Stress relaxation: Difference between revisions

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Maxwell and Voigt are the simplest models for liquids and solids respectively. the slsm is a rather simple derivative of a voigt model, and surely not "most accurate".
en>Bhupender singh chugh
Definition and basics of topic
 
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{{Refimprove|date=February 2009}}
In [[mathematics]], in the area of classical [[potential theory]], '''polar sets''' are the "negligible sets", similar to the way in which sets of measure zero are the [[negligible set]]s in [[measure theory]].
 
== Definition ==
 
A set <math>Z</math> in <math>\R^n</math> (where <math>n\ge 2</math>) is a polar set if there is a non-constant [[subharmonic function]]
 
:<math>u</math> on <math>\R^n</math>
 
such that
 
:<math>Z \subseteq \{x: u(x) = -\infty\}.</math>
 
Note that there are other (equivalent) ways in which polar sets may be defined, such as by replacing "subharmonic" by "superharmonic", and <math>-\infty</math> by <math>\infty</math> in the definition above.
 
== Properties ==
 
The most important properties of polar sets are:
 
*A singleton set in <math>\R^n</math> is polar.
*A countable set in <math>\R^n</math> is polar.
*The union of a countable collection of polar sets is polar.
*A polar set has Lebesgue measure zero in <math>\R^n.</math>
 
==Nearly everywhere==
A property holds '''nearly everywhere''' in a set ''S'' if it holds on ''S''−''E'' where ''E'' is a Borel polar set. If ''P'' holds nearly everywhere then it holds [[almost everywhere]].<ref name=Ran56>Ransford (1995) p.56</ref>
 
==See also==
* [[Pluripolar set]]
 
== References ==
{{reflist}}
* {{cite book | first=Joseph L. | last=Doob | authorlink=Joseph L. Doob | title=Classical Potential Theory and Its Probabilistic Counterpart | publisher=[[Springer-Verlag]] | location=Berlin Heidelberg New York | isbn=3-540-41206-9 | zbl=0549.31001 | series=Grundlehren der Mathematischen Wissenschaften | volume=262 | year=1984 }}
* {{cite book | first=L. L. | last=Helms | year=1975 | title=Introduction to potential theory | publisher=R. E. Krieger | isbn=0-88275-224-3 }}
* {{cite book | last=Ransford | first=Thomas | title=Potential theory in the complex plane | series=London Mathematical Society Student Texts | volume=28 | location=Cambridge | publisher=[[Cambridge University Press]] | year=1995 | isbn=0-521-46654-7 | zbl=0828.31001 }}
 
==External links==
* {{planetmath reference|id=6020|title=Polar set}}
 
[[Category:Subharmonic functions]]
 
 
{{mathanalysis-stub}}

Latest revision as of 07:38, 12 January 2015

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