By Yi Lin
This e-book summarizes the most medical achievements of the blown-up conception of evolution technological know-how, which was once first noticeable in released shape in 1994. It explores - utilizing the perspective and method of the blown-up thought - attainable generalizations of Newtonian particle mechanics and computational schemes, built on Newton's and Leibniz's calculus, in addition to the medical platforms and the corresponding epistemological propositions, brought and polished long ago 300 years.
The authors in short clarify the elemental suggestions, then examine a chain of issues and difficulties of the present, lively learn greatly performed within the traditional sciences. alongside the traces of the analyses, they introduce new issues of view and the corresponding tools. additionally, they indicate that the blown-up concept originated from the assumption of mutual slavings of fabrics' constructions in order that ''numbers are reworked into forms''. This discovery finds that nonlinearity isn't really an issue solvable within the first-push process, and that the fabrics' estate of rotation is not just an epistemology but additionally a technique. The authors then element to the truth that nonlinearity is a moment stir of mutual slavings of fabrics.
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Extra info for Beyond Nonstructural Quantitative Analysis - Blown-Ups, Spinning Currents and Modern Science
It is because they were the first who proposed practical solutions to the problem of stability. Especially Lagrange, whose m e t h o d was different of those of Newton, Euler and Laplace, h a d placed his attention on the study of the structure of t h e kinematic equations. W i t h his method, m a n y analytically unsolvable differential equations can be determined being stable based on their structures. On this basis, Poincare (J. H. Poincare, 1854 1912) later enriched and further developed Lagrange's energy criteria and laid t h e foundation for Russian m a t h e m a t i c i a n Lyapunov (1857 - 1918) to establish his general stability theory.
All the analysis methods, developed for well posed systems (linear differential equations) in the past, should be employed in studies of not well posed systems (nonlinear differential equations) with discrimination. Otherwise, we would be led to misleading conclusions as evidenced in mistakes widely existing in prediction activities. Singularities of differential equations in form can be classified into two classes: One class contains the differential equations, other than its coefficient of the highest order derivative term, at least one other coefficient of derivative and the original function terms blows up at one or several points of a certain region, or the initial or boundary conditions are discontinuous (such as the boundary of the terrain).
14) Discontinuity: The Mathematical Characteristic of Nonlinear Evolutions 41 where z = —-. Now, the general solution of eq. 15) where C is the integration constant and F — J(l —n)f(t)dt. So, the solution of Eq. 16) z for y. Now, let us look at more specific cases. , and a and /? are constant, then Eq. 17) From Eqs. 16), the following can be solved: y (2 18) =cJ^p ' Evidently, a discontinuity occurs at t = tb = —In P in Eq. 18). 5 If n = 3, f(t) = 2t, g(t) = 2at , and a is a constant, then Eq. 19) From Eqs.