Structural complexity (applied mathematics): Difference between revisions

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{{ThisAbout| structural complexity in applied mathematics|structural complexity theory in [[computational complexity theory]] of [[computer science]] |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.{{sfn | Ricca | 2005 | p=}}

Structural complexity emerges from all systems that display morphological organization.{{sfn | Nicolis | 1989 | p=}}. Filamentary structures, for instance, are an example of [[Lagrangian_coherent_structuresLagrangian coherent structures|coherent structures]] that emerge, interact and evolve in many physical and biological systems, such as mass distribution in the [[Shape of the universe|Universe]], [[Vortex |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 [[Order_and_disorder_Order and disorder (physics) |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]].{{sfn | Abraham | 1992 | p=}}{{sfn | Ricca | 2009 | p=}} 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]].

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==Literature==

* {{cite book | last1=Abraham | first1=Ralph |authorlink1=Ralph Abraham (mathematician)|first2=C.D.|last2=Shaw|authorlink2=Robert_Shaw_(Physicist)#Illustrations | title=Dynamics--the geometry of behavior | publisher=Addison-Wesley, Advanced Book Program | ___location=Redwood City, Calif | year=1992 | isbn=978-0-201-56717-5 | oclc=24374484}}

* {{cite book | last=Nicolis | first=G | authorlinkauthor-link= Grégoire Nicolis|title=Exploring complexity : an introduction | publisher=W.H. Freeman | ___location=New York | year=1989 | isbn=978-0-7167-1859-8 | oclc=18989681}}

*{{cite book|last=Ricca|first=R.L.|authorlinkauthor-link=Renzo_L._Ricca |year=2005|chapter=Structural complexity|title=Encyclopedia of Nonlinear Science|editor= A. Scott|pages= 885–887|publisher=Routledge, New York and London|isbn=9781579583859}}

*{{cite book|author=Ricca, R.L.|year=2009|chapter=Detecting structural complexity: from visiometrics to genomics and brain research|url=https://www.springer.com/mathematics/applications/book/978-88-470-1121-2|title=Mathknow|editor= M. Emmer & A. Quarteroni|pages=167–181|publisher=Springer-Verlag|isbn=9788847011212}}