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__NOTOC__<nonumbers/>
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{{Stub}}
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{| style="float:right; margin-left:20px; max-width:20em; border:solid thin black; padding:.5ex;"
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#LyX 2.1 created this file. For more info see http://www.lyx.org/
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|<!-- The 10 --> Recent changes:
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|-
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== Welcome to the Manifold Atlas Project:<br>- a scientific Wiki and<br>- an on-line journal, the [http://www.boma.mpim-bonn.mpg.de Bulletin of the Manifold Atlas] ==
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\font_typewriter default
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The [[Manifold Atlas:About|Manifold Atlas]] has six chapters:
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\index Index
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\shortcut idx
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\end_index
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* '''[[:Category:Manifolds|Manifolds]]:''' with pages about '''constructions''' and '''examples''' of manifolds as well as their '''invariants''' and '''properties'''.
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\begin_body
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* '''[[:Category:Theory|Theory]]:''' with pages summarising useful '''general facts''' and '''theorems'''.
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* '''[[:Category:Definitions|Definitions]]:''' with pages giving succinct and precise '''definitions''' of important concepts in the theory of manifolds.
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* '''[[:Category:Problems|Problems]]:''' with pages listing '''persistent and deep open problems''', their status and their history.
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* '''[[:Category:History|History]]:''' with pages about the '''history''' of the study of manifolds.
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<!--* '''[[:Category:Questions|Questions]]:''' with pages containing '''questions''' and '''answers''' about any aspect of the study of manifolds.-->
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* '''[[:Category:Exercises|Exercises]]:''' with pages containing exercises about the study of manifolds.
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<!-- * '''[[:Category:Philosophy|Philosophy]]:''' with pages on the philosophy of mathematics as it relates to manifolds. -->
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You will find the pages listed alphabetically by clicking each chapter.
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\begin_layout Standard
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The state determinted by two temperatures (refering
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\begin_inset Formula $T_{0}$
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\end_inset
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[[Manifold Atlas:Page evolution|Atlas pages]] offer a workspace for collaboratively organising and creating knowledge about manifolds.
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as constant heat sink) :
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\begin_inset Formula
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$$
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\begin{cases}
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C\frac{dT}{dt}+K\left(T-T_{0}\right)=K_{h}\left(T_{h}-T\right) & \,
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[[Manifold Atlas:Editorial Policy#Editorial criteria|Mature pages]] are [[Manifold Atlas: Editorial Policy|refereed]] by the [[Manifold Atlas:Editorial Board|editorial board]]: if approved they are published in the [http://www.boma.mpim-bonn.mpg.de BoMA].
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C_{h}\frac{dT_{h}}{dt}=P_{h}-K_{h}\left(T_{h}-T\right) & \,
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\end{cases}
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$$
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== General information ==
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\end_inset
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For more information please follow these links: [[Manifold Atlas:About|About the Atlas]] and the [[Manifold Atlas:Editorial Board|Editorial Board]].
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For information about how to contribute please follows these links: [[Manifold Atlas:Writing in the Atlas|Writing in the Atlas]] and [[Manifold Atlas:Community Portal|Community Portal]].
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Matrix representation:
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\begin_inset Formula
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$$
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$$
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== Using the Manifold Atlas ==
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\end_inset
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The page [[Manifold Atlas:User orientation|User orientation]] gives information for registered users.
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The page [[Manifold Atlas:Author orientation|Author orientation]] gives information about writing in the Atlas.
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* [[Manifold Atlas:User orientation#Writing_on_discussion_pages|Join the discussion]] about a page.
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* [[Manifold Atlas:Writing groups#Open-editing pages|Add to or edit ]] existing open-editing articles.
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* [[Manifold Atlas:Create_a_new open-editing page|Create a new open-editing page]].
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* [[Manifold Atlas:Create a new restricted-editing page|Create a new restricted-editing page]].
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<!--
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The '''[[Manifold Atlas:Community Portal|community portal]]''' is a general discussion forum.
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You can either '''[[Manifold Atlas:Create_a_new page|create a new page]]''' and '''discuss''' or '''modify''' all articles by clicking on <b>'discussion'</b> or <b>'edit'</b> at the top of each page. Don't forget to [[Special:UserLogin|log in]]!
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\end_layout
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The [[Manifold Atlas:Community Portal|Community Portal]] is a general discussion forum.
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\begin_layout Standard
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Eigenvalues of
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\begin_inset Formula $M$
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\end_inset
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Read the [[Manifold Atlas:Author information|author information]] for important information about writing articles in the Manifold Atlas.-->
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:
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\begin_inset Formula
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$$\begin{array}{rcl}
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\lambda & = & -\frac{CK_{h}+C_{h}K+C_{h}K_{h}\pm\sqrt{C^{2}K_{h}^{2}-2CC_{h}KK_{h}+2CC_{h}K_{h}^{2}+C_{h}^{2}K^{2}+2C_{h}^{2}KK_{h}+C_{h}^{2}K_{h}^{2}}}{2CC_{h}}
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== Registering for the Manifold Atlas==
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\lambda & = & -\frac{1}{2}\left[\frac{K_{h}}{C_{h}}+\frac{K}{C}+\frac{K_{h}}{C}\pm\sqrt{\frac{K_{h}^{2}}{C_{h}^{2}}-2\frac{KK_{h}}{CC_{h}}+2\frac{K_{h}^{2}}{CC_{h}}+\frac{K^{2}}{C^{2}}+2\frac{KK_{h}}{C^{2}}+\frac{K_{h}^{2}}{C^{2}}}\right]
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* Please '''[[Manifold Atlas:Registration|register]]''' with a verified email address in order to write and discuss in the Manifold Atlas.
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* To sign up for email up-dates about changes at the Atlas visit the [[Manifold Atlas:The Atlas Observer|Atlas Observer]].
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\, & \, & \mbox{denoting}\;\begin{cases}
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\gamma & =\frac{K}{C}
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\gamma_{m} & =\frac{K_{h}}{C}
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\gamma_{h} & =\frac{K_{h}}{C_{h}}
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\end{cases}
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\lambda & = & -\frac{1}{2}\left[\gamma_{h}+\gamma+\gamma_{m}\pm\sqrt{\gamma_{h}^{2}-2\gamma\gamma_{h}+2\gamma_{m}\gamma_{h}+\gamma^{2}+2\gamma\gamma_{m}+\gamma_{m}^{2}}\right]
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\lambda & = & -\frac{1}{2}\left[\gamma+\gamma_{m}+\gamma_{h}\pm\sqrt{\left(\gamma+\gamma_{m}+\gamma_{h}\right)^{2}-4\gamma\gamma_{h}}\right]
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\, & \, & \mbox{denotinng}\begin{cases}
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\gamma+\gamma_{m}+\gamma_{h}=\gamma_{+} & \,
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\gamma\gamma_{h}=\gamma_{*}^{2} & \,
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\end{cases}
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\, & \, & \gamma_{+}\gg\gamma_{*}
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\lambda & = & -\frac{1}{2}\left[\gamma_{+}\pm\sqrt{\gamma_{+}^{2}-4\gamma_{*}^{2}}\right]
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\lambda & \approx & -\frac{1}{2}\left[\gamma_{+}\pm\left(\gamma_{+}-2\frac{\gamma_{*}^{2}}{\gamma_{+}}\right)\right]
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\, & \, & \begin{cases}
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\lambda_{+} & =-\gamma_{+}
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\lambda_{-} & =-\frac{\gamma_{*}^{2}}{\gamma_{+}}=-\frac{\gamma\gamma_{h}}{\gamma+\gamma_{m}+\gamma_{h}}=-\frac{\gamma\gamma_{h}}{\gamma+\gamma_{m}+\gamma_{h}}
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\end{cases}
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\end{array}$$
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\end_body
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\end_document
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</wikitex>
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== References ==
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{{#RefList:}}
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[[Category:]]

Revision as of 07:59, 7 June 2015

This page has not been refereed. The information given here might be incomplete or provisional.

  1. LyX 2.1 created this file. For more info see http://www.lyx.org/ 
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\begin_body


\begin_layout Standard 

The state determinted by two temperatures (refering 
\begin_inset Formula $T_{0}$
\end_inset

as constant heat sink) : 

\begin_inset Formula 
$$
\begin{cases}
C\frac{dT}{dt}+K\left(T-T_{0}\right)=K_{h}\left(T_{h}-T\right) & \,

C_{h}\frac{dT_{h}}{dt}=P_{h}-K_{h}\left(T_{h}-T\right) & \, 

\end{cases}
$$

\end_inset


Matrix representation: 

\begin_inset Formula 
$$
$$

\end_inset 


\end_layout

\begin_layout Standard 

Eigenvalues of 
\begin_inset Formula $M$
\end_inset


\begin_inset Formula 
$$\begin{array}{rcl}
\lambda & = & -\frac{CK_{h}+C_{h}K+C_{h}K_{h}\pm\sqrt{C^{2}K_{h}^{2}-2CC_{h}KK_{h}+2CC_{h}K_{h}^{2}+C_{h}^{2}K^{2}+2C_{h}^{2}KK_{h}+C_{h}^{2}K_{h}^{2}}}{2CC_{h}}

\lambda & = & -\frac{1}{2}\left[\frac{K_{h}}{C_{h}}+\frac{K}{C}+\frac{K_{h}}{C}\pm\sqrt{\frac{K_{h}^{2}}{C_{h}^{2}}-2\frac{KK_{h}}{CC_{h}}+2\frac{K_{h}^{2}}{CC_{h}}+\frac{K^{2}}{C^{2}}+2\frac{KK_{h}}{C^{2}}+\frac{K_{h}^{2}}{C^{2}}}\right]

\, & \, & \mbox{denoting}\;\begin{cases} 

\gamma & =\frac{K}{C} 
\gamma_{m} & =\frac{K_{h}}{C} 
\gamma_{h} & =\frac{K_{h}}{C_{h}}
\end{cases} 
\lambda & = & -\frac{1}{2}\left[\gamma_{h}+\gamma+\gamma_{m}\pm\sqrt{\gamma_{h}^{2}-2\gamma\gamma_{h}+2\gamma_{m}\gamma_{h}+\gamma^{2}+2\gamma\gamma_{m}+\gamma_{m}^{2}}\right]

\lambda & = & -\frac{1}{2}\left[\gamma+\gamma_{m}+\gamma_{h}\pm\sqrt{\left(\gamma+\gamma_{m}+\gamma_{h}\right)^{2}-4\gamma\gamma_{h}}\right]

\, & \, & \mbox{denotinng}\begin{cases} 

\gamma+\gamma_{m}+\gamma_{h}=\gamma_{+} & \,

\gamma\gamma_{h}=\gamma_{*}^{2} & \, 

\end{cases} 
\, & \, & \gamma_{+}\gg\gamma_{*} 
\lambda & = & -\frac{1}{2}\left[\gamma_{+}\pm\sqrt{\gamma_{+}^{2}-4\gamma_{*}^{2}}\right]

\lambda & \approx & -\frac{1}{2}\left[\gamma_{+}\pm\left(\gamma_{+}-2\frac{\gamma_{*}^{2}}{\gamma_{+}}\right)\right]

\, & \, & \begin{cases} 

\lambda_{+} & =-\gamma_{+} 
\lambda_{-} & =-\frac{\gamma_{*}^{2}}{\gamma_{+}}=-\frac{\gamma\gamma_{h}}{\gamma+\gamma_{m}+\gamma_{h}}=-\frac{\gamma\gamma_{h}}{\gamma+\gamma_{m}+\gamma_{h}}
\end{cases}
\end{array}$$

\end_inset 


\end_layout

\end_body 

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</wikitex>

References

[[Category:]]

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