Ph. D. & Dr. Sc. Lev Gelimson's Critical Distance Power and Moment Order Theories in Fundamental Sciences of Estimation, Approximation, Data Modeling and Processing

Critical Distance Power and Moment Order Theories in Fundamental Sciences of Estimation, Approximation, Data Modeling and Processing

© 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), 14

By data modeling, processing, estimation, and approximation in classical mathematics [1], namely the 2nd power of distances is typically used. The same holds for moments of inertia (of the 2nd order) of continual areas and volumes in mechanics. This power is the least even one providing the simplest analytic expressions with avoiding the absolute values. For the 2nd power, there are well-known theorems on the existence and uniqueness of the principal central axes of continual areas and volumes.

In overmathematics [2, 3] and fundamental sciences of estimation [4], approximation [5], data modeling and processing [6], it is shown that the second power can be insufficient at all not only by the least square method [8]. Hence the 4th power is also used [9]. All the more, any positive powers p of distances, or, equivalently, general moments J of inertia of order p , are introduced.

Critical distance power and moment order theories in these sciences investigate the influence of the distance power and moment order on bisecting (approximating) data sets, determine the critical values of the distance power and moment order, and discover new phenomena.

The 2nd distance power and moment order is not sensitive enough because in this case, e.g., by continual areas, the equality between the two extreme principal central moments of inertia provides that each central axis is principal even if there is no central symmetry of such areas, which contradicts intuition.

To investigate this problem, consider one of such areas, e.g. a square (without central symmetry) whose side is L . The square central moment of inertia of order p , or, equivalently, the sum (in the discrete case) or integral (in the continual case) of the powers p of distances of data set points, with respect to a straight line containing the middle points of a pair of opposite sides of this square is

^pJ₀ = ^pS₀ = 2∫₀^L/2 Lx^pdx = L^p+2/[2^p(p + 1)].

Note that there is a mirror symmetry of a square with respect to such a straight line. To designate an axis with respect to which we determine a central moment of inertia, use an index that means the angle with respect to a minimax axis providing the least distance (from the axis) of an area point which is the farest from the axis.

For a square, namely a straight line containing the middle points of a pair of opposite sides of this square is a minimax axis. Therefore, we use index 0. And in the general case of a mixed data set with both discrete and continual parts, use universal uniquantities in overmathematics [2, 3].

The square central moment of inertia of order p with respect to a diagonal (with index π/4) of this square is

^pJ_π/4 = ^pS_π/4 = 2∫₀^L/√2(L√2 - 2t)t^pdt = L^p+2/[2^p^/2-1(p + 1)(p + 2)].

Note that there is a mirror symmetry of a square with respect to any of its two diagonal. And this axis provides the greatest distance (from the axis) of an area point which is the farest from the axis.

Now consider ratio

c_p = ^pJ_π/4 / ^pJ₀ = 2^{(p +
2)}^/2/(p + 2).

Its derivative

dc_p/dp = 2^{(p + 2)}^/2[(p + 2)ln2 / 2 - 1]/(p + 2)²

vanishes by

p = 2/ln2 - 2 ≈ 0.8854

(this critical value of p is individual for a square) changing its sign from negative to positive, which provides here namely the least value about 0.9421 of c_p taking its universal critical value 1 at p = 2 (this critical value of p is universal) only.

Note that c_p = 1 (its universal critical value) also at p = 0 but we consider positive values of p only. This restriction is reasonable because at p = 0 and by any distance d we have identity d^p = 1 without any sensitivity.

Therefore, we have (Figure 1)

c_p < 1 at p < 2,

c_p = 1 at p = 2,

c_p > 1 at p > 2.

Figure 1

This particular case alone shows that the 2nd distance power and moment order is the only universal which provides illusion that all the central axes can be principal even without central symmetry.

Additionally, it is interesting to similarly investigate some other regular polygons. They are equiangular (i.e. all their angles are equal in measure) and equilateral (all their sides are of the same length). Also here compare results for some characteristic central coordinate systems.

For a regular triangle, any of its three bisectors, medians, and heights simultaneously is its minimax axis. Denote the common length of the three triangle sides with L and determine the corresponding central moment of inertia of order p :

^pJ₀ = ^pS₀ = 2∫₀^L/2 (- 3^1/2x + 3^1/2L/2)x^pdx = 3^1/2L^p+2/[2^p+1(p + 1)(p + 2)].

The central axis which is orthogonal to such a minimax axis provides the greatest distance (from the axis) of an area point which is the farest from the axis:

^pJ_π/2 = ^pS_π/2 = ∫_-L/(2√3)^L/√3(2L/3 - 2y/√3)|y|^pdy = (2^p⁺² + 3p + 5)L^p+2/[2^p⁺¹3^(p^+3)/2(p + 1)(p + 2)].

Now consider (Figure 2) ratio

c_p = ^pJ_π/2 / ^pJ₀ = (2^p⁺² + 3p + 5)/3^(p^+4)/2

and its derivative

dc_p/dp = [2^{p + 2}ln(2/3^1/2) - 3/2 p ln3 - 5/2 ln3 + 3]/3^(p^+4)/2.

Nota bene: c_p takes its critical value value 1 at p = 0 (this critical value of p is nonsensitive and hence not considered), p = 2 (this critical value of p is universal), and p = 4 (this critical value of p is common for some types of data sets) only.

Figure 2

For a regular hexagon, any of the three straight lines containing its opposite vertices is its minimax axis. Denote the common length of the six hexagon sides with L and determine the corresponding central moment of inertia of order p :

^pJ₀ = ^pS₀ = 2∫₀^{L/2 √3} (2L - 2/3^1/2 x)x^pdx = 3^(p+1)/2(p + 3)L^p+2/[2^p(p + 1)(p + 2)].

Any of three central axes which are orthogonal to such minimax axes and contain the middle points of a pair of opposite sides of this hexagon provides the greatest distance (from the axis) of an area point which is the farest from the axis:

^pJ_π/2 = ^pS_π/2 = 2∫₀^L/23^1/2Ly^pdy + 4∫_L/2^L 3^1/2(L - y)y^pdy = 3^1/2(2^p⁺² - 1)L^p+2/[2^p(p + 1)(p + 2)].

Now consider (Figure 3) ratio

c_p = ^pJ_π/2 / ^pJ₀ = (2^p⁺² - 1)/[3^p^/2(p + 3)]

and its derivative

dc_p/dp = {2^{p + 2}ln2 - (2^p⁺² - 1)[1/2 ln3 + 1/(p + 3)]}/[3^p^/2(p + 3)].

Figure 3

These theories are very efficient in data estimation, approximation, and processing.

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

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

[4] Lev Gelimson. Fundamental Science of Estimation. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010

[5] Lev Gelimson. Fundamental Science of Approximation. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010

[6] Lev Gelimson. Fundamental Science of Data Modeling and Processing. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010

[7] Lev Gelimson. General Data Direction, as well as Scatter and Trend Measure and Estimation Theories (Essential). Mathematical Journal of the “Collegium” All World Academy of Sciences, Munich (Germany), 11 (2011), 10

[8] 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

[9] Lev Gelimson. Least Biquadratic Distance Theories in Fundamental Sciences of Estimation, Approximation, Data Modeling and Processing (Essential). Mathematical Journal of the “Collegium” All World Academy of Sciences, Munich (Germany), 11 (2011), 11