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<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a>  <a href="chap1.html">1</a>  <a href="chap2.html">2</a>  <a href="chap3.html">3</a>  <a href="chap4.html">4</a>  <a href="chap5.html">5</a>  <a href="chapBib.html">Bib</a>  <a href="chapInd.html">Ind</a>  </div>

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<h1>Thelma</h1>


<h2>THreshold ELements, Modeling and Applications</h2>

<p>
    1.3</p>

<p>
    3 March 2022
  </p>

</div>
<p><b>
    Victor Bovdi



  </b>
<br />Email: <span class="URL"><a href="mailto:vbovdi@gmail.com">vbovdi@gmail.com</a></span>
<br />Address: <br />Department of Mathematical Sciences<br /> UAEU<br /> Al Ain, United Arab Emirates<br />
</p><p><b>
    Vasyl Laver



  </b>
<br />Email: <span class="URL"><a href="mailto:vasyl.laver@uzhnu.edu.ua">vasyl.laver@uzhnu.edu.ua</a></span>
<br />Address: <br />Department of Mathematical Sciences<br /> UAEU<br /> Al Ain, United Arab Emirates<br />
</p>

<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
<h3>Copyright</h3>
<p>© 2018 by the authors</p>

<p>This package may be distributed under the terms and conditions of the GNU Public License Version 2 or higher.</p>

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<h3>Contents<a id="contents" name="contents"></a></h3>

<div class="ContChap"><a href="chap1.html#X7DFB63A97E67C0A1">1 <span class="Heading">Introduction</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X786BACDB82918A65">1.1 <span class="Heading">Overview over this manual</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X8360C04082558A12">1.2 <span class="Heading">Installation</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X80D704CC7EBFDF7A">1.3 <span class="Heading">Feedback</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap2.html#X830F6677819A6656">2 <span class="Heading">Boolean Functions</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X82EB5BE77F9F686A">2.1 <span class="Heading">Basic Operations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7D8A971A84186BF0">2.1-1 LogicFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7C34DEE27EC3A009">2.1-2 IsLogicFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X857A4F117BDDF1B6">2.1-3 PolynomialToBooleanFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X78B9DD8E81EE65A8">2.1-4 IsUnateInVariable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X8203046F7C97EAFE">2.1-5 IsUnateBooleanFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7D0DEF0B7E9FE027">2.1-6 InfluenceOfVariable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X86C000FE831CFEDE">2.1-7 SelfDualExtensionOfBooleanFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7C09D66E84201A77">2.1-8 SplitBooleanFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X8108DE5280908841">2.1-9 KernelOfBooleanFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7B1B54877B354F23">2.1-10 ReducedKernelOfBooleanFunction</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap3.html#X83DFEF607CDA3594">3 <span class="Heading">Threshold Elements</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X82EB5BE77F9F686A">3.1 <span class="Heading">Basic Operations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X853FF8AC85E5C4CE">3.1-1 ThresholdElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7E7F95D57EB87BB4">3.1-2 IsThresholdElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7F293D2D810ADBBB">3.1-3 OutputOfThresholdElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8116985278C60D33">3.1-4 StructureOfThresholdElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X800C6898856CD0E9">3.1-5 RandomThresholdElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7C9E64E17B07DF58">3.1-6 Comparison of Threshold Elements</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7DB8E3E87D8AD820">3.2 <span class="Heading">Single Threshold Element Realizability</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8351E15B7CCDEB62">3.2-1 CharacteristicVectorOfFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8502A0827ECF1FFE">3.2-2 IsCharacteristicVectorOfSTE</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7B0855E28717AF12">3.2-3 IsInverseInKernel</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X83B234DB7F260879">3.2-4 IsKernelContainingPrecedingVectors</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7877118A7A9C51CB">3.2-5 IsRKernelBiggerOfCombSum</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X83D3B18486C5D0B7">3.2-6 BooleanFunctionBySTE</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8451E4EA7F06222F">3.2-7 PDBooleanFunctionBySTE</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7D0584857E0D2265">3.3 <span class="Heading">Iterative Training Methods</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X83567B7D79958EBF">3.3-1 ThresholdElementTraining</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X83CD889E8125E949">3.3-2 ThresholdElementBatchTraining</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X81FEA8AF80573A12">3.3-3 WinnowAlgorithm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X85162EF17EE09E50">3.3-4 Winnow2Algorithm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7C4B705884A78907">3.3-5 STESynthesis</a></span>
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<div class="ContChap"><a href="chap4.html#X7BA69E617DE465FE">4 <span class="Heading">Networks of Threshold Elements</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X82EB5BE77F9F686A">4.1 <span class="Heading">Basic Operations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X780C027786A8E34B">4.1-1 NeuralNetwork</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X79B24B8F79161064">4.1-2 IsNeuralNetwork</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7C93C473784D1676">4.1-3 OutputOfNeuralNetwork</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7BA69E617DE465FE">4.2 <span class="Heading">Networks of Threshold Elements</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X786FB7237D875561">4.2-1 BooleanFunctionByNeuralNetwork</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7C6E4221828E6609">4.2-2 BooleanFunctionByNeuralNetworkDASG</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap5.html#X8735CBCC80897D83">5 <span class="Heading">Multi-Valued Threshold Elements</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X83640D4B7FFE0F9F">5.1 <span class="Heading">Threshold Elements over <span class="SimpleMath">GF(p^k)</span></span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X818FF66D7919D8B1">5.1-1 MVThresholdElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7C4A761F864B8EFA">5.1-2 IsMVThresholdElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7E64FB7684207AED">5.1-3 OutputOfMVThresholdElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7B2520F37C18CC81">5.1-4 StructureOfMVThresholdElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7B103C6B7EB903EF">5.1-5 MVBooleanFunctionBySTE</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chapBib.html"><span class="Heading">References</span></a></div>
<div class="ContChap"><a href="chapInd.html"><span class="Heading">Index</span></a></div>
<br />
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