Centerfocus main - Revision history

CfWikiSysop at 19:18, 21 January 2011

2011-01-21T19:18:14Z

2011-01-21T19:16:39Z

2011-01-21T19:15:49Z

2011-01-17T10:38:06Z

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2011-01-12T23:02:17Z

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 == Our work ==
-Let <math>Z_i</math> be the locus  where the first i focal values vanish. We obtain heuristic information about the variety <math>Z_{13}</math> and <math>Z_{14}</math> respectively by evaluating its defining equations at random points over a small finite field.  For this we calculate the focal values of a plane autonomous system using an C++ implementation of a modified algorithm from Kay Moritzen (see his [http://www.uni-bayreuth.de/departments/math/org/mathe6/publ/da/moritzen/diplom.html diploma thesis], in german) who in turn build on the work of Frommer 1934. To speedup the calculation we parametrized the first three focal values.
+Let <math>Z_i</math> be the locus  where the first <math>i</math> focal values vanish. We obtain heuristic information about the variety <math>Z_{13}</math> and <math>Z_{14}</math> respectively by evaluating its defining equations at random points over a small finite field.  For this we calculate the focal values of a plane autonomous system using an C++ implementation of a modified algorithm from Kay Moritzen (see his [http://www.uni-bayreuth.de/departments/math/org/mathe6/publ/da/moritzen/diplom.html diploma thesis], in german) who in turn build on the work of Frommer 1934. To speedup the calculation we parametrized the first three focal values.
 The interested reader will find more details in our [[centerfocus_findings | findings ]].

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 == Some history ==
-Poincaré showed in 1885, that there exists an infinite number of so called focal polynomials s_i in the coefficients of P and Q that vanish if and only if the system above has a center.
+Poincaré showed in 1885, that there exists an infinite number of so called focal polynomials <math>s_i</math> in the coefficients of P and Q that vanish if and only if the system above has a center.
 The values of these polynomials are called focal values. If the first n-1 focal values vanish, but the n'th doesn't, the system has a n-th order focus.

@@ Line 36: / Line 36: @@
 == Our work ==
-Let Z_i be the locus  where the first i focal values vanish. We obtain heuristic information about the variety Z_{13} and Z_{14} respectively by evaluating its defining equations at random points over a small finite field.  For this we calculate the focal values of a plane autonomous system using an C++ implementation of a modified algorithm from Kay Moritzen (see his [http://www.uni-bayreuth.de/departments/math/org/mathe6/publ/da/moritzen/diplom.html diploma thesis], in german) who in turn build on the work of Frommer 1934. To speedup the calculation we parametrized the first three focal values.
+Let <math>Z_i</math> be the locus  where the first i focal values vanish. We obtain heuristic information about the variety <math>Z_{13}</math> and <math>Z_{14}</math> respectively by evaluating its defining equations at random points over a small finite field.  For this we calculate the focal values of a plane autonomous system using an C++ implementation of a modified algorithm from Kay Moritzen (see his [http://www.uni-bayreuth.de/departments/math/org/mathe6/publ/da/moritzen/diplom.html diploma thesis], in german) who in turn build on the work of Frommer 1934. To speedup the calculation we parametrized the first three focal values.
 The interested reader will find more details in our [[centerfocus_findings | findings ]].

@@ Line 38: / Line 38: @@
 Let Z_i be the locus  where the first i focal values vanish. We obtain heuristic information about the variety Z_{13} and Z_{14} respectively by evaluating its defining equations at random points over a small finite field.  For this we calculate the focal values of a plane autonomous system using an C++ implementation of a modified algorithm from Kay Moritzen (see his [http://www.uni-bayreuth.de/departments/math/org/mathe6/publ/da/moritzen/diplom.html diploma thesis], in german) who in turn build on the work of Frommer 1934. To speedup the calculation we parametrized the first three focal values.
-The interested reader will find more details in our [[centerfocus_publications | publications]].
+The interested reader will find more details in our [[centerfocus_findings | findings ]].
 The statistics from our experiments and some of the analyzed points are stored in a public accessible [[cf:QueryDatabase.html | database]].<br>
 Our program centerfocus is freely available at [http://sourceforge.net/projects/centerfocus/files/ sourceforge].

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 In 1885 Poincaré asked when the solutions of the plane autonomous system<br>
 &nbsp;x' = y + q(x,y) = Q(x,y)<br>
-&nbsp;y' = -x - p(x,y) = -P(x,y)<br><math>fg</math>
+&nbsp;y' = -x - p(x,y) = -P(x,y)<br><math></math>
 are stable in the neighbourhood of the equilibrium (x,y)=(0,0).