wpmath0000012_5

Granulometry_(morphology).html

  1. { B k } \{B_{k}\}
  2. k = 0 , 1 , k=0,1,\ldots
  3. B k = B B k times B_{k}=\underbrace{B\oplus\ldots\oplus B}_{k\mbox{ times}~{}}
  4. \oplus
  5. B 0 B_{0}
  6. B 1 = B B_{1}=B
  7. { γ k ( X ) } \{\gamma_{k}(X)\}
  8. k = 0 , 1 , k=0,1,\ldots
  9. γ k ( X ) = X B k \gamma_{k}(X)=X\circ B_{k}
  10. \circ
  11. G k ( X ) G_{k}(X)
  12. γ k ( X ) \gamma_{k}(X)
  13. G k ( X ) = | γ k ( X ) | G_{k}(X)=|\gamma_{k}(X)|
  14. { P S k ( X ) } \{PS_{k}(X)\}
  15. k = 0 , 1 , k=0,1,\ldots
  16. P S k ( X ) = G k ( X ) - G k + 1 ( X ) PS_{k}(X)=G_{k}(X)-G_{k+1}(X)
  17. P S k ( X ) PS_{k}(X)
  18. P S k ( X ) PS_{k}(X)
  19. Ψ k ( X ) \Psi_{k}(X)
  20. Ψ k ( X ) X \Psi_{k}(X)\subseteq X
  21. X Y Ψ k ( X ) Ψ k ( Y ) X\subseteq Y\Rightarrow\Psi_{k}(X)\subseteq\Psi_{k}(Y)
  22. Ψ k Ψ m ( X ) = Ψ m Ψ k ( X ) = Ψ max ( k , m ) ( X ) \Psi_{k}\Psi_{m}(X)=\Psi_{m}\Psi_{k}(X)=\Psi_{\max(k,m)}(X)
  23. Ψ k ( X ) = γ k ( X ) = X B k \Psi_{k}(X)=\gamma_{k}(X)=X\circ B_{k}
  24. Ψ k ( X ) = i = 1 N X ( B ( i ) ) k \Psi_{k}(X)=\bigcup_{i=1}^{N}X\circ(B^{(i)})_{k}
  25. { B ( i ) } \{B^{(i)}\}

Graph_continuous_function.html

  1. N N
  2. i i
  3. A i A_{i}\subseteq\mathbb{R}
  4. 𝐚 \mathbf{a}
  5. 𝐚 j = 1 N A j \mathbf{a}\in\prod_{j=1}^{N}A_{j}
  6. 𝐚 - i = ( a 1 , a 2 , , a i - 1 , a i + 1 , , a N ) \mathbf{a}_{-i}=(a_{1},a_{2},\ldots,a_{i-1},a_{i+1},\ldots,a_{N})
  7. i i
  8. U i : A i U_{i}:A_{i}\longrightarrow\mathbb{R}
  9. i i
  10. [ ( A i , U i ) ; i = 1 , , N ] [(A_{i},U_{i});i=1,\ldots,N]
  11. U i : A U_{i}:A\longrightarrow\mathbb{R}
  12. 𝐚 A \mathbf{a}\in A
  13. F i : A - i A i F_{i}:A_{-i}\longrightarrow A_{i}
  14. U i ( F i ( 𝐚 - i ) , 𝐚 - i ) U_{i}(F_{i}(\mathbf{a}_{-i}),\mathbf{a}_{-i})
  15. 𝐚 - i \mathbf{a}_{-i}
  16. 1 i N 1\leq i\leq N
  17. A i m A_{i}\subseteq\mathbb{R}^{m}
  18. U i : A U_{i}:A\longrightarrow\mathbb{R}
  19. a i a_{i}
  20. 𝐚 \mathbf{a}
  21. [ ( A i , U i ) ; i = 1 , , N ] [(A_{i},U_{i});i=1,\ldots,N]

Graph_dynamical_system.html

  1. k \mathbb{Z}^{k}
  2. \mathbb{R}
  3. F ( x ) v = f v ( x [ v ] ) . F(x)_{v}=f_{v}(x[v])\;.
  4. π \pi
  5. π 1 \pi_{1}
  6. π 2 , , π n \pi_{2},\dots,\pi_{n}
  7. F i ( x ) = ( x 1 , x 2 , , x i - 1 , f i ( x [ i ] ) , x i + 1 , , x n ) . F_{i}(x)=(x_{1},x_{2},\ldots,x_{i-1},f_{i}(x[i]),x_{i+1},\ldots,x_{n})\;.
  8. [ F Y , w ] = F w ( m ) F w ( m - 1 ) F w ( 2 ) F w ( 1 ) . [F_{Y},w]=F_{w(m)}\circ F_{w(m-1)}\circ\cdots\circ F_{w(2)}\circ F_{w(1)}\;.

Graphical_game_theory.html

  1. n n
  2. m m
  3. G G
  4. i i
  5. j j
  6. i i
  7. G G
  8. u i : { 1 m } d i + 1 u_{i}:\{1\ldots m\}^{d_{i}+1}\rightarrow\mathbb{R}
  9. d i d_{i}
  10. i i
  11. u i u_{i}
  12. i i
  13. n n
  14. m m
  15. O ( m n ) O(m^{n})
  16. O ( m d ) O(m^{d})
  17. d d
  18. d n d\ll n
  19. n n
  20. m 2 m^{2}
  21. n m 2 nm^{2}

Gravitational_lensing_formalism.html

  1. b b~{}
  2. α ^ = 4 G M c 2 b \hat{\alpha}=\frac{4GM}{c^{2}b}
  3. ρ \rho~{}
  4. α ^ ( ξ ) = 4 G c 2 d 2 ξ d z ρ ( ξ , z ) b | b | 2 , b ξ - ξ \vec{\hat{\alpha}}(\vec{\xi})=\frac{4G}{c^{2}}\int d^{2}\xi^{\prime}\int dz% \rho(\vec{\xi}^{\prime},z)\frac{\vec{b}}{|\vec{b}|^{2}},~{}\vec{b}\equiv\vec{% \xi}-\vec{\xi^{\prime}}
  5. z z
  6. b \vec{b}
  7. d 2 ξ d z ρ ( ξ , z ) d^{2}\xi^{\prime}dz\rho(\vec{\xi}^{\prime},z)
  8. ( ξ , z ) (\vec{\xi}^{\prime},z)
  9. Σ ( ξ ) = ρ ( ξ , z ) d z \Sigma(\vec{\xi})=\int\rho(\vec{\xi},z)dz
  10. ξ \vec{\xi}
  11. α ^ = 4 G c 2 ( ξ - ξ ) Σ ( ξ ) | ξ - ξ | 2 d 2 ξ 2 \vec{\hat{\alpha}}=\frac{4G}{c^{2}}\int\frac{(\vec{\xi}-\vec{\xi}^{\prime})% \Sigma(\vec{\xi}^{\prime})}{|\vec{\xi}-\vec{\xi}^{\prime}|^{2}}d^{2}\xi^{% \prime 2}
  12. β \vec{\beta}
  13. θ \vec{\theta}
  14. β = θ - α ( θ ) = θ - D d s D s α ^ ( D d θ ) \vec{\beta}=\vec{\theta}-\vec{\alpha}(\vec{\theta})=\vec{\theta}-\frac{D_{ds}}% {D_{s}}\vec{\hat{\alpha}}(\vec{D_{d}\theta})
  15. D d s D_{ds}~{}
  16. D s D_{s}~{}
  17. D d D_{d}~{}
  18. β \vec{\beta}
  19. κ ( θ ) = Σ ( D d θ ) Σ c r \kappa(\vec{\theta})=\frac{\Sigma(D_{d}\vec{\theta})}{\Sigma_{cr}}
  20. Σ c r = c 2 D s 4 π G D d s D d \Sigma_{cr}=\frac{c^{2}D_{s}}{4\pi GD_{ds}D_{d}}
  21. α ( θ ) \vec{\alpha}(\vec{\theta})
  22. α ( θ ) = 1 π d 2 θ ( θ - θ ) κ ( θ ) | θ - θ | 2 \vec{\alpha}(\vec{\theta})=\frac{1}{\pi}\int d^{2}\theta^{\prime}\frac{(\vec{% \theta}-\vec{\theta}^{\prime})\kappa(\vec{\theta}^{\prime})}{|\vec{\theta}-% \vec{\theta}^{\prime}|^{2}}
  23. ψ ( θ ) = 1 π d 2 θ κ ( θ ) ln | θ - θ | \psi(\vec{\theta})=\frac{1}{\pi}\int d^{2}\theta^{\prime}\kappa(\vec{\theta}^{% \prime})\ln|\vec{\theta}-\vec{\theta}^{\prime}|
  24. α ( θ ) = ψ ( θ ) \vec{\alpha}(\vec{\theta})=\vec{\nabla}\psi(\vec{\theta})
  25. κ ( θ ) = 1 2 2 ψ ( θ ) \kappa(\vec{\theta})=\frac{1}{2}\nabla^{2}\psi(\vec{\theta})
  26. Φ \Phi~{}
  27. ψ ( θ ) = 2 D d s D d D s c 2 Φ ( D d θ , z ) d z \psi(\vec{\theta})=\frac{2D_{ds}}{D_{d}D_{s}c^{2}}\int\Phi(D_{d}\vec{\theta},z% )dz
  28. A i j = β i θ j = δ i j - α i θ j = δ i j - 2 ψ θ i θ j A_{ij}=\frac{\partial\beta_{i}}{\partial\theta_{j}}=\delta_{ij}-\frac{\partial% \alpha_{i}}{\partial\theta_{j}}=\delta_{ij}-\frac{\partial^{2}\psi}{\partial% \theta_{i}\partial\theta_{j}}
  29. δ i j \delta_{ij}~{}
  30. γ \gamma~{}
  31. A = ( 1 - κ ) [ 1 0 0 1 ] - γ [ cos 2 ϕ sin 2 ϕ sin 2 ϕ - cos 2 ϕ ] A=(1-\kappa)\left[\begin{array}[]{ c c }1&0\\ 0&1\end{array}\right]-\gamma\left[\begin{array}[]{ c c }\cos 2\phi&\sin 2\phi% \\ \sin 2\phi&-\cos 2\phi\end{array}\right]
  32. ϕ \phi~{}
  33. α \vec{\alpha}
  34. t = 0 z s n d z c cos α ( z ) t=\int_{0}^{z_{s}}{ndz\over c\cos\alpha(z)}
  35. d z / c dz/c
  36. 1 / cos ( α ( z ) ) 1 + α ( z ) 2 2 1/\cos(\alpha(z))\approx 1+{\alpha(z)^{2}\over 2}
  37. d l = d z c cos α ( z ) dl={dz\over c\cos\alpha(z)}
  38. α ( z ) , \alpha(z),
  39. n n
  40. d s 2 = 0 = c 2 d t 2 ( 1 + 2 Φ c 2 ) - ( 1 + 2 Φ c 2 ) - 1 d l 2 ds^{2}=0=c^{2}dt^{2}\left(1+{2\Phi\over c^{2}}\right)-\left(1+{2\Phi\over c^{2% }}\right)^{-1}dl^{2}
  41. Φ c 2 \Phi\ll c^{2}
  42. c = d l / d t = ( 1 + 2 Φ c 2 ) c . c^{\prime}={dl/dt}=\left(1+{2\Phi\over c^{2}}\right)c.
  43. n c c ( 1 - 2 Φ c 2 ) . n\equiv{c\over c^{\prime}}\approx\left(1-{2\Phi\over c^{2}}\right).
  44. Φ \Phi
  45. t 0 z s d z c + 0 z s d z c α ( z ) 2 2 - 0 z s d z c 2 Φ c 2 . t\approx\int_{0}^{z_{s}}{dz\over c}+\int_{0}^{z_{s}}{dz\over c}{\alpha(z)^{2}% \over 2}-\int_{0}^{z_{s}}{dz\over c}{2\Phi\over c^{2}}.
  46. α ( z ) = θ - β \alpha(z)=\theta-\beta
  47. α ( z ) ( θ - β ) D d D d s \alpha(z)\approx(\theta-\beta){D_{d}\over D_{ds}}
  48. D d c ( θ - β ) 2 2 + D d s c [ ( θ - β ) D d D d s ] 2 2 = D d D s D d s ( θ - β ) 2 2 . {D_{d}\over c}{(\vec{\theta}-\vec{\beta})^{2}\over 2}+{D_{ds}\over c}{\left[(% \vec{\theta}-\vec{\beta}){D_{d}\over D_{ds}}\right]^{2}\over 2}={D_{d}D_{s}% \over D_{ds}}{(\vec{\theta}-\vec{\beta})^{2}\over 2}.
  49. t = c o n s t a n t + D d D s D d s c τ , τ [ ( θ - β ) 2 2 - ψ ] t=constant+{D_{d}D_{s}\over D_{ds}c}\tau,~{}\tau\equiv\left[{(\vec{\theta}-% \vec{\beta})^{2}\over 2}-\psi\right]
  50. τ \tau
  51. ψ ( θ ) = 2 D d s D d D s c 2 Φ ( D d θ , z ) d z . \psi(\vec{\theta})=\frac{2D_{ds}}{D_{d}D_{s}c^{2}}\int\Phi(D_{d}\vec{\theta},z% )dz.
  52. θ \vec{\theta}
  53. 0 = θ τ = θ - β - θ ψ ( θ ) 0=\nabla_{\vec{\theta}}\tau=\vec{\theta}-\vec{\beta}-\nabla_{\vec{\theta}}\psi% (\vec{\theta})
  54. Φ ( ξ ) = - d 3 ξ ρ ( ξ ) | ξ - ξ | \Phi(\vec{\xi})=-\int\frac{d^{3}\xi^{\prime}\rho(\vec{\xi}^{\prime})}{|\vec{% \xi}-\vec{\xi}^{\prime}|}
  55. ψ ( θ ) = - 2 G D d s D d D s c 2 d z d 3 ξ ρ ( ξ ) | ξ - ξ | = - i 2 G M i D i s D s D i c 2 [ sinh - 1 | z - D i | D i | θ - θ i | ] | D i D s + | D i 0 . \psi(\vec{\theta})=-\frac{2GD_{ds}}{D_{d}D_{s}c^{2}}\int dz\int\frac{d^{3}\xi^% {\prime}\rho(\vec{\xi}^{\prime})}{|\vec{\xi}-\vec{\xi}^{\prime}|}=-\sum_{i}% \frac{2GM_{i}D_{is}}{D_{s}D_{i}c^{2}}\left[\sinh^{-1}{|z-D_{i}|\over D_{i}|% \vec{\theta}-\vec{\theta}_{i}|}\right]|_{D_{i}}^{D_{s}}+|_{D_{i}}^{0}.
  56. M i M_{i}
  57. θ i \vec{\theta}_{i}
  58. z = D i . z=D_{i}.
  59. sinh - 1 1 / x = ln ( 1 / x + 1 / x 2 + 1 ) - ln ( x / 2 ) \sinh^{-1}1/x=\ln(1/x+\sqrt{1/x^{2}+1})\approx-\ln(x/2)
  60. x x
  61. ψ ( θ ) i 2 G M i D i s D s D i c 2 [ ln ( | θ - θ i | 2 4 D i D i s ) ] . \psi(\vec{\theta})\approx\sum_{i}\frac{2GM_{i}D_{is}}{D_{s}D_{i}c^{2}}\left[% \ln\left({|\vec{\theta}-\vec{\theta}_{i}|^{2}\over 4}{D_{i}\over D_{is}}\right% )\right].
  62. κ ( θ ) = 1 2 θ 2 ψ ( θ ) = 4 π G D d s D d c 2 D s d z ρ ( D d θ , z ) = = Σ Σ c r = = i 4 π G M i D i s c 2 D i D s δ ( θ - θ i ) \kappa(\vec{\theta})=\frac{1}{2}\nabla_{\vec{\theta}}^{2}\psi(\vec{\theta})=% \frac{4\pi GD_{ds}D_{d}}{c^{2}D_{s}}\int dz\rho(D_{d}\vec{\theta},z)=={\Sigma% \over\Sigma_{cr}}==\sum_{i}{4\pi GM_{i}D_{is}\over c^{2}D_{i}D_{s}}\delta(\vec% {\theta}-\vec{\theta}_{i})
  63. κ ( θ ) = Σ Σ c r \kappa(\vec{\theta})={\Sigma\over\Sigma_{cr}}
  64. 2 1 / r = - 4 π δ ( r ) \nabla^{2}1/r=-4\pi\delta(r)
  65. θ = D d . \nabla_{\vec{\theta}}=D_{d}\nabla.
  66. θ - β = θ ψ ( θ ) = i θ E i 2 | θ - θ i | , π θ E i 2 4 π G M i D i s c 2 D s D i \vec{\theta}-\vec{\beta}=\nabla_{\vec{\theta}}\psi(\vec{\theta})=\sum_{i}{% \theta_{Ei}^{2}\over|\vec{\theta}-\vec{\theta}_{i}|},~{}\pi\theta_{Ei}^{2}% \equiv{4\pi GM_{i}D_{is}\over c^{2}D_{s}D_{i}}
  67. θ E i \theta_{Ei}
  68. θ - β = θ E 2 | θ | . \vec{\theta}-\vec{\beta}={\theta_{E}^{2}\over|\vec{\theta}|}.
  69. A i j = β j θ i = τ θ i θ j = δ i j - ψ θ i θ j = [ 1 - κ - γ 1 γ 2 γ 2 1 - κ + γ 1 ] A_{ij}={\partial\beta_{j}\over\partial\theta_{i}}={\partial\tau\over\partial% \theta_{i}\partial\theta_{j}}=\delta_{ij}-{\partial\psi\over\partial\theta_{i}% \partial\theta_{j}}=\left[\begin{array}[]{ c c }1-\kappa-\gamma_{1}&\gamma_{2}% \\ \gamma_{2}&1-\kappa+\gamma_{1}\end{array}\right]
  70. κ = ψ 2 θ 1 θ 1 + ψ 2 θ 2 θ 2 , γ 1 ψ 2 θ 1 θ 1 - ψ 2 θ 2 θ 2 , γ 2 ψ θ 1 θ 2 \kappa={\partial\psi\over 2\partial\theta_{1}\partial\theta_{1}}+{\partial\psi% \over 2\partial\theta_{2}\partial\theta_{2}},~{}\gamma_{1}\equiv{\partial\psi% \over 2\partial\theta_{1}\partial\theta_{1}}-{\partial\psi\over 2\partial% \theta_{2}\partial\theta_{2}},~{}\gamma_{2}\equiv{\partial\psi\over\partial% \theta_{1}\partial\theta_{2}}
  71. A = 1 / d e t ( A i j ) = 1 ( 1 - κ ) 2 - γ 1 2 - γ 2 2 A=1/det(A_{ij})={1\over(1-\kappa)^{2}-\gamma_{1}^{2}-\gamma_{2}^{2}}
  72. κ = 0 , γ = γ 1 2 + γ 2 2 = θ E 2 | θ | 2 , θ E 2 = 4 G M D d s c 2 D d D s . \kappa=0,~{}\gamma=\sqrt{\gamma_{1}^{2}+\gamma_{2}^{2}}={\theta_{E}^{2}\over|% \theta|^{2}},~{}\theta_{E}^{2}={4GMD_{ds}\over c^{2}D_{d}D_{s}}.
  73. A = ( 1 - θ E 4 θ 4 ) - 1 . A=\left(1-{\theta_{E}^{4}\over\theta^{4}}\right)^{-1}.
  74. θ E . \theta_{E}.
  75. Σ cr κ smooth , \Sigma_{\rm cr}\kappa_{\rm smooth},
  76. ψ ( θ ) 1 2 κ smooth | θ | 2 + i θ E 2 [ ln ( | θ - θ i | 2 4 D d D d s ) ] . \psi(\vec{\theta})\approx{1\over 2}\kappa_{\rm smooth}|\theta|^{2}+\sum_{i}% \theta_{E}^{2}\left[\ln\left({|\vec{\theta}-\vec{\theta}_{i}|^{2}\over 4}{D_{d% }\over D_{ds}}\right)\right].
  77. ( θ x i , θ y i ) (\theta_{xi},\theta_{yi})
  78. A = [ ( 1 - κ smooth ) 2 - ( i ( θ x i 2 - θ y i 2 ) θ E 2 ( θ x i 2 + θ y i 2 ) 2 ) 2 - ( i ( 2 θ x i θ y i ) θ E 2 ( θ x i 2 + θ y i 2 ) 2 ) 2 ] - 1 A=\left[(1-\kappa_{\rm smooth})^{2}-\left(\sum_{i}{(\theta_{xi}^{2}-\theta_{yi% }^{2})\theta_{E}^{2}\over(\theta_{xi}^{2}+\theta_{yi}^{2})^{2}}\right)^{2}-% \left(\sum_{i}{(2\theta_{xi}\theta_{yi})\theta_{E}^{2}\over(\theta_{xi}^{2}+% \theta_{yi}^{2})^{2}}\right)^{2}\right]^{-1}
  79. β \vec{\beta}
  80. θ \vec{\theta}
  81. Φ \Phi~{}
  82. β i θ j = δ i j + 0 r d r g ( r ) 2 Φ ( x ( r ) ) x i x j \frac{\partial\beta_{i}}{\partial\theta_{j}}=\delta_{ij}+\int_{0}^{r_{\infty}}% drg(r)\frac{\partial^{2}\Phi(\vec{x}(r))}{\partial x^{i}\partial x^{j}}
  83. r r~{}
  84. x i x^{i}~{}
  85. g ( r ) = 2 r r r ( 1 - r r ) W ( r ) g(r)=2r\int^{r_{\infty}}_{r}\left(1-\frac{r^{\prime}}{r}\right)W(r^{\prime})
  86. W ( r ) W(r)~{}
  87. A i j A_{ij}~{}
  88. 1 - q 1-q~{}
  89. q = b a q=\frac{b}{a}
  90. ϕ \phi~{}
  91. χ = 1 - q 2 1 + q 2 e 2 i ϕ = a 2 - b 2 a 2 + b 2 e 2 i ϕ \chi=\frac{1-q^{2}}{1+q^{2}}e^{2i\phi}=\frac{a^{2}-b^{2}}{a^{2}+b^{2}}e^{2i\phi}
  92. ϵ = 1 - q 1 + q e 2 i ϕ = a - b a + b e 2 i ϕ \epsilon=\frac{1-q}{1+q}e^{2i\phi}=\frac{a-b}{a+b}e^{2i\phi}
  93. χ = { | χ | cos 2 ϕ , | χ | sin 2 ϕ } \chi=\{\left|\chi\right|\cos 2\phi,\left|\chi\right|\sin 2\phi\}
  94. ϵ = { | ϵ | cos 2 ϕ , | ϵ | sin 2 ϕ } \epsilon=\{\left|\epsilon\right|\cos 2\phi,\left|\epsilon\right|\sin 2\phi\}
  95. ( x ¯ , y ¯ ) (\bar{x},\bar{y})
  96. q x x = ( x - x ¯ ) 2 I ( x , y ) I ( x , y ) q_{xx}=\frac{\sum(x-\bar{x})^{2}I(x,y)}{\sum I(x,y)}
  97. q y y = ( y - y ¯ ) 2 I ( x , y ) I ( x , y ) q_{yy}=\frac{\sum(y-\bar{y})^{2}I(x,y)}{\sum I(x,y)}
  98. q x y = ( x - x ¯ ) ( y - y ¯ ) I ( x , y ) I ( x , y ) q_{xy}=\frac{\sum(x-\bar{x})(y-\bar{y})I(x,y)}{\sum I(x,y)}
  99. χ = q x x - q y y + 2 i q x y q x x + q y y \chi=\frac{q_{xx}-q_{yy}+2iq_{xy}}{q_{xx}+q_{yy}}
  100. ϵ = q x x - q y y + 2 i q x y q x x + q y y + 2 q x x q y y - q x y 2 \epsilon=\frac{q_{xx}-q_{yy}+2iq_{xy}}{q_{xx}+q_{yy}+2\sqrt{q_{xx}q_{yy}-q_{xy% }^{2}}}
  101. q x x = a 2 cos 2 θ + b 2 sin 2 θ q_{xx}=a^{2}\cos^{2}\theta+b^{2}\sin^{2}\theta\,
  102. q y y = a 2 sin 2 θ + b 2 cos 2 θ q_{yy}=a^{2}\sin^{2}\theta+b^{2}\cos^{2}\theta\,
  103. q x y = ( a 2 - b 2 ) sin θ cos θ q_{xy}=(a^{2}-b^{2})\sin\theta\cos\theta\,
  104. a 2 = q x x + q y y + ( q x x - q y y ) 2 + 4 q x y 2 2 a^{2}=\frac{q_{xx}+q_{yy}+\sqrt{(q_{xx}-q_{yy})^{2}+4q_{xy}^{2}}}{2}
  105. b 2 = q x x + q y y - ( q x x - q y y ) 2 + 4 q x y 2 2 b^{2}=\frac{q_{xx}+q_{yy}-\sqrt{(q_{xx}-q_{yy})^{2}+4q_{xy}^{2}}}{2}
  106. tan 2 θ = 2 q x y q x x - q y y \tan 2\theta=\frac{2q_{xy}}{q_{xx}-q_{yy}}
  107. q x x = ( x - x ¯ ) 2 w ( x - x ¯ , y - y ¯ ) I ( x , y ) w ( x - x ¯ , y - y ¯ ) I ( x , y ) q_{xx}=\frac{\sum(x-\bar{x})^{2}w(x-\bar{x},y-\bar{y})I(x,y)}{\sum w(x-\bar{x}% ,y-\bar{y})I(x,y)}
  108. q y y = ( y - y ¯ ) 2 w ( x - x ¯ , y - y ¯ ) I ( x , y ) w ( x - x ¯ , y - y ¯ ) I ( x , y ) q_{yy}=\frac{\sum(y-\bar{y})^{2}w(x-\bar{x},y-\bar{y})I(x,y)}{\sum w(x-\bar{x}% ,y-\bar{y})I(x,y)}
  109. q x y = ( x - x ¯ ) ( y - y ¯ ) w ( x - x ¯ , y - y ¯ ) I ( x , y ) w ( x - x ¯ , y - y ¯ ) I ( x , y ) q_{xy}=\frac{\sum(x-\bar{x})(y-\bar{y})w(x-\bar{x},y-\bar{y})I(x,y)}{\sum w(x-% \bar{x},y-\bar{y})I(x,y)}
  110. w ( x , y ) w(x,y)~{}
  111. γ \gamma~{}
  112. κ \kappa~{}
  113. R R~{}
  114. a = R 1 - κ - γ a=\frac{R}{1-\kappa-\gamma}
  115. b = R 1 - κ + γ b=\frac{R}{1-\kappa+\gamma}
  116. γ = | γ | e 2 i ϕ \gamma=\left|\gamma\right|e^{2i\phi}
  117. g γ 1 - κ g\equiv\frac{\gamma}{1-\kappa}
  118. A = [ 1 - κ - Re [ γ ] - Im [ γ ] - Im [ γ ] 1 - κ + Re [ γ ] ] = ( 1 - κ ) [ 1 - Re [ g ] - Im [ g ] - Im [ g ] 1 + Re [ g ] ] A=\left[\begin{array}[]{ c c }1-\kappa-\mathrm{Re}[\gamma]&-\mathrm{Im}[\gamma% ]\\ -\mathrm{Im}[\gamma]&1-\kappa+\mathrm{Re}[\gamma]\end{array}\right]=(1-\kappa)% \left[\begin{array}[]{ c c }1-\mathrm{Re}[g]&-\mathrm{Im}[g]\\ -\mathrm{Im}[g]&1+\mathrm{Re}[g]\end{array}\right]
  119. g g~{}
  120. χ s \chi_{s}~{}
  121. ϵ s \epsilon_{s}~{}
  122. χ = χ s + 2 g + g 2 χ s * 1 + | g | 2 + 2 R e ( g χ s * ) \chi=\frac{\chi_{s}+2g+g^{2}\chi_{s}^{*}}{1+|g|^{2}+2\mathrm{Re}(g\chi_{s}^{*})}
  123. ϵ = ϵ s + g 1 + g * ϵ \epsilon=\frac{\epsilon_{s}+g}{1+g^{*}\epsilon}
  124. γ 1 \gamma\ll 1
  125. κ 1 \kappa\ll 1
  126. χ χ s + 2 g χ s + 2 γ \chi\approx\chi_{s}+2g\approx\chi_{s}+2\gamma
  127. ϵ ϵ s + g ϵ s + γ \epsilon\approx\epsilon_{s}+g\approx\epsilon_{s}+\gamma
  128. χ = 2 γ \langle\chi\rangle=2\langle\gamma\rangle
  129. ϵ = γ \langle\epsilon\rangle=\langle\gamma\rangle
  130. μ = θ β d θ d β \mu=\frac{\theta}{\beta}\frac{d\theta}{d\beta}
  131. μ = 1 det A = 1 [ ( 1 - κ ) 2 - γ 2 ] \mu=\frac{1}{\det A}=\frac{1}{[(1-\kappa)^{2}-\gamma^{2}]}
  132. A A~{}
  133. A A~{}
  134. λ \lambda~{}
  135. 1 - κ = λ ( 1 - κ ) 1-\kappa^{\prime}=\lambda(1-\kappa)
  136. γ = λ γ \gamma^{\prime}=\lambda\gamma
  137. κ \kappa
  138. κ λ κ + ( 1 - λ ) \kappa\rightarrow\lambda\kappa+(1-\lambda)
  139. μ \mu~{}
  140. λ \lambda~{}
  141. μ λ - 2 \mu\propto\lambda^{-2}

Gravity_tractor.html

  1. m 2 = F r 2 G m 1 = 0.032 [ N ] × ( 200 [ m ] ) 2 6.674 × 10 - 11 [ N m 2 k g - 2 ] × 10 9 [ k g ] 19200 k g m_{2}=\frac{Fr^{2}}{Gm_{1}}=\frac{0.032[N]\times(200[m])^{2}}{6.674\times 10^{% -11}[Nm^{2}kg^{-2}]\times 10^{9}[kg]}\approx 19200kg

Greenhouse_gas.html

  1. τ \tau
  2. τ \tau
  3. m m
  4. F o u t F_{out}
  5. L L
  6. D D
  7. τ = m F o u t + L + D \tau=\frac{m}{F_{out}+L+D}
  8. τ \tau

Greenwood–Hercowitz–Huffman_preferences.html

  1. u ( c , l ) = c 1 - γ 1 - γ - ψ l 1 + θ 1 + θ u(c,l)=\frac{c^{1-\gamma}}{1-\gamma}-\psi\frac{l^{1+\theta}}{1+\theta}
  2. c c
  3. l l
  4. u ( c , l ) = 1 1 - γ ( c - ψ l 1 + θ 1 + θ ) 1 - γ u(c,l)=\frac{1}{1-\gamma}\left(c-\psi\frac{l^{1+\theta}}{1+\theta}\right)^{1-\gamma}
  5. u ( c , l ) = U ( c - G ( l ) ) , U > 0 , U ′′ < 0 , G > 0 , G ′′ > 0. u(c,l)=U\left(c-G(l)\right),U^{\prime}>0,U^{\prime\prime}<0,G^{\prime}>0,G^{% \prime\prime}>0.
  6. u ( c , l ) u(c,l)
  7. l l
  8. U ( c - G ( l ) ) ( d c d l - G ( l ) ) = 0 U^{\prime}\left(c-G(l)\right)\left(\frac{dc}{dl}-G^{\prime}(l)\right)=0
  9. d c d l = G ( l ) . \frac{dc}{dl}=G^{\prime}(l).
  10. d c / d l dc/dl
  11. w w
  12. l l
  13. l = G - 1 ( w ) l=G^{\prime-1}(w)

Gregory_number.html

  1. G x = i = 0 ( - 1 ) i 1 ( 2 i + 1 ) x 2 i + 1 G_{x}=\sum_{i=0}^{\infty}(-1)^{i}\frac{1}{(2i+1)x^{2i+1}}
  2. G x = arctan 1 x . G_{x}=\arctan\frac{1}{x}.

Grigorchuk_group.html

  1. \infty
  2. exp ( n ) \exp(\sqrt{n})
  3. s = log 32 31 0.991 s=\log_{32}31\approx 0.991
  4. s = log 2 / ( log 2 - log η ) 0.7675 s=\log 2/(\log 2-\log\eta)\approx 0.7675
  5. η 3 + η 2 + η = 2 \eta^{3}+\eta^{2}+\eta=2
  6. exp ( n 0.504 ) \exp(n^{0.504})
  7. lim n log n log b ( n ) \lim_{n\to\infty}\log_{n}\log b(n)
  8. \neq
  9. \sqcup

Group_contribution_method.html

  1. T b = 198.2 + G i T_{b}\,=\,198.2+\sum{G_{i}}
  2. T c = T b [ 0.584 + 0.965 G i - G i 2 ] - 1 T_{c}\,=\,T_{b}\left[0.584+0.965\sum{G_{i}}-{G_{i}}^{2}\right]^{-1}
  3. P = f ( G i j ) P\,=\,f(G_{ij})

Growth_elasticity_of_poverty.html

  1. GEP = - % d PR % d y \mathrm{GEP}=-\frac{\%d\mathrm{PR}}{\%dy}\,

Grunwald–Wang_theorem.html

  1. K 𝔭 K_{\mathfrak{p}}
  2. 𝔭 \mathfrak{p}
  3. X 8 - 16 X^{8}-16
  4. X 8 - 16 = ( X 4 - 4 ) ( X 4 + 4 ) = ( X 2 - 2 ) ( X 2 + 2 ) ( X 2 - 2 X + 2 ) ( X 2 + 2 X + 2 ) . X^{8}-16=(X^{4}-4)(X^{4}+4)=(X^{2}-2)(X^{2}+2)(X^{2}-2X+2)(X^{2}+2X+2).
  5. p \mathbb{Q}_{p}
  6. ( 7 ) \mathbb{Q}(\sqrt{7})
  7. p ( 7 ) \mathbb{Q}_{p}(\sqrt{7})
  8. 2 ( 7 ) = 2 ( - 1 ) \mathbb{Q}_{2}(\sqrt{7})=\mathbb{Q}_{2}(\sqrt{-1})
  9. K / K/\mathbb{Q}
  10. K 2 / 2 K_{2}/\mathbb{Q}_{2}
  11. s 2 s\geq 2
  12. η s := exp ( 2 π i 2 s ) + exp ( - 2 π i 2 s ) = 2 cos ( 2 π 2 s ) . \eta_{s}:=\exp\left(\frac{2\pi i}{2^{s}}\right)+\exp\left(-\frac{2\pi i}{2^{s}% }\right)=2\cos\left(\frac{2\pi}{2^{s}}\right).
  13. 2 s 2^{s}
  14. 2 s = ( i , η s ) . \mathbb{Q}_{2^{s}}=\mathbb{Q}(i,\eta_{s}).
  15. η s \eta_{s}
  16. i i
  17. η s + 1 \eta_{s+1}
  18. i η s + 1 i\eta_{s+1}
  19. K ( n , S ) := { x K x K 𝔭 n for all 𝔭 S } . K(n,S):=\{x\in K\mid x\in K_{\mathfrak{p}}^{n}\mathrm{\ for\ all\ }\mathfrak{p% }\not\in S\}.
  20. K ( n , S ) = K n K(n,S)=K^{n}
  21. K K
  22. s s
  23. 2 s + 1 2^{s+1}
  24. S S
  25. S 0 S_{0}
  26. 𝔭 \mathfrak{p}
  27. K 𝔭 K_{\mathfrak{p}}
  28. K × / K × n 𝔭 S K 𝔭 × / K 𝔭 × n K^{\times}/K^{\times n}\to\prod_{\mathfrak{p}\not\in S}K_{\mathfrak{p}}^{% \times}/K_{\mathfrak{p}}^{\times n}
  29. K = K=\mathbb{Q}
  30. η 2 = 0 \eta_{2}=0
  31. i i
  32. η 3 = 2 \eta_{3}=\sqrt{2}
  33. i η 3 = - 2 i\eta_{3}=\sqrt{-2}
  34. S 0 = { 2 } S_{0}=\{2\}
  35. \mathbb{Q}
  36. K = ( 7 ) K=\mathbb{Q}(\sqrt{7})
  37. S 0 = S_{0}=\emptyset

Grushko_theorem.html

  1. 𝐘 ~ \tilde{\mathbf{Y}}
  2. Z ~ 0 \tilde{Z}_{0}
  3. r 0 : Z ~ 0 𝐘 ~ r_{0}:\tilde{Z}_{0}\to\tilde{\mathbf{Y}}
  4. ϕ : F G 1 G 2 \phi:F\rightarrow G_{1}\ast G_{2}
  5. ϕ ( F 1 ) = G 1 \phi(F_{1})=G_{1}
  6. ϕ ( F 2 ) = G 2 \phi(F_{2})=G_{2}
  7. F = F 1 F 2 F=F_{1}\ast F_{2}
  8. G 1 G 2 G_{1}\ast G_{2}
  9. F 1 F_{1}
  10. F 2 F_{2}
  11. X = X 1 X 2 X=X_{1}\cup X_{2}
  12. { p } = X 1 X 2 \{p\}=X_{1}\cap X_{2}
  13. G 1 G 2 G_{1}\ast G_{2}
  14. f : Z X f:Z\rightarrow X
  15. f f_{\ast}
  16. ϕ \phi
  17. f - 1 ( X 1 ) = : Z 1 f^{-1}(X_{1})=:Z_{1}
  18. f - 1 ( X 2 ) = : Z 2 f^{-1}(X_{2})=:Z_{2}
  19. Z 1 Z 2 Z_{1}\cap Z_{2}
  20. p X p\in X
  21. Z 1 Z 2 Z_{1}\cap Z_{2}
  22. Z 1 Z 2 Z_{1}\cap Z_{2}
  23. F = Π 1 ( Z 1 ) Π 1 ( Z 2 ) F=\Pi_{1}(Z_{1})\ast\Pi_{1}(Z_{2})
  24. Z 1 Z 2 Z_{1}\cap Z_{2}
  25. Z 1 Z 2 Z_{1}\cap Z_{2}
  26. γ : [ 0 , 1 ] Z \gamma:[0,1]\rightarrow Z
  27. γ ( [ 0 , 1 ] ) Z 1 \gamma([0,1])\subseteq Z_{1}
  28. γ ( [ 0 , 1 ] ) Z 2 \gamma([0,1])\subseteq Z_{2}
  29. γ ( 0 ) \gamma(0)
  30. γ ( 1 ) \gamma(1)
  31. Z 1 Z 2 Z_{1}\cap Z_{2}
  32. f γ ( [ 0 , 1 ] ) f\circ\gamma([0,1])
  33. γ \gamma
  34. g : [ 0 , 1 ] D 2 g:[0,1]\rightarrow D^{2}
  35. g ( t ) = e i t g(t)=e^{it}
  36. Z Z^{^{\prime}}
  37. Z = Z D 2 / Z^{\prime}=Z\coprod\!D^{2}/\!\sim
  38. x y iff { x = y , or x = γ ( t ) and y = g ( t ) for some t [ 0 , 1 ] or x = g ( t ) and y = γ ( t ) for some t [ 0 , 1 ] x\!\!\sim y\,\text{ iff}\begin{cases}x=y,\mbox{ or }\\ x=\gamma(t)\,\text{ and }y=g(t)\,\text{ for some }t\in[0,1]\mbox{ or }\\ x=g(t)\,\text{ and }y=\gamma(t)\,\text{ for some }t\in[0,1]\end{cases}
  39. f ′′ : Z D 2 / f^{{}^{\prime\prime}}:Z\coprod\partial D^{2}/\!\sim
  40. f ′′ ( x ) = { f ( x ) , x Z p otherwise. f^{{}^{\prime\prime}}(x)=\begin{cases}f(x),\ x\in Z\\ p\,\text{ otherwise.}\end{cases}
  41. f ( γ ) f(\gamma)
  42. f ′′ f^{\prime\prime}
  43. Z Z^{^{\prime}}
  44. Z i = f - 1 ( X i ) Z_{i}^{^{\prime}}=f^{{}^{\prime}-1}(X_{i})
  45. γ ( 0 ) \gamma(0)
  46. γ ( 1 ) \gamma(1)
  47. Z 1 Z 2 Z_{1}\cap Z_{2}
  48. Z 1 Z 2 Z_{1}^{^{\prime}}\cap Z_{2}^{^{\prime}}
  49. Z 1 Z 2 Z_{1}\cap Z_{2}
  50. γ : [ 0 , 1 ] Z \gamma^{\prime}:[0,1]\rightarrow Z
  51. γ ( 0 ) \gamma^{\prime}(0)
  52. γ ( 1 ) \gamma^{\prime}(1)
  53. Z 1 Z 2 Z_{1}\cap Z_{2}
  54. f f_{\ast}
  55. λ \!\lambda
  56. f ( γ ) \!f(\gamma^{\prime})
  57. f ( λ ) \!f(\lambda)
  58. γ : [ 0 , 1 ] Z \gamma:[0,1]\rightarrow Z
  59. γ ( t ) = γ λ ( t ) \gamma(t)=\gamma^{\prime}\ast\lambda(t)
  60. t [ 0 , 1 ] t\in[0,1]
  61. γ \!\gamma
  62. γ \!\gamma
  63. γ 1 γ 2 γ m \gamma_{1}\ast\gamma_{2}\ast\cdots\ast\gamma_{m}
  64. γ i \gamma_{i}
  65. Z 1 Z_{1}
  66. Z 2 Z_{2}
  67. γ i \gamma_{i}
  68. Z 1 Z_{1}
  69. γ i + 1 \gamma_{i+1}
  70. Z 2 Z_{2}
  71. f ( γ i ) f(\gamma_{i})
  72. [ e ] = [ f ( γ ) ] = [ f ( γ 1 ) ] [ f ( γ m ) ] [e]=[f(\gamma)]=[f(\gamma_{1})]\ast\cdots\ast[f(\gamma_{m})]
  73. [ f ( γ j ) ] = [ e ] [f(\gamma_{j})]=[e]
  74. γ j \!\gamma_{j}
  75. γ j \!\gamma_{j}
  76. γ j \!\gamma_{j}
  77. Z 1 Z 2 Z_{1}\cap Z_{2}
  78. γ j \!\gamma_{j}
  79. Z 1 Z 2 Z_{1}\cap Z_{2}
  80. γ j \!\gamma_{j}^{\prime}
  81. γ j - 1 \!\gamma_{j-1}
  82. γ ′′ = γ 1 γ j - 1 γ j + 1 γ m \gamma^{\prime\prime}=\gamma_{1}\ast\cdots\ast\gamma_{j-1}^{\prime}\ast\gamma_% {j+1}\cdots\gamma_{m}
  83. γ j - 1 = γ j - 1 γ j \!\gamma_{j-1}^{\prime}=\gamma_{j-1}\ast\gamma_{j}^{\prime}
  84. G = A * B G=A*B
  85. { g 1 , g 2 , , g n } \{g_{1},g_{2},\ldots,g_{n}\}
  86. F F
  87. n n
  88. { f 1 , f 2 , , f n } \{f_{1},f_{2},\ldots,f_{n}\}
  89. h : F G h:F\rightarrow G
  90. h ( f i ) = g i h(f_{i})=g_{i}
  91. i = 1 , , n i=1,\ldots,n
  92. F 1 F_{1}
  93. F 2 F_{2}
  94. F = F 1 F 2 F=F_{1}\ast F_{2}
  95. h ( F 1 ) = A h(F_{1})=A
  96. h ( F 2 ) = B h(F_{2})=B
  97. Rank ( A ) Rank ( F 1 ) \,\text{Rank }(A)\leq\,\text{Rank }(F_{1})
  98. Rank ( B ) Rank ( F 2 ) \,\text{Rank }(B)\leq\,\text{Rank }(F_{2})
  99. Rank ( A ) + Rank ( B ) Rank ( F 1 ) + Rank ( F 2 ) = Rank ( F ) = Rank ( A B ) . \,\text{Rank }(A)+\,\text{Rank }(B)\leq\,\text{Rank }(F_{1})+\,\text{Rank }(F_% {2})=\,\text{Rank }(F)=\,\text{Rank }(A\ast B).

Gurney_equations.html

  1. 2 E \sqrt{2E}
  2. 2 E \sqrt{2E}
  3. 2 E \sqrt{2E}
  4. D 2 E \frac{D}{\sqrt{2E}}
  5. 2 E \sqrt{2E}
  6. 2 E \sqrt{2E}
  7. g c m 3 \frac{g}{cm^{3}}
  8. m m μ s \frac{mm}{\mu s}
  9. m m μ s \frac{mm}{\mu s}
  10. m m μ s \frac{mm}{\mu s}
  11. M C \frac{M}{C}
  12. M C [ ( 4 N C ) + 1 ] < 1 2 \frac{M}{C}\left[\left(4\frac{N}{C}\right)+1\right]<\frac{1}{2}
  13. M C \frac{M}{C}
  14. N C 1.0 \frac{N}{C}\geq 1.0
  15. M C \frac{M}{C}
  16. V 2 E = ( M C + 1 2 ) - 1 / 2 \frac{V}{\sqrt{2E}}=\left(\frac{M}{C}+\frac{1}{2}\right)^{-1/2}
  17. V 2 E = ( M C + 3 5 ) - 1 / 2 \frac{V}{\sqrt{2E}}=\left(\frac{M}{C}+\frac{3}{5}\right)^{-1/2}
  18. V 2 E = ( 2 M C + 1 3 ) - 1 / 2 \frac{V}{\sqrt{2E}}=\left(2\frac{M}{C}+\frac{1}{3}\right)^{-1/2}
  19. A = 1 + 2 M C 1 + 2 N C A=\frac{1+2\frac{M}{C}}{1+2\frac{N}{C}}
  20. V M 2 E = ( 1 + A 3 3 ( 1 + A ) + A 2 N C + M C ) - 1 / 2 \frac{V_{M}}{\sqrt{2E}}=\left(\frac{1+A^{3}}{3(1+A)}+A^{2}\frac{N}{C}+\frac{M}% {C}\right)^{-1/2}
  21. V M 2 E = ( M C + 1 3 ) - 1 / 2 \frac{V_{M}}{\sqrt{2E}}=\left(\frac{M}{C}+\frac{1}{3}\right)^{-1/2}
  22. N = 0 N=0
  23. A = 1 + 2 ( M C ) A=1+2\left(\frac{M}{C}\right)
  24. V 2 E = [ 1 + ( 1 + 2 M C ) 3 6 ( 1 + M C ) + M C ] - 1 / 2 \frac{V}{\sqrt{2E}}=\left[\frac{1+\left(1+2\frac{M}{C}\right)^{3}}{6\left(1+% \frac{M}{C}\right)}+\frac{M}{C}\right]^{-1/2}
  25. β = R o R i \beta=\frac{R_{o}}{R_{i}}
  26. a = 1 a=1
  27. A = V o V i = ( M C + a ( M C ) ( β - 1 ) + β + 2 3 ( β + 1 ) ) ( N C + 2 β + 1 3 ( β + 1 ) ) A=\frac{V_{o}}{V_{i}}=\frac{\left(\frac{M}{C}+a\left(\frac{M}{C}\right)\left(% \beta-1\right)+\frac{\beta+2}{3\left(\beta+1\right)}\right)}{\left(\frac{N}{C}% +\frac{2\beta+1}{3\left(\beta+1\right)}\right)}
  28. V m 2 E = \frac{V_{m}}{\sqrt{2E}}=
  29. [ A { ( M C + β + 3 6 ( β + 1 ) ) A + A ( N C + 3 β + 1 6 ( β + 1 ) ) - 1 / 3 } ] - 1 / 2 \left[A\left\{\frac{\left(\frac{M}{C}+\frac{\beta+3}{6\left(\beta+1\right)}% \right)}{A}+A\left(\frac{N}{C}+\frac{3\beta+1}{6\left(\beta+1\right)}\right)-1% /3\right\}\right]^{-1/2}
  30. a a
  31. β = R o R i \beta=\frac{R_{o}}{R_{i}}
  32. a = 1 a=1
  33. A = V o V i = [ M C + ( a M C ) ( β 2 - 1 ) + β 2 + 2 β + 3 4 ( β 2 + β + 1 ) ] ( N C + 3 β 2 + 2 β + 1 4 ( β 2 + β + 1 ) ) A=\frac{V_{o}}{V_{i}}=\frac{\left[\frac{M}{C}+\left(a\frac{M}{C}\right)\left(% \beta^{2}-1\right)+\frac{\beta^{2}+2\beta+3}{4\left(\beta^{2}+\beta+1\right)}% \right]}{\left(\frac{N}{C}+\frac{3\beta^{2}+2\beta+1}{4\left(\beta^{2}+\beta+1% \right)}\right)}
  34. V m 2 E = \frac{V_{m}}{\sqrt{2E}}=
  35. [ A { ( M C + β 2 + 3 β + 6 10 ( β 2 + β + 1 ) ) A + A ( N C + 6 β 2 + 3 β + 1 10 ( β 2 + β + 1 ) ) - 3 β 2 + 4 β + 3 10 ( β 2 + β + 1 ) } ] - 1 / 2 \left[A\left\{\frac{\left(\frac{M}{C}+\frac{\beta^{2}+3\beta+6}{10\left(\beta^% {2}+\beta+1\right)}\right)}{A}+A\left(\frac{N}{C}+\frac{6\beta^{2}+3\beta+1}{1% 0\left(\beta^{2}+\beta+1\right)}\right)-\frac{3\beta^{2}+4\beta+3}{10\left(% \beta^{2}+\beta+1\right)}\right\}\right]^{-1/2}

Gyrokinetics.html

  1. ρ i L p l a s m a \rho_{i}\ll L_{plasma}
  2. k ρ i 1 k_{\perp}\rho_{i}\sim 1
  3. ω Ω i Ω e \omega\ll\Omega_{i}\ll\Omega_{e}
  4. f s = f s 0 + f s 1 + f_{s}=f_{s0}+f_{s1}+\ldots
  5. f s 1 f s 0 f_{s1}\ll f_{s0}
  6. r \vec{r}
  7. R \vec{R}
  8. ( v x , v y , v z ) (v_{x},v_{y},v_{z})
  9. v | | v b ^ v_{||}\equiv\vec{v}\cdot\hat{b}
  10. μ m s v 2 2 B \mu\equiv\frac{m_{s}v_{\perp}^{2}}{2B}
  11. φ \varphi
  12. b B / B \vec{b}\equiv\vec{B}/B
  13. m s m_{s}
  14. φ \left\langle\ldots\right\rangle_{\varphi}
  15. h s t + ( v | | b ^ + V d s + V ϕ φ ) R h s - s C [ h s , h s ] φ = Z s e f s 0 T s ϕ φ t - f s 0 ψ V ϕ φ ψ \frac{\partial h_{s}}{\partial t}+\left(v_{||}\hat{b}+\vec{V}_{ds}+\left% \langle\vec{V}_{\phi}\right\rangle_{\varphi}\right)\cdot\vec{\nabla}_{\vec{R}}% h_{s}-\sum_{s^{\prime}}\left\langle C\left[h_{s},h_{s^{\prime}}\right]\right% \rangle_{\varphi}=\frac{Z_{s}ef_{s0}}{T_{s}}\frac{\partial\left\langle\phi% \right\rangle_{\varphi}}{\partial t}-\frac{\partial f_{s0}}{\partial\psi}\left% \langle\vec{V}_{\phi}\right\rangle_{\varphi}\cdot\vec{\nabla}\psi
  16. h s f s 1 + Z s e ϕ T s f s 0 h_{s}\equiv f_{s1}+\frac{Z_{s}e\phi}{T_{s}}f_{s0}
  17. E × B \vec{E}\times\vec{B}
  18. ψ \psi
  19. s Z s e B d v | | d μ d φ h s ( R ) = s Z s 2 e 2 n s ϕ T s \sum_{s}Z_{s}eB\int dv_{||}d\mu d\varphi h_{s}\left(\vec{R}\right)=\sum_{s}% \frac{Z_{s}^{2}e^{2}n_{s}\phi}{T_{s}}

Gyrovector_space.html

  1. \oplus
  2. \oplus
  3. \ominus
  4. \ominus
  5. \oplus
  6. \oplus
  7. \oplus
  8. \oplus
  9. \oplus
  10. \oplus
  11. \oplus
  12. \oplus
  13. \oplus
  14. \oplus
  15. gyr [ 𝐮 , 𝐯 ] 𝐰 = ( 𝐮 𝐯 ) ( 𝐮 ( 𝐯 𝐰 ) ) \mathrm{gyr}[\mathbf{u},\mathbf{v}]\mathbf{w}=\ominus(\mathbf{u}\oplus\mathbf{% v})\oplus(\mathbf{u}\oplus(\mathbf{v}\oplus\mathbf{w}))
  16. 𝐮 ( 𝐯 𝐰 ) = ( 𝐮 𝐯 ) gyr [ 𝐮 , 𝐯 ] 𝐰 \mathbf{u}\oplus(\mathbf{v}\oplus\mathbf{w})=(\mathbf{u}\oplus\mathbf{v})% \oplus\mathrm{gyr}[\mathbf{u},\mathbf{v}]\mathbf{w}
  17. ( 𝐮 𝐯 ) 𝐰 = 𝐮 ( 𝐯 gyr [ 𝐯 , 𝐮 ] 𝐰 ) (\mathbf{u}\oplus\mathbf{v})\oplus\mathbf{w}=\mathbf{u}\oplus(\mathbf{v}\oplus% \mathrm{gyr}[\mathbf{v},\mathbf{u}]\mathbf{w})
  18. \oplus
  19. \oplus
  20. \oplus
  21. \boxplus
  22. \oplus
  23. \ominus
  24. 𝐮 \mathbf{u}
  25. 𝐯 \mathbf{v}
  26. 𝐮 E 𝐯 = 1 1 + 𝐮 𝐯 c 2 { 𝐮 + 1 γ 𝐮 𝐯 + 1 c 2 γ 𝐮 1 + γ 𝐮 ( 𝐮 𝐯 ) 𝐮 } \mathbf{u}\oplus_{E}\mathbf{v}=\frac{1}{1+\frac{\mathbf{u}\cdot\mathbf{v}}{c^{% 2}}}\left\{\mathbf{u}+\frac{1}{\gamma_{\mathbf{u}}}\mathbf{v}+\frac{1}{c^{2}}% \frac{\gamma_{\mathbf{u}}}{1+\gamma_{\mathbf{u}}}(\mathbf{u}\cdot\mathbf{v})% \mathbf{u}\right\}
  27. γ 𝐮 \gamma_{\mathbf{u}}
  28. γ 𝐮 = 1 1 - | 𝐮 | 2 c 2 \gamma_{\mathbf{u}}=\frac{1}{\sqrt{1-\frac{|\mathbf{u}|^{2}}{c^{2}}}}
  29. ( w 1 w 2 w 3 ) = 1 1 + u 1 v 1 + u 2 v 2 + u 3 v 3 c 2 { [ 1 + 1 c 2 γ 𝐮 1 + γ 𝐮 ( u 1 v 1 + u 2 v 2 + u 3 v 3 ) ] ( u 1 u 2 u 3 ) + 1 γ 𝐮 ( v 1 v 2 v 3 ) } \begin{pmatrix}w_{1}\\ w_{2}\\ w_{3}\\ \end{pmatrix}=\frac{1}{1+\frac{u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3}}{c^{2}}}\left% \{\left[1+\frac{1}{c^{2}}\frac{\gamma_{\mathbf{u}}}{1+\gamma_{\mathbf{u}}}(u_{% 1}v_{1}+u_{2}v_{2}+u_{3}v_{3})\right]\begin{pmatrix}u_{1}\\ u_{2}\\ u_{3}\\ \end{pmatrix}+\frac{1}{\gamma_{\mathbf{u}}}\begin{pmatrix}v_{1}\\ v_{2}\\ v_{3}\\ \end{pmatrix}\right\}
  30. γ 𝐮 = 1 1 - u 1 2 + u 2 2 + u 3 2 c 2 \gamma_{\mathbf{u}}=\frac{1}{\sqrt{1-\frac{u_{1}^{2}+u_{2}^{2}+u_{3}^{2}}{c^{2% }}}}
  31. 𝐮 \mathbf{u}
  32. 𝐯 \mathbf{v}
  33. 𝐮 𝐯 = gyr [ 𝐮 , 𝐯 ] ( 𝐯 𝐮 ) \mathbf{u}\oplus\mathbf{v}=\mathrm{gyr}[\mathbf{u},\mathbf{v}](\mathbf{v}% \oplus\mathbf{u})
  34. 𝐮 ( 𝐯 𝐰 ) = ( 𝐮 𝐯 ) gyr [ 𝐮 , 𝐯 ] 𝐰 \mathbf{u}\oplus(\mathbf{v}\oplus\mathbf{w})=(\mathbf{u}\oplus\mathbf{v})% \oplus\mathrm{gyr}[\mathbf{u},\mathbf{v}]\mathbf{w}
  35. gyr [ 𝐮 , 𝐯 ] 𝐰 = ( 𝐮 𝐯 ) ( 𝐮 ( 𝐯 𝐰 ) ) \mathrm{gyr}[\mathbf{u},\mathbf{v}]\mathbf{w}=\ominus(\mathbf{u}\oplus\mathbf{% v})\oplus(\mathbf{u}\oplus(\mathbf{v}\oplus\mathbf{w}))
  36. Gyr [ 𝐮 , 𝐯 ] = ( 1 0 0 gyr [ 𝐮 , 𝐯 ] ) \mathrm{Gyr}[\mathbf{u},\mathbf{v}]=\begin{pmatrix}1&0\\ 0&\mathrm{gyr}[\mathbf{u},\mathbf{v}]\end{pmatrix}
  37. B ( 𝐮 ) B ( 𝐯 ) = B ( 𝐮 𝐯 ) Gyr [ 𝐮 , 𝐯 ] = Gyr [ 𝐮 , 𝐯 ] B ( 𝐯 𝐮 ) B(\mathbf{u})B(\mathbf{v})=B(\mathbf{u}\oplus\mathbf{v})\mathrm{Gyr}[\mathbf{u% },\mathbf{v}]=\mathrm{Gyr}[\mathbf{u},\mathbf{v}]B(\mathbf{v}\oplus\mathbf{u})
  38. \oplus
  39. \oplus
  40. L ( 𝐮 , U ) L ( 𝐯 , V ) = L ( 𝐮 U 𝐯 , gyr [ 𝐮 , U 𝐯 ] U V ) L(\mathbf{u},U)L(\mathbf{v},V)=L(\mathbf{u}\oplus U\mathbf{v},\mathrm{gyr}[% \mathbf{u},U\mathbf{v}]UV)
  41. \oplus
  42. \oplus
  43. \oplus
  44. \oplus
  45. \oplus
  46. \otimes
  47. \oplus
  48. \otimes
  49. \otimes
  50. \otimes
  51. \otimes
  52. \otimes
  53. \oplus
  54. \oplus
  55. \otimes
  56. \otimes
  57. \oplus
  58. \otimes
  59. \otimes
  60. \otimes
  61. \otimes
  62. \otimes
  63. \otimes
  64. \oplus
  65. \otimes
  66. \otimes
  67. \otimes
  68. \oplus
  69. \otimes
  70. \otimes
  71. z e i θ a + z 1 + a z ¯ z\to{e^{i\theta}}{\frac{a+z}{1+a\bar{z}}}
  72. e i θ ( a M z ) e^{i\theta}{(a\oplus_{M}{z})}
  73. a M z = a + z 1 + a z ¯ {a\oplus_{M}{z}}=\frac{a+z}{1+a\bar{z}}
  74. 𝐮 M 𝐯 = ( 1 + 2 s 2 𝐮 𝐯 + 1 s 4 | 𝐯 | 2 ) 𝐮 + ( 1 - 1 s 2 | 𝐮 | 2 ) 𝐯 1 + 2 s 2 𝐮 𝐯 + 1 s 4 | 𝐮 | 2 | 𝐯 | 2 \mathbf{u}\oplus_{M}\mathbf{v}=\frac{(1+\frac{2}{s^{2}}\mathbf{u}\cdot\mathbf{% v}+\frac{1}{s^{4}}|\mathbf{v}|^{2})\mathbf{u}+(1-\frac{1}{s^{2}}|\mathbf{u}|^{% 2})\mathbf{v}}{1+\frac{2}{s^{2}}\mathbf{u}\cdot\mathbf{v}+\frac{1}{s^{4}}|% \mathbf{u}|^{2}|\mathbf{v}|^{2}}
  75. \oplus
  76. \otimes
  77. \oplus
  78. \otimes
  79. \otimes
  80. \otimes
  81. \otimes
  82. 𝐮 U 𝐯 = 𝐮 + 𝐯 + { β 𝐮 1 + β 𝐮 𝐮 𝐯 c 2 + 1 - β 𝐯 β 𝐯 } 𝐮 \mathbf{u}\oplus_{U}\mathbf{v}=\mathbf{u}+\mathbf{v}+\left\{{\frac{\beta_{% \mathbf{u}}}{1+\beta_{\mathbf{u}}}}{\frac{\mathbf{u}\cdot\mathbf{v}}{c^{2}}}+{% \frac{1-\beta_{\mathbf{v}}}{\beta_{\mathbf{v}}}}\right\}\mathbf{u}
  83. β 𝐰 \beta_{\mathbf{w}}
  84. β 𝐰 = 1 1 + | 𝐰 | 2 c 2 \beta_{\mathbf{w}}=\frac{1}{\sqrt{1+\frac{|\mathbf{w}|^{2}}{c^{2}}}}
  85. U \oplus_{U}
  86. \otimes
  87. \otimes
  88. \otimes
  89. \otimes
  90. \rightarrow
  91. γ 𝐯 e 𝐯 e \gamma_{\mathbf{v}_{e}}\mathbf{v}_{e}
  92. \rightarrow
  93. β 𝐯 u 𝐯 u \beta_{\mathbf{v}_{u}}\mathbf{v}_{u}
  94. \rightarrow
  95. 1 2 E 𝐯 e \frac{1}{2}\otimes_{E}\mathbf{v}_{e}
  96. \rightarrow
  97. 2 M 𝐯 m 2\otimes_{M}\mathbf{v}_{m}
  98. \rightarrow
  99. 2 γ 2 𝐯 m 𝐯 m 2{{{\gamma}^{2}}_{\mathbf{v}_{m}}}\mathbf{v}_{m}
  100. \rightarrow
  101. β 𝐯 u 1 + β 𝐯 u 𝐯 u \frac{\beta_{\mathbf{v}_{u}}}{1+\beta_{\mathbf{v}_{u}}}\mathbf{v}_{u}
  102. E \oplus_{E}
  103. M \oplus_{M}
  104. 𝐮 E 𝐯 = 2 ( 1 2 𝐮 M 1 2 𝐯 ) \mathbf{u}\oplus_{E}\mathbf{v}=2\otimes\left({\frac{1}{2}\otimes\mathbf{u}% \oplus_{M}\frac{1}{2}\otimes\mathbf{v}}\right)
  105. 𝐮 M 𝐯 = 1 2 ( 2 𝐮 E 2 𝐯 ) \mathbf{u}\oplus_{M}\mathbf{v}=\frac{1}{2}\otimes\left({2\otimes\mathbf{u}% \oplus_{E}2\otimes\mathbf{v}}\right)

H_square.html

  1. n = - a n e i n φ \sum_{n=-\infty}^{\infty}a_{n}e^{in\varphi}
  2. n = 0 a n e i n φ . \sum_{n=0}^{\infty}a_{n}e^{in\varphi}.
  3. [ f ] ( s ) = 0 e - s t f ( t ) d t [\mathcal{L}f](s)=\int_{0}^{\infty}e^{-st}f(t)dt
  4. : L 2 ( 0 , ) H 2 ( + ) \mathcal{L}:L^{2}(0,\infty)\to H^{2}\left(\mathbb{C}^{+}\right)
  5. L 2 ( 0 , ) L^{2}(0,\infty)
  6. + \mathbb{C}^{+}
  7. f H 2 = 2 π f L 2 . \|\mathcal{L}f\|_{H^{2}}=\sqrt{2\pi}\|f\|_{L^{2}}.
  8. L 2 ( ) = L 2 ( - , 0 ) L 2 ( 0 , ) L^{2}(\mathbb{R})=L^{2}(-\infty,0)\oplus L^{2}(0,\infty)
  9. L 2 ( ) L^{2}(\mathbb{R})
  10. L 2 ( ) = H 2 ( - ) H 2 ( + ) . L^{2}(\mathbb{R})=H^{2}\left(\mathbb{C}^{-}\right)\oplus H^{2}\left(\mathbb{C}% ^{+}\right).

Haagerup_property.html

  1. G G
  2. Ψ : G + \Psi\colon G\to\mathbb{R}^{+}
  3. G G
  4. C 0 C_{0}
  5. ϕ n \phi_{n}
  6. G G
  7. G G
  8. G G
  9. G G
  10. G G
  11. \mathbb{R}

Hadamard's_lemma.html

  1. f ( x ) = f ( a ) + i = 1 n ( x i - a i ) g i ( x ) , f(x)=f(a)+\sum_{i=1}^{n}(x_{i}-a_{i})g_{i}(x),
  2. h ( t ) = f ( a + t ( x - a ) ) . h(t)=f(a+t(x-a)).\,
  3. h ( t ) = i = 1 n f x i ( a + t ( x - a ) ) ( x i - a i ) , h^{\prime}(t)=\sum_{i=1}^{n}\frac{\partial f}{\partial x_{i}}(a+t(x-a))(x_{i}-% a_{i}),
  4. h ( 1 ) - h ( 0 ) = 0 1 h ( t ) d t = 0 1 i = 1 n f x i ( a + t ( x - a ) ) ( x i - a i ) d t = i = 1 n ( x i - a i ) 0 1 f x i ( a + t ( x - a ) ) d t . h(1)-h(0)=\int_{0}^{1}h^{\prime}(t)\,dt=\int_{0}^{1}\sum_{i=1}^{n}\frac{% \partial f}{\partial x_{i}}(a+t(x-a))(x_{i}-a_{i})\,dt=\sum_{i=1}^{n}(x_{i}-a_% {i})\int_{0}^{1}\frac{\partial f}{\partial x_{i}}(a+t(x-a))\,dt.
  5. g i ( x ) = 0 1 f x i ( a + t ( x - a ) ) d t , g_{i}(x)=\int_{0}^{1}\frac{\partial f}{\partial x_{i}}(a+t(x-a))\,dt,

Hadamard_regularization.html

  1. 𝒞 a b f ( t ) t - x d t ( for a < x < b ) \mathcal{C}\int_{a}^{b}\frac{f(t)}{t-x}\,dt\quad(\hbox{for }a<x<b)
  2. x x
  3. d d x ( 𝒞 a b f ( t ) t - x d t ) = a b f ( t ) ( t - x ) 2 d t ( for a < x < b ) . \frac{d}{dx}\left(\mathcal{C}\int_{a}^{b}\frac{f(t)}{t-x}\,dt\right)=\mathcal{% H}\int_{a}^{b}\frac{f(t)}{(t-x)^{2}}\,dt\quad(\hbox{for }a<x<b).
  4. 𝒞 \mathcal{C}
  5. \mathcal{H}
  6. a Align l t ; x Align l t ; b a&lt;x&lt;b
  7. a b f ( t ) ( t - x ) 2 d t = lim ε 0 + { a x - ε f ( t ) ( t - x ) 2 d t + x + ε b f ( t ) ( t - x ) 2 d t - 2 f ( x ) ε } , \mathcal{H}\int_{a}^{b}\frac{f(t)}{(t-x)^{2}}\,dt=\lim_{\varepsilon\to 0^{+}}% \left\{\int_{a}^{x-\varepsilon}\frac{f(t)}{(t-x)^{2}}\,dt+\int_{x+\varepsilon}% ^{b}\frac{f(t)}{(t-x)^{2}}\,dt-\frac{2f(x)}{\varepsilon}\right\},
  8. a b f ( t ) ( t - x ) 2 d t = lim ε 0 + { a b ( t - x ) 2 f ( t ) ( ( t - x ) 2 + ε 2 ) 2 d t - π f ( x ) 2 ε - f ( x ) 2 ( 1 b - x - 1 a - x ) } . \mathcal{H}\int_{a}^{b}\frac{f(t)}{(t-x)^{2}}\,dt=\lim_{\varepsilon\to 0^{+}}% \left\{\int_{a}^{b}\frac{(t-x)^{2}f(t)}{((t-x)^{2}+\varepsilon^{2})^{2}}\,dt-% \frac{\pi f(x)}{2\varepsilon}-\frac{f(x)}{2}(\frac{1}{b-x}-\frac{1}{a-x})% \right\}.
  9. f ( t ) f(t)
  10. t = x f o r a Align l t ; x Align l t ; b t=xfora&lt;x&lt;b
  11. f ( t ) f(t)
  12. t = x t=x
  13. f ( x ( 1 b 1 a ) −f\frac{(}{x}(1\frac{b}{−}−1\frac{a}{−})
  14. f ( t ) f(t)

Hadamard–Rybczynski_equation.html

  1. W b = 2 3 R 2 g ( ρ b - ρ o ) μ o μ o + μ b 2 μ o + 3 μ b W_{b}=\frac{2}{3}\frac{R^{2}g(\rho_{b}-\rho_{o})}{\mu_{o}}\frac{\mu_{o}+\mu_{b% }}{2\mu_{o}+3\mu_{b}}
  2. R R
  3. g g
  4. ρ b \rho_{b}
  5. ρ o \rho_{o}
  6. μ b \mu_{b}
  7. μ o \mu_{o}

Hadjicostas's_formula.html

  1. 0 1 0 1 1 - x 1 - x y ( - log ( x y ) ) s d x d y = Γ ( s + 2 ) ( ζ ( s + 2 ) - 1 s + 1 ) . \int_{0}^{1}\int_{0}^{1}\frac{1-x}{1-xy}(-\log(xy))^{s}\,dx\,dy=\Gamma(s+2)% \left(\zeta(s+2)-\frac{1}{s+1}\right).
  2. γ = 0 1 0 1 1 - x ( 1 - x y ) ( - log ( x y ) ) d x d y . \gamma=\int_{0}^{1}\int_{0}^{1}\frac{1-x}{(1-xy)(-\log(xy))}\,dx\,dy.

Haefliger_structure.html

  1. Ψ γ α ( u ) = Ψ γ β ( u ) Ψ β α ( u ) \displaystyle\Psi_{\gamma\alpha}(u)=\Psi_{\gamma\beta}(u)\Psi_{\beta\alpha}(u)
  2. u U α U β U γ . u\in U_{\alpha}\cap U_{\beta}\cap U_{\gamma}.
  3. ϕ α ( v ) = Φ α , β ( u ) ( ϕ β ( v ) ) \phi_{\alpha}(v)=\Phi_{\alpha,\beta}(u)(\phi_{\beta}(v))
  4. Ψ α , β ( u ) = \Psi_{\alpha,\beta}(u)=
  5. Φ α , β ( u ) \Phi_{\alpha,\beta}(u)

Hagen–Poiseuille_equation.html

  1. Δ P = 8 μ L Q π r 4 \Delta P=\frac{8\mu LQ}{\pi r^{4}}
  2. Δ P = 128 μ L Q π d 4 \Delta P=\frac{128\mu LQ}{\pi d^{4}}
  3. Δ P \Delta P
  4. L L
  5. μ \mu
  6. Q Q
  7. r r
  8. d d
  9. π \pi
  10. Φ = d V d t = v π R 2 = π R 4 8 η ( - Δ P Δ x ) = π R 4 8 η | Δ P | L \Phi=\frac{dV}{dt}=v\pi R^{2}=\frac{\pi R^{4}}{8\eta}\left(\frac{-\Delta P}{% \Delta x}\right)=\frac{\pi R^{4}}{8\eta}\frac{|\Delta P|}{L}
  11. Φ \Phi
  12. Q Q
  13. V ( t ) V(t)
  14. t t
  15. v v
  16. x x
  17. R R
  18. Δ P \Delta P
  19. η \eta
  20. L L
  21. Φ m a x = π R 2 2 Δ P / ρ \Phi_{max}=\pi R^{2}\sqrt{2\Delta P/\rho}
  22. Λ = 64 Re , Re = 2 ρ v r η , \Lambda={64\over{\it\mathrm{Re}}}\;,\quad\quad\mathrm{Re}={2\rho vr\over\eta}\;,
  23. F = - Δ P A F=-\Delta PA
  24. Δ P = P end - P top < 0 \Delta P=P\text{end}-P\text{top}<0
  25. Δ v x / Δ y {\Delta v_{x}}/{\Delta y}
  26. F viscosity, top = - η A Δ v x Δ y . F_{\,\text{viscosity, top}}=-\eta A\frac{\Delta v_{x}}{\Delta y}.
  27. s s
  28. r r
  29. d r dr
  30. A = 2 π r Δ x A=2\pi r\,\Delta x
  31. r r
  32. F viscosity, fast = - η 2 π r Δ x d v d r | r F_{\,\text{viscosity, fast}}=-\eta 2\pi r\,\Delta x\left.\frac{dv}{dr}\right|_% {r}
  33. r r
  34. F viscosity, slow = η 2 π ( r + d r ) Δ x d v d r | r + d r F_{\,\text{viscosity, slow}}=\eta 2\pi(r+dr)\Delta x\left.\frac{dv}{dr}\right|% _{r+dr}
  35. 0 = F pressure + F viscosity, fast + F viscosity, slow 0=F_{\,\text{pressure}}+F_{\,\text{viscosity, fast}}+F_{\,\text{viscosity, % slow}}
  36. 0 = Δ P 2 π r d r - η 2 π r Δ x d v d r | r + η 2 π ( r + d r ) Δ x d v d r | r + d r . 0=\Delta P2\pi rdr-\eta 2\pi r\Delta x\left.\frac{dv}{dr}\right|_{r}+\eta 2\pi% (r+dr)\Delta x\left.\frac{dv}{dr}\right|_{r+dr}.
  37. d v d r | r + d r = d v d r | r + d 2 v d r 2 | r d r . \left.\frac{dv}{dr}\right|_{r+dr}=\left.\frac{dv}{dr}\right|_{r}+\left.\frac{d% ^{2}v}{dr^{2}}\right|_{r}dr.
  38. 0 = Δ P 2 π r d r + η 2 π d r Δ x d v d r + η 2 π r d r Δ x d 2 v d r 2 + η 2 π ( d r ) 2 Δ x d 2 v d r 2 . 0=\Delta P2\pi r\,dr+\eta 2\pi\,dr\,\Delta x\frac{dv}{dr}+\eta 2\pi r\,dr\,% \Delta x\frac{d^{2}v}{dr^{2}}+\eta 2\pi(dr)^{2}\,\Delta x\frac{d^{2}v}{dr^{2}}.
  39. - 1 η Δ P Δ x = d 2 v d r 2 + 1 r d v d r -\frac{1}{\eta}\frac{\Delta P}{\Delta x}=\frac{d^{2}v}{dr^{2}}+\frac{1}{r}% \frac{dv}{dr}
  40. - 1 η Δ P Δ x = 1 r d d r ( r d v d r ) . -\frac{1}{\eta}\frac{\Delta P}{\Delta x}=\frac{1}{r}\frac{d}{dr}\left(r\frac{% dv}{dr}\right).
  41. v ( r ) = 0 v(r)=0
  42. r = R r=R
  43. d v d r = 0 \frac{dv}{dr}=0
  44. r = 0 r=0
  45. d v d r \frac{dv}{dr}
  46. v ( r ) = - 1 4 η r 2 Δ P Δ x + A ln ( r ) + B . v(r)=-\frac{1}{4\eta}r^{2}\frac{\Delta P}{\Delta x}+A\ln(r)+B.
  47. d v d r = - 1 2 η r Δ P Δ x + A 1 r = 0 \frac{dv}{dr}=-\frac{1}{2\eta}r\frac{\Delta P}{\Delta x}+A\frac{1}{r}=0
  48. v ( R ) = - 1 4 η R 2 Δ P Δ x + B = 0 v(R)=-\frac{1}{4\eta}R^{2}\frac{\Delta P}{\Delta x}+B=0
  49. B = 1 4 η R 2 Δ P Δ x . B=\frac{1}{4\eta}R^{2}\frac{\Delta P}{\Delta x}.
  50. v = 1 4 η Δ P Δ x ( R 2 - r 2 ) v=\frac{1}{4\eta}\frac{\Delta P}{\Delta x}(R^{2}-r^{2})
  51. v m a x = 1 4 η Δ P Δ x R 2 . v_{max}=\frac{1}{4\eta}\frac{\Delta P}{\Delta x}R^{2}.
  52. Φ ( r ) d r = 1 4 η | Δ P | Δ x ( R 2 - r 2 ) 2 π r d r = π 2 η | Δ P | Δ x ( r R 2 - r 3 ) d r \Phi(r)\,dr=\frac{1}{4\eta}\frac{|\Delta P|}{\Delta x}(R^{2}-r^{2})2\pi r\,dr=% \frac{\pi}{2\eta}\frac{|\Delta P|}{\Delta x}(rR^{2}-r^{3})\,dr
  53. Φ = π 2 η | Δ P | Δ x 0 R ( r R 2 - r 3 ) d r = | Δ P | π R 4 8 η Δ x \Phi=\frac{\pi}{2\eta}\frac{|\Delta P|}{\Delta x}\int_{0}^{R}(rR^{2}-r^{3})\,% dr=\frac{|\Delta P|\pi R^{4}}{8\eta\Delta x}
  54. Φ = d V d t = v π R 2 = π R 4 ( P i - P o ) 8 η L × P i + P o 2 P o = π R 4 16 η L ( P i 2 - P o 2 P o ) \Phi=\frac{dV}{dt}=v\pi R^{2}=\frac{\pi R^{4}\left(P_{i}-P_{o}\right)}{8\eta L% }\times\frac{P_{i}+P_{o}}{2P_{o}}=\frac{\pi R^{4}}{16\eta L}\left(\frac{P_{i}^% {2}-P_{o}^{2}}{P_{o}}\right)
  55. P i P_{i}
  56. P o P_{o}
  57. L L
  58. η \eta
  59. R R
  60. V V
  61. v v
  62. P i + P o 2 × 1 P o \frac{P_{i}+P_{o}}{2}\times\frac{1}{P_{o}}
  63. V = I R V=IR
  64. Δ F = S Δ P \Delta F=S\Delta P
  65. S = π r 2 S=\pi r^{2}
  66. Δ F = π r 2 Δ P \Delta F=\pi r^{2}\Delta P
  67. Δ P = 8 μ L Q π r 4 \Delta P=\frac{8\mu LQ}{\pi r^{4}}
  68. Δ F = 8 μ L Q r 2 \Delta F=\frac{8\mu LQ}{r^{2}}
  69. n n
  70. [ n ] = m - 3 [n]=m^{-3}
  71. q * q^{*}
  72. [ q * ] = C [q^{*}]=C
  73. q * = e = 1 , 6.10 - 19 C q^{*}=e=1,6.10^{-19}C
  74. n Q nQ
  75. Q Q
  76. n Q q * nQq^{*}
  77. I I
  78. I = n Q q * I=nQq^{*}
  79. Q = I n q * Q=\frac{I}{nq^{*}}
  80. Δ F = 8 μ L I n r 2 q * \Delta F=\frac{8\mu LI}{nr^{2}q^{*}}
  81. Δ F = E q \Delta F=Eq
  82. q q
  83. π r 2 L \pi r^{2}L
  84. n π r 2 L n\pi r^{2}L
  85. q = n π r 2 L q * q=n\pi r^{2}Lq^{*}
  86. E = Δ F q = 8 μ I n 2 π r 4 ( q * ) 2 E=\frac{\Delta F}{q}=\frac{8\mu I}{n^{2}\pi r^{4}(q^{*})^{2}}
  87. V = E L V=EL
  88. V = 8 μ L I n 2 π r 4 ( q * ) 2 V=\frac{8\mu LI}{n^{2}\pi r^{4}(q^{*})^{2}}
  89. R = V I R=\frac{V}{I}
  90. R = 8 μ L n 2 π r 4 ( q * ) 2 R=\frac{8\mu L}{n^{2}\pi r^{4}(q^{*})^{2}}
  91. R R
  92. L L
  93. R R
  94. r r
  95. R R
  96. S = π r 2 S=\pi r^{2}
  97. R = ρ L S R=\frac{\rho L}{S}
  98. ρ \rho
  99. R R
  100. S S
  101. R R

Hahn_polynomials.html

  1. Q n ( x ; α , β , N ) = F 2 3 ( - n , - x , n + α + β + 1 ; α + 1 , - N + 1 ; 1 ) . Q_{n}(x;\alpha,\beta,N)={}_{3}F_{2}(-n,-x,n+\alpha+\beta+1;\alpha+1,-N+1;1).
  2. a ¯ \overline{a}
  3. b ¯ \overline{b}
  4. x = 0 N - 1 Q n ( x ) Q m ( x ) ρ ( x ) = 1 π n δ m , n , \sum_{x=0}^{N-1}Q_{n}(x)Q_{m}(x)\rho(x)=\frac{1}{\pi_{n}}\delta_{m,n},
  5. n = 0 N - 1 Q n ( x ) Q n ( y ) π n = 1 ρ ( x ) δ x , y \sum_{n=0}^{N-1}Q_{n}(x)Q_{n}(y)\pi_{n}=\frac{1}{\rho(x)}\delta_{x,y}
  6. ρ ( x ) = ρ ( x ; α ; β , N ) = ( α + x x ) ( β + N - 1 - x N - 1 - x ) / ( N + α + β N - 1 ) \rho(x)=\rho(x;\alpha;\beta,N)={\left({{\alpha+x}\atop{x}}\right)}{\left({{% \beta+N-1-x}\atop{N-1-x}}\right)}/{\left({{N+\alpha+\beta}\atop{N-1}}\right)}
  7. π n = π n ( α , β , N ) = ( N - 1 n ) 2 n + α + β + 1 α + β + 1 Γ ( β + 1 , n + α + 1 , n + α + β + 1 ) Γ ( α + 1 , α + β + 1 , n + β + 1 , n + 1 ) / ( N + α + β + n n ) \pi_{n}=\pi_{n}(\alpha,\beta,N)={\left({{N-1}\atop{n}}\right)}\frac{2n+\alpha+% \beta+1}{\alpha+\beta+1}\frac{\Gamma(\beta+1,n+\alpha+1,n+\alpha+\beta+1)}{% \Gamma(\alpha+1,\alpha+\beta+1,n+\beta+1,n+1)}/{\left({{N+\alpha+\beta+n}\atop% {n}}\right)}

Hahn_series.html

  1. \mathbb{Q}
  2. \mathbb{R}
  3. K [ [ T Γ ] ] K[[T^{\Gamma}]]
  4. f = e Γ c e T e f=\sum_{e\in\Gamma}c_{e}T^{e}
  5. c e K c_{e}\in K
  6. { e Γ : c e 0 } \{e\in\Gamma:c_{e}\neq 0\}
  7. f = e Γ c e T e f=\sum_{e\in\Gamma}c_{e}T^{e}
  8. g = e Γ d e T e g=\sum_{e\in\Gamma}d_{e}T^{e}
  9. f + g = e Γ ( c e + d e ) T e f+g=\sum_{e\in\Gamma}(c_{e}+d_{e})T^{e}
  10. f g = e Γ e + e ′′ = e c e d e ′′ T e fg=\sum_{e\in\Gamma}\sum_{e^{\prime}+e^{\prime\prime}=e}c_{e^{\prime}}d_{e^{% \prime\prime}}T^{e}
  11. e + e ′′ = e \sum_{e^{\prime}+e^{\prime\prime}=e}\cdot
  12. ( e , e ′′ ) (e^{\prime},e^{\prime\prime})
  13. c e 0 c_{e^{\prime}}\neq 0
  14. d e ′′ 0 d_{e^{\prime\prime}}\neq 0
  15. T - 1 / p + T - 1 / p 2 + T - 1 / p 3 + T^{-1/p}+T^{-1/p^{2}}+T^{-1/p^{3}}+\cdots
  16. { - 1 p , - 1 p 2 , - 1 p 3 , } \{-\frac{1}{p},-\frac{1}{p^{2}},-\frac{1}{p^{3}},\ldots\}
  17. X p - X = T - 1 X^{p}-X=T^{-1}
  18. K ( T ) K(T)
  19. v ( f ) v(f)
  20. f = e Γ c e T e f=\sum_{e\in\Gamma}c_{e}T^{e}
  21. c e 0 c_{e}\neq 0
  22. K [ [ T Γ ] ] K[[T^{\Gamma}]]
  23. K [ [ T Γ ] ] K[[T^{\Gamma}]]
  24. Γ \Gamma\subseteq\mathbb{R}
  25. | f | = exp ( - v ( f ) ) |f|=\exp(-v(f))
  26. K [ [ T Γ ] ] K[[T^{\Gamma}]]
  27. T - 1 / p + T - 1 / p 2 + T - 1 / p 3 + T^{-1/p}+T^{-1/p^{2}}+T^{-1/p^{3}}+\cdots
  28. K [ [ T Γ ] ] K[[T^{\Gamma}]]
  29. K ( ( T ) ) K((T))
  30. K [ [ T ] ] K[[T^{\mathbb{Q}}]]
  31. K ( ( T ) ) K((T))
  32. K [ [ T Γ ] ] K[[T^{\Gamma}]]
  33. K [ [ T Γ ] ] K[[T^{\Gamma}]]
  34. K [ [ T Γ ] ] K[[T^{\Gamma}]]
  35. K [ [ T Γ ] ] K[[T^{\Gamma}]]
  36. { e Γ : c e 0 } \{e\in\Gamma:c_{e}\neq 0\}
  37. K [ [ T Γ ] ] K[[T^{\Gamma}]]
  38. e Γ c e p e \sum_{e\in\Gamma}c_{e}p^{e}
  39. c e c_{e}
  40. p \mathbb{C}_{p}

Half-normal_distribution.html

  1. F ( x ; σ ) = erf ( x σ 2 ) F(x;\sigma)=\mbox{erf}~{}\left(\frac{x}{\sigma\sqrt{2}}\right)
  2. Q ( F ; σ ) = σ 2 erf ( F ) - 1 Q(F;\sigma)=\sigma\sqrt{2}\mbox{erf}~{}^{-1}(F)
  3. σ 2 π \frac{\sigma\sqrt{2}}{\sqrt{\pi}}
  4. σ 2 erf ( 1 / 2 ) - 1 \sigma\sqrt{2}\mbox{erf}~{}^{-1}(1/2)
  5. σ 2 ( 1 - 2 π ) \sigma^{2}\left(1-\frac{2}{\pi}\right)
  6. 1 2 log ( π σ 2 2 ) + 1 2 \frac{1}{2}\log\left(\frac{\pi\sigma^{2}}{2}\right)+\frac{1}{2}
  7. X X
  8. N ( 0 , σ 2 ) N(0,\sigma^{2})
  9. Y = | X | Y=|X|
  10. σ \sigma
  11. f Y ( y ; σ ) = 2 σ π exp ( - y 2 2 σ 2 ) y > 0 f_{Y}(y;\sigma)=\frac{\sqrt{2}}{\sigma\sqrt{\pi}}\exp\left(-\frac{y^{2}}{2% \sigma^{2}}\right)\quad y>0
  12. E [ Y ] = μ = σ 2 π E[Y]=\mu=\frac{\sigma\sqrt{2}}{\sqrt{\pi}}
  13. σ \sigma
  14. θ = π σ 2 \theta=\frac{\sqrt{\pi}}{\sigma\sqrt{2}}
  15. f Y ( y ; θ ) = 2 θ π exp ( - y 2 θ 2 π ) y > 0 f_{Y}(y;\theta)=\frac{2\theta}{\pi}\exp\left(-\frac{y^{2}\theta^{2}}{\pi}% \right)\quad y>0
  16. E [ Y ] = μ = 1 θ E[Y]=\mu=\frac{1}{\theta}
  17. F Y ( y ; σ ) = 0 y 1 σ 2 π exp ( - x 2 2 σ 2 ) d x F_{Y}(y;\sigma)=\int_{0}^{y}\frac{1}{\sigma}\sqrt{\frac{2}{\pi}}\,\exp\left(-% \frac{x^{2}}{2\sigma^{2}}\right)\,dx
  18. z = x / ( 2 σ ) z=x/(\sqrt{2}\sigma)
  19. F Y ( y ; σ ) = 2 π 0 y / ( 2 σ ) exp ( - z 2 ) d z = erf ( y 2 σ ) , F_{Y}(y;\sigma)=\frac{2}{\sqrt{\pi}}\,\int_{0}^{y/(\sqrt{2}\sigma)}\exp\left(-% z^{2}\right)dz=\mbox{erf}~{}\left(\frac{y}{\sqrt{2}\sigma}\right),
  20. Q ( F ; σ ) = σ 2 erf ( F ) - 1 Q(F;\sigma)=\sigma\sqrt{2}\,\mbox{erf}~{}^{-1}(F)
  21. 0 F 1 0\leq F\leq 1
  22. erf ( ) - 1 \mbox{erf}~{}^{-1}()
  23. E ( Y ) = σ 2 / π , E(Y)=\sigma\sqrt{2/\pi},
  24. Var ( Y ) = σ 2 ( 1 - 2 π ) . \operatorname{Var}(Y)=\sigma^{2}\left(1-\frac{2}{\pi}\right).
  25. H ( Y ) = 1 2 log ( π σ 2 2 ) + 1 2 H(Y)=\frac{1}{2}\log\left(\frac{\pi\sigma^{2}}{2}\right)+\frac{1}{2}
  26. { σ 2 f ( x ) + x f ( x ) = 0 , f ( 1 ) = 2 π e - 1 2 σ 2 σ } \left\{\sigma^{2}f^{\prime}(x)+xf(x)=0,f(1)=\frac{\sqrt{\frac{2}{\pi}}e^{-% \frac{1}{2\sigma^{2}}}}{\sigma}\right\}
  27. { π f ( x ) + 2 θ 2 x f ( x ) = 0 , f ( 1 ) = 2 e - θ 2 π θ π } \left\{\pi f^{\prime}(x)+2\theta^{2}xf(x)=0,f(1)=\frac{2e^{-\frac{\theta^{2}}{% \pi}}\theta}{\pi}\right\}
  28. { x i } i = 1 n \{x_{i}\}_{i=1}^{n}
  29. σ \sigma
  30. σ ^ = 1 n i = 1 n x i 2 \hat{\sigma}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}x_{i}^{2}}
  31. Y H N ( σ ) Y\sim HN(\sigma)
  32. Y / σ χ 1 Y/\sigma\sim\chi_{1}
  33. θ = 1 σ π / 2 \theta=\frac{1}{\sigma}\sqrt{\pi/2}

Halin_graph.html

  1. T T
  2. T T
  3. T T

Hall–Littlewood_polynomials.html

  1. P λ ( x 1 , , x n ; t ) = ( i 0 j = 1 m ( i ) 1 - t 1 - t j ) w S n w ( x 1 λ 1 x n λ n i < j x i - t x j x i - x j ) , P_{\lambda}(x_{1},\ldots,x_{n};t)=\left(\prod_{i\geq 0}\prod_{j=1}^{m(i)}\frac% {1-t}{1-t^{j}}\right){\sum_{w\in S_{n}}w\left(x_{1}^{\lambda_{1}}\cdots x_{n}^% {\lambda_{n}}\prod_{i<j}\frac{x_{i}-tx_{j}}{x_{i}-x_{j}}\right)},
  2. P 42 ( x 1 , x 2 ; t ) = x 1 4 x 2 2 + x 1 2 x 2 4 + ( 1 - t ) x 1 3 x 2 3 P_{42}(x_{1},x_{2};t)=x_{1}^{4}x_{2}^{2}+x_{1}^{2}x_{2}^{4}+(1-t)x_{1}^{3}x_{2% }^{3}
  3. P λ ( x ; 1 ) = m λ ( x ) P_{\lambda}(x;1)=m_{\lambda}(x)
  4. P λ ( x ; 0 ) = s λ ( x ) P_{\lambda}(x;0)=s_{\lambda}(x)
  5. P λ ( x ; - 1 ) = P λ ( x ) P_{\lambda}(x;-1)=P_{\lambda}(x)
  6. s λ ( x ) = μ K λ μ ( t ) P μ ( x , t ) s_{\lambda}(x)=\sum_{\mu}K_{\lambda\mu}(t)P_{\mu}(x,t)
  7. K λ μ ( t ) K_{\lambda\mu}(t)
  8. t = 1 t=1
  9. K λ μ ( t ) = T S S Y T ( λ , μ ) t c h a r g e ( T ) K_{\lambda\mu}(t)=\sum_{T\in SSYT(\lambda,\mu)}t^{charge(T)}

Haloform_reaction.html

  1. Br + 2 2 OH - Br + - BrO + - H O 2 \mbox{Br}~{}_{2}+2\mbox{OH}~{}^{{}-{}}~{}\rightarrow~{}\mbox{Br}~{}^{{}-{}}+% \mbox{BrO}~{}^{{}-{}}+\mbox{H}~{}_{2}\mbox{O}~{}

Halting_problem.html

  1. h ( i , x ) = { 1 if program i halts on input x , 0 otherwise. h(i,x)=\begin{cases}1&\,\text{if }\,\text{ program }i\,\text{ halts on input }% x,\\ 0&\,\text{otherwise.}\end{cases}
  2. g ( i ) = { 0 if f ( i , i ) = 0 , undefined otherwise. g(i)=\begin{cases}0&\,\text{if }f(i,i)=0,\\ \,\text{undefined}&\,\text{otherwise.}\end{cases}

Hanna_Neumann_conjecture.html

  1. i = 1 n [ rank ( H a i K a i - 1 ) - 1 ] ( rank ( H ) - 1 ) ( rank ( K ) - 1 ) . \sum_{i=1}^{n}[{\rm rank}(H\cap a_{i}Ka_{i}^{-1})-1]\leq({\rm rank}(H)-1)({\rm rank% }(K)-1).
  2. K K
  3. F ( X ) F(X)

Hard_hexagon_model.html

  1. 𝒵 ( z ) = n z n g ( n , N ) = 1 + N z + 1 2 N ( N - 7 ) z 2 + \displaystyle\mathcal{Z}(z)=\sum_{n}z^{n}g(n,N)=1+Nz+\tfrac{1}{2}N(N-7)z^{2}+\cdots
  2. κ ( z ) = lim N 𝒵 ( z ) 1 / N = 1 + z - 3 z 2 + \kappa(z)=\lim_{N\rightarrow\infty}\mathcal{Z}(z)^{1/N}=1+z-3z^{2}+\cdots
  3. ρ = z d log ( κ ) d z = z - 7 z 2 + 58 z 3 - 519 z 4 + 4856 z 5 + . \rho=z\frac{d\log(\kappa)}{dz}=z-7z^{2}+58z^{3}-519z^{4}+4856z^{5}+\cdots.
  4. ρ 1 = 1 - z - 1 - 5 z - 2 - 34 z - 3 - 267 z - 4 - 2037 z - 5 - \rho_{1}=1-z^{-1}-5z^{-2}-34z^{-3}-267z^{-4}-2037z^{-5}-\cdots
  5. ρ 2 = ρ 3 = z - 2 + 9 z - 3 + 80 z - 4 + 965 z - 5 - . \rho_{2}=\rho_{3}=z^{-2}+9z^{-3}+80z^{-4}+965z^{-5}-\cdots.
  6. z = - x H ( x ) 5 G ( x ) 5 \displaystyle z=\frac{-xH(x)^{5}}{G(x)^{5}}
  7. κ = H ( x ) 3 Q ( x 5 ) 2 G ( x ) 2 n 1 ( 1 - x 6 n - 4 ) ( 1 - x 6 n - 3 ) 2 ( 1 - x 6 n - 2 ) ( 1 - x 6 n - 5 ) ( 1 - x 6 n - 1 ) ( 1 - x 6 n ) 2 \kappa=\frac{H(x)^{3}Q(x^{5})^{2}}{G(x)^{2}}\prod_{n\geq 1}\frac{(1-x^{6n-4})(% 1-x^{6n-3})^{2}(1-x^{6n-2})}{(1-x^{6n-5})(1-x^{6n-1})(1-x^{6n})^{2}}
  8. ρ = ρ 1 = ρ 2 = ρ 3 = - x G ( x ) H ( x 6 ) P ( x 3 ) P ( x ) \rho=\rho_{1}=\rho_{2}=\rho_{3}=\frac{-xG(x)H(x^{6})P(x^{3})}{P(x)}
  9. G ( x ) = n 1 1 ( 1 - x 5 n - 4 ) ( 1 - x 5 n - 1 ) G(x)=\prod_{n\geq 1}\frac{1}{(1-x^{5n-4})(1-x^{5n-1})}
  10. H ( x ) = n 1 1 ( 1 - x 5 n - 3 ) ( 1 - x 5 n - 2 ) H(x)=\prod_{n\geq 1}\frac{1}{(1-x^{5n-3})(1-x^{5n-2})}
  11. P ( x ) = n 1 ( 1 - x 2 n - 1 ) = Q ( x ) / Q ( x 2 ) P(x)=\prod_{n\geq 1}(1-x^{2n-1})=Q(x)/Q(x^{2})
  12. Q ( x ) = n 1 ( 1 - x n ) . Q(x)=\prod_{n\geq 1}(1-x^{n}).
  13. z = G ( x ) 5 x H ( x ) 5 \displaystyle z=\frac{G(x)^{5}}{xH(x)^{5}}
  14. κ = G ( x ) 3 Q ( x 5 ) 2 H ( x ) 2 n 1 ( 1 - x 3 n - 2 ) ( 1 - x 3 n - 1 ) ( 1 - x 3 n ) 2 \kappa=\frac{G(x)^{3}Q(x^{5})^{2}}{H(x)^{2}}\prod_{n\geq 1}\frac{(1-x^{3n-2})(% 1-x^{3n-1})}{(1-x^{3n})^{2}}
  15. ρ 1 = H ( x ) Q ( x ) ( G ( x ) Q ( x ) + x 2 H ( x 9 ) Q ( x 9 ) ) Q ( x 3 ) 2 \rho_{1}=\frac{H(x)Q(x)(G(x)Q(x)+x^{2}H(x^{9})Q(x^{9}))}{Q(x^{3})^{2}}
  16. ρ 2 = ρ 3 = x 2 H ( x ) Q ( x ) H ( x 9 ) Q ( x 9 ) Q ( x 3 ) 2 \rho_{2}=\rho_{3}=\frac{x^{2}H(x)Q(x)H(x^{9})Q(x^{9})}{Q(x^{3})^{2}}
  17. R = ρ 1 - ρ 2 = Q ( x ) Q ( x 5 ) Q ( x 3 ) 2 . R=\rho_{1}-\rho_{2}=\frac{Q(x)Q(x^{5})}{Q(x^{3})^{2}}.

Hardhead_catfish.html

  1. W = c L b W=cL^{b}\!\,

Hardy_Cross_method.html

  1. h f = k Q n h_{f}=k\cdot Q^{n}
  2. h f = L 10.67 Q 1.85 C 1.85 d 4.87 h_{f}=L\cdot\frac{10.67\quad Q^{1.85}}{C^{1.85}\quad d^{4.87}}
  3. L 10.67 C 1.85 d 4.87 L\cdot\frac{10.67}{C^{1.85}\quad d^{4.87}}
  4. h f = 8 f L Q 2 g π 2 d 5 h_{f}=\frac{8fLQ^{2}}{g\pi^{2}d^{5}}
  5. L 8 f g π 2 d 5 L\cdot\frac{8f}{g\pi^{2}d^{5}}
  6. V = K I V=K\cdot I
  7. h f = k Q n h_{f}=k\cdot Q^{n}
  8. Σ r Q n \Sigma rQ^{n}
  9. Δ Q \Delta Q
  10. Q = Q 0 + Δ Q Q=Q_{0}+\Delta Q
  11. Σ r ( Q 0 + Δ Q ) n = 0 \Sigma r(Q_{0}+\Delta Q)^{n}=0
  12. Σ r ( Q 0 + Δ Q ) n \Sigma r(Q_{0}+\Delta Q)^{n}
  13. Σ r ( Q 0 + Δ Q ) n = Σ r ( Q 0 n + n Q 0 n - 1 Δ Q + ) = 0 \Sigma r(Q_{0}+\Delta Q)^{n}=\Sigma r(Q_{0}^{n}+nQ_{0}^{n-1}\Delta Q+...)=0
  14. Δ Q \Delta Q
  15. Q 0 Q_{0}
  16. Σ r ( Q 0 n + n Q 0 n - 1 Δ Q ) = 0 \Sigma r(Q_{0}^{n}+nQ_{0}^{n-1}\Delta Q)=0
  17. Δ Q \Delta Q
  18. Σ r Q 0 n = - Σ n r Q 0 n - 1 Δ Q \Sigma rQ_{0}^{n}=-\Sigma nrQ_{0}^{n-1}\Delta Q
  19. Δ Q = - Σ r Q 0 n Σ n r Q 0 n - 1 \Delta Q=-\frac{\Sigma rQ_{0}^{n}}{\Sigma nrQ_{0}^{n-1}}
  20. Δ Q = - Σ r Q 0 n Σ n r Q 0 n - 1 \Delta Q=-\frac{\Sigma rQ_{0}^{n}}{\Sigma nrQ_{0}^{n-1}}
  21. Δ Q \Delta Q
  22. h f = r Q n h_{f}=rQ^{n}
  23. Σ r Q n \Sigma rQ^{n}
  24. Σ n r Q n - 1 \Sigma nrQ^{n-1}
  25. Σ r Q n Σ n r Q n - 1 \frac{\Sigma rQ^{n}}{\Sigma nrQ^{n-1}}
  26. r Q 2 rQ^{2}
  27. r Q 2 rQ^{2}
  28. 25 - 125 = - 100 25-125=-100
  29. 125 - 25 = 100 125-25=100
  30. Σ n r Q n - 1 \Sigma nrQ^{n-1}
  31. Σ r Q n Σ n r Q n - 1 \frac{\Sigma rQ^{n}}{\Sigma nrQ^{n-1}}
  32. - 100 / 60 = - 1.66 -100/60=-1.66
  33. 100 / 60 = 1.66 100/60=1.66
  34. r Q 2 rQ^{2}
  35. r Q 2 rQ^{2}

Harmonics_(electrical_power).html

  1. THD = V 2 2 + V 3 2 + V 4 2 + + V n 2 V 1 \mathrm{THD}=\frac{\sqrt{V_{2}^{2}+V_{3}^{2}+V_{4}^{2}+\cdots+V_{n}^{2}}}{V_{1}}

Harry_Kesten.html

  1. d d
  2. 2 d - 1 2\sqrt{d-1}

Hartree_equation.html

  1. v ( r ) v(r)
  2. P ( r ) / r P(r)/r
  3. \ell
  4. ψ = ( 1 / r ) P ( r ) S ( θ , ϕ ) \psi=(1/r)P(r)S_{\ell}(\theta,\phi)
  5. d 2 P ( r ) / d r 2 + [ 2 ( E - v ( r ) ) - ( + 1 ) / r 2 ] P ( r ) = 0. d^{2}P(r)/dr^{2}+[2(E-v(r))-\ell(\ell+1)/r^{2}]P(r)=0.
  6. i t u + 2 u = V ( u ) u i\,\partial_{t}u+\nabla^{2}u=V(u)u
  7. d + 1 \mathbb{R}^{d+1}
  8. V ( u ) = ± | x | - n * | u | 2 V(u)=\pm|x|^{-n}*|u|^{2}
  9. 0 < n < d 0<n<d

Hawking_energy.html

  1. ( 3 , g a b ) (\mathcal{M}^{3},g_{ab})
  2. Σ 3 \Sigma\subset\mathcal{M}^{3}
  3. m H ( Σ ) m_{H}(\Sigma)
  4. Σ \Sigma
  5. m H ( Σ ) := Area Σ 16 π ( 1 - 1 16 π Σ H 2 d a ) , m_{H}(\Sigma):=\sqrt{\frac{\,\text{Area}\,\Sigma}{16\pi}}\left(1-\frac{1}{16% \pi}\int_{\Sigma}H^{2}da\right),
  6. H H
  7. Σ \Sigma
  8. S r S_{r}
  9. m m
  10. 3 \mathcal{M}^{3}
  11. Σ \Sigma
  12. Σ \Sigma
  13. Σ t \Sigma_{t}
  14. d x d t = 1 H ν ( x ) , \frac{dx}{dt}=\frac{1}{H}\nu(x),
  15. H H
  16. Σ t \Sigma_{t}
  17. ν \nu
  18. d d t m H ( Σ t ) 0. \frac{d}{dt}m_{H}(\Sigma_{t})\geq 0.

HD_181433.html

  1. Distance in parsecs = 1000 parallax in milliarcseconds \textstyle\mathrm{Distance\ in\ parsecs}=\frac{1000}{\mathrm{parallax\ in\ % milliarcseconds}}

HD_40307.html

  1. M V = 4.83 \scriptstyle M_{V_{\odot}}=4.83
  2. L V L V = 10 0.4 ( M V - M V ) \scriptstyle\frac{L_{V_{\ast}}}{L_{V_{\odot}}}=10^{0.4\left(M_{V_{\odot}}-M_{V% _{\ast}}\right)}

Heat.html

  1. Q ˙ \dot{Q}
  2. Δ U ΔU
  3. Q Q
  4. W W
  5. Δ U = Q - W . \Delta U=Q-W\,.
  6. Δ U ΔU
  7. Q = Δ U + W . Q=\Delta U+W.
  8. Q = Δ U + W boundary + W isochoric . Q=\Delta U+W\text{boundary}+W\text{isochoric}.
  9. U U
  10. d U dU
  11. d d
  12. δ Q δQ
  13. δ W δW
  14. δ δ
  15. T T
  16. δ Q δQ
  17. T T
  18. d S = δ Q T , \mathrm{d}S=\frac{\delta Q}{T},
  19. S S
  20. P P
  21. δ W δW
  22. P P
  23. d V = δ W P , \mathrm{d}V=\frac{\delta W}{P},
  24. V V
  25. d U = T d S - P d V . \mathrm{d}U=T\mathrm{d}S-P\mathrm{d}V.
  26. U ( S , V ) U(S,V)
  27. S S
  28. V V
  29. U = U ( S , V ) . U=U(S,V).
  30. V V
  31. T d S = d U ( V constant) T\mathrm{d}S=\mathrm{d}U\,\,\,\,\,\,\,\,\,\,\,\,(V\,\,\,\text{constant)}
  32. P P
  33. T d S = d H ( P constant) T\mathrm{d}S=\mathrm{d}H\,\,\,\,\,\,\,\,\,\,\,\,(P\,\,\,\text{constant)}
  34. H H
  35. H = U + P V . H=U+PV.
  36. H ( S , P ) H(S,P)
  37. S S
  38. P P
  39. H = H ( S , P ) . H=H(S,P).
  40. Q Q
  41. W W
  42. Δ H = Δ U + Δ ( P V ) . \Delta H=\Delta U+\Delta(PV)\,.
  43. Δ P = 0 ΔP=0
  44. W W
  45. W = P Δ V W=PΔV
  46. Δ U = Q - W = Q - P Δ V and Δ ( P V ) = P Δ V . \Delta U=Q-W=Q-P\,\Delta V\,\text{ and }\Delta(PV)=P\,\Delta V\,.
  47. Δ H = Q - P Δ V + P Δ V \Delta H=Q-P\,\Delta V+P\,\Delta V
  48. = Q at constant pressure. =Q\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,% \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{at constant pressure.}
  49. S a n d P SandP
  50. H H
  51. Δ H = S 1 S 2 ( H S ) P d S + P 1 P 2 ( H P ) S d P \Delta H=\int_{S_{1}}^{S_{2}}\left(\frac{\partial H}{\partial S}\right)_{P}% \mathrm{d}S+\int_{P_{1}}^{P_{2}}\left(\frac{\partial H}{\partial P}\right)_{S}% \mathrm{d}P
  52. = S 1 S 2 ( H S ) P d S at constant pressure. =\int_{S_{1}}^{S_{2}}\left(\frac{\partial H}{\partial S}\right)_{P}\mathrm{d}S% \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{at constant pressure.}
  53. T ( S , P ) T(S,P)
  54. ( H S ) P T ( S , P ) . \left(\frac{\partial H}{\partial S}\right)_{P}\equiv T(S,P)\,.
  55. Δ H = S 1 S 2 T ( S , P ) d S at constant pressure. \Delta H=\int_{S_{1}}^{S_{2}}T(S,P)\mathrm{d}S\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,% \,\,\,\,\,\text{at constant pressure.}
  56. Q T . {}\frac{Q}{T}.
  57. Δ S = Q T ( 1 ) \Delta S=\frac{Q}{T}\,\,\,\,\,\,\,\,\,\,\,\,\,(1)
  58. Δ S ΔS
  59. Δ S ΔS′
  60. Δ S −ΔS′
  61. Δ S ΔS′′
  62. Δ S = Δ S + Δ S ′′ . \Delta S=\Delta S^{\prime}+\Delta S^{\prime\prime}.
  63. Δ S system = Δ S compensated + Δ S uncompensated with Δ S compensated = - Δ S surroundings . \Delta S_{\mathrm{system}}=\Delta S_{\mathrm{compensated}}+\Delta S_{\mathrm{% uncompensated}}\,\,\,\,\,\text{with}\,\,\,\,\Delta S_{\mathrm{compensated}}=-% \Delta S_{\mathrm{surroundings}}.
  64. Δ S overall = Δ S + Δ S ′′ - Δ S = Δ S ′′ . \Delta S_{\mathrm{overall}}=\Delta S^{\prime}+\Delta S^{\prime\prime}-\Delta S% ^{\prime}=\Delta S^{\prime\prime}.
  65. Δ S overall = Δ S compensated + Δ S uncompensated + Δ S surroundings = Δ S uncompensated . \Delta S_{\mathrm{overall}}=\Delta S_{\mathrm{compensated}}+\Delta S_{\mathrm{% uncompensated}}+\Delta S_{\mathrm{surroundings}}=\Delta S_{\mathrm{% uncompensated}}.
  66. Δ S ΔS′
  67. Δ S overall > 0. \Delta S_{\mathrm{overall}}>0.
  68. T T
  69. δ Q δQ
  70. T d S TdS
  71. d S dS
  72. T d S = δ Q . T\,\mathrm{d}S=\delta Q.
  73. T d S = δ Q + T d S uncompensated > δ Q . T\,\mathrm{d}S=\delta Q+T\,\mathrm{d}S_{\mathrm{uncompensated}}>\delta Q.
  74. T d S δ Q ( second law ) . T\,\mathrm{d}S\geq\delta Q\quad{\rm{(second\,\,law)}}\,.
  75. T T
  76. d S = d S e + d S i , \mathrm{d}S=\mathrm{d}S_{\mathrm{e}}+\mathrm{d}S_{\mathrm{i}}\,,
  77. δ Q = T d S e and d S i d S uncompensated . \delta Q=T\,\mathrm{d}S_{\mathrm{e}}\,\,\,\,\,\,\text{and}\,\,\,\,\,\mathrm{d}% S_{\mathrm{i}}\equiv\mathrm{d}S_{\mathrm{uncompensated}}.
  78. d S i > 0. \mathrm{d}S_{\mathrm{i}}>0.
  79. U < s u b > X U<sub>X
  80. V V

Heaviside_cover-up_method.html

  1. x 2 + m x + n ( x - a ) ( x - b ) ( x - c ) = A ( x - a ) + B ( x - b ) + C ( x - c ) \frac{\ell x^{2}+mx+n}{(x-a)(x-b)(x-c)}=\frac{A}{(x-a)}+\frac{B}{(x-b)}+\frac{% C}{(x-c)}
  2. A = a 2 + m a + n ( a - b ) ( a - c ) ; A=\frac{\ell a^{2}+ma+n}{(a-b)(a-c)};
  3. B = b 2 + m b + n ( b - c ) ( b - a ) ; B=\frac{\ell b^{2}+mb+n}{(b-c)(b-a)};
  4. C = c 2 + m c + n ( c - a ) ( c - b ) . C=\frac{\ell c^{2}+mc+n}{(c-a)(c-b)}.
  5. 3 x 2 + 12 x + 11 ( x + 1 ) ( x + 2 ) ( x + 3 ) = A x + 1 + B x + 2 + C x + 3 \frac{3x^{2}+12x+11}{(x+1)(x+2)(x+3)}=\frac{A}{x+1}+\frac{B}{x+2}+\frac{C}{x+3}
  6. 3 x 2 + 12 x + 11 ( x + 2 ) ( x + 3 ) = 3 - 12 + 11 ( 1 ) ( 2 ) = 2 2 = 1 = A . \frac{3x^{2}+12x+11}{(x+2)(x+3)}=\frac{3-12+11}{(1)(2)}=\frac{2}{2}=1=A.
  7. 3 x 2 + 12 x + 11 ( x + 1 ) ( x + 3 ) = 12 - 24 + 11 ( - 1 ) ( 1 ) = - 1 ( - 1 ) = + 1 = B . \frac{3x^{2}+12x+11}{(x+1)(x+3)}=\frac{12-24+11}{(-1)(1)}=\frac{-1}{(-1)}=+1=B.
  8. 3 x 2 + 12 x + 11 ( x + 1 ) ( x + 2 ) = 27 - 36 + 11 ( - 2 ) ( - 1 ) = 2 ( + 2 ) = + 1 = C . \frac{3x^{2}+12x+11}{(x+1)(x+2)}=\frac{27-36+11}{(-2)(-1)}=\frac{2}{(+2)}=+1=C.
  9. 3 x 2 + 12 x + 11 ( x + 1 ) ( x + 2 ) ( x + 3 ) = 1 x + 1 + 1 x + 2 + 1 x + 3 \frac{3x^{2}+12x+11}{(x+1)(x+2)(x+3)}=\frac{1}{x+1}+\frac{1}{x+2}+\frac{1}{x+3}
  10. 3 x + 5 ( 1 - 2 x ) 2 = A ( 1 - 2 x ) 2 + B 1 - 2 x \frac{3x+5}{(1-2x)^{2}}=\frac{A}{(1-2x)^{2}}+\frac{B}{1-2x}
  11. ( 1 - 2 x ) (1-2x)
  12. ( 1 - 2 x ) (1-2x)
  13. 3 x + 5 = A + B ( 1 - 2 x ) 3x+5=A+B(1-2x)
  14. n n
  15. n n
  16. A k A_{k}
  17. ( 1 - 2 x ) k (1-2x)^{k}
  18. k k
  19. n n
  20. A n A_{n}
  21. A 1 A_{1}
  22. n = 2 n=2
  23. A = A 2 A=A_{2}
  24. B = A 1 B=A_{1}
  25. A A
  26. A A
  27. 1 - 2 x = 0 1-2x=0
  28. x x
  29. A A
  30. x = 1 / 2 x=1/2
  31. x = 1 / 2 x=1/2
  32. 3 ( 1 2 ) + 5 = A + B ( 0 ) 3\left(\frac{1}{2}\right)+5=A+B(0)
  33. A = 3 2 + 5 = 13 2 A=\frac{3}{2}+5=\frac{13}{2}
  34. B B
  35. 3 x + 5 = A + B ( 1 - 2 x ) 3x+5=A+B(1-2x)
  36. x x
  37. x x
  38. B B
  39. A A
  40. A = 13 / 2 A=13/2
  41. B B
  42. x = 0 x=0
  43. A = 13 / 2 A=13/2
  44. B B
  45. 3 x + 5 \displaystyle 3x+5
  46. x = 1 x=1
  47. B B
  48. 3 x + 5 \displaystyle 3x+5
  49. x = - 1 x=-1
  50. B B
  51. 3 x + 5 \displaystyle 3x+5
  52. 3 x + 5 ( 1 - 2 x ) 2 = 13 / 2 ( 1 - 2 x ) 2 + - 3 / 2 ( 1 - 2 x ) , \frac{3x+5}{(1-2x)^{2}}=\frac{13/2}{(1-2x)^{2}}+\frac{-3/2}{(1-2x)},
  53. 3 x + 5 ( 1 - 2 x ) 2 = 13 2 ( 1 - 2 x ) 2 - 3 2 ( 1 - 2 x ) \frac{3x+5}{(1-2x)^{2}}=\frac{13}{2(1-2x)^{2}}-\frac{3}{2(1-2x)}

Heckman_correction.html

  1. Prob ( D = 1 | Z ) = Φ ( Z γ ) , \operatorname{Prob}(D=1|Z)=\Phi(Z\gamma),\,
  2. γ \gamma
  3. w * = X β + u w^{*}=X\beta+u\,
  4. w * w^{*}
  5. E [ w | X , D = 1 ] = X β + E [ u | X , D = 1 ] . E[w|X,D=1]=X\beta+E[u|X,D=1].\,
  6. E [ w | X , D = 1 ] = X β + ρ σ u λ ( Z γ ) , E[w|X,D=1]=X\beta+\rho\sigma_{u}\lambda(Z\gamma),\,
  7. ε \varepsilon
  8. u u
  9. λ \lambda
  10. Z γ Z\gamma
  11. λ \lambda
  12. γ \gamma
  13. λ \lambda
  14. σ u > 0 \sigma_{u}>0
  15. λ \lambda
  16. ρ = 0 \rho=0
  17. λ \lambda

Hedonic_index.html

  1. n t n_{t}
  2. ( z 1 i t , , z k i t ) T (z_{1it},...,z_{kit}\;)^{T}
  3. P i t = c 0 t + j = 1 k c i t z j i t + ξ i t , P_{it}=c_{0t}+\sum_{j=1}^{k}c_{it}z_{jit}+\xi_{it},
  4. c i t c_{it}
  5. ξ i t \xi_{it}
  6. N ( 0 , σ 2 ) N(0,\sigma^{2})
  7. c i t c_{it}
  8. t = 0 T P ^ t + 1 ( z τ ) P ^ t ( z τ ) , \prod_{t=0}^{T}\frac{\widehat{P}_{t+1}(z^{\tau})}{\widehat{P}_{t}(z^{\tau})},
  9. P ^ t + 1 ( z τ ) \widehat{P}_{t+1}(z^{\tau})
  10. τ : z τ \tau:\ z^{\tau}
  11. H P I ( 0 , T ) = P ^ T ( z τ ) P ^ 0 ( z τ ) . HPI(0,T)=\frac{\widehat{P}_{T}(z^{\tau})}{\widehat{P}_{0}(z^{\tau})}.
  12. z τ \ z^{\tau}
  13. z τ \ z^{\tau}
  14. t : z t t:\ z^{t}
  15. z τ \ z^{\tau}
  16. t + 1 : z t + 1 t+1:\ z^{t+1}
  17. z 0 \ z^{0}
  18. P ^ η ( z t + 1 ) P ^ η ( z t ) , \frac{\widehat{P}_{\eta}(z^{t+1})}{\widehat{P}_{\eta}(z^{t})},
  19. η \ \eta
  20. t = 0 T P ^ η ( z t + 1 ) P ^ η ( z t ) \prod_{t=0}^{T}\frac{\widehat{P}_{\eta}(z^{t+1})}{\widehat{P}_{\eta}(z^{t})}
  21. P ^ η ( z T ) P ^ η ( z 0 ) . \frac{\widehat{P}_{\eta}(z^{T})}{\widehat{P}_{\eta}(z^{0})}.

Hele-Shaw_flow.html

  1. x x
  2. y y
  3. z z
  4. 2 H 2H
  5. z = ± H z=\pm H
  6. H 0 , H\rightarrow 0,\,
  7. z z
  8. 𝐮 = p z 2 - H 2 2 μ {\mathbf{u}}={\mathbf{\nabla}}p\frac{z^{2}-H^{2}}{2\mu}\,
  9. u {u}
  10. p ( x , y , t ) p(x,y,t)
  11. μ \mu
  12. z z
  13. z z
  14. x x
  15. y y
  16. z z
  17. 2 p x 2 + 2 p y 2 = 0. \frac{\partial^{2}p}{\partial x^{2}}+\frac{\partial^{2}p}{\partial y^{2}}=0.
  18. p n ^ = 0 {\mathbf{\nabla}}p\cdot\hat{n}=0\,
  19. n ^ \hat{n}

Heliocentric_Julian_Day.html

  1. r \vec{r}
  2. n ^ \hat{n}
  3. c c
  4. H J D = J D + r n ^ c HJD=JD+\frac{\vec{r}\cdot\hat{n}}{c}
  5. α \alpha
  6. δ \delta
  7. \odot
  8. H J D = J D - r c [ s i n ( δ ) s i n ( δ ) + c o s ( δ ) c o s ( δ ) c o s ( α - α ) ] HJD=JD-\frac{r}{c}\cdot[sin(\delta)\cdot sin(\delta_{\odot})+cos(\delta)\cdot cos% (\delta_{\odot})\cdot cos(\alpha-\alpha_{\odot})]
  9. r r
  10. H J D = J D - r c c o s ( β ) c o s ( λ - λ ) HJD=JD-\frac{r}{c}\cdot cos(\beta)\cdot cos(\lambda-\lambda_{\odot})

Helium_atom.html

  1. H ψ ( r 1 , r 2 ) = [ i = 1 , 2 ( - 2 2 μ r i 2 - Z e 2 4 π ϵ 0 r i ) - 2 M r 1 r 2 + e 2 4 π ϵ 0 r 12 ] ψ ( r 1 , r 2 ) H\psi(\vec{r}_{1},\,\vec{r}_{2})=\Bigg[\sum_{i=1,2}\Bigg(-\frac{\hbar^{2}}{2% \mu}\nabla^{2}_{r_{i}}-\frac{Ze^{2}}{4\pi\epsilon_{0}r_{i}}\Bigg)-\frac{\hbar^% {2}}{M}\nabla_{r_{1}}\cdot\nabla_{r_{2}}+\frac{e^{2}}{4\pi\epsilon_{0}r_{12}}% \Bigg]\psi(\vec{r}_{1},\,\vec{r}_{2})
  2. μ = m M m + M \mu=\frac{mM}{m+M}
  3. r 1 \vec{r}_{1}
  4. r 2 \vec{r}_{2}
  5. r 12 = | r 1 - r 2 | r_{12}=|\vec{r_{1}}-\vec{r_{2}}|
  6. Z Z
  7. M = M=\infty
  8. μ = m \mu=m
  9. 2 M r 1 r 2 \frac{\hbar^{2}}{M}\nabla_{r_{1}}\cdot\nabla_{r_{2}}
  10. H ψ ( r 1 , r 2 ) = [ - 1 2 r 1 2 - 1 2 r 2 2 - Z r 1 - Z r 2 + 1 r 12 ] ψ ( r 1 , r 2 ) . H\psi(\vec{r}_{1},\,\vec{r}_{2})=\Bigg[-\frac{1}{2}\nabla^{2}_{r_{1}}-\frac{1}% {2}\nabla^{2}_{r_{2}}-\frac{Z}{r_{1}}-\frac{Z}{r_{2}}+\frac{1}{r_{12}}\Bigg]% \psi(\vec{r}_{1},\,\vec{r}_{2}).
  11. ψ 0 ( r 1 , r 2 ) \psi_{0}(\vec{r}_{1},\,\vec{r}_{2})
  12. H = i = 1 2 h ( i ) = H 0 + H H=\sum_{i=1}^{2}h(i)=H_{0}+H^{\prime}
  13. H 0 = - 1 2 r 1 2 - 1 2 r 2 2 - Z r 1 - Z r 2 H_{0}=-\frac{1}{2}\nabla_{r_{1}}^{2}-\frac{1}{2}\nabla_{r_{2}}^{2}-\frac{Z}{r_% {1}}-\frac{Z}{r_{2}}
  14. H = 1 r 12 H^{\prime}=\frac{1}{r_{12}}
  15. H 0 = h ^ 1 + h ^ 2 H_{0}=\hat{h}_{1}+\hat{h}_{2}
  16. h ^ i = - 1 2 r i 2 - Z r i , i = 1 , 2 \hat{h}_{i}=-\frac{1}{2}\nabla_{r_{i}}^{2}-\frac{Z}{r_{i}},i=1,2
  17. ψ n , l , m ( r i ) \psi_{n,l,m}(\vec{r}_{i})
  18. h ^ i ψ n , l , m ( r i ) = E n 1 ψ n , l , m ( r i ) \hat{h}_{i}\psi_{n,l,m}(\vec{r_{i}})=E_{n_{1}}\psi_{n,l,m}(\vec{r_{i}})
  19. E n 1 = - 1 2 Z 2 n i 2 in a.u. E_{n_{1}}=-\frac{1}{2}\frac{Z^{2}}{n_{i}^{2}}\,\text{ in a.u.}
  20. H 0 ψ ( 0 ) ( r 1 , r 2 ) = E ( 0 ) ψ ( 0 ) ( r 1 , r 2 ) H_{0}\psi^{(0)}(\vec{r}_{1},\vec{r}_{2})=E^{(0)}\psi^{(0)}(\vec{r}_{1},\vec{r}% _{2})
  21. ψ ( 0 ) ( r 1 , r 2 ) = ψ n 1 , l 1 , m 1 ( r 1 ) ψ n 2 , l 2 , m 2 ( r 2 ) \psi^{(0)}(\vec{r}_{1},\vec{r}_{2})=\psi_{n_{1},l_{1},m_{1}}(\vec{r}_{1})\psi_% {n_{2},l_{2},m_{2}}(\vec{r}_{2})
  22. E n 1 , n 2 ( 0 ) = E n 1 + E n 2 = - Z 2 2 [ 1 n 1 2 + 1 n 2 2 ] E^{(0)}_{n_{1},n_{2}}=E_{n_{1}}+E_{n_{2}}=-\frac{Z^{2}}{2}\Bigg[\frac{1}{n_{1}% ^{2}}+\frac{1}{n_{2}^{2}}\Bigg]
  23. ψ ( 0 ) ( r 2 , r 1 ) = ψ n 2 , l 2 , m 2 ( r 1 ) ψ n 1 , l 1 , m 1 ( r 2 ) \psi^{(0)}(\vec{r}_{2},\vec{r}_{1})=\psi_{n_{2},l_{2},m_{2}}(\vec{r}_{1})\psi_% {n_{1},l_{1},m_{1}}(\vec{r}_{2})
  24. E n 1 , n 2 ( 0 ) E^{(0)}_{n_{1},n_{2}}
  25. r 1 \vec{r}_{1}
  26. r 2 \vec{r}_{2}
  27. ψ ± ( 0 ) ( r 1 , r 2 ) = 1 2 [ ψ n 1 , l 1 , m 1 ( r 1 ) ψ n 2 , l 2 , m 2 ( r 2 ) ± ψ n 2 , l 2 , m 2 ( r 1 ) ψ n 1 , l 1 , m 1 ( r 2 ) ] \psi^{(0)}_{\pm}(\vec{r}_{1},\vec{r}_{2})=\frac{1}{\sqrt{2}}[\psi_{n_{1},l_{1}% ,m_{1}}(\vec{r}_{1})\psi_{n_{2},l_{2},m_{2}}(\vec{r}_{2})\pm\psi_{n_{2},l_{2},% m_{2}}(\vec{r}_{1})\psi_{n_{1},l_{1},m_{1}}(\vec{r}_{2})]
  28. 1 2 \frac{1}{\sqrt{2}}
  29. ψ ± ( 0 ) \psi^{(0)}_{\pm}
  30. n 1 = n 2 = 1 , l 1 = l 2 = 0 , m 1 = m 2 = 0 n_{1}=n_{2}=1,\,l_{1}=l_{2}=0,\,m_{1}=m_{2}=0
  31. ψ - ( 0 ) \psi^{(0)}_{-}
  32. ψ 0 ( 0 ) ( r 1 , r 2 ) = ψ 1 ( r 1 ) ψ 1 ( r 2 ) = Z 3 π e - Z ( r 1 + r 2 ) \psi^{(0)}_{0}(\vec{r}_{1},\vec{r}_{2})=\psi_{1}(\vec{r_{1}})\psi_{1}(\vec{r_{% 2}})=\frac{Z^{3}}{\pi}e^{-Z(r_{1}+r_{2})}
  33. ψ 1 \psi_{1}
  34. ψ 2 \psi_{2}
  35. a 0 {a_{0}}
  36. ψ ( r ) = 1 π Z 3 2 e - Z r \psi_{(}r)=\frac{1}{\sqrt{\pi}}Z^{\frac{3}{2}}\ e^{-{\textstyle Zr}}\;
  37. E 0 ( 0 ) = E n 1 = 1 , n 2 = 1 ( 0 ) = - Z 2 a.u. E^{(0)}_{0}=E^{(0)}_{n_{1}=1,\,n_{2}=1}=-Z^{2}\,\text{ a.u.}
  38. = 0 ( 0 ) - 4 {}^{(0)}_{0}=-4
  39. = P ( 0 ) 2 {}_{P}^{(0)}=2
  40. 54.4 \simeq 54.4
  41. = 0 - 2.90 {}_{0}=-2.90
  42. - 79.0 \simeq-79.0
  43. = p .90 {}_{p}=.90
  44. 24.6 \simeq 24.6
  45. H = H 0 ¯ + H ¯ H=\bar{H_{0}}+\bar{H^{\prime}}
  46. H 0 ¯ = - 1 2 r 1 2 + V ( r 1 ) - 1 2 r 2 2 + V ( r 2 ) \bar{H_{0}}=-\frac{1}{2}\nabla^{2}_{r_{1}}+V(r_{1})-\frac{1}{2}\nabla^{2}_{r_{% 2}}+V(r_{2})
  47. H ¯ = 1 r 12 - Z r 1 - V ( r 1 ) - Z r 2 - V ( r 2 ) \bar{H^{\prime}}=\frac{1}{r_{12}}-\frac{Z}{r_{1}}-V(r_{1})-\frac{Z}{r_{2}}-V(r% _{2})
  48. H ¯ \bar{H^{\prime}}
  49. V ( r ) = - Z - S r = - Z e r V(r)=-\frac{Z-S}{r}=-\frac{Z_{e}}{r}
  50. E 0 = - ( Z - S ) 2 = - Z e 2 E_{0}=-(Z-S)^{2}=-Z_{e}^{2}
  51. ψ 0 ( r 1 r 2 ) = Z e 3 π e - Z e ( r 1 + r 2 ) \psi_{0}(r_{1}\,r_{2})=\frac{Z_{e}^{3}}{\pi}e^{-Z_{e}(r_{1}+r_{2})}
  52. 1 3 \frac{1}{3}
  53. ρ ( r ) , r ϵ \reals 3 \rho(\vec{r}),\,r\,\epsilon\,\reals^{3}
  54. ξ = 3 5 γ \reals 3 ρ 5 / 3 ( r ) d 3 r + \reals 3 V ( r ) ρ ( r ) d 3 r + e 2 2 \reals 3 ρ ( r ) ρ ( r ) | r - r | d 3 r d 3 r \xi=\frac{3}{5}\gamma\int_{\reals^{3}}\rho^{5/3}(\vec{r})d^{3}\vec{r}\,+\int_{% \reals^{3}}V(\vec{r})\rho(\vec{r})d^{3}\vec{r}\,+\frac{e^{2}}{2}\int_{\reals^{% 3}}\frac{\rho(\vec{r})\rho(\vec{r^{\prime}})}{|\vec{r}-\vec{r^{\prime}}|}\,d^{% 3}\vec{r}d^{3}\vec{r^{\prime}}
  55. γ = ( 3 π 2 ) 2 / 3 2 2 m \gamma=(3\pi^{2})^{2/3}\frac{\hbar^{2}}{2m}
  56. \reals 3 ρ = N \int_{\reals^{3}}\rho=N
  57. ρ ξ \rho\rightarrow\xi
  58. ρ ( x ) \rho(\vec{x})
  59. V ( r ) V(\vec{r})
  60. H = i = 1 N [ - 2 2 m i 2 + V ( r i ) ] + e 2 ρ ( r ) | r - r | d 3 r H=\sum_{i=1}^{N}\Bigg[-\frac{\hbar^{2}}{2m}\nabla_{i}^{2}+V(\vec{r_{i}})\Bigg]% +\int\frac{e^{2}\rho(\vec{r^{\prime}})}{|\vec{r}-\vec{r^{\prime}}|}d^{3}r^{\prime}
  61. H = - 2 2 m ( 1 2 + 2 2 ) + V ( r 1 , r 2 ) + e 2 ρ ( r ) | r - r | d 3 r H=-\frac{\hbar^{2}}{2m}(\nabla_{1}^{2}+\nabla_{2}^{2})+V(\vec{r_{1}},\,\vec{r_% {2}})+\int\frac{e^{2}\rho(\vec{r^{\prime}})}{|\vec{r}-\vec{r^{\prime}}|}d^{3}r% ^{\prime}
  62. e 2 ρ ( r ) | r - r | d 3 r = e 2 4 π ϵ 0 1 | r 1 - r 2 | , and V ( r 1 , r 2 ) = e 2 4 π ϵ 0 [ 2 r 1 + 2 r 2 ] \int\frac{e^{2}\rho(\vec{r^{\prime}})}{|\vec{r}-\vec{r^{\prime}}|}d^{3}r^{% \prime}=\frac{e^{2}}{4\pi\epsilon_{0}}\frac{1}{|\vec{r}_{1}-\vec{r}_{2}|},\,\,% \text{ and }\,V(\vec{r_{1}},\,\vec{r_{2}})=\frac{e^{2}}{4\pi\epsilon_{0}}\Bigg% [\frac{2}{r_{1}}+\frac{2}{r_{2}}\Bigg]
  63. H = - 2 2 m ( 1 2 + 2 2 ) + e 2 4 π ϵ 0 [ 2 r 1 + 2 r 2 - 1 | r 1 - r 2 | ] H=-\frac{\hbar^{2}}{2m}(\nabla_{1}^{2}+\nabla_{2}^{2})+\frac{e^{2}}{4\pi% \epsilon_{0}}\Bigg[\frac{2}{r_{1}}+\frac{2}{r_{2}}-\frac{1}{|\vec{r}_{1}-\vec{% r}_{2}|}\Bigg]
  64. ψ 0 ( r 1 , r 2 ) = 8 π a 3 e - 2 ( r 1 + r 2 ) / a \psi_{0}(\vec{r}_{1},\,\vec{r}_{2})=\frac{8}{\pi a^{3}}e^{-2(r_{1}+r_{2})/a}
  65. H = 8 E 1 + V e e = 8 E 1 + ( e 2 4 π ϵ 0 ) ( 8 π a 3 ) 2 e - 4 ( r 1 + r 2 ) / a | r 1 - r 2 | d 3 r 1 d 3 r 2 \langle H\rangle=8E_{1}+\langle V_{ee}\rangle=8E_{1}+\Bigg(\frac{e^{2}}{4\pi% \epsilon_{0}}\Bigg)\Bigg(\frac{8}{\pi a^{3}}\Bigg)^{2}\int\frac{e^{-4(r_{1}+r_% {2})/a}}{|\vec{r_{1}}-\vec{r_{2}}|}\,d^{3}\vec{r}_{1}\,d^{3}\vec{r}_{2}
  66. H = 8 E 1 + 5 4 a ( e 2 4 π ϵ 0 ) = 8 E 1 - 5 2 E 1 = - 109 + 34 = - 75 e V \langle H\rangle=8E_{1}+\frac{5}{4a}\Bigg(\frac{e^{2}}{4\pi\epsilon_{0}}\Bigg)% =8E_{1}-\frac{5}{2}E_{1}=-109+34=-75eV
  67. ψ ( r 1 , r 2 ) = Z 3 π a 3 e - Z ( r 1 + r 2 ) / a \psi(\vec{r}_{1},\vec{r}_{2})=\frac{Z^{3}}{\pi a^{3}}e^{-Z(r_{1}+r_{2})/a}
  68. H = 2 Z 2 E 1 + 2 ( Z - 2 ) ( e 2 4 π ϵ 0 ) 1 r + V e e \langle H\rangle=2Z^{2}E_{1}+2(Z-2)\Bigg(\frac{e^{2}}{4\pi\epsilon_{0}}\Bigg)% \langle\frac{1}{r}\rangle+\langle V_{ee}\rangle
  69. 1 r \frac{1}{r}
  70. H = [ - 2 Z 2 + 27 4 Z ] E 1 \langle H\rangle=[-2Z^{2}+\frac{27}{4}Z]E_{1}
  71. d d Z ( [ - 2 Z 2 + 27 4 Z ] E 1 ) = 0 \frac{d}{dZ}\Bigg([-2Z^{2}+\frac{27}{4}Z]E_{1}\Bigg)=0
  72. Z = 1.69 Z=1.69
  73. 1 2 ( 3 2 ) 6 E 1 = - 77.5 e V \frac{1}{2}\Bigg(\frac{3}{2}\Bigg)^{6}E_{1}=-77.5eV
  74. - 49 17 ( 1 + α ) = - 2.903386486 a . u . = - 79.005153 e V -\frac{49}{17}\cdot(1+\alpha)=-2.903386486a.u.=-79.005153eV
  75. α \alpha

Helly_metric.html

  1. Γ = 𝔛 , 𝔜 , H \Gamma=\left\langle\mathfrak{X},\mathfrak{Y},H\right\rangle
  2. 𝔛 \mathfrak{X}
  3. 𝔜 \mathfrak{Y}
  4. H = H ( , ) H=H(\cdot,\cdot)
  5. x 𝔛 x\in\mathfrak{X}
  6. y 𝔜 y\in\mathfrak{Y}
  7. H ( x , y ) H(x,y)
  8. ρ ( x 1 , x 2 ) \rho(x_{1},x_{2})
  9. ρ ( x 1 , x 2 ) = sup y 𝔜 | H ( x 1 , y ) - H ( x 2 , y ) | . \rho(x_{1},x_{2})=\sup_{y\in\mathfrak{Y}}\left|H(x_{1},y)-H(x_{2},y)\right|.
  10. ρ ( x 1 , x 2 ) = 0 \rho(x_{1},x_{2})=0
  11. x 1 = x 2 x_{1}=x_{2}
  12. x 1 x_{1}
  13. x 2 x_{2}
  14. ρ ( x 1 , x 2 ) = 0 \rho(x_{1},x_{2})=0
  15. x 1 = x 2 x_{1}=x_{2}
  16. ρ ( y 1 , y 2 ) = sup x 𝔛 | H ( x , y 1 ) - H ( x , y 2 ) | . \rho(y_{1},y_{2})=\sup_{x\in\mathfrak{X}}\left|H(x,y_{1})-H(x,y_{2})\right|.
  17. Γ \Gamma
  18. ϵ \epsilon
  19. X ϵ X_{\epsilon}
  20. ϵ \epsilon
  21. X X
  22. ρ \rho
  23. x X x\in X
  24. x ϵ X ϵ x_{\epsilon}\in X_{\epsilon}
  25. ρ ( x , x ϵ ) < ϵ \rho(x,x_{\epsilon})<\epsilon
  26. P P
  27. ϵ > 0 \epsilon>0
  28. ϵ \epsilon
  29. P P
  30. ϵ \epsilon
  31. ϵ > 0 \epsilon>0

Hermite_number.html

  1. H 0 = 1 H_{0}=1\,
  2. H 1 = 0 H_{1}=0\,
  3. H 2 = - 2 H_{2}=-2\,
  4. H 3 = 0 H_{3}=0\,
  5. H 4 = + 12 H_{4}=+12\,
  6. H 5 = 0 H_{5}=0\,
  7. H 6 = - 120 H_{6}=-120\,
  8. H 7 = 0 H_{7}=0\,
  9. H 8 = + 1680 H_{8}=+1680\,
  10. H 9 = 0 H_{9}=0\,
  11. H 10 = - 30240 H_{10}=-30240\,
  12. H n = - 2 ( n - 1 ) H n - 2 . H_{n}=-2(n-1)H_{n-2}.\,\!
  13. H n = { 0 , if n is odd ( - 1 ) n / 2 2 n / 2 ( n - 1 ) ! ! , if n is even H_{n}=\begin{cases}0,&\mbox{if }~{}n\mbox{ is odd}\\ (-1)^{n/2}2^{n/2}(n-1)!!,&\mbox{if }~{}n\mbox{ is even}\end{cases}
  14. exp ( - t 2 ) = n = 0 H n t n n ! \exp(-t^{2})=\sum_{n=0}^{\infty}H_{n}\frac{t^{n}}{n!}\,\!
  15. H n ( x ) = ( H + 2 x ) n H_{n}(x)=(H+2x)^{n}\,\!

Hermite–Hadamard_inequality.html

  1. f ( a + b 2 ) 1 b - a a b f ( x ) d x f ( a ) + f ( b ) 2 . f\left(\frac{a+b}{2}\right)\leq\frac{1}{b-a}\int_{a}^{b}f(x)\,dx\leq\frac{f(a)% +f(b)}{2}.
  2. F ( 0 ) ( s ) \displaystyle F^{(0)}(s)
  3. F ( 0 ) ( s ) \displaystyle F^{(0)}(s)
  4. F ( 0 ) ( s ) \displaystyle F^{(0)}(s)
  5. i = 1 n F ( n - 1 ) ( x i ) Π i ( x 1 , , x n ) 1 n ! i = 1 n f ( x i ) \sum_{i=1}^{n}\frac{F^{(n-1)}(x_{i})}{\Pi_{i}(x_{1},\dots,x_{n})}\leq\frac{1}{% n!}\sum_{i=1}^{n}f(x_{i})
  6. Π i ( x 1 , , x n ) := ( x i - x 1 ) ( x i - x 2 ) ( x i - x i - 1 ) ( x i - x i + 1 ) ( x i - x n ) , i = 1 , , n . \Pi_{i}(x_{1},\dots,x_{n}):=(x_{i}-x_{1})(x_{i}-x_{2})\cdots(x_{i}-x_{i-1})(x_% {i}-x_{i+1})\cdots(x_{i}-x_{n}),\ \ i=1,\dots,n.
  7. a < α < b . \ a<\alpha<b.
  8. lim x ¯ α ¯ i = 1 n F ( n - 1 ) ( x i ) Π i ( x 1 , , x n ) = lim x ¯ α ¯ 1 n ! i = 1 n f ( x i ) = f ( α ) ( n - 1 ) ! \lim_{\underline{x}\to\underline{\alpha}}\sum_{i=1}^{n}\frac{F^{(n-1)}(x_{i})}% {\Pi_{i}(x_{1},\ldots,x_{n})}=\lim_{\underline{x}\to\underline{\alpha}}\frac{1% }{n!}\sum_{i=1}^{n}f(x_{i})=\frac{f(\alpha)}{(n-1)!}

Hermitian_connection.html

  1. \nabla

Herschel–Bulkley_fluid.html

  1. τ 0 \tau_{0}
  2. τ = τ 0 + k γ ˙ n \tau=\tau_{0}+k\dot{\gamma}^{n}
  3. τ \tau
  4. γ ˙ \dot{\gamma}
  5. τ 0 \tau_{0}
  6. k k
  7. n n
  8. τ < τ 0 \tau<\tau_{0}
  9. n < 1 n<1
  10. n > 1 n>1
  11. n = 1 n=1
  12. τ 0 = 0 \tau_{0}=0
  13. μ eff = { μ 0 , | γ ˙ | γ ˙ 0 k | γ ˙ | n - 1 + τ 0 | γ ˙ | - 1 , | γ ˙ | γ ˙ 0 \mu_{\operatorname{eff}}=\begin{cases}\mu_{0},&|\dot{\gamma}|\leq\dot{\gamma}_% {0}\\ k|\dot{\gamma}|^{n-1}+\tau_{0}|\dot{\gamma}|^{-1},&|\dot{\gamma}|\geq\dot{% \gamma}_{0}\end{cases}
  14. μ 0 \mu_{0}
  15. μ 0 = k γ ˙ 0 n - 1 + τ 0 γ ˙ 0 - 1 \mu_{0}=k\dot{\gamma}_{0}^{n-1}+\tau_{0}\dot{\gamma}_{0}^{-1}
  16. τ i j = 2 μ eff ( | γ ˙ | ) E i j = μ eff ( | γ ˙ | ) ( u i x j + u j x i ) , \tau_{ij}=2\mu_{\operatorname{eff}}(|\dot{\gamma}|)E_{ij}=\mu_{\operatorname{% eff}}(|\dot{\gamma}|)\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{% \partial u_{j}}{\partial x_{i}}\right),
  17. | γ ˙ | = 2 E i j E i j |\dot{\gamma}|=\sqrt{2E_{ij}E^{ij}}
  18. I I E = t r ( E i j E j k ) = E i j E i j II_{E}=tr(E_{ij}E^{jk})=E_{ij}E^{ij}
  19. π 0 = - 10 \pi_{0}=-10
  20. p x = z ( μ u z ) = { μ 0 2 u z 2 , | u z | < γ 0 z [ ( k | u z | n - 1 + τ 0 | u z | - 1 ) u z ] , | u z | γ 0 \frac{\partial p}{\partial x}=\frac{\partial}{\partial z}\left(\mu\frac{% \partial u}{\partial z}\right)\,\,\,=\begin{cases}\mu_{0}\frac{\partial^{2}u}{% \partial{z}^{2}},&\left|\frac{\partial u}{\partial z}\right|<\gamma_{0}\\ \\ \frac{\partial}{\partial z}\left[\left(k\left|\frac{\partial u}{\partial z}% \right|^{n-1}+\tau_{0}\left|\frac{\partial u}{\partial z}\right|^{-1}\right)% \frac{\partial u}{\partial z}\right],&\left|\frac{\partial u}{\partial z}% \right|\geq\gamma_{0}\end{cases}
  21. P 0 = k ( V / H ) n P_{0}=k\left(V/H\right)^{n}
  22. π 0 = H P 0 p x , \pi_{0}=\frac{H}{P_{0}}\frac{\partial p}{\partial x},
  23. B n = τ 0 k ( H V ) n . Bn=\frac{\tau_{0}}{k}\left(\frac{H}{V}\right)^{n}.
  24. u / z > γ 0 \partial u/\partial z>\gamma_{0}
  25. | u / z | < γ 0 |\partial u/\partial z|<\gamma_{0}
  26. u / z < - γ 0 \partial u/\partial z<-\gamma_{0}
  27. u ( z ) = { n n + 1 1 π 0 [ ( π 0 ( z - z 1 ) + γ 0 n ) 1 + ( 1 / n ) - ( - π 0 z 1 + γ 0 n ) 1 + ( 1 / n ) ] , z [ 0 , z 1 ] π 0 2 μ 0 ( z 2 - z ) + k , z [ z 1 , z 2 ] , n n + 1 1 π 0 [ ( - π 0 ( z - z 2 ) + γ 0 n ) 1 + ( 1 / n ) - ( - π 0 ( 1 - z 2 ) + γ 0 n ) 1 + ( 1 / n ) ] , z [ z 2 , 1 ] u\left(z\right)=\begin{cases}\frac{n}{n+1}\frac{1}{\pi_{0}}\left[\left(\pi_{0}% \left(z-z_{1}\right)+\gamma_{0}^{n}\right)^{1+\left(1/n\right)}-\left(-\pi_{0}% z_{1}+\gamma_{0}^{n}\right)^{1+\left(1/n\right)}\right],&z\in\left[0,z_{1}% \right]\\ \frac{\pi_{0}}{2\mu_{0}}\left(z^{2}-z\right)+k,&z\in\left[z_{1},z_{2}\right],% \\ \frac{n}{n+1}\frac{1}{\pi_{0}}\left[\left(-\pi_{0}\left(z-z_{2}\right)+\gamma_% {0}^{n}\right)^{1+\left(1/n\right)}-\left(-\pi_{0}\left(1-z_{2}\right)+\gamma_% {0}^{n}\right)^{1+\left(1/n\right)}\right],&z\in\left[z_{2},1\right]\\ \end{cases}
  28. u ( z 1 ) u\left(z_{1}\right)
  29. u ( 0 ) = u ( 1 ) = 0 , u(0)=u(1)=0,
  30. z 1 , 2 = 1 2 ± δ z_{1,2}=\tfrac{1}{2}\pm\delta
  31. δ = γ 0 μ 0 | π 0 | 1 2 . \delta=\frac{\gamma_{0}\mu_{0}}{|\pi_{0}|}\leq\tfrac{1}{2}.
  32. μ 0 = γ 0 n - 1 + B n / γ 0 \mu_{0}=\gamma_{0}^{n-1}+Bn/\gamma_{0}
  33. ( γ 0 , B n ) \left(\gamma_{0},Bn\right)
  34. | π 0 , c | = 2 ( γ 0 + B n ) . |\pi_{0,\mathrm{c}}|=2\left(\gamma_{0}+Bn\right).
  35. Δ P L = 4 K D ( 8 V D ) n ( 3 n + 1 4 n ) n 1 1 - X ( 1 1 - a X - b X 2 - c X 3 ) n \frac{\Delta P}{L}=\frac{4K}{D}\left(\frac{8V}{D}\right)^{n}\left(\frac{3n+1}{% 4n}\right)^{n}\frac{1}{1-X}\left(\frac{1}{1-aX-bX^{2}-cX^{3}}\right)^{n}
  36. X = 4 L τ y D Δ P X=\frac{4L\tau_{y}}{D\Delta P}
  37. a = 1 2 n + 1 a=\frac{1}{2n+1}
  38. b = 2 n ( n + 1 ) ( 2 n + 1 ) b=\frac{2n}{\left(n+1\right)\left(2n+1\right)}
  39. c = 2 n 2 ( n + 1 ) ( 2 n + 1 ) c=\frac{2n^{2}}{\left(n+1\right)\left(2n+1\right)}
  40. R = 4 n ρ V D ( 1 - a X - b X 2 - c X 3 ) μ W a l l ( 3 n + 1 ) R=\frac{4n\rho VD\left(1-aX-bX^{2}-cX^{3}\right)}{\mu_{Wall}\left(3n+1\right)}
  41. μ W a l l = τ W a l l 1 - 1 / n ( K 1 - X ) 1 / n \mu_{Wall}=\tau_{Wall}^{1-1/n}\left(\frac{K}{1-X}\right)^{1/n}
  42. τ W a l l = D Δ P 4 L \tau_{Wall}=\frac{D\Delta P}{4L}
  43. Δ P \Delta P
  44. L L
  45. D D
  46. V V
  47. m / s m/s
  48. R e = R n 2 ( 1 - X ) 4 Re=\frac{R}{n^{2}\left(1-X\right)^{4}}

Hesse_normal_form.html

  1. 2 \mathbb{R}^{2}
  2. 3 \mathbb{R}^{3}
  3. r n 0 - d = 0. \vec{r}\cdot\vec{n}_{0}-d=0.\,
  4. r \vec{r}
  5. n 0 \vec{n}_{0}
  6. d 0 d\geq 0
  7. \cdot
  8. ( r - a ) n = 0 (\vec{r}-\vec{a})\cdot\vec{n}=0\,
  9. n \vec{n}
  10. a \vec{a}
  11. A E A\in E
  12. n \vec{n}
  13. a n 0 \vec{a}\cdot\vec{n}\geq 0\,
  14. n \vec{n}
  15. | n | |\vec{n}|
  16. n 0 = n | n | \vec{n}_{0}={{\vec{n}}\over{|\vec{n}|}}\,
  17. ( r - a ) n 0 = 0. (\vec{r}-\vec{a})\cdot\vec{n}_{0}=0.\,
  18. d = a n 0 0 d=\vec{a}\cdot\vec{n}_{0}\geq 0\,
  19. r n 0 - d = 0. \vec{r}\cdot\vec{n}_{0}-d=0.\,
  20. r n 0 = d \vec{r}\cdot\vec{n}_{0}=d
  21. r = r s \vec{r}=\vec{r}_{s}
  22. d = r s n 0 = | r s | | n 0 | cos ( 0 ) = | r s | 1 = | r s | . d=\vec{r}_{s}\cdot\vec{n}_{0}=|\vec{r}_{s}|\cdot|\vec{n}_{0}|\cdot\cos(0^{% \circ})=|\vec{r}_{s}|\cdot 1=|\vec{r}_{s}|.\,
  23. | r s | |\vec{r}_{s}|
  24. r s {\vec{r}_{s}}

Hierarchical_RBF.html

  1. 𝐏 = { 𝐜 i = ( 𝐱 i , 𝐲 i , 𝐳 i ) | i = 0 N 3 } \mathbf{P}=\{\mathbf{c}_{i}=(\mathbf{x}_{i},\mathbf{y}_{i},\mathbf{z}_{i})|^{N% }_{i=0}\subset\mathbb{R}^{3}\}
  2. 𝐇 = { 𝐡 i | i = 0 N } \mathbf{H}=\{\mathbf{h}_{i}|^{N}_{i=0}\subset\mathbb{R}\}
  3. 𝐟 ( 𝐱 ) \mathbf{f}(\mathbf{x})
  4. 𝐟 ( 𝐱 ) = 1 \mathbf{f}(\mathbf{x})=1
  5. 𝐟 ( 𝐱 ) 1 \mathbf{f}(\mathbf{x})\neq 1
  6. 𝐟 ( 𝐱 ) = i = 1 N λ i φ ( 𝐱 , 𝐜 i ) \mathbf{f}(\mathbf{x})=\sum_{i=1}^{N}\lambda_{i}\varphi(\mathbf{x},\mathbf{c}_% {i})
  7. φ \varphi
  8. λ \lambda
  9. 𝐟 ( 𝐱 ) \mathbf{f}(\mathbf{x})
  10. 𝐎 ( 𝐧 2 ) \mathbf{O}(\mathbf{n}^{2})
  11. 𝐎 ( 𝐧 2 ) \mathbf{O}(\mathbf{n}^{2})
  12. 𝐎 ( 𝐦𝐧 ) \mathbf{O}(\mathbf{m}\mathbf{n})
  13. 𝐎 ( 𝐧 log 𝐧 ) \mathbf{O}(\mathbf{n}\log{\mathbf{n}})
  14. 𝐎 ( 𝐧 1.2..1.5 ) \mathbf{O}(\mathbf{n}^{1.2..1.5})
  15. 𝐎 ( 𝐦 log 𝐧 ) \mathbf{O}(\mathbf{m}\log{\mathbf{n}})
  16. 𝐎 ( 𝐧 2 ) \mathbf{O}(\mathbf{n}^{2})
  17. 𝐎 ( 𝐧 log 𝐧 ) \mathbf{O}(\mathbf{n}\log{\mathbf{n}})
  18. 𝐎 ( 𝐦 + 𝐧 log 𝐧 ) \mathbf{O}(\mathbf{m}+\mathbf{n}\log{\mathbf{n}})

Higgs_boson.html

  1. ϕ \phi
  2. ϕ = 1 2 ( ϕ 1 + i ϕ 2 ϕ 0 + i ϕ 3 ) , \phi=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}\phi^{1}+i\phi^{2}\\ \phi^{0}+i\phi^{3}\end{array}\right)\;,
  3. ϕ 0 \phi^{0}
  4. ϕ 3 \phi^{3}
  5. H = | ( μ - i g W μ a τ a - i g 2 B μ ) ϕ | 2 + μ 2 ϕ ϕ - λ ( ϕ ϕ ) 2 , \mathcal{L}_{H}=\left|\left(\partial_{\mu}-igW_{\mu}^{a}\tau^{a}-i\frac{g^{% \prime}}{2}B_{\mu}\right)\phi\right|^{2}+\mu^{2}\phi^{\dagger}\phi-\lambda(% \phi^{\dagger}\phi)^{2},
  6. W μ a W_{\mu}^{a}
  7. B μ B_{\mu}
  8. g g
  9. g g^{\prime}
  10. τ a = σ a / 2 \tau^{a}=\sigma^{a}/2
  11. σ a \sigma^{a}
  12. λ > 0 \lambda>0
  13. μ 2 > 0 \mu^{2}>0
  14. ϕ 1 = ϕ 2 = ϕ 3 = 0 \phi^{1}=\phi^{2}=\phi^{3}=0
  15. ϕ 0 \phi^{0}
  16. ϕ 0 = v \langle\phi^{0}\rangle=v
  17. v = | μ | λ v=\tfrac{|\mu|}{\sqrt{\lambda}}
  18. W μ W_{\mu}
  19. B μ B_{\mu}
  20. M W = v | g | 2 , M_{W}=\frac{v|g|}{2},
  21. M Z = v g 2 + g 2 2 , M_{Z}=\frac{v\sqrt{g^{2}+{g^{\prime}}^{2}}}{2},
  22. cos θ W = M W M Z = | g | g 2 + g 2 \cos\theta_{W}=\frac{M_{W}}{M_{Z}}=\frac{|g|}{\sqrt{g^{2}+{g^{\prime}}^{2}}}
  23. γ \gamma
  24. Y = \displaystyle\mathcal{L}_{Y}=
  25. ( d , u , e , ν ) L , R i (d,u,e,\nu)_{L,R}^{i}
  26. λ u , d , e i j \lambda_{u,d,e}^{ij}
  27. ϕ 0 \phi^{0}
  28. m = - m u i u ¯ L i u R i - m d i d ¯ L i d R i - m e i e ¯ L i e R i + h.c. , \mathcal{L}_{m}=-m_{u}^{i}\overline{u}_{L}^{i}u_{R}^{i}-m_{d}^{i}\overline{d}_% {L}^{i}d_{R}^{i}-m_{e}^{i}\overline{e}_{L}^{i}e_{R}^{i}+\textrm{h.c.},
  29. m u , d , e i = λ u , d , e i v / 2 m_{u,d,e}^{i}=\lambda_{u,d,e}^{i}v/\sqrt{2}
  30. λ u , d , e i \lambda_{u,d,e}^{i}
  31. ψ \psi
  32. - m ψ ¯ ψ -m\bar{\psi}\psi
  33. ψ \psi
  34. - m ψ ¯ ψ = - m ( ψ ¯ L ψ R + ψ ¯ R ψ L ) -m\bar{\psi}\psi\;=\;-m(\bar{\psi}_{L}\psi_{R}+\bar{\psi}_{R}\psi_{L})
  35. ψ ¯ L ψ L \bar{\psi}_{L}\psi_{L}
  36. ψ ¯ R ψ R \bar{\psi}_{R}\psi_{R}

High-dimensional_statistics.html

  1. p = O ( exp ( n a ) ) p=O(\exp(n^{a}))
  2. 0 < a < 1 0<a<1
  3. p exp ( n 1 / 2 ) p\gg\exp(n^{1/2})

Higher-order_singular_value_decomposition.html

  1. x i 1 i N x_{i_{1}\cdots i_{N}}
  2. X = r = 1 R D ( r ) = r = 1 R a ( r ) z ( r ) X=\sum_{r=1}^{R}D^{(r)}=\sum_{r=1}^{R}a^{(r)}\otimes\cdots\otimes z^{(r)}
  3. \otimes
  4. D ( r ) D^{(r)}
  5. a ( r ) , , z ( r ) a^{(r)},\cdots,z^{(r)}
  6. x i 1 i N = r = 1 R a i 1 ( r ) z i N ( r ) x_{i_{1}\cdots i_{N}}=\sum_{r=1}^{R}a^{(r)}_{i_{1}}\cdots z^{(r)}_{i_{N}}
  7. a i ( r ) a^{(r)}_{i}
  8. a ( r ) a^{(r)}
  9. A = U S V T A=USV^{T}
  10. a i 1 , i 2 = j 1 j 2 s j 1 , j 2 u i 1 , j 1 v i 2 , j 2 . a_{i_{1},i_{2}}=\sum_{j_{1}}\sum_{j_{2}}s_{j_{1},j_{2}}u_{i_{1},j_{1}}v_{i_{2}% ,j_{2}}.
  11. U U
  12. V V
  13. S S
  14. a i 1 , i 2 , , i N = j 1 j 2 j N s j 1 , j 2 , , j N u i 1 , j 1 ( 1 ) u i 2 , j 2 ( 2 ) u i N , j N ( N ) , a_{i_{1},i_{2},\dots,i_{N}}=\sum_{j_{1}}\sum_{j_{2}}\cdots\sum_{j_{N}}s_{j_{1}% ,j_{2},\dots,j_{N}}u^{(1)}_{i_{1},j_{1}}u^{(2)}_{i_{2},j_{2}}\dots u^{(N)}_{i_% {N},j_{N}},
  15. U ( n ) = [ u i , j ( n ) ] I n × I n U^{(n)}=[u^{(n)}_{i,j}]_{I_{n}\times I_{n}}
  16. 𝒮 = [ s j 1 , , j N ] I 1 × I 2 × × I N \mathcal{S}=[s_{j_{1},\dots,j_{N}}]_{I_{1}\times I_{2}\times\cdots\times I_{N}}
  17. U ( n ) U^{(n)}
  18. 𝒮 \mathcal{S}
  19. 𝒮 i n = p , 𝒮 i n = q = 0 \langle\mathcal{S}_{i_{n}=p},\mathcal{S}_{i_{n}=q}\rangle=0
  20. p q p\neq q
  21. 𝒮 \mathcal{S}
  22. 𝒮 i n = 1 𝒮 i n = 2 𝒮 i n = I n \|\mathcal{S}_{i_{n}=1}\|\geq\|\mathcal{S}_{i_{n}=2}\|\geq\dots\geq\|\mathcal{% S}_{i_{n}=I_{n}}\|
  23. 𝒜 = 𝒮 × n = 1 N U ( n ) \mathcal{A}=\mathcal{S}\times_{n=1}^{N}U^{(n)}
  24. 𝒜 I 1 × I 2 × × I N \mathcal{A}\in\mathbb{R}^{I_{1}\times I_{2}\times\cdots\times I_{N}}
  25. 𝒜 ( k ) \mathcal{A}_{(k)}
  26. I k × ( j k I j ) I_{k}\times(\prod_{j\neq k}I_{j})
  27. 𝒜 \mathcal{A}
  28. 𝒜 ( k ) = U k Σ k V k T \mathcal{A}_{(k)}=U_{k}\Sigma_{k}V^{T}_{k}
  29. U k U_{k}
  30. 𝒮 \mathcal{S}
  31. 𝒜 \mathcal{A}
  32. { U n } n = 1 N \{U_{n}\}_{n=1}^{N}
  33. 𝒮 = 𝒜 × n = 1 N U n T . \mathcal{S}=\mathcal{A}\times_{n=1}^{N}U_{n}^{T}.
  34. x x
  35. k k
  36. h h
  37. T = i = 1 k P r ( h = k ) E [ x | h = k ] 3 T=\sum_{i=1}^{k}Pr(h=k)E[x|h=k]^{\otimes 3}
  38. E [ x | h = k ] E[x|h=k]

Hilbert_C*-module.html

  1. , : E × E A \langle\cdot,\cdot\rangle:E\times E\rightarrow A
  2. x , α y + β z = α x , y + β x , z \langle x,\alpha y+\beta z\rangle=\alpha\langle x,y\rangle+\beta\langle x,z\rangle
  3. x , y a = x , y a \langle x,ya\rangle=\langle x,y\rangle a
  4. x , y = y , x * , \langle x,y\rangle=\langle y,x\rangle^{*},
  5. x , x 0 \langle x,x\rangle\geq 0
  6. x , x = 0 x = 0. \langle x,x\rangle=0\iff x=0.
  7. x , y y , x x , x y , y \langle x,y\rangle\langle y,x\rangle\leq\|\langle x,x\rangle\|\langle y,y\rangle
  8. x = x , x 1 2 . \|x\|=\|\langle x,x\rangle\|^{\frac{1}{2}}.
  9. a λ a x a λ x a . a_{\lambda}\rightarrow a\Rightarrow xa_{\lambda}\rightarrow xa.
  10. x e λ x xe_{\lambda}\rightarrow x
  11. E , E = span { x , y | x , y E } , \langle E,E\rangle=\operatorname{span}\{\langle x,y\rangle|x,y\in E\},
  12. f , h ( x ) := g ( f ( x ) , h ( x ) ) . \langle f,h\rangle(x):=g(f(x),h(x)).
  13. A n = 1 n A A^{n}=\oplus_{1}^{n}A
  14. ( a i ) , ( b i ) = a i * b i . \langle(a_{i}),(b_{i})\rangle=\sum a_{i}^{*}b_{i}.
  15. A = { ( a i ) | a i * a i converges in A } . \mathcal{H}_{A}=\{(a_{i})|\sum a_{i}^{*}a_{i}\,\text{ converges in }A\}.

Hilbert_space.html

  1. ( x 1 , x 2 , x 3 ) ( y 1 , y 2 , y 3 ) = x 1 y 1 + x 2 y 2 + x 3 y 3 . (x_{1},x_{2},x_{3})\cdot(y_{1},y_{2},y_{3})=x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}.
  2. 𝐱 𝐲 = 𝐱 𝐲 cos θ . \mathbf{x}\cdot\mathbf{y}=\|\mathbf{x}\|\,\|\mathbf{y}\|\,\cos\theta.
  3. n = 0 𝐱 n \sum_{n=0}^{\infty}\mathbf{x}_{n}
  4. k = 0 𝐱 k < . \sum_{k=0}^{\infty}\|\mathbf{x}_{k}\|<\infty.
  5. 𝐋 - k = 0 N 𝐱 k 0 as N . \left\|\mathbf{L}-\sum_{k=0}^{N}\mathbf{x}_{k}\right\|\to 0\quad\,\text{as }N% \to\infty.
  6. x , y \langle x,y\rangle
  7. y , x = x , y ¯ . \langle y,x\rangle=\overline{\langle x,y\rangle}.
  8. a x 1 + b x 2 , y = a x 1 , y + b x 2 , y . \langle ax_{1}+bx_{2},y\rangle=a\langle x_{1},y\rangle+b\langle x_{2},y\rangle.
  9. x , x 0 \langle x,x\rangle\geq 0
  10. x , a y 1 + b y 2 = a ¯ x , y 1 + b ¯ x , y 2 . \langle x,ay_{1}+by_{2}\rangle=\bar{a}\langle x,y_{1}\rangle+\bar{b}\langle x,% y_{2}\rangle.
  11. x = x , x , \|x\|=\sqrt{\langle x,x\rangle},
  12. d ( x , y ) = x - y = x - y , x - y . d(x,y)=\|x-y\|=\sqrt{\langle x-y,x-y\rangle}.
  13. d ( x , z ) d ( x , y ) + d ( y , z ) . d(x,z)\leq d(x,y)+d(y,z).
  14. | x , y | x y |\langle x,y\rangle|\leq\|x\|\,\|y\|
  15. k = 0 u k \textstyle{\sum_{k=0}^{\infty}u_{k}}
  16. k = 0 u k < , \sum_{k=0}^{\infty}\|u_{k}\|<\infty,
  17. n = 1 | z n | 2 \sum_{n=1}^{\infty}|z_{n}|^{2}
  18. 𝐳 , 𝐰 = n = 1 z n w n ¯ , \langle\mathbf{z},\mathbf{w}\rangle=\sum_{n=1}^{\infty}z_{n}\overline{w_{n}},
  19. f , g = a b f ( x ) g ( x ) d x \langle f,g\rangle=\int_{a}^{b}f(x)g(x)\,dx
  20. f ( x ) a b K ( x , y ) f ( y ) d y f(x)\mapsto\int_{a}^{b}K(x,y)f(y)\,dy
  21. K ( x , y ) = n λ n φ n ( x ) φ n ( y ) K(x,y)=\sum_{n}\lambda_{n}\varphi_{n}(x)\varphi_{n}(y)\,
  22. X | f | 2 d μ < , \int_{X}|f|^{2}d\mu<\infty,
  23. f , g = X f ( t ) g ( t ) ¯ d μ ( t ) . \langle f,g\rangle=\int_{X}f(t)\overline{g(t)}\ d\mu(t).
  24. 0 1 | f ( t ) | 2 w ( t ) d t < \int_{0}^{1}|f(t)|^{2}w(t)\,dt<\infty
  25. f , g = 0 1 f ( t ) g ( t ) ¯ w ( t ) d t . \langle f,g\rangle=\int_{0}^{1}f(t)\overline{g(t)}w(t)\,dt.
  26. μ ( A ) = A w ( t ) d t . \mu(A)=\int_{A}w(t)\,dt.
  27. f , g = Ω f ( x ) g ¯ ( x ) d x + Ω D f ( x ) D g ¯ ( x ) d x + + Ω D s f ( x ) D s g ¯ ( x ) d x \langle f,g\rangle=\int_{\Omega}f(x)\bar{g}(x)\,dx+\int_{\Omega}Df(x)\cdot D% \bar{g}(x)\,dx+\cdots+\int_{\Omega}D^{s}f(x)\cdot D^{s}\bar{g}(x)\,dx
  28. H s ( Ω ) = { ( 1 - Δ ) - s / 2 f | f L 2 ( Ω ) } . H^{s}(\Omega)=\{(1-\Delta)^{-s/2}f|f\in L^{2}(\Omega)\}.
  29. M r ( f ) = 1 2 π 0 2 π | f ( r e i θ ) | 2 d θ M_{r}(f)=\frac{1}{2\pi}\int_{0}^{2\pi}|f(re^{i\theta})|^{2}\,d\theta
  30. f 2 = lim r 1 M r ( f ) . \|f\|_{2}=\lim_{r\to 1}\sqrt{M_{r}(f)}.
  31. f ( z ) = n = 0 a n z n f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}
  32. n = 0 | a n | 2 < . \sum_{n=0}^{\infty}\,|a_{n}|^{2}<\infty.
  33. f 2 = D | f ( z ) | 2 d μ ( z ) < , \|f\|^{2}=\int_{D}|f(z)|^{2}\,d\mu(z)<\infty,
  34. sup z K | f ( z ) | C K f 2 , \sup_{z\in K}|f(z)|\leq C_{K}\|f\|_{2},
  35. f ( z ) = D f ( ζ ) η z ( ζ ) ¯ d μ ( ζ ) f(z)=\int_{D}f(\zeta)\overline{\eta_{z}(\zeta)}\,d\mu(\zeta)
  36. K ( ζ , z ) = η z ( ζ ) ¯ K(\zeta,z)=\overline{\eta_{z}(\zeta)}
  37. f ( z ) = D f ( ζ ) K ( ζ , z ) d μ ( ζ ) . f(z)=\int_{D}f(\zeta)K(\zeta,z)\,d\mu(\zeta).
  38. - d d x [ p ( x ) d y d x ] + q ( x ) y = λ w ( x ) y -\frac{d}{dx}\left[p(x)\frac{dy}{dx}\right]+q(x)y=\lambda w(x)y
  39. { α y ( a ) + α y ( a ) = 0 β y ( b ) + β y ( b ) = 0. \begin{cases}\alpha y(a)+\alpha^{\prime}y^{\prime}(a)=0\\ \beta y(b)+\beta^{\prime}y^{\prime}(b)=0.\end{cases}
  40. Ω u v = Ω g v . \int_{\Omega}\nabla u\cdot\nabla v=\int_{\Omega}gv.
  41. a ( u , v ) = b ( v ) a(u,v)=b(v)
  42. a ( u , v ) = Ω u v , b ( v ) = Ω g v . a(u,v)=\int_{\Omega}\nabla u\cdot\nabla v,\quad b(v)=\int_{\Omega}gv.
  43. f ( T t w ) = f ( w ) f(T_{t}w)=f(w)\,
  44. f , g L 2 ( Ω E , μ ) = E f g ¯ d μ . \langle f,g\rangle_{L^{2}(\Omega_{E},\mu)}=\int_{E}f\bar{g}\,d\mu.
  45. P x = lim T 1 T 0 T U t x d t . Px=\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}U_{t}x\,dt.
  46. L 2 - lim T 1 T 0 T f ( T t w ) d t = Ω E f ( y ) d μ ( y ) . \underset{T\to\infty}{L^{2}\!-\!\lim}\frac{1}{T}\int_{0}^{T}f(T_{t}w)\,dt=\int% _{\Omega_{E}}f(y)\,d\mu(y).
  47. n = - a n e 2 π i n θ \sum_{n=-\infty}^{\infty}a_{n}e^{2\pi in\theta}
  48. a n = 0 1 f ( θ ) e - 2 π i n θ d θ . a_{n}=\int_{0}^{1}f(\theta)e^{-2\pi in\theta}\,d\theta.
  49. f ( θ ) = n a n e n ( θ ) , a n = f , e n f(\theta)=\sum_{n}a_{n}e_{n}(\theta),\quad a_{n}=\langle f,e_{n}\rangle
  50. f ( x ) f n ( x ) = a 1 e 1 ( x ) + a 2 e 2 ( x ) + + a n e n ( x ) . f(x)\approx f_{n}(x)=a_{1}e_{1}(x)+a_{2}e_{2}(x)+\cdots+a_{n}e_{n}(x).
  51. a j = 0 1 e j ( x ) ¯ f ( x ) d x . a_{j}=\int_{0}^{1}\overline{e_{j}(x)}f(x)\,dx.
  52. u , v \langle u,v\rangle
  53. u + v 2 = u + v , u + v = u , u + 2 Re u , v + v , v = u 2 + v 2 . \|u+v\|^{2}=\langle u+v,u+v\rangle=\langle u,u\rangle+2\,\mathrm{Re}\langle u,% v\rangle+\langle v,v\rangle=\|u\|^{2}+\|v\|^{2}.
  54. u 1 + + u n 2 = u 1 2 + + u n 2 . \|u_{1}+\cdots+u_{n}\|^{2}=\|u_{1}\|^{2}+\cdots+\|u_{n}\|^{2}.
  55. k = 0 u k 2 = k = 0 u k 2 . \bigl\|\sum_{k=0}^{\infty}u_{k}\bigr\|^{2}=\sum_{k=0}^{\infty}\|u_{k}\|^{2}.
  56. u + v 2 + u - v 2 = 2 ( u 2 + v 2 ) . \|u+v\|^{2}+\|u-v\|^{2}=2(\|u\|^{2}+\|v\|^{2}).
  57. u , v = 1 4 ( u + v 2 - u - v 2 ) . \langle u,v\rangle=\frac{1}{4}\left(\|u+v\|^{2}-\|u-v\|^{2}\right).
  58. u , v = 1 4 ( u + v 2 - u - v 2 + i u + i v 2 - i u - i v 2 ) . \langle u,v\rangle=\frac{1}{4}\left(\|u+v\|^{2}-\|u-v\|^{2}+i\|u+iv\|^{2}-i\|u% -iv\|^{2}\right).
  59. y C , x - y = dist ( x , C ) = min { x - z : z C } . y\in C,\ \ \ \|x-y\|=\mathrm{dist}(x,C)=\min\{\|x-z\|:z\in C\}.
  60. y F , x - y F . y\in F,\ \ x-y\perp F.
  61. φ = sup x = 1 , x H | φ ( x ) | . \|\varphi\|=\sup_{\|x\|=1,x\in H}|\varphi(x)|.
  62. φ u ( x ) = x , u . \varphi_{u}(x)=\langle x,u\rangle.
  63. u φ u u\mapsto\varphi_{u}
  64. x , u φ = φ ( x ) \langle x,u_{\varphi}\rangle=\varphi(x)
  65. φ , ψ = u ψ , u φ . \langle\varphi,\psi\rangle=\langle u_{\psi},u_{\varphi}\rangle.
  66. u = v , v - 1 φ ( v ) ¯ v . u=\langle v,v\rangle^{-1}\,\overline{\varphi(v)}\,v.
  67. x | y = y , x . \langle x|y\rangle=\langle y,x\rangle.
  68. lim n x n , v = x , v \lim_{n}\langle x_{n},v\rangle=\langle x,v\rangle
  69. A = sup { A x : x 1 } . \lVert A\rVert=\sup\left\{\,\lVert Ax\rVert:\lVert x\rVert\leq 1\,\right\}.
  70. x , A * y = A x , y \langle x,A^{*}y\rangle=\langle Ax,y\rangle
  71. A = B 2 = B * B . A=B^{2}=B^{*}B.\,
  72. A = A + A * 2 + i A - A * 2 i A=\frac{A+A^{*}}{2}+i\frac{A-A^{*}}{2i}
  73. index T = dim ker T - dim coker T . \operatorname{index}\,T=\dim\ker T-\dim\operatorname{coker}\,T.
  74. ( A f ) ( x ) = - i d d x f ( x ) , (Af)(x)=-i\frac{d}{dx}f(x),\,
  75. ( B f ) ( x ) = x f ( x ) . (Bf)(x)=xf(x).\,
  76. H 1 H 2 , H_{1}\oplus H_{2},
  77. ( x 1 , x 2 ) , ( y 1 , y 2 ) H 1 H 2 = x 1 , y 1 H 1 + x 2 , y 2 H 2 . \langle(x_{1},x_{2}),(y_{1},y_{2})\rangle_{H_{1}\oplus H_{2}}=\langle x_{1},y_% {1}\rangle_{H_{1}}+\langle x_{2},y_{2}\rangle_{H_{2}}.
  78. i I H i \bigoplus_{i\in I}H_{i}
  79. x = ( x i H i | i I ) i I H i x=(x_{i}\in H_{i}|i\in I)\in\prod_{i\in I}H_{i}
  80. i I x i 2 < . \sum_{i\in I}\|x_{i}\|^{2}<\infty.
  81. x , y = i I x i , y i H i . \langle x,y\rangle=\sum_{i\in I}\langle x_{i},y_{i}\rangle_{H_{i}}.
  82. E i E j = 0 , i j . E_{i}E_{j}=0,\quad i\not=j.
  83. x 1 x 2 , y 1 y 2 = x 1 , y 1 x 2 , y 2 . \langle x_{1}\otimes x_{2},\,y_{1}\otimes y_{2}\rangle=\langle x_{1},y_{1}% \rangle\,\langle x_{2},y_{2}\rangle.
  84. H 1 ^ H 2 H_{1}\widehat{\otimes}H_{2}
  85. f 1 f 2 f_{1}\otimes f_{2}
  86. ( s , t ) f 1 ( s ) f 2 ( t ) (s,t)\mapsto f_{1}(s)\,f_{2}(t)
  87. x * H 1 * x^{*}\in H^{*}_{1}
  88. x * x * ( x 1 ) x 2 . x^{*}\mapsto x^{*}(x_{1})\,x_{2}.
  89. H 1 ^ H 2 H_{1}\widehat{\otimes}H_{2}
  90. e k , e j = 0 \langle e_{k},e_{j}\rangle=0
  91. ( c 1 , c 2 , c 3 , ) (c_{1},c_{2},c_{3},\dots)\,
  92. | c 1 | 2 + | c 2 | 2 + | c 3 | 2 + < . |c_{1}|^{2}+|c_{2}|^{2}+|c_{3}|^{2}+\cdots<\infty.\,
  93. e 1 \displaystyle e_{1}
  94. 2 ( B ) = { x : B 𝑥 b B | x ( b ) | 2 < } . \ell^{2}(B)=\big\{x:B\xrightarrow{x}\mathbb{C}\mid\sum_{b\in B}\left|x(b)% \right|^{2}<\infty\big\}.
  95. b B | x ( b ) | 2 = sup n = 1 N | x ( b n ) | 2 \sum_{b\in B}\left|x(b)\right|^{2}=\sup\sum_{n=1}^{N}|x(b_{n})|^{2}
  96. x , y = b B x ( b ) y ( b ) ¯ \langle x,y\rangle=\sum_{b\in B}x(b)\overline{y(b)}
  97. e b ( b ) = { 1 if b = b 0 otherwise. e_{b}(b^{\prime})=\begin{cases}1&\,\text{if }b=b^{\prime}\\ 0&\,\text{otherwise.}\end{cases}
  98. y = j = 1 n x , f j f j . y=\sum_{j=1}^{n}\,\langle x,f_{j}\rangle\,f_{j}.
  99. x 2 = x - y 2 + y 2 y 2 = j = 1 n | x , f j | 2 . \|x\|^{2}=\|x-y\|^{2}+\|y\|^{2}\geq\|y\|^{2}=\sum_{j=1}^{n}|\langle x,f_{j}% \rangle|^{2}.
  100. i I | x , f i | 2 x 2 , x H \sum_{i\in I}|\langle x,f_{i}\rangle|^{2}\leq\|x\|^{2},\quad x\in H
  101. x = k B x , e k e k . x=\sum_{k\in B}\,\langle x,e_{k}\rangle\,e_{k}.
  102. x 2 = k B | x , e k | 2 . \|x\|^{2}=\sum_{k\in B}|\langle x,e_{k}\rangle|^{2}.
  103. Φ ( x ) , Φ ( y ) 2 ( B ) = x , y H \langle\Phi\left(x\right),\Phi\left(y\right)\rangle_{\ell^{2}(B)}=\langle x,y% \rangle_{H}
  104. S = { x H : x , s = 0 s S } . S^{\perp}=\left\{x\in H:\langle x,s\rangle=0\ \forall s\in S\right\}.
  105. i : V H , i:V\to H,
  106. i x , y H = x , π y V \langle ix,y\rangle_{H}=\langle x,\pi y\rangle_{V}
  107. P V = sup x H , x 0 P V x x = 1. \|P_{V}\|=\sup_{x\in H,x\not=0}\frac{\|P_{V}x\|}{\|x\|}=1.
  108. P V 2 = P V . P_{V}^{2}=P_{V}.
  109. V U V^{\perp}\subseteq U^{\perp}
  110. V V^{\bot\bot}
  111. ( i V i ) = i V i \textstyle{\left(\sum_{i}V_{i}\right)^{\perp}=\bigcap_{i}V_{i}^{\perp}}
  112. i V i ¯ = ( i V i ) \textstyle{\overline{\sum_{i}V_{i}^{\perp}}=\left(\bigcap_{i}V_{i}\right)^{% \perp}}
  113. | z | T . \scriptstyle{|z|\leq\|T\|.}
  114. m = inf x = 1 T x , x , M = sup x = 1 T x , x . m=\inf_{\|x\|=1}\langle Tx,x\rangle,\quad M=\sup_{\|x\|=1}\langle Tx,x\rangle.
  115. H λ = ker ( T - λ ) . H_{\lambda}=\ker(T-\lambda).
  116. H = λ σ ( T ) H λ . H=\bigoplus_{\lambda\in\sigma(T)}H_{\lambda}.
  117. T = λ σ ( T ) λ E λ , T=\sum_{\lambda\in\sigma(T)}\lambda E_{\lambda},
  118. A + = 1 2 ( A 2 + A ) . A^{+}=\frac{1}{2}\left(\sqrt{A^{2}}+A\right).
  119. T = λ d E λ . T=\int_{\mathbb{R}}\lambda\,dE_{\lambda}.
  120. T x , y = λ d E λ x , y . \langle Tx,y\rangle=\int_{\mathbb{R}}\lambda\,d\langle E_{\lambda}x,y\rangle.
  121. f ( T ) = σ ( T ) f ( λ ) d E λ . f(T)=\int_{\sigma(T)}f(\lambda)\,dE_{\lambda}.
  122. R λ = ( T - λ ) - 1 R_{\lambda}=(T-\lambda)^{-1}
  123. T x , y = λ d E x , y ( λ ) \langle Tx,y\rangle=\int_{\mathbb{R}}\lambda\,dE_{x,y}(\lambda)

Hilbert_spectral_analysis.html

  1. ω = d θ ( t ) d t . \omega=\frac{d\theta(t)}{dt}.\,
  2. X ( t ) = j = 1 n a j ( t ) exp ( i ω j ( t ) d t ) . X(t)=\sum_{j=1}^{n}a_{j}(t)\exp\left(i\int\omega_{j}(t)dt\right).\,

Hilbert–Poincaré_series.html

  1. V = i 𝐍 V i V=\textstyle\bigoplus_{i\in\mathbf{N}}V_{i}
  2. i 𝐍 dim K ( V i ) t i . \sum_{i\in\mathbf{N}}\dim_{K}(V_{i})t^{i}.
  3. ( n + k n ) {\left({{n+k}\atop{n}}\right)}
  4. X 0 , , X n X_{0},\dots,X_{n}
  5. ( 1 - t ) - n - 1 (1-t)^{-n-1}
  6. A [ x 0 , , x n ] , deg x i = d i A[x_{0},\dots,x_{n}],\operatorname{deg}x_{i}=d_{i}
  7. ( 1 - t d i ) \prod(1-t^{d_{i}})
  8. n = 0 n=0
  9. M k = 0 M_{k}=0
  10. n - 1 n-1
  11. N ( l ) k = N k + l N(l)_{k}=N_{k+l}
  12. 0 K ( - d n ) M ( - d n ) x n M C 0 0\to K(-d_{n})\to M(-d_{n})\overset{x_{n}}{\to}M\to C\to 0
  13. P ( M , t ) = - P ( K ( - d n ) , t ) + P ( M ( - d n ) , t ) - P ( C , t ) P(M,t)=-P(K(-d_{n}),t)+P(M(-d_{n}),t)-P(C,t)
  14. P ( M ( - d n ) , t ) = t d n P ( M , t ) P(M(-d_{n}),t)=t^{d_{n}}P(M,t)
  15. x n x_{n}
  16. A [ x 0 , , x n - 1 ] A[x_{0},\dots,x_{n-1}]
  17. 0 C 0 d 0 C 1 d 1 C 2 d 2 d n - 1 C n 0. 0\to C^{0}\stackrel{d_{0}}{\longrightarrow}C^{1}\stackrel{d_{1}}{% \longrightarrow}C^{2}\stackrel{d_{2}}{\longrightarrow}\cdots\stackrel{d_{n-1}}% {\longrightarrow}C^{n}\longrightarrow 0.
  18. i C i \bigoplus_{i}C^{i}
  19. P C ( t ) = j = 0 n dim ( C j ) t j . P_{C}(t)=\sum_{j=0}^{n}\dim(C^{j})t^{j}.
  20. P H ( t ) = j = 0 n dim ( H j ) t j . P_{H}(t)=\sum_{j=0}^{n}\dim(H^{j})t^{j}.
  21. Q ( t ) Q(t)
  22. P C ( t ) - P H ( t ) = ( 1 + t ) Q ( t ) . P_{C}(t)-P_{H}(t)=(1+t)Q(t).

Hill_differential_equation.html

  1. d 2 y d t 2 + f ( t ) y = 0 , \frac{d^{2}y}{dt^{2}}+f(t)y=0,
  2. d 2 y d t 2 + ( θ 0 + 2 n = 1 θ n cos ( 2 n t ) + m = 1 ϕ m sin ( 2 m t ) ) y = 0. \frac{d^{2}y}{dt^{2}}+\left(\theta_{0}+2\sum_{n=1}^{\infty}\theta_{n}\cos(2nt)% +\sum_{m=1}^{\infty}\phi_{m}\sin(2mt)\right)y=0.

Hill_tetrahedron.html

  1. α ( 0 , 2 π / 3 ) \alpha\in(0,2\pi/3)
  2. v 1 , v 2 , v 3 3 v_{1},v_{2},v_{3}\in\mathbb{R}^{3}
  3. α \alpha
  4. Q ( α ) Q(\alpha)
  5. Q ( α ) = { c 1 v 1 + c 2 v 2 + c 3 v 3 0 c 1 c 2 c 3 1 } . Q(\alpha)\,=\,\{c_{1}v_{1}+c_{2}v_{2}+c_{3}v_{3}\mid 0\leq c_{1}\leq c_{2}\leq c% _{3}\leq 1\}.
  6. Q = Q ( π / 2 ) Q=Q(\pi/2)
  7. 2 \sqrt{2}
  8. 3 \sqrt{3}
  9. Q Q
  10. Q Q
  11. Q ( α ) Q(\alpha)
  12. Q ( w ) = { c 1 v 1 + + c n v n 0 c 1 c n 1 } , Q(w)\,=\,\{c_{1}v_{1}+\cdots+c_{n}v_{n}\mid 0\leq c_{1}\leq\cdots\leq c_{n}% \leq 1\},
  13. v 1 , , v n v_{1},\ldots,v_{n}
  14. ( v i , v j ) = w (v_{i},v_{j})=w
  15. 1 i < j n 1\leq i<j\leq n
  16. - 1 / ( n - 1 ) < w < 1 -1/(n-1)<w<1

Hill_yield_criterion.html

  1. F ( σ 22 - σ 33 ) 2 + G ( σ 33 - σ 11 ) 2 + H ( σ 11 - σ 22 ) 2 + 2 L σ 23 2 + 2 M σ 31 2 + 2 N σ 12 2 = 1 . F(\sigma_{22}-\sigma_{33})^{2}+G(\sigma_{33}-\sigma_{11})^{2}+H(\sigma_{11}-% \sigma_{22})^{2}+2L\sigma_{23}^{2}+2M\sigma_{31}^{2}+2N\sigma_{12}^{2}=1~{}.
  2. σ i j \sigma_{ij}
  3. ( G + H ) ( σ 1 y ) 2 = 1 ; ( F + H ) ( σ 2 y ) 2 = 1 ; ( F + G ) ( σ 3 y ) 2 = 1 (G+H)~{}(\sigma_{1}^{y})^{2}=1~{};~{}~{}(F+H)~{}(\sigma_{2}^{y})^{2}=1~{};~{}~% {}(F+G)~{}(\sigma_{3}^{y})^{2}=1
  4. σ 1 y , σ 2 y , σ 3 y \sigma_{1}^{y},\sigma_{2}^{y},\sigma_{3}^{y}
  5. F = 1 2 [ 1 ( σ 2 y ) 2 + 1 ( σ 3 y ) 2 - 1 ( σ 1 y ) 2 ] F=\cfrac{1}{2}\left[\cfrac{1}{(\sigma_{2}^{y})^{2}}+\cfrac{1}{(\sigma_{3}^{y})% ^{2}}-\cfrac{1}{(\sigma_{1}^{y})^{2}}\right]
  6. G = 1 2 [ 1 ( σ 3 y ) 2 + 1 ( σ 1 y ) 2 - 1 ( σ 2 y ) 2 ] G=\cfrac{1}{2}\left[\cfrac{1}{(\sigma_{3}^{y})^{2}}+\cfrac{1}{(\sigma_{1}^{y})% ^{2}}-\cfrac{1}{(\sigma_{2}^{y})^{2}}\right]
  7. H = 1 2 [ 1 ( σ 1 y ) 2 + 1 ( σ 2 y ) 2 - 1 ( σ 3 y ) 2 ] H=\cfrac{1}{2}\left[\cfrac{1}{(\sigma_{1}^{y})^{2}}+\cfrac{1}{(\sigma_{2}^{y})% ^{2}}-\cfrac{1}{(\sigma_{3}^{y})^{2}}\right]
  8. τ 12 y , τ 23 y , τ 31 y \tau_{12}^{y},\tau_{23}^{y},\tau_{31}^{y}
  9. L = 1 2 ( τ 23 y ) 2 ; M = 1 2 ( τ 31 y ) 2 ; N = 1 2 ( τ 12 y ) 2 L=\cfrac{1}{2~{}(\tau_{23}^{y})^{2}}~{};~{}~{}M=\cfrac{1}{2~{}(\tau_{31}^{y})^% {2}}~{};~{}~{}N=\cfrac{1}{2~{}(\tau_{12}^{y})^{2}}
  10. σ 1 2 + R 0 ( 1 + R 90 ) R 90 ( 1 + R 0 ) σ 2 2 - 2 R 0 1 + R 0 σ 1 σ 2 = ( σ 1 y ) 2 \sigma_{1}^{2}+\cfrac{R_{0}~{}(1+R_{90})}{R_{90}~{}(1+R_{0})}~{}\sigma_{2}^{2}% -\cfrac{2~{}R_{0}}{1+R_{0}}~{}\sigma_{1}\sigma_{2}=(\sigma_{1}^{y})^{2}
  11. σ 1 , σ 2 \sigma_{1},\sigma_{2}
  12. σ 1 \sigma_{1}
  13. σ 2 \sigma_{2}
  14. σ 3 = 0 \sigma_{3}=0
  15. R 0 R_{0}
  16. R 90 R_{90}
  17. R = R 0 = R 90 R=R_{0}=R_{90}
  18. σ 1 2 + σ 2 2 - 2 R 1 + R σ 1 σ 2 = ( σ 1 y ) 2 \sigma_{1}^{2}+\sigma_{2}^{2}-\cfrac{2~{}R}{1+R}~{}\sigma_{1}\sigma_{2}=(% \sigma_{1}^{y})^{2}
  19. f := F ( σ 2 - σ 3 ) 2 + G ( σ 3 - σ 1 ) 2 + H ( σ 1 - σ 2 ) 2 - 1 = 0 f:=F(\sigma_{2}-\sigma_{3})^{2}+G(\sigma_{3}-\sigma_{1})^{2}+H(\sigma_{1}-% \sigma_{2})^{2}-1=0\,
  20. σ 1 , σ 2 , σ 3 \sigma_{1},\sigma_{2},\sigma_{3}
  21. ϵ ˙ i p = λ ˙ f σ i d ϵ i p d λ = f σ i . \dot{\epsilon}^{p}_{i}=\dot{\lambda}~{}\cfrac{\partial f}{\partial\sigma_{i}}% \qquad\implies\qquad\cfrac{d\epsilon^{p}_{i}}{d\lambda}=\cfrac{\partial f}{% \partial\sigma_{i}}~{}.
  22. d ϵ 1 p d λ = 2 ( G + H ) σ 1 - 2 H σ 2 - 2 G σ 3 d ϵ 2 p d λ = 2 ( F + H ) σ 2 - 2 H σ 1 - 2 F σ 3 d ϵ 3 p d λ = 2 ( F + G ) σ 3 - 2 G σ 1 - 2 F σ 2 . \begin{aligned}\displaystyle\cfrac{d\epsilon^{p}_{1}}{d\lambda}&\displaystyle=% 2(G+H)\sigma_{1}-2H\sigma_{2}-2G\sigma_{3}\\ \displaystyle\cfrac{d\epsilon^{p}_{2}}{d\lambda}&\displaystyle=2(F+H)\sigma_{2% }-2H\sigma_{1}-2F\sigma_{3}\\ \displaystyle\cfrac{d\epsilon^{p}_{3}}{d\lambda}&\displaystyle=2(F+G)\sigma_{3% }-2G\sigma_{1}-2F\sigma_{2}~{}.\end{aligned}
  23. σ 3 = 0 \sigma_{3}=0
  24. d ϵ 1 p d λ = 2 ( G + H ) σ 1 - 2 H σ 2 d ϵ 2 p d λ = 2 ( F + H ) σ 2 - 2 H σ 1 d ϵ 3 p d λ = - 2 G σ 1 - 2 F σ 2 . \begin{aligned}\displaystyle\cfrac{d\epsilon^{p}_{1}}{d\lambda}&\displaystyle=% 2(G+H)\sigma_{1}-2H\sigma_{2}\\ \displaystyle\cfrac{d\epsilon^{p}_{2}}{d\lambda}&\displaystyle=2(F+H)\sigma_{2% }-2H\sigma_{1}\\ \displaystyle\cfrac{d\epsilon^{p}_{3}}{d\lambda}&\displaystyle=-2G\sigma_{1}-2% F\sigma_{2}~{}.\end{aligned}
  25. R 0 R_{0}
  26. σ 1 \sigma_{1}
  27. R 90 R_{90}
  28. σ 2 \sigma_{2}
  29. R 0 = d ϵ 2 p d ϵ 3 p = H G ; R 90 = d ϵ 1 p d ϵ 3 p = H F . R_{0}=\cfrac{d\epsilon^{p}_{2}}{d\epsilon^{p}_{3}}=\cfrac{H}{G}~{};~{}~{}R_{90% }=\cfrac{d\epsilon^{p}_{1}}{d\epsilon^{p}_{3}}=\cfrac{H}{F}~{}.
  30. H = R 0 G H=R_{0}G
  31. σ 3 = 0 \sigma_{3}=0
  32. f := F σ 2 2 + G σ 1 2 + R 0 G ( σ 1 - σ 2 ) 2 - 1 = 0 f:=F\sigma_{2}^{2}+G\sigma_{1}^{2}+R_{0}G(\sigma_{1}-\sigma_{2})^{2}-1=0\,
  33. σ 1 2 + F + R 0 G G ( 1 + R 0 ) σ 2 2 - 2 R 0 1 + R 0 σ 1 σ 2 = 1 ( 1 + R 0 ) G . \sigma_{1}^{2}+\cfrac{F+R_{0}G}{G(1+R_{0})}~{}\sigma_{2}^{2}-\cfrac{2R_{0}}{1+% R_{0}}~{}\sigma_{1}\sigma_{2}=\cfrac{1}{(1+R_{0})G}~{}.
  34. F , G F,G
  35. σ 1 y \sigma_{1}^{y}
  36. F = 1 2 [ 1 ( σ 2 y ) 2 + 1 ( σ 3 y ) 2 - 1 ( σ 1 y ) 2 ] G = 1 2 [ 1 ( σ 3 y ) 2 + 1 ( σ 1 y ) 2 - 1 ( σ 2 y ) 2 ] H = 1 2 [ 1 ( σ 1 y ) 2 + 1 ( σ 2 y ) 2 - 1 ( σ 3 y ) 2 ] \begin{aligned}\displaystyle F&\displaystyle=\cfrac{1}{2}\left[\cfrac{1}{(% \sigma_{2}^{y})^{2}}+\cfrac{1}{(\sigma_{3}^{y})^{2}}-\cfrac{1}{(\sigma_{1}^{y}% )^{2}}\right]\\ \displaystyle G&\displaystyle=\cfrac{1}{2}\left[\cfrac{1}{(\sigma_{3}^{y})^{2}% }+\cfrac{1}{(\sigma_{1}^{y})^{2}}-\cfrac{1}{(\sigma_{2}^{y})^{2}}\right]\\ \displaystyle H&\displaystyle=\cfrac{1}{2}\left[\cfrac{1}{(\sigma_{1}^{y})^{2}% }+\cfrac{1}{(\sigma_{2}^{y})^{2}}-\cfrac{1}{(\sigma_{3}^{y})^{2}}\right]\end{aligned}
  37. R 0 = H G ( 1 + R 0 ) 1 ( σ 3 y ) 2 - ( 1 + R 0 ) 1 ( σ 2 y ) 2 = ( 1 - R 0 ) 1 ( σ 1 y ) 2 R 90 = H F ( 1 + R 90 ) 1 ( σ 3 y ) 2 - ( 1 - R 90 ) 1 ( σ 2 y ) 2 = ( 1 + R 90 ) 1 ( σ 1 y ) 2 \begin{aligned}\displaystyle R_{0}=\cfrac{H}{G}&\displaystyle\implies(1+R_{0})% \cfrac{1}{(\sigma_{3}^{y})^{2}}-(1+R_{0})\cfrac{1}{(\sigma_{2}^{y})^{2}}=(1-R_% {0})\cfrac{1}{(\sigma_{1}^{y})^{2}}\\ \displaystyle R_{90}=\cfrac{H}{F}&\displaystyle\implies(1+R_{90})\cfrac{1}{(% \sigma_{3}^{y})^{2}}-(1-R_{90})\cfrac{1}{(\sigma_{2}^{y})^{2}}=(1+R_{90})% \cfrac{1}{(\sigma_{1}^{y})^{2}}\end{aligned}
  38. 1 ( σ 3 y ) 2 , 1 ( σ 2 y ) 2 \cfrac{1}{(\sigma_{3}^{y})^{2}},\cfrac{1}{(\sigma_{2}^{y})^{2}}
  39. 1 ( σ 3 y ) 2 = R 0 + R 90 ( 1 + R 0 ) R 90 1 ( σ 1 y ) 2 ; 1 ( σ 2 y ) 2 = R 0 ( 1 + R 90 ) ( 1 + R 0 ) R 90 1 ( σ 1 y ) 2 \cfrac{1}{(\sigma_{3}^{y})^{2}}=\cfrac{R_{0}+R_{90}}{(1+R_{0})~{}R_{90}}~{}% \cfrac{1}{(\sigma_{1}^{y})^{2}}~{};~{}~{}\cfrac{1}{(\sigma_{2}^{y})^{2}}=% \cfrac{R_{0}(1+R_{90})}{(1+R_{0})~{}R_{90}}~{}\cfrac{1}{(\sigma_{1}^{y})^{2}}
  40. F , G F,G
  41. F = R 0 ( 1 + R 0 ) R 90 1 ( σ 1 y ) 2 ; G = 1 1 + R 0 1 ( σ 1 y ) 2 F=\cfrac{R_{0}}{(1+R_{0})~{}R_{90}}~{}\cfrac{1}{(\sigma_{1}^{y})^{2}}~{};~{}~{% }G=\cfrac{1}{1+R_{0}}~{}\cfrac{1}{(\sigma_{1}^{y})^{2}}
  42. 1 G ( 1 + R 0 ) = ( σ 1 y ) 2 ; F + R 0 G G ( 1 + R 0 ) = R 0 ( 1 + R 90 ) R 90 ( 1 + R 0 ) . \cfrac{1}{G(1+R_{0})}=(\sigma_{1}^{y})^{2}~{};~{}~{}\cfrac{F+R_{0}G}{G(1+R_{0}% )}=\cfrac{R_{0}(1+R_{90})}{R_{90}(1+R_{0})}~{}.
  43. σ 1 2 + R 0 ( 1 + R 90 ) R 90 ( 1 + R 0 ) σ 2 2 - 2 R 0 1 + R 0 σ 1 σ 2 = ( σ 1 y ) 2 \sigma_{1}^{2}+\cfrac{R_{0}~{}(1+R_{90})}{R_{90}~{}(1+R_{0})}~{}\sigma_{2}^{2}% -\cfrac{2~{}R_{0}}{1+R_{0}}~{}\sigma_{1}\sigma_{2}=(\sigma_{1}^{y})^{2}
  44. F | σ 2 - σ 3 | m + G | σ 3 - σ 1 | m + H | σ 1 - σ 2 | m + L | 2 σ 1 - σ 2 - σ 3 | m + M | 2 σ 2 - σ 3 - σ 1 | m + N | 2 σ 3 - σ 1 - σ 2 | m = σ y m . \begin{aligned}\displaystyle F|\sigma_{2}-\sigma_{3}|^{m}&\displaystyle+G|% \sigma_{3}-\sigma_{1}|^{m}+H|\sigma_{1}-\sigma_{2}|^{m}+L|2\sigma_{1}-\sigma_{% 2}-\sigma_{3}|^{m}\\ &\displaystyle+M|2\sigma_{2}-\sigma_{3}-\sigma_{1}|^{m}+N|2\sigma_{3}-\sigma_{% 1}-\sigma_{2}|^{m}=\sigma_{y}^{m}~{}.\end{aligned}
  45. σ i \sigma_{i}
  46. σ y \sigma_{y}
  47. 1 - 2 1-2
  48. F = G F=G
  49. L = M L=M
  50. f := F | σ 2 - σ 3 | m + F | σ 3 - σ 1 | m + H | σ 1 - σ 2 | m + L | 2 σ 1 - σ 2 - σ 3 | m + L | 2 σ 2 - σ 3 - σ 1 | m + N | 2 σ 3 - σ 1 - σ 2 | m - σ y m 0 \begin{aligned}\displaystyle f:=&\displaystyle F|\sigma_{2}-\sigma_{3}|^{m}+F|% \sigma_{3}-\sigma_{1}|^{m}+H|\sigma_{1}-\sigma_{2}|^{m}+L|2\sigma_{1}-\sigma_{% 2}-\sigma_{3}|^{m}\\ &\displaystyle+L|2\sigma_{2}-\sigma_{3}-\sigma_{1}|^{m}+N|2\sigma_{3}-\sigma_{% 1}-\sigma_{2}|^{m}-\sigma_{y}^{m}\leq 0\end{aligned}
  51. σ 1 > ( σ 2 = σ 3 = 0 ) \sigma_{1}>(\sigma_{2}=\sigma_{3}=0)
  52. R = ( 2 m - 1 + 2 ) L - N + H ( 2 m - 1 - 1 ) L + 2 N + F . R=\cfrac{(2^{m-1}+2)L-N+H}{(2^{m-1}-1)L+2N+F}~{}.
  53. L = 0 , H = 0. L=0,H=0.
  54. f := 1 + 2 R 1 + R ( | σ 1 | m + | σ 2 | m ) - R 1 + R | σ 1 + σ 2 | m - σ y m 0 f:=\cfrac{1+2R}{1+R}(|\sigma_{1}|^{m}+|\sigma_{2}|^{m})-\cfrac{R}{1+R}|\sigma_% {1}+\sigma_{2}|^{m}-\sigma_{y}^{m}\leq 0
  55. N = 0 , F = 0. N=0,F=0.
  56. f := 2 m - 1 ( 1 - R ) + ( R + 2 ) ( 1 - 2 m - 1 ) ( 1 + R ) | σ 1 - σ 2 | m - 1 ( 1 - 2 m - 1 ) ( 1 + R ) ( | 2 σ 1 - σ 2 | m + | 2 σ 2 - σ 1 | m ) - σ y m 0 f:=\cfrac{2^{m-1}(1-R)+(R+2)}{(1-2^{m-1})(1+R)}|\sigma_{1}-\sigma_{2}|^{m}-% \cfrac{1}{(1-2^{m-1})(1+R)}(|2\sigma_{1}-\sigma_{2}|^{m}+|2\sigma_{2}-\sigma_{% 1}|^{m})-\sigma_{y}^{m}\leq 0
  57. N = 0 , H = 0. N=0,H=0.
  58. f := 2 m - 1 ( 1 - R ) + ( R + 2 ) ( 2 + 2 m - 1 ) ( 1 + R ) ( | σ 1 | m - | σ 2 | m ) + R ( 2 + 2 m - 1 ) ( 1 + R ) ( | 2 σ 1 - σ 2 | m + | 2 σ 2 - σ 1 | m ) - σ y m 0 f:=\cfrac{2^{m-1}(1-R)+(R+2)}{(2+2^{m-1})(1+R)}(|\sigma_{1}|^{m}-|\sigma_{2}|^% {m})+\cfrac{R}{(2+2^{m-1})(1+R)}(|2\sigma_{1}-\sigma_{2}|^{m}+|2\sigma_{2}-% \sigma_{1}|^{m})-\sigma_{y}^{m}\leq 0
  59. L = 0 , F = 0. L=0,F=0.
  60. f := 1 + 2 R 2 ( 1 + R ) | σ 1 - σ 2 | m + 1 2 ( 1 + R ) | σ 1 + σ 2 | m - σ y m 0 f:=\cfrac{1+2R}{2(1+R)}|\sigma_{1}-\sigma_{2}|^{m}+\cfrac{1}{2(1+R)}|\sigma_{1% }+\sigma_{2}|^{m}-\sigma_{y}^{m}\leq 0
  61. L = 0 , N = 0. L=0,N=0.
  62. f := 1 1 + R ( | σ 1 | m + | σ 2 | m ) + R 1 + R | σ 1 - σ 2 | m - σ y m 0 f:=\cfrac{1}{1+R}(|\sigma_{1}|^{m}+|\sigma_{2}|^{m})+\cfrac{R}{1+R}|\sigma_{1}% -\sigma_{2}|^{m}-\sigma_{y}^{m}\leq 0
  63. R R
  64. m m
  65. ( σ 1 σ 0 ) 2 + ( σ 2 σ 90 ) 2 + [ ( p + q - c ) - p σ 1 + q σ 2 σ b ] ( σ 1 σ 2 σ 0 σ 90 ) = 1 \left(\cfrac{\sigma_{1}}{\sigma_{0}}\right)^{2}+\left(\cfrac{\sigma_{2}}{% \sigma_{90}}\right)^{2}+\left[(p+q-c)-\cfrac{p\sigma_{1}+q\sigma_{2}}{\sigma_{% b}}\right]\left(\cfrac{\sigma_{1}\sigma_{2}}{\sigma_{0}\sigma_{90}}\right)=1
  66. σ 0 \sigma_{0}
  67. σ 90 \sigma_{90}
  68. σ b \sigma_{b}
  69. c , p , q c,p,q
  70. c = σ 0 σ 90 + σ 90 σ 0 - σ 0 σ 90 σ b 2 ( 1 σ 0 + 1 σ 90 - 1 σ b ) p = 2 R 0 ( σ b - σ 90 ) ( 1 + R 0 ) σ 0 2 - 2 R 90 σ b ( 1 + R 90 ) σ 90 2 + c σ 0 ( 1 σ 0 + 1 σ 90 - 1 σ b ) q = 2 R 90 ( σ b - σ 0 ) ( 1 + R 90 ) σ 90 2 - 2 R 0 σ b ( 1 + R 0 ) σ 0 2 + c σ 90 \begin{aligned}\displaystyle c&\displaystyle=\cfrac{\sigma_{0}}{\sigma_{90}}+% \cfrac{\sigma_{90}}{\sigma_{0}}-\cfrac{\sigma_{0}\sigma_{90}}{\sigma_{b}^{2}}% \\ \displaystyle\left(\cfrac{1}{\sigma_{0}}+\cfrac{1}{\sigma_{90}}-\cfrac{1}{% \sigma_{b}}\right)~{}p&\displaystyle=\cfrac{2R_{0}(\sigma_{b}-\sigma_{90})}{(1% +R_{0})\sigma_{0}^{2}}-\cfrac{2R_{90}\sigma_{b}}{(1+R_{90})\sigma_{90}^{2}}+% \cfrac{c}{\sigma_{0}}\\ \displaystyle\left(\cfrac{1}{\sigma_{0}}+\cfrac{1}{\sigma_{90}}-\cfrac{1}{% \sigma_{b}}\right)~{}q&\displaystyle=\cfrac{2R_{90}(\sigma_{b}-\sigma_{0})}{(1% +R_{90})\sigma_{90}^{2}}-\cfrac{2R_{0}\sigma_{b}}{(1+R_{0})\sigma_{0}^{2}}+% \cfrac{c}{\sigma_{90}}\end{aligned}
  71. R 0 R_{0}
  72. R 90 R_{90}
  73. F ( σ 22 - σ 33 ) 2 + G ( σ 33 - σ 11 ) 2 + H ( σ 11 - σ 22 ) 2 + 2 L σ 23 2 + 2 M σ 31 2 + 2 N σ 12 2 + I σ 11 + J σ 22 + K σ 33 = 1 . F(\sigma_{22}-\sigma_{33})^{2}+G(\sigma_{33}-\sigma_{11})^{2}+H(\sigma_{11}-% \sigma_{22})^{2}+2L\sigma_{23}^{2}+2M\sigma_{31}^{2}+2N\sigma_{12}^{2}+I\sigma% _{11}+J\sigma_{22}+K\sigma_{33}=1~{}.
  74. F ( σ 22 - σ 33 ) 2 + G ( σ 33 - σ 11 ) 2 + H ( σ 11 - σ 22 ) 2 + 2 L σ 23 2 + 2 M σ 31 2 + 2 N σ 12 2 + K ( σ 11 + σ 22 + σ 33 ) 2 = 1 . F(\sigma_{22}-\sigma_{33})^{2}+G(\sigma_{33}-\sigma_{11})^{2}+H(\sigma_{11}-% \sigma_{22})^{2}+2L\sigma_{23}^{2}+2M\sigma_{31}^{2}+2N\sigma_{12}^{2}+K(% \sigma_{11}+\sigma_{22}+\sigma_{33})^{2}=1~{}.

Himmelblau's_function.html

  1. f ( x , y ) = ( x 2 + y - 11 ) 2 + ( x + y 2 - 7 ) 2 . f(x,y)=(x^{2}+y-11)^{2}+(x+y^{2}-7)^{2}.\quad
  2. x = - 0.270845 x=-0.270845\,
  3. y = - 0.923039 y=-0.923039\,
  4. f ( x , y ) = 181.617 f(x,y)=181.617\,
  5. f ( 3.0 , 2.0 ) = 0.0 , f(3.0,2.0)=0.0,\quad
  6. f ( - 2.805118 , 3.131312 ) = 0.0 , f(-2.805118,3.131312)=0.0,\quad
  7. f ( - 3.779310 , - 3.283186 ) = 0.0 , f(-3.779310,-3.283186)=0.0,\quad
  8. f ( 3.584428 , - 1.848126 ) = 0.0. f(3.584428,-1.848126)=0.0.\quad

Hindmarsh–Rose_model.html

  1. d x d t = y + ϕ ( x ) - z + I , d y d t = ψ ( x ) - y , d z d t = r [ s ( x - x R ) - z ] , \begin{aligned}\displaystyle\frac{dx}{dt}&\displaystyle=y+\phi(x)-z+I,\\ \displaystyle\frac{dy}{dt}&\displaystyle=\psi(x)-y,\\ \displaystyle\frac{dz}{dt}&\displaystyle=r[s(x-x_{R})-z],\end{aligned}
  2. ϕ ( x ) = - a x 3 + b x 2 , ψ ( x ) = c - d x 2 . \begin{aligned}\displaystyle\phi(x)&\displaystyle=-ax^{3}+bx^{2},\\ \displaystyle\psi(x)&\displaystyle=c-dx^{2}.\end{aligned}
  3. d z d t \displaystyle\frac{dz}{dt}

Hit-or-miss_transform.html

  1. d \mathbb{R}^{d}
  2. d \mathbb{Z}^{d}
  3. C C
  4. D D
  5. C D = C\cap D=\emptyset
  6. A B = ( A C ) ( A c D ) A\odot B=(A\ominus C)\cap(A^{c}\ominus D)
  7. A c A^{c}
  8. E = 2 E=\mathbb{Z}^{2}
  9. C 1 = { ( 0 , 0 ) , ( - 1 , - 1 ) , ( 0 , - 1 ) , ( 1 , - 1 ) } C_{1}=\{(0,0),(-1,-1),(0,-1),(1,-1)\}
  10. D 1 = { ( - 1 , 1 ) , ( 0 , 1 ) , ( 1 , 1 ) } D_{1}=\{(-1,1),(0,1),(1,1)\}
  11. C 2 = { ( - 1 , 0 ) , ( 0 , 0 ) , ( - 1 , - 1 ) , ( 0 , - 1 ) } C_{2}=\{(-1,0),(0,0),(-1,-1),(0,-1)\}
  12. D 2 = { ( 0 , 1 ) , ( 1 , 1 ) , ( 1 , 0 ) } D_{2}=\{(0,1),(1,1),(1,0)\}
  13. B 1 , , B 8 B_{1},\ldots,B_{8}
  14. X B i = X ( X B i ) , X\otimes B_{i}=X\setminus(X\odot B_{i}),
  15. \setminus
  16. A B 1 B 2 B 8 B 1 B 2 A\otimes B_{1}\otimes B_{2}\otimes\ldots\otimes B_{8}\otimes B_{1}\otimes B_{2% }\otimes\ldots

Hobby–Rice_theorem.html

  1. k + 1 k+1
  2. 0 = z 0 < z 1 < < z k < z k + 1 = 1 0=z_{0}<z_{1}<\cdots<z_{k}<z_{k+1}=1
  3. i i
  4. δ i \delta_{i}
  5. δ 1 , , δ k + 1 { + 1 , - 1 } \delta_{1},\ldots,\delta_{k+1}\in\left\{+1,-1\right\}
  6. g 1 , , g k : [ 0 , 1 ] g_{1},\ldots,g_{k}\colon[0,1]\longrightarrow\mathbb{R}
  7. i = 1 k + 1 δ i z i - 1 z i g j ( z ) d z = 0 for 1 j k . \sum_{i=1}^{k+1}\delta_{i}\!\int_{z_{i-1}}^{z_{i}}g_{j}(z)\,dz=0\,\text{ for }% 1\leq j\leq k.

Hong–Ou–Mandel_effect.html

  1. a ^ \hat{a}
  2. a ^ \hat{a}^{\dagger}
  3. b ^ \hat{b}
  4. b ^ \hat{b}^{\dagger}
  5. a ^ b ^ | 0 , 0 a b = | 1 , 1 a b , \hat{a}^{\dagger}\hat{b}^{\dagger}|0,0\rangle_{ab}=|1,1\rangle_{ab},\,
  6. | 1 |1\rangle
  7. a ^ c ^ + d ^ 2 and b ^ c ^ - d ^ 2 . \hat{a}^{\dagger}\rightarrow\frac{\hat{c}^{\dagger}+\hat{d}^{\dagger}}{\sqrt{2% }}\quad\,\text{and}\quad\hat{b}^{\dagger}\rightarrow\frac{\hat{c}^{\dagger}-% \hat{d}^{\dagger}}{\sqrt{2}}.
  8. ( a ^ b ^ ) 1 2 ( 1 1 1 - 1 ) ( c ^ d ^ ) . \begin{pmatrix}\hat{a}\\ \hat{b}\end{pmatrix}\rightarrow\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\ 1&-1\end{pmatrix}\begin{pmatrix}\hat{c}\\ \hat{d}\end{pmatrix}.
  9. | 1 , 1 a b = a ^ b ^ | 0 , 0 a b 1 2 ( c ^ + d ^ ) ( c ^ - d ^ ) | 0 , 0 c d = 1 2 ( c ^ 2 - d ^ 2 ) | 0 , 0 c d = | 2 , 0 c d - | 0 , 2 c d 2 . |1,1\rangle_{ab}=\hat{a}^{\dagger}\hat{b}^{\dagger}|0,0\rangle_{ab}\rightarrow% \frac{1}{2}\left(\hat{c}^{\dagger}+\hat{d}^{\dagger}\right)\left(\hat{c}^{% \dagger}-\hat{d}^{\dagger}\right)|0,0\rangle_{cd}=\frac{1}{2}\left(\hat{c}^{% \dagger 2}-\hat{d}^{\dagger 2}\right)|0,0\rangle_{cd}=\frac{|2,0\rangle_{cd}-|% 0,2\rangle_{cd}}{\sqrt{2}}.
  10. c ^ \hat{c}^{\dagger}
  11. d ^ \hat{d}^{\dagger}
  12. c ^ 2 \hat{c}^{\dagger 2}
  13. d ^ 2 \hat{d}^{\dagger 2}
  14. ρ a \rho_{a}
  15. ρ b \rho_{b}
  16. V = Tr ( ρ a ρ b ) . V=\operatorname{Tr}\left(\rho_{a}\rho_{b}\right).\,
  17. ρ a = ρ b = ρ \rho_{a}=\rho_{b}=\rho
  18. P = Tr ( ρ 2 ) P=\operatorname{Tr}(\rho^{2})
  19. | 2 , 0 + | 0 , 2 |2,0\rangle+|0,2\rangle

Hooke's_atom.html

  1. H ^ = - 1 2 1 2 - 1 2 2 2 + 1 2 k ( r 1 2 + r 2 2 ) + 1 | 𝐫 1 - 𝐫 2 | . \hat{H}=-\frac{1}{2}\nabla^{2}_{1}-\frac{1}{2}\nabla^{2}_{2}+\frac{1}{2}k(r^{2% }_{1}+r^{2}_{2})+\frac{1}{|\mathbf{r}_{1}-\mathbf{r}_{2}|}.
  2. - 2 r 1 2 k r 2 . -\frac{2}{r}\rightarrow\frac{1}{2}kr^{2}.
  3. H ^ Ψ ( 𝐫 1 , 𝐫 2 ) = E Ψ ( 𝐫 1 , 𝐫 2 ) . \hat{H}\Psi(\mathbf{r}_{1},\mathbf{r}_{2})=E\Psi(\mathbf{r}_{1},\mathbf{r}_{2}).
  4. 𝐑 = 1 2 ( 𝐫 1 + 𝐫 2 ) , 𝐮 = 𝐫 2 - 𝐫 1 . \mathbf{R}=\frac{1}{2}(\mathbf{r}_{1}+\mathbf{r}_{2}),\mathbf{u}=\mathbf{r}_{2% }-\mathbf{r}_{1}.
  5. Ψ ( 𝐫 1 , 𝐫 2 ) = χ ( 𝐑 ) Φ ( 𝐮 ) \Psi(\mathbf{r}_{1},\mathbf{r}_{2})=\chi(\mathbf{R})\Phi(\mathbf{u})
  6. ( - 1 4 𝐑 2 + k R 2 ) χ ( 𝐑 ) = E 𝐑 χ ( 𝐑 ) , \left(-\frac{1}{4}\nabla^{2}_{\mathbf{R}}+kR^{2}\right)\chi(\mathbf{R})=E_{% \mathbf{R}}\chi(\mathbf{R}),
  7. ( - 𝐮 2 + 1 4 k u 2 + 1 u ) Φ ( 𝐮 ) = E 𝐮 Φ ( 𝐮 ) . \left(-\nabla^{2}_{\mathbf{u}}+\frac{1}{4}ku^{2}+\frac{1}{u}\right)\Phi(% \mathbf{u})=E_{\mathbf{u}}\Phi(\mathbf{u}).
  8. χ ( 𝐑 ) \chi(\mathbf{R})
  9. E 𝐑 = ( 3 / 2 ) k E h E_{\mathbf{R}}=(3/2)\sqrt{k}E_{\mathrm{h}}
  10. χ ( 𝐑 ) = e - k R 2 . \chi(\mathbf{R})=e^{-\sqrt{k}R^{2}}.
  11. exp ( - ( k / 4 ) u 2 ) \exp(-(\sqrt{k}/4)u^{2})\,
  12. Φ ( 𝐮 ) = f ( u ) exp ( - ( k / 4 ) u 2 ) \Phi(\mathbf{u})=f(u)\exp(-(\sqrt{k}/4)u^{2})\,
  13. f ( u ) f(u)\,
  14. Φ ( 𝐮 ) = R l ( u ) Y l m \Phi(\mathbf{u})=R_{l}(u)Y_{lm}
  15. ( - 1 u 2 u ( u 2 u ) + L ^ 2 u 2 + 1 4 k u 2 + 1 u ) R l ( u ) Y l m ( 𝐮 ^ ) = E l R l ( u ) Y l m ( 𝐮 ^ ) , \left(-\frac{1}{u^{2}}\frac{\partial}{\partial u}\left(u^{2}\frac{\partial}{% \partial u}\right)+\frac{\hat{L}^{2}}{u^{2}}+\frac{1}{4}ku^{2}+\frac{1}{u}% \right)R_{l}(u)Y_{lm}(\hat{\mathbf{u}})=E_{l}R_{l}(u)Y_{lm}(\hat{\mathbf{u}}),
  16. R l ( u ) = S l ( u ) / u R_{l}(u)=S_{l}(u)/u\,
  17. - 2 S l ( u ) u 2 + ( l ( l + 1 ) u 2 + 1 4 k u 2 + 1 u ) S l ( u ) = E l S l ( u ) . -\frac{\partial^{2}S_{l}(u)}{\partial u^{2}}+\left(\frac{l(l+1)}{u^{2}}+\frac{% 1}{4}ku^{2}+\frac{1}{u}\right)S_{l}(u)=E_{l}S_{l}(u).
  18. S l ( u ) e - k 4 u 2 S_{l}(u)\sim e^{-\frac{\sqrt{k}}{4}u^{2}}\,
  19. S l ( u ) = e - k 4 u 2 T l ( u ) . S_{l}(u)=e^{-\frac{\sqrt{k}}{4}u^{2}}T_{l}(u).
  20. T l ( u ) T_{l}(u)\,
  21. - 2 T l ( u ) u 2 + k u T l ( u ) u + ( l ( l + 1 ) u 2 + 1 u + ( k 2 - E l ) ) T l ( u ) = 0. -\frac{\partial^{2}T_{l}(u)}{\partial u^{2}}+\sqrt{k}u\frac{\partial T_{l}(u)}% {\partial u}+\left(\frac{l(l+1)}{u^{2}}+\frac{1}{u}+\left(\frac{\sqrt{k}}{2}-E% _{l}\right)\right)T_{l}(u)=0.
  22. T l ( u ) T_{l}(u)\,
  23. T l ( u ) = u m k = 0 a k u k . T_{l}(u)=u^{m}\sum_{k=0}^{\infty}\ a_{k}u^{k}.
  24. m m\,
  25. { a k } k = 0 k = \{a_{k}\}_{k=0}^{k=\infty}\,
  26. m ( m - 1 ) = l ( l + 1 ) , m(m-1)=l(l+1)\,,
  27. a 0 0 a_{0}\neq 0\,
  28. a 1 = a 0 2 ( l + 1 ) , a_{1}=\frac{a_{0}}{2(l+1)},
  29. a 2 = a 1 + ( k ( l + 3 2 ) - E l ) a 0 2 ( 2 l + 3 ) = a 0 2 ( 2 l + 3 ) ( 1 2 ( l + 1 ) + k ( l + 3 2 ) - E l ) , a_{2}=\frac{a_{1}+\left(\sqrt{k}(l+\frac{3}{2})-E_{l}\right)a_{0}}{2(2l+3)}=% \frac{a_{0}}{2(2l+3)}\left(\frac{1}{2(l+1)}+\sqrt{k}\left(l+\frac{3}{2}\right)% -E_{l}\right),
  30. a 3 = a 2 + ( k ( l + 5 2 ) - E l ) a 1 6 ( l + 2 ) , a_{3}=\frac{a_{2}+\left(\sqrt{k}(l+\frac{5}{2})-E_{l}\right)a_{1}}{6(l+2)},
  31. a n + 1 = a n + ( k ( l + 1 2 + n ) - E l ) a n - 1 ( n + 1 ) ( 2 l + 2 + n ) . a_{n+1}=\frac{a_{n}+\left(\sqrt{k}(l+\frac{1}{2}+n)-E_{l}\right)a_{n-1}}{(n+1)% (2l+2+n)}.
  32. m = l + 1 m=l+1
  33. m = - l m=-l
  34. 1 2 ( l + 1 ) + k ( l + 3 2 ) - E l = 0 , \frac{1}{2(l+1)}+\sqrt{k}\left(l+\frac{3}{2}\right)-E_{l}=0,
  35. k ( l + 5 2 ) = E l . \sqrt{k}(l+\frac{5}{2})=E_{l}.
  36. k \sqrt{k}\,
  37. E l E_{l}\,
  38. k = 1 2 ( l + 1 ) , \sqrt{k}=\frac{1}{2(l+1)},
  39. E l = 2 l + 5 4 ( l + 1 ) , E_{l}=\frac{2l+5}{4(l+1)},
  40. T l = u l + 1 ( a 0 + a 0 2 ( l + 1 ) u ) . T_{l}=u^{l+1}\left(a_{0}+\frac{a_{0}}{2(l+1)}u\right).
  41. R l ( u ) R_{l}(u)\,
  42. R l ( u ) = T l ( u ) e - k 4 u 2 u = u l ( 1 + 1 2 ( l + 1 ) u ) e - k 4 u 2 , R_{l}(u)=\frac{T_{l}(u)e^{-\frac{\sqrt{k}}{4}u^{2}}}{u}=u^{l}\left(1+\frac{1}{% 2(l+1)}u\right)e^{-\frac{\sqrt{k}}{4}u^{2}},
  43. l = 0 l=0\,
  44. 5 / 4 E h 5/4E_{\mathrm{h}}\,
  45. Φ ( 𝐮 ) = ( 1 + u 2 ) e - u 2 / 4 . \Phi(\mathbf{u})=\left(1+\frac{u}{2}\right)e^{-u^{2}/4}.
  46. Ψ ( 𝐫 1 , 𝐫 2 ) = 1 2 8 π 5 / 2 + 5 π 3 ( 1 + 1 2 | 𝐫 1 - 𝐫 2 | ) exp ( - 1 4 ( r 1 2 + r 2 2 ) ) . \Psi(\mathbf{r}_{1},\mathbf{r}_{2})=\frac{1}{2\sqrt{8\pi^{5/2}+5\pi^{3}}}\left% (1+\frac{1}{2}|\mathbf{r}_{1}-\mathbf{r}_{2}|\right)\exp\left(-\frac{1}{4}\big% (r_{1}^{2}+r_{2}^{2}\big)\right).
  47. E = E R + E u = 3 4 + 5 4 = 2 E h E=E_{R}+E_{u}=\frac{3}{4}+\frac{5}{4}=2E_{\mathrm{h}}
  48. ρ ( 𝐫 ) = 2 π 3 / 2 ( 8 + 5 π ) e - ( 1 / 2 ) r 2 ( ( π 2 ) 1 / 2 ( 7 4 + 1 4 r 2 + ( r + 1 r ) erf ( r 2 ) ) + e - ( 1 / 2 ) r 2 ) . \rho(\mathbf{r})=\frac{2\pi^{3/2}}{(8+5\sqrt{\pi})}e^{-(1/2)r^{2}}\left(\left(% \frac{\pi}{2}\right)^{1/2}\left(\frac{7}{4}+\frac{1}{4}r^{2}+\left(r+\frac{1}{% r}\right)\mathrm{erf}\left(\frac{r}{\sqrt{2}}\right)\right)+e^{-(1/2)r^{2}}% \right).

Hopf_maximum_principle.html

  1. L u = i j a i j ( x ) 2 u x i x j + i b i u x i 0 Lu=\sum_{ij}a_{ij}(x)\frac{\partial^{2}u}{\partial x_{i}\partial x_{j}}+\sum_{% i}b_{i}\frac{\partial u}{\partial x_{i}}\geq 0

Hosford_yield_criterion.html

  1. 1 2 | σ 2 - σ 3 | n + 1 2 | σ 3 - σ 1 | n + 1 2 | σ 1 - σ 2 | n = σ y n \tfrac{1}{2}|\sigma_{2}-\sigma_{3}|^{n}+\tfrac{1}{2}|\sigma_{3}-\sigma_{1}|^{n% }+\tfrac{1}{2}|\sigma_{1}-\sigma_{2}|^{n}=\sigma_{y}^{n}\,
  2. σ i \sigma_{i}
  3. n n
  4. σ y \sigma_{y}
  5. σ y = ( 1 2 | σ 2 - σ 3 | n + 1 2 | σ 3 - σ 1 | n + 1 2 | σ 1 - σ 2 | n ) 1 / n . \sigma_{y}=\left(\tfrac{1}{2}|\sigma_{2}-\sigma_{3}|^{n}+\tfrac{1}{2}|\sigma_{% 3}-\sigma_{1}|^{n}+\tfrac{1}{2}|\sigma_{1}-\sigma_{2}|^{n}\right)^{1/n}\,.
  6. x p = ( | x 1 | p + | x 2 | p + + | x n | p ) 1 / p . \ \|x\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\cdots+|x_{n}|^{p}\right)^{1/p}\,.
  7. p = p=\infty
  8. x = max { | x 1 | , | x 2 | , , | x n | } \ \|x\|_{\infty}=\max\left\{|x_{1}|,|x_{2}|,\ldots,|x_{n}|\right\}
  9. ( σ y ) n = max ( | σ 2 - σ 3 | , | σ 3 - σ 1 | , | σ 1 - σ 2 | ) . (\sigma_{y})_{n\rightarrow\infty}=\max\left(|\sigma_{2}-\sigma_{3}|,|\sigma_{3% }-\sigma_{1}|,|\sigma_{1}-\sigma_{2}|\right)\,.
  10. 1 2 ( | σ 1 | n + | σ 2 | n ) + 1 2 | σ 1 - σ 2 | n = σ y n \cfrac{1}{2}\left(|\sigma_{1}|^{n}+|\sigma_{2}|^{n}\right)+\cfrac{1}{2}|\sigma% _{1}-\sigma_{2}|^{n}=\sigma_{y}^{n}\,
  11. n 1 n\geq 1
  12. F | σ 2 - σ 3 | n + G | σ 3 - σ 1 | n + H | σ 1 - σ 2 | n = 1 F|\sigma_{2}-\sigma_{3}|^{n}+G|\sigma_{3}-\sigma_{1}|^{n}+H|\sigma_{1}-\sigma_% {2}|^{n}=1\,
  13. σ i \sigma_{i}
  14. n n
  15. 1 1 + R ( | σ 1 | n + | σ 2 | n ) + R 1 + R | σ 1 - σ 2 | n = σ y n \cfrac{1}{1+R}(|\sigma_{1}|^{n}+|\sigma_{2}|^{n})+\cfrac{R}{1+R}|\sigma_{1}-% \sigma_{2}|^{n}=\sigma_{y}^{n}
  16. R R
  17. σ y \sigma_{y}
  18. n n

HR_8799.html

  1. 4.9 \scriptstyle\sqrt{4.9}

Hrushovski_construction.html

  1. \leq
  2. \subseteq
  3. \leq
  4. 0 \aleph_{0}
  5. \leq
  6. A B A\leq B
  7. A B A\subseteq B
  8. A B C A\subseteq B\subseteq C
  9. A C A\leq C
  10. A B A\leq B
  11. A \varnothing\leq A
  12. A \C A\in\C
  13. A B A\leq B
  14. A C B C A\cap C\leq B\cap C
  15. C \C C\in\C
  16. f : A A f:A\rightarrow A^{\prime}
  17. A B A\leq B
  18. f f
  19. B B B\rightarrow B^{\prime}
  20. B B
  21. A B A^{\prime}\leq B^{\prime}
  22. f : A D f:A\hookrightarrow D
  23. f ( A ) D f(A)\leq D
  24. \leq
  25. A B 1 , A B 2 A\leq B_{1},A\leq B_{2}
  26. D \C D\in\C
  27. B i B_{i}
  28. D D
  29. A A
  30. D D
  31. A \C A\in\C
  32. A D A\leq D
  33. A X A\leq X
  34. A X D A\subseteq X\subseteq D
  35. X \C X\in\C
  36. A D A\subseteq D
  37. A A
  38. D D
  39. cl D ( A ) \operatorname{cl}_{D}(A)
  40. A A
  41. cl ( A ) D \operatorname{cl}(A)\leq D
  42. G G
  43. \leq
  44. A ω G A\subseteq_{\omega}G
  45. A \C A\in\C
  46. A G A\leq G
  47. A B A\leq B
  48. B B
  49. B B
  50. G G
  51. A A
  52. G G
  53. A ω G A\subseteq_{\omega}G
  54. cl G ( A ) \operatorname{cl}_{G}(A)
  55. \leq
  56. \leq
  57. 0 \aleph_{0}

Hu_Washizu_principle.html

  1. V e [ 1 2 ϵ T C ϵ - σ T ϵ + σ T ( u ) - p ¯ T u ] d V - S σ e T ¯ T u d S \int_{V^{e}}\left[\frac{1}{2}\epsilon^{T}C\epsilon-\sigma^{T}\epsilon+\sigma^{% T}(\nabla u)-\bar{p}^{T}u\right]dV-\int_{S_{\sigma}^{e}}\bar{T}^{T}u\ dS
  2. C C

Human_resource_accounting.html

  1. E ( V y ) = ( P y ( t + 1 ) ) ( < m t p l > I ( t ) ( I + R ) ( t - y ) ) E(V_{y})=\left(\sum P_{y}(t+1)\right)\left(\sum\frac{<}{m}tpl>{{I(t)}}{{(I+R)^% {(t-y)}}}\right)

Human_sex_ratio.html

  1. χ 2 \chi^{2}

Hunter–Saxton_equation.html

  1. ( u t + u u x ) x = 1 2 u x 2 (u_{t}+uu_{x})_{x}=\frac{1}{2}\,u_{x}^{2}
  2. W ( 𝐧 , 𝐧 ) = 1 2 ( α ( 𝐧 ) 2 + β ( 𝐧 ( × 𝐧 ) ) 2 + γ | 𝐧 × ( × 𝐧 ) | 2 ) , W(\mathbf{n},\nabla\mathbf{n})=\frac{1}{2}\left(\alpha(\nabla\cdot\mathbf{n})^% {2}+\beta(\mathbf{n}\cdot(\nabla\times\mathbf{n}))^{2}+\gamma|\mathbf{n}\times% (\nabla\times\mathbf{n})|^{2}\right),
  3. α \alpha
  4. β \beta
  5. γ \gamma
  6. = 1 2 | 𝐧 t | 2 - W ( 𝐧 , 𝐧 ) - λ 2 ( 1 - | 𝐧 | 2 ) , \mathcal{L}=\frac{1}{2}\left|\frac{\partial\mathbf{n}}{\partial t}\right|^{2}-% W(\mathbf{n},\nabla\mathbf{n})-\frac{\lambda}{2}(1-|\mathbf{n}|^{2}),
  7. λ \lambda
  8. 𝐧 ( x , y , z , t ) = ( cos φ ( x , t ) , sin φ ( x , t ) , 0 ) . \mathbf{n}(x,y,z,t)=(\cos\varphi(x,t),\sin\varphi(x,t),0).
  9. = 1 2 ( φ t 2 - a 2 ( φ ) φ x 2 ) , a ( φ ) := α sin 2 φ + γ cos 2 φ , \mathcal{L}=\frac{1}{2}\left(\varphi_{t}^{2}-a^{2}(\varphi)\varphi_{x}^{2}% \right),\qquad a(\varphi):=\sqrt{\alpha\sin^{2}\varphi+\gamma\cos^{2}\varphi},
  10. φ t t = a ( φ ) [ a ( φ ) φ x ] x . \varphi_{tt}=a(\varphi)[a(\varphi)\varphi_{x}]_{x}.
  11. a 0 := a ( φ 0 ) a_{0}:=a(\varphi_{0})
  12. φ ( x , t ; ϵ ) = φ 0 + ϵ φ 1 ( θ , τ ) + O ( ϵ 2 ) , \varphi(x,t;\epsilon)=\varphi_{0}+\epsilon\varphi_{1}(\theta,\tau)+O(\epsilon^% {2}),
  13. θ := x - a 0 t , τ := ϵ t . \theta:=x-a_{0}t,\qquad\tau:=\epsilon t.
  14. ϵ 2 \epsilon^{2}
  15. ( φ 1 τ + a ( φ 0 ) φ 1 φ 1 θ ) θ = 1 2 a ( φ 0 ) φ 1 θ 2 . (\varphi_{1\tau}+a^{\prime}(\varphi_{0})\varphi_{1}\varphi_{1\theta})_{\theta}% =\frac{1}{2}a^{\prime}(\varphi_{0})\varphi_{1\theta}^{2}.
  16. a ( φ 0 ) 0 a^{\prime}(\varphi_{0})\neq 0
  17. 𝐧 ( x , y , z , t ) = ( cos φ ( x , t ) , sin φ ( x , t ) cos ψ ( x , t ) , sin φ ( x , t ) sin ψ ( x , t ) ) . \mathbf{n}(x,y,z,t)=(\cos\varphi(x,t),\sin\varphi(x,t)\cos\psi(x,t),\sin% \varphi(x,t)\sin\psi(x,t)).
  18. = 1 2 ( φ t 2 - a 2 ( φ ) φ x 2 + sin 2 φ [ ψ t 2 - b 2 ( φ ) ψ x 2 ] ) , a ( φ ) := α sin 2 φ + γ cos 2 φ , b ( φ ) := β sin 2 φ + γ cos 2 φ . \mathcal{L}=\frac{1}{2}\left(\varphi_{t}^{2}-a^{2}(\varphi)\varphi_{x}^{2}+% \sin^{2}\varphi\left[\psi_{t}^{2}-b^{2}(\varphi)\psi_{x}^{2}\right]\right),% \qquad a(\varphi):=\sqrt{\alpha\sin^{2}\varphi+\gamma\cos^{2}\varphi},\quad b(% \varphi):=\sqrt{\beta\sin^{2}\varphi+\gamma\cos^{2}\varphi}.
  19. ( v t + u v x ) x = 0 , u x x = v x 2 , (v_{t}+uv_{x})_{x}=0,\qquad u_{xx}=v_{x}^{2},
  20. [ ( u t + u u x ) x - 1 2 u x 2 ] x = 0 , \left[(u_{t}+uu_{x})_{x}-\frac{1}{2}\,u_{x}^{2}\right]_{x}=0,
  21. ( u t + u u x ) x x = u x u x x , (u_{t}+uu_{x})_{xx}=u_{x}u_{xx},
  22. u t - u x x t + 3 u u x = 2 u x u x x + u u x x x u_{t}-u_{xxt}+3uu_{x}=2u_{x}u_{xx}+uu_{xxx}
  23. ( x , t ) ( ϵ x , ϵ t ) , ϵ 0. (x,t)\mapsto(\epsilon x,\epsilon t),\qquad\epsilon\to 0.
  24. 2 = 1 2 u x 2 + w ( v t + u v x ) \mathcal{L}_{2}=\frac{1}{2}u_{x}^{2}+w(v_{t}+uv_{x})
  25. 1 = u x u t + u u x 2 , \mathcal{L}_{1}=u_{x}u_{t}+uu_{x}^{2},

Hurwitz_quaternion_order.html

  1. K K
  2. \mathbb{Q}
  3. ( ρ ) (\rho)
  4. ρ \rho
  5. K K
  6. [ η ] \mathbb{Z}[\eta]
  7. η = ρ + ρ ¯ \eta=\rho+\bar{\rho}
  8. 2 cos ( 2 π 7 ) 2\cos(\tfrac{2\pi}{7})
  9. D D
  10. D := ( η , η ) K , D:=\,(\eta,\eta)_{K},
  11. i 2 = j 2 = η i^{2}=j^{2}=\eta
  12. i j = - j i ij=-ji
  13. D . D.
  14. τ = 1 + η + η 2 \tau=1+\eta+\eta^{2}
  15. j = 1 2 ( 1 + η i + τ j ) j^{\prime}=\tfrac{1}{2}(1+\eta i+\tau j)
  16. 𝒬 Hur = [ η ] [ i , j , j ] . \mathcal{Q}_{\mathrm{Hur}}=\mathbb{Z}[\eta][i,j,j^{\prime}].
  17. 𝒬 Hur \mathcal{Q}_{\mathrm{Hur}}
  18. D D
  19. Q Hur Q_{\mathrm{Hur}}
  20. g 2 = 1 η i j g_{2}=\tfrac{1}{\eta}ij
  21. g 3 = 1 2 ( 1 + ( η 2 - 2 ) j + ( 3 - η 2 ) i j ) . g_{3}=\tfrac{1}{2}(1+(\eta^{2}-2)j+(3-\eta^{2})ij).
  22. [ η ] \mathbb{Z}[\eta]
  23. 1 , g 2 , g 3 , g 2 g 3 \,1,g_{2},g_{3},g_{2}g_{3}
  24. g 2 2 = g 3 3 = ( g 2 g 3 ) 7 = - 1 , g_{2}^{2}=g_{3}^{3}=(g_{2}g_{3})^{7}=-1,
  25. I [ η ] I\subset\mathbb{Z}[\eta]
  26. 𝒬 Hur 1 ( I ) = { x 𝒬 Hur 1 : x 1 ( \mathcal{Q}^{1}_{\mathrm{Hur}}(I)=\{x\in\mathcal{Q}_{\mathrm{Hur}}^{1}:x\equiv 1(
  27. I 𝒬 Hur ) } , I\mathcal{Q}_{\mathrm{Hur}})\},
  28. 𝒬 Hur \mathcal{Q}_{\mathrm{Hur}}
  29. I 𝒬 Hur I\mathcal{Q}_{\mathrm{Hur}}
  30. s y s > 4 3 log g sys>\frac{4}{3}\log g

Husimi_Q_representation.html

  1. Q ( α ) = 1 π α | ρ ^ | α , Q(\alpha)=\frac{1}{\pi}\langle\alpha|\hat{\rho}|\alpha\rangle,
  2. { | α } \{|\alpha\rangle\}
  3. Q ( α ) = 2 π W ( β ) e - 2 | α - β | 2 d 2 β . Q(\alpha)=\frac{2}{\pi}\int W(\beta)e^{-2|\alpha-\beta|^{2}}\,d^{2}\beta.
  4. Q ( α ) d α 2 = 1 \int Q(\alpha)\,d\alpha^{2}=1
  5. 0 Q ( α ) 1 π . 0\leq Q(\alpha)\leq\frac{1}{\pi}.
  6. Q Q
  7. α α

Hutchinson_operator.html

  1. { f i : X X | 1 i N } \{f_{i}:X\to X\ |\ 1\leq i\leq N\}
  2. X X
  3. H H
  4. S X S\subset X
  5. H ( S ) = i = 1 N f i ( S ) . H(S)=\bigcup_{i=1}^{N}f_{i}(S).\,
  6. A = H ( A ) A=H(A)
  7. S 0 X S_{0}\subset X
  8. H H
  9. S n + 1 = H ( S n ) = i = 1 N f i ( S n ) S_{n+1}=H(S_{n})=\bigcup_{i=1}^{N}f_{i}(S_{n})
  10. A = lim n S n . A=\lim_{n\to\infty}S_{n}.
  11. A A
  12. X X
  13. f i f_{i}

Hybrid_theory_for_photon_transport_in_tissue.html

  1. 1 - g 1-g
  2. g g
  3. g g
  4. z b = 2 C R D z_{b}=2C_{R}D
  5. z z
  6. z ± i = - z b + 2 i ( d + 2 z b ) ± ( z + z b ) z_{\pm i}=-z_{b}+2i(d+2z_{b})\pm(z^{\prime}+z_{b})
  7. z z^{^{\prime}}
  8. z z
  9. d d
  10. z c z_{c}
  11. R M C R_{MC}
  12. z c z - z c z_{c}\leq z\leq-z_{c}
  13. l t l_{t}^{\prime}
  14. S ( r , z ) S(r,z)
  15. S d S_{d}
  16. S d [ i r , i z ] = S [ i r , i z ] N Δ V ( i r ) S_{d}[i_{r},i_{z}]=\frac{S[i_{r},i_{z}]}{N\Delta V(i_{r})}
  17. Δ V \Delta V
  18. N N
  19. R D T R_{DT}
  20. R D T ( r ) = 0 0 0 2 π S d ( r , z ) R ( r , 0 , 0 ; r , ϕ , z ) r d ϕ d r d z R_{DT}(r)=\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{2\pi}S_{d}(r^{\prime},z^% {\prime})R(r,0,0;r^{\prime},\phi^{\prime},z^{\prime})r^{\prime}d\phi^{\prime}% dr^{\prime}dz^{\prime}
  21. R R
  22. ϕ \phi
  23. R D T R_{DT}
  24. R M C R_{MC}
  25. z c z_{c}
  26. n r e l n_{rel}
  27. d ( c m ) d(cm)
  28. μ a ( c m - 1 ) \mu_{a}(cm^{-1})
  29. T M C ( s ) T_{MC}(s)
  30. T H ( s ) T_{H}(s)
  31. T M C ( s ) / T H ( s ) T_{MC}(s)/T_{H}(s)
  32. n r e l n_{rel}
  33. d d
  34. μ a \mu_{a}
  35. T T
  36. μ a \mu_{a}
  37. c m - 1 cm^{-1}

Hydrostatic_weighing.html

  1. density of body density of water = weight of body weight of body - weight of immersed body \frac{\mbox{density of body}~{}}{\mbox{density of water}}=\frac{\mbox{weight % of body}~{}}{\mbox{weight of body}~{}-\mbox{weight of immersed body}~{}}\,
  2. density of body = density of water weight of body weight of body - weight of immersed body \mbox{density of body}~{}=\mbox{density of water}\frac{\mbox{weight of body}~{% }}{\mbox{weight of body}~{}-\mbox{weight of immersed body}~{}}\,
  3. density of body = density of water * weight of body ( weight of body - weight of immersed body ) - density of water * ( residual lung volume + 100 ml ) \mbox{density of body}~{}=\frac{\mbox{density of water}*\mbox{weight of body}~% {}}{(\mbox{weight of body}~{}-\mbox{weight of immersed body}~{})-\mbox{density% of water}*(\mbox{residual lung volume}~{}+\mbox{100 ml}~{})}\,

Hyperbolic_law_of_cosines.html

  1. - 1 k 2 -\frac{1}{k^{2}}
  2. cosh a k = cosh b k cosh c k - sinh b k sinh c k cos α , \cosh\frac{a}{k}=\cosh\frac{b}{k}\cosh\frac{c}{k}-\sinh\frac{b}{k}\sinh\frac{c% }{k}\cos\alpha,\,
  3. cos α = - cos β cos γ + sin β sin γ cosh a k , \cos\alpha=-\cos\beta\cos\gamma+\sin\beta\sin\gamma\cosh\frac{a}{k},\,

Hyperinteger.html

  1. [ x ] [x]
  2. [ ] * {}^{*}[\,\cdot\,]
  3. x = [ x ] * x={}^{*}\![x]
  4. * {}^{*}\mathbb{Z}
  5. * {}^{*}\mathbb{R}
  6. \mathbb{Z}
  7. * {}^{*}\mathbb{Z}\setminus\mathbb{Z}
  8. \mathbb{N}
  9. * {}^{*}\mathbb{N}

Hyperstability.html

  1. k 1 0 , k 2 0 k_{1}\geq 0,k_{2}\geq 0
  2. x ( t ) < k 1 x ( 0 ) + k 2 , t 0 \|x(t)\|<k_{1}\|x(0)\|+k_{2},\,\forall t\geq 0

Hypograph_(mathematics).html

  1. hyp f = { ( x , μ ) : x n , μ , μ f ( x ) } n + 1 \mbox{hyp}~{}f=\{(x,\mu)\,:\,x\in\mathbb{R}^{n},\,\mu\in\mathbb{R},\,\mu\leq f% (x)\}\subseteq\mathbb{R}^{n+1}
  2. hyp f S = { ( x , μ ) : x n , μ , μ < f ( x ) } n + 1 . \mbox{hyp}~{}_{S}f=\{(x,\mu)\,:\,x\in\mathbb{R}^{n},\,\mu\in\mathbb{R},\,\mu<f% (x)\}\subseteq\mathbb{R}^{n+1}.
  3. f - f\equiv-\infty
  4. n \mathbb{R}^{n}

Hypot.html

  1. r = x 2 + y 2 r=\sqrt{x^{2}+y^{2}}\,
  2. r \displaystyle r

Hypoxic_Training_Index.html

  1. H T i = 1 60 0 t [ 90 - S p O 2 ( t ) ] d t HTi=\frac{1}{60}\int\limits_{0}^{t}[90-SpO_{2}(t)]dt

Ice-sheet_dynamics.html

  1. Σ = k τ n , \Sigma=k\tau^{n},\,
  2. Σ \Sigma\,
  3. τ \tau\,
  4. n n\,
  5. k k\,

Ice-type_model.html

  1. E E
  2. E = n 1 ϵ 1 + n 2 ϵ 2 + + n 6 ϵ 6 , E=n_{1}\epsilon_{1}+n_{2}\epsilon_{2}+\ldots+n_{6}\epsilon_{6},
  3. ϵ 1 , , ϵ 6 \epsilon_{1},\ldots,\epsilon_{6}
  4. n i n_{i}
  5. i i
  6. ϵ i \epsilon_{i}
  7. i i
  8. Z Z
  9. Z = exp ( - E / k B T ) , Z=\sum\exp(-E/k_{B}T),
  10. E E
  11. k B k_{B}
  12. T T
  13. N N
  14. f f
  15. N N\to\infty
  16. f f
  17. f = - k B T N - 1 log Z . f=-k_{B}TN^{-1}\log Z.
  18. W W
  19. W = Z 1 / N . W=Z^{1/N}.
  20. f f
  21. W W
  22. f = - k B T log W . f=-k_{B}T\log W.
  23. ϵ 1 , , ϵ 6 \epsilon_{1},\ldots,\epsilon_{6}
  24. ϵ 1 = ϵ 2 = = ϵ 6 = 0 \epsilon_{1}=\epsilon_{2}=\ldots=\epsilon_{6}=0
  25. Z Z
  26. ϵ 1 = ϵ 2 = 0 , ϵ 3 = ϵ 4 = ϵ 5 = ϵ 6 > 0 \epsilon_{1}=\epsilon_{2}=0,\epsilon_{3}=\epsilon_{4}=\epsilon_{5}=\epsilon_{6% }>0
  27. F F
  28. ϵ 1 = ϵ 2 = ϵ 3 = ϵ 4 > 0 , ϵ 5 = ϵ 6 = 0. \epsilon_{1}=\epsilon_{2}=\epsilon_{3}=\epsilon_{4}>0,\epsilon_{5}=\epsilon_{6% }=0.
  29. ϵ 1 = ϵ 2 , ϵ 3 = ϵ 4 , ϵ 5 = ϵ 6 \epsilon_{1}=\epsilon_{2},\quad\epsilon_{3}=\epsilon_{4},\quad\epsilon_{5}=% \epsilon_{6}
  30. S S
  31. S = k B log Z = k B N log W , S=k_{B}\log Z=k_{B}\,N\,\log W,
  32. k B k_{B}
  33. N N
  34. Z = W N Z=W^{N}
  35. W = 4 W=4
  36. 2 N 2N
  37. W = 1.5 W=1.5
  38. S S
  39. S S
  40. W W
  41. W = 1.50685 ± 0.00015 W=1.50685\pm 0.00015
  42. W = 1.540 ± 0.001 W=1.540\pm 0.001
  43. F F
  44. W W
  45. W 2 D = ( 4 3 ) 3 / 2 = 1.5396007.... W_{2D}=\left(\frac{4}{3}\right)^{3/2}=1.5396007....
  46. W W
  47. W W
  48. W W
  49. W W
  50. W 2 D W_{2D}

Icosahedral_prism.html

  1. s { 3 3 2 } s\left\{\begin{array}[]{l}3\\ 3\\ 2\end{array}\right\}

Idealized_greenhouse_model.html

  1. F = σ T 4 F=\sigma T^{4}
  2. F = ϵ σ T a 4 + ( 1 - ϵ ) σ T s 4 F\uparrow=\epsilon\sigma T_{a}^{4}+(1-\epsilon)\sigma T_{s}^{4}
  3. - 1 4 S 0 ( 1 - α p ) + ϵ σ T a 4 + ( 1 - ϵ ) σ T s 4 = 0 -\frac{1}{4}S_{0}(1-\alpha_{p})+\epsilon\sigma T_{a}^{4}+(1-\epsilon)\sigma T_% {s}^{4}=0
  4. 1 4 S 0 ( 1 - α p ) + ϵ σ T a 4 - σ T s 4 = 0 \frac{1}{4}S_{0}(1-\alpha_{p})+\epsilon\sigma T_{a}^{4}-\sigma T_{s}^{4}=0
  5. 2 ϵ σ T a 4 - ϵ σ T s 4 = 0 2\epsilon\sigma T_{a}^{4}-\epsilon\sigma T_{s}^{4}=0
  6. T a = T s 2 1 / 4 = T s 1.189 T_{a}={T_{s}\over 2^{1/4}}={T_{s}\over 1.189}
  7. 1 4 S 0 ( 1 - α p ) = ( 1 - ϵ 2 ) σ T s 4 \frac{1}{4}S_{0}(1-\alpha_{p})=\left(1-\frac{\epsilon}{2}\right)\sigma T_{s}^{4}
  8. T s = [ S 0 ( 1 - α p ) 4 σ 1 1 - ϵ 2 ] 1 / 4 T_{s}=\left[\frac{S_{0}(1-\alpha_{p})}{4\sigma}\frac{1}{1-{\epsilon\over 2}}% \right]^{1/4}
  9. T e [ S 0 ( 1 - α p ) 4 σ ] 1 / 4 T_{e}\equiv\left[\frac{S_{0}(1-\alpha_{p})}{4\sigma}\right]^{1/4}
  10. T s = T e [ 1 1 - ϵ 2 ] 1 / 4 T_{s}=T_{e}\left[\frac{1}{1-{\epsilon\over 2}}\right]^{1/4}
  11. T s = T e 2 1 / 4 = 1.189 T e T a = T e T_{s}=T_{e}2^{1/4}=1.189T_{e}\qquad T_{a}=T_{e}
  12. T e = 255 K = - 18 C T_{e}=255~{}\mathrm{K}=-18~{}\mathrm{C}
  13. T s = 303 K = 30 C T_{s}=303~{}\mathrm{K}=30~{}\mathrm{C}
  14. T s = 288.3 K T a = 242.5 K T_{s}=288.3~{}\mathrm{K}\qquad T_{a}=242.5~{}\mathrm{K}
  15. F F\uparrow
  16. Δ F = Δ ϵ ( σ T a 4 - σ T s 4 ) \Delta F\uparrow=\Delta\epsilon\left(\sigma T_{a}^{4}-\sigma T_{s}^{4}\right)
  17. Δ F \Delta F\uparrow
  18. T s = 289.5 K T_{s}=289.5~{}\mathrm{K}

Idempotent_measure.html

  1. ( μ * ν ) ( A ) = X μ ( A x - 1 ) d ν ( x ) = X ν ( x - 1 A ) d μ ( x ) (\mu*\nu)(A)=\int_{X}\mu(Ax^{-1})\,\mathrm{d}\nu(x)=\int_{X}\nu(x^{-1}A)\,% \mathrm{d}\mu(x)

Image_filter_end_terminations.html

  1. Z I 1 Z_{I1}\,\!
  2. Z I 2 Z_{I2}\,\!
  3. Z I Z_{I}\,\!
  4. R 1 R_{1}\,\!
  5. R 2 R_{2}\,\!
  6. R R\,\!
  7. r I 1 r_{I1}\,\!
  8. r I 2 r_{I2}\,\!
  9. r I r_{I}\,\!
  10. τ I 1 \tau_{I1}\,\!
  11. τ I 2 \tau_{I2}\,\!
  12. γ \gamma\,\!
  13. α \alpha\,\!
  14. β \beta\,\!
  15. A ( i ω ) = V o V i = Z I 2 Z I 1 e - γ [ τ I 1 τ I 2 1 - e - 2 γ r I 1 r I 2 ] A(i\omega)=\frac{V_{o}}{V_{i}}=\sqrt{\frac{Z_{I2}}{Z_{I1}}}e^{-\gamma}\left[% \frac{\tau_{I1}\tau_{I2}}{1-e^{-2\gamma}r_{I1}r_{I2}}\right]
  16. r I 1 = R 1 - Z I 1 R 1 + Z I 1 r_{I1}=\frac{R_{1}-Z_{I1}}{R_{1}+Z_{I1}}
  17. r I 2 = R 2 - Z I 2 R 2 + Z I 2 r_{I2}=\frac{R_{2}-Z_{I2}}{R_{2}+Z_{I2}}
  18. τ I 1 = 2 Z I 1 R 1 + Z I 1 \tau_{I1}=\frac{2Z_{I1}}{R_{1}+Z_{I1}}
  19. τ I 2 = 2 R 2 R 2 + Z I 2 \tau_{I2}=\frac{2R_{2}}{R_{2}+Z_{I2}}
  20. Z I 2 Z I 1 e - γ \sqrt{\frac{Z_{I2}}{Z_{I1}}}e^{-\gamma}
  21. [ τ I 1 τ I 2 1 - e - 2 γ r I 1 r I 2 ] . \left[\frac{\tau_{I1}\tau_{I2}}{1-e^{-2\gamma}r_{I1}r_{I2}}\right].
  22. A ( i ω ) = e - γ [ 4 Z I R ( R + Z I ) 2 - e - 2 γ ( R - Z I ) 2 ] A(i\omega)=e^{-\gamma}\left[\frac{4Z_{I}R}{(R+Z_{I})^{2}-e^{-2\gamma}(R-Z_{I})% ^{2}}\right]
  23. | A | = 1 1 + ξ 2 \left|A\right|=\frac{1}{\sqrt{1+\xi^{2}}}
  24. ξ = 1 2 ( R I R - R R I ) sin β \xi=\frac{1}{2}\left(\frac{R_{I}}{R}-\frac{R}{R_{I}}\right)\sin\beta
  25. ξ = 1 2 ( X I R - R X I ) sinh β . \xi=\frac{1}{2}\left(\frac{X_{I}}{R}-\frac{R}{X_{I}}\right)\sinh\beta.
  26. Z I 1 Z I 2 = R 1 R 2 = R o 2 Z_{I1}Z_{I2}=R_{1}R_{2}=R_{o}^{2}
  27. ξ = 1 2 ( R I 1 R - R R I 1 ) cos β . \xi=\frac{1}{2}\left(\frac{R_{I1}}{R}-\frac{R}{R_{I1}}\right)\cos\beta.
  28. ξ = 1 2 ( X I 1 R - R X I 1 ) sinh β . \xi=\frac{1}{2}\left(\frac{X_{I1}}{R}-\frac{R}{X_{I1}}\right)\sinh\beta.

Image_impedance.html

  1. Z i 1 = Z + 1 2 Y + 1 Z + Z i 1 Z_{i1}=Z+\frac{1}{2Y+\frac{1}{Z+Z_{i1}}}
  2. Z i 1 2 = Z 2 + Z Y Z_{i1}^{2}=Z^{2}+\frac{Z}{Y}
  3. Y i 2 2 = Y 2 + Y Z Y_{i2}^{2}=Y^{2}+\frac{Y}{Z}
  4. Z i 1 Y i 2 = Z Y \frac{Z_{i1}}{Y_{i2}}=\frac{Z}{Y}
  5. Z i 1 = Z SC Z OC Z_{i1}=\sqrt{Z_{\mathrm{SC}}Z_{\mathrm{OC}}}
  6. Z i 2 Z Y Z_{i}^{2}\rightarrow\frac{Z}{Y}
  7. A ( i ω ) = Z I 2 Z I 1 e - γ A(i\omega)=\sqrt{\frac{Z_{I2}}{Z_{I1}}}e^{-\gamma}
  8. γ γ
  9. γ = sinh - 1 Z Y \gamma=\sinh^{-1}{\sqrt{ZY}}
  10. Z I 2 Z I 1 \sqrt{\frac{Z_{I2}}{Z_{I1}}}
  11. γ γ
  12. A ( i ω ) = e - γ A(i\omega)=e^{-\gamma}\,\!
  13. γ γ
  14. γ = α + i β \gamma=\alpha+i\beta\,\!
  15. γ γ
  16. α α
  17. β β
  18. γ n = n γ \gamma_{n}=n\gamma\,\!
  19. γ Z Y \gamma\rightarrow\sqrt{ZY}
  20. α α
  21. β β
  22. γ γ
  23. Z Z
  24. Y Y
  25. A D B C = 1 AD−BC=1
  26. Z I 1 = A B C D Z_{I1}=\sqrt{\frac{AB}{CD}}
  27. Z I 2 = D B C A Z_{I2}=\sqrt{\frac{DB}{CA}}
  28. γ γ
  29. γ = cosh - 1 A D \gamma=\cosh^{-1}\sqrt{AD}

Imaginary_hyperelliptic_curve.html

  1. g 1 g\geq 1
  2. K K
  3. K K
  4. K ¯ \overline{K}
  5. g g
  6. K K
  7. C : y 2 + h ( x ) y = f ( x ) K [ x , y ] C:y^{2}+h(x)y=f(x)\in K[x,y]
  8. h ( x ) K [ x ] h(x)\in K[x]
  9. g g
  10. f ( x ) K [ x ] f(x)\in K[x]
  11. 2 g + 1 2g+1
  12. ( x , y ) K ¯ × K ¯ (x,y)\in\overline{K}\times\overline{K}
  13. y 2 + h ( x ) y = f ( x ) y^{2}+h(x)y=f(x)
  14. 2 y + h ( x ) = 0 2y+h(x)=0
  15. h ( x ) y = f ( x ) h^{\prime}(x)y=f^{\prime}(x)
  16. f f
  17. 2 g + 2 2g+2
  18. g = 1 g=1
  19. f f
  20. 2 ( K ) \mathbb{P}^{2}(K)
  21. ( X : Y : Z ) (X:Y:Z)
  22. ( 0 : 1 : 0 ) (0:1:0)
  23. O O
  24. C = { ( x , y ) K 2 | y 2 + h ( x ) y = f ( x ) } { O } C=\{(x,y)\in K^{2}|y^{2}+h(x)y=f(x)\}\cup\{O\}
  25. P = ( a , b ) P=(a,b)
  26. O O
  27. P ¯ = ( a , - b - h ( a ) ) \overline{P}=(a,-b-h(a))
  28. ( - b - h ( a ) ) 2 + h ( a ) ( - b - h ( a ) ) (-b-h(a))^{2}+h(a)(-b-h(a))
  29. b 2 + h ( a ) b b^{2}+h(a)b
  30. P ¯ \overline{P}
  31. P ¯ \overline{P}
  32. P P
  33. P P
  34. P = P ¯ P=\overline{P}
  35. h ( a ) = - 2 b h(a)=-2b
  36. O O
  37. O ¯ = O \overline{O}=O
  38. K K
  39. x x x\rightarrow x
  40. y y - h ( x ) 2 y\rightarrow y-\frac{h(x)}{2}
  41. ( K ) 2 (K)\not=2
  42. y 2 + h ( x ) y = f ( x ) y^{2}+h(x)y=f(x)
  43. ( y - h ( x ) 2 ) 2 + h ( x ) ( y - h ( x ) 2 ) = f ( x ) \left(y-\frac{h(x)}{2}\right)^{2}+h(x)\left(y-\frac{h(x)}{2}\right)=f(x)
  44. y 2 = f ( x ) + h ( x ) 2 4 y^{2}=f(x)+\frac{h(x)^{2}}{4}
  45. deg ( h ) g \deg(h)\leq g
  46. deg ( h 2 ) 2 g \deg(h^{2})\leq 2g
  47. f ( x ) + h ( x ) 2 4 f(x)+\frac{h(x)^{2}}{4}
  48. 2 g + 1 2g+1
  49. K K
  50. ( K ) 2 (K)\not=2
  51. g g
  52. C : y 2 = f ( x ) C:y^{2}=f(x)
  53. f f
  54. 2 g + 1 2g+1
  55. P = ( a , b ) P=(a,b)
  56. b = 0 b=0
  57. f ( a ) = 0 f^{\prime}(a)=0
  58. b = 0 b=0
  59. b 2 = f ( a ) b^{2}=f(a)
  60. f ( a ) = 0 f(a)=0
  61. a a
  62. f f
  63. C : y 2 = f ( x ) C:y^{2}=f(x)
  64. f f
  65. ( K ) 2 (K)\not=2
  66. C : y 2 = f ( x ) C:y^{2}=f(x)
  67. f ( x ) = x 5 - 2 x 4 - 7 x 3 + 8 x 2 + 12 x = x ( x + 1 ) ( x - 3 ) ( x + 2 ) ( x - 2 ) f(x)=x^{5}-2x^{4}-7x^{3}+8x^{2}+12x=x(x+1)(x-3)(x+2)(x-2)
  68. \mathbb{R}
  69. f f
  70. C C
  71. g = 2 g=2
  72. O O
  73. C C
  74. C C
  75. C C
  76. K K
  77. K [ C ] = K [ x , y ] / ( y 2 + h ( x ) y - f ( x ) ) \;K[C]=K[x,y]/(y^{2}+h(x)y-f(x))
  78. r ( x , y ) = y 2 + h ( x ) y - f ( x ) \,r(x,y)=y^{2}+h(x)y-f(x)
  79. K ¯ \overline{K}
  80. K ¯ [ C ] = K ¯ [ x , y ] / ( y 2 + h ( x ) y - f ( x ) ) \overline{K}[C]=\overline{K}[x,y]/(y^{2}+h(x)y-f(x))
  81. r ( x , y ) r(x,y)
  82. K ¯ \overline{K}
  83. ( y - u ( x ) ) [ u \xb 7 ] ( y - v ( x ) ) (y-u(x))[u^{\prime}\xb 7^{\prime}](y-v(x))
  84. u , v u,v
  85. K ¯ \overline{K}
  86. u ( x ) [ u \xb 7 ] v ( x ) = f ( x ) u(x)[u^{\prime}\xb 7^{\prime}]v(x)=f(x)
  87. 2 g + 1 2g+1
  88. u ( x ) + v ( x ) = h ( x ) u(x)+v(x)=h(x)
  89. g g
  90. G ( x , y ) K ¯ [ C ] G(x,y)\in\overline{K}[C]
  91. G ( x , y ) = u ( x ) - v ( x ) y \,G(x,y)=u(x)-v(x)y
  92. u ( x ) u(x)
  93. v ( x ) v(x)
  94. K ¯ [ x ] \overline{K}[x]
  95. G ( x , y ) = u ( x ) - v ( x ) y G(x,y)=u(x)-v(x)y
  96. K ¯ [ C ] \overline{K}[C]
  97. G ¯ ( x , y ) = u ( x ) + v ( x ) ( h ( x ) + y ) \overline{G}(x,y)=u(x)+v(x)(h(x)+y)
  98. G G
  99. N ( G ) = G G ¯ N(G)=G\overline{G}
  100. N ( G ) N(G)
  101. G ( x , y ) = u ( x ) - v ( x ) [ u \xb 7 ] y G(x,y)=u(x)-v(x)[u^{\prime}\xb 7^{\prime}]y
  102. G G
  103. deg ( G ) = max [ 2 deg ( u ) , 2 g + 1 + 2 deg ( v ) ] \,\deg(G)=\max[2\deg(u),2g+1+2\deg(v)]
  104. deg ( G ) = deg x ( N ( G ) ) \;\deg(G)=\deg_{x}(N(G))
  105. deg ( G H ) = deg ( G ) + deg ( H ) \;\deg(GH)=\deg(G)+\deg(H)
  106. deg ( G ) = deg ( G ¯ ) \deg(G)=\deg(\overline{G})
  107. K ( C ) K(C)
  108. C C
  109. K K
  110. K C C KCC
  111. K ¯ ( C ) \overline{K}(C)
  112. C C
  113. K ¯ \overline{K}
  114. K ¯ [ C ] \overline{K}[C]
  115. K ¯ ( C ) \overline{K}(C)
  116. C C
  117. R R
  118. P P
  119. C C
  120. R R
  121. P P
  122. G , H G,H
  123. R = G / H R=G/H
  124. H ( P ) 0 H(P)≠0
  125. R R
  126. P P
  127. R ( P ) = G ( P ) / H ( P ) \,R(P)=G(P)/H(P)
  128. P P
  129. C C
  130. P P
  131. O O
  132. R ( P ) R(P)
  133. deg ( G ) < deg ( H ) \;\deg(G)<\deg(H)
  134. R ( O ) = 0 R(O)=0
  135. deg ( G ) > deg ( H ) \;\deg(G)>\deg(H)
  136. R ( O ) R(O)
  137. deg ( G ) = deg ( H ) \;\deg(G)=\deg(H)
  138. R ( O ) R(O)
  139. G G
  140. H H
  141. R K ¯ ( C ) * R\in\overline{K}(C)^{*}
  142. P C P\in C
  143. R ( P ) = 0 \;R(P)=0
  144. R R
  145. P P
  146. R R
  147. P P
  148. R R
  149. P P
  150. R ( P ) = R(P)=\infty
  151. G = u ( x ) - v ( x ) y K ¯ [ C ] 2 G=u(x)-v(x)\cdot y\in\overline{K}[C]^{2}
  152. P C P\in C
  153. G G
  154. P P
  155. o r d P ( G ) = r + s \;ord_{P}(G)=r+s
  156. P = ( a , b ) P=(a,b)
  157. r r
  158. ( x - a ) (x-a)
  159. u ( x ) u(x)
  160. v ( x ) v(x)
  161. s s
  162. ( x - a ) (x-a)
  163. s = 0 s=0
  164. o r d P ( G ) = 2 r + s \;ord_{P}(G)=2r+s
  165. P = ( a , b ) P=(a,b)
  166. r r
  167. s s
  168. o r d P ( G ) = - deg ( G ) \;ord_{P}(G)=-\deg(G)
  169. P = O P=O
  170. C C
  171. K K
  172. D D
  173. C C
  174. D = P C c P [ P ] D=\sum_{P\in C}{c_{P}[P]}
  175. c P \Z c_{P}\in\Z
  176. { c P | c P 0 } \{c_{P}|c_{P}\not=0\}
  177. c P [ P ] c_{P}[P]
  178. D D
  179. deg ( D ) = P C c P \Z \deg(D)=\sum_{P\in C}{c_{P}}\in\Z
  180. Div ( C ) \mathrm{Div}(C)
  181. C C
  182. P C c P [ P ] + P C d P [ P ] = P C ( c P + d P ) [ P ] \sum_{P\in C}{c_{P}[P]}+\sum_{P\in C}{d_{P}[P]}=\sum_{P\in C}{(c_{P}+d_{P})[P]}
  183. 0 = P C 0 [ P ] 0=\sum_{P\in C}{0[P]}
  184. P C c P [ P ] \sum_{P\in C}{c_{P}[P]}
  185. P C - c P [ P ] \sum_{P\in C}{-c_{P}[P]}
  186. Div 0 ( C ) = { D Div ( C ) | deg ( D ) = 0 } \mathrm{Div}^{0}(C)=\{D\in\mathrm{Div}(C)|\deg(D)=0\}
  187. Div ( C ) \mathrm{Div}(C)
  188. φ : Div ( C ) \varphi:\mathrm{Div}(C)\rightarrow\mathbb{Z}
  189. φ ( D ) = deg ( D ) \varphi(D)=\deg(D)
  190. \mathbb{Z}
  191. φ ( P C c P [ P ] + P C d P [ P ] ) = φ ( P C ( c P + d P ) [ P ] ) = P C c P + d P = P C c p + P C d p = φ ( P C c P [ P ] ) + φ ( P C d P [ P ] ) \varphi(\sum_{P\in C}{c_{P}[P]}+\sum_{P\in C}{d_{P}[P]})=\varphi(\sum_{P\in C}% {(c_{P}+d_{P})[P]})=\sum_{P\in C}{c_{P}+d_{P}}=\sum_{P\in C}{c_{p}}+\sum_{P\in C% }{d_{p}}=\varphi(\sum_{P\in C}{c_{P}[P]})+\varphi(\sum_{P\in C}{d_{P}[P]})
  192. φ \varphi
  193. Div 0 ( C ) \mathrm{Div}^{0}(C)
  194. Div ( C ) \mathrm{Div}(C)
  195. f K ¯ ( C ) * f\in\overline{K}(C)^{*}
  196. ( f ) = P C ord P ( f ) [ P ] (f)=\sum_{P\in C}{\mathrm{ord}_{P}(f)[P]}
  197. ( f ) P {}_{P}(f)
  198. f f
  199. P P
  200. ( f ) P < 0 {}_{P}(f)<0
  201. f f
  202. ( f ) P {}_{P}(f)
  203. P P
  204. ( f ) P = 0 {}_{P}(f)=0
  205. f f
  206. P P
  207. ( f ) P > 0 {}_{P}(f)>0
  208. f f
  209. ( f ) P {}_{P}(f)
  210. P P
  211. f f
  212. ( f ) P {}_{P}(f)
  213. ( f ) (f)
  214. P C ord P ( f ) = 0 \sum_{P\in C}{\mathrm{ord}_{P}(f)}=0
  215. ( f ) (f)
  216. f f
  217. Princ ( C ) \mathrm{Princ}(C)
  218. Div 0 ( C ) \mathrm{Div}^{0}(C)
  219. 0 = P C 0 [ P ] 0=\sum_{P\in C}{0[P]}
  220. D 1 = P C ord P ( f ) [ P ] , D 2 = P C ord P ( g ) [ P ] Princ ( C ) D_{1}=\sum_{P\in C}{\mathrm{ord}_{P}(f)[P]},D_{2}=\sum_{P\in C}{\mathrm{ord}_{% P}(g)[P]}\in\mathrm{Princ}(C)
  221. f f
  222. g g
  223. D 1 - D 2 = P C ( ord P ( f ) - ord P ( g ) ) [ P ] D_{1}-D_{2}=\sum_{P\in C}{(\mathrm{ord}_{P}(f)-\mathrm{ord}_{P}(g))[P]}
  224. f / g f/g
  225. D 1 - D 2 D_{1}-D_{2}
  226. Princ ( C ) \mathrm{Princ}(C)
  227. J ( C ) := Div 0 ( C ) / Princ ( C ) J(C):=\mathrm{Div}^{0}(C)/\mathrm{Princ}(C)
  228. C C
  229. D 1 , D 2 Div 0 ( C ) D_{1},D_{2}\in\mathrm{Div}^{0}(C)
  230. J ( C ) J(C)
  231. D 1 - D 2 D_{1}-D_{2}
  232. C : y 2 + h ( x ) y = f ( x ) C:y^{2}+h(x)y=f(x)
  233. K K
  234. P = ( a , b ) P=(a,b)
  235. C C
  236. n > 0 n\in\mathbb{Z}_{>0}
  237. f ( x ) = ( x - a ) n f(x)=(x-a)^{n}
  238. n n
  239. P P
  240. P ¯ \overline{P}
  241. 2 n 2n
  242. O O
  243. ( f ) = n P + n P ¯ - 2 n O (f)=nP+n\overline{P}-2nO
  244. ( f ) = 2 n P - 2 n O (f)=2nP-2nO
  245. P P
  246. E E
  247. K K
  248. D D
  249. P P
  250. P P
  251. E E
  252. D P = 1 [ P ] - 1 [ O ] D_{P}=1[P]-1[O]
  253. O O
  254. O - O = 0 O-O=0
  255. J ( E ) J(E)
  256. P P
  257. P P
  258. D P D_{P}
  259. [ D P ] [D_{P}]
  260. φ : E J ( E ) \varphi:E\rightarrow J(E)
  261. E E
  262. E E
  263. φ ( P ) = [ D P ] \varphi(P)=[D_{P}]
  264. E E
  265. O O
  266. P , Q , R E P,Q,R\in E
  267. P + Q + R = O P+Q+R=O
  268. P + Q = - R P+Q=-R
  269. P P
  270. Q Q
  271. P P
  272. Q Q
  273. P + Q P+Q
  274. P + Q + R = O P+Q+R=O
  275. f K [ x , y ] f\in K[x,y]
  276. P P
  277. Q Q
  278. R R
  279. f ( x , y ) = 0 f(x,y)=0
  280. φ ( P ) + φ ( Q ) + φ ( R ) = [ D P ] + [ D Q ] + [ D R ] = [ [ P ] + [ Q ] + [ R ] - 3 [ O ] ] \varphi(P)+\varphi(Q)+\varphi(R)=[D_{P}]+[D_{Q}]+[D_{R}]=[[P]+[Q]+[R]-3[O]]
  281. J ( E ) J(E)
  282. [ P ] + [ Q ] + [ R ] - 3 [ O ] [P]+[Q]+[R]-3[O]
  283. f f
  284. φ ( P ) + φ ( Q ) + φ ( R ) = φ ( O ) = φ ( P + Q + R ) \varphi(P)+\varphi(Q)+\varphi(R)=\varphi(O)=\varphi(P+Q+R)
  285. D = P E c P [ P ] D=\sum_{P\in E}{c_{P}[P]}
  286. D D
  287. P E c P P = O \sum_{P\in E}{c_{P}P}=O
  288. D 1 , D 2 Div 0 ( E ) D_{1},D_{2}\in\mathrm{Div}^{0}(E)
  289. D 1 - D 2 D_{1}-D_{2}
  290. D 1 = P E c P [ P ] D_{1}=\sum_{P\in E}{c_{P}[P]}
  291. D 2 = P E d P [ P ] D_{2}=\sum_{P\in E}{d_{P}[P]}
  292. P E c P P = P E d P P \sum_{P\in E}{c_{P}P}=\sum_{P\in E}{d_{P}P}
  293. [ P ] - [ O ] [P]-[O]
  294. E E
  295. [ D P ] [D_{P}]
  296. [ D P ] = [ 1 [ P ] - 1 [ O ] ] [D_{P}]=[1[P]-1[O]]
  297. ( P - O ) + O = P (P-O)+O=P
  298. 0 + O = O 0+O=O
  299. ψ : J ( E ) E \psi:J(E)\rightarrow E
  300. ψ ( [ D P ] ) = P \psi([D_{P}])=P
  301. φ \varphi
  302. φ \varphi
  303. E E
  304. J ( E ) J(E)
  305. C : y 2 + h ( x ) y = f ( x ) C:y^{2}+h(x)y=f(x)
  306. g g
  307. K K
  308. D D
  309. C C
  310. D = i = 1 k [ P i ] - k [ O ] D=\sum_{i=1}^{k}{[P_{i}]}-k[O]
  311. k g k\leq g
  312. P i O P_{i}\not=O
  313. i = 1 , , k i=1,...,k
  314. P i P j ¯ P_{i}\not=\overline{P_{j}}
  315. i j i\not=j
  316. P i = P j P_{i}=P_{j}
  317. i j i\not=j
  318. P i P_{i}
  319. D Div 0 ( C ) D\in\mathrm{Div}^{0}(C)
  320. D D^{\prime}
  321. D D
  322. D D^{\prime}
  323. J ( C ) J(C)
  324. J ( C ) J(C)
  325. u , v K [ x ] u,v\in K[x]
  326. u u
  327. deg ( v ) < deg ( u ) g \deg(v)<\deg(u)\leq g
  328. u | v 2 + v h - f u|v^{2}+vh-f
  329. u ( x ) = i = 1 k ( x - x i ) u(x)=\prod_{i=1}^{k}{(x-x_{i})}
  330. K ¯ \overline{K}
  331. u u
  332. u u
  333. v v
  334. P i = ( x i , v ( x i ) ) P_{i}=(x_{i},v(x_{i}))
  335. C C
  336. i = 1 , , k i=1,...,k
  337. D = i = 1 k [ P i ] - k [ O ] D=\sum_{i=1}^{k}{[P_{i}]}-k[O]
  338. deg ( u ) g \deg(u)\leq g
  339. k g k\leq g
  340. 0 = P C 0 [ P ] 0=\sum_{P\in C}{0[P]}
  341. J ( C ) J(C)
  342. u ( x ) = 1 u(x)=1
  343. v ( x ) = 0 v(x)=0
  344. C : y 2 = x 5 - 4 x 4 - 14 x 3 + 36 x 2 + 45 x = x ( x + 1 ) ( x - 3 ) ( x + 3 ) ( x - 5 ) C:y^{2}=x^{5}-4x^{4}-14x^{3}+36x^{2}+45x=x(x+1)(x-3)(x+3)(x-5)
  345. P = ( 1 , 8 ) P=(1,8)
  346. Q = ( 3 , 0 ) Q=(3,0)
  347. R = ( 5 , 0 ) R=(5,0)
  348. D = [ P ] + [ Q ] - 2 [ O ] D=[P]+[Q]-2[O]
  349. D = [ P ] + [ R ] - 2 [ O ] D^{\prime}=[P]+[R]-2[O]
  350. D D
  351. u u
  352. v v
  353. deg ( v ) < deg ( u ) g = 2 \deg(v)<\deg(u)\leq g=2
  354. P P
  355. Q Q
  356. u u
  357. u ( x ) = ( x - 1 ) ( x - 3 ) = x 2 - 4 x + 3 u(x)=(x-1)(x-3)=x^{2}-4x+3
  358. P = ( 1 , v ( 1 ) ) P=(1,v(1))
  359. Q = ( 3 , v ( 3 ) ) Q=(3,v(3))
  360. v ( 1 ) = 8 v(1)=8
  361. v ( 3 ) = 0 v(3)=0
  362. v v
  363. v ( x ) = - 4 x + 12 v(x)=-4x+12
  364. D D
  365. u ( x ) = x 2 - 4 x + 3 u(x)=x^{2}-4x+3
  366. v ( x ) = - 4 x + 12 v(x)=-4x+12
  367. ( u , v ) (u^{\prime},v^{\prime})
  368. D D^{\prime}
  369. u ( x ) = ( x - 1 ) ( x - 5 ) = x 2 - 6 x + 5 u^{\prime}(x)=(x-1)(x-5)=x^{2}-6x+5
  370. v ( x ) = - 2 x + 10 v(x)=-2x+10
  371. P i = ( x i , y i ) P_{i}=(x_{i},y_{i})
  372. ( d d t ) j [ v ( x ) 2 + v ( x ) h ( x ) - f ( x ) ] | x = x i = 0 \left(\frac{d}{dt}\right)^{j}\left[v(x)^{2}+v(x)h(x)-f(x)\right]_{|_{x=x_{i}}}=0
  373. 0 j n i - 1 0\leq j\leq n_{i}-1
  374. D 1 D_{1}
  375. D 2 D_{2}
  376. D D
  377. D D
  378. D 1 + D 2 D_{1}+D_{2}
  379. h ( x ) = 0 h(x)=0
  380. D 1 = ( u 1 , v 1 ) D_{1}=(u_{1},v_{1})
  381. D 2 = ( u 2 , v 2 ) D_{2}=(u_{2},v_{2})
  382. C : y 2 + h ( x ) y = f ( x ) C:y^{2}+h(x)y=f(x)
  383. g g
  384. K K
  385. d 1 , e 1 , e 2 K [ x ] d_{1},e_{1},e_{2}\in K[x]
  386. d 1 = gcd ( u 1 , u 2 ) d_{1}=\gcd(u_{1},u_{2})
  387. d 1 = e 1 u 1 + e 2 u 2 d_{1}=e_{1}u_{1}+e_{2}u_{2}
  388. d , c 1 , c 2 K [ x ] d,c_{1},c_{2}\in K[x]
  389. d = gcd ( d 1 , v 1 + v 2 + h ) d=\gcd(d_{1},v_{1}+v_{2}+h)
  390. d = c 1 d 1 + c 2 ( v 1 + v 2 + h ) d=c_{1}d_{1}+c_{2}(v_{1}+v_{2}+h)
  391. s 1 = c 1 e 1 s_{1}=c_{1}e_{1}
  392. s 2 = c 1 e 2 s_{2}=c_{1}e_{2}
  393. s 3 = c 2 s_{3}=c_{2}
  394. d = s 1 u 1 + s 2 u 2 + s 3 ( v 1 + v 2 + h ) d=s_{1}u_{1}+s_{2}u_{2}+s_{3}(v_{1}+v_{2}+h)
  395. u = u 1 u 2 d 2 u=\frac{u_{1}u_{2}}{d^{2}}
  396. v = s 1 u 1 v 2 + s 2 u 2 v 1 + s 3 ( v 1 v 2 + f ) d mod u v=\frac{s_{1}u_{1}v_{2}+s_{2}u_{2}v_{1}+s_{3}(v_{1}v_{2}+f)}{d}\mod u
  397. u = f - v h - v 2 u u^{\prime}=\frac{f-vh-v^{2}}{u}
  398. v = - h - v mod u v^{\prime}=-h-v\mod u^{\prime}
  399. deg ( u ) > g \deg(u^{\prime})>g
  400. u = u u=u^{\prime}
  401. v = v v=v^{\prime}
  402. deg ( u ) g \deg(u^{\prime})\leq g
  403. u u^{\prime}
  404. D = ( u , v ) D=(u^{\prime},v^{\prime})
  405. C : y 2 = x 5 - 4 x 4 - 14 x 3 + 36 x 2 + 45 x = x ( x + 1 ) ( x - 3 ) ( x + 3 ) ( x - 5 ) C:y^{2}=x^{5}-4x^{4}-14x^{3}+36x^{2}+45x=x(x+1)(x-3)(x+3)(x-5)
  406. P = ( 1 , 8 ) P=(1,8)
  407. Q = ( 3 , 0 ) Q=(3,0)
  408. R = ( 5 , 0 ) R=(5,0)
  409. D 1 = [ P ] + [ Q ] - 2 [ O ] D_{1}=[P]+[Q]-2[O]
  410. D 2 = [ P ] + [ R ] - 2 [ O ] D_{2}=[P]+[R]-2[O]
  411. ( u 1 = x 2 - 4 x + 3 , v 1 = - 4 x + 12 ) (u_{1}=x^{2}-4x+3,v_{1}=-4x+12)
  412. ( u 2 = x 2 - 6 x + 5 , v 2 = - 2 x + 10 ) (u_{2}=x^{2}-6x+5,v_{2}=-2x+10)
  413. D 1 D_{1}
  414. D 2 D_{2}
  415. d 1 = gcd ( u 1 , u 2 ) = x - 1 d_{1}=\gcd(u_{1},u_{2})=x-1
  416. d 1 = e 1 u 1 + e 2 u 2 d_{1}=e_{1}u_{1}+e_{2}u_{2}
  417. e 1 = 1 2 e_{1}=\frac{1}{2}
  418. e 2 = - 1 2 e_{2}=-\frac{1}{2}
  419. d = gcd ( d 1 , v 1 + v 2 + h ) = gcd ( x - 1 , - 6 x + 22 ) = 1 d=\gcd(d_{1},v_{1}+v_{2}+h)=\gcd(x-1,-6x+22)=1
  420. 1 = c 1 d 1 + c 2 ( v 1 + v 2 + h ) 1=c_{1}d_{1}+c_{2}(v_{1}+v_{2}+h)
  421. c 1 = 3 8 c_{1}=\frac{3}{8}
  422. c 2 = 1 16 c_{2}=\frac{1}{16}
  423. s 1 = c 1 e 1 = 3 16 s_{1}=c_{1}e_{1}=\frac{3}{16}
  424. s 2 = c 1 e 2 = - 3 16 s_{2}=c_{1}e_{2}=-\frac{3}{16}
  425. s 3 = c 2 = 1 16 s_{3}=c_{2}=\frac{1}{16}
  426. u = u 1 u 2 d 2 = u 1 u 2 = x 4 - 10 x 3 + 32 x 2 - 38 x + 15 u=\frac{u_{1}u_{2}}{d^{2}}=u_{1}u_{2}=x^{4}-10x^{3}+32x^{2}-38x+15
  427. v = s 1 u 1 v 2 + s 2 u 2 v 1 + s 3 ( v 1 v 2 + f ) d mod u v=\frac{s_{1}u_{1}v_{2}+s_{2}u_{2}v_{1}+s_{3}(v_{1}v_{2}+f)}{d}\mod u
  428. = 1 16 ( x 5 - 4 x 4 - 8 x 3 - 10 x 2 + 119 x + 30 ) mod u =\frac{1}{16}(x^{5}-4x^{4}-8x^{3}-10x^{2}+119x+30)\mod u
  429. = 1 4 ( 5 x 3 - 41 x 2 + 83 x - 15 ) =\frac{1}{4}(5x^{3}-41x^{2}+83x-15)
  430. u = f - v 2 u = 1 16 ( - 25 x 2 + 176 x - 15 ) u^{\prime}=\frac{f-v^{2}}{u}=\frac{1}{16}(-25x^{2}+176x-15)
  431. v = - v mod u = - 72 125 ( 17 x - 5 ) v^{\prime}=-v\mod u^{\prime}=-\frac{72}{125}(17x-5)
  432. u u^{\prime}
  433. D = ( - 25 x 2 + 176 x - 15 , - 72 125 ( 17 x - 5 ) ) D=(-25x^{2}+176x-15,-\frac{72}{125}(17x-5))
  434. D 1 + D 2 D_{1}+D_{2}
  435. C : y 2 = f ( x ) C:y^{2}=f(x)
  436. D 1 = [ P ] + [ Q ] - 2 [ O ] D_{1}=[P]+[Q]-2[O]
  437. D 2 = [ R ] + [ S ] - 2 [ O ] D_{2}=[R]+[S]-2[O]
  438. P , Q R ¯ , S ¯ P,Q\not=\overline{R},\overline{S}
  439. a ( x ) = a 0 x 3 + a 1 x 2 + a 2 x + a 3 a(x)=a_{0}x^{3}+a_{1}x^{2}+a_{2}x+a_{3}
  440. P , Q , R , S P,Q,R,S
  441. P = Q P=Q
  442. y = a ( x ) y=a(x)
  443. y 2 = f ( x ) = a 2 ( x ) y^{2}=f(x)=a^{2}(x)
  444. f ( x ) - a 2 ( x ) = 0 f(x)-a^{2}(x)=0
  445. f ( x ) - a 2 ( x ) f(x)-a^{2}(x)
  446. f ( x ) - a 2 ( x ) f(x)-a^{2}(x)
  447. a ( x ) a(x)
  448. P , Q , R , S P,Q,R,S
  449. C C
  450. T T
  451. U U
  452. T U ¯ T\not=\overline{U}
  453. P , Q , R , S , T , U P,Q,R,S,T,U
  454. C C
  455. D = [ P ] + [ Q ] + [ R ] + [ S ] + [ T ] + [ U ] - 6 [ O ] D=[P]+[Q]+[R]+[S]+[T]+[U]-6[O]
  456. [ P ] + [ Q ] + [ R ] + [ S ] - 4 [ O ] [P]+[Q]+[R]+[S]-4[O]
  457. - ( [ T ] + [ U ] - 2 [ O ] ) -([T]+[U]-2[O])
  458. [ P ] + [ P ¯ ] - 2 [ O ] [P]+[\overline{P}]-2[O]
  459. P = ( a , b ) P=(a,b)
  460. C C
  461. b ( x ) = x - a b(x)=x-a
  462. - ( [ P ] - [ O ] ) -([P]-[O])
  463. [ P ¯ ] - [ O ] [\overline{P}]-[O]
  464. D 1 + D 2 = ( [ P ] + [ Q ] - 2 [ O ] ) + ( [ R ] + [ S ] - 2 [ O ] ) D_{1}+D_{2}=([P]+[Q]-2[O])+([R]+[S]-2[O])
  465. [ T ¯ ] + [ U ¯ ] - 2 [ O ] [\overline{T}]+[\overline{U}]-2[O]
  466. a ( x ) a(x)