User:Miraceti/sandbox

This is the user sandbox of Miraceti. A user sandbox is a subpage of the user's user page. It serves as a testing spot and page development space for the user and is not an encyclopedia article. Create or edit your own sandbox here.

Other sandboxes: Main sandbox | Template sandbox

Finished writing a draft article? Are you ready to request review of it by an experienced editor for possible inclusion in Wikipedia? Submit your draft for review!

$N=R\cdot f_{\mathrm {p} }\cdot n_{\mathrm {e} }\cdot f_{\mathrm {l} }\cdot f_{\mathrm {i} }\cdot f_{\mathrm {c} }\cdot L.$

$N=N_{\mathrm {s} }\cdot f_{\mathrm {p} }\cdot n_{\mathrm {e} }\cdot f_{\mathrm {l} }\cdot f_{\mathrm {i} }\cdot f_{\mathrm {c} }\cdot f_{\mathrm {L} },$

$N=\prod _{k=0}^{7}D_{k},$

$Y=\sum _{k=0}^{7}Y_{k}.$

${\begin{aligned}\langle D_{k}\rangle &={\frac {a_{k}+b_{k}}{2}},\\\sigma _{D_{k}}&={\frac {b_{k}-a_{k}}{2{\sqrt {3}}}}.\\\end{aligned}}$

$f_{Y_{k}}(y)={\frac {\mathrm {e} ^{y}}{b_{k}-a_{k}}}\qquad i=k,\ldots ,7\qquad \ln {a_{k}}\leq y\leq \ln {b_{k}}.$

${\begin{aligned}\langle Y_{k}\rangle &={\frac {b_{k}[\ln {b_{k}}-1]-a_{k}[\ln {a_{k}}-1]}{b_{k}-a_{k}}},\\\sigma _{Y_{k}}&={\sqrt {1-{\frac {a_{k}b_{k}[\ln {b_{k}}-\ln {a_{k}}]^{2}}{\left(b_{k}-a_{k}\right)^{2}}}}}.\\\end{aligned}}$

$f_{N}(y)={\frac {1}{\left|y\right|}}f_{Y}\left(\ln {y}\right).$

${\mathit {\Phi }}_{Y_{k}}\left(\zeta \right)={\frac {b_{k}^{1+i\zeta }-a_{k}^{1+i\zeta }}{\left(b_{k}-a_{k}\right)\left(1+i\zeta \right)}}.$

${\mathit {\Phi }}_{Y}\left(\zeta \right)=\prod _{k=1}^{7}{\frac {b_{k}^{1+i\zeta }-a_{k}^{1+i\zeta }}{\left(b_{k}-a_{k}\right)\left(1+i\zeta \right)}}.$

${\begin{aligned}f_{N}(n)&={\frac {1}{n{\sqrt {2\pi \sigma ^{2}}}}}\,\mathrm {e} ^{-{\frac {\left(\ln n-\mu \right)^{2}}{2\sigma ^{2}}}},\\\langle N\rangle &=\mathrm {e} ^{\mu +\sigma ^{2}/2},\\{\tilde {N}}&=\mathrm {e} ^{\mu },\\{\hat {N}}&=\mathrm {e} ^{\mu -\sigma ^{2}},\\\sigma _{N}^{2}&=(\mathrm {e} ^{\sigma ^{2}}\!\!-1)\mathrm {e} ^{2\mu +\sigma ^{2}}.\\\end{aligned}}$

${\begin{aligned}S_{\mathrm {civ} }&={\frac {C}{\sqrt[{3}]{N}}},\\C&={\sqrt[{3}]{6{R_{G}}^{2}h_{G}}}.\\\end{aligned}}$

$f_{S_{\mathrm {civ} }}(r)={\frac {3}{{\sqrt {2\pi }}\sigma r}}\mathrm {e} ^{-{\frac {\left(\ln {\frac {c^{3}}{r^{3}}}-\mu \right)^{2}}{2\sigma ^{2}}}}.$

${\begin{aligned}\langle S_{\mathrm {civ} }\rangle &=C\mathrm {e} ^{-{\frac {\mu }{3}}+{\frac {\sigma ^{2}}{18}}},\\{\tilde {S}}_{\mathrm {civ} }&=C\mathrm {e} ^{-{\frac {\mu }{3}}},\\{\hat {S}}_{\mathrm {civ} }&=C\mathrm {e} ^{-{\frac {\mu }{3}}-{\frac {\sigma ^{2}}{9}}},\\\sigma _{S_{\mathrm {civ} }}^{2}&=C^{2}\mathrm {e} ^{-{\frac {2\mu }{3}}+{\frac {\sigma ^{2}}{9}}}\left(\mathrm {e} ^{\frac {\sigma ^{2}}{9}}-1\right).\\\end{aligned}}$