wpmath0000016_7

Kernel_function_for_solving_integral_equation_of_surface_radiation_exchanges.html

  1. q ( r i ) = λ = 0 ψ i = 0 2 π θ i = 0 π 2 ε λ , i ( λ , ψ i , θ i , r i ) I b λ , i ( cos θ i sin θ i ) d θ i d ψ i d λ - j = 1 N λ = 0 ρ λ , i ( λ , ψ r , j , θ r , j , ψ j , θ j , r i ) I λ , k ( λ , ψ k , θ k , r i ) cos θ j cos θ k | r k - r j | 2 d A k q(r_{i})=\int_{\lambda=0}^{\infty}\int_{\psi_{i}=0}^{2\pi}\int_{\theta_{i}=0}^% {\frac{\pi}{2}}\varepsilon_{\lambda,i}(\lambda,\psi_{i},\theta_{i},r_{i})I_{b% \lambda,i}(\cos\theta_{i}\sin\theta_{i})\,d\theta_{i}\,d\psi_{i}\,d\lambda-% \sum_{j=1}^{N}\int_{\lambda=0}^{\infty}\rho_{\lambda,i}(\lambda,\psi_{r,j},% \theta_{r,j},\psi_{j},\theta_{j},r_{i})I_{\lambda,k}(\lambda,\psi_{k},\theta_{% k},r_{i})\frac{\cos\theta_{j}\cos\theta_{k}}{|r_{k}-r_{j}|^{2}}\,dA_{k}
  2. λ = wavelength of radiation rays , \lambda=\,\text{wavelength of radiation rays},
  3. I = radiation intensity , I=\,\text{radiation intensity},
  4. ε = emissivity . \varepsilon=\,\text{emissivity}.
  5. r = reflectivity r=\,\text{reflectivity}
  6. θ = angle between the normal of the surface and radiation exchange direction \theta=\,\text{angle between the normal of the surface and radiation exchange direction}
  7. ψ = azimuthal angle \psi=\,\text{azimuthal angle}
  8. q ( r ) + ε ( r ) E b = ε ( r ) K ( r , r ) [ E b ( r ) + 1 - ε ( r ) ε ( r ) d Γ ( r ) ] q(r)+\varepsilon(r)E_{b}=\varepsilon(r)\oint K(r,r^{\prime})\left[E_{b}(r^{% \prime})+1-\frac{\varepsilon(r^{\prime})}{\varepsilon(r)}d\Gamma(r^{\prime})\right]
  9. E b E_{b}
  10. E b ( r ) = σ T 4 ( r ) E_{b}(r)=\sigma T^{4}(r)\,
  11. σ \sigma
  12. K ( r , r ) = - n ( r - r ) n ( r - r ) π | r - r | 4 F = cos θ r cos θ r π | r - r | 4 F K(r,r^{\prime})=-\frac{n(r-r^{\prime})n^{\prime}(r-r^{\prime})}{\pi|r-r^{% \prime}|^{4}}F=\frac{\cos\theta_{r}\cos\theta_{r}^{\prime}}{\pi|r-r^{\prime}|^% {4}}F
  13. K ( r , r ) = - F n ( r - r ) n ( r - r ) π | r - r | 4 d z d z = n ( r - r ) n ( r - r ) π | r - r | 4 F K(r,r^{\prime})=-\int\int\ F\frac{n(r-r^{\prime})n^{\prime}(r-r^{\prime})}{\pi% |r-r^{\prime}|^{4}}\,dz^{\prime}\,dz=\frac{n(r-r^{\prime})n^{\prime}(r-r^{% \prime})}{\pi|r-r^{\prime}|^{4}}F
  14. n = ( cos θ , 0 , sin θ ) n=(\cos\theta,0,\sin\theta)
  15. n = ( cos θ sin ϕ , cos θ sin ϕ , sin θ ) n^{\prime}=(\cos\theta^{\prime}\sin\phi^{\prime},\cos\theta^{\prime}\sin\phi^{% \prime},\sin\theta^{\prime})
  16. K ( ϕ ) = ( c + d cos ϕ ) ( c ′′ + d cos ϕ ) π ( c + d cos ϕ ) 2 F K(\phi^{\prime})=\frac{(c^{\prime}+d\cos\phi^{\prime})(c^{\prime\prime}+d\cos% \phi^{\prime})}{\pi(c+d\cos\phi^{\prime})^{2}}F
  17. c = r i 2 + r j 2 + Z j 2 c=r_{i}^{2}+r_{j}^{2}+Z_{j}^{2}
  18. d = - 2 r i r j d=-2r_{i}r_{j}
  19. c = Z j sin θ - r i cos θ c^{\prime}=Z_{j}\sin\theta-r_{i}\cos\theta
  20. d = r j cos θ d^{\prime}=r_{j}\cos\theta
  21. c ′′ = Z j sin θ + r j cos θ c^{\prime\prime}=Z_{j}\sin\theta^{\prime}+r_{j}\cos\theta^{\prime}
  22. d ′′ = - r i cos θ d^{\prime\prime}=-r_{i}\cos\theta^{\prime}
  23. d { arctan ( c - d c + d tan ϕ 2 ) } d x = c 2 - d 2 2 ( c + d cos ϕ ) \frac{d\left\{\arctan\left(\sqrt{\frac{c-d}{c+d}}\tan\frac{\phi}{2}\right)% \right\}}{dx}=\frac{\sqrt{c^{2}-d^{2}}}{2(c+d\cos\phi)}
  24. k ¯ ( ϕ ) = 2 0 ϕ k ( ϕ ) d ϕ \bar{k}(\phi)=2\int\limits_{0}^{\phi}k(\phi^{\prime})\,d\phi^{\prime}
  25. k ¯ ( ϕ ) = - 2 π [ A ϕ + b arctan ( c - d c + d tan ϕ 2 ) + C sin ϕ c + d cos ϕ ] \bar{k}(\phi)=-\frac{2}{\pi}\left[A\phi+b\arctan\left(\sqrt{\frac{c-d}{c+d}}% \tan\frac{\phi}{2}\right)+C\frac{\sin\phi}{c+d\cos\phi}\right]
  26. A = d d ′′ d 2 A=\frac{d^{\prime}d^{\prime\prime}}{d^{2}}
  27. B = 2 ( c 2 - d 2 ) ( d f + e d ′′ ) + c d e f d ( c 2 - d 2 ) c 2 - d 2 B=2\frac{(c^{2}-d^{2})(d^{\prime}f+ed^{\prime\prime})+cdef}{d(c^{2}-d^{2})% \sqrt{c^{2}-d^{2}}}
  28. C = d e f d 2 - c 2 C=\frac{def}{d^{2}-c^{2}}
  29. e = d c - c d d e=\frac{dc^{\prime}-cd^{\prime}}{d}
  30. f = d c ′′ - c d ′′ d f=\frac{dc^{\prime\prime}-cd^{\prime\prime}}{d}

Kernel_methods_for_vector_output.html

  1. k k
  2. f f
  3. 𝐲 𝐢 \mathbf{y_{i}}
  4. 𝐱 𝐢 \mathbf{x_{i}}
  5. f ( 𝐱 𝐢 ) = 𝐲 𝐢 f(\mathbf{x_{i}})=\mathbf{y_{i}}
  6. i = 1 , , N i=1,\ldots,N
  7. 𝐱 𝐢 𝒳 \mathbf{x_{i}}\in\mathcal{X}
  8. 𝒳 = p \mathcal{X}=\mathbb{R}^{p}
  9. 𝐲 𝐢 D \mathbf{y_{i}}\in\mathbb{R}^{D}
  10. 𝐲 𝐢 \mathbf{y_{i}}
  11. 𝐱 𝐝 , 𝐢 \mathbf{x_{d,i}}
  12. p p
  13. 𝒳 \mathcal{X}
  14. f * f_{*}
  15. \mathcal{H}
  16. 𝐊 : 𝒳 × 𝒳 D × D \mathbf{K}:\mathcal{X}\times\mathcal{X}\rightarrow\mathbb{R}^{D\times D}
  17. k : 𝒳 × 𝒳 k:\mathcal{X}\times\mathcal{X}\rightarrow\mathbb{R}
  18. f * = argmin j = 1 D 1 N i = 1 N ( f j ( 𝐱 𝐢 ) - y j , i ) 2 + λ 𝐟 𝐊 2 f_{*}=\operatorname{argmin}\sum\limits_{j=1}^{D}\frac{1}{N}\sum\limits_{i=1}^{% N}(f_{j}(\mathbf{x_{i}})-y_{j,i})^{2}+\lambda\|\mathbf{f}\|_{\mathbf{K}}^{2}
  19. f * = argmin 1 N i = 1 N ( f ( 𝐱 𝐢 ) - y i ) 2 + λ 𝐟 k 2 f_{*}=\operatorname{argmin}\frac{1}{N}\sum\limits_{i=1}^{N}(f(\mathbf{x_{i}})-% y_{i})^{2}+\lambda\|\mathbf{f}\|_{k}^{2}
  20. {}^{\dagger}
  21. f * ( 𝐱 ) = i = 1 N 𝐊 ( 𝐱 𝐢 , 𝐱 ) c i f_{*}(\mathbf{x})=\sum\limits_{i=1}^{N}\mathbf{K}(\mathbf{x_{i}},\mathbf{x})c_% {i}
  22. 𝐜 ¯ = ( 𝐊 ( 𝐗 , 𝐗 ) + λ N ( I ) ) - 1 𝐲 ¯ \bar{\mathbf{c}}=(\mathbf{K}(\mathbf{X},\mathbf{X})+\lambda N\mathbf{(}I))^{-1% }\bar{\mathbf{y}}
  23. 𝐜 ¯ and 𝐲 ¯ \bar{\mathbf{c}}\,\text{ and }\bar{\mathbf{y}}
  24. N D ND
  25. 𝐊 ( 𝐗 , 𝐗 ) is an N D × N D \mathbf{K}(\mathbf{X},\mathbf{X})\,\text{ is an }ND\times ND
  26. N × N N\times N
  27. ( 𝐊 ( 𝐱 𝐢 , 𝐱 𝐣 ) ) d , d (\mathbf{K}(\mathbf{x_{i}},\mathbf{x_{j}}))_{d,d^{\prime}}
  28. f * ( 𝐱 ) = i = 1 N k ( 𝐱 𝐢 , 𝐱 ) c i = 𝐤 𝐱 𝐜 f_{*}(\mathbf{x})=\sum\limits_{i=1}^{N}k(\mathbf{x_{i}},\mathbf{x})c_{i}=% \mathbf{k}_{\mathbf{x}}^{\intercal}\mathbf{c}
  29. 𝐜 \mathbf{c}
  30. f * f_{*}
  31. 𝐜 = ( 𝐊 + λ I ) - 1 𝐲 \mathbf{c}=(\mathbf{K}+\lambda I)^{-1}\mathbf{y}
  32. 𝐊 i j = k ( 𝐱 𝐢 , 𝐱 𝐣 ) = i th element of 𝐤 𝐱 𝐣 \mathbf{K}_{ij}=k(\mathbf{x_{i}},\mathbf{x_{j}})=i^{\,\text{th}}\,\text{ % element of }\mathbf{k}_{\mathbf{x_{j}}}
  33. {}^{\dagger}
  34. 𝐊 \mathbf{K}
  35. R R
  36. 𝒳 × { 1 , , D } \mathcal{X}\times\{1,\ldots,D\}
  37. ( 𝐊 ( x , x ) ) d , d = R ( ( x , d ) , ( x , d ) ) (\mathbf{K}(x,x^{\prime}))_{d,d^{\prime}}=R((x,d),(x^{\prime},d^{\prime}))
  38. 𝐟 \,\textbf{f}
  39. D D
  40. { f d } d = 1 D \left\{f_{d}\right\}_{d=1}^{D}
  41. 𝐟 𝒢 𝒫 ( 𝐦 , 𝐊 ) \,\textbf{f}\sim\mathcal{GP}(\,\textbf{m},\,\textbf{K})
  42. 𝐦 : 𝒳 𝐑 D \,\textbf{m}:\mathcal{X}\to\,\textbf{R}^{D}
  43. { m d ( 𝐱 ) } d = 1 D \left\{m_{d}(\,\textbf{x})\right\}_{d=1}^{D}
  44. 𝐊 \,\textbf{K}
  45. ( 𝐊 ( 𝐱 , 𝐱 ) ) d , d (\,\textbf{K}(\,\textbf{x},\,\textbf{x}^{\prime}))_{d,d^{\prime}}
  46. f d ( 𝐱 ) f_{d}(\,\textbf{x})
  47. f d ( 𝐱 ) f_{d^{\prime}}(\,\textbf{x}^{\prime})
  48. 𝐗 \,\textbf{X}
  49. 𝐟 ( 𝐗 ) \,\textbf{f}(\,\textbf{X})
  50. 𝒩 ( 𝐦 ( 𝐗 ) , 𝐊 ( 𝐗 , 𝐗 ) ) \mathcal{N}(\,\textbf{m}(\,\textbf{X}),\,\textbf{K}(\,\textbf{X},\,\textbf{X}))
  51. 𝐦 ( 𝐗 ) \,\textbf{m}(\,\textbf{X})
  52. 𝐊 ( 𝐗 , 𝐗 ) \,\textbf{K}(\,\textbf{X},\,\textbf{X})
  53. p ( 𝐲 𝐟 , 𝐱 , Σ ) = 𝒩 ( 𝐟 ( 𝐱 ) , Σ ) p(\,\textbf{y}\mid\,\textbf{f},\,\textbf{x},\Sigma)=\mathcal{N}(\,\textbf{f}(% \,\textbf{x}),\Sigma)
  54. Σ 𝐑 D × D \Sigma\in\mathcal{\,\textbf{R}}^{D\times D}
  55. { σ d 2 } d = 1 D \left\{\sigma_{d}^{2}\right\}_{d=1}^{D}
  56. 𝐱 * \,\textbf{x}_{*}
  57. p ( 𝐟 ( 𝐱 * ) 𝐒 , 𝐟 , 𝐱 * , ϕ ) = 𝒩 ( 𝐟 * ( 𝐱 * ) , 𝐊 * ( 𝐱 * , 𝐱 * ) ) p(\,\textbf{f}(\,\textbf{x}_{*})\mid\,\textbf{S},\,\textbf{f},\,\textbf{x}_{*}% ,\phi)=\mathcal{N}(\,\textbf{f}_{*}(\,\textbf{x}_{*}),\,\textbf{K}_{*}(\,% \textbf{x}_{*},\,\textbf{x}_{*}))
  58. 𝐒 \,\textbf{S}
  59. ϕ \phi
  60. 𝐊 ( 𝐱 , 𝐱 ) \,\textbf{K}(\,\textbf{x},\,\textbf{x}^{\prime})
  61. Σ \Sigma
  62. 𝐟 * \,\textbf{f}_{*}
  63. 𝐊 * \,\textbf{K}_{*}
  64. 𝐟 * ( 𝐱 * ) = 𝐊 𝐱 * T ( 𝐊 ( 𝐗 , 𝐗 ) + s y m b o l Σ ) - 1 𝐲 ¯ \,\textbf{f}_{*}(\,\textbf{x}_{*})=\,\textbf{K}_{\,\textbf{x}_{*}}^{T}(\,% \textbf{K}(\,\textbf{X},\,\textbf{X})+symbol\Sigma)^{-1}\bar{\,\textbf{y}}
  65. 𝐊 * ( 𝐱 * , 𝐱 * ) = 𝐊 ( 𝐱 * , 𝐱 * ) - 𝐊 𝐱 * ( 𝐊 ( 𝐗 , 𝐗 ) + s y m b o l Σ ) - 1 𝐊 𝐱 * T \,\textbf{K}_{*}(\,\textbf{x}_{*},\,\textbf{x}_{*})=\,\textbf{K}(\,\textbf{x}_% {*},\,\textbf{x}_{*})-\,\textbf{K}_{\,\textbf{x}_{*}}(\,\textbf{K}(\,\textbf{X% },\,\textbf{X})+symbol\Sigma)^{-1}\,\textbf{K}_{\,\textbf{x}_{*}}^{T}
  66. s y m b o l Σ = Σ 𝐈 N , 𝐊 𝐱 * 𝐑 D × N D symbol\Sigma=\Sigma\otimes\,\textbf{I}_{N},\,\textbf{K}_{\,\textbf{x}_{*}}\in% \mathcal{\,\textbf{R}}^{D\times ND}
  67. ( 𝐊 ( 𝐱 * , 𝐱 j ) ) d , d (\,\textbf{K}(\,\textbf{x}_{*},\,\textbf{x}_{j}))_{d,d^{\prime}}
  68. j = 1 , , N j=1,\cdots,N
  69. d , d = 1 , , D d,d^{\prime}=1,\cdots,D
  70. 𝐟 * \,\textbf{f}^{*}
  71. ( 𝐊 ( 𝐱 , 𝐱 ) ) d , d = k ( 𝐱 , 𝐱 ) k T ( d , d ) (\mathbf{K}(\mathbf{x},\mathbf{x^{\prime}}))_{d,d^{\prime}}=k(\mathbf{x},% \mathbf{x^{\prime}})k_{T}(d,d^{\prime})
  72. k k
  73. 𝒳 × 𝒳 \mathcal{X}\times\mathcal{X}
  74. k T k_{T}
  75. { 1 , , D } × { 1 , , D } \{1,\ldots,D\}\times\{1,\ldots,D\}
  76. 𝐊 ( 𝐱 , 𝐱 ) = k ( 𝐱 , 𝐱 ) 𝐁 \mathbf{K}(\mathbf{x},\mathbf{x^{\prime}})=k(\mathbf{x},\mathbf{x^{\prime}})% \mathbf{B}
  77. 𝐁 \mathbf{B}
  78. D × D D\times D
  79. 𝐁 \mathbf{B}
  80. k T k_{T}
  81. f f
  82. R ( 𝐟 ) = A ω ( C ω l = 1 D f l k 2 + ω D l = 1 D f l - f ¯ k 2 ) R(\mathbf{f})=A_{\omega}(C_{\omega}\sum\limits_{l=1}^{D}\|f_{l}\|_{k}^{2}+% \omega D\sum\limits_{l=1}^{D}\|f_{l}-\bar{f}\|_{k}^{2})
  83. A ω = 1 2 ( 1 - ω ) ( 1 - ω + ω D ) A_{\omega}=\frac{1}{2(1-\omega)(1-\omega+\omega D)}
  84. C ω = ( 2 - 2 ω + ω D ) C_{\omega}=(2-2\omega+\omega D)
  85. f ¯ = 1 D q = 1 D f q \bar{f}=\frac{1}{D}\sum\limits_{q=1}^{D}f_{q}
  86. K ω ( x , x ) = k ( x , x ) ( ω 𝟏 + ( 1 - ω ) 𝐈 D K_{\omega}(x,x^{\prime})=k(x,x^{\prime})(\omega\mathbf{1}+(1-\omega)\mathbf{I}% _{D}
  87. 𝟏 is a D × D \mathbf{1}\,\text{ is a }D\times D
  88. f l f_{l}
  89. ω = 0 \omega=0
  90. ω = 1 \omega=1
  91. R ( 𝐟 ) = ε 1 c = 1 r l I ( c ) f l - f c ¯ k 2 + ε 2 c = 1 r m c f c ¯ k 2 R(\mathbf{f})=\varepsilon_{1}\sum_{c=1}^{r}\sum_{l\in I(c)}\|f_{l}-\bar{f_{c}}% \|_{k}^{2}+\varepsilon_{2}\sum\limits_{c=1}^{r}m_{c}\|\bar{f_{c}}\|_{k}^{2}
  92. I ( c ) I(c)
  93. c c
  94. m c m_{c}
  95. c c
  96. f c ¯ = 1 m c q I ( c ) f q \bar{f_{c}}=\frac{1}{m_{c}}\sum\limits_{q\in I(c)}f_{q}
  97. 𝐌 l , q = 1 m c \mathbf{M}_{l,q}=\frac{1}{m_{c}}
  98. l l
  99. q q
  100. c c
  101. 𝐌 l , q = 0 \mathbf{M}_{l,q}=0
  102. K ( x , x ) = k ( x , x ) 𝐆 K(x,x^{\prime})=k(x,x^{\prime})\mathbf{G}^{\dagger}
  103. 𝐆 l , q = ε 1 δ l q + ( ε 2 - ε 1 ) 𝐌 l , q \mathbf{G}_{l,q}=\varepsilon_{1}\delta_{lq}+(\varepsilon_{2}-\varepsilon_{1})% \mathbf{M}_{l,q}
  104. r r
  105. R ( 𝐟 ) = 1 2 l , q = 1 D f l - f q k 2 𝐌 l q + l = 1 D f l k 2 𝐌 l , l R(\mathbf{f})=\frac{1}{2}\sum\limits_{l,q=1}^{D}\|f_{l}-f_{q}\|_{k}^{2}\mathbf% {M}_{lq}+\sum\limits_{l=1}^{D}\|f_{l}\|_{k}^{2}\mathbf{M}_{l,l}
  106. 𝐌 is a D × D \mathbf{M}\,\text{ is a }D\times D
  107. K ( x , x ) = k ( x , x ) 𝐋 K(x,x^{\prime})=k(x,x^{\prime})\mathbf{L}^{\dagger}
  108. 𝐋 = 𝐃 - 𝐌 \mathbf{L}=\mathbf{D}-\mathbf{M}
  109. 𝐃 l , q = δ l , q ( h = 1 D 𝐌 l , h + 𝐌 l , q ) \mathbf{D}_{l,q}=\delta_{l,q}(\sum\limits_{h=1}^{D}\mathbf{M}_{l,h}+\mathbf{M}% _{l,q})
  110. 𝐋 \mathbf{L}
  111. 𝐁 \mathbf{B}
  112. 𝐁 \mathbf{B}
  113. 𝐁 \mathbf{B}
  114. 𝐟 \mathbf{f}
  115. D D
  116. { f d ( 𝐱 ) } d = 1 D \left\{f_{d}(\,\textbf{x})\right\}_{d=1}^{D}
  117. 𝐱 𝐑 p \,\textbf{x}\in\mathcal{\,\textbf{R}}^{p}
  118. f d f_{d}
  119. f d ( 𝐱 ) = q = 1 Q a d , q u q ( 𝐱 ) f_{d}(\,\textbf{x})=\sum_{q=1}^{Q}{a_{d,q}u_{q}(\,\textbf{x})}
  120. a d , q a_{d,q}
  121. u q ( 𝐱 ) u_{q}(\,\textbf{x})
  122. [ u q ( 𝐱 ) , u q ( 𝐱 ) ] = k q ( 𝐱 , 𝐱 ) [u_{q}(\,\textbf{x}),u_{q^{\prime}}(\,\textbf{x}^{\prime})]=k_{q}(\,\textbf{x}% ,\,\textbf{x}^{\prime})
  123. q = q q=q^{\prime}
  124. f d ( 𝐱 ) f_{d}(\,\textbf{x})
  125. f d ( 𝐱 ) f_{d^{\prime}}(\,\textbf{x})
  126. cov [ f d ( 𝐱 ) , f d ( 𝐱 ) ] = q = 1 Q i = 1 R q a d , q i a d , q i k q ( 𝐱 , 𝐱 ) = q = 1 Q b d , d q k q ( 𝐱 , 𝐱 ) \operatorname{cov}[f_{d}(\,\textbf{x}),f_{d^{\prime}}(\,\textbf{x}^{\prime})]=% \sum_{q=1}^{Q}{\sum_{i=1}^{R_{q}}{a_{d,q}^{i}a_{d^{\prime},q}^{i}k_{q}(\,% \textbf{x},\,\textbf{x}^{\prime})}}=\sum_{q=1}^{Q}{b_{d,d^{\prime}}^{q}k_{q}(% \,\textbf{x},\,\textbf{x}^{\prime})}
  127. u q i ( 𝐱 ) u_{q}^{i}(\,\textbf{x})
  128. q = 1 , , Q q=1,\cdots,Q
  129. i = 1 , , R q i=1,\cdots,R_{q}
  130. [ u q i ( 𝐱 ) , u q i ( 𝐱 ) ] = k q ( 𝐱 , 𝐱 ) [u_{q}^{i}(\,\textbf{x}),u_{q^{\prime}}^{i^{\prime}}(\,\textbf{x})^{\prime}]=k% _{q}(\,\textbf{x},\,\textbf{x}^{\prime})
  131. i = i i=i^{\prime}
  132. q = q q=q^{\prime}
  133. cov [ f d ( 𝐱 ) , f d ( 𝐱 ) ] \operatorname{cov}[f_{d}(\,\textbf{x}),f_{d^{\prime}}(\,\textbf{x}^{\prime})]
  134. ( 𝐊 ( 𝐱 , 𝐱 ) ) d , d (\,\textbf{K}(\,\textbf{x},\,\textbf{x}^{\prime}))_{d,d^{\prime}}
  135. 𝐊 ( 𝐱 , 𝐱 ) \,\textbf{K}(\,\textbf{x},\,\textbf{x}^{\prime})
  136. 𝐊 ( 𝐱 , 𝐱 ) = q = 1 Q 𝐁 q k q ( 𝐱 , 𝐱 ) \,\textbf{K}(\,\textbf{x},\,\textbf{x}^{\prime})=\sum_{q=1}^{Q}{\,\textbf{B}_{% q}k_{q}(\,\textbf{x},\,\textbf{x}^{\prime})}
  137. 𝐁 q 𝐑 D × D \,\textbf{B}_{q}\in\mathcal{\,\textbf{R}}^{D\times D}
  138. 𝐱 \,\textbf{x}
  139. 𝐁 q \,\textbf{B}_{q}
  140. { f d ( 𝐱 ) } d = 1 D \left\{f_{d}(\,\textbf{x})\right\}_{d=1}^{D}
  141. k q ( 𝐱 , 𝐱 ) k_{q}(\,\textbf{x},\,\textbf{x}^{\prime})
  142. Q = 1 Q=1
  143. b d , d q b_{d,d^{\prime}}^{q}
  144. 𝐁 q \,\textbf{B}_{q}
  145. b d , d q = v d , d b q b_{d,d^{\prime}}^{q}=v_{d,d^{\prime}}b_{q}
  146. v d , d v_{d,d^{\prime}}
  147. b d , d q b_{d,d^{\prime}}^{q}
  148. cov [ f d ( 𝐱 ) , f d ( 𝐱 ) ] = q = 1 Q v d , d b q k q ( 𝐱 , 𝐱 ) = v d , d q = 1 Q b q k q ( 𝐱 , 𝐱 ) = v d , d k ( 𝐱 , 𝐱 ) \operatorname{cov}[f_{d}(\,\textbf{x}),f_{d^{\prime}}(\,\textbf{x}^{\prime})]=% \sum_{q=1}^{Q}{v_{d,d^{\prime}}b_{q}k_{q}(\,\textbf{x},\,\textbf{x}^{\prime})}% =v_{d,d^{\prime}}\sum_{q=1}^{Q}{b_{q}k_{q}(\,\textbf{x},\,\textbf{x}^{\prime})% }=v_{d,d^{\prime}}k(\,\textbf{x},\,\textbf{x}^{\prime})
  149. k ( 𝐱 , 𝐱 ) = q = 1 Q b q k q ( 𝐱 , 𝐱 ) k(\,\textbf{x},\,\textbf{x}^{\prime})=\sum_{q=1}^{Q}{b_{q}k_{q}(\,\textbf{x},% \,\textbf{x}^{\prime})}
  150. v d , d = i = 1 R 1 a d , 1 i a d , 1 i = b d , d 1 v_{d,d^{\prime}}=\sum_{i=1}^{R_{1}}{a_{d,1}^{i}a_{d^{\prime},1}^{i}}=b_{d,d^{% \prime}}^{1}
  151. 𝐊 ( 𝐱 , 𝐱 ) = k ( 𝐱 , 𝐱 ) 𝐁 \,\textbf{K}(\,\textbf{x},\,\textbf{x}^{\prime})=k(\,\textbf{x},\,\textbf{x}^{% \prime})\,\textbf{B}
  152. k q ( 𝐱 , 𝐱 ) k_{q}(\,\textbf{x},\,\textbf{x}^{\prime})
  153. R q = 1 R_{q}=1
  154. Q = 1 Q=1
  155. u q u_{q}
  156. 𝐱 \,\textbf{x}
  157. 𝐱 \,\textbf{x}
  158. 𝐊 ( 𝐗 , 𝐗 ) \mathbf{K}(\mathbf{X},\mathbf{X})
  159. 𝐁 \mathbf{B}
  160. 𝐜 ¯ \bar{\mathbf{c}}
  161. 𝐜 ¯ d = ( k ( 𝐗 , 𝐗 ) + λ N σ d 𝐈 ) - 1 𝐲 ¯ d σ d \bar{\mathbf{c}}^{d}=(k(\mathbf{X},\mathbf{X})+\frac{\lambda_{N}}{\sigma_{d}}% \mathbf{I})^{-1}\frac{\bar{\mathbf{y}}^{d}}{\sigma_{d}}
  162. ϕ \phi
  163. ϕ \phi
  164. 𝐊 ( 𝐗 , 𝐗 ) ¯ = 𝐊 ( 𝐗 , 𝐗 ) + s y m b o l Σ \overline{\,\textbf{K}(\,\textbf{X},\,\textbf{X})}=\,\textbf{K}(\,\textbf{X},% \,\textbf{X})+symbol\Sigma

Kernel_perceptron.html

  1. 𝐰 \mathbf{w}
  2. b b
  3. 𝐱 \mathbf{x}
  4. y ^ = sgn ( 𝐰 𝐱 ) \hat{y}=\operatorname{sgn}(\mathbf{w}^{\top}\mathbf{x})
  5. ŷ ŷ
  6. 𝐰 \mathbf{w}
  7. p p
  8. 𝐱 \mathbf{x}ᵢ
  9. ŷ y ŷ≠yᵢ
  10. 𝐰 𝐰 + y 𝐱 \mathbf{w}←\mathbf{w}+yᵢ\mathbf{x}ᵢ
  11. 𝐱 \mathbf{x}
  12. sgn i α i y i K ( 𝐱 i , 𝐱 ) \operatorname{sgn}\sum_{i}\alpha_{i}y_{i}K(\mathbf{x}_{i},\mathbf{x^{\prime}})
  13. K K
  14. Φ Φ
  15. K ( 𝐱 , 𝐱 ) = Φ ( 𝐱 ) · Φ ( 𝐱 ) ) K(\mathbf{x},\mathbf{x})=Φ(\mathbf{x})·Φ(\mathbf{x}))
  16. 𝐱 K ( 𝐱 , 𝐱 ) ) \mathbf{x}↦K(\mathbf{x}ᵢ,\mathbf{x}))
  17. 𝐰 \mathbf{w}
  18. n n
  19. 𝐰 = i n α i y i 𝐱 i \mathbf{w}=\sum_{i}^{n}\alpha_{i}y_{i}\mathbf{x}_{i}
  20. 𝐰 \mathbf{w}
  21. α \mathbf{α}
  22. 𝐰 \mathbf{w}
  23. y ^ \displaystyle\hat{y}
  24. Φ Φ
  25. Φ ( 𝐱 ) Φ(\mathbf{x})
  26. α \mathbf{α}
  27. n n
  28. 𝐱 , y \mathbf{x},y
  29. y ^ = sgn i n α i y i K ( 𝐱 i , 𝐱 ) \hat{y}=\operatorname{sgn}\sum_{i}^{n}\alpha_{i}y_{i}K(\mathbf{x}_{i},\mathbf{% x})
  30. ŷ y ŷ≠y
  31. α < s u b > i α i + 1 α<sub>i←α_{i}+1

Kernel_random_forest.html

  1. k k
  2. k k\in\mathbb{N}
  3. 𝒟 n = { ( 𝐗 i , Y i ) } i = 1 n \mathcal{D}_{n}=\{(\mathbf{X}_{i},Y_{i})\}_{i=1}^{n}
  4. [ 0 , 1 ] p × [0,1]^{p}\times\mathbb{R}
  5. ( 𝐗 , Y ) (\mathbf{X},Y)
  6. 𝔼 [ Y 2 ] < \mathbb{E}[Y^{2}]<\infty
  7. Y Y
  8. 𝐗 \mathbf{X}
  9. m ( 𝐱 ) = 𝔼 [ Y | 𝐗 = 𝐱 ] m(\mathbf{x})=\mathbb{E}[Y|\mathbf{X}=\mathbf{x}]
  10. M M
  11. m n ( 𝐱 , 𝚯 j ) m_{n}(\mathbf{x},\mathbf{\Theta}_{j})
  12. 𝐱 \mathbf{x}
  13. j j
  14. 𝚯 1 , , 𝚯 M \mathbf{\Theta}_{1},\cdots,\mathbf{\Theta}_{M}
  15. 𝚯 \mathbf{\Theta}
  16. 𝒟 n \mathcal{D}_{n}
  17. m M , n ( 𝐱 , Θ 1 , , Θ M ) = 1 M j = 1 M m n ( 𝐱 , Θ j ) m_{M,n}(\mathbf{x},\Theta_{1},\cdots,\Theta_{M})=\frac{1}{M}\sum_{j=1}^{M}m_{n% }(\mathbf{x},\Theta_{j})
  18. m n = i = 1 n Y i 𝟏 𝐗 i A n ( 𝐱 , Θ j ) N n ( 𝐱 , Θ j ) m_{n}=\sum_{i=1}^{n}\frac{Y_{i}\mathbf{1}_{\mathbf{X}_{i}\in A_{n}(\mathbf{x},% \Theta_{j})}}{N_{n}(\mathbf{x},\Theta_{j})}
  19. A n ( 𝐱 , Θ j ) A_{n}(\mathbf{x},\Theta_{j})
  20. 𝐱 \mathbf{x}
  21. Θ j \Theta_{j}
  22. 𝒟 n \mathcal{D}_{n}
  23. N n ( 𝐱 , Θ j ) = i = 1 n 𝟏 𝐗 i A n ( 𝐱 , Θ j ) N_{n}(\mathbf{x},\Theta_{j})=\sum_{i=1}^{n}\mathbf{1}_{\mathbf{X}_{i}\in A_{n}% (\mathbf{x},\Theta_{j})}
  24. 𝐱 [ 0 , 1 ] d \mathbf{x}\in[0,1]^{d}
  25. m M , n ( 𝐱 , Θ 1 , , Θ M ) = 1 M j = 1 M ( i = 1 n Y i 𝟏 𝐗 i A n ( 𝐱 , Θ j ) N n ( 𝐱 , Θ j ) ) m_{M,n}(\mathbf{x},\Theta_{1},\ldots,\Theta_{M})=\frac{1}{M}\sum_{j=1}^{M}% \left(\sum_{i=1}^{n}\frac{Y_{i}\mathbf{1}_{\mathbf{X}_{i}\in A_{n}(\mathbf{x},% \Theta_{j})}}{N_{n}(\mathbf{x},\Theta_{j})}\right)
  26. m ~ M , n ( 𝐱 , Θ 1 , , Θ M ) = 1 j = 1 M N n ( 𝐱 , Θ j ) j = 1 M i = 1 n Y i 𝟏 𝐗 i A n ( 𝐱 , Θ j ) \tilde{m}_{M,n}(\mathbf{x},\Theta_{1},\ldots,\Theta_{M})=\frac{1}{\sum_{j=1}^{% M}N_{n}(\mathbf{x},\Theta_{j})}\sum_{j=1}^{M}\sum_{i=1}^{n}Y_{i}\mathbf{1}_{% \mathbf{X}_{i}\in A_{n}(\mathbf{x},\Theta_{j})}
  27. Y i Y_{i}
  28. 𝐱 \mathbf{x}
  29. K M , n ( 𝐱 , 𝐳 ) = 1 M j = 1 M 𝟏 𝐗 i A n ( 𝐱 , Θ j ) K_{M,n}(\mathbf{x},\mathbf{z})=\frac{1}{M}\sum_{j=1}^{M}\mathbf{1}_{\mathbf{X}% _{i}\in A_{n}(\mathbf{x},\Theta_{j})}
  30. M M
  31. m ~ M , n ( 𝐱 , Θ 1 , , Θ M ) = i = 1 n Y i K M , n ( 𝐱 , 𝐗 i ) = 1 n K M , n ( 𝐱 , 𝐗 ) \tilde{m}_{M,n}(\mathbf{x},\Theta_{1},\ldots,\Theta_{M})=\frac{\sum_{i=1}^{n}Y% _{i}K_{M,n}(\mathbf{x},\mathbf{X}_{i})}{\sum_{\ell=1}^{n}K_{M,n}(\mathbf{x},% \mathbf{X}_{\ell})}
  32. k k
  33. m ~ M , n ( 𝐱 , Θ 1 , , Θ M ) \tilde{m}_{M,n}(\mathbf{x},\Theta_{1},\ldots,\Theta_{M})
  34. K k c c ( 𝐱 , 𝐳 ) = k 1 , , k d , j = 1 d k j = k k ! k 1 ! k d ! ( 1 d ) k j = 1 d 𝟏 2 k j x j = 2 k j z j K_{k}^{cc}(\mathbf{x},\mathbf{z})=\sum_{k_{1},\ldots,k_{d},\sum_{j=1}^{d}k_{j}% =k}\frac{k!}{k_{1}!\ldots k_{d}!}\left(\frac{1}{d}\right)^{k}\prod_{j=1}^{d}% \mathbf{1}_{\lceil 2^{k_{j}}x_{j}\rceil=\lceil 2^{k_{j}}z_{j}\rceil}
  35. 𝐱 , 𝐳 [ 0 , 1 ] d \mathbf{x},\mathbf{z}\in[0,1]^{d}
  36. m ~ M , n ( 𝐱 , Θ 1 , , Θ M ) \tilde{m}_{M,n}(\mathbf{x},\Theta_{1},\ldots,\Theta_{M})
  37. K k u f ( 𝟎 , 𝐱 ) = k 1 , , k d , j = 1 d k j = k k ! k 1 ! k d ! ( 1 d ) k m = 1 d ( 1 - | x m | j = 0 k m - 1 ( - ln | x m | ) j j ! ) K_{k}^{uf}(\mathbf{0},\mathbf{x})=\sum_{k_{1},\ldots,k_{d},\sum_{j=1}^{d}k_{j}% =k}\frac{k!}{k_{1}!\ldots k_{d}!}\left(\frac{1}{d}\right)^{k}\prod_{m=1}^{d}% \left(1-|x_{m}|\sum_{j=0}^{k_{m}-1}\frac{(-\ln|x_{m}|)^{j}}{j!}\right)
  38. 𝐱 [ 0 , 1 ] d \mathbf{x}\in[0,1]^{d}
  39. Assume that there exist sequences ( a n ) , ( b n ) such that, a.s. , a n N n ( 𝐱 , Θ ) b n and a n 1 M m = 1 M N n 𝐱 , Θ m b n , then almost surely , \,\text{Assume that there exist sequences }(a_{n}),(b_{n})\,\text{ such that, % a.s.},a_{n}\leq N_{n}(\mathbf{x},\Theta)\leq b_{n}\,\text{ and }a_{n}\leq\frac% {1}{M}\sum_{m=1}^{M}N_{n}{\mathbf{x},\Theta_{m}}\leq b_{n},\,\text{ then % almost surely},
  40. | m M , n ( 𝐱 ) - m ~ M , n ( 𝐱 ) | b n - a n a n m ~ M , n ( 𝐱 ) |m_{M,n}(\mathbf{x})-\tilde{m}_{M,n}(\mathbf{x})|\leq\frac{b_{n}-a_{n}}{a_{n}}% \tilde{m}_{M,n}(\mathbf{x})
  41. M M
  42. Assume that there exist sequences ( ε n ) , ( a n ) , ( b n ) such that, a.s. \,\text{Assume that there exist sequences }(\varepsilon_{n}),(a_{n}),(b_{n})\,% \text{ such that, a.s.}
  43. 𝔼 [ N n ( 𝐱 , Θ ) ] 1 \mathbb{E}[N_{n}(\mathbf{x},\Theta)]\geq 1
  44. [ a n N n ( 𝐱 , Θ ) b n 𝒟 n ] 1 - ε n / 2 \mathbb{P}[a_{n}\leq N_{n}(\mathbf{x},\Theta)\leq b_{n}\mid\mathcal{D}_{n}]% \geq 1-\varepsilon_{n}/2
  45. [ a n 𝔼 Θ [ N n ( 𝐱 , Θ ) ] b n 𝒟 n ] 1 - ε n / 2 \mathbb{P}[a_{n}\leq\mathbb{E}_{\Theta}[N_{n}(\mathbf{x},\Theta)]\leq b_{n}% \mid\mathcal{D}_{n}]\geq 1-\varepsilon_{n}/2
  46. Then almost surely, | m , n ( 𝐱 - m ~ , n ( 𝐱 ) | b n - a n a n m ~ , n ( 𝐱 ) + n ε n ( max 1 i n Y i ) \,\text{Then almost surely,}|m_{\infty,n}(\mathbf{x}-\tilde{m}_{\infty,n}(% \mathbf{x})|\leq\frac{b_{n}-a_{n}}{a_{n}}\tilde{m}_{\infty,n}(\mathbf{x})+n% \varepsilon_{n}\left(\max_{1\leq i\leq n}Y_{i}\right)
  47. Y = m ( 𝐗 ) + ε Y=m(\mathbf{X})+\varepsilon
  48. ε \varepsilon
  49. 𝐗 \mathbf{X}
  50. σ 2 < \sigma^{2}<\infty
  51. 𝐗 \mathbf{X}
  52. [ 0 , 1 ] d [0,1]^{d}
  53. m m
  54. k k\rightarrow\infty
  55. n / 2 k n/2^{k}\rightarrow\infty
  56. C 1 > 0 C_{1}>0
  57. n n
  58. 𝔼 [ m ~ n c c ( 𝐗 ) - m ( 𝐗 ) ] 2 C 1 n - 1 / ( 3 + d log 2 ) ( log n ) 2 \mathbb{E}[\tilde{m}_{n}^{cc}(\mathbf{X})-m(\mathbf{X})]^{2}\leq C_{1}n^{-1/(3% +d\log 2)}(\log n)^{2}
  59. k k\rightarrow\infty
  60. n / 2 k n/2^{k}\rightarrow\infty
  61. C > 0 C>0
  62. 𝔼 [ m ~ n u f ( 𝐗 ) - m ( 𝐗 ) ] 2 C n - 2 / ( 6 + 3 d log 2 ) ( log n ) 2 \mathbb{E}[\tilde{m}_{n}^{uf}(\mathbf{X})-m(\mathbf{X})]^{2}\leq Cn^{-2/(6+3d% \log 2)}(\log n)^{2}

Ket_(software).html

  1. a ( b + c ) = a b + a c a(b+c)=ab+ac

Key-independent_optimality.html

  1. m m
  2. X = x 1 , x 2 , , x m X=x_{1},x_{2},\cdots,x_{m}
  3. x i x_{i}
  4. 1 1
  5. n n
  6. X X
  7. 𝑂𝑃𝑇 ( X ) \,\textit{OPT}(X)
  8. X X
  9. b b
  10. n ! n!
  11. 1 , 2 , , n 1,2,\cdots,n
  12. b ( i ) b(i)
  13. i i
  14. b b
  15. b ( X ) = b ( x 1 ) , b ( x 2 ) , , b ( x m ) b(X)=b(x_{1}),b(x_{2}),\cdots,b(x_{m})
  16. X X
  17. 𝐾𝐼𝑂𝑃𝑇 ( X ) = E [ 𝑂𝑃𝑇 ( b ( X ) ) ] \,\textit{KIOPT}(X)=E[\,\textit{OPT}(b(X))]
  18. X X
  19. O ( 𝐾𝐼𝑂𝑃𝑇 ( X ) ) O(\,\textit{KIOPT}(X))

Kingman's_subadditive_ergodic_theorem.html

  1. T T
  2. ( Ω , Σ , μ ) (\Omega,\Sigma,\mu)
  3. { g n } n \{g_{n}\}_{n\in\mathbb{N}}
  4. L 1 L^{1}
  5. g n + m ( x ) g n ( x ) + g m ( T n x ) g_{n+m}(x)\leq g_{n}(x)+g_{m}(T^{n}x)
  6. lim n g n ( x ) n = g ( x ) - \lim_{n\to\infty}\frac{g_{n}(x)}{n}=g(x)\geq-\infty
  7. μ \mu
  8. g n ( x ) := j = 0 n - 1 f ( T n x ) g_{n}(x):=\sum_{j=0}^{n-1}f(T^{n}x)

Kirkwood–Buff_solution_theory.html

  1. i i
  2. j j
  3. s y m b o l r i symbol{r}_{i}
  4. s y m b o l r i symbol{r}_{i}
  5. g i j ( s y m b o l R ) = ρ i j ( s y m b o l R ) ρ i j bulk g_{ij}(symbol{R})=\frac{\rho_{ij}(symbol{R})}{\rho_{ij}\text{bulk}}
  6. ρ i j ( s y m b o l R ) \rho_{ij}(symbol{R})
  7. j j
  8. i i
  9. ρ i j bulk \rho_{ij}\text{bulk}
  10. j j
  11. s y m b o l R = | s y m b o l r i - s y m b o l r j | symbol{R}=|symbol{r}_{i}-symbol{r}_{j}|
  12. g i j ( s y m b o l R ) = g j i ( s y m b o l R ) g_{ij}(symbol{R})=g_{ji}(symbol{R})
  13. g i j ( r ) = ρ i j ( r ) ρ i j bulk g_{ij}(r)=\frac{\rho_{ij}(r)}{\rho_{ij}\text{bulk}}
  14. r = | s y m b o l R | r=|symbol{R}|
  15. P M F i j ( r ) = - k T ln ( g i j ) PMF_{ij}(r)=-kT\ln(g_{ij})
  16. i i
  17. j j
  18. G i j = V [ g i j ( s y m b o l R ) - 1 ] d s y m b o l R G_{ij}=\int\limits_{V}[g_{ij}(symbol{R})-1]\,dsymbol{R}
  19. G i j = 4 π r = 0 [ g i j ( r ) - 1 ] r 2 d r G_{ij}=4\pi\int_{r=0}^{\infty}[g_{ij}(r)-1]r^{2}\,dr
  20. G 11 G_{11}
  21. G 22 G_{22}
  22. G 12 G_{12}
  23. ν ¯ 1 = 1 + c 2 ( G 22 + G 12 ) c 1 + c 2 + c 1 c 2 ( G 11 + G 22 - 2 G 12 ) \bar{\nu}_{1}=\frac{1+c_{2}(G_{22}+G_{12})}{c_{1}+c_{2}+c_{1}c_{2}(G_{11}+G_{2% 2}-2G_{12})}
  24. c c
  25. c 1 ν ¯ 1 + c 2 ν ¯ 2 = 1 c_{1}\bar{\nu}_{1}+c_{2}\bar{\nu}_{2}=1
  26. κ \kappa
  27. κ k T = 1 + c 1 G 11 + c 2 G 22 + c 1 c 2 ( G 11 G 22 - G 12 ) 2 c 1 + c 2 + c 1 c 2 ( G 11 + G 22 - 2 G 12 ) \kappa kT=\frac{1+c_{1}G_{11}+c_{2}G_{22}+c_{1}c_{2}(G_{11}G_{22}-G_{12})^{2}}% {c_{1}+c_{2}+c_{1}c_{2}(G_{11}+G_{22}-2G_{12})}
  28. k k
  29. T T
  30. Π \Pi
  31. ( Π c 2 ) T , μ 1 = k T 1 + c 2 G 22 \left(\frac{\partial\Pi}{\partial c_{2}}\right)_{T,\mu_{1}}=\frac{kT}{1+c_{2}G% _{22}}
  32. μ 1 \mu_{1}
  33. T T
  34. P P
  35. 1 k T ( μ 1 c 1 ) T , P = 1 c 2 + G 12 - G 11 1 + c 1 ( G 11 - G 12 ) \frac{1}{kT}\left(\frac{\partial\mu_{1}}{\partial c_{1}}\right)_{T,P}=\frac{1}% {c_{2}}+\frac{G_{12}-G_{11}}{1+c1(G_{11}-G_{12})}
  36. 1 k T ( μ 2 c 2 ) T , P = 1 c 2 + G 12 - G 22 1 + c 2 ( G 22 - G 12 ) \frac{1}{kT}\left(\frac{\partial\mu_{2}}{\partial c_{2}}\right)_{T,P}=\frac{1}% {c_{2}}+\frac{G_{12}-G_{22}}{1+c2(G_{22}-G_{12})}
  37. 1 k T ( μ 2 χ 2 ) T , P = 1 χ 2 + c 1 ( 2 G 12 - G 11 - G 22 ) 1 + c 1 χ 2 ( G 11 + G 22 - 2 G 12 ) \frac{1}{kT}\left(\frac{\partial\mu_{2}}{\partial\chi_{2}}\right)_{T,P}=\frac{% 1}{\chi_{2}}+\frac{c_{1}(2G_{12}-G_{11}-G_{22})}{1+c1\chi_{2}(G_{11}+G_{22}-2G% _{12})}
  38. Γ \Gamma
  39. Γ W \Gamma_{W}
  40. Γ W = M W ( G W S - G C S ) \Gamma_{W}=M_{W}\left(G_{WS}-G_{CS}\right)
  41. M W M_{W}
  42. Γ W = - M W G C S \Gamma_{W}=-M_{W}G_{CS}
  43. G C S G_{CS}

Kirwan_map.html

  1. H G * ( M ) H * ( M / / p G ) H^{*}_{G}(M)\to H^{*}(M/\!/_{p}G)
  2. μ : M 𝔤 * \mu:M\to{\mathfrak{g}}^{*}
  3. H G * ( M ) H^{*}_{G}(M)
  4. E G × G M EG\times_{G}M
  5. M / / p G = μ - 1 ( p ) / G M/\!/_{p}G=\mu^{-1}(p)/G
  6. p Z ( 𝔤 * ) p\in Z({\mathfrak{g}}^{*})
  7. μ \mu
  8. μ - 1 ( p ) M \mu^{-1}(p)\hookrightarrow M
  9. H G * ( μ - 1 ( p ) ) = H * ( M / / p G ) H_{G}^{*}(\mu^{-1}(p))=H^{*}(M/\!/_{p}G)

Klam_value.html

  1. O ( n c ) + f ( k ) O(n^{c})+f(k)
  2. n n
  3. k k
  4. c c
  5. n n
  6. k k
  7. f f
  8. k k
  9. f ( k ) f(k)
  10. O ( n c ) O(n^{c})
  11. c c
  12. f ( k ) = O ( k 2 1.3248 k ) f(k)=O(k^{2}1.3248^{k})
  13. k 2 1.3248 k = 10 20 k^{2}1.3248^{k}=10^{20}
  14. k 2 k^{2}
  15. O ( 1.3248 k ) O(1.3248^{k})
  16. f ( k ) = c k f(k)=c^{k}
  17. k = 20 / log 10 c k=20/\log_{10}c
  18. f ( k ) = O ( 1.2738 k ) f(k)=O(1.2738^{k})

Kleinian_integer.html

  1. m + n 1 + - 7 2 m+n\frac{1+\sqrt{-7}}{2}
  2. ( - 7 ) \mathbb{Q}(-7)

Kleitman–Wang_algorithms.html

  1. S = ( ( a 1 , b 1 ) , , ( a n , b n ) ) S=((a_{1},b_{1}),\dots,(a_{n},b_{n}))
  2. ( a i , b i ) (a_{i},b_{i})
  3. b i > 0 b_{i}>0
  4. S S
  5. S = ( ( a 1 - 1 , b 1 ) , , ( a b i - 1 - 1 , b b i - 1 ) , ( a b i , 0 ) , ( a b i + 1 - 1 , b b i + 1 ) , ( a b i + 2 , b b i + 2 ) , , ( a n , b n ) ) S^{\prime}=((a_{1}-1,b_{1}),\dots,(a_{b_{i}-1}-1,b_{b_{i}-1}),(a_{b_{i}},0),(a% _{b_{i}+1}-1,b_{b_{i}+1}),(a_{b_{i}+2},b_{b_{i}+2}),\dots,(a_{n},b_{n}))
  6. ( a i , b i ) (a_{i},b_{i})
  7. ( a j , 0 ) (a_{j},0)
  8. S S
  9. n n
  10. S := S S:=S^{\prime}
  11. S S^{\prime}
  12. ( 0 , 0 ) (0,0)
  13. v 1 , , v n v_{1},\dots,v_{n}
  14. S S
  15. S S^{\prime}
  16. ( v i , v 1 ) , ( v i , v 2 ) , , ( v i , v b i - 1 ) , ( v i , v b i + 1 ) (v_{i},v_{1}),(v_{i},v_{2}),\dots,(v_{i},v_{b_{i}-1}),(v_{i},v_{b_{i}+1})
  17. S S
  18. S S^{\prime}
  19. S S
  20. S = ( ( a 1 , b 1 ) , , ( a n , b n ) ) S=((a_{1},b_{1}),\dots,(a_{n},b_{n}))
  21. a 1 a 2 a n a_{1}\geq a_{2}\geq\cdots\geq a_{n}
  22. ( a i , b i ) (a_{i},b_{i})
  23. ( b i , a i ) (b_{i},a_{i})
  24. ( b 1 , a 1 ) , , ( b n , a n ) (b_{1},a_{1}),\dots,(b_{n},a_{n})
  25. S S
  26. S = ( ( a 1 - 1 , b 1 ) , , ( a b i - 1 - 1 , b b i - 1 ) , ( a b i , 0 ) , ( a b i + 1 - 1 , b b i + 1 ) , ( a b i + 2 , b b i + 2 ) , , ( a n , b n ) ) S^{\prime}=((a_{1}-1,b_{1}),\cdots,(a_{b_{i}-1}-1,b_{b_{i}-1}),(a_{b_{i}},0),(% a_{b_{i}+1}-1,b_{b_{i}+1}),(a_{b_{i}+2},b_{b_{i}+2}),\dots,(a_{n},b_{n}))
  27. S S
  28. S S
  29. n n
  30. S := S S:=S^{\prime}
  31. S S^{\prime}
  32. ( 0 , 0 ) (0,0)
  33. v 1 , , v n v_{1},\dots,v_{n}
  34. S S
  35. S S^{\prime}
  36. ( v i , v 1 ) , ( v i , v 2 ) , , ( v i , v b i - 1 ) , ( v i , v b i + 1 ) (v_{i},v_{1}),(v_{i},v_{2}),\dots,(v_{i},v_{b_{i}-1}),(v_{i},v_{b_{i}+1})
  37. S S
  38. S S^{\prime}
  39. S S

Klingen_Eisenstein_series.html

  1. ( a b c d ) \isin P r Γ g f ( a τ + b c τ + d ) det ( c τ + d ) - k . \sum_{{\left({{ab}\atop{cd}}\right)}\isin P_{r}\setminus\Gamma_{g}}f\left(% \frac{a\tau+b}{c\tau+d}\right)\det(c\tau+d)^{-k}.

Kneser–Ney_smoothing.html

  1. p K N ( w i | w i - 1 ) = m a x ( c ( w i - 1 , w i ) - δ , 0 ) w c ( w i - 1 , w ) + p K N ( w i ) p_{KN}(w_{i}|w_{i-1})=\frac{max(c(w_{i-1},w_{i})-\delta,0)}{\sum_{w^{\prime}}c% (w_{i-1},w^{\prime})}+p_{KN}(w_{i})
  2. p K N ( w i | w i - n + 1 i - 1 ) = m a x ( c ( w i - n + 1 i , w i ) - δ , 0 ) w i c ( w i - n + 1 i ) + δ w i c ( w i - n + 1 i ) ( c ( w i - n + 1 i - 1 ) ) p K N ( w i | w i - n + 2 i - 1 ) p_{KN}(w_{i}|w_{i-n+1}^{i-1})=\frac{max(c(w_{i-n+1}^{i},w_{i})-\delta,0)}{\sum% _{w_{i}}c(w_{i-n+1}^{i})}+\frac{\delta}{\sum_{w_{i}}c(w_{i-n+1}^{i})}(c(w_{i-n% +1}^{i-1}))p_{KN}(w_{i}|w_{i-n+2}^{i-1})
  3. p K N ( w i ) p_{KN}(w_{i})
  4. c ( w i - 1 , w i ) c(w_{i-1},w_{i})
  5. δ \delta

Kosnita's_theorem.html

  1. A B C ABC
  2. O O
  3. O a , O b , O c O_{a},O_{b},O_{c}
  4. O B C OBC
  5. O C A OCA
  6. O A B OAB
  7. A O a AO_{a}
  8. B O b BO_{b}
  9. C O c CO_{c}
  10. X ( 54 ) X(54)

Koszul_duality.html

  1. A = T ( V ) / R , A=T(V)/R,
  2. T ( V ) T(V)
  3. V V ( = T 2 ( V ) ) V\otimes V(=T^{2}(V))
  4. A ! := T ( V * ) / R A^{!}:=T(V^{*})/R^{\prime}
  5. V * V^{*}
  6. R V * V * R^{\prime}\subset V^{*}\otimes V^{*}
  7. A = S ( V ) A=S(V)
  8. A ! = Λ ( V * ) A^{!}=\Lambda(V^{*})
  9. ( A ! ) opp = Ext * ( k , k ) . (A^{!})^{\,\text{opp}}=\operatorname{Ext}^{*}(k,k).
  10. A A
  11. A ! A^{!}
  12. D ( A - 𝐌𝐨𝐝 ) D ( A ! - 𝐌𝐨𝐝 ) D(A\,\text{-}\mathbf{Mod})\rightleftarrows D(A^{!}\,\text{-}\mathbf{Mod})
  13. Ω X \Omega_{X}
  14. D X D_{X}
  15. A A ! A\mapsto A^{!}
  16. P n P^{n}

KPP-type_equation.html

  1. u t - α u - β u 2 + γ u 3 = 0 u_{t}-\alpha u-\beta u^{2}+\gamma u^{3}=0

Kuratowski–Ulam_theorem.html

  1. A X × Y A\subset X\times Y
  2. { x X : A x is meager (resp. comeager) in Y } \{x\in X:A_{x}\,\text{ is meager (resp. comeager) in }Y\}
  3. A x = π Y [ A { x } × Y ] A_{x}=\pi_{Y}[A\cap\{x\}\times Y]
  4. π Y \pi_{Y}

Kuwahara_filter.html

  1. I ( x , y ) I(x,y)
  2. 2 a + 1 2a+1
  3. ( x , y ) (x,y)
  4. Q i = 1 4 Q_{i=1\cdots 4}
  5. Q i ( x , y ) = { [ x , x + a ] × [ y , y + a ] if i = 1 [ x - a , x ] × [ y , y + a ] if i = 2 [ x - a , x ] × [ y - a , y ] if i = 3 [ x , x + a ] × [ y - a , y ] if i = 4 Q_{i}(x,y)=\begin{cases}\left[x,x+a\right]\times\left[y,y+a\right]&\mbox{ if }% ~{}i=1\\ \left[x-a,x\right]\times\left[y,y+a\right]&\mbox{ if }~{}i=2\\ \left[x-a,x\right]\times\left[y-a,y\right]&\mbox{ if }~{}i=3\\ \left[x,x+a\right]\times\left[y-a,y\right]&\mbox{ if }~{}i=4\\ \end{cases}
  6. × \times
  7. m i ( x , y ) m_{i}(x,y)
  8. σ i ( x , y ) \sigma_{i}(x,y)
  9. Φ ( x , y ) \Phi(x,y)
  10. ( x , y ) (x,y)
  11. Φ ( x , y ) = { m 1 ( x , y ) if σ 1 ( x , y ) = m i n i σ i ( x , y ) m 2 ( x , y ) if σ 2 ( x , y ) = m i n i σ i ( x , y ) m 3 ( x , y ) if σ 3 ( x , y ) = m i n i σ i ( x , y ) m 4 ( x , y ) if σ 4 ( x , y ) = m i n i σ i ( x , y ) \Phi(x,y)=\begin{cases}m_{1}(x,y)&\mbox{ if }~{}\sigma_{1}(x,y)=min_{i}\mbox{ % }~{}\sigma_{i}(x,y)\\ m_{2}(x,y)&\mbox{ if }~{}\sigma_{2}(x,y)=min_{i}\mbox{ }~{}\sigma_{i}(x,y)\\ m_{3}(x,y)&\mbox{ if }~{}\sigma_{3}(x,y)=min_{i}\mbox{ }~{}\sigma_{i}(x,y)\\ m_{4}(x,y)&\mbox{ if }~{}\sigma_{4}(x,y)=min_{i}\mbox{ }~{}\sigma_{i}(x,y)\\ \end{cases}
  12. m 1 + m 2 2 \frac{m_{1}+m_{2}}{2}
  13. D D
  14. 2 d - 1 × 2 d - 1 2d-1\times 2d-1
  15. d × d d\times d
  16. d × d d\times d
  17. d 2 d^{2}

Ky_Fan_lemma.html

  1. B n B_{n}
  2. S n - 1 S_{n-1}
  3. B n B_{n}
  4. S n - 1 S_{n-1}
  5. S n - 1 S_{n-1}
  6. L : V ( T ) { 0 } L:V(T)\to\mathbb{Z}\setminus\{0\}
  7. v S n - 1 v\in S_{n-1}
  8. L ( - v ) = - L ( v ) L(-v)=-L(v)
  9. B n B_{n}
  10. L : V ( T ) { + 1 , - 1 , + 2 , - 2 , , + n , - n } L:V(T)\to\{+1,-1,+2,-2,\ldots,+n,-n\}
  11. n = 1 n=1
  12. B n B_{n}
  13. [ - 1 , 1 ] [-1,1]
  14. { - 1 , 1 } \{-1,1\}
  15. L ( - 1 ) = - L ( + 1 ) L(-1)=-L(+1)
  16. L ( - 1 ) = - 1 L(-1)=-1
  17. L ( + 1 ) = + 1 L(+1)=+1
  18. n = 2 n=2
  19. B n B_{n}
  20. L ( - v ) = - L ( v ) L(-v)=-L(v)

K–omega_turbulence_model.html

  1. ( ρ k ) t + ( ρ u j k ) x j = P - β * ρ ω k + x j [ ( μ + σ k ρ k ω ) k x j ] , with P = τ i j u i x j , \displaystyle\frac{\partial(\rho k)}{\partial t}+\frac{\partial(\rho u_{j}k)}{% \partial x_{j}}=P-\beta^{*}\rho\omega k+\frac{\partial}{\partial x_{j}}\left[% \left(\mu+\sigma_{k}\frac{\rho k}{\omega}\right)\frac{\partial k}{\partial x_{% j}}\right],\qquad\,\text{with }P=\tau_{ij}\frac{\partial u_{i}}{\partial x_{j}},

L-cysteine:1D-myo-inositol_2-amino-2-deoxy-alpha-D-glucopyranoside_ligase.html

  1. \rightleftharpoons

L-infinity.html

  1. L L_{\infty}
  2. x = sup n | x n | , \|x\|_{\infty}=\sup_{n}|x_{n}|,
  3. L L_{\infty}
  4. L L_{\infty}
  5. ( S , Σ , μ ) (S,Σ,μ)
  6. S S
  7. 𝐑 \mathbf{R}
  8. f f
  9. f inf { C 0 : | f ( x ) | C for almost every x } . \|f\|_{\infty}\equiv\inf\{C\geq 0:|f(x)|\leq C\,\text{ for almost every }x\}.
  10. L L_{\infty}
  11. L L_{\infty}

L-lysine_cyclodeaminase.html

  1. \rightleftharpoons

Labdatriene_synthase.html

  1. \rightleftharpoons

Lagrangian_(field_theory).html

  1. \scriptstyle\mathcal{L}
  2. δ 𝒮 δ φ i = 0 , \frac{\delta\mathcal{S}}{\delta\varphi_{i}}=0,\,
  3. 𝒮 \scriptstyle\mathcal{S}
  4. 𝒮 [ φ i ] = ( φ i ( s ) , φ i ( s ) s α , s α ) d n s \mathcal{S}\left[\varphi_{i}\right]=\int{\mathcal{L}\left(\varphi_{i}(s),\frac% {\partial\varphi_{i}(s)}{\partial s^{\alpha}},s^{\alpha}\right)\,\mathrm{d}^{n% }s}
  5. \scriptstyle\mathcal{L}\,
  6. φ φ
  7. ( ϕ , ϕ , ϕ / t , 𝐱 , t ) \mathcal{L}(\phi,\nabla\phi,\partial\phi/\partial t,\mathbf{x},t)
  8. ( ϕ 1 , ϕ 1 , ϕ 1 / t , , ϕ 2 , ϕ 2 , ϕ 2 / t , , 𝐱 , t ) \mathcal{L}(\phi_{1},\nabla\phi_{1},\partial\phi_{1}/\partial t,\ldots,\phi_{2% },\nabla\phi_{2},\partial\phi_{2}/\partial t,\ldots,\mathbf{x},t)
  9. 𝒮 = L d t , \mathcal{S}=\int L\,\mathrm{d}t\,,
  10. \mathcal{L}
  11. 𝒮 [ ϕ ] = ( ϕ , ϕ , ϕ / t , 𝐱 , t ) d 3 𝐱 d t . \mathcal{S}[\phi]=\int\mathcal{L}(\phi,\nabla\phi,\partial\phi/\partial t,% \mathbf{x},t)\,\mathrm{d}^{3}\mathbf{x}\mathrm{d}t.
  12. L = d 3 x . L=\int\mathcal{L}\,d^{3}x\,.
  13. \mathcal{L}
  14. \mathcal{L}
  15. g \sqrt{g}
  16. 𝒞 \scriptstyle\mathcal{C}
  17. m \scriptstyle\mathbb{R}^{m}
  18. n \scriptstyle\mathbb{R}^{n}
  19. 𝒮 : 𝒞 \mathcal{S}:\mathcal{C}\rightarrow\mathbb{R}
  20. \scriptstyle\mathbb{R}
  21. \scriptstyle\mathbb{C}
  22. φ 𝒞 \scriptstyle\varphi\ \in\ \mathcal{C}
  23. 𝒮 [ φ ] \scriptstyle\mathcal{S}[\varphi]
  24. φ \scriptstyle\varphi
  25. ( φ , φ , φ , , x ) \mathcal{L}(\varphi,\partial\varphi,\partial\partial\varphi,...,x)
  26. φ 𝒞 , 𝒮 [ φ ] M d n x ( φ ( x ) , φ ( x ) , φ ( x ) , , x ) . \forall\varphi\in\mathcal{C},\ \ \mathcal{S}[\varphi]\equiv\int_{M}\mathrm{d}^% {n}x\mathcal{L}\big(\varphi(x),\partial\varphi(x),\partial\partial\varphi(x),.% ..,x\big).
  27. φ \scriptstyle\varphi
  28. φ \scriptstyle\varphi
  29. 𝒞 \scriptstyle\mathcal{C}
  30. φ \scriptstyle\varphi
  31. φ \scriptstyle\varphi
  32. φ \scriptstyle\varphi
  33. δ 𝒮 δ φ = - μ ( ( μ φ ) ) + φ = 0. \frac{\delta\mathcal{S}}{\delta\varphi}=-\partial_{\mu}\left(\frac{\partial% \mathcal{L}}{\partial(\partial_{\mu}\varphi)}\right)+\frac{\partial\mathcal{L}% }{\partial\varphi}=0.
  34. φ \scriptstyle\varphi
  35. ( 𝐱 , t ) \scriptstyle(\mathbf{x},t)
  36. Φ ( 𝐱 , t ) \scriptstyle\Phi(\mathbf{x},t)
  37. Φ ( 𝐱 , t ) \scriptstyle\Phi(\mathbf{x},t)
  38. \scriptstyle\mathcal{L}
  39. ( 𝐱 , t ) = - ρ ( 𝐱 , t ) Φ ( 𝐱 , t ) - 1 8 π G ( Φ ( 𝐱 , t ) ) 2 \mathcal{L}(\mathbf{x},t)=-\rho(\mathbf{x},t)\Phi(\mathbf{x},t)-{1\over 8\pi G% }(\nabla\Phi(\mathbf{x},t))^{2}
  40. δ ( 𝐱 , t ) = - ρ ( 𝐱 , t ) δ Φ ( 𝐱 , t ) - 2 8 π G ( Φ ( 𝐱 , t ) ) ( δ Φ ( 𝐱 , t ) ) . \delta\mathcal{L}(\mathbf{x},t)=-\rho(\mathbf{x},t)\delta\Phi(\mathbf{x},t)-{2% \over 8\pi G}(\nabla\Phi(\mathbf{x},t))\cdot(\nabla\delta\Phi(\mathbf{x},t)).
  41. 0 = - ρ ( 𝐱 , t ) + 1 4 π G Φ ( 𝐱 , t ) 0=-\rho(\mathbf{x},t)+{1\over 4\pi G}\nabla\cdot\nabla\Phi(\mathbf{x},t)
  42. 4 π G ρ ( 𝐱 , t ) = 2 Φ ( 𝐱 , t ) 4\pi G\rho(\mathbf{x},t)=\nabla^{2}\Phi(\mathbf{x},t)
  43. GR = EH + matter = c 4 16 π G ( R - 2 Λ ) + matter \mathcal{L}\text{GR}=\mathcal{L}\text{EH}+\mathcal{L}\text{matter}=\frac{c^{4}% }{16\pi G}\left(R-2\Lambda\right)+\mathcal{L}\text{matter}
  44. R \scriptstyle R
  45. EH \mathcal{L}\text{EH}
  46. g μ ν g_{\mu\nu}
  47. R μ ν - 1 2 R g μ ν + g μ ν Λ = 8 π G c 4 T μ ν R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+g_{\mu\nu}\Lambda=\frac{8\pi G}{c^{4}}T_{\mu\nu}
  48. T μ ν - 2 - g δ ( matter - g ) δ g μ ν = - 2 δ matter δ g μ ν + g μ ν matter . T_{\mu\nu}\equiv\frac{-2}{\sqrt{-g}}\frac{\delta(\mathcal{L}_{\mathrm{matter}}% \sqrt{-g})}{\delta g^{\mu\nu}}=-2\frac{\delta\mathcal{L}_{\mathrm{matter}}}{% \delta g^{\mu\nu}}+g_{\mu\nu}\mathcal{L}_{\mathrm{matter}}.
  49. g \scriptstyle g
  50. Λ \Lambda
  51. - g d 4 x \sqrt{-g}d^{4}x
  52. - q ϕ ( 𝐱 ( t ) , t ) + q 𝐱 ˙ ( t ) 𝐀 ( 𝐱 ( t ) , t ) -q\phi(\mathbf{x}(t),t)+q\dot{\mathbf{x}}(t)\cdot\mathbf{A}(\mathbf{x}(t),t)
  53. 𝐣 \scriptstyle\mathbf{j}
  54. ( 𝐱 , t ) = - ρ ( 𝐱 , t ) ϕ ( 𝐱 , t ) + 𝐣 ( 𝐱 , t ) 𝐀 ( 𝐱 , t ) + ϵ 0 2 E 2 ( 𝐱 , t ) - 1 2 μ 0 B 2 ( 𝐱 , t ) . \mathcal{L}(\mathbf{x},t)=-\rho(\mathbf{x},t)\phi(\mathbf{x},t)+\mathbf{j}(% \mathbf{x},t)\cdot\mathbf{A}(\mathbf{x},t)+{\epsilon_{0}\over 2}{E}^{2}(% \mathbf{x},t)-{1\over{2\mu_{0}}}{B}^{2}(\mathbf{x},t).
  55. 0 = - ρ ( 𝐱 , t ) + ϵ 0 𝐄 ( 𝐱 , t ) 0=-\rho(\mathbf{x},t)+\epsilon_{0}\nabla\cdot\mathbf{E}(\mathbf{x},t)
  56. 𝐀 \scriptstyle\mathbf{A}
  57. 0 = 𝐣 ( 𝐱 , t ) + ϵ 0 𝐄 ˙ ( 𝐱 , t ) - 1 μ 0 × 𝐁 ( 𝐱 , t ) 0=\mathbf{j}(\mathbf{x},t)+\epsilon_{0}\dot{\mathbf{E}}(\mathbf{x},t)-{1\over% \mu_{0}}\nabla\times\mathbf{B}(\mathbf{x},t)
  58. - ρ ϕ ( 𝐱 , t ) + 𝐣 𝐀 -\rho\phi(\mathbf{x},t)+\mathbf{j}\cdot\mathbf{A}
  59. j μ = ( ρ , 𝐣 ) and A μ = ( - ϕ , 𝐀 ) j^{\mu}=(\rho,\mathbf{j})\quad\,\text{and}\quad A_{\mu}=(-\phi,\mathbf{A})
  60. - ρ ϕ + 𝐣 𝐀 = j μ A μ -\rho\phi+\mathbf{j}\cdot\mathbf{A}=j^{\mu}A_{\mu}
  61. F μ ν F_{\mu\nu}
  62. F μ ν = μ A ν - ν A μ F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}
  63. ϵ 0 2 E 2 - 1 2 μ 0 B 2 = - 1 4 μ 0 F μ ν F μ ν = - 1 4 μ 0 F μ ν F ρ σ η μ ρ η ν σ {\epsilon_{0}\over 2}{E}^{2}-{1\over{2\mu_{0}}}{B}^{2}=-\frac{1}{4\mu_{0}}F_{% \mu\nu}F^{\mu\nu}=-\frac{1}{4\mu_{0}}F_{\mu\nu}F_{\rho\sigma}\eta^{\mu\rho}% \eta^{\nu\sigma}
  64. μ F μ ν = - μ 0 j ν and ϵ μ ν λ σ ν F λ σ = 0 \partial_{\mu}F^{\mu\nu}=-\mu_{0}j^{\nu}\quad\,\text{and}\quad\epsilon^{\mu\nu% \lambda\sigma}\partial_{\nu}F_{\lambda\sigma}=0
  65. ( x ) = j μ ( x ) A μ ( x ) - 1 4 μ 0 F μ ν ( x ) F μ ν ( x ) \mathcal{L}(x)=j^{\mu}(x)A_{\mu}(x)-\frac{1}{4\mu_{0}}F_{\mu\nu}(x)F^{\mu\nu}(x)
  66. matter \mathcal{L}\text{matter}
  67. ( x ) = j μ ( x ) A μ ( x ) - 1 4 μ 0 F μ ν ( x ) F ρ σ ( x ) g μ ρ ( x ) g ν σ ( x ) + c 4 16 π G R ( x ) = Maxwell + Einstein-Hilbert . \begin{aligned}\displaystyle\mathcal{L}(x)&\displaystyle=j^{\mu}(x)A_{\mu}(x)-% {1\over 4\mu_{0}}F_{\mu\nu}(x)F_{\rho\sigma}(x)g^{\mu\rho}(x)g^{\nu\sigma}(x)+% \frac{c^{4}}{16\pi G}R(x)\\ &\displaystyle=\mathcal{L}\text{Maxwell}+\mathcal{L}\text{Einstein-Hilbert}.% \end{aligned}
  68. g μ ν ( x ) g_{\mu\nu}(x)
  69. T μ ν ( x ) = 2 - g ( x ) δ δ g μ ν ( x ) 𝒮 Maxwell = 1 μ 0 ( F λ μ ( x ) F ν λ ( x ) - 1 4 g μ ν ( x ) F ρ σ ( x ) F ρ σ ( x ) ) T^{\mu\nu}(x)=\frac{2}{\sqrt{-g(x)}}\frac{\delta}{\delta g_{\mu\nu}(x)}% \mathcal{S}\text{Maxwell}=\frac{1}{\mu_{0}}\left(F^{\mu}_{\,\text{ }\lambda}(x% )F^{\nu\lambda}(x)-\frac{1}{4}g^{\mu\nu}(x)F_{\rho\sigma}(x)F^{\rho\sigma}(x)\right)
  70. T = g μ ν T μ ν = 0 T=g_{\mu\nu}T^{\mu\nu}=0
  71. R = - 8 π G c 4 T R=-\frac{8\pi G}{c^{4}}T
  72. R μ ν = 8 π G c 4 1 μ 0 ( F λ μ ( x ) F ν λ ( x ) - 1 4 g μ ν ( x ) F ρ σ ( x ) F ρ σ ( x ) ) R^{\mu\nu}=\frac{8\pi G}{c^{4}}\frac{1}{\mu_{0}}\left(F^{\mu}_{\,\text{ }% \lambda}(x)F^{\nu\lambda}(x)-\frac{1}{4}g^{\mu\nu}(x)F_{\rho\sigma}(x)F^{\rho% \sigma}(x)\right)
  73. D μ F μ ν = - μ 0 j ν D_{\mu}F^{\mu\nu}=-\mu_{0}j^{\nu}
  74. D μ D_{\mu}
  75. j μ = 0 j^{\mu}=0
  76. d s 2 = ( 1 - 2 M r + Q 2 r 2 ) d t 2 - ( 1 - 2 M r + Q 2 r 2 ) - 1 d r 2 - r 2 d Ω 2 ds^{2}=\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\right)dt^{2}-\left(1-\frac{2M}% {r}+\frac{Q^{2}}{r^{2}}\right)^{-1}dr^{2}-r^{2}d\Omega^{2}
  77. \scriptstyle\mathcal{M}
  78. 𝒮 [ 𝐀 ] = ( - 1 2 𝐅 𝐅 + 𝐀 𝐉 ) . \mathcal{S}[\mathbf{A}]=\int_{\mathcal{M}}\left(-\frac{1}{2}\,\mathbf{F}\wedge% \star\mathbf{F}+\mathbf{A}\wedge\star\mathbf{J}\right).
  79. d 𝐅 = 𝐉 . \mathrm{d}{\star}\mathbf{F}=\mathbf{J}.
  80. d 𝐅 = 0 \mathrm{d}\mathbf{F}=0
  81. = i c ψ ¯ / ψ - m c 2 ψ ¯ ψ \mathcal{L}=i\hbar c\bar{\psi}{\partial}\!\!/\ \psi-mc^{2}\bar{\psi}\psi
  82. ψ ¯ = ψ γ 0 \scriptstyle\bar{\psi}=\psi^{\dagger}\gamma^{0}
  83. / {\partial}\!\!/
  84. γ σ σ \scriptstyle\gamma^{\sigma}\partial_{\sigma}\!
  85. QED = i c ψ ¯ D / ψ - m c 2 ψ ¯ ψ - 1 4 μ 0 F μ ν F μ ν \mathcal{L}_{\mathrm{QED}}=i\hbar c\bar{\psi}{D}\!\!\!\!/\ \psi-mc^{2}\bar{% \psi}\psi-{1\over 4\mu_{0}}F_{\mu\nu}F^{\mu\nu}
  86. F μ ν \scriptstyle F^{\mu\nu}\!
  87. D / {D}\!\!\!\!/
  88. γ σ D σ \scriptstyle\gamma^{\sigma}D_{\sigma}\!
  89. QCD = n ( i c ψ ¯ n D / ψ n - m n c 2 ψ ¯ n ψ n ) - 1 4 G α G α μ ν μ ν \mathcal{L}_{\mathrm{QCD}}=\sum_{n}\left(i\hbar c\bar{\psi}_{n}{D}\!\!\!\!/\ % \psi_{n}-m_{n}c^{2}\bar{\psi}_{n}\psi_{n}\right)-{1\over 4}G^{\alpha}{}_{\mu% \nu}G_{\alpha}{}^{\mu\nu}
  90. G α μ ν \scriptstyle G^{\alpha}{}_{\mu\nu}\!
  91. ( ϕ , μ ϕ , x μ ) \mathcal{L}(\phi,\partial_{\mu}\phi,x_{\mu})
  92. μ μ
  93. ( ϕ , ϕ x , ϕ y , ϕ z , ϕ t , x , y , z , t ) \mathcal{L}\left(\phi,\frac{\partial\phi}{\partial x},\frac{\partial\phi}{% \partial y},\frac{\partial\phi}{\partial z},\frac{\partial\phi}{\partial t},x,% y,z,t\right)

Laguerre_formula.html

  1. ϕ \phi
  2. ϕ = | 1 2 i Log Cr ( I 1 , I 2 , P 1 , P 2 ) | \phi=|\frac{1}{2i}\operatorname{Log}\operatorname{Cr}(I_{1},I_{2},P_{1},P_{2})|
  3. Log \operatorname{Log}
  4. Cr \operatorname{Cr}
  5. P 1 P_{1}
  6. P 2 P_{2}
  7. I 1 I_{1}
  8. I 2 I_{2}
  9. x 0 = x 1 2 + x 2 2 + x 3 2 = 0 x_{0}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=0
  10. P 1 P_{1}
  11. P 2 P_{2}
  12. ( 0 , 1 , i , 0 ) , ( 0 , 1 , - i , 0 ) , ( 0 , 1 , 0 , 0 ) , ( 0 , cos ϕ , ± sin ϕ , 0 ) , (0,1,i,0),\ (0,1,-i,0),\ (0,1,0,0),\ (0,\cos\phi,\pm\sin\phi,0),
  13. i i
  14. - i -i
  15. ± sin ϕ / cos ϕ \pm\sin\phi/\cos\phi
  16. I 1 I_{1}
  17. I 2 I_{2}
  18. ϕ \phi
  19. Cr ( I 1 , I 2 , P 1 , P 2 ) = - - i cos ϕ ± sin ϕ i cos ϕ ± sin ϕ = e ± 2 i ϕ . \operatorname{Cr}(I_{1},I_{2},P_{1},P_{2})=-\frac{-i\cos\phi\pm\sin\phi}{i\cos% \phi\pm\sin\phi}=e^{\pm 2i\phi}.

Lai-Massey_scheme.html

  1. F \mathrm{F}
  2. H \mathrm{H}
  3. K 0 , K 1 , , K n K_{0},K_{1},\ldots,K_{n}
  4. 0 , 1 , , n 0,1,\ldots,n
  5. L 0 L_{0}
  6. R 0 R_{0}
  7. i = 0 , 1 , , n i=0,1,\dots,n
  8. ( L i + 1 , R i + 1 ) = H ( L i + T i , R i + T i ) (L_{i+1}^{\prime},R_{i+1}^{\prime})=\mathrm{H}(L_{i}^{\prime}+T_{i},R_{i}^{% \prime}+T_{i})
  9. T i = F ( L i - R i , K i ) T_{i}=\mathrm{F}(L_{i}^{\prime}-R_{i}^{\prime},K_{i})
  10. ( L 0 , R 0 ) = H ( L 0 , R 0 ) (L_{0}^{\prime},R_{0}^{\prime})=\mathrm{H}(L_{0},R_{0})
  11. ( L n + 1 , R n + 1 ) = ( L n + 1 , R n + 1 ) (L_{n+1},R_{n+1})=(L_{n+1}^{\prime},R_{n+1}^{\prime})
  12. ( L n + 1 , R n + 1 ) (L_{n+1},R_{n+1})
  13. i = n , n - 1 , , 0 i=n,n-1,\ldots,0
  14. ( L i , R i ) = H - 1 ( L i + 1 - T i , R i + 1 - T i ) (L_{i}^{\prime},R_{i}^{\prime})=\mathrm{H}^{-1}(L_{i+1}^{\prime}-T_{i},R_{i+1}% ^{\prime}-T_{i})
  15. T i = F ( L i + 1 - R i + 1 , K i ) T_{i}=\mathrm{F}(L_{i+1}^{\prime}-R_{i+1}^{\prime},K_{i})
  16. ( L n + 1 , R n + 1 ) = H - 1 ( L n + 1 , R n + 1 ) (L_{n+1}^{\prime},R_{n+1}^{\prime})=\mathrm{H}^{-1}(L_{n+1},R_{n+1})
  17. ( L 0 , R 0 ) = ( L 0 , R 0 ) (L_{0},R_{0})=(L_{0}^{\prime},R_{0}^{\prime})
  18. F \mathrm{F}
  19. L 0 - R 0 = L n + 1 - R n + 1 L_{0}-R_{0}=L_{n+1}-R_{n+1}
  20. σ \sigma
  21. H ( L , R ) = ( σ ( L ) , R ) \mathrm{H}(L,R)=(\sigma(L),R)
  22. σ \sigma
  23. x σ ( x ) - x x\mapsto\sigma(x)-x
  24. 2 n 2^{n}
  25. H \mathrm{H}
  26. n .5 n.5
  27. n n

Lambda2_method.html

  1. 𝐉 \mathbf{J}
  2. 𝐉 u = [ x u x y u x z u x x u y y u y z u y x u z y u z z u z ] , \mathbf{J}\equiv\nabla\vec{u}=\begin{bmatrix}\partial_{x}u_{x}&\partial_{y}u_{% x}&\partial_{z}u_{x}\\ \partial_{x}u_{y}&\partial_{y}u_{y}&\partial_{z}u_{y}\\ \partial_{x}u_{z}&\partial_{y}u_{z}&\partial_{z}u_{z}\end{bmatrix},
  3. u \vec{u}
  4. 𝐒 = 𝐉 + 𝐉 T 2 \mathbf{S}=\frac{\mathbf{J}+\mathbf{J}\text{T}}{2}
  5. 𝛀 = 𝐉 - 𝐉 T 2 , \mathbf{\Omega}=\frac{\mathbf{J}-\mathbf{J}\text{T}}{2},
  6. 𝐒 2 + 𝛀 2 \mathbf{S}^{2}+\mathbf{\Omega}^{2}
  7. u \vec{u}
  8. λ 1 \lambda_{1}
  9. λ 2 \lambda_{2}
  10. λ 3 \lambda_{3}
  11. λ 1 λ 2 λ 3 \lambda_{1}\geq\lambda_{2}\geq\lambda_{3}
  12. λ 2 < 0 \lambda_{2}<0
  13. λ 2 \lambda_{2}

Lambda_calculus_definition.html

  1. λ x . y z = λ x . ( y z ) \lambda x.y\ z=\lambda x.(y\ z)
  2. x y z = ( x y ) z x\ y\ z=(x\ y)\ z
  3. λ x , y . z = λ x . λ y . z \lambda x,y.z=\lambda x.\lambda y.z
  4. alpha - con ( a ) canonym [ A , P ] = canonym [ a [ A ] , P ] \operatorname{alpha-con}(a)\to\operatorname{canonym}[A,P]=\operatorname{% canonym}[a[A],P]
  5. beta - redex [ λ p . b v ] = b [ p := v ] \operatorname{beta-redex}[\lambda p.b\ v]=b[p:=v]
  6. x FV ( f ) eta - redex [ λ x . ( f x ) ] = f x\not\in\operatorname{FV}(f)\to\operatorname{eta-redex}[\lambda x.(f\ x)]=f
  7. b [ p := v ] b[p:=v]
  8. p p
  9. v v
  10. b b
  11. FV ( f ) \operatorname{FV}(f)
  12. f f
  13. canonym [ L , Q ] = canonym [ L , O , Q ] \operatorname{canonym}[L,Q]=\operatorname{canonym}[L,O,Q]
  14. canonym [ λ p . b , M , Q ] = λ name ( Q ) . canonym [ b , M [ p := Q ] , Q + N ] \operatorname{canonym}[\lambda p.b,M,Q]=\lambda\operatorname{name}(Q).% \operatorname{canonym}[b,M[p:=Q],Q+N]
  15. canonym [ X Y , x , Q ] = canonym [ X , x , Q + F ] canonym [ Y , x , E + S ] \operatorname{canonym}[X\ Y,x,Q]=\operatorname{canonym}[X,x,Q+F]\ % \operatorname{canonym}[Y,x,E+S]
  16. canonym [ x , M , Q ] = name ( M [ x ] ) \operatorname{canonym}[x,M,Q]=\operatorname{name}(M[x])
  17. O [ x ] = x O[x]=x
  18. M [ x := y ] [ x ] = y M[x:=y][x]=y
  19. x z M [ x := y ] [ z ] = M [ z ] x\neq z\to M[x:=y][z]=M[z]
  20. L [ x := y ] L[x:=y]
  21. ( λ p . b ) [ x := y ] = λ p . b [ x := y ] (\lambda p.b)[x:=y]=\lambda p.b[x:=y]
  22. ( X Y ) [ x := y ] = X [ x := y ] Y [ x := y ] (X\ Y)[x:=y]=X[x:=y]\ Y[x:=y]
  23. z = x ( z ) [ x := y ] = y z=x\to(z)[x:=y]=y
  24. z x ( z ) [ x := y ] = z z\neq x\to(z)[x:=y]=z
  25. FV ( M ) \operatorname{FV}(M)
  26. BV ( M ) \operatorname{BV}(M)
  27. FV ( x ) = { x } \operatorname{FV}(x)=\{x\}
  28. BV ( x ) = { } \operatorname{BV}(x)=\{\}
  29. FV ( λ x . M ) = FV ( M ) ¬ { x } \operatorname{FV}(\lambda x.M)=\operatorname{FV}(M)\cap\neg\{x\}
  30. BV ( λ x . M ) = BV ( M ) { x } \operatorname{BV}(\lambda x.M)=\operatorname{BV}(M)\cup\{x\}
  31. FV ( M N ) = FV ( M ) FV ( N ) \operatorname{FV}(M\ N)=\operatorname{FV}(M)\cup\operatorname{FV}(N)
  32. BV ( M N ) = BV ( M ) BV ( N ) \operatorname{BV}(M\ N)=\operatorname{BV}(M)\cup\operatorname{BV}(N)
  33. eval [ canonym [ L ] , Q ] \operatorname{eval}[\operatorname{canonym}[L],Q]
  34. eval [ x y ] = eval [ apply [ eval [ x ] strategy [ y ] ] ] \operatorname{eval}[x\ y]=\operatorname{eval}[\operatorname{apply}[% \operatorname{eval}[x]\ \operatorname{strategy}[y]]]
  35. apply [ ( λ x . y ) z ] = canonym [ beta - redex [ ( λ x . y ) z ] , x ] \operatorname{apply}[(\lambda x.y)\ z]=\operatorname{canonym}[\operatorname{% beta-redex}[(\lambda x.y)\ z],x]
  36. apply [ x ] = x \operatorname{apply}[x]=x
  37. eval [ λ x . ( f x ) ] = eval [ eta - redex [ λ x . ( f x ) ] ] \operatorname{eval}[\lambda x.(f\ x)]=\operatorname{eval}[\operatorname{eta-% redex}[\lambda x.(f\ x)]]
  38. eval [ L ] = L \operatorname{eval}[L]=L
  39. lazy [ X ] = X \operatorname{lazy}[X]=X
  40. eager [ X ] = eval [ X ] \operatorname{eager}[X]=\operatorname{eval}[X]
  41. strategy = lazy \operatorname{strategy}=\operatorname{lazy}
  42. strategy = eager \operatorname{strategy}=\operatorname{eager}
  43. normal [ λ x . y ) z ] = false \operatorname{normal}[\lambda x.y)\ z]=\operatorname{false}
  44. normal [ λ x . ( f x ) ] = false \operatorname{normal}[\lambda x.(f\ x)]=\operatorname{false}
  45. normal [ x y ] = normal [ x ] and normal [ y ] \operatorname{normal}[x\ y]=\operatorname{normal}[x]\and\operatorname{normal}[y]
  46. normal [ x ] = true \operatorname{normal}[x]=\operatorname{true}
  47. whnf [ λ x . y ) z ] = false \operatorname{whnf}[\lambda x.y)\ z]=\operatorname{false}
  48. whnf [ λ x . ( f x ) ] = false \operatorname{whnf}[\lambda x.(f\ x)]=\operatorname{false}
  49. whnf [ x y ] = whnf [ x ] \operatorname{whnf}[x\ y]=\operatorname{whnf}[x]
  50. whnf [ x ] = true \operatorname{whnf}[x]=\operatorname{true}
  51. λ y . x x y \lambda y.x\ x\ y
  52. λ x . y ( λ x . z x ) \lambda x.y\ (\lambda x.z\ x)
  53. ( λ p . b ) [ x := y ] = λ p . b [ x := y ] (\lambda p.b)[x:=y]=\lambda p.b[x:=y]
  54. ( λ x . b ) [ x := y ] = λ x . b (\lambda x.b)[x:=y]=\lambda x.b
  55. z x ( λ z . b ) [ x := y ] = λ z . b [ x := y ] z\neq x\ \to(\lambda z.b)[x:=y]=\lambda z.b[x:=y]
  56. ( λ x . x z ) [ x := y ] = ( λ x . y z ) (\lambda x.x\ z)[x:=y]=(\lambda x.y\ z)
  57. ( λ x . x z ) [ x := y ] = ( λ x . x z ) (\lambda x.x\ z)[x:=y]=(\lambda x.x\ z)
  58. λ x . x \lambda x.x
  59. λ y . y \lambda y.y
  60. ( y F V ( b ) and a ( λ x . b ) = λ y . b [ x := y ] ) alpha - con ( a ) (y\not\in FV(b)\and a(\lambda x.b)=\lambda y.b[x:=y])\to\operatorname{alpha-% con}(a)
  61. x x
  62. λ z . λ y . ( z y ) \lambda z.\lambda y.(z\ y)
  63. λ P . λ PN . ( P PN ) \lambda\operatorname{P}.\lambda\operatorname{PN}.(\operatorname{P}% \operatorname{PN})
  64. λ z . λ k . ( z k ) \lambda z.\lambda k.(z\ k)
  65. λ P . λ PN . ( P PN ) \lambda\operatorname{P}.\lambda\operatorname{PN}.(\operatorname{P}% \operatorname{PN})
  66. λ y . ( z y ) \lambda y.(z\ y)
  67. λ z . λ z . ( z z ) \lambda z.\lambda z.(z\ z)
  68. λ P . λ PN . ( \color R e d PN PN ) \lambda\operatorname{P}.\lambda\operatorname{PN}.({\color{Red}\operatorname{PN% }}\operatorname{PN})
  69. z z
  70. ( z : z F V ( y ) z B V ( b ) ) beta - redex [ λ x . b y ] = b [ x := y ] (\forall z:z\not\in FV(y)z\not\in BV(b))\to\operatorname{beta-redex}[\lambda x% .b\ y]=b[x:=y]
  71. b b
  72. y y
  73. b b
  74. ( λ x . λ y . ( λ z . ( λ x . z x ) ( λ y . z y ) ) ( x y ) ) (\lambda x.\lambda y.(\lambda z.(\lambda x.z\ x)(\lambda y.z\ y))(x\ y))
  75. ( λ P . λ PN . ( λ PNF . ( λ PNFNF . PNF PNFNF ) ( λ PNFNS . PNF PNFNS ) ) ( P PN ) ) (\lambda\operatorname{P}.\lambda\operatorname{PN}.(\lambda\operatorname{PNF}.(% \lambda\operatorname{PNFNF}.\operatorname{PNF}\operatorname{PNFNF})(\lambda% \operatorname{PNFNS}.\operatorname{PNF}\operatorname{PNFNS}))(\operatorname{P}% \operatorname{PN}))
  76. ( λ x . λ y . ( ( λ x . ( x y ) x ) ( λ y . ( x y ) y ) ) ) (\lambda x.\lambda y.((\lambda x.(x\ y)x)(\lambda y.(x\ y)y)))
  77. ( λ P . λ PN . ( ( λ PNF . ( \color B l u e P PN ) PNF ) ( λ PNS . ( P \color B l u e PN ) PNS ) ) ) (\lambda\operatorname{P}.\lambda\operatorname{PN}.((\lambda\operatorname{PNF}.% ({\color{Blue}\operatorname{P}}\operatorname{PN})\operatorname{PNF})(\lambda% \operatorname{PNS}.(\operatorname{P}{\color{Blue}\operatorname{PN}})% \operatorname{PNS})))
  78. ( λ P . λ PN . ( ( λ PNF . ( \color R e d PNF PN ) PNF ) ( λ PNS . ( P \color R e d PNS ) PNS ) ) ) (\lambda\operatorname{P}.\lambda\operatorname{PN}.((\lambda\operatorname{PNF}.% ({\color{Red}\operatorname{PNF}}\operatorname{PN})\operatorname{PNF})(\lambda% \operatorname{PNS}.(\operatorname{P}{\color{Red}\operatorname{PNS})}% \operatorname{PNS})))
  79. ( λ x . λ y . ( λ z . ( λ x . z a ) ( λ b . z b ) ) ( x y ) ) (\lambda x.\lambda y.(\lambda z.(\lambda x.z\ a)(\lambda b.z\ b))(x\ y))
  80. ( λ P . λ PN . ( λ PNF . ( λ PNFNF . PNF PNFNF ) ( λ PNFNS . PNF PNFNS ) ) ( P PN ) ) (\lambda\operatorname{P}.\lambda\operatorname{PN}.(\lambda\operatorname{PNF}.(% \lambda\operatorname{PNFNF}.\operatorname{PNF}\operatorname{PNFNF})(\lambda% \operatorname{PNFNS}.\operatorname{PNF}\operatorname{PNFNS}))(\operatorname{P}% \operatorname{PN}))
  81. ( λ x . λ y . ( ( λ a . ( x y ) a ) ( λ b . ( x y ) b ) ) ) (\lambda x.\lambda y.((\lambda a.(x\ y)a)(\lambda b.(x\ y)b)))
  82. ( λ P . λ PN . ( ( λ PNF . ( P PN ) PNF ) ( λ PNS . ( P PN ) PNS ) ) ) (\lambda\operatorname{P}.\lambda\operatorname{PN}.((\lambda\operatorname{PNF}.% (\operatorname{P}\operatorname{PN})\operatorname{PNF})(\lambda\operatorname{% PNS}.(\operatorname{P}\operatorname{PN})\operatorname{PNS})))
  83. ( λ P . λ PN . ( ( λ PNF . ( P PN ) PNF ) ( λ PNS . ( P PN ) PNS ) ) ) (\lambda\operatorname{P}.\lambda\operatorname{PN}.((\lambda\operatorname{PNF}.% (\operatorname{P}\operatorname{PN})\operatorname{PNF})(\lambda\operatorname{% PNS}.(\operatorname{P}\operatorname{PN})\operatorname{PNS})))
  84. ( ( λ x . z x ) ( λ y . z y ) ) [ z := ( x y ) ] ((\lambda x.z\ x)(\lambda y.z\ y))[z:=(x\ y)]
  85. ( ( λ a . z a ) ( λ b . z b ) ) [ z := ( x y ) ] ((\lambda a.z\ a)(\lambda b.z\ b))[z:=(x\ y)]
  86. F V ( x y ) = { x , y } FV(x\ y)=\{x,y\}
  87. B V ( ( λ x . z x ) ( λ y . z y ) ) = { x , y } BV((\lambda x.z\ x)(\lambda y.z\ y))=\{x,y\}
  88. FV ( x y ) = { x , y } \operatorname{FV}(x\ y)=\{x,y\}
  89. BV ( ( λ a . z a ) ( λ b . z b ) ) = { a , b } \operatorname{BV}((\lambda a.z\ a)(\lambda b.z\ b))=\{a,b\}
  90. x FV ( f ) eta - redex [ λ x . ( f x ) ] = f x\not\in\operatorname{FV}(f)\to\operatorname{eta-redex}[\lambda x.(fx)]=f
  91. ( λ x . ( λ y . y x ) x ) a (\lambda x.(\lambda y.y\ x)\ x)\ a
  92. λ a . a a \lambda a.a\ a
  93. ( λ y . y x ) a (\lambda y.y\ x)\ a
  94. λ a . a x \lambda a.a\ x
  95. x x
  96. λ y . y x \lambda y.y\ x

Lambda_g_conjecture.html

  1. λ g \lambda_{g}
  2. ¯ g , n \overline{\mathcal{M}}_{g,n}
  3. λ g \lambda_{g}
  4. ψ i \psi_{i}
  5. λ g \lambda_{g}
  6. ¯ g , n ψ 1 a i ψ n a n λ g = ( 2 g + n - 3 a 1 , , a n ) ¯ g , 1 ψ 1 2 g - 2 λ g . \int\limits_{\overline{\mathcal{M}}_{g,n}}\psi_{1}^{a_{i}}\cdot\dots\cdot\psi_% {n}^{a_{n}}\lambda_{g}={\left({{2g+n-3}\atop{a_{1},\dots,a_{n}}}\right)}\int% \limits_{\overline{\mathcal{M}}_{g,1}}\psi_{1}^{2g-2}\lambda_{g}.
  7. ¯ g , 1 ψ 1 2 g - 2 λ g = 2 2 g - 1 - 1 2 2 g - 1 | B 2 g | ( 2 g ) ! , \int\limits_{\overline{\mathcal{M}}_{g,1}}\psi_{1}^{2g-2}\lambda_{g}=\frac{2^{% 2g-1}-1}{2^{2g-1}}\frac{|B_{2g}|}{(2g)!},
  8. λ g \lambda_{g}
  9. ¯ g , n \overline{\mathcal{M}}_{g,n}
  10. ψ \psi
  11. λ g \lambda_{g}

Laminar_flamelet_model.html

  1. Y k t = χ 2 1 L e k ( 2 Y k Z 2 ) + ω k ρ {\partial Y_{k}\over\partial t}={\chi\over 2}{1\over Le_{k}}\biggl({\partial^{% 2}Y_{k}\over\partial Z^{2}}\biggr)+{\omega_{k}\over\rho}
  2. c p T t = χ 2 ( 2 h Z 2 ) - k = 1 N h k { χ 2 2 Y k Z 2 + ω k ρ } {c_{p}}{\partial T\over\partial t}={\chi\over 2}\biggl({\partial^{2}h\over% \partial Z^{2}}\biggr)-{\sum_{k=1}^{N}h_{k}}\biggl\{{\chi\over 2}{\partial^{2}% Y_{k}\over\partial Z^{2}}+{\omega_{k}\over\rho}\biggr\}
  3. χ s t ~ = χ ~ 0 1 F ( Z ) F ( Z s t ) P ( Z ) d Z {\tilde{\chi_{st}}}={\tilde{\chi}\over\int\limits_{0}^{1}\frac{F(Z)}{F(Z_{st})% }\ P(Z)dZ}
  4. F ( Z ) = e x p ( - 2 [ e r f c - 1 ( 2 Z ) ] 2 ) {F(Z)=exp\bigl(-2\bigl[erfc^{-1}(2Z)\bigr]^{2}\bigr)}

Lancichinetti-Fortunato-Radicchi_Benchmark.html

  1. k m i n k_{min}
  2. k m a x k_{max}
  3. s m i n s_{min}
  4. s m a x s_{max}
  5. s m i n > k m i n s_{min}>k_{min}
  6. s m a x > k m a x s_{max}>k_{max}
  7. I n = 1 I_{n}=1
  8. I n = 0 I_{n}=0

Lander,_Parkin,_and_Selfridge_conjecture.html

  1. i = 1 n a i k = b k \sum_{i=1}^{n}a_{i}^{k}=b^{k}
  2. a 1 , a 2 , , a n , b a_{1},a_{2},\dots,a_{n},b
  3. i = 1 n a i k = j = 1 m b j k \sum_{i=1}^{n}a_{i}^{k}=\sum_{j=1}^{m}b_{j}^{k}
  4. i = 1 n a i k = b k \sum_{i=1}^{n}a_{i}^{k}=b^{k}

Lang's_theorem.html

  1. 𝐅 q \mathbf{F}_{q}
  2. σ : G G , x x q \sigma:G\to G,\,x\mapsto x^{q}
  3. G G , x x - 1 σ ( x ) G\to G,\,x\mapsto x^{-1}\sigma(x)
  4. G = G ( 𝐅 q ¯ ) G ( 𝐅 q ¯ ) G=G(\overline{\mathbf{F}_{q}})\to G(\overline{\mathbf{F}_{q}})
  5. G ( 𝐅 q ) G(\mathbf{F}_{q})
  6. H 1 ( 𝐅 q , G ) = H e ´ t 1 ( Spec 𝐅 q , G ) H^{1}(\mathbf{F}_{q},G)=H_{\mathrm{\acute{e}t}}^{1}(\operatorname{Spec}\mathbf% {F}_{q},G)
  7. Spec 𝐅 q \operatorname{Spec}\mathbf{F}_{q}
  8. σ \sigma
  9. σ \sigma
  10. f a : G G , f a ( x ) = x - 1 a σ ( x ) . f_{a}:G\to G,\quad f_{a}(x)=x^{-1}a\sigma(x).
  11. ( d f a ) e = d ( h ( x ( x - 1 , a , σ ( x ) ) ) ) e = d h ( e , a , e ) ( - 1 , 0 , d σ e ) = - 1 + d σ e (df_{a})_{e}=d(h\circ(x\mapsto(x^{-1},a,\sigma(x))))_{e}=dh_{(e,a,e)}\circ(-1,% 0,d\sigma_{e})=-1+d\sigma_{e}
  12. h ( x , y , z ) = x y z h(x,y,z)=xyz
  13. ( d f a ) e (df_{a})_{e}
  14. σ \sigma
  15. f a ( b x ) = f f a ( b ) ( x ) f_{a}(bx)=f_{f_{a}(b)}(x)
  16. ( d f a ) b (df_{a})_{b}
  17. f 1 f_{1}
  18. f 1 ( b ) f_{1}(b)
  19. f 1 ( b ) f_{1}(b)
  20. f 1 f_{1}
  21. f a f_{a}
  22. U V U\cap V
  23. f 1 f_{1}
  24. H 1 ( 𝐅 q , G ) = H 1 ( Gal ( 𝐅 q ¯ / 𝐅 q ) , G ( 𝐅 q ¯ ) ) H^{1}(\mathbf{F}_{q},G)=H^{1}(\operatorname{Gal}(\overline{\mathbf{F}_{q}}/% \mathbf{F}_{q}),G(\overline{\mathbf{F}_{q}}))
  25. f a f_{a}

Laplace_equation_for_irrotational_flow.html

  1. × v = 0 \nabla\times v=0
  2. Δ x b + Δ x a = u y d y Δ t \Delta x_{b}+\Delta x_{a}={\partial u\over\partial y}dy\Delta t
  3. - Δ θ = ( Δ x b + Δ x a ) Δ t = u y Δ t -\Delta\theta={(\Delta x_{b}+\Delta x_{a})\over\Delta t}={\partial u\over% \partial y}\Delta t
  4. d θ d t = - u y {d\theta\over dt}=-{\partial u\over\partial y}
  5. ω z = 1 2 ( v x - u y ) \omega_{z}={1\over 2}\left({\partial v\over\partial x}-{\partial u\over% \partial y}\right)
  6. ω x = 1 2 ( w y - v z ) \omega_{x}={1\over 2}\left({\partial w\over\partial y}-{\partial v\over% \partial z}\right)
  7. ω y = 1 2 ( u z - w x ) \omega_{y}={1\over 2}\left({\partial u\over\partial z}-{\partial w\over% \partial x}\right)
  8. v x = u y {\partial v\over\partial x}={\partial u\over\partial y}
  9. w y = v z {\partial w\over\partial y}={\partial v\over\partial z}
  10. u z = w x {\partial u\over\partial z}={\partial w\over\partial x}
  11. v d x + v d y = - d vdx+vdy=-d\varnothing
  12. v = - v=-\nabla\varnothing
  13. × ( - ) = 0 \nabla\times(-\nabla\varnothing)=0
  14. u x + v y + w z = 0 {\partial u\over\partial x}+{\partial v\over\partial y}+{\partial w\over% \partial z}=0
  15. ( 2 ) ( x 2 ) + ( 2 ) ( y 2 ) + ( 2 ) ( z 2 ) = 0 {(\partial^{2}\varnothing)\over(\partial x^{2})}+{(\partial^{2}\varnothing)% \over(\partial y^{2})}+{(\partial^{2}\varnothing)\over(\partial z^{2})}=0
  16. 2 = 0 \nabla^{2}\varnothing=0

Last_diminisher.html

  1. n × ( n - 1 ) / 2 = O ( n 2 ) n\times(n-1)/2=O(n^{2})
  2. 1 / n 1/n
  3. ϵ \epsilon
  4. 1 / ϵ 1/\epsilon
  5. ϵ \epsilon
  6. ϵ \epsilon
  7. ϵ \epsilon
  8. ϵ \epsilon
  9. ϵ \epsilon
  10. n 2 / ϵ n^{2}/\epsilon
  11. n / ϵ n/\epsilon
  12. n n

Lattice_path.html

  1. 2 \mathbb{Z}^{2}
  2. S = { ( 2 , 0 ) , ( 1 , 1 ) , ( 0 , - 1 ) } S=\left\{(2,0),(1,1),(0,-1)\right\}
  3. L L
  4. d \mathbb{Z}^{d}
  5. k k
  6. S S
  7. v 0 , v 1 , , v k d v_{0},v_{1},\ldots,v_{k}\in\mathbb{Z}^{d}
  8. v i - v i - 1 v_{i}-v_{i-1}
  9. S S
  10. d \mathbb{R}^{d}
  11. d \mathbb{Z}^{d}
  12. 2 \mathbb{Z}^{2}
  13. S = { ( 2 , 0 ) , ( 1 , 1 ) , ( 0 , - 1 ) } S=\{(2,0),(1,1),(0,-1)\}
  14. L = { ( - 1 , - 2 ) , ( 0 , - 1 ) , ( 2 , - 1 ) , ( 2 , - 2 ) , ( 2 , - 3 ) , ( 4 , - 3 ) } L=\{(-1,-2),(0,-1),(2,-1),(2,-2),(2,-3),(4,-3)\}
  15. 2 \mathbb{Z}^{2}
  16. S = { ( 0 , 1 ) , ( 1 , 0 ) } S=\{(0,1),(1,0)\}
  17. ( 0 , 1 ) (0,1)
  18. N N
  19. ( 1 , 0 ) (1,0)
  20. E E
  21. L L
  22. k k
  23. N N
  24. E E
  25. L L
  26. N N
  27. E E
  28. L L
  29. n n
  30. N N
  31. e e
  32. E E
  33. ( e , n ) (e,n)
  34. n n
  35. e e
  36. ( 0 , 0 ) (0,0)
  37. ( 0 , 0 ) (0,0)
  38. N N
  39. E E
  40. ( 3 , 1 ) (3,1)
  41. n th n^{\,\text{th}}
  42. C n C_{n}
  43. 2 \mathbb{Z}^{2}
  44. ( 0 , 0 ) (0,0)
  45. ( 2 n , 0 ) (2n,0)
  46. S = { ( 1 , 1 ) , ( 1 , - 1 ) } S=\{(1,1),(1,-1)\}
  47. x x
  48. ( 0 , 0 ) (0,0)
  49. ( n , n ) (n,n)
  50. y = x y=x
  51. ( 0 , 0 ) (0,0)
  52. ( n , n ) (n,n)
  53. ( 1 , 0 ) , ( 0 , 1 ) (1,0),(0,1)
  54. ( 1 , 1 ) (1,1)
  55. y = x y=x
  56. ( 0 , 0 ) (0,0)
  57. ( a , b ) (a,b)
  58. a a
  59. a + b a+b
  60. ( 0 , 0 ) (0,0)
  61. ( 2 , 3 ) (2,3)
  62. ( 2 + 3 2 ) = ( 5 2 ) = 10 {\left({{2+3}\atop{2}}\right)}={\left({{5}\atop{2}}\right)}=10
  63. ( 0 , 0 ) (0,0)
  64. ( n , k ) (n,k)
  65. ( n + k n ) {\left({{n+k}\atop{n}}\right)}
  66. 0 k n = 4 0\leq k\leq n=4
  67. n , k { 0 } n,k\in\mathbb{N}\cup\{0\}
  68. k th k^{\,\text{th}}
  69. n th n^{\,\text{th}}
  70. ( n k ) {\left({{n}\atop{k}}\right)}
  71. k = 0 n ( n k ) 2 = ( 2 n n ) \sum_{k=0}^{n}{\left({{n}\atop{k}}\right)}^{2}={\left({{2n}\atop{n}}\right)}
  72. ( 0 , 0 ) (0,0)
  73. ( n , n ) (n,n)
  74. ( x , n - x ) (x,n-x)
  75. x { 0 , 1 , , n } x\in\{0,1,\ldots,n\}
  76. n = 4 n=4
  77. ( 0 , 0 ) (0,0)
  78. ( 4 , 4 ) (4,4)
  79. ( n k ) 2 {\left({{n}\atop{k}}\right)}^{2}
  80. ( 0 , 0 ) (0,0)
  81. ( k , n - k ) (k,n-k)
  82. ( n k ) = ( n n - k ) {\left({{n}\atop{k}}\right)}={\left({{n}\atop{n-k}}\right)}
  83. ( 0 , 0 ) (0,0)
  84. ( n , n ) (n,n)
  85. \Box

Laver_property.html

  1. M M
  2. N N
  3. N N
  4. M M
  5. g M g\in M
  6. ω \omega
  7. ω { 0 } \omega\setminus\{0\}
  8. g g
  9. f N f\in N
  10. ω \omega
  11. ω \omega
  12. h M h\in M
  13. f f
  14. T M T\in M
  15. T T
  16. h h
  17. n n
  18. n th n\text{th}
  19. T T
  20. g ( n ) g(n)
  21. f f
  22. T T
  23. ω ω {}^{\omega}\omega

Laves_graph.html

  1. ( 0 , 0 , 0 ) , ( 1 , 2 , 3 ) , ( 2 , 3 , 1 ) , ( 3 , 1 , 2 ) , (0,0,0),\quad(1,2,3),\quad(2,3,1),\quad(3,1,2),
  2. ( 2 , 2 , 2 ) , ( 3 , 0 , 1 ) , ( 0 , 1 , 3 ) , ( 1 , 3 , 0 ) , (2,2,2),\quad(3,0,1),\quad(0,1,3),\quad(1,3,0),
  3. 3 \mathbb{Z}^{3}
  4. d \mathbb{Z}^{d}
  5. d \mathbb{Z}^{d}

Lawrence_C._Washington.html

  1. Z p Z_{p}
  2. μ \mu
  3. Z p Z_{p}

Lax_functor.html

  1. P : C D P:C\to D
  2. P x D P_{x}\in D
  3. P x , y : C ( x , y ) D ( P x , P y ) P_{x,y}:C(x,y)\to D(P_{x},P_{y})
  4. P id x : id P x P x , x ( id x ) P_{\,\text{id}_{x}}:\,\text{id}_{P_{x}}\to P_{x,x}(\,\text{id}_{x})
  5. P x , y , z ( f , g ) : P x , y ( f ) ; P y , z ( g ) P x , z ( f ; g ) P_{x,y,z}(f,g):P_{x,y}(f);P_{y,z}(g)\to P_{x,z}(f;g)
  6. P id x P_{\,\text{id}_{x}}
  7. P x , y , z P_{x,y,z}

Lax_natural_transformation.html

  1. F , G : C D F,G\colon C\to D
  2. α : F G \alpha\colon F\to G
  3. α c : F ( c ) G ( c ) \alpha_{c}\colon F(c)\to G(c)
  4. c C c\in C
  5. α f : G ( f ) α c α c F ( f ) \alpha_{f}\colon G(f)\circ\alpha_{c}\to\alpha_{c^{\prime}}\circ F(f)
  6. f : c c f\colon c\to c^{\prime}

Le_bail.html

  1. I o b s ( 1 ) = y i ( o b s ) y i ( 1 ) y i ( c a l c ) I_{obs}(1)=\frac{\sum y_{i}(obs)y_{i}(1)}{y_{i}(calc)}

Least_squares_adjustment.html

  1. X ^ \hat{X}
  2. Y ^ \hat{Y}
  3. f ( X ^ , Y ^ ) = 0 f\left(\hat{X},\hat{Y}\right)=0
  4. Y ~ \tilde{Y}
  5. X ~ \tilde{X}
  6. w ~ = f ( X ~ , Y ~ ) . \tilde{w}=f\left(\tilde{X},\tilde{Y}\right).
  7. A = f / X ; A=\partial{f}/\partial{X};
  8. B = f / Y . B=\partial{f}/\partial{Y}.
  9. w ~ + A x ^ + B y ^ = 0 , \tilde{w}+A\hat{x}+B\hat{y}=0,
  10. x ^ = X ^ - X ~ \hat{x}=\hat{X}-\tilde{X}
  11. y ^ = Y ^ - Y ~ \hat{y}=\hat{Y}-\tilde{Y}
  12. y ~ = w ~ = h ( X ~ ) - Y ~ \tilde{y}=\tilde{w}=h(\tilde{X})-\tilde{Y}
  13. A x ^ = y ^ - y ~ , A\hat{x}=\hat{y}-\tilde{y},
  14. X ^ \hat{X}
  15. Y ^ \hat{Y}

Legendre's_formula.html

  1. n ! n!
  2. ν p ( n ) \nu_{p}(n)
  3. ν p ( n ! ) = i = 1 n p i , \nu_{p}(n!)=\sum_{i=1}^{\infty}\left\lfloor\frac{n}{p^{i}}\right\rfloor,
  4. x \lfloor x\rfloor
  5. s p ( n ) s_{p}(n)
  6. ν p ( n ! ) = n - s p ( n ) p - 1 . \nu_{p}(n!)=\frac{n-s_{p}(n)}{p-1}.
  7. n ! n!
  8. n ! n!
  9. { 1 , 2 , , n } \{1,2,\dots,n\}
  10. n p \textstyle\left\lfloor\frac{n}{p}\right\rfloor
  11. p 2 p^{2}
  12. p 3 p^{3}
  13. ν p ( n ! ) \nu_{p}(n!)
  14. n p i = 0 \textstyle\left\lfloor\frac{n}{p^{i}}\right\rfloor=0
  15. p i > n p^{i}>n
  16. n = n p + + n 1 p + n 0 n=n_{\ell}p^{\ell}+\cdots+n_{1}p+n_{0}
  17. n p i = n p - i + + n i + 1 p + n i \textstyle\left\lfloor\frac{n}{p^{i}}\right\rfloor=n_{\ell}p^{\ell-i}+\cdots+n% _{i+1}p+n_{i}
  18. ν p ( n ! ) = i = 1 n p i = i = 1 ( n p - i + + n i + 1 p + n i ) = i = 1 j = i n j p j - i = j = 1 i = 1 j n j p j - i = j = 1 n j p j - 1 p - 1 = j = 0 n j p j - 1 p - 1 = 1 p - 1 j = 0 ( n j p j - n j ) = 1 p - 1 ( n - s p ( n ) ) . \begin{aligned}\displaystyle\nu_{p}(n!)&\displaystyle=\sum_{i=1}^{\ell}\left% \lfloor\frac{n}{p^{i}}\right\rfloor\\ &\displaystyle=\sum_{i=1}^{\ell}\left(n_{\ell}p^{\ell-i}+\cdots+n_{i+1}p+n_{i}% \right)\\ &\displaystyle=\sum_{i=1}^{\ell}\sum_{j=i}^{\ell}n_{j}p^{j-i}\\ &\displaystyle=\sum_{j=1}^{\ell}\sum_{i=1}^{j}n_{j}p^{j-i}\\ &\displaystyle=\sum_{j=1}^{\ell}n_{j}\cdot\frac{p^{j}-1}{p-1}\\ &\displaystyle=\sum_{j=0}^{\ell}n_{j}\cdot\frac{p^{j}-1}{p-1}\\ &\displaystyle=\frac{1}{p-1}\sum_{j=0}^{\ell}\left(n_{j}p^{j}-n_{j}\right)\\ &\displaystyle=\frac{1}{p-1}\left(n-s_{p}(n)\right).\end{aligned}
  19. p = 2 p=2
  20. ν 2 ( n ! ) = n - s 2 ( n ) \nu_{2}(n!)=n-s_{2}(n)
  21. s 2 ( n ) s_{2}(n)
  22. p - 1 / ( p - 1 ) p^{-1/(p-1)}

Legendre's_theorem_on_spherical_triangles.html

  1. π \pi
  2. A - A B - B C - C 1 3 E = 1 3 Δ , a , b , c 1. \displaystyle A-A^{\prime}\;\approx\;B-B^{\prime}\;\approx\;C-C^{\prime}\;% \approx\;\frac{1}{3}E\;=\;\frac{1}{3}\Delta,\qquad a,\;b,\;c\,\ll\,1.
  3. 1 / 2 {1}/{2}
  4. π \pi
  5. Δ \displaystyle\Delta

Legendre's_three-square_theorem.html

  1. n = x 2 + y 2 + z 2 n=x^{2}+y^{2}+z^{2}
  2. n n
  3. n = 4 a ( 8 b + 7 ) n=4^{a}(8b+7)
  4. a a
  5. b b
  6. n = 4 a ( 8 b + 7 ) n=4^{a}(8b+7)

Legg's_equation.html

  1. R ac μ L = a B max f + c f + e f 2 \frac{R_{\,\text{ac}}}{\mu L}=aB_{\,\text{max}}f+cf+ef^{2}

Level_(logarithmic_quantity).html

  1. L Q = log r ( Q Q 0 ) , L_{Q}=\log_{r}\!\left(\frac{Q}{Q_{0}}\right)\!,
  2. L F = ln ( F F 0 ) , L_{F}=\ln\!\left(\frac{F}{F_{0}}\right)\!,
  3. L F = ln ( F F 0 ) , L_{F}=\ln\!\left(\frac{F}{F_{0}}\right)\!,
  4. L P = log e 2 ( P P 0 ) = 1 2 ln ( P P 0 ) , L_{P}=\log_{e^{2}}\!\left(\frac{P}{P_{0}}\right)=\frac{1}{2}\ln\!\left(\frac{P% }{P_{0}}\right)\!,
  5. L F = ln ( F F 0 ) Np = 2 log 10 ( F F 0 ) B = 20 log 10 ( F F 0 ) dB . L_{F}=\ln\!\left(\frac{F}{F_{0}}\right)\!~{}\mathrm{Np}=2\log_{10}\!\left(% \frac{F}{F_{0}}\right)\!~{}\mathrm{B}=20\log_{10}\!\left(\frac{F}{F_{0}}\right% )\!~{}\mathrm{dB}.
  6. L P = 1 2 ln ( P P 0 ) Np = log 10 ( P P 0 ) B = 10 log 10 ( P P 0 ) dB . L_{P}=\frac{1}{2}\ln\!\left(\frac{P}{P_{0}}\right)\!~{}\mathrm{Np}=\log_{10}\!% \left(\frac{P}{P_{0}}\right)\!~{}\mathrm{B}=10\log_{10}\!\left(\frac{P}{P_{0}}% \right)\!~{}\mathrm{dB}.

Levopimaradiene_synthase.html

  1. \rightleftharpoons

Liberation_(pharmacology).html

  1. d W d t = D A ( C s - C ) L \frac{dW}{dt}=\frac{DA(C_{s}-C)}{L}
  2. d W d t \frac{dW}{dt}
  3. C s C_{s}

Lie_algebra_extension.html

  1. 𝐞 \mathbf{e}
  2. 𝐠 \mathbf{g}
  3. 𝐡 \mathbf{h}
  4. G G
  5. H H
  6. E E
  7. N N
  8. G G
  9. E / N E/N
  10. H H
  11. e x p exp
  12. g g
  13. G × H G×H
  14. H H
  15. g + h g+h
  16. ( g , h ) (g,h)
  17. G G
  18. H H
  19. G G
  20. H H
  21. g g
  22. h h
  23. 𝐠 \mathbf{g}
  24. 𝐡 \mathbf{h}
  25. 𝐠 \mathbf{g}
  26. 𝐡 \mathbf{h}
  27. G G
  28. H H
  29. i i
  30. s s
  31. k e r s = i m i kers=imi
  32. 𝐡 \mathbf{h}
  33. 𝐞 \mathbf{e}
  34. 𝔤 𝔢 / Im i = 𝔢 / Ker s , \mathfrak{g}\cong\mathfrak{e}/\operatorname{Im}i=\mathfrak{e}/\operatorname{% Ker}s,
  35. 𝐠 \mathbf{g}
  36. 𝐞 \mathbf{e}
  37. 𝐞 \mathbf{e}
  38. 𝐠 \mathbf{g}
  39. 𝐡 \mathbf{h}
  40. 𝐞 \mathbf{e}
  41. 𝐠 \mathbf{g}
  42. 𝐡 \mathbf{h}
  43. ί ί
  44. σ σ
  45. 𝐠 \mathbf{g}
  46. i i
  47. s s
  48. 𝐠 \mathbf{g}
  49. 𝐡 \mathbf{h}
  50. f i = i , s f = s , f\circ i=i^{\prime},\quad s^{\prime}\circ f=s,
  51. 𝐞 \mathbf{e}
  52. 𝔥 𝑖 𝔱 𝑠 𝔤 , \mathfrak{h}\;\overset{i}{\hookrightarrow}\;\mathfrak{t}\;\overset{s}{% \twoheadrightarrow}\;\mathfrak{g},
  53. 𝐢 \mathbf{i}
  54. 𝐭 = 𝐢 k e r s \mathbf{t}=\mathbf{i}⊕kers
  55. 𝐢 \mathbf{i}
  56. 𝐭 \mathbf{t}
  57. 𝔥 𝑖 𝔰 𝑠 𝔤 , \mathfrak{h}\;\overset{i}{\hookrightarrow}\;\mathfrak{s}\;\overset{s}{% \twoheadrightarrow}\;\mathfrak{g},
  58. 𝐮 \mathbf{u}
  59. 𝐬 = 𝐮 k e r s \mathbf{s}=\mathbf{u}⊕kers
  60. 𝐮 \mathbf{u}
  61. 𝐬 \mathbf{s}
  62. 𝐠 \mathbf{g}
  63. 𝐚 \mathbf{a}
  64. 𝐠 \mathbf{g}
  65. 𝔥 𝑖 𝔠 𝑠 𝔤 , \mathfrak{h}\;\overset{i}{\hookrightarrow}\;\mathfrak{c}\;\overset{s}{% \twoheadrightarrow}\;\mathfrak{g},
  66. k e r s kers
  67. Z ( 𝐜 ) Z(\mathbf{c})
  68. 𝐞 \mathbf{e}
  69. 𝐡 i m i = k e r s \mathbf{h}≅imi=kers
  70. 𝐞 \mathbf{e}
  71. 𝐠 \mathbf{g}
  72. 𝐠 \mathbf{g}
  73. 𝐞 \mathbf{e}
  74. 𝐠 \mathbf{g}
  75. 𝐡 \mathbf{h}
  76. l l
  77. 𝐞 \mathbf{e}
  78. 𝐠 \mathbf{g}
  79. l l
  80. s s
  81. ε : 𝐠 × 𝐠 𝐞 ε:\mathbf{g}×\mathbf{g}→\mathbf{e}
  82. ϵ ( G 1 , G 2 ) = l ( [ G 1 , G 2 ] ) - [ l ( G 1 ) , l ( G 2 ) ] , G 1 , G 2 𝔤 . \epsilon(G_{1},G_{2})=l([G_{1},G_{2}])-[l(G_{1}),l(G_{2})],\quad G_{1},G_{2}% \in\mathfrak{g}.
  83. ε ε
  84. ϵ ( G 1 , [ G 2 , G 3 ] ) + ϵ ( G 2 , [ G 3 , G 1 ] ) + ϵ ( G 3 , [ G 1 , G 2 ] ) = 0 𝔢 . \epsilon(G_{1},[G_{2},G_{3}])+\epsilon(G_{2},[G_{3},G_{1}])+\epsilon(G_{3},[G_% {1},G_{2}])=0\in\mathfrak{e}.
  85. ε ε
  86. l l
  87. 𝐠 \mathbf{g}
  88. ε ε
  89. 𝐞 \mathbf{e}
  90. I m ε k e r s Imε⊂kers
  91. k e r s Z ( 𝐞 ) kers⊂Z(\mathbf{e})
  92. 𝐡 \mathbf{h}
  93. φ φ
  94. 𝐠 \mathbf{g}
  95. 𝐡 \mathbf{h}
  96. 0 𝜄 𝔥 𝑖 𝔢 𝑠 𝔤 𝜎 0 0\;\overset{\iota}{\hookrightarrow}\mathfrak{h}\;\overset{i}{\hookrightarrow}% \;\mathfrak{e}\;\overset{s}{\twoheadrightarrow}\;\mathfrak{g}\;\overset{\sigma% }{\twoheadrightarrow}\;0
  97. 0 𝜄 𝔥 i 𝔢 s 𝔤 𝜎 0 0\;\overset{\iota}{\hookrightarrow}\mathfrak{h}^{\prime}\;\overset{i^{\prime}}% {\hookrightarrow}\;\mathfrak{e}^{\prime}\;\overset{s^{\prime}}{% \twoheadrightarrow}\;\mathfrak{g}\;\overset{\sigma}{\twoheadrightarrow}\;0
  98. Ψ Ψ
  99. Φ Φ
  100. 𝐠 , 𝐡 \mathbf{g},\mathbf{h}
  101. K K
  102. 𝔢 = 𝔥 × 𝔤 , \mathfrak{e}=\mathfrak{h}\times\mathfrak{g},
  103. 𝐞 \mathbf{e}
  104. α ( H , G ) = ( α G , α G ) , α F , H 𝔥 , G 𝔤 . \alpha(H,G)=(\alpha G,\alpha G),\alpha\in F,H\in\mathfrak{h},G\in\mathfrak{g}.
  105. 𝐡 × 𝐠 𝐡 𝐠 \mathbf{h}×\mathbf{g}≡\mathbf{h}⊕\mathbf{g}
  106. F F
  107. 𝐞 \mathbf{e}
  108. i : 𝔥 𝔢 ; H ( H , 0 ) , s : 𝔢 𝔤 ; ( H , G ) G . i:\mathfrak{h}\hookrightarrow\mathfrak{e};H\mapsto(H,0),\quad s:\mathfrak{e}% \twoheadrightarrow\mathfrak{g};(H,G)\mapsto G.
  109. 𝐠 \mathbf{g}
  110. 𝐡 \mathbf{h}
  111. 𝐞 \mathbf{e}
  112. 𝐡 \mathbf{h}
  113. 𝐠 \mathbf{g}
  114. 𝐡 𝐠 𝐠 𝐡 \mathbf{h}⊕\mathbf{g}≠\mathbf{g}⊕\mathbf{h}
  115. 0 𝐠 0⊕\mathbf{g}
  116. G A u t ( H ) G→Aut(H)
  117. ψ : 𝐠 d e r 𝐡 ψ:\mathbf{g}→der\mathbf{h}
  118. 𝐞 = 𝐡 𝐠 \mathbf{e}=\mathbf{h}⊕\mathbf{g}
  119. [ ( H , G ) , ( H , G ) ] = [ H , H ] + [ G , G ] + ψ G ( H ) - ψ G ( H ) , H , H 𝔥 , G , G 𝔤 . [(H,G),(H^{\prime},G^{\prime})]=[H,H^{\prime}]+[G,G^{\prime}]+\psi_{G}(H)-\psi% _{G^{\prime}}(H^{\prime}),\quad H,H^{\prime}\in\mathfrak{h},G,G^{\prime}\in% \mathfrak{g}.
  120. 𝐡 \mathbf{h}
  121. 𝐠 \mathbf{g}
  122. 0 𝐠 0⊕\mathbf{g}
  123. 𝐞 \mathbf{e}
  124. 𝐡 0 \mathbf{h}⊕0
  125. 𝐞 \mathbf{e}
  126. i : 𝐡 𝐞 i:\mathbf{h}→\mathbf{e}
  127. H H 0 H↦H⊕0
  128. s : 𝐞 𝐠 s:\mathbf{e}→\mathbf{g}
  129. H G G , H 𝐡 , G 𝐠 H⊕G↦G,H∈\mathbf{h},G∈\mathbf{g}
  130. k e r s = i m i kers=imi
  131. 𝐞 \mathbf{e}
  132. 𝐠 \mathbf{g}
  133. 𝐡 \mathbf{h}
  134. G G
  135. O ( 3 , 1 ) O(3,1)
  136. T T
  137. P P
  138. ( a 2 , Λ 2 ) ( a 1 , Λ 1 ) = ( a 2 + Λ 2 a 1 , Λ 2 Λ 1 ) , a 1 , a 2 T P , Λ 1 , Λ 2 O ( 3 , 1 ) P , (a_{2},\Lambda_{2})(a_{1},\Lambda_{1})=(a_{2}+\Lambda_{2}a_{1},\Lambda_{2}% \Lambda_{1}),\quad a_{1},a_{2}\in\mathrm{T}\subset P,\Lambda_{1},\Lambda_{2}% \in\mathrm{O}(3,1)\subset\mathrm{P},
  139. T T
  140. S O ( 3 , 1 ) SO(3,1)
  141. P P
  142. Λ Λ
  143. T T
  144. P ¯ = T S O ( 3 , 1 ) , \overline{\mathrm{P}}=\mathrm{T}\otimes_{S}\mathrm{O}(3,1),
  145. P ¯ = P \overline{P}=P
  146. δ δ
  147. 𝐠 \mathbf{g}
  148. 𝐡 \mathbf{h}
  149. δ δ
  150. 𝐞 = 𝐡 𝐠 \mathbf{e}=\mathbf{h}⊗\mathbf{g}
  151. [ H 1 + G 1 , H 2 + G 2 ] = [ G 1 , G 2 ] + δ ( G 1 ) - δ ( G 2 ) . [H_{1}+G_{1},H_{2}+G_{2}]=[G_{1},G_{2}]+\delta(G_{1})-\delta(G_{2}).
  152. 𝐠 \mathbf{g}
  153. 𝐞 \mathbf{e}
  154. 𝐡 \mathbf{h}
  155. 𝐞 \mathbf{e}
  156. 𝐡 \mathbf{h}
  157. 𝐠 \mathbf{g}
  158. 𝐞 \mathbf{e}
  159. i : 𝐡 𝐞 i:\mathbf{h}→\mathbf{e}
  160. H ( H , 0 ) H↦(H,0)
  161. s : 𝐞 𝐠 s:\mathbf{e}→\mathbf{g}
  162. ( H , G ) G (H,G)↦G
  163. i m i = k e r s imi=kers
  164. 𝐞 \mathbf{e}
  165. 𝐠 \mathbf{g}
  166. 𝐡 \mathbf{h}
  167. ε ε
  168. 𝐠 \mathbf{g}
  169. 𝐡 \mathbf{h}
  170. 𝐞 = 𝐡 𝐠 \mathbf{e}=\mathbf{h}⊕\mathbf{g}
  171. 𝐞 \mathbf{e}
  172. [ μ H + G 1 , ν H + G 2 ] = [ G 1 , G 2 ] + ϵ ( G 1 , G 2 ) H , μ , ν F . [\mu H+G_{1},\nu H+G_{2}]=[G_{1},G_{2}]+\epsilon(G1,G2)H,\quad\mu,\nu\in F.
  173. H H
  174. 𝐡 \mathbf{h}
  175. 𝐠 \mathbf{g}
  176. 𝐠 \mathbf{g}
  177. ε ε
  178. 𝐞 \mathbf{e}
  179. μ H Z ( 𝐞 ) μH∈Z(\mathbf{e})
  180. i : μ H ( μ H , 0 ) i:μH↦(μH,0)
  181. s : ( μ H , G ) G s:(μH,G)↦G
  182. 𝐞 \mathbf{e}
  183. 𝐠 \mathbf{g}
  184. h h
  185. 𝐠 \mathbf{g}
  186. ψ : G + μ c 𝔢 1 G + μ c + f ( G ) c 𝔢 2 . \psi:G+\mu c\in\mathfrak{e}_{1}\mapsto G+\mu c+f(G)c\in\mathfrak{e}_{2}.
  187. ψ ψ
  188. 𝐠 \mathbf{g}
  189. φ φ
  190. φ φ
  191. 𝐠 \mathbf{g}
  192. f f
  193. φ = δ f φ=δf
  194. K K
  195. ν ν
  196. d : 𝐠 𝐠 d:\mathbf{g}→\mathbf{g}
  197. K K
  198. d d
  199. d d
  200. φ ( G 1 , G 2 ) ρ G 1 ( G 2 ) = K ( ν - 1 ( ρ G 1 ) , G 2 ) K ( d ( G 1 ) , G 2 ) = K ( ad G d ( G 1 ) , G 2 ) = K ( [ G d , G 1 ] , G 2 ) = K ( G d , [ G 1 , G 2 ] ) . \varphi(G_{1},G_{2})\equiv\rho_{G_{1}}(G_{2})=K(\nu^{-1}(\rho_{G_{1}}),G_{2})% \equiv K(d(G_{1}),G_{2})=K(\mathrm{ad}_{G_{d}}(G_{1}),G_{2})=K([G_{d},G_{1}],G% _{2})=K(G_{d},[G_{1},G_{2}]).
  201. f f
  202. f ( G ) = K ( G d , G ) . f(G)=K(G_{d},G).
  203. δ f ( G 1 , G 2 ) = f ( [ G 1 , G 2 ] ) = K ( G d , [ G 1 , G 2 ] ) = φ ( G 1 , G 2 ) , \delta f(G_{1},G_{2})=f([G_{1},G_{2}])=K(G_{d},[G_{1},G_{2}])=\varphi(G_{1},G_% {2}),
  204. φ φ
  205. d d
  206. ν ν
  207. K ( d ( [ G 1 , G 2 ] ) , G 3 ) ) = φ ( [ G 1 , G 2 ] ) , G 3 ) ) = φ ( G 1 , [ G 2 , G 3 ] ) + φ ( G 2 , [ G 3 , G 1 ] ) = K ( d ( G 1 ) , [ G 2 , G 3 ] ) + K ( d ( G 1 ) , ( G 3 , G 1 ) ) = K ( [ d ( G 1 ) , G 2 ] , G 3 ) + K ( [ G 1 , d ( G 2 ) ] , G 3 ) ) = K ( [ d ( G 1 ) , G 2 ] + [ G 1 , d ( G 2 ) ] , G 3 ) . \begin{aligned}\displaystyle K(d([G_{1},G_{2}]),G_{3}))&\displaystyle=\varphi(% [G_{1},G_{2}]),G_{3}))=\varphi(G_{1},[G_{2},G_{3}])+\varphi(G_{2},[G_{3},G_{1}% ])\\ &\displaystyle=K(d(G_{1}),[G_{2},G_{3}])+K(d(G_{1}),(G_{3},G_{1}))=K([d(G_{1})% ,G_{2}],G_{3})+K([G_{1},d(G_{2})],G_{3}))\\ &\displaystyle=K([d(G_{1}),G_{2}]+[G_{1},d(G_{2})],G_{3}).\end{aligned}
  208. K K
  209. K K
  210. d d
  211. K K
  212. φ φ
  213. K ( ν - 1 ( ρ G 1 ) , G 2 ) K ( d ( G 1 ) , G 2 ) , K(\nu^{-1}(\rho_{G_{1}}),G_{2})\equiv K(d(G_{1}),G_{2}),
  214. K K
  215. φ φ
  216. K ( d ( G 1 ) , G 2 ) = - K ( G 1 , d ( G 2 ) ) , K(d(G_{1}),G_{2})=-K(G_{1},d(G_{2})),
  217. 𝐠 : 𝐅 \mathbf{g:→F}
  218. d d
  219. L ( d ( G 1 ) , G 2 ) = - L ( G 1 , d ( G 2 ) ) , L(d(G_{1}),G_{2})=-L(G_{1},d(G_{2})),
  220. φ φ
  221. φ ( G 1 , G 2 ) = L ( d ( G 1 ) , G 2 ) \varphi(G_{1},G_{2})=L(d(G_{1}),G_{2})
  222. d d
  223. φ φ
  224. φ ( [ G 1 , G 2 ] , G 3 ) = L ( d [ G 1 , G 2 ] , G 3 ) = L ( [ d ( G 1 ) , G 2 ] , G 3 ) + L ( [ G 1 , d ( G 2 ) ] , G 3 ) , \varphi([G1,G_{2}],G_{3})=L(d[G1,G_{2}],G_{3})=L([d(G1),G_{2}],G_{3})+L([G1,d(% G_{2})],G_{3}),
  225. φ φ
  226. L L
  227. 𝐠 \mathbf{g}
  228. G G
  229. 𝐞 \mathbf{e}
  230. 𝐠 \mathbf{g}
  231. E E
  232. 𝐞 \mathbf{e}
  233. E E
  234. G G
  235. 𝐞 \mathbf{e}
  236. 𝐠 \mathbf{g}
  237. 𝔤 = C [ λ , λ - 1 ] 𝔤 0 , \mathfrak{g}=C[\lambda,\lambda^{-1}]\otimes\mathfrak{g}_{0},
  238. 𝐠 \mathbf{g}
  239. d k λ k + 1 d d k , k . d_{k}\equiv\lambda^{k+1}\frac{d}{dk},\quad k\in\mathbb{Z}.
  240. L L
  241. 𝐠 \mathbf{g}
  242. m , n m,n
  243. K K
  244. L ( λ l G 1 , λ m G 2 ) = γ l m K ( G 1 , G 2 ) . L(\lambda^{l}\otimes G_{1},\lambda^{m}\otimes G_{2})=\gamma_{lm}K(G_{1},G_{2}).
  245. K K
  246. γ m n = γ n m , \gamma_{mn}=\gamma_{nm},
  247. γ k + l , m = γ k , l + m . \gamma_{k+l,m}=\gamma_{k,l+m}.
  248. l = 0 l=0
  249. L ( λ m G 1 , λ m G 2 ) = f ( l + m ) K ( G 1 , G 2 ) . L(\lambda^{m}\otimes G_{1},\lambda^{m}\otimes G_{2})=f(l+m)K(G_{1},G_{2}).
  250. i i∈ℤ
  251. f ( n ) = δ n i γ l m = δ l + m , i f(n)=\delta_{ni}\Leftrightarrow\gamma_{lm}=\delta_{l+m,i}
  252. L i ( λ l G 1 , λ m G 2 ) = δ l + m , i K ( G 1 , G 2 ) . L_{i}(\lambda^{l}\otimes G_{1},\lambda^{m}\otimes G_{2})=\delta_{l+m,i}K(G_{1}% ,G_{2}).
  253. L i ( d k ( λ l G 1 ) , λ m G 2 ) = - L i ( λ l G 1 , d k ( λ m G 2 ) ) , L_{i}(d_{k}(\lambda^{l}\otimes G_{1}),\lambda^{m}\otimes G_{2})=-L_{i}(\lambda% ^{l}\otimes G_{1},d_{k}(\lambda^{m}\otimes G_{2})),
  254. l δ k + l + m , i = - m δ k + l + m , i , l\delta_{k+l+m,i}=-m\delta_{k+l+m,i},
  255. n = l + m n=l+m
  256. n δ k + n , i = 0. n\delta_{k+n,i}=0.
  257. k = i k=i
  258. k = i = 0 k=i=0
  259. φ φ
  260. φ ( P ( λ ) G 1 ) , Q ( λ ) G 2 ) ) = L ( λ d P d λ G 1 , Q ( λ ) G 2 ) \varphi(P(\lambda)\otimes G_{1}),Q(\lambda)\otimes G_{2}))=L(\lambda\frac{dP}{% d\lambda}\otimes G_{1},Q(\lambda)\otimes G_{2})
  261. 𝐠 \mathbf{g}
  262. 𝔢 = 𝔤 C , \mathfrak{e}=\mathfrak{g}\oplus\mathbb{C}C,
  263. [ P ( λ ) G 1 + μ C , Q ( λ ) G 2 + ν C ] = P ( λ ) Q ( λ ) [ G 1 , G 2 ] + φ ( P ( λ ) G 1 , Q ( λ ) G 2 ) C . [P(\lambda)\otimes G_{1}+\mu C,Q(\lambda)\otimes G_{2}+\nu C]=P(\lambda)Q(% \lambda)\otimes[G_{1},G_{2}]+\varphi(P(\lambda)\otimes G_{1},Q(\lambda)\otimes G% _{2})C.
  264. [ λ l G i + μ C , λ m G j + ν C ] = λ l + m [ G i , G j ] + φ ( λ l G i , λ m G j ) C = λ l + m C i j k G k + L ( λ d λ l d λ G i , λ m G j ) C = λ l + m C i j k G k + l L ( λ l G i , λ m G j ) C = λ l + m C i j k G k + l δ l + m , 0 K ( G i , G j ) C = λ l + m C i j k G k + l δ l + m , 0 C i k m C j m k C = λ l + m C i j k G k + l δ l + m , 0 δ i j C . \begin{aligned}\displaystyle{}[\lambda^{l}\otimes G_{i}+\mu C,\lambda^{m}% \otimes G_{j}+\nu C]&\displaystyle=\lambda^{l+m}\otimes[G_{i},G_{j}]+\varphi(% \lambda^{l}\otimes G_{i},\lambda^{m}\otimes G_{j})C\\ &\displaystyle=\lambda^{l+m}\otimes{C_{ij}}^{k}G_{k}+L(\lambda\frac{d\lambda^{% l}}{d\lambda}\otimes G_{i},\lambda^{m}\otimes G_{j})C\\ &\displaystyle=\lambda^{l+m}\otimes{C_{ij}}^{k}G_{k}+lL(\lambda^{l}\otimes G_{% i},\lambda^{m}\otimes G_{j})C\\ &\displaystyle=\lambda^{l+m}\otimes{C_{ij}}^{k}G_{k}+l\delta_{l+m,0}K(G_{i},G_% {j})C\\ &\displaystyle=\lambda^{l+m}\otimes{C_{ij}}^{k}G_{k}+l\delta_{l+m,0}{C_{ik}}^{% m}{C_{jm}}^{k}C=\lambda^{l+m}\otimes{C_{ij}}^{k}G_{k}+l\delta_{l+m,0}\delta^{% ij}C.\end{aligned}
  265. [ J a 0 ( t , 𝐱 ) , J b i ( t , 𝐲 ) ] = i C a b c J c i ( t , 𝐱 ) δ ( 𝐱 - 𝐲 ) + S a b i j j δ ( 𝐱 - 𝐲 ) + , [J_{a}^{0}(t,\mathbf{x}),J_{b}^{i}(t,\mathbf{y})]=i{C_{ab}}^{c}J_{c}^{i}(t,% \mathbf{x})\delta(\mathbf{x}-\mathbf{y})+S_{ab}^{ij}\partial_{j}\delta(\mathbf% {x}-\mathbf{y})+...,
  266. 𝐠 \mathbf{g}
  267. [ λ l G i + μ C , λ m G j + ν C ] = λ l + m C i j k G k + l δ l + m , 0 δ i j C [\lambda^{l}\otimes G_{i}+\mu C,\lambda^{m}\otimes G_{j}+\nu C]=\lambda^{l+m}% \otimes{C_{ij}}^{k}G_{k}+l\delta_{l+m,0}\delta_{ij}C
  268. i i
  269. [ T a m , T b n ] = i C a b c T c m + n + m δ m + n , 0 δ a b C . [T^{m}_{a},T^{n}_{b}]=i{C_{ab}}^{c}T^{m+n}_{c}+m\delta_{m+n,0}\delta_{ab}C.
  270. 𝐠 \mathbf{g}
  271. J a ( x ) = L n = - e 2 π i n x L T a - n , x . J_{a}(x)=\frac{\hbar}{L}\sum_{n=-\infty}^{\infty}e^{\frac{2\pi inx}{L}}T_{a}^{% -n},x\in\mathbb{R}.
  272. J a ( x + L ) = J a ( x ) J_{a}(x+L)=J_{a}(x)
  273. [ ] [ J a ( x ) , J b ( y ) ] = ( L ) 2 [ n = - e 2 π i n x L T a - n , m = - e 2 π i m y L T b - m ] = ( L ) 2 m , n = - e 2 π i n x L e 2 π i m y L [ T a - n , T b - m ] . \begin{aligned}\displaystyle[][J_{a}(x),J_{b}(y)]&\displaystyle=\left(\frac{% \hbar}{L}\right)^{2}\left[\sum_{n=-\infty}^{\infty}e^{\frac{2\pi inx}{L}}T_{a}% ^{-n},\sum_{m=-\infty}^{\infty}e^{\frac{2\pi imy}{L}}T_{b}^{-m}\right]\\ &\displaystyle=\left(\frac{\hbar}{L}\right)^{2}\sum_{m,n=-\infty}^{\infty}e^{% \frac{2\pi inx}{L}}e^{\frac{2\pi imy}{L}}[T_{a}^{-n},T_{b}^{-m}].\end{aligned}
  274. y 0 , x x y z y→0,x→x−y≡z
  275. [ ] [ J a ( z ) , J b ( 0 ) ] = ( L ) 2 m , n = - e 2 π i n z L [ i C a b c T c - m - n + m δ m + n , 0 δ a b C ] = ( L ) 2 m = - e 2 π i ( - m ) z L l = - i e 2 π i ( l ) z L C a b c T c - l + ( L ) 2 m , n = - e 2 π i n z L m δ m + n , 0 δ a b C = ( L ) m = - e 2 π i m z L i C a b c J c ( z ) - ( L ) 2 n = - e 2 π i n z L n δ a b C \begin{aligned}\displaystyle[][J_{a}(z),J_{b}(0)]&\displaystyle=\left(\frac{% \hbar}{L}\right)^{2}\sum_{m,n=-\infty}^{\infty}e^{\frac{2\pi inz}{L}}[i{C_{ab}% }^{c}T^{-m-n}_{c}+m\delta_{m+n,0}\delta_{ab}C]\\ &\displaystyle=\left(\frac{\hbar}{L}\right)^{2}\sum_{m=-\infty}^{\infty}e^{% \frac{2\pi i(-m)z}{L}}\sum_{l=-\infty}^{\infty}ie^{\frac{2\pi i(l)z}{L}}{C_{ab% }}^{c}T^{-l}_{c}+\left(\frac{\hbar}{L}\right)^{2}\sum_{m,n=-\infty}^{\infty}e^% {\frac{2\pi inz}{L}}m\delta_{m+n,0}\delta_{ab}C\\ &\displaystyle=\left(\frac{\hbar}{L}\right)\sum_{m=-\infty}^{\infty}e^{\frac{2% \pi imz}{L}}i{C_{ab}}^{c}J_{c}(z)-\left(\frac{\hbar}{L}\right)^{2}\sum_{n=-% \infty}^{\infty}e^{\frac{2\pi inz}{L}}n\delta_{ab}C\end{aligned}
  276. 1 L n = - e - 2 π i n z L = 1 L n = - δ ( z + n L ) = δ ( z ) \frac{1}{L}\sum_{n=-\infty}^{\infty}e^{\frac{-2\pi inz}{L}}=\frac{1}{L}\sum_{n% =-\infty}^{\infty}\delta(z+nL)=\delta(z)
  277. z z
  278. ( 0 , L ) (0,L)
  279. - 2 π i L 2 n = - n e - 2 π i n z L = δ ( z ) , -\frac{2\pi i}{L^{2}}\sum_{n=-\infty}^{\infty}ne^{\frac{-2\pi inz}{L}}=\delta^% {\prime}(z),
  280. [ J a ( x - y ) , J b ( 0 ) ] = i C a b c J c ( x - y ) δ ( x - y ) + i 2 2 π δ a b C δ ( x - y ) , [J_{a}(x-y),J_{b}(0)]=i\hbar{C_{ab}}^{c}J_{c}(x-y)\delta(x-y)+\frac{i\hbar^{2}% }{2\pi}\delta_{ab}C\delta^{\prime}(x-y),
  281. [ J a ( x ) , J b ( y ) ] = i C a b c J c ( x ) δ ( x - y ) + i 2 2 π δ a b C δ ( x - y ) , [J_{a}(x),J_{b}(y)]=i\hbar{C_{ab}}^{c}J_{c}(x)\delta(x-y)+\frac{i\hbar^{2}}{2% \pi}\delta_{ab}C\delta^{\prime}(x-y),
  282. δ ( z ) = δ ( z 0 ) δ ( ( x y ) 0 ) = δ ( x y ) δ(z)=δ(z−0)↦δ((x−y)−0)=δ(x−y)
  283. φ φ
  284. D D
  285. 𝐠 \mathbf{g}
  286. D ( P ( λ ) G + μ C ) = λ d P ( λ ) d λ G ) . D(P(\lambda)\otimes G+\mu C)=\lambda\frac{dP(\lambda)}{d\lambda}\otimes G).
  287. 𝔢 = d + 𝔤 . \mathfrak{e}=\mathbb{C}d+\mathfrak{g}.
  288. 𝐞 \mathbf{e}
  289. [ λ m G 1 + μ C + ν D , λ n G 2 + μ C + ν D ] = λ m + n [ G 1 , G 2 ] + m δ m + n , 0 K ( G 1 , G 2 ) C + ν D ( λ n G 1 ) - ν D ( λ m G 2 ) = λ m + n [ G 1 , G 2 ] + m δ m + n , 0 K ( G 1 , G 2 ) C + ν n λ n G 1 - ν m λ m G 2 . \begin{aligned}\displaystyle{}[\lambda^{m}\otimes G_{1}+\mu C+\nu D,\lambda^{n% }\otimes G_{2}+\mu^{\prime}C+\nu^{\prime}D]&\displaystyle=\lambda^{m+n}\otimes% [G_{1},G_{2}]+m\delta_{m+n,0}K(G_{1},G_{2})C+\nu D(\lambda^{n}\otimes G_{1})-% \nu^{\prime}D(\lambda^{m}\otimes G_{2})\\ &\displaystyle=\lambda^{m+n}\otimes[G_{1},G_{2}]+m\delta_{m+n,0}K(G_{1},G_{2})% C+\nu n\lambda^{n}\otimes G_{1}-\nu^{\prime}m\lambda^{m}\otimes G_{2}.\end{aligned}
  290. G i m λ m G i . G_{i}^{m}\leftrightarrow\lambda^{m}\otimes G_{i}.
  291. [ G i m , G j n ] = C i j k G k m + n + m δ i j δ m + n , 0 C , [ C , G i m ] = 0 , 1 i , j , N , m , n [ D , G i m ] = m G i m [ D , C ] = 0. \begin{aligned}\displaystyle{}[G_{i}^{m},G_{j}^{n}]&\displaystyle={C_{ij}}^{k}% G_{k}^{m+n}+m\delta_{ij}\delta^{m+n,0}C,\\ \displaystyle{}[C,G_{i}^{m}]&\displaystyle=0,\quad 1\leq i,j,N,\quad m,n\in% \mathbb{Z}\\ \displaystyle{}[D,G_{i}^{m}]&\displaystyle=mG_{i}^{m}\\ \displaystyle{}[D,C]&\displaystyle=0.\end{aligned}
  292. φ φ
  293. W W
  294. ( l - m ) η n + m , p + ( m - n ) η m + n , l + ( n - l ) η l + n , m = 0 , η i j = φ ( d i , d j ) . (l-m)\eta_{n+m,p}+(m-n)\eta_{m+n,l}+(n-l)\eta_{l+n,m}=0,\quad\eta_{ij}=\varphi% (d_{i},d_{j}).
  295. l = 0 l=0
  296. η η
  297. ( m + p ) η m p = ( m - p ) η m + p , 0 . (m+p)\eta_{mp}=(m-p)\eta_{m+p,0}.
  298. [ d 0 + μ C , d m + ν C ] φ = - m d m + η 0 m C = - m ( d m - η 0 m m C ) . [d_{0}+\mu C,d_{m}+\nu C]_{\varphi}=-md_{m}+\eta_{0m}C=-m(d_{m}-\frac{\eta_{0m% }}{m}C).
  299. f : W ; d m φ ( d 0 , d m ) m = η 0 m m . f:W\to\mathbb{C};d_{m}\to\frac{\varphi(d_{0},d_{m})}{m}=\frac{\eta_{0m}}{m}.
  300. f f
  301. η 0 n = φ ( d 0 , d n ) = φ ( d 0 , d n ) + δ f ( [ d 0 , d n ] ) = φ ( d 0 , d n ) - n η 0 n n = 0 , \eta^{\prime}_{0n}=\varphi^{\prime}(d_{0},d_{n})=\varphi(d_{0},d_{n})+\delta f% ([d_{0},d_{n}])=\varphi(d_{0},d_{n})-n\frac{\eta^{0n}}{n}=0,
  302. [ d 0 + μ C , d m + ν C ] φ = - m d m . [d_{0}+\mu C,d_{m}+\nu C]_{\varphi^{\prime}}=-md_{m}.
  303. ( n + p ) η m p = ( n - p ) η m + p , 0 = 0 , (n+p)\eta_{mp}=(n-p)\eta_{m+p,0}=0,
  304. η m p = a ( m ) δ m . - p , a ( - m ) = - a ( m ) , \eta_{mp}=a(m)\delta_{m.-p},\quad a(-m)=-a(m),
  305. l + m + p = 0 l+m+p=0
  306. ( 2 m + p ) a ( p ) + ( m - p ) a ( m + p ) + ( m + 2 p ) a ( m ) = 0 , (2m+p)a(p)+(m-p)a(m+p)+(m+2p)a(m)=0,
  307. p = 1 p=1
  308. ( m - 1 ) a ( m + 1 ) - ( m + 2 ) a ( m ) + ( 2 m + 1 ) a ( 1 ) = 0. (m-1)a(m+1)-(m+2)a(m)+(2m+1)a(1)=0.
  309. a ( m ) = α m + β m 3 . a(m)=\alpha m+\beta m^{3}.
  310. W W
  311. [ d l , d m ] = ( l - m ) d l + m + ( α m + β m 3 ) δ l , - m C . [d_{l},d_{m}]=(l-m)d_{l+m}+(\alpha m+\beta m^{3})\delta_{l,-m}C.
  312. β β
  313. [ d l , d m ] = ( l - m ) d l + m , [d^{\prime}_{l},d^{\prime}_{m}]=(l-m)d_{l+m},
  314. β 0 β≠0
  315. d l = d l + δ 0 l α + γ 2 C , d^{\prime}_{l}=d_{l}+\delta_{0l}\frac{\alpha+\gamma}{2}C,
  316. [ d l , d m ] = ( l - m ) d l + m + ( γ m + β m 3 ) δ l , - m C , [d^{\prime}_{l},d^{\prime}_{m}]=(l-m)d^{\prime}_{l+m}+(\gamma m+\beta m^{3})% \delta_{l,-m}C,
  317. m m
  318. β β
  319. α = β = 1 / 12 α=−β={1}/{12}
  320. C C
  321. V V
  322. 𝒱 = 𝒲 + C , \mathcal{V}=\mathcal{W}+\mathbb{C}C,
  323. [ d l + μ C , d m + ν C ] = ( l - m ) d l + m + ( m - m 3 ) 12 δ l , - m C . [d_{l}+\mu C,d_{m}+\nu C]=(l-m)d_{l+m}+\frac{(m-m^{3})}{12}\delta_{l,-m}C.
  324. σ σ
  325. [ X I ( τ , σ ) , 𝒫 τ J ( τ , σ ) ] = i η I J δ ( σ - σ ) , [ x 0 - ( τ ) , p + ( τ ) ] = - i . \begin{aligned}\displaystyle{}[X^{I}(\tau,\sigma),\mathcal{P}^{\tau J}(\tau,% \sigma)]&\displaystyle=i\eta^{IJ}\delta(\sigma-\sigma^{\prime}),\\ \displaystyle{}[x_{0}^{-}(\tau),p^{+}(\tau)]&\displaystyle=-i.\end{aligned}
  326. [ α m I , α n J ] = m η I J δ m + n , 0 [\alpha_{m}^{I},\alpha_{n}^{J}]=m\eta^{IJ}\delta_{m+n,0}
  327. L n = 1 2 p α n - p I α p I . L_{n}=\frac{1}{2}\sum_{p\in\mathbb{Z}}\alpha_{n-p}^{I}\alpha_{p}^{I}.
  328. m + n = 0 m+n=0
  329. L 0 = 1 2 α 0 I α 0 I + p = 1 α - p I α p I , = α p I p I + p = 1 p α p I α p I + c L_{0}=\frac{1}{2}\alpha_{0}^{I}\alpha_{0}^{I}+\sum_{p=1}^{\infty}\alpha_{-p}^{% I}\alpha_{p}^{I},=\alpha^{\prime}p^{I}p^{I}+\sum_{p=1}^{\infty}p\alpha_{p}^{I% \dagger}\alpha_{p}^{I}+c
  330. c c
  331. [ L m , L n ] = ( m - n ) L m + n , m + n 0. [L_{m},L_{n}]=(m-n)L_{m+n},\quad m+n\neq 0.
  332. m + n = 0 m+n=0
  333. [ L m , L n ] = ( m - n ) L m + n + D - 2 12 ( m 3 - m ) δ m + n , 0 , m , n . [L_{m},L_{n}]=(m-n)L_{m+n}+\frac{D-2}{12}(m^{3}-m)\delta_{m+n,0},\quad\forall m% ,n\in\mathbb{Z}.
  334. ( D 2 ) (D−2)
  335. Π ( G ) Π(G)
  336. G G
  337. G G
  338. G G
  339. S O ( 3 ) SO(3)
  340. O ( 3 , 1 ) O(3,1)
  341. ω ω
  342. G G
  343. Π Π
  344. G ex = * × G = { ( λ , g ) | λ , g G } G_{\mathrm{ex}}=\mathbb{C}^{*}\times G=\{(\lambda,g)|\lambda\in\mathbb{C},g\in G\}
  345. ( λ 1 , g 1 ) ( λ 2 , g 2 ) = ( λ 1 λ 2 ω ( g 1 , g 2 ) , g 1 g 2 ) . (\lambda_{1},g_{1})(\lambda_{2},g_{2})=(\lambda_{1}\lambda_{2}\omega(g_{1},g_{% 2}),g_{1}g_{2}).
  346. ω ω
  347. G G
  348. ( 1 , e ) ( λ , g ) = ( λ ω ( e , g ) , g ) = ( λ , g ) = ( λ , g ) ( 1 , e ) , (1,e)(\lambda,g)=(\lambda\omega(e,g),g)=(\lambda,g)=(\lambda,g)(1,e),
  349. ( λ , g ) - 1 = ( 1 λ ω ( g , g - 1 ) , g - 1 ) . (\lambda,g)^{-1}=\left(\frac{1}{\lambda\omega(g,g^{-1})},g^{-1}\right).
  350. G G
  351. 𝔤 ex = C 𝔤 , \mathfrak{g}_{\mathrm{ex}}=\mathbb{C}C\oplus\mathfrak{g},
  352. [ μ C + G 1 , ν C + G 2 ] = [ G 1 , G 2 ] + η ( G 1 , G 2 ) C . [\mu C+G_{1},\nu C+G_{2}]=[G_{1},G_{2}]+\eta(G_{1},G_{2})C.
  353. η η
  354. 𝐠 \mathbf{g}
  355. ω ω
  356. Π Π
  357. Π ex ( ( λ , g ) ) = λ Π ( g ) . \Pi_{\mathrm{ex}}((\lambda,g))=\lambda\Pi(g).
  358. Π ex ( ( λ 1 , g 1 ) ) Π ex ( ( λ 2 , g 2 ) ) = λ 1 λ 2 Π ( g 1 ) Π ( g 2 ) = λ 1 λ 2 ω ( g 1 , g 2 ) Π ( g 1 g 2 ) = Π ex ( λ 1 λ 2 ω ( g 1 , g 2 ) , g 1 g 2 ) = Π ex ( ( λ 1 , g 1 ) ( λ 2 , g 2 ) ) , \Pi_{\mathrm{ex}}((\lambda_{1},g_{1}))\Pi_{\mathrm{ex}}((\lambda_{2},g_{2}))=% \lambda_{1}\lambda_{2}\Pi(g_{1})\Pi(g_{2})=\lambda_{1}\lambda_{2}\omega(g_{1},% g_{2})\Pi(g_{1}g_{2})=\Pi_{\mathrm{ex}}(\lambda_{1}\lambda_{2}\omega(g_{1},g_{% 2}),g_{1}g_{2})=\Pi_{\mathrm{ex}}((\lambda_{1},g_{1})(\lambda_{2},g_{2})),
  359. U ( 1 ) U(1)
  360. S H SH
  361. H H
  362. ( · , · ) (·,·)
  363. P H PH
  364. · , · · ·,··
  365. π 2 Π ex ( ( λ , g ) ) ( ψ ) = Π π ( g ) ( π 1 ( ψ ) ) , ψ S . \pi_{2}\circ\Pi_{\mathrm{ex}}((\lambda,g))(\psi)=\Pi\circ\pi(g)(\pi_{1}(\psi))% ,\quad\psi\in S\mathcal{H}.
  366. G G
  367. P H PH
  368. · , · · ·,··
  369. S H SH
  370. ( · , · ) (·,·)
  371. U ( 1 ) U(1)
  372. U ( 1 ) U(1)
  373. S H SH
  374. P H PH
  375. U ( 1 ) U(1)
  376. δ δ
  377. 𝐠 \mathbf{g}
  378. δ : 𝔤 × 𝔤 𝔤 \delta:\mathfrak{g}\times\mathfrak{g}\rightarrow\mathfrak{g}
  379. δ [ G 1 , G 2 ] = [ δ G 1 , G 2 ] + [ G 1 , δ G 2 ] , \delta[G_{1},G_{2}]=[\delta G_{1},G_{2}]+[G_{1},\delta G_{2}],
  380. 𝐠 \mathbf{g}
  381. d e r 𝐠 der\mathbf{g}
  382. [ δ 1 , δ 2 ] = δ 1 δ 2 - δ 2 δ 1 . [\delta_{1},\delta_{2}]=\delta_{1}\circ\delta_{2}-\delta_{2}\circ\delta_{1}.
  383. A u t 𝐠 Aut\mathbf{g}
  384. 𝐠 \mathbf{g}
  385. δ [ G 1 , G 1 ] = [ δ G 1 , G 2 ] + [ G 1 , δ G 2 ] e t δ [ G 1 , G 2 ] = [ e t δ G 1 , e t δ G 2 ] , t . \delta[G_{1},G_{1}]=[\delta G_{1},G_{2}]+[G_{1},\delta G_{2}]\Leftrightarrow e% ^{t\delta}[G_{1},G_{2}]=[e^{t\delta}G_{1},e^{t\delta}G_{2}],\quad\forall t\in% \mathbb{R}.
  386. t = 0 t=0
  387. ( 𝐀 ) (\mathbf{A})
  388. [ G 1 , G 2 ] = ? e - t δ [ e t δ G 1 , e t δ G 2 ] , [G_{1},G_{2}]\;\overset{?}{=}\;e^{-t\delta}[e^{t\delta}G_{1},e^{t\delta}G_{2}],
  389. ( 𝐀 ) (\mathbf{A})
  390. t t
  391. t = 0 t=0
  392. G 𝐠 G∈\mathbf{g}
  393. G G
  394. H H
  395. A u t H AutH
  396. H H
  397. H H
  398. Φ : G A u t H Φ:G→AutH
  399. g G g∈G
  400. H × G H×G
  401. E E
  402. H H
  403. E E
  404. E E
  405. H H
  406. E E
  407. g G g∈G
  408. G G
  409. H H
  410. E E
  411. Ψ Ψ
  412. A d Ad
  413. E E
  414. 𝐡 \mathbf{h}
  415. G G
  416. Ψ : G A u t 𝐡 Ψ:G→Aut\mathbf{h}
  417. G L ( 𝐡 ) ⊂GL(\mathbf{h})
  418. 𝐡 \mathbf{h}
  419. ψ : 𝐠 L i e ( A u t 𝐡 ) = D e r 𝐡 g l ( 𝐡 ) ψ:\mathbf{g}→Lie(Aut\mathbf{h})=Der\mathbf{h}⊂gl(\mathbf{h})
  420. Ψ = A d Ψ=Ad
  421. A d Ad
  422. a d ad
  423. 𝐞 = 𝐡 𝐠 \mathbf{e}=\mathbf{h}⊕\mathbf{g}
  424. G H GH
  425. E E
  426. [ H 1 + G 1 , H 2 + G 2 ] 𝔢 = [ H 1 , H 2 ] 𝔥 + ψ G 1 ( H 2 ) - ψ G 2 ( H 1 ) + [ G 1 , G 2 ] 𝔤 . [H_{1}+G_{1},H_{2}+G_{2}]_{\mathfrak{e}}=[H_{1},H_{2}]_{\mathfrak{h}}+\psi_{G_% {1}}(H_{2})-\psi_{G_{2}}(H_{1})+[G_{1},G_{2}]_{\mathfrak{g}}.
  427. E E
  428. s s
  429. t t
  430. 𝐡 \mathbf{h}
  431. 𝐞 = 𝐡 𝐠 \mathbf{e}=\mathbf{h}⊕\mathbf{g}
  432. 𝐠 \mathbf{g}
  433. e e t G ¯ s H ¯ e - t G ¯ = e t G ¯ e s H ¯ e - t G ¯ = ( 1 , e t G ) ( e s H , 1 ) ( 1 , e - t G ) = ( ϕ e t G ( e s H ) , e t G ) ( 1 , e - t G ) = ( ϕ e t G ( e s H ) ϕ e t G ( 1 ) , 1 ) = ( ϕ e t G ( e s H ) , 1 ) \begin{aligned}\displaystyle e^{e^{t\overline{G}}s\overline{H}e^{-t\overline{G% }}}&\displaystyle=e^{t\overline{G}}e^{s\overline{H}}e^{-t\overline{G}}=(1,e^{% tG})(e^{sH},1)(1,e^{-tG})\\ &\displaystyle=(\phi_{e^{tG}}(e^{sH}),e^{tG})(1,e^{-tG})=(\phi_{e^{tG}}(e^{sH}% )\phi_{e^{tG}}(1),1)\\ &\displaystyle=(\phi_{e^{tG}}(e^{sH}),1)\end{aligned}
  434. d d s e A d e t G ¯ s H ¯ | s = 0 = A d e t G ¯ H ¯ \frac{d}{ds}\left.e^{Ad_{e^{t\overline{G}}}s\overline{H}}\right|_{s=0}=Ad_{e^{% t\overline{G}}}\overline{H}
  435. d d s ( ϕ e t G ( e s H ) , 1 ) | s = 0 = ( Ψ e t G ( H ) , 0 ) \frac{d}{ds}\left.(\phi_{e^{tG}}(e^{sH}),1)\right|_{s=0}=(\Psi_{e^{tG}}(H),0)
  436. A d e t G ¯ H ¯ = ( Ψ e t G ( H ) , 0 ) . Ad_{e^{t\overline{G}}}\overline{H}=(\Psi_{e^{tG}}(H),0).
  437. t t
  438. t = 0 t=0
  439. d d t e t G ¯ H ¯ e - t G ¯ | t = 0 = [ G ¯ , H ¯ ] \frac{d}{dt}\left.e^{t\overline{G}}\overline{H}e^{-t\overline{G}}\right|_{t=0}% =[\overline{G},\overline{H}]
  440. d d t ( Ψ e t G ( H ) , 0 ) | t = 0 = ( ψ G ( H ) , 0 ) \frac{d}{dt}\left.(\Psi_{e^{tG}}(H),0)\right|_{t=0}=(\psi_{G}(H),0)
  441. [ H 1 + G 1 , H 2 + G 2 ] 𝔢 = [ H 1 , H 2 ] 𝔥 + [ G 1 , H 2 ] + [ H 1 , G 2 ] + [ G 1 , G 2 ] 𝔤 = [ H 1 , H 2 ] 𝔥 + ψ G 1 ( H 2 ) - ψ G 2 ( H 1 ) + [ G 1 , G 2 ] 𝔤 . [H_{1}+G_{1},H_{2}+G_{2}]_{\mathfrak{e}}=[H_{1},H_{2}]_{\mathfrak{h}}+[G_{1},H% _{2}]+[H_{1},G_{2}]+[G_{1},G_{2}]_{\mathfrak{g}}=[H_{1},H_{2}]_{\mathfrak{h}}+% \psi_{G_{1}}(H_{2})-\psi_{G_{2}}(H_{1})+[G_{1},G_{2}]_{\mathfrak{g}}.
  442. 𝐠 \mathbf{g}
  443. ϕ : 𝔤 × 𝔤 F , \phi:\mathfrak{g}\times\mathfrak{g}\rightarrow F,
  444. ϕ ( G 1 , G 2 ) = - ϕ ( G 2 , G 1 ) , \phi(G_{1},G_{2})=-\phi(G_{2},G_{1}),
  445. ϕ ( G 1 , [ G 2 , G 3 ] ) + ϕ ( G 2 , [ G 3 , G 1 ] ) + ϕ ( G 3 , [ G 1 , G 2 ] ) = 0. \phi(G_{1},[G_{2},G_{3}])+\phi(G_{2},[G_{3},G_{1}])+\phi(G_{3},[G_{1},G_{2}])=0.
  446. 𝐠 \mathbf{g}
  447. 𝐠 \mathbf{g}
  448. f : 𝔤 F f:\mathfrak{g}\rightarrow F
  449. f f
  450. δ f δf
  451. δ f ( G 1 , G 2 ) = f ( [ G 1 , G 2 ] ) . \delta f(G_{1},G_{2})=f([G_{1},G_{2}]).
  452. 𝐠 \mathbf{g}
  453. f f
  454. δ δ
  455. H 2 ( 𝔤 , 𝔽 ) = Z 2 ( 𝔤 , 𝔽 ) / B 2 ( 𝔤 , 𝔽 ) H^{2}(\mathfrak{g},\mathbb{F})=Z^{2}(\mathfrak{g},\mathbb{F})/B^{2}(\mathfrak{% g},\mathbb{F})
  456. 𝐠 \mathbf{g}
  457. B B
  458. 𝐠 \mathbf{g}
  459. G 𝐠 G∈\mathbf{g}
  460. G = α A c α G α , c α F , G α B G=\sum_{\alpha\in A}c_{\alpha}G_{\alpha},\quad c_{\alpha}\in F,G_{\alpha}\in B
  461. A A
  462. [ G i , G j ] = C i j k G k [G_{i},G_{j}]={C_{ij}}^{k}G_{k}
  463. 𝐮 ( 1 ) \mathbf{u}(1)
  464. g g
  465. 𝐠 \mathbf{g}
  466. g α β C β γ δ = - g γ β C β α δ . g_{\alpha\beta}{C^{\beta}}_{\gamma\delta}=-g_{\gamma\beta}{C^{\beta}}_{\alpha% \delta}.
  467. 𝐬𝐥 ( n , ) 𝐬𝐮 ( n ) \mathbf{sl}(n,ℂ)→\mathbf{su}(n)
  468. U ( 1 ) × S U ( 2 ) × S U ( 3 ) U(1)×SU(2)×SU(3)
  469. 𝐮 ( 1 ) 𝐬𝐮 ( 2 ) 𝐬𝐮 ( 3 ) \mathbf{u}(1)⊕\mathbf{su}(2)⊕\mathbf{su}(3)
  470. 𝐠 \mathbf{g}
  471. K ( G 1 , G 2 ) = trace ( ad G 1 ad G 2 ) . K(G_{1},G_{2})=\mathrm{trace}(\mathrm{ad}_{G_{1}}\mathrm{ad}_{G_{2}}).
  472. 𝐠 \mathbf{g}
  473. 𝐠 \mathbf{g}
  474. K K
  475. K K
  476. 𝐠 \mathbf{g}
  477. λ , G = K ( G λ , G ) G 𝔤 . \langle\lambda,G\rangle=K(G_{\lambda},G)\quad\forall G\in\mathfrak{g}.
  478. K ( [ G 1 , G 2 ] , G 3 ) = K ( G 1 , [ G 2 , G 3 ] ) , K([G_{1},G_{2}],G_{3})=K(G_{1},[G_{2},G_{3}]),
  479. G G
  480. G G
  481. 𝐠 \mathbf{g}
  482. G G
  483. { h : S 1 𝔤 | h ( λ ) = λ n G n , n , λ = e i θ S 1 , G n 𝔤 } . \{h:S^{1}\to\mathfrak{g}|h(\lambda)=\sum\lambda^{n}G_{n},n\in\mathbb{Z},% \lambda=e^{i\theta}\in S^{1},G_{n}\in\mathfrak{g}\}.
  484. H ( λ ) H(λ)
  485. G G
  486. H H
  487. 𝐠 \mathbf{g}
  488. H ( λ ) = e h k ( λ ) G k = e G + h k ( λ ) G k + , H(\lambda)=e^{h^{k}(\lambda)G_{k}}=e_{G}+h^{k}(\lambda)G_{k}+\ldots,
  489. K K
  490. 𝐠 \mathbf{g}
  491. h k ( λ ) = n = - θ - n k λ n h^{k}(\lambda)=\sum_{n=-\infty}^{\infty}\theta^{k}_{-n}\lambda^{n}
  492. e h k ( λ ) G k = 1 G + n = - θ - n k λ n G k + . e^{h^{k}(\lambda)G_{k}}=1_{G}+\sum_{n=-\infty}^{\infty}\theta^{k}_{-n}\lambda^% {n}G_{k}+\ldots.
  493. h : S 1 𝔤 ; h ( λ ) = n = - k = 1 K θ - n k λ n G k n = - λ n G n h:S^{1}\to\mathfrak{g};h(\lambda)=\sum_{n=-\infty}^{\infty}\sum_{k=1}^{K}% \theta^{k}_{-n}\lambda^{n}G_{k}\equiv\sum_{n=-\infty}^{\infty}\lambda^{n}G_{n}
  494. 𝐠 \mathbf{g}
  495. θ θ
  496. 0
  497. 2 π
  498. 𝐠 \mathbf{g}
  499. C [ λ , λ - 1 ] 𝔤 , C[\lambda,\lambda^{-1}]\otimes\mathfrak{g},
  500. λ k G k λ k G k . \sum\lambda^{k}G_{k}\leftrightarrow\sum\lambda^{k}\otimes G_{k}.
  501. [ P ( λ ) G 1 , Q ( λ ) G 2 ] = P ( λ ) Q ( λ ) [ G 1 , G 2 ] . [P(\lambda)\otimes G_{1},Q(\lambda)\otimes G_{2}]=P(\lambda)Q(\lambda)\otimes[% G_{1},G_{2}].
  502. 𝐠 \mathbf{g}
  503. [ λ m G i , λ n G j ] = C i j k λ m + n G k . [\lambda^{m}\otimes G_{i},\lambda^{n}\otimes G_{j}]={C_{ij}}^{k}\lambda^{m+n}% \otimes G_{k}.
  504. λ m G i λ m G i T i m ( λ ) T i m , \lambda^{m}\otimes G_{i}\cong\lambda^{m}G_{i}\leftrightarrow T^{m}_{i}(\lambda% )\equiv T^{m}_{i},
  505. λ λ
  506. [ T i m , T j n ] = C i j k T k m + n , [T^{m}_{i},T^{n}_{j}]={C_{ij}}^{k}T^{m+n}_{k},
  507. m = n = 0 m=n=0
  508. 𝐠 \mathbf{g}
  509. G G
  510. G G
  511. e x p exp
  512. G G
  513. G G
  514. 𝐠 \mathbf{g}
  515. T i n = ( λ n G i ) = - λ - n G i = - T i - n . T_{i}^{n\dagger}=(\lambda^{n}G_{i})^{\dagger}=-\lambda^{-n}G_{i}=-T_{i}^{-n}.
  516. H ( λ ) = e θ n k T k - n G H(\lambda)=e^{\theta_{n}^{k}T_{k}^{-n}}\in G
  517. T T
  518. λ λ
  519. T T
  520. i i
  521. Φ Φ
  522. = μ ϕ μ ϕ - m 2 ϕ ϕ . \mathcal{L}=\partial_{\mu}\phi^{\dagger}\partial^{\mu}\phi-m^{2}\phi^{\dagger}\phi.
  523. ϕ e - i a = 1 r α a F a ϕ , \phi\mapsto e^{-i\sum_{a=1}^{r}\alpha^{a}F_{a}}\phi,
  524. U ( N ) U(N)
  525. [ F a , F b ] = i C a b c F c . [F_{a},F_{b}]=i{C_{ab}}^{c}F_{c}.
  526. r r
  527. J a μ = - π μ i F a ϕ , π k μ = ( μ ϕ k ) , J_{a}^{\mu}=-\pi^{\mu}iF_{a}\phi,\quad\pi^{k\mu}=\frac{\partial\mathcal{L}}{% \partial(\partial_{\mu}\phi_{k})},
  528. μ J a μ = 0 , \partial_{\mu}J^{\mu}_{a}=0,
  529. Q a ( t ) = J a 0 d 3 x = const Q a , Q_{a}(t)=\int J^{0}_{a}d^{3}x=\mathrm{const}\equiv Q_{a},
  530. [ ϕ k ( t , x ) , π l ( t , x ) ] = i δ ( x - y ) δ k l , [ ϕ k ( t , x ) , ϕ l ( t , x ) ] = [ π k ( t , x ) , π l ( t , x ) ] = 0. \begin{aligned}\displaystyle{}[\phi_{k}(t,x),\pi^{l}(t,x)]&\displaystyle=i% \delta(x-y)\delta_{k}^{l},\\ \displaystyle{}[\phi_{k}(t,x),\phi_{l}(t,x)]&\displaystyle=[\pi^{k}(t,x),\pi^{% l}(t,x)]=0.\end{aligned}
  531. [ J a 0 ( t , 𝐱 ) , J b 0 ( t , 𝐲 ) ] = i δ ( 𝐱 - 𝐲 ) C a b c J c 0 ( c t , 𝐱 ) [ Q a , Q b ] = i Q a b c Q c [ Q a , J b μ ( t , 𝐱 ) ] = i C a b c J c μ ( t , 𝐱 ) , \begin{aligned}\displaystyle{}[J_{a}^{0}(t,\mathbf{x}),J_{b}^{0}(t,\mathbf{y})% ]&\displaystyle=i\delta(\mathbf{x}-\mathbf{y}){C_{ab}}^{c}J_{c}^{0}(ct,\mathbf% {x})\\ \displaystyle{}[Q_{a},Q_{b}]&\displaystyle=i{Q_{ab}}^{c}Q_{c}\\ \displaystyle{}[Q_{a},J_{b}^{\mu}(t,\mathbf{x})]&\displaystyle=i{C_{ab}}^{c}J_% {c}^{\mu}(t,\mathbf{x}),\end{aligned}
  532. μ = 0 μ=0
  533. μ = 1 , 2 , 3 μ=1,2,3
  534. [ J a 0 ( t , 𝐱 ) , J b i ( t , 𝐲 ) ] = i C a b c J c i ( t , 𝐱 ) δ ( 𝐱 - 𝐲 ) + S a b i j j δ ( 𝐱 - 𝐲 ) + . [J_{a}^{0}(t,\mathbf{x}),J_{b}^{i}(t,\mathbf{y})]=i{C_{ab}}^{c}J_{c}^{i}(t,% \mathbf{x})\delta(\mathbf{x}-\mathbf{y})+S_{ab}^{ij}\partial_{j}\delta(\mathbf% {x}-\mathbf{y})+....
  535. 𝐱 𝐲 \mathbf{x}≠\mathbf{y}
  536. 𝐱 = 𝐲 \mathbf{x}=\mathbf{y}
  537. 𝐗 \mathbf{X}
  538. 0 J 0 + i J i = 0 , 0 | J i | 0 = 0 , J 0 J 0 = J 0 J 0 = I . \partial_{0}J^{0}+\partial_{i}J^{i}=0,\quad\langle 0|J^{i}|0\rangle=0,\quad J^% {0\dagger}J^{0}=J^{0}J^{0\dagger}=I.
  539. [ J 0 ( t , 𝐱 ) , J i ( t , 𝐲 ) ] = i δ ( 𝐱 - 𝐲 ) J i ( t , 𝐱 ) + C i ( 𝐱 , 𝐲 ) . [J^{0}(t,\mathbf{x}),J^{i}(t,\mathbf{y})]=i\delta(\mathbf{x}-\mathbf{y})J^{i}(% t,\mathbf{x})+C^{i}(\mathbf{x},\mathbf{y}).
  540. 0 | C i ( 𝐱 , 𝐲 ) | 0 = 0 | [ J 0 ( t , 𝐱 ) , J i ( t , 𝐲 ) ] | 0 , \langle 0|C^{i}(\mathbf{x},\mathbf{y})|0\rangle=\langle 0|[J^{0}(t,\mathbf{x})% ,J^{i}(t,\mathbf{y})]|0\rangle,
  541. 0 | C i ( 𝐱 , 𝐲 ) y i | 0 = 0 | [ J 0 ( t , 𝐱 ) , J i ( t , 𝐲 ) y i ] | 0 = - 0 | [ J 0 ( t , 𝐱 ) , J 0 ( t , 𝐲 ) t ] | 0 = i 0 | [ J 0 ( t , 𝐱 ) , [ J 0 ( t , 𝐲 ) , H ] ] | 0 = - i 0 | J 0 ( t , 𝐱 ) H J 0 ( t , 𝐲 ) + J 0 ( t , 𝐱 ) H J 0 ( t , 𝐱 ) | 0 , \begin{aligned}\displaystyle\langle 0|\frac{\partial C^{i}(\mathbf{x},\mathbf{% y})}{\partial_{y^{i}}}|0\rangle&\displaystyle=\langle 0|[J^{0}(t,\mathbf{x}),% \frac{\partial J^{i}(t,\mathbf{y})}{\partial_{y^{i}}}]|0\rangle\\ &\displaystyle=-\langle 0|[J^{0}(t,\mathbf{x}),\frac{\partial J^{0}(t,\mathbf{% y})}{\partial_{t}}]|0\rangle=i\langle 0|[J^{0}(t,\mathbf{x}),[J^{0}(t,\mathbf{% y}),H]]|0\rangle\\ &\displaystyle=-i\langle 0|J^{0}(t,\mathbf{x})HJ^{0}(t,\mathbf{y})+J^{0}(t,% \mathbf{x})HJ^{0}(t,\mathbf{x})|0\rangle,\end{aligned}
  542. H | 0 = 0 H|0⟩=0
  543. f ( 𝐱 ) f ( 𝐲 ) f(\mathbf{x})f(\mathbf{y})
  544. 𝐱 \mathbf{x}
  545. 𝐲 \mathbf{y}
  546. - i d 𝐱 d 𝐲 0 | C i ( 𝐱 , 𝐲 ) | 0 f ( 𝐱 ) f y i f ( 𝐱 ) = 2 0 | F H F | , F = J 0 ( 𝐱 ) f ( 𝐱 ) . -i\int\int d\mathbf{x}d\mathbf{y}\langle 0|C^{i}(\mathbf{x},\mathbf{y})|0% \rangle f(\mathbf{x})\frac{\partial f}{\partial y^{i}}f(\mathbf{x})=2\langle 0% |FHF|\rangle,\quad F=\int J^{0}(\mathbf{x})f(\mathbf{x}).
  547. | n |n⟩
  548. 0 | F H F | = m n 0 | F | m m | H | n n | F | 0 = m n 0 | F | m E n δ m n n | F | 0 ) n 0 | 0 | F | n | 2 E n > 0 C i ( 𝐱 , 𝐲 ) 0. \langle 0|FHF|\rangle=\sum_{mn}\langle 0|F|m\rangle\langle m|H|n\rangle\langle n% |F|0\rangle=\sum_{mn}\langle 0|F|m\rangle E_{n}\delta_{mn}\langle n|F|0\rangle% )\sum_{n\neq 0}|\langle 0|F|n\rangle|^{2}E_{n}>0\Rightarrow C^{i}(\mathbf{x},% \mathbf{y})\neq 0.
  549. F F
  550. F F
  551. 𝐠 \mathbf{g}
  552. N N
  553. [ G i , G j ] = C i j k G k , 1 i , j , N . [G_{i},G_{j}]={C_{ij}}^{k}G_{k},\quad 1\leq i,j,N.
  554. 𝐠 ¯ \overline{\mathbf{g}}
  555. n n∈ℤ
  556. 𝔤 ¯ = F C F D 1 i 𝒩 , m F G m i \overline{\mathfrak{g}}=FC\oplus FD\oplus\bigoplus_{1\leq i\leq\mathcal{N},m% \in\mathbb{Z}}FG^{i}_{m}
  557. [ G i m , G j n ] = C i j k G k m + n + m δ i j δ m + n , 0 C , [ C , G i m ] = 0 , 1 i , j , N , m , n [ D , G i m ] = m G i m [ D , C ] = 0. \begin{aligned}\displaystyle{}[G_{i}^{m},G_{j}^{n}]&\displaystyle={C_{ij}}^{k}% G_{k}^{m+n}+m\delta_{ij}\delta^{m+n,0}C,\\ \displaystyle{}[C,G_{i}^{m}]&\displaystyle=0,\quad 1\leq i,j,N,\quad m,n\in% \mathbb{Z}\\ \displaystyle{}[D,G_{i}^{m}]&\displaystyle=mG_{i}^{m}\\ \displaystyle{}[D,C]&\displaystyle=0.\end{aligned}
  558. C = D = 0 C=D=0
  559. X = f ( φ ) d d φ , X=f(\varphi)\frac{d}{d\varphi},
  560. [ X , Y ] = [ f d d φ , g d d φ ] = ( f d g d φ - g d f d φ ) d d φ . [X,Y]=\left[f\frac{d}{d\varphi},g\frac{d}{d\varphi}\right]=\left(f\frac{dg}{d% \varphi}-g\frac{df}{d\varphi}\right)\frac{d}{d\varphi}.
  561. W W
  562. { d n , n } = { i e i n φ d d φ = - z n + 1 d d z | n } . \{d_{n},n\in\mathbb{Z}\}=\left\{\left.ie^{in\varphi}\frac{d}{d\varphi}=-z^{n+1% }\frac{d}{dz}\right|n\in\mathbb{Z}\right\}.
  563. [ d l , d m ] = ( l - m ) d l + m C l m n d n = ( l - m ) δ l + m n d n , l , m , n . [d_{l},d_{m}]=(l-m)d_{l+m}\equiv{C_{lm}}^{n}d_{n}=(l-m)\delta_{l+m}^{n}d_{n},% \quad l,m,n\in\mathbb{Z}.
  564. 3 - 3-
  565. 𝐬𝐮 ( 1 , 1 ) \mathbf{su}(1,1)
  566. 𝐬𝐥 ( 2 , ) \mathbf{sl}(2,ℝ)
  567. n 0 n≠0
  568. 𝐬𝐮 ( 1 , 1 ) 𝐬𝐥 ( 2 , ) \mathbf{su}(1,1)≅\mathbf{sl}(2,ℝ)
  569. [ d 0 , d - 1 ] = d - 1 , [ d 0 , d 1 ] = - d 1 , [ d 1 , d - 1 ] = 2 d 0 . [d_{0},d_{-1}]=d_{-1},\quad[d_{0},d_{1}]=-d_{1},\quad[d_{1},d_{-1}]=2d_{0}.
  570. 𝐬𝐥 ( 2 , ) \mathbf{sl}(2,ℝ)
  571. d 0 H = ( 1 0 0 - 1 ) , d - 1 X = ( 0 1 0 0 ) , d 1 Y = ( 0 0 1 0 ) , H , X , Y 𝔰 𝔩 ( 2 , ) . d_{0}\leftrightarrow H=\left(\begin{smallmatrix}1&0\\ 0&-1\end{smallmatrix}\right),\quad d_{-1}\leftrightarrow X=\left(\begin{% smallmatrix}0&1\\ 0&0\end{smallmatrix}\right),\quad d_{1}\leftrightarrow Y=\left(\begin{% smallmatrix}0&0\\ 1&0\end{smallmatrix}\right),\quad H,X,Y\in\mathfrak{sl}(2,\mathbb{R}).
  572. S U ( 1 , 1 ) SU(1,1)
  573. S L ( 2 , ) SL(2,ℝ)
  574. S U ( 1 , 1 ) = ( 1 - i 1 i ) S L ( 2 , ) ( 1 - i 1 i ) - 1 , SU(1,1)=\left(\begin{smallmatrix}1&-i\\ 1&i\end{smallmatrix}\right)SL(2,\mathbb{R})\left(\begin{smallmatrix}1&-i\\ 1&i\end{smallmatrix}\right)^{-1},
  575. 𝐬𝐮 ( 1 , 1 ) \mathbf{su}(1,1)
  576. U 0 = ( 0 1 1 0 ) , U 1 = ( 0 - i i 0 ) , U 2 = ( i 0 0 - i ) U_{0}=\left(\begin{smallmatrix}0&1\\ 1&0\end{smallmatrix}\right),\quad U_{1}=\left(\begin{smallmatrix}0&-i\\ i&0\end{smallmatrix}\right),\quad U_{2}=\left(\begin{smallmatrix}i&0\\ 0&-i\end{smallmatrix}\right)
  577. H 𝔰 𝔲 ( 1 , 1 ) = ( 1 - i 1 i ) H ( 1 - i 1 i ) - 1 = ( 0 1 1 0 ) = U 0 , X 𝔰 𝔲 ( 1 , 1 ) = ( 1 - i 1 i ) X ( 1 - i 1 i ) - 1 = 1 2 ( i - i i - i ) = 1 2 ( U 1 + U 2 ) , Y 𝔰 𝔲 ( 1 , 1 ) = ( 1 - i 1 i ) Y ( 1 - i 1 i ) - 1 = 1 2 ( - i - i i i ) = 1 2 ( U 1 - U 2 ) . \begin{aligned}\displaystyle H_{\mathfrak{su}(1,1)}&\displaystyle=\left(\begin% {smallmatrix}1&-i\\ 1&i\end{smallmatrix}\right)H\left(\begin{smallmatrix}1&-i\\ 1&i\end{smallmatrix}\right)^{-1}=\left(\begin{smallmatrix}0&1\\ 1&0\end{smallmatrix}\right)=U_{0},\\ \displaystyle X_{\mathfrak{su}(1,1)}&\displaystyle=\left(\begin{smallmatrix}1&% -i\\ 1&i\end{smallmatrix}\right)X\left(\begin{smallmatrix}1&-i\\ 1&i\end{smallmatrix}\right)^{-1}=\frac{1}{2}\left(\begin{smallmatrix}i&-i\\ i&-i\end{smallmatrix}\right)=\frac{1}{2}(U_{1}+U_{2}),\\ \displaystyle Y_{\mathfrak{su}(1,1)}&\displaystyle=\left(\begin{smallmatrix}1&% -i\\ 1&i\end{smallmatrix}\right)Y\left(\begin{smallmatrix}1&-i\\ 1&i\end{smallmatrix}\right)^{-1}=\frac{1}{2}\left(\begin{smallmatrix}-i&-i\\ i&i\end{smallmatrix}\right)=\frac{1}{2}(U_{1}-U_{2}).\end{aligned}
  578. W W
  579. 𝐬𝐥 ( 2 , ) \mathbf{sl}(2,ℝ)
  580. 𝐬𝐮 ( 1 , 1 ) \mathbf{su}(1,1)
  581. G G
  582. G G
  583. G = d d t ( g ( t ) ) | t = 0 , G=\frac{d}{dt}\left.(g(t))\right|_{t=0},
  584. α α
  585. G G
  586. t = 0 t=0
  587. [ G 1 , G 2 ] = d d t g 1 ( t ) g 2 ( t ) g 1 ( t ) - 1 g 2 ( t ) - 1 | t = 0 , G 1 = g 1 ( 0 ) , G 2 = g 2 ( 0 ) . [G_{1},G_{2}]=\frac{d}{dt}\left.g_{1}(t)g_{2}(t)g_{1}(t)^{-1}g_{2}(t)^{-1}% \right|_{t=0},\quad G_{1}=g_{1}^{\prime}(0),G_{2}=g_{2}^{\prime}(0).
  588. U ( G ) U(G)
  589. 𝐮 ( 𝐠 ) \mathbf{u}(\mathbf{g})
  590. [ ] [ U 1 , U 2 ] = d d t U ( g 1 ( t ) ) U ( g 2 ( t ) ) U ( g 1 ( t ) ) - 1 U ( g 2 ( t ) ) - 1 | t = 0 = d d t U ( g 1 ( t ) g 2 ( t ) g 1 ( t ) - 1 g 2 ( t ) - 1 ) | t = 0 , G 1 = g 1 ( 0 ) , G 2 = g 2 ( 0 ) . \begin{aligned}\displaystyle[][U_{1},U_{2}]&\displaystyle=\frac{d}{dt}\left.U(% g_{1}(t))U(g_{2}(t))U(g_{1}(t))^{-1}U(g_{2}(t))^{-1}\right|_{t=0}\\ &\displaystyle=\frac{d}{dt}\left.U(g_{1}(t)g_{2}(t)g_{1}(t)^{-1}g_{2}(t)^{-1})% \right|_{t=0},\quad G_{1}=g_{1}^{\prime}(0),G_{2}=g_{2}^{\prime}(0).\end{aligned}
  591. 𝐠 \mathbf{g}
  592. 𝐮 ( 𝐠 ) \mathbf{u}(\mathbf{g})
  593. 𝐮 \mathbf{u}
  594. 𝐠 \mathbf{g}
  595. U ( G ) U(G)
  596. 𝐠 \mathbf{g}
  597. U ( g 1 ) U ( g 2 ) = ω ( g 1 , g 2 ) U ( g 1 g 2 ) = e i ξ ( g 1 , g 2 ) U ( g 1 g 2 ) . U(g_{1})U(g_{2})=\omega(g_{1},g_{2})U(g_{1}g_{2})=e^{i\xi(g_{1},g_{2})}U(g_{1}% g_{2}).
  598. ω ω
  599. ω ( g , e ) = ω ( e , g ) = 1 , ω ( g 1 , g 2 g 3 ) ω ( g 2 , g 3 ) = ω ( g 1 , g 2 ) ω ( g 1 g 2 , g 3 ) ω ( g , g - 1 ) = ω ( g - 1 , g ) . \begin{aligned}\displaystyle\omega(g,e)&\displaystyle=\omega(e,g)=1,\\ \displaystyle\omega(g_{1},g_{2}g_{3})\omega(g_{2},g_{3})&\displaystyle=\omega(% g_{1},g_{2})\omega(g_{1}g_{2},g_{3})\\ \displaystyle\omega(g,g^{-1})&\displaystyle=\omega(g^{-1},g).\end{aligned}
  600. G G
  601. [ ] [ U 1 , U 2 ] = d d t U ( g 1 ( t ) ) U ( g 2 ( t ) ) U ( g 1 ( t ) ) - 1 U ( g 2 ( t ) ) - 1 | t = 0 = d d t e i ξ ( g 1 , g 2 ) ξ ( g 1 - 1 , g 2 - 1 ) ξ ( g 1 g 2 , g 1 - 1 g 2 - 1 ) U ( g 1 ( t ) g 2 ( t ) g 1 ( t ) - 1 g 2 ( t ) - 1 ) | t = 0 d d t Ω ( g 1 , g 2 ) U ( g 1 ( t ) g 2 ( t ) g 1 ( t ) - 1 g 2 ( t ) - 1 ) | t = 0 = d U ( g 1 ( t ) g 2 ( t ) g 1 ( t ) - 1 g 2 ( t ) - 1 ) d t | t = 0 + d Ω ( g 1 , g 2 ) d t | t = 0 I , G 1 = g 1 ( 0 ) , G 2 = g 2 ( 0 ) , \begin{aligned}\displaystyle[][U_{1},U_{2}]&\displaystyle=\frac{d}{dt}\left.U(% g_{1}(t))U(g_{2}(t))U(g_{1}(t))^{-1}U(g_{2}(t))^{-1}\right|_{t=0}\\ &\displaystyle=\frac{d}{dt}\left.e^{i\xi(g_{1},g_{2})\xi(g_{1}^{-1},g_{2}^{-1}% )\xi(g_{1}g_{2},g_{1}^{-1}g_{2}^{-1})}U(g_{1}(t)g_{2}(t)g_{1}(t)^{-1}g_{2}(t)^% {-1})\right|_{t=0}\\ &\displaystyle\equiv\frac{d}{dt}\left.\Omega(g_{1},g_{2})U(g_{1}(t)g_{2}(t)g_{% 1}(t)^{-1}g_{2}(t)^{-1})\right|_{t=0}\\ &\displaystyle=\left.\frac{dU(g_{1}(t)g_{2}(t)g_{1}(t)^{-1}g_{2}(t)^{-1})}{dt}% \right|_{t=0}+\left.\frac{d\Omega(g_{1},g_{2})}{dt}\right|_{t=0}I,\quad G_{1}=% g_{1}^{\prime}(0),G_{2}=g_{2}^{\prime}(0),\end{aligned}
  602. Ω Ω
  603. U U
  604. t = 0 t=0
  605. ξ ξ
  606. 𝐠 \mathbf{g}
  607. [ G i , G j ] = C i j k G k [G_{i},G_{j}]={C_{ij}^{k}}G_{k}
  608. 𝐮 \mathbf{u}
  609. [ U i , U j ] = C i j k U k + D i j I , [U_{i},U_{j}]={C_{ij}^{k}}U_{k}+D_{ij}I,
  610. 𝐮 \mathbf{u}
  611. I I
  612. σ σ
  613. τ τ
  614. X X
  615. 𝒫 μ τ τ + 𝒫 μ σ σ = 0 , 𝒫 μ τ = - T 0 c ( X ˙ X ) X μ - ( X ) 2 X ˙ μ ( X ˙ X ) 2 - ( X ˙ ) 2 ( X ) 2 , 𝒫 μ σ = - T 0 c ( X ˙ X ) X μ - ( X ˙ ) 2 X μ ( X ˙ X ) 2 - ( X ˙ ) 2 ( X ) 2 . \frac{\partial\mathcal{P}_{\mu}^{\tau}}{\partial\tau}+\frac{\partial\mathcal{P% }_{\mu}^{\sigma}}{\partial\sigma}=0,\quad\mathcal{P}_{\mu}^{\tau}=-\frac{T_{0}% }{c}\frac{(\dot{X}\cdot X^{\prime})X^{\prime}_{\mu}-(X^{\prime})^{2}\dot{X}_{% \mu}}{\sqrt{(\dot{X}\cdot X^{\prime})^{2}-(\dot{X})^{2}(X^{\prime})^{2}}},% \quad\mathcal{P}_{\mu}^{\sigma}=-\frac{T_{0}}{c}\frac{(\dot{X}\cdot X^{\prime}% )X^{\prime}_{\mu}-(\dot{X})^{2}X^{\prime}_{\mu}}{\sqrt{(\dot{X}\cdot X^{\prime% })^{2}-(\dot{X})^{2}(X^{\prime})^{2}}}.
  616. τ τ
  617. σ σ
  618. X ¨ μ - X μ ′′ = 0 , \ddot{X}^{\mu}-{X^{\mu}}^{\prime\prime}=0,
  619. X ˙ μ X μ = 0 , ( X ˙ ) 2 + ( X ) 2 = 0 , \dot{X}^{\mu}\cdot{X^{\mu}}^{\prime}=0,\quad(\dot{X})^{2}+(X^{\prime})^{2}=0,
  620. X μ ( σ , τ ) = x 0 μ + 2 α p 0 μ τ - i 2 α n = 1 ( a n μ * e i n τ - a n μ e - i n τ ) cos n σ n , X^{\mu}(\sigma,\tau)=x_{0}^{\mu}+2\alpha^{\prime}p_{0}^{\mu}\tau-i\sqrt{2% \alpha^{\prime}}\sum_{n=1}\left(a_{n}^{\mu*}e^{in\tau}-a_{n}^{\mu}e^{-in\tau}% \right)\frac{\cos n\sigma}{\sqrt{n}},
  621. α 0 μ = 2 α a μ , α n μ = a n μ n , α - n μ = a n μ * n , \alpha_{0}^{\mu}=\sqrt{2\alpha^{\prime}}a_{\mu},\quad\alpha_{n}^{\mu}=a_{n}^{% \mu}\sqrt{n},\quad\alpha_{-n}^{\mu}=a_{n}^{\mu*}\sqrt{n},
  622. X μ ( σ , τ ) = x 0 μ + 2 α α 0 μ τ + i 2 α n 0 1 n α n μ e - i n τ cos n σ . X^{\mu}(\sigma,\tau)=x_{0}^{\mu}+\sqrt{2\alpha^{\prime}}\alpha_{0}^{\mu}\tau+i% \sqrt{2\alpha^{\prime}}\sum_{n\neq 0}\frac{1}{n}\alpha_{n}^{\mu}e^{-in\tau}% \cos n\sigma.
  623. I = 2 , 3 , d I=2,3,...d
  624. d d
  625. X I ( σ , τ ) = x 0 I + 2 α α 0 I τ + i 2 α n 0 1 n α n I e - i n τ cos n σ , X + ( σ , τ ) = 2 α α 0 + τ , X - ( σ , τ ) = x 0 - + 2 α α 0 - τ + i 2 α n 0 1 n α n - e - i n τ cos n σ . \begin{aligned}\displaystyle X^{I}(\sigma,\tau)&\displaystyle=x_{0}^{I}+\sqrt{% 2\alpha^{\prime}}\alpha_{0}^{I}\tau+i\sqrt{2\alpha^{\prime}}\sum_{n\neq 0}% \frac{1}{n}\alpha_{n}^{I}e^{-in\tau}\cos n\sigma,\\ \displaystyle X^{+}(\sigma,\tau)&\displaystyle=\sqrt{2\alpha^{\prime}}\alpha_{% 0}^{+}\tau,\\ \displaystyle X^{-}(\sigma,\tau)&\displaystyle=x_{0}^{-}+\sqrt{2\alpha^{\prime% }}\alpha_{0}^{-}\tau+i\sqrt{2\alpha^{\prime}}\sum_{n\neq 0}\frac{1}{n}\alpha_{% n}^{-}e^{-in\tau}\cos n\sigma.\end{aligned}
  626. 2 α α n - = 1 2 p + p α n - p I α p I . \sqrt{2\alpha^{\prime}}\alpha_{n}^{-}=\frac{1}{2p^{+}}\sum_{p\in\mathbb{Z}}% \alpha_{n-p}^{I}\alpha_{p}^{I}.
  627. 2 α α n - 1 p + L n , L n = 1 2 p α n - p I α p I , \sqrt{2\alpha^{\prime}}\alpha_{n}^{-}\equiv\frac{1}{p^{+}}L_{n},\quad L_{n}=% \frac{1}{2}\sum_{p\in\mathbb{Z}}\alpha_{n-p}^{I}\alpha_{p}^{I},
  628. L < s u b > n L<sub>n
  629. δ X X , Y = δ δ ( X ) , Y + X X , δ ( Y ) δXX,Y=δδ(X),Y+XX,δ(Y)
  630. W W
  631. U ( 1 ) U(1)
  632. < s u p > * ℂ<sup>*
  633. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  634. A u t 𝐡 ) Aut\mathbf{h})
  635. D e r 𝐡 Der\mathbf{h}
  636. 𝐡 \mathbf{h}
  637. i i
  638. i i
  639. i i
  640. [ - F o r m u l a E r r o r - ] [-FormulaError-]

Lie_bialgebroid.html

  1. [ ϕ , f ψ ] = ρ ( ϕ ) [ f ] ψ + f [ ϕ , ψ ] , [\phi,f\cdot\psi]=\rho(\phi)[f]\cdot\psi+f\cdot[\phi,\psi],
  2. [ ϕ , [ ψ 1 , ψ 2 ] ] = [ [ ϕ , ψ 1 ] , ψ 2 ] + [ ψ 1 , [ ϕ , ψ 2 ] ] [\phi,[\psi_{1},\psi_{2}]]=[[\phi,\psi_{1}],\psi_{2}]+[\psi_{1},[\phi,\psi_{2}]]
  3. [ Φ Ψ , \Chi ] A = Φ [ Ψ , \Chi ] A + ( - 1 ) | Ψ | ( | \Chi | - 1 ) [ Φ , \Chi ] A Ψ [\Phi\wedge\Psi,\Chi]_{A}=\Phi\wedge[\Psi,\Chi]_{A}+(-1)^{|\Psi|(|\Chi|-1)}[% \Phi,\Chi]_{A}\wedge\Psi
  4. d A ( α β ) = ( d A α ) β + ( - 1 ) | α | α d A β d_{A}(\alpha\wedge\beta)=(d_{A}\alpha)\wedge\beta+(-1)^{|\alpha|}\alpha\wedge d% _{A}\beta
  5. ( d A f ) ( ϕ ) = ρ ( ϕ ) [ f ] (d_{A}f)(\phi)=\rho(\phi)[f]
  6. ( d A α ) [ ϕ , ψ ] = ρ ( ϕ ) [ α ( ψ ) ] - ρ ( ψ ) [ α ( ϕ ) ] - α [ ϕ , ψ ] (d_{A}\alpha)[\phi,\psi]=\rho(\phi)[\alpha(\psi)]-\rho(\psi)[\alpha(\phi)]-% \alpha[\phi,\psi]
  7. d * [ ϕ , ψ ] A = [ d * ϕ , ψ ] A + [ ϕ , d * ψ ] A d_{*}[\phi,\psi]_{A}=[d_{*}\phi,\psi]_{A}+[\phi,d_{*}\psi]_{A}
  8. π \pi
  9. [ X + α , Y + β ] = [ X , Y ] g + ad α Y - ad β X + [ α , β ] * + ad X * β - ad Y * α [X+\alpha,Y+\beta]=[X,Y]_{g}+\mathrm{ad}_{\alpha}Y-\mathrm{ad}_{\beta}X+[% \alpha,\beta]_{*}+\mathrm{ad}^{*}_{X}\beta-\mathrm{ad}^{*}_{Y}\alpha

Lie_group_action.html

  1. σ : G × M M , ( g , x ) g x \sigma:G\times M\to M,(g,x)\to g\cdot x
  2. σ x : G M , g x \sigma_{x}:G\to M,g\cdot x
  3. 𝔤 T x M \mathfrak{g}\to T_{x}M
  4. 𝔤 \mathfrak{g}
  5. X # X^{\#}
  6. 𝔤 Γ ( T M ) \mathfrak{g}\to\Gamma(TM)
  7. 𝔤 x \mathfrak{g}_{x}
  8. G x G_{x}
  9. P M P\to M
  10. a a u # : 𝔤 T u P a\mapsto a^{\#}_{u}:\mathfrak{g}\to T_{u}P
  11. M / G M/G
  12. M / G M/G
  13. M / G M/G
  14. M M / G M\to M/G
  15. E G EG
  16. E G × M EG\times M
  17. M G = ( E G × M ) / G M_{G}=(EG\times M)/G
  18. H G * ( M ) = H dr * ( M G ) H^{*}_{G}(M)=H^{*}_{\,\text{dr}}(M_{G})
  19. M G M_{G}

Lie_group–Lie_algebra_correspondence.html

  1. Lie ( G ) \operatorname{Lie}(G)
  2. ( d l g ) h ( X h ) = X g h (dl_{g})_{h}(X_{h})=X_{gh}
  3. l g : G G , x g x l_{g}:G\to G,x\mapsto gx
  4. ( d l g ) h : T h G T g h G (dl_{g})_{h}:T_{h}G\to T_{gh}G
  5. l g l_{g}
  6. ϕ : G × Lie ( G ) T G \phi:G\times\operatorname{Lie}(G)\to TG
  7. ϕ g ( X ) = X g \phi_{g}(X)=X_{g}
  8. T G TG
  9. Lie ( G ) T e G , X X e \operatorname{Lie}(G)\to T_{e}G,\,X\mapsto X_{e}
  10. Lie ( G ) \operatorname{Lie}(G)
  11. T e G T_{e}G
  12. Lie ( G ) \operatorname{Lie}(G)
  13. Lie ( G ) \operatorname{Lie}(G)
  14. Lie ( G ) = Lie ( G 0 ) \operatorname{Lie}(G)=\operatorname{Lie}(G^{0})
  15. G 0 G^{0}
  16. f : G H f:G\to H
  17. d f = d f e : Lie ( G ) Lie ( H ) df=df_{e}:\operatorname{Lie}(G)\to\operatorname{Lie}(H)
  18. Lie ( ker ( f ) ) = ker ( d f ) \operatorname{Lie}(\operatorname{ker}(f))=\operatorname{ker}(df)
  19. Lie ( im ( f ) ) = im ( d f ) \operatorname{Lie}(\operatorname{im}(f))=\operatorname{im}(df)
  20. G / ker ( f ) im ( f ) G/\operatorname{ker}(f)\to\operatorname{im}(f)
  21. f , g : G H f,g:G\to H
  22. d ( f g ) ( X ) = d f ( X ) + d g ( X ) . d(fg)(X)=df(X)+dg(X).
  23. f : G H f:G\to H
  24. g : H K g:H\to K
  25. d ( g f ) = ( d g ) ( d f ) . d(g\circ f)=(dg)\circ(df).
  26. d ( g g - 1 ) ( X ) = - X . d(g\mapsto g^{-1})(X)=-X.
  27. Lie ( H ) \operatorname{Lie}(H)
  28. Lie ( G ) \operatorname{Lie}(G)
  29. G / ker ( f ) G/\operatorname{ker}(f)
  30. G = G 1 × × G r G=G_{1}\times\cdots\times G_{r}
  31. p i : G G i p_{i}:G\to G_{i}
  32. d p i : Lie ( G ) Lie ( G i ) dp_{i}:\operatorname{Lie}(G)\to\operatorname{Lie}(G_{i})
  33. Lie ( G 1 × × G r ) = Lie ( G 1 ) Lie ( G r ) \operatorname{Lie}(G_{1}\times\cdots\times G_{r})=\operatorname{Lie}(G_{1})% \oplus\cdots\oplus\operatorname{Lie}(G_{r})
  34. H , H H,H^{\prime}
  35. Lie ( H H ) = Lie ( H ) Lie ( H ) . \operatorname{Lie}(H\cap H^{\prime})=\operatorname{Lie}(H)\cap\operatorname{% Lie}(H^{\prime}).
  36. f : G H f:G\to H
  37. d f df
  38. exp : Lie ( G ) G \operatorname{exp}:\operatorname{Lie}(G)\to G
  39. f ( exp ( X ) ) = exp ( d f ( X ) ) f(\operatorname{exp}(X))=\operatorname{exp}(df(X))
  40. Lie ( G ) \operatorname{Lie}(G)
  41. exp ( X ) = e X = 0 X j / j ! \displaystyle\exp(X)=e^{X}=\sum_{0}^{\infty}{X^{j}/j!}
  42. Lie ( H ) = { X Lie ( G ) | exp ( t X ) H for all t in } . \operatorname{Lie}(H)=\{X\in\operatorname{Lie}(G)|\operatorname{exp}(tX)\in H% \,\text{ for all }t\,\text{ in }\mathbb{\mathbb{R}}\}.
  43. Lie \operatorname{Lie}
  44. Lie \operatorname{Lie}
  45. ϕ : Lie ( G ) Lie ( H ) \phi:\operatorname{Lie}(G)\to\operatorname{Lie}(H)
  46. f : G H f:G\to H
  47. ϕ = d f \phi=df
  48. 𝔤 𝔩 n \mathfrak{gl}_{n}
  49. 𝔤 𝔤 𝔩 n ( ) = Lie ( G L n ( ) ) \mathfrak{g}\subset\mathfrak{gl}_{n}(\mathbb{R})=\operatorname{Lie}(GL_{n}(% \mathbb{R}))
  50. G L n ( ) GL_{n}(\mathbb{R})
  51. e 𝔤 e^{\mathfrak{g}}
  52. G ~ \widetilde{G}
  53. G ~ \widetilde{G}
  54. T e G ~ = T e G = 𝔤 T_{e}\widetilde{G}=T_{e}G=\mathfrak{g}
  55. 𝔤 = Lie ( G ) \mathfrak{g}=\operatorname{Lie}(G)
  56. 𝔤 , t t X . \mathbb{R}\to\mathfrak{g},\,t\mapsto tX.
  57. Lie ( ) = T 0 = \operatorname{Lie}(\mathbb{R})=T_{0}\mathbb{R}=\mathbb{R}
  58. H \mathbb{R}\to H
  59. t exp ( t X ) t\mapsto\operatorname{exp}(tX)
  60. 𝔤 \mathfrak{g}
  61. G L n ( ) GL_{n}(\mathbb{C})
  62. π : G G L n ( ) \pi:G\to GL_{n}(\mathbb{C})
  63. d π : 𝔤 𝔤 𝔩 n ( ) d\pi:\mathfrak{g}\to\mathfrak{gl}_{n}(\mathbb{C})
  64. π \pi
  65. e X e^{X}
  66. π ( exp ( X ) ) = I + π ( X ) + π ( X ) 2 2 ! + π ( X ) 3 3 ! + \pi(\exp(X))=I+\pi(X)+{\pi(X)^{2}\over 2!}+{\pi(X)^{3}\over 3!}+\dots
  67. c g ( h ) = g h g - 1 c_{g}(h)=ghg^{-1}
  68. d c g dc_{g}
  69. 𝔤 \mathfrak{g}
  70. Ad : G G L ( 𝔤 ) , g d c g \operatorname{Ad}:G\to GL(\mathfrak{g}),\,g\mapsto dc_{g}
  71. 𝔤 𝔤 𝔩 ( 𝔤 ) \mathfrak{g}\to\mathfrak{gl}(\mathfrak{g})
  72. 𝔤 \mathfrak{g}
  73. ad \operatorname{ad}
  74. ad ( X ) ( Y ) = [ X , Y ] \operatorname{ad}(X)(Y)=[X,Y]
  75. 𝔤 \mathfrak{g}
  76. Int ( 𝔤 ) \operatorname{Int}(\mathfrak{g})
  77. G L ( 𝔤 ) GL(\mathfrak{g})
  78. ad ( 𝔤 ) \operatorname{ad}(\mathfrak{g})
  79. Int ( 𝔤 ) \operatorname{Int}(\mathfrak{g})
  80. 𝔤 \mathfrak{g}
  81. 0 Z ( G ) G Ad Int ( 𝔤 ) 0 0\to Z(G)\to G\overset{\operatorname{Ad}}{\to}\operatorname{Int}(\mathfrak{g})\to 0
  82. Z ( G ) Z(G)
  83. det ( Ad ( g ) ) = 1 \operatorname{det}(\operatorname{Ad}(g))=1
  84. ρ ( x ) : G X , g g x \rho(x):G\to X,\,g\mapsto g\cdot x
  85. Lie ( G x ) = ker ( d ρ ( x ) : T e G T x X ) \operatorname{Lie}(G_{x})=\operatorname{ker}(d\rho(x):T_{e}G\to T_{x}X)
  86. G x G\cdot x
  87. T x ( G x ) = im ( d ρ ( x ) : T e G T x X ) T_{x}(G\cdot x)=\operatorname{im}(d\rho(x):T_{e}G\to T_{x}X)
  88. 𝔤 \mathfrak{g}
  89. 𝔷 𝔤 ( A ) = { X 𝔤 | ad ( a ) X = 0 or Ad ( a ) X = 0 for all a in A } \mathfrak{z}_{\mathfrak{g}}(A)=\{X\in\mathfrak{g}|\operatorname{ad}(a)X=0\,% \text{ or }\operatorname{Ad}(a)X=0\,\text{ for all }a\,\text{ in }A\}
  90. Z G ( A ) = { g G | Ad ( g ) a = 0 or g a = a g for all a in A } Z_{G}(A)=\{g\in G|\operatorname{Ad}(g)a=0\,\text{ or }ga=ag\,\text{ for all }a% \,\text{ in }A\}
  91. Lie ( Z G ( A ) ) = 𝔷 𝔤 ( A ) \operatorname{Lie}(Z_{G}(A))=\mathfrak{z}_{\mathfrak{g}}(A)
  92. Lie ( H ) \operatorname{Lie}(H)
  93. Lie ( G / H ) = Lie ( G ) / Lie ( H ) \operatorname{Lie}(G/H)=\operatorname{Lie}(G)/\operatorname{Lie}(H)
  94. exp : 𝔤 G \operatorname{exp}:\mathfrak{g}\to G
  95. Γ \Gamma
  96. exp \operatorname{exp}
  97. 𝔤 / Γ G \mathfrak{g}/\Gamma\to G
  98. π 1 ( G ) \pi_{1}(G)
  99. G ~ \widetilde{G}
  100. 1 π 1 ( G ) G ~ 𝑝 G 1. 1\to\pi_{1}(G)\to\widetilde{G}\overset{p}{\to}G\to 1.
  101. 𝔤 \mathfrak{g}
  102. G ~ \widetilde{G}
  103. 𝔤 \mathfrak{g}
  104. G ~ \widetilde{G}
  105. 𝔤 \mathfrak{g}
  106. G ~ \widetilde{G}
  107. Int 𝔤 \operatorname{Int}\mathfrak{g}
  108. G O ( n , ) G\hookrightarrow O(n,\mathbb{R})
  109. 𝔤 \mathfrak{g}
  110. 𝔤 \mathfrak{g}
  111. ad ( X ) \operatorname{ad}(X)
  112. 𝔤 \mathfrak{g}
  113. = { A M n + 1 ( ) | A ¯ T A = I , det ( A ) = 1 } =\{A\in M_{n+1}(\mathbb{C})|{\overline{A}}^{T}A=I,\operatorname{det}(A)=1\}
  114. 𝔰 𝔩 ( n + 1 , ) \mathfrak{sl}(n+1,\mathbb{C})
  115. = { X M n + 1 ( ) | tr X = 0 } =\{X\in M_{n+1}(\mathbb{C})|\operatorname{tr}X=0\}
  116. = { A M 2 n + 1 ( ) | A T A = I , det ( A ) = 1 } =\{A\in M_{2n+1}(\mathbb{R})|A^{T}A=I,\operatorname{det}(A)=1\}
  117. 𝔰 𝔬 ( 2 n + 1 , ) \mathfrak{so}(2n+1,\mathbb{C})
  118. = { X M 2 n + 1 ( ) | X T + X = 0 } =\{X\in M_{2n+1}(\mathbb{C})|X^{T}+X=0\}
  119. = { A U ( 2 n ) | A T J A = J } , J = [ 0 I n - I n 0 ] =\{A\in U(2n)|A^{T}JA=J\},\,J=\begin{bmatrix}0&I_{n}\\ -I_{n}&0\end{bmatrix}
  120. 𝔰 𝔭 ( n , ) \mathfrak{sp}(n,\mathbb{C})
  121. = { X M 2 n ( ) | X T J + J X = 0 } =\{X\in M_{2n}(\mathbb{C})|X^{T}J+JX=0\}
  122. = { A M 2 n ( ) | A T A = I , det ( A ) = 1 } =\{A\in M_{2n}(\mathbb{R})|A^{T}A=I,\operatorname{det}(A)=1\}
  123. 𝔰 𝔬 ( 2 n , ) \mathfrak{so}(2n,\mathbb{C})
  124. = { X M 2 n ( ) | X T + X = 0 } =\{X\in M_{2n}(\mathbb{C})|X^{T}+X=0\}
  125. H k ( 𝔤 ; ) = H DR ( G ) H^{k}(\mathfrak{g};\mathbb{R})=H_{\,\text{DR}}(G)
  126. 𝔤 \mathfrak{g}
  127. Lie ( G ) \operatorname{Lie}(G)
  128. A ( G ) A(G)
  129. A ( G ) A(G)
  130. 𝔤 = Lie ( G ) = P ( A ( G ) ) \mathfrak{g}=\operatorname{Lie}(G)=P(A(G))
  131. A ( G ) A(G)
  132. U ( 𝔤 ) = A ( G ) U(\mathfrak{g})=A(G)
  133. 𝔤 \mathfrak{g}
  134. A ( G ) A(G)
  135. l g l_{g}
  136. Lie ( f - 1 ( H ) ) = ( d f ) - 1 ( Lie ( H ) ) . \operatorname{Lie}(f^{-1}(H^{\prime}))=(df)^{-1}(\operatorname{Lie}(H^{\prime}% )).
  137. n > 0 U n \bigcup_{n>0}U^{n}
  138. exp : Lie ( G ) G \operatorname{exp}:\operatorname{Lie}(G)\to G
  139. exp ( 𝔤 ) n = exp ( 𝔤 ) \operatorname{exp}(\mathfrak{g})^{n}=\operatorname{exp}(\mathfrak{g})
  140. 𝔤 \mathfrak{g}

Lieb-Robinson_bounds.html

  1. d d
  2. d = 1 , 2 d=1,2
  3. 3 3
  4. Γ = d \Gamma=\mathbb{Z}^{d}
  5. x \mathcal{H}_{x}
  6. x Γ x\in\Gamma
  7. X Γ X\subset\Gamma
  8. X = x X x \mathcal{H}_{X}=\otimes_{x\in X}\mathcal{H}_{x}
  9. X \mathcal{H}_{X}
  10. Y \mathcal{H}_{Y}
  11. X Y X\subset Y
  12. A A
  13. X Γ X\subset\Gamma
  14. X \mathcal{H}_{X}
  15. x \mathcal{H}_{x}
  16. x \mathcal{H}_{x}
  17. x \mathcal{H}_{x}
  18. x \mathcal{H}_{x}
  19. Φ ( ) \Phi(\cdot)
  20. X Γ X\subset\Gamma
  21. Φ ( X ) \Phi(X)
  22. X X
  23. Φ ( X ) = 0 \Phi(X)=0
  24. X X
  25. R R
  26. X X
  27. R R
  28. Φ ( X ) = 0 \Phi(X)=0
  29. Φ \Phi
  30. H Φ = X Γ Φ ( X ) H_{\Phi}=\sum_{X\subset\Gamma}\Phi(X)
  31. A A
  32. A A
  33. τ t \tau_{t}
  34. τ t \tau_{t}
  35. τ t ( A ) = e i t H Φ A e - i t H Φ . \tau_{t}(A)=e^{itH_{\Phi}}Ae^{-itH_{\Phi}}.
  36. t t
  37. τ t ( A ) = A + i t [ H , A ] + ( i t ) 2 2 ! [ H , [ H , A ] ] + \tau_{t}(A)=A+it[H,A]+\frac{(it)^{2}}{2!}[H,[H,A]]+\cdots
  38. A A
  39. B B
  40. X Γ X\subset\Gamma
  41. Y Γ Y\subset\Gamma
  42. t t\in\mathbb{R}
  43. a , c a,c
  44. v v
  45. d ( X , Y ) d(X,Y)
  46. X X
  47. Y Y
  48. [ A , B ] = A B - B A [A,B]=AB-BA
  49. A A
  50. B B
  51. O \|O\|
  52. O O
  53. c c
  54. A A
  55. B B
  56. X X
  57. Y Y
  58. x \mathcal{H}_{x}
  59. v v
  60. a > 0 a>0
  61. d ( X , Y ) / v | t | d(X,Y)/v|t|
  62. d ( X , Y ) - v | t | d(X,Y)-v|t|
  63. a a
  64. v v
  65. | t | < d ( X , Y ) / v |t|<d(X,Y)/v
  66. A A
  67. B B
  68. A A
  69. B B
  70. X X
  71. A A
  72. X X
  73. ϵ > 0 \epsilon>0
  74. [ A , B ] ϵ B \|[A,B]\|\leq\epsilon\|B\|
  75. A A
  76. B B
  77. X X
  78. A ( ϵ ) A(\epsilon)
  79. X X
  80. A A
  81. A - A ( ϵ ) ϵ \|A-A(\epsilon)\|\leq\epsilon
  82. A A
  83. X X
  84. δ \delta
  85. X X
  86. δ > v | t | \delta>v|t|
  87. v v
  88. A A
  89. c c
  90. H x H_{x}
  91. Λ Γ \Lambda\subset\Gamma
  92. H Λ = x Λ H x + X Λ Φ ( X ) , H_{\Lambda}=\sum_{x\in\Lambda}H_{x}+\sum_{X\subset\Lambda}\Phi(X),
  93. H x H_{x}
  94. x \mathcal{H}_{x}
  95. Γ L = ( - L , L ) d d , \Gamma_{L}=(-L,L)^{d}\cap\mathbb{Z}^{d},
  96. L , d L,d
  97. x Γ L p x 2 + ω 2 q x 2 + x Γ L j = 1 ν λ j ( q x - q x + e j ) 2 , \sum_{x\in\Gamma_{L}}p_{x}^{2}+\omega^{2}q_{x}^{2}+\sum_{x\in\Gamma_{L}}\sum_{% j=1}^{\nu}\lambda_{j}(q_{x}-q_{x+e_{j}})^{2},
  98. λ j 0 \lambda_{j}\geq 0
  99. ω > 0 \omega>0
  100. { e j } \{e_{j}\}
  101. d \mathbb{Z}^{d}
  102. τ t \tau_{t}
  103. τ t Γ \tau_{t}^{\Gamma}
  104. Γ \Gamma
  105. Λ n Γ \Lambda_{n}\subset\Gamma
  106. n < m n<m
  107. Λ n Λ m \Lambda_{n}\subset\Lambda_{m}
  108. τ t Γ \tau_{t}^{\Gamma}
  109. { τ t Λ n } n \{\tau_{t}^{\Lambda_{n}}\}_{n}
  110. < A > Ω <A>_{\Omega}
  111. A A
  112. Ω \Omega
  113. A A
  114. B B
  115. < A B > Ω - < A > Ω < B > Ω . <AB>_{\Omega}-<A>_{\Omega}<B>_{\Omega}.
  116. Ω \Omega
  117. | < A B > Ω - < A > Ω < B > Ω | K A B min ( | X | , | Y | ) e - a d ( X , Y ) , |<AB>_{\Omega}-<A>_{\Omega}<B>_{\Omega}|\leq K\|A\|\|B\|\min(|X|,|Y|)\ e^{-a\,% d(X,Y)},
  118. A A
  119. B B
  120. X X
  121. Y Y
  122. K K
  123. a a
  124. Ω \Omega
  125. γ L \gamma_{L}
  126. γ L c / L , \gamma_{L}\leq c/L,
  127. L L
  128. c c
  129. d > 1 d>1
  130. γ L c log ( L ) / L . \gamma_{L}\leq c\log(L)/L.

Lieb–Oxford_inequality.html

  1. ρ ρ
  2. ρ ρ
  3. x 3 x∈ℝ^{3}
  4. 1 2 3 3 ρ ( x ) ρ ( y ) | x - y | d 3 x d 3 y . \frac{1}{2}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\frac{\rho(x)\rho(y)}{|x-% y|}\,\mathrm{d}^{3}x\,\mathrm{d}^{3}y.
  5. x x
  6. y y
  7. x x
  8. y y
  9. 1 / 2 {1}/{2}
  10. ρ ρ
  11. x 3 x∈ℝ^{3}
  12. ρ ρ
  13. N N
  14. e e
  15. N N
  16. P ( x 1 , , x N ) . P(x_{1},\dots,x_{N}).
  17. P P
  18. N N
  19. ψ L 2 ( 3 N ) , \psi\in L^{2}(\mathbb{R}^{3N}),
  20. P ( x 1 , , x N ) = | ψ ( x 1 , , x N ) | 2 . P(x_{1},\dots,x_{N})=|\psi(x_{1},\dots,x_{N})|^{2}.
  21. q q
  22. ψ ( x 1 , σ 1 , , x N , σ N ) \psi(x_{1},\sigma_{1},\dots,x_{N},\sigma_{N})
  23. N N
  24. P ( x 1 , , x N ) = σ 1 = 1 q σ N = 1 q | ψ ( x 1 , σ 1 , , x N , σ N ) | 2 . P(x_{1},\dots,x_{N})=\sum_{\sigma_{1}=1}^{q}\cdots\sum_{\sigma_{N}=1}^{q}|\psi% (x_{1},\sigma_{1},\dots,x_{N},\sigma_{N})|^{2}.
  25. γ γ
  26. P P
  27. γ ( x 1 , , x N ; x 1 , , x N ) . \gamma(x_{1},...,x_{N};x_{1},...,x_{N}).
  28. I P = e 2 1 i < j N 3 N P ( x 1 , , x i , , x j , , x N ) | x i - x j | d 3 x 1 d 3 x N . I_{P}=e^{2}\sum_{1\leq i<j\leq N}\int_{\mathbb{R}^{3N}}\frac{P(x_{1},\dots,x_{% i},\dots,x_{j},\dots,x_{N})}{|x_{i}-x_{j}|}\,\mathrm{d}^{3}x_{1}\cdots\mathrm{% d}^{3}x_{N}.
  29. x 3 x∈ℝ^{3}
  30. ρ ( x ) = | e | i = 1 N 3 ( N - 1 ) P ( x 1 , , x i - 1 , x , x i + 1 , , x N ) d 3 x 1 d 3 x i - 1 d 3 x i + 1 d 3 x N \rho(x)=|e|\sum_{i=1}^{N}\int_{\mathbb{R}^{3(N-1)}}P(x_{1},\dots,x_{i-1},x,x_{% i+1},\dots,x_{N})\,\mathrm{d}^{3}x_{1}\cdots\mathrm{d}^{3}x_{i-1}\,\mathrm{d}^% {3}x_{i+1}\cdots\mathrm{d}^{3}x_{N}
  31. N N
  32. ρ ρ
  33. D ( ρ ) = 1 2 3 3 ρ ( x ) ρ ( y ) | x - y | d 3 x d 3 y . D(\rho)=\frac{1}{2}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\frac{\rho(x)\rho% (y)}{|x-y|}\,\mathrm{d}^{3}x\,\mathrm{d}^{3}y.
  34. I P I_{P}
  35. D ( ρ ) D(ρ)
  36. E P = I P - D ( ρ ) - C | e | 2 3 3 | ρ ( x ) | 4 3 d 3 x , E_{P}=I_{P}-D(\rho)\geq-C|e|^{\frac{2}{3}}\int_{\mathbb{R}^{3}}|\rho(x)|^{% \frac{4}{3}}\,\mathrm{d}^{3}x,
  37. C 1.68 C≤1.68
  38. N N
  39. E P E_{P}
  40. e 1 , , e N e_{1},...,e_{N}
  41. E P E_{P}
  42. C = 8.52 C=8.52
  43. C = 1.68 C=1.68
  44. C = 1.64 C=1.64
  45. N N
  46. N N
  47. N = 1 N=1
  48. I P = 0 I_{P}=0
  49. C 1 = 1.092 C_{1}=1.092
  50. ρ ρ
  51. N = 2 N=2
  52. C 2 1.234 C_{2}≥1.234
  53. C N C_{N}
  54. C N C N + 1 C_{N}≤C_{N+1}
  55. N N
  56. C C
  57. N = 30 N=30
  58. C 30 1.31 C_{30}≥1.31
  59. C C
  60. 1.31 C 1.64 1.31≤C≤1.64
  61. E P E_{P}
  62. ψ ( x 1 , σ 1 , , x N , σ N ) = det ( φ i ( x j , σ j ) ) N ! . \psi(x_{1},\sigma_{1},\dots,x_{N},\sigma_{N})=\frac{\det(\varphi_{i}(x_{j},% \sigma_{j}))}{\sqrt{N!}}.
  63. | Λ | |Λ|
  64. φ α , k ( x , σ ) = χ α ( σ ) e 2 π i k x | Λ | , \varphi_{\alpha,k}(x,\sigma)=\frac{\chi_{\alpha}(\sigma)\mathrm{e}^{2\pi% \mathrm{i}k\cdot x}}{\sqrt{|\Lambda|}},
  65. χ 1 , , χ q χ_{1},...,χ_{q}
  66. q ℂ^{q}
  67. k 3 k∈ℝ^{3}
  68. n / | Λ | 1 / 3 n/|Λ|^{{1}/{3}}
  69. n [ u s u , u p = 3 , u b = + ] n∈ℤ[u^{\prime}su^{\prime},u^{\prime}p=3^{\prime},u^{\prime}b=+^{\prime}]
  70. N N
  71. | Λ | |Λ|
  72. ρ = N | e | / | Λ | ρ=N|e|/|Λ|
  73. E P ( Dirac ) = - C | e | 2 / 3 q - 1 / 3 ρ 4 / 3 | Λ | , E_{P}(\mathrm{Dirac})=-C|e|^{2/3}q^{-1/3}\rho^{4/3}|\Lambda|,
  74. C = 0.93 C=0.93
  75. q q
  76. q q
  77. C C
  78. ρ ρ
  79. C C
  80. 1.45 1.45
  81. C 1.45 C≤1.45

Lieb–Thirring_inequality.html

  1. N N
  2. - Δ + V ( x ) = - 2 + V ( x ) -\Delta+V(x)=-\nabla^{2}+V(x)
  3. n \mathbb{R}^{n}
  4. V ( x ) : n V(x):\mathbb{R}^{n}\to\mathbb{R}
  5. λ 1 λ 2 0 \lambda_{1}\leq\lambda_{2}\leq\dots\leq 0
  6. γ \gamma
  7. n n
  8. γ 1 2 , n = 1 , γ > 0 , n = 2 , γ 0 , n 3 , \begin{aligned}\displaystyle\gamma\geq\frac{1}{2}&\displaystyle,\,n=1,\\ \displaystyle\gamma>0&\displaystyle,\,n=2,\\ \displaystyle\gamma\geq 0&\displaystyle,\,n\geq 3,\end{aligned}
  9. L γ , n L_{\gamma,n}
  10. γ \gamma
  11. n n
  12. j 1 | λ j | γ L γ , n \R n V ( x ) - γ + n 2 d n x \sum_{j\geq 1}|\lambda_{j}|^{\gamma}\leq L_{\gamma,n}\int_{\R^{n}}V(x)_{-}^{% \gamma+\frac{n}{2}}\mathrm{d}^{n}x
  13. V ( x ) - := max ( - V ( x ) , 0 ) V(x)_{-}:=\max(-V(x),0)
  14. V V
  15. γ > 1 / 2 , n = 1 \gamma>1/2,n=1
  16. γ > 0 , n 2 \gamma>0,n\geq 2
  17. γ = 0 , n 3 \gamma=0,n\geq 3
  18. γ = 0 \gamma=0
  19. γ = 1 / 2 , n = 1 \gamma=1/2,n=1
  20. γ \gamma
  21. n n
  22. ( p , x ) 2 n (p,x)\in\mathbb{R}^{2n}
  23. - i -\mathrm{i}\nabla
  24. p p
  25. ( 2 π ) n (2\pi)^{n}
  26. 2 n 2n
  27. j 1 | λ j | γ 1 ( 2 π ) n n n ( p 2 + V ( x ) ) - γ d n p d n x = L γ , n cl n V ( x ) - γ + n 2 d n x \sum_{j\geq 1}|\lambda_{j}|^{\gamma}\approx\frac{1}{(2\pi)^{n}}\int_{\mathbb{R% }^{n}}\int_{\mathbb{R}^{n}}\big(p^{2}+V(x)\big)_{-}^{\gamma}\mathrm{d}^{n}p% \mathrm{d}^{n}x=L^{\mathrm{cl}}_{\gamma,n}\int_{\mathbb{R}^{n}}V(x)_{-}^{% \gamma+\frac{n}{2}}\mathrm{d}^{n}x
  28. L γ , n cl = ( 4 π ) - n 2 Γ ( γ + 1 ) Γ ( γ + 1 + n 2 ) . L_{\gamma,n}^{\mathrm{cl}}=(4\pi)^{-\frac{n}{2}}\frac{\Gamma(\gamma+1)}{\Gamma% (\gamma+1+\frac{n}{2})}\,.
  29. γ > 0 \gamma>0
  30. γ \gamma
  31. L γ , n L_{\gamma,n}
  32. β V \beta V
  33. lim β 1 β γ + n 2 tr ( - Δ + β V ) - γ = L γ , n cl n V ( x ) - γ + n 2 d n x \lim_{\beta\to\infty}\frac{1}{\beta^{\gamma+\frac{n}{2}}}\mathrm{tr}(-\Delta+% \beta V)_{-}^{\gamma}=L^{\mathrm{cl}}_{\gamma,n}\int_{\mathbb{R}^{n}}V(x)_{-}^% {\gamma+\frac{n}{2}}\mathrm{d}^{n}x
  34. L γ , n cl L γ , n L_{\gamma,n}^{\mathrm{cl}}\leq L_{\gamma,n}
  35. L γ , n = L γ , n cl L_{\gamma,n}=L_{\gamma,n}^{\mathrm{cl}}
  36. γ 3 / 2 , n = 1 \gamma\geq 3/2,n=1
  37. n n
  38. L γ , n / L γ , n cl L_{\gamma,n}/L_{\gamma,n}^{\mathrm{cl}}
  39. γ \gamma
  40. L γ , n = L γ , n cl L_{\gamma,n}=L_{\gamma,n}^{\mathrm{cl}}
  41. n n
  42. γ 3 / 2 \gamma\geq 3/2
  43. γ = 1 / 2 , n = 1 \gamma=1/2,\,n=1
  44. L 1 / 2 , 1 = 2 L 1 / 2 , 1 cl = 1 / 2 L_{1/2,1}=2L_{1/2,1}^{\mathrm{cl}}=1/2
  45. L γ , n cl < L γ , n L^{\mathrm{cl}}_{\gamma,n}<L_{\gamma,n}
  46. 1 / 2 γ < 3 / 2 , n = 1 1/2\leq\gamma<3/2,n=1
  47. γ < 1 , d 1 \gamma<1,d\geq 1
  48. L γ , 1 = 2 L γ , 1 cl ( γ - 1 2 γ + 1 2 ) γ - 1 2 . L_{\gamma,1}=2L^{\mathrm{cl}}_{\gamma,1}\left(\frac{\gamma-\frac{1}{2}}{\gamma% +\frac{1}{2}}\right)^{\gamma-\frac{1}{2}}.
  49. L 1 , 3 L_{1,3}
  50. π L 1 , 3 cl / 3 \pi L_{1,3}^{\mathrm{cl}}/\sqrt{3}
  51. 6.869 L 0 , n cl 6.869L_{0,n}^{\mathrm{cl}}
  52. L γ , n L_{\gamma,n}
  53. γ = 1 \gamma=1
  54. N N
  55. ψ L 2 ( N n ) \psi\in L^{2}(\mathbb{R}^{Nn})
  56. ψ ( x 1 , , x i , , x j , , x N ) = - ψ ( x 1 , , x j , , x i , , x N ) \psi(x_{1},\dots,x_{i},\dots,x_{j},\dots,x_{N})=-\psi(x_{1},\dots,x_{j},\dots,% x_{i},\dots,x_{N})
  57. 1 i , j N 1\leq i,j\leq N
  58. ρ ψ ( x ) = N ( N - 1 ) n | ψ ( x , x 2 , x N ) | 2 d n x 2 d n x N , x n . \rho_{\psi}(x)=N\int_{\mathbb{R}^{(N-1)n}}|\psi(x,x_{2}\dots,x_{N})|^{2}% \mathrm{d}^{n}x_{2}\cdots\mathrm{d}^{n}x_{N},\,x\in\mathbb{R}^{n}.
  59. γ = 1 \gamma=1
  60. i = 1 N n | i ψ | 2 d n x i K n n ρ ψ ( x ) 1 + 2 n d n x \sum_{i=1}^{N}\int_{\mathbb{R}^{n}}|\nabla_{i}\psi|^{2}\mathrm{d}^{n}x_{i}\geq K% _{n}\int_{\mathbb{R}^{n}}{\rho_{\psi}(x)^{1+\frac{2}{n}}}\mathrm{d}^{n}x
  61. K n K_{n}
  62. ( ( 1 + 2 n ) K n ) 1 + n 2 ( ( 1 + n 2 ) L 1 , n ) 1 + 2 n = 1 . \left(\left(1+\frac{2}{n}\right)K_{n}\right)^{1+\frac{n}{2}}\left(\left(1+% \frac{n}{2}\right)L_{1,n}\right)^{1+\frac{2}{n}}=1\,.
  63. K n K_{n}
  64. K n / q 2 / n K_{n}/q^{2/n}
  65. q q
  66. q = 2 q=2
  67. ψ ( x 1 , , x i , , x j , , x n ) = ψ ( x 1 , , x j , , x i , , x n ) \psi(x_{1},\dots,x_{i},\dots,x_{j},\dots,x_{n})=\psi(x_{1},\dots,x_{j},\dots,x% _{i},\dots,x_{n})
  68. 1 i , j N 1\leq i,j\leq N
  69. K n K_{n}
  70. K n / N 2 / n K_{n}/N^{2/n}
  71. ρ ψ \rho_{\psi}
  72. N N
  73. n n
  74. L 1 , 3 = L 1 , 3 cl L_{1,3}=L^{\mathrm{cl}}_{1,3}
  75. n = 3 n=3
  76. N N
  77. q q
  78. M M
  79. R j 3 R_{j}\in\mathbb{R}^{3}
  80. Z j > 0 Z_{j}>0
  81. ψ \psi
  82. K n / q 2 / n K_{n}/q^{2/n}
  83. K n / N 2 / n K_{n}/N^{2/n}
  84. E N , M ( Z 1 , , Z M ) E_{N,M}(Z_{1},\dots,Z_{M})
  85. Z max Z_{\max}
  86. E N , M ( Z 1 , , Z M ) - C ( Z max ) ( M + N ) . E_{N,M}(Z_{1},\dots,Z_{M})\geq-C(Z_{\max})(M+N)\,.
  87. ψ \psi
  88. K n / N 2 / n K_{n}/N^{2/n}
  89. - C N 5 / 3 -CN^{5/3}
  90. 5 / 3 5/3
  91. - Δ = - 2 -\Delta=-\nabla^{2}
  92. ( i + A ( x ) ) 2 (\mathrm{i}\nabla+A(x))^{2}
  93. A ( x ) A(x)
  94. n \mathbb{R}^{n}
  95. L γ , n L_{\gamma,n}
  96. - Δ -\Delta
  97. - Δ \sqrt{-\Delta}
  98. L γ , n L_{\gamma,n}
  99. γ + n \gamma+n
  100. 1 + 2 / n 1+2/n
  101. 1 + 1 / n 1+1/n
  102. Z k Z_{k}
  103. λ j \lambda_{j}
  104. [ 0 , ) [0,\infty)
  105. V V
  106. L γ , 1 L_{\gamma,1}
  107. γ 1 / 2 \gamma\geq 1/2

Liénard–Chipart_criterion.html

  1. f ( z ) = a 0 z n + a 1 z n - 1 + + a n ( a 0 > 0 ) f(z)=a_{0}z^{n}+a_{1}z^{n-1}+\cdots+a_{n}\,(a_{0}>0)
  2. f f
  3. Δ 1 > 0 , Δ 2 > 0 , , Δ n > 0 , \Delta_{1}>0,\,\Delta_{2}>0,\ldots,\Delta_{n}>0,
  4. Δ i \Delta_{i}
  5. f f
  6. f f
  7. a n > 0 , a n - 2 > 0 , ; Δ 1 > 0 , Δ 3 > 0 , a_{n}>0,a_{n-2}>0,\ldots;\,\Delta_{1}>0,\Delta_{3}>0,\ldots
  8. a n > 0 , a n - 2 > 0 , ; Δ 2 > 0 , Δ 4 > 0 , a_{n}>0,a_{n-2}>0,\ldots;\,\Delta_{2}>0,\Delta_{4}>0,\ldots
  9. a n > 0 , a n - 1 > 0 , a n - 3 > 0 , ; Δ 1 > 0 , Δ 3 > 0 , a_{n}>0,a_{n-1}>0,a_{n-3}>0,\ldots;\,\Delta_{1}>0,\Delta_{3}>0,\ldots
  10. a n > 0 , a n - 1 > 0 , a n - 3 > 0 , ; Δ 2 > 0 , Δ 4 > 0 , a_{n}>0,a_{n-1}>0,a_{n-3}>0,\ldots;\,\Delta_{2}>0,\Delta_{4}>0,\ldots

Lifshitz_Theory_of_Van_der_Waals_Force.html

  1. α \alpha
  2. Q Q
  3. α 2 \alpha_{2}
  4. r r
  5. p p
  6. U ( r ) = - Q p 4 π ϵ 0 ϵ 3 r 2 U(r)=\frac{-Qp}{4\pi\epsilon_{0}\epsilon_{3}r^{2}}
  7. p = α 2 E p=\alpha_{2}E
  8. E E
  9. r r
  10. E = Q 4 π ϵ 0 ϵ 3 1 r 2 E=\frac{Q}{4\pi\epsilon_{0}\epsilon_{3}}\frac{1}{r^{2}}
  11. U ( r ) = - Q 2 α 2 ( 4 π ϵ 0 ϵ 3 ) 2 r 4 U(r)=\frac{-Q^{2}\alpha_{2}}{(4\pi\epsilon_{0}\epsilon_{3})^{2}r^{4}}
  12. ρ 2 \rho_{2}
  13. U ( D ) = - 2 π Q 2 α 2 ( 4 π ϵ 0 ϵ 3 ) 2 ρ 2 z = D d z x = 0 d x x ( z 2 + x 2 ) 2 = - π Q 2 α 2 ρ 2 ( 4 π ϵ 0 ϵ 3 ) 2 1 D \begin{aligned}\displaystyle U(D)&\displaystyle=-2\pi\frac{Q^{2}\alpha_{2}}{(4% \pi\epsilon_{0}\epsilon_{3})^{2}}\rho_{2}\int\limits_{z=D}^{\infty}dz\int% \limits_{x=0}^{\infty}dx\frac{x}{(z^{2}+x^{2})^{2}}\\ &\displaystyle=-\frac{\pi Q^{2}\alpha_{2}\rho_{2}}{(4\pi\epsilon_{0}\epsilon_{% 3})^{2}}\frac{1}{D}\end{aligned}
  14. Q Q
  15. ϵ 3 \epsilon_{3}
  16. D D
  17. Q = - Q ( ϵ 2 - ϵ 3 ) / ( ϵ 2 + ϵ 3 ) Q^{\prime}=-Q(\epsilon_{2}-\epsilon_{3})/(\epsilon_{2}+\epsilon_{3})
  18. F ( D ) = - Q 2 ( 4 π ϵ 0 ϵ 3 ) 2 ( 2 D ) 2 ϵ 2 - ϵ 3 ϵ 2 + ϵ 3 F(D)=\frac{-Q^{2}}{(4\pi\epsilon_{0}\epsilon_{3})^{2}(2D)^{2}}\frac{\epsilon_{% 2}-\epsilon_{3}}{\epsilon_{2}+\epsilon_{3}}
  19. U ( D ) = - Q 2 ( 4 π ϵ 0 ϵ 3 ) 2 4 D ϵ 2 - ϵ 3 ϵ 2 + ϵ 3 U(D)=\frac{-Q^{2}}{(4\pi\epsilon_{0}\epsilon_{3})^{2}4D}\frac{\epsilon_{2}-% \epsilon_{3}}{\epsilon_{2}+\epsilon_{3}}
  20. ρ 2 α 2 = 2 ϵ 2 ϵ 2 ϵ 2 - ϵ 3 ϵ 2 + ϵ 3 \rho_{2}\alpha_{2}=2\epsilon_{2}\epsilon_{2}\frac{\epsilon_{2}-\epsilon_{3}}{% \epsilon_{2}+\epsilon_{3}}
  21. Q Q
  22. ρ 1 \rho_{1}
  23. α 1 \alpha_{1}
  24. α 1 ρ 1 \alpha_{1}\rho_{1}
  25. U ( r ) = - C / r n U(r)=-C/r^{n}
  26. T T
  27. A = π 2 C ρ 1 ρ 2 = 6 π 2 k B T ρ 1 ρ 2 ( 4 π ϵ 0 ) 2 n = 0 , 1... α 1 ( i ν n ) α 2 ( i ν n ) ϵ 3 2 A=\pi^{2}C\rho_{1}\rho_{2}=\frac{6\pi^{2}k_{B}T\rho_{1}\rho_{2}}{(4\pi\epsilon% _{0})^{2}}\sum_{n=0,1...}^{\infty}\frac{\alpha_{1}(i\nu_{n})\alpha_{2}(i\nu_{n% })}{\epsilon_{3}^{2}}
  28. ν n = 2 π n k B T / h \nu_{n}=2\pi nk_{B}T/h
  29. k B k_{B}
  30. h h
  31. ρ α \rho\alpha
  32. k B T n = 0 , 1... h 2 π ν 1 d ν k_{B}T\sum_{n=0,1...}^{\infty}\rightarrow\frac{h}{2\pi}\int\limits_{\nu_{1}}^{% \infty}d\nu
  33. A 3 4 k B T ( ϵ 1 - ϵ 3 ϵ 1 + ϵ 3 ) ( ϵ 2 - ϵ 3 ϵ 2 + ϵ 3 ) + 3 h 4 π ν 1 d ν ( ϵ 1 ( i ν ) - ϵ 3 ( i ν ) ϵ 1 ( i ν ) + ϵ 3 ( i ν ) ) ( ϵ 2 ( i ν ) - ϵ 3 ( i ν ) ϵ 2 ( i ν ) + ϵ 3 ( i ν ) ) A\approx\frac{3}{4}k_{B}T\left(\frac{\epsilon_{1}-\epsilon_{3}}{\epsilon_{1}+% \epsilon_{3}}\right)\left(\frac{\epsilon_{2}-\epsilon_{3}}{\epsilon_{2}+% \epsilon_{3}}\right)+\frac{3h}{4\pi}\int\limits_{\nu_{1}}^{\infty}d\nu\left(% \frac{\epsilon_{1}(i\nu)-\epsilon_{3}(i\nu)}{\epsilon_{1}(i\nu)+\epsilon_{3}(i% \nu)}\right)\left(\frac{\epsilon_{2}(i\nu)-\epsilon_{3}(i\nu)}{\epsilon_{2}(i% \nu)+\epsilon_{3}(i\nu)}\right)
  34. ϵ ( i ν ) \epsilon(i\nu)

Ligand_binding_assay.html

  1. p m = C P M / S A ( C P M / f m o l ) V o l u m e ( m l ) × 0.001 ( p m o l / f m o l ) 0.001 ( l i t e r / m l ) = ( C P M / S A ) ( V o l ) pm=\frac{CPM/SA(CPM/fmol)}{Volume(ml)}\times{0.001(pmol/fmol)\over 0.001(liter% /ml)}={(CPM/SA)\over(Vol)}

Light-front_computational_methods.html

  1. x + c t + z x^{+}\equiv ct+z
  2. x - c t - z x^{-}\equiv ct-z
  3. t t
  4. z z
  5. c c
  6. x x
  7. y y
  8. x = ( x , y ) \vec{x}_{\perp}=(x,y)
  9. t t
  10. z z
  11. P + = 0 P^{+}=0
  12. { | n : p i + , p i } \{|n:p_{i}^{+},\vec{p}_{\perp i}\rangle\}
  13. 𝒫 + \mathcal{P}^{+}
  14. 𝒫 \vec{\mathcal{P}}_{\perp}
  15. | P ¯ |\underline{P}\rangle
  16. | P ¯ = n [ d x ] n [ d 2 k ] n ψ n ( x , k ) | n : x P + , x P + k , |\underline{P}\rangle=\sum_{n}\int[dx]_{n}\,[d^{2}k_{\perp}]_{n}\,\psi_{n}(x,% \vec{k}_{\perp})|n:xP^{+},x\vec{P}_{\perp}+\vec{k}_{\perp}\rangle\,,
  17. [ d x ] n = 4 π δ ( 1 - i = 1 n x i ) i = 1 n d x i 4 π x i , [ d 2 k ] n = 4 π 2 δ ( i = 1 n k i ) i = 1 n d 2 k i 4 π 2 . [dx]_{n}=4\pi\delta(1-\sum_{i=1}^{n}x_{i})\prod_{i=1}^{n}\frac{dx_{i}}{4\pi% \sqrt{x_{i}}}\,,\;\;\;[d^{2}k_{\perp}]_{n}=4\pi^{2}\delta(\sum_{i=1}^{n}\vec{k% }_{\perp i})\prod_{i=1}^{n}\frac{d^{2}k_{\perp i}}{4\pi^{2}}\,.
  18. ψ n \psi_{n}
  19. n n
  20. 𝒫 - | P ¯ = M 2 + P 2 P + | P ¯ \mathcal{P}^{-}|\underline{P}\rangle=\frac{M^{2}+P_{\perp}^{2}}{P^{+}}|% \underline{P}\rangle
  21. p + 2 π L n , p ( π L n x , π L n y ) , p^{+}\rightarrow\frac{2\pi}{L}n\,,\;\;\vec{p}_{\perp}\rightarrow(\frac{\pi}{L_% {\perp}}n_{x},\frac{\pi}{L_{\perp}}n_{y}),
  22. - L < x - < L -L<x^{-}<L
  23. - L < x , y < L -L_{\perp}<x,y<L_{\perp}
  24. L L
  25. L L_{\perp}
  26. L L\rightarrow\infty
  27. K L 2 π P + K\equiv\frac{L}{2\pi}P^{+}
  28. H LC = P + 𝒫 - H_{\rm LC}=P^{+}\mathcal{P}^{-}
  29. L L
  30. x i p i + / P + x_{i}\equiv p_{i}^{+}/P^{+}
  31. n i / K n_{i}/K
  32. n i n_{i}
  33. K K
  34. Q - Q^{-}
  35. 𝒫 - \mathcal{P}^{-}
  36. 𝒫 - = { Q - , Q - } / 2 2 \mathcal{P}^{-}=\{Q^{-},Q^{-}\}/2\sqrt{2}
  37. T + + ( x ) T + + ( y ) \langle T^{++}(x)T^{++}(y)\rangle
  38. x + x^{+}
  39. x - x^{-}
  40. p - p^{-}
  41. N c N_{c}
  42. x - x^{-}
  43. T T
  44. Z e T | ϕ \sqrt{Z}e^{T}|\phi\rangle
  45. Z \sqrt{Z}
  46. | ϕ |\phi\rangle
  47. T T
  48. T T
  49. T T
  50. T T
  51. | ϕ |\phi\rangle
  52. T T
  53. P v 𝒫 - ¯ | ϕ = M 2 + P 2 P + | ϕ , ( 1 - P v ) 𝒫 - ¯ | ϕ = 0. P_{v}\overline{\mathcal{P}^{-}}|\phi\rangle=\frac{M^{2}+P_{\perp}^{2}}{P^{+}}|% \phi\rangle,\;\;\;\;(1-P_{v})\overline{\mathcal{P}^{-}}|\phi\rangle=0.
  54. P v P_{v}
  55. 𝒫 - ¯ e - T 𝒫 - e T \overline{\mathcal{P}^{-}}\equiv e^{-T}\mathcal{P}^{-}e^{T}
  56. 1 - P v 1-P_{v}
  57. T T
  58. 𝒫 - ¯ = 𝒫 - + [ 𝒫 - , T ] + 1 2 [ [ 𝒫 - , T ] , T ] + \overline{\mathcal{P}^{-}}=\mathcal{P}^{-}+[\mathcal{P}^{-},T]+\frac{1}{2}[[% \mathcal{P}^{-},T],T]+\cdots
  59. 1 - P v 1-P_{v}
  60. T T
  61. T T
  62. | q q ¯ g |q\bar{q}g\rangle
  63. T T
  64. q q ¯ q\bar{q}
  65. | q q ¯ |q\bar{q}\rangle
  66. T T
  67. e T e^{T}
  68. X observable ( μ ) X_{\rm observable}(\mu)
  69. μ \mu
  70. p p
  71. Λ \Lambda
  72. μ \mu
  73. X observable ( μ ) = X theory ( p , Λ , μ ) . X_{\rm observable}(\mu)=X_{\rm theory}(p,\Lambda,\mu)\,.
  74. Λ \Lambda
  75. X theory ( p , Λ , μ ) X_{\rm theory}(p,\Lambda,\mu)
  76. μ \mu
  77. μ \mu
  78. μ \mu
  79. p p
  80. Λ \Lambda
  81. X ( p , Λ , μ ) X(p,\Lambda,\mu)
  82. Λ \Lambda
  83. p p
  84. X ( p , Λ , μ ) X(p,\Lambda,\mu)
  85. μ \mu
  86. p ( Λ ) p(\Lambda)
  87. X observable ( μ ) = lim Λ X theory [ p ( Λ ) , Λ , μ ] . X_{\rm observable}(\mu)=\lim_{\Lambda\rightarrow\infty}X_{\rm theory}[p(% \Lambda),\Lambda,\mu]\,.
  88. n n
  89. n n
  90. μ \mu
  91. Λ \Lambda
  92. n n
  93. μ \mu
  94. n n
  95. μ \mu
  96. Λ \Lambda
  97. μ \mu
  98. n n
  99. μ \mu
  100. μ \mu
  101. n n
  102. p p
  103. Λ \Lambda
  104. n n
  105. n n
  106. Λ \Lambda
  107. μ \mu
  108. Λ \Lambda
  109. p ( Λ ) p(\Lambda)
  110. p ( Λ ) p(\Lambda)
  111. p ( Λ ) p(\Lambda)
  112. μ \mu
  113. μ \mu
  114. p ( Λ ) p(\Lambda)
  115. p p
  116. Λ \Lambda
  117. Λ / 2 \Lambda/2
  118. Λ \Lambda
  119. T T
  120. T T
  121. T T
  122. T T
  123. x - x^{-}
  124. p + p^{+}
  125. x x^{\perp}
  126. p p^{\perp}
  127. Λ \Lambda
  128. Λ / 2 \Lambda/2
  129. H H
  130. H ψ = E ψ , H\psi=E\psi,
  131. H = H 0 + H I H=H_{0}+H_{I}
  132. H 0 H_{0}
  133. H I H_{I}
  134. ψ \psi
  135. H 0 H_{0}
  136. P P
  137. Q Q
  138. P P
  139. H 0 H_{0}
  140. Λ / 2 \Lambda/2
  141. Q Q
  142. H 0 H_{0}
  143. Λ / 2 \Lambda/2
  144. Λ \Lambda
  145. H H
  146. P P
  147. Q Q
  148. H 0 Q ψ + Q H I Q ψ + Q H I P ψ = E Q ψ , H_{0}Q\psi+QH_{I}Q\psi+QH_{I}P\psi=EQ\psi\,,
  149. H 0 P ψ + P H I Q ψ + P H I P ψ = E P ψ . H_{0}P\psi+PH_{I}Q\psi+PH_{I}P\psi=EP\psi\,.
  150. Q ψ Q\psi
  151. P ψ P\psi
  152. Q ψ = 1 E - H 0 - Q H I Q Q H I P ψ . Q\psi=\frac{1}{E-H_{0}-QH_{I}Q}\ QH_{I}P\psi\,.
  153. P ψ P\psi
  154. H eff P ψ = E P ψ , H_{\rm eff}P\psi=EP\psi\,,
  155. H eff = H 0 + P H I P + P H I Q 1 E - H 0 - Q H I Q Q H I P . H_{\rm eff}=H_{0}+PH_{I}P+PH_{I}Q\frac{1}{E-H_{0}-QH_{I}Q}QH_{I}P.
  156. P ψ P\psi
  157. H eff H_{\rm eff}
  158. Λ / 2 \Lambda/2
  159. H eff H_{\rm eff}
  160. E E
  161. Λ / 2 \Lambda/2
  162. E E
  163. E E
  164. Q H 0 Q QH_{0}Q
  165. Q H I Q QH_{I}Q
  166. Q H 0 Q QH_{0}Q
  167. H 0 H_{0}
  168. H eff H_{\rm eff}
  169. E E
  170. Λ \Lambda
  171. E E
  172. Q Q
  173. E E
  174. P P
  175. H 0 H_{0}
  176. H eff H_{\rm eff}
  177. H 0 H_{0}
  178. Λ / 2 \Lambda/2
  179. Λ / 2 \Lambda/2
  180. Λ / 2 \Lambda/2
  181. H 0 H_{0}
  182. Λ \Lambda
  183. Λ / 2 \Lambda/2
  184. Λ / 2 \Lambda/2
  185. Λ / 4 \Lambda/4
  186. Q H I Q QH_{I}Q
  187. H 0 H_{0}
  188. E E
  189. E E
  190. H 0 H_{0}
  191. H I H_{I}
  192. H 0 H_{0}
  193. λ \lambda
  194. H 0 H_{0}
  195. λ \lambda
  196. E E
  197. Λ \Lambda
  198. H eff H_{\rm eff}
  199. Λ \Lambda
  200. H eff H_{\rm eff}
  201. λ \lambda
  202. λ \lambda
  203. λ \lambda
  204. ψ s = U s ψ 0 U s , \psi_{s}=U_{s}\,\psi_{0}\,U_{s}^{\dagger}\,,
  205. ψ s \psi_{s}
  206. s 1 / λ s\sim 1/\lambda
  207. ψ 0 \psi_{0}
  208. ψ 0 \psi_{0}
  209. s s
  210. ψ s \psi_{s}
  211. H s ( ψ s ) = H 0 ( ψ 0 ) , H_{s}(\psi_{s})=H_{0}(\psi_{0})\,,
  212. s s
  213. c s c_{s}
  214. ψ s \psi_{s}
  215. s s
  216. s s
  217. s s
  218. c s c_{s}
  219. s s
  220. s s
  221. s s
  222. λ 1 / s \lambda\sim 1/s
  223. s s
  224. x 1 , 2 = ( c t 1 , 2 , x 1 , 2 ) x_{1,2}=(ct_{1,2},\vec{x}_{1,2})
  225. Φ = Φ ( x 1 , x 2 ; p ) \Phi=\Phi(x_{1},x_{2};p)
  226. p p
  227. Φ ( k 1 , k 2 ; p ) = d 4 x 1 d 4 x 2 Φ ( x 1 , x 2 ; p ) exp ( i k 1 x 1 + i k 2 x 2 ) \Phi(k_{1},k_{2};p)=\int d^{4}x_{1}d^{4}x_{2}\Phi(x_{1},x_{2};p)\exp(ik_{1}x_{% 1}+ik_{2}x_{2})
  228. Φ ( k 1 , k 2 ; p ) \Phi(k_{1},k_{2};p)
  229. k 1 + k 2 = p k_{1}+k_{2}=p
  230. x 1 , x 2 x_{1},x_{2}
  231. ω x 1 = ω x 2 = 0 \omega\cdot x_{1}=\omega\cdot x_{2}=0
  232. δ ( ω x 1 , 2 ) \delta(\omega\cdot x_{1,2})
  233. ψ L F d 4 x 1 d 4 x 2 δ ( ω x 1 ) δ ( ω x 2 ) Φ ( x 1 , x 2 ; p ) exp ( i k 1 x 1 + i k 2 x 2 ) . \psi_{LF}\propto\int d^{4}x_{1}d^{4}x_{2}\delta(\omega\cdot x_{1})\delta(% \omega\cdot x_{2})\Phi(x_{1},x_{2};p)\exp(ik_{1}x_{1}+ik_{2}x_{2}).
  234. ψ L F \psi_{LF}
  235. l l
  236. ψ l m ( k , n ^ ) = f 1 ( k , k n ^ ) Y l m ( k ^ ) + f 2 ( k , k n ^ ) Y l m ( n ^ ) , \psi_{lm}(\vec{k},\hat{n})=f_{1}(k,\vec{k}\cdot\hat{n})Y_{lm}(\hat{k})+f_{2}(k% ,\vec{k}\cdot\hat{n})Y_{lm}(\hat{n}),
  237. g ( γ , z ) g(\gamma,z)
  238. Φ ( k , p ) = 1 4 π - 1 1 d z 0 d γ g ( γ , z ) [ γ + m 2 - 1 4 M 2 - k 2 - p k z - i ϵ ] 3 , \Phi(k,p)=\frac{1}{\sqrt{4\pi}}\int_{-1}^{1}dz^{\prime}\int_{0}^{\infty}d% \gamma^{\prime}\frac{g(\gamma^{\prime},z^{\prime})}{\left[\gamma^{\prime}+m^{2% }-\frac{1}{4}M^{2}-k^{2}-p\cdot k\;z^{\prime}-i\epsilon\right]^{3}},
  239. k = ( k 1 - k 2 ) / 2 k=(k_{1}-k_{2})/2
  240. g ( γ , z ) g(\gamma,z)
  241. g ( γ , z = ± 1 ) = 0 g(\gamma,z=\pm 1)=0
  242. g ( γ , z ) 0 g(\gamma\to\infty,z)\to 0
  243. ψ L F ( k , x ) = 1 4 π 0 x ( 1 - x ) g ( γ , 1 - 2 x ) d γ [ γ + k 2 + m 2 - x ( 1 - x ) M 2 ] 2 . \psi_{LF}(k_{\perp},x)=\frac{1}{\sqrt{4\pi}}\int_{0}^{\infty}\frac{x(1-x)g(% \gamma^{\prime},1-2x)d\gamma^{\prime}}{\Bigl[\gamma^{\prime}+k_{\perp}^{2}+m^{% 2}-x(1-x)M^{2}\Bigr]^{2}}.
  244. Φ ( k , p ) \Phi(k,p)
  245. ψ L F ( k , x ) \psi_{LF}(k_{\perp},x)
  246. p i + p_{i}^{+}
  247. ϕ 1 + 1 4 \phi^{4}_{1+1}
  248. k + = 0 k^{+}=0
  249. ϕ 4 \phi^{4}
  250. ϕ 4 \phi^{4}
  251. ϕ 4 \phi^{4}

Light-front_quantization_applications.html

  1. x + c t + z x^{+}\equiv ct+z
  2. x - c t - z x^{-}\equiv ct-z
  3. t t
  4. z z
  5. c c
  6. x x
  7. y y
  8. x = ( x , y ) \vec{x}_{\perp}=(x,y)
  9. t t
  10. z z
  11. p p
  12. y y
  13. P P
  14. y = p + / P + y=p^{+}/P^{+}
  15. e p e p . ep\to e^{\prime}p^{\prime}.
  16. e p e Δ + ep\to e^{\prime}\Delta^{+}
  17. γ p γ p \gamma p\to\gamma^{\prime}p^{\prime}
  18. γ p π + n \gamma p\to\pi^{+}n
  19. π + p π + p \pi^{+}p\to{\pi^{+}}^{\prime}p^{\prime}
  20. B B
  21. e H e H eH\to eH^{\prime}
  22. F H ( q 2 ) = < H ( p + q ) | j + | H ( p ) > F_{H}(q^{2})=<H^{\prime}(p+q)|j^{+}|H(p)>
  23. q μ q^{\mu}
  24. | H ( p ) > |H(p)>
  25. H H
  26. p μ p^{\mu}
  27. q + = 0 , q = Q , q - = 2 q p P + q^{+}=0,q_{\perp}=Q,q^{-}=\frac{2q\cdot p}{P^{+}}
  28. q 2 = Q 2 = - q 2 . q^{2}_{\perp}=Q^{2}=-q^{2}.
  29. Ψ H ( x i , k , λ i ) \Psi_{H}(x_{i},\vec{k}_{\perp},\lambda_{i})
  30. Ψ H ( x i , k , λ i ) \Psi_{H}(x_{i},\vec{k}^{\prime}_{\perp},\lambda_{i})
  31. x x
  32. k = k + ( 1 - x i ) q k_{\perp}^{\prime}=\vec{k}_{\perp}+(1-x_{i})\vec{q}_{\perp}
  33. k = k - x 1 q \vec{k}_{\perp}^{\prime}=\vec{k}_{\perp}-x_{1}\vec{q}_{\perp}
  34. q + = 0 q^{+}=0
  35. Q 2 Q^{2}
  36. p p
  37. p + q p+q
  38. k + k^{+}
  39. k + = 0 k^{+}=0
  40. + +
  41. F H ( Q 2 ) ( 1 Q 2 ) n - 1 F_{H}(Q^{2})\propto\left(\frac{1}{Q^{2}}\right)^{n-1}
  42. n n
  43. n = 3 n=3
  44. K + p K + p K^{+}p\to K^{+}p
  45. u u
  46. K + ( u s ¯ ) K^{+}(u\bar{s})
  47. ( u u d ) (uud)
  48. A + B C + D A+B\to C+D
  49. P X P_{X}
  50. s = ( P A + P B ) 2 = E C M 2 , t = ( P D + P B ) 2 , u = ( P A - P D ) 2 s=(P_{A}+P_{B})^{2}=E_{CM}^{2},t=(P_{D}+P_{B})^{2},u=(P_{A}-P_{D})^{2}
  51. M 1 u t 2 M\propto\frac{1}{ut^{2}}
  52. p T 2 = t u s p^{2}_{T}=\frac{tu}{s}
  53. cos θ C M = t - u 2 s \cos\theta_{CM}=\frac{t-u}{2s}
  54. 1 u \frac{1}{u}
  55. t t
  56. u α ( t ) - 1 u^{\alpha}(t)\to-1
  57. t t
  58. s - 8 s^{-8}
  59. K + p K + p K^{+}p\to K^{+}p
  60. d σ d t ( A + B C + D ) F ( θ C M s n A + n B + n C + n D - 2 \frac{d\sigma}{dt}(A+B\to C+D)\propto\frac{F(\theta_{CM}}{s^{n_{A}+n_{B}+n_{C}% +n_{D}-2}}
  61. n A n_{A}
  62. T T
  63. k + = x i P + k^{+}=x_{i}P^{+}
  64. k = x i P \vec{k}_{\perp}=x_{i}\vec{P}_{\perp}
  65. ϕ H ( x i , Q ) \phi_{H}(x_{i},Q)
  66. α s ( q 2 ) \alpha_{s}(q^{2})
  67. ϕ H ( x i , Q ) \phi_{H}(x_{i},Q)
  68. x i = k i + P + x_{i}=\frac{k^{+}_{i}}{P^{+}}
  69. A + = 0 A^{+}=0
  70. Π i Q d 2 k i ψ H ( x i , k i ) \Pi_{i}\int^{Q}d^{2}\vec{k}_{\perp i}\psi_{H}(x_{i},\vec{k}_{\perp i})
  71. k 2 < Q 2 k^{2}_{\perp}<Q^{2}
  72. Q 2 Q^{2}
  73. log Q 2 \log Q^{2}
  74. ϕ π 3 f π x ( 1 - x ) \phi_{\pi}\to\sqrt{3}f_{\pi}x(1-x)
  75. f π f_{\pi}
  76. π + W * μ + ν μ \pi^{+}\to W^{*}\to\mu^{+}\nu_{\mu}
  77. d n p d\to np
  78. 5 × 5 5\times 5
  79. Q 2 . Q^{2}\to\infty.
  80. E lab / Q 2 E_{\rm lab}/Q^{2}
  81. π A J e t J e t A \pi A\to JetJetA^{\prime}
  82. x x
  83. 3 + 1 3+1
  84. ζ 2 = b 2 x ( 1 - x ) \zeta^{2}_{\perp}=b^{2}_{\perp}x(1-x)
  85. τ \tau
  86. M q q ¯ 2 {M^{2}_{q\bar{q}}}
  87. H L F H_{LF}
  88. ζ 2 = b 2 x ( 1 - x ) \zeta^{2}=b^{2}_{\perp}x(1-x)
  89. M q q ¯ 2 {M^{2}_{q\bar{q}}}
  90. U ( ζ 2 ) U(\zeta^{2})
  91. Λ QCD \Lambda_{\rm QCD}
  92. H H
  93. | p = n = 1 ψ n ( k 1 , , k n , p ) | n D k . |p\rangle=\sum_{n=1}^{\infty}\int\psi_{n}(k_{1},\ldots,k_{n},p)|n\rangle D_{k}.
  94. ψ n \psi_{n}
  95. n n
  96. D k D_{k}
  97. H H
  98. | p |p\rangle
  99. H | p = M | p , H|p\rangle=M|p\rangle,
  100. ψ n \psi_{n}