Next: Appendix C: Adjoint-state tomography Up: Li et al.: DSR Previous: Appendix A: Causal discretization

Appendix B: Frechét derivative of DSR tomography

To derive the Frechét derivative, we start from equations 13 and 14. Applying $\partial / \partial w_s$ to both sides of equation 13 results in

$\displaystyle D_z \frac{\partial t^{dsr}}{\partial w_s}$	$\textstyle =$	$\displaystyle - \frac{1}{2 \sqrt{w_s - D_s t^{dsr} \cdot D_s t^{dsr}}}$
	$\textstyle +$	$\displaystyle \left(\frac{D_s t^{dsr} \cdot D_s}{\sqrt{w_s - D_s t^{dsr} \cdot ... ..._r t^{dsr} \cdot D_r t^{dsr}}}\right) \frac{\partial t^{dsr}}{\partial w_s}\;.$	(35)

Analogously

$\displaystyle D_z \frac{\partial t^{dsr}}{\partial w_r}$	$\textstyle =$	$\displaystyle - \frac{1}{2 \sqrt{w_r - D_r t^{dsr} \cdot D_r t^{dsr}}}$
	$\textstyle +$	$\displaystyle \left(\frac{D_s t^{dsr} \cdot D_s}{\sqrt{w_s - D_s t^{dsr} \cdot ... ..._r t^{dsr} \cdot D_r t^{dsr}}}\right) \frac{\partial t^{dsr}}{\partial w_r}\;.$	(36)

Inserting equations B-1 and B-2 into 14 and regrouping the terms, we prove equation 15

$\begin{displaymath} J^{dsr} = B^{-1} (C_s + C_r)\;, \end{displaymath}$

(37)

where

$\begin{displaymath} B = D_z - \left( \frac{D_s t^{dsr} \cdot D_s}{\sqrt{w_s - D... ...ot D_r}{\sqrt{w_r - D_r t^{dsr} \cdot D_r t^{dsr}}} \right)\;, \end{displaymath}$

(38)

and

$\begin{displaymath} C_s = - \frac{1}{2 \sqrt{w_s - D_s t^{dsr} \cdot D_s t^{dsr}}}\, \frac{\partial w_s}{\partial w}\;; \end{displaymath}$

(39)

$\begin{displaymath} C_r = - \frac{1}{2 \sqrt{w_r - D_r t^{dsr} \cdot D_r t^{dsr}}}\, \frac{\partial w_r}{\partial w}\;. \end{displaymath}$

(40)

At the singularity of DSR eikonal equation, the operators , and take simpler forms and can be derived directly from equation 3.

Next: Appendix C: Adjoint-state tomography Up: Li et al.: DSR Previous: Appendix A: Causal discretization

2013-10-16