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Measurement Theory in Physical Mathematics Monograph

by

Leo Himmelsohn

Second Edition (2001)

The “Collegium” International Academy of Sciences Publishers

First Edition (2001)

0. Abstract

The purpose of this monograph is to give many new very effective ideas and tools to the specialist in measurement technology in order to help him solve his urgent problems. Details, grounds, and developments are available in the monographs and articles by the author at his Web site.

Traditional 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 measurement technology. 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 measurement technology 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 measurement data processing, loss of time and costs, dangers, accidents, and catastrophes.

Object-, task- and problem-oriented mathematics by the author within his physical mathematics by the principles of new scientific thought brings many natural, universal, and effective basic concepts and solving methods, in particular, for many typical problems in measurement technology. The hypernumbers 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 hypersets with any quantity of each element perfectly express arbitrary collections without structure. The hyperquantities build a universal hypermeasure independent of dimensions. Hypererrors 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, hypererror 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 measurement technology 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. Proposing general strength theory 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 inventions having, in particular, measurement technology orientation.

1. Basic Principles

1.1. Principles of Object-, Task-, and Problem-Oriented Thought

Exclusively practical purpose arrangement.

Unrestricted freedom in foundations research and development.

Inheritance (effective using all the known concepts, methods, and results).

Purposeful creation (of all the necessary and useful objects but no hindering ones such as counterexamples leading to artificial contradictions typical in traditional mathematics).

Exact discrimination (by holding universal conservation laws) of noncoinciding possibly infinitely great or small objects.

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

Unrestricted flexibility (even creating an individual mathematics for an urgent problem).

Reasonable fuzziness (intuitive ideas without axiomatic strictness).

Symbolic existence (of all the necessary and useful even contradictory objects).

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 effectively and adequately enough).

1.2. Principles of Object-, Task-, and Problem-Oriented Mathematics

Universal expressibility (of each urgent object).

Intuitivism (instead of axiomatic approaches).

Nonstrict essence (without strictness in presentations).

Decision delays (by estimating sense and existence).

Symbolic sense (of each urgent contradictory object).

Unrestricted operability (each operation on an arbitrary set of operands).

Quasiequivalent reduction (by an equivalent parameter).

Natural generalizations (not formal but essential and useful).

Structural considering (systems as objects with structures).

Quasicritical phenomena (in the structures of systems).

Practical tendencies (of infinite processes).

Simplest corrections (if they are reasonable in a certain hypersense).

Subtle estimation (to discriminate noncoinciding objects).

Reasonable control (by possibly inexact knowledge).

Quasisolving and supersolving (a problem by determining its quasisolution as the best possibly inexact pseudosolution, its supersolution as the best exact solution, and its contradictoriness measure).

Reserve determination (estimating the exactness of each exact object or model by its confidence and reliability).

Unified presentation (of bounded concepts).

2. Numbers, Operations, Sets, and Measures

2.1. Real Numbers and Hypernumbers

Traditional mathematics believes there are no gaps between the real numbers whose set is denoted by R. The probabilities of many typical reasonable events do not exist, for example, the same probability pn = p of the choice of a certain natural number n (e.g., 7) in the set N = {0, 1, 2, ...}. This might be proved as follows. If that probability were positive, the infinite sum of those probabilities for all natural numbers would be +¥. If that probability were zero then the countable sum of those probabilities for all natural numbers would vanish because each partial sum would be zero and the limit of the sequence of these sums would vanish. But this infinite countable sum is the probability of the reliable event, that some natural number is chosen, and must be exactly 1. Further the probabilities of many typical possible events vanish (e.g., that of the choice of a certain point on a segment of a straight line or curve), as if those were impossible events. +¥ is a lot of many different infinities hardly distinguishable. In ancient times, it was counted as follows: 1, 2, many; the concept „many“ played the role of a lot. Any measure of each segment or interval on a straight line or a curve is independent of whether or not that includes its endpoints. Bolzano [1] has stated his dissatisfaction with such circumstances and tried to do something in the particular case of a natural-number length. Cantor [2] has introduced cardinal numbers to roughly discriminate very different infinities only. For example, the cardinal numbers of the segment [0, 1] and of the complete three-dimensional space are both equal to the same continuum cardinality c. The cause is that, in contrast to the real numbers, each infinite cardinal number absorbs all the less and even equal cardinal numbers. Infinitely small infinitesimal numbers have been already introduced by Leibniz [3] and justified (together with hypernumbers) by Conway [4], Klaua [5], Robinson [6], and Sikorski [7] (see also [8, 9]). But these do not allow to conveniently express each required amount, e.g., that probability of the choice of a certain natural number. Belonging and existence answers („This is an infinitesimal number“; „There is such an infinitesimal number“) bring no satisfaction because all the necessary operations should be also applicable to the hypernumbers. If there were no equality such as i2 = –1, the situation with the imaginary numbers would be similar. It would be impossible to both determine specific solutions to many very important equations also in measurement technology and operate on such numbers. Instead of this, one could say: „That is an imaginary number“, „There is such an imaginary number“. The conclusion is obvious. Also in measurement technology, it is an urgent problem, in a suitable extension of the real numbers, to exactly express all the infinitesimal, finite, infinite, and combined pure (dimensionless) amounts and conveniently operate on them.

In object-oriented mathematics, namely in hyperanalysis [10], this problem is solved as follows. First, the real numbers with all operations, relations, and their properties are taken. All infinite cardinal numbers are then included in such a way that in this extension called the hypernumbers, the same operations, relations, and their properties (also the conservation laws without absorption) hold. For an essentially unique interpretation of infinite cardinal numbers, each of them is expressed by means of the so-called hyperquantity Q of a certain standard set reasonably chosen, for example:

À = Q(…, – 2, –1, 0, 1, 2,...)/2 = Q(0, 1, 2,...) – ½ = Q(N) – ½,

c = Q[0, 1[ (incl. 0 and excl. 1) = Q]0, 1] (excl. 0 and incl. 1).

This allows to exactly discriminate all different hypernumbers incl. infinitesimals, finite numbers, and infinities because, for instance, no infinitely large hypernumber can absorb any infinitesimal, and the conservation laws unrestrictedly hold also at infinities. Further each pure (dimensionless) amount, also an infinitesimal or infinite one, is exactly expressed by a unique suitable hypernumber.

Only some representative examples follow:

Q(N+) = Q(1, 2,...) = Q(N) – Q{0} = (À + ½) – 1 = À – ½;

Q{z, z + 1, z + 2, ...} = Q{z + 1× n | n Î N} = À - z + ½ (z Î R);

that probability pn = p of the choice of a certain natural number n (e.g., 7) in the set N = {0, 1, 2, ...} is the unique solution 1/(À + ½) to the equation Q(N)p = 1;

Q[a, b[ = Q]a, b] = |b – a|c,

Q]a, b[= |b – a|c – 1,

Q[a, b] = |b – a|c + 1,

Q(R) = 2Àc;

the probability p of the choice of a certain point on a straight line is the unique solution 1/(2Àc) to the equation Q(R)p = 1;

for the segment [a, b], a plane, and the complete three-dimensional space, the similar probabilities are 1/(|b – a|c + 1), 1/(2Àc)2, and 1/(2Àc)3, respectively.

This realizes Bolzano’s dream [1] on discovering the mysteries and paradoxes of the infinity with effective operations at it like those on the usual numbers.

2.2. Concrete (Mixed) Magnitudes, Hyperelements, and Quantification

In each concrete (mixed) physical magnitude, e.g., 5 liter petrol, the unifying operation is not obvious. „Petrol multiplied by 5 liter“ or, all the more, „5 liter multiplied by petrol“ would be not reasonable. There is no known suitable operation.

For such a purpose, object-oriented mathematics proposes hyperelements, quantifying, and quantity definition [10].

Hyperelements are hyperunions of equal objects.

Hyperelement means an object of the form qa (read: deep q of a) for any two objects a (its basis, unique element) and q (the quantity of a in qa).

Examples:

2.5 kgmeat, loafbread, -100 DM + $ 60money;

1a =° {a}° = {a}

where a little circle ° means that namely hyperobjects incl. hyperrelations are considered and is optional if those can be replaced with the corresponding usual ones;

na =° {a, a, ..., a}° ( n times, n Î N+ = {1, 2, 3, ...}).

Quantifying means the following hyperoperation with one parameter q:

Qq: a ® qa, Qq(a) = qa.

Quantity determination means the hyperoperation

Q: qa ® q, Q(qa) = Q(a Î qa) = q

with

Q(b Î qa) = 0 (b ¹ a).

Nonempty hyperelemens qa, rb are called similar (qa @° rb) by a = b and hyperequal (qa =° rb) by a = b, q = r. The empty hyperelement # Î Æ, whose uncertain intermediate forms 0a and q# have to be reduced to its final certain canonic form 0#, coincides with the empty set Æ and is considered to be similar to each hyperelement and hyperequal to itself only.

2.3. Sets and Hypersets

Cantor sets [2] may contain any object as an element either once or not at all; its repetitions are ignored also by Cantor set relations and operations [11 – 15]. Those set operations are only restrictedly invertible. The simplest equations

X È A = B

and

X Ç A = B

in X are solvable only by A Í B and A Ê B, respectively (uniquely by A = Æ and A = B = U = a universal set [2], respectively). The equations

X È A = B

and

X = B \ A

are equivalent only by A = Æ. In a fuzzy set [16, 17], the membership function of each element may also lie strictly between these ultimate values 1 and 0 in the case of uncertainty only. Element repetitions are taken into account in multisets [18] with any cardinal numbers as multiplicities and in ordered sets (tuples, sequences, vectors, permutations, arrangements, etc.) [11 – 13]. Unordered combinations with repetitions are considered in [13] to be no sets but equivalence classes of equipartite arrangements with repetitions. In many unordered collections (for example, of bonds and coins), namely the quantity (multiplicity, number, amount) of each element is decisive and can be arbitrary, for example, 3.5 kg, 1 loaf (bread), – 45 DM by buying and those with opposite signs by selling.

For such a purpose, object-oriented mathematics proposes hypersets [10, 19, 20] as possibly the most general collections (divisible objects) without structure.

Hyperset means a hyperelement or a collection and, equally, a hypersum of hyperelements:

A =° {..., qa, ... , rb, ...}° =° ... +° qa +° ... +° rb +° ... .

Excepting irreducible hypersets of the form

A° =° {..., qa, ... , rb, ...}°°,

a hyperset must be reduced through hyperadding all similar hyperelements.

In a hyperset, the quantity of any element means its quantity in the unique similar hyperelement of the hyperset after its reducing:

Q(a Î A) = q.

If in a hyperset there is no hyperelement similar to an element, its quantity in the hyperset vanishes:

Q(c Î A) = 0.

The (reduction-independent) hyperquantity of a hyperset means the hypersum of the quantities of all elements of the hyperset as the result of its quantity determination:

Q(A) = ... +° q +° ... +° r +° ... .

Quantifying a hyperset is multiplying its element quantities:

sA =° s{..., qa, ... , rb, ...}° =° {... , sqa, ... , srb, ...}°,

Q(sA) = sQ(A).

The graph of a reduced hyperset A means the pair and possibly point hyperset

G(A) = {..., (a, q), ... , (b, r), ...}.

Examples:

{0a, q#}° =° 0# (=° Æ);

{qa, -ra, s#}° =° q - ra;

{20, p# , -10, e1}° =° {(1)0, e1}°;

Q(1/20, 1, 2,...) = Q(N) – Q{1/20} = (À + ½) – ½ = À;


 * a, b| = 1/2a +° ]a, b[ +° 1/2b.

A unit quantity is optional:

a = 1a =° {1a}° =° {a}° = {a}.

The hypersets naturally generalize the Cantor sets [2] whose particular case corresponds to either zero or unit element quantities only. The linear notation of a hyperset without structure is used as the most convenient one. The Euler–Wenn diagrams [11] are also applicable to hypersets. Even by ..., q, ... , r, ... Î [0, 1], the fuzzy sets [16, 17] are generalized by the hypersets that can and may also be entirely certain even by such limited element quantities. The hyperset

{¼apple, ½pear}°

can consist of exactly a quarter of an apple and a half of a pear.

In hypersets and hyperelements, negative, rational, irrational, imaginary, and concrete (mixed) quantities naturally appear by subtraction (e.g., as losses, debts, deliveries, sales, expenses), division (shares), taking roots, and measuring.

Examples:

10 DM money - 15 DM money =° -5 DM money

(e.g., an account to be settled in the case of a zero personal account; therefore, the empty set Æ is not the (nonexistent) „emptiest“ hyperset but the neutral one only like zero that is the neutral number but not the (nonexistent) smallest one);

(34)1/2 =° Ö32;

(-11)1/2 =° ±i(±1)

where i is an imaginary unit;

a housewife purchase result might be

{loafbread, 1.5 kgmeat, case + 2water-melons, 0.1 literlinseed-oil, 2 packetsdetergent, 3 boxesmatches, -58.74 DM - $ 5money, -2 htime, -2.5 literpetrol}°.

If necessary, different units in a quantity can be (exactly, approximately, conditionally, time-dependently, etc.) transformed into a common unit with corresponding factors (here, e.g., through

1 case Û 8,

$ 1 Û 2.15 DM, more exactly ca. 2.2 DM Þ $ 1 Þ 2.1 DM).

Hypersets are called hyperequal if they in the reduced forms contain the same hyperelements.

An algebraic hypersum as a result of algebraic hyperadding certain hypersets means a hyperset that is the corresponding algebraic hypersum of all the hyperelements of the hypersets-operands:

...+° {..., qa, ...}° -° ... -° {... , rb, ...}° +° ... =° {... , qa, ... , - rb, ...}°.

Example:

{31, 0, p2}° -° {q3, e1, -i0}° +° {ei, -p2, i1}° =° {1+ i 0, 3 - e + i1, - q3, ei}°.

A hyperproduct as a result of hypermultiplying certain hypersets means a hyperset that is the hypersum of all the hyperproducts of exactly one hyperelement of each of the hypersets-operands.

Example:

{21, -12}° ´° {-32, 64}° =° {21 ´° -32, -12 ´° -32, 21 ´° 64, -12 ´° 64}° =° {-62, 34, 124, -68}° =° {-62, 154, -68}°.

A hyperquotient as the result of hyperdividing a hyperset by a hyperelement means a hyperset that is the hypersum of all the hyperquotients by dividing each hyperelement of the hyperset-dividend by the hyperelement-divisor.

Example:

{-4-2, 69}° /° -8-6 =° {1/2(1/3), -3/4(-3/2)}°.

In object-oriented mathematics [10, 19, 20], it is possible to practically unrestrictedly operate on hypersets like numbers.

2.4. Measures and Hyperquantities

Each known measure [11 – 13] is sensitive only within a certain dimension. For example, the area measure gives 0 for a segment of any length, nontrivial results for areas only, and +¥ for a spatial body of any sizes. Also by matching the dimensions of an object to be measured and of a measure, the last is not sensitive to those changes of the object that have a lower dimension in comparison with the object. As mentioned, the linear measure gives the same result for a segment and an interval with pairwise coinciding endpoints independently of whether or not these belong to the set to be measured. And in the case of the mixed (combined) dimensions of a set to be measured, which, for instance, consists of some points, lines, areas, and spatial bodies, no known measure can take all this into account.

In object-oriented mathematics, the hyperquantities [10] build a universal hypermeasure.

Example:

Q((0, 0, 0) È° {0} ´ |–1/2, 1/2] ´ {0} È° [3, +¥[2 ´ {4} È° ]–¥, –1|3) =

1 + (c + 1/2) + ((À – 3)c + 1/2)2 + (– 1 + À)3c3 =

7/4 + (À – 2)c + (À – 3)2c2 + (À – 1)3c3.

3. Errors and Reserves

3.1. Errors and Hypererrors

For estimating measurement data, the absolute error and the relative one [11 – 13] are usually determined.

The absolute error alone offers no sufficient quality estimation giving, for example, the same result 1 for the acceptable formal (correct or not) equality 1000 =? 999 and for the inadmissible one 1 =? 0. Further the absolute error is not invariant by equivalent transformations of a problem because, for instance, when multiplying a formal equality by a nonzero number, the absolute error is multiplied by the absolute value of that number.

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

Object-oriented mathematics proposes a hypererror [19, 21] irreproachably correcting the relative error and generalizing it possibly for any conceivable range of applicability.

For the same simplest formal equality a =? b of two numbers, a hypererror can be represented by the estimating fraction

da =? b = |a – b|/(|a| + |b|) (3.1.1)

in the case |a| + |b| > 0, which should simply vanish by a = b = 0.

We can get a shorter and more explicit notation by introducing extended division

a//b = a/b by a ¹ 0;

a//b = 0 by a = 0

independently of the existence and value of b.

The estimating fraction (3.1.1) is then represented without words:

da =? b = |a – b|//(|a| + |b|). (3.1.2)

To generalize (3.1.2), it is possible to add a positive hypernumber p suitable in a specific problem to the denominator and/or to replace the origin 0 with a hypernumber h:

da =? b (p, h) = |a – b|//(|a – h| + |b – h| + p). (3.1.3)

Another possibility is using the quadratic estimating fraction

2da =? b = |a – b|//[2(a2 + b2)]1/2 (3.1.4)

instead of the previous linear one (3.1.2).

The outputs (return values) of such hypererrors always belong to the segment [0, 1], which should hold for the relative error.

Examples:

d0 =? 0 = 0;

d1 =? 0 = 1;

d100 =? 99 = 1/199;

2d0 =? 0 = 0;

2d1 =? 0 = 1/21/2;

2d1 =? –1 = 1.

We shall use the linear estimating fraction (3.1.2) alone if it suffices.

For a formal vector equality

åwÎWzw =? 0,

a hypererror can be represented by the estimating fraction (3.1.2)

d(åwÎWzw =? 0) = ||åwÎWzw||//åwÎW||zw||

whose denominator contains all elements that have been initially in the equality, i.e., before any transformations. If all the vectors are replaced with numbers, the norms can be replaced with the moduli (absolute values).

The quadratic estimating fraction (3.1.4) is

2d(åwÎWzw =? 0) = ||åwÎWzw||//(Q(W)åwÎW||zw||2)1/2.

Examples:

d100 – 99 =? 0 = 1/199 = d100 =? 99;

d1 – 2 + 3 – 4 =? –1= |1 – 2 + 3 – 4 + 1|/(1 + 2 + 3 + 4 + 1) = 1/11.

For a formal functional equality

g[wÎWzw] =? 0

with a domain of definition Z and a range of values in a normed vector space, a natural generalization idea is to define a hypererror (3.1.2) by the estimating fraction

dg =? 0 = medZ||g[wÎWzw]||//supZ||g[wÎWzw]||.

Its numerator is a mean value of the norm of the function in its domain of definition Z. The denominator is the least upper bound on the norm of this function in the same domain of definition. If the range of values of the function contains numbers only, the norms can be replaced with the moduli (absolute values).

For a formal equality

g[wÎWzw] =? h[wÎWzw]

of two functions with a common domain of definition Z and with ranges of values in a common normed vector space, the following holds. A further natural generalization idea is to define a hypererror (3.1.2) by the estimating fraction

dg =? h = supZ||g[wÎWzw] – h[wÎWzw]||//supZ(||g[wÎWzw]|| + ||h[wÎWzw]||).

Its numerator is the least upper bound on the norm of the difference of these functions in their domain of definition Z. The denominator is the least upper bound on the sum of the norms of these functions in the same domain of definition. If the ranges of values of the functions consist of numbers only, the norms can be replaced with the moduli (absolute values).

If a formal functional equality in its initial form contains more than two functions, the numerator of the estimating fraction is the least upper bound on the norm of the difference of the sides of the equality. The denominator is then the least upper bound on the sum of the norms of all the functions available in the initial form. For example, in the case of a formal equality

g[wÎWzw] – h[wÎWzw] =? k[wÎWzw],

the estimating fraction (3.1.2) is

dg – h =? k = supZ||g[wÎWzw] – h[wÎWzw] – k[wÎWzw]||//supZ(||g[wÎWzw]|| + ||h[wÎWzw]|| + ||k[wÎWzw]||).

Let us consider a hyperset of equations over indexed functions (dependent variables) fj of indexed independent variables zw, all of them belonging to their possibly individual vector spaces. We may gather all the available functions in the left-hand sides of the equalities without further transformations. The unique natural exception is changing the signs of the functions by moving them to the other sides of the same equalities. The hyperset can be brought to the form

w(l)(Ll[jÎF fj[wÎW zw]] = 0) (lÎL) (3.1.5)

where

Ll – an operator with index l from an index set L;

fj – a function (dependent variable) with index j from an index set F;

zw – an independent variable with index w from an index set W;

[wÎW zw] – a set of indexed elements;

w(l) – the quantity as a weight of the equation with index l.

When replacing all the unknowns (unknown functions) with their possible “values” (some known functions), the hyperset is transformed into the corresponding hyperset of formal functional equalities. To conserve the hyperset form, let us use for these known functions the same designations fj. For the equality with index l

w(l)(Ll[jÎF fj[wÎW zw]] =? 0) (lÎL), (3.1.6)

an estimating fraction may be

dl(m(l)) = {lim [V(zl’)]-1ò(||Ll[jÎF fj[wÎW zw]]||l//sup||Ll[jÎF fj’[wÎW zw]]||l)m(l) dV(zl’)}1/m(l) (zl’ ® zl) (3.1.7)

where

m(l) – a positive number, we shall take 1;

in the denominator, a direct (not composite) function of independent variables is used and by determining the least upper bound, all different isometric transformations (conserving the norms)


 * fj’ [wÎW zw]||j = || fj [wÎW zw]||j

of even equal elements are considered.

For the complete hyperset of the equalities, an estimating fraction can be chosen in the forms

d(m) = ålÎL w(l)dl(m) // ålÎL w(l) (3.1.8)

(the arithmetic mean value) by the linear law,

2d(m) = {ålÎL w(l)[dl(m)]2 // ålÎL w(l)}1/2 (3.1.9)

(the quadratic mean value) by the quadratic law, and

nd(m) = {ålÎL w(l)[dl(m)]n // ålÎL w(l)}1/n (3.1.10)

by the law of the nth power (n > 0).

3.2. Reserves

The absolute error, the relative error, and the hypererror of any exact object or model always vanish. But not only in measurement technology, it is often reasonable to additionally discriminate exact objects or models by the confidence in their exactness reliability. For example, both

x1 = 1 + 10-10

and

x2 = 1 + 1010

are exact solutions to the inequation x > 1, x1 practically unreliable and x2 guaranteed. Their discrimination is especially important by approximate data, which is always the case in measurement technology. Traditional mathematics cannot provide this.

Object-oriented mathematics proposes for this purpose the basic concept of the reserve [19, 21], which is quite new in mathematics and extends the hypererror in the following sense. The values of a hypererror H belong to the segment [0, 1], those of a reserve R to [–1, 1]. For each inexact object I, H(I) > 0 and we can take R(I) = –H(I). For each exact object E, H(E) = 0 and R(E) ³ 0. A proposition to determine the reserve of an inexact object as its hypererror with the opposite sign is at once evident. For an exact object, it seems to be reasonable, to first define a suitable mapping of the object with respect to its exactness boundary and to further take the hypererror of the mapped object. It is exact only if the object itself precisely lies on its exactness boundary where the reserve vanishes. Otherwise, the mapped object is inexact and the object itself has a positive reserve.

For inequalities, such a mapping can be replaced with negating inequality relations and conserving equality ones. In our example, we have (see (3.1.2))

Rx > 1(x1) = Rx > 1(1 + 10 -10) = Hx <? 1(1 + 10 -10) = 10-10/(2 + 10 -10)

and

Rx > 1(x2) = Rx > 1(1 + 1010) = Hx <? 1(1 + 1010) = 1010/(2 + 1010).

Further we need some useful agreements. Let a problem be a hypersystem of relations containing both known elements and unknown ones, which can be regarded as values and variables, respectively.

A pseudosolution to such a problem is an arbitrary hypersystem of such values of all the corresponding variables that, after replacing each variable with its value, the problem becomes a hypersystem of relations containing known elements only, and each of its relations becomes determinable (e.g., true or false).

If each of the last relations is true, such a pseudosolution is also called a solution to the problem.

If a pseudosolution to such a problem has the least hypererror or the greatest reserve, respectively, by a specific realization of a certain method among all pseudosolutions to the problem, then this pseudosolution is called the quasisolution to the problem by this realization of that method. A quasisolution is not necessarily a solution, which is especially important in contradictory problems that have no solutions in principle but can possess quasisolutions.

If a solution to such a problem has the greatest reserve (note that all solution hypererrors are zero) by a specific realization of a certain method among all the solutions to the problem, this solution is called the supersolution to the problem by this realization of that method. The supersolution to such a problem not necessarily coincides with its quasisolution because the set of the solutions is a subset of the set of the pseudosolutions. If the both exist then the quasisolution, which is not necessarily a solution, has a not less reserve in comparison with the supersolution. If in the last comparison, namely the strict inequality holds, then the quasisolution is certainly no solution.

If a pseudosolution to such a problem has the greatest hypererror or the least reserve, respectively, by a specific realization of a certain method among all the pseudosolutions to the problem then this pseudosolution is called the antisolution to the problem by this realization of that method.

Quasisolutions and supersolutions, as well as antisolutions, not necessarily exist because a set of hypererrors or reserves not necessarily contains its greatest lower bound and its least upper one, respectively.

Let us determine the supersolution to the compound inequation

1 < x < 2

as the set of two simple inequations. The simplest way is to solve the equation

Rx > 1(x) = Rx < 2(x)

on the interval defined by the compound inequation itself. We receive (see (3.1.2)):

dx 2,


 * x – 1|/(|x| + |1|) = |x – 2|/(|x| + |2|),

(x – 1)/(x + 1) = – (x – 2)/(x + 2),

x = 21/2.

Let us further determine the supersolution to the generalized compound inequation

a < x < b (3.2.1)

in the case a < b also by solving the equation

Rx > a(x) = Rx < b(x)

on the interval defined by the compound inequation (3.2.1). We receive (see (3.1.2)):

dx b,


 * x – a|//(|x| + |a|) = |x – b|//(|x| + |b|),

by a > 0 (x – a)/(x + a) = – (x – b)/(x + b), x = (ab)1/2,

by b < 0 (x – a)/(– x – a) = – (x – b)/(– x – b), x = – (ab)1/2,

by a < 0 < b x = 0,

by ab = 0 no supersolution.

For a ³ b, the same is the quasisolution only.

Similarly, in the overdetermined set of equations

x = a,

x = b, (3.2.2)

where variable x is an unknown real number, a and b known real numbers, for the quasisolution holds the following:

if a and b have the same sign, it is their geometric mean value with the same sign;

it vanishes if a and b have distinct signs;

it does not exist if ab = 0.

If in the case (3.2.1) we are dissatisfied by a < 0 < b and ab = 0, it is also possible to use the estimating fraction (3.1.3) where normally one parameter only (either p > 0 or h) suffices. Let us first use p:


 * x – a|/(|x| + |a| + p) = |x – b|/(|x| + |b| + p),

by a ³ 0 (x – a)/(x + a + p) = – (x – b)/(x + b + p), x = – p/2 + [(p/2 + a)(p/2 + b)]1/2,

by b £ 0 (x – a)/(– x – a + p) = – (x – b)/(– x – b + p), x = p/2 – [(p/2 – a)( p/2 – b)]1/2.

By p = 1, for the compound inequation 0 < x < 1, we receive the supersolution

x = (31/2 – 1)/2 » 0.36603

and for –1 < x < 0

x = – (31/2 – 1)/2 » – 0.36603.

Let us now use the parameter h and introduce

X = x – h,

A = a – h,

B = b – h.

We have


 * X – A|//(|X| + |A|) = |X – B|//(|X| + |B|)

and, for example, by h < a

(X – A)/(X + A) = – (X – B)/(X + B),

X = (AB)1/2,

x = [(a – h)( b – h)]1/2 + h.

Let us take h = –1 for the compound inequation 0 < x < 1. We then receive the supersolution

x = 21/2 – 1 » 0.41421.

By h ® –¥ in the general case of the compound inequation

a < x < b,

we receive the supersolution

x = (a + b)/2

by a < b, which is the quasisolution only by a ³ b, corresponding to the use of the absolute errors or the least-square method.

4. Methods

4.1. Least-Square Method and Its Defects

The least-square method by Legendre [22] and Gauß [23] is practically the unique known one applicable to contradictory problems, e.g., overdetermined sets of equations, no of which can be derived from the other ones and the amount of which is greater than that of the unknowns (independent variables or dependent ones called functions). Experimental data are inexact, and their amount is always taken greater than that of the parameters in an approximating function often geometrically interpretable by a straight line or curve, plane or surface. That is why this method was possibly the most important one for measurement technology and seemed to be irreplaceable.

For the hyperset of equations above (see (3.1.5) – (3.1.7), (3.1.9)), the method minimizes the sum

ålÎL w(l) lim {[V(zl’)]-1ò||Ll[jÎF fj[wÎW zw]]||l2 dV(zl’)} (zl’ ® zl). (4.1.1)

A deep analysis of the least-square method by the principles of object-oriented thought and object-oriented mathematics has discovered many fundamental defects in both the essence (as causes) and applicability (as effects) of this method that is adequate only in some rare special cases and at least needs attention.

The most important defects in the essence of the least-square method itself are the following:

1. The method is based on the absolute error alone not invariant by equivalent transformations of a problem.

2. The method ignores the possibly noncoinciding physical dimensions (units) of relations in a problem.

3. The method does not correlate the deviations of the approximation desired from the objects approximated with these objects themselves.

4. The method simply mixes those deviations without their adequate weighing.

5. The method considers equal changes of the squares of those deviations with relatively less and greater moduli (absolute values) as equivalent ones.

6. The method foresees no iterating.

7. The method is based on a fixed algorithm accepting no a priori flexibility.

8. The method provides no own a posteriori adapting.

9. The method uses no invariant estimation of approximation.

10. The method considers no different approximations.

11. The method foresees no comparing different approximations.

12. The method considers no choosing the best approximation among different ones.

These defects in the essence of the least-square method result in many fundamental shortcomings in its applicability. The most important of them are the following:

1. Loss of applicability sense.

The method loses any sense and is not applicable at all to problems simulated by a set of equations with different physical dimensions (units), e.g., to a mechanics problem with one equation by the impulse conservation law and another equation by the energy conservation law.

2. No objective sense of the result.

3. No result invariance by equivalent transformations of a problem.

The result returned by the method has no objective sense because the result depends on choosing or introducing accidental parameters inessential for a problem. Let us unify two determined subsets of linear equations with the same unknown variables and different unique solutions in one overdetermined set of equations and use the method. If all the factors by the unknown variables in the equations are of the same order, the result lies more or less properly between the solutions of the subsets. But let us first multiply each equation of the first subset by a number greater than 1 by an order (e.g., 10) and use the method. The result then becomes very close to the solution of the first subset with relatively greater factors. Let us secondly take once more the initial set of equations before that multiplication, multiply each equation of the second subset by a number greater than 1 by an order (e.g., 10), and use the method. The result then becomes very close to the solution of the second subset with relatively greater factors.

For example, let us consider the set of two equations

x = 1,

x = 2. (4.1.2)

The method returns

x = 3/2

that is the arithmetical mean value of 1 and 2. Let us first multiply the first equation by 10. For the set

10x = 10,

x = 2,

the method returns

x = 102/101

that is very close to the solution of the first equation.

Let us secondly take once more the initial set (4.1.2) of equations and multiply only the second equation by 10. For the set

x = 1,

10x = 20,

the method returns

x = 201/101

that is very close to the solution of the second equation.

Those are equivalent transformations of a set of equations, and by each adequate method, the result may not change.

4. Restriction of the class of acceptable equivalent transformations of a problem.

For contradictory sets of linear equations, the following holds. Only so-called simple multiplications of all the equations by a common nonzero number, whose sign can be individually chosen for each equation because namely the squares of these additional factors have to coincide, do not change the result and are acceptable by using the method. But all the other so-called complicated multiplications of those equations by nonzero numbers without a common modulus (absolute value) are also equivalent transformations of such a set, which are unacceptable by using the method.

5. No essentially unique correction of the loss of applicability sense.

If the equations of a set have different physical dimensions (units), there are different proper possibilities, to bring all the equations to a common physical dimension (unit), and the result essentially depends on such a choice. In that mechanics problem with one equation by the impulse conservation law and another one by the energy conservation law, it is equally proper, to multiply the first equation by either a whole velocity or half of it, nothing to say about choosing one of several velocities.

6. Possibly ignoring some part of a problem.

In an overdetermined set of equations, this takes place for ones whose factors by the unknowns are relatively small in comparison with those in the other equations of the set. Such ignoring holds by the squared factor law and is often strong enough.

7. Paradoxical approximation.

For less values, the method brings greater (even absolute) errors and, in turn, for greater values, less (even absolute) errors. By relative errors and hypererrors, such a paradoxicality is still much stronger.

For example, let us consider the problem on the best approximation to the two points

x1 = 1, y1 = 1,

x2 = 10, y2 = 15

of the Cartesian plane x0y by a straight line

y = kx

where namely the best value of a real-number parameter k is required. For this purpose, the method minimizes the sum of the squares of the deviations in the direction 0y, namely

(1k – 1)2 + (10k – 15)2,

which brings

k = 151/101.

For the first point with relatively less coordinates, the absolute error, relative ones, and hypererror are

D1 = |k – 1| = 51/101,

d1’ = |k – 1|/|k| = 51/151,

d1’’ = |k – 1|/|1| = 51/101,

H1 = |k – 1|/(|k| + |1|) = 50/252 = 25/126,

respectively, and for the second point with relatively greater coordinates,

D2 = |10k – 15| = 5/101,

d2’ = |10k – 15|/|10k| = 5/1510 = 1/302,

d2’’ = |10k – 15|/|15| = 5/1515 = 1/303,

H2 = |10k – 15|/(|10k| + |15|) = 5/3025 = 1/605.

8. No analyzing the deviations of the result.

The deviations of the result obtained from the objects approximated are not analyzed by the method at all.

9. No adequate estimating and evaluating the quality of the result.

No tools to adequately estimate and evaluate the quality of the result obtained are used by the method at all.

10. The highest truth ungrounded.

Each result is presented by the method as the highest truth without justification.

11. No refining the results.

The results obtained cannot be refined by the method itself.

12. No choice.

Even if the result returned by the method by solving a specific problem is obviously unsatisfactory, the method provides no other possibility and practically requires accepting such a result without criticism.

4.2. Iteration Method of the Least Normed Powers

Basic ideas of the method are the following:

1. Instead of absolute errors, hypererrors are minimized.

2. Results are corrected by iterating.

3. Both the power and the hypererror parameters may be freely chosen.

The method minimizes [24, 25] the mean value (see (3.1.6), (3.1.7), (3.1.10), (4.1.1))

d(f(k), f(k+1), m) = {[ålÎL w(l)]-1ålÎL w(l) (lim [V(zl’)]-1ò(||Ll[jÎF fj(k+1)[wÎW zw]]||l//sup||Ll[jÎF fj’[wÎW zw]]||l)m dV(zl’)}1/m (zl’ ® zl) (4.2.1)

of the hypererror of a pseudosolution to the hyperset of equations (3.1.5)

where

m – a positive number that is 2 in the method of the least normed squares;

f(k+1), f(k) – the (k+1)st and kth approximations to the quasisolution to the hyperset of equations (3.1.5);

in the denominator, a direct (not composite) function of independent variables stands, and by determining the least upper bound, all different isometric transformations (conserving norms) even of equal elements


 * fj’ [wÎW zw]||j = || fj(k) [wÎW zw]||j

are used.

This ensures that each next approximation to the quasisolution to the hyperset of equations can be expressed over the prior approximation already known, which provides iterating. Another possibility (especially if the set L is infinite) is minimizing the hypererror

d(f(k), f(k+1)) = suplÎLlim [V(zl’)]-1ò(||Ll[jÎF fj(k+1)[wÎW zw]]||l//sup||Ll[jÎF fj’[wÎW zw]]||l)dV(zl’) (zl’ ® zl) (4.2.2)

where sup M – the least upper hyperbound on an ordered hyperset M. This hyperbound is the hyperset of the least upper bounds on the subsets of M reduced from above. The least upper hyperbounds on two hypersets are ordered by ordering the usual least upper bounds on their hypersubsets minimally equally reduced from above to discriminate them.

Example:

One pseudosolution to a set of four equations brings for them the hypererrors 0, 1, 1, and 1, respectively; another one 1, 0, 0, and 0. Intuitively, the second pseudosolution is better than the first. But their Cantor sets of the hypererrors are both {0, 1} and hence provide no discriminating the pseudosolutions by their quality. The hypersets of the hypererrors are {31, 0}° and {1, 30}°, respectively. Again the usual least upper bounds are both 1. The minimal reducing the hypersets from above is subtracting the hyperset {1}° and brings the required discrimination:

sup({31, 0}° –° {1}°) = sup{21, 0}° = 1 >

sup({1, 30}° –° {1}°) = sup{30}° = 0

and therefore

sup{31, 0} > sup{1, 30}°.

The known formulas in the least-square method can be used also here by m = 2 to determine all the currently unknown variables in the next approximation in the numerators after replacing all the variables in the prior approximation with their values already known in the denominators.

Replacing the hypererrors with the reserves that naturally have to be maximized brings a further method generalization also applicable to determining the supersolution to a set of relations.

4.3. Hypererror and Reserve Equalizing Iteration Methods

A hypererror equalizing iteration method is still more effective. At any point [wÎW zw], the hypererror (see also (3.1.7))

dl[wÎW zw] = ||Ll[jÎF fj[wÎW zw]]||l//sup||Ll[jÎF fj’[wÎW zw]]||l

of each (lth) equation in the hyperset (3.1.5) for a pseudosolution

[jÎF fj[wÎW zw]]

is determined.

Now the lth equation (3.1.5) can be equivalently transformed as follows:

(Ll[jÎF fj(k+1)[wÎW zw]]//sup||Ll[jÎF fj’[wÎW zw]]||l) ´

(||Ll[jÎF fj(k)[wÎW zw]]||l//Ll[jÎF fj(k)[wÎW zw]]) = 0 (lÎL) (4.3.1)

where the second fraction drops out if its denominator vanishes.

The mathematical sense of this transformation (4.3.1) is following. If the kth approximation were substituted for the (k+1)st one, the left-hand side of the equation (4.3.1) would be equal to the hypererror dl(k) of the kth approximation.

Let us order the hyperset of the hypererrors dl(k). The left-hand sides of two equations with the greatest absolute value of the difference of their hypererrors dl(k) with corresponding indexes l are setting equal to each other. Such a procedure is repeated for the other equations, and so on. By every step of designing such an equalizing hyperset of equations, each initial equation is used at most once if possible. A new equation drops out if it can be derived from the already designed equations of the equalizing hyperset. Independently of ending each step, this designing is finished when the equalizing hyperset of equations has exactly one solution. If this hyperset contains all the possible independent new equations and nevertheless has more than one solution, the required amount of additional new equations that express equalizing the greatest hypererrors dl(k) to zero must be also designed.

This algorithm for designing the equalizing hyperset of equations provides the following:

1) each solution to an initial hyperset of equations is a solution to the corresponding equalizing hyperset of equations;

2) if an initial hyperset of equations is consistent, the passage from it to the equalizing hyperset of equations is an equivalent transformation;

3) if an initial hyperset of equations is inconsistent, the equalizing hyperset of equations is consistent;

4) if an initial hyperset of equations is overdetermined, the equalizing hyperset of equations is determined;

5) the complication of an equalizing hyperset of equations is the same as that of the corresponding initial hyperset of equations. In particular, if all the initial equations are linear algebraic, all the equalizing equations are also linear algebraic.

A solution to an equalizing hyperset of equations expresses the (k+1)st approximation to the quasisolution to the corresponding initial hyperset of equations through the kth approximation. If for their hypererrors, the inequality

d(f(k), f(k+1)) £ d(f(k–1), f(k))

holds then the equalizing hyperset of equations for the (k+2)nd approximation is not redesigned but only modified where k+1 is substituted for k. If this inequality is not valid, it is necessary to return to the kth approximation and redesign the equalizing hyperset of equations.

The sequence d(f(k), f(k+1)) is nonnegative, monotonically nonincreasing in k and therefore has a limit

d = lim d(f(k), f(k+1)) ³ 0 (k ® ¥)

as a measure of contradictions in the initial hyperset of equations. This measure remains invariant if the initial equations are equivalently transformed by complicated multiplication. Earlier it was only stated that some specific set of equations is contradictory without estimating and, all the more, exactly measuring this contradictoriness, which is proposed by the author perhaps for the first time.

A criterion for finishing the iterations can be a stabilization (within an acceptable inaccuracy chosen) of the equalizing hyperset of equations, of its quasisolution approximation as well as of the hypererrors of both the separate initial equations and the complete initial hyperset of equations.

Replacing the hypererrors to be minimized with the corresponding reserves to be maximized leads to a reserve equalizing iteration method generalizing the described hypererror equalizing iteration method and additionally allowing supersolutions determination.

These methods are very effective by solving problems simulated by overdetermined hypersets of equations, which is always the case by best approximating experimental data in measurement technology.

Let us consider the same problem on the best approximation to the two points

x1 = 1, y1 = 1

x2 = 10, y2 = 15

of the Cartesian plane x0y by a straight line

y = kx

where namely the best value of a real-number parameter k is required. Equalizing the hypererrors (3.1.2) gives the condition


 * k – 1|/(|k| + 1) = |10k – 15|/(|10k| + |15|)

that can be solved by the interval method opening the modulus signs also without iterating, i.e., finitely. The best solution under this condition is chosen by the criteria of the least hypererrors of the equations


 * k – 1|/|k| = 0,


 * 10k – 15|/|10k| = 0,

which in this case simply requires the least value of |k – 1|/|k| on the interval [1, 15/10] = [1, 3/2]. The equalizing condition then gives

(k – 1)/k = –(10k – 15)/(10k),

k = 1.51/2 » 1.22474.

In this case, the quadratic estimating fraction (3.1.4) would lead to the same value in another way:


 * k – 1|/[2(k2 + 12)]1/2 = |10k – 15|/{2[(10k)2 + 152]}1/2,

k = 1.51/2 » 1.22474.

For the first point with relatively less coordinates, the absolute error, relative ones, and hypererror (3.1.2) are

D1 = |k – 1| = 1.51/2 – 1 » 0.22474,

d1’ = |k – 1|/|k| = (1.51/2 – 1)/1.51/2 » 0.18350,

d1’’ = |k – 1|/|1| = (1.51/2 – 1)/1 » 0.22474,

H1 = |k – 1|/(|k| + |1|) = (1.51/2 – 1)/(1.51/2 + 1) » 0.10102

and for the second point with relatively greater coordinates

D2 = |10k – 15| = 15 – 10 × 1.51/2 » 2.75255,

d2’ = |10k – 15|/|10k| = (15 – 10 × 1.51/2)/(10 × 1,51/2) » 0.22474,

d2’’ = |10k – 15|/|15| = (15 – 10 × 1.51/2)/15 » 0.18350,

H2 = |10k – 15|/(|10k| + |15|) = (15 – 10 × 1.51/2)/(10 × 1.51/2 + 15) » 0.10102.

Let us now use the hypererror equalizing iteration method itself to show what to do if the equalizing hyperset of equations cannot be finitely solved, which is typical. To prepare iterating, let us transform the equalizing condition unique in our case on the interval already chosen as follows:

(k(n+1) – 1)/(k(n) + 1) = (15 – 10k(n+1))/(10k(n) + 15),

(k(n+1) – 1)/(k(n) + 1) = (3 – 2k(n+1))/(2k(n) + 3)

where k(n) means the nth approximation to the best value of k.

Let us first take k(0) = 1. Then we receive

(k(1) – 1)/(k(0) + 1) = (3 – 2k(1))/(2k(0) + 3), k(1) = 11/9 » 1.22222,

(k(2) – 1)/(k(1) + 1) = (3 – 2k(2))/(2k(1) + 3), k(2) = 109/89 » 1.22472

that is very close to the exact value k = 1.51/2 » 1,22474.

The quadratic estimating fraction (3.1.4) brings

(k(n+1) – 1)/{2[(k(n))2 + 12]}1/2 = (3 – 2k(n+1))/{2[(2k(n))2 + 32]}1/2,

(k(1) – 1)/{2[(k(0))2 + 12]}1/2 = (3 – 2k(1))/{2[(2k(0))2 + 32]}1/2,

k(1) = (131/2 + 3 × 21/2)/ (131/2 + 2 × 21/2) » 1.21980,

which is in this case more complicated and somewhat worse.

4.4. Direct-Solution Method

Earlier we considered the overdetermined set (3.2.2) of equations

x = a,

x = b,

where variable x is an unknown real number, a and b known real numbers with the same sign, and their geometric mean value with the same sign is the quasisolution. The last coincides with the quasisolution to the compound inequation (3.2.1)

a < x < b

and is also the supersolution if a < b.

The idea to use the geometric mean values with the same sign can be generalized for more than two elements as follows.

Let us consider a hyperset

A = {..., aa, ... , bb, ... , g0, ...}° (4.4.1)

whose elements ..., a, ... , b, ... , 0, ... belong to a common normed vector space and their quantities ... , a, ... , b, ... , g, ... are nonnegative real numbers, together with its hypersubset

A’ = {..., aa, ... , bb, ... }° (4.4.2)

after excluding exactly all the hyperelements (4.4.1) with zero elements. The complete hyperset A (4.4.1) is said to be approximated by such an element x of the same vector space that x is the quasisolution to the hyperset of equations

.................

a(x = a),

.................

b(x = b),

.................

g(x = 0),

................. (4.4.3)

with the corresponding quantities and so can be regarded as the best approximation to all the elements (with their quantities as weights) of A (4.4.1). Of course, this quasisolution x is not necessarily an element of the hyperset A (4.4.1) itself.

The essence of the direct-solution method is that the element

x = Q(A’)//Q(A)(... + aa/||a||n + ... + bb/||b||n + ...)//(... + a/||a||n + ... + b/||b||n + ...) (4.4.4)

of the same vector space is at once proposed as a proper approximation to this quasisolution generally unknown where n is a nonnegative real number. All zero elements are also taken into account (of course, not geometrically but arithmetically) due to the first fraction

Q(A’)//Q(A) = ( ... + a + ... + b + ...)//( ... + a + ... + b + ... + g + ...) (4.4.5)

together with all the quantities. The case n = 0 in (4.4.4) corresponds to the usual arithmetical mean value by using the least-square method (and the absolute error) with quantities as weights where zero elements have not to be initially excluded and relatively small elements are also adequately taken into account. The case n = ½ in (4.4.4) corresponds to our idea to use geometric mean values, and by further essential increase of n (e.g., for n ³ 1), relatively great elements are weakly taken into account. In the following examples, we shall therefore take namely n = ½ in (4.4.4).

Let us return to the same overdetermined set of equations (3.2.2)

x = a,

x = b

by ab > 0. The method (see (4.4.4), (4.4.5)) proposes

x = (1 + 1)/(1 + 1)(1a//|a|1/2 + 1b//|b|1/2)//(1/|a|1/2 + 1/|b|1/2) = (ab)1/2 sign a (4.4.6)

as a quasisolution, which is already known. (4.4.6) is also the quasisolution to the compound inequation (3.2.1)

a < x < b

by ab > 0 and is additionally the supersolution if a < b.

Let us add to the same set (3.2.2) of equations the third one

x = 0

with a zero element. Only the denominator of the first fraction (see (4.4.4), (4.4.5)) then naturally changes:

x = (1 + 1)/(1 + 1 + 1)(1a//|a|1/2 + 1b//|b|1/2)//(1/|a|1/2 + 1/|b|1/2) = 2/3(ab)1/2 sign a.

For the set of equations

x = 1,

y = 1,

x = 3,

y = 2,

we receive (see (4.4.4), (4.4.5))

x = (1 + 1)/(1 + 1)(1×1/|1|1/2 + 1×3/|3|1/2)/(1/|1|1/2 + 1/|3|1/2) = 31/2,

y = (1 + 1)/(1 + 1)(1×1/|1|1/2 + 1×2/|2|1/2)/(1/|1|1/2 + 1/|2|1/2) = 21/2.

For the set of equations

x = 1,

x = 0,

y = 1,

x = 3,

y = 0,

y = 2,

we receive (see (4.4.4), (4.4.5))

x = (1 + 1)/(1 + 1 + 1)(1×1/|1|1/2 + 1×3/|3|1/2)/(1/|1|1/2 + 1/|3|1/2) = 2/31/2,

y = (1 + 1)/(1 + 1 + 1)(1×1/|1|1/2 + 1×2/|2|1/2)/(1/|1|1/2 + 1/|2|1/2) = 2/3×21/2.

Earlier we considered hypersets of equations with only one unknown variable in each of them. Such hypersets of equations are equivalent to problems on the quasisolution to the corresponding point hyperset or also on the supersolution to the corresponding hyperset of inequations because the obtained equations for the hypererrors and reserves, respectively, in these cases are the same.

Let us consider a more complicated overdetermined set of equations such that each of them might include two or more unknown variables, e.g.,

x + 2y = 3,

2x – y = 1,

x – y = 1,

2x – 3y = 0. (4.4.7)

These four straight lines (4.4.7) have 4×3/2 = 6 intersections

x = 1, y = 1,

x = 3, y = 2,

x = 5/3, y = 2/3,

x = 3/4, y = 1/2,

x = 9/7, y = 6/7,

x = 0, y = –1. (4.4.8)

Each of them belongs to exactly two straight lines and therefore is taken with quantity (2×1)/2 = 1. If an intersection belongs to exactly three equation lines or, generally, m ones then it has to be taken with quantity (3×2)/2 = 3 or m(m – 1)/2, respectively, because this means a limiting case of a triangle or m-gon, respectively, where m is a natural number. By such counting, the conservation law for the amount of intersections holds independently of their coincidences. It would be also possible to duplicate all these quantities because this would change no results. But this would bring many superfluous factors, and an intersection of m equation lines would have to be taken with quantity m(m – 1) and not simply m, otherwise the conservation law for the number of intersections could be violated. Therefore, let us choose the formula m(m – 1)/2. Naturally, this has to be additionally multiplied by the product of the quantities (weights) w(l) of all the equations whose lines contain that intersection.

In contrast to the hypersets of equations with only one unknown variable in each of them, every equation with at least two unknown variables brings another hypererror and reserve in comparison with point hypersets and hypersets of inequations. Note that pseudosolutions have to be estimated namely in the initial hyperset of equations. In this general case, an equivalent problem on a point hyperset or a hyperset of inequations cannot be proposed. To use the same effective method by searching for quasisolution approximations, we can consider an approximately equivalent problem on a certain point hyperset. The hypererrors and reserves of all the pseudosolutions including quasisolution approximations have to be exactly estimated in the initial hyperset of equations itself in principle.

Naturally, let us first take the hyperset of all the intersections (with the mentioned quantities) of the equation interpretations, in this case equation straight lines. Generally, it is not admissible to equivalently use all the intersections by determining quasisolution approximations because there might be misleading intersections. Some straight lines can be parallel to each other, and their intersection then lies at infinity in one of two directions, which would bring an ambiguous and false result infinite as well. By almost parallel straight lines, their intersections lie so far from the area of possible quasisolution approximations that taking those intersections into account would fake quasisolution approximations. Because the last ones have to be estimated anyway by means of the initial equations themselves, namely the shortest distances to all the equation straight lines and not their remote intersections are of great importance. There are at least two ways for our choice.

1. The distances of each intersection from the geometrical interpretations of all the equations are taken into account due to its quantity suitably transformed. For example, the last one can be divided by the sum (possibly of the squares) of the distances of the intersection from all the equation interpretations, in this case equation straight lines. Each infinitely remote intersection then brings exactly 0, which is reasonable. And generally, the farther an intersection from all the equation interpretations that do not contain it, the less its influence on the result. This approach takes all intersections with their proper weights into account, needs no individual intersection hyperset analysis, and seems to be very suitable for computer-aided investigations.

2. Some hypersubset of the intersections for their further considering is chosen, which, of course, needs an individual intersection hyperset analysis. A proper choice algorithm might be following. Let us first exclude each intersection such that on each straight line containing this intersection, all the other intersections lie on one half-line whose origin is this intersection that is hence no intermediate one. A second step is possibly adding some first excluded intersections by the rule that each equation straight line has to be represented by at least one intersection belonging to this line, otherwise the corresponding equation would be ignored. For example, by two intersecting straight lines, their intersection is first excluded and then taken into account. Further, in many cases it can be reasonable to take some additional intersections first excluded into account. By choosing each additional intersection, the first and main criterion is the representativity one by which namely the equation lines or other interpretations, containing the least number of intersections already chosen, have to be represented by additional intersections. The second criterion of choice is certainly the distance criterion requiring to choose such an intersection that the sum (of nonnegative powers, e.g., squares analytically especially convenient in many cases) of the distances of this intersection from all the equation interpretations is the least one among such sums for all the intersections first excluded. For example, by three straight lines with three intersections, all three ones are first excluded, two of them with the least distance are then included and finally the third possibly remote intersection is first ignored and then taken into account by comparing both results or also using iterations. We can consider many different possibilities, estimate the quasisolutions to the corresponding point hyperset problems as pseudosolutions to our problem, choose the best of them or also consider all such quasisolutions as corrected point hypersets, etc. In the last case, the direct-solution method is iteration one. By each step, also the first approach, e.g., with dividing the quantity of each intersection or pseudosolution approximation by the sum (of the squares) of the distances of the intersection or pseudosolution approximation from all the geometrical equation interpretations, in this case equation straight lines, can be used. This second approach has also flexibility advantages with many choice possibilities and comparison ones.

Let us return to our problem and first determine the distances between each intersection (4.4.8) and each straight line (4.4.7).

intersection straight line

x + 2y = 3 straight line

2x – y = 1 straight line

x – y = 1 straight line

2x – 3y = 0 sum of the squares x = 1,

y = 1 0 0 2-1/2 13-1/2 15/26 x = 3,

y = 2 4/51/2 3/51/2 0 0 5 x = 5/3,

y = 2/3 0 51/2/3 0 4/3/131/2 9/13 x = 3/4,

y = 1/2 51/2/4 0 3/4/21/2 0 19/32 x = 9/7,

y = 6/7 0 51/2/7 4/7/21/2 0 13/49 x = 0,

y = –1 51/2 0 0 3/131/2 74/13

In our problem by the second approach, first only three intermediate intersections

x = 5/3, y = 2/3,

x = 3/4, y = 1/2,

x = 9/7, y = 6/7 (4.4.9)

are chosen, which represent all the four equation straight lines, and no completion is necessary. In a second consideration by the representativity criterion, we can also take the additional intersection

x = 0, y = –1 (4.4.10)

into account, which represents the straight lines to the equations

2x – y = 1,

x – y = 1

as a second intersection, and each of the two remaining straight lines contains two intersections earlier chosen. But this intersection lies relatively far, and by its consideration, the distances have to be taken into account by quantity determination. The intersection

x = 1, y = 1 (4.4.11)

lies much nearer than the additional intersection to the three intersections already chosen and would be chosen by the distance criterion, but it is only the second one. Nevertheless, we can also take (4.4.11) as an additional intersection instead of (4.4.10) into account in a third consideration. The more approximation attempts, the better results.

In the first consideration of the three points (4.4.9) without taking the distances into account in the quantities, the method (see (4.4.4), (4.4.5)) brings the quasisolution

x = 3/3[(5/3)/|5/3|1/2 + (3/4)/|3/4|1/2 + (9/7)/|9/7|1/2]/[1/|5/3|1/2 + 1/|3/4|1/2 + 1/|9/7|1/2) » 1.170,

y = 3/3[(2/3)/|2/3|1/2 + (1/2)/|1/2|1/2 + (6/7)/|6/7|1/2]/[1/|2/3|1/2 + 1/|1/2|1/2 + 1/|6/7|1/2) » 0.659

to the corresponding point set problem. The hypererrors (see (3.1.2), (3.1.7)) of the equations in their initial set (4.4.7) by this quasisolution approximation are:

dx + 2y =? 3 = |x + 2y – 3|/(|x| + 2|y| + 3) » 0.0933,

d2x – y = ? 1 = |2x – y – 1|/(2|x| + |y| + 1) » 0.1703,

dx – y = ? 1 = |x – y – 1|/(|x| + |y| + 1) » 0.1729,

d2x – 3y = ? 0 = |2x – 3y|/(2|x| + 3|y|) » 0.0841.

The hypererror of the whole set (4.4.7) of equations itself as a quantitative measure of its contradictoriness can be approximately determined namely by this quasisolution approximation and is

1d = (dx + 2y =? 3 + d2x – y =? 1 + dx – y =? 1 + d2x – 3y =? 0)/4 » 0.1302

(see (3.1.8)) as an arithmetical mean value by the linear law,

2d = {[(dx + 2y =? 3)2 + (d2x – y =? 1)2 + (dx – y =? 1)2 + (d2x – 3y =? 0)2]/4}1/2 » 0.1366

(see (3.1.9)) by the quadratic law, and

4d = {[(dx + 2y =? 3)4 + (d2x – y =? 1)4 + (dx – y =? 1)4 + (d2x – 3y =? 0)4]/4}1/4 » 0.1469

(see (3.1.10)) by the law of the fourth power (four is the number of the equations in their set).

In the first consideration of the three points (4.4.9) with taking the distances into account in the quantities, the method (see (4.4.4), (4.4.5)) brings the quasisolution

x = (13/9 + 32/19 + 49/13)/(13/9 + 32/19 + 49/13)[13/9×(5/3)/|5/3|1/2 + 32/19×(3/4)/|3/4|1/2 + 49/13×(9/7)/|9/7|1/2]/[13/9/|5/3|1/2 + 32/19/|3/4|1/2 + 49/13/|9/7|1/2) » 1.189,

y = (13/9 + 32/19 + 49/13)/(13/9 + 32/19 + 49/13)[13/9×(2/3)/|2/3|1/2 + 32/19×(1/2)/|1/2|1/2 + 49/13×(6/7)/|6/7|1/2]/[13/9/|2/3|1/2 + 32/19/|1/2|1/2 + 49/13/|6/7|1/2) » 0.713

to the corresponding point set problem. The hypererrors (see (3.1.2), (3.1.7)) of the equations in their initial set (4.4.7) by this quasisolution approximation are:

dx + 2y =? 3 = |x + 2y – 3|/(|x| + 2|y| + 3) » 0.0686,

d2x – y =? 1 = |2x – y – 1|/(2|x| + |y| + 1) » 0.1626,

dx – y =? 1 = |x – y – 1|/(|x| + |y| + 1) » 0.1806,

d2x – 3y =? 0 = |2x – 3y|/(2|x| + 3|y|) » 0.0529.

The hypererror of the whole set (4.4.7) of equations itself is

1d = (dx + 2y =? 3 + d2x – y =? 1 + dx – y =? 1 + d2x – 3y =? 0)/4 » 0.1162

(see (3.1.8)) as an arithmetical mean value by the linear law,

2d = {[(dx + 2y =? 3)2 + (d2x – y =? 1)2 + (dx – y =? 1)2 + (d2x – 3y =? 0)2]/4}1/2 » 0.1290

(see (3.1.9)) by the quadratic law, and

4d = {[(dx + 2y =? 3)4 + (d2x – y =? 1)4 + (dx – y =? 1)4 + (d2x – 3y =? 0)4]/4}1/4 » 0.1455

(see (3.1.10)) by the law of the fourth power.

In the second consideration of the four points (4.4.9) and (4.4.10) with taking the distances into account in the quantities, the method (see (4.4.4), (4.4.5)) brings the quasisolution

x = (13/9 + 32/19 + 49/13)/(13/9 + 32/19 + 49/13 + 13/74)[13/9×(5/3)/|5/3|1/2 + 32/19×(3/4)/|3/4|1/2 + 49/13×(9/7)/|9/7|1/2]/[13/9/|5/3|1/2 + 32/19/|3/4|1/2 + 49/13/|9/7|1/2) » 1.159,

y = (13/9 + 32/19 + 49/13 + 13/74)/(13/9 + 32/19 + 49/13 + 13/74)[13/9×(2/3)/|2/3|1/2 + 32/19×(1/2)/|1/2|1/2 + 49/13×(6/7)/|6/7|1/2 + 13/74×(–1)/|–1|1/2]/[13/9/|2/3|1/2 + 32/19/|1/2|1/2 + 49/13/|6/7|1/2 + 13/74/|–1|1/2) » 0.677

to the corresponding point set problem. The hypererrors (see (3.1.2), (3.1.7)) of the equations in their initial set (4.4.7) by this quasisolution approximation are:

dx + 2y =? 3 = |x + 2y – 3|/(|x| + 2|y| + 3) » 0.0833,

d2x – y =? 1 = |2x – y – 1|/(2|x| + |y| + 1) » 0.1605,

dx – y =? 1 = |x – y – 1|/(|x| + |y| + 1) » 0.1827,

d2x – 3y =? 0 = |2x – 3y|/(2|x| + 3|y|) » 0.0660.

The hypererror of the whole set (4.4.7) of equations itself is

1d = (dx + 2y =? 3 + d2x – y =? 1 + dx – y =? 1 + d2x – 3y =? 0)/4 » 0.1231

(see (3.1.8)) as an arithmetical mean value by the linear law,

2d = {[(dx + 2y =? 3)2 + (d2x – y =? 1)2 + (dx – y =? 1)2 + (d2x – 3y =? 0)2]/4}1/2 » 0.1327

(see (3.1.9)) by the quadratic law, and

4d = {[(dx + 2y =? 3)4 + (d2x – y =? 1)4 + (dx – y =? 1)4 + (d2x – 3y =? 0)4]/4}1/4 » 0.1465

(see (3.1.10)) by the law of the fourth power.

In the third consideration of the four points (4.4.9) and (4.4.11) without taking the distances into account in the quantities, the method (see (4.4.4), (4.4.5)) brings the quasisolution

x = 3/3[(5/3)/|5/3|1/2 + (3/4)/|3/4|1/2 + (9/7)/|9/7|1/2 + 1/|1|1/2]/[1/|5/3|1/2 + 1/|3/4|1/2 + 1/|9/7|1/2 + 1/|1|1/2) » 1.126,

y = 3/3[ (2/3)/|2/3|1/2 + (1/2)/|1/2|1/2 + (6/7)/|6/7|1/2 + 1/|1|1/2]/[1/|2/3|1/2 + 1/|1/2|1/2 + 1/|6/7|1/2 + 1/|1|1/2) » 0.731

to the corresponding point set problem. The hypererrors (see (3.1.2), (3.1.7)) of the equations in their initial set (4.4.7) by this quasisolution approximation are:

dx + 2y =? 3 = |x + 2y – 3|/(|x| + 2|y| + 3) » 0.0737,

d2x – y =? 1 = |2x – y – 1|/(2|x| + |y| + 1) » 0.1308,

dx – y =? 1 = |x – y – 1|/(|x| + |y| + 1) » 0.2118,

d2x – 3y =? 0 = |2x – 3y|/(2|x| + 3|y|) » 0.0133.

The hypererror of the whole set (4.4.7) of equations itself is

1d = (dx + 2y =? 3 + d2x – y =? 1 + dx – y =? 1 + d2x – 3y =? 0)/4 » 0.1074

(see (3.1.8)) as an arithmetical mean value by the linear law,

2d = {[(dx + 2y =? 3)2 + (d2x – y =? 1)2 + (dx – y =? 1)2 + (d2x – 3y =? 0)2]/4}1/2 » 0.1300

(see (3.1.9)) by the quadratic law, and

4d = {[(dx + 2y =? 3)4 + (d2x – y =? 1)4 + (dx – y =? 1)4 + (d2x – 3y =? 0)4]/4}1/4 » 0.1554

(see (3.1.10)) by the law of the fourth power.

In the consideration of all the six points (4.4.8) by the first approach with taking the distances into account in the quantities, the method (see (4.4.4), (4.4.5)) brings the quasisolution

x = (26/15 + 1/5 + 13/9 + 32/19 + 49/13)/(26/15 + 1/5 + 13/9 + 32/19 + 49/13 + 13/74)[26/15×1/|1|1/2 + 1/5×3/|3|1/2 + 13/9×(5/3)/|5/3|1/2 + 32/19×(3/4)/|3/4|1/2 + 49/13×(9/7)/|9/7|1/2]/[26/15/|1|1/2 + 1/5/|3|1/2 + 13/9/|5/3|1/2 + 32/19/|3/4|1/2 + 49/13/|9/7|1/2) » 1.151,

y = (26/15 +1/5 + 13/9 + 32/19 + 49/13 + 13/74)/(26/15 + 1/5 + 13/9 + 32/19 + 49/13 +13/74)[26/15×1/|1|1/2 + 1/5×2/|2|1/2 + 13/9×(2/3)/|2/3|1/2 + 32/19×(1/2)/|1/2|1/2 + 49/13×(6/7)/|6/7|1/2 + 13/74×(–1)/|–1|1/2]/[26/15/|1|1/2 + 1/5/|2|1/2 + 13/9/|2/3|1/2 + 32/19/|1/2|1/2 + 49/13/|6/7|1/2 + 13/74/|–1|1/2) » 0.750

to the corresponding point set problem. The hypererrors (see (3.1.2), (3.1.7)) of the equations in their initial set (4.4.7) by this quasisolution approximation are:

dx + 2y =? 3 = |x + 2y – 3|/(|x| + 2|y| + 3) » 0.0618,

d2x – y =? 1 = |2x – y – 1|/(2|x| + |y| + 1) » 0.1362,

dx – y =? 1 = |x – y – 1|/(|x| + |y| + 1) » 0.2065,

d2x – 3y =? 0 = |2x – 3y|/(2|x| + 3|y|) » 0.0114.

The hypererror of the whole set (4.4.7) of equations itself is

1d = (dx + 2y =? 3 + d2x – y =? 1 + dx – y =? 1 + d2x – 3y =? 0)/4 » 0.1040

(see (3.1.8)) as an arithmetical mean value by the linear law,

2d = {[(dx + 2y =? 3)2 + (d2x – y =? 1)2 + (dx – y =? 1)2 + (d2x – 3y =? 0)2]/4}1/2 » 0.1276

(see (3.1.9)) by the quadratic law, and

4d = {[(dx + 2y =? 3)4 + (d2x – y =? 1)4 + (dx – y =? 1)4 + (d2x – 3y =? 0)4]/4}1/4 » 0.1527

(see (3.1.10)) by the law of the fourth power.

All these quasisolutions to the corresponding point set problems seem to be sufficient and roughly equally suitable quasisolution approximations to our problem on the set of equations. Its unknown quasisolution should be close enough to these approximations. The last result with taking all the intersections and distances into account seems to be the best one. The hypererror of the whole set of equations by the quadratic law might be the most suitable for both the analytic comparison of different pseudosolutions by quasisolutions search within one approach and the numeric comparison of the results by different approaches. The linear law and more likely the law of the fourth power can play a completion role. Note that in simple examples with two points and the geometrical mean quasisolution, the linear law brings not the global least value but a local maximum.

4.5. Typical Examples for Comparing Different Methods

Let us first consider the problem on determining the best linear approximation to the four points

(x, y) = (–1, –1); (1, 1); (8, 10); (12, 10)

of the Cartesian plane x0y by a straight line with a general equation

y = ax + b

where namely the real-number coefficients a and b are required.

The least-square method (see (4.1.1)) gives

y » 0.927 x + 0.364

as a final and nonimprovable result (essentially deviating from the intuitively evident analytic and graphical choice

y = x)

with

1d » 0.128,

2d » 0.135,

4d » 0.146

(see (3.1.8) – (3.1.10)) as hypererrors.

Let us use the proposed method of the least normed squares (powers by m = 2) (see (4.2.1)) and take a(0) = b(0) = 1 (which is obviously false) as the 0th approximation. Then at once the first approximation brings a very good result

a(1) » 0.99789;

b(1) » 0.00817;

1d(1) » 0.0525,

2d(1) » 0.0718,

4d(1) » 0.0864

(see (3.1.8) – (3.1.10)).

In the problem on the quasisolution to the overdetermined set of linear equations

29x + 21y = 50,

50x – 17y = 33,

x + 2y = 7,

2x – 3y = 0, (4.5.1)

this set can also be considered as a union of the two following determined subsets. The first of them

29x + 21y = 50,

50x – 17y = 33 (4.5.2)

has relatively greater factors by the unknown variables x and y and

x = 1,

y = 1 (4.5.3)

as the unique exact solution. The second subset

x + 2y = 7,

2x – 3y = 0 (4.5.4)

with relatively less factors by the unknown variables x and y has another unique exact solution, namely

x = 3,

y = 2. (4.5.5)

The least-square method (see (4.1.1)) gives

x » 1.00234;

y » 1.00750;

1d » 0.151,

2d » 0.223,

4d » 0.286

(see (3.1.8) – (3.1.10)) as a final and nonimprovable result practically ignoring the second subset (4.5.4) with relatively less factors.

Let us use the proposed method of the least normed squares (powers by m = 2) (see (4.2.1)), again taking the units

x (0) = 1,

y(0) = 1

as the 0th approximation, which accidentally coincides with the exact solution (4.5.3) to the first subset (4.5.2), is obviously false as applied to the whole set (4.5.1) of equations, and should be no especially suitable choice according to the experience with the least-square method. The first approximation then gives at once the much better result

x (1) » 1.263;

y(1) » 1.051;

1d(1) » 0.163,

2d(1) » 0.195,

4d(1) » 0.250

(see (3.1.8) – (3.1.10)).

The hypererror equalizing iteration method (4.3.1) brings the good further approximations

x (2) » 1.5974; y(2) » 1.4374; 1d(2) » 0.1855, 2d(2) » 0.1879, 4d(2) » 0.1925;

x (3) » 1.5993; y(3) » 1.4674; 1d(3) » 0.1864, 2d(3) » 0.1884, 4d(3) » 0.19219;

x (4) » 1.5981; y(4) » 1.4684; 1d(4) » 0.1864, 2d(4) » 0.1884, 4d(4) » 0.19216

and also the desired result

x » 1.60; y » 1.47; 1d » 0,186, 2d » 0,188, 4d » 0,192

(see (3.1.8) – (3.1.10)).

Let us apply to this problem also the second approach of the direct-solution method (see (4.4.4), (4.4.5)) without taking the distances into account. First three intermediate intersections

x = 1, y = 1,

x = 150/129, y = 100/129,

x = 185/117, y = 317/117 (4.5.6)

only are chosen that represent all the four equation straight lines (4.5.1), hence no completion is necessary. In a second consideration by the representativity criterion, we can also take the additional intersection

x = 3, y = 2 (4.5.7)

into account, which represents the straight lines to the equations

x + 2y = 7,

2x – 3y = 0

as a second intersection, each of the remaining two straight lines containing two intersections already chosen.

In the first consideration of the three points (4.5.6) without taking the distances into account, the method (see (4.4.4), (4.4.5)) brings the quasisolution

x = 3/3[1/|1|1/2 + (150/129)/|150/129|1/2 + (185/117)/|185/117|1/2]/[1/|1|1/2 + 1/|150/129|1/2 + 1/|185/117|1/2) » 1.225,

y = 3/3[1/|1|1/2 + (100/129)/|100/129|1/2 + (317/117)/|317/117|1/2]/[1/|1|1/2 + 1/|100/129|1/2 + 1/|317/117|1/2) » 1.285

to the corresponding point set problem. The hypererrors (see (3.1.2), (3.1.7)) of the equations in their initial set (4.5.1) by this quasisolution approximation are:

d29x + 21y =? 50 = |29x + 21y – 50|/(29|x| + 21|y| + 50) » 0.111,

d50x – 17y =? 33 = |50x – 17y – 33|/(50|x| + 17|y| + 33) » 0.055,

dx + 2y =? 7 = |x + 2y – 7|/(|x| + 2|y| + 7) » 0.297,

d2x – 3y =? 0 = |2x – 3y|/(2|x| + 3|y|) » 0.223.

The hypererror of the whole set of equations (4.5.1) itself is

1d = (d29x + 21y =? 50 + d50x – 17y =? 33 + dx + 2y =? 7 + d2x – 3y =? 0)/4 » 0.172

(see (3.1.8)) as an arithmetical mean value by the linear law,

2d = {[(d29x + 21y =? 50)2 + (d50x – 17y =? 33)2 + (dx + 2y =? 7)2 + (d2x – 3y =? 0)2]/4}1/2 » 0.196

(see (3.1.9)) by the quadratic law, and

4d = {[(d29x + 21y =? 50)4 + (d50x – 17y =? 33)4 + (dx + 2y =? 7)4 + (d2x – 3y =? 0)4]/4}1/4 » 0.226

(see (3.1.10)) by the law of the fourth power.

In the second consideration of the four points (4.5.6) and (4.5.7) without taking the distances into account, the method (see (4.4.4), (4.4.5)) brings the quasisolution

x = 4/4[1/|1|1/2 + (150/129)/|150/129|1/2 + (185/117)/|185/117|1/2 + 3/|3|1/2]/[1/|1|1/2 + 1/|150/129|1/2 + 1/|185/117|1/2 + 1/|3|1/2) » 1.536,

y = 4/4[1/|1|1/2 + (100/129)/|100/129|1/2 + (317/117)/|317/117|1/2 + 2/|2|1/2]/[1/|1|1/2 + 1/|100/129|1/2 + 1/|317/117|1/2 + 1/|2|1/2) » 1.432

to the corresponding point set problem. The hypererrors (see (3.1.2), (3.1.7)) of the equations in their initial set (4.5.1) by this quasisolution approximation are:

d29x + 21y =? 50 = |29x + 21y – 50|/(29|x| + 21|y| + 50) » 0.198,

d50x – 17y =? 33 = |50x – 17y – 33|/(50|x| + 17|y| + 33) » 0.145,

dx + 2y =? 7 = |x + 2y – 7|/(|x| + 2|y| + 7) » 0.228,

d2x – 3y =? 0 = |2x – 3y|/(2|x| + 3|y|) » 0.166.

The hypererror of the whole set of equations (4.5.1) itself is

1d = (d29x + 21y =? 50 + d50x – 17y =? 33 + dx + 2y =? 7 + d2x – 3y =? 0)/4 » 0.184

(see (3.1.8)) as an arithmetical mean value by the linear law,

2d = {[(d29x + 21y =? 50)2 + (d50x – 17y =? 33)2 + (dx + 2y =? 7)2 + (d2x – 3y =? 0)2]/4}1/2 » 0.187

(see (3.1.9)) by the quadratic law, and

4d = {[(d29x + 21y =? 50)4 + (d50x – 17y =? 33)4 + (dx + 2y =? 7)4 + (d2x – 3y =? 0)4]/4}1/4 » 0.192

(see (3.1.10)) by the law of the fourth power.

In the second consideration we have directly received a certainly better result that is very close to the result of the hypererror equalizing iteration method. This problem has also shown that namely the quadratic hypererror 2d has especial advantages as applied to the whole set of equations. The linear hypererror 1d seems to be not very suitable for this purpose but can play a completion role together with 4d.

The more methods, approaches, and trials, the better quasisolution approximations. Naturally, the hypererror of each quasisolution approximation has to be determined namely by means of the initial set of equations.

4.6. Demodulation Methods

Measuring each physical magnitude not uniformly distributed in space and/or time [26] by means of a real physical device with nonzero sizes and inertia results in inadequate measurement data. These are distorted (modulated) by a certain law and/or delayed and hence differ from the true distribution. In measurement technology, especially by very heterogeneous objects and rapidly changeable processes, it is very important to recover the true distributions by using measurement data [27]. Such a delay can be considered stable in many cases and compensated by a corresponding displacement of the measurement data in the positive direction of the time axis. But modulation much more complicated depends on the properties of both the distribution and device. Therefore, suitable demodulation methods are very important for measurement technology.

For example, let us consider a one-variable function p(s) continuously distributed and integral-mean modulated on each segment of a constant length D:

MD: p(s) ® p(s) = D-1òs-D/2 s+D/2 p(t)dt (4.6.1)

where MD is the corresponding measurement operator. The domain of definition of p(s) can be restricted by not greater than D from each boundary point of the domain. It is important to know whether and under what conditions an inversion of the measurement operator (4.6.1) exists and is unambiguous and how that can be determined.

Let us first consider the unambiguity problem. If some functions p(s) and q(s) are solutions to the equation (4.6.1), r(s) = q(s) – p(s) is a solution to the equation

òs-D/2s+D/2 r(t)dt = 0.

Its differentiating by s shows that for each s from the domain of definition, the equality

r(s + D/2) = r(s – D/2)

is an identity. Therefore, r(s) is a periodic function where D is a period. The integral mean value of the function on each segment, whose length is exactly the period, vanishes. The inversion of MD (4.6.1) is therefore ambiguous and determined up to such a function r(s), e.g., a finite or infinite sum of harmonic functions with divisors of D as periods. If the device is such suitable one that the variation of the distribution p(s) on each segment of length D is small enough in comparison with the least value of the distribution itself on the same segment, such a function r(s) can be considered vanishing and the inversion practically unambiguous. Otherwise, it is necessary and sufficient to additionally use another device with such a constant D’ that D’/D is theoretically irrational and practically an uncancelled fraction with the sum of the numerator and denominator great enough.

If there is a continuous inversion of MD (4.6.1), the function p(s) (4.6.1) is continuously differentiable, and vice versa. The thorough proof brings also an inversion algorithm that could be used if the measurement data would be an exactly known continuous distribution. But this is only seldom the case. We usually know some discrete values of the function p(s) (4.6.1) with measurement errors. In such a situation, interpolating and, all the more, numerically differentiating would lead to very great errors. Therefore, the following method is advantageous. We often know the domain of definition and some properties of a distribution (whether or not it is periodic or monotone, what order of magnitude, extreme values and signs under what conditions it has, etc.). In many cases this allows to a priori choose standard functions (e.g., power, trigonometric, exponential, hyperbolic, logarithmic ones, etc.) whose linear combinations can be considered proper. By the principle of tolerable simplicity, the least sufficient number of such functions has to be used. When a priori choosing the coefficients by the arguments of those functions, due to the linearity of the measurement operator (4.6.1) we receive linear equations with unknown factors in the combinations, whose set is usually overdetermined and thus contradictory. The proposed methods are very suitable and quickly lead to quasisolutions. What is more, this brings no increasing but correcting measurement errors.

By many most important standard functions, using the measurement operator (4.6.1) can be replaced with multiplication by a suitable factor. For example, this factor is:

1 for each linear function;

sh(0.5nD)/(0.5nD) for exp(ns), sh ns, and ch ns;

sin(0.5nD)/(0.5nD) for sin ns and cos ns.

For example, by a continuous periodic function p(s) and a suitable scale, the period can be considered to be 2p. The Fourier series [11]

p(s) = 0.5c0 + ån=1¥ (cn cos ns + sn sin ns)

is then uniformly summable and therefore integrable. We receive

p(s) = 0.5c0 + ån=1¥ [sin(0.5nD)/(0.5nD)](cn cos ns + sn sin ns).

The inversion algorithm is now obvious and very simple. Let us first determine the Fourier series

p(s) = 0.5c0 + ån=1¥ (cn cos ns + sn sin ns)

for the known function p(s) (4.6.1) and then

p(s) = 0.5c0 + ån=1¥ [0.5nD/sin(0.5nD)](cn cos ns + sn sin ns)

for the unknown function p(s) (4.6.1) by means of the known factors inverted.

Similar ideas are also useful in more complicated cases.

The essence of the Kirsch problem [28] is determining the greatest stress concentration on the boundary of a round hole in a theoretically infinite plate extended in one direction at infinity. The maximum stress holds on two generatrixes of the hole with radius r. But a device best placed on the surface of the hole has a measuring mesh of a nonzero length l on which averaging takes place. To determine the true greatest stress, the stress measured has to be multiplied by the corresponding factor

K = 3/[1 + (2r/l)sin(l/r)].

For a round plate of radius r, which is supported on the boundary and loaded by a one-side pressure uniformly distributed [28], by determining the true greatest stress at the center of the opposite side, a factor

K = 1/{1 – [l/(2r)]2}

can be analogously used.

If the same plate is fixed on the boundary [28], the factor is

K = 1/{1 – [(1 + m)/(3 + m)][l/(2r)]2}

where m is the Poisson ratio of the plate material.

Analogous formulas are available for a generalized Kirsch problem with two extensions in orthogonal directions, under a uniform pressure in the hole, and by considering the transverse sensitivity of the measuring mesh of the device placed on a side of a plate with taking the device width and distance from the hole into account. In additional, similar formulas are obtained for cylindrical shells [28].

5. Some Applications to Measurement Technology and Corresponding Results

5.1. Mechanics and Optics

The proposed basic concepts and solving methods were extensively used for measuring stresses and strains in spatial solids whose sizes were of the same order and which possibly included concentration areas. This allowed to test an analytic macroelement method [29, 30] with advantages in comparison with the finite-element method. Further the strength and optical properties of circular portholes suitable for any depth in the world ocean, as well as high-pressure vessels, test stands, and other objects in high-pressure engineering [31 – 35] were complexly (multiparametrically) optimized. This resulted in many effective inventions in high-pressure engineering and measurement technology and provided discovering many new phenomena in mechanic systems, e.g., the following ones:

a boundary value problem can limit the degree of a power representation of its pseudosolution also from above;

the whole problem type can be overdetermined;

continuously varying a system can lead to spasmodical varying its structure;

systems can have critical relations possibly with bifurcations and boundary relations;

for a system, there can exist a determining parameter and an equivalent one;

rational control by changing the determining parameter of a system can raise its equivalent parameter by an order;

coincidence of the rational control and the critical relation in a system can depend on this relation;

an equivalent parameter can be uniform if a determining parameter is (a)symmetrically non-uniform.

5.2. Strength Theories

The proposed principles, concepts, and solving methods were extensively used by analyzing strength data on many solids. Their materials have been different isotropic and anisotropic as well as ductile and brittle ones with equal or unequal strength in tension and compression. Extreme loads have been stationary and arbitrary variable ones of mechanic, thermal, and radiation nature. Object-oriented mathematics allowed to propose transforming stresses and stress processes into relative ones [29]. Using them simplifies many problems in measurement technology to be correctly solved. This is a very promising basis for many engineering areas (aviation, space flight, machine-building, etc.), mechanics, and physics. In particular, general strength theory [36 – 40] has been created as a source to discover many strength laws of nature.

A usual stress depends on the choice of physical dimensions (units), is not numerically invariant by unit transformations, and alone represents no danger degree even by static loading.

In the stationary case, the proposed stress transformation simply means dividing each stress by the modulus of its limiting value (e.g., by elasticity, plasticity, or failure) with the same sign in the same direction at the same point of the same solid by vanishing all the other stresses and the same remaining loading conditions. By arbitrarily changeable loads, the complete process of each stress alone stationarily indexed is represented by an equidangerous cycle and finally by a stationary vector. The relative stresses always numerically invariant are reciprocal uniaxial reserves, alone show danger degree better than usual reserves, allow additional measurement and investigation possibilities, and open many new ways in discovering mechanical and physical laws of nature.

General strength theory is such an example. It is very hard, expensive, and, all the more, simply impossible to experimentally determine the strength at a point of a spatial solid with all three arbitrarily changeable principal normal stresses (by vanishing all the shearing stresses). The duty of strength criteria is to theoretically determine that strength using measurement data in simple tests (tension, compression, bending, and torsion) with one or two principal stresses only in bars of the same material. But the known strength criteria are applicable only to rare special cases, sometimes have contradictions as well as vague ranges of applicability, and not always bring a suitable equivalent stress. By arbitrarily changeable loads, there are neither applicable strength criteria nor equivalent stress propositions at all. The known strength criteria explain not all important phenomena (e.g., an essential role of hydrostatic stresses), often lose physical sense, contain material parameters with dimensions (units), and cannot lead to laws of nature in principle. Such strength criteria are only some simple special cases of general strength theory represented in the relative stresses with stress transformations independent of both specific strength criteria and this theory itself. It determines the range of applicability of known strength criteria, transforms those into universal ones, and proposes many new laws of nature. These are themselves as always very simple and only by complicated specifying bring results with corresponding complexity like for example the Archimedean law does. This theory naturally extends the physical sense of initial criteria, considers and explains many known phenomena (e.g., an essential role of hydrostatic stresses set by Bridgman) and new ones (nonzero initial states, deviations from Drucker’s postulate, etc.). This theory is very suitable, allows general representing and processing measurement data, reduces time and cost expense, and brings many further advantages of mutual completion of difficultly obtained experimental data on the strength of different materials in spatial stress states under arbitrarily changeable loading. Further this theory proposes many good tools, e.g., an additive and a multiplicative approach [36, 41] to determining true strength reserves additional to known ones in subcritical states of materials. Note that the known reserves are uncertain in principle, alone show real danger degree only in rare special cases of simple loading proportional to a common factor, and otherwise underestimate those often by an order. This all means a further generalization of strength laws of nature from critical solid states to subcritical and supercritical ones.

5.3. Nuclear Physics

The proposed principles, basic concepts, and solving methods were extensively used for processing measurement data on implanting distinct ions into metals with different physical properties. This allowed proposing general implantation theory [42 – 45] with universal estimating and natural discriminating arbitrary implantation doses as well as discovering and explaining many new phenomena in this area.

Implantation doses are usually measured in 1/cm2 and thus depend on the choice of physical units (dimensions). Further distinct ions have unequal sizes. This complicates processing and comparing measurement data for different ions as well as searching for unknown phenomena.

A new basic concept of implantation multiplicity is introduced. This naturally discriminates low, middle, and high implantation doses with orders of 1013, 1015, and 1017 cm-2 by taking real-number values with orders of 0.01, 1, and 100, respectively.

General implantation theory can also be used for strengthening the surfaces of solids extremely loaded and allowed to discover the following phenomena:

existence of a first critical implantation multiplicity;

existence of a second critical implantation multiplicity;

existence of the critical value of implantation energy, whose excess results in the non-equiresistibility of the surface layer of a solid-target;

abrupt supercritical decrease of the resistance of a solid-target;

the coincidence and the common displacement of all the depths of the principal implantation maximums for different particles (ions having diverse sizes, initial energies, etc.).

Conclusions

Traditional mathematics brings key concepts of the numbers, sets, measures, and errors very often insufficient, in particular, for measurement technology. In each concrete (mixed) physical magnitude, no known operation holds. The relative errors of distributions, exactness confidence, and problem contradictoriness are not estimated at all. The known least-square method uniquely applicable in many cases has a very restricted range of applicability and, all the more, of adequacy. Otherwise, it brings false and misleading results that cannot be corrected and improved by the method itself. There are no suitable methods for recovering true distributions by using modulated discrete measurement data.

Object-oriented mathematics by the author within his physical mathematics by the principles of object-oriented thought brings many new key concepts suitable, in particular, for measurement technology. Among them are the hypernumbers, hypersets, quantity operations also holding in concrete (mixed) physical magnitudes, hyperquantities, hypererrors, and reserves estimating and measuring exactness, contradictoriness, and distributions. Even in contradictory problems, the method of the least normed powers, the direct-solution method, as well as the hypererror and reserve equalizing methods give quasisolution approximations improvable by iterations and comparable with one another. Using modulated and inexact discrete measurement data due to the proposed demodulation methods can approximately recover the true continuous space and time distributions of possibly very heterogeneous objects and quickly changeable processes.

The basic concepts and solving methods in object-oriented mathematics were successfully used for analyzing different measurement data in high-pressure engineering, mechanics, optics, and nuclear physics. This allowed to test the analytic macroelement method with advantages in comparison with the finite-element method, to achieve complex optimizing the strength and optical properties of different technical objects, and to propose many inventions also in measurement technology. General strength theory is a source for discovering many strength criteria for any anisotropic solid under arbitrarily changeable extreme loading, which are universal laws of nature. General implantation theory brings the basic concept of implantation multiplicity to naturally discriminate low, middle, and high implantation doses. All this allowed discovering many new phenomena in these areas.

The basic concepts and solving methods in object-oriented mathematics are from the beginning exclusively practice-oriented, effective, universal, and successfully applicable to any measurement technology problem mathematically simulated.

Further developing object-oriented mathematics will bring new results also in the directions of this work.

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[10] Himmelsohn, L. G.: Hyperanalysis: Hypernumbers, Hyperoperations, Hypersets, and Hyperquantities. Second Edition. The “Collegium” International Academy of Sciences Publishers, 2001. First Edition, 2000.

[11] Encyclopaedia of Mathematics / Managing editor M. Hazewinkel. Dordrecht, etc., 1988.

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

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

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

[15] Hausdorff, F.: Mengenlehre. Berlin, 1935.

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

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

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

[19] Himmelsohn, L. G.: Basic New Mathematics. Drukar Publishers, Sumy, 1995.

[20] Himmelsohn, L. G.: Mengen mit beliebiger Quantität von jedem Element. Third Edition. The “Collegium” International Academy of Sciences Publishers, 2001. Second Edition, 2000. First Edition, 1997.

[21] Himmelsohn, L. G.: General Estimation Theory. Transactions of the Ukrainian Glass Institute 1 (1994), p. 214-221.

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

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

[24] Himmelsohn, L. G.: Generalized Methods for Solving Functional Equations and Their Sets. International Scientific and Technical Conference “Glass Technology and Quality”. Mathematical Methods in Applying Glass Materials (1993), Theses of Reports, p. 106-108.

[25] Himmelsohn, L. G.: The method of least normalized powers and the method of equalizing errors to solve functional equations. Transactions of the Ukrainian Glass Institute 1 (1994), p. 209-214.

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

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

[28] Timoshenko, S. P., Goodier, J. N.: Theory of Elasticity. McGrow-Hill, N. Y., 1970.

[29] Himmelsohn, L. G.: Generalization of Analytic Methods for Solving Strength Problems [In Russian]. Drukar Publishers, Sumy, 1992.

[30] Himmelsohn, L. G.: Analytic Macroelement Method in Axially Symmetric Elasticity. International Scientific and Technical Conference “Glass Technology and Quality”. Mathematical Methods in Applying Glass Materials (1993), Theses of Reports, p. 104-106.

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

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

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

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

[35] Himmelsohn, L. G.: Applying the Analytic Macroelement Method and General Strength Theory to Three-Dimensional Cylindrical Glass Elements of High-Pressure Illuminators (Deep-Sea Portholes). Second Edition. The “Collegium” International Academy of Sciences Publishers, 2001. First Edition, 2000.

[36] Himmelsohn, L. G.: General Strength Theory. Drukar Publishers, Sumy, 1993.

[37] Himmelsohn, L. G.: Generalization Method for Limiting Criteria. International Scientific and Technical Conference “Glass Technology and Quality”. Mathematical Methods in Applying Glass Materials (1993), Theses of Reports, p. 98-100.

[38] Himmelsohn, L. G.: Linear Correction Method for Limiting Criteria. International Scientific and Technical Conference “Glass Technology and Quality”. Mathematical Methods in Applying Glass Materials (1993), Theses of Reports, p. 100-101.

[39] Himmelsohn, L. G.: The Generalized Structure for Critical State Criteria. Transactions of the Ukrainian Glass Institute 1 (1994), p. 204-209.

[40] Himmelsohn, L. G.: Yield and Fracture Laws of Nature (Universal Yield and Failure Criteria in the Relative Stresses). Fourth Edition. The “Collegium” International Academy of Sciences Publishers, 2001. Third Edition, 2001. Second Edition, 2000. First Edition, 1998.

[41] Himmelsohn, L. G.: Generalized Reserve Determination Methods. International Scientific and Technical Conference “Glass Technology and Quality”. Mathematical Methods in Applying Glass Materials (1993), Theses of Reports, p. 102-103.

[42] Pogrebnjak, A. D., Himmelsohn, L. G.: Fundamentals of the general theory of ion implantation. First International Symposium ‘Beam Technologies’. Dubna, 1995.

[43] Pogrebnjak, A. D., Himmelsohn, L. G.: Fundamentals of the general theory of ion implantation. 9th International Conference on Surface Modification of Metals by Ion Beams. San Sebastian, 1995.

[44] Himmelsohn, L. G.: 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.

[45] Himmelsohn, L. G.: Fundamentals of General Theory of Ion Implantation. Second Edition. The “Collegium” International Academy of Sciences Publishers, 2001. First Edition, 2000.

[46] Himmelsohn, L. G.: Objektorientierte Mathematik in der Messtechnik. Fachgebiet: „Informatik und Mathematik in der Messtechnik“. Manuskript der Arbeit zur Teilnahme am Wettbewerb um den Helmholtz-Preis 2001. Physikalisch-Technische Bundesanstalt, Braunschweig.

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http://www.tbns.net/leo/ReferTxt.htm

USSR STATE COMMITTEE OF SCIENCE AND ENGINEERING SCIENTIFIC COUNCIL ON PROBLEM “NEW INORGANIC MATERIALS AND COVERINGS ON THE BASIS OF REFRACTORY COMPOUNDS” Address: 11 GORKY STREET, MOSCOW 103009 USSR

Tel. (7095)2297214

September 25, 1989 No. 3

EXTRACT from the Decision of the Strong Glass Department of the Scientific Council of the State Committee of Science and Engineering on the subject: “Rising Methods for Bearing Capacity of Constructions from Glass and Glass-Crystalline Materials”

On hearing and discussing the report on the subject “Elaboration of Methods and Resources for Strengthening High-Pressure Glass Elements” made by Ph. D. in Engineering L. G. Himmelsohn, the Department considers it necessary:

– to acknowledge a high scientific level and degree of completeness of theoretical research of the stress state and strength of glass elements; – to note progressiveness of the new constructions of deep-sea portholes elaborated on the basis of these investigations; – to admit that the present course of scientific work is urgent and it needs development and realization of further results.

Chairman of the Department, Dr. Sc. in Engineering, Professor I. A. Boguslavsky Scientific Secretary E. M. Akimova

June 6, 1991 No. 60/U-328

Sumy Physical and Technological Institute Scientific Manager of Economic Contracts Mr. L. G. Himmelsohn 2 Rimsky-Korsakov Street, Sumy 244007 USSR

DECISION of the All-Union Seminar (Scientific and Technical Conference) “The Strength Problems of Glass and Glass-Crystalline Materials”

On hearing and thoroughly discussing three reports:

1) “On the Connection of Strength of Glass with the Quantity of Cracks by Destruction”; 2) “The Stress State of Glass Elements of Illuminators (Deep-Sea Portholes)”; 3) “The Strength of Glass Elements of Illuminators (Deep-Sea Portholes)”

made by Ph. D. in Engineering, Senior Scientist L. G. Himmelsohn and aroused the interest of experts, the All-Union Seminar considers it necessary:

– to ascertain scientific newness, trustworthiness, approbation and applied value of the invented analytical method for designing the stress state, strength, and defocusing of glass elements of deep-sea illuminators (portholes) as well as the method and the algorithm for the complex optimization of their strength and optical characteristics; that is qualified as the creation of the scientific foundations of designing illuminators (deep-sea portholes);

- to note the newness of principle and scientific and applied value of the obtained conception of deformation, fracture and optical characteristics of glass elements of deep-sea illuminators (portholes) as well as the originality of a number of their new rational constructions elaborated on the basis of these theoretical conceptions;

– to solicit for the StateBudget financing, having a special purpose, of the work of the present scientific level, degree of completeness, and applied value.

Vice-Chairman of the Organizing Committee of the All-Union Seminar, Chairman of the Glass Department, Manager of the Scientific Department of the Scientific and Research Institute AUTOSTEKLO (Auto Glass),

Ph. D. in Engineering, Senior Scientist V. I. Borulko Secretary of the Glass Department, Senior Scientist G. G. Zhivenkova

Academician M. G. VORONKOV

Professor, Dr. Chem. Sc., Dr. h. c.

Director

Institute of Organic Chemistry

Siberian Branch of the Academy of Sciences of the USSR

1 Favorsky Street, 664033 Irkutsk, USSR

Tel. 46-24-00

July 9, 1992

A Letter of Recommendation

I have a pleasure to recommend you Dr. Leo G. Himmelsohn, Senior Scientist at the Sumy Division, Kharkov Polytechnical Institute, Ukraine.

Dr. Leo Himmelsohn (b. 1952) is a recognized specialist in the field of analytical methods of calculation of stressed-deformational state of light-transparent elements as essentially three-dimensional bodies and the determination of the influence of this state on the strength and optical properties of deep-water port-lights. A great contribution has been made by him to the problem of designing these port-lights as well as those widely used in chemical industry, physico-chemical experiments, and other areas of high-pressure engineering. For his Thesis dealing with the problems above he was awarded the degree of Ph. D. (1987). This research was highly appreciated by the Council of Experts. He continues to hardly work in this field for his Dr. Sc. Thesis. He is the author of 14 scientific publications and 13 inventor's certificates on the design of deep-water port-lights.

He is well known for his high eruditions and ability for work. He is good-natured and communicable.

I am sure that Dr. Leo Himmelsohn will soon adapt himself to new conditions and be able to continue his research activities.

Thank you very much for your attention and consideration.

Sincerely yours,

Academician M. G. Voronkov, Dr. h. c.

UKRAINE BUILDING MATERIALS CORPORATION THE STATE GLASS INSTITUTE OF UKRAINE UkrDIS “AUTOSTEKLO” (AUTO GLASS) 20 Shmidt Street, Konstantinovka 342000 Ukraine

26.11.1993 N° 60/A – 660

DECISION of the International Scientific and Technical Conference “THE TECHNOLOGY AND QUALITY OF GLASS” (Konstantinovka, 24–26.11.1993)

On hearing and discussing the report of Leading Scientist L. G. Himmelsohn on his Dr. Sc. Dissertation “Generalization of Analytic Methods for Solving Strength Problems for Typical Structure Elements in High-Pressure Engineering’’, the conference decided:

1) to note the urgency, newness, high level, broad perspectives, and great scientific value of the generalized analytic methods for solving sets of functional equations, for determining the relative errors of their pseudosolutions, the analytic macroelement method, methods for generalizing and correcting the limiting state criteria for solids from any material under nonstationary loading, the method generalizing reserves, and the methods estimating and correcting measurement errors;

2) to ascertain the scientific and practical value of the applications of the generalized analytic methods to rational controlling the strength and other working properties of glass constructions and other elements for high hydrostatic pressure engineering;

3) to ascertain that the review report made and five theses published to the present conference show that L. G. Himmelsohn’s work completely satisfies the modern Dr. Sc. dissertations demands;

4) to solicit for the State Budget financing, having a special purpose, of L. G. Himmelsohn’s investigations.

Vice-Chairman of the Organizing Committee,

Vice-Director of the State Glass Institute for Scientific Work,

Ph. D. in Engineering A. M. Rayhel

Scientific Secretary T. N. Aver’yanova

R E F E R E N C E (translation) Dr. Sc. Dissertation by Himmelsohn L.G. “Generalization of Analytic Methods for Solving Strength Problems for Typical Structure Elements in High-Pressure Engineering’’

The dissertation is dedicated to the elaboration of mathematical methods for effective solving urgent strength problems for spatial elastic bodies.

The named methods and solutions as well as the introduced concepts of the generalized least upper and greatest lower bounds on a set, the concept of the complete linear independence of a system, and the methods for determining the relative errors of equalities and pseudosolutions are new in the science.

The generalization and correction of the least square method have been proposed.

The general solutions to the harmonic and biharmonic equations in power series have been obtained and applied to stress functions for spatial and axially symmetric elasticity problems in the Papkovich-Neuber and Love forms. For the latter function, the question on the necessity of its biharmonicity to satisfy the equations of equilibrium and continuity has been set and solved positively.

The validity of the results is ensured by using modern approaches and criteria, by methods multivariance and solutions verifiability, by estimating their accuracy, and by comparison them with the known, numeric, and experimental data, the agreement having been achieved.

The methods for solving sets of functional equations, for estimating errors, for typifying spatial body loading diagrams, for individualizing the reserves of the input data of a problem, for correcting measurement errors, the analytic macroelement method, and the methods for the rational control of the stress-strain states and strength of spatial axially symmetric elastic bodies are significant in practice.

By urgency, newness, extent, the scientific level of research, and completeness degree, the work satisfies the modern Dr. Sc. dissertation demands. L. G. Himmelsohn is worthy of the Dr. Sc. degree.

Head of Laboratory

of the Institute for Mathematics of the National Academy of Sciences of Ukraine,

Ukraine State Prize Winner,

Professor, Dr. Sc. in Physics and Mathematics L. P. Nizhnik

R E F E R E N C E (translation) Dr. Sc. Dissertation by Himmelsohn, L. G. “Generalization of Analytic Methods for Solving Strength Problems for Typical Structure Elements in High-Pressure Engineering’’ Specialty 01.02.06 – Dynamics, Strength of Machines, Instruments, and Apparatus

In L. G. Himmelsohn’s work, some new analytic methods for solving strength problems as applied to structure elements under high pressure have been elaborated. Their essential distinction is the simplicity of the design formulas obtained, which is of great importance for designing real structures in high-pressure engineering. At the same time, the author uses the complicated mathematical apparatus of functional analysis, which made it possible, to devise generalized analytic methods for determining the stress and strain states of spatial structure elements. The results of applying these methods to specific structures in high-pressure engineering essentially improve their strength and operational performance corroborating the economical efficiency of the investigations carried out.

It should also be noted that, by using the analytic methods elaborated, the author has designed combined plungers, cylinders, and high-pressure vessels working at both industrial enterprises and scientific and technical institutes and defended by many invention certificates. The results presented are published by well known scientific publishing houses.

I consider that L. G. Himmelsohn’s dissertation corresponds to the demands of the Supreme Attestation Commission, and the candidate for a degree is undoubtedly worthy of awarding the degree of Dr. Sc. in Engineering.

Academician of the Ukrainian National Academy of Sciences,

Professor, Dr. Sc. in Engineering N. V. Novikov

26.05.1994

Ph. D. in Engineering O. O. Leshchuk

The signatures of

N. V. Novikov, Director of the Institute for Hyperhard Materials

of the Ukrainian National Academy of Sciences,

and of

O. O. Leshchuk, Senior Scientist,

are witnessed

Scientific Secretary of the Institute,

Ph. D. in Engineering N. F. Kolesnichenko

R E F E R E N C E (translation)

I have familiarized myself with all the materials prepared by Leo Gregory Himmelsohn, Ph. D. in Engineering, for the defense of his dissertation “Generalization of Analytic Methods for Solving Strength Problems for Typical Structure Elements in High-Pressure Engineering’’ presented for a Dr. Sc. degree in Engineering and can say the following:

The subject of the dissertation dealing with the strength problems for apparatus connected with high pressure is one of the most complicated strength problems in engineering. Almost all structural elements have dimensions of equal orders in all the directions, i.e., strength problems should be solved for spatial components with complicated boundary conditions, tolerances, assembly accuracy, and other peculiarities. Hence it is clear how complex problems are raised in the dissertation.

The candidate for the degree possesses outstanding grounding in elasticity theory and has thoroughly studied the classical works in general and its parts dealing with thick and thin plates. He compares his propositions and solutions with known similar and close ones and tests the accuracy of his solutions by known and experimental data. The approximate approaches adopted can be useful for solving applied problems in stress, strain, and strength analysis when the finite-element method and other ones cannot give results desired.

The material presented for the defense shows that its author is a very good master in elasticity theory, which is proved by the irreproachableness of the dissertation. The results of the investigations prove their practical benefit and show that the author of the work is a skilled and sufficiently experienced scientist who is able to set and solve the most complicated problems in modern mechanics of deformable solids.

There can be no doubt that the dissertation presented possesses all the characteristics of a modern qualification work, and its author, Leo Gregory Himmelsohn, is entirely worthy of awarding the degree of Dr. Sc. in Engineering.

09.06.1994

Scientific Consultant,

Academician of the Ukrainian National Academy of Sciences,

Professor, Dr. Sc. in Engineering G. S. Pisarenko

The signature of G. S. PISARENKO,

Councilor of the Presidium of the Ukrainian National Academy of Sciences,

Honorary Director of the Institute for Strength Problems

of the Ukrainian National Academy of Sciences,

Academician of the Ukrainian National Academy of Sciences,

is witnessed

Scientific Secretary of the Institute for Strength Problems

of the Ukrainian National Academy of Sciences

Ph. D. in Engineering R. I. Kuriat

REFERENCE Monograph “General Strength Theory” by Leo Himmelsohn “Drukar”, Sumy, 1993, Ukraine

The monograph presents one of the urgent problems of the contemporary strength theory of anisotropic materials. As the field of using composition materials has been lately increased, very often demonstrating record and irreplaceable characteristics, the work seems of great interest and urgent.

As a rule these materials have complex granular, fibrous or schistose structure. At present the theory of their destruction is not well developed and it does not include all the experimental data.

The author considering the given problem uses a completely new approach, namely strength criteria correlation with the maximum strength of the material in the given direction. And the generalized strength criteria correlate with the type of stress state at the given point of the element investigated (the monograph, formula 25).

The suggested generalization fully reflects the peculiarities of anisontropic materials and, in fact, develops the fundamental results that were obtained by other researchers for deformation and destruction of isotropic materials.

The efficiency of the problem under consideration is proved by the author at interpretation of well-known experimental data that were obtained during the experiments on strength of composite materials, and those data cannot be kept within the frame of well-known theoretical schemes (Fig. 1 & 2).

Thus, the generalized strength criteria suggested by Leo Himmelsohn reflect real regularity of static and cycle destruction of the materials that have complex structure; criteria are convenient for usage in practical calculation and they are perspective in the valuation of strength of different constructions, made of up-to-date construction materials.

Head of the Strength Laboratory, Irkutsk, Scientific & Research Institute for Chemical Machine Building,

Professor, Dr. Sc. in Engineering,

Specialty “Dynamics and Strength of Machines, Devices, and Apparatus”,

P. G. Pimshtein

Head of the Numerical Method Department,

Ph. D. in Physics and Mathematics, Specialty “Mechanics of the Deformed Solid”

Leo Tsvik

REFERENCE Monograph “Basic New Mathematics” by Leo Himmelsohn “Drukar”, Sumy, 1995

The monograph deals with the system of basic mathematical concepts in some mathematical theories and methods that are used in solving the most urgent problems of the present mechanics and mathematical physics.

The author touches upon such general mathematical concepts as objects, operations and sets, and he is trying to summarize their definition. It all resulted in some deep generalization (sets with multiplicities of elements, the new scale of finite and infinite numbers, etc.) that brought to a new approach to solving several tasks. Thus the multiplicities of elements suggested by the author in creating the theory of sets immensely enlarge the scale of finite and even infinite numbers in comparison with non-standard analysis, etc.

In addition to it there is the generalization of some well-known mathematical methods and description of new ways in solving some urgent tasks. The author suggests a generalizing correction to the method of least squares, the error minimean method to correctly determine relative errors and the row (series) methods using the function or system closed concerning the operator, analytic method of macroelements in elasticity theory, etc.

The conclusions in the monograph are formulated in a very short form that makes it sometimes difficult to understand. That is why it is recommended the monograph be published in more detailed way, with more examples illustrating new methods.

In general the monograph contains new ways and approaches of theoretical generalizations, some practical directions for applying the suggested methods and is of great interest for the specialists in algebra, applied mathematics, and mathematical physics and may be a good help for the post-graduates dealing with mathematics and exact sciences.

Head of the Irkutsk Branch of the Institute of Laser Physics of the Siberian Branch of the Russian Academy of Sciences, Professor, Dr. Sc. in Physics and Mathematics P. I. Ostromensky

R E F E R E N C E

Monograph "Basic New Mathematics" by Ph. D. & Dr. Sc. Leo G. Himmelsohn

The author has proposed dozens of concepts and methods, which generalize known ones and are fundamentals of a certain "new mathematics". Now he calls it "object-oriented mathematics" which deals only with chosen but basic parts of mathematics as its additional alternative: not "instead of" but "together with".

It is well known that the usual mathematics with its axioms' systems is not a language of science and life whose many urgent problems permit no rigorous but only fuzzy approach. The author consequently uses it by formulating his principles of the new scientific thought and of the introduced mathematics by creating it.

Leo G. Himmelsohn considers also uncountable general operations with any objects and possibly a symbolic intermediate sense, which can lead to an actual final one. In his multielements and general sets generalizing Cantor sets, multisets, and fuzzy sets, elements and their quantities are arbitrary objects. Even if an element quantity is a multiplicity (a pure number) between 0 and 1, it can be also nonfuzzy (e.g. a half an apple). Like numbers, general sets are suitable for many useful operations, and their general quantities as possibly uncountable sums of element quantities are sensitive without absorption also to intersecting infinite sets of different dimensions unlike cardinalities and measures. General numbers as pure general quantities exactly express different infinities and infinitesimals by means of general sets.

Each possibly structured object can be expressed by a general system with quantified elements, general relations between them being introduced. For such variable objects, their possibly critical general states and processes are also considered. A critical parameter reduction leads to a general strength theory for an arbitrarily anisotropic nonequiresistant solid nonstationarily loaded.

On the base of such a hyperanalysis, the author has introduced general estimators for general objects and systems. For example, differently possible events always have positive general probabilities naturally different from each other. Distributions and distribution functions on infinite sets are interpreted in the Lobachevskian geometry.

It is shown that the relative error is indeterminate and unbounded contrary to intuition. An introduced general error correctly applicable to any general objects and systems completely sensitively estimates different inexactness, a general reserve discriminates also exact ones by reliability degree.

For inconsistent problems, it is shown that the unique known applicable least square method loses any sense by noncoincidence of physical units. Its result depends on equivalent transformations of a system of equations by their multiplication by different positive numbers. Such an approximation is paradoxical: the greater an absolute value, the less even an absolute error. The result cannot be specified by the method itself. The author has proposed universal and effective iterative methods optimizing general errors and reserves. Another method corrects the error of taking a mean value by measuring, other ones give general solutions to partial differential equations, general sums of divergent series etc.

The proposed object-oriented mathematics was applied also to ion implantation theory, low, medium, and high doses being naturally discriminated by an introduced multiplicity, some new phenomena being discovered.

The presentation of the monograph is too brief. The author should prepare a whole treatise whose publication could be very interesting and important for the development of pure and applied mathematics inclusive fuzzy one and other sciences.

Ph. D. & Dr. Sc. Valery Cherniaev

Professor, Dr. Sc. in Physics and Mathematics Alexander D. Pogrebnjak Director of the Sumy Institute for Surface Modification Sumy 244030 Ukraine Fax 08215977425 Tel. (0542) 220338

LETTER OF RECOMMENDATION

Dear Sir:

I recommend Professor, Dr. Sc. in Engineering Leo G. Himmelsohn as a prominent scientist in mathematics and physics, the author of a new scientific thought, a new mathematics, a general strength theory, and a general theory of ion implantation. He has 3 scientific monographs and manuals, 30 inventions, and 70 articles.

The new mathematics includes over 70 theories and 30 methods for effective solving many fundamental and applied problems and permits to discover new phenomena and general laws of nature and science.

The general strength theory for any states of any solids under arbitrary loading conditions gives known critical state criteria as its very special cases.

The general theory of ion implantation is proposed by L. G. Himmelsohn and is developed in our collaboration by his theoretical analysis of my experimental data and my long-term experience. His key concept of an implantation multiplicity naturally distinguishes low, medium, and high doses of implantation. His simple and adequate formula gives the expected depth of implanted ions. This theory permits to discover new phenomena:

the existence of the critical specific implantation energy,

the abrupt supercritical decrease of the specific resistance of the surface layer of a solid,

the fuzziness of the boundary between the damaged and the strengthened sublayers of the surface layer of a solid,

the coincidence and the common displacement of the principal maximums depths of implanted ions having diverse sizes and initial energies.

I hope that L. G. Himmelsohn can be useful for you and your Institute as a scientist.

Sincerely yours,

Alexander D. Pogrebnjak

R E F E R E N C E

Article "Hyperanalysis: Hypernumbers, Hypersets, and Hyperquantities" by Ph. D. & Dr. Sc. Leo G. Himmelsohn

The author has proposed so-called hyperanalysis as a fundament of his object-oriented mathematics whose hypernumbers, hypersets, and hyperquantities generalize the respective known mathematical objects.

It is shown that the real numbers cannot express many even bounded pure quantities, e.g. the probability of the random sampling of a certain element of an infinite set. The known attempts to introduce infinitesimals justify such introducing but give no instrument to express every given infinite or infinitesimal quantity, for instance such a probability or distributions and distribution functions on sets of infinite measure. Leo G. Himmelsohn has solved this problem by extending the properties of the operations and relations for finite numbers without absorption to infinite ones and infinitesimals expressed by using infinite cardinals.

The Cantor sets, fuzzy sets, and multisets cannot express many nonstructured collections, e.g. a half an apple and 5 l fuel. For such concrete quantities, there is no suitable mathematical model and no known operation. Set operations with absorption are only restrictedly reversible. The author has introduced hyperoperations such as quantifying which gives hyperelements and hypersets whose quantities together with elements can be any (possibly fuzzy) objects and generalize known multiplicities that can be also nonfuzzy if they are between 0 and 1. The hypersets can express any collection and, like numbers, can be operands by many operations.

The cardinality is sensitive only to non-intersecting finite sets. The measures (probabilities) are sensitive only to a certain dimension and cannot discriminate the empty set and null sets (impossible and differently possible events). The known algebraic operations require at most countable sets of operands. Leo G. Himmelsohn has introduced the hyperquantity of a hyperset as the possibly uncountable sum of the quantities of all the elements of the hyperset. If a hyperquantity is pure then it can be expressed by a hypernumber. Such hyperquantities exactly estimate even the quantity of the elements of any infinite set, naturally vary together with it, and give a universal and even uncountably algebraically additive degree of quantity.

The article sets an unexpected connection between probability theory and non-Euclidean geometry. It is interesting that distributions and distribution functions on the real numbers can be interpreted in the Lobachevskian geometry.

It is important that all the interconnected concepts introduced in the article do not replace usual ones but are alternative and additional. The author answers many difficult questions set by Bolzano and other famous mathematicians and gives different examples of applying his hyperanalysis to many urgent problems that cannot be even considered in the known mathematics. The article can be very useful for researchers not only in pure and applied mathematics but also in other sciences and is recommended for a publication.

Ph. D. & Dr. Sc. Valery Cherniaev