- testwiki

General

Display information for equation id:math.930.99 on revision:930

* Page found: Verallgemeinerte kanonische Verteilung (eq math.930.99)

Occurrences on the following pages:

Verallgemeinerte kanonische Verteilung Eq: math.940.99

Hash: 88682300bea1fce8a7a9bbe0132e7ff7

TeX (original user input):

\begin{align}
  & K\left( P,{{P}^{0}} \right)=\sum\limits_{i}^{{}}{{}}{{P}_{i}}\ln {{P}_{i}}-{{P}_{i}}\ln {{P}_{i}}^{0}+{{P}_{i}}^{0}\ln {{P}_{i}}^{0}-{{P}_{i}}^{0}\ln {{P}_{i}}^{0} \\ 
 & \sum\limits_{i}^{{}}{{}}{{P}_{i}}\ln {{P}_{i}}=I(P) \\ 
 & \sum\limits_{i}^{{}}{{}}{{P}_{i}}^{0}\ln {{P}_{i}}^{0}=I({{P}^{0}}) \\ 
 & -{{P}_{i}}\ln {{P}_{i}}^{0}+{{P}_{i}}^{0}\ln {{P}_{i}}^{0}=-\sum\limits_{i}^{{}}{{}}\left( {{P}_{i}}-{{P}_{i}}^{0} \right)\ln {{P}_{i}}^{0} \\ 
 & \ln {{P}_{i}}^{0}=\Psi -{{\lambda }_{n}}{{M}_{i}}^{n} \\ 
 & -\sum\limits_{i}^{{}}{{}}\left( {{P}_{i}}-{{P}_{i}}^{0} \right)\left( \Psi -{{\lambda }_{n}}{{M}_{i}}^{n} \right)={{\lambda }_{n}}\left( \sum\limits_{i}^{{}}{{}}\left( {{P}_{i}}{{M}_{i}}^{n} \right)-\sum\limits_{i}^{{}}{{}}\left( {{P}_{i}}^{0}{{M}_{i}}^{n} \right) \right) \\ 
 & \sum\limits_{i}^{{}}{{}}\left( {{P}_{i}}{{M}_{i}}^{n} \right)=\left\langle {{M}^{n}} \right\rangle  \\ 
 & \sum\limits_{i}^{{}}{{}}\left( {{P}_{i}}^{0}{{M}_{i}}^{n} \right)={{\left\langle {{M}^{n}} \right\rangle }_{0}} \\ 
\end{align}

TeX (checked):

{\begin{aligned}&K\left(P,{{P}^{0}}\right)=\sum \limits _{i}^{}{}{{P}_{i}}\ln {{P}_{i}}-{{P}_{i}}\ln {{P}_{i}}^{0}+{{P}_{i}}^{0}\ln {{P}_{i}}^{0}-{{P}_{i}}^{0}\ln {{P}_{i}}^{0}\\&\sum \limits _{i}^{}{}{{P}_{i}}\ln {{P}_{i}}=I(P)\\&\sum \limits _{i}^{}{}{{P}_{i}}^{0}\ln {{P}_{i}}^{0}=I({{P}^{0}})\\&-{{P}_{i}}\ln {{P}_{i}}^{0}+{{P}_{i}}^{0}\ln {{P}_{i}}^{0}=-\sum \limits _{i}^{}{}\left({{P}_{i}}-{{P}_{i}}^{0}\right)\ln {{P}_{i}}^{0}\\&\ln {{P}_{i}}^{0}=\Psi -{{\lambda }_{n}}{{M}_{i}}^{n}\\&-\sum \limits _{i}^{}{}\left({{P}_{i}}-{{P}_{i}}^{0}\right)\left(\Psi -{{\lambda }_{n}}{{M}_{i}}^{n}\right)={{\lambda }_{n}}\left(\sum \limits _{i}^{}{}\left({{P}_{i}}{{M}_{i}}^{n}\right)-\sum \limits _{i}^{}{}\left({{P}_{i}}^{0}{{M}_{i}}^{n}\right)\right)\\&\sum \limits _{i}^{}{}\left({{P}_{i}}{{M}_{i}}^{n}\right)=\left\langle {{M}^{n}}\right\rangle \\&\sum \limits _{i}^{}{}\left({{P}_{i}}^{0}{{M}_{i}}^{n}\right)={{\left\langle {{M}^{n}}\right\rangle }_{0}}\\\end{aligned}}

LaTeXML (experimental; uses MathML) rendering

MathML (0 B / 8 B) :

SVG image empty. Force Re-Rendering

SVG (0 B / 8 B) :

MathML (experimental; no images) rendering

MathML (8.044 KB / 621 B) :

\begin{aligned} K (P, P^{0}) = \sum_{i}^{} P_{i} \ln P_{i} - P_{i} \ln {P_{i}}^{0} + {P_{i}}^{0} \ln {P_{i}}^{0} - {P_{i}}^{0} \ln {P_{i}}^{0} \\ \sum_{i}^{} P_{i} \ln P_{i} = I (P) \\ \sum_{i}^{} {P_{i}}^{0} \ln {P_{i}}^{0} = I (P^{0}) \\ - P_{i} \ln {P_{i}}^{0} + {P_{i}}^{0} \ln {P_{i}}^{0} = - \sum_{i}^{} (P_{i} - {P_{i}}^{0}) \ln {P_{i}}^{0} \\ \ln {P_{i}}^{0} = Ψ - λ_{n} {M_{i}}^{n} \\ - \sum_{i}^{} (P_{i} - {P_{i}}^{0}) (Ψ - λ_{n} {M_{i}}^{n}) = λ_{n} (\sum_{i}^{} (P_{i} {M_{i}}^{n}) - \sum_{i}^{} ({P_{i}}^{0} {M_{i}}^{n})) \\ \sum_{i}^{} (P_{i} {M_{i}}^{n}) = ⟨ M^{n} ⟩ \\ \sum_{i}^{} ({P_{i}}^{0} {M_{i}}^{n}) = {⟨ M^{n} ⟩}_{0} \end{aligned}

<math class="mwe-math-element" xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt"><mtr><mtd></mtd><mtd><mi>K</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>P</mi><mo>,</mo><msup><mi>P</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><munderover><mo form="prefix" texclass="OP">&#x2211;</mo><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow><mrow data-mjx-texclass="ORD"></mrow></munderover><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mi>ln</mi><mo>&#x2061;</mo><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mo>&#x2212;</mo><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mi>ln</mi><mo>&#x2061;</mo><msup><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msup><mo>+</mo><msup><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msup><mi>ln</mi><mo>&#x2061;</mo><msup><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msup><mo>&#x2212;</mo><msup><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msup><mi>ln</mi><mo>&#x2061;</mo><msup><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msup></mtd></mtr><mtr><mtd></mtd><mtd><munderover><mo form="prefix" texclass="OP">&#x2211;</mo><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow><mrow data-mjx-texclass="ORD"></mrow></munderover><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mi>ln</mi><mo>&#x2061;</mo><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mo>=</mo><mi>I</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><munderover><mo form="prefix" texclass="OP">&#x2211;</mo><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow><mrow data-mjx-texclass="ORD"></mrow></munderover><msup><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msup><mi>ln</mi><mo>&#x2061;</mo><msup><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msup><mo>=</mo><mi>I</mi><mo stretchy="false">(</mo><msup><mi>P</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msup><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mo>&#x2212;</mo><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mi>ln</mi><mo>&#x2061;</mo><msup><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msup><mo>+</mo><msup><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msup><mi>ln</mi><mo>&#x2061;</mo><msup><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msup><mo>=</mo><mo>&#x2212;</mo><munderover><mo form="prefix" texclass="OP">&#x2211;</mo><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow><mrow data-mjx-texclass="ORD"></mrow></munderover><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mo>&#x2212;</mo><msup><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mi>ln</mi><mo>&#x2061;</mo><msup><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msup></mtd></mtr><mtr><mtd></mtd><mtd><mi>ln</mi><mo>&#x2061;</mo><msup><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msup><mo>=</mo><mi mathvariant="normal">&#x03A8;</mi><mo>&#x2212;</mo><msub><mi>&#x03BB;</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub><msup><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msup></mtd></mtr><mtr><mtd></mtd><mtd><mo>&#x2212;</mo><munderover><mo form="prefix" texclass="OP">&#x2211;</mo><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow><mrow data-mjx-texclass="ORD"></mrow></munderover><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mo>&#x2212;</mo><msup><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi mathvariant="normal">&#x03A8;</mi><mo>&#x2212;</mo><msub><mi>&#x03BB;</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub><msup><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><msub><mi>&#x03BB;</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><munderover><mo form="prefix" texclass="OP">&#x2211;</mo><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow><mrow data-mjx-texclass="ORD"></mrow></munderover><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><msup><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>&#x2212;</mo><munderover><mo form="prefix" texclass="OP">&#x2211;</mo><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow><mrow data-mjx-texclass="ORD"></mrow></munderover><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msup><msup><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><munderover><mo form="prefix" texclass="OP">&#x2211;</mo><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow><mrow data-mjx-texclass="ORD"></mrow></munderover><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><msup><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">&#x27E8;</mo><msup><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msup><mo data-mjx-texclass="CLOSE">&#x27E9;</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><munderover><mo form="prefix" texclass="OP">&#x2211;</mo><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow><mrow data-mjx-texclass="ORD"></mrow></munderover><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msup><msup><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">&#x27E8;</mo><msup><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msup><mo data-mjx-texclass="CLOSE">&#x27E9;</mo></mrow><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>

Translations to Computer Algebra Systems

Translation to Maple

In Maple:

Translation to Mathematica

In Mathematica:

Identifiers

$K$

$P$

$P$

$i$

$P_{i}$

$P_{i}$

$P_{i}$

$P_{i}$

$P_{i}$

$P_{i}$

$P_{i}$

$P_{i}$

$i$

$P_{i}$

$P_{i}$

$I$

$P$

$i$

$P_{i}$

$P_{i}$

$I$

$P$

$P_{i}$

$P_{i}$

$P_{i}$

$P_{i}$

$i$

$P_{i}$

$P_{i}$

$P_{i}$

$P_{i}$

$Ψ$

$λ_{n}$

$M_{i}$

$n$

$i$

$P_{i}$

$P_{i}$

$Ψ$

$λ_{n}$

$M_{i}$

$n$

$λ_{n}$

$i$

$P_{i}$

$M_{i}$

$n$

$i$

$P_{i}$

$M_{i}$

$n$

$i$

$P_{i}$

$M_{i}$

$n$

$M$

$n$

$i$

$P_{i}$

$M_{i}$

$n$

$M$

$n$

MathML observations

0results

no statistics present please run the maintenance script ExtractFeatures.php

0 results

General

LaTeXML (experimental; uses MathML) rendering

MathML (experimental; no images) rendering

Translations to Computer Algebra Systems

Translation to Maple

Translation to Mathematica

Similar pages

Identifiers

MathML observations

Navigation menu

General

LaTeXML (experimental; uses MathML) rendering

MathML (experimental; no images) rendering

Translations to Computer Algebra Systems

Translation to Maple

Translation to Mathematica

Similar pages

Identifiers

MathML observations

Navigation menu

Search