Commit 7b25fff7 authored by Guenter Paul Peters's avatar Guenter Paul Peters
Browse files

documentation moved to package.html

git-svn-id: https://svn.math.tu-berlin.de/svn/jtem@186 f5b180c5-49ee-4939-b20e-b6ed35f0f7b7
parent 67575170
......@@ -19,5 +19,5 @@ package de.jtem.riemann;
class DocletDummy {
// this class is to trick javadoc.
// this class is to trick javadoc, so that it reads the package.html
}
<html xmlns:v="urn:schemas-microsoft-com:vml" xmlns:o="urn:schemas-microsoft-com:office:office" xmlns="http://www.w3.org/TR/REC-html40">
<head>
<meta http-equiv="Content-Language" content="de">
<meta name="GENERATOR" content="Microsoft FrontPage 5.0">
<meta name="ProgId" content="FrontPage.Editor.Document">
<meta http-equiv="Content-Type" content="text/html; charset=windows-1252">
<title>The riemann Project</title>
<style>
<!--
code { }
-->
</style>
</head>
<body>
<p></p>
<p>The riemann project currently contains two public sub-packages:</p>
<blockquote>
<p>&nbsp;<a href="#sec: schottky"><i><font face="Courier">de.jtem.riemann.schottky</font></i><font face="Courier">&nbsp;
</font>
</a></p>
</blockquote>
<p>and </p>
<blockquote>
<p><i><font face="Courier">
<!-- teaser start -->
<p>The <code>riemann</code> project currently contains two public sub-packages:
<a href="#sec: schottky"><i><font face="Courier">de.jtem.riemann.schottky</font></i></a>
and
<i><font face="Courier">
<a href="#sec: theta">de.jtem.riemann.theta</a></font></i><font face="Courier">.
</font> </p>
</blockquote>
<p>Both packages implement the current state of research in this field. A third
</font></p>
<p> Both packages implement the current state of research in this field. A third
package concerning algebraic curves and coverings is subject of a major
refactoring and will be published as soon as possible.</p>
<!-- teaser end -->
<p>This tutorial gives an introduction to the major functionality of these
packages and closes with an example: <a href="#sec: KP2 Equation">KP2 Equation -
Shallow Water Waves</a>.</p>
......@@ -126,7 +102,7 @@ Poincare theta series:
<tr VALIGN="MIDDLE">
<td ALIGN="CENTER" NOWRAP>
<a NAME="eq:_series_for_differential_of_1st_kind"></a>
<img border="0" src="index.9.gif" width="280" height="47"></td>
<img border="0" src="doc-files/index.9.gif" width="280" height="47"></td>
<td WIDTH="10" ALIGN="RIGHT">(2)</td>
</tr>
</table>
......@@ -146,7 +122,7 @@ Elementary computations deliver further formulas for the integrals of first kind
<table WIDTH="100%" ALIGN="CENTER">
<tr VALIGN="MIDDLE">
<td ALIGN="CENTER" NOWRAP><a NAME="eq:_integrals_of_1st_kind"></a>
<img border="0" src="index.17.gif" width="212" height="47"></td>
<img border="0" src="doc-files/index.17.gif" width="212" height="47"></td>
<td WIDTH="10" ALIGN="RIGHT">(3)</td>
</tr>
</table>
......@@ -162,13 +138,13 @@ B_{nm}=\delta_{nm}\log\mu_{n}+\sum_{\sigma\in\GmGGn,\sigma\neq id}\log\left\{ B_
<table WIDTH="100%" ALIGN="CENTER">
<tr VALIGN="MIDDLE">
<td ALIGN="CENTER" NOWRAP><a NAME="eq:_series_for_period_matrix"></a>
<img border="0" src="index.1.gif" width="301" height="36"></td>
<img border="0" src="doc-files/index.1.gif" width="301" height="36"></td>
<td WIDTH="10" ALIGN="RIGHT">(4)</td>
</tr>
</table>
</div>
<p>where the <i>curly</i> brackets indicate the cross-ratio</p>
<p align="center">&nbsp;<img border="0" src="index.2.gif" width="148" height="38">. </p>
<p align="center">&nbsp;<img border="0" src="doc-files/index.2.gif" width="148" height="38">. </p>
<p>The series above do not always converge and it is challenging to evaluate
them in a stable manner. </p>
<p>The numerics in this package allow the evaluation of several other series:
......@@ -187,7 +163,7 @@ them in a stable manner. </p>
<tr VALIGN="MIDDLE">
<td ALIGN="CENTER" NOWRAP>
<a NAME="eq:_series_for_differential_of_3rd_kind"></a>
<img border="0" src="index.3.gif" width="245" height="44"></td>
<img border="0" src="doc-files/index.3.gif" width="245" height="44"></td>
<td WIDTH="10" ALIGN="RIGHT">(5)</td>
</tr>
</table>
......@@ -204,7 +180,7 @@ them in a stable manner. </p>
<tr VALIGN="MIDDLE">
<td ALIGN="CENTER" NOWRAP>
<a NAME="eq:_series_for_integrals_of_3rd_kind"></a>
<img border="0" src="index.4.gif" width="176" height="46"></td>
<img border="0" src="doc-files/index.4.gif" width="176" height="46"></td>
<td WIDTH="10" ALIGN="RIGHT">(6)</td>
</tr>
</table>
......@@ -221,7 +197,7 @@ them in a stable manner. </p>
<table WIDTH="100%" ALIGN="CENTER">
<tr VALIGN="MIDDLE">
<td ALIGN="CENTER" NOWRAP><a NAME="eq:_series_sigma"></a>
<img border="0" src="index.5.gif" width="168" height="34"></td>
<img border="0" src="doc-files/index.5.gif" width="168" height="34"></td>
<td WIDTH="10" ALIGN="RIGHT">(7)</td>
</tr>
</table>
......@@ -238,7 +214,7 @@ V_{n,k}=\sum_{\sigma\in\GnG}\sigma\left(A_{n}\right)^{k}-\sigma\left(B_{n}\right
<table WIDTH="100%" ALIGN="CENTER">
<tr VALIGN="MIDDLE">
<td ALIGN="CENTER" NOWRAP><a NAME="eq:_series_for_vector_V"></a>
<img border="0" src="index.6.gif" width="186" height="37"></td>
<img border="0" src="doc-files/index.6.gif" width="186" height="37"></td>
<td WIDTH="10" ALIGN="RIGHT">(8)</td>
</tr>
</table>
......@@ -255,7 +231,7 @@ V_{n,k}=\sum_{\sigma\in\GnG}\sigma\left(A_{n}\right)^{k}-\sigma\left(B_{n}\right
<table WIDTH="100%" ALIGN="CENTER">
<tr VALIGN="MIDDLE">
<td ALIGN="CENTER" NOWRAP><a NAME="eq:_series_for_gamma"></a>
<img border="0" src="index.7.gif" width="66" height="42"></td>
<img border="0" src="doc-files/index.7.gif" width="66" height="42"></td>
<td WIDTH="10" ALIGN="RIGHT">(9)</td>
</tr>
</table>
......@@ -274,7 +250,7 @@ and (<a href="file:///H:/dr/documents/dr.jtem/jtemBook.bak/jtemBook.html#eq:_ser
converge if the limit <i>
<!-- MATH
$q_{\infty}^{\Omega}$
-->q<font face="Symbol"><sup>W</sup><sub>¥</sub></font>&nbsp;</i> of the
-->q<font face="Symbol"><sup>W</sup><sub></sub></font>&nbsp;</i> of the
monotonously decreasing series
<!-- MATH
$q_{\infty}^{\Omega}$
......@@ -309,8 +285,8 @@ the evaluation class <i>Schottky</i>. To create the Schottky data <br>
S_{-,2}=\left\{ A,-A,\mu,-\overline{A},\overline{A},\mu\right\}
\end{displaymath}
-->
<p><img border="0" src="index.8.gif" width="157" height="26"></div>
<p>with <i>A, </i> <font face="Symbol"><i>m Î</i></font><i><span style="font-family: Monotype Corsiva">C</span></i><font face="Symbol">
<p><img border="0" src="doc-files/index.8.gif" width="157" height="26"></div>
<p>with <i>A, </i> <font face="Symbol"><i>m </i></font><i><span style="font-family: Monotype Corsiva">C</span></i><font face="Symbol">
</font>&nbsp;for in the Helicoid <i><span style="font-family: Monotype Corsiva">
He</span></i><sub>2 </sub>you can either write
</p>
......@@ -482,7 +458,7 @@ instance for the result, which helps to relieve the garbage collector. The call
<p>This package implements the methods and algorithms presented in [<a href="#[Sch05]">Sch05</a>,
Chapter 3] for computing riemann theta functions including those with
characteristics. An important ingredient for the computation of Riemann theta
function are modular transformations and Siegel´s Reduction algorithm for which
function are modular transformations and Siegel�s Reduction algorithm for which
the package also offers public interfaces.</p>
<p></p>
<h3><a NAME="SECTION00162100000000000000">Computing Riemann theta functions</a>
......@@ -498,9 +474,9 @@ complex variables and defined by <br>
\end{displaymath}
-->
<p>
<img border="0" src="index.10.gif" width="144" height="46">
<img border="0" src="doc-files/index.10.gif" width="144" height="46">
</div>
<p>where <i>z<font face="Symbol">Î</font>C<sup>g</sup></i>
<p>where <i>z<font face="Symbol"></font>C<sup>g</sup></i>
and <i>B</i>
is a symmetric <i>g</i>-dimensional
matrix with strictly negative definite real part. Be aware that there are many
......@@ -543,7 +519,7 @@ evaluating instance. </p>
</blockquote>
<p>evaluates the Riemann theta function at the argument <i>z<font face="Symbol">Î</font>C<sup>g</sup></i>.
<p>evaluates the Riemann theta function at the argument <i>z<font face="Symbol"></font>C<sup>g</sup></i>.
Following the philosophy of the whole project, there exist also overloaded
methods which allow to prescribe the result as a parameter: </p>
......@@ -566,7 +542,7 @@ exponential growth from the oscillating part of the function: <br>
\theta\left(z\left|B\right.\right)=e^{f\left(z\left|B\right.\right)}\cdot\theta_{\Sigma}\left(z\left|B\right.\right)\,,
\end{displaymath}
-->
<p><img border="0" src="index.11.gif" width="142" height="27">,</div>
<p><img border="0" src="doc-files/index.11.gif" width="142" height="27">,</div>
<p>with
<!-- MATH
$f\left(z\left|B\right.\right)$
......@@ -602,13 +578,13 @@ f\left(z\right)=\frac{\theta\left(z+a\left|B\right.\right)}{\theta\left(z+b\left
\end{displaymath}
-->
<p>
<img border="0" src="index.12.gif" width="286" height="47">
<img border="0" src="doc-files/index.12.gif" width="286" height="47">
</div>
<p>Since the vectors
<!-- MATH
$a,b\in\mathbb{C}^{g}$
-->
<i>a,b<font face="Symbol">Î</font>C<sup>g</sup></i>
<i>a,b<font face="Symbol"></font>C<sup>g</sup></i>
are relatively small the exponential factor is about <i>1</i>. An implementation of
this function could look like:
</p>
......@@ -674,7 +650,7 @@ D_{X}\theta\left(z\left|B\right.\right)=e^{f\left(z\left|B\right.\right)}\cdot\l
\end{displaymath}
-->
<p>
<img border="0" src="index.13.gif" width="193" height="27">
<img border="0" src="doc-files/index.13.gif" width="193" height="27">
</div>
<p>with <i>f</i>&nbsp;
being the same quadratic function as for the Riemann theta function itself.
......@@ -811,7 +787,7 @@ S=\left\{ A_{1},B_{1},\mu_{1},\ldots,A_{N},B_{N},\mu_{N}\right\}
-->
B<sub>i</sub>=A<sub>i</sub>
and&nbsp; <i><font face="Symbol">
m</font></i><sub>i</sub><font face="Symbol">Î</font><b>R</b> yield all real
m</font></i><sub>i</sub><font face="Symbol"></font><b>R</b> yield all real
non-singular finite-gap solutions of the KP2 equation and can be described by
Krichever's formula<br>
</p>
......@@ -823,16 +799,16 @@ u\left(x,y,t\right)=2\frac{\partial^{2}}{\partial x^{2}}\log\theta\left(U\, x+V\
\end{displaymath}
-->
<p>
<img border="0" src="index.15.gif" width="285" height="41"></div>
<img border="0" src="doc-files/index.15.gif" width="285" height="41"></div>
<p>The parameters in this formula can be given by
Poincare series. It is</p>
<p><i>U=V<sub>n,1</sub>, V=V<sub>n,2</sub>, </i>and<i>&nbsp; W=V<sub>n,3 </sub>&nbsp;with
</i>
</p>
<p align="center"><span style="position: relative; top: 16.0pt">
<img border="0" src="index.18.gif" width="186" height="37"></span></p>
<img border="0" src="doc-files/index.18.gif" width="186" height="37"></span></p>
<p align="center">and</p>
<p align="center"><img border="0" src="index.16.gif" width="88" height="42"><br>
<p align="center"><img border="0" src="doc-files/index.16.gif" width="88" height="42"><br>
</p>
<p></p>
<div ALIGN="CENTER">
......@@ -928,7 +904,7 @@ Krichever. Algebraic curves and non-linear difference equations. Russ. Math.
surv., 33(4):255-256, 1978.</p>
<p><a name="[Sch05]"></a>[Sch05]&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; M.
Schmies. Computational Methods for Riemann Surfaces and Helicoids with Handles.
PhD thesis, Technische Universität Berlin, 2005.&nbsp;&nbsp;&nbsp; (<a href="http://opus.kobv.de/tuberlin/volltexte/2005/1105/pdf/schmies_markus.pdf">.pdf</a>)</p>
PhD thesis, Technische Universit�t Berlin, 2005.&nbsp;&nbsp;&nbsp; (<a href="http://opus.kobv.de/tuberlin/volltexte/2005/1105/pdf/schmies_markus.pdf">.pdf</a>)</p>
<address>
Markus Schmies 2006-01-14
</address>
......
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