Talk:DWBA - Revision history

KevinYager: /* Breaking into components */

2018-04-05T14:21:01Z

‎Breaking into components

KevinYager: /* Breaking into components */

2018-04-05T14:17:35Z

‎Breaking into components

KevinYager: /* Expansion */

2018-03-13T13:38:50Z

‎Expansion

KevinYager: /* Expansion */

2018-03-13T13:15:01Z

‎Expansion

KevinYager: /* Expansion */

2018-03-13T13:11:57Z

‎Expansion

KevinYager: /* Breaking into components */

2018-03-13T13:09:18Z

‎Breaking into components

KevinYager: /* Expansion */

2018-03-13T13:04:48Z

‎Expansion

KevinYager: /* Expansion */

2018-03-13T13:01:32Z

‎Expansion

KevinYager: /* TBD */

2018-03-13T12:56:31Z

‎TBD

KevinYager: /* Expansion */

2018-03-13T12:55:18Z

‎Expansion

@@ Line 288: / Line 288: @@
 I_d(q_{z}) & = |Tc|^2 ( I_{d,\mathrm{Tc}}(q_z) ) + |Rc|^2 ( I_{d,\mathrm{Rc}}(q_z) ) \\
      & = |Tc|^2 \left( \frac{w}{1-w}  I_{d,\mathrm{Rc}}(q_z) \right) + |Rc|^2 ( I_{d,\mathrm{Rc}}(q_z) ) \\
+I_{d,\mathrm{Rc}}(q_z) & = \frac{ I_d(q_{z}) }{|Tc|^2 \frac{w}{1-w} + |Rc|^2}
 \end{align}
 </math>

@@ Line 240: / Line 240: @@
 ==Breaking into components==
-The experimental data <math>I_d(q_z)</math> can be broken into contributions from the transmitted channel <math>I_{d,Tc}(qz)</math> and reflected channel <math>I_{d,Rc}(qz)</math>:
+The experimental data <math>I_d(q_z)</math> can be broken into contributions from the transmitted channel <math>I_{d,\mathrm{Tc}}(qz)</math> and reflected channel <math>I_{d,\mathrm{Rc}}(qz)</math>:
 <math>
@@ Line 246: / Line 246: @@
 I_d(q_{z})
    & = [ | T_i T_f|^2 + |R_i R_f|^2 ] I_{R}(+Q_{z1}) + [ |T_i R_f|^2 + |R_i T_f|^2 ] I_{R}(+Q_{z2}) \\
-   & = [ | T_i T_f|^2 + |R_i R_f|^2 ] I_{R}(q_z-\Delta q_{z,Tc}) + [ |T_i R_f|^2 + |R_i T_f|^2 ] I_{R}(q_z-\Delta q_{z,Rc}) \\
+   & = [ | T_i T_f|^2 + |R_i R_f|^2 ] I_{R}(q_z-\Delta q_{z,\mathrm{Tc}}) + [ |T_i R_f|^2 + |R_i T_f|^2 ] I_{R}(q_z-\Delta q_{z,\mathrm{Rc}}) \\
-   & = [ | T_i T_f|^2 + |R_i R_f|^2 ] I_{d,Tc}(q_z) + [ |T_i R_f|^2 + |R_i T_f|^2 ] I_{d,Rc}(q_z) \\
+   & = [ | T_i T_f|^2 + |R_i R_f|^2 ] I_{d,\mathrm{Tc}}(q_z) + [ |T_i R_f|^2 + |R_i T_f|^2 ] I_{d,\mathrm{Rc}}(q_z) \\
-   & = |Tc|^2 I_{d,Tc}(q_z) + |Rc|^2 I_{d,Rc}(q_z) \\
+   & = |Tc|^2 I_{d,\mathrm{Tc}}(q_z) + |Rc|^2 I_{d,\mathrm{Rc}}(q_z) \\
 \end{align}
 </math>
@@ Line 256: / Line 256: @@
 <math>
 \begin{align}
+w (q_z)
-   & = \frac{ I_{d,Tc}(q_z) }{ I_{d,Tc}(q_z) + I_{d,Rc}(q_z) }
+   & = \frac{ I_{d,\mathrm{Tc}}(q_z) }{ I_{d,\mathrm{Tc}}(q_z) + I_{d,\mathrm{Rc}}(q_z) }
 \end{align}
 </math>
@@ Line 265: / Line 265: @@
 <math>
 \begin{align}
-I_d(q_{z}) & = |Tc|^2 ( I_{d,Tc}(q_z) ) + |Rc|^2 ( I_{d,Rc}(q_z) ) \\
+I_d(q_{z}) & = |Tc|^2 ( I_{d,\mathrm{Tc}}(q_z) ) + |Rc|^2 ( I_{d,\mathrm{Rc}}(q_z) ) \\
-I_d(q_{z}) & = |Tc|^2 ( I_{d,Tc}(q_z) ) + |Rc|^2 \left ( \frac{ I_{d,Tc}(q_z) - w I_{d,Tc}(q_z) }{w}  \right ) \\
+I_d(q_{z}) & = |Tc|^2 ( I_{d,\mathrm{Tc}}(q_z) ) + |Rc|^2 \left ( \frac{ I_{d,\mathrm{Tc}}(q_z) - w I_{d,\mathrm{Tc}}(q_z) }{w}  \right ) \\
-I_d(q_{z}) & = I_{d,Tc}(q_z) \times \left ( |Tc|^2  +  |Rc|^2 \frac{ 1}{w}  - |Rc|^2 \frac{w }{w}  \right ) \\
+I_d(q_{z}) & = I_{d,\mathrm{Tc}}(q_z) \times \left ( |Tc|^2  +  |Rc|^2 \frac{ 1}{w}  - |Rc|^2 \frac{w }{w}  \right ) \\
-I_{d,Tc}(q_z)  & = \frac{ I_d(q_{z}) }{  |Tc|^2  +  \frac{ |Rc|^2 }{w}  - |Rc|^2  } \\
+I_{d,\mathrm{Tc}}(q_z)  & = \frac{ I_d(q_{z}) }{  |Tc|^2  +  \frac{ |Rc|^2 }{w}  - |Rc|^2  } \\
 \end{align}
 </math>
@@ Line 277: / Line 277: @@
 <math>
 \begin{align}
-I_{d,Rc}(q_z) & = \frac{ I_d(q_{z}) - |Tc|^2 I_{d,Tc}(q_z) }{|Rc|^2}
+I_{d,\mathrm{Rc}}(q_z) & = \frac{ I_d(q_{z}) - |Tc|^2 I_{d,\mathrm{Tc}}(q_z) }{|Rc|^2} \\
 \end{align}
 </math>

@@ Line 177: / Line 177: @@
 ===Simplification===
 We can rearrange to:
 ==Breaking into components==

@@ Line 158: / Line 158: @@
      & \begin{align}
-         = \,\, &  |T_i T_f|^2 | F_{+1} |^2 && + |T_i|^2 T_f R_f F_{+1}F_{-2}^* \\
+         = \,\, &  |T_i T_f|^2 | F_{+1} |^2 && + |T_i|^2 T_f R_f^* F_{+1}F_{-2}^* \\
-           & && + T_i R_i |T_f|^2 F_{+1}F_{+2}^* + T_i R_i T_f R_f F_{+1} F_{-1}^*  \\
+           & && + T_i R_i^* |T_f|^2 F_{+1}F_{+2}^* + T_i R_i^* T_f R_f^* F_{+1} F_{-1}^*  \\
-           & + |T_i R_f|^2 | F_{-2} |^2 && + |T_i|^2T_fR_f F_{+1}^* F_{-2} \\
+           & + |T_i R_f|^2 | F_{-2} |^2 && + |T_i|^2T_f^*R_f F_{+1}^* F_{-2} \\
-           & && + T_i R_i T_f R_f F_{+2}^*F_{-2} + T_i R_i |R_f|^2 F_{-1}^* F_{-2}  \\
+           & && + T_i R_i^* T_f^* R_f F_{+2}^*F_{-2} + T_i R_i^* |R_f|^2 F_{-1}^* F_{-2}  \\
-           & + |R_i T_f|^2 | F_{+2} |^2 && + T_i R_i |T_f|^2 F_{+1}^* F_{+2} \\
+           & + |R_i T_f|^2 | F_{+2} |^2 && + T_i^* R_i |T_f|^2 F_{+1}^* F_{+2} \\
-           & && + T_i R_i T_f R_f F_{+2}^*F_{-2} + |R_i|^2 T_f R_f F_{-1}^* F_{+2}  \\
+           & && + T_i^* R_i T_f R_f^* F_{+2}^*F_{-2} + |R_i|^2 T_f R_f^* F_{-1}^* F_{+2}  \\
-           & + |R_i R_f|^2 | F_{-1} |^2 && + T_i R_i T_f R_f F_{+1}^* F_{-1} \\
+           & + |R_i R_f|^2 | F_{-1} |^2 && + T_i^* R_i T_f^* R_f F_{+1}^* F_{-1} \\
-           & && + T_i R_i |R_f|^2 F_{-1}F_{-2}^* + |R_i|^2 T_f R_f F_{-1} F_{+2}^*  \\
+           & && + T_i^* R_i |R_f|^2 F_{-1}F_{-2}^* + |R_i|^2 T_f^* R_f F_{-1} F_{+2}^*  \\
          \end{align} \\

@@ Line 158: / Line 158: @@
      & \begin{align}
-         = \,\, &  |T_i T_f|^2 | F_{+1} |^2 && + T_i^2 T_f R_f F_{+1}F_{-2}^* \\
+         = \,\, &  |T_i T_f|^2 | F_{+1} |^2 && + |T_i|^2 T_f R_f F_{+1}F_{-2}^* \\
-           & && + T_i R_i T_f^2 F_{+1}F_{+2}^* + T_i R_i T_f R_f F_{+1} F_{-1}^*  \\
+           & && + T_i R_i |T_f|^2 F_{+1}F_{+2}^* + T_i R_i T_f R_f F_{+1} F_{-1}^*  \\
-           & + |T_i R_f|^2 | F_{-2} |^2 && + T_i^2T_fR_f F_{+1}^* F_{-2} \\
+           & + |T_i R_f|^2 | F_{-2} |^2 && + |T_i|^2T_fR_f F_{+1}^* F_{-2} \\
-           & && + T_i R_i T_f R_f F_{+2}^*F_{-2} + T_i R_i R_f^2 F_{-1}^* F_{-2}  \\
+           & && + T_i R_i T_f R_f F_{+2}^*F_{-2} + T_i R_i |R_f|^2 F_{-1}^* F_{-2}  \\
-           & + |R_i T_f|^2 | F_{+2} |^2 && + T_i R_i T_f^2 F_{+1}^* F_{+2} \\
+           & + |R_i T_f|^2 | F_{+2} |^2 && + T_i R_i |T_f|^2 F_{+1}^* F_{+2} \\
-           & && + T_i R_i T_f R_f F_{+2}^*F_{-2} + R_i^2 T_f R_f F_{-1}^* F_{+2}  \\
+           & && + T_i R_i T_f R_f F_{+2}^*F_{-2} + |R_i|^2 T_f R_f F_{-1}^* F_{+2}  \\
            & + |R_i R_f|^2 | F_{-1} |^2 && + T_i R_i T_f R_f F_{+1}^* F_{-1} \\
-           & && + T_i R_i R_f^2 F_{-1}F_{-2}^* + R_i^2 T_f R_f F_{-1} F_{+2}^*  \\
+           & && + T_i R_i |R_f|^2 F_{-1}F_{-2}^* + |R_i|^2 T_f R_f F_{-1} F_{+2}^*  \\
          \end{align} \\
@@ Line 174: / Line 174: @@
 \end{align}
 </math>
 ==Breaking into components==

@@ Line 158: / Line 158: @@
      & \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 T_f|^2 | F_{+1} |^2 && + T_i^2 T_f R_f F_{+1}F_{-2} \\
-           & && + T_i R_i T_f^2 F(+Q_{z1})F(+Q_{z2}) + T_i R_i T_f R_f F(+Q_{z1}) F(-Q_{z1})  \\
+           & && + T_i R_i T_f^2 F_{+1}F_{+2} + T_i R_i T_f R_f F_{+1} F_{-1}  \\
-           & + T_i^2 R_f^2 | F(-Q_{z2}) |^2 && + T_i^2T_fR_f F(+Q_{z1}) F(-Q_{z2}) \\
+           & + |T_i R_f|^2 | F_{-2} |^2 && + T_i^2T_fR_f F_{+1} F_{-2} \\
-           & && + 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 R_f F_{+2}F_{-2} + T_i R_i R_f^2 F_{-1} F_{-2}  \\
-           & + R_i^2 T_f^2 | F(+Q_{z2}) |^2 && + T_i R_i T_f^2 F(+Q_{z1}) F(+Q_{z2}) \\
+           & + |R_i T_f|^2 | F_{+2} |^2 && + T_i R_i T_f^2 F_{+1} F_{+2} \\
-           & && + 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_{+2}F_{-2} + R_i^2 T_f R_f F_{-1} F_{+2}  \\
-           & + R_i^2 R_f^2 | F(-Q_{z1}) |^2 && + T_i R_i T_f R_f F(+Q_{z1}) F(-Q_{z1}) \\
+           & + |R_i R_f|^2 | F_{-1} |^2 && + T_i R_i T_f R_f F_{+1} F_{-1} \\
-           & && + 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 R_i R_f^2 F_{-1}F_{-2} + R_i^2 T_f R_f F_{-1} F_{+2}  \\
          \end{align} \\

@@ Line 143: / Line 143: @@
 </math>
-===TBD===
 We take advantage of a more compact form using the notation <math>T_i = T(\alpha_i)</math> and <math>F_{+1} = F(+Q_{z1})</math>:
 ==Breaking into components==

← Older revision		Revision as of 12:55, 13 March 2018
Line 142:		Line 142:

	</math>		</math>
		+
		+	===TBD===
		+	We take advantage of a more compact form using the notation <math>T_i = T(\alpha_i)</math> and <math>F_{+1} = F(+Q_{z1})</math>:

	==Breaking into components==		==Breaking into components==