wpmath0000008_11

Schramm–Loewner_evolution.html

  1. f t ( z ) t = - z f t ( z ) ζ ( t ) + z ζ ( t ) - z \frac{\partial f_{t}(z)}{\partial t}=-zf^{\prime}_{t}(z)\frac{\zeta(t)+z}{% \zeta(t)-z}
  2. g t ( z ) t = g t ( z ) ζ ( t ) + g t ( z ) ζ ( t ) - g t ( z ) . \dfrac{\partial g_{t}(z)}{\partial t}=g_{t}(z)\dfrac{\zeta(t)+g_{t}(z)}{\zeta(% t)-g_{t}(z)}.
  3. f t ( z ) t = 2 f t ( z ) ζ ( t ) - z \frac{\partial f_{t}(z)}{\partial t}=\frac{2f_{t}^{\prime}(z)}{\zeta(t)-z}
  4. g t ( z ) t = 2 g t ( z ) - ζ ( t ) . \dfrac{\partial g_{t}(z)}{\partial t}=\dfrac{2}{g_{t}(z)-\zeta(t)}.
  5. f t ( ζ ( t ) ) = γ ( t ) \displaystyle f_{t}(\zeta(t))=\gamma(t)
  6. ζ ( t ) = g t ( γ ( t ) ) \displaystyle\zeta(t)=g_{t}(\gamma(t))
  7. f t ( z ) = z 2 - 4 t f_{t}(z)=\sqrt{z^{2}-4t}
  8. g t ( z ) = z 2 + 4 t g_{t}(z)=\sqrt{z^{2}+4t}
  9. γ ( t ) = 2 i t \gamma(t)=2i\sqrt{t}
  10. D t D_{t}
  11. 2 i t 2i\sqrt{t}
  12. ζ ( t ) = κ B ( t ) \displaystyle\zeta(t)=\sqrt{\kappa}B(t)
  13. c = ( 8 - 3 κ ) ( κ - 6 ) 2 κ . c=\frac{(8-3\kappa)(\kappa-6)}{2\kappa}.
  14. d = 1 + κ 8 . d=1+\frac{\kappa}{8}.

Schröder's_equation.html

  1. h ( x ) h(x)
  2. Ψ ( x ) Ψ(x)
  3. C < s u b > h C<sub>h
  4. h ( Φ ( y ) ) = Φ ( s y ) h(Φ(y))=Φ(sy)
  5. α ( x ) = l o g ( Ψ ( x ) ) / l o g ( s ) α(x)=log(Ψ(x))/log(s)
  6. α ( h ( x ) ) = α ( x ) + 1 α(h(x))=α(x)+1
  7. Ψ ( x ) = l o g ( φ ( x ) ) Ψ(x)=log(φ(x))
  8. β ( x ) = Ψ / Ψ β(x)=Ψ/Ψ
  9. β ( f ( x ) ) = f ( x ) β ( x ) β(f(x))=f(x)β(x)
  10. s < s u p > n s<sup>n
  11. h ( a ) ) h(a))
  12. x h ( x ) x→h(x)
  13. t t
  14. h ( x ) h(x)
  15. h ( x ) h(x)
  16. h ( x ) h(x)
  17. Ψ ( x ) Ψ(x)
  18. x h ( x ) x→h(x)
  19. h ( x ) h(x)
  20. v = d h < s u b > t / d t v=dh<sub>t/dt
  21. x x
  22. h ( x ) = 4 x ( 1 x ) h(x)=4x(1−x)
  23. Ψ ( x ) = a r c s i n ² ( x ) Ψ(x)=arcsin²(√x)
  24. s = 4 s=4
  25. h ( x ) = 2 x ( 1 x ) h(x)=2x(1−x)
  26. Ψ ( x ) = ½ l n ( 1 2 x ) Ψ(x)=−½ln(1−2x)
  27. h ( x ) = x / ( 2 x ) h(x)=x/(2−x)
  28. Ψ ( x ) = x / ( 1 x ) Ψ(x)=x/(1−x)
  29. h t ( x ) = Ψ - 1 ( 2 - t Ψ ( x ) ) = x 2 t + x ( 1 - 2 t ) . h_{t}(x)=\Psi^{-1}(2^{-t}\Psi(x))=\frac{x}{2^{t}+x(1-2^{t})}~{}.

Schur_orthogonality_relations.html

  1. α , β := 1 | G | g G α ( g ) β ( g ) ¯ \left\langle\alpha,\beta\right\rangle:=\frac{1}{\left|G\right|}\sum_{g\in G}% \alpha(g)\overline{\beta(g)}
  2. β ( g ) ¯ \overline{\beta(g)}
  3. β \beta
  4. χ i , χ j = { 0 if i j , 1 if i = j . \left\langle\chi_{i},\chi_{j}\right\rangle=\begin{cases}0&\mbox{ if }~{}i\neq j% ,\\ 1&\mbox{ if }~{}i=j.\end{cases}
  5. g , h G g,h\in G
  6. χ i χ i ( g ) χ i ( h ) ¯ = { | C G ( g ) | , if g , h are conjugate 0 otherwise. \sum_{\chi_{i}}\chi_{i}(g)\overline{\chi_{i}(h)}=\begin{cases}\left|C_{G}(g)% \right|,&\mbox{ if }~{}g,h\mbox{ are conjugate }\\ 0&\mbox{ otherwise.}\end{cases}
  7. χ i \chi_{i}
  8. | C G ( g ) | \left|C_{G}(g)\right|
  9. g g
  10. Γ ( λ ) ( R ) m n \Gamma^{(\lambda)}(R)_{mn}
  11. Γ ( λ ) \Gamma^{(\lambda)}
  12. G = { R } G=\{R\}
  13. Γ ( λ ) \Gamma^{(\lambda)}
  14. n = 1 l λ Γ ( λ ) ( R ) n m * Γ ( λ ) ( R ) n k = δ m k for all R G , \sum_{n=1}^{l_{\lambda}}\;\Gamma^{(\lambda)}(R)_{nm}^{*}\;\Gamma^{(\lambda)}(R% )_{nk}=\delta_{mk}\quad\hbox{for all}\quad R\in G,
  15. l λ l_{\lambda}
  16. Γ ( λ ) \Gamma^{(\lambda)}
  17. R G | G | Γ ( λ ) ( R ) n m * Γ ( μ ) ( R ) n m = δ λ μ δ n n δ m m | G | l λ . \sum_{R\in G}^{|G|}\;\Gamma^{(\lambda)}(R)_{nm}^{*}\;\Gamma^{(\mu)}(R)_{n^{% \prime}m^{\prime}}=\delta_{\lambda\mu}\delta_{nn^{\prime}}\delta_{mm^{\prime}}% \frac{|G|}{l_{\lambda}}.
  18. Γ ( λ ) ( R ) n m * \Gamma^{(\lambda)}(R)_{nm}^{*}
  19. Γ ( λ ) ( R ) n m \Gamma^{(\lambda)}(R)_{nm}\,
  20. δ λ μ \delta_{\lambda\mu}
  21. Γ ( λ ) = Γ ( μ ) \Gamma^{(\lambda)}=\Gamma^{(\mu)}
  22. Γ ( λ ) \Gamma^{(\lambda)}
  23. Γ ( μ ) \Gamma^{(\mu)}
  24. n = n n=n^{\prime}
  25. m = m m=m^{\prime}
  26. R G | G | Γ ( μ ) ( R ) n m = 0 \sum_{R\in G}^{|G|}\;\Gamma^{(\mu)}(R)_{nm}=0
  27. n , m = 1 , , l μ n,m=1,\ldots,l_{\mu}
  28. Γ ( μ ) \Gamma^{(\mu)}\,
  29. S 3 S_{3}
  30. C 3 v C_{3v}
  31. S 3 S_{3}
  32. λ = [ 2 , 1 ] \lambda=[2,1]
  33. C 3 v C_{3v}
  34. λ = E \lambda=E
  35. ( 1 0 0 1 ) ( 1 0 0 - 1 ) ( - 1 2 3 2 3 2 1 2 ) ( - 1 2 - 3 2 - 3 2 1 2 ) ( - 1 2 3 2 - 3 2 - 1 2 ) ( - 1 2 - 3 2 3 2 - 1 2 ) \begin{pmatrix}1&0\\ 0&1\\ \end{pmatrix}\quad\begin{pmatrix}1&0\\ 0&-1\\ \end{pmatrix}\quad\begin{pmatrix}-\frac{1}{2}&\frac{\sqrt{3}}{2}\\ \frac{\sqrt{3}}{2}&\frac{1}{2}\\ \end{pmatrix}\quad\begin{pmatrix}-\frac{1}{2}&-\frac{\sqrt{3}}{2}\\ -\frac{\sqrt{3}}{2}&\frac{1}{2}\\ \end{pmatrix}\quad\begin{pmatrix}-\frac{1}{2}&\frac{\sqrt{3}}{2}\\ -\frac{\sqrt{3}}{2}&-\frac{1}{2}\\ \end{pmatrix}\quad\begin{pmatrix}-\frac{1}{2}&-\frac{\sqrt{3}}{2}\\ \frac{\sqrt{3}}{2}&-\frac{1}{2}\\ \end{pmatrix}
  36. R G 6 Γ ( R ) 11 * Γ ( R ) 11 = 1 2 + 1 2 + ( - 1 2 ) 2 + ( - 1 2 ) 2 + ( - 1 2 ) 2 + ( - 1 2 ) 2 = 3. \sum_{R\in G}^{6}\;\Gamma(R)_{11}^{*}\;\Gamma(R)_{11}=1^{2}+1^{2}+\left(-% \tfrac{1}{2}\right)^{2}+\left(-\tfrac{1}{2}\right)^{2}+\left(-\tfrac{1}{2}% \right)^{2}+\left(-\tfrac{1}{2}\right)^{2}=3.
  37. R G 6 Γ ( R ) 11 * Γ ( R ) 22 = 1 2 + ( 1 ) ( - 1 ) + ( - 1 2 ) ( 1 2 ) + ( - 1 2 ) ( 1 2 ) + ( - 1 2 ) 2 + ( - 1 2 ) 2 = 0. \sum_{R\in G}^{6}\;\Gamma(R)_{11}^{*}\;\Gamma(R)_{22}=1^{2}+(1)(-1)+\left(-% \tfrac{1}{2}\right)\left(\tfrac{1}{2}\right)+\left(-\tfrac{1}{2}\right)\left(% \tfrac{1}{2}\right)+\left(-\tfrac{1}{2}\right)^{2}+\left(-\tfrac{1}{2}\right)^% {2}=0.
  38. Tr ( Γ ( R ) ) = m = 1 l Γ ( R ) m m . \operatorname{Tr}\big(\Gamma(R)\big)=\sum_{m=1}^{l}\Gamma(R)_{mm}.
  39. χ { Tr ( Γ ( R ) ) | R G } \chi\equiv\{\operatorname{Tr}\big(\Gamma(R)\big)\;|\;R\in G\}
  40. χ ( λ ) \chi^{(\lambda)}
  41. χ ( λ ) ( R ) Tr ( Γ ( λ ) ( R ) ) . \chi^{(\lambda)}(R)\equiv\operatorname{Tr}\left(\Gamma^{(\lambda)}(R)\right).
  42. R G | G | χ ( λ ) ( R ) * χ ( μ ) ( R ) = δ λ μ | G | , \sum_{R\in G}^{|G|}\chi^{(\lambda)}(R)^{*}\,\chi^{(\mu)}(R)=\delta_{\lambda\mu% }|G|,
  43. R G | G | χ ( λ ) ( R ) * χ ( R ) = n ( λ ) | G | , \sum_{R\in G}^{|G|}\chi^{(\lambda)}(R)^{*}\,\chi(R)=n^{(\lambda)}|G|,
  44. Γ ( λ ) \Gamma^{(\lambda)}
  45. Γ \Gamma\,
  46. χ ( R ) \chi(R)
  47. n ( λ ) | G | = 96 n^{(\lambda)}\,|G|=96
  48. | G | = 24 |G|=24\,
  49. Γ ( λ ) \Gamma^{(\lambda)}\,
  50. Γ \Gamma\,
  51. n ( λ ) = 4 . n^{(\lambda)}=4\,.
  52. G G
  53. d g dg
  54. ( π α ) (\pi^{\alpha})
  55. G G
  56. ϕ v , w α ( g ) = v , π α ( g ) w \phi^{\alpha}_{v,w}(g)=\langle v,\pi^{\alpha}(g)w\rangle
  57. π α \pi^{\alpha}
  58. π α π β \pi^{\alpha}\ncong\pi^{\beta}
  59. G ϕ v , w α ( g ) ϕ v , w β ( g ) d g = 0 \int_{G}\phi^{\alpha}_{v,w}(g)\phi^{\beta}_{v^{\prime},w^{\prime}}(g)dg=0
  60. { e i } \{e_{i}\}
  61. π α \pi^{\alpha}
  62. d α - 1 G ϕ e i , e j α ( g ) ϕ e m , e n α ( g ) ¯ d g = δ i , m δ j , n {d^{\alpha}}^{-1}\int_{G}\phi^{\alpha}_{e_{i},e_{j}}(g)\overline{\phi^{\alpha}% _{e_{m},e_{n}}(g)}dg=\delta_{i,m}\delta_{j,n}
  63. d α d^{\alpha}
  64. π α \pi^{\alpha}
  65. 𝐱 = ( α , β , γ ) \mathbf{x}=(\alpha,\beta,\gamma)
  66. 0 α , γ 2 π 0\leq\alpha,\gamma\leq 2\pi
  67. 0 β π 0\leq\beta\leq\pi
  68. ω ( 𝐱 ) d x 1 d x 2 d x r \omega(\mathbf{x})\,dx_{1}dx_{2}\cdots dx_{r}
  69. ω ( 𝐱 ) \omega(\mathbf{x})
  70. ω ( α , β , γ ) = sin β , \omega(\alpha,\beta,\gamma)=\sin\!\beta\,,
  71. ω ( ψ , θ , ϕ ) = 2 ( 1 - cos ψ ) sin θ \omega(\psi,\theta,\phi)=2(1-\cos\psi)\sin\!\theta\,
  72. 0 ψ π , 0 ϕ 2 π , 0 θ π . 0\leq\psi\leq\pi,\;\;0\leq\phi\leq 2\pi,\;\;0\leq\theta\leq\pi.
  73. Γ ( λ ) ( R - 1 ) = Γ ( λ ) ( R ) - 1 = Γ ( λ ) ( R ) with Γ ( λ ) ( R ) m n Γ ( λ ) ( R ) n m * . \Gamma^{(\lambda)}(R^{-1})=\Gamma^{(\lambda)}(R)^{-1}=\Gamma^{(\lambda)}(R)^{% \dagger}\quad\hbox{with}\quad\Gamma^{(\lambda)}(R)^{\dagger}_{mn}\equiv\Gamma^% {(\lambda)}(R)^{*}_{nm}.
  74. Γ ( λ ) ( 𝐱 ) = Γ ( λ ) ( R ( 𝐱 ) ) \Gamma^{(\lambda)}(\mathbf{x})=\Gamma^{(\lambda)}\Big(R(\mathbf{x})\Big)
  75. x 1 0 x 1 1 x r 0 x r 1 Γ ( λ ) ( 𝐱 ) n m * Γ ( μ ) ( 𝐱 ) n m ω ( 𝐱 ) d x 1 d x r = δ λ μ δ n n δ m m | G | l λ , \int_{x_{1}^{0}}^{x_{1}^{1}}\cdots\int_{x_{r}^{0}}^{x_{r}^{1}}\;\Gamma^{(% \lambda)}(\mathbf{x})^{*}_{nm}\Gamma^{(\mu)}(\mathbf{x})_{n^{\prime}m^{\prime}% }\;\omega(\mathbf{x})dx_{1}\cdots dx_{r}\;=\delta_{\lambda\mu}\delta_{nn^{% \prime}}\delta_{mm^{\prime}}\frac{|G|}{l_{\lambda}},
  76. | G | = x 1 0 x 1 1 x r 0 x r 1 ω ( 𝐱 ) d x 1 d x r . |G|=\int_{x_{1}^{0}}^{x_{1}^{1}}\cdots\int_{x_{r}^{0}}^{x_{r}^{1}}\omega(% \mathbf{x})dx_{1}\cdots dx_{r}.
  77. D ( α β γ ) D^{\ell}(\alpha\beta\gamma)
  78. 2 + 1 2\ell+1
  79. | S O ( 3 ) | = 0 2 π d α 0 π sin β d β 0 2 π d γ = 8 π 2 , |SO(3)|=\int_{0}^{2\pi}d\alpha\int_{0}^{\pi}\sin\!\beta\,d\beta\int_{0}^{2\pi}% d\gamma=8\pi^{2},
  80. 0 2 π 0 π 0 2 π D ( α β γ ) n m * D ( α β γ ) n m sin β d α d β d γ = δ δ n n δ m m 8 π 2 2 + 1 . \int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{2\pi}D^{\ell}(\alpha\beta\gamma)^{*}_{% nm}\;D^{\ell^{\prime}}(\alpha\beta\gamma)_{n^{\prime}m^{\prime}}\;\sin\!\beta% \,d\alpha\,d\beta\,d\gamma=\delta_{\ell\ell^{\prime}}\delta_{nn^{\prime}}% \delta_{mm^{\prime}}\frac{8\pi^{2}}{2\ell+1}.
  81. l λ l_{\lambda}

Schur_test.html

  1. L 2 L 2 L^{2}\to L^{2}
  2. X , Y X,\,Y
  3. n \mathbb{R}^{n}
  4. T \,T
  5. K ( x , y ) \,K(x,y)
  6. x X x\in X
  7. y Y y\in Y
  8. T f ( x ) = Y K ( x , y ) f ( y ) d y . Tf(x)=\int_{Y}K(x,y)f(y)\,dy.
  9. p ( x ) > 0 \,p(x)>0
  10. q ( x ) > 0 \,q(x)>0
  11. α , β > 0 \,\alpha,\beta>0
  12. ( 1 ) Y K ( x , y ) q ( y ) d y α p ( x ) (1)\qquad\int_{Y}K(x,y)q(y)\,dy\leq\alpha p(x)
  13. x \,x
  14. ( 2 ) X p ( x ) K ( x , y ) d x β q ( y ) (2)\qquad\int_{X}p(x)K(x,y)\,dx\leq\beta q(y)
  15. y \,y
  16. T \,T
  17. T : L 2 L 2 T:L^{2}\to L^{2}
  18. T L 2 L 2 α β . \|T\|_{L^{2}\to L^{2}}\leq\sqrt{\alpha\beta}.
  19. p ( x ) \,p(x)
  20. q ( x ) \,q(x)
  21. T \,T
  22. α = β = 1 \,\alpha=\beta=1
  23. p ( x ) = q ( x ) = 1. \,p(x)=q(x)=1.
  24. T L 2 L 2 2 sup x X Y | K ( x , y ) | d y sup y Y X | K ( x , y ) | d x . \|T\|^{2}_{L^{2}\to L^{2}}\leq\sup_{x\in X}\int_{Y}|K(x,y)|\,dy\cdot\sup_{y\in Y% }\int_{X}|K(x,y)|\,dx.
  25. K ( x , y ) \,K(x,y)
  26. L p L q L^{p}\to L^{q}
  27. sup x ( Y | K ( x , y ) | r d y ) 1 / r + sup y ( X | K ( x , y ) | r d x ) 1 / r C , \sup_{x}\Big(\int_{Y}|K(x,y)|^{r}\,dy\Big)^{1/r}+\sup_{y}\Big(\int_{X}|K(x,y)|% ^{r}\,dx\Big)^{1/r}\leq C,
  28. r r\,
  29. 1 r = 1 - ( 1 p - 1 q ) \frac{1}{r}=1-\Big(\frac{1}{p}-\frac{1}{q}\Big)
  30. 1 p q 1\leq p\leq q\leq\infty
  31. T f ( x ) = Y K ( x , y ) f ( y ) d y Tf(x)=\int_{Y}K(x,y)f(y)\,dy
  32. T : L p ( Y ) L q ( X ) T:L^{p}(Y)\to L^{q}(X)
  33. T L p L q C . \|T\|_{L^{p}\to L^{q}}\leq C.
  34. | T f ( x ) | 2 = | Y K ( x , y ) f ( y ) d y | 2 \displaystyle|Tf(x)|^{2}=\left|\int_{Y}K(x,y)f(y)\,dy\right|^{2}
  35. x x
  36. T f L 2 2 α Y ( X p ( x ) K ( x , y ) d x ) f ( y ) 2 q ( y ) d y α β Y f ( y ) 2 d y = α β f L 2 2 . \|Tf\|_{L^{2}}^{2}\leq\alpha\int_{Y}\left(\int_{X}p(x)K(x,y)\,dx\right)\frac{f% (y)^{2}}{q(y)}\,dy\leq\alpha\beta\int_{Y}f(y)^{2}dy=\alpha\beta\|f\|_{L^{2}}^{% 2}.
  37. T f L 2 α β f L 2 \|Tf\|_{L^{2}}\leq\sqrt{\alpha\beta}\|f\|_{L^{2}}
  38. f L 2 ( Y ) f\in L^{2}(Y)
  39. L 2 L^{2}

Schur–Zassenhaus_theorem.html

  1. G G
  2. N N
  3. G / N G/N
  4. G G
  5. N N
  6. G / N G/N
  7. N N
  8. G G
  9. G G
  10. N N
  11. G / N G/N
  12. N N
  13. N N
  14. G / N G/N
  15. N N
  16. G / N G/N
  17. C 4 C_{4}
  18. C 2 C_{2}
  19. C 4 C_{4}
  20. C 2 C_{2}
  21. C 4 / C 2 C 2 C_{4}/C_{2}\cong C_{2}
  22. C 4 C_{4}
  23. C 4 C_{4}
  24. C 2 C_{2}
  25. C 2 C_{2}
  26. C 4 C_{4}
  27. S 3 S_{3}
  28. C 3 C_{3}
  29. S 3 S_{3}
  30. S 3 / C 3 C 2 S_{3}/C_{3}\cong C_{2}
  31. S 3 C 3 C 2 S_{3}\cong C_{3}\rtimes C_{2}
  32. C 3 C_{3}
  33. C 2 C_{2}
  34. C 3 C_{3}
  35. S 3 S_{3}
  36. C 3 C_{3}
  37. S 3 S_{3}
  38. C 3 C_{3}
  39. S 3 S_{3}
  40. V V
  41. V V
  42. V V
  43. V V
  44. V V
  45. V V
  46. C 4 C_{4}

Scleronomous.html

  1. m m\,\!
  2. 𝐯 \mathbf{v}\,\!
  3. T = 1 2 m v 2 . T=\frac{1}{2}mv^{2}\,\!.
  4. 𝐯 = d 𝐫 d t = i 𝐫 q i q ˙ i + 𝐫 t . \mathbf{v}=\frac{d\mathbf{r}}{dt}=\sum_{i}\ \frac{\partial\mathbf{r}}{\partial q% _{i}}\dot{q}_{i}+\frac{\partial\mathbf{r}}{\partial t}\,\!.
  5. T = 1 2 m ( i 𝐫 q i q ˙ i + 𝐫 t ) 2 . T=\frac{1}{2}m\left(\sum_{i}\ \frac{\partial\mathbf{r}}{\partial q_{i}}\dot{q}% _{i}+\frac{\partial\mathbf{r}}{\partial t}\right)^{2}\,\!.
  6. T = T 0 + T 1 + T 2 : T=T_{0}+T_{1}+T_{2}\,\!:
  7. T 0 = 1 2 m ( 𝐫 t ) 2 , T_{0}=\frac{1}{2}m\left(\frac{\partial\mathbf{r}}{\partial t}\right)^{2}\,\!,
  8. T 1 = i m 𝐫 t 𝐫 q i q ˙ i , T_{1}=\sum_{i}\ m\frac{\partial\mathbf{r}}{\partial t}\cdot\frac{\partial% \mathbf{r}}{\partial q_{i}}\dot{q}_{i}\,\!,
  9. T 2 = i , j 1 2 m 𝐫 q i 𝐫 q j q ˙ i q ˙ j , T_{2}=\sum_{i,j}\ \frac{1}{2}m\frac{\partial\mathbf{r}}{\partial q_{i}}\cdot% \frac{\partial\mathbf{r}}{\partial q_{j}}\dot{q}_{i}\dot{q}_{j}\,\!,
  10. T 0 T_{0}\,\!
  11. T 1 T_{1}\,\!
  12. T 2 T_{2}\,\!
  13. 𝐫 t = 0 . \frac{\partial\mathbf{r}}{\partial t}=0\,\!.
  14. T 2 T_{2}\,\!
  15. T = T 2 . T=T_{2}\,\!.
  16. x 2 + y 2 - L = 0 , \sqrt{x^{2}+y^{2}}-L=0\,\!,
  17. ( x , y ) (x,y)\,\!
  18. L L\,\!
  19. x t = x 0 cos ω t , x_{t}=x_{0}\cos\omega t\,\!,
  20. x 0 x_{0}\,\!
  21. ω \omega\,\!
  22. t t\,\!
  23. ( x - x 0 cos ω t ) 2 + y 2 - L = 0 . \sqrt{(x-x_{0}\cos\omega t)^{2}+y^{2}}-L=0\,\!.

Screw_(simple_machine).html

  1. d d\,
  2. α \alpha\,
  3. d = l α 360 d=l\frac{\alpha}{360^{\circ}}\,
  4. l l\,
  5. distance ratio d i n d o u t = 2 π r l \mbox{distance ratio}~{}\equiv\frac{d_{in}}{d_{out}}=\frac{2\pi r}{l}\,
  6. W i n = W o u t W_{in}=W_{out}\,
  7. W i n = 2 π r F i n W_{in}=2\pi rF_{in}\,
  8. W o u t = l F o u t W_{out}=lF_{out}\,
  9. l l\,
  10. T i n = F i n r T_{in}=F_{in}r\,
  11. F o u t T i n = 2 π l \frac{F_{out}}{T_{in}}=\frac{2\pi}{l}\,
  12. W i n = W o u t + W f r i c W_{in}=W_{out}+W_{fric}\,
  13. η = W o u t / W i n \eta=W_{out}/W_{in}\,
  14. W o u t = η W i n W_{out}=\eta W_{in}\,
  15. W i n = F i n d i n W_{in}=F_{in}d_{in}\,
  16. W o u t = F o u t d o u t W_{out}=F_{out}d_{out}\,
  17. F o u t d o u t = η F i n d i n F_{out}d_{out}=\eta F_{in}d_{in}\,
  18. F o u t F i n = η d i n d o u t \frac{F_{out}}{F_{in}}=\eta\frac{d_{in}}{d_{out}}\,
  19. F o u t T i n = 2 π η l \frac{F_{out}}{T_{in}}=\frac{2\pi\eta}{l}\qquad\,
  20. η \eta\,
  21. η \eta\,
  22. η = F o u t / F i n d i n / d o u t = F o u t F i n l 2 π r < 0.50 \eta=\frac{F_{out}/F_{in}}{d_{in}/d_{out}}=\frac{F_{out}}{F_{in}}\frac{l}{2\pi r% }<0.50\,

Second-harmonic_generation.html

  1. E ( ω ) E(\omega)
  2. P ( 2 ) ( 2 ω ) P^{(2)}(2\omega)
  3. E ( 2 ω ) P ( 2 ) ( 2 ω ) = χ ( 2 ) E ( ω ) E ( ω ) E(2\omega)\sim P^{(2)}(2\omega)=\chi^{(2)}E(\omega)E(\omega)
  4. χ ( 2 ) \chi^{(2)}
  5. E ( 2 ω ) E(2\omega)
  6. χ ( 2 ) \chi^{(2)}
  7. χ ( 2 ) \chi^{(2)}
  8. χ z z z ( 2 ) = N s c o s 3 θ α z z z ( 2 ) \chi^{(2)}_{zzz}=N_{s}\langle cos^{3}\theta\rangle\alpha^{(2)}_{zzz}
  9. χ x z x ( 2 ) = 1 2 N s c o s θ s i n 2 θ α z z z ( 2 ) \chi^{(2)}_{xzx}=\frac{1}{2}N_{s}\langle cos\theta sin^{2}\theta\rangle\alpha^% {(2)}_{zzz}
  10. α z z z ( 2 ) \alpha^{(2)}_{zzz}
  11. I 2 ω t o t a l I^{total}_{2\omega}
  12. I 2 ω t o t a l j = 1 n ( E j 2 ω ) 2 = n ( E 2 ω ) 2 = n I 2 ω I^{total}_{2\omega}\propto\sum\limits_{j=1}^{n}(E^{2\omega}_{j})^{2}=n(E^{2% \omega})^{2}=nI_{2\omega}
  13. E j 2 ω E^{2\omega}_{j}
  14. E ( 2 ω ) E(2\omega)
  15. P ( 2 ω ) = 2 ϵ 0 d eff ( 2 ω ; ω , ω ) E 2 ( ω ) , P(2\omega)=2\epsilon_{0}d_{\,\text{eff}}(2\omega;\omega,\omega)E^{2}(\omega),\,
  16. E ( 2 ω ) z = - i ω n 2 ω c d eff E 2 ( ω ) e i Δ k z \frac{\partial E(2\omega)}{\partial z}=-\frac{i\omega}{n_{2\omega}c}d_{\,\text% {eff}}E^{2}(\omega)e^{i\Delta kz}
  17. Δ k = k ( 2 ω ) - 2 k ( ω ) \Delta k=k(2\omega)-2k(\omega)
  18. l l
  19. E ( 2 ω , z = 0 ) = 0 E(2\omega,z=0)=0
  20. E ( 2 ω , z = l ) = - i ω d eff n 2 ω c E 2 ( ω ) 0 l e i Δ k z d z = - i ω d eff n 2 ω c E 2 ( ω ) l sin ( Δ k l / 2 ) Δ k l / 2 e i Δ k l / 2 E(2\omega,z=l)=-\frac{i\omega d_{\,\text{eff}}}{n_{2\omega}c}E^{2}(\omega)\int% _{0}^{l}{e^{i\Delta kz}dz}=-\frac{i\omega d_{\,\text{eff}}}{n_{2\omega}c}E^{2}% (\omega)l\,\frac{\sin{(\Delta kl/2)}}{\Delta kl/2}e^{i\Delta kl/2}
  21. I = n / 2 ϵ 0 / μ 0 | E | 2 I=n/2\sqrt{\epsilon_{0}/\mu_{0}}|E|^{2}
  22. I ( 2 ω , l ) = 2 ω 2 d eff 2 l 2 n 2 ω n ω 2 c 3 ϵ 0 ( sin ( Δ k l / 2 ) Δ k l / 2 ) 2 I 2 ( ω ) I(2\omega,l)=\frac{2\omega^{2}d^{2}_{\,\text{eff}}l^{2}}{n_{2\omega}n_{\omega}% ^{2}c^{3}\epsilon_{0}}\left(\frac{\sin{(\Delta kl/2)}}{\Delta kl/2}\right)^{2}% I^{2}(\omega)
  23. l c = π Δ k l_{c}=\frac{\pi}{\Delta k}
  24. E ( 2 ω ) z = - i ω n 2 ω c d eff E 2 ( ω ) e i Δ k z \frac{\partial E(2\omega)}{\partial z}=-\frac{i\omega}{n_{2\omega}c}d_{\,\text% {eff}}E^{2}(\omega)e^{i\Delta kz}
  25. E ( ω ) z = - i ω n ω c d eff * E ( 2 ω ) E * ( ω ) e - i Δ k z \frac{\partial E(\omega)}{\partial z}=-\frac{i\omega}{n_{\omega}c}d_{\,\text{% eff}}^{*}E(2\omega)E^{*}(\omega)e^{-i\Delta kz}
  26. * *
  27. Δ k = 0 \Delta k=0
  28. n 2 ω [ E * ( 2 ω ) E ( 2 ω ) z + c . c . ] = - n ω [ E ( ω ) E * ( ω ) z + c . c . ] n_{2\omega}[E^{*}(2\omega)\frac{\partial E(2\omega)}{\partial z}+c.c.]=-n_{% \omega}[E(\omega)\frac{\partial E^{*}(\omega)}{\partial z}+c.c.]
  29. c . c . c.c.
  30. n 2 ω | E ( 2 ω ) | 2 + n ω | E ( ω ) | 2 = n 2 ω E 0 2 n_{2\omega}|E(2\omega)|^{2}+n_{\omega}|E(\omega)|^{2}=n_{2\omega}E_{0}^{2}
  31. E ( ω ) = | E ( ω ) | e i ϕ ( ω ) E(\omega)=|E(\omega)|e^{i\phi(\omega)}
  32. E ( 2 ω ) = | E ( 2 ω ) | e i ϕ ( 2 ω ) E(2\omega)=|E(2\omega)|e^{i\phi(2\omega)}
  33. d | E ( 2 ω ) | d z = - i ω d eff n ω c [ E 0 2 - | E ( 2 ω ) | 2 ] e 2 i ϕ ( ω ) - i ϕ ( 2 ω ) \frac{d|E(2\omega)|}{dz}=-\frac{i\omega d_{\,\text{eff}}}{n_{\omega}c}\left[E_% {0}^{2}-|E(2\omega)|^{2}\right]e^{2i\phi(\omega)-i\phi(2\omega)}
  34. 0 | E ( 2 ω ) | l d | E ( 2 ω ) | E 0 2 - | E ( 2 ω ) | 2 = - 0 l i ω d eff n ω c d z \int_{0}^{|E(2\omega)|l}{\frac{d|E(2\omega)|}{E_{0}^{2}-|E(2\omega)|^{2}}}=-% \int_{0}^{l}{\frac{i\omega d_{\,\text{eff}}}{n_{\omega}c}dz}
  35. d x a 2 - x 2 = 1 a tanh - 1 x a \int{\frac{dx}{a^{2}-x^{2}}}=\frac{1}{a}\tanh^{-1}{\frac{x}{a}}
  36. | E ( 2 ω ) | z = l = E 0 tanh ( - i E 0 l ω d eff n ω c e 2 i ϕ ( ω ) - i ϕ ( 2 ω ) ) |E(2\omega)|_{z=l}=E_{0}\tanh{\left(\frac{-iE_{0}l\omega d_{\,\text{eff}}}{n_{% \omega}c}e^{2i\phi(\omega)-i\phi(2\omega)}\right)}
  37. d eff d_{\,\text{eff}}
  38. e 2 i ϕ ( ω ) - i ϕ ( 2 ω ) = i e^{2i\phi(\omega)-i\phi(2\omega)}=i
  39. I ( 2 ω , l ) = I ( ω , 0 ) tanh 2 ( E 0 ω d eff l n ω c ) I(2\omega,l)=I(\omega,0)\tanh^{2}{\left(\frac{E_{0}\omega d_{\,\text{eff}}l}{n% _{\omega}c}\right)}
  40. I ( 2 ω , l ) = I ( ω , 0 ) tanh 2 ( Γ l ) , I(2\omega,l)=I(\omega,0)\tanh^{2}{(\Gamma l)},
  41. Γ = ω d eff E 0 / n c \Gamma=\omega d_{\,\text{eff}}E_{0}/nc
  42. I ( 2 ω , l ) + I ( ω , l ) = I ( ω , 0 ) I(2\omega,l)+I(\omega,l)=I(\omega,0)
  43. I ( ω , l ) = I ( ω , 0 ) sech 2 ( Γ l ) . I(\omega,l)=I(\omega,0)\mathrm{sech}^{2}{(\Gamma l)}.

Second-order_cone_programming.html

  1. f T x \ f^{T}x
  2. A i x + b i 2 c i T x + d i , i = 1 , , m \lVert A_{i}x+b_{i}\rVert_{2}\leq c_{i}^{T}x+d_{i},\quad i=1,\dots,m
  3. F x = g Fx=g
  4. f n , A i n i × n , b i n i , c i n , d i , F p × n f\in\mathbb{R}^{n},\ A_{i}\in\mathbb{R}^{{n_{i}}\times n},\ b_{i}\in\mathbb{R}% ^{n_{i}},\ c_{i}\in\mathbb{R}^{n},\ d_{i}\in\mathbb{R},\ F\in\mathbb{R}^{p% \times n}
  5. g p g\in\mathbb{R}^{p}
  6. x n x\in\mathbb{R}^{n}
  7. A i = 0 A_{i}=0
  8. i = 1 , , m i=1,\dots,m
  9. c i = 0 c_{i}=0
  10. i = 1 , , m i=1,\dots,m
  11. x T A T A x + b T x + c 0. x^{T}A^{T}Ax+b^{T}x+c\leq 0.
  12. ( 1 + b T x + c ) / 2 A x 2 ( 1 - b T x - c ) / 2. \left\|\begin{matrix}(1+b^{T}x+c)/2\\ Ax\end{matrix}\right\|_{2}\leq(1-b^{T}x-c)/2.
  13. c T x \ c^{T}x
  14. P ( a i T x b i ) p , i = 1 , , m P(a_{i}^{T}x\leq b_{i})\geq p,\quad i=1,\dots,m
  15. a i a_{i}
  16. a ¯ i \bar{a}_{i}
  17. Σ i \Sigma_{i}
  18. p 0.5 p\geq 0.5
  19. c T x \ c^{T}x
  20. a ¯ i T x + Φ - 1 ( p ) Σ i 1 / 2 x 2 b i , i = 1 , , m \bar{a}_{i}^{T}x+\Phi^{-1}(p)\lVert\Sigma_{i}^{1/2}x\rVert_{2}\leq b_{i},\quad i% =1,\dots,m
  21. Φ - 1 \Phi^{-1}

Security_market_line.html

  1. SML : E ( R i ) = R f + β i [ E ( R M ) - R f ] \mathrm{SML}:E(R_{i})=R_{f}+\beta_{i}[E(R_{M})-R_{f}]\,
  2. E ( R i ) - R f β i = E ( R M ) - R f . \frac{E(R_{i})-R_{f}}{\beta_{i}}=E(R_{M})-R_{f}.
  3. β M = 1 \beta_{M}=1

Sediment_transport.html

  1. τ b \tau_{b}
  2. τ c \tau_{c}
  3. τ b = τ c \tau_{b}=\tau_{c}
  4. τ b * \tau_{b}*
  5. τ c * \tau_{c}*
  6. τ * \tau*
  7. τ * = τ ( ρ s - ρ f ) ( g ) ( D ) \tau*=\frac{\tau}{(\rho_{s}-\rho_{f})(g)(D)}
  8. τ b * = τ c * \tau_{b}*=\tau_{c}*
  9. Re p \mathrm{Re}_{p}
  10. Re p * \mathrm{Re}_{p}*
  11. τ b * = f ( Re p * ) \tau_{b}*=f\left(\mathrm{Re}_{p}*\right)
  12. τ c * \tau_{c}*
  13. Re p = U p D ν \mathrm{Re}_{p}=\frac{U_{p}D}{\nu}
  14. U p U_{p}
  15. D D
  16. ν \nu
  17. μ \mu
  18. ρ f {\rho_{f}}
  19. ν = μ ρ f \nu=\frac{\mu}{\rho_{f}}
  20. u * u_{*}
  21. u * = τ b ρ f = κ z u z u_{*}=\sqrt{\frac{\tau_{b}}{\rho_{f}}}=\kappa z\frac{\partial u}{\partial z}
  22. τ b \tau_{b}
  23. κ \kappa
  24. κ = 0.407 \kappa={0.407}
  25. Re p * = u * D ν \mathrm{Re}_{p}*=\frac{u_{*}D}{\nu}
  26. τ c * = f ( Re p * ) \tau_{c}*=f\left(\mathrm{Re}_{p}*\right)
  27. τ b * = τ c * \tau_{b}*=\tau_{c}*
  28. τ b * = τ b ( ρ s - ρ f ) ( g ) ( D ) \tau_{b}*=\frac{\tau_{b}}{(\rho_{s}-\rho_{f})(g)(D)}
  29. τ b {\tau_{b}}
  30. τ b = ρ g h sin ( θ ) \tau_{b}=\rho gh\sin(\theta)
  31. sin ( θ ) \sin(\theta)
  32. tan ( θ ) \tan(\theta)
  33. S S
  34. τ b = ρ g h S \tau_{b}=\rho ghS
  35. τ b = ρ g h S \tau_{b}=\rho ghS
  36. u * = ( τ b ρ ) u_{*}=\sqrt{\left(\frac{\tau_{b}}{\rho}\right)}
  37. τ b = ρ u * 2 \tau_{b}=\rho u_{*}^{2}
  38. u * u*
  39. u ¯ \bar{u}
  40. C f C_{f}
  41. τ b = ρ C f ( u ¯ ) 2 \tau_{b}=\rho C_{f}\left(\bar{u}\right)^{2}
  42. τ b * = τ c * \tau_{b}*=\tau_{c}*
  43. τ * = τ ( ρ s - ρ ) ( g ) ( D ) \tau*=\frac{\tau}{(\rho_{s}-\rho)(g)(D)}
  44. τ b ( ρ s - ρ ) ( g ) ( D ) = τ c ( ρ s - ρ ) ( g ) ( D ) \frac{\tau_{b}}{(\rho_{s}-\rho)(g)(D)}=\frac{\tau_{c}}{(\rho_{s}-\rho)(g)(D)}
  45. τ c * \tau_{c}*
  46. τ c * = f ( R e p * ) \tau_{c}*=f\left(Re_{p}*\right)
  47. τ c * \tau_{c}*
  48. τ b ( ρ s - ρ ) ( g ) ( D ) = f ( R e p * ) \frac{\tau_{b}}{(\rho_{s}-\rho)(g)(D)}=f\left(Re_{p}*\right)
  49. ρ g h S = 0.06 ( ρ s - ρ ) ( g ) ( D ) {\rho ghS}=0.06{(\rho_{s}-\rho)(g)(D)}
  50. h S = ( ρ s - ρ ) ρ ( D ) ( f ( Re p * ) ) = R D ( f ( Re p * ) ) {hS}={\frac{(\rho_{s}-\rho)}{\rho}(D)}\left(f\left(\mathrm{Re}_{p}*\right)% \right)=RD\left(f\left(\mathrm{Re}_{p}*\right)\right)
  51. τ c * = 0.06 \tau_{c}*=0.06
  52. τ c * = 0.03 \tau_{c}*=0.03
  53. τ c * = f ( Re p * ) \tau_{c}*=f\left(\mathrm{Re}_{p}*\right)
  54. h S = R D τ c * {hS}=RD\tau_{c}*
  55. ( ρ s = 2650 k g m 3 ) \left(\rho_{s}=2650\frac{kg}{m^{3}}\right)
  56. ( ρ = 1000 k g m 3 ) \left(\rho=1000\frac{kg}{m^{3}}\right)
  57. R = ( ρ s - ρ ) ρ = 1.65 R=\frac{(\rho_{s}-\rho)}{\rho}=1.65
  58. h S = 1.65 ( D ) τ c * {hS}=1.65(D)\tau_{c}*
  59. τ c * = 0.06 \tau_{c}*=0.06
  60. h S = 0.1 ( D ) {hS}={0.1(D)}
  61. τ c * = 0.03 \tau_{c}*=0.03
  62. h S = 0.05 ( D 50 ) {hS}={0.05(D_{50})}
  63. P = w s κ u P=\frac{w_{s}}{\kappa u_{\ast}}
  64. w s = R g D 2 C 1 ν + ( 0.75 C 2 R g D 3 ) ( 0.5 ) w_{s}=\frac{RgD^{2}}{C_{1}\nu+(0.75C_{2}RgD^{3})^{(0.5)}}
  65. ν \nu
  66. C 1 C_{1}
  67. C 2 C_{2}
  68. C 1 C_{1}
  69. C 2 C_{2}
  70. g = 9.8 g=9.8
  71. ν \nu
  72. R R
  73. w s = 16.17 D 2 1.8 10 - 5 + ( 12.1275 D 3 ) ( 0.5 ) w_{s}=\frac{16.17D^{2}}{1.8\cdot 10^{-5}+(12.1275D^{3})^{(0.5)}}
  74. ( τ b * - τ c * ) (\tau^{*}_{b}-\tau^{*}_{c})
  75. ( T s or ϕ ) (T_{s}\,\text{ or }\phi)
  76. T s = ϕ = τ b τ c T_{s}=\phi=\frac{\tau_{b}}{\tau_{c}}
  77. b b
  78. q s = Q s b q_{s}=\frac{Q_{s}}{b}
  79. τ * = τ ( ρ s - ρ ) ( g ) ( D ) \tau*=\frac{\tau}{(\rho_{s}-\rho)(g)(D)}
  80. q s * = q s D ρ s - ρ ρ g D = q s R e p ν q_{s}*=\frac{q_{s}}{D\sqrt{\frac{\rho_{s}-\rho}{\rho}gD}}=\frac{q_{s}}{Re_{p}\nu}
  81. q s * = 8 ( τ * - τ * c ) 3 / 2 q_{s}*=8\left(\tau*-\tau*_{c}\right)^{3/2}
  82. τ * c \tau*_{c}
  83. T s 2 q s * = 5.7 ( τ * - 0.047 ) 3 / 2 T_{s}\approx 2\rightarrow q_{s}*=5.7\left(\tau*-0.047\right)^{3/2}
  84. T s 100 q s * = 12.1 ( τ * - 0.047 ) 3 / 2 T_{s}\approx 100\rightarrow q_{s}*=12.1\left(\tau*-0.047\right)^{3/2}
  85. { q s * = α s ( τ * - τ c * ) n n = 3 2 α s = 1.6 ln ( τ * ) + 9.8 9.64 τ * 0.166 \begin{cases}q_{s}*=\alpha_{s}\left(\tau*-\tau_{c}*\right)^{n}\\ n=\frac{3}{2}\\ \alpha_{s}=1.6\ln\left(\tau*\right)+9.8\approx 9.64\tau*^{0.166}\end{cases}
  86. F i F_{i}
  87. i i
  88. i {}_{i}
  89. F i F_{i}
  90. f s f_{s}
  91. f s f_{s}
  92. f s f_{s}
  93. ϕ \phi
  94. τ r i \tau_{ri}
  95. i {}_{i}
  96. W i * W_{i}^{*}
  97. * *
  98. i {}_{i}
  99. W i * = R g q b i F i u * 3 W_{i}^{*}=\frac{Rgq_{bi}}{F_{i}u*^{3}}
  100. q b i q_{bi}
  101. i i
  102. b b
  103. F i F_{i}
  104. i i
  105. ϕ \phi
  106. ϕ < ϕ \phi<\phi^{^{\prime}}
  107. W i * = 0.002 ϕ 7.5 W_{i}^{*}=0.002\phi^{7.5}
  108. ϕ ϕ \phi\geq\phi^{^{\prime}}
  109. W i * = A ( 1 - χ ϕ 0.5 ) 4.5 W_{i}^{*}=A\left(1-\frac{\chi}{\phi^{0.5}}\right)^{4.5}
  110. W i * W_{i}^{*}
  111. ϕ \phi
  112. A , ϕ , χ A,\phi^{^{\prime}},\chi
  113. A = 70 , ϕ = 1.19 , χ = 0.908 , l a b o r a t o r y A=70,\phi^{^{\prime}}=1.19,\chi=0.908,laboratory
  114. A = 115 , ϕ = 1.27 , χ = 0.923 , f i e l d A=115,\phi^{^{\prime}}=1.27,\chi=0.923,field
  115. D s D_{s}
  116. D g D_{g}
  117. F s F_{s}
  118. F g F_{g}
  119. u * u_{*}
  120. q s * = 2.29 * 10 - 5 A ( z s ) 2.14 ( τ b τ c s ) 3.49 q^{*}_{s}=2.29*10^{-5}A(z_{s})^{2.14}\left(\frac{\tau_{b}}{\tau_{cs}}\right)^{% 3.49}
  121. q s * = q s [ ( s - 1 ) g D s ] 0.5 ρ s D s q^{*}_{s}=\frac{q_{s}}{[(s-1)gD_{s}]^{0.5}\rho_{s}D_{s}}
  122. s {}_{s}
  123. ρ s / ρ w \rho_{s}/\rho_{w}
  124. ρ s \rho_{s}
  125. A ( z s ) A(z_{s})
  126. z s z_{s}
  127. τ b \tau_{b}
  128. τ c s \tau_{cs}
  129. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  130. c 0 c_{0}
  131. z 0 z_{0}
  132. c s c 0 = [ z ( h - z 0 ) z 0 ( h - z ) ] - P / α \frac{c_{s}}{c_{0}}=\left[\frac{z\left(h-z_{0}\right)}{z_{0}\left(h-z\right)}% \right]^{-P/\alpha}
  133. z z
  134. c s c_{s}
  135. h h
  136. P P
  137. α \alpha
  138. K m K_{m}
  139. α = K s K m 1 \alpha=\frac{K_{s}}{K_{m}}\approx 1
  140. α \alpha
  141. q s * = 0.05 c f τ * 2.5 q_{s}*=\frac{0.05}{c_{f}}\tau*^{2.5}
  142. q s * q_{s}*
  143. c f c_{f}
  144. τ * \tau*

Seismic_source.html

  1. s ( t ) = β e - α t 2 sin ( 2 π f m a x t ) s(t)=\beta e^{-\alpha t^{2}}\sin(2\pi f_{max}t)
  2. f m a x f_{max}

Self-averaging.html

  1. X X

Self-consistent_mean_field_(biology).html

  1. p p
  2. k t h kth
  3. r k A r_{k}^{A}
  4. 1 / p 1/p
  5. r k r_{k}
  6. N N
  7. p p
  8. i i
  9. M i M_{i}
  10. E E
  11. P i ( r k A ) P_{i}(r_{k}^{A})
  12. M i ( r k A ) = E k ( r k A ) + x = 1 N y = 1 p P i - 1 ( r x y ) E x y ( r k A , r x y ) M_{i}(r_{k}^{A})=E_{k}(r_{k}^{A})+\sum_{x=1}^{N}\sum_{y=1}^{p}P_{i-1}(r_{x}^{y% })E_{xy}(r_{k}^{A},r_{x}^{y})
  13. P i ( r k A ) = ( exp ( - M i ( r k A ) k T ) ) ( y = 1 p exp ( - M i ( r k y ) k T ) ) - 1 P_{i}(r_{k}^{A})=\left(\exp\left(-\frac{M_{i}(r_{k}^{A})}{kT}\right)\right)% \left(\sum_{y=1}^{p}\exp\left(-\frac{M_{i}(r_{k}^{y})}{kT}\right)\right)^{-1}
  14. k k
  15. T T
  16. M s y s M_{sys}
  17. M s y s = M s i n g l e + M p a i r M_{sys}=M_{single}+M_{pair}
  18. M s i n g l e = x = 1 N y = 1 p P ( r x y ) E x ( r x y ) M_{single}=\sum_{x=1}^{N}\sum_{y=1}^{p}P(r_{x}^{y})E_{x}(r_{x}^{y})
  19. M p a i r = x = 1 N y = 1 p a = x + 1 N b = 1 p ( P ( r x y ) P ( r a b ) E x y ( r x y , r a b ) ) M_{pair}=\sum_{x=1}^{N}\sum_{y=1}^{p}\sum_{a=x+1}^{N}\sum_{b=1}^{p}\left(P(r_{% x}^{y})P(r_{a}^{b})E_{xy}(r_{x}^{y},r_{a}^{b})\right)
  20. k k

Self-organizing_list.html

  1. T a v g = 1 * p ( 1 ) + 2 * p ( 2 ) + 3 * p ( 3 ) + + n * p ( n ) . Tavg=1*p(1)+2*p(2)+3*p(3)+...+n*p(n).
  2. T ( n ) = 1 / n + 2 / n + 3 / n + + n / n = ( 1 + 2 + 3 + + n ) / n = ( n + 1 ) / 2 T(n)=1/n+2/n+3/n+...+n/n=(1+2+3+...+n)/n=(n+1)/2
  3. T ( n ) = 1 * 0.1 + 2 * 0.1 + 3 * 0.3 + 4 * 0.1 + 5 * 0.4 = 3.6 T(n)=1*0.1+2*0.1+3*0.3+4*0.1+5*0.4=3.6
  4. T ( n ) = 1 * 0.4 + 2 * 0.3 + 3 * 0.1 + 4 * 0.1 + 5 * 0.1 = 2.2 T(n)=1*0.4+2*0.3+3*0.1+4*0.1+5*0.1=2.2

Semipermutable_subgroup.html

  1. H H
  2. G G
  3. H H
  4. K K
  5. H H

Semiprime_ring.html

  1. n n\mathbb{Z}
  2. 30 30\mathbb{Z}
  3. 12 12\mathbb{Z}\,
  4. B := { P R B P , P a prime ideal } { x R x n B for some n + } \sqrt{B}:=\bigcap\{P\subseteq R\mid B\subseteq P,P\mbox{ a prime ideal}~{}\}% \subseteq\{x\in R\mid x^{n}\in B\mbox{ for some }~{}n\in\mathbb{N}^{+}\}\,
  5. B \sqrt{B}
  6. A = A \sqrt{A}=A
  7. { 0 } = { 0 } \sqrt{\{0\}}=\{0\}
  8. { 0 } \sqrt{\{0\}}
  9. N i l * ( R ) Nil_{*}(R)\,

Sensitivity_and_specificity.html

  1. 𝑇𝑃𝑅 = 𝑇𝑃 / P = 𝑇𝑃 / ( 𝑇𝑃 + 𝐹𝑁 ) \mathit{TPR}=\mathit{TP}/P=\mathit{TP}/(\mathit{TP}+\mathit{FN})
  2. 𝑆𝑃𝐶 = 𝑇𝑁 / N = 𝑇𝑁 / ( 𝑇𝑁 + 𝐹𝑃 ) \mathit{SPC}=\mathit{TN}/N=\mathit{TN}/(\mathit{TN}+\mathit{FP})
  3. 𝑃𝑃𝑉 = 𝑇𝑃 / ( 𝑇𝑃 + 𝐹𝑃 ) \mathit{PPV}=\mathit{TP}/(\mathit{TP}+\mathit{FP})
  4. 𝑁𝑃𝑉 = 𝑇𝑁 / ( 𝑇𝑁 + 𝐹𝑁 ) \mathit{NPV}=\mathit{TN}/(\mathit{TN}+\mathit{FN})
  5. 𝐹𝑃𝑅 = 𝐹𝑃 / N = 𝐹𝑃 / ( 𝐹𝑃 + 𝑇𝑁 ) = 1 - 𝑆𝑃𝐶 \mathit{FPR}=\mathit{FP}/N=\mathit{FP}/(\mathit{FP}+\mathit{TN})=1-\mathit{SPC}
  6. 𝐹𝑁𝑅 = 𝐹𝑁 / ( 𝑇𝑃 + 𝐹𝑁 ) = 1 - 𝑇𝑃𝑅 \mathit{FNR}=\mathit{FN}/(\mathit{TP}+\mathit{FN})=1-\mathit{TPR}
  7. 𝐹𝐷𝑅 = 𝐹𝑃 / ( 𝑇𝑃 + 𝐹𝑃 ) = 1 - 𝑃𝑃𝑉 \mathit{FDR}=\mathit{FP}/(\mathit{TP}+\mathit{FP})=1-\mathit{PPV}
  8. 𝐴𝐶𝐶 = ( 𝑇𝑃 + 𝑇𝑁 ) / ( 𝑇𝑃 + 𝐹𝑃 + 𝐹𝑁 + 𝑇𝑁 ) \mathit{ACC}=(\mathit{TP}+\mathit{TN})/(\mathit{TP}+\mathit{FP}+\mathit{FN}+% \mathit{TN})
  9. F1 = 2 𝑇𝑃 / ( 2 𝑇𝑃 + 𝐹𝑃 + 𝐹𝑁 ) \mathit{F1}=2\mathit{TP}/(2\mathit{TP}+\mathit{FP}+\mathit{FN})
  10. 𝑇𝑃 × 𝑇𝑁 - 𝐹𝑃 × 𝐹𝑁 ( 𝑇𝑃 + 𝐹𝑃 ) ( 𝑇𝑃 + 𝐹𝑁 ) ( 𝑇𝑁 + 𝐹𝑃 ) ( 𝑇𝑁 + 𝐹𝑁 ) \frac{\mathit{TP}\times\mathit{TN}-\mathit{FP}\times\mathit{FN}}{\sqrt{(% \mathit{TP}+\mathit{FP})(\mathit{TP}+\mathit{FN})(\mathit{TN}+\mathit{FP})(% \mathit{TN}+\mathit{FN})}}
  11. 𝑇𝑃𝑅 + 𝑆𝑃𝐶 - 1 \mathit{TPR}+\mathit{SPC}-1
  12. 𝑃𝑃𝑉 + 𝑁𝑃𝑉 - 1 \mathit{PPV}+\mathit{NPV}-1
  13. sensitivity \displaystyle\,\text{sensitivity}
  14. specificity \displaystyle\,\text{specificity}
  15. μ S \mu_{S}
  16. σ S \sigma_{S}
  17. μ N \mu_{N}
  18. σ N \sigma_{N}
  19. d = μ S - μ N 1 2 ( σ S 2 + σ N 2 ) d^{\prime}=\frac{\mu_{S}-\mu_{N}}{\sqrt{\frac{1}{2}(\sigma_{S}^{2}+\sigma_{N}^% {2})}}
  20. F = 2 × precision × recall precision + recall F=2\times\frac{\,\text{precision}\times\,\text{recall}}{\,\text{precision}+\,% \text{recall}}

Sensitivity_index.html

  1. μ S \mu_{S}
  2. σ S \sigma_{S}
  3. μ N \mu_{N}
  4. σ N \sigma_{N}
  5. d = μ S - μ N 1 2 ( σ S 2 + σ N 2 ) d^{\prime}=\frac{\mu_{S}-\mu_{N}}{\sqrt{\frac{1}{2}(\sigma_{S}^{2}+\sigma_{N}^% {2})}}

Separable_partial_differential_equation.html

  1. R R
  2. n {\mathbb{R}}^{n}
  3. R R
  4. [ - 2 + V ( 𝐱 ) ] ψ ( 𝐱 ) = E ψ ( 𝐱 ) [-\nabla^{2}+V(\mathbf{x})]\psi(\mathbf{x})=E\psi(\mathbf{x})
  5. ψ ( 𝐱 ) \psi(\mathbf{x})
  6. V ( 𝐱 ) V(\mathbf{x})
  7. V ( x 1 , x 2 , x 3 ) = V 1 ( x 1 ) + V 2 ( x 2 ) + V 3 ( x 3 ) , V(x_{1},x_{2},x_{3})=V_{1}(x_{1})+V_{2}(x_{2})+V_{3}(x_{3}),
  8. ψ 1 ( x 1 ) \psi_{1}(x_{1})
  9. ψ 2 ( x 2 ) \psi_{2}(x_{2})
  10. ψ 3 ( x 3 ) \psi_{3}(x_{3})
  11. ψ ( 𝐱 ) = ψ 1 ( x 1 ) ψ 2 ( x 2 ) ψ 3 ( x 3 ) \psi(\mathbf{x})=\psi_{1}(x_{1})\cdot\psi_{2}(x_{2})\cdot\psi_{3}(x_{3})

Sequential_probability_ratio_test.html

  1. H 0 H_{0}
  2. H 1 H_{1}
  3. H 0 : p = p 0 H_{0}:p=p_{0}
  4. H 1 : p = p 1 H_{1}:p=p_{1}
  5. log Λ i \log\Lambda_{i}
  6. S 0 = 0 S_{0}=0
  7. S i = S i - 1 + log Λ i S_{i}=S_{i-1}+\log\Lambda_{i}
  8. a < S i < b a<S_{i}<b
  9. S i b S_{i}\geq b
  10. H 1 H_{1}
  11. S i a S_{i}\leq a
  12. H 0 H_{0}
  13. a < 0 < b < a<0<b<\infty
  14. α \alpha
  15. β \beta
  16. a log β 1 - α a\approx\log\frac{\beta}{1-\alpha}
  17. b log 1 - β α b\approx\log\frac{1-\beta}{\alpha}
  18. α \alpha
  19. β \beta
  20. f θ ( x ) = θ - 1 exp ( - x / θ ) , x , θ > 0 f_{\theta}(x)=\theta^{-1}\exp\left(-x/\theta\right),x,\theta>0
  21. H 0 : θ = θ 0 H_{0}:\theta=\theta_{0}
  22. H 1 : θ = θ 1 H_{1}:\theta=\theta_{1}
  23. θ 1 > θ 0 \theta_{1}>\theta_{0}
  24. log Λ ( x ) = log [ θ 1 - 1 exp ( - x / θ 1 ) θ 0 - 1 exp ( - x / θ 0 ) ] = log [ θ 0 θ 1 exp ( x / θ 0 - x / θ 1 ) ] = θ 1 - θ 0 θ 0 θ 1 x - log θ 1 θ 0 \begin{matrix}\log\Lambda(x)&=&\log\left[\frac{\theta_{1}^{-1}\exp\left(-x/% \theta_{1}\right)}{\theta_{0}^{-1}\exp\left(-x/\theta_{0}\right)}\right]\\ &=&\log\left[\frac{\theta_{0}}{\theta_{1}}\exp\left(x/\theta_{0}-x/\theta_{1}% \right)\right]\\ &=&\frac{\theta_{1}-\theta_{0}}{\theta_{0}\theta_{1}}x-\log\frac{\theta_{1}}{% \theta_{0}}\end{matrix}
  25. S n = i = 1 n log Λ ( x i ) = θ 1 - θ 0 θ 0 θ 1 i = 1 n x i - n log θ 1 θ 0 S_{n}=\sum_{i=1}^{n}\log\Lambda(x_{i})=\frac{\theta_{1}-\theta_{0}}{\theta_{0}% \theta_{1}}\sum_{i=1}^{n}x_{i}-n\log\frac{\theta_{1}}{\theta_{0}}
  26. a < θ 1 - θ 0 θ 0 θ 1 i = 1 n x i - n log θ 1 θ 0 < b a<\frac{\theta_{1}-\theta_{0}}{\theta_{0}\theta_{1}}\sum_{i=1}^{n}x_{i}-n\log% \frac{\theta_{1}}{\theta_{0}}<b
  27. a + n log θ 1 θ 0 < θ 1 - θ 0 θ 0 θ 1 i = 1 n x i < b + n log θ 1 θ 0 a+n\log\frac{\theta_{1}}{\theta_{0}}<\frac{\theta_{1}-\theta_{0}}{\theta_{0}% \theta_{1}}\sum_{i=1}^{n}x_{i}<b+n\log\frac{\theta_{1}}{\theta_{0}}
  28. log ( θ 1 / θ 0 ) \log(\theta_{1}/\theta_{0})

Sequential_quadratic_programming.html

  1. min x f ( x ) s.t. b ( x ) 0 c ( x ) = 0. \begin{array}[]{rl}\min\limits_{x}&f(x)\\ \mbox{s.t.}&b(x)\geq 0\\ &c(x)=0.\end{array}
  2. ( x , λ , σ ) = f ( x ) - λ T b ( x ) - σ T c ( x ) , \mathcal{L}(x,\lambda,\sigma)=f(x)-\lambda^{T}b(x)-\sigma^{T}c(x),
  3. λ \lambda
  4. σ \sigma
  5. x k x_{k}
  6. d k d_{k}
  7. min d f ( x k ) + f ( x k ) T d + 1 2 d T x x 2 ( x k , λ k , σ k ) d s . t . b ( x k ) + b ( x k ) T d 0 c ( x k ) + c ( x k ) T d = 0. \begin{array}[]{rl}\min\limits_{d}&f(x_{k})+\nabla f(x_{k})^{T}d+\tfrac{1}{2}d% ^{T}\nabla_{xx}^{2}\mathcal{L}(x_{k},\lambda_{k},\sigma_{k})d\\ \mathrm{s.t.}&b(x_{k})+\nabla b(x_{k})^{T}d\geq 0\\ &c(x_{k})+\nabla c(x_{k})^{T}d=0.\end{array}
  8. f ( x k ) f(x_{k})

Sergey_Stechkin.html

  1. L p L_{p}

Sextic_equation.html

  1. a x 6 + b x 5 + c x 4 + d x 3 + e x 2 + f x + g = 0 , ax^{6}+bx^{5}+cx^{4}+dx^{3}+ex^{2}+fx+g=0,\,
  2. a 0 a≠0
  3. a , b , c , d , e , f , g a,b,c,d,e,f,g
  4. a a

Sexual_dimorphism_measures.html

  1. X ¯ m X ¯ f , \frac{\bar{X}_{m}}{\bar{X}_{f}},
  2. X ¯ m \bar{X}_{m}
  3. X ¯ f \bar{X}_{f}
  4. log X ¯ m X ¯ f , \operatorname{log}\frac{\bar{X}_{m}}{\bar{X}_{f}},
  5. 100 X ¯ m - X ¯ f X ¯ f , 100\frac{\bar{X}_{m}-\bar{X}_{f}}{\bar{X}_{f}},
  6. 100 X ¯ m - X ¯ f X ¯ f + X ¯ f , 100\frac{\bar{X}_{m}-\bar{X}_{f}}{\bar{X}_{f}+\bar{X}_{f}},
  7. μ m μ f , \frac{\mu_{m}}{\mu_{f}},
  8. T = X ¯ 1 - X ¯ 2 - ( μ 1 - μ 2 ) S 1 2 n 1 + S 2 2 n 2 : t ν , T=\frac{\bar{X}_{1}-\bar{X}_{2}-(\mu_{1}-\mu_{2})}{\sqrt{\frac{S^{2}_{1}}{n_{1% }}+\frac{S^{2}_{2}}{n_{2}}}}:t_{\nu},
  9. ν = ( S 1 2 n 1 + S 2 2 n 2 ) 2 S 1 2 n 1 ( n 1 - 1 ) + S 2 2 n 2 ( n 2 - 1 ) , \nu=\frac{(\frac{S^{2}_{1}}{n_{1}}+\frac{S^{2}_{2}}{n_{2}})^{2}}{\frac{S^{2}_{% 1}}{n_{1}(n_{1}-1)}+\frac{S^{2}_{2}}{n_{2}(n_{2}-1)}},
  10. S i 2 , n i , i = 1 , 2 S^{2}_{i},n_{i},i=1,2
  11. μ 0 = μ 1 - μ 2 . \mu_{0}=\mu_{1}-\mu_{2}.
  12. μ i , σ i 2 , i = 1 , 2 \mu_{i},\sigma^{2}_{i},i=1,2
  13. var ( D ^ ) = p ^ m ( 1 - p ^ m ) n m + p ^ f ( 1 - p ^ f ) n f , \operatorname{var}(\widehat{D})=\frac{\widehat{p}_{m}(1-\widehat{p}_{m})}{n_{m% }}+\frac{\widehat{p}_{f}(1-\widehat{p}_{f})}{n_{f}},
  14. D ^ \widehat{D}
  15. p ^ i , n i , i = m , f \widehat{p}_{i},n_{i},i=m,f
  16. i i
  17. n f n_{f}
  18. X X
  19. f ( x ) = i = 1 n π i f i ( x ) , - < x < , f(x)=\sum_{i=1}^{n}\pi_{i}f_{i}(x),-\infty<x<\infty,
  20. f i , π i , i = 1 , 2 f_{i},\pi_{i},i=1,2
  21. π i f i \pi_{i}f_{i}
  22. π i f i \pi_{i}f_{i}
  23. M I MI
  24. π 1 f 1 \pi_{1}f_{1}
  25. π 2 f 2 \pi_{2}f_{2}
  26. M I MI
  27. M I = R min [ π 1 f 1 ( x ) , ( 1 - π 1 ) f 2 ( x ) ] d x , MI=\int_{R}\operatorname{min}[\pi_{1}f_{1}(x),(1-\pi_{1})f_{2}(x)]\,dx,
  28. R R
  29. π 1 f 1 \pi_{1}f_{1}
  30. π 2 f 2 \pi_{2}f_{2}
  31. μ i , σ i 2 , π 1 , i = 1 , 2 ) \mu_{i},\sigma^{2}_{i},\pi_{1},i=1,2)
  32. ( 0 , 0.5 ] (0,0.5]
  33. max x | F 1 ( x ) - F 2 ( x ) | , \operatorname{max}_{x}|F_{1}(x)-F_{2}(x)|,
  34. F i , i = 1 , 2 F_{i},i=1,2

Seymour_Narrows.html

  1. 10 9 10^{9}

Shadows_of_the_Mind.html

  1. \scriptstyle\hbar
  2. G \scriptstyle G

Shape_correction_function.html

  1. V V
  2. V d V_{d}
  3. M ( V ) M(V)
  4. M ( V ) = ( V / V d ) - 2 / 3 M(V)=(V/V_{d})^{-2/3}
  5. M ( V ) = ( V / V d ) 1 / 3 M(V)=(V/V_{d})^{1/3}
  6. M ( V ) = ( V / V d ) 0 = 1 M(V)=(V/V_{d})^{0}=1
  7. M ( V ) = w ( V / V d ) - 2 / 3 + ( 1 - w ) ( V / V d ) 1 / 3 M(V)=w(V/V_{d})^{-2/3}+(1-w)(V/V_{d})^{1/3}
  8. 0 < w < 1 0<w<1

Shaping_codes.html

  1. C = B log 2 ( 1 + S N ) C=B\log_{2}\left(1+\frac{S}{N}\right)

Shapiro_inequality.html

  1. x 1 , x 2 , , x n x_{1},x_{2},\dots,x_{n}
  2. i = 1 n x i x i + 1 + x i + 2 n 2 \sum_{i=1}^{n}\frac{x_{i}}{x_{i+1}+x_{i+2}}\geq\frac{n}{2}
  3. x n + 1 = x 1 , x n + 2 = x 2 x_{n+1}=x_{1},x_{n+2}=x_{2}
  4. γ n 2 \gamma\frac{n}{2}
  5. γ 0.9891 \gamma\approx 0.9891\dots
  6. 1 2 ψ ( 0 ) \frac{1}{2}\psi(0)
  7. g ( x ) = 2 e x + e x 2 g(x)=\frac{2}{e^{x}+e^{\frac{x}{2}}}
  8. x 20 = ( 1 + 5 ϵ , 6 ϵ , 1 + 4 ϵ , 5 ϵ , 1 + 3 ϵ , 4 ϵ , 1 + 2 ϵ , 3 ϵ , 1 + ϵ , 2 ϵ , 1 + 2 ϵ , ϵ , 1 + 3 ϵ , 2 ϵ , 1 + 4 ϵ , 3 ϵ , 1 + 5 ϵ , 4 ϵ , 1 + 6 ϵ , 5 ϵ ) x_{20}=(1+5\epsilon,\ 6\epsilon,\ 1+4\epsilon,\ 5\epsilon,\ 1+3\epsilon,\ 4% \epsilon,\ 1+2\epsilon,\ 3\epsilon,\ 1+\epsilon,\ 2\epsilon,\ 1+2\epsilon,\ % \epsilon,\ 1+3\epsilon,\ 2\epsilon,\ 1+4\epsilon,\ 3\epsilon,\ 1+5\epsilon,\ 4% \epsilon,\ 1+6\epsilon,\ 5\epsilon)
  9. ϵ \epsilon
  10. 10 - ϵ 2 + O ( ϵ 3 ) 10-\epsilon^{2}+O(\epsilon^{3})
  11. ϵ \epsilon
  12. x 14 x_{14}

Shear_band.html

  1. σ ˙ \dot{\sigma}
  2. ε ˙ \dot{\varepsilon}
  3. \mathbb{C}
  4. σ ˙ = ε ˙ , ( 1 ) \dot{\sigma}=\mathbb{C}\dot{\varepsilon},\qquad{(1)}
  5. \mathbb{C}
  6. 𝐧 \,\textbf{n}
  7. σ ˙ + 𝐧 = σ ˙ - 𝐧 , ( 2 ) \dot{\sigma}^{+}\,\textbf{n}=\dot{\sigma}^{-}\,\textbf{n},\qquad{(2)}
  8. ε ˙ + - ε ˙ - = 1 2 ( 𝐠 𝐧 + 𝐧 𝐠 ) , ( 3 ) \dot{\varepsilon}^{+}-\dot{\varepsilon}^{-}=\frac{1}{2}\left(\,\textbf{g}% \otimes\,\textbf{n}+\,\textbf{n}\otimes\,\textbf{g}\right),\qquad{(3)}
  9. \otimes
  10. 𝐠 \,\textbf{g}
  11. 𝐧 \,\textbf{n}
  12. ( 𝐠 𝐧 ) 𝐧 = 0. ( 4 ) \mathbb{C}\left(\,\textbf{g}\otimes\,\textbf{n}\right)\,\textbf{n}=0.\qquad{(4)}
  13. 𝔸 ( 𝐧 ) \mathbb{A}(\,\textbf{n})
  14. 𝐠 \,\textbf{g}
  15. 𝔸 ( 𝐧 ) 𝐠 = ( 𝐠 𝐧 ) 𝐧 \mathbb{A}(\,\textbf{n})\,\textbf{g}=\mathbb{C}\left(\,\textbf{g}\otimes\,% \textbf{n}\right)\,\textbf{n}

Shear_matrix.html

  1. S = ( 1 0 0 λ 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 ) . S=\begin{pmatrix}1&0&0&\lambda&0\\ 0&1&0&0&0\\ 0&0&1&0&0\\ 0&0&0&1&0\\ 0&0&0&0&1\end{pmatrix}.
  2. λ \lambda
  3. ( x y ) = ( 1 λ 0 1 ) ( x y ) . \begin{pmatrix}x^{\prime}\\ y^{\prime}\end{pmatrix}=\begin{pmatrix}1&\lambda\\ 0&1\end{pmatrix}\begin{pmatrix}x\\ y\end{pmatrix}.
  4. λ \lambda
  5. ( x y ) = ( 1 0 λ 1 ) ( x y ) . \begin{pmatrix}x^{\prime}\\ y^{\prime}\end{pmatrix}=\begin{pmatrix}1&0\\ \lambda&1\end{pmatrix}\begin{pmatrix}x\\ y\end{pmatrix}.
  6. λ \lambda
  7. λ \lambda

Shift_matrix.html

  1. U i j = δ i + 1 , j , L i j = δ i , j + 1 , U_{ij}=\delta_{i+1,j},\quad L_{ij}=\delta_{i,j+1},
  2. δ i j \delta_{ij}
  3. U 5 = ( 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 ) L 5 = ( 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 ) . U_{5}=\begin{pmatrix}0&1&0&0&0\\ 0&0&1&0&0\\ 0&0&0&1&0\\ 0&0&0&0&1\\ 0&0&0&0&0\end{pmatrix}\quad L_{5}=\begin{pmatrix}0&0&0&0&0\\ 1&0&0&0&0\\ 0&1&0&0&0\\ 0&0&1&0&0\\ 0&0&0&1&0\end{pmatrix}.
  4. p U ( λ ) = ( - 1 ) n λ n . p_{U}(\lambda)=(-1)^{n}\lambda^{n}.
  5. N ( U ) = span { ( 1 , 0 , , 0 ) T } , N(U)=\operatorname{span}\{(1,0,\ldots,0)^{T}\},
  6. N ( L ) = span { ( 0 , , 0 , 1 ) T } . N(L)=\operatorname{span}\{(0,\ldots,0,1)^{T}\}.
  7. { 0 } \{0\}
  8. ( 1 , 0 , , 0 ) T (1,0,\ldots,0)^{T}
  9. ( 0 , , 0 , 1 ) T (0,\ldots,0,1)^{T}
  10. U L = I - diag ( 0 , , 0 , 1 ) , UL=I-\operatorname{diag}(0,\ldots,0,1),
  11. L U = I - diag ( 1 , 0 , , 0 ) . LU=I-\operatorname{diag}(1,0,\ldots,0).
  12. ( S 1 0 0 0 S 2 0 0 0 S r ) \begin{pmatrix}S_{1}&0&\ldots&0\\ 0&S_{2}&\ldots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\ldots&S_{r}\end{pmatrix}
  13. S = ( 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 ) ; A = ( 1 1 1 1 1 1 2 2 2 1 1 2 3 2 1 1 2 2 2 1 1 1 1 1 1 ) . S=\begin{pmatrix}0&0&0&0&0\\ 1&0&0&0&0\\ 0&1&0&0&0\\ 0&0&1&0&0\\ 0&0&0&1&0\end{pmatrix};\quad A=\begin{pmatrix}1&1&1&1&1\\ 1&2&2&2&1\\ 1&2&3&2&1\\ 1&2&2&2&1\\ 1&1&1&1&1\end{pmatrix}.
  14. S A = ( 0 0 0 0 0 1 1 1 1 1 1 2 2 2 1 1 2 3 2 1 1 2 2 2 1 ) ; A S = ( 1 1 1 1 0 2 2 2 1 0 2 3 2 1 0 2 2 2 1 0 1 1 1 1 0 ) . SA=\begin{pmatrix}0&0&0&0&0\\ 1&1&1&1&1\\ 1&2&2&2&1\\ 1&2&3&2&1\\ 1&2&2&2&1\end{pmatrix};\quad AS=\begin{pmatrix}1&1&1&1&0\\ 2&2&2&1&0\\ 2&3&2&1&0\\ 2&2&2&1&0\\ 1&1&1&1&0\end{pmatrix}.
  15. S T A S S^{T}AS
  16. S T A S = ( 2 2 2 1 0 2 3 2 1 0 2 2 2 1 0 1 1 1 1 0 0 0 0 0 0 ) . S^{T}AS=\begin{pmatrix}2&2&2&1&0\\ 2&3&2&1&0\\ 2&2&2&1&0\\ 1&1&1&1&0\\ 0&0&0&0&0\end{pmatrix}.

Ship_resistance_and_propulsion.html

  1. V = 1.34 L V=1.34\sqrt{L}

Shock_factor.html

  1. S F = ( W ) R ( 1 + s i n ϕ ) 2 SF={\sqrt{(}W)\over R}{{(1+sin\phi)}\over 2}
  2. ϕ \phi

Shubnikov–de_Haas_effect.html

  1. I m = 2 e i h ( μ m - l m T m l μ l ) , I_{m}=2\frac{e\cdot i}{h}\left(\mu_{m}-\sum_{l\neq m}T_{ml}\mu_{l}\right),\,
  2. l m l≠m
  3. l m l≠m
  4. T 21 = T 32 = T 43 = T 14 = 1 , T_{21}=T_{32}=T_{43}=T_{14}=1,
  5. T m l = 0 T_{ml}=0
  6. ( I 1 I 2 I 3 I 4 ) = 2 e i h ( 1 0 0 - 1 - 1 1 0 0 0 - 1 1 0 0 0 - 1 1 ) ( μ 1 μ 2 μ 3 μ 4 ) . \left(\begin{matrix}I_{1}\\ I_{2}\\ I_{3}\\ I_{4}\end{matrix}\right)=\frac{2e\cdot i}{h}\left(\begin{matrix}1&0&0&-1\\ -1&1&0&0\\ 0&-1&1&0\\ 0&0&-1&1\end{matrix}\right)\left(\begin{matrix}\mu_{1}\\ \mu_{2}\\ \mu_{3}\\ \mu_{4}\end{matrix}\right).
  7. I 3 = 0 = 2 e i h ( - μ 2 + μ 3 ) , I_{3}=0=\frac{2e\cdot i}{h}\left(-\mu_{2}+\mu_{3}\right),
  8. μ 2 = μ 3 . \mu_{2}=\mu_{3}.
  9. R SdH = μ 2 - μ 3 e I 1 = 0. R_{\mathrm{SdH}}=\frac{\mu_{2}-\mu_{3}}{e\cdot I_{1}}=0.
  10. D = ( 2 S + 1 ) ϕ ϕ 0 = 2 ϕ ϕ 0 . D=\left(2S+1\right)\frac{\phi}{\phi_{0}}=2\frac{\phi}{\phi_{0}}.\,
  11. D = 2 e B A h D=2\frac{e\cdot B\cdot A}{h}
  12. N = 2 e B h . N=2\frac{e\cdot B}{h}.
  13. n = i N = 2 i e B h . n=i\cdot N=2\cdot i\cdot\frac{e\cdot B}{h}.
  14. B i = n h 2 e i , B_{i}=\frac{n\cdot h}{2\cdot e\cdot i},
  15. 1 B i = 2 e i n h , \frac{1}{B_{i}}=\frac{2\cdot e\cdot i}{n\cdot h},
  16. Δ ( 1 B ) = 1 B i + 1 - 1 B i = 2 e n h . \Delta\left(\frac{1}{B}\right)=\frac{1}{B_{i+1}}-\frac{1}{B_{i}}=\frac{2\cdot e% }{n\cdot h}.\,
  17. n = 2 e 0.00618 / T h 7.82 10 14 / m 2 . n=\frac{2\cdot e}{0.00618/\mathrm{T}\cdot h}\approx 7.82\cdot 10^{14}/\mathrm{% m}^{2}.
  18. < V A R > ν = 2 i <VAR>ν=2∙i
  19. Δ ( 1 B ) = 2 e n h c . \Delta\left(\frac{1}{B}\right)=\frac{2\cdot e}{n\cdot h\cdot c}.

Siegel_modular_form.html

  1. g , N g,N\in\mathbb{N}
  2. g = { τ M g × g ( ) | τ T = τ , Im ( τ ) positive definite } , \mathcal{H}_{g}=\left\{\tau\in M_{g\times g}(\mathbb{C})\ \big|\ \tau^{\mathrm% {T}}=\tau,\textrm{Im}(\tau)\,\text{ positive definite}\right\},
  3. N N
  4. Γ g ( N ) , \Gamma_{g}(N),
  5. Γ g ( N ) = { γ G L 2 g ( ) | γ T ( 0 I g - I g 0 ) γ = ( 0 I g - I g 0 ) , γ I 2 g mod N } , \Gamma_{g}(N)=\left\{\gamma\in GL_{2g}(\mathbb{Z})\ \big|\ \gamma^{\mathrm{T}}% \begin{pmatrix}0&I_{g}\\ -I_{g}&0\end{pmatrix}\gamma=\begin{pmatrix}0&I_{g}\\ -I_{g}&0\end{pmatrix},\ \gamma\equiv I_{2g}\mod N\right\},
  6. I g I_{g}
  7. g × g g\times g
  8. ρ : GL g ( ) GL ( V ) \rho:\textrm{GL}_{g}(\mathbb{C})\rightarrow\textrm{GL}(V)
  9. V V
  10. γ = ( A B C D ) \gamma=\begin{pmatrix}A&B\\ C&D\end{pmatrix}
  11. γ Γ g ( N ) , \gamma\in\Gamma_{g}(N),
  12. ( f | γ ) ( τ ) = ( ρ ( C τ + D ) ) - 1 f ( γ τ ) . (f\big|\gamma)(\tau)=(\rho(C\tau+D))^{-1}f(\gamma\tau).
  13. f : g V f:\mathcal{H}_{g}\rightarrow V
  14. g g
  15. ρ \rho
  16. N N
  17. ( f | γ ) = f . (f\big|\gamma)=f.
  18. g = 1 g=1
  19. f f
  20. g > 1 g>1
  21. ρ \rho
  22. g g
  23. N N
  24. M ρ ( Γ g ( N ) ) . M_{\rho}(\Gamma_{g}(N)).
  25. f f
  26. ρ \rho
  27. g > 1 g>1
  28. f f
  29. g \mathcal{H}_{g}
  30. { τ g | Im ( τ ) > ϵ I g } , \left\{\tau\in\mathcal{H}_{g}\ |\textrm{Im}(\tau)>\epsilon I_{g}\right\},
  31. ϵ > 0 \epsilon>0
  32. g > 1 g>1

Siegel–Walfisz_theorem.html

  1. ψ ( x ; q , a ) = n x n a ( mod q ) Λ ( n ) , \psi(x;q,a)=\sum_{n\leq x\atop n\equiv a\;\;(\mathop{{\rm mod}}q)}\Lambda(n),
  2. Λ \Lambda
  3. ψ ( x ; q , a ) = x φ ( q ) + O ( x exp ( - C N ( log x ) 1 2 ) ) , \psi(x;q,a)=\frac{x}{\varphi(q)}+O\left(x\exp\left(-C_{N}(\log x)^{\frac{1}{2}% }\right)\right),
  4. q ( log x ) N . q\leq(\log x)^{N}.
  5. π ( x ; q , a ) \pi(x;q,a)
  6. π ( x ; q , a ) = Li ( x ) φ ( q ) + O ( x exp ( - C N 2 ( log x ) 1 2 ) ) , \pi(x;q,a)=\frac{{\rm Li}(x)}{\varphi(q)}+O\left(x\exp\left(-\frac{C_{N}}{2}(% \log x)^{\frac{1}{2}}\right)\right),

Sieve_analysis.html

  1. W S i e v e W T o t a l \frac{W_{Sieve}}{W_{Total}}
  2. S i e v e L a r g e s t A g g r e g a t e m a x - s i z e \frac{Sieve_{Largest}}{Aggregate_{max-size}}
  3. W B e l o w W T o t a l \frac{W_{Below}}{W_{Total}}

Sigma-ring.html

  1. \mathcal{R}
  2. \mathcal{R}
  3. n = 1 A n \bigcup_{n=1}^{\infty}A_{n}\in\mathcal{R}
  4. A n A_{n}\in\mathcal{R}
  5. n n\in\mathbb{N}
  6. A B A\setminus B\in\mathcal{R}
  7. A , B A,B\in\mathcal{R}
  8. n = 1 A n \bigcap_{n=1}^{\infty}A_{n}\in\mathcal{R}
  9. A n A_{n}\in\mathcal{R}
  10. n n\in\mathbb{N}
  11. n = 1 A n = A 1 n = 1 ( A 1 A n ) \cap_{n=1}^{\infty}A_{n}=A_{1}\setminus\cup_{n=1}^{\infty}(A_{1}\setminus A_{n})
  12. A B A\cup B\in\mathcal{R}
  13. A , B A,B\in\mathcal{R}
  14. \mathcal{R}
  15. \mathcal{R}
  16. X X
  17. X X
  18. 𝒜 \mathcal{A}
  19. X X
  20. \mathcal{R}
  21. \mathcal{R}
  22. 𝒜 \mathcal{A}
  23. X X
  24. 𝒜 \mathcal{A}
  25. \mathcal{R}
  26. \mathcal{R}

Sign_(mathematics).html

  1. sgn ( x ) = { - 1 if x < 0 , 0 if x = 0 , 1 if x > 0. \operatorname{sgn}(x)=\begin{cases}-1&\,\text{if }x<0,\\ ~{}~{}\,0&\,\text{if }x=0,\\ ~{}~{}\,1&\,\text{if }x>0.\end{cases}
  2. sgn ( x ) = x | x | = | x | x \operatorname{sgn}(x)=\frac{x}{|x|}=\frac{|x|}{x}
  3. Δ x = x final - x initial . \Delta x=x\text{final}-x\text{initial}.\,
  4. q q
  5. q = Q q=Q
  6. q = Q q=−Q
  7. Q Q
  8. q = Q q=Q
  9. | q | |q|
  10. Q Q
  11. Q −Q
  12. q q
  13. | q | = | Q | |q|=|Q|

Signal-to-noise_statistic.html

  1. μ a \mu_{a}
  2. μ b \mu_{b}
  3. σ a \sigma_{a}
  4. σ b \sigma_{b}
  5. D s n = ( μ a - μ b ) ( σ a + σ b ) D_{sn}={(\mu_{a}-\mu_{b})\over(\sigma_{a}+\sigma_{b})}

Signature_(logic).html

  1. \cup
  2. 0 \mathbb{N}_{0}
  3. \cup
  4. \in

Signature_matrix.html

  1. A = ( ± 1 0 0 0 0 ± 1 0 0 0 0 ± 1 0 0 0 0 ± 1 ) A=\begin{pmatrix}\pm 1&0&\cdots&0&0\\ 0&\pm 1&\cdots&0&0\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&\cdots&\pm 1&0\\ 0&0&\cdots&0&\pm 1\end{pmatrix}

Signed_distance_function.html

  1. f ( x ) = { d ( x , Ω c ) if x Ω - d ( x , Ω ) if x Ω c f(x)=\begin{cases}d(x,\Omega^{c})&\mbox{ if }~{}x\in\Omega\\ -d(x,\Omega)&\mbox{ if }~{}x\in\Omega^{c}\end{cases}
  2. d ( x , Ω ) = inf y Ω d ( x , y ) d(x,\Omega)=\inf_{y\in\Omega}d(x,y)
  3. | f | = 1. |\nabla f|=1.
  4. f ( x ) = N ( x ) , \nabla f(x)=N(x),
  5. T ( Ω , μ ) g ( x ) d x = Ω - μ μ g ( u + λ N ( u ) ) det ( I - λ W u ) d λ d S u , \int_{T(\partial\Omega,\mu)}g(x)\,dx=\int_{\partial\Omega}\int_{-\mu}^{\mu}g(u% +\lambda N(u))\,\det(I-\lambda W_{u})\,d\lambda\,dS_{u},

Silanes.html

  1. 2 arccos ( - 1 3 ) 109.47 \scriptstyle 2\arccos\left(-\frac{1}{3}\right)\approx 109.47^{\circ}

Silver_machine.html

  1. α \alpha
  2. ϕ ( μ 0 , μ 1 , , μ n ) \phi(\mu_{0},\mu_{1},\ldots,\mu_{n})
  3. β 1 , , β n , γ X \exists\beta_{1},\ldots,\beta_{n},\gamma\in X
  4. α \alpha
  5. L γ ϕ ( α , β 1 , , β n ) \models_{L_{\gamma}}\phi(\alpha^{\circ},\beta_{1}^{\circ},\ldots,\beta^{\circ}% _{n})
  6. α \alpha
  7. α \alpha^{\circ}
  8. α \alpha
  9. L γ L_{\gamma}
  10. X , < , ( h i ) i < ω \langle X,<,(h_{i})_{i<\omega}\rangle
  11. X O n X\subseteq On
  12. i , h i \forall i,h_{i}
  13. X k ( i ) X^{k(i)}
  14. N = X , < , ( h i ) i < ω N=\langle X,<,(h_{i})_{i<\omega}\rangle
  15. N λ N_{\lambda}
  16. X λ X\cap\lambda
  17. N 1 , N 2 N^{1},N^{2}
  18. N 1 N 2 N^{1}\triangleleft N^{2}
  19. i ω \forall i\in\omega
  20. x 1 , , x k ( i ) X 1 \forall x_{1},\ldots,x_{k(i)}\in X^{1}
  21. h i 1 ( x 1 , , x k ( i ) ) h i 2 ( x 1 , , x k ( i ) ) h_{i}^{1}(x_{1},\ldots,x_{k(i)})\cong h_{i}^{2}(x_{1},\ldots,x_{k(i)})
  22. M = O n , < , ( h i ) i < ω M=\langle On,<,(h_{i})_{i<\omega}\rangle
  23. N M λ N\triangleleft M_{\lambda}
  24. α \alpha
  25. N M α N\cong M_{\alpha}
  26. λ \lambda
  27. H λ H\subseteq\lambda
  28. A λ + 1 A\subseteq\lambda+1
  29. M λ + 1 [ A ] M λ [ ( A λ ) H ] { λ } M_{\lambda+1}[A]\subseteq M_{\lambda}[(A\cap\lambda)\cup H]\cup\{\lambda\}
  30. α \alpha
  31. X O n X\subseteq On
  32. α M [ X ] \alpha\in M[X]
  33. λ < [ s u p ( X ) α ] + \lambda<[sup(X)\cup\alpha]^{+}
  34. Σ 1 \Sigma_{1}
  35. X { α } X\cup\{\alpha\}
  36. α M λ [ X ] \alpha\in M_{\lambda}[X]

Simplex_noise.html

  1. O ( n 2 ) O(n^{2})
  2. n n
  3. O ( 2 n ) O(2^{n})
  4. n n
  5. n n
  6. 2 n 2^{n}
  7. n n
  8. n + 1 n+1
  9. x = x + ( x + y + ) * F x^{\prime}=x+(x+y+...)*F
  10. y = y + ( x + y + ) * F y^{\prime}=y+(x+y+...)*F
  11. ...
  12. F = n + 1 - 1 n F=\frac{\sqrt{n+1}-1}{n}
  13. n * {}^{*}_{n}
  14. x = x + ( x + y + ) * G x=x^{\prime}+(x^{\prime}+y^{\prime}+...)*G
  15. y = y + ( x + y + ) * G y=y^{\prime}+(x^{\prime}+y^{\prime}+...)*G
  16. ...
  17. G = 1 / n + 1 - 1 n G=\frac{1/\sqrt{n+1}-1}{n}
  18. ( r 2 - d 2 ) 4 * ( < Δ x , Δ y , > < g r a d . x , g r a d . y , > ) (r^{2}-d^{2})^{4}*(<\Delta x,\Delta y,...>\cdot<grad.x,grad.y,...>)

Simul.html

  1. Y r , b = X r , b . a r , b + ε r , b Y_{r,b}=X_{r,b}.a_{r,b}+\varepsilon_{r,b}
  2. ε \varepsilon

Single-ended_primary-inductor_converter.html

  1. V I N = V L 1 + V C 1 + V L 2 V_{IN}=V_{L1}+V_{C1}+V_{L2}
  2. I D 1 = I L 1 - I L 2 I_{D1}=I_{L1}-I_{L2}

Single-index_model.html

  1. r i t - r f = α i + β i ( r m t - r f ) + ϵ i t r_{it}-r_{f}=\alpha_{i}+\beta_{i}(r_{mt}-r_{f})+\epsilon_{it}\,
  2. ϵ i t N ( 0 , σ i ) \epsilon_{it}\sim N(0,\sigma_{i})\,
  3. α i \alpha_{i}
  4. β i \beta_{i}
  5. r i t - r f r_{it}-r_{f}
  6. r m t - r f r_{mt}-r_{f}
  7. ϵ i t \epsilon_{it}
  8. σ i \sigma_{i}
  9. β i ( r m - r f ) \beta_{i}(r_{m}-r_{f})
  10. ϵ i \epsilon_{i}
  11. E ( ( R i , t - β i m t ) ( R k , t - β k m t ) ) = 0 , E((R_{i,t}-\beta_{i}m_{t})(R_{k,t}-\beta_{k}m_{t}))=0,
  12. C o v ( R i , R k ) = β i β k σ 2 . Cov(R_{i},R_{k})=\beta_{i}\beta_{k}\sigma^{2}.

Single_vegetative_obstruction_model.html

  1. A = { d γ , frequency < 3 G H z R f d + k [ 1 - e ( R f - R i ) d k ] , frequency > 5 G H z A=\begin{cases}d\gamma\mbox{ , frequency}~{}<3GHz\\ R_{f}d\;+\;k[1-e^{(R_{f}-R_{i})\frac{d}{k}}]\mbox{ , frequency}~{}>5GHz\end{cases}
  2. γ \gamma
  3. R i = a f R_{i}\;=\;af
  4. R f = b f c R_{f}\;=\;bf^{c}
  5. k = k 0 - 10 log [ A 0 ( 1 - e - A i A 0 ) ( 1 - e R f f ) ] k=k_{0}\;-\;10\;\log{[A_{0}\;(1\;-\;e^{\frac{-A^{i}}{A_{0}}})(1-e^{R_{f}f})]}
  6. A i = m i n ( w T , w R , w ) x m i n ( h T , h R , h ) A_{i}\;=\;min(w_{T},w_{R},w)\;x\;min(h_{T},h_{R},h)
  7. A i = m i n ( 2 d T tan a T 2 , 2 d R tan a R 2 , w ) x m i n ( 2 d T tan e T 2 , 2 d R tan e R 2 , h ) A_{i}\;=\;min(2d_{T}\;\tan{\frac{a_{T}}{2}},2d_{R}\tan{\frac{a_{R}}{2}},w)\;x% \;min(2d_{T}\tan{\frac{e_{T}}{2}},2d_{R}\tan{\frac{e_{R}}{2}},h)

Singular_control.html

  1. u u
  2. H ( u ) = ϕ ( x , λ , t ) u + H(u)=\phi(x,\lambda,t)u+\cdots
  3. a u ( t ) b a\leq u(t)\leq b
  4. H ( u ) H(u)
  5. u u
  6. ϕ ( x , λ , t ) \phi(x,\lambda,t)
  7. u ( t ) = { b , ϕ ( x , λ , t ) < 0 ? , ϕ ( x , λ , t ) = 0 a , ϕ ( x , λ , t ) > 0. u(t)=\begin{cases}b,&\phi(x,\lambda,t)<0\\ ?,&\phi(x,\lambda,t)=0\\ a,&\phi(x,\lambda,t)>0.\end{cases}
  8. ϕ \phi
  9. b b
  10. a a
  11. ϕ \phi
  12. ϕ \phi
  13. t 1 t t 2 t_{1}\leq t\leq t_{2}
  14. t 1 t_{1}
  15. t 2 t_{2}
  16. H / u \partial H/\partial u
  17. t 1 t_{1}
  18. t 2 t_{2}
  19. u u
  20. ( - 1 ) k u [ ( d d t ) 2 k H u ] 0 , k = 0 , 1 , (-1)^{k}\frac{\partial}{\partial u}\left[{\left(\frac{d}{dt}\right)}^{2k}H_{u}% \right]\geq 0,\,k=0,1,\cdots

Sinusoidal_plane-wave_solutions_of_the_electromagnetic_wave_equation.html

  1. 𝐄 ( 𝐫 , t ) = ( E x 0 cos ( k z - ω t + α x ) E y 0 cos ( k z - ω t + α y ) 0 ) = E x 0 cos ( k z - ω t + α x ) 𝐱 ^ + E y 0 cos ( k z - ω t + α y ) 𝐲 ^ \mathbf{E}(\mathbf{r},t)=\begin{pmatrix}E_{x}^{0}\cos\left(kz-\omega t+\alpha_% {x}\right)\\ E_{y}^{0}\cos\left(kz-\omega t+\alpha_{y}\right)\\ 0\end{pmatrix}=E_{x}^{0}\cos\left(kz-\omega t+\alpha_{x}\right)\hat{\mathbf{x}% }\;+\;E_{y}^{0}\cos\left(kz-\omega t+\alpha_{y}\right)\hat{\mathbf{y}}
  2. c 𝐁 ( 𝐫 , t ) = 𝐳 ^ × 𝐄 ( 𝐫 , t ) = ( - E y 0 cos ( k z - ω t + α y ) E x 0 cos ( k z - ω t + α x ) 0 ) = - E y 0 cos ( k z - ω t + α y ) 𝐱 ^ + E x 0 cos ( k z - ω t + α x ) 𝐲 ^ c\,\mathbf{B}(\mathbf{r},t)=\hat{\mathbf{z}}\times\mathbf{E}(\mathbf{r},t)=% \begin{pmatrix}-E_{y}^{0}\cos\left(kz-\omega t+\alpha_{y}\right)\\ E_{x}^{0}\cos\left(kz-\omega t+\alpha_{x}\right)\\ 0\end{pmatrix}=-E_{y}^{0}\cos\left(kz-\omega t+\alpha_{y}\right)\hat{\mathbf{x% }}\;+\;E_{x}^{0}\cos\left(kz-\omega t+\alpha_{x}\right)\hat{\mathbf{y}}
  3. ω = c k \omega=ck
  4. c c
  5. 𝐫 = ( x , y , z ) \mathbf{r}=(x,y,z)
  6. E x 0 = 𝐄 cos θ E_{x}^{0}=\mid\mathbf{E}\mid\cos\theta
  7. E y 0 = 𝐄 sin θ E_{y}^{0}=\mid\mathbf{E}\mid\sin\theta
  8. α x , α y \alpha_{x},\alpha_{y}
  9. θ = def tan - 1 ( E y 0 E x 0 ) \theta\ \stackrel{\mathrm{def}}{=}\ \tan^{-1}\left({E_{y}^{0}\over E_{x}^{0}}\right)
  10. 𝐄 2 = def ( E x 0 ) 2 + ( E y 0 ) 2 \mid\mathbf{E}\mid^{2}\ \stackrel{\mathrm{def}}{=}\ \left(E_{x}^{0}\right)^{2}% +\left(E_{y}^{0}\right)^{2}
  11. 𝐄 ( 𝐫 , t ) = 𝐄 Re { | ψ exp [ i ( k z - ω t ) ] } \mathbf{E}(\mathbf{r},t)=\mid\mathbf{E}\mid\mathrm{Re}\left\{|\psi\rangle\exp% \left[i\left(kz-\omega t\right)\right]\right\}
  12. | ψ = def ( ψ x ψ y ) = ( cos θ exp ( i α x ) sin θ exp ( i α y ) ) |\psi\rangle\ \stackrel{\mathrm{def}}{=}\ \begin{pmatrix}\psi_{x}\\ \psi_{y}\end{pmatrix}=\begin{pmatrix}\cos\theta\exp\left(i\alpha_{x}\right)\\ \sin\theta\exp\left(i\alpha_{y}\right)\end{pmatrix}
  13. ψ | = def ( ψ x * ψ y * ) = ( cos θ exp ( - i α x ) sin θ exp ( - i α y ) ) \langle\psi|\ \stackrel{\mathrm{def}}{=}\ \begin{pmatrix}\psi_{x}^{*}&\psi_{y}% ^{*}\end{pmatrix}=\begin{pmatrix}\quad\cos\theta\exp\left(-i\alpha_{x}\right)&% \sin\theta\exp\left(-i\alpha_{y}\right)\end{pmatrix}
  14. ψ | ψ = ( ψ x * ψ y * ) ( ψ x ψ y ) = 1 \langle\psi|\psi\rangle=\begin{pmatrix}\psi_{x}^{*}&\psi_{y}^{*}\end{pmatrix}% \begin{pmatrix}\psi_{x}\\ \psi_{y}\end{pmatrix}=1
  15. α x , α y \alpha_{x},\alpha_{y}
  16. α x = α y = def α \alpha_{x}=\alpha_{y}\ \stackrel{\mathrm{def}}{=}\ \alpha
  17. θ \theta
  18. | ψ = ( cos θ sin θ ) exp ( i α ) |\psi\rangle=\begin{pmatrix}\cos\theta\\ \sin\theta\end{pmatrix}\exp\left(i\alpha\right)
  19. | ψ = ( ψ x ψ y ) = ( cos θ exp ( i α x ) sin θ exp ( i α y ) ) |\psi\rangle=\begin{pmatrix}\psi_{x}\\ \psi_{y}\end{pmatrix}=\begin{pmatrix}\cos\theta\exp\left(i\alpha_{x}\right)\\ \sin\theta\exp\left(i\alpha_{y}\right)\end{pmatrix}
  20. = exp ( i α ) ( cos θ sin θ exp ( i Δ α ) ) =\exp\left(i\alpha\right)\begin{pmatrix}\cos\theta\\ \sin\theta\exp\left(i\Delta\alpha\right)\end{pmatrix}
  21. | ψ = ( ψ x ψ y ) = exp ( i α ) 2 2 ( 1 ± i ) |\psi\rangle=\begin{pmatrix}\psi_{x}\\ \psi_{y}\end{pmatrix}=\exp\left(i\alpha\right){\sqrt{2}\over 2}\begin{pmatrix}% 1\\ \pm i\end{pmatrix}

Site-specific_DNA-methyltransferase_(adenine-specific).html

  1. \rightleftharpoons

Site-specific_DNA-methyltransferase_(cytosine-N4-specific).html

  1. \rightleftharpoons

Skorokhod's_representation_theorem.html

  1. x n 𝑑 μ n x_{n}\xrightarrow{d}\ \mu_{n}
  2. x 𝑑 μ x\xrightarrow{d}\ \mu
  3. x n a . s . x x_{n}\xrightarrow{\mathrm{a.s.}}x

Slack_variable.html

  1. 𝐲 0 \mathbf{y}\geq 0
  2. 𝐀𝐱 𝐛 \mathbf{A}\mathbf{x}\leq\mathbf{b}
  3. 𝐀𝐱 + 𝐲 = 𝐛 \mathbf{A}\mathbf{x}+\mathbf{y}=\mathbf{b}
  4. P ( 𝐑 0 ) f P\hookrightarrow(\mathbf{R}_{\geq 0})^{f}
  5. ( n - 1 ) (n-1)
  6. Δ n - 1 P , \Delta^{n-1}\twoheadrightarrow P,

Slant_height.html

  1. r 2 + h 2 \sqrt{r^{2}+h^{2}}
  2. r r
  3. h h
  4. r r
  5. h h
  6. r 2 + h 2 = d 2 r^{2}+h^{2}=d^{2}
  7. d = r 2 + h 2 d=\sqrt{r^{2}+h^{2}}

Slice_genus.html

  1. g s ( K ) ( TB ( K ) + 1 ) / 2. g_{s}(K)\geq({\rm TB}(K)+1)/2.\,

Slice_sampling.html

  1. Y Y
  2. [ 0 , f ( x ) ] [0,f(x)]
  3. f ( x ) f(x)
  4. f ( x ) > Y f(x)>Y
  5. Y Y
  6. Y Y
  7. Y Y
  8. f - 1 [ y , + ) f^{-1}[y,+\infty)
  9. x 0 x_{0}
  10. x 1 x_{1}
  11. x 1 x_{1}
  12. x 1 x_{1}
  13. x x
  14. x i x_{i}
  15. p ( x i | x 0 x n ) p(x_{i}|x_{0}...x_{n})
  16. g ( x ) N ( 0 , 5 ) g(x)\sim N(0,5)
  17. f ( x ) = 1 2 π 5 2 e - ( x - 0 ) 2 2 5 2 f(x)=\frac{1}{\sqrt{2\pi\cdot 5^{2}}}\ e^{-\frac{(x-0)^{2}}{2\cdot 5^{2}}}
  18. N ( 0 , 1 ) N(0,1)
  19. ( 0 , e - x 2 / 2 / 2 π ] (0,e^{-x^{2}/2}/\sqrt{2\pi}]
  20. N ( 0 , 1 ) N(0,1)
  21. [ - α , α ] [-\alpha,\alpha]
  22. α = - 2 ln ( y 2 π ) \alpha=\sqrt{-2\ln(y\sqrt{2\pi})}
  23. f ( x ) > y f(x)>y

Sliding_glass_door.html

  1. - -\leftarrow
  2. - \rightarrow-
  3. - - -\leftarrow-
  4. - - -\rightarrow-
  5. - - \rightarrow--\leftarrow
  6. - - -\leftarrow\rightarrow-

Slip_ratio.html

  1. S l i p R a t i o % = V e h i c l e S p e e d - W h e e l S p e e d V e h i c l e S p e e d × 100 % Slip\ Ratio\%=\frac{Vehicle\ Speed-Wheel\ Speed}{Vehicle\ Speed}\times 100\%

Slutsky's_theorem.html

  1. X n + Y n 𝑑 X + c ; X_{n}+Y_{n}\ \xrightarrow{d}\ X+c;
  2. X n Y n 𝑑 c X ; X_{n}Y_{n}\ \xrightarrow{d}\ cX;
  3. X n / Y n 𝑑 X / c , X_{n}/Y_{n}\ \xrightarrow{d}\ X/c,
  4. 𝑑 \xrightarrow{d}

Small-angle_scattering.html

  1. q = 4 π sin ( θ ) / λ q=4\pi\sin(\theta)/\lambda
  2. 2 θ 2\theta
  3. λ \lambda
  4. I ( q ) q 2 d x \int I(q)q^{2}\,dx
  5. I ( q ) S q - 4 I(q)\sim Sq^{-4}
  6. I ( q ) S q - ( 6 - d ) I(q)\sim S^{\prime}q^{-(6-d)}
  7. I ( q ) = P ( q ) S ( q ) , I(q)=P(q)S(q),
  8. I ( q ) I(q)
  9. q q
  10. P ( q ) P(q)
  11. S ( q ) S(q)
  12. P ( q ) P(q)
  13. p ( r ) p(r)
  14. p ( r ) = r 2 2 π 2 0 I ( q ) sin q r q r q 2 d q . p(r)=\frac{r^{2}}{2\pi^{2}}\int_{0}^{\infty}I(q)\frac{\sin qr}{qr}q^{2}dq.
  15. p ( r ) p(r)
  16. r r
  17. r = 0 r=0
  18. r 2 r^{2}
  19. p ( r ) p(r)

Small-world_routing.html

  1. v v
  2. 1 / d ( v , w ) q 1/d(v,w)^{q}
  3. q q
  4. O ( n ) O(n)
  5. q = 0 q=0
  6. q = 2 q=2
  7. n / ( p i * r 2 ) n/(pi*r^{2})
  8. r 2 r^{2}
  9. 1 / r 2 1/r^{2}
  10. q = 2 q=2

Small_area_estimation.html

  1. y i j = x i j β + μ i + ϵ i j y_{ij}=x_{ij}^{\prime}\beta+\mu_{i}+\epsilon_{ij}\,
  2. β \beta\,
  3. μ \mu\,

Smearing_retransformation.html

  1. log ( y ) = f ( X ) + er \log(y)=f(X)+\mathrm{er}
  2. y = exp ( f ( X ) ) exp ( er ) . y=\exp(f(X))\exp(\mathrm{er}).

SMODEM.html

  1. m a x p a c k e t s i z e m a x p a c k e t s i z e + f r a m e s i z e = 1024 1024 + 5 = 0.9951 = 99.5 % \frac{max\ packetsize}{max\ packetsize\ +\ frame\ size}=\frac{1024}{1024+5}=0.% 9951=99.5\%

SMOG.html

  1. grade = 1.0430 number of polysyllables × 30 number of sentences + 3.1291 \mbox{grade}~{}=1.0430\sqrt{\mbox{number of polysyllables}~{}\times{30\over% \mbox{number of sentences}~{}}}+3.1291

Smooth_morphism.html

  1. f : X S f:X\to S
  2. s ¯ S \overline{s}\to S
  3. X s ¯ = X × S s ¯ X_{\overline{s}}=X\times_{S}{\overline{s}}
  4. s S s\in S
  5. f - 1 ( s ) f^{-1}(s)
  6. f : X S f:X\to S
  7. Ω X / S \Omega_{X/S}
  8. X / S X/S
  9. s S s\in S
  10. Spec B \operatorname{Spec}B
  11. Spec A \operatorname{Spec}A
  12. f ( s ) f(s)
  13. B = A [ t 1 , , t n ] / ( P 1 , , P m ) B=A[t_{1},\dots,t_{n}]/(P_{1},\dots,P_{m})
  14. ( P i / t j ) (\partial P_{i}/\partial t_{j})
  15. X 𝑔 𝔸 S n S X\overset{g}{\to}\mathbb{A}^{n}_{S}\to S
  16. X 𝑔 𝔸 S n 𝔸 S n - 1 𝔸 S 1 S X\overset{g}{\to}\mathbb{A}^{n}_{S}\to\mathbb{A}^{n-1}_{S}\to\cdots\to\mathbb{% A}^{1}_{S}\to S
  17. T 0 T_{0}
  18. X ( T ) X ( T 0 ) X(T)\to X(T_{0})
  19. X ( T ) = Hom S ( T , X ) X(T)=\operatorname{Hom}_{S}(T,X)
  20. char ( S ) \operatorname{char}(S)
  21. S Spec S\to\operatorname{Spec}\mathbb{Z}
  22. f : X S f:X\to S
  23. g : S S g:S^{\prime}\to S
  24. \mathcal{F}
  25. X et X\text{et}
  26. 0 p 0\neq p
  27. char ( S ) \operatorname{char}(S)
  28. p : p:\mathcal{F}\to\mathcal{F}
  29. g * ( R i f * ) R i f * ( g * ) g^{*}(R^{i}f_{*}\mathcal{F})\to R^{i}f^{\prime}_{*}(g^{\prime*}\mathcal{F})

Social_Choice_and_Individual_Values.html

  1. R i R_{i}
  2. R i R_{i}
  3. R i R_{i}
  4. R 1 R_{1}
  5. R n R_{n}
  6. R 1 R_{1}
  7. R n R_{n}
  8. R i R_{i}
  9. R i R_{i}
  10. R 1 R_{1}
  11. R n R_{n}
  12. R 1 R_{1}
  13. R n R_{n}
  14. R i R_{i}
  15. R i R_{i}
  16. P i P_{i}
  17. P i P_{i}
  18. R 1 R_{1}
  19. R n R_{n}

Soddy's_hexlet.html

  1. K = 3 ( a 1 a 2 + a 2 c 1 + c 1 a 1 - ( a 1 + a 2 + c 1 + 1 2 ) 2 ) K=\sqrt{3\left(a_{1}a_{2}+a_{2}c_{1}+c_{1}a_{1}-\left(\frac{a_{1}+a_{2}+c_{1}+% 1}{2}\right)^{2}\right)}
  2. c 2 = ( a 1 + a 2 + c 1 - 1 ) / 2 - K c 3 = ( 3 a 1 + 3 a 2 - c 1 - 3 ) / 2 - K c 4 = 2 a 1 + 2 a 2 - c 1 - 2 c 5 = ( 3 a 1 + 3 a 2 - c 1 - 3 ) / 2 + K c 6 = ( a 1 + a 2 + c 1 - 1 ) / 2 + K . \begin{aligned}\displaystyle c_{2}&\displaystyle=(a_{1}+a_{2}+c_{1}-1)/2-K\\ \displaystyle c_{3}&\displaystyle=(3a_{1}+3a_{2}-c_{1}-3)/2-K\\ \displaystyle c_{4}&\displaystyle=2a_{1}+2a_{2}-c_{1}-2\\ \displaystyle c_{5}&\displaystyle=(3a_{1}+3a_{2}-c_{1}-3)/2+K\\ \displaystyle c_{6}&\displaystyle=(a_{1}+a_{2}+c_{1}-1)/2+K.\end{aligned}
  3. 1 r 1 + 1 r 4 = 1 r 2 + 1 r 5 = 1 r 3 + 1 r 6 . \frac{1}{r_{1}}+\frac{1}{r_{4}}=\frac{1}{r_{2}}+\frac{1}{r_{5}}=\frac{1}{r_{3}% }+\frac{1}{r_{6}}.

Soft_body_dynamics.html

  1. s y m b o l ϵ symbol{\epsilon}
  2. s y m b o l σ symbol{\sigma}
  3. s y m b o l σ = 𝖢 s y m b o l ε , symbol{\sigma}=\mathsf{C}symbol{\varepsilon}\ ,
  4. 𝖢 \mathsf{C}
  5. O [ n 2 ] O[n^{2}]

Softmax_function.html

  1. K K
  2. 𝐳 \mathbf{z}
  3. K K
  4. σ ( 𝐳 ) \sigma(\mathbf{z})
  5. σ ( 𝐳 𝐣 ) = e z j k = 1 K e z k \sigma(\mathbf{z_{j}})=\frac{e^{z_{j}}}{\sum_{k=1}^{K}e^{z_{k}}}
  6. K K
  7. j j
  8. 𝐱 \mathbf{x}
  9. P ( y = j | 𝐱 ) = e 𝐱 𝖳 𝐰 j k = 1 K e 𝐱 𝖳 𝐰 k P(y=j|\mathbf{x})=\frac{e^{\mathbf{x}^{\mathsf{T}}\mathbf{w}_{j}}}{\sum_{k=1}^% {K}e^{\mathbf{x}^{\mathsf{T}}\mathbf{w}_{k}}}
  10. K K
  11. 𝐱 𝐱 𝖳 𝐰 1 , , 𝐱 𝐱 𝖳 𝐰 K \mathbf{x}\mapsto\mathbf{x}^{\mathsf{T}}\mathbf{w}_{1},\ldots,\mathbf{x}% \mapsto\mathbf{x}^{\mathsf{T}}\mathbf{w}_{K}
  12. q k σ ( 𝐪 , i ) = = σ ( 𝐪 , i ) ( δ i k - σ ( 𝐪 , k ) ) \frac{\partial}{\partial q_{k}}\sigma(\,\textbf{q},i)=\dots=\sigma(\,\textbf{q% },i)(\delta_{ik}-\sigma(\,\textbf{q},k))
  13. P t ( a ) = exp ( q t ( a ) / τ ) i = 1 n exp ( q t ( i ) / τ ) , P_{t}(a)=\frac{\exp(q_{t}(a)/\tau)}{\sum_{i=1}^{n}\exp(q_{t}(i)/\tau)}\,\text{,}
  14. q t ( a ) q_{t}(a)
  15. τ \tau
  16. τ \tau\to\infty
  17. τ 0 + \tau\to 0^{+}
  18. x i 1 1 + e - ( x i - μ i σ i ) x_{i}^{\prime}\equiv\frac{1}{1+e^{-(\frac{x_{i}-\mu_{i}}{\sigma_{i}})}}
  19. x i 1 - e - ( x i - μ i σ i ) 1 + e - ( x i - μ i σ i ) x_{i}^{\prime}\equiv\frac{1-e^{-(\frac{x_{i}-\mu_{i}}{\sigma_{i}})}}{1+e^{-(% \frac{x_{i}-\mu_{i}}{\sigma_{i}})}}

Soil_gradation.html

  1. C u = D 60 D 10 C_{u}=\frac{D_{60}}{D_{10}}
  2. C c = ( D 30 ) 2 D 10 × D 60 C_{c}=\frac{(D_{30})^{2}}{D_{10}\times\ D_{60}}

Soil_production_function.html

  1. d e / d t = P 0 exp [ - k h ] de/dt=P_{0}\exp{[-kh]}

Soil_thermal_properties.html

  1. Q = - λ d T / d z Q=-\lambda dT/dz\,

Sol-air_temperature.html

  1. q A = h o ( T o - T s ) \frac{q}{A}=h_{o}(T_{o}-T_{s})
  2. q q
  3. A A
  4. h o h_{o}
  5. T o T_{o}
  6. T s T_{s}
  7. T sol - air = T o + ( a I - Δ Q i r ) h o T_{\mathrm{sol-air}}=T_{o}+\frac{(a\cdot I-\Delta Q_{ir})}{h_{o}}
  8. a a
  9. I I
  10. Δ Q i r \Delta Q_{ir}
  11. Δ Q i r = F r * h r * Δ T o - s k y \Delta Q_{ir}=F_{r}*h_{r}*\Delta T_{o-sky}
  12. T sol - air T_{\mathrm{sol-air}}
  13. q A = h o ( T sol - air - T s ) \frac{q}{A}=h_{o}(T_{\mathrm{sol-air}}-T_{s})
  14. q A = U c ( T i - T sol - air ) \frac{q}{A}=U_{c}(T_{i}-T_{\mathrm{sol-air}})
  15. U c U_{c}
  16. T i T_{i}
  17. Δ T o - s k y \Delta T_{o-sky}
  18. F r F_{r}
  19. F r F_{r}
  20. F r F_{r}
  21. h r h_{r}
  22. T sol - air T_{\mathrm{sol-air}}
  23. q A = U c ( T i - T o ) - U c h o [ a I - F r h r Δ T o - s k y ] \frac{q}{A}=U_{c}(T_{i}-T_{o})-\frac{U_{c}}{h_{o}}{[a\cdot I-F_{r}\cdot h_{r}% \cdot\Delta T_{o-sky}]}
  24. T sol - air T_{\mathrm{sol-air}}

Solid_immersion_lens.html

  1. n n
  2. ( 1 + 1 / n ) r (1+1/n)r
  3. n 2 n^{2}

Solution_of_Schrödinger_equation_for_a_step_potential.html

  1. ψ ( x ) \psi(x)
  2. H ψ ( x ) = [ - 2 2 m d 2 d x 2 + V ( x ) ] ψ ( x ) = E ψ ( x ) , H\psi(x)=\left[-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}+V(x)\right]\psi(x)=E% \psi(x),
  3. V ( x ) = { 0 , x < 0 V 0 , x 0 V(x)=\begin{cases}0,&x<0\\ V_{0},&x\geq 0\end{cases}
  4. - 2 2 m d 2 d x 2 ψ -\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}\psi
  5. k 1 = 2 m E / 2 k_{1}=\sqrt{2mE/\hbar^{2}}
  6. k 2 = 2 m ( E - V 0 ) / 2 k_{2}=\sqrt{2m(E-V_{0})/\hbar^{2}}
  7. p = k p=\hbar k
  8. ψ 1 ( 0 ) = ψ 2 ( 0 ) \psi_{1}(0)=\psi_{2}(0)
  9. d d x ψ 1 ( 0 ) = d d x ψ 2 ( 0 ) \frac{d}{dx}\psi_{1}(0)=\frac{d}{dx}\psi_{2}(0)
  10. ( A + A ) = ( B + B ) (A_{\rightarrow}+A_{\leftarrow})=(B_{\rightarrow}+B_{\leftarrow})
  11. k 1 ( A - A ) = k 2 ( B - B ) k_{1}(A_{\rightarrow}-A_{\leftarrow})=k_{2}(B_{\rightarrow}-B_{\leftarrow})
  12. T = 2 k 1 k 2 k 1 + k 2 \sqrt{T}=\frac{2\sqrt{k_{1}k_{2}}}{k_{1}+k_{2}}
  13. R = k 1 - k 2 k 1 + k 2 . \sqrt{R}=\frac{k_{1}-k_{2}}{k_{1}+k_{2}}.
  14. T = T = 2 k 1 k 2 k 1 + k 2 \sqrt{T^{\prime}}=\sqrt{T}=\frac{2\sqrt{k_{1}k_{2}}}{k_{1}+k_{2}}
  15. R = - R = k 2 - k 1 k 1 + k 2 . \sqrt{R^{\prime}}=-\sqrt{R}=\frac{k_{2}-k_{1}}{k_{1}+k_{2}}.
  16. 1 / ( k 2 ) 1/(k_{2})
  17. T = | T | = 4 k 1 k 2 ( k 1 + k 2 ) 2 T=|T^{\prime}|=\frac{4k_{1}k_{2}}{(k_{1}+k_{2})^{2}}
  18. R = | R | = 1 - T = ( k 1 - k 2 ) 2 ( k 1 + k 2 ) 2 R=|R^{\prime}|=1-T=\frac{(k_{1}-k_{2})^{2}}{(k_{1}+k_{2})^{2}}
  19. w k 0 wk\rightarrow 0

Sommerfeld_radiation_condition.html

  1. ( 2 + k 2 ) u = - f in n (\nabla^{2}+k^{2})u=-f\mbox{ in }~{}\mathbb{R}^{n}
  2. n = 2 , 3 n=2,3
  3. f f
  4. k > 0 k>0
  5. u u
  6. lim | x | | x | n - 1 2 ( | x | - i k ) u ( x ) = 0 \lim_{|x|\to\infty}|x|^{\frac{n-1}{2}}\left(\frac{\partial}{\partial|x|}-ik% \right)u(x)=0
  7. x ^ = x | x | \hat{x}=\frac{x}{|x|}
  8. i i
  9. | | |\cdot|
  10. e - i ω t u . e^{-i\omega t}u.
  11. e i ω t u , e^{i\omega t}u,
  12. - i -i
  13. + i +i
  14. x 0 x_{0}
  15. f f
  16. f ( x ) = δ ( x - x 0 ) , f(x)=\delta(x-x_{0}),
  17. δ \delta
  18. u = c u + + ( 1 - c ) u - u=cu_{+}+(1-c)u_{-}\,
  19. c c
  20. u ± ( x ) = e ± i k | x - x 0 | 4 π | x - x 0 | . u_{\pm}(x)=\frac{e^{\pm ik|x-x_{0}|}}{4\pi|x-x_{0}|}.
  21. u + u_{+}
  22. x 0 . x_{0}.
  23. u - u_{-}
  24. x 0 . x_{0}.

Sommerfeld–Kossel_displacement_law.html

  1. 1 s 2 1s^{2}\,
  2. 2 s 2 2s^{2}\,
  3. 2 p 2 2p^{2}\,
  4. 3 {}^{3}
  5. P 0 P_{0}\,

Sound_from_ultrasound.html

  1. p 2 ( x , t ) = K P c 2 2 t 2 E 2 ( x , t ) p_{2}(x,t)=K\cdot P_{c}^{2}\cdot\frac{\partial^{2}}{\partial t^{2}}E^{2}(x,t)
  2. p 2 ( x , t ) = p_{2}(x,t)=\,
  3. K = K=\,
  4. P c = P_{c}=\,
  5. E ( x , t ) = E(x,t)=\,

Sound_masking.html

  1. A I = i = 160 8000 S N R i * W 1 i \displaystyle AI=\sum\limits_{i=160}^{8000}{SN{{R}_{i}}*W{{1}_{i}}}
  2. A I = i = 160 8000 S N R i * W 1 i \displaystyle AI=\sum\limits_{i=160}^{8000}{SN{{R}_{i}}*W{{1}_{i}}}
  3. A C = i = 200 5000 T L i * 10 * W 1 i S L = i = 200 5000 T L i * W 1 i E S L = i = 200 5000 T L i * W 1 i T L i = T L i + ( M S i - B S ) i \begin{aligned}&\displaystyle AC=\sum\limits_{i=200}^{5000}{T{{L}_{i}}*10*{{W1% }_{i}}}\\ &\displaystyle SL=\sum\limits_{i=200}^{5000}{T{{L}_{i}}*{{W1}_{i}}}\\ &\displaystyle ESL=\sum\limits_{i=200}^{5000}{TL_{i}^{{}^{\prime}}*{{W1}_{i}}}% \\ &\displaystyle TL_{i}^{{}^{\prime}}=T{{L}_{i}}+\left(M{{S}_{i}}-BS{}_{i}\right% )\\ \end{aligned}

Sound_ranging.html

  1. r 5 = 1267.9 r_{5}=1267.9
  2. r 4 = 499.1 r_{4}=499.1
  3. \approx
  4. \Rightarrow
  5. \Rightarrow
  6. ( r 1 + r 2 ) 2 = ( r 1 + r 3 ) 2 + r 4 2 - 2 ( r 1 + r 3 ) r 4 cos θ \left(r_{1}+r_{2}\right)^{2}=\left(r_{1}+r_{3}\right)^{2}+r_{4}^{2}-2\cdot% \left(r_{1}+r_{3}\right)\cdot r_{4}\cos\theta
  7. \triangle
  8. r 1 2 = ( r 1 + r 3 ) 2 + r 5 2 - 2 ( r 1 + r 3 ) r 5 cos ( θ - ϕ ) r_{1}^{2}=\left(r_{1}+r_{3}\right)^{2}+r_{5}^{2}-2\cdot\left(r_{1}+r_{3}\right% )\cdot r_{5}\cos(\theta-\phi)
  9. \triangle
  10. θ \theta
  11. r 1 r_{1}

Source_function.html

  1. S λ = def j λ κ λ S_{\lambda}\ \stackrel{\mathrm{def}}{=}\ \frac{j_{\lambda}}{\kappa_{\lambda}}
  2. j λ j_{\lambda}
  3. κ λ \kappa_{\lambda}
  4. - 1 κ λ ρ d I λ d s = I λ - S λ -\frac{1}{\kappa_{\lambda}\rho}\frac{dI_{\lambda}}{ds}=I_{\lambda}-S_{\lambda}

Space-oblique_Mercator_projection.html

  1. x R \displaystyle\frac{x}{R}

Space-time_adaptive_processing.html

  1. N M NM
  2. N N
  3. M M
  4. N = M = 10 N=M=10
  5. s s
  6. W 1 W_{1}
  7. W N M W_{NM}
  8. M N × M N MN\times MN
  9. 𝐑 \mathbf{R}
  10. 𝐖 = κ 𝐑 - 1 s \mathbf{W}=\kappa\mathbf{R}^{-1}s
  11. κ \kappa
  12. y = 𝐖 x y=\mathbf{W}x
  13. x x
  14. 𝐑 \mathbf{R}
  15. s s
  16. 𝐑 𝐤 = E [ x k x k H ] | H 0 \mathbf{R_{k}}=\mbox{E}~{}\left[x_{k}x^{H}_{k}\right]\Bigr|_{H_{0}}
  17. x k x_{k}
  18. k t h k^{th}
  19. H 0 H_{0}
  20. k t h k^{th}
  21. 𝐑 𝐤 ^ = 1 P m = 0 P - 1 x m x m H \mathbf{\hat{R_{k}}}=\frac{1}{P}\sum_{m=0}^{P-1}x_{m}x^{H}_{m}
  22. x m x_{m}
  23. m t h m^{th}
  24. 𝐑 \mathbf{R}
  25. 2 N M 2NM
  26. 𝐑 𝐑 ~ \mathbf{R}\Rightarrow\mathbf{\widetilde{R}}
  27. M N × M N MN\times MN
  28. M M
  29. N N
  30. N × N N\times N
  31. L s , 2 = SINR | W = W ^ SINR | W = W o p t L_{s,2}=\frac{\mbox{SINR}~{}\Bigr|_{W=\hat{W}}}{\mbox{SINR}~{}\Bigr|_{W=W_{opt% }}}
  32. W ^ \hat{W}
  33. W o p t W_{opt}
  34. 𝐑 ~ 𝐏𝐂 - 𝐂𝐌𝐓 = ( m = 0 P - 1 λ m v m v m H ) T + σ n 2 \mathbf{\widetilde{R}_{PC-CMT}}=\left(\sum_{m=0}^{P-1}\lambda_{m}v_{m}v^{H}_{m% }\right)\circ T+\sigma^{2}_{n}
  35. λ m \lambda_{m}
  36. m t h m^{th}
  37. v m v_{m}
  38. m t h m^{th}
  39. A B A\circ B
  40. A A
  41. B B
  42. T T
  43. σ n 2 \sigma^{2}_{n}
  44. T T
  45. 𝐑 ~ 𝐒𝐌𝐈 - 𝐂𝐌𝐓 = 𝐑 ~ 𝐒𝐌𝐈 T + δ I \mathbf{\widetilde{R}_{SMI-CMT}}=\mathbf{\widetilde{R}_{SMI}}\circ T+\delta I
  46. 𝐑 ~ 𝐒𝐌𝐈 \mathbf{\widetilde{R}_{SMI}}
  47. δ \delta
  48. I I
  49. 𝐒 ^ \mathbf{\hat{S}}
  50. 𝐙 ~ \mathbf{\widetilde{Z}}
  51. 𝐖 ~ \mathbf{\widetilde{W}}
  52. 𝐒 ^ = 𝐖 ~ T 𝐙 ~ \mathbf{\hat{S}}=\mathbf{\widetilde{W}}^{\mathrm{T}}\mathbf{\widetilde{Z}}
  53. 𝐒 ~ \mathbf{\widetilde{S}}
  54. 𝐖 ~ ( 𝐙 ~ 𝐙 ~ T ) 𝐙 ~ 𝐒 ~ T \mathbf{\widetilde{W}}\approx\left(\mathbf{\widetilde{Z}}\mathbf{\widetilde{Z}% }^{\mathrm{T}}\right)\mathbf{\widetilde{Z}}\mathbf{\widetilde{S}}^{\mathrm{T}}
  55. M M
  56. N L NL
  57. M + N L M+NL
  58. M N L MNL

Space_(mathematics).html

  1. x = x , x . \|x\|=\sqrt{\langle x,x\rangle}.
  2. n n
  3. n n
  4. n n
  5. n n
  6. 2 n 2n
  7. 2 n 2n
  8. ( 0 , 1 ) (0,1)
  9. ( - , ) (-\infty,\infty)
  10. [ 0 , 1 ] [0,1]
  11. [ 0 , 1 ] [0,1]
  12. [ - , ] [-\infty,\infty]
  13. ( 0 , 1 ) (0,1)
  14. ( - , ) (-\infty,\infty)
  15. n n
  16. ( n + 1 ) (n+1)
  17. n n
  18. n n
  19. ( n + 1 ) (n+1)
  20. ( n + 1 ) (n+1)
  21. ( n + 1 ) (n+1)
  22. n n
  23. n n
  24. ( n + 1 ) (n+1)
  25. ( 0 , 1 ) (0,1)
  26. A A
  27. C C
  28. B B
  29. A A
  30. C C
  31. A A
  32. B B
  33. B B
  34. C C
  35. x x
  36. x \|x\|
  37. x - y 2 + x + y 2 = 2 x 2 + 2 y 2 \|x-y\|^{2}+\|x+y\|^{2}=2\|x\|^{2}+2\|y\|^{2}
  38. n n
  39. n n
  40. n n
  41. n n
  42. n n
  43. mod 0 \mod 0
  44. mod 0 ; \mod 0;
  45. ( 0 , 1 ) (0,1)
  46. 2 2^{\mathbb{R}}

Spalart–Allmaras_turbulence_model.html

  1. ν t = ν ~ f v 1 , f v 1 = χ 3 χ 3 + C v 1 3 , χ := ν ~ ν \nu_{t}=\tilde{\nu}f_{v1},\quad f_{v1}=\frac{\chi^{3}}{\chi^{3}+C^{3}_{v1}},% \quad\chi:=\frac{\tilde{\nu}}{\nu}
  2. ν ~ t + u j ν ~ x j = C b 1 [ 1 - f t 2 ] S ~ ν ~ + 1 σ { [ ( ν + ν ~ ) ν ~ ] + C b 2 | ν | 2 } - [ C w 1 f w - C b 1 κ 2 f t 2 ] ( ν ~ d ) 2 + f t 1 Δ U 2 \frac{\partial\tilde{\nu}}{\partial t}+u_{j}\frac{\partial\tilde{\nu}}{% \partial x_{j}}=C_{b1}[1-f_{t2}]\tilde{S}\tilde{\nu}+\frac{1}{\sigma}\{\nabla% \cdot[(\nu+\tilde{\nu})\nabla\tilde{\nu}]+C_{b2}|\nabla\nu|^{2}\}-\left[C_{w1}% f_{w}-\frac{C_{b1}}{\kappa^{2}}f_{t2}\right]\left(\frac{\tilde{\nu}}{d}\right)% ^{2}+f_{t1}\Delta U^{2}
  3. S ~ S + ν ~ κ 2 d 2 f v 2 , f v 2 = 1 - χ 1 + χ f v 1 \tilde{S}\equiv S+\frac{\tilde{\nu}}{\kappa^{2}d^{2}}f_{v2},\quad f_{v2}=1-% \frac{\chi}{1+\chi f_{v1}}
  4. f w = g [ 1 + C w 3 6 g 6 + C w 3 6 ] 1 / 6 , g = r + C w 2 ( r 6 - r ) , r ν ~ S ~ κ 2 d 2 f_{w}=g\left[\frac{1+C_{w3}^{6}}{g^{6}+C_{w3}^{6}}\right]^{1/6},\quad g=r+C_{w% 2}(r^{6}-r),\quad r\equiv\frac{\tilde{\nu}}{\tilde{S}\kappa^{2}d^{2}}
  5. f t 1 = C t 1 g t exp ( - C t 2 ω t 2 Δ U 2 [ d 2 + g t 2 d t 2 ] ) f_{t1}=C_{t1}g_{t}\exp\left(-C_{t2}\frac{\omega_{t}^{2}}{\Delta U^{2}}[d^{2}+g% ^{2}_{t}d^{2}_{t}]\right)
  6. f t 2 = C t 3 exp ( - C t 4 χ 2 ) f_{t2}=C_{t3}\exp\left(-C_{t4}\chi^{2}\right)
  7. S = 2 Ω i j Ω i j S=\sqrt{2\Omega_{ij}\Omega_{ij}}
  8. Ω i j = 1 2 ( u i / x j - u j / x i ) \Omega_{ij}=\frac{1}{2}(\partial u_{i}/\partial x_{j}-\partial u_{j}/\partial x% _{i})
  9. σ = 2 / 3 C b 1 = 0.1355 C b 2 = 0.622 κ = 0.41 C w 1 = C b 1 / κ 2 + ( 1 + C b 2 ) / σ C w 2 = 0.3 C w 3 = 2 C v 1 = 7.1 C t 1 = 1 C t 2 = 2 C t 3 = 1.1 C t 4 = 2 \begin{matrix}\sigma&=&2/3\\ C_{b1}&=&0.1355\\ C_{b2}&=&0.622\\ \kappa&=&0.41\\ C_{w1}&=&C_{b1}/\kappa^{2}+(1+C_{b2})/\sigma\\ C_{w2}&=&0.3\\ C_{w3}&=&2\\ C_{v1}&=&7.1\\ C_{t1}&=&1\\ C_{t2}&=&2\\ C_{t3}&=&1.1\\ C_{t4}&=&2\end{matrix}
  10. C t 3 = 1.2 C t 4 = 0.5 \begin{matrix}C_{t3}&=&1.2\\ C_{t4}&=&0.5\end{matrix}
  11. μ t = ρ ν ~ f v 1 \mu_{t}=\rho\tilde{\nu}f_{v1}
  12. ρ \rho
  13. ν ~ \tilde{\nu}
  14. ν ~ t + x j ( ν ~ u j ) = RHS \frac{\partial\tilde{\nu}}{\partial t}+\frac{\partial}{\partial x_{j}}(\tilde{% \nu}u_{j})=\mbox{RHS}~{}
  15. ν ~ = 0 \tilde{\nu}=0
  16. ν ~ = 0 \tilde{\nu}=0
  17. ν ~ ν 2 \tilde{\nu}<=\frac{\nu}{2}
  18. ν ~ = 5 ν \tilde{\nu}=5{\nu}

Spatial_descriptive_statistics.html

  1. K ^ ( t ) = λ - 1 i j I ( d i j < t ) / n , \hat{K}(t)=\lambda^{-1}\sum_{i\neq j}I(d_{ij}<t)/n,
  2. K ^ ( s ) \hat{K}(s)
  3. L ^ ( t ) = ( K ^ ( t ) / π ) 1 / 2 . \hat{L}(t)=\Big(\hat{K}(t)/\pi\Big)^{1/2}.
  4. t - L ^ ( t ) t-\hat{L}(t)

Spatial_multiplexing.html

  1. N t N_{t}
  2. N r N_{r}
  3. N s = min ( N t , N r ) N_{s}=\min(N_{t},N_{r})\!
  4. N s N_{s}
  5. N s N_{s}
  6. N t N_{t}
  7. N r N_{r}
  8. 𝐲 = 𝐇𝐱 + 𝐧 \mathbf{y}=\mathbf{Hx}+\mathbf{n}
  9. 𝐱 = [ x 1 , x 2 , , x N t ] T \mathbf{x}=[x_{1},x_{2},\ldots,x_{N_{t}}]^{T}
  10. N t × 1 N_{t}\times 1
  11. 𝐲 , 𝐧 \mathbf{y,n}
  12. N r × 1 N_{r}\times 1
  13. 𝐇 \mathbf{H}
  14. N r × N t N_{r}\times N_{t}
  15. 𝐲 = 𝐇𝐖𝐬 + 𝐧 \mathbf{y}=\mathbf{HWs}+\mathbf{n}
  16. 𝐬 = [ s 1 , s 2 , , s N s ] T \mathbf{s}=[s_{1},s_{2},\ldots,s_{N_{s}}]^{T}
  17. N s × 1 N_{s}\times 1
  18. 𝐲 , 𝐧 \mathbf{y,n}
  19. N r × 1 N_{r}\times 1
  20. 𝐇 \mathbf{H}
  21. N r × N t N_{r}\times N_{t}
  22. 𝐖 \mathbf{W}
  23. N t × N s N_{t}\times N_{s}
  24. 𝐖 \mathbf{W}
  25. N s N_{s}
  26. 𝐖 \mathbf{W}
  27. N t N_{t}
  28. N s ( N t ) N_{s}(\neq N_{t})

Specht_module.html

  1. { T } \{T\}
  2. T T
  3. E T = σ Q T ϵ ( σ ) { σ ( T ) } V E_{T}=\sum_{\sigma\in Q_{T}}\epsilon(\sigma)\{\sigma(T)\}\in V
  4. ϵ ( σ ) \epsilon(\sigma)

Species_distribution.html

  1. m e a n d i s t a n c e 1 2 density \frac{mean\ distance}{\frac{1}{2}\sqrt{\,\text{density}}}

Specific_speed.html

  1. N s = n Q ( g H ) 3 / 4 N_{s}=\frac{n\sqrt{Q}}{(gH)^{3/4}}
  2. N s N_{s}
  3. n n
  4. Q Q
  5. H H
  6. g g
  7. N s s = N Q N P S H R 0.75 N_{ss}=\frac{N\sqrt{Q}}{{NPSH}_{R}^{0.75}}
  8. N s s = N_{ss}=
  9. N = N=
  10. Q = Q=
  11. N P S H R = {NPSH}_{R}=
  12. n s = n P / H 5 / 4 n_{s}=n\sqrt{P}/H^{5/4}
  13. n n
  14. Ω \Omega
  15. H n H_{n}
  16. Q Q
  17. N N
  18. P P
  19. H H
  20. N s = 2.294 H n 0.486 N_{s}=\frac{2.294}{H_{n}^{0.486}}
  21. D e = 84.5 ( 0.79 + 1.602 N s ) H n 60 * Ω D_{e}=84.5(0.79+1.602N_{s})\frac{\sqrt{H_{n}}}{60*\Omega}
  22. D e D_{e}

Spectral_flatness.html

  1. Flatness = n = 0 N - 1 x ( n ) N n = 0 N - 1 x ( n ) N = exp ( 1 N n = 0 N - 1 ln x ( n ) ) 1 N n = 0 N - 1 x ( n ) \mathrm{Flatness}=\frac{\sqrt[N]{\prod_{n=0}^{N-1}x(n)}}{\frac{\sum_{n=0}^{N-1% }x(n)}{N}}=\frac{\exp\left(\frac{1}{N}\sum_{n=0}^{N-1}\ln x(n)\right)}{\frac{1% }{N}\sum_{n=0}^{N-1}x(n)}

Spectral_index.html

  1. ν \nu
  2. S S
  3. α \alpha
  4. S ν α . S\propto\nu^{\alpha}.
  5. α ( ν ) = log S ( ν ) log ν . \alpha\!\left(\nu\right)=\frac{\partial\log S\!\left(\nu\right)}{\partial\log% \nu}.
  6. λ \lambda
  7. α \alpha
  8. S λ α , S\propto\lambda^{\alpha},
  9. α ( λ ) = log S ( λ ) log λ . \alpha\!\left(\lambda\right)=\frac{\partial\log S\!\left(\lambda\right)}{% \partial\log\lambda}.
  10. S ν - α . S\propto\nu^{-\alpha}.
  11. B ν ( T ) 2 ν 2 k T c 2 . B_{\nu}(T)\simeq\frac{2\nu^{2}kT}{c^{2}}.
  12. log ν \log\,\nu
  13. log B ν ( T ) log ν 2. \frac{\partial\log B_{\nu}(T)}{\partial\log\nu}\simeq 2.
  14. α 2 \alpha\simeq 2
  15. S ν α T . S\propto\nu^{\alpha}T.

Spectral_slope.html

  1. S = R F 1 - R F 0 λ 1 - λ 0 S=\frac{R_{F_{1}}-R_{F_{0}}}{\lambda_{1}-\lambda_{0}}
  2. R F 0 , R F 1 R_{F_{0}},R_{F_{1}}

Spherical_model.html

  1. 𝕃 \mathbb{L}
  2. 𝕃 \mathbb{L}
  3. σ j \sigma_{j}
  4. σ j \sigma_{j}
  5. σ j { 1 , - 1 } \sigma_{j}\in\{1,-1\}
  6. j = 1 N σ j 2 = N \sum_{j=1}^{N}\sigma_{j}^{2}=N
  7. Z N = - - d σ 1 d σ N e x p [ K j l σ j σ l + h j σ j ] δ [ N - j σ j 2 ] Z_{N}=\int_{-\infty}^{\infty}\ldots\int_{-\infty}^{\infty}d\sigma_{1}\ldots d% \sigma_{N}exp\left[K\sum_{\langle jl\rangle}\sigma_{j}\sigma_{l}+h\sum_{j}% \sigma_{j}\right]\delta\left[N-\sum_{j}\sigma_{j}^{2}\right]
  8. δ \delta
  9. j l \langle jl\rangle
  10. K = J / k T K=J/kT
  11. h = H / k T h=H/kT
  12. σ \sigma
  13. σ \sigma
  14. 2 J ( 1 - M 2 ) = k T g ( H / 2 J M ) 2J(1-M^{2})=kTg^{\prime}(H/2JM)
  15. g ( z ) = ( 2 π ) - d 0 2 π 0 2 π d ω 1 d ω d l n [ z + d - c o s ω 1 - - c o s ω d ] g(z)=(2\pi)^{-d}\int_{0}^{2\pi}\ldots\int_{0}^{2\pi}d\omega_{1}\ldots d\omega_% {d}\ ln[z+d-cos\omega_{1}-\ldots-cos\omega_{d}]
  16. u = 1 2 k T - J d - 1 2 H ( M + M - 1 ) u=\frac{1}{2}kT-Jd-\frac{1}{2}H(M+M^{-1})
  17. d 2 d\leq 2
  18. α , β , γ \alpha,\beta,\gamma
  19. γ \gamma^{\prime}
  20. α = { - 4 - d d - 2 if 2 < d < 4 0 i f d > 4 \alpha=\begin{cases}-\frac{4-d}{d-2}&\ \mathrm{if}\ 2<d<4\\ 0&\ if\ d>4\end{cases}
  21. β = 1 2 \beta=\frac{1}{2}
  22. γ = { 2 d - 2 i f 2 < d < 4 1 i f d > 4 \gamma=\begin{cases}\frac{2}{d-2}&\ if\ 2<d<4\\ 1&if\ d>4\end{cases}
  23. δ = { d + 2 d - 2 i f 2 < d < 4 3 i f d > 4 \delta=\begin{cases}\frac{d+2}{d-2}&\ if\ 2<d<4\\ 3&if\ d>4\end{cases}

Spheroidal_wave_equation.html

  1. ( 1 - t 2 ) d 2 y d t 2 - 2 ( b + 1 ) t d y d t + ( c - 4 q t 2 ) y = 0 (1-t^{2})\frac{d^{2}y}{dt^{2}}-2(b+1)t\,\frac{dy}{dt}+(c-4qt^{2})\,y=0

Sphingomyelin_phosphodiesterase_D.html

  1. \rightleftharpoons

Spike-triggered_average.html

  1. 𝐱 𝐢 \mathbf{x_{i}}
  2. i i
  3. y i y_{i}
  4. E [ 𝐱 ] = 0 E[\mathbf{x}]=0
  5. STA = 1 n s p i = 1 T y i 𝐱 𝐢 , \mathrm{STA}=\tfrac{1}{n_{sp}}\sum_{i=1}^{T}y_{i}\mathbf{x_{i}},
  6. n s p = y i n_{sp}=\sum y_{i}
  7. X X
  8. i i
  9. 𝐱 𝐢 𝐓 \mathbf{x_{i}^{T}}
  10. 𝐲 \mathbf{y}
  11. i i
  12. y i y_{i}
  13. STA = 1 n s p X T 𝐲 . \mathrm{STA}=\tfrac{1}{n_{sp}}X^{T}\mathbf{y}.
  14. STA w = ( 1 T i = 1 T 𝐱 𝐢 𝐱 𝐢 T ) - 1 ( 1 n s p i = 1 T y i 𝐱 𝐢 ) , \mathrm{STA}_{w}=\left(\tfrac{1}{T}\sum_{i=1}^{T}\mathbf{x_{i}}\mathbf{x_{i}}^% {T}\right)^{-1}\left(\tfrac{1}{n_{sp}}\sum_{i=1}^{T}y_{i}\mathbf{x_{i}}\right),
  15. STA w = T n s p ( X T X ) - 1 X T 𝐲 . \mathrm{STA}_{w}=\tfrac{T}{n_{sp}}\left(X^{T}X\right)^{-1}X^{T}\mathbf{y}.
  16. STA r i d g e = T n s p ( X T X + λ I ) - 1 X T 𝐲 , \mathrm{STA}_{ridge}=\tfrac{T}{n_{sp}}\left(X^{T}X+\lambda I\right)^{-1}X^{T}% \mathbf{y},
  17. I I
  18. λ \lambda
  19. P ( 𝐱 ) P(\mathbf{x})
  20. P ( 𝐱 ) P(\mathbf{x})
  21. e x p ( x ) exp(x)

Spin-weighted_spherical_harmonics.html

  1. Y m Y_{\ell m}
  2. Y m s {}_{s}Y_{\ell m}
  3. Y m 0 = Y m . {}_{0}Y_{\ell m}=Y_{\ell m}\ .
  4. 𝐱 𝐚 = 𝐱 𝐛 = 0 𝐚 𝐚 = 𝐛 𝐛 = 1 𝐚 𝐛 = 0 𝐱 ( 𝐚 × 𝐛 ) > 0 , \begin{aligned}\displaystyle\mathbf{x}\cdot\mathbf{a}&\displaystyle=\mathbf{x}% \cdot\mathbf{b}=0\\ \displaystyle\mathbf{a}\cdot\mathbf{a}&\displaystyle=\mathbf{b}\cdot\mathbf{b}% =1\\ \displaystyle\mathbf{a}\cdot\mathbf{b}&\displaystyle=0\\ \displaystyle\mathbf{x}\cdot(\mathbf{a}\times\mathbf{b})&\displaystyle>0,\end{aligned}
  5. f ( 𝐱 , cos θ 𝐚 - sin θ 𝐛 , sin θ 𝐚 + cos θ 𝐛 ) = e i s θ f ( 𝐱 , 𝐚 , 𝐛 ) f(\mathbf{x},\cos\theta\mathbf{a}-\sin\theta\mathbf{b},\sin\theta\mathbf{a}+% \cos\theta\mathbf{b})=e^{is\theta}f(\mathbf{x},\mathbf{a},\mathbf{b})
  6. f ( λ z , λ ¯ z ¯ ) = ( λ ¯ λ ) s f ( z , z ¯ ) . f(\lambda z,\overline{\lambda}\bar{z})=\left(\frac{\overline{\lambda}}{\lambda% }\right)^{s}f(z,\bar{z}).
  7. 𝐎 ( 2 s ) ¯ \overline{\mathbf{O}(2s)}
  8. g ( λ z , λ ¯ z ¯ ) = λ ¯ 2 s g ( z , z ¯ ) . g(\lambda z,\overline{\lambda}\bar{z})=\overline{\lambda}^{2s}g(z,\bar{z}).
  9. P ( z , z ¯ ) = z z ¯ . P(z,\bar{z})=z\cdot\bar{z}.
  10. ð \eth
  11. : 𝐎 ( 2 s ) ¯ 1 , 0 𝐎 ( 2 s ) ¯ 𝐎 ( 2 s ) ¯ 𝐎 ( - 2 ) . \partial:\overline{\mathbf{O}(2s)}\to\mathcal{E}^{1,0}\otimes\overline{\mathbf% {O}(2s)}\cong\overline{\mathbf{O}(2s)}\otimes\mathbf{O}(-2).
  12. ð f = def P - s + 1 ( P s f ) \eth f\stackrel{\,\text{def}}{=}P^{-s+1}\partial(P^{s}f)
  13. ( s ) \mathcal{E}(s)
  14. η \eta
  15. η e i s ψ η \eta\rightarrow e^{is\psi}\eta
  16. ð \eth
  17. η \eta
  18. ð η = - ( sin θ ) s { θ + i sin θ ϕ } [ ( sin θ ) - s η ] . \eth\eta=-(\sin{\theta})^{s}\left\{\frac{\partial}{\partial\theta}+\frac{i}{% \sin{\theta}}\frac{\partial}{\partial\phi}\right\}\left[(\sin{\theta})^{-s}% \eta\right].
  19. θ \theta
  20. ϕ \phi
  21. ð \eth
  22. ð η \eth\eta
  23. η \eta
  24. s s
  25. ð η \eth\eta
  26. s + 1 s+1
  27. ð ¯ η = - ( sin θ ) - s { θ - i sin θ ϕ } [ ( sin θ ) s η ] . \bar{\eth}\eta=-(\sin{\theta})^{-s}\left\{\frac{\partial}{\partial\theta}-% \frac{i}{\sin{\theta}}\frac{\partial}{\partial\phi}\right\}\left[(\sin{\theta}% )^{s}\eta\right].
  28. Y m s = ( - s ) ! ( + s ) ! ð s Y m , 0 s ; {}_{s}Y_{\ell m}=\sqrt{\frac{(\ell-s)!}{(\ell+s)!}}\ \eth^{s}Y_{\ell m},\ \ 0% \leq s\leq\ell;
  29. Y m s = ( + s ) ! ( - s ) ! ( - 1 ) s ð ¯ - s Y m , - s 0 ; {}_{s}Y_{\ell m}=\sqrt{\frac{(\ell+s)!}{(\ell-s)!}}\ (-1)^{s}\bar{\eth}^{-s}Y_% {\ell m},\ \ -\ell\leq s\leq 0;
  30. Y m s = 0 , < | s | . {}_{s}Y_{\ell m}=0,\ \ \ell<|s|.
  31. Y m s {}_{s}Y_{\ell m}
  32. ð ( Y m s ) = + ( - s ) ( + s + 1 ) Y m s + 1 ; \eth\left({}_{s}Y_{\ell m}\right)=+\sqrt{(\ell-s)(\ell+s+1)}\ {}_{s+1}Y_{\ell m};
  33. ð ¯ ( Y m s ) = - ( + s ) ( - s + 1 ) Y m s - 1 ; \bar{\eth}\left({}_{s}Y_{\ell m}\right)=-\sqrt{(\ell+s)(\ell-s+1)}\ {}_{s-1}Y_% {\ell m};
  34. S 2 Y m s Y ¯ m s d S = δ δ m m , \int_{S^{2}}{}_{s}Y_{\ell m}\ {}_{s}\bar{Y}_{\ell^{\prime}m^{\prime}}\ dS=% \delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}},
  35. m Y ¯ m s ( θ , ϕ ) Y m s ( θ , ϕ ) = δ ( ϕ - ϕ ) δ ( cos θ - cos θ ) \sum_{\ell m}{}_{s}\bar{Y}_{\ell m}(\theta^{\prime},\phi^{\prime}){}_{s}Y_{% \ell m}(\theta,\phi)=\delta(\phi^{\prime}-\phi)\delta(\cos\theta^{\prime}-\cos\theta)
  36. Y m s ( θ , ϕ ) = ( - 1 ) m ( + m ) ! ( - m ) ! ( 2 + 1 ) 4 π ( + s ) ! ( - s ) ! sin 2 ( θ 2 ) {}_{s}Y_{\ell m}(\theta,\phi)=(-1)^{m}\sqrt{\frac{(\ell+m)!(\ell-m)!(2\ell+1)}% {4\pi(\ell+s)!(\ell-s)!}}\sin^{2\ell}\left(\frac{\theta}{2}\right)
  37. × r = 0 - s ( - s r ) ( + s r + s - m ) ( - 1 ) - r - s e i m ϕ cot 2 r + s - m ( θ 2 ) . \times\sum_{r=0}^{\ell-s}{\ell-s\choose r}{\ell+s\choose r+s-m}(-1)^{\ell-r-s}% e^{im\phi}\cot^{2r+s-m}\left(\frac{\theta}{2}\right)\ .
  38. Y ¯ m s = ( - 1 ) s + m Y ( - m ) - s {}_{s}\bar{Y}_{\ell m}=(-1)^{s+m}{}_{-s}Y_{\ell(-m)}
  39. Y m s ( π - θ , ϕ + π ) = ( - 1 ) Y m - s ( θ , ϕ ) {}_{s}Y_{\ell m}(\pi-\theta,\phi+\pi)=(-1)^{\ell}{}_{-s}Y_{\ell m}(\theta,\phi)
  40. Y 10 1 ( θ , ϕ ) = 3 8 π sin θ {}_{1}Y_{10}(\theta,\phi)=\sqrt{\frac{3}{8\pi}}\,\sin\theta
  41. Y 1 ± 1 1 ( θ , ϕ ) = - 3 16 π ( 1 cos θ ) e ± i ϕ {}_{1}Y_{1\pm 1}(\theta,\phi)=-\sqrt{\frac{3}{16\pi}}(1\mp\cos\theta)\,e^{\pm i\phi}
  42. D - m s ( ϕ , θ , - ψ ) = ( - 1 ) m 4 π 2 + 1 Y m s ( θ , ϕ ) e i s ψ D^{\ell}_{-ms}(\phi,\theta,-\psi)=(-1)^{m}\sqrt{\frac{4\pi}{2\ell+1}}{}_{s}Y_{% \ell m}(\theta,\phi)e^{is\psi}

Spin_echo.html

  1. τ \tau
  2. T T
  3. τ \tau

Spin_pumping.html

  1. j s = 4 π g 1 M S 2 M ( t ) × d M ( t ) ) d t \vec{j}_{s}={\hbar\over{4\pi}}g^{\uparrow\downarrow}{1\over M_{S}^{2}}\langle% \vec{M}(t)\times{{d\vec{M}(t))}\over{dt}}\rangle
  2. j s \vec{j}_{s}
  3. g g^{\uparrow\downarrow}
  4. M S M_{S}
  5. M ( t ) \vec{M}(t)

Spin_spherical_harmonics.html

  1. 𝐣 2 Y l , s , j , m \displaystyle\mathbf{j}^{2}Y_{l,s,j,m}
  2. 𝐣 = 𝐥 + 𝐬 \mathbf{j}=\mathbf{l}+\mathbf{s}
  3. Y j ± 1 2 , 1 2 , j , m = 1 2 ( j ± 1 2 ) + 1 ( j ± 1 2 m + 1 2 Y j ± 1 2 m - 1 2 j ± 1 2 ± m + 1 2 Y j ± 1 2 m + 1 2 ) Y_{j\pm\frac{1}{2},\frac{1}{2},j,m}=\frac{1}{\sqrt{2\bigl(j\pm\frac{1}{2}\bigr% )+1}}\begin{pmatrix}\mp\sqrt{j\pm\frac{1}{2}\mp m+\frac{1}{2}}Y_{j\pm\frac{1}{% 2}}^{m-\frac{1}{2}}\\ \sqrt{j\pm\frac{1}{2}\pm m+\frac{1}{2}}Y_{j\pm\frac{1}{2}}^{m+\frac{1}{2}}\end% {pmatrix}

Spinning_dust.html

  1. P ( ω ) = 2 3 s y m b o l μ ¨ 2 c 3 = 2 3 ω 4 μ 2 c 3 , P(\omega)=\frac{2}{3}\frac{\ddot{symbol{\mu}}^{2}}{c^{3}}=\frac{2}{3}\frac{% \omega^{4}\mu_{\bot}^{2}}{c^{3}},
  2. μ < s u b > μ<sub>⊥

Spiric_section.html

  1. ( x 2 + y 2 ) 2 = d x 2 + e y 2 + f . (x^{2}+y^{2})^{2}=dx^{2}+ey^{2}+f.\,
  2. ( x 2 + y 2 + z 2 + b 2 - a 2 ) 2 = 4 b 2 ( x 2 + y 2 ) . (x^{2}+y^{2}+z^{2}+b^{2}-a^{2})^{2}=4b^{2}(x^{2}+y^{2}).\,
  3. ( x 2 + y 2 - a 2 + b 2 + c 2 ) 2 = 4 b 2 ( x 2 + c 2 ) . (x^{2}+y^{2}-a^{2}+b^{2}+c^{2})^{2}=4b^{2}(x^{2}+c^{2}).\,
  4. ( x 2 + y 2 ) 2 = d x 2 + e y 2 + f (x^{2}+y^{2})^{2}=dx^{2}+ey^{2}+f\,
  5. d = 2 ( a 2 + b 2 - c 2 ) , e = 2 ( a 2 - b 2 - c 2 ) , f = - ( a + b + c ) ( a + b - c ) ( a - b + c ) ( a - b - c ) . d=2(a^{2}+b^{2}-c^{2}),\ e=2(a^{2}-b^{2}-c^{2}),\ f=-(a+b+c)(a+b-c)(a-b+c)(a-b% -c).\,
  6. ( r 2 - a 2 + b 2 + c 2 ) 2 = 4 b 2 ( r 2 cos 2 θ + c 2 ) (r^{2}-a^{2}+b^{2}+c^{2})^{2}=4b^{2}(r^{2}\cos^{2}\theta+c^{2})\,
  7. r 4 = d r 2 cos 2 θ + e r 2 sin 2 θ + f . r^{4}=dr^{2}\cos^{2}\theta+er^{2}\sin^{2}\theta+f.\,

Split-radix_FFT_algorithm.html

  1. X k = n = 0 N - 1 x n ω N n k X_{k}=\sum_{n=0}^{N-1}x_{n}\omega_{N}^{nk}
  2. k k
  3. 0
  4. N - 1 N-1
  5. ω N \omega_{N}
  6. ω N = e - 2 π i N , \omega_{N}=e^{-\frac{2\pi i}{N}},
  7. ω N N = 1 \omega_{N}^{N}=1
  8. x 2 n 2 x_{2n_{2}}
  9. x 4 n 4 + 1 x_{4n_{4}+1}
  10. x 4 n 4 + 3 x_{4n_{4}+3}
  11. n m n_{m}
  12. N / m - 1 N/m-1
  13. X k = n 2 = 0 N / 2 - 1 x 2 n 2 ω N / 2 n 2 k + ω N k n 4 = 0 N / 4 - 1 x 4 n 4 + 1 ω N / 4 n 4 k + ω N 3 k n 4 = 0 N / 4 - 1 x 4 n 4 + 3 ω N / 4 n 4 k X_{k}=\sum_{n_{2}=0}^{N/2-1}x_{2n_{2}}\omega_{N/2}^{n_{2}k}+\omega_{N}^{k}\sum% _{n_{4}=0}^{N/4-1}x_{4n_{4}+1}\omega_{N/4}^{n_{4}k}+\omega_{N}^{3k}\sum_{n_{4}% =0}^{N/4-1}x_{4n_{4}+3}\omega_{N/4}^{n_{4}k}
  14. ω N m n k = ω N / m n k \omega_{N}^{mnk}=\omega_{N/m}^{nk}
  15. U k U_{k}
  16. k = 0 , , N / 2 - 1 k=0,\ldots,N/2-1
  17. Z k Z_{k}
  18. Z k Z^{\prime}_{k}
  19. k = 0 , , N / 4 - 1 k=0,\ldots,N/4-1
  20. X k X_{k}
  21. X k = U k + ω N k Z k + ω N 3 k Z k . X_{k}=U_{k}+\omega_{N}^{k}Z_{k}+\omega_{N}^{3k}Z^{\prime}_{k}.
  22. k N / 4 k\geq N/4
  23. k < N / 4 k<N/4
  24. ω N k \omega_{N}^{k}
  25. ω N 3 k \omega_{N}^{3k}
  26. ω N k + N / 4 = - i ω N k \omega_{N}^{k+N/4}=-i\omega_{N}^{k}
  27. ω N 3 ( k + N / 4 ) = i ω N 3 k \omega_{N}^{3(k+N/4)}=i\omega_{N}^{3k}
  28. X k = U k + ( ω N k Z k + ω N 3 k Z k ) , X_{k}=U_{k}+\left(\omega_{N}^{k}Z_{k}+\omega_{N}^{3k}Z^{\prime}_{k}\right),
  29. X k + N / 2 = U k - ( ω N k Z k + ω N 3 k Z k ) , X_{k+N/2}=U_{k}-\left(\omega_{N}^{k}Z_{k}+\omega_{N}^{3k}Z^{\prime}_{k}\right),
  30. X k + N / 4 = U k + N / 4 - i ( ω N k Z k - ω N 3 k Z k ) , X_{k+N/4}=U_{k+N/4}-i\left(\omega_{N}^{k}Z_{k}-\omega_{N}^{3k}Z^{\prime}_{k}% \right),
  31. X k + 3 N / 4 = U k + N / 4 + i ( ω N k Z k - ω N 3 k Z k ) , X_{k+3N/4}=U_{k+N/4}+i\left(\omega_{N}^{k}Z_{k}-\omega_{N}^{3k}Z^{\prime}_{k}% \right),
  32. X k X_{k}
  33. k k
  34. 0
  35. N / 4 - 1 N/4-1
  36. k = 0 k=0
  37. k = N / 8 k=N/8
  38. ( 1 ± i ) / 2 (1\pm i)/\sqrt{2}
  39. ± 1 \pm 1
  40. ± i \pm i
  41. X 0 = x 0 X_{0}=x_{0}
  42. X 0 = x 0 + x 1 X_{0}=x_{0}+x_{1}
  43. X 1 = x 0 - x 1 X_{1}=x_{0}-x_{1}
  44. 4 N log 2 N - 6 N + 8 4N\log_{2}N-6N+8

Split-step_method.html

  1. A z = - i β 2 2 2 A t 2 + i γ | A | 2 A = [ D ^ + N ^ ] A , {\partial A\over\partial z}=-{i\beta_{2}\over 2}{\partial^{2}A\over\partial t^% {2}}+i\gamma|A|^{2}A=[\hat{D}+\hat{N}]A,
  2. A ( t , z ) A(t,z)
  3. t t
  4. z z
  5. A D z = - i β 2 2 2 A t 2 = D ^ A , {\partial A_{D}\over\partial z}=-{i\beta_{2}\over 2}{\partial^{2}A\over% \partial t^{2}}=\hat{D}A,
  6. A N z = i γ | A | 2 A = N ^ A . {\partial A_{N}\over\partial z}=i\gamma|A|^{2}A=\hat{N}A.
  7. h h
  8. z z
  9. A N ( t , z + h ) = exp [ i γ | A | 2 h ] A ( t , z ) , A_{N}(t,z+h)=\exp\left[i\gamma|A|^{2}h\right]A(t,z),
  10. A N A_{N}
  11. A ~ N ( ω , z ) = - A N ( t , z ) exp [ i ( ω - ω 0 ) t ] d t \tilde{A}_{N}(\omega,z)=\int_{-\infty}^{\infty}A_{N}(t,z)\exp[i(\omega-\omega_% {0})t]dt
  12. ω 0 \omega_{0}
  13. A ~ ( ω , z + h ) = exp [ i β 2 2 ( ω - ω 0 ) 2 h ] A ~ N ( ω , z + h ) . \tilde{A}(\omega,z+h)=\exp\left[{i\beta_{2}\over 2}(\omega-\omega_{0})^{2}h% \right]\tilde{A}_{N}(\omega,z+h).
  14. A ~ ( ω , z + h ) \tilde{A}(\omega,z+h)
  15. A ( t , z + h ) A\left(t,z+h\right)
  16. h h
  17. N N
  18. N h Nh
  19. i ψ t = - 2 2 m 2 ψ x 2 + γ | ψ | 2 ψ = [ D ^ + N ^ ] ψ , i\hbar{\partial\psi\over\partial t}=-{{\hbar}^{2}\over{2m}}{\partial^{2}\psi% \over\partial x^{2}}+\gamma|\psi|^{2}\psi=[\hat{D}+\hat{N}]\psi,
  20. ψ ( x , t ) \psi(x,t)
  21. x x
  22. t t
  23. D ^ = - 2 2 m 2 x 2 \hat{D}=-{{\hbar}^{2}\over{2m}}{\partial^{2}\over\partial x^{2}}
  24. N ^ = γ | ψ | 2 \hat{N}=\gamma|\psi|^{2}
  25. m m
  26. \hbar
  27. 2 π 2\pi
  28. ψ ( x , t ) = e i t ( D ^ + N ^ ) ψ ( x , 0 ) \psi(x,t)=e^{it(\hat{D}+\hat{N})}\psi(x,0)
  29. D ^ \hat{D}
  30. N ^ \hat{N}
  31. d t 2 dt^{2}
  32. d t dt
  33. ψ ( x , t + d t ) e i d t D ^ e i d t N ^ ψ ( x , t ) \psi(x,t+dt)\approx e^{idt\hat{D}}e^{idt\hat{N}}\psi(x,t)
  34. N ^ \hat{N}
  35. t t
  36. D ^ \hat{D}
  37. i k ik
  38. x \partial\over\partial x
  39. k k
  40. e i d t N ^ ψ ( x , t ) e^{idt\hat{N}}\psi(x,t)
  41. e - i d t k 2 e^{-idtk^{2}}
  42. N ^ \hat{N}
  43. D ^ \hat{D}
  44. e - i d t k 2 F [ e i d t N ^ ψ ( x , t ) ] e^{-idtk^{2}}F[e^{idt\hat{N}}\psi(x,t)]
  45. F F
  46. ψ ( x , t + d t ) = F - 1 [ e - i d t k 2 F [ e i d t N ^ ψ ( x , t ) ] ] \psi(x,t+dt)=F^{-1}[e^{-idtk^{2}}F[e^{idt\hat{N}}\psi(x,t)]]
  47. d t 3 dt^{3}
  48. d t dt

Spot–future_parity.html

  1. F = S e ( r + y - q - u ) T F=Se^{(r+y-q-u)T}
  2. F = S e r T F=Se^{rT}
  3. S 0 S_{0}
  4. P 0 P_{0}
  5. P 0 = S 0 - K e - r T P_{0}=S_{0}-Ke^{-rT}
  6. P 0 = ( F 0 - K ) e - r T P_{0}=(F_{0}-K)e^{-rT}
  7. F 0 F_{0}

Spouge's_approximation.html

  1. Γ ( z + 1 ) = ( z + a ) z + 1 / 2 e - ( z + a ) [ c 0 + k = 1 a - 1 c k z + k + ε a ( z ) ] \Gamma(z+1)=(z+a)^{z+1/2}e^{-(z+a)}\left[c_{0}+\sum_{k=1}^{a-1}\frac{c_{k}}{z+% k}+\varepsilon_{a}(z)\right]
  2. c 0 = 2 π c_{0}=\sqrt{2\pi}\,
  3. c k = ( - 1 ) k - 1 ( k - 1 ) ! ( - k + a ) k - 1 / 2 e - k + a k { 1 , 2 , , a - 1 } . c_{k}=\frac{(-1)^{k-1}}{(k-1)!}(-k+a)^{k-1/2}e^{-k+a}\quad k\in\{1,2,\dots,a-1\}.
  4. a - 1 / 2 ( 2 π ) - ( a + 1 / 2 ) . \,a^{-1/2}(2\pi)^{-(a+1/2)}.
  5. c k c_{k}

Spread_of_a_matrix.html

  1. A A
  2. λ 1 , , λ n \lambda_{1},\ldots,\lambda_{n}
  3. λ i \lambda_{i}
  4. v i v_{i}
  5. A A
  6. A v i = λ i v i . Av_{i}=\lambda_{i}v_{i}.
  7. A A
  8. s ( A ) = max { | λ i - λ j | : i , j = 1 , n } . s(A)=\max\{|\lambda_{i}-\lambda_{j}|:i,j=1,\ldots n\}.
  9. 0
  10. 1 1
  11. A A
  12. B B
  13. B A B - 1 BAB^{-1}
  14. A A
  15. A A

Spread_option.html

  1. C = max ( 0 , S 1 - S 2 - K ) C=\max(0,S_{1}-S_{2}-K)
  2. P = max ( 0 , K - S 1 + S 2 ) P=\max(0,K-S_{1}+S_{2})

SQ-universal_group.html

  1. P = a , b , c , d | a 2 = b 2 = c 2 = d 2 = ( a b ) 3 = ( b c ) 3 = ( a c ) 3 = ( a d ) 3 = ( c d ) 3 = ( b d ) 3 = 1 P=\left\langle a,b,c,d\,|\,a^{2}=b^{2}=c^{2}=d^{2}=(ab)^{3}=(bc)^{3}=(ac)^{3}=% (ad)^{3}=(cd)^{3}=(bd)^{3}=1\right\rangle
  2. h 1 * , h 2 * , , h n * G h^{*}_{1},h^{*}_{2},\dots,h^{*}_{n}\dots\in G
  3. h n * h n h^{*}_{n}\mapsto h_{n}
  4. N G N\triangleleft G
  5. M N / M N / ( M N ) MN/M\cong N/(M\cap N)
  6. M N / M G / M MN/M\triangleleft G/M
  7. M N / M S 1 MN/M\cap S\cong 1
  8. S M N / M N / ( M N ) S\subseteq MN/M\cong N/(M\cap N)
  9. 𝒫 \mathcal{P}
  10. 𝒫 \mathcal{P}
  11. G 𝒫 G\in\mathcal{P}
  12. 𝒫 \mathcal{P}
  13. n > 10 78 n>10^{78}
  14. 𝒫 \mathcal{P}
  15. 𝒫 \mathcal{P}
  16. 𝒫 \mathcal{P}
  17. G 𝒫 G\in\mathcal{P}
  18. 𝒢 \mathcal{G}
  19. 𝒢 \mathcal{G}
  20. 𝒢 \mathcal{G}
  21. 𝒫 \mathcal{P}
  22. F , G 𝒫 F,G\in\mathcal{P}
  23. H 𝒫 H\in\mathcal{P}
  24. 𝒫 \mathcal{P}
  25. 𝒫 \mathcal{P}
  26. N G N\triangleleft G
  27. 𝒫 \mathcal{P}
  28. 𝒫 \mathcal{P}
  29. 𝒫 \mathcal{P}
  30. 𝒫 \mathcal{P}
  31. 𝒫 \mathcal{P}
  32. 𝒫 \mathcal{P}
  33. 𝒫 \mathcal{P}
  34. 𝒫 \mathcal{P}
  35. 𝒫 \mathcal{P}
  36. 𝒲 \mathcal{W}
  37. 𝒲 \mathcal{W}
  38. 𝒲 \mathcal{W}
  39. 𝒲 \mathcal{W}
  40. 𝒲 \mathcal{W}
  41. \mathcal{F}
  42. \mathcal{F}
  43. 𝒲 \mathcal{W}
  44. \mathcal{F}
  45. 𝒲 \mathcal{W}
  46. 𝒲 \mathcal{W}
  47. 𝒲 \mathcal{W}
  48. 𝒞 \mathcal{C}
  49. 𝒫 \mathcal{P}
  50. 𝒞 \mathcal{C}
  51. 𝒫 \mathcal{P}
  52. 𝒞 \mathcal{C}
  53. 𝒫 \mathcal{P}
  54. 𝒞 \mathcal{C}

Square-lattice_Ising_model.html

  1. H 0 H\neq 0
  2. Λ \Lambda
  3. K = β J K=\beta J
  4. L = β J * L=\beta J*
  5. β = 1 / ( k T ) \beta=1/(kT)
  6. Z N ( K , L ) Z_{N}(K,L)
  7. Z N ( K , L ) = { σ } exp ( K i j H σ i σ j + L i j V σ i σ j ) . Z_{N}(K,L)=\sum_{\{\sigma\}}\exp\left(K\sum_{\langle ij\rangle_{H}}\sigma_{i}% \sigma_{j}+L\sum_{\langle ij\rangle_{V}}\sigma_{i}\sigma_{j}\right).
  8. T c T_{c}
  9. F ( K , L ) F(K,L)
  10. β F ( K * , L * ) = β F ( K , L ) + 1 2 log [ sinh ( 2 K ) sinh ( 2 L ) ] \beta F\left(K^{*},L^{*}\right)=\beta F\left(K,L\right)+\frac{1}{2}\log\left[% \sinh\left(2K\right)\sinh\left(2L\right)\right]
  11. sinh ( 2 K * ) sinh ( 2 L ) = 1 \sinh\left(2K^{*}\right)\sinh\left(2L\right)=1
  12. sinh ( 2 L * ) sinh ( 2 K ) = 1 \sinh\left(2L^{*}\right)\sinh\left(2K\right)=1
  13. sinh ( 2 K ) sinh ( 2 L ) = 1 \sinh\left(2K\right)\sinh\left(2L\right)=1
  14. J = J * J=J^{*}
  15. T c T_{c}
  16. k B T c / J = 2 ln ( 1 + 2 ) 2.26918531421 k_{B}T_{c}/J=\frac{2}{\ln(1+\sqrt{2})}\approx 2.26918531421
  17. { σ } \{\sigma\}
  18. Λ \Lambda
  19. Z N Z_{N}
  20. { σ } \{\sigma\}
  21. e K ( N - 2 s ) + L ( N - 2 r ) e^{K(N-2s)+L(N-2r)}
  22. Λ D \Lambda_{D}
  23. { σ } \{\sigma\}
  24. Λ \Lambda
  25. Z N ( K , L ) = 2 e N ( K + L ) P Λ D e - 2 L r - 2 K s Z_{N}(K,L)=2e^{N(K+L)}\sum_{P\subset\Lambda_{D}}e^{-2Lr-2Ks}
  26. T 0 , e - K , e - L 0 T\rightarrow 0,\ \ e^{-K},e^{-L}\rightarrow 0
  27. Z N ( K , L ) = 2 e N ( K + L ) P Λ D e - 2 L r - 2 K s Z_{N}(K,L)=2e^{N(K+L)}\sum_{P\subset\Lambda_{D}}e^{-2Lr-2Ks}
  28. Z N ( K , L ) Z_{N}(K,L)
  29. σ σ = ± 1 \sigma\sigma^{\prime}=\pm 1
  30. e K σ σ = cosh K + sinh K ( σ σ ) = cosh K ( 1 + tanh K ( σ σ ) ) . e^{K\sigma\sigma^{\prime}}=\cosh K+\sinh K(\sigma\sigma^{\prime})=\cosh K(1+% \tanh K(\sigma\sigma^{\prime})).
  31. Z N ( K , L ) = ( cosh K cosh L ) N { σ } i j H ( 1 + v σ i σ j ) i j V ( 1 + w σ i σ j ) Z_{N}(K,L)=(\cosh K\cosh L)^{N}\sum_{\{\sigma\}}\prod_{\langle ij\rangle_{H}}(% 1+v\sigma_{i}\sigma_{j})\prod_{\langle ij\rangle_{V}}(1+w\sigma_{i}\sigma_{j})
  32. v = tanh K v=\tanh K
  33. w = tanh L w=\tanh L
  34. 2 2 N 2^{2N}
  35. v σ i σ j v\sigma_{i}\sigma_{j}
  36. w σ i σ j ) w\sigma_{i}\sigma_{j})
  37. σ i = ± 1 σ i n = { 0 for n odd 2 for n even \sum_{\sigma_{i}=\pm 1}\sigma_{i}^{n}=\begin{cases}0&\mbox{for }~{}n\mbox{ odd% }\\ 2&\mbox{for }~{}n\mbox{ even}\end{cases}
  38. Z N ( K , L ) = 2 N ( cosh K cosh L ) N P Λ v r w s Z_{N}(K,L)=2^{N}(\cosh K\cosh L)^{N}\sum_{P\subset\Lambda}v^{r}w^{s}
  39. 0 \rightarrow 0
  40. T T\rightarrow\infty
  41. Z N ( K , L ) Z_{N}(K,L)
  42. N N\to\infty
  43. k = 1 sinh ( 2 K ) sinh ( 2 L ) k=\frac{1}{\sinh\left(2K\right)\sinh\left(2L\right)}
  44. - β F = log ( 2 ) 2 + 1 2 π 0 π log [ cosh ( 2 K ) cosh ( 2 L ) + 1 k 1 + k 2 - 2 k cos ( 2 θ ) ] d θ -\beta F=\frac{\log(2)}{2}+\frac{1}{2\pi}\int_{0}^{\pi}\log\left[\cosh\left(2K% \right)\cosh\left(2L\right)+\frac{1}{k}\sqrt{1+k^{2}-2k\cos(2\theta)}\right]d\theta
  45. J = J * J=J^{*}
  46. U = - J coth ( 2 β J ) [ 1 + 2 π ( 2 tanh 2 ( 2 β J ) - 1 ) 0 π / 2 1 1 - 4 k ( 1 + k ) - 2 sin 2 ( θ ) d θ ] U=-J\coth(2\beta J)\left[1+\frac{2}{\pi}(2\tanh^{2}(2\beta J)-1)\int_{0}^{\pi/% 2}\frac{1}{\sqrt{1-4k(1+k)^{-2}\sin^{2}(\theta)}}d\theta\right]
  47. T < T c T<T_{c}
  48. M = [ 1 - sinh - 4 ( 2 β J ) ] 1 / 8 M=\left[1-\sinh^{-4}(2\beta J)\right]^{1/8}

Squashed_entanglement.html

  1. ϱ A , B \varrho_{A,B}
  2. ( A , B ) (A,B)
  3. A A
  4. B B
  5. E C M I E_{CMI}
  6. ( A , B ) (A,B)
  7. K K
  8. ϱ A , B , Λ \varrho_{A,B,\Lambda}
  9. ( A , B , Λ ) (A,B,\Lambda)
  10. ϱ A , B = t r Λ ( ϱ A , B , Λ ) \varrho_{A,B}=tr_{\Lambda}(\varrho_{A,B,\Lambda})
  11. S ( A : B | Λ ) S(A:B|\Lambda)
  12. ϱ A , B , Λ \varrho_{A,B,\Lambda}
  13. S ( A : B | Λ ) S(A:B|\Lambda)
  14. ϱ A , B \varrho_{A,B}
  15. E C M I ( ϱ A , B ) = S ( ϱ A ) = S ( ϱ B ) E_{CMI}(\varrho_{A,B})=S(\varrho_{A})=S(\varrho_{B})
  16. S ( ϱ ) S(\varrho)
  17. ϱ \varrho
  18. A , B A,B
  19. A , B , Λ A,B,\Lambda
  20. H ( A : B | Λ ) 0 H(A:B|\Lambda)\geq 0
  21. ϱ A , B , Λ \varrho_{A,B,\Lambda}
  22. ( A , B , Λ ) (A,B,\Lambda)
  23. ϱ A , B , Λ \varrho_{A,B,\Lambda}
  24. ϱ A , B , Λ \varrho_{A,B,\Lambda}
  25. ϱ A , B = t r a c e Λ ( ϱ A , B , Λ ) \varrho_{A,B}=trace_{\Lambda}(\varrho_{A,B,\Lambda})
  26. S ( A : B | Λ ) 0 S(A:B|\Lambda)\geq 0
  27. A , B , Λ A,B,\Lambda
  28. P A , B , Λ ( a , b , λ ) P_{A,B,\Lambda}(a,b,\lambda)
  29. P ( a , b , λ ) P(a,b,\lambda)
  30. P ( a , b , λ ) P(a,b,\lambda)
  31. H ( A : B | Λ ) = 0 H(A:B|\Lambda)=0
  32. K K
  33. P A , B , Λ P_{A,B,\Lambda}
  34. A , B , Λ A,B,\Lambda
  35. λ P A , B , Λ ( a , b , λ ) = P A , B ( a , b ) \sum_{\lambda}P_{A,B,\Lambda}(a,b,\lambda)=P_{A,B}(a,b)\,
  36. a , b a,b
  37. P A , B P_{A,B}
  38. P A , B , Λ P_{A,B,\Lambda}
  39. E C M I ( P A , B ) E_{CMI}(P_{A,B})
  40. P A , B P_{A,B}
  41. E C M I ( P A , B ) E_{CMI}(P_{A,B})
  42. E C M I ( ϱ A , B ) E_{CMI}(\varrho_{A,B})
  43. w λ w_{\lambda}
  44. λ = 1 , 2 , , d i m ( Λ ) \lambda=1,2,...,dim(\Lambda)
  45. | λ |\lambda\rangle
  46. λ = 1 , 2 , , d i m ( Λ ) \lambda=1,2,...,dim(\Lambda)
  47. Λ \Lambda
  48. ϱ A λ \varrho_{A}^{\lambda}
  49. ϱ B λ \varrho_{B}^{\lambda}
  50. λ = 1 , 2 , , d i m ( Λ ) \lambda=1,2,...,dim(\Lambda)
  51. A A
  52. B B
  53. S ( A : B | Λ ) = 0 S(A:B|\Lambda)=0
  54. Λ \Lambda
  55. ϱ A , B = λ ϱ A λ ϱ B λ w λ \varrho_{A,B}=\sum_{\lambda}\varrho_{A}^{\lambda}\varrho_{B}^{\lambda}w_{% \lambda}\,
  56. E C M I ( ϱ A , B ) E_{CMI}(\varrho_{A,B})
  57. ϱ A , B \varrho_{A,B}
  58. E C M I ( ϱ A , B ) = S ( ϱ A ) = S ( ϱ B ) E_{CMI}(\varrho_{A,B})=S(\varrho_{A})=S(\varrho_{B})
  59. | ψ A , B λ |\psi_{A,B}^{\lambda}\rangle
  60. λ = 1 , 2 , , d i m ( Λ ) \lambda=1,2,...,dim(\Lambda)
  61. ( A , B ) (A,B)
  62. K K
  63. K o K_{o}
  64. ϱ A , B , Λ \varrho_{A,B,\Lambda}
  65. K K
  66. ϱ A , B , Λ = λ | ψ A , B λ ψ A , B λ | w λ | λ λ | \varrho_{A,B,\Lambda}=\sum_{\lambda}|\psi_{A,B}^{\lambda}\rangle\langle\psi_{A% ,B}^{\lambda}|w_{\lambda}|\lambda\rangle\langle\lambda|\,
  67. K K
  68. K o K_{o}
  69. K K
  70. K o K_{o}
  71. ϱ A , B , Λ \varrho_{A,B,\Lambda}
  72. K K
  73. K o K_{o}
  74. K K
  75. S ( A : B | Λ ) 0 S(A:B|\Lambda)\geq 0
  76. S ( A | B ) S(A|B)

Squeeze_operator.html

  1. S ^ ( z ) = exp ( 1 2 ( z * a ^ 2 - z a ^ 2 ) ) , z = r e i θ \hat{S}(z)=\exp\left({1\over 2}(z^{*}\hat{a}^{2}-z\hat{a}^{\dagger 2})\right),% \qquad z=re^{i\theta}
  2. S ( ζ ) S ( ζ ) = S ( ζ ) S ( ζ ) = 1 ^ S(\zeta)S^{\dagger}(\zeta)=S^{\dagger}(\zeta)S(\zeta)=\hat{1}
  3. 1 ^ \hat{1}
  4. S ^ ( z ) a ^ S ^ ( z ) = a ^ cosh r - e i θ a ^ sinh r and S ^ ( z ) a ^ S ^ ( z ) = a ^ cosh r - e - i θ a ^ sinh r \hat{S}^{\dagger}(z)\hat{a}\hat{S}(z)=\hat{a}\cosh r-e^{i\theta}\hat{a}^{% \dagger}\sinh r\qquad\,\text{and}\qquad\hat{S}^{\dagger}(z)\hat{a}^{\dagger}% \hat{S}(z)=\hat{a}^{\dagger}\cosh r-e^{-i\theta}\hat{a}\sinh r
  5. S ^ ( z ) D ^ ( α ) D ^ ( α ) S ^ ( z ) , \hat{S}(z)\hat{D}(\alpha)\neq\hat{D}(\alpha)\hat{S}(z),
  6. D ^ ( α ) S ^ ( r ) | 0 = | α , r \hat{D}(\alpha)\hat{S}(r)|0\rangle=|\alpha,r\rangle

Squircle.html

  1. ( x - a ) 4 + ( y - b ) 4 = r 4 \left(x-a\right)^{4}+\left(y-b\right)^{4}=r^{4}
  2. | x - a | n + | y - b | n = | r | n . \left|x-a\right|^{n}+\left|y-b\right|^{n}=|r|^{n}.\,
  3. | x - a r a | n + | y - b r b | n = 1 , \left|\frac{x-a}{r_{a}}\right|^{n}\!+\left|\frac{y-b}{r_{b}}\right|^{n}\!=1,\,
  4. Area = 4 r 2 ( Γ ( 1 + 1 4 ) ) 2 Γ ( 1 + 2 4 ) = 4 r 2 1 4 ! 2 1 2 ! , \mathrm{Area}=4r^{2}\frac{\left(\Gamma\left(1+\tfrac{1}{4}\right)\right)^{2}}{% \Gamma\left(1+\tfrac{2}{4}\right)}=4r^{2}\frac{\tfrac{1}{4}!^{2}}{\tfrac{1}{2}% !},
  5. 1 4 ! \frac{1}{4}!
  6. 1 2 ! \frac{1}{2}!
  7. Area = S 2 r 2 , \mathrm{Area}=\frac{S}{\sqrt{2}}r^{2},

Stability_derivatives.html

  1. ψ \psi
  2. β \beta
  3. u = U cos β u=U\cos\beta
  4. v = U sin β v=U\sin\beta
  5. ( β ) (\beta)
  6. u f = U cos ( β ) cos ( ψ ) - U sin ( β ) sin ( ψ ) = U cos ( β + ψ ) u_{f}=U\cos(\beta)\cos(\psi)-U\sin(\beta)\sin(\psi)=U\cos(\beta+\psi)
  7. v f = U sin ( β ) cos ( ψ ) + U cos ( β ) sin ( ψ ) = U sin ( β + ψ ) v_{f}=U\sin(\beta)\cos(\psi)+U\cos(\beta)\sin(\psi)=U\sin(\beta+\psi)
  8. d u f d t = d U d t cos ( β + ψ ) - U d ( β + ψ ) d t sin ( β + ψ ) \frac{du_{f}}{dt}=\frac{dU}{dt}\cos(\beta+\psi)-U\frac{d(\beta+\psi)}{dt}\sin(% \beta+\psi)
  9. d v f d t = d U d t sin ( β + ψ ) + U d ( β + ψ ) d t cos ( β + ψ ) \frac{dv_{f}}{dt}=\frac{dU}{dt}\sin(\beta+\psi)+U\frac{d(\beta+\psi)}{dt}\cos(% \beta+\psi)
  10. X f = X cos ( ψ ) - Y sin ( ψ ) X_{f}=X\cos(\psi)-Y\sin(\psi)
  11. Y f = Y cos ( ψ ) + X sin ( ψ ) Y_{f}=Y\cos(\psi)+X\sin(\psi)
  12. X f = m d u f d t X_{f}=m\frac{du_{f}}{dt}
  13. Y f = m d v f d t Y_{f}=m\frac{dv_{f}}{dt}
  14. X = m d U d t cos ( β ) - m U d ( β + ψ ) d t sin ( β ) X=m\frac{dU}{dt}\cos(\beta)-mU\frac{d(\beta+\psi)}{dt}\sin(\beta)
  15. Y = m d U d t sin ( β ) + m U d ( β + ψ ) d t cos ( β ) Y=m\frac{dU}{dt}\sin(\beta)+mU\frac{d(\beta+\psi)}{dt}\cos(\beta)
  16. β \beta
  17. X = m d U d t X=m\frac{dU}{dt}
  18. Y = m U d ( β + ψ ) d t Y=mU\frac{d(\beta+\psi)}{dt}
  19. N = C d 2 ψ d t 2 N=C\frac{d^{2}\psi}{dt^{2}}
  20. β \beta
  21. d ψ d t \frac{d\psi}{dt}
  22. β \beta
  23. β \beta
  24. Y = Y 0 + Y β β + Y r r Y=Y_{0}+\frac{\partial Y}{\partial\beta}\beta+\frac{\partial Y}{\partial r}r
  25. Y 0 Y_{0}
  26. Y β = Y β \frac{\partial Y}{\partial\beta}=Y_{\beta}
  27. Y β \frac{\partial Y}{\partial\beta}
  28. Y r \frac{\partial Y}{\partial r}
  29. d β d t = Y β m U β - r \frac{d\beta}{dt}=\frac{Y_{\beta}}{mU}\beta-r
  30. d r d t = N β C β + N r C r \frac{dr}{dt}=\frac{N_{\beta}}{C}\beta+\frac{N_{r}}{C}r
  31. Y β Y_{\beta}
  32. β \beta
  33. Y β Y_{\beta}
  34. x c p x_{cp}
  35. x c p x_{cp}
  36. N β N_{\beta}
  37. N β N_{\beta}
  38. N r N_{r}
  39. Y r Y_{r}
  40. Y β Y_{\beta}
  41. β \beta
  42. d 2 β d t 2 - ( Y β m U + N r C ) d β d t + ( N β C + Y β m U N r C ) β = 0 \frac{d^{2}\beta}{dt^{2}}-\left(\frac{Y_{\beta}}{mU}+\frac{N_{r}}{C}\right)% \frac{d\beta}{dt}+\left(\frac{N_{\beta}}{C}+\frac{Y_{\beta}}{mU}\frac{N_{r}}{C% }\right)\beta=0
  43. Y β Y_{\beta}
  44. N r N_{r}
  45. N β N_{\beta}
  46. β \beta
  47. Y u Y_{u}
  48. ζ \zeta
  49. d β d t = Y β m U β - r + Y ζ m U ζ \frac{d\beta}{dt}=\frac{Y_{\beta}}{mU}\beta-r+\frac{Y_{\zeta}}{mU}\zeta
  50. d r d t = N β C β + N r C r + N ζ C ζ \frac{dr}{dt}=\frac{N_{\beta}}{C}\beta+\frac{N_{r}}{C}r+\frac{N_{\zeta}}{C}\zeta
  51. β \beta
  52. β \beta

Stability_theory.html

  1. f e f_{e}
  2. ϵ > 0 \epsilon>0
  3. δ > 0 \delta>0
  4. f ( t ) f(t)
  5. δ \delta
  6. f ( t 0 ) - f e < δ \|f(t_{0})-f_{e}\|<\delta
  7. ϵ \epsilon
  8. f ( t ) - f e < ϵ \|f(t)-f_{e}\|<\epsilon
  9. t t 0 t\geq t_{0}
  10. δ 0 > 0 \delta_{0}>0
  11. δ 0 > f ( t 0 ) - f e \delta_{0}>\|f(t_{0})-f_{e}\|
  12. f ( t ) f e f(t)\rightarrow f_{e}
  13. t t\rightarrow\infty
  14. x n + 1 = f ( x n ) , n = 0 , 1 , 2 , . x_{n+1}=f(x_{n}),\quad n=0,1,2,\ldots.
  15. f ( x ) f ( a ) + f ( a ) ( x - a ) . f(x)\approx f(a)+f^{\prime}(a)(x-a).
  16. x n + 1 - a x n - a = f ( x n ) - a x n - a f ( a ) ( x n - a ) x n - a = f ( a ) , \frac{x_{n+1}-a}{x_{n}-a}=\frac{f(x_{n})-a}{x_{n}-a}\approx\frac{f^{\prime}(a)% (x_{n}-a)}{x_{n}-a}=f^{\prime}(a),
  17. x = A x , x^{\prime}=Ax,
  18. x ( t ) = 0. x(t)=0.
  19. x = v ( x ) x^{\prime}=v(x)
  20. x ( t ) = p . x(t)=p.

Stable_manifold.html

  1. X X
  2. f : X X f\colon X\to X
  3. p p
  4. f f
  5. p p
  6. W s ( f , p ) = { q X : f n ( q ) p as n } W^{s}(f,p)=\{q\in X:f^{n}(q)\to p\mbox{ as }~{}n\to\infty\}
  7. p p
  8. W u ( f , p ) = { q X : f - n ( q ) p as n } . W^{u}(f,p)=\{q\in X:f^{-n}(q)\to p\mbox{ as }~{}n\to\infty\}.
  9. f - 1 f^{-1}
  10. f f
  11. f f - 1 = f - 1 f = i d X f\circ f^{-1}=f^{-1}\circ f=id_{X}
  12. i d X id_{X}
  13. X X
  14. p p
  15. k k
  16. f k f^{k}
  17. p p
  18. W s ( f , p ) = W s ( f k , p ) W^{s}(f,p)=W^{s}(f^{k},p)
  19. W u ( f , p ) = W u ( f k , p ) . W^{u}(f,p)=W^{u}(f^{k},p).
  20. U U
  21. p p
  22. p p
  23. W loc s ( f , p , U ) = { q U : f n ( q ) U for each n 0 } W^{s}_{\mathrm{loc}}(f,p,U)=\{q\in U:f^{n}(q)\in U\mbox{ for each }~{}n\geq 0\}
  24. W loc u ( f , p , U ) = W loc s ( f - 1 , p , U ) . W^{u}_{\mathrm{loc}}(f,p,U)=W^{s}_{\mathrm{loc}}(f^{-1},p,U).
  25. X X
  26. W s ( f , p ) = { q X : d ( f n ( q ) , f n ( p ) ) 0 for n } W^{s}(f,p)=\{q\in X:d(f^{n}(q),f^{n}(p))\to 0\mbox{ for }n\to\infty\}
  27. W u ( f , p ) = W s ( f - 1 , p ) , W^{u}(f,p)=W^{s}(f^{-1},p),
  28. d d
  29. X X
  30. p p
  31. X X
  32. f f
  33. 𝒞 k \mathcal{C}^{k}
  34. k 1 k\geq 1
  35. p p
  36. U U
  37. p p
  38. 𝒞 k \mathcal{C}^{k}
  39. p p
  40. E s E^{s}
  41. E u E^{u}
  42. D f ( p ) Df(p)
  43. f f
  44. 𝒞 k \mathcal{C}^{k}
  45. Diff k ( X ) \mathrm{Diff}^{k}(X)
  46. 𝒞 k \mathcal{C}^{k}
  47. X X
  48. 𝒞 k \mathcal{C}^{k}
  49. X X
  50. f f

Stable_manifold_theorem.html

  1. f : U n n f:U\subset\mathbb{R}^{n}\to\mathbb{R}^{n}
  2. W s ( p ) W^{s}(p)
  3. W u ( p ) W^{u}(p)
  4. W s ( p ) W^{s}(p)
  5. W u ( p ) W^{u}(p)
  6. W s ( p ) W^{s}(p)
  7. W u ( p ) W^{u}(p)

Stagnation_enthalpy.html

  1. h t h_{t}
  2. v v
  3. h t = h + v 2 2 h_{t}=h+\frac{v^{2}}{2}
  4. h h
  5. v v

Standard_illuminant.html

  1. M e , λ ( λ , T ) = c 1 λ - 5 exp ( c 2 λ T ) - 1 M_{e,\lambda}(\lambda,T)=\frac{c_{1}\lambda^{-5}}{\exp\left(\frac{c_{2}}{% \lambda T}\right)-1}
  2. c 1 = 2 π h c 2 c_{1}=2\pi\cdot h\cdot c^{2}
  3. c 2 = h c / k c_{2}=h\cdot c/k
  4. T n e w = T o l d × 1.4388 1.435 = 2848 K × 1.002648 = 2855.54 K T_{new}=T_{old}\times\frac{1.4388}{1.435}=2848\ \,\text{K}\times 1.002648=2855% .54\ \,\text{K}
  5. S A ( λ ) = 100 ( 560 λ ) 5 exp 1.435 × 10 7 2848 × 560 - 1 exp 1.435 × 10 7 2848 λ - 1 S_{A}(\lambda)=100\left(\frac{560}{\lambda}\right)^{5}\frac{\exp\frac{1.435% \times 10^{7}}{2848\times 560}-1}{\exp\frac{1.435\times 10^{7}}{2848\lambda}-1}
  6. y = 2.870 x - 3.000 x 2 - 0.275 y=2.870x-3.000x^{2}-0.275
  7. S ( λ ) = S 0 ( λ ) + M 1 S 1 ( λ ) + M 2 S 2 ( λ ) S(\lambda)=S_{0}(\lambda)+M_{1}S_{1}(\lambda)+M_{2}S_{2}(\lambda)
  8. x = X 0 + M 1 X 1 + M 2 X 2 S 0 + M 1 S 1 + M 2 S 2 x=\frac{X_{0}+M_{1}X_{1}+M_{2}X_{2}}{S_{0}+M_{1}S_{1}+M_{2}S_{2}}
  9. y = Y 0 + M 1 Y 1 + M 2 Y 2 S 0 + M 1 S 1 + M 2 S 2 y=\frac{Y_{0}+M_{1}Y_{1}+M_{2}Y_{2}}{S_{0}+M_{1}S_{1}+M_{2}S_{2}}
  10. M 1 = - 1.3515 - 1.7703 x + 5.9114 y 0.0241 + 0.2562 x - 0.7341 y M_{1}=\frac{-1.3515-1.7703x+5.9114y}{0.0241+0.2562x-0.7341y}
  11. M 2 = 0.0300 - 31.4424 x + 30.0717 y 0.0241 + 0.2562 x - 0.7341 y M_{2}=\frac{0.0300-31.4424x+30.0717y}{0.0241+0.2562x-0.7341y}
  12. ( x , y ) (x,y)
  13. S D ( λ ) S_{D}(\lambda)
  14. ( x D , y D ) (x_{D},y_{D})
  15. x D = { 0.244063 + 0.09911 10 3 T + 2.9678 10 6 T 2 - 4.6070 10 9 T 3 4000 K T 7000 K 0.237040 + 0.24748 10 3 T + 1.9018 10 6 T 2 - 2.0064 10 9 T 3 7000 K < T 25000 K x_{D}=\begin{cases}0.244063+0.09911\frac{10^{3}}{T}+2.9678\frac{10^{6}}{T^{2}}% -4.6070\frac{10^{9}}{T^{3}}&4000K\leq T\leq 7000K\\ 0.237040+0.24748\frac{10^{3}}{T}+1.9018\frac{10^{6}}{T^{2}}-2.0064\frac{10^{9}% }{T^{3}}&7000K<T\leq 25000K\end{cases}
  16. y D = - 3.000 x D 2 + 2.870 x D - 0.275 y_{D}=-3.000x_{D}^{2}+2.870x_{D}-0.275
  17. Δ u v = 0.003 \Delta_{uv}=0.003
  18. S D ( λ ) = S 0 ( λ ) + M 1 S 1 ( λ ) + M 2 S 2 ( λ ) S_{D}(\lambda)=S_{0}(\lambda)+M_{1}S_{1}(\lambda)+M_{2}S_{2}(\lambda)
  19. M 1 = ( - 1.3515 - 1.7703 x D + 5.9114 y D ) / M M_{1}=(-1.3515-1.7703x_{D}+5.9114y_{D})/M
  20. M 2 = ( 0.03000 - 31.4424 x D + 30.0717 y D ) / M M_{2}=(0.03000-31.4424x_{D}+30.0717y_{D})/M
  21. M = 0.0241 + 0.2562 x D - 0.7341 y D M=0.0241+0.2562x_{D}-0.7341y_{D}
  22. S 0 ( λ ) , S 1 ( λ ) , S 2 ( λ ) S_{0}(\lambda),S_{1}(\lambda),S_{2}(\lambda)

Standard_linear_solid_model.html

  1. σ s = E ε \sigma_{s}=E\varepsilon
  2. σ D = η d ε d t \sigma_{D}=\eta\frac{d\varepsilon}{dt}
  3. E = E 2 E=E_{2}
  4. η \eta
  5. E = E 1 E=E_{1}
  6. σ t o t = σ 1 + σ 2 \sigma_{tot}=\sigma_{1}+\sigma_{2}
  7. ε t o t = ε 1 = ε 2 \varepsilon_{tot}=\varepsilon_{1}=\varepsilon_{2}
  8. σ t o t = σ 1 = σ 2 \sigma_{tot}=\sigma_{1}=\sigma_{2}
  9. ε t o t = ε 1 + ε 2 \varepsilon_{tot}=\varepsilon_{1}+\varepsilon_{2}
  10. σ t o t = σ m + σ s 1 \sigma_{tot}=\sigma_{m}+\sigma_{s_{1}}
  11. ε t o t = ε m = ε s 1 \varepsilon_{tot}=\varepsilon_{m}=\varepsilon_{s_{1}}
  12. σ m = σ D = σ s 2 \sigma_{m}=\sigma_{D}=\sigma_{s_{2}}
  13. ε m = ε D + ε s 2 \varepsilon_{m}=\varepsilon_{D}+\varepsilon_{s_{2}}
  14. m m
  15. D D
  16. S 1 S_{1}
  17. S 2 S_{2}
  18. d ε ( t ) d t = E 2 η ( η E 2 d σ ( t ) d t + σ ( t ) - E 1 ε ( t ) ) E 1 + E 2 \frac{d\varepsilon(t)}{dt}=\frac{\frac{E_{2}}{\eta}\left(\frac{\eta}{E_{2}}% \frac{d\sigma(t)}{dt}+\sigma(t)-E_{1}\varepsilon(t)\right)}{E_{1}+E_{2}}
  19. d ε ( t ) d t = ( E 1 + E 2 ) - 1 [ d σ ( t ) d t + E 2 η σ ( t ) - E 1 E 2 η ε ( t ) ] \frac{d\varepsilon(t)}{dt}=\left(E_{1}+E_{2}\right)^{-1}\cdot\left[\frac{d% \sigma(t)}{dt}+\frac{E_{2}}{\eta}\sigma(t)-\frac{E_{1}E_{2}}{\eta}\varepsilon(% t)\right]
  20. τ \tau
  21. η E 2 = τ \frac{\eta}{E_{2}}=\tau

Standard_solar_model.html

  1. d L d r = 4 π r 2 ρ ( ϵ - ϵ ν ) \frac{dL}{dr}=4\pi r^{2}\rho\left(\epsilon-\epsilon_{\nu}\right)
  2. ρ ( r ) \scriptstyle\rho(r)
  3. d T d r = - 3 κ ρ l 64 π r 2 σ T 3 , {\mbox{d}~{}T\over\mbox{d}~{}r}=-{3\kappa\rho l\over 64\pi r^{2}\sigma T^{3}},
  4. d T d r = ( 1 - 1 γ ) T P d P d r , {\mbox{d}~{}T\over\mbox{d}~{}r}=\left(1-{1\over\gamma}\right){T\over P}{\mbox{% d}~{}P\over\mbox{d}~{}r},
  5. ϕ ( 8 B ) T 25 \phi(^{8}B)\propto T^{25}
  6. T sun = 15.7 × 10 6 K ± 1 % T\text{sun}=15.7\times 10^{6}\;\,\text{K}\;\pm 1\%

Stanley–Wilf_conjecture.html

  1. lim n | S n ( β ) | n . \lim_{n\to\infty}\sqrt[n]{|S_{n}(\beta)|}.
  2. lim sup n a n n , \limsup_{n\to\infty}\sqrt[n]{a_{n}},
  3. x k + 1 - 2 x k + 1 x^{k+1}-2x^{k}+1

Star_domain.html

  1. S S
  2. B = { t a : a A , t [ 0 , 1 ] } B=\{ta:a\in A,t\in[0,1]\}

STAR_model.html

  1. z t z_{t}
  2. ζ \zeta
  3. y t = γ 0 + γ 1 y t - 1 + γ 2 y t - 2 + + γ p y t - p + ϵ t . y_{t}=\gamma_{0}+\gamma_{1}y_{t-1}+\gamma_{2}y_{t-2}+...+\gamma_{p}y_{t-p}+% \epsilon_{t}.\,
  4. γ i \gamma_{i}\,
  5. ϵ t 𝑖𝑖𝑑 W N ( 0 ; σ 2 ) \epsilon_{t}\stackrel{\mathit{iid}}{\sim}WN(0;\sigma^{2})\,
  6. y t = 𝐗 𝐭 γ + σ ϵ t . y_{t}=\mathbf{X_{t}\gamma}+\sigma\epsilon_{t}.\,
  7. 𝐗 𝐭 = ( 1 , y t - 1 , y t - 2 , , y t - p ) \mathbf{X_{t}}=(1,y_{t-1},y_{t-2},\ldots,y_{t-p})\,
  8. γ \gamma\,
  9. γ 0 , γ 1 , γ 2 , , γ p \gamma_{0},\gamma_{1},\gamma_{2},...,\gamma_{p}\,
  10. ϵ t 𝑖𝑖𝑑 W N ( 0 ; 1 ) \epsilon_{t}\stackrel{\mathit{iid}}{\sim}WN(0;1)\,
  11. z t z_{t}
  12. ζ \zeta
  13. c 1 c_{1}
  14. c 2 c_{2}
  15. y t = 𝐗 𝐭 + G ( z t , ζ , c ) 𝐗 𝐭 + σ ( j ) ϵ t y_{t}=\mathbf{X_{t}}+G(z_{t},\zeta,c)\mathbf{X_{t}}+\sigma^{(j)}\epsilon_{t}\,
  16. X t = ( 1 , y t - 1 , y t - 2 , , y t - p ) X_{t}=(1,y_{t-1},y_{t-2},...,y_{t-p})\,
  17. G ( z t , ζ , c ) G(z_{t},\zeta,c)
  18. G ( z t , ζ , c ) = ( 1 + e x p ( - ζ ( z t - c ) ) ) - 1 , ζ > 0 G(z_{t},\zeta,c)=(1+exp(-\zeta(z_{t}-c)))^{-1},\zeta>0
  19. G ( z t , ζ , c ) = 1 - e x p ( - ζ ( z t - c ) 2 ) , ζ > 0 G(z_{t},\zeta,c)=1-exp(-\zeta(z_{t}-c)^{2}),\zeta>0
  20. G ( z t , ζ , c ) = ( 1 + e x p ( - ζ ( z t - c 1 ) ( z t - c 2 ) ) - 1 , ζ > 0 G(z_{t},\zeta,c)=(1+exp(-\zeta(z_{t}-c_{1})(z_{t}-c_{2}))^{-1},\zeta>0

State_prices.html

  1. c 0 = k q k × c k c_{0}=\sum_{k}q_{k}\times c_{k}

Stationary_sequence.html

  1. F X n , X n + 1 , , X n + N - 1 ( x n , x n + 1 , , x n + N - 1 ) \displaystyle{}\quad F_{X_{n},X_{n+1},\dots,X_{n+N-1}}(x_{n},x_{n+1},\dots,x_{% n+N-1})
  2. E ( X [ n ] ) = μ for all n . E(X[n])=\mu\quad\,\text{for all }n.

Statistical_potential.html

  1. ϕ , ψ \phi,\psi
  2. P ( r ) = 1 Z e - F ( r ) k T P\left(r\right)=\frac{1}{Z}e^{-\frac{F\left(r\right)}{kT}}
  3. r r
  4. k k
  5. T T
  6. Z Z
  7. Z = e - F ( r ) k T d r Z=\int e^{-\frac{F(r)}{kT}}dr
  8. F ( r ) F(r)
  9. F ( r ) F(r)
  10. P ( r ) P(r)
  11. F ( r ) = - k T ln P ( r ) - k T ln Z F\left(r\right)=-kT\ln P\left(r\right)-kT\ln Z
  12. Q R Q_{R}
  13. Z R Z_{R}
  14. Δ F ( r ) = - k T ln P ( r ) Q R ( r ) - k T ln Z Z R \Delta F\left(r\right)=-kT\ln\frac{P\left(r\right)}{Q_{R}\left(r\right)}-kT\ln% \frac{Z}{Z_{R}}
  15. Z Z
  16. Z R Z_{R}
  17. P ( r ) P(r)
  18. Q R ( r ) Q_{R}(r)
  19. P ( r ) P(r)
  20. C β C\beta
  21. r r
  22. Δ F \Delta F
  23. Δ F T \Delta F_{\textrm{T}}
  24. Δ F T = i < j Δ F ( r i j a i , a j ) = - k T i < j ln P ( r i j a i , a j ) Q R ( r i j a i , a j ) \Delta F_{\textrm{T}}=\sum_{i<j}\Delta F(r_{ij}\mid a_{i},a_{j})=-kT\sum_{i<j}% \ln\frac{P\left(r_{ij}\mid a_{i},a_{j}\right)}{Q_{R}\left(r_{ij}\mid a_{i},a_{% j}\right)}
  25. a i , a j a_{i},a_{j}
  26. i < j i<j
  27. r i j r_{ij}
  28. Q R Q_{R}
  29. Δ F T \Delta F_{\textrm{T}}
  30. g ( r ) g(r)
  31. g ( r ) = P ( r ) Q R ( r ) g(r)=\frac{P(r)}{Q_{R}(r)}
  32. P ( r ) P(r)
  33. Q R ( r ) Q_{R}(r)
  34. r r
  35. W ( r ) W(r)
  36. g ( r ) g(r)
  37. W ( r ) = - k T log g ( r ) = - k T log P ( r ) Q R ( r ) W(r)=-kT\log g(r)=-kT\log\frac{P(r)}{Q_{R}(r)}
  38. W ( r ) W(r)
  39. r r
  40. g ( r ) g(r)
  41. P ( r ) P(r)
  42. P ( X A ) P(X\mid A)
  43. X X
  44. A A
  45. P ( X A ) = P ( A X ) P ( X ) P ( A ) P ( A X ) P ( X ) P\left(X\mid A\right)=\frac{P\left(A\mid X\right)P\left(X\right)}{P\left(A% \right)}\propto P\left(A\mid X\right)P\left(X\right)
  46. P ( X A ) P(X\mid A)
  47. P ( A X ) P\left(A\mid X\right)
  48. P ( X ) P\left(X\right)
  49. P ( A X ) i < j P ( a i , a j r i j ) i < j P ( r i j a i , a j ) P ( r i j ) P\left(A\mid X\right)\approx\prod_{i<j}P\left(a_{i},a_{j}\mid r_{ij}\right)% \propto\prod_{i<j}\frac{P\left(r_{ij}\mid a_{i},a_{j}\right)}{P(r_{ij})}
  50. a i , a j a_{i},a_{j}
  51. i < j i<j
  52. r i j r_{ij}
  53. i i
  54. j j
  55. Q ( X ) Q(X)
  56. Q ( X ) Q(X)
  57. P ( Y ) P(Y)
  58. P ( Y ) P(Y)
  59. Q ( X ) Q(X)
  60. P ( Y ) P(Y)
  61. Q ( X ) Q(X)
  62. Y Y
  63. Q ( X ) Q(X)
  64. X X
  65. P ( Y ) P(Y)
  66. Y Y
  67. Y = f ( X ) Y=f(X)
  68. X X
  69. Y Y
  70. Q ( X ) Q(X)
  71. P ( Y ) P(Y)
  72. X X
  73. Q ( X ) Q(X)
  74. P ( Y ) P(Y)
  75. P ( X , Y ) = P ( Y ) Q ( Y ) Q ( X ) P(X,Y)=\frac{P(Y)}{Q(Y)}Q(X)
  76. Q ( Y ) Q(Y)
  77. Y Y
  78. Q ( X ) Q(X)
  79. Q ( X ) Q(X)
  80. Q ( X ) Q(X)
  81. Q ( X ) Q(X)

Statistical_signal_processing.html

  1. y ( t ) y(t)
  2. x ( t ) x(t)
  3. w ( t ) w(t)
  4. y ( t ) = x ( t ) + w ( t ) y(t)=x(t)+w(t)\,
  5. w ( t ) 𝒩 ( 0 , σ 2 ) w(t)\sim\mathcal{N}(0,\sigma^{2})
  6. R w w ( τ ) = σ 2 δ ( τ ) R_{ww}(\tau)=\sigma^{2}\delta(\tau)\,
  7. δ ( τ ) \delta(\tau)\,
  8. y ¯ = 1 n i y ( t ) i = x ( t ) + 1 n i w ( t ) i \bar{y}=\frac{1}{n}\sum_{i}y(t)_{i}=x(t)+\frac{1}{n}\sum_{i}w(t)_{i}
  9. σ ( y ¯ ) = 1 n σ \sigma(\bar{y})=\frac{1}{\sqrt{n}}\sigma

Statisticians'_and_engineers'_cross-reference_of_statistical_terms.html

  1. P F A P_{FA}
  2. P D P_{D}
  3. 1 - P D 1-P_{D}

Steane_code.html

  1. [ H 0 0 H ] \begin{bmatrix}H&0\\ 0&H\end{bmatrix}
  2. H = [ 1 0 0 1 0 1 1 0 1 0 1 1 0 1 0 0 1 0 1 1 1 ] . H=\begin{bmatrix}1&0&0&1&0&1&1\\ 0&1&0&1&1&0&1\\ 0&0&1&0&1&1&1\end{bmatrix}.

Steinmetz_solid.html

  1. 16 3 r 3 \frac{16}{3}r^{3}
  2. 8 × 1 3 r 2 × r = 8 3 r 3 \textstyle 8\times\frac{1}{3}r^{2}\times r=\frac{8}{3}r^{3}
  3. ( 2 r ) 3 - 8 3 r 3 = 16 3 r 3 \textstyle(2r)^{3}-\frac{8}{3}r^{3}=\frac{16}{3}r^{3}
  4. r 3 \tfrac{r}{3}
  5. ( 16 - 8 2 ) r 3 (16-8\sqrt{2})r^{3}\,
  6. 3 ( 16 - 8 2 ) r 2 . 3(16-8\sqrt{2})r^{2}.\,
  7. 12 ( 2 2 - 6 ) r 3 12(2\sqrt{2}-\sqrt{6})r^{3}\,
  8. 16 3 ( 3 + 2 3 - 4 2 ) r 3 \frac{16}{3}(3+2\sqrt{3}-4\sqrt{2})r^{3}\,

Stereoscopic_spectroscopy.html

  1. ( x , y , λ ) (x,y,\lambda)
  2. ( x , y , z ) (x,y,z)
  3. ( x , y , λ ) (x,y,\lambda)

Steroid_17a-monooxygenase.html

  1. \rightleftharpoons

Steryl-sulfatase.html

  1. \rightleftharpoons

Stewart's_theorem.html

  1. a a
  2. b b
  3. c c
  4. d d
  5. a a
  6. a a
  7. m m
  8. n n
  9. m m
  10. c c
  11. n n
  12. b b
  13. b 2 m + c 2 n = a ( d 2 + m n ) . b^{2}m+c^{2}n=a(d^{2}+mn).\,
  14. d a d + m a n = b m b + c n c . dad+man=bmb+cnc.\,
  15. d d
  16. P A 2 B C + P B 2 C A + P C 2 A B + B C C A A B = 0. PA^{2}\cdot BC+PB^{2}\cdot CA+PC^{2}\cdot AB+BC\cdot CA\cdot AB=0.\,
  17. c 2 = m 2 + d 2 - 2 d m cos θ b 2 = n 2 + d 2 - 2 d n cos θ = n 2 + d 2 + 2 d n cos θ . \begin{aligned}\displaystyle c^{2}&\displaystyle=m^{2}+d^{2}-2dm\cos\theta\\ \displaystyle b^{2}&\displaystyle=n^{2}+d^{2}-2dn\cos\theta^{\prime}\\ &\displaystyle=n^{2}+d^{2}+2dn\cos\theta.\end{aligned}
  18. b 2 m + c 2 n = n m 2 + n 2 m + ( m + n ) d 2 = ( m + n ) ( m n + d 2 ) = a ( m n + d 2 ) , \begin{aligned}&\displaystyle b^{2}m+c^{2}n\\ &\displaystyle=nm^{2}+n^{2}m+(m+n)d^{2}\\ &\displaystyle=(m+n)(mn+d^{2})\\ &\displaystyle=a(mn+d^{2}),\\ \end{aligned}

Stewart–Walker_lemma.html

  1. Δ δ T = 0 \Delta\delta T=0
  2. T 0 = 0 T_{0}=0
  3. T 0 T_{0}
  4. T 0 T_{0}
  5. δ a b \delta_{a}^{b}
  6. ϵ \mathcal{M}_{\epsilon}
  7. 0 = 4 \mathcal{M}_{0}=\mathcal{M}^{4}
  8. g i k = η i k + ϵ h i k g_{ik}=\eta_{ik}+\epsilon h_{ik}
  9. 𝒩 \mathcal{N}
  10. γ \gamma
  11. 𝒩 \mathcal{N}
  12. X X
  13. ϵ \mathcal{M}_{\epsilon}
  14. X X
  15. h t h_{t}
  16. 𝒩 𝒩 \mathcal{N}\to\mathcal{N}
  17. p 0 0 p_{0}\in\mathcal{M}_{0}
  18. p ϵ ϵ p_{\epsilon}\in\mathcal{M}_{\epsilon}
  19. h ϵ ( p 0 ) h_{\epsilon}(p_{0})
  20. h ϵ * h_{\epsilon}^{*}
  21. T ϵ ϵ T_{\epsilon}\in\mathcal{M}_{\epsilon}
  22. 0 \mathcal{M}_{0}
  23. h ϵ * ( T ϵ ) = T 0 + ϵ h ϵ * ( X T ϵ ) + O ( ϵ 2 ) h_{\epsilon}^{*}(T_{\epsilon})=T_{0}+\epsilon\,h_{\epsilon}^{*}(\mathcal{L}_{X% }T_{\epsilon})+O(\epsilon^{2})
  24. δ T = ϵ h ϵ * ( X T ϵ ) ϵ ( X T ϵ ) 0 \delta T=\epsilon h_{\epsilon}^{*}(\mathcal{L}_{X}T_{\epsilon})\equiv\epsilon(% \mathcal{L}_{X}T_{\epsilon})_{0}
  25. T T
  26. X X
  27. Δ δ T = ϵ ( X T ϵ ) 0 - ϵ ( Y T ϵ ) 0 = ϵ ( X - Y T ϵ ) 0 \Delta\delta T=\epsilon(\mathcal{L}_{X}T_{\epsilon})_{0}-\epsilon(\mathcal{L}_% {Y}T_{\epsilon})_{0}=\epsilon(\mathcal{L}_{X-Y}T_{\epsilon})_{0}
  28. X a = ( ξ μ , 1 ) X^{a}=(\xi^{\mu},1)
  29. Y a = ( 0 , 1 ) Y^{a}=(0,1)
  30. X a - Y a = ( ξ μ , 0 ) X^{a}-Y^{a}=(\xi^{\mu},0)
  31. ϵ \mathcal{M}_{\epsilon}
  32. Δ δ T = ϵ ξ T 0 . \Delta\delta T=\epsilon\mathcal{L}_{\xi}T_{0}.\,

Stiles–Crawford_effect.html

  1. η ( d ) = a m o u n t o f l i g h t e n t e r i n g t h r o u g h t h e c e n t e r o f t h e p u p i l t o p r o d u c e a c e r t a i n r e s p o n s e a m o u n t o f l i g h t e n t e r i n g a t a d i s t a n c e d a w a y f r o m t h e c e n t e r t o p r o d u c e t h e s a m e r e s p o n s e \eta\,\!(d)=\frac{amount\ of\ light\ entering\ through\ the\ center\ of\ the\ % pupil\ to\ produce\ a\ certain\ response}{amount\ of\ light\ entering\ at\ a\ % distance\ d\ away\ from\ the\ center\ to\ produce\ the\ same\ response}
  2. η ( d ) = η ( d m ) 10 - p ( λ ) ( d - d m ) 2 \eta\,\!(d)=\eta\,\!(d_{m})10^{-p(\lambda\,\!)(d-d_{m})^{2}}

Stinespring_factorization_theorem.html

  1. Φ : A B ( H ) , \Phi:A\to B(H),
  2. π : A B ( K ) \pi:A\to B(K)
  3. Φ ( a ) = V π ( a ) V , \Phi(a)=V^{\ast}\pi(a)V,
  4. V : H K V:H\to K
  5. Φ ( 1 ) = V 2 . \|\Phi(1)\|=\|V\|^{2}.
  6. Φ \Phi
  7. V ( ) V * V(\cdot)V^{*}
  8. K = A H K=A\otimes H
  9. a h , b g K a\otimes h,b\otimes g\in K
  10. a h , b g K := Φ ( b * a ) h , g H = h , Φ ( a * b ) g H \langle a\otimes h,b\otimes g\rangle_{K}:=\langle\Phi(b^{*}a)h,g\rangle_{H}=% \langle h,\Phi(a^{*}b)g\rangle_{H}
  11. Φ \Phi
  12. Φ \Phi
  13. K = { x K | x , x K = 0 } K K^{\prime}=\{x\in K|\langle x,x\rangle_{K}=0\}\subset K
  14. K / K K/K^{\prime}
  15. K K
  16. π ( a ) ( b g ) = a b g \pi(a)(b\otimes g)=ab\otimes g
  17. V h = 1 A h Vh=1_{A}\otimes h
  18. π \pi
  19. V V
  20. V V
  21. V ( a h ) = Φ ( a ) h V^{\ast}(a\otimes h)=\Phi(a)h
  22. V V = Φ ( 1 ) V^{\ast}V=\Phi(1)
  23. V V
  24. Φ ( 1 ) = 1 \Phi(1)=1
  25. V V^{\ast}
  26. Φ ( a ) = P H π ( a ) | H . \Phi(a)=P_{H}\;\pi(a)|_{H}.
  27. Φ ( a ) \Phi(a)
  28. π ( a ) \pi(a)
  29. π 1 ( a ) = π ( a ) | K 1 . \pi_{1}(a)=\pi(a)|_{K_{1}}.
  30. π 1 ( a ) π 1 ( b ) = π ( a ) | K 1 π ( b ) | K 1 = π ( a ) π ( b ) | K 1 = π ( a b ) | K 1 = π 1 ( a b ) \pi_{1}(a)\pi_{1}(b)=\pi(a)|_{K_{1}}\pi(b)|_{K_{1}}=\pi(a)\pi(b)|_{K_{1}}=\pi(% ab)|_{K_{1}}=\pi_{1}(ab)
  31. π 1 ( a * ) k , l = π ( a * ) k , l = π ( a ) * k , l = k , π ( a ) l = k , π 1 ( a ) l = π 1 ( a ) * k , l . \langle\pi_{1}(a^{*})k,l\rangle=\langle\pi(a^{*})k,l\rangle=\langle\pi(a)^{*}k% ,l\rangle=\langle k,\pi(a)l\rangle=\langle k,\pi_{1}(a)l\rangle=\langle\pi_{1}% (a)^{*}k,l\rangle.
  32. W π 1 ( a ) V 1 h = π 2 ( a ) V 2 h . \;W\pi_{1}(a)V_{1}h=\pi_{2}(a)V_{2}h.
  33. W π 1 = π 2 W . \;W\pi_{1}=\pi_{2}W.
  34. V * : H K V^{*}:H\to K
  35. V * 1 = ξ V^{*}1=\xi
  36. ξ K \xi\in K
  37. Φ ( a ) = V π ( a ) V * = V π ( a ) V * 1 , 1 H = π ( a ) V * 1 , V * 1 K = π ( a ) ξ , ξ K \Phi(a)=V\pi(a)V^{*}=\langle V\pi(a)V^{*}1,1\rangle_{H}=\langle\pi(a)V^{*}1,V^% {*}1\rangle_{K}=\langle\pi(a)\xi,\xi\rangle_{K}
  38. Φ : B ( G ) B ( H ) \Phi:B(G)\to B(H)
  39. Φ ( a ) = i = 1 n m V i a V i * . \Phi(a)=\sum_{i=1}^{nm}V_{i}aV_{i}^{*}.
  40. C n × n C m C^{n\times n}\otimes C^{m}
  41. K = i = 1 n m C i n . K=\oplus_{i=1}^{nm}C_{i}^{n}.
  42. C i n C_{i}^{n}
  43. π ( a ) ( b g ) = a b g \pi(a)(b\otimes g)=ab\otimes g
  44. P i π ( a ) P i = a \;P_{i}\pi(a)P_{i}=a
  45. C i n C_{i}^{n}
  46. V i * = P i V * V_{i}^{*}=P_{i}V^{*}
  47. Φ ( a ) = i = 1 n m ( V P i ) ( P i π ( a ) P i ) ( P i V * ) = i = 1 n m V i a V i * \Phi(a)=\sum_{i=1}^{nm}(VP_{i})(P_{i}\pi(a)P_{i})(P_{i}V^{*})=\sum_{i=1}^{nm}V% _{i}aV_{i}^{*}
  48. { V i } \{V_{i}\}
  49. Φ ( a ) = i = 1 n m V i a V i * . \Phi(a)=\sum_{i=1}^{nm}V_{i}aV_{i}^{*}.
  50. i = 1 n m V i ( ) V i * \sum_{i=1}^{nm}V_{i}(\cdot)V_{i}^{*}

Stochastic_simulation.html

  1. f X ( x ) = 1 2 π σ 2 e - ( x - μ ) 2 2 σ 2 . f_{X}(x)=\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{(x-\mu)^{2}}{2\sigma^{2}}}.
  2. f ( t ) = Γ ( ν + 1 2 ) ν π Γ ( ν 2 ) ( 1 + t 2 ν ) - ν + 1 2 , f(t)=\frac{\Gamma(\frac{\nu+1}{2})}{\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})}\left% (1+\frac{t^{2}}{\nu}\right)^{-\frac{\nu+1}{2}},\!
  3. ν \nu
  4. Γ \Gamma

Stochastic_volatility.html

  1. d S t = μ S t d t + σ S t d W t dS_{t}=\mu S_{t}\,dt+\sigma S_{t}\,dW_{t}\,
  2. μ \mu\,
  3. S t S_{t}\,
  4. σ \sigma\,
  5. d W t dW_{t}\,
  6. S t = S 0 e ( μ - 1 2 σ 2 ) t + σ W t S_{t}=S_{0}e^{(\mu-\frac{1}{2}\sigma^{2})t+\sigma W_{t}}
  7. σ \sigma\,
  8. S t S_{t}\,
  9. t i t_{i}\,
  10. σ ^ 2 = ( 1 n i = 1 n ( ln S t i - ln S t i - 1 ) 2 t i - t i - 1 ) - 1 n ( ln S t n - ln S t 0 ) 2 t n - t 0 = 1 n i = 1 n ( t i - t i - 1 ) ( ln S t i S t i - 1 t i - t i - 1 - ln S t n S t 0 t n - t 0 ) 2 ; \begin{aligned}\displaystyle\hat{\sigma}^{2}&\displaystyle=\left(\frac{1}{n}% \sum_{i=1}^{n}\frac{(\ln S_{t_{i}}-\ln S_{t_{i-1}})^{2}}{t_{i}-t_{i-1}}\right)% -\frac{1}{n}\frac{(\ln S_{t_{n}}-\ln S_{t_{0}})^{2}}{t_{n}-t_{0}}\\ &\displaystyle=\frac{1}{n}\sum_{i=1}^{n}(t_{i}-t_{i-1})\left(\frac{\ln\frac{S_% {t_{i}}}{S_{t_{i-1}}}}{t_{i}-t_{i-1}}-\frac{\ln\frac{S_{t_{n}}}{S_{t_{0}}}}{t_% {n}-t_{0}}\right)^{2};\end{aligned}
  11. E [ σ 2 ] = n n - 1 σ ^ 2 E\left[\sigma^{2}\right]=\frac{n}{n-1}\hat{\sigma}^{2}
  12. σ \sigma\,
  13. σ \sigma\,
  14. ν t \nu_{t}\,
  15. S t S_{t}\,
  16. ν t \nu_{t}\,
  17. d S t = μ S t d t + ν t S t d W t dS_{t}=\mu S_{t}\,dt+\sqrt{\nu_{t}}S_{t}\,dW_{t}\,
  18. d ν t = α S , t d t + β S , t d B t d\nu_{t}=\alpha_{S,t}\,dt+\beta_{S,t}\,dB_{t}\,
  19. α S , t \alpha_{S,t}\,
  20. β S , t \beta_{S,t}\,
  21. ν \nu\,
  22. d B t dB_{t}\,
  23. d W t dW_{t}\,
  24. ρ \rho\,
  25. d ν t = θ ( ω - ν t ) d t + ξ ν t d B t d\nu_{t}=\theta(\omega-\nu_{t})dt+\xi\sqrt{\nu_{t}}\,dB_{t}\,
  26. ω \omega
  27. θ \theta
  28. ξ \xi
  29. d B t dB_{t}
  30. d W t dW_{t}
  31. d t \sqrt{dt}
  32. d W t dW_{t}
  33. d B t dB_{t}
  34. ρ \rho
  35. ω \omega
  36. θ \theta
  37. ρ \rho
  38. d S t = μ S t d t + σ S t γ d W t dS_{t}=\mu S_{t}\,dt+\sigma S_{t}^{\gamma}\,dW_{t}
  39. γ > 1 \gamma>1
  40. γ < 1 \gamma<1
  41. F F
  42. σ \sigma
  43. d F t = σ t F t β d W t , dF_{t}=\sigma_{t}F^{\beta}_{t}\,dW_{t},
  44. d σ t = α σ t d Z t , d\sigma_{t}=\alpha\sigma_{t}\,dZ_{t},
  45. F 0 F_{0}
  46. σ 0 \sigma_{0}
  47. W t W_{t}
  48. Z t Z_{t}
  49. - 1 < ρ < 1 -1<\rho<1
  50. β , α \beta,\;\alpha
  51. 0 β 1 , α 0 0\leq\beta\leq 1,\;\alpha\geq 0
  52. d ν t = θ ( ω - ν t ) d t + ξ ν t d B t d\nu_{t}=\theta(\omega-\nu_{t})\,dt+\xi\nu_{t}\,dB_{t}\,
  53. ν t 3 / 2 \nu_{t}^{3/2}
  54. d ν t = ν t ( ω - θ ν t ) d t + ξ ν t 3 2 d B t . d\nu_{t}=\nu_{t}(\omega-\theta\nu_{t})\,dt+\xi\nu_{t}^{\frac{3}{2}}\,dB_{t}.\,
  55. θ ν t \theta\nu_{t}
  56. ξ ν t \xi\nu_{t}
  57. d r t = ( θ t - α t ) d t + r t σ t , d W t , dr_{t}=(\theta_{t}-\alpha_{t})\,dt+\sqrt{r_{t}}\,\sigma_{t},\,dW_{t},
  58. d α t = ( ζ t - α t ) d t + α t σ t d W t , d\alpha_{t}=(\zeta_{t}-\alpha_{t})\,dt+\sqrt{\alpha_{t}}\,\sigma_{t}\,dW_{t},
  59. d σ t = ( β t - σ t ) d t + σ t η t d W t . d\sigma_{t}=(\beta_{t}-\sigma_{t})\,dt+\sqrt{\sigma_{t}}\,\eta_{t}\,dW_{t}.
  60. Ψ 0 = { ω , θ , ξ , ρ } \Psi_{0}=\{\omega,\theta,\xi,\rho\}\,
  61. Ψ 0 \Psi_{0}\,
  62. Ψ \Psi\,

Stolarsky_mean.html

  1. S p ( x , y ) \displaystyle S_{p}(x,y)
  2. f f
  3. ( x , f ( x ) ) (x,f(x))
  4. ( y , f ( y ) ) (y,f(y))
  5. ξ \xi
  6. [ x , y ] [x,y]
  7. ξ [ x , y ] f ( ξ ) = f ( x ) - f ( y ) x - y \exists\xi\in[x,y]\ f^{\prime}(\xi)=\frac{f(x)-f(y)}{x-y}
  8. ξ = f - 1 ( f ( x ) - f ( y ) x - y ) \xi=f^{\prime-1}\left(\frac{f(x)-f(y)}{x-y}\right)
  9. f ( x ) = x p f(x)=x^{p}
  10. lim p - S p ( x , y ) \lim_{p\to-\infty}S_{p}(x,y)
  11. S - 1 ( x , y ) S_{-1}(x,y)
  12. lim p 0 S p ( x , y ) \lim_{p\to 0}S_{p}(x,y)
  13. f ( x ) = ln x f(x)=\ln x
  14. S 1 2 ( x , y ) S_{\frac{1}{2}}(x,y)
  15. 1 2 \frac{1}{2}
  16. lim p 1 S p ( x , y ) \lim_{p\to 1}S_{p}(x,y)
  17. f ( x ) = x ln x f(x)=x\cdot\ln x
  18. S 2 ( x , y ) S_{2}(x,y)
  19. S 3 ( x , y ) = Q M ( x , y , G M ( x , y ) ) S_{3}(x,y)=QM(x,y,GM(x,y))
  20. lim p S p ( x , y ) \lim_{p\to\infty}S_{p}(x,y)
  21. S p ( x 0 , , x n ) = f ( n ) - 1 ( n ! f [ x 0 , , x n ] ) S_{p}(x_{0},\dots,x_{n})={f^{(n)}}^{-1}(n!\cdot f[x_{0},\dots,x_{n}])
  22. f ( x ) = x p f(x)=x^{p}

Stopped_process.html

  1. ( Ω , , ) (\Omega,\mathcal{F},\mathbb{P})
  2. ( 𝕏 , 𝒜 ) (\mathbb{X},\mathcal{A})
  3. X : [ 0 , + ) × Ω 𝕏 X:[0,+\infty)\times\Omega\to\mathbb{X}
  4. τ : Ω [ 0 , + ] \tau:\Omega\to[0,+\infty]
  5. { t | t 0 } \{\mathcal{F}_{t}|t\geq 0\}
  6. {}\mathcal{F}
  7. X τ X^{\tau}
  8. t 0 t\geq 0
  9. ω Ω \omega\in\Omega
  10. X t τ ( ω ) := X min { t , τ ( ω ) } ( ω ) . X_{t}^{\tau}(\omega):=X_{\min\{t,\tau(\omega)\}}(\omega).
  11. τ ( ω ) := inf { t 0 | Y t ( ω ) = 0 } \tau(\omega):=\inf\{t\geq 0|Y_{t}(\omega)=0\}
  12. B : [ 0 , + ) × Ω B:[0,+\infty)\times\Omega\to\mathbb{R}
  13. T > 0 T>0
  14. τ ( ω ) T \tau(\omega)\equiv T
  15. B τ B^{\tau}
  16. T T
  17. B t τ ( ω ) B T ( ω ) B_{t}^{\tau}(\omega)\equiv B_{T}(\omega)
  18. t T t\geq T
  19. τ \tau
  20. { x | x a } \{x\in\mathbb{R}|x\geq a\}
  21. τ ( ω ) := inf { t > 0 | B t ( ω ) a } . \tau(\omega):=\inf\{t>0|B_{t}(\omega)\geq a\}.
  22. B τ B^{\tau}
  23. τ \tau
  24. a a
  25. B t τ ( ω ) a B_{t}^{\tau}(\omega)\equiv a
  26. t τ ( ω ) t\geq\tau(\omega)

Strahler_number.html

  1. n i n i + 1 \frac{n_{i}}{n_{i+1}}

Strain_energy_density_function.html

  1. W = W ^ ( s y m b o l C ) = W ^ ( s y m b o l F T \cdotsymbol F ) = W ¯ ( s y m b o l F ) = W ¯ ( s y m b o l B 1 / 2 \cdotsymbol R ) = W ~ ( s y m b o l B , s y m b o l R ) W=\hat{W}(symbol{C})=\hat{W}(symbol{F}^{T}\cdotsymbol{F})=\bar{W}(symbol{F})=% \bar{W}(symbol{B}^{1/2}\cdotsymbol{R})=\tilde{W}(symbol{B},symbol{R})
  2. W = W ^ ( s y m b o l C ) = W ^ ( s y m b o l R T \cdotsymbol B \cdotsymbol R ) = W ~ ( s y m b o l B , s y m b o l R ) W=\hat{W}(symbol{C})=\hat{W}(symbol{R}^{T}\cdotsymbol{B}\cdotsymbol{R})=\tilde% {W}(symbol{B},symbol{R})
  3. s y m b o l F symbol{F}
  4. s y m b o l C symbol{C}
  5. s y m b o l B symbol{B}
  6. s y m b o l R symbol{R}
  7. s y m b o l F symbol{F}
  8. W ^ ( s y m b o l C ) \hat{W}(symbol{C})
  9. W ~ ( s y m b o l B , s y m b o l R ) \tilde{W}(symbol{B},symbol{R})
  10. s y m b o l R symbol{R}
  11. s y m b o l C symbol{C}
  12. s y m b o l B symbol{B}
  13. W = W ^ ( λ 1 , λ 2 , λ 3 ) = W ~ ( I 1 , I 2 , I 3 ) = W ¯ ( I ¯ 1 , I ¯ 2 , J ) = U ( I 1 c , I 2 c , I 3 c ) W=\hat{W}(\lambda_{1},\lambda_{2},\lambda_{3})=\tilde{W}(I_{1},I_{2},I_{3})=% \bar{W}(\bar{I}_{1},\bar{I}_{2},J)=U(I_{1}^{c},I_{2}^{c},I_{3}^{c})
  14. I ¯ 1 = J - 2 / 3 I 1 ; I 1 = λ 1 2 + λ 2 2 + λ 3 2 ; J = det ( s y m b o l F ) I ¯ 2 = J - 4 / 3 I 2 ; I 2 = λ 1 2 λ 2 2 + λ 2 2 λ 3 2 + λ 3 2 λ 1 2 \begin{aligned}\displaystyle\bar{I}_{1}&\displaystyle=J^{-2/3}~{}I_{1}~{};~{}~% {}I_{1}=\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}~{};~{}~{}J=\det(symbol% {F})\\ \displaystyle\bar{I}_{2}&\displaystyle=J^{-4/3}~{}I_{2}~{};~{}~{}I_{2}=\lambda% _{1}^{2}\lambda_{2}^{2}+\lambda_{2}^{2}\lambda_{3}^{2}+\lambda_{3}^{2}\lambda_% {1}^{2}\end{aligned}
  15. W = 1 2 i = 1 3 j = 1 3 σ i j ϵ i j = 1 2 ( σ x ϵ x + σ y ϵ y + σ z ϵ z + τ x y γ x y + τ y z γ y z + τ x z γ x z ) W=\frac{1}{2}\sum_{i=1}^{3}\sum_{j=1}^{3}\sigma_{ij}\epsilon_{ij}=\frac{1}{2}(% \sigma_{x}\epsilon_{x}+\sigma_{y}\epsilon_{y}+\sigma_{z}\epsilon_{z}+\tau_{xy}% \gamma_{xy}+\tau_{yz}\gamma_{yz}+\tau_{xz}\gamma_{xz})
  16. W W
  17. ψ \psi
  18. W = ρ 0 ψ . W=\rho_{0}\psi\;.
  19. u u
  20. W = ρ 0 u . W=\rho_{0}u\;.

Strange_B_meson.html

  1. B r ( B s μ + μ - ) = 2.8 - 0.6 + 0.7 × 10 - 9 Br(B_{s}\rightarrow\mu^{+}\mu^{-})=2.8^{+0.7}_{-0.6}\times 10^{-9}

Streamline_diffusion.html

  1. u = 0 \nabla\cdot{u}=0
  2. || u || = 1 ||{u}||=1
  3. ψ t + u ψ = 0. \frac{\partial\psi}{\partial t}+{u}\cdot\nabla\psi=0.
  4. D 2 ψ D\nabla^{2}\psi
  5. ψ t + u ψ + D 2 ψ = 0 \frac{\partial\psi}{\partial t}+{u}\cdot\nabla\psi+D\nabla^{2}\psi=0
  6. ψ t + u ψ + u ( u D 2 ψ ) + ( D 2 ψ - u ( u D 2 ψ ) ) = 0 \frac{\partial\psi}{\partial t}+{u}\cdot\nabla\psi+{u}({u}\cdot D\nabla^{2}% \psi)+(D\nabla^{2}\psi-{u}({u}\cdot D\nabla^{2}\psi))=0
  7. u ( u D 2 ψ ) {u}({u}\cdot D\nabla^{2}\psi)
  8. ( D 2 ψ - u ( u D 2 ψ ) ) (D\nabla^{2}\psi-{u}({u}\cdot D\nabla^{2}\psi))

Stress_relaxation.html

  1. σ 0 \sigma_{0}
  2. t 0 t_{0}
  3. t f t_{f}
  4. t f t_{f}
  5. σ ( t ) = σ 0 1 - [ 1 - ( t / t * ) ( 1 1 - n ) ] \sigma(t)=\frac{\sigma_{0}}{1-[1-(t/t*)(1^{1-n})]}
  6. σ 0 \sigma_{0}
  7. σ ( t ) = m n A m n [ ln ( 1 + t ) ] m ( ϵ 0 ) n \sigma(t)=\sum_{mn}{A_{mn}[\ln(1+t)]^{m}(\epsilon^{\prime}_{0})^{n}}
  8. σ ( t ) = 1 b * l o g 10 α ( t - t n ) + 1 10 α ( t - t n ) - 1 \sigma(t)=\frac{1}{b}*log{\frac{10^{\alpha}(t-t_{n})+1}{10^{\alpha}(t-t_{n})-1}}
  9. α \alpha
  10. t n t_{n}

Stress–energy–momentum_pseudotensor.html

  1. t L L μ ν t_{LL}^{\mu\nu}\,
  2. t L L μ ν = t L L ν μ t_{LL}^{\mu\nu}=t_{LL}^{\nu\mu}\,
  3. T μ ν T^{\mu\nu}\,
  4. t L L μ ν = - c 4 8 π G G μ ν + c 4 16 π G ( - g ) ( ( - g ) ( g μ ν g α β - g μ α g ν β ) ) , α β t_{LL}^{\mu\nu}=-\frac{c^{4}}{8\pi G}G^{\mu\nu}+\frac{c^{4}}{16\pi G(-g)}((-g)% (g^{\mu\nu}g^{\alpha\beta}-g^{\mu\alpha}g^{\nu\beta}))_{,\alpha\beta}
  5. , α β = 2 x α x β ,_{\alpha\beta}=\frac{\partial^{2}}{\partial x^{\alpha}\partial x^{\beta}}\,
  6. G μ ν G^{\mu\nu}\,
  7. t L L μ ν t_{LL}^{\mu\nu}
  8. G μ ν G^{\mu\nu}\,
  9. t L L μ ν t_{LL}^{\mu\nu}
  10. T μ ν T^{\mu\nu}\,
  11. ( ( - g ) ( T μ ν + t L L μ ν ) ) , μ = 0 ((-g)(T^{\mu\nu}+t_{LL}^{\mu\nu}))_{,\mu}=0
  12. G μ ν G^{\mu\nu}\,
  13. T μ ν T^{\mu\nu}\,
  14. G μ ν G^{\mu\nu}\,
  15. t L L μ ν = 0 t_{LL}^{\mu\nu}=0
  16. Λ \Lambda\,
  17. Λ \Lambda\,
  18. t L L μ ν = - c 4 8 π G ( G μ ν + Λ g μ ν ) + c 4 16 π G ( - g ) ( ( - g ) ( g μ ν g α β - g μ α g ν β ) ) , α β t_{LL}^{\mu\nu}=-\frac{c^{4}}{8\pi G}(G^{\mu\nu}+\Lambda g^{\mu\nu})+\frac{c^{% 4}}{16\pi G(-g)}((-g)(g^{\mu\nu}g^{\alpha\beta}-g^{\mu\alpha}g^{\nu\beta}))_{,% \alpha\beta}
  19. ( - g ) ( t L L μ ν + c 4 Λ g μ ν 8 π G ) = c 4 16 π G ( ( - g g μ ν ) , α ( - g g α β ) , β - (-g)(t_{LL}^{\mu\nu}+\frac{c^{4}\Lambda g^{\mu\nu}}{8\pi G})=\frac{c^{4}}{16% \pi G}((\sqrt{-g}g^{\mu\nu}),_{\alpha}(\sqrt{-g}g^{\alpha\beta}),_{\beta}-
  20. - ( - g g μ α ) , α ( - g g ν β ) , β + 1 2 g μ ν g α β ( - g g α σ ) , ρ ( - g g ρ β ) , σ - -(\sqrt{-g}g^{\mu\alpha}),_{\alpha}(\sqrt{-g}g^{\nu\beta}),_{\beta}+\frac{1}{2% }g^{\mu\nu}g_{\alpha\beta}(\sqrt{-g}g^{\alpha\sigma}),_{\rho}(\sqrt{-g}g^{\rho% \beta}),_{\sigma}-
  21. - ( g μ α g β σ ( - g g ν σ ) , ρ ( - g g β ρ ) , α + g ν α g β σ ( - g g μ σ ) , ρ ( - g g β ρ ) , α ) + -(g^{\mu\alpha}g_{\beta\sigma}(\sqrt{-g}g^{\nu\sigma}),_{\rho}(\sqrt{-g}g^{% \beta\rho}),_{\alpha}+g^{\nu\alpha}g_{\beta\sigma}(\sqrt{-g}g^{\mu\sigma}),_{% \rho}(\sqrt{-g}g^{\beta\rho}),_{\alpha})+
  22. + g α β g σ ρ ( - g g μ α ) , σ ( - g g ν β ) , ρ + +g_{\alpha\beta}g^{\sigma\rho}(\sqrt{-g}g^{\mu\alpha}),_{\sigma}(\sqrt{-g}g^{% \nu\beta}),_{\rho}+\,
  23. + 1 8 ( 2 g μ α g ν β - g μ ν g α β ) ( 2 g σ ρ g λ ω - g ρ λ g σ ω ) ( - g g σ ω ) , α ( - g g ρ λ ) , β ) +\frac{1}{8}(2g^{\mu\alpha}g^{\nu\beta}-g^{\mu\nu}g^{\alpha\beta})(2g_{\sigma% \rho}g_{\lambda\omega}-g_{\rho\lambda}g_{\sigma\omega})(\sqrt{-g}g^{\sigma% \omega}),_{\alpha}(\sqrt{-g}g^{\rho\lambda}),_{\beta})
  24. t L L μ ν + c 4 Λ g μ ν 8 π G = c 4 16 π G ( ( 2 Γ α β σ Γ σ ρ ρ - Γ α ρ σ Γ β σ ρ - Γ α σ σ Γ β ρ ρ ) ( g μ α g ν β - g μ ν g α β ) + t_{LL}^{\mu\nu}+\frac{c^{4}\Lambda g^{\mu\nu}}{8\pi G}=\frac{c^{4}}{16\pi G}((% 2\Gamma^{\sigma}_{\alpha\beta}\Gamma^{\rho}_{\sigma\rho}-\Gamma^{\sigma}_{% \alpha\rho}\Gamma^{\rho}_{\beta\sigma}-\Gamma^{\sigma}_{\alpha\sigma}\Gamma^{% \rho}_{\beta\rho})(g^{\mu\alpha}g^{\nu\beta}-g^{\mu\nu}g^{\alpha\beta})+
  25. + g μ α g β σ ( Γ α ρ ν Γ β σ ρ + Γ β σ ν Γ α ρ ρ - Γ σ ρ ν Γ α β ρ - Γ α β ν Γ σ ρ ρ ) + +g^{\mu\alpha}g^{\beta\sigma}(\Gamma^{\nu}_{\alpha\rho}\Gamma^{\rho}_{\beta% \sigma}+\Gamma^{\nu}_{\beta\sigma}\Gamma^{\rho}_{\alpha\rho}-\Gamma^{\nu}_{% \sigma\rho}\Gamma^{\rho}_{\alpha\beta}-\Gamma^{\nu}_{\alpha\beta}\Gamma^{\rho}% _{\sigma\rho})+
  26. + g ν α g β σ ( Γ α ρ μ Γ β σ ρ + Γ β σ μ Γ α ρ ρ - Γ σ ρ μ Γ α β ρ - Γ α β μ Γ σ ρ ρ ) + +g^{\nu\alpha}g^{\beta\sigma}(\Gamma^{\mu}_{\alpha\rho}\Gamma^{\rho}_{\beta% \sigma}+\Gamma^{\mu}_{\beta\sigma}\Gamma^{\rho}_{\alpha\rho}-\Gamma^{\mu}_{% \sigma\rho}\Gamma^{\rho}_{\alpha\beta}-\Gamma^{\mu}_{\alpha\beta}\Gamma^{\rho}% _{\sigma\rho})+
  27. + g α β g σ ρ ( Γ α σ μ Γ β ρ ν - Γ α β μ Γ σ ρ ν ) ) +g^{\alpha\beta}g^{\sigma\rho}(\Gamma^{\mu}_{\alpha\sigma}\Gamma^{\nu}_{\beta% \rho}-\Gamma^{\mu}_{\alpha\beta}\Gamma^{\nu}_{\sigma\rho}))
  28. t μ ν = c 4 16 π G - g ( ( g α β - g ) , μ ( Γ α β ν - δ β ν Γ α σ σ ) - δ μ ν g α β ( Γ α β σ Γ σ ρ ρ - Γ α σ ρ Γ β ρ σ ) - g ) {t_{\mu}}^{\nu}=\frac{c^{4}}{16\pi G\sqrt{-g}}((g^{\alpha\beta}\sqrt{-g})_{,% \mu}(\Gamma^{\nu}_{\alpha\beta}-\delta^{\nu}_{\beta}\Gamma^{\sigma}_{\alpha% \sigma})-\delta_{\mu}^{\nu}g^{\alpha\beta}(\Gamma^{\sigma}_{\alpha\beta}\Gamma% ^{\rho}_{\sigma\rho}-\Gamma^{\rho}_{\alpha\sigma}\Gamma^{\sigma}_{\beta\rho})% \sqrt{-g})
  29. ( ( T μ ν + t μ ν ) - g ) , ν = 0. (({T_{\mu}}^{\nu}+{t_{\mu}}^{\nu})\sqrt{-g})_{,\nu}=0.

Stress–strain_index.html

  1. S S I = i = 0 n r i 2 a ( C D N D ) r m a x SSI=\sum_{i=0}^{n}{{r_{i}^{2}a({{CD}\over{ND}})}\over{r_{max}}}

Stretched_exponential_function.html

  1. f β ( t ) = e - t β f_{\beta}(t)=e^{-t^{\beta}}
  2. τ 0 d t e - ( t / τ K ) β = τ K β Γ ( 1 β ) \langle\tau\rangle\equiv\int_{0}^{\infty}dt\,e^{-\left({t/\tau_{K}}\right)^{% \beta}}={\tau_{K}\over\beta}\Gamma({1\over\beta})
  3. τ n 0 d t t n - 1 e - ( t / τ K ) β = τ K n β Γ ( n β ) \langle\tau^{n}\rangle\equiv\int_{0}^{\infty}dt\,t^{n-1}\,e^{-\left({t/\tau_{K% }}\right)^{\beta}}={{\tau_{K}}^{n}\over\beta}\Gamma({n\over\beta})
  4. e - t β = 0 d u ρ ( u ) e - t / u e^{-t^{\beta}}=\int_{0}^{\infty}du\,\rho(u)\,e^{-t/u}
  5. G = u ρ ( u ) G=u\rho(u)\,
  6. ρ ( u ) = - 1 π u k = 0 ( - 1 ) k k ! sin ( π β k ) Γ ( β k + 1 ) u β k \rho(u)=-{1\over{\pi u}}\sum\limits_{k=0}^{\infty}{{(-1)^{k}}\over{k!}}\sin(% \pi\beta k)\Gamma(\beta k+1)u^{\beta k}
  7. G G
  8. t / τ t/\tau
  9. τ n = Γ ( n ) 0 d τ t n ρ ( τ ) \langle\tau^{n}\rangle=\Gamma(n)\int_{0}^{\infty}d\tau\,t^{n}\,\rho(\tau)
  10. ln τ = ( 1 - 1 β ) Eu + ln τ K \langle\ln\tau\rangle=\left(1-{1\over\beta}\right){\rm Eu}+\ln\tau_{K}
  11. p ( τ λ , β ) d τ = λ Γ ( 1 + β - 1 ) e - ( τ λ ) β d τ p(\tau\mid\lambda,\beta)~{}d\tau=\frac{\lambda}{\Gamma(1+\beta^{-1})}~{}e^{-(% \tau\lambda)^{\beta}}~{}d\tau
  12. f β ( t ) = e - t β ( t ) f_{\beta}(t)=e^{-t^{\beta(t)}}

Strict_differentiability.html

  1. lim ( x , y ) ( a , a ) f ( x ) - f ( y ) x - y \lim_{(x,y)\to(a,a)}\frac{f(x)-f(y)}{x-y}
  2. ( x , y ) ( a , a ) (x,y)\to(a,a)
  3. 𝐑 2 \mathbf{R}^{2}
  4. x y x\neq y
  5. f ( x ) = x 2 sin 1 x , f ( 0 ) = 0 , x n = 1 ( n + 1 2 ) π , y n = x n + 1 f(x)=x^{2}\sin\tfrac{1}{x},\ f(0)=0,~{}x_{n}=\tfrac{1}{(n+\frac{1}{2})\pi},\ y% _{n}=x_{n+1}
  6. C 1 ( I ) C^{1}(I)
  7. f ( x ) - f ( y ) = L ( x - y ) + o ( ( x , y ) - ( a , a ) ) f(x)-f(y)=L(x-y)+o((x,y)-(a,a))
  8. o ( ) o(\cdot)
  9. F ( x ) = { p 2 if x p ( mod p 3 ) p 4 if x p 2 ( mod p 5 ) p 6 if x p 3 ( mod p 7 ) 0 otherwise . F(x)=\begin{cases}p^{2}&\mbox{if }~{}x\equiv p\;\;(\mathop{{\rm mod}}p^{3})\\ p^{4}&\mbox{if }~{}x\equiv p^{2}\;\;(\mathop{{\rm mod}}p^{5})\\ p^{6}&\mbox{if }~{}x\equiv p^{3}\;\;(\mathop{{\rm mod}}p^{7})\\ \vdots&\vdots\\ 0&\mbox{otherwise}~{}.\end{cases}
  10. lim h 0 F ( x + h ) - F ( x ) h = 0. \lim_{h\to 0}\frac{F(x+h)-F(x)}{h}=0.
  11. F ( y ) - F ( x ) y - x \frac{F(y)-F(x)}{y-x}
  12. F ( y ) - F ( x ) y - x = p 2 n - 0 p n - ( p n - p 2 n ) = 1 , \frac{F(y)-F(x)}{y-x}=\frac{p^{2n}-0}{p^{n}-(p^{n}-p^{2n})}=1,
  13. lim ( x , y ) ( a , a ) F ( y ) - F ( x ) y - x \lim_{(x,y)\to(a,a)}\frac{F(y)-F(x)}{y-x}

Strictly_positive_measure.html

  1. U T s.t. U , μ ( U ) > 0. \forall U\in T\mbox{ s.t. }~{}U\neq\emptyset,\mu(U)>0.

Strongly_positive_bilinear_form.html

  1. a ( u , u ) c u 2 a(u,u)\geq c\cdot\|u\|^{2}
  2. u V u\in V
  3. \|\cdot\|

Structural_dynamics.html

  1. D A F = u m a x u s t a t i c DAF=\frac{{u_{max}}}{{u_{static}}}
  2. M x ¨ + k x = F ( t ) M{\ddot{x}}+kx=F(t)
  3. x ¨ \ddot{x}
  4. x = < m t p l > F 0 k [ 1 - c o s ( ω t ) ] x=\frac{<}{m}tpl>{{F_{0}}}{{k}}[1-cos{(\omega t)}]
  5. ω = < m t p l > k M \omega=\sqrt{\frac{<}{m}tpl>{{k}}{{M}}}
  6. f = ω 2 π f=\frac{\omega}{2\pi}
  7. x s t a t i c = < m t p l > F 0 k x_{static}=\frac{<}{m}tpl>{{F_{0}}}{{k}}
  8. x = x s t a t i c [ 1 - c o s ( ω t ) ] x=x_{static}[1-cos(\omega t)]
  9. D A F = 1 + e - c π DAF=1+e^{-c\pi}
  10. c = Damping Coefficient Critical Damping Coefficient c=\frac{{\text{Damping Coefficient}}}{{\,\text{Critical Damping Coefficient}}}
  11. u ¯ ( x ) \bar{u}(x)
  12. Equivalent mass, M e q = M u ¯ 2 d u \,\text{Equivalent mass, }M_{eq}=\int{M\bar{u}^{2}}du
  13. Equivalent stiffness, k e q = E I ( d 2 u ¯ < m t p l > d x 2 ) 2 d x \,\text{Equivalent stiffness, }k_{eq}=\int{EI\bigg(\frac{{d^{2}\bar{u}}}{<}% mtpl>{{dx^{2}}}\bigg)^{2}}dx
  14. Equivalent force, F e q = F u ¯ d x \,\text{Equivalent force, }F_{eq}=\int{F\bar{u}}dx
  15. ω = k e q M e q \omega=\sqrt{\frac{{k_{eq}}}{{M_{eq}}}}
  16. v ( x , t ) = u n ( x , t ) v(x,t)=\sum{u_{n}(x,t)}
  17. u s t a t i c = F 1 , e q k 1 , e q u_{static}=\frac{F_{1,eq}}{k_{1,eq}}
  18. u m a x = u s t a t i c D A F u_{max}=u_{static}DAF
  19. = M n u ¯ n M n u ¯ n 2 =\frac{\sum{M_{n}\bar{u}_{n}}}{\sum{M_{n}\bar{u}_{n}^{2}}}

Structure_tensor.html

  1. I I
  2. S w ( p ) = [ w ( r ) ( I x ( p - r ) ) 2 d r w ( r ) I x ( p - r ) I y ( p - r ) d r w ( r ) I x ( p - r ) I y ( p - r ) d r w ( r ) ( I y ( p - r ) ) 2 d r ] S_{w}(p)=\begin{bmatrix}\int w(r)(I_{x}(p-r))^{2}\,dr&\int w(r)I_{x}(p-r)I_{y}% (p-r)\,dr\\ \int w(r)I_{x}(p-r)I_{y}(p-r)\,dr&\int w(r)(I_{y}(p-r))^{2}\,dr\end{bmatrix}
  3. I x I_{x}
  4. I y I_{y}
  5. I I
  6. 2 \mathbb{R}^{2}
  7. S w S_{w}
  8. S w ( p ) = w ( r ) S 0 ( p - r ) d r S_{w}(p)=\int w(r)S_{0}(p-r)\,dr
  9. S 0 S_{0}
  10. S 0 ( p ) = [ ( I x ( p ) ) 2 I x ( p ) I y ( p ) I x ( p ) I y ( p ) ( I y ( p ) ) 2 ] S_{0}(p)=\begin{bmatrix}(I_{x}(p))^{2}&I_{x}(p)I_{y}(p)\\ I_{x}(p)I_{y}(p)&(I_{y}(p))^{2}\end{bmatrix}
  11. I = ( I x , I y ) \nabla I=(I_{x},I_{y})
  12. I I
  13. S 0 S_{0}
  14. ( I ) ( I ) (\nabla I)^{\prime}(\nabla I)
  15. ( I ) (\nabla I)^{\prime}
  16. S w ( p ) S_{w}(p)
  17. I I
  18. I [ p ] I[p]
  19. S w [ p ] = [ r w [ r ] ( I x [ p - r ] ) 2 r w [ r ] I x [ p - r ] I y [ p - r ] r w [ r ] I x [ p - r ] I y [ p - r ] r w [ r ] ( I y [ p - r ] ) 2 ] S_{w}[p]=\begin{bmatrix}\sum_{r}w[r](I_{x}[p-r])^{2}&\sum_{r}w[r]I_{x}[p-r]I_{% y}[p-r]\\ \sum_{r}w[r]I_{x}[p-r]I_{y}[p-r]&\sum_{r}w[r](I_{y}[p-r])^{2}\end{bmatrix}
  20. { - m . . + m } × { - m . . + m } \{-m..+m\}\times\{-m..+m\}
  21. I x [ p ] , I y [ p ] I_{x}[p],I_{y}[p]
  22. I I
  23. S w [ p ] = r w [ r ] S 0 [ p - r ] S_{w}[p]=\sum_{r}w[r]S_{0}[p-r]
  24. S 0 S_{0}
  25. S 0 [ p ] = [ ( I x [ p ] ) 2 I x [ p ] I y [ p ] I x [ p ] I y [ p ] ( I y [ p ] ) 2 ] S_{0}[p]=\begin{bmatrix}(I_{x}[p])^{2}&I_{x}[p]I_{y}[p]\\ I_{x}[p]I_{y}[p]&(I_{y}[p])^{2}\end{bmatrix}
  26. S w S_{w}
  27. λ 1 , λ 2 \lambda_{1},\lambda_{2}
  28. λ 1 λ 2 0 \lambda_{1}\geq\lambda_{2}\geq 0
  29. e 1 , e 2 e_{1},e_{2}
  30. I = ( I x , I y ) \nabla I=(I_{x},I_{y})
  31. I I
  32. w w
  33. p p
  34. λ 1 > λ 2 \lambda_{1}>\lambda_{2}
  35. e 1 e_{1}
  36. - e 1 -e_{1}
  37. λ 1 > 0 , λ 2 = 0 \lambda_{1}>0,\lambda_{2}=0
  38. e 1 e_{1}
  39. I I
  40. e 1 e_{1}
  41. e 2 e_{2}
  42. λ 1 = λ 2 \lambda_{1}=\lambda_{2}
  43. λ 1 = λ 2 = 0 \lambda_{1}=\lambda_{2}=0
  44. I I
  45. I = ( 0 , 0 ) \nabla I=(0,0)
  46. W W
  47. λ k \lambda_{k}
  48. w w
  49. I I
  50. e k e_{k}
  51. S w S_{w}
  52. c w = ( λ 1 - λ 2 λ 1 + λ 2 ) 2 c_{w}=\left(\frac{\lambda_{1}-\lambda_{2}}{\lambda_{1}+\lambda_{2}}\right)^{2}
  53. λ 2 > 0 \lambda_{2}>0
  54. λ 1 = λ 2 = 0 \lambda_{1}=\lambda_{2}=0
  55. I \nabla I
  56. w w
  57. I I
  58. S w ( p ) = w ( r ) S 0 ( p - r ) d r S_{w}(p)=\int w(r)S_{0}(p-r)\,dr
  59. S 0 ( p ) = [ ( I x ( p ) ) 2 I x ( p ) I y ( p ) I x ( p ) I z ( p ) I x ( p ) I y ( p ) ( I y ( p ) ) 2 I y ( p ) I z ( p ) I x ( p ) I z ( p ) I y ( p ) I z ( p ) ( I z ( p ) ) 2 ] S_{0}(p)=\begin{bmatrix}(I_{x}(p))^{2}&I_{x}(p)I_{y}(p)&I_{x}(p)I_{z}(p)\\ I_{x}(p)I_{y}(p)&(I_{y}(p))^{2}&I_{y}(p)I_{z}(p)\\ I_{x}(p)I_{z}(p)&I_{y}(p)I_{z}(p)&(I_{z}(p))^{2}\end{bmatrix}
  60. I x , I y , I z I_{x},I_{y},I_{z}
  61. I I
  62. 3 \mathbb{R}^{3}
  63. S w [ p ] = r w [ r ] S 0 [ p - r ] S_{w}[p]=\sum_{r}w[r]S_{0}[p-r]
  64. S 0 [ p ] = [ ( I x [ p ] ) 2 I x [ p ] I y [ p ] I x [ p ] I z [ p ] I x [ p ] I y [ p ] ( I y [ p ] ) 2 I y [ p ] I z [ p ] I x [ p ] I z [ p ] I y [ p ] I z [ p ] ( I z [ p ] ) 2 ] S_{0}[p]=\begin{bmatrix}(I_{x}[p])^{2}&I_{x}[p]I_{y}[p]&I_{x}[p]I_{z}[p]\\ I_{x}[p]I_{y}[p]&(I_{y}[p])^{2}&I_{y}[p]I_{z}[p]\\ I_{x}[p]I_{z}[p]&I_{y}[p]I_{z}[p]&(I_{z}[p])^{2}\end{bmatrix}
  65. { - m . . + m } × { - m . . + m } × { - m . . + m } \{-m..+m\}\times\{-m..+m\}\times\{-m..+m\}
  66. λ 1 , λ 2 , λ 3 \lambda_{1},\lambda_{2},\lambda_{3}
  67. S w [ p ] S_{w}[p]
  68. e 1 , e 2 , e 3 e_{1},e_{2},e_{3}
  69. w w
  70. λ 1 \lambda_{1}
  71. λ 2 \lambda_{2}
  72. λ 3 \lambda_{3}
  73. e 1 e_{1}
  74. I I
  75. λ 3 \lambda_{3}
  76. λ 1 \lambda_{1}
  77. λ 2 \lambda_{2}
  78. e 3 e_{3}
  79. λ 1 λ 2 λ 3 \lambda_{1}\approx\lambda_{2}\approx\lambda_{3}
  80. I I
  81. I I
  82. I I
  83. t t
  84. ( I ) ( x ; t ) (\nabla I)(x;t)
  85. s s
  86. w ( ξ ; s ) w(\xi;s)
  87. ( I ) ( I ) T (\nabla I)(\nabla I)^{T}
  88. I I
  89. k \mathbb{R}^{k}
  90. t > 0 t>0
  91. I ( x ; t ) I(x;t)
  92. I ( x ; t ) = h ( x ; t ) * I ( x ) I(x;t)=h(x;t)*I(x)
  93. h ( x ; t ) h(x;t)
  94. ( I ) ( x ; t ) (\nabla I)(x;t)
  95. μ ( x ; t , s ) = ξ k ( I ) ( x - ξ ; t ) ( I ) T ( x - ξ ; t ) w ( ξ ; s ) d ξ \mu(x;t,s)=\int_{\xi\in\mathbb{R}^{k}}(\nabla I)(x-\xi;t)\,(\nabla I)^{T}(x-% \xi;t)\,w(\xi;s)\,d\xi
  96. h ( x ; t ) h(x;t)
  97. w ( ξ ; s ) w(\xi;s)
  98. t t
  99. s s
  100. t t
  101. s s
  102. t t
  103. r 1 r\geq 1
  104. s = r t s=rt
  105. r r
  106. r 1 r\geq 1
  107. g ( x ; t ) g(x;t)
  108. T ( n ; t ) T(n;t)
  109. μ ( x ; t , s ) = n k ( I ) ( x - n ; t ) ( I ) T ( x - n ; t ) w ( n ; s ) \mu(x;t,s)=\sum_{n\in\mathbb{Z}^{k}}(\nabla I)(x-n;t)\,(\nabla I)^{T}(x-n;t)\,% w(n;s)
  110. t t
  111. s s
  112. α i \alpha^{i}
  113. λ 2 \lambda_{2}
  114. v = ( v x , v y ) T v=(v_{x},v_{y})^{T}
  115. [ x y t ] = G [ x y t ] = G [ x - v x t y - v y t t ] \begin{bmatrix}x^{\prime}\\ y^{\prime}\\ t^{\prime}\end{bmatrix}=G\begin{bmatrix}x\\ y\\ t\end{bmatrix}=G\begin{bmatrix}x-v_{x}\,t\\ y-v_{y}\,t\\ t\end{bmatrix}
  116. S S
  117. S = R s p a c e - T G - T S G - 1 R s p a c e - 1 = [ ν 1 ν 2 ν 3 ] S^{\prime}=R_{space}^{-T}\,G^{-T}\,S\,G^{-1}\,R_{space}^{-1}=\begin{bmatrix}% \nu_{1}&&\\ &\nu_{2}&\\ &&\nu_{3}\end{bmatrix}
  118. G G
  119. R s p a c e R_{space}
  120. S ′′ = R s p a c e - t i m e - T S R s p a c e - t i m e - 1 = [ λ 1 λ 2 λ 3 ] S^{\prime\prime}=R_{space-time}^{-T}\,S\,R_{space-time}^{-1}=\begin{bmatrix}% \lambda_{1}&&\\ &\lambda_{2}&\\ &&\lambda_{3}\end{bmatrix}