- testwiki

General

Display information for equation id:math.2270.77 on revision:2270

* Page found: Vorurteilsfreie Schätzung des statistischen Operators zu einem festen Zeitpunkt (eq math.2270.77)

Occurrences on the following pages:

Vorurteilsfreie Schätzung des statistischen Operators zu einem festen Zeitpunkt Eq: math.2273.77

Hash: 83d37b6c2c2582a29a268c7971c1b3f5

TeX (original user input):

\begin{align}
  & S=-k\operatorname{Tr}\left( \frac{1}{Z}{{e}^{-\sum\limits_{\nu }{{{\lambda }_{\nu }}{{G}_{\nu }}}}}\ln \left( \frac{1}{Z}{{e}^{-\sum\limits_{\nu }{{{\lambda }_{\nu }}{{G}_{\nu }}}}} \right) \right) \\
 & =-k\operatorname{Tr}\left( \frac{1}{Z}{{e}^{-\sum\limits_{\nu }{{{\lambda }_{\nu }}{{G}_{\nu }}}}}\left( -\ln Z-\sum\limits_{\nu }{{{\lambda }_{\nu }}{{G}_{\nu }}} \right) \right) \\
 & =\underbrace{k\sum\limits_{\nu }{{{\lambda }_{\nu }}\left\langle {{G}_{\nu }} \right\rangle }}_{f\left( {{\lambda }_{\nu }},\left\langle {{G}_{\nu }} \right\rangle  \right)}+\underbrace{k\ln Z}_{g\left( {{\lambda }_{\nu }},{{G}_{\nu }}\left( {{h}_{\alpha }} \right) \right)} \\
 & S=S\left( \left\langle {{G}_{\nu }} \right\rangle ,{{h}_{\alpha }} \right) 
\end{align}

TeX (checked):

{\begin{aligned}&S=-k\operatorname {Tr} \left({\frac {1}{Z}}{{e}^{-\sum \limits _{\nu }{{{\lambda }_{\nu }}{{G}_{\nu }}}}}\ln \left({\frac {1}{Z}}{{e}^{-\sum \limits _{\nu }{{{\lambda }_{\nu }}{{G}_{\nu }}}}}\right)\right)\\&=-k\operatorname {Tr} \left({\frac {1}{Z}}{{e}^{-\sum \limits _{\nu }{{{\lambda }_{\nu }}{{G}_{\nu }}}}}\left(-\ln Z-\sum \limits _{\nu }{{{\lambda }_{\nu }}{{G}_{\nu }}}\right)\right)\\&=\underbrace {k\sum \limits _{\nu }{{{\lambda }_{\nu }}\left\langle {{G}_{\nu }}\right\rangle }} _{f\left({{\lambda }_{\nu }},\left\langle {{G}_{\nu }}\right\rangle \right)}+\underbrace {k\ln Z} _{g\left({{\lambda }_{\nu }},{{G}_{\nu }}\left({{h}_{\alpha }}\right)\right)}\\&S=S\left(\left\langle {{G}_{\nu }}\right\rangle ,{{h}_{\alpha }}\right)\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 (5.44 KB / 594 B) :

\begin{aligned} S = - k Tr (\frac{1}{Z} e^{- \sum_{ν} λ_{ν} G_{ν}} \ln (\frac{1}{Z} e^{- \sum_{ν} λ_{ν} G_{ν}})) \\ = - k Tr (\frac{1}{Z} e^{- \sum_{ν} λ_{ν} G_{ν}} (- \ln Z - \sum_{ν} λ_{ν} G_{ν})) \\ = \underset{f (λ_{ν}, ⟨ G_{ν} ⟩)}{\underset{⏟}{k \sum_{ν} λ_{ν} ⟨ G_{ν} ⟩}} + \underset{g (λ_{ν}, G_{ν} (h_{α}))}{\underset{⏟}{k \ln Z}} \\ S = S (⟨ G_{ν} ⟩, h_{α}) \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>S</mi><mo>=</mo><mo>&#x2212;</mo><mi>k</mi><mi data-mjx-texclass="OP" mathvariant="normal">Tr</mi><mo>&#x2061;</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>Z</mi></mrow></mfrac></mrow><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo>&#x2212;</mo><munder><mo form="prefix" texclass="OP">&#x2211;</mo><mrow data-mjx-texclass="ORD"><mi>&#x03BD;</mi></mrow></munder><mrow data-mjx-texclass="ORD"><msub><mi>&#x03BB;</mi><mrow data-mjx-texclass="ORD"><mi>&#x03BD;</mi></mrow></msub><msub><mi>G</mi><mrow data-mjx-texclass="ORD"><mi>&#x03BD;</mi></mrow></msub></mrow></mrow></mrow></msup><mi>ln</mi><mo>&#x2061;</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>Z</mi></mrow></mfrac></mrow><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo>&#x2212;</mo><munder><mo form="prefix" texclass="OP">&#x2211;</mo><mrow data-mjx-texclass="ORD"><mi>&#x03BD;</mi></mrow></munder><mrow data-mjx-texclass="ORD"><msub><mi>&#x03BB;</mi><mrow data-mjx-texclass="ORD"><mi>&#x03BD;</mi></mrow></msub><msub><mi>G</mi><mrow data-mjx-texclass="ORD"><mi>&#x03BD;</mi></mrow></msub></mrow></mrow></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mo>&#x2212;</mo><mi>k</mi><mi data-mjx-texclass="OP" mathvariant="normal">Tr</mi><mo>&#x2061;</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>Z</mi></mrow></mfrac></mrow><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo>&#x2212;</mo><munder><mo form="prefix" texclass="OP">&#x2211;</mo><mrow data-mjx-texclass="ORD"><mi>&#x03BD;</mi></mrow></munder><mrow data-mjx-texclass="ORD"><msub><mi>&#x03BB;</mi><mrow data-mjx-texclass="ORD"><mi>&#x03BD;</mi></mrow></msub><msub><mi>G</mi><mrow data-mjx-texclass="ORD"><mi>&#x03BD;</mi></mrow></msub></mrow></mrow></mrow></msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mo>&#x2212;</mo><mi>ln</mi><mo>&#x2061;</mo><mi>Z</mi><mo>&#x2212;</mo><munder><mo form="prefix" texclass="OP">&#x2211;</mo><mrow data-mjx-texclass="ORD"><mi>&#x03BD;</mi></mrow></munder><mrow data-mjx-texclass="ORD"><msub><mi>&#x03BB;</mi><mrow data-mjx-texclass="ORD"><mi>&#x03BD;</mi></mrow></msub><msub><mi>G</mi><mrow data-mjx-texclass="ORD"><mi>&#x03BD;</mi></mrow></msub></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><munder><mrow data-mjx-texclass="OP"><munder><mrow data-mjx-texclass="ORD"><mi>k</mi><munder><mo form="prefix" texclass="OP">&#x2211;</mo><mrow data-mjx-texclass="ORD"><mi>&#x03BD;</mi></mrow></munder><mrow data-mjx-texclass="ORD"><msub><mi>&#x03BB;</mi><mrow data-mjx-texclass="ORD"><mi>&#x03BD;</mi></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">&#x27E8;</mo><msub><mi>G</mi><mrow data-mjx-texclass="ORD"><mi>&#x03BD;</mi></mrow></msub><mo data-mjx-texclass="CLOSE">&#x27E9;</mo></mrow></mrow></mrow><mo>&#x23DF;</mo></munder></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>f</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msub><mi>&#x03BB;</mi><mrow data-mjx-texclass="ORD"><mi>&#x03BD;</mi></mrow></msub><mo>,</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">&#x27E8;</mo><msub><mi>G</mi><mrow data-mjx-texclass="ORD"><mi>&#x03BD;</mi></mrow></msub><mo data-mjx-texclass="CLOSE">&#x27E9;</mo></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow></mrow></mrow></munder><mo>+</mo><munder><mrow data-mjx-texclass="OP"><munder><mrow data-mjx-texclass="ORD"><mi>k</mi><mi>ln</mi><mo>&#x2061;</mo><mi>Z</mi></mrow><mo>&#x23DF;</mo></munder></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>g</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msub><mi>&#x03BB;</mi><mrow data-mjx-texclass="ORD"><mi>&#x03BD;</mi></mrow></msub><mo>,</mo><msub><mi>G</mi><mrow data-mjx-texclass="ORD"><mi>&#x03BD;</mi></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msub><mi>h</mi><mrow data-mjx-texclass="ORD"><mi>&#x03B1;</mi></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow></mrow></mrow></munder></mtd></mtr><mtr><mtd></mtd><mtd><mi>S</mi><mo>=</mo><mi>S</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">&#x27E8;</mo><msub><mi>G</mi><mrow data-mjx-texclass="ORD"><mi>&#x03BD;</mi></mrow></msub><mo data-mjx-texclass="CLOSE">&#x27E9;</mo></mrow><mo>,</mo><msub><mi>h</mi><mrow data-mjx-texclass="ORD"><mi>&#x03B1;</mi></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr></mtable></mrow></mstyle></mrow></math>

Translations to Computer Algebra Systems

Translation to Maple

In Maple:

Translation to Mathematica

In Mathematica:

Identifiers

$S$

$k$

$Z$

$e$

$ν$

$λ_{ν}$

$G_{ν}$

$Z$

$e$

$ν$

$λ_{ν}$

$G_{ν}$

$k$

$Z$

$e$

$ν$

$λ_{ν}$

$G_{ν}$

$Z$

$ν$

$λ_{ν}$

$G_{ν}$

$k$

$ν$

$λ_{ν}$

$G_{ν}$

$f$

$λ_{ν}$

$G_{ν}$

$k$

$Z$

$g$

$λ_{ν}$

$G_{ν}$

$h_{α}$

$S$

$S$

$G_{ν}$

$h_{α}$

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