Total electron content: Difference between revisions

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Changed the hyperlink of scintillation. Scintillation of radio waves is astronomical scintillation, not physical scintillation.
 
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In [[mathematics]], the notion of '''polyconvexity''' is a generalization of the notion of [[convex function|convexity]] for [[function (mathematics)|functions]] defined on spaces of [[matrix (mathematics)|matrices]]. Let ''M''<sub>''m''&times;''n''</sub>(''K'') denote the space of all ''m''&nbsp;&times;&nbsp;''n'' matrices over the [[field (algebra)|field]] ''K'', which may be either the [[real number]]s '''R''', or the [[complex number]]s '''C'''. A function ''f''&nbsp;:&nbsp;''M''<sub>''m''&times;''n''</sub>(''K'')&nbsp;&rarr;&nbsp;'''R'''&nbsp;&cup;&nbsp;{&plusmn;&infin;} is said to be '''polyconvex''' if
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:<math>A \mapsto f(A)</math>
 
can be written as a convex function of the ''p''&nbsp;&times;&nbsp;''p'' sub[[determinant]]s of ''A'', for 1&nbsp;&le;&nbsp;''p''&nbsp;&le;&nbsp;min{''m'',&nbsp;''n''}.
 
Polyconvexity is a weaker property than convexity. For example, the function ''f'' given by
 
:<math>f(A) = \begin{cases} \frac1{\det (A)}, & \det (A) > 0; \\ + \infty, & \det (A) \leq 0; \end{cases}</math>
 
is polyconvex but not convex.
 
==References==
 
* {{cite book
|  author = Renardy, Michael and Rogers, Robert C.
|    title = An introduction to partial differential equations
|  series = Texts in Applied Mathematics 13
|  edition = Second edition
|publisher = Springer-Verlag
| location = New York
|    year = 2004
|    pages = 353
|      id = ISBN 0-387-00444-0
}} (Definition 9.25)
 
 
[[Category:Convex analysis]]
[[Category:Matrices]]
[[Category:Types of functions]]

Latest revision as of 13:05, 5 October 2014

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