Johnson circles

2014-01-03T21:49:34Z

68.231.124.189: /* Properties */

== Summary ==
Image 2 (state at t=10) of a sequence showing a Turing bifurcation from a noisy ground state to a hexagonal state in a two-component reaction-diffusion system of Fitzhugh-Nagumo type, generated by Dr. H. U. Bödeker.

The system reads:

:<math>
\begin{array}{rl}
\partial_t u &= d_u^2 \Delta u + u -u^3 - v + \kappa,\\
\tau \partial_t v &= d_v^2 \Delta v + u - v
\end{array}
</math>

with ''τ'' = 0.1, ''du2'' = 0.00028, ''dv2'' = 0.005, ''κ'' = - 0.05

and was solved using a finite-element algorithm.

== Licensing ==
{{self|GFDL|cc-by-sa-2.5,2.0,1.0|migration=relicense}}

== Licensing ==
{{self|GFDL|cc-by-sa-2.5,2.0,1.0|migration=relicense}}

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Squared deviations

2012-11-01T06:15:25Z

68.231.212.2:

The '''Ragsdale conjecture''' is a [[mathematics|mathematical]] [[conjecture]] that concerns the possible arrangements of real [[algebraic curves]] embedded in the [[projective plane]]. It was proposed by [[Virginia Ragsdale]] several years after 1900 and was disproved in 1979.<ref>{{cite journal| last=Viro | first=Oleg Ya. | authorlink=Oleg Viro | year=1980 | title=Кривые степени 7, кривые степени 8 и гипотеза Рэгсдейл | trans_title=Curves of degree 7, curves of degree 8 and the hypothesis of Ragsdale | journal=[[Doklady Akademii Nauk SSSR]] | volume=254 | issue=6 | pages=1306–1309}} Translated in {{cite journal| journal=Soviet Mathematics - Doklady | volume=22 | pages=566–570 | year=1980}}</ref>

==Background==
Her dissertation dealt with [[Hilbert's sixteenth problem]], which was proposed in the year 1900, along with [[Hilbert's problems|22 other unsolved problems of the 19th century]]. Ragsdale conjectured a particular upper bound on the number of topological circles of a certain type, along with the basis of evidence. The conjecture was held of high importance in the field of real algebraic geometry for nearly a century. Later [[Oleg Viro]] and [[Ilya Itenberg]] produced [[counterexamples]] to the Ragsdale conjecture, although the problem of finding a sharp upper bound remains unsolved.

==Conjecture==
Ragsdale's main conjecture is as follows.

Assume that an [[algebraic curve]] of degree 2''k'' contains ''p'' even and ''n'' odd ovals. Ragsdale conjectured that

:<math> p \le \tfrac32 k(k-1) + 1 \quad\text{and}\quad n \le \tfrac32 k(k-1). </math>

She also posed the inequality

:<math> | 2(p-n)-1 | \le 3k^2 - 3k + 1, </math>

and showed that the inequality could not be further improved. This inequality was later proved by [[Ivan Petrovsky|Petrovsky]].

==Notes==
<references/>

==References==
* {{cite web |url=http://www.agnesscott.edu/LRiddle/women/ragsdale.htm |title=Virginia Ragsdale |accessdate=2007-03-09 |last=De Loera |first=Jesús |coauthors=Frederick J. Wicklin |year=2006 |work=Biographies of Women Mathematicians }}

[[Category:Conjectures]]
[[Category:Real algebraic geometry]]

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Johnson circles

Squared deviations