Ph. D. & Dr. Sc. Lev Gelimson's Power Distance Transformation Theories in Fundamental Science of General Problem Transformation

Power Distance Transformation Theories in Fundamental Science of General Problem Transformation

© 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), 60

Introduction. Distance Transformation Theory

Finite real linear equations sets, or simply linear equations sets,

Σ_i=1^m a_ijx_i = c_j(j = 1, 2, ... , n)

consisting of n equations with m unknowns x_i (i = 1, 2, ... , m) and any given real numbers a_ij and c_jare typical in classical mathematics [1] but like any Cantor set [1] ignore equations quantities. They are very important by contradictory (e.g. overdetermined) problems without precise solutions by n > m (m , n ∈ N⁺ = {1, 2, ...}). Besides that, without equations quantities, by subjoining an equation coinciding with one of the already given equations of such a set, this subjoined equation is simply ignored whereas any (even infinitely small) changing this subjoined equation alone at once makes this subjoining essential and changes the given set of equations. Therefore, the concept of a set of equations is ill-defined [1].

Therefore, let us consider a finite overdetermined quantiset [2-5] of n (n > m; m , n ∈ N⁺ = {1, 2, ...}) linear equations

_q(j)(Σ_i=1^m a_ijx_i = c_j) (j = 1, 2, ... , n)

with their own quantities q(j) and m unknowns x_i (i = 1, 2, ... , m) by any given real numbers q(j) > 0, a_i_j , and c_j.

Let us use the concepts of a nonzero, or positive, or negative proportional transformation of a quantiset or set of equations with multiplying each equation by a nonzero, or positive, or negative number individual for this equation, respectively.

Classical mathematics [1] considers a nonzero proportional transformation as an equivalent transformation of a set of equations. However, this holds for exact solutions only. Otherwise, namely by contradictory (e.g. overdetermined) problems without precise solutions, this also holds for any pseudosolutions but only by nonzero proportional transformation invariant theories and methods of solving problems and estimating their pseudosolutions [2-5].

Nota bene: The least square method (LSM) [1] by Legendre and Gauss is the only method well-known in classical mathematics [1] and applcable to contradictory (e.g. overdetermined) problems. Overmathematics [2-4] and the system of fundamental sciences on general problems [5] have discovered many principal shortcomings [2-6] of this method (and all theories and methods based on this method) which is nonzero proportional transformation noninvariant and hence gives results without any objective sense.

Distance transformation theory (DTT) in fundamental science of general problem transformation [5] applies a specific positive proportional transformation to a quantiset or set of equations with dividing each equation by the denominator of the canonic fraction expressing the distance between the graph of this equation and any point.

The distance between the jth m-1-dimensional "plane" (straight line by m = 2)

Σ_i=1^m a_ijx_i = c_j (j = 1, 2, ... , n)

in an m-dimensional space of points

[_i=1^m x_i] = (x₁ , x₂ , ... , x_m)

and namely this point is

d_j = |Σ_i=1^m a_ijx_i - c_j|/(Σ_i=1^m a_ij²)^1/2

which is the above canonic fraction.

Now divide each equation

Σ_i=1^m a_ijx_i = c_j (j = 1, 2, ... , n)

by the denominator

(Σ_i=1^m a_ij²)^1/2

of the above canonic fraction and obtain the distance-transformed finite overdetermined quantiset [2-5] of n linear equations

_q(j)[Σ_i=1^m a_ij / (Σ_i=1^m a_ij²)^1/2 x_i = c_j / (Σ_i=1^m a_ij²)^1/2] (j = 1, 2, ... , n).

with their own quantities q(j) and m unknowns x_k (k = 1, 2, ... , m) by any given real numbers q(j) > 0, a_ij , and c_j.

Nota bene: Distance transformation of a finite overdetermined quantiset [2-5] of n linear equations is both nonzero proportional transformation invariant and rotation invariant.

Some furter generalizations are possible. Namely, all a_ij in the jth equation can have any common (independent of i) physical dimension (unit) [a_j], all x_k can have any common (independent of i) physical dimension (unit) [x], and then each c_j must have the corresponding physical dimension (unit) [a_j][x].

For example, by m = 2, replacing x₁ with x , x₂ with y , a_1j with a_j , and a_2j with b_j , we have:

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

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

_q(j)[a_j/(a_j² + b_j²)^1/2 x + b_j/(a_j² + b_j²)^1/2 y = c_j/(a_j² + b_j²)^1/2] (j = 1, 2, ... , n).

Some furter generalizations are possible. Namely, a_j and b_j in the jth equation can have any common physical dimension (unit) [a_j], x and y can have any common physical dimension (unit) [x], and then each c_j must have the corresponding physical dimension (unit) [a_j][x].

Power Distance Transformation Theories

Power distance transformation theories (PDTT) in fundamental science of general problem transformation [5] with any power exponent t > 1 include distance transformation theory (DTT) and are its natural generalizations.

Power distance transformation theories (PDTT) apply a specific positive proportional transformation to a quantiset or set of equations with dividing each equation by the denominator of the canonic fraction expressing the power distance between the graph of this equation and any point.

The power distance (with any power exponent t > 1) between the jth m-1-dimensional "plane" (straight line by m = 2)

Σ_i=1^m a_ijx_i = c_j (j = 1, 2, ... , n)

in an m-dimensional space of points

[_i=1^m x_i] = (x₁ , x₂ , ... , x_m)

and namely this point is defined and determined as

^td_j = |Σ_i=1^m a_ijx_i - c_j|/(Σ_i=1^m |a_ij|^t)^1/t

which is the above canonic fraction.

Now divide each equation

Σ_i=1^m a_ijx_i = c_j (j = 1, 2, ... , n)

by the denominator

(Σ_i=1^m |a_ij|^t)^1/t

of the above canonic fraction and obtain the power-distance-transformed finite overdetermined quantiset [2-5] of n linear equations

_q(j)[Σ_i=1^m a_ij / (Σ_i=1^m |a_ij|^t)^1/t x_i = c_j / (Σ_i=1^m |a_ij|^t)^1/t] (j = 1, 2, ... , n).

with their own quantities q(j) and m unknowns x_k (k = 1, 2, ... , m) by any given real numbers q(j) > 0, a_ij , and c_j.

Nota bene: Power distance transformation of a finite overdetermined quantiset [2-5] of n linear equations is both nonzero proportional transformation invariant and rotation invariant.

For example, by m = 2, replacing x₁ with x , x₂ with y , a_1j with a_j , and a_2j with b_j , we have:

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

^td_j = |a_jx + b_jy - c_j|/(|a_j|^t + |b_j|^t)^1/t ,

_q(j)[a_j/(|a_j|^t + |b_j|^t)^1/t x + b_j/(|a_j|^t + |b_j|^t)^1/t y = c_j/(|a_j|^t + |b_j|^t)^1/t] (j = 1, 2, ... , n).

Power distance transformation theories (PDTT) in fundamental science of general problem transformation [5] apply to any linear equations quantiset, provide its power distance transformation which is both nonzero proportional transformation invariant and rotation invariant, and are very efficient by solving many urgent (even contradictory) general 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

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