Linear bottleneck assignment problem: Difference between revisions

Revision as of 18:10, 7 August 2013

Template:Distinguish In potential theory, a branch of mathematics, Cartan's lemma, named after Henri Cartan, is a bound on the measure and complexity of the set on which a logarithmic Newtonian potential is small. The following statement can be found in Levin's book.^[1]

Let μ be a finite positive Borel measure on the complex plane C with μ(C) = n. Let u(z) be the logarithmic potential of μ:

u (z) = \frac{1}{2 π} \int_{C} \log | z - ζ | d μ (ζ)

Given H ∈ (0, 1), there exist discs of radius r_i such that

\sum_{i} r_{i} < 5 H

and

u (z) \geq \frac{n}{2 π} \log \frac{H}{e}

for all z outside the union of these discs.

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+{{distinguish|Cartan's theorem}}
+In [[potential theory]], a branch of [[mathematics]], '''Cartan's lemma''', named after [[Henri Cartan]], is a bound on the measure and complexity of the set on which a logarithmic [[Newtonian potential]] is small.  The following statement can be found in Levin's book.<ref>B.Ya. Levin, ''Lectures on Entire Functions''</ref>
+Let ''&mu;'' be a finite positive [[Borel measure]] on the complex plane '''C''' with ''&mu;''('''C''')&nbsp;=&nbsp;''n''.  Let ''u''(''z'') be the logarithmic potential of&nbsp;''&mu;'':
+:<math>u(z) = \frac{1}{2\pi}\int_\mathbf{C} \log|z-\zeta|\,d\mu(\zeta)</math>
+Given ''H''&nbsp;&isin;&nbsp;(0,&nbsp;1), there exist discs of radius ''r''<sub>''i''</sub> such that
+:<math>\sum_i r_i < 5H\,</math>
+and
+:<math>u(z) \ge \frac{n}{2\pi}\log \frac{H}{e}</math>
+for all ''z'' outside the union of these discs.
+==Notes==
+<references/>
+{{mathanalysis-stub}}
+[[Category:Complex analysis]]