en>IronGargoyle: unclear, unhelpful (HG)

2014-05-07T10:56:59Z

unclear, unhelpful (HG)

en>Dthomsen8: /* External links */

2014-01-22T17:15:25Z

External links

← Older revision		Revision as of 19:15, 22 January 2014
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	~~Oscar~~ is ~~what my wife enjoys~~ to ~~call me~~ and I ~~completely dig that~~ title. ~~My day occupation~~ is a ~~librarian~~. ~~North Dakota~~ is ~~her beginning place but she will have~~ to ~~move~~ 1 ~~day or~~ an ~~additional~~. ~~To collect cash~~ is ~~one~~ of the ~~things I adore most~~.<br><br>~~Visit my web~~-~~site;~~ [http://www.~~buzzbit~~.~~net~~/~~user~~/~~AFilson std testing at home~~]		[[Image:Amoeba of p=w-2z-1.svg\|right\|thumb\|The amoeba of<br>
			''P(''z'',''w'')=''w''-2''z''-1.]]
			[[Image:Amoeba2.svg\|right\|thumb\|The amoeba of<br>
			''P''(''z'', ''w'')=3''z''<sup>2</sup>+5''zw''+''w<sup>3</sup>+1.<br>
			Notice the "[[vacuole]]" in the middle of the amoeba.]]

			[[Image:Amoeba3.svg\|right\|thumb\|The amoeba of <br>
			''P''(''z'', ''w'') = 1 + ''z''+''z''<sup>2</sup>
			+ ''z''<sup>3</sup> + ''z''<sup>2</sup>''w''<sup>3</sup>
			+ 10''zw'' + 12''z''<sup>2</sup>''w''
			+10''z''<sup>2</sup>''w''<sup>2</sup>.
			]]
			[[Image:Amoeba4 400.svg\|right\|thumb\|The amoeba of<br>
			''P''(''z'', ''w'')=50 ''z''<sup>3</sup>
			+83 ''z''<sup>2</sup> ''w''+24 ''z w''<sup>2</sup>
			+''w''<sup>3</sup>+392 ''z''<sup>2</sup>
			+414 ''z w''+50 ''w''<sup>2</sup>
			-28 ''z'' +59 ''w''-100.
			]]

			[[Image:Amoeba of x+y+z-1.png\|right\|thumb\|Points in the amoeba of<br>
			''P''(''x'',''y'',''z'')=''x''+''y''+''z''-1.
			Note that the amoeba is actually 3-dimensional, and not a surface,
			(this is not entirely evident from the image).
			]]

			In [[complex analysis]], a branch of [[mathematics]], an '''amoeba''' is a [[Set (mathematics)\|set]] associated with a [[polynomial]] in one or more [[several complex variables\|complex variables]]. Amoebas have applications in [[algebraic geometry]]. There is independently a concept of "amoeba order" in [[set theory]].

			==Definition==
			Consider the function

			:<math>\mathrm{Log}: \left({\mathbb C}\backslash\{0\}\right)^n \to \mathbb R^n</math>

			defined on the set of all ''n''-[[tuple]]s <math>z=(z_1, z_2, \dots, z_n)</math> of non-zero [[complex number]]s with values in the [[Euclidean space]] <math>\mathbb R^n,</math> given by the formula
			:<math>\mathrm{Log}(z_1, z_2, \dots, z_n)= (\log \|z_1\|, \log\|z_2\|, \dots, \log \|z_n\|).\,</math>

			Here, 'log' denotes the [[natural logarithm]]. If ''p''(''z'') is a polynomial in <math>n</math> complex variables, its '''amoeba''' <math>{\mathcal A}_p</math> is defined as the [[image (mathematics)\|image]] of the set of [[root of a function\|zeros]] of ''p'' under Log, so

			: <math>{\mathcal A}_p = \left\{\mathrm{Log} (z) \, : \, z\in \left({\mathbb C}\backslash\{0\}\right)^n, p(z)=0\right\}.\,</math>

			Amoebas were introduced in 1994 in a book by [[Israel Gelfand\|Gelfand]], Kapranov, and [[Andrei Zelevinsky\|Zelevinsky]].<ref>
			{{cite book
			\| last = Gelfand
			\| first = I. M.
			\| coauthors = M.M. Kapranov, A.V. Zelevinsky
			\| title = Discriminants, resultants, and multidimensional determinants
			\| publisher = Boston: Birkhäuser
			\| year = 1994
			\| pages =
			\| isbn = 0-8176-3660-9
			}}</ref>

			==Properties==

			* Any amoeba is a [[closed set]].
			* Any [[connected space\|connected component]] of the [[complement (set theory)\|complement]] <math>\mathbb R^n\backslash {\mathcal A}_p</math> is [[convex set\|convex]].
			* The area of an amoeba of a not identically zero polynomial in two complex variables is finite.
			* A two-dimensional amoeba has a number of "tentacles" which are infinitely long and exponentially narrowing towards infinity.

			==Ronkin function==

			A useful tool in studying amoebas is the '''Ronkin function'''<!--, named after ??????? Ronkin-->. For ''p''(''z'') a polynomial in ''n'' complex variables, one defines the Ronkin function

			: <math>N_p:\mathbb R^n \to \mathbb R</math>

			by the formula

			: <math>N_p(x)=\frac{1}{(2\pi i)^n}\int_{\mathrm{Log}^{-1}(x)}\log\|p(z)\| \,\frac{dz_1}{z_1} \wedge \frac{d z_2}{z_2}\wedge\cdots \wedge \frac{d z_n}{z_n},</math>

			where <math>x</math> denotes <math>x=(x_1, x_2, \dots, x_n).</math> Equivalently, <math>N_p</math> is given by the integral

			: <math>N_p(x)=\frac{1}{(2\pi)^n}\int_{[0, 2\pi]^n}\log\|p(z)\| \,d\theta_1\,d\theta_2 \cdots d\theta_n,</math>

			where

			: <math>z=\left(e^{x_1+i\theta_1}, e^{x_2+i\theta_2}, \dots, e^{x_n+i\theta_n}\right).</math>

			The Ronkin function is convex, and it is [[affine function\|affine]] on each connected component of the complement of the amoeba of <math>p(z).</math>

			As an example, the Ronkin function of a [[monomial]]

			: <math>p(z)=az_1^{k_1}z_2^{k_2}\dots z_n^{k_n}\, </math>

			with <math>a\ne 0</math> is

			: <math>N_p(x) = \log\|a\|+k_1x_1+k_2x_2+\cdots+k_nx_n.\,</math>

			==Set theory==
			In [[set theory]], the amoeba order is the set of pairs <math>\langle P,\varepsilon\rangle</math> where <math>P</math> is an open subset of the Euclidean unit square <math>[0,1]\times[0,1]</math> with [[Lebesgue measure]] <math>\mu(P) < \varepsilon</math>. We order the elements of the amoeba order by <math>\langle P,\varepsilon\rangle\le\langle Q,\varepsilon^\rangle \iff P\supseteq Q \hbox{ and } \varepsilon\le\varepsilon^</math>.<ref>This definition is from [[Benedikt Löwe]], ''What is ... An Amoeba (2)?'' [http://www.math.uni-bonn.de/people/loewe/Publ/amoebaR.ps].</ref>

			==References==
			<references/>

			==External links==
			{{Commons category\|Amoeba (mathematics)}}
			* [http://www.dm.unipi.it/~bertrand/amoeb-geotrop/node1.html Amoebas of algebraic varieties]

			{{DEFAULTSORT:Amoeba (Mathematics)}}
			[[Category:Algebraic geometry]]

en>David Eppstein: stub sort

2012-08-29T07:19:58Z

stub sort

New page

Oscar is what my wife enjoys to call me and I completely dig that title. My day occupation is a librarian. North Dakota is her beginning place but she will have to move 1 day or an additional. To collect cash is one of the things I adore most.<br><br>Visit my web-site; [http://www.buzzbit.net/user/AFilson std testing at home]

BCJR algorithm - Revision history

en>IronGargoyle: unclear, unhelpful (HG)

en>Dthomsen8: /* External links */

en>David Eppstein: stub sort