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| In [[real algebraic geometry]], the ''' Łojasiewicz inequality''', named after [[Stanisław Łojasiewicz]], gives an upper bound for the distance of a point to the nearest zero of a given [[real analytic function]]. Specifically, let ƒ : ''U'' → '''R''' be a real-analytic function on an [[open set]] ''U'' in '''R'''<sup>''n''</sup>, and let ''Z'' be the zero [[locus (mathematics)|locus]] of ƒ. Assume that ''Z'' is not empty. Then for any [[compact set]] ''K'' in ''U'', there exist positive constants α and ''C'' such that, for all ''x'' in ''K''
| | My name is Numbers and I am studying Social Service and Modern Languages and Classics at Fruitport / United States.<br><br>my webpage: [http://bit.ly/1tn2bLa Best treadmills 2014] |
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| :<math>\operatorname{dist}(x,Z)^\alpha \le C|f(x)|. \, </math>
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| Here α can be large.
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| The following form of this inequality is often seen in more analytic contexts: with the same assumptions on ƒ, for every ''p'' ∈ ''U'' there is a possibly smaller open neighborhood ''W'' of ''p'' and constants θ ∈ (0,1) and ''c'' > 0 such that
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| :<math>|f(x)-f(p)|^\theta\le c|\nabla f(x)|. \, </math>
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| ==References==
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| *{{Citation | last1=Bierstone | first1=Edward | last2=Milman | first2=Pierre D. | title=Semianalytic and subanalytic sets | id={{MathSciNet | id = 972342}} | year=1988 | journal=[[Publications Mathématiques de l'IHÉS]] | issn=1618-1913 | issue=67 | pages=5–42|url=http://www.numdam.org/item?id=PMIHES_1988__67__5_0}}
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| *{{Citation | doi=10.2307/2153965 | last1=Ji | first1=Shanyu | last2=Kollár | first2=János | last3=Shiffman | first3=Bernard | title=A global Łojasiewicz inequality for algebraic varieties | id={{MathSciNet | id = 1046016}} | url=http://www.ams.org/journals/tran/1992-329-02/S0002-9947-1992-1046016-6/ | year=1992 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=329 | issue=2 | pages=813–818 | jstor=2153965}}
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| ==External links==
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| *[http://www.encyclopediaofmath.org/index.php/Lojasiewicz_inequality Lojasiewicz inequality] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
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| {{mathanalysis-stub}}
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| {{DEFAULTSORT:Lojasiewicz inequality}}
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| [[Category:Inequalities]]
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| [[Category:Mathematical analysis]]
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| [[Category:Real algebraic geometry]]
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Latest revision as of 18:59, 8 October 2014
My name is Numbers and I am studying Social Service and Modern Languages and Classics at Fruitport / United States.
my webpage: Best treadmills 2014