wpmath0000004_7

Hamilton–Jacobi_equation.html

  1. 𝐪 \mathbf{q}
  2. N N
  3. 𝐪 ( q 1 , q 2 , , q N - 1 , q N ) \mathbf{q}\equiv(q_{1},q_{2},\ldots,q_{N-1},q_{N})
  4. 𝐪 ˙ d 𝐪 d t \dot{\mathbf{q}}\equiv\frac{d\mathbf{q}}{dt}
  5. 𝐩 𝐪 k = 1 N p k q k . \mathbf{p}\cdot\mathbf{q}\equiv\sum_{k=1}^{N}p_{k}q_{k}.
  6. H = H ( q 1 , , q N ; S q 1 , , S q N ; t ) H=H\left(q_{1},\cdots,q_{N};\frac{\partial S}{\partial q_{1}},\cdots,\frac{% \partial S}{\partial q_{N}};t\right)\,\!
  7. S = S ( q 1 , q 2 q N , t ) S=S(q_{1},q_{2}\cdots q_{N},t)
  8. H = H ( q 1 , q 2 q N ; p 1 , p 2 p N ; t ) . H=H(q_{1},q_{2}\cdots q_{N};p_{1},p_{2}\cdots p_{N};t).
  9. p k = S q k . p_{k}=\frac{\partial S}{\partial q_{k}}.
  10. S / t \scriptstyle\partial S/\partial t
  11. β k = S α k , k = 1 , 2 N \beta_{k}=\frac{\partial S}{\partial\alpha_{k}},\quad k=1,2\cdots N
  12. p = G 2 q , Q = G 2 P , K ( Q , P , t ) = H ( q , p , t ) + G 2 t {p}={\partial G_{2}\over\partial{q}},\quad{Q}={\partial G_{2}\over\partial{P}}% ,\quad K({Q},{P},t)=H({q},{p},t)+{\partial G_{2}\over\partial t}
  13. 𝐏 ˙ = - K Q , 𝐐 ˙ = + K P . \dot{\mathbf{P}}=-{\partial K\over\partial{Q}},\quad\dot{\mathbf{Q}}=+{% \partial K\over\partial{P}}.
  14. 𝐏 ˙ = 𝐐 ˙ = 0 \dot{\mathbf{P}}=\dot{\mathbf{Q}}=0
  15. G 2 ( q , s y m b o l α , t ) = S ( q , t ) + A , G_{2}({q},symbol{\alpha},t)=S({q},t)+A,
  16. p = G 2 q = S q H ( q , p , t ) + G 2 t = 0 H ( q , S q , t ) + S t = 0. {p}=\frac{\partial G_{2}}{\partial{q}}=\frac{\partial S}{\partial{q}}\,% \rightarrow\,H({q},{p},t)+{\partial G_{2}\over\partial t}=0\,\rightarrow\,H% \left({q},\frac{\partial S}{\partial{q}},t\right)+{\partial S\over\partial t}=0.
  17. Q = s y m b o l β = S s y m b o l α {Q}=symbol\beta={\partial S\over\partial symbol\alpha}
  18. Q m = β m = S ( q , s y m b o l α , t ) α m . Q_{m}=\beta_{m}=\frac{\partial S({q},symbol\alpha,t)}{\partial\alpha_{m}}.
  19. d S = i S q i d q i + S t d t \mathrm{d}S=\sum_{i}\frac{\partial S}{\partial q_{i}}\mathrm{d}q_{i}+\frac{% \partial S}{\partial t}\mathrm{d}t
  20. d S d t = i S q i q ˙ i + S t = i p i q ˙ i - H = L . \frac{\mathrm{d}S}{\mathrm{d}t}=\sum_{i}\frac{\partial S}{\partial q_{i}}\dot{% q}_{i}+\frac{\partial S}{\partial t}=\sum_{i}p_{i}\dot{q}_{i}-H=L.
  21. S = L d t , S=\int L\,\mathrm{d}t,
  22. W = S + E t = S + H t = ( L + H ) d t = p d q , W=S+Et=S+Ht=\int(L+H)\,\mathrm{d}t=\int{p}\cdot\mathrm{d}{q},
  23. S / t \scriptstyle\partial S/\partial t
  24. S = W ( q 1 , q 2 q N ) - E t S=W(q_{1},q_{2}\cdots q_{N})-Et
  25. H ( q , S q ) = E . H\left({q},\frac{\partial S}{\partial{q}}\right)=E.
  26. S / q k \scriptstyle\partial S/\partial q_{k}
  27. ψ ( q k , S q k ) \psi\left(q_{k},\frac{\partial S}{\partial q_{k}}\right)
  28. H = H ( q 1 , q 2 q k - 1 , q k + 1 q N ; p 1 , p 2 p k - 1 , p k + 1 p N ; ψ ; t ) . H=H(q_{1},q_{2}\cdots q_{k-1},q_{k+1}\cdots q_{N};p_{1},p_{2}\cdots p_{k-1},p_% {k+1}\cdots p_{N};\psi;t).
  29. S = S k ( q k ) + S r e m ( q 1 q k - 1 , q k + 1 q N , t ) . S=S_{k}(q_{k})+S_{rem}(q_{1}\cdots q_{k-1},q_{k+1}\cdots q_{N},t).
  30. ψ ( q k , d S k d q k ) = Γ k . \psi\left(q_{k},\frac{\mathrm{d}S_{k}}{\mathrm{d}q_{k}}\right)=\Gamma_{k}.
  31. S = S 1 ( q 1 ) + S 2 ( q 2 ) + + S N ( q N ) - E t . S=S_{1}(q_{1})+S_{2}(q_{2})+\cdots+S_{N}(q_{N})-Et.
  32. H = 1 2 m [ p r 2 + p θ 2 r 2 + p ϕ 2 r 2 sin 2 θ ] + U ( r , θ , ϕ ) . H=\frac{1}{2m}\left[p_{r}^{2}+\frac{p_{\theta}^{2}}{r^{2}}+\frac{p_{\phi}^{2}}% {r^{2}\sin^{2}\theta}\right]+U(r,\theta,\phi).
  33. U ( r , θ , ϕ ) = U r ( r ) + U θ ( θ ) r 2 + U ϕ ( ϕ ) r 2 sin 2 θ . U(r,\theta,\phi)=U_{r}(r)+\frac{U_{\theta}(\theta)}{r^{2}}+\frac{U_{\phi}(\phi% )}{r^{2}\sin^{2}\theta}.
  34. S = S r ( r ) + S θ ( θ ) + S ϕ ( ϕ ) - E t S=S_{r}(r)+S_{\theta}(\theta)+S_{\phi}(\phi)-Et
  35. 1 2 m ( d S r d r ) 2 + U r ( r ) + 1 2 m r 2 [ ( d S θ d θ ) 2 + 2 m U θ ( θ ) ] + 1 2 m r 2 sin 2 θ [ ( d S ϕ d ϕ ) 2 + 2 m U ϕ ( ϕ ) ] = E . \frac{1}{2m}\left(\frac{\mathrm{d}S_{r}}{\mathrm{d}r}\right)^{2}+U_{r}(r)+% \frac{1}{2mr^{2}}\left[\left(\frac{\mathrm{d}S_{\theta}}{\mathrm{d}\theta}% \right)^{2}+2mU_{\theta}(\theta)\right]+\frac{1}{2mr^{2}\sin^{2}\theta}\left[% \left(\frac{\mathrm{d}S_{\phi}}{\mathrm{d}\phi}\right)^{2}+2mU_{\phi}(\phi)% \right]=E.
  36. ( d S ϕ d ϕ ) 2 + 2 m U ϕ ( ϕ ) = Γ ϕ \left(\frac{\mathrm{d}S_{\phi}}{\mathrm{d}\phi}\right)^{2}+2mU_{\phi}(\phi)=% \Gamma_{\phi}
  37. 1 2 m ( d S r d r ) 2 + U r ( r ) + 1 2 m r 2 [ ( d S θ d θ ) 2 + 2 m U θ ( θ ) + Γ ϕ sin 2 θ ] = E . \frac{1}{2m}\left(\frac{\mathrm{d}S_{r}}{\mathrm{d}r}\right)^{2}+U_{r}(r)+% \frac{1}{2mr^{2}}\left[\left(\frac{\mathrm{d}S_{\theta}}{\mathrm{d}\theta}% \right)^{2}+2mU_{\theta}(\theta)+\frac{\Gamma_{\phi}}{\sin^{2}\theta}\right]=E.
  38. ( d S θ d θ ) 2 + 2 m U θ ( θ ) + Γ ϕ sin 2 θ = Γ θ \left(\frac{\mathrm{d}S_{\theta}}{\mathrm{d}\theta}\right)^{2}+2mU_{\theta}(% \theta)+\frac{\Gamma_{\phi}}{\sin^{2}\theta}=\Gamma_{\theta}
  39. 1 2 m ( d S r d r ) 2 + U r ( r ) + Γ θ 2 m r 2 = E \frac{1}{2m}\left(\frac{\mathrm{d}S_{r}}{\mathrm{d}r}\right)^{2}+U_{r}(r)+% \frac{\Gamma_{\theta}}{2mr^{2}}=E
  40. H = p μ 2 + p ν 2 2 m a 2 ( sinh 2 μ + sin 2 ν ) + p z 2 2 m + U ( μ , ν , z ) H=\frac{p_{\mu}^{2}+p_{\nu}^{2}}{2ma^{2}\left(\sinh^{2}\mu+\sin^{2}\nu\right)}% +\frac{p_{z}^{2}}{2m}+U(\mu,\nu,z)
  41. U ( μ , ν , z ) = U μ ( μ ) + U ν ( ν ) sinh 2 μ + sin 2 ν + U z ( z ) U(\mu,\nu,z)=\frac{U_{\mu}(\mu)+U_{\nu}(\nu)}{\sinh^{2}\mu+\sin^{2}\nu}+U_{z}(z)
  42. S = S μ ( μ ) + S ν ( ν ) + S z ( z ) - E t S=S_{\mu}(\mu)+S_{\nu}(\nu)+S_{z}(z)-Et
  43. 1 2 m ( d S z d z ) 2 + U z ( z ) + 1 2 m a 2 ( sinh 2 μ + sin 2 ν ) [ ( d S μ d μ ) 2 + ( d S ν d ν ) 2 + 2 m a 2 U μ ( μ ) + 2 m a 2 U ν ( ν ) ] = E . \frac{1}{2m}\left(\frac{\mathrm{d}S_{z}}{\mathrm{d}z}\right)^{2}+U_{z}(z)+% \frac{1}{2ma^{2}\left(\sinh^{2}\mu+\sin^{2}\nu\right)}\left[\left(\frac{% \mathrm{d}S_{\mu}}{\mathrm{d}\mu}\right)^{2}+\left(\frac{\mathrm{d}S_{\nu}}{% \mathrm{d}\nu}\right)^{2}+2ma^{2}U_{\mu}(\mu)+2ma^{2}U_{\nu}(\nu)\right]=E.
  44. 1 2 m ( d S z d z ) 2 + U z ( z ) = Γ z \frac{1}{2m}\left(\frac{\mathrm{d}S_{z}}{\mathrm{d}z}\right)^{2}+U_{z}(z)=% \Gamma_{z}
  45. ( d S μ d μ ) 2 + ( d S ν d ν ) 2 + 2 m a 2 U μ ( μ ) + 2 m a 2 U ν ( ν ) = 2 m a 2 ( sinh 2 μ + sin 2 ν ) ( E - Γ z ) \left(\frac{\mathrm{d}S_{\mu}}{\mathrm{d}\mu}\right)^{2}+\left(\frac{\mathrm{d% }S_{\nu}}{\mathrm{d}\nu}\right)^{2}+2ma^{2}U_{\mu}(\mu)+2ma^{2}U_{\nu}(\nu)=2% ma^{2}\left(\sinh^{2}\mu+\sin^{2}\nu\right)\left(E-\Gamma_{z}\right)
  46. ( d S μ d μ ) 2 + 2 m a 2 U μ ( μ ) + 2 m a 2 ( Γ z - E ) sinh 2 μ = Γ μ \left(\frac{\mathrm{d}S_{\mu}}{\mathrm{d}\mu}\right)^{2}+2ma^{2}U_{\mu}(\mu)+2% ma^{2}\left(\Gamma_{z}-E\right)\sinh^{2}\mu=\Gamma_{\mu}
  47. ( d S ν d ν ) 2 + 2 m a 2 U ν ( ν ) + 2 m a 2 ( Γ z - E ) sin 2 ν = Γ ν \left(\frac{\mathrm{d}S_{\nu}}{\mathrm{d}\nu}\right)^{2}+2ma^{2}U_{\nu}(\nu)+2% ma^{2}\left(\Gamma_{z}-E\right)\sin^{2}\nu=\Gamma_{\nu}
  48. H = p σ 2 + p τ 2 2 m ( σ 2 + τ 2 ) + p z 2 2 m + U ( σ , τ , z ) . H=\frac{p_{\sigma}^{2}+p_{\tau}^{2}}{2m\left(\sigma^{2}+\tau^{2}\right)}+\frac% {p_{z}^{2}}{2m}+U(\sigma,\tau,z).
  49. U ( σ , τ , z ) = U σ ( σ ) + U τ ( τ ) σ 2 + τ 2 + U z ( z ) U(\sigma,\tau,z)=\frac{U_{\sigma}(\sigma)+U_{\tau}(\tau)}{\sigma^{2}+\tau^{2}}% +U_{z}(z)
  50. S = S σ ( σ ) + S τ ( τ ) + S z ( z ) - E t S=S_{\sigma}(\sigma)+S_{\tau}(\tau)+S_{z}(z)-Et
  51. 1 2 m ( d S z d z ) 2 + U z ( z ) + 1 2 m ( σ 2 + τ 2 ) [ ( d S σ d σ ) 2 + ( d S τ d τ ) 2 + 2 m U σ ( σ ) + 2 m U τ ( τ ) ] = E . \frac{1}{2m}\left(\frac{\mathrm{d}S_{z}}{\mathrm{d}z}\right)^{2}+U_{z}(z)+% \frac{1}{2m\left(\sigma^{2}+\tau^{2}\right)}\left[\left(\frac{\mathrm{d}S_{% \sigma}}{\mathrm{d}\sigma}\right)^{2}+\left(\frac{\mathrm{d}S_{\tau}}{\mathrm{% d}\tau}\right)^{2}+2mU_{\sigma}(\sigma)+2mU_{\tau}(\tau)\right]=E.
  52. 1 2 m ( d S z d z ) 2 + U z ( z ) = Γ z \frac{1}{2m}\left(\frac{\mathrm{d}S_{z}}{\mathrm{d}z}\right)^{2}+U_{z}(z)=% \Gamma_{z}
  53. ( d S σ d σ ) 2 + ( d S τ d τ ) 2 + 2 m U σ ( σ ) + 2 m U τ ( τ ) = 2 m ( σ 2 + τ 2 ) ( E - Γ z ) \left(\frac{\mathrm{d}S_{\sigma}}{\mathrm{d}\sigma}\right)^{2}+\left(\frac{% \mathrm{d}S_{\tau}}{\mathrm{d}\tau}\right)^{2}+2mU_{\sigma}(\sigma)+2mU_{\tau}% (\tau)=2m\left(\sigma^{2}+\tau^{2}\right)\left(E-\Gamma_{z}\right)
  54. ( d S σ d σ ) 2 + 2 m U σ ( σ ) + 2 m σ 2 ( Γ z - E ) = Γ σ \left(\frac{\mathrm{d}S_{\sigma}}{\mathrm{d}\sigma}\right)^{2}+2mU_{\sigma}(% \sigma)+2m\sigma^{2}\left(\Gamma_{z}-E\right)=\Gamma_{\sigma}
  55. ( d S τ d τ ) 2 + 2 m U τ ( τ ) + 2 m τ 2 ( Γ z - E ) = Γ τ \left(\frac{\mathrm{d}S_{\tau}}{\mathrm{d}\tau}\right)^{2}+2mU_{\tau}(\tau)+2m% \tau^{2}\left(\Gamma_{z}-E\right)=\Gamma_{\tau}
  56. ψ = ψ 0 e i S / \psi=\psi_{0}e^{iS/\hbar}
  57. 2 2 m ψ ( ψ ) 2 - U ψ = i ψ t \frac{\hbar^{2}}{2m\psi}\left(\nabla\psi\right)^{2}-U\psi=\frac{\hbar}{i}\frac% {\partial\psi}{\partial t}
  58. 1 2 m ( S ) 2 + U + S t = i 2 m 2 S . \frac{1}{2m}\left(\nabla S\right)^{2}+U+\frac{\partial S}{\partial t}=\frac{i% \hbar}{2m}\nabla^{2}S.
  59. 1 2 m ( S ) 2 + U + S t = 0. \frac{1}{2m}\left(\nabla S\right)^{2}+U+\frac{\partial S}{\partial t}=0.
  60. g α β P α P β + ( m c ) 2 = 0 , g^{\alpha\beta}P_{\alpha}P_{\beta}+(mc)^{2}=0\,,
  61. P α = S x α P_{\alpha}=\frac{\partial S}{\partial x^{\alpha}}
  62. g α β S x α S x β + ( m c ) 2 = 0 , g^{\alpha\beta}\frac{\partial S}{\partial x^{\alpha}}\frac{\partial S}{% \partial x^{\beta}}+(mc)^{2}=0\,,
  63. m m
  64. e e
  65. A i = ( ϕ , \Alpha ) A_{i}=(\phi,\Alpha)
  66. g i k = g i k g^{ik}=g_{ik}
  67. g i k ( S x i + e c A i ) ( S x k + e c A k ) = m 2 c 2 g^{ik}\left(\frac{\partial S}{\partial x^{i}}+\frac{e}{c}A_{i}\right)\left(% \frac{\partial S}{\partial x^{k}}+\frac{e}{c}A_{k}\right)=m^{2}c^{2}
  68. S S
  69. x = - e c γ A z d ξ x=-\frac{e}{c\gamma}\int A_{z}d\xi
  70. y = - e c γ A y d ξ y=-\frac{e}{c\gamma}\int A_{y}d\xi
  71. z = - e 2 2 c 2 γ 2 ( \Alpha 2 - \Alpha 2 ¯ ) d ξ z=-\frac{e^{2}}{2c^{2}\gamma^{2}}\int(\Alpha^{2}-\overline{\Alpha^{2}})d\xi
  72. γ = m 2 c 2 + e 2 \Alpha 2 ¯ \gamma=m^{2}c^{2}+e^{2}\overline{\Alpha^{2}}
  73. ξ = c t - e 2 2 γ 2 c 2 ( \Alpha 2 - \Alpha 2 ¯ ) d ξ \xi=ct-\frac{e^{2}}{2\gamma^{2}c^{2}}\int(\Alpha^{2}-\overline{\Alpha^{2}})d\xi
  74. p x = - e c A x p_{x}=-\frac{e}{c}A_{x}
  75. p y = - e c A y p_{y}=-\frac{e}{c}A_{y}
  76. p z = e 2 2 γ c ( \Alpha 2 - \Alpha 2 ¯ ) p_{z}=\frac{e^{2}}{2\gamma c}(\Alpha^{2}-\overline{\Alpha^{2}})
  77. = c γ + e 2 2 γ c ( \Alpha 2 - \Alpha 2 ¯ ) \mathcal{E}=c\gamma+\frac{e^{2}}{2\gamma c}(\Alpha^{2}-\overline{\Alpha^{2}})
  78. ξ = c t - z \xi=ct-z
  79. E x = E 0 sin ω ξ 1 E_{x}=E_{0}\sin\omega\xi_{1}
  80. E y = E 0 cos ω ξ 1 E_{y}=E_{0}\cos\omega\xi_{1}
  81. A x = c E 0 ω cos ω ξ 1 A_{x}=\frac{cE_{0}}{\omega}\cos\omega\xi_{1}
  82. A y = - c E 0 ω sin ω ξ 1 A_{y}=-\frac{cE_{0}}{\omega}\sin\omega\xi_{1}
  83. x = - e c E 0 ω sin ω ξ 1 x=-\frac{ecE_{0}}{\omega}\sin\omega\xi_{1}
  84. y = - e c E 0 ω cos ω ξ 1 y=-\frac{ecE_{0}}{\omega}\cos\omega\xi_{1}
  85. p x = - e E 0 ω cos ω ξ 1 p_{x}=-\frac{eE_{0}}{\omega}\cos\omega\xi_{1}
  86. p y = e E 0 ω sin ω ξ 1 p_{y}=\frac{eE_{0}}{\omega}\sin\omega\xi_{1}
  87. ξ 1 = ξ / c \xi_{1}=\xi/c
  88. e c E 0 / γ ω 2 ecE_{0}/\gamma\omega^{2}
  89. e E 0 / ω 2 eE_{0}/\omega^{2}
  90. E E
  91. y y
  92. E y = E 0 cos ω ξ 1 E_{y}=E_{0}\cos\omega\xi_{1}
  93. A y = - c E 0 ω sin ω ξ 1 A_{y}=-\frac{cE_{0}}{\omega}\sin\omega\xi_{1}
  94. x = c o n s t x=const
  95. y 0 = e c E 0 γ ω 2 y_{0}=\frac{ecE_{0}}{\gamma\omega^{2}}
  96. y = y 0 cos ω ξ 1 y=y_{0}\cos\omega\xi_{1}
  97. z = - C z y 0 sin 2 ω ξ 1 z=-C_{z}y_{0}\sin 2\omega\xi_{1}
  98. C z = e E 0 γ ω 2 C_{z}=\frac{eE_{0}}{\gamma\omega^{2}}
  99. γ 2 = m 2 c 2 + e 2 E 0 2 2 ω 2 \gamma^{2}=m^{2}c^{2}+\frac{e^{2}E_{0}^{2}}{2\omega^{2}}
  100. p x = 0 p_{x}=0
  101. p y , 0 = e E 0 ω p_{y,0}=\frac{eE_{0}}{\omega}
  102. p y = p y , 0 sin ω ξ 1 p_{y}=p_{y,0}\sin\omega\xi_{1}
  103. p z = - 2 C z p y , 0 cos 2 ω ξ 1 p_{z}=-2C_{z}p_{y,0}\cos 2\omega\xi_{1}
  104. E E
  105. E = E ϕ = ω ρ 0 c B 0 cos ω ξ 1 E=E_{\phi}=\frac{\omega\rho_{0}}{c}B_{0}\cos\omega\xi_{1}
  106. A ϕ = - ρ 0 B 0 sin ω ξ 1 = - L s π ρ 0 N s I 0 sin ω ξ 1 A_{\phi}=-\rho_{0}B_{0}\sin\omega\xi_{1}=-\frac{L_{s}}{\pi\rho_{0}N_{s}}I_{0}% \sin\omega\xi_{1}
  107. x = c o n s t x=const
  108. y 0 = e ρ 0 B 0 γ ω y_{0}=\frac{e\rho_{0}B_{0}}{\gamma\omega}
  109. y = y 0 cos ω ξ 1 y=y_{0}\cos\omega\xi_{1}
  110. z = - C z y 0 sin 2 ω ξ 1 z=-C_{z}y_{0}\sin 2\omega\xi_{1}
  111. C z = e ρ 0 B 0 c γ C_{z}=\frac{e\rho_{0}B_{0}}{c\gamma}
  112. γ 2 = m 2 c 2 + e 2 ρ 0 2 B 0 2 2 c 2 \gamma^{2}=m^{2}c^{2}+\frac{e^{2}\rho_{0}^{2}B_{0}^{2}}{2c^{2}}
  113. p x = 0 p_{x}=0
  114. p y , 0 = e ρ 0 B 0 c p_{y,0}=\frac{e\rho_{0}B_{0}}{c}
  115. p y = p y , 0 sin ω ξ 1 p_{y}=p_{y,0}\sin\omega\xi_{1}
  116. p z = - 2 C z p y , 0 cos 2 ω ξ 1 p_{z}=-2C_{z}p_{y,0}\cos 2\omega\xi_{1}
  117. B 0 B_{0}
  118. ρ 0 \rho_{0}
  119. L s L_{s}
  120. N s N_{s}
  121. I 0 I_{0}
  122. y z yz
  123. φ \varphi

Hamilton–Jacobi–Bellman_equation.html

  1. [ 0 , T ] [0,T]
  2. V ( x ( 0 ) , 0 ) = min u { 0 T C [ x ( t ) , u ( t ) ] d t + D [ x ( T ) ] } V(x(0),0)=\min_{u}\left\{\int_{0}^{T}C[x(t),u(t)]\,dt+D[x(T)]\right\}
  3. x ˙ ( t ) = F [ x ( t ) , u ( t ) ] \dot{x}(t)=F[x(t),u(t)]\,
  4. V ˙ ( x , t ) + min u { V ( x , t ) F ( x , u ) + C ( x , u ) } = 0 \dot{V}(x,t)+\min_{u}\left\{\nabla V(x,t)\cdot F(x,u)+C(x,u)\right\}=0
  5. V ( x , T ) = D ( x ) , V(x,T)=D(x),\,
  6. a b a\cdot b
  7. \nabla
  8. V ( x , t ) V(x,t)
  9. x x
  10. t t
  11. T T
  12. V ( x ( t ) , t ) V(x(t),t)
  13. V ( x ( t ) , t ) = min u { t t + d t C ( x ( t ) , u ( t ) ) d t + V ( x ( t + d t ) , t + d t ) } . V(x(t),t)=\min_{u}\left\{\int_{t}^{t+dt}C(x(t),u(t))\,dt+V(x(t+dt),t+dt)\right\}.
  14. V ( x ( t + d t ) , t + d t ) = V ( x ( t ) , t ) + V ˙ ( x ( t ) , t ) d t + V ( x ( t ) , t ) x ˙ ( t ) d t + o ( d t ) , V(x(t+dt),t+dt)=V(x(t),t)+\dot{V}(x(t),t)\,dt+\nabla V(x(t),t)\cdot\dot{x}(t)% \,dt+o(dt),
  15. t = T t=T
  16. t = 0 t=0
  17. V V
  18. u u
  19. min { 0 T C ( t , X t , u t ) d t + D ( X T ) } \min\left\{\int_{0}^{T}C(t,X_{t},u_{t})\,dt+D(X_{T})\right\}
  20. ( X t ) t [ 0 , T ] (X_{t})_{t\in[0,T]}\,\!
  21. ( u t ) t [ 0 , T ] (u_{t})_{t\in[0,T]}\,\!
  22. V ( X t , t ) V(X_{t},t)
  23. min u { 𝒜 V ( x , t ) + C ( t , x , u ) } = 0 , \min_{u}\left\{\mathcal{A}V(x,t)+C(t,x,u)\right\}=0,
  24. 𝒜 \mathcal{A}
  25. V ( x , T ) = D ( x ) . V(x,T)=D(x)\,\!.
  26. V V\,\!
  27. d x t = ( a x t + b u t ) d t + σ d w t , dx_{t}=(ax_{t}+bu_{t})dt+\sigma dw_{t},
  28. C ( x t , u t ) = r ( t ) u t 2 / 2 + q ( t ) x t 2 / 2 C(x_{t},u_{t})=r(t)u_{t}^{2}/2+q(t)x_{t}^{2}/2
  29. - V ( x , t ) t = 1 2 q ( t ) x 2 + V ( x , t ) x a x - b 2 2 r ( t ) ( V ( x , t ) x ) 2 + σ 2 V ( x , t ) x 2 . -\frac{\partial V(x,t)}{\partial t}=\frac{1}{2}q(t)x^{2}+\frac{\partial V(x,t)% }{\partial x}ax-\frac{b^{2}}{2r(t)}\left(\frac{\partial V(x,t)}{\partial x}% \right)^{2}+\sigma\frac{\partial^{2}V(x,t)}{\partial x^{2}}.

Handicap_(golf).html

  1. Handicap differential = ( Equitable Stroke Control - course rating ) × 113 slope rating \mbox{Handicap differential}~{}=\frac{(\mbox{Equitable Stroke Control}~{}-% \mbox{course rating}~{})\times 113}{\mbox{slope rating}~{}}
  2. Course Handicap = ( Handicap index × Slope Rating ) 113 \mbox{Course Handicap}~{}=\frac{(\mbox{Handicap index}~{}\times\mbox{Slope % Rating}~{})}{113}
  3. ( 10.5 × 117 ) 113 = 11 \frac{(\mbox{10.5}~{}\times\mbox{117}~{})}{113}=11
  4. ( 10.5 × 124 ) 113 = 12 \frac{(\mbox{10.5}~{}\times\mbox{124}~{})}{113}=12

Handle_decomposition.html

  1. = M - 1 M 0 M 1 M 2 M m - 1 M m = M \emptyset=M_{-1}\subset M_{0}\subset M_{1}\subset M_{2}\subset\dots\subset M_{% m-1}\subset M_{m}=M
  2. M i M_{i}
  3. M i - 1 M_{i-1}
  4. i i
  5. S n S^{n}
  6. χ : D n S n \chi:D^{n}\to S^{n}
  7. S n - 1 S^{n-1}
  8. N p N_{p}
  9. D m D^{m}
  10. N p N_{p}
  11. M int ( N p ) M\setminus\operatorname{int}(N_{p})
  12. M int ( N p ) M\setminus\operatorname{int}(N_{p})
  13. I × D m - 1 I\times D^{m-1}
  14. M M
  15. D m D^{m}
  16. I × D m - 1 I\times D^{m-1}
  17. M int ( N p ) M\setminus\operatorname{int}(N_{p})
  18. I × D m - 1 I\times D^{m-1}
  19. D m D^{m}
  20. ( I ) × D m - 1 (\partial I)\times D^{m-1}
  21. D m \partial D^{m}
  22. f : S j - 1 × D m - j M f:S^{j-1}\times D^{m-j}\to\partial M
  23. H j = D j × D m - j H^{j}=D^{j}\times D^{m-j}
  24. M f H j M\cup_{f}H^{j}
  25. M M
  26. H j H^{j}
  27. S j - 1 × D m - j S^{j-1}\times D^{m-j}
  28. M \partial M
  29. M f H j = ( M ( D j × D m - j ) ) / M\cup_{f}H^{j}=\left(M\sqcup(D^{j}\times D^{m-j})\right)/\sim
  30. \sim
  31. ( p , x ) f ( p , x ) (p,x)\sim f(p,x)
  32. ( p , x ) S j - 1 × D m - j D j × D m - j (p,x)\in S^{j-1}\times D^{m-j}\subset D^{j}\times D^{m-j}
  33. H j H^{j}
  34. M f H j = ( M ( D j × D m - j ) ) / M\cup_{f}H^{j}=\left(M\sqcup(D^{j}\times D^{m-j})\right)/\sim
  35. f ( S j - 1 × { 0 } ) M f(S^{j-1}\times\{0\})\subset M
  36. f f
  37. { 0 } j × S m - j - 1 D j × D m - j = H j \{0\}^{j}\times S^{m-j-1}\subset D^{j}\times D^{m-j}=H^{j}
  38. H j H^{j}
  39. M f H j M\cup_{f}H^{j}
  40. D m D^{m}
  41. W = M 0 M 1 \partial W=M_{0}\cup M_{1}
  42. W - 1 W 0 W 1 W m + 1 = W W_{-1}\subset W_{0}\subset W_{1}\subset\cdots\subset W_{m+1}=W
  43. W - 1 W_{-1}
  44. M 0 × [ 0 , 1 ] M_{0}\times[0,1]
  45. W i W_{i}
  46. W i - 1 W_{i-1}
  47. f : M f:M\to\mathbb{R}
  48. { p 1 , , p k } M \{p_{1},\ldots,p_{k}\}\subset M
  49. f ( p 1 ) < f ( p 2 ) < < f ( p k ) f(p_{1})<f(p_{2})<\cdots<f(p_{k})
  50. t 0 < f ( p 1 ) < t 1 < f ( p 2 ) < < t k - 1 < f ( p k ) < t k t_{0}<f(p_{1})<t_{1}<f(p_{2})<\cdots<t_{k-1}<f(p_{k})<t_{k}
  51. f - 1 [ t j - 1 , t j ] f^{-1}[t_{j-1},t_{j}]
  52. ( f - 1 ( t j - 1 ) × [ 0 , 1 ] ) H I ( j ) (f^{-1}(t_{j-1})\times[0,1])\cup H^{I(j)}
  53. p j p_{j}
  54. T p j M T_{p_{j}}M
  55. I ( 1 ) I ( 2 ) I ( k ) I(1)\leq I(2)\leq\cdots\leq I(k)
  56. W W
  57. W = M 0 M 1 \partial W=M_{0}\cup M_{1}
  58. f : W f:W\to\mathbb{R}
  59. T 1 T^{1}
  60. ( M f H i ) g H j (M\cup_{f}H^{i})\cup_{g}H^{j}
  61. j i j\leq i
  62. ( M H j ) H i (M\cup H^{j})\cup H^{i}
  63. M f H j M\cup_{f}H^{j}
  64. M \partial M
  65. f f
  66. S m S^{m}
  67. S m S^{m}

Handlebody.html

  1. ( W , W ) (W,\partial W)
  2. n n
  3. S r - 1 × D n - r W S^{r-1}\times D^{n-r}\subset\partial W
  4. n n
  5. ( W , W ) = ( ( W ( D r × D n - r ) ) , ( W - S r - 1 × D n - r ) ( D r × S n - r - 1 ) ) (W^{\prime},\partial W^{\prime})=((W\cup(D^{r}\times D^{n-r})),(\partial W-S^{% r-1}\times D^{n-r})\cup(D^{r}\times S^{n-r-1}))
  6. ( W , W ) (W,\partial W)
  7. r r
  8. W \partial W^{\prime}
  9. W \partial W
  10. ( W , W ) (W,\partial W)
  11. W \partial W

Harada–Norton_group.html

  1. × 10 1 4 \times 10^{1}4
  2. T 5 A ( τ ) T_{5A}(\tau)
  3. j 5 A ( τ ) = T 5 A ( τ ) - 6 = ( η ( τ ) η ( 5 τ ) ) 6 + 5 3 ( η ( 5 τ ) η ( τ ) ) 6 = 1 q - 6 + 134 q + 760 q 2 + 3345 q 3 + 12256 q 4 + 39350 q 5 + \begin{aligned}\displaystyle j_{5A}(\tau)&\displaystyle=T_{5A}(\tau)-6\\ &\displaystyle=\big(\tfrac{\eta(\tau)}{\eta(5\tau)}\big)^{6}+5^{3}\big(\tfrac{% \eta(5\tau)}{\eta(\tau)}\big)^{6}\\ &\displaystyle=\frac{1}{q}-6+134q+760q^{2}+3345q^{3}+12256q^{4}+39350q^{5}+% \dots\end{aligned}

Hard-core_predicate.html

  1. x j x_{j}
  2. r j r_{j}
  3. b ( x , r ) = j x j r j b(x,r)=\bigoplus_{j}x_{j}r_{j}
  4. b ( x , r ) = x , r b(x,r)=\langle x,r\rangle
  5. , \langle\cdot,\cdot\rangle
  6. ( \Z / 2 \Z ) n (\Z/2\Z)^{n}
  7. O ( log | x | ) O(\log|x|)
  8. { b ( f n ( s ) ) } n \left\{b(f^{n}(s))\right\}_{n}

Hard_spheres.html

  1. σ \sigma
  2. V ( 𝐫 1 , 𝐫 2 ) = { 0 if | 𝐫 1 - 𝐫 2 | σ if | 𝐫 1 - 𝐫 2 | < σ V(\mathbf{r}_{1},\mathbf{r}_{2})=\left\{\begin{matrix}0&\mbox{if}~{}\quad|% \mathbf{r}_{1}-\mathbf{r}_{2}|\geq\sigma\\ \infty&\mbox{if}~{}\quad|\mathbf{r}_{1}-\mathbf{r}_{2}|<\sigma\end{matrix}\right.
  3. 𝐫 1 \mathbf{r}_{1}
  4. 𝐫 2 \mathbf{r}_{2}
  5. B 2 v 0 \frac{B_{2}}{v_{0}}
  6. 4 4{\frac{}{}}
  7. B 3 v 0 2 \frac{B_{3}}{{v_{0}}^{2}}
  8. 10 10{\frac{}{}}
  9. B 4 v 0 3 \frac{B_{4}}{{v_{0}}^{3}}
  10. - 712 35 + 219 2 35 π + 4131 35 π arccos 1 3 18.365 -\frac{712}{35}+\frac{219\sqrt{2}}{35\pi}+\frac{4131}{35\pi}\arccos{\frac{1}{% \sqrt{3}}}\approx 18.365
  11. B 5 v 0 4 \frac{B_{5}}{{v_{0}}^{4}}
  12. 28.24 ± 0.08 28.24\pm 0.08
  13. B 6 v 0 5 \frac{B_{6}}{{v_{0}}^{5}}
  14. 39.5 ± 0.4 39.5\pm 0.4
  15. B 7 v 0 6 \frac{B_{7}}{{v_{0}}^{6}}
  16. 56.5 ± 1.6 56.5\pm 1.6
  17. P P
  18. η \eta
  19. η f 0.494 \eta_{\mathrm{f}}\approx 0.494
  20. η m 0.545 \eta_{\mathrm{m}}\approx 0.545
  21. η rcp 0.644 \eta_{\mathrm{rcp}}\approx 0.644
  22. η cp = 2 π / 6 0.74048 \eta_{\mathrm{cp}}=\sqrt{2}\pi/6\approx 0.74048

Hare_quota.html

  1. total votes total seats \frac{\mbox{total}~{}\;\mbox{votes}~{}}{\mbox{total}~{}\;\mbox{seats}~{}}
  2. 100 2 = 50 \frac{100}{2}=50

Harmonic_conjugate.html

  1. u ( x , y ) u(x,\,y)
  2. Ω \R 2 \Omega\subset\R^{2}
  3. v ( x , y ) v(x,\,y)
  4. f ( z ) f(z)
  5. z := x + i y Ω . z:=x+iy\in\Omega.
  6. v v
  7. u u
  8. f ( z ) := u ( x , y ) + i v ( x , y ) f(z):=u(x,y)+iv(x,y)
  9. Ω . \Omega.
  10. Ω \Omega
  11. u , u,
  12. u u
  13. v v
  14. v v
  15. - u -u
  16. v v
  17. u u
  18. Ω \Omega
  19. u u
  20. v v
  21. Ω . \Omega.
  22. u u
  23. Ω \R 2 , \Omega\subset\R^{2},
  24. u y u_{y}
  25. - u x -u_{x}
  26. Δ u = 0 \Delta u=0
  27. u x y = u y x . u_{xy}=u_{yx}.
  28. u u
  29. g ( z ) := u x ( x , y ) - i u y ( x , y ) g(z):=u_{x}(x,y)-iu_{y}(x,y)
  30. f ( z ) f(z)
  31. Ω , \Omega,
  32. u u
  33. - Im f ( x + i y ) . -\scriptstyle\mathrm{Im}\,f(x+iy).
  34. u ( x , y ) = e x sin y . u(x,y)=e^{x}\sin y.\,
  35. u x = e x sin y , 2 u x 2 = e x sin y {\partial u\over\partial x}=e^{x}\sin y,{\partial^{2}u\over\partial x^{2}}=e^{% x}\sin y
  36. u y = e x cos y , 2 u y 2 = - e x sin y , {\partial u\over\partial y}=e^{x}\cos y,{\partial^{2}u\over\partial y^{2}}=-e^% {x}\sin y,
  37. Δ u = 2 u = 0 \Delta u=\nabla^{2}u=0\,
  38. Δ \Delta
  39. v ( x , y ) v(x,y)
  40. u x = v y = e x sin y {\partial u\over\partial x}={\partial v\over\partial y}=e^{x}\sin y\,
  41. u y = - v x = e x cos y . {\partial u\over\partial y}=-{\partial v\over\partial x}=e^{x}\cos y.\,
  42. v y = e x sin y {\partial v\over\partial y}=e^{x}\sin y
  43. v x = - e x cos y {\partial v\over\partial x}=-e^{x}\cos y
  44. v = - e x cos y + C . v=-e^{x}\cos y+C.\!\;

Harmonic_spectrum.html

  1. ω \omega\,
  2. { , - 2 ω , - ω , 0 , ω , 2 ω , } . \{\dots,-2\omega,-\omega,0,\omega,2\omega,\dots\}.

Harold_Davenport.html

  1. Y 2 = X ( X - 1 ) ( X - 2 ) ( X - k ) Y^{2}=X(X-1)(X-2)\ldots(X-k)
  2. χ ( X ( X - 1 ) ( X - 2 ) ( X - k ) ) \sum\chi(X(X-1)(X-2)\ldots(X-k))

Harshad_number.html

  1. X = i = 0 m - 1 a i n i . X=\sum_{i=0}^{m-1}a_{i}n^{i}.
  2. X = A i = 0 m - 1 a i . X=A\sum_{i=0}^{m-1}a_{i}.
  3. x 1 - ε N ( x ) x log log x log x x^{1-\varepsilon}\ll N(x)\ll\frac{x\log\log x}{\log x}
  4. N ( x ) = ( c + o ( 1 ) ) x log x N(x)=(c+o(1))\frac{x}{\log x}
  5. 6804 / 18 = 378 378 / 18 = 21 21 / 3 = 7 \begin{array}[]{l}6804/18=378\\ 378/18=21\\ 21/3=7\end{array}

Hasse's_theorem_on_elliptic_curves.html

  1. | N - ( q + 1 ) | 2 q . |N-(q+1)|\leq 2\sqrt{q}.
  2. 𝔽 q \mathbb{F}_{q}
  3. # C ( 𝔽 q ) \#C(\mathbb{F}_{q})
  4. | # C ( 𝔽 q ) - ( q + 1 ) | 2 g q . |\#C(\mathbb{F}_{q})-(q+1)|\leq 2g\sqrt{q}.

Hasse_norm_theorem.html

  1. 𝐍 L / K ( l ) = k \mathbf{N}_{L/K}(l)=k
  2. 𝐐 ( - 3 , 13 ) / 𝐐 {\mathbf{Q}}(\sqrt{-3},\sqrt{13})/{\mathbf{Q}}
  3. 𝐐 ( 13 , 17 ) / 𝐐 {\mathbf{Q}}(\sqrt{13},\sqrt{17})/{\mathbf{Q}}
  4. 5 2 5^{2}

Hasse–Weil_zeta_function.html

  1. Z V , Q ( s ) Z_{V,Q}(s)
  2. ζ V , p ( p - s ) . \zeta_{V,p}\left(p^{-s}\right).
  3. p - s p^{-s}
  4. ρ ( Frob ( p ) ) , \rho(\operatorname{Frob}(p)),
  5. Z E , Q ( s ) = ζ ( s ) ζ ( s - 1 ) L ( s , E ) . Z_{E,Q}(s)=\frac{\zeta(s)\zeta(s-1)}{L(s,E)}.\,
  6. L ( s , E ) = p L p ( s , E ) - 1 L(s,E)=\prod_{p}L_{p}(s,E)^{-1}\,
  7. L p ( s , E ) = { ( 1 - a p p - s + p 1 - 2 s ) , if p N ( 1 - a p p - s ) , if p N 1 , if p 2 | N L_{p}(s,E)=\begin{cases}(1-a_{p}p^{-s}+p^{1-2s}),&\,\text{if }p\nmid N\\ (1-a_{p}p^{-s}),&\,\text{if }p\|N\\ 1,&\,\text{if }p^{2}|N\end{cases}

Hawaiian_earring.html

  1. G / N i = 0 \textstyle G/N\approx\prod_{i=0}^{\infty}\mathbb{Z}

Hearing_the_shape_of_a_drum.html

  1. { Δ u + λ u = 0 u | D = 0 \begin{cases}\Delta u+\lambda u=0\\ u|_{\partial D}=0\end{cases}
  2. V = ( 2 π ) d lim R N ( R ) R d / 2 V=(2\pi)^{d}\lim_{R\to\infty}\frac{N(R)}{R^{d/2}}\,
  3. N ( R ) = ( 2 π ) - d ω d V R d / 2 + 1 4 ( 2 π ) - d + 1 ω d - 1 A R ( d - 1 ) / 2 + o ( R ( d - 1 ) / 2 ) . \,N(R)=(2\pi)^{-d}\omega_{d}VR^{d/2}+\frac{1}{4}(2\pi)^{-d+1}\omega_{d-1}AR^{(% d-1)/2}+o(R^{(d-1)/2}).\,
  4. ω d \omega_{d}
  5. R D / 2 R^{D/2}\,

Heat_of_combustion.html

  1. Δ H c \Delta H_{c}^{\circ}
  2. Δ H f \Delta H_{f}^{\circ}

Heegaard_splitting.html

  1. M = V f W . M=V\cup_{f}W.
  2. α \alpha
  3. α \alpha
  4. β \beta
  5. α \alpha
  6. β \beta
  7. α \alpha
  8. β \beta
  9. α \alpha
  10. β \beta
  11. α \alpha
  12. β \beta
  13. V i , W i , i = 1 , , n V_{i},W_{i},i=1,\dots,n
  14. H i , i = 1 , , n H_{i},i=1,\dots,n
  15. + V i = + W i = H i \partial_{+}V_{i}=\partial_{+}W_{i}=H_{i}
  16. - W i = - V i + 1 \partial_{-}W_{i}=\partial_{-}V_{i+1}
  17. M M
  18. H i H_{i}
  19. V i W i V_{i}\cup W_{i}
  20. M M
  21. V i W i V_{i}\cup W_{i}
  22. max { 0 , 1 - χ ( S ) } \operatorname{max}\{0,1-\chi(S)\}
  23. S 3 S^{3}
  24. 4 \mathbb{R}^{4}
  25. x y z xyz
  26. S 3 S^{3}
  27. S 3 S^{3}
  28. 4 \mathbb{R}^{4}
  29. 2 \mathbb{C}^{2}
  30. S 3 S^{3}
  31. 2 \mathbb{C}^{2}
  32. 1 / 2 1/\sqrt{2}
  33. T 2 T^{2}
  34. S 3 S^{3}
  35. ( M , H ) (M,H)
  36. ( S 3 , T 2 ) (S^{3},T^{2})
  37. S 3 S^{3}
  38. T 3 T^{3}
  39. S 1 S^{1}
  40. x 0 x_{0}
  41. S 1 S^{1}
  42. Γ = S 1 × { x 0 } × { x 0 } { x 0 } × S 1 × { x 0 } { x 0 } × { x 0 } × S 1 \Gamma=S^{1}\times\{x_{0}\}\times\{x_{0}\}\cup\{x_{0}\}\times S^{1}\times\{x_{% 0}\}\cup\{x_{0}\}\times\{x_{0}\}\times S^{1}
  43. Γ \Gamma
  44. T 3 - V T^{3}-V
  45. T 3 T^{3}
  46. T 3 T^{3}
  47. S 3 S^{3}
  48. S 3 S^{3}
  49. H 1 H_{1}
  50. H 2 H_{2}
  51. H H
  52. S 1 S_{1}
  53. S 2 S_{2}
  54. S 3 S^{3}
  55. R 3 R^{3}
  56. R 3 R^{3}
  57. g t h g^{th}

Heilbronn_triangle_problem.html

  1. c 1 f ( n ) Δ ( n ) c 2 f ( n ) c_{1}f(n)\leq\Delta(n)\leq c_{2}f(n)
  2. Δ ( n ) = O ( 1 n 2 ) . \Delta(n)=O\left(\frac{1}{n^{2}}\right).
  3. Δ ( n ) = Ω ( log n n 2 ) . \Delta(n)=\Omega\left(\frac{\log n}{n^{2}}\right).
  4. Δ ( n ) = O ( 1 n log log n ) . \Delta(n)=O\left(\frac{1}{n\sqrt{\log\log n}}\right).
  5. Δ ( n ) exp ( c log n ) n 8 / 7 , \Delta(n)\leq\frac{\exp{\left(c\sqrt{\log n}\right)}}{n^{8/7}},

Heine–Cantor_theorem.html

  1. f : M N f:M\to N
  2. ϵ > 0 \epsilon>0
  3. δ > 0 \delta>0
  4. d M ( x , y ) < δ d_{M}(x,y)<\delta
  5. d N ( f ( x ) , f ( y ) ) < ϵ d_{N}(f(x),f(y))<\epsilon
  6. ϵ > 0 \epsilon>0
  7. δ x > 0 \delta_{x}>0
  8. d M ( f ( x ) , f ( y ) ) < ϵ / 2 d_{M}(f(x),f(y))<\epsilon/2
  9. δ x \delta_{x}
  10. δ x / 2 \delta_{x}/2
  11. U x = { y d M ( x , y ) < 1 2 δ x } U_{x}=\left\{y\mid d_{M}(x,y)<\frac{1}{2}\delta_{x}\right\}
  12. { U x x M } \{U_{x}\mid x\in M\}
  13. U x 1 , U x 2 , , U x n U_{x_{1}},U_{x_{2}},\ldots,U_{x_{n}}
  14. { x 1 , x 2 , , x n } M \{x_{1},x_{2},\ldots,x_{n}\}\subset M
  15. δ x i / 2 \delta_{x_{i}}/2
  16. δ = min 1 i n 1 2 δ x i \delta=\min_{1\leq i\leq n}\frac{1}{2}\delta_{x_{i}}
  17. δ \delta
  18. δ \delta
  19. d M ( x , y ) < δ d_{M}(x,y)<\delta
  20. U x i U_{x_{i}}
  21. U x i U_{x_{i}}
  22. d M ( x , x i ) < 1 2 δ x i d_{M}(x,x_{i})<\frac{1}{2}\delta_{x_{i}}
  23. d M ( x i , y ) d M ( x i , x ) + d M ( x , y ) < 1 2 δ x i + δ δ x i d_{M}(x_{i},y)\leq d_{M}(x_{i},x)+d_{M}(x,y)<\frac{1}{2}\delta_{x_{i}}+\delta% \leq\delta_{x_{i}}
  24. δ x i \delta_{x_{i}}
  25. δ x i \delta_{x_{i}}
  26. d N ( f ( x i ) , f ( x ) ) d_{N}(f(x_{i}),f(x))
  27. d N ( f ( x i ) , f ( y ) ) d_{N}(f(x_{i}),f(y))
  28. ϵ / 2 \epsilon/2
  29. d N ( f ( x ) , f ( y ) ) d N ( f ( x i ) , f ( x ) ) + d N ( f ( x i ) , f ( y ) ) < 1 2 ϵ + 1 2 ϵ = ϵ d_{N}(f(x),f(y))\leq d_{N}(f(x_{i}),f(x))+d_{N}(f(x_{i}),f(y))<\frac{1}{2}% \epsilon+\frac{1}{2}\epsilon=\epsilon

Held_group.html

  1. × 10 9 \times 10^{9}
  2. T 7 A ( τ ) T_{7A}(\tau)
  3. j 7 A ( τ ) = T 7 A ( τ ) + 10 = ( ( η ( τ ) η ( 7 τ ) ) 2 + 7 ( η ( 7 τ ) η ( τ ) ) 2 ) 2 = 1 q + 10 + 51 q + 204 q 2 + 681 q 3 + 1956 q 4 + 5135 q 5 + \begin{aligned}\displaystyle j_{7A}(\tau)&\displaystyle=T_{7A}(\tau)+10\\ &\displaystyle=\Big(\big(\tfrac{\eta(\tau)}{\eta(7\tau)}\big)^{2}+7\big(\tfrac% {\eta(7\tau)}{\eta(\tau)}\big)^{2}\Big)^{2}\\ &\displaystyle=\frac{1}{q}+10+51q+204q^{2}+681q^{3}+1956q^{4}+5135q^{5}+\dots% \end{aligned}
  4. a 2 = b 7 = ( a b ) 17 = [ a , b ] 6 = [ a , b 3 ] 5 = [ a , b a b a b - 1 a b a b ] = ( a b ) 4 a b 2 a b - 3 a b a b a b - 1 a b 3 a b - 2 a b 2 = 1. a^{2}=b^{7}=(ab)^{17}=[a,b]^{6}=\left[a,b^{3}\right]^{5}=\left[a,babab^{-1}% abab\right]=(ab)^{4}ab^{2}ab^{-3}ababab^{-1}ab^{3}ab^{-2}ab^{2}=1.

Helicoid.html

  1. x = ρ cos ( α θ ) , x=\rho\cos(\alpha\theta),
  2. y = ρ sin ( α θ ) , y=\rho\sin(\alpha\theta),
  3. z = θ , z=\theta,
  4. ± 1 / ( 1 + ρ 2 ) \pm 1/(1+\rho^{2})
  5. 2 \mathbb{R}^{2}
  6. π [ R ( R 2 + h 2 ) + h 2 * ln ( ( R + ( R 2 + h 2 ) / h ) ] \pi[R\sqrt{(}R^{2}+h^{2})+h^{2}*\ln((R+\sqrt{(}R^{2}+h^{2})/h)]

Helly_family.html

  1. k k
  2. X G X = , \bigcap_{X\in G}X=\varnothing,
  3. X H X = \bigcap_{X\in H}X=\varnothing
  4. | H | k . \left|H\right|\leq k.

Helmholtz_decomposition.html

  1. - grad Φ + curl 𝐀 -\operatorname{grad}\Phi+\operatorname{curl}\mathbf{A}
  2. Φ Φ
  3. 𝐀 \mathbf{A}
  4. 𝐅 \mathbf{F}
  5. S S
  6. V V
  7. 𝐅 \mathbf{F}
  8. 𝐅 = - Φ + × 𝐀 , \mathbf{F}=-\nabla\Phi+\nabla\times\mathbf{A},
  9. Φ ( 𝐫 ) = 1 4 π V 𝐅 ( 𝐫 ) | 𝐫 - 𝐫 | d V - 1 4 π S 𝐧 ^ 𝐅 ( 𝐫 ) | 𝐫 - 𝐫 | d S \Phi(\mathbf{r})=\frac{1}{4\pi}\int_{V}\frac{\nabla^{\prime}\cdot\mathbf{F}% \left(\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}% \mathrm{d}V^{\prime}-\frac{1}{4\pi}\oint_{S}\mathbf{\hat{n}}^{\prime}\cdot% \frac{\mathbf{F}\left(\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^% {\prime}\right|}\mathrm{d}S^{\prime}
  10. 𝐀 ( 𝐫 ) = 1 4 π V × 𝐅 ( 𝐫 ) | 𝐫 - 𝐫 | d V - 1 4 π S 𝐧 ^ × 𝐅 ( 𝐫 ) | 𝐫 - 𝐫 | d S \mathbf{A}(\mathbf{r})=\frac{1}{4\pi}\int_{V}\frac{\nabla^{\prime}\times% \mathbf{F}\left(\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^{% \prime}\right|}\mathrm{d}V^{\prime}-\frac{1}{4\pi}\oint_{S}\mathbf{\hat{n}}^{% \prime}\times\frac{\mathbf{F}\left(\mathbf{r}^{\prime}\right)}{\left|\mathbf{r% }-\mathbf{r}^{\prime}\right|}\mathrm{d}S^{\prime}
  11. 𝐅 \mathbf{F}
  12. 1 / r 1/r
  13. r r→∞
  14. Φ ( 𝐫 ) = 1 4 π all space 𝐅 ( 𝐫 ) | 𝐫 - 𝐫 | d V \Phi(\mathbf{r})=\frac{1}{4\pi}\int_{\,\text{all space}}\frac{\nabla^{\prime}% \cdot\mathbf{F}\left(\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^{% \prime}\right|}\mathrm{d}V^{\prime}
  15. 𝐀 ( 𝐫 ) = 1 4 π all space × 𝐅 ( 𝐫 ) | 𝐫 - 𝐫 | d V \mathbf{A}(\mathbf{r})=\frac{1}{4\pi}\int_{\,\text{all space}}\frac{\nabla^{% \prime}\times\mathbf{F}\left(\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-% \mathbf{r}^{\prime}\right|}\mathrm{d}V^{\prime}
  16. 𝐅 ( 𝐫 ) \mathbf{F}(\mathbf{r})
  17. × 𝐅 ∇×\mathbf{F}
  18. 𝐅 ∇⋅\mathbf{F}
  19. δ 3 ( 𝐫 - 𝐫 ) = - 1 4 π 2 1 | 𝐫 - 𝐫 | , \delta^{3}(\mathbf{r}-\mathbf{r}^{\prime})=-\frac{1}{4\pi}\nabla^{2}\frac{1}{% \left|\mathbf{r}-\mathbf{r}^{\prime}\right|},
  20. 𝐅 ( 𝐫 ) = V 𝐅 ( 𝐫 ) δ 3 ( 𝐫 - 𝐫 ) d V [ 6 p t ] = V 𝐅 ( 𝐫 ) ( - 1 4 π 2 1 | 𝐫 - 𝐫 | ) d V [ 6 p t ] = - 1 4 π 2 V 𝐅 ( 𝐫 ) | 𝐫 - 𝐫 | d V [ 6 p t ] = - 1 4 π [ ( V 𝐅 ( 𝐫 ) | 𝐫 - 𝐫 | d V ) - × ( × V 𝐅 ( 𝐫 ) | 𝐫 - 𝐫 | d V ) ] 2 𝐚 = ( 𝐚 ) - × ( × 𝐚 ) [ 6 p t ] = - 1 4 π [ ( V 𝐅 ( 𝐫 ) 1 | 𝐫 - 𝐫 | d V ) + × ( V 𝐅 ( 𝐫 ) × 1 | 𝐫 - 𝐫 | d V ) ] [ 6 p t ] = - 1 4 π [ - ( V 𝐅 ( 𝐫 ) 1 | 𝐫 - 𝐫 | d V ) - × ( V 𝐅 ( 𝐫 ) × 1 | 𝐫 - 𝐫 | d V ) ] 1 | 𝐫 - 𝐫 | = - 1 | 𝐫 - 𝐫 | \begin{aligned}\displaystyle\mathbf{F}(\mathbf{r})&\displaystyle=\int_{V}% \mathbf{F}\left(\mathbf{r}^{\prime}\right)\delta^{3}(\mathbf{r}-\mathbf{r}^{% \prime})\mathrm{d}V^{\prime}\\ \displaystyle[6pt]&\displaystyle=\int_{V}\mathbf{F}(\mathbf{r}^{\prime})\left(% -\frac{1}{4\pi}\nabla^{2}\frac{1}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}% \right)\mathrm{d}V^{\prime}\\ \displaystyle[6pt]&\displaystyle=-\frac{1}{4\pi}\nabla^{2}\int_{V}\frac{% \mathbf{F}(\mathbf{r}^{\prime})}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}% \mathrm{d}V^{\prime}\\ \displaystyle[6pt]&\displaystyle=-\frac{1}{4\pi}\left[\nabla\left(\nabla\cdot% \int_{V}\frac{\mathbf{F}(\mathbf{r}^{\prime})}{\left|\mathbf{r}-\mathbf{r}^{% \prime}\right|}\mathrm{d}V^{\prime}\right)-\nabla\times\left(\nabla\times\int_% {V}\frac{\mathbf{F}(\mathbf{r}^{\prime})}{\left|\mathbf{r}-\mathbf{r}^{\prime}% \right|}\mathrm{d}V^{\prime}\right)\right]&&\displaystyle\nabla^{2}\mathbf{a}=% \nabla(\nabla\cdot\mathbf{a})-\nabla\times(\nabla\times\mathbf{a})\\ \displaystyle[6pt]&\displaystyle=-\frac{1}{4\pi}\left[\nabla\left(\int_{V}% \mathbf{F}(\mathbf{r}^{\prime})\cdot\nabla\frac{1}{\left|\mathbf{r}-\mathbf{r}% ^{\prime}\right|}\mathrm{d}V^{\prime}\right)+\nabla\times\left(\int_{V}\mathbf% {F}(\mathbf{r}^{\prime})\times\nabla\frac{1}{\left|\mathbf{r}-\mathbf{r}^{% \prime}\right|}\mathrm{d}V^{\prime}\right)\right]\\ \displaystyle[6pt]&\displaystyle=-\frac{1}{4\pi}\left[-\nabla\left(\int_{V}% \mathbf{F}(\mathbf{r}^{\prime})\cdot\nabla^{\prime}\frac{1}{\left|\mathbf{r}-% \mathbf{r}^{\prime}\right|}\mathrm{d}V^{\prime}\right)-\nabla\times\left(\int_% {V}\mathbf{F}(\mathbf{r}^{\prime})\times\nabla^{\prime}\frac{1}{\left|\mathbf{% r}-\mathbf{r}^{\prime}\right|}\mathrm{d}V^{\prime}\right)\right]&&% \displaystyle\nabla\frac{1}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}=-% \nabla^{\prime}\frac{1}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}\\ \end{aligned}
  21. 𝐚 ψ = - ψ ( 𝐚 ) + ( ψ 𝐚 ) 𝐚 × ψ = ψ ( × 𝐚 ) - × ( ψ 𝐚 ) \begin{aligned}\displaystyle\mathbf{a}\cdot\nabla\psi&\displaystyle=-\psi(% \nabla\cdot\mathbf{a})+\nabla\cdot(\psi\mathbf{a})\\ \displaystyle\mathbf{a}\times\nabla\psi&\displaystyle=\psi(\nabla\times\mathbf% {a})-\nabla\times(\psi\mathbf{a})\end{aligned}
  22. 𝐅 ( 𝐫 ) = - 1 4 π [ - ( - V 𝐅 ( 𝐫 ) | 𝐫 - 𝐫 | d V + V 𝐅 ( 𝐫 ) | 𝐫 - 𝐫 | d V ) - × ( V × 𝐅 ( 𝐫 ) | 𝐫 - 𝐫 | d V - V × 𝐅 ( 𝐫 ) | 𝐫 - 𝐫 | d V ) ] . \mathbf{F}(\mathbf{r})=-\frac{1}{4\pi}\left[-\nabla\left(-\int_{V}\frac{\nabla% ^{\prime}\cdot\mathbf{F}\left(\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-% \mathbf{r}^{\prime}\right|}\mathrm{d}V^{\prime}+\int_{V}\nabla^{\prime}\cdot% \frac{\mathbf{F}\left(\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^% {\prime}\right|}\mathrm{d}V^{\prime}\right)-\nabla\times\left(\int_{V}\frac{% \nabla^{\prime}\times\mathbf{F}\left(\mathbf{r}^{\prime}\right)}{\left|\mathbf% {r}-\mathbf{r}^{\prime}\right|}\mathrm{d}V^{\prime}-\int_{V}\nabla^{\prime}% \times\frac{\mathbf{F}\left(\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-% \mathbf{r}^{\prime}\right|}\mathrm{d}V^{\prime}\right)\right].
  23. 𝐅 ( 𝐫 ) = - 1 4 π [ - ( - V 𝐅 ( 𝐫 ) | 𝐫 - 𝐫 | d V + S 𝐧 ^ 𝐅 ( 𝐫 ) | 𝐫 - 𝐫 | d S ) - × ( V × 𝐅 ( 𝐫 ) | 𝐫 - 𝐫 | d V - S 𝐧 ^ × 𝐅 ( 𝐫 ) | 𝐫 - 𝐫 | d S ) ] = - [ 1 4 π V 𝐅 ( 𝐫 ) | 𝐫 - 𝐫 | d V - 1 4 π S 𝐧 ^ 𝐅 ( 𝐫 ) | 𝐫 - 𝐫 | d S ] + × [ 1 4 π V × 𝐅 ( 𝐫 ) | 𝐫 - 𝐫 | d V - 1 4 π S 𝐧 ^ × 𝐅 ( 𝐫 ) | 𝐫 - 𝐫 | d S ] \begin{aligned}\displaystyle\mathbf{F}(\mathbf{r})&\displaystyle=-\frac{1}{4% \pi}\left[-\nabla\left(-\int_{V}\frac{\nabla^{\prime}\cdot\mathbf{F}\left(% \mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}% \mathrm{d}V^{\prime}+\oint_{S}\mathbf{\hat{n}}^{\prime}\cdot\frac{\mathbf{F}% \left(\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}% \mathrm{d}S^{\prime}\right)-\nabla\times\left(\int_{V}\frac{\nabla^{\prime}% \times\mathbf{F}\left(\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^% {\prime}\right|}\mathrm{d}V^{\prime}-\oint_{S}\mathbf{\hat{n}}^{\prime}\times% \frac{\mathbf{F}\left(\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^% {\prime}\right|}\mathrm{d}S^{\prime}\right)\right]\\ &\displaystyle=-\nabla\left[\frac{1}{4\pi}\int_{V}\frac{\nabla^{\prime}\cdot% \mathbf{F}\left(\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^{% \prime}\right|}\mathrm{d}V^{\prime}-\frac{1}{4\pi}\oint_{S}\mathbf{\hat{n}}^{% \prime}\cdot\frac{\mathbf{F}\left(\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}% -\mathbf{r}^{\prime}\right|}\mathrm{d}S^{\prime}\right]+\nabla\times\left[% \frac{1}{4\pi}\int_{V}\frac{\nabla^{\prime}\times\mathbf{F}\left(\mathbf{r}^{% \prime}\right)}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}\mathrm{d}V^{% \prime}-\frac{1}{4\pi}\oint_{S}\mathbf{\hat{n}}^{\prime}\times\frac{\mathbf{F}% \left(\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}% \mathrm{d}S^{\prime}\right]\end{aligned}
  24. 𝐧 ^ \mathbf{\hat{n}}^{\prime}
  25. Φ ( 𝐫 ) 1 4 π V 𝐅 ( 𝐫 ) | 𝐫 - 𝐫 | d V - 1 4 π S 𝐧 ^ 𝐅 ( 𝐫 ) | 𝐫 - 𝐫 | d S \Phi(\mathbf{r})\equiv\frac{1}{4\pi}\int_{V}\frac{\nabla^{\prime}\cdot\mathbf{% F}\left(\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right% |}\mathrm{d}V^{\prime}-\frac{1}{4\pi}\oint_{S}\mathbf{\hat{n}}^{\prime}\cdot% \frac{\mathbf{F}\left(\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^% {\prime}\right|}\mathrm{d}S^{\prime}
  26. 𝐀 ( 𝐫 ) 1 4 π V × 𝐅 ( 𝐫 ) | 𝐫 - 𝐫 | d V - 1 4 π S 𝐧 ^ × 𝐅 ( 𝐫 ) | 𝐫 - 𝐫 | d S \mathbf{A}(\mathbf{r})\equiv\frac{1}{4\pi}\int_{V}\frac{\nabla^{\prime}\times% \mathbf{F}\left(\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^{% \prime}\right|}\mathrm{d}V^{\prime}-\frac{1}{4\pi}\oint_{S}\mathbf{\hat{n}}^{% \prime}\times\frac{\mathbf{F}\left(\mathbf{r}^{\prime}\right)}{\left|\mathbf{r% }-\mathbf{r}^{\prime}\right|}\mathrm{d}S^{\prime}
  27. 𝐅 = - Φ + × 𝐀 \mathbf{F}=-\nabla\Phi+\nabla\times\mathbf{A}
  28. 𝐅 \mathbf{F}
  29. 𝐅 ( r ) = 𝐆 ( ω ) e i ω r d ω \vec{\mathbf{F}}(\vec{r})=\iiint\vec{\mathbf{G}}(\vec{\omega})e^{i\vec{\omega}% \cdot\vec{r}}d\vec{\omega}
  30. G Φ ( ω ) \displaystyle G_{\Phi}(\vec{\omega})
  31. 𝐆 ( ω ) = - i ω G Φ ( ω ) + i ω × 𝐆 𝐀 ( ω ) [ 6 p t ] 𝐅 ( r ) = - i ω G Φ ( ω ) e i ω r d ω + i ω × 𝐆 𝐀 ( ω ) e i ω r d ω = - Φ ( r ) + × 𝐀 ( r ) \begin{aligned}\displaystyle\vec{\mathbf{G}}(\vec{\omega})&\displaystyle=-i% \vec{\omega}G_{\Phi}(\vec{\omega})+i\vec{\omega}\times\vec{\mathbf{G}}_{% \mathbf{A}}(\vec{\omega})\\ \displaystyle[6pt]\vec{\mathbf{F}}(\vec{r})&\displaystyle=-\iiint i\vec{\omega% }G_{\Phi}(\vec{\omega})e^{i\vec{\omega}\cdot\vec{r}}d\vec{\omega}+\iiint i\vec% {\omega}\times\vec{\mathbf{G}}_{\mathbf{A}}(\vec{\omega})e^{i\vec{\omega}\cdot% \vec{r}}d\vec{\omega}\\ &\displaystyle=-\nabla\Phi(\vec{r})+\nabla\times\vec{\mathbf{A}}(\vec{r})\end{aligned}
  32. 𝐂 \mathbf{C}
  33. 𝐅 \mathbf{F}
  34. 𝐅 = d \nabla\cdot\mathbf{F}=d
  35. × 𝐅 = 𝐂 ; \nabla\times\mathbf{F}=\mathbf{C};
  36. 𝐅 \mathbf{F}
  37. r r→∞
  38. 𝐅 \mathbf{F}
  39. 𝐅 = - ( 𝒢 ( d ) ) + × ( 𝒢 ( 𝐂 ) ) , \mathbf{F}=-\nabla(\mathcal{G}(d))+\nabla\times(\mathcal{G}(\mathbf{C})),
  40. 𝒢 \mathcal{G}
  41. × 𝐅 ∇×\mathbf{F}
  42. Ω Ω
  43. 𝐮 = φ + × 𝐀 \mathbf{u}=\nabla\varphi+\nabla\times\mathbf{A}
  44. φ φ
  45. Ω Ω
  46. 𝐀 H ( c u r l , Ω ) \mathbf{A}∈H(curl,Ω)
  47. 𝐮 H ( c u r l , Ω ) \mathbf{u}∈H(curl,Ω)
  48. 𝐮 = φ + 𝐯 \mathbf{u}=\nabla\varphi+\mathbf{v}
  49. 𝐅 ( 𝐤 ) = 𝐅 t ( 𝐤 ) + 𝐅 l ( 𝐤 ) \mathbf{F}(\mathbf{k})=\mathbf{F}_{t}(\mathbf{k})+\mathbf{F}_{l}(\mathbf{k})
  50. 𝐤 𝐅 t ( 𝐤 ) = 0. \mathbf{k}\cdot\mathbf{F}_{t}(\mathbf{k})=0.
  51. 𝐤 × 𝐅 l ( 𝐤 ) = 0. \mathbf{k}\times\mathbf{F}_{l}(\mathbf{k})=\mathbf{0}.
  52. 𝐅 ( 𝐫 ) = 𝐅 t ( 𝐫 ) + 𝐅 l ( 𝐫 ) \mathbf{F}(\mathbf{r})=\mathbf{F}_{t}(\mathbf{r})+\mathbf{F}_{l}(\mathbf{r})
  53. 𝐅 t ( 𝐫 ) = 0 \nabla\cdot\mathbf{F}_{t}(\mathbf{r})=0
  54. × 𝐅 l ( 𝐫 ) = 𝟎 \nabla\times\mathbf{F}_{l}(\mathbf{r})=\mathbf{0}
  55. × ( Φ ) = 0 \nabla\times(\nabla\Phi)=0
  56. ( × 𝐀 ) = 0 \nabla\cdot(\nabla\times\mathbf{A})=0
  57. 𝐅 t = × 𝐀 = 1 4 π × V × 𝐅 | 𝐫 - 𝐫 | d V \mathbf{F}_{t}=\nabla\times\mathbf{A}=\frac{1}{4\pi}\nabla\times\int_{V}\frac{% \nabla^{\prime}\times\mathbf{F}}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}% \mathrm{d}V^{\prime}
  58. 𝐅 l = - Φ = - 1 4 π V 𝐅 | 𝐫 - 𝐫 | d V \mathbf{F}_{l}=-\nabla\Phi=-\frac{1}{4\pi}\nabla\int_{V}\frac{\nabla^{\prime}% \cdot\mathbf{F}}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}\mathrm{d}V^{\prime}
  59. × 𝐀 \nabla\times\mathbf{A}

Helmholtz_equation.html

  1. 2 A + k 2 A = 0 \nabla^{2}A+k^{2}A=0
  2. ( 2 - 1 c 2 2 t 2 ) u ( 𝐫 , t ) = 0. \left(\nabla^{2}-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial{t}^{2}}\right)u(% \mathbf{r},t)=0.
  3. u ( 𝐫 , t ) = A ( 𝐫 ) T ( t ) . u(\mathbf{r},t)=A(\mathbf{r})T(t).
  4. 2 A A = 1 c 2 T d 2 T d t 2 . {\nabla^{2}A\over A}={1\over c^{2}T}{d^{2}T\over dt^{2}}.
  5. 2 A A = - k 2 {\nabla^{2}A\over A}=-k^{2}
  6. 1 c 2 T d 2 T d t 2 = - k 2 {1\over c^{2}T}{d^{2}T\over dt^{2}}=-k^{2}
  7. 2 A + k 2 A = ( 2 + k 2 ) A = 0. \nabla^{2}A+k^{2}A=(\nabla^{2}+k^{2})A=0.
  8. ω = def k c \omega\stackrel{\mathrm{def}}{=}kc
  9. d 2 T d t 2 + ω 2 T = ( d 2 d t 2 + ω 2 ) T = 0 , \frac{d^{2}{T}}{d{t}^{2}}+\omega^{2}T=\left({d^{2}\over dt^{2}}+\omega^{2}% \right)T=0,
  10. ( 2 + k 2 ) A = 0 (\nabla^{2}+k^{2})A=0
  11. A r r + 1 r A r + 1 r 2 A θ θ + k 2 A = 0. A_{rr}+\frac{1}{r}A_{r}+\frac{1}{r^{2}}A_{\theta\theta}+k^{2}A=0.
  12. A ( a , θ ) = 0. A(a,\theta)=0.\,
  13. A ( r , θ ) = R ( r ) Θ ( θ ) , A(r,\theta)=R(r)\Theta(\theta),\,
  14. Θ ′′ + n 2 Θ = 0 , \Theta^{\prime\prime}+n^{2}\Theta=0,\,
  15. r 2 R ′′ + r R + r 2 k 2 R - n 2 R = 0. r^{2}R^{\prime\prime}+rR^{\prime}+r^{2}k^{2}R-n^{2}R=0.\,
  16. Θ = α cos n θ + β sin n θ , \Theta=\alpha\cos n\theta+\beta\sin n\theta,\,
  17. R ( r ) = γ J n ( ρ ) , R(r)=\gamma J_{n}(\rho),\,
  18. ρ 2 J n ′′ + ρ J n + ( ρ 2 - n 2 ) J n = 0 , \rho^{2}J_{n}^{\prime\prime}+\rho J_{n}^{\prime}+(\rho^{2}-n^{2})J_{n}=0,
  19. k m , n = 1 a ρ m , n . k_{m,n}=\frac{1}{a}\rho_{m,n}.\,
  20. sin ( n θ ) or cos ( n θ ) , and J n ( k m , n r ) . \sin(n\theta)\,\hbox{or}\,\cos(n\theta),\,\text{ and }J_{n}(k_{m,n}r).
  21. A ( r , θ , φ ) = = 0 m = - ( a m j ( k r ) + b m y ( k r ) ) Y m ( θ , φ ) . A(r,\theta,\varphi)=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}(a_{\ell m}j_{% \ell}(kr)+b_{\ell m}y_{\ell}(kr))Y^{m}_{\ell}({\theta,\varphi}).
  22. j ( k r ) j_{\ell}(kr)
  23. y ( k r ) y_{\ell}(kr)
  24. Y m ( θ , φ ) Y^{m}_{\ell}({\theta,\varphi})
  25. 𝐫 𝟎 = ( x , y , z ) \mathbf{r_{0}}=(x,y,z)
  26. A ( r 0 ) A(r_{0})
  27. A ( r 0 ) = e i k r 0 r 0 f ( 𝐫 0 / r 0 , k , u 0 ) + o ( 1 / r 0 ) as r 0 A(r_{0})=\frac{e^{ikr_{0}}}{r_{0}}f(\mathbf{r}_{0}/r_{0},k,u_{0})+o(1/r_{0})\,% \text{ as }r_{0}\to\infty
  28. u 0 ( r 0 ) u_{0}(r_{0})
  29. r 0 r_{0}
  30. A ( 𝐫 ) = u ( 𝐫 ) e i k z A(\mathbf{r})=u(\mathbf{r})e^{ikz}
  31. 2 u + 2 i k u z = 0 , \nabla_{\perp}^{2}u+2ik\frac{\partial u}{\partial z}=0,
  32. 2 = def 2 x 2 + 2 y 2 \textstyle\nabla_{\perp}^{2}\stackrel{\mathrm{def}}{=}\frac{\partial^{2}}{% \partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}
  33. | 2 u z 2 | | k u z | . \bigg|{\partial^{2}u\over\partial z^{2}}\bigg|\ll\bigg|{k{\partial u\over% \partial z}}\bigg|.
  34. θ 1 \theta\ll 1
  35. 2 ( u ( x , y , z ) e i k z ) + k 2 u ( x , y , z ) e i k z = 0. \nabla^{2}(u\left(x,y,z\right)e^{ikz})+k^{2}u\left(x,y,z\right)e^{ikz}=0.
  36. ( 2 x 2 + 2 y 2 ) u ( x , y , z ) e i k z + ( 2 z 2 u ( x , y , z ) ) e i k z + 2 ( z u ( x , y , z ) ) i k e i k z = 0. \left({\frac{\partial^{2}}{\partial{x}^{2}}}+{\frac{\partial^{2}}{\partial{y}^% {2}}}\right)u\left(x,y,z\right)e^{ikz}+\left({\frac{\partial^{2}}{\partial{z}^% {2}}}u\left(x,y,z\right)\right){e^{ikz}}+2\,\left({\frac{\partial}{\partial z}% }u\left(x,y,z\right)\right)ik{e^{ikz}}=0.
  37. u ( 𝐫 ) = A ( 𝐫 ) e - i k z u(\mathbf{r})=A(\mathbf{r})e^{-ikz}
  38. 2 A + 2 i k A z + 2 k 2 A = 0. \nabla_{\perp}^{2}A+2ik\frac{\partial A}{\partial z}+2k^{2}A=0.
  39. 2 A ( x ) + k 2 A ( x ) = - f ( x ) in n \nabla^{2}A(x)+k^{2}A(x)=-f(x)\mbox{ in }\mathbb{R}^{n}
  40. lim r r n - 1 2 ( r - i k ) A ( r x ^ ) = 0 \lim_{r\to\infty}r^{\frac{n-1}{2}}\left(\frac{\partial}{\partial r}-ik\right)A% (r\hat{x})=0
  41. x ^ \hat{x}
  42. | x ^ | = 1 |\hat{x}|=1
  43. A ( x ) = ( G * f ) ( x ) = n G ( x - y ) f ( y ) d y A(x)=(G*f)(x)=\int\limits_{\mathbb{R}^{n}}\!G(x-y)f(y)\,dy
  44. f f
  45. G G
  46. 2 G ( x ) + k 2 G ( x ) = - δ ( x ) in n . \nabla^{2}G(x)+k^{2}G(x)=-\delta(x)\,\text{ in }\mathbb{R}^{n}.\,
  47. n n
  48. G ( x ) = i e i k | x | 2 k G(x)=\frac{ie^{ik|x|}}{2k}
  49. G ( x ) = i 4 H 0 ( 1 ) ( k | x | ) G(x)=\frac{i}{4}H^{(1)}_{0}(k|x|)
  50. H 0 ( 1 ) H^{(1)}_{0}
  51. G ( x ) = e i k | x | 4 π | x | G(x)=\frac{e^{ik|x|}}{4\pi|x|}
  52. | x | |x|\to\infty

Herbrand–Ribet_theorem.html

  1. σ a ( ζ ) = ζ a \sigma_{a}(\zeta)=\zeta^{a}
  2. p \mathbb{Z}_{p}
  3. p \mathbb{Z}_{p}
  4. p \mathbb{Z}_{p}
  5. p [ Δ ] \mathbb{Z}_{p}[\Delta]
  6. ϵ n = 1 p - 1 a = 1 p - 1 ω ( a ) n σ a - 1 . \epsilon_{n}=\frac{1}{p-1}\sum_{a=1}^{p-1}\omega(a)^{n}\sigma_{a}^{-1}.
  7. ϵ n = 1 \sum\epsilon_{n}=1
  8. ϵ i ϵ j = δ i j ϵ i \epsilon_{i}\epsilon_{j}=\delta_{ij}\epsilon_{i}
  9. δ i j \delta_{ij}
  10. G = G n G=\oplus G_{n}

Hermitian_adjoint.html

  1. A A
  2. A A
  3. A * A*
  4. H H
  5. , \langle\cdot,\cdot\rangle
  6. A : H H A:H→H
  7. A A
  8. A * : H H A*:H→H
  9. A x , y = x , A * y for all x , y H . \langle Ax,y\rangle=\langle x,A^{*}y\rangle\quad\mbox{for all }~{}x,y\in H.
  10. A * * = A A**=A
  11. A A
  12. A * A*
  13. ( A + B ) * = A * + B * (A+B)*=A*+B*
  14. ( λ A ) * = λ ¯ A * (λA)*=\overline{λ}A*
  15. λ ¯ \overline{λ}
  16. λ λ
  17. ( A B ) * = B * A * (AB)*=B*A*
  18. A A
  19. A o p := sup { A x : x 1 } \|A\|_{op}:=\sup\{\|Ax\|:\|x\|\leq 1\}
  20. A * o p = A o p . \|A^{*}\|_{op}=\|A\|_{op}.
  21. A * A o p = A o p 2 . \|A^{*}A\|_{op}=\|A\|_{op}^{2}.
  22. H H
  23. A A
  24. H H
  25. D ( A ) D(A)
  26. H H
  27. H H
  28. D ( A * ) D(A*)
  29. A * A*
  30. y H y∈H
  31. z H z∈H
  32. A x , y = x , z for all x D ( A ) , \langle Ax,y\rangle=\langle x,z\rangle\quad\mbox{for all }~{}x\in D(A),
  33. A * ( y ) A*(y)
  34. z z
  35. ( A B ) * (AB)*
  36. B * A * B*A*
  37. A A
  38. B B
  39. A B AB
  40. A A
  41. ker A * = ( im A ) \ker A^{*}=\left(\operatorname{im}\ A\right)^{\bot}
  42. ( ker A * ) = im A ¯ \left(\ker A^{*}\right)^{\bot}=\overline{\operatorname{im}\ A}
  43. \bot
  44. A * x = 0 A * x , y = 0 y H x , A y = 0 y H x im A \begin{aligned}\displaystyle A^{*}x=0&\displaystyle\iff\langle A^{*}x,y\rangle% =0\quad\forall y\in H\\ &\displaystyle\iff\langle x,Ay\rangle=0\quad\forall y\in H\\ &\displaystyle\iff x\ \bot\ \operatorname{im}\ A\end{aligned}
  45. A : H H A:H→H
  46. A = A * A=A^{*}
  47. A x , y = x , A y for all x , y H . \langle Ax,y\rangle=\langle x,Ay\rangle\mbox{ for all }~{}x,y\in H.
  48. A A
  49. H H
  50. A * : H H A*:H→H
  51. A x , y = x , A * y ¯ for all x , y H . \langle Ax,y\rangle=\overline{\langle x,A^{*}y\rangle}\quad\,\text{for all }x,% y\in H.
  52. A x , y = x , A * y \langle Ax,y\rangle=\langle x,A^{*}y\rangle

Hermitian_hat_wavelet.html

  1. Ψ ( t ) = 2 5 π - 1 4 ( 1 - t 2 + i t ) e - 1 2 t 2 . \Psi(t)=\frac{2}{\sqrt{5}}\pi^{-\frac{1}{4}}(1-t^{2}+it)e^{-\frac{1}{2}t^{2}}.
  2. Ψ ^ ( ω ) = 2 5 π - 1 4 ω ( 1 + ω ) e - 1 2 ω 2 . \hat{\Psi}(\omega)=\frac{2}{\sqrt{5}}\pi^{-\frac{1}{4}}\omega(1+\omega)e^{-% \frac{1}{2}\omega^{2}}.
  3. C Ψ C_{\Psi}
  4. C Ψ = 16 5 π . C_{\Psi}=\frac{16}{5}\sqrt{\pi}.

Hermitian_wavelet.html

  1. n th n^{\textrm{th}}
  2. n th n^{\textrm{th}}
  3. Ψ n ( t ) = ( 2 n ) - n 2 c n H n ( t n ) e - 1 2 n t 2 \Psi_{n}(t)=(2n)^{-\frac{n}{2}}c_{n}H_{n}\left(\frac{t}{\sqrt{n}}\right)e^{-% \frac{1}{2n}t^{2}}
  4. H n ( x ) H_{n}\left({x}\right)
  5. n th n^{\textrm{th}}
  6. c n c_{n}
  7. c n = ( n 1 2 - n Γ ( n + 1 2 ) ) - 1 2 = ( n 1 2 - n π 2 - n ( 2 n - 1 ) ! ! ) - 1 2 n . c_{n}=\left(n^{\frac{1}{2}-n}\Gamma(n+\frac{1}{2})\right)^{-\frac{1}{2}}=\left% (n^{\frac{1}{2}-n}\sqrt{\pi}2^{-n}(2n-1)!!\right)^{-\frac{1}{2}}\quad n\in% \mathbb{Z}.
  8. C Ψ C_{\Psi}
  9. C Ψ = 4 π n 2 n - 1 C_{\Psi}=\frac{4\pi n}{2n-1}
  10. n n
  11. μ = 0 , σ = 1 \mu=0,\sigma=1
  12. f ( t ) = π - 1 / 4 e ( - t 2 / 2 ) f(t)=\pi^{-1/4}e^{(-t^{2}/2)}
  13. f ( t ) = - π - 1 / 4 t e ( - t 2 / 2 ) f ′′ ( t ) = π - 1 / 4 ( t 2 - 1 ) e ( - t 2 / 2 ) f ( 3 ) ( t ) = π - 1 / 4 ( 3 t - t 3 ) e ( - t 2 / 2 ) \begin{aligned}\displaystyle f^{\prime}(t)&\displaystyle=-\pi^{-1/4}te^{(-t^{2% }/2)}\\ \displaystyle f^{\prime\prime}(t)&\displaystyle=\pi^{-1/4}(t^{2}-1)e^{(-t^{2}/% 2)}\\ \displaystyle f^{(3)}(t)&\displaystyle=\pi^{-1/4}(3t-t^{3})e^{(-t^{2}/2)}\end{aligned}
  14. L 2 L^{2}
  15. || f || = 2 / 2 , || f ′′ || = 3 / 2 , || f ( 3 ) || = 30 / 4 ||f^{\prime}||=\sqrt{2}/2,||f^{\prime\prime}||=\sqrt{3}/2,||f^{(3)}||=\sqrt{30% }/4
  16. Ψ 1 ( t ) \displaystyle\Psi_{1}(t)

Heronian_triangle.html

  1. A = 1 2 ( b + d ) a A=\frac{1}{2}(b+d)a
  2. a = n ( m 2 + k 2 ) a=n(m^{2}+k^{2})\,
  3. b = m ( n 2 + k 2 ) b=m(n^{2}+k^{2})\,
  4. c = ( m + n ) ( m n - k 2 ) c=(m+n)(mn-k^{2})\,
  5. Semiperimeter = s = ( a + b + c ) / 2 = m n ( m + n ) \,\text{Semiperimeter}=s=(a+b+c)/2=mn(m+n)\,
  6. Area = m n k ( m + n ) ( m n - k 2 ) \,\text{Area}=mnk(m+n)(mn-k^{2})\,
  7. Inradius = k ( m n - k 2 ) \,\text{Inradius}=k(mn-k^{2})\,
  8. s - a = n ( m n - k 2 ) s-a=n(mn-k^{2})\,
  9. s - b = m ( m n - k 2 ) s-b=m(mn-k^{2})\,
  10. s - c = ( m + n ) k 2 s-c=(m+n)k^{2}\,
  11. gcd ( m , n , k ) = 1 \gcd{(m,n,k)}=1\,
  12. m n > k 2 m 2 n / ( 2 m + n ) mn>k^{2}\geq m^{2}n/(2m+n)\,
  13. m n 1 m\geq n\geq 1\,
  14. p q \frac{p}{q}
  15. q = gcd ( a , b , c ) q=\gcd{(a,b,c)}
  16. p p
  17. n t = 4 n t - 1 - n t - 2 , n_{t}=4n_{t-1}-n_{t-2}\,,
  18. ( 2 + 3 ) t + ( 2 - 3 ) t (2+\sqrt{3})^{t}+(2-\sqrt{3})^{t}
  19. ( ( n - 1 ) 2 + n 2 + ( n + 1 ) 2 ) 2 - 2 ( ( n - 1 ) 4 + n 4 + ( n + 1 ) 4 ) = ( 6 n y ) 2 = ( 4 A ) 2 \big((n-1)^{2}+n^{2}+(n+1)^{2}\big)^{2}-2\big((n-1)^{4}+n^{4}+(n+1)^{4}\big)=(% 6ny)^{2}=(4A)^{2}
  20. n = 2 + 2 k n=\sqrt{2+2k}

Heterojunction.html

  1. Δ E C / Δ E V \Delta E_{C}/\Delta E_{V}
  2. Δ E C / Δ E V = 0.73 / 0.27 \Delta E_{C}/\Delta E_{V}=0.73/0.27
  3. l w l_{w}
  4. ψ ( z ) , 1 m * z ψ ( z ) \psi(z),{\frac{1}{m^{*}}}{\partial\over{\partial z}}\psi(z)\,
  5. l w l_{w}
  6. - 2 2 m b * d 2 ψ ( z ) d z 2 + V ψ ( z ) = E ψ ( z ) for z < - l w 2 ( 1 ) -\frac{\hbar^{2}}{2m_{b}^{*}}\frac{\mathrm{d}^{2}\psi(z)}{\mathrm{d}z^{2}}+V% \psi(z)=E\psi(z)\quad\quad\,\text{ for }z<-\frac{l_{w}}{2}\quad\quad(1)
  7. - 2 2 m w * d 2 ψ ( z ) d z 2 = E ψ ( z ) for - l w 2 < z < + l w 2 ( 2 ) \quad\quad-\frac{\hbar^{2}}{2m_{w}^{*}}\frac{\mathrm{d}^{2}\psi(z)}{\mathrm{d}% z^{2}}=E\psi(z)\quad\quad\,\text{ for }-\frac{l_{w}}{2}<z<+\frac{l_{w}}{2}% \quad\quad(2)
  8. - 2 2 m b * d 2 ψ ( z ) d z 2 + V ψ ( z ) = E ψ ( z ) for z > + l w 2 ( 3 ) -\frac{\hbar^{2}}{2m_{b}^{*}}\frac{\mathrm{d}^{2}\psi(z)}{\mathrm{d}z^{2}}+V% \psi(z)=E\psi(z)\quad\,\text{ for }z>+\frac{l_{w}}{2}\quad\quad(3)
  9. κ \kappa
  10. k = 2 m w E κ = 2 m b ( V - E ) ( 4 ) k=\frac{\sqrt{2m_{w}E}}{\hbar}\quad\quad\kappa=\frac{\sqrt{2m_{b}(V-E)}}{\hbar% }\quad\quad(4)
  11. + l w 2 +\frac{l_{w}}{2}
  12. A cos ( k l w 2 ) = B exp ( - κ l w 2 ) ( 5 ) A\cos(\frac{kl_{w}}{2})=B\exp(-\frac{\kappa l_{w}}{2})\quad\quad(5)
  13. 1 m * \frac{1}{m^{*}}
  14. - k A m w * sin ( k l w 2 ) = - κ B m b * exp ( - κ l w 2 ) ( 6 ) -\frac{kA}{m_{w}^{*}}\sin(\frac{kl_{w}}{2})=-\frac{\kappa B}{m_{b}^{*}}\exp(-% \frac{\kappa l_{w}}{2})\quad\quad(6)
  15. f ( E ) = - k m w * tan ( k l w 2 ) - κ m b * = 0 ( 7 ) f(E)=-\frac{k}{m_{w}^{*}}\tan(\frac{kl_{w}}{2})-\frac{\kappa}{m_{b}^{*}}=0% \quad\quad(7)
  16. f ( E ) = - k m w * cot ( k l w 2 ) + κ m b * = 0 ( 8 ) f(E)=-\frac{k}{m_{w}^{*}}\cot(\frac{kl_{w}}{2})+\frac{\kappa}{m_{b}^{*}}=0% \quad\quad(8)
  17. d f d E = 1 m w * d k d E tan ( k l w 2 ) + k m w * sec 2 ( k l w 2 ) × l w 2 d k d E - 1 m b * d κ d E ( 9 - 1 ) \frac{df}{dE}=\frac{1}{m_{w}^{*}}\frac{dk}{dE}\tan(\frac{kl_{w}}{2})+\frac{k}{% m_{w}^{*}}\sec^{2}(\frac{kl_{w}}{2})\times\frac{l_{w}}{2}\frac{dk}{dE}-\frac{1% }{m_{b}^{*}}\frac{d\kappa}{dE}\quad\quad(9-1)
  18. d f d E = 1 m w * d k d E cot ( k l w 2 ) - k m w * csc 2 ( k l w 2 ) × l w 2 d k d E + 1 m b * d κ d E ( 9 - 2 ) \frac{df}{dE}=\frac{1}{m_{w}^{*}}\frac{dk}{dE}\cot(\frac{kl_{w}}{2})-\frac{k}{% m_{w}^{*}}\csc^{2}(\frac{kl_{w}}{2})\times\frac{l_{w}}{2}\frac{dk}{dE}+\frac{1% }{m_{b}^{*}}\frac{d\kappa}{dE}\quad\quad(9-2)
  19. d k d E = 2 m w * 2 E d κ d E = - 2 m b * 2 V - E \frac{dk}{dE}=\frac{\sqrt{2m_{w}^{*}}}{2\sqrt{E}\hbar}\quad\quad\quad\frac{d% \kappa}{dE}=-\frac{\sqrt{2m_{b}^{*}}}{2\sqrt{V-E}\hbar}

Hexapawn.html

  1. N N
  2. N N

Hick's_law.html

  1. T = b log 2 ( n + 1 ) T=b\cdot\log_{2}(n+1)
  2. T = b H T=bH
  3. H = i n p i log 2 ( 1 / p i + 1 ) H=\sum_{i}^{n}p_{i}\log_{2}(1/p_{i}+1)
  4. Reaction Time = Movement Time + log 2 ( n ) Processing Speed \,\text{Reaction Time}=\,\text{Movement Time}+\frac{\log_{2}(n)}{\,\text{% Processing Speed}}
  5. Processing Speed log 2 ( n ) \,\text{Processing Speed}\cdot\log_{2}(n)

Hierarchy_problem.html

  1. Yukawa = - λ f ψ ¯ H ψ \mathcal{L}_{\mathrm{Yukawa}}=-\lambda_{f}\bar{\psi}H\psi
  2. ψ \psi
  3. H H
  4. Δ m H 2 = - | λ f | 2 8 π 2 [ Λ UV 2 + ] . \Delta m_{H}^{2}=-\frac{\left|\lambda_{f}\right|^{2}}{8\pi^{2}}[\Lambda_{% \mathrm{UV}}^{2}+...].
  5. Λ UV \Lambda_{\mathrm{UV}}
  6. Δ m H 2 = 2 × λ S 16 π 2 [ Λ UV 2 + ] . \Delta m_{H}^{2}=2\times\frac{\lambda_{S}}{16\pi^{2}}[\Lambda_{\mathrm{UV}}^{2% }+...].
  7. 𝐠 ( 𝐫 ) = - G m 𝐞 𝐫 r 2 \mathbf{g}(\mathbf{r})=-Gm\frac{\mathbf{e_{r}}}{r^{2}}
  8. 1 M Pl 2 \frac{1}{M_{\mathrm{Pl}}^{2}}
  9. δ \delta
  10. 𝐠 ( 𝐫 ) = - m 𝐞 𝐫 M Pl 3 + 1 + δ 2 + δ r 2 + δ \mathbf{g}(\mathbf{r})=-m\frac{\mathbf{e_{r}}}{M_{\mathrm{Pl}_{3+1+\delta}}^{2% +\delta}r^{2+\delta}}
  11. M Pl 3 + 1 + δ M_{\mathrm{Pl}_{3+1+\delta}}
  12. δ \delta
  13. n δ n^{\delta}
  14. 𝐠 ( 𝐫 ) = - m 𝐞 𝐫 M Pl 3 + 1 + δ 2 + δ r 2 n δ \mathbf{g}(\mathbf{r})=-m\frac{\mathbf{e_{r}}}{M_{\mathrm{Pl}_{3+1+\delta}}^{2% +\delta}r^{2}n^{\delta}}
  15. - m 𝐞 𝐫 M Pl 2 r 2 = - m 𝐞 𝐫 M Pl 3 + 1 + δ 2 + δ r 2 n δ -m\frac{\mathbf{e_{r}}}{M_{\mathrm{Pl}}^{2}r^{2}}=-m\frac{\mathbf{e_{r}}}{M_{% \mathrm{Pl}_{3+1+\delta}}^{2+\delta}r^{2}n^{\delta}}
  16. 1 M Pl 2 r 2 = 1 M Pl 3 + 1 + δ 2 + δ r 2 n δ \frac{1}{M_{\mathrm{Pl}}^{2}r^{2}}=\frac{1}{M_{\mathrm{Pl}_{3+1+\delta}}^{2+% \delta}r^{2}n^{\delta}}\Rightarrow
  17. M Pl 2 = M Pl 3 + 1 + δ 2 + δ n δ . M_{\mathrm{Pl}}^{2}=M_{\mathrm{Pl}_{3+1+\delta}}^{2+\delta}n^{\delta}.
  18. 48. | M | - 1 / 3 = 125.5 GeV 48.|M|^{-1/3}=125.5\;\mathrm{GeV}
  19. W e y l ( E 8 × E 8 ) Weyl(E_{8}\times E_{8})

Higgs_mechanism.html

  1. ( 0 , v e - i θ / 2 ) . (0,ve^{-i\theta/2}).
  2. 1 2 \frac{1}{2}
  3. 1 2 \frac{1}{2}
  4. Q = T z + Y 2 Q=T_{z}+\frac{Y}{2}
  5. Fermion ( ϕ , A , ψ ) = ψ ¯ γ μ D μ ψ + G ψ ψ ¯ ϕ ψ , \mathcal{L}_{\mathrm{Fermion}}(\phi,A,\psi)=\overline{\psi}\gamma^{\mu}D_{\mu}% \psi+G_{\psi}\overline{\psi}\phi\psi,
  6. | ϕ | |\langle\phi\rangle|
  7. i t ψ = ( - i q A ) 2 2 m ψ . i{\partial\over\partial t}\psi={(\nabla-iqA)^{2}\over 2m}\psi.
  8. ψ \displaystyle\psi
  9. ψ ( x ) = ρ ( x ) e i θ ( x ) , \psi(x)=\rho(x)\,e^{i\theta(x)},
  10. H = 1 2 m | ( q A + ) ψ | 2 , H={1\over 2m}|{(qA+\nabla)\psi|^{2}},
  11. H ρ 2 2 m ( q A + θ ) 2 . H\approx{\rho^{2}\over 2m}(qA+\nabla\theta)^{2}.
  12. q 2 ρ 2 2 m A 2 . {q^{2}\rho^{2}\over 2m}A^{2}.
  13. E A ˙ 2 2 + q 2 ρ 2 2 m A 2 . E\approx{{\dot{A}}^{2}\over 2}+{q^{2}\rho^{2}\over 2m}A^{2}.
  14. 1 m q 2 ρ 2 . \sqrt{\frac{1}{m}q^{2}\rho^{2}}.
  15. S ( ϕ ) = 1 2 | ϕ | 2 - λ ( | ϕ | 2 - Φ 2 ) 2 , S(\phi)=\int{1\over 2}|\partial\phi|^{2}-\lambda\left(|\phi|^{2}-\Phi^{2}% \right)^{2},
  16. H ( ϕ ) = 1 2 | ϕ ˙ | 2 + | ϕ | 2 + V ( | ϕ | ) . H(\phi)={1\over 2}|\dot{\phi}|^{2}+|\nabla\phi|^{2}+V(|\phi|).
  17. ϕ ( x ) = Φ e i θ ( x ) \phi(x)=\Phi e^{i\theta(x)}
  18. S ( ϕ , A ) = - 1 4 F μ ν F μ ν + | ( - i q A ) ϕ | 2 - λ ( | ϕ | 2 - Φ 2 ) 2 . S(\phi,A)=\int-{1\over 4}F^{\mu\nu}F_{\mu\nu}+|(\partial-iqA)\phi|^{2}-\lambda% (|\phi|^{2}-\Phi^{2})^{2}.
  19. E = 1 2 | ( Φ e i q A x ) | 2 = 1 2 q 2 Φ 2 A 2 . E={1\over 2}\left|\partial\left(\Phi e^{iqAx}\right)\right|^{2}={1\over 2}q^{2% }\Phi^{2}A^{2}.
  20. 1 2 \frac{1}{2}
  21. S ( ϕ , 𝐀 ) = 1 4 g 2 tr ( F μ ν F μ ν ) + | D ϕ | 2 + V ( | ϕ | ) S(\phi,\mathbf{A})=\int{1\over 4g^{2}}\mathop{\textrm{tr}}(F^{\mu\nu}F_{\mu\nu% })+|D\phi|^{2}+V(|\phi|)
  22. F μ ν F^{\mu\nu}
  23. F μ ν F_{\mu\nu}
  24. D ϕ = ϕ - i A k t k ϕ D\phi=\partial\phi-iA^{k}t_{k}\phi
  25. c = = 1 c=\hbar=1
  26. θ θ + e α \theta\rightarrow\theta+e\alpha\,
  27. A A + α . A\rightarrow A+\alpha.\,
  28. D θ = θ - e A . D\theta=\partial\theta-eA.\,
  29. ϕ = H e 1 H i θ \phi\;=\;He^{\frac{1}{H}i\theta}
  30. S = 1 4 F 2 + 1 2 ( D θ ) 2 = 1 4 F 2 + 1 2 ( θ - H e A ) 2 = 1 4 F 2 + 1 2 ( θ - m A ) 2 S=\int{1\over 4}F^{2}+{1\over 2}(D\theta)^{2}=\int{1\over 4}F^{2}+{1\over 2}(% \partial\theta-HeA)^{2}=\int{1\over 4}F^{2}+{1\over 2}(\partial\theta-mA)^{2}
  31. S = 1 4 F 2 + 1 2 m 2 A 2 . S=\int{1\over 4}F^{2}+{1\over 2}m^{2}A^{2}.\,

High_energy_nuclear_physics.html

  1. 10 6 10^{6}
  2. 10 8 10^{8}
  3. T = 300 MeV/k = 3.3 × 10 12 K T=300\mbox{MeV/k}~{}=3.3\times 10^{12}\mbox{K}~{}
  4. ϵ = 10 GeV/fm = 3 1.8 × 10 16 g cm - 3 \epsilon=10\mbox{GeV/fm}~{}^{3}=1.8\times 10^{16}\mbox{g cm}~{}^{-3}
  5. P 1 3 ϵ = 0.52 × 10 31 bar . P\simeq\frac{1}{3}\epsilon=0.52\times 10^{31}\,\mbox{bar}~{}.

Highly_totient_number.html

  1. x = i p i e i x=\prod_{i}p_{i}^{e_{i}}
  2. ϕ ( x ) = i ( p i - 1 ) p i e i - 1 . \phi(x)=\prod_{i}(p_{i}-1)p_{i}^{e_{i}-1}.

Higman–Sims_graph.html

  1. 2 , 6 , 6 2,\sqrt{6},\sqrt{6}
  2. 6 \sqrt{6}

Hilbert's_Theorem_90.html

  1. ( i ) / \mathbb{Q}(i)/\mathbb{Q}
  2. s : c - d i c + d i . s:\,\,c-di\mapsto c+di\ .
  3. x = a + b i x=a+bi
  4. x x s = a 2 + b 2 xx^{s}=a^{2}+b^{2}
  5. a 2 + b 2 = 1 a^{2}+b^{2}=1
  6. y = c + d i c - d i = c 2 - d 2 c 2 + d 2 + 2 c d c 2 + d 2 i y=\frac{c+di}{c-di}=\frac{c^{2}-d^{2}}{c^{2}+d^{2}}+\frac{2cd}{c^{2}+d^{2}}i
  7. ( x , y ) = ( a / c , b / c ) \,(x,y)=(a/c,b/c)
  8. x 2 + y 2 = 1 x^{2}+y^{2}=1
  9. ( a , b , c ) \,(a,b,c)
  10. a 2 + b 2 = c 2 \,a^{2}+b^{2}=c^{2}
  11. H e ´ t 1 ( X , 𝐆 m ) = H 1 ( X , 𝒪 X × ) = Pic ( X ) H^{1}_{\acute{e}t}(X,\mathbf{G}_{m})=H^{1}(X,\mathcal{O}_{X}^{\times})=\mathrm% {Pic}(X)

Hilbert–Pólya_conjecture.html

  1. 1 2 + i t \tfrac{1}{2}+it
  2. 1 / 2 + i H \scriptstyle 1/2+iH
  3. H \scriptstyle H
  4. m m
  5. V ( x ) \scriptstyle V(x)
  6. V \scriptstyle V
  7. E n = E n 0 + φ n 0 | V | φ n 0 E_{n}=E_{n}^{0}+\langle\varphi^{0}_{n}|V|\varphi^{0}_{n}\rangle
  8. E n 0 \scriptstyle E^{0}_{n}
  9. φ n 0 \scriptstyle\varphi^{0}_{n}
  10. E n \scriptstyle E_{n}
  11. V ( x ) = A - ( g ( k ) + g ( k ) ¯ - E k 0 ) R ( x , k ) d k V(x)=A\int_{-\infty}^{\infty}(g(k)+\overline{g(k)}-E_{k}^{0})\,R(x,k)\,dk
  12. R ( x , k ) \scriptstyle R(x,k)
  13. A \scriptstyle A
  14. g ( k ) = i n = 0 ( 1 2 - ρ n ) δ ( k - n ) g(k)=i\sum_{n=0}^{\infty}\left(\frac{1}{2}-\rho_{n}\right)\delta(k-n)
  15. δ ( k - n ) \scriptstyle\delta(k-n)
  16. ρ n \scriptstyle\rho_{n}
  17. ζ ( ρ n ) = 0 \scriptstyle\zeta(\rho_{n})=0
  18. H = 1 2 ( x p + p x ) = - i ( x d d x + 1 2 ) . H=\tfrac{1}{2}(xp+px)=-i\left(x\frac{\mathrm{d}}{\mathrm{d}x}+\frac{1}{2}% \right).
  19. 1 2 + i 2 π n log n . \frac{1}{2}+i\frac{2\pi n}{\log n}.
  20. exp ( i γ ) \exp(i\gamma)

Hilbert–Schmidt_operator.html

  1. A H S 2 = Tr ( A * A ) := i I A e i 2 \|A\|^{2}_{HS}={\rm Tr}(A^{{}^{*}}A):=\sum_{i\in I}\|Ae_{i}\|^{2}
  2. \|\ \|
  3. { e i : i I } \{e_{i}:i\in I\}
  4. A H S 2 = i , j | A i , j | 2 = A 2 2 \|A\|^{2}_{HS}=\sum_{i,j}|A_{i,j}|^{2}=\|A\|^{2}_{2}
  5. A i , j = e i , A e j A_{i,j}=\langle e_{i},Ae_{j}\rangle
  6. A 2 \|A\|_{2}
  7. A A
  8. H S \|\ \|_{HS}
  9. A , B HS = Tr ( A * B ) = i A e i , B e i . \langle A,B\rangle_{\mathrm{HS}}=\operatorname{Tr}(A^{*}B)=\sum_{i}\langle Ae_% {i},Be_{i}\rangle.
  10. H * H , H^{*}\otimes H,\,

Hill_cipher.html

  1. ( 6 24 1 13 16 10 20 17 15 ) \begin{pmatrix}6&24&1\\ 13&16&10\\ 20&17&15\end{pmatrix}
  2. ( 0 2 19 ) \begin{pmatrix}0\\ 2\\ 19\end{pmatrix}
  3. ( 6 24 1 13 16 10 20 17 15 ) ( 0 2 19 ) = ( 67 222 319 ) ( 15 14 7 ) ( mod 26 ) \begin{pmatrix}6&24&1\\ 13&16&10\\ 20&17&15\end{pmatrix}\begin{pmatrix}0\\ 2\\ 19\end{pmatrix}=\begin{pmatrix}67\\ 222\\ 319\end{pmatrix}\equiv\begin{pmatrix}15\\ 14\\ 7\end{pmatrix}\;\;(\mathop{{\rm mod}}26)
  4. ( 2 0 19 ) \begin{pmatrix}2\\ 0\\ 19\end{pmatrix}
  5. ( 6 24 1 13 16 10 20 17 15 ) ( 2 0 19 ) ( 31 216 325 ) ( 5 8 13 ) ( mod 26 ) \begin{pmatrix}6&24&1\\ 13&16&10\\ 20&17&15\end{pmatrix}\begin{pmatrix}2\\ 0\\ 19\end{pmatrix}\equiv\begin{pmatrix}31\\ 216\\ 325\end{pmatrix}\equiv\begin{pmatrix}5\\ 8\\ 13\end{pmatrix}\;\;(\mathop{{\rm mod}}26)
  6. ( 6 24 1 13 16 10 20 17 15 ) - 1 ( 8 5 10 21 8 21 21 12 8 ) ( mod 26 ) \begin{pmatrix}6&24&1\\ 13&16&10\\ 20&17&15\end{pmatrix}^{-1}\equiv\begin{pmatrix}8&5&10\\ 21&8&21\\ 21&12&8\end{pmatrix}\;\;(\mathop{{\rm mod}}26)
  7. ( 8 5 10 21 8 21 21 12 8 ) ( 15 14 7 ) ( 260 574 539 ) ( 0 2 19 ) ( mod 26 ) \begin{pmatrix}8&5&10\\ 21&8&21\\ 21&12&8\end{pmatrix}\begin{pmatrix}15\\ 14\\ 7\end{pmatrix}\equiv\begin{pmatrix}260\\ 574\\ 539\end{pmatrix}\equiv\begin{pmatrix}0\\ 2\\ 19\end{pmatrix}\;\;(\mathop{{\rm mod}}26)
  8. | 6 24 1 13 16 10 20 17 15 | 6 ( 16 15 - 10 17 ) - 24 ( 13 15 - 10 20 ) + 1 ( 13 17 - 16 20 ) 441 25 ( mod 26 ) \begin{vmatrix}6&24&1\\ 13&16&10\\ 20&17&15\end{vmatrix}\equiv 6(16\cdot 15-10\cdot 17)-24(13\cdot 15-10\cdot 20)% +1(13\cdot 17-16\cdot 20)\equiv 441\equiv 25\;\;(\mathop{{\rm mod}}26)
  9. K = ( 3 3 2 5 ) K=\begin{pmatrix}3&3\\ 2&5\end{pmatrix}
  10. H E L P ( H E ) , ( L P ) ( 7 4 ) , ( 11 15 ) HELP\to\begin{pmatrix}H\\ E\end{pmatrix},\begin{pmatrix}L\\ P\end{pmatrix}\to\begin{pmatrix}7\\ 4\end{pmatrix},\begin{pmatrix}11\\ 15\end{pmatrix}
  11. ( 3 3 2 5 ) ( 7 4 ) ( 7 8 ) ( mod 26 ) , \begin{pmatrix}3&3\\ 2&5\end{pmatrix}\begin{pmatrix}7\\ 4\end{pmatrix}\equiv\begin{pmatrix}7\\ 8\end{pmatrix}\;\;(\mathop{{\rm mod}}26),
  12. ( 3 3 2 5 ) ( 11 15 ) ( 0 19 ) ( mod 26 ) \begin{pmatrix}3&3\\ 2&5\end{pmatrix}\begin{pmatrix}11\\ 15\end{pmatrix}\equiv\begin{pmatrix}0\\ 19\end{pmatrix}\;\;(\mathop{{\rm mod}}26)
  13. ( 7 8 ) , ( 0 19 ) ( H I ) , ( A T ) \begin{pmatrix}7\\ 8\end{pmatrix},\begin{pmatrix}0\\ 19\end{pmatrix}\to\begin{pmatrix}H\\ I\end{pmatrix},\begin{pmatrix}A\\ T\end{pmatrix}
  14. K - 1 K^{-1}
  15. K K - 1 = K - 1 K = I 2 KK^{-1}=K^{-1}K=I_{2}
  16. ( a b c d ) - 1 = ( a d - b c ) - 1 ( d - b - c a ) \begin{pmatrix}a&b\\ c&d\end{pmatrix}^{-1}=(ad-bc)^{-1}\begin{pmatrix}d&-b\\ -c&a\end{pmatrix}
  17. ( a d - b c ) - 1 (ad-bc)^{-1}
  18. K - 1 9 - 1 ( 5 23 24 3 ) 3 ( 5 23 24 3 ) ( 15 17 20 9 ) ( mod 26 ) K^{-1}\equiv 9^{-1}\begin{pmatrix}5&23\\ 24&3\end{pmatrix}\equiv 3\begin{pmatrix}5&23\\ 24&3\end{pmatrix}\equiv\begin{pmatrix}15&17\\ 20&9\end{pmatrix}\;\;(\mathop{{\rm mod}}26)
  19. H I A T ( H I ) , ( A T ) ( 7 8 ) , ( 0 19 ) HIAT\to\begin{pmatrix}H\\ I\end{pmatrix},\begin{pmatrix}A\\ T\end{pmatrix}\to\begin{pmatrix}7\\ 8\end{pmatrix},\begin{pmatrix}0\\ 19\end{pmatrix}
  20. ( 15 17 20 9 ) ( 7 8 ) ( 7 4 ) ( mod 26 ) , \begin{pmatrix}15&17\\ 20&9\end{pmatrix}\begin{pmatrix}7\\ 8\end{pmatrix}\equiv\begin{pmatrix}7\\ 4\end{pmatrix}\;\;(\mathop{{\rm mod}}26),
  21. ( 15 17 20 9 ) ( 0 19 ) ( 11 15 ) ( mod 26 ) \begin{pmatrix}15&17\\ 20&9\end{pmatrix}\begin{pmatrix}0\\ 19\end{pmatrix}\equiv\begin{pmatrix}11\\ 15\end{pmatrix}\;\;(\mathop{{\rm mod}}26)
  22. ( 7 4 ) , ( 11 15 ) ( H E ) , ( L P ) H E L P \begin{pmatrix}7\\ 4\end{pmatrix},\begin{pmatrix}11\\ 15\end{pmatrix}\to\begin{pmatrix}H\\ E\end{pmatrix},\begin{pmatrix}L\\ P\end{pmatrix}\to HELP
  23. n 2 n^{2}
  24. 26 n 2 26^{n^{2}}
  25. log 2 ( 26 n 2 ) \log_{2}(26^{n^{2}})
  26. 4.7 n 2 4.7n^{2}
  27. 2 n 2 ( 1 - 1 / 2 ) ( 1 - 1 / 2 2 ) ( 1 - 1 / 2 n ) . 2^{n^{2}}(1-1/2)(1-1/2^{2})\cdots(1-1/2^{n}).
  28. 13 n 2 ( 1 - 1 / 13 ) ( 1 - 1 / 13 2 ) ( 1 - 1 / 13 n ) . 13^{n^{2}}(1-1/13)(1-1/13^{2})\cdots(1-1/13^{n}).
  29. 26 n 2 ( 1 - 1 / 2 ) ( 1 - 1 / 2 2 ) ( 1 - 1 / 2 n ) ( 1 - 1 / 13 ) ( 1 - 1 / 13 2 ) ( 1 - 1 / 13 n ) . 26^{n^{2}}(1-1/2)(1-1/2^{2})\cdots(1-1/2^{n})(1-1/13)(1-1/13^{2})\cdots(1-1/13% ^{n}).
  30. 4.64 n 2 - 1.7 4.64n^{2}-1.7

Hill_sphere.html

  1. r a ( 1 - e ) m 3 M 3 . r\approx a(1-e)\sqrt[3]{\frac{m}{3M}}.
  2. r a m 3 M 3 . r\approx a\sqrt[3]{\frac{m}{3M}}.
  3. 3 r 3 a 3 m M . 3\frac{r^{3}}{a^{3}}\approx\frac{m}{M}.
  4. r R secondary a R primary ρ secondary 3 ρ primary 3 a R primary , \frac{r}{R_{\mathrm{secondary}}}\approx\frac{a}{R_{\mathrm{primary}}}\sqrt[3]{% \frac{\rho_{\mathrm{secondary}}}{3\rho_{\mathrm{primary}}}}\approx\frac{a}{R_{% \mathrm{primary}}},
  5. ρ second \rho_{\mathrm{second}}
  6. ρ primary \rho_{\mathrm{primary}}
  7. R secondary R_{\mathrm{secondary}}
  8. R primary R_{\mathrm{primary}}
  9. ρ secondary 3 ρ primary 3 \sqrt[3]{\frac{\rho_{\mathrm{secondary}}}{3\rho_{\mathrm{primary}}}}

History_of_calculus.html

  1. x {x}
  2. y {y}
  3. x ˙ \dot{x}
  4. y ˙ \dot{y}
  5. \scriptstyle\int
  6. f ˙ \dot{f}
  7. \int
  8. d y d x \frac{dy}{dx}
  9. 0 1 x n - 1 ( 1 - x ) n - 1 d x \int_{0}^{1}x^{n-1}(1-x)^{n-1}\,dx
  10. 0 e - x x n - 1 d x \int_{0}^{\infty}e^{-x}x^{n-1}\,dx
  11. 0 e - x x n - 1 d x = ( n - 1 ) ! , \int_{0}^{\infty}e^{-x}x^{n-1}dx=(n-1)!,
  12. n n
  13. Γ \Gamma
  14. Γ \Gamma
  15. log Γ \log\Gamma
  16. Γ ( x ) \Gamma(x)
  17. log Γ ( x ) \log\Gamma(x)
  18. f ( x ) , f^{\prime}\left(x\right),

Hodrick–Prescott_filter.html

  1. λ \lambda
  2. y t y_{t}\,
  3. t = 1 , 2 , , T t=1,2,...,T\,
  4. y t y_{t}\,
  5. τ \tau\,
  6. c c\,
  7. y t = τ t + c t + ϵ t y_{t}\ =\tau_{t}\ +c_{t}\ +\epsilon_{t}\,
  8. λ \lambda
  9. min τ ( t = 1 T ( y t - τ t ) 2 + λ t = 2 T - 1 [ ( τ t + 1 - τ t ) - ( τ t - τ t - 1 ) ] 2 ) . \min_{\tau}\left(\sum_{t=1}^{T}{(y_{t}-\tau_{t})^{2}}+\lambda\sum_{t=2}^{T-1}{% [(\tau_{t+1}-\tau_{t})-(\tau_{t}-\tau_{t-1})]^{2}}\right).\,
  10. d t = y t - τ t d_{t}=y_{t}-\tau_{t}
  11. λ \lambda
  12. λ \lambda
  13. λ \lambda
  14. λ \lambda
  15. λ \lambda
  16. λ \lambda
  17. t + i , i > 0 t+i,i>0
  18. t t
  19. λ \lambda

Hoeffding's_inequality.html

  1. p p
  2. 1 p 1−p
  3. n n
  4. p n pn
  5. k k
  6. ( H ( n ) k ) = i = 0 k ( n i ) p i ( 1 - p ) n - i , \mathbb{P}(H(n)\leq k)=\sum_{i=0}^{k}{\left({{n}\atop{i}}\right)}p^{i}(1-p)^{n% -i},
  7. H ( n ) H(n)
  8. n n
  9. k = ( p ε ) n k=(p−ε)n
  10. ε > 0 ε>0
  11. ( H ( n ) ( p - ε ) n ) exp ( - 2 ε 2 n ) . \mathbb{P}(H(n)\leq(p-\varepsilon)n)\leq\exp\left(-2\varepsilon^{2}n\right).
  12. k = ( p + ε ) n k=(p+ε)n
  13. ε > 0 ε>0
  14. ε n εn
  15. ( H ( n ) ( p + ε ) n ) exp ( - 2 ε 2 n ) . \mathbb{P}(H(n)\geq(p+\varepsilon)n)\leq\exp\left(-2\varepsilon^{2}n\right).
  16. ( ( p - ϵ ) n H ( n ) ( p + ε ) n ) 1 - 2 exp ( - 2 ε 2 n ) . \mathbb{P}((p-\epsilon)n\leq H(n)\leq(p+\varepsilon)n)\geq 1-2\exp\left(-2% \varepsilon^{2}n\right).
  17. ( X i [ a i , b i ] ) = 1 , 1 i n . \mathbb{P}(X_{i}\in[a_{i},b_{i}])=1,\qquad 1\leq i\leq n.
  18. X ¯ = 1 n ( X 1 + + X n ) . \overline{X}=\frac{1}{n}(X_{1}+\cdots+X_{n}).
  19. ( X ¯ - E [ X ¯ ] t ) e - 2 n t 2 \displaystyle\mathbb{P}(\overline{X}-\mathrm{E}[\overline{X}]\geq t)\leq e^{-2% nt^{2}}
  20. ( X ¯ - E [ X ¯ ] t ) \displaystyle\mathbb{P}\left(\overline{X}-\mathrm{E}\left[\overline{X}\right]% \geq t\right)
  21. t t
  22. E o v e r l i n e i n e X EoverlineineX
  23. X ¯ \overline{X}
  24. S n = X 1 + + X n S_{n}=X_{1}+\cdots+X_{n}
  25. ( S n - E [ S n ] t ) exp ( - 2 t 2 i = 1 n ( b i - a i ) 2 ) , \mathbb{P}(S_{n}-\mathrm{E}[S_{n}]\geq t)\leq\exp\left(-\frac{2t^{2}}{\sum_{i=% 1}^{n}(b_{i}-a_{i})^{2}}\right),
  26. ( | S n - E [ S n ] | t ) 2 exp ( - 2 t 2 i = 1 n ( b i - a i ) 2 ) . \mathbb{P}(|S_{n}-\mathrm{E}[S_{n}]|\geq t)\leq 2\exp\left(-\frac{2t^{2}}{\sum% _{i=1}^{n}(b_{i}-a_{i})^{2}}\right).
  27. X X
  28. ( X [ a , b ] ) = 1 \textstyle\mathbb{P}\left(X\in\left[a,b\right]\right)=1
  29. E [ e s X ] exp ( 1 8 s 2 ( b - a ) 2 ) . \mathrm{E}\left[e^{sX}\right]\leq\exp\left(\tfrac{1}{8}s^{2}(b-a)^{2}\right).
  30. a a
  31. b b
  32. ( X = 0 ) = 1 \textstyle\mathbb{P}\left(X=0\right)=1
  33. a a
  34. b b
  35. x [ a , b ] : e s x b - x b - a e s a + x - a b - a e s b . \forall x\in[a,b]:\qquad e^{sx}\leq\frac{b-x}{b-a}e^{sa}+\frac{x-a}{b-a}e^{sb}.
  36. E E⋅
  37. E [ e s X ] \displaystyle\mathrm{E}\left[e^{sX}\right]
  38. u = s ( b a ) u=s(b−a)
  39. { φ : 𝐑 𝐑 φ ( u ) = - θ u + log ( 1 - θ + θ e u ) \begin{cases}\varphi:\mathbf{R}\to\mathbf{R}\\ \varphi(u)=-\theta u+\log\left(1-\theta+\theta e^{u}\right)\end{cases}
  40. φ φ
  41. 𝐑 \mathbf{R}
  42. 1 - θ + θ e u \displaystyle 1-\theta+\theta e^{u}
  43. φ φ
  44. E [ e s X ] e φ ( u ) . \mathrm{E}\left[e^{sX}\right]\leq e^{\varphi(u)}.
  45. u u
  46. v v
  47. 0
  48. u u
  49. φ ( u ) = φ ( 0 ) + u φ ( 0 ) + 1 2 u 2 φ ′′ ( v ) . \varphi(u)=\varphi(0)+u\varphi^{\prime}(0)+\tfrac{1}{2}u^{2}\varphi^{\prime% \prime}(v).
  50. φ ( 0 ) \displaystyle\varphi(0)
  51. φ ( u ) 0 + u 0 + 1 2 u 2 1 4 = 1 8 u 2 = 1 8 s 2 ( b - a ) 2 . \varphi(u)\leq 0+u\cdot 0+\tfrac{1}{2}u^{2}\cdot\tfrac{1}{4}=\tfrac{1}{8}u^{2}% =\tfrac{1}{8}s^{2}(b-a)^{2}.
  52. E [ e s X ] exp ( 1 8 s 2 ( b - a ) 2 ) . \mathrm{E}\left[e^{sX}\right]\leq\exp\left(\tfrac{1}{8}s^{2}(b-a)^{2}\right).
  53. n n
  54. ( X i [ a i , b i ] ) = 1 , 1 i n . \mathbb{P}\left(X_{i}\in[a_{i},b_{i}]\right)=1,\qquad 1\leq i\leq n.
  55. S n = X 1 + + X n . S_{n}=X_{1}+\cdots+X_{n}.
  56. s , t 0 s,t≥0
  57. ( S n - E [ S n ] t ) \displaystyle\mathbb{P}\left(S_{n}-\mathrm{E}\left[S_{n}\right]\geq t\right)
  58. s s
  59. { g : 𝐑 + 𝐑 g ( s ) = - s t + s 2 8 i = 1 n ( b i - a i ) 2 \begin{cases}g:\mathbf{R_{+}}\to\mathbf{R}\\ g(s)=-st+\frac{s^{2}}{8}\sum_{i=1}^{n}(b_{i}-a_{i})^{2}\end{cases}
  60. g g
  61. s = 4 t i = 1 n ( b i - a i ) 2 . s=\frac{4t}{\sum_{i=1}^{n}(b_{i}-a_{i})^{2}}.
  62. ( S n - E [ S n ] t ) exp ( - 2 t 2 i = 1 n ( b i - a i ) 2 ) . \mathbb{P}\left(S_{n}-\mathrm{E}\left[S_{n}\right]\geq t\right)\leq\exp\left(-% \frac{2t^{2}}{\sum_{i=1}^{n}(b_{i}-a_{i})^{2}}\right).
  63. ( X ¯ - E [ X ¯ ] t ) e - 2 n t 2 \mathbb{P}(\overline{X}-\mathrm{E}[\overline{X}]\geq t)\leq e^{-2nt^{2}}
  64. t t
  65. ( - X ¯ + E [ X ¯ ] t ) e - 2 n t 2 \mathbb{P}(-\overline{X}+\mathrm{E}[\overline{X}]\geq t)\leq e^{-2nt^{2}}
  66. ( | X ¯ - E [ X ¯ ] | t ) 2 e - 2 n t 2 \mathbb{P}(|\overline{X}-\mathrm{E}[\overline{X}]|\geq t)\leq 2e^{-2nt^{2}}
  67. α \alpha
  68. E [ X ¯ ] \mathrm{E}[\overline{X}]
  69. 2 t 2t
  70. α = ( X ¯ [ E [ X ¯ ] - t , E [ X ¯ ] + t ] ) 2 e - 2 n t 2 \alpha=\mathbb{P}(\overline{X}\notin[\mathrm{E}[\overline{X}]-t,\mathrm{E}[% \overline{X}]+t])\leq 2e^{-2nt^{2}}
  71. n n
  72. n - log ( α / 2 ) 2 t 2 n\geq-\frac{\log(\alpha/2)}{2t^{2}}
  73. - log ( α / 2 ) 2 t 2 \textstyle-\frac{\log(\alpha/2)}{2t^{2}}
  74. ( 1 - α ) \textstyle(1-\alpha)
  75. E [ X ¯ ] ± t \textstyle\mathrm{E}[\overline{X}]\pm t

Holonomy.html

  1. Hol x ( ) = { P γ GL ( E x ) γ is a loop based at x } . \mathrm{Hol}_{x}(\nabla)=\{P_{\gamma}\in\mathrm{GL}(E_{x})\mid\gamma\,\text{ % is a loop based at }x\}.
  2. Hol x 0 ( ) \mathrm{Hol}^{0}_{x}(\nabla)
  3. Hol y ( ) = P γ Hol x ( ) P γ - 1 . \mathrm{Hol}_{y}(\nabla)=P_{\gamma}\mathrm{Hol}_{x}(\nabla)P_{\gamma}^{-1}.
  4. γ ~ : [ 0 , 1 ] P \tilde{\gamma}\colon[0,1]\to P
  5. γ ~ ( 0 ) = p \tilde{\gamma}(0)=p
  6. γ ~ ( 1 ) \tilde{\gamma}(1)
  7. Hol p ( ω ) = { g G p p g } . \mathrm{Hol}_{p}(\omega)=\{g\in G\mid p\sim p\cdot g\}.\,
  8. Hol q ( ω ) = g - 1 Hol p ( ω ) g . \mathrm{Hol}_{q}(\omega)=g^{-1}\mathrm{Hol}_{p}(\omega)g.\,
  9. Hol p g ( ω ) = g - 1 Hol p ( ω ) g , \mathrm{Hol}_{p\cdot g}(\omega)=g^{-1}\mathrm{Hol}_{p}(\omega)g,
  10. Hol p 0 ( ω , U ) Hol p 0 ( ω , V ) . \mathrm{Hol}_{p}^{0}(\omega,U)\subset\mathrm{Hol}_{p}^{0}(\omega,V).
  11. Hol * ( ω ) = k = 1 Hol 0 ( ω , U k ) \mathrm{Hol}^{*}(\omega)=\cap_{k=1}^{\infty}\mathrm{Hol}^{0}(\omega,U_{k})
  12. k U k = π ( p ) \cap_{k}U_{k}=\pi(p)
  13. D d x D d y V - D d y D d x V = R ( σ x , σ y ) V \frac{D}{dx}\frac{D}{dy}V-\frac{D}{dy}\frac{D}{dx}V=R\left(\frac{\partial% \sigma}{\partial x},\frac{\partial\sigma}{\partial y}\right)V
  14. V 0 × V 1 × × V k , V_{0}\times V_{1}\times\dots\times V_{k},
  15. Sp ( 2 , 𝐂 ) Sp ( 2 n , 𝐂 ) \displaystyle\mathrm{Sp}(2,\mathbf{C})\cdot\mathrm{Sp}(2n,\mathbf{C})
  16. Z 𝐂 SL ( m , 𝐂 ) SL ( n , 𝐂 ) \displaystyle Z_{\mathbf{C}}\cdot\mathrm{SL}(m,\mathbf{C})\cdot\mathrm{SL}(n,% \mathbf{C})
  17. Sp ( 2 , 𝐂 ) SO ( n , 𝐂 ) \displaystyle\mathrm{Sp}(2,\mathbf{C})\cdot\mathrm{SO}(n,\mathbf{C})

Homentropic_flow.html

  1. d P = c 2 d ρ dP=c^{2}d\rho

Homes's_law.html

  1. ρ s 0 \rho_{s0}
  2. ρ d c \rho_{dc}
  3. ρ d c α ρ s 0 α / 8 4.4 T c \rho_{dc}^{\alpha}\,\rho_{s0}^{\alpha}/8\simeq 4.4\,T_{c}
  4. ρ s 0 α / 8 4.4 σ d c α T c \rho_{s0}^{\alpha}/8\simeq 4.4\,\sigma_{dc}^{\alpha}\,T_{c}
  5. α \alpha

HOMFLY_polynomial.html

  1. P ( unknot ) = 1 , P(\mathrm{unknot})=1,\,
  2. P ( L + ) + - 1 P ( L - ) + m P ( L 0 ) = 0 , \ell P(L_{+})+\ell^{-1}P(L_{-})+mP(L_{0})=0,\,
  3. L + , L - , L 0 L_{+},L_{-},L_{0}
  4. L 1 L_{1}
  5. L 2 L_{2}
  6. P ( L ) = - ( + - 1 ) m P ( L 1 ) P ( L 2 ) . P(L)=\frac{-(\ell+\ell^{-1})}{m}P(L_{1})P(L_{2}).
  7. α P ( L + ) - α - 1 P ( L - ) = z P ( L 0 ) , \alpha P(L_{+})-\alpha^{-1}P(L_{-})=zP(L_{0}),\,
  8. x P ( L + ) + y P ( L - ) + z P ( L 0 ) = 0 , xP(L_{+})+yP(L_{-})+zP(L_{0})=0,\,
  9. P ( L 1 # L 2 ) = P ( L 1 ) P ( L 2 ) , P(L_{1}\#L_{2})=P(L_{1})P(L_{2}),\,
  10. P K ( , m ) = P Mirror Image ( K ) ( - 1 , m ) , P_{K}(\ell,m)=P_{\,\text{Mirror Image}(K)}(\ell^{-1},m),\,
  11. Δ ( t ) \Delta(t)\,
  12. V ( t ) = P ( α = t - 1 , z = t 1 / 2 - t - 1 / 2 ) , V(t)=P(\alpha=t^{-1},z=t^{1/2}-t^{-1/2}),\,
  13. Δ ( t ) = P ( α = 1 , z = t 1 / 2 - t - 1 / 2 ) , \Delta(t)=P(\alpha=1,z=t^{1/2}-t^{-1/2}),\,

Homotopy_groups_of_spheres.html

  1. S 1 S 3 S 2 . S^{1}\hookrightarrow S^{3}\rightarrow S^{2}.\,\!
  2. π i ( F ) π i ( E ) π i ( B ) π i - 1 ( F ) . \cdots\to\pi_{i}(F)\to\pi_{i}(E)\to\pi_{i}(B)\to\pi_{i-1}(F)\to\cdots.\,\!
  3. 0 π i ( S 3 ) π i ( S 2 ) π i - 1 ( S 1 ) 0. 0\rightarrow\pi_{i}(S^{3})\rightarrow\pi_{i}(S^{2})\rightarrow\pi_{i-1}(S^{1})% \rightarrow 0.\,\!
  4. π i ( S 2 ) = π i ( S 3 ) π i - 1 ( S 1 ) . \pi_{i}(S^{2})=\pi_{i}(S^{3})\oplus\pi_{i-1}(S^{1}).\,\!
  5. S 3 S 7 S 4 S^{3}\hookrightarrow S^{7}\rightarrow S^{4}\,\!
  6. S 7 S 15 S 8 S^{7}\hookrightarrow S^{15}\rightarrow S^{8}\,\!
  7. π i ( S 4 ) = π i ( S 7 ) π i - 1 ( S 3 ) , \pi_{i}(S^{4})=\pi_{i}(S^{7})\oplus\pi_{i-1}(S^{3}),\,\!
  8. π i ( S 8 ) = π i ( S 15 ) π i - 1 ( S 7 ) . \pi_{i}(S^{8})=\pi_{i}(S^{15})\oplus\pi_{i-1}(S^{7}).\,\!
  9. π 30 ( S 16 ) π 30 ( S 31 ) π 29 ( S 15 ) . \pi_{30}(S^{16})\neq\pi_{30}(S^{31})\oplus\pi_{29}(S^{15}).\,\!
  10. S 15 S 31 S 16 , S^{15}\hookrightarrow S^{31}\rightarrow S^{16},\,\!
  11. M k = f - 1 ( 1 , 0 , , 0 ) S n + k M^{k}=f^{-1}(1,0,\dots,0)\subset S^{n+k}
  12. f : S n S n f:S^{n}\to S^{n}
  13. S 3 S 2 S^{3}\rightarrow S^{2}
  14. S 1 S 3 S^{1}\subset S^{3}
  15. π 2 m + k ( S 2 m ) ( p ) = π 2 m + k - 1 ( S 2 m - 1 ) ( p ) π 2 m + k ( S 4 m - 1 ) ( p ) \pi_{2m+k}(S^{2m})(p)=\pi_{2m+k-1}(S^{2m-1})(p)\oplus\pi_{2m+k}(S^{4m-1})(p)
  16. π S = k 0 π k S \pi_{\ast}^{S}=\bigoplus_{k\geq 0}\pi_{k}^{S}
  17. Θ n / b P n + 1 π n S / J , \Theta_{n}/bP_{n+1}\to\pi_{n}^{S}/J,\,\!

Hopfield_network.html

  1. w i j w_{ij}
  2. G = V , f G=\langle V,f\rangle
  3. V V
  4. f : V 2 R f:V^{2}\rightarrow R
  5. w i i = 0 , i w_{ii}=0,\forall i
  6. w i j = w j i , i , j w_{ij}=w_{ji},\forall i,j
  7. s i { + 1 if j w i j s j θ i , - 1 otherwise. s_{i}\leftarrow\left\{\begin{array}[]{ll}+1&\mbox{if }\sum_{j}{w_{ij}s_{j}}% \geq\theta_{i},\\ -1&\mbox{otherwise.}\end{array}\right.
  8. w i j w_{ij}
  9. s j s_{j}
  10. θ i \theta_{i}
  11. w i j w_{ij}
  12. w i j > 0 w_{ij}>0
  13. s j = 1 s_{j}=1
  14. s i s_{i}
  15. s j = 1 s_{j}=1
  16. s j = - 1 s_{j}=-1
  17. s i s_{i}
  18. s j = - 1 s_{j}=-1
  19. E = - 1 2 i , j w i j s i s j + i θ i s i E=-\frac{1}{2}\sum_{i,j}{w_{ij}{s_{i}}{s_{j}}}+\sum_{i}{\theta_{i}}{s_{i}}
  20. n n
  21. w i j = 1 n μ = 1 n ϵ i μ ϵ j μ w_{ij}=\frac{1}{n}\sum_{\mu=1}^{n}\epsilon_{i}^{\mu}\epsilon_{j}^{\mu}
  22. ϵ i μ \epsilon_{i}^{\mu}
  23. μ \mu
  24. μ \mu
  25. ϵ i μ ϵ j μ \epsilon_{i}^{\mu}\epsilon_{j}^{\mu}
  26. w i j w_{ij}
  27. w i j ν = w i j ν - 1 + 1 n ϵ i ν ϵ j ν - 1 n ϵ i ν h j i ν - 1 n ϵ j ν h i j ν w_{ij}^{\nu}=w_{ij}^{\nu-1}+\frac{1}{n}\epsilon_{i}^{\nu}\epsilon_{j}^{\nu}-% \frac{1}{n}\epsilon_{i}^{\nu}h_{ji}^{\nu}-\frac{1}{n}\epsilon_{j}^{\nu}h_{ij}^% {\nu}
  28. h i j ν = k = 1 , k i , j n w i k ν - 1 ϵ k ν h_{ij}^{\nu}=\sum_{k=1,k\neq i,j}^{n}w_{ik}^{\nu-1}\epsilon_{k}^{\nu}
  29. μ 1 , μ 2 , μ 3 \mu_{1},\mu_{2},\mu_{3}
  30. ϵ i m i x = ± s g n ( ± ϵ i μ 1 ± ϵ i μ 2 ± ϵ i μ 3 ) \epsilon_{i}^{mix}=\pm sgn(\pm\epsilon_{i}^{\mu_{1}}\pm\epsilon_{i}^{\mu_{2}}% \pm\epsilon_{i}^{\mu_{3}})

Hotelling's_T-squared_distribution.html

  1. T p , m 2 T^{2}_{p,m}
  2. X T p , m 2 X\sim T^{2}_{p,m}
  3. m - p + 1 p m X F p , m - p + 1 \frac{m-p+1}{pm}X\sim F_{p,m-p+1}
  4. F p , m - p + 1 F_{p,m-p+1}
  5. 𝒩 p ( s y m b o l μ , 𝚺 ) \mathcal{N}_{p}(symbol{\mu},{\mathbf{\Sigma}})
  6. s y m b o l μ symbol{\mu}
  7. 𝚺 {\mathbf{\Sigma}}
  8. 𝐱 1 , , 𝐱 n 𝒩 p ( s y m b o l μ , 𝚺 ) {\mathbf{x}}_{1},\dots,{\mathbf{x}}_{n}\sim\mathcal{N}_{p}(symbol{\mu},{% \mathbf{\Sigma}})
  9. p × 1 p\times 1
  10. 𝐱 ¯ = 𝐱 1 + + 𝐱 n n \overline{\mathbf{x}}=\frac{\mathbf{x}_{1}+\cdots+\mathbf{x}_{n}}{n}
  11. n ( 𝐱 ¯ - s y m b o l μ ) 𝚺 - 1 ( 𝐱 ¯ - s y m b o l μ ) χ p 2 , n(\overline{\mathbf{x}}-symbol{\mu})^{\prime}{\mathbf{\Sigma}}^{-1}(\overline{% \mathbf{x}}-symbol{\mathbf{\mu}})\sim\chi^{2}_{p},
  12. χ p 2 \chi^{2}_{p}
  13. 𝐱 ¯ 𝒩 p ( s y m b o l μ , 𝚺 / n ) \overline{\mathbf{x}}\sim\mathcal{N}_{p}(symbol{\mu},{\mathbf{\Sigma}}/n)
  14. 𝐲 = n ( 𝐱 ¯ - s y m b o l μ ) 𝚺 - 1 ( 𝐱 ¯ - s y m b o l μ ) \mathbf{y}=n(\overline{\mathbf{x}}-symbol{\mu})^{\prime}{\mathbf{\Sigma}}^{-1}% (\overline{\mathbf{x}}-symbol{\mathbf{\mu}})
  15. ϕ 𝐲 ( θ ) = E e i θ 𝐲 , \phi_{\mathbf{y}}(\theta)=\operatorname{E}e^{i\theta\mathbf{y}},
  16. = E e i θ n ( 𝐱 ¯ - s y m b o l μ ) 𝚺 - 1 ( 𝐱 ¯ - s y m b o l μ ) =\operatorname{E}e^{i\theta n(\overline{\mathbf{x}}-symbol{\mu})^{\prime}{% \mathbf{\Sigma}}^{-1}(\overline{\mathbf{x}}-symbol{\mathbf{\mu}})}
  17. = e i θ n ( 𝐱 ¯ - s y m b o l μ ) 𝚺 - 1 ( 𝐱 ¯ - s y m b o l μ ) ( 2 π ) - p 2 | s y m b o l Σ / n | - 1 2 e - 1 2 n ( 𝐱 ¯ - s y m b o l μ ) s y m b o l Σ - 1 ( 𝐱 ¯ - s y m b o l μ ) d x 1 d x p =\int e^{i\theta n(\overline{\mathbf{x}}-symbol{\mu})^{\prime}{\mathbf{\Sigma}% }^{-1}(\overline{\mathbf{x}}-symbol{\mathbf{\mu}})}(2\pi)^{-\frac{p}{2}}|% symbol\Sigma/n|^{-\frac{1}{2}}\,e^{-\frac{1}{2}n(\overline{\mathbf{x}}-symbol% \mu)^{\prime}symbol\Sigma^{-1}(\overline{\mathbf{x}}-symbol\mu)}\,dx_{1}...dx_% {p}
  18. = ( 2 π ) - p 2 | s y m b o l Σ / n | - 1 2 e - 1 2 n ( 𝐱 ¯ - s y m b o l μ ) ( s y m b o l Σ - 1 - 2 i θ s y m b o l Σ - 1 ) ( 𝐱 ¯ - s y m b o l μ ) d x 1 d x p , =\int(2\pi)^{-\frac{p}{2}}|symbol\Sigma/n|^{-\frac{1}{2}}\,e^{-\frac{1}{2}n(% \overline{\mathbf{x}}-symbol\mu)^{\prime}(symbol\Sigma^{-1}-2i\theta symbol% \Sigma^{-1})(\overline{\mathbf{x}}-symbol\mu)}\,dx_{1}...dx_{p},
  19. = | ( s y m b o l Σ - 1 - 2 i θ s y m b o l Σ - 1 ) - 1 / n | 1 2 | s y m b o l Σ / n | - 1 2 ( 2 π ) - p 2 | ( s y m b o l Σ - 1 - 2 i θ s y m b o l Σ - 1 ) - 1 / n | - 1 2 e - 1 2 n ( 𝐱 ¯ - s y m b o l μ ) ( s y m b o l Σ - 1 - 2 i θ s y m b o l Σ - 1 ) ( 𝐱 ¯ - s y m b o l μ ) d x 1 d x p , =|(symbol\Sigma^{-1}-2i\theta symbol\Sigma^{-1})^{-1}/n|^{\frac{1}{2}}|symbol% \Sigma/n|^{-\frac{1}{2}}\int(2\pi)^{-\frac{p}{2}}|(symbol\Sigma^{-1}-2i\theta symbol% \Sigma^{-1})^{-1}/n|^{-\frac{1}{2}}\,e^{-\frac{1}{2}n(\overline{\mathbf{x}}-% symbol\mu)^{\prime}(symbol\Sigma^{-1}-2i\theta symbol\Sigma^{-1})(\overline{% \mathbf{x}}-symbol\mu)}\,dx_{1}...dx_{p},
  20. = | ( 𝐈 p - 2 i θ 𝐈 p ) | - 1 2 , =|(\mathbf{I}_{p}-2i\theta\mathbf{I}_{p})|^{-\frac{1}{2}},
  21. = ( 1 - 2 i θ ) - p 2 . =(1-2i\theta)^{-\frac{p}{2}}.~{}~{}\blacksquare
  22. 𝚺 {\mathbf{\Sigma}}
  23. s y m b o l μ symbol{\mu}
  24. 𝐖 = 1 n - 1 i = 1 n ( 𝐱 i - 𝐱 ¯ ) ( 𝐱 i - 𝐱 ¯ ) {\mathbf{W}}=\frac{1}{n-1}\sum_{i=1}^{n}(\mathbf{x}_{i}-\overline{\mathbf{x}})% (\mathbf{x}_{i}-\overline{\mathbf{x}})^{\prime}
  25. 𝐖 \mathbf{W}
  26. ( n - 1 ) 𝐖 (n-1)\mathbf{W}
  27. t 2 = n ( 𝐱 ¯ - s y m b o l μ ) 𝐖 - 1 ( 𝐱 ¯ - s y m b o l μ ) t^{2}=n(\overline{\mathbf{x}}-symbol{\mu})^{\prime}{\mathbf{W}}^{-1}(\overline% {\mathbf{x}}-symbol{\mathbf{\mu}})
  28. t 2 T p , n - 1 2 t^{2}\sim T^{2}_{p,n-1}
  29. n - p p ( n - 1 ) t 2 F p , n - p , \frac{n-p}{p(n-1)}t^{2}\sim F_{p,n-p},
  30. F p , n - p F_{p,n-p}
  31. 𝐱 1 , , 𝐱 n x N p ( s y m b o l μ , 𝐕 ) {\mathbf{x}}_{1},\dots,{\mathbf{x}}_{n_{x}}\sim N_{p}(symbol{\mu},{\mathbf{V}})
  32. 𝐲 1 , , 𝐲 n y N p ( s y m b o l μ , 𝐕 ) {\mathbf{y}}_{1},\dots,{\mathbf{y}}_{n_{y}}\sim N_{p}(symbol{\mu},{\mathbf{V}})
  33. 𝐱 ¯ = 1 n x i = 1 n x 𝐱 i 𝐲 ¯ = 1 n y i = 1 n y 𝐲 i \overline{\mathbf{x}}=\frac{1}{n_{x}}\sum_{i=1}^{n_{x}}\mathbf{x}_{i}\qquad% \overline{\mathbf{y}}=\frac{1}{n_{y}}\sum_{i=1}^{n_{y}}\mathbf{y}_{i}
  34. 𝐖 = i = 1 n x ( 𝐱 i - 𝐱 ¯ ) ( 𝐱 i - 𝐱 ¯ ) + i = 1 n y ( 𝐲 i - 𝐲 ¯ ) ( 𝐲 i - 𝐲 ¯ ) n x + n y - 2 {\mathbf{W}}=\frac{\sum_{i=1}^{n_{x}}(\mathbf{x}_{i}-\overline{\mathbf{x}})(% \mathbf{x}_{i}-\overline{\mathbf{x}})^{\prime}+\sum_{i=1}^{n_{y}}(\mathbf{y}_{% i}-\overline{\mathbf{y}})(\mathbf{y}_{i}-\overline{\mathbf{y}})^{\prime}}{n_{x% }+n_{y}-2}
  35. t 2 = n x n y n x + n y ( 𝐱 ¯ - 𝐲 ¯ ) 𝐖 - 1 ( 𝐱 ¯ - 𝐲 ¯ ) T 2 ( p , n x + n y - 2 ) t^{2}=\frac{n_{x}n_{y}}{n_{x}+n_{y}}(\overline{\mathbf{x}}-\overline{\mathbf{y% }})^{\prime}{\mathbf{W}}^{-1}(\overline{\mathbf{x}}-\overline{\mathbf{y}})\sim T% ^{2}(p,n_{x}+n_{y}-2)
  36. n x + n y - p - 1 ( n x + n y - 2 ) p t 2 F ( p , n x + n y - 1 - p ) . \frac{n_{x}+n_{y}-p-1}{(n_{x}+n_{y}-2)p}t^{2}\sim F(p,n_{x}+n_{y}-1-p).
  37. n x + n y - p - 1 ( n x + n y - 2 ) p t 2 F ( p , n x + n y - 1 - p ; δ ) , \frac{n_{x}+n_{y}-p-1}{(n_{x}+n_{y}-2)p}t^{2}\sim F(p,n_{x}+n_{y}-1-p;\delta),
  38. δ = n x n y n x + n y s y m b o l ν 𝐕 - 1 s y m b o l ν , \delta=\frac{n_{x}n_{y}}{n_{x}+n_{y}}symbol{\nu}^{\prime}\mathbf{V}^{-1}symbol% {\nu},
  39. s y m b o l ν symbol{\nu}
  40. r r
  41. t 2 t^{2}
  42. d 1 = x ¯ .1 - y ¯ .1 , d 2 = x ¯ .2 - y ¯ .2 d_{1}=\overline{x}_{.1}-\overline{y}_{.1},\qquad d_{2}=\overline{x}_{.2}-% \overline{y}_{.2}
  43. S D 1 = W 11 S D 2 = W 22 SD_{1}=\sqrt{W_{11}}\qquad SD_{2}=\sqrt{W_{22}}
  44. t 2 = n x n y ( n x + n y ) ( 1 - r 2 ) [ ( d 1 S D 1 ) 2 + ( d 2 S D 2 ) 2 - 2 r ( d 1 S D 1 ) ( d 2 S D 2 ) ] t^{2}=\frac{n_{x}n_{y}}{(n_{x}+n_{y})(1-r^{2})}\left[\left(\frac{d_{1}}{SD_{1}% }\right)^{2}+\left(\frac{d_{2}}{SD_{2}}\right)^{2}-2r\left(\frac{d_{1}}{SD_{1}% }\right)\left(\frac{d_{2}}{SD_{2}}\right)\right]
  45. ( 𝐱 ¯ - 𝐲 ¯ ) (\overline{\mathbf{x}}-\overline{\mathbf{y}})
  46. t 2 t^{2}
  47. r r
  48. t 2 t^{2}
  49. r r

Hounsfield_scale.html

  1. μ \mu
  2. H U = 1000 × μ - μ w a t e r μ w a t e r - μ a i r HU=1000\times\frac{\mu-\mu_{water}}{\mu_{water}-\mu_{air}}
  3. μ w a t e r \mu_{water}
  4. μ a i r \mu_{air}

How_Long_Is_the_Coast_of_Britain?_Statistical_Self-Similarity_and_Fractional_Dimension.html

  1. L ( G ) = M G 1 - D L(G)=MG^{1-D}
  2. G 1 - D G^{1-D}

Howard_T._Odum.html

  1. J J
  2. X X
  3. C C
  4. J = C X J=CX

Howland_will_forgery_trial.html

  1. 1 2.666 × 10 21 \textstyle\frac{1}{2.666\times 10^{21}}

HSAB_theory.html

  1. η = 1 2 ( 2 E N 2 ) Z . \eta=\frac{1}{2}\left(\frac{\partial^{2}E}{\partial N^{2}}\right)_{Z}.
  2. η E ( N + 1 ) - 2 E ( N ) + E ( N - 1 ) 2 , = ( E ( N - 1 ) - E ( N ) ) - ( E ( N ) - E ( N + 1 ) ) 2 , = 1 2 ( I - A ) , \begin{aligned}\displaystyle\eta&\displaystyle\approx\frac{E(N+1)-2E(N)+E(N-1)% }{2},\\ &\displaystyle=\frac{(E(N-1)-E(N))-(E(N)-E(N+1))}{2},\\ &\displaystyle=\frac{1}{2}(I-A),\end{aligned}
  3. μ = ( E N ) Z , \mu=\left(\frac{\partial E}{\partial N}\right)_{Z},
  4. μ E ( N + 1 ) - E ( N - 1 ) 2 , = - ( E ( N - 1 ) - E ( N ) ) - ( E ( N ) - E ( N + 1 ) ) 2 , = - 1 2 ( I + A ) , \begin{aligned}\displaystyle\mu&\displaystyle\approx\frac{E(N+1)-E(N-1)}{2},\\ &\displaystyle=\frac{-(E(N-1)-E(N))-(E(N)-E(N+1))}{2},\\ &\displaystyle=-\frac{1}{2}(I+A),\end{aligned}
  5. 2 η = ( μ N ) Z - ( χ N ) Z , 2\eta=\left(\frac{\partial\mu}{\partial N}\right)_{Z}\approx-\left(\frac{% \partial\chi}{\partial N}\right)_{Z},

Human_torpedo.html

  1. \infty

Hurwitz's_automorphisms_theorem.html

  1. a 2 = b 3 = ( a b ) 7 = 1 , a^{2}=b^{3}=(ab)^{7}=1,\,

Hurwitz_quaternion.html

  1. H = { a + b i + c j + d k a , b , c , d or a , b , c , d + 1 2 } . H=\left\{a+bi+cj+dk\in\mathbb{H}\mid a,b,c,d\in\mathbb{Z}\;\mbox{ or }~{}\,a,b% ,c,d\in\mathbb{Z}+\tfrac{1}{2}\right\}.
  2. L = { a + b i + c j + d k a , b , c , d } L=\left\{a+bi+cj+dk\in\mathbb{H}\mid a,b,c,d\in\mathbb{Z}\right\}
  3. a 2 + b 2 + c 2 + d 2 a^{2}+b^{2}+c^{2}+d^{2}
  4. 2 E 2 ( 2 τ ) - E 2 ( τ ) = n c ( n ) q n = 1 + 24 q + 24 q 2 + 96 q 3 + 24 q 4 + 144 q 5 + 2E_{2}(2\tau)-E_{2}(\tau)=\sum_{n}c(n)q^{n}=1+24q+24q^{2}+96q^{3}+24q^{4}+144q% ^{5}+\cdots
  5. q = e 2 π i τ q=e^{2\pi i\tau}
  6. E 2 ( τ ) = 1 - 24 n σ 1 ( n ) q n E_{2}(\tau)=1-24\sum_{n}\sigma_{1}(n)q^{n}

Hydrodynamical_helicity.html

  1. 𝐮 ( x , t ) \mathbf{u}(x,t)
  2. × 𝐮 \nabla\times\mathbf{u}
  3. 𝐮 = 0 \nabla\cdot\mathbf{u}=0
  4. p = p ( ρ ) p=p(\rho)
  5. p p
  6. ρ \rho
  7. S S
  8. n ( × 𝐮 ) = 0 n\cdot(\nabla\times\mathbf{u})=0
  9. V V
  10. V V
  11. H = V 𝐮 ( × 𝐮 ) d V . H=\int_{V}\mathbf{u}\cdot\left(\nabla\times\mathbf{u}\right)\,dV\;.
  12. V V
  13. H H
  14. H H
  15. κ 1 \kappa_{1}
  16. κ 2 \kappa_{2}
  17. H = \plusmn 2 n κ 1 κ 2 H=\plusmn 2n\kappa_{1}\kappa_{2}
  18. n n
  19. κ \kappa
  20. H = κ 2 ( W r + T w ) H=\kappa^{2}(Wr+Tw)
  21. W r Wr
  22. T w Tw
  23. W r + T w Wr+Tw
  24. H = V h ζ h d 𝐙 = V h × V h d 𝐙 { Z = A l t i t u d e V h = H o r i z o n t a l v e l o c i t y ζ h = H o r i z o n t a l v o r t i c i t y H=\int{\vec{V}_{h}}\cdot\vec{\zeta}_{h}\,d{\mathbf{Z}}=\int{\vec{V}_{h}}\cdot% \nabla\times\vec{V}_{h}\,d{\mathbf{Z}}\qquad\qquad\begin{cases}Z=Altitude\\ \vec{V}_{h}=Horizontal\ velocity\\ \vec{\zeta}_{h}=Horizontal\ vorticity\end{cases}
  25. V h V_{h}
  26. × V h \nabla\times V_{h}
  27. m 2 / s 2 {m^{2}}/{s^{2}}
  28. S R H = ( V h - C ) × V h d 𝐙 { C = C l o u d m o t i o n t o t h e g r o u n d SRH=\int{\left(\vec{V}_{h}-\vec{C}\right)}\cdot\nabla\times\vec{V}_{h}\,d{% \mathbf{Z}}\qquad\qquad\begin{cases}\vec{C}=Cloud\ motion\ to\ the\ ground\end% {cases}

Hyperbolic_angle.html

  1. cos ( i x ) = cosh ( x ) and sin ( i x ) = i sinh ( x ) \cos(ix)=\cosh(x)\quad\,\text{and}\quad\sin(ix)=i\sinh(x)
  2. e x = cosh x + sinh x e^{x}=\cosh x+\sinh x\!
  3. cosh x = n = 0 x 2 n ( 2 n ) ! \textstyle\cosh x=\sum_{n=0}^{\infty}\frac{x^{2n}}{(2n)!}
  4. sinh x = n = 0 x 2 n + 1 ( 2 n + 1 ) ! \textstyle\sinh x=\sum_{n=0}^{\infty}\frac{x^{2n+1}}{(2n+1)!}

Hyperbolic_discounting.html

  1. f H ( D ) = 1 1 + k D f_{H}(D)=\frac{1}{1+kD}\,
  2. f E ( D ) = e - k D f_{E}(D)=e^{-kD}\,
  3. f ( n ) = 2 - n f(n)=2^{-n}\,
  4. g ( n ) = 1 1 + n g(n)=\frac{1}{1+n}\,
  5. f ( 1 ) f ( 0 ) = 1 2 \frac{f(1)}{f(0)}=\frac{1}{2}\,
  6. f ( n + 1 ) f ( n ) = 1 2 \frac{f(n+1)}{f(n)}=\frac{1}{2}\,
  7. g ( 1 ) g ( 0 ) = 1 2 \frac{g(1)}{g(0)}=\frac{1}{2}\,
  8. g ( n + 1 ) g ( n ) = 1 - 1 n + 2 \frac{g(n+1)}{g(n)}=1-\frac{1}{n+2}\,
  9. g ( n + 1 ) g ( n ) \frac{g(n+1)}{g(n)}\,
  10. f Q H ( 0 ) = 1 , f_{QH}(0)=1,\,
  11. f Q H ( D ) = β × δ D , f_{QH}(D)=\beta\times\delta^{D},\,
  12. P ( R t | λ ) = exp ( - λ t ) P(R_{t}|\lambda)=\exp(-\lambda t)\,
  13. p ( λ ) = exp ( - λ / k ) / k p(\lambda)=\exp(-\lambda/k)/k\,
  14. P ( R t ) = 0 P ( R t | λ ) p ( λ ) d λ = 1 1 + k t P(R_{t})=\int_{0}^{\infty}P(R_{t}|\lambda)p(\lambda)d\lambda=\frac{1}{1+kt}\,
  15. V = P ln ( 1 + k D ) k V=P\frac{\ln(1+kD)}{k}\,

Hyperbolic_orthogonality.html

  1. a c , b d a\rVert c,\ b\rVert d
  2. x , y , z , t and x 1 , y 1 , z 1 , t 1 x,y,z,t\quad\text{and}\quad x_{1},y_{1},z_{1},t_{1}
  3. c 2 t t 1 - x x 1 - y y 1 - z z 1 = 0. c^{2}t\ t_{1}-x\ x_{1}-y\ y_{1}-z\ z_{1}=0.
  4. t x = x 1 t 1 \frac{t}{x}=\frac{x_{1}}{t_{1}}
  5. g g = - b 2 a 2 gg^{\prime}=-\frac{b^{2}}{a^{2}}
  6. g g = b 2 a 2 gg^{\prime}=\frac{b^{2}}{a^{2}}

Hyperbolic_quaternion.html

  1. q = a + b i + c j + d k , a , b , c , d q=a+bi+cj+dk,\quad a,b,c,d\in\mathbb{R}\!
  2. q = a + b i + c j + d k q=a+bi+cj+dk
  3. a , b , c , a,b,c,
  4. d d
  5. { 1 , i , j , k } \{1,i,j,k\}
  6. i j = k = - j i ij=k=-ji
  7. j k = i = - k j jk=i=-kj
  8. k i = j = - i k ki=j=-ik
  9. i 2 = + 1 = j 2 = k 2 i^{2}=+1=j^{2}=k^{2}
  10. ( i j ) j = k j = - i (ij)j=kj=-i
  11. i ( j j ) = i i(jj)=i
  12. { 1 , i , j , k , - 1 , - i , - j , - k } \{1,i,j,k,-1,-i,-j,-k\}
  13. q * = a - b i - c j - d k q^{*}=a-bi-cj-dk
  14. q q
  15. q ( q * ) = a 2 - b 2 - c 2 - d 2 q(q^{*})=a^{2}-b^{2}-c^{2}-d^{2}
  16. - p 0 q 0 + p 1 q 1 + p 2 q 2 + p 3 q 3 -p_{0}q_{0}+p_{1}q_{1}+p_{2}q_{2}+p_{3}q_{3}
  17. D r = { t + x r : t , x R } D_{r}=\{t+xr:t,x\in R\}
  18. exp ( a r ) = cosh ( a ) + r sinh ( a ) \exp(ar)=\cosh(a)+r\ \sinh(a)
  19. t + x r exp ( a r ) ( t + x r ) = t+xr\quad\mapsto\quad\exp(ar)(t+xr)=
  20. ( cosh ( a ) t + x sinh ( a ) ) + ( sinh ( a ) t + x cosh ( a ) ) r . (\cosh(a)t+x\sinh(a))+(\sinh(a)t+x\cosh(a))r.\!
  21. H 3 = { q M : q ( q * ) = 1 } H^{3}=\{q\in M:q(q^{*})=1\}\!
  22. { 1 , i , j , k } \{1,\,i,\,j,\,k\}
  23. j i = - k ji=-\!k
  24. { 1 , i , j , k , - 1 , - i , - j , - k } \{1,\,i,\,j,\,k,\,-\!1,\,-\!i,\,-\!j,\,-\!k\}
  25. i i
  26. j j
  27. k k
  28. + 1 +1
  29. - 1 -1
  30. { 1 , i , j , k } \{1,\,i,\,j,\,k\}
  31. i 2 = - 1 = j 2 = k 2 i^{2}=-\!1=j^{2}=k^{2}
  32. j = k i = ( - j i ) i = - j ( i i ) j=ki=(-ji)i=-j(ii)
  33. i 2 = - 1 i^{2}=-1
  34. i i
  35. j j
  36. k k
  37. i 2 = - 1 = j 2 = k 2 i^{2}=-1=j^{2}=k^{2}
  38. i j = k - 1 ij=k\sqrt{-1}
  39. j k = i - 1 jk=i\sqrt{-1}
  40. k i = j - 1 ki=j\sqrt{-1}
  41. σ 1 σ 2 = σ 3 - 1 \sigma_{1}\sigma_{2}=\sigma_{3}\sqrt{-1}
  42. σ 2 σ 3 = σ 1 - 1 \sigma_{2}\sigma_{3}=\sigma_{1}\sqrt{-1}
  43. σ 3 σ 1 = σ 2 - 1 \sigma_{3}\sigma_{1}=\sigma_{2}\sqrt{-1}

Hyperbolic_sector.html

  1. x y = 1 , xy=1,\,
  2. 2 cosh u , \sqrt{2}\cosh u,\,
  3. 2 sinh u , \sqrt{2}\sinh u,\,
  4. y = 1 + x 2 y=\sqrt{1+x^{2}}

Hyperbolic_trajectory.html

  1. v v_{\infty}\,\!
  2. v = μ - a v_{\infty}=\sqrt{\mu\over{-a}}\,\!
  3. μ = G m \mu=Gm\,\!
  4. a a\,\!
  5. 2 ϵ = C 3 = v 2 2\epsilon=C_{3}=v_{\infty}^{2}\,\!
  6. C 3 C_{3}
  7. v v\,
  8. v = μ ( 2 r - 1 a ) v=\sqrt{\mu\left({2\over{r}}-{1\over{a}}\right)}
  9. μ \mu\,
  10. r r\,
  11. a a\,\!
  12. v v\,
  13. v e s c {v_{esc}}\,
  14. v v_{\infty}\,\!
  15. v 2 = v e s c 2 + v 2 v^{2}={v_{esc}}^{2}+{v_{\infty}}^{2}
  16. 11.6 2 - 11.2 2 = 3.02 \sqrt{11.6^{2}-11.2^{2}}=3.02
  17. 2 θ 2\theta\,
  18. θ = cos - 1 ( 1 / e ) \theta=\cos^{-1}(1/e)\,
  19. e = 1 / cos θ e=1/\cos\theta\,
  20. e e\,
  21. - a ( e - 1 ) -a(e-1)\,
  22. ϵ \epsilon\,
  23. ϵ = v 2 2 - μ r = μ - 2 a \epsilon={v^{2}\over 2}-{\mu\over{r}}={\mu\over{-2a}}
  24. v v\,
  25. r r\,
  26. a a\,
  27. μ \mu\,

Hyperfinite_type_II_factor.html

  1. 1 - 1 / n 1-1/n

Hyperfocal_distance.html

  1. H = f 2 N c + f H=\frac{f^{2}}{Nc}+f
  2. H H
  3. f f
  4. N N
  5. f / D f/D
  6. D D
  7. c c
  8. H f 2 N c H\approx\frac{f^{2}}{Nc}
  9. H H
  10. H H
  11. x - f c / 2 = f D / 2 x - f = c f D x = f + c f D \begin{array}[]{crcl}&\dfrac{x-f}{c/2}&=&\dfrac{f}{D/2}\\ \therefore&x-f&=&\dfrac{cf}{D}\\ \therefore&x&=&f+\dfrac{cf}{D}\end{array}
  12. H D / 2 = x c / 2 H = D x c = D c ( f + c f D ) = D f c + f = f 2 N c + f \begin{array}[]{crclcl}&\dfrac{H}{D/2}&=&\dfrac{x}{c/2}\\ \therefore&H&=&\dfrac{Dx}{c}&=&\dfrac{D}{c}\Big(f+\dfrac{cf}{D}\Big)\\ &&=&\dfrac{Df}{c}+f&=&\dfrac{f^{2}}{Nc}+f\end{array}
  13. H D / 2 = f c / 2 H = D f c = f 2 N c \begin{array}[]{crclcl}&\dfrac{H}{D/2}&=&\dfrac{f}{c/2}\\ \therefore&H&=&\dfrac{Df}{c}&=&\dfrac{f^{2}}{Nc}\end{array}
  14. f / 8 f/8
  15. H = ( 50 ) 2 ( 8 ) ( 0.03 ) + ( 50 ) = 10467 mm H=\frac{(50)^{2}}{(8)(0.03)}+(50)=10467\mbox{ mm}~{}
  16. p p
  17. D D
  18. d d
  19. f f
  20. p = ( D + d ) f d p=\frac{(D+d)f}{d}
  21. D D
  22. f f
  23. N N
  24. c = d c=d
  25. p = ( f N + c ) f c = f 2 N c + f p=\frac{(\tfrac{f}{N}+c)f}{c}=\frac{f^{2}}{Nc}+f

Hypergeometric_identity.html

  1. i = 0 n ( n i ) = 2 n \sum_{i=0}^{n}{n\choose i}=2^{n}
  2. i = 0 n ( n i ) 2 = ( 2 n n ) \sum_{i=0}^{n}{n\choose i}^{2}={2n\choose n}
  3. k = 0 n k ( n k ) = n 2 n - 1 \sum_{k=0}^{n}k{n\choose k}=n2^{n-1}
  4. i = n N i ( i n ) = ( n + 1 ) ( N + 2 n + 2 ) - ( N + 1 n + 1 ) \sum_{i=n}^{N}i{i\choose n}=(n+1){N+2\choose n+2}-{N+1\choose n+1}
  5. t k + 1 t k \frac{t_{k+1}}{t_{k}}
  6. F ( n , k + 1 ) F ( n , k ) \frac{F(n,k+1)}{F(n,k)}
  7. k t k . \sum_{k}t_{k}.\,
  8. k = 0 n F ( n , k ) . \sum_{k=0}^{n}F(n,k).

Hyperkähler_manifold.html

  1. k k
  2. I 2 = J 2 = K 2 = I J K = - 1. I^{2}=J^{2}=K^{2}=IJK=-1.\,
  3. a I + b J + c K aI+bJ+cK\,
  4. a , b , c a,b,c
  5. a 2 + b 2 + c 2 = 1 a^{2}+b^{2}+c^{2}=1\,
  6. 4 k = k \mathbb{R}^{4k}=\mathbb{H}^{k}
  7. T 4 T^{4}

Ibn_Yunus.html

  1. 2 c o s ( a ) c o s ( b ) = c o s ( a + b ) + c o s ( a - b ) 2cos(a)cos(b)=cos(a+b)+cos(a-b)

Ice_Ih.html

  1. ( 4 2 ) = 6 {\textstyle\left({{4}\atop{2}}\right)}=6
  2. 2 4 = 16 2^{4}=16
  3. 6 N / 2 ( 6 / 16 ) N / 2 = ( 3 / 2 ) N . 6^{N/2}(6/16)^{N/2}=(3/2)^{N}.
  4. S 0 = N k ln ( 3 / 2 ) , S_{0}=Nk\ln(3/2),
  5. k k

Ideal_quotient.html

  1. ( I : J ) = { r R | r J I } (I:J)=\{r\in R|rJ\subset I\}
  2. I J K IJ\subset K
  3. I K : J I\subset K:J
  4. ( I : J ) = Ann R ( ( J + I ) / I ) (I:J)=\mathrm{Ann}_{R}((J+I)/I)
  5. R R
  6. Ann R ( M ) \mathrm{Ann}_{R}(M)
  7. M M
  8. R R
  9. J I I : J = R J\subset I\Rightarrow I:J=R
  10. I : R = I I:R=I
  11. R : I = R R:I=R
  12. I : ( J + K ) = ( I : J ) ( I : K ) I:(J+K)=(I:J)\cap(I:K)
  13. I : ( r ) = 1 r ( I ( r ) ) I:(r)=\frac{1}{r}(I\cap(r))
  14. I : J = ( I : ( g 1 ) ) ( I : ( g 2 ) ) = ( 1 g 1 ( I ( g 1 ) ) ) ( 1 g 2 ( I ( g 2 ) ) ) I:J=(I:(g_{1}))\cap(I:(g_{2}))=\left(\frac{1}{g_{1}}(I\cap(g_{1}))\right)\cap% \left(\frac{1}{g_{2}}(I\cap(g_{2}))\right)
  15. I ( g 1 ) = t I + ( 1 - t ) ( g 1 ) k [ x 1 , , x n ] , I ( g 2 ) = t I + ( 1 - t ) ( g 1 ) k [ x 1 , , x n ] I\cap(g_{1})=tI+(1-t)(g_{1})\cap k[x_{1},\dots,x_{n}],\quad I\cap(g_{2})=tI+(1% -t)(g_{1})\cap k[x_{1},\dots,x_{n}]
  16. I ( g 1 ) I\cap(g_{1})
  17. I ( V ) : I ( W ) = I ( V W ) I(V):I(W)=I(V\setminus W)
  18. I ( ) I(\bullet)
  19. Z ( I : J ) = cl ( Z ( I ) Z ( J ) ) Z(I:J)=\mathrm{cl}(Z(I)\setminus Z(J))
  20. cl ( ) \mathrm{cl}(\bullet)
  21. Z ( ) Z(\bullet)
  22. Z ( I : J ) = cl ( Z ( I ) Z ( J ) ) Z(I:J^{\infty})=\mathrm{cl}(Z(I)\setminus Z(J))
  23. J = J + J 2 + + J n + J^{\infty}=J+J^{2}+\cdots+J^{n}+\cdots

Ideal_solution.html

  1. p i = x i p i * p_{i}=x_{i}p_{i}^{*}
  2. p i p_{i}
  3. x i x_{i}
  4. p i * p_{i}^{*}
  5. μ ( T , p i ) = g ( T , p i ) = g u ( T , p u ) + R T ln p i p u \mu(T,p_{i})=g(T,p_{i})=g^{\mathrm{u}}(T,p^{u})+RT\ln{\frac{p_{i}}{p^{u}}}
  6. p u p^{u}
  7. P 0 P^{0}
  8. p i p_{i}
  9. μ ( T , p i ) = g u ( T , p u ) + R T ln p i * p u + R T ln x i = μ i * + R T ln x i \mu(T,p_{i})=g^{\mathrm{u}}(T,p^{u})+RT\ln{\frac{p_{i}^{*}}{p^{u}}}+RT\ln x_{i% }=\mu_{i}^{*}+RT\ln x_{i}
  10. f i = x i f i * f_{i}=x_{i}f_{i}^{*}
  11. f i f_{i}
  12. i i
  13. f i * f_{i}^{*}
  14. i i
  15. μ ( T , P ) = g ( T , P ) = g u ( T , p u ) + R T ln f i p u \mu(T,P)=g(T,P)=g^{\mathrm{u}}(T,p^{u})+RT\ln{\frac{f_{i}}{p^{u}}}
  16. P P
  17. T T
  18. ( g ( T , P ) P ) T = R T ( ln f P ) T \left(\frac{\partial g(T,P)}{\partial P}\right)_{T}=RT\left(\frac{\partial\ln f% }{\partial P}\right)_{T}
  19. ( g ( T , P ) P ) T = v \left(\frac{\partial g(T,P)}{\partial P}\right)_{T}=v
  20. ( ln f P ) T = v R T \left(\frac{\partial\ln f}{\partial P}\right)_{T}=\frac{v}{RT}
  21. i i
  22. v v
  23. v i ¯ \bar{v_{i}}
  24. ( ln f i P ) T , x i = v i ¯ R T \left(\frac{\partial\ln f_{i}}{\partial P}\right)_{T,x_{i}}=\frac{\bar{v_{i}}}% {RT}
  25. v i * = v i ¯ v_{i}^{*}=\bar{v_{i}}
  26. T T
  27. g ( T , P ) - g gas ( T , p u ) R T = ln f p u \frac{g(T,P)-g^{\mathrm{gas}}(T,p^{u})}{RT}=\ln\frac{f}{p^{u}}
  28. ( g T T ) P = - h T 2 \left(\frac{\partial\frac{g}{T}}{\partial T}\right)_{P}=-\frac{h}{T^{2}}
  29. - h i ¯ - h i gas R = - h i * - h i gas R -\frac{\bar{h_{i}}-h_{i}^{\mathrm{gas}}}{R}=-\frac{h_{i}^{*}-h_{i}^{\mathrm{% gas}}}{R}
  30. h i ¯ = h i * \bar{h_{i}}=h_{i}^{*}
  31. u i ¯ = h i ¯ - p v i ¯ \bar{u_{i}}=\bar{h_{i}}-p\bar{v_{i}}
  32. u i * = h i * - p v i * u_{i}^{*}=h_{i}^{*}-pv_{i}^{*}
  33. u i * = u i ¯ u_{i}^{*}=\bar{u_{i}}
  34. C p i * = C p i ¯ C_{pi}^{*}=\bar{C_{pi}}
  35. g i ¯ = μ i = g i gas + R T ln f i p u = g i gas + R T ln f i * p u + R T ln x i = μ i * + R T ln x i \bar{g_{i}}=\mu_{i}=g_{i}^{\mathrm{gas}}+RT\ln\frac{f_{i}}{p^{u}}=g_{i}^{% \mathrm{gas}}+RT\ln\frac{f_{i}^{*}}{p^{u}}+RT\ln x_{i}=\mu_{i}^{*}+RT\ln x_{i}
  36. Δ g i , mix = R T ln x i \Delta g_{i,\mathrm{mix}}=RT\ln x_{i}
  37. G = i x i g i G=\sum_{i}x_{i}{g_{i}}
  38. Δ G mix = R T i x i ln x i \Delta G_{\mathrm{mix}}=RT\sum_{i}{x_{i}\ln x_{i}}
  39. g i * = h i * - T s i * g_{i}^{*}=h_{i}^{*}-Ts_{i}^{*}
  40. g i ¯ = h i ¯ - T s i ¯ \bar{g_{i}}=\bar{h_{i}}-T\bar{s_{i}}
  41. Δ s i , mix = - R i ln x i \Delta s_{i,\mathrm{mix}}=-R\sum_{i}\ln x_{i}
  42. Δ S mix = - R i x i ln x i \Delta S_{\mathrm{mix}}=-R\sum_{i}x_{i}\ln x_{i}
  43. Δ G m , mix = R T i x i ln x i \Delta G_{\mathrm{m,mix}}=RT\sum_{i}x_{i}\ln x_{i}
  44. Δ G m , mix = R T ( x A ln x A + x B ln x B ) \Delta G_{\mathrm{m,mix}}=RT(x_{A}\ln x_{A}+x_{B}\ln x_{B})
  45. x i x_{i}
  46. i i
  47. x i x_{i}
  48. ln x i \ln x_{i}
  49. Δ G m , mix = i x i Δ μ i , mix \Delta G_{\mathrm{m,mix}}=\sum_{i}x_{i}\Delta\mu_{i,\mathrm{mix}}
  50. Δ μ i , mix = R T ln x i \Delta\mu_{i,\mathrm{mix}}=RT\ln x_{i}
  51. i i
  52. i i
  53. μ i * \mu_{i}^{*}
  54. i i
  55. μ i = μ i * + Δ μ i , mix = μ i * + R T ln x i \mu_{i}=\mu_{i}^{*}+\Delta\mu_{i,\mathrm{mix}}=\mu_{i}^{*}+RT\ln x_{i}
  56. i i
  57. P i = ( P i ) p u r e x i \ P_{i}=(P_{i})_{pure}x_{i}
  58. ( P i ) p u r e (P_{i})_{pure}\,
  59. x i x_{i}\,

Identity_component.html

  1. π 0 ( G , e ) . \pi_{0}(G,e).

Illuminance.html

  1. E v = 10 ( - 14.18 - M v ) / 2.5 E_{\mathrm{v}}=10^{(-14.18-M_{\mathrm{v}})/2.5}
  2. M v = - 14.18 - 2.5 log ( E v ) M_{\mathrm{v}}=-14.18-2.5\log(E_{\mathrm{v}})

Immirzi_parameter.html

  1. S = A / 4 \,S=A/4\!
  2. S = γ 0 A / 4 γ . \,S=\gamma_{0}A/4\gamma.\!
  3. γ \gamma
  4. γ 0 = ln ( 2 ) / 3 π \gamma_{0}=\ln(2)/\sqrt{3}\pi
  5. γ 0 = ln ( 3 ) / 8 π , \gamma_{0}=\ln(3)/\sqrt{8}\pi,
  6. γ 0 \,\gamma_{0}
  7. ln ( 3 ) / 8 π , \ln(3)/\sqrt{8}\pi,

Impedance_bridging.html

  1. Z load Z source Z_{\mathrm{load}}>>Z_{\mathrm{source}}\,
  2. D F = Z load Z source DF=\frac{Z_{\mathrm{load}}}{Z_{\mathrm{source}}}\,
  3. Z source = Z load D F Z_{\mathrm{source}}=\frac{Z_{\mathrm{load}}}{DF}\,

Imperiali_quota.html

  1. total votes total seats + 2 \frac{\mbox{total}~{}\;\mbox{votes}~{}}{\mbox{total}~{}\;\mbox{seats}~{}+2}
  2. 100 2 + 2 = 25 \frac{100}{2+2}=25

Implicant.html

  1. = Align g t ; =&gt;
  2. f ( x , y , z , w ) = x y + y z + w f(x,y,z,w)=xy+yz+w
  3. x y xy
  4. x y z xyz
  5. x y z w xyzw
  6. w w
  7. f f
  8. x y xy
  9. x y z xyz
  10. x y z w xyzw
  11. x x
  12. y y
  13. z z
  14. w w
  15. z z
  16. w w
  17. x y xy
  18. x x
  19. w w
  20. y z yz
  21. x y z xyz
  22. x y xy
  23. y z yz
  24. f f

Importance_sampling.html

  1. X : Ω X:\Omega\to\mathbb{R}
  2. ( Ω , , P ) (\Omega,\mathcal{F},P)
  3. x 1 , , x n x_{1},\ldots,x_{n}
  4. 𝐄 ^ n [ X ; P ] = 1 n i = 1 n x i . \widehat{\mathbf{E}}_{n}[X;P]=\frac{1}{n}\sum_{i=1}^{n}x_{i}.
  5. L 0 L\geq 0
  6. L ( ω ) 0 L(\omega)\neq 0
  7. P ( L ) := L P P^{(L)}:=L\,P
  8. 𝐄 [ X ; P ] = 𝐄 [ X L ; P ( L ) ] . \mathbf{E}[X;P]=\mathbf{E}\left[\frac{X}{L};P^{(L)}\right].
  9. 𝐄 [ X ; P ] \mathbf{E}[X;P]
  10. var [ X L ; P ( L ) ] < var [ X ; P ] \operatorname{var}\left[\frac{X}{L};P^{(L)}\right]<\operatorname{var}[X;P]
  11. L * = X 𝐄 [ X ; P ] 0 L^{*}=\frac{X}{\mathbf{E}[X;P]}\geq 0
  12. a , P ( L * ) ( X [ a ; a + d a ] ) \displaystyle\forall a\in\mathbb{R},\;P^{(L^{*})}(X\in[a;a+da])
  13. a P ( X [ a ; a + d a ] ) a\,P(X\in[a;a+da])
  14. E [ X ; P ] = a = - + a P ( X [ a ; a + d a ] ) E[X;P]=\int_{a=-\infty}^{+\infty}a\,P(X\in[a;a+da])
  15. P P
  16. Ω = \Omega=\mathbb{R}
  17. X : X:\mathbb{R}\to\mathbb{R}
  18. X X
  19. f ( X ) g ( X ) \frac{f(X)}{g(X)}
  20. f f
  21. g g
  22. g g
  23. g * = min g Var g ( X f ( X ) g ( X ) ) g^{*}=\min_{g}\operatorname{Var}_{g}\left(X\frac{f(X)}{g(X)}\right)
  24. g * ( X ) = | X | f ( X ) | x | f ( x ) d x . g^{*}(X)=\frac{|X|f(X)}{\int|x|f(x)dx}.
  25. X 0 X\geq 0
  26. p t p_{t}\,
  27. X t {X\geq t\ }
  28. X X
  29. F F
  30. f ( x ) = F ( x ) f(x)=F^{\prime}(x)\,
  31. K K
  32. X i X_{i}\,
  33. F F
  34. k t k_{t}
  35. t t
  36. k t k_{t}
  37. P ( k t = k ) = ( K k ) p t k ( 1 - p t ) K - k , k = 0 , 1 , , K . P(k_{t}=k)={K\choose k}p_{t}^{k}(1-p_{t})^{K-k},\,\quad\quad k=0,1,\dots,K.
  38. E [ k t / K ] = p t \operatorname{E}[k_{t}/K]=p_{t}
  39. var [ k t / K ] = p t ( 1 - p t ) / K \operatorname{var}[k_{t}/K]=p_{t}(1-p_{t})/K
  40. K K\to\infty
  41. p t p_{t}
  42. p t 1 p_{t}\approx 1
  43. f * f_{*}\,
  44. X t {X\geq t\ }
  45. K K
  46. K K
  47. p t p_{t}\,
  48. f * f_{*}\,
  49. p t = E [ 1 ( X t ) ] = 1 ( x t ) f ( x ) f * ( x ) f * ( x ) d x = E * [ 1 ( X t ) W ( X ) ] \begin{aligned}\displaystyle p_{t}&\displaystyle{}={E}[1(X\geq t)]\\ &\displaystyle{}=\int 1(x\geq t)\frac{f(x)}{f_{*}(x)}f_{*}(x)\,dx\\ &\displaystyle{}={E_{*}}[1(X\geq t)W(X)]\end{aligned}
  50. W ( ) f ( ) f * ( ) W(\cdot)\equiv\frac{f(\cdot)}{f_{*}(\cdot)}
  51. p ^ t = 1 K i = 1 K 1 ( X i t ) W ( X i ) , X i f * \hat{p}_{t}=\frac{1}{K}\,\sum_{i=1}^{K}1(X_{i}\geq t)W(X_{i}),\,\quad\quad X_{% i}\sim f_{*}
  52. p t p_{t}\,
  53. f * f_{*}\,
  54. t t\,
  55. W W\,
  56. K K\,
  57. var * p ^ t = 1 K var * [ 1 ( X t ) W ( X ) ] = 1 K { E * [ 1 ( X t ) 2 W 2 ( X ) ] - p t 2 } = 1 K { E [ 1 ( X t ) W ( X ) ] - p t 2 } \begin{aligned}\displaystyle\operatorname{var}_{*}\hat{p}_{t}&\displaystyle{}=% \frac{1}{K}\operatorname{var}_{*}[1(X\geq t)W(X)]\\ &\displaystyle{}=\frac{1}{K}\left\{{E_{*}}[1(X\geq t)^{2}W^{2}(X)]-p_{t}^{2}% \right\}\\ &\displaystyle{}=\frac{1}{K}\left\{{E}[1(X\geq t)W(X)]-p_{t}^{2}\right\}\end{aligned}
  58. f * f_{*}\,
  59. X t {X\geq t\ }
  60. X X\,
  61. a X aX\,
  62. a > 1 a>1
  63. f * ( x ) = 1 a f ( x a ) f_{*}(x)=\frac{1}{a}f\bigg(\frac{x}{a}\bigg)\,
  64. W ( x ) = a f ( x ) f ( x / a ) W(x)=a\frac{f(x)}{f(x/a)}\,
  65. X < t X<t\,
  66. X X\,
  67. n n\,
  68. n n\,
  69. n n\,
  70. f * ( x ) = f ( x - c ) , c > 0 f_{*}(x)=f(x-c),\quad c>0\,
  71. c c\,
  72. σ M C 2 / σ I S 2 \sigma^{2}_{MC}/\sigma^{2}_{IS}\,
  73. σ M C 2 / σ I S 2 \sigma^{2}_{MC}/\sigma^{2}_{IS}\,
  74. σ M C 2 / σ I S 2 \sigma^{2}_{MC}/\sigma^{2}_{IS}\,

Incidence_structure.html

  1. n n
  2. P , L , I P,L,I
  3. P P
  4. L L
  5. I P × L I⊆P×L
  6. I I
  7. p , l p,l
  8. I I
  9. p p
  10. l l
  11. l l
  12. p p
  13. p p
  14. l l
  15. l l
  16. p p
  17. p I l pIl
  18. ( p , l ) I (p,l)∈I
  19. L L
  20. P P
  21. I I
  22. p I l pIl
  23. p p
  24. l l
  25. P P
  26. L L
  27. v v
  28. M M
  29. C = ( P , L , I ) C=(P,L,I)
  30. I I
  31. C C
  32. X X
  33. F F
  34. X X
  35. X X
  36. F F
  37. F F
  38. F F
  39. X X
  40. P = { 1 , 2 , 3 , 4 , 5 , 6 , 7 } P=\left\{1,2,3,4,5,6,7\right\}
  41. L = { { 1 , 2 , 3 } , { 1 , 4 , 5 } , { 1 , 6 , 7 } , { 2 , 4 , 6 } , { 2 , 5 , 7 } , { 3 , 4 , 7 } , { 3 , 5 , 6 } } L=\left\{\{1,2,3\},\{1,4,5\},\{1,6,7\},\{2,4,6\},\{2,5,7\},\{3,4,7\},\{3,5,6\}\right\}
  42. P P
  43. L L
  44. ( 0 0 0 1 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 1 1 1 0 1 ) \left(\begin{matrix}0&0&0&1&1&0\\ 0&0&0&0&1&1\\ 1&0&0&0&0&0\\ 0&0&1&0&0&0\\ 1&0&0&0&0&0\\ 1&1&1&1&0&1\end{matrix}\right)
  45. I I
  46. l l
  47. m m
  48. n n
  49. o o
  50. p p
  51. q q
  52. A A
  53. B B
  54. C C
  55. D D
  56. E E
  57. P P
  58. C C
  59. M M
  60. M M
  61. A , B , C , D , G , F , E A,B,C,D,G,F,E
  62. l , p , n , s , r , m , q l,p,n,s,r,m,q
  63. ( 1 1 1 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 0 1 1 0 1 0 1 0 1 0 0 1 0 0 1 0 1 0 0 1 1 0 0 1 0 0 1 0 1 1 0 ) . \left(\begin{matrix}1&1&1&0&0&0&0\\ 1&0&0&1&1&0&0\\ 1&0&0&0&0&1&1\\ 0&1&0&1&0&1&0\\ 0&1&0&0&1&0&1\\ 0&0&1&1&0&0&1\\ 0&0&1&0&1&1&0\end{matrix}\right).
  64. n n
  65. n n
  66. π / 4 \pi/4
  67. k k
  68. k k
  69. k k
  70. k + 1 k+1
  71. I i < j P i × P j . I\subseteq\bigcup_{i<j}P_{i}\times P_{j}.
  72. n < s u b > 3 n<sub>3

Inclusive_fitness.html

  1. c < r b c<rb
  2. r r
  3. b b
  4. c c
  5. r r
  6. r r
  7. r r
  8. r r
  9. r r
  10. r r

Incompressible_surface.html

  1. D S = D . D\cap S=\partial\!D.
  2. D \partial\!D
  3. ι : S M \iota:S\rightarrow M
  4. π 1 \pi_{1}
  5. ι : π 1 ( S ) π 1 ( M ) \iota_{\star}:\pi_{1}(S)\rightarrow\pi_{1}(M)
  6. π 1 \pi_{1}
  7. L ( 4 , 1 ) L(4,1)
  8. π 1 \pi_{1}
  9. S S
  10. S S
  11. π 1 \pi_{1}

Indecomposable_module.html

  1. N < M N<M
  2. N P = M N\oplus P=M
  3. R / p n R/p^{n}
  4. 𝐙 / p n 𝐙 \mathbf{Z}/p^{n}\mathbf{Z}

Industrial_noise.html

  1. d 2 d_{2}
  2. d 1 d_{1}
  3. 10 l o g 10 [ ( d 2 d 1 ) 2 ] = 20 l o g 10 [ d 2 d 1 ] 10log_{10}\left[\left(\frac{d_{2}}{d_{1}}\right)^{2}\right]=20log_{10}\left[% \frac{d_{2}}{d_{1}}\right]
  4. 20 * l o g 10 ( 2 ) 20*log_{10}(2)

Infinitesimal_transformation.html

  1. I + ε A I+\varepsilon A
  2. Θ F = r F \Theta F=rF\,
  3. Θ = i x i x i , \Theta=\sum_{i}x_{i}{\partial\over\partial x_{i}},
  4. F ( λ x 1 , , λ x n ) = λ r F ( x 1 , , x n ) F(\lambda x_{1},\dots,\lambda x_{n})=\lambda^{r}F(x_{1},\dots,x_{n})\,
  5. e t D f ( x ) = f ( x + t ) e^{tD}f(x)=f(x+t)\,
  6. D = d d x D={d\over dx}

Influence_diagram.html

  1. d d
  2. X X
  3. Y Y
  4. Z Z
  5. Y Y
  6. Y Y
  7. X X
  8. Z Z

Information-theoretic_security.html

  1. c = E k ( m ) c=E_{k}(m)
  2. Π \Pi
  3. Π \Pi^{\prime}

Information_cascade.html

  1. P ( A | H ) \displaystyle P\left(A|H\right)
  2. P ( A | Previous , Personal signal ) = p q a ( 1 - q ) b p q a ( 1 - q ) b + ( 1 - p ) ( 1 - q ) a q b P(A|\,\text{Previous},\,\text{Personal signal})=\frac{pq^{a}(1-q)^{b}}{pq^{a}(% 1-q)^{b}+(1-p)(1-q)^{a}q^{b}}

Infrared_fixed_point.html

  1. μ μ y y 16 π 2 ( 9 2 y 2 - 8 g 3 2 ) \mu\frac{\partial}{\partial\mu}y\approx\frac{y}{16\pi^{2}}\left(\frac{9}{2}y^{% 2}-8g_{3}^{2}\right)
  2. g 3 g_{3}
  3. μ \mu
  4. y y
  5. μ \mu
  6. μ 10 15 \mu\approx 10^{15}
  7. y 2 y^{2}
  8. y y
  9. μ 100 \mu\approx 100
  10. y y
  11. y y
  12. g 3 g_{3}

Initial_condition.html

  1. X t + 1 = A X t X_{t+1}=AX_{t}
  2. X t = A t X 0 X_{t}=A^{t}X_{0}
  3. X 0 X_{0}
  4. X 0 X_{0}
  5. x t = a 1 x t - 1 + a 2 x t - 2 + + a k x t - k . x_{t}=a_{1}x_{t-1}+a_{2}x_{t-2}+\cdots+a_{k}x_{t-k}.
  6. λ k - a 1 λ k - 1 - a 2 λ k - 2 - - a k - 1 λ - a k = 0 \lambda^{k}-a_{1}\lambda^{k-1}-a_{2}\lambda^{k-2}-\cdots-a_{k-1}\lambda-a_{k}=0
  7. λ 1 , , λ k , \lambda_{1},\dots,\lambda_{k},
  8. x t = c 1 λ 1 t + + c k λ k t . x_{t}=c_{1}\lambda_{1}^{t}+\cdots+c_{k}\lambda_{k}^{t}.
  9. c 1 , , c k c_{1},\dots,c_{k}
  10. x t x_{t}
  11. d X d t = A X . \frac{dX}{dt}=AX.
  12. X 0 X_{0}
  13. d k x d t k + a k - 1 d k - 1 x d t k - 1 + + a 1 d x d t + a 0 x = 0. \frac{d^{k}x}{dt^{k}}+a_{k-1}\frac{d^{k-1}x}{dt^{k-1}}+\cdots+a_{1}\frac{dx}{% dt}+a_{0}x=0.
  14. λ k + a k - 1 λ k - 1 + + a 1 λ + a 0 = 0 , \lambda^{k}+a_{k-1}\lambda^{k-1}+\cdots+a_{1}\lambda+a_{0}=0,
  15. λ 1 , , λ k ; \lambda_{1},\dots,\lambda_{k};
  16. x ( t ) = c 1 e λ 1 t + + c k e λ k t . x(t)=c_{1}e^{\lambda_{1}t}+\cdots+c_{k}e^{\lambda_{k}t}.
  17. c 1 , , c k , c_{1},\dots,c_{k},

Initial_topology.html

  1. X X
  2. X X
  3. f i : X Y i f_{i}:X\to Y_{i}
  4. X X
  5. f i : ( X , τ ) Y i f_{i}:(X,\tau)\to Y_{i}
  6. f i - 1 ( U ) f_{i}^{-1}(U)
  7. U U
  8. Y i Y_{i}
  9. f i - 1 ( U ) f_{i}^{-1}(U)
  10. ( X , τ ) (X,\tau)
  11. g g
  12. Z Z
  13. X X
  14. f i g f_{i}\circ g
  15. f : X i Y i . f\colon X\to\prod_{i}Y_{i}\,.
  16. f i ( x ) cl ( f i ( A ) ) f_{i}(x)\notin\operatorname{cl}(f_{i}(A))
  17. f i - 1 ( U ) f_{i}^{-1}(U)

Injector.html

  1. P 2 / P 1 P_{2}/P_{1}
  2. P 2 P_{2}
  3. P 1 P_{1}
  4. W s / W v W_{s}/W_{v}
  5. W s W_{s}
  6. W v W_{v}

Inner_model.html

  1. L = L=\langle\in\rangle
  2. L L
  3. N , M , \langle N,\in_{M},\ldots\rangle
  4. \exists

Input_impedance.html

  1. Z i n Z o u t Z_{in}\gg Z_{out}
  2. Z i n = X - Im ( Z o u t ) × j \begin{aligned}\displaystyle Z_{in}&\displaystyle=X-\operatorname{Im}(Z_{out})% \times j\\ \end{aligned}
  3. Z i n = Z o u t * = | Z o u t | e - Θ o u t × j = Re ( Z o u t ) - Im ( Z o u t ) × j \begin{aligned}\displaystyle Z_{in}&\displaystyle=Z_{out}^{*}\\ &\displaystyle=\left|Z_{out}\right|e^{-\Theta_{out}\times j}\\ &\displaystyle=\operatorname{Re}(Z_{out})-\operatorname{Im}(Z_{out})\times j\\ \end{aligned}
  4. Z i n = Z o u t Z_{in}=Z_{out}
  5. Z o u t Z_{out}
  6. Z i n = Z l i n e Z_{in}=Z_{line}

Input–output_model.html

  1. n n
  2. x i x_{i}
  3. i i
  4. a i j a_{ij}
  5. j j
  6. i i
  7. d i d_{i}
  8. x i = a i 1 x 1 + a i 2 x 2 + + a i n x n + d i , x_{i}=a_{i1}x_{1}+a_{i2}x_{2}+\ldots+a_{in}x_{n}+d_{i},
  9. A A
  10. a i j a_{ij}
  11. x x
  12. d d
  13. x = A x + d x=Ax+d
  14. ( I - A ) x = d (I-A)x=d
  15. I - A I-A
  16. I - A I-A
  17. x x
  18. A = ( 0.5 0.2 0.4 0.1 ) and d = ( 7 4 ) . A=\left(\begin{array}[]{cc}0.5&0.2\\ 0.4&0.1\end{array}\right)\,\text{ and }d=\left(\begin{array}[]{c}7\\ 4\end{array}\right).
  19. x = ( I - A ) - 1 d = ( 19.19 12.97 ) . x=(I-A)^{-1}d=\left(\begin{array}[]{c}19.19\\ 12.97\end{array}\right).

Integer_square_root.html

  1. isqrt ( n ) = n . \mbox{isqrt}~{}(n)=\lfloor\sqrt{n}\rfloor.
  2. isqrt ( 27 ) = 5 \mbox{isqrt}~{}(27)=5
  3. 5 5 = 25 27 5\cdot 5=25\leq 27
  4. 6 6 = 36 > 27 6\cdot 6=36>27
  5. n \sqrt{n}
  6. isqrt ( n ) \mbox{isqrt}~{}(n)
  7. x 2 - n = 0 x^{2}-n=0
  8. x k + 1 = 1 2 ( x k + n x k ) , k 0 , x 0 > 0. {x}_{k+1}=\frac{1}{2}\left(x_{k}+\frac{n}{x_{k}}\right),\quad k\geq 0,\quad x_% {0}>0.
  9. { x k } \{x_{k}\}
  10. n \sqrt{n}
  11. k k\to\infty
  12. x 0 = n x_{0}=n
  13. | x k + 1 - x k | < 1 |x_{k+1}-x_{k}|<1
  14. x k + 1 = n . \lfloor x_{k+1}\rfloor=\lfloor\sqrt{n}\rfloor.
  15. n \lfloor\sqrt{n}\rfloor
  16. x k + 1 = 1 2 ( x k + n x k ) , k 0 , x 0 > 0 , x 0 . {x}_{k+1}=\left\lfloor\frac{1}{2}\left(x_{k}+\left\lfloor\frac{n}{x_{k}}\right% \rfloor\right)\right\rfloor,\quad k\geq 0,\quad x_{0}>0,\quad x_{0}\in\mathbb{% Z}.
  17. 1 2 ( x k + n x k ) = 1 2 ( x k + n x k ) , \left\lfloor\frac{1}{2}\left(x_{k}+\left\lfloor\frac{n}{x_{k}}\right\rfloor% \right)\right\rfloor=\left\lfloor\frac{1}{2}\left(x_{k}+\frac{n}{x_{k}}\right)% \right\rfloor,
  18. n \lfloor\sqrt{n}\rfloor
  19. n \lfloor\sqrt{n}\rfloor
  20. n \lfloor\sqrt{n}\rfloor
  21. n + 1 n+1
  22. n + 1 n+1
  23. n \lfloor\sqrt{n}\rfloor
  24. n + 1 \lfloor\sqrt{n}\rfloor+1
  25. n \sqrt{n}
  26. n n
  27. { x k } \{x_{k}\}
  28. x 0 x_{0}
  29. isqrt ( n ) \mbox{isqrt}~{}(n)
  30. c = 1 c=1
  31. | x k + 1 - x k | < c |x_{k+1}-x_{k}|<c
  32. x k + 1 = n \lfloor x_{k+1}\rfloor=\lfloor\sqrt{n}\rfloor

Integrability_conditions_for_differential_systems.html

  1. i : N M i:N\subset M
  2. i * : Ω p 1 ( M ) Ω p 1 ( N ) i^{*}:\Omega_{p}^{1}(M)\rightarrow\Omega_{p}^{1}(N)
  3. \mathcal{I}
  4. d , d{\mathcal{I}}\subset{\mathcal{I}},
  5. θ = x d y + y d z + z d x . \theta=x\,dy+y\,dz+z\,dx.
  6. θ d θ = 0. \theta\wedge d\theta=0.
  7. θ d θ = ( x + y + z ) d x d y d z \theta\wedge d\theta=(x+y+z)\,dx\wedge dy\wedge dz
  8. x = t , y = c , z = e - t c , t > 0 x=t,\quad y=c,\qquad z=e^{-{t\over c}},\quad t>0
  9. θ i , θ j = δ i j \langle\theta^{i},\theta^{j}\rangle=\delta^{ij}
  10. Θ = ( θ 1 , , θ n ) \Theta=(\theta^{1},\dots,\theta^{n})
  11. Φ = ( ϕ 1 , , ϕ n ) \Phi=(\phi^{1},\dots,\phi^{n})
  12. Φ = M Θ \Phi=M\Theta
  13. d Φ = ω Φ d\Phi=\omega\wedge\Phi
  14. d Φ \displaystyle d\Phi
  15. ω = ( d M ) M - 1 \omega=(dM)M^{-1}
  16. d ω + ω ω = 0 , d\omega+\omega\wedge\omega=0,
  17. Ω = d ω + ω ω = 0. \Omega=d\omega+\omega\wedge\omega=0.

Interaction_energy.html

  1. Q 1 Q 2 / ( 4 π ϵ 0 Δ r ) Q_{1}Q_{2}/(4\pi\epsilon_{0}\Delta r)
  2. Q 1 Q_{1}
  3. Q 2 Q_{2}
  4. Δ E i n t = E ( A , B ) - ( E ( A ) + E ( B ) ) \Delta E_{int}=E(A,B)-\left(E(A)+E(B)\right)
  5. E ( A ) E(A)
  6. E ( B ) E(B)
  7. E ( A , B ) E(A,B)
  8. Δ E i n t = E ( A 1 , A 2 , . . , A N ) - i = 1 N E ( A i ) \Delta E_{int}=E(A_{1},A_{2},..,A_{N})-\sum_{i=1}^{N}E(A_{i})

Interaction_picture.html

  1. e ± i H 0 , S t / e^{\pm iH_{0,S}t/\hbar}
  2. t t
  3. t t
  4. H 0 , I ( t ) = e i H 0 , S t / H 0 , S e - i H 0 , S t / = H 0 , S . H_{0,I}(t)=e^{iH_{0,S}t/\hbar}H_{0,S}e^{-iH_{0,S}t/\hbar}=H_{0,S}.
  5. H 1 , I ( t ) = e i H 0 , S t / H 1 , S e - i H 0 , S t / , H_{1,I}(t)=e^{iH_{0,S}t/\hbar}H_{1,S}e^{-iH_{0,S}t/\hbar},
  6. ρ I ( t ) = n p n ( t ) | ψ n , I ( t ) ψ n , I ( t ) | = n p n ( t ) e i H 0 , S t / | ψ n , S ( t ) ψ n , S ( t ) | e - i H 0 , S t / = e i H 0 , S t / ρ S ( t ) e - i H 0 , S t / . \rho_{I}(t)=\sum_{n}p_{n}(t)|\psi_{n,I}(t)\rangle\langle\psi_{n,I}(t)|=\sum_{n% }p_{n}(t)e^{iH_{0,S}t/\hbar}|\psi_{n,S}(t)\rangle\langle\psi_{n,S}(t)|e^{-iH_{% 0,S}t/\hbar}=e^{iH_{0,S}t/\hbar}\rho_{S}(t)e^{-iH_{0,S}t/\hbar}.
  7. | ψ I ( t ) = e i H 0 , S t / | ψ S ( t ) |\psi_{I}(t)\rangle=e^{iH_{0,S}~{}t/\hbar}|\psi_{S}(t)\rangle
  8. | ψ S ( t ) = e - i H S t / | ψ S ( 0 ) |\psi_{S}(t)\rangle=e^{-iH_{S}~{}t/\hbar}|\psi_{S}(0)\rangle
  9. A H ( t ) = e i H S t / A S e - i H S t / A_{H}(t)=e^{iH_{S}~{}t/\hbar}A_{S}e^{-iH_{S}~{}t/\hbar}
  10. A I ( t ) = e i H 0 , S t / A S e - i H 0 , S t / A_{I}(t)=e^{iH_{0,S}~{}t/\hbar}A_{S}e^{-iH_{0,S}~{}t/\hbar}
  11. ρ I ( t ) = e i H 0 , S t / ρ S ( t ) e - i H 0 , S t / \rho_{I}(t)=e^{iH_{0,S}~{}t/\hbar}\rho_{S}(t)e^{-iH_{0,S}~{}t/\hbar}
  12. ρ S ( t ) = e - i H S t / ρ S ( 0 ) e i H S t / \rho_{S}(t)=e^{-iH_{S}~{}t/\hbar}\rho_{S}(0)e^{iH_{S}~{}t/\hbar}
  13. i d d t ψ I ( t ) = H 1 , I ( t ) ψ I ( t ) . i\hbar\frac{d}{dt}\mid\psi_{I}(t)\rangle=H_{1,I}(t)\mid\psi_{I}(t)\rangle.
  14. i d d t A I ( t ) = [ A I ( t ) , H 0 ] . i\hbar\frac{d}{dt}A_{I}(t)=\left[A_{I}(t),H_{0}\right].\;
  15. i d d t ρ I ( t ) = [ H 1 , I ( t ) , ρ I ( t ) ] . i\hbar\frac{d}{dt}\rho_{I}(t)=\left[H_{1,I}(t),\rho_{I}(t)\right].

Intermediate_logic.html

  1. 𝒫 ( X ) { X } , \langle\mathcal{P}(X)\setminus\{X\},\subseteq\rangle
  2. 𝐈𝐏𝐂 + i = 0 n ( j < i p j p i ) \textstyle\mathbf{IPC}+\bigvee_{i=0}^{n}\bigl(\bigwedge_{j<i}p_{j}\to p_{i}\bigr)
  3. 𝐈𝐏𝐂 + i = 0 n ( j i p j p i ) \textstyle\mathbf{IPC}+\bigvee_{i=0}^{n}\bigl(\bigwedge_{j\neq i}p_{j}\to p_{i% }\bigr)
  4. 𝐈𝐏𝐂 + i = 0 n ( j < i p j ¬ ¬ p i ) \textstyle\mathbf{IPC}+\bigvee_{i=0}^{n}\bigl(\bigwedge_{j<i}p_{j}\to\neg\neg p% _{i}\bigr)
  5. 𝐈𝐏𝐂 + i = 0 n ( ( p i j i p j ) j i p j ) i = 0 n p i \textstyle\mathbf{IPC}+\bigwedge_{i=0}^{n}\bigl(\bigl(p_{i}\to\bigvee_{j\neq i% }p_{j}\bigr)\to\bigvee_{j\neq i}p_{j}\bigr)\to\bigvee_{i=0}^{n}p_{i}
  6. { x M , x p } \{x\mid M,x\Vdash p\}
  7. T ( p n ) = p n T(p_{n})=\Box p_{n}
  8. T ( ¬ A ) = ¬ T ( A ) T(\neg A)=\Box\neg T(A)
  9. T ( A and B ) = T ( A ) and T ( B ) T(A\and B)=T(A)\and T(B)
  10. T ( A B ) = T ( A ) T ( B ) T(A\vee B)=T(A)\vee T(B)
  11. T ( A B ) = ( T ( A ) T ( B ) ) T(A\to B)=\Box(T(A)\to T(B))

Internal_resistance.html

  1. R int = ( V NL V FL - 1 ) R L R_{\,\text{int}}=\left({\frac{V_{\,\text{NL}}}{V_{\,\text{FL}}}-1}\right){R_{% \,\text{L}}}

Internal_set_theory.html

  1. st x ϕ ( x ) = x ( standard ( x ) ϕ ( x ) ) , st x ϕ ( x ) = x ( standard ( x ) ϕ ( x ) ) . \begin{aligned}\displaystyle\exists^{\mathrm{st}}x\,\phi(x)&\displaystyle=% \exists x\,(\operatorname{standard}(x)\land\phi(x)),\\ \displaystyle\forall^{\mathrm{st}}x\,\phi(x)&\displaystyle=\forall x\,(% \operatorname{standard}(x)\to\phi(x)).\end{aligned}
  2. ϕ \phi
  3. st z ( z is finite y x z ϕ ( x , y , u 1 , , u n ) ) y st x ϕ ( x , y , u 1 , , u n ) . \forall^{\mathrm{st}}z\,(z\,\text{ is finite}\to\exists y\,\forall x\in z\,% \phi(x,y,u_{1},\dots,u_{n}))\leftrightarrow\exists y\,\forall^{\mathrm{st}}x\,% \phi(x,y,u_{1},\dots,u_{n}).
  4. ϕ \phi
  5. st x st y st t ( t y ( t x ϕ ( t , u 1 , , u n ) ) ) \forall^{\mathrm{st}}x\,\exists^{\mathrm{st}}y\,\forall^{\mathrm{st}}t\,(t\in y% \leftrightarrow(t\in x\land\phi(t,u_{1},\dots,u_{n})))
  6. ϕ ( x , u 1 , , u n ) \phi(x,u_{1},\dots,u_{n})
  7. st u 1 st u n ( st x ϕ ( x , u 1 , , u n ) x ϕ ( x , u 1 , , u n ) ) \forall^{\mathrm{st}}u_{1}\dots\forall^{\mathrm{st}}u_{n}\,(\forall^{\mathrm{% st}}x\,\phi(x,u_{1},\dots,u_{n})\to\forall x\,\phi(x,u_{1},\dots,u_{n}))

Internal_wave.html

  1. ρ \rho
  2. ρ 0 \rho_{0}
  3. g ( ρ - ρ 0 ) g(\rho-\rho_{0})
  4. g g
  5. ρ 00 \rho_{00}
  6. g g ρ - ρ 0 ρ 00 g^{\prime}\equiv g\frac{\rho-\rho_{0}}{\rho_{00}}
  7. ρ > ρ 0 \rho>\rho_{0}
  8. g g^{\prime}
  9. g g
  10. g g g^{\prime}\sim g
  11. ρ 0 ( z ) \rho_{0}(z)
  12. ρ 0 ( z 0 ) \rho_{0}(z_{0})
  13. Δ z \Delta z
  14. d 2 Δ z d t 2 = - g = - g ( ρ 0 ( z 0 ) - ρ 0 ( z 0 + Δ z ) ) / ρ 0 ( z 0 ) - g ( - d ρ 0 d z Δ z ) / ρ 0 ( z 0 ) \frac{d^{2}\Delta z}{dt^{2}}=-g^{\prime}=-g(\rho_{0}(z_{0})-\rho_{0}(z_{0}+% \Delta z))/\rho_{0}(z_{0})\simeq-g\left(-\frac{d\rho_{0}}{dz}\Delta z\right)/% \rho_{0}(z_{0})
  15. z 0 z_{0}
  16. N = ( - g ρ 0 d ρ 0 d z ) 1 / 2 . N=\left(-\frac{g}{\rho_{0}}\frac{d\rho_{0}}{dz}\right)^{1/2}.
  17. ω \omega
  18. Θ \Theta
  19. ω = N cos Θ \omega=N\cos\Theta
  20. Θ \Theta
  21. ρ 1 \rho_{1}
  22. ρ 2 \rho_{2}
  23. z = 0. z=0.
  24. u = ϕ , {\vec{u}=\nabla\phi,}
  25. 2 ϕ = 0. \nabla^{2}\phi=0.
  26. x - z x-z
  27. x x
  28. k > 0 , k>0,
  29. z z
  30. ϕ 1 ( x , z , t ) = A e - k z cos ( k x - ω t ) \phi_{1}(x,z,t)=Ae^{-kz}\cos(kx-\omega t)
  31. ϕ 2 ( x , z , t ) = A e k z cos ( k x - ω t ) , \phi_{2}(x,z,t)=Ae^{kz}\cos(kx-\omega t),
  32. A A
  33. ω \omega
  34. ω 2 = g k \omega^{2}=g^{\prime}k
  35. g g^{\prime}
  36. g = ρ 2 - ρ 1 ρ 2 + ρ 1 g , g^{\prime}=\frac{\rho_{2}-\rho_{1}}{\rho_{2}+\rho_{1}}\,g,
  37. g g
  38. g = g . g^{\prime}=g.
  39. x u + z w = 0 \partial_{x}u+\partial_{z}w=0
  40. ρ 00 t u = - x p \rho_{00}\partial_{t}u=-\partial_{x}p
  41. ρ 00 t w = - z p - ρ g \rho_{00}\partial_{t}w=-\partial_{z}p-\rho g
  42. t ρ = - w d ρ 0 / d z \partial_{t}\rho=-wd\rho_{0}/dz
  43. ρ \rho
  44. p p
  45. ( u , w ) (u,w)
  46. ρ 0 ( z ) \rho_{0}(z)
  47. ρ 00 \rho_{00}
  48. exp [ i ( k x + m z - ω t ) ] \exp[i(kx+mz-\omega t)]
  49. ω 2 = N 2 k 2 k 2 + m 2 = N 2 cos 2 Θ \omega^{2}=N^{2}\frac{k^{2}}{k^{2}+m^{2}}=N^{2}\cos^{2}\Theta
  50. N N
  51. Θ = tan - 1 ( m / k ) \Theta=\tan^{-1}(m/k)

Intersection_homology.html

  1. H i ( X , ) × H n - i ( X , ) H 0 ( X , ) . H_{i}(X,\mathbb{Q})\times H_{n-i}(X,\mathbb{Q})\to H_{0}(X,\mathbb{Q})\cong% \mathbb{Q}.
  2. = X - 1 X 0 X 1 X n = X \emptyset=X_{-1}\subset X_{0}\subset X_{1}\subset\cdots\subset X_{n}=X
  3. U X U\subset X
  4. U i × C L U\cong\mathbb{R}^{i}\times CL
  5. C L CL
  6. p ( k ) + q ( k ) = k - 2 p(k)+q(k)=k-2\,
  7. σ - 1 ( X n - k - X n - k - 1 ) \sigma^{-1}(X_{n-k}-X_{n-k-1})
  8. I p H i ( X ) I^{p}H_{i}(X)\,
  9. f : X Y f:X\rightarrow Y
  10. I p H n - i ( X ) = I p H i ( X ) = H c i ( I C p ( X ) ) I^{p}H_{n-i}(X)=I^{p}H^{i}(X)=H^{i}_{c}(IC_{p}(X))
  11. I C p ( X ) = τ p ( n ) - n R i n * τ p ( n - 1 ) - n R i n - 1 * τ p ( 2 ) - n R i 2 * X - X n - 2 IC_{p}(X)=\tau_{\leq p(n)-n}Ri_{n*}\tau_{\leq p(n-1)-n}Ri_{n-1*}\cdots\tau_{% \leq p(2)-n}Ri_{2*}{\mathbb{C}}_{X-X_{n-2}}
  12. H i ( j x * I C p ) H^{i}(j_{x}^{*}IC_{p})
  13. H i ( j x * I C p ) H^{i}(j_{x}^{*}IC_{p})
  14. H - i ( j x * I C p ) H^{-i}(j_{x}^{*}IC_{p})
  15. H - i ( j x ! I C p ) H^{-i}(j_{x}^{!}IC_{p})

Invariant_(mathematics).html

  1. M K d μ \textstyle{\int_{M}K\,d\mu}
  2. x S T ( x ) S . x\in S\Rightarrow T(x)\in S.

Invariant_subspace_problem.html

  1. x x
  2. A A

Inverse_Galois_problem.html

  1. 𝐐 \mathbf{Q}
  2. 𝐐 \mathbf{Q}
  3. 5 5
  4. 8 8
  5. G G
  6. K K
  7. L / K L/K
  8. G G
  9. G G
  10. K K
  11. L L
  12. 𝐂 \mathbf{C}
  13. 𝐐 \mathbf{Q}
  14. 𝐐 \mathbf{Q}
  15. G G
  16. K K
  17. 𝐐 \mathbf{Q}
  18. G G
  19. 𝐐 \mathbf{Q}
  20. G G
  21. 𝐐 \mathbf{Q}
  22. 𝐐 \mathbf{Q}
  23. G G
  24. 𝐐 ( t ) \mathbf{Q}(t)
  25. t t
  26. t t
  27. 𝐐 \mathbf{Q}
  28. 𝐙 / n 𝐙 \mathbf{Z}/n\mathbf{Z}
  29. n n
  30. p p
  31. p 1 ( m o d n ) p≡1(modn)
  32. 𝐐 ( μ ) \mathbf{Q}(μ)
  33. 𝐐 \mathbf{Q}
  34. μ μ
  35. μ μ
  36. 𝐐 ( μ ) / 𝐐 \mathbf{Q}(μ)/\mathbf{Q}
  37. p 1 p−1
  38. n n
  39. p 1 p−1
  40. H H
  41. ( p 1 ) / n (p−1)/n
  42. 𝐙 / n 𝐙 \mathbf{Z}/n\mathbf{Z}
  43. 𝐐 \mathbf{Q}
  44. μ μ
  45. α α
  46. F F
  47. F F
  48. 𝐐 \mathbf{Q}
  49. 𝐐 \mathbf{Q}
  50. n = 3 n=3
  51. p = 7 p=7
  52. G a l ( 𝐐 ( μ ) / 𝐐 ) Gal(\mathbf{Q}(μ)/\mathbf{Q})
  53. η η
  54. μ μ
  55. α α
  56. H H
  57. 𝐐 \mathbf{Q}
  58. α + β + γ = 1 α+β+γ=−1
  59. α β + β γ + γ α = 2 αβ+βγ+γα=−2
  60. α β γ = 1 αβγ=1
  61. α α
  62. 𝐙 / 3 𝐙 \mathbf{Z}/3\mathbf{Z}
  63. 𝐐 \mathbf{Q}
  64. ( - 1 ) n ( n - 1 ) 2 ( n n b n - 1 + ( - 1 ) 1 - n ( n - 1 ) n - 1 a n ) . (-1)^{\frac{n(n-1)}{2}}\left(n^{n}b^{n-1}+(-1)^{1-n}(n-1)^{n-1}a^{n}\right).
  65. s s
  66. f ( x , s ) f(x,s)
  67. f ( x , s ) f(x,s)
  68. f ( x , s ) f(x,s)
  69. 𝐐 ( s ) \mathbf{Q}(s)
  70. f ( x , s ) f(x,s)
  71. x n - x 2 - 1 2 - ( s - 1 2 ) ( x + 1 ) x^{n}-\tfrac{x}{2}-\tfrac{1}{2}-\left(s-\tfrac{1}{2}\right)(x+1)
  72. f ( x , 1 / 2 ) f(x,1/2)
  73. 1 2 ( x - 1 ) ( 1 + 2 x + 2 x 2 + + 2 x n - 1 ) \tfrac{1}{2}(x-1)\left(1+2x+2x^{2}+\cdots+2x^{n-1}\right)
  74. G a l ( f ( x , s ) / 𝐐 ( s ) ) Gal(f(x,s)/\mathbf{Q}(s))
  75. ( 1 n ) x = n y (1−n)x=ny
  76. y n - { s ( 1 - n n ) n - 1 } y - { s ( 1 - n n ) n } y^{n}-\left\{s\left(\frac{1-n}{n}\right)^{n-1}\right\}y-\left\{s\left(\frac{1-% n}{n}\right)^{n}\right\}
  77. t = s ( 1 - n ) n - 1 n n , t=\frac{s(1-n)^{n-1}}{n^{n}},
  78. g ( y , 1 ) g(y,1)
  79. 1 1
  80. n 2 n−2
  81. G a l ( f ( x , s ) / 𝐐 ( s ) ) Gal(f(x,s)/\mathbf{Q}(s))
  82. f ( x , t ) f(x,t)
  83. 𝐐 \mathbf{Q}
  84. 𝐐 \mathbf{Q}
  85. g ( y , t ) g(y,t)
  86. ( - 1 ) n ( n - 1 ) 2 n n ( n - 1 ) n - 1 t n - 1 ( 1 - t ) , (-1)^{\frac{n(n-1)}{2}}n^{n}(n-1)^{n-1}t^{n-1}(1-t),
  87. t = 1 - ( - 1 ) n ( n - 1 ) 2 n u 2 t=1-(-1)^{\tfrac{n(n-1)}{2}}nu^{2}
  88. g ( y , t ) g(y,t)
  89. ( - 1 ) n ( n - 1 ) 2 n n ( n - 1 ) n - 1 t n - 1 ( 1 - t ) = ( - 1 ) n ( n - 1 ) 2 n n ( n - 1 ) n - 1 t n - 1 ( 1 - ( 1 - ( - 1 ) n ( n - 1 ) 2 n u 2 ) ) = ( - 1 ) n ( n - 1 ) 2 n n ( n - 1 ) n - 1 t n - 1 ( ( - 1 ) n ( n - 1 ) 2 n u 2 ) = n n + 1 ( n - 1 ) n - 1 t n - 1 u 2 \begin{aligned}\displaystyle(-1)^{\frac{n(n-1)}{2}}n^{n}(n-1)^{n-1}t^{n-1}(1-t% )&\displaystyle=(-1)^{\frac{n(n-1)}{2}}n^{n}(n-1)^{n-1}t^{n-1}\left(1-\left(1-% (-1)^{\tfrac{n(n-1)}{2}}nu^{2}\right)\right)\\ &\displaystyle=(-1)^{\frac{n(n-1)}{2}}n^{n}(n-1)^{n-1}t^{n-1}\left((-1)^{% \tfrac{n(n-1)}{2}}nu^{2}\right)\\ &\displaystyle=n^{n+1}(n-1)^{n-1}t^{n-1}u^{2}\end{aligned}
  90. n n
  91. t = 1 1 + ( - 1 ) n ( n - 1 ) 2 ( n - 1 ) u 2 t=\frac{1}{1+(-1)^{\tfrac{n(n-1)}{2}}(n-1)u^{2}}
  92. g ( y , t ) g(y,t)
  93. ( - 1 ) n ( n - 1 ) 2 n n ( n - 1 ) n - 1 t n - 1 ( 1 - t ) \displaystyle(-1)^{\frac{n(n-1)}{2}}n^{n}(n-1)^{n-1}t^{n-1}(1-t)
  94. n n
  95. C < s u b > 1 , , C n C<sub>1,...,C_{n}

Inverse_system.html

  1. \to
  2. \to
  3. \to
  4. Hom C ( F ( i ) , G ( j ) ) \mathrm{Hom}_{C}(F(i),G(j))
  5. \to

Involute.html

  1. r : n r:\mathbb{R}\to\mathbb{R}^{n}
  2. | r ( s ) | = 1 |r^{\prime}(s)|=1
  3. s r ( s ) - s r ( s ) s\mapsto r(s)-sr^{\prime}(s)
  4. ( x ( t ) , y ( t ) ) (x(t),y(t))
  5. X ( t ) = x ( t ) - x ( t ) x ( t ) 2 + y ( t ) 2 a t x ( t ) 2 + y ( t ) 2 d t X(t)=x(t)-\frac{x^{\prime}(t)}{\sqrt{x^{\prime}(t)^{2}+y^{\prime}(t)^{2}}}\int% _{a}^{t}\sqrt{x^{\prime}(t)^{2}+y^{\prime}(t)^{2}}\operatorname{d}t
  6. Y ( t ) = y ( t ) - y ( t ) x ( t ) 2 + y ( t ) 2 a t x ( t ) 2 + y ( t ) 2 d t Y(t)=y(t)-\frac{y(t)^{\prime}}{\sqrt{x^{\prime}(t)^{2}+y^{\prime}(t)^{2}}}\int% _{a}^{t}\sqrt{x^{\prime}(t)^{2}+y^{\prime}(t)^{2}}\operatorname{d}t
  7. x = r ( cos θ + θ sin θ ) \,x=r\left(\cos\theta+\theta\sin\theta\right)
  8. y = r ( sin θ - θ cos θ ) \,y=r\left(\sin\theta-\theta\cos\theta\right)
  9. r \,r
  10. θ \theta
  11. θ \ \theta\in\mathbb{R}
  12. θ \theta
  13. θ \theta
  14. r , φ \,r,\varphi
  15. r = a sec α = a cos α \,r=a\cdot\sec\alpha=\frac{a}{\cos\alpha}
  16. φ = tan α - α \,\varphi=\tan\alpha-\alpha
  17. a \,a
  18. α \,\alpha
  19. α \alpha\in\mathbb{R}
  20. t - φ \,t-\varphi
  21. tan α = t \tan\alpha=t
  22. t ( \ \ t\ (
  23. t = tan α ; t ) \ \ t=\tan\alpha\ ;\ \ t\in\mathbb{R}\ )
  24. r = a 1 + t 2 \,r=a\cdot\sqrt{1+\ t^{2}}
  25. φ = t - arctan ( t ) \,\varphi=\ t-\arctan(\ t)
  26. 0 t t 1 \ \quad 0\leq t\leq t_{1}\quad
  27. L = a 2 t 1 2 \ L=\frac{a}{2}\cdot t_{1}^{2}
  28. x = t - tanh ( t ) x=t-\mathrm{tanh}(t)\,
  29. y = sech ( t ) = 1 cosh ( t ) y=\mathrm{sech}(t)\ =\ \frac{1}{\mathrm{cosh}(t)}
  30. r ( s ) = ( sinh - 1 ( s ) , cosh ( sinh - 1 ( s ) ) ) r(s)=(\sinh^{-1}(s),\cosh(\sinh^{-1}(s)))\,
  31. r ( s ) = ( 1 , s ) / 1 + s 2 r^{\prime}(s)=(1,s)/\sqrt{1+s^{2}}\,
  32. r ( t ) - t r ( t ) = ( sinh - 1 ( t ) - t / 1 + t 2 , 1 / 1 + t 2 ) r(t)-tr^{\prime}(t)=(\sinh^{-1}(t)-t/\sqrt{1+t^{2}},1/\sqrt{1+t^{2}})
  33. t = 1 - y 2 / y t=\sqrt{1-y^{2}}/y
  34. ( sech - 1 ( y ) - 1 - y 2 , y ) ({\rm sech}^{-1}(y)-\sqrt{1-y^{2}},y)
  35. x = r ( t - sin ( t ) ) x=r(t-\sin(t))\,
  36. y = r ( 1 - cos ( t ) ) y=r(1-\cos(t))\,

Irreducibility_(mathematics).html

  1. P 2 \mathbb{R}P^{2}

Isabella_piercing.html

  1. 5 16 \tfrac{5}{16}
  2. 7 16 \tfrac{7}{16}

Isoperimetric_dimension.html

  1. area ( D ) C vol ( D ) ( d - 1 ) / d . \mathrm{area}\,(\partial D)\geq C\,\mathrm{vol}\,(D)^{(d-1)/d}.\,
  2. area ( D ) C vol ( D ) , \mathrm{area}\,(\partial D)\geq C\,\mathrm{vol}\,(D),
  3. A \partial A
  4. G A G\setminus A
  5. | A | C ( min ( | A | , | G A | ) ) ( d - 1 ) / d . |\partial A|\geq C\left(\min\left(|A|,|G\setminus A|\right)\right)^{(d-1)/d}.\,
  6. A A
  7. min ( | A | , | G A | ) \min(|A|,|G\setminus A|)
  8. vol B ( x , r ) C r d \mathrm{vol}\,B(x,r)\geq Cr^{d}
  9. p n ( x , y ) C n - d / 2 p_{n}(x,y)\leq Cn^{-d/2}\,
  10. p n ( x , y ) \scriptstyle p_{n}(x,y)\,

Isospin.html

  1. 1 / 2 {1}/{2}
  2. 1 / 2 {1}/{2}
  3. 1 / 2 {1}/{2}
  4. 1 / 2 {1}/{2}
  5. 1 / 2 {1}/{2}
  6. - 1 / 2 {-1}/{2}
  7. 1 / 2 {1}/{2}
  8. 3 / 2 {3}/{2}
  9. 3 / 2 {3}/{2}
  10. 2 / 3 {2}/{3}
  11. 1 / 3 {1}/{3}
  12. 3 / 2 {3}/{2}
  13. 3 / 2 {3}/{2}
  14. 1 / 2 {1}/{2}
  15. 1 / 2 {1}/{2}
  16. 3 / 2 {3}/{2}
  17. 3 / 2 {3}/{2}
  18. 3 / 2 {3}/{2}
  19. I 3 = 1 2 [ ( n u - n u ¯ ) - ( n d - n d ¯ ) ] I_{\mathrm{3}}=\frac{1}{2}\left[\left(n_{\mathrm{u}}-n_{\mathrm{\bar{u}}})-(n_% {\mathrm{d}}-n_{\mathrm{\bar{d}}}\right)\right]
  20. 1 / 2 {1}/{2}
  21. 𝐈 \mathbf{I}
  22. 1 / 2 {1}/{2}
  23. 𝐈 \mathbf{I}
  24. σ 3 = ( 1 0 0 - 1 ) \sigma_{3}=\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}
  25. S U ( 2 ) SU(2)
  26. S U ( 2 ) SU(2)
  27. 1 / 2 {1}/{2}
  28. 1 / 2 {1}/{2}
  29. 1 / 2 {1}/{2}
  30. I 3 = 1 2 ( n u - n d ) . I_{3}=\frac{1}{2}(n_{u}-n_{d}).
  31. 3 / 2 {3}/{2}
  32. 1 / 2 {1}/{2}
  33. | p = 1 3 2 ( | d u u | u d u | u u d ) ( 2 - 1 - 1 - 1 2 - 1 - 1 - 1 2 ) ( | | | ) |p\uparrow\rangle=\frac{1}{3\sqrt{2}}\left(\begin{array}[]{ccc}|duu\rangle&|% udu\rangle&|uud\rangle\end{array}\right)\left(\begin{array}[]{ccc}2&-1&-1\\ -1&2&-1\\ -1&-1&2\end{array}\right)\left(\begin{array}[]{c}|\downarrow\uparrow\uparrow% \rangle\\ |\uparrow\downarrow\uparrow\rangle\\ |\uparrow\uparrow\downarrow\rangle\end{array}\right)
  34. | n = 1 3 2 ( | u d d | d u d | d d u ) ( 2 - 1 - 1 - 1 2 - 1 - 1 - 1 2 ) ( | | | ) |n\uparrow\rangle=\frac{1}{3\sqrt{2}}\left(\begin{array}[]{ccc}|udd\rangle&|% dud\rangle&|ddu\rangle\end{array}\right)\left(\begin{array}[]{ccc}2&-1&-1\\ -1&2&-1\\ -1&-1&2\end{array}\right)\left(\begin{array}[]{c}|\downarrow\uparrow\uparrow% \rangle\\ |\uparrow\downarrow\uparrow\rangle\\ |\uparrow\uparrow\downarrow\rangle\end{array}\right)
  35. | u |u\rangle
  36. | d |d\rangle
  37. | |\uparrow\rangle
  38. | |\downarrow\rangle
  39. S z S_{z}
  40. | π + \displaystyle|\pi^{+}\rangle

Iterated_logarithm.html

  1. log * n := { 0 if n 1 ; 1 + log * ( log n ) if n > 1 \log^{*}n:=\begin{cases}0&\mbox{if }~{}n\leq 1;\\ 1+\log^{*}(\log n)&\mbox{if }~{}n>1\end{cases}
  2. log * n = slog e ( n ) \log^{*}n=\lceil\mathrm{slog}_{e}(n)\rceil
  3. slog e ( - x ) = - 1 \lceil\,\text{slog}_{e}(-x)\rceil=-1
  4. e 1 / e 1.444667 e^{1/e}\approx 1.444667
  5. n log * n . n\sqrt{\log^{*}n}.

Iterative_reconstruction.html

  1. f ( r ) f(r)
  2. 𝐀 x + ϵ \mathbf{A}x+\epsilon
  3. ϵ \epsilon

Iwasawa_decomposition.html

  1. 𝔤 0 \mathfrak{g}_{0}
  2. 𝔤 \mathfrak{g}
  3. 𝔤 0 \mathfrak{g}_{0}
  4. 𝔤 0 \mathfrak{g}_{0}
  5. 𝔤 0 = 𝔨 0 𝔭 0 \mathfrak{g}_{0}=\mathfrak{k}_{0}\oplus\mathfrak{p}_{0}
  6. 𝔞 0 \mathfrak{a}_{0}
  7. 𝔭 0 \mathfrak{p}_{0}
  8. 𝔞 0 \mathfrak{a}_{0}
  9. 𝔞 0 \mathfrak{a}_{0}
  10. 𝔤 0 \mathfrak{g}_{0}
  11. 𝔫 0 \mathfrak{n}_{0}
  12. 𝔨 0 , 𝔞 0 \mathfrak{k}_{0},\mathfrak{a}_{0}
  13. 𝔫 0 \mathfrak{n}_{0}
  14. 𝔤 0 \mathfrak{g}_{0}
  15. 𝔤 0 = 𝔨 0 𝔞 0 𝔫 0 \mathfrak{g}_{0}=\mathfrak{k}_{0}\oplus\mathfrak{a}_{0}\oplus\mathfrak{n}_{0}
  16. G = K A N G=KAN
  17. 𝔞 0 \mathfrak{a}_{0}
  18. 𝔤 0 = 𝔪 0 𝔞 0 𝔤 λ \mathfrak{g}_{0}=\mathfrak{m}_{0}\oplus\mathfrak{a}_{0}\oplus_{\lambda\in% \Sigma}\mathfrak{g}_{\lambda}
  19. 𝔪 0 \mathfrak{m}_{0}
  20. 𝔞 0 \mathfrak{a}_{0}
  21. 𝔨 0 \mathfrak{k}_{0}
  22. 𝔤 λ = { X 𝔤 0 : [ H , X ] = λ ( H ) X H 𝔞 0 } \mathfrak{g}_{\lambda}=\{X\in\mathfrak{g}_{0}:[H,X]=\lambda(H)X\;\;\forall H% \in\mathfrak{a}_{0}\}
  23. m λ = dim 𝔤 λ m_{\lambda}=\,\text{dim}\,\mathfrak{g}_{\lambda}
  24. λ \lambda
  25. F F
  26. G L n ( F ) GL_{n}(F)
  27. G L n ( O F ) GL_{n}(O_{F})
  28. O F O_{F}
  29. F F

Jacobi_field.html

  1. γ \gamma
  2. γ τ \gamma_{\tau}
  3. γ 0 = γ \gamma_{0}=\gamma
  4. J ( t ) = γ τ ( t ) τ | τ = 0 J(t)=\left.\frac{\partial\gamma_{\tau}(t)}{\partial\tau}\right|_{\tau=0}
  5. γ \gamma
  6. γ \gamma
  7. D 2 d t 2 J ( t ) + R ( J ( t ) , γ ˙ ( t ) ) γ ˙ ( t ) = 0 , \frac{D^{2}}{dt^{2}}J(t)+R(J(t),\dot{\gamma}(t))\dot{\gamma}(t)=0,
  8. γ ˙ ( t ) = d γ ( t ) / d t \dot{\gamma}(t)=d\gamma(t)/dt
  9. γ τ \gamma_{\tau}
  10. J J
  11. D d t J \frac{D}{dt}J
  12. γ \gamma
  13. γ ˙ ( t ) \dot{\gamma}(t)
  14. t γ ˙ ( t ) t\dot{\gamma}(t)
  15. γ τ ( t ) = γ ( τ + t ) \gamma_{\tau}(t)=\gamma(\tau+t)
  16. γ τ ( t ) = γ ( ( 1 + τ ) t ) \gamma_{\tau}(t)=\gamma((1+\tau)t)
  17. J J
  18. T + I T+I
  19. T = a γ ˙ ( t ) + b t γ ˙ ( t ) T=a\dot{\gamma}(t)+bt\dot{\gamma}(t)
  20. I ( t ) I(t)
  21. γ ˙ ( t ) \dot{\gamma}(t)
  22. t t
  23. I I
  24. J J
  25. γ 0 \gamma_{0}
  26. γ τ \gamma_{\tau}
  27. t [ 0 , π ] t\in[0,\pi]
  28. τ \tau
  29. d ( γ 0 ( t ) , γ τ ( t ) ) d(\gamma_{0}(t),\gamma_{\tau}(t))\,
  30. d ( γ 0 ( t ) , γ τ ( t ) ) = sin - 1 ( sin t sin τ 1 + cos 2 t tan 2 ( τ / 2 ) ) . d(\gamma_{0}(t),\gamma_{\tau}(t))=\sin^{-1}\bigg(\sin t\sin\tau\sqrt{1+\cos^{2% }t\tan^{2}(\tau/2)}\bigg).
  31. d ( γ 0 ( π ) , γ τ ( π ) ) = 0 d(\gamma_{0}(\pi),\gamma_{\tau}(\pi))=0\,
  32. τ \tau
  33. τ \tau
  34. τ = 0 \tau=0
  35. τ | τ = 0 d ( γ 0 ( t ) , γ τ ( t ) ) = | J ( t ) | = sin t . \frac{\partial}{\partial\tau}\bigg|_{\tau=0}d(\gamma_{0}(t),\gamma_{\tau}(t))=% |J(t)|=\sin t.
  36. t = π t=\pi
  37. d ( γ 0 ( t ) , γ τ ( t ) ) d(\gamma_{0}(t),\gamma_{\tau}(t))\,
  38. y ′′ + y = 0 y^{\prime\prime}+y=0\,
  39. e 1 ( 0 ) = γ ˙ ( 0 ) / | γ ˙ ( 0 ) | e_{1}(0)=\dot{\gamma}(0)/|\dot{\gamma}(0)|
  40. { e i ( 0 ) } \big\{e_{i}(0)\big\}
  41. T γ ( 0 ) M T_{\gamma(0)}M
  42. { e i ( t ) } \{e_{i}(t)\}
  43. γ \gamma
  44. e 1 ( t ) = γ ˙ ( t ) / | γ ˙ ( t ) | e_{1}(t)=\dot{\gamma}(t)/|\dot{\gamma}(t)|
  45. J ( t ) = y k ( t ) e k ( t ) J(t)=y^{k}(t)e_{k}(t)
  46. D d t J = k d y k d t e k ( t ) , D 2 d t 2 J = k d 2 y k d t 2 e k ( t ) , \frac{D}{dt}J=\sum_{k}\frac{dy^{k}}{dt}e_{k}(t),\quad\frac{D^{2}}{dt^{2}}J=% \sum_{k}\frac{d^{2}y^{k}}{dt^{2}}e_{k}(t),
  47. d 2 y k d t 2 + | γ ˙ | 2 j y j ( t ) R ( e j ( t ) , e 1 ( t ) ) e 1 ( t ) , e k ( t ) = 0 \frac{d^{2}y^{k}}{dt^{2}}+|\dot{\gamma}|^{2}\sum_{j}y^{j}(t)\langle R(e_{j}(t)% ,e_{1}(t))e_{1}(t),e_{k}(t)\rangle=0
  48. k k
  49. t t
  50. y k ( 0 ) y^{k}(0)
  51. y k ( 0 ) {y^{k}}^{\prime}(0)
  52. k k
  53. γ ( t ) \gamma(t)
  54. e i ( t ) e_{i}(t)
  55. e 1 ( t ) = γ ˙ ( t ) / | γ ˙ | e_{1}(t)=\dot{\gamma}(t)/|\dot{\gamma}|
  56. γ \gamma
  57. γ ˙ ( t ) \dot{\gamma}(t)
  58. t γ ˙ ( t ) t\dot{\gamma}(t)
  59. t t
  60. - k 2 -k^{2}
  61. γ ˙ ( t ) \dot{\gamma}(t)
  62. t γ ˙ ( t ) t\dot{\gamma}(t)
  63. exp ( ± k t ) e i ( t ) \exp(\pm kt)e_{i}(t)
  64. i > 1 i>1
  65. k 2 k^{2}
  66. γ ˙ ( t ) \dot{\gamma}(t)
  67. t γ ˙ ( t ) t\dot{\gamma}(t)
  68. sin ( k t ) e i ( t ) \sin(kt)e_{i}(t)
  69. cos ( k t ) e i ( t ) \cos(kt)e_{i}(t)
  70. i > 1 i>1
  71. T M TM
  72. M M

Jacobian_conjecture.html

  1. J F = | f 1 X 1 f 1 X N f N X 1 f N X N | , J_{F}=\left|\begin{matrix}\frac{\partial f_{1}}{\partial X_{1}}&\cdots&\frac{% \partial f_{1}}{\partial X_{N}}\\ \vdots&\ddots&\vdots\\ \frac{\partial f_{N}}{\partial X_{1}}&\cdots&\frac{\partial f_{N}}{\partial X_% {N}}\end{matrix}\right|,

Jakaltek_language.html

  1. 𝐧 ¨ \mathbf{\ddot{n}}

James_Hopwood_Jeans.html

  1. λ J = 15 k B T 4 π G m ρ \lambda_{J}=\sqrt{\frac{15k_{B}T}{4\pi Gm\rho}}
  2. f ( λ ) = 8 π c k B T λ 4 f(\lambda)=8\pi c\frac{k_{B}T}{\lambda^{4}}

James_Meade.html

  1. Y = F ( K , L , N , t ) Y=F(K,L,N,t)
  2. Y = Y=
  3. K = K=
  4. L = L=
  5. N = N=
  6. t = t=
  7. K K
  8. L L
  9. N N
  10. Y Y
  11. Δ Y \Delta Y
  12. Δ Y \Delta Y
  13. V Δ K V\Delta K
  14. V V
  15. L L
  16. Δ L \Delta L
  17. W W
  18. W Δ L W\Delta L
  19. Δ Y \Delta Y^{\prime}
  20. Δ Y = V Δ K + W Δ L + Δ Y \Delta Y=V\Delta K+W\Delta L+\Delta Y^{\prime}
  21. Y Y
  22. Δ Y Y = V Y . Δ K + W Y Δ L + Δ Y Y \frac{\Delta Y}{Y}=\frac{V}{Y}.\Delta K+\frac{W}{Y}\Delta L+\frac{\Delta Y^{% \prime}}{Y}
  23. Δ Y Y = V K Y . Δ K K + W L Y . Δ L L + Δ Y Y \frac{\Delta Y}{Y}=\frac{VK}{Y}.\frac{\Delta K}{K}+\frac{WL}{Y}.\frac{\Delta L% }{L}+\frac{\Delta Y^{\prime}}{Y}
  24. Δ Y Y \frac{\Delta Y}{Y}
  25. Δ K K \frac{\Delta K}{K}
  26. Δ L L \frac{\Delta L}{L}
  27. Δ Y Y \frac{\Delta Y^{\prime}}{Y}
  28. y , k , l y,k,l
  29. r r
  30. V K Y \frac{VK}{Y}
  31. U U
  32. W L Y \frac{WL}{Y}
  33. Q Q
  34. Y = U K + Q l + r Y=UK+Ql+r
  35. y y
  36. k k
  37. U U
  38. I I
  39. Q Q
  40. r r
  41. y - l = U k - ( 1 - Q ) L + r y-l=Uk-(1-Q)L+r
  42. y - l y-l
  43. k k
  44. U U
  45. l l
  46. l - Q l-Q
  47. r r
  48. U K UK
  49. V K Y . k \frac{VK}{Y}.k
  50. S Y K \frac{SY}{K}
  51. S S
  52. U k = U S . Y K = S V Uk=US.\frac{Y}{K}=SV
  53. y - l = U k - ( 1 - Q ) l + r y-l=Uk-(1-Q)l+r
  54. y - l = U S . Y K - ( 1 - Q ) l + r y-l=US.\frac{Y}{K}-(1-Q)l+r
  55. y - l = S V - ( 1 - Q ) l + r y-l=SV-(1-Q)l+r
  56. S = S=
  57. V = 5 V=5
  58. S V SV
  59. Y = 1000 Y=1000
  60. K = 2000 K=2000
  61. U = V K Y = ( 5 / 100 ) × 2000 1000 = 1 10 U=\frac{VK}{Y}=\frac{(5/100)\times 2000}{1000}=\frac{1}{10}
  62. U k = 1 10 × 5 = 1 2 Uk=\frac{1}{10}\times 5=\frac{1}{2}
  63. U U
  64. S S
  65. Y K \frac{Y}{K}
  66. U S . Y K = 1 10 × 1 10 × 1 2 = 1 2 US.\frac{Y}{K}=\frac{1}{10}\times\frac{1}{10}\times\frac{1}{2}=\frac{1}{2}

Jefferson_disk.html

  1. 10 ! = 3 , 628 , 800 10!=3,628,800

Jensen's_alpha.html

  1. α J = R i - [ R f + β i M ( R M - R f ) ] \alpha_{J}=R_{i}-[R_{f}+\beta_{iM}\cdot(R_{M}-R_{f})]
  2. α J = ( R i - R f ) - β i M ( R M - R f ) \alpha_{J}=(R_{i}-R_{f})-\beta_{iM}\cdot(R_{M}-R_{f})
  3. Δ R ( R i - R f ) \Delta_{R}\equiv(R_{i}-R_{f})
  4. Δ M ( R M - R f ) \Delta_{M}\equiv(R_{M}-R_{f})
  5. α J = Δ R - β i M Δ M \alpha_{J}=\Delta_{R}-\beta_{iM}\Delta_{M}

Jet_bundle.html

  1. | I | := i = 1 m I ( i ) |I|:=\sum_{i=1}^{m}I(i)
  2. | I | x I := i = 1 m ( x i ) I ( i ) . \frac{\partial^{|I|}}{\partial x^{I}}:=\prod_{i=1}^{m}\left(\frac{\partial}{% \partial x^{i}}\right)^{I(i)}.
  3. | I | σ α x I | p = | I | η α x I | p , 0 | I | r . \left.\frac{\partial^{|I|}\sigma^{\alpha}}{\partial x^{I}}\right|_{p}=\left.% \frac{\partial^{|I|}\eta^{\alpha}}{\partial x^{I}}\right|_{p},\quad 0\leq|I|% \leq r.
  4. j p r σ j^{r}_{p}\sigma
  5. { j p r σ : p M , σ Γ ( π ) } \{j^{r}_{p}\sigma:p\in M,\sigma\in\Gamma(\pi)\}
  6. { π r : J r ( π ) M j p r σ p \begin{cases}\pi_{r}:J^{r}(\pi)\to M\\ j^{r}_{p}\sigma\mapsto p\end{cases}
  7. { π r , 0 : J r ( π ) E j p r σ σ ( p ) \begin{cases}\pi_{r,0}:J^{r}(\pi)\to E\\ j^{r}_{p}\sigma\mapsto\sigma(p)\end{cases}
  8. { π r , k : J r ( π ) J k ( π ) j p r σ j p k σ \begin{cases}\pi_{r,k}:J^{r}(\pi)\to J^{k}(\pi)\\ j^{r}_{p}\sigma\mapsto j^{k}_{p}\sigma\end{cases}
  9. U r = { j p r σ : σ ( p ) U } U^{r}=\{j^{r}_{p}\sigma:\sigma(p)\in U\}\,
  10. u r = ( x i , u α , u I α ) u^{r}=(x^{i},u^{\alpha},u^{\alpha}_{I})\,
  11. x i ( j p r σ ) = x i ( p ) x^{i}(j^{r}_{p}\sigma)=x^{i}(p)
  12. u α ( j p r σ ) = u α ( σ ( p ) ) u^{\alpha}(j^{r}_{p}\sigma)=u^{\alpha}(\sigma(p))
  13. n ( C r m + r - 1 ) n\left({}^{m+r}C_{r}-1\right)\,
  14. u I α : U k 𝐑 u^{\alpha}_{I}:U^{k}\to\mathbf{R}\,
  15. u I α ( j p r σ ) = | I | σ α x I | p u^{\alpha}_{I}(j^{r}_{p}\sigma)=\left.\frac{\partial^{|I|}\sigma^{\alpha}}{% \partial x^{I}}\right|_{p}
  16. J r ( π | π - 1 ( W ) ) π r - 1 ( W ) . J^{r}\left(\pi|_{\pi^{-1}(W)}\right)\cong\pi^{-1}_{r}(W).\,
  17. π r - 1 ( p ) \pi^{-1}_{r}(p)\,
  18. J p r ( π ) J^{r}_{p}(\pi)
  19. ( j r σ ) ( p ) = j p r σ . (j^{r}\sigma)(p)=j^{r}_{p}\sigma.\,
  20. ( σ α , | I | σ α x | I | ) 1 | I | r . \left(\sigma^{\alpha},\frac{\partial^{|I|}\sigma^{\alpha}}{\partial x^{|I|}}% \right)\qquad 1\leq|I|\leq r.\,
  21. σ ¯ = p r 2 σ C ( M ) \bar{\sigma}=pr_{2}\circ\sigma\in C^{\infty}(M)\,
  22. j p 1 σ = { ψ : ψ Γ p ( π ) ; ψ ¯ ( p ) = σ ¯ ( p ) ; d ψ ¯ p = d σ ¯ p } . j^{1}_{p}\sigma=\{\psi:\psi\in\Gamma_{p}(\pi);\bar{\psi}(p)=\bar{\sigma}(p);d% \bar{\psi}_{p}=d\bar{\sigma}_{p}\}.\,
  23. { J 1 ( π ) T * M × 𝐑 j p 1 σ ( d σ ¯ p , σ ¯ ( p ) ) \begin{cases}J^{1}(\pi)\to T^{*}M\times\mathbf{R}\\ j^{1}_{p}\sigma\mapsto(d\bar{\sigma}_{p},\bar{\sigma}(p))\end{cases}
  24. Λ 1 J r ( π ) \Lambda^{1}J^{r}(\pi)
  25. Λ C r π \Lambda_{C}^{r}\pi
  26. θ Λ 1 J r π \theta\in\Lambda^{1}J^{r}\pi
  27. ( j r + 1 σ ) * θ = 0 (j^{r+1}\sigma)^{*}\theta=0
  28. x ( j p 1 σ ) x(j^{1}_{p}\sigma)
  29. = x ( p ) = x =x(p)=x\,
  30. u ( j p 1 σ ) u(j^{1}_{p}\sigma)
  31. = u ( σ ( p ) ) = u ( σ ( x ) ) = σ ( x ) =u(\sigma(p))=u(\sigma(x))=\sigma(x)\,
  32. u 1 ( j p 1 σ ) u_{1}(j^{1}_{p}\sigma)
  33. = σ x | p = σ ( x ) =\left.\frac{\partial\sigma}{\partial x}\right|_{p}=\sigma^{\prime}(x)
  34. θ = a ( x , u , u 1 ) d x + b ( x , u , u 1 ) d u + c ( x , u , u 1 ) d u 1 \theta=a(x,u,u_{1})dx+b(x,u,u_{1})du+c(x,u,u_{1})du_{1}\,
  35. j 1 σ = ( u , u 1 ) = ( σ ( p ) , σ x | p ) . j^{1}\sigma=(u,u_{1})=\left(\sigma(p),\left.\frac{\partial\sigma}{\partial x}% \right|_{p}\right).
  36. ( j p 1 σ ) * θ (j^{1}_{p}\sigma)^{*}\theta\,
  37. = θ j p 1 σ =\theta\circ j^{1}_{p}\sigma\,
  38. = a ( x , σ ( x ) , σ ( x ) ) d x + b ( x , σ ( x ) , σ ( x ) ) d ( σ ( x ) ) + c ( x , σ ( x ) , σ ( x ) ) d ( σ ( x ) ) =a(x,\sigma(x),\sigma^{\prime}(x))dx+b(x,\sigma(x),\sigma^{\prime}(x))d(\sigma% (x))+c(x,\sigma(x),\sigma^{\prime}(x))d(\sigma^{\prime}(x))\,
  39. = a ( x , σ ( x ) , σ ( x ) ) d x + b ( x , σ ( x ) , σ ( x ) ) σ ( x ) d x + c ( x , σ ( x ) , σ ( x ) ) σ ′′ ( x ) d x =a(x,\sigma(x),\sigma^{\prime}(x))dx+b(x,\sigma(x),\sigma^{\prime}(x))\sigma^{% \prime}(x)dx+c(x,\sigma(x),\sigma^{\prime}(x))\sigma^{\prime\prime}(x)dx\,
  40. = [ a ( x , σ ( x ) , σ ( x ) ) + b ( x , σ ( x ) , σ ( x ) ) σ ( x ) + c ( x , σ ( x ) , σ ( x ) ) σ ′′ ( x ) ] d x =[\,a(x,\sigma(x),\sigma^{\prime}(x))+b(x,\sigma(x),\sigma^{\prime}(x))\sigma^% {\prime}(x)+c(x,\sigma(x),\sigma^{\prime}(x))\sigma^{\prime\prime}(x)\,]dx\,
  41. u 2 ( j p 2 σ ) = 2 σ x 2 | p = σ ′′ ( x ) u_{2}(j^{2}_{p}\sigma)=\left.\frac{\partial^{2}\sigma}{\partial x^{2}}\right|_% {p}=\sigma^{\prime\prime}(x)\,
  42. θ = a ( x , u , u 1 , u 2 ) d x + b ( x , u , u 1 , u 2 ) d u + c ( x , u , u 1 , u 2 ) d u 1 + e ( x , u , u 1 , u 2 ) d u 2 \theta=a(x,u,u_{1},u_{2})dx+b(x,u,u_{1},u_{2})du+c(x,u,u_{1},u_{2})du_{1}+e(x,% u,u_{1},u_{2})du_{2}\,
  43. ( j p 2 σ ) * θ (j^{2}_{p}\sigma)^{*}\theta\,
  44. = θ j p 2 σ =\theta\circ j^{2}_{p}\sigma\,
  45. = a ( x , σ ( x ) , σ ( x ) , σ ′′ ( x ) ) d x + b ( x , σ ( x ) , σ ( x ) , σ ′′ ( x ) ) d ( σ ( x ) ) + =a(x,\sigma(x),\sigma^{\prime}(x),\sigma^{\prime\prime}(x))dx+b(x,\sigma(x),% \sigma^{\prime}(x),\sigma^{\prime\prime}(x))d(\sigma(x))+\,
  46. + c ( x , σ ( x ) , σ ( x ) , σ ( x ) ) d ( σ ( x ) ) + e ( x , σ ( x ) , σ ( x ) , σ ′′ ( x ) ) d ( σ ′′ ( x ) ) +c(x,\sigma(x),\sigma^{\prime}(x),\sigma^{\prime}(x))d(\sigma^{\prime}(x))+e(x% ,\sigma(x),\sigma^{\prime}(x),\sigma^{\prime\prime}(x))d(\sigma^{\prime\prime}% (x))\,
  47. = a d x + b σ ( x ) d x + c σ ′′ ( x ) d x + e σ ′′′ ( x ) d x =adx+b\sigma^{\prime}(x)dx+c\sigma^{\prime\prime}(x)dx+e\sigma^{\prime\prime% \prime}(x)dx\,
  48. = [ a + b σ ( x ) + c σ ′′ ( x ) + e σ ′′′ ( x ) ] d x =[\,a+b\sigma^{\prime}(x)+c\sigma^{\prime\prime}(x)+e\sigma^{\prime\prime% \prime}(x)\,]dx\,
  49. = 0 =0\,
  50. θ = b ( x , σ ( x ) , σ ( x ) ) θ 0 + c ( x , σ ( x ) , σ ( x ) ) θ 1 \theta=b(x,\sigma(x),\sigma^{\prime}(x))\theta_{0}+c(x,\sigma(x),\sigma^{% \prime}(x))\theta_{1}\,
  51. ( π 2 , 1 ) * θ 0 (\pi_{2,1})^{*}\theta_{0}\,
  52. θ k = d u k - u k + 1 d x k = 0 , , r - 1 \theta_{k}=du_{k}-u_{k+1}dx\qquad k=0,\ldots,r-1\,
  53. u k ( j k σ ) = k σ x k | p u_{k}(j^{k}\sigma)=\left.\frac{\partial^{k}\sigma}{\partial x^{k}}\right|_{p}\,
  54. θ = | I | = 0 r P α I θ I α \theta=\sum_{|I|=0}^{r}P_{\alpha}^{I}\theta_{I}^{\alpha}\,
  55. P I α ( x i , u α ) P^{\alpha}_{I}(x^{i},u^{\alpha})\,
  56. θ I α = d u I α - u I , i α d x i \theta_{I}^{\alpha}=du^{\alpha}_{I}-u^{\alpha}_{I,i}dx^{i}\,
  57. θ I α \theta_{I}^{\alpha}
  58. ψ * ( θ | W ) = 0 , θ Λ C 1 π r + 1 , r . \psi^{*}(\theta|_{W})=0,\forall\theta\in\Lambda_{C}^{1}\pi_{r+1,r}.\,
  59. ( x , u ) = def ( x i , u α ) (x,u)\ \stackrel{\mathrm{def}}{=}\ (x^{i},u^{\alpha})\,
  60. V = def ρ i ( x , u ) x i + ϕ α ( x , u ) u α . V\ \stackrel{\mathrm{def}}{=}\ \rho^{i}(x,u)\frac{\partial}{\partial x^{i}}+% \phi^{\alpha}(x,u)\frac{\partial}{\partial u^{\alpha}}.\,
  61. V ( x u ) = def ρ i ( x , u ) x i + ϕ α ( x , u ) u α V_{(xu)}\ \stackrel{\mathrm{def}}{=}\ \rho^{i}(x,u)\frac{\partial}{\partial x^% {i}}+\phi^{\alpha}(x,u)\frac{\partial}{\partial u^{\alpha}}\,
  62. { ψ : E T E ( x , u ) ψ ( x , u ) = V \begin{cases}\psi:E\to TE\\ (x,u)\mapsto\psi(x,u)=V\end{cases}
  63. V = ρ i ( x , u ) x i + ϕ α ( x , u ) u α V=\rho^{i}(x,u)\frac{\partial}{\partial x^{i}}+\phi^{\alpha}(x,u)\frac{% \partial}{\partial u^{\alpha}}\,
  64. ( x , u , w ) = def ( x i , u α , w i α ) (x,u,w)\ \stackrel{\mathrm{def}}{=}\ (x^{i},u^{\alpha},w_{i}^{\alpha})\,
  65. V ( x u w ) = def V_{(xuw)}\ \stackrel{\mathrm{def}}{=}\ \,
  66. V i ( x , u , w ) x i + V α ( x , u , w ) u α + V i α ( x , u , w ) w i α + V^{i}(x,u,w)\frac{\partial}{\partial x^{i}}+V^{\alpha}(x,u,w)\frac{\partial}{% \partial u^{\alpha}}\ +\ V^{\alpha}_{i}(x,u,w)\frac{\partial}{\partial w^{% \alpha}_{i}}+\,
  67. + V i 1 i 2 α ( x , u , w ) w i 1 i 2 α + + + V i 1 i 2 i r α ( x , u , w ) w i 1 i 2 i r α \qquad+\ V^{\alpha}_{i_{1}i_{2}}(x,u,w)\frac{\partial}{\partial w^{\alpha}_{i_% {1}i_{2}}}+\cdots\ +\ \cdots+V^{\alpha}_{i_{1}i_{2}\cdots i_{r}}(x,u,w)\frac{% \partial}{\partial w^{\alpha}_{i_{1}i_{2}\cdots i_{r}}}\,
  68. ( x , u , w , v i α , v i 1 i 2 α , , v i 1 i 2 i r α ) (x,u,w,v^{\alpha}_{i},v^{\alpha}_{i_{1}i_{2}},\ldots,v^{\alpha}_{i_{1}i_{2}% \cdots i_{r}})\,
  69. T x u w ( J r π ) T_{xuw}(J^{r}\pi)\,
  70. v i α , v i 1 i 2 α , , v i 1 i 2 i r α v^{\alpha}_{i},v^{\alpha}_{i_{1}i_{2}},\ldots,v^{\alpha}_{i_{1}i_{2}\cdots i_{% r}}\,
  71. { Ψ : J r ( π ) T J r ( π ) ( x , u , w ) Ψ ( u , w ) = V \begin{cases}\Psi:J^{r}(\pi)\to TJ^{r}(\pi)\\ (x,u,w)\mapsto\Psi(u,w)=V\end{cases}
  72. Ψ Γ ( T ( J r π ) ) \Psi\in\Gamma(T(J^{r}\pi))\,
  73. j p r σ S j^{r}_{p}\sigma\in S
  74. F = u 1 1 u 2 1 - 2 x 2 u 1 F=u^{1}_{1}u^{1}_{2}-2x^{2}u^{1}\,
  75. S = { j p 1 σ J 1 π : ( u 1 1 u 2 1 - 2 x 2 u 1 ) ( j p 1 σ ) = 0 } S=\{j^{1}_{p}\sigma\in J^{1}\pi:(u^{1}_{1}u^{1}_{2}-2x^{2}u^{1})(j^{1}_{p}% \sigma)=0\}\,
  76. σ x 1 σ x 2 - 2 x 2 σ = 0. \frac{\partial\sigma}{\partial x^{1}}\frac{\partial\sigma}{\partial x^{2}}-2x^% {2}\sigma=0.\,
  77. σ ( p 1 , p 2 ) = ( p 1 , p 2 , p 1 ( p 2 ) 2 ) \sigma(p_{1},p_{2})=(p^{1},p^{2},p^{1}(p^{2})^{2})\,
  78. j 1 σ ( p 1 , p 2 ) = ( p 1 , p 2 , p 1 ( p 2 ) 2 , ( p 2 ) 2 , 2 p 1 p 2 ) j^{1}\sigma(p_{1},p_{2})=(p^{1},p^{2},p^{1}(p^{2})^{2},(p^{2})^{2},2p^{1}p^{2})\,
  79. ( u 1 1 u 2 1 - 2 x 2 u 1 ) ( j p 1 σ ) (u^{1}_{1}u^{1}_{2}-2x^{2}u^{1})(j^{1}_{p}\sigma)\,
  80. = u 1 1 ( j p 1 σ ) u 2 1 ( j p 1 σ ) - 2 x 2 ( j p 1 σ ) u 1 ( j p 1 σ ) =u^{1}_{1}(j^{1}_{p}\sigma)u^{1}_{2}(j^{1}_{p}\sigma)-2x^{2}(j^{1}_{p}\sigma)u% ^{1}(j^{1}_{p}\sigma)\,
  81. = ( p 2 ) 2 2 p 1 p 2 - 2 p 2 p 1 ( p 2 ) 2 =(p^{2})^{2}\cdot 2p^{1}p^{2}-2\cdot p^{2}\cdot p^{1}(p^{2})^{2}\,
  82. = 2 p 1 ( p 2 ) 3 - 2 p 1 ( p 2 ) 3 =2p^{1}(p^{2})^{3}-2p^{1}(p^{2})^{3}\,
  83. = 0 =0\,
  84. j p 1 σ S j^{1}_{p}\sigma\in S
  85. V r ( θ ) \mathcal{L}_{V^{r}}(\theta)
  86. V 1 = def ρ i ( u 1 ) x i + ϕ α ( u 1 ) u α + χ i α ( u 1 ) u i α . V^{1}\ \stackrel{\mathrm{def}}{=}\ \rho^{i}(u^{1})\frac{\partial}{\partial x^{% i}}+\phi^{\alpha}(u^{1})\frac{\partial}{\partial u^{\alpha}}+\chi^{\alpha}_{i}% (u^{1})\frac{\partial}{\partial u^{\alpha}_{i}}.
  87. V 1 \mathcal{L}_{V^{1}}
  88. θ α = d u α - u i α d x i \theta^{\alpha}=du^{\alpha}-u_{i}^{\alpha}dx^{i}\,
  89. V 1 ( θ α ) \mathcal{L}_{V^{1}}(\theta^{\alpha})
  90. = V 1 ( d u α - u i α d x i ) =\mathcal{L}_{V^{1}}(du^{\alpha}-u_{i}^{\alpha}dx^{i})
  91. = V 1 d u α - ( V 1 u i α ) d x i - u i α ( V 1 d x i ) =\mathcal{L}_{V^{1}}du^{\alpha}-(\mathcal{L}_{V^{1}}u_{i}^{\alpha})dx^{i}-u_{i% }^{\alpha}(\mathcal{L}_{V^{1}}dx^{i})\,
  92. = d ( V 1 u α ) - V 1 u i α d x i - u i α d ( V 1 x i ) =d(V^{1}u^{\alpha})-V^{1}u_{i}^{\alpha}dx^{i}-u_{i}^{\alpha}d(V^{1}x^{i})\,
  93. = d ϕ α - χ i α d x i - u i α d ρ i =d\phi^{\alpha}-\chi^{\alpha}_{i}dx^{i}-u_{i}^{\alpha}d\rho^{i}\,
  94. = ϕ α x i d x i + ϕ α u k d u k + ϕ α u i k d u i k - χ i α d x i - u i α [ ρ i x m d x m + ρ i u k d u k + ρ i u m k d u m k ] =\frac{\partial\phi^{\alpha}}{\partial x^{i}}\,dx^{i}+\frac{\partial\phi^{% \alpha}}{\partial u^{k}}\,du^{k}+\frac{\partial\phi^{\alpha}}{\partial u^{k}_{% i}}\,du^{k}_{i}-\chi^{\alpha}_{i}dx^{i}-u_{i}^{\alpha}\left[\frac{\partial\rho% ^{i}}{\partial x^{m}}\,dx^{m}+\frac{\partial\rho^{i}}{\partial u^{k}}\,du^{k}+% \frac{\partial\rho^{i}}{\partial u^{k}_{m}}\,du^{k}_{m}\right]\,
  95. θ k = d u k - u i k d x i d u k = θ k + u i k d x i \theta^{k}=du^{k}-u_{i}^{k}dx^{i}\quad\Longrightarrow\quad du^{k}=\theta^{k}+u% _{i}^{k}dx^{i}\,
  96. V 1 ( θ α ) \mathcal{L}_{V^{1}}(\theta^{\alpha})\,
  97. = ϕ α x i d x i + ϕ α u k ( θ k + u i k d x i ) + ϕ α u i k d u i k - χ i α d x i - =\frac{\partial\phi^{\alpha}}{\partial x^{i}}\,dx^{i}+\frac{\partial\phi^{% \alpha}}{\partial u^{k}}\,(\theta^{k}+u_{i}^{k}dx^{i})+\frac{\partial\phi^{% \alpha}}{\partial u^{k}_{i}}\,du^{k}_{i}-\chi^{\alpha}_{i}dx^{i}-\,
  98. - u l α [ ρ l x i d x i + ρ l u k ( θ k + u i k d x i ) + ρ l u i k d u i k ] -u_{l}^{\alpha}\left[\frac{\partial\rho^{l}}{\partial x^{i}}\,dx^{i}+\frac{% \partial\rho^{l}}{\partial u^{k}}\,(\theta^{k}+u_{i}^{k}dx^{i})+\frac{\partial% \rho^{l}}{\partial u^{k}_{i}}\,du^{k}_{i}\right]\,
  99. = [ ϕ α x i + ϕ α u k u i k - u l α ( ρ l x i + ρ l u k u i k ) - χ i α ] d x i + [ ϕ α u i k - u l α ρ l u i k ] d u i k + =\left[\frac{\partial\phi^{\alpha}}{\partial x^{i}}+\frac{\partial\phi^{\alpha% }}{\partial u^{k}}u_{i}^{k}-u_{l}^{\alpha}\left(\frac{\partial\rho^{l}}{% \partial x^{i}}+\frac{\partial\rho^{l}}{\partial u^{k}}u_{i}^{k}\right)-\chi^{% \alpha}_{i}\right]\,dx^{i}+\left[\frac{\partial\phi^{\alpha}}{\partial u^{k}_{% i}}-u_{l}^{\alpha}\frac{\partial\rho^{l}}{\partial u^{k}_{i}}\right]\,du^{k}_{% i}+\,
  100. + ( ϕ α u k - u l α ρ l u k ) θ k . +\left(\frac{\partial\phi^{\alpha}}{\partial u^{k}}-u_{l}^{\alpha}\frac{% \partial\rho^{l}}{\partial u^{k}}\right)\theta^{k}.\,
  101. d u i k du^{k}_{i}\,
  102. ϕ α u i k - u l α ρ l u i k = 0 \frac{\partial\phi^{\alpha}}{\partial u^{k}_{i}}-u^{\alpha}_{l}\frac{\partial% \rho^{l}}{\partial u^{k}_{i}}=0\,
  103. χ i α = D ^ i ϕ α - u l α ( D ^ i ρ l ) \chi^{\alpha}_{i}=\widehat{D}_{i}\phi^{\alpha}-u^{\alpha}_{l}(\widehat{D}_{i}% \rho^{l})
  104. D ^ i = x i + u i k u k \widehat{D}_{i}=\frac{\partial}{\partial x^{i}}+u^{k}_{i}\frac{\partial}{% \partial u^{k}}
  105. V r \mathcal{L}_{V^{r}}\,
  106. x ( j p 1 σ ) x(j^{1}_{p}\sigma)\,
  107. = x ( p ) = x =x(p)=x\,
  108. u ( j p 1 σ ) u(j^{1}_{p}\sigma)\,
  109. = u ( σ ( p ) ) = u ( σ ( x ) ) = σ ( x ) =u(\sigma(p))=u(\sigma(x))=\sigma(x)\,
  110. u 1 ( j p 1 σ ) u_{1}(j^{1}_{p}\sigma)\,
  111. = σ x | p = σ ˙ ( x ) =\left.\frac{\partial\sigma}{\partial x}\right|_{p}=\dot{\sigma}(x)\,
  112. θ = d u - u 1 d x \theta=du-u_{1}dx\,
  113. V = x u - u x V=x\frac{\partial}{\partial u}-u\frac{\partial}{\partial x}\,
  114. V 1 V^{1}\,
  115. = V + Z =V+Z\,
  116. = x u - u x + Z =x\frac{\partial}{\partial u}-u\frac{\partial}{\partial x}+Z\,
  117. = x u - u x + ρ ( x , u , u 1 ) u 1 =x\frac{\partial}{\partial u}-u\frac{\partial}{\partial x}+\rho(x,u,u_{1})% \frac{\partial}{\partial u_{1}}\,
  118. V 1 ( θ ) \mathcal{L}_{V^{1}}(\theta)\,
  119. V 1 ( θ ) \mathcal{L}_{V^{1}}(\theta)\,
  120. = V 1 ( d u - u 1 d x ) =\mathcal{L}_{V^{1}}(du-u_{1}dx)\,
  121. = V 1 d u - ( V 1 u 1 ) d x - u 1 ( V 1 d x ) =\mathcal{L}_{V^{1}}du-(\mathcal{L}_{V^{1}}u_{1})dx-u_{1}(\mathcal{L}_{V^{1}}% dx)\,
  122. = d ( V 1 u ) - V 1 u 1 d x - u 1 d ( V 1 x ) =d(V^{1}u)-V^{1}u_{1}dx-u_{1}d(V^{1}x)\,
  123. = d x - ρ ( x , u , u 1 ) d x + u 1 d u =dx-\rho(x,u,u_{1})dx+u_{1}du\,
  124. = ( 1 - ρ ( x , u , u 1 ) ) d x + u 1 d u =(1-\rho(x,u,u_{1}))dx+u_{1}du\,
  125. V 1 ( θ ) \mathcal{L}_{V^{1}}(\theta)\,
  126. = [ 1 - ρ ( x , u , u 1 ) ] d x + u 1 ( θ + u 1 d x ) =[\,1-\rho(x,u,u_{1})\,]dx+u_{1}(\theta+u_{1}dx)\,
  127. = [ 1 + u 1 u 1 - ρ ( x , u , u 1 ) ] d x + u 1 θ =[\,1+u_{1}u_{1}-\rho(x,u,u_{1})\,]dx+u_{1}\theta\,
  128. V 1 ( θ ) \mathcal{L}_{V^{1}}(\theta)\,
  129. 1 + u 1 u 1 - ρ ( x , u , u 1 ) = 0 1+u_{1}u_{1}-\rho(x,u,u_{1})=0\,
  130. \Longrightarrow\quad\,
  131. ρ ( x , u , u 1 ) = 1 + u 1 u 1 \rho(x,u,u_{1})=1+u_{1}u_{1}\,
  132. V 1 = x u - u x + ( 1 + u 1 u 1 ) u 1 V^{1}=x\frac{\partial}{\partial u}-u\frac{\partial}{\partial x}+(1+u_{1}u_{1})% \frac{\partial}{\partial u_{1}}\,
  133. { x , u , u 1 , y 2 } \{x,u,u_{1},y_{2}\}\,
  134. V 2 = x u - u x + ρ ( x , u , u 1 , u 2 ) u 1 + ϕ ( x , u , u 1 , u 2 ) u 2 V^{2}=x\frac{\partial}{\partial u}-u\frac{\partial}{\partial x}+\rho(x,u,u_{1}% ,u_{2})\frac{\partial}{\partial u_{1}}+\phi(x,u,u_{1},u_{2})\frac{\partial}{% \partial u_{2}}\,
  135. θ \theta\,
  136. = d u - u 1 d x =du-u_{1}dx\,
  137. θ 1 \theta_{1}\,
  138. = d u 1 - u 2 d x =du_{1}-u_{2}dx\,
  139. V 2 ( θ ) \mathcal{L}_{V^{2}}(\theta)\,
  140. = 0 =0\,
  141. V 2 ( θ 1 ) \mathcal{L}_{V^{2}}(\theta_{1})\,
  142. = 0 =0\,
  143. ρ ( x , u , u 1 ) = 1 + u 1 u 1 \rho(x,u,u_{1})=1+u_{1}u_{1}\,
  144. V 2 V^{2}\,
  145. = V 1 + ϕ ( x , u , u 1 , u 2 ) u 2 =V^{1}+\phi(x,u,u_{1},u_{2})\frac{\partial}{\partial u_{2}}\,
  146. = x u - u x + ( 1 + u 1 u 1 ) u 1 + ϕ ( x , u , u 1 , u 2 ) u 2 =x\frac{\partial}{\partial u}-u\frac{\partial}{\partial x}+(1+u_{1}u_{1})\frac% {\partial}{\partial u_{1}}+\phi(x,u,u_{1},u_{2})\frac{\partial}{\partial u_{2}}\,
  147. V 2 ( θ 1 ) \mathcal{L}_{V^{2}}(\theta_{1})\,
  148. = V 2 ( d u 1 - u 2 d x ) =\mathcal{L}_{V^{2}}(du_{1}-u_{2}dx)\,
  149. = V 2 d u 1 - ( V 2 u 2 ) d x - u 2 ( V 2 d x ) =\mathcal{L}_{V^{2}}du_{1}-(\mathcal{L}_{V^{2}}u_{2})dx-u_{2}(\mathcal{L}_{V^{% 2}}dx)\,
  150. = d ( V 2 u 1 ) - V 2 u 2 d x - u 2 d ( V 2 x ) =d(V^{2}u_{1})-V^{2}u_{2}dx-u_{2}d(V^{2}x)\,
  151. = d ( 1 - u 1 u 1 ) - ϕ ( x , u , u 1 , u 2 ) d x + u 2 d u =d(1-u_{1}u_{1})-\phi(x,u,u_{1},u_{2})dx+u_{2}du\,
  152. = 2 u 1 d u 1 - ϕ ( x , u , u 1 , u 2 ) d x + u 2 d u =2u_{1}du_{1}-\phi(x,u,u_{1},u_{2})dx+u_{2}du\,
  153. V 2 ( θ 1 ) \mathcal{L}_{V^{2}}(\theta_{1})\,
  154. = 2 u 1 ( θ 1 + u 2 d x ) - ϕ ( x , u , u 1 , u 2 ) d x + u 2 ( θ + u 1 d x ) =2u_{1}(\theta_{1}+u_{2}dx)-\phi(x,u,u_{1},u_{2})dx+u_{2}(\theta+u_{1}dx)\,
  155. = [ 3 u 1 u 2 - ϕ ( x , u , u 1 , u 2 ) ] d x + u 2 θ + 2 u 1 θ 1 =[\,3u_{1}u_{2}-\phi(x,u,u_{1},u_{2})\,]dx+u_{2}\theta+2u_{1}\theta_{1}\,
  156. V 2 ( θ 1 ) \mathcal{L}_{V^{2}}(\theta_{1})\,
  157. 3 u 1 u 2 - ϕ ( x , u , u 1 , u 2 ) = 0 3u_{1}u_{2}-\phi(x,u,u_{1},u_{2})=0\,
  158. \Longrightarrow\quad\,
  159. ϕ ( x , u , u 1 , u 2 ) = 3 u 1 u 2 \phi(x,u,u_{1},u_{2})=3u_{1}u_{2}\,
  160. V 2 = x u - u x + ( 1 + u 1 u 1 ) u 1 + 3 u 1 u 2 u 2 V^{2}=x\frac{\partial}{\partial u}-u\frac{\partial}{\partial x}+(1+u_{1}u_{1})% \frac{\partial}{\partial u_{1}}+3u_{1}u_{2}\frac{\partial}{\partial u_{2}}\,
  161. π k + 1 , k : J k + 1 ( π ) J k ( π ) \pi_{k+1,k}:J^{k+1}(\pi)\to J^{k}(\pi)
  162. j p ( σ ) j_{p}^{\infty}(\sigma)
  163. j p ( σ ) j_{p}^{\infty}(\sigma)
  164. π k + 1 , k : J k + 1 ( π ) J k ( π ) \pi_{k+1,k}:J^{k+1}(\pi)\to J^{k}(\pi)
  165. π k + 1 , k * : C ( J k ( π ) ) C ( J k + 1 ( π ) ) \pi_{k+1,k}^{*}:C^{\infty}(J^{k}(\pi))\to C^{\infty}(J^{k+1}(\pi))
  166. C ( J k ( π ) ) C^{\infty}(J^{k}(\pi))
  167. k ( π ) \mathcal{F}_{k}(\pi)
  168. ( π ) \mathcal{F}(\pi)
  169. k ( π ) \mathcal{F}_{k}(\pi)
  170. ( π ) \mathcal{F}(\pi)
  171. φ ( π ) \varphi\in\mathcal{F}(\pi)
  172. k ( π ) \mathcal{F}_{k}(\pi)
  173. k ( π ) \mathcal{F}_{k}(\pi)
  174. ( π ) \mathcal{F}(\pi)
  175. ( π ) \mathcal{F}(\pi)
  176. j p ( σ ) j_{p}^{\infty}(\sigma)
  177. j p k ( σ ) j_{p}^{k}(\sigma)
  178. φ j k ( σ ) \varphi\circ j^{k}(\sigma)
  179. 𝒞 \mathcal{C}
  180. 𝒞 \mathcal{C}
  181. ( E ( ) , 𝒞 | E ( ) ) (E_{(\infty)},\mathcal{C}|_{E_{(\infty)}})

Johann_Heinrich_von_Thünen.html

  1. R = Y ( p - c ) - Y F m R=Y(p-c)-YFm\,
  2. L = Y ( P - C ) - Y D F L=Y(P-C)-YDF
  3. k m 2 km^{2}
  4. t / k m 2 t/km^{2}
  5. D M / t DM/t
  6. D M / t DM/t
  7. k m km
  8. D M / t / k m DM/t/km
  9. t / k m 2 t/km^{2}

Johann_Jakob_Balmer.html

  1. λ = h m 2 m 2 - n 2 \lambda\ =\frac{hm^{2}}{m^{2}-n^{2}}
  2. 1 λ = 4 h ( 1 n 1 2 - 1 n 2 2 ) = R H ( 1 n 1 2 - 1 n 2 2 ) \frac{1}{\lambda}\ =\frac{4}{h}\left(\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}}% \right)=R_{H}\left(\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}}\right)
  3. R H R_{H}
  4. n 1 = 2 n_{1}=2
  5. n 2 > n 1 n_{2}>n_{1}

John_James_Waterston.html

  1. v 2 ¯ \bar{v^{2}}
  2. P = N M v 2 ¯ P=NM\bar{v^{2}}
  3. P V T = a constant \frac{PV}{T}=\mbox{a constant}~{}

Join_(SQL).html

  1. \bowtie
  2. \bowtie
  3. \bowtie
  4. R S = { t s | t R s S 𝐹𝑢𝑛 ( t s ) } R\bowtie S=\left\{t\cup s\ |\ t\in R\ \land\ s\in S\ \land\ \mathit{Fun}(t\cup s% )\right\}
  5. T = ρ x 1 / c 1 , , x m / c m ( S ) = ρ x 1 / c 1 ( ρ x 2 / c 2 ( ρ x m / c m ( S ) ) ) T=\rho_{x_{1}/c_{1},\ldots,x_{m}/c_{m}}(S)=\rho_{x_{1}/c_{1}}(\rho_{x_{2}/c_{2% }}(\ldots\rho_{x_{m}/c_{m}}(S)\ldots))
  6. P = σ c 1 = x 1 , , c m = x m ( R × T ) = σ c 1 = x 1 ( σ c 2 = x 2 ( σ c m = x m ( R × T ) ) ) P=\sigma_{c_{1}=x_{1},\ldots,c_{m}=x_{m}}(R\times T)=\sigma_{c_{1}=x_{1}}(% \sigma_{c_{2}=x_{2}}(\ldots\sigma_{c_{m}=x_{m}}(R\times T)\ldots))
  7. U = π r 1 , , r n , c 1 , , c m , s 1 , , s k ( P ) U=\pi_{r_{1},\ldots,r_{n},c_{1},\ldots,c_{m},s_{1},\ldots,s_{k}}(P)

Joint_entropy.html

  1. X X
  2. Y Y
  3. H ( X , Y ) = - x y P ( x , y ) log 2 [ P ( x , y ) ] H(X,Y)=-\sum_{x}\sum_{y}P(x,y)\log_{2}[P(x,y)]\!
  4. x x
  5. y y
  6. X X
  7. Y Y
  8. P ( x , y ) P(x,y)
  9. P ( x , y ) log 2 [ P ( x , y ) ] P(x,y)\log_{2}[P(x,y)]
  10. P ( x , y ) = 0 P(x,y)=0
  11. X 1 , , X n X_{1},...,X_{n}
  12. H ( X 1 , , X n ) = - x 1 x n P ( x 1 , , x n ) log 2 [ P ( x 1 , , x n ) ] H(X_{1},...,X_{n})=-\sum_{x_{1}}...\sum_{x_{n}}P(x_{1},...,x_{n})\log_{2}[P(x_% {1},...,x_{n})]\!
  13. x 1 , , x n x_{1},...,x_{n}
  14. X 1 , , X n X_{1},...,X_{n}
  15. P ( x 1 , , x n ) P(x_{1},...,x_{n})
  16. P ( x 1 , , x n ) log 2 [ P ( x 1 , , x n ) ] P(x_{1},...,x_{n})\log_{2}[P(x_{1},...,x_{n})]
  17. P ( x 1 , , x n ) = 0 P(x_{1},...,x_{n})=0
  18. H ( X , Y ) max [ H ( X ) , H ( Y ) ] H(X,Y)\geq\max[H(X),H(Y)]
  19. H ( X 1 , , X n ) max [ H ( X 1 ) , , H ( X n ) ] H(X_{1},...,X_{n})\geq\max[H(X_{1}),...,H(X_{n})]
  20. X X
  21. Y Y
  22. H ( X , Y ) H ( X ) + H ( Y ) H(X,Y)\leq H(X)+H(Y)
  23. H ( X 1 , , X n ) H ( X 1 ) + + H ( X n ) H(X_{1},...,X_{n})\leq H(X_{1})+...+H(X_{n})
  24. H ( X | Y ) = H ( Y , X ) - H ( Y ) H(X|Y)=H(Y,X)-H(Y)\,
  25. I ( X ; Y ) = H ( X ) + H ( Y ) - H ( X , Y ) I(X;Y)=H(X)+H(Y)-H(X,Y)\,

Joint_Institute_for_Nuclear_Research.html

  1. π - + C Σ ¯ - + K 0 + K ¯ 0 + K - + p + + π + + π - + nucleus recoil \pi^{-}+C\to\bar{\Sigma}^{-}+K^{0}+\bar{K}^{0}+K^{-}+p^{+}+\pi^{+}+\pi^{-}+% \hbox{nucleus recoil}
  2. Σ ¯ - n ¯ 0 + π - \bar{\Sigma}^{-}\to\bar{n}^{0}+\pi^{-}

Joint_probability_distribution.html

  1. ( A = 0 , B = 0 ) , ( A = 0 , B = 1 ) , ( A = 1 , B = 0 ) , ( A = 1 , B = 1 ) (A=0,B=0),(A=0,B=1),(A=1,B=0),(A=1,B=1)
  2. P ( A , B ) = 1 / 4 P(A,B)=1/4
  3. A , B { 0 , 1 } A,B\in\{0,1\}
  4. P ( A , B ) = P ( A ) P ( B ) P(A,B)=P(A)P(B)
  5. P ( A = 0 , B = 0 ) = P { 1 } = 1 6 , P ( A = 1 , B = 0 ) = P { 4 , 6 } = 2 6 , \mathrm{P}(A=0,B=0)=P\{1\}=\frac{1}{6},\;\mathrm{P}(A=1,B=0)=P\{4,6\}=\frac{2}% {6},
  6. P ( A = 0 , B = 1 ) = P { 3 , 5 } = 2 6 , P ( A = 1 , B = 1 ) = P { 2 } = 1 6 . \mathrm{P}(A=0,B=1)=P\{3,5\}=\frac{2}{6},\;\mathrm{P}(A=1,B=1)=P\{2\}=\frac{1}% {6}.
  7. X , Y X,Y
  8. P ( X = x and Y = y ) = P ( Y = y X = x ) P ( X = x ) = P ( X = x Y = y ) P ( Y = y ) . \begin{aligned}\displaystyle\mathrm{P}(X=x\ \mathrm{and}\ Y=y)=\mathrm{P}(Y=y% \mid X=x)\cdot\mathrm{P}(X=x)=\mathrm{P}(X=x\mid Y=y)\cdot\mathrm{P}(Y=y)\end{% aligned}.
  9. n n\,
  10. X 1 , X 2 , , X n X_{1},X_{2},\dots,X_{n}
  11. P ( X 1 = x 1 , , X n = x n ) \displaystyle\mathrm{P}(X_{1}=x_{1},\dots,X_{n}=x_{n})
  12. i j P ( X = x i and Y = y j ) = 1 , \sum_{i}\sum_{j}\mathrm{P}(X=x_{i}\ \mathrm{and}\ Y=y_{j})=1,\,
  13. n n\,
  14. X 1 , X 2 , , X n X_{1},X_{2},\dots,X_{n}
  15. i j k P ( X 1 = x 1 i , X 2 = x 2 j , , X n = x n k ) = 1. \sum_{i}\sum_{j}\dots\sum_{k}\mathrm{P}(X_{1}=x_{1i},X_{2}=x_{2j},\dots,X_{n}=% x_{nk})=1.\;
  16. f X , Y ( x , y ) = f Y X ( y x ) f X ( x ) = f X Y ( x y ) f Y ( y ) f_{X,Y}(x,y)=f_{Y\mid X}(y\mid x)f_{X}(x)=f_{X\mid Y}(x\mid y)f_{Y}(y)\;
  17. x y f X , Y ( x , y ) d y d x = 1. \int_{x}\int_{y}f_{X,Y}(x,y)\;dy\;dx=1.
  18. f X , Y ( x , y ) = f X Y ( x y ) P ( Y = y ) = P ( Y = y X = x ) f X ( x ) \displaystyle f_{X,Y}(x,y)=f_{X\mid Y}(x\mid y)\mathrm{P}(Y=y)=\mathrm{P}(Y=y% \mid X=x)f_{X}(x)
  19. F X , Y ( x , y ) \displaystyle F_{X,Y}(x,y)
  20. X X
  21. Y Y
  22. P ( X = x and Y = y ) = P ( X = x ) P ( Y = y ) \ P(X=x\ \mbox{and}~{}\ Y=y)=P(X=x)\cdot P(Y=y)
  23. f X , Y ( x , y ) = f X ( x ) f Y ( y ) \ f_{X,Y}(x,y)=f_{X}(x)\cdot f_{Y}(y)
  24. A A
  25. X 1 , , X n X_{1},\cdots,X_{n}
  26. B B
  27. P ( X 1 , , X n ) \mathrm{P}(X_{1},\ldots,X_{n})
  28. P ( B ) P ( A B ) P(B)\cdot P(A\mid B)
  29. P ( B ) P(B)
  30. P ( A B ) P(A\mid B)
  31. F ( x , y ) = P ( X x , Y y ) . F(x,y)=P(X\leq x,Y\leq y).

Joint_quantum_entropy.html

  1. ρ \rho
  2. σ \sigma
  3. S ( ρ , σ ) S(\rho,\sigma)
  4. H ( ρ , σ ) H(\rho,\sigma)
  5. S ( ρ , σ ) S(\rho,\sigma)
  6. X X
  7. H ( X ) H(X)
  8. X X
  9. X X
  10. X X
  11. H ( X ) = 0 H(X)=0
  12. X X
  13. n n
  14. X X
  15. H ( X ) = log 2 ( n ) H(X)=\log_{2}(n)
  16. ρ \rho
  17. - Tr ρ log ρ . -\operatorname{Tr}\rho\log\rho.
  18. S ( ρ ) S(\rho)
  19. H ( ρ ) H(\rho)
  20. ρ A B \rho^{AB}
  21. S ( ρ A , ρ B ) = S ( ρ A B ) = - Tr ( ρ A B log ( ρ A B ) ) . S(\rho^{A},\rho^{B})=S(\rho^{AB})=-\operatorname{Tr}(\rho^{AB}\log(\rho^{AB})).
  22. ρ A B \rho^{AB}
  23. ρ A B \rho^{AB}
  24. | Ψ = 1 2 ( | 00 + | 11 ) , \left|\Psi\right\rangle=\frac{1}{\sqrt{2}}\left(|00\rangle+|11\rangle\right),
  25. log 2 = 1 \log 2=1
  26. S ( ρ A B ) S(\rho^{AB})
  27. S ( ρ A | ρ B ) = def S ( ρ A , ρ B ) - S ( ρ B ) S(\rho^{A}|\rho^{B})\ \stackrel{\mathrm{def}}{=}\ S(\rho^{A},\rho^{B})-S(\rho^% {B})
  28. I ( ρ A : ρ B ) = def S ( ρ A ) + S ( ρ B ) - S ( ρ A , ρ B ) I(\rho^{A}:\rho^{B})\ \stackrel{\mathrm{def}}{=}\ S(\rho^{A})+S(\rho^{B})-S(% \rho^{A},\rho^{B})

Jones_polynomial.html

  1. t 1 / 2 t^{1/2}
  2. L L
  3. V ( L ) V(L)
  4. \langle~{}\rangle
  5. A A
  6. X ( L ) = ( - A 3 ) - w ( L ) L X(L)=(-A^{3})^{-w(L)}\langle L\rangle
  7. w ( L ) w(L)
  8. L L
  9. L + L_{+}
  10. L - L_{-}
  11. X ( L ) X(L)
  12. L L
  13. - A ± 3 -A^{\pm 3}
  14. X X
  15. A = t - 1 / 4 A=t^{-1/4}
  16. X ( L ) X(L)
  17. V ( L ) V(L)
  18. t 1 / 2 t^{1/2}
  19. ρ \rho
  20. [ A , A - 1 ] \mathbb{Z}[A,A^{-1}]
  21. δ = - A 2 - A - 2 \delta=-A^{2}-A^{-2}
  22. σ i \sigma_{i}
  23. A e i + A - 1 1 A\cdot e_{i}+A^{-1}\cdot 1
  24. 1 , e 1 , , e n - 1 1,e_{1},\dots,e_{n-1}
  25. σ \sigma
  26. δ n - 1 t r ρ ( σ ) \delta^{n-1}tr\rho(\sigma)
  27. L \langle L\rangle
  28. \langle
  29. \rangle
  30. ( t 1 / 2 - t - 1 / 2 ) V ( L 0 ) = t - 1 V ( L + ) - t V ( L - ) (t^{1/2}-t^{-1/2})V(L_{0})=t^{-1}V(L_{+})-tV(L_{-})\,
  31. L + L_{+}
  32. L - L_{-}
  33. L 0 L_{0}
  34. K K
  35. t - 1 t^{-1}
  36. t t
  37. V ( K ) V(K)
  38. V N ( L , t ) V_{N}(L,t)
  39. L L
  40. S U ( 2 ) SU(2)
  41. V N ( L , t ) V_{N}(L,t)
  42. V X Y ( L , t ) = V X ( L , t ) + V Y ( L , t ) V_{X\oplus Y}(L,t)=V_{X}(L,t)+V_{Y}(L,t)
  43. X , Y X,Y
  44. V X Y ( L , t ) V_{X\otimes Y}(L,t)
  45. X X
  46. Y Y
  47. L L
  48. V ( L , t ) = V 1 ( L , t ) V(L,t)=V_{1}(L,t)
  49. γ \gamma
  50. W F ( γ ) W_{F}(\gamma)
  51. γ \gamma
  52. F F
  53. SU ( 2 ) \mathrm{SU}(2)
  54. e h e^{h}
  55. t t
  56. s l 2 sl_{2}