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By Bronstein I.N., Semendjajew K.A., et al.

ISBN-10: 3817120060

ISBN-13: 9783817120062

ISBN-10: 3817120168

ISBN-13: 9783817120161

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Competing factors often generates critical points. In a system where multiple factors are interacting, the graph of the governing law is a complex manifold immersed in a high-dimensional space, which is accompanied by various deformations, plenty of pleats, and multiple overlaps. Such a complicated manifold gives a geometric representation of a complex system. For example, the system “earth”—being an extremely high-dimensional manifold connecting materials in many phases, various types of energies and many species creating ecological chains—should have many branches of possible states.

44) Here, t1 is an arbitrary number (not smaller than t0 ). 42) loses its uniqueness when it touches the singularity (see Fig. 10). 2). 4 Nonlinearity Escaping from Linearity—Criticality There is another possibility of abrupt nonlinearity, in addition to the aforementioned singularity (irregularity), that prevents any linear approximation. It is the case where the law is regular (Taylor-expandable) but its proportionality coefficient (the first derivative) vanishes. The point x0 where the first-order term of the Taylor expansion vanishes is called a critical point.

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Taschenbuch der Mathematik by Bronstein I.N., Semendjajew K.A., et al.


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