Dilution (equation)

2014-01-24T15:37:19Z

129.81.28.123: changed increasing solute concentration to decreasing solute concentration in solution

{{Distinguish|Cauchy condensation test}}

The '''Cauchy convergence test''' is a method used to test [[Series (mathematics)|infinite series]] for [[convergent series|convergence]]. A series

:<math>\sum_{i=0}^\infty a_i</math>

with [[real number|real]] or [[complex number|complex]] summands ''a''''i'' is convergent if and only if for every <math>\varepsilon>0</math> there is a [[natural number]] ''N'' such that

:<math>|a_{n+1}+a_{n+2}+\cdots+a_{n+p}|<\varepsilon</math>

holds for all {{nobreak|''n'' > ''N''}} and {{nobreak|''p'' ≥ 1}}.<ref>Abbott, Stephen (2001). ''Understanding Analysis'', p.63. Springer, New York. ISBN 9781441928665</ref>

The test works because the space '''R''' of real numbers and the space '''C''' of complex numbers (with the metric given by the absolute value) are both [[complete metric space|complete]], so that the series is convergent [[if and only if]] the partial sum

: <math>s_n:=\sum_{i=0}^n a_i</math>

is a [[Cauchy sequence]]: for every <math>\varepsilon>0</math> there is a number ''N'', such that for all ''n'', ''m'' > ''N'' holds

<math>|s_m-s_n|<\varepsilon.</math>

We can assume ''m'' > ''n'' and thus set ''p'' = ''m'' − ''n''.

:<math>|s_{n+p}-s_n|=|a_{n+1}+a_{n+2}+\cdots+a_{n+p}|<\varepsilon.</math>

{{PlanetMath attribution|id=3894|title=Cauchy criterion for convergence}}

== References ==
{{reflist}}
==External links==
* [http://www.encyclopediaofmath.org/index.php/Cauchy_criteria Cauchy criteria] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]

[[Category:Convergence tests]]

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2013-12-04T22:53:34Z

129.81.30.54:

{{Infobox enzyme
| Name = Delta11-fatty-acid desaturase
| EC_number = 1.14.19.5
| CAS_number =
| IUBMB_EC_number = 1/14/19/5
| GO_code =
| image =
| width =
| caption =
}}
'''Delta11-fatty-acid desaturase''' ({{EC number|1.14.19.5}}, ''Delta11 desaturase'', ''fatty acid Delta11-desaturase'', ''TpDESN'', ''Cro-PG'', ''Delta11 fatty acid desaturase'', ''Z/E11-desaturase'', ''Delta11-palmitoyl-CoA desaturase'') is an [[enzyme]] with system name ''acyl-CoA,hydrogen donor:oxygen Delta11-oxidoreductase''.<ref>{{cite journal | title = Identification of a fatty acid Δ11-desaturase from the microalga ''Thalassiosira pseudonana'' |author = Tonon, T., Harvey, D., Qing, R., Li, Y., Larson, T.R. and Graham, I.A. |journal = FEBS Lett. |year = 2004 |volume = 563 |pages = 28–34 |pmid = 15063718 |doi=10.1016/S0014-5793(04)00241-8 |issue=1-3}}</ref><ref>{{cite journal | title = Characterization of ''Z''/''E''11- and ''Z''9-desaturases from the obliquebanded leafroller moth, ''Choristoneura rosaceana'' |author = Hao, G., O'Connor, M., Liu, W. and Roelofs, W.L. |journal = J. Insect Sci. |year = 2002 |volume = 2:26 |pages = 1–7 |pmid = }}</ref><ref>{{cite journal | title = Characterization of the Δ11-palmitoyl-CoA-desaturase from ''Spodoptera littoralis'' (Lepidoptera:Noctuidae) |author = Rodriguez, F., Hallahan, D.L., Pickett, J.A. and Camps, F. |journal = Insect Biochem. Mol. Biol. |year = 1992 |volume = 22 |pages = 143–148 |pmid = }}</ref> This enzyme [[catalysis|catalyses]] the following [[chemical reaction]]

: [[acyl-CoA]] + reduced acceptor + O2 <math>\rightleftharpoons</math> Delta11-acyl-CoA + acceptor + 2 H2O

The enzyme from the marine [[microalga]] ''[[Thalassiosira pseudonana]]'' desaturates [[palmitic acid]] 16:0 to 16:1Delta11, whereas that from the leafroller [[moth]] ''[[Choristoneura rosaceana]]'' desaturates [[myristic acid]] 14:0 to 14:1Delta11.

== References ==
{{reflist}}

== External links ==
* {{MeshName|Delta11-fatty-acid+desaturase}}

[[Category:EC 1.14.19]]

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Dilution (equation)

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