does not hold so that addition is not distributive with respect to multiplication. Then the remaining distributivity law for multiplication over addition should not be regarded as inviolable dogma. The decisive argument in favor of introducing alternative negativity-preserving multiplication is that it is introduced additionally only to common multiplication and is effectively based on it without any attempts to eliminate and replace it. If common multiplication is applicable, suitable, and adequate, then there is no reason to use any alternative multiplication. On the contrary, such additional multiplication even essentially helps ordinary multiplication in the cases difficult for it and greatly expands the palette of techniques for solving many classes of typical urgent problems;
2) the fundamental science of alternative raising to the power with preserving the base sign (and the absolute value of the common power function value), which removes any restrictions for raising arbitrary negative bases to any power. Usual exponentiation does not allow to raise any negative base to any power and leads to the inacceptable restrictions of the domains of power and exponential functions to the cases of nonnegative bases only. In many classes of typical urgent problems, this advantage is necessary for their successful solving, and base sign preserving exponentiation does not create any difficulties. The decisive argument in favor of introducing alternative base sign preserving exponentiation is that it is introduced namely and only additionally to common exponentiation and is effectively based on it without any attempts to eliminate and replace it. If common exponentiation is applicable, suitable, and adequate, then there is no reason to use any alternative exponentiation. On the contrary, such additional exponentiation even essentially helps ordinary exponentiation in the cases difficult for it and greatly expands the palette of techniques for solving many classes of typical urgent problems;
3) the fundamental science of overfficient power and exponential functions which (unlike standard power and exponential functions) are useful for representing numbers both with very large and with very small absolute values everywhere, namely on the whole real-number axis and not only on its part (e.g., for the equal values of the base and its exponent not less than one). To provide such overfficiency for any relations between the base and its exponents whose number may be also nonintegral, replace each exponent either with its absolute value if this modulus is not less than 1 or with the reciprocal of this modulus if it is less than 1 and further apply base sign preserving exponentiation with the central symmetry of the graphs of these overfficient functions with respect to the origin naturally belonging to these graphs;
4) the fundamental science of root-logarithmic overfunctions inverse to overfficient power and exponential functions in which each exponent equals the base;
5) the fundamental science of self-root-logarithmic overfunctions inverse to overfficient power and exponential functions in which both each exponent and the total number of the base and all the exponents equal the base;
6) the fundamental science of commuting composite (combined) overoperations including the theory of coherently pairing exponentiation with multiplication or addition in the direct or reverse orders.
The system of fundamental unimodeling sciences includes:
1) the fundamental science of the mathematical and physical nature and strategy of universal modeling including theories of setting, methodologies, strategies, and tactics of unitransforming and unisolving problems of universal mathematical and physical modeling;
2) the fundamental science of analysis and synthesis of universal mathematical and physical models including theories of their analysis and synthesis;
3) the fundamental science of the universality and symmetry of mathematical and physical models including theories of their universality and symmetry;
4) the fundamental science of 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).
5) the fundamental science of uniforming and unigrouping data in universal mathematical and physical models including theories of uniforming and unigrouping such data;
6) the fundamental science of unistructuring and unirestructuring data in universal mathematical and physical models including theories of unistructuring and unirestructuring such data;
7) the fundamental science of data scatter and trends in universal mathematical and physical models including theories of the orientation and spread of such data, as well as of unimeasuring and uniestimating such directions and scatter;
8) the fundamental science of data outliers in universal mathematical and physical models including theories of separating, transforming (including optionally dividing a single point into parts if useful), centralizing, compensating, evaluating, measuring, estimating, and best accounting such outliers.
The system of fundamental unimeasurement and unicomparison sciences includes fundamental sciences on universally measuring and comparing objects and systems and their mathematical and physical models including general theories and methods of:
1. Developing and applying uniquantity as the universal perfectly sensitive measure of uniobjects, unisystems, and their mathematical and physical unimodels.
2. 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).
3. 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).
Foreword. Megamathematics as Revolutions in Mathematics
There are separate scientific achievements of mankind but many of them often bring rather unsolvable problems than really improving human life quality. One of the reasons of such situation is that the available level of classical science is clearly insufficient to adequately solve and even consider many urgent human problems. To provide creating and developing applicable and, moreover, adequate methods, theories, and sciences, we need their testing (Lev Tsvik [1975, 1978, 1995, 2001, 2002], Alexey Borisenko [2002]) 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.
[Wikipedia Mathematics] notes that mathematics can be subdivided into arithmetic, algebra, geometry, and analysis building pure mathematics and studying quantity, structure, space, and change, respectively. Among additional subdivisions are logic, set theory (foundations), empirical mathematics of the various sciences (applied mathematics including computational mathematics), and the rigorous study of uncertainty.
It seems to be logical to fuzzily divide mathematics into fundamental mathematics, advanced mathematics, applied mathematics, and computational mathematics (with dividing pure mathematics into fundamental mathematics and advanced mathematics). But mathematics remains unified. Mark Burgin [2004] proposed named sets as unified foundations for mathematics.
There were a number of scientific revolutions (qualitatively radical constructive jump-like changes) in mathematics (Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], [Encyclopaedia of Mathematics 1988]) such as:
discovering the existence of irrational numbers via proving the irrationality of 21/2 or, equivalently, that the diagonal of a square is incommensurable with its side by Pythagoreans (possibly by Hippasus in the 5th century BC);
creating axiomatic geometry and number theory by Euclid [1482] in c. 300 BC;
introducing negative numbers in the "Nine Chapters on the Mathematical Art" in China (100 BC - 50 BC) and using them by Fibonacci in the 13th century as debts and losses whereas many European mathematicians ignored them as meaningless even in the 17th-18th centuries;
using square roots of negative numbers by Heron in the 1st century AD, Niccolo Fontana Tartaglia and Gerolamo Cardano in the 16th century, naming imaginary numbers by René Descartes in the 17th century, introducing complex analysis by Abraham de Moivre and Leonhard Euler and geometrically interpreting complex numbers by Caspar Wessel in the 18th century, their general acceptance due to Carl Friedrich Gauss, Augustin Louis Cauchy, and Niels Henrik Abel, as well as introducing quaternions by Sir William Rowan Hamilton and octonions by John Thomas Graves in the 18th century;
creating analysis (calculus) by Gottfried Wilhelm Leibniz [1684] and Isaac Newton [1687];
creating non-Euclidean hyperbolic geometry by Nikolai Lobachevsky [1829] and non-Euclidean elliptic geometry in 1854 by Bernhard Riemann [1990] who constructed an infinite family of non-Euclidean geometries via Riemannian metrics on the unit ball in Euclidean space;
creating the foundations of real analysis by Augustin Louis Cauchy [1882], Karl Theodor Wilhelm Weierstrass [1894], and Richard Dedekind [1930];
creating set theory by Georg Cantor [1932];
research axiomatization by David Hilbert [1899, 1932].
Mathematics has very many achievements and is both a universal scientific language and a basis for future research.
Rigorously axiomatized standard (classical) mathematics created by many famous mathematicians has already successfully solved very many scientific, engineering, educational, and life problems. Its future development will provide its very important role.
But there are very many typical urgent problems of our complicated world and time for which standard (classical) mathematics cannot propose its available adequate methods.
It occurs that in a whole series of key directions, by the level of thinking, modern classical mathematics corresponds to physics from the antique times to the 19th century, which also considered its atoms as indivisible. In the 20th and 21st centuries, physics slowly deepens their final division into the component parts. It requires such research monsters as the Large Hadron Collider. And physics is indeed quite foremost natural science…
Practice let a number of scientists recognize and explicitly express such understanding.
Ruggero Maria Santilli [2008] wrote: "...there cannot be really new physical theories without really new mathematics, and there cannot be really new mathematics without new numbers".
Ivan Gandzha and Jerdsey Kadeisvily [2011] noted: "Santilli has repeatedly stated that: The origin of protracted controversies or unsolved problems in physics, chemistry, biology, and other sciences, is generally due to the use of mathematics basically insufficient for the quantitative treatment of the problem at hand, with consequential need to develop new appropriate mathematics".
Jakub Czajko [2004a] wrote: "Physics needs new mathematical foundations. Some problems of physics could be traced to hidden, unresolved issues in pure mathematics (PM), some of which are almost as old as the, allegedly impossible and therefore prohibited, division by zero... Mathematics may need an upgrade after a discovery is made in physics, for some old ideas may be irrelevant to new aspects of the physical reality. We need a synthetic mathematics (SM) to complement the classical analytic methods in mathematical research. Induction alone is insufficient for the syntheses needed to comprehend physics. Yet some new aspects of the physical reality could be deduced from experiments backed by the SM. Its inductive abstraction and strict rules of inference for deduction made mathematics the most exact of all exact sciences. Yet the PM emphasized its apriorical character and its ‘‘statutory’’ independence of any experimental evidence almost to the point of self-destruction. For by allowing the use of postulative method to define its primitive notions and fundamental objects, its splendid exactness is practically defeated... Many results of recent experiments and observations remain unexplained, because the PM still operates within the perimeter outlined by some ancient and medieval paradigms. Therefore we must upgrade abstract mathematics after significant breakthroughs in physics. It is not enough to justify former physical achievements by showing that mathematics complies with them. We should create quite new mathematics that goes far beyond and above of what past physics may have suggested. It is imperative thus to keep mathematics in sync with developments of new ideas in physics."
Classical mathematics [Encyclopaedia of Mathematics 1988] 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. Lev Gelimson [1995a-g, 2001a, 2001b, 2001h, 2003f, 2004a, 2009a, 2009b] discovered that many classical fundamental mathematical theories, methods, and concepts [Encyclopaedia of Mathematics 1988] are insufficient for adequately solving and even considering many typical urgent problems.
It is clear that further scientific revolutions in mathematics must follow. There can be different ways for them, e.g.:
internal revolutions in standard (classical) mathematics itself;
scientific revolutions external with respect to standard (classical) mathematics itself but internal with respect to whole mathematics via creating a number of alternative mathematics so that whole mathematics becomes megamathematics.
Already either accepting or rejecting the axiom of choice bisects standard (classical) mathematics into two partially different alternative mathematics even if they have very much in common.
Along with classical (standard) mathematics, a number of other (alternative, nonstandard, nonclassical) mathematics can be possible and useful. Among them are, e.g.:
megamathematics based on alternative set theories by Petr Vopěnka [1979], Karel Hrbacek [2009b], etc.;
megamathematics based on mereology by Edmund Husserl [1901], Stanisław Leśniewski [1916], etc.;
non-Archimedean megamathematics by Abraham Robinson [1966], John Horton Conway [1976], Mark Burgin [2002, 2012], etc.;
megamathematics based on multisets by Richard Dedekind [1930], fuzzy sets by Lotfi Zadeh [1965] and Dieter Klaua [1965, 1966a, 1966b, 1967], rough sets by Zdzisław Pawlak [1982], fuzzy multisets by Ronald R. Yager [1986], etc.;
megamathematics by Ruggero Maria Santilli [1985a, 1985b, 1993a, 1993b, 1999]: "We therefore outline three sequential generalized mathematics introduced by the author under the name of iso-, geno- and hyper-mathematics which are based on generalized, Hermitean, non-Hermitean and multi-valued units, respectively";
mega-overmathematics by Lev Gelimson [1995a-g, 2001a, 2001b, 2001h, 2003f, 2004a, 2009a, 2009b, 2011c] first named "basic new mathematics" [1995a, 1995b] and then [ [2003f, 2004a] "elastic mathematics" are based on their uninumbers, quantielements, quantisets, and uniquantities with quantioperations and quantirelations. They provide universally and adequately modeling, expressing, measuring, evaluating, and estimating general objects. This all creates the basis for science unimathematical test fundamental metasciences systems by Lev Gelimson [2011b] which are universal.
The present monograph is dedicated to mega-overmathematics as revolutions in fundamental mathematics dealing with entirely abstract fundamental concepts [Encyclopaedia of Mathematics 1988] such as numbers, sets, their cardinalities and measures. These revolutions mean (in their general sense) creating fundamental mega-overmathematics alternative to classical (standard) fundamental mathematics and (in their special sense) creating own unified fundamentals for mega-overmathematics but DO NOT mean any change in classical mathematics itself.
Uniarithmetics, quantialgebra, and quantianalysis revolutionarily replace the very fundamentals of classical mathematics in mega-overmathematics.
Unfortunately, there is no possibility to refer to all the thousands of monographs and articles used by the author. He presents his sincere apologies and is much obliged to all those scientists as well as to all the readers for their attention and desire to understand and use the following concepts that can help them solve their urgent problems and possibly give them the joy of touching some new knowledge.
The purpose of this work is a very "naive" joint presentations of some interconnected new concepts with many examples of their applications useful for the reader. No existence, consistancy, and uniqueness question is considered; sets, semisets, and classes are not distinguished; each complex uninumber is regarded as a separated number system. The following constructions are based on the real numbers, sets, and infinite cardinal numbers.
Introduction. Testing Advanced Mathematics
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 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 that classical fundamental mathematical theories, methods, and concepts are insufficient for adequately solving and even considering many typical urgent problems.
Mega-overmathematics 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 mega-overmathematics fundamental sciences systems developing, extending, and applying overmathematics. Among them are, in particular, science unimathematical test fundamental metasciences systems which are universal.
Advanced Science Unimathematical Test Fundamental Metasciences System
Advanced science unimathematical test fundamental metasciences system in mega-overmathematics [2] is one of such systems and can efficiently, universally and adequately strategically unimathematically test any pure science. This system includes:
fundamental metascience of advanced science test philosophy, strategy, and tactic including advanced science test philosophy metatheory, advanced science test strategy metatheory, and advanced science test tactic metatheory;
fundamental metascience of advanced science consideration including advanced science fundamentals determination metatheory, advanced science approaches determination metatheory, advanced science methods determination metatheory, and advanced science conclusions determination metatheory;
fundamental metascience of advanced science analysis including advanced subscience analysis metatheory, advanced science fundamentals analysis metatheory, advanced science approaches analysis metatheory, advanced science methods analysis metatheory, and advanced science conclusions analysis metatheory;
fundamental metascience of advanced science synthesis including advanced science fundamentals synthesis metatheory, advanced science approaches synthesis metatheory, advanced science methods synthesis metatheory, and advanced science conclusions synthesis metatheory;
fundamental metascience of advanced science objects, operations, relations, and criteria including advanced science object metatheory, advanced science operation metatheory, advanced science relation metatheory, and advanced science criterion metatheory;
fundamental metascience of advanced science evaluation, measurement, and estimation including advanced science evaluation metatheory, advanced science measurement metatheory, and advanced science estimation metatheory;
fundamental metascience of advanced science expression, modeling, and processing including advanced science expression metatheory, advanced science modeling metatheory, and advanced science processing metatheory;
fundamental metascience of advanced science symmetry and invariance including advanced science symmetry metatheory and advanced science invariance metatheory;
fundamental metascience of advanced science bounds and levels including advanced science bound metatheory and advanced science level metatheory;
fundamental metascience of advanced science directed test systems including advanced science test direction metatheory and advanced science test step metatheory;
fundamental metascience of advanced science tolerably simplest limiting, critical, and worst cases analysis and synthesis including advanced science tolerably simplest limiting cases analysis and synthesis metatheories, advanced science tolerably simplest critical cases analysis and synthesis metatheories, advanced science tolerably simplest worst cases analysis and synthesis metatheories, and advanced science tolerably simplest limiting, critical, and worst cases counterexamples building metatheories;
fundamental metascience of advanced science defects, mistakes, errors, reserves, reliability, and risk including advanced science defect metatheory, advanced science mistake metatheory, advanced science error metatheory, advanced science reserve metatheory, advanced science reliability metatheory, and advanced science risk metatheory;
fundamental metascience of advanced science test result evaluation, measurement, estimation, and conclusion including advanced science test result evaluation metatheory, advanced science test result measurement metatheory, advanced science test result estimation metatheory, and advanced science test result conclusion metatheory;
fundamental metascience of advanced science supplement, improvement, modernization, variation, modification, correction, transformation, generalization, and replacement including advanced science supplement metatheory, advanced science improvement metatheory, advanced science modernization metatheory, advanced science variation metatheory, advanced science modification metatheory, advanced science correction metatheory, advanced science transformation metatheory, advanced science generalization metatheory, and advanced science replacement metatheory.
Advanced science unimathematical test fundamental metasciences system in megamathematics is universal and very efficient.
Fundamental Defects of Advanced Mathematics
In particular, apply the advanced science unimathematical test fundamental metasciences system by Lev Gelimson [2011b] in his mega-overmathematics to classical fundamental mathematics (Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], [Encyclopaedia of Mathematics 1988]).
Even the very fundamentals of classical fundamental mathematics have evident cardinal defects of principle.
1. Operations
In classical mathematics [Encyclopaedia of Mathematics 1988], in each concrete (mixed) physical magnitude, e.g. 5 liter fuel, the operation unifying "5 L" and "fuel" is not obvious.
It is impossible to consider either "fuel multiplied by 5 liter" or, all the more, "5 liter multiplied by fuel". These both pure theoretical possibilities are not reasonable at all. And classical mathematics [Encyclopaedia of Mathematics 1988] cannot propose nothing else.
Hence for any concrete (mixed) physical magnitude (quantity with a measurement unit), there is no suitable mathematical model and no known suitable operation.
Nota bene: Multiplication is the evident operation between the number "5" and the measurement unit "L".
Further even the pure number operations in classical mathematics [Encyclopaedia of Mathematics 1988] are considered to be at most countable, which makes the range of mathematical models very narrow.
John Wallis [1656] extended power exponents from positive integers to rational numbers.
But such a finite pure number operation as raising a negative number to a power 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 .
In classical mathematics [Encyclopaedia of Mathematics 1988], division by zero is undefined and hence avoided in real-number arithmetic, algebra, and analysis.
[Wikipedia Division_by_zero] notes that by formal operations in formal calculation using rules of arithmetic without consideration of whether the result of the calculation is well-defined, it is sometimes useful to think of a/0, where a ≠ 0, as being ∞ . This infinity can be either positive, negative, or unsigned, depending on context. The real projective line, the Riemann sphere, the extended non-negative real number line, and complex analysis bring here nothing new. In the known hyperreal and surreal numbers, division by zero is still impossible unlike division by nonzero infinitesimals. In computer arithmetic, the IEEE floating-point standard specifies that every floating point arithmetic operation, including division by zero, has a well-defined result. To provide this, the standard supports signed zeroes (positive zero +0 and negative zero -0), signed infinities, and NaN (not a number):
dividing a by +0 is positive infinity when a is positive, negative infinity when a is negative, and NaN when a = ±0;
dividing a by -0 is negative infinity when a is positive, positive infinity when a is negative, and NaN when a = ±0.
For 1/0, most calculators return either an error or undefined state. Some of them give (1/0)2 = ∞ .
Nota bene: Here a/0 = ∞ independently of a or, at least, of its modulus (absolute value) |a| , e.g. 1/0 = ∞ = 1010/0.
[Wikipedia Signed_zero] gives further rules of the IEEE floating-point standard:
-0/|x| = -0 (x ≠ 0),
(-0)(-0) = +0,
|x|(-0) = -0,
x + (-0) = x + (+0) = x ,
(-0) + (-0) = (-0) - (+0) = -0,
(+0) + (+0) = (+0) - (-0) = +0,
x - x = x + (-x) = +0
(for any finite x , -0 when rounding toward negative).
Because of negative zero (and only because of it), the statements
z = -(x - y)
and
z = (-x) - (-y),
for floating-point variables x , y , and z , cannot be optimized to
z = y - x .
Some other special rules:
(-0)1/2 = -0,
(-0)/(-∞) = +0 (follows the sign rule for division),
|x|/(-0) = -∞ (for non-zero x, follows the sign rule for division),
(±0)(±∞) = NaN ,
(±0)/(±0) = NaN .
Nota bene: Conservation law does not hold here.
Jakub Czajko [2004a] proposed distinguishing a/0 and b/0 by a ≠ b but rejected 0 × 0 = 0 with replacing this precise equality via approximate equality 0 × 0 ≈ 0 because he considered
0 × 0 = 1/∞ × 1/∞ = 1/∞2 ≈ 0.
Robert Goldblatt [1998] built the hyperreals *R including both infinitesimal and infinite elements via extending standard arithmetic operations to R∞ , see also Jonathan W. Hoyle [2007]. For a = <a(0), a(1), ...> and b = <b(0), b(1), ...>, they defined, in particular,
a/b = <a(0)/b(0), a(1)/b(1), …>,
ab = <a(0)b(0), a(1)b(1), …>
with undefined sequences indices for which these operations are not defined. Therefore, also by building the nonstandard universe, division by zero and raising a negative number to a power can lead to problems.
It is usual to consider in classical mathematics [Encyclopaedia of Mathematics 1988] that the empty sum equals 0 whereas the empty product equals 1. The both rules work in the best cases only because both adding a number a to the empty sum and multiplying the empty product with a number a give the correct result (value, output) a . Otherwise, the results are incorrect as a rule. It would be better to consider:
the empty sum equals 0 if and only if namely addition is the only further operation;
the empty product equals 1 if and only if namely multiplication is the only further operation.
But any dependence of the already performed operations output on any further operation proves that such a result makes no objective sense at all.
Moreover, both the empty sum and the empty product are particular cases of the "result" of performing no operations (hence on no operands, arguments, or inputs). This "result" may not depend on any particular operation and must be universal. But classical mathematics [Encyclopaedia of Mathematics 1988] cannot provide any universal value of this output. In particular, neither 0 nor 1 can provide such universality.
Conclusions
1. The system of operations in classical mathematics has gaps because it cannot mathematically model any concrete (mixed) physical magnitude (quantity with a measurement unit).
2. Even the pure number operations in classical mathematics are considered to be at most countable.
3. Even finite pure number operations in classical mathematics can have very narrow correct definition domains.
4. It is urgent to define raising a negative number to a power.
5. It is urgent to exactly express (in some suitable extension of the real numbers) division by zero.
6. Lev Gelimson [1994c, 1995a] first explicitly directly expressed division by zero to further extend all the infinitesimal, finite, infinite, and combined pure (dimensionless) amounts and to conveniently operate on them with holding conservation law and introduced the empting (voiding) operation transforming any object to the empty (void) object (element) # (or the empty set ∅ so that # ∈ ∅ and # = ∅). Using the empty (void) operand # (or ∅) excludes (drops) any operation on this operand so that this operand neutralizes any operation. Then the "result" of performing no operations (hence on no operands, arguments, or inputs) equals namely # (or ∅), which is universal. Further zero 0 may be considered to be nonnumber which does not belong to the natural numbers N , to the integer numbers Z , to the real numbers R , to the complex numbers C , etc.
2. Measures
Measure theory in classical mathematics ([Bourbaki 1949], [Encyclopaedia of Mathematics 1988]) typically uses measures of Johann Radon and Henri Lebesgue along with integrals of Bernhard Riemann and Henri Lebesgue. Usual measures may take nonnegative values or +∞ . Signed measures including charges may also take negative values or -∞ .
Felix Hausdorff [1919] proposed his measure already before [1935]. It always exists along with his dimension which can be noninteger and is widely used in fractal theory (Benoît Mandelbrot [1975, 1977, 1982], Alexey Stakhov [2009], and Sergey Abachiev [2012]), as well as Cantor sets [Encyclopaedia of Mathematics 1988] and Cantorian spacetime theory (Mohammed El Naschie [2009] and Jakub Czajko [2004a]).
The counting measure slightly generalizes counting and is defined either as the number of elements in any finite Cantor set or +∞ for any infinite Cantor set, which is natural but brings nothing new.
Therefore, well known measures are sensitive only restrictedly, namely in the limits of the specific dimensionality, within them completely ignore even uncountable zero-measure changes, and give either 0 or +∞ for distinct point sets between two parallel lines or planes differently distant from one another.
Any measure of each segment or interval on a straight line or a curve is independent of whether or not that includes its endpoints.
In classical mathematics ([Bourbaki 1949], [Encyclopaedia of Mathematics 1988]), there are no sensitive common measures for any even bounded sets of mixed dimensions, i.e. sets simultaneously including parts of different dimensions such as separate points, intervals, as well as bounded parts of surfaces and spaces.
Bernard Bolzano [1851] stated his dissatisfaction with such circumstances and tried to do something in the particular case of a natural-number length.
Conclusions
1. Well-known measures are only 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.
2. There are no sensitive common measures for any even bounded sets of mixed dimensions.
3. It is urgent to exactly measure any possibly infinitely great or small objects or models by holding universal conservation laws.
4. Lev Gelimson [1994c, 1995a] first generalized exactly counting all the elements of any infinite set via its uniquantity with universalization even for infinitesimal differences of element quantities of any infinite quantisets. This provided exactly (perfectly sensitively) measuring any possibly infinitely great or small objects or models by holding universal conservation laws for setting and solving many typical urgent problems. He introduced canonic infinitesimals (namely signed zeroes) [1994d] and canonic infinities [1994c, 1995a, 1995c, 1997a, 1997b] exactly expressing the also uncountably algebraically additive quantities of the elements of some suitable canonic sets. From the beginning of 2001, the further editions [Gelimson 2001a, 2001b, 2001c, 2001d, 2001e, 2001h] of these and many other scientific publications are available on the Internet. Giovanni Giuseppe Nicosia placed in his Doctor Thesis [2001] in February, 2001, a reference to [Gelimson 2001d], as well as in [Nicosia 2005]. Scientist Vuara has published [Gelimson 2001h] along with the references of some academicians and professors to Lev Gelimson's monographs, articles, and other scientific publications in [Wikibooks Hyperanalysis-MeasurementTheory]. [Hypernumber Blogspot] noted: "Other kinds of hypernumber are defined differently by Mark Burgin, Rugerro Maria Santilli and" the author. Armahedi Mahzar [Hypernumbers Group] wrote: "Other kinds of hypernumbers are the internal extensions of real numbers created by making the axis of real number more dense. Examples of such internal hypernumbers are the hyperreal numbers of Robinson, surreal numbers of Conway, hypernumbers of Mark Burgin and" the author. "This group will explore such existing kinds of hypernumbers and beyond." [Gelimson 2003a, 2004a] further developed his uniquantities building an explicit universal exact counting measure perfectly sensitive even to infinitesimal differences of element quantities of any infinite quantisets.
Addition
Alexey Petrovsky [2003] investigated new types of the spaces of measurable sets and multisets, as well as the general properties of set and multiset measures.
Mark Burgin [2005] proposed his hypermeasures via adding infinitely big and oscillating numbers as external objects to the real and complex numbers without changing the inner structure of the real and complex numbers spaces via injecting into them infinitely small numbers and other nonstandard entities as nonstandard analysis does. His real hypernumbers are sets of equivalent sequences of real numbers like real numbers are sets of equivalent fundamental sequences of rational numbers. In the universe of his hypernumbers, all sequences and series of real and complex numbers, as well as definite integrals of continuous functions, have values. They are either ordinary (real and complex) numbers for convergent sequences, series, and integrals, or (infinite and oscillating) hypernumbers for divergent sequences, series, and integrals.
3. Probabilities
Any probability measure takes values from the closed unit interval [0, 1] only and is a particular case of a measure.
In classical mathematics [Encyclopaedia of Mathematics 1988], real numbers having gaps between them cannot express not only unlimited, but also many limited quantities (e.g. the probability of selecting one given number from all natural numbers).
Let us assume that there are 10 balls with ciphers (digits) 0, 1, ... , 9, respectively, in a bag. Precisely one of the balls is picked out blindly (randomly, without any extrasensory abilities). What is the probability that the picked ball has namely a given (predefined) cipher (digit), for example 7? The total number of all the possible outcomes is 10. By only one of them, the desired event occurs. That is why the desired probability by its classical definition [Encyclopaedia of Mathematics 1988] is 1/10.
Let us now consider a more complicated problem. Imagine that we select exactly one number from countably many nonnegative integers 0, 1, 2, ... , 10, ... , 100, ... , 1000, ... with equal probability of selecting any of them. What is this probability or, equally, the probability that we have selected namely a given (predefined) number, for example, 7?
Classical mathematics [Encyclopaedia of Mathematics 1988] declares that this probability does not exist at all because of the following "proof". Ad absurdum, suppose that this probability exists. Then it has to be either 0 or positive. If it were 0, then the total probability of selecting any of nonnegative integers whose total number is finite would also vanish. The same would hold for all the nonnegative integers, which is proved via the corresponding limiting process. But the total probability of selecting any of all the nonnegative integers has to be precisely 1 as the probability of a certain event. Indeed, exactly one of all the nonnegative integers is selected. If the desired probability were, on the contrary, any positive number, then divide 1 by this number and take any nonnegative integer which is greater than this quotient. The so-called axiom of Archimedes [Encyclopaedia of Mathematics 1988] provides that there exist infinitely many such nonnegative integers. For definity, take the least from these nonnegative integers. Then the total probability of selecting any of nonnegative integers from 0 to the taken nonnegative integer would be greater than 1, which is impossible for any probability at all. Moreover, in this case of any positive desired probability, the corresponding limiting process would even give plus infinity for the total probability of selecting any of all the nonnegative integers instead of 1 as the probability of this certain event. In this way, classical mathematics [Encyclopaedia of Mathematics 1988] leads to the conclusion that the desired probability does not exist at all.
Note that the Archimedes axiom [Encyclopaedia of Mathematics 1988] in mathematics is an invention, which is typical for entire mathematics itself, whereas the Archimedes law on a buoyant force [Encyclopaedia of Mathematics 1988] much better known is a law of nature fully objective and hence is a discovery, which is typical for natural sciences.
Classical mathematics [Encyclopaedia of Mathematics 1988] also declares without any explanations that if the probability of selecting anyone of the elements of any uncountable set, for example an interval, a straight line, a rectangle, a plane, or a space, is the same, then it vanishes, as if that would be an impossible event.
But these and many other typical events are fully reasonable and possible and hence must have certain positive probabilities. And if classical mathematics cannot indicate them, then its real number system is clearly insufficient and has gaps.
It is possible to assume that such probabilities are indeterminate infinitesimals.
Let us give the following analogy. It is necessary to solve a certain equation. How much benefit would we achieve due to the conclusion that the solutions to this equation are some undetermined imaginary numbers without their determination and clear indication? It is not difficult to guess the school grade for such an answer…
Conclusions
1. Any probability measure is a particular case of a measure.
2. The probabilities cannot discriminate impossible and some differently possible events.
3. The probabilities of reasonable possible events can be nonexising at all.
4. There is no known universal exact perfectly sensitive measure for any possibly infinitely great or small objects or models by holding universal conservation laws.
5. The real numbers having gaps between them cannot express not only unlimited, but also many limited quantities.
6. It is urgent to exactly measure any probability so that each fully reasonable and possible event has a certain namely positive probability.
7. Lev Gelimson [1994c, 1995a] first generalized exactly counting all the elements of any infinite set with universalization even for infinitesimal differences of element quantities of any infinite quantisets and with providing a certain namely positive probability of any fully reasonable and possible event.
4. Contradictory Objects, Systems, and Models
In classical mathematics [Encyclopaedia of Mathematics 1988], contradictory objects, systems, and models are declared nonexisting at all and are completely ignored along with contradictory problems even if they are urgent. 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.
Nota bene: All existing objects and systems in nature, society, and thinking have complications, e.g. contradictoriness, and hence exist without adequate models in classical mathematics [Encyclopaedia of Mathematics 1988].
Conclusions
1. Classical mathematics intentionally avoids, ignores, and cannot (and possibly hence does not want to) adequately consider, model, express, measure, evaluate, and estimate many complications such as contradictions, infringements, damages, hindrances, obstacles, restrictions, mistakes, distortions, errors, information incompleteness, multivariant approach, etc.
2. All existing objects and systems in nature, society, and thinking have complications, e.g. contradictoriness.
3. It is urgent to adequately model also contradictory objects and systems.
4. Lev Gelimson [1995a, 1995b] created his own mathematics named "basic new mathematics" allowing and efficiently using also contradictory objects, systems, and models existing at least in the symbolic sense as "black boxes". He also unified the membership, inclusion, and part-whole relations, which alone can often avoid contradictions and excludes many paradoxes. From the beginning of 2001, the further editions [Gelimson 2001a, 2001b, 2001c, 2001d, 2001h] of these and many other scientific publications are available on the Internet. Giovanni Giuseppe Nicosia placed in his Doctor Thesis [2001] in February, 2001, a reference to [Gelimson 2001d], as well as in [Nicosia 2005]. Scientist Vuara has published [Gelimson 2001h] along with the references of some academicians and professors to Lev Gelimson's monographs, articles, and other scientific publications in [Wikibooks Hyperanalysis-MeasurementTheory]. [Gelimson 2003a, 2003b, 2003d, 2003e, 2003f, 2004a] further developed his own mathematics then named "elastic mathematics".
Addition
Alexey Petrovsky [2001a, 2001b] constructed a general decision rule for contradictory expert classification of multiattribute objects and multiple criteria project selection based on contradictory sorting rules.
Classical Fundamental Mathematics Revolutions Necessity
Naturally, along with the above, there are very many further lacks and shortcomings of classical fundamental mathematics (Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], [Encyclopaedia of Mathematics 1988]).
Therefore, the very fundamentals of classical fundamental mathematics have a lot of obviously deep and even cardinal defects of principle.
Consequently, to make classical fundamental mathematics adequate, its evolutionarily locally correcting, improving, and developing which can be useful are, unfortunately, fully insufficient. Classical fundamental mathematics needs revolutionarily replacing its inadequate very fundamentals via adequate very fundamentals.
Conclusions
1. The very fundamentals of classical fundamental mathematics have a lot of obviously deep and even cardinal defects of principle.
2. Mathematics is the only science whose very fundamentals remain almost unchanged for more than a century whereas there were many revolutions in all natural, technical, and humanitarian sciences natural, technical, and humanitarian sciences in this time.
3. It is urgent to create and develop alternative mathematics with revolutionarily replacing the inadequate very fundamentals of classical fundamental mathematics via adequate very fundamentals.
4. Lev Gelimson [1995a, 1995b] created his own mathematics named "basic new mathematics" with revolutionarily replacing the inadequate very fundamentals of classical fundamental mathematics via adequate very fundamentals. From the beginning of 2001, the further editions [Gelimson 2001a, 2001b, 2001c, 2001d, 2001h] of these and many other scientific publications are available on the Internet. Giovanni Giuseppe Nicosia placed in his Doctor Thesis [2001] in February, 2001, a reference to [Gelimson 2001d], as well as in [Nicosia 2005]. Scientist Vuara has published [Gelimson 2001h] along with the references of some academicians and professors to Lev Gelimson's monographs, articles, and other scientific publications in [Wikibooks Hyperanalysis-MeasurementTheory]. [Gelimson 2003a, 2003b, 2003d, 2003e, 2003f, 2004a] further developed his own mathematics then named "elastic mathematics".
Classical Fundamental Mathematics Revolutions Possibility
Natural, technical, and humanitarian sciences discover and model real, objective laws of nature, technology, thinking, and society. It is impossible and inadmissible to devise any imaginable dependencies instead of such real laws and to try to impose these dependencies to the real nature, technology, and society. Each of these sciences (with its object) is unique. It is difficult to imagine any alternative general physics another than classical general physics because the nature is the same and unique. However, also in general physics, there are different particular approaches, methods, theories, and especially hypotheses, for example, the corpuscular and wave theories of light.
On the contrary, mathematics is an expedient fabrication, absolute invention, and a result of fantasy sufficiently free. Where can be found, e.g., number 2 or a rectangle themselves (NOT their images completely material, for example, drawn on a chalk board) in the nature? Symbol 2 is relative at all. It is inherent in the numeration system with the base not less than 3 in the Arab (Indian) numeration. In the binary system, this is symbol 10. In the Roman numeration, this is symbol II. I.e., symbol 2 is conditional. In general, it is not possible to noticeably depict the contour of a rectangle. Indeed, a one-dimensional line possesses only a length and has zero width. I.e., mathematics completely consists of inventions. But it is not completely freely devised. It is the universal language of sciences which simulates real objects and their relations. Any real two-element sets such as 2 apples, 2 lions, etc. (which can be placed in the one-to-one correspondence), have their real quantity in common. And number 2 has been devised to express this real quantity which is objective, i.e., independent of our consciousness. And a rectangle is an ideal model for the description of, e.g., the faces of bricks. Carl Friedrich Gauss [1863] said and wrote: "Mathematics is the queen of sciences, and number theory is the queen of mathematics".
The sole limitations of fantasy by creating mathematics are its convenience and suitibility for sufficiently adequate mathematical simulation of real (natural, technical, and social) objects and systems, as well as for solving other problems important for these objects and systems.
In principle, each mathematical science allows entire full-valuable alternatives. A classical example could be of Euclidean, Lobachevskian, and Riemannian geometries. Different whole and complete mathematics (plural) as distinct sciences are also well-possible, well-considerable, and well-creatable. In particular, this holds for overmathematics advanced, created, and developed by Lev Gelimson [1995a-2012]. Consequently, questions of the type "What right has the author to advance, create, and develop his overmathematics?" are fully absurd. The sole limitation should be furthermore the expedience: "Why is overmathematics necessary and useful under conditions that classical mathematics is already available, well-known, and well-developed for millenia?"
Conclusions
1. There is a fundamental difference of mathematics as a pure invention of imaginable models from natural, technical, and humanitarian sciences discovering and modeling true laws of nature, engineering, thinking, and society.
2. Pluralism in mathematics already takes place and is always admissible.
3. It is possible and admissible to create and develop alternative mathematics.
4. Lev Gelimson [1995a, 1995b] created his own mathematics named "basic new mathematics" with revolutionarily replacing the inadequate very fundamentals of classical fundamental mathematics via adequate very fundamentals. From the beginning of 2001, the further editions [Gelimson 2001a, 2001b, 2001c, 2001d, 2001h] of these and many other scientific publications are available on the Internet. Giovanni Giuseppe Nicosia placed in his Doctor Thesis [2001] in February, 2001, a reference to [Gelimson 2001d], as well as in [Nicosia 2005]. Scientist Vuara has published [Gelimson 2001h] along with the references of some academicians and professors to Lev Gelimson's monographs, articles, and other scientific publications in [Wikibooks Hyperanalysis-MeasurementTheory]. [Gelimson 2003a, 2003b, 2003d, 2003e, 2003f, 2004a] further developed his own mathematics then named "elastic mathematics".
Classical Fundamental Mathematics Revolutions Usefulness
First of all, it is necessary to show essential differences of revolutionary fundamental mathematics from classical mathematics [1]. Moreover, these differences have to take place namely in the fundamentals of revolutionary fundamental mathematics versus classical fundamental mathematics. The reason is that it is NOT a matter of any particular theory. Revolutionary fundamental mathematics has to be entire science based on its own principles and to include a whole synergistic system of many concepts, theories, and methods.
Further, revolutionary fundamental mathematics has to give many new possibilities (as compared to those in classical fundamental mathematics) to consider and solve new classes of problems (including the simulation of objects) very important for real life and science but NOT considerable and NOT solvable by classical mathematics. Or, at least, revolutionary mathematics has to do this essentially better than classical mathematics does.
Conclusions
1. Revolutionary fundamental mathematics should both have a fundamental difference from classicalal fundamental mathematics and has to give many new possibilities (as compared to those in classicala fundamental mathematics) to consider and solve new classes of problems (including the simulation of objects) very important for real life and science but NOT considerable and NOT solvable by classical mathematics. Or, at least, revolutionary fundamental mathematics has to do this essentially better than classical fundamental mathematics does.
2. Lev Gelimson [1995a, 1995b] created his own mathematics named "basic new mathematics" with revolutionarily replacing the inadequate very fundamentals of classical fundamental mathematics via adequate very fundamentals. From the beginning of 2001, the further editions [Gelimson 2001a, 2001b, 2001c, 2001d, 2001h] of these and many other scientific publications are available on the Internet. Giovanni Giuseppe Nicosia placed in his Doctor Thesis [2001] in February, 2001, a reference to [Gelimson 2001d], as well as in [Nicosia 2005]. Scientist Vuara has published [Gelimson 2001h] along with the references of some academicians and professors to Lev Gelimson's monographs, articles, and other scientific publications in [Wikibooks Hyperanalysis-MeasurementTheory]. [Gelimson 2003a, 2003b, 2003d, 2003e, 2003f, 2004a] further developed his own mathematics then named "elastic mathematics".
Revolutions in Advanced Mathematics
Overmathematics [2-10] in fundamental megamathematics [2-10] revolutionarily replaces the inadequate very fundamentals of classical fundamental mathematics [1] via adequate very fundamentals.
V. Unimathematical Modeling 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 modeling fundamental sciences system [14] including:
the fundamental science of universal mathematical and physical modeling essence and strategy including universal mathematical and physical modeling problem setting theories, universal mathematical and physical modeling problem pseudosolution theories, universal mathematical and physical modeling problem solving strategy theories, and universal mathematical and physical modeling problem transformation theories;
the fundamental science of universal mathematical and physical model analysis and synthesis including universal mathematical and physical model analysis theories and universal mathematical and physical model synthesis theories;
the fundamental science of universal mathematical and physical model invariance and symmetry including universal mathematical and physical model data invariance theories, universal mathematical and physical model problem invariance theories, universal mathematical and physical model method invariance theories, universal mathematical and physical model result invariance theories, and universal mathematical and physical model symmetry theories;
the fundamental science of universal mathematical and physical model data unification and grouping including universal mathematical and physical model data unification theories and universal mathematical and physical model data grouping theories;
the fundamental science of universal mathematical and physical model data structuring and restructuring including universal mathematical and physical model data structuring theories and universal mathematical and physical model data restructuring theories;
the fundamental science of universal mathematical and physical model data scatter and trend including universal mathematical and physical model data direction theories, universal mathematical and physical model data scatter theories, universal mathematical and physical model data trend theories, and general power universal mathematical and physical model data scatter and trend measure and estimation theories;
the fundamental science of unimathematically considering mathematical and physical model data outliers including universal mathematical and physical model data outlier determination theories, universal mathematical and physical model data outlier centralization theories, universal mathematical and physical model data outlier transformation theories, universal mathematical and physical model data outlier compensation theories, and universal mathematical and physical model data outlier estimation theories;
the fundamental science of universal mathematical and physical model measurement which includes general theories and methods of developing and applying unimathematical uniquantity as universal perfectly sensitive quantimeasure of universal mathematical and physical models with possibly recovering true measurement information using incomplete changed data;
the fundamental science of measuring universal mathematical and physical model concessions which for the first time regularly applies and develops unimathematical theories and methods of measuring universal mathematical and physical model 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 measuring universal mathematical and physical model reserves further naturally generalizing the fundamental science of measuring mathematical and physical model concessions and for the first time regularly applying and developing unimathematical theories and methods of measuring not only universal mathematical and physical model 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 mathematical and physical models, as well as for solving general problems;
the fundamental sciences of measuring universal mathematical and physical model reliability and risk for the first time regularly applying and developing unimathematical theories and methods of quantitatively measuring the reliabilities and risks of universal mathematical and physical models with avoiding unjustified artificial randomization in deterministic problems;
the fundamental science of measuring universal mathematical and physical model deviation for the first time regularly applying unimathematics to measuring deviations of real general objects and systems from their ideal universal mathematical and physical models, and also of universal mathematical and physical 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 data scatter and trend give corresponding invariant and universal measures concerning linear and nonlinear models. Group center theories sharply reduce this scatter, raise data scatter and trend, and for the first time also consider their outliers. Overmathematics even allows to divide a point into parts and to refer them to different groups. Coordinate division theories and especially principal bisector (as a model) division theories efficiently form such groups. Note that there are many reasonable deviation arts, e.g., the following:
the value of a nonnegative binary function (e.g., the norm of the difference of the parts of an equation as a subproblem in a problem after substituting a pseudosolution to this problem, distance from the graph of this equation, its absolute error [1], relative error [1], unierror [2], etc.) of this object and each of all the given objects;
the value of a nonnegative function (e.g., the power mean value) of these values for all the equations in a general problem by some positive power exponent.
Along with the usual straight line square distance, we may also use, e.g., other possibly curvilinear (by additional limitations and other conditions such as using curves lying in a certain surface, etc.) power distances. By point objects and the usual straight line square distance, e.g., we obtain the only quasisolution by two points on a straight line, three points in a plane, or four points in the three-dimensional space. Using distances only makes this criterion invariant by coordinate system translation and rotation.
The unimathematical modeling fundamental sciences system is universal and very efficient.
VI. Unimathematical Measurement 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 measurement fundamental sciences system [15] including:
the fundamental science of unimathematical object measurement which includes general theories and methods of developing and applying overmathematical uniquantity as universal perfectly sensitive quantimeasure of general objects with possibly recovering true measurement information using incomplete changed data;
the fundamental science of unimathematical system measurement which includes general theories and methods of developing and applying overmathematical uniquantity as universal perfectly sensitive quantimeasure of general systems with possibly recovering true measurement information using incomplete changed data;
the fundamental science of unimathematical physical model measurement which includes general theories and methods of developing and applying overmathematical uniquantity as universal perfectly sensitive quantimeasure of physical models with possibly recovering true measurement information using incomplete changed data;
the fundamental science of unimathematical model measurement which includes general theories and methods of developing and applying overmathematical uniquantity as universal perfectly sensitive quantimeasure of mathematical models with possibly recovering true measurement information using incomplete changed data;
the fundamental science of measuring concessions which for the first time regularly applies and develops universal overmathematical theories and methods of measuring 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 measuring reserves further naturally generalizing the fundamental science of measuring concessions and for the first time regularly applying and developing universal overmathematical theories and methods of measuring 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 measuring reliability and risk for the first time regularly applying and developing universal overmathematical theories and methods of quantitatively measuring 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 measuring deviation for the first time regularly applying overmathematics to measuring 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 data scatter and trend give corresponding invariant and universal measures concerning linear and nonlinear models. Group center theories sharply reduce this scatter, raise data scatter and trend, and for the first time also consider their outliers. Overmathematics even allows to divide a point into parts and to refer them to different groups. Coordinate division theories and especially principal bisector (as a model) division theories efficiently form such groups. Note that there are many reasonable deviation arts, e.g., the following:
the value of a nonnegative binary function (e.g., the norm of the difference of the parts of an equation as a subproblem in a problem after substituting a pseudosolution to this problem, distance from the graph of this equation, its absolute error [1], relative error [1], unierror [2], etc.) of this object and each of all the given objects;
the value of a nonnegative function (e.g., the power mean value) of these values for all the equations in a general problem by some positive power exponent.
Along with the usual straight line square distance, we may also use, e.g., other possibly curvilinear (by additional limitations and other conditions such as using curves lying in a certain surface, etc.) power distances. By point objects and the usual straight line square distance, e.g., we obtain the only quasisolution by two points on a straight line, three points in a plane, or four points in the three-dimensional space. Using distances only makes this criterion invariant by coordinate system translation and rotation.
The unimathematical measurement fundamental sciences system is universal and very efficient.
0.6.4. Mega-Overmathematics Operations
0.6.4.1. Sign-Conserving Multiplication
In classical mathematics (Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], G. A. Korn, T. M. Korn [1968], I. N. Bronstein, K. A. Semendjajew [1989], [Encyclopaedia of Mathematics 1988]), usual multiplication is well-known. If all the factors to be multiplied are positive, then their product is positive, too, which is natural. If some factors to be multiplied are positive whereas the remaining factors to be multiplied are negative, then their product is positive when the number of negative factors is even whereas their product is negative when the number of negative factors is odd. This seems to be rather artificial than natural. Its origin (source) is that in the (real or complex) numbers, multiplication of real numbers distributes over addition in the well-known concepts of a ring and a field which both are commutative.
Nota bene: Both in mathematical logic and in set theory, the both distributive laws (of multiplication over addition and of addition over multiplication) hold. In mathematical logic, namely in Boolean algebra, logical disjunction ∨ plays the role of addition whereas logical conjunction ∧ plays the role of multiplication, and for any sentences (propositions that may be true or false) A , B , and C , we have both
(A ∨ B) ∧ C = (A ∧ C) ∨ (B ∧ C)
and
(A ∧ B) ∨ C = (A ∨ C) ∧ (B ∨ C).
In set theory, namely in set algebra, unification ∪ plays the role of addition whereas intersection ∩ plays the role of multiplication, for any sets A , B , and C , we have both
(A ∪ B) ∩ C = (A ∩ C) ∪ (B ∩ C)
and
(A ∩ B) ∪ C = (A ∪ C) ∩ (B ∪ C).
In real or complex algebra, for any numbers A , B , and C , multiplication distributes over addition
(A + B)C = AC + BC
whereas addition does not distribute over multiplication because generally (as a rule)
AB + C ≠ (A + C)(B + C).
Therefore, already in the well-known concepts of a ring and a field which both are commutative, one of the the both distributive laws (of multiplication over addition and of addition over multiplication) does not hold. Hence it is possible and at least equally natural to additionally consider completely nondistributive rings and fields along with well-known rings and fields which are partially nondistributive.
Nota bene: Common multiplication naturally leads to noncommutative common power and exponential functions well-defined by negative bases if and only if exponents are integer whereas sign-conserving multiplication naturally leads to commutative sign-conserving power and exponential functions well-defined by any real-number bases and exponents.
The fundamental ideas of sign-conserving multiplication of real numbers are as follows:
the modulus (absolute value) of the sign-conserving product (as a result of sign-conserving multiplication) of real numbers equals the modulus (absolute value) of the usual product of these numbers;
the value of the sign function of the sign-conserving product of real numbers vanishes if and only if the value of the sign function of the usual product of these numbers vanishes, i.e. if and only if at least one of these numbers vanishes;
the value of the sign function of the sign-conserving product of real numbers equals 1 if and only if all these numbers are positive;
the value of the sign function of the sign-conserving product of real numbers equals -1 if and only if at least one of these numbers is negative and none of these numbers vanishes.
To denote sign-conserving multiplication, simply use the parenthesis " either instead of a multiplication sign if it is implicit (i.e. omitped) or to the left of a multiplication sign (e.g. × , • , Π , etc.) if it is explicitly used.
Analytically, for any index set J , all indices j ∈ J , and any indexed real numbers building a set {aj | j ∈ J}, their sign-conserving product
"Πj∈J aj = min(sign aj | j ∈ J) |Πj∈J aj|
so that
|"Πj∈J aj| = |Πj∈J aj|
and
sign(Πj∈J aj|) = sign|Πj∈J aj| min(sign aj | j ∈ J) = min(sign|aj| | j ∈ J) min(sign aj | j ∈ J).
Example: For any real numbers a , b , and c ,
a"b"c = a "× b "× c = a "• b "• c
= min(sign a , sign b , sign c) |abc|
so that
|a"b"c| = |abc|
and
sign(a"b"c) = sign|abc| min(sign a , sign b , sign c)
= min(sign|a|, sign|b|, sign|c|) min(sign a , sign b , sign c).
Nota bene: Introducing an additional factor, e.g.
sign|abc| = min(sign|a|, sign|b|, sign|c|),
is here necessary to provide
sign(a"b"c) = 0
if at least one of these numbers a , b , and c vanishes whereas then
a"b"c = 0
due to vanishing the factor |abc| which is absent in sign(a"b"c).
0.6.4.2. Negative Base Power Theory
If a < 0 and we want to consider the real numbers R only, then we may consider by m ∈ Z , n ∈ N
a(2m + 1)/(2n) := a2(2m + 1)/(4n)
giving real sense to ab by any irrational b , too. Using modulus (absolute value) |a| gives the same results but is much less natural because
a ≠ |a|,
(2m + 1)/(2n) = 2(2m + 1)/(4n).
Mega-overmathematics by Lev Gelimson [1987-2011c] naturally introduces many further (also uncountable) quantioperations and quantirelations. Among them is sign-conserving power function
a"b = |a|b sign a
defined by any real numbers a ≠ 0 and b , as well as by a = 0 and any b > 0. Then we have, e.g.,
a"2 = a2 sign a ,
(-1)"3 = -1 = [(-1)"6]"1/2 = (-1)"6/2 ,
(-1)"1/3 = -1 = [(-1)"2]"1/6 = (-1)"2/6 .
Nota bene: Fundamental, advanced, applied, and/or computational mathematical considerations also belong to fundamental, advanced, applied, and/or computational overmathematics, respectively, if and only if such considerations directly and explicitly use namely overmathematical very fundamentals revolutionarily replacing the inadequate very fundamentals of classical mathematics.
Examples:
1. Sign-conserving raising a specific nonnegative number to a real power belongs to mathematics but not to overmathematics because simply raising this specific nonnegative number to this real power in classical mathematics brings the same result. Hence replacing simply raising a specific nonnegative number to a real power with sign-conserving raising a specific nonnegative number to a real power brings nothing new. Therefore, raising a specific nonnegative number to a real power does not require overmathematical sign-conserving raising to a real power.
2. Sign-conserving raising a specific negative number to a real power belongs both to mathematics and to overmathematics because simply raising any negative number to any real power in classical mathematics is always ill-defined [Encyclopaedia of Mathematics 1988] because there are infinitely many irrational power exponents arbitrarily near to the given real power exponent so that for them, such a power is indefinite at all. All the more, for any even integer power exponent considered isolated, such a power is well-defined but brings the opposite result:
ab = |a|b ,
a"b = |a|b sign a = -|a|b (a < 0).
For any odd integer power exponent, such a power (also considered isolated) is always ill-defined:
a2z+1 = -|a|2z+1 ≠ |a|2z+1 = a2(2z+1)/2 = [a2(2z+1)]1/2 (a < 0; z =0, ±1, ±2, ...).
Hence replacing simply raising any negative number to any real power with sign-conserving raising this negative number to this real power always brings a new result. Therefore, raising any negative number to any real power does require namely overmathematical sign-conserving raising to a real power.
3. Sign-conserving real power function as whole belongs both to mathematics and to overmathematics because simply raising any negative number to any real power in classical mathematics is always ill-defined [Encyclopaedia of Mathematics 1988] because there are infinitely many real power exponents arbitrarily near to the given real power exponent so that for them, such a power is indefinite at all. All the more, for any even integer power exponent, such a power is well-defined but brings the opposite result. For any odd integer power exponent, such a power is ill-defined. Hence replacing simply raising any negative number to any real power with sign-conserving raising this negative number to this real power always brings a new result. Therefore, raising any negative number to any real power does require namely overmathematical sign-conserving raising to a real power.
Notata bene:
1. For any polarly represented complex (also imaginary) power base
a = reiφ
where r is a nonnegative number (modulus, or polar radius), unique polar argument φ belongs to half-opened segment [0, 2π[ (0 included but 2π excluded), and
i2 = -1,
naturally generalize the sign function with direction function
dir a = eiφ = cos φ + i sin φ
and the above sign-conserving real power function with direction-conserving complex-base real-exponent power function
a"b = |a|b dir a = rb dir a
with a complex power base a and a real power exponent b .
2. For any polarly represented complex (also imaginary) power base
a = reiφ
where r is a nonnegative number (modulus, or polar radius), unique polar argument φ belongs to half-opened segment [0, 2π[ (0 included but 2π excluded), and
i2 = -1,
as well as for any complex (also imaginary) power exponent
b = c + di
where c and d are real numbers,
further naturally generalize the above direction-conserving complex-base real-exponent power function
a"b = |a|b dir a = rb dir a
with direction-adding complex power function
a"b = a"c+di = |a|c+di dir a = rc+di eiφ = rcrdi eiφ = rceid ln r eiφ = rcei(d ln r + φ) .
Nota bene: Use " in a"b if necessary only.
0.6.4.3. General Power-Exponential Function Hyperefficiency Theory
Introduction
Numbers with very small and very large absolute values [Wikipedia Large_numbers] are extremely important for real world modeling. Moreover, their role exponentially increases because of computer science evolution which requires the so-called scientific number representation, as well as the storage and handling of such numbers to avoid the permanent danger of "computing overflow".
In classical mathematics (Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], G. A. Korn, T. M. Korn [1968], I. N. Bronstein, K. A. Semendjajew [1989], [Encyclopaedia of Mathematics 1988]), exponentiation as raising numbers to powers by Michael Stifel [1544], as well as power functions y = xn with constant exponents n and exponential functions y = ax with constant bases a are widely used. Some power-exponential functions with variable bases and variable exponents such as
y = xx
and also iterated (nested) exponentials (power towers)
a^b^c^... = a^{b^[c^(...)]}
with multiply (repeatedly) raising bases to powers so that power exponents are powers themselves are well-known, see also [Wikipedia Tetration]. Leonhard Euler [1777] introduced the notation
expa(x) = a^x = ax ,
which can be combined with function iteration notation fn(x) giving
expan(x) = a^a^...^a^x
(with a used n times on the right-hand side). He also showed that the infinite power tower
a^a^...
defined as the limit of
a^a^...^a
(with a used n times), converges for e-e ≤ x ≤ e1/e as n goes to infinity, which roughly gives the interval from 0.066 to 1.44. In particular, at a = 21/2 , this limit equals 2. Hans Maurer [1901] already used modern tetration notation
na = a^a^...^a (with a used n times on the right-hand side).
Donald Ervin Knuth [1976] introduced his up-arrow notation
a↑n = a^n = an ,
a↑↑n = a^a^...^a (with a used n times on the right-hand side),
a↑↑n(x) = expan(x) = a^a^...^a^x (with a used n times on the right-hand side)
interpreting super-powers and super-exponential functions via using m arrows in expression a↑m n(x). John Horton Conway [1996] chained arrow notation
a→n→2 = a^a^...^a (with a used n times on the right-hand side)
provides similar generalization via increasing the number 2 and, more powerfully, by extending the chain.
Albert Arnold Bennett [1915] proposed commutative hyperoperations sequence defined by the recursion rule
Fn+1(a , b) = exp(Fn(ln(a), ln(b))
beginning with
F0(a , b) = ln(e^a + e^b) = ln(ea + eb),
addition (I)
F1(a , b) = a + b ,
multiplication (II)
F2(a , b) = ab = eln(a) + ln(b) ,
a commutative form of exponentiation (III)
F3(a , b) = eln(a) ln(b) ,
F4(a , b) = e^{e^[ln(ln(a))ln(ln(b))]}
not to be confused with tetration [Wikipedia Hyperoperation].
Wilhelm Ackermann [1928] defined the function
φ(m , n , p)
resembling the hyperoperation sequence with reproducing such basic operations as addition, multiplication, and exponentiation at p = 0, 1, 2, respectively:
φ(m , n , 0) = m + n ,
φ(m , n , 1) = mn ,
φ(m , n , 2) = m^n = mn ,
φ(m , n , p) = m↑p-1 (n + 1)
for p > 2 with extending these basic operations using Knuth's up-arrow notation.
Reuben Louis Goodstein [1947] introduced the hyperoperations sequence of operations extending succession (the 0th) 1 + b , addition (the 1st) a + b , multiplication (the 2nd) ab , and exponentiation (the 3rd) ab and gave the extended operations beyond exponentiation the Greek names tetration (the 4th)
a↑↑b ,
pentation (the 5th)
a↑↑↑b = a↑3 b ,
hexation (the 6th)
a↑↑↑↑b = a↑4 b ,
etc., where each operation is defined by iterating the previous one.
All this is used for numbers with so-called very small and very large absolute values [Wikipedia Large_numbers].
But common approaches have many disadvantages:
1) investigating already available possibilities is much less efficient than concertedly creating new possibilities;
2) positive number bases only are usually considered;
3) bases between 0 and 1 are not efficiently used for representing numbers with so-called very small and very large absolute values;
4) a uniform number scale is not suitable for creating hyperoperation hierarchy;
5) known number scale transformations such as using logarithmic scales cannot provide suitably simultaneously representing numbers both with very small and very large absolute values of the both signs;
6) natural numbers (positive integers) of multiple (combined, composite) power exponents only are usually considered;
7) multilevel placing multiple power exponents brings many typesetting difficulties and misunderstanding, especially by text transformation via software including browsers;
8) already usual exponentiation ab is noncommutative and nonassociative, e.g.
23 = 8 ≠ 9 = 32,
2^3^4 = 2^(3^4) = 281 ≠ 212 = (2^3)^4,
because in
ab = eb ln(a)
the roles of a and b are very different;
9) a commutative form of exponentiation (III)
F3(a , b) = eln(a) ln(b) = aln(b)
by Albert Arnold Bennett [1915] provides noninteger values by natural a , b > 1 and growth much more slower than that of ab by great a , b , which is a very important disadvantage when applying this commutative form of exponentiation to representing great numbers;
10) individual quantities of operands and operation results are not considered at all.
Therefore, in classical mathematics, both power functions and exponential functions providing often useful high orders of growth especially by multiply (repeatedly) raising bases to powers have very bounded domains of definition and efficiency.
Hence classical mathematics cannot (and does not want to) regard (adequately solve and even consider) very many typical urgent 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 [Encyclopaedia of Physics 1973]. This is also very important for chaos theory (Ilya Prigogine [1993, 1997]) and fractal theory (Benoît Mandelbrot [1975, 1977, 1982]).
Mega-overmathematics by Lev Gelimson [1987-2012] 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 developing, extending, and applying mega-overmathematics fundamental sciences systems. Among them are, in particular, negative base power theory which defines raising a negative number to a power and present general power-exponential function hyperefficiency theory which creates principally new possibilities providing number scale transformation. It is suitable for developing hyperoperation hierarchy via simultaneously representing numbers both with very small and very large absolute values of the both signs due to efficiently using bases also between 0 and 1.
Principal Ideas
One possible idea is very natural: to explicitly or implicitly compress a uniform number scale between -1 and 1 and extend it by (-∞ , -1] and [1, +∞).
For a function y = f(x) which has an inverse function and the whole real-number axis (-∞ , +∞) both as a domain (of definition) and range, we want to have the following properties:
|y| < |x| by x ∈ (-1, 0) ∪ (0, 1),
|y| > |x| by x ∈ (-∞ , -1) ∪ (1, +∞),
limx→±∞ |y|/|x| = +∞ ,
limx→0 |y|/|x| = 0.
It is also natural and desirable that such a function y = f(x) is:
a) sign-conserving, i.e.
sign y = sign x , x ∈ (-∞ , +∞);
b) continuously differentiable if possible;
c) strictly increasing:
x1 < x2 implies f(x1) < f(x2);
d) strictly convex by [0, +∞) and strictly concave by (-∞ , 0].
There are well-known power functions with odd exponents greater than 1
y = x2n+1 , n ∈ N = {1, 2, 3, ...}
which have all the above properties. The same holds for sign-conserving power functions with positive even exponents
y = x"2n , n ∈ N = {1, 2, 3, ...}
where
a"b = |a|b sign a
due to negative base power theory by Lev Gelimson [1987-2012].
But no constant exponent by power functions
y = xn , n ∈ N = {1, 2, 3, ...}
can provide such growth by x→+∞ as by any exponential functions (which could be satisfactory by x > 1 only)
y = ax , a > 1:
limx→+∞ xn/ax = 0.
Therefore, these power functions are sufficient but not very efficient.
The well-known power-exponential function
y = f(x) = xx = 2x = ex ln x
is still more suitable by x > 1 only because the exponent x here grows together with the base x . On the contrary, by 0 < x < 1, we have
xx > x
and even
limx→0+ xx = 1
instead of the required relations
y < x ,
limx→0 |y|/|x| = 0.
Therefore, by 0 < x < 1, the equality of the base x and the exponent x plays a negative role, and any power function
y = xn , n ∈ N = {1, 2, 3, ...}
works here better even if
y = x
(by n = 1) does not provide
y < x ,
limx→0 |y|/|x| = 0
and is insufficient. Hence it seems to be natural to construct piecewise power-exponential functions differently defined on (-∞ , -1], (-1, 1), and [1, +∞), namely with distinct relations between the bases and exponents.
Tetration with Possibly Noninteger Multiplicity
To begin with, use power-exponential function definition
y = f(x) = ax = x^^a = x^2 a = expx[a]+1({a}) = x^x^...^x^{a}
with any positive a and x used [a] + 1 times on the right-hand side
where
a = [a] + {a}
is a positive (possibly noninteger) number,
[a] = floor(a) = entier(a) = max{z ∈ Z | z ≤ a} ≤ a
(Z is the set of all the integers)
is the integer part of a as the greatest integer not exceeding a , and
{a} = a - [a] ∈ [0, 1),
i.e.
0 ≤ {a} < 1,
is the fractional part of a as a sawtooth function.
In particular, by 0 ≤ a ≤ 1, we simply have
y = f(x) = ax = x^^a = expx[a]+1({a}) = x^a = xa ,
which behaves not better than y = x and is hence uninteresting.
To understand the naturalness of this sophisticated definition, consider the following example for
a = 1.5
with
[a] = 1,
{a} = 0.5:
ax = 1.5x = x^x^0.5.
This is natural because
1x = x1 = x = x^x^0,
2x = xx = x^x = x^x^1
and the power exponent
x^0.5 = x0.5
in
1.5x = x^x^0.5
is the geometric mean value of the power exponents x^0 = 1 in
1x = x1 = x = x^x^0
and x^1 = x in
2x = xx = x^x = x^x^1.
To further generalize this result, take a with the same [a] = 1 (which is here inessential because the only two last power exponents are relevant) and any {a} with
0 ≤ {a} < 1:
a = 1 + {a}
where
1 ≤ {a} < 2.
1+{a}x = x^x^{a}.
This is natural because
1x = x1 = x = x^x^0,
2x = xx = x^x = x^x^1
and the power exponent
x^{a}= x{a}
in
1+{a}x = x^x^{a}
is the weighted geometric mean value of the power exponents
x^0 = 1
with its natural weight 1 - {a} in
1x = x1 = x = x^x^0
and
x^1= x
with its natural weight {a} in
2x = xx = x^x = x^x^1.
In fact,
[(x^0)1-{a}(x^1){a}]1/[(1-{a})+{a}] = [11-{a}x{a}]1 = x{a} ,
quod erat demonstrandum.
Tetration Transformation Algorithm
To begin with, consider the well-known tetration notation
nx = x^^n = x^x^...^x
with x used n times (n ∈ N = {1, 2, 3, ...}) on the right-hand side, namely always one (the first) time as a base and (if n > 1) further n - 1 times as exponents.
Namely, use the following algorithm:
1) separate the sign of argument (variable) x from its modulus (absolute value) |x|;
2) directly and explicitly assign sign x to the function value itself;
3) replace the argument (variable) x with its modulus (absolute value) |x|;
4) replace each exponent |x| with the maximum max(|x|, 1/|x|) of |x| and its inverse 1/|x|;
5) consider the product whose first factor
sign x = 0 (x = 0)
to vanish independently of the second factor, or, alternatively, which is sufficient, take its (zero) limit as its value.
General Power-Exponential Function Notation and Sense
Let us introduce:
a notation for powers and exponentials with single-level placing multiple power exponents via separating them with the backslash sign \ , e.g.
a^b^c^d = ab\c\d ;
a space-saving notation for the sign function
a° = sign a ;
a space-saving notation for the function
a? = max(a , 1/a).
Also consider the well-known tetration notation
nx = x^^n = x^x^...^x
(with x used n times on the right-hand side) by n = 2:
2x = x^^2 = x^x = xx .
The above tetration transformation algorithm leads to the function
y = f(x) = x°|x||x|? = (sign x) |x|max(|x|, 1/|x|) = (sign x) |x|(|x|+1/|x|+||x|-1/|x||)/2
where by x° = sign x = 0 (x = 0), the second factor
|x||x|? = |x|max(|x|, 1/|x|) = |x|(|x|+1/|x|+||x|-1/|x||)/2
is not considered at all, or, alternatively, its (zero) limit is taken as its value.
We have piecewise y = f(x) =
xx by x ∈ [1, +∞),
x1/x by x ∈ (0, 1],
0 by x = 0,
-(-x)1/(-x) by x ∈ [-1, 0),
-(-x)(-x) by x ∈ (-∞, -1].
Now, beginning with the tetration notation nx and using the parenthesis " (as well as in negative base power theory by Lev Gelimson [1987-2012]) between the base x and the number n of the base and the exponents each of which equals the base x , naturally introduce the simple notation for this function
y = f(x) = 2"x = x"^^2
and, more generally, for a function
y = f(x) = n"x = x"^^n
with giving it sense further.
Notata bene:
1. In negative base power theory, e.g. in
x"x = x"^x = x°|x|x = (sign x) |x|x ,
the parenthesis " (placed to the right from the base x between the base x and the exponent which equals the base x in this case only) designates the following:
1.1) in the base x only, separate the sign of argument (variable) x from its modulus (absolute value) |x|;
1.2) directly and explicitly assign x° = sign x to the function value itself;
1.3) in the base x only, replace the argument (variable) x with its modulus (absolute value) |x|.
2. In power-exponential function hyperefficiency theory, e.g. in
n"x = x"^^n ,
the parenthesis " (placed to the left or to the right, respectively, from the base x between the base x and the optional operation signs with the number n of the base and the exponents each of which equals the base x) designates the following:
2.1) separate the sign of argument (variable) x from its modulus (absolute value) |x|;
2.2) directly and explicitly assign sign x to the function value itself;
2.3) replace the argument (variable) x with its modulus (absolute value) |x|;
2.4) replace each exponent |x| with the maximum
|x|? = max(|x|, 1/|x|)
of |x| and its inverse 1/|x|;
2.5) consider the product whose first factor
x° = sign x = 0 (x = 0)
to vanish independently of the second factor, or, alternatively, which is sufficient, take its (zero) limit as its value.
Power-Exponential Functions y = x"^^2 = x"^x = (sign x) |x|^max(|x|, 1/|x|), y = x^x , and y = x^(1/x)
Now systematically consider once more function
y = f(x) = 2"x = x"^^2.
To begin with, take the well-known power-exponential function
y = f(x) = xx = 2x = x^^2 = x^x = ex ln x .
Its first two derivatives are
y' = df(x)/dx = ex ln x (ln x + x/x) = xx (1 + ln x), f'(1) = 1;
y'' = d2f(x)/dx2 = (xx + xx ln x)' = xx (1 + ln x) + xx (1 + ln x) ln x + xx/x = xx [(1 + ln x)2 + 1/x], f''(1) = 2.
By x ∈ [1, +∞), this function behaves as desired (and much better than y = x2n-1 and y = ax) and has a suitable inverse.
By x ∈ [0, 1], the function y = f(x) = xx brings nothing:
takes value 1 at x = 1;
has limit
limx→0+ f(x) = 1;
has the minimum
f(1/e) = (1/e)(1/e).
To provide a suitable extension of this function to x ∈ [0, 1], it is logical to use
y = f(x) = x1/x = e(ln x)/x
with
y' = df(x)/dx = e(ln x)/x [1/x2 - (ln x)/x2] = x1/x (1 - ln x)/x2 , f'(1) = 1;
y'' = d2f(x)/dx2 = [e(ln x)/x (1 - ln x)/x2]' = e(ln x)/x (1 - ln x)/x2 (1 - ln x)/x2 + e(ln x)/x [(-2)/x3(1 - ln x) + 1/x2(-1/x)]
= x1/x [(1 - ln x)2 + 2x ln x - 3x]/x4 , f''(1) = -2.
Therefore, function y = f(x) = 2"x = x"^^2 = x"^x =
xx by x ∈ [1, +∞),
x1/x by x ∈ (0, 1],
0 by x = 0,
- (-x)1/(-x) by x ∈ [-1, 0),
- (-x)(-x) by x ∈ (-∞, -1]
compresses a uniform number scale between -1 and 1 and extends it by x ∈ (-∞ , -1] and by x ∈ [1, +∞), see Fig. 1:
Fig. 1
Using the sign function and the maximum function provides unifying the above piecewise representations as
y = f(x) = 2"x = x"^^2 = x"^x
= x°|x|^|x|? = (sign x) |x|^max(|x|, 1/|x|) =(sign x) |x|^[(|x|+1/|x|+||x|-1/|x||)/2]
= (sign x) |x|max(|x|, 1/|x|) = (sign x) |x|(|x|+1/|x|+||x|-1/|x||)/2
where by
x° = sign x = 0 (x = 0),
the second factor
|x|^|x|? = |x|^max(|x|, 1/|x|) =|x|^[(|x|+1/|x|+||x|-1/|x||)/2] = |x|max(|x|, 1/|x|) = |x|(|x|+1/|x|+||x|-1/|x||)/2
is not considered at all, or, alternatively, its (zero) limit is taken as its value.
Examples:
2"3 = 3"^^2 = 3"^3 = 23 = 3^^2 = 3^3 = 33 = 27,
2"(1/3) = (1/3)"^^2 = (1/3)"^(1/3) = (1/3)^[1/(1/3)] = (1/3)1/(1/3) = (1/3)3 = 1/33 = 1/27,
2"0 = 0,
2"(-1/3) = (-1/3)"^^2 = (-1/3)"^(-1/3) = - (1/3)^[1/(1/3)] = - (1/3)1/(1/3) = - (1/3)3 = - 1/33 = -1/27,
2"(-3) = (-3)"^^2 = (-3)"^3 = - 23 = - 3^^2 = - 3^3 = - 33 = -27.
Notata bene:
1. The exponent is
1/|x| = |x|-1 by 0 < |x| ≤ 1,
|x| = |x|1 by 1 ≤ |x| < +∞
with the clear mirror symmetry (-1 and 1) of the exponent in the first additional level about |x| = 1.
2. The whole exponent |x|? = max(|x|, 1/|x|) of the base |x| has its minimum 1 by |x| = 1 whereas
limx→0 |x|? = limx→0 max(|x|, 1/|x|) = limx→±∞ |x|? = limx→±∞ max(|x|, 1/|x|) = +∞ ,
which provides much more efficiency than it is possible due to using both power and exponential functions.
3. This function y = f(x) is continuous together with its first derivative whereas the second derivative has discontinuity jumps at x = -1 and x = 1.
4. This function y = f(x) has three inflection points (with changes from being convex to concave or vice versa) at 0 (which is natural) and about ±0.582 where
y'' = d2f(x)/dx2 = [e(ln x)/x (1 - ln x)/x2]' = x1/x [(1 - ln x)2 + 2x ln x - 3x]/x4 = 0.
In particular, at x = 1,
(1 - ln x)2 + 2x ln x - 3x = -2.
5. It is reasonable to try to avoid such two additional inflection points via increasing the complexity of power-exponential functions.
Power-Exponential Functions y = (sign x) |x|^[a max(|x|, 1/|x|)], y = x^(ax), and y = x^(a/x)
Let us investigate whether natural extensions
y = f(x) = xax ,
y = f(x) = xb/x
of the above functions
y = f(x) = xx = 2x ,
y = f(x) = x1/x ,
respectively, by any positive a and b can bring something useful.
First take the power-exponential function
y = f(x) = xax = eax ln x .
Its first two derivatives are
y' = df(x)/dx = eax ln x (a ln x + ax/x) = axax (1 + ln x), f'(1) = a ;
y'' = d2f(x)/dx2 = (axax + axax ln x)' = a2xax (1 + ln x) + a2xax (1 + ln x) ln x + axax/x = axax [a(1 + ln x)2 + 1/x], f''(1) = a(1 + a).
By x ∈ [1, +∞), this function behaves as desired (and much better than y = x2n-1 and y = ax).
By x ∈ [0, 1], y = f(x) = xax brings nothing:
takes value 1 at x = 1;
has limit
limx→0+ f(x) = 1;
has the minimum
f(1/e) = (1/e)a/e .
To provide a suitable extension of this function to x ∈ [0, 1], it is logical to use
y = f(x) = xb/x = eb(ln x)/x
with
y' = df(x)/dx = eb(ln x)/x [b/x2 - b(ln x)/x2] = bxb/x (1 - ln x)/x2 , f'(1) = b ;
y'' = d2f(x)/dx2 = [beb(ln x)/x (1 - ln x)/x2]' = b2eb(ln x)/x (1 - ln x)/x2 (1 - ln x)/x2 + beb(ln x)/x [(-2)/x3(1 - ln x) + 1/x2(-1/x)]
= bxb/x [b(1 - ln x)2 + 2x ln x - 3x]/x4 , f''(1) = b(b - 3).
To provide that unifying these two functions is continuous together with its first derivative whereas the second derivative has discontinuity jumps at x = -1 and x = 1, take b = a by any positive a .
Therefore, transformation (function) y = f(x) =
xax by x ∈ [1, +∞),
xa/x by x ∈ (0, 1],
0 by x = 0,
-(-x)a/(-x) by x ∈ [-1, 0),
-(-x)a(-x) by x ∈ (-∞, -1]
compresses a uniform number scale between -1 and 1 and extends it by x ∈ (-∞ , -1] and by x ∈ [1, +∞), see Fig. 2:
Fig. 2. a = 3
Using the sign and maximum functions provides unifying the above piecewise representations as
y = f(x) = x°|x|^(a|x|?) = (sign x) |x|^[a max(|x|, 1/|x|)] = (sign x) |x|^[a(|x|+1/|x|+||x|-1/|x||)/2]
= x°|x|a|x|? = (sign x) |x|a max(|x|, 1/|x|) = (sign x) |x|a(|x|+1/|x|+||x|-1/|x||)/2
where by sign x = 0 (x = 0), the second factor
|x|a|x|? = |x|a max(|x|, 1/|x|) = |x|a(|x|+1/|x|+||x|-1/|x||)/2
is not considered at all, or, alternatively, its (zero) limit is taken as its value.
Notata bene:
1. For analytic simplicity, calculation suitability, and qualitative analysis, it is reasonable to explicitly apply the parameter a to max(|x|, 1/|x|) in the exponent as above.
2. For essence understanding and qualitative analysis, it is reasonable to explicitly apply the parameter a to the initial base |x| with obtaining the new whole base |x|a :
y = f(x) = x°(|x|a)|x|? = (sign x)(|x|a)max(|x|, 1/|x|) = (sign x)(|x|a)(|x|+1/|x|+||x|-1/|x||)/2 .
3. The new exponent |x|? = max(|x|, 1/|x|) is
1/|x| = |x|-1 by 0 < |x| ≤ 1,
|x| = |x|1 by 1 ≤ |x| < +∞
with the clear mirror symmetry (-1 and 1) of the exponent in the first additional level about |x| = 1.
4. The new exponent |x|? = max(|x|, 1/|x|) of the base |x| has its minimum 1 by |x| = 1 whereas
limx→0 |x|? = limx→0 max(|x|, 1/|x|) = limx→±∞ |x|? = limx→±∞ max(|x|, 1/|x|) = +∞ ,
which provides much more efficiency than it is possible due to using both power and exponential functions.
5. This function y = f(x) is continuous together with its first derivative whereas the second derivative has discontinuity jumps at x = -1 and x = 1.
6. By 0 < a < 3, this function y = f(x) has three inflection points (with changes from being convex to concave or vice versa) at 0 (which is natural) and at two additional inflection points where
y'' = d2f(x)/dx2 = [aea(ln x)/x (1 - ln x)/x2]' = axa/x [a(1 - ln x)2 + 2x ln x - 3x]/x4 = 0.
7. By a ≥ 3, this function y = f(x) has the only inflection point at 0 (which is natural). The both additional inflection points drop namely by a = 3 because then, in particular, at x = 1,
a(1 - ln x)2 + 2x ln x - 3x = 0.
8. It is reasonable to also try other possibilities to avoid such two additional inflection points via increasing the complexity of power-exponential functions.
Power-Exponential Functions y = (sign x) (|x|^a)^max{|x|, 1/|x|}^b , y = (x^a)^x^b , and y = (x^a)^[1/(x^b)]
Let us investigate naturally generalizing the above functions
y = f(x) = xx = 2x ,
y = f(x) = x1/x ,
respectively, via replacing variable x both in the base and in the exponent with its powers xa and xb , respectively, by any positive a and b .
To begin with, take power-exponential function
y = f(x) = (xa)^xb = ex^b ln x^a = ex^b a ln x .
Its first two derivatives are
y' = df(x)/dx = ex^b a ln x a(xb/x + bxb-1ln x) = (xa)^xb axb-1(1 + b ln x),
f'(1) = a ;
y'' = d2f(x)/dx2 = [ex^b a ln x (axb-1 + abxb-1 ln x)]'
= ex^b a ln x [(axb-1 + abxb-1 ln x)2 + a(b - 1)xb-2 + ab(b - 1)xb-2 ln x + abxb-2],
f''(1) = a2 + 2ab - a .
By x ∈ [1, +∞), this function behaves as desired (and much better than y = x2n-1 and y = ax) and has a suitable inverse.
By x ∈ [0, 1], the function y = f(x) = (xa)^xb brings nothing:
takes value 1 at x = 1;
has limit
limx→0+ f(x) = 1;
has the minimum
f(1/e1/b) = (1/ea/b)^(1/e) = 1/ea/(be).
To provide a suitable extension of this function to x ∈ [0, 1], it is logical to use
y = f(x) = (x^a)^[1/(x^b)] = (x^a)^x^(-b) = ex^(-b) ln x^a = ex^(-b) a ln x .
with
y' = df(x)/dx = ex^(-b) a ln x a(x-b/x + bx-b-1ln x) = (xa)^x(-b) ax-b-1(1 - b ln x),
f'(1) = a ;
y'' = d2f(x)/dx2 = [ex^(-b) a ln x (ax-b-1 - abx-b-1 ln x)]'
= ex^(-b) a ln x [(ax-b-1 - abx-b-1 ln x)2 - a(b + 1)x-b-2 - ab(b + 1)x-b-2 ln x - abx-b-2],
f''(1) = a2 - 2ab - a .
Therefore, function y = f(x) =
(x^a)^(x^b) by x ∈ [1, +∞),
(x^a)^[1/(x^b)] by x ∈ (0, 1],
0 by x = 0,
-[(-x)^a]^[1/(-x)^b] by x ∈ [-1, 0),
-[(-x)^a]^[1/(-x)^b] by x ∈ (-∞, -1]
compresses a uniform number scale between -1 and 1 and extends it by x ∈ (-∞ , -1] and by x ∈ [1, +∞), see Fig. 3:
Fig. 3. a = 2, b = 3
Using the sign and maximum functions provides unifying the above piecewise representations as
y = f(x) = x°|x|a^|x|?b = (sign x)(|x|^a)^max(|x|, 1/|x|)^b = (sign x) (|x|^a)^[(|x|+1/|x|+||x|-1/|x||)/2]^b
where by sign x = 0 (x = 0), the second factor
|x|a^|x|?b = (|x|^a)^max(|x|, 1/|x|)^b = (|x|^a)^[(|x|+1/|x|+||x|-1/|x||)/2]^b
is not considered at all, or, alternatively, its (zero) limit is taken as its value.
Notata bene:
1. The new exponent |x|? = max(|x|, 1/|x|) is
1/|x| = |x|-1 by 0 < |x| ≤ 1,
|x| = |x|1 by 1 ≤ |x| < +∞
with the clear mirror symmetry (-1 and 1) of the exponent in the first additional level about |x| = 1.
2. The new exponent |x|? = max(|x|, 1/|x|) of the base |x| has its minimum 1 by |x| = 1 whereas
limx→0 |x|? = limx→0 max(|x|, 1/|x|) = limx→±∞ |x|? = limx→±∞ max(|x|, 1/|x|) = +∞ ,
which provides much more efficiency than it is possible due to using both power and exponential functions.
3. This function y = f(x) is continuous together with its first derivative.
4. The second derivative has discontinuity jumps at x = -1 and x = 1.
5. Introducing the second constant b cannot provide the continuity of this function y = f(x) because
a2 + 2ab - a = a2 - 2ab - a
requires b = 0.
6. It is reasonable to also try other possibilities via increasing the complexity of power-exponential functions.
Power-Exponential Functions y = x"^^a = (sign x) |x|^max(|x|, 1/|x|)^^(a-1), y = x^^a = x^[x^^(a - 1)], and y = x^[(1/x)^^(a - 1)]
Let us investigate naturally generalizing the above functions
y = f(x) = xx = 2x = x^x ,
y = f(x) = x1/x ,
respectively, via iterating the exponents x and 1/x , respectively, any fixed (possibly noninteger) number of times.
To begin with, take power-exponential function
y = f(x) = ax = x^^a = expx[a]+1({a}) = x^x^...^x^{a}
with x used [a] + 1 times on the right-hand side
where
a = [a] + {a}
is a positive (possibly noninteger) number,
[a] = floor(a) = entier(a) = max{z ∈ Z | z ≤ a} ≤ a
(Z is the set of all the integers)
is the integer part of a as the greatest integer not exceeding a , and
{a} = a - [a] ∈ [0, 1),
i.e.
0 ≤ {a} < 1,
is the fractional part of a as a sawtooth function.
In particular, by 0 ≤ a ≤ 1, we simply have
y = f(x) = ax = x^^a = expx[a]+1({a}) = x^a = xa ,
which behaves not better than y = x and is hence not interesting.
Further, e.g., by 1 < a ≤ 2, we simply have
y = f(x) = ax = x^^a = expx[a]+1({a}) = x^x^(a-1) = exp[e(a-1) ln x ln x].
Its first derivative is
y' = df(x)/dx = exp[e(a-1) ln x ln x] [e(a-1) ln x ln x]' = exp[e(a-1) ln x ln x] e(a-1) ln x /x [1 + (a - 1)ln x] = x^x^(a-1) xa-2 [1 + (a - 1)ln x],
f'(1) = 1.
By x ∈ [1, +∞), this function behaves as desired (and much better than y = x2n-1 and y = ax) and has a suitable inverse.
By x ∈ [0, 1], this function brings nothing:
takes value 1 at x = 1;
has limit
limx→0+ f(x) = 1.
To provide a suitable extension of this function to x ∈ [0, 1], it is logical to use
y = f(x) = x^(1/x)^(a-1) = exp[e(1-a) ln x ln x].
with
y' = df(x)/dx = exp[e(1-a) ln x ln x] [e(1-a) ln x ln x]' = exp[e(1-a) ln x ln x] e(1-a) ln x /x [1 - (a - 1)ln x] = x^(1/x)^(a-1) /xa [1 - (a - 1)ln x],
f'(1) = 1.
Therefore, function y = f(x) = a"x = x"^^a =
x^x^(a-1) by x ∈ [1, +∞),
x^(1/x)^(a-1) by x ∈ (0, 1],
0 by x = 0,
-(-x)^[1/(-x)]^(a-1) by x ∈ [-1, 0),
-(-x)^(-x)^(a-1) by x ∈ (-∞, -1]
compresses a uniform number scale between -1 and 1 and extends it by x ∈ (-∞ , -1] and by x ∈ [1, +∞), see Fig. 4:
Fig. 4. a = 2.9
Using the sign and maximum functions provides unifying the above piecewise representations as
y = f(x) = a"x = x"^^a = x°|x|^[|x|?^^(a-1)] = (sign x)|x|^max(|x|, 1/|x|)^(a-1) = (sign x)|x|^[(|x|+1/|x|+||x|-1/|x||)/2]^(a-1)
where by x° = sign x = 0 (x = 0), the second factor
|x|^[|x|?^^(a-1)] = |x|^max(|x|, 1/|x|)^(a-1) = |x|^[(|x|+1/|x|+||x|-1/|x||)/2]^(a-1)
is not considered at all, or, alternatively, its (zero) limit is taken as its value.
Note that the last formulae hold by 1 < a ≤ 2 only.
Generally, by a > 1,
y = f(x) = ax = x^^a = expx[a]+1({a}) = x^x^...^x^{a}
with x used [a] + 1 times on the right-hand side.
By x ∈ [1, +∞), this function behaves as desired (and much better than y = x2n-1 and y = ax).
By x ∈ [0, 1], this function brings nothing:
takes value 1 at x = 1;
has limit
limx→0+ f(x) = 1.
To provide a suitable extension of this function to x ∈ [0, 1], it is logical to use
y = f(x) = x^(a-1)(1/x) = x^exp1/x[a]({a}).
Therefore, function y = f(x) = a"x = x"^^a =
ax = expx[a]+1({a}) by x ∈ [1, +∞),
x^(a-1)(1/x) = x^exp1/x[a]({a}) by x ∈ (0, 1],
0 by x = 0,
- (-x)^(a-1)[1/(-x)] = - (-x)^exp1(-x)[a]({a}) by x ∈ [-1, 0),
- a(-x) = - exp-x[a]+1({a}) by x ∈ (-∞, -1]
compesses a uniform number scale between -1 and 1 and extends it by (-∞ , -1] and [1, +∞).
Using the sign and maximum functions provides unifying the above piecewise representations as
y = f(x) = a"x = x"^^a = x°|x|^[|x|?^^(a-1)] = (sign x) |x|^[max(|x|, 1/|x|)^^(a-1)] = (sign x) |x|^{[(|x|+1/|x|+||x|-1/|x||)/2]^^(a-1)}
= (sign x) |x|^(a-1)max(|x|, 1/|x|) = (sign x) |x|^(a-1)[(|x|+1/|x|+||x|-1/|x||)/2]
= (sign x) |x|^expmax(|x|, 1/|x|)[a]({a}) = (sign x) |x|^exp(|x|+1/|x|+||x|-1/|x||)/2[a]({a})
where by sign x = 0 (x = 0), the second factor
|x|^[|x|?^^(a-1)] = |x|^[max(|x|, 1/|x|)^^(a-1)] = |x|^{[(|x|+1/|x|+||x|-1/|x||)/2]^^(a-1)} =
|x|^(a-1)max(|x|, 1/|x|) = |x|^(a-1)[(|x|+1/|x|+||x|-1/|x||)/2]
= |x|^expmax(|x|, 1/|x|)[a]({a}) = |x|^exp(|x|+1/|x|+||x|-1/|x||)/2[a]({a})
is not considered at all, or, alternatively, its (zero) limit is taken as its value.
Examples for a = 4:
4"3 = 3"^^4 = 43 = 3^^4 = 3^3^3^3 = 43 = 3^327 ,
4"(1/3) = (1/3)"^^4 = (1/3)^(4-1)[1/(1/3)] = (1/3)^33 = (1/3)^3^3^3 = (1/3)^327 = 1/(3^327),
4"0 = 0,
4"(-1/3) = - 4"(1/3) = - (1/3)"^^4 = - (1/3)^(4-1)[1/(1/3)] = - (1/3)^33 = - (1/3)^3^3^3 = - (1/3)^327 = - 1/(3^327),
4"(-3) = - 3"^^4 = - 43 = - 3^^4 = - 3^3^3^3 = - 43 = - 3^327 .
Notata bene:
1. The new exponent |x|? = max(|x|, 1/|x|) is
1/|x| = |x|-1 by 0 < |x| ≤ 1,
|x| = |x|1 by 1 ≤ |x| < +∞
with the clear mirror symmetry (-1 and 1) of the exponent in the first additional level about |x| = 1.
2. The new exponent |x|? = max(|x|, 1/|x|) of the base |x| has its minimum 1 by |x| = 1 whereas
limx→0 |x|? = limx→0 max(|x|, 1/|x|) = limx→±∞ |x|? = limx→±∞ max(|x|, 1/|x|) = +∞ ,
which provides much more efficiency than it is possible due to using both power and exponential functions.
3. This function y = f(x) is continuous together with its first derivative.
4. It is reasonable to also try other possibilities via increasing the complexity of power-exponential functions.
Power-Exponential Functions y = x"^^x = (sign x) |x|^max(|x|, 1/|x|)^^(x-1), y = x^^x = x^[x^^(x - 1)], and y = x^[(1/x)^^(x - 1)]
Let us investigate naturally generalizing the above functions
y = f(x) = xx = 2x = x^x ,
y = f(x) = x1/x = x^(1/x),
respectively, via iterating the exponents x and 1/x namely x - 1 and 1/x - 1 times, respectively, and these numbers of times are possibly noninteger.
To begin with, take power-exponential function
y = f(x) = xx = x^^x = expx[x]+1({x}) = x^x^...^x^{x}
with x used [x] + 1 times on the right-hand side
where
x = [x] + {x}
is a positive (possibly noninteger) number,
[x] = floor(x) = entier(x) = max{z ∈ Z | z ≤ x} ≤ x
(Z is the set of all the integers)
is the integer part of x as the greatest integer not exceeding x , and
{x} = x - [x] ∈ [0, 1),
i.e.
0 ≤ {x} < 1,
is the fractional part of x as a sawtooth function.
By x ∈ [1, +∞), power-exponential function
y = f(x) = xx = x^^x = expx[x]+1({x}) = x^x^...^x^{x}
behaves as desired (and much better than y = x2n-1 and y = ax) and has a suitable inverse.
For example, by 1 ≤ x < 2, we obtain
y = f(x) = xx = x^^x = expx[x]+1({x}) = x^x^(x-1) = exp[e(x-1) ln x ln x].
Its first derivative is
y' = df(x)/dx = exp[e(x-1) ln x ln x] [e(x-1) ln x ln x]' = exp[e(x-1) ln x ln x] e(x-1) ln x /x [1 + (x - 1)ln x + x ln2 x]
= x^x^(x-1) xx-2 [1 + (x - 1)ln x + x ln2 x],
f'(1) = 1.
By 0 ≤ x ≤ 1, we simply have
y = f(x) = xx = x^^x = expx[x]+1({x}) = x^x = xx ,
which brings nothing:
takes value 1 at x = 1;
has limit
limx→0+ f(x) = 1;
has the minimum
f(1/e) = (1/e)(1/e).
To provide a suitable extension of this function to x ∈ [0, 1], it is logical to replace
y = f(x) = xx = x^^x = x^(x-1x)
with
y = f(x) = x^[(1/x-1)(1/x)] = x^[(1/x)^^(1/x-1)] = x^exp1/x[1/x]({1/x}).
For example, by 1/2 < x ≤ 1, we obtain
1 ≤ 1/x < 2,
[1/x] = 1,
{1/x} = 1/x - 1,
y = f(x) = x^[(1/x-1)(1/x)] = x^[(1/x)^^(1/x-1)] = x^exp1/x[1/x]({1/x}) = x^(1/x)1/x-1 = x^(1/x)^(1/x - 1) = exp[e(1-1/x) ln x ln x].
Its first derivative is
y' = df(x)/dx = exp[e(1-1/x) ln x ln x] [e(1-1/x) ln x ln x]' = exp[e(1-1/x) ln x ln x] e(1-1/x) ln x /x2 [x + (x - 1)ln x - ln2 x]
= x^x^(1-1/x) /x1+1/x [x + (x - 1)ln x - ln2 x],
f'(1) = 1.
Therefore, function y = f(x) = x"x = x"^^x =
xx = expx[x]+1({x}) by x ∈ [1, +∞),
x^[(1/x-1)(1/x)] = x^exp1/x[1/x]({1/x}) by x ∈ (0, 1],
0 by x = 0,
- (-x)^{[1/(-x)-1][1/(-x)]} = (-x)^exp1/(-x)[1/(-x)]({1/(-x)}) by x ∈ [-1, 0),
- (-x)(-x) = exp(-x)[(-x)]+1({-x}) by x ∈ (-∞, -1]
compresses a uniform number scale between -1 and 1 and extends it by x ∈ (-∞ , -1] and by x ∈ [1, +∞), see Fig. 5:
Fig. 5
Using the sign and maximum functions provides unifying the above piecewise representations as
y = f(x) = x"x = x"^^x = x°|x|^[|x|?^^(x-1)] = (sign x) |x|^[max(|x|, 1/|x|)^^(x-1)] = (sign x) |x|^{[(|x|+1/|x|+||x|-1/|x||)/2]^^(x-1)}
= (sign x) |x|^(x-1)max(|x|, 1/|x|) = (sign x) |x|^(x-1)[(|x|+1/|x|+||x|-1/|x||)/2]
= x°|x|^exp|x|?[x]({x}) = (sign x) |x|^expmax(|x|, 1/|x|)[x]({x}) = (sign x) |x|^exp(|x|+1/|x|+||x|-1/|x||)/2[x]({x})
where by sign x = 0 (x = 0), the second factor
|x|^[|x|?^^(x-1)] = |x|^[max(|x|, 1/|x|)^^(x-1)] = |x|^{[(|x|+1/|x|+||x|-1/|x||)/2]^^(x-1)}
= |x|^(x-1)max(|x|, 1/|x|) = |x|^(x-1)[(|x|+1/|x|+||x|-1/|x||)/2]
= x°|x|^exp|x|?[x]({x}) = |x|^expmax(|x|, 1/|x|)[x]({x}) = |x|^exp(|x|+1/|x|+||x|-1/|x||)/2[x]({x})
is not considered at all, or, alternatively, its (zero) limit is taken as its value.
Examples:
10"10 = 10"^^10 = 1010 = 10^^10 = 10^10^10^10^10^10^10^10^10^10,
1/10"(1/10) = (1/10)"^^(1/10) = (1/10)^(10-1)[1/(1/10)] = (1/10)^910 = (1/10)^10^10^10^10^10^10^10^10^10 = 1/1010,
0"0 = 0,
-1/10"(-1/10) = - 1/10"(1/10) = - (1/10)"^^(1/10) = - (1/10)^(10-1)[1/(1/10)] = - (1/10)^910 = - (1/10)^10^10^10^10^10^10^10^10^10 = - 1/1010,
(-10)"(-10) = - 10"10 = - 10"^^10 = - 1010 = - 10^^10 = - 10^10^10^10^10^10^10^10^10^10.
And also generally, it is natural to define
-nx = - nx (n > 0).
Notata bene:
1. The new exponent |x|? = max(|x|, 1/|x|) is
1/|x| = |x|-1 by 0 < |x| ≤ 1,
|x| = |x|1 by 1 ≤ |x| < +∞
with the clear mirror symmetry (-1 and 1) of the exponent in the first additional level about |x| = 1.
2. The new exponent |x|? = max(|x|, 1/|x|) of the base |x| has its minimum 1 by |x| = 1 whereas
limx→0 |x|? = limx→0 max(|x|, 1/|x|) = limx→±∞ |x|? = limx→±∞ max(|x|, 1/|x|) = +∞ ,
which provides much more efficiency than it is possible due to using both power and exponential functions.
3. This function y = f(x) is continuous together with its first derivative.
4. It is reasonable to also try other possibilities via increasing the complexity of power-exponential functions.
Basic Results and Conclusions
1. General quanti-exponential theory is advanced on the base of the proposed ideas.
2. Well-known power, root, logarithmic, exponential, and power-exponential functions cannot provide suitably simultaneously representing numbers both with very small and with very large absolute values of the both signs and have very bounded domains of definition and efficiency.
3. General power-exponential functions algorithmically transformate tetrations, compress a uniform number scale between -1 and 1, extend it by x ∈ (-∞ , -1] and by x ∈ [1, +∞), and are sign-conserving, continuously differentiable, and strictly increasing.
4. General power-exponential functions provide often useful high orders of growth especially by multiply (repeatedly) raising bases to powers with suitably simultaneously representing numbers both with very small and with very large absolute values of the both signs.
5. General power-exponential functions are suitable for creating hyperoperation hierarchy.
6. General power-exponential theory in mega-overmathematics by Lev Gelimson [1987-2012] is universal and very efficient.
0.6.4.4. Quanti-Hyper-Root-Logarithm Function Theory
Introduction
Numbers with very small and very large absolute values [Wikipedia Large_numbers] are extremely important for real world modeling. Moreover, their role exponentially increases because of computer science evolution which requires the so-called scientific number representation, as well as the storage and handling of such numbers to avoid the permanent danger of "computing overflow".
Classical mathematics (Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], G. A. Korn, T. M. Korn [1968], I. N. Bronstein, K. A. Semendjajew [1989], [Encyclopaedia of Mathematics 1988]) widely uses
exponentiation as raising numbers to powers by Michael Stifel [1544],
power functions y = xn with constant exponents n and
exponential functions y = ax with constant bases a ,
as well as their inverse functions, namely
root functions y = x1/n (the nth root of a number x is a number y which, when raised to the power of n , equals x , i.e. yn = x) and
logarithmic functions y = loga x (the logarithm of a number x to a base a is a number y so that raising a base a to the power of y gives x , i.e. ay = x) introduced by John Napier [1614, 1619] and as a notion and notation by Leonhard Euler.
[Wikipedia Tetration, Super-logarithm, Iterated_logarithm] also represents
super-root functions which can be denoted as y = srtn(x) = srtn(x) (the nth super-root of a number x is a number y so that tetration ny equals x , i.e. ny = x), e.g. the 2nd-order super-root, square super-root, or super square root ssrt(x) which has no real values by 0 < x < e-1/e , two positive real values by e-1/e < x < 1, and one positive real value by x ≥ 1,
super-logarithm functions y = sloga x (the super-logarithm of a number x to the base a is a number y so that tetration ya equals x , i.e. ya = x), and
iterated logarithm functions y = log*a x (the iterated logarithm of a number x to the base a is the number of times the logarithm function must be iteratively applied before the result is less than or equal to 1) so that by positive x ,
log*a x = [sloga x].
Nota bene: Unlike the possibility to represent the usual nth root of a number x as x1/n , it is generally inadmissible to represent the nth super-root of a number x as 1/nx even if
srtn(x) = srtn(x) = 1/nx
holds for n = 2 and x = 4 as an exception.
Examples:
srt2(4) = srt2(4) = 2 because 22 = 22 = 4,
1/24 = 41/2 = 2 = srt2(4) = srt2(4)
whereas
srt2(27) = srt2(27) = 3 because 23 = 33 = 27,
1/227 = 271/2 ≠ 91/2 = 3 = srt2(27) = srt2(27),
and
srt3 4256 = srt3 4256= 4 because 34 = 4^4^4= 4^256= 4256 ,
1/3(4256) = (4256)1/3 = 4256/3 ≠ 4 = srt3 4256 = srt3 4256.
Mega-overmathematics by Lev Gelimson [1987-2012] 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 developing, extending, and applying mega-overmathematics fundamental sciences systems. Among them are, in particular,
negative base power theory which defines raising a negative number to a power,
general power-exponential function hyperefficiency theory which creates principally new possibilities providing number scale transformation,
and present quanti-hyper-root-logarithm function theory which creates sign-conserving quanti-hyper-root-logarithm functions unifying the advantages of super-root, super-logarithm, and iterated logarithm functions and providing number scale transformation with suitably simultaneously representing numbers both with very small and very large absolute values of the both signs due to efficiently using bases also between 0 and 1.
Principal Ideas
A quanti-hyper-root-logarithm function y = lh(x) has to explicitly or implicitly extend a uniform number scale between -1 and 1 and compress it by (-∞ , -1] and [1, +∞).
For a function y = lh(x) which has the whole real-number axis (-∞ , +∞) both as a domain (of definition) and range, we want to have the following properties:
|y| > |x| by x ∈ (-1, 0) ∪ (0, 1),
|y| < |x| by x ∈ (-∞ , -1) ∪ (1, +∞),
limx→±∞ |y|/|x| = 0,
limx→0 |y|/|x| = +∞ .
It is also natural and desirable that such a function y = f(x) is:
a) sign-conserving, i.e.
sign y = sign x , x ∈ (-∞ , +∞);
b) continuously differentiable if possible;
c) strictly increasing:
x1 < x2 implies f(x1) < f(x2);
d) strictly convex by [0, +∞) and strictly concave by (-∞ , 0].
One possible idea is very natural: to search for such a function y = lh(x) to be an inverse function to a function y = f(x) (in general power-exponential function hyperefficiency theory) which explicitly or implicitly compresses a uniform number scale between -1 and 1 and extends it by (-∞ , -1] and [1, +∞).
For a function y = f(x) which has an inverse function and the whole real-number axis (-∞ , +∞) both as a domain (of definition) and range, we want to have the following properties:
|y| < |x| by x ∈ (-1, 0) ∪ (0, 1),
|y| > |x| by x ∈ (-∞ , -1) ∪ (1, +∞),
limx→±∞ |y|/|x| = +∞ ,
limx→0 |y|/|x| = 0.
It is also natural and desirable that such a function y = f(x) is:
a) sign-conserving, i.e.
sign y = sign x , x ∈ (-∞ , +∞);
b) continuously differentiable if possible;
c) strictly increasing:
x1 < x2 implies f(x1) < f(x2);
d) strictly convex by [0, +∞) and strictly concave by (-∞ , 0].
There are well-known power functions with odd exponents greater than 1
y = x2n+1 , n ∈ N = {1, 2, 3, ...}
which have all the above properties. The same holds for sign-conserving power functions with positive even exponents
y = x"2n , n ∈ N = {1, 2, 3, ...}
where
a"b = |a|b sign a
due to negative base power theory by Lev Gelimson [1987-2012].
But no constant exponent by power functions
y = xn , n ∈ N = {1, 2, 3, ...}
can provide such growth by x→+∞ as by any exponential functions (which could be satisfactory by x > 1 only)
y = ax , a > 1:
limx→+∞ xn/ax = 0.
Therefore, these power functions are sufficient but not very efficient.
The well-known power-exponential function
y = f(x) = xx = 2x = ex ln x
is still more suitable by x > 1 only because the exponent x here grows together with the base x . On the contrary, by 0 < x < 1, we have
xx > x
and even
limx→0+ xx = 1
instead of the required relations
y < x ,
limx→0 |y|/|x| = 0.
Therefore, by 0 < x < 1, the equality of the base x and the exponent x plays a negative role, and any power function
y = xn , n ∈ N = {1, 2, 3, ...}
works here better even if
y = x
(by n = 1) does not provide
y < x ,
limx→0 |y|/|x| = 0
and is insufficient. Hence it seems to be natural to construct piecewise power-exponential functions differently defined on (-∞ , -1], (-1, 1), and [1, +∞), namely with distinct relations between the bases and exponents.
Quanti-Hyper-Root-Logarithm Function y = lh2\x Inverse to Power-Exponential Function y = x"^^2 = x"^x = (sign x) |x|^max(|x|, 1/|x|)
Now systematically consider quanti-hyper-root-logarithm function
y = lh2\x = lh2(x)
to be inverse to power-exponential function
y = f(x) = x"^^2 = x"^x = (sign x) |x|^max(|x|, 1/|x|).
Therefore, quanti-hyper-root-logarithm function
y = lh2\x = lh2(x)
extends a uniform number scale between -1 and 1 and compresses it by x ∈ (-∞ , -1] and by x ∈ [1, +∞), see Fig. 6:
Fig. 6
Examples:
lh2\27 = lh227 = 3 because 2"3 = 3"^^2 = 3"^3 = 23 = 3^^2 = 3^3 = 33 = 27,
lh2\(1/27) = lh2(1/27) = 1/3 because 2"(1/3) = (1/3)"^^2 = (1/3)"^(1/3) = (1/3)^[1/(1/3)] = (1/3)1/(1/3) = (1/3)3 = 1/33 = 1/27,
lh2\0 = lh2(0) = 0 because 2"0 = 0,
lh2\(-1/27) = lh2(-1/27) = -1/3 because 2"(-1/3) = (-1/3)"^^2 = (-1/3)"^(-1/3) = - (1/3)^[1/(1/3)] = - (1/3)1/(1/3) = - (1/3)3 = - 1/33 = -1/27,
lh2\(-1/27) = lh2(-1/27) = -3 because 2"(-3) = (-3)"^^2 = (-3)"^3 = - 23 = - 3^^2 = - 3^3 = - 33 = -27.
Notata bene:
1. This function y = lh2\x is continuous together with its first derivative whereas the second derivative has discontinuity jumps at x = -1 and x = 1.
2. This function y = lh2\x has three inflection points (with changes from being convex to concave or vice versa) at 0 (which is natural) and about ±0.395.
Quanti-Hyper-Root-Logarithm Function y = lha\x Inverse to Power-Exponential Function y = x"^^a = (sign x) |x|^max(|x|, 1/|x|)^^(a-1)
Now systematically consider quanti-hyper-root-logarithm function
y = lha\x = lha(x)
to be inverse to power-exponential function
y = x"^^a = (sign x) |x|^max(|x|, 1/|x|)^^(a-1).
Therefore, quanti-hyper-root-logarithm function
y = lha\x = lha(x)
extends a uniform number scale between -1 and 1 and compresses it by x ∈ (-∞ , -1] and by x ∈ [1, +∞), see Fig. 7:
Fig. 7
Examples:
lh4\(3^327) = lh4(3^327) = 3 because 4"3 = 3"^^4 = 43 = 3^^4 = 3^3^3^3 = 43 = 3^327 ,
lh4\[1/(3^327)] = lh4[1/(3^327)] = 1/3 because 4"(1/3) = (1/3)"^^4 = (1/3)^(4-1)[1/(1/3)] = (1/3)^33 = (1/3)^3^3^3 = (1/3)^327 = 1/(3^327),
lh4\0 = lh40 = 0 because 4"0 = 0,
lh4\[-1/(3^327)] = lh4[-1/(3^327)] = -1/3 because 4"(-1/3) = - 4"(1/3) = - (1/3)"^^4 = - (1/3)^(4-1)[1/(1/3)] = - (1/3)^33 = - (1/3)^3^3^3 = - (1/3)^327 = - 1/(3^327),
lh4\(- 3^327) = lh4(- 3^327) = -3 because 4"(-3) = - 3"^^4 = - 43 = - 3^^4 = - 3^3^3^3 = - 43 = - 3^327 .
Basic Results and Conclusions
1. Quanti-hyper-root-logarithm function theory is advanced on the base of the proposed ideas.
2. Well-known super-root, super-logarithm, and iterated logarithm functions cannot provide suitably simultaneously representing numbers both with very small and with very large absolute values of the both signs and have very bounded domains of definition and efficiency.
3. Quanti-hyper-root-logarithm functions (inverse to general power-exponential functions which algorithmically transformate tetrations) extend a uniform number scale between -1 and 1, compress it by x ∈ (-∞ , -1] and by x ∈ [1, +∞), and are sign-conserving, continuously differentiable, and strictly increasing.
4. Quanti-hyper-root-logarithm functions provide suitably simultaneously representing numbers both with very small and with very large absolute values of the both signs.
5. Quanti-hyper-root-logarithm function theory in mega-overmathematics by Lev Gelimson [1987-2012] is universal and very efficient.
0.6.4.5. Self-Hyper-Root-Logarithm Function Theory
Mega-overmathematics by Lev Gelimson [1987-2012] 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 developing, extending, and applying mega-overmathematics fundamental sciences systems. Among them are, in particular,
negative base power theory which defines raising a negative number to a power,
general power-exponential function hyperefficiency theory which creates principally new possibilities providing number scale transformation,
and present self-hyper-root-logarithm function theory which creates sign-conserving self-hyper-root-logarithm functions unifying the advantages of super-root, super-logarithm, and iterated logarithm functions and providing number scale transformation with suitably simultaneously representing numbers both with very small and very large absolute values of the both signs due to efficiently using bases also between 0 and 1.
Self-Hyper-Root-Logarithm Function y = lh x Inverse to Power-Exponential Function y = x"^^x = (sign x) |x|^max(|x|, 1/|x|)^^(x-1)
Now systematically consider self-hyper-root-logarithm function
y = lh x
to be inverse to power-exponential function
y = f(x) = x"x = x"^^x = (sign x) |x|^max(|x|, 1/|x|)^^(x-1).
Therefore, quanti-hyper-root-logarithm function
y = lh x
extends a uniform number scale between -1 and 1 and compresses it by x ∈ (-∞ , -1] and by x ∈ [1, +∞), see Fig. 8:
Fig. 8
Examples:
lh 1010 = 10 because 10"10 = 10"^^10 = 1010 = 10^^10 = 10^10^10^10^10^10^10^10^10^10,
lh 1/1010 = 1/10 because 1/10"(1/10) = (1/10)"^^(1/10) = (1/10)^(10-1)[1/(1/10)] = (1/10)^910 = (1/10)^10^10^10^10^10^10^10^10^10 = 1/1010,
lh 0 = 0 because 0"0 = 0,
lh(-1/1010) = -1/10 because -1/10"(-1/10) = - 1/10"(1/10) = - (1/10)"^^(1/10) = - (1/10)^(10-1)[1/(1/10)] = - (1/10)^910 = - (1/10)^10^10^10^10^10^10^10^10^10 = - 1/1010,
lh(-1010) = -10 because (-10)"(-10) = - 10"10 = - 10"^^10 = - 1010 = - 10^^10 = - 10^10^10^10^10^10^10^10^10^10.
Basic Results and Conclusions
1. Self-hyper-root-logarithm function theory is advanced on the base of the proposed ideas.
2. Well-known super-root, super-logarithm, and iterated logarithm functions cannot provide suitably simultaneously representing numbers both with very small and with very large absolute values of the both signs and have very bounded domains of definition and efficiency.
3. Self-hyper-root-logarithm functions (inverse to general power-exponential functions which algorithmically transformate tetrations) extend a uniform number scale between -1 and 1, compress it by x ∈ (-∞ , -1] and by x ∈ [1, +∞), and are sign-conserving, continuously differentiable, and strictly increasing.
4. Self-hyper-root-logarithm functions provide suitably simultaneously representing numbers both with very small and with very large absolute values of the both signs.
5. Self-hyper-root-logarithm function theory in mega-overmathematics by Lev Gelimson [1987-2012] is universal and very efficient.
Overmathematics: Fundamental Principles, Theories, Methods, and Laws of Science
(former Elastic Mathematics: Theoretical Fundamentals)
Monograph
Parts 1, 2, and 3
Abstract
It is discovered that classical fundamental mathematical theories, methods, and even concepts are not sufficient for solving many scientific and life problems. Overmathematics advanced by the author is based on the principles of constructive philosophy and includes dozens of new general theories and methods. Their substantive conceptual fundamentals illustrated mainly in mathematics and physics allow discovering new phenomena and laws of nature, society, and consciousness.
This monograph is the quintessence of the treatise “Overmathematics” and is intended for scientists and engineers, lecturers and students, teachers and pupils. It does not claim any exhaustive account and is readily available due to its quasielementary presentation.
Preface
There are many urgent scientific and life problems that permit no rigorous approach and call for creating constructive philosophy and overmathematics as its universal language. Their fundamental principles and presentation must be fuzzy unlike any axioms’ system. Such a fuzziness and considering no optional relation in any problem eliminate any destruction in overmathematics that seems to be not only science but also art and even life and can also be named, e.g. as follows:
1) natural, physical, substantial, essential,
2) object-oriented, task-oriented, problem-oriented, purpose-oriented,
3) life-oriented, practical, adequate, sensitive,
4) alternative, intuitive, nonrigorous, tolerable,
5) fuzzy, free, flexible, creative,
6) nondestructive, constructive, useful, effective,
7) reasonable, rational, hyper-, quanti-,
8) all-embracing, unified, general, and universal.
It is not indisputable and can be completely estimated, corrected, and developed only in future. Now it seems to be a certain alternative, a useful addition, and possibly a new stage in mathematics development because it promises many essential perspectives.
Unlike many well-known scientific monographs, textbooks, and manuals, this book mainly contains theories and methods advanced by the author only. Many examples can help the reader in his initial mastering overmathematics, possibly by the method of successive approximations very effective both in solving and in learning.
Of course, some of the proposed concepts can seem to be partially known, and there is no possibility for references to all the thousands of scientific monographs and articles used by the author. He presents his sincere apologies and is very much obliged to all those scientists, as well as to all the readers of this book for their attention and desire to understand and to use overmathematics. The author believes that it will help the readers in solving their urgent problems and possibly will give them the joy of touching some new knowledge.
Remark Parentheses can include optional words.
Remark The words “a basic theory” reduce the words “a conceptual basis of a general theory”.
Remark Any general concept in this book can be found through its contents.
Notation
N , Z+, Z- , Z , Q , R , C
are the ordinary sets of all the usual (or common, i.e. noninfinite and noninfinitesimal) natural numbers, positive integers, negative integers, integers, rational, real, and complex numbers, respectively. In particular,
N = Z+ = {1, 2, 3, 4, 5, ...},
Z- = {-1, -2, -3, -4, -5, ...}.
Notation A´ and A` denote a successible set A in ascending and descending order, respectively.
Part 1. Introduction. Fundamental Principles
1.1. Introduction
The purpose of this monograph is to present some fundamentals and especially applications of the author’s overmathematics and other fundamental mathematical and strength sciences giving the reader many new very effective ideas, principles, concepts, methods, and other tools in order to help her or him solve her or his urgent problems. Details, grounds, and developments are available in the other monographs and articles by the author, in particular, on his website.
Classical mathematics with hardened systems of axioms, intended search for contradictions and even their creation suggests no suitable basic concepts and solving methods for many typical 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. There are gaps between the real numbers, and the probabilities of many reasonable possible events vanish or do not exist at all. In each concrete (mixed) physical magnitude, there is no known operation. Sets are no models for many collections without structure. Each measure has a very restricted domain of sensitivity. The absolute error alone is not sufficient for quality estimation. The relative error is uncertain in principle and has a very restricted domain of applicability. The unique known method applicable to overdetermined problems typical in any data processing is the least-square method that has narrow applicability and adequacy domains as well as many fundamental defects. For estimating the quality of distribution approximations, there is no applicable proposition. This results in the unreliability of results obtained, loss of time and costs, dangers, accidents, and catastrophes.
Overmathematics and other fundamental mathematical and strength sciences by the author’s principles of constructive philosophy bring many natural, universal, and effective basic concepts and solving methods for many typical problems. The uninumbers fill the gaps of the real numbers, perfectly precisely discriminate noncoinciding infinitely great or small objects, and each possible event has a positive probability. An introduced quantifying operation holds in every concrete (mixed) physical magnitude. The quantisets with any individual quantity of each element perfectly express arbitrary collections without structure. The uniquantities build a universal quantimeasure independent of dimensions. Unierrors and reserves bring reliable estimations of approximation quality and of the confidence in the exactness of any precise possibly distributed object. An iteration method of the least normed powers, unierror and reserve equalizing iteration methods, and a direct-solution method give both quasisolutions to contradictory problems and for the first time their contradictoriness measures. Demodulation methods provide recovering true distributions (in space and time) of very inhomogeneous objects and rapidly changeable processes by using discrete inexact measurement data.
The proposed basic concepts and solving methods are universal and very effective perhaps by solving any problem in science, engineering, and life and especially urgent in the case of responsible objects under dynamically changeable extreme conditions. For example, in elasticity theory, it becomes possible to estimate experimental stress and strain distributions in many spatial solids possibly with stress concentration. Mechanical and optical properties of many objects in high-pressure engineering were complexly optimized. Proposed fundamental mechanical and strength sciences and general implantation theory in solid physics allowed to discover many new phenomena and laws of nature in these areas as well as to create a lot of inventions.
1.2. Creative Philosophy Principles
Peaceful scientific and life pluralism.
Unrestrictedly free exclusively constructive and useful self-determination and activity (in particular, in knowledge and information research and development).
Exclusively practically purposeful creativity, intuition and phantasy flight.
Methodology priority based on the unity of matter, information, and measure.
Structurization, systematization, and generalization of objects and models.
Creating and considering exclusively necessary and useful objects and models (with completely ignoring any hindering ones such as counterexamples leading to artificial contradictions typical in classical mathematics).
Creating, considering, and efficiently utilizing useful also worst-case and contradictory objects and models, as well as difficulties, problems, and other complications.
Symbolic existence (of all the necessary and useful even contradictory objects and models).
Efficient quality (adequacy, deepness, exactness, structurality, systematization, etc.) and quantity (volume, completeness, etc.) of knowledge, information, data, and intelligent activity management.
Adequate and efficient analysis, synthesis, testing, verification, modeling, generalization, hierarchization, systematization, universalization, measurement, estimation, utilization, producing, improvement, and development of knowledge, information, and data.
Knowledge fundamentals priority.
Inheritance (efficiently using already available knowledge and information).
Perfect sensitivity = universal conservation law (no change of a general object conserves it).
Exact discrimination (by holding universal conservation laws) of noncoinciding possibly infinitely great or small objects and models.
Tolerable simplicity (choosing the best in the not evidently unacceptable simplest).
Unrestricted flexibility (even creating some appropriate individual science to set, consider, and solve an urgent problem).
Reasonable fuzziness (intuitive ideas without axiomatic strictness if necessary and useful).
Bounded consecutive generalizations (of concepts in definitions).
Unity and relativity of opposites (real/ideal, specific/abstract, exact/inexact, definitively/possibly, pure/applied, theory/experiment/practice, nature/life/science).
Sufficient partial laws (if there are no known general laws).
Scientific optimism (each urgent problem can be solved adequately and efficiently enough).
1.3. Mega-Overmathematics Principles
Peaceful pluralism in mathematics.
Methodology priority based on the unity of matter, information, and measure.
Structurization, systematization, and generalization of objects, models, and problems.
Urgent problems exclusiveness (setting and solving urgent problems only with avoiding unnecessary considerations), priority, and efficiently using.
Adequacy, universality, invariance, and flexibility of concepts, methods, theories, and sciences.
Creating and efficiently using unified knowledge foundation via fundamental general systems including objects, models, and intuitive fuzzy principles.
Exclusively practically useful axiomatization, rigor, and proving.
Creating and considering exclusively necessary and useful objects and models (with completely ignoring any hindering ones such as counterexamples leading to artificial contradictions typical in classical mathematics).
Efficient quality (adequacy, deepness, exactness, structurality, systematization, etc.) and quantity (volume, completeness, etc.) of objects, knowledge, information, data, and their consideration.
Creating, considering, and efficiently utilizing exclusively necessary and useful also worst-case and contradictory objects and models, as well as difficulties, problems, and other complications.
Symbolic existence (of all the necessary and useful even contradictory objects and models).
Decision delay (by estimating sense and existence).
Adequate and efficient analysis, synthesis, testing, verification, modeling, generalization, hierarchization, systematization, universalization, measurement, estimation, utilization, producing, improvement, and development of knowledge, information, and data.
Inheritance (efficiently using already available knowledge and information).
Counting measurement.
Perfectly sensitive, invariant, and universal modeling, evaluation, measurement, and estimation of each urgent object.
Universal conservation law.
Exact discrimination (by holding universal conservation laws) of noncoinciding possibly infinitely great or small objects or models.
Tolerable simplicity (choosing the best in the not evidently unacceptable simplest).
Unrestricted flexibility (even creating some appropriate individual science to set, consider, and solve an urgent problem).
Reasonable fuzziness (intuitive ideas without axiomatic strictness if necessary and useful).
Unrestricted general operability (each operation on an arbitrary set of operands).
Perfectly sensitive, invariant, and universal infinite and infinitesimal generalization of numbers and counting.
General (nonlogical) quantification and quantity determination with creating quantielements and quantisets.
Identifying membership, inclusion, and part-whole relations.
Essential and useful universal generalization of relations, structures, systems, and their contents extending sets and quantisets.
General problem systematization.
Quasisolving, solving, and supersolving a general problem by determining its quasisolution as the best possibly inexact pseudosolution, its exact solution, and its supersolution as the best exact solution, as well as its contradictoriness measure.
Existence and expressibility of the best quasisolution, solution, and supersolution to any urgent problem.
Corrected error universalization.
Exactness confidence measurement and estimation.
Avoiding artificial randomization.
Precise measurement and infinitely subtle estimation and management of reserve, reliability, and risk.
Free intuitive intelligent iterativity.
Quasiequivalent reduction (by an equivalent parameter).
Structural considering (systems as objects with structures).
Quasicritical phenomena (in the structures of systems).
Practical tendencies (of infinite processes).
Reasonable control (by possibly inexact knowledge).
Convenient convention (in denoting general images).
Unified presentation (of bounded concepts).
Part 2. General Objects, Quantioperations, Quantisets, and Uninumbers
Chapter 2.1. General Object Theories
2.1.1. Basic Theories of General Objects and Quasielements
Classical mathematics (Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], G. A. Korn, T. M. Korn [1968], I. N. Bronstein, K. A. Semendjajew [1989], [Encyclopaedia of Mathematics 1988]) usually considers that only noncontradictory (in some inflexible axioms’ system and even intentionally introduced relations) usual objects exist. But in reality there are many contradictory objects that also need mathematical modeling.
Usual objects, e.g. in set theory by Georg Cantor [1932], often break conservation laws.
Definition A general object is anything (natural or artificial, possibly many-valued, contradictory, fuzzy (Lotfi Zadeh [1965], Dieter Klaua [1965, 1966a, 1966b, 1967]), inexact, etc.) under consideration.
Definition A general value means a possible realization of a general object.
Definition A general process means a possibly many-valued general object generally changing in time.
Definition A quasielement means a general object conventionally indivisible in a given consideration.
Definition The general position of a general object means the general image of the general distribution (configuration, location, time-dependence, etc.) of the general object.
Definition The general structurality of a general object means its general position together with all the general relations between that object and other general objects in a given consideration.
Definition The quantirange of a general object means the quantiset of all its general values in a present consideration.
Definition The quantirange system of a general object means the quantisystem of all its general values in a present consideration.
Example The quantirange and the quantirange system of all
x ∈ [0, 1]
in ascending order both simply mean [0, 1] that might be presented in other forms, e.g.
[1/2, 1] ∪ [0, 1/2[.
Definition General objects are said to be equal (or unequal) if they can (or cannot, respectively) be presently considered indistinguishable.
Definition General objects are said to be equalizable if for each pair of them, there is their common value. Otherwise, they are said to be unequalizable.
Definition General objects are said to be equivalent if they can be transposed without changing a general result in a given general consideration. Otherwise, they are said to be nonequivalent.
Remark These relations are fuzzy in discrete finite-operation computing.
Remark These relations can be:
irreflexive (e.g. in physics, cybernetics, biology, psychology, and philosophy),
nonsymmetric (e.g. if they are latent relations of membership, assignment, or simplification), and
intransitive (e.g. if some bounded error is tolerable).
General Conservation Law
A general object changed by any not self-compensating means is not invariant.
Remark This is also a cardinal principle of creating new concepts, methods, theories, and sciences.
Remark The general conservation law for a general object yields the corresponding general laws for a quasielement, quantioperation, quantiset, quantimapping, quantirelation, quantisystem, and superuniverse. Those are explicitly formulated here only if they are nontrivial.
2.1.2. Basic Theory of Quantioperations
Usual operations in classical mathematics (Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], G. A. Korn, T. M. Korn [1968], I. N. Bronstein, K. A. Semendjajew [1989], [Encyclopaedia of Mathematics 1988]) have very restricted domains of definition [see, e.g., 1:0,
(1 + 10-16 cm2)
raised to the power with exponent
1016 cm-2,
etc. usually considered to make no sense] and not always provide inverting. Even the simplest equations
a + b = c
and
ab = c
with any (one) unknown are considered and, moreover, solvable in some very scarce special cases only (when
a , b , c
are complex numbers or their systems, etc.).
Besides that, for some of those operations, no general conservation law holds. By Cantor’s concepts of sets and set operations such as unification, subtraction, and intersection, as well as equality relation (Georg Cantor [1932], Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], G. A. Korn, T. M. Korn [1968], I. N. Bronstein, K. A. Semendjajew [1989], [Encyclopaedia of Mathematics 1988]), the simplest equations
X ∪ A = B
and
X ∩ A = B
in X are solvable only by
A ⊆ B
and
A ⊇ B ,
respectively (uniquely by
A = ∅
and
A = B = U
(U is a universal set (Georg Cantor [1932], Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], G. A. Korn, T. M. Korn [1968], I. N. Bronstein, K. A. Semendjajew [1989], [Encyclopaedia of Mathematics 1988])), respectively).
The equations
X ∪ A = B
and
X = B \ A
are equivalent by
A = ∅
only.
Definition A quantioperation is a simultaneous (joint, combined) consideration of a quantisystem (of general objects as quantioperands) that is possibly replaced by a general object as a general result.
Definition A quantioperation is called systematically commutative if its general result is independent of the general positions of any quantioperands.
Remark In the definition of a quantioperation, considering namely a quantisystem (and not simply a quantiset) is essential in the case of the systematic noncommutativity of the quantioperation.
Author's Creative Philosophy Principle
In each consideration, it is possible, useful, and therefore necessary to select the most creative (constructive) point of view with giving senses to all necessary and useful general objects, operations, relations, systems, processes, etc.
Example
If a < 0 and we want to consider the real numbers R only, then we may consider by m ∈ Z , n ∈ N
a(2m + 1)/(2n) := a2(2m + 1)/(4n)
giving real sense to ab by any irrational b , too. Using modulus (absolute value) |a| gives the same results but is much less natural because
a ≠ |a|,
(2m + 1)/(2n) = 2(2m + 1)/(4n).
Mega-overmathematics by Lev Gelimson [1987-2011c] naturally introduces many further (also uncountable) quantioperations and quantirelations. Among them is sign-conserving power function
a"b = |a|b sign a
defined by any real numbers a ≠ 0 and b , as well as by a = 0 and any b > 0. Then we have, e.g.,
a"2 = a2 sign a ,
(-1)"3 = -1 = [(-1)"6]"1/2 = (-1)"6/2 ,
(-1)"1/3 = -1 = [(-1)"2]"1/6 = (-1)"2/6 .
Nota bene: Fundamental, advanced, applied, and/or computational mathematical considerations also belong to fundamental, advanced, applied, and/or computational overmathematics, respectively, if and only if such considerations directly and explicitly use namely overmathematical very fundamentals revolutionarily replacing the inadequate very fundamentals of classical mathematics.
Examples:
1. Sign-conserving raising a specific nonnegative number to a real power belongs to mathematics but not to overmathematics because simply raising this specific nonnegative number to this real power in classical mathematics brings the same result. Hence replacing simply raising a specific nonnegative number to a real power with sign-conserving raising a specific nonnegative number to a real power brings nothing new. Therefore, raising a specific nonnegative number to a real power does not require overmathematical sign-conserving raising to a real power.
2. Sign-conserving raising a specific negative number to a real power belongs both to mathematics and to overmathematics because simply raising any negative number to any real power in classical mathematics is always ill-defined [Encyclopaedia of Mathematics 1988] because there are infinitely many irrational power exponents arbitrarily near to the given real power exponent so that for them, such a power is indefinite at all. All the more, for any even integer power exponent considered isolated, such a power is well-defined but brings the opposite result:
ab = |a|b ,
a"b = |a|b sign a = -|a|b (a < 0).
For any odd integer power exponent, such a power (also considered isolated) is always ill-defined:
a2z+1 = -|a|2z+1 ≠ |a|2z+1 = a2(2z+1)/2 = [a2(2z+1)]1/2 (a < 0; z =0, ±1, ±2, ...).
Hence replacing simply raising any negative number to any real power with sign-conserving raising this negative number to this real power always brings a new result. Therefore, raising any negative number to any real power does require namely overmathematical sign-conserving raising to a real power.
3. Sign-conserving real power function as whole belongs both to mathematics and to overmathematics because simply raising any negative number to any real power in classical mathematics is always ill-defined [Encyclopaedia of Mathematics 1988] because there are infinitely many real power exponents arbitrarily near to the given real power exponent so that for them, such a power is indefinite at all. All the more, for any even integer power exponent, such a power is well-defined but brings the opposite result. For any odd integer power exponent, such a power is ill-defined. Hence replacing simply raising any negative number to any real power with sign-conserving raising this negative number to this real power always brings a new result. Therefore, raising any negative number to any real power does require namely overmathematical sign-conserving raising to a real power.
Notata bene:
1. For any polarly represented complex (also imaginary) power base
a = reiφ
where r is a nonnegative number (modulus, or polar radius), unique polar argument φ belongs to half-opened segment [0, 2π[ (0 included but 2π excluded), and
i2 = -1,
naturally generalize the sign function with direction function
dir a = eiφ = cos φ + i sin φ
and the above sign-conserving real power function with direction-conserving complex-base real-exponent power function
a"b = |a|b dir a = rb dir a
with a complex power base a and a real power exponent b .
2. For any polarly represented complex (also imaginary) power base
a = reiφ
where r is a nonnegative number (modulus, or polar radius), unique polar argument φ belongs to half-opened segment [0, 2π[ (0 included but 2π excluded), and
i2 = -1,
as well as for any complex (also imaginary) power exponent
b = c + di
where c and d are real numbers,
further naturally generalize the above direction-conserving complex-base real-exponent power function
a"b = |a|b dir a = rb dir a
with direction-adding complex power function
a"b = a"c+di = |a|c+di dir a = rc+di eiφ = rcrdi eiφ = rceid ln r eiφ = rcei(d ln r + φ) .
Nota bene: Use " in a"b if necessary only.
Remark Intermediate symbolic senses can finally lead to an actual sense.
Examples
a table + a chair - the same table = the same chair;
1/a + a - 1/a = a
for any general object a (zero, a table, etc.).
Remark A linear structure of usual notation is conventional only. It is completely adequate by generally successive quantioperations only.
Remark A quantioperation is often a natural generalization (conserving the former sign, if necessary possibly with adding ° from the right as well as by relations, e.g. =°) of the corresponding usual operation (if it is known) on a quantisystem of general objects. A quantioperation might also be a sufficiently simple and useful quantimapping similar to another one already known.
Examples (with general result names after the sign “/”, here often with unity quantities)
A quantireplacement (quantisubstitution), e.g.
a ::= b .
A quantisimplification, e.g.
2 + 3 :=> 5.
A quantiassignment, e.g.
x := 1.
A quantiannihilation, e.g.
a ::= 0.
A quantiemptification, e.g.
x ::= #
(# is the empty element,
# ∈ ∅).
A quantinormalization, e.g.
a ::= ||a||,
a ::= |a|
(the quantinorm and quantimodulus of a , respectively).
A quantimetric/quantidistance, e.g.
{a, b}° ::= d{a, b}°.
A quantiaddition/quantisum of quantiaddends, e.g.
... + a + ... + b + ... .
A quantisubtraction/quantidifference, e.g.
a - b .
An algebraic quantiaddition/quantisum, e.g.
... + a + … - b + ... = ... + a + ... + (-b) + ... .
A quantimultiplication/quantiproduct, e.g.
...a...b... .
A quanti-inversion, e.g.
1/a (= a-1).
A quantidivision/quantiquotient (quantifraction), e.g.
a/b = ab-1.
An algebraic quantimultiplication/quantiproduct, e.g.
… a ... : b ... = ... a ... b-1 ... .
A quantisummary, e.g.
{..., a, ..., b, ...}°.
A quantiunification/quantiunion, e.g.
A ∪° B .
A quantireiteration, e.g.
αa =° {... , a , ... , a , ...}°
(α times).
A quantidiscrimination/quantidistinction, e.g.
a -° b =° {a , -1b}°.
An algebraic quantisummary, e.g.
{... , a , ... , -1b , ...}°.
An algebraic quantiunification/quantiunion, e.g.
A ∪° B \° C .
A quantisummary/quantiset, e.g.
{... , αa , ..., βb , ...}°.
An algebraic Cartesian quantimultiplication/quantiproduct, e.g.
Πι∈Ι α(ι)Aιβ(ι).
A quantimapping
f(... , x , ... , y , ...).
Remark
αa = --αa ,
0a = # = ∅
(therefore, the empty element # and the empty set ∅ including empty elements only can be identified).
Remark For
a ∈ [0, 1],
αa can be a fuzzy set (Lotfi Zadeh [1965], Dieter Klaua [1965, 1966a, 1966b, 1967]) in which a is its unique element whose "membership function" is α . But even by
α ∈ [0, 1],
αa can also be nonfuzzy.
Example
0.5apple
can also mean exactly half an apple.
Remark Relations in the pairs “quantiunification / quantireiteration” and “quantiaddition / quantimultiplication” are similar.
Remark Introducing both negative numbers and α < 0 in αa is equinatural.
2.1.3. Basic Theory of Quantielements
In classical mathematics (Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], G. A. Korn, T. M. Korn [1968], I. N. Bronstein, K. A. Semendjajew [1989], [Encyclopaedia of Mathematics 1988]), the quantity of an element in a set is usually ignored and there is no space-saving notation even for sequences of equal elements whose quantities are finite or countably infinite positive integers.
Definition A quantielement αa is the general result of an α-fold quantireiteration of a quasielement (radix) a associated with its quasielement (own, or individual, quantity) α .
Examples
3(-2) = {-2, -2, -2}° ≠ {-2, -2, -2} = {-2},
-π-e ,
a chaira table;
αa = # = ∅
if and only if
a = #
or/and
a = 0.
Definition Nonempty quantielements αa and βb are said to be equal
αa = βb
if and only if
a = b
and
α = β .
Definition Nonempty quantielements
αa , βb
are said to be similar if and only if they can be represented with
a = b .
Definition
... + αa - ... - βa + … = ... + α - ... - β + …a .
Definition
... αa ... : βb … = ... α ... : β …(… a … : b …).
Definition The elemental, multiple, and complete quantinorms (quantimoduli) of
αa
are
α||a||,
||α||a ,
and
||α||||a||,
respectively.
Definition The left, right, and two-sided quanti-integrals of
αa
are
αa ,
aα ,
and
αa = aα ,
respectively.
Examples
1a = a ,
1(-1)αa = α(-a),
-11αa = -αa ,
1/αa = 1/α(1/a).
(αa)β
is the quantiradix
aβ
with the own, or individual, quantity
αβ .
2.1.4. Basic Theory of Quantisets
Set concept by Georg Cantor [1932] in classical mathematics (Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], G. A. Korn, T. M. Korn [1968], I. N. Bronstein, K. A. Semendjajew [1989], [Encyclopaedia of Mathematics 1988]) contrasts the relations of membership and inclusion and leads to antinomies, ignores the quantities of set elements, and contradicts intuition because
1) any nonempty set is indeterminate, e.g.
{0, 1} = {0, 1, 1, 0, 0, 0, ...};
2) a sequence including some equal elements differs from their set;
3) even the simplest equations
X ∪ A = B
and
X ∩ A = B
in X are solvable only by
A ⊆ B
and
A ⊇ B ,
respectively (uniquely by
A = ∅
and
A = B = U
(a universal set (Georg Cantor [1932], Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], G. A. Korn, T. M. Korn [1968], I. N. Bronstein, K. A. Semendjajew [1989], [Encyclopaedia of Mathematics 1988])), respectively);
4) the equations
X ∪ A = B
and
X = B \ A
are equivalent by
A = ∅
only;
5) the concepts of sets and their relations satisfy no conservation laws;
6) the Cantor concept of a set is ill-posed [Encyclopaedia of Mathematics 1988] because, e.g.,
limε→+0{1 - ε , 1, 1 + ε} ≠ {1, 1, 1} = {1};
7) many urgent sets whose elements have quantities are inexpressible.
Definition A quantiset is a quantisummary of quantielements and is considered to be collected by collecting all similar and omitting all empty quasielements.
Notation
A =° ... +° αa +° ... +° α’a’ +° ... =°
{... , αa, ... , α’a’, ...}°.
Remark A quantiset is nonstructured, its linear notation is inessential, its quasielements are nonstructural, but it is structurable and might become structured.
Remark Belonging
αa ∈° A
and including
αa ⊆° A
are indistinguishable. Therefore, the Russell-Zermelo antinomy [Encyclopaedia of Mathematics 1988] on the set of all the sets that are not their own elements has no contradiction because that set is empty. Cantor’s antinomy [Encyclopaedia of Mathematics 1988] on the set of all the sets only shows that such a set has an indeterminate “volume” and cardinality but might be used as the universe of sets in some considerations.
Remark A quantiset generalizes a set and a fuzzy set (Lotfi Zadeh [1965], Dieter Klaua [1965, 1966a, 1966b, 1967]) even if
... , α , ... , α’, ... ∈ ]0, 1[
because such a quantiset may also be nonfuzzy like a quantielement
{0.5an apple}°
might consist of half an apple.
Remark Quantisets only can naturally and adequately express any collection of nonpositional general objects having any own, or individual, quantities.
Example
{0.5an apple, 1.5 loavesbread, -1 kgflour, $ -5.5money}°
might express the purchases, debt (deficiency), and expense of a housewife.
Definition A one-quantielement quantiset is
{αa}° =° αa .
Definition Quantisets are said to be equal if they coincide after collection.
Examples
{20, 3-1, 30, -2-1, 0e , π}° =° {π , -1, 50}°,
{1, 1, 1, 0}° =° {31, 0}° ≠° {30, 1}° =° {1, 0, 0, 0}°.
Definition An ordinary quantiset is a quantiset expressible with
... = α = … = α’ = ... = 1.
Definition The general distribution of A is A itself, i.e.,
{... , αa , ... , α’a’, ...}°.
Definition The quantiassortment of A is
{... , a , ... , a’, ...}°.
Definition The quanticlusterment of A is
{..., α, ..., α’, ...}°.
Definition The quantidifferential of A is
... + a + ... + a’ + ... .
Definition The uniquantity of A is
... + α + ... + α’ + … .
Remark It is a further generalization of an own, or individual, quantity.
Definition The left, right, and two-sided quanti-integrals of A are
... + αa + ... + α’a’+... ,
... + aα + ... + a’α’+ ... ,
and their common general value if they are equal to one another, respectively.
Definition The (logical) quantiunion of quantisets is
... ∪° {... , αa , ...}° ∪° ... ∪° {... , α’a , ...}° ∪° ... =°
{... , sup{... , α , ... , α’, ...}a , ...}°.
Definition The quanti-intersection of quantisets is
... ∩° {... , αa , ...}° ∩° ... ∩° {... , α’a , ...}° ∩° ... =°
{... , inf{... , α , ... , α’, ...}a , ...}°.
Definition The quantisum of quantisets is the quantiset of all their quantielements.
Example
{31, 0, -π2}° +° {α3, e1, -i0}° +° {ei , π2, i1}° =°
{1- i0, 3+e+i1, α3, ei}°.
Definition The algebraic quantisum of quantisets is the quantiset of all their quantielements after changing the sign of the quantity of each quantielement in every quantiset-subtrahend.
Definition The quantiproduct of quantisets is the quantiset of all the corresponding quantiproducts every of which includes exactly one quantielement of each of the given quantisets of their quasielements in any combinations.
Definition The elemental βth power of A is
{... , αaβ , ...}°,
i.e. each quasielement (radix) is raised to the βth power. In the multiple βth power of A , each own, or individual, quantity is raised to the βth power. In the complete βth power of A , both each quasielement (radix) and each own, or individual, quantity are raised to the βth power.
Definition The quantiopposition of A is
-A =° -1A =° -11 ×° A .
Corollary An algebraic quantisum
... + A + ... - B + …
of quantisets is the quantiunion
... +° A +° ... +° (-B) +° … .
Definition Left, right, and two-sided quantisets have the forms
β1 ×° A ,
A ×° β1,
and
β1 ×° A = A ×° β1,
respectively.
Definition For any a > b :
]a, b[ ::=° -1]b, a[,
]a, b] ::=° -1[b, a[,
[a, b[ ::=° -1]b, a],
[a, b] ::=° -1[b, a].
Definition For any a :
[a, a] ::=° {a},
[a, a[ ::=° ]a, a] ::=° ∅ =° 0# ,
]a, a[ ::=° {-1a}°.
Definition A simple multiplication of a quantiset is its quantimultiplication by a quantielement.
Definition A complicated multiplication of a quantiset is a quantimultiplication of each of its quantielements by an individual (not obligatorily common) quantielement.
Definition Quantisets are said to be left-similar if they might be represented as quantiproducts of a common quantiset (similarity basis) by quantielements from the left.
Definition Quantisets are said to be right-similar if they might be represented as quantiproducts of a common quantiset (similarity basis) by quantielements from the right.
Definition Quantisets are said to be similar if they are both left-similar and right-similar.
Definition Quantisets are said to be generally left-similar if they might be represented as quantiproducts of a common quantiset (similarity basis) by quantisets of quantielements from the left.
Definition Quantisets are said to be generally right-similar if they might be represented as quantiproducts of a common quantiset (similarity basis) by quantisets of quantielements from the right.
Definition Quantisets are said to be generally similar if they are both generally left-similar and generally right-similar.
Definition A left similarity class of quantisets is the quantiunion of left-similar quantisets.
Definition A right similarity class of quantisets is the quantiunion of right-similar quantisets.
Definition A similarity class of quantisets is the quantiunion of similar quantisets.
Definition A left similarity quanticlass of quantisets is the quantiunion of generally left-similar quantisets.
Definition A right similarity quanticlass of quantisets is the quantiunion of generally right-similar quantisets.
Definition A similarity quanticlass of quantisets is the quantiunion of generally similar quantisets.
Definition The left-equivalent quantielement to A is
AeL =° ... + α + ...[(... + αa + ...)/(... + α + ...)].
Definition The right-equivalent quantielement to A is
AeR =° ... + α + ...[(... + aα + ...)/(... + α + ...)].
Definition The equivalent quantielement to A is
AeL ,
or
AeR ,
if
AeL =° AeR .
Remark Such a quantielement is generally multiplicative in a quantimultiplication of quantisets if own, or individual, quantities commute with quantiradices in any quantiproduct of them.
Definition A quantisubset of A is a quantiset B (denoted:
B ⊆° A)
for which every own, or individual, quantity in
A -° B
is nonnegative.
Example
{-π0, a tablea chair, 31, -13}° ⊆°
{-30, that tablethat chair, π1, 2}°.
Definition
B ⊂° A
if and only if
B ⊆° A
and
B ≠° A .
Definition The complete ordinary subset of A is the quantiunion of all a for which
α ≥ 1.
Definition The quanti-inversion of A is the general object A-1 such that
A-1A = AA-1 = 1.
Remark The quanti-inversion of A with respect to nonempty nonzero
αa
is the quantiset
(1 - A1 + A12 - ...) 1/α(1/a)
where
A1 =° 1/α(1/a) A - {1}.
Example
{1}/{1, αa , βb}° =°
{1, -αa , -βb , αα(a2), αβ(ab), βα(ba), ββ(b2), ...}°.
Example
exp A =° {1} +° A +° A2/2! +° A3/3! +° ... .
Remark For quantisets, all the basic quantioperations and mappings are realizable and invertible.
2.1.5. Basic Theory of Linear Quanticombinations
In classical mathematics (Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], G. A. Korn, T. M. Korn [1968], I. N. Bronstein, K. A. Semendjajew [1989], [Encyclopaedia of Mathematics 1988]), a linear combination:
1) is only finite even in considering infinite sets;
2) is homogeneous;
3) implicitly includes only different elements without iterations;
4) includes usual objects, addition, and multiplication only; no unification or Cartesian multiplication, and no general objects and operations are considered.
Definition An ordinary left homogeneous linear quanticombination of the quasielements-bases of its basic quantiset
A = {... , αa , ... , α’a’, ...}°
over the quasielements-coefficients of its coefficient quantiset
C = {... , γc , ... , γ’c’, ...}°
is a quantisum/quantiunion of quantiproducts
... + ca + ... + c’a’ + ... .
Definition An ordinary right homogeneous linear quanticombination of the quasielements-bases of its basic quantiset
A = {... , αa , ... , α’a’, ...}°
over its coefficient quantiset
C = {... , γc , ... , γ’c’, ...}°
is a quantisum/quantiunion of quantiproducts
... + ac + ... + a’c’ + ... .
Definition An ordinary homogeneous linear quanticombination of the quasielements-bases of its basic quantiset
A = {... , αa , ... , α’a’, ...}°
over the quasielements-coefficients of its coefficient quantiset
C = {... , γc , ... , γ’c’, ...}°
is their ordinary left, or right, homogeneous linear quanticombination if
... + ca + ... + c’a’ + ... =
... + ac + ... + a’c’ + ... .
Definition A left homogeneous linear quanticombination of its basic quantiset
A = {... , αa , ... , α’a’, ...}°
of quantielements-bases over its coefficient quantiset
C = {... , γc , ... , γ’c’, ...}°
of quantielements-coefficients is a quantisum/quantiunion of quantiproducts
... + γcαa + ... + γ’c’α’a’ + ... .
Definition A right homogeneous linear quanticombination of its basic quantiset
A = {... , αa , ... , α’a’, ...}°
of quantielements-bases over its coefficient quantiset
C = {... , γc , ... , γ’c’, ...}°
of quantielements-coefficients is a quantisum/quantiunion of quantiproducts
... + αaγc + ... + α’a’γ’c’ + ... .
Definition A homogeneous linear quanticombination of its basic quantiset
A = {... , αa , ... , α’a’, ...}°
of quantielements-bases over its coefficient quantiset
C = {... , γc , ... , γ’c’, ...}°
of quantielements-coefficients is their left, or right, homogeneous linear quanticombination if
... + γcαa + ... + γ’c’α’a’ + ... =
... + αaγc + ... + α’a’γ’c’ + ... .
Definition An ordinary left homogeneous linear Cartesian quanticombination of the quasielements-bases of its basic quantiset
A = {... , αa , ... , α’a’, ...}°
over the quasielements-coefficients of its coefficient quantiset
C = {... , γc , ... , γ’c’, ...}°
is a quantisum/quantiunion of Cartesian products
... + c × a + ... + c’ × a’ + ... .
Definition An ordinary right homogeneous linear Cartesian quanticombination of the quasielements-bases of its basic quantiset
A = {... , αa , ... , α’a’, ...}°
over the quasielements-coefficients of its coefficient quantiset
C = {... , γc , ... , γ’c’, ...}°
is a quantisum/quantiunion of Cartesian products
... + a × c + ... + a’ × c’ + ... .
Definition An ordinary homogeneous linear Cartesian quanticombination of the quasielements-bases of its basic quantiset
A = {... , αa , ... , α’a’, ...}°
over the quasielements-coefficients of its coefficient quantiset
C = {... , γc , ... , γ’c’, ...}°
is their ordinary left, or right, homogeneous linear Cartesian quanticombination if
... + c × a + ... + c’ × a’ + ... =
... + a × c + ... + a’ × c’ + ... .
Definition A left homogeneous linear Cartesian quanticombination of its basic quantiset
A = {... , αa , ... , α’a’, ...}°
of quantielements-bases over its coefficient quantiset
C = {... , γc , ... , γ’c’, ...}°
of quantielements-coefficients is a quantisum/quantiunion of Cartesian products
... + γc × αa + ... + γ’c’ × α’a’ + ... .
Definition A right homogeneous linear Cartesian quanticombination of its basic quantiset
A = {... , αa , ... , α’a’, ...}°
of quantielements-bases over its coefficient quantiset
C = {... , γc , ... , γ’c’, ...}°
of quantielements-coefficients is a quantisum/quantiunion of Cartesian products
... + αa × γc + ... + α’a’ × γ’c’ + ... .
Definition A homogeneous linear Cartesian quanticombination of its basic quantiset
A = {... , αa , ... , α’a’, ...}°
of quantielements-bases over its coefficient quantiset
C = {... , γc , ... , γ’c’, ...}°
of quantielements-coefficients is their left, or right, homogeneous linear Cartesian quanticombination if
... + γc × αa + ... + γ’c’ × α’a’ + ... =
... + αa × γc + ... + α’a’ × γ’c’ + ... .
Definition An ordinary (optional: left or right) linear quanticombination of A over C is the quantisum/quantiunion of the corresponding homogeneous ordinary linear quanticombination and of some
c ∈ C .
Definition A (optional: left or right) linear quanticombination of A over C is the quantisum/quantiunion of the corresponding homogeneous linear quanticombination and of some
γc ∈ C .
Definition An ordinary (optional: left or right) linear Cartesian quanticombination of A over C is the quantisum/quantiunion of the corresponding homogeneous ordinary Cartesian linear quanticombination and of some
c ∈ C .
Definition A (optional: left or right) linear Cartesian quanticombination of A over C is the quantisum/quantiunion of the corresponding homogeneous linear Cartesian quanticombination and of some
γc ∈ C .
Definition A quantiset A is called linearly quantidependent over C if there exists some zero-value (or empty-value) homogeneous linear combination of A over C using nonzero (or nonempty, respectively) quanticoefficients. Otherwise, a quantiset A is called generally linearly quanti-independent over C .
Definition A quantibasis B of A over C is a generally linearly independent (over C) quantisubset (of A) such that any
a ∈ A
is an ordinary homogeneous linear quanticombination of B over C .
Example Let H be a Hilbert space (Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], G. A. Korn, T. M. Korn [1968], I. N. Bronstein, K. A. Semendjajew [1989], [Encyclopaedia of Mathematics 1988]) including a countable orthonormal basis
F = {fk | k ∈ N}
and let
f-1 = ∑k∈N akfk
(by
∑k∈N |ak|2
finite)
be an essentially infinite (Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], G. A. Korn, T. M. Korn [1968], I. N. Bronstein, K. A. Semendjajew [1989], [Encyclopaedia of Mathematics 1988]) series. Then ordinary set
{f-1} + F
is linearly independent but generally linearly dependent.
Chapter 2.2. Uninumber Theories
2.2.1. Basic Theory of Uniquantities
A known set cardinality in classical mathematics (Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], G. A. Korn, T. M. Korn [1968], I. N. Bronstein, K. A. Semendjajew [1989], [Encyclopaedia of Mathematics 1988]) based on a bijection is
1) determined for an ordinary set only;
2) sensitive for a finite ordinary set only.
Definition A bijection is called isometric if it conserves a quantinorm.
Definition The uniquantity of a quantiset
A = {... , αa , ... , α’a’, ...}°
is defined to be such a generally algebraically union-to-sum additive and Cartesian-product-to-product multiplicative (see 3.1.3) [non-positional and possibly uncountable] quantisum
Q(A) = ... + α + ... + α’ + ...
of the own [inside], or individual, quantityties of all quasielements [bases] of A, which extends a point [zero-dimensional] measure [the number of elements] by satisfying the following axioms:
(A1) a uniquantity is completely algebraically additive, say
Q(... +° A -° ... -° B +° ... +° C -° ... -° D +° ...) =
... + Q(A) - ... - Q(B) + ... + Q(C) - ... - Q(D) + ... ,
commutative, and associative;
(A2) quantifying a quantiset implies multiplying its uniquantity:
Q(tA) = tQ(A);
(A3) the uniquantity of a Cartesian product of quantisets is the product of their uniquantities:
Q(... × A × ... × B × ...) = ... Q(A) ... Q(B) ... ,
(A4) norming any bases in a quantiset does not affect its uniquantity, say
Q{... , qa , ..., rb , ... , sc , ...}° =
Q{... , q||a||, ... , rb , ... , s||c||, ...}°;
(A5) if all mutual distances between the bases of a quantiset are bounded above in common and a mapping of the quantiset preserves both the distances [is an isometry] and the own quantities of the bases in the quantiset, the mapping preserves its uniquantity;
(A6) uniquantity determination commutes with a passage to the limit;
(A7) the uniquantities of some chosen canonical sets coincide with their cardinalities, in particular:
Q{a} = 1
for any object a ;
Q(N) = ω
(in place of ℵ0 for simplicity of notation);
Q|0, 1| = Ω
(the continuum cardinality).
Examples
Q{a + bn | n ∈ N} = ω/|b| - a/b - 1/2 + 1/(2|b|)
where
a , b
are unireal numbers,
b ≠ 0;
in particular,
Q{z + 1, z + 2, ...} = Q({1, 2, ...} \ {1, 2, ... , z}) = Q(N \ {1, 2, ... , z}) = ω - z ,
Q{... , z - 3, z - 2, z - 1} = Q({... , -3, -2, -1} +° {0, 1, 2, ... , z - 1}) = Q(Z-) + z = ω + z ,
Q{a + bn | n ∈ Z-} = Q{a + (-b)n | n ∈ Z+} = Q{a + (-b)n | n ∈ N} =
ω/|-b| - a/(-b) - 1/2 + 1/(2|-b|) =
ω/|b| + a/b - 1/2 + 1/(2|b|),
Q{a + bn | n ∈ Z±} =
Q{a + bn | n ∈ Z-} + Q{a + bn | n ∈ Z+} =
ω/|b| + a/b - 1/2 + 1/(2|b|) +
ω/|b| - a/b - 1/2 + 1/(2|b|) =
2ω/|b| - 1 + 1/(|b|),
Q{a + bn | n ∈ Z} =
Q{a + bn | n ∈ Z-} + Q{a + bn | n = 0} + Q{a + bn | n ∈ Z+} =
ω/|b| + a/b - 1/2 + 1/(2|b|) +
1 +
ω/|b| - a/b - 1/2 + 1/(2|b|) =
2ω/|b| + 1/(|b|);
for the sets of all odd natural numbers,
Q{1, 3, 5, ...} = Q{-1 + 2n | n ∈ N} = ω/|2| - (-1)/2 - 1/2 + 1/(2|2|) = ω/2 + 1/4;
for the sets of all even natural numbers,
Q{2, 4, 6, ...} = Q{0 + 2n | n ∈ N} = ω/|2| - 0/2 - 1/2 + 1/(2|2|) = ω/2 - 1/4;
for the sets of all odd natural numbers excluding 1,
Q{3, 5, 7, ...} = Q{1 + 2n | n ∈ N} = ω/|2| - 1/2 - 1/2 + 1/(2|2|) = ω/2 - 3/4;
for the sets of all even natural numbers excluding 2,
Q{4, 6, 8, ...} = Q{2 + 2n | n ∈ N} = ω/|2| - 2/2 - 1/2 + 1/(2|2|) = ω/2 - 5/4;
for the sets of all natural numbers giving remainder 1 if divided by 3,
Q{1, 4, 7, ...} = Q{-2 + 3n | n ∈ N} = ω/|3| - (-2)/3 - 1/2 + 1/(2|3|) = ω/3 + 1/3;
for the sets of all natural numbers giving remainder 2 if divided by 3,
Q{2, 5, 8, ...} = Q{-1 + 3n | n ∈ N} = ω/|3| - (-1)/3 - 1/2 + 1/(2|3|) = ω/3;
for the sets of all natural numbers divisible by 3,
Q{3, 6, 9, ...} = Q{0 + 3n | n ∈ N} = ω/|3| - 0/3 - 1/2 + 1/(2|3|) = ω/3 - 1/3;
for the sets of all natural numbers giving remainder 1 if divided by 3 excluding 1,
Q{4, 7, 10, ...} = Q{1 + 3n | n ∈ N} = ω/|3| - 1/3 - 1/2 + 1/(2|3|) = ω/3 - 2/3;
for the sets of all natural numbers giving remainder 2 if divided by 3 excluding 2,
Q{5, 8, 11, ...} = Q{2 + 3n | n ∈ N} = ω/|3| - 2/3 - 1/2 + 1/(2|3|) = ω/3 - 1;
for the sets of all natural numbers divisible by 3 excluding 3,
Q{6, 9, 12, ...} = Q{3 + 3n | n ∈ N} = ω/|3| - 3/3 - 1/2 + 1/(2|3|) = ω/3 - 4/3;
for the set H- of all negative half-integers,
Q(H-) = Q{-1/2, -3/2, -5/2, ...} = Q{1/2 + (-1)n | n ∈ N} = ω/|-1| - (1/2)/(-1) - 1/2 + 1/(2|-1|) = ω + 1/2;
for the set H+ of all positive half-integers,
Q(H+) = Q{1/2, 3/2, 5/2, ...} = Q{-1/2 + 1 × n | n ∈ N} = ω/|1| - (-1/2)/1 - 1/2 + 1/(2|1|) = ω + 1/2;
for the set H of all half-integers,
Q(H) = Q{... , -5/2, -3/2, -1/2, 1/2, 3/2, 5/2, ...} = Q(H- +° H+) = ω + 1/2 + ω + 1/2 = 2ω + 1.
Definition A uninumber is a pure uniquantity.
2.2.2. Basic Theory of Quantidimensions
In classical mathematics (Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], G. A. Korn, T. M. Korn [1968], I. N. Bronstein, K. A. Semendjajew [1989], [Encyclopaedia of Mathematics 1988]), physical dimensions (units) [Encyclopaedia of Physics 1973] and the prehistory of artificially pure numbers are not adequately considered or even ignored.
So a known quantity
1) is not invariant when changing a physical dimension;
2) satisfies no conservation law;
3) cannot express and distinguish many urgent actual infinities.
Definition A quantidimension is a physical dimension generally multiplied by the uniquantity of the corresponding canonical dimension.
Examples The quantivolume of the n-dimensional cube
[0, 1[n
is
Ωn.
∑v∈V Ωn = ΩmΩn = Ωm+n
where
V = [0, 1[m.
2.2.3. Basic Theory of Quanti-Infinities
For all the known actually infinite cardinalities (Georg Cantor [1932], Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], G. A. Korn, T. M. Korn [1968], I. N. Bronstein, K. A. Semendjajew [1989], [Encyclopaedia of Mathematics 1988]), the following holds:
1) scarce positive and no negative uniquantities are expressible;
2) usual addition and multiplication only are realizable;
3) no conservation law holds.
In modern nonstandard analysis by Abraham Robinson [1966], actual number infinities as the inversions of infinitesimals are indeterminate along with them.
Definition A number infinity (overnumber) is a uninumber whose quantinorm is greater than any finite integer.
Definition A quanti-infinity is a general object whose quantinorm is greater than any finite integer.
Notation For any number set S , an S-number is any element of S , an S-operation is an operation in S under which S is closed, and an S-relation is a relation in S .
Examples
-3 is a Z-number [Z the integers];
-2/11 a Q-number [Q the rational numbers];
+ , × , and -≥ [subtracting a relatively not greater number] are N-operations;
= , > , and < [as well as their “unions’’ ≥ and ≤] are N-, Z-, Q-, and R-relations.
Notation An S-number system is a number set S with all S-operations and S-relations.
Example The rational-number system [linearly ordered field] is the set Q with the four usual arithmetic operations
+ , - , × , /
[division] as well as with the usual relations
= , > , < .
Definition An S-uninumber algebraic system is the smallest structure-preserving extension of an S-number system with including all the infinite cardinal numbers. They are regarded as some symbols [letters] with certain number values for which the usual strict inequalities
0 < 1 < 2 < ... < ω < ω1 < ω2 < ...
between the finite and infinite ordinal numbers identified with the corresponding cardinal numbers for which
0 < 1 < 2 < ... < ℵ0 < ℵ1 < ℵ2 < ...
are valid. All the properties of the S-operations and of the S-relations, except the Archimedean property of multiplication by [finite] S-numbers, [formally] hold for any terms possibly including infinite cardinal numbers.
Remark For S = Q , the field and order properties hold. Relations valid in set theory but formally contradicting the properties of an S-number system do not hold in an S-uninumber system. For example, relations
ℵ0 = ℵ0 + 1 = 2ℵ0 = nℵ0 = (ℵ0)2 = (ℵ0)n (n ∈ N),
∁2 ≤ ∁ ,
or, by another notation regarding the finite and infinite ordinal numbers namely as the corresponding cardinal numbers,
ω = ω + 1 = 2ω = nω = ω2 = ωn (n ∈ N),
Ω2 ≤ Ω
(where
Ω = ω1 = ℵ1 = ∁),
respectively, hold in classical mathematics but are false in the natural-uninumber system [monoid] N , the uniinteger system [integral domain] Z , the rational-uninumber system Q , and the real-uninumber system R . In our constructing a new sensitive degree of quantity by using the known objects [the numbers and infinite cardinal numbers] only, we have to change some familiar relations between them. Every measure also gives other results than the cardinality does. Each other relation between terms possibly including infinite cardinal numbers [which satisfies Definition 1.1.1.5] holds in an S-uninumber system, e.g.
(1 + ω)2 = ω2 + 2ω + 1
or the nonstrict inequality
Ω + 1 ≥ Ω
whose strict "subinequality"
Ω + 1 > Ω
satisfies the algebraic rules for the usual numbers and hence Definition 1.1.1.5 [in set theory, on the contrary,
Ω + 1 = Ω].
Each S-uninumber system is clearly non-Archimedean but hyper-Archimedean with respect to the four arithmetic operations by
Definitions
A partially ordered set M with a partial binary operation
f : M2 → M ,
where
M2 ⊆ M × M ,
is called hyper-Archimedean with respect to f if for any a ∈ M such that there are nonidentical elements of M in the Cantorian set union
{f(m , a) | m∈M} ∪ {f(a , m) | m ∈ M}
and for any b ∈ M , there exists a c ∈ M for which
f(c , a) > b
or
f(a , c) > b .
A partially ordered set M is called hyper-Archimedean over a Cantorian set L with respect to a partial binary operation
g : L1 × M1 ∪ M2 × L2 → M ,
where
M1 , M2 ⊆ M ,
L1 , L2 ⊆ L ,
if for any a ∈ M with nonidentical elements of M in the Cantorian set union
{g(l , a) | l ∈ L1} ∪ {g(a , l) | l ∈ L2}
and for any b ∈ M , there exists an l ∈ L with
g(l , a) > b
or
g(a , l) > b .
A variable, mapping, or correspondence with range M is called hyper-Archimedean over a Cantorian set L with respect to g if for M the same holds.
Remark Such a hyper-Archimedean extension of S to S depends on a choice of the S-operations [forming an S-uninumber set S] as well as of the S-relations and can be various.
Examples The addition, subtraction, and multiplication of any numbers, division by finite numbers (or also including the limits of all the Cauchy sequences [1]) lead to the ordered commutative ring [no field] Q (or R, respectively). Such a uninumber is a sum of a uniinteger and a rational or real number, respectively, giving a [translational] number scale extension without condensation. The addition, subtraction, multiplication, and division of any numbers [without restriction] make the inclusion of these limits unnecessary and give the same ordered commutative field of all rational or, equivalently, real uninumbers, whose carrier is the set
Q° = R°
with condensation on any interval in comparison with Q and R.
Definition An S-uninumber is an element from the set S being the carrier [support] of an S-uninumber system and the extension of S [see Definition 1.1.1.5].
Definition An S-quantioperation is the extension of an S-operation from S to S in an S-uninumber system.
Definition An S-quantirelation is the extension of an S-relation from S to S in an S-uninumber system.
Definition An S-ultranumber is an S-uninumber that is no S-number. The set of all S-ultranumbers
S = S \ S .
Definition An S-overnumber is an S-uninumber [namely an S-ultranumber] with modulus greater than that of any S-number. The set of all S-overnumbers is denoted by S .
Definition An S-undernumber is an
S-uninumber [0 or an S-ultranumber] with modulus less than that of any nonzero
S-number. The set of all S-undernumbers is denoted by
S.
Definition An S-internumber is an
S-ultranumber that is no S-overnumber and no S-undernumber. The set of all
S-internumbers is denoted by S .
Corollary
S =
S \ S \ S
but
S
≠ S ∪ S ∪ S
[since 0 ∈ S and 0 ∉ S
].
Examples Ω and other infinite cardinal numbers, their sums and products are N-, Z-, Q-, and R-overnumbers;
their reciprocals such as Ω-1 [belonging to
Q° = R°]
are undernumbers;
2/Ω - 1, other sums of a nonzero S-undernumber and of a nonzero S-number are S-internumbers.
Examples For
S ∈ {N , Z , Q , R} :
S = S = S \ S ,
S
= {0} ,
S =
∅ .
2.2.4. Basic Theory of Quanti-Infinitesimals
In modern nonstandard analysis by Abraham Robinson [1966],
1) infinitesimals and nontrivial ultrafilters are indeterminate;
2) urgent infinitesimals are inexpressible (because there is no relation like
i2 = -1).
Definition A number infinitesimal is defined to be zero or the quanti-inversion of a number infinity from their complete ordinary set IA of a quantiset A .
Definition A quanti-infinitesimal is defined to be zero or the quanti-inversion of a quanti-infinity from their complete ordinary set
∞A-1
of a quantitype/quantiset A .
Examples
Ω-1;
ω-1;
ln(1 + 1/(Ω - ω));
the equation
∑n∈N x := 1
has the unique solution
x := ω-1.
Notation
∞-1
is defined to be the ordinary set of all the quanti-infinitesimals.
2.2.5. Basic Additive Theory of Quasinumbers
Usual finite real numbers in classical mathematics (Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], G. A. Korn, T. M. Korn [1968], I. N. Bronstein, K. A. Semendjajew [1989], [Encyclopaedia of Mathematics 1988]) without infinitesimal parts satisfy no conservation law. A finite nonstandard number by Abraham Robinson [1966]
1) is indeterminate in principle along with any infinitesimal;
2) cannot explicitly express any urgent number.
Definition A quasinumber
a, B
is a usual number
a ∈ A
possibly increased or decreased by a number infinitesimal belonging to IB .
Notation
NR = N + IR ,
ZC = Z + IC ,
QQ = Q + IQ ,
RR = R + IR ,
CC = C + IC .
2.2.6. Basic Additive Theory of Uninumbers
Usual finite real numbers in classical mathematics (Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], G. A. Korn, T. M. Korn [1968], I. N. Bronstein, K. A. Semendjajew [1989], [Encyclopaedia of Mathematics 1988]) without infinite and infinitesimal parts:
1) cannot express many urgent uniquantities;
2) satisfy no conservation law like indeterminate nonstandard numbers by Abraham Robinson [1966].
Definition A superinteger is defined to be a quantisum of finite integers.
Example Q(A) is a superinteger if any element quantity a in quantiset A is a superinteger, e.g.:
Q(-3ω[0, 1] - 2N) = - 3ω(Ω + 1) - 2(ω + 1/2).
Definition A uninumber, or a superquasinumber, is a quasinumber possibly increased or decreased by a superinteger.
Notation
N+R , C = N + R + IC ,
CR , C = C + R + IC ,
N(+) = N(+) + N(+) + {0},
Z = Z + Z + {0},
Q = Q + Q + IQ ,
R = R + R + IR ,
C = C + C + IC .
Remark All the quantioperations, quantimappings, and quantirelations are applicable and all their properties including conservation laws are satisfied.
Definition Uninumbers
a , b
are called equal
(a = b)
if
a - b = 0.
Definition Uninumbers
a , b ∈ A
are called equivalent
(a ↔ b)
if
(a - b) / (a + b) ∈ IA .
Definition Uninumbers
a , b ∈ A
are called equiorderal
(a ≅ b)
if
a/b , b/a ∉ IA .
Definition The quantimonad a of a uninumber a is the ordinary set of all the uninumbers b for which
b - a
is a quanti-infinitesimal.
Remark A monad by Gottfried Wilhelm Leibniz [1714] is considered by Martin Davis [1977] as a class, i.e. a nonset.
Notation
[aIb] ::= ∑d∈[a , b] d ,
]aIb[ ::= [aIb] - a - b ,
g := Q(a) (a ∈ R),
G := Q[0I1[ = cg ,
[aIb] ::= {a} + a+ + ]aIb[ + b- +{b},
]aIb[ ::= [aIb] - {a} - {b}.
Definition A uninumber basis is an ordinary set including a (nonzero) non-infinitesimal finite uninumber and such that any quotient of its elements is an infinity or infinitesimal.
Definition Uninumber bases are called equivalent if there exists their bijection in which any preimage and its image are equivalent.
Definition Uninumber bases are called equiorderal if there exists their bijection in which any preimage and its image are equiorderal.
Definition A reduced quantioperation on a quantiset of uninumber bases is the composition of the corresponding quantioperation and of further leaving anyone of any equiorderal uninumbers in a general result.
Definition A reduced quantimapping of a quantiset of uninumber bases is the quanticomposition of the corresponding quantimapping and of further leaving anyone of any equiorderal uninumbers in a general result.
Definition A uninumber of order 0 is a finite usual number.
Definition A uninumber of order n + 1 (n ∈ N) over a uninumber basis B is an ordinary homogeneous linear quanticombination (of B) whose coefficients are uninumbers of orders not greater than n over B.
Definition A uninumber basis is orderal-successive if it is a successive ordinary set whose quasielements' orders are nonstrictly monotone.
Definition A uninumber basis is strictly orderal-successive if it is a successive ordinary set whose quasielements' orders are strictly monotone.
Definition A uninumber basis is orderal-ascending if it is a successive ordinary set whose quasielements' orders are nondecreasing.
Definition A uninumber basis is strictly orderal-ascending if it is a successive ordinary set whose quasielements' orders are strictly increasing.
Definition A uninumber basis is orderal-descending if it is a successive ordinary set whose quasielements' orders are nonincreasing.
Definition A uninumber basis is strictly orderal-descending if it is a successive ordinary set whose quasielements' orders are strictly decreasing.
Example
(Ωαωβ | α , β ∈ R)
(in succession by α and for the same α by β) is a strictly orderal-successive, namely a strictly orderal-ascending, uninumber basis.
Definition A (common) G-uninumber basis for a quantiset A of uninumbers is a uninumber basis B for which any
a ∈ A
is an ordinary homogeneous linear quanticombination of B over G .
Remark A common basis is a reduced quantiunion of individual bases.
2.2.7. Basic Theory of Uninumber Representations
There are different number systems (Solomon Feferman [1964]) including a number system with an irrational base, namely the golden ratio
a = (51/2 + 1)/2,
by George Bergman [1957] with generalizations by Alexey Stakhov [2009] and Sergey Abachiev [2012], as well as binomial enumeration theory by Alexey Borisenko [1991, 2004a, 2004b, 2009].
Known number representations in classical mathematics (Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], G. A. Korn, T. M. Korn [1968], I. N. Bronstein, K. A. Semendjajew [1989], [Encyclopaedia of Mathematics 1988]) ignore infinities and infinitesimals, can be non-unique (e.g.
1 = 0.99...),
and satisfy no conservation law.
Indeterminate nonstandard numbers by Abraham Robinson [1966] have no explicit representation, even if nonstandard analysis can give new proofs.
Definition The separation of a uninumber a is the quantioperation whose result (of the same name) is
a = }a{ + )a( = [a] + {a} + )a( = [a].{a}.)a(
where
}a{ is the noninfinitesimal (maybe infinite) part of a ,
[a] is the quantiintegral part of a , or [}a{] (using the entier function [1]),
{a} is the fractional part of a ,
{a} ∈ [0, 1[ ,
)a( is the infinitesimal part of a .
Examples Using canonical infinities ω , Ω and the general neutralizator ⅁ which equals the empty element # :
a = ω - Ω2 - Ω-1 :
}a{ = [a] = ω - Ω2,
{a} = 0 ,
)a( = Ω-1;
a = ω/b (b ∈ R):
}a{ = [a] = ω/b ,
{a} = 0 ,
)a( = ⅁ .
Definition A nonfuzzy [m-ary (m ∈ N \ {1})] continued quantifraction is the quantilimit of the corresponding sequence if there is some explicit or latent law of possibly correcting the last digit/denominator taken into account in truncating.
Definition A fuzzy [m-ary (m ∈ N \ {1})] continued quantifraction is the quantilimit of the corresponding sequence if there is no explicit or latent law of possibly correcting the last digit/denominator taken into account in truncating.
Notation
ω.1-1+... = (ω.1, ω.12, …);
-Ω.1928... = (-Ω.2 , -Ω.19, -Ω.193, …).
Examples
0.111...- = (0.1, 0.11, 0.111, ...) =
(1/9-10-n/9 | n ∈ N) = 1/9 - 10-ω/9 < 1/9;
0.23999...- = (0.2, 0.23, 0.239, …) = 0.24 - 10-ω < 0.24 < 0.24+ =
(0.3, 0.25, 0.241, …) = 0.24 + 10-ω ;
0.11...
is a fuzzy number in
[1/9 - 10-ω/9, 1/9 + 8/9 × 10-ω]
(included into the quantimonade 1/9 of 1/9 and is asymmetric);
am- = sign a × [|a| × mω]/mω ;
am+ = sign a × [|a| × mω + 1]/mω ;
am is a fuzzy number in
sign a[am- , am+] ⊆ a
(the unimonade of a);
by using the Fibonacci numbers
F1 = 1, F2 = 1, Fn+2 = Fn+1 + Fn (n ∈ N = {1, 2, …})
and the Binet formula
Fn = (αn + (-1)n+1/αn)/51/2
already known by Abraham de Moivre [Encyclopaedia of Mathematics 1988]
where
α = (51/2 + 1)/2,
the continued fractions
Ω + 1/(1 + 1/(1 + …))- =
(Ω + 1/(1 + 1), Ω + 1/(1 + 1/(1 + 1)), Ω + 1/(1 + 1/(1 + 1/(1 + 1))), …) =
(Ω + F2/F3 , Ω + F3/F4 , Ω + F4/F5 , … , Ω + Fn+1/Fn+2 , …) =
Ω + (αω+1 + (-1)ω+2/αω+1) / (αω+2 + (-1)ω+3/αω+2),
Ω + 1/(1 + 1/(1 + …))+ =
(Ω + 1/1, Ω + 1/(1 + 1/1), Ω + 1/(1 + 1/(1 + 1/1)), …) =
(Ω + F1/F2 , Ω + F2/F3 , Ω + F3/F4 , … , Ω + Fn/Fn+1 , …) =
Ω + (αω + (-1)ω+1/αω) / (αω+1 + (-1)ω+2/αω+1).
Definition The total harmonic expansion of a uninumber a is
aht = [a] + ∑n∈N\{1} cn/n
where
cn = [n({a} - ∑k=2n-1 ck/k)] ∈ {0, 1}
with using the entier function [Encyclopaedia of Mathematics 1988].
Definition The selective harmonic expansion of a uninumber a is
ahs = [a] + ∑k∈N 1/nk
where
nk = ]1/{a} - ∑l=1 k-1 1/nl[
and
]x[ ::= -[-x]
with using the entier function [Encyclopaedia of Mathematics 1988].
Remark In the general case, aht and ahs are distinct as a nonordinary and an ordinary successive sets and do not coincide with a .
Examples
(2/3)ht = 1/2 + 0/3 + 0/4 + 0/5 + 1/6 =
2/3 = 1/2 + 1/6 = (2/3)hs ;
ehs = 2 + 1/2! + 1/3! + … ;
(√2)hs = 1 + 1/3 + 1/13 + 1/253 + 1/218204 + … ;
((2ω + 3)/(5ω - 5))hs = 1/3 + 1/5 + 1/(ω - 1) =
(2ω + 3)/(5ω - 5).
Definition A common quantifraction is a uninumber expressible by a uniquotient of uniintegers.
Remark
aht = ahs = a
if and only if a is a common quantifraction.
Definition The canonical expansion of a uninumber is the corresponding homogeneous linear quanticombination of an R-uninumber basis including unit and canonical infinities and infinitesimals.
Example
(5ωΩ - Ω - 5ω3 + ω2 + 2ω + 1)/(5ω - 1) =
Ω - ω2 + 2/5 + 7/10 × ω-1 + 7/50 × ω-2 + 7/250 × ω-3 + … .
Definition The geometric representation of an R-uninumber
a = ∑j∈J aj bj
as a possible element of a separated S-uninumber system over an R-uninumber basis
B = (bj | j ∈ J)
is the unipoint
A = (aj | j ∈ J)
in the frame of reference
B = (bj | j ∈ J).
Example
0.111…-
is represented by the unipoint
(1/9, -1/9)
in the frame of reference
(1, 10-ω).
Definition The positional representation of an S-uninumber
a = ∑j∈J aj bj
over a fixed positional S-uninumber basis
B = (bj | j ∈ J)
is the positional set
A = (aj | j ∈ J).
Example
0.111…- = 9-1.-9-1 || (1, 10-ω).
Remark The unireal a and uniimaginary b parts of a complex uninumber a + bi can be separately represented.
Part 3. Quantisystems, States, and Processes
Chapter 3.1. Quantisystem Theories
3.1.1. Basic Theory of Positional Quantisets
The indeterminate usual concept of a set in classical mathematics (Georg Cantor [1932], Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], G. A. Korn, T. M. Korn [1968], I. N. Bronstein, K. A. Semendjajew [1989], [Encyclopaedia of Mathematics 1988]) also ignores their elements’ factual positions essential for sequences and series that both are usually considered as nonsets.
Definition A positional quantiset is a quantiunion of positional general objects.
Definition A partially positional quantiset is a quantiunion of nonpositional and possibly positional general objects.
Examples
(αa , b , βb , a) ≠ (α+1a , β+1b),
N´ = (1, 2, ...) ≠
N = {1, 2, ...}.
Definition The quantiapportionment of a (partially) positional quantiset is the positional quantiset of the general positions (with the corresponding own, or individual, quantities) of all the quasielements in a given quantiset.
Definition The general positionality of a quasielement in a (partially) positional set is a general object expressing the degree of occupying each possible general position by that element.
Definition The general positionality of a quantisystem is the quantisystem of the general positionalities (with the corresponding own, or individual, quantities) of all its quasielements.
3.1.2. Basic Theory of Quantimappings
A usual mapping in classical mathematics (Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], G. A. Korn, T. M. Korn [1968], I. N. Bronstein, K. A. Semendjajew [1989], [Encyclopaedia of Mathematics 1988]) binds usual sets of usual objects.
Definition A quantimapping f is a correspondence between some quantisubsets of given quantisets (X , Y , etc.).
Remark It can be fuzzy, generally inexact, inexactly known, etc.
Notation
f : X — Y
is a quantimapping from X in Y ,
f : X → Y
is a quantimapping/quanti-injection of X in Y ,
f : X ← Y
is a quantimapping/quantisurjection from X on Y , and
f : X ↔ Y
is a quantimapping/quantibijection of X on Y .
Definition A general image is the quantiset of all the corresponding images (with the corresponding own, or individual, quantities) in a quantimapping.
Definition A quantipreimage is the quantiset of all the corresponding preimages (with the corresponding own, or individual, quantities) in a quantimapping.
Definition The proper domain of a quantimapping is the quantiset of all its factual quantipreimages (with the corresponding own, or individual, quantities).
Definition The improper domain of a quantimapping is the quantiset of all its nominal quantipreimages (with the corresponding own, or individual, quantities).
Definition The proper range of a quantimapping is the quantiset of all its factual general images (with the corresponding own, or individual, quantities).
Definition The improper range of a quantimapping is the quantiset of all its nominal general images (with the corresponding own, or individual, quantities).
Remark A quantimapping becomes single-valued if any its general image is considered as a quasielement.
Examples
A quantifunctional if
Y ⊆ R .
A quantioperator if any general image is a unique quasielement.
A point quantimapping if
g(x0) = h(x0)
implies
f(g(x0)) = f(h(x0)).
A neighborhood quantimapping if
g(x) ≡ h(x)
in a quantineighborhood of x0 implies
f(g(x0)) = f(h(x0)).
A total quantimapping if it is not a neighborhood quantimapping.
An n-monotone quantimapping if X can be subdivided into not more than n quantisubsets where f is monotone.
A quantihomogeneous power combination
∑j∈J cj Πk∈K(j) xkα(k)
over C
where
cj ∈ C ,
α(k) ≥ 0,
∑k∈K(j) α(k)
is independent of j .
A power quanticombination over C as a quantisum of homogeneous power quanticombinations over C .
Notation
y = f(j∈J xj) ≡ f(xj | j ∈ J).
Notation It is proposed to use omitting a summation sign for doubled indices by Albert Einstein [1961] in tensor calculus [Encyclopaedia of Mathematics 1988] if and only if they are absent in one of the sides of equality.
3.1.3. Basic Theory of General Algebraic Additivity and Multiplicativity
Known finite or countable additivity and multiplicativity in classical mathematics are not algebraic and general enough, concern addition and multiplication only, and ignore unification and Cartesian multiplication.
Definition A quantimapping is called generally sum-to-sum additive if the general image of a quantisum of quantipreimages identically equals the quantisum of the general images of those quantipreimages.
Definition A quantimapping is called generally algebraically sum-to-sum additive if the general image of an algebraic quantisum (with a certain distribution of the plus and minus signs) of quantipreimages identically equals the corresponding algebraic quantisum (with the same distribution of the plus and minus signs) of the general images of those quantipreimages.
Definition A quantimapping is called generally union-to-sum additive if the general image of a quantiunion of quantipreimages identically equals the quantisum of the general images of those quantipreimages.
Definition A quantimapping is called generally algebraically union-to-sum additive if the general image of an algebraic quantiunion (with a certain distribution of the unification and subtraction signs) of quantipreimages identically equals the corresponding algebraic quantisum (with the corresponding distribution of the plus and minus signs) of the general images of those quantipreimages.
Definition A quantimapping is called generally sum-to-union additive if the general image of a quantisum of quantipreimages identically equals the quantiunion of the general images of those quantipreimages.
Definition A quantimapping is called generally algebraically sum-to-union additive if the general image of an algebraic quantisum (with a certain distribution of the plus and minus signs) of quantipreimages identically equals the corresponding algebraic quantiunion (with the corresponding distribution of the unification and subtraction signs) of the general images of those quantipreimages.
Definition A quantimapping is called generally union-to-union additive if the general image of a quantiunion of quantipreimages identically equals the quantiunion of the general images of those quantipreimages.
Definition A quantimapping is called generally algebraically union-to-union additive if the general image of an algebraic quantiunion (with a certain distribution of the unification and subtraction signs) of quantipreimages identically equals the corresponding algebraic quantiunion (with the same distribution of the unification and subtraction signs) of the general images of those quantipreimages.
Definition A quantimapping is called generally product-to-product multiplicative if the general image of a quantiproduct of quantipreimages identically equals the quantiproduct of the general images of those quantipreimages.
Definition A quantimapping is called algebraically product-to-product quantimultiplicative if the general image of an algebraic quantiproduct (with a certain distribution of the multiplication and division signs) of quantipreimages identically equals the corresponding algebraic quantiproduct (with the same distribution of the multiplication and division signs) of the general images of those quantipreimages.
Definition A quantimapping is called product-to-Cartesian-product quantimultiplicative if the general image of a quantiproduct of quantipreimages identically equals the Cartesian quantiproduct of the general images of those quantipreimages.
Definition A quantimapping is called Cartesian-product-to-product quantimultiplicative if the general image of a Cartesian quantiproduct of quantipreimages identically equals the quantiproduct of the general images of those quantipreimages.
Definition A quantimapping is called Cartesian-product-to-Cartesian-product quantimultiplicative if the general image of a Cartesian quantiproduct of quantipreimages identically equals the Cartesian quantiproduct of the general images of those quantipreimages.
Example A uniquantity mapping
Q : A → Q(A)
is generally algebraically union-to-sum additive and generally Cartesian-product-to-product multiplicative.
3.1.4. Basic Theory of Quantilinearity
Known linearity in classical mathematics (Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], G. A. Korn, T. M. Korn [1968], I. N. Bronstein, K. A. Semendjajew [1989], [Encyclopaedia of Mathematics 1988]) is only finite and homogeneous (a linear function
f(x) ≡ ax + b
is not a linear operator if
b ≠ 0)
over usual addition and multiplication but not over unification and Cartesian multiplication.
Definition A quantimapping is called addition-multiplication quantilinear if the image of any ordinary linear addition-multiplication quanticombination of preimages is the corresponding ordinary linear addition-multiplication quanticombination of their images.
Definition A quantimapping is called homogeneously addition-multiplication quantilinear if the image of any ordinary homogeneous linear addition-multiplication quanticombination of preimages is the corresponding ordinary homogeneous linear addition-multiplication quanticombination of their images.
Definition A quantimapping is called generally unification-multiplication linear if the image of any ordinary linear unification-multiplication quanticombination of preimages is the corresponding ordinary linear unification-multiplication quanticombination of their images.
Definition A quantimapping is called generally homogeneously unification-multiplication linear if the image of any ordinary homogeneous linear unification-multiplication quanticombination of preimages is the corresponding ordinary homogeneous linear unification-multiplication quanticombination of their images.
Definition A quantimapping is called generally addition-Cartesian-multiplication linear if the image of any ordinary linear addition-Cartesian-multiplication quanticombination of preimages is the corresponding ordinary linear addition-Cartesian-multiplication quanticombination of their images.
Definition A quantimapping is called homogeneously addition-Cartesian-multiplication quantilinear if the image of any ordinary homogeneous linear addition-Cartesian-multiplication quanticombination of preimages is the corresponding ordinary homogeneous linear addition-Cartesian-multiplication quanticombination of their images.
Definition A quantimapping is called generally unification-Cartesian-multiplication linear if the image of any ordinary linear unification-Cartesian-multiplication quanticombination of preimages is the corresponding ordinary linear unification-Cartesian-multiplication quanticombination of their images.
Definition A quantimapping is called generally homogeneously unification-Cartesian-multiplication linear if the image of any ordinary homogeneous linear unification-Cartesian-multiplication quanticombination of preimages is the corresponding ordinary homogeneous linear unification-Cartesian-multiplication quanticombination of their images.
Remark Quantiaddition and quantimultiplication can be generally transformed into quantiunification and Cartesian quantimultiplication, and vice versa.
3.1.5. Basic Theory of Quantirelations
Usual relations in classical mathematics (Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], G. A. Korn, T. M. Korn [1968], I. N. Bronstein, K. A. Semendjajew [1989], [Encyclopaedia of Mathematics 1988]) bind indeterminate usual sets of usual objects.
Definition A quantirelation is a quantisubset of an algebraic Cartesian quantiproduct of quantisets.
Remark A quantirelation can be invariant one-valued (e.g.
2 > 1),
variable many-valued (e.g.
x >=< y),
impulse (games in a championships),
generally inexact (inexactly known),
fuzzy, etc.
3.1.6. Basic Theory of Quantisuccessions
Usual nonstrict
≥ , ≤
and strict
> , <
"order" relations in classical mathematics (Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], G. A. Korn, T. M. Korn [1968], I. N. Bronstein, K. A. Semendjajew [1989], [Encyclopaedia of Mathematics 1988]) fit usual real numbers and functions only.
Definition A quantisuccession is a reflexive, skew-symmetric, and transitive quantirelation.
Definition A strict quantisuccession is a (skew-)reflexive, skew-symmetric, and transitive quantirelation.
Definition The {-1, 1} truncation of a unireal quasielement a is quasielement 'a'
where
'a' := -1
if
a < -1,
'a' := a
if
-1 ≤ a ≤ 1,
and
'a' := 1
if
a > 1.
Definition The left strict quantisuccession between quantielements
αa <L βb
holds if and only if
1) α < 0 ≤ β , or
2) α ≤ 0 < β , or
3) else when
3.1) 'α'a < 'β'b , or
3.2) 'α'a = 'β'b and
3.2.1) a < b and α ≥ β > 0, or
3.2.2) a > b and α ≤ β < 0, or
3.2.3) a = b and α < β .
Definition The right strict quantisuccession between quantielements
αa <R βb
holds if and only if
1) α < 0 ≤ β, or
2) α ≤ 0 < β, or
3) else when
3.1) a'α' < b'β', or
3.2) a'α' = b'β' and
3.2.1) a < b and α ≥ β > 0, or
3.2.2) a > b and α ≤ β < 0, or
3.2.3) a = b and α < β .
Definition The (two-sided) strict quantisuccession between quantielements
αa < βb
holds if and only if both
αa <L βb
and
αa <R βb .
Examples
-2 < 0 < 1,
102 < 2/33,
e/π-π < -2,
-eπ < # < e-π ,
-π10 < -e/ππ2,
0.51 < 0.15,
-0.51 > -0.15,
0.5-1 > 0.1-5,
-0.5-1 < -0.1-5,
21 < 31.
Definition A quantielement αa is called generally left-positive if and only if
αa >L 0.
Definition A quantielement αa is called generally right-positive if and only if
αa >R 0.
Definition A quantielement αa is called generally left-nonnegative if and only if
αa ≥L 0.
Definition A quantielement αa is called generally right-nonnegative if and only if
αa ≥R 0.
Definition A quantielement αa is called generally left-nonpositive if and only if
αa ≤L 0.
Definition A quantielement αa is called generally right-nonpositive if and only if
αa ≤R 0.
Definition A quantielement αa is called generally left-negative if and only if
αa <L 0.
Definition A quantielement αa is called generally right-negative if and only if
αa <R 0.
Definition A quantielement αa is called generally (two-sided) positive if and only if both
αa >L 0
and
αa >R 0.
Definition A quantielement αa is called generally (two-sided) nonnegative if and only if both
αa ≥L 0
and
αa ≥R 0.
Definition A quantielement αa is called generally (two-sided) nonpositive if and only if both
αa ≤L 0
and
αa ≤R 0.
Definition A quantielement αa is called generally (two-sided) negative if and only if both
αa <L 0
and
αa <R 0.
Definition A quantiset A is called generally left-positive, left-nonnegative, left-nonpositive, or left-negative if its left-equivalent quantielement is such.
Definition A quantiset A is called generally right-positive, right-nonnegative, right-nonpositive, or right-negative if its right-equivalent quantielement is such.
Definition A quantiset A is called generally (two-sided) positive, (two-sided) nonnegative, (two-sided) nonpositive, or (two-sided) negative if its (two-sided) equivalent quantielement is such.
Definition The left, right, or two-sided strict or nonstrict quantisuccession between quantisets coincides with that between their equivalent quantielements.
Remark If
0 ≤ A ≤ B
and
0 ≤ D ≤ F ,
then
0 ≤ AD ≤ BF .
If
0 < A < B
and
0 < D < F ,
then
0 < AD < BF .
If
A ≤ B
and
D ≥ 0,
then
AD ≤ BD .
If
A < B
and
D > 0,
then
AD < BD .
If
A ≤ B
and
D ≤ 0,
then
AD ≥ BD .
If
A < B
and
D < 0,
then
AD > BD .
Definition A general position quantisuccession is the linear quantiapportionment of some positional quantiset.
3.1.7. Basic Theory of Successible Quantisets
The true positions even of usual objects in 'ordered' usual sets (Georg Cantor [1932], Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], G. A. Korn, T. M. Korn [1968], I. N. Bronstein, K. A. Semendjajew [1989], [Encyclopaedia of Mathematics 1988]) are usually ignored in classical mathematics.
Definition A successive quantiset, or a quantisequence, is a positional quantiset whose quantiapportionment is an admissible quantisuccession.
Examples
N(+)´, Z´, Q´, R´
for relation '<';
((-1)n n | n ∈ N´)
for relation '<' between moduli,
or for relation '<' between latent (implicit) indices n .
Definition A successible quantiset is a partially positional quantiset which can be generally successed.
Example
(π , -1) + {2, e}
can be generally successed by the relation '< ':
(-1, 2, e , π).
Definition A quantimapping
f : X — X
is (strictly) regressive if for any x ∈ X ,
f(x) < x .
Definition A quantimapping
f : X — X
is (strictly) progressive if for any x ∈ X ,
f(x) > x .
Definition A quantimapping
f : X — X
is nonprogressive if for any x ∈ X ,
f(x) ≤ x .
Definition A quantimapping
f: X — X
is nonregressive if for any x ∈ X ,
f(x) ≥ x .
3.1.8. Basic Theory of Quantiorders
Only linear orders (usual successions) are considered in classical mathematics (Georg Cantor [1932], Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], G. A. Korn, T. M. Korn [1968], I. N. Bronstein, K. A. Semendjajew [1989], [Encyclopaedia of Mathematics 1988]).
Definition A quantiorder is the quantiapportionment of a positional quantiset.
Examples
The quantiorder of all the points in a solid;
the quantiorder of a matrix or any its Q-dimensional discrete or continual spatial generalization where Q is any uniquantity.
3.1.9. Basic Theory of Orderable Sets
Definition An ordered quantiset is a positional quantiset whose quantiapportionment is an admissible (in a given consideration) quantiorder.
Definition An orderable quantiset is a partially positional quantiset that can be generally ordered.
3.1.10. Basic Theory of Quantisystems
Known algebraic systems and structures in classical mathematics (Georg Cantor [1932], Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], G. A. Korn, T. M. Korn [1968], I. N. Bronstein, K. A. Semendjajew [1989], [Encyclopaedia of Mathematics 1988]) are not general enough.
Definition An isolated quantisystem is a quantiunion of some general objects along with some quantirelations between them and no other general objects.
Definition A nonisolated quantisystem is a quantiunion of some general objects along with some quantirelations between them and some other general objects.
Remark It is the most general possibly structurized object.
Examples
A general object, set, mapping, and order;
a consideration;
a solid;
a cybernetic system;
a plant;
an animal;
nature;
the universe;
a human;
society;
thought;
science;
a general consideration.
Definition A generative quantiset of a quantisystem is the quantiset of all its quasielements in a given general consideration.
Remark A quantirelation (of belonging, etc.) can differ in them.
Definition A quantisubsystem of a quantisystem is a quantisubset of its generative quantiset along with all its quantirelations with participating some quasielements of that subset.
3.1.11. Basic Theories of Quantistructures
Known structures in classical mathematics (Georg Cantor [1932], Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], G. A. Korn, T. M. Korn [1968], I. N. Bronstein, K. A. Semendjajew [1989], [Encyclopaedia of Mathematics 1988]) are systems not sufficiently general.
Definition The quantistructure of a quantisystem is the quantisystem of all the quantirelations in a given quantisystem.
Examples A quantisuccession, an order, an apportionment, a relation, as well as a mapping.
Definition Quantisystems are (generally) equistructured if their quantistructures coincide.
3.1.12. Basic Theory of Quanticorrespondences
Even quantirelations and mappings are based on nonstructured sets.
Definition A quanticorrespondence between its (domain and range) quantisystems is a quantisubset of a Cartesian quantiproduct of quantisubsystems in the given quantisystems and has its generative quantimapping/quantirelation between the generative quanti(sub)sets of those quanti(sub)systems.
Examples
f(]α , β][a , b] , N´Q`, Ωω) = (R`Z , {a , β}°{b , α}°, NN);
the generative quantimapping
f : X — Y
of a quanticorrespondence
f :
X — Y .
3.1.13. Basic Theory of Quantisystem Operations
Even quantioperations are applicable to nonstructured quantisets only.
Definition A quantisystem operation is a quanticorrespondence between equiquantistructured quantisystems in which the corresponding quantioperation is realized separately at all their equiquantistructural quasielements.
Example
{a , b , (c , d)}° ⊕ {a' , b' , (c' , d')}° =
{a + a', b + b', (c + c', d + d')}°.
3.1.14. Basic Theory of Quantirelation Systems
Definition A quantirelation system is the quantisystem of the quantirelations between all the quantisubsystems in the given quantisystems.
3.1.15. Basic Theory of Linear Quantisystems
Known linear systems in classical mathematics (Georg Cantor [1932], Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], G. A. Korn, T. M. Korn [1968], I. N. Bronstein, K. A. Semendjajew [1989], [Encyclopaedia of Mathematics 1988]) are not sufficiently general.
Definition A quantisystem is generally linear in a given general consideration if all its general objects (quantimappings, quantiorders, etc.) are generally linear in some reasonable sense.
Examples
A (homogeneous) linear quanticombination;
a uniquantity.
3.1.16. Basic Theory of Uninumber Systems
Separate (even general) numbers cannot adequately express many urgent quantisystems.
Definition A quantisystem is a uninumber quantisystem if all its quasielements are uninumbers and can be quantitransformed, quantisimplified, and quantiestimated.
Examples
A complex uninumber, vector, matrix, and set;
an order-type, an ordinal number;
a system of relations after substituting number values;
a general process or sequence with uninumber elements or values.
Chapter 3.2. Theories of General States and Processes
3.2.1. Basic Theory of Quantitimes
Known concepts of time in classical mathematics [Encyclopaedia of Mathematics 1988] are not general enough.
Definition A quantitime is a general object with taking into account its general values in a conventionally uniform quantisuccession.
Examples
N(+)´, Z`, R´, R`;
physical time.
Definition A quantitime system is a quantisystem of some quantitimes.
Remark A general process is a general image of a quantitime system.
3.2.2. Basic Theory of Potential Quanti-Infinities
The indeterminate known concept of potential infinities in classical mathematics (Georg Cantor [1932], Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], G. A. Korn, T. M. Korn [1968], I. N. Bronstein, K. A. Semendjajew [1989], [Encyclopaedia of Mathematics 1988])
1) ignores the factual degree of divergence: for
∑n∈N 1/n = '+∞'
and the computer pseudozero
0.5 × 10-19 ,
the final result would be about 45 after conditional computing during 600 centuries (I. I. Blekhman, A. D. Myshkis, and Ya. G. Panovko [1990]) by calculating on the computers of that time (about 1990);
2) unifies diverse potential infinities by common symbols
'∞', '-∞', '+∞';
3) gives only very rough estimations by relative orders;
4) considers only few usual operations without conservation laws.
Definition A potential quanti-infinity is a successive set of general objects whose quantinorms are finite but, beginning with some of them, are greater than any preassigned finite uninumber.
Definition The canonical continual potential quanti-infinity is
Tc = ({0} + R+)´;
the canonical discrete potential quanti-infinity
Td = N´.
The quantitimes
tc ∈ Tc
and
td ∈ Td
are considered to have ω as a limit.
Examples
e2t - t3 + 2 → e2ω - ω3 + 2
(t → ω);
x ∈ (0, 2, 4, ...),
x → 2ω - 2;
x ∈ (1, 3, 5, ...),
x → 2ω - 1.
Remark Such a rarefaction [Encyclopaedia of Mathematics 1988] can increase potential infinities and simultaneously decrease actual infinities.
3.2.3. Basic Theory of General Infinities
Actual and potential infinities and quanti-infinities can be also joint (combined).
Definition A general infinity is a quantisystem of general objects without any finite common upper bound of their quantinorms.
Example
(1, c-1, ω-1, ω , c , c-2, ω-2, ω2, c2, ...).
Remark All the quantioperations, quantimappings, and quantirelations are applicable to the general infinity with satisfying all their general properties (the conservation law etc.).
Example
(0, -1, 2, -3, ... ) → (-1)ω-1 (ω - 1).
3.2.4. Basic Theory of Quantistates and Quantistate Processes
Definition The quantistate of a general object is its quanti-image in a quantimapping.
Definition A quantistate parameter, or coordinate, of a general object is a quasielement of its (multiparameter) quantistate.
Definition The quantistate process of a general object is a quantimapping from a quantitime system into a quantistate system.
Definition A quantistate parameter is equivalent to a multiparameter quantistate if the corresponding uniparameter quantistate and the given quantistate are indistinguishable by the quantistructure of a given quantisystem.
Definition A quantistate parameter process is equivalent to a multiparameter quantistate process if the corresponding uniparameter quantistate process and the given quantistate process are indistinguishable by the quantistructure of a given quantisystem.
3.2.5. Basic Theory of Subcritical States and Processes
Definition A quantistate of a quantisystem is subcritical if for any sufficiently small variations of the quantistate it conserves the quantistructure of that quantisystem.
Definition A quantistate process of a quantisystem is subcritical if for any sufficiently small variations of the quantistate process it conserves the quantistructure of that quantisystem.
3.2.6. Basic Theory of Supercritical States and Processes
Definition A quantistate of a quantisystem is supercritical if for any sufficiently small variations of the quantistate it changes the quantistructure of that quantisystem.
Definition A quantistate process of a quantisystem is supercritical if for any sufficiently small variations of the quantistate process it changes the quantistructure of that quantisystem.
Remark Such a state (or a state process) is possible in flexible overmathematics and other fundamental mathematical and strength sciences unlike classical mathematics, strength of materials, etc.
3.2.7. Basic Theory of Critical States and Processes
Definition A quantistate of a quantisystem is critical if some sufficiently small variations of the quantistate conserve and some other ones change the quantistructure of that quantisystem.
Definition A quantistate process of a quantisystem is critical if some sufficiently small variations of the quantistate process conserve and some other ones change the quantistructure of that quantisystem.
3.2.8. Basic Theory of Parameter Reductions
Definition A local reduction of a quantiparameter p is a point quantimapping of its general values.
Definition A neighborhood reduction of a quantiparameter p is a neighborhood quantimapping of its general values.
Definition A total reduction of a quantiparameter p is a total quantimapping of its general values.
Definition A local critical reduction of p is its local reduction with respect to its critical general values corresponding to its general values.
Example For the general case of an anisotropic nonequiresistant solid nonstationarily loaded, the synchronous reduction (with respect to a possibly displaced origin) of any (unregulated) principal stress by Lev Gelimson [1993a, 1993b, 1994a, 2001f, 2001i, 2003c, 2004a, 2005b]
σj(u)(t)
at a given point of that solid at the instant t of time gives
σ°j(u)(t) = (σj(u)(t) - σm0j(u)(t)) / |σLj(u)(t) - σm0j(u)(t)|
where
σLj(u)(t)
is the limiting value of the uniaxial stress
σj(u)(t)
of the same direction and sign,
σm0j(u)(t)
is the mean cycle stress in the maximum-amplitude corresponding uniaxial stress cycle.
Definition The total critical reduction of p is a total reduction of p locally critically reduced by a (local) equivalent reduced general value.
Example The vectorial reduced parameter by Lev Gelimson [1993a, 1993b, 1994a, 2001f, 2001i, 2003c, 2004a, 2005b]
σ°j(u) = (σmj(u) , σaj(u))
including the quasiformer mean stress σmj(u) and the equidangerous-cycle amplitude reduced stress σaj(u) .
3.2.9. Basic Theory of Quasicritical Quantirelations
Definition A quasicritical quantirelation is a quantirelation explicitly distinguishing subcritical, critical, and supercritical states and processes.
Example
σ°e = max{supt∈Tmaxju(t)F[j=13σ°ju(t)], maxju|F[j=13σ°ju]|} </=/> 1
by Lev Gelimson [1993a, 1993b, 1994a, 2001f, 2001i, 2003c, 2004a, 2005b]
where
σ°e is an equivalent reduced stress,
σe = F[j=13σj] = σL
is an initial critical state criterion in usual principal stresses σj (j = 1, 2, 3) and usual limiting stress σL .
Remark Critical relations can be based on the postulates by Lev Gelimson [1993a, 1993b, 1994a, 2001f, 2001i, 2003c, 2004a, 2005b] for which either the equivalent critically reduced quantiparameter in a critical multiparameter state or state process is an ordinary linear quanticombination (or quantimapping) of its reduced quantiparameters or such quantimappings satisfy a quantirelation.
Remark A quasicritical quantirelation can be based on generally inexact and/or fuzzy estimations.
Discovering New Phenomena in General Systems
1) Continuously varying a general system can lead to spasmodically varying its general structure.
2) A critical relation [4] in a general system can exist and bifurcate.
3) A critical relation in a general system can be invariant when a general system varies.
4) There can exist total critical relations forming the general boundary subsystem of the general system of critical relations.
5) The skips of the general structure of a general system can be both successed and reversive.
6) There can exist the determining general parameter and the equivalent general parameter for a general system.
7) A rational control by the determining general parameter of a general system can raise its equivalent general parameter by an order.
8) A coincidence of a rational control and of a critical relation can depend on that relation.
9) An equivalent general parameter can be uniform if a determining general parameter is (symmetrically or asymmetrically) nonuniform.
10) A general state can be initial out of the general center of a general system.
11) A general state of a general center of a general system is initial for some critical relation.
12) The general measure of a changed-structured general subsystem of a general system can be invariant.
13) The total general measure of all the changed-structured general subsystems with variable general measures can be invariant.
14) The well-posedness and the ill-posedness [1] of a general system are relative.
Basic Results and Conclusions
1. Fundamental mega-overmathematics as a universal language of megascience is advanced on the base of the proposed fundamental principles of creative philosophy and mega-overmathematics.
2. Fundamental mega-overmathematics is based on uniarithmetics, quantialgebra, and quantianalysis revolutionarily replacing the inadequate very fundamentals of classical fundamental mathematics.
3. Uninumbers, quantielements, quantisets, and quantisystems satisfying the general conservation law can adequately express many urgent general objects in nature, engineering, society, and thinking with completing the universal scale for perfectly expressing arbitrary uniquantities.
4. These fundamental concepts provide principally new possibilities for modeling, evaluating, measuring, and estimating general objects and systems, as well as for solving very many urgent scientific and life problems.
The System of Revolutions in Advanced Mathematics
The system of revolutions in advanced mathematics includes:
the uniobject subsystem of revolutions, in particular, universalizing unification, uniobjects, unielements, unisets, mereological uniaggregates (unicontents), and unisystems as further generalizations of general (not logical) quantification, quantiobjects, quantielements, quantisets, mereological quantiaggregates (quanticontents), and quantisystems, respectively;
the overoperation subsystem of revolutions, in particular, exponentiation overefficiency, composite (combined) commutative overoperations, root-logarithmic overfunctions, and self-root-logarithmic overfunctions;
the unisystem subsystem of revolutions, in particular, unipositional unisets, unimappings, unisuccessions, unisuccessible unisets, uniorders, uniorderable unisets, unistructures, unicorrespondences, unisystems, and unirelation unisystems as further generalizations of quantipositional quantisets, quantimappings, quantisuccessions, quantisuccessible quantisets, quantiorders, quantiorderable quantisets, quantistructures, quanticorrespondences, quantisystems, and quantirelation quantisystems, respectively;
the uniprocess subsystem of revolutions, in particular, unitimes, potential uninfinities, actual uninfinities, subcritical, critical, supercritical, underlimiting, limiting, and overlimiting unistates and uniprocesses, as well as generally noncritical and generally nonlimiting unirelations as further generalizations of quantitimes, potential quantiinfinities, actual quantiinfinities, subcritical, critical, supercritical, underlimiting, limiting, and overlimiting quantistates and quantiprocesses, as well as generally noncritical and generally nonlimiting quantirelations, respectively;
the unimeasuring and uniestimating subsystem of revolutions, in particular, 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 as further generalizations of quantidestructurizators, quantidiscriminators, quanticontrollers, quantimeaners, quantimean quantisystems, quantibounders, quantibound quantisystems, quantitruncators, quantilevelers, quantilevel quantisystems, quantilimiters, quantiseries quantiestimators, quantimeasurers, quantimeasure quantisystems, quantiintegrators, quantiintegral quantisystems, quantiprobabilers, quantiprobability quantisystems, and quanticentral quantiestimators, respectively;
the unicomparison subsystem of revolutions, in particular:
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);
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);
the subsystem of discovered new possible not obligatorily universal phenomena in unisystems (including quantisistems) as qualitative leaps of the principal novelty, in particular:
1) natural self-restraint, in particular, self-exclusion of excessive useless generalization;
2) the multiple overdeterminacy of the whole type of uniproblems as unisystems with unknown unisubsystems;
3) the existence of a critical relationship in the unistructure of a unisystem;
4) splitting (bifurcating) a critical relationship in the unistructure of a unisystem;
5) the invariance of a critical relationship in the unistructure of a variable unisystem;
6) the existence of principal and/or limiting critical relationships in the unistructure of a unisystem;
7) the possibility of abruptly changing the unistructure of a unisystem by continuously changing the unisystem itself;
8) unidirectional subsequent jumps in the unistructure of a unisystem;
9) multidirectional subsequent jumps in the unistructure of a unisystem;
10) reversal subsequent jumps in the unistructure of a unisystem;
11) the existence of the equivalent uniparameter of a unisystem;
12) the existence of the determining uniparameter of a unisystem;
13) multiply increasing the equivalent uniparameter of a unisystem via reasonably controlling its determining uniparameter;
14) the dependence of the coincidence of reasonably controlling a unisystem and of its critical relation from the choice of this relation;
15) the uniformity of the equivalent uniparameter of a unisystem by the symmetric nonuniformity of the determining uniparameter of the unisystem;
16) the uniformity of the equivalent uniparameter of a unisystem by the asymmetric nonuniformity of the determining uniparameter of the unisystem;
17) the initial eccentricity of a unisystem;
18) the unisystem origin centrality by the unisystem criticality;
19) the invariance of the unimeasure of a unistructurably variable unisubsystem of a unisystem;
20) the invariance of the total unimeasure of a set of unistructurably variable unisubsystems of a unisystem;
21) the relative correctness and incorrectness of setting a uniproblem as a unisystem with unknown unisubsystems.
References
[Abachiev 2012] Sergey Abachiev. Concepts of Modern Natural Sciences. Phoenix, Rostov-on-Don, 2012
[Aigner 1979] Martin Aigner. Combinatorial theory. Springer-Verlag, Berlin, Heidelberg, New York, 1979
[Archimedes 1912] Archimedes. The Method of Mechanical Theorems. The method of Archimedes recently discovered by Heiberg; a supplement to the Works of Archimedes. Translated by Thomas Little Heath. Cambridge University Press, 1912
[Bell 1998] John Lane Bell. A Primer of Infinitesimal Analysis. Cambridge University Press, 1998
[Bell 2009] John Lane Bell. Continuity and Infinitesimals. Stanford Encyclopedia of Philosophy, 2009
[Bergman 1957] George Bergman. A number system with an irrational base. Mathematical Magazine, vol. 31, pp. 98-119, 1957
[Blekhman Myshkis Panovko 1990] I. I. Blekhman, A. D. Myshkis, and Ya. G. Panovko. Mechanics and Applied Mathematics [In Russian]. Nauka, Moscow, 1990
[Blizard 1991] Wayne D. Blizard. The development of multiset theory. Modern Logic 1 (1991), no. 4, 319-352
[Bolzano 1851] Bernard Bolzano. Paradoxien des Unendlichen. Bei C. H. Reclam Sen., Leipzig, 1851
[Borisenko 1991] Alexey Borisenko. Methods of Synthezing Information Systems on the Base of Positional Numbers with Inhomogeneous Structure [In Russian]. Dr. Sc. Dissertation, Kharkov, 1991
[Borisenko 2002] Alexey Borisenko. Lectures on Discrete Mathematics: Sets and Logic [In Russian]. Sumy State University Publishers, Sumy, 1998. The same in Ukrainian. University Book Publishers, Sumy, 2002
[Borisenko 2004a] Alexey Borisenko. Introduction to Binomial Enumeration Theory [In Russian]. University Book Publishers, Sumy, 2004
[Borisenko 2004b] Alexey Borisenko. Binomial Enumeration: Theory and Practice [In Russian]. University Book Publishers, Sumy, 2004
[Borisenko 2009] Alexey Borisenko. Numeration Systems in Computer Technology [In Russian]. Trinity Academy Publishers, 2009
[Bourbaki 1949] Nicolas Bourbaki. Elements de mathematique. Hermann, Paris, 1949 etc.
[Bronstein Semendjajew 1989] I. N. Bronstein, K. A. Semendjajew. Taschenbuch der Mathematik. Frankfurt/M., 1989
[Burgin 2002] Mark Burgin. Theory of Hypernumbers and Extrafunctions: Functional Spaces and Differentiation. Discrete Dynamics in Nature and Society, 2002, Vol. 7 (3), pp. 201-212
[Burgin 2004] Mark Burgin. Unified Foundations for Mathematics. Logic, 2004/03/10, 1-39
[Burgin 2005] Mark Burgin. Hypermeasures in General Spaces. International Journal of Pure and Applied Mathematics, Volume 24, No. 3, 2005, 299-322
[Burgin 2012] Mark Burgin. Hypernumbers and Extrafunctions. Extending the Classical Calculus. Springer New York, 2012
[Cantor 1932] Georg Cantor. Gesammelte Abhandlungen mathematischen und philosophischen Inhalts. Springer-Verlag, Berlin, 1932
[Cauchy 1882] Augustin Louis Cauchy. Oeuvres completes. Gauthier-Villars, 1882
[Conway 1976] John Horton Conway. On Numbers and Games. Academic Press, London, 1976
[Czajko 2004a] Jakub Czajko. On Cantorian spacetime over number systems with division by zero. Chaos, Solitons and Fractals, 21 (2004), 261-271
[Czajko 2004b] Jakub Czajko. Cantor and generalized continuum hypotheses may be false. Chaos, Solitons and Fractals, 21 (2004), 501-512
[Dales Woodin 1996] H. Garth Dales, William Hugh Woodin. Super-real fields. London Mathematical Society Monographs. New Series, 14, The Clarendon Press, Oxford University Press, 1996
[Davis 1977] Martin Davis. Applied Nonstandard Analysis. Wiley, New York, London, Sydney, Toronto, 1977
[Dedekind 1930] Richard Dedekind. Gesammelte mathematische Werke. Herausgegeben von Robert Fricke, Emmy Noether, Öystein Ore. Braunschweig, Friedr. Vieweg & Sohn, 1930-1932
[Dubois Prade 1980] Didier Dubois, Henri Prade. Fuzzy Sets and Systems: Theory and Applications. New York, 1980
[Dubois Prade 1990] Didier Dubois, Henri Prade. Rough fuzzy sets and fuzzy rough sets. International Journal of General Systems 17 (2-3), 1990, 191-209
[Ehrlich 1994] Real Numbers, Generalizations of the Reals, and Theories of Continua (ed. by Ph. Ehrlich). Kluwer Academic Publisher Group, Dordrecht, 1994
[Ehrlich 2006] Philip Ehrlich. The rise of non-Archimedean mathematics and the roots of a misconception. I. The emergence of non-Archimedean systems of magnitudes. Archive for History of Exact Sciences 60 (2006), no. 1, 1-121
[Ehrlich 2012] Philip Ehrlich. The absolute arithmetic continuum and the unification of all numbers great and small. Bull. Symbolic Logic, Volume 18, Issue 1 (2012), 1-45
[Einstein 1961] Albert Einstein. Relativity: The Special and the General Theory. Crown Publishers, N.Y., 1961
[El Naschie 2009] Mohammed El Naschie. The theory of Cantorian spacetime and high energy particle physics (an informal review). Chaos, Solitons and Fractals, 41 (2009), 2635-2646
[Encyclopaedia of Mathematics 1988] Encyclopaedia of Mathematics / Managing editor M. Hazewinkel. Volumes 1 to 10. Kluwer Academic Publ., Dordrecht, 1988-1994
[Encyclopaedia of Physics 1973] Encyclopaedia of Physics / Chief Editor S. Flugge. Springer, Berlin etc., 1973 etc.
[Euclid 1482] Euclid. The Thirteen Books of the Elements. Erhard Ratdolt, Venice, 1482
[Euler 1748] Leonhard Euler. Introductio in Analysis infitorum, 1748
[Feferman 1964] Solomon Feferman. The Number Systems. Foundations of Algebra and Analysis. Addison-Wesley, Reading, 1964
[Fontenelle 1727] Bernard le Bovier de Fontenelle. Elements de la geometrie de l'infini. L'Imprimerie Royal, Paris, 1727
[Gandzha Kadeisvily 2011] Ivan Gandzha, Jerdsey Kadeisvily. New Sciences for a New Era: Mathematical, Physical and Chemical Discoveries of Ruggero Maria Santilli. Sankata Printing Press, Nepal, 2011
[Gauss 1863] Carl Friedrich Gauss. Werke. Herausgegeben von der Königlichen Gesellschaft der Wissenschaften zu Göttingen. Band 1 bis 6. Dieterich, Göttingen, 1863-1874. Band 7 bis 12. B. G. Teubner, Leipzig, 1900–1917; Julius Springer, Berlin, 1922–1933
[Gelimson 1987] 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
[Gelimson 1992] Lev Gelimson. Generalization of Analytic Methods for Solving Strength Problems [In Russian]. Drukar Publishers, Sumy, 1992
[Gelimson 1993a] Lev Gelimson. General Strength Theory. Drukar Publishers, Sumy, 1993
[Gelimson 1993b] Lev Gelimson. Generalization Method for Limiting Criteria [In Russian]. International Scientific and Technical Conference "Glass Technology and Quality". Mathematical Methods in Applying Glass Materials (1993), Theses of Reports, p. 98-100
[Gelimson 1993c] Lev Gelimson. Linear Correction Method for Limiting Criteria [In Russian]. International Scientific and Technical Conference "Glass Technology and Quality". Mathematical Methods in Applying Glass Materials (1993), Theses of Reports, p. 100-101
[Gelimson 1993d] 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
[Gelimson 1993e] Lev Gelimson. Analytic Macroelement Method in Axially Symmetric Elasticity [In Russian]. International Scientific and Technical Conference “Glass Technology and Quality”. Mathematical Methods in Applying Glass Materials (1993), Theses of Reports, p. 104-106
[Gelimson 1993f] 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
[Gelimson 1993g] Lev Gelimson. Generalization of Analytic Methods for Solving Strength Problems for Typical Structure Elements in High-Pressure Engineering [In Russian]. Dr. Sc. dissertation. Institute for Strength Problems, National Academy of Sciences of Ukraine, Kiev, 1994
[Gelimson 1994a] Lev Gelimson. The Generalized Structure for Critical State Criteria. Transactions of the Ukrainian Glass Institute 1 (1994), p. 204-209
[Gelimson 1994b] 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
[Gelimson 1994c] Lev Gelimson. General Estimation Theory. Transactions of the Ukrainian Glass Institute 1 (1994), p. 214-221 (both this article and a further mathematical monograph have been also translated from English into Japanese)
[Gelimson 1995a] Lev Gelimson. Basic New Mathematics. Drukar Publishers, Sumy, 1995
[Gelimson 1995b] Lev Gelimson. New mathematics as new scientific thinking language [in Russian]. International Scientific and Technical Conference "Energy and Resource Saving Technologies in Glass Production", General Problems, Theses of Reports, 1995, p. 67-68
[Gelimson 1995c] Lev Gelimson. General objects, operations, sets, and numbers [in Russian]. International Scientific and Technical Conference "Energy and Resource Saving Technologies in Glass Production", General Problems, Theses of Reports, 1995, p. 68-70
[Gelimson 1995d] Lev Gelimson. General systems, states, and processes [in Russian]. International Scientific and Technical Conference "Energy and Resource Saving Technologies in Glass Production", General Problems, Theses of Reports, 1995, p. 71-72
[Gelimson 1995e] 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
[Gelimson 1995f] Lev Gelimson. General problems and methods of solving them [in Russian]. International Scientific and Technical Conference "Energy and Resource Saving Technologies in Glass Production", General Problems, Theses of Reports, 1995, p. 74-76
[Gelimson 1995g] Lev Gelimson. New phenomena and general laws of nature and science [in Russian]. International Scientific and Technical Conference "Energy and Resource Saving Technologies in Glass Production", General Problems, Theses of Reports, 1995, p. 76-78
[Gelimson 1996] 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
[Gelimson 1997a] Lev Gelimson. Hyperanalisis: Hypernumbers, Hyperoperations, Hypersets and Hyperquantities. Collegium International Academy of Sciences Publishers, 1997
[Gelimson 1997b] Lev Gelimson. Mengen mit beliebiger Quantität von jedem Element. Collegium International Academy of Sciences Publishers, 1997
[Gelimson 2001a] Lev Gelimson. Elastic Mathematics: Theoretical Fundamentals. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2001
[Gelimson 2001b] Lev Gelimson. Elastic Mathematics: Principles, Theories, Methods, and Applications. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2001
[Gelimson 2001c] Lev Gelimson. General Estimation Theory. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2001
[Gelimson 2001d] Lev Gelimson. Hyperanalisis: Hypernumbers, Hyperoperations, Hypersets and Hyperquantities. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2001
[Gelimson 2001e] Lev Gelimson. Mengen mit beliebiger Quantität von jedem Element. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2001
[Gelimson 2001f] Lev Gelimson. General Strength Theory: Fundamentals and Applications. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2001
[Gelimson 2001g] Lev Gelimson. Objektorientierte Mathematik in der Messtechnik. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2001
[Gelimson 2001h] Lev Gelimson. Measurement Theory in Physical Mathematics. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2001. Also published by Vuara along with a number of references to Lev Gelimson's scientific works.
[Gelimson 2001i] Lev Gelimson. Yield and Fracture Laws of Nature (Universal Yield and Failure Criteria in the Relative Stresses). The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2001
[Gelimson 2002] Lev Gelimson. Applying the Analytic Macroelement Method and Fundamental Strength Sciences to Three-Dimensional Cylindrical Glass Elements of High-Pressure Illuminators (Deep-Sea Portholes). The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2002
[Gelimson 2003a] Lev Gelimson. Quantianalysis: Uninumbers, Quantioperations, Quantisets, and Multiquantities (now Uniquantities). Abhandlungen der WIGB (Wissenschaftlichen Gesellschaft zu Berlin), 3 (2003), Berlin, 15-21
[Gelimson 2003b] Lev Gelimson. General Problem Theory. Abhandlungen der WIGB (Wissenschaftlichen Gesellschaft zu Berlin), 3 (2003), Berlin, 26-32
[Gelimson 2003c] Lev Gelimson. General Strength Theory. Abhandlungen der WIGB (Wissenschaftlichen Gesellschaft zu Berlin), 3 (2003), Berlin, 56-62
[Gelimson 2003d] Lev Gelimson. General Analytic Methods. Abhandlungen der WIGB (Wissenschaftlichen Gesellschaft zu Berlin), 3 (2003), Berlin, 260-261
[Gelimson 2003e] Lev Gelimson. Quantisets Algebra. Abhandlungen der WIGB (Wissenschaftlichen Gesellschaft zu Berlin), 3 (2003), Berlin, 262-263
[Gelimson 2003f] Lev Gelimson. Elastic Mathematics. Abhandlungen der WIGB (Wissenschaftlichen Gesellschaft zu Berlin), 3 (2003), Berlin, 264-265
[Gelimson 2004a] Lev Gelimson. Elastic Mathematics. General Strength Theory. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2004, 496 pp.
[Gelimson 2004b] Lev Gelimson. General Problem Theory. The Second International Science Conference "Contemporary methods of coding in electronic systems", Sumy, 26-27 October 2004
[Gelimson 2004c] Lev Gelimson. Quantisets Algebra. The Second International Science Conference “Contemporary methods of coding in electronic systems”, Sumy, 26-27 October 2004
[Gelimson 2005a] 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
[Gelimson 2005b] Lev Gelimson. Providing Helicopter Fatigue Strength: Unit Loads [Fundamental Mechanical and Strength Sciences]. In: Structural Integrity of Advanced Aircraft and Life Extension for Current Fleets – Lessons Learned in 50 Years After the Comet Accidents, Proceedings of the 23rd ICAF Symposium, Dalle Donne, C. (Ed.), 2005, Hamburg, Vol. II, 589-600
[Gelimson 2006a] Lev Gelimson. Quantisets and Their Quantirelations. The Third International Science Conference "Contemporary methods of coding in electronic systems", Sumy, 24-25 October 2006
[Gelimson 2006b] Lev Gelimson. Quantiintervals and Semiquantiintervals. The Third International Science Conference "Contemporary methods of coding in electronic systems", Sumy, 24-25 October 2006
[Gelimson 2006c] Lev Gelimson. Multiquantities (now Uniquantities). The Third International Science Conference "Contemporary methods of coding in electronic systems", Sumy, 24-25 October 2006
[Gelimson 2006d] Lev Gelimson. Sets with Any Quantity of Each Element. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2006
[Gelimson 2009a] Lev Gelimson. Overmathematics: Fundamental Principles, Theories, Methods, and Laws of Science. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2009
[Gelimson 2009b] Lev Gelimson. Overmathematics: Principles, Theories, Methods, and Applications. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2009
[Gelimson 2010] Lev Gelimson. Uniarithmetics, Quantialgebra, and Quantianalysis: Uninumbers, Quantielements, Quantisets, and Uniquantities with Quantioperations and Quantirelations. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2010
[Gelimson 2011a] 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
[Gelimson 2011b] Lev Gelimson. Science Unimathematical Test Fundamental Metasciences Systems. Monograph. The “Collegium” All World Academy of Sciences, Munich (Germany), 2011
[Gelimson 2011c] Lev Gelimson. Overmathematics Essence. Mathematical Journal of the "Collegium" All World Academy of Sciences, Munich (Germany), 11 (2011), 25
[Gödel 1929] Kurt Gödel. Über die Vollständigkeit des Logikkalküls. Doctoral dissertation. University of Vienna, 1929
[Gödel 1930] Kurt Gödel. Die Vollständigkeit der Axiome des logischen Funktionenkalküls. Monatshefte für Mathematik, 37 (1), 1930, 349-360
[Goldblatt 1998] Robert Goldblatt. Lectures on the hyperreals. An introduction to nonstandard analysis. Graduate Texts in Mathematics, 188. Springer-Verlag, New York, 1998
[Hahn 1907] Hans Hahn. Über die nichtarchimedischen Größensysteme. Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Wien, Mathematisch-Naturwissenschaftliche Klasse, 116, 1907, 601-655
[Hausdorff 1919] Felix Hausdorff. Dimension und äußeres Maß. Mathematische Annalen, 79 (1919), 157-179
[Hausdorff 1935] Felix Hausdorff. Mengenlehre. Berlin, 1935
[Hilbert 1899] David Hilbert. Grundlagen der Geometrie. Teubner, Leipzig, 1899
[Hilbert 1932] David Hilbert. Gesammelte Abhandlungen. Julius Springer, Berlin, 1932-1935
[Hewitt 1948] Edwin Hewitt. Rings of real-valued continuous functions. I. Trans. Amer. Math. Soc. 64, 1948, 4599
[Hoyle 2007] Jonathan W. Hoyle. Infinitesimals in Modern Mathematics. Mathematical Association of America, Seaway Section Conference, Rochester, New York, October 20, 2007, Updated: October 30, 2007, Online Edition, http://www.jonhoyle.com/MAAseaway
[Hrbacek 2006] Karel Hrbacek. Nonstandard objects in set theory. In: Nonstandard Methods and Applications in Mathematics, ed. by NJ Cutland, M Di Nasso and D A Ross. Lecture Notes in Logic 25, Association for Symbolic Logic, 2006, 80-120
[Hrbacek 2009a] Karel Hrbacek. Relative set theory: Internal view. Journal of Logic & Analysis, 1:8 (2009), 1-108
[Hrbacek 2009b] Karel Hrbacek. Mathematics in relative set theory. Department of Mathematics. The City College of CUNY. New York, NY 10031. July 31, 2009
[Husserl 1901] Edmund Husserl. Logische Untersuchungen. Teil 2. Untersuchungen zur Phänomenologie und Theorie der Erkenntnis. Niemeyer, Halle a. S., 1901
[Hypernumber Blogspot] Hypernumber. Blogspot. Web Portal entry: http://hypernumber.blogspot.com/
[Hypernumbers Group] Hypernumbers Group. Web Resource entry: http://tech.groups.yahoo.com/group/hypernumber/
[Kaufmann Denis-Papin Faure 1964] Arnold Kaufmann, Maurice Denis-Papin, Robert Faure. Mathématiques nouvelles. – Dunod (impr. Jouve), Paris, 1964
[Keisler 1976a] Howard Jerome Keisler. Foundations of infinitesimal calculus. Prindle, Weber and Schmidt, Boston, 1976
[Keisler 1976b] Howard Jerome Keisler. Elementary Calculus: An Infinitesimal Approach. Prindle,Weber and Schmidt (First Edition 1976, Second Edition 1986)
[Klaua 1965] Dieter Klaua. Über einen Ansatz zur mehrwertigen Mengenlehre. Monatsber. Deutsch. Akad. Wiss. Berlin, 7, 859-867, 1965
[Klaua 1966a] Dieter Klaua. Grundbegriffe einer mehrwertigen Mengenlehre. Monatsber. Deutsch. Akad. Wiss. Berlin, 8, 782-802, 1966
[Klaua 1966b] Dieter Klaua. Über einen zweiten Ansatz zur mehrwertigen Mengenlehre. Monatsber. Deutsch. Akad. Wiss. Berlin, 8, 161-177, 1966
[Klaua 1967] Dieter Klaua. Ein Ansatz zur mehrwertigen Mengenlehre. Math. Nachr., 33, 273-296, 1967
[Knuth 1997] Donald Ervin Knuth. The Art of Computer Programming. Volumes 1-4. Addison-Wesley, 1997-2011
[Korn 1968] G. A. Korn, T. M. Korn. Mathematical Handbook for Scientists and Engineers. McGraw-Hill, N.Y. etc., 1968
[Lagrange 1811] Joseph-Louis Lagrange. Mecanique Analytique. 1811
[Leibniz 1684] Gottfried Wilhelm Leibniz. Nova methodus pro maximis et minimis, itemque tangentibus. quae ne fractas nec irrationales quantitates moratur, et singulare pro ilk calculi genus. Acta Eruditorum 3, 467-473
[Leibniz 1686] Gottfried Wilhelm Leibniz. De geometriae recondite et analysi indivisibilium atque infinitorum. Acta Eruditorum 5, 1686, 292-300
[Leibniz 1714] Gottfried Wilhelm Leibniz. La Monadologie (original title: Principes de la Nature et de la Grâce fondés en Raison – Monadologie). 1714
[Leibniz 1716] Gottfried Wilhelm 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
[Leibniz 1846] Gottfried Wilhelm Leibniz. Leibnizens gesammelte Werke. Hansch, Hannover, 1846
[Leśniewski 1916] Stanisław Leśniewski. Podstawy ogólnej teoryi mnogosci. I, Prace Polskiego Kola Naukowego w Moskwie, Sekcya matematyczno-przyrodnicza, 1916 (Foundations of the General Theory of Manifolds I. Eng. trans. by D. I. Barnett. In: S. Leśniewski. Collected Works, ed. S. J. Surma, J. Srzednicki, D. I. Barnett, and F. V. Rickey. Dordrecht, Kluwer, Vol. 1, 1992, pp. 129-173
[Levi-Civita 1892] Tullio Levi-Civita. Sugli infiniti ed infinitesimi attuali quali elementi analitici. Atti Ist. Veneto di Sc., Lett. ed Art., 7a, 4:1765, 1892
[Levi-Civita 1898] Tullio Levi-Civita. Sui numeri transfiniti. Rend. Acc. Lincei, 5a, 7:91,113, 1898
[Lobachevsky 1829] Nikolai Lobachevsky. A concise outline of the foundations of geometry. Geometrical investigations on the theory of parallel lines. Kazan University Messenger, Kazan, 1829-1830
[Loeb 1992] Daniel Loeb. Sets with a negative number of elements. Advances in Mathematics, 91 (1992), 64-74
[Łoś 1955] Jerzy Łoś. Quelques remarques, théorèmes et problèmes sur les classes définissables d'algèbres. Mathematical interpretation of formal systems. North-Holland Publishing Co., Amsterdam, 1955, pp. 98-113
[Magnani 2001] Lorenzo Magnani. Abduction, Reason, and Science. Processes of Discovery and Explanation. Kluwer Academic/Plenum Publishers, New York, 2001
[Magnani 2002] Lorenzo Magnani. Epistemic mediators and model-based discovery in science. In: Model-Based Reasoning: Science, Technology, Values, L. Magnani and N.J. Nersessian, eds., Kluwer Academic/Plenum Publishers, New York, 2002, pp. 305-329
[Maltsev 1936] Anatoly Maltsev. The Metamathematics of Algebraic Systems. Collected Papers: 1936-1967. Studies in Logic and the Foundations of Mathematics, Volume 66. translated, edited, and provided with supplementary notes by Benjamin Franklin Wells. North-Holland Publishing Co., Amsterdam, 1971
[Mandelbrot 1975] Benoît Mandelbrot. Les objets fractals. Flammarion, Paris, 1975, 1984
[Mandelbrot 1977] Benoît Mandelbrot. Fractals: Form, Chance and Dimension. Freeman, New York, 1977
[Mandelbrot 1982] Benoît Mandelbrot. The Fractal Geometry of Nature. Freeman, New York, 1982
[Miyamoto 2001] Sadaaki Miyamoto. Fuzzy Multisets and their Generalizations. In: C. Calude, G. P˘aun, G. Rozenberg, A. Salomaa (Eds.), Multiset Processing, no. 2235 in Lecture Notes in Computer Science. Springer-Verlag, Berlin, 2001, pp. 225-235
[Miyamoto 2004] Sadaaki Miyamoto. Data Structure and Operations for Fuzzy Multisets. In: J. F. Peters, A. Skowron, D. Dubois, J. Grzymala-Busse, M. Inuiguchi (Eds.). Transactions on Rough Sets II: Rough Sets and Fuzzy Sets, no. 3135 in Lecture Notes in Computer Science. Springer-Verlag, Berlin, 2004, pp. 189-200
[Nelson 1977] Edward Nelson. Internal set theory: a new approach to Nonstandard Analysis. Bull. Amer. Math. Soc. 83 (1977), 1165-1198
[Newton 1687] Isaac Newton. Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy). London, 1687
[Nicosia 2001] Giovanni Giuseppe Nicosia. L'autoriferimento in teoria degli insiemi. Thesis. Bologna, Università degli Studi di Bologna, 2001
[Nicosia 2005] Giovanni Giuseppe Nicosia. Insiemi, scatole, iperinsiemi e grafi. Rivista – Zeitschrift Annali Italo-Austriaci di Scienza e Filosofia, 01/2005, 1-19
[Pawlak 1982] Zdzisław Pawlak. Rough sets. International Journal of Parallel Programming, 11 (5), 1982, 341-356
[Petrovsky 1992] Alexey Petrovsky. An axiomatic approach to metrization of multiset space. The 10th International Conference on Multiple Criteria Decision Making. Proceedings. – Taipei, Taiwan, R.O.C. July 19-24, 1992, V.1, 381-390
[Petrovsky 1994] Alexey Petrovsky. An axiomatic approach to metrization of multiset space. G.H.Tzeng, H.F.Wang, U.P.Wen, P.L.Yu (Eds.). Multiple Criteria Decision Making. Springer-Verlag, New York, 1994, 129-140
[Petrovsky 2001a] Alexey Petrovsky. Constructing a general decision rule for contradictory expert classification of multiattribute objects. Pattern Recognition and Image Analysis, 2001, V.11, No. 1, 73-76
[Petrovsky 2001b] Alexey Petrovsky. Multiple criteria project selection based on contradictory sorting rules. M.Goldevsky, H.Mayr (Eds.). Information Systems Technology and its Applications. Gesellschaft fur Informatik, Bonn, 2001, 199-206
[Petrovsky 2003] Alexey Petrovsky. Spaces of Sets and Multisets [in Russian]. Editorial URSS, Мoscow, 2003
[Prigogine 1993] Ilya Prigogine. Chaotic Dynamics and Transport in Fluids and Plasmas: Research Trends in Physics Series. American Institute of Physics, New York, 1993
[Prigogine 1997] Ilya Prigogine. End of Certainty. The Free Press, 1997
[Riemann 1990] Bernhard Riemann. Gesammelte Mathematische Werke, Wissenschaftlicher Nachlass und Nachträge. Based on the edition by Heinrich Weber and Richard Dedekind. Edited and with a preface by Raghavan Narasimhan. BSB B. G. Teubner Verlagsgesellschaft, Leipzig, Springer-Verlag, Berlin, 1990
[Robinson 1966] Abraham Robinson. Non-Standard Analysis. North-Holland, Amsterdam, 1966
[Santilli 1985a] Ruggero Maria Santilli. Lie-isotopic liftings of Lie symmetries. I: General considerations. Hadronic J. Vol. 8 (1985), 25-35
[Santilli 1985b] Ruggero Maria Santilli. Lie-isotopic liftings of Lie symmetries. II: Lifting of rotations. Hadronic J. Vol. 8 (1985), 36-51
[Santilli 1993a] Ruggero Maria Santilli. Isonumbers and genonumbers of dimension 1, 2, 4, 8, their isoduals and pseudoduals, and "hidden numbers" of dimension 3, 5, 6, 7. Algebras, Groups and Geometries, Vol. 10 (1993), 273-322
[Santilli 1993b] Ruggero Maria Santilli. Elements of Hadronic Mechanics. Vol. I: Mathematical Foundations. 1st edition. Ukraine Academy of Sciences, Kiev, 1993
[Santilli 1999] Ruggero Maria Santilli. Isotopic, Genotopic and Hyperstructural Liftings of Lie's Theory and their Isoduals. Algebras, Groups and Geometries, 15 (1999), 473-498
[Santilli 2008] Ruggero Maria Santilli. Hadronic Mathematics, Mechanics and Chemistry. Volumes I, II, III, IV and V. International Academic Press, America-Europe-Asia, 2008
[Schwartz 1967] L. Schwartz. Analyse mathematique. Hermann, Paris, 1967
[Skolem 1970] Albert Thoralf Skolem. Selected works in logic. J. E. Fenstad (ed.). Scandinavian University Books, Oslo, 1970
[Stakhov 2009] Alexey Stakhov. The Mathematics of Harmony. From Euclid to Contemporary Mathematics and Computer Science. World Scientific Publishing Company, 2009
[Stroyan 1972] Keith Duncan Stroyan. Uniform continuity and rates of growth of meromorphic functions. In: Contributions to Non-Standard Analysis. W.J. Luxemburg and A. Robinson (eds.). North-Holland, Amsterdam, 1972, pp. 47–64
[Stroyan 1997] Keith Duncan Stroyan. Mathematical Background: Foundations of Infinitesimal Calculus. Second Edition, Academic Press, 1997
[Syropoulos 2001] Apostolos Syropoulos. Mathematics of Multisets. In: Multiset Processing. C. Calude, G. P˘aun, G. Rozenberg, A. Salomaa(Eds.). Lecture Notes in Computer Science. 2235/2001. Springer, Berlin, 2001, p. 347-358
[Tall 1979] David Tall. Standard Infinitesimal Calculus Using the Superreal Numbers. Preprint, Warwick University, 1979
[Tall 1980a] David Tall. The notion of infinite measuring number and its relevance in the intuition of infinity. Educational Studies in Mathematics, 11, 1980, 271-284
[Tall 1980b] David Tall. Looking at graphs through infinitesimal microscopes, windows and telescopes. The Math. Gazette 64 (1980) 22-46
[Tall 2001] David Tall. Tall, David (2001) Natural and formal infinities. Educational Studies in Mathematics, Vol. 48, ,No. 2, 2001, pp. 199-238
[Tsvik 1975] Lev Tsvik. Schwarz's algorithm generalization for areas interfaced without overlapping [In Russian]. Proceedings of the USSR Academy of Sciences, 1975, 224 (2), 309-312
[Tsvik 1978] Lev 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
[Tsvik 1995] Lev Tsvik. Finite Element Method Application to Static Deformation [In Russian]. Irkutsk State University Publishers, Irkutsk, 1995
[Tsvik 2001] Lev Tsvik. Triaxial Stress and Strength of Single-Layered and Multilayered High Pressure Vessels with Branch Pipes [In Russian]. Dr. Sc. Dissertation, Irkutsk, 2001
[Tsvik 2002] Lev Tsvik. Mathematical educatuon and intellectual activity improvement [In Russian]. Scientific and practical conference "Higher education in Siberia and the Far East", Sib-GUPS Publishers, Novosibirsk, 2002, 379
[Veronese 1891] Guiseppe Veronese. Fondamenti di geometria a più dimensioni e a più specie di unità rettilinee esposti in forma elementare. Tipografia del Seminario, Padova, 1891
[Vopěnka 1979] Petr Vopěnka. Mathematics in the Alternative Set Theory. Teubner, Leipzig, 1979
[Wallis 1655] John Wallis. De sectionibus conicis. Academix Typographi, 1655
[Wallis 1656] John Wallis. Arithmetica Infinitorum. Academix Typographi, 1656
[Weierstrass 1894] Karl Theodor Wilhelm Weierstrass (German: Weierstraß). Mathematische Werke. Bd. 1-7. Königliche Preussische Akademie der Wissenschaften, Berlin, 1894-1927
[Wikibooks Hyperanalysis-MeasurementTheory] http://en.wikibooks.org/wiki/User:Vuara/Hyperanalysis-MeasurementTheory . Scientist Vuara has published in the English Wikipedia Lev Gelimson's Monograph "Measurement Theory in Physical Mathematics" (unfortunately with unproperly showing some symbols due to using NOT the Internet Explorer Web Browser without adding its character sets) along with the references of some academicians and professors to Lev Gelimson's monographs, articles, and other scientific publications, later "Measurement Theory in Elastic Mathematics", see http://scie.freehostia.com/MeaEnTxt.pdf
[Wikipedia Division_by_zero] Wikipedia entry: http://en.wikipedia.org/wiki/Division_by_zero
[Wikipedia Infinitesimal] Wikipedia entry: http://en.wikipedia.org/wiki/Infinitesimal
[Wikipedia Infinity] Wikipedia entry: http://en.wikipedia.org/wiki/Infinity
[Wikipedia Mathematics] Wikipedia entry: http://en.wikipedia.org/wiki/Mathematics
[Wikipedia Mereology] Wikipedia entry: http://en.wikipedia.org/wiki/Mereology
[Wikipedia Signed_zero] Wikipedia entry: http://en.wikipedia.org/wiki/Signed_zero
[Yager 1986] Ronald R. Yager. On the theory of bags. Int. J. General Systems, 13 (1986), 23-37
[Zadeh 1965] Lotfi Zadeh. Fuzzy sets. Information and Control 8 (1965), 338-353