In the theory of junction conditions, the Riemann tensor in terms of the Heaviside theta function is expressed as follows
$$R^{\alpha}_{\beta\lambda\mu} = \Theta(l)R^{\alpha^{+}}_{\beta\lambda\mu} + \Theta(-l)R^{\alpha^{-}}_{\beta\lambda\mu} + \delta(l)A^{\alpha}_{\beta\lambda\mu}.$$
Now, the tr...
Consider a harmonic oscillator defined by the coupled differential equations
\begin{align}
\begin{split}
\dot{X} &= \omega Y \\
\dot{Y} &= - \omega X \, .
\end{split} \tag{1}
\end{align}
Defining new variables $a = X + i Y$ and $a^* = X - i Y$, produces a new uncoupled system of equations
\begin{...
