wpmath0000006_12

Species-area_curve.html

  1. S = c A z S=cA^{z}
  2. S = c + z l o g ( A ) = l o g ( c A z ) , S=c+zlog(A)=log(cA^{z}),

Specific_volume.html

  1. ( m 3 / kg ) (\mathrm{m^{3}}/\mathrm{kg})
  2. v = V m = ρ - 1 \ v=\frac{V}{m}\ ={\rho}^{-1}
  3. P V = m R T \ PV={mRT}
  4. v = V m \ v=\frac{V}{m}
  5. v = R T P \ v=\frac{RT}{P}
  6. m 3 kg \frac{\mathrm{m^{3}}}{\mathrm{kg}}
  7. ft 3 lbm \frac{\mathrm{ft^{3}}}{\mathrm{lbm}}
  8. ft 3 slug \frac{\mathrm{ft^{3}}}{\mathrm{slug}}
  9. mL g \frac{\mathrm{mL}}{\mathrm{g}}
  10. v = R T P \ v=\frac{RT}{P}

Spectrum_continuation_analysis.html

  1. f ( x ) = n = - F n e i n x f(x)=\sum_{n=-\infty}^{\infty}F_{n}\,e^{inx}
  2. F n F_{n}
  3. f ( x ) = n = - F n e i ω n x . f(x)=\sum_{n=-\infty}^{\infty}F_{n}\,e^{i\omega_{n}x}.
  4. 2 π 2\pi
  5. f ( x ) = n = 0 [ A n cos ( ω n x ) + B n sin ( ω n x ) ] + C ( x ) f(x)=\sum_{n=0}^{\infty}\left[A_{n}\cos(\omega_{n}x)+B_{n}\sin(\omega_{n}x)% \right]+C(x)
  6. ω n \omega_{n}
  7. C ( x ) C(x)

SpeedStep.html

  1. P = C V 2 f . P=CV^{2}f.

Spence's_function.html

  1. Li 2 ( z ) = - 0 z ln ( 1 - u ) u d u , z [ 1 , ) \operatorname{Li}_{2}(z)=-\int_{0}^{z}{\ln(1-u)\over u}\,\mathrm{d}u\,\text{, % }z\in\mathbb{C}\setminus[1,\infty)
  2. | z | < 1 |z|<1
  3. Li 2 ( z ) = k = 1 z k k 2 . \operatorname{Li}_{2}(z)=\sum_{k=1}^{\infty}{z^{k}\over k^{2}}.
  4. 1 v ln t 1 - t d t = Li 2 ( 1 - v ) . \int_{1}^{v}\frac{\ln t}{1-t}\mathrm{d}t=\operatorname{Li}_{2}(1-v).
  5. Li 2 ( z ) \operatorname{Li}_{2}(z)
  6. z z
  7. Li 2 ( z ) + Li 2 ( - z ) = 1 2 Li 2 ( z 2 ) \operatorname{Li}_{2}(z)+\operatorname{Li}_{2}(-z)=\frac{1}{2}\operatorname{Li% }_{2}(z^{2})
  8. Li 2 ( 1 - z ) + Li 2 ( 1 - 1 z ) = - ln 2 z 2 \operatorname{Li}_{2}(1-z)+\operatorname{Li}_{2}\left(1-\frac{1}{z}\right)=-% \frac{\ln^{2}z}{2}
  9. Li 2 ( z ) + Li 2 ( 1 - z ) = π 2 6 - ln z ln ( 1 - z ) \operatorname{Li}_{2}(z)+\operatorname{Li}_{2}(1-z)=\frac{{\pi}^{2}}{6}-\ln z% \cdot\ln(1-z)
  10. Li 2 ( - z ) - Li 2 ( 1 - z ) + 1 2 Li 2 ( 1 - z 2 ) = - π 2 12 - ln z ln ( z + 1 ) \operatorname{Li}_{2}(-z)-\operatorname{Li}_{2}(1-z)+\frac{1}{2}\operatorname{% Li}_{2}(1-z^{2})=-\frac{{\pi}^{2}}{12}-\ln z\cdot\ln(z+1)
  11. Li 2 ( z ) + Li 2 ( 1 z ) = - π 2 6 - 1 2 ln 2 ( - z ) \operatorname{Li}_{2}(z)+\operatorname{Li}_{2}(\frac{1}{z})=-\frac{\pi^{2}}{6}% -\frac{1}{2}\ln^{2}(-z)
  12. Li 2 ( 1 3 ) - 1 6 Li 2 ( 1 9 ) = π 2 18 - ln 2 3 6 \operatorname{Li}_{2}\left(\frac{1}{3}\right)-\frac{1}{6}\operatorname{Li}_{2}% \left(\frac{1}{9}\right)=\frac{{\pi}^{2}}{18}-\frac{\ln^{2}3}{6}
  13. Li 2 ( - 1 2 ) + 1 6 Li 2 ( 1 9 ) = - π 2 18 + ln 2 ln 3 - ln 2 2 2 - ln 2 3 3 \operatorname{Li}_{2}\left(-\frac{1}{2}\right)+\frac{1}{6}\operatorname{Li}_{2% }\left(\frac{1}{9}\right)=-\frac{{\pi}^{2}}{18}+\ln 2\cdot\ln 3-\frac{\ln^{2}2% }{2}-\frac{\ln^{2}3}{3}
  14. Li 2 ( 1 4 ) + 1 3 Li 2 ( 1 9 ) = π 2 18 + 2 ln 2 ln 3 - 2 ln 2 2 - 2 3 ln 2 3 \operatorname{Li}_{2}\left(\frac{1}{4}\right)+\frac{1}{3}\operatorname{Li}_{2}% \left(\frac{1}{9}\right)=\frac{{\pi}^{2}}{18}+2\ln 2\ln 3-2\ln^{2}2-\frac{2}{3% }\ln^{2}3
  15. Li 2 ( - 1 3 ) - 1 3 Li 2 ( 1 9 ) = - π 2 18 + 1 6 ln 2 3 \operatorname{Li}_{2}\left(-\frac{1}{3}\right)-\frac{1}{3}\operatorname{Li}_{2% }\left(\frac{1}{9}\right)=-\frac{{\pi}^{2}}{18}+\frac{1}{6}\ln^{2}3
  16. Li 2 ( - 1 8 ) + Li 2 ( 1 9 ) = - 1 2 ln 2 9 8 \operatorname{Li}_{2}\left(-\frac{1}{8}\right)+\operatorname{Li}_{2}\left(% \frac{1}{9}\right)=-\frac{1}{2}\ln^{2}{\frac{9}{8}}
  17. 36 Li 2 ( 1 2 ) - 36 Li 2 ( 1 4 ) - 12 Li 2 ( 1 8 ) + 6 Li 2 ( 1 64 ) = π 2 36\operatorname{Li}_{2}\left(\frac{1}{2}\right)-36\operatorname{Li}_{2}\left(% \frac{1}{4}\right)-12\operatorname{Li}_{2}\left(\frac{1}{8}\right)+6% \operatorname{Li}_{2}\left(\frac{1}{64}\right)={\pi}^{2}
  18. Li 2 ( - 1 ) = - π 2 12 \operatorname{Li}_{2}(-1)=-\frac{{\pi}^{2}}{12}
  19. Li 2 ( 0 ) = 0 \operatorname{Li}_{2}(0)=0
  20. Li 2 ( 1 2 ) = π 2 12 - ln 2 2 2 \operatorname{Li}_{2}\left(\frac{1}{2}\right)=\frac{{\pi}^{2}}{12}-\frac{\ln^{% 2}2}{2}
  21. Li 2 ( 1 ) = π 2 6 \operatorname{Li}_{2}(1)=\frac{{\pi}^{2}}{6}
  22. Li 2 ( 2 ) = π 2 4 - i π ln 2 \operatorname{Li}_{2}(2)=\frac{{\pi}^{2}}{4}-i\pi\ln 2
  23. Li 2 ( - 5 - 1 2 ) = - π 2 15 + 1 2 ln 2 5 - 1 2 \operatorname{Li}_{2}\left(-\frac{\sqrt{5}-1}{2}\right)=-\frac{{\pi}^{2}}{15}+% \frac{1}{2}\ln^{2}\frac{\sqrt{5}-1}{2}
  24. = - π 2 15 + 1 2 arcsch 2 2 =-\frac{{\pi}^{2}}{15}+\frac{1}{2}\operatorname{arcsch}^{2}2
  25. Li 2 ( - 5 + 1 2 ) = - π 2 10 - ln 2 5 + 1 2 \operatorname{Li}_{2}\left(-\frac{\sqrt{5}+1}{2}\right)=-\frac{{\pi}^{2}}{10}-% \ln^{2}\frac{\sqrt{5}+1}{2}
  26. = - π 2 10 - arcsch 2 2 =-\frac{{\pi}^{2}}{10}-\operatorname{arcsch}^{2}2
  27. Li 2 ( 3 + 5 2 ) = π 2 15 - 1 2 ln 2 5 - 1 2 \operatorname{Li}_{2}\left(\frac{3+\sqrt{5}}{2}\right)=\frac{{\pi}^{2}}{15}-% \frac{1}{2}\ln^{2}\frac{\sqrt{5}-1}{2}
  28. = π 2 15 - 1 2 arcsch 2 2 =\frac{{\pi}^{2}}{15}-\frac{1}{2}\operatorname{arcsch}^{2}2
  29. Li 2 ( 5 + 1 2 ) = π 2 10 - ln 2 5 - 1 2 \operatorname{Li}_{2}\left(\frac{\sqrt{5}+1}{2}\right)=\frac{{\pi}^{2}}{10}-% \ln^{2}\frac{\sqrt{5}-1}{2}
  30. = π 2 10 - arcsch 2 2 =\frac{{\pi}^{2}}{10}-\operatorname{arcsch}^{2}2
  31. Φ ( x ) = - 0 x ln | 1 - u | u d u = { Li 2 ( x ) , x 1 ; π 2 3 - 1 2 ln 2 ( x ) - Li 2 ( 1 x ) , x > 1. \operatorname{\Phi}(x)=-\int_{0}^{x}{\ln|1-u|\over u}\mathrm{d}u=\begin{cases}% \operatorname{Li}_{2}(x),&x\leq 1;\\ \frac{{\pi}^{2}}{3}-\frac{1}{2}\ln^{2}(x)-\operatorname{Li}_{2}(\frac{1}{x}),&% x>1.\end{cases}

Spherically_symmetric_spacetime.html

  1. M M
  2. dim K ( M ) = 3 \dim K(M)=3

Sphericity.html

  1. Ψ \Psi
  2. Ψ = π 1 3 ( 6 V p ) 2 3 A p \Psi=\frac{\pi^{\frac{1}{3}}(6V_{p})^{\frac{2}{3}}}{A_{p}}
  3. V p V_{p}
  4. A p A_{p}
  5. Ψ \Psi
  6. Ψ = π 1 3 ( 6 V p ) 2 3 A p = 2 a b 2 3 a + b 2 a 2 - b 2 ln ( a + a 2 - b 2 b ) , \Psi=\frac{\pi^{\frac{1}{3}}(6V_{p})^{\frac{2}{3}}}{A_{p}}=\frac{2\sqrt[3]{ab^% {2}}}{a+\frac{b^{2}}{\sqrt{a^{2}-b^{2}}}\ln{\left(\frac{a+\sqrt{a^{2}-b^{2}}}{% b}\right)}},
  7. A s A_{s}
  8. V p V_{p}
  9. A s 3 = ( 4 π r 2 ) 3 = 4 3 π 3 r 6 = 4 π ( 4 2 π 2 r 6 ) = 4 π 3 2 ( 4 2 π 2 3 2 r 6 ) = 36 π ( 4 π 3 r 3 ) 2 = 36 π V p 2 A_{s}^{3}=\left(4\pi r^{2}\right)^{3}=4^{3}\pi^{3}r^{6}=4\pi\left(4^{2}\pi^{2}% r^{6}\right)=4\pi\cdot 3^{2}\left(\frac{4^{2}\pi^{2}}{3^{2}}r^{6}\right)=36\pi% \left(\frac{4\pi}{3}r^{3}\right)^{2}=36\,\pi V_{p}^{2}
  10. A s = ( 36 π V p 2 ) 1 3 = 36 1 3 π 1 3 V p 2 3 = 6 2 3 π 1 3 V p 2 3 = π 1 3 ( 6 V p ) 2 3 A_{s}=\left(36\,\pi V_{p}^{2}\right)^{\frac{1}{3}}=36^{\frac{1}{3}}\pi^{\frac{% 1}{3}}V_{p}^{\frac{2}{3}}=6^{\frac{2}{3}}\pi^{\frac{1}{3}}V_{p}^{\frac{2}{3}}=% \pi^{\frac{1}{3}}\left(6V_{p}\right)^{\frac{2}{3}}
  11. Ψ \Psi
  12. Ψ = A s A p = π 1 3 ( 6 V p ) 2 3 A p \Psi=\frac{A_{s}}{A_{p}}=\frac{\pi^{\frac{1}{3}}\left(6V_{p}\right)^{\frac{2}{% 3}}}{A_{p}}
  13. 2 12 s 3 \frac{\sqrt{2}}{12}\,s^{3}
  14. 3 s 2 \sqrt{3}\,s^{2}
  15. ( π 6 3 ) 1 3 0.671 \left(\frac{\pi}{6\sqrt{3}}\right)^{\frac{1}{3}}\approx 0.671
  16. s 3 \,s^{3}
  17. 6 s 2 6\,s^{2}
  18. ( π 6 ) 1 3 0.806 \left(\frac{\pi}{6}\right)^{\frac{1}{3}}\approx 0.806
  19. 1 3 2 s 3 \frac{1}{3}\sqrt{2}\,s^{3}
  20. 2 3 s 2 2\sqrt{3}\,s^{2}
  21. ( π 3 3 ) 1 3 0.846 \left(\frac{\pi}{3\sqrt{3}}\right)^{\frac{1}{3}}\approx 0.846
  22. 1 4 ( 15 + 7 5 ) s 3 \frac{1}{4}\left(15+7\sqrt{5}\right)\,s^{3}
  23. 3 25 + 10 5 s 2 3\sqrt{25+10\sqrt{5}}\,s^{2}
  24. ( ( 15 + 7 5 ) 2 π 12 ( 25 + 10 5 ) 3 2 ) 1 3 0.910 \left(\frac{\left(15+7\sqrt{5}\right)^{2}\pi}{12\left(25+10\sqrt{5}\right)^{% \frac{3}{2}}}\right)^{\frac{1}{3}}\approx 0.910
  25. 5 12 ( 3 + 5 ) s 3 \frac{5}{12}\left(3+\sqrt{5}\right)\,s^{3}
  26. 5 3 s 2 5\sqrt{3}\,s^{2}
  27. ( ( 3 + 5 ) 2 π 60 3 ) 1 3 0.939 \left(\frac{\left(3+\sqrt{5}\right)^{2}\pi}{60\sqrt{3}}\right)^{\frac{1}{3}}% \approx 0.939
  28. ( h = 2 2 r ) (h=2\sqrt{2}r)
  29. 1 3 π r 2 h \frac{1}{3}\pi\,r^{2}h
  30. = 2 2 3 π r 3 =\frac{2\sqrt{2}}{3}\pi\,r^{3}
  31. π r ( r + r 2 + h 2 ) \pi\,r(r+\sqrt{r^{2}+h^{2}})
  32. = 4 π r 2 =4\pi\,r^{2}
  33. ( 1 2 ) 1 3 0.794 \left(\frac{1}{2}\right)^{\frac{1}{3}}\approx 0.794
  34. 2 3 π r 3 \frac{2}{3}\pi\,r^{3}
  35. 3 π r 2 3\pi\,r^{2}
  36. ( 16 27 ) 1 3 0.840 \left(\frac{16}{27}\right)^{\frac{1}{3}}\approx 0.840
  37. ( h = 2 r ) (h=2\,r)
  38. π r 2 h = 2 π r 3 \pi r^{2}h=2\pi\,r^{3}
  39. 2 π r ( r + h ) = 6 π r 2 2\pi r(r+h)=6\pi\,r^{2}
  40. ( 2 3 ) 1 3 0.874 \left(\frac{2}{3}\right)^{\frac{1}{3}}\approx 0.874
  41. ( R = r ) (R=r)
  42. 2 π 2 R r 2 = 2 π 2 r 3 2\pi^{2}Rr^{2}=2\pi^{2}\,r^{3}
  43. 4 π 2 R r = 4 π 2 r 2 4\pi^{2}Rr=4\pi^{2}\,r^{2}
  44. ( 9 4 π ) 1 3 0.894 \left(\frac{9}{4\pi}\right)^{\frac{1}{3}}\approx 0.894
  45. 4 3 π r 3 \frac{4}{3}\pi r^{3}
  46. 4 π r 2 4\pi\,r^{2}
  47. 1 1\,

Spherometer.html

  1. R = ( h / 2 ) + ( a 2 / 6 h ) R=(h/2)+(a^{2}/6h)
  2. R = ( h / 2 ) + ( D 2 / 8 h ) R=(h/2)+(D^{2}/8h)

Spider_diagram.html

  1. , , ¬ \land,\lor,\lnot
  2. A B A\land B
  3. B C B\land C
  4. F E F\land E
  5. G F G\land F
  6. ( F E ) ( G ) ( D ) (F\land E)\lor(G)\lor(D)
  7. ( A ) ( C B ) ( F ) (A)\lor(C\land B)\lor(F)

Spin-½.html

  1. S = 1 2 ( 1 2 + 1 ) = 3 2 . S={{\sqrt{\frac{1}{2}\left(\frac{1}{2}+1\right)}}}\ \hbar=\frac{\sqrt{3}}{2}\hbar.
  2. 1 / 2 {1}/{2}
  3. S z = 2 σ z = 2 [ 1 0 0 - 1 ] S_{z}=\frac{\hbar}{2}\sigma_{z}=\frac{\hbar}{2}\begin{bmatrix}1&0\\ 0&-1\end{bmatrix}
  4. χ + = [ 1 0 ] = | s z = + 1 2 = | = | 0 \chi_{+}=\begin{bmatrix}1\\ 0\end{bmatrix}=\left|{s_{z}=+\textstyle\frac{1}{2}}\right\rangle=|{\uparrow}% \rangle=|0\rangle
  5. χ - = [ 0 1 ] = | s z = - 1 2 = | = | 1 . \chi_{-}=\begin{bmatrix}0\\ 1\end{bmatrix}=\left|{s_{z}=-\textstyle\frac{1}{2}}\right\rangle=|{\downarrow}% \rangle=|1\rangle.
  6. S + = [ 0 1 0 0 ] , S - = [ 0 0 1 0 ] S_{+}=\hbar\begin{bmatrix}0&1\\ 0&0\end{bmatrix},S_{-}=\hbar\begin{bmatrix}0&0\\ 1&0\end{bmatrix}
  7. S x = 2 σ x = 2 [ 0 1 1 0 ] S_{x}=\frac{\hbar}{2}\sigma_{x}=\frac{\hbar}{2}\begin{bmatrix}0&1\\ 1&0\end{bmatrix}
  8. S y = 2 σ y = 2 [ 0 - i i 0 ] S_{y}=\frac{\hbar}{2}\sigma_{y}=\frac{\hbar}{2}\begin{bmatrix}0&-i\\ i&0\end{bmatrix}
  9. χ + ( x ) = 1 2 [ 1 1 ] = | s x = + 1 2 \chi^{(x)}_{+}=\frac{1}{\sqrt{2}}\begin{bmatrix}1\\ 1\end{bmatrix}=\left|{s_{x}=+\textstyle\frac{1}{2}}\right\rangle
  10. χ - ( x ) = 1 2 [ 1 - 1 ] = | s x = - 1 2 \chi^{(x)}_{-}=\frac{1}{\sqrt{2}}\begin{bmatrix}1\\ -1\end{bmatrix}=\left|{s_{x}=-\textstyle\frac{1}{2}}\right\rangle
  11. χ + ( y ) = 1 2 [ 1 i ] = | s y = + 1 2 \chi^{(y)}_{+}=\frac{1}{\sqrt{2}}\begin{bmatrix}1\\ i\end{bmatrix}=\left|{s_{y}=+\textstyle\frac{1}{2}}\right\rangle
  12. χ - ( y ) = 1 2 [ 1 - i ] = | s y = - 1 2 \chi^{(y)}_{-}=\frac{1}{\sqrt{2}}\begin{bmatrix}1\\ -i\end{bmatrix}=\left|{s_{y}=-\textstyle\frac{1}{2}}\right\rangle

Spin_connection.html

  1. e μ a e_{\mu}^{a}
  2. g μ ν = e μ a e ν b η a b , g_{\mu\nu}=e_{\mu}^{a}e_{\nu}^{b}\eta_{ab},
  3. g μ ν g_{\mu\nu}
  4. η a b \eta_{ab}
  5. g μ ν g_{\mu\nu}
  6. e μ a e^{a}_{\mu}
  7. g μ ν g^{\mu\nu}
  8. η a b \eta^{ab}
  9. e μ a = g μ ν e ν a e^{\mu a}=g^{\mu\nu}e_{\nu}^{a}
  10. ω μ a b = e ν a μ e ν b + e ν a Γ μ σ ν e σ b , \omega_{\mu}^{ab}=e^{a}_{\nu}\partial_{\mu}e^{\nu b}+e^{a}_{\nu}\Gamma^{\nu}_{% \mu\sigma}e^{\sigma b},
  11. Γ μ ν σ \Gamma^{\sigma}_{\mu\nu}
  12. ω μ a b = 1 2 e ν a ( μ e ν b - ν e μ b ) - 1 2 e ν b ( μ e ν a - ν e μ a ) - 1 2 e ρ a e σ b ( ρ e σ c - σ e ρ c ) e μ c , \omega_{\mu}^{ab}=\frac{1}{2}e^{\nu a}(\partial_{\mu}e_{\nu}^{b}-\partial_{\nu% }e_{\mu}^{b})-\frac{1}{2}e^{\nu b}(\partial_{\mu}e_{\nu}^{a}-\partial_{\nu}e_{% \mu}^{a})-\frac{1}{2}e^{\rho a}e^{\sigma b}(\partial_{\rho}e_{\sigma c}-% \partial_{\sigma}e_{\rho c})e_{\mu}^{c},
  13. a , b a,b
  14. ω μ a b \omega^{ab}_{\mu}
  15. D μ D_{\mu}
  16. V ν a V_{\nu}^{a}
  17. D μ V ν a = μ V ν a + ω μ b a V ν b - Γ μ ν σ V σ a D_{\mu}V_{\nu}^{a}=\partial_{\mu}V_{\nu}^{a}+\omega_{\mu b}^{a}V^{b}_{\nu}-% \Gamma_{\mu\nu}^{\sigma}V_{\sigma}^{a}
  18. D μ e ν a μ e ν a + ω μ b a e ν b - Γ μ ν σ e σ a = 0. D_{\mu}e^{a}_{\nu}\equiv\partial_{\mu}e^{a}_{\nu}+\omega^{a}_{\mu b}e_{\nu}^{b% }-\Gamma^{\sigma}_{\mu\nu}e_{\sigma}^{a}=0.
  19. ω μ a b \omega^{ab}_{\mu}
  20. ω μ a b = e ν a μ e ν b + e ν a Γ μ σ ν e σ b , \omega_{\mu}^{ab}=e^{a}_{\nu}\partial_{\mu}e^{\nu b}+e^{a}_{\nu}\Gamma^{\nu}_{% \mu\sigma}e^{\sigma b},
  21. e a μ e_{a}^{\mu}
  22. e μ a e b μ = δ b a e^{a}_{\mu}e_{b}^{\mu}=\delta^{a}_{b}
  23. e μ b e b ν = δ μ ν e^{b}_{\mu}e_{b}^{\nu}=\delta^{\nu}_{\mu}
  24. D μ D_{\mu}
  25. η a b \eta_{ab}
  26. D μ η a b = μ η a b + ω μ a c η c b + ω μ b c η a c = 0. D_{\mu}\eta_{ab}=\partial_{\mu}\eta_{ab}+\omega_{\mu}^{ac}\eta_{cb}+\omega_{% \mu}^{bc}\eta_{ac}=0.
  27. ω μ a b = - ω μ b a \omega_{\mu}^{ab}=-\omega_{\mu}^{ba}
  28. Γ σ μ ν = 1 2 g ν δ ( σ g δ μ + μ g σ δ - δ g σ μ ) \Gamma^{\nu}_{\sigma\mu}={1\over 2}g^{\nu\delta}(\partial_{\sigma}g_{\delta\mu% }+\partial_{\mu}g_{\sigma\delta}-\partial_{\delta}g_{\sigma\mu})
  29. e μ a e^{a}_{\mu}
  30. e μ a e^{a}_{\mu}
  31. ω μ a b = 1 2 e ν [ a ( e ν , μ b ] - e μ , ν b ] + e b ] σ e μ c e ν c , σ ) . \omega_{\mu}^{ab}=\frac{1}{2}e^{\nu[a}(e_{\nu,\mu}^{b]}-e_{\mu,\nu}^{b]}+e^{b]% \sigma}e_{\mu}^{c}e_{\nu c,\sigma}).
  32. ω μ a b \omega_{\mu}^{ab}
  33. ρ g α β = 0 \nabla_{\rho}g_{\alpha\beta}=0
  34. Γ α β γ \Gamma_{\alpha\beta}^{\gamma}
  35. e b α e c β ( [ α e β ] a + ω [ α a d e β ] d ) = 0 e_{b}^{\alpha}e_{c}^{\beta}(\partial_{[\alpha}e_{\beta]a}+\omega_{[\alpha a}^{% \;\;\;\;d}e_{\beta]d})=0
  36. a , b , a,b,
  37. c c
  38. Ω b c a + Ω a b c - Ω c a b + 2 e b α ω α a c = 0 \Omega_{bca}+\Omega_{abc}-\Omega_{cab}+2e_{b}^{\alpha}\omega_{\alpha ac}=0
  39. Ω b c a := e b α e c β [ α e β ] a \Omega_{bca}:=e_{b}^{\alpha}e_{c}^{\beta}\partial_{[\alpha}e_{\beta]a}
  40. ω α c a = 1 2 e α b ( Ω b c a + Ω a b c - Ω c a b ) \omega_{\alpha ca}={1\over 2}e_{\alpha}^{b}(\Omega_{bca}+\Omega_{abc}-\Omega_{% cab})
  41. γ a \gamma^{a}
  42. γ a e a μ ( x ) = γ μ ( x ) \gamma^{a}e_{a}^{\mu}(x)=\gamma^{\mu}(x)
  43. ψ e i ϵ a b ( x ) σ a b ψ \psi\mapsto e^{i\epsilon^{ab}(x)\sigma_{ab}}\psi
  44. ϵ a b \epsilon_{ab}
  45. ω μ a b \omega_{\mu}^{ab}
  46. μ ψ = ( μ - i 4 ω μ a b σ a b ) ψ = ( μ - i 4 e ν a μ e ν b σ a b ) ψ \nabla_{\mu}\psi=(\partial_{\mu}-{i\over 4}\omega_{\mu}^{ab}\sigma_{ab})\psi=(% \partial_{\mu}-{i\over 4}e^{\nu a}\partial_{\mu}e_{\nu}^{b}\sigma_{ab})\psi
  47. ( i γ μ μ - m ) ψ = 0 (i\gamma^{\mu}\nabla_{\mu}-m)\psi=0
  48. = - 1 2 κ 2 e e a μ e b ν Ω μ ν a b [ ω ] + e ψ ¯ ( i γ μ μ - m ) ψ \mathcal{L}=-{1\over 2\kappa^{2}}ee_{a}^{\mu}e_{b}^{\nu}\Omega_{\mu\nu}^{\;\;% \;\;ab}[\omega]+e\overline{\psi}(i\gamma^{\mu}\nabla_{\mu}-m)\psi
  49. e := det e μ a e:=\det e_{\mu}^{a}
  50. Ω μ ν a b \Omega_{\mu\nu}^{\;\;\;\;ab}
  51. q a b ( x ) q_{ab}(x)
  52. e a i e_{a}^{i}
  53. D a q b c = 0 D_{a}q_{bc}=0
  54. e a i e_{a}^{i}
  55. D a e b i = 0 D_{a}e_{b}^{i}=0
  56. Γ a i j \Gamma_{a}^{ij}
  57. S U ( 2 ) SU(2)
  58. Γ a i = ϵ i j k Γ a j k \Gamma_{a}^{i}=\epsilon^{ijk}\Gamma_{a}^{jk}
  59. A a i = Γ a i + β c a i A_{a}^{i}=\Gamma_{a}^{i}+\beta c_{a}^{i}
  60. c a i = c a b e b i c_{a}^{i}=c_{ab}e^{bi}
  61. c a b c_{ab}
  62. β \beta
  63. A a i A_{a}^{i}
  64. E a i = | d e t ( e ) | e a i E_{a}^{i}=|det(e)|e_{a}^{i}
  65. S U ( 2 ) SU(2)

Spin_structure.html

  1. π F 𝐏 = π 𝐏 \pi\circ F_{\mathbf{P}}=\pi_{\mathbf{P}}
  2. F 𝐏 2 f = F 𝐏 1 F_{\mathbf{P}_{2}}\circ f=F_{\mathbf{P}_{1}}
  3. 𝐩 𝐏 1 {\mathbf{p}}\in{\mathbf{P}_{1}}
  4. F 𝐏 1 F_{\mathbf{P}_{1}}
  5. F 𝐏 2 F_{\mathbf{P}_{2}}
  6. ϕ ( p g ) = ϕ ( p ) ρ ( g ) \phi(pg)=\phi(p)\rho(g)
  7. 1 2 Spin ( n ) SO ( n ) × U ( 1 ) 1. 1\to\mathbb{Z}_{2}\to{\mathrm{Spin}}^{\mathbb{C}}(n)\to{\mathrm{SO}}(n)\times{% \mathrm{U}}(1)\to 1.
  8. κ × i : Spin ( n ) × U ( 1 ) U ( N ) . \kappa\times i\colon{\mathrm{Spin}}(n)\times{\mathrm{U}}(1)\to{\mathrm{U}}(N).
  9. Spin ( n ) = Spin ( n ) × 2 U ( 1 ) , {\mathrm{Spin}}^{\mathbb{C}}(n)={\mathrm{Spin}}(n)\times_{\mathbb{Z}_{2}}{% \mathrm{U}}(1)\,,
  10. H 2 ( M ; 𝐙 ) 2 H 2 ( M ; 𝐙 ) H 2 ( M ; 𝐙 2 ) β H 3 ( M ; 𝐙 ) \dots\longrightarrow\textrm{H}^{2}(M;\mathbf{Z})\stackrel{2}{\longrightarrow}% \textrm{H}^{2}(M;\mathbf{Z})\longrightarrow\textrm{H}^{2}(M;\mathbf{Z}_{2})% \stackrel{\beta}{\longrightarrow}\textrm{H}^{3}(M;\mathbf{Z})\longrightarrow\dots
  11. W 3 = β w 2 = 0 W_{3}=\beta w_{2}=0
  12. w ^ 2 ( F ) \hat{w}_{2}(F)
  13. w ^ 2 \hat{w}_{2}
  14. w ^ 2 \hat{w}_{2}

Spin_tensor.html

  1. 𝔰 𝔢 ( d ) \mathfrak{se}(d)
  2. ν T μ ν = 0 , \partial_{\nu}T^{\mu\nu}=0\,,
  3. T μ ν T^{\mu\nu}\,
  4. d 4 x T μ 0 ( x , t ) = P μ \int d^{4}xT^{\mu 0}(\vec{x},t)=P^{\mu}
  5. M y α β μ M^{\alpha\beta\mu}_{y}
  6. M y α β μ ( x ) = M 0 α β μ ( x ) + y α T β μ ( x ) - y β T α μ ( x ) , M^{\alpha\beta\mu}_{y}(x)=M^{\alpha\beta\mu}_{0}(x)+y^{\alpha}T^{\beta\mu}(x)-% y^{\beta}T^{\alpha\mu}(x)\,,
  7. d 4 x M 0 μ ν ( x , t ) \int d^{4}xM^{\mu\nu}_{0}(\vec{x},t)
  8. M μ ν M^{\mu\nu}\,
  9. S α β μ ( 𝐱 ) = def M x α β μ ( 𝐱 ) = M 0 α β μ ( 𝐱 ) + x α T β μ ( 𝐱 ) - x β T α μ ( 𝐱 ) S^{\alpha\beta\mu}(\mathbf{x})\ \stackrel{\mathrm{def}}{=}\ M^{\alpha\beta\mu}% _{x}(\mathbf{x})=M^{\alpha\beta\mu}_{0}(\mathbf{x})+x^{\alpha}T^{\beta\mu}(% \mathbf{x})-x^{\beta}T^{\alpha\mu}(\mathbf{x})
  10. μ M 0 α β μ = 0 , \partial_{\mu}M^{\alpha\beta\mu}_{0}=0\,,
  11. μ S α β μ = T β α - T α β 0 \partial_{\mu}S^{\alpha\beta\mu}=T^{\beta\alpha}-T^{\alpha\beta}\neq 0
  12. T i j - T j i T_{ij}-T_{ji}

Spin_wave.html

  1. μ e V μeV
  2. \mathcal{H}
  3. = - 1 2 J i , j 𝐒 i 𝐒 j - g μ B i 𝐇 𝐒 i \mathcal{H}=-\frac{1}{2}J\sum_{i,j}\mathbf{S}_{i}\cdot\mathbf{S}_{j}-g\mu_{B}% \sum_{i}\mathbf{H}\cdot\mathbf{S}_{i}
  4. J J
  5. S S
  6. g g
  7. g g
  8. 𝐇 \mathbf{H}
  9. 1 + 1 1+1
  10. 𝐒 t = 𝐒 × 𝐒 x x . \mathbf{S}_{t}=\mathbf{S}\times\mathbf{S}_{xx}.
  11. 1 + 1 , 2 + 1 1+1,2+1
  12. 3 + 1 3+1
  13. J > 0 J>0
  14. | 0 |0\rangle
  15. 𝐇 \mathbf{H}
  16. | 0 |0\rangle
  17. \mathcal{H}
  18. S ± = S x ± i S y S^{\pm}=S^{x}\pm iS^{y}
  19. = - 1 2 J i , j S i z S j z - g μ B H i S i z - 1 2 J i , j S i - S j + \mathcal{H}=-\frac{1}{2}J\sum_{i,j}S^{z}_{i}S^{z}_{j}-g\mu_{B}H\sum_{i}S^{z}_{% i}-\frac{1}{2}J\sum_{i,j}S^{-}_{i}S^{+}_{j}
  20. z z
  21. z z
  22. z z
  23. S i z | 0 = s | 0 S^{z}_{i}|0\rangle=s|0\rangle
  24. | 0 = ( - J s 2 - g μ B H s ) N | 0 \mathcal{H}|0\rangle=\left(-Js^{2}-g\mu_{B}Hs\right)N|0\rangle
  25. i i
  26. S i z | 1 = ( s - 1 ) | 1 , S^{z}_{i}|1\rangle=(s-1)|1\rangle,
  27. S i + S^{+}_{i}
  28. z z
  29. i i
  30. S j - S^{-}_{j}
  31. z z
  32. j j
  33. | 0 |0\rangle
  34. \mathcal{H}
  35. M = N μ B g s V M=\frac{N\mu_{B}gs}{V}
  36. V V
  37. d 𝐌 d t = - γ 𝐌 × 𝐇 - λ 𝐌 × 𝐌 × 𝐇 M 2 \frac{d\mathbf{M}}{dt}=-\gamma\mathbf{M}\times\mathbf{H}-\frac{\lambda\mathbf{% M}\times\mathbf{M}\times\mathbf{H}}{M^{2}}
  38. γ γ
  39. λ λ
  40. λ λ
  41. k k
  42. ώ = c k ώ=ck
  43. ω ω
  44. c c
  45. ώ = A k < s u p > 2 ώ=Ak<sup>2

Split_supersymmetry.html

  1. g ~ \tilde{g}
  2. 1 2 \frac{1}{2}
  3. ( 8 , 1 ) 0 (8,1)_{0}
  4. W ~ \tilde{W}
  5. 1 2 \frac{1}{2}
  6. ( 1 , 3 ) 0 (1,3)_{0}
  7. B ~ \tilde{B}
  8. 1 2 \frac{1}{2}
  9. ( 1 , 1 ) 0 (1,1)_{0}
  10. H ~ u \tilde{H}_{u}
  11. 1 2 \frac{1}{2}
  12. ( 1 , 2 ) 1 2 (1,2)_{\frac{1}{2}}
  13. H ~ d \tilde{H}_{d}
  14. 1 2 \frac{1}{2}
  15. ( 1 , 2 ) - 1 2 (1,2)_{-\frac{1}{2}}
  16. tan β \tan\beta
  17. m g 5 m s q 4 {{m_{g}}^{5}\over{m_{sq}}^{4}}
  18. m g m_{g}
  19. m s q m_{sq}
  20. 10 9 10^{9}

Spontaneous_magnetization.html

  1. M ( T ) = M ( 0 ) ( 1 - ( T / T c ) 3 / 2 ) , M(T)=M(0)\left(1-(T/T_{c}\right)^{3/2}),
  2. M ( 0 ) M(0)
  3. T 0 T\rightarrow 0
  4. M ( T ) ( T - T c ) β , M(T)\propto\left(T-T_{c}\right)^{\beta},
  5. 0.34 0.34
  6. 0.51 0.51
  7. M ( T ) M ( 0 ) = ( 1 - ( T / T c ) α ) β , \frac{M(T)}{M(0)}=\left(1-(T/T_{c}\right)^{\alpha})^{\beta},
  8. T 0 T\rightarrow 0
  9. T T C T\rightarrow T_{C}

Sports_rating_system.html

  1. Pythagorean wins = Points For 2.37 Points For 2.37 + Points Against 2.37 × Games Played . \,\text{Pythagorean wins}=\frac{\,\text{Points For}^{2.37}}{\,\text{Points For% }^{2.37}+\,\text{Points Against}^{2.37}}\times\,\text{Games Played}.

Spring_(mathematics).html

  1. x ( u , v ) = ( R + r cos v ) cos u , x(u,v)=\left(R+r\cos{v}\right)\cos{u},
  2. y ( u , v ) = ( R + r cos v ) sin u , y(u,v)=\left(R+r\cos{v}\right)\sin{u},
  3. z ( u , v ) = r sin v + P u π , z(u,v)=r\sin{v}+{P\cdot u\over\pi},
  4. u [ 0 , 2 n π ) ( n ) , u\in[0,\ 2n\pi)\ \left(n\in\mathbb{R}\right),
  5. v [ 0 , 2 π ) , v\in[0,\ 2\pi),
  6. R R\,
  7. r r\,
  8. P P\,
  9. n n\,
  10. n n
  11. ( R - x 2 + y 2 ) 2 + ( z + P arctan ( x / y ) π ) 2 = r 2 . \left(R-\sqrt{x^{2}+y^{2}}\right)^{2}+\left(z+{P\arctan(x/y)\over\pi}\right)^{% 2}=r^{2}.
  12. V = 2 π 2 n R r 2 = ( π r 2 ) ( 2 π n R ) . V=2\pi^{2}nRr^{2}=\left(\pi r^{2}\right)\left(2\pi nR\right).\,
  13. P P\,

Springfield_Lake.html

  1. m 3 m^{3}
  2. k m - 1 km^{-1}
  3. 3 / 4 {3}/{4}

Sprue_(manufacturing).html

  1. t f = 2 A m ( h t - h t - h m ) A g 2 g t_{f}=\frac{2A_{m}(\sqrt{h_{t}}-\sqrt{h_{t}-h_{m}})}{A_{g}\sqrt{2g}}
  2. t f t_{f}
  3. A m A_{m}
  4. A g A_{g}
  5. g g
  6. h t h_{t}
  7. h m h_{m}
  8. t f = 2 A m h t - ( h t - h m ) A g 2 g t_{f}=\frac{2A_{m}\sqrt{h_{t}-(h_{t}-h_{m})}}{A_{g}\sqrt{2g}}
  9. t f = 2 A m h m A g 2 g t_{f}=\frac{2A_{m}\sqrt{h_{m}}}{A_{g}\sqrt{2g}}
  10. t f t_{f}
  11. A m A_{m}
  12. A g A_{g}
  13. g g

Square-free.html

  1. s 2 r s^{2}\mid r
  2. r = p 1 p 2 p n r=p_{1}p_{2}\cdots p_{n}

Square_degree.html

  1. × 10 - 4 \times 10^{-}4
  2. 4 π ( 180 π ) 2 = 129 600 π , 4\pi\left(\frac{180}{\pi}\right)^{2}=\frac{129\,600}{\pi},
  3. 0.5 2 \frac{0.5}{2}

Square_principle.html

  1. ( C β ) β Sing (C_{\beta})_{\beta\in\mathrm{Sing}}
  2. C β C_{\beta}
  3. β \beta
  4. ( C β ) < β (C_{\beta})<\beta
  5. γ \gamma
  6. C β C_{\beta}
  7. γ Sing \gamma\in\mathrm{Sing}
  8. C γ = C β γ C_{\gamma}=C_{\beta}\cap\gamma
  9. κ \kappa
  10. κ \Box_{\kappa}
  11. ( C β β a limit point of κ + ) (C_{\beta}\mid\beta\,\text{ a limit point of }\kappa^{+})
  12. C β C_{\beta}
  13. β \beta
  14. c f β < κ cf\beta<\kappa
  15. | C β | < κ |C_{\beta}|<\kappa
  16. γ \gamma
  17. C β C_{\beta}
  18. C γ = C β γ C_{\gamma}=C_{\beta}\cap\gamma

Square_root_of_a_matrix.html

  1. B B
  2. A A
  3. B B
  4. B B
  5. A A
  6. ( 33 24 48 57 ) \left(\begin{smallmatrix}33&24\\ 48&57\end{smallmatrix}\right)
  7. ( 1 4 8 5 ) \left(\begin{smallmatrix}1&4\\ 8&5\end{smallmatrix}\right)
  8. ( 5 2 4 7 ) \left(\begin{smallmatrix}5&2\\ 4&7\end{smallmatrix}\right)
  9. ( 1 0 0 1 ) , \bigl(\begin{smallmatrix}1&0\\ 0&1\end{smallmatrix}\bigr),
  10. 1 t ( s r r - s ) , 1 t ( s - r - r - s ) , 1 t ( - s r r s ) , 1 t ( - s - r - r s ) , ( 1 0 0 ± 1 ) , and ( - 1 0 0 ± 1 ) , \frac{1}{t}\left(\begin{matrix}s&r\\ r&-s\end{matrix}\right),\quad\frac{1}{t}\left(\begin{matrix}s&-r\\ -r&-s\end{matrix}\right),\quad\frac{1}{t}\left(\begin{matrix}-s&r\\ r&s\end{matrix}\right),\quad\frac{1}{t}\left(\begin{matrix}-s&-r\\ -r&s\end{matrix}\right),\quad\left(\begin{matrix}1&0\\ 0&\pm 1\end{matrix}\right),\quad\,\text{and}\quad\left(\begin{matrix}-1&0\\ 0&\pm 1\end{matrix}\right),
  11. ( r , s , t ) (r,s,t)
  12. r 2 + s 2 = t 2 r^{2}+s^{2}=t^{2}
  13. ( 0 4 - 1 5 ) \left(\begin{smallmatrix}~{}\;0&4\\ -1&5\end{smallmatrix}\right)
  14. ( 2 3 4 3 - 1 3 7 3 ) \left(\begin{smallmatrix}~{}\;\frac{2}{3}&\frac{4}{3}\\ -\frac{1}{3}&\frac{7}{3}\end{smallmatrix}\right)
  15. ( 2 - 4 1 - 3 ) \left(\begin{smallmatrix}2&-4\\ 1&-3\end{smallmatrix}\right)
  16. n × n n×n
  17. n n
  18. A A
  19. V V
  20. A A
  21. D D
  22. n n
  23. A A
  24. D D
  25. D D
  26. D D
  27. D D
  28. D D
  29. D D
  30. ( 0 1 0 0 ) \left(\begin{smallmatrix}\\ 0&1\\ 0&0\end{smallmatrix}\right)
  31. D D
  32. R R
  33. D D
  34. R R
  35. D D
  36. A A
  37. V V
  38. D D
  39. A A
  40. V V
  41. A A
  42. R = V S V - 1 , R=VSV^{-1}~{},
  43. S S
  44. D D
  45. ( V D 1 / 2 V - 1 ) 2 = V D 1 / 2 ( V - 1 V ) D 1 / 2 V - 1 = V D V - 1 = A . (VD^{1/2}V^{-1})^{2}=VD^{1/2}(V^{-1}V)D^{1/2}V^{-1}=VDV^{-1}=A~{}.
  46. A = ( 33 24 48 57 ) A=\bigl(\begin{smallmatrix}\\ 33&24\\ 48&57\end{smallmatrix}\bigr)
  47. V = ( 1 1 2 - 1 ) V=\bigl(\begin{smallmatrix}\\ 1&~{}\;1\\ 2&-1\end{smallmatrix}\bigr)
  48. D = ( 81 0 0 9 ) D=\bigl(\begin{smallmatrix}\\ 81&0\\ ~{}\;0&9\end{smallmatrix}\bigr)
  49. D D
  50. D 1 / 2 = ( 9 0 0 3 ) D^{1/2}=\bigl(\begin{smallmatrix}\\ 9&0\\ 0&3\end{smallmatrix}\bigr)
  51. A 1 / 2 = V D 1 / 2 V - 1 = ( 5 2 4 7 ) A^{1/2}=VD^{1/2}V^{-1}=\bigl(\begin{smallmatrix}\\ 5&2\\ 4&7\end{smallmatrix}\bigr)
  52. A A
  53. V V
  54. V V
  55. R = V S V T . R=VSV^{T}~{}.
  56. λ √λ
  57. M M
  58. M M
  59. L + M L+M
  60. I + ( L + M ) / 2 I+(L+M)/2
  61. M M
  62. A A
  63. p ( t ) p(t)
  64. A A
  65. A A
  66. A A
  67. λ ( I + N ) λ(I+N)
  68. A A
  69. Y k + 1 = 1 2 ( Y k + Z k - 1 ) , Z k + 1 = 1 2 ( Z k + Y k - 1 ) . \begin{aligned}\displaystyle Y_{k+1}&\displaystyle=\tfrac{1}{2}(Y_{k}+Z_{k}^{-% 1}),\\ \displaystyle Z_{k+1}&\displaystyle=\tfrac{1}{2}(Z_{k}+Y_{k}^{-1}).\end{aligned}
  70. X n + 1 = 2 X n - X n B X n . X_{n+1}=2X_{n}-X_{n}BX_{n}.
  71. k k
  72. X 0 = Z k - 1 - 1 X_{0}=Z_{k-1}^{-1}
  73. B = Z k , B=Z_{k},
  74. Z k - 1 = X n Z_{k}^{-1}=X_{n}
  75. Y k - 1 . Y_{k}^{-1}.
  76. Y k Y_{k}
  77. A A
  78. Z k Z_{k}
  79. A A
  80. X k + 1 = 1 2 ( X k + A X k - 1 ) . X_{k+1}=\tfrac{1}{2}(X_{k}+AX_{k}^{-1})\,.
  81. X k X_{k}
  82. U * U \displaystyle U^{*}U
  83. A = U P ; A=UP;\,
  84. Φ : C n × n C m × m \Phi:C^{n\times n}\rightarrow C^{m\times m}
  85. Φ ( A ) = i k V i A V i * \Phi(A)=\sum_{i}^{k}V_{i}AV_{i}^{*}
  86. M Φ = ( Φ ( E p q ) ) p q C n m × n m M_{\Phi}=(\Phi(E_{pq}))_{pq}\in C^{nm\times nm}
  87. ρ = i p i v i v i * \rho=\sum_{i}p_{i}v_{i}v_{i}^{*}
  88. { p i , v i } \{p_{i},v_{i}\}\,
  89. ρ = j a j a j * . \rho=\sum_{j}a_{j}a_{j}^{*}.
  90. j a j * a j = 1. \sum_{j}a_{j}^{*}a_{j}=1.
  91. p i = a i * a i , p_{i}=a_{i}^{*}a_{i},
  92. { p i , v i } \{p_{i},v_{i}\}\,

Stability_(probability).html

  1. X 1 + X 2 + + X n = d c n X + d n , X_{1}+X_{2}+\ldots+X_{n}\stackrel{d}{=}c_{n}X+d_{n},
  2. c n = n 1 / α c_{n}=n^{1/\alpha}\,
  3. 0 < α 2. 0<\alpha\leq 2.

Stability_radius.html

  1. p ^ \hat{p}
  2. P P
  3. p p
  4. P ( s ) P(s)
  5. ρ ^ ( p ^ ) := max { ρ 0 : p P ( s ) , p B ( ρ , p ^ ) } \hat{\rho}(\hat{p}):=\max\ \{\rho\geq 0:p\in P(s),\forall p\in B(\rho,\hat{p})\}
  6. B ( ρ , p ^ ) B(\rho,\hat{p})
  7. ρ \rho
  8. P P
  9. p ^ \hat{p}
  10. max { ρ 0 : p P ( s ) , p B ( ρ , p ^ ) } max ρ 0 min p B ( ρ , p ^ ) f ( ρ , p ) \max\ \{\rho\geq 0:p\in P(s),\forall p\in B(\rho,\hat{p})\}\equiv\max_{\rho% \geq 0}\min_{p\in B(\rho,\hat{p})}f(\rho,p)
  11. f ( ρ , p ) = { ρ , p P ( s ) - , p P ( s ) f(\rho,p)=\left\{\begin{array}[]{cc}\rho&,\ p\in P(s)\\ -\infty&,\ p\notin P(s)\end{array}\right.
  12. - -\infty
  13. max \max
  14. α ^ ( q , u ~ ) := max { α 0 : r c R ( q , u ) , u U ( α , u ~ ) } \hat{\alpha}(q,\tilde{u}):=\max\ \{\alpha\geq 0:r_{c}\leq R(q,u),\forall u\in U% (\alpha,\tilde{u})\}
  15. r c R ( q , u ) r_{c}\leq R(q,u)
  16. q q
  17. u u
  18. u ~ \tilde{u}
  19. u u
  20. U ( α , u ~ ) U(\alpha,\tilde{u})
  21. α \alpha
  22. u ~ \tilde{u}
  23. u ~ \tilde{u}
  24. ρ ^ ( q ) := min p P ( s ) d i s t ( p , p ^ ) \hat{\rho}(q):=\min_{p\notin P(s)}dist(p,\hat{p})
  25. d i s t ( p , p ^ ) dist(p,\hat{p})
  26. p P p\in P
  27. p ^ \hat{p}
  28. r ( f , D ) = inf g C { g : f + g S ( D ) } , r(f,D)=\inf_{g\in C}\{\|g\|:f+g\notin S(D)\},
  29. n + 1 \mathbb{C}^{n+1}
  30. r ( f , D ) = inf z D | f ( z ) | q ( z ) , r(f,D)=\inf_{z\in\partial D}\frac{|f(z)|}{\|q(z)\|},
  31. q ( z ) = ( 1 , z , , z n ) q(z)=(1,z,\ldots,z^{n})
  32. f ( z ) = z 8 - 9 / 10 f(z)=z^{8}-9/10
  33. g ( z ) = - 1 / 90 i = 0 8 z i g(z)=-1/90\sum_{i=0}^{8}z^{i}

Stable_homotopy_theory.html

  1. π k S \pi_{k}^{S}
  2. π * S \pi_{*}^{S}
  3. π 0 S \pi_{0}^{S}
  4. π * S \pi_{*}^{S}

Stable_map.html

  1. X X
  2. ω \omega
  3. g g
  4. n n
  5. A A
  6. X X
  7. ( ( C , j ) , f , ( x 1 , , x n ) ) ((C,j),f,(x_{1},\ldots,x_{n}))\,
  8. ( C , j ) (C,j)
  9. g g
  10. n n
  11. x 1 , , x n x_{1},\ldots,x_{n}
  12. f : C X f:C\to X\,
  13. ω \omega
  14. J J
  15. ν \nu
  16. ¯ j , J f := 1 2 ( d f + J d f j ) = ν . \bar{\partial}_{j,J}f:=\frac{1}{2}(df+J\circ df\circ j)=\nu.
  17. g g
  18. n n
  19. 2 - 2 g - n 2-2g-n
  20. C C
  21. C C
  22. ¯ j , J \bar{\partial}_{j,J}
  23. ω \omega
  24. J J
  25. ν \nu
  26. ( j , J , ν ) (j,J,\nu)
  27. g g
  28. n n
  29. A A
  30. M g , n J , ν ( X , A ) M_{g,n}^{J,\nu}(X,A)
  31. d := dim M g , n ( X , A ) = 2 c 1 X ( A ) + ( dim X - 6 ) ( 1 - g ) + 2 n . d:=\dim_{\mathbb{R}}M_{g,n}(X,A)=2c_{1}^{X}(A)+(\dim_{\mathbb{R}}X-6)(1-g)+2n.
  32. f f
  33. st ( C ) \mathrm{st}(C)
  34. C C
  35. g g
  36. n n
  37. M ¯ g , n J , ν ( X , A ) . \bar{M}_{g,n}^{J,\nu}(X,A).
  38. M ¯ g , n \bar{M}_{g,n}
  39. ω \omega
  40. B B
  41. ω ( B ) \omega(B)
  42. ω ( B ) 1 2 | d f | 2 , \omega(B)\leq\frac{1}{2}\int|df|^{2},
  43. M g , n J , ν ( X , A ) M ¯ g , n × X n , M_{g,n}^{J,\nu}(X,A)\to\bar{M}_{g,n}\times X^{n},
  44. ( ( C , j ) , f , ( x 1 , , x n ) ) ( st ( C , j ) , f ( x 1 ) , , f ( x n ) ) ((C,j),f,(x_{1},\ldots,x_{n}))\mapsto(\mathrm{st}(C,j),f(x_{1}),\ldots,f(x_{n}))
  45. G W g , n X , A H d ( M ¯ g , n × X n , ) . GW_{g,n}^{X,A}\in H_{d}(\bar{M}_{g,n}\times X^{n},\mathbb{Q}).
  46. ω \omega
  47. J J
  48. ν \nu
  49. X X
  50. g g
  51. n n
  52. A A
  53. ω \omega
  54. X X

Stable_polynomial.html

  1. Q ( z ) = ( z - 1 ) d P ( z + 1 z - 1 ) Q(z)=(z-1)^{d}P\left({{z+1}\over{z-1}}\right)
  2. z z + 1 z - 1 z\mapsto{{z+1}\over{z-1}}
  3. P ( 1 ) 0 P(1)\neq 0
  4. f ( z ) = a 0 + a 1 z + + a n z n f(z)=a_{0}+a_{1}z+\cdots+a_{n}z^{n}
  5. a n > a n - 1 > > a 0 > 0 , a_{n}>a_{n-1}>\cdots>a_{0}>0,
  6. 4 z 3 + 3 z 2 + 2 z + 1 4z^{3}+3z^{2}+2z+1
  7. z 10 z^{10}
  8. z 2 - z - 2 z^{2}-z-2
  9. z 2 + 3 z + 2 z^{2}+3z+2
  10. z 4 + z 3 + z 2 + z + 1 z^{4}+z^{3}+z^{2}+z+1
  11. z k = cos ( 2 π k 5 ) + i sin ( 2 π k 5 ) , k = 1 , , 4 . z_{k}=\cos\left({{2\pi k}\over 5}\right)+i\sin\left({{2\pi k}\over 5}\right),% \,k=1,\ldots,4\ .
  12. cos ( 2 π / 5 ) = 5 - 1 4 > 0. \cos({{2\pi}/5})={{\sqrt{5}-1}\over 4}>0.

Stagnation_point.html

  1. C p C_{p}
  2. C p = p - p q C_{p}={p-p_{\infty}\over q_{\infty}}
  3. C p C_{p}
  4. p p
  5. p p_{\infty}
  6. q q_{\infty}
  7. C p C_{p}

Stagnation_pressure.html

  1. P stagnation = 1 2 ρ v 2 + P static P\text{stagnation}=\tfrac{1}{2}\rho v^{2}+P\text{static}
  2. P stagnation P\text{stagnation}
  3. ρ \rho\;
  4. v v
  5. P static P\text{static}
  6. P total = 0 + P stagnation P\text{total}=0+P\text{stagnation}\;
  7. p t p = ( 1 + γ - 1 2 M 2 ) γ γ - 1 \frac{p_{t}}{p}=\left(1+\frac{\gamma-1}{2}M^{2}\right)^{\frac{\gamma}{\gamma-1% }}\,
  8. p t p = ( T t T ) γ γ - 1 \frac{p_{t}}{p}=\left(\frac{T_{t}}{T}\right)^{\frac{\gamma}{\gamma-1}}\,
  9. p t p_{t}
  10. p p
  11. T t T_{t}
  12. T T
  13. γ \gamma
  14. γ \gamma

Stagnation_temperature.html

  1. h 0 = h + V 2 2 h_{0}=h+\frac{V^{2}}{2}\,
  2. h 0 = h_{0}=\,
  3. h = h=\,
  4. V = V=\,
  5. h = C p T h=C_{p}T
  6. T 0 = T + V 2 2 C p T_{0}=T+\frac{V^{2}}{2C_{p}}\,
  7. T 0 T = 1 + γ - 1 2 M 2 \frac{T_{0}}{T}=1+\frac{\gamma-1}{2}M^{2}\,
  8. C p = C_{p}=\,
  9. T 0 = T_{0}=\,
  10. T = T=\,
  11. V = V=\,
  12. M = M=\,
  13. γ = \gamma=\,
  14. C p / C v C_{p}/C_{v}
  15. h 02 = h 01 + q h_{02}=h_{01}+q
  16. T 02 = T 01 + q C p T_{02}=T_{01}+\frac{q}{C_{p}}

Standard_cubic_feet_per_minute.html

  1. P 1 V 1 T 1 = P 2 V 2 T 2 \frac{P_{1}V_{1}}{T_{1}}=\frac{P_{2}V_{2}}{T_{2}}
  2. SCF = ACF ( P actual P standard ) ( T standard T actual ) {\rm SCF}={\rm ACF}\,\cdot\,\left(\frac{P_{\rm actual}}{P_{\rm standard}}% \right)\,\left(\frac{T_{\rm standard}}{T_{\rm actual}}\right)
  3. P P
  4. T T

Standard_normal_table.html

  1. Z = X - μ σ Z=\frac{X-\mu}{\sigma}
  2. t = X ¯ - μ S / n t=\frac{\overline{X}-\mu}{S/\sqrt{n}}
  3. P ( X 82 ) = P ( Z 82 - 80 5 ) = P ( Z 0.40 ) P(X\leq 82)=P\left(Z\leq\frac{82-80}{5}\right)=P(Z\leq 0.40)
  4. = 0.15542 + 0.5 = 0.65542 =0.15542+0.5=0.65542
  5. P ( X 90 ) = P ( Z 90 - 80 5 ) = P ( Z 2.00 ) P(X\geq 90)=P\left(Z\geq\frac{90-80}{5}\right)=P(Z\geq 2.00)
  6. = 1 - P ( Z 2.00 ) = 1 - ( 0.47725 + 0.5 ) = 0.02275 =1-P(Z\leq 2.00)=1-(0.47725+0.5)=0.02275
  7. P ( X 74 ) = P ( Z 74 - 80 5 ) = P ( Z - 1.20 ) P(X\leq 74)=P\left(Z\leq\frac{74-80}{5}\right)=P(Z\leq-1.20)
  8. P ( Z - 1.20 ) = P ( Z 1.20 ) P(Z\leq-1.20)=P(Z\geq 1.20)
  9. = 1 - ( 0.38493 + 0.5 ) = 0.11507 =1-(0.38493+0.5)=0.11507
  10. P ( 74 X 82 ) = P ( X 82 ) - P ( X 74 ) = 0.65542 - 0.11507 P(74\leq X\leq 82)=P(X\leq 82)-P(X\leq 74)=0.65542-0.11507
  11. = 0.54035 =0.54035
  12. P ( X 82 ) = P ( Z 82 - 80 5 / 3 ) P(X\leq 82)=P\left(Z\leq\frac{82-80}{5/\sqrt{3}}\right)
  13. = P ( Z 0.69 ) = 0.2549 + 0.5 = 0.7549 =P(Z\leq 0.69)=0.2549+0.5=0.7549

Standard_Reference_Method.html

  1. S R M = 12.7 × D × A 430 SRM=12.7\times D\times A_{430}
  2. D D
  3. D = 1 D=1
  4. D = 2 D=2
  5. A 430 A_{430}
  6. S R M = 1.3546 × L - 0.76 SRM=1.3546\times{{}^{\circ}L}-0.76
  7. EBC = SRM × 1.97 \mbox{EBC}~{}=\mbox{SRM}~{}\times 1.97
  8. SRM = EBC × .508 \mbox{SRM}~{}=\mbox{EBC}~{}\times.508
  9. H L H_{L}
  10. A 430 = A 530 × 10 H L / 10 A_{430}=A_{530}\times 10^{H_{L}/10}
  11. λ \lambda
  12. A ( λ ) = S R M 12.7 ( 0.018747 e - ( λ - 430 ) 13.374 + 0.98226 e - ( λ - 430 ) 80.514 ) A(\lambda)={SRM\over 12.7}(0.018747e^{-{(\lambda-430)\over 13.374}}+0.98226e^{% -{(\lambda-430)\over 80.514}})
  13. A ( λ ) A(\lambda)
  14. T ( λ ) = l o g - 1 ( - S R M 12.7 ( 0.018747 e - ( λ - 430 ) 13.374 + 0.98226 e - ( λ - 430 ) 80.514 ) ) T(\lambda)=log^{-1}(-{SRM\over 12.7}(0.018747e^{-{(\lambda-430)\over 13.374}}+% 0.98226e^{-{(\lambda-430)\over 80.514}}))
  15. T ( λ ) = l o g - 1 ( - S R M 12.7 ( 0.018747 e - ( λ - 430 ) 13.374 + 0.98226 e - ( λ - 430 ) 80.514 + c 1 ξ 1 + c 2 ξ 2 + ) ) T(\lambda)=log^{-1}(-{SRM\over 12.7}(0.018747e^{-{(\lambda-430)\over 13.374}}+% 0.98226e^{-{(\lambda-430)\over 80.514}}+c_{1}\xi_{1}+c_{2}\xi_{2}+...))
  16. ξ i \xi_{i}
  17. A ( λ ) A(\lambda)
  18. c 1 c_{1}
  19. c 2 c_{2}
  20. c i c_{i}

Standardized_Kt::V.html

  1. V d C d t = - K C + m ˙ ( 1 ) V\frac{dC}{dt}=-K\cdot C+\dot{m}\qquad(1)
  2. m ˙ \dot{m}
  3. d C d t \frac{dC}{dt}
  4. C = m ˙ K + ( C o - m ˙ K ) e - K t V ( 2 ) C=\frac{\dot{m}}{K}+\left(C_{o}-\frac{\dot{m}}{K}\right)e^{-\frac{K\cdot t}{V}% }\qquad(2)
  5. C = m ˙ K ( 3 a ) C_{\infty}=\frac{\dot{m}}{K}\qquad(3a)
  6. K = m ˙ C ( 3 b ) K=\frac{\dot{m}}{C_{\infty}}\qquad(3b)
  7. K = m ˙ C o ( 4 ) {K^{\prime}}=\frac{\dot{m}}{C_{o}}\qquad(4)
  8. m ˙ \dot{m}
  9. K V = m ˙ C o V ( 5 ) \frac{K^{\prime}}{V}=\frac{\dot{m}}{C_{o}\cdot V}\qquad(5)
  10. const K V = const m ˙ C o V ( 6 ) \mbox{const}~{}\cdot\frac{K^{\prime}}{V}=\mbox{const}~{}\cdot\frac{\dot{m}}{C_% {o}\cdot V}\qquad(6)
  11. std K t V = def const m ˙ C o V ( 7 ) \mbox{std}~{}\frac{K\cdot t}{V}\ \stackrel{\mathrm{def}}{=}\ \mbox{const}~{}% \cdot\frac{\dot{m}}{C_{o}\cdot V}\qquad(7)
  12. std K t V = def const m ˙ V 1 C o ( 8 ) \mbox{std}~{}\frac{K\cdot t}{V}\ \stackrel{\mathrm{def}}{=}\mbox{ const}~{}% \cdot\frac{\dot{m}}{V}\frac{1}{C_{o}}\qquad(8)
  13. [ s t d K t V ] - 1 C o m ˙ / V ( 9 ) \left[std\frac{K\cdot t}{V}\right]^{-1}\propto\frac{C_{o}}{\dot{m}/V}\qquad(9)
  14. K t V = ln C o C ( 10 ) \frac{K\cdot t}{V}=\ln\frac{C_{o}}{C}\qquad(10)
  15. m ˙ \dot{m}
  16. s t d K t / V = 10080 ( 1 - e - e K t / V ) t 1 - e - e K t V s p K t / V + 10080 N t - 1 stdKt/V=\frac{\frac{10080\cdot(1-e^{-eKt/V})}{t}}{\frac{1-e^{-eKtV}}{spKt/V}+% \frac{10080}{N\cdot t}-1}
  17. e k t / V = s p K t / V t t + C ekt/V=spKt/V\cdot\frac{t}{t+C}

Star_product.html

  1. ( P , P ) (P,\leq_{P})
  2. ( Q , Q ) (Q,\leq_{Q})
  3. P P
  4. 1 ^ \widehat{1}
  5. Q Q
  6. 0 ^ \widehat{0}
  7. P * Q P*Q
  8. ( P { 1 ^ } ) ( Q { 0 ^ } ) (P\setminus\{\widehat{1}\})\cup(Q\setminus\{\widehat{0}\})
  9. P * Q \leq_{P*Q}
  10. x y x\leq y
  11. { x , y } P \{x,y\}\subset P
  12. x P y x\leq_{P}y
  13. { x , y } Q \{x,y\}\subset Q
  14. x Q y x\leq_{Q}y
  15. x P x\in P
  16. y Q y\in Q
  17. P P
  18. Q Q
  19. P P
  20. Q Q
  21. P P
  22. Q Q
  23. P * Q P*Q
  24. f f
  25. 𝐜𝐝 \mathbf{cd}

Static_universe.html

  1. Λ E = 4 π G ρ / c 2 \Lambda_{E}=4\pi G\rho/c^{2}
  2. G G
  3. ρ \rho
  4. c c
  5. R E = Λ E - 1 / 2 = c 4 π G ρ . R_{E}=\Lambda_{E}^{-1/2}={c\over\sqrt{4\pi G\rho}}.
  6. ρ \rho
  7. Λ E \Lambda_{E}
  8. R E R_{E}

Stationary_spacetime.html

  1. g μ ν g_{\mu\nu}
  2. ( i , j = 1 , 2 , 3 ) (i,j=1,2,3)
  3. d s 2 = λ ( d t - ω i d y i ) 2 - λ - 1 h i j d y i d y j , ds^{2}=\lambda(dt-\omega_{i}\,dy^{i})^{2}-\lambda^{-1}h_{ij}\,dy^{i}\,dy^{j},
  4. t t
  5. y i y^{i}
  6. h i j h_{ij}
  7. ξ μ \xi^{\mu}
  8. ξ μ = ( 1 , 0 , 0 , 0 ) \xi^{\mu}=(1,0,0,0)
  9. λ \lambda
  10. λ = g μ ν ξ μ ξ ν \lambda=g_{\mu\nu}\xi^{\mu}\xi^{\nu}
  11. ω i \omega_{i}
  12. ω μ = e μ ν ρ σ ξ ν ρ ξ σ \omega_{\mu}=e_{\mu\nu\rho\sigma}\xi^{\nu}\nabla^{\rho}\xi^{\sigma}
  13. ξ μ \xi^{\mu}
  14. ω μ ξ μ = 0 \omega_{\mu}\xi^{\mu}=0
  15. G G
  16. M M
  17. V = M / G V=M/G
  18. V V
  19. M M
  20. π : M V \pi:M\rightarrow V
  21. M M
  22. V V
  23. h = - λ π * g h=-\lambda\pi*g
  24. V V
  25. λ \lambda
  26. ω i \omega_{i}
  27. h i j h_{ij}
  28. V V
  29. ω i = 0 \omega_{i}=0
  30. R μ ν = 0 R_{\mu\nu}=0
  31. ω μ \omega_{\mu}
  32. μ ω ν - ν ω μ = 0 , \nabla_{\mu}\omega_{\nu}-\nabla_{\nu}\omega_{\mu}=0,\,
  33. ω \omega
  34. ω μ = μ ω . \omega_{\mu}=\nabla_{\mu}\omega.\,
  35. λ \lambda
  36. ω \omega
  37. Φ M \Phi_{M}
  38. Φ J \Phi_{J}
  39. Φ M = 1 4 λ - 1 ( λ 2 + ω 2 - 1 ) , \Phi_{M}=\frac{1}{4}\lambda^{-1}(\lambda^{2}+\omega^{2}-1),
  40. Φ J = 1 2 λ - 1 ω . \Phi_{J}=\frac{1}{2}\lambda^{-1}\omega.
  41. Φ M \Phi_{M}
  42. Φ J \Phi_{J}
  43. Φ A \Phi_{A}
  44. A = M A=M
  45. J J
  46. h i j h_{ij}
  47. ( h i j i j - 2 R ( 3 ) ) Φ A = 0 , (h^{ij}\nabla_{i}\nabla_{j}-2R^{(3)})\Phi_{A}=0,\,
  48. R i j ( 3 ) = 2 [ i Φ A j Φ A - ( 1 + 4 Φ 2 ) - 1 i Φ 2 j Φ 2 ] , R^{(3)}_{ij}=2[\nabla_{i}\Phi_{A}\nabla_{j}\Phi_{A}-(1+4\Phi^{2})^{-1}\nabla_{% i}\Phi^{2}\nabla_{j}\Phi^{2}],
  49. Φ 2 = Φ A Φ A = ( Φ M 2 + Φ J 2 ) \Phi^{2}=\Phi_{A}\Phi_{A}=(\Phi_{M}^{2}+\Phi_{J}^{2})
  50. R i j ( 3 ) R^{(3)}_{ij}
  51. R ( 3 ) = h i j R i j ( 3 ) R^{(3)}=h^{ij}R^{(3)}_{ij}

Stationary_wavelet_transform.html

  1. 2 ( j - 1 ) 2^{(j-1)}
  2. j j

Steady_state_(chemistry).html

  1. U 239 N 239 p P 239 u {}^{239}U\rightarrow\;{}^{239}Np\rightarrow\;{}^{239}Pu\!
  2. k 1 k_{1}
  3. k 2 k_{2}
  4. A B C A\rightarrow\;B\rightarrow\;C
  5. d [ A ] d t = - k 1 [ A ] \frac{d[A]}{dt}=-k_{1}[A]
  6. d [ B ] d t = k 1 [ A ] - k 2 [ B ] \frac{d[B]}{dt}=k_{1}[A]-k_{2}[B]
  7. A B A\rightarrow\;B
  8. B C B\rightarrow\;C
  9. d [ C ] d t = k 2 [ B ] \frac{d[C]}{dt}=k_{2}[B]
  10. [ A ] = [ A ] 0 e - k 1 t [A]=[A]_{0}e^{-k_{1}t}
  11. [ B ] = { [ A ] 0 k 1 k 2 - k 1 ( e - k 1 t - e - k 2 t ) ; k 1 k 2 [ A ] 0 k 1 t e - k 1 t otherwise \left[B\right]=\left\{\begin{matrix}\left[A\right]_{0}\frac{k_{1}}{k_{2}-k_{1}% }\left(e^{-k_{1}t}-e^{-k_{2}t}\right);\,\,k_{1}\neq k_{2}\\ \\ \left[A\right]_{0}k_{1}te^{-k_{1}t}\;\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,% \,\,\,\,\,\,\,\,\,\,\text{otherwise}\\ \end{matrix}\right.
  12. [ C ] = { [ A ] 0 ( 1 + k 1 e - k 2 t - k 2 e - k 1 t k 2 - k 1 ) ; k 1 k 2 [ A ] 0 ( 1 - e - k 1 t - k 1 t e - k 1 t ) ; otherwise \left[C\right]=\left\{\begin{matrix}\left[A\right]_{0}\left(1+\frac{k_{1}e^{-k% _{2}t}-k_{2}e^{-k_{1}t}}{k_{2}-k_{1}}\right);\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,k_{% 1}\neq k_{2}\\ \\ \left[A\right]_{0}\left(1-e^{-k_{1}t}-k_{1}te^{-k_{1}t}\right);\,\,\,\,\,\,\,% \text{otherwise}\\ \end{matrix}\right.
  13. d [ B ] d t = 0 = k 1 [ A ] - k 2 [ B ] [ B ] = k 1 k 2 [ A ] \frac{d[B]}{dt}=0=k_{1}[A]-k_{2}[B]\Rightarrow\;[B]=\frac{k_{1}}{k_{2}}[A]
  14. d [ C ] d t = k 1 [ A ] \frac{d[C]}{dt}=k_{1}[A]
  15. [ C ] = [ A ] 0 ( 1 - e - k 1 t ) [C]=[A]_{0}\left(1-e^{-k_{1}t}\right)
  16. k 2 k 1 k_{2}\gg k_{1}
  17. e - k 2 t e - k 1 t e^{-k_{2}t}\ll e^{-k_{1}t}
  18. k 2 - k 1 k 2 k_{2}-k_{1}\approx\;k_{2}
  19. t = { ln ( k 1 k 2 ) k 1 - k 2 k 1 k 2 1 k 1 otherwise t=\left\{\begin{matrix}\frac{\ln\left(\frac{k_{1}}{k_{2}}\right)}{k_{1}-k_{2}}% &\,k_{1}\neq k_{2}\\ \\ \frac{1}{k_{1}}&\,\,\text{otherwise}\\ \end{matrix}\right.

Steenrod_algebra.html

  1. H n ( X ; R ) H 2 n ( X ; R ) H^{n}(X;R)\to H^{2n}(X;R)
  2. x x x . x\mapsto x\smile x.
  3. S q i : H n ( X ; 𝐙 / 2 ) H n + i ( X ; 𝐙 / 2 ) Sq^{i}\colon H^{n}(X;\mathbf{Z}/2)\to H^{n+i}(X;\mathbf{Z}/2)
  4. P i : H n ( X ; 𝐙 / p ) H n + 2 i ( p - 1 ) ( X ; 𝐙 / p ) P^{i}\colon H^{n}(X;\mathbf{Z}/p)\to H^{n+2i(p-1)}(X;\mathbf{Z}/p)
  5. 0 𝐙 / p 𝐙 / p 2 𝐙 / p 0. 0\to\mathbf{Z}/p\to\mathbf{Z}/p^{2}\to\mathbf{Z}/p\to 0.
  6. S q n ( x y ) = i + j = n ( S q i x ) ( S q j y ) Sq^{n}(x\smile y)=\sum_{i+j=n}(Sq^{i}x)\smile(Sq^{j}y)
  7. 0 𝐙 / 2 𝐙 / 4 𝐙 / 2 0. 0\to\mathbf{Z}/2\to\mathbf{Z}/4\to\mathbf{Z}/2\to 0.
  8. P n ( x y ) = i + j = n ( P i x ) ( P j y ) P^{n}(x\smile y)=\sum_{i+j=n}(P^{i}x)\smile(P^{j}y)
  9. S q i S q j = k = 0 [ i / 2 ] ( j - k - 1 i - 2 k ) S q i + j - k S q k Sq^{i}Sq^{j}=\sum_{k=0}^{[i/2]}{j-k-1\choose i-2k}Sq^{i+j-k}Sq^{k}
  10. P ( t ) = i 0 t i Sq i P(t)=\sum_{i\geq 0}t^{i}\,\text{Sq}^{i}
  11. P ( s 2 + s t ) P ( t 2 ) = P ( t 2 + s t ) P ( s 2 ) P(s^{2}+st)\cdot P(t^{2})=P(t^{2}+st)\cdot P(s^{2})
  12. P ( t ) = i 0 t i P i P(t)=\sum_{i\geq 0}t^{i}\,\text{P}^{i}
  13. ( 1 + s Ad β ) P ( t p + t p - 1 s + + t s p - 1 ) P ( s p ) (1+s\,\text{Ad}\beta)P(t^{p}+t^{p-1}s+\cdots+ts^{p-1})P(s^{p})
  14. i 1 , i 2 , , i n i_{1},i_{2},\ldots,i_{n}
  15. S q I = S q i 1 S q i n , Sq^{I}=Sq^{i_{1}}\ldots Sq^{i_{n}},
  16. S q p I = S q p i 1 S q p i n , Sq_{p}^{I}=Sq_{p}^{i_{1}}\ldots Sq_{p}^{i_{n}},
  17. i j p i j + 1 i_{j}\geq pi_{j+1}
  18. i j 0 , 1 mod 2 ( p - 1 ) i_{j}\equiv 0,1\bmod 2(p-1)
  19. S q p 2 k ( p - 1 ) = P k Sq_{p}^{2k(p-1)}=P^{k}
  20. S q p 2 k ( p - 1 ) + 1 = β P k Sq_{p}^{2k(p-1)+1}=\beta P^{k}
  21. ψ : A A A . \psi\colon A\to A\otimes A.
  22. ψ ( S q k ) = i + j = k S q i S q j \psi(Sq^{k})=\sum_{i+j=k}Sq^{i}\otimes Sq^{j}
  23. ψ ( P k ) = i + j = k P i P j \psi(P^{k})=\sum_{i+j=k}P^{i}\otimes P^{j}
  24. ψ ( β ) = β 1 + 1 β . \psi(\beta)=\beta\otimes 1+1\otimes\beta.
  25. ψ ( ξ n ) = i = 0 n ξ n - i p i ξ i . \psi(\xi_{n})=\sum_{i=0}^{n}\xi_{n-i}^{p^{i}}\otimes\xi_{i}.
  26. ψ ( τ n ) = τ n 1 + i = 0 n ξ n - i p i τ i \psi(\tau_{n})=\tau_{n}\otimes 1+\sum_{i=0}^{n}\xi_{n-i}^{p^{i}}\otimes\tau_{i}
  27. ξ 1 2 i \xi_{1}^{2^{i}}
  28. S q 2 i Sq^{2^{i}}
  29. x x + ξ 1 x 2 + ξ 2 x 4 + ξ 3 x 8 + x\rightarrow x+\xi_{1}x^{2}+\xi_{2}x^{4}+\xi_{3}x^{8}+\cdots
  30. P ( x ) : S V [ [ x ] ] S V [ [ x ] ] P(x):SV[[x]]\rightarrow SV[[x]]
  31. P ( x ) ( v ) = v + F ( v ) x = v + v q x \displaystyle P(x)(v)=v+F(v)x=v+v^{q}x
  32. P ( x ) ( f ) = P i ( f ) x i P(x)(f)=\sum P^{i}(f)x^{i}
  33. P ( x ) ( f ) = S q 2 i ( f ) x i P(x)(f)=\sum Sq^{2i}(f)x^{i}
  34. Ext A s , t ( 𝐅 p , 𝐅 p ) . \mathrm{Ext}^{s,t}_{A}(\mathbf{F}_{p},\mathbf{F}_{p}).

Steinhart–Hart_equation.html

  1. 1 T = A + B ln ( R ) + C [ ln ( R ) ] 3 , {1\over T}=A+B\ln(R)+C[\ln(R)]^{3},
  2. T T
  3. R R
  4. A A
  5. B B
  6. C C
  7. [ ln ( R ) ] 2 [\ln(R)]^{2}
  8. R = exp ( y - x 2 3 - y + x 2 3 ) , R=\exp\left(\sqrt[3]{y-{x\over 2}}-\sqrt[3]{y+{x\over 2}}\right),
  9. x = 1 C ( A - 1 T ) , y = ( B 3 C ) 3 + ( x 2 ) 2 . \begin{aligned}\displaystyle x&\displaystyle=\frac{1}{C}\left(A-\frac{1}{T}% \right),\\ \displaystyle y&\displaystyle=\sqrt{\left({B\over 3C}\right)^{3}+\left(\frac{x% }{2}\right)^{2}}.\end{aligned}
  10. [ 1 ln ( R 1 ) ln ( R 1 ) 3 1 ln ( R 2 ) ln ( R 2 ) 3 1 ln ( R 3 ) ln ( R 3 ) 3 ] [ A B C ] = [ 1 T 1 1 T 2 1 T 3 ] \begin{bmatrix}1&\ln\left(R_{1}\right)&\ln\left(R_{1}\right)^{3}\\ 1&\ln\left(R_{2}\right)&\ln\left(R_{2}\right)^{3}\\ 1&\ln\left(R_{3}\right)&\ln\left(R_{3}\right)^{3}\end{bmatrix}\begin{bmatrix}A% \\ B\\ C\end{bmatrix}=\begin{bmatrix}\frac{1}{T_{1}}\\ \frac{1}{T_{2}}\\ \frac{1}{T_{3}}\end{bmatrix}
  11. R 1 R_{1}
  12. R 2 R_{2}
  13. R 3 R_{3}
  14. T 1 T_{1}
  15. T 2 T_{2}
  16. T 3 T_{3}
  17. A A
  18. B B
  19. C C
  20. L 1 \displaystyle L_{1}

Stepper.html

  1. CD = k λ NA \mathrm{CD}=k\frac{\lambda}{\mathrm{NA}}
  2. CD \mathrm{CD}
  3. k k
  4. λ \lambda
  5. NA \mathrm{NA}
  6. k k

Sterile_neutrino.html

  1. ϕ \phi
  2. ν \nu
  3. ( ψ ) = ψ ¯ ( i / ) ψ - G ψ ¯ L ϕ ψ R \mathcal{L}(\psi)=\bar{\psi}(i\partial\!\!\!/)\psi-G\bar{\psi}_{L}\phi\psi_{R}
  4. 2 / 3 {2}/{3}
  5. 2 / 3 {2}/{3}
  6. m ν = ( 0 m D m D M N H L ) m_{\nu}=\begin{pmatrix}0&m_{D}\\ m_{D}&M_{NHL}\end{pmatrix}
  7. M N H L M_{NHL}
  8. m D m_{D}
  9. m ν m D M N H L m_{\nu}\ll m_{D}\ll M_{NHL}
  10. m ν m D 2 M N H L m_{\nu}\approx\frac{m_{D}^{2}}{M_{NHL}}

Stern–Brocot_tree.html

  1. q = a 0 + 1 a 1 + 1 a 2 + 1 a 3 + 1 + 1 a k = [ a 0 ; a 1 , a 2 , , a k ] q=a_{0}+\cfrac{1}{a_{1}+\cfrac{1}{a_{2}+\cfrac{1}{a_{3}+\cfrac{1}{\ddots+% \cfrac{1}{a_{k}}}}}}=[a_{0};a_{1},a_{2},\ldots,a_{k}]
  2. [ a 0 ; a 1 , a 2 , , a k - 1 , 1 ] = [ a 0 ; a 1 , a 2 , , a k - 1 + 1 ] [a_{0};a_{1},a_{2},\ldots,a_{k-1},1]=[a_{0};a_{1},a_{2},\ldots,a_{k-1}+1]
  3. [ a 0 ; a 1 , a 2 , , a k - 1 ] , [a_{0};a_{1},a_{2},\ldots,a_{k}-1],
  4. [ a 0 ; a 1 , a 2 , , a k - 1 + 1 ] [a_{0};a_{1},a_{2},\ldots,a_{k-1}+1]
  5. 23 16 = 1 + 1 2 + 1 3 + 1 2 = [ 1 ; 2 , 3 , 2 ] , \frac{23}{16}=1+\cfrac{1}{2+\cfrac{1}{3+\frac{1}{2}}}=[1;2,3,2],
  6. [ 1 ; 2 , 3 , 1 ] = [ 1 ; 2 , 4 ] = 1 + 1 2 + 1 4 = 13 9 . [1;2,3,1]=[1;2,4]=1+\cfrac{1}{2+\frac{1}{4}}=\frac{13}{9}.
  7. 1 1 \tfrac{1}{1}
  8. q = [ a 0 ; a 1 , a 2 , , a k ] = [ a 0 ; a 1 , a 2 , , a k - 1 , 1 ] q=[a_{0};a_{1},a_{2},\ldots,a_{k}]=[a_{0};a_{1},a_{2},\ldots,a_{k}-1,1]
  9. [ a 0 ; a 1 , a 2 , , a k + 1 ] \displaystyle[a_{0};a_{1},a_{2},\ldots,a_{k}+1]
  10. [ a 0 ; a 1 , a 2 , , a k - 1 , 2 ] . [a_{0};a_{1},a_{2},\ldots,a_{k}-1,2].
  11. p 1 q 1 , p 2 q 2 , , p n q n {\frac{p_{1}}{q_{1}},\frac{p_{2}}{q_{2}},\dots,\frac{p_{n}}{q_{n}}}
  12. k = 1 n 1 p k q k = 1. \sum_{k=1}^{n}\frac{1}{p_{k}q_{k}}=1.

Steven_Anson_Coons.html

  1. z = f ( d ) where d = x - origin range z=f(d)\,\text{ where }d=\frac{x-\,\text{origin}}{\,\text{range}}
  2. z = a 0 + a d + a 2 d 2 + + a 7 d 7 z=a_{0}+ad+a_{2}d^{2}+\cdots+a_{7}d^{7}
  3. c = f ( Φ , u , w , θ ) all mapped into the unit square. c=f(\Phi,u,w,\theta)\,\text{ all mapped into the unit square.}\,
  4. Φ u ( w - 1 ) + ( w - u ) 2 = 0 \Phi u(w-1)+(w-u)^{2}=0\,
  5. u = 1 Φ + θ + ( θ + 1 ) , w = 1 - θ ( u ) u=\frac{1}{\sqrt{\Phi+\sqrt{\theta+(\theta+1)}}},\,w=1-\theta(u)

Stickelberger's_theorem.html

  1. ζ m a {ζ}_{m}^{a}
  2. θ ( K m ) = 1 m a = 1 m ( a , m ) = 1 a σ a - 1 𝐐 [ G m ] . \theta(K_{m})=\frac{1}{m}\underset{(a,m)=1}{\sum_{a=1}^{m}}a\cdot\sigma_{a}^{-% 1}\in\mathbf{Q}[G_{m}].
  3. I ( K m ) = θ ( K m ) 𝐙 [ G m ] 𝐙 [ G m ] . I(K_{m})=\theta(K_{m})\mathbf{Z}[G_{m}]\cap\mathbf{Z}[G_{m}].
  4. θ ( F ) = 1 m a = 1 m ( a , m ) = 1 a res m σ a - 1 𝐐 [ G F ] . \theta(F)=\frac{1}{m}\underset{(a,m)=1}{\sum_{a=1}^{m}}a\cdot\mathrm{res}_{m}% \sigma_{a}^{-1}\in\mathbf{Q}[G_{F}].
  5. I ( F ) = θ ( F ) 𝐙 [ G F ] 𝐙 [ G F ] . I(F)=\theta(F)\mathbf{Z}[G_{F}]\cap\mathbf{Z}[G_{F}].
  6. θ ( F ) = ϕ ( m ) 2 [ F : 𝐐 ] σ G F σ , \theta(F)=\frac{\phi(m)}{2[F:\mathbf{Q}]}\sum_{\sigma\in G_{F}}\sigma,
  7. α θ ( F ) = σ G F a σ σ 𝐙 [ G F ] \alpha\theta(F)=\sum_{\sigma\in G_{F}}a_{\sigma}\sigma\in\mathbf{Z}[G_{F}]
  8. σ G F σ ( J a σ ) \prod_{\sigma\in G_{F}}\sigma(J^{a_{\sigma}})

Stiff_equation.html

  1. y ( t ) = e - 15 t y(t)=e^{-15t}\,
  2. y ( t ) 0 y(t)\to 0
  3. t . t\to\infty.
  4. t [ 0 , 40 ] t\in[0,40]
  5. y , f n y,f\in\mathbb{R}^{n}
  6. A A
  7. n × n n\times n
  8. λ t , t = 1 , 2 , , n \lambda_{t}\in\mathbb{C},t=1,2,\ldots,n
  9. c t n , t = 1 , 2 , , n c_{t}\in\mathbb{C}^{n},t=1,2,\ldots,n
  10. y ( x ) = t = 1 n κ t exp ( λ t x ) c t + g ( x ) , y(x)=\sum_{t=1}^{n}\kappa_{t}\exp(\lambda_{t}x)c_{t}+g(x),
  11. g ( x ) g(x)
  12. exp ( λ t x ) c t 0 \exp(\lambda_{t}x)c_{t}\rightarrow 0
  13. x x\rightarrow\infty
  14. y ( x ) y(x)
  15. g ( x ) g(x)
  16. x x\rightarrow\infty
  17. exp ( λ t x ) c t \exp(\lambda_{t}x)c_{t}
  18. Σ t = 1 n κ t exp ( λ t x ) c t \Sigma_{t=1}^{n}\kappa_{t}\exp(\lambda_{t}x)c_{t}
  19. g ( x ) g(x)
  20. | R e ( λ t ) | |Re(\lambda_{t})|
  21. κ t exp ( λ t x ) c t \kappa_{t}\exp(\lambda_{t}x)c_{t}
  22. | R e ( λ t ) | |Re(\lambda_{t})|
  23. κ t exp ( λ t x ) c t \kappa_{t}\exp(\lambda_{t}x)c_{t}
  24. λ ¯ , λ ¯ { λ t , t = 1 , 2 , , n } \overline{\lambda},\underline{\lambda}\in\{\lambda_{t},t=1,2,\ldots,n\}
  25. | R e ( λ ¯ ) | | R e ( λ t ) | | R e ( λ ¯ ) | , t = 1 , 2 , , n |Re(\overline{\lambda})|\geq|Re(\lambda_{t})|\geq|Re(\underline{\lambda})|,% \qquad t=1,2,\ldots,n
  26. κ t exp ( λ ¯ x ) c t \kappa_{t}\exp(\overline{\lambda}x)c_{t}
  27. κ t exp ( λ ¯ x ) c t \kappa_{t}\exp(\underline{\lambda}x)c_{t}
  28. A = ( 0 1 - 1000 - 1001 ) , A=\left(\begin{array}[]{rr}0&1\\ -1000&-1001\end{array}\right),
  29. f ( t ) = ( 0 0 ) , f(t)=\left(\begin{array}[]{c}0\\ 0\end{array}\right),
  30. x ( 0 ) = ( x 0 0 ) , x(0)=\left(\begin{array}[]{c}x_{0}\\ 0\end{array}\right),
  31. λ ¯ = - 1000 , λ ¯ = - 1 \overline{\lambda}=-1000,\underline{\lambda}=-1
  32. | - 1000 | | - 1 | = 1000 , \frac{|-1000|}{|-1|}=1000,
  33. x ( t ) = x 0 ( - 1 999 e - 1000 t + 1000 999 e - t ) x 0 e - t . x(t)=x_{0}\left(-\frac{1}{999}e^{-1000t}+\frac{1000}{999}e^{-t}\right)\approx x% _{0}e^{-t}.
  34. y = k y y=ky
  35. y ( 0 ) = 1 y(0)=1
  36. k k\in\mathbb{C}
  37. t t\to\infty
  38. Re ( k ) < 0. \mathrm{Re}\,(k)<0.
  39. y = k y y^{\prime}=k\cdot y
  40. y n + 1 = ϕ ( h k ) y n y_{n+1}=\phi(hk)\cdot y_{n}
  41. y n = ( ϕ ( h k ) ) n y 0 y_{n}=\left(\phi(hk)\right)^{n}\cdot y_{0}
  42. ϕ \phi
  43. y n 0 y_{n}\to 0
  44. n n\to\infty
  45. | ϕ ( h k ) | < 1 |\phi(hk)|<1
  46. { z | | ϕ ( z ) | < 1 } \{z\in\mathbb{C}|\,|\phi(z)|<1\}
  47. { z \C | Re ( z ) < 0 } \{z\in\C|\mathrm{Re}(z)<0\}
  48. y = k y y^{\prime}=k\cdot y
  49. y n + 1 = y n + h f ( t n , y n ) = y n + h ( k y n ) = y n + h k y n = ( 1 + h k ) y n . y_{n+1}=y_{n}+h\cdot f(t_{n},y_{n})=y_{n}+h\cdot(ky_{n})=y_{n}+h\cdot k\cdot y% _{n}=(1+h\cdot k)y_{n}.
  50. y n = ( 1 + h k ) n y 0 y_{n}=(1+hk)^{n}\cdot y_{0}
  51. ϕ ( z ) = 1 + z \phi(z)=1+z
  52. { z | | 1 + z | < 1 } \{z\in\mathbb{C}||1+z|<1\}
  53. k = - 15 k=-15
  54. h = 1 / 4 h=1/4
  55. z = - 15 * 1 / 4 = - 3.75 z=-15*1/4=-3.75
  56. h = 1 / 8 h=1/8
  57. z = - 1.875 z=-1.875
  58. y n + 1 = y n + 1 2 h ( f ( t n , y n ) + f ( t n + 1 , y n + 1 ) ) , y_{n+1}=y_{n}+\tfrac{1}{2}h\cdot\left(f(t_{n},y_{n})+f(t_{n+1},y_{n+1})\right),
  59. y = k y y^{\prime}=k\cdot y
  60. y n + 1 = y n + 1 2 h ( k y n + k y n + 1 ) . y_{n+1}=y_{n}+\tfrac{1}{2}h\cdot\left(ky_{n}+ky_{n+1}\right).
  61. y n + 1 y_{n+1}
  62. y n + 1 = 1 + 1 2 h k 1 - 1 2 h k y n . y_{n+1}=\frac{1+\frac{1}{2}hk}{1-\frac{1}{2}hk}\cdot y_{n}.
  63. ϕ ( z ) = 1 + 1 2 z 1 - 1 2 z \phi(z)={1+{1\over 2}z\over 1-{1\over 2}z}
  64. { z | | 1 + 1 2 z 1 - 1 2 z | < 1 } . \left\{z\in\mathbb{C}\left|\ \left|{1+{1\over 2}z\over 1-{1\over 2}z}\right|<1% \right.\right\}.
  65. y = k y y^{\prime}=k\cdot y
  66. ϕ ( z ) 1 \phi(z)\to 1
  67. z - z\to-\infty
  68. | ϕ ( z ) | 0 |\phi(z)|\to 0
  69. z z\to\infty
  70. A A
  71. b b
  72. ϕ ( z ) = det ( I - z A + z e b T ) det ( I - z A ) , \phi(z)=\frac{\det(I-zA+zeb^{T})}{\det(I-zA)},
  73. e e
  74. A A
  75. ϕ \phi
  76. { z \C | | ϕ ( z ) | > | e z | } \{z\in\C||\phi(z)|>|\mathrm{e}^{z}|\}
  77. y n + 1 = i = 0 s a i y n - i + h j = - 1 s b j f ( t n - j , y n - j ) . y_{n+1}=\sum_{i=0}^{s}a_{i}y_{n-i}+h\sum_{j=-1}^{s}b_{j}f(t_{n-j},y_{n-j}).
  78. y n + 1 = i = 0 s a i y n - i + h k j = - 1 s b j y n - j , y_{n+1}=\sum_{i=0}^{s}a_{i}y_{n-i}+hk\sum_{j=-1}^{s}b_{j}y_{n-j},
  79. ( 1 - b - 1 z ) y n + 1 - j = 0 s ( a j + b j z ) y n - j = 0 (1-b_{-1}z)y_{n+1}-\sum_{j=0}^{s}(a_{j}+b_{j}z)y_{n-j}=0
  80. z z\in\mathbb{C}
  81. Φ ( w , z ) = w 2 - ( 1 + 3 2 z ) w + 1 2 z = 0 \Phi(w,z)=w^{2}-(1+\tfrac{3}{2}z)w+\tfrac{1}{2}z=0
  82. w = 1 2 ( 1 + 3 2 z ± 1 + z + 9 4 z 2 ) , w=\tfrac{1}{2}\Big(1+\tfrac{3}{2}z\pm\sqrt{1+z+\tfrac{9}{4}z^{2}}\Big),
  83. { z | | 1 2 ( 1 + 3 2 z ± 1 + z + 9 4 z 2 ) | < 1 } . \left\{z\in\mathbb{C}\left|\ \left|\tfrac{1}{2}\Big(1+\tfrac{3}{2}z\pm\sqrt{1+% z+\tfrac{9}{4}z^{2}}\Big)\right|<1\right.\right\}.

Stirling_numbers_of_the_first_kind.html

  1. c ( n , k ) , c(n,k),
  2. | s ( n , k ) | |s(n,k)|
  3. [ n k ] \left[{n\atop k}\right]
  4. 3 ! = 6 3!=6
  5. 123 123
  6. ( 1 ) ( 2 ) ( 3 ) (1)(2)(3)
  7. 132 = ( 1 ) ( 23 ) 132=(1)(23)
  8. 213 = ( 3 ) ( 12 ) 213=(3)(12)
  9. 321 = ( 2 ) ( 13 ) 321=(2)(13)
  10. 312 = ( 132 ) 312=(132)
  11. 231 = ( 123 ) 231=(123)
  12. [ 3 3 ] = 1 \left[{3\atop 3}\right]=1
  13. [ 3 2 ] = 3 \left[{3\atop 2}\right]=3
  14. [ 3 1 ] = 2 \left[{3\atop 1}\right]=2
  15. [ 4 2 ] = 11 \left[{4\atop 2}\right]=11
  16. ( ) ( ) (\bullet\bullet)(\bullet\bullet)
  17. ( ) ( ) (\bullet\bullet\bullet)(\bullet)
  18. ( x ) ( n ) = x ( x + 1 ) ( x + n - 1 ) = k = 0 n [ n k ] x k (x)^{(n)}=x(x+1)\cdots(x+n-1)=\sum_{k=0}^{n}\left[{n\atop k}\right]x^{k}
  19. ( x ) ( 3 ) = x ( x + 1 ) ( x + 2 ) = 1 x 3 + 3 x 2 + 2 x (x)^{(3)}=x(x+1)(x+2)=1\cdot x^{3}+3\cdot x^{2}+2\cdot x
  20. s ( n , k ) = ( - 1 ) n - k [ n k ] . s(n,k)=(-1)^{n-k}\left[{n\atop k}\right].
  21. ( x ) n = k = 0 n s ( n , k ) x k , (x)_{n}=\sum_{k=0}^{n}s(n,k)x^{k},
  22. ( x ) n (x)_{n}
  23. ( x ) n = x ( x - 1 ) ( x - 2 ) ( x - n + 1 ) . (x)_{n}=x(x-1)(x-2)\cdots(x-n+1).
  24. [ n k ] = | s ( n , k ) | . \left[{n\atop k}\right]=\left|s(n,k)\right|.
  25. [ n + 1 k ] = n [ n k ] + [ n k - 1 ] \left[{n+1\atop k}\right]=n\left[{n\atop k}\right]+\left[{n\atop k-1}\right]
  26. k > 0 k>0
  27. [ 0 0 ] = 1 and [ n 0 ] = [ 0 n ] = 0 \left[{0\atop 0}\right]=1\quad\mbox{and}~{}\quad\left[{n\atop 0}\right]=\left[% {0\atop n}\right]=0
  28. s ( n + 1 , k ) = - n s ( n , k ) + s ( n , k - 1 ) s(n+1,k)=-ns(n,k)+s(n,k-1)
  29. ( x ) ( n + 1 ) = x ( x + 1 ) ( x + n - 1 ) ( x + n ) = n ( x ) ( n ) + x ( x ) ( n ) . (x)^{(n+1)}=x(x+1)\cdots(x+n-1)(x+n)=n(x)^{(n)}+x(x)^{(n)}.
  30. [ n + 1 k ] \left[{n+1\atop k}\right]
  31. n [ n k ] n\cdot\left[{n\atop k}\right]
  32. [ n k - 1 ] \left[{n\atop k-1}\right]
  33. [ n k - 1 ] \left[{n\atop k-1}\right]
  34. ( a 1 a j 1 ) ( a j 1 + 1 a j 2 ) ( a j k - 1 + 1 a n ) k cycles . \displaystyle\underbrace{(a_{1}\ldots a_{j_{1}})(a_{j_{1}+1}\ldots a_{j_{2}})% \ldots(a_{j_{k-1}+1}\ldots a_{n})}_{k\ \mathrm{cycles}}.
  35. n [ n k ] n\left[{n\atop k}\right]
  36. [ 0 0 ] = 1 \left[{0\atop 0}\right]=1
  37. [ n 0 ] = 0 \left[{n\atop 0}\right]=0
  38. [ 0 k ] = 0 \left[{0\atop k}\right]=0
  39. [ n k ] = 0 \left[{n\atop k}\right]=0
  40. [ n 1 ] = ( n - 1 ) ! , [ n n ] = 1 , [ n n - 1 ] = ( n 2 ) , \left[{n\atop 1}\right]=(n-1)!,\quad\left[{n\atop n}\right]=1,\quad\left[{n% \atop n-1}\right]={n\choose 2},
  41. [ n n - 2 ] = 1 4 ( 3 n - 1 ) ( n 3 ) and [ n n - 3 ] = ( n 2 ) ( n 4 ) . \left[{n\atop n-2}\right]=\frac{1}{4}(3n-1){n\choose 3}\quad\mbox{ and }~{}% \quad\left[{n\atop n-3}\right]={n\choose 2}{n\choose 4}.
  42. ( n 6 ) ( 6 2 , 2 , 2 ) 1 6 {n\choose 6}{6\choose 2,2,2}\frac{1}{6}
  43. ( n 5 ) ( 5 3 ) × 2 {n\choose 5}{5\choose 3}\times 2
  44. ( n 4 ) × 6. {n\choose 4}\times 6.
  45. ( n 6 ) ( 6 2 , 2 , 2 ) 1 6 + ( n 5 ) ( 5 3 ) × 2 + ( n 4 ) × 6 = ( n 2 ) ( n 4 ) . {n\choose 6}{6\choose 2,2,2}\frac{1}{6}+{n\choose 5}{5\choose 3}\times 2+{n% \choose 4}\times 6={n\choose 2}{n\choose 4}.
  46. [ n 2 ] = ( n - 1 ) ! H n - 1 , \left[{n\atop 2}\right]=(n-1)!\;H_{n-1},
  47. [ n 3 ] = 1 2 ( n - 1 ) ! [ ( H n - 1 ) 2 - H n - 1 ( 2 ) ] \left[{n\atop 3}\right]=\frac{1}{2}(n-1)!\left[(H_{n-1})^{2}-H_{n-1}^{(2)}\right]
  48. [ n 4 ] = 1 3 ! ( n - 1 ) ! [ ( H n - 1 ) 3 - 3 H n - 1 H n - 1 ( 2 ) + 2 H n - 1 ( 3 ) ] \left[{n\atop 4}\right]=\frac{1}{3!}(n-1)!\left[(H_{n-1})^{3}-3H_{n-1}H_{n-1}^% {(2)}+2H_{n-1}^{(3)}\right]
  49. H ( z , u ) = ( 1 + z ) u = n = 0 ( u n ) z n = n = 0 z n n ! k = 0 n s ( n , k ) u k = k = 0 u k n = k z n n ! s ( n , k ) . H(z,u)=(1+z)^{u}=\sum_{n=0}^{\infty}{u\choose n}z^{n}=\sum_{n=0}^{\infty}\frac% {z^{n}}{n!}\sum_{k=0}^{n}s(n,k)u^{k}=\sum_{k=0}^{\infty}u^{k}\sum_{n=k}^{% \infty}\frac{z^{n}}{n!}s(n,k).
  50. ( 1 + z ) u = e u log ( 1 + z ) = k = 0 ( log ( 1 + z ) ) k u k k ! , (1+z)^{u}=e^{u\log(1+z)}=\sum_{k=0}^{\infty}(\log(1+z))^{k}\frac{u^{k}}{k!},
  51. n = k ( - 1 ) n - k [ n k ] z n n ! = ( log ( 1 + z ) ) k k ! . \sum_{n=k}^{\infty}(-1)^{n-k}\left[{n\atop k}\right]\frac{z^{n}}{n!}=\frac{% \left(\log(1+z)\right)^{k}}{k!}.
  52. n = i [ n i ] n ( n ! ) = ζ ( i + 1 ) \sum_{n=i}^{\infty}\frac{\left[{n\atop i}\right]}{n(n!)}=\zeta(i+1)
  53. ζ ( k ) \zeta(k)
  54. k = 0 n [ n k ] = n ! \sum_{k=0}^{n}\left[{n\atop k}\right]=n!
  55. k = 0 a [ n k ] = n ! - k = 0 n [ n k + a + 1 ] . \sum_{k=0}^{a}\left[{n\atop k}\right]=n!-\sum_{k=0}^{n}\left[{n\atop k+a+1}% \right].
  56. p = k n [ n p ] ( p k ) = [ n + 1 k + 1 ] \sum_{p=k}^{n}{\left[{n\atop p}\right]{\left({{p}\atop{k}}\right)}}=\left[{n+1% \atop k+1}\right]
  57. s ( n , n - p ) = 1 ( n - p - 1 ) ! 0 k 1 , , k p : 2 p m k m = p ( - 1 ) K ( n + K - 1 ) ! k 2 ! k p ! 1 ! k 1 2 ! k 2 3 ! k 3 p ! k p , \begin{aligned}\displaystyle s(n,n-p)&\displaystyle=\frac{1}{(n-p-1)!}\sum_{0% \leq k_{1},\ldots,k_{p}:\sum_{2}^{p}mk_{m}=p}(-1)^{K}\frac{(n+K-1)!}{k_{2}!% \cdots k_{p}!~{}1!^{k_{1}}2!^{k_{2}}3!^{k_{3}}\cdots p!^{k_{p}}},\end{aligned}
  58. K = k 2 + + k p . K=k_{2}+\cdots+k_{p}.

Stirling_numbers_of_the_second_kind.html

  1. S ( n , k ) S(n,k)
  2. { n k } \textstyle\{{n\atop k}\}
  3. S ( n , k ) S(n,k)
  4. { n k } \{\textstyle{n\atop k}\}
  5. { n n } = 1 \left\{{n\atop n}\right\}=1
  6. n 1 , { n 1 } = 1. n\geq 1,\left\{{n\atop 1}\right\}=1.
  7. { n k } = 1 k ! j = 0 k ( - 1 ) k - j ( k j ) j n . \left\{{n\atop k}\right\}=\frac{1}{k!}\sum_{j=0}^{k}(-1)^{k-j}{\left({{k}\atop% {j}}\right)}j^{n}.
  8. { n k } \textstyle\{{n\atop k}\}
  9. B n = k = 0 n { n k } B_{n}=\sum_{k=0}^{n}\left\{{n\atop k}\right\}
  10. ( x ) n = x ( x - 1 ) ( x - 2 ) ( x - n + 1 ) (x)_{n}=x(x-1)(x-2)\cdots(x-n+1)\,
  11. k = 0 n { n k } ( x ) k = x n . \sum_{k=0}^{n}\left\{{n\atop k}\right\}(x)_{k}=x^{n}.
  12. a n = k = 0 n k ! { n k } a_{n}=\sum_{k=0}^{n}k!\left\{{n\atop k}\right\}
  13. { n + 1 k } = k { n k } + { n k - 1 } \left\{{n+1\atop k}\right\}=k\left\{{n\atop k}\right\}+\left\{{n\atop k-1}\right\}
  14. { 0 0 } = 1 and { n 0 } = { 0 n } = 0 \left\{{0\atop 0}\right\}=1\quad\mbox{ and }~{}\quad\left\{{n\atop 0}\right\}=% \left\{{0\atop n}\right\}=0
  15. { n k - 1 } \left\{{n\atop k-1}\right\}
  16. n n
  17. k { n k } k\left\{{n\atop k}\right\}
  18. { n + 1 k + 1 } = j = k n ( n j ) { j k } , \left\{{n+1\atop k+1}\right\}=\sum_{j=k}^{n}{n\choose j}\left\{{j\atop k}% \right\},
  19. { n + 1 k + 1 } = j = k n ( k + 1 ) n - j { j k } , \left\{{n+1\atop k+1}\right\}=\sum_{j=k}^{n}(k+1)^{n-j}\left\{{j\atop k}\right\},
  20. { n + k + 1 k } = j = 0 k j { n + j j } . \left\{{n+k+1\atop k}\right\}=\sum_{j=0}^{k}j\left\{{n+j\atop j}\right\}.
  21. n 2 n\geq 2
  22. 1 k n - 1 1\leq k\leq n-1
  23. L ( n , k ) { n k } U ( n , k ) L(n,k)\leq\left\{{n\atop k}\right\}\leq U(n,k)
  24. L ( n , k ) = 1 2 ( k 2 + k + 2 ) k n - k - 1 - 1 L(n,k)=\frac{1}{2}(k^{2}+k+2)k^{n-k-1}-1
  25. U ( n , k ) = 1 2 ( n k ) k n - k . U(n,k)=\frac{1}{2}{n\choose k}k^{n-k}.
  26. n n
  27. { n k } \left\{{n\atop k}\right\}
  28. K n K_{n}
  29. { n 1 } < { n 2 } < < { n K n } , \left\{{n\atop 1}\right\}<\left\{{n\atop 2}\right\}<\cdots<\left\{{n\atop K_{n% }}\right\},
  30. { n K n } { n K n + 1 } > > { n n } . \left\{{n\atop K_{n}}\right\}\geq\left\{{n\atop K_{n}+1}\right\}>\cdots>\left% \{{n\atop n}\right\}.
  31. n n
  32. K n n log n , K_{n}\sim\frac{n}{\log n},
  33. log { n K n } = n log n - n log log n - n + O ( n log log n / log n ) . \log\left\{{n\atop K_{n}}\right\}=n\log n-n\log\log n-n+O(n\log\log n/\log n).
  34. { n k } ( z w ) ( mod 2 ) , \left\{{n\atop k}\right\}\equiv{\left({{z}\atop{w}}\right)}\ \;\;(\mathop{{\rm mod% }}2),
  35. z = n - k + 1 2 , w = k - 1 2 . z=n-\left\lceil\displaystyle\frac{k+1}{2}\right\rceil,\ w=\left\lfloor% \displaystyle\frac{k-1}{2}\right\rfloor.
  36. 𝔸 : i 𝔸 2 i \displaystyle\mathbb{A}:\ \sum_{i\in\mathbb{A}}2^{i}
  37. { n k } mod 2 = { 0 , 𝔸 𝔹 ; 1 , 𝔸 𝔹 = ; \begin{Bmatrix}n\\ k\end{Bmatrix}\,\bmod\,2=\begin{cases}0,&\mathbb{A}\cap\mathbb{B}\neq;\\ 1,&\mathbb{A}\cap\mathbb{B}=;\end{cases}
  38. { n k } mod 2 := ( ( n - k ) & ( ( k - 1 ) div 2 ) = = 0 ) . \begin{Bmatrix}n\\ k\end{Bmatrix}\,\bmod\,2:=\left(\left(n-k\right)\ \And\ \left(\left(k-1\right)% \,\mathrm{div}\,2\right)==0\right).
  39. { n n - 1 } = ( n 2 ) . \left\{{n\atop n-1}\right\}={\left({{n}\atop{2}}\right)}.
  40. { n 2 } = 2 n - 1 - 1. \left\{{n\atop 2}\right\}=2^{n-1}-1.
  41. { n 2 } = 1 1 ( 2 n - 1 - 1 n - 1 ) 0 ! { n 3 } = 1 1 ( 3 n - 1 - 2 n - 1 ) - 1 2 ( 3 n - 1 - 1 n - 1 ) 1 ! { n 4 } = 1 1 ( 4 n - 1 - 3 n - 1 ) - 2 2 ( 4 n - 1 - 2 n - 1 ) + 1 3 ( 4 n - 1 - 1 n - 1 ) 2 ! { n 5 } = 1 1 ( 5 n - 1 - 4 n - 1 ) - 3 2 ( 5 n - 1 - 3 n - 1 ) + 3 3 ( 5 n - 1 - 2 n - 1 ) - 1 4 ( 5 n - 1 - 1 n - 1 ) 3 ! \begin{aligned}\displaystyle\left\{{n\atop 2}\right\}&\displaystyle=\frac{% \frac{1}{1}(2^{n-1}-1^{n-1})}{0!}\\ \displaystyle\left\{{n\atop 3}\right\}&\displaystyle=\frac{\frac{1}{1}(3^{n-1}% -2^{n-1})-\frac{1}{2}(3^{n-1}-1^{n-1})}{1!}\\ \displaystyle\left\{{n\atop 4}\right\}&\displaystyle=\frac{\frac{1}{1}(4^{n-1}% -3^{n-1})-\frac{2}{2}(4^{n-1}-2^{n-1})+\frac{1}{3}(4^{n-1}-1^{n-1})}{2!}\\ \displaystyle\left\{{n\atop 5}\right\}&\displaystyle=\frac{\frac{1}{1}(5^{n-1}% -4^{n-1})-\frac{3}{2}(5^{n-1}-3^{n-1})+\frac{3}{3}(5^{n-1}-2^{n-1})-\frac{1}{4% }(5^{n-1}-1^{n-1})}{3!}\\ &\displaystyle{}\ \ \vdots\end{aligned}
  42. { n k } = j = 1 k ( - 1 ) k - j j n - 1 ( j - 1 ) ! ( k - j ) ! = 1 k ! j = 0 k ( - 1 ) k - j ( k j ) j n . \left\{{n\atop k}\right\}=\sum_{j=1}^{k}(-1)^{k-j}\frac{j^{n-1}}{(j-1)!(k-j)!}% =\frac{1}{k!}\sum_{j=0}^{k}(-1)^{k-j}{k\choose j}j^{n}.
  43. x n x^{n}
  44. Δ k x n = j = 0 k ( - 1 ) k - j ( k j ) ( x + j ) n . \Delta^{k}x^{n}=\sum_{j=0}^{k}(-1)^{k-j}{k\choose j}(x+j)^{n}.
  45. B m ( 0 ) = k = 0 m ( - 1 ) k k ! k + 1 { m k } . B_{m}(0)=\sum_{k=0}^{m}\frac{(-1)^{k}k!}{k+1}\left\{{m\atop k}\right\}.
  46. { n 0 } , { n 1 } , \left\{{n\atop 0}\right\},\left\{{n\atop 1}\right\},\ldots
  47. k = 0 n { n k } ( x ) k = x n , \sum_{k=0}^{n}\left\{{n\atop k}\right\}(x)_{k}=x^{n},
  48. k = 0 n { n k } x k = T n ( x ) , \sum_{k=0}^{n}\left\{{n\atop k}\right\}x^{k}=T_{n}(x),
  49. T n ( x ) T_{n}(x)
  50. { 0 k } , { 1 k } , \left\{{0\atop k}\right\},\left\{{1\atop k}\right\},\ldots
  51. n = 0 { n k } x n - k = r = 1 k 1 1 - r x , \sum_{n=0}^{\infty}\left\{{n\atop k}\right\}x^{n-k}=\prod_{r=1}^{k}\frac{1}{1-% rx},
  52. n = 0 { n k } x n + 1 = 1 ( k + 1 ) ! ( 1 x k + 1 ) \sum_{n=0}^{\infty}\left\{{n\atop k}\right\}x^{n+1}=\frac{1}{(k+1)!{\left({{% \frac{1}{x}}\atop{k+1}}\right)}}
  53. n = 0 { n k } x n n ! = ( e x - 1 ) k k ! . \sum_{n=0}^{\infty}\left\{{n\atop k}\right\}\frac{x^{n}}{n!}=\frac{(e^{x}-1)^{% k}}{k!}.
  54. { n k } = 0 \left\{{n\atop k}\right\}=0
  55. n < k n<k
  56. n , k = 0 { n k } x n n ! y k = n = 0 T n ( y ) x n n ! = e y ( e x - 1 ) . \sum_{n,k=0}^{\infty}\left\{{n\atop k}\right\}\frac{x^{n}}{n!}y^{k}=\sum_{n=0}% ^{\infty}T_{n}(y)\frac{x^{n}}{n!}=e^{y(e^{x}-1)}.
  57. k , k,
  58. { n k } k n k ! . \left\{{n\atop k}\right\}\sim\frac{k^{n}}{k!}.
  59. n n\rightarrow\infty
  60. k o ( n ) k\sim o(\sqrt{n})
  61. { n n - k } ( n - k ) 2 k 2 k k ! ( 1 + 1 3 2 k 2 + k n - k + 1 18 4 k 4 - k 2 - 3 k ( n - k ) 2 + ) . \left\{{n\atop n-k}\right\}\sim\frac{(n-k)^{2k}}{2^{k}k!}\left(1+\frac{1}{3}% \frac{2k^{2}+k}{n-k}+\frac{1}{18}\frac{4k^{4}-k^{2}-3k}{(n-k)^{2}}+\cdots% \right).
  62. { n k } n - k n ( 1 - G ) G k ( v - G ) n - k ( n - k e ) n - k ( n k ) k , 1 < k < n \left\{{n\atop k}\right\}\sim\frac{\sqrt{n-k}}{\sqrt{n(1-G)}\ G^{k}\ (v-G)^{n-% k}}\left(\frac{n-k}{e}\right)^{n-k}\left({n\atop k}\right)\quad\forall k,1<k<n
  63. G = - W 0 ( - v e - v ) , v = n / k G=-W_{0}(-ve^{-v}),v=n/k
  64. W 0 ( z ) \ W_{0}(z)
  65. 0.06 / n 0.06/n
  66. E ( X n ) = k = 1 n { n k } λ k . E(X^{n})=\sum_{k=1}^{n}\left\{{n\atop k}\right\}\lambda^{k}.
  67. E ( X n ) = k = 1 m { n k } . E(X^{n})=\sum_{k=1}^{m}\left\{{n\atop k}\right\}.
  68. S ( n , k ) S(n,k)
  69. S r ( n , k ) S_{r}(n,k)
  70. S r ( n + 1 , k ) = k S r ( n , k ) + ( n r - 1 ) S r ( n - r + 1 , k - 1 ) S_{r}(n+1,k)=k\ S_{r}(n,k)+{\left({{n}\atop{r-1}}\right)}S_{r}(n-r+1,k-1)
  71. S d ( n , k ) S^{d}(n,k)
  72. | i - j | d |i-j|\geq d
  73. S d ( n , k ) = S ( n - d + 1 , k - d + 1 ) , n k d S^{d}(n,k)=S(n-d+1,k-d+1),n\geq k\geq d
  74. S 1 ( n , k ) = S ( n , k ) S^{1}(n,k)=S(n,k)

Stochastic_drift.html

  1. n n
  2. y t = f ( t ) + e t y_{t}=f(t)+e_{t}
  3. y t y_{t}
  4. t t
  5. y t = y t - 1 + c + u t y_{t}=y_{t-1}+c+u_{t}
  6. u t u_{t}
  7. z t = y t - y t - 1 z_{t}=y_{t}-y_{t-1}

Stock_dilution.html

  1. ( O × O P ) + ( N × I P ) O + N \frac{(O\times OP)+(N\times IP)}{O+N}

Stokes_flow.html

  1. 𝑅𝑒 1 \,\textit{Re}\ll 1
  2. s y m b o l + 𝐟 = 0 symbol{\nabla}\cdot\mathbb{P}+\mathbf{f}=0
  3. \scriptstyle\mathbb{P}
  4. 𝐟 \scriptstyle\mathbf{f}
  5. d ρ d t + ρ 𝐮 = 0 \frac{d\rho}{dt}+\rho\nabla\cdot\mathbf{u}=0
  6. ρ \scriptstyle\rho
  7. 𝐮 \scriptstyle\mathbf{u}
  8. ρ \scriptstyle\rho
  9. ρ 𝐮 t \scriptstyle\rho\frac{\partial\mathbf{u}}{\partial t}
  10. R e 0. Re\to 0.
  11. μ 2 𝐮 - s y m b o l p + 𝐟 \displaystyle\mu\nabla^{2}\mathbf{u}-symbol{\nabla}p+\mathbf{f}
  12. 𝐮 \scriptstyle\mathbf{u}
  13. s y m b o l p \scriptstyle symbol{\nabla}p
  14. μ \scriptstyle\mu
  15. 𝐟 \scriptstyle\mathbf{f}
  16. 𝐮 = ( u , v , w ) \scriptstyle\mathbf{u}=(u,v,w)
  17. 𝐟 = ( f x , f y , f z ) \scriptstyle\mathbf{f}=(f_{x},f_{y},f_{z})
  18. μ ( 2 u x 2 + 2 u y 2 + 2 u z 2 ) - p x + f x = 0 μ ( 2 v x 2 + 2 v y 2 + 2 v z 2 ) - p y + f y = 0 μ ( 2 w x 2 + 2 w y 2 + 2 w z 2 ) - p z + f z = 0 u x + v y + w z = 0 \begin{aligned}\displaystyle\mu\left(\frac{\partial^{2}u}{\partial x^{2}}+% \frac{\partial^{2}u}{\partial y^{2}}+\frac{\partial^{2}u}{\partial z^{2}}% \right)-\frac{\partial p}{\partial x}+f_{x}&\displaystyle=0\\ \displaystyle\mu\left(\frac{\partial^{2}v}{\partial x^{2}}+\frac{\partial^{2}v% }{\partial y^{2}}+\frac{\partial^{2}v}{\partial z^{2}}\right)-\frac{\partial p% }{\partial y}+f_{y}&\displaystyle=0\\ \displaystyle\mu\left(\frac{\partial^{2}w}{\partial x^{2}}+\frac{\partial^{2}w% }{\partial y^{2}}+\frac{\partial^{2}w}{\partial z^{2}}\right)-\frac{\partial p% }{\partial z}+f_{z}&\displaystyle=0\\ \displaystyle{\partial u\over\partial x}+{\partial v\over\partial y}+{\partial w% \over\partial z}&\displaystyle=0\end{aligned}
  19. = 1 2 ( s y m b o l 𝐮 + ( s y m b o l 𝐮 ) T ) - p 𝕀 \scriptstyle\mathbb{P}=\frac{1}{2}(symbol{\nabla}\mathbf{u}+(symbol{\nabla}% \mathbf{u})^{T})-p\mathbb{I}
  20. ρ \rho
  21. ( ψ ) (\psi)
  22. 4 ψ = 0 \nabla^{4}\psi=0
  23. Δ 2 ψ = 0 \Delta^{2}\psi=0
  24. Δ \Delta
  25. ( Ψ ) (\Psi)
  26. E 2 Ψ = 0 , E^{2}\Psi=0,
  27. E = 2 r 2 + sin θ r 2 θ ( 1 sin θ θ ) E={\partial^{2}\over\partial r^{2}}+{\sin{\theta}\over r^{2}}{\partial\over% \partial\theta}\left({1\over\sin{\theta}}{\partial\over\partial\theta}\right)
  28. E E
  29. ( Ψ ) (\Psi)
  30. L - 1 2 Ψ = 0 , L_{-1}^{2}\Psi=0,
  31. L - 1 = 2 z 2 + 2 ρ 2 - 1 ρ ρ L_{-1}=\frac{\partial^{2}}{\partial z^{2}}+\frac{\partial^{2}}{\partial\rho^{2% }}-\frac{1}{\rho}\frac{\partial}{\partial\rho}
  32. L - 1 L_{-1}
  33. 𝕁 ( 𝐫 ) \mathbb{J}(\mathbf{r})
  34. μ 2 𝐮 - s y m b o l p \displaystyle\mu\nabla^{2}\mathbf{u}-symbol{\nabla}p
  35. δ ( 𝐫 ) \mathbf{\delta}(\mathbf{r})
  36. 𝐅 δ ( 𝐫 ) \mathbf{F}\cdot\delta(\mathbf{r})
  37. 𝐮 ( 𝐫 ) = 𝐅 𝕁 ( 𝐫 ) , p ( 𝐫 ) = 𝐅 𝐫 4 π | 𝐫 | 3 \mathbf{u}(\mathbf{r})=\mathbf{F}\cdot\mathbb{J}(\mathbf{r}),\qquad p(\mathbf{% r})=\frac{\mathbf{F}\cdot\mathbf{r}}{4\pi|\mathbf{r}|^{3}}
  38. 𝕁 ( 𝐫 ) = 1 8 π μ ( 𝕀 | 𝐫 | + 𝐫𝐫 | 𝐫 | 3 ) \mathbb{J}(\mathbf{r})={1\over 8\pi\mu}\left(\frac{\mathbb{I}}{|\mathbf{r}|}+% \frac{\mathbf{r}\mathbf{r}}{|\mathbf{r}|^{3}}\right)
  39. 𝐅 𝕁 ( 𝐫 ) \mathbf{F}\cdot\mathbb{J}(\mathbf{r})
  40. 𝐅 \mathbf{F}
  41. 𝐟 ( 𝐫 ) \mathbf{f}(\mathbf{r})
  42. 𝐮 ( 𝐫 ) = 𝐟 ( 𝐫 ) 𝕁 ( 𝐫 - 𝐫 ) d 𝐫 , p ( 𝐫 ) = 𝐟 ( 𝐫 ) ( 𝐫 - 𝐫 ) 4 π | 𝐫 - 𝐫 | 3 d 𝐫 \mathbf{u}(\mathbf{r})=\int\mathbf{f}(\mathbf{r^{\prime}})\cdot\mathbb{J}(% \mathbf{r}-\mathbf{r^{\prime}})\mathrm{d}\mathbf{r^{\prime}},\qquad p(\mathbf{% r})=\int\frac{\mathbf{f}(\mathbf{r^{\prime}})\cdot(\mathbf{r}-\mathbf{r^{% \prime}})}{4\pi|\mathbf{r}-\mathbf{r^{\prime}}|^{3}}\,\mathrm{d}\mathbf{r^{% \prime}}
  43. p p
  44. 𝐮 \displaystyle\mathbf{u}
  45. p n , Φ n , p_{n},\Phi_{n},
  46. χ n \chi_{n}
  47. n n
  48. p n \displaystyle p_{n}
  49. P n m P_{n}^{m}
  50. n < 0 n<0
  51. n > 0 n>0
  52. n - n - 1 n\to-n-1
  53. a a
  54. U U
  55. μ \mu
  56. F D F_{D}
  57. F D = 6 π μ a U F_{D}=6\pi\mu aU
  58. V V
  59. S S
  60. 𝐮 \mathbf{u}
  61. 𝐮 \mathbf{u}^{\prime}
  62. V V
  63. σ \mathbf{\sigma}
  64. σ \mathbf{\sigma}^{\prime}
  65. S 𝐮 ( σ 𝐧 ) d S = S 𝐮 ( σ 𝐧 ) d S \int_{S}\mathbf{u}\cdot(\mathbf{\sigma}^{\prime}\cdot\mathbf{n})dS=\int_{S}% \mathbf{u}^{\prime}\cdot(\mathbf{\sigma}\cdot\mathbf{n})dS
  66. 𝐧 \mathbf{n}
  67. S S
  68. 𝐅 \mathbf{F}
  69. 𝐓 \mathbf{T}
  70. 𝐅 \displaystyle\mathbf{F}
  71. μ \mu
  72. a a
  73. 𝐯 \mathbf{v}^{\infty}
  74. 𝐔 \mathbf{U}
  75. 𝛀 \mathbf{\Omega}^{\infty}
  76. ω \mathbf{\omega}

Stokes_phenomenon.html

  1. y ′′ - x y = 0 , y^{\prime\prime}-xy=0,\,
  2. e ± 2 3 x 3 / 2 x 1 / 4 . \frac{e^{\pm\frac{2}{3}x^{3/2}}}{x^{1/4}}.
  3. Ai ( x ) \displaystyle\mathrm{Ai}(x)
  4. arg x = π / 3 \operatorname{arg}\,x=\pi/3
  5. e - 2 3 x 3 / 2 x 1 / 4 \frac{e^{-\frac{2}{3}x^{3/2}}}{x^{1/4}}
  6. e + 2 3 x 3 / 2 x 1 / 4 \frac{e^{+\frac{2}{3}x^{3/2}}}{x^{1/4}}
  7. d 2 w d z 2 = f ( z ) w \frac{d^{2}w}{dz^{2}}=f(z)w
  8. exp ± a z f ( z ) d z f ( z ) 1 / 4 \frac{\exp{\pm\int_{a}^{z}\sqrt{f(z)}dz}}{f(z)^{1/4}}
  9. a z f ( z ) d z . \int_{a}^{z}\sqrt{f(z)}dz.
  10. f ( a ) ( z - a ) f^{\prime}(a)(z-a)

Stokes_radius.html

  1. f f
  2. F d r a g = f s = ( 6 π η a ) s F_{drag}=fs=(6\pi\eta a)s
  3. η \eta
  4. s s
  5. a a
  6. μ \mu
  7. μ = z e f \mu=\frac{ze}{f}
  8. z e ze
  9. D D
  10. D = μ k B T q = k b T f D=\frac{\mu k_{B}T}{q}=\frac{k_{b}T}{f}
  11. k B k_{B}
  12. q q
  13. D = k b T 6 π η a D=\frac{k_{b}T}{6\pi\eta a}
  14. a a
  15. R H = a = k b T 6 π η D R_{H}=a=\frac{k_{b}T}{6\pi\eta D}

Stolz–Cesàro_theorem.html

  1. ( a n ) n 1 (a_{n})_{n\geq 1}
  2. ( b n ) n 1 (b_{n})_{n\geq 1}
  3. ( b n ) n 1 (b_{n})_{n\geq 1}
  4. + +\infty
  5. - -\infty
  6. lim n a n + 1 - a n b n + 1 - b n = . \lim_{n\to\infty}\frac{a_{n+1}-a_{n}}{b_{n+1}-b_{n}}=\ell.
  7. lim n a n b n \lim_{n\to\infty}\frac{a_{n}}{b_{n}}
  8. ( a n ) n 1 (a_{n})_{n\geq 1}
  9. ( b n ) n 1 (b_{n})_{n\geq 1}
  10. ( b n ) n 1 (b_{n})_{n\geq 1}
  11. lim inf n a n + 1 - a n b n + 1 - b n lim inf n a n b n lim sup n a n b n lim sup n a n + 1 - a n b n + 1 - b n . \liminf_{n\to\infty}\frac{a_{n+1}-a_{n}}{b_{n+1}-b_{n}}\leq\liminf_{n\to\infty% }\frac{a_{n}}{b_{n}}\leq\limsup_{n\to\infty}\frac{a_{n}}{b_{n}}\leq\limsup_{n% \to\infty}\frac{a_{n+1}-a_{n}}{b_{n+1}-b_{n}}.

Strategic_dominance.html

  1. i i
  2. s * S i s^{*}\in S_{i}
  3. s S i s^{\prime}\in S_{i}
  4. s - i S - i [ u i ( s * , s - i ) u i ( s , s - i ) ] \forall s_{-i}\in S_{-i}\left[u_{i}(s^{*},s_{-i})\geq u_{i}(s^{\prime},s_{-i})\right]
  5. s - i s_{-i}
  6. s * s^{*}
  7. s s^{\prime}
  8. s - i S - i [ u i ( s * , s - i ) > u i ( s , s - i ) ] \forall s_{-i}\in S_{-i}\left[u_{i}(s^{*},s_{-i})>u_{i}(s^{\prime},s_{-i})\right]
  9. S - i S_{-i}
  10. i i

Stratonovich_integral.html

  1. W : [ 0 , T ] × Ω W:[0,T]\times\Omega\to\mathbb{R}
  2. X : [ 0 , T ] × Ω X:[0,T]\times\Omega\to\mathbb{R}
  3. 0 T X t d W t \int_{0}^{T}X_{t}\circ\mathrm{d}W_{t}
  4. : Ω :\Omega\to\mathbb{R}
  5. i = 0 k - 1 X t i + 1 + X t i 2 ( W t i + 1 - W t i ) \sum_{i=0}^{k-1}{X_{t_{i+1}}+X_{t_{i}}\over 2}\left(W_{t_{i+1}}-W_{t_{i}}\right)
  6. 0 = t 0 < t 1 < < t k = T 0=t_{0}<t_{1}<\dots<t_{k}=T
  7. [ 0 , T ] [0,T]
  8. 0 T f ( W t ) d W t = f ( W T ) - f ( W 0 ) \int_{0}^{T}f^{\prime}(W_{t})\circ\mathrm{d}W_{t}=f(W_{T})-f(W_{0})
  9. 0 T f W ( W t , t ) d W t + 0 T f t ( W t , t ) d t = f ( W T , T ) - f ( W 0 , 0 ) . \int_{0}^{T}{\partial f\over\partial W}(W_{t},t)\circ\mathrm{d}W_{t}+\int_{0}^% {T}{\partial f\over\partial t}(W_{t},t)\,\mathrm{d}t=f(W_{T},T)-f(W_{0},0).
  10. X T - X 0 = 0 T Y t d W t + 0 T Z t d t X_{T}-X_{0}=\int_{0}^{T}Y_{t}\circ\mathrm{d}W_{t}+\int_{0}^{T}Z_{t}\,\mathrm{d}t
  11. d X = Y d W + Z d t . \mathrm{d}X=Y\circ\mathrm{d}W+Z\,\mathrm{d}t.
  12. d ( t 2 W 3 ) = 3 t 2 W 2 d W + 2 t W 3 d t . \mathrm{d}(t^{2}\,W^{3})=3t^{2}W^{2}\circ\mathrm{d}W+2tW^{3}\,\mathrm{d}t.
  13. 0 T X t d W t \int_{0}^{T}X_{t}\,\mathrm{d}W_{t}
  14. X X
  15. X t i X_{t_{i}}
  16. X ( t i + 1 + t i ) / 2 X_{(t_{i+1}+t_{i})/2}
  17. 0 T f ( W t , t ) d W t = 1 2 0 T f W ( W t , t ) d t + 0 T f ( W t , t ) d W t , \int_{0}^{T}f(W_{t},t)\circ\mathrm{d}W_{t}=\frac{1}{2}\int_{0}^{T}{\partial f% \over\partial W}(W_{t},t)\,\mathrm{d}t+\int_{0}^{T}f(W_{t},t)\,\mathrm{d}W_{t},
  18. d X t = μ ( X t ) d t + σ ( X t ) d W t \mathrm{d}X_{t}=\mu(X_{t})\,\mathrm{d}t+\sigma(X_{t})\,\mathrm{d}W_{t}
  19. 0 T σ ( X t ) d W t = 1 2 0 T σ ( X t ) σ ( X t ) d t + 0 T σ ( X t ) d W t . \int_{0}^{T}\sigma(X_{t})\circ\mathrm{d}W_{t}=\frac{1}{2}\int_{0}^{T}\sigma^{% \prime}(X_{t})\sigma(X_{t})\,\mathrm{d}t+\int_{0}^{T}\sigma(X_{t})\,\mathrm{d}% W_{t}.
  20. 0 T X s - d Y s = 0 T X s - d Y s + 1 2 [ X , Y ] T c , \int_{0}^{T}X_{s-}\circ\mathrm{d}Y_{s}=\int_{0}^{T}X_{s-}\,\mathrm{d}Y_{s}+% \frac{1}{2}[X,Y]_{T}^{c},
  21. [ X , Y ] T c [X,Y]_{T}^{c}

Strength_of_ships.html

  1. I y = b h 3 12 + A d 2 I_{y}=\frac{bh^{3}}{12}+Ad^{2}
  2. I y I_{y}
  3. b b
  4. h h
  5. A A
  6. d d

Stress_intensity_factor.html

  1. K K
  2. K K
  3. σ i j \sigma_{ij}
  4. r , θ r,\theta
  5. σ i j ( r , θ ) = K 2 π r f i j ( θ ) + higher order terms \sigma_{ij}(r,\theta)=\frac{K}{\sqrt{2\pi r}}\,f_{ij}(\theta)+\,\,\rm{higher\,% order\,terms}
  6. K K
  7. × \times
  8. f i j f_{ij}
  9. r r
  10. r r
  11. σ i j \sigma_{ij}
  12. \infty
  13. K I K_{\rm I}
  14. K II K_{\rm II}
  15. K III K_{\rm III}
  16. K I = lim r 0 2 π r σ y y ( r , 0 ) K II = lim r 0 2 π r σ y x ( r , 0 ) K III = lim r 0 2 π r σ y z ( r , 0 ) . \begin{aligned}\displaystyle K_{\rm I}&\displaystyle=\lim_{r\rightarrow 0}% \sqrt{2\pi r}\,\sigma_{yy}(r,0)\\ \displaystyle K_{\rm II}&\displaystyle=\lim_{r\rightarrow 0}\sqrt{2\pi r}\,% \sigma_{yx}(r,0)\\ \displaystyle K_{\rm III}&\displaystyle=\lim_{r\rightarrow 0}\sqrt{2\pi r}\,% \sigma_{yz}(r,0)\,.\end{aligned}
  17. G G
  18. G = K I 2 ( 1 - ν 2 E ) G=K_{\rm I}^{2}\left(\frac{1-\nu^{2}}{E}\right)
  19. E E
  20. ν \nu
  21. G = K I 2 ( 1 E ) . G=K_{\rm I}^{2}\left(\frac{1}{E}\right)\,.
  22. G = K II 2 ( 1 - ν 2 E ) or G = K II 2 ( 1 E ) . G=K_{\rm II}^{2}\left(\frac{1-\nu^{2}}{E}\right)\quad\,\text{or}\quad G=K_{\rm II% }^{2}\left(\frac{1}{E}\right)\,.
  23. G = K III 2 ( 1 2 μ ) G=K_{\rm III}^{2}\left(\frac{1}{2\mu}\right)
  24. μ \mu
  25. G = K I 2 ( 1 - ν 2 E ) + K II 2 ( 1 - ν 2 E ) + K III 2 ( 1 2 μ ) . G=K_{\rm I}^{2}\left(\frac{1-\nu^{2}}{E}\right)+K_{\rm II}^{2}\left(\frac{1-% \nu^{2}}{E}\right)+K_{\rm III}^{2}\left(\frac{1}{2\mu}\right)\,.
  26. G = J = Γ ( W d x 2 - 𝐭 𝐮 x 1 d s ) . G=J=\int_{\Gamma}\left(W~{}dx_{2}-\mathbf{t}\cdot\cfrac{\partial\mathbf{u}}{% \partial x_{1}}~{}ds\right)\,.
  27. K K
  28. Y Y
  29. K I K_{\mathrm{I}}
  30. K Ic K_{\mathrm{Ic}}
  31. K Ic K_{\mathrm{Ic}}
  32. K Ic K_{\mathrm{Ic}}
  33. K Ic K_{\mathrm{Ic}}
  34. K Ic K_{\mathrm{Ic}}
  35. K Ic 2 = K I 2 + K II 2 + E 2 μ K III 2 K_{\rm Ic}^{2}=K_{\rm I}^{2}+K_{\rm II}^{2}+\frac{E^{\prime}}{2\mu}\,K_{\rm III% }^{2}
  36. K Ic K_{\rm Ic}
  37. E = E / ( 1 - ν 2 ) E^{\prime}=E/(1-\nu^{2})
  38. E = E E^{\prime}=E
  39. K c K_{\rm c}
  40. 2 a 2a
  41. σ \sigma
  42. K I = σ π a K_{\mathrm{I}}=\sigma\sqrt{\pi a}
  43. 2 b 2b
  44. 2 h 2h
  45. K I = σ π a [ 1 - a 2 b + 0.326 ( a b ) 2 1 - a b ] . K_{\rm I}=\sigma\sqrt{\pi a}\left[\cfrac{1-\frac{a}{2b}+0.326\left(\frac{a}{b}% \right)^{2}}{\sqrt{1-\frac{a}{b}}}\right]\,.
  46. d b d\neq b
  47. K IA = σ π a [ 1 + n = 2 M C n ( a b ) n ] K_{\rm IA}=\sigma\sqrt{\pi a}\left[1+\sum_{n=2}^{M}C_{n}\left(\frac{a}{b}% \right)^{n}\right]
  48. C n C_{n}
  49. d d
  50. K IA = σ π a Φ A , K IB = σ π a Φ B K_{\rm IA}=\sigma\sqrt{\pi a}\,\Phi_{A}\,\,,K_{\rm IB}=\sigma\sqrt{\pi a}\,% \Phi_{B}
  51. Φ A : = [ β + ( 1 - β 4 ) ( 1 + 1 4 sec α A ) 2 ] sec α A Φ B : = 1 + [ sec α A B - 1 1 + 0.21 sin { 8 tan - 1 [ ( α A - α B α A + α B ) 0.9 ] } ] \begin{aligned}\displaystyle\Phi_{A}&\displaystyle:=\left[\beta+\left(\frac{1-% \beta}{4}\right)\left(1+\frac{1}{4\sqrt{\sec\alpha_{A}}}\right)^{2}\right]% \sqrt{\sec\alpha_{A}}\\ \displaystyle\Phi_{B}&\displaystyle:=1+\left[\frac{\sqrt{\sec\alpha_{AB}}-1}{1% +0.21\sin\left\{8\,\tan^{-1}\left[\left(\frac{\alpha_{A}-\alpha_{B}}{\alpha_{A% }+\alpha_{B}}\right)^{0.9}\right]\right\}}\right]\end{aligned}
  52. β := sin ( π α B α A + α B ) , α A := π a 2 d , α B := π a 4 b - 2 d ; α A B := 4 7 α A + 3 7 α B . \beta:=\sin\left(\frac{\pi\alpha_{B}}{\alpha_{A}+\alpha_{B}}\right)~{},~{}~{}% \alpha_{A}:=\frac{\pi a}{2d}~{},~{}~{}\alpha_{B}:=\frac{\pi a}{4b-2d}~{};~{}~{% }\alpha_{AB}:=\frac{4}{7}\,\alpha_{A}+\frac{3}{7}\,\alpha_{B}\,.
  53. d d
  54. d = b d=b
  55. h × b h\times b
  56. a a
  57. h / b 1 h/b\geq 1
  58. a / b 0.6 a/b\leq 0.6
  59. σ \sigma
  60. K I = σ π a [ 1.12 - 0.23 ( a b ) + 10.6 ( a b ) 2 - 21.7 ( a b ) 3 + 30.4 ( a b ) 4 ] . K_{\rm I}=\sigma\sqrt{\pi a}\left[1.12-0.23\left(\frac{a}{b}\right)+10.6\left(% \frac{a}{b}\right)^{2}-21.7\left(\frac{a}{b}\right)^{3}+30.4\left(\frac{a}{b}% \right)^{4}\right]\,.
  61. h / b 1 h/b\geq 1
  62. a / b 0.3 a/b\geq 0.3
  63. K I = σ π a [ 1 + 3 a b 2 π a b ( 1 - a b ) 3 / 2 ] . K_{\rm I}=\sigma\sqrt{\pi a}\left[\frac{1+3\frac{a}{b}}{2\sqrt{\pi\frac{a}{b}}% \left(1-\frac{a}{b}\right)^{3/2}}\right]\,.
  64. 2 a 2a
  65. σ \sigma
  66. y y
  67. α σ \alpha\sigma
  68. x x
  69. K I = σ π a ( cos 2 β + α sin 2 β ) K II = σ π a ( 1 - α ) sin β cos β \begin{aligned}\displaystyle K_{\rm I}&\displaystyle=\sigma\sqrt{\pi a}\left(% \cos^{2}\beta+\alpha\sin^{2}\beta\right)\\ \displaystyle K_{\rm II}&\displaystyle=\sigma\sqrt{\pi a}\left(1-\alpha\right)% \sin\beta\cos\beta\end{aligned}
  70. β \beta
  71. x x
  72. a a
  73. σ \sigma
  74. K I = 2 σ a π . K_{\rm I}=2\sigma\sqrt{\frac{a}{\pi}}\,.
  75. 2 h × 2 b 2h\times 2b
  76. 2 a 2a
  77. F x F_{x}
  78. F y F_{y}
  79. x , y x,y
  80. h a h\gg a
  81. b a b\gg a
  82. x b x\ll b
  83. y h y\ll h
  84. F x F_{x}
  85. x = a x=a
  86. K I = F x 2 π a ( κ - 1 κ + 1 ) [ G 1 + 1 κ - 1 H 1 ] K II = F x 2 π a [ G 2 + 1 κ + 1 H 2 ] \begin{aligned}\displaystyle K_{\rm I}&\displaystyle=\frac{F_{x}}{2\sqrt{\pi a% }}\left(\frac{\kappa-1}{\kappa+1}\right)\left[G_{1}+\frac{1}{\kappa-1}H_{1}% \right]\\ \displaystyle K_{\rm II}&\displaystyle=\frac{F_{x}}{2\sqrt{\pi a}}\left[G_{2}+% \frac{1}{\kappa+1}H_{2}\right]\end{aligned}
  87. G 1 = 1 - Re [ a + z z 2 - a 2 ] , G 2 = - Im [ a + z z 2 - a 2 ] H 1 = Re [ a ( z ¯ - z ) ( z ¯ - a ) z ¯ 2 - a 2 ] , H 2 = - Im [ a ( z ¯ - z ) ( z ¯ - a ) z ¯ 2 - a 2 ] \begin{aligned}\displaystyle G_{1}&\displaystyle=1-\,\text{Re}\left[\frac{a+z}% {\sqrt{z^{2}-a^{2}}}\right]\,,\,\,G_{2}=-\,\text{Im}\left[\frac{a+z}{\sqrt{z^{% 2}-a^{2}}}\right]\\ \displaystyle H_{1}&\displaystyle=\,\text{Re}\left[\frac{a(\bar{z}-z)}{(\bar{z% }-a)\sqrt{{\bar{z}}^{2}-a^{2}}}\right]\,,\,\,H_{2}=-\,\text{Im}\left[\frac{a(% \bar{z}-z)}{(\bar{z}-a)\sqrt{{\bar{z}}^{2}-a^{2}}}\right]\end{aligned}
  88. z = x + i y z=x+iy
  89. z ¯ = x - i y \bar{z}=x-iy
  90. κ = 3 - 4 ν \kappa=3-4\nu
  91. κ = ( 3 - ν ) / ( 1 + ν ) \kappa=(3-\nu)/(1+\nu)
  92. ν \nu
  93. F y F_{y}
  94. K I = F y 2 π a [ G 2 - 1 κ + 1 H 2 ] K II = - F y 2 π a ( κ - 1 κ + 1 ) [ G 1 - 1 κ - 1 H 1 ] . \begin{aligned}\displaystyle K_{\rm I}&\displaystyle=\frac{F_{y}}{2\sqrt{\pi a% }}\left[G_{2}-\frac{1}{\kappa+1}H_{2}\right]\\ \displaystyle K_{\rm II}&\displaystyle=-\frac{F_{y}}{2\sqrt{\pi a}}\left(\frac% {\kappa-1}{\kappa+1}\right)\left[G_{1}-\frac{1}{\kappa-1}H_{1}\right]\,.\end{aligned}
  95. x = - a x=-a
  96. F x F_{x}
  97. ( x , y ) (x,y)
  98. K I ( - a ; x , y ) = - K I ( a ; - x , y ) , K II ( - a ; x , y ) = K II ( a ; - x , y ) . K_{\rm I}(-a;x,y)=-K_{\rm I}(a;-x,y)\,,\,\,K_{\rm II}(-a;x,y)=K_{\rm II}(a;-x,% y)\,.
  99. F y F_{y}
  100. K I ( - a ; x , y ) = K I ( a ; - x , y ) , K II ( - a ; x , y ) = - K II ( a ; - x , y ) . K_{\rm I}(-a;x,y)=K_{\rm I}(a;-x,y)\,,\,\,K_{\rm II}(-a;x,y)=-K_{\rm II}(a;-x,% y)\,.
  101. F x F_{x}
  102. F y F_{y}
  103. F y F_{y}
  104. y = 0 y=0
  105. - a < x < a -a<x<a
  106. K I = F y 2 π a a + x a - x , K II = - F x 2 π a ( κ - 1 κ + 1 ) . K_{\rm I}=\frac{F_{y}}{2\sqrt{\pi a}}\sqrt{\frac{a+x}{a-x}}\,,\,\,K_{\rm II}=-% \frac{F_{x}}{2\sqrt{\pi a}}\left(\frac{\kappa-1}{\kappa+1}\right)\,.
  107. - a < x < a -a<x<a
  108. K I = 1 2 π a - a a F y ( x ) a + x a - x d x , K II = - 1 2 π a ( κ - 1 κ + 1 ) - a a F y ( x ) d x , . K_{\rm I}=\frac{1}{2\sqrt{\pi a}}\int_{-a}^{a}F_{y}(x)\,\sqrt{\frac{a+x}{a-x}}% \,{\rm d}x\,,\,\,K_{\rm II}=-\frac{1}{2\sqrt{\pi a}}\left(\frac{\kappa-1}{% \kappa+1}\right)\int_{-a}^{a}F_{y}(x)\,{\rm d}x,\,.
  109. K I = P B π W [ 16.7 ( a W ) 1 / 2 - 104.7 ( a W ) 3 / 2 + 369.9 ( a W ) 5 / 2 - 573.8 ( a W ) 7 / 2 + 360.5 ( a W ) 9 / 2 ] \begin{aligned}\displaystyle K_{\rm I}&\displaystyle=\frac{P}{B}\sqrt{\frac{% \pi}{W}}\left[16.7\left(\frac{a}{W}\right)^{1/2}-104.7\left(\frac{a}{W}\right)% ^{3/2}+369.9\left(\frac{a}{W}\right)^{5/2}\right.\\ &\displaystyle\qquad\left.-573.8\left(\frac{a}{W}\right)^{7/2}+360.5\left(% \frac{a}{W}\right)^{9/2}\right]\end{aligned}
  110. P P
  111. B B
  112. a a
  113. W W
  114. K I = 4 P B π W [ 1.6 ( a W ) 1 / 2 - 2.6 ( a W ) 3 / 2 + 12.3 ( a W ) 5 / 2 - 21.2 ( a W ) 7 / 2 + 21.8 ( a W ) 9 / 2 ] \begin{aligned}\displaystyle K_{\rm I}&\displaystyle=\frac{4P}{B}\sqrt{\frac{% \pi}{W}}\left[1.6\left(\frac{a}{W}\right)^{1/2}-2.6\left(\frac{a}{W}\right)^{3% /2}+12.3\left(\frac{a}{W}\right)^{5/2}\right.\\ &\displaystyle\qquad\left.-21.2\left(\frac{a}{W}\right)^{7/2}+21.8\left(\frac{% a}{W}\right)^{9/2}\right]\end{aligned}
  115. P P
  116. B B
  117. a a
  118. W W

Strictness_analysis.html

  1. { } \{\bot\}
  2. { } \{\bot\}
  3. f f
  4. f = f π f=f\circ\pi
  5. π \pi

String_cosmology.html

  1. S 2 = 1 4 π α d 2 z γ [ γ a b G μ ν ( X ) a X μ b X ν + α R ( 2 ) Φ ( X ) ] , S_{2}=\frac{1}{4\pi\alpha^{\prime}}\int d^{2}z\sqrt{\gamma}\left[\gamma^{ab}G_% {\mu\nu}(X)\partial_{a}X^{\mu}\partial_{b}X^{\nu}+\alpha^{\prime}\ {}^{(2)}R% \Phi(X)\right],
  2. R ( 2 ) \ {}^{(2)}R
  3. Φ \Phi
  4. α \alpha^{\prime}
  5. a , b a,b
  6. μ , ν \mu,\nu
  7. 1 , , D 1,\ldots,D
  8. β μ ν G = R μ ν + 2 α μ Φ ν Φ + O ( α 2 ) , \beta^{G}_{\mu\nu}=R_{\mu\nu}+2\alpha^{\prime}\nabla_{\mu}\Phi\nabla_{\nu}\Phi% +O(\alpha^{\prime 2}),
  9. β Φ = D - 26 6 - α 2 2 Φ + α κ Φ κ Φ + O ( α 2 ) . \beta^{\Phi}=\frac{D-26}{6}-\frac{\alpha^{\prime}}{2}\nabla^{2}\Phi+\alpha^{% \prime}\nabla_{\kappa}\Phi\nabla^{\kappa}\Phi+O(\alpha^{\prime 2}).
  10. β μ ν G = β Φ = 0 , \beta^{G}_{\mu\nu}=\beta^{\Phi}=0,
  11. β Φ \beta^{\Phi}
  12. D 26 D\neq 26
  13. S = 1 2 κ 0 2 d D x - G e - 2 Φ [ - 2 ( D - 26 ) 3 α + R + 4 μ Φ μ Φ + O ( α ) ] , S=\frac{1}{2\kappa_{0}^{2}}\int d^{D}x\sqrt{-G}e^{-2\Phi}\left[-\frac{2(D-26)}% {3\alpha^{\prime}}+R+4\partial_{\mu}\Phi\partial^{\mu}\Phi+O(\alpha^{\prime})% \right],
  14. κ 0 2 \kappa_{0}^{2}
  15. g μ ν = e 2 ω G μ ν , \,g_{\mu\nu}=e^{2\omega}G_{\mu\nu}\!,
  16. ω = 2 ( Φ 0 - Φ ) D - 2 , \omega=\frac{2(\Phi_{0}-\Phi)}{D-2},
  17. Φ ~ = Φ - Φ 0 \tilde{\Phi}=\Phi-\Phi_{0}
  18. S = 1 2 κ 2 d D x - g [ - 2 ( D - 26 ) 3 α e 4 Φ ~ D - 2 + R ~ - 4 D - 2 μ Φ ~ μ Φ ~ + O ( α ) ] , S=\frac{1}{2\kappa^{2}}\int d^{D}x\sqrt{-g}\left[-\frac{2(D-26)}{3\alpha^{% \prime}}e^{\frac{4\tilde{\Phi}}{D-2}}+\tilde{R}-\frac{4}{D-2}\partial_{\mu}% \tilde{\Phi}\partial^{\mu}\tilde{\Phi}+O(\alpha^{\prime})\right],
  19. R ~ = e - 2 ω [ R - ( D - 1 ) 2 ω - ( D - 2 ) ( D - 1 ) μ ω μ ω ] . \tilde{R}=e^{-2\omega}[R-(D-1)\nabla^{2}\omega-(D-2)(D-1)\partial_{\mu}\omega% \partial^{\mu}\omega].
  20. κ = κ 0 e 2 Φ 0 = ( 8 π G D ) 1 2 = 8 π M p , \kappa=\kappa_{0}e^{2\Phi_{0}}=(8\pi G_{D})^{\frac{1}{2}}=\frac{\sqrt{8\pi}}{M% _{p}},
  21. G D G_{D}
  22. M p M_{p}
  23. D = 4 D=4

String_theory_landscape.html

  1. P ( x ) P(x)
  2. x x
  3. P ( x ) = P prior ( x ) × P selection ( x ) , P(x)=P_{\mathrm{prior}}(x)\times P_{\mathrm{selection}}(x),
  4. P prior P_{\mathrm{prior}}
  5. x x
  6. P selection P_{\mathrm{selection}}
  7. x x
  8. P prior P_{\mathrm{prior}}
  9. P selection P_{\mathrm{selection}}

Stroke_(engine).html

  1. T 3 T 2 = ( V 2 V 3 ) γ - 1 \frac{T3}{T2}=\left(\frac{V2}{V3}\right)^{\gamma-1}

Strophoid.html

  1. r = f ( θ ) r=f(\theta)
  2. K = ( r cos θ , r sin θ ) K=(r\cos\theta,\ r\sin\theta)
  3. d = ( r cos θ - a ) 2 + ( r sin θ - b ) 2 = ( f ( θ ) cos θ - a ) 2 + ( f ( θ ) sin θ - b ) 2 d=\sqrt{(r\cos\theta-a)^{2}+(r\sin\theta-b)^{2}}=\sqrt{(f(\theta)\cos\theta-a)% ^{2}+(f(\theta)\sin\theta-b)^{2}}
  4. θ \theta
  5. f ( θ ) ± d f(\theta)\pm d
  6. r = f ( θ ) ± ( f ( θ ) cos θ - a ) 2 + ( f ( θ ) sin θ - b ) 2 r=f(\theta)\pm\sqrt{(f(\theta)\cos\theta-a)^{2}+(f(\theta)\sin\theta-b)^{2}}
  7. u ( t ) = p + ( x ( t ) - p ) ( 1 ± n ( t ) ) , v ( t ) = q + ( y ( t ) - q ) ( 1 ± n ( t ) ) u(t)=p+(x(t)-p)(1\pm n(t)),\ v(t)=q+(y(t)-q)(1\pm n(t))
  8. n ( t ) = ( x ( t ) - a ) 2 + ( y ( t ) - b ) 2 ( x ( t ) - p ) 2 + ( y ( t ) - q ) 2 n(t)=\sqrt{\frac{(x(t)-a)^{2}+(y(t)-b)^{2}}{(x(t)-p)^{2}+(y(t)-q)^{2}}}
  9. θ \theta
  10. ϑ \vartheta
  11. ϑ \vartheta
  12. θ \theta
  13. ϑ = l ( θ ) \vartheta=l(\theta)
  14. ψ \psi
  15. ψ = ϑ - θ \psi=\vartheta-\theta
  16. r sin ϑ = a sin ψ , r = a sin ϑ sin ψ = a sin l ( θ ) sin ( l ( θ ) - θ ) {r\over\sin\vartheta}={a\over\sin\psi},\ r=a\frac{\sin\vartheta}{\sin\psi}=a% \frac{\sin l(\theta)}{\sin(l(\theta)-\theta)}
  17. ψ = P 1 K A \psi=\angle P_{1}KA
  18. π - ψ = A K P 2 \pi-\psi=\angle AKP_{2}
  19. P 1 K A \triangle P_{1}KA
  20. ψ \psi
  21. A P 1 K \angle AP_{1}K
  22. K A P 1 \angle KAP_{1}
  23. ( π - ψ ) / 2 (\pi-\psi)/2
  24. l 1 ( θ ) = ϑ + K A P 1 = ϑ + ( π - ψ ) / 2 = ϑ + ( π - ϑ + θ ) / 2 = ( ϑ + θ + π ) / 2 l_{1}(\theta)=\vartheta+\angle KAP_{1}=\vartheta+(\pi-\psi)/2=\vartheta+(\pi-% \vartheta+\theta)/2=(\vartheta+\theta+\pi)/2
  25. l 2 ( θ ) = ( ϑ + θ ) / 2 l_{2}(\theta)=(\vartheta+\theta)/2
  26. r 1 = a sin l 1 ( θ ) sin ( l 1 ( θ ) - θ ) = a sin ( ( l ( θ ) + θ + π ) / 2 ) sin ( ( l ( θ ) + θ + π ) / 2 - θ ) = a cos ( ( l ( θ ) + θ ) / 2 ) cos ( ( l ( θ ) - θ ) / 2 ) r_{1}=a\frac{\sin l_{1}(\theta)}{\sin(l_{1}(\theta)-\theta)}=a\frac{\sin((l(% \theta)+\theta+\pi)/2)}{\sin((l(\theta)+\theta+\pi)/2-\theta)}=a\frac{\cos((l(% \theta)+\theta)/2)}{\cos((l(\theta)-\theta)/2)}
  27. r 2 = a sin l 2 ( θ ) sin ( l 2 ( θ ) - θ ) = a sin ( ( l ( θ ) + θ ) / 2 ) sin ( ( l ( θ ) + θ ) / 2 - θ ) = a sin ( ( l ( θ ) + θ ) / 2 ) sin ( ( l ( θ ) - θ ) / 2 ) r_{2}=a\frac{\sin l_{2}(\theta)}{\sin(l_{2}(\theta)-\theta)}=a\frac{\sin((l(% \theta)+\theta)/2)}{\sin((l(\theta)+\theta)/2-\theta)}=a\frac{\sin((l(\theta)+% \theta)/2)}{\sin((l(\theta)-\theta)/2)}
  28. q θ + θ 0 q\theta+\theta_{0}
  29. l ( θ ) = α l(\theta)=\alpha
  30. α \alpha
  31. l 1 ( θ ) = ( θ + α + π ) / 2 l_{1}(\theta)=(\theta+\alpha+\pi)/2
  32. l 2 ( θ ) = ( θ + α ) / 2 l_{2}(\theta)=(\theta+\alpha)/2
  33. r = a cos ( ( α + θ ) / 2 ) cos ( ( α - θ ) / 2 ) r=a\frac{\cos((\alpha+\theta)/2)}{\cos((\alpha-\theta)/2)}
  34. r = a sin ( ( α + θ ) / 2 ) sin ( ( α - θ ) / 2 ) r=a\frac{\sin((\alpha+\theta)/2)}{\sin((\alpha-\theta)/2)}
  35. r = a sin ( 2 θ - α ) sin ( θ - α ) r=a\frac{\sin(2\theta-\alpha)}{\sin(\theta-\alpha)}
  36. α \alpha
  37. r = a sin ( 2 θ + α ) sin ( θ ) r=a\frac{\sin(2\theta+\alpha)}{\sin(\theta)}
  38. y ( x 2 + y 2 ) = b ( x 2 - y 2 ) + 2 c x y y(x^{2}+y^{2})=b(x^{2}-y^{2})+2cxy
  39. α = π / 2 \alpha=\pi/2
  40. r = a sin ( 2 θ - α ) sin ( θ - α ) r=a\frac{\sin(2\theta-\alpha)}{\sin(\theta-\alpha)}
  41. r = a cos 2 θ cos θ = a ( 2 cos θ - sec θ ) r=a\frac{\cos 2\theta}{\cos\theta}=a(2\cos\theta-\sec\theta)
  42. y 2 = x 2 ( a - x ) / ( a + x ) y^{2}=x^{2}(a-x)/(a+x)
  43. x ± i y = - a x\pm iy=-a
  44. l ( θ ) = α + θ l(\theta)=\alpha+\theta
  45. α \alpha
  46. l 1 ( θ ) = θ + ( α + π ) / 2 l_{1}(\theta)=\theta+(\alpha+\pi)/2
  47. l 2 ( θ ) = θ + α / 2 l_{2}(\theta)=\theta+\alpha/2
  48. r = a cos ( θ + α / 2 ) cos ( α / 2 ) r=a\frac{\cos(\theta+\alpha/2)}{\cos(\alpha/2)}
  49. r = a sin ( θ + α / 2 ) sin ( α / 2 ) r=a\frac{\sin(\theta+\alpha/2)}{\sin(\alpha/2)}
  50. π / 4 \pi/4

Structural_similarity.html

  1. x x
  2. y y
  3. SSIM ( x , y ) = ( 2 μ x μ y + c 1 ) ( 2 σ x y + c 2 ) ( μ x 2 + μ y 2 + c 1 ) ( σ x 2 + σ y 2 + c 2 ) \hbox{SSIM}(x,y)=\frac{(2\mu_{x}\mu_{y}+c_{1})(2\sigma_{xy}+c_{2})}{(\mu_{x}^{% 2}+\mu_{y}^{2}+c_{1})(\sigma_{x}^{2}+\sigma_{y}^{2}+c_{2})}
  4. μ x \scriptstyle\mu_{x}
  5. x \scriptstyle x
  6. μ y \scriptstyle\mu_{y}
  7. y \scriptstyle y
  8. σ x 2 \scriptstyle\sigma_{x}^{2}
  9. x \scriptstyle x
  10. σ y 2 \scriptstyle\sigma_{y}^{2}
  11. y \scriptstyle y
  12. σ x y \scriptstyle\sigma_{xy}
  13. x \scriptstyle x
  14. y \scriptstyle y
  15. c 1 = ( k 1 L ) 2 \scriptstyle c_{1}=(k_{1}L)^{2}
  16. c 2 = ( k 2 L ) 2 \scriptstyle c_{2}=(k_{2}L)^{2}
  17. L \scriptstyle L
  18. 2 # b i t s p e r p i x e l - 1 \scriptstyle 2^{\#bits\ per\ pixel}-1
  19. k 1 = 0.01 \scriptstyle k_{1}=0.01
  20. k 2 = 0.03 \scriptstyle k_{2}=0.03
  21. DSSIM ( x , y ) = 1 - SSIM ( x , y ) 2 \hbox{DSSIM}(x,y)=\frac{1-\hbox{SSIM}(x,y)}{2}

Stub_(electronics).html

  1. Z SC = j Z 0 tan ( β l ) Z_{\mathrm{SC}}=jZ_{0}\tan(\beta l)\,\!
  2. Z 0 Z_{0}
  3. β = 2 π / λ \beta=2\pi/\lambda\,
  4. l l
  5. tan ( β l ) \tan(\beta l)
  6. ω \omega
  7. l = 1 β [ ( n + 1 ) π - arctan ( 1 ω C Z 0 ) ] l=\frac{1}{\beta}\left[(n+1)\pi-\arctan\left(\frac{1}{\omega CZ_{0}}\right)\right]
  8. l = 1 β [ n π + arctan ( ω L Z 0 ) ] l=\frac{1}{\beta}\left[n\pi+\arctan\left(\frac{\omega L}{Z_{0}}\right)\right]
  9. Z OC = - j Z 0 cot ( β l ) Z_{\mathrm{OC}}=-jZ_{0}\cot(\beta l)\,\!
  10. cot ( β l ) \cot(\beta l)
  11. ω \omega
  12. l = 1 β [ ( n + 1 ) π - \arccot ( ω L Z 0 ) ] l=\frac{1}{\beta}\left[(n+1)\pi-\arccot\left(\frac{\omega L}{Z_{0}}\right)\right]
  13. l = 1 β [ n π + \arccot ( 1 ω C Z 0 ) ] l=\frac{1}{\beta}\left[n\pi+\arccot\left(\frac{1}{\omega CZ_{0}}\right)\right]
  14. l \scriptstyle l
  15. β l < π / 2 \scriptstyle\beta l<\pi/2
  16. β l = π / 2 \scriptstyle\beta l=\pi/2
  17. β = ω v \beta={\omega\over v}
  18. ω 0 = π v 2 l \omega_{0}=\frac{\pi v}{2l}
  19. ω 0 \scriptstyle\omega_{0}\,
  20. n ω 0 \scriptstyle n\omega_{0}\,
  21. β l = π \scriptstyle\beta l=\pi
  22. β l = 3 π / 2 \scriptstyle\beta l=3\pi/2\,
  23. π / 2 \scriptstyle\pi/2
  24. 3 π / 2 \scriptstyle 3\pi/2

SU(6)_(physics).html

  1. 15 6 ¯ 6 ¯ 15\oplus\bar{6}\oplus\bar{6}
  2. 6 ¯ H \bar{6}_{H}
  3. S U ( 6 ) 6 H / 6 ¯ H S U ( 5 ) × U ( 1 ) SU(6)\begin{matrix}6_{H}/\bar{6}_{H}\\ \longrightarrow\end{matrix}SU(5)\times U(1)
  4. 15 10 5 15\rightarrow 10\oplus 5
  5. 6 ¯ 5 ¯ 1 \bar{6}\rightarrow\bar{5}\oplus 1
  6. 6 ¯ 6 ¯ H 15 \bar{6}\bar{6}_{H}15
  7. 5 ¯ \bar{5}

Subfactor.html

  1. \subseteq
  2. \cap
  3. X X * X , X * X X * , X * X X , X X * X * . X\boxtimes X^{*}\boxtimes\cdots\boxtimes X,\,\,X^{*}\boxtimes X\boxtimes\cdots% \boxtimes X^{*},\,\,X^{*}\boxtimes X\boxtimes\cdots\boxtimes X,\,\,X\boxtimes X% ^{*}\boxtimes\cdots\boxtimes X^{*}.
  4. \subseteq
  5. ( 𝐂 End X * X X * ) ′′ ( End X X * X X * ) ′′ , (\mathbf{C}\otimes\mathrm{End}\,X^{*}\boxtimes X\boxtimes X^{*}\boxtimes\cdots% )^{\prime\prime}\subseteq(\mathrm{End}\,X\boxtimes X^{*}\boxtimes X\boxtimes X% ^{*}\boxtimes\cdots)^{\prime\prime},

Subjective_video_quality.html

  1. 2 - 4 + 5 = 3 2-4+5=3

Subnormal_subgroup.html

  1. H H
  2. k k
  3. G G
  4. H = H 0 , H 1 , H 2 , , H k = G H=H_{0},H_{1},H_{2},\ldots,H_{k}=G
  5. G G
  6. H i H_{i}
  7. H i + 1 H_{i+1}
  8. i i
  9. k k
  10. k k

Substring.html

  1. S S
  2. S S^{\prime}
  3. S S
  4. S S
  5. S S
  6. S S
  7. S S
  8. S S
  9. T = t 1 t n T=t_{1}\dots t_{n}
  10. T ^ = t 1 + i t m + i \hat{T}=t_{1+i}\dots t_{m+i}
  11. 0 i 0\leq i
  12. m + i n m+i\leq n
  13. T ^ \hat{T}
  14. T T
  15. P P
  16. T T
  17. n n
  18. n + 1 n+1
  19. ( n + 1 2 ) = n ( n + 1 ) 2 {\textstyle\left({{n+1}\atop{2}}\right)}=\tfrac{n(n+1)}{2}
  20. T = t 1 t n T=t_{1}\dots t_{n}
  21. T ^ = t 1 t m \widehat{T}=t_{1}\dots t_{m}
  22. m n m\leq n
  23. 0 m < n 0\leq m<n
  24. 0 < m < n 0<m<n
  25. T ^ T \widehat{T}\sqsubseteq T
  26. T ^ \widehat{T}
  27. T T
  28. k k
  29. P = { s 1 , s 2 , s 3 , s k } P=\{s_{1},s_{2},s_{3},\dots s_{k}\}
  30. P P
  31. P P
  32. P P
  33. P P
  34. P = { abcc , efab , bccla } P=\{\,\text{abcc},\,\text{efab},\,\text{bccla}\}
  35. bcclabccefab \,\text{bcclabccefab}
  36. P P
  37. efabccla \,\text{efabccla}
  38. P P

Substring_index.html

  1. S S
  2. n n
  3. D = { S 1 , S 2 , , S d } D=\{S^{1},S^{2},\dots,S^{d}\}
  4. n n
  5. P P
  6. o ( n ) o(n)

Substructure.html

  1. \subseteq
  2. \cap
  3. \cap

Superconformal_algebra.html

  1. 𝒩 = 1 \mathcal{N}=1
  2. P μ P_{\mu}
  3. D D
  4. M μ ν M_{\mu\nu}
  5. K μ K_{\mu}
  6. A A
  7. T j i T^{i}_{j}
  8. Q α i Q^{\alpha i}
  9. Q ¯ i α ˙ \overline{Q}^{\dot{\alpha}}_{i}
  10. S i α S^{\alpha}_{i}
  11. S ¯ α ˙ i {\overline{S}}^{\dot{\alpha}i}
  12. μ , ν , ρ , \mu,\nu,\rho,\dots
  13. α , β , \alpha,\beta,\dots
  14. α ˙ , β ˙ , \dot{\alpha},\dot{\beta},\dots
  15. i , j , i,j,\dots
  16. [ M μ ν , M ρ σ ] = η ν ρ M μ σ - η μ ρ M ν σ + η ν σ M ρ μ - η μ σ M ρ ν [M_{\mu\nu},M_{\rho\sigma}]=\eta_{\nu\rho}M_{\mu\sigma}-\eta_{\mu\rho}M_{\nu% \sigma}+\eta_{\nu\sigma}M_{\rho\mu}-\eta_{\mu\sigma}M_{\rho\nu}
  17. [ M μ ν , P ρ ] = η ν ρ P μ - η μ ρ P ν [M_{\mu\nu},P_{\rho}]=\eta_{\nu\rho}P_{\mu}-\eta_{\mu\rho}P_{\nu}
  18. [ M μ ν , K ρ ] = η ν ρ K μ - η μ ρ K ν [M_{\mu\nu},K_{\rho}]=\eta_{\nu\rho}K_{\mu}-\eta_{\mu\rho}K_{\nu}
  19. [ M μ ν , D ] = 0 [M_{\mu\nu},D]=0
  20. [ D , P ρ ] = - P ρ [D,P_{\rho}]=-P_{\rho}
  21. [ D , K ρ ] = + K ρ [D,K_{\rho}]=+K_{\rho}
  22. [ P μ , K ν ] = - 2 M μ ν + 2 η μ ν D [P_{\mu},K_{\nu}]=-2M_{\mu\nu}+2\eta_{\mu\nu}D
  23. [ K n , K m ] = 0 [K_{n},K_{m}]=0
  24. [ P n , P m ] = 0 [P_{n},P_{m}]=0
  25. { Q α i , Q ¯ β ˙ j } = 2 δ i j σ α β ˙ μ P μ \left\{Q_{\alpha i},\overline{Q}_{\dot{\beta}}^{j}\right\}=2\delta^{j}_{i}% \sigma^{\mu}_{\alpha\dot{\beta}}P_{\mu}
  26. { Q , Q } = { Q ¯ , Q ¯ } = 0 \left\{Q,Q\right\}=\left\{\overline{Q},\overline{Q}\right\}=0
  27. { S α i , S ¯ β ˙ j } = 2 δ j i σ α β ˙ μ K μ \left\{S_{\alpha}^{i},\overline{S}_{\dot{\beta}j}\right\}=2\delta^{i}_{j}% \sigma^{\mu}_{\alpha\dot{\beta}}K_{\mu}
  28. { S , S } = { S ¯ , S ¯ } = 0 \left\{S,S\right\}=\left\{\overline{S},\overline{S}\right\}=0
  29. { Q , S } = \left\{Q,S\right\}=
  30. { Q , S ¯ } = { Q ¯ , S } = 0 \left\{Q,\overline{S}\right\}=\left\{\overline{Q},S\right\}=0
  31. [ A , M ] = [ A , D ] = [ A , P ] = [ A , K ] = 0 [A,M]=[A,D]=[A,P]=[A,K]=0
  32. [ T , M ] = [ T , D ] = [ T , P ] = [ T , K ] = 0 [T,M]=[T,D]=[T,P]=[T,K]=0
  33. [ A , Q ] = - 1 2 Q [A,Q]=-\frac{1}{2}Q
  34. [ A , Q ¯ ] = 1 2 Q ¯ [A,\overline{Q}]=\frac{1}{2}\overline{Q}
  35. [ A , S ] = 1 2 S [A,S]=\frac{1}{2}S
  36. [ A , S ¯ ] = - 1 2 S ¯ [A,\overline{S}]=-\frac{1}{2}\overline{S}
  37. [ T j i , Q k ] = - δ k i Q j [T^{i}_{j},Q_{k}]=-\delta^{i}_{k}Q_{j}
  38. [ T j i , Q ¯ k ] = δ j k Q ¯ i [T^{i}_{j},{\overline{Q}}^{k}]=\delta^{k}_{j}{\overline{Q}}^{i}
  39. [ T j i , S k ] = δ j k S i [T^{i}_{j},S^{k}]=\delta^{k}_{j}S^{i}
  40. [ T j i , S ¯ k ] = - δ k i S ¯ j [T^{i}_{j},\overline{S}_{k}]=-\delta^{i}_{k}\overline{S}_{j}
  41. [ D , Q ] = - 1 2 Q [D,Q]=-\frac{1}{2}Q
  42. [ D , Q ¯ ] = - 1 2 Q ¯ [D,\overline{Q}]=-\frac{1}{2}\overline{Q}
  43. [ D , S ] = 1 2 S [D,S]=\frac{1}{2}S
  44. [ D , S ¯ ] = 1 2 S ¯ [D,\overline{S}]=\frac{1}{2}\overline{S}
  45. [ P , Q ] = [ P , Q ¯ ] = 0 [P,Q]=[P,\overline{Q}]=0
  46. [ K , S ] = [ K , S ¯ ] = 0 [K,S]=[K,\overline{S}]=0

Supercontinent_cycle.html

  1. d d
  2. t t
  3. d ( t ) = ( 2 / π ) a eff T 1 κ t + d r d(t)=(2/\sqrt{\pi})a_{\rm eff}T_{1}\sqrt{\kappa t}+d_{r}
  4. κ \kappa
  5. a eff a_{\rm eff}
  6. T 1 T_{1}
  7. d r d_{r}
  8. d ( t ) = 350 t + 2500 d(t)=350\sqrt{t}+2500
  9. d ( t ) = 390 t + 2500 d(t)=390\sqrt{t}+2500
  10. d d
  11. t t

Superdense_coding.html

  1. | B 00 = 1 2 ( | 0 A | 0 B + | 1 A | 1 B ) |B_{00}\rangle=\frac{1}{\sqrt{2}}(|0\rangle_{A}\otimes|0\rangle_{B}+|1\rangle_% {A}\otimes|1\rangle_{B})
  2. | B 00 = 1 2 ( | 0 A 0 B + | 1 A 1 B ) |B_{00}\rangle=\frac{1}{\sqrt{2}}(|0_{A}0_{B}\rangle+|1_{A}1_{B}\rangle)
  3. X X
  4. X = [ 0 1 1 0 ] X=\begin{bmatrix}0&1\\ 1&0\\ \end{bmatrix}
  5. X X
  6. X X
  7. | B 01 = 1 2 ( | 1 A 0 B + | 0 A 1 B ) |B_{01}\rangle=\frac{1}{\sqrt{2}}(|1_{A}0_{B}\rangle+|0_{A}1_{B}\rangle)
  8. Z Z
  9. Z = [ 1 0 0 - 1 ] Z=\begin{bmatrix}1&0\\ 0&-1\\ \end{bmatrix}
  10. Z Z
  11. | B 10 = 1 2 ( | 0 A 0 B - | 1 A 1 B ) |B_{10}\rangle=\frac{1}{\sqrt{2}}(|0_{A}0_{B}\rangle-|1_{A}1_{B}\rangle)
  12. X Z XZ
  13. X Z XZ
  14. | B 11 = 1 2 ( | 1 A 0 B - | 0 A 1 B ) |B_{11}\rangle=\frac{1}{\sqrt{2}}(|1_{A}0_{B}\rangle-|0_{A}1_{B}\rangle)
  15. X , Z , I , X Z ( = - i Y ) X,Z,I,XZ(=-iY)
  16. B 00 , B 01 , B 10 , B 11 B_{00},B_{01},B_{10},B_{11}
  17. C N O T CNOT
  18. H I H\otimes I
  19. B 00 B_{00}
  20. | 00 |00\rangle
  21. B 01 B_{01}
  22. | 01 |01\rangle
  23. B 10 B_{10}
  24. | 10 |10\rangle
  25. B 11 B_{11}
  26. | 11 |11\rangle
  27. Φ x \;\Phi_{x}
  28. ω ( Φ x I ) ( ω ) \omega\rightarrow(\Phi_{x}\otimes I)(\omega)
  29. Tr ( Φ x I ) ( ω ) F y . \operatorname{Tr}\;(\Phi_{x}\otimes I)(\omega)\cdot F_{y}.
  30. Tr ( Φ x I ) ( ω ) F y = δ x y \operatorname{Tr}\;(\Phi_{x}\otimes I)(\omega)\cdot F_{y}=\delta_{xy}

Superfield.html

  1. θ 1 , θ 2 , θ ¯ 1 , θ ¯ 2 \theta^{1},\theta^{2},\bar{\theta}^{1},\bar{\theta}^{2}
  2. F θ 1 θ 2 F\theta^{1}\theta^{2}

Superlens.html

  1. E ( x , y , z , t ) = k x , k y A ( k x , k y ) e i ( k z z + k y y + k x x - ω t ) E(x,y,z,t)=\sum_{k_{x},k_{y}}A(k_{x},k_{y})e^{i\left(k_{z}z+k_{y}y+k_{x}x-% \omega t\right)}
  2. k z k_{z}
  3. k x , k y k_{x},k_{y}
  4. k z = ω 2 c 2 - ( k x 2 + k y 2 ) k_{z}=\sqrt{\frac{\omega^{2}}{c^{2}}-\left(k_{x}^{2}+k_{y}^{2}\right)}
  5. k z k_{z}
  6. k x 2 + k y 2 > ω 2 c 2 k_{x}^{2}+k_{y}^{2}>\frac{\omega^{2}}{c^{2}}
  7. k z k_{z}
  8. k m a x ω c = 2 π λ k_{max}\approx\frac{\omega}{c}=\frac{2\pi}{\lambda}
  9. Δ x m i n λ \Delta x_{min}\approx\lambda
  10. k z = - ω 2 c 2 - ( k x 2 + k y 2 ) k^{\prime}_{z}=-\sqrt{\frac{\omega^{2}}{c^{2}}-\left(k_{x}^{2}+k_{y}^{2}\right)}

Supermatrix.html

  1. X = [ X 00 X 01 X 10 X 11 ] X=\begin{bmatrix}X_{00}&X_{01}\\ X_{10}&X_{11}\end{bmatrix}
  2. [ even odd odd even ] \begin{bmatrix}\mathrm{even}&\mathrm{odd}\\ \mathrm{odd}&\mathrm{even}\end{bmatrix}
  3. [ odd even even odd ] \begin{bmatrix}\mathrm{odd}&\mathrm{even}\\ \mathrm{even}&\mathrm{odd}\end{bmatrix}
  4. [ X 00 X 01 X 10 X 11 ] [ Y 00 Y 01 Y 10 Y 11 ] = [ X 00 Y 00 + X 01 Y 10 X 00 Y 01 + X 01 Y 11 X 10 Y 00 + X 11 Y 10 X 10 Y 01 + X 11 Y 11 ] . \begin{bmatrix}X_{00}&X_{01}\\ X_{10}&X_{11}\end{bmatrix}\begin{bmatrix}Y_{00}&Y_{01}\\ Y_{10}&Y_{11}\end{bmatrix}=\begin{bmatrix}X_{00}Y_{00}+X_{01}Y_{10}&X_{00}Y_{0% 1}+X_{01}Y_{11}\\ X_{10}Y_{00}+X_{11}Y_{10}&X_{10}Y_{01}+X_{11}Y_{11}\end{bmatrix}.
  5. Z i j = X i 0 Y 0 j + X i 1 Y 1 j . Z_{ij}=X_{i0}Y_{0j}+X_{i1}Y_{1j}.\,
  6. α X = [ α X 00 α X 01 α ^ X 10 α ^ X 11 ] \alpha\cdot X=\begin{bmatrix}\alpha\,X_{00}&\alpha\,X_{01}\\ \hat{\alpha}\,X_{10}&\hat{\alpha}\,X_{11}\end{bmatrix}
  7. α ^ \hat{\alpha}
  8. α ^ = ( - 1 ) | α | α . \hat{\alpha}=(-1)^{|\alpha|}\alpha.
  9. X α = [ X 00 α X 01 α ^ X 10 α X 11 α ^ ] . X\cdot\alpha=\begin{bmatrix}X_{00}\,\alpha&X_{01}\,\hat{\alpha}\\ X_{10}\,\alpha&X_{11}\,\hat{\alpha}\end{bmatrix}.
  10. α ^ = α \hat{\alpha}=\alpha
  11. α X = ( - 1 ) | α | | X | X α . \alpha\cdot X=(-1)^{|\alpha||X|}X\cdot\alpha.
  12. T : R p | q R r | s T:R^{p|q}\to R^{r|s}\,
  13. T : M N T:M\to N\,
  14. T ( e i ) = k = 1 r + s f k T k i . T(e_{i})=\sum_{k=1}^{r+s}f_{k}\,{T^{k}}_{i}.
  15. M = M 0 M 1 N = N 0 N 1 . M=M_{0}\oplus M_{1}\qquad N=N_{0}\oplus N_{1}.
  16. X = [ A B C D ] X=\begin{bmatrix}A&B\\ C&D\end{bmatrix}
  17. X s t = [ A t ( - 1 ) | X | C t - ( - 1 ) | X | B t D t ] X^{st}=\begin{bmatrix}A^{t}&(-1)^{|X|}C^{t}\\ -(-1)^{|X|}B^{t}&D^{t}\end{bmatrix}
  18. ( X s t ) s t = [ A - B - C D ] . (X^{st})^{st}=\begin{bmatrix}A&-B\\ -C&D\end{bmatrix}.
  19. ( X Y ) s t = ( - 1 ) | X | | Y | Y s t X s t . (XY)^{st}=(-1)^{|X||Y|}Y^{st}X^{st}.\,
  20. X = [ A B C D ] X=\begin{bmatrix}A&B\\ C&D\end{bmatrix}
  21. X π = [ D C B A ] . X^{\pi}=\begin{bmatrix}D&C\\ B&A\end{bmatrix}.
  22. ( X + Y ) π = X π + Y π (X+Y)^{\pi}=X^{\pi}+Y^{\pi}\,
  23. ( X Y ) π = X π Y π (XY)^{\pi}=X^{\pi}Y^{\pi}\,
  24. ( α X ) π = α ^ X π (\alpha\cdot X)^{\pi}=\hat{\alpha}\cdot X^{\pi}
  25. ( X α ) π = X π α ^ (X\cdot\alpha)^{\pi}=X^{\pi}\cdot\hat{\alpha}
  26. π 2 = 1 \pi^{2}=1\,
  27. π s t π = ( s t ) 3 \pi\circ st\circ\pi=(st)^{3}
  28. str ( X ) = tr ( X 00 ) - ( - 1 ) | X | tr ( X 11 ) \mathrm{str}(X)=\mathrm{tr}(X_{00})-(-1)^{|X|}\mathrm{tr}(X_{11})\,
  29. str ( X Y ) = ( - 1 ) | X | | Y | str ( Y X ) \mathrm{str}(XY)=(-1)^{|X||Y|}\mathrm{str}(YX)\,
  30. Ber ( X ) = det ( X 00 - X 01 X 11 - 1 X 10 ) det ( X 11 ) - 1 . \mathrm{Ber}(X)=\det(X_{00}-X_{01}X_{11}^{-1}X_{10})\det(X_{11})^{-1}.
  31. Ber ( e X ) = e str ( X ) . \mathrm{Ber}(e^{X})=e^{\mathrm{str(X)}}.\,

Supersingular_elliptic_curve.html

  1. K ¯ \overline{K}
  2. K ¯ \overline{K}
  3. E ( K ¯ ) E(\overline{K})
  4. [ n ] : E E [n]:E\to E
  5. E [ n ] E[n]
  6. E [ p r ] ( K ¯ ) { 0 or / p r E[p^{r}](\overline{K})\cong\begin{cases}0&\mbox{or}\\ \mathbb{Z}/p^{r}\mathbb{Z}\end{cases}
  7. K ¯ \overline{K}
  8. K ¯ \overline{K}
  9. K ¯ \overline{K}
  10. F : E E F:E\to E
  11. F * : H 1 ( E , 𝒪 E ) H 1 ( E , 𝒪 E ) F^{*}:H^{1}(E,\mathcal{O}_{E})\to H^{1}(E,\mathcal{O}_{E})
  12. F * F^{*}
  13. F : E E F:E\to E
  14. F * : H 0 ( E , Ω E 1 ) H 0 ( E , Ω E 1 ) F^{*}:H^{0}(E,\Omega^{1}_{E})\to H^{0}(E,\Omega^{1}_{E})
  15. F * F^{*}
  16. y 2 = x ( x - 1 ) ( x - λ ) y^{2}=x(x-1)(x-\lambda)
  17. i = 0 n ( n i ) 2 λ i \sum_{i=0}^{n}{n\choose{i}}^{2}\lambda^{i}
  18. n = ( p - 1 ) / 2 n=(p-1)/2
  19. y 2 + a 3 y = x 3 + a 4 x + a 6 y^{2}+a_{3}y=x^{3}+a_{4}x+a_{6}
  20. y 2 + y = x 3 + x + 1 y^{2}+y=x^{3}+x+1
  21. y 2 + y = x 3 + 1 y^{2}+y=x^{3}+1
  22. y 2 + y = x 3 + x y^{2}+y=x^{3}+x
  23. y 2 + y = x 3 y^{2}+y=x^{3}
  24. x x + ω x\rightarrow x+\omega
  25. y y + x + ω , x x + 1 y\rightarrow y+x+\omega,x\rightarrow x+1
  26. ω 2 + ω + 1 = 0 \omega^{2}+\omega+1=0
  27. y 2 = x 3 + a 4 x + a 6 y^{2}=x^{3}+a_{4}x+a_{6}
  28. y 2 = x 3 - x y^{2}=x^{3}-x
  29. y 2 = x 3 - x + 1 y^{2}=x^{3}-x+1
  30. y 2 = x 3 - x + 2 y^{2}=x^{3}-x+2
  31. y 2 = x 3 + x y^{2}=x^{3}+x
  32. y 2 = x 3 - x y^{2}=x^{3}-x
  33. x x + 1 x\rightarrow x+1
  34. y i y , x - x y\rightarrow iy,x\rightarrow-x
  35. 𝔽 p \mathbb{F}_{p}
  36. y 2 = x 3 + 1 y^{2}=x^{3}+1
  37. p 2 (mod 3) p\equiv 2\,\text{(mod 3)}
  38. y 2 = x 3 + x y^{2}=x^{3}+x
  39. p 3 (mod 4) p\equiv 3\,\text{(mod 4)}
  40. y 2 = x ( x - 1 ) ( x + 2 ) y^{2}=x(x-1)(x+2)
  41. 𝔽 p \mathbb{F}_{p}
  42. p 2 , 3 p\neq 2,3
  43. p 73 p\leq 73

Supersymmetric_gauge_theory.html

  1. Q Q
  2. Q | boson = fermion Q|\,\text{boson}\rangle=\,\text{fermion}
  3. Q | fermion = boson Q|\,\text{fermion}\rangle=\,\text{boson}
  4. 2 {\mathbb{Z}_{2}}
  5. 𝒲 = 𝒲 0 𝒲 1 \mathcal{W}=\mathcal{W}^{0}\oplus\mathcal{W}^{1}
  6. 𝒲 0 \mathcal{W}^{0}
  7. 𝒲 1 \mathcal{W}^{1}
  8. V μ V μ + μ A V_{\mu}\rightarrow V_{\mu}+\partial_{\mu}A
  9. V μ V_{\mu}
  10. A A
  11. N = 1 N=1
  12. N = 1 N=1
  13. θ 1 , θ 2 , θ ¯ 1 , θ ¯ 2 \theta^{1},\theta^{2},\bar{\theta}^{1},\bar{\theta}^{2}
  14. θ θ
  15. D ¯ f = 0 \overline{D}f=0
  16. V = C + i θ χ - i θ ¯ χ ¯ + i 2 θ 2 ( M + i N ) - i 2 θ 2 ¯ ( M - i N ) - θ σ μ θ ¯ v μ + i θ 2 θ ¯ ( λ ¯ + 1 2 σ ¯ μ μ χ ) - i θ ¯ 2 θ ( λ + i 2 σ μ μ χ ¯ ) + 1 2 θ 2 θ ¯ 2 ( D + 1 2 C ) V=C+i\theta\chi-i\overline{\theta}\overline{\chi}+\tfrac{i}{2}\theta^{2}(M+iN)% -\tfrac{i}{2}\overline{\theta^{2}}(M-iN)-\theta\sigma^{\mu}\overline{\theta}v_% {\mu}+i\theta^{2}\overline{\theta}\left(\overline{\lambda}+\tfrac{1}{2}% \overline{\sigma}^{\mu}\partial_{\mu}\chi\right)-i\overline{\theta}^{2}\theta% \left(\lambda+\tfrac{i}{2}\sigma^{\mu}\partial_{\mu}\overline{\chi}\right)+% \tfrac{1}{2}\theta^{2}\overline{\theta}^{2}\left(D+\tfrac{1}{2}\Box C\right)
  17. V V
  18. V ¯ = V \overline{V}=V
  19. V V + Λ + Λ ¯ V\to V+\Lambda+\overline{\Lambda}
  20. Λ Λ
  21. W α - 1 4 D ¯ 2 D α V W_{\alpha}\equiv-\tfrac{1}{4}\overline{D}^{2}D_{\alpha}V
  22. W ¯ α ˙ \overline{W}_{\dot{\alpha}}
  23. C , χ , M C,χ,M
  24. N N
  25. X X
  26. q q
  27. X e q Λ X , X ¯ e q Λ ¯ X X\to e^{q\Lambda}X,\qquad\overline{X}\to e^{q\overline{\Lambda}}X
  28. Λ Λ
  29. Λ ¯ \overline{Λ}
  30. G G
  31. U ( 1 ) U(1)
  32. D M = d M + i q A M D_{M}=d_{M}+iqA_{M}
  33. D ¯ α ˙ X = 0 \overline{D}_{\dot{\alpha}}X=0
  34. { D ¯ α ˙ , D ¯ β ˙ } = F α ˙ β ˙ = 0. \left\{\overline{D}_{\dot{\alpha}},\overline{D}_{\dot{\beta}}\right\}=F_{\dot{% \alpha}\dot{\beta}}=0.
  35. A α ˙ = 0 A_{\dot{\alpha}}=0
  36. d ¯ α ˙ X = 0 \overline{d}_{\dot{\alpha}}X=0
  37. X ¯ X \overline{X}X
  38. X X
  39. d ¯ α ˙ Λ = 0 \overline{d}_{\dot{\alpha}}\Lambda=0
  40. e - V e - Λ ¯ - V - Λ . e^{-V}\to e^{-\overline{\Lambda}-V-\Lambda}.
  41. F α ˙ β F_{\dot{\alpha}\beta}
  42. N > 1 N>1

Surface_charge.html

  1. σ = q A \sigma=\frac{q}{A}
  2. σ = E ϵ 0 \sigma=E\epsilon_{0}
  3. ϵ 0 \epsilon_{0}
  4. C o C_{o}
  5. C = C 0 e - ( ψ z e k T ) C=C_{0}e^{-(\frac{\psi ze}{kT})}
  6. σ = 8 c 0 ϵ ϵ 0 R T sinh ( z e ψ 0 2 k B T ) \sigma=\sqrt{8c_{0}\epsilon\epsilon_{0}RT}\sinh\left(\frac{ze\psi_{0}}{2k_{B}T% }\right)
  7. sinh ( x ) \sinh(x)
  8. sinh ( x ) \sinh(x)
  9. x + x 3 / 3 ! + x+x^{3}/3!+...
  10. \approx
  11. x x
  12. λ D \lambda_{D}
  13. σ = ϵ ϵ 0 ψ 0 λ D \sigma=\frac{\epsilon\epsilon_{0}\psi_{0}}{\lambda_{D}}

Surface_metrology.html

  1. R a = 1 n i = 1 n | y i | R_{a}=\frac{1}{n}\sum_{i=1}^{n}\left|y_{i}\right|
  2. R q = 1 n i = 1 n y i 2 R_{q}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}y_{i}^{2}}
  3. R v = min i y i R_{v}=\min_{i}y_{i}
  4. R p = max i y i R_{p}=\max_{i}y_{i}
  5. R t = R p - R v R_{t}=R_{p}-R_{v}
  6. R s k = 1 n R q 3 i = 1 n y i 3 R_{sk}=\frac{1}{nR_{q}^{3}}\sum_{i=1}^{n}y_{i}^{3}
  7. R k u = 1 n R q 4 i = 1 n y i 4 R_{ku}=\frac{1}{nR_{q}^{4}}\sum_{i=1}^{n}y_{i}^{4}
  8. R S m = 1 n i = 1 n S i RS_{m}=\frac{1}{n}\sum_{i=1}^{n}S_{i}

Surface_of_general_type.html

  1. c 1 2 + c 2 0 ( mod 12 ) c_{1}^{2}+c_{2}\equiv 0\;\;(\mathop{{\rm mod}}12)
  2. c 1 2 0 , c 2 0 c_{1}^{2}\geq 0,c_{2}\geq 0
  3. c 1 2 3 c 2 c_{1}^{2}\leq 3c_{2}
  4. 5 c 1 2 - c 2 + 36 12 q 0 5c_{1}^{2}-c_{2}+36\geq 12q\geq 0
  5. c 1 2 + c 2 0 ( mod 12 ) , c_{1}^{2}+c_{2}\equiv 0\;\;(\mathop{{\rm mod}}12),

Surface_power_density.html

  1. π r 2 \pi r^{2}

Surface_roughness.html

  1. R a R\text{a}
  2. R a R\text{a}
  3. R z R\text{z}
  4. R q R\text{q}
  5. R sk R\text{sk}
  6. R k R\text{k}
  7. R a R\text{a}
  8. n n
  9. y i y_{i}
  10. i th i\text{th}
  11. R a R\text{a}
  12. R a R\text{a}
  13. R a = 1 n i = 1 n | y i | R\text{a}=\frac{1}{n}\sum_{i=1}^{n}\left|y_{i}\right|
  14. R q = 1 n i = 1 n y i 2 R\text{q}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}y_{i}^{2}}
  15. R v = min i y i R\text{v}=\min_{i}y_{i}
  16. R p = max i y i R\text{p}=\max_{i}y_{i}
  17. R t = R p - R v R\text{t}=R\text{p}-R\text{v}
  18. R sk = 1 n R q 3 i = 1 n y i 3 R\text{sk}=\frac{1}{nR\text{q}^{3}}\sum_{i=1}^{n}y_{i}^{3}
  19. R ku = 1 n R q 4 i = 1 n y i 4 R\text{ku}=\frac{1}{nR\text{q}^{4}}\sum_{i=1}^{n}y_{i}^{4}
  20. R zDIN = 1 s i = 1 s R t i R\text{zDIN}=\frac{1}{s}\sum_{i=1}^{s}R_{\,\text{t}i}
  21. s s
  22. R t i R_{\,\text{t}i}
  23. R t R\text{t}
  24. i th i\text{th}
  25. R z R\text{z}
  26. R zJIS = 1 5 i = 1 5 R p i - R v i R\text{zJIS}=\frac{1}{5}\sum_{i=1}^{5}R_{\,\text{p}i}-R_{\,\text{v}i}
  27. R p i R_{\,\text{p}i}
  28. R v i R_{\,\text{v}i}
  29. i th i\text{th}
  30. Δ \Delta
  31. R d q = 1 N i = 1 N Δ i 2 R_{dq}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}\Delta_{i}^{2}}
  32. Δ \Delta
  33. R d a = 1 N i = 1 N | Δ i | R_{da}=\frac{1}{N}\sum_{i=1}^{N}|\Delta_{i}|
  34. Δ i \Delta_{i}\!
  35. Δ i = 1 60 d x ( y i + 3 - 9 y i + 2 + 45 y i + 1 - 45 y i - 1 + 9 y i - 2 - y i - 3 ) \Delta_{i}=\frac{1}{60dx}(y_{i+3}-9y_{i+2}+45y_{i+1}-45y_{i-1}+9y_{i-2}-y_{i-3})
  36. λ \lambda
  37. λ \lambda
  38. λ \lambda
  39. λ \lambda
  40. Δ \Delta
  41. λ \lambda
  42. R a R_{a}
  43. R s k R_{sk}

Swain_equation.html

  1. k H k T = ( k H k D ) 1.442 \frac{k_{H}}{k_{T}}=(\frac{k_{H}}{k_{D}})^{1.442}

Swash.html

  1. Z b e r m = 0.125 H b 5 / 8 ( g T 2 ) 3 / 8 , Zberm=0.125Hb^{5/8}(gT^{2})^{3/8},
  2. Z s t e p = H b T w s , Zstep=\sqrt{HbTws},
  3. λ = g π T 2 t a n β , \lambda=\frac{g}{\pi}T^{2}tan\beta,
  4. λ = f S , \lambda=fS,

Swell_(ocean).html

  1. t = 4 π X / ( g T ) t=4\pi X/(gT)

Sylvester_matrix.html

  1. p ( z ) = p 0 + p 1 z + p 2 z 2 + + p m z m , q ( z ) = q 0 + q 1 z + q 2 z 2 + + q n z n . p(z)=p_{0}+p_{1}z+p_{2}z^{2}+\cdots+p_{m}z^{m},\;q(z)=q_{0}+q_{1}z+q_{2}z^{2}+% \cdots+q_{n}z^{n}.
  2. ( n + m ) × ( n + m ) (n+m)\times(n+m)
  3. ( p m p m - 1 p 1 p 0 0 0 ) . \begin{pmatrix}p_{m}&p_{m-1}&\cdots&p_{1}&p_{0}&0&\cdots&0\end{pmatrix}.
  4. ( q n q n - 1 q 1 q 0 0 0 ) . \begin{pmatrix}q_{n}&q_{n-1}&\cdots&q_{1}&q_{0}&0&\cdots&0\end{pmatrix}.
  5. S p , q = ( p 4 p 3 p 2 p 1 p 0 0 0 0 p 4 p 3 p 2 p 1 p 0 0 0 0 p 4 p 3 p 2 p 1 p 0 q 3 q 2 q 1 q 0 0 0 0 0 q 3 q 2 q 1 q 0 0 0 0 0 q 3 q 2 q 1 q 0 0 0 0 0 q 3 q 2 q 1 q 0 ) . S_{p,q}=\begin{pmatrix}p_{4}&p_{3}&p_{2}&p_{1}&p_{0}&0&0\\ 0&p_{4}&p_{3}&p_{2}&p_{1}&p_{0}&0\\ 0&0&p_{4}&p_{3}&p_{2}&p_{1}&p_{0}\\ q_{3}&q_{2}&q_{1}&q_{0}&0&0&0\\ 0&q_{3}&q_{2}&q_{1}&q_{0}&0&0\\ 0&0&q_{3}&q_{2}&q_{1}&q_{0}&0\\ 0&0&0&q_{3}&q_{2}&q_{1}&q_{0}\end{pmatrix}.
  6. 2 max ( n , m ) × 2 max ( n , m ) 2\,\max(n,m)\times 2\,\max(n,m)
  7. max ( n , m ) \max(n,m)
  8. m > n , m>n,
  9. ( p m p m - 1 p n p 1 p 0 0 0 0 0 q n q 1 q 0 0 0 ) . \begin{pmatrix}p_{m}&p_{m-1}&\cdots&p_{n}&\cdots&p_{1}&p_{0}&0&\cdots&0\\ 0&\cdots&0&q_{n}&\cdots&q_{1}&q_{0}&0&\cdots&0\end{pmatrix}.
  10. m a x ( n , m ) - 2 max(n,m)-2
  11. ( p 4 p 3 p 2 p 1 p 0 0 0 0 0 q 3 q 2 q 1 q 0 0 0 0 0 p 4 p 3 p 2 p 1 p 0 0 0 0 0 q 3 q 2 q 1 q 0 0 0 0 0 p 4 p 3 p 2 p 1 p 0 0 0 0 0 q 3 q 2 q 1 q 0 0 0 0 0 p 4 p 3 p 2 p 1 p 0 0 0 0 0 q 3 q 2 q 1 q 0 ) . \begin{pmatrix}p_{4}&p_{3}&p_{2}&p_{1}&p_{0}&0&0&0\\ 0&q_{3}&q_{2}&q_{1}&q_{0}&0&0&0\\ 0&p_{4}&p_{3}&p_{2}&p_{1}&p_{0}&0&0\\ 0&0&q_{3}&q_{2}&q_{1}&q_{0}&0&0\\ 0&0&p_{4}&p_{3}&p_{2}&p_{1}&p_{0}&0\\ 0&0&0&q_{3}&q_{2}&q_{1}&q_{0}&0\\ 0&0&0&p_{4}&p_{3}&p_{2}&p_{1}&p_{0}\\ 0&0&0&0&q_{3}&q_{2}&q_{1}&q_{0}\\ \end{pmatrix}.
  12. p m m - n p_{m}^{m-n}
  13. m n m\geq n
  14. S p , q T ( x y ) = ( 0 0 ) {S_{p,q}}^{\mathrm{T}}\cdot\begin{pmatrix}x\\ y\end{pmatrix}=\begin{pmatrix}0\\ 0\end{pmatrix}
  15. x x
  16. n n
  17. y y
  18. m m
  19. x , y x,y
  20. n - 1 n-1
  21. m - 1 m-1
  22. x ( z ) p ( z ) + y ( z ) q ( z ) = 0 , x(z)\cdot p(z)+y(z)\cdot q(z)=0,
  23. deg x < deg q \deg x<\deg q
  24. deg y < deg p \deg y<\deg p
  25. deg ( gcd ( p , q ) ) = m + n - rank S p , q \deg(\gcd(p,q))=m+n-\mathrm{rank}~{}S_{p,q}

Symbol_rate.html

  1. T s = 1 f s T_{s}={1\over f_{s}}
  2. f s = R N f_{s}={R\over N}
  3. R = f s log 2 ( M ) R=f_{s}\log_{2}(M)
  4. symbol rate = 18096263 6 3 4 204 188 = 18096263 6 4 3 204 188 = 4363638 \,\text{symbol rate}=\cfrac{18096263}{6\cdot\frac{3}{4}}~{}\cfrac{204}{188}=% \cfrac{18096263}{6}~{}\cfrac{4}{3}~{}\cfrac{204}{188}=4363638

Symmetric_game.html

  1. ( A i , U i ) (A_{i},U_{i})
  2. U i : A i U_{i}:A_{i}\longrightarrow\mathbb{R}
  3. U i , i = 1 , N U_{i},i=1,\ldots N
  4. i i
  5. A 1 = A 2 = = A N A_{1}=A_{2}=\ldots=A_{N}
  6. i i
  7. π \pi
  8. U i ( a 1 , , a i , , a N ) = U π ( i ) ( a π ( 1 ) , , a π ( i ) , , a π ( N ) ) . U_{i}(a_{1},\ldots,a_{i},\ldots,a_{N})=U_{\pi(i)}(a_{\pi(1)},\ldots,a_{\pi(i)}% ,\ldots,a_{\pi(N)}).

Symmetric_product_of_an_algebraic_curve.html

  1. n = 0 i = 0 2 n b i ( Σ n C ) y n u i - n = ( 1 + y ) 2 g ( 1 - u y ) ( 1 - u - 1 y ) \sum_{n=0}^{\infty}\sum_{i=0}^{2n}b_{i}(\Sigma^{n}C)y^{n}u^{i-n}=\frac{(1+y)^{% 2g}}{(1-uy)(1-u^{-1}y)}
  2. n = 0 e ( Σ n C ) p n = ( 1 - p ) 2 g - 2 . \sum_{n=0}^{\infty}e(\Sigma^{n}C)p^{n}=(1-p)^{2g-2}.

Symmetry_(physics).html

  1. δ t \delta t
  2. t t + a t\,\rightarrow t+a
  3. m g h \,mgh
  4. h h
  5. t 0 t_{0}
  6. t 0 + 3 t_{0}+3
  7. r r + a \vec{r}\,\rightarrow\vec{r}+\vec{a}
  8. t - t t\,\rightarrow-t
  9. F = m r ¨ F\,=m\ddot{r}
  10. t t
  11. - t -t
  12. r - r \vec{r}\,\rightarrow-\vec{r}
  13. S O ( 3 ) \,SO(3)
  14. S O ( 3 ) \,SO(3)
  15. S 3 \,S_{3}
  16. δ ϕ ( x ) = h μ ( x ) μ ϕ ( x ) \delta\phi(x)=h^{\mu}(x)\partial_{\mu}\phi(x)
  17. δ ψ α ( x ) = h μ ( x ) μ ψ α ( x ) + μ h ν ( x ) σ μ ν α β ψ β ( x ) \delta\psi^{\alpha}(x)=h^{\mu}(x)\partial_{\mu}\psi^{\alpha}(x)+\partial_{\mu}% h_{\nu}(x)\sigma_{\mu\nu}^{\alpha\beta}\psi^{\beta}(x)
  18. δ A μ ( x ) = h ν ( x ) ν A μ ( x ) + A ν ( x ) ν h μ ( x ) \delta A_{\mu}(x)=h^{\nu}(x)\partial_{\nu}A_{\mu}(x)+A_{\nu}(x)\partial_{\nu}h% _{\mu}(x)
  19. h ( x ) h(x)
  20. h ( x ) h(x)
  21. h μ ( x ) = M μ ν x ν + P μ h^{\mu}(x)=M^{\mu\nu}x_{\nu}+P^{\mu}
  22. δ ψ α ( x ) = λ ( x ) . τ α β ψ β ( x ) \delta\psi^{\alpha}(x)=\lambda(x).\tau^{\alpha\beta}\psi^{\beta}(x)
  23. δ A μ ( x ) = μ λ ( x ) \delta A_{\mu}(x)=\partial_{\mu}\lambda(x)
  24. τ \tau
  25. δ ϕ ( x ) = Ω ( x ) ϕ ( x ) \delta\phi(x)=\Omega(x)\phi(x)
  26. h μ ( x ) = M μ ν x ν + P μ + D x μ + K μ | x | 2 - 2 K ν x ν x μ h^{\mu}(x)=M^{\mu\nu}x_{\nu}+P^{\mu}+Dx_{\mu}+K^{\mu}|x|^{2}-2K^{\nu}x_{\nu}x_% {\mu}

Symmetry_in_mathematics.html

  1. f ( x ) = f ( - x ) . f(x)=f(-x).\,
  2. - f ( x ) = f ( - x ) , -f(x)=f(-x)\,,
  3. f ( x ) + f ( - x ) = 0 . f(x)+f(-x)=0\,.
  4. A = A A=A^{\top}
  5. [ 1 7 3 7 4 - 5 3 - 5 6 ] . \begin{bmatrix}1&7&3\\ 7&4&-5\\ 3&-5&6\end{bmatrix}.
  6. X 1 3 + X 2 3 - 7 X_{1}^{3}+X_{2}^{3}-7
  7. 4 X 1 2 X 2 2 + X 1 3 X 2 + X 1 X 2 3 + ( X 1 + X 2 ) 4 4X_{1}^{2}X_{2}^{2}+X_{1}^{3}X_{2}+X_{1}X_{2}^{3}+(X_{1}+X_{2})^{4}
  8. X 1 X 2 X 3 - 2 X 1 X 2 - 2 X 1 X 3 - 2 X 2 X 3 X_{1}X_{2}X_{3}-2X_{1}X_{2}-2X_{1}X_{3}-2X_{2}X_{3}\,
  9. T ( v 1 , v 2 , , v r ) = T ( v σ 1 , v σ 2 , , v σ r ) T(v_{1},v_{2},\dots,v_{r})=T(v_{\sigma 1},v_{\sigma 2},\dots,v_{\sigma r})
  10. T i 1 i 2 i r = T i σ 1 i σ 2 i σ r . T_{i_{1}i_{2}\dots i_{r}}=T_{i_{\sigma 1}i_{\sigma 2}\dots i_{\sigma r}}.
  11. | x \scriptstyle|x\rangle
  12. | y \scriptstyle|y\rangle
  13. | ψ = x , y A ( x , y ) | x , y |\psi\rangle=\sum_{x,y}A(x,y)|x,y\rangle
  14. A ( x , y ) = ψ | x , y = ψ | ( | x | y ) A(x,y)=\langle\psi|x,y\rangle=\langle\psi|(|x\rangle\otimes|y\rangle)
  15. ψ | ( ( | x + | y ) ( | x + | y ) ) \langle\psi|((|x\rangle+|y\rangle)\otimes(|x\rangle+|y\rangle))\,
  16. | x + | y \scriptstyle|x\rangle+|y\rangle
  17. ψ | x , x + ψ | x , y + ψ | y , x + ψ | y , y \langle\psi|x,x\rangle+\langle\psi|x,y\rangle+\langle\psi|y,x\rangle+\langle% \psi|y,y\rangle\,
  18. ψ | x , y + ψ | y , x = 0 \langle\psi|x,y\rangle+\langle\psi|y,x\rangle=0\,
  19. A ( x , y ) = - A ( y , x ) A(x,y)=-A(y,x)\,

Symplectic_integrator.html

  1. p ˙ = - H q and q ˙ = H p , \dot{p}=-\frac{\partial H}{\partial q}\quad\mbox{and}~{}\quad\dot{q}=\frac{% \partial H}{\partial p},
  2. q q
  3. p p
  4. H H
  5. ( q , p ) (q,p)
  6. d p d q dp\wedge dq
  7. H ( p , q ) = T ( p ) + V ( q ) . ( 1 ) H(p,q)=T(p)+V(q).\qquad\qquad(1)
  8. z = ( q , p ) z=(q,p)
  9. z ˙ = { z , H ( z ) } ( 2 ) \dot{z}=\{z,H(z)\}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)
  10. { , } \{\cdot,\cdot\}
  11. D H = { , H } D_{H}\cdot=\{\cdot,H\}
  12. z ˙ = D H z . \dot{z}=D_{H}z.
  13. z ( τ ) = exp ( τ D H ) z ( 0 ) . ( 3 ) z(\tau)=\exp(\tau D_{H})z(0).\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)
  14. τ D H \tau D_{H}
  15. z ( τ ) = exp [ τ ( D T + D V ) ] z ( 0 ) . ( 4 ) z(\tau)=\exp[\tau(D_{T}+D_{V})]z(0).\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (4)
  16. exp [ τ ( D T + D V ) ] \exp[\tau(D_{T}+D_{V})]
  17. exp [ τ ( D T + D V ) ] = i = 1 k exp ( c i τ D T ) exp ( d i τ D V ) + O ( τ k + 1 ) = exp ( c 1 τ D T ) exp ( d 1 τ D V ) exp ( c k τ D T ) exp ( d k τ D V ) + O ( τ k + 1 ) , ( 5 ) \begin{array}[]{rl}\exp[\tau(D_{T}+D_{V})]&=\prod_{i=1}^{k}\exp(c_{i}\tau D_{T% })\exp(d_{i}\tau D_{V})+O(\tau^{k+1})\\ \\ &=\exp(c_{1}\tau D_{T})\exp(d_{1}\tau D_{V})\dots\exp(c_{k}\tau D_{T})\exp(d_{% k}\tau D_{V})+O(\tau^{k+1})\end{array},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (5)
  18. c i c_{i}
  19. d i d_{i}
  20. k k
  21. i = 1 k c i = i = 1 k d i = 1 \sum_{i=1}^{k}c_{i}=\sum_{i=1}^{k}d_{i}=1
  22. exp ( c i τ D T ) \exp(c_{i}\tau D_{T})
  23. exp ( d i τ D V ) \exp(d_{i}\tau D_{V})
  24. D T 2 z = { { z , T } , T } = { ( q ˙ , 0 ) , T } = ( 0 , 0 ) D_{T}^{2}z=\{\{z,T\},T\}=\{(\dot{q},0),T\}=(0,0)
  25. z z
  26. D T 2 = 0. ( 6 ) D_{T}^{2}=0.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (6)
  27. exp ( a D T ) \exp(aD_{T})
  28. exp ( a D T ) = n = 0 ( a D T ) n n ! , ( 7 ) \exp(aD_{T})=\sum_{n=0}^{\infty}\frac{(aD_{T})^{n}}{n!},\ \ \ \ \ \ \ \ \ \ \ % \ \ \ \ \ (7)
  29. a a
  30. D V D_{V}
  31. D T D_{T}
  32. { exp ( a D T ) = 1 + a D T exp ( a D V ) = 1 + a D V . ( 8 ) \left\{\begin{array}[]{c}\exp(aD_{T})=1+aD_{T}\\ \exp(aD_{V})=1+aD_{V}\end{array}\right..\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (8)
  33. exp ( c i τ D T ) \exp(c_{i}\tau D_{T})
  34. ( q p ) ( q + τ c i T p ( p ) p ) \begin{pmatrix}q\\ p\end{pmatrix}\mapsto\begin{pmatrix}q+\tau c_{i}\frac{\partial T}{\partial p}(% p)\\ p\end{pmatrix}
  35. exp ( d i τ D V ) \exp(d_{i}\tau D_{V})
  36. ( q p ) ( q p - τ d i V q ( q ) ) . \begin{pmatrix}q\\ p\end{pmatrix}\mapsto\begin{pmatrix}q\\ p-\tau d_{i}\frac{\partial V}{\partial q}(q)\\ \end{pmatrix}.
  37. k = 1 k=1
  38. c 1 = d 1 = 1. c_{1}=d_{1}=1.\
  39. k = 2 k=2
  40. c 1 = c 2 = 1 2 , d 1 = 1 , d 2 = 0. c_{1}=c_{2}=\tfrac{1}{2},\qquad d_{1}=1,\qquad d_{2}=0.
  41. k = 3 k=3
  42. c 1 = 1 , c 2 = - 2 3 , c 3 = 2 3 c_{1}=1,\qquad c_{2}=-\tfrac{2}{3},\qquad c_{3}=\tfrac{2}{3}
  43. d 1 = - 1 24 , d 2 = 3 4 , d 3 = 7 24 . d_{1}=-\tfrac{1}{24},\qquad d_{2}=\tfrac{3}{4},\qquad d_{3}=\tfrac{7}{24}.
  44. k = 4 k=4
  45. c 1 = c 4 = 1 2 ( 2 - 2 1 / 3 ) , c 2 = c 3 = 1 - 2 1 / 3 2 ( 2 - 2 1 / 3 ) , c_{1}=c_{4}=\frac{1}{2(2-2^{1/3})},\ \ c_{2}=c_{3}=\frac{1-2^{1/3}}{2(2-2^{1/3% })},
  46. d 1 = d 3 = 1 2 - 2 1 / 3 , d 2 = - 2 1 / 3 2 - 2 1 / 3 , d 4 = 0. d_{1}=d_{3}=\frac{1}{2-2^{1/3}},\ \ d_{2}=-\frac{2^{1/3}}{2-2^{1/3}},\ \ d_{4}% =0.

Synthetic_fuel.html

  1. n C + ( n + 1 ) H 2 C n H 2 n + 2 n{\rm C}+(n+1){\rm H}_{2}\rightarrow{\rm C}_{n}{\rm H}_{2n+2}

Synthetic_setae.html

  1. μ \mu
  2. Φ ( R , D ) = - ρ 1 ρ 2 π 2 α R 6 D \Phi(R,D)=-\frac{\rho_{1}\rho_{2}\pi^{2}\alpha R}{6D}
  3. ρ 1 \rho_{1}
  4. ρ 2 \rho_{2}
  5. μ = F a d h e s i o n / F p r e l o a d \mu^{^{\prime}}=F_{adhesion}/F_{preload}
  6. F p r e l o a d F_{preload}
  7. F a d h e s i o n F_{adhesion}
  8. μ \mu
  9. μ \mu
  10. μ \mu
  11. μ \mu
  12. μ \mu

Syntractrix.html

  1. x + b 2 - y 2 = a ln b + b 2 - y 2 y . x+\sqrt{b^{2}-y^{2}}=a\ln\frac{b+\sqrt{b^{2}-y^{2}}}{y}.

Systolic_geometry.html

  1. L 2 π 4 area ( P ) . L^{2}\leq\frac{\pi}{4}\mathrm{area}(\partial P).
  2. π A \sqrt{\pi A}
  3. sys 2 2 3 area \mathrm{sys}^{2}\leq\frac{2}{\sqrt{3}}\cdot\mathrm{area}
  4. sys 2 π 2 area \mathrm{sys}^{2}\leq\frac{\pi}{2}\cdot\mathrm{area}
  5. area - 3 2 sys 2 var ( f ) , \mathrm{area}-\frac{\sqrt{3}}{2}\mathrm{sys}^{2}\geq\mathrm{var}(f),
  6. sys π 1 n C n vol ( M ) , \operatorname{sys\pi}_{1}{}^{n}\leq C_{n}\operatorname{vol}(M),
  7. H n ( M ; A ) = A H_{n}(M;A)=A
  8. FillRad ( M E ) = inf { ϵ > 0 | ι ϵ ( [ M ] ) = 0 H n ( U ϵ M ) } , \mathrm{FillRad}(M\subset E)=\inf\left\{\epsilon>0\left|\;\iota_{\epsilon}([M]% )=0\in H_{n}(U_{\epsilon}M)\right.\right\},
  9. \|\;\|
  10. d ( x , y ) = f x - f y , d(x,y)=\|f_{x}-f_{y}\|,
  11. FillRad ( M ) = FillRad ( M L ( M ) ) . \mathrm{FillRad}(M)=\mathrm{FillRad}\left(M\subset L^{\infty}(M)\right).
  12. sys π 1 6 FillRad ( M ) , \mathrm{sys\pi}_{1}\leq 6\;\mathrm{FillRad}(M),
  13. FillRad C n vol n ( M ) 1 / n , \mathrm{FillRad}\leq C_{n}\mathrm{vol}_{n}{}^{1/n}(M),
  14. stsys 2 n n ! vol 2 n ( n ) \mathrm{stsys}_{2}{}^{n}\leq n!\;\mathrm{vol}_{2n}(\mathbb{CP}^{n})
  15. stsys 2 = λ 1 ( H 2 ( M , ) , ) , \mathrm{stsys}_{2}=\lambda_{1}\left(H_{2}(M,\mathbb{Z})_{\mathbb{R}},\|\;\|% \right),
  16. \|\;\|
  17. sys π 1 ( Σ g ) 4 3 log g , \mathrm{sys}\pi_{1}(\Sigma_{g})\geq\frac{4}{3}\log g,
  18. M ¯ \bar{M}
  19. M ~ \tilde{M}
  20. h c h h\to\int_{c}h
  21. M ~ E * = H 1 ( M , 𝐑 ) \tilde{M}\to E^{*}=H_{1}(M,\mathbf{R})
  22. A ¯ M : M ¯ E * , c ( h c h ) , \overline{A}_{M}:\overline{M}\to E^{*},\;\;c\mapsto\left(h\mapsto\int_{c}h% \right),
  23. M ¯ \overline{M}
  24. A M : M J 1 ( M ) , A_{M}:M\to J_{1}(M),
  25. stsys 1 n γ n vol n ( M ) , \mathrm{stsys}_{1}{}^{n}\leq\gamma_{n}\mathrm{vol}_{n}(M),
  26. γ n \gamma_{n}

Sørensen_formol_titration.html

  1. 4 NH 4 + + 6 HCHO + 4 H 2 O ( CH2 ) 6 N 4 + 6 H 2 O + 4 H 3 O + \mathrm{4\ NH_{4}^{+}\ +\ 6\ HCHO\ +\ 4\ H_{2}O\longrightarrow\ (CH2)_{6}N_{4}% +\ 6\ H_{2}O\ +\ 4\ H_{3}O^{+}}

T-antenna.html

  1. R R = 80 π 2 ( h λ ) 2 R_{R}=80\pi^{2}\left(\frac{h}{\lambda}\right)^{2}\,
  2. P = 80 π 2 ( h I 0 λ ) 2 P=80\pi^{2}\left(\frac{hI_{0}}{\lambda}\right)^{2}\,
  3. z = R C + R D + R L + R G + R R + j ω L - 1 j ω C z=R_{C}+R_{D}+R_{L}+R_{G}+R_{R}+j\omega L-\frac{1}{j\omega C}\,
  4. z 0 = R C + R D + R L + R G + R R z_{0}=R_{C}+R_{D}+R_{L}+R_{G}+R_{R}\,
  5. η = R R R C + R D + R L + R G + R R \eta=\frac{R_{R}}{R_{C}+R_{D}+R_{L}+R_{G}+R_{R}}\,

T-norm.html

  1. * *
  2. * *
  3. * *
  4. * *
  5. * *
  6. * *
  7. min ( a , b ) = min { a , b } , \top_{\mathrm{min}}(a,b)=\min\{a,b\},
  8. prod ( a , b ) = a b \top_{\mathrm{prod}}(a,b)=a\cdot b
  9. Luk ( a , b ) = max { 0 , a + b - 1 } . \top_{\mathrm{Luk}}(a,b)=\max\{0,a+b-1\}.
  10. H 0 ( a , b ) = { 0 if a = b = 0 a b a + b - a b otherwise \top_{\mathrm{H}_{0}}(a,b)=\begin{cases}0&\mbox{if }~{}a=b=0\\ \frac{ab}{a+b-ab}&\mbox{otherwise}\end{cases}
  11. D ( a , b ) ( a , b ) min ( a , b ) , \top_{\mathrm{D}}(a,b)\leq\top(a,b)\leq\mathrm{\top_{min}}(a,b),
  12. \top
  13. ( x , y ) = f - 1 ( Luk ( f ( x ) , f ( y ) ) ) . \top(x,y)=f^{-1}(\top_{\mathrm{Luk}}(f(x),f(y))).
  14. \top
  15. \Rightarrow
  16. ( z , x ) y \top(z,x)\leq y
  17. z ( x y ) z\leq(x\Rightarrow y)
  18. \top
  19. \vec{\top}
  20. \Rightarrow
  21. \top
  22. ( x y ) = sup { z ( z , x ) y } . (x\Rightarrow y)=\sup\{z\mid\top(z,x)\leq y\}.
  23. ( x y ) = 1 (x\Rightarrow y)=1
  24. x y x\leq y
  25. ( 1 y ) = y . (1\Rightarrow y)=y.
  26. * *
  27. \Rightarrow
  28. min ( x , y ) x * ( x y ) max ( x , y ) = min ( ( x y ) y , ( y x ) x ) . \begin{array}[]{rcl}\min(x,y)&\geq&x*(x\Rightarrow y)\\ \max(x,y)&=&\min((x\Rightarrow y)\Rightarrow y,(y\Rightarrow x)\Rightarrow x).% \end{array}
  29. * *
  30. ( a , b ) = 1 - ( 1 - a , 1 - b ) . \bot(a,b)=1-\top(1-a,1-b).
  31. max ( a , b ) = max { a , b } \bot_{\mathrm{max}}(a,b)=\max\{a,b\}
  32. sum ( a , b ) = a + b - a b \bot_{\mathrm{sum}}(a,b)=a+b-a\cdot b
  33. Luk ( a , b ) = min { a + b , 1 } \bot_{\mathrm{Luk}}(a,b)=\min\{a+b,1\}
  34. D ( a , b ) = { b if a = 0 a if b = 0 1 otherwise, \bot_{\mathrm{D}}(a,b)=\begin{cases}b&\mbox{if }~{}a=0\\ a&\mbox{if }~{}b=0\\ 1&\mbox{otherwise,}\end{cases}
  35. H 2 ( a , b ) = a + b 1 + a b \bot_{\mathrm{H}_{2}}(a,b)=\frac{a+b}{1+ab}
  36. max ( a , b ) ( a , b ) D ( a , b ) \mathrm{\bot_{max}}(a,b)\leq\bot(a,b)\leq\bot_{\mathrm{D}}(a,b)
  37. \bot

T_cell_receptor.html

  1. K D K_{D}

Table_of_Clebsch–Gordan_coefficients.html

  1. j 1 j_{1}
  2. j 2 j_{2}
  3. j j
  4. | ( j 1 j 2 ) j m = m 1 = - j 1 j 1 m 2 = - j 2 j 2 | j 1 m 1 j 2 m 2 j 1 j 2 ; m 1 m 2 | j 1 j 2 ; j m |(j_{1}j_{2})jm\rangle=\sum_{m_{1}=-j_{1}}^{j_{1}}\sum_{m_{2}=-j_{2}}^{j_{2}}|% j_{1}m_{1}j_{2}m_{2}\rangle\langle j_{1}j_{2};m_{1}m_{2}|j_{1}j_{2};jm\rangle
  5. j 1 j 2 ; m 1 m 2 | j 1 j 2 ; j m = \langle j_{1}j_{2};m_{1}m_{2}|j_{1}j_{2};jm\rangle=
  6. δ m , m 1 + m 2 ( 2 j + 1 ) ( j + j 1 - j 2 ) ! ( j - j 1 + j 2 ) ! ( j 1 + j 2 - j ) ! ( j 1 + j 2 + j + 1 ) ! × \delta_{m,m_{1}+m_{2}}\sqrt{\frac{(2j+1)(j+j_{1}-j_{2})!(j-j_{1}+j_{2})!(j_{1}% +j_{2}-j)!}{(j_{1}+j_{2}+j+1)!}}\ \times
  7. ( j + m ) ! ( j - m ) ! ( j 1 - m 1 ) ! ( j 1 + m 1 ) ! ( j 2 - m 2 ) ! ( j 2 + m 2 ) ! × \sqrt{(j+m)!(j-m)!(j_{1}-m_{1})!(j_{1}+m_{1})!(j_{2}-m_{2})!(j_{2}+m_{2})!}\ \times
  8. k ( - 1 ) k k ! ( j 1 + j 2 - j - k ) ! ( j 1 - m 1 - k ) ! ( j 2 + m 2 - k ) ! ( j - j 2 + m 1 + k ) ! ( j - j 1 - m 2 + k ) ! . \sum_{k}\frac{(-1)^{k}}{k!(j_{1}+j_{2}-j-k)!(j_{1}-m_{1}-k)!(j_{2}+m_{2}-k)!(j% -j_{2}+m_{1}+k)!(j-j_{1}-m_{2}+k)!}.
  9. j 1 j 2 ; m 1 m 2 | j 1 j 2 ; j m = ( - 1 ) j - j 1 - j 2 j 1 j 2 ; - m 1 , - m 2 | j 1 j 2 ; j , - m \langle j_{1}j_{2};m_{1}m_{2}|j_{1}j_{2};jm\rangle=(-1)^{j-j_{1}-j_{2}}\langle j% _{1}j_{2};-m_{1},-m_{2}|j_{1}j_{2};j,-m\rangle
  10. j 1 j 2 ; m 1 m 2 | j 1 j 2 ; j m = ( - 1 ) j - j 1 - j 2 j 2 j 1 ; m 2 m 1 | j 2 j 1 ; j m \langle j_{1}j_{2};m_{1}m_{2}|j_{1}j_{2};jm\rangle=(-1)^{j-j_{1}-j_{2}}\langle j% _{2}j_{1};m_{2}m_{1}|j_{2}j_{1};jm\rangle
  11. δ j , j 1 δ m , m 1 \delta_{j,j_{1}}\delta_{m,m_{1}}
  12. 1 1\!\,
  13. 1 2 \sqrt{\frac{1}{2}}\!\,
  14. 1 2 \sqrt{\frac{1}{2}}\!\,
  15. 1 2 \sqrt{\frac{1}{2}}\!\,
  16. - 1 2 -\sqrt{\frac{1}{2}}\!\,
  17. 1 1\!\,
  18. 1 3 \sqrt{\frac{1}{3}}\!\,
  19. 2 3 \sqrt{\frac{2}{3}}\!\,
  20. 2 3 \sqrt{\frac{2}{3}}\!\,
  21. - 1 3 -\sqrt{\frac{1}{3}}\!\,
  22. 1 1\!\,
  23. 1 2 \sqrt{\frac{1}{2}}\!\,
  24. 1 2 \sqrt{\frac{1}{2}}\!\,
  25. 1 2 \sqrt{\frac{1}{2}}\!\,
  26. - 1 2 -\sqrt{\frac{1}{2}}\!\,
  27. 1 6 \sqrt{\frac{1}{6}}\!\,
  28. 1 2 \sqrt{\frac{1}{2}}\!\,
  29. 1 3 \sqrt{\frac{1}{3}}\!\,
  30. 2 3 \sqrt{\frac{2}{3}}\!\,
  31. 0 0\!\,
  32. - 1 3 -\sqrt{\frac{1}{3}}\!\,
  33. 1 6 \sqrt{\frac{1}{6}}\!\,
  34. - 1 2 -\sqrt{\frac{1}{2}}\!\,
  35. 1 3 \sqrt{\frac{1}{3}}\!\,
  36. 1 1\!\,
  37. 1 2 \frac{1}{2}\!\,
  38. 3 4 \sqrt{\frac{3}{4}}\!\,
  39. 3 4 \sqrt{\frac{3}{4}}\!\,
  40. - 1 2 -\frac{1}{2}\!\,
  41. 1 2 \sqrt{\frac{1}{2}}\!\,
  42. 1 2 \sqrt{\frac{1}{2}}\!\,
  43. 1 2 \sqrt{\frac{1}{2}}\!\,
  44. - 1 2 -\sqrt{\frac{1}{2}}\!\,
  45. 1 1\!\,
  46. 2 5 \sqrt{\frac{2}{5}}\!\,
  47. 3 5 \sqrt{\frac{3}{5}}\!\,
  48. 3 5 \sqrt{\frac{3}{5}}\!\,
  49. - 2 5 -\sqrt{\frac{2}{5}}\!\,
  50. 1 10 \sqrt{\frac{1}{10}}\!\,
  51. 2 5 \sqrt{\frac{2}{5}}\!\,
  52. 1 2 \sqrt{\frac{1}{2}}\!\,
  53. 3 5 \sqrt{\frac{3}{5}}\!\,
  54. 1 15 \sqrt{\frac{1}{15}}\!\,
  55. - 1 3 -\sqrt{\frac{1}{3}}\!\,
  56. 3 10 \sqrt{\frac{3}{10}}\!\,
  57. - 8 15 -\sqrt{\frac{8}{15}}\!\,
  58. 1 6 \sqrt{\frac{1}{6}}\!\,
  59. 1 1\!\,
  60. 1 2 \sqrt{\frac{1}{2}}\!\,
  61. 1 2 \sqrt{\frac{1}{2}}\!\,
  62. 1 2 \sqrt{\frac{1}{2}}\!\,
  63. - 1 2 -\sqrt{\frac{1}{2}}\!\,
  64. 1 5 \sqrt{\frac{1}{5}}\!\,
  65. 1 2 \sqrt{\frac{1}{2}}\!\,
  66. 3 10 \sqrt{\frac{3}{10}}\!\,
  67. 3 5 \sqrt{\frac{3}{5}}\!\,
  68. 0 0\!\,
  69. - 2 5 -\sqrt{\frac{2}{5}}\!\,
  70. 1 5 \sqrt{\frac{1}{5}}\!\,
  71. - 1 2 -\sqrt{\frac{1}{2}}\!\,
  72. 3 10 \sqrt{\frac{3}{10}}\!\,
  73. 1 20 \sqrt{\frac{1}{20}}\!\,
  74. 1 2 \frac{1}{2}\!\,
  75. 9 20 \sqrt{\frac{9}{20}}\!\,
  76. 1 2 \frac{1}{2}\!\,
  77. 9 20 \sqrt{\frac{9}{20}}\!\,
  78. 1 2 \frac{1}{2}\!\,
  79. - 1 20 -\sqrt{\frac{1}{20}}\!\,
  80. - 1 2 -\frac{1}{2}\!\,
  81. 9 20 \sqrt{\frac{9}{20}}\!\,
  82. - 1 2 -\frac{1}{2}\!\,
  83. - 1 20 -\sqrt{\frac{1}{20}}\!\,
  84. 1 2 \frac{1}{2}\!\,
  85. 1 20 \sqrt{\frac{1}{20}}\!\,
  86. - 1 2 -\frac{1}{2}\!\,
  87. 9 20 \sqrt{\frac{9}{20}}\!\,
  88. - 1 2 -\frac{1}{2}\!\,
  89. 1 1\!\,
  90. 1 5 \sqrt{\frac{1}{5}}\!\,
  91. 4 5 \sqrt{\frac{4}{5}}\!\,
  92. 4 5 \sqrt{\frac{4}{5}}\!\,
  93. - 1 5 -\sqrt{\frac{1}{5}}\!\,
  94. 2 5 \sqrt{\frac{2}{5}}\!\,
  95. 3 5 \sqrt{\frac{3}{5}}\!\,
  96. 3 5 \sqrt{\frac{3}{5}}\!\,
  97. - 2 5 -\sqrt{\frac{2}{5}}\!\,
  98. 1 1\!\,
  99. 1 3 \sqrt{\frac{1}{3}}\!\,
  100. 2 3 \sqrt{\frac{2}{3}}\!\,
  101. 2 3 \sqrt{\frac{2}{3}}\!\,
  102. - 1 3 -\sqrt{\frac{1}{3}}\!\,
  103. 1 15 \sqrt{\frac{1}{15}}\!\,
  104. 1 3 \sqrt{\frac{1}{3}}\!\,
  105. 3 5 \sqrt{\frac{3}{5}}\!\,
  106. 8 15 \sqrt{\frac{8}{15}}\!\,
  107. 1 6 \sqrt{\frac{1}{6}}\!\,
  108. - 3 10 -\sqrt{\frac{3}{10}}\!\,
  109. 2 5 \sqrt{\frac{2}{5}}\!\,
  110. - 1 2 -\sqrt{\frac{1}{2}}\!\,
  111. 1 10 \sqrt{\frac{1}{10}}\!\,
  112. 1 5 \sqrt{\frac{1}{5}}\!\,
  113. 1 2 \sqrt{\frac{1}{2}}\!\,
  114. 3 10 \sqrt{\frac{3}{10}}\!\,
  115. 3 5 \sqrt{\frac{3}{5}}\!\,
  116. 0 0\!\,
  117. - 2 5 -\sqrt{\frac{2}{5}}\!\,
  118. 1 5 \sqrt{\frac{1}{5}}\!\,
  119. - 1 2 -\sqrt{\frac{1}{2}}\!\,
  120. 3 10 \sqrt{\frac{3}{10}}\!\,
  121. 1 1\!\,
  122. 3 7 \sqrt{\frac{3}{7}}\!\,
  123. 4 7 \sqrt{\frac{4}{7}}\!\,
  124. 4 7 \sqrt{\frac{4}{7}}\!\,
  125. - 3 7 -\sqrt{\frac{3}{7}}\!\,
  126. 1 7 \sqrt{\frac{1}{7}}\!\,
  127. 16 35 \sqrt{\frac{16}{35}}\!\,
  128. 2 5 \sqrt{\frac{2}{5}}\!\,
  129. 4 7 \sqrt{\frac{4}{7}}\!\,
  130. 1 35 \sqrt{\frac{1}{35}}\!\,
  131. - 2 5 -\sqrt{\frac{2}{5}}\!\,
  132. 2 7 \sqrt{\frac{2}{7}}\!\,
  133. - 18 35 -\sqrt{\frac{18}{35}}\!\,
  134. 1 5 \sqrt{\frac{1}{5}}\!\,
  135. 1 35 \sqrt{\frac{1}{35}}\!\,
  136. 6 35 \sqrt{\frac{6}{35}}\!\,
  137. 2 5 \sqrt{\frac{2}{5}}\!\,
  138. 2 5 \sqrt{\frac{2}{5}}\!\,
  139. 12 35 \sqrt{\frac{12}{35}}\!\,
  140. 5 14 \sqrt{\frac{5}{14}}\!\,
  141. 0 0\!\,
  142. - 3 10 -\sqrt{\frac{3}{10}}\!\,
  143. 18 35 \sqrt{\frac{18}{35}}\!\,
  144. - 3 35 -\sqrt{\frac{3}{35}}\!\,
  145. - 1 5 -\sqrt{\frac{1}{5}}\!\,
  146. 1 5 \sqrt{\frac{1}{5}}\!\,
  147. 4 35 \sqrt{\frac{4}{35}}\!\,
  148. - 27 70 -\sqrt{\frac{27}{70}}\!\,
  149. 2 5 \sqrt{\frac{2}{5}}\!\,
  150. - 1 10 -\sqrt{\frac{1}{10}}\!\,
  151. 1 1\!\,
  152. 1 2 \sqrt{\frac{1}{2}}\!\,
  153. 1 2 \sqrt{\frac{1}{2}}\!\,
  154. 1 2 \sqrt{\frac{1}{2}}\!\,
  155. - 1 2 -\sqrt{\frac{1}{2}}\!\,
  156. 3 14 \sqrt{\frac{3}{14}}\!\,
  157. 1 2 \sqrt{\frac{1}{2}}\!\,
  158. 2 7 \sqrt{\frac{2}{7}}\!\,
  159. 4 7 \sqrt{\frac{4}{7}}\!\,
  160. 0 0\!\,
  161. - 3 7 -\sqrt{\frac{3}{7}}\!\,
  162. 3 14 \sqrt{\frac{3}{14}}\!\,
  163. - 1 2 -\sqrt{\frac{1}{2}}\!\,
  164. 2 7 \sqrt{\frac{2}{7}}\!\,
  165. 1 14 \sqrt{\frac{1}{14}}\!\,
  166. 3 10 \sqrt{\frac{3}{10}}\!\,
  167. 3 7 \sqrt{\frac{3}{7}}\!\,
  168. 1 5 \sqrt{\frac{1}{5}}\!\,
  169. 3 7 \sqrt{\frac{3}{7}}\!\,
  170. 1 5 \sqrt{\frac{1}{5}}\!\,
  171. - 1 14 -\sqrt{\frac{1}{14}}\!\,
  172. - 3 10 -\sqrt{\frac{3}{10}}\!\,
  173. 3 7 \sqrt{\frac{3}{7}}\!\,
  174. - 1 5 -\sqrt{\frac{1}{5}}\!\,
  175. - 1 14 -\sqrt{\frac{1}{14}}\!\,
  176. 3 10 \sqrt{\frac{3}{10}}\!\,
  177. 1 14 \sqrt{\frac{1}{14}}\!\,
  178. - 3 10 -\sqrt{\frac{3}{10}}\!\,
  179. 3 7 \sqrt{\frac{3}{7}}\!\,
  180. - 1 5 -\sqrt{\frac{1}{5}}\!\,
  181. 1 70 \sqrt{\frac{1}{70}}\!\,
  182. 1 10 \sqrt{\frac{1}{10}}\!\,
  183. 2 7 \sqrt{\frac{2}{7}}\!\,
  184. 2 5 \sqrt{\frac{2}{5}}\!\,
  185. 1 5 \sqrt{\frac{1}{5}}\!\,
  186. 8 35 \sqrt{\frac{8}{35}}\!\,
  187. 2 5 \sqrt{\frac{2}{5}}\!\,
  188. 1 14 \sqrt{\frac{1}{14}}\!\,
  189. - 1 10 -\sqrt{\frac{1}{10}}\!\,
  190. - 1 5 -\sqrt{\frac{1}{5}}\!\,
  191. 18 35 \sqrt{\frac{18}{35}}\!\,
  192. 0 0\!\,
  193. - 2 7 -\sqrt{\frac{2}{7}}\!\,
  194. 0 0\!\,
  195. 1 5 \sqrt{\frac{1}{5}}\!\,
  196. 8 35 \sqrt{\frac{8}{35}}\!\,
  197. - 2 5 -\sqrt{\frac{2}{5}}\!\,
  198. 1 14 \sqrt{\frac{1}{14}}\!\,
  199. 1 10 \sqrt{\frac{1}{10}}\!\,
  200. - 1 5 -\sqrt{\frac{1}{5}}\!\,
  201. 1 70 \sqrt{\frac{1}{70}}\!\,
  202. - 1 10 -\sqrt{\frac{1}{10}}\!\,
  203. 2 7 \sqrt{\frac{2}{7}}\!\,
  204. - 2 5 -\sqrt{\frac{2}{5}}\!\,
  205. 1 5 \sqrt{\frac{1}{5}}\!\,
  206. 1 1\!\,
  207. 1 6 \sqrt{\frac{1}{6}}\!\,
  208. 5 6 \sqrt{\frac{5}{6}}\!\,
  209. 5 6 \sqrt{\frac{5}{6}}\!\,
  210. - 1 6 -\sqrt{\frac{1}{6}}\!\,
  211. 1 3 \sqrt{\frac{1}{3}}\!\,
  212. 2 3 \sqrt{\frac{2}{3}}\!\,
  213. 2 3 \sqrt{\frac{2}{3}}\!\,
  214. - 1 3 -\sqrt{\frac{1}{3}}\!\,
  215. 1 2 \sqrt{\frac{1}{2}}\!\,
  216. 1 2 \sqrt{\frac{1}{2}}\!\,
  217. 1 2 \sqrt{\frac{1}{2}}\!\,
  218. - 1 2 -\sqrt{\frac{1}{2}}\!\,
  219. 1 1\!\,
  220. 2 7 \sqrt{\frac{2}{7}}\!\,
  221. 5 7 \sqrt{\frac{5}{7}}\!\,
  222. 5 7 \sqrt{\frac{5}{7}}\!\,
  223. - 2 7 -\sqrt{\frac{2}{7}}\!\,
  224. 1 21 \sqrt{\frac{1}{21}}\!\,
  225. 2 7 \sqrt{\frac{2}{7}}\!\,
  226. 2 3 \sqrt{\frac{2}{3}}\!\,
  227. 10 21 \sqrt{\frac{10}{21}}\!\,
  228. 9 35 \sqrt{\frac{9}{35}}\!\,
  229. - 4 15 -\sqrt{\frac{4}{15}}\!\,
  230. 10 21 \sqrt{\frac{10}{21}}\!\,
  231. - 16 35 -\sqrt{\frac{16}{35}}\!\,
  232. 1 15 \sqrt{\frac{1}{15}}\!\,
  233. 1 7 \sqrt{\frac{1}{7}}\!\,
  234. 16 35 \sqrt{\frac{16}{35}}\!\,
  235. 2 5 \sqrt{\frac{2}{5}}\!\,
  236. 4 7 \sqrt{\frac{4}{7}}\!\,
  237. 1 35 \sqrt{\frac{1}{35}}\!\,
  238. - 2 5 -\sqrt{\frac{2}{5}}\!\,
  239. 2 7 \sqrt{\frac{2}{7}}\!\,
  240. - 18 35 -\sqrt{\frac{18}{35}}\!\,
  241. 1 5 \sqrt{\frac{1}{5}}\!\,
  242. 1 1\!\,
  243. 3 8 \sqrt{\frac{3}{8}}\!\,
  244. 5 8 \sqrt{\frac{5}{8}}\!\,
  245. 5 8 \sqrt{\frac{5}{8}}\!\,
  246. - 3 8 -\sqrt{\frac{3}{8}}\!\,
  247. 3 28 \sqrt{\frac{3}{28}}\!\,
  248. 5 12 \sqrt{\frac{5}{12}}\!\,
  249. 10 21 \sqrt{\frac{10}{21}}\!\,
  250. 15 28 \sqrt{\frac{15}{28}}\!\,
  251. 1 12 \sqrt{\frac{1}{12}}\!\,
  252. - 8 21 -\sqrt{\frac{8}{21}}\!\,
  253. 5 14 \sqrt{\frac{5}{14}}\!\,
  254. - 1 2 -\sqrt{\frac{1}{2}}\!\,
  255. 1 7 \sqrt{\frac{1}{7}}\!\,
  256. 1 56 \sqrt{\frac{1}{56}}\!\,
  257. 1 8 \sqrt{\frac{1}{8}}\!\,
  258. 5 14 \sqrt{\frac{5}{14}}\!\,
  259. 1 2 \sqrt{\frac{1}{2}}\!\,
  260. 15 56 \sqrt{\frac{15}{56}}\!\,
  261. 49 120 \sqrt{\frac{49}{120}}\!\,
  262. 1 42 \sqrt{\frac{1}{42}}\!\,
  263. - 3 10 -\sqrt{\frac{3}{10}}\!\,
  264. 15 28 \sqrt{\frac{15}{28}}\!\,
  265. - 1 60 -\sqrt{\frac{1}{60}}\!\,
  266. - 25 84 -\sqrt{\frac{25}{84}}\!\,
  267. 3 20 \sqrt{\frac{3}{20}}\!\,
  268. 5 28 \sqrt{\frac{5}{28}}\!\,
  269. - 9 20 -\sqrt{\frac{9}{20}}\!\,
  270. 9 28 \sqrt{\frac{9}{28}}\!\,
  271. - 1 20 -\sqrt{\frac{1}{20}}\!\,
  272. 1 14 \sqrt{\frac{1}{14}}\!\,
  273. 3 10 \sqrt{\frac{3}{10}}\!\,
  274. 3 7 \sqrt{\frac{3}{7}}\!\,
  275. 1 5 \sqrt{\frac{1}{5}}\!\,
  276. 3 7 \sqrt{\frac{3}{7}}\!\,
  277. 1 5 \sqrt{\frac{1}{5}}\!\,
  278. - 1 14 -\sqrt{\frac{1}{14}}\!\,
  279. - 3 10 -\sqrt{\frac{3}{10}}\!\,
  280. 3 7 \sqrt{\frac{3}{7}}\!\,
  281. - 1 5 -\sqrt{\frac{1}{5}}\!\,
  282. - 1 14 -\sqrt{\frac{1}{14}}\!\,
  283. 3 10 \sqrt{\frac{3}{10}}\!\,
  284. 1 14 \sqrt{\frac{1}{14}}\!\,
  285. - 3 10 -\sqrt{\frac{3}{10}}\!\,
  286. 3 7 \sqrt{\frac{3}{7}}\!\,
  287. - 1 5 -\sqrt{\frac{1}{5}}\!\,
  288. 1 1\!\,
  289. 2 3 \frac{2}{3}\!\,
  290. 5 9 \sqrt{\frac{5}{9}}\!\,
  291. 5 9 \sqrt{\frac{5}{9}}\!\,
  292. - 2 3 -\frac{2}{3}\!\,
  293. 1 6 \sqrt{\frac{1}{6}}\!\,
  294. 10 21 \sqrt{\frac{10}{21}}\!\,
  295. 5 14 \sqrt{\frac{5}{14}}\!\,
  296. 5 9 \sqrt{\frac{5}{9}}\!\,
  297. 1 63 \sqrt{\frac{1}{63}}\!\,
  298. - 3 7 -\sqrt{\frac{3}{7}}\!\,
  299. 5 18 \sqrt{\frac{5}{18}}\!\,
  300. - 32 63 -\sqrt{\frac{32}{63}}\!\,
  301. 3 14 \sqrt{\frac{3}{14}}\!\,
  302. 1 21 \sqrt{\frac{1}{21}}\!\,
  303. 5 21 \sqrt{\frac{5}{21}}\!\,
  304. 3 7 \sqrt{\frac{3}{7}}\!\,
  305. 2 7 \sqrt{\frac{2}{7}}\!\,
  306. 5 14 \sqrt{\frac{5}{14}}\!\,
  307. 2 7 \sqrt{\frac{2}{7}}\!\,
  308. - 1 70 -\sqrt{\frac{1}{70}}\!\,
  309. - 12 35 -\sqrt{\frac{12}{35}}\!\,
  310. 10 21 \sqrt{\frac{10}{21}}\!\,
  311. - 2 21 -\sqrt{\frac{2}{21}}\!\,
  312. - 6 35 -\sqrt{\frac{6}{35}}\!\,
  313. 9 35 \sqrt{\frac{9}{35}}\!\,
  314. 5 42 \sqrt{\frac{5}{42}}\!\,
  315. - 8 21 -\sqrt{\frac{8}{21}}\!\,
  316. 27 70 \sqrt{\frac{27}{70}}\!\,
  317. - 4 35 -\sqrt{\frac{4}{35}}\!\,
  318. 1 126 \sqrt{\frac{1}{126}}\!\,
  319. 4 63 \sqrt{\frac{4}{63}}\!\,
  320. 3 14 \sqrt{\frac{3}{14}}\!\,
  321. 8 21 \sqrt{\frac{8}{21}}\!\,
  322. 1 3 \sqrt{\frac{1}{3}}\!\,
  323. 10 63 \sqrt{\frac{10}{63}}\!\,
  324. 121 315 \sqrt{\frac{121}{315}}\!\,
  325. 6 35 \sqrt{\frac{6}{35}}\!\,
  326. - 2 105 -\sqrt{\frac{2}{105}}\!\,
  327. - 4 15 -\sqrt{\frac{4}{15}}\!\,
  328. 10 21 \sqrt{\frac{10}{21}}\!\,
  329. 4 105 \sqrt{\frac{4}{105}}\!\,
  330. - 8 35 -\sqrt{\frac{8}{35}}\!\,
  331. - 2 35 -\sqrt{\frac{2}{35}}\!\,
  332. 1 5 \sqrt{\frac{1}{5}}\!\,
  333. 20 63 \sqrt{\frac{20}{63}}\!\,
  334. - 14 45 -\sqrt{\frac{14}{45}}\!\,
  335. 0 0\!\,
  336. 5 21 \sqrt{\frac{5}{21}}\!\,
  337. - 2 15 -\sqrt{\frac{2}{15}}\!\,
  338. 5 126 \sqrt{\frac{5}{126}}\!\,
  339. - 64 315 -\sqrt{\frac{64}{315}}\!\,
  340. 27 70 \sqrt{\frac{27}{70}}\!\,
  341. - 32 105 -\sqrt{\frac{32}{105}}\!\,
  342. 1 15 \sqrt{\frac{1}{15}}\!\,
  343. j 1 j_{1}
  344. j 2 j_{2}

Table_of_spherical_harmonics.html

  1. θ \theta
  2. φ \varphi\,
  3. x \displaystyle x
  4. Y 0 0 ( θ , φ ) = 1 2 1 π Y_{0}^{0}(\theta,\varphi)={1\over 2}\sqrt{1\over\pi}
  5. Y 1 - 1 ( θ , φ ) = 1 2 3 2 π e - i φ sin θ = 1 2 3 2 π ( x - i y ) r Y 1 0 ( θ , φ ) = 1 2 3 π cos θ = 1 2 3 π z r Y 1 1 ( θ , φ ) = - 1 2 3 2 π e i φ sin θ = - 1 2 3 2 π ( x + i y ) r \begin{aligned}\displaystyle Y_{1}^{-1}(\theta,\varphi)&\displaystyle={1\over 2% }\sqrt{3\over 2\pi}\cdot e^{-i\varphi}\cdot\sin\theta\quad={1\over 2}\sqrt{3% \over 2\pi}\cdot{(x-iy)\over r}\\ \displaystyle Y_{1}^{0}(\theta,\varphi)&\displaystyle={1\over 2}\sqrt{3\over% \pi}\cdot\cos\theta\quad\quad={1\over 2}\sqrt{3\over\pi}\cdot{z\over r}\\ \displaystyle Y_{1}^{1}(\theta,\varphi)&\displaystyle={-1\over 2}\sqrt{3\over 2% \pi}\cdot e^{i\varphi}\cdot\sin\theta\quad={-1\over 2}\sqrt{3\over 2\pi}\cdot{% (x+iy)\over r}\end{aligned}
  6. Y 2 - 2 ( θ , φ ) = 1 4 15 2 π e - 2 i φ sin 2 θ = 1 4 15 2 π ( x - i y ) 2 r 2 Y_{2}^{-2}(\theta,\varphi)={1\over 4}\sqrt{15\over 2\pi}\cdot e^{-2i\varphi}% \cdot\sin^{2}\theta\quad={1\over 4}\sqrt{15\over 2\pi}\cdot{(x-iy)^{2}\over r^% {2}}
  7. Y 2 - 1 ( θ , φ ) = 1 2 15 2 π e - i φ sin θ cos θ = 1 2 15 2 π ( x - i y ) z r 2 Y_{2}^{-1}(\theta,\varphi)={1\over 2}\sqrt{15\over 2\pi}\cdot e^{-i\varphi}% \cdot\sin\theta\cdot\cos\theta\quad={1\over 2}\sqrt{15\over 2\pi}\cdot{(x-iy)z% \over r^{2}}
  8. Y 2 0 ( θ , φ ) = 1 4 5 π ( 3 cos 2 θ - 1 ) = 1 4 5 π ( 2 z 2 - x 2 - y 2 ) r 2 Y_{2}^{0}(\theta,\varphi)={1\over 4}\sqrt{5\over\pi}\cdot(3\cos^{2}\theta-1)% \quad={1\over 4}\sqrt{5\over\pi}\cdot{(2z^{2}-x^{2}-y^{2})\over r^{2}}
  9. Y 2 1 ( θ , φ ) = - 1 2 15 2 π e i φ sin θ cos θ = - 1 2 15 2 π ( x + i y ) z r 2 Y_{2}^{1}(\theta,\varphi)={-1\over 2}\sqrt{15\over 2\pi}\cdot e^{i\varphi}% \cdot\sin\theta\cdot\cos\theta\quad={-1\over 2}\sqrt{15\over 2\pi}\cdot{(x+iy)% z\over r^{2}}
  10. Y 2 2 ( θ , φ ) = 1 4 15 2 π e 2 i φ sin 2 θ = 1 4 15 2 π ( x + i y ) 2 r 2 Y_{2}^{2}(\theta,\varphi)={1\over 4}\sqrt{15\over 2\pi}\cdot e^{2i\varphi}% \cdot\sin^{2}\theta\quad={1\over 4}\sqrt{15\over 2\pi}\cdot{(x+iy)^{2}\over r^% {2}}
  11. Y 3 - 3 ( θ , φ ) = 1 8 35 π e - 3 i φ sin 3 θ = 1 8 35 π ( x - i y ) 3 r 3 Y_{3}^{-3}(\theta,\varphi)={1\over 8}\sqrt{35\over\pi}\cdot e^{-3i\varphi}% \cdot\sin^{3}\theta\quad={1\over 8}\sqrt{35\over\pi}\cdot{(x-iy)^{3}\over r^{3}}
  12. Y 3 - 2 ( θ , φ ) = 1 4 105 2 π e - 2 i φ sin 2 θ cos θ = 1 4 105 2 π ( x - i y ) 2 z r 3 Y_{3}^{-2}(\theta,\varphi)={1\over 4}\sqrt{105\over 2\pi}\cdot e^{-2i\varphi}% \cdot\sin^{2}\theta\cdot\cos\theta\quad={1\over 4}\sqrt{105\over 2\pi}\cdot{(x% -iy)^{2}z\over r^{3}}
  13. Y 3 - 1 ( θ , φ ) = 1 8 21 π e - i φ sin θ ( 5 cos 2 θ - 1 ) = 1 8 21 π ( x - i y ) ( 4 z 2 - x 2 - y 2 ) r 3 Y_{3}^{-1}(\theta,\varphi)={1\over 8}\sqrt{21\over\pi}\cdot e^{-i\varphi}\cdot% \sin\theta\cdot(5\cos^{2}\theta-1)\quad={1\over 8}\sqrt{21\over\pi}\cdot{(x-iy% )(4z^{2}-x^{2}-y^{2})\over r^{3}}
  14. Y 3 0 ( θ , φ ) = 1 4 7 π ( 5 cos 3 θ - 3 cos θ ) = 1 4 7 π z ( 2 z 2 - 3 x 2 - 3 y 2 ) r 3 Y_{3}^{0}(\theta,\varphi)={1\over 4}\sqrt{7\over\pi}\cdot(5\cos^{3}\theta-3% \cos\theta)\quad={1\over 4}\sqrt{7\over\pi}\cdot{z(2z^{2}-3x^{2}-3y^{2})\over r% ^{3}}
  15. Y 3 1 ( θ , φ ) = - 1 8 21 π e i φ sin θ ( 5 cos 2 θ - 1 ) = - 1 8 21 π ( x + i y ) ( 4 z 2 - x 2 - y 2 ) r 3 Y_{3}^{1}(\theta,\varphi)={-1\over 8}\sqrt{21\over\pi}\cdot e^{i\varphi}\cdot% \sin\theta\cdot(5\cos^{2}\theta-1)\quad={-1\over 8}\sqrt{21\over\pi}\cdot{(x+% iy)(4z^{2}-x^{2}-y^{2})\over r^{3}}
  16. Y 3 2 ( θ , φ ) = 1 4 105 2 π e 2 i φ sin 2 θ cos θ = 1 4 105 2 π ( x + i y ) 2 z r 3 Y_{3}^{2}(\theta,\varphi)={1\over 4}\sqrt{105\over 2\pi}\cdot e^{2i\varphi}% \cdot\sin^{2}\theta\cdot\cos\theta\quad={1\over 4}\sqrt{105\over 2\pi}\cdot{(x% +iy)^{2}z\over r^{3}}
  17. Y 3 3 ( θ , φ ) = - 1 8 35 π e 3 i φ sin 3 θ = - 1 8 35 π ( x + i y ) 3 r 3 Y_{3}^{3}(\theta,\varphi)={-1\over 8}\sqrt{35\over\pi}\cdot e^{3i\varphi}\cdot% \sin^{3}\theta\quad={-1\over 8}\sqrt{35\over\pi}\cdot{(x+iy)^{3}\over r^{3}}
  18. Y 4 - 4 ( θ , φ ) = 3 16 35 2 π e - 4 i φ sin 4 θ = 3 16 35 2 π ( x - i y ) 4 r 4 Y_{4}^{-4}(\theta,\varphi)={3\over 16}\sqrt{35\over 2\pi}\cdot e^{-4i\varphi}% \cdot\sin^{4}\theta=\frac{3}{16}\sqrt{\frac{35}{2\pi}}\cdot\frac{(x-iy)^{4}}{r% ^{4}}
  19. Y 4 - 3 ( θ , φ ) = 3 8 35 π e - 3 i φ sin 3 θ cos θ = 3 8 35 π ( x - i y ) 3 z r 4 Y_{4}^{-3}(\theta,\varphi)={3\over 8}\sqrt{35\over\pi}\cdot e^{-3i\varphi}% \cdot\sin^{3}\theta\cdot\cos\theta=\frac{3}{8}\sqrt{\frac{35}{\pi}}\cdot\frac{% (x-iy)^{3}z}{r^{4}}
  20. Y 4 - 2 ( θ , φ ) = 3 8 5 2 π e - 2 i φ sin 2 θ ( 7 cos 2 θ - 1 ) = 3 8 5 2 π ( x - i y ) 2 ( 7 z 2 - r 2 ) r 4 Y_{4}^{-2}(\theta,\varphi)={3\over 8}\sqrt{5\over 2\pi}\cdot e^{-2i\varphi}% \cdot\sin^{2}\theta\cdot(7\cos^{2}\theta-1)=\frac{3}{8}\sqrt{\frac{5}{2\pi}}% \cdot\frac{(x-iy)^{2}\cdot(7z^{2}-r^{2})}{r^{4}}
  21. Y 4 - 1 ( θ , φ ) = 3 8 5 π e - i φ sin θ ( 7 cos 3 θ - 3 cos θ ) = 3 8 5 π ( x - i y ) z ( 7 z 2 - 3 r 2 ) r 4 Y_{4}^{-1}(\theta,\varphi)={3\over 8}\sqrt{5\over\pi}\cdot e^{-i\varphi}\cdot% \sin\theta\cdot(7\cos^{3}\theta-3\cos\theta)=\frac{3}{8}\sqrt{\frac{5}{\pi}}% \cdot\frac{(x-iy)\cdot z\cdot(7z^{2}-3r^{2})}{r^{4}}
  22. Y 4 0 ( θ , φ ) = 3 16 1 π ( 35 cos 4 θ - 30 cos 2 θ + 3 ) = 3 16 1 π ( 35 z 4 - 30 z 2 r 2 + 3 r 4 ) r 4 Y_{4}^{0}(\theta,\varphi)={3\over 16}\sqrt{1\over\pi}\cdot(35\cos^{4}\theta-30% \cos^{2}\theta+3)=\frac{3}{16}\sqrt{\frac{1}{\pi}}\cdot\frac{(35z^{4}-30z^{2}r% ^{2}+3r^{4})}{r^{4}}
  23. Y 4 1 ( θ , φ ) = - 3 8 5 π e i φ sin θ ( 7 cos 3 θ - 3 cos θ ) = - 3 8 5 π ( x + i y ) z ( 7 z 2 - 3 r 2 ) r 4 Y_{4}^{1}(\theta,\varphi)={-3\over 8}\sqrt{5\over\pi}\cdot e^{i\varphi}\cdot% \sin\theta\cdot(7\cos^{3}\theta-3\cos\theta)=\frac{-3}{8}\sqrt{\frac{5}{\pi}}% \cdot\frac{(x+iy)\cdot z\cdot(7z^{2}-3r^{2})}{r^{4}}
  24. Y 4 2 ( θ , φ ) = 3 8 5 2 π e 2 i φ sin 2 θ ( 7 cos 2 θ - 1 ) = 3 8 5 2 π ( x + i y ) 2 ( 7 z 2 - r 2 ) r 4 Y_{4}^{2}(\theta,\varphi)={3\over 8}\sqrt{5\over 2\pi}\cdot e^{2i\varphi}\cdot% \sin^{2}\theta\cdot(7\cos^{2}\theta-1)=\frac{3}{8}\sqrt{\frac{5}{2\pi}}\cdot% \frac{(x+iy)^{2}\cdot(7z^{2}-r^{2})}{r^{4}}
  25. Y 4 3 ( θ , φ ) = - 3 8 35 π e 3 i φ sin 3 θ cos θ = - 3 8 35 π ( x + i y ) 3 z r 4 Y_{4}^{3}(\theta,\varphi)={-3\over 8}\sqrt{35\over\pi}\cdot e^{3i\varphi}\cdot% \sin^{3}\theta\cdot\cos\theta=\frac{-3}{8}\sqrt{\frac{35}{\pi}}\cdot\frac{(x+% iy)^{3}z}{r^{4}}
  26. Y 4 4 ( θ , φ ) = 3 16 35 2 π e 4 i φ sin 4 θ = 3 16 35 2 π ( x + i y ) 4 r 4 Y_{4}^{4}(\theta,\varphi)={3\over 16}\sqrt{35\over 2\pi}\cdot e^{4i\varphi}% \cdot\sin^{4}\theta=\frac{3}{16}\sqrt{\frac{35}{2\pi}}\cdot\frac{(x+iy)^{4}}{r% ^{4}}
  27. Y 5 - 5 ( θ , φ ) = 3 32 77 π e - 5 i φ sin 5 θ Y_{5}^{-5}(\theta,\varphi)={3\over 32}\sqrt{77\over\pi}\cdot e^{-5i\varphi}% \cdot\sin^{5}\theta
  28. Y 5 - 4 ( θ , φ ) = 3 16 385 2 π e - 4 i φ sin 4 θ cos θ Y_{5}^{-4}(\theta,\varphi)={3\over 16}\sqrt{385\over 2\pi}\cdot e^{-4i\varphi}% \cdot\sin^{4}\theta\cdot\cos\theta
  29. Y 5 - 3 ( θ , φ ) = 1 32 385 π e - 3 i φ sin 3 θ ( 9 cos 2 θ - 1 ) Y_{5}^{-3}(\theta,\varphi)={1\over 32}\sqrt{385\over\pi}\cdot e^{-3i\varphi}% \cdot\sin^{3}\theta\cdot(9\cos^{2}\theta-1)
  30. Y 5 - 2 ( θ , φ ) = 1 8 1155 2 π e - 2 i φ sin 2 θ ( 3 cos 3 θ - cos θ ) Y_{5}^{-2}(\theta,\varphi)={1\over 8}\sqrt{1155\over 2\pi}\cdot e^{-2i\varphi}% \cdot\sin^{2}\theta\cdot(3\cos^{3}\theta-\cos\theta)
  31. Y 5 - 1 ( θ , φ ) = 1 16 165 2 π e - i φ sin θ ( 21 cos 4 θ - 14 cos 2 θ + 1 ) Y_{5}^{-1}(\theta,\varphi)={1\over 16}\sqrt{165\over 2\pi}\cdot e^{-i\varphi}% \cdot\sin\theta\cdot(21\cos^{4}\theta-14\cos^{2}\theta+1)
  32. Y 5 0 ( θ , φ ) = 1 16 11 π ( 63 cos 5 θ - 70 cos 3 θ + 15 cos θ ) Y_{5}^{0}(\theta,\varphi)={1\over 16}\sqrt{11\over\pi}\cdot(63\cos^{5}\theta-7% 0\cos^{3}\theta+15\cos\theta)
  33. Y 5 1 ( θ , φ ) = - 1 16 165 2 π e i φ sin θ ( 21 cos 4 θ - 14 cos 2 θ + 1 ) Y_{5}^{1}(\theta,\varphi)={-1\over 16}\sqrt{165\over 2\pi}\cdot e^{i\varphi}% \cdot\sin\theta\cdot(21\cos^{4}\theta-14\cos^{2}\theta+1)
  34. Y 5 2 ( θ , φ ) = 1 8 1155 2 π e 2 i φ sin 2 θ ( 3 cos 3 θ - cos θ ) Y_{5}^{2}(\theta,\varphi)={1\over 8}\sqrt{1155\over 2\pi}\cdot e^{2i\varphi}% \cdot\sin^{2}\theta\cdot(3\cos^{3}\theta-\cos\theta)
  35. Y 5 3 ( θ , φ ) = - 1 32 385 π e 3 i φ sin 3 θ ( 9 cos 2 θ - 1 ) Y_{5}^{3}(\theta,\varphi)={-1\over 32}\sqrt{385\over\pi}\cdot e^{3i\varphi}% \cdot\sin^{3}\theta\cdot(9\cos^{2}\theta-1)
  36. Y 5 4 ( θ , φ ) = 3 16 385 2 π e 4 i φ sin 4 θ cos θ Y_{5}^{4}(\theta,\varphi)={3\over 16}\sqrt{385\over 2\pi}\cdot e^{4i\varphi}% \cdot\sin^{4}\theta\cdot\cos\theta
  37. Y 5 5 ( θ , φ ) = - 3 32 77 π e 5 i φ sin 5 θ Y_{5}^{5}(\theta,\varphi)={-3\over 32}\sqrt{77\over\pi}\cdot e^{5i\varphi}% \cdot\sin^{5}\theta
  38. Y 6 - 6 ( θ , φ ) = 1 64 3003 π e - 6 i φ sin 6 θ Y_{6}^{-6}(\theta,\varphi)={1\over 64}\sqrt{3003\over\pi}\cdot e^{-6i\varphi}% \cdot\sin^{6}\theta
  39. Y 6 - 5 ( θ , φ ) = 3 32 1001 π e - 5 i φ sin 5 θ cos θ Y_{6}^{-5}(\theta,\varphi)={3\over 32}\sqrt{1001\over\pi}\cdot e^{-5i\varphi}% \cdot\sin^{5}\theta\cdot\cos\theta
  40. Y 6 - 4 ( θ , φ ) = 3 32 91 2 π e - 4 i φ sin 4 θ ( 11 cos 2 θ - 1 ) Y_{6}^{-4}(\theta,\varphi)={3\over 32}\sqrt{91\over 2\pi}\cdot e^{-4i\varphi}% \cdot\sin^{4}\theta\cdot(11\cos^{2}\theta-1)
  41. Y 6 - 3 ( θ , φ ) = 1 32 1365 π e - 3 i φ sin 3 θ ( 11 cos 3 θ - 3 cos θ ) Y_{6}^{-3}(\theta,\varphi)={1\over 32}\sqrt{1365\over\pi}\cdot e^{-3i\varphi}% \cdot\sin^{3}\theta\cdot(11\cos^{3}\theta-3\cos\theta)
  42. Y 6 - 2 ( θ , φ ) = 1 64 1365 π e - 2 i φ sin 2 θ ( 33 cos 4 θ - 18 cos 2 θ + 1 ) Y_{6}^{-2}(\theta,\varphi)={1\over 64}\sqrt{1365\over\pi}\cdot e^{-2i\varphi}% \cdot\sin^{2}\theta\cdot(33\cos^{4}\theta-18\cos^{2}\theta+1)
  43. Y 6 - 1 ( θ , φ ) = 1 16 273 2 π e - i φ sin θ ( 33 cos 5 θ - 30 cos 3 θ + 5 cos θ ) Y_{6}^{-1}(\theta,\varphi)={1\over 16}\sqrt{273\over 2\pi}\cdot e^{-i\varphi}% \cdot\sin\theta\cdot(33\cos^{5}\theta-30\cos^{3}\theta+5\cos\theta)
  44. Y 6 0 ( θ , φ ) = 1 32 13 π ( 231 cos 6 θ - 315 cos 4 θ + 105 cos 2 θ - 5 ) Y_{6}^{0}(\theta,\varphi)={1\over 32}\sqrt{13\over\pi}\cdot(231\cos^{6}\theta-% 315\cos^{4}\theta+105\cos^{2}\theta-5)
  45. Y 6 1 ( θ , φ ) = - 1 16 273 2 π e i φ sin θ ( 33 cos 5 θ - 30 cos 3 θ + 5 cos θ ) Y_{6}^{1}(\theta,\varphi)={-1\over 16}\sqrt{273\over 2\pi}\cdot e^{i\varphi}% \cdot\sin\theta\cdot(33\cos^{5}\theta-30\cos^{3}\theta+5\cos\theta)
  46. Y 6 2 ( θ , φ ) = 1 64 1365 π e 2 i φ sin 2 θ ( 33 cos 4 θ - 18 cos 2 θ + 1 ) Y_{6}^{2}(\theta,\varphi)={1\over 64}\sqrt{1365\over\pi}\cdot e^{2i\varphi}% \cdot\sin^{2}\theta\cdot(33\cos^{4}\theta-18\cos^{2}\theta+1)
  47. Y 6 3 ( θ , φ ) = - 1 32 1365 π e 3 i φ sin 3 θ ( 11 cos 3 θ - 3 cos θ ) Y_{6}^{3}(\theta,\varphi)={-1\over 32}\sqrt{1365\over\pi}\cdot e^{3i\varphi}% \cdot\sin^{3}\theta\cdot(11\cos^{3}\theta-3\cos\theta)
  48. Y 6 4 ( θ , φ ) = 3 32 91 2 π e 4 i φ sin 4 θ ( 11 cos 2 θ - 1 ) Y_{6}^{4}(\theta,\varphi)={3\over 32}\sqrt{91\over 2\pi}\cdot e^{4i\varphi}% \cdot\sin^{4}\theta\cdot(11\cos^{2}\theta-1)
  49. Y 6 5 ( θ , φ ) = - 3 32 1001 π e 5 i φ sin 5 θ cos θ Y_{6}^{5}(\theta,\varphi)={-3\over 32}\sqrt{1001\over\pi}\cdot e^{5i\varphi}% \cdot\sin^{5}\theta\cdot\cos\theta
  50. Y 6 6 ( θ , φ ) = 1 64 3003 π e 6 i φ sin 6 θ Y_{6}^{6}(\theta,\varphi)={1\over 64}\sqrt{3003\over\pi}\cdot e^{6i\varphi}% \cdot\sin^{6}\theta
  51. Y 7 - 7 ( θ , φ ) = 3 64 715 2 π e - 7 i φ sin 7 θ Y_{7}^{-7}(\theta,\varphi)={3\over 64}\sqrt{715\over 2\pi}\cdot e^{-7i\varphi}% \cdot\sin^{7}\theta
  52. Y 7 - 6 ( θ , φ ) = 3 64 5005 π e - 6 i φ sin 6 θ cos θ Y_{7}^{-6}(\theta,\varphi)={3\over 64}\sqrt{5005\over\pi}\cdot e^{-6i\varphi}% \cdot\sin^{6}\theta\cdot\cos\theta
  53. Y 7 - 5 ( θ , φ ) = 3 64 385 2 π e - 5 i φ sin 5 θ ( 13 cos 2 θ - 1 ) Y_{7}^{-5}(\theta,\varphi)={3\over 64}\sqrt{385\over 2\pi}\cdot e^{-5i\varphi}% \cdot\sin^{5}\theta\cdot(13\cos^{2}\theta-1)
  54. Y 7 - 4 ( θ , φ ) = 3 32 385 2 π e - 4 i φ sin 4 θ ( 13 cos 3 θ - 3 cos θ ) Y_{7}^{-4}(\theta,\varphi)={3\over 32}\sqrt{385\over 2\pi}\cdot e^{-4i\varphi}% \cdot\sin^{4}\theta\cdot(13\cos^{3}\theta-3\cos\theta)
  55. Y 7 - 3 ( θ , φ ) = 3 64 35 2 π e - 3 i φ sin 3 θ ( 143 cos 4 θ - 66 cos 2 θ + 3 ) Y_{7}^{-3}(\theta,\varphi)={3\over 64}\sqrt{35\over 2\pi}\cdot e^{-3i\varphi}% \cdot\sin^{3}\theta\cdot(143\cos^{4}\theta-66\cos^{2}\theta+3)
  56. Y 7 - 2 ( θ , φ ) = 3 64 35 π e - 2 i φ sin 2 θ ( 143 cos 5 θ - 110 cos 3 θ + 15 cos θ ) Y_{7}^{-2}(\theta,\varphi)={3\over 64}\sqrt{35\over\pi}\cdot e^{-2i\varphi}% \cdot\sin^{2}\theta\cdot(143\cos^{5}\theta-110\cos^{3}\theta+15\cos\theta)
  57. Y 7 - 1 ( θ , φ ) = 1 64 105 2 π e - i φ sin θ ( 429 cos 6 θ - 495 cos 4 θ + 135 cos 2 θ - 5 ) Y_{7}^{-1}(\theta,\varphi)={1\over 64}\sqrt{105\over 2\pi}\cdot e^{-i\varphi}% \cdot\sin\theta\cdot(429\cos^{6}\theta-495\cos^{4}\theta+135\cos^{2}\theta-5)
  58. Y 7 0 ( θ , φ ) = 1 32 15 π ( 429 cos 7 θ - 693 cos 5 θ + 315 cos 3 θ - 35 cos θ ) Y_{7}^{0}(\theta,\varphi)={1\over 32}\sqrt{15\over\pi}\cdot(429\cos^{7}\theta-% 693\cos^{5}\theta+315\cos^{3}\theta-35\cos\theta)
  59. Y 7 1 ( θ , φ ) = - 1 64 105 2 π e i φ sin θ ( 429 cos 6 θ - 495 cos 4 θ + 135 cos 2 θ - 5 ) Y_{7}^{1}(\theta,\varphi)={-1\over 64}\sqrt{105\over 2\pi}\cdot e^{i\varphi}% \cdot\sin\theta\cdot(429\cos^{6}\theta-495\cos^{4}\theta+135\cos^{2}\theta-5)
  60. Y 7 2 ( θ , φ ) = 3 64 35 π e 2 i φ sin 2 θ ( 143 cos 5 θ - 110 cos 3 θ + 15 cos θ ) Y_{7}^{2}(\theta,\varphi)={3\over 64}\sqrt{35\over\pi}\cdot e^{2i\varphi}\cdot% \sin^{2}\theta\cdot(143\cos^{5}\theta-110\cos^{3}\theta+15\cos\theta)
  61. Y 7 3 ( θ , φ ) = - 3 64 35 2 π e 3 i φ sin 3 θ ( 143 cos 4 θ - 66 cos 2 θ + 3 ) Y_{7}^{3}(\theta,\varphi)={-3\over 64}\sqrt{35\over 2\pi}\cdot e^{3i\varphi}% \cdot\sin^{3}\theta\cdot(143\cos^{4}\theta-66\cos^{2}\theta+3)
  62. Y 7 4 ( θ , φ ) = 3 32 385 2 π e 4 i φ sin 4 θ ( 13 cos 3 θ - 3 cos θ ) Y_{7}^{4}(\theta,\varphi)={3\over 32}\sqrt{385\over 2\pi}\cdot e^{4i\varphi}% \cdot\sin^{4}\theta\cdot(13\cos^{3}\theta-3\cos\theta)
  63. Y 7 5 ( θ , φ ) = - 3 64 385 2 π e 5 i φ sin 5 θ ( 13 cos 2 θ - 1 ) Y_{7}^{5}(\theta,\varphi)={-3\over 64}\sqrt{385\over 2\pi}\cdot e^{5i\varphi}% \cdot\sin^{5}\theta\cdot(13\cos^{2}\theta-1)
  64. Y 7 6 ( θ , φ ) = 3 64 5005 π e 6 i φ sin 6 θ cos θ Y_{7}^{6}(\theta,\varphi)={3\over 64}\sqrt{5005\over\pi}\cdot e^{6i\varphi}% \cdot\sin^{6}\theta\cdot\cos\theta
  65. Y 7 7 ( θ , φ ) = - 3 64 715 2 π e 7 i φ sin 7 θ Y_{7}^{7}(\theta,\varphi)={-3\over 64}\sqrt{715\over 2\pi}\cdot e^{7i\varphi}% \cdot\sin^{7}\theta
  66. Y 8 - 8 ( θ , φ ) = 3 256 12155 2 π e - 8 i φ sin 8 θ Y_{8}^{-8}(\theta,\varphi)={3\over 256}\sqrt{12155\over 2\pi}\cdot e^{-8i% \varphi}\cdot\sin^{8}\theta
  67. Y 8 - 7 ( θ , φ ) = 3 64 12155 2 π e - 7 i φ sin 7 θ cos θ Y_{8}^{-7}(\theta,\varphi)={3\over 64}\sqrt{12155\over 2\pi}\cdot e^{-7i% \varphi}\cdot\sin^{7}\theta\cdot\cos\theta
  68. Y 8 - 6 ( θ , φ ) = 1 128 7293 π e - 6 i φ sin 6 θ ( 15 cos 2 θ - 1 ) Y_{8}^{-6}(\theta,\varphi)={1\over 128}\sqrt{7293\over\pi}\cdot e^{-6i\varphi}% \cdot\sin^{6}\theta\cdot(15\cos^{2}\theta-1)
  69. Y 8 - 5 ( θ , φ ) = 3 64 17017 2 π e - 5 i φ sin 5 θ ( 5 cos 3 θ - cos θ ) Y_{8}^{-5}(\theta,\varphi)={3\over 64}\sqrt{17017\over 2\pi}\cdot e^{-5i% \varphi}\cdot\sin^{5}\theta\cdot(5\cos^{3}\theta-\cos\theta)
  70. Y 8 - 4 ( θ , φ ) = 3 128 1309 2 π e - 4 i φ sin 4 θ ( 65 cos 4 θ - 26 cos 2 θ + 1 ) Y_{8}^{-4}(\theta,\varphi)={3\over 128}\sqrt{1309\over 2\pi}\cdot e^{-4i% \varphi}\cdot\sin^{4}\theta\cdot(65\cos^{4}\theta-26\cos^{2}\theta+1)
  71. Y 8 - 3 ( θ , φ ) = 1 64 19635 2 π e - 3 i φ sin 3 θ ( 39 cos 5 θ - 26 cos 3 θ + 3 cos θ ) Y_{8}^{-3}(\theta,\varphi)={1\over 64}\sqrt{19635\over 2\pi}\cdot e^{-3i% \varphi}\cdot\sin^{3}\theta\cdot(39\cos^{5}\theta-26\cos^{3}\theta+3\cos\theta)
  72. Y 8 - 2 ( θ , φ ) = 3 128 595 π e - 2 i φ sin 2 θ ( 143 cos 6 θ - 143 cos 4 θ + 33 cos 2 θ - 1 ) Y_{8}^{-2}(\theta,\varphi)={3\over 128}\sqrt{595\over\pi}\cdot e^{-2i\varphi}% \cdot\sin^{2}\theta\cdot(143\cos^{6}\theta-143\cos^{4}\theta+33\cos^{2}\theta-1)
  73. Y 8 - 1 ( θ , φ ) = 3 64 17 2 π e - i φ sin θ ( 715 cos 7 θ - 1001 cos 5 θ + 385 cos 3 θ - 35 cos θ ) Y_{8}^{-1}(\theta,\varphi)={3\over 64}\sqrt{17\over 2\pi}\cdot e^{-i\varphi}% \cdot\sin\theta\cdot(715\cos^{7}\theta-1001\cos^{5}\theta+385\cos^{3}\theta-35% \cos\theta)
  74. Y 8 0 ( θ , φ ) = 1 256 17 π ( 6435 cos 8 θ - 12012 cos 6 θ + 6930 cos 4 θ - 1260 cos 2 θ + 35 ) Y_{8}^{0}(\theta,\varphi)={1\over 256}\sqrt{17\over\pi}\cdot(6435\cos^{8}% \theta-12012\cos^{6}\theta+6930\cos^{4}\theta-1260\cos^{2}\theta+35)
  75. Y 8 1 ( θ , φ ) = - 3 64 17 2 π e i φ sin θ ( 715 cos 7 θ - 1001 cos 5 θ + 385 cos 3 θ - 35 cos θ ) Y_{8}^{1}(\theta,\varphi)={-3\over 64}\sqrt{17\over 2\pi}\cdot e^{i\varphi}% \cdot\sin\theta\cdot(715\cos^{7}\theta-1001\cos^{5}\theta+385\cos^{3}\theta-35% \cos\theta)
  76. Y 8 2 ( θ , φ ) = 3 128 595 π e 2 i φ sin 2 θ ( 143 cos 6 θ - 143 cos 4 θ + 33 cos 2 θ - 1 ) Y_{8}^{2}(\theta,\varphi)={3\over 128}\sqrt{595\over\pi}\cdot e^{2i\varphi}% \cdot\sin^{2}\theta\cdot(143\cos^{6}\theta-143\cos^{4}\theta+33\cos^{2}\theta-1)
  77. Y 8 3 ( θ , φ ) = - 1 64 19635 2 π e 3 i φ sin 3 θ ( 39 cos 5 θ - 26 cos 3 θ + 3 cos θ ) Y_{8}^{3}(\theta,\varphi)={-1\over 64}\sqrt{19635\over 2\pi}\cdot e^{3i\varphi% }\cdot\sin^{3}\theta\cdot(39\cos^{5}\theta-26\cos^{3}\theta+3\cos\theta)
  78. Y 8 4 ( θ , φ ) = 3 128 1309 2 π e 4 i φ sin 4 θ ( 65 cos 4 θ - 26 cos 2 θ + 1 ) Y_{8}^{4}(\theta,\varphi)={3\over 128}\sqrt{1309\over 2\pi}\cdot e^{4i\varphi}% \cdot\sin^{4}\theta\cdot(65\cos^{4}\theta-26\cos^{2}\theta+1)
  79. Y 8 5 ( θ , φ ) = - 3 64 17017 2 π e 5 i φ sin 5 θ ( 5 cos 3 θ - cos θ ) Y_{8}^{5}(\theta,\varphi)={-3\over 64}\sqrt{17017\over 2\pi}\cdot e^{5i\varphi% }\cdot\sin^{5}\theta\cdot(5\cos^{3}\theta-\cos\theta)
  80. Y 8 6 ( θ , φ ) = 1 128 7293 π e 6 i φ sin 6 θ ( 15 cos 2 θ - 1 ) Y_{8}^{6}(\theta,\varphi)={1\over 128}\sqrt{7293\over\pi}\cdot e^{6i\varphi}% \cdot\sin^{6}\theta\cdot(15\cos^{2}\theta-1)
  81. Y 8 7 ( θ , φ ) = - 3 64 12155 2 π e 7 i φ sin 7 θ cos θ Y_{8}^{7}(\theta,\varphi)={-3\over 64}\sqrt{12155\over 2\pi}\cdot e^{7i\varphi% }\cdot\sin^{7}\theta\cdot\cos\theta
  82. Y 8 8 ( θ , φ ) = 3 256 12155 2 π e 8 i φ sin 8 θ Y_{8}^{8}(\theta,\varphi)={3\over 256}\sqrt{12155\over 2\pi}\cdot e^{8i\varphi% }\cdot\sin^{8}\theta
  83. Y 9 - 9 ( θ , φ ) = 1 512 230945 π e - 9 i φ sin 9 θ Y_{9}^{-9}(\theta,\varphi)={1\over 512}\sqrt{230945\over\pi}\cdot e^{-9i% \varphi}\cdot\sin^{9}\theta
  84. Y 9 - 8 ( θ , φ ) = 3 256 230945 2 π e - 8 i φ sin 8 θ cos θ Y_{9}^{-8}(\theta,\varphi)={3\over 256}\sqrt{230945\over 2\pi}\cdot e^{-8i% \varphi}\cdot\sin^{8}\theta\cdot\cos\theta
  85. Y 9 - 7 ( θ , φ ) = 3 512 13585 π e - 7 i φ sin 7 θ ( 17 cos 2 θ - 1 ) Y_{9}^{-7}(\theta,\varphi)={3\over 512}\sqrt{13585\over\pi}\cdot e^{-7i\varphi% }\cdot\sin^{7}\theta\cdot(17\cos^{2}\theta-1)
  86. Y 9 - 6 ( θ , φ ) = 1 128 40755 π e - 6 i φ sin 6 θ ( 17 cos 3 θ - 3 cos θ ) Y_{9}^{-6}(\theta,\varphi)={1\over 128}\sqrt{40755\over\pi}\cdot e^{-6i\varphi% }\cdot\sin^{6}\theta\cdot(17\cos^{3}\theta-3\cos\theta)
  87. Y 9 - 5 ( θ , φ ) = 3 256 2717 π e - 5 i φ sin 5 θ ( 85 cos 4 θ - 30 cos 2 θ + 1 ) Y_{9}^{-5}(\theta,\varphi)={3\over 256}\sqrt{2717\over\pi}\cdot e^{-5i\varphi}% \cdot\sin^{5}\theta\cdot(85\cos^{4}\theta-30\cos^{2}\theta+1)
  88. Y 9 - 4 ( θ , φ ) = 3 128 95095 2 π e - 4 i φ sin 4 θ ( 17 cos 5 θ - 10 cos 3 θ + cos θ ) Y_{9}^{-4}(\theta,\varphi)={3\over 128}\sqrt{95095\over 2\pi}\cdot e^{-4i% \varphi}\cdot\sin^{4}\theta\cdot(17\cos^{5}\theta-10\cos^{3}\theta+\cos\theta)
  89. Y 9 - 3 ( θ , φ ) = 1 256 21945 π e - 3 i φ sin 3 θ ( 221 cos 6 θ - 195 cos 4 θ + 39 cos 2 θ - 1 ) Y_{9}^{-3}(\theta,\varphi)={1\over 256}\sqrt{21945\over\pi}\cdot e^{-3i\varphi% }\cdot\sin^{3}\theta\cdot(221\cos^{6}\theta-195\cos^{4}\theta+39\cos^{2}\theta% -1)
  90. Y 9 - 2 ( θ , φ ) = 3 128 1045 π e - 2 i φ sin 2 θ ( 221 cos 7 θ - 273 cos 5 θ + 91 cos 3 θ - 7 cos θ ) Y_{9}^{-2}(\theta,\varphi)={3\over 128}\sqrt{1045\over\pi}\cdot e^{-2i\varphi}% \cdot\sin^{2}\theta\cdot(221\cos^{7}\theta-273\cos^{5}\theta+91\cos^{3}\theta-% 7\cos\theta)
  91. Y 9 - 1 ( θ , φ ) = 3 256 95 2 π e - i φ sin θ ( 2431 cos 8 θ - 4004 cos 6 θ + 2002 cos 4 θ - 308 cos 2 θ + 7 ) Y_{9}^{-1}(\theta,\varphi)={3\over 256}\sqrt{95\over 2\pi}\cdot e^{-i\varphi}% \cdot\sin\theta\cdot(2431\cos^{8}\theta-4004\cos^{6}\theta+2002\cos^{4}\theta-% 308\cos^{2}\theta+7)
  92. Y 9 0 ( θ , φ ) = 1 256 19 π ( 12155 cos 9 θ - 25740 cos 7 θ + 18018 cos 5 θ - 4620 cos 3 θ + 315 cos θ ) Y_{9}^{0}(\theta,\varphi)={1\over 256}\sqrt{19\over\pi}\cdot(12155\cos^{9}% \theta-25740\cos^{7}\theta+18018\cos^{5}\theta-4620\cos^{3}\theta+315\cos\theta)
  93. Y 9 1 ( θ , φ ) = - 3 256 95 2 π e i φ sin θ ( 2431 cos 8 θ - 4004 cos 6 θ + 2002 cos 4 θ - 308 cos 2 θ + 7 ) Y_{9}^{1}(\theta,\varphi)={-3\over 256}\sqrt{95\over 2\pi}\cdot e^{i\varphi}% \cdot\sin\theta\cdot(2431\cos^{8}\theta-4004\cos^{6}\theta+2002\cos^{4}\theta-% 308\cos^{2}\theta+7)
  94. Y 9 2 ( θ , φ ) = 3 128 1045 π e 2 i φ sin 2 θ ( 221 cos 7 θ - 273 cos 5 θ + 91 cos 3 θ - 7 cos θ ) Y_{9}^{2}(\theta,\varphi)={3\over 128}\sqrt{1045\over\pi}\cdot e^{2i\varphi}% \cdot\sin^{2}\theta\cdot(221\cos^{7}\theta-273\cos^{5}\theta+91\cos^{3}\theta-% 7\cos\theta)
  95. Y 9 3 ( θ , φ ) = - 1 256 21945 π e 3 i φ sin 3 θ ( 221 cos 6 θ - 195 cos 4 θ + 39 cos 2 θ - 1 ) Y_{9}^{3}(\theta,\varphi)={-1\over 256}\sqrt{21945\over\pi}\cdot e^{3i\varphi}% \cdot\sin^{3}\theta\cdot(221\cos^{6}\theta-195\cos^{4}\theta+39\cos^{2}\theta-1)
  96. Y 9 4 ( θ , φ ) = 3 128 95095 2 π e 4 i φ sin 4 θ ( 17 cos 5 θ - 10 cos 3 θ + cos θ ) Y_{9}^{4}(\theta,\varphi)={3\over 128}\sqrt{95095\over 2\pi}\cdot e^{4i\varphi% }\cdot\sin^{4}\theta\cdot(17\cos^{5}\theta-10\cos^{3}\theta+\cos\theta)
  97. Y 9 5 ( θ , φ ) = - 3 256 2717 π e 5 i φ sin 5 θ ( 85 cos 4 θ - 30 cos 2 θ + 1 ) Y_{9}^{5}(\theta,\varphi)={-3\over 256}\sqrt{2717\over\pi}\cdot e^{5i\varphi}% \cdot\sin^{5}\theta\cdot(85\cos^{4}\theta-30\cos^{2}\theta+1)
  98. Y 9 6 ( θ , φ ) = 1 128 40755 π e 6 i φ sin 6 θ ( 17 cos 3 θ - 3 cos θ ) Y_{9}^{6}(\theta,\varphi)={1\over 128}\sqrt{40755\over\pi}\cdot e^{6i\varphi}% \cdot\sin^{6}\theta\cdot(17\cos^{3}\theta-3\cos\theta)
  99. Y 9 7 ( θ , φ ) = - 3 512 13585 π e 7 i φ sin 7 θ ( 17 cos 2 θ - 1 ) Y_{9}^{7}(\theta,\varphi)={-3\over 512}\sqrt{13585\over\pi}\cdot e^{7i\varphi}% \cdot\sin^{7}\theta\cdot(17\cos^{2}\theta-1)
  100. Y 9 8 ( θ , φ ) = 3 256 230945 2 π e 8 i φ sin 8 θ cos θ Y_{9}^{8}(\theta,\varphi)={3\over 256}\sqrt{230945\over 2\pi}\cdot e^{8i% \varphi}\cdot\sin^{8}\theta\cdot\cos\theta
  101. Y 9 9 ( θ , φ ) = - 1 512 230945 π e 9 i φ sin 9 θ Y_{9}^{9}(\theta,\varphi)={-1\over 512}\sqrt{230945\over\pi}\cdot e^{9i\varphi% }\cdot\sin^{9}\theta
  102. Y 10 - 10 ( θ , φ ) = 1 1024 969969 π e - 10 i φ sin 10 θ Y_{10}^{-10}(\theta,\varphi)={1\over 1024}\sqrt{969969\over\pi}\cdot e^{-10i% \varphi}\cdot\sin^{10}\theta
  103. Y 10 - 9 ( θ , φ ) = 1 512 4849845 π e - 9 i φ sin 9 θ cos θ Y_{10}^{-9}(\theta,\varphi)={1\over 512}\sqrt{4849845\over\pi}\cdot e^{-9i% \varphi}\cdot\sin^{9}\theta\cdot\cos\theta
  104. Y 10 - 8 ( θ , φ ) = 1 512 255255 2 π e - 8 i φ sin 8 θ ( 19 cos 2 θ - 1 ) Y_{10}^{-8}(\theta,\varphi)={1\over 512}\sqrt{255255\over 2\pi}\cdot e^{-8i% \varphi}\cdot\sin^{8}\theta\cdot(19\cos^{2}\theta-1)
  105. Y 10 - 7 ( θ , φ ) = 3 512 85085 π e - 7 i φ sin 7 θ ( 19 cos 3 θ - 3 cos θ ) Y_{10}^{-7}(\theta,\varphi)={3\over 512}\sqrt{85085\over\pi}\cdot e^{-7i% \varphi}\cdot\sin^{7}\theta\cdot(19\cos^{3}\theta-3\cos\theta)
  106. Y 10 - 6 ( θ , φ ) = 3 1024 5005 π e - 6 i φ sin 6 θ ( 323 cos 4 θ - 102 cos 2 θ + 3 ) Y_{10}^{-6}(\theta,\varphi)={3\over 1024}\sqrt{5005\over\pi}\cdot e^{-6i% \varphi}\cdot\sin^{6}\theta\cdot(323\cos^{4}\theta-102\cos^{2}\theta+3)
  107. Y 10 - 5 ( θ , φ ) = 3 256 1001 π e - 5 i φ sin 5 θ ( 323 cos 5 θ - 170 cos 3 θ + 15 cos θ ) Y_{10}^{-5}(\theta,\varphi)={3\over 256}\sqrt{1001\over\pi}\cdot e^{-5i\varphi% }\cdot\sin^{5}\theta\cdot(323\cos^{5}\theta-170\cos^{3}\theta+15\cos\theta)
  108. Y 10 - 4 ( θ , φ ) = 3 256 5005 2 π e - 4 i φ sin 4 θ ( 323 cos 6 θ - 255 cos 4 θ + 45 cos 2 θ - 1 ) Y_{10}^{-4}(\theta,\varphi)={3\over 256}\sqrt{5005\over 2\pi}\cdot e^{-4i% \varphi}\cdot\sin^{4}\theta\cdot(323\cos^{6}\theta-255\cos^{4}\theta+45\cos^{2% }\theta-1)
  109. Y 10 - 3 ( θ , φ ) = 3 256 5005 π e - 3 i φ sin 3 θ ( 323 cos 7 θ - 357 cos 5 θ + 105 cos 3 θ - 7 cos θ ) Y_{10}^{-3}(\theta,\varphi)={3\over 256}\sqrt{5005\over\pi}\cdot e^{-3i\varphi% }\cdot\sin^{3}\theta\cdot(323\cos^{7}\theta-357\cos^{5}\theta+105\cos^{3}% \theta-7\cos\theta)
  110. Y 10 - 2 ( θ , φ ) = 3 512 385 2 π e - 2 i φ sin 2 θ ( 4199 cos 8 θ - 6188 cos 6 θ + 2730 cos 4 θ - 364 cos 2 θ + 7 ) Y_{10}^{-2}(\theta,\varphi)={3\over 512}\sqrt{385\over 2\pi}\cdot e^{-2i% \varphi}\cdot\sin^{2}\theta\cdot(4199\cos^{8}\theta-6188\cos^{6}\theta+2730% \cos^{4}\theta-364\cos^{2}\theta+7)
  111. Y 10 - 1 ( θ , φ ) = 1 256 1155 2 π e - i φ sin θ ( 4199 cos 9 θ - 7956 cos 7 θ + 4914 cos 5 θ - 1092 cos 3 θ + 63 cos θ ) Y_{10}^{-1}(\theta,\varphi)={1\over 256}\sqrt{1155\over 2\pi}\cdot e^{-i% \varphi}\cdot\sin\theta\cdot(4199\cos^{9}\theta-7956\cos^{7}\theta+4914\cos^{5% }\theta-1092\cos^{3}\theta+63\cos\theta)
  112. Y 10 0 ( θ , φ ) = 1 512 21 π ( 46189 cos 10 θ - 109395 cos 8 θ + 90090 cos 6 θ - 30030 cos 4 θ + 3465 cos 2 θ - 63 ) Y_{10}^{0}(\theta,\varphi)={1\over 512}\sqrt{21\over\pi}\cdot(46189\cos^{10}% \theta-109395\cos^{8}\theta+90090\cos^{6}\theta-30030\cos^{4}\theta+3465\cos^{% 2}\theta-63)
  113. Y 10 1 ( θ , φ ) = - 1 256 1155 2 π e i φ sin θ ( 4199 cos 9 θ - 7956 cos 7 θ + 4914 cos 5 θ - 1092 cos 3 θ + 63 cos θ ) Y_{10}^{1}(\theta,\varphi)={-1\over 256}\sqrt{1155\over 2\pi}\cdot e^{i\varphi% }\cdot\sin\theta\cdot(4199\cos^{9}\theta-7956\cos^{7}\theta+4914\cos^{5}\theta% -1092\cos^{3}\theta+63\cos\theta)
  114. Y 10 2 ( θ , φ ) = 3 512 385 2 π e 2 i φ sin 2 θ ( 4199 cos 8 θ - 6188 cos 6 θ + 2730 cos 4 θ - 364 cos 2 θ + 7 ) Y_{10}^{2}(\theta,\varphi)={3\over 512}\sqrt{385\over 2\pi}\cdot e^{2i\varphi}% \cdot\sin^{2}\theta\cdot(4199\cos^{8}\theta-6188\cos^{6}\theta+2730\cos^{4}% \theta-364\cos^{2}\theta+7)
  115. Y 10 3 ( θ , φ ) = - 3 256 5005 π e 3 i φ sin 3 θ ( 323 cos 7 θ - 357 cos 5 θ + 105 cos 3 θ - 7 cos θ ) Y_{10}^{3}(\theta,\varphi)={-3\over 256}\sqrt{5005\over\pi}\cdot e^{3i\varphi}% \cdot\sin^{3}\theta\cdot(323\cos^{7}\theta-357\cos^{5}\theta+105\cos^{3}\theta% -7\cos\theta)
  116. Y 10 4 ( θ , φ ) = 3 256 5005 2 π e 4 i φ sin 4 θ ( 323 cos 6 θ - 255 cos 4 θ + 45 cos 2 θ - 1 ) Y_{10}^{4}(\theta,\varphi)={3\over 256}\sqrt{5005\over 2\pi}\cdot e^{4i\varphi% }\cdot\sin^{4}\theta\cdot(323\cos^{6}\theta-255\cos^{4}\theta+45\cos^{2}\theta% -1)
  117. Y 10 5 ( θ , φ ) = - 3 256 1001 π e 5 i φ sin 5 θ ( 323 cos 5 θ - 170 cos 3 θ + 15 cos θ ) Y_{10}^{5}(\theta,\varphi)={-3\over 256}\sqrt{1001\over\pi}\cdot e^{5i\varphi}% \cdot\sin^{5}\theta\cdot(323\cos^{5}\theta-170\cos^{3}\theta+15\cos\theta)
  118. Y 10 6 ( θ , φ ) = 3 1024 5005 π e 6 i φ sin 6 θ ( 323 cos 4 θ - 102 cos 2 θ + 3 ) Y_{10}^{6}(\theta,\varphi)={3\over 1024}\sqrt{5005\over\pi}\cdot e^{6i\varphi}% \cdot\sin^{6}\theta\cdot(323\cos^{4}\theta-102\cos^{2}\theta+3)
  119. Y 10 7 ( θ , φ ) = - 3 512 85085 π e 7 i φ sin 7 θ ( 19 cos 3 θ - 3 cos θ ) Y_{10}^{7}(\theta,\varphi)={-3\over 512}\sqrt{85085\over\pi}\cdot e^{7i\varphi% }\cdot\sin^{7}\theta\cdot(19\cos^{3}\theta-3\cos\theta)
  120. Y 10 8 ( θ , φ ) = 1 512 255255 2 π e 8 i φ sin 8 θ ( 19 cos 2 θ - 1 ) Y_{10}^{8}(\theta,\varphi)={1\over 512}\sqrt{255255\over 2\pi}\cdot e^{8i% \varphi}\cdot\sin^{8}\theta\cdot(19\cos^{2}\theta-1)
  121. Y 10 9 ( θ , φ ) = - 1 512 4849845 π e 9 i φ sin 9 θ cos θ Y_{10}^{9}(\theta,\varphi)={-1\over 512}\sqrt{4849845\over\pi}\cdot e^{9i% \varphi}\cdot\sin^{9}\theta\cdot\cos\theta
  122. Y 10 10 ( θ , φ ) = 1 1024 969969 π e 10 i φ sin 10 θ Y_{10}^{10}(\theta,\varphi)={1\over 1024}\sqrt{969969\over\pi}\cdot e^{10i% \varphi}\cdot\sin^{10}\theta
  123. Y 00 = s = Y 0 0 = 1 2 1 π \begin{aligned}\displaystyle Y_{00}&\displaystyle=s=Y_{0}^{0}=\frac{1}{2}\sqrt% {\frac{1}{\pi}}\end{aligned}
  124. Y 1 , - 1 = p y = i 1 2 ( Y 1 - 1 + Y 1 1 ) = 3 4 π y r Y 10 = p z = Y 1 0 = 3 4 π z r Y 11 = p x = 1 2 ( Y 1 - 1 - Y 1 1 ) = 3 4 π x r \begin{aligned}\displaystyle Y_{1,-1}&\displaystyle=p_{y}=i\sqrt{\frac{1}{2}}% \left(Y_{1}^{-1}+Y_{1}^{1}\right)=\sqrt{\frac{3}{4\pi}}\cdot\frac{y}{r}\\ \displaystyle Y_{10}&\displaystyle=p_{z}=Y_{1}^{0}=\sqrt{\frac{3}{4\pi}}\cdot% \frac{z}{r}\\ \displaystyle Y_{11}&\displaystyle=p_{x}=\sqrt{\frac{1}{2}}\left(Y_{1}^{-1}-Y_% {1}^{1}\right)=\sqrt{\frac{3}{4\pi}}\cdot\frac{x}{r}\end{aligned}
  125. Y 2 , - 2 = d x y = i 1 2 ( Y 2 - 2 - Y 2 2 ) = 1 2 15 π x y r 2 Y 2 , - 1 = d y z = i 1 2 ( Y 2 - 1 + Y 2 1 ) = 1 2 15 π y z r 2 Y 20 = d z 2 = Y 2 0 = 1 4 5 π - x 2 - y 2 + 2 z 2 r 2 Y 21 = d x z = 1 2 ( Y 2 - 1 - Y 2 1 ) = 1 2 15 π z x r 2 Y 22 = d x 2 - y 2 = 1 2 ( Y 2 - 2 + Y 2 2 ) = 1 4 15 π x 2 - y 2 r 2 \begin{aligned}\displaystyle Y_{2,-2}&\displaystyle=d_{xy}=i\sqrt{\frac{1}{2}}% \left(Y_{2}^{-2}-Y_{2}^{2}\right)=\frac{1}{2}\sqrt{\frac{15}{\pi}}\cdot\frac{% xy}{r^{2}}\\ \displaystyle Y_{2,-1}&\displaystyle=d_{yz}=i\sqrt{\frac{1}{2}}\left(Y_{2}^{-1% }+Y_{2}^{1}\right)=\frac{1}{2}\sqrt{\frac{15}{\pi}}\cdot\frac{yz}{r^{2}}\\ \displaystyle Y_{20}&\displaystyle=d_{z^{2}}=Y_{2}^{0}=\frac{1}{4}\sqrt{\frac{% 5}{\pi}}\cdot\frac{-x^{2}-y^{2}+2z^{2}}{r^{2}}\\ \displaystyle Y_{21}&\displaystyle=d_{xz}=\sqrt{\frac{1}{2}}\left(Y_{2}^{-1}-Y% _{2}^{1}\right)=\frac{1}{2}\sqrt{\frac{15}{\pi}}\cdot\frac{zx}{r^{2}}\\ \displaystyle Y_{22}&\displaystyle=d_{x^{2}-y^{2}}=\sqrt{\frac{1}{2}}\left(Y_{% 2}^{-2}+Y_{2}^{2}\right)=\frac{1}{4}\sqrt{\frac{15}{\pi}}\cdot\frac{x^{2}-y^{2% }}{r^{2}}\end{aligned}
  126. Y 3 , - 3 = f y ( 3 x 2 - y 2 ) = i 1 2 ( Y 3 - 3 + Y 3 3 ) = 1 4 35 2 π ( 3 x 2 - y 2 ) y r 3 Y 3 , - 2 = f x y z = i 1 2 ( Y 3 - 2 - Y 3 2 ) = 1 2 105 π x y z r 3 Y 3 , - 1 = f y z 2 = i 1 2 ( Y 3 - 1 + Y 3 1 ) = 1 4 21 2 π y ( 4 z 2 - x 2 - y 2 ) r 3 Y 30 = f z 3 = Y 3 0 = 1 4 7 π z ( 2 z 2 - 3 x 2 - 3 y 2 ) r 3 Y 31 = f x z 2 = 1 2 ( Y 3 - 1 - Y 3 1 ) = 1 4 21 2 π x ( 4 z 2 - x 2 - y 2 ) r 3 Y 32 = f z ( x 2 - y 2 ) = 1 2 ( Y 3 - 2 + Y 3 2 ) = 1 4 105 π ( x 2 - y 2 ) z r 3 Y 33 = f x ( x 2 - 3 y 2 ) = 1 2 ( Y 3 - 3 - Y 3 3 ) = 1 4 35 2 π ( x 2 - 3 y 2 ) x r 3 \begin{aligned}\displaystyle Y_{3,-3}&\displaystyle=f_{y(3x^{2}-y^{2})}=i\sqrt% {\frac{1}{2}}\left(Y_{3}^{-3}+Y_{3}^{3}\right)=\frac{1}{4}\sqrt{\frac{35}{2\pi% }}\cdot\frac{\left(3x^{2}-y^{2}\right)y}{r^{3}}\\ \displaystyle Y_{3,-2}&\displaystyle=f_{xyz}=i\sqrt{\frac{1}{2}}\left(Y_{3}^{-% 2}-Y_{3}^{2}\right)=\frac{1}{2}\sqrt{\frac{105}{\pi}}\cdot\frac{xyz}{r^{3}}\\ \displaystyle Y_{3,-1}&\displaystyle=f_{yz^{2}}=i\sqrt{\frac{1}{2}}\left(Y_{3}% ^{-1}+Y_{3}^{1}\right)=\frac{1}{4}\sqrt{\frac{21}{2\pi}}\cdot\frac{y(4z^{2}-x^% {2}-y^{2})}{r^{3}}\\ \displaystyle Y_{30}&\displaystyle=f_{z^{3}}=Y_{3}^{0}=\frac{1}{4}\sqrt{\frac{% 7}{\pi}}\cdot\frac{z(2z^{2}-3x^{2}-3y^{2})}{r^{3}}\\ \displaystyle Y_{31}&\displaystyle=f_{xz^{2}}=\sqrt{\frac{1}{2}}\left(Y_{3}^{-% 1}-Y_{3}^{1}\right)=\frac{1}{4}\sqrt{\frac{21}{2\pi}}\cdot\frac{x(4z^{2}-x^{2}% -y^{2})}{r^{3}}\\ \displaystyle Y_{32}&\displaystyle=f_{z(x^{2}-y^{2})}=\sqrt{\frac{1}{2}}\left(% Y_{3}^{-2}+Y_{3}^{2}\right)=\frac{1}{4}\sqrt{\frac{105}{\pi}}\cdot\frac{\left(% x^{2}-y^{2}\right)z}{r^{3}}\\ \displaystyle Y_{33}&\displaystyle=f_{x(x^{2}-3y^{2})}=\sqrt{\frac{1}{2}}\left% (Y_{3}^{-3}-Y_{3}^{3}\right)=\frac{1}{4}\sqrt{\frac{35}{2\pi}}\cdot\frac{\left% (x^{2}-3y^{2}\right)x}{r^{3}}\end{aligned}
  127. Y 4 , - 4 = g x y ( x 2 - y 2 ) = i 1 2 ( Y 4 - 4 - Y 4 4 ) = 3 4 35 π x y ( x 2 - y 2 ) r 4 Y 4 , - 3 = g z y 3 = i 1 2 ( Y 4 - 3 + Y 4 3 ) = 3 4 35 2 π ( 3 x 2 - y 2 ) y z r 4 Y 4 , - 2 = g z 2 x y = i 1 2 ( Y 4 - 2 - Y 4 2 ) = 3 4 5 π x y ( 7 z 2 - r 2 ) r 4 Y 4 , - 1 = g z 3 y = i 1 2 ( Y 4 - 1 + Y 4 1 ) = 3 4 5 2 π y z ( 7 z 2 - 3 r 2 ) r 4 Y 40 = g z 4 = Y 4 0 = 3 16 1 π ( 35 z 4 - 30 z 2 r 2 + 3 r 4 ) r 4 Y 41 = g z 3 x = 1 2 ( Y 4 - 1 - Y 4 1 ) = 3 4 5 2 π x z ( 7 z 2 - 3 r 2 ) r 4 Y 42 = g z 2 x y = 1 2 ( Y 4 - 2 + Y 4 2 ) = 3 8 5 π ( x 2 - y 2 ) ( 7 z 2 - r 2 ) r 4 Y 43 = g z x 3 = 1 2 ( Y 4 - 3 - Y 4 3 ) = 3 4 35 2 π ( x 2 - 3 y 2 ) x z r 4 Y 44 = g x 4 + y 4 = 1 2 ( Y 4 - 4 + Y 4 4 ) = 3 16 35 π x 2 ( x 2 - 3 y 2 ) - y 2 ( 3 x 2 - y 2 ) r 4 \begin{aligned}\displaystyle Y_{4,-4}&\displaystyle=g_{xy(x^{2}-y^{2})}=i\sqrt% {\frac{1}{2}}\left(Y_{4}^{-4}-Y_{4}^{4}\right)=\frac{3}{4}\sqrt{\frac{35}{\pi}% }\cdot\frac{xy\left(x^{2}-y^{2}\right)}{r^{4}}\\ \displaystyle Y_{4,-3}&\displaystyle=g_{zy^{3}}=i\sqrt{\frac{1}{2}}\left(Y_{4}% ^{-3}+Y_{4}^{3}\right)=\frac{3}{4}\sqrt{\frac{35}{2\pi}}\cdot\frac{(3x^{2}-y^{% 2})yz}{r^{4}}\\ \displaystyle Y_{4,-2}&\displaystyle=g_{z^{2}xy}=i\sqrt{\frac{1}{2}}\left(Y_{4% }^{-2}-Y_{4}^{2}\right)=\frac{3}{4}\sqrt{\frac{5}{\pi}}\cdot\frac{xy\cdot(7z^{% 2}-r^{2})}{r^{4}}\\ \displaystyle Y_{4,-1}&\displaystyle=g_{z^{3}y}=i\sqrt{\frac{1}{2}}\left(Y_{4}% ^{-1}+Y_{4}^{1}\right)=\frac{3}{4}\sqrt{\frac{5}{2\pi}}\cdot\frac{yz\cdot(7z^{% 2}-3r^{2})}{r^{4}}\\ \displaystyle Y_{40}&\displaystyle=g_{z^{4}}=Y_{4}^{0}=\frac{3}{16}\sqrt{\frac% {1}{\pi}}\cdot\frac{(35z^{4}-30z^{2}r^{2}+3r^{4})}{r^{4}}\\ \displaystyle Y_{41}&\displaystyle=g_{z^{3}x}=\sqrt{\frac{1}{2}}\left(Y_{4}^{-% 1}-Y_{4}^{1}\right)=\frac{3}{4}\sqrt{\frac{5}{2\pi}}\cdot\frac{xz\cdot(7z^{2}-% 3r^{2})}{r^{4}}\\ \displaystyle Y_{42}&\displaystyle=g_{z^{2}xy}=\sqrt{\frac{1}{2}}\left(Y_{4}^{% -2}+Y_{4}^{2}\right)=\frac{3}{8}\sqrt{\frac{5}{\pi}}\cdot\frac{(x^{2}-y^{2})% \cdot(7z^{2}-r^{2})}{r^{4}}\\ \displaystyle Y_{43}&\displaystyle=g_{zx^{3}}=\sqrt{\frac{1}{2}}\left(Y_{4}^{-% 3}-Y_{4}^{3}\right)=\frac{3}{4}\sqrt{\frac{35}{2\pi}}\cdot\frac{(x^{2}-3y^{2})% xz}{r^{4}}\\ \displaystyle Y_{44}&\displaystyle=g_{x^{4}+y^{4}}=\sqrt{\frac{1}{2}}\left(Y_{% 4}^{-4}+Y_{4}^{4}\right)=\frac{3}{16}\sqrt{\frac{35}{\pi}}\cdot\frac{x^{2}% \left(x^{2}-3y^{2}\right)-y^{2}\left(3x^{2}-y^{2}\right)}{r^{4}}\end{aligned}

Tannaka–Krein_duality.html

  1. G ^ , \hat{G},
  2. i d Π ( G ) id_{\Pi(G)}
  3. T O b Π ( G ) T\in Ob\Pi(G)
  4. ϕ ( T U ) = ϕ ( T ) ϕ ( U ) \phi(T\otimes U)=\phi(T)\otimes\phi(U)
  5. f ϕ ( T ) = ϕ ( U ) f f\circ\phi(T)=\phi(U)\circ f
  6. ϕ a ϕ \phi_{a}\to\phi
  7. ϕ a ( T ) ϕ ( T ) \phi_{a}(T)\to\phi(T)
  8. T O b Π ( G ) T\in Ob\Pi(G)
  9. τ τ V \tau\mapsto\tau_{V}
  10. τ V W = τ V τ W \tau_{V\otimes W}=\tau_{V}\otimes\tau_{W}
  11. τ ¯ = τ \overline{\tau}=\tau
  12. 𝒯 ( G ) \mathcal{T}(G)
  13. G 𝒯 ( G ) G\to\mathcal{T}(G)
  14. I A A I\otimes A\approx A