Difference between revisions of "Talk:DWBA"

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(Created page with "<math> \begin{align} I_d(q_{z}) & = | T_i T_f F(+Q_{z1}) + T_i R_f F(-Q_{z2}) + R_i T_f F(+Q_{z2}) + R_i R_f F(-Q_{z1}) | ^{2} \\ & \begin{align} ...")
 
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Using the notation <math>T_i = T(\alpha_i)</math> for compactness, the DWBA equation can be written as:
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<math>
 
<math>
 
\begin{align}
 
\begin{align}
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         \end{align} \\
 
         \end{align} \\
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\end{align}
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</math>
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The DWBA equation is:
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<math>
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I_d(q_{z}) = T_i^2 T_f^2 | F(+Q_{z1}) |^2 + T_i^2 R_f^2 | F(-Q_{z2}) |^2 + R_i^2 T_f^2 | F(+Q_{z2}) |^2 + R_i^2 R_f^2 | F(-Q_{z1}) |^2
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</math>
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Taking advantage of the fact that <math>|F(Q)|^2 = I(Q)</math> and <math>I(+Q)=I(-Q)</math>, we can rearrange to:
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<math>
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\begin{align}
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I_d(q_{z}) = \,\, & T_i^2 T_f^2 | F(+Q_{z1}) |^2 + T_i^2 R_f^2 | F(-Q_{z2}) |^2 + R_i^2 T_f^2 | F(+Q_{z2}) |^2 + R_i^2 R_f^2 | F(-Q_{z1}) |^2 \\
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        & + T_i^2 T_f R_f F(+Q_{z1})F(-Q_{z2}) \\
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        & + T_i R_i T_f R_f F(+Q_{z1})F(+Q_{z2}) + T_i R_i T_f R_f F(+Q_{z1})F(-Q_{z1})  \\
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        & + T_i^2T_fR_f F(+Q_{z1}) F(-Q_{z2}) \\
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        & + T_i R_i T_f R_f F(+Q_{z2})F(-Q_{z2}) + T_i R_i R_f^2 F(-Q_{z1}) F(-Q_{z2})  \\
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        & + T_i R_i T_f^2 F(+Q_{z1}) F(+Q_{z2}) \\
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        & + T_i R_i T_f R_f F(+Q_{z2})F(-Q_{z2}) + R_i^2 T_f R_f F(-Q_{z1}) F(+Q_{z2})  \\
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        & + T_i R_i T_f R_f F(+Q_{z1}) F(-Q_{z1}) \\
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        & + T_i R_i R_f^2 F(-Q_{z1})F(-Q_{z2}) + R_i^2 T_f R_f F(-Q_{z1}) F(+Q_{z2})  \\     
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= \,\, & T_i^2 T_f^2 | F(+Q_{z1}) |^2 + T_i^2 R_f^2 | F(-Q_{z2}) |^2 + R_i^2 T_f^2 | F(+Q_{z2}) |^2 + R_i^2 R_f^2 | F(-Q_{z1}) |^2 \\
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        & + 2 \times T_i^2 T_f R_f F(+Q_{z1})F(-Q_{z2}) \\
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        & + T_i R_i T_f R_f [ 2 F(+Q_{z1})F(-Q_{z1}) + F(+Q_{z1})F(+Q_{z2}) + 2 F(+Q_{z2})F(-Q_{z2}) ]  \\
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        & + 2 \times T_i R_i R_f^2 F(-Q_{z1}) F(-Q_{z2})  \\
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        & + T_i R_i T_f^2 F(+Q_{z1}) F(+Q_{z2}) \\
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        & + 2 \times R_i^2 T_f R_f F(-Q_{z1}) F(+Q_{z2}) \\     
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 +
 +
  
 
\end{align}
 
\end{align}
 
</math>
 
</math>

Revision as of 18:02, 6 March 2018

Using the notation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_i = T(\alpha_i)} for compactness, the DWBA equation can be written as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} I_d(q_{z}) & = | T_i T_f F(+Q_{z1}) + T_i R_f F(-Q_{z2}) + R_i T_f F(+Q_{z2}) + R_i R_f F(-Q_{z1}) | ^{2} \\ & \begin{align} = \,\, & T_i^2 T_f^2 | F(+Q_{z1}) |^2 && + T_i^2 T_f R_f F(+Q_{z1})F(-Q_{z2}) \\ & && + T_i R_i T_f R_f F(+Q_{z1})F(+Q_{z2}) + T_i R_i T_f R_f F(+Q_{z1}) F(-Q_{z1}) \\ & + T_i^2 R_f^2 | F(-Q_{z2}) |^2 && + T_i^2T_fR_f F(+Q_{z1}) F(-Q_{z2}) \\ & && + T_i R_i T_f R_f F(+Q_{z2})F(-Q_{z2}) + T_i R_i R_f^2 F(-Q_{z1}) F(-Q_{z2}) \\ & + R_i^2 T_f^2 | F(+Q_{z2}) |^2 && + T_i R_i T_f^2 F(+Q_{z1}) F(+Q_{z2}) \\ & && + T_i R_i T_f R_f F(+Q_{z2})F(-Q_{z2}) + R_i^2 T_f R_f F(-Q_{z1}) F(+Q_{z2}) \\ & + R_i^2 R_f^2 | F(-Q_{z1}) |^2 && + T_i R_i T_f R_f F(+Q_{z1}) F(-Q_{z1}) \\ & && + T_i R_i R_f^2 F(-Q_{z1})F(-Q_{z2}) + R_i^2 T_f R_f F(-Q_{z1}) F(+Q_{z2}) \\ \end{align} \\ \end{align} } The DWBA equation is: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_d(q_{z}) = T_i^2 T_f^2 | F(+Q_{z1}) |^2 + T_i^2 R_f^2 | F(-Q_{z2}) |^2 + R_i^2 T_f^2 | F(+Q_{z2}) |^2 + R_i^2 R_f^2 | F(-Q_{z1}) |^2 }

Taking advantage of the fact that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |F(Q)|^2 = I(Q)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I(+Q)=I(-Q)} , we can rearrange to:


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} I_d(q_{z}) = \,\, & T_i^2 T_f^2 | F(+Q_{z1}) |^2 + T_i^2 R_f^2 | F(-Q_{z2}) |^2 + R_i^2 T_f^2 | F(+Q_{z2}) |^2 + R_i^2 R_f^2 | F(-Q_{z1}) |^2 \\ & + T_i^2 T_f R_f F(+Q_{z1})F(-Q_{z2}) \\ & + T_i R_i T_f R_f F(+Q_{z1})F(+Q_{z2}) + T_i R_i T_f R_f F(+Q_{z1})F(-Q_{z1}) \\ & + T_i^2T_fR_f F(+Q_{z1}) F(-Q_{z2}) \\ & + T_i R_i T_f R_f F(+Q_{z2})F(-Q_{z2}) + T_i R_i R_f^2 F(-Q_{z1}) F(-Q_{z2}) \\ & + T_i R_i T_f^2 F(+Q_{z1}) F(+Q_{z2}) \\ & + T_i R_i T_f R_f F(+Q_{z2})F(-Q_{z2}) + R_i^2 T_f R_f F(-Q_{z1}) F(+Q_{z2}) \\ & + T_i R_i T_f R_f F(+Q_{z1}) F(-Q_{z1}) \\ & + T_i R_i R_f^2 F(-Q_{z1})F(-Q_{z2}) + R_i^2 T_f R_f F(-Q_{z1}) F(+Q_{z2}) \\ = \,\, & T_i^2 T_f^2 | F(+Q_{z1}) |^2 + T_i^2 R_f^2 | F(-Q_{z2}) |^2 + R_i^2 T_f^2 | F(+Q_{z2}) |^2 + R_i^2 R_f^2 | F(-Q_{z1}) |^2 \\ & + 2 \times T_i^2 T_f R_f F(+Q_{z1})F(-Q_{z2}) \\ & + T_i R_i T_f R_f [ 2 F(+Q_{z1})F(-Q_{z1}) + F(+Q_{z1})F(+Q_{z2}) + 2 F(+Q_{z2})F(-Q_{z2}) ] \\ & + 2 \times T_i R_i R_f^2 F(-Q_{z1}) F(-Q_{z2}) \\ & + T_i R_i T_f^2 F(+Q_{z1}) F(+Q_{z2}) \\ & + 2 \times R_i^2 T_f R_f F(-Q_{z1}) F(+Q_{z2}) \\ \end{align} }

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