Ph. D. & Dr. Sc. Lev Gelimson's Triangle Theory in Fundamental Science of General Problem Testing

Triangle Theory in Fundamental Science of General Problem Testing

© Ph. D. & Dr. Sc. Lev Gelimson

Academic Institute for Creating Fundamental Sciences (Munich, Germany)

Mathematical Journal

of the "Collegium" All World Academy of Sciences

Munich (Germany)

11 (2011), 54

Introduction

By solving contradictory (e.g., overdetermined [1]) problems without precise solutions, it is necessary to find the best pseudosolutions, so-called quasisolutions [2-5]. If such a problem is a set of equations, then their graphs in a Cartesian coordinate system have no point in common but in many cases determine a certain (limited if possible) point set whose center (in some reasonable sense) could be considered as the desired quasisolution. The straightforward basic idea is as follows. If it is impossible to precisely satisfy all the given equations and each point (pseudosolution) gives deviations (e.g., errors), then it is logical to try to equally (uniformly, homogeneously) distribute them among all the given equations. Such an approach corresponds to intuition and leads to the intuitive concept of the center (in some reasonable sense) of that point set.

In classical mathematics [1], to solve such overdetermined sets of equations, the least square method (LSM) [1] by Legendre and Gauss only usually applies. Overmathematics [2-4] and the system of fundamental sciences on general problems [5] have discovered many principal shortcomings of this method, by methods of finite elements, points, etc. Additionally consider its simplest approach which is typical. Minimizing the sum of the squared differences of the alone preselected coordinates (e.g., ordinates in a two-dimensional problem) of the graph of the desired approximation function and of everyone among the given functions depends on this preselection, ignores the remaining coordinates, and provides no coordinate system rotation invariance and hence no objective sense of the result.

The implicit center criterion of the least square method (LSM) [1] is based, in particular, on the following:

1) determining the componentwise deviation by a separate equation via the absolute error;

2) determining the total deviation by a whole set of equations as the quadratic mean value of the componentwise deviations by all the separate equations.

Overmathematics [2-4] and the system of fundamental sciences on general problems [5] have discovered many principal shortcomings of the absolute error. And the second power is often insufficient to find realistic point set centers.

General center theory (GCT) in fundamental science of general problem testing is based, in particular, on the following:

1) determining the componentwise deviation by a separate equation via adequate estimators such as distances which are invariant by coordinate system rotations, unierrors and reserves [2-5];

2) determining the total deviation by a whole set of equations via much more general and adequate functions of the componentwise deviations by all the separate equations.

Nota bene: Uncountable operations and their results are not considered in classical mathematics [1] at all. In particular, this holds both for addition (and its result, namely a sum) and a set of equations (also with completely ignoring their quantities). On the contrary, overmathematics [2-4] considers any (also uncountable) sets, quantisets, systems, and quantisystems of any objects, operations, and relations. In particular, this holds both for addition (and its result, namely a sum) and a quantiset of equations (also with completely taking their quantities into account).

The essence of a general problem includes, in particular, its origin (source) which can give very different settings (and hence both mathematical models and results) of a general problem even if graphical interpretations seem to be very similar or almost identical. For example, in the two-dimensional case, the same graphical interpretation with a triangle corresponds to many very different general problem settings and, moreover, to many very different general problems and even their systems (sets, families, etc.). Among them are, e.g., the following with determining:

1) the point nearest to the set or to the quantiset (with own quantities, which is very important by coinciding straight lines) of the three straight lines including the three sides, respectively, of the given triangle by different nearness criteria;

2) the point nearest to the triangle boundary, i.e. either to the set or to the quantiset (with counting the vertices twice) of all the points of the three sides of the given triangle by different nearness criteria;

3) the incenter and/or all the three excenters [1] of the given triangle;

4) the circumference (circle containing all the three vertices) of the given triangle;

5) the gravity (mass, length, uniquantity [2-5]) center of the triangle boundary, e.g., either of the set or of the quantiset (with counting the vertices twice) of all the points of the three sides of the given triangle;

6) the gravity (mass, area, uniquantity [2-5]) center of the triangle area including its interior and either including or not including its boundary, e.g., either of the set or of the quantiset (with counting the vertices twice) of all the points of the three sides of the given triangle.

The similar holds for a tetrahedron in the three-dimensional case with natural additional possibilities (the incenter/excenters for its flat faces along with the carcass incenter/excenters for its straight edges etc.).

By curvilinearity, the usual distance from any selected point to a certain point which lies on the curve or in the curvilinear surface is not the only. It is also possible to consider the distance from the selected point to the tangent (straight line or plane, respectively, if it exists) to the curve or curvilinear surface at that certain point if this tangent is the only. Otherwise, consider a certain suitable nonnegative function of the distances from the selected point to all the tangents. Additionally, if the selected point lies on the same curve or in the same curvilinear surface, then the usual straight line distance is not the only. It is also possible to consider the curvilinear distance as the greatest lower bound of the lengths of the curves lying on that curve or in that curvilinear surface and connecting those both points (simply the length of the shortest curve lying on that curve or in that curvilinear surface and connecting the both points if it exists). The similar can hold for polygons and polyhedra. Naturally, it is also possible to consider other conditions and limitations.

General Distance Center

The general center of a general problem depends on a general problem estimator which can be simply a distance. If for the graphs of all the subproblems, there is an inscribed general sphere (usual circumference in the two-dimensional case or usual sphere in the three-dimensional case), then regard its center (so-called incenter [1]) as the proper center (of a general problem) which is naturally the only and is simultaneously the general center of a general problem. Otherwise, define and determine the general center of a general problem via a reasonable and suitable general problem estimator as a criterion. Naturally, there can be many reasonable and suitable general problem estimators and hence many reasonable and suitable general centers of a general problem. To test general problem estimators for their reasonability, adequacy, usability, and suitability, apply them to general problems whose proper centers exist and can be relatively simply defined and determined. In the two-dimensional case, consider (naturally, convex) polygons with existing inscribed circles (so-called incircles [1]) (which is the case by any triangle) and the set of the equations of the straight lines each of which includes a certain side of such a polygon. In the three-dimensional case, there are two possibilities:

1) consider (naturally, convex) polyhedrons with existing incenters (centers of inspheres, i.e., spheres inscribed in all their flat faces) (which is the case by any tetrahedron) and the set of the equations of the planes each of which includes a certain flat face of such a polyhedron;

2) consider (naturally, convex) polyhedrons with existing carcass incenters (centers of carcass inspheres, i.e., spheres inscribed in all their straight edges) (which is the case by any tetrahedron) and the set of the equations of the straight lines each of which includes a certain straight edge of such a polyhedron.

Similarly consider further multidimensional spaces if necessary.

Nota bene: If there is no incenter but there are excenters [1], it is inadmissible to simply replace above the incenter with one of the excenters. The reason is that at an excenter, a general problem estimator takes equal values by all the subproblems but a suitable nonnegative subproblems estimations unification function F can take smaller values at other points than its value at an excenter. For example, consider a circle arc with a relatively small central angle (e.g., π/1800), divide this arc via 9 points into 10 equal parts, and add the both arc endpoints. Build 11 straight lines touching this circle (its tangents) at these 11 points. Compare the power mean distances (by any common power p ≥ 1) both of the circle center (which is here an excenter) and of the arc midpoint (which lies here near the general center of the set of these 11 straight lines) from these 11 straight lines. Now consider the circle radius infinitely increasing. Then the limit of that arc is a straight line segment included in each of these 11 straight lines. The limit of the general center of the set of these 11 straight lines is the midpoint of that segment. The limit of the power mean distance of the general center of the set of these 11 straight lines from these 11 straight lines vanishes. But the power mean distance of the circle center (which is here an excenter) from these 11 straight lines equals the circle radius and infinitely increases together with it. The critical value of the central angle of the arc is namely π . By central angles not greater than π , the excenter keeps this role whereas by central angles greater than π , the excenter becomes the incenter and, naturally, the general center jumps by precisely half a circle to this incenter (former excenter).

General Unierror Center

The general unierror center of a general problem depends on a general problem unierror as a general problem estimator. If for the graphs of all the subproblems, there is a point at which both all the subproblems unierrors are equal to one another and a general problem unierror as a general problem estimator takes its minimum value, then regard this point (so-called unierror incenter) as the proper unierror center (of a general problem) which is naturally the only and is simultaneously the general unierror center of a general problem. Otherwise, define and determine the general unierror center of a general problem via a reasonable and suitable general problem unierror as a general problem estimator and a criterion. Naturally, there can be many reasonable and suitable general problem unierrors and hence many reasonable and suitable general unierror centers of a general problem. To test general problem unierrors for their reasonability, adequacy, usability, and suitability, apply them to general problems whose proper unierror centers exist and can be relatively simply defined and determined. In the two-dimensional case, consider specially constructed (naturally, convex) polygons with existing unierror incenters (which is the case by any triangle) and the set of the equations of the straight lines each of which includes a certain side of such a polygon. In the three-dimensional case, there are two possibilities:

1) consider (naturally, convex) polyhedrons with existing unierror incenters (with respect to all their flat faces) (which is the case by any tetrahedron) and the set of the equations of the planes each of which includes a certain flat face of such a polyhedron;

2) consider (naturally, convex) polyhedrons with existing carcass unierror incenters (with respect to all their straight edges) (which is the case by any tetrahedron) and the set of the equations of the straight lines each of which includes a certain straight edge of such a polyhedron.

Similarly consider further multidimensional spaces if necessary.

Nota bene: If there is no unierror incenter but there are unierror excenters, it is inadmissible to simply replace above the unierror incenter with one of the unierror excenters. The reason is that at a unierror excenter, a general problem estimator takes equal values by all the subproblems but a suitable nonnegative subproblems estimations unification function F can take smaller values at other points than its value at a unierror excenter.

Triangle Theory

Triangle theory (TT) in fundamental science of general problem testing provides testing general problem solving theories and methods via applying them namely to a triangle. The reasons are that for any triangle, their proper center exists and can be relatively simply determined. To provide result comparison, this theory applies them to triangles of some specific forms. To investigate result invariance, this theory also considers coordinate system linear transformations including translation, rotation, and axes units variation.

To begin with, deal with the triangle form only and consider an isosceles triangle (with two equal sides and their opposite angles). It has a symmetry axis to which triangle centers practically in any reasonable senses have to belong. Use this symmetry axis as one of the two-dimensional coordinate system axes, e.g., x-axis. Take the common vertex of two equal sides as the coordinate system origin. Preselect namely three (quasi)critical cases:

1) a relatively wide isosceles triangle namely if the common vertex of two equal sides is relatively close to the midpoint of the remaining third side. In this case, triangle centers practically in any reasonable senses are also relatively close to this midpoint;

2) an equilateral triangle (whose sides have the same length and all three angles measure 60°) which is a regular triangle. In this case, triangle centers practically in any reasonable senses coincide with the triangle gravity center as the intersection of all the three triangle medians coinciding with its altitudes (whose intersection is a triangle orthocenter), angle bisectors (whose intersection is a triangle incenter), and perpendicular bisectors (whose intersection is a triangle circumcenter);

3) a relatively narrow isosceles triangle namely if the common length of the two equal sides is greater than the length of the remaining third side by magnitude order, which provides relatively great distances between triangle centers in distinct reasonable senses.

Nota bene: A relatively narrow isosceles triangle is hence the best to truly test general problem solving theories and methods. Both a relatively wide isosceles triangle and an equilateral triangle are trivial cases which can be additionally used to test and possibly prove namely the extreme inadequacy of problem solving theories and methods giving pseudosolutions with relatively great deviations from relatively narrow isosceles triangles centers practically in any reasonable senses.

Isosceles Triangles in Two-Dimensional Coordinate Systems

Use the symmetry axis of an isosceles triangle as one of the two-dimensional coordinate system axes, e.g., x-axis. Consider two test triangles:

1) an origin-top isosceles triangle TOP with vertices (and their coordinates x , y) T(a , -b), O(0, 0), P(a , b) where a , b are any positive numbers. Then isosceles triangle TOP sides (edges) equations are

OT: bx + ay = 0,

OP: bx - ay = 0,

TP: x - a = 0;

2) an origin-base isosceles triangle BAS with vertices (and their coordinates x , y) B(0, -b), A(a , 0), S(0 , b) where a , b are any positive numbers. Then isosceles triangle BAS sides (edges) equations are

AB: bx + ay - ab = 0,

AS: bx - ay - ab = 0,

BS: x = 0.

Nota bene: To begin with, we have composed these three equations namely in their natural forms "as is" typical in practically solving problems with avoiding any additional equivalent (admissible but unnecessary, artificial, etc.) equation transformation, e.g.,

OT: 100bx + 100ay = 0,

OP: 10bx - 10ay = 0,

TP: x - a = 0.

Such transformations would be unrealistic in practically solving problems but can be very useful to test and possibly prove not only namely the extreme inadequacy of problem solving theories and methods giving pseudosolutions with relatively great deviations from relatively narrow isosceles triangles centers practically in any reasonable senses but also, moreover, problem solving theories and methods objective sense loss.

Triangle TOP Center

Isosceles triangle TOP sides (edges) lengths are

L(OT) = (a² + b²)^1/2 ,

L(OP) = (a² + b²)^1/2 ,

L(TP) = 2b .

Incenter C coordinates [1] are

x(C) = [L(OP) x(T) + L(TP) x(O) + L(OT) x(P)]/[L(OP) + L(TP) + L(OT)] =

[(a² + b²)^1/2 a + 2b 0 + (a² + b²)^1/2 a]/[(a² + b²)^1/2 + 2b + (a² + b²)^1/2] =

a(a² + b²)^1/2/[(a² + b²)^1/2 + b] =

a - ab/[(a² + b²)^1/2 + b]

and (which is natural because the x-axis is a symmetry axis)

y(C) = [L(OP) y(T) + L(TP) y(O) + L(OT) y(P)]/[L(OP) + L(TP) + L(OT)] =

[(a² + b²)^1/2 (-b) + 2b 0 + (a² + b²)^1/2 b]/[(a² + b²)^1/2 + 2b + (a² + b²)^1/2] = 0.

Nota bene: Because of proportionality invariance, NOT the values of a and b themselves but the values of ratios a/b and b/a only are important.

Form Invariance Test

x(C)/a = (a² + b²)^1/2/[(a² + b²)^1/2 + b] =

[(a/b)² + 1]^1/2/{[(a/b)² + 1]^1/2 + 1},

quod erat demonstrandum.

We obtain the above three (quasi)critical cases via selecting some suitable values of these ratios:

1) a relatively wide isosceles triangle if a << b ;

2) an equilateral triangle if a = 3^1/2b ;

3) a relatively narrow isosceles triangle if a >> b .

Triangle BAS Center

Isosceles triangle BAS sides (edges) lengths are

L(AB) = (a² + b²)^1/2 ,

L(AS) = (a² + b²)^1/2 ,

L(BS) = 2b .

Incenter C coordinates [1] are

x(C) = [L(AS) x(B) + L(BS) x(A) + L(AB) x(S)]/[L(AS) + L(BS) + L(AB)] =

[(a² + b²)^1/2 0 + 2b a + (a² + b²)^1/2 0]/[(a² + b²)^1/2 + 2b + (a² + b²)^1/2] =

ab/[(a² + b²)^1/2 + b]

and (which is natural because the x-axis is a symmetry axis)

y(C) = [L(AS) y(B) + L(BS) y(A) + L(AB) y(S)]/[L(AS) + L(BS) + L(AB)] =

[(a² + b²)^1/2 (-b) + 2b 0 + (a² + b²)^1/2 b]/[(a² + b²)^1/2 + 2b + (a² + b²)^1/2] = 0.

Nota bene: Because of proportionality invariance, NOT the values of a and b themselves but the values of ratios a/b and b/a only are important.

Form Invariance Test

x(C)/a = (b/a)/{[1 + (b/a)²]^1/2 + b/a},

quod erat demonstrandum.

We obtain the above three (quasi)critical cases via selecting some suitable values of these ratios:

1) a relatively wide isosceles triangle if a << b ;

2) an equilateral triangle if a = 3^1/2b ;

3) a relatively narrow isosceles triangle if a >> b .

Testing the Least Square Method (LSM)

The least square method (LSM) [1] minimizes the sum of the squares of the differences of the both parts of all the equations in a problem to be solved.

OT: bx + ay = 0,

OP: bx - ay = 0,

TP: x - a = 0.

In this problem

²S = (bx + ay)² + (bx - ay)² + (x - a)² .

Vanishing derivatives gives

²S'_x = 2(bx + ay)b + 2(bx - ay)b + 2(x - a) = 0,

²S'_y = 2(bx + ay)a + 2(bx - ay)(-a) = 0;

4b²x + 2x - 2a = 0,

4a²y = 0;

x = a/(2b² + 1),

y = 0.

Form Invariance Test

x/a = 1/(2b² + 1)

very strongly depends on b itself without proportionality invariance, which is fully false and means objective sense loss.

Analytic Test

We have, e.g., by b = 1

x/a = 1/3

by any positive a , in particular by

a = 3^1/2

giving an equilateral triangle for which clearly

x/a = 2/3.

Nota bene: b = 1/2 is the only value of b giving

x/a = 2/3

for an equilateral triangle with

a = 3^1/2/2

only.

AB: bx + ay - ab = 0,

AS: bx - ay - ab = 0,

BS: x = 0.

In this problem

²S = (bx + ay - ab)² + (bx - ay - ab)² + x² .

Vanishing derivatives gives

²S'_x = 2(bx + ay - ab)b + 2(bx - ay - ab)b + 2x = 0,

²S'_y = 2(bx + ay - ab)a + 2(bx - ay - ab)(-a) = 0;

4b²x + 2x - 4ab² = 0,

4a²y = 0;

x = 2ab²/(2b² + 1),

y = 0.

Form Invariance Test

x/a = 2b²/(2b² + 1)

very strongly depends on b itself without proportionality invariance, which is fully false and means objective sense loss.

Analytic Test

We have, e.g., by b = 1

x/a = 2/3

by any positive a , in particular by

a = 3^1/2

giving an equilateral triangle for which clearly

x/a = 1/3.

Nota bene: b = 1/2 is the only value of B giving

x/a = 1/3

for an equilateral triangle with

a = 3^1/2/2

only.

Hence the least square method (LSM) [1] gives as a rule fully false results with objective sense loss.

Testing Unit Factor Power Theories (UFPT)

Unit factor power theories (UFPT) (with any power exponent t > 1) [5] are based on the following algorithm for a set of linear equations:

Stage I. Linear Equation Set Preparation via Its Transformation

Step I.1) for every preselected unknown variable separately, select all the equations with initial nonzero factors at this preselected unknown variable and transform each of these equations via its division by the initial nonzero factor at this unknown variable in this equation so that the transformed factor becomes 1 (unit);

Step I.2) for this preselected unknown variable and for each of the remaining equations (with initial zero factor at this preselected unknown variable in this equation) separately,

determine all the nonzero factor unknown variables with initial nonzero factors in this equation;

Step I.3) for this preselected unknown variable, for each of the remaining equations (with initial zero factor at this preselected unknown variable in this equation), and for each of these nonzero factor unknown variables separately,

determine the ratio of a mean (e.g., a power mean) of the moduli (absolute values) of the factors at this nonzero factor unknown variable in all the already transformed equations with initial nonzero factors at this preselected unknown variable to the factor at this nonzero factor unknown variable in this equation;

Step I.4) for this preselected unknown variable, for each of the remaining equations (with initial zero factor at this preselected unknown variable in this equation), and for all the nonzero factor unknown variables separately,

determine a mean value (e.g., a geometric mean) of the moduli (absolute values) of the above ratios for each of all these nonzero factor unknown variables separately, give this mean value a sign which is the product of the signs of the above ratios, and multiply this equation with this signed mean value.

Stage II. Linear Equation Set Center Determination via Difference Module Power Mean Minimization

Step II.5) for every preselected unknown variable separately, compose the initial separate sum of the powers (with the same power exponent t > 1) of the moduli (absolute values) of the differences of the both parts of all the transformed equations in a problem to be solved;

Step II.6) for every preselected unknown variable separately, compose the initial separate sum of the powers (with the same power exponent t > 1) of the moduli (absolute values) of the free factors in all these transformed equations in a problem to be solved;

Step II.7) for every preselected unknown variable separately, compose the free-factor normalized separate sum of the powers (with the same power exponent t > 1) of the moduli (absolute values) of the differences of the both parts of all the transformed equations via dividing the initial separate sum of the powers of the moduli (absolute values) of the differences of the both parts of all the transformed equations by the initial separate sum of the powers of the moduli (absolute values) of the free factors in all these transformed equations;

Step II.8) compose the total free-factor normalized sum of the powers (with the same power exponent t > 1) of the moduli (absolute values) of the differences of the both parts of all the transformed equations via adding the separate free-factor normalized sums for all the preselected unknown variables;

Step II.9) minimize the total free-factor normalized sum of the powers (with the same power exponent t > 1) of the moduli (absolute values) of the differences of the both parts of all the transformed equations in a problem to be solved.

Apply this algorithm to

OT: bx + ay = 0,

OP: bx - ay = 0,

TP: x - a = 0.

Stage I. Linear Equation Set Preparation via Its Transformation

Step I.2) for this preselected unknown variable and for each of the remaining equations (with initial zero factor at this preselected unknown variable in this equation) separately,

determine all the nonzero factor unknown variables with initial nonzero factors in this equation;

determine the ratio of a mean (e.g., a power mean) of the moduli (absolute values) of the factors at this nonzero factor unknown variable in all the already transformed equations with initial nonzero factors at this preselected unknown variable to the factor at this nonzero factor unknown variable in this equation;

Step I.4) for this preselected unknown variable, for each of the remaining equations (with initial zero factor at this preselected unknown variable in this equation), and for all the nonzero factor unknown variables separately,

For preselected unknown variable x separately,

in all the equations, the initial factors at this preselected unknown variable x are nonzero. Hence simply omit steps I.2, I.3, and I.4.

Step I.1) Simply select all the equations (with initial nonzero factors at this preselected unknown variable x).

Transform each of these equations via its division by the initial nonzero factor at this unknown variable in this equation so that the transformed factor becomes 1 (unit).

Compose the initial separate set of linear equations for preselected unknown variable x :

x + a/b y = 0,

x - a/b y = 0,

x - a = 0.

Hence transforming the initial separate set of linear equations for this preselected unknown variable x is completed.

For preselected unknown variable y separately,

in the equations

bx + ay = 0,

bx - ay = 0,

the initial factors at this preselected unknown variable y are nonzero. Hence simply omit steps I.2, I.3, and I.4.

Step I.1) Simply select all the equations (with initial nonzero factors at this preselected unknown variable y).

Transform each of these equations via its division by the initial nonzero factor at this unknown variable in this equation so that the transformed factor becomes 1 (unit).

Compose the nonzero factor subset

b/a x + y = 0,

- b/a x + y = 0

of the initial separate set of linear equations for preselected unknown variable y .

The only remaining equation (with initial zero factor at this preselected unknown variable y in this equation) is

x - a = 0.

Step I.2) for this preselected unknown variable y and for each of the remaining equations (with initial zero factor at this preselected unknown variable y in this equation) separately,

determine all the nonzero factor unknown variables with initial nonzero factors in this equation:

Here we determine nonzero factor unknown variable x only.

Step I.3) for this preselected unknown variable y , for each of the remaining equations (with initial zero factor at this preselected unknown variable y in this equation)

x - a = 0,

and for each of these nonzero factor unknown variables x separately,

determine the ratio b/a of a mean (e.g., a power mean) b/a of the moduli (absolute values) of the factors b/a and - b/a at this nonzero factor unknown variable x in all the already transformed equations with initial nonzero factors at this preselected unknown variable y

b/a x + y = 0,

- b/a x + y = 0

to the factor 1 at this nonzero factor unknown variable x in this equation

x - a = 0.

Here the moduli (absolute values) of the factors b/a and - b/a are both b/a (coinciding).

Hence any reasonable mean of them has to equal b/a , too.

Step I.4) for this preselected unknown variable y , for each of the remaining equations (with initial zero factor at this preselected unknown variable in this equation)

x - a = 0,

and for all the nonzero factor unknown variables separately,

determine a mean value b/a (e.g., a geometric mean) of the moduli (absolute values) of the above ratios b/a for each of all these nonzero factor unknown variables x separately, give this mean value b/a the sign + 1 which is the product of the signs + 1 of the above ratios b/a , and multiply this equation

x - a = 0

with this signed mean value b/a :

b/a x - b = 0.

This only equation forms the zero factor subset of the initial separate set

b/a x + y = 0,

- b/a x + y = 0,

b/a x - b = 0

of linear equations for preselected unknown variable y .

Hence transforming the initial separate set of linear equations for this preselected unknown variable y is completed.

Stage II. Linear Equation Set Center Determination via Difference Module Power Mean Minimization

For preselected unknown variable x separately,

the initial separate set of linear equations is

x + a/b y = 0,

x - a/b y = 0,

x - a = 0.

The initial separate sum of the powers (with the same power exponent t > 1) of the moduli (absolute values) of the differences of the both parts of all the transformed equations in a problem to be solved for preselected unknown variable x is

^{t ,
x}^{, d}S = |x + a/b y|^t + |x - a/b y|^t + |x - a|^t .

For preselected unknown variable y separately,

the initial separate set of linear equations is

b/a x + y = 0,

- b/a x + y = 0,

b/a x - b = 0.

^{t , y ,
d}S = |b/a x + y|^t + |- b/a x + y|^t + |b/a x - b|^t .

For preselected unknown variable x separately,

the initial separate sum of the powers (with the same power exponent t > 1) of the moduli (absolute values) of the free factors in all these transformed equations in a problem to be solved is

^{t , x ,
f}S = 0^t + 0^t + a^t = a^t .

For preselected unknown variable y separately,

the initial separate sum of the powers (with the same power exponent t > 1) of the moduli (absolute values) of the free factors in all these transformed equations in a problem to be solved is

^{t , y ,
f}S = 0^t + 0^t + b^t = b^t .

For preselected unknown variable x separately,

the free-factor normalized separate sum of the powers (with the same power exponent t > 1) of the differences of the both parts of all the transformed equations is

^{t , x ,
n}S = ^{t , x , d}S / ^{t , x , f}S = [|x + a/b y|^t + |x - a/b y|^t + |x - a|^t] / a^t = |x/a + y/b|^t + |x/a - y/b|^t + |x/a - 1|^t .

For preselected unknown variable y separately,

the free-factor normalized separate sum of the powers (with the same power exponent t > 1) of the differences of the both parts of all the transformed equations is

^{t , y ,
n}S = ^{t , y , d}S / ^{t , y , f}S = [|b/a x + y|^t + |- b/a x + y|^t + |b/a x - b|^t] / b^t = |x/a + y/b|^t + |x/a - y/b|^t + |x/a - 1|^t .

Nota bene: In this simple problem we obtain

^{t , x ,
n}S = ^{t , y , n}S .

Such coinciding tests and proves the analytic correctness of this theory but is accidental. Generally, we can expect more complicated correlation between free-factor normalized separate sums for different preselected unknown variables.

The total free-factor normalized sum of the powers (with the same power exponent t > 1) of the moduli (absolute values) of the differences of the both parts of all the transformed equations is

^tS = ^{t , x , n}S + ^{t , y , n}S = 2[|x/a + y/b|^t + |x/a - y/b|^t + |x/a - 1|^t] .

To begin with, let us consider the simplest case t = 2.

Testing Unit Factor Quadrat Theory

Unit factor quadrat theory (UFQT) [5] minimizes the total sum of the squares of the differences of the both parts of all the normalized equations of their separate sets for all the preselected unknown variables in a problem to be solved, namely in this problem

²S = 2[(x/a + y/b)² + (x/a - y/b)² + (x/a - 1)²] .

Vanishing derivatives gives

²S'_x = 4(x/a + y/b)/a + 4(x/a - y/b)/a + 4(x/a - 1)/a = 0,

²S'_y = 4(x/a + y/b)/b + 4(x/a - y/b)(-1)/b = 0;

x = a/3,

y = 0.

Form Invariance Test

x/a = 1/3

provides proportionality invariance, which is correct.

Analytic Test

We have

x/a = 1/3

by any positive a , in particular by

a = 3^1/2b

giving an equilateral triangle for which clearly

x/a = 2/3.

Now apply the same algorithm of unit factor power theory (UFPT) (with any power exponent t > 1) [5] to

AB: bx + ay - ab = 0,

AS: bx - ay - ab = 0,

BS: x = 0.

Stage I. Linear Equation Set Preparation via Its Transformation

Step I.2) for this preselected unknown variable and for each of the remaining equations (with initial zero factor at this preselected unknown variable in this equation) separately,

determine all the nonzero factor unknown variables with initial nonzero factors in this equation;

determine the ratio of a mean (e.g., a power mean) of the moduli (absolute values) of the factors at this nonzero factor unknown variable in all the already transformed equations with initial nonzero factors at this preselected unknown variable to the factor at this nonzero factor unknown variable in this equation;

For preselected unknown variable x separately,

in all the equations, the initial factors at this preselected unknown variable x are nonzero. Hence simply omit steps I.2, I.3, and I.4.

Step I.1) Simply select all the equations (with initial nonzero factors at this preselected unknown variable x).

Transform each of these equations via its division by the initial nonzero factor at this unknown variable in this equation so that the transformed factor becomes 1 (unit).

Compose the initial separate set of linear equations for preselected unknown variable x :

x + a/b y - a = 0,

x - a/b y - a = 0,

x = 0.

Hence transforming the initial separate set of linear equations for this preselected unknown variable x is completed.

For preselected unknown variable y separately,

in the equations

bx + ay - ab = 0,

bx - ay - ab = 0,

the initial factors at this preselected unknown variable y are nonzero. Hence simply omit steps I.2, I.3, and I.4.

Step I.1) Simply select all the equations (with initial nonzero factors at this preselected unknown variable y).

Transform each of these equations via its division by the initial nonzero factor at this unknown variable in this equation so that the transformed factor becomes 1 (unit).

Compose the nonzero factor subset

b/a x + y - b = 0,

- b/a x + y + b = 0

of the initial separate set of linear equations for preselected unknown variable y .

The only remaining equation (with initial zero factor at this preselected unknown variable y in this equation) is

x = 0.

Step I.2) for this preselected unknown variable y and for each of the remaining equations (with initial zero factor at this preselected unknown variable y in this equation) separately,

determine all the nonzero factor unknown variables with initial nonzero factors in this equation:

Here we determine nonzero factor unknown variable x only.

Step I.3) for this preselected unknown variable y , for each of the remaining equations (with initial zero factor at this preselected unknown variable y in this equation)

x = 0,

and for each of these nonzero factor unknown variables x separately,

b/a x + y - b = 0,

- b/a x + y + b = 0

to the factor 1 at this nonzero factor unknown variable x in this equation

x = 0.

Here the moduli (absolute values) of the factors b/a and - b/a are both b/a (coinciding).

Hence any reasonable mean of them has to equal b/a , too.

Step I.4) for this preselected unknown variable y , for each of the remaining equations (with initial zero factor at this preselected unknown variable in this equation)

x = 0,

and for all the nonzero factor unknown variables separately,

x = 0

with this signed mean value b/a :

b/a x = 0.

This only equation forms the zero factor subset of the initial separate set

b/a x + y - b = 0,

- b/a x + y + b = 0,

b/a x = 0

of linear equations for preselected unknown variable y .

Hence transforming the initial separate set of linear equations for this preselected unknown variable y is completed.

Stage II. Linear Equation Set Center Determination via Difference Module Power Mean Minimization

For preselected unknown variable x separately,

the initial separate set of linear equations is

x + a/b y - a = 0,

x - a/b y - a = 0,

x = 0.

^{t ,
x}^{, d}S = |x + a/b y - a|^t + |x - a/b y - a|^t + |x|^t .

For preselected unknown variable y separately,

the initial separate set of linear equations is

b/a x + y - b = 0,

- b/a x + y + b = 0,

b/a x = 0.

^{t , y ,
d}S = |b/a x + y - b|^t + |- b/a x + y + b|^t + |b/a x|^t .

For preselected unknown variable x separately,

the initial separate sum of the powers (with the same power exponent t > 1) of the moduli (absolute values) of the free factors in all these transformed equations in a problem to be solved is

^{t , x ,
f}S = a^t + a^t + 0^t = 2a^t .

For preselected unknown variable y separately,

the initial separate sum of the powers (with the same power exponent t > 1) of the moduli (absolute values) of the free factors in all these transformed equations in a problem to be solved is

^{t , y ,
f}S = b^t + b^t + 0^t = 2b^t .

For preselected unknown variable x separately,

the free-factor normalized separate sum of the powers (with the same power exponent t > 1) of the differences of the both parts of all the transformed equations is

^{t , x ,
n}S = ^{t , x , d}S / ^{t , x , f}S = [|x + a/b y - a|^t + |x - a/b y - a|^t + |x|^t] / (2a^t) = (|x/a + y/b - 1|^t + |x/a - y/b - 1|^t + |x/a|^t)/2.

For preselected unknown variable y separately,

the free-factor normalized separate sum of the powers (with the same power exponent t > 1) of the differences of the both parts of all the transformed equations is

^{t , y ,
n}S = ^{t , y , d}S / ^{t , y , f}S = [|b/a x + y - b|^t + |- b/a x + y + b|^t + |b/a x|^t] / (2b^t) = (|x/a + y/b - 1|^t + |x/a - y/b - 1|^t + |x/a|^t)/2.

Nota bene: In this simple problem we obtain

^{t , x ,
n}S = ^{t , y , n}S .

The total free-factor normalized sum of the powers (with the same power exponent t > 1) of the moduli (absolute values) of the differences of the both parts of all the transformed equations is

^tS = ^{t , x , n}S + ^{t , y , n}S = |x/a + y/b - 1|^t + |x/a - y/b - 1|^t + |x/a|^t.

To begin with, let us consider the simplest case t = 2.

Testing Unit Factor Quadrat Theory

²S = (x/a + y/b - 1)² + (x/a - y/b - 1)² + (x/a)².

Vanishing derivatives gives

²S'_x = 2(x/a + y/b - 1)/a + 2(x/a - y/b - 1)/a + 2(x/a)/a = 0,

²S'_y = 2(x/a + y/b - 1)/b + 2(x/a - y/b - 1)(-1)/b = 0;

x = 2a/3,

y = 0.

Form Invariance Test

x/a = 2/3

provides proportionality invariance, which is correct.

Analytic Test

We have

x/a = 2/3

by any positive a , in particular by

a = 3^1/2b

giving an equilateral triangle for which clearly

x/a = 1/3.

Now consider two modifications of unit factor quadrat theory (UFQT) [5] which have additional subalgorithms regarding the remaining equations (with initial zero factors at variables) and show their essences in our case of an origin-base isosceles triangle BAS with vertices (and their coordinates x , y)

B(0, -b), A(a , 0), S(0 , b)

where a , b are any positive numbers. Then isosceles triangle BAS sides (edges) equations are

AB: bx + ay - ab = 0,

AS: bx - ay - ab = 0,

BS: x = 0.

The only remaining equation (with initial zero factor at preselected unknown variable y in this equation) is

x = 0.

Its above transformation for preselected unknown variable y gave

b/a x = 0.

Our goal is to artificially introduce preselected unknown variable y into this equation in which this preselected unknown variable y is absent.

Due to overmathematics and other fundamental mathematical sciences systems [2-5], consider this equation as a quantielement

₁(b/a x = 0)

with naturally implicit quantity 1 and element

b/a x = 0.

Represent this quantielement as a quantisum (symbol ° shows quantioperations and their quantiresults)

₁(b/a x = 0) =° _1/2(b/a x = 0) +° _1/2(b/a x = 0)

of two equal half-quantielements

_1/2(b/a x = 0),

_1/2(b/a x = 0).

Now artificially introduce a positive number c and symmetrically replace these two equal half-quantielements with the following two quantielements:

_1/2(b/a x - b/c y = 0),

_1/2(b/a x + b/c y = 0).

As the result, replace quantielement

₁(b/a x = 0)

with the quantisum

_1/2(b/a x + b/c y = 0) +° _1/2(b/a x - b/c y = 0).

Now consider two natural approaches:

1) take these two half-equations "as is" with keeping "true" factors b/a (at x) obtained before (and independently of) artificially introducing c ;

2) provide namely unit factors at preselected unknown variable y in these two half-equations like the both already obtained (basic, nonzero-factor) equations

b/a x + y - b = 0,

- b/a x + y + b = 0

via dividing these two half-equations by nonzero numbers - b/c and b/c , respectively.

First apply approach

1) take these two half-equations "as is" with keeping "true" factors b/a (at x) obtained before (and independently of) artificially introducing c .

We have the initial separate quantiset

b/a x + y - b = 0,

- b/a x + y + b = 0,

_1/2(b/a x + b/c y = 0),

_1/2(b/a x - b/c y = 0)

of linear equations for preselected unknown variable y .

Hence transforming the initial separate set of linear equations for this preselected unknown variable y is completed.

Stage II. Linear Equation Set Center Determination via Difference Module Power Mean Minimization

For preselected unknown variable x separately,

the initial separate set of linear equations is

x + a/b y - a = 0,

x - a/b y - a = 0,

x = 0.

^{t ,
x}^{, d}S = |x + a/b y - a|^t + |x - a/b y - a|^t + |x|^t .

For preselected unknown variable y separately,

the initial separate quantiset of linear equations is

b/a x + y - b = 0,

- b/a x + y + b = 0,

_1/2(b/a x + b/c y = 0),

_1/2(b/a x - b/c y = 0).

^{t , y ,
d}S = |b/a x + y - b|^t + |- b/a x + y + b|^t + |b/a x + b/c y|^t/2 + |b/a x - b/c y|^t/2.

For preselected unknown variable x separately,

the initial separate sum of the powers (with the same power exponent t > 1) of the moduli (absolute values) of the free factors in all these transformed equations in a problem to be solved is

^{t , x ,
f}S = a^t + a^t + 0^t = 2a^t .

For preselected unknown variable y separately,

the initial separate sum of the powers (with the same power exponent t > 1) of the moduli (absolute values) of the free factors in all these transformed equations in a problem to be solved is

^{t , y ,
f}S = b^t + b^t + 0^t = 2b^t .

For preselected unknown variable x separately,

the free-factor normalized separate sum of the powers (with the same power exponent t > 1) of the differences of the both parts of all the transformed equations is

^{t , x ,
n}S = ^{t , x , d}S / ^{t , x , f}S = [|x + a/b y - a|^t + |x - a/b y - a|^t + |x|^t] / (2a^t) = (|x/a + y/b - 1|^t + |x/a - y/b - 1|^t + |x/a|^t)/2.

For preselected unknown variable y separately,

the free-factor normalized separate sum of the powers (with the same power exponent t > 1) of the differences of the both parts of all the transformed equations is

^{t , y ,
n}S = ^{t , y , d}S / ^{t , y , f}S = [|b/a x + y - b|^t + |- b/a x + y + b|^t + |b/a x + b/c y|^t/2 + |b/a x - b/c y|^t/2] / (2b^t) =

(|x/a + y/b - 1|^t + |x/a - y/b - 1|^t)/2 + (|x/a + y/c|^t + |x/a - y/c|^t)/4.

Nota bene: In this simple problem we obtain

^{t , x ,
n}S ≠ ^{t , y , n}S .

now without coinciding which is accidental. Generally, we can expect more complicated similarity correlation between free-factor normalized separate sums for different preselected unknown variables.

The total free-factor normalized sum of the powers (with the same power exponent t > 1) of the moduli (absolute values) of the differences of the both parts of all the transformed equations is

^tS = ^{t , x , n}S + ^{t , y , n}S = |x/a + y/b - 1|^t + |x/a - y/b - 1|^t + |x/a|^t/2 + (|x/a + y/c|^t + |x/a - y/c|^t)/4.

To begin with, let us consider the simplest case t = 2.

Testing Unit Factor Quadrat Theory

²S = (x/a + y/b - 1)² + (x/a - y/b - 1)² + (x/a)²/2 + (x/a + y/c)²/4 + (x/a - y/c)²/4.

Vanishing derivatives gives

²S'_x = 2(x/a + y/b - 1)/a + 2(x/a - y/b - 1)/a + (x/a)/a + (x/a + y/c)/(2a) + (x/a - y/c)/(2a) = 0,

²S'_y = 2(x/a + y/b - 1)/b + 2(x/a - y/b - 1)(-1)/b + (x/a + y/c)/(2c) + (x/a - y/c)(-1)/(2c) = 0;

x = 2a/3,

y = 0.

Form Invariance Test

x/a = 2/3

provides proportionality invariance, which is correct.

Analytic Test

We have

x/a = 2/3

by any positive a , in particular by

a = 3^1/2b

giving an equilateral triangle for which clearly

x/a = 1/3.

Now apply approach

2) provide namely unit factors at preselected unknown variable y in these two half-equations like the both already obtained (basic, nonzero-factor) equations

b/a x + y - b = 0,

- b/a x + y + b = 0

via dividing these two half-equations by nonzero numbers - b/c and b/c , respectively.

We have the initial separate quantiset

b/a x + y - b = 0,

- b/a x + y + b = 0,

_1/2(c/a x + y = 0),

_1/2(- c/a x + y = 0)

of linear equations for preselected unknown variable y .

Hence transforming the initial separate set of linear equations for this preselected unknown variable y is completed.

Stage II. Linear Equation Set Center Determination via Difference Module Power Mean Minimization

For preselected unknown variable x separately,

the initial separate set of linear equations is

x + a/b y - a = 0,

x - a/b y - a = 0,

x = 0.

^{t ,
x}^{, d}S = |x + a/b y - a|^t + |x - a/b y - a|^t + |x|^t .

For preselected unknown variable y separately,

the initial separate quantiset of linear equations is

b/a x + y - b = 0,

- b/a x + y + b = 0,

_1/2(c/a x + y = 0),

_1/2(- c/a x + y = 0).

^{t , y ,
d}S = |b/a x + y - b|^t + |- b/a x + y + b|^t + |c/a x + y|^t/2 + |- c/a x + y|^t/2.

For preselected unknown variable x separately,

the initial separate sum of the powers (with the same power exponent t > 1) of the moduli (absolute values) of the free factors in all these transformed equations in a problem to be solved is

^{t , x ,
f}S = a^t + a^t + 0^t = 2a^t .

For preselected unknown variable y separately,

the initial separate sum of the powers (with the same power exponent t > 1) of the moduli (absolute values) of the free factors in all these transformed equations in a problem to be solved is

^{t , y ,
f}S = b^t + b^t + 0^t = 2b^t .

For preselected unknown variable x separately,

the free-factor normalized separate sum of the powers (with the same power exponent t > 1) of the differences of the both parts of all the transformed equations is

^{t , x ,
n}S = ^{t , x , d}S / ^{t , x , f}S = [|x + a/b y - a|^t + |x - a/b y - a|^t + |x|^t] / (2a^t) = (|x/a + y/b - 1|^t + |x/a - y/b - 1|^t + |x/a|^t)/2.

For preselected unknown variable y separately,

the free-factor normalized separate sum of the powers (with the same power exponent t > 1) of the differences of the both parts of all the transformed equations is

^{t , y ,
n}S = ^{t , y , d}S / ^{t , y , f}S = [|b/a x + y - b|^t + |- b/a x + y + b|^t + |c/a x + y|^t/2 + |- c/a x + y|^t/2] / (2b^t) =

(|x/a + y/b - 1|^t + |x/a - y/b - 1|^t)/2 + (|c/(ab) x + y/b|^t + |c/(ab) x - y/b|^t)/4.

Nota bene: In this simple problem we obtain

^{t , x ,
n}S ≠ ^{t , y , n}S .

The total free-factor normalized sum of the powers (with the same power exponent t > 1) of the moduli (absolute values) of the differences of the both parts of all the transformed equations is

^tS = ^{t , x , n}S + ^{t , y , n}S = |x/a + y/b - 1|^t + |x/a - y/b - 1|^t + |x/a|^t/2 + [|c/(ab) x + y/b|^t + |c/(ab) x - y/b|^t]/4.

To begin with, let us consider the simplest case t = 2.

Testing Unit Factor Quadrat Theory

²S = (x/a + y/b - 1)² + (x/a - y/b - 1)² + (x/a)²/2 + [c/(ab) x + y/b]²/4 + [c/(ab) x - y/b]²/4.

Vanishing derivatives gives

²S'_x = 2(x/a + y/b - 1)/a + 2(x/a - y/b - 1)/a + (x/a)/a + [c/(ab) x + y/b]c/(2ab) + [c/(ab) x - y/b]c/(2ab) = 0,

²S'_y = 2(x/a + y/b - 1)/b + 2(x/a - y/b - 1)(-1)/b + [c/(ab) x + y/b]/(2b) + [c/(ab) x - y/b](-1)/(2b) = 0;

[5/a² + c²/(ab)²]x = 4/a ,

(5b² + c²)x = 4ab² ,

x = 4ab²/(5b² + c²),

y = 0.

Form Invariance Test

x/a = 4b²/(5b² + c²)

provides proportionality invariance, which is correct.

Analytic Test

We have for incenter C x-coordinate

x(C) = ab/[(a² + b²)^1/2 + b],

x(C)/a = b/[(a² + b²)^1/2 + b].

Unit factor quadrat theory (UFQT) [5] can give this result if

x/a = x(C)/a ,

4b²/(5b² + c²) = b/[(a² + b²)^1/2 + b] ,

c = [4b(a² + b²)^1/2 - b²]^1/2 .

Hence unit factor quadrat theory (UFQT) [5] analytically corrects the least square method (LSM) [1] but can give both adequate and inadequate results. The reason is that both the least square method (LSM) [1] and unit factor quadrat theory (UFQT) [5] use coordinate differences rather than distances and hence ignore straight line inclination. Further both the least square method (LSM) [1] and unit factor quadrat theory (UFQT) [5] provide no coordinate system rotation invariance.

Testing Distance Quadrat Theory

Distance quadrat theory (DQT) [5] minimizes the sum of the squares of the deviations (or distances as their moduli, or absolute values) of a general problem pseudosolution from the graphs of all the equations in a general problem to be solved.

Apply preliminary distance normalization to a general linear m-dimensional problem (m = 2, 3, ...). Namely preliminarily divide each linear equation by the denominator of the formula for the distance of a point from a straight line (by m = 2) or an (m-1)-dimensional plane (by m = 3, 4, ...) to provide that the difference of the both parts of this equation namely expresses the deviation of a point as a general problem pseudosolution from this straight line or plane.

Consider a finite overdetermined set of n (n > m; m , n ∈ N⁺ = {1, 2, ...}) linear equations

Σ_k=1^m a_kjx_k = c_j(j = 1, 2, ... , n)

with m unknowns x_k (k = 1, 2, ... , m) and any given real numbers a_kj and c_j. The distance between the jth m-1-dimensional "plane" (1) in an m-dimensional space of points

[_k=1^m x_k] = (x₁ , x₂ , ... , x_m)

and this point is [1]

d_j = |Σ_k=1^m a_kjx_k - c_j|/(Σ_k=1^m a_kj²)^1/2 .

Hence preliminarily divide jth linear equation namely by the denominator

(Σ_k=1^m a_kj²)^1/2

of this formula.

For example, in a two-dimensional problem (by m = 2), replacing x₁ with x , x₂ with y , a_1j with a_j , and a_2j with b_j , we have a finite overdetermined set of n (n ∈ {3, 4, ...}) linear equations

a_jx + b_jy = c_j (j = 1, 2, ... , n)

and distance

d_j = |a_jx + b_jy - c_j|/(a_j² + b_j²)^1/2 .

Hence preliminarily divide jth linear equation namely by the denominator

(a_j² + b_j²)^1/2

of this formula.

In our practically given set of n = 3 linear equations

bx + ay = 0,

bx - ay = 0,

x - a = 0.

Preliminary distance normalization gives

b/(a² + b²)^1/2 x + a/(a² + b²)^1/2 y = 0,

b/(a² + b²)^1/2 x - a/(a² + b²)^1/2 y = 0,

x - a = 0.

The sum of the squares of the deviations (or distances as their moduli, or absolute values) of a general problem pseudosolution (x , y) from the graphs of all the equations in this problem is

²S = [(bx + ay)² + (bx - ay)²]/(a² + b²) + (x - a)² .

Vanishing derivatives gives

²S'_x = [2(bx + ay)b + 2(bx - ay)b]/(a² + b²) + 2(x - a) = 0,

²S'_y = [2(bx + ay)a + 2(bx - ay)(-a)]/(a² + b²) = 0;

x = a(a² + b²)/(a² + 3b²) = a - 2ab²/(a² + 3b²),

y = 0.

Form Invariance Test

x/a = [1 + (b/a)²]/[1 + 3(b/a)²]

provides proportionality invariance, which is correct.

Analytic Test

We have for an equilateral triangle

a = 3^1/2b ,

distance quadrat theory (DQT) [5] gives

x/a = (1 + 1/3)/2 = 2/3,

which is correct.

For a relatively narrow isosceles triangle

a >> b ,

distance quadrat theory (DQT) [5] gives

x/a = 1 - 2b²/(a² + 3b²) ≈ 1 - 2(b/a)² .

The correct x-coordinate of the triangle center then gives

x(C)/a = 1 - (b/a)/{[(1 + (b/a)²]^1/2 + b/a} ≈ 1 - b/a .

Distance quadrat theory (DQT) [5] provides coordinate system rotation invariance.

Triangle theory (TT) in fundamental science of general problem testing is very efficient by solving many urgent (including contradictory) problems.

Acknowledgements to Anatolij Gelimson for our constructive discussions on coordinate system transformation invariances and his very useful remarks.

References

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

[2] Lev Gelimson. Elastic Mathematics. General Strength Theory. The ”Collegium” International Academy of Sciences Publishers, Munich, 2004

[3] Lev Gelimson. Providing Helicopter Fatigue Strength: Flight Conditions. In: Structural Integrity of Advanced Aircraft and Life Extension for Current Fleets – Lessons Learned in 50 Years After the Comet Accidents, Proceedings of the 23rd ICAF Symposium, Dalle Donne, C. (Ed.), 2005, Hamburg, Vol. II, 405-416

[4] Lev Gelimson. Overmathematics: Fundamental Principles, Theories, Methods, and Laws of Science. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010

[5] Lev Gelimson. General Problem Fundamental Sciences System. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2011