By Yi Lin
This e-book summarizes the most medical achievements of the blown-up idea of evolution technological know-how, which was once first obvious 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 clinical structures and the corresponding epistemological propositions, brought and polished long ago 300 years.
The authors in short clarify the basic ideas, then examine a sequence of subject matters and difficulties of the present, lively learn greatly performed within the average 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 remodeled into forms''. This discovery unearths that nonlinearity isn't really an issue solvable within the first-push method, and that the fabrics' estate of rotation isn't just an epistemology but in addition a strategy. The authors then aspect 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
Lin (1990), Y. Lin, Y. Ma and R. Port (1990), S. C. OuYang (1994), I. Prigogine (1967), R. Thorn (1975), L. Y. Xu (1983), Y. Z. Zhu (1985). For more details, please consult with these references. This page is intentionally left blank Chapter 2 Nonlinearity: The Conclusion of Calculus In this chapter, we will look at a brief history of calculus, its achievements, fundamental concepts and results of the differential and integral analysis. As soon as the concepts of well-psedness and singularity of differential equations are looked at, one starts to see the limitations of calculus and all theories developed on calculus.
The organization of this book can be outlined as follows: Chapter two is devoted to the presentation of the so-called mystery of nonlinearity and how the modern study, entitled "nonlinear science", has been originated, after a brief introduction to an array of basic concepts of calculus. After showing that discontinuity is a fundamental characteristic of nonlinear evolutions, it is concluded that in order to revolve problems of nonlinearity, new methodology and thinking logic are needed. At the end, some traditional treatments of nonlinearity are analyzed.
For example, n- - x 2 y = sina;, y = W — — , y = x + x + 1 V 1 +x or the following piecewise defined function v -— fix) 1 — x, 1 1 + x, when x < 1 when x > 1 etc. One main advantage of employing analytic representation of a given function is that such a representation is convenient in the study and analysis of the differential method. (2) Graphical Representation. This is a representation of a given function as a graph in the Cartesian coordinate system, where the horizontal axis stands for the independent variable x and the vertical axis for the dependent variable y.