General Inertia Moment Theories in Fundamental Sciences of Estimation, Approximation, Data Modeling and Processing

by

© Ph. D. & Dr. Sc. Lev Gelimson

Academic Institute for Creating Fundamental Sciences (Munich, Germany)

Mechanical and Physical Journal

of the “Collegium” All World Academy of Sciences

Munich (Germany)

10 (2010), 3

Area moments of inertia apply to continual sections in bending and torsion in mechanics, i.e. in two-dimensional problems. Volume and mass moments of inertia apply to rotating continual objects in mechanics, too, i.e. in three-dimensional problems.

In data processing, estimation, and approximation, the least square method [1] by Legendre and Gauss only usually applies to contradictory (e.g., overdetermined) problems. Overmathematics [2, 3] and fundamental sciences of estimation [4], approximation [5], and data processing [6] have discovered a lot of principal shortcomings [7] of this method.

Linear two-dimensional and three-dimensional theories of moments of inertia in these sciences apply the approaches and methods of using area, volume, and mass moments of inertia to linear two-dimensional and three-dimensional discrete data processing, estimation, and approximation and create some theoretical foundations for them.

General (also nonlinear) theories of moments of inertia in these sciences apply both to discrete and to continual objects by coordinate system rotation invariance and create further theoretical foundations for their processing, estimation, and approximation. Show the essence and algorithm of applying these theories.

By separate data points, use the quantity of everyone of them, e.g., its integer multiplicity (1 for a single point, 2 for a double point, 3 for a triple point, etc.) instead of area, volume, and/or mass.

By coordinate system translation invariance of a given data point set, centralize it by using centralization transformation (subtracting every coordinate of the data center from the corresponding coordinate of every data point) clearly invariant to provide coordinate system central for this set.

First, following the principle of tolerable simplicity [3-6], apply namely linear theories of moments of inertia to the given data point set already centralized and determine its principal central directions (also planes in the three-dimensional case, generally all principal central objects whose dimensionality is smaller than the dimensionality of the given data point set) and moments of inertia.

Test the values of these principal central moments of inertia for coincidence. If all these values are distinct, then all the principal central directions are unique. Generally, each principal central direction with a single value of a principal central moment of inertia is unique, whereas each group of principal central directions with a common multiple value of their principal central moments of inertia allows any rotations of these principal central directions about the remaining principal central directions.

Compare the values of these principal central moments of inertia for their relations, first of all their ratios. Order these values, determine the greatest of them, and divide each value by this greatest value to obtain the relative values of these principal central moments of inertia (with the same relations between coincidence, uniqueness, and rotations). The relative value of a principal central moment of inertia, the square root and other appropriate functions of this value are estimates, or measures, of data scatter with respect to linear approximation. The differences between 1 and such relative values are estimates, or measures, of the two-sided trend of the already centralized given data point set in the corresponding two-sided principal central direction. These measures allow introducing the natural concepts of a data point set clearly linearly directed, conditionally linearly directed, and linearly undirected in a certain principal central direction. Generally, divide the moment of inertia of a possibly noncentralized data point set in any two-sided possibly nonprincipal and noncentral direction by the greatest principal central moment of inertia. The obtained ratio (relative value), the square root and other appropriate functions of this ratio are estimates, or measures, of data scatter with respect to linear approximation. The differences between 1 and such relative values are estimates, or measures, of the two-sided trend of a possibly noncentralized data point set in any two-sided possibly nonprincipal and noncentralized direction.

To linearly centrally approximate the already centralized given data point set, select principal central directions (also planes in the three-dimensional case, generally all principal objects whose dimensionality is smaller than the dimensionality of the given data point set) with the least moments of inertia.

To linearly approximate the given data point set possibly noncentralized, determine the transformation inverse to the previous centralization transformation and apply this inverse transformation to the already determined linear central approximation.

Further apply relatively simple piecewise linear theories of moments of inertia to decide whether nonlinear theories of moments of inertia could be useful.

Finally, investigate whether nonlinear theories of moments of inertia have essential estimation and approximation advantages compared to linear theories of moments of inertia.

In nonlinear theories of moments of inertia, naturally use curvilinear axes (instead of straight lines), surfaces (instead of planes), etc., with clearly generalizing static moments and moments of inertia.

If a given data point set is clearly linearly directed, then it is clearly directed because nonlinearity brings possibilities additional to those by linearity.

But data point sets clearly directed, e.g., building substantial parts of circumferences or spherical surfaces, can be linearly undirected in any direction. In such a case, linear theories of moments of inertia are clearly inadequate.

These theories bring deep theoretical fundamentals for data estimation, approximation, and processing, namely as applied to the problems of the existence and uniqueness of solutions, whereas least squared distance theories give explicit formulae more suitable.

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. 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 Processing. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010

[7] 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