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Informationsmaße Eq: math.2392.61

Hash: 161943c282e0fa2fc1085c22d80f2153

TeX (original user input):

\begin{align}

& {{P}_{i}}=\rho \left( {{x}^{i}} \right){{\Delta }^{d}}x \\

& I(P)=\sum\limits_{i}^{{}}{{}}{{\Delta }^{d}}x\rho \left( {{x}^{i}} \right)\ln \left( {{\Delta }^{d}}x\rho \left( {{x}^{i}} \right) \right)=\sum\limits_{i}^{{}}{{}}{{\Delta }^{d}}x\rho \left( {{x}^{i}} \right)\ln \left( \rho \left( {{x}^{i}} \right) \right)+\sum\limits_{i}^{{}}{{}}{{\Delta }^{d}}x\rho \left( {{x}^{i}} \right)\ln \left( {{\Delta }^{d}}x \right) \\

& \sum\limits_{i}^{{}}{{}}{{\Delta }^{d}}x\rho \left( {{x}^{i}} \right)=1 \\

& \sum\limits_{i}^{{}}{{}}{{\Delta }^{d}}x\rho \left( {{x}^{i}} \right)\ln \left( {{\Delta }^{d}}x \right)=const. \\

\end{align}

TeX (checked):

{\begin{aligned}&{{P}_{i}}=\rho \left({{x}^{i}}\right){{\Delta }^{d}}x\\&I(P)=\sum \limits _{i}^{}{}{{\Delta }^{d}}x\rho \left({{x}^{i}}\right)\ln \left({{\Delta }^{d}}x\rho \left({{x}^{i}}\right)\right)=\sum \limits _{i}^{}{}{{\Delta }^{d}}x\rho \left({{x}^{i}}\right)\ln \left(\rho \left({{x}^{i}}\right)\right)+\sum \limits _{i}^{}{}{{\Delta }^{d}}x\rho \left({{x}^{i}}\right)\ln \left({{\Delta }^{d}}x\right)\\&\sum \limits _{i}^{}{}{{\Delta }^{d}}x\rho \left({{x}^{i}}\right)=1\\&\sum \limits _{i}^{}{}{{\Delta }^{d}}x\rho \left({{x}^{i}}\right)\ln \left({{\Delta }^{d}}x\right)=const.\\\end{aligned}}

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\begin{matrix} P_{i} = ρ (x^{i}) Δ^{d} x \\ I (P) = \sum_{i}^{} Δ^{d} x ρ (x^{i}) \ln (Δ^{d} x ρ (x^{i})) = \sum_{i}^{} Δ^{d} x ρ (x^{i}) \ln (ρ (x^{i})) + \sum_{i}^{} Δ^{d} x ρ (x^{i}) \ln (Δ^{d} x) \\ \sum_{i}^{} Δ^{d} x ρ (x^{i}) = 1 \\ \sum_{i}^{} Δ^{d} x ρ (x^{i}) \ln (Δ^{d} x) = c o n s t . \end{matrix}

<math xmlns="http://www.w3.org/1998/Math/MathML" class="mwe-math-element mwe-math-element-inline"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="true" scriptlevel="0"><mtable displaystyle="true"><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mo stretchy="false">=</mo><mi>ρ</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><mi>x</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow><msup><mi>Δ</mi><mrow data-mjx-texclass="ORD"><mi>d</mi></mrow></msup><mi>x</mi></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mi>I</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><munderover><mo form="prefix" movablelimits="false" stretchy="false">∑</mo><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow><mrow data-mjx-texclass="ORD"></mrow></munderover><msup><mi>Δ</mi><mrow data-mjx-texclass="ORD"><mi>d</mi></mrow></msup><mi>x</mi><mi>ρ</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><mi>x</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mi>ln</mi><mo>&#x2061;</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><mi>Δ</mi><mrow data-mjx-texclass="ORD"><mi>d</mi></mrow></msup><mi>x</mi><mi>ρ</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><mi>x</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">=</mo><munderover><mo form="prefix" movablelimits="false" stretchy="false">∑</mo><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow><mrow data-mjx-texclass="ORD"></mrow></munderover><msup><mi>Δ</mi><mrow data-mjx-texclass="ORD"><mi>d</mi></mrow></msup><mi>x</mi><mi>ρ</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><mi>x</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mi>ln</mi><mo>&#x2061;</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>ρ</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><mi>x</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">+</mo><munderover><mo form="prefix" movablelimits="false" stretchy="false">∑</mo><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow><mrow data-mjx-texclass="ORD"></mrow></munderover><msup><mi>Δ</mi><mrow data-mjx-texclass="ORD"><mi>d</mi></mrow></msup><mi>x</mi><mi>ρ</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><mi>x</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mi>ln</mi><mo>&#x2061;</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><mi>Δ</mi><mrow data-mjx-texclass="ORD"><mi>d</mi></mrow></msup><mi>x</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><munderover><mo form="prefix" movablelimits="false" stretchy="false">∑</mo><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow><mrow data-mjx-texclass="ORD"></mrow></munderover><msup><mi>Δ</mi><mrow data-mjx-texclass="ORD"><mi>d</mi></mrow></msup><mi>x</mi><mi>ρ</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><mi>x</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">=</mo><mn>1</mn></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><munderover><mo form="prefix" movablelimits="false" stretchy="false">∑</mo><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow><mrow data-mjx-texclass="ORD"></mrow></munderover><msup><mi>Δ</mi><mrow data-mjx-texclass="ORD"><mi>d</mi></mrow></msup><mi>x</mi><mi>ρ</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><mi>x</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mi>ln</mi><mo>&#x2061;</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><mi>Δ</mi><mrow data-mjx-texclass="ORD"><mi>d</mi></mrow></msup><mi>x</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">=</mo><mi>c</mi><mi>o</mi><mi>n</mi><mi>s</mi><mi>t</mi><mo stretchy="false">.</mo></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd></mtr></mtable></mstyle></mrow></math>

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Identifiers

$P_{i}$

$ρ$

$x$

$i$

$Δ$

$d$

$x$

$I$

$P$

$i$

$Δ$

$d$

$x$

$ρ$

$x$

$i$

$Δ$

$d$

$x$

$ρ$

$x$

$i$

$i$

$Δ$

$d$

$x$

$ρ$

$x$

$i$

$ρ$

$x$

$i$

$i$

$Δ$

$d$

$x$

$ρ$

$x$

$i$

$Δ$

$d$

$x$

$i$

$Δ$

$d$

$x$

$ρ$

$x$

$i$

$i$

$Δ$

$d$

$x$

$ρ$

$x$

$i$

$Δ$

$d$

$x$

$c$

$o$

$n$

$s$

$t$

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