Basic Reliability 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), 1

Classical mathematics [1] brings key concepts that are very often insufficient, have evident lacks, and are not suitable for solving many typical problems. For example, to estimating reliability, a stochastic approach [1] often applies. Also in deterministic problems, their parameters are often artificially randomized. The corresponding distributions are considered known (even if they are really unknown) and most suitable for calculation. But even such a simplification brings complicated formulae and difficulties by analysis.

Overmathematics [2] based on the principles of natural thinking brings many new general sciences, theories, concepts, and methods suitable for research, engineering, and life. Among them is general reliability science.

The reliability of an object (e.g., element, structure, etc.) can be naturally defined as the probability that the object holds in some reasonable sense (e.g., exists and successfully functions 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 actual value of the object can deviate from its nominal value corresponding to available information always incomplete, usually inexact, and sometimes partially unreliable. This can be not only quantitatively, but also qualitatively significant. The last can especially hold for the limiting values and values near to those in some reasonable sense. For example, by such an admissible, limiting, or inadmissible nominal value, the actual one can be admissible, limiting, or inadmissible in any combination of these properties.

The introduced reserve [2, 3] makes it possible for the first time, to also deterministically estimate reliability in many types of problems and to provide relatively simple formulae very suitable for analysis. Let us determine reliability S via reserve R (whose values belong to the closed interval [-1; 1]) as S = (1 + R)/2. Then the range of S is the closed interval [0; 1] like that of probability, which is natural. For admissible values (states etc.) of the object that are infinitely far from their boundary (its limiting values) in certain reasonable sense, S = 1. For the limiting values themselves, S = 1/2. For inadmissible values infinitely far from their boundary (its limiting values) in certain reasonable sense, S = 0. All these values of reliability 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 also be either admissible or inadmissible. It is reasonable to consider equal the measures, probabilities, and some similar estimates (e.g., uniquantities [2]) of the both kinds of the deviations. The same estimates of the limiting values only can be regarded as infinitely small in comparison with the previous both because the dimension of the above boundary (the set of the limiting values) is usually less than the dimension of the set of the admissible values and that of the inadmissible ones. In the one-dimensional case, e.g., for a unique real parameter, the boundary typically consists of some discrete values and is zero-dimensional whereas both the set of the admissible values and the set of the inadmissible values are one-dimensional.

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∈Α a_α ⇐ x ⇐ b_{β β∈Β}]

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

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

Then

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

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

a little greater than 1/2 and

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

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

a little smaller than 1, both in accordance with intuition.

Introduced reliability brings (also by distributions) adequate estimations of approximation quality, exactness confidence, and risk. General reliability 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. General Reserve Theory. In: Review of Aeronautical Fatigue Investigations in Germany During the Period March 2003 to May 2005, Ed. Dr. Claudio Dalle Donne, SC/IRT/LG-MT-2005-039 Technical Report, Aeronautical fatigue, ICAF2007, EADS Corporate Research Center Germany, 2005, 55-56