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Structural complexity (applied mathematics)

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Structural complexity (applied mathematics)

This page is about structural complexity in applied mathematics. For structural complexity theory in computational complexity theory of computer science see structural complexity theory.

Structural complexity is a science of applied mathematics, that aims at relating fundamental physical or biological aspects of a complex system with the mathematical description of the morphological complexity that the system exhibits, by establishing rigorous relations between mathematical and physical properties of such system (Ricca 2005).

Structural complexity emerges from all systems that display morphological organization (Nicolis & Prigogine 1989). Filamentary structures, for instance, are an example of coherent structures that emerge, interact and evolve in many physical and biological systems, such as mass distribution in the Universe, vortex filaments in turbulent flows, neural networks in our brain and genetic material (such as DNA) in a cell. In general information on the degree of morphological disorder present in the system tells us something important about fundamental physical or biological processes.

Structural complexity methods are based on applications of differential geometry and topology (and in particular knot theory) to interpret physical properties of dynamical systems (Abraham & Shaw 1992; Ricca 2009), such as relations between kinetic energy and tangles of vortex filaments in a turbulent flow or magnetic energy and braiding of magnetic fields in the solar corona, including aspects of topological fluid dynamics.

References

  • Abraham, R.H. & Shaw, C.D. (1992) Dynamics - the Geometry of Behavior. Addison-Wesley. ISBN 978-0201567175
  • Nicolis, G. & Prigogine, I. (1989) Exploring Complexity. W.H. Freeman & Co., New York. ISBN 9780716718598
  • Ricca, R.L. (2005) Structural complexity, in Encyclopedia of Nonlinear Science (ed. A. Scott), pp. 885-887. Routledge, New York and London. ISBN 9781579583859
  • Ricca, R.L. (2009) Detecting structural complexity: from visiometrics to genomics and brain research, in Mathknow, (ed. M. Emmer & A. Quarteroni), pp. 167-181. Springer-Verlag. ISBN 9788847011212
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