wpmath0000012_12

Single-entry_matrix.html

  1. 𝐉 23 = [ 0 0 0 0 0 1 0 0 0 ] . \mathbf{J}^{23}=\left[\begin{matrix}0&0&0\\ 0&0&1\\ 0&0&0\end{matrix}\right].
  2. 𝐉 23 𝐀 = [ 0 0 0 a 31 a 32 a 33 0 0 0 ] \mathbf{J}^{23}\mathbf{A}=\left[\begin{matrix}0&0&0\\ a_{31}&a_{32}&a_{33}\\ 0&0&0\end{matrix}\right]
  3. 𝐀𝐉 23 = [ 0 0 a 12 0 0 a 22 0 0 a 32 ] \mathbf{A}\mathbf{J}^{23}=\left[\begin{matrix}0&0&a_{12}\\ 0&0&a_{22}\\ 0&0&a_{32}\end{matrix}\right]

Singular_isothermal_sphere_profile.html

  1. ρ ( r ) = σ V 2 2 π G r 2 \rho(r)=\frac{\sigma_{V}^{2}}{2\pi Gr^{2}}

Sinuosity.html

  1. S = π 2 1.5708... S=\tfrac{\pi}{2}\approx 1.5708...
  2. S = 1 n π 0 n π 1 + ( cos x ) 2 d x 1.216... S=\textstyle\tfrac{1}{n\pi}\int_{0}^{n\pi}\sqrt{1+(\cos x)^{2}}dx\approx 1.216% ...
  3. π 6 \frac{\pi}{6}
  4. π 3 ( 6 - 2 ) \frac{\pi}{3(\sqrt{6}-\sqrt{2})}
  5. π 3 \frac{\pi}{3}
  6. π 3 \frac{\pi}{3}
  7. π 2 \frac{\pi}{2}
  8. π 2 2 \frac{\pi}{2\sqrt{2}}
  9. 2 π 3 \frac{2\cdot\pi}{3}
  10. 2 π 3 3 \frac{2\cdot\pi}{3\sqrt{3}}
  11. 5 π 6 \frac{5\cdot\pi}{6}
  12. 5 π 3 ( 6 + 2 ) \frac{5\cdot\pi}{3(\sqrt{6}+\sqrt{2})}
  13. π \pi
  14. π 2 \frac{\pi}{2}
  15. 7 π 6 \frac{7\cdot\pi}{6}
  16. 7 π 3 ( 6 + 2 ) \frac{7\cdot\pi}{3(\sqrt{6}+\sqrt{2})}
  17. 4 π 3 \frac{4\cdot\pi}{3}
  18. 4 π 3 3 \frac{4\cdot\pi}{3\sqrt{3}}
  19. 3 π 2 \frac{3\cdot\pi}{2}
  20. 3 π 2 2 \frac{3\cdot\pi}{2\sqrt{2}}
  21. 5 π 3 \frac{5\cdot\pi}{3}
  22. 5 π 3 \frac{5\cdot\pi}{3}
  23. 11 π 6 \frac{11\cdot\pi}{6}
  24. 11 π 3 ( 6 - 2 ) \frac{11\cdot\pi}{3(\sqrt{6}-\sqrt{2})}
  25. SI = channel length downvalley length \,\text{SI}=\frac{{\,\text{channel length}}}{{\,\text{downvalley length}}}
  26. π \pi

Sinusoidal_model.html

  1. Y i = C + α sin ( ω T i + ϕ ) + E i Y_{i}=C+\alpha\sin(\omega T_{i}+\phi)+E_{i}
  2. Y i = ( B 0 + B 1 T i ) + α sin ( 2 π ω T i + ϕ ) + E i Y_{i}=(B_{0}+B_{1}T_{i})+\alpha\sin(2\pi\omega T_{i}+\phi)+E_{i}
  3. Y i = ( B 0 + B 1 T i + B 2 T i 2 ) + α sin ( 2 π ω T i + ϕ ) + E i Y_{i}=(B_{0}+B_{1}T_{i}+B_{2}T_{i}^{2})+\alpha\sin(2\pi\omega T_{i}+\phi)+E_{i}
  4. Y i = C + ( B 0 + B 1 T i ) sin ( 2 π ω T i + ϕ ) + E i Y_{i}=C+(B_{0}+B_{1}T_{i})\sin(2\pi\omega T_{i}+\phi)+E_{i}

Sinusoidal_spiral.html

  1. r n = a n cos ( n θ ) r^{n}=a^{n}\cos(n\theta)\,
  2. r n = a n sin ( n θ ) . r^{n}=a^{n}\sin(n\theta).\,
  3. r n = a n cos ( n θ ) r^{n}=a^{n}\cos(n\theta)\,
  4. d r d θ cos n θ + r sin n θ = 0 \frac{dr}{d\theta}\cos n\theta+r\sin n\theta=0
  5. ( d r d s , r d θ d s ) cos n θ d s d θ = ( - r sin n θ , r cos n θ ) = r ( - sin n θ , cos n θ ) \left(\frac{dr}{ds},\ r\frac{d\theta}{ds}\right)\cos n\theta\frac{ds}{d\theta}% =\left(-r\sin n\theta,\ r\cos n\theta\right)=r\left(-\sin n\theta,\ \cos n% \theta\right)
  6. ψ = n θ ± π / 2 \psi=n\theta\pm\pi/2
  7. φ = ( n + 1 ) θ ± π / 2 \varphi=(n+1)\theta\pm\pi/2
  8. ( d r d s , r d θ d s ) \left(\frac{dr}{ds},\ r\frac{d\theta}{ds}\right)
  9. d s d θ = r cos - 1 n θ = a cos - 1 + 1 n n θ \frac{ds}{d\theta}=r\cos^{-1}n\theta=a\cos^{-1+\tfrac{1}{n}}n\theta
  10. n > 0 n>0
  11. a - π 2 n π 2 n cos - 1 + 1 n n θ d θ a\int_{-\tfrac{\pi}{2n}}^{\tfrac{\pi}{2n}}\cos^{-1+\tfrac{1}{n}}n\theta\ d\theta
  12. d φ d s = ( n + 1 ) d θ d s = n + 1 a cos 1 - 1 n n θ \frac{d\varphi}{ds}=(n+1)\frac{d\theta}{ds}=\frac{n+1}{a}\cos^{1-\tfrac{1}{n}}n\theta

Sion's_minimax_theorem.html

  1. X X
  2. Y Y
  3. f f
  4. X × Y X\times Y
  5. f ( x , ) f(x,\cdot)
  6. Y Y
  7. x X \forall x\in X
  8. f ( , y ) f(\cdot,y)
  9. X X
  10. y Y \forall y\in Y
  11. min x X sup y Y f ( x , y ) = sup y Y min x X f ( x , y ) . \min_{x\in X}\sup_{y\in Y}f(x,y)=\sup_{y\in Y}\min_{x\in X}f(x,y).

Size_function.html

  1. x < y x<y
  2. ( M , φ ) : Δ + = { ( x , y ) 2 : x < y } \ell_{(M,\varphi)}:\Delta^{+}=\{(x,y)\in\mathbb{R}^{2}:x<y\}\to\mathbb{N}
  3. ( M , φ : M ) (M,\varphi:M\to\mathbb{R})
  4. ( x , y ) Δ + (x,y)\in\Delta^{+}
  5. ( M , φ ) ( x , y ) \ell_{(M,\varphi)}(x,y)
  6. { p M : φ ( p ) y } \{p\in M:\varphi(p)\leq y\}
  7. M M
  8. k \mathbb{R}^{k}
  9. φ \varphi
  10. x x
  11. φ : M k \varphi:M\to\mathbb{R}^{k}
  12. k \mathbb{R}^{k}
  13. M M
  14. C 1 C^{1}
  15. C C^{\infty}
  16. M M
  17. C 0 C^{0}
  18. φ \varphi
  19. γ M \gamma\in M
  20. M M
  21. k k
  22. M M
  23. d ( ( P 1 , , P k ) , ( Q 1 , Q k ) ) = max 1 i k P i - Q i d((P_{1},\ldots,P_{k}),(Q_{1}\ldots,Q_{k}))=\max_{1\leq i\leq k}\|P_{i}-Q_{i}\|
  24. k \mathbb{R}^{k}
  25. 0
  26. Δ + \Delta^{+}
  27. M M
  28. ( M , φ ) ( x , y ) \ell_{(M,\varphi)}(x,y)
  29. x x
  30. y y
  31. ( M , φ ) ( x , y ) \ell_{(M,\varphi)}(x,y)
  32. x < y x<y
  33. ( M , φ ) ( x , y ) \ell_{(M,\varphi)}(x,y)
  34. x < min φ x<\min\varphi
  35. y > x y>x
  36. ( M , φ ) ( x , y ) = 0 \ell_{(M,\varphi)}(x,y)=0
  37. y max φ y\geq\max\varphi
  38. x < y x<y
  39. ( M , φ ) ( x , y ) \ell_{(M,\varphi)}(x,y)
  40. M M
  41. φ \varphi
  42. x x
  43. M M
  44. φ \varphi
  45. C 1 C^{1}
  46. ( x , y ) (x,y)
  47. ( M , φ ) \ell_{(M,\varphi)}
  48. x x
  49. y y
  50. φ \varphi
  51. d ( ( M , φ ) , ( N , ψ ) ) d((M,\varphi),(N,\psi))
  52. ( M , φ ) , ( N , ψ ) (M,\varphi),\ (N,\psi)
  53. ( N , ψ ) ( x ¯ , y ¯ ) > ( M , φ ) ( x ~ , y ~ ) \ell_{(N,\psi)}(\bar{x},\bar{y})>\ell_{(M,\varphi)}(\tilde{x},\tilde{y})
  54. d ( ( M , φ ) , ( N , ψ ) ) min { x ~ - x ¯ , y ¯ - y ~ } d((M,\varphi),(N,\psi))\geq\min\{\tilde{x}-\bar{x},\bar{y}-\tilde{y}\}
  55. p = ( x , y ) p=(x,y)
  56. x < y x<y
  57. μ ( p ) m < t p l > def = min α > 0 , β > 0 ( M , φ ) ( x + α , y - β ) - ( M , φ ) ( x + α , y + β ) - ( M , φ ) ( x - α , y - β ) + ( M , φ ) ( x - α , y + β ) \mu(p){\stackrel{<}{m}tpl>{{\rm def}}{=}}\min_{\alpha>0,\beta>0}\ell_{({M},% \varphi)}(x+\alpha,y-\beta)-\ell_{({M},\varphi)}(x+\alpha,y+\beta)-\ell_{({M},% \varphi)}(x-\alpha,y-\beta)+\ell_{({M},\varphi)}(x-\alpha,y+\beta)
  58. μ ( p ) \mu(p)
  59. p p
  60. r : x = k r:x=k
  61. μ ( r ) = def min α > 0 , k + α < y ( M , φ ) ( k + α , y ) - ( M , φ ) ( k - α , y ) > 0. \mu(r){\stackrel{\rm def}{=}}\min_{\alpha>0,k+\alpha<y}\ell_{({M},\varphi)}(k+% \alpha,y)-\ell_{({M},\varphi)}(k-\alpha,y)>0.
  62. μ ( r ) \mu(r)
  63. r r
  64. x ¯ < y ¯ {\bar{x}}<{\bar{y}}
  65. ( M , φ ) ( x ¯ , y ¯ ) = p = ( x , y ) x x ¯ , y > y ¯ μ ( p ) + r : x = k k x ¯ μ ( r ) \ell_{({M},\varphi)}({\bar{x}},{\bar{y}})=\sum_{p=(x,y)\atop x\leq{\bar{x}},y>% \bar{y}}\mu\big(p\big)+\sum_{r:x=k\atop k\leq{\bar{x}}}\mu\big(r\big)

Size_functor.html

  1. ( M , f ) (M,f)
  2. M M
  3. n n
  4. f f
  5. i i
  6. i = 0 , , n i=0,\ldots,n
  7. F i F_{i}
  8. F u n ( Rord , Ab ) Fun(\mathrm{Rord},\mathrm{Ab})
  9. Rord \mathrm{Rord}
  10. Ab \mathrm{Ab}
  11. x y x\leq y
  12. M x = { p M : f ( p ) x } M_{x}=\{p\in M:f(p)\leq x\}
  13. M y = { p M : f ( p ) y } M_{y}=\{p\in M:f(p)\leq y\}
  14. j x y j_{xy}
  15. M x M_{x}
  16. M y M_{y}
  17. k x y k_{xy}
  18. Rord \mathrm{Rord}
  19. x x
  20. y y
  21. x \R x\in\R
  22. F i ( x ) = H i ( M x ) ; F_{i}(x)=H_{i}(M_{x});
  23. F i ( k x y ) = H i ( j x y ) . F_{i}(k_{xy})=H_{i}(j_{xy}).
  24. M M
  25. f f
  26. F 0 F_{0}
  27. H 0 H_{0}
  28. ( M , f ) ( x , y ) \ell_{(M,f)}(x,y)
  29. H 0 ( j x y ) : H 0 ( M x ) H 0 ( M y ) H_{0}(j_{xy}):H_{0}(M_{x})\rightarrow H_{0}(M_{y})
  30. i i
  31. F i ( k x y ) = H i ( j x y ) : H i ( M x ) H i ( M y ) F_{i}(k_{xy})=H_{i}(j_{xy}):H_{i}(M_{x})\rightarrow H_{i}(M_{y})

Size_homotopy_group.html

  1. ( M , φ ) (M,\varphi)
  2. M M
  3. C 0 C^{0}
  4. φ : M k \varphi:M\to\mathbb{R}^{k}
  5. \preceq
  6. k \mathbb{R}^{k}
  7. ( x 1 , , x k ) ( y 1 , , y k ) (x_{1},\ldots,x_{k})\preceq(y_{1},\ldots,y_{k})
  8. x 1 y 1 , , x k y k x_{1}\leq y_{1},\ldots,x_{k}\leq y_{k}
  9. Y k Y\in\mathbb{R}^{k}
  10. M Y = { Z k : Z Y } M_{Y}=\{Z\in\mathbb{R}^{k}:Z\preceq Y\}
  11. P M X P\in M_{X}
  12. X Y X\preceq Y
  13. α \alpha
  14. β \beta
  15. P P
  16. P P
  17. α \alpha
  18. β \beta
  19. P P
  20. M Y M_{Y}
  21. α Y β \alpha\approx_{Y}\beta
  22. ( M , φ ) (M,\varphi)
  23. ( X , Y ) (X,Y)
  24. P P
  25. P P
  26. M X M_{X}
  27. Y \approx_{Y}
  28. ( M , φ ) (M,\varphi)
  29. ( X , Y ) (X,Y)
  30. P P
  31. h X Y ( π 1 ( M X , P ) ) h_{XY}(\pi_{1}(M_{X},P))
  32. π 1 ( M X , P ) \pi_{1}(M_{X},P)
  33. P P
  34. M X M_{X}
  35. h X Y h_{XY}
  36. M X M_{X}
  37. M Y M_{Y}
  38. n n
  39. P P
  40. α : S n M \alpha:S^{n}\to M
  41. S n S^{n}
  42. P P

Size_pair.html

  1. ( M , φ ) (M,\varphi)
  2. M M
  3. φ : M k \varphi:M\to\mathbb{R}^{k}
  4. φ \varphi
  5. M M

Sketch_(mathematics).html

  1. M : D C M:D\rightarrow C

Skew-symmetric_graph.html

  1. u v \scriptstyle u\lor v
  2. ( ¬ u ) v \scriptstyle(\lnot u)\Rightarrow v
  3. ( ¬ v ) u \scriptstyle(\lnot v)\Rightarrow u

Skew_normal_distribution.html

  1. Φ ( x - ξ ω ) - 2 T ( x - ξ ω , α ) \Phi\left(\frac{x-\xi}{\omega}\right)-2T\left(\frac{x-\xi}{\omega},\alpha\right)
  2. T ( h , a ) T(h,a)
  3. ξ + ω δ 2 π \xi+\omega\delta\sqrt{\frac{2}{\pi}}
  4. δ = α 1 + α 2 \delta=\frac{\alpha}{\sqrt{1+\alpha^{2}}}
  5. ω 2 ( 1 - 2 δ 2 π ) \omega^{2}\left(1-\frac{2\delta^{2}}{\pi}\right)
  6. γ 1 = 4 - π 2 ( δ 2 / π ) 3 ( 1 - 2 δ 2 / π ) 3 / 2 \gamma_{1}=\frac{4-\pi}{2}\frac{\left(\delta\sqrt{2/\pi}\right)^{3}}{\left(1-2% \delta^{2}/\pi\right)^{3/2}}
  7. 2 ( π - 3 ) ( δ 2 / π ) 4 ( 1 - 2 δ 2 / π ) 2 2(\pi-3)\frac{\left(\delta\sqrt{2/\pi}\right)^{4}}{\left(1-2\delta^{2}/\pi% \right)^{2}}
  8. M X ( t ) = 2 exp ( ξ t + ω 2 t 2 2 ) Φ ( ω δ t ) M_{X}\left(t\right)=2\exp\left(\xi t+\frac{\omega^{2}t^{2}}{2}\right)\Phi\left% (\omega\delta t\right)
  9. M X ( i δ ω t ) M_{X}\left(i\delta\omega t\right)
  10. ϕ ( x ) \phi(x)
  11. ϕ ( x ) = 1 2 π e - x 2 2 \phi(x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{x^{2}}{2}}
  12. Φ ( x ) = - x ϕ ( t ) d t = 1 2 [ 1 + erf ( x 2 ) ] \Phi(x)=\int_{-\infty}^{x}\phi(t)\ dt=\frac{1}{2}\left[1+\operatorname{erf}% \left(\frac{x}{\sqrt{2}}\right)\right]
  13. α \alpha
  14. f ( x ) = 2 ϕ ( x ) Φ ( α x ) . f(x)=2\phi(x)\Phi(\alpha x).\,
  15. x x - ξ ω x\rightarrow\frac{x-\xi}{\omega}
  16. α = 0 \alpha=0
  17. α \alpha
  18. α > 0 \alpha>0
  19. α < 0 \alpha<0
  20. ξ \xi
  21. ω \omega
  22. α \alpha
  23. f ( x ) = 2 ω ϕ ( x - ξ ω ) Φ ( α ( x - ξ ω ) ) . f(x)=\frac{2}{\omega}\phi\left(\frac{x-\xi}{\omega}\right)\Phi\left(\alpha% \left(\frac{x-\xi}{\omega}\right)\right).\,
  24. ( - 1 , 1 ) (-1,1)
  25. ξ \xi
  26. ω \omega
  27. α \alpha
  28. α = 0 \alpha=0
  29. α \alpha
  30. | δ | = π 2 | γ ^ 3 | 2 3 | γ ^ 3 | 2 3 + ( ( 4 - π ) / 2 ) 2 3 |\delta|=\sqrt{\frac{\pi}{2}\frac{|\hat{\gamma}_{3}|^{\frac{2}{3}}}{|\hat{% \gamma}_{3}|^{\frac{2}{3}}+((4-\pi)/2)^{\frac{2}{3}}}}
  31. δ = α 1 + α 2 \delta=\frac{\alpha}{\sqrt{1+\alpha^{2}}}
  32. γ ^ 3 \hat{\gamma}_{3}
  33. δ \delta
  34. γ ^ 3 \hat{\gamma}_{3}
  35. α ^ = δ / 1 - δ 2 \hat{\alpha}=\delta/\sqrt{1-\delta^{2}}
  36. δ = 1 {\delta=1}
  37. γ 3 0.9952717 \gamma_{3}\approx 0.9952717
  38. α \alpha
  39. | γ ^ 3 | = min ( 0.99 , | ( 1 / n ) ( ( x i - x ¯ ) / s ) 3 | ) |\hat{\gamma}_{3}|=\min(0.99,|(1/n)\sum{((x_{i}-\bar{x})/s)^{3}}|)
  40. ω 4 f ′′ ( x ) + ( α 2 + 2 ) ω 2 ( x - ζ ) f ( x ) + f ( x ) ( ( α 2 + 1 ) ( x - ζ ) 2 + ω 2 ) = 0 \omega^{4}f^{\prime\prime}(x)+\left(\alpha^{2}+2\right)\omega^{2}(x-\zeta)f^{% \prime}(x)+f(x)\left(\left(\alpha^{2}+1\right)(x-\zeta)^{2}+\omega^{2}\right)=0
  41. f ( 0 ) = exp ( - ζ 2 2 ω 2 ) erfc ( α ζ 2 ω ) 2 π ω and f ( 0 ) = exp ( - ( α 2 + 1 ) ζ 2 2 ω 2 ) ( 2 α ω + 2 π ζ exp ( α 2 ζ 2 2 ω 2 ) erfc ( α ζ 2 ω ) ) 2 π ω 3 . \begin{array}[]{l}\displaystyle f(0)=\frac{\exp\left(-\frac{\zeta^{2}}{2\omega% ^{2}}\right)\operatorname{erfc}\left(\frac{\alpha\zeta}{\sqrt{2}\omega}\right)% }{\sqrt{2\pi}\omega}\,\text{ and}\\ \displaystyle f^{\prime}(0)=\frac{\exp\left(-\frac{\left(\alpha^{2}+1\right)% \zeta^{2}}{2\omega^{2}}\right)\left(2\alpha\omega+\sqrt{2\pi}\zeta\exp\left(% \frac{\alpha^{2}\zeta^{2}}{2\omega^{2}}\right)\operatorname{erfc}\left(\frac{% \alpha\zeta}{\sqrt{2}\omega}\right)\right)}{2\pi\omega^{3}}.\end{array}

Ski_rental_problem.html

  1. p i = { ( b - 1 b ) b - i 1 b ( 1 - ( 1 - ( 1 / b ) ) b ) i b 0 i > b , p_{i}=\left\{\begin{array}[]{ll}(\frac{b-1}{b})^{b-i}\frac{1}{b(1-(1-(1/b))^{b% })}&i\leq b\\ 0&i>b\end{array}\right.,
  2. \approx

Skorokhod_integral.html

  1. 𝐄 [ X ] := Ω X ( ω ) d 𝐏 ( ω ) . \mathbf{E}[X]:=\int_{\Omega}X(\omega)\,\mathrm{d}\mathbf{P}(\omega).
  2. 𝐄 [ W ( h ) ] = 0 , \mathbf{E}[W(h)]=0,
  3. 𝐄 [ W ( g ) W ( h ) ] = g , h H , \mathbf{E}[W(g)W(h)]=\langle g,h\rangle_{H},
  4. F = f ( W ( h 1 ) , , W ( h n ) ) , F=f(W(h_{1}),\ldots,W(h_{n})),
  5. D F := i = 1 n f x i ( W ( h 1 ) , , W ( h n ) ) h i . \mathrm{D}F:=\sum_{i=1}^{n}\frac{\partial f}{\partial x_{i}}(W(h_{1}),\ldots,W% (h_{n}))h_{i}.
  6. F 1 , p := ( 𝐄 [ | F | p ] + 𝐄 [ D F H p ] ) 1 / p . \|F\|_{1,p}:=\big(\mathbf{E}[|F|^{p}]+\mathbf{E}[\|\mathrm{D}F\|_{H}^{p}]\big)% ^{1/p}.
  7. | 𝐄 [ D F , u H ] | C ( u ) F L 2 ( Ω ) . \big|\mathbf{E}[\langle\mathrm{D}F,u\rangle_{H}]\big|\leq C(u)\|F\|_{L^{2}(% \Omega)}.
  8. 𝐄 [ F δ u ] = 𝐄 [ D F , u H ] . \mathbf{E}[F\,\delta u]=\mathbf{E}[\langle\mathrm{D}F,u\rangle_{H}].
  9. u = j = 1 n F j h j u=\sum_{j=1}^{n}F_{j}h_{j}
  10. δ u = j = 1 n ( F j W ( h j ) - D F j , h j H ) . \delta u=\sum_{j=1}^{n}\left(F_{j}W(h_{j})-\langle\mathrm{D}F_{j},h_{j}\rangle% _{H}\right).
  11. 𝐄 [ ( δ u ) 2 ] = 𝐄 [ u H 2 ] + 𝐄 [ D u H H 2 ] . \mathbf{E}\big[(\delta u)^{2}\big]=\mathbf{E}\big[\|u\|_{H}^{2}\big]+\mathbf{E% }\big[\|\mathrm{D}u\|_{H\otimes H}^{2}\big].
  12. D h ( δ u ) = u , h H + δ ( D h u ) , \mathrm{D}_{h}(\delta u)=\langle u,h\rangle_{H}+\delta(\mathrm{D}_{h}u),
  13. δ ( F u ) = F δ u - D F , u H . \delta(Fu)=F\,\delta u-\langle\mathrm{D}F,u\rangle_{H}.

Slater–Condon_rules.html

  1. 𝒜 \mathcal{A}
  2. | Ψ = 𝒜 ( ϕ 1 ( 𝐫 1 σ 1 ) ϕ 2 ( 𝐫 2 σ 2 ) ϕ m ( 𝐫 m σ m ) ϕ n ( 𝐫 n σ n ) ϕ N ( 𝐫 N σ N ) ) . |\Psi\rangle=\mathcal{A}(\phi_{1}(\mathbf{r}_{1}\sigma_{1})\phi_{2}(\mathbf{r}% _{2}\sigma_{2})\cdots\phi_{m}(\mathbf{r}_{m}\sigma_{m})\phi_{n}(\mathbf{r}_{n}% \sigma_{n})\cdots\phi_{N}(\mathbf{r}_{N}\sigma_{N})).
  3. | Ψ m p = 𝒜 ( ϕ 1 ( 𝐫 1 σ 1 ) ϕ 2 ( 𝐫 2 σ 2 ) ϕ p ( 𝐫 m σ m ) ϕ n ( 𝐫 n σ n ) ϕ N ( 𝐫 N σ N ) ) , |\Psi_{m}^{p}\rangle=\mathcal{A}(\phi_{1}(\mathbf{r}_{1}\sigma_{1})\phi_{2}(% \mathbf{r}_{2}\sigma_{2})\cdots\phi_{p}(\mathbf{r}_{m}\sigma_{m})\phi_{n}(% \mathbf{r}_{n}\sigma_{n})\cdots\phi_{N}(\mathbf{r}_{N}\sigma_{N})),
  4. | Ψ m n p q = 𝒜 ( ϕ 1 ( 𝐫 1 σ 1 ) ϕ 2 ( 𝐫 2 σ 2 ) ϕ p ( 𝐫 m σ m ) ϕ q ( 𝐫 n σ n ) ϕ N ( 𝐫 N σ N ) ) . |\Psi_{mn}^{pq}\rangle=\mathcal{A}(\phi_{1}(\mathbf{r}_{1}\sigma_{1})\phi_{2}(% \mathbf{r}_{2}\sigma_{2})\cdots\phi_{p}(\mathbf{r}_{m}\sigma_{m})\phi_{q}(% \mathbf{r}_{n}\sigma_{n})\cdots\phi_{N}(\mathbf{r}_{N}\sigma_{N})).
  5. Ψ | O ^ | Ψ , Ψ | O ^ | Ψ m p , and Ψ | O ^ | Ψ m n p q . \langle\Psi|\hat{O}|\Psi\rangle,\langle\Psi|\hat{O}|\Psi_{m}^{p}\rangle,\ % \mathrm{and}\ \langle\Psi|\hat{O}|\Psi_{mn}^{pq}\rangle.
  6. F ^ = i = 1 N f ^ ( i ) . \hat{F}=\sum_{i=1}^{N}\ \hat{f}(i).
  7. Ψ | F ^ | Ψ = i = 1 N ϕ i | f ^ | ϕ i , Ψ | F ^ | Ψ m p = ϕ m | f ^ | ϕ p , Ψ | F ^ | Ψ m n p q = 0. \begin{aligned}\displaystyle\langle\Psi|\hat{F}|\Psi\rangle&\displaystyle=\sum% _{i=1}^{N}\ \langle\phi_{i}|\hat{f}|\phi_{i}\rangle,\\ \displaystyle\langle\Psi|\hat{F}|\Psi_{m}^{p}\rangle&\displaystyle=\langle\phi% _{m}|\hat{f}|\phi_{p}\rangle,\\ \displaystyle\langle\Psi|\hat{F}|\Psi_{mn}^{pq}\rangle&\displaystyle=0.\end{aligned}
  8. G ^ = 1 2 i = 1 N j = 1 j i N g ^ ( i , j ) . \hat{G}=\frac{1}{2}\sum_{i=1}^{N}\sum_{{j=1}\atop{j\neq i}}^{N}\ \hat{g}(i,j).
  9. Ψ | G ^ | Ψ = 1 2 i = 1 N j = 1 j i N ( ϕ i ϕ j | g ^ | ϕ i ϕ j - ϕ i ϕ j | g ^ | ϕ j ϕ i ) , Ψ | G ^ | Ψ m p = i = 1 N ( ϕ m ϕ i | g ^ | ϕ p ϕ i - ϕ m ϕ i | g ^ | ϕ i ϕ p ) , Ψ | G ^ | Ψ m n p q = ϕ m ϕ n | g ^ | ϕ p ϕ q - ϕ m ϕ n | g ^ | ϕ q ϕ p , \begin{aligned}\displaystyle\langle\Psi|\hat{G}|\Psi\rangle&\displaystyle=% \frac{1}{2}\sum_{i=1}^{N}\sum_{j=1\atop{j\neq i}}^{N}\ \bigg(\langle\phi_{i}% \phi_{j}|\hat{g}|\phi_{i}\phi_{j}\rangle-\langle\phi_{i}\phi_{j}|\hat{g}|\phi_% {j}\phi_{i}\rangle\bigg),\\ \displaystyle\langle\Psi|\hat{G}|\Psi_{m}^{p}\rangle&\displaystyle=\sum_{i=1}^% {N}\ \bigg(\langle\phi_{m}\phi_{i}|\hat{g}|\phi_{p}\phi_{i}\rangle-\langle\phi% _{m}\phi_{i}|\hat{g}|\phi_{i}\phi_{p}\rangle\bigg),\\ \displaystyle\langle\Psi|\hat{G}|\Psi_{mn}^{pq}\rangle&\displaystyle=\langle% \phi_{m}\phi_{n}|\hat{g}|\phi_{p}\phi_{q}\rangle-\langle\phi_{m}\phi_{n}|\hat{% g}|\phi_{q}\phi_{p}\rangle,\end{aligned}
  10. ϕ i ϕ j | g ^ | ϕ k ϕ l = d 𝐫 d 𝐫 ϕ i * ( 𝐫 ) ϕ j * ( 𝐫 ) g ( 𝐫 , 𝐫 ) ϕ k ( 𝐫 ) ϕ l ( 𝐫 ) . \langle\phi_{i}\phi_{j}|\hat{g}|\phi_{k}\phi_{l}\rangle=\int\mathrm{d}\mathbf{% r}\int\mathrm{d}\mathbf{r}^{\prime}\ \phi_{i}^{*}(\mathbf{r})\phi_{j}^{*}(% \mathbf{r}^{\prime})g(\mathbf{r},\mathbf{r}^{\prime})\phi_{k}(\mathbf{r})\phi_% {l}(\mathbf{r}^{\prime}).

Slave_boson.html

  1. f f
  2. b b
  3. f f b f^{\dagger}\rightarrow f^{\dagger}b

Sliding_window_based_part-of-speech_tagging.html

  1. Γ = { γ 1 , γ 2 , , γ | Γ | } \Gamma=\{\gamma_{1},\gamma_{2},\ldots,\gamma_{|\Gamma|}\}
  2. W = { w 1 , w 2 , } W=\{w1,w2,\ldots\}
  3. T : W P ( Γ ) T:W\rightarrow P(\Gamma)
  4. w w
  5. T ( w ) Γ T(w)\subseteq\Gamma
  6. Σ = { σ 1 , σ 2 , , σ | Σ | } \Sigma=\{\sigma_{1},\sigma_{2},\ldots,\sigma_{|\Sigma|}\}
  7. W W
  8. σ Σ \sigma\in\Sigma
  9. w Σ σ w\,\Sigma\,\sigma
  10. σ \sigma
  11. Σ \Sigma
  12. w [ 1 ] w [ 2 ] w [ L ] W * w[1]w[2]\ldots w[L]\in W^{*}
  13. w [ t ] w[t]
  14. T ( w [ t ] ) Σ T(w[t])\in\Sigma
  15. σ [ 1 ] σ [ 2 ] σ [ L ] W * \sigma[1]\sigma[2]\ldots\sigma[L]\in W^{*}
  16. γ [ 1 ] γ [ 2 ] γ [ L ] \gamma[1]\gamma[2]\ldots\gamma[L]
  17. γ [ t ] T ( σ [ t ] ) \gamma[t]\in T(\sigma[t])
  18. σ [ 1 ] σ [ 2 ] σ [ L ] \sigma[1]\sigma[2]\ldots\sigma[L]
  19. γ * [ 1 ] γ * [ L ] = arg max γ [ t ] T ( σ [ t ] ) p ( γ [ 1 ] γ [ L ] σ [ 1 ] σ [ L ] ) \gamma^{*}[1]\ldots\gamma^{*}[L]=\operatorname*{arg\,max}\limits_{\gamma[t]\in T% (\sigma[t])}p(\gamma[1]\ldots\gamma[L]\sigma[1]\ldots\sigma[L])
  20. γ * [ 1 ] γ * [ L ] = arg max γ [ t ] T ( σ [ t ] ) p ( γ [ 1 ] γ [ L ] ) p ( σ [ 1 ] σ [ L ] γ [ 1 ] γ [ L ] ) \gamma^{*}[1]\ldots\gamma^{*}[L]=\operatorname*{arg\,max}\limits_{\gamma[t]\in T% (\sigma[t])}p(\gamma[1]\ldots\gamma[L])p(\sigma[1]\ldots\sigma[L]\gamma[1]% \ldots\gamma[L])
  21. p ( γ [ 1 ] γ [ 2 ] γ [ L ] ) p(\gamma[1]\gamma[2]\ldots\gamma[L])
  22. p ( σ [ 1 ] σ [ L ] γ [ 1 ] γ [ L ] ) p(\sigma[1]\dots\sigma[L]\gamma[1]\ldots\gamma[L])
  23. σ [ 1 ] σ [ L ] \sigma[1]\ldots\sigma[L]
  24. p ( γ [ 1 ] γ [ 2 ] γ [ L ] ) = t = 1 t = L p ( γ [ t + 1 ] γ [ t ] ) p(\gamma[1]\gamma[2]\ldots\gamma[L])=\prod_{t=1}^{t=L}p(\gamma[t+1]\gamma[t])
  25. γ [ 0 ] \gamma[0]
  26. γ [ L + 1 ] \gamma[L+1]
  27. p ( σ [ 1 ] σ [ 2 ] σ [ L ] γ [ 1 ] γ [ 2 ] γ [ L ] ) = t = 1 t = L p ( σ [ t ] γ [ t ] ) p(\sigma[1]\sigma[2]\ldots\sigma[L]\gamma[1]\gamma[2]\ldots\gamma[L])=\prod_{t% =1}^{t=L}p(\sigma[t]\gamma[t])
  28. p ( σ [ 1 ] σ [ 2 ] σ [ L ] γ [ 1 ] γ [ 2 ] γ [ L ] ) = t = 1 t = L p ( γ [ t ] C ( - ) [ t ] σ [ t ] C ( + ) [ t ] ) p(\sigma[1]\sigma[2]\ldots\sigma[L]\gamma[1]\gamma[2]\ldots\gamma[L])=\prod_{t% =1}^{t=L}p(\gamma[t]C_{(-)}[t]\sigma[t]C_{(+)}[t])
  29. C ( - ) [ t ] = σ [ t - N ( - ) ] σ [ t - N ( - ) ] σ [ t - 1 ] C_{(-)}[t]=\sigma[t-N_{(-)}]\sigma[t-N_{(-)}]\ldots\sigma[t-1]
  30. N ( + ) N_{(+)}
  31. N ( - ) + N ( + ) + 1 N_{(-)}+N_{(+)}+1
  32. N ( - ) = N ( + ) = 1 N_{(-)}=N_{(+)}=1

Slowly_varying_function.html

  1. lim x L ( a x ) L ( x ) = 1. \lim_{x\to\infty}\frac{L(ax)}{L(x)}=1.
  2. g ( a ) = lim x L ( a x ) L ( x ) g(a)=\lim_{x\to\infty}\frac{L(ax)}{L(x)}
  3. lim x L ( x ) = b ( 0 , ) , \lim_{x\to\infty}L(x)=b\in(0,\infty),
  4. L ( x ) = exp ( η ( x ) + B x ε ( t ) t d t ) L(x)=\exp\left(\eta(x)+\int_{B}^{x}\frac{\varepsilon(t)}{t}\,dt\right)

Small-gain_theorem.html

  1. \mathcal{L}
  2. S 1 S_{1}
  3. S 2 S_{2}
  4. S 1 S 2 < 1 \|S_{1}\|\cdot\|S_{2}\|<1

Small_cancellation_theory.html

  1. G = X | R ( * ) G=\langle X|R\rangle\qquad(*)
  2. G = X | S G=\langle X|S\rangle
  3. G = X | R G=\langle X|R\rangle
  4. G = a , b | a b a - 1 b - 1 G=\langle a,b|aba^{-1}b^{-1}\rangle
  5. G = a 1 , b 1 , , a k , b k | [ a 1 , b 1 ] [ a k , b k ] G=\langle a_{1},b_{1},\dots,a_{k},b_{k}|[a_{1},b_{1}]\cdot\dots\cdot[a_{k},b_{% k}]\rangle
  6. G = a , b | a b a b 2 a b 3 a b 100 G=\langle a,b|abab^{2}ab^{3}\dots ab^{100}\rangle
  7. G = X | r n G=\langle X|r^{n}\rangle
  8. G = X | S G=\langle X|S\rangle
  9. X | R \langle X|R\rangle
  10. X | S \langle X|S\rangle
  11. 1 K G Q 1 1\to K\to G\to Q\to 1
  12. n n\to\infty

Small_Veblen_ordinal.html

  1. ϕ < m t p l > Ω ω ( 0 ) \phi_{<}mtpl>{{\Omega^{\omega}}}(0)
  2. θ ( Ω ω ) \theta(\Omega^{\omega})
  3. ψ ( Ω Ω ω ) \psi(\Omega^{\Omega^{\omega}})

Smith_predictor.html

  1. G ( z ) G(z)
  2. z - k z^{-k}
  3. G ( z ) G(z)
  4. C ( z ) C(z)
  5. H ( z ) = C ( z ) G ( z ) 1 + C ( z ) G ( z ) H(z)=\frac{C(z)G(z)}{1+C(z)G(z)}
  6. C ¯ ( z ) \bar{C}(z)
  7. G ( z ) z - k G(z)z^{-k}
  8. H ¯ ( z ) \bar{H}(z)
  9. H ( z ) z - k H(z)z^{-k}
  10. C ¯ G z - k 1 + C ¯ G z - k = z - k C G 1 + C G \frac{\bar{C}Gz^{-k}}{1+\bar{C}Gz^{-k}}=z^{-k}\frac{CG}{1+CG}
  11. C ¯ = C 1 + C G ( 1 - z - k ) \bar{C}=\frac{C}{1+CG(1-z^{-k})}
  12. G ( z ) G(z)
  13. G ^ ( z ) \hat{G}(z)
  14. G ^ ( z ) z - k \hat{G}(z)z^{-k}
  15. G ( z ) z - k G(z)z^{-k}

Smoothing_spline.html

  1. ( x i , Y i ) ; x 1 < x 2 < < x n , i (x_{i},Y_{i});x_{1}<x_{2}<\dots<x_{n},i\in\mathbb{Z}
  2. Y i = μ ( x i ) Y_{i}=\mu(x_{i})
  3. μ ^ \hat{\mu}
  4. μ \mu
  5. i = 1 n ( Y i - μ ^ ( x i ) ) 2 + λ x 1 x n μ ^ ′′ ( x ) 2 d x . \sum_{i=1}^{n}(Y_{i}-\hat{\mu}(x_{i}))^{2}+\lambda\int_{x_{1}}^{x_{n}}\hat{\mu% }^{\prime\prime}(x)^{2}\,dx.
  6. λ 0 \lambda\geq 0
  7. x i x_{i}
  8. λ 0 \lambda\to 0
  9. λ \lambda\to\infty
  10. x i x_{i}
  11. μ ^ ( x i ) ; i = 1 , , n \hat{\mu}(x_{i});i=1,\ldots,n
  12. μ ^ ( x ) \hat{\mu}(x)
  13. m ^ = ( μ ^ ( x 1 ) , , μ ^ ( x n ) ) T \hat{m}=(\hat{\mu}(x_{1}),\ldots,\hat{\mu}(x_{n}))^{T}
  14. μ ^ ′′ ( x ) 2 d x \int\hat{\mu}^{\prime\prime}(x)^{2}\,dx
  15. ( x i , μ ^ ( x i ) ) (x_{i},\hat{\mu}(x_{i}))
  16. μ ^ ( x ) = i = 1 n μ ^ ( x i ) f i ( x ) \hat{\mu}(x)=\sum_{i=1}^{n}\hat{\mu}(x_{i})f_{i}(x)
  17. f i ( x ) f_{i}(x)
  18. μ ^ ′′ ( x ) 2 d x = m ^ T A m ^ . \int\hat{\mu}^{\prime\prime}(x)^{2}dx=\hat{m}^{T}A\hat{m}.
  19. f i ′′ ( x ) f j ′′ ( x ) d x \int f_{i}^{\prime\prime}(x)f_{j}^{\prime\prime}(x)dx
  20. x i x_{i}
  21. Y i Y_{i}
  22. m ^ \hat{m}
  23. Y - m ^ 2 + λ m ^ T A m ^ , \|Y-\hat{m}\|^{2}+\lambda\hat{m}^{T}A\hat{m},
  24. Y = ( Y 1 , , Y n ) T Y=(Y_{1},\ldots,Y_{n})^{T}
  25. m ^ \hat{m}
  26. m ^ = ( I + λ A ) - 1 Y . \hat{m}=(I+\lambda A)^{-1}Y.
  27. p i = 1 n ( Y i - μ ^ ( x i ) δ i ) 2 + ( 1 - p ) ( μ ^ ( m ) ( x ) ) 2 d x p\sum_{i=1}^{n}\left(\frac{Y_{i}-\hat{\mu}\left(x_{i}\right)}{\delta_{i}}% \right)^{2}+\left(1-p\right)\int\left(\hat{\mu}^{\left(m\right)}\left(x\right)% \right)^{2}\,dx
  28. p p
  29. [ 0 , 1 ] [0,1]
  30. δ i ; i = 1 , , n \delta_{i};i=1,\dots,n
  31. δ i - 2 \delta_{i}^{-2}
  32. Y i Y_{i}
  33. m m
  34. 2 2
  35. m = 2 m=2
  36. m = 2 m=2
  37. p p
  38. 1 1
  39. μ ^ \hat{\mu}
  40. p p
  41. 0
  42. μ ^ \hat{\mu}
  43. p p
  44. S S
  45. S S
  46. p p
  47. μ ^ \hat{\mu}
  48. i = 1 n ( Y i - μ ^ ( x i ) δ i ) 2 S \sum_{i=1}^{n}\left(\frac{Y_{i}-\hat{\mu}\left(x_{i}\right)}{\delta_{i}}\right% )^{2}\leq S
  49. p = 0 p=0
  50. p p
  51. δ i \delta_{i}
  52. Y i Y_{i}
  53. S S
  54. [ n - 2 n , n + 2 n ] \left[n-\sqrt{2n},n+\sqrt{2n}\right]
  55. S = 0 S=0
  56. S S
  57. x 1 < x 2 < < x n x_{1}<x_{2}<\dots<x_{n}
  58. x x
  59. y y
  60. x ( t ) x(t)
  61. y ( t ) y(t)
  62. t 1 < t 2 < < t n t_{1}<t_{2}<\dots<t_{n}
  63. t t
  64. t i + 1 = t i + ( x i + 1 - x i ) 2 + ( y i + 1 - y i ) 2 t_{i+1}=t_{i}+\sqrt{(x_{i+1}-x_{i})^{2}+(y_{i+1}-y_{i})^{2}}
  65. t 1 = 0 t_{1}=0

Socialist_millionaire.html

  1. x \scriptstyle x
  2. y \scriptstyle y
  3. x = y \scriptstyle x~{}=~{}y
  4. x \scriptstyle x
  5. y \scriptstyle y
  6. x = y \scriptstyle x~{}=~{}y
  7. x = y \scriptstyle x~{}=~{}y
  8. x \scriptstyle x
  9. y \scriptstyle y
  10. p \scriptstyle p
  11. h \scriptstyle h
  12. ( / p ) * \scriptstyle(\mathbb{Z}/p\mathbb{Z})^{*}
  13. p \scriptstyle p
  14. h \scriptstyle h
  15. ( / p ) * \scriptstyle(\mathbb{Z}/p\mathbb{Z})^{*}
  16. p \scriptstyle p
  17. ( / p ) * \scriptstyle(\mathbb{Z}/p\mathbb{Z})^{*}
  18. h | a , b \scriptstyle\langle h|a,\,b\rangle
  19. a , b \scriptstyle a,\,b
  20. h a b \scriptstyle h^{ab}
  21. h a \scriptstyle h^{a}
  22. ( h a ) b h a b \scriptstyle\left(h^{a}\right)^{b}~{}\equiv~{}h^{ab}
  23. h b \scriptstyle h^{b}
  24. ( h b ) a h b a \scriptstyle\left(h^{b}\right)^{a}~{}\equiv~{}h^{ba}
  25. h a b h b a \scriptstyle h^{ab}~{}\equiv~{}h^{ba}
  26. ( / p ) * \scriptstyle(\mathbb{Z}/p\mathbb{Z})^{*}
  27. a \scriptstyle a
  28. b \scriptstyle b
  29. α \scriptstyle\alpha
  30. β \scriptstyle\beta
  31. h a , h b , h α , \scriptstyle h^{a},\,h^{b},\,h^{\alpha},
  32. h β \scriptstyle h^{\beta}
  33. P a P b \scriptstyle P_{a}~{}\neq~{}P_{b}
  34. Q a Q b \scriptstyle Q_{a}~{}\neq~{}Q_{b}
  35. x x
  36. a , α , r a,\alpha,r
  37. p , h p,h
  38. y y
  39. b , β , s b,\beta,s
  40. g = h | a , b g=\langle h|a,b\rangle
  41. γ = h | α , β \gamma=\langle h|\alpha,\beta\rangle
  42. h b 1 h^{b}\neq 1
  43. h β 1 h^{\beta}\neq 1
  44. h a 1 h^{a}\neq 1
  45. h α 1 h^{\alpha}\neq 1
  46. P a = γ r Q a = h r g x \begin{aligned}\displaystyle P_{a}&\displaystyle=\gamma^{r}\\ \displaystyle Q_{a}&\displaystyle=h^{r}g^{x}\end{aligned}
  47. P b = γ s Q b = h s g y \begin{aligned}\displaystyle P_{b}&\displaystyle=\gamma^{s}\\ \displaystyle Q_{b}&\displaystyle=h^{s}g^{y}\end{aligned}
  48. P a , Q a , P b , Q b P_{a},Q_{a},P_{b},Q_{b}
  49. c = Q a Q b - 1 | α , β c=\left\langle\left.Q_{a}Q_{b}^{-1}\right|\alpha,\beta\right\rangle
  50. P a P b P_{a}\neq P_{b}
  51. Q a Q b Q_{a}\neq Q_{b}
  52. P a P b P_{a}\neq P_{b}
  53. Q a Q b Q_{a}\neq Q_{b}
  54. c = P a P b - 1 c=P_{a}P_{b}^{-1}
  55. c = P a P b - 1 c=P_{a}P_{b}^{-1}
  56. P a P b - 1 \displaystyle P_{a}P_{b}^{-1}
  57. c = ( Q a Q b - 1 ) α β = ( h r g x h - s g - y ) α β = ( h r - s g x - y ) α β = ( h r - s h a b ( x - y ) ) α β = h α β ( r - s ) h α β a b ( x - y ) = ( P a P b - 1 ) h α β a b ( x - y ) \begin{aligned}\displaystyle c&\displaystyle=\left(Q_{a}Q_{b}^{-1}\right)^{% \alpha\beta}\\ &\displaystyle=\left(h^{r}g^{x}h^{-s}g^{-y}\right)^{\alpha\beta}=\left(h^{r-s}% g^{x-y}\right)^{\alpha\beta}\\ &\displaystyle=\left(h^{r-s}h^{ab(x-y)}\right)^{\alpha\beta}=h^{\alpha\beta(r-% s)}h^{\alpha\beta ab(x-y)}\\ &\displaystyle=\left(P_{a}P_{b}^{-1}\right)h^{\alpha\beta ab(x-y)}\end{aligned}
  58. c \scriptstyle c
  59. P a P b - 1 \scriptstyle P_{a}P_{b}^{-1}
  60. x \scriptstyle x
  61. y \scriptstyle y
  62. h α β a b ( x - y ) = h 0 = 1 \scriptstyle h^{\alpha\beta ab(x-y)}~{}=~{}h^{0}~{}=~{}1

Sodium_adsorption_ratio.html

  1. S.A.R. = N a + 1 2 ( C a 2 + + M g 2 + ) \,\text{S.A.R.}=\frac{{Na^{+}}}{\sqrt{\tfrac{1}{2}({Ca^{2+}+Mg^{2+}})}}

Software_development_effort_estimation.html

  1. | actual effort - estimated effort | actual effort \frac{|\,\text{actual effort}-\,\text{estimated effort}|}{\,}\text{actual effort}

Sommerfeld_number.html

  1. S = ( r c ) 2 μ N P \mathrm{S}=\left(\frac{r}{c}\right)^{2}\frac{\mu N}{P}
  2. S = ( r c ) 2 μ ω P = ( r c ) 2 μ ω L D W \mathrm{S}=\left(\frac{r}{c}\right)^{2}\frac{\mu\omega}{P}=\left(\frac{r}{c}% \right)^{2}\frac{\mu\omega LD}{W}
  3. ω \omega
  4. U = 2 π r N U=2\pi rN
  5. τ = μ u y | y = 0 \tau=\mu\left.\frac{\partial u}{\partial y}\right|_{y=0}
  6. τ = μ U h = 2 π r μ N c \tau=\mu\frac{U}{h}=\frac{2\pi r\mu N}{c}
  7. T = ( τ A ) ( r ) = ( 2 π r μ N c ) ( 2 π r l ) ( r ) = 4 π 2 r 3 l μ N c T=\left(\tau A\right)\left(r\right)=\left(\frac{2\pi r\mu N}{c}\right)\left(2% \pi rl\right)\left(r\right)=\frac{4\pi^{2}r^{3}l\mu N}{c}
  8. T = f W r = 2 r 2 f l P T=fWr=2r^{2}flP
  9. f = 2 π 2 μ N P r c f=2\pi^{2}\frac{\mu N}{P}\frac{r}{c}

Sound.html

  1. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  2. c = p ρ c=\sqrt{{p\over\rho}}\,
  3. γ \sqrt{\gamma}\,
  4. p ρ \sqrt{{p\over\rho}}\,
  5. c = γ p ρ c=\sqrt{\gamma\cdot{p\over\rho}}\,
  6. K = γ p K=\gamma\cdot p\,
  7. c = K ρ c=\sqrt{\frac{K}{\rho}}\,
  8. ρ {\rho}\,
  9. - 2 -\sqrt{2}
  10. + 2 +\sqrt{2}
  11. L p = 10 log 10 ( p 2 p ref 2 ) = 20 log 10 ( p p ref ) dB L_{\mathrm{p}}=10\,\log_{10}\left(\frac{{p}^{2}}{{p_{\mathrm{ref}}}^{2}}\right% )=20\,\log_{10}\left(\frac{p}{p_{\mathrm{ref}}}\right)\mbox{ dB}~{}\,
  12. p ref p_{\mathrm{ref}}

Sound_energy.html

  1. W = W potential + W kinetic = V p 2 2 ρ 0 c 2 d V + V ρ v 2 2 d V , W=W_{\mathrm{potential}}+W_{\mathrm{kinetic}}=\int_{V}\frac{p^{2}}{2\rho_{0}c^% {2}}\,\mathrm{d}V+\int_{V}\frac{\rho v^{2}}{2}\,\mathrm{d}V,

Sołtan_argument.html

  1. L L
  2. L = ϵ M ˙ c 2 L=\epsilon\dot{M}c^{2}
  3. ϵ \epsilon
  4. M ˙ \dot{M}
  5. c c

Space-time_block_coding_based_transmit_diversity.html

  1. h 1 , h 2 h_{1},h_{2}
  2. { h 1 S 1 - h 2 S 2 * , h 1 S 2 + h 2 S 1 * } \{h_{1}S_{1}-h_{2}S_{2}^{*},\;\;h_{1}S_{2}+h_{2}S_{1}^{*}\}
  3. { n 1 , n 2 } \{n_{1},\;n_{2}\}
  4. [ x 1 x 2 * ] = [ h 1 - h 2 h 2 * h 1 * ] [ S 1 S 2 * ] + [ n 1 n 2 * ] \begin{bmatrix}x_{1}\\ x_{2}^{*}\end{bmatrix}=\begin{bmatrix}h_{1}&-h_{2}\\ h_{2}^{*}&h_{1}^{*}\end{bmatrix}\begin{bmatrix}S_{1}\\ S_{2}^{*}\end{bmatrix}+\begin{bmatrix}n_{1}\\ n_{2}^{*}\end{bmatrix}
  5. [ S ^ 1 S ^ 2 * ] = [ h 1 - h 2 h 2 * h 1 * ] - 1 [ x 1 x 2 * ] = 1 h 1 h 1 * + h 2 h 2 * [ h 1 * h 2 - h 2 * h 1 ] [ x 1 x 2 * ] \begin{bmatrix}\hat{S}_{1}\\ \hat{S}_{2}^{*}\end{bmatrix}=\begin{bmatrix}h_{1}&-h_{2}\\ h_{2}^{*}&h_{1}^{*}\end{bmatrix}^{-1}\begin{bmatrix}x_{1}\\ x_{2}^{*}\end{bmatrix}={1\over{h_{1}h_{1}^{*}+h_{2}h_{2}^{*}}}\begin{bmatrix}h% _{1}^{*}&h_{2}\\ -h_{2}^{*}&h_{1}\end{bmatrix}\begin{bmatrix}x_{1}\\ x_{2}^{*}\end{bmatrix}
  6. { b 0 , b 1 , b 2 , b 3 } \{b_{0},\;b_{1},\;b_{2},\;b_{3}\}
  7. { S 1 = ( 2 b 0 - 1 ) + i ( 2 b 1 - 1 ) , S 2 = ( 2 b 2 - 1 ) + i ( 2 b 3 - 1 ) } \{S_{1}=(2b_{0}-1)+i(2b_{1}-1),\;S_{2}=(2b_{2}-1)+i(2b_{3}-1)\}
  8. - S 1 * = ( 2 b 0 ¯ - 1 ) + i ( 2 b 1 - 1 ) , S 2 * = ( 2 b 2 - 1 ) + i ( 2 b 3 ¯ - 1 ) } \;-S_{1}^{*}=(2\overline{b_{0}}-1)+i(2b_{1}-1),\;\;S_{2}^{*}=(2b_{2}-1)+i(2% \overline{b_{3}}-1)\}

Space_vector_modulation.html

  1. α β γ \alpha\beta\gamma
  2. α β γ \alpha\beta\gamma
  3. α β γ \alpha\beta\gamma

Spacecraft_magnetometer.html

  1. μ a = μ 1 + N μ \ {\mu}_{a}=\frac{\mu}{1+N\mu}

Spark_(mathematics).html

  1. spark ( A ) = min d 0 d 0 s.t. A d = 0. \mathrm{spark}(A)=\min_{d\neq 0}\|d\|_{0}\,\text{ s.t. }Ad=0.

Sparse_approximation.html

  1. x = D α x=D\alpha
  2. D D
  3. m × p m\times p
  4. ( m p ) (m\ll p)
  5. x m , α p x\in\mathbb{R}^{m},\alpha\in\mathbb{R}^{p}
  6. D D
  7. α \alpha
  8. ( m ) (m)
  9. ( k m ) (k\ll m)
  10. α \alpha
  11. x x
  12. m × 1 m\times 1
  13. D D
  14. D D
  15. ( m p ) (m\ll p)
  16. x x
  17. min α p α 0 such that x = D α , \min_{\alpha\in\mathbb{R}^{p}}\|\alpha\|_{0}\,\text{ such that }x=D\alpha,
  18. α 0 = # { i : α i 0 , i = 1 , , p } \|\alpha\|_{0}=\#\{i:\alpha_{i}\neq 0,\,i=1,\ldots,p\}
  19. l 0 l_{0}
  20. α = [ α 1 , , α p ] T \alpha=[\alpha_{1},\ldots,\alpha_{p}]^{T}
  21. l 1 l_{1}
  22. l 0 l_{0}
  23. α 1 = i = 1 p | α i | \|\alpha\|_{1}=\sum_{i=1}^{p}|\alpha_{i}|
  24. l 1 l_{1}
  25. x x
  26. l 2 l_{2}
  27. min α p 1 2 x - D α 2 2 + λ α 1 , \min_{\alpha\in\mathbb{R}^{p}}\frac{1}{2}\|x-D\alpha\|_{2}^{2}+\lambda\|\alpha% \|_{1},
  28. λ \lambda
  29. α 1 \|\alpha\|_{1}
  30. x x
  31. x x
  32. l 2 l_{2}
  33. l 0 l_{0}
  34. D D
  35. x x
  36. l 0 l_{0}
  37. l 1 l_{1}
  38. k k
  39. 0 \ell_{0}
  40. 1 \ell_{1}

Sparse_PCA.html

  1. v p v\in\mathbb{R}^{p}
  2. max \displaystyle\max
  3. v 0 \left\|v\right\|_{0}
  4. Σ 1 = Σ - ( v T Σ v ) v v T , \Sigma_{1}=\Sigma-(v^{T}\Sigma v)vv^{T},
  5. V V
  6. max T r ( Σ V ) subject to T r ( V ) = 1 V 0 k 2 R a n k ( V ) = 1 , V 0. \begin{aligned}\displaystyle\max&\displaystyle Tr(\Sigma V)\\ \displaystyle\,\text{subject to}&\displaystyle Tr(V)=1\\ &\displaystyle\|V\|_{0}\leq k^{2}\\ &\displaystyle Rank(V)=1,V\succeq 0.\end{aligned}
  7. V 0 \|V\|_{0}
  8. V = v v T V=vv^{T}
  9. max \displaystyle\max
  10. 𝟏 \mathbf{1}
  11. V V
  12. V V
  13. p n . \scriptstyle p\gg n.
  14. X X
  15. X X
  16. θ \theta
  17. H 0 : X N ( 0 , I p ) , H 1 : X N ( 0 , I p + θ v v T ) , H_{0}:X\sim N(0,I_{p}),\quad H_{1}:X\sim N(0,I_{p}+\theta vv^{T}),
  18. v p v\in\mathbb{R}^{p}
  19. θ > Θ ( k log ( p ) / n ) \theta>\Theta(\sqrt{k\log(p)/n})
  20. θ > Θ ( k 2 log ( p ) / n ) \theta>\Theta(\sqrt{k^{2}\log(p)/n})
  21. k \sqrt{k}

Spatial_acceleration.html

  1. ω \vec{\omega}
  2. v P \vec{v}_{P}
  3. v C \vec{v}_{C}
  4. v P = v C + ω × ( r P - r C ) \vec{v}_{P}=\vec{v}_{C}+\vec{\omega}\times(\vec{r}_{P}-\vec{r}_{C})
  5. ω \vec{\omega}
  6. a P = d v P d t \vec{a}_{P}=\frac{{\rm d}\vec{v}_{P}}{{\rm d}t}
  7. a P = a C + α × ( r P - r C ) + ω × ( v P - v C ) \vec{a}_{P}=\vec{a}_{C}+\vec{\alpha}\times(\vec{r}_{P}-\vec{r}_{C})+\vec{% \omega}\times(\vec{v}_{P}-\vec{v}_{C})
  8. α \vec{\alpha}
  9. ψ P \vec{\psi}_{P}
  10. ψ C \vec{\psi}_{C}
  11. ψ P = v P t \vec{\psi}_{P}=\frac{\partial\vec{v}_{P}}{\partial t}
  12. ψ P = ψ C + α × ( r P - r C ) \vec{\psi}_{P}=\vec{\psi}_{C}+\vec{\alpha}\times(\vec{r}_{P}-\vec{r}_{C})
  13. ψ P \vec{\psi}_{P}
  14. v P \vec{v}_{P}
  15. a P \vec{a}_{P}
  16. ψ P = a P - ω × v P \vec{\psi}_{P}=\vec{a}_{P}-\vec{\omega}\times\vec{v}_{P}

Spatial_correlation.html

  1. N t N_{t}
  2. N r N_{r}
  3. 𝐲 = 𝐇𝐱 + 𝐧 \mathbf{y}=\mathbf{H}\mathbf{x}+\mathbf{n}
  4. 𝐲 \scriptstyle\mathbf{y}
  5. 𝐱 \scriptstyle\mathbf{x}
  6. N r × 1 \scriptstyle N_{r}\times 1
  7. N t × 1 \scriptstyle N_{t}\times 1
  8. N r × 1 \scriptstyle N_{r}\times 1
  9. 𝐧 \scriptstyle\mathbf{n}
  10. i j ij
  11. N r × N t \scriptstyle N_{r}\times N_{t}
  12. 𝐇 \scriptstyle\mathbf{H}
  13. j j
  14. i i
  15. 𝐇 = 𝐑 R 1 / 2 𝐇 w ( 𝐑 T 1 / 2 ) T \mathbf{H}=\mathbf{R}_{R}^{1/2}\mathbf{H}_{w}(\mathbf{R}_{T}^{1/2})^{T}
  16. 𝐇 w \scriptstyle\mathbf{H}_{w}
  17. 𝐇 w \scriptstyle\mathbf{H}_{w}
  18. 𝐑 R \scriptstyle\mathbf{R}_{R}
  19. 𝐑 T \scriptstyle\mathbf{R}_{T}
  20. 𝐇 𝒞 𝒩 ( 𝟎 , 𝐑 T 𝐑 R ) \mathbf{H}\sim\mathcal{CN}(\mathbf{0},\mathbf{R}_{T}\otimes\mathbf{R}_{R})
  21. \otimes
  22. 𝐑 T \scriptstyle\mathbf{R}_{T}
  23. 𝐑 R \scriptstyle\mathbf{R}_{R}
  24. 𝐑 T \scriptstyle\mathbf{R}_{T}
  25. 𝐑 R \scriptstyle\mathbf{R}_{R}
  26. 𝐑 T \scriptstyle\mathbf{R}_{T}
  27. 𝐑 R \scriptstyle\mathbf{R}_{R}
  28. 𝐑 T \scriptstyle\mathbf{R}_{T}
  29. 𝐑 R \scriptstyle\mathbf{R}_{R}
  30. 𝐑 T \scriptstyle\mathbf{R}_{T}
  31. 𝐑 R \scriptstyle\mathbf{R}_{R}
  32. 𝐑 T \scriptstyle\mathbf{R}_{T}
  33. 𝐑 R \scriptstyle\mathbf{R}_{R}

Special_affine_group.html

  1. 𝐱 A 𝐱 + 𝐛 \mathbf{x}\mapsto A\mathbf{x}+\mathbf{b}

Special_cases_of_Apollonius'_problem.html

  1. 𝐏 1 𝐏 2 ¯ \overline{\mathbf{P}_{1}\mathbf{P}_{2}}
  2. 𝐏 1 𝐏 3 ¯ \overline{\mathbf{P}_{1}\mathbf{P}_{3}}
  3. 𝐏 2 𝐏 3 ¯ \overline{\mathbf{P}_{2}\mathbf{P}_{3}}
  4. 𝐆𝐓 1 ¯ \overline{\mathbf{GT}_{1}}
  5. 𝐆𝐓 2 ¯ \overline{\mathbf{GT}_{2}}
  6. 𝐆𝐏 ¯ 𝐆𝐐 ¯ = 𝐆𝐓 1 ¯ 𝐆𝐓 1 ¯ = 𝐆𝐓 2 ¯ 𝐆𝐓 2 ¯ \overline{\mathbf{GP}}\cdot\overline{\mathbf{GQ}}=\overline{\mathbf{GT}_{1}}% \cdot\overline{\mathbf{GT}_{1}}=\overline{\mathbf{GT}_{2}}\cdot\overline{% \mathbf{GT}_{2}}
  7. 𝐆𝐓 1 - 2 ¯ \overline{\mathbf{GT}_{1-2}}
  8. 𝐆𝐏 ¯ \overline{\mathbf{GP}}
  9. 𝐆𝐐 ¯ \overline{\mathbf{GQ}}
  10. P Q ¯ \overline{PQ}
  11. 𝐆𝐓 ¯ 𝐆𝐓 ¯ = 𝐆𝐏 ¯ 𝐆𝐐 ¯ \overline{\mathbf{GT}}\cdot\overline{\mathbf{GT}}=\overline{\mathbf{GP}}\cdot% \overline{\mathbf{GQ}}
  12. 𝐆𝐓 ¯ \overline{\mathbf{GT}}

Special_linear_Lie_algebra.html

  1. 𝔰 𝔩 n ( F ) \mathfrak{sl}_{n}(F)
  2. n × n n\times n
  3. [ X , Y ] := X Y - Y X [X,Y]:=XY-YX
  4. 𝔰 𝔩 2 ( ) \mathfrak{sl}_{2}(\mathbb{C})
  5. 𝔰 𝔩 2 ( ) \mathfrak{sl}_{2}(\mathbb{R})
  6. 𝔰 𝔩 2 \mathfrak{sl}_{2}
  7. 𝔰 𝔩 2 ( ) . \mathfrak{sl}_{2}(\mathbb{C}).
  8. e e
  9. f f
  10. h h
  11. e = ( 0 1 0 0 ) e=\left(\begin{array}[]{cc}0&1\\ 0&0\end{array}\right)
  12. f = ( 0 0 1 0 ) f=\left(\begin{array}[]{cc}0&0\\ 1&0\end{array}\right)
  13. h = ( 1 0 0 - 1 ) h=\left(\begin{array}[]{cc}1&0\\ 0&-1\end{array}\right)
  14. [ e , f ] = h [e,f]=h
  15. [ h , f ] = - 2 f [h,f]=-2f
  16. [ h , e ] = 2 e [h,e]=2e
  17. V V
  18. 𝔰 𝔩 2 \mathfrak{sl}_{2}
  19. v v
  20. h h
  21. λ \lambda
  22. [ h , e ] v = h e v - e h v = 2 e v [h,e]v=hev-ehv=2ev
  23. h e v = ( λ + 2 ) e v hev=(\lambda+2)ev
  24. v v
  25. e v = 0 ev=0
  26. h f v = ( λ - 2 ) f v hfv=(\lambda-2)fv
  27. N N
  28. f N v = 0 f^{N}v=0
  29. N N
  30. e f k v = ( - k 2 + ( λ + 1 ) k ) f k - 1 v ef^{k}v=(-k^{2}+(\lambda+1)k)f^{k-1}v
  31. e k f k v = k ! h ( h - 1 ) ( h - k + 1 ) v e^{k}f^{k}v=k!h(h-1)...(h-k+1)v
  32. k = N k=N
  33. 0 = e N f N v = N ! h ( h - 1 ) ( h - N + 1 ) v 0=e^{N}f^{N}v=N!h(h-1)...(h-N+1)v
  34. N N
  35. f N v = 0 f^{N}v=0
  36. λ = N - 1 \lambda=N-1
  37. 0 = e f N v = ( - N 2 + ( λ + 1 ) N ) f N - 1 v 0=ef^{N}v=(-N^{2}+(\lambda+1)N)f^{N-1}v
  38. v v
  39. f v fv
  40. f λ v f^{\lambda}v
  41. N N
  42. V V
  43. N N
  44. v v
  45. f v fv
  46. f λ v f^{\lambda}v
  47. 𝔰 𝔩 2 \mathfrak{sl}_{2}
  48. 𝔰 𝔩 2 \mathfrak{sl}_{2}

Special_relativity_(alternative_formulations).html

  1. c = 1 μ 0 ε 0 . c=\frac{1}{\sqrt{\mu_{0}\varepsilon_{0}}}\ .
  2. 1 1 - β 2 \frac{1}{\sqrt{1-\beta^{2}}}
  3. s 2 = w 2 - r 2 s^{2}=w^{2}-r^{2}
  4. 1 ( 1 - β 2 ) 1\over\sqrt{(1-\beta^{2})}
  5. s 2 = c 2 t 2 - r 2 s^{2}=c^{2}t^{2}-r^{2}
  6. s 2 = w 2 - r 2 s^{2}=w^{2}-r^{2}
  7. s 2 = c 2 t 2 - r 2 s^{2}=c^{2}t^{2}-r^{2}
  8. ( c d τ ) 2 = ( c d t ) 2 - d x 2 - d y 2 - d z 2 (cd\tau)^{2}=(cdt)^{2}-dx^{2}-dy^{2}-dz^{2}
  9. ( c d t ) 2 = d x 2 + d y 2 + d z 2 + ( c d τ ) 2 (cdt)^{2}=dx^{2}+dy^{2}+dz^{2}+(cd\tau)^{2}
  10. τ \tau
  11. τ \tau
  12. c c
  13. c 2 = ( d x / d t ) 2 + ( d y / d t ) 2 + ( d z / d t ) 2 + ( c d τ / d t ) 2 c^{2}=(dx/dt)^{2}+(dy/dt)^{2}+(dz/dt)^{2}+(cd\tau/dt)^{2}
  14. t t
  15. i t it
  16. τ \tau
  17. τ \tau
  18. c 2 = ( d x / d t ) 2 + ( d y / d t ) 2 + ( d z / d t ) 2 + ( c d τ / d t ) 2 c^{2}=(dx/dt)^{2}+(dy/dt)^{2}+(dz/dt)^{2}+(cd\tau/dt)^{2}
  19. τ {\tau}
  20. τ \tau

Specific_force.html

  1. Specific Force = Force non - gravitational Mass \mbox{Specific Force}~{}=\frac{\mathrm{Force_{non-gravitational}}}{{\mathrm{% Mass}}}
  2. F s F_{s}
  3. F s = Q 2 g A + z A F_{s}=\frac{Q^{2}}{gA}+zA

Spectral_concentration_problem.html

  1. w t w_{t}
  2. t = 1 , 2 , 3 , 4 , , T t=1,2,3,4,...,T
  3. U ( f ) = t = 1 T w t e - 2 π i f t . U(f)=\sum_{t=1}^{T}w_{t}e^{-2\pi ift}.
  4. λ ( T , W ) = - W W U ( f ) 2 d f - 1 / 2 1 / 2 U ( f ) 2 d f . \lambda(T,W)=\frac{\int_{-W}^{W}{\|U(f)\|}^{2}\,df}{\int_{-1/2}^{1/2}{\|U(f)\|% }^{2}\,df}.
  5. 0 < λ ( T , W ) < 1 0<\lambda(T,W)<1
  6. w t w_{t}
  7. { w t } \{w_{t}\}
  8. λ ( T , W ) \lambda(T,W)
  9. - W W U ( f ) 2 d f e \int_{-W}^{W}{\|U(f)\|}^{2}\,dfe
  10. - 1 / 2 1 / 2 U ( f ) 2 d f = 1 , \int_{-1/2}^{1/2}{\|U(f)\|}^{2}\,df=1,
  11. w t w_{t}
  12. t = 1 T sin 2 π W ( t - t ) π ( t - t ) w t = λ w t . \sum_{t^{{}^{\prime}}=1}^{T}\frac{\sin 2\pi W(t-t^{{}^{\prime}})}{\pi(t-t^{{}^% {\prime}})}w_{t^{{}^{\prime}}}=\lambda w_{t}.
  13. M t , t = sin 2 π W ( t - t ) π ( t - t ) . M_{t,t^{{}^{\prime}}}=\frac{\sin 2\pi W(t-t^{{}^{\prime}})}{\pi(t-t^{{}^{% \prime}})}.
  14. w t w_{t}
  15. λ \lambda
  16. λ 1 > λ 2 > λ 3 > > λ N \lambda_{1}>\lambda_{2}>\lambda_{3}>...>\lambda_{N}
  17. λ n + 1 \lambda_{n+1}
  18. ( 0 , 1 , 2 , 3.... , n - 1 ) (0,1,2,3....,n-1)

Spectral_theory_of_ordinary_differential_equations.html

  1. D f ( x ) = - p ( x ) f ′′ ( x ) + r ( x ) f ( x ) + q ( x ) f ( x ) , Df(x)=-p(x)f^{\prime\prime}(x)+r(x)f^{\prime}(x)+q(x)f(x),
  2. ψ ( x ) = x 0 x p ( t ) - 1 / 2 d t \psi(x)=\int_{x_{0}}^{x}p(t)^{-1/2}\,dt
  3. U : L 2 ( a , b ) L 2 ( ψ ( a ) , ψ ( b ) ) U:L^{2}(a,b)\mapsto L^{2}(\psi(a),\psi(b))
  4. ( U f ) ( ψ ( x ) ) = f ( x ) × ( ψ ( x ) ) - 1 / 2 , x ( a , b ) (Uf)(\psi(x))=f(x)\times\left(\psi^{\prime}(x)\right)^{-1/2},\ \ \forall x\in(% a,b)
  5. U d d x U - 1 g = g ψ + 1 2 g ψ ′′ ψ U\frac{\mathrm{d}}{\mathrm{d}x}U^{-1}g=g^{\prime}\psi^{\prime}+\frac{1}{2}g% \frac{\psi^{\prime\prime}}{\psi^{\prime}}
  6. U d 2 d x 2 U - 1 g = ( U d d x U - 1 ) × ( U d d x U - 1 ) g = d d ψ [ g ψ + 1 2 g ψ ′′ ψ ] ψ + 1 2 [ g ψ + 1 2 g ψ ′′ ψ ] × ψ ′′ ψ = g ′′ ψ 2 + 2 g ψ ′′ + 1 2 g × [ ψ ′′′ ψ + ψ ′′ 2 ψ 2 ] \begin{aligned}\displaystyle U\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}U^{-1}g&% \displaystyle=\left(U\frac{\mathrm{d}}{\mathrm{d}x}U^{-1}\right)\times\left(U% \frac{\mathrm{d}}{\mathrm{d}x}U^{-1}\right)g\\ &\displaystyle=\frac{\mathrm{d}}{\mathrm{d}\psi}\left[g^{\prime}\psi^{\prime}+% \frac{1}{2}g\frac{\psi^{\prime\prime}}{\psi^{\prime}}\right]\cdot\psi^{\prime}% +\frac{1}{2}\left[g^{\prime}\psi^{\prime}+\frac{1}{2}g\frac{\psi^{\prime\prime% }}{\psi^{\prime}}\right]\times\frac{\psi^{\prime\prime}}{\psi^{\prime}}\\ &\displaystyle=g^{\prime\prime}\psi^{\prime 2}+2g^{\prime}\psi^{\prime\prime}+% \frac{1}{2}g\times\left[\frac{\psi^{\prime\prime\prime}}{\psi^{\prime}}+\frac{% \psi^{\prime\prime 2}}{\psi^{\prime 2}}\right]\end{aligned}
  7. U D U - 1 g = - g ′′ + R g + Q g , UDU^{-1}g=-g^{\prime\prime}+Rg^{\prime}+Qg,
  8. R = p + r p 1 / 2 R=\frac{p^{\prime}+r}{p^{1/2}}
  9. Q = q - r p 4 p + p ′′ 4 - p 2 8 p Q=q-\frac{rp^{\prime}}{4p}+\frac{p^{\prime\prime}}{4}-\frac{p^{\prime 2}}{8p}
  10. ( S U D U - 1 S - 1 ) h = - h ′′ + V h , (SUDU^{-1}S^{-1})h=-h^{\prime\prime}+Vh,
  11. V = Q + S ′′ S V=Q+\frac{S^{\prime\prime}}{S}
  12. D f = - f ′′ + q f . Df=-f^{\prime\prime}+qf.
  13. T f ( x ) = c x K ( x , y ) f ( y ) d y , Tf(x)=\int_{c}^{x}K(x,y)f(y)\,dy,
  14. x y ( D f ) g - f ( D g ) d t = W ( f , g ) ( y ) - W ( f , g ) ( x ) . \int_{x}^{y}(Df)g-f(Dg)\,dt=W(f,g)(y)-W(f,g)(x).
  15. d d x W ( f , g ) = 0 , {d\over dx}W(f,g)=0,
  16. cos α f ( a ) - sin α f ( a ) = 0 , cos β f ( b ) - sin β f ( b ) = 0 , \cos\alpha\,f(a)-\sin\alpha\,f^{\prime}(a)=0,\qquad\cos\beta\,f(b)-\sin\beta\,% f^{\prime}(b)=0,
  17. ( f , g ) = a b f ( x ) g ( x ) ¯ d x . (f,g)=\int_{a}^{b}f(x)\overline{g(x)}\,dx.
  18. D f = - f ′′ + q f Df=-f^{\prime\prime}+qf
  19. ( D f , f ) M ( f , f ) (Df,f)\geq M(f,f)
  20. M M
  21. ( D f , f ) = [ - f f ¯ ] a b + | f | 2 + q | f | 2 . (Df,f)=[-f^{\prime}\overline{f}]_{a}^{b}+\int|f^{\prime}|^{2}+\int q|f|^{2}.
  22. ( G λ f ) ( x ) = a b G λ ( x , y ) f ( y ) d y . (G_{\lambda}f)(x)=\int_{a}^{b}G_{\lambda}(x,y)f(y)\,dy.
  23. λ k = max dim G = k - 1 min f G ( D f , f ) ( f , f ) . \lambda_{k}=\max_{{\rm dim}\,G=k-1}\,\min_{f\perp G}{(Df,f)\over(f,f)}.
  24. λ k ( D 1 ) λ k ( D 2 ) . \lambda_{k}(D_{1})\leq\lambda_{k}(D_{2}).
  25. λ ( G ) = min f G ( D f , f ) ( f , f ) , \lambda(G)=\min_{f\perp G}{(Df,f)\over(f,f)},
  26. n 2 + m λ n ( D ) n 2 + M . n^{2}+m\leq\lambda_{n}(D)\leq n^{2}+M.
  27. ψ ( x ) = λ φ λ ( x ) | λ = μ \psi(x)=\partial_{\lambda}\varphi_{\lambda}(x)|_{\lambda=\mu}
  28. ω ( λ ) = C ( 1 - λ / λ n ) , \omega(\lambda)=C\prod(1-\lambda/\lambda_{n}),
  29. det ( I - μ ( D - λ ) - 1 ) = ( 1 - μ λ n - λ ) = 1 - ( λ + μ ) / λ n 1 - λ / λ n = ω ( λ + μ ) ω ( λ ) . {\rm det}\,(I-\mu(D-\lambda)^{-1})=\prod\left(1-{\mu\over\lambda_{n}-\lambda}% \right)=\prod{1-(\lambda+\mu)/\lambda_{n}\over 1-\lambda/\lambda_{n}}={\omega(% \lambda+\mu)\over\omega(\lambda)}.
  30. ω ( μ ) = ω ( 0 ) det ( I - μ D - 1 ) . \omega(\mu)=\omega(0)\cdot{\rm det}\,(I-\mu D^{-1}).
  31. r = 0 k - 1 | ρ ( x r + 1 ) - ρ ( x r ) | \sum_{r=0}^{k-1}|\rho(x_{r+1})-\rho(x_{r})|
  32. a = x 0 < x 1 < < x k = b a=x_{0}<x_{1}<\dots<x_{k}=b
  33. ρ ( x ) = ρ + ( x ) - ρ - ( x ) , \rho(x)=\rho_{+}(x)-\rho_{-}(x),
  34. a b f ( x ) d ρ ( x ) \int_{a}^{b}f(x)\,d\rho(x)
  35. r = 0 k - 1 f ( x r ) ( ρ ( x r + 1 ) - ρ ( x r ) ) \sum_{r=0}^{k-1}f(x_{r})(\rho(x_{r+1})-\rho(x_{r}))
  36. | a b f ( x ) d ρ ( x ) | V ( ρ ) f \left|\int_{a}^{b}f(x)\,d\rho(x)\right|\leq V(\rho)\cdot\|f\|_{\infty}
  37. | μ | ( f ) = sup 0 | g | f | μ ( g ) | . |\mu|(f)=\sup_{0\leq|g|\leq f}|\mu(g)|.
  38. μ = | μ | - ( | μ | - μ ) μ + - μ - \mu=|\mu|-(|\mu|-\mu)\equiv\mu_{+}-\mu_{-}
  39. μ ( g ) = lim μ ( f n ) , \mu(g)=\lim\mu(f_{n}),
  40. ρ ( x ) = μ ( χ [ a , x ] ) , \rho(x)=\mu(\chi_{[a,x]}),
  41. T T
  42. 0 T I 0\leq T\leq I
  43. σ ( T ) \sigma(T)
  44. T T
  45. [ 0 , 1 ] [0,1]
  46. p ( t ) p(t)
  47. σ ( p ( T ) ) = p ( σ ( T ) ) \sigma(p(T))=p(\sigma(T))
  48. p ( T ) p \|p(T)\|\leq\|p\|_{\infty}
  49. \|\,\|_{\infty}
  50. C [ 0 , 1 ] C[0,1]
  51. C [ 0 , 1 ] C[0,1]
  52. f ( T ) f(T)
  53. f C [ 0 , 1 ] \forall f\in C[0,1]
  54. σ ( f ( T ) ) = f ( σ ( T ) ) \sigma(f(T))=f(\sigma(T))
  55. f ( T ) f \|f(T)\|\leq\|f\|_{\infty}
  56. 0 g 1 0\leq g\leq 1
  57. [ 0 , 1 ] [0,1]
  58. χ [ 0 , α ] \chi_{[0,\alpha]}
  59. [ 0 , 1 ] [0,1]
  60. g g
  61. f n C [ 0 , 1 ] f_{n}\in C[0,1]
  62. ξ \xi
  63. η n = f n ( T ) ξ \eta_{n}=f_{n}(T)\xi
  64. n m n\geq m
  65. η n - η m 2 ( η n , ξ ) - ( η m , ξ ) , \|\eta_{n}-\eta_{m}\|^{2}\leq(\eta_{n},\xi)-(\eta_{m},\xi),
  66. ( η n , ξ ) = ( f n ( T ) ξ , ξ ) (\eta_{n},\xi)=(f_{n}(T)\xi,\xi)
  67. g ( T ) g(T)
  68. g ( T ) ξ = lim f n ( T ) ξ g(T)\xi=\lim f_{n}(T)\xi
  69. ξ \xi
  70. η \eta
  71. μ ξ , η ( f ) = ( f ( T ) ξ , η ) \mu_{\xi,\eta}(f)=(f(T)\xi,\eta)
  72. μ ξ , η \mu_{\xi,\eta}
  73. μ ξ , η = d ρ ξ , η \mu_{\xi,\eta}=d\rho_{\xi,\eta}
  74. ρ ξ , η \rho_{\xi,\eta}
  75. [ 0 , 1 ] [0,1]
  76. d ρ ξ , η d\rho_{\xi,\eta}
  77. ρ ξ , η \rho_{\xi,\eta}
  78. ξ \xi
  79. η \eta
  80. g ( T ) g(T)
  81. ( g ( T ) ξ , η ) = μ ξ , η ( g ) = 0 1 g ( λ ) d ρ ξ , η ( λ ) . (g(T)\xi,\eta)=\mu_{\xi,\eta}(g)=\int_{0}^{1}g(\lambda)\,d\rho_{\xi,\eta}(% \lambda).
  82. E ( λ ) E(\lambda)
  83. E ( λ ) = χ [ 0 , λ ] ( T ) , E(\lambda)=\chi_{[0,\lambda]}(T),
  84. ρ ξ , η ( λ ) = ( E ( λ ) ξ , η ) . \rho_{\xi,\eta}(\lambda)=(E(\lambda)\xi,\eta).
  85. g ( T ) = 0 1 g ( λ ) d E ( λ ) , g(T)=\int_{0}^{1}g(\lambda)\,dE(\lambda),
  86. ξ \xi
  87. η \eta
  88. ( g ( T ) ξ , η ) = 0 1 g ( λ ) d ( E ( λ ) ξ , η ) = 0 1 g ( λ ) d ρ ξ , η ( λ ) . (g(T)\xi,\eta)=\int_{0}^{1}g(\lambda)\,d(E(\lambda)\xi,\eta)=\int_{0}^{1}g(% \lambda)\,d\rho_{\xi,\eta}(\lambda).
  89. ξ , μ ξ = μ ξ , ξ \xi,\,\mu_{\xi}=\mu_{\xi,\xi}
  90. [ 0 , 1 ] [0,1]
  91. [ 0 , 1 ] [0,1]
  92. ρ ξ = ρ ξ , ξ \rho_{\xi}=\rho_{\xi,\xi}
  93. μ ξ , η \mu_{\xi,\eta}
  94. μ ξ , η = 1 4 ( μ ξ + η + i μ ξ + i η - μ ξ - η - i μ ξ - i η ) \mu_{\xi,\eta}=\frac{1}{4}\bigg(\mu_{\xi+\eta}+i\mu_{\xi+i\eta}-\mu_{\xi-\eta}% -i\mu_{\xi-i\eta}\bigg)
  95. ξ \xi
  96. ( T n ξ ) (T^{n}\xi)
  97. ξ \xi
  98. T T
  99. U U
  100. U ( f ) = f ( T ) ξ , C [ 0 , 1 ] H U(f)=f(T)\xi,\,C[0,1]\rightarrow H
  101. ( U f 1 , U f 2 ) = 0 1 f 1 ( λ ) f 2 ( λ ) ¯ d ρ ξ ( λ ) . (Uf_{1},Uf_{2})=\int_{0}^{1}f_{1}(\lambda)\overline{f_{2}(\lambda)}\,d\rho_{% \xi}(\lambda).
  102. L 2 ( [ 0 , 1 ] , d ρ ξ ) L_{2}([0,1],d\rho_{\xi})
  103. C [ 0 , 1 ] C[0,1]
  104. U U
  105. L 2 ( [ 0 , 1 ] , ρ ξ ) L_{2}([0,1],\rho_{\xi})
  106. U T U UTU^{\ast}
  107. λ \lambda
  108. L 2 ( [ 0 , 1 ] , d ρ ξ ) L_{2}([0,1],d\rho_{\xi})
  109. U f ( T ) U Uf(T)U^{\ast}
  110. f ( λ ) f(\lambda)
  111. d ρ ξ d\rho_{\xi}
  112. σ ( T ) \sigma(T)
  113. D f = - ( p f ) + q f Df=-(pf^{\prime})^{\prime}+qf
  114. E ( λ ) = χ [ λ - 1 , 1 ] ( T ) E(\lambda)=\chi_{[\lambda^{-1},1]}(T)
  115. f ( x , λ ) = ( E ( λ ) f ) ( x ) . f(x,\lambda)=(E(\lambda)f)(x).
  116. x ( d λ f ) ( x ) x\mapsto(d_{\lambda}f)(x)
  117. D ( d λ f ) = λ d λ f . D(d_{\lambda}f)=\lambda\cdot d_{\lambda}f.
  118. ( d λ f ) ( c ) = d λ f ( c , ) , ( d λ f ) ( c ) = d λ f x ( c , ) . (d_{\lambda}f)(c)=d_{\lambda}f(c,\cdot),\quad(d_{\lambda}f)^{\prime}(c)=d_{% \lambda}f_{x}(c,\cdot).
  119. f ( c , λ ) = ( f , ξ 1 ( λ ) ) , f x ( c , λ ) = ( f , ξ 2 ( λ ) ) , f(c,\lambda)=(f,\xi_{1}(\lambda)),\quad f_{x}(c,\lambda)=(f,\xi_{2}(\lambda)),
  120. σ i j ( λ ) = ( ξ i ( λ ) , ξ j ( λ ) ) \sigma_{ij}(\lambda)=(\xi_{i}(\lambda),\xi_{j}(\lambda))
  121. D f = - f ′′ + q f Df=-f^{\prime\prime}+qf
  122. φ λ , θ λ \varphi_{\lambda},\theta_{\lambda}
  123. ( D - λ ) φ λ = 0 , ( D - λ ) θ λ = 0 (D-\lambda)\varphi_{\lambda}=0,\quad(D-\lambda)\theta_{\lambda}=0
  124. φ λ ( c ) = 1 , φ λ ( c ) = 0 , θ λ ( c ) = 0 , θ λ ( c ) = 1. \varphi_{\lambda}(c)=1,\,\varphi_{\lambda}^{\prime}(c)=0,\,\theta_{\lambda}(c)% =0,\,\theta_{\lambda}^{\prime}(c)=1.
  125. W ( φ λ , θ λ ) = φ λ θ λ - θ λ φ λ 1 , W(\varphi_{\lambda},\theta_{\lambda})=\varphi_{\lambda}\theta_{\lambda}^{% \prime}-\theta_{\lambda}\varphi_{\lambda}^{\prime}\equiv 1,
  126. f = φ + μ θ f=\varphi+\mu\theta
  127. cos β f ( x ) - sin β f ( x ) = 0 \cos\beta\,f(x)-\sin\beta\,f^{\prime}(x)=0
  128. β \beta
  129. f ( x ) / f ( x ) f^{\prime}(x)/f(x)
  130. Im ( λ ) c x | φ + μ θ | 2 = Im ( μ ) . {\rm Im}(\lambda)\int_{c}^{x}|\varphi+\mu\theta|^{2}={\rm Im}(\mu).
  131. μ \mu
  132. μ \mu
  133. μ \mu
  134. c x | φ + μ θ | 2 < Im ( μ ) Im ( λ ) \int_{c}^{x}|\varphi+\mu\theta|^{2}<{{\rm Im}(\mu)\over{\rm Im}(\lambda)}
  135. x c | φ + μ θ | 2 < Im ( μ ) Im ( λ ) \int_{x}^{c}|\varphi+\mu\theta|^{2}<{{\rm Im}(\mu)\over{\rm Im}(\lambda)}
  136. μ \mu
  137. f = φ + μ θ f=\varphi+\mu\theta
  138. μ \mu
  139. c x | φ + μ θ | 2 < Im ( μ ) Im ( λ ) \int_{c}^{x}|\varphi+\mu\theta|^{2}<{{\rm Im}(\mu)\over{\rm Im}(\lambda)}
  140. h ( x ) = g ( x ) - ( λ - λ ) c x ( φ λ ( x ) θ λ ( y ) - θ λ ( x ) φ λ ( y ) ) g ( y ) d y h(x)=g(x)-(\lambda^{\prime}-\lambda)\int_{c}^{x}(\varphi_{\lambda}(x)\theta_{% \lambda}(y)-\theta_{\lambda}(x)\varphi_{\lambda}(y))g(y)\,dy
  141. g ( x ) = c 1 φ λ + c 2 θ λ + ( λ - λ ) c x ( φ λ ( x ) θ λ ( y ) - θ λ ( x ) φ λ ( y ) ) g ( y ) d y . g(x)=c_{1}\varphi_{\lambda}+c_{2}\theta_{\lambda}+(\lambda^{\prime}-\lambda)% \int_{c}^{x}(\varphi_{\lambda}(x)\theta_{\lambda}(y)-\theta_{\lambda}(x)% \varphi_{\lambda}(y))g(y)\,dy.
  142. g ( x ) = c 1 φ λ + c 2 θ λ - c x ( φ λ ( x ) θ λ ( y ) - θ λ ( x ) φ λ ( y ) ) r ( y ) g ( y ) d y . g(x)=c_{1}\varphi_{\lambda}+c_{2}\theta_{\lambda}-\int_{c}^{x}(\varphi_{% \lambda}(x)\theta_{\lambda}(y)-\theta_{\lambda}(x)\varphi_{\lambda}(y))r(y)g(y% )\,dy.
  143. D 0 f = - ( p 0 f ) + q 0 f D_{0}f=-(p_{0}f^{\prime})^{\prime}+q_{0}f
  144. D f = - f ′′ + q f Df=-f^{\prime\prime}+qf
  145. W ( f , Φ 0 ) ( 0 ) = 0. W(f,\Phi_{0})(0)=0.
  146. ω ( λ ) = W ( Φ λ , \Chi λ ) , \omega(\lambda)=W(\Phi_{\lambda},\Chi_{\lambda}),
  147. G λ ( x , y ) = Φ λ ( x ) \Chi λ ( y ) / ω ( λ ) ( x y ) , \Chi λ ( x ) Φ λ ( y ) / ω ( λ ) ( x y ) . G_{\lambda}(x,y)=\Phi_{\lambda}(x)\Chi_{\lambda}(y)/\omega(\lambda)\,\,(x\leq y% ),\,\,\,\,\Chi_{\lambda}(x)\Phi_{\lambda}(y)/\omega(\lambda)\,\,(x\geq y).
  148. ( T f ) ( x ) = 0 G 0 ( x , y ) f ( y ) d y . (Tf)(x)=\int_{0}^{\infty}G_{0}(x,y)f(y)\,dy.
  149. ( D f , f ) ( f , f ) . (Df,f)\geq(f,f).
  150. ( D - λ ) - 1 = T ( I - λ T ) - 1 (D-\lambda)^{-1}=T(I-\lambda T)^{-1}
  151. ρ ( λ ) = lim δ 0 lim ε 0 1 π δ λ + δ Im m ( t + i ε ) d t . \rho(\lambda)=\lim_{\delta\downarrow 0}\lim_{\varepsilon\downarrow 0}{1\over% \pi}\int_{\delta}^{\lambda+\delta}{\rm Im}\,m(t+i\varepsilon)\,dt.
  152. ( U f ) ( λ ) = 0 f ( x ) Φ ( x , λ ) d x , (Uf)(\lambda)=\int_{0}^{\infty}f(x)\Phi(x,\lambda)\,dx,
  153. ( U - 1 g ) ( x ) = 1 g ( λ ) Φ ( x , λ ) d ρ ( λ ) . (U^{-1}g)(x)=\int_{1}^{\infty}g(\lambda)\Phi(x,\lambda)\,d\rho(\lambda).
  154. E ( λ ) = χ [ λ - 1 , 1 ] ( T ) E(\lambda)=\chi_{[\lambda^{-1},1]}(T)
  155. f ( x , λ ) = ( E ( λ ) f ) ( x ) . f(x,\lambda)=(E(\lambda)f)(x).
  156. x ( d λ f ) ( x ) x\mapsto(d_{\lambda}f)(x)
  157. D ( d λ f ) = λ d λ f , D(d_{\lambda}f)=\lambda\cdot d_{\lambda}f,
  158. ( d λ f ) ( c ) = d λ f ( c , ) = μ ( 0 ) , ( d λ f ) ( c ) = d λ f x ( c , ) = μ ( 1 ) . (d_{\lambda}f)(c)=d_{\lambda}f(c,\cdot)=\mu^{(0)},\quad(d_{\lambda}f)^{\prime}% (c)=d_{\lambda}f_{x}(c,\cdot)=\mu^{(1)}.
  159. d λ f ( x ) = φ λ ( x ) μ ( 0 ) + χ λ ( x ) μ ( 1 ) . d_{\lambda}f(x)=\varphi_{\lambda}(x)\mu^{(0)}+\chi_{\lambda}(x)\mu^{(1)}.
  160. μ ( k ) = d λ ( f , ξ λ ( k ) ) , \mu^{(k)}=d_{\lambda}(f,\xi^{(k)}_{\lambda}),
  161. ξ λ ( k ) = D E ( λ ) η ( k ) , \xi^{(k)}_{\lambda}=DE(\lambda)\eta^{(k)},
  162. η z ( 0 ) ( y ) = G z ( c , y ) , η z ( 1 ) ( x ) = x G z ( c , y ) , ( z [ 1 , ) ) . \eta_{z}^{(0)}(y)=G_{z}(c,y),\,\,\,\,\eta_{z}^{(1)}(x)=\partial_{x}G_{z}(c,y),% \,\,\,\,(z\notin[1,\infty)).
  163. σ i j ( λ ) = ( ξ λ ( i ) , ξ λ ( j ) ) , \sigma_{ij}(\lambda)=(\xi^{(i)}_{\lambda},\xi^{(j)}_{\lambda}),
  164. d λ ( E ( λ ) η z ( i ) , η z ( j ) ) = | λ - z | - 2 d λ σ i j ( λ ) . d_{\lambda}(E(\lambda)\eta_{z}^{(i)},\eta_{z}^{(j)})=|\lambda-z|^{-2}\cdot d_{% \lambda}\sigma_{ij}(\lambda).
  165. G λ ( x , y ) = ( φ λ ( x ) + a ( λ ) χ λ ( x ) ) ( φ λ ( y ) + b ( λ ) χ λ ( y ) ) b ( λ ) - a ( λ ) ( x y ) , ( φ λ ( x ) + b ( λ ) χ λ ( x ) ) ( φ λ ( y ) + a ( λ ) χ λ ( y ) ) b ( λ ) - a ( λ ) ( y x ) . G_{\lambda}(x,y)={(\varphi_{\lambda}(x)+a(\lambda)\chi_{\lambda}(x))(\varphi_{% \lambda}(y)+b(\lambda)\chi_{\lambda}(y))\over b(\lambda)-a(\lambda)}\,\,(x\leq y% ),\,\,\,\,{(\varphi_{\lambda}(x)+b(\lambda)\chi_{\lambda}(x))(\varphi_{\lambda% }(y)+a(\lambda)\chi_{\lambda}(y))\over b(\lambda)-a(\lambda)}\,\,(y\leq x).
  166. ( η z ( i ) , η z ( j ) ) = Im M i j ( z ) / Im z , (\eta_{z}^{(i)},\eta_{z}^{(j)})={\rm Im}\,M_{ij}(z)/{\rm Im}\,z,
  167. M 00 ( z ) = a ( z ) b ( z ) a ( z ) - b ( z ) , M 01 ( z ) = M 10 ( z ) = a ( z ) + b ( z ) 2 ( a ( z ) - b ( z ) ) , M 11 ( z ) = 1 a ( z ) - b ( z ) . M_{00}(z)={a(z)b(z)\over a(z)-b(z)},\,\,M_{01}(z)=M_{10}(z)={a(z)+b(z)\over 2(% a(z)-b(z))},\,\,M_{11}(z)={1\over a(z)-b(z)}.
  168. - ( Im z ) | λ - z | - 2 d σ i j ( λ ) = Im M i j ( z ) , \int_{-\infty}^{\infty}({\rm Im}\,z)\cdot|\lambda-z|^{-2}\,d\sigma_{ij}(% \lambda)={\rm Im}M_{ij}(z),
  169. σ i j ( λ ) = lim δ 0 lim ε 0 δ λ + δ Im M i j ( t + i ε ) d t . \sigma_{ij}(\lambda)=\lim_{\delta\downarrow 0}\lim_{\varepsilon\downarrow 0}% \int_{\delta}^{\lambda+\delta}{\rm Im}\,M_{ij}(t+i\varepsilon)\,dt.
  170. ( E ( μ ) f ) ( x ) = i . j 0 μ 0 ψ λ ( i ) ( x ) ψ λ ( j ) ( y ) f ( y ) d y d σ i j ( λ ) = 0 μ 0 Φ λ ( x ) Φ λ ( y ) f ( y ) d y d ρ ( λ ) . (E(\mu)f)(x)=\sum_{i.j}\int_{0}^{\mu}\int_{0}^{\infty}\psi^{(i)}_{\lambda}(x)% \psi^{(j)}_{\lambda}(y)f(y)\,dy\,d\sigma_{ij}(\lambda)=\int_{0}^{\mu}\int_{0}^% {\infty}\Phi_{\lambda}(x)\Phi_{\lambda}(y)f(y)\,dy\,d\rho(\lambda).
  171. D f = - ( ( x 2 - 1 ) f ) = - ( x 2 - 1 ) f ′′ - 2 x f Df=-((x^{2}-1)f^{\prime})^{\prime}=-(x^{2}-1)f^{\prime\prime}-2xf^{\prime}
  172. P - 1 / 2 + i λ ( cosh r ) = 1 2 π 0 2 π ( sin θ + i e - r cos θ cos θ - i e - r sin θ ) 1 2 + i λ d θ P_{-1/2+i\sqrt{\lambda}}(\cosh r)={1\over 2\pi}\int_{0}^{2\pi}\left({\sin% \theta+ie^{-r}\cos\theta\over\cos\theta-ie^{-r}\sin\theta}\right)^{{1\over 2}+% i\sqrt{\lambda}}\,d\theta
  173. U f ( λ ) = 1 f ( x ) P - 1 / 2 + i λ ( x ) d x Uf(\lambda)=\int_{1}^{\infty}f(x)\,P_{-1/2+i\sqrt{\lambda}}(x)\,dx
  174. U - 1 g ( x ) = 0 g ( λ ) 1 2 tanh π λ d λ . U^{-1}g(x)=\int_{0}^{\infty}g(\lambda)\,{1\over 2}\tanh\pi\sqrt{\lambda}\,d\lambda.
  175. ( α β β ¯ α ¯ ) \left(\begin{matrix}\alpha&\beta\\ \overline{\beta}&\overline{\alpha}\end{matrix}\right)
  176. a a
  177. D f = - f ′′ + l ( l + 1 ) x 2 f + V ( x ) f , x ( 0 , ) Df=-f^{\prime\prime}+\frac{l(l+1)}{x^{2}}f+V(x)f,\qquad x\in(0,\infty)
  178. f f
  179. μ ξ ( | f | 2 ) = 0 \mu_{\xi}(|f|_{2})=0
  180. σ ( T ) \sigma(T)
  181. C ( σ ( T ) ) C(\sigma(T))

Speed_of_light_(cellular_automaton).html

  1. c c
  2. c / 4 c/4
  3. c / 2 c/2
  4. c c
  5. c / 2 c/2

Spekkens_Toy_Model.html

  1. 1 2 | 0 1\lor 2\iff|0\rangle
  2. 3 4 | 1 3\lor 4\iff|1\rangle
  3. 1 3 | + 1\lor 3\iff|+\rangle
  4. 2 4 | - 2\lor 4\iff|-\rangle
  5. 1 4 | i 1\lor 4\iff|i\rangle
  6. 2 3 | - i 2\lor 3\iff|-i\rangle
  7. 1 2 3 4 I / 2 1\lor 2\lor 3\lor 4\iff I/2
  8. ( ( 12 ) ( 34 ) ) ( 1 2 ) 1 2 ((12)(34))(1\lor 2)\to 1\lor 2
  9. ( ( 12 ) ( 34 ) ) ( 1 3 ) 2 4 ((12)(34))(1\lor 3)\to 2\lor 4
  10. ( ( 12 ) ( 3 ) ( 4 ) ) ( 1 3 ) 2 3. ((12)(3)(4))(1\lor 3)\to 2\lor 3.
  11. S ( 1 2 ) 3 4 S ( 3 4 ) 1 2 S(1\lor 2)\to 3\lor 4\qquad S(3\lor 4)\to 1\lor 2
  12. S ( 1 3 ) 2 4 S ( 2 4 ) 1 3 S(1\lor 3)\to 2\lor 4\qquad S(2\lor 4)\to 1\lor 3
  13. S ( 1 4 ) 2 3 S ( 2 3 ) 1 4. S(1\lor 4)\to 2\lor 3\qquad S(2\lor 3)\to 1\lor 4.

Sphere_mapping.html

  1. ( x , y ) (x,y)
  2. ( x , y , z ) (x,y,z)
  3. 1 - x 2 - y 2 \sqrt{1-x^{2}-y^{2}}
  4. < x , y , z > <x,y,z>
  5. < x , y , z > <x,y,z>
  6. ( x , y ) (x,y)

Sphere_of_influence_(black_hole).html

  1. r h = G M BH σ 2 r_{h}=\frac{GM_{\rm BH}}{\sigma^{2}}
  2. M ( r < r h ) = 2 M BH M_{\star}(r<r_{h})=2M_{\rm BH}
  3. σ \sigma

Sphere_spectrum.html

  1. S ( p ) S_{(p)}

Spin-exchange_interaction.html

  1. A A
  2. B B
  3. A ( ) + B ( ) A ( ) + B ( ) A(\uparrow)+B(\downarrow)\rightarrow A(\downarrow)+B(\uparrow)
  4. T s e = ( σ s e n v ¯ ) - 1 T_{se}=(\sigma_{se}n\bar{v})^{-1}
  5. σ s e = 2 × 10 - 14 cm 2 \sigma_{se}=2\times 10^{-14}\ \mathrm{cm}^{2}
  6. n n
  7. v ¯ \bar{v}
  8. v ¯ = 8 R T π m \bar{v}=\sqrt{\frac{8RT}{\pi m}}
  9. R R
  10. T T
  11. m m

Spin-flip.html

  1. M 1 M_{1}
  2. M 2 M_{2}
  3. J J
  4. L {L}
  5. S 1 , 2 = S 1 + S 2 {S}_{1,2}={S}_{1}+{S}_{2}
  6. 𝐌 𝟏 , 𝐌 𝟐 \mathbf{M_{1}},\mathbf{M_{2}}
  7. 𝐚 𝟏 , 𝐚 𝟐 \mathbf{a_{1}},\mathbf{a_{2}}
  8. θ \theta
  9. 𝐒 1 = { 𝐚 1 * 𝐌 1 * cos ( π / 2 - θ ) , 𝐚 1 * 𝐌 1 * sin ( π / 2 - θ ) } \mathbf{S}_{1}=\{\mathbf{a}_{1}*\mathbf{M}_{1}*\cos(\pi/2-\theta),\mathbf{a}_{% 1}*\mathbf{M}_{1}*\sin(\pi/2-\theta)\}
  10. 𝐒 2 = { 𝐚 2 * 𝐌 2 * cos ( π / 2 - θ ) , 𝐚 2 * 𝐌 2 * sin ( π / 2 - θ ) } \mathbf{S}_{2}=\{\mathbf{a}_{2}*\mathbf{M}_{2}*\cos(\pi/2-\theta),\mathbf{a}_{% 2}*\mathbf{M}_{2}*\sin(\pi/2-\theta)\}
  11. 𝐉 init = 𝐋 orb + 𝐒 1 + 𝐒 2 . \mathbf{J}_{\rm init}=\mathbf{L}_{\rm orb}+\mathbf{S}_{1}+\mathbf{S}_{2}.
  12. M 2 M_{2}
  13. 𝐉 final = 𝐒 , \mathbf{J}_{\rm final}=\mathbf{S},
  14. 𝐒 𝐋 ISCO + 𝐒 1 + 𝐒 2 . \mathbf{S}\approx\mathbf{L}_{\rm ISCO}+\mathbf{S}_{1}+\mathbf{S}_{2}.
  15. S 2 S_{2}
  16. ( M 2 / M 1 ) 2 (M_{2}/M_{1})^{2}
  17. S 1 S_{1}
  18. M 2 M_{2}
  19. M 1 M_{1}
  20. 𝐒 𝐋 ISCO + 𝐒 1 . \mathbf{S}\approx\mathbf{L}_{\rm ISCO}+\mathbf{S}_{1}.
  21. S 1 S_{1}
  22. L L
  23. S S
  24. S 1 S_{1}
  25. L ISCO L_{\rm ISCO}
  26. S 1 S_{1}
  27. S 1 S_{1}
  28. L ISCO L_{\rm ISCO}
  29. S 1 G M 1 2 / c . S_{1}\approx GM_{1}^{2}/c.
  30. L ISCO G M 1 M 2 / c . L_{\rm ISCO}\approx GM_{1}M_{2}/c.

Spin_(physics).html

  1. S = h 2 π s ( s + 1 ) = h 4 π n ( n + 2 ) , S=\frac{h}{2\pi}\,\sqrt{s(s+1)}=\frac{h}{4\pi}\,\sqrt{n(n+2)},
  2. s y m b o l μ = g s q 2 m 𝐒 symbol{\mu}=\frac{g_{s}q}{2m}\mathbf{S}
  3. μ ν 3 × 10 - 19 μ B m ν eV \mu_{\nu}\approx 3\times 10^{-19}\mu_{\mathrm{B}}\frac{m_{\nu}}{\,\text{eV}}
  4. S i = s i , s i { - s , - ( s - 1 ) , , s - 1 , s } S_{i}=\hbar s_{i},\quad s_{i}\in\{-s,-(s-1),\dots,s-1,s\}\,\!
  5. S z = s z , s z { - s , - ( s - 1 ) , , s - 1 , s } S_{z}=\hbar s_{z},\quad s_{z}\in\{-s,-(s-1),\dots,s-1,s\}\,\!
  6. S \langle S\rangle
  7. S = [ S x , S y , S z ] \langle S\rangle=[\langle S_{x}\rangle,\langle S_{y}\rangle,\langle S_{z}\rangle]
  8. [ S i , S j ] = i ϵ i j k S k [S_{i},S_{j}]=i\hbar\epsilon_{ijk}S_{k}
  9. ϵ i j k \epsilon_{ijk}
  10. S 2 | s , m \displaystyle S^{2}|s,m\rangle
  11. S ± | s , m = s ( s + 1 ) - m ( m ± 1 ) | s , m ± 1 S_{\pm}|s,m\rangle=\hbar\sqrt{s(s+1)-m(m\pm 1)}|s,m\pm 1\rangle
  12. S ± = S x ± i S y . S_{\pm}=S_{x}\pm iS_{y}.
  13. ψ = ψ ( 𝐫 ) \psi=\psi(\mathbf{r})
  14. ψ = ψ ( 𝐫 , σ ) \psi=\psi(\mathbf{r},\sigma)\,
  15. σ \sigma
  16. σ { - s , - ( s - 1 ) , , + ( s - 1 ) , + s } . \sigma\in\{-s\hbar,-(s-1)\hbar,\cdots,+(s-1)\hbar,+s\hbar\}.
  17. 1 2 \tfrac{1}{2}
  18. 𝐒 ^ = 2 s y m b o l σ \hat{\mathbf{S}}=\frac{\hbar}{2}symbol{\sigma}
  19. S x = 2 σ x , S y = 2 σ y , S z = 2 σ z . S_{x}={\hbar\over 2}\sigma_{x},\quad S_{y}={\hbar\over 2}\sigma_{y},\quad S_{z% }={\hbar\over 2}\sigma_{z}\,.
  20. σ x = ( 0 1 1 0 ) σ y = ( 0 - i i 0 ) σ z = ( 1 0 0 - 1 ) . \sigma_{x}=\begin{pmatrix}0&1\\ 1&0\end{pmatrix}\,\quad\sigma_{y}=\begin{pmatrix}0&-i\\ i&0\end{pmatrix}\,\quad\sigma_{z}=\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}\,.
  21. ψ ( 𝐫 i , σ i 𝐫 j , σ j ) = ( - 1 ) 2 s ψ ( 𝐫 j , σ j 𝐫 i , σ i ) . \psi(\cdots\mathbf{r}_{i},\sigma_{i}\cdots\mathbf{r}_{j},\sigma_{j}\cdots)=(-1% )^{2s}\psi(\cdots\mathbf{r}_{j},\sigma_{j}\cdots\mathbf{r}_{i},\sigma_{i}% \cdots).
  22. | a 1 2 | 2 + | a - 1 2 | 2 = 1. \left|a_{\frac{1}{2}}\right|^{2}+\left|a_{-\frac{1}{2}}\right|^{2}\,=1.
  23. m = - j j a m * b m = m = - j j ( n = - j j U n m a n ) * ( k = - j j U k m b k ) \sum_{m=-j}^{j}a_{m}^{*}b_{m}=\sum_{m=-j}^{j}\left(\sum_{n=-j}^{j}U_{nm}a_{n}% \right)^{*}\left(\sum_{k=-j}^{j}U_{km}b_{k}\right)
  24. n = - j j k = - j j U n p * U k q = δ p q . \sum_{n=-j}^{j}\sum_{k=-j}^{j}U_{np}^{*}U_{kq}=\delta_{pq}.
  25. s y m b o l θ ^ \hat{symbol{\theta}}
  26. U = e - i s y m b o l θ 𝐒 , U=e^{-\frac{i}{\hbar}symbol{\theta}\cdot\mathbf{S}},
  27. s y m b o l θ = θ s y m b o l θ ^ symbol{\theta}=\theta\hat{symbol{\theta}}
  28. ( α , β , γ ) = e - i α S x e - i β S y e - i γ S z \mathcal{R}(\alpha,\beta,\gamma)=e^{-i\alpha S_{x}}e^{-i\beta S_{y}}e^{-i% \gamma S_{z}}
  29. D m m s ( α , β , γ ) s m | ( α , β , γ ) | s m = e - i m α d m m s ( β ) e - i m γ , D^{s}_{m^{\prime}m}(\alpha,\beta,\gamma)\equiv\langle sm^{\prime}|\mathcal{R}(% \alpha,\beta,\gamma)|sm\rangle=e^{-im^{\prime}\alpha}d^{s}_{m^{\prime}m}(\beta% )e^{-im\gamma},
  30. d m m s ( β ) = s m | e - i β s y | s m d^{s}_{m^{\prime}m}(\beta)=\langle sm^{\prime}|e^{-i\beta s_{y}}|sm\rangle
  31. D m m s ( 0 , 0 , 2 π ) = d m m s ( 0 ) e - i m 2 π = δ m m ( - 1 ) 2 m . D^{s}_{m^{\prime}m}(0,0,2\pi)=d^{s}_{m^{\prime}m}(0)e^{-im2\pi}=\delta_{m^{% \prime}m}(-1)^{2m}.
  32. ψ \psi
  33. ψ = exp ( 1 8 ω μ ν [ γ μ , γ ν ] ) ψ \psi^{\prime}=\exp{\left(\frac{1}{8}\omega_{\mu\nu}[\gamma_{\mu},\gamma_{\nu}]% \right)}\psi
  34. γ μ \gamma_{\mu}
  35. ω μ ν \omega_{\mu\nu}
  36. ψ | ϕ = ψ ¯ ϕ = ψ γ 0 ϕ \langle\psi|\phi\rangle=\bar{\psi}\phi=\psi^{\dagger}\gamma_{0}\phi
  37. ψ x + = 1 2 ( 1 1 ) , ψ x - = 1 2 ( 1 - 1 ) , ψ y + = 1 2 ( 1 i ) , ψ y - = 1 2 ( 1 - i ) , ψ z + = ( 1 0 ) , ψ z - = ( 0 1 ) . \begin{array}[]{lclc}\psi_{x+}=\displaystyle\frac{1}{\sqrt{2}}&\begin{pmatrix}% {1}\\ {1}\end{pmatrix},&\psi_{x-}=\displaystyle\frac{1}{\sqrt{2}}&\begin{pmatrix}{1}% \\ {-1}\end{pmatrix},\\ \psi_{y+}=\displaystyle\frac{1}{\sqrt{2}}&\begin{pmatrix}{1}\\ {i}\end{pmatrix},&\psi_{y-}=\displaystyle\frac{1}{\sqrt{2}}&\begin{pmatrix}{1}% \\ {-i}\end{pmatrix},\\ \psi_{z+}=&\begin{pmatrix}{1}\\ {0}\end{pmatrix},&\psi_{z-}=&\begin{pmatrix}{0}\\ {1}\end{pmatrix}.\end{array}
  38. ψ = ( a + b i c + d i ) . \psi=\begin{pmatrix}{a+bi}\\ {c+di}\end{pmatrix}.
  39. | ψ x + | ψ | 2 \left|\langle\psi_{x+}|\psi\rangle\right|^{2}
  40. | ψ x - | ψ | 2 \left|\langle\psi_{x-}|\psi\rangle\right|^{2}
  41. | ψ x + | ψ x + | 2 = 1 \left|\langle\psi_{x+}|\psi_{x+}\rangle\right|^{2}=1
  42. S u = 2 ( u x σ x + u y σ y + u z σ z ) S_{u}=\frac{\hbar}{2}(u_{x}\sigma_{x}+u_{y}\sigma_{y}+u_{z}\sigma_{z})
  43. 1 2 + 2 u z ( 1 + u z u x + i u y ) . \frac{1}{\sqrt{2+2u_{z}}}\begin{pmatrix}1+u_{z}\\ u_{x}+iu_{y}\end{pmatrix}.
  44. σ u \sigma_{u}
  45. ψ x ± ψ y ± 2 = ψ x ± ψ z ± 2 = ψ y ± ψ z ± 2 = 1 2 . \mid\langle\psi_{x\pm}\mid\psi_{y\pm}\rangle\mid^{2}=\mid\langle\psi_{x\pm}% \mid\psi_{z\pm}\rangle\mid^{2}=\mid\langle\psi_{y\pm}\mid\psi_{z\pm}\rangle% \mid^{2}=\frac{1}{2}.
  46. ψ x + \mid\psi_{x+}\rangle
  47. ψ y + \mid\psi_{y+}\rangle
  48. ψ y - \mid\psi_{y-}\rangle
  49. ψ x + ψ y - 2 \mid\langle\psi_{x+}\mid\psi_{y-}\rangle\mid^{2}
  50. ψ x - ψ y - 2 \mid\langle\psi_{x-}\mid\psi_{y-}\rangle\mid^{2}
  51. S x \displaystyle S_{x}
  52. 3 2 \frac{3}{2}
  53. S x = 2 ( 0 3 0 0 3 0 2 0 0 2 0 3 0 0 3 0 ) S y = 2 ( 0 - i 3 0 0 i 3 0 - 2 i 0 0 2 i 0 - i 3 0 0 i 3 0 ) S z = 2 ( 3 0 0 0 0 1 0 0 0 0 - 1 0 0 0 0 - 3 ) \begin{aligned}\displaystyle S_{x}&\displaystyle=\frac{\hbar}{2}\begin{pmatrix% }0&\sqrt{3}&0&0\\ \sqrt{3}&0&2&0\\ 0&2&0&\sqrt{3}\\ 0&0&\sqrt{3}&0\end{pmatrix}\\ \displaystyle S_{y}&\displaystyle=\frac{\hbar}{2}\begin{pmatrix}0&-i\sqrt{3}&0% &0\\ i\sqrt{3}&0&-2i&0\\ 0&2i&0&-i\sqrt{3}\\ 0&0&i\sqrt{3}&0\end{pmatrix}\\ \displaystyle S_{z}&\displaystyle=\frac{\hbar}{2}\begin{pmatrix}3&0&0&0\\ 0&1&0&0\\ 0&0&-1&0\\ 0&0&0&-3\end{pmatrix}\end{aligned}
  54. 5 2 \frac{5}{2}
  55. S x = 2 ( 0 5 0 0 0 0 5 0 2 2 0 0 0 0 2 2 0 3 0 0 0 0 3 0 2 2 0 0 0 0 2 2 0 5 0 0 0 0 5 0 ) S y = 2 ( 0 - i 5 0 0 0 0 i 5 0 - 2 i 2 0 0 0 0 2 i 2 0 - 3 i 0 0 0 0 3 i 0 - 2 i 2 0 0 0 0 2 i 2 0 - i 5 0 0 0 0 i 5 0 ) S z = 2 ( 5 0 0 0 0 0 0 3 0 0 0 0 0 0 1 0 0 0 0 0 0 - 1 0 0 0 0 0 0 - 3 0 0 0 0 0 0 - 5 ) . \begin{aligned}\displaystyle S_{x}&\displaystyle=\frac{\hbar}{2}\begin{pmatrix% }0&\sqrt{5}&0&0&0&0\\ \sqrt{5}&0&2\sqrt{2}&0&0&0\\ 0&2\sqrt{2}&0&3&0&0\\ 0&0&3&0&2\sqrt{2}&0\\ 0&0&0&2\sqrt{2}&0&\sqrt{5}\\ 0&0&0&0&\sqrt{5}&0\end{pmatrix}\\ \displaystyle S_{y}&\displaystyle=\frac{\hbar}{2}\begin{pmatrix}0&-i\sqrt{5}&0% &0&0&0\\ i\sqrt{5}&0&-2i\sqrt{2}&0&0&0\\ 0&2i\sqrt{2}&0&-3i&0&0\\ 0&0&3i&0&-2i\sqrt{2}&0\\ 0&0&0&2i\sqrt{2}&0&-i\sqrt{5}\\ 0&0&0&0&i\sqrt{5}&0\end{pmatrix}\\ \displaystyle S_{z}&\displaystyle=\frac{\hbar}{2}\begin{pmatrix}5&0&0&0&0&0\\ 0&3&0&0&0&0\\ 0&0&1&0&0&0\\ 0&0&0&-1&0&0\\ 0&0&0&0&-3&0\\ 0&0&0&0&0&-5\end{pmatrix}\,.\end{aligned}
  56. ( S x ) a b = 2 ( δ a , b + 1 + δ a + 1 , b ) ( s + 1 ) ( a + b - 1 ) - a b ( S y ) a b = 2 i ( δ a , b + 1 - δ a + 1 , b ) ( s + 1 ) ( a + b - 1 ) - a b 1 a , b 2 s + 1 ( S z ) a b = ( s + 1 - a ) δ a , b = ( s + 1 - b ) δ a , b . \begin{aligned}\displaystyle\left(S_{x}\right)_{ab}&\displaystyle=\frac{\hbar}% {2}(\delta_{a,b+1}+\delta_{a+1,b})\sqrt{(s+1)(a+b-1)-ab}\\ \displaystyle\left(S_{y}\right)_{ab}&\displaystyle=\frac{\hbar}{2i}(\delta_{a,% b+1}-\delta_{a+1,b})\sqrt{(s+1)(a+b-1)-ab}\,\quad 1\leq a,b\leq 2s+1\\ \displaystyle\left(S_{z}\right)_{ab}&\displaystyle=\hbar(s+1-a)\delta_{a,b}=% \hbar(s+1-b)\delta_{a,b}\,.\end{aligned}
  57. e i θ ( 𝐧 ^ s y m b o l σ ) = I cos θ + i ( 𝐧 ^ s y m b o l σ ) sin θ e^{i\theta(\hat{\mathbf{n}}\cdot symbol{\sigma})}=I\cos\theta+i(\hat{\mathbf{n% }}\cdot symbol{\sigma})\sin\theta\,

Spin_contamination.html

  1. Ψ HF ( 𝐫 1 σ ( 1 ) 𝐫 N σ ( N ) ) = 𝒜 ( ψ 1 α ( 𝐫 1 α 1 ) ψ N α α ( 𝐫 N α α N α ) ψ N α + 1 β ( 𝐫 N α + 1 β N α + 1 ) ψ N β ( 𝐫 N β N ) ) . \Psi^{\mathrm{HF}}(\mathbf{r}_{1}\sigma(1)\cdots\mathbf{r}_{N}\sigma(N))=% \mathcal{A}\left(\psi_{1}^{\alpha}(\mathbf{r}_{1}\alpha_{1})\cdots\psi_{N_{% \alpha}}^{\alpha}(\mathbf{r}_{N_{\alpha}}\alpha_{N_{\alpha}})\psi_{N_{\alpha}+% 1}^{\beta}(\mathbf{r}_{N_{\alpha}+1}\beta_{N_{\alpha}+1})\cdots\psi_{N}^{\beta% }(\mathbf{r}_{N}\beta_{N})\right).
  2. 𝒜 \mathcal{A}
  3. ψ j α ( 𝐫 j ) = ψ N α + j β ( 𝐫 N α + j ) , 1 j N β . \psi^{\alpha}_{j}(\mathbf{r}_{j})=\psi^{\beta}_{N_{\alpha}+j}(\mathbf{r}_{N_{% \alpha}+j}),\ \ \ 1\leq j\leq N_{\beta}.
  4. S 2 ROHF = S 2 exact = ( N α - N β 2 ) ( N α - N β 2 + 1 ) . \langle S^{2}\rangle_{\mathrm{ROHF}}=\langle S^{2}\rangle_{\mathrm{exact}}=% \left(\frac{N_{\alpha}-N_{\beta}}{2}\right)\left(\frac{N_{\alpha}-N_{\beta}}{2% }+1\right).
  5. S 2 UHF = S 2 exact + N β - i , j all | ψ i α | ψ j β | 2 . \langle S^{2}\rangle_{\mathrm{UHF}}=\langle S^{2}\rangle_{\mathrm{exact}}+N_{% \beta}-\sum_{i,j}^{\mathrm{all}}|\langle\psi_{i}^{\alpha}|\psi_{j}^{\beta}% \rangle|^{2}.

Spin_transition.html

  1. d n d^{n}
  2. n = 4 n=4
  3. 7 7
  4. Δ \Delta
  5. e g e_{g}
  6. t 2 g t_{2g}
  7. P P
  8. Δ P \Delta>>P
  9. d d
  10. t 2 g t_{2g}
  11. e g e_{g}
  12. Δ P \Delta<<P
  13. P P
  14. Δ \Delta
  15. d n d^{n}
  16. d n d^{n}
  17. d 5 d^{5}
  18. d 6 d^{6}
  19. Δ S = 2 \Delta S=2
  20. e g e_{g}
  21. t 2 g t_{2g}
  22. ρ H = f ( T ) \rho_{H}=f(T)
  23. ρ H \rho_{H}
  24. ρ H \rho_{H}
  25. ρ H = f ( T ) \rho_{H}=f(T)

Spinning_drop_method.html

  1. E = E k + γ s E=E_{k}+\gamma_{s}
  2. E k = 1 2 I ω 2 = 1 4 m R 2 ω 2 E_{k}=\frac{1}{2}I\omega^{2}=\frac{1}{4}mR^{2}\omega^{2}
  3. I = 1 2 m R 2 I=\frac{1}{2}mR^{2}
  4. γ s = 2 π L R σ = 2 V R σ \gamma_{s}=2\pi LR\sigma=\frac{2V}{R}\sigma
  5. E = E k + γ s = 1 4 Δ ρ V R 2 ω 2 + 2 V R σ E=E_{k}+\gamma_{s}=\frac{1}{4}\Delta\rho VR^{2}\omega^{2}+\frac{2V}{R}\sigma
  6. d E d R = 0 = 1 2 Δ ρ V R ω 2 - 2 V R 2 σ \frac{dE}{dR}=0=\frac{1}{2}\Delta\rho VR\omega^{2}-\frac{2V}{R^{2}}\sigma
  7. V = π L R 2 V=\pi LR^{2}
  8. σ = Δ ρ ω 2 4 R 3 \sigma=\frac{\Delta\rho\omega^{2}}{4}R^{3}

Spiral_array_model.html

  1. P ( k ) = [ x k y k z k ] = [ r s i n ( k π / 2 ) r c o s ( k π / 2 ) k h ] P(k)=\begin{bmatrix}x_{k}\\ y_{k}\\ z_{k}\\ \end{bmatrix}=\begin{bmatrix}rsin(k\cdot\pi/2)\\ rcos(k\cdot\pi/2)\\ kh\end{bmatrix}
  2. C M ( k ) = w 1 P ( k ) + w 2 P ( k + 1 ) + w 3 P ( k + 4 ) C_{M}(k)=w_{1}\cdot P(k)+w_{2}\cdot P(k+1)+w_{3}\cdot P(k+4)
  3. w 1 w 2 w 3 > 0 w_{1}\geq w_{2}\geq w_{3}>0
  4. i = 1 3 w i = 1 \sum_{i=1}^{3}w_{i}=1
  5. C m ( k ) = u 1 P ( k ) + u 2 P ( k + 1 ) + u 3 P ( k - 3 ) C_{m}(k)=u_{1}\cdot P(k)+u_{2}\cdot P(k+1)+u_{3}\cdot P(k-3)
  6. u 1 u 2 u 3 > 0 u_{1}\geq u_{2}\geq u_{3}>0
  7. i = 1 3 u i = 1 \sum_{i=1}^{3}u_{i}=1
  8. T M ( k ) = W 1 P ( k ) + W 2 P ( k + 1 ) + W 3 P ( k - 1 ) T_{M}(k)=W_{1}\cdot P(k)+W_{2}\cdot P(k+1)+W_{3}\cdot P(k-1)
  9. W 1 W 2 W 3 > 0 W_{1}\geq W_{2}\geq W_{3}>0
  10. i = 1 3 W i = 1 \sum_{i=1}^{3}W_{i}=1
  11. T m ( k ) = V 1 C M ( k ) + V 2 ( α C M ( k + 1 ) + ( 1 - α ) C m ( k + 1 ) ) + V 3 ( β * C m ( k - 1 ) + ( 1 - β ) C M ( k - 1 ) ) T_{m}(k)=V_{1}\cdot C_{M}(k)+V_{2}\cdot(\alpha\cdot C_{M}(k+1)+(1-\alpha)\cdot C% _{m}(k+1))+V_{3}\cdot(\beta*C_{m}(k-1)+(1-\beta)\cdot C_{M}(k-1))
  12. V 1 V 2 V 3 > 0 V_{1}\geq V_{2}\geq V_{3}>0
  13. V 1 + V 2 + V 3 = 1 V_{1}+V_{2}+V_{3}=1
  14. 0 α 1 0\geq\alpha\geq 1
  15. β 1 \beta\geq 1

Spiral_of_Theodorus.html

  1. i + 1 \sqrt{i}{+1}
  2. 16 \sqrt{16}
  3. 17 \sqrt{17}
  4. tan ( φ n ) = 1 n . \tan\left(\varphi_{n}\right)=\frac{1}{\sqrt{n}}.
  5. φ n = arctan ( 1 n ) . \varphi_{n}=\arctan\left(\frac{1}{\sqrt{n}}\right).
  6. φ ( k ) = n = 1 k φ n = 2 k + c 2 ( k ) \varphi\left(k\right)=\sum_{n=1}^{k}\varphi_{n}=2\sqrt{k}+c_{2}(k)
  7. lim k c 2 ( k ) = - 2.157782996659 . \lim_{k\to\infty}c_{2}(k)=-2.157782996659\ldots.
  8. Δ r = n + 1 - n . \Delta r=\sqrt{n+1}-\sqrt{n}.
  9. T ( x ) = k = 1 1 + i / k 1 + i / x + k ( - 1 < x < ) T(x)=\prod_{k=1}^{\infty}\frac{1+i/\sqrt{k}}{1+i/\sqrt{x+k}}\qquad(-1<x<\infty)
  10. f ( x + 1 ) = ( 1 + i x + 1 ) f ( x ) , f(x+1)=\left(1+\frac{i}{\sqrt{x+1}}\right)\cdot f(x),
  11. f ( 0 ) = 1 , f(0)=1,

Splitting_lemma_(functions).html

  1. f : ( n , 0 ) ( , 0 ) \scriptstyle f:(\mathbb{R}^{n},0)\to(\mathbb{R},0)
  2. ( f / x i ) ( 0 ) = 0 , ( i = 1 , , n ) \scriptstyle(\partial f/\partial x_{i})(0)=0,\;(i=1,\dots,n)
  3. n \scriptstyle\mathbb{R}^{n}
  4. Φ ( x , y ) \Phi(x,y)
  5. Φ ( x , y ) = ( ϕ ( x , y ) , y ) \Phi(x,y)=(\phi(x,y),y)
  6. x V , y W \scriptstyle x\in V,\;y\in W
  7. f Φ ( x , y ) = 1 2 x T B x + h ( y ) . f\circ\Phi(x,y)=\textstyle\frac{1}{2}x^{T}Bx+h(y).

Splitting_principle.html

  1. 2 \mathbb{Z}_{2}
  2. L i L_{i}
  3. p * : H * ( X ) H * ( Y ) p^{*}\colon H^{*}(X)\rightarrow H^{*}(Y)
  4. H * ( Y ) H^{*}(Y)
  5. H * ( X ) H^{*}(X)
  6. Y Y
  7. X X

Spreading_resistance_profiling.html

  1. R < m t p l ρ 2 a R<mtpl>{{=}}\frac{\rho}{2a}
  2. R R
  3. ρ \rho
  4. a a
  5. R < m t p l ρ 2 a R<mtpl>{{=}}\frac{\rho}{2a}
  6. 2 a 2a
  7. ρ \rho

Square_thread_form.html

  1. S q 22 × 5 Sq\,22\times 5
  2. S q 60 × 18 ( P 9 ) L H Sq\,60\times 18(P9)LH

Srivastava_code.html

  1. [ α 1 μ α 1 - w 1 α n μ α n - w 1 α 1 μ α 1 - w s α n μ α n - w s ] \begin{bmatrix}\frac{\alpha_{1}^{\mu}}{\alpha_{1}-w_{1}}&\cdots&\frac{\alpha_{% n}^{\mu}}{\alpha_{n}-w_{1}}\\ \vdots&\ddots&\vdots\\ \frac{\alpha_{1}^{\mu}}{\alpha_{1}-w_{s}}&\cdots&\frac{\alpha_{n}^{\mu}}{% \alpha_{n}-w_{s}}\\ \end{bmatrix}

Stability_constants_of_complexes.html

  1. [ M ( H 2 O ) n ] + L [ M ( H 2 O ) n - 1 L ] + H 2 O \mathrm{[M(H_{2}O)_{n}]+L\leftrightharpoons[M(H_{2}O)_{n-1}L]+H_{2}O}
  2. β = [ M ( H 2 O ) n - 1 L ] [ H 2 O ] [ M ( H 2 O ) n ] [ L ] \beta^{\prime}=\mathrm{\frac{[M(H_{2}O)_{n-1}L][H_{2}O]}{[M(H_{2}O)_{n}][L]}}
  3. β = [ ML ] [ M ] [ L ] . \beta=\mathrm{\frac{[ML]}{[M][L]}}.
  4. p M + q L M p L q pM+qL...\leftrightharpoons M_{p}L_{q}...
  5. β p q = [ M p L q ] [ M ] p [ L ] q \beta_{pq...}=\mathrm{\frac{[M_{p}L_{q}...]}{[M]^{p}[L]^{q}...}}
  6. M + 2 L ML 2 ; β 12 = [ ML 2 ] [ M ] [ L ] 2 \mathrm{M+2L\rightleftharpoons ML_{2};\beta_{12}=\mathrm{\frac{[ML_{2}]}{[M][L% ]^{2}}}}
  7. M + L ML ; K 1 = [ ML ] [ M ] [ L ] \mathrm{M+L\rightleftharpoons ML;\mathit{K}_{1}=\frac{[ML]}{[M][L]}}
  8. ML + L ML 2 ; K 2 = [ ML 2 ] [ ML ] [ L ] \mathrm{ML+L\rightleftharpoons ML_{2};\mathit{K}_{2}=\frac{[ML_{2}]}{[ML][L]}}
  9. β 12 = K 1 K 2 \beta_{12}=K_{1}K_{2}\,
  10. K = [ M ( OH ) ] [ M ] [ OH ] K=\mathrm{\frac{[M(OH)]}{[M][OH]}}
  11. K = [ M ( OH ) ] [ M ] K w [ H ] - 1 K=\frac{[\,\text{M}(\,\text{OH})]}{[\,\text{M}]K\text{w}[\,\text{H}]^{-1}}
  12. β 1 - 1 * = K K w = [ M ( OH ) ] [ M ] [ H ] - 1 \beta^{*}_{1-1}=\frac{K}{K\text{w}}=\frac{[\,\text{M}(\,\text{OH})]}{[\,\text{% M}][\,\text{H}]^{-1}}
  13. A + B AB : K = [ AB ] [ A ] [ B ] \mathrm{A+B\rightleftharpoons AB}:K=\mathrm{\frac{[AB]}{[A][B]}}
  14. K = { ML } { M } { L } K^{\ominus}=\mathrm{\frac{\{ML\}}{\{M\}\{L\}}}
  15. K = [ ML ] [ M ] [ L ] × γ ML γ M γ L = [ ML ] [ M ] [ L ] × Γ K^{\ominus}=\mathrm{\frac{[ML]}{[M][L]}\times\frac{\gamma_{ML}}{\gamma_{M}% \gamma_{L}}=\mathrm{\frac{[ML]}{[M][L]}}\times\Gamma}
  16. β p q = [ M p L q ] [ M ] p [ L ] q × Γ \beta_{pq...}^{\ominus}=\mathrm{\frac{[M_{p}L_{q}...]}{[M]^{p}[L]^{q}...}% \times\Gamma}
  17. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  18. K 1 = [ [ Ag ( NH 3 ) ] + ] [ Ag + ] [ NH 3 ] K_{1}=\mathrm{\frac{[[Ag(NH_{3})]^{+}]}{[Ag^{+}][NH_{3}]}}
  19. K 2 = [ [ Ag ( NH 3 ) 2 ] + ] [ [ Ag ( NH 3 ) ] + ] [ NH 3 ] K_{2}=\mathrm{\frac{[[Ag(NH_{3})_{2}]^{+}]}{[[Ag(NH_{3})]^{+}][NH_{3}]}}

Stable_module_category.html

  1. Hom ¯ ( M , N ) \underline{\mathrm{Hom}}(M,N)
  2. Hom ¯ ( M , N ) \underline{\mathrm{Hom}}(M,N)
  3. p : P M p\colon P\to M
  4. Ω ( M ) \Omega(M)
  5. f : M N f\colon M\to N
  6. q : Q N q\colon Q\to N
  7. P Q P\to Q
  8. Ω ( M ) \Omega(M)
  9. Ω ( N ) \Omega(N)
  10. Ω \Omega
  11. Ω \Omega
  12. Ω - 1 \Omega^{-1}
  13. i : M I i\colon M\to I
  14. Ω - 1 ( M ) \Omega^{-1}(M)
  15. Hom ¯ ( Ω n ( M ) , N ) Ext k G n ( M , N ) Hom ¯ ( M , Ω - n ( N ) ) \underline{\mathrm{Hom}}(\Omega^{n}(M),N)\cong\mathrm{Ext}^{n}_{kG}(M,N)\cong% \underline{\mathrm{Hom}}(M,\Omega^{-n}(N))
  16. H n ( G ; M ) = Ext k G n ( k , M ) \mathrm{H}^{n}(G;M)=\mathrm{Ext}^{n}_{kG}(k,M)
  17. 0 X E Y 0 0\to X\to E\to Y\to 0\,
  18. Ext k G 1 ( Y , X ) \mathrm{Ext}^{1}_{kG}(Y,X)
  19. Hom ¯ ( Y , Ω - 1 ( X ) ) \underline{\mathrm{Hom}}(Y,\Omega^{-1}(X))
  20. X E Y Ω - 1 ( X ) . X\to E\to Y\to\Omega^{-1}(X).\,
  21. Ω - 1 \Omega^{-1}

Stable_vector_bundle.html

  1. deg ( V ) rank ( V ) < deg ( W ) rank ( W ) \displaystyle\frac{\deg(V)}{\hbox{rank}(V)}<\frac{\deg(W)}{\hbox{rank}(W)}
  2. deg ( V ) rank ( V ) deg ( W ) rank ( W ) \displaystyle\frac{\deg(V)}{\hbox{rank}(V)}\leq\frac{\deg(W)}{\hbox{rank}(W)}
  3. χ ( V ( n H ) ) rank ( V ) < χ ( W ( n H ) ) rank ( W ) for n large \frac{\chi(V(nH))}{\hbox{rank}(V)}<\frac{\chi(W(nH))}{\hbox{rank}(W)}\,\text{ % for }n\,\text{ large}
  4. χ \chi
  5. V ( n H ) V(nH)

Stably_free_module.html

  1. M F = G . M\oplus F=G.\,

Stalagmometric_method.html

  1. m g = 2 π r σ \ mg=2\pi r\sigma
  2. m 1 σ 1 = m 2 σ 2 \frac{m_{1}}{\sigma_{1}}=\frac{m_{2}}{\sigma_{2}}
  3. σ = σ H 2 O × m m H 2 O \sigma=\sigma_{H_{2}O}\times\frac{m}{m_{H_{2}O}}

Stalk_(sheaf).html

  1. \mathcal{F}
  2. \mathcal{F}
  3. \mathcal{F}
  4. x \mathcal{F}_{x}
  5. x := lim U x ( U ) . \mathcal{F}_{x}:=\underrightarrow{\lim}_{U\ni x}\mathcal{F}(U).
  6. U < V U<V
  7. U V U\supset V
  8. x U ( U ) x_{U}\in\mathcal{F}(U)
  9. x U x_{U}
  10. x V x_{V}
  11. x \mathcal{F}_{x}
  12. i - 1 i^{-1}\mathcal{F}
  13. i - 1 ( { x } ) = lim U { x } ( U ) = lim U x ( U ) = x . i^{-1}\mathcal{F}(\{x\})=\underrightarrow{\lim}_{U\supseteq\{x\}}\mathcal{F}(U% )=\underrightarrow{\lim}_{U\ni x}\mathcal{F}(U)=\mathcal{F}_{x}.
  14. S ¯ \underline{S}
  15. S ¯ \underline{S}
  16. 1 + e - 1 / x 2 1+e^{-1/x^{2}}
  17. 1 + e - 1 / x 2 1+e^{-1/x^{2}}

Stallings_theorem_about_ends_of_groups.html

  1. e ( ) = 2. e(\mathbb{Z})=2.
  2. 2 \mathbb{Z}^{2}
  3. e ( 2 ) = 1. e(\mathbb{Z}^{2})=1.
  4. G = H , t | t - 1 C 1 t = C 2 \scriptstyle G=\langle H,t|t^{-1}C_{1}t=C_{2}\rangle
  5. G = H , t | t - 1 C 1 t = C 2 \scriptstyle G=\langle H,t|t^{-1}C_{1}t=C_{2}\rangle

Stand_density_index.html

  1. 150 400 × .005454 = 8.29 \sqrt{\frac{150}{400\times.005454}}=8.29

Standard_array.html

  1. q n - k q^{n-k}
  2. q k q^{k}
  3. 𝔽 q n \mathbb{F}_{q}^{n}
  4. q n - k q^{n-k}
  5. q k q^{k}
  6. C 3 C_{3}
  7. C 3 C_{3}
  8. C C
  9. Z 2 Z_{2}
  10. C C
  11. q n - k = 2 4 - 2 = 2 2 = 4 q^{n-k}=2^{4-2}=2^{2}=4
  12. C C
  13. 2 32 2^{32}

Standard_part_function.html

  1. x x
  2. x 0 x_{0}
  3. x - x 0 x-x_{0}
  4. x {}^{\circ}x
  5. x x
  6. \mathbb{R}\subset{}^{\ast}\mathbb{R}
  7. {}^{\ast}\mathbb{R}
  8. \mathbb{R}
  9. st ( x ) = x 0 . \,\mathrm{st}(x)=x_{0}.
  10. u u
  11. u n : n \langle u_{n}:n\in\mathbb{N}\rangle
  12. st ( u ) = lim n u n . \,\text{st}(u)=\lim_{n\to\infty}u_{n}.
  13. u u\in{}^{\ast}\mathbb{R}
  14. \mathbb{R}\subset{}^{\ast}\mathbb{R}
  15. {}^{\ast}\mathbb{R}
  16. * \mathbb{R}\subset{}^{*}\mathbb{R}
  17. {}^{\ast}\mathbb{R}
  18. \mathbb{R}
  19. f ( x ) = st ( f ( x + h ) - f ( x ) h ) . f^{\prime}(x)=\operatorname{st}\left(\frac{f(x+h)-f(x)}{h}\right).
  20. y = f ( x ) y=f(x)
  21. Δ x \Delta x
  22. Δ y = f ( x + Δ x ) - f ( x ) \Delta y=f(x+\Delta x)-f(x)
  23. Δ y Δ x \frac{\Delta y}{\Delta x}
  24. d y d x = st ( Δ y Δ x ) \frac{dy}{dx}=\mathrm{st}\left(\frac{\Delta y}{\Delta x}\right)
  25. f f
  26. [ a , b ] [a,b]
  27. a b f ( x ) d x \int_{a}^{b}f(x)dx
  28. S ( f , a , b , Δ x ) S(f,a,b,\Delta x)
  29. Δ x \Delta x
  30. ( u n ) (u_{n})
  31. lim n u n = st ( u H ) \lim_{n\to\infty}u_{n}=\,\text{st}(u_{H})
  32. H H\in{}^{\ast}\mathbb{N}\setminus\mathbb{N}
  33. f f
  34. x x
  35. st f \,\text{st}\circ f
  36. x x

Standard_wire_gauge.html

  1. Diameter Ratio = 1 - ( 1 - 0.2 ) 1 2 10.6 % \mbox{Diameter Ratio}~{}=1-(1-0.2)^{\frac{1}{2}}\approx 10.6\%

Star_coloring.html

  1. χ s ( G ) \chi_{s}(G)
  2. χ a ( G ) \chi_{a}(G)
  3. χ a ( G ) χ s ( G ) \chi_{a}(G)\leq\chi_{s}(G)
  4. χ s ( G ) 3 \chi_{s}(G)\leq 3
  5. χ s ( G ) \chi_{s}(G)
  6. χ s ( G ) \chi_{s}(G)

State-transition_matrix.html

  1. x x
  2. t 0 t_{0}
  3. x x
  4. t t
  5. 𝐱 ˙ ( t ) = 𝐀 ( t ) 𝐱 ( t ) + 𝐁 ( t ) 𝐮 ( t ) , 𝐱 ( t 0 ) = 𝐱 0 \dot{\mathbf{x}}(t)=\mathbf{A}(t)\mathbf{x}(t)+\mathbf{B}(t)\mathbf{u}(t),% \mathbf{x}(t_{0})=\mathbf{x}_{0}
  6. 𝐱 ( t ) \mathbf{x}(t)
  7. 𝐮 ( t ) \mathbf{u}(t)
  8. 𝐱 0 \mathbf{x}_{0}
  9. t 0 t_{0}
  10. 𝚽 ( t , τ ) \mathbf{\Phi}(t,\tau)
  11. 𝐱 ( t ) = 𝚽 ( t , t 0 ) 𝐱 ( t 0 ) + t 0 t 𝚽 ( t , τ ) 𝐁 ( τ ) 𝐮 ( τ ) d τ \mathbf{x}(t)=\mathbf{\Phi}(t,t_{0})\mathbf{x}(t_{0})+\int_{t_{0}}^{t}\mathbf{% \Phi}(t,\tau)\mathbf{B}(\tau)\mathbf{u}(\tau)d\tau
  12. 𝚽 ( t , τ ) = 𝐈 + τ t 𝐀 ( σ 1 ) d σ 1 + τ t 𝐀 ( σ 1 ) τ σ 1 𝐀 ( σ 2 ) d σ 2 d σ 1 + τ t 𝐀 ( σ 1 ) τ σ 1 𝐀 ( σ 2 ) τ σ 2 𝐀 ( σ 3 ) d σ 3 d σ 2 d σ 1 + \mathbf{\Phi}(t,\tau)=\mathbf{I}+\int_{\tau}^{t}\mathbf{A}(\sigma_{1})\,d% \sigma_{1}+\int_{\tau}^{t}\mathbf{A}(\sigma_{1})\int_{\tau}^{\sigma_{1}}% \mathbf{A}(\sigma_{2})\,d\sigma_{2}\,d\sigma_{1}+\int_{\tau}^{t}\mathbf{A}(% \sigma_{1})\int_{\tau}^{\sigma_{1}}\mathbf{A}(\sigma_{2})\int_{\tau}^{\sigma_{% 2}}\mathbf{A}(\sigma_{3})\,d\sigma_{3}\,d\sigma_{2}\,d\sigma_{1}+...
  13. 𝐈 \mathbf{I}
  14. 𝚽 ( t , τ ) \mathbf{\Phi}(t,\tau)
  15. 𝚽 ( t , τ ) 𝐔 ( t ) 𝐔 - 1 ( τ ) \mathbf{\Phi}(t,\tau)\equiv\mathbf{U}(t)\mathbf{U}^{-1}(\tau)
  16. 𝐔 ( t ) \mathbf{U}(t)
  17. 𝐔 ˙ ( t ) = 𝐀 ( t ) 𝐔 ( t ) \dot{\mathbf{U}}(t)=\mathbf{A}(t)\mathbf{U}(t)
  18. n × n n\times n
  19. 𝐮 ( t ) = 0 \mathbf{u}(t)=0
  20. 𝐱 ( τ ) \mathbf{x}(\tau)
  21. τ \tau
  22. t t
  23. 𝐱 ( t ) = 𝚽 ( t , τ ) 𝐱 ( τ ) \mathbf{x}(t)=\mathbf{\Phi}(t,\tau)\mathbf{x}(\tau)
  24. 𝚽 ( t , t 0 ) t = 𝐀 ( t ) 𝚽 ( t , t 0 ) \frac{\partial\mathbf{\Phi}(t,t_{0})}{\partial t}=\mathbf{A}(t)\mathbf{\Phi}(t% ,t_{0})
  25. 𝚽 ( τ , τ ) = I \mathbf{\Phi}(\tau,\tau)=I
  26. τ \tau
  27. I I
  28. 𝚽 \mathbf{\Phi}
  29. 𝚽 ( t 2 , t 1 ) Φ ( t 1 , t 0 ) = Φ ( t 2 , t 0 ) \mathbf{\Phi}(t_{2},t_{1})\Phi(t_{1},t_{0})=\Phi(t_{2},t_{0})
  30. 𝚽 - 1 ( t , τ ) = Φ ( τ , t ) \mathbf{\Phi}^{-1}(t,\tau)=\Phi(\tau,t)
  31. 𝚽 - 1 ( t , τ ) Φ ( t , τ ) = I \mathbf{\Phi}^{-1}(t,\tau)\Phi(t,\tau)=I
  32. d 𝚽 ( t , t 0 ) d t = 𝐀 ( t ) Φ ( t , t 0 ) \frac{d\mathbf{\Phi}(t,t_{0})}{dt}=\mathbf{A}(t)\Phi(t,t_{0})
  33. 𝚽 \mathbf{\Phi}
  34. 𝚽 ( t , t 0 ) = e 𝐀 ( t - t 0 ) \mathbf{\Phi}(t,t_{0})=e^{\mathbf{A}(t-t_{0})}

Statistical_coupling_analysis.html

  1. Δ G i s t a t = x ( ln P i x ) 2 \Delta G_{i}^{stat}=\sqrt{\sum_{x}(\ln P_{i}^{x})^{2}}
  2. P i x = N ! n x ! ( N - n x ) ! p x n x ( 1 - p x ) N - n x P_{i}^{x}=\frac{N!}{n_{x}!(N-n_{x})!}p_{x}^{n_{x}}(1-p_{x})^{N-n_{x}}
  3. Δ Δ G i , j s t a t = Δ G i | δ j s t a t - Δ G i s t a t \Delta\Delta G_{i,j}^{stat}=\Delta G_{i|\delta j}^{stat}-\Delta G_{i}^{stat}
  4. Δ Δ G i , j s t a t = x ( ln P i | δ j x - ln P i x ) 2 \Delta\Delta G_{i,j}^{stat}=\sqrt{\sum_{x}(\ln P_{i|\delta j}^{x}-\ln P_{i}^{x% })^{2}}

Statistical_proof.html

  1. { Pr ( h ) 0 } \bigg\{\Pr(h)\geqq 0\bigg\}
  2. { Pr ( t ) = 1 } \bigg\{\Pr(t)=1\bigg\}
  3. { Pr ( h 1 ) + Pr ( h 2 ) = Pr ( h 1 o r h 2 ) } \bigg\{\Pr\left(h_{1}\right)+\Pr\left(h_{2}\right)=\Pr\left(h_{1}orh_{2}\right% )\bigg\}
  4. { Pr ( h 1 | h 2 ) } \Bigg\{\Pr(h_{1}|h_{2})\Bigg\}
  5. { Pr ( h 1 & h 2 ) } \bigg\{\Pr(h_{1}\And h_{2})\bigg\}
  6. { Pr ( h 2 ) } \bigg\{\Pr(h_{2})\bigg\}
  7. { Pr ( h 1 | h 2 ) = Pr ( h 1 & h 2 ) Pr ( h 2 ) } \bigg\{\frac{\Pr(h_{1}|h_{2})=\Pr(h_{1}\And h_{2})}{\Pr(h_{2})}\bigg\}
  8. { Pr ( h 2 ) > 0 } \bigg\{\Pr(h_{2})>0\bigg\}
  9. P r [ P a r a m e t e r | D a t a ] = P r [ D a t a | P a r a m e t e r ] × P r [ P a r a m e t e r ] P r [ D a t a ] Pr[Parameter|Data]=\frac{Pr[Data|Parameter]\times Pr[Parameter]}{Pr[Data]}

Stein's_unbiased_risk_estimate.html

  1. μ d \mu\in{\mathbb{R}}^{d}
  2. x d x\in{\mathbb{R}}^{d}
  3. μ \mu
  4. σ 2 \sigma^{2}
  5. h ( x ) h(x)
  6. μ \mu
  7. x x
  8. h ( x ) = x + g ( x ) h(x)=x+g(x)
  9. g g
  10. SURE ( h ) = d σ 2 + g ( x ) 2 + 2 σ 2 i = 1 d x i g i ( x ) , \mathrm{SURE}(h)=d\sigma^{2}+\|g(x)\|^{2}+2\sigma^{2}\sum_{i=1}^{d}\frac{% \partial}{\partial x_{i}}g_{i}(x),
  11. g i ( x ) g_{i}(x)
  12. i i
  13. g ( x ) g(x)
  14. \|\cdot\|
  15. h ( x ) h(x)
  16. E μ { SURE ( h ) } = MSE ( h ) , E_{\mu}\{\mathrm{SURE}(h)\}=\mathrm{MSE}(h),\,\!
  17. MSE ( h ) = E μ h ( x ) - μ 2 . \mathrm{MSE}(h)=E_{\mu}\|h(x)-\mu\|^{2}.
  18. μ \mu
  19. μ \mu
  20. E μ h ( x ) - μ 2 = E μ { SURE ( h ) } E_{\mu}\|h(x)-\mu\|^{2}=E_{\mu}\{\mathrm{SURE}(h)\}
  21. E μ h ( x ) - μ 2 \displaystyle E_{\mu}\|h(x)-\mu\|^{2}
  22. E μ g ( x ) T ( x - μ ) \displaystyle E_{\mu}g(x)^{T}(x-\mu)
  23. E μ h ( x ) - μ 2 = E μ ( d σ 2 + g ( x ) 2 + 2 σ 2 i = 1 d d g i d x i ) . E_{\mu}\|h(x)-\mu\|^{2}=E_{\mu}\left(d\sigma^{2}+\|g(x)\|^{2}+2\sigma^{2}\sum_% {i=1}^{d}\frac{dg_{i}}{dx_{i}}\right).

Steiner_chain.html

  1. sin θ = ρ r + ρ \sin\theta=\frac{\rho}{r+\rho}
  2. ρ = r sin θ 1 - sin θ \rho=\frac{r\sin\theta}{1-\sin\theta}
  3. R r = 1 + 2 sin θ 1 - sin θ = 1 + sin θ 1 - sin θ = [ sec θ + tan θ ] 2 \frac{R}{r}=1+\frac{2\sin\theta}{1-\sin\theta}=\frac{1+\sin\theta}{1-\sin% \theta}=\left[\sec\theta+\tan\theta\right]^{2}
  4. δ = ln R r \delta=\ln\frac{R}{r}
  5. δ = 2 ln ( sec θ + tan θ ) . \delta=2\ln\left(\sec\theta+\tan\theta\right).
  6. θ = m n 180 \theta=\frac{m}{n}180^{\circ}
  7. 𝐏 k 𝐀 ¯ + 𝐏 k 𝐁 ¯ = ( r α - r k ) + ( r β + r k ) = r α + r β \overline{\mathbf{P}_{k}\mathbf{A}}+\overline{\mathbf{P}_{k}\mathbf{B}}=\left(% r_{\alpha}-r_{k}\right)+\left(r_{\beta}+r_{k}\right)=r_{\alpha}+r_{\beta}
  8. 2 a = r α + r β 2a=r_{\alpha}+r_{\beta}
  9. e = p 2 a = p r α + r β e=\frac{p}{2a}=\frac{p}{r_{\alpha}+r_{\beta}}
  10. b 2 = a 2 ( 1 - e 2 ) = a 2 - p 2 4 b^{2}=a^{2}\left(1-e^{2}\right)=a^{2}-\frac{p^{2}}{4}
  11. L = b 2 a = a - p 2 4 a L=\frac{b^{2}}{a}=a-\frac{p^{2}}{4a}
  12. d = L 1 - e cos θ d=\frac{L}{1-e\cos\theta}
  13. m n + p q = 1 2 . \frac{m}{n}+\frac{p}{q}=\frac{1}{2}.
  14. 1 4 + 1 4 = 1 2 . \frac{1}{4}+\frac{1}{4}=\frac{1}{2}.

Steiner_ellipse.html

  1. 4 π 3 3 , \frac{4\pi}{3\sqrt{3}},
  2. b c y z + c a z x + a b x y = 0 bcyz+cazx+abxy=0
  3. 1 3 a 2 + b 2 + c 2 ± 2 Z , \frac{1}{3}\sqrt{a^{2}+b^{2}+c^{2}\pm 2Z},
  4. 2 3 Z \frac{2}{3}\sqrt{Z}
  5. Z = a 4 + b 4 + c 4 - a 2 b 2 - b 2 c 2 - c 2 a 2 . Z=\sqrt{a^{4}+b^{4}+c^{4}-a^{2}b^{2}-b^{2}c^{2}-c^{2}a^{2}}.
  6. p 1 = [ x 1 y 1 ] , p 2 = [ x 2 y 2 ] , p 3 = [ x 3 y 3 ] p_{1}=\begin{bmatrix}x_{1}\\ y_{1}\end{bmatrix},p_{2}=\begin{bmatrix}x_{2}\\ y_{2}\end{bmatrix},p_{3}=\begin{bmatrix}x_{3}\\ y_{3}\end{bmatrix}
  7. [ ( x 1 - x 2 ) 2 ( x 1 - x 2 ) ( y 1 - y 2 ) ( y 1 - y 2 ) 2 ( x 1 - x 3 ) 2 ( x 1 - x 3 ) ( y 1 - y 3 ) ( y 1 - y 3 ) 2 ( x 2 - x 3 ) 2 ( x 2 - x 3 ) ( y 2 - y 3 ) ( y 2 - y 3 ) 2 ] [ s x x s x y s y y ] = [ 1 1 1 ] \begin{bmatrix}(x_{1}-x_{2})^{2}&(x_{1}-x_{2})\cdot(y_{1}-y_{2})&(y_{1}-y_{2})% ^{2}\\ (x_{1}-x_{3})^{2}&(x_{1}-x_{3})\cdot(y_{1}-y_{3})&(y_{1}-y_{3})^{2}\\ (x_{2}-x_{3})^{2}&(x_{2}-x_{3})\cdot(y_{2}-y_{3})&(y_{2}-y_{3})^{2}\end{% bmatrix}\begin{bmatrix}s_{xx}\\ s_{xy}\\ s_{yy}\end{bmatrix}=\begin{bmatrix}1\\ 1\\ 1\end{bmatrix}
  8. S ¯ ¯ = [ s x x s x y s x y s y y ] \underline{\underline{S}}=\begin{bmatrix}s_{xx}&s_{xy}\\ s_{xy}&s_{yy}\end{bmatrix}

Steiner_inellipse.html

  1. a 2 x 2 + b 2 y 2 + c 2 z 2 - 2 a b x y - 2 b c y z - 2 c a z x = 0 a^{2}x^{2}+b^{2}y^{2}+c^{2}z^{2}-2abxy-2bcyz-2cazx=0
  2. π 3 3 \tfrac{\pi}{3\sqrt{3}}
  3. 1 6 a 2 + b 2 + c 2 ± 2 Z , \frac{1}{6}\sqrt{a^{2}+b^{2}+c^{2}\pm 2Z},
  4. Z = a 4 + b 4 + c 4 - a 2 b 2 - b 2 c 2 - c 2 a 2 . Z=\sqrt{a^{4}+b^{4}+c^{4}-a^{2}b^{2}-b^{2}c^{2}-c^{2}a^{2}}.
  5. P A ¯ Q A ¯ C A ¯ A B ¯ + P B ¯ Q B ¯ A B ¯ B C ¯ + P C ¯ Q C ¯ B C ¯ C A ¯ = 1. \frac{\overline{PA}\cdot\overline{QA}}{\overline{CA}\cdot\overline{AB}}+\frac{% \overline{PB}\cdot\overline{QB}}{\overline{AB}\cdot\overline{BC}}+\frac{% \overline{PC}\cdot\overline{QC}}{\overline{BC}\cdot\overline{CA}}=1.

Steinitz_exchange_lemma.html

  1. k k
  2. 0 k m 0\leq k\leq m
  3. k = m k=m
  4. { v 1 , , v k , w k + 1 , , w n } \{v_{1},\ldots,v_{k},w_{k+1},\ldots,w_{n}\}
  5. V V
  6. w j w_{j}
  7. k k
  8. k k
  9. k = 0 k=0
  10. k k
  11. 0 k < m 0\leq k<m
  12. v k + 1 V v_{k+1}\in V
  13. { v 1 , , v k , w k + 1 , , w n } \{v_{1},\ldots,v_{k},w_{k+1},\ldots,w_{n}\}
  14. V V
  15. μ 1 , , μ n \mu_{1},\ldots,\mu_{n}
  16. v k + 1 = j = 1 k μ j v j + j = k + 1 n μ j w j . v_{k+1}=\sum_{j=1}^{k}\mu_{j}v_{j}+\sum_{j=k+1}^{n}\mu_{j}w_{j}.
  17. { μ k + 1 , , μ n } \{\mu_{k+1},\ldots,\mu_{n}\}
  18. { v 1 , , v m } \{v_{1},\ldots,v_{m}\}
  19. k < n k<n
  20. w k + 1 , , w n w_{k+1},\ldots,w_{n}
  21. μ k + 1 \mu_{k+1}
  22. w k + 1 = 1 μ k + 1 ( v k + 1 - j = 1 k μ j v j - j = k + 2 n μ j w j ) w_{k+1}=\frac{1}{\mu_{k+1}}\left(v_{k+1}-\sum_{j=1}^{k}\mu_{j}v_{j}-\sum_{j=k+% 2}^{n}\mu_{j}w_{j}\right)
  23. w k + 1 w_{k+1}
  24. { v 1 , , v k + 1 , w k + 2 , , w n } \{v_{1},\ldots,v_{k+1},w_{k+2},\ldots,w_{n}\}
  25. V V
  26. k + 1 k+1

Stencil_code.html

  1. ( I , S , S 0 , s , T ) (I,S,S_{0},s,T)
  2. I = i = 1 k [ 0 , , n i ] I=\prod_{i=1}^{k}[0,\ldots,n_{i}]
  3. S S
  4. S 0 : \Z k S S_{0}\colon\Z^{k}\to S
  5. s i = 1 l \Z k s\in\prod_{i=1}^{l}\Z^{k}
  6. l l
  7. T : S l S T\colon S^{l}\to S
  8. c I c\in I
  9. l l
  10. c c
  11. I c I_{c}
  12. I c = { j x s : j = c + x } I_{c}=\{j\mid\exists x\in s:j=c+x\}\,
  13. I c I_{c}
  14. N i ( c ) N_{i}(c)
  15. N i : I S l N_{i}\colon I\to S^{l}
  16. N i ( c ) = ( s 1 , , s l ) with s j = S i ( I c ( j ) ) N_{i}(c)=(s_{1},\ldots,s_{l})\,\text{ with }s_{j}=S_{i}(I_{c}(j))\,
  17. S i + 1 : \Z k S S_{i+1}\colon\Z^{k}\to S
  18. i 𝒩 i\in\mathcal{N}
  19. S i + 1 ( c ) = { T ( N i ( c ) ) , c I S i ( c ) , c \Z k I S_{i+1}(c)=\begin{cases}T(N_{i}(c)),&c\in I\\ S_{i}(c),&c\in\Z^{k}\setminus I\end{cases}
  20. S i S_{i}
  21. \Z k \Z^{k}
  22. I I
  23. I c I_{c}
  24. I c = { j x s : j = ( ( c + x ) mod ( n 1 , , n k ) ) } I_{c}=\{j\mid\exists x\in s:j=((c+x)\mod(n_{1},\ldots,n_{k}))\}
  25. I \displaystyle I

Stencil_lithography.html

  1. h ( x ) h(x)
  2. h ( x ) = c t ( x ) M ( x - x ) d x h(x)=c\int t(x^{\prime})M(x-x^{\prime})dx^{\prime}
  3. t ( x ) t(x)
  4. x x
  5. c c
  6. M ( x ) M(x)

Stereology.html

  1. 4 / π 4/\pi

Stilb_(unit).html

  1. 1 cd m 2 = 10 - 4 sb \mathrm{1\,\frac{cd}{m^{2}}=10^{-4}\,sb}
  2. 1 sb = 1 cd cm 2 = 10 4 cd m 2 \mathrm{1\,sb=1\,\frac{cd}{cm^{2}}=10^{4}\,\frac{cd}{m^{2}}}
  3. 1 sb = 10 4 nit = 10 7 millinit \mathrm{1\,sb=10^{4}\,nit=10^{7}\,millinit}
  4. 1 sb = 1 π L = 10 3 π mL = 10 4 π asb = 10 4 π blondel = 10 7 π sk = 10 11 π bril \mathrm{1\,sb=1\pi\,L=10^{3}\pi\,mL=10^{4}\,\pi\,asb=10^{4}\pi\,blondel=10^{7}% \pi\,sk=10^{11}\pi\,bril}
  5. 1 sb = 10 4 cd m 2 0.3048 2 10 4 π fL = 2918.6... fL \mathrm{1\,sb=10^{4}\,\frac{cd}{m^{2}}\approx 0.3048^{2}\cdot 10^{4}\cdot\pi\,% \,fL=2918.6...\,fL}

Stochastic_control.html

  1. E 1 t = 1 S [ y t T Q y t + u t T R u t ] \,\text{E}_{1}\sum_{t=1}^{S}[y_{t}^{T}Qy_{t}+u_{t}^{T}Ru_{t}]
  2. y t = A t y t - 1 + B t u t , y_{t}=A_{t}y_{t-1}+B_{t}u_{t},
  3. u t * = - [ E ( B T X t B + R ) ] - 1 E ( B T X t A ) y t - 1 , u_{t}^{*}=-[\,\text{E}(B^{T}X_{t}B+R)]^{-1}\,\text{E}(B^{T}X_{t}A)y_{t-1},
  4. X S = Q X_{S}=Q
  5. X t - 1 = Q + E [ A T X t A ] - E [ A T X t B ] [ E ( B T X t B + R ) ] - 1 E ( B T X t A ) , X_{t-1}=Q+\,\text{E}[A^{T}X_{t}A]-\,\text{E}[A^{T}X_{t}B][\,\text{E}(B^{T}X_{t% }B+R)]^{-1}\,\text{E}(B^{T}X_{t}A),\,

Stokes_boundary_layer.html

  1. u t = - 1 ρ p x + ν 2 u z 2 , \frac{\partial u}{\partial t}=-\frac{1}{\rho}\frac{\partial p}{\partial x}+\nu% \frac{\partial^{2}u}{\partial z^{2}},
  2. ω = u z . \omega=\frac{\partial u}{\partial z}.
  3. ω t = ν 2 ω z 2 . \frac{\partial\omega}{\partial t}=\nu\frac{\partial^{2}\omega}{\partial z^{2}}.
  4. u 0 ( t ) = U 0 cos ( Ω t ) , u_{0}(t)=U_{0}\,\cos\left(\Omega\,t\right),\,
  5. u 1 ( 0 , t ) = u 0 ( t ) = U 0 cos ( Ω t ) at z = 0. u_{1}(0,t)=u_{0}(t)=U_{0}\,\cos\left(\Omega\,t\right)\quad\,\text{ at }\;z=0.
  6. u 1 t = ν 2 u 1 z 2 . \frac{\partial u_{1}}{\partial t}=\nu\frac{\partial^{2}u_{1}}{\partial z^{2}}.
  7. u 1 = { F ( z ) e - i Ω t } , u_{1}=\Re\left\{F(z)\;\,\text{e}^{-i\,\Omega\,t}\right\},
  8. - i Ω F = ν d 2 F d z 2 -i\,\Omega\,F=\nu\frac{\,\text{d}^{2}F}{\,\text{d}z^{2}}
  9. F \displaystyle F
  10. F ( z ) = U 0 e ( - 1 + i ) Ω 2 ν z . F(z)=U_{0}\,\,\text{e}^{(-1+i)\,\sqrt{\frac{\Omega}{2\nu}}\,z}.
  11. u 1 ( z , t ) = U 0 e - κ z cos ( Ω t - κ z ) with κ = Ω 2 ν . u_{1}(z,t)=U_{0}\,\,\text{e}^{-\kappa\,z}\,\cos\left(\Omega\,t\,-\,\kappa\,z% \right)\quad\,\text{ with }\;\kappa\,=\,\sqrt{\frac{\Omega}{2\nu}}.
  12. δ = 2 π κ = 2 π 2 ν Ω \delta=\frac{2\pi}{\kappa}=2\pi\,\sqrt{\frac{2\nu}{\Omega}}
  13. ω 1 ( z , t ) = u 1 z = - κ U 0 e - κ z [ cos ( Ω t - κ z ) - sin ( Ω t - κ z ) ] \omega_{1}(z,t)=\frac{\partial u_{1}}{\partial z}=-\kappa\,U_{0}\,\,\text{e}^{% -\kappa\,z}\,\Bigl[\,\cos\left(\Omega\,t\,-\,\kappa\,z\right)\,-\,\sin\left(% \Omega\,t\,-\,\kappa\,z\right)\,\Bigr]
  14. u ( z , t ) = U 0 cos ( Ω t ) , u_{\infty}(z,t)=U_{0}\,\cos\left(\Omega\,t\right),\,
  15. p 2 x = ρ Ω U 0 sin ( Ω t ) . \frac{\partial p_{2}}{\partial x}=\rho\,\Omega\,U_{0}\,\sin\left(\Omega\,t% \right).
  16. u 2 ( z , t ) = U 0 [ cos ( Ω t ) - e - κ z cos ( Ω t - κ z ) ] , u_{2}(z,t)=U_{0}\,\Bigl[\,\cos\left(\Omega\,t\right)\,-\,\,\text{e}^{-\kappa\,% z}\,\cos\left(\Omega\,t\,-\,\kappa\,z\right)\,\Bigr],

Stokes_stream_function.html

  1. Ψ \Psi
  2. u ρ = - 1 ρ Ψ z , u z = + 1 ρ Ψ ρ . \begin{aligned}\displaystyle u_{\rho}&\displaystyle=-\frac{1}{\rho}\,\frac{% \partial\Psi}{\partial z},\\ \displaystyle u_{z}&\displaystyle=+\frac{1}{\rho}\,\frac{\partial\Psi}{% \partial\rho}.\end{aligned}
  3. Ψ \Psi
  4. u r = + 1 r 2 sin ( θ ) Ψ θ , u θ = - 1 r sin ( θ ) Ψ r . \begin{aligned}\displaystyle u_{r}&\displaystyle=+\frac{1}{r^{2}\,\sin(\theta)% }\,\frac{\partial\Psi}{\partial\theta},\\ \displaystyle u_{\theta}&\displaystyle=-\frac{1}{r\,\sin(\theta)}\,\frac{% \partial\Psi}{\partial r}.\end{aligned}
  5. s y m b o l ω = × s y m b o l u = × × s y m b o l ψ symbol{\omega}=\nabla\times symbol{u}=\nabla\times\nabla\times symbol{\psi}
  6. s y m b o l ψ = - Ψ r sin θ s y m b o l ϕ ^ , symbol{\psi}=-\frac{\Psi}{r\sin\theta}symbol{\hat{\phi}},
  7. s y m b o l ϕ ^ symbol{\hat{\phi}}
  8. ϕ \phi\,
  9. s y m b o l ω symbol{\omega}
  10. s y m b o l ω = × s y m b o l u . symbol{\omega}=\nabla\times symbol{u}.
  11. ω r \displaystyle\omega_{r}
  12. r r
  13. θ \theta
  14. u r u_{r}
  15. u θ u_{\theta}
  16. ω ϕ . \omega_{\phi}.
  17. ω r \displaystyle\omega_{r}
  18. ω ϕ \displaystyle\omega_{\phi}
  19. s y m b o l ω = ( 0 0 - 1 r sin θ ( 2 Ψ r 2 + sin θ r 2 θ ( 1 sin θ Ψ θ ) ) ) . symbol{\omega}=\begin{pmatrix}0\\ 0\\ \displaystyle-\frac{1}{r\sin\theta}\left(\frac{\partial^{2}\Psi}{\partial r^{2% }}+\frac{\sin\theta}{r^{2}}{\partial\over\partial\theta}\left(\frac{1}{\sin% \theta}\frac{\partial\Psi}{\partial\theta}\right)\right)\end{pmatrix}.
  20. z = r cos ( θ ) z=r\,\cos(\theta)\,
  21. ρ = r sin ( θ ) . \rho=r\,\sin(\theta).\,
  22. s y m b o l u = 1 ρ ρ ( ρ u ρ ) + u z z = 1 ρ ρ ( - Ψ z ) + z ( 1 ρ Ψ ρ ) = 0 , \begin{aligned}\displaystyle\nabla\cdot symbol{u}&\displaystyle=\frac{1}{\rho}% \frac{\partial}{\partial\rho}\Bigl(\rho\,u_{\rho}\Bigr)+\frac{\partial u_{z}}{% \partial z}\\ &\displaystyle=\frac{1}{\rho}\frac{\partial}{\partial\rho}\left(-\frac{% \partial\Psi}{\partial z}\right)+\frac{\partial}{\partial z}\left(\frac{1}{% \rho}\frac{\partial\Psi}{\partial\rho}\right)=0,\end{aligned}
  23. s y m b o l u = 1 r sin ( θ ) θ ( u θ sin ( θ ) ) + 1 r 2 r ( r 2 u r ) = 1 r sin ( θ ) θ ( - 1 r Ψ r ) + 1 r 2 r ( 1 sin ( θ ) Ψ θ ) = 0. \begin{aligned}\displaystyle\nabla\cdot symbol{u}&\displaystyle=\frac{1}{r\,% \sin(\theta)}\frac{\partial}{\partial\theta}\Bigl(u_{\theta}\,\sin(\theta)% \Bigr)+\frac{1}{r^{2}}\frac{\partial}{\partial r}\Bigl(r^{2}\,u_{r}\Bigr)\\ &\displaystyle=\frac{1}{r\,\sin(\theta)}\frac{\partial}{\partial\theta}\left(-% \frac{1}{r}\frac{\partial\Psi}{\partial r}\right)+\frac{1}{r^{2}}\frac{% \partial}{\partial r}\left(\frac{1}{\sin(\theta)}\frac{\partial\Psi}{\partial% \theta}\right)=0.\end{aligned}
  24. Ψ \nabla\Psi
  25. Ψ = C \Psi=C
  26. s y m b o l u Ψ = 0 , symbol{u}\cdot\nabla\Psi=0,
  27. s y m b o l u symbol{u}
  28. Ψ , \Psi,
  29. Ψ \Psi
  30. Ψ = Ψ ρ s y m b o l e ρ + Ψ z s y m b o l e z \nabla\Psi={\partial\Psi\over\partial\rho}symbol{e}_{\rho}+{\partial\Psi\over% \partial z}symbol{e}_{z}
  31. s y m b o l u = u ρ s y m b o l e ρ + u z s y m b o l e z = - 1 ρ Ψ z s y m b o l e ρ + 1 ρ Ψ ρ s y m b o l e z . symbol{u}=u_{\rho}symbol{e}_{\rho}+u_{z}symbol{e}_{z}=-{1\over\rho}{\partial% \Psi\over\partial z}symbol{e}_{\rho}+{1\over\rho}{\partial\Psi\over\partial% \rho}symbol{e}_{z}.
  32. Ψ s y m b o l u = Ψ ρ ( - 1 ρ Ψ z ) + Ψ z 1 ρ Ψ ρ = 0. \nabla\Psi\cdot symbol{u}={\partial\Psi\over\partial\rho}(-{1\over\rho}{% \partial\Psi\over\partial z})+{\partial\Psi\over\partial z}{1\over\rho}{% \partial\Psi\over\partial\rho}=0.
  33. Ψ = Ψ r s y m b o l e r + 1 r Ψ θ s y m b o l e θ \nabla\Psi={\partial\Psi\over\partial r}symbol{e}_{r}+{1\over r}{\partial\Psi% \over\partial\theta}symbol{e}_{\theta}
  34. s y m b o l u = u r s y m b o l e r + u θ s y m b o l e θ = 1 r 2 sin ( θ ) Ψ θ s y m b o l e r - 1 r sin ( θ ) Ψ r s y m b o l e θ . symbol{u}=u_{r}symbol{e}_{r}+u_{\theta}symbol{e}_{\theta}={1\over r^{2}\sin(% \theta)}{\partial\Psi\over\partial\theta}symbol{e}_{r}-{1\over r\sin(\theta)}{% \partial\Psi\over\partial r}symbol{e}_{\theta}.
  35. Ψ s y m b o l u = Ψ r 1 r 2 sin ( θ ) Ψ θ + 1 r Ψ θ ( - 1 r sin ( θ ) Ψ r ) = 0. \nabla\Psi\cdot symbol{u}={\partial\Psi\over\partial r}\cdot{1\over r^{2}\sin(% \theta)}{\partial\Psi\over\partial\theta}+{1\over r}{\partial\Psi\over\partial% \theta}\cdot\Big(-{1\over r\sin(\theta)}{\partial\Psi\over\partial r}\Big)=0.

Strain_energy_release_rate.html

  1. G := - ( U - V ) A G:=-\cfrac{\partial(U-V)}{\partial A}
  2. U U
  3. V V
  4. A A
  5. G G
  6. G G
  7. G c G_{c}
  8. G G c G\geq G_{c}
  9. G c G_{c}
  10. K I K_{I}
  11. G G
  12. G = K I 2 E G=\cfrac{K_{I}^{2}}{E^{\prime}}
  13. E E
  14. E = E E^{\prime}=E
  15. E = E / ( 1 - ν 2 ) E^{\prime}=E/(1-\nu^{2})
  16. K I K I c K_{I}\geq K_{Ic}
  17. K I c K_{Ic}

Stratified_Morse_theory.html

  1. f : M f:M\to\mathbb{R}
  2. f - 1 ( - , c ] f^{-1}(-\infty,c]
  3. c c\in\mathbb{R}

Stress_measures.html

  1. s y m b o l τ symbol{\tau}
  2. s y m b o l N symbol{N}
  3. s y m b o l P symbol{P}
  4. s y m b o l P = s y m b o l N T symbol{P}=symbol{N}^{T}
  5. s y m b o l S symbol{S}
  6. s y m b o l T symbol{T}
  7. Ω 0 \Omega_{0}
  8. d Γ 0 d\Gamma_{0}
  9. 𝐍 𝐧 0 \mathbf{N}\equiv\mathbf{n}_{0}
  10. 𝐭 0 \mathbf{t}_{0}
  11. d 𝐟 0 d\mathbf{f}_{0}
  12. Ω \Omega
  13. d Γ d\Gamma
  14. 𝐧 \mathbf{n}
  15. 𝐭 \mathbf{t}
  16. d 𝐟 d\mathbf{f}
  17. s y m b o l F symbol{F}
  18. d 𝐟 = 𝐭 d Γ = s y m b o l σ T 𝐧 d Γ d\mathbf{f}=\mathbf{t}~{}d\Gamma=symbol{\sigma}^{T}\cdot\mathbf{n}~{}d\Gamma
  19. 𝐭 = s y m b o l σ T 𝐧 \mathbf{t}=symbol{\sigma}^{T}\cdot\mathbf{n}
  20. 𝐭 \mathbf{t}
  21. 𝐧 \mathbf{n}
  22. s y m b o l τ = J s y m b o l σ symbol{\tau}=J~{}symbol{\sigma}
  23. s y m b o l N = s y m b o l P T symbol{N}=symbol{P}^{T}
  24. s y m b o l P symbol{P}
  25. d 𝐟 = 𝐭 0 d Γ 0 = s y m b o l N T 𝐧 0 d Γ 0 = s y m b o l P 𝐧 0 d Γ 0 d\mathbf{f}=\mathbf{t}_{0}~{}d\Gamma_{0}=symbol{N}^{T}\cdot\mathbf{n}_{0}~{}d% \Gamma_{0}=symbol{P}\cdot\mathbf{n}_{0}~{}d\Gamma_{0}
  26. 𝐭 0 = s y m b o l N T 𝐧 0 = s y m b o l P 𝐧 0 \mathbf{t}_{0}=symbol{N}^{T}\cdot\mathbf{n}_{0}=symbol{P}\cdot\mathbf{n}_{0}
  27. d 𝐟 d\mathbf{f}
  28. d 𝐟 0 = s y m b o l F - 1 d 𝐟 d\mathbf{f}_{0}=symbol{F}^{-1}\cdot d\mathbf{f}
  29. d 𝐟 0 = s y m b o l F - 1 s y m b o l N T 𝐧 0 d Γ 0 = s y m b o l F - 1 𝐭 0 d Γ 0 d\mathbf{f}_{0}=symbol{F}^{-1}\cdot symbol{N}^{T}\cdot\mathbf{n}_{0}~{}d\Gamma% _{0}=symbol{F}^{-1}\cdot\mathbf{t}_{0}~{}d\Gamma_{0}
  30. s y m b o l S symbol{S}
  31. d 𝐟 0 = s y m b o l S T 𝐧 0 d Γ 0 = s y m b o l F - 1 𝐭 0 d Γ 0 d\mathbf{f}_{0}=symbol{S}^{T}\cdot\mathbf{n}_{0}~{}d\Gamma_{0}=symbol{F}^{-1}% \cdot\mathbf{t}_{0}~{}d\Gamma_{0}
  32. s y m b o l S T 𝐧 0 = s y m b o l F - 1 𝐭 0 symbol{S}^{T}\cdot\mathbf{n}_{0}=symbol{F}^{-1}\cdot\mathbf{t}_{0}
  33. s y m b o l U symbol{U}
  34. s y m b o l P T \cdotsymbol R symbol{P}^{T}\cdotsymbol{R}
  35. s y m b o l R symbol{R}
  36. s y m b o l T = 1 2 ( s y m b o l R T \cdotsymbol P + s y m b o l P T \cdotsymbol R ) . symbol{T}=\tfrac{1}{2}(symbol{R}^{T}\cdotsymbol{P}+symbol{P}^{T}\cdotsymbol{R}% )~{}.
  37. s y m b o l T symbol{T}
  38. s y m b o l R T d 𝐟 = ( s y m b o l P T \cdotsymbol R ) T 𝐧 0 d Γ 0 symbol{R}^{T}~{}d\mathbf{f}=(symbol{P}^{T}\cdotsymbol{R})^{T}\cdot\mathbf{n}_{% 0}~{}d\Gamma_{0}
  39. 𝐧 d Γ = J s y m b o l F - T 𝐧 0 d Γ 0 \mathbf{n}~{}d\Gamma=J~{}symbol{F}^{-T}\cdot\mathbf{n}_{0}~{}d\Gamma_{0}
  40. s y m b o l σ T 𝐧 d Γ = d 𝐟 = s y m b o l N T 𝐧 0 d Γ 0 symbol{\sigma}^{T}\cdot\mathbf{n}~{}d\Gamma=d\mathbf{f}=symbol{N}^{T}\cdot% \mathbf{n}_{0}~{}d\Gamma_{0}
  41. s y m b o l σ T ( J s y m b o l F - T 𝐧 0 d Γ 0 ) = s y m b o l N T 𝐧 0 d Γ 0 symbol{\sigma}^{T}\cdot(J~{}symbol{F}^{-T}\cdot\mathbf{n}_{0}~{}d\Gamma_{0})=% symbol{N}^{T}\cdot\mathbf{n}_{0}~{}d\Gamma_{0}
  42. s y m b o l N T = J ( s y m b o l F - 1 \cdotsymbol σ ) T = J s y m b o l σ T \cdotsymbol F - T symbol{N}^{T}=J~{}(symbol{F}^{-1}\cdotsymbol{\sigma})^{T}=J~{}symbol{\sigma}^{% T}\cdotsymbol{F}^{-T}
  43. s y m b o l N = J s y m b o l F - 1 \cdotsymbol σ and s y m b o l N T = s y m b o l P = J s y m b o l σ T \cdotsymbol F - T symbol{N}=J~{}symbol{F}^{-1}\cdotsymbol{\sigma}\qquad\,\text{and}\qquad symbol% {N}^{T}=symbol{P}=J~{}symbol{\sigma}^{T}\cdotsymbol{F}^{-T}
  44. N I j = J F I k - 1 σ k j and P i J = J σ k i F J k - 1 N_{Ij}=J~{}F_{Ik}^{-1}~{}\sigma_{kj}\qquad\,\text{and}\qquad P_{iJ}=J~{}\sigma% _{ki}~{}F^{-1}_{Jk}
  45. J s y m b o l σ = s y m b o l F \cdotsymbol N = s y m b o l P \cdotsymbol F T . J~{}symbol{\sigma}=symbol{F}\cdotsymbol{N}=symbol{P}\cdotsymbol{F}^{T}~{}.
  46. s y m b o l N symbol{N}
  47. s y m b o l P symbol{P}
  48. s y m b o l F symbol{F}
  49. s y m b o l N T 𝐧 0 d Γ 0 = d 𝐟 symbol{N}^{T}\cdot\mathbf{n}_{0}~{}d\Gamma_{0}=d\mathbf{f}
  50. d 𝐟 = s y m b o l F d 𝐟 0 = s y m b o l F ( s y m b o l S T 𝐧 0 d Γ 0 ) d\mathbf{f}=symbol{F}\cdot d\mathbf{f}_{0}=symbol{F}\cdot(symbol{S}^{T}\cdot% \mathbf{n}_{0}~{}d\Gamma_{0})
  51. s y m b o l N T 𝐧 0 = s y m b o l F \cdotsymbol S T 𝐧 0 symbol{N}^{T}\cdot\mathbf{n}_{0}=symbol{F}\cdotsymbol{S}^{T}\cdot\mathbf{n}_{0}
  52. s y m b o l S symbol{S}
  53. s y m b o l N = s y m b o l S \cdotsymbol F T and s y m b o l P = s y m b o l F \cdotsymbol S symbol{N}=symbol{S}\cdotsymbol{F}^{T}\qquad\,\text{and}\qquad symbol{P}=symbol% {F}\cdotsymbol{S}
  54. N I j = S I K F j K T and P i J = F i K S K J N_{Ij}=S_{IK}~{}F^{T}_{jK}\qquad\,\text{and}\qquad P_{iJ}=F_{iK}~{}S_{KJ}
  55. s y m b o l S = s y m b o l N \cdotsymbol F - T and s y m b o l S = s y m b o l F - 1 \cdotsymbol P symbol{S}=symbol{N}\cdotsymbol{F}^{-T}\qquad\,\text{and}\qquad symbol{S}=% symbol{F}^{-1}\cdotsymbol{P}
  56. s y m b o l N = J s y m b o l F - 1 \cdotsymbol σ symbol{N}=J~{}symbol{F}^{-1}\cdotsymbol{\sigma}
  57. s y m b o l S \cdotsymbol F T = J s y m b o l F - 1 \cdotsymbol σ symbol{S}\cdotsymbol{F}^{T}=J~{}symbol{F}^{-1}\cdotsymbol{\sigma}
  58. s y m b o l S = J s y m b o l F - 1 \cdotsymbol σ \cdotsymbol F - T = s y m b o l F - 1 \cdotsymbol τ \cdotsymbol F - T symbol{S}=J~{}symbol{F}^{-1}\cdotsymbol{\sigma}\cdotsymbol{F}^{-T}=symbol{F}^{% -1}\cdotsymbol{\tau}\cdotsymbol{F}^{-T}
  59. S I J = F I k - 1 τ k l F J l - 1 S_{IJ}=F_{Ik}^{-1}~{}\tau_{kl}~{}F_{Jl}^{-1}
  60. s y m b o l σ = J - 1 s y m b o l F \cdotsymbol S \cdotsymbol F T symbol{\sigma}=J^{-1}~{}symbol{F}\cdotsymbol{S}\cdotsymbol{F}^{T}
  61. s y m b o l τ = s y m b o l F \cdotsymbol S \cdotsymbol F T . symbol{\tau}=symbol{F}\cdotsymbol{S}\cdotsymbol{F}^{T}~{}.
  62. s y m b o l S = φ * [ s y m b o l τ ] = s y m b o l F - 1 \cdotsymbol τ \cdotsymbol F - T symbol{S}=\varphi^{*}[symbol{\tau}]=symbol{F}^{-1}\cdotsymbol{\tau}\cdotsymbol% {F}^{-T}
  63. s y m b o l τ = φ * [ s y m b o l S ] = s y m b o l F \cdotsymbol S \cdotsymbol F T . symbol{\tau}=\varphi_{*}[symbol{S}]=symbol{F}\cdotsymbol{S}\cdotsymbol{F}^{T}~% {}.
  64. s y m b o l S symbol{S}
  65. s y m b o l τ symbol{\tau}
  66. s y m b o l F symbol{F}
  67. s y m b o l τ symbol{\tau}
  68. s y m b o l S symbol{S}

Strict-feedback_form.html

  1. { 𝐱 ˙ = f 0 ( 𝐱 ) + g 0 ( 𝐱 ) z 1 z ˙ 1 = f 1 ( 𝐱 , z 1 ) + g 1 ( 𝐱 , z 1 ) z 2 z ˙ 2 = f 2 ( 𝐱 , z 1 , z 2 ) + g 2 ( 𝐱 , z 1 , z 2 ) z 3 z ˙ i = f i ( 𝐱 , z 1 , z 2 , , z i - 1 , z i ) + g i ( 𝐱 , z 1 , z 2 , , z i - 1 , z i ) z i + 1 for 1 i < k - 1 z ˙ k - 1 = f k - 1 ( 𝐱 , z 1 , z 2 , , z k - 1 ) + g k - 1 ( 𝐱 , z 1 , z 2 , , z k - 1 ) z k z ˙ k = f k ( 𝐱 , z 1 , z 2 , , z k - 1 , z k ) + g k ( 𝐱 , z 1 , z 2 , , z k - 1 , z k ) u \begin{cases}\dot{\mathbf{x}}=f_{0}(\mathbf{x})+g_{0}(\mathbf{x})z_{1}\\ \dot{z}_{1}=f_{1}(\mathbf{x},z_{1})+g_{1}(\mathbf{x},z_{1})z_{2}\\ \dot{z}_{2}=f_{2}(\mathbf{x},z_{1},z_{2})+g_{2}(\mathbf{x},z_{1},z_{2})z_{3}\\ \vdots\\ \dot{z}_{i}=f_{i}(\mathbf{x},z_{1},z_{2},\ldots,z_{i-1},z_{i})+g_{i}(\mathbf{x% },z_{1},z_{2},\ldots,z_{i-1},z_{i})z_{i+1}\quad\,\text{ for }1\leq i<k-1\\ \vdots\\ \dot{z}_{k-1}=f_{k-1}(\mathbf{x},z_{1},z_{2},\ldots,z_{k-1})+g_{k-1}(\mathbf{x% },z_{1},z_{2},\ldots,z_{k-1})z_{k}\\ \dot{z}_{k}=f_{k}(\mathbf{x},z_{1},z_{2},\ldots,z_{k-1},z_{k})+g_{k}(\mathbf{x% },z_{1},z_{2},\dots,z_{k-1},z_{k})u\end{cases}
  2. 𝐱 n \mathbf{x}\in\mathbb{R}^{n}
  3. n 1 n\geq 1
  4. z 1 , z 2 , , z i , , z k - 1 , z k z_{1},z_{2},\ldots,z_{i},\ldots,z_{k-1},z_{k}
  5. u u
  6. f 0 , f 1 , f 2 , , f i , , f k - 1 , f k f_{0},f_{1},f_{2},\ldots,f_{i},\ldots,f_{k-1},f_{k}
  7. f i ( 0 , 0 , , 0 ) = 0 f_{i}(0,0,\dots,0)=0
  8. g 1 , g 2 , , g i , , g k - 1 , g k g_{1},g_{2},\ldots,g_{i},\ldots,g_{k-1},g_{k}
  9. g i ( 𝐱 , z 1 , , z k ) 0 g_{i}(\mathbf{x},z_{1},\ldots,z_{k})\neq 0
  10. 1 i k 1\leq i\leq k
  11. f i f_{i}
  12. g i g_{i}
  13. z ˙ i \dot{z}_{i}
  14. x , z 1 , , z i x,z_{1},\ldots,z_{i}
  15. 𝐱 ˙ = f 0 ( 𝐱 ) + g 0 ( 𝐱 ) u x ( 𝐱 ) \dot{\mathbf{x}}=f_{0}(\mathbf{x})+g_{0}(\mathbf{x})u_{x}(\mathbf{x})
  16. u x ( 𝐱 ) u_{x}(\mathbf{x})
  17. u x ( 𝟎 ) = 0 u_{x}(\mathbf{0})=0
  18. u x u_{x}
  19. V x V_{x}
  20. u 1 ( 𝐱 , z 1 ) u_{1}(\mathbf{x},z_{1})
  21. z ˙ 1 = f 1 ( 𝐱 , z 1 ) + g 1 ( 𝐱 , z 1 ) u 1 ( 𝐱 , z 1 ) \dot{z}_{1}=f_{1}(\mathbf{x},z_{1})+g_{1}(\mathbf{x},z_{1})u_{1}(\mathbf{x},z_% {1})
  22. z 1 z_{1}
  23. u x u_{x}
  24. V 1 ( 𝐱 , z 1 ) = V x ( 𝐱 ) + 1 2 ( z 1 - u x ( 𝐱 ) ) 2 V_{1}(\mathbf{x},z_{1})=V_{x}(\mathbf{x})+\frac{1}{2}(z_{1}-u_{x}(\mathbf{x}))% ^{2}
  25. u 1 u_{1}
  26. V ˙ 1 \dot{V}_{1}
  27. u 2 ( 𝐱 , z 1 , z 2 ) u_{2}(\mathbf{x},z_{1},z_{2})
  28. z ˙ 2 = f 2 ( 𝐱 , z 1 , z 2 ) + g 2 ( 𝐱 , z 1 , z 2 ) u 2 ( 𝐱 , z 1 , z 2 ) \dot{z}_{2}=f_{2}(\mathbf{x},z_{1},z_{2})+g_{2}(\mathbf{x},z_{1},z_{2})u_{2}(% \mathbf{x},z_{1},z_{2})
  29. z 2 z_{2}
  30. u 1 u_{1}
  31. V 2 ( 𝐱 , z 1 , z 2 ) = V 1 ( 𝐱 , z 1 ) + 1 2 ( z 2 - u 1 ( 𝐱 , z 1 ) ) 2 V_{2}(\mathbf{x},z_{1},z_{2})=V_{1}(\mathbf{x},z_{1})+\frac{1}{2}(z_{2}-u_{1}(% \mathbf{x},z_{1}))^{2}
  32. u 2 u_{2}
  33. V ˙ 2 \dot{V}_{2}
  34. u u
  35. u u
  36. z k z_{k}
  37. u k - 1 u_{k-1}
  38. u k - 1 u_{k-1}
  39. z k - 1 z_{k-1}
  40. u k - 2 u_{k-2}
  41. u k - 2 u_{k-2}
  42. z k - 2 z_{k-2}
  43. u k - 3 u_{k-3}
  44. u 2 u_{2}
  45. z 2 z_{2}
  46. u 1 u_{1}
  47. u 1 u_{1}
  48. z 1 z_{1}
  49. u x u_{x}
  50. u x u_{x}
  51. 𝐱 \mathbf{x}
  52. f i f_{i}
  53. 0 i k 0\leq i\leq k
  54. g i g_{i}
  55. 1 i k 1\leq i\leq k
  56. u x u_{x}
  57. u x ( 𝟎 ) = 0 u_{x}(\mathbf{0})=0
  58. 𝐱 = 𝟎 \mathbf{x}=\mathbf{0}\,
  59. z 1 = 0 z_{1}=0
  60. z 2 = 0 z_{2}=0
  61. z k - 1 = 0 z_{k-1}=0
  62. z k = 0 z_{k}=0

Strominger's_equations.html

  1. ω \omega
  2. g = η g=\eta
  3. N = 0 N=0
  4. ω \omega
  5. ¯ ω = i Tr F ( h ) F ( h ) - i Tr R - ( ω ) R - ( ω ) , \partial\bar{\partial}\omega=i\,\text{Tr}F(h)\wedge F(h)-i\,\text{Tr}R^{-}(% \omega)\wedge R^{-}(\omega),
  6. d ω = i ( - ¯ ) ln || Ω || , d^{\dagger}\omega=i(\partial-\bar{\partial})\,\text{ln}||\Omega||,
  7. R - R^{-}
  8. ω \omega
  9. Ω \Omega
  10. ω \omega
  11. d ( || Ω || ω ω 2 ) = 0 d(||\Omega||_{\omega}\omega^{2})=0
  12. ω a b ¯ F a b ¯ = 0 , \omega^{a\bar{b}}F_{a\bar{b}}=0,
  13. F a b = F a ¯ b ¯ = 0. F_{ab}=F_{\bar{a}\bar{b}}=0.
  14. c 2 ( M ) = c 2 ( F ) c_{2}(M)=c_{2}(F)
  15. Ω \Omega
  16. h n , 0 = 1 h^{n,0}=1
  17. c 1 = 0 c_{1}=0
  18. T Y T_{Y}
  19. ω \omega
  20. Y Y
  21. T Y T_{Y}
  22. Δ \Delta
  23. ϕ \phi
  24. Δ ( y ) = ϕ ( y ) + constant \Delta(y)=\phi(y)+\,\text{constant}
  25. ϕ ( y ) = 1 8 ln || Ω || + constant \phi(y)=\frac{1}{8}\,\text{ln}||\Omega||+\,\text{constant}
  26. H = i 2 ( ¯ - ) ω . H=\frac{i}{2}(\bar{\partial}-\partial)\omega.

Strong_gravitational_lensing.html

  1. Σ c r \Sigma_{cr}\,

Strong_orientation.html

  1. G G
  2. G G
  3. u u
  4. v v
  5. G G
  6. u u
  7. v v
  8. k 2 \lfloor\frac{k}{2}\rfloor
  9. k k
  10. u u
  11. v v
  12. 2 k 2k
  13. k k
  14. G G
  15. G G
  16. G G
  17. G G
  18. G G
  19. G G
  20. G G
  21. G G
  22. T < s u b > G ( 0 , 2 ) T<sub>G(0,2)

Structural_complexity_theory.html

  1. f Ω ( log ( n ) ) f\in\Omega(\log(n))
  2. NSPACE ( f ( n ) ) DSPACE ( ( f ( n ) ) 2 ) . \,\text{NSPACE}\left(f\left(n\right)\right)\subseteq\,\text{DSPACE}\left(\left% (f\left(n\right)\right)^{2}\right).

Structure_constants.html

  1. T i T^{i}
  2. f a b c f^{abc}
  3. [ T a , T b ] = c f a b c T c [T^{a},T^{b}]=\sum_{c}f^{abc}T^{c}
  4. X , Y X,Y
  5. exp ( X ) exp ( Y ) exp ( X + Y + 1 2 [ X , Y ] ) \exp(X)\exp(Y)\approx\exp(X+Y+\tfrac{1}{2}[X,Y])
  6. σ i \sigma_{i}
  7. ϵ a b c \epsilon^{abc}
  8. [ σ a , σ b ] = i ϵ a b c σ c [\sigma_{a},\sigma_{b}]=i\epsilon^{abc}\sigma_{c}\,
  9. f a b c = i ϵ a b c f^{abc}=i\epsilon^{abc}
  10. δ a b \delta_{ab}
  11. T a = λ a 2 . T^{a}=\frac{\lambda^{a}}{2}.\,
  12. λ \lambda\,
  13. λ 1 = ( 0 1 0 1 0 0 0 0 0 ) \lambda^{1}=\begin{pmatrix}0&1&0\\ 1&0&0\\ 0&0&0\end{pmatrix}
  14. λ 2 = ( 0 - i 0 i 0 0 0 0 0 ) \lambda^{2}=\begin{pmatrix}0&-i&0\\ i&0&0\\ 0&0&0\end{pmatrix}
  15. λ 3 = ( 1 0 0 0 - 1 0 0 0 0 ) \lambda^{3}=\begin{pmatrix}1&0&0\\ 0&-1&0\\ 0&0&0\end{pmatrix}
  16. λ 4 = ( 0 0 1 0 0 0 1 0 0 ) \lambda^{4}=\begin{pmatrix}0&0&1\\ 0&0&0\\ 1&0&0\end{pmatrix}
  17. λ 5 = ( 0 0 - i 0 0 0 i 0 0 ) \lambda^{5}=\begin{pmatrix}0&0&-i\\ 0&0&0\\ i&0&0\end{pmatrix}
  18. λ 6 = ( 0 0 0 0 0 1 0 1 0 ) \lambda^{6}=\begin{pmatrix}0&0&0\\ 0&0&1\\ 0&1&0\end{pmatrix}
  19. λ 7 = ( 0 0 0 0 0 - i 0 i 0 ) \lambda^{7}=\begin{pmatrix}0&0&0\\ 0&0&-i\\ 0&i&0\end{pmatrix}
  20. λ 8 = 1 3 ( 1 0 0 0 1 0 0 0 - 2 ) . \lambda^{8}=\frac{1}{\sqrt{3}}\begin{pmatrix}1&0&0\\ 0&1&0\\ 0&0&-2\end{pmatrix}.
  21. [ T a , T b ] = i f a b c T c \left[T^{a},T^{b}\right]=if^{abc}T^{c}\,
  22. { T a , T b } = 1 3 δ a b + d a b c T c . \{T^{a},T^{b}\}=\frac{1}{3}\delta^{ab}+d^{abc}T^{c}.\,
  23. f 123 = 1 f^{123}=1\,
  24. f 147 = - f 156 = f 246 = f 257 = f 345 = - f 367 = 1 2 f^{147}=-f^{156}=f^{246}=f^{257}=f^{345}=-f^{367}=\frac{1}{2}\,
  25. f 458 = f 678 = 3 2 , f^{458}=f^{678}=\frac{\sqrt{3}}{2},\,
  26. f a b c f^{abc}
  27. d 118 = d 228 = d 338 = - d 888 = 1 3 d^{118}=d^{228}=d^{338}=-d^{888}=\frac{1}{\sqrt{3}}\,
  28. d 448 = d 558 = d 668 = d 778 = - 1 2 3 d^{448}=d^{558}=d^{668}=d^{778}=-\frac{1}{2\sqrt{3}}\,
  29. d 146 = d 157 = - d 247 = d 256 = d 344 = d 355 = - d 366 = - d 377 = 1 2 . d^{146}=d^{157}=-d^{247}=d^{256}=d^{344}=d^{355}=-d^{366}=-d^{377}=\frac{1}{2}.\,
  30. G μ ν a G^{a}_{\mu\nu}\,
  31. G μ ν a = μ 𝒜 ν a - ν 𝒜 μ a + g f a b c 𝒜 μ b 𝒜 ν c , G^{a}_{\mu\nu}=\partial_{\mu}\mathcal{A}^{a}_{\nu}-\partial_{\nu}\mathcal{A}^{% a}_{\mu}+gf^{abc}\mathcal{A}^{b}_{\mu}\mathcal{A}^{c}_{\nu}\,,

Structured_support_vector_machine.html

  1. \ell
  2. ( s y m b o l x n , y n ) 𝒳 × 𝒴 (symbol{x}_{n},y_{n})\in\mathcal{X}\times\mathcal{Y}
  3. n = 1 , , n=1,\dots,\ell
  4. 𝒳 \mathcal{X}
  5. 𝒴 \mathcal{Y}
  6. min s y m b o l w s y m b o l w 2 + C n = 1 max y 𝒴 ( Δ ( y n , y ) + s y m b o l w Ψ ( s y m b o l x n , y ) - s y m b o l w Ψ ( s y m b o l x n , y n ) ) \underset{symbol{w}}{\min}\quad\|symbol{w}\|^{2}+C\sum_{n=1}^{\ell}\underset{y% \in\mathcal{Y}}{\max}\left(\Delta(y_{n},y)+symbol{w}^{\prime}\Psi(symbol{x}_{n% },y)-symbol{w}^{\prime}\Psi(symbol{x}_{n},y_{n})\right)
  7. s y m b o l w symbol{w}
  8. Δ : 𝒴 × 𝒴 + \Delta:\mathcal{Y}\times\mathcal{Y}\to\mathbb{R}_{+}
  9. Δ ( y , z ) 0 \Delta(y,z)\geq 0
  10. Δ ( y , y ) = 0 y , z 𝒴 \Delta(y,y)=0\;\;\forall y,z\in\mathcal{Y}
  11. Ψ : 𝒳 × 𝒴 d \Psi:\mathcal{X}\times\mathcal{Y}\to\mathbb{R}^{d}
  12. ξ n \xi_{n}
  13. min s y m b o l w , s y m b o l ξ s y m b o l w 2 + C n = 1 ξ n s.t. s y m b o l w Ψ ( s y m b o l x n , y n ) - s y m b o l w Ψ ( s y m b o l x n , y ) + ξ n Δ ( y n , y ) , n = 1 , , , y 𝒴 \begin{array}[]{cl}\underset{symbol{w},symbol{\xi}}{\min}&\|symbol{w}\|^{2}+C% \sum_{n=1}^{\ell}\xi_{n}\\ \textrm{s.t.}&symbol{w}^{\prime}\Psi(symbol{x}_{n},y_{n})-symbol{w}^{\prime}% \Psi(symbol{x}_{n},y)+\xi_{n}\geq\Delta(y_{n},y),\qquad n=1,\dots,\ell,\quad% \forall y\in\mathcal{Y}\end{array}
  14. s y m b o l x 𝒳 symbol{x}\in\mathcal{X}
  15. f : 𝒳 𝒴 f:\mathcal{X}\to\mathcal{Y}
  16. 𝒴 \mathcal{Y}
  17. s y m b o l w symbol{w}
  18. f ( s y m b o l x ) = argmax y 𝒴 s y m b o l w Ψ ( s y m b o l x , y ) f(symbol{x})=\underset{y\in\mathcal{Y}}{\textrm{argmax}}\quad symbol{w}^{% \prime}\Psi(symbol{x},y)
  19. Ψ \Psi
  20. s y m b o l w symbol{w}
  21. ( s y m b o l x n , y n ) (symbol{x}_{n},y_{n})
  22. y n * = argmax y 𝒴 ( Δ ( y n , y ) + s y m b o l w Ψ ( s y m b o l x n , y ) - s y m b o l w Ψ ( s y m b o l x n , y n ) - ξ n ) y_{n}^{*}=\underset{y\in\mathcal{Y}}{\textrm{argmax}}\left(\Delta(y_{n},y)+% symbol{w}^{\prime}\Psi(symbol{x}_{n},y)-symbol{w}^{\prime}\Psi(symbol{x}_{n},y% _{n})-\xi_{n}\right)
  23. - s y m b o l w Ψ ( s y m b o l x n , y n ) - ξ n -symbol{w}^{\prime}\Psi(symbol{x}_{n},y_{n})-\xi_{n}
  24. Δ ( y n , y ) + s y m b o l w Ψ ( s y m b o l x n , y ) \Delta(y_{n},y)+symbol{w}^{\prime}\Psi(symbol{x}_{n},y)
  25. y n * = argmax y 𝒴 ( Δ ( y n , y ) + s y m b o l w Ψ ( s y m b o l x n , y ) ) y_{n}^{*}=\underset{y\in\mathcal{Y}}{\textrm{argmax}}\left(\Delta(y_{n},y)+% symbol{w}^{\prime}\Psi(symbol{x}_{n},y)\right)
  26. Δ ( y n , y ) \Delta(y_{n},y)
  27. Δ \Delta

Struve_function.html

  1. y ( x ) y(x)
  2. x 2 d 2 y d x 2 + x d y d x + ( x 2 - α 2 ) y = 4 ( x 2 ) α + 1 π Γ ( α + 1 2 ) x^{2}\frac{d^{2}y}{dx^{2}}+x\frac{dy}{dx}+\left(x^{2}-\alpha^{2}\right)y=\frac% {4\left(\frac{x}{2}\right)^{\alpha+1}}{\sqrt{\pi}\Gamma\left(\alpha+\frac{1}{2% }\right)}
  3. 𝐇 α ( x ) = m = 0 ( - 1 ) m Γ ( m + 3 2 ) Γ ( m + α + 3 2 ) ( x 2 ) 2 m + α + 1 \mathbf{H}_{\alpha}(x)=\sum_{m=0}^{\infty}\frac{(-1)^{m}}{\Gamma\left(m+\frac{% 3}{2}\right)\Gamma\left(m+\alpha+\frac{3}{2}\right)}\left({\frac{x}{2}}\right)% ^{2m+\alpha+1}
  4. Γ ( z ) Γ(z)
  5. 𝐋 ν ( z ) = ( z 2 ) ν + 1 k = 0 1 Γ ( 3 2 + k ) Γ ( 3 2 + k + ν ) ( z 2 ) 2 k \mathbf{L}_{\nu}(z)=\left({\frac{z}{2}}\right)^{\nu+1}\sum_{k=0}^{\infty}\frac% {1}{\Gamma\left(\frac{3}{2}+k\right)\Gamma\left(\frac{3}{2}+k+\nu\right)}\left% (\frac{z}{2}\right)^{2k}
  6. α α
  7. R e ( α ) > 1 2 Re(α)>−\frac{1}{2}
  8. 𝐇 α ( x ) = 2 ( x 2 ) α π Γ ( α + 1 2 ) 0 π 2 sin ( x cos τ ) sin 2 α ( τ ) d τ . \mathbf{H}_{\alpha}(x)=\frac{2\left(\frac{x}{2}\right)^{\alpha}}{\sqrt{\pi}% \Gamma\left(\alpha+\frac{1}{2}\right)}\int_{0}^{\frac{\pi}{2}}\sin(x\cos\tau)% \sin^{2\alpha}(\tau)d\tau.
  9. x x
  10. x x
  11. 𝐇 α ( x ) - Y α ( x ) ( x 2 ) α - 1 π Γ ( α + 1 2 ) + O ( ( x 2 ) α - 3 ) , \mathbf{H}_{\alpha}(x)-Y_{\alpha}(x)\to\frac{\left(\frac{x}{2}\right)^{\alpha-% 1}}{\sqrt{\pi}\Gamma\left(\alpha+\frac{1}{2}\right)}+O\left(\left(\tfrac{x}{2}% \right)^{\alpha-3}\right),
  12. 𝐇 α - 1 ( x ) + 𝐇 α + 1 ( x ) \displaystyle\mathbf{H}_{\alpha-1}(x)+\mathbf{H}_{\alpha+1}(x)
  13. n n
  14. 𝐄 n ( z ) = 1 π k = 0 n - 1 2 Γ ( k + 1 2 ) ( z 2 ) n - 2 k - 1 Γ ( n - k - 1 2 ) 𝐇 n 𝐄 - n ( z ) = ( - 1 ) n + 1 π k = 0 n - 1 2 Γ ( n - k - 1 2 ) ( z 2 ) - n + 2 k + 1 Γ ( k + 3 2 ) 𝐇 - n . \begin{aligned}\displaystyle\mathbf{E}_{n}(z)&\displaystyle=\frac{1}{\pi}\sum_% {k=0}^{\left\lfloor\frac{n-1}{2}\right\rfloor}\frac{\Gamma\left(k+\frac{1}{2}% \right)\left(\frac{z}{2}\right)^{n-2k-1}}{\Gamma\left(n-k-\frac{1}{2}\right)}% \mathbf{H}_{n}\\ \displaystyle\mathbf{E}_{-n}(z)&\displaystyle=\frac{(-1)^{n+1}}{\pi}\sum_{k=0}% ^{\left\lfloor\frac{n-1}{2}\right\rfloor}\frac{\Gamma(n-k-\frac{1}{2})\left(% \frac{z}{2}\right)^{-n+2k+1}}{\Gamma\left(k+\frac{3}{2}\right)}\mathbf{H}_{-n}% .\end{aligned}
  15. n + 1 2 n+\frac{1}{2}
  16. n n
  17. n n
  18. 𝐇 - n - 1 2 ( z ) = ( - 1 ) n J n + 1 2 ( z ) \mathbf{H}_{-n-\frac{1}{2}}(z)=(-1)^{n}J_{n+\frac{1}{2}}(z)
  19. 𝐇 α ( z ) = ( z 2 ) α + 1 2 2 π Γ ( α + 3 2 ) F 2 1 ( 1 , 3 2 , α + 3 2 , - z 2 4 ) . \mathbf{H}_{\alpha}(z)=\frac{\left(\frac{z}{2}\right)^{\alpha+\frac{1}{2}}}{% \sqrt{2\pi}\Gamma\left(\alpha+\tfrac{3}{2}\right)}{}_{1}F_{2}\left(1,\tfrac{3}% {2},\alpha+\tfrac{3}{2},-\tfrac{z^{2}}{4}\right).

Student_governments_in_the_United_States.html

  1. \swarrow
  2. \downarrow
  3. \downarrow
  4. \searrow

Stumpff_function.html

  1. c k ( x ) = 1 k ! - x ( k + 2 ) ! + x 2 ( k + 4 ) ! - = i = 0 ( - 1 ) i x i ( k + 2 i ) ! c_{k}(x)=\frac{1}{k!}-\frac{x}{(k+2)!}+\frac{x^{2}}{(k+4)!}-\cdots=\sum_{i=0}^% {\infty}{\frac{(-1)^{i}x^{i}}{(k+2i)!}}
  2. k = 0 , 1 , 2 , 3 , k=0,1,2,3,\ldots
  3. c 0 ( x ) = cos x , for x > 0 c_{0}(x)=\cos{\sqrt{x}},\,\text{ for }x>0
  4. c 1 ( x ) = sin x x , for x > 0 c_{1}(x)=\frac{\sin{\sqrt{x}}}{\sqrt{x}},\,\text{ for }x>0
  5. c 0 ( x ) = cosh - x , for x < 0 c_{0}(x)=\cosh{\sqrt{-x}},\,\text{ for }x<0
  6. c 1 ( x ) = sinh - x - x , for x < 0 c_{1}(x)=\frac{\sinh{\sqrt{-x}}}{\sqrt{-x}},\,\text{ for }x<0
  7. x c k + 2 ( x ) = 1 k ! - c k ( x ) , for k = 0 , 1 , 2 , . xc_{k+2}(x)=\frac{1}{k!}-c_{k}(x),\,\text{ for }k=0,1,2,\ldots\,.

Størmer_number.html

  1. G a / b = arctan b a G_{a/b}=\arctan\frac{b}{a}
  2. G a / b G_{a/b}
  3. a + b i a+bi
  4. n ± i n\pm i
  5. n n
  6. n 2 + 1 n^{2}+1
  7. p p

Success_likelihood_index_method.html

  1. SLI = j i R i j W i \mbox{SLI}~{}_{j}=\sum_{i}R_{ij}W_{i}
  2. log P = a SLI + b \log P=a\,\mbox{SLI}~{}+b
  3. log 10 HEP = a SLI + b \log_{10}\mbox{HEP}~{}=a\,\mbox{SLI}~{}+b
  4. log HEP = - 1.85 SLI + 7.1 \log\mbox{HEP}~{}=-1.85\,\mbox{SLI}~{}+7.1

Sulston_score.html

  1. α \alpha
  2. β \beta
  3. m m
  4. n n
  5. m n m\geq n
  6. α \alpha
  7. β \beta
  8. h h
  9. β \beta
  10. α \alpha
  11. n n
  12. h h
  13. x x
  14. ± t \pm t
  15. x ± t x\pm t
  16. j j
  17. β \beta
  18. i i
  19. α \alpha
  20. i { 1 , 2 , , m } i\in\{1,2,\dots,m\}
  21. j { 1 , 2 , , n } j\in\{1,2,\dots,n\}
  22. j j
  23. E i j E_{ij}
  24. i i
  25. j j
  26. i i
  27. j j
  28. 2 t 2t
  29. j j
  30. i i
  31. G G
  32. P E i j = 2 t / G P\langle E_{ij}\rangle=2t/G
  33. i i
  34. j j
  35. P E i , j C = 1 - 2 t / G P\langle E_{i,j}^{C}\rangle=1-2t/G
  36. α \alpha
  37. j j
  38. β \beta
  39. i { 1 , 2 , , m } i\in\{1,2,\dots,m\}
  40. E i , j C E_{i,j}^{C}
  41. P E 1 , j C E 2 , j C E m , j C P\langle E_{1,j}^{C}\cap E_{2,j}^{C}\cap\cdots\cap E_{m,j}^{C}\rangle
  42. α \alpha
  43. j j
  44. β \beta
  45. j j
  46. P E 1 , j C × P E 2 , j C × × P E m , j C = ( 1 - 2 t / G ) m . P\langle E_{1,j}^{C}\rangle\times P\langle E_{2,j}^{C}\rangle\times\cdots% \times P\langle E_{m,j}^{C}\rangle=\left(1-2t/G\right)^{m}.
  47. p = 1 - ( 1 - 2 t / G ) m p=1-\left(1-2t/G\right)^{m}
  48. p p
  49. α \alpha
  50. j j
  51. β \beta
  52. p p
  53. j j
  54. β \beta
  55. p p
  56. h h
  57. S S
  58. h h
  59. S = j = h n C n , j p j ( 1 - p ) n - j , S=\sum_{j=h}^{n}C_{n,j}p^{j}(1-p)^{n-j},
  60. C n , j C_{n,j}

Sum_frequency_generation_spectroscopy.html

  1. P \overrightarrow{P}
  2. P = ϵ 0 ( χ ( 1 ) E + χ ( 2 ) E 2 + χ ( 3 ) E 3 + + χ ( n ) E n ) = ϵ 0 i = 1 n χ ( i ) E i \overrightarrow{P}=\epsilon_{0}\left(\chi^{(1)}\overrightarrow{E}+\chi^{(2)}% \overrightarrow{E}^{2}+\chi^{(3)}\overrightarrow{E}^{3}+\dots+\chi^{(n)}% \overrightarrow{E}^{n}\right)=\epsilon_{0}\sum_{i=1}^{n}\chi^{(i)}% \overrightarrow{E}^{i}
  3. χ ( i ) \chi^{(i)}
  4. i i
  5. i [ 1 , 2 , 3 , , n ] i\in[1,2,3,\dots,n]
  6. I i n v I_{inv}
  7. I i n v L = - L I_{inv}\overrightarrow{L}=-\overrightarrow{L}
  8. L \overrightarrow{L}
  9. I i n v I_{inv}
  10. I i n v P = - P = I i n v ( ϵ 0 i = 1 n χ ( i ) E i ) = ϵ 0 i = 1 n χ ( i ) ( I i n v E ) i = ϵ 0 i = 1 n ( - 1 ) i χ ( i ) E i . I_{inv}\overrightarrow{P}=-\overrightarrow{P}=I_{inv}\left(\epsilon_{0}\sum_{i% =1}^{n}\chi^{(i)}\overrightarrow{E}^{i}\right)=\epsilon_{0}\sum_{i=1}^{n}\chi^% {(i)}\left(I_{inv}\overrightarrow{E}\right)^{i}=\epsilon_{0}\sum_{i=1}^{n}(-1)% ^{i}\chi^{(i)}\overrightarrow{E}^{i}.
  11. P - P = 0 = ϵ 0 i = 1 n ( 1 + ( - 1 ) i ) χ ( i ) E i = 2 ϵ 0 i = 1 n / 2 χ ( 2 i ) E ( 2 i ) \overrightarrow{P}-\overrightarrow{P}=\overrightarrow{0}=\epsilon_{0}\sum_{i=1% }^{n}\left(1+(-1)^{i}\right)\chi^{(i)}\overrightarrow{E}^{i}=2\epsilon_{0}\sum% _{i=1}^{n/2}\chi^{(2i)}\overrightarrow{E}^{(2i)}
  12. χ ( 2 i ) = 0 \chi^{(2i)}=0
  13. i [ 1 , 2 , 3 , , n / 2 ] i\in[1,2,3,\dots,n/2]
  14. k k
  15. k k
  16. n n
  17. m = n - 1 m=n-1
  18. χ ( 2 ) \chi^{(2)}
  19. I I
  20. I ( ω 3 ; ω 1 , ω 2 ) | χ ( 2 ) | 2 I 1 ( ω 1 ) I 2 ( ω 2 ) I(\omega_{3};\omega_{1},\omega_{2})\propto|\chi^{(2)}|^{2}I_{1}(\omega_{1})I_{% 2}(\omega_{2})
  21. ω 1 \omega_{1}
  22. ω 2 \omega_{2}
  23. ω 3 = ω 1 + ω 2 \omega_{3}=\omega_{1}+\omega_{2}
  24. ω 2 \omega_{2}
  25. sec 2 β \sec^{2}\beta
  26. χ = χ n r + χ r \chi=\chi_{nr}+\chi_{r}
  27. χ n r \chi_{nr}
  28. χ r \chi_{r}
  29. q A q ω 2 - ω 0 q + i Γ q \sum_{q}\frac{A_{q}}{\omega_{2}-\omega_{0_{q}}+i\Gamma_{q}}
  30. A A
  31. ω 0 \omega_{0}
  32. Γ \Gamma
  33. q > 1 q>1
  34. μ \mu
  35. α \alpha
  36. χ = | χ n r | e i ϕ + q A q ω 2 - ω 0 q + i Γ q \chi=|\chi_{nr}|e^{i\phi}+\sum_{q}\frac{A_{q}}{\omega_{2}-\omega_{0_{q}}+i% \Gamma_{q}}
  37. χ ( 2 ) \chi^{(2)}
  38. χ ( 2 ) \chi^{(2)}
  39. C C_{\infty}
  40. χ z z z ( 2 ) , \chi^{(2)}_{zzz},
  41. χ x x z ( 2 ) = χ y y z ( 2 ) , \chi^{(2)}_{xxz}=\chi^{(2)}_{yyz},
  42. χ x z x ( 2 ) = χ y z y ( 2 ) , \chi^{(2)}_{xzx}=\chi^{(2)}_{yzy},
  43. χ z x x ( 2 ) = χ z y y ( 2 ) . \chi^{(2)}_{zxx}=\chi^{(2)}_{zyy}.
  44. I P P P = | f z f z f z χ z z z ( 2 ) + f z f i f i χ z i i ( 2 ) + f i f z f i χ z i i ( 2 ) + f i f i f z χ i i z ( 2 ) | 2 , I_{PPP}=|f^{\prime}_{z}f_{z}f_{z}\chi_{zzz}^{(2)}+f^{\prime}_{z}f_{i}f_{i}\chi% _{zii}^{(2)}+f^{\prime}_{i}f_{z}f_{i}\chi_{zii}^{(2)}+f^{\prime}_{i}f_{i}f_{z}% \chi_{iiz}^{(2)}|^{2},
  45. I S S P = | f i f i f z χ i i z ( 2 ) | 2 , I_{SSP}=|f^{\prime}_{i}f_{i}f_{z}\chi_{iiz}^{(2)}|^{2},
  46. I S P S = | f i f z f i χ z i i ( 2 ) | 2 , I_{SPS}=|f^{\prime}_{i}f_{z}f_{i}\chi_{zii}^{(2)}|^{2},
  47. I P S S = | f z f i f i χ z i i ( 2 ) | 2 I_{PSS}=|f^{\prime}_{z}f_{i}f_{i}\chi_{zii}^{(2)}|^{2}
  48. i i
  49. x y xy
  50. f f
  51. f f^{\prime}

Sum_rule_in_quantum_mechanics.html

  1. H ^ \hat{H}
  2. | n |n\rangle
  3. E n E_{n}
  4. H ^ | n = E n | n . \hat{H}|n\rangle=E_{n}|n\rangle.
  5. A ^ \hat{A}
  6. C ^ ( k ) \hat{C}^{(k)}
  7. C ^ ( 0 ) \displaystyle\hat{C}^{(0)}
  8. C ^ ( 0 ) \hat{C}^{(0)}
  9. A ^ \hat{A}
  10. C ^ ( 1 ) \hat{C}^{(1)}
  11. ( C ^ ( 1 ) ) = ( H ^ A ^ ) - ( A ^ H ^ ) = A ^ H ^ - H ^ A ^ = - C ^ ( 1 ) . \left(\hat{C}^{(1)}\right)^{\dagger}=(\hat{H}\hat{A})^{\dagger}-(\hat{A}\hat{H% })^{\dagger}=\hat{A}\hat{H}-\hat{H}\hat{A}=-\hat{C}^{(1)}.
  12. ( C ^ ( k ) ) = ( - 1 ) k C ^ ( k ) \left(\hat{C}^{(k)}\right)^{\dagger}=(-1)^{k}\hat{C}^{(k)}
  13. m | C ^ ( k ) | n = ( E m - E n ) k m | A ^ | n . \langle m|\hat{C}^{(k)}|n\rangle=(E_{m}-E_{n})^{k}\langle m|\hat{A}|n\rangle.
  14. | m | A ^ | n | 2 = m | A ^ | n m | A ^ | n = m | A ^ | n n | A ^ | m . |\langle m|\hat{A}|n\rangle|^{2}=\langle m|\hat{A}|n\rangle\langle m|\hat{A}|n% \rangle^{\ast}=\langle m|\hat{A}|n\rangle\langle n|\hat{A}|m\rangle.
  15. m | [ A ^ , C ^ ( k ) ] | m \displaystyle\langle m|[\hat{A},\hat{C}^{(k)}]|m\rangle
  16. m | [ A ^ , C ^ ( k ) ] | m = { 0 , if k is even 2 n ( E n - E m ) k | m | A ^ | n | 2 , if k is odd . \langle m|[\hat{A},\hat{C}^{(k)}]|m\rangle=\begin{cases}0,&\mbox{if }~{}k\mbox% { is even}\\ 2\sum_{n}(E_{n}-E_{m})^{k}|\langle m|\hat{A}|n\rangle|^{2},&\mbox{if }~{}k% \mbox{ is odd}~{}.\end{cases}
  17. k = 1 k=1
  18. m | [ A ^ , [ H ^ , A ^ ] ] | m = 2 n ( E n - E m ) | m | A ^ | n | 2 . \langle m|[\hat{A},[\hat{H},\hat{A}]]|m\rangle=2\sum_{n}(E_{n}-E_{m})|\langle m% |\hat{A}|n\rangle|^{2}.

Summed_area_table.html

  1. I ( x , y ) = x x y y i ( x , y ) I(x,y)=\sum_{\begin{smallmatrix}x^{\prime}\leq x\\ y^{\prime}\leq y\end{smallmatrix}}i(x^{\prime},y^{\prime})
  2. I ( x , y ) = i ( x , y ) + I ( x - 1 , y ) + I ( x , y - 1 ) - I ( x - 1 , y - 1 ) I(x,y)=i(x,y)+I(x-1,y)+I(x,y-1)-I(x-1,y-1)\,
  3. i ( x , y ) i(x,y)
  4. x 0 < x x 1 y 0 < y y 1 i ( x , y ) = I ( D ) + I ( A ) - I ( B ) - I ( C ) . \sum_{\begin{smallmatrix}x0<x\leq x1\\ y0<y\leq y1\end{smallmatrix}}i(x,y)=I(D)+I(A)-I(B)-I(C).
  5. x p x^{p}
  6. p p
  7. { 0 , 1 } d \{0,1\}^{d}
  8. p { 0 , 1 } d ( - 1 ) d - p 1 I ( x p ) \sum_{p\in\{0,1\}^{d}}(-1)^{d-\|p\|_{1}}I(x^{p})\,
  9. I ( x ) I(x)
  10. x x
  11. d d
  12. x p x^{p}
  13. d = 2 d=2
  14. A = x ( 0 , 0 ) A=x^{(0,0)}
  15. B = x ( 1 , 0 ) B=x^{(1,0)}
  16. C = x ( 1 , 1 ) C=x^{(1,1)}
  17. D = x ( 0 , 1 ) D=x^{(0,1)}
  18. d = 3 d=3
  19. d = 4 d=4
  20. I ( x , y ) = x x y y i ( x , y ) I(x,y)=\sum_{\begin{smallmatrix}x^{\prime}\leq x\\ y^{\prime}\leq y\end{smallmatrix}}i(x^{\prime},y^{\prime})
  21. I 2 ( x , y ) = x x y y i 2 ( x , y ) I^{2}(x,y)=\sum_{\begin{smallmatrix}x^{\prime}\leq x\\ y^{\prime}\leq y\end{smallmatrix}}i^{2}(x^{\prime},y^{\prime})
  22. Var ( X ) = 1 n i = 1 n ( x i - μ ) 2 . \operatorname{Var}(X)=\frac{1}{n}\sum_{i=1}^{n}(x_{i}-\mu)^{2}.
  23. S 1 S_{1}
  24. S 2 S_{2}
  25. A B C D ABCD
  26. I I
  27. I 2 I^{2}
  28. S 1 S_{1}
  29. S 2 S_{2}
  30. Var ( X ) = 1 n i = 1 n ( x i 2 - 2 μ x i + μ 2 ) = 1 n ( i = 1 n ( x i ) 2 - 2 i = 1 n ( μ x i ) + i = 1 n ( μ 2 ) ) = 1 n ( i = 1 n ( x i ) 2 - 2 i = 1 n ( μ x i ) + n ( μ 2 ) ) \operatorname{Var}(X)=\frac{1}{n}\sum_{i=1}^{n}(x_{i}^{2}-2\cdot\mu\cdot x_{i}% +\mu^{2})=\frac{1}{n}(\sum_{i=1}^{n}(x_{i})^{2}-2\cdot\sum_{i=1}^{n}(\mu\cdot x% _{i})+\sum_{i=1}^{n}(\mu^{2}))=\frac{1}{n}(\sum_{i=1}^{n}(x_{i})^{2}-2\cdot% \sum_{i=1}^{n}(\mu\cdot x_{i})+n\cdot(\mu^{2}))
  31. = 1 n ( i = 1 n ( x i ) 2 - 2 μ i = 1 n ( x i ) + n ( μ 2 ) ) = 1 n ( S 2 - 2 S 1 / n S 1 + n ( ( S 1 / n ) 2 ) ) = 1 n ( S 2 - ( S 1 ) 2 / n ) =\frac{1}{n}(\sum_{i=1}^{n}(x_{i})^{2}-2\cdot\mu\cdot\sum_{i=1}^{n}(x_{i})+n% \cdot(\mu^{2}))=\frac{1}{n}(S_{2}-2\cdot S_{1}/n\cdot S_{1}+n\cdot((S_{1}/n)^{% 2}))=\frac{1}{n}(S_{2}-(S_{1})^{2}/n)
  32. μ = S 1 / n \mu=S_{1}/n
  33. S 2 = i = 1 n ( x i 2 ) S_{2}=\sum_{i=1}^{n}(x_{i}^{2})
  34. μ \mu
  35. V a r Var
  36. I , I 2 I,I^{2}
  37. I 3 ( x , y ) , I 4 ( x , y ) I^{3}(x,y),I^{4}(x,y)

Sun's_curious_identity.html

  1. ( x + m + 1 ) i = 0 m ( - 1 ) i ( x + y + i m - i ) ( y + 2 i i ) - i = 0 m ( x + i m - i ) ( - 4 ) i = ( x - m ) ( x m ) . (x+m+1)\sum_{i=0}^{m}(-1)^{i}{\displaystyle\left({{x+y+i}\atop{m-i}}\right)}{% \displaystyle\left({{y+2i}\atop{i}}\right)}-\sum_{i=0}^{m}{\displaystyle\left(% {{x+i}\atop{m-i}}\right)}(-4)^{i}=(x-m){\displaystyle\left({{x}\atop{m}}\right% )}.

Supercritical_adsorption.html

  1. T c T_{c}
  2. T c T_{c}
  3. T c T_{c}
  4. T c T_{c}
  5. T c T_{c}
  6. T c T_{c}
  7. T c T_{c}
  8. P s P_{s}
  9. P s P_{s}
  10. V r e f V_{ref}
  11. T r e f T_{ref}
  12. V r e f V_{ref}
  13. T a d T_{ad}
  14. V t V_{t}
  15. T a d T_{ad}
  16. V a d V_{ad}
  17. V r e f V_{ref}
  18. p 1 p_{1}
  19. V r e f V_{ref}
  20. n 1 = p 1 V r e f z f 1 R T r e f n_{1}=\frac{p_{1}V_{ref}}{z_{f1}RT_{ref}}
  21. p 2 p_{2}
  22. V r e f V_{ref}
  23. V t V_{t}
  24. V a d V_{ad}
  25. n 2 = p 2 V r e f z f 2 R T r e f n_{2}=\frac{p_{2}V_{ref}}{z_{f2}RT_{ref}}
  26. n 3 = p 2 V t z f 2 R T r e f n_{3}=\frac{p_{2}V_{t}}{z_{f2}RT_{ref}}
  27. n 4 = p 2 V a d z d 2 R T a d n_{4}=\frac{p_{2}V_{ad}}{z_{d2}RT_{ad}}
  28. N = n 1 + n 3 + n 4 - n 2 - n 3 - n 4 N=n_{1}+n_{3}^{\prime}+n_{4}^{\prime}-n_{2}-n_{3}-n_{4}
  29. n 3 n_{3}^{\prime}
  30. n 4 n_{4}^{\prime}
  31. V t V_{t}
  32. V a d V_{ad}
  33. ρ H e \rho_{He}
  34. Δ V = Δ W ρ H e ( p , T ) \Delta V=\frac{\Delta W}{\rho_{He}(p,T)}
  35. W = W e x p - Δ V ρ b ( p , T ) W=W_{exp}-\Delta V\rho_{b}(p,T)
  36. W e x p W_{exp}
  37. V ( r ) = 4 ϵ f f [ ( σ f f r ) 12 - ( σ f f r ) 6 ] V(r)=4\epsilon_{ff}\left[\left(\frac{\sigma_{ff}}{r}\right)^{12}-\left(\frac{% \sigma_{ff}}{r}\right)^{6}\right]
  38. σ f f \sigma_{ff}
  39. ϵ f f \epsilon_{ff}
  40. σ f f \sigma_{ff}
  41. ϵ f f \epsilon_{ff}

Superellipsoid.html

  1. ( | x | r + | y | r ) t / r + | z | t 1 \left(\left|x\right|^{r}+\left|y\right|^{r}\right)^{t/r}+\left|z\right|^{t}\leq 1
  2. a = ( 1 - | z | t ) 1 / t a=(1-\left|z\right|^{t})^{1/t}
  3. | x a | r + | y a | r 1 \left|\frac{x}{a}\right|^{r}+\left|\frac{y}{a}\right|^{r}\leq 1
  4. | u w | t + | z | t 1 \left|\frac{u}{w}\right|^{t}+\left|z\right|^{t}\leq 1
  5. w = ( | cos θ | r + | sin θ | r ) - 1 / r . w=(\left|\cos\theta\right|^{r}+\left|\sin\theta\right|^{r})^{-1/r}.
  6. ( | x A | r + | y B | r ) t / r + | z C | t 1 \left(\left|\frac{x}{A}\right|^{r}+\left|\frac{y}{B}\right|^{r}\right)^{t/r}+% \left|\frac{z}{C}\right|^{t}\leq 1
  7. x ( u , v ) = A c ( v , 2 t ) c ( u , 2 r ) x(u,v)=Ac\left(v,\frac{2}{t}\right)c\left(u,\frac{2}{r}\right)
  8. y ( u , v ) = B c ( v , 2 t ) s ( u , 2 r ) y(u,v)=Bc\left(v,\frac{2}{t}\right)s\left(u,\frac{2}{r}\right)
  9. z ( u , v ) = C s ( v , 2 t ) z(u,v)=Cs\left(v,\frac{2}{t}\right)
  10. c ( ω , m ) = sgn ( cos ω ) | cos ω | m c(\omega,m)=\operatorname{sgn}(\cos\omega)|\cos\omega|^{m}
  11. s ( ω , m ) = sgn ( sin ω ) | sin ω | m s(\omega,m)=\operatorname{sgn}(\sin\omega)|\sin\omega|^{m}
  12. sgn ( x ) = { - 1 , x < 0 0 , x = 0 + 1 , x > 0. \operatorname{sgn}(x)=\begin{cases}-1,&x<0\\ 0,&x=0\\ +1,&x>0.\end{cases}
  13. V = 2 3 A B C 4 r t β ( 1 r , 1 r ) β ( 2 t , 1 t ) . V=\frac{2}{3}ABC\frac{4}{rt}\beta\left(\frac{1}{r},\frac{1}{r}\right)\beta% \left(\frac{2}{t},\frac{1}{t}\right).

Superexchange.html

  1. 1 , 2 = + 2 t M n , O 2 U S ^ 1 S ^ 2 , \mathcal{H}_{1,\,2}=+\frac{2t_{Mn,\,O}^{2}\,}{U}\hat{S}_{1}\cdot\hat{S}_{2}\,,
  2. t M n , O \,t_{Mn,O}
  3. S ^ 1 S ^ 2 , \hat{S}_{1}\cdot\hat{S}_{2}\,,

Superincreasing_sequence.html

  1. 𝐬 𝟏 , 𝐬 𝟐 , \mathbf{s_{1},s_{2},...}
  2. s n + 1 > j = 1 n s j s_{n+1}>\sum_{j=1}^{n}s_{j}

Supernatural_number.html

  1. ω \omega
  2. ω = p p n p , \omega=\prod_{p}p^{n_{p}},
  3. p p
  4. n p n_{p}
  5. v p ( ω ) v_{p}(\omega)
  6. n p n_{p}
  7. n p = n_{p}=\infty
  8. n p n_{p}
  9. n p n_{p}
  10. \infty
  11. ω \omega
  12. \infty
  13. p p n p p p m p = p p n p + m p \prod_{p}p^{n_{p}}\cdot\prod_{p}p^{m_{p}}=\prod_{p}p^{n_{p}+m_{p}}
  14. ω 1 ω 2 \omega_{1}\mid\omega_{2}
  15. v p ( ω 1 ) v p ( ω 2 ) v_{p}(\omega_{1})\leq v_{p}(\omega_{2})
  16. p p
  17. lcm ( { ω i } ) = p p sup ( v p ( ω i ) ) \displaystyle\operatorname{lcm}(\{\omega_{i}\})\displaystyle=\prod_{p}p^{\sup(% v_{p}(\omega_{i}))}
  18. gcd ( { ω i } ) = p p inf ( v p ( ω i ) ) \displaystyle\operatorname{gcd}(\{\omega_{i}\})\displaystyle=\prod_{p}p^{\inf(% v_{p}(\omega_{i}))}
  19. p p
  20. v p ( ω ) = n p v_{p}(\omega)=n_{p}
  21. p p

Superperfect_number.html

  1. σ 2 ( n ) = σ ( σ ( n ) ) = 2 n , \sigma^{2}(n)=\sigma(\sigma(n))=2n\,,
  2. σ m ( n ) = 2 n , \sigma^{m}(n)=2n,
  3. σ m ( n ) = k n . \sigma^{m}(n)=kn\,.

Superstring_theory.html

  1. z z + i A z ( z , z ¯ ) \partial_{z}\rightarrow\partial_{z}+iA_{z}(z,\overline{z})
  2. z X μ - i θ L ¯ Γ μ z θ L \partial_{z}X^{\mu}-i\overline{\theta_{L}}\Gamma^{\mu}\partial_{z}\theta_{L}
  3. z X μ - i θ L ¯ Γ μ z θ L - i θ R ¯ Γ μ z θ R \partial_{z}X^{\mu}-i\overline{\theta_{L}}\Gamma^{\mu}\partial_{z}\theta_{L}-i% \overline{\theta_{R}}\Gamma^{\mu}\partial_{z}\theta_{R}
  4. z X μ - i θ L 1 ¯ Γ μ z θ L 1 - i θ L 2 ¯ Γ μ z θ L 2 \partial_{z}X^{\mu}-i\overline{\theta^{1}_{L}}\Gamma^{\mu}\partial_{z}\theta^{% 1}_{L}-i\overline{\theta^{2}_{L}}\Gamma^{\mu}\partial_{z}\theta^{2}_{L}
  5. - exp ( a x 4 + b x 3 + c x 2 + d x + f ) d x = e f n , m , p = 0 b 4 n ( 4 n ) ! c 2 m ( 2 m ) ! d 4 p ( 4 p ) ! Γ ( 3 n + m + p + 1 4 ) a 3 n + m + p + 1 4 \int_{-\infty}^{\infty}\exp({ax^{4}+bx^{3}+cx^{2}+dx+f})\,dx=e^{f}\sum_{n,m,p=% 0}^{\infty}\frac{b^{4n}}{(4n)!}\frac{c^{2m}}{(2m)!}\frac{d^{4p}}{(4p)!}\frac{% \Gamma(3n+m+p+\frac{1}{4})}{a^{3n+m+p+\frac{1}{4}}}

Supersymmetry_nonrenormalization_theorems.html

  1. 𝒩 \mathcal{N}
  2. 𝒩 \mathcal{N}
  3. 𝒩 \mathcal{N}
  4. 𝒩 = 1 \mathcal{N}=1
  5. 𝒩 = 2 \mathcal{N}=2
  6. 𝒩 = 2 \mathcal{N}=2
  7. 𝒩 = 2 \mathcal{N}=2
  8. 𝒩 = 4 \mathcal{N}=4
  9. 𝒩 \mathcal{N}
  10. 𝒩 = 1 \mathcal{N}=1
  11. 𝒩 = 2 \mathcal{N}=2
  12. 𝒩 = 3 \mathcal{N}=3
  13. 𝒩 = 2 \mathcal{N}=2
  14. 𝒩 = 3 \mathcal{N}=3
  15. 𝒩 = 4 \mathcal{N}=4
  16. 𝒩 = 3 \mathcal{N}=3
  17. 𝒩 = ( 2 , 2 ) \mathcal{N}=(2,2)

Supertoroid.html

  1. P ( u , v ) = ( X ( u , v ) Y ( u , v ) Z ( u , v ) ) = ( ( a + C u s ) C v t ( b + C u s ) S v t S u s ) P(u,v)=\left(\begin{array}[]{c}X(u,v)\\ Y(u,v)\\ Z(u,v)\end{array}\right)=\left(\begin{array}[]{c}(a+C_{u}^{s})C_{v}^{t}\\ (b+C_{u}^{s})S_{v}^{t}\\ S_{u}^{s}\end{array}\right)
  2. C θ ϵ = sgn ( cos θ ) | cos θ | ϵ C_{\theta}^{\epsilon}=\operatorname{sgn}(\cos\theta)\left|\cos\theta\right|^{\epsilon}
  3. S θ ϵ = sgn ( sin θ ) | sin θ | ϵ S_{\theta}^{\epsilon}=\operatorname{sgn}(\sin\theta)\left|\sin\theta\right|^{\epsilon}
  4. [ - ( a + 1 ) , + ( a + 1 ) ] [-(a+1),+(a+1)]
  5. [ - ( b + 1 ) , + ( b + 1 ) ] [-(b+1),+(b+1)]
  6. [ - 1 , + 1 ] [-1,+1]
  7. [ - ( a - 1 ) , + ( a - 1 ) ] [-(a-1),+(a-1)]
  8. [ - ( b - 1 ) , + ( b - 1 ) ] [-(b-1),+(b-1)]
  9. [ a - 1 , a + 1 ] [a-1,a+1]
  10. [ - 1 , + 1 ] [-1,+1]

Support_of_a_module.html

  1. 𝔭 \mathfrak{p}
  2. M 𝔭 0 M_{\mathfrak{p}}\neq 0
  3. Supp ( M ) \operatorname{Supp}(M)
  4. M = 0 M=0
  5. 0 M M M ′′ 0 0\to M^{\prime}\to M\to M^{\prime\prime}\to 0
  6. Supp ( M ) = Supp ( M ) Supp ( M ′′ ) . \operatorname{Supp}(M)=\operatorname{Supp}(M^{\prime})\cup\operatorname{Supp}(% M^{\prime\prime}).
  7. M M
  8. M λ M_{\lambda}
  9. Supp ( M ) = λ supp ( M λ ) . \operatorname{Supp}(M)=\cup_{\lambda}\operatorname{supp}(M_{\lambda}).
  10. M M
  11. Supp ( M ) \operatorname{Supp}(M)
  12. M , N M,N
  13. Supp ( M A N ) = Supp ( M ) Supp ( N ) . \operatorname{Supp}(M\otimes_{A}N)=\operatorname{Supp}(M)\cap\operatorname{% Supp}(N).
  14. M M
  15. Supp ( M / I M ) \operatorname{Supp}(M/IM)
  16. I + Ann ( M ) . I+\operatorname{Ann}(M).
  17. V ( I ) Supp ( M ) V(I)\cap\operatorname{Supp}(M)

Surface-extended_X-ray_absorption_fine_structure.html

  1. μ = ln ( I ) ln ( I o ) , \begin{aligned}\displaystyle\mu=\frac{\ln(I)}{\ln(I_{o})},\end{aligned}
  2. S N = ( Ω 4 π ϵ n μ A ) ( 1 + I b I n ( μ T + n ) ) ( δ μ A μ A I o 1 / 2 ) , \frac{S}{N}=\sqrt{\frac{(\frac{\Omega}{4\pi}\epsilon_{n}\mu_{A})}{(1+\frac{I_{% b}}{I_{n}}(\mu_{T}+n))}}\left(\frac{\delta\mu_{A}}{\mu_{A}}I_{o}^{1/2}\right),
  3. P = 2 π f | M f s | 2 δ ( E i + ω - E f ) , P=\frac{2\pi}{\hbar}\sum_{f}|M_{fs}|^{2}\delta(E_{i}+\hbar\omega-E_{f}),
  4. M f s = f | e ϵ 𝐫 | i , M_{fs}=\langle f|e\mathbf{\epsilon}\cdot\mathbf{r}|i\rangle,
  5. k = 1 [ 2 m ( ( ω - ω T ) + V o ) ] , k=\frac{1}{\hbar}\sqrt{[2m(\hbar(\omega-\omega_{T})+V_{o})]},
  6. χ ( k ) = k - 1 | f ( k , π ) | j W j sin [ 2 k R j + α ( k ) ] exp ( - γ R j - 2 σ j 2 k 2 ) , \chi(k)=k^{-1}|f(k,\pi)|\sum_{j}\ W_{j}\sin[2kR_{j}+\alpha(k)]\exp(-\gamma R_{% j}-2\sigma_{j}^{2}k^{2}),
  7. f ( k , θ ) = ( 1 / k ) l = 0 ( 2 l + 1 ) [ exp ( 2 i δ l ( k ) ) - 1 ] P l ( cos θ ) . f(k,\theta)=(1/k)\sum_{l=0}^{\infty}\ (2l+1)[\exp(2i\delta_{l}(k))-1]P_{l}(% \cos\theta).
  8. W j = N j R j 2 . W_{j}=\frac{N_{j}}{R_{j}^{2}}.
  9. N T = N W α X Y ( ω ) + N B ( ω ) , N_{T}=N_{W_{\alpha}XY}(\hbar\omega)+N_{B}(\hbar\omega),
  10. N W α X Y ( ω ) N_{W_{\alpha}XY}(\hbar\omega)
  11. N W α X Y ( ω ) = ( 4 π ) - 1 ψ W α X Y [ 1 - κ ] Ω 0 ρ α ( z ) P W α ( ω ; z ) exp [ - z λ ( W α X Y ) cos θ ] d z d Ω , N_{W_{\alpha}XY}(\hbar\omega)=(4\pi)^{-1}\psi_{W_{\alpha}XY}[1-\kappa]\int_{% \Omega}\int_{0}^{\infty}\ \rho_{\alpha}(z)\,\,P_{W_{\alpha}}(\hbar\omega;z)% \exp\left[\frac{-z}{\lambda(W_{\alpha}XY)}\cos\theta\right]\ dzd\Omega,
  12. ψ W α X Y \psi_{W_{\alpha}XY}
  13. P W α ( ω ; z ) P_{W_{\alpha}}(\hbar\omega;z)
  14. N W α X Y ( ω ) N_{W_{\alpha}XY}(\hbar\omega)

Surface_stress.html

  1. d w dw
  2. d A dA
  3. d w = γ d A dw=\gamma dA
  4. γ \gamma
  5. f i j f_{ij}
  6. γ A \gamma A
  7. d e i j de_{ij}
  8. d ( γ A ) = A f i j d ϵ i j d(\gamma A)=Af_{ij}d\epsilon_{ij}
  9. W 1 = 2 γ 0 A 0 W_{1}=2\gamma_{0}A_{0}
  10. γ 0 \gamma_{0}
  11. A 0 A_{0}
  12. w 2 w_{2}
  13. w 1 w_{1}
  14. w 2 - w 1 w_{2}-w_{1}
  15. A 0 A_{0}
  16. A ( e i j ) A(e_{ij})
  17. w 2 - w 1 = 2 ( f i j d ( A ( ϵ i j ) ) = 2 ( A f i j d ϵ i j ) w_{2}-w_{1}=2\int(f_{ij}d(A(\epsilon_{ij}))=2\int(Af_{ij}d\epsilon_{ij})
  18. W 2 = 2 γ ( e i j ) A ( e i j ) W_{2}=2\gamma(e_{ij})A(e_{ij})
  19. W 2 - W 1 = 2 [ γ ( e i j ) A ( e i j ) - γ 0 A 0 ] W_{2}-W_{1}=2[\gamma(e_{ij})A(e_{ij})-\gamma_{0}\ A_{0}]
  20. 2 [ γ ( ϵ i j ) A ( ϵ i j ) - γ 0 A 0 ] = 2 ( A f i j d ϵ i j ) 2[\gamma(\epsilon_{ij})A(\epsilon_{ij})-\gamma_{0}A_{0}]=2\int(Af_{ij}d% \epsilon_{ij})
  21. f i j = γ δ i j + γ / e i j f_{ij}=\gamma\delta_{ij}+\partial\gamma/\partial e_{ij}
  22. f = γ + γ / e f=\gamma+\partial\gamma/\partial e
  23. Δ d \Delta d

Surface_wave_magnitude.html

  1. M s M_{s}
  2. M s M_{s}
  3. M L M_{L}
  4. M = log 10 ( A T ) max + σ ( Δ ) M=\log_{10}\left(\frac{A}{T}\right)_{\,\text{max}}+\sigma(\Delta)
  5. σ ( Δ ) = 1.66 log 10 ( Δ ) + 3.5 \sigma(\Delta)=1.66\cdot\log_{10}(\Delta)+3.5
  6. T = T N A N + T E A E A N + A E T=\frac{T_{N}A_{N}+T_{E}A_{E}}{A_{N}+A_{E}}
  7. M s = - 3.2 + 1.45 M L M_{s}=-3.2+1.45M_{L}
  8. M s = log 10 ( A m a x T ) + 1.54 log 10 ( Δ ) + 3.53 M_{s}=\log_{10}\left(\frac{A_{max}}{T}\right)+1.54\cdot\log_{10}(\Delta)+3.53
  9. M s = log 10 ( A m a x T ) + 1.73 log 10 ( Δ ) + 3.27 M_{s}=\log_{10}\left(\frac{A_{max}}{T}\right)+1.73\cdot\log_{10}(\Delta)+3.27
  10. M s = log 10 ( A m a x T ) - 6.2 log 10 ( Δ ) + 20.6 M_{s}=\log_{10}\left(\frac{A_{max}}{T}\right)-6.2\cdot\log_{10}(\Delta)+20.6

Surgery_exact_sequence.html

  1. > 4 >4
  2. 𝒮 ( X ) \mathcal{S}(X)
  3. n n
  4. X X
  5. n n
  6. X X
  7. 𝒮 ( X ) \mathcal{S}(X)
  8. X \mathcal{}X
  9. s s
  10. h h
  11. 𝒩 ( X × I ) L n + 1 ( π 1 ( X ) ) 𝒮 ( X ) 𝒩 ( X ) L n ( π 1 ( X ) ) \cdots\to\mathcal{N}_{\partial}(X\times I)\to L_{n+1}(\pi_{1}(X))\to\mathcal{S% }(X)\to\mathcal{N}(X)\to L_{n}(\pi_{1}(X))
  12. 𝒩 ( X × I ) \mathcal{N}_{\partial}(X\times I)
  13. 𝒩 ( X ) \mathcal{N}(X)
  14. L n + 1 ( π 1 ( X ) ) \mathcal{}L_{n+1}(\pi_{1}(X))
  15. L n ( π 1 ( X ) ) \mathcal{}L_{n}(\pi_{1}(X))
  16. [ π 1 ( X ) ] \mathbb{Z}[\pi_{1}(X)]
  17. θ : 𝒩 ( X × I ) L n + 1 ( π 1 ( X ) ) \theta\colon\mathcal{N}_{\partial}(X\times I)\to L_{n+1}(\pi_{1}(X))
  18. θ : 𝒩 ( X ) L n ( π 1 ( X ) ) \theta\colon\mathcal{N}(X)\to L_{n}(\pi_{1}(X))
  19. : L n + 1 ( π 1 ( X ) ) 𝒮 ( X ) \partial\colon L_{n+1}(\pi_{1}(X))\to\mathcal{S}(X)
  20. η : 𝒮 ( X ) 𝒩 ( X ) \eta\colon\mathcal{S}(X)\to\mathcal{N}(X)
  21. s s
  22. h h
  23. ( f , b ) : M X (f,b)\colon M\to X
  24. n n
  25. M M
  26. f f
  27. f * ( [ M ] ) = [ X ] f_{*}([M])=[X]
  28. b : T M ε k ξ b\colon TM\oplus\varepsilon^{k}\to\xi
  29. M M
  30. ξ \xi
  31. X X
  32. 𝒩 ( X ) \mathcal{N}(X)
  33. ( i d , i d ) (id,id)
  34. 𝒩 ( X ) \mathcal{N}(X)
  35. 𝒩 ( X ) [ X , G / O ] \mathcal{N}(X)\cong[X,G/O]
  36. G / O G/O
  37. J : B O B G J\colon BO\to BG
  38. [ X , G / P L ] [X,G/PL]
  39. [ X , G / T O P ] [X,G/TOP]
  40. L L
  41. θ : 𝒩 ( X ) L n ( π 1 ( X ) ) \theta\colon\mathcal{N}(X)\to L_{n}(\pi_{1}(X))
  42. n 5 n\geq 5
  43. ( f , b ) : M X (f,b)\colon M\to X
  44. θ ( f , b ) = 0 \theta(f,b)=0
  45. L n ( [ π 1 ( X ) ] ) L_{n}(\mathbb{Z}[\pi_{1}(X)])
  46. η : 𝒮 ( X ) 𝒩 ( X ) \eta\colon\mathcal{S}(X)\to\mathcal{N}(X)
  47. f : M X f\colon M\to X
  48. : L n + 1 ( π 1 ( X ) ) 𝒮 ( X ) \partial\colon L_{n+1}(\pi_{1}(X))\to\mathcal{S}(X)
  49. L n + 1 ( π 1 ( X ) ) L_{n+1}(\pi_{1}(X))
  50. 𝒮 ( X ) \mathcal{S}(X)
  51. L L
  52. M M
  53. n n
  54. π 1 ( M ) π 1 ( X ) \pi_{1}(M)\cong\pi_{1}(X)
  55. x L n + 1 ( π 1 ( X ) ) x\in L_{n+1}(\pi_{1}(X))
  56. ( F , B ) : ( W , M , M ) ( M × I , M × 0 , M × 1 ) (F,B)\colon(W,M,M^{\prime})\to(M\times I,M\times 0,M\times 1)
  57. θ ( F , B ) = x L n + 1 ( π 1 ( X ) ) \theta(F,B)=x\in L_{n+1}(\pi_{1}(X))
  58. F 0 : M M × 0 F_{0}\colon M\to M\times 0
  59. F 1 : M M × 1 F_{1}\colon M^{\prime}\to M\times 1
  60. f : M X f\colon M\to X
  61. 𝒮 ( X ) \mathcal{S}(X)
  62. x L n + 1 ( π 1 ( X ) ) x\in L_{n+1}(\pi_{1}(X))
  63. ( f , x ) \partial(f,x)
  64. f F 1 : M X f\circ F_{1}\colon M^{\prime}\to X
  65. θ \theta
  66. z 𝒩 ( X ) z\in\mathcal{N}(X)
  67. z Im ( η ) z\in\mathrm{Im}(\eta)
  68. θ ( z ) = 0 \theta(z)=0
  69. x 1 , x 2 𝒮 ( X ) x_{1},x_{2}\in\mathcal{S}(X)
  70. η ( x 1 ) = η ( x 2 ) \eta(x_{1})=\eta(x_{2})
  71. u L n + 1 ( π 1 ( X ) ) u\in L_{n+1}(\pi_{1}(X))
  72. ( u , x 1 ) = x 2 \partial(u,x_{1})=x_{2}
  73. u L n + 1 ( π 1 ( X ) ) u\in L_{n+1}(\pi_{1}(X))
  74. ( u , id ) = id \partial(u,\mathrm{id})=\mathrm{id}
  75. u Im ( θ ) u\in\mathrm{Im}(\theta)
  76. 𝒮 ( X ) \mathcal{S}(X)
  77. X X
  78. X X
  79. 𝒩 ( X ) \mathcal{N}(X)
  80. x 𝒩 ( X ) x\in\mathcal{N}(X)
  81. θ ( x ) = 0 \theta(x)=0
  82. θ : 𝒩 ( X ) L n ( π 1 ( X ) ) \theta\colon\mathcal{N}(X)\rightarrow L_{n}(\pi_{1}(X))
  83. 0 L n ( π 1 ( X ) ) 0\in L_{n}(\pi_{1}(X))
  84. f : M X f\colon M\to X
  85. f : M X f^{\prime}\colon M^{\prime}\to X
  86. 𝒮 ( X ) \mathcal{S}(X)
  87. f \mathcal{}f
  88. f \mathcal{}f^{\prime}
  89. η ( f ) = η ( f ) \mathcal{}\eta(f)=\eta(f^{\prime})
  90. 𝒩 ( X ) \mathcal{N}(X)
  91. ( F , B ) : ( W , M , M ) ( X × I , X × 0 , X × 1 ) (F,B)\colon(W,M,M^{\prime})\to(X\times I,X\times 0,X\times 1)
  92. θ ( F , B ) \mathcal{}\theta(F,B)
  93. L n + 1 ( π 1 ( X ) ) \mathcal{}L_{n+1}(\pi_{1}(X))
  94. f \mathcal{}f
  95. f \mathcal{}f^{\prime}
  96. n 5 n\geq 5
  97. 𝒮 D I F F ( S n ) = Θ n \mathcal{S}^{DIFF}(S^{n})=\Theta^{n}
  98. 𝒩 D I F F ( S n ) = Ω n a l m \mathcal{N}^{DIFF}(S^{n})=\Omega^{alm}_{n}
  99. n n
  100. 𝒩 D I F F ( S n × I ) = Ω n + 1 a l m \mathcal{N}^{DIFF}_{\partial}(S^{n}\times I)=\Omega^{alm}_{n+1}
  101. L n ( 1 ) = , 0 , 2 , 0 L_{n}(1)=\mathbb{Z},0,\mathbb{Z}_{2},0
  102. n 0 , 1 , 2 , 3 n\equiv 0,1,2,3
  103. 4 4
  104. 4 4
  105. b P n + 1 = ker ( η : 𝒮 D I F F ( S n ) 𝒩 D I F F ( S n ) ) = coker ( θ : 𝒩 D I F F ( S n × I ) L n + 1 ( 1 ) ) bP^{n+1}=\mathrm{ker}(\eta\colon\mathcal{S}^{DIFF}(S^{n})\to\mathcal{N}^{DIFF}% (S^{n}))=\mathrm{coker}(\theta\colon\mathcal{N}^{DIFF}_{\partial}(S^{n}\times I% )\to L_{n+1}(1))
  106. 0 Θ 4 i Ω 4 i a l m b P 4 i 0 0\to\Theta^{4i}\to\Omega^{alm}_{4i}\to\mathbb{Z}\to bP^{4i}\to 0
  107. 0 Θ 4 i - 2 Ω 4 i - 2 a l m / 2 b P 4 i - 2 0 0\to\Theta^{4i-2}\to\Omega^{alm}_{4i-2}\to\mathbb{Z}/2\to bP^{4i-2}\to 0
  108. 0 b P 2 j Θ 2 j - 1 Ω 2 j - 1 a l m 0 0\to bP^{2j}\to\Theta^{2j-1}\to\Omega^{alm}_{2j-1}\to 0
  109. Ω i a l m \Omega_{i}^{alm}
  110. n n
  111. 𝒮 T O P ( S n ) = 0 \mathcal{S}^{TOP}(S^{n})=0
  112. n n
  113. S n S^{n}
  114. n 5 n\geq 5
  115. θ : 𝒩 T O P ( S n ) L n ( 1 ) \theta\colon\mathcal{N}^{TOP}(S^{n})\to L_{n}(1)
  116. n 1 n\geq 1
  117. P n \mathbb{C}P^{n}
  118. ( 2 n ) (2n)
  119. π 1 ( P n ) = 1 \pi_{1}(\mathbb{C}P^{n})=1
  120. π 1 ( X ) = 1 \pi_{1}(X)=1
  121. θ \theta
  122. 0 𝒮 T O P ( P n ) 𝒩 T O P ( P n ) L 2 n ( 1 ) 0 0\to\mathcal{S}^{TOP}(\mathbb{C}P^{n})\to\mathcal{N}^{TOP}(\mathbb{C}P^{n})\to L% _{2n}(1)\to 0
  123. 𝒩 ( P n ) i = 1 n / 2 i = 1 ( n + 1 ) / 2 2 \mathcal{N}(\mathbb{C}P^{n})\cong\oplus_{i=1}^{\lfloor n/2\rfloor}\mathbb{Z}% \oplus\oplus_{i=1}^{\lfloor(n+1)/2\rfloor}\mathbb{Z}_{2}
  124. 𝒮 ( P n ) i = 1 ( n - 1 ) / 2 i = 1 n / 2 2 \mathcal{S}(\mathbb{C}P^{n})\cong\oplus_{i=1}^{\lfloor(n-1)/2\rfloor}\mathbb{Z% }\oplus\oplus_{i=1}^{\lfloor n/2\rfloor}\mathbb{Z}_{2}
  125. n n
  126. X X
  127. n n
  128. π i ( X ) = 0 \pi_{i}(X)=0
  129. i 2 i\geq 2
  130. π 1 ( X ) \pi_{1}(X)
  131. X X
  132. W h ( π 1 ( X ) ) Wh(\pi_{1}(X))
  133. 𝒮 ( X ) = 0 \mathcal{S}(X)=0
  134. π 1 ( X ) \pi_{1}(X)
  135. n \mathbb{Z}^{n}

Surgery_obstruction.html

  1. θ : 𝒩 ( X ) L n ( π 1 ( X ) ) \theta\colon\mathcal{N}(X)\to L_{n}(\pi_{1}(X))
  2. n 5 n\geq 5
  3. ( f , b ) : M X (f,b)\colon M\to X
  4. θ ( f , b ) = 0 \theta(f,b)=0
  5. L n ( [ π 1 ( X ) ] ) L_{n}(\mathbb{Z}[\pi_{1}(X)])
  6. ( f , b ) : M X (f,b)\colon M\to X
  7. ( f , b ) (f,b)
  8. f f
  9. m m
  10. π * ( f ) = 0 \pi_{*}(f)=0
  11. * m *\leq m
  12. m m
  13. m > n / 2 m>\lfloor n/2\rfloor
  14. f f
  15. M M
  16. π i ( f ) \pi_{i}(f)
  17. f f
  18. K i ( M ~ ) := ker { f * : H i ( M ~ ) H i ( X ~ ) } K_{i}(\tilde{M}):=\mathrm{ker}\{f_{*}\colon H_{i}(\tilde{M})\rightarrow H_{i}(% \tilde{X})\}
  19. [ π 1 ( X ) ] \mathbb{Z}[\pi_{1}(X)]
  20. f f
  21. M M
  22. X X
  23. [ π 1 ( X ) ] \mathbb{Z}[\pi_{1}(X)]
  24. K n - i ( M ~ ) K i ( M ~ ) K^{n-i}(\tilde{M})\cong K_{i}(\tilde{M})
  25. i n / 2 i\leq\lfloor n/2\rfloor
  26. n / 2 \lfloor n/2\rfloor
  27. K i ( M ~ ) K_{i}(\tilde{M})
  28. i < n / 2 i<\lfloor n/2\rfloor
  29. p + q = n p+q=n
  30. i = p < n / 2 i=p<\lfloor n/2\rfloor
  31. n = 2 k n=2k
  32. K k ( M ~ ) := ker { f * : H k ( M ~ ) H k ( X ~ ) } K_{k}(\tilde{M}):=\mathrm{ker}\{f_{*}\colon H_{k}(\tilde{M})\rightarrow H_{k}(% \tilde{X})\}
  33. M M
  34. X X
  35. K k ( M ~ ) K_{k}(\tilde{M})
  36. k = 2 l k=2l
  37. k = 2 l + 1 k=2l+1
  38. ε \varepsilon
  39. ε = ( - 1 ) k \varepsilon=(-1)^{k}
  40. ε \varepsilon
  41. L n ( π 1 ( X ) ) L_{n}(\pi_{1}(X))
  42. n = 2 k + 1 n=2k+1
  43. L n ( π 1 ( X ) ) L_{n}(\pi_{1}(X))
  44. θ ( f , b ) \theta(f,b)
  45. M M
  46. f f
  47. K k ( M ~ ) K_{k}(\tilde{M})
  48. K k - 1 ( M ~ ) K_{k-1}(\tilde{M})
  49. n = 2 k n=2k
  50. K k ( M ~ ) K_{k}(\tilde{M})
  51. n = 2 k + 1 n=2k+1
  52. θ ( f , b ) \theta(f,b)
  53. n = 2 k + 1 n=2k+1
  54. n = 4 l n=4l
  55. n = 4 l + 2 n=4l+2
  56. 2 \mathbb{Z}_{2}

Surgery_structure_set.html

  1. 𝒮 ( X ) \mathcal{S}(X)
  2. f i : M i X f_{i}:M_{i}\to X
  3. M i M_{i}
  4. n n
  5. X X
  6. i = 0 , 1 i=0,1
  7. ( W ; M 0 , M 1 ) \mathcal{}(W;M_{0},M_{1})
  8. ( F ; f 0 , f 1 ) : ( W ; M 0 , M 1 ) ( X × [ 0 , 1 ] ; X × { 0 } , X × { 1 } ) (F;f_{0},f_{1}):(W;M_{0},M_{1})\to(X\times[0,1];X\times\{0\},X\times\{1\})
  9. F F
  10. f 0 f_{0}
  11. f 1 f_{1}
  12. 𝒮 h ( X ) \mathcal{S}^{h}(X)
  13. f : M X f:M\to X
  14. i d : X X id:X\to X
  15. f 0 f_{0}
  16. f 1 f_{1}
  17. 𝒮 s ( X ) \mathcal{S}^{s}(X)
  18. ( W ; M 0 , M 1 ) (W;M_{0},M_{1})
  19. 𝒮 h ( X ) \mathcal{S}^{h}(X)
  20. 𝒮 s ( X ) \mathcal{S}^{s}(X)
  21. 𝒮 s ( X ) \mathcal{S}^{s}(X)
  22. 𝒮 s ( X ) \mathcal{S}^{s}(X)
  23. f : M X f:M\to X
  24. M M
  25. f i : M i X f_{i}:M_{i}\to X
  26. g : M 0 M 1 g:M_{0}\to M_{1}
  27. f 1 g f_{1}\circ g
  28. f 0 f_{0}
  29. 𝒮 s ( X ) \mathcal{S}^{s}(X)
  30. 𝒮 s ( X ) \mathcal{S}^{s}(X)
  31. ϕ : M X \phi:M\to X
  32. 𝒮 s ( X ) \mathcal{S}^{s}(X)
  33. ϕ : M X \phi:M\to X
  34. 𝒮 s ( X ) \mathcal{S}^{s}(X)
  35. 𝒮 s ( S n ) \mathcal{S}^{s}(S^{n})
  36. 𝒮 s ( S n ) = θ n = π n ( P L / O ) \mathcal{S}^{s}(S^{n})=\theta_{n}=\pi_{n}(PL/O)

Swap_regret.html

  1. swap-regret = i = 1 n max j n 1 T t = 1 T x i t ( p j t - p i t ) . \mbox{swap-regret}~{}=\sum_{i=1}^{n}\max_{j\leq n}\frac{1}{T}\sum_{t=1}^{T}x^{% t}_{i}\cdot(p^{t}_{j}-p^{t}_{i}).\,

Swing_(jazz_performance_style).html

  1. \approx

Symmetric_convolution.html

  1. f * g = h f*g=h
  2. * *
  3. f f
  4. g g
  5. f f
  6. g g
  7. h h

Symmetric_inverse_semigroup.html

  1. X \mathcal{I}_{X}
  2. 𝒮 X \mathcal{IS}_{X}
  3. X \mathcal{I}_{X}

Symplectic_representation.html

  1. ω : V × V 𝔽 \omega\colon V\times V\to\mathbb{F}
  2. ω ( g v , g w ) = ω ( v , w ) \omega(g\cdot v,g\cdot w)=\omega(v,w)
  3. ω ( ξ v , w ) + ω ( v , ξ w ) = 0 \omega(\xi\cdot v,w)+\omega(v,\xi\cdot w)=0

Syncategorematic_term.html

  1. \land
  2. ϕ ψ = 1 \lVert\phi\land\psi\rVert=1
  3. ϕ = ψ = 1 \lVert\phi\rVert=\lVert\psi\rVert=1
  4. ϕ \phi
  5. ψ \psi
  6. \lVert\land\rVert
  7. \land
  8. ( λ b . ( λ v . b ( v ) ( b ) ) ) (\lambda b.(\lambda v.b(v)(b)))
  9. ( λ x . ( λ y . x ) ) (\lambda x.(\lambda y.x))
  10. ( λ x . ( λ y . y ) ) (\lambda x.(\lambda y.y))
  11. t , t , t \langle\langle t,t\rangle,t\rangle
  12. \land
  13. λ \lambda
  14. λ \lambda

Synergetics_coordinates.html

  1. ( x , y , z ) (x,y,z)
  2. x , y , z x,y,z
  3. ( w , x , y , z ) (w,x,y,z)
  4. w , x , y , z w,x,y,z

Systoles_of_surfaces.html

  1. 2 / 3 2/\sqrt{3}
  2. π / 8 \pi/\sqrt{8}
  3. SR ( K ) π 8 , \mathrm{SR}(K)\leq\frac{\pi}{\sqrt{8}},
  4. SR ( 2 ) 2 3 \mathrm{SR}(2)\leq\tfrac{2}{\sqrt{3}}
  5. ( log g ) 2 g . \frac{(\log g)^{2}}{g}.
  6. log ( g ) \log(g)
  7. ( log g ) 2 g \tfrac{(\log g)^{2}}{g}
  8. g g
  9. Σ g \Sigma_{g}
  10. sys ( Σ g ) 4 3 log g , \mathrm{sys}(\Sigma_{g})\geq\frac{4}{3}\log g,
  11. \mathbb{Q}
  12. 4 9 π ( log g ) 2 g . \frac{4}{9\pi}\frac{(\log g)^{2}}{g}.
  13. ( log g ) 2 π g , \frac{(\log g)^{2}}{\pi g},
  14. 1 / 2 3 1/2\sqrt{3}

Szpiro's_conjecture.html

  1. | Δ | C ( ε ) f 6 + ε . |\Delta|\leq C(\varepsilon)\cdot f^{6+\varepsilon}.\,
  2. max { | c 4 | 3 , | c 6 | 2 } C ( ε ) f 6 + ε . \max\{|c_{4}|^{3},|c_{6}|^{2}\}\leq C(\varepsilon)\cdot f^{6+\varepsilon}.\,

T-model.html

  1. ( 1 ) T = g + R O E - g P B + Δ P B P B ( 1 + g ) (1)\mathit{T}=\mathit{g}+\frac{\mathit{R}OE-\mathit{g}}{\mathit{P}B}+\frac{% \Delta PB}{PB}\mathit{(}1+g)
  2. ( 2 ) T = D P + Δ P P (2)\mathit{T}=\frac{\mathit{D}}{\mathit{P}}+\frac{\Delta P}{P}
  3. P \mathit{P}
  4. Δ P \Delta P
  5. D \mathit{D}
  6. X C F = E - D i v - g B V \mathit{X}CF=\mathit{E}-\mathit{D}iv-\mathit{g}BV\,
  7. D i v P + X C F P \frac{\mathit{D}iv}{\mathit{P}}+\frac{\mathit{X}CF}{\mathit{P}}
  8. R O E = E B V \mathit{R}OE=\frac{\mathit{E}}{\mathit{B}V}
  9. P B = P B V \mathit{P}B=\frac{\mathit{P}}{\mathit{B}V}
  10. ( 3 ) D P = R O E - g P B (3)\frac{\mathit{D}}{\mathit{P}}=\frac{\mathit{R}OE-\mathit{g}}{\mathit{P}B}
  11. \equiv
  12. P + Δ P = ( A + Δ A ) ( B + Δ B ) = A B + B Δ A + A Δ B + Δ A Δ B \mathit{P}+\Delta\mathit{P}=(\mathit{A}+\Delta\mathit{A})(\mathit{B}+\Delta% \mathit{B})\,=\mathit{A}B+\mathit{B}\Delta\mathit{A}+\mathit{A}\Delta\mathit{B% }+\Delta\mathit{A}\Delta\mathit{B}\,
  13. \equiv
  14. \equiv
  15. Δ P P = Δ B B + Δ A A ( 1 + Δ B B ) \frac{\Delta P}{P}=\frac{\Delta\mathit{B}}{\mathit{B}}+\frac{\Delta\mathit{A}}% {\mathit{A}}\left(\mathit{1}+\frac{\Delta\mathit{B}}{\mathit{B}}\right)
  16. Δ B B = g \frac{\Delta\mathit{B}}{\mathit{B}}=\mathit{g}
  17. ( 4 ) Δ P P = g + Δ P B P B ( 1 + g ) (4)\frac{\Delta P}{P}=\mathit{g}+\frac{\Delta PB}{PB}\mathit{(}1+g)
  18. T = C F P + s y m b o l Φ g + Δ P B P B ( 1 + g ) \mathit{T}=\frac{\mathit{C}F}{\mathit{P}}+symbol{\Phi}g+\frac{\Delta PB}{PB}% \mathit{(}1+g)
  19. C F = c a s h f l o w \mathit{C}F=cashflow\,
  20. (net income + depreciation + all other non-cash charges), \mbox{(net income + depreciation + all other non-cash charges),}~{}\,
  21. s y m b o l Φ = M k t C a p - g r o s s a s s e t s + t o t a l l i a b i l i t i e s M k t C a p symbol{\Phi}=\frac{\mathit{M}ktCap-grossassets+totalliabilities}{\mathit{M}ktCap}
  22. T = g + R O E - g 1 = R O E \mathit{T}=\mathit{g}+\frac{\mathit{R}OE-\mathit{g}}{1}=ROE
  23. R O E = E B V \mathit{R}OE=\frac{\mathit{E}}{\mathit{B}V}
  24. B V = P \mathit{B}V=\mathit{P}\,
  25. T = E P \mathit{T}=\frac{\mathit{E}}{\mathit{P}}
  26. T = g + R O E - R O E ( 1 - D / E ) P B \mathit{T}=\mathit{g}+\frac{\mathit{R}OE-\mathit{R}OE(1-D/E)}{\mathit{P}B}
  27. T = g + D P \mathit{T}=\mathit{g}+\frac{D}{\mathit{P}}
  28. P E = R O E - g R O E ( T - g ) \frac{\mathit{P}}{\mathit{E}}=\frac{\mathit{R}OE-\mathit{g}}{\mathit{R}OE(% \mathit{T}-\mathit{g})}
  29. P / E g \frac{\mathit{P}/E}{g}

Table_of_the_largest_known_graphs_of_a_given_diameter_and_maximal_degree.html

  1. d d
  2. k k

Talbot_effect.html

  1. z T = 2 a 2 λ , z_{T}=\frac{2a^{2}}{\lambda},
  2. a a
  3. λ \lambda
  4. λ \lambda
  5. a a
  6. z T z_{T}
  7. z T = λ 1 - 1 - λ 2 a 2 . z_{T}=\frac{\lambda}{1-\sqrt{1-\frac{\lambda^{2}}{a^{2}}}}.
  8. λ d B \lambda_{dB}
  9. λ d B \lambda_{dB}
  10. i ψ z + 1 2 2 ψ x 2 + | ψ | 2 ψ = 0 i\frac{\partial\psi}{\partial z}+\frac{1}{2}\frac{\partial^{2}\psi}{\partial x% ^{2}}+|\psi|^{2}\psi=0

Tangent_lines_to_circles.html

  1. A B ¯ + C D ¯ = B C ¯ + D A ¯ . \overline{AB}+\overline{CD}=\overline{BC}+\overline{DA}.
  2. A B ¯ + C D ¯ = ( a + b ) + ( c + d ) = B C ¯ + D A ¯ = ( b + c ) + ( d + a ) \overline{AB}+\overline{CD}=(a+b)+(c+d)=\overline{BC}+\overline{DA}=(b+c)+(d+a)
  3. a x + b y + c = 0 , ax+by+c=0,
  4. ( a , b , c ) (a,b,c)
  5. d = ( Δ x ) 2 + ( Δ y ) 2 d=\sqrt{(\Delta x)^{2}+(\Delta y)^{2}}
  6. cos θ \cos\theta
  7. θ \theta
  8. sin θ \sin\theta
  9. ± 1 - R 2 \pm\sqrt{1-R^{2}}
  10. θ \theta
  11. ± θ , \pm\theta,
  12. ( R 1 - R 2 ± 1 - R 2 R ) \begin{pmatrix}R&\mp\sqrt{1-R^{2}}\\ \pm\sqrt{1-R^{2}}&R\end{pmatrix}
  13. ( t 2 - v 2 ) ( t 2 - t 1 ) \displaystyle(t_{2}-v_{2})\cdot(t_{2}-t_{1})
  14. t 2 - t 1 , t_{2}-t_{1},
  15. d > r 1 + r 2 d>r_{1}+r_{2}
  16. d = r 1 + r 2 d=r_{1}+r_{2}
  17. | r 1 - r 2 | < d < r 1 + r 2 |r_{1}-r_{2}|<d<r_{1}+r_{2}
  18. d = | r 1 - r 2 | d=|r_{1}-r_{2}|
  19. d = | r 1 - r 2 | d=|r_{1}-r_{2}|
  20. x 2 + y 2 = ( - r ) 2 , x^{2}+y^{2}=(-r)^{2},

Tangential_angle.html

  1. ( x ( t ) , y ( t ) ) (x(t),\ y(t))
  2. φ \varphi
  3. t t
  4. 2 π 2\pi
  5. ( x ( t ) , y ( t ) ) | x ( t ) , y ( t ) | = ( cos φ , sin φ ) . \frac{(x^{\prime}(t),\ y^{\prime}(t))}{|x^{\prime}(t),\ y^{\prime}(t)|}=(\cos% \varphi,\ \sin\varphi).
  6. ( x ( t ) , y ( t ) ) (x^{\prime}(t),\ y^{\prime}(t))
  7. ( x ( t ) , y ( t ) ) | x ( t ) , y ( t ) | \frac{(x^{\prime}(t),\ y^{\prime}(t))}{|x^{\prime}(t),\ y^{\prime}(t)|}
  8. t t
  9. φ \varphi
  10. ( cos φ , sin φ ) (\cos\varphi,\ \sin\varphi)
  11. t t
  12. s s
  13. | x ( s ) , y ( s ) | = 1 |x^{\prime}(s),\ y^{\prime}(s)|=1
  14. ( x ( s ) , y ( s ) ) = ( cos φ , sin φ ) (x^{\prime}(s),\ y^{\prime}(s))=(\cos\varphi,\ \sin\varphi)
  15. κ \kappa
  16. φ ( s ) \varphi^{\prime}(s)
  17. κ \kappa
  18. y = f ( x ) y=f(x)
  19. ( x , f ( x ) ) (x,\ f(x))
  20. φ \varphi
  21. - π / 2 -\pi/2
  22. π / 2 \pi/2
  23. φ = arctan f ( x ) \varphi=\arctan f^{\prime}(x)
  24. ψ \psi
  25. ψ = φ - θ \psi=\varphi-\theta
  26. φ \varphi
  27. θ \theta
  28. r = f ( θ ) r=f(\theta)
  29. ψ \psi
  30. θ \theta
  31. 2 π 2\pi
  32. ( f ( θ ) , f ( θ ) ) | f ( θ ) , f ( θ ) | = ( cos ψ , sin ψ ) \frac{(f^{\prime}(\theta),\ f(\theta))}{|f^{\prime}(\theta),\ f(\theta)|}=(% \cos\psi,\ \sin\psi)
  33. s s
  34. r = r ( s ) , θ = θ ( s ) r=r(s),\ \theta=\theta(s)
  35. | r ( s ) , r θ ( s ) | = 1 |r^{\prime}(s),\ r\theta^{\prime}(s)|=1
  36. ( r ( s ) , r θ ( s ) ) = ( cos ψ , sin ψ ) (r^{\prime}(s),\ r\theta^{\prime}(s))=(\cos\psi,\ \sin\psi)

Tarski's_exponential_function_problem.html

  1. Th ( \R ) φ . \operatorname{Th}(\R)\models\varphi.
  2. f 1 ( x 1 , , x n , e x 1 , , e x n ) = = f n ( x 1 , , x n , e x 1 , , e x n ) = 0 f_{1}(x_{1},\ldots,x_{n},e^{x_{1}},\ldots,e^{x_{n}})=\ldots=f_{n}(x_{1},\ldots% ,x_{n},e^{x_{1}},\ldots,e^{x_{n}})=0

Tarski's_high_school_algebra_problem.html

  1. ( ( 1 + x ) y + ( 1 + x + x 2 ) y ) x ( ( 1 + x 3 ) x + ( 1 + x 2 + x 4 ) x ) y = ( ( 1 + x ) x + ( 1 + x + x 2 ) x ) y ( ( 1 + x 3 ) y + ( 1 + x 2 + x 4 ) y ) x . \begin{aligned}&\displaystyle\left((1+x)^{y}+(1+x+x^{2})^{y}\right)^{x}\cdot% \left((1+x^{3})^{x}+(1+x^{2}+x^{4})^{x}\right)^{y}\\ \displaystyle=&\displaystyle\left((1+x)^{x}+(1+x+x^{2})^{x}\right)^{y}\cdot% \left((1+x^{3})^{y}+(1+x^{2}+x^{4})^{y}\right)^{x}.\end{aligned}
  2. ( 1 - x + x 2 ) x y (1-x+x^{2})^{xy}
  3. 1 - x + x 2 1-x+x^{2}
  4. - x -x

Tarski's_plank_problem.html

  1. C P 1 P m \R n , C\subseteq P_{1}\cup\ldots\cup P_{m}\subset\R^{n},
  2. i = 1 m w ( P i ) w ( C ) . \sum_{i=1}^{m}w(P_{i})\geq w(C).

Tarski–Seidenberg_theorem.html

  1. p ( x 1 , , x n ) = 0 p(x_{1},\ldots,x_{n})=0\,
  2. q ( x 1 , , x n ) > 0 q(x_{1},\ldots,x_{n})>0\,
  3. x 2 + y 2 - 1 = 0. x^{2}+y^{2}-1=0.\,

Taub–NUT_space.html

  1. d s 2 = - d t 2 / U ( t ) + 4 l 2 U ( t ) ( d ψ + cos θ d ϕ ) 2 + ( t 2 + l 2 ) ( d θ 2 + ( sin θ ) 2 d ϕ 2 ) ds^{2}=-dt^{2}/U(t)+4l^{2}U(t)(d\psi+\cos\theta d\phi)^{2}+(t^{2}+l^{2})(d% \theta^{2}+(\sin\theta)^{2}d\phi^{2})
  2. U ( t ) = 2 m t + l 2 - t 2 t 2 + l 2 U(t)=\frac{2mt+l^{2}-t^{2}}{t^{2}+l^{2}}

Taylor_state.html

  1. S S
  2. β 0 \beta\rightarrow 0
  3. B d s = 0 \vec{B}\cdot\vec{ds}=0
  4. S S
  5. A | | = 0 \vec{A}_{||}=0
  6. δ B d s = 0 \delta\vec{B}\cdot\vec{ds}=0
  7. δ A | | = 0 \delta\vec{A}_{||}=0
  8. S S
  9. W = d 3 r B 2 / 2 μ W=\int d^{3}rB^{2}/2\mu_{\circ}
  10. K = d 3 r A B K=\int d^{3}r\vec{A}\cdot\vec{B}
  11. δ W - λ δ K = 0 \delta W-\lambda\delta K=0
  12. × B = λ B \nabla\times\vec{B}=\lambda\vec{B}