Basic Risk Science in Overmathematics

by

© Ph. D. & Dr. Sc. Lev Gelimson

Academic Institute for Creating Fundamental Sciences (Munich, Germany)

RUAG Aerospace Services GmbH, Germany

Mathematical Journal

of the "Collegium" All World Academy of Sciences

Munich (Germany)

7 (2007), 2

For many typical problems in science and engineering including aeronautical fatigue, there are no concepts and methods adequate and general enough. Even artificially simplifying by randomizing parameters in deterministic problems to apply stochastic approaches in classical mathematics [1] to estimating risk brings complicated formulae and analysis difficulties. Overmathematics [2] is very suitable for solving such problems and includes many general theories and methods. Among its new concepts are the unierrors and reserves estimating and measuring also exactness, contradictoriness, and distributions. The unierrors [2, 3] correct and generalize the relative error. The reserves [2] new in principle estimate exactness reliability. They both form a basis for general risk science.

The risk of an object (e.g., element, structure, etc.) can be naturally defined as the probability of the event that the object does not hold in some reasonable sense (e.g., cannot exist and/or successfully function during a given time interval under certain conditions such as loading, temperature, etc.). The above object can take so called values as its particular states, realizations, etc. which are any objects, too. A value is called admissible if the object taking this value holds. A value is called inadmissible if the object taking this value does not hold. A value is called limiting if it belongs to the common boundary of the admissible values and inadmissible ones. In other words, in each, arbitrarily small, neighborhood (in a certain reasonable sense) of such a value, there are both admissible and inadmissible values. A limiting value is either admissible or inadmissible.

The reserve [2] makes it possible for the first time, to also deterministically estimate risk in many types of problems in science, engineering, etc. and to provide relatively simple formulae very suitable for analysis.

Let us determine risk r via reserve R (whose values belong to the closed interval [-1; 1]) [2] as

r = (1 - R)/2.

Then the range of r is the closed interval [0; 1] like that of the unierror (with certain sense similarity). By no risk we have r = 0. For a limiting value, r = 1/2. For inadmissible values infinitely far from a limiting value, r = 1. All these values of risk are natural and correspond to intuition. For a limiting value, due to possible deviations from it in the reality, the value of a parameter with risk can be either greater or less than the limiting value. It is reasonable to consider equal the probabilities, measures, and some similar estimates (e.g, uniquantities [2]) of the both kinds of the deviations whereas the same estimates of the only limiting value can be regarded as infinitely small in comparison with the previous both. The remaining cases r = 0 and r = 1 are trivial.

Example. Both

x₁ = 1 + 10^-10

and

x₂ = 1 + 10¹⁰

are exact solutions to the inequation

x > 1,

x₁ practically unreliable and x₂ guaranteed. Their discrimination is especially important by any inexact data. Classical mathematics [1] cannot provide this at all. In overmathematics [2, 3], the reserve of any pseudosolution x to the combined inequations

 [_α∈Αa_α⇐x ⇐b_ββ∈Β]

where a, b, x are real numbers, ⇐ is one of the two inequality signs < , ≤ , can be defined as [2]

R(x, [_α∈Αa_α⇐x⇐b_ββ∈Β]) = inf_{α∈Α, β∈Β}((x - a_α)/(|x|+2|a_α|+1), (b_β- x)/(|x|+2|b_β|+1)).

Then we obtain

r_{x > 1}(x₁) = r_{x > 1}(1 + 10^-10) = (1 - R_{x > 1}(1 + 10^-10))/2 =

(1 - 10^-10/(2 + 10^-10))/2 = 1/(2 + 10^-10)

a little smaller than 1/2 and

r_{x > 1}(x₂) = r_{x > 1}(1 + 10¹⁰) = (1 - R_{x > 1}(1 + 10¹⁰))/2 =

(1 - 10¹⁰/(2 + 10¹⁰))/2 = 1/(2 + 10¹⁰)

very small, both in accordance with intuition.

Introduced risk brings (also by distributions) adequate estimations of approximation quality, exactness confidence, and reliability. General risk science is very effective by setting, reasonably simulating, and solving many types of problems in research, engineering, and life.

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

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

[3]Lev Gelimson. Corrections and Generalizations of the Absolute and Relative Errors. In: Review of Aeronautical Fatigue Investigations in Germany During the Period May 2005 to April 2007, Ed. Dr. Claudio Dalle Donne, Pascal Vermeer, CTO/IW/MS-2007-042 Technical Report, Aeronautical fatigue, ICAF2007, EADS Innovation Works Germany, 2007, 49-50