Ph. D. & Dr. Sc. Lev Gelimson's Unit Factor Power Theories in Fundamental Science of Solving General Problems

Unit Factor Power Theories in Fundamental Science of Solving General Problems

© 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), 59

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.

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.

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.

Unit factor power theories (UFPT) [5] in fundamental science of solving general problems can provide increasing power exponent as compared to 2 by unit factor quadrat theory (UFQT) [5] as a particular case of unit factor power theories (UFPT) [5], which can be necessary to adequately solvee 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