wpmath0000004_8

Jordan_algebra.html

  1. x y = y x xy=yx
  2. ( x y ) ( x x ) = x ( y ( x x ) ) (xy)(xx)=x(y(xx))
  3. ( x m y ) x n = x m ( y x n ) (x^{m}y)x^{n}=x^{m}(yx^{n})
  4. x y = x y + y x 2 . x\circ y=\frac{xy+yx}{2}.
  5. σ ( x y + y x ) = x y + y x . \sigma(xy+yx)=xy+yx.
  6. ( x y + y x ) / 2 (xy+yx)/2
  7. ( x y + y x ) / 2 , (xy+yx)/2,
  8. x 2 = x , x x^{2}=\langle x,x\rangle
  9. a b a b , a 2 = a 2 , a 2 a 2 + b 2 . \displaystyle{\|a\circ b\|\leq\|a\|\cdot\|b\|,\,\,\,\|a^{2}\|=\|a\|^{2},\,\,\,% \|a^{2}\|\leq\|a^{2}+b^{2}\|.}
  10. / 2 \mathbb{Z}/2
  11. J 0 J 1 J_{0}\oplus J_{1}
  12. J 0 J_{0}
  13. J 1 J_{1}
  14. J 0 J_{0}
  15. / 2 \mathbb{Z}/2
  16. A 0 A 1 A_{0}\oplus A_{1}
  17. { x i , y j } = x i y j + ( - 1 ) i j y j x i . \{x_{i},y_{j}\}=x_{i}y_{j}+(-1)^{ij}y_{j}x_{i}\ .
  18. K 3 K_{3}
  19. K 10 K_{10}

K1.html

  1. K 1 ( R ) K_{1}(R)
  2. R R

K3_surface.html

  1. q = h 0 , 1 = dim H 1 ( S , 𝒪 S ) = 0 q=h^{0,1}=\,\text{dim}H^{1}(S,\mathcal{O}_{S})=0
  2. h 2 ( S , 𝒪 S ) = h 0 ( S , K S ) = 1. h^{2}(S,\mathcal{O}_{S})=h^{0}(S,K_{S})=1.
  3. χ ( S , 𝒪 S ) := i ( - 1 ) i h i ( S , 𝒪 S ) = 2. \chi(S,\mathcal{O}_{S}):=\sum_{i}(-1)^{i}h^{i}(S,\mathcal{O}_{S})=2.
  4. χ ( S , 𝒪 S ) = 1 12 ( c 1 ( S ) 2 + c 2 ( S ) ) \chi(S,\mathcal{O}_{S})=\frac{1}{12}(c_{1}(S)^{2}+c_{2}(S))
  5. H 2 ( S , ) H^{2}(S,\mathbb{Z})

K4.html

  1. K 4 K_{4}

Kac–Moody_algebra.html

  1. 𝔥 \mathfrak{h}
  2. α i \alpha_{i}^{\vee}
  3. 𝔥 \mathfrak{h}
  4. α i \alpha_{i}
  5. 𝔥 * \mathfrak{h}^{*}
  6. α i ( α j ) = c j i \alpha_{i}(\alpha_{j}^{\vee})=c_{ji}
  7. α i \alpha_{i}
  8. α i \alpha_{i}^{\vee}
  9. 𝔤 \mathfrak{g}
  10. e i e_{i}
  11. f i f_{i}
  12. i { 1 , , n } i\in\{1,\ldots,n\}
  13. 𝔥 \mathfrak{h}
  14. [ h , h ] = 0 [h,h^{\prime}]=0
  15. h , h 𝔥 h,h^{\prime}\in\mathfrak{h}
  16. [ h , e i ] = α i ( h ) e i [h,e_{i}]=\alpha_{i}(h)e_{i}
  17. h 𝔥 h\in\mathfrak{h}
  18. [ h , f i ] = - α i ( h ) f i [h,f_{i}]=-\alpha_{i}(h)f_{i}
  19. h 𝔥 h\in\mathfrak{h}
  20. [ e i , f j ] = δ i j α i [e_{i},f_{j}]=\delta_{ij}\alpha_{i}^{\vee}
  21. δ i j \delta_{ij}
  22. i j i\neq j
  23. c i j 0 c_{ij}\leq 0
  24. ad ( e i ) 1 - c i j ( e j ) = 0 \textrm{ad}(e_{i})^{1-c_{ij}}(e_{j})=0
  25. ad ( f i ) 1 - c i j ( f j ) = 0 \textrm{ad}(f_{i})^{1-c_{ij}}(f_{j})=0
  26. ad : 𝔤 End ( 𝔤 ) , ad ( x ) ( y ) = [ x , y ] , \textrm{ad}:\mathfrak{g}\to\textrm{End}(\mathfrak{g}),\textrm{ad}(x)(y)=[x,y],
  27. 𝔤 \mathfrak{g}
  28. 𝔥 \mathfrak{h}
  29. 𝔤 \mathfrak{g}
  30. x 0 x\neq 0
  31. 𝔤 \mathfrak{g}
  32. h 𝔥 , [ h , x ] = λ ( h ) x \forall h\in\mathfrak{h},[h,x]=\lambda(h)x
  33. λ 𝔥 * \ { 0 } \lambda\in\mathfrak{h}^{*}\backslash\{0\}
  34. x x
  35. λ \lambda
  36. 𝔤 \mathfrak{g}
  37. 𝔤 \mathfrak{g}
  38. Δ \Delta
  39. R R
  40. λ \lambda
  41. 𝔤 λ \mathfrak{g}_{\lambda}
  42. λ \lambda
  43. 𝔤 λ = { x 𝔤 : h 𝔥 , [ h , x ] = λ ( h ) x } \mathfrak{g}_{\lambda}=\{x\in\mathfrak{g}:\forall h\in\mathfrak{h},[h,x]=% \lambda(h)x\}
  44. 𝔤 \mathfrak{g}
  45. e i 𝔤 α i e_{i}\in\mathfrak{g}_{\alpha_{i}}
  46. f i 𝔤 - α i f_{i}\in\mathfrak{g}_{-\alpha_{i}}
  47. x 1 𝔤 λ 1 x_{1}\in\mathfrak{g}_{\lambda_{1}}
  48. x 2 𝔤 λ 2 x_{2}\in\mathfrak{g}_{\lambda_{2}}
  49. [ x 1 , x 2 ] 𝔤 λ 1 + λ 2 [x_{1},x_{2}]\in\mathfrak{g}_{\lambda_{1}+\lambda_{2}}
  50. 𝔥 \mathfrak{h}
  51. 𝔤 = 𝔥 λ Δ 𝔤 λ \mathfrak{g}=\mathfrak{h}\oplus\bigoplus_{\lambda\in\Delta}\mathfrak{g}_{\lambda}
  52. λ \lambda
  53. λ = i = 1 n z i α i \lambda=\sum_{i=1}^{n}z_{i}\alpha_{i}
  54. z i z_{i}
  55. 𝔤 ( C ) 𝔤 ( C 1 ) 𝔤 ( C 2 ) , \mathfrak{g}(C)\simeq\mathfrak{g}(C_{1})\oplus\mathfrak{g}(C_{2}),

Kantorovich_inequality.html

  1. p i 0 , 0 < a x i b for i = 1 , , n . p_{i}\geq 0,\quad 0<a\leq x_{i}\leq b\,\text{ for }i=1,\dots,n.
  2. A n = { 1 , 2 , , n } . A_{n}=\{1,2,\dots,n\}.
  3. ( i = 1 n p i x i ) ( i = 1 n p i x i ) \displaystyle{}\qquad\left(\sum_{i=1}^{n}p_{i}x_{i}\right)\left(\sum_{i=1}^{n}% \frac{p_{i}}{x_{i}}\right)

Kappa_curve.html

  1. x 2 ( x 2 + y 2 ) = a 2 y 2 x^{2}(x^{2}+y^{2})=a^{2}y^{2}
  2. x = a sin t y = a sin t tan t \begin{matrix}x&=&a\sin t\\ y&=&a\sin t\tan t\end{matrix}
  3. r = a tan θ r=a\tan\theta
  4. x = ± a x=\pm a
  5. κ ( θ ) = 8 ( 3 - sin 2 θ ) sin 4 θ a [ sin 2 ( 2 θ ) + 4 ] 3 2 \kappa(\theta)={8\left(3-\sin^{2}\theta\right)\sin^{4}\theta\over a\left[\sin^% {2}(2\theta)+4\right]^{3\over 2}}
  6. ϕ ( θ ) = - arctan [ 1 2 sin ( 2 θ ) ] \phi(\theta)=-\arctan\left[{1\over 2}\sin(2\theta)\right]
  7. x 2 ( x 2 + y 2 ) - a 2 y 2 = 0 x^{2}(x^{2}+y^{2})-a^{2}y^{2}=0
  8. d ( x 2 ( x 2 + y 2 ) - a 2 y 2 ) = 0 d(x^{2}(x^{2}+y^{2})-a^{2}y^{2})=0
  9. d ( x 2 ( x 2 + y 2 ) ) - d ( a 2 y 2 ) = 0 d(x^{2}(x^{2}+y^{2}))-d(a^{2}y^{2})=0
  10. ( 2 x d x ) ( x 2 + y 2 ) + x 2 ( 2 x d x + 2 y d y ) - a 2 2 y d y = 0 (2xdx)(x^{2}+y^{2})+x^{2}(2xdx+2ydy)-a^{2}2ydy=0
  11. ( 4 x 3 + 2 x y 2 ) d x + ( 2 y x 2 - 2 a 2 y ) d y = 0 (4x^{3}+2xy^{2})dx+(2yx^{2}-2a^{2}y)dy=0
  12. x ( 2 x 2 + y 2 ) d x + y ( x 2 - a 2 ) d y = 0 x(2x^{2}+y^{2})dx+y(x^{2}-a^{2})dy=0
  13. x ( 2 x 2 + y 2 ) y ( a 2 - x 2 ) = d y d x \frac{x(2x^{2}+y^{2})}{y(a^{2}-x^{2})}=\frac{dy}{dx}
  14. 2 x ( x 2 + y 2 ) + x 2 ( 2 x + 2 y d y d x ) = 2 a 2 y d y d x 2x(x^{2}+y^{2})+x^{2}(2x+2y\frac{dy}{dx})=2a^{2}y\frac{dy}{dx}
  15. 2 x ( x 2 + y 2 ) + x 2 ( 2 x + 2 y d y d x ) = 2 a 2 y d y d x 2x(x^{2}+y^{2})+x^{2}(2x+2y\frac{dy}{dx})=2a^{2}y\frac{dy}{dx}
  16. 2 x 3 + 2 x y 2 + 2 x 3 = 2 a 2 y d y d x - 2 x 2 y d y d x 2x^{3}+2xy^{2}+2x^{3}=2a^{2}y\frac{dy}{dx}-2x^{2}y\frac{dy}{dx}
  17. 4 x 3 + 2 x y 2 = ( 2 a 2 y - 2 x 2 y ) d y d x 4x^{3}+2xy^{2}=(2a^{2}y-2x^{2}y)\frac{dy}{dx}
  18. 2 x 3 + x y 2 a 2 y - x 2 y = d y d x \frac{2x^{3}+xy^{2}}{a^{2}y-x^{2}y}=\frac{dy}{dx}

Karhunen–Loève_theorem.html

  1. a a , b aa,b
  2. t a a , b t∈aa,b
  3. X t = k = 1 Z k e k ( t ) X_{t}=\sum_{k=1}^{\infty}Z_{k}e_{k}(t)
  4. a a , b aa,b
  5. 0 , 11 0,11
  6. ( Ω , F , 𝐏 ) (Ω,F,\mathbf{P})
  7. a a , b aa,b
  8. t [ a , b ] X t L 2 ( Ω , F , 𝐏 ) , \forall t\in[a,b]\qquad X_{t}\in L^{2}(\Omega,F,\mathbf{P}),
  9. t [ a , b ] 𝐄 [ X t ] = 0 , \forall t\in[a,b]\qquad\mathbf{E}[X_{t}]=0,
  10. t , s [ a , b ] K X ( s , t ) = 𝐄 [ X s X t ] . \forall t,s\in[a,b]\qquad K_{X}(s,t)=\mathbf{E}[X_{s}X_{t}].
  11. { T K X : L 2 ( [ a , b ] ) L 2 ( [ a , b ] ) f a b K X ( s , ) f ( s ) d s \begin{cases}T_{K_{X}}:L^{2}([a,b])\to L^{2}([a,b])\\ f\mapsto\int_{a}^{b}K_{X}(s,\cdot)f(s)ds\end{cases}
  12. a b K X ( s , t ) e k ( s ) d s = λ k e k ( t ) \int_{a}^{b}K_{X}(s,t)e_{k}(s)\,ds=\lambda_{k}e_{k}(t)
  13. ( Ω , F , 𝐏 ) (Ω,F,\mathbf{P})
  14. X t = k = 1 Z k e k ( t ) X_{t}=\sum_{k=1}^{\infty}Z_{k}e_{k}(t)
  15. Z k = a b X t e k ( t ) d t Z_{k}=\int_{a}^{b}X_{t}e_{k}(t)\,dt
  16. 𝐄 [ Z k ] = 0 , k and 𝐄 [ Z i Z j ] = δ i j λ j , i , j \mathbf{E}[Z_{k}]=0,~{}\forall k\in\mathbb{N}\qquad\mbox{and}~{}\qquad\mathbf{% E}[Z_{i}Z_{j}]=\delta_{ij}\lambda_{j},~{}\forall i,j\in\mathbb{N}
  17. K X ( s , t ) = k = 1 λ k e k ( s ) e k ( t ) K_{X}(s,t)=\sum_{k=1}^{\infty}\lambda_{k}e_{k}(s)e_{k}(t)
  18. X t = k = 1 Z k e k ( t ) X_{t}=\sum_{k=1}^{\infty}Z_{k}e_{k}(t)
  19. Z k = a b X t e k ( t ) d t Z_{k}=\int_{a}^{b}X_{t}e_{k}(t)\,dt
  20. 𝐄 [ Z k ] \displaystyle\mathbf{E}[Z_{k}]
  21. S N = k = 1 N Z k e k ( t ) . S_{N}=\sum_{k=1}^{N}Z_{k}e_{k}(t).
  22. 𝐄 [ | X t - S N | 2 ] = 𝐄 [ X t 2 ] + 𝐄 [ S N 2 ] - 2 𝐄 [ X t S N ] = K X ( t , t ) + 𝐄 [ k = 1 N l = 1 N Z k Z l e k ( t ) e l ( t ) ] - 2 𝐄 [ X t k = 1 N Z k e k ( t ) ] = K X ( t , t ) + k = 1 N λ k e k ( t ) 2 - 2 𝐄 [ k = 1 N a b X t X s e k ( s ) e k ( t ) d s ] = K X ( t , t ) - k = 1 N λ k e k ( t ) 2 \begin{aligned}\displaystyle\mathbf{E}\left[\left|X_{t}-S_{N}\right|^{2}\right% ]&\displaystyle=\mathbf{E}\left[X_{t}^{2}\right]+\mathbf{E}\left[S_{N}^{2}% \right]-2\mathbf{E}\left[X_{t}S_{N}\right]\\ &\displaystyle=K_{X}(t,t)+\mathbf{E}\left[\sum_{k=1}^{N}\sum_{l=1}^{N}Z_{k}Z_{% l}e_{k}(t)e_{l}(t)\right]-2\mathbf{E}\left[X_{t}\sum_{k=1}^{N}Z_{k}e_{k}(t)% \right]\\ &\displaystyle=K_{X}(t,t)+\sum_{k=1}^{N}\lambda_{k}e_{k}(t)^{2}-2\mathbf{E}% \left[\sum_{k=1}^{N}\int_{a}^{b}X_{t}X_{s}e_{k}(s)e_{k}(t)ds\right]\\ &\displaystyle=K_{X}(t,t)-\sum_{k=1}^{N}\lambda_{k}e_{k}(t)^{2}\end{aligned}
  23. lim N i = 1 N e i ( t ) Z i ( ω ) = X t ( ω ) \lim_{N\to\infty}\sum_{i=1}^{N}e_{i}(t)Z_{i}(\omega)=X_{t}(\omega)
  24. X t ( ω ) = k = 1 A k ( ω ) f k ( t ) X_{t}(\omega)=\sum_{k=1}^{\infty}A_{k}(\omega)f_{k}(t)
  25. A k ( ω ) = a b X t ( ω ) f k ( t ) d t A_{k}(\omega)=\int_{a}^{b}X_{t}(\omega)f_{k}(t)\,dt
  26. X ^ t ( ω ) = k = 1 N A k ( ω ) f k ( t ) \hat{X}_{t}(\omega)=\sum_{k=1}^{N}A_{k}(\omega)f_{k}(t)
  27. ε N ( t ) = k = N + 1 A k ( ω ) f k ( t ) \varepsilon_{N}(t)=\sum_{k=N+1}^{\infty}A_{k}(\omega)f_{k}(t)
  28. ε N 2 ( t ) \displaystyle\varepsilon_{N}^{2}(t)
  29. a b ε N 2 ( t ) d t = k = N + 1 a b a b K X ( s , t ) f k ( t ) f k ( s ) d s d t \int_{a}^{b}\varepsilon_{N}^{2}(t)dt=\sum_{k=N+1}^{\infty}\int_{a}^{b}\int_{a}% ^{b}K_{X}(s,t)f_{k}(t)f_{k}(s)ds\,dt
  30. E r [ f k ( t ) , k { N + 1 , } ] = k = N + 1 a b a b K X ( s , t ) f k ( t ) f k ( s ) d s d t - β k ( a b f k ( t ) f k ( t ) d t - 1 ) Er[f_{k}(t),k\in\{N+1,\ldots\}]=\sum_{k=N+1}^{\infty}\int_{a}^{b}\int_{a}^{b}K% _{X}(s,t)f_{k}(t)f_{k}(s)dsdt-\beta_{k}\left(\int_{a}^{b}f_{k}(t)f_{k}(t)dt-1\right)
  31. E r f i ( t ) = a b ( a b K X ( s , t ) f i ( s ) d s - β i f i ( t ) ) d t = 0 \frac{\partial Er}{\partial f_{i}(t)}=\int_{a}^{b}\left(\int_{a}^{b}K_{X}(s,t)% f_{i}(s)ds-\beta_{i}f_{i}(t)\right)dt=0
  32. a b K X ( s , t ) f i ( s ) d s = β i f i ( t ) . \int_{a}^{b}K_{X}(s,t)f_{i}(s)\,ds=\beta_{i}f_{i}(t).
  33. Var [ X t ] = k = 0 e k ( t ) 2 Var [ Z k ] = k = 1 λ k e k ( t ) 2 \mbox{Var}~{}[X_{t}]=\sum_{k=0}^{\infty}e_{k}(t)^{2}\mbox{Var}~{}[Z_{k}]=\sum_% {k=1}^{\infty}\lambda_{k}e_{k}(t)^{2}
  34. a b Var [ X t ] d t = k = 1 λ k \int_{a}^{b}\mbox{Var}~{}[X_{t}]dt=\sum_{k=1}^{\infty}\lambda_{k}
  35. k = 1 N λ k . \sum_{k=1}^{N}\lambda_{k}.
  36. k = 1 N λ k k = 1 λ k \frac{\sum_{k=1}^{N}\lambda_{k}}{\sum_{k=1}^{\infty}\lambda_{k}}
  37. N N\in\mathbb{N}
  38. k = 1 N λ k k = 1 λ k 0.95 \frac{\sum_{k=1}^{N}\lambda_{k}}{\sum_{k=1}^{\infty}\lambda_{k}}\geq 0.95
  39. M M
  40. Y n n Ynn
  41. N N
  42. Y Y
  43. Y Y
  44. { g m } 0 m N \left\{g_{m}\right\}_{0\leq m\leq N}
  45. Y = m = 0 N - 1 Y , g m g m , Y=\sum_{m=0}^{N-1}\left\langle Y,g_{m}\right\rangle g_{m},
  46. Y , g m = n = 0 N - 1 Y [ n ] g m * [ n ] \left\langle Y,g_{m}\right\rangle=\sum_{n=0}^{N-1}{Y[n]}g_{m}^{*}[n]
  47. M N M≤N
  48. Y M = m = 0 M - 1 Y , g m g m Y_{M}=\sum_{m=0}^{M-1}\left\langle Y,g_{m}\right\rangle g_{m}
  49. ε [ M ] = 𝐄 { Y - Y M 2 } = m = M N - 1 𝐄 { | Y , g m | 2 } \varepsilon[M]=\mathbf{E}\left\{\left\|Y-Y_{M}\right\|^{2}\right\}=\sum_{m=M}^% {N-1}\mathbf{E}\left\{\left|\left\langle Y,g_{m}\right\rangle\right|^{2}\right\}
  50. Y Y
  51. R [ n , m ] = 𝐄 { Y [ n ] Y * [ m ] } R[n,m]=\mathbf{E}\left\{Y[n]Y^{*}[m]\right\}
  52. x n n xnn
  53. K K
  54. 𝐄 { | Y , x | 2 } = K x , x = n = 0 N - 1 m = 0 N - 1 R [ n , m ] x [ n ] x * [ m ] \mathbf{E}\left\{\left|\langle Y,x\rangle\right|^{2}\right\}=\langle Kx,x% \rangle=\sum_{n=0}^{N-1}\sum_{m=0}^{N-1}R[n,m]x[n]x^{*}[m]
  55. ε M M εMM
  56. N M N−M
  57. ε [ M ] = m = M N - 1 K g m , g m \varepsilon[M]=\sum_{m=M}^{N-1}{\left\langle Kg_{m},g_{m}\right\rangle}
  58. K K
  59. K K
  60. M 1 M≥1
  61. ε [ M ] = m = M N - 1 K g m , g m \varepsilon[M]=\sum_{m=M}^{N-1}\left\langle Kg_{m},g_{m}\right\rangle
  62. { g m } 0 m < N \left\{g_{m}\right\}_{0\leq m<N}
  63. K g m , g m K g m + 1 , g m + 1 , 0 m < N - 1. \left\langle Kg_{m},g_{m}\right\rangle\geq\left\langle Kg_{m+1},g_{m+1}\right% \rangle,\qquad 0\leq m<N-1.
  64. f \Eta f\in\Eta
  65. \Eta \Eta
  66. B = { g m } m B=\left\{g_{m}\right\}_{m\in\mathbb{N}}
  67. f M f_{M}
  68. f M = m I M f , g m g m f_{M}=\sum_{m\in I_{M}}\left\langle f,g_{m}\right\rangle g_{m}
  69. ε [ M ] = { f - f M 2 } = m I M N - 1 { | f , g m | 2 } \varepsilon[M]=\left\{\left\|f-f_{M}\right\|^{2}\right\}=\sum_{m\notin I_{M}}^% {N-1}\left\{\left|\left\langle f,g_{m}\right\rangle\right|^{2}\right\}
  70. | f , g m | . \left|\left\langle f,g_{m}\right\rangle\right|.
  71. { | f , g m | } m \left\{\left|\left\langle f,g_{m}\right\rangle\right|\right\}_{m\in\mathbb{N}}
  72. | f , g m k | | f , g m k + 1 | . \left|\left\langle f,g_{m_{k}}\right\rangle\right|\geq\left|\left\langle f,g_{% m_{k+1}}\right\rangle\right|.
  73. f M = k = 1 M f , g m k g m k f_{M}=\sum_{k=1}^{M}\left\langle f,g_{m_{k}}\right\rangle g_{m_{k}}
  74. f M = m = 0 θ T ( f , g m ) g m f_{M}=\sum_{m=0}^{\infty}\theta_{T}\left(\left\langle f,g_{m}\right\rangle% \right)g_{m}
  75. T = | f , g m M | , θ T ( x ) = { x | x | T 0 | x | < T T=\left|\left\langle f,g_{m_{M}}\right\rangle\right|,\qquad\theta_{T}(x)=% \begin{cases}x&|x|\geq T\\ 0&|x|<T\end{cases}
  76. ε [ M ] = { f - f M 2 } = k = M + 1 { | f , g m k | 2 } \varepsilon[M]=\left\{\left\|f-f_{M}\right\|^{2}\right\}=\sum_{k=M+1}^{\infty}% \left\{\left|\left\langle f,g_{m_{k}}\right\rangle\right|^{2}\right\}
  77. | f , g m k | \left|\left\langle f,g_{m_{k}}\right\rangle\right|
  78. \Iota \Rho \Iota^{\Rho}
  79. f B , p = ( m = 0 | f , g m | p ) 1 p \|f\|_{B,p}=\left(\sum_{m=0}^{\infty}\left|\left\langle f,g_{m}\right\rangle% \right|^{p}\right)^{\frac{1}{p}}
  80. ε M M εMM
  81. f B , p \|f\|_{B,p}
  82. f B , p < \|f\|_{B,p}<\infty
  83. ε [ M ] f B , p 2 2 p - 1 M 1 - 2 p \varepsilon[M]\leq\frac{\|f\|_{B,p}^{2}}{\frac{2}{p}-1}M^{1-\frac{2}{p}}
  84. ε [ M ] = o ( M 1 - 2 p ) . \varepsilon[M]=o\left(M^{1-\frac{2}{p}}\right).
  85. ε [ M ] = o ( M 1 - 2 p ) \varepsilon[M]=o\left(M^{1-\frac{2}{p}}\right)
  86. f B , q < \|f\|_{B,q}<\infty
  87. q > p q>p
  88. n = 0 N - 1 f [ n ] = 0 \sum_{n=0}^{N-1}f[n]=0
  89. Y [ n ] = f [ ( n - p ) mod N ] Y[n]=f[(n-p)\bmod N]
  90. Pr ( P = p ) = 1 N , 0 p < N \Pr(P=p)=\frac{1}{N},\qquad 0\leq p<N
  91. 𝐄 { Y [ n ] } = 1 N p = 0 N - 1 f [ ( n - p ) mod N ] = 0 \mathbf{E}\{Y[n]\}=\frac{1}{N}\sum_{p=0}^{N-1}f[(n-p)\bmod N]=0
  92. R [ n , k ] = 𝐄 { Y [ n ] Y [ k ] } = 1 N p = 0 N - 1 f [ ( n - p ) mod N ] f [ ( k - p ) mod N ] = 1 N f Θ f ¯ [ n - k ] , f ¯ [ n ] = f [ - n ] R[n,k]=\mathbf{E}\{Y[n]Y[k]\}=\frac{1}{N}\sum_{p=0}^{N-1}f[(n-p)\bmod N]f[(k-p% )\bmod N]=\frac{1}{N}f\Theta\bar{f}[n-k],\quad\bar{f}[n]=f[-n]
  93. R [ n , k ] = R Y [ n - k ] , R Y [ k ] = 1 N f Θ f ¯ [ k ] R[n,k]=R_{Y}[n-k],\qquad R_{Y}[k]=\frac{1}{N}f\Theta\bar{f}[k]
  94. { 1 N e i 2 π m n N } 0 m < N . \left\{\frac{1}{\sqrt{N}}e^{\frac{i2\pi mn}{N}}\right\}_{0\leq m<N}.
  95. P Y [ m ] R ^ Y [ m ] = 1 N | f ^ [ m ] | 2 P_{Y}[m]\hat{R}_{Y}[m]=\frac{1}{N}\left|\hat{f}[m]\right|^{2}
  96. f [ n ] = δ [ n ] - δ [ n - 1 ] f[n]=\delta[n]-\delta[n-1]
  97. { g m [ n ] = δ [ n - m ] } 0 m < N \left\{g_{m}[n]=\delta[n-m]\right\}_{0\leq m<N}
  98. 𝐄 { | Y [ n ] , 1 N e i 2 π m n N | 2 } = P Y [ m ] = 4 N sin 2 ( π k N ) \mathbf{E}\left\{\left|\left\langle Y[n],\frac{1}{\sqrt{N}}e^{\frac{i2\pi mn}{% N}}\right\rangle\right|^{2}\right\}=P_{Y}[m]=\frac{4}{N}\sin^{2}\left(\frac{% \pi k}{N}\right)
  99. f [ n ] = δ [ n ] - δ [ n - 1 ] f[n]=\delta[n]-\delta[n-1]
  100. M 2 M≥2
  101. a b K X ( s , t ) e k ( s ) d s = λ k e k ( t ) . \int_{a}^{b}K_{X}(s,t)e_{k}(s)\,ds=\lambda_{k}e_{k}(t).
  102. ( X n ) n { 1 , , N } \left(X_{n}\right)_{n\in\{1,\ldots,N\}}
  103. X = ( X 1 X 2 X N ) T X=\left(X_{1}~{}X_{2}~{}\ldots~{}X_{N}\right)^{T}
  104. X := X - μ X X:=X-\mu_{X}
  105. μ X \mu_{X}
  106. Σ i j = 𝐄 [ X i X j ] , i , j { 1 , , N } \Sigma_{ij}=\mathbf{E}[X_{i}X_{j}],\qquad\forall i,j\in\{1,\ldots,N\}
  107. i = 1 N Σ i j e j = λ e i Σ e = λ e \sum_{i=1}^{N}\Sigma_{ij}e_{j}=\lambda e_{i}\quad\Leftrightarrow\quad\Sigma e=\lambda e
  108. e = ( e 1 e 2 e N ) T e=(e_{1}~{}e_{2}~{}\ldots~{}e_{N})^{T}
  109. \R N \R^{N}
  110. { λ i , ϕ i } i { 1 , , N } \{\lambda_{i},\phi_{i}\}_{i\in\{1,\ldots,N\}}
  111. Φ Φ
  112. Φ \displaystyle\Phi
  113. X = i = 1 N ϕ i , X ϕ i = i = 1 N ϕ i T X ϕ i X=\sum_{i=1}^{N}\langle\phi_{i},X\rangle\phi_{i}=\sum_{i=1}^{N}\phi_{i}^{T}X% \phi_{i}
  114. { Y = Φ T X X = Φ Y \begin{cases}Y=\Phi^{T}X\\ X=\Phi Y\end{cases}
  115. Y i = ϕ i T X Y_{i}=\phi_{i}^{T}X
  116. ϕ i \phi_{i}
  117. X = Φ Y X=ΦY
  118. X X
  119. ϕ i \phi_{i}
  120. X = i = 1 N Y i ϕ i = i = 1 N ϕ i , X ϕ i X=\sum_{i=1}^{N}Y_{i}\phi_{i}=\sum_{i=1}^{N}\langle\phi_{i},X\rangle\phi_{i}
  121. K { 1 , , N } K\in\{1,\ldots,N\}
  122. i = 1 K λ i i = 1 N λ i α \frac{\sum_{i=1}^{K}\lambda_{i}}{\sum_{i=1}^{N}\lambda_{i}}\geq\alpha
  123. K W ( t , s ) = Cov ( W t , W s ) = min ( s , t ) . K_{W}(t,s)=\operatorname{Cov}(W_{t},W_{s})=\min(s,t).
  124. e k ( t ) = 2 sin ( ( k - 1 2 ) π t ) e_{k}(t)=\sqrt{2}\sin\left(\left(k-\tfrac{1}{2}\right)\pi t\right)
  125. λ k = 1 ( k - 1 2 ) 2 π 2 . \lambda_{k}=\frac{1}{(k-\frac{1}{2})^{2}\pi^{2}}.
  126. a b K W ( s , t ) e ( s ) d s \displaystyle\int_{a}^{b}K_{W}(s,t)e(s)ds
  127. t 1 e ( s ) d s = λ e ( t ) \int_{t}^{1}e(s)ds=\lambda e^{\prime}(t)
  128. - e ( t ) = λ e ′′ ( t ) -e(t)=\lambda e^{\prime\prime}(t)
  129. e ( t ) = A sin ( t λ ) + B cos ( t λ ) e(t)=A\sin\left(\frac{t}{\sqrt{\lambda}}\right)+B\cos\left(\frac{t}{\sqrt{% \lambda}}\right)
  130. cos ( 1 λ ) = 0 \cos\left(\frac{1}{\sqrt{\lambda}}\right)=0
  131. λ k = ( 1 ( k - 1 2 ) π ) 2 , k 1 \lambda_{k}=\left(\frac{1}{(k-\frac{1}{2})\pi}\right)^{2},\qquad k\geq 1
  132. e k ( t ) = A sin ( ( k - 1 2 ) π t ) , k 1 e_{k}(t)=A\sin\left((k-\frac{1}{2})\pi t\right),\qquad k\geq 1
  133. 0 1 e k 2 ( t ) d t = 1 A = 2 \int_{0}^{1}e_{k}^{2}(t)dt=1\quad\implies\quad A=\sqrt{2}
  134. W t = 2 k = 1 Z k sin ( ( k - 1 2 ) π t ) ( k - 1 2 ) π . W_{t}=\sqrt{2}\sum_{k=1}^{\infty}Z_{k}\frac{\sin\left(\left(k-\frac{1}{2}% \right)\pi t\right)}{\left(k-\frac{1}{2}\right)\pi}.
  135. t [ 0 , 1 ] . t\in[0,1].
  136. B t = W t - t W 1 B_{t}=W_{t}-tW_{1}
  137. K B ( t , s ) = min ( t , s ) - t s K_{B}(t,s)=\min(t,s)-ts
  138. B t = k = 1 Z k 2 sin ( k π t ) k π B_{t}=\sum_{k=1}^{\infty}Z_{k}\frac{\sqrt{2}\sin(k\pi t)}{k\pi}
  139. R N ( t , s ) = E [ N ( t ) N ( s ) ] R_{N}(t,s)=E[N(t)N(s)]
  140. H : X ( t ) = N ( t ) , H:X(t)=N(t),
  141. K : X ( t ) = N ( t ) + s ( t ) , t ( 0 , T ) K:X(t)=N(t)+s(t),\quad t\in(0,T)
  142. R N ( t ) = 1 2 N 0 δ ( t ) , R_{N}(t)=\tfrac{1}{2}N_{0}\delta(t),
  143. S N ( f ) = { N 0 2 | f | < w 0 | f | > w S_{N}(f)=\begin{cases}\frac{N_{0}}{2}&|f|<w\\ 0&|f|>w\end{cases}
  144. n 2 ω , n 𝐙 . \frac{n}{2\omega},n\in\mathbf{Z}.
  145. Δ t = n 2 ω \Delta t=\frac{n}{2\omega}
  146. X i = X ( i Δ t ) X_{i}=X(i\Delta t)
  147. n = T Δ t = T ( 2 ω ) = 2 ω T n=\frac{T}{\Delta t}=T(2\omega)=2\omega T
  148. { X 1 , X 2 , , X n } \{X_{1},X_{2},...,X_{n}\}
  149. S i = S ( i Δ t ) S_{i}=S(i\Delta t)
  150. H : X i = N i H:X_{i}=N_{i}
  151. K : X i = N i + S i , i = 1 , 2... n . K:X_{i}=N_{i}+S_{i},i=1,2...n.
  152. ( x ¯ ) = log i = 1 n ( 2 S i x i - S i 2 ) 2 σ 2 Δ t i = 1 n S i x i = i = 1 n S ( i Δ t ) x ( i Δ t ) Δ t λ 2 \mathcal{L}(\underline{x})=\log\frac{\sum^{n}_{i=1}(2S_{i}x_{i}-S_{i}^{2})}{2% \sigma^{2}}\Leftrightarrow\Delta t\sum^{n}_{i=1}S_{i}x_{i}=\sum^{n}_{i=1}S(i% \Delta t)x(i\Delta t)\Delta t\gtrless\lambda_{2}
  153. t 0 t→0
  154. G = 0 T S ( t ) x ( t ) d t . G=\int^{T}_{0}S(t)x(t)dt.
  155. G ( x ¯ ) > G 0 K < G 0 H . G(\underline{x})>G_{0}\Rightarrow K<G_{0}\Rightarrow H.
  156. H : G N ( 0 , 1 2 N 0 E ) H:G\sim N\left(0,\tfrac{1}{2}N_{0}E\right)
  157. K : G N ( E , 1 2 N 0 E ) K:G\sim N\left(E,\tfrac{1}{2}N_{0}E\right)
  158. 𝐄 = 0 T S 2 ( t ) d t \mathbf{E}=\int^{T}_{0}S^{2}(t)dt
  159. α = G 0 N ( 0 , 1 2 N 0 E ) d G G 0 = 1 2 N 0 E Φ - 1 ( 1 - α ) \alpha=\int^{\infty}_{G_{0}}N\left(0,\tfrac{1}{2}N_{0}E\right)dG\Rightarrow G_% {0}=\sqrt{\tfrac{1}{2}N_{0}E}\Phi^{-1}(1-\alpha)
  160. β = G 0 N ( E , 1 2 N 0 E ) d G = 1 - Φ ( G 0 - E 1 2 N 0 E ) = Φ ( 2 E N 0 - Φ - 1 ( 1 - α ) ) , \beta=\int^{\infty}_{G_{0}}N\left(E,\tfrac{1}{2}N_{0}E\right)dG=1-\Phi\left(% \frac{G_{0}-E}{\sqrt{\tfrac{1}{2}N_{0}E}}\right)=\Phi\left(\sqrt{\frac{2E}{N_{% 0}}}-\Phi^{-1}(1-\alpha)\right),
  161. R N ( t , s ) = E [ X ( t ) X ( s ) ] , R_{N}(t,s)=E[X(t)X(s)],
  162. N ( t ) = i = 1 N i Φ i ( t ) , 0 < t < T N(t)=\sum^{\infty}_{i=1}N_{i}\Phi_{i}(t),\quad 0<t<T
  163. N i = N ( t ) Φ i ( t ) d t N_{i}=\int N(t)\Phi_{i}(t)dt
  164. { Φ i t } \{\Phi_{i}{t}\}
  165. R N ( t , s ) R_{N}(t,s)
  166. 0 T R N ( t , s ) Φ i ( s ) d s = λ i Φ i ( t ) , v a r [ N i ] = λ i \int^{T}_{0}R_{N}(t,s)\Phi_{i}(s)ds=\lambda_{i}\Phi_{i}(t),var[N_{i}]=\lambda_% {i}
  167. S ( t ) = i = 1 S i Φ i ( t ) S(t)=\sum^{\infty}_{i=1}S_{i}\Phi_{i}(t)
  168. S i = 0 T S ( t ) Φ i ( t ) d t , 0 < t < T . S_{i}=\int^{T}_{0}S(t)\Phi_{i}(t)dt,0<t<T.
  169. X i = 0 T X ( t ) Φ i ( t ) d t = N i X_{i}=\int^{T}_{0}X(t)\Phi_{i}(t)dt=N_{i}
  170. N i + S i N_{i}+S_{i}
  171. X ¯ = { X 1 , X 2 , } \overline{X}=\{X_{1},X_{2},\dots\}
  172. N i N_{i}
  173. λ i \lambda_{i}
  174. { X i } \{X_{i}\}
  175. f H [ x ( t ) | 0 < t < T ] = f H ( x ¯ ) = i = 1 1 2 π λ i exp ( - x i 2 2 λ i ) f_{H}[x(t)|0<t<T]=f_{H}(\underline{x})=\prod^{\infty}_{i=1}\frac{1}{\sqrt{2\pi% \lambda_{i}}}\exp\left(-\frac{x_{i}^{2}}{2\lambda_{i}}\right)
  176. { X i - S i } \{X_{i}-S_{i}\}
  177. f K [ x ( t ) | 0 < t < T ] = f K ( x ¯ ) = i = 1 1 2 π λ i exp ( - ( x i - S i ) 2 2 λ i ) f_{K}[x(t)|0<t<T]=f_{K}(\underline{x})=\prod^{\infty}_{i=1}\frac{1}{\sqrt{2\pi% \lambda_{i}}}\exp\left(-\frac{(x_{i}-S_{i})^{2}}{2\lambda_{i}}\right)
  178. ( x ¯ ) = i = 1 2 S i x i - S i 2 2 λ i \mathcal{L}(\underline{x})=\sum^{\infty}_{i=1}\frac{2S_{i}x_{i}-S_{i}^{2}}{2% \lambda_{i}}
  179. G = i = 1 S i x i λ i > G 0 K , < G 0 H . G=\sum^{\infty}_{i=1}S_{i}x_{i}\lambda_{i}>G_{0}\Rightarrow K,<G_{0}% \Rightarrow H.
  180. k ( t ) = i = 1 λ i S i Φ i ( t ) , 0 < t < T , k(t)=\sum^{\infty}_{i=1}\lambda_{i}S_{i}\Phi_{i}(t),0<t<T,
  181. G = 0 T k ( t ) x ( t ) d t G=\int^{T}_{0}k(t)x(t)dt
  182. 0 T R N ( t , s ) k ( s ) d s = i = 1 λ i S i 0 T R N ( t , s ) Φ i ( s ) d s = i = 1 S i Φ i ( t ) = S ( t ) \int^{T}_{0}R_{N}(t,s)k(s)ds=\sum^{\infty}_{i=1}\lambda_{i}S_{i}\int^{T}_{0}R_% {N}(t,s)\Phi_{i}(s)ds=\sum^{\infty}_{i=1}S_{i}\Phi_{i}(t)=S(t)
  183. 0 T R N ( t , s ) k ( s ) d s = S ( t ) \int^{T}_{0}R_{N}(t,s)k(s)ds=S(t)
  184. 0 T R N ( t - s ) k ( s ) d s = S ( t ) \int^{T}_{0}R_{N}(t-s)k(s)ds=S(t)
  185. 0 T N 0 2 δ ( t - s ) k ( s ) d s = S ( t ) k ( t ) = C S ( t ) , 0 < t < T \int^{T}_{0}\frac{N_{0}}{2}\delta(t-s)k(s)ds=S(t)\Rightarrow k(t)=CS(t),0<t<T
  186. G = 0 T k ( t ) x ( t ) d t , G=\int^{T}_{0}k(t)x(t)dt,
  187. 𝐄 [ G | H ] = 0 T k ( t ) 𝐄 [ x ( t ) | H ] d t = 0 𝐄 [ G | K ] = 0 T k ( t ) 𝐄 [ x ( t ) | K ] d t = 0 T k ( t ) S ( t ) d t ρ 𝐄 [ G 2 | H ] = 0 T 0 T k ( t ) k ( s ) R N ( t , s ) d t d s = 0 T k ( t ) ( 0 T k ( s ) R N ( t , s ) d s ) = 0 T k ( t ) S ( t ) d t = ρ Var [ G | H ] = 𝐄 [ G 2 | H ] - ( 𝐄 [ G | H ] ) 2 = ρ 𝐄 [ G 2 | K ] = 0 T 0 T k ( t ) k ( s ) 𝐄 [ x ( t ) x ( s ) ] d t d s = 0 T 0 T k ( t ) k ( s ) ( R N ( t , s ) + S ( t ) S ( s ) ) d t d s = ρ + ρ 2 Var [ G | K ] = 𝐄 [ G 2 | K ] - ( 𝐄 [ G | K ] ) 2 = ρ + ρ 2 - ρ 2 = ρ \begin{aligned}\displaystyle\mathbf{E}[G|H]&\displaystyle=\int^{T}_{0}k(t)% \mathbf{E}[x(t)|H]dt=0\\ \displaystyle\mathbf{E}[G|K]&\displaystyle=\int^{T}_{0}k(t)\mathbf{E}[x(t)|K]% dt=\int^{T}_{0}k(t)S(t)dt\equiv\rho\\ \displaystyle\mathbf{E}[G^{2}|H]&\displaystyle=\int^{T}_{0}\int^{T}_{0}k(t)k(s% )R_{N}(t,s)dtds=\int^{T}_{0}k(t)\left(\int^{T}_{0}k(s)R_{N}(t,s)ds\right)=\int% ^{T}_{0}k(t)S(t)dt=\rho\\ \displaystyle\,\text{Var}[G|H]&\displaystyle=\mathbf{E}[G^{2}|H]-(\mathbf{E}[G% |H])^{2}=\rho\\ \displaystyle\mathbf{E}[G^{2}|K]&\displaystyle=\int^{T}_{0}\int^{T}_{0}k(t)k(s% )\mathbf{E}[x(t)x(s)]dtds=\int^{T}_{0}\int^{T}_{0}k(t)k(s)(R_{N}(t,s)+S(t)S(s)% )dtds=\rho+\rho^{2}\\ \displaystyle\,\text{Var}[G|K]&\displaystyle=\mathbf{E}[G^{2}|K]-(\mathbf{E}[G% |K])^{2}=\rho+\rho^{2}-\rho^{2}=\rho\end{aligned}
  188. H : G N ( 0 , ρ ) H:G\sim N(0,\rho)
  189. K : G N ( ρ , ρ ) K:G\sim N(\rho,\rho)
  190. α = G 0 N ( 0 , ρ ) d G = 1 - Φ ( G 0 ρ ) . \alpha=\int^{\infty}_{G_{0}}N(0,\rho)dG=1-\Phi\left(\frac{G_{0}}{\sqrt{\rho}}% \right).
  191. G 0 = ρ Φ - 1 ( 1 - α ) G_{0}=\sqrt{\rho}\Phi^{-1}(1-\alpha)
  192. β = G 0 N ( ρ , ρ ) d G = Φ ( ρ - Φ - 1 ( 1 - α ) ) \beta=\int^{\infty}_{G_{0}}N(\rho,\rho)dG=\Phi\left(\sqrt{\rho}-\Phi^{-1}(1-% \alpha)\right)
  193. ρ = 0 T k ( t ) S ( t ) d t = 0 T S ( t ) 2 d t = E \rho=\int^{T}_{0}k(t)S(t)dt=\int^{T}_{0}S(t)^{2}dt=E
  194. R N ( τ ) = B N 0 4 e - B | τ | R_{N}(\tau)=\frac{BN_{0}}{4}e^{-B|\tau|}
  195. S N ( f ) = N 0 2 ( 1 + ( w B ) 2 ) S_{N}(f)=\frac{N_{0}}{2(1+(\frac{w}{B})^{2})}
  196. H ( f ) = 1 + j w B . H(f)=1+j\frac{w}{B}.
  197. H 0 : Y ( t ) = N ( t ) H_{0}:Y(t)=N(t)
  198. H 1 : Y ( t ) = N ( t ) + X ( t ) , 0 < t < T . H_{1}:Y(t)=N(t)+X(t),0<t<T.
  199. R X ( t , s ) = E { X [ t ] X [ s ] } R_{X}(t,s)=E\{X[t]X[s]\}
  200. X ( t ) = i = 1 X i Φ i ( t ) X(t)=\sum^{\infty}_{i=1}X_{i}\Phi_{i}(t)
  201. X i = 0 T X ( t ) Φ i ( t ) . Φ ( t ) X_{i}=\int^{T}_{0}X(t)\Phi_{i}(t).\Phi(t)
  202. 0 T R X ( t , s ) Φ i ( s ) d s = λ i Φ i ( t ) \int^{T}_{0}R_{X}(t,s)\Phi_{i}(s)ds=\lambda_{i}\Phi_{i}(t)
  203. X i X_{i}
  204. λ i \lambda_{i}
  205. Φ i ( t ) \Phi_{i}(t)
  206. Y i = 0 T Y ( t ) Φ i ( t ) d t = 0 T [ N ( t ) + X ( t ) ] Φ i ( t ) = N i + X i Y_{i}=\int^{T}_{0}Y(t)\Phi_{i}(t)dt=\int^{T}_{0}[N(t)+X(t)]\Phi_{i}(t)=N_{i}+X% _{i}
  207. N i = 0 T N ( t ) Φ i ( t ) d t . N_{i}=\int^{T}_{0}N(t)\Phi_{i}(t)dt.
  208. N i N_{i}
  209. 1 2 N 0 \tfrac{1}{2}N_{0}
  210. H 0 : Y i = N i H_{0}:Y_{i}=N_{i}
  211. H 1 : Y i = N i + X i H_{1}:Y_{i}=N_{i}+X_{i}
  212. Λ = f Y | H 1 f Y | H 0 = C e - i = 1 y i 2 2 λ i 1 2 N 0 ( 1 2 N 0 + λ i ) , \Lambda=\frac{f_{Y}|H_{1}}{f_{Y}|H_{0}}=Ce^{-\sum^{\infty}_{i=1}\frac{y_{i}^{2% }}{2}\frac{\lambda_{i}}{\tfrac{1}{2}N_{0}(\tfrac{1}{2}N_{0}+\lambda_{i})}},
  213. = ln ( Λ ) = K - i = 1 1 2 y i 2 λ i N 0 2 ( N 0 2 + λ i ) \mathcal{L}=\ln(\Lambda)=K-\sum^{\infty}_{i=1}\tfrac{1}{2}y_{i}^{2}\frac{% \lambda_{i}}{\frac{N_{0}}{2}(\frac{N_{0}}{2}+\lambda_{i})}
  214. X i ^ = λ i N 0 2 ( N 0 2 + λ i ) \hat{X_{i}}=\frac{\lambda_{i}}{\frac{N_{0}}{2}(\frac{N_{0}}{2}+\lambda_{i})}
  215. X i X_{i}
  216. Y i Y_{i}
  217. = K + 1 N 0 i = 1 Y i X i ^ \mathcal{L}=K+\frac{1}{N_{0}}\sum^{\infty}_{i=1}Y_{i}\hat{X_{i}}
  218. f ( t ) = f i Φ i ( t ) , g ( t ) = g i Φ i ( t ) f(t)=\sum f_{i}\Phi_{i}(t),g(t)=\sum g_{i}\Phi_{i}(t)
  219. f i = 0 T f ( t ) Φ i ( t ) , g i = 0 T g ( t ) Φ i ( t ) . f_{i}=\int_{0}^{T}f(t)\Phi_{i}(t),g_{i}=\int_{0}^{T}g(t)\Phi_{i}(t).
  220. i = 1 f i g i = 0 T g ( t ) f ( t ) d t \sum^{\infty}_{i=1}f_{i}g_{i}=\int^{T}_{0}g(t)f(t)dt
  221. X ( t | T ) ^ = i = 1 X i ^ Φ i ( t ) , = K + 1 N 0 0 T Y ( t ) X ( t | T ) ^ d t \hat{X(t|T)}=\sum^{\infty}_{i=1}\hat{X_{i}}\Phi_{i}(t),\quad\mathcal{L}=K+% \frac{1}{N_{0}}\int^{T}_{0}Y(t)\hat{X(t|T)}dt
  222. X ( t | T ) ^ = 0 T Q ( t , s ) Y ( s ) d s \hat{X(t|T)}=\int^{T}_{0}Q(t,s)Y(s)ds
  223. 0 T Q ( t , s ) R X ( s , t ) d s + N 0 2 Q ( t , λ ) = R X ( t , λ ) , 0 < λ < T , 0 < t < T . \int^{T}_{0}Q(t,s)R_{X}(s,t)ds+\tfrac{N_{0}}{2}Q(t,\lambda)=R_{X}(t,\lambda),0% <\lambda<T,0<t<T.
  224. X ( t | t ) ^ \hat{X(t|t)}
  225. Q ( t , s ) = h ( t , s ) + h ( s , t ) - 0 T h ( λ , t ) h ( s , λ ) d λ Q(t,s)=h(t,s)+h(s,t)-\int^{T}_{0}h(\lambda,t)h(s,\lambda)d\lambda

Karoubi_envelope.html

  1. e : A A e:A\rightarrow A
  2. e e = e e\circ e=e
  3. e : A A e:A\rightarrow A
  4. ( e , f , e ) : ( A , e ) ( A , e ) (e,f,e^{\prime}):(A,e)\rightarrow(A^{\prime},e^{\prime})
  5. f : A A f:A\rightarrow A^{\prime}
  6. e f = f = f e e^{\prime}\circ f=f=f\circ e
  7. f = e f e f=e^{\prime}\circ f\circ e
  8. ( A , e ) (A,e)
  9. ( e , e , e ) (e,e,e)
  10. A A
  11. 𝐂 ^ \hat{\mathbf{C}}
  12. ( e , f , e ) : ( A , e ) ( A , e ) (e,f,e):(A,e)\rightarrow(A,e)
  13. ( e , g , e ) : ( A , e ) ( A , e ) (e,g,e):(A,e)\rightarrow(A,e)
  14. g f = e = f g g\circ f=e=f\circ g
  15. g f g = g g\circ f\circ g=g
  16. f g f = f f\circ g\circ f=f
  17. g f = f g g\circ f=f\circ g
  18. f : A B f:A\rightarrow B
  19. f × f - 1 : A × B B × A f\times f^{-1}:A\times B\rightarrow B\times A
  20. γ : B × A A × B \gamma:B\times A\rightarrow A\times B

Kármán_line.html

  1. L = 1 2 ρ v 2 S C L L=\tfrac{1}{2}\rho v^{2}SC_{L}
  2. L = 1 2 ρ v 0 2 S C L = m g L=\tfrac{1}{2}\rho v_{0}^{2}SC_{L}=mg

Kármán_vortex_street.html

  1. Re = V d ν \mathrm{Re}=\frac{Vd}{\nu}
  2. d d
  3. V V
  4. ν \nu\,
  5. Re = ρ V d μ \mathrm{Re}=\frac{\rho_{\infty}V_{\infty}d}{\mu_{\infty}}
  6. ρ \rho_{\infty}
  7. V V_{\infty}
  8. d d
  9. μ \mu_{\infty}
  10. f d V = 0.198 ( 1 - 19.7 R e ) \frac{fd}{V}=0.198\left(1-\frac{19.7}{Re}\right)

KCDSA.html

  1. G F ( p ) GF(p)
  2. p p
  3. | p | = 512 + 256 i |p|=512+256i
  4. i = 0 , 1 , , 6 i=0,1,\dots,6
  5. q q
  6. p - 1 p-1
  7. | q | = 128 + 32 j |q|=128+32j
  8. j = 0 , 1 , , 4 j=0,1,\dots,4
  9. g g
  10. q G F ( p ) q\in GF(p)
  11. x x
  12. x Z q x\in Z_{q}
  13. y y
  14. y = g x ¯ ( mod p ) , y=g^{\bar{x}}\;\;(\mathop{{\rm mod}}p),
  15. x ¯ = x - 1 ( mod q ) \bar{x}=x^{-1}\;\;(\mathop{{\rm mod}}q)
  16. z z
  17. z = h ( C e r t D a t a ) z=h(CertData)
  18. k Z q k\in Z_{q}
  19. w = g k mod p w=g^{k}\mod{p}
  20. r = h ( w ) r=h(w)
  21. s = x ( k - r h ( z | | m ) ) ( mod q ) s=x(k-r\oplus h(z||m))\;\;(\mathop{{\rm mod}}q)
  22. ( r , s ) (r,s)
  23. e = r h ( z | | m ) e=r\oplus h(z||m)
  24. r = h ( y s g e mod p ) r=h(y^{s}\cdot g^{e}\mod{p})

Kelvin_wave.html

  1. u x + v y = - 1 H η t \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=\frac{-1}{H}\frac{% \partial\eta}{\partial t}
  2. u t = - g η x + f v \frac{\partial u}{\partial t}=-g\frac{\partial\eta}{\partial x}+fv
  3. v t = - g η y - f u . \frac{\partial v}{\partial t}=-g\frac{\partial\eta}{\partial y}-fu.
  4. v y = - 1 H η t \frac{\partial v}{\partial y}=\frac{-1}{H}\frac{\partial\eta}{\partial t}
  5. g η x = f v g\frac{\partial\eta}{\partial x}=fv
  6. v t = - g η y \frac{\partial v}{\partial t}=-g\frac{\partial\eta}{\partial y}
  7. f = β y , f=\beta y,

Kelvin–Helmholtz_mechanism.html

  1. U = - G m 1 m 2 r , U=-\frac{Gm_{1}m_{2}}{r},
  2. U = - G 0 R m ( r ) 4 π r 2 ρ r d r , U=-G\int_{0}^{R}\frac{m(r)4\pi r^{2}\rho}{r}\,dr,
  3. U = - G 0 R 4 π r 3 ρ 4 π r 2 ρ 3 r d r = - 16 15 G π 2 ρ 2 R 5 . U=-G\int_{0}^{R}\frac{4\pi r^{3}\rho 4\pi r^{2}\rho}{3r}\,dr=-\frac{16}{15}G% \pi^{2}\rho^{2}R^{5}.
  4. U = - 3 M 2 G 5 R . U=-\frac{3M^{2}G}{5R}.
  5. U r = 3 M 2 G 10 R . U\text{r}=\frac{3M^{2}G}{10R}.
  6. U r L 1.1 × 10 41 J 3.9 × 10 26 W 8 900 000 years , \frac{U\text{r}}{L_{\odot}}\approx\frac{1.1\times 10^{41}~{}\,\text{J}}{3.9% \times 10^{26}~{}\,\text{W}}\approx 8\,900\,000~{}\,\text{years},
  7. L L_{\odot}

Kerma_(physics).html

  1. K = d E t r / d m K=dE_{tr}/dm
  2. k c o l k_{col}
  3. k r a d k_{rad}
  4. K = k c o l + k r a d K=k_{col}+k_{rad}
  5. k c o l k_{col}
  6. k c o l = K ( 1 - g ) , k_{col}=K(1-g),

Kernel_(linear_algebra).html

  1. ker ( L ) = { 𝐯 V | L ( 𝐯 ) = 𝟎 } . \ker(L)=\left\{\mathbf{v}\in V|L(\mathbf{v})=\mathbf{0}\right\}\,\text{.}
  2. L ( 𝐯 1 ) = L ( 𝐯 2 ) L ( 𝐯 1 - 𝐯 2 ) = 𝟎 . L(\mathbf{v}_{1})=L(\mathbf{v}_{2})\;\;\;\;\Leftrightarrow\;\;\;\;L(\mathbf{v}% _{1}-\mathbf{v}_{2})=\mathbf{0}\,\text{.}
  3. im ( L ) V / ker ( L ) . \mathop{\mathrm{im}}(L)\cong V/\ker(L)\,\text{.}
  4. dim ( ker L ) + dim ( im L ) = dim ( V ) . \dim(\ker L)+\dim(\mathop{\mathrm{im}}L)=\dim(V)\,\text{.}\,
  5. N ( A ) = Null ( A ) = ker ( A ) = { 𝐱 K n | A 𝐱 = 𝟎 } . \operatorname{N}(A)=\operatorname{Null}(A)=\operatorname{ker}(A)=\left\{% \mathbf{x}\in K^{n}|A\mathbf{x}=\mathbf{0}\right\}.
  6. A 𝐱 = 𝟎 a 11 x 1 + a 12 x 2 + + a 1 n x n = 0 a 21 x 1 + a 22 x 2 + + a 2 n x n = 0 a m 1 x 1 + a m 2 x 2 + + a m n x n = 0 . A\mathbf{x}=\mathbf{0}\;\;\Leftrightarrow\;\;\begin{aligned}\displaystyle a_{1% 1}x_{1}&&\displaystyle\;+&&\displaystyle a_{12}x_{2}&&\displaystyle\;+\;\cdots% \;+&&\displaystyle a_{1n}x_{n}&&\displaystyle\;=&&&\displaystyle 0\\ \displaystyle a_{21}x_{1}&&\displaystyle\;+&&\displaystyle a_{22}x_{2}&&% \displaystyle\;+\;\cdots\;+&&\displaystyle a_{2n}x_{n}&&\displaystyle\;=&&&% \displaystyle 0\\ \displaystyle\vdots&&&&\displaystyle\vdots&&&&\displaystyle\vdots&&&&&% \displaystyle\;\vdots\\ \displaystyle a_{m1}x_{1}&&\displaystyle\;+&&\displaystyle a_{m2}x_{2}&&% \displaystyle\;+\;\cdots\;+&&\displaystyle a_{mn}x_{n}&&\displaystyle\;=&&&% \displaystyle 0\,\text{.}\\ \end{aligned}
  7. A 𝐱 = [ 𝐚 1 𝐱 𝐚 2 𝐱 𝐚 m 𝐱 ] . A\mathbf{x}=\begin{bmatrix}\mathbf{a}_{1}\cdot\mathbf{x}\\ \mathbf{a}_{2}\cdot\mathbf{x}\\ \vdots\\ \mathbf{a}_{m}\cdot\mathbf{x}\end{bmatrix}.
  8. rank ( A ) + nullity ( A ) = n . \operatorname{rank}(A)+\operatorname{nullity}(A)=n.
  9. A 𝐱 = 𝐛 or a 11 x 1 + a 12 x 2 + + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2 n x n = b 2 a m 1 x 1 + a m 2 x 2 + + a m n x n = b m A\mathbf{x}=\mathbf{b}\;\;\;\;\;\;\,\text{or}\;\;\;\;\;\;\begin{aligned}% \displaystyle a_{11}x_{1}&&\displaystyle\;+&&\displaystyle a_{12}x_{2}&&% \displaystyle\;+\;\cdots\;+&&\displaystyle a_{1n}x_{n}&&\displaystyle\;=&&&% \displaystyle b_{1}\\ \displaystyle a_{21}x_{1}&&\displaystyle\;+&&\displaystyle a_{22}x_{2}&&% \displaystyle\;+\;\cdots\;+&&\displaystyle a_{2n}x_{n}&&\displaystyle\;=&&&% \displaystyle b_{2}\\ \displaystyle\vdots&&&&\displaystyle\vdots&&&&\displaystyle\vdots&&&&&% \displaystyle\;\vdots\\ \displaystyle a_{m1}x_{1}&&\displaystyle\;+&&\displaystyle a_{m2}x_{2}&&% \displaystyle\;+\;\cdots\;+&&\displaystyle a_{mn}x_{n}&&\displaystyle\;=&&&% \displaystyle b_{m}\\ \end{aligned}
  10. A ( 𝐮 - 𝐯 ) = A 𝐮 - A 𝐯 = 𝐛 - 𝐛 = 𝟎 A(\mathbf{u}-\mathbf{v})=A\mathbf{u}-A\mathbf{v}=\mathbf{b}-\mathbf{b}=\mathbf% {0}\,
  11. { 𝐯 + 𝐱 | A 𝐯 = 𝐛 𝐱 Null ( A ) } , \left\{\mathbf{v}+\mathbf{x}|A\mathbf{v}=\mathbf{b}\land\mathbf{x}\in% \operatorname{Null}(A)\right\},
  12. A = [ 2 3 5 - 4 2 3 ] . A=\begin{bmatrix}\,\,\,2&3&5\\ -4&2&3\end{bmatrix}.
  13. [ 2 3 5 - 4 2 3 ] [ x y z ] = [ 0 0 ] , \begin{bmatrix}\,\,\,2&3&5\\ -4&2&3\end{bmatrix}\begin{bmatrix}x\\ y\\ z\end{bmatrix}=\begin{bmatrix}0\\ 0\end{bmatrix},
  14. 2 x \displaystyle 2x
  15. [ 2 3 5 0 - 4 2 3 0 ] . \left[\begin{array}[]{ccc|c}2&3&5&0\\ -4&2&3&0\end{array}\right].
  16. [ 1 0 1 / 16 0 0 1 13 / 8 0 ] . \left[\begin{array}[]{ccc|c}1&0&1/16&0\\ 0&1&13/8&0\end{array}\right].
  17. x = \displaystyle x=
  18. [ x y z ] = c [ - 1 / 16 - 13 / 8 1 ] . \begin{bmatrix}x\\ y\\ z\end{bmatrix}=c\begin{bmatrix}-1/16\\ -13/8\\ 1\end{bmatrix}.
  19. [ x y z ] = c [ - 1 - 26 16 ] . \begin{bmatrix}x\\ y\\ z\end{bmatrix}=c\begin{bmatrix}-1\\ -26\\ 16\end{bmatrix}.
  20. [ 2 3 5 ] [ - 1 - 26 16 ] = 0 and [ - 4 2 3 ] [ - 1 - 26 16 ] = 0 , \left[\begin{array}[]{ccc}2&3&5\end{array}\right]\cdot\begin{bmatrix}-1\\ -26\\ 16\end{bmatrix}=0\quad\mathrm{and}\quad\left[\begin{array}[]{ccc}-4&2&3\end{% array}\right]\cdot\begin{bmatrix}-1\\ -26\\ 16\end{bmatrix}=0\mathrm{,}
  21. L ( x 1 , x 2 , x 3 ) = ( 2 x 1 + 3 x 2 + 5 x 3 , - 4 x 1 + 2 x 2 + 3 x 3 ) L(x_{1},x_{2},x_{3})=(2x_{1}+3x_{2}+5x_{3},\;-4x_{1}+2x_{2}+3x_{3})
  22. 2 x 1 + 3 x 2 + 5 x 3 = 0 - 4 x 1 + 2 x 2 + 3 x 3 = 0 \begin{aligned}\displaystyle 2x_{1}&\displaystyle\;+&\displaystyle 3x_{2}&% \displaystyle\;+&\displaystyle 5x_{3}&\displaystyle\;=&\displaystyle 0\\ \displaystyle-4x_{1}&\displaystyle\;+&\displaystyle 2x_{2}&\displaystyle\;+&% \displaystyle 3x_{3}&\displaystyle\;=&\displaystyle 0\end{aligned}
  23. L ( f ) = f ( 0.3 ) . L(f)=f(0.3)\,\text{.}\,
  24. D ( f ) = d f d x . D(f)=\frac{df}{dx}\,\text{.}
  25. s ( x 1 , x 2 , x 3 , x 4 , ) = ( x 2 , x 3 , x 4 , ) . s(x_{1},x_{2},x_{3},x_{4},\ldots)=(x_{2},x_{3},x_{4},\ldots)\,\text{.}
  26. [ A I ] , \left[\begin{array}[]{c}A\\ \hline I\end{array}\right],
  27. I I
  28. [ B C ] . \left[\begin{array}[]{c}B\\ \hline C\end{array}\right].
  29. A = [ 1 0 - 3 0 2 - 8 0 1 5 0 - 1 4 0 0 0 1 7 - 9 0 0 0 0 0 0 ] . A=\left[\begin{array}[]{cccccc}1&0&-3&0&2&-8\\ 0&1&5&0&-1&4\\ 0&0&0&1&7&-9\\ 0&0&0&0&0&0\end{array}\,\right].
  30. [ A I ] = [ 1 0 - 3 0 2 - 8 0 1 5 0 - 1 4 0 0 0 1 7 - 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 ] . \left[\begin{array}[]{c}A\\ \hline I\end{array}\right]=\left[\begin{array}[]{cccccc}1&0&-3&0&2&-8\\ 0&1&5&0&-1&4\\ 0&0&0&1&7&-9\\ 0&0&0&0&0&0\\ \hline 1&0&0&0&0&0\\ 0&1&0&0&0&0\\ 0&0&1&0&0&0\\ 0&0&0&1&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1\end{array}\right].
  31. [ B C ] = [ 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 3 - 2 8 0 1 0 - 5 1 - 4 0 0 0 1 0 0 0 0 1 0 - 7 9 0 0 0 0 1 0 0 0 0 0 0 1 ] . \left[\begin{array}[]{c}B\\ \hline C\end{array}\right]=\left[\begin{array}[]{cccccc}1&0&0&0&0&0\\ 0&1&0&0&0&0\\ 0&0&1&0&0&0\\ 0&0&0&0&0&0\\ \hline 1&0&0&3&-2&8\\ 0&1&0&-5&1&-4\\ 0&0&0&1&0&0\\ 0&0&1&0&-7&9\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1\end{array}\right].
  32. [ 3 - 5 1 0 0 0 ] , [ - 2 1 0 - 7 1 0 ] , [ 8 - 4 0 9 0 1 ] \left[\!\!\begin{array}[]{r}3\\ -5\\ 1\\ 0\\ 0\\ 0\end{array}\right],\;\left[\!\!\begin{array}[]{r}-2\\ 1\\ 0\\ -7\\ 1\\ 0\end{array}\right],\;\left[\!\!\begin{array}[]{r}8\\ -4\\ 0\\ 9\\ 0\\ 1\end{array}\right]
  33. [ A I ] \left[\begin{array}[]{c}A\\ \hline I\end{array}\right]
  34. [ B C ] \left[\begin{array}[]{c}B\\ \hline C\end{array}\right]
  35. A C = B AC=B
  36. A A
  37. C C
  38. B B
  39. B B
  40. B B
  41. B B
  42. A A
  43. C C
  44. C C
  45. A A
  46. A A

Kerr_effect.html

  1. Δ n = λ K E 2 , \Delta n=\lambda KE^{2},
  2. 𝐏 = ε 0 χ ( 1 ) : 𝐄 + ε 0 χ ( 2 ) : 𝐄𝐄 + ε 0 χ ( 3 ) : 𝐄𝐄𝐄 + \mathbf{P}=\varepsilon_{0}\chi^{(1)}:\mathbf{E}+\varepsilon_{0}\chi^{(2)}:% \mathbf{EE}+\varepsilon_{0}\chi^{(3)}:\mathbf{EEE}+\cdots
  3. P i = ε 0 j = 1 3 χ i j ( 1 ) E j + ε 0 j = 1 3 k = 1 3 χ i j k ( 2 ) E j E k + ε 0 j = 1 3 k = 1 3 l = 1 3 χ i j k l ( 3 ) E j E k E l + P_{i}=\varepsilon_{0}\sum_{j=1}^{3}\chi^{(1)}_{ij}E_{j}+\varepsilon_{0}\sum_{j% =1}^{3}\sum_{k=1}^{3}\chi^{(2)}_{ijk}E_{j}E_{k}+\varepsilon_{0}\sum_{j=1}^{3}% \sum_{k=1}^{3}\sum_{l=1}^{3}\chi^{(3)}_{ijkl}E_{j}E_{k}E_{l}+\cdots
  4. i = 1 , 2 , 3 i=1,2,3
  5. P 1 = P x P_{1}=P_{x}
  6. E 2 = E y E_{2}=E_{y}
  7. 𝐄 = 𝐄 0 + 𝐄 ω cos ( ω t ) , \mathbf{E}=\mathbf{E}_{0}+\mathbf{E}_{\omega}\cos(\omega t),
  8. χ ( 3 ) | 𝐄 0 | 2 𝐄 ω \chi^{(3)}|\mathbf{E}_{0}|^{2}\mathbf{E}_{\omega}
  9. 𝐏 ε 0 ( χ ( 1 ) + 3 χ ( 3 ) | 𝐄 0 | 2 ) 𝐄 ω cos ( ω t ) , \mathbf{P}\simeq\varepsilon_{0}\left(\chi^{(1)}+3\chi^{(3)}|\mathbf{E}_{0}|^{2% }\right)\mathbf{E}_{\omega}\cos(\omega t),
  10. Δ n = λ 0 K | 𝐄 0 | 2 , \Delta n=\lambda_{0}K|\mathbf{E}_{0}|^{2},
  11. 𝐄 = 𝐄 ω cos ( ω t ) , \mathbf{E}=\mathbf{E}_{\omega}\cos(\omega t),
  12. 𝐏 ε 0 ( χ ( 1 ) + 3 4 χ ( 3 ) | 𝐄 ω | 2 ) 𝐄 ω cos ( ω t ) . \mathbf{P}\simeq\varepsilon_{0}\left(\chi^{(1)}+\frac{3}{4}\chi^{(3)}|\mathbf{% E}_{\omega}|^{2}\right)\mathbf{E}_{\omega}\cos(\omega t).
  13. χ = χ LIN + χ NL = χ ( 1 ) + 3 χ ( 3 ) 4 | 𝐄 ω | 2 , \chi=\chi_{\mathrm{LIN}}+\chi_{\mathrm{NL}}=\chi^{(1)}+\frac{3\chi^{(3)}}{4}|% \mathbf{E}_{\omega}|^{2},
  14. n = ( 1 + χ ) 1 / 2 = ( 1 + χ LIN + χ NL ) 1 / 2 n 0 ( 1 + 1 2 n 0 2 χ NL ) n=(1+\chi)^{1/2}=\left(1+\chi_{\mathrm{LIN}}+\chi_{\mathrm{NL}}\right)^{1/2}% \simeq n_{0}\left(1+\frac{1}{2{n_{0}}^{2}}\chi_{\mathrm{NL}}\right)
  15. n = n 0 + 3 χ ( 3 ) 8 n 0 | 𝐄 ω | 2 = n 0 + n 2 I n=n_{0}+\frac{3\chi^{(3)}}{8n_{0}}|\mathbf{E}_{\omega}|^{2}=n_{0}+n_{2}I

Killing_spinor.html

  1. ψ \psi
  2. X ψ = λ X ψ \nabla_{X}\psi=\lambda X\cdot\psi
  3. \nabla
  4. \cdot
  5. λ \lambda
  6. ψ \psi\,
  7. λ = 0 \lambda=0

Killing_vector_field.html

  1. X g = 0 . \mathcal{L}_{X}g=0\,.
  2. g ( Y X , Z ) + g ( Y , Z X ) = 0 g(\nabla_{Y}X,Z)+g(Y,\nabla_{Z}X)=0\,
  3. μ X ν + ν X μ = 0 . \nabla_{\mu}X_{\nu}+\nabla_{\nu}X_{\mu}=0\,.
  4. g μ ν g_{\mu\nu}\,
  5. d x a dx^{a}\,
  6. x κ x^{\kappa}\,
  7. K μ = δ κ μ K^{\mu}=\delta^{\mu}_{\kappa}\,
  8. δ κ μ \delta^{\mu}_{\kappa}\,
  9. g μ ν , 0 = 0 g_{\mu\nu},_{0}=0\,
  10. K μ = δ 0 μ K^{\mu}=\delta^{\mu}_{0}\,
  11. K μ = g μ ν K ν = g μ ν δ 0 ν = g μ 0 K_{\mu}=g_{\mu\nu}K^{\nu}=g_{\mu\nu}\delta^{\nu}_{0}=g_{\mu 0}\,
  12. K μ ; ν + K ν ; μ = K μ , ν + K ν , μ - 2 Γ μ ν ρ K ρ = g μ 0 , ν + g ν 0 , μ - g ρ σ ( g σ μ , ν + g σ ν , μ - g μ ν , σ ) g ρ 0 K_{\mu;\nu}+K_{\nu;\mu}=K_{\mu,\nu}+K_{\nu,\mu}-2\Gamma^{\rho}_{\mu\nu}K_{\rho% }=g_{\mu 0,\nu}+g_{\nu 0,\mu}-g^{\rho\sigma}(g_{\sigma\mu,\nu}+g_{\sigma\nu,% \mu}-g_{\mu\nu,\sigma})g_{\rho 0}\,
  13. g ρ 0 g ρ σ = δ 0 σ g_{\rho 0}g^{\rho\sigma}=\delta_{0}^{\sigma}\,
  14. g μ 0 , ν + g ν 0 , μ - ( g 0 μ , ν + g 0 ν , μ - g μ ν , 0 ) = 0 g_{\mu 0,\nu}+g_{\nu 0,\mu}-(g_{0\mu,\nu}+g_{0\nu,\mu}-g_{\mu\nu,0})=0\,
  15. g μ ν , 0 = 0 g_{\mu\nu,0}=0
  16. X X
  17. Y Y
  18. g ( X , Y ) g(X,Y)
  19. X X
  20. ω \omega
  21. X ω = 0 . \mathcal{L}_{X}\omega=0\,.
  22. λ \lambda
  23. d d λ ( K μ d x μ d λ ) = 0 \frac{d}{d\lambda}(K_{\mu}\frac{dx^{\mu}}{d\lambda})=0
  24. X g = λ g \mathcal{L}_{X}g=\lambda g\,
  25. λ . \lambda\,.
  26. T \nabla T\,
  27. 𝔤 \mathfrak{g}

Kinetic_isotope_effect.html

  1. K I E = k L k H KIE=\frac{k_{L}}{k_{H}}
  2. k H k D = ( s H s D s D s H ) ( M H M D M D M H ) 3 / 2 ( I A H I B H I C H I A D I B D I C D I A D I B D I C D I A H I B H I C H ) 1 / 2 × ( i = 1 3 N - 7 1 - e - u i D 1 - e - u i H i = 1 3 N - 6 1 - e - u i H 1 - e - u i D ) e - 1 / 2 ( i = 1 3 N - 7 ( u i H - u i D ) - i = 1 3 N - 6 ( u i H - u i D ) ) \begin{aligned}\displaystyle\frac{k_{H}}{k_{D}}=&\displaystyle\left(\frac{s^{% \ddagger}_{H}s_{D}}{s^{\ddagger}_{D}s_{H}}\right)\left(\frac{M^{\ddagger}_{H}M% _{D}}{M^{\ddagger}_{D}M_{H}}\right)^{3/2}\left(\frac{I^{\ddagger}_{AH}I^{% \ddagger}_{BH}I^{\ddagger}_{CH}}{I^{\ddagger}_{AD}I^{\ddagger}_{BD}I^{\ddagger% }_{CD}}\frac{I_{AD}I_{BD}I_{CD}}{I_{AH}I_{BH}I_{CH}}\right)^{1/2}\\ &\displaystyle\times\left(\frac{\prod\limits_{i=1}^{3N^{\ddagger}-7}\frac{1-e^% {-u^{\ddagger}_{iD}}}{1-e^{-u^{\ddagger}_{iH}}}}{\prod\limits_{i=1}^{3N-6}% \frac{1-e^{-u_{iH}}}{1-e^{-u_{iD}}}}\right)e^{-1/2(\sum\limits_{i=1}^{3N^{% \ddagger}-7}(u^{\ddagger}_{iH}-u^{\ddagger}_{iD})-\sum\limits_{i=1}^{3N-6}(u_{% iH}-u_{iD}))}\end{aligned}
  3. k H k D = S × M M I × E X C × Z P E \frac{k_{H}}{k_{D}}=S\times MMI\times EXC\times ZPE
  4. k H k D e - 1 / 2 { i = 1 3 N - 7 ( u i H - u i D ) - i = 1 3 N - 6 ( u i H - u i D ) } = e 1 / 2 { i = 1 3 N - 6 Δ u i - i = 1 3 N - 7 Δ u i } \begin{aligned}\displaystyle\frac{k_{H}}{k_{D}}&\displaystyle\cong e^{-1/2\{% \sum\limits_{i=1}^{3N^{\ddagger}-7}(u^{\ddagger}_{iH}-u^{\ddagger}_{iD})-\sum% \limits_{i=1}^{3N-6}(u_{iH}-u_{iD})\}}\\ &\displaystyle=e^{1/2\{\sum\limits_{i=1}^{3N-6}\Delta u_{i}-\sum\limits_{i=1}^% {3N^{\ddagger}-7}\Delta u^{\ddagger}_{i}\}}\end{aligned}
  5. ( ln ( k H k T ) ln ( k H k D ) ) s = 1 - m H / m T 1 - m H / m D = 1 - 1 / 3 1 - 1 / 2 1.44 \left(\frac{\ln(\frac{k_{H}}{k_{T}})}{\ln(\frac{k_{H}}{k_{D}})}\right)_{s}=% \frac{1-\sqrt{m_{H}/m_{T}}}{1-\sqrt{m_{H}/m_{D}}}=\frac{1-\sqrt{1/3}}{1-\sqrt{% 1/2}}\cong 1.44
  6. ( k H k T ) s = ( k H k D ) s 1.44 \left(\frac{k_{H}}{k_{T}}\right)_{s}=\left(\frac{k_{H}}{k_{D}}\right)_{s}^{1.44}
  7. k = Q A e - E / R T k=QAe^{-E/RT}
  8. Q = e α β - α ( β e - α - α e - β ) Q=\frac{e^{\alpha}}{\beta-\alpha}(\beta e^{-\alpha}-\alpha e^{-\beta})
  9. α = E R T \alpha=\frac{E}{RT}
  10. β = 2 a π 2 ( 2 m E ) 1 / 2 h \beta=\frac{2a\pi^{2}(2mE)^{1/2}}{h}
  11. K I E = k 1 k 2 = ln ( 1 - F 1 ) ln ( 1 - F 2 ) KIE={k_{1}\over k_{2}}=\frac{\ln(1-F_{1})}{\ln(1-F_{2})}
  12. k 1 k 2 = ln ( 1 - F 1 ) ln [ 1 - ( F 1 R P / R 0 ) ] {k_{1}\over k_{2}}=\frac{\ln(1-F_{1})}{\ln[1-(F_{1}R_{P}/R_{0})]}
  13. k ( X = D ) k ( X = H ) = 1.15 \frac{k_{(X=D)}}{k_{(X=H)}}=1.15

Kinetic_term.html

  1. T = 1 2 x ˙ 2 = 1 2 ( x t ) 2 . T=\frac{1}{2}\dot{x}^{2}=\frac{1}{2}\left(\frac{\partial x}{\partial t}\right)% ^{2}.
  2. T = 1 2 μ Φ μ Φ + 1 4 g 2 F μ ν F μ ν + i ψ ¯ γ μ μ ψ . T=\frac{1}{2}\partial_{\mu}\Phi\partial^{\mu}\Phi+\frac{1}{4g^{2}}F_{\mu\nu}F^% {\mu\nu}+i\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi.

Klein–Nishina_formula.html

  1. E γ E_{\gamma}
  2. d σ d Ω = α 2 r c 2 P ( E γ , θ ) 2 [ P ( E γ , θ ) + P ( E γ , θ ) - 1 - 1 + cos 2 ( θ ) ] / 2 \frac{d\sigma}{d\Omega}=\alpha^{2}r_{c}^{2}P(E_{\gamma},\theta)^{2}[P(E_{% \gamma},\theta)+P(E_{\gamma},\theta)^{-1}-1+\cos^{2}(\theta)]/2
  3. d σ d Ω \frac{d\sigma}{d\Omega}
  4. d Ω d\Omega
  5. α \alpha
  6. θ \theta
  7. r c = / m e c r_{c}=\hbar/m_{e}c
  8. m e m_{e}
  9. / c 2 /c^{2}
  10. P ( E γ , θ ) P(E_{\gamma},\theta)
  11. P ( E γ , θ ) = 1 1 + ( E γ / m e c 2 ) ( 1 - cos θ ) P(E_{\gamma},\theta)=\frac{1}{1+(E_{\gamma}/m_{e}c^{2})(1-\cos\theta)}
  12. r e = α r c r_{e}=\alpha r_{c}
  13. θ \theta
  14. r e = α r c r_{e}=\alpha r_{c}
  15. r e 2 r_{e}^{2}
  16. E γ m e c 2 E_{\gamma}\ll m_{e}c^{2}
  17. P ( E γ , θ ) 1 P(E_{\gamma},\theta)\rightarrow 1
  18. E γ E_{\gamma}^{\prime}
  19. E γ ( E γ , θ ) = E γ P ( E γ , θ ) E_{\gamma}^{\prime}(E_{\gamma},\theta)=E_{\gamma}\cdot P(E_{\gamma},\theta)\,

Knight_shift.html

  1. B \vec{B}
  2. ^ KS = - i < m t p l > N γ i I ^ i 𝐊 ^ i B {{\hat{\mathcal{H}}}_{\,\text{KS}}}=-\sum\limits_{\mathit{i}}^{<}mtpl>{{N}}{{{% \gamma}_{\mathit{i}}}\cdot{{{\hat{\vec{I}}}}_{\mathit{i}}}\cdot{{{\hat{\mathbf% {K}}}}_{\mathit{i}}}\cdot\vec{B}}
  3. γ i {\gamma}_{\mathit{i}}
  4. I ^ i {{{\hat{\vec{I}}}}_{\mathit{i}}}
  5. 𝐊 ^ i = ( K x x K x y K x z K y x K y y K y z K z x K z y K z z ) {{{\hat{\mathbf{K}}}}_{i}}=\left(\begin{matrix}{{K}_{xx}}&{{K}_{xy}}&{{K}_{xz}% }\\ {{K}_{yx}}&{{K}_{yy}}&{{K}_{yz}}\\ {{K}_{zx}}&{{K}_{zy}}&{{K}_{zz}}\\ \end{matrix}\right)

Knot_group.html

  1. π 1 ( 3 K ) . \pi_{1}(\mathbb{R}^{3}\setminus K).
  2. S 3 S^{3}
  3. 3 \mathbb{R}^{3}
  4. x , y x 2 = y 3 \langle x,y\mid x^{2}=y^{3}\rangle
  5. a , b a b a = b a b . \langle a,b\mid aba=bab\rangle.
  6. x , y x p = y q . \langle x,y\mid x^{p}=y^{q}\rangle.
  7. x , y y x y - 1 x y = x y x - 1 y x \langle x,y\mid yxy^{-1}xy=xyx^{-1}yx\rangle

Kochanek–Bartels_spline.html

  1. 𝐝 i = ( 1 - t ) ( 1 + b ) ( 1 + c ) 2 ( 𝐩 i - 𝐩 i - 1 ) + ( 1 - t ) ( 1 - b ) ( 1 - c ) 2 ( 𝐩 i + 1 - 𝐩 i ) \mathbf{d}_{i}=\frac{(1-t)(1+b)(1+c)}{2}(\mathbf{p}_{i}-\mathbf{p}_{i-1})+% \frac{(1-t)(1-b)(1-c)}{2}(\mathbf{p}_{i+1}-\mathbf{p}_{i})
  2. 𝐝 i + 1 = ( 1 - t ) ( 1 + b ) ( 1 - c ) 2 ( 𝐩 i + 1 - 𝐩 i ) + ( 1 - t ) ( 1 - b ) ( 1 + c ) 2 ( 𝐩 i + 2 - 𝐩 i + 1 ) \mathbf{d}_{i+1}=\frac{(1-t)(1+b)(1-c)}{2}(\mathbf{p}_{i+1}-\mathbf{p}_{i})+% \frac{(1-t)(1-b)(1+c)}{2}(\mathbf{p}_{i+2}-\mathbf{p}_{i+1})
  3. t t
  4. b b
  5. c c

Konami_SCC.html

  1. f t o n e = f c l o c k 32 ( P + 1 ) f_{tone}=\frac{f_{clock}}{32(P+1)}
  2. f c l o c k f_{clock}
  3. y = 2 - ( ( 15 - n ) / 2 ) y=2^{-((15-n)/2)}

Krasnikov_tube.html

  1. c 2 d t 2 c^{2}dt^{2}
  2. d x 2 + d y 2 + d z 2 dx^{2}+dy^{2}+dz^{2}

Kripke_semantics.html

  1. \to
  2. ¬ \neg
  3. \Box
  4. \Diamond
  5. \Box
  6. A := ¬ ¬ A \Diamond A:=\neg\Box\neg A
  7. W , R \langle W,R\rangle
  8. W , R , \langle W,R,\Vdash\rangle
  9. W , R \langle W,R\rangle
  10. \Vdash
  11. w ¬ A w\Vdash\neg A
  12. w A w\nVdash A
  13. w A B w\Vdash A\to B
  14. w A w\nVdash A
  15. w B w\Vdash B
  16. w A w\Vdash\Box A
  17. u A u\Vdash A
  18. u u
  19. w R u w\;R\;u
  20. w A w\Vdash A
  21. \Vdash
  22. W , R , \langle W,R,\Vdash\rangle
  23. w A w\Vdash A
  24. W , R \langle W,R\rangle
  25. W , R , \langle W,R,\Vdash\rangle
  26. \Vdash
  27. A A \Box A\to A
  28. W , R \langle W,R\rangle
  29. w A w\Vdash\Box A
  30. w A w\Vdash A
  31. u p u\Vdash p
  32. w p w\Vdash\Box p
  33. w p w\Vdash p
  34. \Vdash
  35. ( A A ) A \Box(A\leftrightarrow\Box A)\to\Box A
  36. A A \Box A\to\Box\Box A
  37. ( A B ) ( A B ) \Box(A\to B)\to(\Box A\to\Box B)
  38. A A \Box A\to A
  39. w R w w\,R\,w
  40. A A \Box A\to\Box\Box A
  41. w R v v R u w R u w\,R\,v\wedge v\,R\,u\Rightarrow w\,R\,u
  42. A A \Box\Box A\to\Box A
  43. w R u v ( w R v v R u ) w\,R\,u\Rightarrow\exists v\,(w\,R\,v\land v\,R\,u)
  44. A A \Box A\to\Diamond A
  45. \Diamond\top
  46. w v ( w R v ) \forall w\,\exists v\,(w\,R\,v)
  47. A A A\to\Box\Diamond A
  48. w R v v R w w\,R\,v\Rightarrow v\,R\,w
  49. A A \Diamond A\to\Box\Diamond A
  50. w R u w R v u R v w\,R\,u\land w\,R\,v\Rightarrow u\,R\,v
  51. ( A A ) A \Box(\Box A\to A)\to\Box A
  52. ( ( A A ) A ) A \Box(\Box(A\to\Box A)\to A)\to A
  53. ( A B ) ( B A ) \Box(\Box A\to B)\lor\Box(\Box B\to A)
  54. w R u w R v u R v v R u w\,R\,u\land w\,R\,v\Rightarrow u\,R\,v\lor v\,R\,u
  55. A A \Box\Diamond A\to\Diamond\Box A
  56. A A \Diamond\Box A\to\Box\Diamond A
  57. w R u w R v x ( u R x v R x ) w\,R\,u\land w\,R\,v\Rightarrow\exists x\,(u\,R\,x\land v\,R\,x)
  58. A A A\to\Box A
  59. w R v w = v w\,R\,v\Rightarrow w=v
  60. A A \Diamond A\to\Box A
  61. w R u w R v u = v w\,R\,u\land w\,R\,v\Rightarrow u=v
  62. A A \Diamond A\leftrightarrow\Box A
  63. w ! u w R u \forall w\,\exists!u\,w\,R\,u
  64. A \Box A
  65. \Box\bot
  66. w u ¬ ( w R u ) \forall w\,\forall u\,\neg(w\,R\,u)
  67. w u ( w R u v ( u R v u = v ) ) \forall w\,\exists u\,(w\,R\,u\land\forall v\,(u\,R\,v\Rightarrow u=v))
  68. W , R , \langle W,R,\Vdash\rangle
  69. \Vdash
  70. X R Y X\;R\;Y
  71. A A
  72. A X \Box A\in X
  73. A Y A\in Y
  74. X A X\Vdash A
  75. A X A\in X
  76. { i i I } \{\Box_{i}\mid\,i\in I\}
  77. w i A w\Vdash\Box_{i}A
  78. u ( w R i u u A ) . \forall u\,(w\;R_{i}\;u\Rightarrow u\Vdash A).
  79. W , R , { D i } i I , \langle W,R,\{D_{i}\}_{i\in I},\Vdash\rangle
  80. w i A w\Vdash\Box_{i}A
  81. u D i ( w R u u A ) . \forall u\in D_{i}\,(w\;R\;u\Rightarrow u\Vdash A).
  82. W , , \langle W,\leq,\Vdash\rangle
  83. W , \langle W,\leq\rangle
  84. \Vdash
  85. w u w\leq u
  86. w p w\Vdash p
  87. u p u\Vdash p
  88. w A B w\Vdash A\land B
  89. w A w\Vdash A
  90. w B w\Vdash B
  91. w A B w\Vdash A\lor B
  92. w A w\Vdash A
  93. w B w\Vdash B
  94. w A B w\Vdash A\to B
  95. u w u\geq w
  96. u A u\Vdash A
  97. u B u\Vdash B
  98. w w\Vdash\bot
  99. W , , { M w } w W \langle W,\leq,\{M_{w}\}_{w\in W}\rangle
  100. W , \langle W,\leq\rangle
  101. w A [ e ] w\Vdash A[e]
  102. w P ( t 1 , , t n ) [ e ] w\Vdash P(t_{1},\dots,t_{n})[e]
  103. P ( t 1 [ e ] , , t n [ e ] ) P(t_{1}[e],\dots,t_{n}[e])
  104. w ( A B ) [ e ] w\Vdash(A\land B)[e]
  105. w A [ e ] w\Vdash A[e]
  106. w B [ e ] w\Vdash B[e]
  107. w ( A B ) [ e ] w\Vdash(A\lor B)[e]
  108. w A [ e ] w\Vdash A[e]
  109. w B [ e ] w\Vdash B[e]
  110. w ( A B ) [ e ] w\Vdash(A\to B)[e]
  111. u w u\geq w
  112. u A [ e ] u\Vdash A[e]
  113. u B [ e ] u\Vdash B[e]
  114. w [ e ] w\Vdash\bot[e]
  115. w ( x A ) [ e ] w\Vdash(\exists x\,A)[e]
  116. a M w a\in M_{w}
  117. w A [ e ( x a ) ] w\Vdash A[e(x\to a)]
  118. w ( x A ) [ e ] w\Vdash(\forall x\,A)[e]
  119. u w u\geq w
  120. a M u a\in M_{u}
  121. u A [ e ( x a ) ] u\Vdash A[e(x\to a)]
  122. W , R \langle W,R\rangle
  123. W , R \langle W^{\prime},R^{\prime}\rangle
  124. f : W W f\colon W\to W^{\prime}
  125. W , R , \langle W,R,\Vdash\rangle
  126. W , R , \langle W^{\prime},R^{\prime},\Vdash^{\prime}\rangle
  127. f : W W f\colon W\to W^{\prime}
  128. w p w\Vdash p
  129. f ( w ) p f(w)\Vdash^{\prime}p
  130. W , R \langle W,R\rangle
  131. W , R \langle W^{\prime},R^{\prime}\rangle
  132. w p w\Vdash p
  133. w p w^{\prime}\Vdash^{\prime}p
  134. W , R , \langle W,R,\Vdash\rangle
  135. W , R , \langle W^{\prime},R^{\prime},\Vdash^{\prime}\rangle
  136. s = w 0 , w 1 , , w n s=\langle w_{0},w_{1},\dots,w_{n}\rangle
  137. w n p w_{n}\Vdash p
  138. w 0 , w 1 , , w n R w 0 , w 1 , , w n , w n + 1 \langle w_{0},w_{1},\dots,w_{n}\rangle\;R^{\prime}\;\langle w_{0},w_{1},\dots,% w_{n},w_{n+1}\rangle
  139. W , R , \langle W,R,\Vdash\rangle
  140. W , R , \langle W^{\prime},R^{\prime},\Vdash^{\prime}\rangle
  141. u A u\Vdash\Box A
  142. A X \Box A\in X
  143. v A v\Vdash A
  144. u A u\Vdash A
  145. v A v\Vdash A

Kripke–Platek_set_theory.html

  1. u v \forall u\in v
  2. \exist u v . \exist u\in v.
  3. \exist r p ( a r and x r ( x = a ) ) and \exist s p ( a s and b s and x s ( x = a x = b ) ) \exist r\in p(a\in r\and\forall x\in r(x=a))\and\exist s\in p(a\in s\and b\in s% \and\forall x\in s(x=ax=b))
  4. t p ( ( a t and x t ( x = a ) ) ( a t and b t and x t ( x = a x = b ) ) ) . \forall t\in p((a\in t\and\forall x\in t(x=a))(a\in t\and b\in t\and\forall x% \in t(x=ax=b))).
  5. ψ ( a , b , p ) . \psi(a,b,p)\!.
  6. \exist a A ψ ( a , b , p ) \exist a\in A\psi(a,b,p)
  7. a A \exist p v ψ ( a , b , p ) and p v \exist a A ψ ( a , b , p ) . \forall a\in A\exist p\in v\psi(a,b,p)\and\forall p\in v\exist a\in A\psi(a,b,% p).
  8. \exist b B \exist b\in B
  9. \cup
  10. A A\,
  11. A , \langle A,\in\rangle

Kronecker_product.html

  1. 𝐀 𝐁 = [ a 11 𝐁 a 1 n 𝐁 a m 1 𝐁 a m n 𝐁 ] , \mathbf{A}\otimes\mathbf{B}=\begin{bmatrix}a_{11}\mathbf{B}&\cdots&a_{1n}% \mathbf{B}\\ \vdots&\ddots&\vdots\\ a_{m1}\mathbf{B}&\cdots&a_{mn}\mathbf{B}\end{bmatrix},
  2. 𝐀 𝐁 = [ a 11 b 11 a 11 b 12 a 11 b 1 q a 1 n b 11 a 1 n b 12 a 1 n b 1 q a 11 b 21 a 11 b 22 a 11 b 2 q a 1 n b 21 a 1 n b 22 a 1 n b 2 q a 11 b p 1 a 11 b p 2 a 11 b p q a 1 n b p 1 a 1 n b p 2 a 1 n b p q a m 1 b 11 a m 1 b 12 a m 1 b 1 q a m n b 11 a m n b 12 a m n b 1 q a m 1 b 21 a m 1 b 22 a m 1 b 2 q a m n b 21 a m n b 22 a m n b 2 q a m 1 b p 1 a m 1 b p 2 a m 1 b p q a m n b p 1 a m n b p 2 a m n b p q ] . {\mathbf{A}\otimes\mathbf{B}}=\begin{bmatrix}a_{11}b_{11}&a_{11}b_{12}&\cdots&% a_{11}b_{1q}&\cdots&\cdots&a_{1n}b_{11}&a_{1n}b_{12}&\cdots&a_{1n}b_{1q}\\ a_{11}b_{21}&a_{11}b_{22}&\cdots&a_{11}b_{2q}&\cdots&\cdots&a_{1n}b_{21}&a_{1n% }b_{22}&\cdots&a_{1n}b_{2q}\\ \vdots&\vdots&\ddots&\vdots&&&\vdots&\vdots&\ddots&\vdots\\ a_{11}b_{p1}&a_{11}b_{p2}&\cdots&a_{11}b_{pq}&\cdots&\cdots&a_{1n}b_{p1}&a_{1n% }b_{p2}&\cdots&a_{1n}b_{pq}\\ \vdots&\vdots&&\vdots&\ddots&&\vdots&\vdots&&\vdots\\ \vdots&\vdots&&\vdots&&\ddots&\vdots&\vdots&&\vdots\\ a_{m1}b_{11}&a_{m1}b_{12}&\cdots&a_{m1}b_{1q}&\cdots&\cdots&a_{mn}b_{11}&a_{mn% }b_{12}&\cdots&a_{mn}b_{1q}\\ a_{m1}b_{21}&a_{m1}b_{22}&\cdots&a_{m1}b_{2q}&\cdots&\cdots&a_{mn}b_{21}&a_{mn% }b_{22}&\cdots&a_{mn}b_{2q}\\ \vdots&\vdots&\ddots&\vdots&&&\vdots&\vdots&\ddots&\vdots\\ a_{m1}b_{p1}&a_{m1}b_{p2}&\cdots&a_{m1}b_{pq}&\cdots&\cdots&a_{mn}b_{p1}&a_{mn% }b_{p2}&\cdots&a_{mn}b_{pq}\end{bmatrix}.
  3. [ 1 2 3 4 ] [ 0 5 6 7 ] = [ 1 0 1 5 2 0 2 5 1 6 1 7 2 6 2 7 3 0 3 5 4 0 4 5 3 6 3 7 4 6 4 7 ] = [ 0 5 0 10 6 7 12 14 0 15 0 20 18 21 24 28 ] . \begin{bmatrix}1&2\\ 3&4\\ \end{bmatrix}\otimes\begin{bmatrix}0&5\\ 6&7\\ \end{bmatrix}=\begin{bmatrix}1\cdot 0&1\cdot 5&2\cdot 0&2\cdot 5\\ 1\cdot 6&1\cdot 7&2\cdot 6&2\cdot 7\\ 3\cdot 0&3\cdot 5&4\cdot 0&4\cdot 5\\ 3\cdot 6&3\cdot 7&4\cdot 6&4\cdot 7\\ \end{bmatrix}=\begin{bmatrix}0&5&0&10\\ 6&7&12&14\\ 0&15&0&20\\ 18&21&24&28\end{bmatrix}.
  4. H Tot = i H i H_{\mathrm{Tot}}=\bigoplus_{i}H^{i}
  5. ( 𝐁 T 𝐀 ) vec ( 𝐗 ) = vec ( 𝐀𝐗𝐁 ) = vec ( 𝐂 ) . (\mathbf{B}^{T}\otimes\mathbf{A})\,\operatorname{vec}(\mathbf{X})=% \operatorname{vec}(\mathbf{AXB})=\operatorname{vec}(\mathbf{C}).
  6. 𝐀 𝐁 = ( 𝐀 i j 𝐁 ) i j = ( ( 𝐀 i j 𝐁 k l ) k l ) i j \mathbf{A}\circ\mathbf{B}=(\mathbf{A}_{ij}\circ\mathbf{B})_{ij}=((\mathbf{A}_{% ij}\otimes\mathbf{B}_{kl})_{kl})_{ij}
  7. 𝐀 = [ 𝐀 11 𝐀 12 𝐀 21 𝐀 22 ] = [ 1 2 3 4 5 6 7 8 9 ] , 𝐁 = [ 𝐁 11 𝐁 12 𝐁 21 𝐁 22 ] = [ 1 4 7 2 5 8 3 6 9 ] , \mathbf{A}=\left[\begin{array}[]{c | c}\mathbf{A}_{11}&\mathbf{A}_{12}\\ \hline\mathbf{A}_{21}&\mathbf{A}_{22}\end{array}\right]=\left[\begin{array}[]{% c c | c}1&2&3\\ 4&5&6\\ \hline 7&8&9\end{array}\right],\quad\mathbf{B}=\left[\begin{array}[]{c | c}% \mathbf{B}_{11}&\mathbf{B}_{12}\\ \hline\mathbf{B}_{21}&\mathbf{B}_{22}\end{array}\right]=\left[\begin{array}[]{% c | c c}1&4&7\\ \hline 2&5&8\\ 3&6&9\end{array}\right],
  8. 𝐀 𝐁 = [ 𝐀 11 𝐁 𝐀 12 𝐁 𝐀 21 𝐁 𝐀 22 𝐁 ] = [ 𝐀 11 𝐁 11 𝐀 11 𝐁 12 𝐀 12 𝐁 11 𝐀 12 𝐁 12 𝐀 11 𝐁 21 𝐀 11 𝐁 22 𝐀 12 𝐁 21 𝐀 12 𝐁 22 𝐀 21 𝐁 11 𝐀 21 𝐁 12 𝐀 22 𝐁 11 𝐀 22 𝐁 12 𝐀 21 𝐁 21 𝐀 21 𝐁 22 𝐀 22 𝐁 21 𝐀 22 𝐁 22 ] \mathbf{A}\circ\mathbf{B}=\left[\begin{array}[]{c | c}\mathbf{A}_{11}\circ% \mathbf{B}&\mathbf{A}_{12}\circ\mathbf{B}\\ \hline\mathbf{A}_{21}\circ\mathbf{B}&\mathbf{A}_{22}\circ\mathbf{B}\end{array}% \right]=\left[\begin{array}[]{c | c | c | c }\mathbf{A}_{11}\otimes\mathbf{B}_% {11}&\mathbf{A}_{11}\otimes\mathbf{B}_{12}&\mathbf{A}_{12}\otimes\mathbf{B}_{1% 1}&\mathbf{A}_{12}\otimes\mathbf{B}_{12}\\ \hline\mathbf{A}_{11}\otimes\mathbf{B}_{21}&\mathbf{A}_{11}\otimes\mathbf{B}_{% 22}&\mathbf{A}_{12}\otimes\mathbf{B}_{21}&\mathbf{A}_{12}\otimes\mathbf{B}_{22% }\\ \hline\mathbf{A}_{21}\otimes\mathbf{B}_{11}&\mathbf{A}_{21}\otimes\mathbf{B}_{% 12}&\mathbf{A}_{22}\otimes\mathbf{B}_{11}&\mathbf{A}_{22}\otimes\mathbf{B}_{12% }\\ \hline\mathbf{A}_{21}\otimes\mathbf{B}_{21}&\mathbf{A}_{21}\otimes\mathbf{B}_{% 22}&\mathbf{A}_{22}\otimes\mathbf{B}_{21}&\mathbf{A}_{22}\otimes\mathbf{B}_{22% }\end{array}\right]
  9. = [ 1 2 4 7 8 14 3 12 21 4 5 16 28 20 35 6 24 42 2 4 5 8 10 16 6 15 24 3 6 6 9 12 18 9 18 27 8 10 20 32 25 40 12 30 48 12 15 24 36 30 45 18 36 54 7 8 28 49 32 56 9 36 63 14 16 35 56 40 64 18 45 72 21 24 42 63 48 72 27 54 81 ] . =\left[\begin{array}[]{c c | c c c c | c | c c}1&2&4&7&8&14&3&12&21\\ 4&5&16&28&20&35&6&24&42\\ \hline 2&4&5&8&10&16&6&15&24\\ 3&6&6&9&12&18&9&18&27\\ 8&10&20&32&25&40&12&30&48\\ 12&15&24&36&30&45&18&36&54\\ \hline 7&8&28&49&32&56&9&36&63\\ \hline 14&16&35&56&40&64&18&45&72\\ 21&24&42&63&48&72&27&54&81\end{array}\right].
  10. 𝐀 𝐁 = ( 𝐀 i j 𝐁 i j ) i j \mathbf{A}\ast\mathbf{B}=(\mathbf{A}_{ij}\otimes\mathbf{B}_{ij})_{ij}
  11. 𝐀 𝐁 = [ 𝐀 11 𝐁 11 𝐀 12 𝐁 12 𝐀 21 𝐁 21 𝐀 22 𝐁 22 ] = [ 1 2 12 21 4 5 24 42 14 16 45 72 21 24 54 81 ] . \mathbf{A}\ast\mathbf{B}=\left[\begin{array}[]{c | c}\mathbf{A}_{11}\otimes% \mathbf{B}_{11}&\mathbf{A}_{12}\otimes\mathbf{B}_{12}\\ \hline\mathbf{A}_{21}\otimes\mathbf{B}_{21}&\mathbf{A}_{22}\otimes\mathbf{B}_{% 22}\end{array}\right]=\left[\begin{array}[]{c c | c c}1&2&12&21\\ 4&5&24&42\\ \hline 14&16&45&72\\ 21&24&54&81\end{array}\right].
  12. 𝐂 = [ 𝐂 1 𝐂 2 𝐂 3 ] = [ 1 2 3 4 5 6 7 8 9 ] , 𝐃 = [ 𝐃 1 𝐃 2 𝐃 3 ] = [ 1 4 7 2 5 8 3 6 9 ] , \mathbf{C}=\left[\begin{array}[]{ c | c | c}\mathbf{C}_{1}&\mathbf{C}_{2}&% \mathbf{C}_{3}\end{array}\right]=\left[\begin{array}[]{c | c | c}1&2&3\\ 4&5&6\\ 7&8&9\end{array}\right],\quad\mathbf{D}=\left[\begin{array}[]{ c | c | c }% \mathbf{D}_{1}&\mathbf{D}_{2}&\mathbf{D}_{3}\end{array}\right]=\left[\begin{% array}[]{ c | c | c }1&4&7\\ 2&5&8\\ 3&6&9\end{array}\right],
  13. 𝐂 𝐃 = [ 𝐂 1 𝐃 1 𝐂 2 𝐃 2 𝐂 3 𝐃 3 ] = [ 1 8 21 2 10 24 3 12 27 4 20 42 8 25 48 12 30 54 7 32 63 14 40 72 21 48 81 ] . \mathbf{C}\ast\mathbf{D}=\left[\begin{array}[]{ c | c | c }\mathbf{C}_{1}% \otimes\mathbf{D}_{1}&\mathbf{C}_{2}\otimes\mathbf{D}_{2}&\mathbf{C}_{3}% \otimes\mathbf{D}_{3}\end{array}\right]=\left[\begin{array}[]{ c | c | c }1&8&% 21\\ 2&10&24\\ 3&12&27\\ 4&20&42\\ 8&25&48\\ 12&30&54\\ 7&32&63\\ 14&40&72\\ 21&48&81\end{array}\right].

Kronecker_symbol.html

  1. ( a n ) \left(\frac{a}{n}\right)
  2. ( a | n ) (a|n)
  3. n n
  4. n n
  5. n = u p 1 e 1 p k e k , n=u\cdot p_{1}^{e_{1}}\cdots p_{k}^{e_{k}},
  6. u u
  7. u = ± 1 u=\pm 1
  8. p i p_{i}
  9. a a
  10. ( a | n ) (a|n)
  11. ( a n ) = ( a u ) i = 1 k ( a p i ) e i . \left(\frac{a}{n}\right)=\left(\frac{a}{u}\right)\prod_{i=1}^{k}\left(\frac{a}% {p_{i}}\right)^{e_{i}}.
  12. p i p_{i}
  13. ( a | p i ) (a|p_{i})
  14. p i = 2 p_{i}=2
  15. ( a | 2 ) (a|2)
  16. ( a 2 ) = { 0 if a is even, 1 if a ± 1 ( mod 8 ) , - 1 if a ± 3 ( mod 8 ) . \left(\frac{a}{2}\right)=\begin{cases}0&\mbox{if }~{}a\mbox{ is even,}\\ 1&\mbox{if }~{}a\equiv\pm 1\;\;(\mathop{{\rm mod}}8),\\ -1&\mbox{if }~{}a\equiv\pm 3\;\;(\mathop{{\rm mod}}8).\end{cases}
  17. ( a | u ) (a|u)
  18. 1 1
  19. u = 1 u=1
  20. u = - 1 u=-1
  21. ( a - 1 ) = { - 1 if a < 0 , 1 if a 0. \left(\frac{a}{-1}\right)=\begin{cases}-1&\mbox{if }~{}a<0,\\ 1&\mbox{if }~{}a\geq 0.\end{cases}
  22. ( a 0 ) = { 1 if a = ± 1 , 0 otherwise. \left(\frac{a}{0}\right)=\begin{cases}1&\,\text{if }a=\pm 1,\\ 0&\,\text{otherwise.}\end{cases}
  23. a , n a,n
  24. a a
  25. 0 , 1 mod 4 0,1\bmod 4
  26. n > 0 n>0
  27. ( a n ) = ± 1 \left(\tfrac{a}{n}\right)=\pm 1
  28. gcd ( a , n ) = 1 \gcd(a,n)=1
  29. ( a n ) = 0 \left(\tfrac{a}{n}\right)=0
  30. ( a b n ) = ( a n ) ( b n ) \left(\tfrac{ab}{n}\right)=\left(\tfrac{a}{n}\right)\left(\tfrac{b}{n}\right)
  31. n = - 1 n=-1
  32. a , b a,b
  33. ( a m n ) = ( a m ) ( a n ) \left(\tfrac{a}{mn}\right)=\left(\tfrac{a}{m}\right)\left(\tfrac{a}{n}\right)
  34. a = - 1 a=-1
  35. m , n m,n
  36. 3 mod 4 3\bmod 4
  37. n > 0 n>0
  38. ( a n ) = ( b n ) \left(\tfrac{a}{n}\right)=\left(\tfrac{b}{n}\right)
  39. a b mod { 4 n , n 2 ( mod 4 ) , n otherwise. a\equiv b\bmod\begin{cases}4n,&n\equiv 2\;\;(\mathop{{\rm mod}}4),\\ n&\,\text{otherwise.}\end{cases}
  40. a , b a,b
  41. n < 0 n<0
  42. a 3 ( mod 4 ) a\not\equiv 3\;\;(\mathop{{\rm mod}}4)
  43. a 0 a\neq 0
  44. ( a m ) = ( a n ) \left(\tfrac{a}{m}\right)=\left(\tfrac{a}{n}\right)
  45. m n mod { 4 | a | , a 2 ( mod 4 ) , | a | otherwise. m\equiv n\bmod\begin{cases}4|a|,&a\equiv 2\;\;(\mathop{{\rm mod}}4),\\ |a|&\,\text{otherwise.}\end{cases}
  46. n n
  47. n n^{\prime}
  48. n = 2 e n n=2^{e}n^{\prime}
  49. n n^{\prime}
  50. n = 0 n=0
  51. 0 = 1 0^{\prime}=1
  52. m , n m,n
  53. gcd ( m , n ) = 1 \gcd(m,n)=1
  54. ( m n ) ( n m ) = ± ( - 1 ) m - 1 2 n - 1 2 , \left(\frac{m}{n}\right)\left(\frac{n}{m}\right)=\pm(-1)^{\frac{m^{\prime}-1}{% 2}\frac{n^{\prime}-1}{2}},
  55. ± \pm
  56. + +
  57. m 0 m\geq 0
  58. n 0 n\geq 0
  59. - -
  60. m < 0 m<0
  61. n < 0 n<0
  62. m , n m,n
  63. ( m n ) = ( - 1 ) m - 1 2 n - 1 2 ( n | m | ) . \left(\frac{m}{n}\right)=(-1)^{\frac{m^{\prime}-1}{2}\frac{n^{\prime}-1}{2}}% \left(\frac{n}{|m|}\right).
  64. n n
  65. n * = ( - 1 ) ( n - 1 ) / 2 n n^{*}=(-1)^{(n^{\prime}-1)/2}n
  66. ( m * n ) = ( n | m | ) \left(\frac{m^{*}}{n}\right)=\left(\frac{n}{|m|}\right)
  67. m , n m,n
  68. n n
  69. ( - 1 n ) = ( - 1 ) n - 1 2 \left(\frac{-1}{n}\right)=(-1)^{\frac{n^{\prime}-1}{2}}
  70. n n
  71. ( 2 n ) = ( - 1 ) n 2 - 1 8 . \left(\frac{2}{n}\right)=(-1)^{\frac{n^{2}-1}{8}}.
  72. a 3 ( mod 4 ) a\not\equiv 3\;\;(\mathop{{\rm mod}}4)
  73. a 0 a\neq 0
  74. χ ( n ) = ( a n ) \chi(n)=\left(\tfrac{a}{n}\right)
  75. { 4 | a | , a 2 ( mod 4 ) , | a | , otherwise. \begin{cases}4|a|,&a\equiv 2\;\;(\mathop{{\rm mod}}4),\\ |a|,&\,\text{otherwise.}\end{cases}
  76. a 0 , 1 ( mod 4 ) a\equiv 0,1\;\;(\mathop{{\rm mod}}4)
  77. a 2 ( mod 4 ) a\equiv 2\;\;(\mathop{{\rm mod}}4)
  78. ( a n ) = ( 4 a n ) \left(\tfrac{a}{n}\right)=\left(\tfrac{4a}{n}\right)
  79. χ \chi
  80. F = ( m ) F=\mathbb{Q}(\sqrt{m})
  81. m m
  82. ( 1 ) = \mathbb{Q}(\sqrt{1})=\mathbb{Q}
  83. χ \chi
  84. ( F / ) \left(\tfrac{F/\mathbb{Q}}{\cdot}\right)
  85. p p
  86. χ ( p ) \chi(p)
  87. ( p ) (p)
  88. O F O_{F}
  89. χ ( p ) = { 0 , ( p ) is ramified, 1 , ( p ) splits, - 1 , ( p ) is inert. \chi(p)=\begin{cases}0,&(p)\,\text{ is ramified,}\\ 1,&(p)\,\text{ splits,}\\ -1,&(p)\,\text{ is inert.}\end{cases}
  90. χ ( n ) \chi(n)
  91. ( D n ) \left(\tfrac{D}{n}\right)
  92. D = { m , m 1 ( mod 4 ) , 4 m , m 2 , 3 ( mod 4 ) D=\begin{cases}m,&m\equiv 1\;\;(\mathop{{\rm mod}}4),\\ 4m,&m\equiv 2,3\;\;(\mathop{{\rm mod}}4)\end{cases}
  93. F F
  94. χ \chi
  95. | D | |D|
  96. n > 0 n>0
  97. χ ( a ) = ( a n ) \chi(a)=\left(\tfrac{a}{n}\right)
  98. { 4 n , n 2 ( mod 4 ) , n , otherwise. \begin{cases}4n,&n\equiv 2\;\;(\mathop{{\rm mod}}4),\\ n,&\,\text{otherwise.}\end{cases}
  99. ( - 4 ) \left(\tfrac{-4}{\cdot}\right)
  100. ( n ) \left(\tfrac{\cdot}{n}\right)
  101. n n
  102. ( n ) = ( n * ) \left(\tfrac{\cdot}{n}\right)=\left(\tfrac{n^{*}}{\cdot}\right)
  103. ( a ) \left(\tfrac{a}{\cdot}\right)
  104. ( n ) \left(\tfrac{\cdot}{n}\right)
  105. a 1 ( mod 4 ) a^{\prime}\equiv 1\;\;(\mathop{{\rm mod}}4)
  106. n = | a | n=|a|

Kronecker–Weber_theorem.html

  1. 5 = e 2 π i / 5 - e 4 π i / 5 - e 6 π i / 5 + e 8 π i / 5 . \sqrt{5}=e^{2\pi i/5}-e^{4\pi i/5}-e^{6\pi i/5}+e^{8\pi i/5}.

Kruskal–Wallis_one-way_analysis_of_variance.html

  1. K = ( N - 1 ) i = 1 g n i ( r ¯ i - r ¯ ) 2 i = 1 g j = 1 n i ( r i j - r ¯ ) 2 , K=(N-1)\frac{\sum_{i=1}^{g}n_{i}(\bar{r}_{i\cdot}-\bar{r})^{2}}{\sum_{i=1}^{g}% \sum_{j=1}^{n_{i}}(r_{ij}-\bar{r})^{2}},
  2. n i n_{i}
  3. i i
  4. r i j r_{ij}
  5. j j
  6. i i
  7. N N
  8. r ¯ i = j = 1 n i r i j n i \bar{r}_{i\cdot}=\frac{\sum_{j=1}^{n_{i}}{r_{ij}}}{n_{i}}
  9. r ¯ = 1 2 ( N + 1 ) \bar{r}=\tfrac{1}{2}(N+1)
  10. r i j r_{ij}
  11. K K
  12. ( N - 1 ) N ( N + 1 ) / 12 (N-1)N(N+1)/12
  13. r ¯ = N + 1 2 \bar{r}=\tfrac{N+1}{2}
  14. K = 12 N ( N + 1 ) i = 1 g n i ( r ¯ i - N + 1 2 ) 2 = 12 N ( N + 1 ) i = 1 g n i r ¯ i 2 - 3 ( N + 1 ) . \begin{aligned}\displaystyle K&\displaystyle=\frac{12}{N(N+1)}\sum_{i=1}^{g}n_% {i}\left(\bar{r}_{i\cdot}-\frac{N+1}{2}\right)^{2}\\ &\displaystyle=\frac{12}{N(N+1)}\sum_{i=1}^{g}n_{i}\bar{r}_{i\cdot}^{2}-\ 3(N+% 1).\end{aligned}
  15. K K
  16. 1 - i = 1 G ( t i 3 - t i ) N 3 - N 1-\frac{\sum_{i=1}^{G}(t_{i}^{3}-t_{i})}{N^{3}-N}
  17. Pr ( χ g - 1 2 K ) \Pr(\chi^{2}_{g-1}\geq K)
  18. n i n_{i}
  19. χ α : g - 1 2 \chi^{2}_{\alpha:g-1}

Kummer_surface.html

  1. 3 \mathbb{P}^{3}
  2. K 3 K\subset\mathbb{P}^{3}
  3. 3 \mathbb{P}^{3}
  4. 2 \mathbb{P}^{2}
  5. 2 \mathbb{P}^{2}
  6. p q ¯ \scriptstyle\overline{pq}
  7. 6 6
  8. p p
  9. p p
  10. 3 \mathbb{P}^{3}
  11. C C
  12. J a c ( C ) Jac(C)
  13. P i c 2 ( C ) Pic^{2}(C)
  14. x x + K C x\mapsto x+K_{C}
  15. C C
  16. S y m 2 C Sym^{2}C
  17. P i c 2 C Pic^{2}C
  18. { p , q } p + q \{p,q\}\mapsto p+q
  19. C | K C | * C\to|K_{C}|^{*}
  20. K u m ( C ) S y m 2 | K C | * Kum(C)\to Sym^{2}|K_{C}|^{*}
  21. J a c ( C ) Jac(C)
  22. T 0 J a c ( C ) | K C | * T_{0}Jac(C)\cong|K_{C}|^{*}
  23. | K C | * |K_{C}|^{*}
  24. C C
  25. ( Θ + w 1 ) ( Θ + w 2 ) = { w 1 - w 2 , 0 } (\Theta+w_{1})\cap(\Theta+w_{2})=\{w_{1}-w_{2},0\}
  26. w 1 , w 2 w_{1},w_{2}
  27. C | K C | * C\mapsto|K_{C}|^{*}
  28. C C
  29. | O J a c ( C ) ( 2 Θ C ) | 2 2 - 1 |O_{Jac(C)}(2\Theta_{C})|\cong\mathbb{P}^{2^{2}-1}
  30. J a c ( C ) Jac(C)
  31. 16 6 16_{6}
  32. J a c ( C ) Jac(C)
  33. { q - w | q C } \{q-w|q\in C\}
  34. C C
  35. w - w w^{\prime}-w
  36. w w^{\prime}
  37. w , w w,w^{\prime}
  38. 0
  39. w - w w-w^{\prime}
  40. | 2 Θ C | |2\Theta_{C}|
  41. J a c ( C ) Jac(C)
  42. 16 16
  43. 3 \mathbb{P}^{3}
  44. 16 6 16_{6}
  45. p 1 - p 2 , p 3 - p 4 = # { p 1 , p 2 } { p 3 , p 4 } \langle p_{1}-p_{2},p_{3}-p_{4}\rangle=\#\{p_{1},p_{2}\}\cap\{p_{3},p_{4}\}
  46. S p 4 ( 2 ) Sp_{4}(2)
  47. 16 6 16_{6}

Künneth_theorem.html

  1. i + j = k H i ( X ; F ) H j ( Y ; F ) H k ( X × Y ; F ) . \bigoplus_{i+j=k}H_{i}(X;F)\otimes H_{j}(Y;F)\cong H_{k}(X\times Y;F).
  2. p X × Y ( t ) = p X ( t ) p Y ( t ) . p_{X\times Y}(t)=p_{X}(t)p_{Y}(t).
  3. 0 i + j = k H i ( X ; R ) R H j ( Y ; R ) H k ( X × Y ; R ) i + j = k - 1 Tor 1 R ( H i ( X ; R ) , H j ( Y ; R ) ) 0. 0\to\bigoplus_{i+j=k}H_{i}(X;R)\otimes_{R}H_{j}(Y;R)\to H_{k}(X\times Y;R)\to% \bigoplus_{i+j=k-1}\mathrm{Tor}_{1}^{R}(H_{i}(X;R),H_{j}(Y;R))\to 0.
  4. Tor 1 𝐙 ( h 1 , h 1 ) Tor 1 𝐙 ( 𝐙 / ( 2 ) , 𝐙 / ( 2 ) ) 𝐙 / ( 2 ) . \mathrm{Tor}^{\mathbf{Z}}_{1}(h_{1},h_{1})\cong\mathrm{Tor}^{\mathbf{Z}}_{1}(% \mathbf{Z}/(2),\mathbf{Z}/(2))\cong\mathbf{Z}/(2).
  5. H 0 ( 𝐏 2 ( 𝐑 ) × 𝐏 2 ( 𝐑 ) ; 𝐙 ) h 0 h 0 𝐙 H 1 ( 𝐏 2 ( 𝐑 ) × 𝐏 2 ( 𝐑 ) ; 𝐙 ) h 0 h 1 h 1 h 0 𝐙 / ( 2 ) 𝐙 / ( 2 ) H 2 ( 𝐏 2 ( 𝐑 ) × 𝐏 2 ( 𝐑 ) ; 𝐙 ) h 1 h 1 𝐙 / ( 2 ) H 3 ( 𝐏 2 ( 𝐑 ) × 𝐏 2 ( 𝐑 ) ; 𝐙 ) Tor 1 𝐙 ( h 1 , h 1 ) 𝐙 / ( 2 ) \begin{aligned}\displaystyle H_{0}\left(\mathbf{P}^{2}(\mathbf{R})\times% \mathbf{P}^{2}(\mathbf{R});\mathbf{Z}\right)&\displaystyle\cong\;h_{0}\otimes h% _{0}\;\cong\;\mathbf{Z}\\ \displaystyle H_{1}\left(\mathbf{P}^{2}(\mathbf{R})\times\mathbf{P}^{2}(% \mathbf{R});\mathbf{Z}\right)&\displaystyle\cong\;h_{0}\otimes h_{1}\;\oplus\;% h_{1}\otimes h_{0}\;\cong\;\mathbf{Z}/(2)\oplus\mathbf{Z}/(2)\\ \displaystyle H_{2}\left(\mathbf{P}^{2}(\mathbf{R})\times\mathbf{P}^{2}(% \mathbf{R});\mathbf{Z}\right)&\displaystyle\cong\;h_{1}\otimes h_{1}\;\cong\;% \mathbf{Z}/(2)\\ \displaystyle H_{3}\left(\mathbf{P}^{2}(\mathbf{R})\times\mathbf{P}^{2}(% \mathbf{R});\mathbf{Z}\right)&\displaystyle\cong\;\mathrm{Tor}^{\mathbf{Z}}_{1% }(h_{1},h_{1})\;\cong\;\mathbf{Z}/(2)\\ \end{aligned}
  6. E p q 2 = q 1 + q 2 = q Tor p R ( H q 1 ( X ; R ) , H q 2 ( Y ; R ) ) H p + q ( X × Y ; R ) . E_{pq}^{2}=\bigoplus_{q_{1}+q_{2}=q}\mathrm{Tor}^{R}_{p}(H_{q_{1}}(X;R),H_{q_{% 2}}(Y;R))\Rightarrow H_{p+q}(X\times Y;R).
  7. C * ( X × Y ) C * ( X ) C * ( Y ) . C_{*}(X\times Y)\cong C_{*}(X)\otimes C_{*}(Y).

K–Ar_dating.html

  1. t = t 1 2 ln ( 2 ) ln ( K f + A r f 0.109 K f ) t=\frac{t_{\frac{1}{2}}}{\ln(2)}\ln\left(\frac{K_{f}+\frac{Ar_{f}}{0.109}}{K_{% f}}\right)

L-theory.html

  1. L * ( R ) L_{*}(R)
  2. L * ( R ) L^{*}(R)
  3. L 2 k ( R ) L_{2k}(R)
  4. ϵ = ( - 1 ) k \epsilon=(-1)^{k}
  5. L 2 k ( R ) L_{2k}(R)
  6. [ ψ ] [\psi]
  7. ψ Q ϵ ( F ) \psi\in Q_{\epsilon}(F)
  8. [ ψ ] = [ ψ ] n , n 0 : ψ H ( - 1 ) k ( R ) n ψ H ( - 1 ) k ( R ) n [\psi]=[\psi^{\prime}]\Longleftrightarrow n,n^{\prime}\in{\mathbb{N}}_{0}:\psi% \oplus H_{(-1)^{k}}(R)^{n}\cong\psi^{\prime}\oplus H_{(-1)^{k}}(R)^{n^{\prime}}
  9. L 2 k ( R ) L_{2k}(R)
  10. [ ψ 1 ] + [ ψ 2 ] := [ ψ 1 ψ 2 ] . [\psi_{1}]+[\psi_{2}]:=[\psi_{1}\oplus\psi_{2}].
  11. H ( - 1 ) k ( R ) n H_{(-1)^{k}}(R)^{n}
  12. n 0 n\in{\mathbb{N}}_{0}
  13. [ ψ ] [\psi]
  14. [ - ψ ] [-\psi]
  15. π \pi
  16. L * ( 𝐙 [ π ] ) L_{*}(\mathbf{Z}[\pi])
  17. 𝐙 [ π ] \mathbf{Z}[\pi]
  18. π \pi
  19. π 1 ( X ) \pi_{1}(X)
  20. X X
  21. L * ( 𝐙 [ π ] ) L_{*}(\mathbf{Z}[\pi])
  22. n n
  23. n > 4 n>4
  24. H * H^{*}
  25. 𝐙 2 \mathbf{Z}_{2}
  26. 𝐙 2 \mathbf{Z}_{2}
  27. H * H_{*}
  28. 𝐙 2 \mathbf{Z}_{2}
  29. X G X^{G}
  30. X G = X / G X_{G}=X/G
  31. L n ( R ) L_{n}(R)
  32. L n ( R ) L^{n}(R)
  33. L n ( R ) L n ( R ) L_{n}(R)\to L^{n}(R)
  34. L L
  35. L * ( 𝐙 [ π ] ) L_{*}(\mathbf{Z}[\pi])
  36. π \pi
  37. π \pi
  38. L ( e ) := L ( 𝐙 [ e ] ) = L ( 𝐙 ) L(e):=L(\mathbf{Z}[e])=L(\mathbf{Z})
  39. L L
  40. L * L^{*}
  41. L * . L_{*}.
  42. L 4 k ( 𝐙 ) \displaystyle L_{4k}(\mathbf{Z})
  43. L 4 k ( 𝐙 ) \displaystyle L^{4k}(\mathbf{Z})

Lagrange_reversion_theorem.html

  1. v = x + y f ( v ) v=x+yf(v)
  2. g ( v ) = g ( x ) + k = 1 y k k ! ( x ) k - 1 ( f ( x ) k g ( x ) ) g(v)=g(x)+\sum_{k=1}^{\infty}\frac{y^{k}}{k!}\left(\frac{\partial}{\partial x}% \right)^{k-1}\left(f(x)^{k}g^{\prime}(x)\right)
  3. v = x + k = 1 y k k ! ( x ) k - 1 ( f ( x ) k ) v=x+\sum_{k=1}^{\infty}\frac{y^{k}}{k!}\left(\frac{\partial}{\partial x}\right% )^{k-1}\left(f(x)^{k}\right)
  4. g ( v ) = δ ( y f ( z ) - z + x ) g ( z ) ( 1 - y f ( z ) ) d z g(v)=\int\delta(yf(z)-z+x)g(z)(1-yf^{\prime}(z))\,dz
  5. g ( v ) \displaystyle g(v)
  6. δ ( x - z ) \delta(x-z)
  7. g ( v ) \displaystyle g(v)
  8. g ( v ) = g ( x ) + k = 1 y k k ! ( x ) k - 1 ( f ( x ) k g ( x ) ) g(v)=g(x)+\sum_{k=1}^{\infty}\frac{y^{k}}{k!}\left(\frac{\partial}{\partial x}% \right)^{k-1}\left(f(x)^{k}g^{\prime}(x)\right)

Laguerre's_method.html

  1. p ( x ) = 0 \ p(x)=0
  2. x 0 x_{0}
  3. p ( x k ) p(x_{k})
  4. G = p ( x k ) p ( x k ) G=\frac{p^{\prime}(x_{k})}{p(x_{k})}
  5. H = G 2 - p ′′ ( x k ) p ( x k ) H=G^{2}-\frac{p^{\prime\prime}(x_{k})}{p(x_{k})}
  6. a = n G \plusmn ( n - 1 ) ( n H - G 2 ) a=\frac{n}{G\plusmn\sqrt{(n-1)(nH-G^{2})}}
  7. x k + 1 = x k - a x_{k+1}=x_{k}-a
  8. p p
  9. p ( x ) = C ( x - x 1 ) ( x - x 2 ) ( x - x n ) , p(x)=C(x-x_{1})(x-x_{2})\cdots(x-x_{n}),
  10. x k x_{k}
  11. ( k = 1 , 2 , , n ) (k=1,2,...,n)
  12. ln | p ( x ) | = ln | C | + ln | x - x 1 | + ln | x - x 2 | + + ln | x - x n | . \ln|p(x)|=\ln|C|+\ln|x-x_{1}|+\ln|x-x_{2}|+\cdots+\ln|x-x_{n}|.
  13. G = d d x ln | p ( x ) | = 1 x - x 1 + 1 x - x 2 + + 1 x - x n , G=\frac{d}{dx}\ln|p(x)|=\frac{1}{x-x_{1}}+\frac{1}{x-x_{2}}+\cdots+\frac{1}{x-% x_{n}},
  14. H = - d 2 d x 2 ln | p ( x ) | = 1 ( x - x 1 ) 2 + 1 ( x - x 2 ) 2 + + 1 ( x - x n ) 2 . H=-\frac{d^{2}}{dx^{2}}\ln|p(x)|=\frac{1}{(x-x_{1})^{2}}+\frac{1}{(x-x_{2})^{2% }}+\cdots+\frac{1}{(x-x_{n})^{2}}.
  15. x 1 x_{1}
  16. x x
  17. a = x - x 1 a=x-x_{1}\,
  18. b = x - x i , i = 2 , 3 , , n b=x-x_{i},\quad i=2,3,\ldots,n
  19. G G
  20. G = 1 a + n - 1 b G=\frac{1}{a}+\frac{n-1}{b}
  21. H H
  22. H = 1 a 2 + n - 1 b 2 . H=\frac{1}{a^{2}}+\frac{n-1}{b^{2}}.
  23. a a
  24. a = n G \plusmn ( n - 1 ) ( n H - G 2 ) a=\frac{n}{G\plusmn\sqrt{(n-1)(nH-G^{2})}}
  25. Re ( G ¯ ( n - 1 ) ( n H - G 2 ) ) > 0 \operatorname{Re}\,(\overline{G}\sqrt{(n-1)(nH-G^{2})}\,)>0
  26. Re \operatorname{Re}
  27. G ¯ \overline{G}
  28. G G
  29. a = p ( x ) p ( x ) ( 1 n + n - 1 n 1 - n n - 1 p ( x ) p ′′ ( x ) p ( x ) 2 ) - 1 a=\frac{p(x)}{p^{\prime}(x)}\cdot\left(\frac{1}{n}+\frac{n-1}{n}\,\sqrt{1-% \frac{n}{n-1}\,\frac{p(x)p^{\prime\prime}(x)}{p^{\prime}(x)^{2}}}\right)^{-1}
  30. p ( x ) p(x)
  31. O ( p ( x ) 3 ) O(p(x)^{3})
  32. p p
  33. p p
  34. r r
  35. w w
  36. q ( z ) = p ( z - w ) q(z)=p(z-w)
  37. r r
  38. q q
  39. r r
  40. p p
  41. r r
  42. G G
  43. x i , i = 2 , 3 , , n x_{i},\quad i=2,3,\ldots,n
  44. x 1 x_{1}

Laguerre_form.html

  1. 𝔏 = ( w 1 ) 2 D a 11 + 2 w 1 w 2 D a 12 + ( w 2 ) 2 D a 22 \mathfrak{L}=(w^{1})^{2}Da_{11}+2w^{1}w^{2}Da_{12}+(w^{2})^{2}Da_{22}

Lamb_shift.html

  1. Δ V = V ( r + δ r ) - V ( r ) = δ r V + 1 2 ( δ r ) 2 V ( r ) + \Delta V=V(\vec{r}+\delta\vec{r})-V(\vec{r})=\delta\vec{r}\cdot\nabla V+\frac{% 1}{2}(\delta\vec{r}\cdot\nabla)^{2}V(\vec{r})+...
  2. δ r v a c = 0 \langle\delta\vec{r}\rangle_{vac}=0
  3. ( δ r ) 2 v a c = 1 3 ( δ r ) 2 v a c 2 \langle(\delta\vec{r}\cdot\nabla)^{2}\rangle_{vac}=\frac{1}{3}\langle(\delta% \vec{r})^{2}\rangle_{vac}\nabla^{2}
  4. Δ V = 1 6 ( δ r ) 2 v a c 2 ( - e 2 4 π ϵ 0 r ) a t \langle\Delta V\rangle=\frac{1}{6}\langle(\delta\vec{r})^{2}\rangle_{vac}\left% \langle\nabla^{2}\left(\frac{-e^{2}}{4\pi\epsilon_{0}r}\right)\right\rangle_{at}
  5. m d 2 d t 2 ( δ r ) k = - e E k m\frac{d^{2}}{dt^{2}}(\delta r)_{\vec{k}}=-eE_{\vec{k}}
  6. δ r ( t ) δ r ( 0 ) e - i ν t + c . c . \delta r(t)\cong\delta r(0)e^{-i\nu t}+c.c.
  7. ( δ r ) k e m c 2 k 2 E k = e m c 2 k 2 k ( a k e - i ν t + i k r + h . c . ) (\delta r)_{\vec{k}}\cong\frac{e}{mc^{2}k^{2}}E_{\vec{k}}=\frac{e}{mc^{2}k^{2}% }\mathcal{E}_{\vec{k}}(a_{\vec{k}}e^{-i\nu t+i\vec{k}\cdot\vec{r}}+h.c.)
  8. k \vec{k}
  9. ( δ r ) 2 v a c = k ( e m c 2 k 2 ) 2 0 | ( E k ) 2 | 0 = k ( e m c 2 k 2 ) 2 ( c k 2 ϵ 0 Ω ) \langle(\delta\vec{r})^{2}\rangle_{vac}=\sum_{\vec{k}}\left(\frac{e}{mc^{2}k^{% 2}}\right)^{2}\langle 0|(E_{\vec{k}})^{2}|0\rangle=\sum_{\vec{k}}\left(\frac{e% }{mc^{2}k^{2}}\right)^{2}\left(\frac{\hbar ck}{2\epsilon_{0}\Omega}\right)
  10. Ω \Omega
  11. k = ( c k / 2 ϵ 0 Ω ) 1 / 2 \mathcal{E}_{\vec{k}}=(\hbar ck/2\epsilon_{0}\Omega)^{1/2}
  12. k 2 Ω ( 2 π ) 3 d 3 k \sum_{\vec{k}}\rightarrow 2\frac{\Omega}{(2\pi)^{3}}\int d^{3}k
  13. ( δ r ) 2 v a c = 2 Ω ( 2 π ) 3 4 π d k k 2 ( e m c 2 k 2 ) 2 ( c k 2 ϵ 0 Ω ) = 1 2 ϵ 0 π 2 ( e 2 c ) ( m c ) 2 d k k \langle(\delta\vec{r})^{2}\rangle_{vac}=2\frac{\Omega}{(2\pi)^{3}}4\pi\int dkk% ^{2}\left(\frac{e}{mc^{2}k^{2}}\right)^{2}\left(\frac{\hbar ck}{2\epsilon_{0}% \Omega}\right)=\frac{1}{2\epsilon_{0}\pi^{2}}\left(\frac{e^{2}}{\hbar c}\right% )\left(\frac{\hbar}{mc}\right)^{2}\int\frac{dk}{k}
  14. 2 ( - e 2 4 π ϵ 0 r ) a t = - e 2 4 π ϵ 0 d r ψ * ( r ) 2 ( 1 r ) ψ ( r ) = e 2 ϵ 0 | ψ ( 0 ) | 2 \left\langle\nabla^{2}\left(\frac{-e^{2}}{4\pi\epsilon_{0}r}\right)\right% \rangle_{at}=\frac{-e^{2}}{4\pi\epsilon_{0}}\int d\vec{r}\psi^{*}(\vec{r})% \nabla^{2}\left(\frac{1}{r}\right)\psi(\vec{r})=\frac{e^{2}}{\epsilon_{0}}|% \psi(0)|^{2}
  15. 2 ( 1 r ) = - 4 π δ ( r ) \nabla^{2}\left(\frac{1}{r}\right)=-4\pi\delta(\vec{r})
  16. ψ 2 S ( 0 ) = 1 ( 8 π a 0 3 ) 1 / 2 \psi_{2S}(0)=\frac{1}{(8\pi a_{0}^{3})^{1/2}}
  17. a 0 = 4 π ϵ 0 2 m e 2 a_{0}=\frac{4\pi\epsilon_{0}\hbar^{2}}{me^{2}}
  18. 2 ( - e 2 4 π ϵ 0 r ) a t = e 2 ϵ 0 | ψ 2 S ( 0 ) | 2 = e 2 8 π ϵ 0 a 0 3 \left\langle\nabla^{2}\left(\frac{-e^{2}}{4\pi\epsilon_{0}r}\right)\right% \rangle_{at}=\frac{e^{2}}{\epsilon_{0}}|\psi_{2S}(0)|^{2}=\frac{e^{2}}{8\pi% \epsilon_{0}a_{0}^{3}}
  19. Δ V = 4 3 e 2 4 π ϵ 0 e 2 4 π ϵ 0 c ( m c ) 2 1 8 π a 0 3 ln 4 ϵ 0 c e 2 = α 5 m c 2 1 6 π ln 4 ϵ 0 c e 2 \langle\Delta V\rangle=\frac{4}{3}\frac{e^{2}}{4\pi\epsilon_{0}}\frac{e^{2}}{4% \pi\epsilon_{0}\hbar c}\left(\frac{\hbar}{mc}\right)^{2}\frac{1}{8\pi a_{0}^{3% }}\ln\frac{4\epsilon_{0}\hbar c}{e^{2}}=\alpha^{5}mc^{2}\frac{1}{6\pi}\ln\frac% {4\epsilon_{0}\hbar c}{e^{2}}
  20. α \alpha
  21. E pot = - Z e 2 4 π ϵ 0 1 r + δ r . \langle E_{\mathrm{pot}}\rangle=-\frac{Ze^{2}}{4\pi\epsilon_{0}}\left\langle% \frac{1}{r+\delta r}\right\rangle.
  22. Δ E Lamb = α 5 m e c 2 k ( n , 0 ) 4 n 3 for = 0 \Delta E_{\mathrm{Lamb}}=\alpha^{5}m_{e}c^{2}\frac{k(n,0)}{4n^{3}}\ \mathrm{% for}\ \ell=0\,
  23. Δ E Lamb = α 5 m e c 2 1 4 n 3 [ k ( n , ) ± 1 π ( j + 1 2 ) ( + 1 2 ) ] for 0 and j = ± 1 2 , \Delta E_{\mathrm{Lamb}}=\alpha^{5}m_{e}c^{2}\frac{1}{4n^{3}}\left[k(n,\ell)% \pm\frac{1}{\pi(j+\frac{1}{2})(\ell+\frac{1}{2})}\right]\ \mathrm{for}\ \ell% \neq 0\ \mathrm{and}\ j=\ell\pm\frac{1}{2},

Lambda-CDM_model.html

  1. p = - ρ c 2 p=-\rho c^{2}
  2. Ω Λ \Omega_{\Lambda}
  3. a ( t ) a(t)
  4. a 0 = 1 a_{0}=1
  5. a a
  6. a ( t e m ) ( 1 + z ( t e m ) ) - 1 a(t_{em})\equiv(1+z(t_{em}))^{-1}\!
  7. t e m t_{em}
  8. H ( a ) H(a)
  9. H ( a ) a ˙ a H(a)\equiv\frac{\dot{a}}{a}\!
  10. a ˙ \dot{a}
  11. ρ \rho\!
  12. k k
  13. Λ \Lambda\!
  14. H 2 = ( a ˙ a ) 2 = 8 π G 3 ρ - k c 2 a 2 + Λ c 2 3 H^{2}=\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8\pi G}{3}\rho-\frac{kc^{2}}{a^% {2}}+\frac{\Lambda c^{2}}{3}
  15. G G
  16. ρ c r i t \rho_{crit}
  17. ρ c r i t = 3 H 2 8 π G = 1.88 × 10 - 26 h 2 kg m - 3 \rho_{crit}=\frac{3H^{2}}{8\pi G}=1.88\times 10^{-26}h^{2}\,\text{ kg m}^{-3}
  18. h H 0 / ( 100 km/s/Mpc ) h\equiv H_{0}/(100\,\text{ km/s/Mpc})
  19. Ω x \Omega_{x}
  20. Ω x ρ x ρ c r i t = 8 π G ρ x ( t = t 0 ) 3 H 0 2 \Omega_{x}\equiv\frac{\rho_{x}}{\rho_{crit}}=\frac{8\pi G\rho_{x}(t=t_{0})}{3H% _{0}^{2}}
  21. Λ \Lambda
  22. a a
  23. a - 3 a^{-3}
  24. H ( a ) a ˙ a = H 0 [ ( Ω c + Ω b ) a - 3 + Ω r a d a - 4 + Ω k a - 2 + Ω D E a - 3 ( 1 + w ) ] H(a)\equiv\frac{\dot{a}}{a}=H_{0}\sqrt{\left[(\Omega_{c}+\Omega_{b})a^{-3}+% \Omega_{rad}a^{-4}+\Omega_{k}a^{-2}+\Omega_{DE}a^{-3(1+w)}\right]}
  25. Ω \Omega
  26. Ω k \Omega_{k}
  27. w = - 1 w=-1
  28. H ( a ) = H 0 [ Ω m a - 3 + Ω r a d a - 4 + Ω Λ ] H(a)=H_{0}\sqrt{\left[\Omega_{m}a^{-3}+\Omega_{rad}a^{-4}+\Omega_{\Lambda}% \right]}
  29. Ω r a d 10 - 4 \Omega_{rad}\sim 10^{-4}
  30. a ( t ) = ( Ω m / Ω Λ ) 1 / 3 sinh 2 / 3 ( t / t Λ ) a(t)=(\Omega_{m}/\Omega_{\Lambda})^{1/3}\,\sinh^{2/3}(t/t_{\Lambda})
  31. t Λ 2 / ( 3 H 0 Ω Λ ) ; t_{\Lambda}\equiv 2/(3H_{0}\sqrt{\Omega_{\Lambda}})\ ;
  32. a ( t ) = 1 a(t)=1
  33. t 0 t_{0}
  34. a ¨ \ddot{a}
  35. a = ( Ω m / 2 Ω Λ ) 1 / 3 a=(\Omega_{m}/2\Omega_{\Lambda})^{1/3}
  36. h h
  37. h h
  38. t t
  39. 9 {}^{9}
  40. n n
  41. 9 {}^{−9}
  42. k k
  43. τ τ
  44. w w
  45. m m
  46. H H
  47. h h
  48. σ σ
  49. z z
  50. t t
  51. z z
  52. r r
  53. w w
  54. r r
  55. σ σ
  56. n n
  57. k k
  58. k k
  59. h h
  60. m m
  61. r r

Lambda_cube.html

  1. F < : ω F^{\omega}_{<:}
  2. F < : ω F^{\omega}_{<:}

Lambertian_reflectance.html

  1. 𝐍 \mathbf{N}
  2. 𝐋 \mathbf{L}
  3. I D = 𝐋 𝐍 C I L I_{D}=\mathbf{L}\cdot\mathbf{N}CI_{L}
  4. I D I_{D}
  5. C C
  6. I L I_{L}
  7. 𝐋 𝐍 = | N | | L | cos α = cos α \mathbf{L}\cdot\mathbf{N}=|N||L|\cos{\alpha}=\cos{\alpha}
  8. α \alpha
  9. cos ( 0 ) = 1 \cos{(0)}=1
  10. cos ( π / 2 ) = 0 \cos{(\pi/2)}=0

Lamé's_special_quartic.html

  1. x 4 + y 4 = r 4 x^{4}+y^{4}=r^{4}
  2. r > 0 r>0
  3. 2 r 2r

Lami's_theorem.html

  1. A sin α = B sin β = C sin γ \frac{A}{\sin\alpha}=\frac{B}{\sin\beta}=\frac{C}{\sin\gamma}
  2. A sin ( π - α ) = B sin ( π - β ) = C sin ( π - γ ) \frac{A}{\sin(\pi-\alpha)}=\frac{B}{\sin(\pi-\beta)}=\frac{C}{\sin(\pi-\gamma)}
  3. A sin α = B sin β = C sin γ \Rightarrow\frac{A}{\sin\alpha}=\frac{B}{\sin\beta}=\frac{C}{\sin\gamma}

Landau's_constants.html

  1. f ( 0 ) = 1. f^{\prime}(0)=1.\,
  2. f ( 0 ) = 1. f^{\prime}(0)=1.\,
  3. 0.4330 + 10 - 14 < B < 0.472 0.4330+10^{-14}<B<0.472\,\!
  4. 0.5 < L 0.5432589... 0.5<L\leq 0.5432589...\,\!
  5. 0.5 < A 0.7853 0.5<A\leq 0.7853

Landau_damping.html

  1. v p h v_{ph}
  2. v p h v_{ph}
  3. v p h v_{ph}
  4. v p h v_{ph}
  5. E E
  6. f f
  7. E E
  8. f = f 0 ( v ) + f 1 ( x , v , t ) + f 2 ( x , v , t ) f=f_{0}(v)+f_{1}(x,v,t)+f_{2}(x,v,t)
  9. E = E 1 ( x , t ) + E 2 ( x , t ) E=E_{1}(x,t)+E_{2}(x,t)
  10. ( t + v x ) f 1 + e m E 1 f 0 = 0 , x E 1 = e ϵ 0 f 1 d v (\partial_{t}+v\partial_{x})f_{1}+{e\over m}E_{1}f^{\prime}_{0}=0,\quad% \partial_{x}E_{1}={e\over\epsilon_{0}}\int f_{1}{\rm d}v
  11. f 1 ( x , v , 0 ) = g ( v ) exp ( i k x ) f_{1}(x,v,0)=g(v)\exp(ikx)
  12. exp [ i k ( x - v p h t ) - γ t ] \exp[ik(x-v_{ph}t)-\gamma t]
  13. k k
  14. γ - π ω p 3 2 k 2 N f 0 ( v p h ) , N = f 0 d v \gamma\approx-{\pi\omega_{p}^{3}\over 2k^{2}N}f^{\prime}_{0}(v_{ph}),\quad N=% \int f_{0}{\rm d}v
  15. ω p \omega_{p}
  16. N N
  17. ω p 2 k N f 0 𝒫 k v - ω + ϵ δ ( v - ω k ) \frac{\omega_{p}^{2}}{kN}f^{\prime}_{0}\frac{\mathcal{P}}{kv-\omega}+\epsilon% \delta(v-\frac{\omega}{k})
  18. 𝒫 \mathcal{P}
  19. δ \delta
  20. ϵ = 1 + ω p 2 k N f 0 𝒫 ω - k v d v \epsilon=1+\frac{\omega_{p}^{2}}{kN}\int f^{\prime}_{0}\frac{\mathcal{P}}{% \omega-kv}{\rm d}v
  21. ω p \omega_{p}
  22. ω ( ω ϵ ) \partial_{\omega}(\omega\epsilon)
  23. f f
  24. x x
  25. v v
  26. f f
  27. v v

Landau_distribution.html

  1. p ( x ) = 1 2 π i c - i c + i e s log s + x s d s , p(x)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\!e^{s\log s+xs}\,ds,
  2. p ( x ) = 1 π 0 e - t log t - x t sin ( π t ) d t . p(x)=\frac{1}{\pi}\int_{0}^{\infty}\!e^{-t\log t-xt}\sin(\pi t)\,dt.
  3. p ( x ) = 1 2 π exp { - 1 2 ( x + e - x ) } . p(x)=\frac{1}{\sqrt{2\pi}}\exp\left\{-\frac{1}{2}(x+e^{-x})\right\}.
  4. φ ( t ; μ , c ) = exp [ i t μ - | c t | ( 1 + 2 i π log ( | t | ) ) ] . \varphi(t;\mu,c)=\exp\!\Big[\;it\mu-|c\,t|(1+\tfrac{2i}{\pi}\log(|t|))\Big].
  5. X Landau ( μ , c ) X\sim\textrm{Landau}(\mu,c)\,
  6. X + m Landau ( μ + m , c ) X+m\sim\textrm{Landau}(\mu+m,c)\,

Landau_pole.html

  1. Λ m Λ≫m
  2. g o b s = g 0 1 + β 2 g 0 ln Λ / m ( 1 ) g_{obs}=\frac{g_{0}}{1+\beta_{2}g_{0}\ln\Lambda/m}\qquad\qquad\qquad(1)
  3. m m
  4. Λ Λ
  5. Λ Λ
  6. Λ L a n d a u = m exp { 1 β 2 g o b s } . ( 3 ) \Lambda_{Landau}=m\exp\left\{\frac{1}{\beta_{2}g_{obs}}\right\}.\qquad\qquad% \qquad(3)
  7. g ( μ ) g(μ)
  8. μ μ
  9. d g d ln μ = β ( g ) = β 2 g 2 + β 3 g 3 + ( 4 ) \frac{dg}{d\ln\mu}=\beta(g)=\beta_{2}g^{2}+\beta_{3}g^{3}+\ldots\qquad\qquad% \qquad(4)
  10. g ( μ ) = g < s u b > o b s g(μ)=g<sub>obs

Landau_theory.html

  1. Ψ \Psi
  2. F = a + r Ψ 2 + s Ψ 4 + H Ψ + F=a+r\Psi^{2}+s\Psi^{4}+H\Psi+...\,
  3. Ψ \Psi
  4. s > 0 s>0
  5. T c T_{c}
  6. Ψ = 0 \Psi=0
  7. Ψ 0 \Psi\neq 0
  8. r r
  9. r r
  10. r = r 0 ( T - T c ) r=r_{0}(T-T_{c})\,
  11. r 0 r_{0}
  12. a a
  13. S i - S i {S_{i}}\rightarrow{-S_{i}}
  14. S i S_{i}
  15. i t h i^{th}
  16. ( H = 0 ) (H=0)
  17. Ψ \Psi
  18. F F
  19. r r
  20. s s
  21. Ψ \Psi
  22. T c T_{c}
  23. Ψ = ± - r 0 ( T - T c ) 2 s . \Psi=\pm\sqrt{\frac{-r_{0}(T-T_{c})}{2s}}.
  24. | T - T c | β |T-T_{c}|^{\beta}
  25. β \beta
  26. Ψ \Psi
  27. H H
  28. F := d D x ( a ( T ) + r ( T ) ψ 2 ( x ) + s ( T ) ψ 4 ( x ) + f ( T ) ( ψ ( x ) ) 2 + h ( x ) ψ ( x ) + 𝒪 ( ψ 6 ; ( ψ ) 4 ) ) F:=\int d^{D}x\ \left(a(T)+r(T)\psi^{2}(x)+s(T)\psi^{4}(x)\ +f(T)(\nabla\psi(x% ))^{2}\ +h(x)\psi(x)\ \ +\mathcal{O}(\psi^{6};(\nabla\psi)^{4})\right)
  29. D D
  30. ψ ( x ) := Tr ψ ( x ) exp - β H Z \langle\psi(x)\rangle:=\frac{\,\text{Tr}\ \psi(x)\exp^{-\beta H}}{Z}
  31. h ( x ) 0 + h 0 δ ( x ) h(x)\rightarrow 0+h_{0}\delta(x)
  32. ψ ( x ) ψ 0 + ϕ ( x ) \psi(x)\rightarrow\psi_{0}+\phi(x)
  33. δ ψ ( x ) δ h ( 0 ) = ϕ ( x ) h 0 = β ( ψ ( x ) ψ ( 0 ) - ψ ( x ) ψ ( 0 ) ) \frac{\delta\langle\psi(x)\rangle}{\delta h(0)}=\frac{\phi(x)}{h_{0}}=\beta% \left(\langle\psi(x)\psi(0)\rangle-\langle\psi(x)\rangle\langle\psi(0)\rangle\right)
  34. ϕ ( x ) \phi(x)
  35. ϕ ( x ) \phi(x)
  36. ν \nu
  37. ξ ( T - T c ) - ν \xi\sim(T-T_{c})^{-\nu}
  38. D 2 + 2 β ν D\geq 2+2\frac{\beta}{\nu}
  39. β = 1 / 2 = ν \beta=1/2=\nu
  40. D = 4 D=4

Laplace's_method.html

  1. a b e M f ( x ) d x \int_{a}^{b}\!e^{Mf(x)}\,dx
  2. e M f ( x ) . e^{Mf(x)}.\,
  3. f ′′ ( x 0 ) < 0 f^{\prime\prime}(x_{0})<0
  4. f ( x ) = f ( x 0 ) + f ( x 0 ) ( x - x 0 ) + 1 2 f ′′ ( x 0 ) ( x - x 0 ) 2 + R f(x)=f(x_{0})+f^{\prime}(x_{0})(x-x_{0})+\frac{1}{2}f^{\prime\prime}(x_{0})(x-% x_{0})^{2}+R
  5. R = O ( ( x - x 0 ) 3 ) . R=O\left((x-x_{0})^{3}\right).
  6. f ( x ) f ( x 0 ) - 1 2 | f ′′ ( x 0 ) | ( x - x 0 ) 2 f(x)\approx f(x_{0})-\frac{1}{2}|f^{\prime\prime}(x_{0})|(x-x_{0})^{2}
  7. a b e M f ( x ) d x e M f ( x 0 ) a b e - M | f ′′ ( x 0 ) | ( x - x 0 ) 2 / 2 d x \int_{a}^{b}\!e^{Mf(x)}\,dx\approx e^{Mf(x_{0})}\int_{a}^{b}e^{-M|f^{\prime% \prime}(x_{0})|(x-x_{0})^{2}/2}\,dx
  8. a b e M f ( x ) d x 2 π M | f ′′ ( x 0 ) | e M f ( x 0 ) as M . \int_{a}^{b}\!e^{Mf(x)}\,dx\approx\sqrt{\frac{2\pi}{M|f^{\prime\prime}(x_{0})|% }}e^{Mf(x_{0})}\text{ as }M\to\infty.\,
  9. f ( x ) f(x)
  10. [ a , b ] [a,b]
  11. x 0 [ a , b ] x_{0}\in[a,b]
  12. f ( x 0 ) = max [ a , b ] f ( x ) f(x_{0})=\max_{[a,b]}f(x)
  13. f ′′ ( x 0 ) < 0 f^{\prime\prime}(x_{0})<0
  14. lim n + ( a b e n f ( x ) d x ( e n f ( x 0 ) 2 π n ( - f ′′ ( x 0 ) ) ) ) = 1 \lim_{n\to+\infty}\left(\frac{\int_{a}^{b}e^{nf(x)}\,dx}{\left(e^{nf(x_{0})}% \sqrt{\frac{2\pi}{n(-f^{\prime\prime}(x_{0}))}}\right)}\right)=1
  15. ε > 0 \varepsilon>0
  16. f ′′ f^{\prime\prime}
  17. δ > 0 \delta>0
  18. | x 0 - c | < δ |x_{0}-c|<\delta
  19. f ′′ ( c ) f ′′ ( x 0 ) - ε . f^{\prime\prime}(c)\geq f^{\prime\prime}(x_{0})-\varepsilon.
  20. x ( x 0 - δ , x 0 + δ ) x\in(x_{0}-\delta,x_{0}+\delta)
  21. f ( x ) f ( x 0 ) + 1 2 ( f ′′ ( x 0 ) - ε ) ( x - x 0 ) 2 f(x)\geq f(x_{0})+\frac{1}{2}(f^{\prime\prime}(x_{0})-\varepsilon)(x-x_{0})^{2}
  22. a b e n f ( x ) d x x 0 - δ x 0 + δ e n f ( x ) d x e n f ( x 0 ) x 0 - δ x 0 + δ e n 2 ( f ′′ ( x 0 ) - ε ) ( x - x 0 ) 2 d x = e n f ( x 0 ) 1 n ( - f ′′ ( x 0 ) + ε ) - δ n ( - f ′′ ( x 0 ) + ε ) δ n ( - f ′′ ( x 0 ) + ε ) e - 1 2 y 2 d y \int_{a}^{b}e^{nf(x)}\,dx\geq\int_{x_{0}-\delta}^{x_{0}+\delta}e^{nf(x)}\,dx% \geq e^{nf(x_{0})}\int_{x_{0}-\delta}^{x_{0}+\delta}e^{\frac{n}{2}(f^{\prime% \prime}(x_{0})-\varepsilon)(x-x_{0})^{2}}\,dx=e^{nf(x_{0})}\sqrt{\frac{1}{n(-f% ^{\prime\prime}(x_{0})+\varepsilon)}}\int_{-\delta\sqrt{n(-f^{\prime\prime}(x_% {0})+\varepsilon)}}^{\delta\sqrt{n(-f^{\prime\prime}(x_{0})+\varepsilon)}}e^{-% \frac{1}{2}y^{2}}\,dy
  23. y = n ( - f ′′ ( x 0 ) + ε ) ( x - x 0 ) y=\sqrt{n(-f^{\prime\prime}(x_{0})+\varepsilon)}(x-x_{0})
  24. f ′′ ( x 0 ) < 0 f^{\prime\prime}(x_{0})<0
  25. e n f ( x 0 ) 2 π n ( - f ′′ ( x 0 ) ) e^{nf(x_{0})}\sqrt{\frac{2\pi}{n(-f^{\prime\prime}(x_{0}))}}
  26. lim n + ( a b e n f ( x ) d x ( e n f ( x 0 ) 2 π n ( - f ′′ ( x 0 ) ) ) ) lim n + 1 2 π - δ n ( - f ′′ ( x 0 ) + ε ) δ n ( - f ′′ ( x 0 ) + ε ) e - 1 2 y 2 d y - f ′′ ( x 0 ) - f ′′ ( x 0 ) + ε = - f ′′ ( x 0 ) - f ′′ ( x 0 ) + ε \lim_{n\to+\infty}\left(\frac{\int_{a}^{b}e^{nf(x)}\,dx}{\left(e^{nf(x_{0})}% \sqrt{\frac{2\pi}{n(-f^{\prime\prime}(x_{0}))}}\right)}\right)\geq\lim_{n\to+% \infty}\frac{1}{\sqrt{2\pi}}\int_{-\delta\sqrt{n(-f^{\prime\prime}(x_{0})+% \varepsilon)}}^{\delta\sqrt{n(-f^{\prime\prime}(x_{0})+\varepsilon)}}e^{-\frac% {1}{2}y^{2}}\,dy\sqrt{\frac{-f^{\prime\prime}(x_{0})}{-f^{\prime\prime}(x_{0})% +\varepsilon}}=\sqrt{\frac{-f^{\prime\prime}(x_{0})}{-f^{\prime\prime}(x_{0})+% \varepsilon}}
  27. ε \varepsilon
  28. lim n + ( a b e n f ( x ) d x ( e n f ( x 0 ) 2 π n ( - f ′′ ( x 0 ) ) ) ) 1 \lim_{n\to+\infty}\left(\frac{\int_{a}^{b}e^{nf(x)}\,dx}{\left(e^{nf(x_{0})}% \sqrt{\frac{2\pi}{n(-f^{\prime\prime}(x_{0}))}}\right)}\right)\geq 1
  29. a = - a=-\infty
  30. b = b=\infty
  31. ε > 0 \varepsilon>0
  32. ε \varepsilon
  33. f ′′ ( x 0 ) + ε < 0 f^{\prime\prime}(x_{0})+\varepsilon<0
  34. f ′′ f^{\prime\prime}
  35. δ > 0 \delta>0
  36. | x - x 0 | < δ |x-x_{0}|<\delta
  37. f ( x ) f ( x 0 ) + 1 2 ( f ′′ ( x 0 ) + ε ) ( x - x 0 ) 2 f(x)\leq f(x_{0})+\frac{1}{2}(f^{\prime\prime}(x_{0})+\varepsilon)(x-x_{0})^{2}
  38. a , b a,b
  39. η > 0 \eta>0
  40. | x - x 0 | δ |x-x_{0}|\geq\delta
  41. f ( x ) f ( x 0 ) - η f(x)\leq f(x_{0})-\eta
  42. a b e n f ( x ) d x a x 0 - δ e n f ( x ) d x + x 0 - δ x 0 + δ e n f ( x ) d x + x 0 + δ b e n f ( x ) d x ( b - a ) e n ( f ( x 0 ) - η ) + x 0 - δ x 0 + δ e n f ( x ) d x \int_{a}^{b}e^{nf(x)}\,dx\leq\int_{a}^{x_{0}-\delta}e^{nf(x)}\,dx+\int_{x_{0}-% \delta}^{x_{0}+\delta}e^{nf(x)}\,dx+\int_{x_{0}+\delta}^{b}e^{nf(x)}\,dx\leq(b% -a)e^{n(f(x_{0})-\eta)}+\int_{x_{0}-\delta}^{x_{0}+\delta}e^{nf(x)}\,dx
  43. ( b - a ) e n ( f ( x 0 ) - η ) + e n f ( x 0 ) x 0 - δ x 0 + δ e n 2 ( f ′′ ( x 0 ) + ε ) ( x - x 0 ) 2 d x ( b - a ) e n ( f ( x 0 ) - η ) + e n f ( x 0 ) - + e n 2 ( f ′′ ( x 0 ) + ε ) ( x - x 0 ) 2 d x \leq(b-a)e^{n(f(x_{0})-\eta)}+e^{nf(x_{0})}\int_{x_{0}-\delta}^{x_{0}+\delta}e% ^{\frac{n}{2}(f^{\prime\prime}(x_{0})+\varepsilon)(x-x_{0})^{2}}\,dx\leq(b-a)e% ^{n(f(x_{0})-\eta)}+e^{nf(x_{0})}\int_{-\infty}^{+\infty}e^{\frac{n}{2}(f^{% \prime\prime}(x_{0})+\varepsilon)(x-x_{0})^{2}}\,dx
  44. ( b - a ) e n ( f ( x 0 ) - η ) + e n f ( x 0 ) 2 π n ( - f ′′ ( x 0 ) - ε ) \leq(b-a)e^{n(f(x_{0})-\eta)}+e^{nf(x_{0})}\sqrt{\frac{2\pi}{n(-f^{\prime% \prime}(x_{0})-\varepsilon)}}
  45. e n f ( x 0 ) 2 π n ( - f ′′ ( x 0 ) ) e^{nf(x_{0})}\sqrt{\frac{2\pi}{n(-f^{\prime\prime}(x_{0}))}}
  46. lim n + ( a b e n f ( x ) d x ( e n f ( x 0 ) 2 π n ( - f ′′ ( x 0 ) ) ) ) lim n + ( ( b - a ) e - η n n ( - f ′′ ( x 0 ) ) 2 π + - f ′′ ( x 0 ) - f ′′ ( x 0 ) - ε ) = - f ′′ ( x 0 ) - f ′′ ( x 0 ) - ε \lim_{n\to+\infty}\left(\frac{\int_{a}^{b}e^{nf(x)}\,dx}{\left(e^{nf(x_{0})}% \sqrt{\frac{2\pi}{n(-f^{\prime\prime}(x_{0}))}}\right)}\right)\leq\lim_{n\to+% \infty}\left((b-a)e^{-\eta n}\sqrt{\frac{n(-f^{\prime\prime}(x_{0}))}{2\pi}}+% \sqrt{\frac{-f^{\prime\prime}(x_{0})}{-f^{\prime\prime}(x_{0})-\varepsilon}}% \right)=\sqrt{\frac{-f^{\prime\prime}(x_{0})}{-f^{\prime\prime}(x_{0})-% \varepsilon}}
  47. ε \varepsilon
  48. lim n + ( a b e n f ( x ) d x ( e n f ( x 0 ) 2 π n ( - f ′′ ( x 0 ) ) ) ) 1 \lim_{n\to+\infty}\left(\frac{\int_{a}^{b}e^{nf(x)}\,dx}{\left(e^{nf(x_{0})}% \sqrt{\frac{2\pi}{n(-f^{\prime\prime}(x_{0}))}}\right)}\right)\leq 1
  49. a = - a=-\infty
  50. b = b=\infty
  51. n = 1 n=1
  52. a b e n f ( x ) d x \int_{a}^{b}e^{nf(x)}\,dx
  53. η \eta
  54. [ a , b ] [a,b]
  55. a x 0 - δ e n f ( x ) d x + x 0 + δ b e n f ( x ) d x \int_{a}^{x_{0}-\delta}e^{nf(x)}\,dx+\int_{x_{0}+\delta}^{b}e^{nf(x)}\,dx
  56. a x 0 - δ e n f ( x ) d x + x 0 + δ b e n f ( x ) d x a b e f ( x ) e ( n - 1 ) ( f ( x 0 ) - η ) d x = e ( n - 1 ) ( f ( x 0 ) - η ) a b e f ( x ) d x \int_{a}^{x_{0}-\delta}e^{nf(x)}\,dx+\int_{x_{0}+\delta}^{b}e^{nf(x)}\,dx\leq% \int_{a}^{b}e^{f(x)}e^{(n-1)(f(x_{0})-\eta)}\,dx=e^{(n-1)(f(x_{0})-\eta)}\int_% {a}^{b}e^{f(x)}\,dx
  57. ( b - a ) e n ( f ( x 0 ) - η ) (b-a)e^{n(f(x_{0})-\eta)}
  58. e n f ( x 0 ) 2 π n ( - f ′′ ( x 0 ) ) e^{nf(x_{0})}\sqrt{\frac{2\pi}{n(-f^{\prime\prime}(x_{0}))}}
  59. e ( n - 1 ) ( f ( x 0 ) - η ) a b e f ( x ) d x e n f ( x 0 ) 2 π n ( - f ′′ ( x 0 ) ) = e - ( n - 1 ) η n e - f ( x 0 ) a b e f ( x ) d x - f ′′ ( x 0 ) 2 π \frac{e^{(n-1)(f(x_{0})-\eta)}\int_{a}^{b}e^{f(x)}\,dx}{e^{nf(x_{0})}\sqrt{% \frac{2\pi}{n(-f^{\prime\prime}(x_{0}))}}}=e^{-(n-1)\eta}\sqrt{n}e^{-f(x_{0})}% \int_{a}^{b}e^{f(x)}\,dx\sqrt{\frac{-f^{\prime\prime}(x_{0})}{2\pi}}
  60. n n\rightarrow\infty
  61. 0
  62. x 0 x_{0}
  63. η > 0 \eta>0
  64. n = 1 n=1
  65. n = N n=N
  66. s = 2 π M | f ′′ ( x 0 ) | s=\sqrt{\frac{2\pi}{M\left|f^{\prime\prime}(x_{0})\right|}}
  67. a b e M f ( x ) d x = s e M f ( x 0 ) 1 s a b e M ( f ( x ) - f ( x 0 ) ) d x = s e M f ( x 0 ) ( a - x 0 ) / s ( b - x 0 ) / s e M ( f ( s y + x 0 ) - f ( x 0 ) ) d y \begin{aligned}\displaystyle\int_{a}^{b}\!e^{Mf(x)}\,dx&\displaystyle=se^{Mf(x% _{0})}\frac{1}{s}\int_{a}^{b}\!e^{M(f(x)-f(x_{0}))}\,dx\\ &\displaystyle=se^{Mf(x_{0})}\int_{(a-x_{0})/s}^{(b-x_{0})/s}\!e^{M(f(sy+x_{0}% )-f(x_{0}))}\,dy\end{aligned}
  68. s s
  69. M M
  70. | ( a - x 0 ) / s ( b - x 0 ) / s e M ( f ( s y + x 0 ) - f ( x 0 ) ) d y - 1 | . \left|\int_{(a-x_{0})/s}^{(b-x_{0})/s}e^{M(f(sy+x_{0})-f(x_{0}))}dy-1\right|.
  71. y [ - D y , D y ] y\in[-D_{y},D_{y}]
  72. e M ( f ( s y + x 0 ) - f ( x 0 ) ) e^{M\left(f(sy+x_{0})-f(x_{0})\right)}
  73. e - π y 2 e^{-\pi y^{2}}
  74. M M
  75. M ( f ( x ) - f ( x 0 ) ) M\left(f(x)-f(x_{0})\right)
  76. M ( f ( x ) - f ( x 0 ) ) = M f ′′ ( x 0 ) 2 s 2 y 2 + M f ′′′ ( x 0 ) 6 s 3 y 3 + = - π y 2 + O ( 1 M ) . \begin{aligned}\displaystyle M\left(f(x)-f(x_{0})\right)&\displaystyle=\frac{% Mf^{\prime\prime}(x_{0})}{2}s^{2}y^{2}+\frac{Mf^{\prime\prime\prime}(x_{0})}{6% }s^{3}y^{3}+\cdots\\ &\displaystyle=-\pi y^{2}+O\left(\frac{1}{\sqrt{M}}\right).\end{aligned}
  77. f ( x 0 ) = 0 f^{\prime}(x_{0})=0
  78. x 0 x_{0}
  79. 1 M \frac{1}{\sqrt{M}}
  80. exp ( M ( f ( x ) - f ( x 0 ) ) ) \exp\left(M\left(f(x)-f(x_{0})\right)\right)
  81. - e - π y 2 d y = 1. \int_{-\infty}^{\infty}e^{-\pi y^{2}}dy=1.
  82. e M [ f ( s y + x 0 ) - f ( x 0 ) ] e^{M[f(sy+x_{0})-f(x_{0})]}
  83. M M
  84. e - π y 2 e^{-\pi y^{2}}
  85. M M
  86. x x
  87. y y
  88. y [ - D y , D y ] y\in[-D_{y},D_{y}]
  89. x [ - s D y , s D y ] x\in[-sD_{y},sD_{y}]
  90. s s
  91. M \sqrt{M}
  92. x x
  93. M M
  94. x 0 x_{0}
  95. M M
  96. M M
  97. M M
  98. m ( x ) m(x)
  99. m ( x ) f ( x ) m(x)\geq f(x)
  100. e M m ( x ) e^{Mm(x)}
  101. M M
  102. M m ( x ) Mm(x)
  103. m ( x ) m(x)
  104. m ( x ) m(x)
  105. e M f ( x ) e^{Mf(x)}
  106. m ( x ) m(x)
  107. x = s D y x=sD_{y}
  108. m ( x ) m(x)
  109. x = ± s D y + x 0 x=\pm sD_{y}+x_{0}
  110. s D y sD_{y}
  111. f ( x ) f(x)
  112. m ( x ) m(x)
  113. M M
  114. e M m ( x ) e^{Mm(x)}
  115. M M
  116. m ( x ) m(x)
  117. f ( x ) f(x)
  118. e M f ( x ) e^{Mf(x)}
  119. f ( x ) = s i n ( x ) x f(x)=\frac{sin(x)}{x}
  120. 0 e M f ( x ) d x \int^{\infty}_{0}e^{Mf(x)}dx
  121. d e M f ( x ) d x \int^{\infty}_{d}e^{Mf(x)}dx
  122. d d
  123. d d
  124. m ( x ) m(x)
  125. f ( x ) f(x)
  126. d e f ( x ) d x \int^{\infty}_{d}e^{f(x)}dx
  127. f ( x ) f(x)
  128. - ln x -\ln x
  129. e f ( x ) = 1 x e^{f(x)}=\frac{1}{x}
  130. M = 2 M=2
  131. e M f ( x ) = 1 x 2 e^{Mf(x)}=\frac{1}{x^{2}}
  132. M M
  133. M M
  134. a b h ( x ) e M g ( x ) d x 2 π M | g ′′ ( x 0 ) | h ( x 0 ) e M g ( x 0 ) as M \int_{a}^{b}\!h(x)e^{Mg(x)}\,dx\approx\sqrt{\frac{2\pi}{M|g^{\prime\prime}(x_{% 0})|}}h(x_{0})e^{Mg(x_{0})}\text{ as }M\to\infty\,
  135. h h
  136. g ( x ) g(x)
  137. h ( x ) h(x)
  138. x 0 = 0 x_{0}=0
  139. | R | \left|R\right|
  140. a b h ( x ) e M g ( x ) d x = h ( 0 ) e M g ( 0 ) s a / s b / s h ( x ) h ( 0 ) e M [ g ( s y ) - g ( 0 ) ] d y 1 + R , \int_{a}^{b}\!h(x)e^{Mg(x)}\,dx=h(0)e^{Mg(0)}s\underbrace{\int_{a/s}^{b/s}% \frac{h(x)}{h(0)}e^{M\left[g(sy)-g(0)\right]}dy}_{1+R},
  141. s 2 π M | g ′′ ( 0 ) | s\equiv\sqrt{\frac{2\pi}{M\left|g^{\prime\prime}(0)\right|}}
  142. A h ( s y ) h ( 0 ) e M [ g ( s y ) - g ( 0 ) ] A\equiv\frac{h(sy)}{h(0)}e^{M\left[g(sy)-g(0)\right]}
  143. A 0 e - π y 2 A_{0}\equiv e^{-\pi y^{2}}
  144. | R | = | a / s b / s A d y - - A 0 d y | \left|R\right|=\left|\int_{a/s}^{b/s}A\,dy-\int_{-\infty}^{\infty}A_{0}\,dy\right|
  145. - A 0 d y = 1 \int_{-\infty}^{\infty}A_{0}\,dy=1
  146. | A + B | | A | + | B | \left|A+B\right|\leq|A|+|B|
  147. | R | < | D y A 0 d y | ( a 1 ) + | D y b / s A d y | ( b 1 ) + | - D y D y ( A - A 0 ) d y | ( c ) + | a / s - D y A d y | ( b 2 ) + | - - D y A 0 d y | ( a 2 ) |R|<\underbrace{\left|\int_{D_{y}}^{\infty}A_{0}dy\right|}_{(a_{1})}+% \underbrace{\left|\int_{D_{y}}^{b/s}Ady\right|}_{(b_{1})}+\underbrace{\left|% \int_{-D_{y}}^{D_{y}}\left(A-A_{0}\right)dy\right|}_{(c)}+\underbrace{\left|% \int_{a/s}^{-D_{y}}Ady\right|}_{(b_{2})}+\underbrace{\left|\int_{-\infty}^{-D_% {y}}A_{0}dy\right|}_{(a_{2})}
  148. ( a 1 ) (a_{1})
  149. ( a 2 ) (a_{2})
  150. ( a 1 ) (a_{1})
  151. ( b 1 ) (b_{1})
  152. ( b 2 ) (b_{2})
  153. ( b 1 ) (b_{1})
  154. ( a 1 ) (a_{1})
  155. z π y 2 z\equiv\pi y^{2}
  156. ( a 1 ) = | 1 2 π π D y 2 e - z z - 1 / 2 d z | < e - π D y 2 2 π D y . (a_{1})=\left|\frac{1}{2\sqrt{\pi}}\int_{\pi D_{y}^{2}}^{\infty}e^{-z}z^{-1/2}% dz\right|<\frac{e^{-\pi D_{y}^{2}}}{2\pi D_{y}}.
  157. D y D_{y}
  158. ( b 1 ) (b_{1})
  159. ( b 1 ) | D y b / s [ h ( s y ) h ( 0 ) ] max e M m ( s y ) d y | (b_{1})\leq\left|\int_{D_{y}}^{b/s}\left[\frac{h(sy)}{h(0)}\right]_{\,\text{% max}}e^{Mm(sy)}dy\right|
  160. m ( x ) 1 M ln h ( x ) h ( 0 ) + g ( x ) - g ( 0 ) as x [ s D y , b ] m(x)\geq\frac{1}{M}\ln{\frac{h(x)}{h(0)}}+g(x)-g(0)\,\,\,\text{as}\,\,x\in[sD_% {y},b]
  161. h ( x ) h(x)
  162. h ( 0 ) h(0)
  163. m ( x ) m(x)
  164. x = s D y x=sD_{y}
  165. m ( s y ) = g ( s D y ) - g ( 0 ) + g ( s D y ) ( s y - s D y ) m(sy)=g(sD_{y})-g(0)+g^{\prime}(sD_{y})\left(sy-sD_{y}\right)
  166. m ( x ) m(x)
  167. x = s D y x=sD_{y}
  168. s s
  169. D y D_{y}
  170. m ( x ) m(x)
  171. f ( x ) f(x)
  172. ( b 1 ) (b_{1})
  173. D y D_{y}
  174. e - α x e^{-\alpha x}
  175. ( b 1 ) (b_{1})
  176. M [ g ( s D y ) - g ( 0 ) ] = M [ g ′′ ( 0 ) 2 s 2 D y 2 + g ′′′ ( ξ ) 6 s 3 D y 3 ] as ξ [ 0 , s D y ] = - π D y 2 + ( 2 π ) 3 / 2 g ′′′ ( ξ ) D y 3 6 M | g ′′ ( 0 ) | 3 / 2 , \begin{aligned}\displaystyle M\left[g(sD_{y})-g(0)\right]&\displaystyle=M\left% [\frac{g^{\prime\prime}(0)}{2}s^{2}D_{y}^{2}+\frac{g^{\prime\prime\prime}(\xi)% }{6}s^{3}D_{y}^{3}\right]\,\,\,\text{as}\,\,\xi\in[0,sD_{y}]\\ &\displaystyle=-\pi D_{y}^{2}+\frac{(2\pi)^{3/2}g^{\prime\prime\prime}(\xi)D_{% y}^{3}}{6\sqrt{M}|g^{\prime\prime}(0)|^{3/2}},\end{aligned}
  177. M s g ( s D y ) = M s ( g ′′ ( 0 ) s D y + g ′′′ ( ζ ) 2 s 2 D y 2 ) , as ζ [ 0 , s D y ] = - 2 π D y + 2 M ( π | g ′′ ( 0 ) | ) 3 / 2 g ′′′ ( ζ ) D y 2 , \begin{aligned}\displaystyle Msg^{\prime}(sD_{y})&\displaystyle=Ms\left(g^{% \prime\prime}(0)sD_{y}+\frac{g^{\prime\prime\prime}(\zeta)}{2}s^{2}D_{y}^{2}% \right),\,\,\,\text{as}\,\,\zeta\in[0,sD_{y}]\\ &\displaystyle=-2\pi D_{y}+\sqrt{\frac{2}{M}}\left(\frac{\pi}{|g^{\prime\prime% }(0)|}\right)^{3/2}g^{\prime\prime\prime}(\zeta)D_{y}^{2},\end{aligned}
  178. ( b 1 ) (b_{1})
  179. M M
  180. ( b 1 ) | [ h ( s y ) h ( 0 ) ] max e - π D y 2 0 b / s - D y e - 2 π D y y d y | | [ h ( s y ) h ( 0 ) ] max e - π D y 2 1 2 π D y | . \begin{aligned}\displaystyle(b_{1})&\displaystyle\leq\left|\left[\frac{h(sy)}{% h(0)}\right]_{\,\text{max}}e^{-\pi D_{y}^{2}}\int_{0}^{b/s-D_{y}}e^{-2\pi D_{y% }y}dy\right|\\ &\displaystyle\leq\left|\left[\frac{h(sy)}{h(0)}\right]_{\,\text{max}}e^{-\pi D% _{y}^{2}}\frac{1}{2\pi D_{y}}\right|.\end{aligned}
  181. D y D_{y}
  182. D y D_{y}
  183. x = 0 x=0
  184. h ( 0 ) 0 h^{\prime}(0)\neq 0
  185. ( c ) - D y D y e - π y 2 | s h ( ξ ) h ( 0 ) y | d y < 2 π M | g ′′ ( 0 ) | | h ( ξ ) h ( 0 ) y | max ( 1 - e - π D y 2 ) \begin{aligned}\displaystyle(c)&\displaystyle\leq\int_{-D_{y}}^{D_{y}}e^{-\pi y% ^{2}}\left|\frac{sh^{\prime}(\xi)}{h(0)}y\right|\,dy\\ &\displaystyle<\sqrt{\frac{2}{\pi M|g^{\prime\prime}(0)|}}\left|\frac{h^{% \prime}(\xi)}{h(0)}y\right|\text{max}\left(1-e^{-\pi D_{y}^{2}}\right)\end{aligned}
  186. M M
  187. ( c ) (c)
  188. h ( x ) h(x)
  189. M \sqrt{M}
  190. D y D_{y}
  191. D y D_{y}
  192. m ( x ) m(x)
  193. g ( x ) - g ( 0 ) g(x)-g(0)
  194. m ( x ) m(x)
  195. D y D_{y}
  196. M \sqrt{M}
  197. m ( x ) m(x)
  198. g ( x ) - g ( 0 ) g(x)-g(0)
  199. x = s D y x=sD_{y}
  200. M M
  201. D y D_{y}
  202. 𝐱 \mathbf{x}
  203. d d
  204. f ( 𝐱 ) f(\mathbf{x})
  205. 𝐱 \mathbf{x}
  206. e M f ( 𝐱 ) d 𝐱 ( 2 π M ) d / 2 | - H ( f ) ( 𝐱 0 ) | - 1 / 2 e M f ( 𝐱 0 ) as M \int e^{Mf(\mathbf{x})}\,d\mathbf{x}\approx\left(\frac{2\pi}{M}\right)^{d/2}|-% H(f)(\mathbf{x}_{0})|^{-1/2}e^{Mf(\mathbf{x}_{0})}\text{ as }M\to\infty\,
  207. H ( f ) ( 𝐱 0 ) H(f)(\mathbf{x}_{0})
  208. f f
  209. 𝐱 0 \mathbf{x}_{0}
  210. | | |\cdot|
  211. 𝐱 \mathbf{x}
  212. d d
  213. d 𝐱 d\mathbf{x}
  214. d 𝐱 := d x 1 d x 2 d x d d\mathbf{x}:=dx_{1}dx_{2}\cdots dx_{d}
  215. a b e M f ( z ) d z 2 π - M f ′′ ( z 0 ) e M f ( z 0 ) as M . \int_{a}^{b}\!e^{Mf(z)}\,dz\approx\sqrt{\frac{2\pi}{-Mf^{\prime\prime}(z_{0})}% }e^{Mf(z_{0})}\,\text{ as }M\to\infty.\,
  216. 1 2 π i c - i c + i g ( s ) e s t d s \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}g(s)e^{st}\,ds
  217. 1 2 π - g ( c + i x ) e - u x e i c u d x . \frac{1}{2\pi}\int_{-\infty}^{\infty}g(c+ix)e^{-ux}e^{icu}\,dx.
  218. N ! 2 π N N N e - N N!\approx\sqrt{2\pi N}N^{N}e^{-N}\,
  219. N ! = Γ ( N + 1 ) = 0 e - x x N d x . N!=\Gamma(N+1)=\int_{0}^{\infty}e^{-x}x^{N}\,dx.
  220. x = N z x=Nz\,
  221. d x = N d z . dx=N\,dz.
  222. N ! \displaystyle N!
  223. f ( z ) = ln z - z f\left(z\right)=\ln{z}-z
  224. f ( z ) = 1 z - 1 , f^{\prime}(z)=\frac{1}{z}-1,\,
  225. f ′′ ( z ) = - 1 z 2 . f^{\prime\prime}(z)=-\frac{1}{z^{2}}.\,
  226. N ! N N + 1 2 π N e - N = 2 π N N N e - N . N!\approx N^{N+1}\sqrt{\frac{2\pi}{N}}e^{-N}=\sqrt{2\pi N}N^{N}e^{-N}.\,

Laplace_distribution.html

  1. f ( x μ , b ) = 1 2 b exp ( - | x - μ | b ) f(x\mid\mu,b)=\frac{1}{2b}\exp\left(-\frac{|x-\mu|}{b}\right)\,\!
  2. = 1 2 b { exp ( - μ - x b ) if x < μ exp ( - x - μ b ) if x μ =\frac{1}{2b}\left\{\begin{matrix}\exp\left(-\frac{\mu-x}{b}\right)&\,\text{if% }x<\mu\\ \exp\left(-\frac{x-\mu}{b}\right)&\,\text{if }x\geq\mu\end{matrix}\right.
  3. { { b f ( x ) + f ( x ) = 0 f ( 0 ) = e μ b 2 b } if x μ { b f ( x ) - f ( x ) = 0 f ( 0 ) = e - μ b 2 b } if x < μ \begin{cases}\left\{\begin{array}[]{l}bf^{\prime}(x)+f(x)=0\\ f(0)=\frac{e^{\frac{\mu}{b}}}{2b}\end{array}\right\}&\,\text{if }x\geq\mu\\ \left\{\begin{array}[]{l}bf^{\prime}(x)-f(x)=0\\ f(0)=\frac{e^{-\frac{\mu}{b}}}{2b}\end{array}\right\}&\,\text{if }x<\mu\end{cases}
  4. F ( x ) \displaystyle F(x)
  5. F - 1 ( p ) = μ - b sgn ( p - 0.5 ) ln ( 1 - 2 | p - 0.5 | ) . F^{-1}(p)=\mu-b\,\operatorname{sgn}(p-0.5)\,\ln(1-2|p-0.5|).
  6. X = μ - b sgn ( U ) ln ( 1 - 2 | U | ) X=\mu-b\,\operatorname{sgn}(U)\,\ln(1-2|U|)
  7. μ ^ \hat{\mu}
  8. b ^ = 1 N i = 1 N | x i - μ ^ | \hat{b}=\frac{1}{N}\sum_{i=1}^{N}|x_{i}-\hat{\mu}|
  9. μ r = ( 1 2 ) k = 0 r [ r ! k ! ( r - k ) ! b k μ ( r - k ) k ! { 1 + ( - 1 ) k } ] \mu_{r}^{\prime}=\bigg({\frac{1}{2}}\bigg)\sum_{k=0}^{r}\bigg[{\frac{r!}{k!(r-% k)!}}b^{k}\mu^{(r-k)}k!\{1+(-1)^{k}\}\bigg]
  10. 2 i = 1 n | X i - μ | b χ 2 ( 2 n ) \frac{\displaystyle 2\sum_{i=1}^{n}|X_{i}-\mu|}{b}\sim\chi^{2}(2n)\,
  11. | X - μ | | Y - μ | F ( 2 , 2 ) \tfrac{|X-\mu|}{|Y-\mu|}\sim\operatorname{F}(2,2)
  12. X = μ + b 2 V Z Laplace ( μ , b ) X=\mu+b\sqrt{2V}Z\sim\mathrm{Laplace}(\mu,b)
  13. λ - i t + λ , λ i t + λ \frac{\lambda}{-it+\lambda},\quad\frac{\lambda}{it+\lambda}
  14. λ 2 ( - i t + λ ) ( i t + λ ) = λ 2 t 2 + λ 2 . \frac{\lambda^{2}}{(-it+\lambda)(it+\lambda)}=\frac{\lambda^{2}}{t^{2}+\lambda% ^{2}}.
  15. 1 1 + t 2 λ 2 . \frac{1}{1+\frac{t^{2}}{\lambda^{2}}}.
  16. f p ( x ) = 1 2 exp ( - α | x | ) 1 + j = 1 p β j α j | x | j 1 + j = 1 p j ! β j , f_{p}(x)=\tfrac{1}{2}\exp(-\alpha|x|)\frac{\displaystyle 1+\sum_{j=1}^{p}\beta% _{j}\alpha^{j}|x|^{j}}{\displaystyle 1+\sum_{j=1}^{p}j!\beta_{j}},

Laplace–Runge–Lenz_vector.html

  1. 𝒜 \mathcal{A}
  2. 𝐅 ( r ) = - k r 2 𝐫 ^ \mathbf{F}(r)=\frac{-k}{r^{2}}\mathbf{\hat{r}}
  3. m m
  4. k k
  5. 𝐫 ^ \mathbf{\hat{r}}\!\,
  6. 𝐫 ^ = 𝐫 r \mathbf{\hat{r}}=\frac{\mathbf{r}}{r}
  7. E E
  8. E = p 2 2 m - k r = 1 2 m v 2 - k r . E=\frac{p^{2}}{2m}-\frac{k}{r}=\frac{1}{2}mv^{2}-\frac{k}{r}~{}.
  9. 𝐞 = 𝐀 m k = 1 m k ( 𝐩 × 𝐋 ) - 𝐫 ^ . \mathbf{e}=\frac{\mathbf{A}}{mk}=\frac{1}{mk}(\mathbf{p}\times\mathbf{L})-% \mathbf{\hat{r}}~{}.
  10. 𝐀 𝐫 = A r cos θ = 𝐫 ( 𝐩 × 𝐋 ) - m k r \mathbf{A}\cdot\mathbf{r}=Ar\cos\theta=\mathbf{r}\cdot\left(\mathbf{p}\times% \mathbf{L}\right)-mkr
  11. 𝐫 ( 𝐩 × 𝐋 ) = ( 𝐫 × 𝐩 ) 𝐋 = 𝐋 𝐋 = L 2 \mathbf{r}\cdot\left(\mathbf{p}\times\mathbf{L}\right)=\left(\mathbf{r}\times% \mathbf{p}\right)\cdot\mathbf{L}=\mathbf{L}\cdot\mathbf{L}=L^{2}
  12. e = A m k = | 𝐀 | m k e=\frac{A}{mk}=\frac{\left|\mathbf{A}\right|}{mk}
  13. | 2 | = 2 L 2 m k . \left|2\ell\right|=\frac{2L^{2}}{mk}~{}.
  14. a a
  15. a ( 1 ± e 2 ) = = L 2 m k , a\left(1\pm e^{2}\right)=\ell=\frac{L^{2}}{mk}~{},
  16. E E
  17. A 2 = m 2 k 2 + 2 m E L 2 , A^{2}=m^{2}k^{2}+2mEL^{2}\,,
  18. e 2 - 1 = 2 L 2 m k 2 E . e^{2}-1=\frac{2L^{2}}{mk^{2}}E~{}.
  19. m k 𝐫 ^ = 𝐩 × 𝐋 - 𝐀 mk~{}\hat{\mathbf{r}}=\mathbf{p}\times\mathbf{L}-\mathbf{A}
  20. ( m k ) 2 = A 2 + p 2 L 2 + 2 𝐋 ( 𝐩 × 𝐀 ) . (mk)^{2}=A^{2}+p^{2}L^{2}+2\mathbf{L}\cdot(\mathbf{p}\times\mathbf{A})~{}.
  21. m k / L = L / mk/L=L/ℓ
  22. ( 0 , A / L ) (0,A/L)
  23. e e
  24. p 0 = 2 m | E | p_{0}=\sqrt{2m\left|E\right|}
  25. h ( r ) h(r)
  26. h ( r ) h(r)
  27. E E
  28. A A
  29. h ( r ) h(r)
  30. h ( r ) h(r)
  31. L h ( r ) \displaystyle\frac{\partial}{\partial L}\langle h(r)\rangle
  32. T T
  33. h ( r ) 〈h(r)〉
  34. h ( r ) = k L 2 m 2 c 2 ( 1 r 3 ) . h(r)=\frac{kL^{2}}{m^{2}c^{2}}\left(\frac{1}{r^{3}}\right)~{}.
  35. 1 r = m k L 2 ( 1 + A m k cos θ ) \frac{1}{r}=\frac{mk}{L^{2}}\left(1+\frac{A}{mk}\cos\theta\right)
  36. r r
  37. θ θ
  38. 6 π k 2 T L 2 c 2 , \frac{6\pi k^{2}}{TL^{2}c^{2}}~{},
  39. { L i , L j } = s = 1 3 ϵ i j s L s , \left\{L_{i},L_{j}\right\}=\sum_{s=1}^{3}\epsilon_{ijs}L_{s}~{},
  40. i i
  41. s s
  42. k k
  43. p 0 = 2 m | E | p_{0}=\sqrt{2m|E|}
  44. { D i , L j } = s = 1 3 ϵ i j s D s . \left\{D_{i},L_{j}\right\}=\sum_{s=1}^{3}\epsilon_{ijs}D_{s}~{}.
  45. { D i , D j } = s = 1 3 ϵ i j s L s ; \left\{D_{i},D_{j}\right\}=\sum_{s=1}^{3}\epsilon_{ijs}L_{s}~{};
  46. { D i , D j } = - s = 1 3 ϵ i j s L s . \left\{D_{i},D_{j}\right\}=-\sum_{s=1}^{3}\epsilon_{ijs}L_{s}~{}.
  47. C 1 = 𝐃 𝐃 + 𝐋 𝐋 = m k 2 2 | E | C_{1}=\mathbf{D}\cdot\mathbf{D}+\mathbf{L}\cdot\mathbf{L}=\frac{mk^{2}}{2\left% |E\right|}
  48. C 2 = 𝐃 𝐋 = 0 , C_{2}=\mathbf{D}\cdot\mathbf{L}=0,
  49. { C 1 , L i } = { C 1 , D i } = { C 2 , L i } = { C 2 , D i } = 0 . \left\{C_{1},L_{i}\right\}=\left\{C_{1},D_{i}\right\}=\left\{C_{2},L_{i}\right% \}=\left\{C_{2},D_{i}\right\}=0~{}.
  50. i ħ
  51. C C
  52. A s = - m k r ^ s + 1 2 i = 1 3 j = 1 3 ϵ s i j ( p i l j + l j p i ) , A_{s}=-mk\hat{r}_{s}+\frac{1}{2}\sum_{i=1}^{3}\sum_{j=1}^{3}\epsilon_{sij}% \left(p_{i}l_{j}+l_{j}p_{i}\right),
  53. J 0 = A 3 J_{0}=A_{3}\,
  54. J ± 1 = 1 2 ( A 1 ± i A 2 ) . J_{\pm 1}=\mp\frac{1}{\sqrt{2}}\left(A_{1}\pm iA_{2}\right)~{}.
  55. C 1 = - m k 2 2 2 H - 1 - I , C_{1}=-\frac{mk^{2}}{2\hbar^{2}}H^{-1}-I~{},
  56. I I
  57. m n mn
  58. C C
  59. m m
  60. E n = - m k 2 2 2 n 2 , E_{n}=-\frac{mk^{2}}{2\hbar^{2}n^{2}}~{},
  61. ± p 0 = ± 2 m | E | \pm p_{0}=\pm\sqrt{2m\left|E\right|}
  62. | 𝐞 | 2 = e 1 2 + e 2 2 + e 3 2 + e 4 2 . \left|\mathbf{e}\right|^{2}=e_{1}^{2}+e_{2}^{2}+e_{3}^{2}+e_{4}^{2}.
  63. d s 2 = e 1 2 + e 2 2 + e 3 2 - e 4 2 . ds^{2}=e_{1}^{2}+e_{2}^{2}+e_{3}^{2}-e_{4}^{2}.
  64. s y m b o l η symbol\eta
  65. s y m b o l η = p 2 - p 0 2 p 2 + p 0 2 𝐰 ^ + 2 p 0 p 2 + p 0 2 𝐩 = m k - r p 0 2 m k 𝐰 ^ + r p 0 m k 𝐩 \begin{aligned}\displaystyle symbol\eta&\displaystyle=\displaystyle\frac{p^{2}% -p_{0}^{2}}{p^{2}+p_{0}^{2}}\mathbf{\hat{w}}+\frac{2p_{0}}{p^{2}+p_{0}^{2}}% \mathbf{p}\\ &\displaystyle=\displaystyle\frac{mk-rp_{0}^{2}}{mk}\mathbf{\hat{w}}+\frac{rp_% {0}}{mk}\mathbf{p}\end{aligned}
  66. 𝐰 ^ \mathbf{\hat{w}}
  67. p x = p 0 η x 1 - η w p_{x}=p_{0}\frac{\eta_{x}}{1-\eta_{w}}
  68. s y m b o l η symbol\eta
  69. s y m b o l η symbol\eta
  70. s y m b o l η symbol\eta
  71. s y m b o l η symbol\eta
  72. η w = cn χ cn ψ \eta_{w}=\mathrm{cn}\,\chi\ \mathrm{cn}\,\psi
  73. η x = sn χ dn ψ cos ϕ \eta_{x}=\mathrm{sn}\,\chi\ \mathrm{dn}\,\psi\ \cos\phi
  74. η y = sn χ dn ψ sin ϕ \eta_{y}=\mathrm{sn}\,\chi\ \mathrm{dn}\,\psi\ \sin\phi
  75. η z = dn χ sn ψ \eta_{z}=\mathrm{dn}\,\chi\ \mathrm{sn}\,\psi
  76. 𝒜 \mathcal{A}
  77. 𝒜 = 𝐀 + m q 2 [ ( 𝐫 × 𝐄 ) × 𝐫 ] , \mathcal{A}=\mathbf{A}+\frac{mq}{2}\left[\left(\mathbf{r}\times\mathbf{E}% \right)\times\mathbf{r}\right],
  78. 𝒜 \mathcal{A}
  79. 𝒜 𝐄 \mathcal{A}\cdot\mathbf{E}
  80. 𝒜 = ( ξ u ) ( 𝐩 × 𝐋 ) + [ ξ - u ( ξ u ) ] L 2 𝐫 ^ \mathcal{A}=\left(\frac{\partial\xi}{\partial u}\right)\left(\mathbf{p}\times% \mathbf{L}\right)+\left[\xi-u\left(\frac{\partial\xi}{\partial u}\right)\right% ]L^{2}\mathbf{\hat{r}}
  81. θ = L u d u m 2 c 2 ( γ 2 - 1 ) - L 2 u 2 \theta=L\int^{u}\frac{du}{\sqrt{m^{2}c^{2}\left(\gamma^{2}-1\right)-L^{2}u^{2}}}
  82. = 𝐋 × 𝒜 . \mathcal{B}=\mathbf{L}\times\mathcal{A}.
  83. 𝒲 = α 𝒜 𝒜 + β \mathcal{W}=\alpha\mathcal{A}\otimes\mathcal{A}+\beta\,\mathcal{B}\otimes% \mathcal{B}
  84. 𝐅 ( r ) = - k 𝐫 , \mathbf{F}(r)=-k\mathbf{r},
  85. 𝒲 = 1 2 m 𝐩 𝐩 + k 2 𝐫 𝐫 , \mathcal{W}=\frac{1}{2m}\mathbf{p}\otimes\mathbf{p}+\frac{k}{2}\,\mathbf{r}% \otimes\mathbf{r}~{},
  86. 𝒜 = 1 m r 2 ω 0 A - m r 2 E + L 2 { ( 𝐩 × 𝐋 ) + ( m r ω 0 A - m r E ) 𝐫 ^ } , \mathcal{A}=\frac{1}{\sqrt{mr^{2}\omega_{0}A-mr^{2}E+L^{2}}}\left\{\left(% \mathbf{p}\times\mathbf{L}\right)+\left(mr\omega_{0}A-mrE\right)\mathbf{\hat{r% }}\right\},
  87. ω 0 = k m \omega_{0}=\sqrt{\frac{k}{m}}
  88. A = ( E 2 - ω 2 L 2 ) 1 / 2 / ω A=(E^{2}-\omega^{2}L^{2})^{1/2}/\omega
  89. 𝐅 \mathbf{F}
  90. 𝐅 = d 𝐩 d t = f ( r ) 𝐫 r = f ( r ) 𝐫 ^ \mathbf{F}=\frac{d\mathbf{p}}{dt}=f(r)\frac{\mathbf{r}}{r}=f(r)\mathbf{\hat{r}}
  91. f ( r ) f(r)
  92. r r
  93. 𝐋 = 𝐫 × 𝐩 \mathbf{L}=\mathbf{r}\times\mathbf{p}
  94. d d t 𝐋 = 0 \frac{d}{dt}\mathbf{L}=0
  95. d d t ( 𝐩 × 𝐋 ) = d 𝐩 d t × 𝐋 = f ( r ) 𝐫 ^ × ( 𝐫 × m d 𝐫 d t ) = f ( r ) m r [ 𝐫 ( 𝐫 d 𝐫 d t ) - r 2 d 𝐫 d t ] \frac{d}{dt}\left(\mathbf{p}\times\mathbf{L}\right)=\frac{d\mathbf{p}}{dt}% \times\mathbf{L}=f(r)\mathbf{\hat{r}}\times\left(\mathbf{r}\times m\frac{d% \mathbf{r}}{dt}\right)=f(r)\frac{m}{r}\left[\mathbf{r}\left(\mathbf{r}\cdot% \frac{d\mathbf{r}}{dt}\right)-r^{2}\frac{d\mathbf{r}}{dt}\right]
  96. 𝐩 = m d 𝐫 d t \mathbf{p}=m\frac{d\mathbf{r}}{dt}
  97. 𝐫 × ( 𝐫 × d 𝐫 d t ) = 𝐫 ( 𝐫 d 𝐫 d t ) - r 2 d 𝐫 d t \mathbf{r}\times\left(\mathbf{r}\times\frac{d\mathbf{r}}{dt}\right)=\mathbf{r}% \left(\mathbf{r}\cdot\frac{d\mathbf{r}}{dt}\right)-r^{2}\frac{d\mathbf{r}}{dt}
  98. d d t ( 𝐫 𝐫 ) = 2 𝐫 d 𝐫 d t = d d t ( r 2 ) = 2 r d r d t \frac{d}{dt}\left(\mathbf{r}\cdot\mathbf{r}\right)=2\mathbf{r}\cdot\frac{d% \mathbf{r}}{dt}=\frac{d}{dt}\left(r^{2}\right)=2r\frac{dr}{dt}
  99. d d t ( 𝐩 × 𝐋 ) = - m f ( r ) r 2 [ 1 r d 𝐫 d t - 𝐫 r 2 d r d t ] = - m f ( r ) r 2 d d t ( 𝐫 r ) \frac{d}{dt}\left(\mathbf{p}\times\mathbf{L}\right)=-mf(r)r^{2}\left[\frac{1}{% r}\frac{d\mathbf{r}}{dt}-\frac{\mathbf{r}}{r^{2}}\frac{dr}{dt}\right]=-mf(r)r^% {2}\frac{d}{dt}\left(\frac{\mathbf{r}}{r}\right)
  100. f ( r ) = - k r 2 f(r)=\frac{-k}{r^{2}}
  101. d d t ( 𝐩 × 𝐋 ) = m k d d t ( 𝐫 r ) = d d t ( m k 𝐫 ^ ) \frac{d}{dt}\left(\mathbf{p}\times\mathbf{L}\right)=mk\frac{d}{dt}\left(\frac{% \mathbf{r}}{r}\right)=\frac{d}{dt}\left(mk\mathbf{\hat{r}}\right)
  102. d d t 𝐀 = d d t ( 𝐩 × 𝐋 ) - d d t ( m k 𝐫 ^ ) = 𝟎 \frac{d}{dt}\mathbf{A}=\frac{d}{dt}\left(\mathbf{p}\times\mathbf{L}\right)-% \frac{d}{dt}\left(mk\mathbf{\hat{r}}\right)=\mathbf{0}
  103. 𝐋 = m r 2 s y m b o l ω \mathbf{L}=mr^{2}symbol{\omega}
  104. 𝐋 \mathbf{L}
  105. 𝐩 × 𝐋 \mathbf{p}\times\mathbf{L}
  106. d d t 𝐩 × 𝐋 = ( - k r 2 𝐫 ^ ) × ( m r 2 s y m b o l ω ) = m k s y m b o l ω × 𝐫 ^ = m k d d t 𝐫 ^ \frac{d}{dt}\mathbf{p}\times\mathbf{L}=\left(\frac{-k}{r^{2}}\mathbf{\hat{r}}% \right)\times\left(mr^{2}symbol{\omega}\right)=mk\,symbol{\omega}\times\mathbf% {\hat{r}}=mk\,\frac{d}{dt}\mathbf{\hat{r}}
  107. s y m b o l ω × 𝐫 ^ symbol{\omega}\times\mathbf{\hat{r}}
  108. 𝒜 \mathcal{A}
  109. 𝒜 \mathcal{A}
  110. 𝒜 \mathcal{A}
  111. ξ = r + x \xi=r+x\,
  112. η = r - x \eta=r-x\,
  113. r = x 2 + y 2 r=\sqrt{x^{2}+y^{2}}
  114. x = 1 2 ( ξ - η ) x=\frac{1}{2}\left(\xi-\eta\right)
  115. y = ξ η y=\sqrt{\xi\eta}
  116. 2 ξ p ξ 2 - m k - m E ξ = - Γ 2\xi p_{\xi}^{2}-mk-mE\xi=-\Gamma
  117. 2 η p η 2 - m k - m E η = Γ 2\eta p_{\eta}^{2}-mk-mE\eta=\Gamma
  118. Γ = p y ( x p y - y p x ) - m k x r = A x \Gamma=p_{y}\left(xp_{y}-yp_{x}\right)-mk\frac{x}{r}=A_{x}
  119. δ q i = ϵ g i ( 𝐪 , 𝐪 ˙ , t ) \delta q_{i}=\epsilon g_{i}(\mathbf{q},\mathbf{\dot{q}},t)
  120. δ L = ϵ d d t G ( 𝐪 , t ) \delta L=\epsilon\frac{d}{dt}G(\mathbf{q},t)
  121. Γ = - G + i g i ( L q ˙ i ) \Gamma=-G+\sum_{i}g_{i}\left(\frac{\partial L}{\partial\dot{q}_{i}}\right)
  122. δ x i = ϵ 2 [ 2 p i x s - x i p s - δ i s ( 𝐫 𝐩 ) ] \delta x_{i}=\frac{\epsilon}{2}\left[2p_{i}x_{s}-x_{i}p_{s}-\delta_{is}\left(% \mathbf{r}\cdot\mathbf{p}\right)\right]
  123. δ L = ϵ m k d d t ( x s r ) \delta L=\epsilon mk\frac{d}{dt}\left(\frac{x_{s}}{r}\right)
  124. A s = [ p 2 x s - p s ( 𝐫 𝐩 ) ] - m k ( x s r ) = [ 𝐩 × ( 𝐫 × 𝐩 ) ] s - m k ( x s r ) A_{s}=\left[p^{2}x_{s}-p_{s}\ \left(\mathbf{r}\cdot\mathbf{p}\right)\right]-mk% \left(\frac{x_{s}}{r}\right)=\left[\mathbf{p}\times\left(\mathbf{r}\times% \mathbf{p}\right)\right]_{s}-mk\left(\frac{x_{s}}{r}\right)
  125. t λ 3 t , 𝐫 λ 2 𝐫 , 𝐩 1 λ 𝐩 . t\rightarrow\lambda^{3}t,\qquad\mathbf{r}\rightarrow\lambda^{2}\mathbf{r},% \qquad\mathbf{p}\rightarrow\frac{1}{\lambda}\mathbf{p}~{}.
  126. L λ L , E 1 λ 2 E , L\rightarrow\lambda L,\qquad E\rightarrow\frac{1}{\lambda^{2}}E~{},
  127. A 2 = m 2 k 2 e 2 = m 2 k 2 + 2 m E L 2 A^{2}=m^{2}k^{2}e^{2}=m^{2}k^{2}+2mEL^{2}
  128. 𝐞 = 1 m k ( 𝐩 × 𝐋 ) - 𝐫 ^ = m k ( 𝐯 × ( 𝐫 × 𝐯 ) ) - 𝐫 ^ \mathbf{e}=\frac{1}{mk}\left(\mathbf{p}\times\mathbf{L}\right)-\mathbf{\hat{r}% }=\frac{m}{k}\left(\mathbf{v}\times\left(\mathbf{r}\times\mathbf{v}\right)% \right)-\mathbf{\hat{r}}
  129. 𝐌 = 𝐯 × 𝐋 - k 𝐫 ^ \mathbf{M}=\mathbf{v}\times\mathbf{L}-k\mathbf{\hat{r}}
  130. 𝐃 = 𝐀 p 0 = 1 2 m | E | { 𝐩 × 𝐋 - m k 𝐫 ^ } \mathbf{D}=\frac{\mathbf{A}}{p_{0}}=\frac{1}{\sqrt{2m\left|E\right|}}\left\{% \mathbf{p}\times\mathbf{L}-mk\mathbf{\hat{r}}\right\}
  131. 𝐁 = 𝐩 - ( m k L 2 r ) ( 𝐋 × 𝐫 ) \mathbf{B}=\mathbf{p}-\left(\frac{mk}{L^{2}r}\right)\ \left(\mathbf{L}\times% \mathbf{r}\right)
  132. 𝐖 = α 𝐀 𝐀 + β 𝐁 𝐁 . \mathbf{W}=\alpha\mathbf{A}\otimes\mathbf{A}+\beta\,\mathbf{B}\otimes\mathbf{B% }~{}.
  133. \otimes
  134. W i j = α A i A j + β B i B j . W_{ij}=\alpha A_{i}A_{j}+\beta B_{i}B_{j}\,.
  135. 𝐋 𝐖 = α ( 𝐋 𝐀 ) 𝐀 + β ( 𝐋 𝐁 ) 𝐁 = 0 , \mathbf{L}\cdot\mathbf{W}=\alpha\left(\mathbf{L}\cdot\mathbf{A}\right)\mathbf{% A}+\beta\left(\mathbf{L}\cdot\mathbf{B}\right)\mathbf{B}=0~{},
  136. ( 𝐋 𝐖 ) j = α ( i = 1 3 L i A i ) A j + β ( i = 1 3 L i B i ) B j = 0 . \left(\mathbf{L}\cdot\mathbf{W}\right)_{j}=\alpha\left(\sum_{i=1}^{3}L_{i}A_{i% }\right)A_{j}+\beta\left(\sum_{i=1}^{3}L_{i}B_{i}\right)B_{j}=0~{}.

Large_diffeomorphism.html

  1. a , b a,b

Large_eddy_simulation.html

  1. ϕ ( s y m b o l x , t ) \phi(symbol{x},t)
  2. ϕ ( s y m b o l x , t ) ¯ = - - ϕ ( s y m b o l r , t ) G ( s y m b o l x - s y m b o l r , t - t ) d t d s y m b o l r \overline{\phi(symbol{x},t)}=\displaystyle{\int_{-\infty}^{\infty}}\int_{-% \infty}^{\infty}\phi(symbol{r},t^{\prime})G(symbol{x}-symbol{r},t-t^{\prime})% dt^{\prime}dsymbol{r}
  3. G G
  4. ϕ ¯ = G ϕ . \overline{\phi}=G\star\phi.
  5. G G
  6. Δ \Delta
  7. τ c \tau_{c}
  8. ϕ ¯ \overline{\phi}
  9. ϕ \phi
  10. ϕ = ϕ ¯ + ϕ . \phi=\bar{\phi}+\phi^{\prime}.
  11. ρ s y m b o l u ( s y m b o l x , t ) \rho symbol{u}(symbol{x},t)
  12. u i ¯ x i = 0 \frac{\partial\bar{u_{i}}}{\partial x_{i}}=0
  13. u i ¯ t + x j ( u i u j ¯ ) = - 1 ρ p ¯ x i + ν x j ( u i ¯ x j + u j ¯ x i ) = - 1 ρ p ¯ x i + 2 ν x j S i j , \frac{\partial\bar{u_{i}}}{\partial t}+\frac{\partial}{\partial x_{j}}\left(% \overline{u_{i}u_{j}}\right)=-\frac{1}{\rho}\frac{\partial\overline{p}}{% \partial x_{i}}+\nu\frac{\partial}{\partial x_{j}}\left(\frac{\partial\bar{u_{% i}}}{\partial x_{j}}+\frac{\partial\bar{u_{j}}}{\partial x_{i}}\right)=-\frac{% 1}{\rho}\frac{\partial\overline{p}}{\partial x_{i}}+2\nu\frac{\partial}{% \partial x_{j}}S_{ij},
  14. p ¯ \bar{p}
  15. S i j S_{ij}
  16. u i u j ¯ \overline{u_{i}u_{j}}
  17. u i u j ¯ = τ i j r + u ¯ i u ¯ j \overline{u_{i}u_{j}}=\tau_{ij}^{r}+\overline{u}_{i}\overline{u}_{j}
  18. τ i j r \tau_{ij}^{r}
  19. u i ¯ t + x j ( u ¯ i u ¯ j ) = - 1 ρ p ¯ x i + 2 ν x j S ¯ i j - τ i j r x j \frac{\partial\bar{u_{i}}}{\partial t}+\frac{\partial}{\partial x_{j}}\left(% \overline{u}_{i}\overline{u}_{j}\right)=-\frac{1}{\rho}\frac{\partial\overline% {p}}{\partial x_{i}}+2\nu\frac{\partial}{\partial x_{j}}\bar{S}_{ij}-\frac{% \partial\tau_{ij}^{r}}{\partial x_{j}}
  20. τ i j r \tau_{ij}^{r}
  21. τ i j r = L i j + C i j + R i j \tau_{ij}^{r}=L_{ij}+C_{ij}+R_{ij}
  22. L i j L_{ij}
  23. R i j R_{ij}
  24. C i j C_{ij}
  25. τ i j r \tau_{ij}^{r}
  26. τ i j r \tau_{ij}^{r}
  27. ϕ \phi
  28. ϕ ¯ t + x j ( u ¯ j ϕ ¯ ) = J ϕ ¯ x j + q i j x j \frac{\partial\overline{\phi}}{\partial t}+\frac{\partial}{\partial x_{j}}% \left(\overline{u}_{j}\overline{\phi}\right)=\frac{\partial\overline{J_{\phi}}% }{\partial x_{j}}+\frac{\partial q_{ij}}{\partial x_{j}}
  29. J ϕ J_{\phi}
  30. ϕ \phi
  31. q i j q_{ij}
  32. ϕ \phi
  33. J ϕ ¯ \overline{J_{\phi}}
  34. J ϕ = D ϕ ϕ x i J_{\phi}=D_{\phi}\frac{\partial\phi}{\partial x_{i}}
  35. q i j q_{ij}
  36. τ i j r \tau_{ij}^{r}
  37. q i j = ϕ ¯ u ¯ j - ϕ u j ¯ q_{ij}=\bar{\phi}\overline{u}_{j}-\overline{\phi u_{j}}
  38. ρ ¯ t + u i ρ ¯ x i = 0 \frac{\partial\overline{\rho}}{\partial t}+\frac{\partial\overline{u_{i}\rho}}% {\partial x_{i}}=0
  39. ϕ \phi
  40. ϕ ~ = ρ ϕ ¯ ρ ¯ \tilde{\phi}=\frac{\overline{\rho\phi}}{\overline{\rho}}
  41. ρ ¯ t + ρ ¯ u i ~ x i = 0. \frac{\partial\overline{\rho}}{\partial t}+\frac{\partial\overline{\rho}\tilde% {u_{i}}}{\partial x_{i}}=0.
  42. ρ ¯ u i ~ t + ρ ¯ u i ~ u j ~ x j + p ¯ x i - σ i j ¯ x j = - ρ ¯ τ i j r x j + x j ( σ ¯ i j - σ ~ i j ) \frac{\partial\overline{\rho}\tilde{u_{i}}}{\partial t}+\frac{\partial% \overline{\rho}\tilde{u_{i}}\tilde{u_{j}}}{\partial x_{j}}+\frac{\partial% \overline{p}}{\partial x_{i}}-\frac{\partial\overline{\sigma_{ij}}}{\partial x% _{j}}=-\frac{\partial\overline{\rho}\tau_{ij}^{r}}{\partial x_{j}}+\frac{% \partial}{\partial x_{j}}\left(\overline{\sigma}_{ij}-\tilde{\sigma}_{ij}\right)
  43. σ i j \sigma_{ij}
  44. σ i j = 2 μ ( T ) S i j - 2 3 μ ( T ) δ i j S k k \sigma_{ij}=2\mu(T)S_{ij}-\frac{2}{3}\mu(T)\delta_{ij}S_{kk}
  45. x j ( σ ¯ i j - σ ~ i j ) \frac{\partial}{\partial x_{j}}\left(\overline{\sigma}_{ij}-\tilde{\sigma}_{ij% }\right)
  46. μ ( T ) \mu(T)
  47. T ~ \tilde{T}
  48. τ i j r = u i u j ~ - u i ~ u j ~ \tau_{ij}^{r}=\widetilde{u_{i}\cdot u_{j}}-\tilde{u_{i}}\tilde{u_{j}}
  49. ρ ϕ ψ ¯ \overline{\rho\phi\psi}
  50. ρ ¯ ϕ ψ ~ \overline{\rho}\widetilde{\phi\psi}
  51. ϕ \phi
  52. ψ \psi
  53. ϕ ~ \tilde{\phi}
  54. ψ ~ \tilde{\psi}
  55. u i u j ¯ \overline{u_{i}u_{j}}
  56. ρ ¯ ( ϕ ψ ~ - ϕ ~ ψ ~ ) \overline{\rho}\left(\widetilde{\phi\psi}-\tilde{\phi}\tilde{\psi}\right)
  57. L i j L_{ij}
  58. C i j C_{ij}
  59. R i j R_{ij}
  60. E ¯ = 1 2 u i u i ¯ \overline{E}=\frac{1}{2}\overline{u_{i}u_{i}}
  61. E f E_{f}
  62. E f = 1 2 u i ¯ u i ¯ E_{f}=\frac{1}{2}\overline{u_{i}}\,\overline{u_{i}}
  63. k r k_{r}
  64. k r = 1 2 u i u i ¯ - 1 2 u i ¯ u i ¯ = 1 2 τ i i r k_{r}=\frac{1}{2}\overline{u_{i}u_{i}}-\frac{1}{2}\overline{u_{i}}\,\overline{% u_{i}}=\frac{1}{2}\tau_{ii}^{r}
  65. E ¯ = E f + k r \overline{E}=E_{f}+k_{r}
  66. E f E_{f}
  67. u i ¯ \overline{u_{i}}
  68. E f t + u j ¯ E f x j + 1 ρ u i ¯ p ¯ x i + u i ¯ τ i j r x j - 2 ν u i ¯ S i j ¯ x j = - ϵ f - Π \frac{\partial E_{f}}{\partial t}+\overline{u_{j}}\frac{\partial E_{f}}{% \partial x_{j}}+\frac{1}{\rho}\frac{\partial\overline{u_{i}}\bar{p}}{\partial x% _{i}}+\frac{\partial\overline{u_{i}}\tau_{ij}^{r}}{\partial x_{j}}-2\nu\frac{% \partial\overline{u_{i}}\bar{S_{ij}}}{\partial x_{j}}=-\epsilon_{f}-\Pi
  69. ϵ f = 2 ν S i j ¯ S i j ¯ \epsilon_{f}=2\nu\bar{S_{ij}}\bar{S_{ij}}
  70. Π = - τ i j r S i j ¯ \Pi=-\tau_{ij}^{r}\bar{S_{ij}}
  71. Π \Pi
  72. Π \Pi
  73. Π \Pi
  74. E f E_{f}
  75. L L
  76. Δ \Delta
  77. Δ x \Delta x
  78. u ¯ ( s y m b o l x ) \overline{u}(symbol{x})
  79. Δ \Delta
  80. Δ x \Delta x
  81. ( Δ x ) 4 (\Delta x)^{4}
  82. k c k_{c}
  83. Δ \Delta
  84. τ i j \tau_{ij}
  85. τ i j r - 1 3 τ i j δ i j = - 2 ν t S ¯ i j \tau_{ij}^{r}-\frac{1}{3}\tau_{ij}\delta_{ij}=-2\nu_{\mathrm{t}}\bar{S}_{ij}
  86. ν t \nu_{\mathrm{t}}
  87. S ¯ i j = 1 2 ( u ¯ i x j + u ¯ j x i ) \bar{S}_{ij}=\frac{1}{2}\left(\frac{\partial\bar{u}_{i}}{\partial x_{j}}+\frac% {\partial\bar{u}_{j}}{\partial x_{i}}\right)
  88. [ ν t ] = m 2 s \left[\nu_{\mathrm{t}}\right]=\frac{\mathrm{m^{2}}}{\mathrm{s}}
  89. ν t = ( C s Δ g ) 2 2 S ¯ i j S ¯ i j = ( C s Δ g ) 2 | S | \nu_{\mathrm{t}}=(C_{s}\Delta_{g})^{2}\sqrt{2\bar{S}_{ij}\bar{S}_{ij}}=(C_{s}% \Delta_{g})^{2}\left|S\right|
  90. Δ g \Delta_{g}
  91. C s C_{s}
  92. ϵ = Π \epsilon=\Pi
  93. C s C_{s}
  94. ¯ \overline{\cdot}
  95. ^ \hat{\cdot}
  96. i j \mathcal{L}_{ij}
  97. i j = T i j r - τ ^ i j r \mathcal{L}_{ij}=T_{ij}^{r}-\hat{\tau}_{ij}^{r}
  98. T i j r = u i u j ¯ ^ - u ¯ ^ i u ¯ ^ j T_{ij}^{r}=\widehat{\overline{u_{i}u_{j}}}-\hat{\bar{u}}_{i}\hat{\bar{u}}_{j}
  99. τ ^ i j r = u i u j ¯ ^ - u ¯ i u ¯ j ^ \hat{\tau}_{ij}^{r}=\widehat{\overline{u_{i}u_{j}}}-\widehat{\overline{u}_{i}% \overline{u}_{j}}
  100. i j \mathcal{L}_{ij}
  101. Δ ^ \hat{\Delta}
  102. Δ ¯ \overline{\Delta}
  103. C s C_{s}
  104. C s 2 = i j i j i j i j C_{s}^{2}=\frac{\mathcal{L}_{ij}\mathcal{M}_{ij}}{\mathcal{M}_{ij}\mathcal{M}_% {ij}}
  105. i j = 2 Δ ¯ 2 ( | S ^ | S ^ i j ¯ - α 2 | S ^ ¯ | S ^ ¯ i j ) \mathcal{M}_{ij}=2\overline{\Delta}^{2}\left(\overline{\left|\hat{S}\right|% \hat{S}_{ij}}-\alpha^{2}\left|\overline{\hat{S}}\right|\overline{\hat{S}}_{ij}\right)
  106. α = Δ ^ / Δ ¯ . \alpha=\hat{\Delta}/\overline{\Delta}.
  107. C s C_{s}
  108. C s 2 = i j i j i j i j C_{s}^{2}=\frac{\left\langle\mathcal{L}_{ij}\mathcal{M}_{ij}\right\rangle}{% \left\langle\mathcal{M}_{ij}\mathcal{M}_{ij}\right\rangle}
  109. C s C_{s}
  110. u i x i = 0 \frac{\partial u_{i}}{\partial x_{i}}=0
  111. u i t + u i u j x j = - 1 ρ p x i + ν 2 u i x j x j . \frac{\partial u_{i}}{\partial t}+\frac{\partial u_{i}u_{j}}{\partial x_{j}}=-% \frac{1}{\rho}\frac{\partial p}{\partial x_{i}}+\nu\frac{\partial^{2}u_{i}}{% \partial x_{j}\partial x_{j}}.
  112. u i t ¯ + u i u j x j ¯ = - 1 ρ p x i ¯ + ν 2 u i x j x j ¯ . \overline{\frac{\partial u_{i}}{\partial t}}+\overline{\frac{\partial u_{i}u_{% j}}{\partial x_{j}}}=-\overline{\frac{1}{\rho}\frac{\partial p}{\partial x_{i}% }}+\overline{\nu\frac{\partial^{2}u_{i}}{\partial x_{j}\partial x_{j}}}.
  113. u i ¯ t + u i u j x j ¯ = - 1 ρ p ¯ x i + ν 2 u i ¯ x j x j . \frac{\partial\bar{u_{i}}}{\partial t}+\overline{\frac{\partial u_{i}u_{j}}{% \partial x_{j}}}=-\frac{1}{\rho}\frac{\partial\bar{p}}{\partial x_{i}}+\nu% \frac{\partial^{2}\bar{u_{i}}}{\partial x_{j}\partial x_{j}}.
  114. u i ¯ \bar{u_{i}}
  115. u i u_{i}
  116. u i u j x j ¯ \overline{\frac{\partial u_{i}u_{j}}{\partial x_{j}}}
  117. u i ¯ u j ¯ x j \frac{\partial\bar{u_{i}}\bar{u_{j}}}{\partial x_{j}}
  118. u i ¯ t + u i ¯ u j ¯ x j = - 1 ρ p ¯ x i + ν 2 u i ¯ x j x j - ( u i u j x j ¯ - u i ¯ u j ¯ x j ) . \frac{\partial\bar{u_{i}}}{\partial t}+\frac{\partial\bar{u_{i}}\bar{u_{j}}}{% \partial x_{j}}=-\frac{1}{\rho}\frac{\partial\bar{p}}{\partial x_{i}}+\nu\frac% {\partial^{2}\bar{u_{i}}}{\partial x_{j}\partial x_{j}}-\left(\overline{\frac{% \partial u_{i}u_{j}}{\partial x_{j}}}-\frac{\partial\bar{u_{i}}\bar{u_{j}}}{% \partial x_{j}}\right).
  119. τ i j = u i u j ¯ - u i ¯ u j ¯ \tau_{ij}=\overline{u_{i}u_{j}}-\bar{u_{i}}\bar{u_{j}}
  120. u i ¯ t + u j ¯ u i ¯ x j = - 1 ρ p ¯ x i + ν 2 u i ¯ x j x j - τ i j x j . \frac{\partial\bar{u_{i}}}{\partial t}+\bar{u_{j}}\frac{\partial\bar{u_{i}}}{% \partial x_{j}}=-\frac{1}{\rho}\frac{\partial\bar{p}}{\partial x_{i}}+\nu\frac% {\partial^{2}\bar{u_{i}}}{\partial x_{j}\partial x_{j}}-\frac{\partial\tau_{ij% }}{\partial x_{j}}.

Latent_semantic_analysis.html

  1. X X
  2. ( i , j ) (i,j)
  3. i i
  4. j j
  5. X X
  6. 𝐝 j 𝐭 i T [ x 1 , 1 x 1 , n x m , 1 x m , n ] \begin{matrix}&\,\textbf{d}_{j}\\ &\downarrow\\ \,\textbf{t}_{i}^{T}\rightarrow&\begin{bmatrix}x_{1,1}&\dots&x_{1,n}\\ \vdots&\ddots&\vdots\\ x_{m,1}&\dots&x_{m,n}\\ \end{bmatrix}\end{matrix}
  7. 𝐭 i T = [ x i , 1 x i , n ] \,\textbf{t}_{i}^{T}=\begin{bmatrix}x_{i,1}&\dots&x_{i,n}\end{bmatrix}
  8. 𝐝 j = [ x 1 , j x m , j ] \,\textbf{d}_{j}=\begin{bmatrix}x_{1,j}\\ \vdots\\ x_{m,j}\end{bmatrix}
  9. 𝐭 i T 𝐭 p \,\textbf{t}_{i}^{T}\,\textbf{t}_{p}
  10. X X T XX^{T}
  11. ( i , p ) (i,p)
  12. ( p , i ) (p,i)
  13. 𝐭 i T 𝐭 p \,\textbf{t}_{i}^{T}\,\textbf{t}_{p}
  14. = 𝐭 p T 𝐭 i =\,\textbf{t}_{p}^{T}\,\textbf{t}_{i}
  15. X T X X^{T}X
  16. 𝐝 j T 𝐝 q = 𝐝 q T 𝐝 j \,\textbf{d}_{j}^{T}\,\textbf{d}_{q}=\,\textbf{d}_{q}^{T}\,\textbf{d}_{j}
  17. X X
  18. U U
  19. V V
  20. Σ \Sigma
  21. X = U Σ V T \begin{matrix}X=U\Sigma V^{T}\end{matrix}
  22. X X T = ( U Σ V T ) ( U Σ V T ) T = ( U Σ V T ) ( V T T Σ T U T ) = U Σ V T V Σ T U T = U Σ Σ T U T X T X = ( U Σ V T ) T ( U Σ V T ) = ( V T T Σ T U T ) ( U Σ V T ) = V Σ T U T U Σ V T = V Σ T Σ V T \begin{matrix}XX^{T}&=&(U\Sigma V^{T})(U\Sigma V^{T})^{T}=(U\Sigma V^{T})(V^{T% ^{T}}\Sigma^{T}U^{T})=U\Sigma V^{T}V\Sigma^{T}U^{T}=U\Sigma\Sigma^{T}U^{T}\\ X^{T}X&=&(U\Sigma V^{T})^{T}(U\Sigma V^{T})=(V^{T^{T}}\Sigma^{T}U^{T})(U\Sigma V% ^{T})=V\Sigma^{T}U^{T}U\Sigma V^{T}=V\Sigma^{T}\Sigma V^{T}\end{matrix}
  23. Σ Σ T \Sigma\Sigma^{T}
  24. Σ T Σ \Sigma^{T}\Sigma
  25. U U
  26. X X T XX^{T}
  27. V V
  28. X T X X^{T}X
  29. Σ Σ T \Sigma\Sigma^{T}
  30. Σ T Σ \Sigma^{T}\Sigma
  31. X U Σ V T ( 𝐝 j ) ( 𝐝 ^ j ) ( 𝐭 i T ) [ x 1 , 1 x 1 , n x m , 1 x m , n ] = ( 𝐭 ^ i T ) [ [ 𝐮 1 ] [ 𝐮 l ] ] [ σ 1 0 0 σ l ] [ [ 𝐯 1 ] [ 𝐯 l ] ] \begin{matrix}&X&&&U&&\Sigma&&V^{T}\\ &(\,\textbf{d}_{j})&&&&&&&(\hat{\,\textbf{d}}_{j})\\ &\downarrow&&&&&&&\downarrow\\ (\,\textbf{t}_{i}^{T})\rightarrow&\begin{bmatrix}x_{1,1}&\dots&x_{1,n}\\ \\ \vdots&\ddots&\vdots\\ \\ x_{m,1}&\dots&x_{m,n}\\ \end{bmatrix}&=&(\hat{\,\textbf{t}}_{i}^{T})\rightarrow&\begin{bmatrix}\begin{% bmatrix}\\ \\ \,\textbf{u}_{1}\\ \\ \end{bmatrix}\dots\begin{bmatrix}\\ \\ \,\textbf{u}_{l}\\ \\ \end{bmatrix}\end{bmatrix}&\cdot&\begin{bmatrix}\sigma_{1}&\dots&0\\ \vdots&\ddots&\vdots\\ 0&\dots&\sigma_{l}\\ \end{bmatrix}&\cdot&\begin{bmatrix}\begin{bmatrix}&&\,\textbf{v}_{1}&&\end{% bmatrix}\\ \vdots\\ \begin{bmatrix}&&\,\textbf{v}_{l}&&\end{bmatrix}\end{bmatrix}\end{matrix}
  32. σ 1 , , σ l \sigma_{1},\dots,\sigma_{l}
  33. u 1 , , u l u_{1},\dots,u_{l}
  34. v 1 , , v l v_{1},\dots,v_{l}
  35. U U
  36. 𝐭 i \,\textbf{t}_{i}
  37. i ’th i\textrm{'th}
  38. t ^ i \hat{\textrm{t}}_{i}
  39. V T V^{T}
  40. 𝐝 j \,\textbf{d}_{j}
  41. j ’th j\textrm{'th}
  42. d ^ j \hat{\textrm{d}}_{j}
  43. k k
  44. U U
  45. V V
  46. k k
  47. X X
  48. 𝐭 ^ i \hat{\,\textbf{t}}_{i}
  49. k k
  50. 𝐝 ^ j \hat{\,\textbf{d}}_{j}
  51. X k = U k Σ k V k T X_{k}=U_{k}\Sigma_{k}V_{k}^{T}
  52. j j
  53. q q
  54. Σ k 𝐝 ^ j \Sigma_{k}\hat{\,\textbf{d}}_{j}
  55. Σ k 𝐝 ^ q \Sigma_{k}\hat{\,\textbf{d}}_{q}
  56. i i
  57. p p
  58. Σ k 𝐭 ^ i \Sigma_{k}\hat{\,\textbf{t}}_{i}
  59. Σ k 𝐭 ^ p \Sigma_{k}\hat{\,\textbf{t}}_{p}
  60. 𝐝 ^ j = Σ k - 1 U k T 𝐝 j \hat{\,\textbf{d}}_{j}=\Sigma_{k}^{-1}U_{k}^{T}\,\textbf{d}_{j}
  61. Σ k \Sigma_{k}
  62. q q
  63. 𝐪 ^ = Σ k - 1 U k T 𝐪 \hat{\,\textbf{q}}=\Sigma_{k}^{-1}U_{k}^{T}\,\textbf{q}
  64. 𝐭 i T = 𝐭 ^ i T Σ k V k T \,\textbf{t}_{i}^{T}=\hat{\,\textbf{t}}_{i}^{T}\Sigma_{k}V_{k}^{T}
  65. 𝐭 ^ i T = 𝐭 i T V k - T Σ k - 1 = 𝐭 i T V k Σ k - 1 \hat{\,\textbf{t}}_{i}^{T}=\,\textbf{t}_{i}^{T}V_{k}^{-T}\Sigma_{k}^{-1}=\,% \textbf{t}_{i}^{T}V_{k}\Sigma_{k}^{-1}
  66. 𝐭 ^ i = Σ k - 1 V k T 𝐭 i \hat{\,\textbf{t}}_{i}=\Sigma_{k}^{-1}V_{k}^{T}\,\textbf{t}_{i}

Latin_hypercube_sampling.html

  1. N N
  2. M M
  3. M M
  4. M M
  5. M M
  6. N N
  7. ( n = 0 M - 1 ( M - n ) ) N - 1 = ( M ! ) N - 1 \left(\prod_{n=0}^{M-1}(M-n)\right)^{N-1}=(M!)^{N-1}
  8. M = 4 M=4
  9. N = 2 N=2
  10. M = 4 M=4
  11. N = 3 N=3

Lattice_gauge_theory.html

  1. a a
  2. a 0 a\to 0
  3. S = F - { χ ( ρ ) ( U ( e 1 ) U ( e n ) ) } . S=\sum_{F}-\Re\{\chi^{(\rho)}(U(e_{1})\cdots U(e_{n}))\}.
  4. a a
  5. a 2 a^{2}
  6. e - β S e^{-\beta S}
  7. S S
  8. β \beta
  9. a a
  10. a a
  11. a 0 a\to 0

Laver_table.html

  1. L n ( p , q ) := p q L_{n}(p,q):=p\star q
  2. p 1 := p + 1 mod 2 n p\star 1:=p+1\mod 2^{n}
  3. p ( q r ) := ( p q ) ( p r ) p\star(q\star r):=(p\star q)\star(p\star r)

Laws_of_Form.html

  1. = \ =
  2. = \ =
  3. J 1 : ( ( A ) A ) = . \ J1:((A)A)=.
  4. J 2 : ( ( A ) ( B ) ) C = ( ( A C ) ( B C ) ) . \ J2:((A)(B))C=((AC)(BC)).
  5. J 0 : ( ( ) ) A = A . \ J0:(())A=A.
  6. J 1 a : ( A ) A = ( ) \ J1a:(A)A=()
  7. C 2 : A ( A B ) = A ( B ) . \ C2:A(AB)=A(B).
  8. B = { \ B=\{
  9. , ,
  10. } \ \}
  11. B , - - , ( - ) , ( ) \langle B,--,(-),()\rangle
  12. 2 , 1 , 0 \langle 2,1,0\rangle
  13. B , ( - - ) , ( ) \langle B,(--),()\rangle
  14. 2 , 0 \langle 2,0\rangle
  15. B , + , ¬ , 1 \langle B,+,\lnot,1\rangle
  16. 2 , 1 , 0 \langle 2,1,0\rangle
  17. B , + , × , ¬ , 1 , 0 \langle B,+,\times,\lnot,1,0\rangle
  18. 2 , 2 , 1 , 0 , 0 \langle 2,2,1,0,0\rangle
  19. ( ( ( A ) B ) ( A ( B ) ) ) , ( ( A ) ( B ) ) ( A B ) \ (((A)B)(A(B))),((A)(B))(AB)
  20. \square

Laws_of_thermodynamics.html

  1. Δ U s y s t e m = Q - W \Delta U_{system}=Q-W
  2. Δ U s y s t e m ( f u l l c y c l e ) = 0 \Delta U_{system\,(full\,cycle)}=0
  3. Q = Q i n - Q o u t = W Q=Q_{in}-Q_{out}=W
  4. Q = 0 Q=0
  5. Δ U s y s t e m = U f i n a l - U i n i t i a l = - W \Delta U_{system}=U_{final}-U_{initial}=-W
  6. E t o t a l = KE s y s t e m + PE s y s t e m + U s y s t e m E_{total}=\mathrm{KE}_{system}+\mathrm{PE}_{system}+U_{system}
  7. Δ U s y s t e m = Q \Delta U_{system}=Q
  8. Q Q
  9. - W = Δ PE s y s t e m -W=\Delta\mathrm{PE}_{system}
  10. - W = Δ KE s y s t e m + Δ PE s y s t e m + Δ U s y s t e m -W=\Delta\mathrm{KE}_{system}+\Delta\mathrm{PE}_{system}+\Delta U_{system}
  11. ( u e x t e r n a l Δ M ) i n = Δ U s y s t e m \left(u_{external}\,\,\Delta M\right)_{in}=\Delta U_{system}
  12. Δ M ΔM
  13. δ Q = T d S . \delta Q=T\,dS\,.
  14. S = k B ln Ω S=k_{\mathrm{B}}\,\mathrm{ln}\,\Omega

Lawson_criterion.html

  1. τ E \tau_{E}
  2. P B = 1.4 10 - 34 N 2 T 1 / 2 W cm 3 P_{B}=1.4\cdot 10^{-34}\cdot N^{2}\cdot T^{1/2}\frac{\mathrm{W}}{\mathrm{cm}^{% 3}}
  3. D 1 2 + 1 3 T 2 4 He ( 3.5 MeV ) + 0 1 n ( 14.1 MeV ) {}^{2}_{1}\mathrm{D}+\,^{3}_{1}\mathrm{T}\rightarrow\,^{4}_{2}\mathrm{He}\left% (3.5\,\mathrm{MeV}\right)+\,^{1}_{0}\mathrm{n}\left(14.1\,\mathrm{MeV}\right)
  4. D 1 2 + 1 2 D 1 3 T ( 1.0 MeV ) + 1 1 p ( 3.0 MeV ) {}^{2}_{1}\mathrm{D}+\,^{2}_{1}\mathrm{D}\rightarrow\,^{3}_{1}\mathrm{T}\left(% 1.0\,\mathrm{MeV}\right)+\,^{1}_{1}\mathrm{p}\left(3.0\,\mathrm{MeV}\right)
  5. τ E \tau_{E}
  6. W W
  7. P loss P_{\mathrm{loss}}
  8. τ E = W P loss \tau_{E}=\frac{W}{P_{\mathrm{loss}}}
  9. W = 3 n k B T W=3nk_{\mathrm{B}}T
  10. k B k_{\mathrm{B}}
  11. n n
  12. f f
  13. f = n d n t σ v = 1 4 n 2 σ v f=n_{\mathrm{d}}n_{\mathrm{t}}\langle\sigma v\rangle=\frac{1}{4}n^{2}\langle% \sigma v\rangle
  14. σ \sigma
  15. v v
  16. \langle\rangle
  17. T T
  18. f f
  19. E ch E_{\mathrm{ch}}
  20. E ch = 3.5 MeV E_{\mathrm{ch}}=3.5\,\mathrm{MeV}
  21. f E ch P loss fE_{\rm ch}\geq P_{\rm loss}
  22. 1 4 n 2 σ v E ch 3 n k B T τ E \frac{1}{4}n^{2}\langle\sigma v\rangle E_{\rm ch}\geq\frac{3nk_{\rm B}T}{\tau_% {E}}
  23. n τ E L 12 E ch k B T σ v n\tau_{\rm E}\geq L\equiv\frac{12}{E_{\rm ch}}\,\frac{k_{\rm B}T}{\langle% \sigma v\rangle}
  24. T / σ v T/\langle\sigma v\rangle
  25. n τ E n\tau_{E}
  26. n τ E 1.5 10 20 s m 3 n\tau_{E}\geq 1.5\cdot 10^{20}\frac{\mathrm{s}}{\mathrm{m}^{3}}
  27. T = 25 keV T=25\,\mathrm{keV}
  28. n T τ E 12 k B E ch T 2 σ v nT\tau_{\rm E}\geq\frac{12k_{\rm B}}{E_{\rm ch}}\,\frac{T^{2}}{\langle\sigma v\rangle}
  29. T 2 σ v \frac{T^{2}}{\langle\sigma v\rangle}
  30. T σ v \frac{T}{\langle\sigma v\rangle}
  31. σ v = 1.1 10 - 24 m 3 s [ T in keV ] 2 , \left\langle\sigma v\right\rangle=1.1\cdot 10^{-24}\frac{{\rm m}^{3}}{\rm s}% \left[T{\rm\,in\,keV}\right]^{2}\,{\rm,}
  32. n T τ E 12 14 2 keV 2 1.1 10 - 24 m 3 s 14 2 3500 keV 3 10 21 keV s / m 3 \begin{matrix}nT\tau_{E}&\geq&\frac{12\cdot 14^{2}\cdot{\rm keV}^{2}}{1.1\cdot 1% 0^{-24}\frac{{\rm m}^{3}}{\rm s}14^{2}\cdot 3500\cdot{\rm keV}}\approx 3\cdot 1% 0^{21}\mbox{keV s}~{}/\mbox{m}~{}^{3}\\ \end{matrix}
  33. n T τ E n T ( n 1 / 3 / P 2 / 3 ) n T ( n 1 / 3 / ( n 2 T 2 ) 2 / 3 ) T - 1 / 3 \begin{matrix}nT\tau_{E}&\propto&nT\left(n^{1/3}/P^{2/3}\right)\\ &\propto&nT\left(n^{1/3}/\left(n^{2}T^{2}\right)^{2/3}\right)\\ &\propto&T^{-1/3}\\ \end{matrix}
  34. τ E \tau_{E}
  35. v t h = k B T m i v_{th}=\sqrt{\frac{k_{\rm B}T}{m_{i}}}
  36. τ E \tau_{E}
  37. τ E R v t h = R k B T m i = R m i k B T . \begin{matrix}\tau_{E}&\approx&\frac{R}{v_{th}}\\ \\ &=&\frac{R}{\sqrt{\frac{k_{\rm B}T}{m_{i}}}}\\ \\ &=&R\cdot\sqrt{\frac{m_{i}}{k_{\rm B}T}}\mbox{ .}\\ \end{matrix}
  38. n τ E n R m i k B T 12 E ch k B T σ v n R 12 E ch ( k B T ) 3 / 2 σ v m i 1 / 2 n R ( k B T ) 3 / 2 σ v . \begin{matrix}n\tau_{E}&\approx&n\cdot R\cdot\sqrt{\frac{m_{i}}{k_{B}T}}\geq% \frac{12}{E_{\rm ch}}\,\frac{k_{\rm B}T}{\langle\sigma v\rangle}\\ \\ n\cdot R&\gtrapprox&\frac{12}{E_{\rm ch}}\,\frac{\left(k_{\rm B}T\right)^{3/2}% }{\langle\sigma v\rangle\cdot m_{i}^{1/2}}\\ \\ n\cdot R&\gtrapprox&\frac{\left(k_{\rm B}T\right)^{3/2}}{\langle\sigma v% \rangle}\mbox{ .}\\ \end{matrix}
  39. ρ R 1 g / cm 2 \rho\cdot R\geq 1\mathrm{g}/\mathrm{cm}^{2}
  40. burn-up fraction n 2 σ v T - 1 / 2 / n ( n T ) σ v / T 3 / 2 \begin{matrix}\mbox{burn-up fraction }&\propto&n^{2}\langle\sigma v\rangle T^{% -1/2}/n\\ &\propto&\left(nT\right)\langle\sigma v\rangle/T^{3/2}\\ \end{matrix}
  41. n n
  42. n n
  43. p p
  44. n = p / 2 T i n=p/2T_{\mathrm{i}}
  45. 2 2
  46. n n
  47. p p
  48. n 1 , 2 n_{1,2}
  49. Z 1 , 2 Z_{1,2}
  50. T i T_{\mathrm{i}}
  51. T e T_{\mathrm{e}}
  52. n 1 / n 2 = ( 1 + Z 2 T e / T i ) / ( 1 + Z 1 T e / T i ) n_{1}/n_{2}=(1+Z_{2}T_{\mathrm{e}}/T_{\mathrm{i}})/(1+Z_{1}T_{\mathrm{e}}/T_{% \mathrm{i}})
  53. n τ n\tau
  54. n T τ nT\tau
  55. ( 1 + Z 1 T e / T i ) ( 1 + Z 2 T e / T i ) / 4 (1+Z_{1}T_{\mathrm{e}}/T_{\mathrm{i}})\cdot(1+Z_{2}T_{\mathrm{e}}/T_{\mathrm{i% }})/4
  56. Z = 5 Z=5
  57. 3 3
  58. 4 4
  59. Z > 1 Z>1

Leaky_bucket.html

  1. M = 1 + τ T - δ M=\left\lfloor 1+\frac{\tau}{T-\delta}\right\rfloor
  2. τ = ( M - 1 ) ( T - δ ) \tau=\left(M-1\right)\left(T-\delta\right)

Learning_vector_quantization.html

  1. M M
  2. W i \vec{W_{i}}
  3. i = 0 , 1 , , M - 1 i=0,1,...,M-1
  4. η \eta
  5. L L
  6. X \vec{X}
  7. L L
  8. W m \vec{Wm}
  9. d ( X , W m ) d(\vec{X},\vec{W_{m}})
  10. d \,d\,
  11. W m \vec{W_{m}}
  12. W m \vec{W_{m}}
  13. X \vec{X}
  14. W m W m + η ( X - W m ) \vec{W_{m}}\leftarrow\vec{W_{m}}+\eta\cdot\left(\vec{X}-\vec{W_{m}}\right)
  15. L L
  16. W i \vec{W_{i}}
  17. X \vec{X}

Lebesgue's_number_lemma.html

  1. ( X , d ) (X,d)
  2. X X
  3. δ > 0 \delta>0
  4. X X
  5. δ \delta
  6. δ \delta
  7. 𝒰 \mathcal{U}
  8. X X
  9. X X
  10. { A 1 , , A n } 𝒰 \{A_{1},\dots,A_{n}\}\subseteq\mathcal{U}
  11. i { 1 , , n } i\in\{1,\dots,n\}
  12. C i := X A i C_{i}:=X\setminus A_{i}
  13. f : X f:X\rightarrow\mathbb{R}
  14. f ( x ) := 1 n i = 1 n d ( x , C i ) f(x):=\frac{1}{n}\sum_{i=1}^{n}d(x,C_{i})
  15. f f
  16. δ \delta
  17. δ > 0 \delta>0
  18. Y Y
  19. X X
  20. δ \delta
  21. x 0 X x_{0}\in X
  22. Y B δ ( x 0 ) Y\subseteq B_{\delta}(x_{0})
  23. B δ ( x 0 ) B_{\delta}(x_{0})
  24. δ \delta
  25. x 0 x_{0}
  26. x 0 x_{0}
  27. Y Y
  28. f ( x 0 ) δ f(x_{0})\geq\delta
  29. i i
  30. d ( x 0 , C i ) δ d(x_{0},C_{i})\geq\delta
  31. B δ ( x 0 ) A i B_{\delta}(x_{0})\subseteq A_{i}
  32. Y A i Y\subseteq A_{i}

Lender_of_last_resort.html

  1. M = [ 1 + C D C D + R D ] B M=\quad\left[\frac{1+\frac{C}{D}}{\frac{C}{D}+\frac{R}{D}}\right]B

Length_of_a_module.html

  1. N 0 N 1 N n N_{0}\subsetneq N_{1}\subsetneq\cdots\subsetneq N_{n}
  2. 0 L M N 0 0\rightarrow L\rightarrow M\rightarrow N\rightarrow 0
  3. 0 = N 0 N 1 N n = M 0=N_{0}\subsetneq N_{1}\subsetneq\cdots\subsetneq N_{n}=M
  4. N i + 1 / N i is simple for i = 0 , , n - 1 N_{i+1}/N_{i}\mbox{ is simple for }~{}i=0,\dots,n-1

Length_scale.html

  1. = / p \ell=\hbar/p
  2. \hbar
  3. p p
  4. a 10 - 10 \ell_{a}\sim 10^{-10}
  5. a 1 / α m e \ell_{a}\sim 1/\alpha m_{e}
  6. s 10 - 15 \ell_{s}\sim 10^{-15}
  7. 10 - 23 10^{-23}
  8. w 10 - 18 \ell_{w}\sim 10^{-18}
  9. P 10 - 35 \ell_{P}\sim 10^{-35}
  10. 10 19 10^{19}
  11. - 1 {}^{-1}

Lens_space.html

  1. S 2 × S 1 S^{2}\times S^{1}
  2. L ( p , q ) L(p,q)
  3. L ( 5 ; 1 ) L(5;1)
  4. L ( 5 ; 2 ) L(5;2)
  5. L ( p ; q ) L(p;q)
  6. S 3 S^{3}
  7. / p \mathbb{Z}/p
  8. p p
  9. q q
  10. S 3 S^{3}
  11. 2 \mathbb{C}^{2}
  12. / p \mathbb{Z}/p
  13. S 3 S^{3}
  14. [ 1 ] . ( z 1 , z 2 ) := ( e 2 π i / p z 1 , e 2 π i q / p z 2 ) [1].(z_{1},z_{2}):=(e^{2\pi i/p}\cdot z_{1},e^{2\pi iq/p}\cdot z_{2})
  15. L ( p ; q ) L(p;q)
  16. p , q 1 , , q n p,q_{1},\ldots,q_{n}
  17. q i q_{i}
  18. p p
  19. S 2 n - 1 S^{2n-1}
  20. n \mathbb{C}^{n}
  21. L ( p ; q 1 , q n ) L(p;q_{1},\ldots q_{n})
  22. S 2 n - 1 S^{2n-1}
  23. / p \mathbb{Z}/p
  24. [ 1 ] . ( z 1 , , z n ) := ( e 2 π i q 1 / p z 1 , , e 2 π i q n / p z n ) . [1].(z_{1},\ldots,z_{n}):=(e^{2\pi iq_{1}/p}\cdot z_{1},\ldots,e^{2\pi iq_{n}/% p}\cdot z_{n}).
  25. L ( p ; q ) = L ( p ; 1 , q ) . L(p;q)=L(p;1,q).
  26. L ( p ; q 1 , , q n ) L(p;q_{1},\ldots,q_{n})
  27. / p \mathbb{Z}/p
  28. q i q_{i}
  29. L ( p , q ) L(p,q)
  30. L ( p ; q 1 ) L(p;q_{1})
  31. L ( p ; q 2 ) L(p;q_{2})
  32. q 1 q 2 ± n 2 ( mod p ) q_{1}q_{2}\equiv\pm n^{2}\;\;(\mathop{{\rm mod}}p)
  33. n n\in\mathbb{N}
  34. q 1 ± q 2 ± 1 ( mod p ) q_{1}\equiv\pm q_{2}^{\pm 1}\;\;(\mathop{{\rm mod}}p)
  35. § \S

Levelling.html

  1. Δ h m e t e r s = 0.067 D k m 2 \Delta h_{meters}=0.067D_{km}^{2}
  2. Δ h f e e t = 0.021 ( D f t 1000 ) 2 \Delta h_{feet}=0.021\left(\frac{D_{ft}}{1000}\right)^{2}
  3. i = 0 n Δ h i = 0 \sum_{i=0}^{n}\Delta h_{i}=0
  4. i = 0 n Δ h i g i , \sum_{i=0}^{n}\Delta h_{i}g_{i},
  5. g i g_{i}
  6. Δ W i = Δ h i g i \Delta W_{i}=\Delta h_{i}g_{i}
  7. W i W_{i}

Levenberg–Marquardt_algorithm.html

  1. S ( s y m b o l β ) = i = 1 m [ y i - f ( x i , s y m b o l β ) ] 2 S(symbol\beta)=\sum_{i=1}^{m}[y_{i}-f(x_{i},\ symbol\beta)]^{2}
  2. f ( x i , s y m b o l β + s y m b o l δ ) f(x_{i},symbol\beta+symbol\delta)
  3. f ( x i , s y m b o l β + s y m b o l δ ) f ( x i , s y m b o l β ) + J i s y m b o l δ f(x_{i},symbol\beta+symbol\delta)\approx f(x_{i},symbol\beta)+J_{i}symbol\delta\!
  4. J i = f ( x i , s y m b o l β ) s y m b o l β J_{i}=\frac{\partial f(x_{i},symbol\beta)}{\partial symbol\beta}
  5. S ( β ) S(\beta)
  6. S S
  7. f ( x i , s y m b o l β + s y m b o l δ ) f(x_{i},symbol\beta+symbol\delta)
  8. S ( s y m b o l β + s y m b o l δ ) i = 1 m ( y i - f ( x i , s y m b o l β ) - J i s y m b o l δ ) 2 S(symbol\beta+symbol\delta)\approx\sum_{i=1}^{m}\left(y_{i}-f(x_{i},symbol% \beta)-J_{i}symbol\delta\right)^{2}
  9. S ( s y m b o l β + s y m b o l δ ) 𝐲 - 𝐟 ( s y m b o l β ) - 𝐉 s y m b o l δ 2 S(symbol\beta+symbol\delta)\approx\|\mathbf{y}-\mathbf{f}(symbol\beta)-\mathbf% {J}symbol\delta\|^{2}
  10. ( 𝐉 𝐓 𝐉 ) 𝐬𝐲𝐦𝐛𝐨𝐥 δ = 𝐉 𝐓 [ 𝐲 - 𝐟 ( 𝐬𝐲𝐦𝐛𝐨𝐥 β ) ] \mathbf{(J^{T}J)symbol\delta=J^{T}[y-f(symbol\beta)]}\!
  11. 𝐉 \mathbf{J}
  12. J i J_{i}
  13. 𝐟 \mathbf{f}
  14. 𝐲 \mathbf{y}
  15. f ( x i , s y m b o l β ) f(x_{i},symbol\beta)
  16. y i y_{i}
  17. ( 𝐉 𝐓 𝐉 + λ 𝐈 ) 𝐬𝐲𝐦𝐛𝐨𝐥 δ = 𝐉 𝐓 [ 𝐲 - 𝐟 ( 𝐬𝐲𝐦𝐛𝐨𝐥 β ) ] \mathbf{(J^{T}J+\lambda I)symbol\delta=J^{T}[y-f(symbol\beta)]}\!
  18. - 2 ( 𝐉 T [ 𝐲 - 𝐟 ( s y m b o l β ) ] ) T -2(\mathbf{J}^{T}[\mathbf{y}-\mathbf{f}(symbol\beta)])^{T}
  19. ( 𝐉 𝐓 𝐉 + λ 𝐝𝐢𝐚𝐠 ( 𝐉 𝐓 𝐉 ) ) 𝐬𝐲𝐦𝐛𝐨𝐥 δ = 𝐉 𝐓 [ 𝐲 - 𝐟 ( 𝐬𝐲𝐦𝐛𝐨𝐥 β ) ] \mathbf{(J^{T}J+\lambda\,diag(J^{T}J))symbol\delta=J^{T}[y-f(symbol\beta)]}\!
  20. y = a cos ( b X ) + b sin ( a X ) y=a\cos(bX)+b\sin(aX)
  21. cos ( β x ) \cos(\beta x)
  22. β ^ \hat{\beta}
  23. β ^ + 2 n π . \hat{\beta}+2n\pi.

Lévy's_constant.html

  1. lim n q n 1 / n = γ \lim_{n\to\infty}{q_{n}}^{1/n}=\gamma
  2. γ = e π 2 / ( 12 ln 2 ) = 3.275822918721811159787681882 . \gamma=e^{\pi^{2}/(12\ln 2)}=3.275822918721811159787681882\ldots.
  3. π 2 / ( 12 ln 2 ) \pi^{2}/(12\ln 2)

Lévy_C_curve.html

  1. f 1 ( z ) = ( 1 - i ) z 2 f_{1}(z)=\frac{(1-i)z}{2}
  2. f 2 ( z ) = 1 + ( 1 + i ) ( z - 1 ) 2 f_{2}(z)=1+\frac{(1+i)(z-1)}{2}
  3. S 0 = { 0 , 1 } S_{0}=\{0,1\}

Lévy_process.html

  1. X = { X t : t 0 } X=\{X_{t}:t\geq 0\}
  2. X 0 = 0 X_{0}=0\,
  3. 0 t 1 < t 2 < < t n < 0\leq t_{1}<t_{2}<\cdots<t_{n}<\infty
  4. X t 2 - X t 1 , X t 3 - X t 2 , , X t n - X t n - 1 X_{t_{2}}-X_{t_{1}},X_{t_{3}}-X_{t_{2}},\dots,X_{t_{n}}-X_{t_{n-1}}
  5. s < t s<t\,
  6. X t - X s X_{t}-X_{s}\,
  7. X t - s . X_{t-s}.\,
  8. ϵ > 0 \epsilon>0
  9. t 0 t\geq 0
  10. lim h 0 P ( | X t + h - X t | > ϵ ) = 0 \lim_{h\rightarrow 0}P(|X_{t+h}-X_{t}|>\epsilon)=0
  11. X X
  12. X X
  13. t X t t\mapsto X_{t}
  14. X X
  15. X X
  16. F F
  17. X X
  18. X 1 X_{1}
  19. F F
  20. μ n ( t ) = E ( X t n ) \mu_{n}(t)=E(X_{t}^{n})
  21. μ n ( t + s ) = k = 0 n ( n k ) μ k ( t ) μ n - k ( s ) . \mu_{n}(t+s)=\sum_{k=0}^{n}{n\choose k}\mu_{k}(t)\mu_{n-k}(s).
  22. X = ( X t ) t 0 X=(X_{t})_{t\geq 0}
  23. ϕ X ( θ ) \phi_{X}(\theta)
  24. ϕ X ( θ ) := 𝔼 [ e i θ X 1 ] = exp ( a i θ - 1 2 σ 2 θ 2 + \ { 0 } ( e i θ x - 1 - i θ x 𝐈 | x | < 1 ) Π ( d x ) ) \phi_{X}(\theta):=\mathbb{E}\Big[e^{i\theta X_{1}}\Big]=\exp\Bigg(ai\theta-% \frac{1}{2}\sigma^{2}\theta^{2}+\int_{\mathbb{R}\backslash\{0\}}\big(e^{i% \theta x}-1-i\theta x\mathbf{I}_{|x|<1}\big)\,\Pi(dx)\Bigg)
  25. a a\in\mathbb{R}
  26. σ 0 \sigma\geq 0
  27. 𝐈 \mathbf{I}
  28. Π \Pi
  29. X X
  30. \ { 0 } 1 x 2 Π ( d x ) < . \int_{\mathbb{R}\backslash\{0\}}1\wedge x^{2}\Pi(dx)<\infty.
  31. Π ( d x ) \Pi(dx)
  32. x x
  33. ( a , σ 2 , Π ) (a,\sigma^{2},\Pi)
  34. ( a , σ 2 , Π ) (a,\sigma^{2},\Pi)
  35. X ( 1 ) X^{(1)}
  36. X ( 2 ) X^{(2)}
  37. X ( 3 ) X^{(3)}
  38. X ( 1 ) X^{(1)}
  39. σ 2 \sigma^{2}
  40. X ( 2 ) X^{(2)}
  41. X ( 3 ) X^{(3)}
  42. X = X ( 1 ) + X ( 2 ) + X ( 3 ) X=X^{(1)}+X^{(2)}+X^{(3)}
  43. ( a , σ 2 , Π ) (a,\sigma^{2},\Pi)
  44. X ( 3 ) X^{(3)}
  45. 1 1

Liar's_dice.html

  1. P ( q ) = C ( n , q ) ( 1 / 6 ) q ( 5 / 6 ) n - q \ P(q)=C(n,q)\cdot(1/6)^{q}\cdot(5/6)^{n-q}
  2. B ( n , 1 6 ) B(n,\tfrac{1}{6})
  3. P ( q ) = x = q n C ( n , x ) ( 1 / 6 ) x ( 5 / 6 ) n - x \ P^{\prime}(q)=\sum_{x=q}^{n}C(n,x)\cdot(1/6)^{x}\cdot(5/6)^{n-x}

Lie–Kolchin_theorem.html

  1. ρ : G G L ( V ) \rho\colon G\to GL(V)
  2. ρ ( G ) ( L ) = L . \rho(G)(L)=L.
  3. ρ ( g ) , g G \rho(g),\,\,g\in G
  4. ρ ( G ) \rho(G)
  5. ρ ( G ) \rho(G)
  6. { x + i y | x 2 + y 2 = 1 } \{x+iy\in\mathbb{C}\,|\,x^{2}+y^{2}=1\}
  7. ρ ( z ) \rho(z)
  8. z = x + i y z=x+iy
  9. ( x y - y x ) . \begin{pmatrix}x&y\\ -y&x\end{pmatrix}.

Light_field.html

  1. θ \theta
  2. ϕ \phi
  3. θ \theta
  4. ϕ \phi
  5. D 1 D_{1}
  6. D 2 D_{2}
  7. I 1 I_{1}
  8. I 2 I_{2}
  9. D D
  10. P 1 , P 2 P_{1},P_{2}

Lights_Out_(game).html

  1. ( 0 1 1 1 0 1 0 1 0 1 1 1 0 1 1 1 0 1 0 1 0 1 1 1 0 ) \begin{pmatrix}0&1&1&1&0\\ 1&0&1&0&1\\ 1&1&0&1&1\\ 1&0&1&0&1\\ 0&1&1&1&0\end{pmatrix}
  2. ( 1 0 1 0 1 1 0 1 0 1 0 0 0 0 0 1 0 1 0 1 1 0 1 0 1 ) \begin{pmatrix}1&0&1&0&1\\ 1&0&1&0&1\\ 0&0&0&0&0\\ 1&0&1&0&1\\ 1&0&1&0&1\end{pmatrix}

Likelihood_ratios_in_diagnostic_testing.html

  1. L R + = sensitivity 1 - specificity LR+=\frac{\,\text{sensitivity}}{1-\,\text{specificity}}
  2. L R + = Pr ( T + | D + ) Pr ( T + | D - ) LR+=\frac{\Pr({T+}|D+)}{\Pr({T+}|D-)}
  3. L R - = 1 - sensitivity specificity LR-=\frac{1-\,\text{sensitivity}}{\,\text{specificity}}
  4. L R - = Pr ( T - | D + ) Pr ( T - | D - ) LR-=\frac{\Pr({T-}|D+)}{\Pr({T-}|D-)}

Limaçon.html

  1. r = b + a cos θ . r=b+a\cos\theta\ .
  2. r 2 = x 2 + y 2 r^{2}=x^{2}+y^{2}
  3. r cos θ = x r\,\cos\theta=x
  4. ( x 2 + y 2 - a x ) 2 = b 2 ( x 2 + y 2 ) . (x^{2}+y^{2}-ax)^{2}=b^{2}(x^{2}+y^{2}).\,
  5. x = a 2 + b cos θ + a 2 cos 2 θ , y = b sin θ + a 2 sin 2 θ . x={a\over 2}+b\cos\theta+{a\over 2}\cos 2\theta,\,y=b\sin\theta+{a\over 2}\sin 2\theta.
  6. z = a 2 + b e i θ + a 2 e 2 i θ . z={a\over 2}+be^{i\theta}+{a\over 2}e^{2i\theta}.
  7. z = b e i t + a 2 e 2 i t . z=be^{it}+{a\over 2}e^{2it}.
  8. r = b ( 1 + cos θ ) = 2 b cos 2 θ 2 r=b(1+\cos\theta)=2b\cos^{2}{\theta\over 2}
  9. r 1 2 = ( 2 b ) 1 2 cos θ 2 r^{1\over 2}=(2b)^{1\over 2}\cos{\theta\over 2}
  10. a = 2 b a=2b
  11. z = b ( e i t + e 2 i t ) = b e 3 i t 2 ( e i t 2 + e - i t 2 ) = 2 b cos t 2 e 3 i t 2 z=b(e^{it}+e^{2it})=be^{3it\over 2}(e^{it\over 2}+e^{-it\over 2})=2b\cos{t% \over 2}e^{3it\over 2}
  12. r = 2 b cos θ 3 r=2b\cos{\theta\over 3}
  13. b > a b>a
  14. b > 2 a b>2a
  15. a < b < 2 a a<b<2a
  16. b = 2 a b=2a
  17. ( - a , 0 ) (-a,0)
  18. b b
  19. a a
  20. b = a b=a
  21. 0 < b < a 0<b<a
  22. b b
  23. r = b + a cos θ r=b+a\cos\theta
  24. ( b 2 + a 2 2 ) π (b^{2}+{{a^{2}}\over 2})\pi
  25. b < a b<a
  26. π ± arccos b a \pi\pm\arccos{b\over a}
  27. ( b 2 + a 2 2 ) arccos b a - 3 2 b a 2 - b 2 (b^{2}+{{a^{2}}\over 2})\arccos{b\over a}-{3\over 2}b\sqrt{{a^{2}}-{b^{2}}}
  28. ( b 2 + a 2 2 ) ( π - arccos b a ) + 3 2 b a 2 - b 2 (b^{2}+{{a^{2}}\over 2})(\pi-\arccos{b\over a})+{3\over 2}b\sqrt{{a^{2}}-{b^{2% }}}
  29. ( b 2 + a 2 2 ) ( π - 2 arccos b a ) + 3 b a 2 - b 2 . (b^{2}+{{a^{2}}\over 2})(\pi-2\arccos{b\over a})+3b\sqrt{{a^{2}}-{b^{2}}}.
  30. b b
  31. ( a , 0 ) (a,0)
  32. r = b + a cos θ r=b+a\cos\theta
  33. r = b + a cos θ r=b+a\cos\theta
  34. r = 1 b + a cos θ r={1\over{b+a\cos\theta}}

Limiting_magnitude.html

  1. 5 log 10 ( D 1 D 0 ) 5\cdot\log_{10}(\frac{D_{1}}{D_{0}})
  2. m v = m n a k e d e y e - 2 + 2.5 log 10 ( D P t ) m\cdot v=m_{nakedeye}-2+2.5\cdot\log_{10}(D\cdot P\cdot t)

Line_(geometry).html

  1. ( A , B ) (A,B)
  2. y = m x + b y=mx+b\,
  3. L = { ( x , y ) a x + b y = c } L=\{(x,y)\mid ax+by=c\}\,
  4. P 0 = ( x 0 , y 0 ) P_{0}=(x_{0},y_{0})
  5. P 1 = ( x 1 , y 1 ) P_{1}=(x_{1},y_{1})
  6. ( y - y 0 ) ( x 1 - x 0 ) = ( y 1 - y 0 ) ( x - x 0 ) (y-y_{0})(x_{1}-x_{0})=(y_{1}-y_{0})(x-x_{0})
  7. y = ( x - x 0 ) y 1 - y 0 x 1 - x 0 + y 0 y=(x-x_{0})\,\frac{y_{1}-y_{0}}{x_{1}-x_{0}}+y_{0}
  8. y = x y 1 - y 0 x 1 - x 0 + x 1 y 0 - x 0 y 1 x 1 - x 0 . y=x\,\frac{y_{1}-y_{0}}{x_{1}-x_{0}}+\frac{x_{1}y_{0}-x_{0}y_{1}}{x_{1}-x_{0}}\,.
  9. x = x 0 + a t x=x_{0}+at\,
  10. y = y 0 + b t y=y_{0}+bt\,
  11. z = z 0 + c t z=z_{0}+ct\,
  12. a 1 x + b 1 y + c 1 z - d 1 = 0 a_{1}x+b_{1}y+c_{1}z-d_{1}=0\,
  13. a 2 x + b 2 y + c 2 z - d 2 = 0 a_{2}x+b_{2}y+c_{2}z-d_{2}=0\,
  14. ( a 1 , b 1 , c 1 ) (a_{1},b_{1},c_{1})
  15. ( a 2 , b 2 , c 2 ) (a_{2},b_{2},c_{2})
  16. a 1 = t a 2 , b 1 = t b 2 , c 1 = t c 2 a_{1}=ta_{2},b_{1}=tb_{2},c_{1}=tc_{2}
  17. t = 0 t=0
  18. y sin θ + x cos θ - p = 0 , y\sin\theta+x\cos\theta-p=0,\,
  19. | c | - c a 2 + b 2 . \frac{|c|}{-c}\sqrt{a^{2}+b^{2}}.
  20. π \pi
  21. π \pi
  22. r = m r cos θ + b sin θ , r=\frac{mr\cos\theta+b}{\sin\theta},
  23. r sin θ = m r cos θ + b . r\sin\theta=mr\cos\theta+b.\,
  24. r = 1 cos θ x o + sin θ y o r=\frac{1}{\frac{\cos\theta}{x_{o}}+\frac{\sin\theta}{y_{o}}}
  25. x o x_{o}
  26. y o y_{o}
  27. x o x_{o}
  28. y o y_{o}
  29. r cos θ = x o . r\cos\theta=x_{o}.
  30. r sin θ = y o . r\sin\theta=y_{o}.
  31. θ = m \theta=m
  32. L = { ( 1 - t ) a + t b t } L=\{(1-t)\,a+t\,b\mid t\in\mathbb{R}\}
  33. [ 1 x 1 x 2 x n 1 y 1 y 2 y n 1 z 1 z 2 z n ] \begin{bmatrix}1&x_{1}&x_{2}&\dots&x_{n}\\ 1&y_{1}&y_{2}&\dots&y_{n}\\ 1&z_{1}&z_{2}&\dots&z_{n}\end{bmatrix}

Line_element.html

  1. d s 2 = d q d q = g ( d q , d q ) ds^{2}=d{q}\cdot d{q}=g(d{q},d{q})
  2. d s 2 = g i j d q i d q j ds^{2}=g_{ij}dq^{i}dq^{j}
  3. s = λ 1 λ 2 d λ g i j d q i d λ d q j d λ s=\int_{\lambda_{1}}^{\lambda_{2}}d\lambda\sqrt{g_{ij}\frac{dq^{i}}{d\lambda}% \frac{dq^{j}}{d\lambda}}
  4. g i j = δ i j g_{ij}=\delta_{ij}
  5. [ g i j ] = ( 1 0 0 0 1 0 0 0 1 ) [g_{ij}]=\begin{pmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{pmatrix}
  6. ( q 1 , q 2 , q 3 ) = ( x , y , z ) d r = ( d x , d y , d z ) (q^{1},q^{2},q^{3})=(x,y,z)\,\Rightarrow\,d{r}=(dx,dy,dz)
  7. d s 2 = g i j d q i d q j = d x 2 + d y 2 + d z 2 ds^{2}=g_{ij}dq^{i}dq^{j}=dx^{2}+dy^{2}+dz^{2}
  8. [ g i j ] = ( h 1 2 0 0 0 h 2 2 0 0 0 h 3 2 ) [g_{ij}]=\begin{pmatrix}h_{1}^{2}&0&0\\ 0&h_{2}^{2}&0\\ 0&0&h_{3}^{2}\end{pmatrix}
  9. h i = | r q i | h_{i}=\left|\frac{\partial{r}}{\partial q^{i}}\right|
  10. d s 2 = h 1 2 ( q 1 ) 2 + h 2 2 ( q 2 ) 2 + h 3 2 ( q 3 ) 2 ds^{2}=h_{1}^{2}(q^{1})^{2}+h_{2}^{2}(q^{2})^{2}+h_{3}^{2}(q^{3})^{2}
  11. [ g i j ] = ( 1 0 0 r 2 ) [g_{ij}]=\begin{pmatrix}1&0\\ 0&r^{2}\\ \end{pmatrix}
  12. d s 2 = d r 2 + r 2 d θ 2 ds^{2}=dr^{2}+r^{2}d\theta\ ^{2}
  13. [ g i j ] = ( 1 0 0 0 r 2 0 0 0 r 2 sin 2 θ ) [g_{ij}]=\begin{pmatrix}1&0&0\\ 0&r^{2}&0\\ 0&0&r^{2}\sin^{2}\theta\\ \end{pmatrix}
  14. d s 2 = d r 2 + r 2 d θ 2 + r 2 sin 2 θ d ϕ 2 ds^{2}=dr^{2}+r^{2}d\theta\ ^{2}+r^{2}\sin^{2}\theta\ d\phi\ ^{2}
  15. [ g i j ] = ( 1 0 0 0 r 2 0 0 0 1 ) [g_{ij}]=\begin{pmatrix}1&0&0\\ 0&r^{2}&0\\ 0&0&1\\ \end{pmatrix}
  16. d s 2 = d r 2 + r 2 d θ 2 + d z 2 ds^{2}=dr^{2}+r^{2}d\theta\ ^{2}+dz^{2}
  17. g i j = r q i r q j g_{ij}=\frac{\partial{r}}{\partial q^{i}}\cdot\frac{\partial{r}}{\partial q^{j}}
  18. d s 2 = g i j d q i d q j = r q i r q j d q i d q j ds^{2}=g_{ij}dq^{i}dq^{j}=\frac{\partial{r}}{\partial q^{i}}\cdot\frac{% \partial{r}}{\partial q^{j}}dq^{i}dq^{j}
  19. [ g i j ] = ( ± 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) [g_{ij}]=\begin{pmatrix}\pm 1&0&0&0\\ 0&\mp 1&0&0\\ 0&0&\mp 1&0\\ 0&0&0&\mp 1\\ \end{pmatrix}
  20. x = ( x 0 , x 1 , x 2 , x 3 ) = ( c t , r ) , d x = ( c d t , d r ) {x}=(x^{0},x^{1},x^{2},x^{3})=(ct,{r})\,\Rightarrow,\,d{x}=(cdt,d{r})
  21. d s 2 = ± c 2 d t 2 d r d r ds^{2}=\pm c^{2}dt^{2}\mp d{r}\cdot d{r}
  22. d s 2 = d x d x = g ( d x , d x ) ds^{2}=d{x}\cdot d{x}=g(d{x},d{x})
  23. d s 2 = g α β d x α d x β ds^{2}=g_{\alpha\beta}dx^{\alpha}dx^{\beta}

Line_graph.html

  1. G , L ( G ) , L ( L ( G ) ) , L ( L ( L ( G ) ) ) , . G,L(G),L(L(G)),L(L(L(G))),\dots.

Linear_approximation.html

  1. f f
  2. n = 1 n=1
  3. f ( x ) = f ( a ) + f ( a ) ( x - a ) + R 2 f(x)=f(a)+f^{\prime}(a)(x-a)+R_{2}
  4. R 2 R_{2}
  5. f ( x ) f ( a ) + f ( a ) ( x - a ) f(x)\approx f(a)+f^{\prime}(a)(x-a)
  6. x x
  7. a a
  8. f f
  9. ( a , f ( a ) ) (a,f(a))
  10. f f
  11. x x
  12. a a
  13. f f
  14. f ( x , y ) f(x,y)
  15. f ( x , y ) f(x,y)
  16. ( x , y ) (x,y)
  17. ( a , b ) (a,b)
  18. f ( x , y ) f ( a , b ) + f x ( a , b ) ( x - a ) + f y ( a , b ) ( y - b ) . f\left(x,y\right)\approx f\left(a,b\right)+\frac{\partial f}{\partial x}\left(% a,b\right)\left(x-a\right)+\frac{\partial f}{\partial y}\left(a,b\right)\left(% y-b\right).
  19. z = f ( x , y ) z=f(x,y)
  20. ( a , b ) . (a,b).
  21. f ( x ) f ( a ) + D f ( a ) ( x - a ) f(x)\approx f(a)+Df(a)(x-a)
  22. D f ( a ) Df(a)
  23. f f
  24. a a
  25. T = 2 π L g ( 1 + 1 16 θ 0 2 + 11 3072 θ 0 4 + ) T=2\pi\sqrt{L\over g}\left(1+\frac{1}{16}\theta_{0}^{2}+\frac{11}{3072}\theta_% {0}^{4}+\cdots\right)
  26. T 2 π L g θ 0 1 ( 1 ) T\approx 2\pi\sqrt{\frac{L}{g}}\qquad\qquad\qquad\theta_{0}\ll 1\qquad(1)\,
  27. ρ ( T ) = ρ 0 [ 1 + α ( T - T 0 ) ] \rho(T)=\rho_{0}[1+\alpha(T-T_{0})]
  28. α \alpha
  29. T 0 T_{0}
  30. ρ 0 \rho_{0}
  31. T 0 T_{0}
  32. α \alpha
  33. α \alpha
  34. α \alpha
  35. α 15 \alpha_{15}

Linear_density.html

  1. M M
  2. L L
  3. λ ¯ m \bar{\lambda}_{m}
  4. M M
  5. L L
  6. λ ¯ m = M L \bar{\lambda}_{m}=\frac{M}{L}
  7. l l
  8. m = m ( l ) m=m(l)
  9. d m dm
  10. λ m \lambda_{m}
  11. d l dl
  12. d m = λ m d l dm=\lambda_{m}dl
  13. l l
  14. λ m = d m d l \lambda_{m}=\frac{dm}{dl}
  15. Q Q
  16. L L
  17. λ ¯ q \bar{\lambda}_{q}
  18. Q Q
  19. L L
  20. λ ¯ q = Q L \bar{\lambda}_{q}=\frac{Q}{L}
  21. l l
  22. q = q ( l ) q=q(l)
  23. d q dq
  24. λ q \lambda_{q}
  25. d l dl
  26. d q = λ q d l dq=\lambda_{q}dl
  27. l l
  28. λ q = d q d l \lambda_{q}=\frac{dq}{dl}

Linear_system.html

  1. H H
  2. x ( t ) x(t)
  3. t t
  4. y ( t ) y(t)
  5. x 1 ( t ) x_{1}(t)\,
  6. x 2 ( t ) x_{2}(t)\,
  7. y 1 ( t ) = H { x 1 ( t ) } y_{1}(t)=H\left\{x_{1}(t)\right\}
  8. y 2 ( t ) = H { x 2 ( t ) } y_{2}(t)=H\left\{x_{2}(t)\right\}
  9. α y 1 ( t ) + β y 2 ( t ) = H { α x 1 ( t ) + β x 2 ( t ) } \alpha y_{1}(t)+\beta y_{2}(t)=H\left\{\alpha x_{1}(t)+\beta x_{2}(t)\right\}
  10. α \alpha\,
  11. β \beta\,
  12. m d 2 ( x ) d t 2 = - k x m\frac{d^{2}(x)}{dt^{2}}=-kx
  13. H ( x ( t ) ) = m d 2 ( x ( t ) ) d t 2 + k x ( t ) H(x(t))=m\frac{d^{2}(x(t))}{dt^{2}}+kx(t)
  14. x ( t ) = δ ( t - t 1 ) x(t)=\delta(t-t_{1})\,
  15. y ( t ) | t = t 2 = h ( t 2 , t 1 ) y(t)|_{t=t_{2}}=h(t_{2},t_{1})\,
  16. y ( t ) = - h ( t , s ) x ( s ) d s y(t)=\int_{-\infty}^{\infty}h(t,s)x(s)ds
  17. y ( t ) = - h ( t , t - τ ) x ( t - τ ) d τ y(t)=\int_{-\infty}^{\infty}h(t,t-\tau)x(t-\tau)d\tau
  18. s = t - τ s=t-\tau
  19. y [ n ] = k = - h [ n , k ] x [ k ] y[n]=\sum_{k=-\infty}^{\infty}{h[n,k]x[k]}
  20. y [ n ] = m = - h [ n , n - m ] x [ n - m ] y[n]=\sum_{m=-\infty}^{\infty}{h[n,n-m]x[n-m]}
  21. k = n - m k=n-m\,
  22. h ( t , s ) = 0 for t < s h(t,s)=0\,\text{ for }t<s\,

Linear_system_of_divisors.html

  1. D = E + ( f ) D=E+(f)
  2. ( Γ ( V , L ) { 0 } ) / k , (\Gamma(V,L)\setminus\{0\})/k^{\ast},
  3. 𝔡 \mathfrak{d}
  4. Γ ( V , L ) . \Gamma(V,L).
  5. 𝔡 \mathfrak{d}
  6. dim 𝔡 = dim W - 1 \dim\mathfrak{d}=\dim W-1
  7. x = a x=a
  8. λ C + μ C = 0 \lambda C+\mu C^{\prime}=0
  9. [ λ : μ ] [\lambda:\mu]
  10. ( 4 2 , 2 ) / 2 = 3 \textstyle{{\left({{4}\atop{2,2}}\right)}/2=3}
  11. ( 4 2 ) \textstyle{{\left({{4}\atop{2}}\right)}}
  12. ( ± 1 , ± 1 ) , (\pm 1,\pm 1),
  13. a x 2 + ( 1 - a ) y 2 = 1 , ax^{2}+(1-a)y^{2}=1,
  14. x 2 = 1 x^{2}=1
  15. y 2 = 1 , y^{2}=1,
  16. 0 , 1 , . 0,1,\infty.
  17. ( 1 + a ) x 2 + ( 1 - a ) y 2 = 2 , (1+a)x^{2}+(1-a)y^{2}=2,
  18. a - a a\mapsto-a
  19. a > 1 : a>1:
  20. a = 1 : a=1:
  21. x = - 1 , x = 1 ; x=-1,x=1;
  22. 0 < a < 1 : 0<a<1:
  23. a = 0 : a=0:
  24. 2 \sqrt{2}
  25. - 1 < a < 0 : -1<a<0:
  26. a = - 1 : a=-1:
  27. y = - 1 , y = 1 ; y=-1,y=1;
  28. a < - 1 : a<-1:
  29. a = : a=\infty:
  30. y = x , y = - x ; y=x,y=-x;
  31. a a
  32. a a\to\infty
  33. x 2 - y 2 = 0 x^{2}-y^{2}=0
  34. a > 1 , a>1,

Linear_temporal_logic.html

  1. \vDash
  2. \vDash
  3. \vDash
  4. \nvDash
  5. \vDash
  6. \vDash
  7. \vDash
  8. \vDash
  9. \vDash
  10. \vDash
  11. \vDash
  12. \vDash
  13. \vDash
  14. \vDash
  15. \vDash
  16. \vDash
  17. ϕ \phi
  18. ϕ \bigcirc\phi
  19. ϕ \phi
  20. ϕ \phi
  21. ϕ \Box\phi
  22. ϕ \phi
  23. ϕ \phi
  24. ϕ \Diamond\phi
  25. ϕ \phi
  26. ψ \psi
  27. ϕ \phi
  28. ψ 𝒰 ϕ \psi\;\mathcal{U}\,\phi
  29. ψ \psi
  30. ϕ \phi
  31. ψ \psi
  32. ϕ \phi
  33. ψ ϕ \psi\;\mathcal{R}\,\phi
  34. ϕ \phi
  35. ψ \psi
  36. ψ \psi
  37. ϕ \phi
  38. ¬ \neg
  39. ϕ \phi
  40. ψ \psi
  41. ( ϕ (\phi\rightarrow
  42. ψ ) \psi)

Linearization.html

  1. y = f ( x ) y=f(x)
  2. x = a x=a
  3. x = b x=b
  4. f ( x ) f(x)
  5. [ a , b ] [a,b]
  6. [ b , a ] [b,a]
  7. a a
  8. b b
  9. x = a x=a
  10. 4 = 2 \sqrt{4}=2
  11. 4.001 = 4 + .001 \sqrt{4.001}=\sqrt{4+.001}
  12. y = f ( x ) y=f(x)
  13. f ( x ) f(x)
  14. L a ( x ) L_{a}(x)
  15. f ( x ) f(x)
  16. x = a x=a
  17. L a ( a ) = f ( a ) L_{a}(a)=f(a)
  18. ( H , K ) (H,K)
  19. M M
  20. y - K = M ( x - H ) y-K=M(x-H)
  21. ( a , f ( a ) ) (a,f(a))
  22. L a ( x ) L_{a}(x)
  23. y = f ( a ) + M ( x - a ) y=f(a)+M(x-a)
  24. f ( x ) f(x)
  25. x = a x=a
  26. x = a x=a
  27. M M
  28. x = a x=a
  29. f ( x ) f(x)
  30. x x
  31. f ( x + h ) f(x+h)
  32. h h
  33. f ( x + h ) f(x+h)
  34. ( x + h , L ( x + h ) ) (x+h,L(x+h))
  35. x = a x=a
  36. y = f ( a ) + f ( a ) ( x - a ) y=f(a)+f^{\prime}(a)(x-a)\,
  37. x = a x=a
  38. f ( a ) = f ( x ) f(a)=f(x)
  39. f ( x ) f(x)
  40. f ( x ) f^{\prime}(x)
  41. f ( x ) f(x)
  42. a a
  43. f ( a ) f^{\prime}(a)
  44. 4.001 \sqrt{4.001}
  45. 4 = 2 \sqrt{4}=2
  46. f ( x ) = x f(x)=\sqrt{x}
  47. x = a x=a
  48. y = a + 1 2 a ( x - a ) y=\sqrt{a}+\frac{1}{2\sqrt{a}}(x-a)
  49. f ( x ) = 1 2 x f^{\prime}(x)=\frac{1}{2\sqrt{x}}
  50. f ( x ) = x f(x)=\sqrt{x}
  51. x x
  52. a = 4 a=4
  53. y = 2 + x - 4 4 y=2+\frac{x-4}{4}
  54. x = 4.001 x=4.001
  55. 4.001 \sqrt{4.001}
  56. 2 + 4.001 - 4 4 = 2.00025 2+\frac{4.001-4}{4}=2.00025
  57. f ( x , y ) f(x,y)
  58. p ( a , b ) p(a,b)
  59. f ( x , y ) f ( a , b ) + < m t p l > f ( x , y ) x | a , b ( x - a ) + f ( x , y ) y | a , b ( y - b ) f(x,y)\approx f(a,b)+\left.{\frac{<}{m}tpl>{{\partial f(x,y)}}{{\partial x}}}% \right|_{a,b}(x-a)+\left.{\frac{{\partial f(x,y)}}{{\partial y}}}\right|_{a,b}% (y-b)
  60. f ( 𝐱 ) f(\mathbf{x})
  61. 𝐩 \mathbf{p}
  62. f ( 𝐱 ) f ( 𝐩 ) + f | 𝐩 ( 𝐱 - 𝐩 ) f({\mathbf{x}})\approx f({\mathbf{p}})+\left.{\nabla f}\right|_{\mathbf{p}}% \cdot({\mathbf{x}}-{\mathbf{p}})
  63. 𝐱 \mathbf{x}
  64. 𝐩 \mathbf{p}
  65. d x d t = F ( x , t ) \frac{d{x}}{dt}={F}({x},t)
  66. d x d t F ( x 0 , t ) + D F ( x 0 , t ) ( x - x 0 ) \frac{d{x}}{dt}\approx{F}({x_{0}},t)+D{F}({x_{0}},t)\cdot({x}-{x_{0}})
  67. x 0 {x_{0}}
  68. D F ( x 0 ) D{F}({x_{0}})
  69. F ( x ) {F}({x})
  70. x 0 {x_{0}}

Link_(knot_theory).html

  1. S n S^{n}
  2. S j S^{j}
  3. T : X 𝐑 2 × I T\colon X\to\mathbf{R}^{2}\times I
  4. ( X , X ) (X,\partial X)
  5. I = [ 0 , 1 ] , I=[0,1],
  6. X \partial X
  7. 𝐑 × { 0 , 1 } \mathbf{R}\times\{0,1\}
  8. { 0 , 1 } = I \{0,1\}=\partial I
  9. X . \partial X.
  10. I = [ 0 , 1 ] I=[0,1]
  11. S 1 S^{1}
  12. ( 0 , 1 ) (0,1)
  13. [ 0 , 1 ) , [0,1),
  14. I = [ 0 , 1 ] I=[0,1]
  15. S 1 . S^{1}.
  16. 𝐑 × { 0 , 1 } \mathbf{R}\times\{0,1\}
  17. 𝐑 × 0 \mathbf{R}\times 0
  18. 𝐑 × 1 \mathbf{R}\times 1
  19. 𝐑 2 \mathbf{R}^{2}

Linkage_disequilibrium.html

  1. p A p_{A}
  2. p A p_{A}
  3. p B p_{B}
  4. p A B p_{AB}
  5. p A B p_{AB}
  6. p A B p_{AB}
  7. p A p B p_{A}p_{B}
  8. p A B p_{AB}
  9. p A p B p_{A}p_{B}
  10. D A B D_{AB}
  11. D A B = p A B - p A p B D_{AB}=p_{AB}-p_{A}p_{B}
  12. D A B 0 D_{AB}\neq 0
  13. D A B = 0 D_{AB}=0
  14. p A B = p A p B p_{AB}=p_{A}p_{B}
  15. D A B D_{AB}
  16. D D
  17. D D
  18. D = D / D max D^{\prime}=D/D_{\max}
  19. D max = { min { p A p B , ( 1 - p A ) ( 1 - p B ) } when D < 0 min { p A ( 1 - p B ) , ( 1 - p A ) p B } when D > 0 D_{\max}=\begin{cases}\min\{p_{A}p_{B},\,(1-p_{A})(1-p_{B})\}&\,\text{when }D<% 0\\ \min\{p_{A}(1-p_{B}),\,(1-p_{A})p_{B}\}&\,\text{when }D>0\end{cases}
  20. D D^{\prime}
  21. r = D p A ( 1 - p A ) p B ( 1 - p B ) r=\frac{D}{\sqrt{p_{A}(1-p_{A})p_{B}(1-p_{B})}}
  22. A 1 B 1 A_{1}B_{1}
  23. x 11 x_{11}
  24. A 1 B 2 A_{1}B_{2}
  25. x 12 x_{12}
  26. A 2 B 1 A_{2}B_{1}
  27. x 21 x_{21}
  28. A 2 B 2 A_{2}B_{2}
  29. x 22 x_{22}
  30. A 1 A_{1}
  31. p 1 = x 11 + x 12 p_{1}=x_{11}+x_{12}
  32. A 2 A_{2}
  33. p 2 = x 21 + x 22 p_{2}=x_{21}+x_{22}
  34. B 1 B_{1}
  35. q 1 = x 11 + x 21 q_{1}=x_{11}+x_{21}
  36. B 2 B_{2}
  37. q 2 = x 12 + x 22 q_{2}=x_{12}+x_{22}
  38. A 1 B 1 A_{1}B_{1}
  39. A 1 A_{1}
  40. B 1 B_{1}
  41. A 1 A_{1}
  42. p 1 p_{1}
  43. B 1 B_{1}
  44. q 1 q_{1}
  45. A 1 B 1 A_{1}B_{1}
  46. x 11 x_{11}
  47. x 11 = p 1 q 1 x_{11}=p_{1}q_{1}
  48. D = x 11 - p 1 q 1 D=x_{11}-p_{1}q_{1}
  49. A 1 A_{1}
  50. A 2 A_{2}
  51. B 1 B_{1}
  52. x 11 = p 1 q 1 + D x_{11}=p_{1}q_{1}+D
  53. x 21 = p 2 q 1 - D x_{21}=p_{2}q_{1}-D
  54. q 1 q_{1}
  55. B 2 B_{2}
  56. x 12 = p 1 q 2 - D x_{12}=p_{1}q_{2}-D
  57. x 22 = p 2 q 2 + D x_{22}=p_{2}q_{2}+D
  58. q 2 q_{2}
  59. p 1 p_{1}
  60. p 2 p_{2}
  61. 1 1
  62. D D
  63. c c
  64. D = x 11 - p 1 q 1 D=x_{11}-p_{1}q_{1}
  65. x 11 x_{11}^{\prime}
  66. A 1 B 1 A_{1}B_{1}
  67. x 11 = ( 1 - c ) x 11 + c p 1 q 1 x_{11}^{\prime}=(1-c)\,x_{11}+c\,p_{1}q_{1}
  68. ( 1 - c ) (1-c)
  69. x 11 x_{11}
  70. A 1 B 1 A_{1}B_{1}
  71. c c
  72. A A
  73. A 1 A_{1}
  74. p 1 p_{1}
  75. B B
  76. B 1 B_{1}
  77. q 1 q_{1}
  78. x 11 - p 1 q 1 = ( 1 - c ) ( x 11 - p 1 q 1 ) x_{11}^{\prime}-p_{1}q_{1}=(1-c)\,(x_{11}-p_{1}q_{1})
  79. D 1 = ( 1 - c ) D 0 D_{1}=(1-c)\;D_{0}
  80. D D
  81. n n
  82. D n D_{n}
  83. D n = ( 1 - c ) n D 0 D_{n}=(1-c)^{n}\;D_{0}
  84. n n\to\infty
  85. ( 1 - c ) n 0 (1-c)^{n}\to 0
  86. D n D_{n}
  87. D D
  88. + +
  89. - -
  90. B 8 + B8^{+}
  91. B 8 - B8^{-}
  92. + +
  93. A 1 + A1^{+}
  94. a = 376 a=376
  95. b = 237 b=237
  96. C C
  97. - -
  98. A 1 - A1^{-}
  99. c = 91 c=91
  100. d = 1265 d=1265
  101. D D
  102. A A
  103. B B
  104. N N
  105. + +
  106. i i
  107. p f i = C / N = 0.311 pf_{i}=C/N=0.311\!
  108. + +
  109. j j
  110. p f j = A / N = 0.237 pf_{j}=A/N=0.237\!
  111. i i
  112. g f i = 1 - 1 - p f i = 0.170 gf_{i}=1-\sqrt{1-pf_{i}}=0.170\!
  113. h f i j = estimated frequency of haplotype i j = g f i g f j = 0.0215 hf_{ij}=\,\text{estimated frequency of haplotype }ij=gf_{i}\;gf_{j}=0.0215\!
  114. o [ h f x y ] = d / N o[hf_{xy}]=\sqrt{d/N}
  115. e [ h f x y ] = ( D / N ) ( B / N ) e[hf_{xy}]=\sqrt{(D/N)(B/N)}
  116. Δ i j \Delta_{ij}
  117. Δ i j = o [ h f x y ] - e [ h f x y ] = N d - B D N = 0.0769 \Delta_{ij}=o[hf_{xy}]-e[hf_{xy}]=\frac{\sqrt{Nd}-\sqrt{BD}}{N}=0.0769
  118. S E s SEs
  119. S E of g f i = C / ( 2 N ) = 0.00628 SE\,\text{ of }gf_{i}=\sqrt{C}/(2N)=0.00628
  120. S E of h f i j = ( 1 - d / B ) ( 1 - d / D ) - h f i j - h f i j 2 / 2 2 N = 0.00514 SE\,\text{ of }hf_{ij}=\sqrt{\frac{(1-\sqrt{d/B})(1-\sqrt{d/D})-hf_{ij}-hf_{ij% }^{2}/2}{2N}}=0.00514
  121. S E of Δ i j = 1 2 N a - 4 N Δ i j ( B + D 2 B D - B D N ) = 0.00367 SE\,\text{ of }\Delta_{ij}=\frac{1}{2N}\sqrt{a-4N\Delta_{ij}\left(\frac{B+D}{2% \sqrt{BD}}-\frac{\sqrt{BD}}{N}\right)}=0.00367
  122. t = Δ i j / ( S E of Δ i j ) t=\Delta_{ij}/(SE\,\text{ of }\Delta_{ij})
  123. Δ i j \Delta_{ij}
  124. Δ i j \Delta_{ij}
  125. t t
  126. Δ 0 \Delta_{0}
  127. Δ n = ( 1 - c ) n Δ 0 \Delta_{n}=(1-c)^{n}\Delta_{0}
  128. n 400 n\approx 400
  129. δ \delta
  130. B 27 + B27^{+}
  131. a = 96 a=96
  132. b = 77 b=77
  133. C C
  134. B 27 - B27^{-}
  135. c = 22 c=22
  136. d = 701 d=701
  137. D D
  138. A A
  139. B B
  140. N N
  141. x x
  142. x = a / b c / d = a d b c ( = 39.7 , in Table 3 ) x=\frac{a/b}{c/d}=\frac{ad}{bc}\;(=39.7,\,\text{ in Table 3 })
  143. y = ln ( x ) ( = 3.68 ) y=\ln(x)\;(=3.68)
  144. 1 w = 1 a + 1 b + 1 c + 1 d ( = 0.0703 ) \frac{1}{w}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}\;(=0.0703)
  145. χ 2 = w y 2 [ = 193 > χ 2 ( p = 0.001 , d f = 1 ) = 10.8 ] \chi^{2}=wy^{2}\;\left[=193>\chi^{2}(p=0.001,\;df=1)=10.8\right]
  146. d f = 1 df=1
  147. a , b , c , and d a,\;b,\;c,\,\text{ and }d
  148. x x
  149. 1 / w 1/w
  150. x = ( a + 1 / 2 ) ( d + 1 / 2 ) ( b + 1 / 2 ) ( c + 1 / 2 ) x=\frac{(a+1/2)(d+1/2)}{(b+1/2)(c+1/2)}
  151. 1 w = 1 a + 1 + 1 b + 1 + 1 c + 1 + 1 d + 1 \frac{1}{w}=\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}+\frac{1}{d+1}
  152. δ \delta
  153. δ \delta
  154. δ = F A D - F A P 1 - F A P , 0 δ 1 \delta=\frac{FAD-FAP}{1-FAP},\;\;0\leq\delta\leq 1
  155. F A D FAD
  156. F A P FAP
  157. δ \delta
  158. δ \delta
  159. = 6 =6
  160. χ 2 \chi^{2}
  161. χ 2 = ( a d - b c ) 2 N A B C D ( = 336 , for data in Table 3; P < 0.001 ) \chi^{2}=\frac{(ad-bc)^{2}N}{ABCD}\;(=336,\,\text{ for data in Table 3; }P<0.0% 01)
  162. d f = 1 df=1
  163. N 30 N\leq 30