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  <region id="0" left="18" top="18" width="496" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p bold="true">Funktionen elementweise anwenden</p>
    </text>
    <text lang="eng">
      <p bold="true">Element-wise application of functions to matrix and vectors</p>
    </text>
  </region>
  <region id="1" left="540" top="18" width="177" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p href="Chapter Matrices.sm">Vektoren und Matrizen</p>
    </text>
    <text lang="eng">
      <p href="Chapter Matrices.sm">Vectors and Matrices</p>
    </text>
  </region>
  <region id="2" left="540" top="36" width="103" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p href="Mathematik.sm">Mathematik</p>
    </text>
    <text lang="eng">
      <p href="Mathematik.sm">Mathematics</p>
    </text>
  </region>
  <region id="3" left="540" top="54" width="78" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p href="Handbook.sm">Inhalt</p>
    </text>
    <text lang="eng">
      <p href="Handbook.sm">Contents</p>
    </text>
  </region>
  <region id="4" left="18" top="90" width="714" height="72" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Diese Seite zeigt, wie Rechenoperationen elementweise auf Vektoren oder Matrizen
angewendet werden können:
- Schleifen über alle Elemente
- Funktionen, die Matrix-Argumente elementweise behandeln</p>
    </text>
    <text lang="eng">
      <p>This page explains the options if  mathematical expressions must be applied to vectors
or matrices of data.  
- You can use loops to handle each element separately
- You can use functions which handle matrix valued arguments </p>
    </text>
  </region>
  <region id="5" left="18" top="180" width="53" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p underline="true">Schleifen</p>
    </text>
    <text lang="eng">
      <p underline="true">Loops</p>
    </text>
  </region>
  <region id="6" left="36" top="216" width="125" height="41" color="#000000" bgColor="#ffffff" fontSize="10">
    <math optimize="2">
      <input>
        <e type="operand">v.x</e>
        <e type="operand">0</e>
        <e type="operand">2</e>
        <e type="operand">π</e>
        <e type="operator" args="2">*</e>
        <e type="operand">π</e>
        <e type="operand">4</e>
        <e type="operator" args="2">/</e>
        <e type="function" preserve="true" args="3">range</e>
        <e type="operator" args="2">:</e>
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  <region id="7" left="216" top="216" width="42" height="35" color="#000000" bgColor="#ffffff">
    <picture>
      <raw format="png" encoding="base64"> </raw>
    </picture>
  </region>
  <region id="8" left="261" top="225" width="82" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>v.x:rang </p>
    </text>
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  <region id="9" left="342" top="225" width="20" height="24" border="true" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="eng">
      <p>↓</p>
    </text>
  </region>
  <region id="10" left="369" top="225" width="37" height="24" border="true" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Tab</p>
    </text>
  </region>
  <region id="11" left="405" top="225" width="20" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>0</p>
    </text>
  </region>
  <region id="12" left="423" top="225" width="20" height="24" border="true" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>→</p>
    </text>
  </region>
  <region id="13" left="441" top="225" width="28" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>2p</p>
    </text>
  </region>
  <region id="14" left="468" top="225" width="61" height="24" border="true" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Strg+G</p>
    </text>
    <text lang="eng">
      <p>Ctrl-G</p>
    </text>
  </region>
  <region id="15" left="531" top="225" width="20" height="24" border="true" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>→</p>
    </text>
  </region>
  <region id="16" left="549" top="225" width="20" height="24" border="true" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>→</p>
    </text>
  </region>
  <region id="17" left="567" top="225" width="20" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>p</p>
    </text>
  </region>
  <region id="18" left="585" top="225" width="61" height="24" border="true" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Strg+G</p>
    </text>
    <text lang="eng">
      <p>Ctrl-G</p>
    </text>
  </region>
  <region id="19" left="648" top="225" width="28" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>/4</p>
    </text>
  </region>
  <region id="20" left="36" top="261" width="96" height="170" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">v.x</e>
      </input>
      <contract>
        <e type="operand">π</e>
      </contract>
      <result action="numeric">
        <e type="operand">0</e>
        <e type="operand">0.25</e>
        <e type="operand">0.5</e>
        <e type="operand">0.75</e>
        <e type="operand">1</e>
        <e type="operand">1.25</e>
        <e type="operand">1.5</e>
        <e type="operand">1.75</e>
        <e type="operand">2</e>
        <e type="operand">9</e>
        <e type="operand">1</e>
        <e type="function" preserve="true" args="11">mat</e>
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  </region>
  <region id="21" left="216" top="324" width="42" height="35" color="#000000" bgColor="#ffffff">
    <picture>
      <raw format="png" encoding="base64"> </raw>
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  <region id="22" left="261" top="333" width="45" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>v.x=</p>
    </text>
  </region>
  <region id="23" left="306" top="333" width="20" height="24" border="true" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>→</p>
    </text>
  </region>
  <region id="24" left="324" top="333" width="20" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>p</p>
    </text>
  </region>
  <region id="25" left="342" top="333" width="61" height="24" border="true" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Strg+G</p>
    </text>
    <text lang="eng">
      <p>Ctrl-G</p>
    </text>
  </region>
  <region id="26" left="18" top="459" width="544" height="40" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Typischerweise benötigt man eine Schleife, deren Zähler von 1 bis zur Länge des 
Vektors läuft.</p>
    </text>
    <text lang="eng">
      <p>Typically loops run through the indices of the a vector from 1 to
it's length.</p>
    </text>
  </region>
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    <picture>
      <raw format="png" encoding="base64"> </raw>
    </picture>
  </region>
  <region id="28" left="36" top="513" width="179" height="50" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">i</e>
        <e type="operand">1</e>
        <e type="operand">v.x</e>
        <e type="function" preserve="true" args="1">length</e>
        <e type="function" preserve="true" args="2">range</e>
        <e type="operand">#</e>
        <e type="function" preserve="true" args="3">for</e>
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    </math>
  </region>
  <region id="29" left="279" top="513" width="37" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>for</p>
    </text>
  </region>
  <region id="30" left="315" top="513" width="37" height="24" border="true" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Tab</p>
    </text>
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  <region id="31" left="351" top="513" width="20" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>i</p>
    </text>
  </region>
  <region id="32" left="369" top="513" width="20" height="24" border="true" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>→</p>
    </text>
  </region>
  <region id="33" left="387" top="513" width="45" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>rang</p>
    </text>
  </region>
  <region id="34" left="432" top="513" width="37" height="24" border="true" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Tab</p>
    </text>
  </region>
  <region id="35" left="468" top="513" width="20" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">1</e>
      </input>
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  </region>
  <region id="36" left="486" top="513" width="20" height="24" border="true" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>→</p>
    </text>
  </region>
  <region id="37" left="504" top="513" width="41" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>len </p>
    </text>
  </region>
  <region id="38" left="540" top="513" width="37" height="24" border="true" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Tab</p>
    </text>
  </region>
  <region id="39" left="576" top="513" width="37" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>v.x</p>
    </text>
  </region>
  <region id="40" left="36" top="567" width="179" height="68" color="#000000" bgColor="#ffffff" fontSize="10">
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      <input>
        <e type="operand">i</e>
        <e type="operand">1</e>
        <e type="operand">v.x</e>
        <e type="function" preserve="true" args="1">length</e>
        <e type="function" preserve="true" args="2">range</e>
        <e type="operand">v.y</e>
        <e type="operand">i</e>
        <e type="function" preserve="true" args="2">el</e>
        <e type="operand">v.x</e>
        <e type="operand">i</e>
        <e type="function" preserve="true" args="2">el</e>
        <e type="function" preserve="true" args="1">sin</e>
        <e type="operator" args="2">:</e>
        <e type="function" preserve="true" args="3">for</e>
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  </region>
  <region id="41" left="234" top="594" width="42" height="35" color="#000000" bgColor="#ffffff">
    <picture>
      <raw format="png" encoding="base64"> </raw>
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  </region>
  <region id="42" left="279" top="603" width="53" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>v.y[i</p>
    </text>
  </region>
  <region id="43" left="333" top="603" width="20" height="24" border="true" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>→</p>
    </text>
  </region>
  <region id="44" left="351" top="603" width="45" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>:sin</p>
    </text>
  </region>
  <region id="45" left="396" top="603" width="37" height="24" border="true" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Tab</p>
    </text>
  </region>
  <region id="46" left="432" top="603" width="53" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>v.x[i</p>
    </text>
  </region>
  <region id="47" left="234" top="639" width="240" height="158" color="#000000" bgColor="#ffffff" fontSize="10">
    <plot type="2d" render="lines" scale_x="3.28307557444403" scale_y="3.28307557444403" scale_z="3.28307557444403" rotate_x="0" rotate_y="0" rotate_z="0" transpose_x="-101" transpose_y="-6" transpose_z="0">
      <input>
        <e type="operand">v.x</e>
        <e type="operand">v.y</e>
        <e type="function" preserve="true" args="2">augment</e>
      </input>
    </plot>
  </region>
  <region id="48" left="36" top="648" width="173" height="179" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">v.y</e>
      </input>
      <result action="numeric">
        <e type="operand">0</e>
        <e type="operand">0.7071</e>
        <e type="operand">1</e>
        <e type="operand">0.7071</e>
        <e type="operand">0</e>
        <e type="operand">0.7071</e>
        <e type="operator" args="1">-</e>
        <e type="operand">1</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.7071</e>
        <e type="operator" args="1">-</e>
        <e type="operand">2.4492</e>
        <e type="operand">10</e>
        <e type="operand">16</e>
        <e type="operator" args="1">-</e>
        <e type="operator" args="2">^</e>
        <e type="operator" args="2">*</e>
        <e type="operator" args="1">-</e>
        <e type="operand">9</e>
        <e type="operand">1</e>
        <e type="function" preserve="true" args="11">mat</e>
      </result>
    </math>
  </region>
  <region id="49" left="18" top="837" width="664" height="40" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Matrizen und Datenvektoren lassen sich aber auch zeilenweise mit der Funktion 
stack() aufbauen, siehe dazu auch </p>
    </text>
    <text lang="eng">
      <p>Matrices and data vectors alternatively can be generated line by line using the 
function stack(). See the following link:</p>
    </text>
  </region>
  <region id="50" left="18" top="873" width="135" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p href="Section matrix handling.sm">Manipulation von Matrizen</p>
    </text>
    <text lang="eng">
      <p href="Section matrix handling.sm">Matrix Handling</p>
    </text>
  </region>
  <region id="51" left="18" top="909" width="365" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p underline="true">Funktionen, die elementweise auf Matrizen angewendet werden können</p>
    </text>
    <text lang="eng">
      <p underline="true">Functions which can be applied element-wise</p>
    </text>
  </region>
  <region id="52" left="18" top="945" width="709" height="40" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Manche Funktionen für skalare Argumente funktionieren auch für vektor- oder matrix-
wertige Argumente. Das ist bei den eingebauten Funktionen in SMath nur in Ausnahme-
fällen möglich. </p>
    </text>
    <text lang="eng">
      <p>Some functions for scalar arguments also work for vector- or matrix-valued arguments.
However, most of the built-in functions do not have that capability.</p>
    </text>
  </region>
  <region id="53" left="36" top="1017" width="448" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p underline="true">Addition und Multiplikation mit einer Zahl</p>
    </text>
    <text lang="eng">
      <p underline="true">Addition and multiplication with scalars and matrices</p>
    </text>
  </region>
  <region id="54" left="36" top="1053" width="124" height="44" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">1</e>
        <e type="operand">2</e>
        <e type="operand">3</e>
        <e type="operand">4</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
        <e type="operand">5</e>
        <e type="operator" args="2">+</e>
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      <result action="numeric">
        <e type="operand">6</e>
        <e type="operand">7</e>
        <e type="operand">8</e>
        <e type="operand">9</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
      </result>
    </math>
  </region>
  <region id="55" left="189" top="1053" width="150" height="44" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">1</e>
        <e type="operand">2</e>
        <e type="operand">3</e>
        <e type="operand">4</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
        <e type="operand">10</e>
        <e type="operator" args="2">-</e>
      </input>
      <result action="numeric">
        <e type="operand">9</e>
        <e type="operator" args="1">-</e>
        <e type="operand">8</e>
        <e type="operator" args="1">-</e>
        <e type="operand">7</e>
        <e type="operator" args="1">-</e>
        <e type="operand">6</e>
        <e type="operator" args="1">-</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
      </result>
    </math>
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  <region id="56" left="351" top="1053" width="126" height="44" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">1</e>
        <e type="operand">2</e>
        <e type="operand">3</e>
        <e type="operand">4</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
        <e type="operand">3</e>
        <e type="operator" args="2">*</e>
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      <result action="numeric">
        <e type="operand">3</e>
        <e type="operand">6</e>
        <e type="operand">9</e>
        <e type="operand">12</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
      </result>
    </math>
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  <region id="57" left="486" top="1053" width="134" height="44" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">5</e>
        <e type="operand">1</e>
        <e type="operand">2</e>
        <e type="operand">3</e>
        <e type="operand">4</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
        <e type="operator" args="2">*</e>
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      <result action="numeric">
        <e type="operand">5</e>
        <e type="operand">10</e>
        <e type="operand">15</e>
        <e type="operand">20</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
      </result>
    </math>
  </region>
  <region id="58" left="36" top="1107" width="431" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p underline="true">Funktionen aus dem Plugin "Matlab C++ Math Library"</p>
    </text>
    <text lang="eng">
      <p underline="true">Functions from the plugin "Matlab C++ Math Library"</p>
    </text>
  </region>
  <region id="59" left="36" top="1143" width="381" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p href="Section math trig matlab.sm">Winkelfunktionen mit Matrixargument</p>
    </text>
    <text lang="eng">
      <p href="Section math trig matlab.sm">Trigonometric Functions with matrix arguments</p>
    </text>
  </region>
  <region id="60" left="36" top="1161" width="356" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p href="Section math hyperbolic matlab.sm">Hyperbelfunktionen mit Matrixargument</p>
    </text>
    <text lang="eng">
      <p href="Section math hyperbolic matlab.sm">Hyperbolic Functions with matrix arguments</p>
    </text>
  </region>
  <region id="61" left="36" top="1179" width="340" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p href="Section math boolean matlab.sm">Boolsche Operationen mit Matrixargument</p>
    </text>
    <text lang="eng">
      <p href="Section math boolean matlab.sm">Boolean operations with matrix arguments</p>
    </text>
  </region>
  <region id="62" left="45" top="1215" width="226" height="188" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">v.x</e>
        <e type="function" preserve="true" args="1">mwsin</e>
      </input>
      <result action="numeric">
        <e type="operand">0</e>
        <e type="operand">0.7071</e>
        <e type="operand">1</e>
        <e type="operand">0.7071</e>
        <e type="operand">1.2246</e>
        <e type="operand">10</e>
        <e type="operand">16</e>
        <e type="operator" args="1">-</e>
        <e type="operator" args="2">^</e>
        <e type="operator" args="2">*</e>
        <e type="operand">0.7071</e>
        <e type="operator" args="1">-</e>
        <e type="operand">1</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.7071</e>
        <e type="operator" args="1">-</e>
        <e type="operand">2.4493</e>
        <e type="operand">10</e>
        <e type="operand">16</e>
        <e type="operator" args="1">-</e>
        <e type="operator" args="2">^</e>
        <e type="operator" args="2">*</e>
        <e type="operator" args="1">-</e>
        <e type="operand">9</e>
        <e type="operand">1</e>
        <e type="function" preserve="true" args="11">mat</e>
      </result>
    </math>
  </region>
  <region id="63" left="36" top="1431" width="210" height="32" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">M</e>
        <e type="operand">3</e>
        <e type="operand">4</e>
        <e type="operand">10</e>
        <e type="operator" args="1">-</e>
        <e type="operand">10</e>
        <e type="function" preserve="true" args="4">Random.N</e>
        <e type="operator" args="2">:</e>
      </input>
    </math>
  </region>
  <region id="64" left="36" top="1458" width="154" height="62" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">M</e>
      </input>
      <result action="numeric">
        <e type="operand">10</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0</e>
        <e type="operand">2</e>
        <e type="operand">3</e>
        <e type="operator" args="1">-</e>
        <e type="operand">3</e>
        <e type="operand">5</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0</e>
        <e type="operand">7</e>
        <e type="operand">7</e>
        <e type="operand">7</e>
        <e type="operator" args="1">-</e>
        <e type="operand">4</e>
        <e type="operator" args="1">-</e>
        <e type="operand">10</e>
        <e type="operand">3</e>
        <e type="operand">4</e>
        <e type="function" preserve="true" args="14">mat</e>
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    </math>
  </region>
  <region id="65" left="216" top="1458" width="179" height="62" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">M</e>
        <e type="function" preserve="true" args="1">mwabs</e>
      </input>
      <result action="numeric">
        <e type="operand">10</e>
        <e type="operand">0</e>
        <e type="operand">2</e>
        <e type="operand">3</e>
        <e type="operand">3</e>
        <e type="operand">5</e>
        <e type="operand">0</e>
        <e type="operand">7</e>
        <e type="operand">7</e>
        <e type="operand">7</e>
        <e type="operand">4</e>
        <e type="operand">10</e>
        <e type="operand">3</e>
        <e type="operand">4</e>
        <e type="function" preserve="true" args="14">mat</e>
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    </math>
  </region>
  <region id="66" left="432" top="1476" width="266" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Betrag der Elemente</p>
    </text>
    <text lang="eng">
      <p>Absolute values of the elements</p>
    </text>
  </region>
  <region id="67" left="216" top="1521" width="207" height="62" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">M</e>
        <e type="function" preserve="true" args="1">mwsign</e>
      </input>
      <result action="numeric">
        <e type="operand">1</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0</e>
        <e type="operand">1</e>
        <e type="operand">1</e>
        <e type="operator" args="1">-</e>
        <e type="operand">1</e>
        <e type="operand">1</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0</e>
        <e type="operand">1</e>
        <e type="operand">1</e>
        <e type="operand">1</e>
        <e type="operator" args="1">-</e>
        <e type="operand">1</e>
        <e type="operator" args="1">-</e>
        <e type="operand">1</e>
        <e type="operand">3</e>
        <e type="operand">4</e>
        <e type="function" preserve="true" args="14">mat</e>
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    </math>
  </region>
  <region id="68" left="432" top="1539" width="176" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Vorzeichen der Elemente</p>
    </text>
    <text lang="eng">
      <p>Sign of the elements</p>
    </text>
  </region>
  <region id="69" left="18" top="1602" width="177" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p href="Chapter Matrices.sm">Vektoren und Matrizen</p>
    </text>
    <text lang="eng">
      <p href="Chapter Matrices.sm">Vectors and Matrices</p>
    </text>
  </region>
  <region id="70" left="18" top="1620" width="103" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p href="Mathematik.sm">Mathematik</p>
    </text>
    <text lang="eng">
      <p href="Mathematik.sm">Mathematics</p>
    </text>
  </region>
  <region id="71" left="18" top="1638" width="78" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p href="Handbook.sm">Inhalt</p>
    </text>
    <text lang="eng">
      <p href="Handbook.sm">Contents</p>
    </text>
  </region>
  <region id="72" left="531" top="1665" width="20" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">.</e>
      </input>
    </math>
  </region>
  <region id="73" left="18" top="1674" width="450" height="40" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p italic="true">$Author: mkraska $
$Date: 2013-12-01 23:13:16 +0100 (So, 01. Dez 2013) $ </p>
    </text>
  </region>
</regions>