Variable

N

$0<N<\frac{k^2{\theta }^3}{16\beta }$

$p^{*}$

$\frac{\theta }{\beta }-{\left(\frac{2N}{k^2{\beta }^2}\right)}^{1/3}$

$a^{*}$

${\left(\frac{N^2}{2k\beta }\right)}^{2/3}$

$D^{*}$

N

${\pi }_{m+r}^{\ast }$

$\frac{N\theta }{\beta }-3{\left(\frac{N^2}{2k\beta }\right)}^{2/3}$