wpmath0000006_3

E-folding.html

  1. A B A\rightarrow B
  2. d [ A ] d t = - k [ A ] \frac{d[A]}{dt}=-k[A]
  3. d [ A ] [ A ] = - k d t \frac{d[A]}{[A]}=-kdt
  4. [ A ] i [ A ] f d [ A ] [ A ] = t = 0 t - k d t \int_{[A]_{i}}^{[A]_{f}}\frac{d[A]}{[A]}=\int_{t=0}^{t}-kdt
  5. l n [ A ] f - l n [ A ] i = - k d t - k ( 0 ) ln[A]_{f}-ln[A]_{i}=-kdt-k(0)
  6. l n [ A ] f l n [ A ] i = - k t \frac{ln[A]_{f}}{ln[A]_{i}}=-kt
  7. e l n [ A ] f l n [ A ] i = e - k t e^{\frac{ln[A]_{f}}{ln[A]_{i}}}=e^{-kt}
  8. [ A ] f [ A ] i = e - k t \frac{[A]_{f}}{[A]_{i}}=e^{-kt}
  9. [ A ] f [ A ] i = e - k ( 1 k ) = e - 1 = 1 e 0.37 \frac{[A]_{f}}{[A]_{i}}=e^{-k(\frac{1}{k})}=e^{-1}=\frac{1}{e}\approx 0.37
  10. [ A ] f [ A ] i = 37 / 100 = 37 % \frac{[A]_{f}}{[A]_{i}}=37/100=37\%
  11. [ A ] f [ A ] i = e - k ( 2 k ) = e - 2 = 1 e 2 0.14 = 14 % \frac{[A]_{f}}{[A]_{i}}=e^{-k(\frac{2}{k})}=e^{-2}=\frac{1}{e^{2}}\approx 0.14% =14\%

E2F.html

  1. α \alpha
  2. δ \delta

Earthing_system.html

  1. 3 \sqrt{3}

Eckert_number.html

  1. Ec = u 2 c p Δ T \mathrm{Ec}=\frac{u^{2}}{c_{p}\Delta T}
  2. Δ T \Delta T

Economic_lot_scheduling_problem.html

  1. D j , j = 1 , , N D_{j},j=1,\cdots,N
  2. Q j Q_{j}
  3. c i j c_{ij}
  4. s i j s_{ij}
  5. h j h_{j}

Edward_Ullman.html

  1. I i , j = p i p j d i , j β I_{i,j}=\frac{p_{i}p_{j}}{d_{i,j}^{\beta}}

Effective_atomic_number.html

  1. Z eff = f 1 × ( Z 1 ) 2.94 + f 2 × ( Z 2 ) 2.94 + f 3 × ( Z 3 ) 2.94 + 2.94 Z_{\,\text{eff}}=\sqrt[2.94]{f_{1}\times(Z_{1})^{2.94}+f_{2}\times(Z_{2})^{2.9% 4}+f_{3}\times(Z_{3})^{2.94}+...}
  2. f n f_{n}
  3. Z n Z_{n}
  4. Z eff = 0.2 × 1 2.94 + 0.8 × 8 2.94 2.94 = 7.42 Z_{\,\text{eff}}=\sqrt[2.94]{0.2\times 1^{2.94}+0.8\times 8^{2.94}}=7.42

Ehrenfest_paradox.html

  1. F S = m v 2 r S < m c s 2 r S m G r S ρ G \frac{F}{S}=\frac{mv^{2}}{rS}<\frac{mc_{s}^{2}}{rS}\approx\frac{mG}{rS\rho}\approx G
  2. c s c_{s}
  3. ρ \rho
  4. G G
  5. ω \omega
  6. ω R \omega R
  7. 1 - ( ω R ) 2 / c 2 \sqrt{1-(\omega R)^{2}/c^{2}}
  8. circumference diameter = 2 π R 1 - ( ω R ) 2 / c 2 2 R = π 1 - ( ω R ) 2 / c 2 \frac{\mathrm{circumference}}{\mathrm{diameter}}=\frac{2\pi R\sqrt{1-(\omega R% )^{2}/c^{2}}}{2R}=\pi\sqrt{1-(\omega R)^{2}/c^{2}}
  9. π \pi
  10. 2 π R 2\pi R
  11. C = 2 π R 1 - v 2 / c 2 C^{\prime}=\frac{2\pi R}{\sqrt{1-v^{2}/c^{2}}}
  12. U / D = π U/D=\pi
  13. U / D > π U/D>\pi

Ehrenfest_theorem.html

  1. F = d V / d x F=−dV/dx
  2. A A
  3. A ⟨A⟩
  4. i ħ
  5. Φ Φ
  6. A A
  7. d d t A = d d t Φ * A Φ d x 3 = ( Φ * t ) A Φ d x 3 + Φ * ( A t ) Φ d x 3 + Φ * A ( Φ t ) d x 3 = ( Φ * t ) A Φ d x 3 + A t + Φ * A ( Φ t ) d x 3 \begin{aligned}\displaystyle\frac{d}{dt}\langle A\rangle&\displaystyle=\frac{d% }{dt}\int\Phi^{*}A\Phi~{}dx^{3}\\ &\displaystyle=\int\left(\frac{\partial\Phi^{*}}{\partial t}\right)A\Phi~{}dx^% {3}+\int\Phi^{*}\left(\frac{\partial A}{\partial t}\right)\Phi~{}dx^{3}+\int% \Phi^{*}A\left(\frac{\partial\Phi}{\partial t}\right)~{}dx^{3}\\ &\displaystyle=\int\left(\frac{\partial\Phi^{*}}{\partial t}\right)A\Phi~{}dx^% {3}+\left\langle\frac{\partial A}{\partial t}\right\rangle+\int\Phi^{*}A\left(% \frac{\partial\Phi}{\partial t}\right)~{}dx^{3}\end{aligned}
  8. Φ t = 1 i H Φ \frac{\partial\Phi}{\partial t}=\frac{1}{i\hbar}H\Phi
  9. Φ * t = - 1 i Φ * H * = - 1 i Φ * H . \frac{\partial\Phi^{*}}{\partial t}=-\frac{1}{i\hbar}\Phi^{*}H^{*}=-\frac{1}{i% \hbar}\Phi^{*}H.
  10. t ϕ | x = - 1 i ϕ | H ^ | x = - 1 i ϕ | x H = - 1 i Φ * H , \frac{\partial}{\partial t}\langle\phi|x\rangle=\frac{-1}{i\hbar}\langle\phi|% \hat{H}|x\rangle=\frac{-1}{i\hbar}\langle\phi|x\rangle H=\frac{-1}{i\hbar}\Phi% ^{*}H,
  11. H ^ \hat{H}
  12. H H
  13. H H
  14. Φ Φ
  15. d d t A = 1 i Φ * ( A H - H A ) Φ d x 3 + A t = 1 i [ A , H ] + A t . \frac{d}{dt}\langle A\rangle=\frac{1}{i\hbar}\int\Phi^{*}(AH-HA)\Phi~{}dx^{3}+% \left\langle\frac{\partial A}{\partial t}\right\rangle=\frac{1}{i\hbar}\langle% [A,H]\rangle+\left\langle\frac{\partial A}{\partial t}\right\rangle.
  16. A A
  17. d d t A ( t ) = A ( t ) t + 1 i [ A ( t ) , H ] , \frac{d}{dt}A(t)=\frac{\partial A(t)}{\partial t}+\frac{1}{i\hbar}[A(t),H],
  18. | Ψ |\Psi\rangle
  19. Ψ | \langle\Psi|
  20. Ψ | d d t A ( t ) | Ψ = Ψ | A ( t ) t | Ψ + Ψ | 1 i [ A ( t ) , H ] | Ψ , \langle\Psi|\frac{d}{dt}A(t)|\Psi\rangle=\langle\Psi|\frac{\partial A(t)}{% \partial t}|\Psi\rangle+\langle\Psi|\frac{1}{i\hbar}[A(t),H]|\Psi\rangle,
  21. d d t \frac{d}{dt}
  22. d d t A ( t ) = A ( t ) t + 1 i [ A ( t ) , H ] \frac{d}{dt}\langle A(t)\rangle=\left\langle\frac{\partial A(t)}{\partial t}% \right\rangle+\frac{1}{i\hbar}\langle[A(t),H]\rangle
  23. H ( x , p , t ) = p 2 2 m + V ( x , t ) H(x,p,t)=\frac{p^{2}}{2m}+V(x,t)
  24. x x
  25. p p
  26. d d t p = 1 i [ p , H ] + p t = 1 i [ p , V ( x , t ) ] , \frac{d}{dt}\langle p\rangle=\frac{1}{i\hbar}\langle[p,H]\rangle+\left\langle% \frac{\partial p}{\partial t}\right\rangle=\frac{1}{i\hbar}\langle[p,V(x,t)]\rangle,
  27. p p
  28. p p
  29. i ħ −iħ∇
  30. d d t p = Φ * V ( x , t ) Φ d x 3 - Φ * ( V ( x , t ) Φ ) d x 3 . \frac{d}{dt}\langle p\rangle=\int\Phi^{*}V(x,t)\nabla\Phi~{}dx^{3}-\int\Phi^{*% }\nabla(V(x,t)\Phi)~{}dx^{3}.
  31. d d t p = Φ * V ( x , t ) Φ d x 3 - Φ * ( V ( x , t ) ) Φ d x 3 - Φ * V ( x , t ) Φ d x 3 = - Φ * ( V ( x , t ) ) Φ d x 3 = - V ( x , t ) = F , \begin{aligned}\displaystyle\frac{d}{dt}\langle p\rangle&\displaystyle=\int% \Phi^{*}V(x,t)\nabla\Phi~{}dx^{3}-\int\Phi^{*}(\nabla V(x,t))\Phi~{}dx^{3}-% \int\Phi^{*}V(x,t)\nabla\Phi~{}dx^{3}\\ &\displaystyle=-\int\Phi^{*}(\nabla V(x,t))\Phi~{}dx^{3}\\ &\displaystyle=\langle-\nabla V(x,t)\rangle=\langle F\rangle,\end{aligned}
  32. d d t x = 1 i [ x , H ] + x t = 1 i [ x , p 2 2 m + V ( x , t ) ] + 0 = 1 i [ x , p 2 2 m ] = 1 i 2 m [ x , p ] d d p p 2 = 1 i 2 m i 2 p = 1 m p \begin{aligned}\displaystyle\frac{d}{dt}\langle x\rangle&\displaystyle=\frac{1% }{i\hbar}\langle[x,H]\rangle+\left\langle\frac{\partial x}{\partial t}\right% \rangle\\ &\displaystyle=\frac{1}{i\hbar}\left\langle\left[x,\frac{p^{2}}{2m}+V(x,t)% \right]\right\rangle+0\\ &\displaystyle=\frac{1}{i\hbar}\left\langle\left[x,\frac{p^{2}}{2m}\right]% \right\rangle\\ &\displaystyle=\frac{1}{i\hbar 2m}\left\langle[x,p]\frac{d}{dp}p^{2}\right% \rangle\\ &\displaystyle=\frac{1}{i\hbar 2m}\langle i\hbar 2p\rangle\\ &\displaystyle=\frac{1}{m}\langle p\rangle\end{aligned}
  33. m d d t Ψ ( t ) | x ^ | Ψ ( t ) = Ψ ( t ) | p ^ | Ψ ( t ) , d d t Ψ ( t ) | p ^ | Ψ ( t ) = Ψ ( t ) | - V ( x ^ ) | Ψ ( t ) . \begin{aligned}\displaystyle m\frac{d}{dt}\left\langle\Psi(t)\right|\hat{x}% \left|\Psi(t)\right\rangle&\displaystyle=\left\langle\Psi(t)\right|\hat{p}% \left|\Psi(t)\right\rangle,\\ \displaystyle\frac{d}{dt}\left\langle\Psi(t)\right|\hat{p}\left|\Psi(t)\right% \rangle&\displaystyle=\left\langle\Psi(t)\right|-V^{\prime}(\hat{x})\left|\Psi% (t)\right\rangle.\end{aligned}
  34. d Ψ d t | x ^ | Ψ + Ψ | x ^ | d Ψ d t = Ψ | p ^ m | Ψ , d Ψ d t | p ^ | Ψ + Ψ | p ^ | d Ψ d t = Ψ | - V ( x ^ ) | Ψ , \begin{aligned}\displaystyle\left\langle\frac{d\Psi}{dt}\Big|\hat{x}\Big|\Psi% \right\rangle+\left\langle\Psi\Big|\hat{x}\Big|\frac{d\Psi}{dt}\right\rangle&% \displaystyle=\left\langle\Psi\Big|\frac{\hat{p}}{m}\Big|\Psi\right\rangle,\\ \displaystyle\left\langle\frac{d\Psi}{dt}\Big|\hat{p}\Big|\Psi\right\rangle+% \left\langle\Psi\Big|\hat{p}\Big|\frac{d\Psi}{dt}\right\rangle&\displaystyle=% \langle\Psi|-V^{\prime}(\hat{x})|\Psi\rangle,\end{aligned}
  35. i | d Ψ d t = H ^ | Ψ ( t ) , i\hbar\left|\frac{d\Psi}{dt}\right\rangle=\hat{H}|\Psi(t)\rangle~{},
  36. ħ ħ
  37. i m [ H ^ , x ^ ] = p ^ , i [ H ^ , p ^ ] = - V ( x ^ ) . im[\hat{H},\hat{x}]=\hbar\hat{p},\qquad i[\hat{H},\hat{p}]=-\hbar V^{\prime}(% \hat{x}).
  38. , = i ħ x̂,p̂p̂=iħ
  39. H ^ = H ( x ^ , p ^ ) \hat{H}=H(\hat{x},\hat{p})
  40. m H ( x , p ) p = p , H ( x , p ) x = V ( x ) , m\frac{\partial H(x,p)}{\partial p}=p,\qquad\frac{\partial H(x,p)}{\partial x}% =V^{\prime}(x),
  41. H ^ = p ^ 2 2 m + V ( x ^ ) . \hat{H}=\frac{\hat{p}^{2}}{2m}+V(\hat{x}).
  42. , x̂,p̂p̂
  43. p ⟨p⟩
  44. p p
  45. x t < s u p > 2 ⟨xt<sup>2⟩

Eigenvalues_and_eigenvectors.html

  1. A 𝐯 = λ 𝐯 , A\mathbf{v}=\lambda\mathbf{v},
  2. v v
  3. A A
  4. n \mathbb{R}^{n}
  5. f ( x ) = e λ x f(x)=e^{\lambda x}
  6. {}^{\prime}
  7. λ \lambda
  8. f ( x ) = λ e λ x = λ f ( x ) f^{\prime}(x)=\lambda e^{\lambda x}=\lambda f(x)
  9. A A
  10. A A
  11. A A
  12. A A
  13. 𝐮 = { 1 3 4 } and 𝐯 = { - 20 - 60 - 80 } . \mathbf{u}=\begin{Bmatrix}1\\ 3\\ 4\end{Bmatrix}\quad\mbox{and}~{}\quad\mathbf{v}=\begin{Bmatrix}-20\\ -60\\ -80\end{Bmatrix}.
  14. 𝐮 = λ 𝐯 . \mathbf{u}=\lambda\mathbf{v}.
  15. A 𝐯 = 𝐰 , A\mathbf{v}=\mathbf{w},
  16. [ A 1 , 1 A 1 , 2 A 1 , n A 2 , 1 A 2 , 2 A 2 , n A n , 1 A n , 2 A n , n ] { v 1 v 2 v n } = { w 1 w 2 w n } \begin{bmatrix}A_{1,1}&A_{1,2}&\ldots&A_{1,n}\\ A_{2,1}&A_{2,2}&\ldots&A_{2,n}\\ \vdots&\vdots&\ddots&\vdots\\ A_{n,1}&A_{n,2}&\ldots&A_{n,n}\\ \end{bmatrix}\begin{Bmatrix}v_{1}\\ v_{2}\\ \vdots\\ v_{n}\end{Bmatrix}=\begin{Bmatrix}w_{1}\\ w_{2}\\ \vdots\\ w_{n}\end{Bmatrix}
  17. i i
  18. w i = A i , 1 v 1 + A i , 2 v 2 + + A i , n v n = j = 1 n A i , j v j w_{i}=A_{i,1}v_{1}+A_{i,2}v_{2}+\cdots+A_{i,n}v_{n}=\sum_{j=1}^{n}A_{i,j}v_{j}
  19. A 𝐯 = λ 𝐯 , A\mathbf{v}=\lambda\mathbf{v},
  20. [ 2 1 1 2 ] \bigl[\begin{smallmatrix}2&1\\ 1&2\end{smallmatrix}\bigr]
  21. A = [ 2 1 1 2 ] . A=\begin{bmatrix}2&1\\ 1&2\end{bmatrix}.
  22. A 𝐯 = λ 𝐯 . A\mathbf{v}=\lambda\mathbf{v}.
  23. ( A - λ I ) 𝐯 = 0 , (A-\lambda I)\mathbf{v}=0,
  24. | A λ I | |A−λI|
  25. p ( λ ) = | A - λ I | = 3 - 4 λ + λ 2 = 0 , p(\lambda)=|A-\lambda I|=3-4\lambda+\lambda^{2}=0,
  26. ( A - I ) 𝐯 = [ 1 1 1 1 ] { v 1 v 2 } = { 0 0 } , (A-I)\mathbf{v}=\begin{bmatrix}1&1\\ 1&1\end{bmatrix}\begin{Bmatrix}v_{1}\\ v_{2}\end{Bmatrix}=\begin{Bmatrix}0\\ 0\end{Bmatrix},
  27. 𝐯 = { 1 - 1 } . \mathbf{v}=\begin{Bmatrix}1\\ -1\end{Bmatrix}.
  28. ( A - 3 I ) 𝐰 = [ - 1 1 1 - 1 ] { w 1 w 2 } = { 0 0 } , (A-3I)\mathbf{w}=\begin{bmatrix}-1&1\\ 1&-1\end{bmatrix}\begin{Bmatrix}w_{1}\\ w_{2}\end{Bmatrix}=\begin{Bmatrix}0\\ 0\end{Bmatrix},
  29. 𝐰 = { 1 1 } . \mathbf{w}=\begin{Bmatrix}1\\ 1\end{Bmatrix}.
  30. A = [ 2 0 1 0 2 0 1 0 2 ] , A=\begin{bmatrix}2&0&1\\ 0&2&0\\ 1&0&2\end{bmatrix},
  31. ( A - λ I ) 𝐯 = 0. (A-\lambda I)\mathbf{v}=0.
  32. | A λ I | |A−λI|
  33. p ( λ ) = | A - λ I | = 6 - 11 λ + 6 λ 2 - λ 3 , p(\lambda)=|A-\lambda I|=6-11\lambda+6\lambda^{2}-\lambda^{3},
  34. 𝐮 = { 1 0 - 1 } , 𝐯 = { 0 1 0 } , and 𝐰 = { 1 0 1 } , \mathbf{u}=\begin{Bmatrix}1\\ 0\\ -1\end{Bmatrix},\quad\mathbf{v}=\begin{Bmatrix}0\\ 1\\ 0\end{Bmatrix},\quad\mbox{and}~{}\quad\mathbf{w}=\begin{Bmatrix}1\\ 0\\ 1\end{Bmatrix},
  35. A = [ 1 0 0 0 2 0 0 0 3 ] . A=\begin{bmatrix}1&0&0\\ 0&2&0\\ 0&0&3\end{bmatrix}.
  36. p ( λ ) = | A - λ I | = ( 1 - λ ) ( 2 - λ ) ( 3 - λ ) = 0 , p(\lambda)=|A-\lambda I|=(1-\lambda)(2-\lambda)(3-\lambda)=0,
  37. 𝐮 = { 1 0 0 } , 𝐯 = { 0 1 0 } , and 𝐰 = { 0 0 1 } , \mathbf{u}=\begin{Bmatrix}1\\ 0\\ 0\end{Bmatrix},\quad\mathbf{v}=\begin{Bmatrix}0\\ 1\\ 0\end{Bmatrix},\quad\mbox{and}~{}\quad\mathbf{w}=\begin{Bmatrix}0\\ 0\\ 1\end{Bmatrix},
  38. A = [ 1 0 0 1 2 0 2 3 3 ] . A=\begin{bmatrix}1&0&0\\ 1&2&0\\ 2&3&3\end{bmatrix}.
  39. p ( λ ) = | A - λ I | = ( 1 - λ ) ( 2 - λ ) ( 3 - λ ) = 0 , p(\lambda)=|A-\lambda I|=(1-\lambda)(2-\lambda)(3-\lambda)=0,
  40. 𝐮 = { 1 - 1 1 / 2 } , 𝐯 = { 0 1 - 3 } , and 𝐰 = { 0 0 1 } , \mathbf{u}=\begin{Bmatrix}1\\ -1\\ 1/2\end{Bmatrix},\quad\mathbf{v}=\begin{Bmatrix}0\\ 1\\ -3\end{Bmatrix},\quad\mbox{and}~{}\quad\mathbf{w}=\begin{Bmatrix}0\\ 0\\ 1\end{Bmatrix},
  41. 𝐗 = A 𝐱 , \mathbf{X}=A\mathbf{x},
  42. A 𝐯 i = λ i 𝐯 i i = 1 , n . A\mathbf{v}_{i}=\lambda_{i}\mathbf{v}_{i}\quad i=1\ldots,n.
  43. 𝐲 = V - 1 𝐱 , and 𝐘 = V - 1 𝐗 . \mathbf{y}=V^{-1}\mathbf{x},\quad\mbox{and}~{}\quad\mathbf{Y}=V^{-1}\mathbf{X}.
  44. K = V - 1 A V . K=V^{-1}AV.
  45. A V ( V - 1 𝐯 i ) = λ i V ( V - 1 𝐯 i ) , AV(V^{-1}\mathbf{v}_{i})=\lambda_{i}V(V^{-1}\mathbf{v}_{i}),
  46. V - 1 A V ( V - 1 𝐯 i ) = λ i ( V - 1 𝐯 i ) , V^{-1}AV(V^{-1}\mathbf{v}_{i})=\lambda_{i}(V^{-1}\mathbf{v}_{i}),
  47. V - 1 𝐯 i = 𝐞 i , V^{-1}\mathbf{v}_{i}=\mathbf{e}_{i},
  48. V - 1 A V 𝐞 i = K 𝐞 i = λ i 𝐞 i , V^{-1}AV\mathbf{e}_{i}=K\mathbf{e}_{i}=\lambda_{i}\mathbf{e}_{i},
  49. A A
  50. A v - λ v = 0 , Av-\lambda v=0,
  51. ( A - λ I ) v = 0 , (A-\lambda I)v=0,
  52. I I
  53. n × n n\times n
  54. M v = 0 Mv=0
  55. v v
  56. det ( M ) \det(M)
  57. M M
  58. A A
  59. λ \lambda
  60. det ( A - λ I ) = 0 \det(A-\lambda I)=0
  61. λ \lambda
  62. n n
  63. A A
  64. n n
  65. ( - 1 ) n λ n (-1)^{n}\lambda^{n}
  66. A A
  67. A A
  68. A A
  69. A = [ 2 0 0 0 3 4 0 4 9 ] A=\begin{bmatrix}2&0&0\\ 0&3&4\\ 0&4&9\end{bmatrix}
  70. A A
  71. det ( A - λ I ) = det ( [ 2 0 0 0 3 4 0 4 9 ] - λ [ 1 0 0 0 1 0 0 0 1 ] ) = det [ 2 - λ 0 0 0 3 - λ 4 0 4 9 - λ ] \det(A-\lambda I)\;=\;\det\left(\begin{bmatrix}2&0&0\\ 0&3&4\\ 0&4&9\end{bmatrix}-\lambda\begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix}\right)\;=\;\det\begin{bmatrix}2-\lambda&0&0\\ 0&3-\lambda&4\\ 0&4&9-\lambda\end{bmatrix}
  72. ( 2 - λ ) [ ( 3 - λ ) ( 9 - λ ) - 16 ] = - λ 3 + 14 λ 2 - 35 λ + 22 (2-\lambda)\bigl[(3-\lambda)(9-\lambda)-16\bigr]=-\lambda^{3}+14\lambda^{2}-35% \lambda+22
  73. A A
  74. [ 1 , 0 , 0 ] , [1,0,0]^{\prime},
  75. [ 0 , 2 , - 1 ] , [0,2,-1]^{\prime},
  76. [ 0 , 1 , 2 ] [0,1,2]^{\prime}
  77. n × n n\times n
  78. n n
  79. n n
  80. A = [ 0 1 0 0 0 1 1 0 0 ] A=\begin{bmatrix}0&1&0\\ 0&0&1\\ 1&0&0\end{bmatrix}
  81. 1 - λ 3 1-\lambda^{3}
  82. λ 1 = 1 \lambda_{1}=1
  83. A [ 5 5 5 ] = [ 5 5 5 ] = 1 [ 5 5 5 ] A\begin{bmatrix}5\\ 5\\ 5\end{bmatrix}=\begin{bmatrix}5\\ 5\\ 5\end{bmatrix}=1\cdot\begin{bmatrix}5\\ 5\\ 5\end{bmatrix}
  84. n × n n\times n
  85. A A
  86. n n
  87. n n
  88. n n
  89. det ( A - λ I ) = ( λ 1 - λ ) ( λ 2 - λ ) ( λ n - λ ) \det(A-\lambda I)=(\lambda_{1}-\lambda)(\lambda_{2}-\lambda)\cdots(\lambda_{n}% -\lambda)
  90. λ i \lambda_{i}
  91. λ 1 \lambda_{1}
  92. λ 2 \lambda_{2}
  93. λ n \lambda_{n}
  94. A A
  95. A A
  96. A A
  97. A A
  98. A A
  99. 1 - λ 3 1-\lambda^{3}
  100. λ 2 = - 1 / 2 + 𝐢 3 / 2 \lambda_{2}=-1/2+\mathbf{i}\sqrt{3}/2\quad\quad
  101. λ 3 = λ 2 * = - 1 / 2 - 𝐢 3 / 2 \quad\quad\lambda_{3}=\lambda_{2}^{*}=-1/2-\mathbf{i}\sqrt{3}/2
  102. 𝐢 = - 1 \mathbf{i}=\sqrt{-1}
  103. λ 2 λ 3 = 1 \lambda_{2}\lambda_{3}=1
  104. λ 2 2 = λ 3 \lambda_{2}^{2}=\lambda_{3}
  105. λ 3 2 = λ 2 \lambda_{3}^{2}=\lambda_{2}
  106. A [ 1 λ 2 λ 3 ] = [ λ 2 λ 3 1 ] = λ 2 [ 1 λ 2 λ 3 ] A\begin{bmatrix}1\\ \lambda_{2}\\ \lambda_{3}\end{bmatrix}=\begin{bmatrix}\lambda_{2}\\ \lambda_{3}\\ 1\end{bmatrix}=\lambda_{2}\cdot\begin{bmatrix}1\\ \lambda_{2}\\ \lambda_{3}\end{bmatrix}\quad\quad
  107. A [ 1 λ 3 λ 2 ] = [ λ 3 λ 2 1 ] = λ 3 [ 1 λ 3 λ 2 ] \quad\quad A\begin{bmatrix}1\\ \lambda_{3}\\ \lambda_{2}\end{bmatrix}=\begin{bmatrix}\lambda_{3}\\ \lambda_{2}\\ 1\end{bmatrix}=\lambda_{3}\cdot\begin{bmatrix}1\\ \lambda_{3}\\ \lambda_{2}\end{bmatrix}
  108. [ 1 , λ 2 , λ 3 ] [1,\lambda_{2},\lambda_{3}]^{\prime}
  109. [ 1 , λ 3 , λ 2 ] [1,\lambda_{3},\lambda_{2}]^{\prime}
  110. A A
  111. λ 2 \lambda_{2}
  112. λ 3 \lambda_{3}
  113. λ i \lambda_{i}
  114. n × n n\times n
  115. A A
  116. μ A ( λ i ) \mu_{A}(\lambda_{i})
  117. λ i \lambda_{i}
  118. k k
  119. ( λ - λ i ) k (\lambda-\lambda_{i})^{k}
  120. γ A ( λ i ) \gamma_{A}(\lambda_{i})
  121. 1 μ A ( λ i ) n 1\leq\mu_{A}(\lambda_{i})\leq n
  122. s y m b o l μ A symbol{\mu}_{A}
  123. μ A ( λ i ) \mu_{A}(\lambda_{i})
  124. n n
  125. s y m b o l μ A symbol{\mu}_{A}
  126. n n
  127. γ A ( λ i ) \gamma_{A}(\lambda_{i})
  128. μ A ( λ i ) \mu_{A}(\lambda_{i})
  129. s y m b o l γ A symbol{\gamma}_{A}
  130. s y m b o l μ A symbol{\mu}_{A}
  131. μ A ( λ i ) = 1 \mu_{A}(\lambda_{i})=1
  132. λ i \lambda_{i}
  133. γ A ( λ i ) = μ A ( λ i ) \gamma_{A}(\lambda_{i})=\mu_{A}(\lambda_{i})
  134. λ i \lambda_{i}
  135. A = [ 2 0 0 0 1 2 0 0 0 1 3 0 0 0 1 3 ] , A=\begin{bmatrix}2&0&0&0\\ 1&2&0&0\\ 0&1&3&0\\ 0&0&1&3\end{bmatrix},
  136. A A
  137. det ( A - λ I ) = det [ 2 - λ 0 0 0 1 2 - λ 0 0 0 1 3 - λ 0 0 0 1 3 - λ ] = ( 2 - λ ) 2 ( 3 - λ ) 2 \det(A-\lambda I)\;=\;\det\begin{bmatrix}2-\lambda&0&0&0\\ 1&2-\lambda&0&0\\ 0&1&3-\lambda&0\\ 0&0&1&3-\lambda\end{bmatrix}=(2-\lambda)^{2}(3-\lambda)^{2}
  138. [ 0 , 1 , - 1 , 1 ] [0,1,-1,1]
  139. [ 0 , 0 , 0 , 1 ] [0,0,0,1]
  140. μ A \mu_{A}
  141. γ A \gamma_{A}
  142. s y m b o l γ A symbol{\gamma}_{A}
  143. n n
  144. A A
  145. n n
  146. Q Q
  147. A Q = Q Λ AQ=Q\Lambda
  148. Λ \Lambda
  149. Λ i i \Lambda_{ii}
  150. i i
  151. Q Q
  152. Q Q
  153. Q Q
  154. Q - 1 Q^{-1}
  155. Q - 1 A Q = Λ Q^{-1}AQ=\Lambda
  156. A A
  157. A A
  158. Q Q
  159. Q - 1 A Q Q^{-1}AQ
  160. D D
  161. Q Q
  162. A Q = Q D AQ=QD
  163. Q Q
  164. A A
  165. D D
  166. Q Q
  167. s y m b o l γ A = n symbol{\gamma}_{A}=n
  168. s y m b o l γ A symbol{\gamma}_{A}
  169. n n
  170. A A
  171. A A
  172. n n
  173. A A
  174. A A
  175. A A
  176. A A
  177. n × n n\times n
  178. λ 1 \lambda_{1}
  179. λ 2 \lambda_{2}
  180. λ n \lambda_{n}
  181. μ \mu
  182. μ \mu
  183. A A
  184. tr ( A ) = i = 1 n A i i = i = 1 n λ i = λ 1 + λ 2 + + λ n \operatorname{tr}(A)=\sum_{i=1}^{n}A_{ii}=\sum_{i=1}^{n}\lambda_{i}=\lambda_{1% }+\lambda_{2}+\cdots+\lambda_{n}
  185. A A
  186. det ( A ) = i = 1 n λ i = λ 1 λ 2 λ n \operatorname{det}(A)=\prod_{i=1}^{n}\lambda_{i}=\lambda_{1}\lambda_{2}\cdots% \lambda_{n}
  187. k k
  188. A A
  189. A k A^{k}
  190. k k
  191. λ 1 k , λ 2 k , , λ n k \lambda_{1}^{k},\lambda_{2}^{k},\dots,\lambda_{n}^{k}
  192. A A
  193. λ i \lambda_{i}
  194. A A
  195. A - 1 A^{-1}
  196. 1 / λ 1 , 1 / λ 2 , , 1 / λ n 1/\lambda_{1},1/\lambda_{2},\dots,1/\lambda_{n}
  197. A A
  198. A * A^{*}
  199. A A
  200. A A
  201. | λ | = 1 |\lambda|=1
  202. A A
  203. A v = λ v Av=\lambda v
  204. A A
  205. u A = λ u uA=\lambda u
  206. u u
  207. A A
  208. A A
  209. A 𝖳 A^{\mathsf{T}}
  210. A 𝖳 u 𝖳 = λ u 𝖳 A^{\mathsf{T}}u^{\mathsf{T}}=\lambda u^{\mathsf{T}}
  211. A A
  212. A A
  213. H H
  214. x 𝖳 H x / x 𝖳 x x^{\mathsf{T}}Hx/x^{\mathsf{T}}x
  215. x x
  216. V V
  217. K K
  218. T T
  219. V V
  220. V V
  221. v v
  222. V V
  223. T T
  224. λ \lambda
  225. K K
  226. T ( v ) = λ v T(v)=\lambda v
  227. T T
  228. λ \lambda
  229. T T
  230. v v
  231. T ( v ) T(v)
  232. T T
  233. v v
  234. λ v \lambda v
  235. λ \lambda
  236. v v
  237. V V
  238. n n
  239. T T
  240. n × n n\times n
  241. A A
  242. v v
  243. T T
  244. λ \lambda
  245. K K
  246. v v
  247. V V
  248. T ( v ) = λ v T(v)=\lambda v
  249. v v
  250. V V
  251. λ \lambda
  252. K K
  253. T ( v ) = λ v T(v)=\lambda v
  254. γ T ( λ ) \gamma_{T}(\lambda)
  255. λ \lambda
  256. λ \lambda
  257. γ T ( λ ) 1. \gamma_{T}(\lambda)\geq 1.
  258. v v
  259. T T
  260. λ \lambda
  261. α v \alpha v
  262. v v
  263. α \alpha
  264. λ \lambda
  265. T ( α v ) = α T ( v ) = α ( λ v ) = λ ( α v ) T(\alpha v)=\alpha T(v)=\alpha(\lambda v)=\lambda(\alpha v)
  266. u u
  267. v v
  268. λ \lambda
  269. u - v u\neq-v
  270. u + v u+v
  271. λ \lambda
  272. λ \lambda
  273. V V
  274. T T
  275. λ \lambda
  276. T T
  277. T T
  278. T T
  279. T T
  280. T T
  281. V V
  282. V V
  283. T T
  284. 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}.
  285. 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,
  286. x t - 1 = x t - 1 , , x t - k + 1 = x t - k + 1 , x_{t-1}=x_{t-1},\dots,x_{t-k+1}=x_{t-k+1},
  287. [ x t , , x t - k + 1 ] [x_{t},\dots,x_{t-k+1}]
  288. λ 1 , , λ k , \lambda_{1},\dots,\lambda_{k},
  289. 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}.
  290. 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.
  291. A A
  292. n n
  293. n n
  294. n n
  295. A = [ 4 1 6 3 ] A=\begin{bmatrix}4&1\\ 6&3\end{bmatrix}
  296. A v = 6 v Av=6v
  297. [ 4 1 6 3 ] [ x y ] = 6 [ x y ] \begin{bmatrix}4&1\\ 6&3\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix}=6\cdot\begin{bmatrix}x\\ y\end{bmatrix}
  298. { 4 x + y = 6 x 6 x + 3 y = 6 y \left\{\begin{matrix}4x+{\ }y&{}=6x\\ 6x+3y&{}=6y\end{matrix}\right.
  299. { - 2 x + y = 0 + 6 x - 3 y = 0 \left\{\begin{matrix}-2x+{\ }y&{}=0\\ +6x-3y&{}=0\end{matrix}\right.
  300. y = 2 x y=2x
  301. [ a , 2 a ] [a,2a]^{\prime}
  302. a a
  303. A A
  304. λ = 6 \lambda=6
  305. A A
  306. λ = 1 \lambda=1
  307. 3 x + y = 0 3x+y=0
  308. [ b , - 3 b ] [b,-3b]^{\prime}
  309. b b
  310. T T
  311. V V
  312. λ \lambda
  313. v v
  314. T ( v ) = λ v T(v)=\lambda v
  315. D D
  316. 𝐂 \mathbf{C^{\infty}}
  317. t t
  318. D D
  319. D f = λ f Df=\lambda f
  320. D D
  321. d / d t d/dt
  322. f ( t ) = A e λ t , f(t)=Ae^{\lambda t},
  323. λ \lambda
  324. λ \lambda
  325. T T
  326. T - λ I T-\lambda I
  327. ( T - λ I ) - 1 (T-\lambda I)^{-1}
  328. T - λ I T-\lambda I
  329. λ \lambda
  330. T T
  331. λ \lambda
  332. T - λ I T-\lambda I
  333. [ k 0 0 k ] \begin{bmatrix}k&0\\ 0&k\end{bmatrix}
  334. [ k 1 0 0 k 2 ] \begin{bmatrix}k_{1}&0\\ 0&k_{2}\end{bmatrix}
  335. [ c - s s c ] \begin{bmatrix}c&-s\\ s&c\end{bmatrix}
  336. c = cos θ c=\cos\theta
  337. s = sin θ s=\sin\theta
  338. [ 1 k 0 1 ] \begin{bmatrix}1&k\\ 0&1\end{bmatrix}
  339. [ c s s c ] \begin{bmatrix}c&s\\ s&c\end{bmatrix}
  340. c = cosh φ c=\cosh\varphi
  341. s = sinh φ s=\sinh\varphi
  342. ( λ - k ) 2 \ (\lambda-k)^{2}
  343. ( λ - k 1 ) ( λ - k 2 ) (\lambda-k_{1})(\lambda-k_{2})
  344. λ 2 - 2 c λ + 1 \lambda^{2}-2c\lambda+1
  345. ( λ - 1 ) 2 \ (\lambda-1)^{2}
  346. λ 2 - 2 c λ + 1 \lambda^{2}-2c\lambda+1
  347. λ i \lambda_{i}
  348. λ 1 = λ 2 = k \lambda_{1}=\lambda_{2}=k
  349. λ 1 = k 1 \lambda_{1}=k_{1}
  350. λ 2 = k 2 \lambda_{2}=k_{2}
  351. λ 1 = e 𝐢 θ = c + s 𝐢 \lambda_{1}=e^{\mathbf{i}\theta}=c+s\mathbf{i}
  352. λ 2 = e - 𝐢 θ = c - s 𝐢 \lambda_{2}=e^{-\mathbf{i}\theta}=c-s\mathbf{i}
  353. λ 1 = λ 2 = 1 \lambda_{1}=\lambda_{2}=1
  354. λ 1 = e φ \lambda_{1}=e^{\varphi}
  355. λ 2 = e - φ \lambda_{2}=e^{-\varphi}
  356. μ i = μ ( λ i ) \mu_{i}=\mu(\lambda_{i})
  357. μ 1 = 2 \mu_{1}=2
  358. μ 1 = 1 \mu_{1}=1
  359. μ 2 = 1 \mu_{2}=1
  360. μ 1 = 1 \mu_{1}=1
  361. μ 2 = 1 \mu_{2}=1
  362. μ 1 = 2 \mu_{1}=2
  363. μ 1 = 1 \mu_{1}=1
  364. μ 2 = 1 \mu_{2}=1
  365. γ i = γ ( λ i ) \gamma_{i}=\gamma(\lambda_{i})
  366. γ 1 = 2 \gamma_{1}=2
  367. γ 1 = 1 \gamma_{1}=1
  368. γ 2 = 1 \gamma_{2}=1
  369. γ 1 = 1 \gamma_{1}=1
  370. γ 2 = 1 \gamma_{2}=1
  371. γ 1 = 1 \gamma_{1}=1
  372. γ 1 = 1 \gamma_{1}=1
  373. γ 2 = 1 \gamma_{2}=1
  374. u 1 = [ 1 0 ] u_{1}=\begin{bmatrix}1\\ 0\end{bmatrix}
  375. u 2 = [ 0 1 ] u_{2}=\begin{bmatrix}0\\ 1\end{bmatrix}
  376. u 1 = [ 1 - 𝐢 ] u_{1}=\begin{bmatrix}{\ }1\\ -\mathbf{i}\end{bmatrix}
  377. u 2 = [ 1 + 𝐢 ] u_{2}=\begin{bmatrix}{\ }1\\ +\mathbf{i}\end{bmatrix}
  378. u 1 = [ 1 0 ] u_{1}=\begin{bmatrix}1\\ 0\end{bmatrix}
  379. u 1 = [ 1 1 ] u_{1}=\begin{bmatrix}{\ }1\\ {\ }1\end{bmatrix}
  380. u 2 = [ 1 - 1 ] . u_{2}=\begin{bmatrix}{\ }1\\ -1\end{bmatrix}.
  381. D = - 4 ( sin θ ) 2 D=-4(\sin\theta)^{2}
  382. θ \theta
  383. cos θ ± 𝐢 sin θ \cos\theta\pm\mathbf{i}\sin\theta
  384. n = 1 , 2 , 3 , n=1,2,3,\ldots
  385. T T
  386. H ψ E = E ψ E H\psi_{E}=E\psi_{E}\,
  387. H H
  388. ψ E \psi_{E}
  389. E E
  390. ψ E \psi_{E}
  391. ψ E \psi_{E}
  392. H H
  393. | Ψ E |\Psi_{E}\rangle
  394. H | Ψ E = E | Ψ E H|\Psi_{E}\rangle=E|\Psi_{E}\rangle
  395. | Ψ E |\Psi_{E}\rangle
  396. H H
  397. E E
  398. H | Ψ E H|\Psi_{E}\rangle
  399. H H
  400. | Ψ E |\Psi_{E}\rangle
  401. v 1 , v 2 , v 3 v_{1},v_{2},v_{3}
  402. E 1 E 2 E 3 E_{1}\geq E_{2}\geq E_{3}
  403. v 1 v_{1}
  404. v 2 v_{2}
  405. v 3 v_{3}
  406. E 1 E_{1}
  407. E 2 E_{2}
  408. E 3 E_{3}
  409. E 1 = E 2 = E 3 E_{1}=E_{2}=E_{3}
  410. E 1 = E 2 > E 3 E_{1}=E_{2}>E_{3}
  411. E 1 > E 2 > E 3 E_{1}>E_{2}>E_{3}
  412. m x ¨ + k x = 0 m\ddot{x}+kx=0
  413. m x ¨ = - k x m\ddot{x}=-kx
  414. x x
  415. n n
  416. m m
  417. k k
  418. - k x = ω 2 m x -kx=\omega^{2}mx
  419. ω 2 \omega^{2}
  420. ω \omega
  421. k k
  422. m x ¨ + c x ˙ + k x = 0 m\ddot{x}+c\dot{x}+kx=0
  423. ( ω 2 m + ω c + k ) x = 0. (\omega^{2}m+\omega c+k)x=0.
  424. A A
  425. T - A T-A
  426. I - T - 1 / 2 A T - 1 / 2 I-T^{-1/2}AT^{-1/2}
  427. T T
  428. T i i T_{ii}
  429. v i v_{i}
  430. T - 1 / 2 T^{-1/2}
  431. i i
  432. 1 / deg ( v i ) 1/\sqrt{\operatorname{deg}(v_{i})}
  433. k k
  434. k k
  435. k k
  436. R 0 R_{0}
  437. R 0 R_{0}
  438. t G t_{G}
  439. t G t_{G}
  440. R 0 R_{0}

Eilenberg–Steenrod_axioms.html

  1. H n H_{n}
  2. : H i ( X , A ) H i - 1 ( A ) \partial:H_{i}(X,A)\to H_{i-1}(A)
  3. g : ( X , A ) ( Y , B ) g:(X,A)\rightarrow(Y,B)
  4. h : ( X , A ) ( Y , B ) h:(X,A)\rightarrow(Y,B)
  5. i : ( X - U , A - U ) ( X , A ) i:(X-U,A-U)\to(X,A)
  6. H n ( P ) = 0 H_{n}(P)=0
  7. n 0 n\neq 0
  8. X = α X α X=\coprod_{\alpha}{X_{\alpha}}
  9. X α X_{\alpha}
  10. H n ( X ) α H n ( X α ) . H_{n}(X)\cong\bigoplus_{\alpha}H_{n}(X_{\alpha}).
  11. i : A X i:A\to X
  12. j : X ( X , A ) j:X\to(X,A)
  13. H n ( A ) i * H n ( X ) j * H n ( X , A ) * H n - 1 ( A ) . \cdots\to H_{n}(A)\to^{\!\!\!\!\!\!i_{*}}H_{n}(X)\to^{\!\!\!\!\!\!j_{*}}H_{n}(% X,A)\to^{\!\!\!\!\!\!\partial_{*}}H_{n-1}(A)\to\cdots.

Einstein_coefficients.html

  1. ν ν
  2. h h
  3. n 2 n_{2}
  4. n 1 n_{1}
  5. ν ν
  6. ϵ \epsilon
  7. d V dV
  8. d t dt
  9. d Ω d\Omega
  10. ϵ = h ν 4 π n 2 A 21 \epsilon=\frac{h\nu}{4\pi}n_{2}A_{21}\,
  11. A 21 A_{21}
  12. κ \kappa
  13. κ = h ν 4 π ( n 1 B 12 - n 2 B 21 ) \kappa^{\prime}=\frac{h\nu}{4\pi}~{}(n_{1}B_{12}-n_{2}B_{21})\,
  14. B 12 B_{12}
  15. B 21 B_{21}
  16. A 21 A_{21}
  17. B 12 B_{12}
  18. B 21 B_{21}
  19. n 2 n_{2}
  20. n 1 n_{1}
  21. n 2 n_{2}
  22. n 1 n_{1}
  23. n 2 n_{2}
  24. n 1 n_{1}
  25. ν ν
  26. ρ ( ν ) ρ(ν)
  27. E 2 E_{2}
  28. E 1 E_{1}
  29. n i n_{i}
  30. ( d n 2 d t ) spontaneous = - A 21 n 2 . \left(\frac{dn_{2}}{dt}\right)_{\mathrm{spontaneous}}=-A_{21}n_{2}\,.
  31. ( d n 1 d t ) spontaneous = A 21 n 2 . \left(\frac{dn_{1}}{dt}\right)_{\mathrm{spontaneous}}=A_{21}n_{2}\,.
  32. B 21 B_{21}
  33. E 2 E_{2}
  34. E 1 E_{1}
  35. ( d n 1 d t ) neg absorb = B 21 n 2 ρ ( ν ) \left(\frac{dn_{1}}{dt}\right)_{\mathrm{neg}\,\mathrm{absorb}}=B_{21}n_{2}\rho% (\nu)
  36. ρ ( ν ) \rho(\nu)
  37. B 12 B_{12}
  38. E 1 E_{1}
  39. E 2 E_{2}
  40. ( d n 1 d t ) pos absorb = - B 12 n 1 ρ ( ν ) \left(\frac{dn_{1}}{dt}\right)_{\mathrm{pos}\,\mathrm{absorb}}=-B_{12}n_{1}% \rho(\nu)
  41. 0 = A 21 n 2 + B 21 n 2 ρ ( ν ) - B 12 n 1 ρ ( ν ) 0=A_{21}n_{2}+B_{21}n_{2}\rho(\nu)-B_{12}n_{1}\rho(\nu)\,
  42. T T
  43. n i n = g i e - E i / k T Z \frac{n_{i}}{n}=\frac{g_{i}e^{-E_{i}/kT}}{Z}
  44. g i g_{i}
  45. T T
  46. ν ν
  47. ρ ν ( ν , T ) = F ( ν ) 1 e h ν / k T - 1 \rho_{\nu}(\nu,T)=F(\nu)\frac{1}{e^{h\nu/kT}-1}
  48. F ( ν ) = 8 π h ν 3 c 3 F(\nu)=\frac{8\pi h\nu^{3}}{c^{3}}
  49. c c
  50. h h
  51. A 21 g 2 e - h ν / k T + B 21 g 2 e - h ν / k T F ( ν ) e h ν / k T - 1 = B 12 g 1 F ( ν ) e h ν / k T - 1 A_{21}g_{2}e^{-h\nu/kT}+B_{21}g_{2}e^{-h\nu/kT}\frac{F(\nu)}{e^{h\nu/kT}-1}=B_% {12}g_{1}\frac{F(\nu)}{e^{h\nu/kT}-1}
  52. A 21 g 2 ( e h ν / k T - 1 ) + B 21 g 2 F ( ν ) = B 12 g 1 e h ν / k T F ( ν ) A_{21}g_{2}(e^{h\nu/kT}-1)+B_{21}g_{2}F(\nu)=B_{12}g_{1}e^{h\nu/kT}F(\nu)\,
  53. A 21 g 2 = B 12 g 1 F ( ν ) A_{21}g_{2}=B_{12}g_{1}F(\nu)\,
  54. - A 21 g 2 + B 21 g 2 F ( ν ) = 0 -A_{21}g_{2}+B_{21}g_{2}F(\nu)=0\,
  55. A 21 B 21 = F ( ν ) \frac{A_{21}}{B_{21}}=F(\nu)
  56. B 21 B 12 = g 1 g 2 \frac{B_{21}}{B_{12}}=\frac{g_{1}}{g_{2}}
  57. A 21 A_{21}
  58. B 12 B_{12}
  59. f 12 f_{12}
  60. a 12 a_{12}
  61. a 12 = π e 2 2 ε 0 m e c f 12 a_{12}=\frac{\pi e^{2}}{2\varepsilon_{0}m_{e}c}\,f_{12}
  62. e e
  63. m e m_{e}
  64. B 12 = 4 π 2 e 2 m e h ν c f 12 B_{12}=\frac{4\pi^{2}e^{2}}{m_{e}h\nu c}\,f_{12}
  65. B 21 = 4 π 2 e 2 m e h ν c g 1 g 2 f 12 B_{21}=\frac{4\pi^{2}e^{2}}{m_{e}h\nu c}~{}\frac{g_{1}}{g_{2}}~{}f_{12}
  66. A 21 = 2 π ν 2 e 2 ε 0 m e c 3 g 1 g 2 f 12 A_{21}=\frac{2\pi\nu^{2}e^{2}}{\varepsilon_{0}m_{e}c^{3}}~{}\frac{g_{1}}{g_{2}% }~{}f_{12}
  67. A A
  68. s - 1 \,\text{s}^{-1}
  69. B B
  70. cm 3 s - 2 erg - 1 \,\text{cm}^{3}\,\text{s}^{-2}\,\text{erg}^{-1}

Einstein_force.html

  1. 𝐅 = - m d 2 𝐂 d t 2 . \,\textbf{F}=-m{d^{2}\,\textbf{C}\over dt^{2}}.

Einstein_relation_(kinetic_theory).html

  1. D = μ k B T D=\mu\,k_{B}T
  2. D = μ q k B T q D={{\mu_{q}\,k_{B}T}\over{q}}
  3. D = k B T 6 π η r D=\frac{k_{B}T}{6\pi\,\eta\,r}
  4. D = μ q k B T q D=\frac{\mu_{q}\,k_{B}T}{q}
  5. ζ \zeta
  6. γ = ζ / m \gamma=\zeta/m
  7. ζ = 6 π η r , \zeta=6\pi\,\eta\,r,
  8. η \eta
  9. D = k B T 6 π η r D=\frac{k_{\mathrm{B}}T}{6\pi\,\eta\,r}
  10. ζ r = 8 π η r 3 \zeta_{\mathrm{r}}=8\pi\eta r^{3}
  11. D r D_{\mathrm{r}}
  12. D r = k B T 8 π η r 3 D_{\mathrm{r}}=\frac{k_{\mathrm{B}}T}{8\pi\,\eta\,r^{3}}
  13. D = μ q p q d p d η = μ p p V p = μ p k B T q D=\frac{\mu_{q}\,p}{q\frac{dp}{d\eta}}=\mu_{p}p\frac{\partial V}{\partial p}=% \mu_{p}\frac{k_{B}T}{q}
  14. μ \mu
  15. k B k_{B}
  16. \nabla
  17. F = - d U / d x F=-dU/dx
  18. v = μ F v=\mu F
  19. ρ ( x ) \rho(x)
  20. J drift ( x ) = μ F ( x ) ρ ( x ) = - ρ ( x ) μ d U d x J_{\mathrm{drift}}(x)=\mu\,F(x)\,\rho(x)=-\rho(x)\mu\frac{dU}{dx}
  21. J diffusion ( x ) = - D d ρ d x J_{\mathrm{diffusion}}(x)=-D\frac{d\rho}{dx}
  22. 0 = J drift + J diffusion = - ρ ( x ) μ d U d x - D d ρ d x 0=J_{\mathrm{drift}}+J_{\mathrm{diffusion}}=-\rho(x)\mu\frac{dU}{dx}-D\frac{d% \rho}{dx}
  23. ρ ( x ) = A e - U / ( k B T ) \rho(x)=Ae^{-U/(k_{B}T)}
  24. d ρ d x = - 1 k B T d U d x ρ ( x ) . \frac{d\rho}{dx}=-\frac{1}{k_{B}T}\frac{dU}{dx}\rho(x).
  25. 0 = J drift + J diffusion = - ρ ( x ) μ d U d x + D k B T d U d x ρ ( x ) = - ρ ( x ) d U d x ( μ - D k B T ) 0=J_{\mathrm{drift}}+J_{\mathrm{diffusion}}=-\rho(x)\mu\frac{dU}{dx}+\frac{D}{% k_{B}T}\frac{dU}{dx}\rho(x)=-\rho(x)\frac{dU}{dx}\left(\mu-\frac{D}{k_{B}T}\right)
  26. μ = D k B T . \mu=\frac{D}{k_{B}T}.

Einstein_synchronisation.html

  1. τ 1 \tau_{1}
  2. τ 2 \tau_{2}
  3. τ 3 \tau_{3}
  4. τ 3 = τ 1 + 1 2 ( τ 2 - τ 1 ) = 1 2 ( τ 1 + τ 2 ) \tau_{3}=\tau_{1}+\tfrac{1}{2}(\tau_{2}-\tau_{1})=\tfrac{1}{2}(\tau_{1}+\tau_{% 2})
  5. v v
  6. x x
  7. t = t - v x V 2 t^{\prime}=t-\tfrac{vx}{V^{2}}

Eisenstein_integer.html

  1. z = a + b ω , z=a+b\omega,
  2. ω = 1 2 ( - 1 + i 3 ) = e 2 π i / 3 \omega=\frac{1}{2}(-1+i\sqrt{3})=e^{2\pi i/3}
  3. z 2 - ( 2 a - b ) z + ( a 2 - a b + b 2 ) . z^{2}-(2a-b)z+(a^{2}-ab+b^{2}).\,\!
  4. ω 2 + ω + 1 = 0. \omega^{2}+\omega+1=0.\,\!
  5. a + b ω a+b\omega
  6. c + d ω c+d\omega
  7. ( a + b ω ) ( c + d ω ) = ( a c - b d ) + ( b c + a d - b d ) ω . (a+b\omega)\cdot(c+d\omega)=(ac-bd)+(bc+ad-bd)\omega.\,\!
  8. | a + b ω | 2 = a 2 - a b + b 2 . |a+b\omega|^{2}=a^{2}-ab+b^{2}.\,\!
  9. 4 a 2 - 4 a b + 4 b 2 = ( 2 a - b ) 2 + 3 b 2 , 4a^{2}-4ab+4b^{2}=(2a-b)^{2}+3b^{2},\,\!
  10. N ( a + b ω ) = a 2 - a b + b 2 . N(a+b\,\omega)=a^{2}-ab+b^{2}.\,\!
  11. N ( a + b ω ) \displaystyle N(a+b\,\omega)

Eisenstein_prime.html

  1. z = a + b ω ( ω = e 2 π i / 3 ) z=a+b\,\omega\qquad(\omega=e^{2\pi i/3})

Elastic_recoil_detection.html

  1. E 2 = ( 4 m 1 m 2 ) ( m 1 + m 2 ) 2 ( E 1 cos 2 ϕ ) E_{2}=\frac{(4m_{1}m_{2})}{(m_{1}+m_{2})^{2}}(E_{1}\cos^{2}\phi)
  2. θ m a x = arcsin m 2 m 1 \theta_{max}=\arcsin\frac{m_{2}}{m_{1}}
  3. σ E R D = ( Z 1 Z 2 e 2 2 E 1 ) 2 ( m 1 + m 2 m 2 ) 2 cos - 3 ϕ \sigma_{ERD}=\left(\frac{Z_{1}Z_{2}e^{2}}{2E_{1}}\right)^{2}\left(\frac{m_{1}+% m_{2}}{m_{2}}\right)^{2}\cos^{-3}\phi
  4. δ x = δ E 2 E 2 S r e l - 1 \delta x=\frac{\delta E_{2}}{E_{2}}S_{rel}^{-1}
  5. S r e l = d E 1 d x E 1 1 sin α + d E 2 d x E 2 1 sin β S_{rel}=\frac{\frac{dE_{1}}{dx}}{E_{1}}\frac{1}{\sin\alpha}+\frac{\frac{dE_{2}% }{dx}}{E_{2}}\frac{1}{\sin\beta}
  6. δ E k i n = 2 E 2 tan ϕ δ ϕ \delta E_{kin}=2E_{2}\tan\phi\ \delta\phi
  7. E K E 0 E\approx KE_{0}
  8. K s = ( cos θ ± r 2 - sin 2 θ ( 1 + r ) ) 2 K_{s}=\left(\frac{\cos\theta\pm\sqrt{r^{2}-\sin^{2}\theta}}{(1+r)}\right)^{2}
  9. K r = 4 r cos 2 ϕ ( 1 + r ) 2 Kr=\frac{4r\,\cos^{2}\phi}{(1+r)^{2}}
  10. r = M 1 M 2 r=\frac{M_{1}}{M_{2}}
  11. x = l - r l + r x=\frac{l-r}{l+r}
  12. y = l + r E t o t y=\frac{l+r}{E_{tot}}
  13. | q | B m \frac{|q|B}{m}
  14. L 1 = x cos θ 1 L_{1}=\frac{x}{\cos\theta_{1}}
  15. L 2 = x cos θ 2 L_{2}=\frac{x}{\cos\theta_{2}}
  16. L 3 = x cos θ 3 L_{3}=\frac{x}{\cos\theta_{3}}
  17. R ( ϕ , α ) = c o s θ 1 c o s α sin ϕ c o s 2 α - c o s 2 θ 1 - c o s θ 1 c o s ϕ R(\phi,\alpha)={cos\theta_{1}cos\alpha}\over{\sin\phi\sqrt{cos^{2}\alpha-cos^{% 2}\theta_{1}}-cos\theta_{1}cos\phi}
  18. E 0 ( x ) = E 0 - 0 ( x / c o s θ 1 ) S ( E ) d L 1 - - - E q u a t i o n ( 1 ) E_{0}(x)=E_{0}-\int_{0}^{(x/cos\theta 1)}S(E)\,dL_{1}---Equation(1)
  19. E 1 ( x ) = K E 0 ( x ) - 0 ( x / c o s θ 2 ) S ( E ) d L 2 - - - E q u a t i o n ( 2 ) E_{1}(x)=KE_{0}(x)-\int_{0}^{(x/cos\theta 2)}S(E)\,dL_{2}---Equation(2)
  20. E 2 ( x ) = K E 0 ( x ) - 0 ( x / c o s θ 3 ) S r ( E ) d L 3 - - - E q u a t i o n ( 3 ) E_{2}(x)=K^{\prime}E_{0}(x)-\int_{0}^{(x/cos\theta 3)}Sr(E)\,dL_{3}---Equation% (3)
  21. S = d E d x S=\frac{dE}{dx}
  22. Δ E = 0 Δ x d E d x d x \Delta E=\int_{0}^{\Delta x}\frac{dE}{dx}\,dx
  23. L 1 = E 0 ( x ) E 0 d E S ( E ) L_{1}=\int_{E_{0}(x)}^{E_{0}}\frac{dE}{S(E)}
  24. L 2 = E 1 ( x ) E 0 1 ( x ) d E S ( E ) L_{2}=\int_{E_{1}(x)}^{E_{0}1(x)}\frac{dE}{S(E)}
  25. L 3 = E 2 ( x ) E 0 2 ( x ) d E S r ( E ) L_{3}=\int_{E_{2}(x)}^{E_{0}2(x)}\frac{dE}{Sr(E)}

Electrodynamic_tether.html

  1. V emf = 0 L ( v orb × B ) d L . V_{\mathrm{emf}}=\int_{0}^{L}\left(\vec{v}_{\mathrm{orb}}\times\vec{B}\right)d% \vec{L}.
  2. F = 0 L I ( L ) m d L × B . \vec{F}=\int_{0}^{L}I(L)m\,d\vec{L}\times\vec{B}.
  3. V anode V_{\mathrm{anode}}
  4. V cathode V_{\mathrm{cathode}}
  5. V - V p V-V_{p}
  6. V anode + A C I ( y ) d R t + R load I C + V emit + V cathode = V emf V_{\mathrm{anode}}+\int_{A}^{C}I(y)\,dR_{t}+R_{\mathrm{load}}I_{C}+V_{\mathrm{% emit}}+V_{\mathrm{cathode}}=V_{\mathrm{emf}}
  7. I A B = I B C + I C I_{AB}=I_{BC}+I_{C}
  8. I A B I_{AB}
  9. I B C I_{BC}
  10. I C I_{C}
  11. A C I ( y ) d R t \textstyle\int_{A}^{C}I(y)\,dR_{t}
  12. d R t dR_{t}
  13. I ( y ) I(y)
  14. V - V p V-V_{p}
  15. R load R_{\mathrm{load}}
  16. V emit V_{\mathrm{emit}}
  17. V cathode V_{\mathrm{cathode}}
  18. V = 𝐄 𝐋 = E L cos τ = v B L cos τ V=\mathbf{E}\cdot\mathbf{L}=EL\cos\tau=vBL\cos\tau
  19. λ De ε 0 T e q n 0 . \lambda_{\mathrm{De}}\cong\sqrt{\frac{\varepsilon_{0}T_{e}}{qn_{0}}}.
  20. J t h e = A R T 2 e - ϕ / ( k t ) . J_{the}=A_{R}T^{2}e^{-\phi/(kt)}.
  21. Δ V t c = [ η I t ρ ] 2 / 3 \Delta V_{tc}=\left[\frac{\eta\cdot I_{t}}{\rho}\right]^{2/3}
  22. J t h e = A F N E F N 2 e - B F N / E F N . J_{the}=A_{FN}\cdot E_{FN}^{2}\cdot e^{-B_{FN}/E_{FN}}.
  23. 𝐁 = B x x ^ + B y y ^ + B z z ^ \mathbf{B}=B_{x}\hat{x}+B_{y}\hat{y}+B_{z}\hat{z}
  24. 𝐋 = L cos α out sin α in x ^ + L sin α out y ^ + L cos α out cos α in z ^ \mathbf{L}=L\cos\alpha_{\mathrm{out}}\sin\alpha_{\mathrm{in}}\hat{x}+L\sin% \alpha_{\mathrm{out}}\hat{y}+L\cos\alpha_{\mathrm{out}}\cos\alpha_{\mathrm{in}% }\hat{z}
  25. 𝐯 orb = v orb cos λ out sin λ in x ^ + v orb sin λ out y ^ + v orb cos λ out cos λ in z ^ \mathbf{v}_{\mathrm{orb}}=v_{\mathrm{orb}}\cos\lambda_{\mathrm{out}}\sin% \lambda_{\mathrm{in}}\hat{x}+v_{\mathrm{orb}}\sin\lambda_{\mathrm{out}}\hat{y}% +v_{\mathrm{orb}}\cos\lambda_{\mathrm{out}}\cos\lambda_{\mathrm{in}}\hat{z}

Electrogastrogram.html

  1. P ( i ) = n = s t i n = f i n i | S ( n ) | 2 , i = 1 , , 5 P(i)=\sum_{n=st_{i}}^{n=fin_{i}}|S(n)|^{2},i=1,\dots,5
  2. P S = i = 1 i = 5 P ( i ) . PS=\sum_{i=1}^{i=5}P(i).
  3. K r i t m ( i ) = 1 f i n i - s t i n = s t i n = f i n i - 1 | S ( n ) - S ( n - 1 ) | , i = 1 , , 5 Kritm(i)=\frac{1}{fin_{i}-st_{i}}\sum_{n=st_{i}}^{n=fin_{i}-1}|S(n)-S(n-1)|,i=% 1,\dots,5

Electromagnetic_electron_wave.html

  1. ω ( 4 π n e e 2 / m e ) 1 / 2 \omega>>(4\pi n_{e}e^{2}/m_{e})^{1/2}
  2. n e m e ω 2 / 4 π e 2 n_{e}<<m_{e}\omega^{2}\,/\,4\pi e^{2}
  3. n c = ε o m e e 2 ω 2 n_{c}=\frac{\varepsilon_{o}\,m_{e}}{e^{2}}\,\omega^{2}
  4. ω R = 1 2 [ ω c + ( ω c 2 + 4 ω p 2 ) 1 / 2 ] \omega_{R}=\frac{1}{2}\left[\omega_{c}+(\omega_{c}^{2}+4\omega_{p}^{2})^{1/2}\right]
  5. ω L = 1 2 [ - ω c + ( ω c 2 + 4 ω p 2 ) 1 / 2 ] \omega_{L}=\frac{1}{2}\left[-\omega_{c}+(\omega_{c}^{2}+4\omega_{p}^{2})^{1/2}\right]
  6. ω c \omega_{c}
  7. ω p \omega_{p}
  8. B 0 = 0 \vec{B}_{0}=0
  9. ω 2 = ω p 2 + k 2 c 2 \omega^{2}=\omega_{p}^{2}+k^{2}c^{2}
  10. k B 0 , E 1 B 0 \vec{k}\perp\vec{B}_{0},\ \vec{E}_{1}\|\vec{B}_{0}
  11. c 2 k 2 ω 2 = 1 - ω p 2 ω 2 \frac{c^{2}k^{2}}{\omega^{2}}=1-\frac{\omega_{p}^{2}}{\omega^{2}}
  12. k B 0 , E 1 B 0 \vec{k}\perp\vec{B}_{0},\ \vec{E}_{1}\perp\vec{B}_{0}
  13. c 2 k 2 ω 2 = 1 - ω p 2 ω 2 ω 2 - ω p 2 ω 2 - ω h 2 \frac{c^{2}k^{2}}{\omega^{2}}=1-\frac{\omega_{p}^{2}}{\omega^{2}}\,\frac{% \omega^{2}-\omega_{p}^{2}}{\omega^{2}-\omega_{h}^{2}}
  14. k B 0 \vec{k}\|\vec{B}_{0}
  15. c 2 k 2 ω 2 = 1 - ω p 2 / ω 2 1 - ( ω c / ω ) \frac{c^{2}k^{2}}{\omega^{2}}=1-\frac{\omega_{p}^{2}/\omega^{2}}{1-(\omega_{c}% /\omega)}
  16. k B 0 \vec{k}\|\vec{B}_{0}
  17. c 2 k 2 ω 2 = 1 - ω p 2 / ω 2 1 + ( ω c / ω ) \frac{c^{2}k^{2}}{\omega^{2}}=1-\frac{\omega_{p}^{2}/\omega^{2}}{1+(\omega_{c}% /\omega)}

Electromagnetic_wave_equation.html

  1. 𝐄 \mathbf{E}
  2. 𝐁 \mathbf{B}
  3. ( c 2 2 - 2 t 2 ) 𝐄 \displaystyle\left(c^{2}\nabla^{2}-\frac{\partial^{2}}{\partial t^{2}}\right)% \mathbf{E}
  4. c = 1 μ 0 ε 0 c=\frac{1}{\sqrt{\mu_{0}\varepsilon_{0}}}
  5. μ μ
  6. ε ε
  7. c = 299 , 792 , 458 c=299,792,458
  8. 𝐁 \mathbf{B}
  9. 𝐄 \displaystyle\nabla\cdot\mathbf{E}
  10. × ( × 𝐄 ) \displaystyle\nabla\times\left(\nabla\times\mathbf{E}\right)
  11. × ( × 𝐕 ) = ( 𝐕 ) - 2 𝐕 \nabla\times\left(\nabla\times\mathbf{V}\right)=\nabla\left(\nabla\cdot\mathbf% {V}\right)-\nabla^{2}\mathbf{V}
  12. 𝐕 \mathbf{V}
  13. 2 𝐕 = ( 𝐕 ) \nabla^{2}\mathbf{V}=\nabla\cdot\left(\nabla\mathbf{V}\right)
  14. 𝐕 ∇\mathbf{V}
  15. ∇⋅
  16. 𝐄 = 0 𝐁 = 0 \begin{aligned}\displaystyle\nabla\cdot\mathbf{E}&\displaystyle=0\\ \displaystyle\nabla\cdot\mathbf{B}&\displaystyle=0\end{aligned}
  17. 2 𝐄 t 2 - c 0 2 2 𝐄 \displaystyle\frac{\partial^{2}\mathbf{E}}{\partial t^{2}}-c_{0}^{2}\cdot% \nabla^{2}\mathbf{E}
  18. c 0 = 1 μ 0 ε 0 = 2.99792458 × 10 8 m/s c_{0}=\frac{1}{\sqrt{\mu_{0}\varepsilon_{0}}}=2.99792458\times 10^{8}\;\textrm% {m/s}
  19. A μ = 0 \Box A^{\mu}=0
  20. A μ = ( ϕ c , 𝐀 ) A^{\mu}=\left(\frac{\phi}{c},\mathbf{A}\right)
  21. μ A μ = 0 , \partial_{\mu}A^{\mu}=0,
  22. - = 2 - 1 c 2 2 t 2 -\Box=\nabla^{2}-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}
  23. - A α ; β ; β + R α β A β = 0 -{A^{\alpha;\beta}}_{;\beta}+{R^{\alpha}}_{\beta}A^{\beta}=0
  24. R α β \scriptstyle{R^{\alpha}}_{\beta}
  25. A μ ; μ = 0. {A^{\mu}}_{;\mu}=0.
  26. 𝐄 ( 𝐫 , t ) = g ( ϕ ( 𝐫 , t ) ) = g ( ω t - 𝐤 𝐫 ) \mathbf{E}(\mathbf{r},t)=g(\phi(\mathbf{r},t))=g(\omega t-\mathbf{k}\cdot% \mathbf{r})
  27. 𝐁 ( 𝐫 , t ) = g ( ϕ ( 𝐫 , t ) ) = g ( ω t - 𝐤 𝐫 ) \mathbf{B}(\mathbf{r},t)=g(\phi(\mathbf{r},t))=g(\omega t-\mathbf{k}\cdot% \mathbf{r})
  28. g g
  29. φ φ
  30. ω ω
  31. g g
  32. g g
  33. k = | 𝐤 | = ω c = 2 π λ k=|\mathbf{k}|={\omega\over c}={2\pi\over\lambda}
  34. k k
  35. λ λ
  36. c c
  37. 𝐄 ( 𝐫 , t ) = { 𝐄 ( 𝐫 ) e i ω t } \mathbf{E}(\mathbf{r},t)=\Re\left\{\mathbf{E}(\mathbf{r})e^{i\omega t}\right\}
  38. i i
  39. ω = 2 π f ω=2πf
  40. f f
  41. e i ω t = cos ( ω t ) + i sin ( ω t ) \scriptstyle e^{i\omega t}=\cos(\omega t)+i\sin(\omega t)
  42. 𝐧 = 𝐤 k . \mathbf{n}={\mathbf{k}\over k}.
  43. 𝐄 ( 𝐫 ) = 𝐄 0 e - i 𝐤 𝐫 \mathbf{E}(\mathbf{r})=\mathbf{E}_{0}e^{-i\mathbf{k}\cdot\mathbf{r}}
  44. 𝐁 ( 𝐫 ) = 𝐁 0 e - i 𝐤 𝐫 \mathbf{B}(\mathbf{r})=\mathbf{B}_{0}e^{-i\mathbf{k}\cdot\mathbf{r}}
  45. 𝐫 = ( x , y , z ) \mathbf{r}=(x,y,z)
  46. 𝐧 \mathbf{n}
  47. 𝐧 \mathbf{n}
  48. 𝐄 \mathbf{E}
  49. c 2 B z = E t . c^{2}{\partial B\over\partial z}={\partial E\over\partial t}.
  50. 𝐄 ( 𝐫 , t ) = 𝐄 0 cos ( ω t - 𝐤 𝐫 + ϕ 0 ) \mathbf{E}(\mathbf{r},t)=\mathbf{E}_{0}\cos(\omega t-\mathbf{k}\cdot\mathbf{r}% +\phi_{0})
  51. 𝐁 ( 𝐫 , t ) = 𝐁 0 cos ( ω t - 𝐤 𝐫 + ϕ 0 ) \mathbf{B}(\mathbf{r},t)=\mathbf{B}_{0}\cos(\omega t-\mathbf{k}\cdot\mathbf{r}% +\phi_{0})
  52. t t
  53. ω ω
  54. ϕ 0 \scriptstyle\phi_{0}
  55. k = | 𝐤 | = ω c = 2 π λ k=|\mathbf{k}|={\omega\over c}={2\pi\over\lambda}
  56. k k
  57. λ λ
  58. e - i ω t e^{-i\omega t}
  59. 𝐁 \mathbf{B}
  60. 𝐄 \mathbf{E}
  61. ( 2 + k 2 ) 𝐄 = 0 , 𝐁 = - i k × 𝐄 , (\nabla^{2}+k^{2})\mathbf{E}=0,\,\mathbf{B}=-\frac{i}{k}\nabla\times\mathbf{E},
  62. 𝐄 \mathbf{E}
  63. 𝐁 \mathbf{B}
  64. ( 2 + k 2 ) 𝐁 = 0 , 𝐄 = - i k × 𝐁 . (\nabla^{2}+k^{2})\mathbf{B}=0,\,\mathbf{E}=-\frac{i}{k}\nabla\times\mathbf{B}.
  65. ω ω
  66. 𝐄 \mathbf{E}
  67. 𝐁 \mathbf{B}
  68. · 𝐄 = · 𝐁 = 0 \mathbf{∇}·\mathbf{E}=\mathbf{∇}·\mathbf{B}=0
  69. 𝐄 \mathbf{E}
  70. 𝐁 \mathbf{B}
  71. 𝐫 · 𝐄 \mathbf{r·E}
  72. 𝐫 · 𝐁 \mathbf{r·B}
  73. 𝐄 \mathbf{E}
  74. 𝐁 \mathbf{B}
  75. 𝐅 \mathbf{F}
  76. 𝐄 = e - i ω t l , m l ( l + 1 ) [ a E ( l , m ) 𝐄 l , m ( E ) + a M ( l , m ) 𝐄 l , m ( M ) ] \mathbf{E}=e^{-i\omega t}\sum_{l,m}\sqrt{l(l+1)}\left[a_{E}(l,m)\mathbf{E}_{l,% m}^{(E)}+a_{M}(l,m)\mathbf{E}_{l,m}^{(M)}\right]
  77. 𝐁 = e - i ω t l , m l ( l + 1 ) [ a E ( l , m ) 𝐁 l , m ( E ) + a M ( l , m ) 𝐁 l , m ( M ) ] \mathbf{B}=e^{-i\omega t}\sum_{l,m}\sqrt{l(l+1)}\left[a_{E}(l,m)\mathbf{B}_{l,% m}^{(E)}+a_{M}(l,m)\mathbf{B}_{l,m}^{(M)}\right]
  78. 𝐄 l , m ( E ) \mathbf{E}_{l,m}^{(E)}
  79. 𝐁 l , m ( E ) \mathbf{B}_{l,m}^{(E)}
  80. 𝐄 l , m ( M ) \mathbf{E}_{l,m}^{(M)}
  81. 𝐁 l , m ( M ) \mathbf{B}_{l,m}^{(M)}
  82. 𝐁 l , m ( E ) = l ( l + 1 ) [ B l ( 1 ) h l ( 1 ) ( k r ) + B l ( 2 ) h l ( 2 ) ( k r ) ] 𝚽 l , m \mathbf{B}_{l,m}^{(E)}=\sqrt{l(l+1)}\left[B_{l}^{(1)}h_{l}^{(1)}(kr)+B_{l}^{(2% )}h_{l}^{(2)}(kr)\right]\mathbf{\Phi}_{l,m}
  83. 𝐄 l , m ( E ) = i k × 𝐁 l , m ( E ) \mathbf{E}_{l,m}^{(E)}=\frac{i}{k}\nabla\times\mathbf{B}_{l,m}^{(E)}
  84. 𝐄 l , m ( M ) = l ( l + 1 ) [ E l ( 1 ) h l ( 1 ) ( k r ) + E l ( 2 ) h l ( 2 ) ( k r ) ] 𝚽 l , m \mathbf{E}_{l,m}^{(M)}=\sqrt{l(l+1)}\left[E_{l}^{(1)}h_{l}^{(1)}(kr)+E_{l}^{(2% )}h_{l}^{(2)}(kr)\right]\mathbf{\Phi}_{l,m}
  85. 𝐁 l , m ( M ) = - i k × 𝐄 l , m ( M ) \mathbf{B}_{l,m}^{(M)}=-\frac{i}{k}\nabla\times\mathbf{E}_{l,m}^{(M)}
  86. 𝚽 l , m = 1 l ( l + 1 ) ( 𝐫 × ) Y l , m \mathbf{\Phi}_{l,m}=\frac{1}{\sqrt{l(l+1)}}(\mathbf{r}\times\nabla)Y_{l,m}
  87. 𝚽 l , m * 𝚽 l , m d Ω = δ l , l δ m , m . \int\mathbf{\Phi}^{*}_{l,m}\cdot\mathbf{\Phi}_{l^{\prime},m^{\prime}}d\Omega=% \delta_{l,l^{\prime}}\delta_{m,m^{\prime}}.
  88. 𝐄 \mathbf{E}
  89. 𝐁 \mathbf{B}
  90. 𝐁 e i ( k r - ω t ) k r l , m ( - i ) l + 1 [ a E ( l , m ) 𝚽 l , m + a M ( l , m ) 𝐫 ^ × 𝚽 l , m ] \mathbf{B}\approx\frac{e^{i(kr-\omega t)}}{kr}\sum_{l,m}(-i)^{l+1}\left[a_{E}(% l,m)\mathbf{\Phi}_{l,m}+a_{M}(l,m)\mathbf{\hat{r}}\times\mathbf{\Phi}_{l,m}\right]
  91. 𝐄 𝐁 × 𝐫 ^ . \mathbf{E}\approx\mathbf{B}\times\mathbf{\hat{r}}.
  92. d P d Ω 1 2 k 2 | l , m ( - i ) l + 1 [ a E ( l , m ) 𝚽 l , m × 𝐫 ^ + a M ( l , m ) 𝚽 l , m ] | 2 . \frac{dP}{d\Omega}\approx\frac{1}{2k^{2}}\left|\sum_{l,m}(-i)^{l+1}\left[a_{E}% (l,m)\mathbf{\Phi}_{l,m}\times\mathbf{\hat{r}}+a_{M}(l,m)\mathbf{\Phi}_{l,m}% \right]\right|^{2}.
  93. 𝐄 ( 𝐫 , t ) = 1 r 𝐄 0 cos ( ω t - k r + ϕ 0 ) , \mathbf{E}(\mathbf{r},t)=\frac{1}{r}\mathbf{E}_{0}\cos(\omega t-k\cdot r+\phi_% {0}),
  94. 𝐄 ( 𝐫 , t ) = 1 r 𝐄 0 sin ( ω t - k r + ϕ 0 ) , \mathbf{E}(\mathbf{r},t)=\frac{1}{r}\mathbf{E}_{0}\sin(\omega t-k\cdot r+\phi_% {0}),
  95. 𝐁 ( 𝐫 , t ) = 1 r 𝐁 0 cos ( ω t - k r + ϕ 0 ) , \mathbf{B}(\mathbf{r},t)=\frac{1}{r}\mathbf{B}_{0}\cos(\omega t-k\cdot r+\phi_% {0}),
  96. 𝐁 ( 𝐫 , t ) = 1 r 𝐁 0 sin ( ω t - k r + ϕ 0 ) . \mathbf{B}(\mathbf{r},t)=\frac{1}{r}\mathbf{B}_{0}\sin(\omega t-k\cdot r+\phi_% {0}).
  97. c < s u b > 0 c<sub>0

Electron-capture_dissociation.html

  1. [ M + n H ] n + + e - [ [ M + n H ] ( n - 1 ) + ] * f r a g m e n t s [M+nH]^{n+}+e^{-}\to\bigg[[M+nH]^{(n-1)+}\bigg]^{*}\to fragments

Electron_acceptor.html

  1. Δ E = A - I {\Delta}E=A-I\,

Electron_avalanche.html

  1. M = 1 1 - X 1 X 2 α d x M=\frac{1}{1-\int_{X_{1}}^{X_{2}}\alpha\,dx}
  2. M = 1 1 - | V V BR | n M=\frac{1}{1-|\frac{V}{V_{\mathrm{BR}}}|^{n}}

Electron_bubble.html

  1. E h 2 8 m R 2 + 4 π R 2 α + 4 3 π R 3 P E\approx\frac{h^{2}}{8mR^{2}}+4\pi R^{2}\alpha+\frac{4}{3}\pi R^{3}P

Electron_donor.html

  1. Δ E = A - I {\Delta}E=A-I\,

Electron_optics.html

  1. r = 2 m c v e H r=\frac{2mcv}{eH}

Electronic_density.html

  1. ρ ( 𝐫 ) = N s 1 s N d 𝐫 2 d 𝐫 N | Ψ ( 𝐫 , s 1 , 𝐫 2 , s 2 , , 𝐫 N , s N ) | 2 , = Ψ | ρ ^ ( 𝐫 ) | Ψ , \begin{aligned}\displaystyle\rho(\mathbf{r})&\displaystyle=N\sum_{{s}_{1}}% \cdots\sum_{{s}_{N}}\int\ \mathrm{d}\mathbf{r}_{2}\ \cdots\int\ \mathrm{d}% \mathbf{r}_{N}\ |\Psi(\mathbf{r},s_{1},\mathbf{r}_{2},s_{2},...,\mathbf{r}_{N}% ,s_{N})|^{2},\\ &\displaystyle=\langle\Psi|\hat{\rho}(\mathbf{r})|\Psi\rangle,\end{aligned}
  2. ρ ^ ( 𝐫 ) = i = 1 N s i δ ( 𝐫 - 𝐫 i ) . \hat{\rho}(\mathbf{r})=\sum_{i=1}^{N}\sum_{s_{i}}\ \delta(\mathbf{r}-\mathbf{r% }_{i}).
  3. ρ ( 𝐫 ) = k = 1 N n k | φ k ( 𝐫 ) | 2 . \rho(\mathbf{r})=\sum_{k=1}^{N}n_{k}|\varphi_{k}(\mathbf{r})|^{2}.
  4. 1 2 d 𝐫 ( ρ ( 𝐫 ) ) 2 T . \frac{1}{2}\int\mathrm{d}\mathbf{r}\ \big(\nabla\sqrt{\rho(\mathbf{r})}\big)^{% 2}\leq T.
  5. 3 2 ( π 2 ) 4 / 3 ( d 𝐫 ρ 3 ( 𝐫 ) ) 1 / 3 T . \frac{3}{2}\left(\frac{\pi}{2}\right)^{4/3}\left(\int\mathrm{d}\mathbf{r}\ % \rho^{3}(\mathbf{r})\right)^{1/3}\leq T.
  6. 𝒥 N = { ρ | ρ ( 𝐫 ) 0 , ρ 1 / 2 ( 𝐫 ) H 1 ( 𝐑 3 ) , d 𝐫 ρ ( 𝐫 ) = N } . \mathcal{J}_{N}=\left\{\rho\left|\rho(\mathbf{r})\geq 0,\ \rho^{1/2}(\mathbf{r% })\in H^{1}(\mathbf{R}^{3}),\ \int\mathrm{d}\mathbf{r}\ \rho(\mathbf{r})=N% \right.\right\}.
  7. 𝒥 N \mathcal{J}_{N}
  8. ρ ¯ \bar{\rho}
  9. r α ρ ¯ ( r α ) | r α = 0 = - 2 Z α ρ ¯ ( 0 ) . \left.\frac{\partial}{\partial r_{\alpha}}\bar{\rho}(r_{\alpha})\right|_{r_{% \alpha}=0}=-2Z_{\alpha}\bar{\rho}(0).
  10. ρ ( r ) e - 2 Z α r . \rho(r)\sim e^{-2Z_{\alpha}r}\,.
  11. ρ ( r ) e - 2 2 I r . \rho(r)\sim e^{-2\sqrt{2\mathrm{I}}r}\,.

Electrospray.html

  1. d d\,
  2. ( μ ) (\mu)\,
  3. ( γ ) (\gamma)\,
  4. ( κ ) (\kappa)\,
  5. ( ϵ r ) (\epsilon_{r})\,
  6. V V\,
  7. E = 2 V r ln ( 4 d / r ) E={2V\over r\ln(4d/r)}
  8. r r\,
  9. R 1 / 2 R^{1/2}\,
  10. V = V 0 + A R 1 / 2 P 1 / 2 ( cos θ 0 ) V=V_{0}+AR^{1/2}P_{1/2}(\cos\theta_{0})\,
  11. V = V 0 V=V_{0}\,
  12. θ 0 \theta_{0}
  13. V = V 0 V=V_{0}\,
  14. P 1 / 2 ( cos θ 0 ) P_{1/2}(\cos\theta_{0})\,
  15. π \pi\,
  16. τ H = μ r γ \tau_{H}={\mu r\over\gamma}
  17. τ C = ϵ r ϵ 0 κ \tau_{C}={\epsilon_{r}\epsilon_{0}\over\kappa}
  18. ( r ) (r)\,
  19. ( ϵ 0 ) (\epsilon_{0})\,

Electrovacuum_solution.html

  1. g a b \,g_{ab}
  2. R a b c d \,R_{abcd}
  3. G a b G^{ab}
  4. F a b F_{ab}
  5. F a b ; c + F b c ; a + F c a ; b = 0 \,F_{ab;c}+F_{bc;a}+F_{ca;b}=0
  6. F j b ; j = 0 {F^{jb}}_{;j}=0
  7. G a b = 2 ( F a F b j j - 1 4 g a b F m n F m n ) G^{ab}=2\,\left(F^{a}{}_{j}F^{bj}-\frac{1}{4}g^{ab}\,F^{mn}\,F_{mn}\right)
  8. A \vec{A}
  9. F = d A F=dA
  10. I = ( F F ) = F a b F a b = - 2 ( E 2 - B 2 ) I=\star(F\wedge\star F)=F_{ab}\,F^{ab}=-2\,\left(\|\vec{E}\|^{2}-\|\vec{B}\|^{% 2}\right)
  11. J = ( F F ) = F a b F a b = - 4 E B J=\star(F\wedge F)=F_{ab}\,{\star F}^{ab}=-4\,\vec{E}\cdot\vec{B}
  12. I < 0 I<0
  13. J = 0 J=0
  14. I > 0 I>0
  15. J = 0 J=0
  16. I = J = 0 I=J=0
  17. e 0 , e 1 , e 2 , e 3 \vec{e}_{0},\;\vec{e}_{1},\;\vec{e}_{2},\;\vec{e}_{3}
  18. G a ^ b ^ = 8 π ϵ [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 - 1 ] G^{\hat{a}\hat{b}}=8\pi\epsilon\,\left[\begin{matrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&-1\end{matrix}\right]
  19. ϵ \epsilon
  20. e 3 \vec{e}_{3}
  21. e 3 \vec{e}_{3}
  22. G a ^ b ^ = 8 π ϵ [ 1 0 0 ± 1 0 0 0 0 0 0 0 0 ± 1 0 0 1 ] G^{\hat{a}\hat{b}}=8\pi\epsilon\,\left[\begin{matrix}1&0&0&\pm 1\\ 0&0&0&0\\ 0&0&0&0\\ \pm 1&0&0&1\end{matrix}\right]
  23. e 3 \vec{e}_{3}
  24. e 3 \vec{e}_{3}
  25. χ ( λ ) = ( λ + 8 π ϵ ) 2 ( λ - 8 π ϵ ) 2 \chi(\lambda)=\left(\lambda+8\pi\epsilon\right)^{2}\,\left(\lambda-8\pi% \epsilon\right)^{2}
  26. t 1 = t 3 = 0 , t 4 = t 2 2 / 4 t_{1}=t_{3}=0,\;t_{4}=t_{2}^{2}/4
  27. t 1 = G a a , t 2 = G a b G b a , t 3 = G a b G b c G c a , t 4 = G a b G b c G c d G d a t_{1}={G^{a}}_{a},\;t_{2}={G^{a}}_{b}\,{G^{b}}_{a},\;t_{3}={G^{a}}_{b}\,{G^{b}% }_{c}\,{G^{c}}_{a},\;t_{4}={G^{a}}_{b}\,{G^{b}}_{c}\,{G^{c}}_{d}\,{G^{d}}_{a}

Elliptic_complex.html

  1. Γ ( E 0 ) P 1 Γ ( E 1 ) P 2 P k Γ ( E k ) \Gamma(E_{0})\stackrel{P_{1}}{\longrightarrow}\Gamma(E_{1})\stackrel{P_{2}}{% \longrightarrow}\ldots\stackrel{P_{k}}{\longrightarrow}\Gamma(E_{k})
  2. 0 π * E 0 σ ( P 1 ) π * E 1 σ ( P 2 ) σ ( P k ) π * E k 0 0\rightarrow\pi^{*}E_{0}\stackrel{\sigma(P_{1})}{\longrightarrow}\pi^{*}E_{1}% \stackrel{\sigma(P_{2})}{\longrightarrow}\ldots\stackrel{\sigma(P_{k})}{% \longrightarrow}\pi^{*}E_{k}\rightarrow 0

Elliptic_gamma_function.html

  1. Γ ( z ; p , q ) = m = 0 n = 0 1 - p m + 1 q n + 1 / z 1 - p m q n z . \Gamma(z;p,q)=\prod_{m=0}^{\infty}\prod_{n=0}^{\infty}\frac{1-p^{m+1}q^{n+1}/z% }{1-p^{m}q^{n}z}.
  2. Γ ( z ; p , q ) = 1 Γ ( p q / z ; p , q ) \Gamma(z;p,q)=\frac{1}{\Gamma(pq/z;p,q)}\,
  3. Γ ( p z ; p , q ) = θ ( z ; q ) Γ ( z ; p , q ) \Gamma(pz;p,q)=\theta(z;q)\Gamma(z;p,q)\,
  4. Γ ( q z ; p , q ) = θ ( z ; p ) Γ ( z ; p , q ) \Gamma(qz;p,q)=\theta(z;p)\Gamma(z;p,q)\,
  5. p = 0 p=0
  6. Γ ( z ; 0 , q ) = 1 ( z ; q ) . \Gamma(z;0,q)=\frac{1}{(z;q)_{\infty}}.

Elliptic_surface.html

  1. ( 1 ν 1 1 ) \begin{pmatrix}1&\nu\\ 1&1\end{pmatrix}
  2. \infty
  3. 𝐙 / ν × 𝐂 * \mathbf{Z}/\nu\times\mathbf{C}^{*}
  4. ( 1 1 - 1 0 ) \begin{pmatrix}1&1\\ -1&0\end{pmatrix}
  5. 𝐂 \mathbf{C}
  6. ( 0 1 - 1 0 ) \begin{pmatrix}0&1\\ -1&0\end{pmatrix}
  7. 𝐙 / 2 × 𝐂 \mathbf{Z}/2\times\mathbf{C}
  8. ( 0 1 - 1 - 1 ) \begin{pmatrix}0&1\\ -1&-1\end{pmatrix}
  9. 𝐙 / 3 × 𝐂 \mathbf{Z}/3\times\mathbf{C}
  10. ( - 1 - ν 0 - 1 ) \begin{pmatrix}-1&-\nu\\ 0&-1\end{pmatrix}
  11. \infty
  12. ( 𝐙 / 2 ) 2 × 𝐂 (\mathbf{Z}/2)^{2}\times\mathbf{C}
  13. 𝐙 / 4 × 𝐂 \mathbf{Z}/4\times\mathbf{C}
  14. ( - 1 - 1 1 0 ) \begin{pmatrix}-1&-1\\ 1&0\end{pmatrix}
  15. 𝐙 / 3 × 𝐂 \mathbf{Z}/3\times\mathbf{C}
  16. ( 0 - 1 1 0 ) \begin{pmatrix}0&-1\\ 1&0\end{pmatrix}
  17. 𝐙 / 2 × 𝐂 \mathbf{Z}/2\times\mathbf{C}
  18. ( 0 - 1 1 1 ) \begin{pmatrix}0&-1\\ 1&1\end{pmatrix}
  19. 𝐂 \mathbf{C}
  20. Γ ~ \tilde{\Gamma}
  21. Γ \Gamma

Empirical_distribution_function.html

  1. F ^ n ( t ) = number of elements in the sample t n = 1 n i = 1 n 𝟏 x i t , \hat{F}_{n}(t)=\frac{\mbox{number of elements in the sample}~{}\leq t}{n}=% \frac{1}{n}\sum_{i=1}^{n}\mathbf{1}_{x_{i}\leq t},
  2. 𝟏 A \mathbf{1}_{A}
  3. 𝟏 x i t \mathbf{1}_{x_{i}\leq t}
  4. n F ^ n ( t ) \scriptstyle n\hat{F}_{n}(t)
  5. F ^ n ( t ) \scriptstyle\hat{F}_{n}(t)
  6. F ^ n ( t ) \scriptstyle\hat{F}_{n}(t)
  7. F ^ n ( t ) a . s . F ( t ) , \hat{F}_{n}(t)\ \xrightarrow{a.s.}\ F(t),
  8. F ^ n ( t ) \scriptstyle\hat{F}_{n}(t)
  9. F ^ n - F sup t | F ^ n ( t ) - F ( t ) | a . s . 0. \|\hat{F}_{n}-F\|_{\infty}\equiv\sup_{t\in\mathbb{R}}\big|\hat{F}_{n}(t)-F(t)% \big|\ \xrightarrow{a.s.}\ 0.
  10. F ^ n ( t ) \scriptstyle\hat{F}_{n}(t)
  11. F ^ n ( t ) \scriptstyle\hat{F}_{n}(t)
  12. n \sqrt{n}
  13. n ( F ^ n ( t ) - F ( t ) ) 𝑑 𝒩 ( 0 , F ( t ) ( 1 - F ( t ) ) ) . \sqrt{n}\big(\hat{F}_{n}(t)-F(t)\big)\ \ \xrightarrow{d}\ \ \mathcal{N}\Big(0,% F(t)\big(1-F(t)\big)\Big).
  14. n ( F ^ n - F ) \scriptstyle\sqrt{n}(\hat{F}_{n}-F)
  15. t \scriptstyle t\in\mathbb{R}
  16. D [ - , + ] \scriptstyle D[-\infty,+\infty]
  17. G F = B F \scriptstyle G_{F}=B\circ F
  18. E [ G F ( t 1 ) G F ( t 2 ) ] = F ( t 1 t 2 ) - F ( t 1 ) F ( t 2 ) . \mathrm{E}[\,G_{F}(t_{1})G_{F}(t_{2})\,]=F(t_{1}\wedge t_{2})-F(t_{1})F(t_{2}).
  19. lim sup n n ln 2 n n ( F ^ n - F ) - G F , n < , a.s. \limsup_{n\to\infty}\frac{\sqrt{n}}{\ln^{2}n}\big\|\sqrt{n}(\hat{F}_{n}-F)-G_{% F,n}\big\|_{\infty}<\infty,\quad\,\text{a.s.}
  20. n ( F ^ n - F ) \scriptstyle\sqrt{n}(\hat{F}_{n}-F)
  21. n F ^ n - F \scriptstyle\sqrt{n}\|\hat{F}_{n}-F\|_{\infty}
  22. Pr ( n F ^ n - F > z ) 2 e - 2 z 2 . \Pr\!\Big(\sqrt{n}\|\hat{F}_{n}-F\|_{\infty}>z\Big)\leq 2e^{-2z^{2}}.
  23. n F ^ n - F \scriptstyle\sqrt{n}\|\hat{F}_{n}-F\|_{\infty}
  24. B \scriptstyle\|B\|_{\infty}
  25. lim sup n n F ^ n - F 2 ln ln n 1 2 , a.s. \limsup_{n\to\infty}\frac{\sqrt{n}\|\hat{F}_{n}-F\|_{\infty}}{\sqrt{2\ln\ln n}% }\leq\frac{1}{2},\quad\,\text{a.s.}
  26. lim inf n 2 n ln ln n F ^ n - F = π 2 , a.s. \liminf_{n\to\infty}\sqrt{2n\ln\ln n}\|\hat{F}_{n}-F\|_{\infty}=\frac{\pi}{2},% \quad\,\text{a.s.}

Empty_domain.html

  1. A x ϕ ( x ) iff there is an a A such that A ϕ [ a ] A\models\exists x\phi(x)\,\text{ iff there is an }a\in A\,\text{ such that }A% \models\phi[a]
  2. A x ϕ ( x ) iff every a A is such that A ϕ [ a ] A\models\forall x\phi(x)\,\text{ iff every }a\in A\,\text{ is such that }A% \models\phi[a]
  3. A ϕ 1 ϕ n ϕ i ( 1 i n ) , A ϕ i A\models\phi_{1}\land\dots\land\phi_{n}\iff\forall\phi_{i}(1\leq i\leq n),A% \models\phi_{i}
  4. A ϕ 1 ϕ n ϕ i ( 1 i n ) , A ϕ i A\models\phi_{1}\lor\dots\lor\phi_{n}\iff\exists\phi_{i}(1\leq i\leq n),A% \models\phi_{i}

Energy_(signal_processing).html

  1. E s E_{s}
  2. E s = x ( t ) , x ( t ) = - | x ( t ) | 2 d t E_{s}\ \ =\ \ \langle x(t),x(t)\rangle\ \ =\int_{-\infty}^{\infty}{|x(t)|^{2}}dt
  3. E = E s Z = 1 Z - | x ( t ) | 2 d t E={E_{s}\over Z}={1\over Z}\int_{-\infty}^{\infty}{|x(t)|^{2}}dt
  4. E s E_{s}
  5. E s E_{s}
  6. E s ( f ) = | X ( f ) | 2 \ E_{s}(f)=|X(f)|^{2}
  7. E s ( f ) E_{s}(f)
  8. E s ( f ) E_{s}(f)

Energy_condition.html

  1. T a b T^{ab}
  2. X \vec{X}
  3. ρ = T a b X a X b \rho=T_{ab}\,X^{a}\,X^{b}
  4. - T a b X b -{T^{a}}_{b}\,X^{b}
  5. k \vec{k}
  6. ν = T a b k a k b \nu=T_{ab}\,k^{a}\,k^{b}
  7. X \vec{X}
  8. E [ X ] m m = R a b X a X b {E[\vec{X}]^{m}}_{m}=R_{ab}\,X^{a}\,X^{b}
  9. 1 8 π E [ X ] m m = 1 8 π R a b X a X b = ( T a b - 1 2 T g a b ) X a X b , \frac{1}{8\pi}\;{E[\vec{X}]^{m}}_{m}=\frac{1}{8\pi}R_{ab}\,X^{a}\,X^{b}=\left(% T_{ab}-\frac{1}{2}\,T\,g_{ab}\right)\,X^{a}\,X^{b},
  10. T = T m m T={T^{m}}_{m}
  11. k \vec{k}
  12. ρ = T a b k a k b 0. \rho=T_{ab}\,k^{a}\,k^{b}\geq 0.
  13. C C
  14. k \vec{k}
  15. C T a b k a k b d λ 0. \int_{C}T_{ab}\,k^{a}\,k^{b}\,d\lambda\geq 0.
  16. X \vec{X}
  17. ρ = T a b X a X b 0. \rho=T_{ab}\,X^{a}\,X^{b}\geq 0.
  18. Y \vec{Y}
  19. - T a b Y b -{T^{a}}_{b}\,Y^{b}
  20. X \vec{X}
  21. ( T a b - 1 2 T g a b ) X a X b 0 \left(T_{ab}-\frac{1}{2}\,T\,g_{ab}\right)\,X^{a}\,X^{b}\geq 0
  22. T a b = ρ u a u b + p h a b , T^{ab}=\rho\,u^{a}\,u^{b}+p\,h^{ab},
  23. u \vec{u}
  24. h a b g a b + u a u b h^{ab}\equiv g^{ab}+u^{a}u^{b}
  25. T a ^ b ^ = [ ρ 0 0 0 0 p 0 0 0 0 p 0 0 0 0 p ] . T^{\hat{a}\hat{b}}=\left[\begin{matrix}\rho&0&0&0\\ 0&p&0&0\\ 0&0&p&0\\ 0&0&0&p\end{matrix}\right].
  26. ρ \rho
  27. p p
  28. ρ 0 , ρ + p 0. \rho\geq 0,\;\;\rho+p\geq 0.
  29. ρ + p 0. \rho+p\geq 0.
  30. ρ + p 0 , ρ + 3 p 0. \rho+p\geq 0,\;\;\rho+3p\geq 0.
  31. ρ | p | . \rho\geq|p|.
  32. ε = - π 2 720 d 4 \varepsilon=\frac{-\pi^{2}}{720}\,\frac{\hbar}{d^{4}}
  33. p = w ρ p=w\rho
  34. ρ \rho
  35. p p
  36. w w
  37. w = - 1 ρ = c o n s t w=-1\Rightarrow\rho=const
  38. 10 12 10^{12}

Energy–momentum_relation.html

  1. E E
  2. p p
  3. E E
  4. m m
  5. E = p c E=pc
  6. v v
  7. c c
  8. v = 0 v=0
  9. p = 0 p=0
  10. E E
  11. 𝐩 \mathbf{p}
  12. E E
  13. 𝐩 \mathbf{p}
  14. E E′
  15. 𝐩 \mathbf{p′}
  16. E E E′≠E
  17. 𝐩 𝐩 \mathbf{p′}≠\mathbf{p}
  18. E 2 - ( p c ) 2 = ( m 0 c 2 ) 2 . {E^{\prime}}^{2}-(p^{\prime}c)^{2}=(m_{0}c^{2})^{2}\,.
  19. E E
  20. 𝐩 \mathbf{p}
  21. E E′
  22. 𝐩 \mathbf{p′}
  23. E 2 - ( p c ) 2 = E 2 - ( p c ) 2 = ( m 0 c 2 ) 2 . {E}^{2}-(pc)^{2}={E^{\prime}}^{2}-(p^{\prime}c)^{2}=(m_{0}c^{2})^{2}\,.
  24. E E′
  25. 𝐩 \mathbf{p′}
  26. E E
  27. 𝐩 \mathbf{p}
  28. | 𝐮 | = u |\mathbf{u}|=u
  29. E = γ ( 𝐮 ) m 0 c 2 E=\gamma_{(\mathbf{u})}m_{0}c^{2}
  30. 𝐩 = γ ( 𝐮 ) m 0 𝐮 \mathbf{p}=\gamma_{(\mathbf{u})}m_{0}\mathbf{u}
  31. | 𝐩 | = p |\mathbf{p}|=p
  32. E E
  33. 𝐩 \mathbf{p}
  34. γ ( 𝐮 ) = 1 1 - 𝐮 𝐮 c 2 = 1 1 - ( u c ) 2 \gamma_{(\mathbf{u})}=\frac{1}{\sqrt{1-\frac{\mathbf{u}\cdot\mathbf{u}}{c^{2}}% }}=\frac{1}{\sqrt{1-\left(\frac{u}{c}\right)^{2}}}
  35. m = γ ( 𝐮 ) m 0 m=\gamma_{(\mathbf{u})}m_{0}
  36. m m
  37. p 2 = 𝐩 𝐩 = m 0 2 𝐮 𝐮 1 - 𝐮 𝐮 c 2 = m 0 2 u 2 1 - ( u c ) 2 p^{2}=\mathbf{p}\cdot\mathbf{p}=\frac{m_{0}^{2}\mathbf{u}\cdot\mathbf{u}}{1-% \frac{\mathbf{u}\cdot\mathbf{u}}{c^{2}}}=\frac{m_{0}^{2}u^{2}}{1-\left(\frac{u% }{c}\right)^{2}}
  38. γ = 1 + ( p m 0 c ) 2 \gamma=\sqrt{1+\left(\frac{p}{m_{0}c}\right)^{2}}
  39. E = m 0 c 2 1 + ( p m 0 c ) 2 E=m_{0}c^{2}\sqrt{1+\left(\frac{p}{m_{0}c}\right)^{2}}
  40. E = 0 E=0
  41. 𝐩 = 𝟎 \mathbf{p}=\mathbf{0}
  42. E E
  43. 𝐩 \mathbf{p}
  44. 𝐩 \mathbf{p′}
  45. 𝐏 \mathbf{P}
  46. 𝐏 = ( E / c , 𝐩 ) , \mathbf{P}=(E/c,\mathbf{p})\,,
  47. , \langle,\rangle
  48. m m
  49. 𝐏 , 𝐏 = | 𝐏 | 2 = ( m 0 c ) 2 , \left\langle\mathbf{P},\mathbf{P}\right\rangle=|\mathbf{P}|^{2}=(m_{0}c)^{2}\,,
  50. η η
  51. ( + ) (+−−−)
  52. 𝐏 , 𝐏 = P α η α β P β = ( E / c p x p y p z ) ( 1 0 0 0 0 - 1 0 0 0 0 - 1 0 0 0 0 - 1 ) ( E / c p x p y p z ) = ( E c ) 2 - p 2 , \left\langle\mathbf{P},\mathbf{P}\right\rangle=P^{\alpha}\eta_{\alpha\beta}P^{% \beta}=\begin{pmatrix}E/c&p_{x}&p_{y}&p_{z}\end{pmatrix}\begin{pmatrix}1&0&0&0% \\ 0&-1&0&0\\ 0&0&-1&0\\ 0&0&0&-1\\ \end{pmatrix}\begin{pmatrix}E/c\\ p_{x}\\ p_{y}\\ p_{z}\end{pmatrix}=\left(\frac{E}{c}\right)^{2}-p^{2}\,,
  53. ( m 0 c ) 2 = ( E c ) 2 - p 2 , (m_{0}c)^{2}=\left(\frac{E}{c}\right)^{2}-p^{2}\,,
  54. ( + + + ) (−+++)
  55. η η
  56. 𝐏 , 𝐏 = | 𝐏 | 2 = - ( m 0 c ) 2 , \left\langle\mathbf{P},\mathbf{P}\right\rangle=|\mathbf{P}|^{2}=-(m_{0}c)^{2}\,,
  57. 𝐏 , 𝐏 = P α η α β P β = ( E / c p x p y p z ) ( - 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) ( E / c p x p y p z ) = - ( E c ) 2 + p 2 , \left\langle\mathbf{P},\mathbf{P}\right\rangle=P^{\alpha}\eta_{\alpha\beta}P^{% \beta}=\begin{pmatrix}E/c&p_{x}&p_{y}&p_{z}\end{pmatrix}\begin{pmatrix}-1&0&0&% 0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{pmatrix}\begin{pmatrix}E/c\\ p_{x}\\ p_{y}\\ p_{z}\end{pmatrix}=-\left(\frac{E}{c}\right)^{2}+p^{2}\,,
  58. - ( m 0 c ) 2 = - ( E c ) 2 + p 2 , -(m_{0}c)^{2}=-\left(\frac{E}{c}\right)^{2}+p^{2}\,,
  59. 𝐏 , 𝐏 = | 𝐏 | 2 = ( m 0 c ) 2 , \left\langle\mathbf{P},\mathbf{P}\right\rangle=|\mathbf{P}|^{2}=(m_{0}c)^{2}\,,
  60. η η
  61. 𝐏 , 𝐏 = | 𝐏 | 2 = P α g α β P β , \left\langle\mathbf{P},\mathbf{P}\right\rangle=|\mathbf{P}|^{2}=P^{\alpha}g_{% \alpha\beta}P^{\beta}\,,
  62. P α g α β P β = ( m 0 c ) 2 . P^{\alpha}g_{\alpha\beta}P^{\beta}=(m_{0}c)^{2}\,.
  63. g 00 ( P 0 ) 2 timelike + 2 g 0 i P 0 P i spacetime-like + g i j P i P j spacelike = ( m 0 c ) 2 . \underbrace{g_{00}{(P^{0})}^{2}}_{\,\text{timelike}}+2\underbrace{g_{0i}P^{0}P% ^{i}}_{\,\text{spacetime-like}}+\underbrace{g_{ij}P^{i}P^{j}}_{\,\text{% spacelike}}=(m_{0}c)^{2}\,.
  64. c = 1 c=1
  65. E 2 = p 2 + m 0 2 . E^{2}=p^{2}+m_{0}^{2}\,.
  66. 𝐄 \mathbf{E}
  67. 𝐁 \mathbf{B}
  68. E 0 = m 0 c 2 , E_{0}=m_{0}c^{2}\,,
  69. E = p c . E=pc\,.
  70. E = h c λ = c k . E=\frac{hc}{\lambda}=\hbar ck\,.
  71. λ λ
  72. k k
  73. E = m 0 c 2 [ 1 + ( p m 0 c ) 2 ] 1 / 2 , E=m_{0}c^{2}\left[1+\left(\frac{p}{m_{0}c}\right)^{2}\right]^{1/2}\,,
  74. E = m 0 c 2 [ 1 + 1 2 ( p m 0 c ) 2 - 1 8 ( p m 0 c ) 4 + ] , E=m_{0}c^{2}\left[1+\frac{1}{2}\left(\frac{p}{m_{0}c}\right)^{2}-\frac{1}{8}% \left(\frac{p}{m_{0}c}\right)^{4}+\cdots\right]\,,
  75. u c u≪c
  76. γ ( u ) 1 γ(u)≈1
  77. n = 1 n=1
  78. n 2 n≥2
  79. E m 0 c 2 [ 1 + 1 2 ( m 0 u m 0 c ) 2 ] , E\approx m_{0}c^{2}\left[1+\frac{1}{2}\left(\frac{m_{0}u}{m_{0}c}\right)^{2}% \right]\,,
  80. E m 0 c 2 + 1 2 m 0 u 2 , E\approx m_{0}c^{2}+\frac{1}{2}m_{0}u^{2}\,,
  81. n = 1 , 2 , n=1,2,...
  82. n 𝐏 n = n ( E n / c , 𝐩 n ) = ( n E n / c , n 𝐩 n ) , \sum_{n}\mathbf{P}_{n}=\sum_{n}(E_{n}/c,\mathbf{p}_{n})=\left(\sum_{n}E_{n}/c,% \sum_{n}\mathbf{p}_{n}\right)\,,
  83. | ( n 𝐏 n ) | 2 = ( n E n / c ) 2 - ( n 𝐩 n ) 2 = ( M 0 c ) 2 , \left|\left(\sum_{n}\mathbf{P}_{n}\right)\right|^{2}=\left(\sum_{n}E_{n}/c% \right)^{2}-\left(\sum_{n}\mathbf{p}_{n}\right)^{2}=(M_{0}c)^{2}\,,
  84. n 𝐩 n = s y m b o l 0 , \sum_{n}\mathbf{p}_{n}=symbol{0}\,,
  85. ( n E n ) 2 = ( M 0 c 2 ) 2 n E COM n = E COM = M 0 c 2 , \left(\sum_{n}E_{n}\right)^{2}=(M_{0}c^{2})^{2}\Rightarrow\sum_{n}E_{\mathrm{% COM}\,n}=E_{\mathrm{COM}}=M_{0}c^{2}\,,
  86. E n 2 - ( 𝐩 n c ) 2 = ( m n c 2 ) 2 , E^{2}_{n}-(\mathbf{p}_{n}c)^{2}=(m_{n}c^{2})^{2}\,,
  87. ( n E n ) 2 = ( n E n ) ( k E k ) = n , k E n E k = 2 n < k E n E k + n E n 2 , \left(\sum_{n}E_{n}\right)^{2}=\left(\sum_{n}E_{n}\right)\left(\sum_{k}E_{k}% \right)=\sum_{n,k}E_{n}E_{k}=2\sum_{n<k}E_{n}E_{k}+\sum_{n}E_{n}^{2}\,,
  88. ( n 𝐩 n ) 2 = ( n 𝐩 n ) ( k 𝐩 k ) = n , k 𝐩 n 𝐩 k = 2 n < k 𝐩 n 𝐩 k + n 𝐩 n 2 , \left(\sum_{n}\mathbf{p}_{n}\right)^{2}=\left(\sum_{n}\mathbf{p}_{n}\right)% \cdot\left(\sum_{k}\mathbf{p}_{k}\right)=\sum_{n,k}\mathbf{p}_{n}\cdot\mathbf{% p}_{k}=2\sum_{n<k}\mathbf{p}_{n}\cdot\mathbf{p}_{k}+\sum_{n}\mathbf{p}_{n}^{2}\,,
  89. n ( m n c 2 ) 2 - 2 n < k ( E n E k - c 2 𝐩 n 𝐩 k ) = ( M 0 c 2 ) 2 . \sum_{n}(m_{n}c^{2})^{2}-2\sum_{n<k}(E_{n}E_{k}-c^{2}\mathbf{p}_{n}\cdot% \mathbf{p}_{k})=(M_{0}c^{2})^{2}\,.
  90. E n = ( 𝐩 n c ) 2 + ( m n c 2 ) 2 , E k = ( 𝐩 k c ) 2 + ( m k c 2 ) 2 , E_{n}=\sqrt{(\mathbf{p}_{n}c)^{2}+(m_{n}c^{2})^{2}}\,,\quad E_{k}=\sqrt{(% \mathbf{p}_{k}c)^{2}+(m_{k}c^{2})^{2}}\,,
  91. 𝐩 n 𝐩 k = | 𝐩 n | | 𝐩 k | cos θ n k , | 𝐩 n | = 1 c E n 2 - ( m n c 2 ) 2 , | 𝐩 k | = 1 c E k 2 - ( m k c 2 ) 2 , \mathbf{p}_{n}\cdot\mathbf{p}_{k}=|\mathbf{p}_{n}||\mathbf{p}_{k}|\cos\theta_{% nk}\,,\quad|\mathbf{p}_{n}|=\frac{1}{c}\sqrt{E_{n}^{2}-(m_{n}c^{2})^{2}}\,,% \quad|\mathbf{p}_{k}|=\frac{1}{c}\sqrt{E_{k}^{2}-(m_{k}c^{2})^{2}}\,,
  92. n ( m n c 2 ) 2 - ( M 0 c 2 ) 2 = 2 n < k ( E n E k - c 2 𝐩 n 𝐩 k ) . \sum_{n}(m_{n}c^{2})^{2}-(M_{0}c^{2})^{2}=2\sum_{n<k}(E_{n}E_{k}-c^{2}\mathbf{% p}_{n}\cdot\mathbf{p}_{k})\,.
  93. E = ω , 𝐩 = 𝐤 , E=\hbar\omega\,,\quad\mathbf{p}=\hbar\mathbf{k}\,,
  94. ω ω
  95. 𝐤 \mathbf{k}
  96. | 𝐤 | = k |\mathbf{k}|=k
  97. ( ω ) 2 = ( c k ) 2 + ( m 0 c 2 ) 2 , (\hbar\omega)^{2}=(c\hbar k)^{2}+(m_{0}c^{2})^{2}\,,
  98. 𝐊 = ( ω / c , 𝐤 ) , \mathbf{K}=(\omega/c,\mathbf{k})\,,
  99. ħ ħ
  100. c c
  101. ħ = c = 1 ħ=c=1
  102. ω 2 = k 2 + m 0 2 . \omega^{2}=k^{2}+m_{0}^{2}\,.

Engel_expansion.html

  1. { a 1 , a 2 , a 3 , } \{a_{1},a_{2},a_{3},\dots\}
  2. x = 1 a 1 + 1 a 1 a 2 + 1 a 1 a 2 a 3 + . x=\frac{1}{a_{1}}+\frac{1}{a_{1}a_{2}}+\frac{1}{a_{1}a_{2}a_{3}}+\cdots.\;
  3. x = 1 + 1 + 1 + a 3 a 2 a 1 . x=\frac{\displaystyle 1+\frac{\displaystyle 1+\frac{\displaystyle 1+\cdots}{% \displaystyle a_{3}}}{\displaystyle a_{2}}}{\displaystyle a_{1}}.
  4. a b c d e f g h = d + c + b + a e f g h . \frac{a\ b\ c\ d}{e\ f\ g\ h}=\dfrac{d+\dfrac{c+\dfrac{b+\dfrac{a}{e}}{f}}{g}}% {h}.
  5. u 1 = x , u_{1}=x,
  6. a k = 1 u k , a_{k}=\left\lceil\frac{1}{u_{k}}\right\rceil,
  7. u k + 1 = u k a k - 1 u_{k+1}=u_{k}a_{k}-1
  8. r \left\lceil r\right\rceil
  9. u i = 0 u_{i}=0
  10. u 1 = 1.175 , a 1 = 1 1.175 = 1 ; u_{1}=1.175,a_{1}=\left\lceil\frac{1}{1.175}\right\rceil=1;\,
  11. u 2 = u 1 a 1 - 1 = 1.175 1 - 1 = 0.175 , a 2 = 1 0.175 = 6 u_{2}=u_{1}a_{1}-1=1.175\cdot 1-1=0.175,a_{2}=\left\lceil\frac{1}{0.175}\right% \rceil=6\,
  12. u 3 = u 2 a 2 - 1 = 0.175 6 - 1 = 0.05 , a 3 = 1 0.05 = 20 u_{3}=u_{2}a_{2}-1=0.175\cdot 6-1=0.05,a_{3}=\left\lceil\frac{1}{0.05}\right% \rceil=20\,
  13. u 4 = u 3 a 3 - 1 = 0.05 20 - 1 = 0 u_{4}=u_{3}a_{3}-1=0.05\cdot 20-1=0\,
  14. 1.175 = 1 1 + 1 1 6 + 1 1 6 20 1.175=\frac{1}{1}+\frac{1}{1\cdot 6}+\frac{1}{1\cdot 6\cdot 20}
  15. 1 n = r = 1 1 ( n + 1 ) r . \frac{1}{n}=\sum_{r=1}^{\infty}\frac{1}{(n+1)^{r}}.
  16. 1.175 = { 1 , 6 , 20 } = { 1 , 6 , 21 , 21 , 21 , } . 1.175=\{1,6,20\}=\{1,6,21,21,21,\dots\}.\;\;
  17. π \pi
  18. 2 \sqrt{2}
  19. e e
  20. e 1 / r - 1 = { 1 r , 2 r , 3 r , 4 r , 5 r , 6 r , } e^{1/r}-1=\{1r,2r,3r,4r,5r,6r,\dots\}\;
  21. lim n a n 1 / n \lim_{n\rightarrow\infty}a_{n}^{1/n}

Engine_balance.html

  1. Δ x = r cos Δ α + l \Delta x=r\cos\Delta\alpha+l\,
  2. Δ x \Delta x
  3. l l
  4. r r
  5. Δ α \Delta\alpha
  6. Δ x = r cos Δ α + l 2 - r 2 sin 2 Δ α \Delta x=r\cos\Delta\alpha+\sqrt{l^{2}-r^{2}\sin^{2}\Delta\alpha}

Enol_ether.html

  1. R 1 R 2 C = C R 3 - O - R 4 R_{1}R_{2}C=CR_{3}-O-R_{4}

Enriques_surface.html

  1. w 2 x 2 y 2 + w 2 x 2 z 2 + w 2 y 2 z 2 + x 2 y 2 z 2 + w x y z Q ( w , x , y , z ) = 0 w^{2}x^{2}y^{2}+w^{2}x^{2}z^{2}+w^{2}y^{2}z^{2}+x^{2}y^{2}z^{2}+wxyzQ(w,x,y,z)=0

Ensemble_average.html

  1. A ¯ = A e - β H ( q 1 , q 2 , q M , p 1 , p 2 , p N ) d τ e - β H ( q 1 , q 2 , q M , p 1 , p 2 , p N ) d τ \bar{A}=\frac{\int{Ae^{-\beta H(q_{1},q_{2},...q_{M},p_{1},p_{2},...p_{N})}d% \tau}}{\int{e^{-\beta H(q_{1},q_{2},...q_{M},p_{1},p_{2},...p_{N})}d\tau}}
  2. A ¯ \bar{A}
  3. β \beta
  4. 1 k T \frac{1}{kT}
  5. q i q_{i}
  6. p i p_{i}
  7. d τ d\tau
  8. A ¯ = i A i e - β E i i e - β E i \bar{A}=\frac{\sum_{i}{A_{i}e^{-\beta E_{i}}}}{\sum_{i}{e^{-\beta E_{i}}}}

Enstrophy.html

  1. \mathcal{E}
  2. ω \omega
  3. ( s y m b o l ω ) 1 2 S s y m b o l ω 2 d S . \mathcal{E}(symbol\omega)\equiv\frac{1}{2}\int_{S}symbol\omega^{2}dS.
  4. ( 𝐮 ) 1 2 S ( × 𝐮 ) 2 d S . \mathcal{E}(\mathbf{u})\equiv\frac{1}{2}\int_{S}(\nabla\times\mathbf{u})^{2}dS.
  5. 𝐮 = 0. \nabla\cdot\mathbf{u}=0.
  6. ( 𝐮 ) = S | ( 𝐮 ) | 2 d S . \mathcal{E}(\mathbf{u})=\int_{S}|\nabla(\mathbf{u})|^{2}dS.
  7. | ( 𝐮 ) | |\nabla(\mathbf{u})|
  8. 𝐮 \mathbf{u}

Entropy_of_mixing.html

  1. Δ S m i x \Delta S_{mix}\,
  2. Δ S m i x = - n R ( x 1 ln x 1 + x 2 ln x 2 ) \Delta S_{mix}=-nR(x_{1}\ln x_{1}+x_{2}\ln x_{2})\,
  3. R R\,
  4. n n\,
  5. x i x_{i}\,
  6. i i\,
  7. V i = x i V V_{i}=x_{i}V\,
  8. n i = n x i n_{i}=nx_{i}\,
  9. i i\,
  10. V V\,
  11. n x i R ln ( V / V i ) = - n R x i ln x i nx_{i}R\ln(V/V_{i})=-nRx_{i}\ln x_{i}\,
  12. Δ G m i x = Δ H m i x - T Δ S m i x \Delta G_{mix}=\Delta H_{mix}-T\Delta S_{mix}\,
  13. T \ T
  14. p \ p
  15. Δ H m i x \Delta H_{mix}\,
  16. Δ G m i x = - T Δ S m i x \Delta G_{mix}=-T\Delta S_{mix}\,
  17. Δ S m i x = - n R ( x 1 ln x 1 + x 2 ln x 2 ) = - n R [ x ln x + ( 1 - x ) ln ( 1 - x ) ] \Delta S_{mix}=-nR(x_{1}\ln x_{1}+x_{2}\ln x_{2})=-nR[x\ln x+(1-x)\ln(1-x)]\,
  18. 0 < x < 1 0<x<1
  19. ln \ln
  20. x x
  21. ln ( 1 - x ) \ln(1-x)
  22. Δ S m i x \Delta S_{mix}\,
  23. Δ S m i x \Delta S_{mix}\,
  24. x x
  25. ( 2 Δ S m i x x 2 ) T , P = - n R ( 1 x + 1 1 - x ) \left(\frac{\partial^{2}\Delta S_{mix}}{\partial x^{2}}\right)_{T,P}=-nR\left(% \frac{1}{x}+\frac{1}{1-x}\right)
  26. ( 0 < x < 1 ) (0<x<1)
  27. - T Δ S m i x -T\Delta S_{mix}\,
  28. Δ G m i x \Delta G_{mix}\,
  29. Δ S m i x = - ( Δ G m i x T ) P \Delta S_{mix}=-\left(\frac{\partial\Delta G_{mix}}{\partial T}\right)_{P}
  30. Δ S m i x = k B ln Ω \Delta S_{mix}=k_{B}\ln\Omega\,
  31. k B k_{B}\,
  32. Ω \Omega\,
  33. N 1 N_{1}\,
  34. N 2 N_{2}\,
  35. N = N 1 + N 2 N=N_{1}+N_{2}\,
  36. N N\,
  37. N 1 N_{1}\,
  38. N 2 N_{2}\,
  39. Ω = N ! / N 1 ! N 2 ! \Omega=N!/N_{1}!N_{2}!\,
  40. ln m ! = k ln k 1 m d k ln k = m ln m - m \ln m!=\sum_{k}\ln k\approx\int_{1}^{m}dk\ln k=m\ln m-m
  41. Δ S m i x = - k B [ N 1 ln ( N 1 / N ) + N 2 ln ( N 2 / N ) ] = - k B N [ x 1 ln x 1 + x 2 ln x 2 ] \Delta S_{mix}=-k_{B}[N_{1}\ln(N_{1}/N)+N_{2}\ln(N_{2}/N)]=-k_{B}N[x_{1}\ln x_% {1}+x_{2}\ln x_{2}]\,
  42. x 1 = N 1 / N = p 1 and x 2 = N 2 / N = p 2 x_{1}=N_{1}/N=p_{1}\;\;\,\text{and}\;\;x_{2}=N_{2}/N=p_{2}\,
  43. k B = R / N A k_{B}=R/N_{A}\,
  44. N A N_{A}\,
  45. N = n N A N=nN_{A}\,
  46. Δ S m i x = - n R [ x 1 ln x 1 + x 2 ln x 2 ] \Delta S_{mix}=-nR[x_{1}\ln x_{1}+x_{2}\ln x_{2}]\,
  47. r r\,
  48. N i N_{i}\,
  49. i = 1 , 2 , 3 , , r i=1,2,3,\ldots,r\,
  50. Δ S m i x = - k B i = 1 r N i ln ( N i / N ) = - N k B i = 1 r x i ln x i = - n R i = 1 r x i ln x i \Delta S_{mix}=-k_{B}\sum_{i=1}^{r}N_{i}\ln(N_{i}/N)=-Nk_{B}\sum_{i=1}^{r}x_{i% }\ln x_{i}=-nR\sum_{i=1}^{r}x_{i}\ln x_{i}\,\!
  51. H = - i = 1 r p i ln ( p i ) H=-\sum_{i=1}^{r}p_{i}\ln(p_{i})
  52. N N\,
  53. p i p_{i}\,
  54. i i\,
  55. x i x_{i}\,
  56. k B k_{B}\,
  57. Δ S m i x = - N k B i = 1 r x i ln x i \Delta S_{mix}=-Nk_{B}\sum_{i=1}^{r}x_{i}\ln x_{i}\,\!

Entropy_of_vaporization.html

  1. Δ G v a p = Δ H v a p - T v a p × Δ S v a p = 0 \Delta G_{vap}=\Delta H_{vap}-T_{vap}\times\Delta S_{vap}=0
  2. Δ H v a p \Delta H_{vap}
  3. Δ S v a p = Δ H v a p T v a p \Delta S_{vap}=\frac{\Delta H_{vap}}{T_{vap}}

Enumerative_geometry.html

  1. X P 4 X\subset P^{4}
  2. X P 4 X\subset P^{4}
  3. d d
  4. d d
  5. X X
  6. d 9 d\leq 9
  7. d d
  8. P 4 P^{4}
  9. X X
  10. d > 0 d>0
  11. d 5 d\leq 5

Envy-free.html

  1. A < i B A<_{i}B
  2. Θ ( 1 ϵ n ) \Theta(\frac{1}{\epsilon^{n}})
  3. ϵ > 0 \epsilon>0
  4. ϵ \epsilon
  5. n 2 / ϵ n^{2}/\epsilon

Enzyme_kinetics.html

  1. V max V_{\max}
  2. E S k c a t E + P ES\overset{k_{cat}}{\longrightarrow}E+P
  3. v 0 = V max [ S ] K M + [ S ] v_{0}=\frac{V_{\max}[\mbox{S}~{}]}{K_{M}+[\mbox{S}~{}]}
  4. K M = def k 2 + k - 1 k 1 K D V max = def k c a t [ E ] t o t \begin{aligned}\displaystyle K_{M}&\displaystyle\stackrel{\mathrm{def}}{=}\ % \frac{k_{2}+k_{-1}}{k_{1}}\approx K_{D}\\ \displaystyle V_{\max}&\displaystyle\stackrel{\mathrm{def}}{=}\ k_{cat}{[}E{]}% _{tot}\end{aligned}
  5. d [ E S ] / d t = ! 0 d{[}ES{]}/{dt}\;\overset{!}{=}\;0
  6. [ E ] tot = [ E ] + [ E S ] = ! const {[}E{]}\text{tot}={[}E{]}+{[}ES{]}\;\overset{!}{=}\;\,\text{const}
  7. k 2 k - 1 k_{2}\ll k_{-1}
  8. [ S ] [S]
  9. K M K_{M}
  10. [ S ] / ( K M + [ S ] ) [ S ] / K M [S]/(K_{M}+[S])\approx[S]/K_{M}
  11. [ E ] 0 [ E ] [E]_{0}\approx[E]
  12. v 0 k c a t K M [ E ] [ S ] if [ S ] K M v_{0}\approx\frac{k_{cat}}{K_{M}}[E][S]\qquad\qquad\,\text{if }[S]\ll K_{M}
  13. k 2 / K M k_{2}/K_{M}
  14. k 2 / K M k_{2}/K_{M}
  15. [ S ] = [ S ] 0 ( 1 - k ) t [S]=[S]_{0}(1-k)^{t}\,
  16. [ S ] = [ S ] 0 ( 1 - v / [ S ] 0 ) t [S]=[S]_{0}(1-v/[S]_{0})^{t}\,
  17. [ S ] = [ S ] 0 ( 1 - ( V max [ S ] 0 / ( K M + [ S ] 0 ) / [ S ] 0 ) ) t [S]=[S]_{0}(1-(V_{\max}[S]_{0}/(K_{M}+[S]_{0})/[S]_{0}))^{t}\,
  18. [ S ] K M = W [ F ( t ) ] \frac{[S]}{K_{M}}=W\left[F(t)\right]\,
  19. F ( t ) = [ S ] 0 K M exp ( [ S ] 0 K M - V max K M t ) F(t)=\frac{[S]_{0}}{K_{M}}\exp\!\left(\frac{[S]_{0}}{K_{M}}-\frac{V_{\max}}{K_% {M}}\,t\right)\,
  20. [ S ] K M = W [ F ( t ) ] - V max k c a t K M W [ F ( t ) ] 1 + W [ F ( t ) ] \frac{[S]}{K_{M}}=W\left[F(t)\right]-\frac{V_{\max}}{k_{cat}K_{M}}\ \frac{W% \left[F(t)\right]}{1+W\left[F(t)\right]}\,
  21. 1 v = K M V max [ S ] + 1 V max \frac{1}{v}=\frac{K_{M}}{V_{\max}[\mbox{S}~{}]}+\frac{1}{V_{\max}}
  22. E + S k 1 k - 1 E S k 2 E I k 3 E + P \displaystyle E+S\underset{k_{-1}}{\overset{k_{1}}{\begin{smallmatrix}% \displaystyle\longrightarrow\\ \displaystyle\longleftarrow\end{smallmatrix}}}ES\overset{k_{2}}{% \longrightarrow}EI\overset{k_{3}}{\longrightarrow}E+P
  23. v 0 \displaystyle v_{0}
  24. K M = def k 3 k 2 + k 3 K M = k 3 k 2 + k 3 k 2 + k - 1 k 1 k c a t = def k 3 k 2 k 2 + k 3 \begin{aligned}\displaystyle K_{M}^{\prime}&\displaystyle\stackrel{\mathrm{def% }}{=}\ \frac{k_{3}}{k_{2}+k_{3}}K_{M}=\frac{k_{3}}{k_{2}+k_{3}}\cdot\frac{k_{2% }+k_{-1}}{k_{1}}\\ \displaystyle k_{cat}&\displaystyle\stackrel{\mathrm{def}}{=}\ \dfrac{k_{3}k_{% 2}}{k_{2}+k_{3}}\end{aligned}
  25. k 3 k 2 k_{3}\gg k_{2}
  26. K M K M K_{M}^{\prime}\approx K_{M}
  27. k c a t k 2 k_{cat}\approx k_{2}
  28. V max 1 + [ I ] K i \cfrac{V_{\max}}{1+\cfrac{[I]}{K_{i}}}
  29. V max [ I ] + K i K i \cfrac{V_{\max}}{\cfrac{[I]+K_{i}}{K_{i}}}
  30. V max [ I ] + K i [ I ] + K i - [ I ] \cfrac{V_{\max}}{\cfrac{[I]+K_{i}}{[I]+K_{i}-[I]}}
  31. V max 1 1 - [ I ] [ I ] + K i \cfrac{V_{\max}}{\cfrac{1}{1-\cfrac{[I]}{[I]+K_{i}}}}
  32. V max - V max [ I ] [ I ] + K i V_{\max}-V_{\max}\cfrac{[I]}{[I]+K_{i}}
  33. [ S ] [ S ] + K m \cfrac{[S]}{[S]+K_{m}}
  34. [ I ] [ I ] + K i \cfrac{[I]}{[I]+K_{i}}
  35. V max - Δ V max [ I ] [ I ] + K i V_{\max}-\Delta V_{\max}\cfrac{[I]}{[I]+K_{i}}
  36. V max 1 - ( V max 1 - V max 2 ) [ I ] [ I ] + K i V_{\max}1-(V_{\max}1-V_{\max}2)\cfrac{[I]}{[I]+K_{i}}
  37. V max 1 - ( V max 1 - V max 2 ) [ X ] [ X ] + K x V_{\max}1-(V_{\max}1-V_{\max}2)\cfrac{[X]}{[X]+K_{x}}
  38. K m 1 - ( K m 1 - K m 2 ) [ X ] [ X ] + K x K_{m}1-(K_{m}1-K_{m}2)\cfrac{[X]}{[X]+K_{x}}

EP_quantum_mechanics.html

  1. S ( q ) S(q)
  2. d S ( q ) = p i ( q ) d q i dS(q)=p_{i}(q)dq^{i}
  3. S ( q ) S(q)
  4. Q ( q ) = 2 4 m { S ( q ) , q } Q(q)=\frac{\hbar^{2}}{4m}\{S(q),q\}
  5. { , } \{,\}

Epsilon-induction.html

  1. \in
  2. x ( y ( y x P [ y ] ) P [ x ] ) x P [ x ] \forall x\Big(\forall y(y\in x\rightarrow P[y])\rightarrow P[x]\Big)% \rightarrow\forall x\,P[x]
  3. \in
  4. \in
  5. ϵ \epsilon

Equation_of_the_center.html

  1. e e
  2. ν \nu
  3. M M
  4. e 3 e^{3}
  5. ν = M + ( 2 e - 1 4 e 3 ) sin M + 5 4 e 2 sin 2 M + 13 12 e 3 sin 3 M + \nu=M+(2e-\frac{1}{4}e^{3})\sin M+\frac{5}{4}e^{2}\sin 2M+\frac{13}{12}e^{3}% \sin 3M+...
  6. r r
  7. a a
  8. r a = ( 1 + e 2 / 2 ) - ( e - 3 8 e 3 ) cos M - 1 2 e 2 cos 2 M - 3 8 e 3 cos 3 M - \frac{r}{a}=(1+e^{2}/2)-(e-\frac{3}{8}e^{3})\cos M-\frac{1}{2}e^{2}\cos 2M-% \frac{3}{8}e^{3}\cos 3M-...
  9. a / r a/r
  10. a r = 1 + ( e - e 3 / 8 ) cos M + e 2 cos 2 M + 9 8 e 3 cos 3 M + \frac{a}{r}=1+(e-e^{3}/8)\cos M+e^{2}\cos 2M+\frac{9}{8}e^{3}\cos 3M+...
  11. sin ( M ) \sin(M)

Equiangular_lines.html

  1. 8 \mathbb{R}^{8}
  2. 8 \mathbb{R}^{8}

Equidistribution_theorem.html

  1. / \mathbb{R}/\mathbb{Z}
  2. μ = d θ 2 π \mu=\frac{d\theta}{2\pi}
  3. lim n 1 n k = 1 n f ( ( x + k a ) mod 1 ) = 0 1 f ( y ) d y \lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}f((x+ka)\mod 1)=\int_{0}^{1}f(y)\,dy
  4. lim n 1 n k = 1 n f ( ( x + b k a ) mod 1 ) = 0 1 f ( y ) d y \lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}f((x+b_{k}a)\mod 1)=\int_{0}^{1}f(y)% \,dy

Equivariant_cohomology.html

  1. Λ \Lambda
  2. E G × G X EG\times_{G}X
  3. H G * ( X ; Λ ) = H * ( E G × G X ; Λ ) . H_{G}^{*}(X;\Lambda)=H^{*}(EG\times_{G}X;\Lambda).
  4. G G
  5. X X
  6. X X
  7. B G BG
  8. G G
  9. E G × G X X / G EG\times_{G}X\to X/G
  10. H G * ( X ; Λ ) = H * ( X / G ; Λ ) . H_{G}^{*}(X;\Lambda)=H^{*}(X/G;\Lambda).
  11. Λ \Lambda
  12. X X
  13. G G
  14. X X
  15. X ( ) X(\mathbb{C})
  16. B G BG
  17. P sm P\text{sm}
  18. X X
  19. P sm P\text{sm}
  20. Ω \Omega
  21. P sm P\text{sm}
  22. Ω \Omega
  23. 𝒢 \mathcal{G}
  24. P sm P\text{sm}
  25. Ω \Omega
  26. 𝒢 \mathcal{G}
  27. B 𝒢 B\mathcal{G}
  28. 𝒢 \mathcal{G}
  29. Bun G ( X ) \operatorname{Bun}_{G}(X)
  30. [ Ω / 𝒢 ] [\Omega/\mathcal{G}]
  31. B 𝒢 B\mathcal{G}
  32. Bun G ( X ) \operatorname{Bun}_{G}(X)
  33. E ~ \widetilde{E}
  34. E G × G M EG\times_{G}M
  35. E G × E E G × M EG\times E\to EG\times M
  36. E ~ \widetilde{E}
  37. H * ( E G × G M ) = H G * ( M ) H^{*}(EG\times_{G}M)=H^{*}_{G}(M)
  38. H 2 ( M ; ) . H^{2}(M;\mathbb{Z}).
  39. H G 2 ( M ; ) H^{2}_{G}(M;\mathbb{Z})
  40. H 1 ( M ; * ) H 2 ( M ; ) H^{1}(M;\mathbb{C}^{*})\simeq H^{2}(M;\mathbb{Z})

Erdős–Burr_conjecture.html

  1. r ( G ) r(G)
  2. r ( G ) r(G)
  3. r ( G ) 2 c p log n n . r(G)\leq 2^{c_{p}\sqrt{\log n}}n.

Erdős–Straus_conjecture.html

  1. 4 n = 1 x + 1 y + 1 z . \frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.
  2. 4 5 = 1 2 + 1 4 + 1 20 = 1 2 + 1 5 + 1 10 . \frac{4}{5}=\frac{1}{2}+\frac{1}{4}+\frac{1}{20}=\frac{1}{2}+\frac{1}{5}+\frac% {1}{10}.
  3. 4 n = 1 n + 1 ( n - 2 ) / 3 + 1 + 1 n ( ( n - 2 ) / 3 + 1 ) . \frac{4}{n}=\frac{1}{n}+\frac{1}{(n-2)/3+1}+\frac{1}{n((n-2)/3+1)}.
  4. 4 4 k + 1 = 1 k - 1 k ( 4 k + 1 ) \frac{4}{4k+1}=\frac{1}{k}-\frac{1}{k(4k+1)}
  5. 4 4 k - 1 = 1 k + 1 k ( 4 k - 1 ) . \frac{4}{4k-1}=\frac{1}{k}+\frac{1}{k(4k-1)}.
  6. 4 n = 1 ( n - 1 ) / 2 + 1 ( n + 1 ) / 2 - 1 n ( n - 1 ) ( n + 1 ) / 4 . \frac{4}{n}=\frac{1}{(n-1)/2}+\frac{1}{(n+1)/2}-\frac{1}{n(n-1)(n+1)/4}.

Erdős–Szekeres_theorem.html

  1. r s - r - s + 2 r - 1 = s . \left\lceil\frac{rs-r-s+2}{r-1}\right\rceil=s.

Ergodic_sequence.html

  1. A = { a j } A=\{a_{j}\}
  2. 1 k q 1\leq k\leq q
  3. lim t N ( A , t , k , q ) N ( A , t ) = 1 q \lim_{t\to\infty}\frac{N(A,t,k,q)}{N(A,t)}=\frac{1}{q}
  4. N ( A , t ) = card { a j A : a j t } N(A,t)=\mbox{card}~{}\{a_{j}\in A:a_{j}\leq t\}
  5. N ( A , t ) N(A,t)
  6. N ( A , t , k , q ) = card { a j A : a j t , a j mod q = k } N(A,t,k,q)=\mbox{card}~{}\{a_{j}\in A:a_{j}\leq t,\,a_{j}\mod q=k\}
  7. N ( A , t , k , q ) N(A,t,k,q)
  8. lim t 1 N ( A , t ) j ; a j t exp 2 π i k a j q = 0 \lim_{t\to\infty}\frac{1}{N(A,t)}\sum_{j;a_{j}\leq t}\exp\frac{2\pi ika_{j}}{q% }=0
  9. k mod q 0 k\mod q\neq 0
  10. ( Ω , P r ) (\Omega,Pr)
  11. { 0 , 1 } \{0,1\}
  12. ω Ω \omega\in\Omega
  13. X j ( ω ) X_{j}(\omega)
  14. ω \omega
  15. ω = { n : X n ( ω ) = 1 } \mathbb{Z}^{\omega}=\{n\in\mathbb{Z}:X_{n}(\omega)=1\}
  16. ω \mathbb{Z}^{\omega}

Error_threshold_(evolution).html

  1. a a
  2. a > 1 a>1
  3. μ \mu
  4. ( L d ) {\textstyle\left({{L}\atop{d}}\right)}
  5. n i n^{\prime}_{i}
  6. n j n_{j}
  7. n i = j = 0 3 w i j n j n^{\prime}_{i}=\sum_{j=0}^{3}w_{ij}n_{j}
  8. 𝐰 = [ a Q 3 a μ 0 0 μ Q 2 μ 0 0 2 μ Q μ 0 0 3 μ Q ] \mathbf{w}=\begin{bmatrix}a\cdot Q&3a\cdot\mu&0&0\\ \mu&Q&2\mu&0\\ 0&2\mu&Q&\mu\\ 0&0&3\mu&Q\end{bmatrix}
  9. Q = ( 1 - μ ) L Q=(1-\mu)^{L}
  10. [ n 0 , n 1 , n 2 , n 3 ] = [ 1 , 0 , 0 , 0 ] [n_{0},n_{1},n_{2},n_{3}]=[1,0,0,0]
  11. [ 0.33 , 0.38 , 0.24 , 0.06 ] [0.33,0.38,0.24,0.06]
  12. [ 0.125 , 0.375 , 0.375 , 0.125 ] [0.125,0.375,0.375,0.125]
  13. ( 100 k ) {\textstyle\left({{100}\atop{k}}\right)}
  14. ( 1 - Q ) (1-Q)
  15. 2 - L 10 - 30 2^{-L}\approx 10^{-30}
  16. Q c = 1 / a . Q_{c}=1/a.
  17. 1 - Q c 1-Q_{c}

Ethanol_(data_page).html

  1. P m m H g = 10 8.04494 - 1554.3 222.65 + T \scriptstyle P_{mmHg}=10^{8.04494-\frac{1554.3}{222.65+T}}
  2. log 10 P m m H g = 8.04494 - 1554.3 222.65 + T \scriptstyle\log_{10}{P_{mmHg}}=8.04494-\frac{1554.3}{222.65+T}

Ettingshausen_effect.html

  1. P = d T / d x d z | B z | I y P=\frac{dT/dx\cdot d_{z}}{|B_{z}|\cdot I_{y}}
  2. P bismuth = 7.5 ± 0.3 × 10 - 4 K T A P_{\mathrm{bismuth}}=7.5\pm 0.3\times 10^{-4}\frac{K}{T\cdot A}

Euclid's_theorem.html

  1. p P 1 1 - 1 / p = p P k 0 1 p k = n 1 n . \prod_{p\in P}\frac{1}{1-1/p}=\prod_{p\in P}\sum_{k\geq 0}\frac{1}{p^{k}}=\sum% _{n}\frac{1}{n}.
  2. p P k 0 1 p k = k 0 1 2 k × k 0 1 3 k × k 0 1 5 k × k 0 1 7 k × = k , l , m , n , 0 1 2 k 3 l 5 m 7 n = n 1 n \prod_{p\in P}\sum_{k\geq 0}\frac{1}{p^{k}}=\sum_{k\geq 0}\frac{1}{2^{k}}% \times\sum_{k\geq 0}\frac{1}{3^{k}}\times\sum_{k\geq 0}\frac{1}{5^{k}}\times% \sum_{k\geq 0}\frac{1}{7^{k}}\times\cdots=\sum_{k,l,m,n,\cdots\geq 0}\frac{1}{% 2^{k}3^{l}5^{m}7^{n}\cdots}=\sum_{n}\frac{1}{n}
  3. r s 2 rs^{2}
  4. 2 k N N . 2^{k}\sqrt{N}\geq N.
  5. 1 + i x p i - i < j x p i p j \displaystyle 1+\sum_{i}\left\lfloor\frac{x}{p_{i}}\right\rfloor-\sum_{i<j}% \left\lfloor\frac{x}{p_{i}p_{j}}\right\rfloor
  6. i 1 p i - i < j 1 p i p j + i < j < k 1 p i p j p k - ± ( - 1 ) N + 1 1 p 1 p N . ( 2 ) \sum_{i}\frac{1}{p_{i}}-\sum_{i<j}\frac{1}{p_{i}p_{j}}+\sum_{i<j<k}\frac{1}{p_% {i}p_{j}p_{k}}-\cdots\pm(-1)^{N+1}\frac{1}{p_{1}\cdots p_{N}}.\qquad(2)
  7. 1 - i = 1 N ( 1 - 1 p i ) . ( 3 ) 1-\prod_{i=1}^{N}\left(1-\frac{1}{p_{i}}\right).\qquad(3)
  8. x \lfloor x\rfloor
  9. k ! = p prime p f ( p , k ) k!=\prod_{p\,\text{ prime}}p^{f(p,k)}\,
  10. f ( p , k ) = k p + k p 2 + . f(p,k)=\left\lfloor\frac{k}{p}\right\rfloor+\left\lfloor\frac{k}{p^{2}}\right% \rfloor+\cdots.
  11. f ( p , k ) < k p + k p 2 + = k p - 1 k . f(p,k)<\frac{k}{p}+\frac{k}{p^{2}}+\cdots=\frac{k}{p-1}\leq k.
  12. lim k ( p p ) k k ! = 0 , \lim_{k\to\infty}\frac{\left(\prod_{p}p\right)^{k}}{k!}=0,
  13. π \pi
  14. π \pi
  15. π 4 = 3 4 × 5 4 × 7 8 × 11 12 × 13 12 × 17 16 × 19 20 × 23 24 × 29 28 × 31 32 × \frac{\pi}{4}=\frac{3}{4}\times\frac{5}{4}\times\frac{7}{8}\times\frac{11}{12}% \times\frac{13}{12}\times\frac{17}{16}\times\frac{19}{20}\times\frac{23}{24}% \times\frac{29}{28}\times\frac{31}{32}\times\cdots\;
  16. π \pi
  17. π \pi
  18. a b a\mid b
  19. a c a\mid c
  20. a ( b - c ) a\mid(b-c)

Euler_class.html

  1. H r ( X ; 𝐙 ) , H^{r}(X;\mathbf{Z}),
  2. H r ( F , F F 0 ; 𝐙 ) H^{r}(F,F\setminus F_{0};\mathbf{Z})
  3. u H r ( E , E E 0 ; 𝐙 ) u\in H^{r}(E,E\setminus E_{0};\mathbf{Z})
  4. ( X , ) ( E , ) ( E , E E 0 ) , (X,\emptyset)\hookrightarrow(E,\emptyset)\hookrightarrow(E,E\setminus E_{0}),
  5. H r ( E , E E 0 ; 𝐙 ) H r ( E ; 𝐙 ) H r ( X ; 𝐙 ) . H^{r}(E,E\setminus E_{0};\mathbf{Z})\to H^{r}(E;\mathbf{Z})\to H^{r}(X;\mathbf% {Z}).
  6. e ( E F ) = e ( E ) e ( F ) . e(E\oplus F)=e(E)\smile e(F).
  7. E ¯ \overline{E}
  8. E ¯ \overline{E}
  9. H r ( X , 𝐙 ) H r ( X , 𝐙 / 2 ) . H^{r}(X,\mathbf{Z})\to H^{r}(X,\mathbf{Z}/2).
  10. e ( E ) e ( E ) = e ( E E ) = e ( E 𝐂 ) = c r ( E 𝐂 ) H 2 r ( X , 𝐙 ) . e(E)\cup e(E)=e(E\oplus E)=e(E\otimes\mathbf{C})=c_{r}(E\otimes\mathbf{C})\in H% ^{2r}(X,\mathbf{Z}).
  11. e ( E ) e ( E ) e(E)\cup e(E)
  12. χ ( 𝐒 n ) = 1 + ( - 1 ) n = { 2 n even 0 n odd . \chi(\mathbf{S}^{n})=1+(-1)^{n}=\begin{cases}2&n\,\text{ even}\\ 0&n\,\text{ odd}.\end{cases}
  13. ( x 2 , - x 1 , x 4 , - x 3 , , x 2 n , - x 2 n - 1 ) (x_{2},-x_{1},x_{4},-x_{3},\dots,x_{2n},-x_{2n-1})

Euler_diagram.html

  1. A = { 1 , 2 , 5 } A=\{1,\,2,\,5\}
  2. B = { 1 , 6 } B=\{1,\,6\}
  3. C = { 4 , 7 } C=\{4,\,7\}

Euler_function.html

  1. ϕ ( q ) = k = 1 ( 1 - q k ) . \phi(q)=\prod_{k=1}^{\infty}(1-q^{k}).
  2. p ( k ) p(k)
  3. 1 / ϕ ( q ) 1/\phi(q)
  4. 1 ϕ ( q ) = k = 0 p ( k ) q k \frac{1}{\phi(q)}=\sum_{k=0}^{\infty}p(k)q^{k}
  5. p ( k ) p(k)
  6. ϕ ( q ) = n = - ( - 1 ) n q ( 3 n 2 - n ) / 2 . \phi(q)=\sum_{n=-\infty}^{\infty}(-1)^{n}q^{(3n^{2}-n)/2}.
  7. ( 3 n 2 - n ) / 2 (3n^{2}-n)/2
  8. ϕ ( q ) = q - 1 24 η ( τ ) \phi(q)=q^{-\frac{1}{24}}\eta(\tau)
  9. q = e 2 π i τ q=e^{2\pi i\tau}
  10. ϕ ( q ) = ( q ; q ) \phi(q)=(q;q)_{\infty}
  11. ln ( ϕ ( q ) ) = - n = 1 1 n q n 1 - q n \ln(\phi(q))=-\sum_{n=1}^{\infty}\frac{1}{n}\,\frac{q^{n}}{1-q^{n}}
  12. ln ( ϕ ( q ) ) = n = 1 b n q n \ln(\phi(q))=\sum_{n=1}^{\infty}b_{n}q^{n}
  13. b n = - d | n 1 d = b_{n}=-\sum_{d|n}\frac{1}{d}=
  14. d | n d = d | n n d \sum_{d|n}d=\sum_{d|n}\frac{n}{d}
  15. ln ( ϕ ( q ) ) = - n = 1 q n n d | n d \ln(\phi(q))=-\sum_{n=1}^{\infty}\frac{q^{n}}{n}\sum_{d|n}d
  16. ϕ ( e - π ) = e π / 24 Γ ( 1 4 ) 2 7 / 8 π 3 / 4 \phi(e^{-\pi})=\frac{e^{\pi/24}\Gamma\left(\frac{1}{4}\right)}{2^{7/8}\pi^{3/4}}
  17. ϕ ( e - 2 π ) = e π / 12 Γ ( 1 4 ) 2 π 3 / 4 \phi(e^{-2\pi})=\frac{e^{\pi/12}\Gamma\left(\frac{1}{4}\right)}{2\pi^{3/4}}
  18. ϕ ( e - 4 π ) = e π / 6 Γ ( 1 4 ) 2 11 / 8 π 3 / 4 \phi(e^{-4\pi})=\frac{e^{\pi/6}\Gamma\left(\frac{1}{4}\right)}{2^{{11}/8}\pi^{% 3/4}}
  19. ϕ ( e - 8 π ) = e π / 3 Γ ( 1 4 ) 2 29 / 16 π 3 / 4 ( 2 - 1 ) 1 / 4 \phi(e^{-8\pi})=\frac{e^{\pi/3}\Gamma\left(\frac{1}{4}\right)}{2^{29/16}\pi^{3% /4}}(\sqrt{2}-1)^{1/4}

Euler_method.html

  1. A 0 , A_{0},
  2. A 0 A_{0}
  3. A 1 . A_{1}.
  4. A 1 A_{1}
  5. A 1 A_{1}
  6. A 0 A_{0}
  7. A 0 A 1 A 2 A 3 A_{0}A_{1}A_{2}A_{3}\dots
  8. y ( t ) = f ( t , y ( t ) ) , y ( t 0 ) = y 0 . y^{\prime}(t)=f(t,y(t)),\qquad\qquad y(t_{0})=y_{0}.
  9. h h
  10. t n = t 0 + n h t_{n}=t_{0}+nh
  11. t n t_{n}
  12. t n + 1 = t n + h t_{n+1}=t_{n}+h
  13. y n + 1 = y n + h f ( t n , y n ) . y_{n+1}=y_{n}+hf(t_{n},y_{n}).
  14. y n y_{n}
  15. t n t_{n}
  16. y n y ( t n ) y_{n}\approx y(t_{n})
  17. y n + 1 y_{n+1}
  18. y i y_{i}
  19. i n i\leq n
  20. y ( N ) ( t ) = f ( t , y ( t ) , y ( t ) , , y ( N - 1 ) ( t ) ) y^{(N)}(t)=f(t,y(t),y^{\prime}(t),\ldots,y^{(N-1)}(t))
  21. z 1 ( t ) = y ( t ) , z 2 ( t ) = y ( t ) , , z N ( t ) = y ( N - 1 ) ( t ) z_{1}(t)=y(t),z_{2}(t)=y^{\prime}(t),\ldots,z_{N}(t)=y^{(N-1)}(t)
  22. 𝐳 ( t ) = ( z 1 ( t ) z N - 1 ( t ) z N ( t ) ) = ( y ( t ) y ( N - 1 ) ( t ) y ( N ) ( t ) ) = ( z 2 ( t ) z N ( t ) f ( t , z 1 ( t ) , , z N ( t ) ) ) \mathbf{z}^{\prime}(t)=\begin{pmatrix}z_{1}^{\prime}(t)\\ \vdots\\ z_{N-1}^{\prime}(t)\\ z_{N}^{\prime}(t)\end{pmatrix}=\begin{pmatrix}y^{\prime}(t)\\ \vdots\\ y^{(N-1)}(t)\\ y^{(N)}(t)\end{pmatrix}=\begin{pmatrix}z_{2}(t)\\ \vdots\\ z_{N}(t)\\ f(t,z_{1}(t),\ldots,z_{N}(t))\end{pmatrix}
  23. 𝐳 ( t ) \mathbf{z}(t)
  24. y = y , y ( 0 ) = 1 , y^{\prime}=y,\quad y(0)=1,
  25. y ( 4 ) y(4)
  26. y n + 1 = y n + h f ( t n , y n ) . y_{n+1}=y_{n}+hf(t_{n},y_{n}).\qquad\qquad
  27. f ( t 0 , y 0 ) f(t_{0},y_{0})
  28. f f
  29. f ( t , y ) = y f(t,y)=y
  30. f ( t 0 , y 0 ) = f ( 0 , 1 ) = 1. f(t_{0},y_{0})=f(0,1)=1.\qquad\qquad
  31. ( 0 , 1 ) (0,1)
  32. y y
  33. t t
  34. Δ y / Δ t \Delta y/\Delta t
  35. h h
  36. h f ( y 0 ) = 1 1 = 1. h\cdot f(y_{0})=1\cdot 1=1.\qquad\qquad
  37. t t
  38. y y
  39. y y
  40. y 0 + h f ( y 0 ) = y 1 = 1 + 1 1 = 2. y_{0}+hf(y_{0})=y_{1}=1+1\cdot 1=2.\qquad\qquad
  41. y 2 y_{2}
  42. y 3 y_{3}
  43. y 4 y_{4}
  44. y 2 = y 1 + h f ( y 1 ) = 2 + 1 2 = 4 , y 3 = y 2 + h f ( y 2 ) = 4 + 1 4 = 8 , y 4 = y 3 + h f ( y 3 ) = 8 + 1 8 = 16. \begin{aligned}\displaystyle y_{2}&\displaystyle=y_{1}+hf(y_{1})=2+1\cdot 2=4,% \\ \displaystyle y_{3}&\displaystyle=y_{2}+hf(y_{2})=4+1\cdot 4=8,\\ \displaystyle y_{4}&\displaystyle=y_{3}+hf(y_{3})=8+1\cdot 8=16.\end{aligned}
  45. n n
  46. y n y_{n}
  47. t n t_{n}
  48. f ( t n , y n ) f(t_{n},y_{n})
  49. h h
  50. Δ y \Delta y
  51. y n + 1 y_{n+1}
  52. y 4 = 16 y_{4}=16
  53. y ( t ) = e t y(t)=e^{t}
  54. y ( 4 ) = e 4 54.598 y(4)=e^{4}\approx 54.598
  55. h h
  56. t = 4 t=4
  57. y y
  58. t 0 t_{0}
  59. y ( t 0 + h ) = y ( t 0 ) + h y ( t 0 ) + 1 2 h 2 y ′′ ( t 0 ) + O ( h 3 ) . y(t_{0}+h)=y(t_{0})+hy^{\prime}(t_{0})+\frac{1}{2}h^{2}y^{\prime\prime}(t_{0})% +O(h^{3}).
  60. y = f ( t , y ) y^{\prime}=f(t,y)
  61. y ( t 0 ) y ( t 0 + h ) - y ( t 0 ) h y^{\prime}(t_{0})\approx\frac{y(t_{0}+h)-y(t_{0})}{h}
  62. y = f ( t , y ) y^{\prime}=f(t,y)
  63. t 0 t_{0}
  64. t 0 + h t_{0}+h
  65. y ( t 0 + h ) - y ( t 0 ) = t 0 t 0 + h f ( t , y ( t ) ) d t . y(t_{0}+h)-y(t_{0})=\int_{t_{0}}^{t_{0}+h}f(t,y(t))\,\mathrm{d}t.
  66. t 0 t 0 + h f ( t , y ( t ) ) d t h f ( t 0 , y ( t 0 ) ) . \int_{t_{0}}^{t_{0}+h}f(t,y(t))\,\mathrm{d}t\approx hf(t_{0},y(t_{0})).
  67. y 1 y_{1}
  68. t 1 = t 0 + h t_{1}=t_{0}+h
  69. y 1 = y 0 + h f ( t 0 , y 0 ) . y_{1}=y_{0}+hf(t_{0},y_{0}).\quad
  70. y ( t 0 + h ) = y ( t 0 ) + h y ( t 0 ) + 1 2 h 2 y ′′ ( t 0 ) + O ( h 3 ) . y(t_{0}+h)=y(t_{0})+hy^{\prime}(t_{0})+\frac{1}{2}h^{2}y^{\prime\prime}(t_{0})% +O(h^{3}).
  71. LTE = y ( t 0 + h ) - y 1 = 1 2 h 2 y ′′ ( t 0 ) + O ( h 3 ) . \mathrm{LTE}=y(t_{0}+h)-y_{1}=\frac{1}{2}h^{2}y^{\prime\prime}(t_{0})+O(h^{3}).
  72. y y
  73. h h
  74. h 2 h^{2}
  75. h h
  76. y y
  77. ξ [ t 0 , t 0 + h ] \xi\in[t_{0},t_{0}+h]
  78. LTE = y ( t 0 + h ) - y 1 = 1 2 h 2 y ′′ ( ξ ) . \mathrm{LTE}=y(t_{0}+h)-y_{1}=\frac{1}{2}h^{2}y^{\prime\prime}(\xi).
  79. y y
  80. y = f ( t , y ) y^{\prime}=f(t,y)
  81. y ′′ ( t 0 ) = f t ( t 0 , y ( t 0 ) ) + f y ( t 0 , y ( t 0 ) ) f ( t 0 , y ( t 0 ) ) . y^{\prime\prime}(t_{0})={\partial f\over\partial t}(t_{0},y(t_{0}))+{\partial f% \over\partial y}(t_{0},y(t_{0}))\,f(t_{0},y(t_{0})).
  82. t t
  83. ( t - t 0 ) / h (t-t_{0})/h
  84. 1 / h 1/h
  85. h 2 h^{2}
  86. h h
  87. y y
  88. f f
  89. | GTE | h M 2 L ( e L ( t - t 0 ) - 1 ) |\,\text{GTE}|\leq\frac{hM}{2L}(e^{L(t-t_{0})}-1)\qquad\qquad
  90. M M
  91. y y
  92. L L
  93. f f
  94. h h
  95. y = - 2.3 y , y ( 0 ) = 1. y^{\prime}=-2.3y,\qquad y(0)=1.
  96. y ( t ) = e - 2.3 t y(t)=e^{-2.3t}
  97. t t\to\infty
  98. h = 1 h=1
  99. h = 0.7 h=0.7
  100. y = k y y^{\prime}=ky
  101. h k hk
  102. { z 𝐂 | z + 1 | 1 } , \{z\in\mathbf{C}\mid|z+1|\leq 1\},
  103. k k
  104. h = 1 h=1
  105. h k = - 2.3 hk=-2.3
  106. ε / h \varepsilon/\sqrt{h}
  107. y n + 1 = y n + h f ( t n + 1 , y n + 1 ) . y_{n+1}=y_{n}+hf(t_{n+1},y_{n+1}).
  108. f f
  109. y n + 1 y_{n+1}
  110. y n + 1 = y n + h f ( t n + 1 2 h , y n + 1 2 h f ( t n , y n ) ) . y_{n+1}=y_{n}+hf\Big(t_{n}+\tfrac{1}{2}h,y_{n}+\tfrac{1}{2}hf(t_{n},y_{n})\Big).
  111. y n + 1 = y n + 3 2 h f ( t n , y n ) - 1 2 h f ( t n - 1 , y n - 1 ) . y_{n+1}=y_{n}+\tfrac{3}{2}hf(t_{n},y_{n})-\tfrac{1}{2}hf(t_{n-1},y_{n-1}).

Euler–Bernoulli_beam_theory.html

  1. d 2 d x 2 ( E I d 2 w d x 2 ) = q \frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}\left(EI\frac{\mathrm{d}^{2}w}{\mathrm{d% }x^{2}}\right)=q\,
  2. w ( x ) w(x)
  3. z z
  4. x x
  5. q q
  6. x x
  7. w w
  8. E E
  9. I I
  10. I I
  11. I = z 2 d y d z , I=\iint z^{2}\;dy\;dz,
  12. E I d 4 w d x 4 = q ( x ) . EI\frac{\mathrm{d}^{4}w}{\mathrm{d}x^{4}}=q(x).\,
  13. w w
  14. 𝐞 𝐳 × 𝐞 𝐱 = 𝐞 𝐲 \mathbf{e_{z}}\times\mathbf{e_{x}}=\mathbf{e_{y}}
  15. 𝐞 𝐱 \mathbf{e_{x}}
  16. 𝐞 𝐲 \mathbf{e_{y}}
  17. 𝐞 𝐳 \mathbf{e_{z}}
  18. x x
  19. z z
  20. d M = Q d x dM=Qdx
  21. Q Q
  22. d Q = q d x dQ=qdx
  23. 𝐞 𝐱 \mathbf{e_{x}}
  24. 𝐞 𝐲 \mathbf{e_{y}}
  25. 𝐞 𝐳 \mathbf{e_{z}}
  26. M = - E I d 2 w d x 2 M=-EI\frac{d^{2}w}{dx^{2}}
  27. Q = - d d x ( E I d 2 w d x 2 ) Q=-\frac{d}{dx}\left(EI\frac{d^{2}w}{dx^{2}}\right)
  28. ρ d θ \rho\ d\theta
  29. ( ρ + e ) d θ \left(\rho+e\right)\ d\theta
  30. ( ρ + e - ρ ) d θ ρ d θ = e ρ . \frac{\left(\rho+e-\rho\right)\ d\theta}{\rho\ d\theta}=\frac{e}{\rho}\ .
  31. E e ρ E\ \frac{e}{\rho}
  32. d 𝐅 , d\mathbf{F}\ ,
  33. d 𝐅 = E e ρ d A 𝐞 𝐱 . d\mathbf{F}=E\ \frac{e}{\rho}\ dA\ \mathbf{e_{x}}\ .
  34. 𝐞 𝐱 \mathbf{e_{x}}
  35. d A dA
  36. d 𝐌 , d\mathbf{M}\ ,
  37. d 𝐅 , d\mathbf{F}\ ,
  38. d 𝐌 = - e 𝐞 𝐳 × d 𝐅 = - 𝐞 𝐲 E e 2 ρ d A . d\mathbf{M}=-e\mathbf{e_{z}}\times d\mathbf{F}\ =-\mathbf{e_{y}}E\ \frac{e^{2}% }{\rho}\ dA\ .
  39. 𝐞 𝐳 × - 𝐞 𝐱 = - 𝐞 𝐲 . \mathbf{e_{z}}\times-\mathbf{e_{x}}=-\mathbf{e_{y}}\ .
  40. 𝐌 \mathbf{M}
  41. 𝐌 = d 𝐌 = - 𝐞 𝐲 E ρ e 2 d A = - 𝐞 𝐲 E I ρ \mathbf{M}=\int d\mathbf{M}=-\mathbf{e_{y}}\frac{E}{\rho}\int{e^{2}}\ dA=\ -% \mathbf{e_{y}}\frac{EI}{\rho}
  42. I I
  43. d w d x {dw\over dx}
  44. 1 ρ = d 2 w d x 2 \frac{1}{\rho}={d^{2}w\over dx^{2}}
  45. ρ \rho
  46. 𝐌 = - 𝐞 𝐲 E I d 2 w d x 2 = - 𝐞 𝐲 E I d 2 z d x 2 \mathbf{M}=\ -\mathbf{e_{y}}EI{d^{2}w\over dx^{2}}=\ -\mathbf{e_{y}}EI{d^{2}z% \over dx^{2}}
  47. S = 0 L [ 1 2 μ ( w t ) 2 - 1 2 E I ( 2 w x 2 ) 2 + q ( x ) w ( x , t ) ] d x . S=\int_{0}^{L}\left[\frac{1}{2}\mu\left(\frac{\partial w}{\partial t}\right)^{% 2}-\frac{1}{2}EI\left(\frac{\partial^{2}w}{\partial x^{2}}\right)^{2}+q(x)w(x,% t)\right]dx.
  48. μ \mu
  49. q ( x ) q(x)
  50. S S
  51. 2 x 2 ( E I 2 w x 2 ) = - μ 2 w t 2 + q ( x ) \cfrac{\partial^{2}}{\partial x^{2}}\left(EI\cfrac{\partial^{2}w}{\partial x^{% 2}}\right)=-\mu\cfrac{\partial^{2}w}{\partial t^{2}}+q(x)
  52. := 1 2 μ ( w t ) 2 - 1 2 E I ( 2 w x 2 ) 2 + q ( x ) w ( x , t ) = μ 2 w ˙ 2 - E I 2 w x x 2 + q w ( x , t , w , w ˙ , w x x ) \mathcal{L}:=\tfrac{1}{2}\mu\left(\frac{\partial w}{\partial t}\right)^{2}-% \tfrac{1}{2}EI\left(\frac{\partial^{2}w}{\partial x^{2}}\right)^{2}+q(x)w(x,t)% =\tfrac{\mu}{2}\dot{w}^{2}-\tfrac{EI}{2}w_{xx}^{2}+qw\equiv\mathcal{L}(x,t,w,% \dot{w},w_{xx})
  53. w - t ( w ˙ ) + 2 x 2 ( w x x ) = 0 \cfrac{\partial\mathcal{L}}{\partial w}-\frac{\partial}{\partial t}\left(\frac% {\partial\mathcal{L}}{\partial\dot{w}}\right)+\frac{\partial^{2}}{\partial x^{% 2}}\left(\frac{\partial\mathcal{L}}{\partial w_{xx}}\right)=0
  54. w = q ; w ˙ = μ w ˙ ; w x x = - E I w x x . \cfrac{\partial\mathcal{L}}{\partial w}=q~{};~{}~{}\frac{\partial\mathcal{L}}{% \partial\dot{w}}=\mu\dot{w}~{};~{}~{}\frac{\partial\mathcal{L}}{\partial w_{xx% }}=-EIw_{xx}~{}.
  55. q - μ w ¨ - ( E I w x x ) x x = 0 q-\mu\ddot{w}-(EIw_{xx})_{xx}=0
  56. 2 x 2 ( E I 2 w x 2 ) = - μ 2 w t 2 + q \cfrac{\partial^{2}}{\partial x^{2}}\left(EI\cfrac{\partial^{2}w}{\partial x^{% 2}}\right)=-\mu\cfrac{\partial^{2}w}{\partial t^{2}}+q
  57. E E
  58. I I
  59. x x
  60. E I 4 w x 4 = - μ 2 w t 2 + q . EI\cfrac{\partial^{4}w}{\partial x^{4}}=-\mu\cfrac{\partial^{2}w}{\partial t^{% 2}}+q\,.
  61. q q
  62. w ( x , t ) = Re [ w ^ ( x ) e - i ω t ] w(x,t)=\,\text{Re}[\hat{w}(x)~{}e^{-i\omega t}]
  63. ω \omega
  64. E I d 4 w ^ d x 4 - μ ω 2 w ^ = 0 . EI~{}\cfrac{\mathrm{d}^{4}\hat{w}}{\mathrm{d}x^{4}}-\mu\omega^{2}\hat{w}=0\,.
  65. w ^ = A 1 cosh ( β x ) + A 2 sinh ( β x ) + A 3 cos ( β x ) + A 4 sin ( β x ) with β := ( μ ω 2 E I ) 1 / 4 \hat{w}=A_{1}\cosh(\beta x)+A_{2}\sinh(\beta x)+A_{3}\cos(\beta x)+A_{4}\sin(% \beta x)\quad\,\text{with}\quad\beta:=\left(\frac{\mu\omega^{2}}{EI}\right)^{1% /4}
  66. A 1 , A 2 , A 3 , A 4 A_{1},A_{2},A_{3},A_{4}
  67. w ^ n = A 1 cosh ( β n x ) + A 2 sinh ( β n x ) + A 3 cos ( β n x ) + A 4 sin ( β n x ) with β n := ( μ ω n 2 E I ) 1 / 4 . \hat{w}_{n}=A_{1}\cosh(\beta_{n}x)+A_{2}\sinh(\beta_{n}x)+A_{3}\cos(\beta_{n}x% )+A_{4}\sin(\beta_{n}x)\quad\,\text{with}\quad\beta_{n}:=\left(\frac{\mu\omega% _{n}^{2}}{EI}\right)^{1/4}\,.
  68. ω n \omega_{n}
  69. L L
  70. x = 0 x=0
  71. w ^ n = 0 , d w ^ n d x = 0 at x = 0 d 2 w ^ n d x 2 = 0 , d 3 w ^ n d x 3 = 0 at x = L . \begin{aligned}&\displaystyle\hat{w}_{n}=0~{},~{}~{}\frac{d\hat{w}_{n}}{dx}=0% \quad\,\text{at}~{}~{}x=0\\ &\displaystyle\frac{d^{2}\hat{w}_{n}}{dx^{2}}=0~{},~{}~{}\frac{d^{3}\hat{w}_{n% }}{dx^{3}}=0\quad\,\text{at}~{}~{}x=L\,.\end{aligned}
  72. cosh ( β n L ) cos ( β n L ) + 1 = 0 . \cosh(\beta_{n}L)\,\cos(\beta_{n}L)+1=0\,.
  73. ω 1 = β 1 2 E I μ = 3.515 L 2 E I μ , \omega_{1}=\beta_{1}^{2}\sqrt{\frac{EI}{\mu}}=\frac{3.515}{L^{2}}\sqrt{\frac{% EI}{\mu}}~{},~{}~{}\dots
  74. w ^ n = A 1 [ cosh β n x - cos β n x + ( cos β n L + cosh β n L ) ( sin β n x - sinh β n x ) sin β n L + sinh β n L ] \hat{w}_{n}=A_{1}\Bigl[\cosh\beta_{n}x-\cos\beta_{n}x+\frac{(\cos\beta_{n}L+% \cosh\beta_{n}L)(\sin\beta_{n}x-\sinh\beta_{n}x)}{\sin\beta_{n}L+\sinh\beta_{n% }L}\Bigr]
  75. n n
  76. A 1 A_{1}
  77. t = 0 t=0
  78. A 1 = 1 A_{1}=1
  79. ω n \omega_{n}
  80. σ = M z I = - z E d 2 w d x 2 . \sigma=\frac{Mz}{I}=-zE~{}\frac{\mathrm{d}^{2}w}{\mathrm{d}x^{2}}.\,
  81. z z
  82. M M
  83. z = c 1 z=c_{1}
  84. z = - c 2 z=-c_{2}
  85. h = c 1 + c 2 h=c_{1}+c_{2}
  86. σ 1 = M c 1 I = M S 1 ; σ 2 = - M c 2 I = - M S 2 \sigma_{1}=\cfrac{Mc_{1}}{I}=\cfrac{M}{S_{1}}~{};~{}~{}\sigma_{2}=-\cfrac{Mc_{% 2}}{I}=-\cfrac{M}{S_{2}}
  87. S 1 , S 2 S_{1},S_{2}
  88. S 1 = I c 1 ; S 2 = I c 2 S_{1}=\cfrac{I}{c_{1}}~{};~{}~{}S_{2}=\cfrac{I}{c_{2}}
  89. c 1 = c 2 c_{1}=c_{2}
  90. S = I / c S=I/c
  91. ρ \rho
  92. d x \mathrm{d}x
  93. ρ \rho
  94. d θ \mathrm{d}\theta
  95. d x = ρ d θ \mathrm{d}x=\rho~{}\mathrm{d}\theta
  96. z z
  97. d x \mathrm{d}x
  98. d x = ( ρ - z ) d θ = d x - z d θ \mathrm{d}x^{\prime}=(\rho-z)~{}\mathrm{d}\theta=\mathrm{d}x-z~{}\mathrm{d}\theta
  99. ε x = d x - d x d x = - z ρ = - κ z \varepsilon_{x}=\cfrac{\mathrm{d}x^{\prime}-\mathrm{d}x}{\mathrm{d}x}=-\cfrac{% z}{\rho}=-\kappa~{}z
  100. κ \kappa
  101. w w
  102. x x
  103. ( x , z ) (x,z)
  104. x x
  105. θ ( x ) = d w d x \theta(x)=\cfrac{\mathrm{d}w}{\mathrm{d}x}
  106. d x \mathrm{d}x
  107. d x = ρ d θ \mathrm{d}x=\rho~{}\mathrm{d}\theta
  108. 1 ρ = d θ d x = d 2 w d x 2 = κ \cfrac{1}{\rho}=\cfrac{\mathrm{d}\theta}{\mathrm{d}x}=\cfrac{\mathrm{d}^{2}w}{% \mathrm{d}x^{2}}=\kappa
  109. ε x = - z d 2 w d x 2 \varepsilon_{x}=-z\cfrac{\mathrm{d}^{2}w}{\mathrm{d}x^{2}}
  110. σ = E ε \sigma=E\varepsilon
  111. E E
  112. σ x = - z E d 2 w d x 2 \sigma_{x}=-zE\cfrac{\mathrm{d}^{2}w}{\mathrm{d}x^{2}}
  113. M = - E I d 2 w d x 2 M=-EI\cfrac{\mathrm{d}^{2}w}{\mathrm{d}x^{2}}
  114. Q = d M / d x Q=\mathrm{d}M/\mathrm{d}x
  115. Q = - E I d 3 w d x 3 Q=-EI\cfrac{\mathrm{d}^{3}w}{\mathrm{d}x^{3}}
  116. x x
  117. w ( x , t ) w(x,t)
  118. w w
  119. d w / d x \mathrm{d}w/\mathrm{d}x
  120. w w
  121. x x
  122. 0
  123. L L
  124. E I EI
  125. w | x = 0 = 0 ; w x | x = 0 = 0 (fixed end) w|_{x=0}=0\quad;\quad\frac{\partial w}{\partial x}\bigg|_{x=0}=0\qquad\mbox{(% fixed end)}~{}\,
  126. 2 w x 2 | x = L = 0 ; 3 w x 3 | x = L = 0 (free end) \frac{\partial^{2}w}{\partial x^{2}}\bigg|_{x=L}=0\quad;\quad\frac{\partial^{3% }w}{\partial x^{3}}\bigg|_{x=L}=0\qquad\mbox{(free end)}~{}\,
  127. λ = F / E I \lambda=F/EI
  128. τ = M / E I \tau=M/EI
  129. Δ \Delta
  130. Δ w ′′ = w ′′ ( x + ) - w ′′ ( x - ) \Delta w^{\prime\prime}=w^{\prime\prime}(x+)-w^{\prime\prime}(x-)
  131. w ′′ ( x + ) w^{\prime\prime}(x+)
  132. w ′′ w^{\prime\prime}
  133. w ′′ ( x - ) w^{\prime\prime}(x-)
  134. w ′′ w^{\prime\prime}
  135. Δ w ′′ = 0 * \Delta w^{\prime\prime}=0^{*}
  136. w ′′ ( x - ) = w ′′ ( x + ) w^{\prime\prime}(x-)=w^{\prime\prime}(x+)
  137. w ′′′ w^{\prime\prime\prime}
  138. w ′′ w^{\prime\prime}
  139. w w^{\prime}
  140. w w
  141. Δ w = 0 * \Delta w^{\prime}=0^{*}
  142. Δ w = 0 * \Delta w=0^{*}
  143. Δ w ′′ = 0 \Delta w^{\prime\prime}=0
  144. Δ w = 0 \Delta w^{\prime}=0
  145. Δ w = 0 * \Delta w=0^{*}
  146. Δ w ′′′ = λ \Delta w^{\prime\prime\prime}=\lambda
  147. Δ w ′′ = 0 \Delta w^{\prime\prime}=0
  148. Δ w = 0 \Delta w^{\prime}=0
  149. Δ w = 0 \Delta w=0
  150. Δ w ′′′ = 0 \Delta w^{\prime\prime\prime}=0
  151. Δ w ′′ = τ \Delta w^{\prime\prime}=\tau
  152. Δ w = 0 \Delta w^{\prime}=0
  153. Δ w = 0 \Delta w=0
  154. w ′′′ = 0 w^{\prime\prime\prime}=0
  155. w ′′ = 0 w^{\prime\prime}=0
  156. w w^{\prime}
  157. w w
  158. w ′′ = 0 w^{\prime\prime}=0
  159. w w
  160. w ′′′ = ± λ w^{\prime\prime\prime}=\pm\lambda
  161. w ′′ = 0 w^{\prime\prime}=0
  162. w ′′′ = 0 w^{\prime\prime\prime}=0
  163. w ′′ = ± τ w^{\prime\prime}=\pm\tau
  164. q ( x , t ) q(x,t)
  165. L L
  166. F F
  167. E I d 4 w d x 4 = 0 \displaystyle EI\frac{\mathrm{d}^{4}w}{\mathrm{d}x^{4}}=0
  168. E I d 4 w d x 4 = F δ ( x - L ) w | x = 0 = 0 ; d w d x | x = 0 = 0 ; d 2 w d x 2 | x = L = 0 \begin{aligned}&\displaystyle EI\frac{\mathrm{d}^{4}w}{\mathrm{d}x^{4}}=F% \delta(x-L)\\ &\displaystyle w|_{x=0}=0\quad;\quad\frac{\mathrm{d}w}{\mathrm{d}x}\bigg|_{x=0% }=0\quad;\quad\frac{\mathrm{d}^{2}w}{\mathrm{d}x^{2}}\bigg|_{x=L}=0\end{aligned}
  169. w = F 6 E I ( 3 L x 2 - x 3 ) . w=\frac{F}{6EI}(3Lx^{2}-x^{3})\,~{}.
  170. w ( x ) w(x)
  171. q ( x , t ) = μ 2 w t 2 q(x,t)=\mu\frac{\partial^{2}w}{\partial t^{2}}\,
  172. μ \mu
  173. 2 x 2 ( E I 2 w x 2 ) = - μ 2 w t 2 . \frac{\partial^{2}}{\partial x^{2}}\left(EI\frac{\partial^{2}w}{\partial x^{2}% }\right)=-\mu\frac{\partial^{2}w}{\partial t^{2}}.
  174. ω \omega
  175. q ( x ) = μ ω 2 w ( x ) q(x)=\mu\omega^{2}w(x)\,
  176. q q
  177. M M
  178. Q Q
  179. w w
  180. M ( x ) = { P x 2 , for 0 x L 2 P ( L - x ) 2 , for L 2 < x L M(x)=\begin{cases}\frac{Px}{2},&\mbox{for }~{}0\leq x\leq\tfrac{L}{2}\\ \frac{P(L-x)}{2},&\mbox{for }~{}\tfrac{L}{2}<x\leq L\end{cases}
  181. M L / 2 = P L 4 M_{L/2}=\tfrac{PL}{4}
  182. Q ( x ) = { P 2 , for 0 x L 2 - P 2 , for L 2 < x L Q(x)=\begin{cases}\frac{P}{2},&\mbox{for }~{}0\leq x\leq\tfrac{L}{2}\\ \frac{-P}{2},&\mbox{for }~{}\tfrac{L}{2}<x\leq L\end{cases}
  183. | Q 0 | = | Q L | = P 2 |Q_{0}|=|Q_{L}|=\tfrac{P}{2}
  184. w ( x ) = { - P x ( 4 x 2 - 3 L 2 ) 48 E I , for 0 x L 2 P ( x - L ) ( L 2 - 8 L x + 4 x 2 ) 48 E I , for L 2 < x L w(x)=\begin{cases}-\frac{Px(4x^{2}-3L^{2})}{48EI},&\mbox{for }~{}0\leq x\leq% \tfrac{L}{2}\\ \frac{P(x-L)(L^{2}-8Lx+4x^{2})}{48EI},&\mbox{for }~{}\tfrac{L}{2}<x\leq L\end{cases}
  185. w L / 2 = P L 3 48 E I w_{L/2}=\tfrac{PL^{3}}{48EI}
  186. M ( x ) = { P b x L , for 0 x a P b x L - P ( x - a ) = P a ( L - x ) L , for a < x L M(x)=\begin{cases}\tfrac{Pbx}{L},&\mbox{for }~{}0\leq x\leq a\\ \tfrac{Pbx}{L}-P(x-a)=\tfrac{Pa(L-x)}{L},&\mbox{for }~{}a<x\leq L\end{cases}
  187. M B = P a b L M_{B}=\tfrac{Pab}{L}
  188. Q ( x ) = { P b L , for 0 x a P b L - P , for a < x L Q(x)=\begin{cases}\tfrac{Pb}{L},&\mbox{for }~{}0\leq x\leq a\\ \tfrac{Pb}{L}-P,&\mbox{for }~{}a<x\leq L\end{cases}
  189. Q A = P b L Q_{A}=\tfrac{Pb}{L}
  190. Q C = P a L Q_{C}=\tfrac{Pa}{L}
  191. w ( x ) = { P b x ( L 2 - b 2 - x 2 ) 6 L E I , 0 x a P b x ( L 2 - b 2 - x 2 ) 6 L E I + P ( x - a ) 3 6 E I , a < x L w(x)=\begin{cases}\tfrac{Pbx(L^{2}-b^{2}-x^{2})}{6LEI},&0\leq x\leq a\\ \tfrac{Pbx(L^{2}-b^{2}-x^{2})}{6LEI}+\tfrac{P(x-a)^{3}}{6EI},&a<x\leq L\end{cases}
  192. w max = 3 P b ( L 2 - b 2 ) 3 2 27 L E I w_{\mathrm{max}}=\tfrac{\sqrt{3}Pb(L^{2}-b^{2})^{\frac{3}{2}}}{27LEI}
  193. x = L 2 - b 2 3 x=\sqrt{\tfrac{L^{2}-b^{2}}{3}}
  194. M M
  195. Q Q
  196. w w
  197. M ( x ) = P ( x - L ) M(x)=P(x-L)
  198. M A = P L M_{A}=PL
  199. Q ( x ) = P Q(x)=P
  200. Q max = P Q_{\mathrm{max}}=P
  201. w ( x ) = P x 2 ( 3 L - x ) 6 E I w(x)=\tfrac{Px^{2}(3L-x)}{6EI}
  202. w C = P L 3 3 E I w_{C}=\tfrac{PL^{3}}{3EI}
  203. M ( x ) = - q ( L 2 - 2 L x + x 2 ) 2 M(x)=-\tfrac{q(L^{2}-2Lx+x^{2})}{2}
  204. M A = q L 2 2 M_{A}=\tfrac{qL^{2}}{2}
  205. Q ( x ) = q ( L - x ) , Q(x)=q(L-x),
  206. Q A = q L Q_{A}=qL
  207. w ( x ) = q x 2 ( 6 L 2 - 4 L x + x 2 ) 24 E I w(x)=\tfrac{qx^{2}(6L^{2}-4Lx+x^{2})}{24EI}
  208. w C = q L 4 8 E I w_{C}=\tfrac{qL^{4}}{8EI}
  209. M max = q L 2 12 ; w max = q L 4 384 E I M_{\mathrm{max}}=\cfrac{qL^{2}}{12}~{};~{}~{}w_{\mathrm{max}}=\cfrac{qL^{4}}{3% 84EI}
  210. M ( x ) = - q 8 ( L 2 - 5 L x + 4 x 2 ) M(x)=-\tfrac{q}{8}(L^{2}-5Lx+4x^{2})
  211. M B = - 9 q L 2 128 at x = 5 L 8 M A = q L 2 8 \begin{aligned}\displaystyle M_{B}&\displaystyle=-\tfrac{9qL^{2}}{128}\mbox{ % at }~{}x=\tfrac{5L}{8}\\ \displaystyle M_{A}&\displaystyle=\tfrac{qL^{2}}{8}\end{aligned}
  212. Q ( x ) = - q 8 ( 8 x - 5 L ) Q(x)=-\tfrac{q}{8}(8x-5L)
  213. Q A = - 5 q L 8 Q_{A}=-\tfrac{5qL}{8}
  214. w ( x ) = q x 2 48 E I ( 3 L 2 - 5 L x + 2 x 2 ) w(x)=\tfrac{qx^{2}}{48EI}(3L^{2}-5Lx+2x^{2})
  215. w max = q L 4 185 E I at x = 0.5785 L w_{\mathrm{max}}=\tfrac{qL^{4}}{185EI}\mbox{ at }~{}x=0.5785L
  216. v 1 = v 0 ( x ) - z d w 0 d x ; v 2 = 0 ; v 3 = w 0 ( x ) v_{1}=v_{0}(x)-z\cfrac{\mathrm{d}w_{0}}{\mathrm{d}x}~{};~{}~{}v_{2}=0~{};~{}~{% }v_{3}=w_{0}(x)
  217. ε 11 = d u 0 d x 1 - x 3 d 2 w 0 d x 1 2 + 1 2 [ ( d u 0 d x 1 - x 3 d 2 w 0 d x 1 2 ) 2 + ( d w 0 d x 1 ) 2 ] ε 22 = 0 ε 33 = 1 2 ( d w 0 d x 1 ) 2 ε 23 = 0 ε 31 = 1 2 ( d w 0 d x 1 - d w 0 d x 1 ) - 1 2 [ ( d u 0 d x 1 - x 3 d 2 w 0 d x 1 2 ) ( d w 0 d x 1 ) ] ε 12 = 0 \begin{aligned}\displaystyle\varepsilon_{11}&\displaystyle=\cfrac{\mathrm{d}u_% {0}}{dx_{1}}-x_{3}\cfrac{\mathrm{d}^{2}w_{0}}{\mathrm{d}x_{1}^{2}}+\frac{1}{2}% \left[\left(\cfrac{\mathrm{d}u_{0}}{\mathrm{d}x_{1}}-x_{3}\cfrac{\mathrm{d}^{2% }w_{0}}{\mathrm{d}x_{1}^{2}}\right)^{2}+\left(\cfrac{\mathrm{d}w_{0}}{\mathrm{% d}x_{1}}\right)^{2}\right]\\ \displaystyle\varepsilon_{22}&\displaystyle=0\\ \displaystyle\varepsilon_{33}&\displaystyle=\frac{1}{2}\left(\cfrac{\mathrm{d}% w_{0}}{\mathrm{d}x_{1}}\right)^{2}\\ \displaystyle\varepsilon_{23}&\displaystyle=0\\ \displaystyle\varepsilon_{31}&\displaystyle=\frac{1}{2}\left(\cfrac{\mathrm{d}% w_{0}}{\mathrm{d}x_{1}}-\cfrac{\mathrm{d}w_{0}}{\mathrm{d}x_{1}}\right)-\frac{% 1}{2}\left[\left(\cfrac{\mathrm{d}u_{0}}{\mathrm{d}x_{1}}-x_{3}\cfrac{\mathrm{% d}^{2}w_{0}}{\mathrm{d}x_{1}^{2}}\right)\left(\cfrac{\mathrm{d}w_{0}}{\mathrm{% d}x_{1}}\right)\right]\\ \displaystyle\varepsilon_{12}&\displaystyle=0\end{aligned}
  218. d N x x d x + f ( x ) = 0 d 2 M x x d x 2 + q ( x ) + d d x ( N x x d w 0 d x ) = 0 \begin{aligned}\displaystyle\cfrac{\mathrm{d}N_{xx}}{\mathrm{d}x}+f(x)&% \displaystyle=0\\ \displaystyle\cfrac{\mathrm{d}^{2}M_{xx}}{\mathrm{d}x^{2}}+q(x)+\cfrac{\mathrm% {d}}{\mathrm{d}x}\left(N_{xx}\cfrac{\mathrm{d}w_{0}}{\mathrm{d}x}\right)&% \displaystyle=0\end{aligned}
  219. f ( x ) f(x)
  220. q ( x ) q(x)
  221. N x x = A σ x x d A ; M x x = A z σ x x d A N_{xx}=\int_{A}\sigma_{xx}~{}\mathrm{d}A~{};~{}~{}M_{xx}=\int_{A}z\sigma_{xx}~% {}\mathrm{d}A
  222. N x x = A x x [ d u 0 d x + 1 2 ( d w 0 d x ) 2 ] - B x x d 2 w 0 d x 2 M x x = B x x [ d u 0 d x + 1 2 ( d w 0 d x ) 2 ] - D x x d 2 w 0 d x 2 \begin{aligned}\displaystyle N_{xx}&\displaystyle=A_{xx}\left[\cfrac{\mathrm{d% }u_{0}}{dx}+\frac{1}{2}\left(\cfrac{\mathrm{d}w_{0}}{\mathrm{d}x}\right)^{2}% \right]-B_{xx}\cfrac{\mathrm{d}^{2}w_{0}}{\mathrm{d}x^{2}}\\ \displaystyle M_{xx}&\displaystyle=B_{xx}\left[\cfrac{du_{0}}{\mathrm{d}x}+% \frac{1}{2}\left(\cfrac{\mathrm{d}w_{0}}{\mathrm{d}x}\right)^{2}\right]-D_{xx}% \cfrac{\mathrm{d}^{2}w_{0}}{\mathrm{d}x^{2}}\end{aligned}
  223. A x x = A E d A ; B x x = A z E d A ; D x x = A z 2 E d A . A_{xx}=\int_{A}E~{}\mathrm{d}A~{};~{}~{}B_{xx}=\int_{A}zE~{}\mathrm{d}A~{};~{}% ~{}D_{xx}=\int_{A}z^{2}E~{}\mathrm{d}A~{}.
  224. A x x A_{xx}
  225. B x x B_{xx}
  226. D x x D_{xx}
  227. E I d 4 w d x 4 - 3 2 E A ( d w d x ) 2 ( d 2 w d x 2 ) = q ( x ) EI~{}\cfrac{\mathrm{d}^{4}w}{\mathrm{d}x^{4}}-\frac{3}{2}~{}EA~{}\left(\cfrac{% \mathrm{d}w}{\mathrm{d}x}\right)^{2}\left(\cfrac{\mathrm{d}^{2}w}{\mathrm{d}x^% {2}}\right)=q(x)

Euro_Stoxx_50.html

  1. I n d e x t = i = 1 n ( p i t s i t f f i t c f i t x i t ) D t = M t D t Index_{t}=\frac{\displaystyle\sum_{i=1}^{n}(p_{it}\cdot s_{it}\cdot ff_{it}% \cdot cf_{it}\cdot x_{it})}{D_{t}}=\frac{M_{t}}{D_{t}}
  2. D t + 1 = D t i = 1 n ( p i t s i t f f i t c f i t x i t ) ± Δ M C t + 1 i = 1 n ( p i t s i t f f i t c f i t x i t ) D_{t+1}=D_{t}\cdot\frac{\displaystyle\sum_{i=1}^{n}(p_{it}\cdot s_{it}\cdot ff% _{it}\cdot cf_{it}\cdot x_{it})\pm\Delta MC_{t+1}}{\displaystyle\sum_{i=1}^{n}% (p_{it}\cdot s_{it}\cdot ff_{it}\cdot cf_{it}\cdot x_{it})}

European_Community_number.html

  1. R = ( N 1 + 2 N 2 + 3 N 3 + 4 N 4 + 5 N 5 + 6 N 6 ) mod 11 R=(N_{1}+2N_{2}+3N_{3}+4N_{4}+5N_{5}+6N_{6})\mod 11
  2. 2 + 2 × 0 + 3 × 0 + 4 × 0 + 5 × 0 + 6 × 3 11 = 20 11 = 1 + 9 11 \frac{2+2\!\times\!0+3\!\times\!0+4\!\times\!0+5\!\times\!0+6\!\times\!3}{11}=% \frac{20}{11}=1+\frac{9}{11}

Evection.html

  1. + 4586.45 ′′ sin ( 2 D - l ) +4586.45^{\prime\prime}\sin(2D-l)
  2. D D
  3. l l
  4. + 22639.55 ′′ sin ( l ) + 4586.45 ′′ sin ( 2 D - l ) +22639.55^{\prime\prime}\sin(l)+4586.45^{\prime\prime}\sin(2D-l)
  5. + ( 22639.55 - 4586.45 ) ′′ sin ( l ) +(22639.55-4586.45)^{\prime\prime}\sin(l)
  6. + ( 22639.55 + 4586.45 ) ′′ sin ( l ) +(22639.55+4586.45)^{\prime\prime}\sin(l)

Event_calculus.html

  1. H o l d s A t HoldsAt
  2. H o l d s A t ( o n ( b o x , t a b l e ) , t ) HoldsAt(on(box,table),t)
  3. t t
  4. H o l d s A t HoldsAt
  5. o n on
  6. I n i t i a t e s Initiates
  7. T e r m i n a t e s Terminates
  8. I n i t i a t e s ( e , f , t ) Initiates(e,f,t)
  9. e e
  10. t t
  11. f f
  12. t t
  13. T e r m i n a t e s Terminates
  14. f f
  15. t t
  16. t t
  17. H o l d s A t ( f , t ) [ H a p p e n s ( e , t 1 ) I n i t i a t e s ( e , f , t 1 ) ( t 1 < t ) ¬ C l i p p e d ( t 1 , f , t ) ] HoldsAt(f,t)\leftarrow[Happens(e,t_{1})\wedge Initiates(e,f,t_{1})\wedge(t_{1}% <t)\wedge\neg Clipped(t_{1},f,t)]
  18. f f
  19. t t
  20. e e
  21. H a p p e n s ( e , t 1 ) Happens(e,t_{1})
  22. t 1 < t t_{1}<t
  23. f f
  24. I n i t i a t e s ( e , f , t 1 ) Initiates(e,f,t_{1})
  25. C l i p p e d ( t 1 , f , t ) Clipped(t_{1},f,t)
  26. H a p p e n s ( e , t 1 ) I n i t i a t e s ( e , f , t 1 ) Happens(e,t_{1})\wedge Initiates(e,f,t_{1})
  27. H o l d s A t ( f , t 1 ) HoldsAt(f,t_{1})
  28. C l i p p e d Clipped
  29. C l i p p e d ( t 1 , f , t 2 ) e , t [ H a p p e n s ( e , t ) ( t 1 t < t 2 ) T e r m i n a t e s ( e , f , t ) ] Clipped(t_{1},f,t_{2})\equiv\exists e,t[Happens(e,t)\wedge(t_{1}\leq t<t_{2})% \wedge Terminates(e,f,t)]
  30. H o l d s A t HoldsAt
  31. I n i t i a t e s Initiates
  32. T e r m i n a t e s Terminates
  33. H o l d s A t ( f , t ) HoldsAt(f,t)
  34. o p e n open
  35. i s o p e n isopen
  36. h a s k e y haskey
  37. I n i t i a t e s ( e , f , t ) [ e = o p e n f = i s o p e n H o l d s A t ( h a s k e y , t ) ] Initiates(e,f,t)\equiv[e=open\wedge f=isopen\wedge HoldsAt(haskey,t)]\vee\cdots
  38. e e
  39. f f
  40. I n i t i a t e s ( e , f , t ) Initiates(e,f,t)
  41. I n i t i a t e s ( o p e n , i s o p e n , t ) H o l d s A t ( h a s k e y , t ) Initiates(open,isopen,t)\leftarrow HoldsAt(haskey,t)
  42. I n i t i a t e s ( b r e a k , i s o p e n , t ) H o l d s A t ( h a s h a m m e r , t ) Initiates(break,isopen,t)\leftarrow HoldsAt(hashammer,t)
  43. I n i t i a t e s ( b r e a k , b r o k e n , t ) H o l d s A t ( h a s h a m m e r , t ) Initiates(break,broken,t)\leftarrow HoldsAt(hashammer,t)
  44. e e
  45. f f
  46. I n i t i a t e s ( e , f , t ) Initiates(e,f,t)
  47. I n i t i a t e s ( e , f , t ) Initiates(e,f,t)
  48. I n i t i a t e s Initiates
  49. I n i t i a t e s Initiates
  50. I n i t i a t e s Initiates
  51. T e r m i n a t e s Terminates
  52. I n i t i a t e s Initiates
  53. H a p p e n s Happens
  54. H a p p e n s ( e , t ) ( e = o p e n t = 0 ) ( e = e x i t t = 1 ) Happens(e,t)\equiv(e=open\wedge t=0)\vee(e=exit\wedge t=1)\vee\cdots
  55. H a p p e n s ( o p e n , 0 ) Happens(open,0)
  56. H a p p e n s ( e x i t , 1 ) Happens(exit,1)
  57. H a p p e n s Happens
  58. F F
  59. I n i t i a t e s ( e , f , t ) Initiates(e,f,t)\leftarrow\cdots
  60. G G
  61. H a p p e n s ( e , t ) Happens(e,t)
  62. H H
  63. C i r c ( F ; I n i t i a t e s , T e r m i n a t e s ) C i r c ( G ; H a p p e n s ) H Circ(F;Initiates,Terminates)\wedge Circ(G;Happens)\wedge H

Evert_Willem_Beth.html

  1. Γ \Gamma\,
  2. φ \varphi\,
  3. Γ \Gamma\,
  4. ¬ φ \neg\varphi
  5. φ \varphi\,
  6. Γ { ¬ φ } \Gamma\cup\{\neg\varphi\}
  7. Γ \Gamma\,
  8. φ \varphi\,

Evolutionarily_stable_state.html

  1. x i ˙ = x i ( ( A x ) i - x T A x ) , \dot{x_{i}}=x_{i}\left(\left(Ax\right)_{i}-x^{T}Ax\right),
  2. x ^ \hat{x}
  3. x x ^ x\neq\hat{x}
  4. x ^ \hat{x}
  5. x T A x < x ^ T A x x^{T}Ax<\hat{x}^{T}Ax

Evolutionary_graph_theory.html

  1. ρ M = 1 - r - 1 1 - r - N \displaystyle\rho_{M}=\frac{1-r^{-1}}{1-r^{-N}}
  2. ρ G \rho_{G}
  3. ρ M \rho_{M}
  4. ρ G = 1 / N \rho_{G}=1/N
  5. ρ G < ρ M \rho_{G}<\rho_{M}

Ex-meridian.html

  1. Hav.  MZD = Hav. TZD  -  Hav.H  Cos.L  Cos.D \textrm{Hav.\, MZD}={\textrm{Hav. TZD\, -\, Hav.H\, Cos.L\, Cos.D\, }}\,

Exact_cover.html

  1. 𝒮 \mathcal{S}
  2. 𝒮 * \mathcal{S}^{*}
  3. 𝒮 \mathcal{S}
  4. 𝒮 * \mathcal{S}^{*}
  5. 𝒮 * \mathcal{S}^{*}
  6. 𝒮 \mathcal{S}
  7. 𝒮 * \mathcal{S}^{*}
  8. 𝒮 \mathcal{S}
  9. 𝒮 * \mathcal{S}^{*}
  10. 𝒮 * \mathcal{S}^{*}
  11. 𝒮 * \mathcal{S}^{*}
  12. 𝒮 * \mathcal{S}^{*}
  13. 𝒮 * \mathcal{S}^{*}
  14. 𝒮 * \mathcal{S}^{*}
  15. 𝒮 * \mathcal{S}^{*}
  16. 𝒮 * \mathcal{S}^{*}
  17. 𝒮 \mathcal{S}
  18. 𝒮 \mathcal{S}
  19. 𝒮 \mathcal{S}
  20. 𝒮 \mathcal{S}
  21. 𝒮 * \mathcal{S}^{*}
  22. 𝒮 * \mathcal{S}^{*}
  23. 𝒮 \mathcal{S}
  24. 𝒮 \bigcup\mathcal{S}
  25. 𝒮 \mathcal{S}
  26. 𝒮 \mathcal{S}
  27. 𝒮 * \mathcal{S}^{*}
  28. 𝒮 \mathcal{S}
  29. 𝒮 * \mathcal{S}^{*}
  30. 𝒮 * \mathcal{S}^{*}
  31. 𝒮 \mathcal{S}
  32. 𝒮 * \mathcal{S}^{*}
  33. 𝒮 \mathcal{S}
  34. 𝒮 \mathcal{S}
  35. 𝒮 * \mathcal{S}^{*}
  36. 𝒮 \mathcal{S}
  37. \subseteq
  38. 𝒮 \mathcal{S}
  39. 𝒮 \mathcal{S}
  40. 𝒮 \mathcal{S}
  41. 𝒮 \mathcal{S}
  42. 𝒮 \mathcal{S}
  43. 𝒮 \mathcal{S}
  44. 𝒮 \mathcal{S}
  45. 𝒮 \mathcal{S}
  46. 𝒮 \mathcal{S}
  47. 𝒮 * \mathcal{S}^{*}

Exact_test.html

  1. Pr ( exact ) = 𝐲 : T ( 𝐲 ) T ( 𝐱 ) Pr ( 𝐲 ) \Pr(\,\text{exact})=\sum_{\mathbf{y}\,:\,T(\mathbf{y})\geq T(\mathbf{x)}}\Pr(% \mathbf{y})
  2. ( observed - expected ) 2 expected = k = 1 6 ( X k - n / 6 ) 2 n / 6 , \sum\frac{(\,\text{observed}-\,\text{expected})^{2}}{\,\text{expected}}=\sum_{% k=1}^{6}\frac{(X_{k}-n/6)^{2}}{n/6},

Exchange_bias.html

  1. E = 1 2 n J e x S F S A F + M F t F H E=\frac{1}{2}nJ_{ex}S_{F}S_{AF}+M_{F}t_{F}H
  2. H b = n J e x S F S A F 2 M F t F H_{b}=\frac{nJ_{ex}S_{F}S_{AF}}{2M_{F}t_{F}}

Expected_value_of_perfect_information.html

  1. EMV = max i j p j R i j . \mbox{EMV}~{}=\max_{i}\sum_{j}p_{j}R_{ij}.\,
  2. j p j R i j . \sum_{j}p_{j}R_{ij}.\,
  3. EMV = max i \mbox{EMV}~{}=\max_{i}\,
  4. EV | PI = j p j ( max i R i j ) , \mbox{EV}~{}|\mbox{PI}~{}=\sum_{j}p_{j}(\max_{i}R_{ij}),\,
  5. p j p_{j}
  6. R i j R_{ij}
  7. ( max i R i j ) , (\max_{i}R_{ij}),\,
  8. EVPI = EV | PI - EMV . \mbox{EVPI}~{}=\mbox{EV}~{}|\mbox{PI}~{}-\mbox{EMV}~{}.\,
  9. Exp = s t o c k 0.5 × 1500 + 0.3 * 300 + 0.2 × ( - 800 ) = 680 \mbox{Exp}~{}_{stock}=0.5\times 1500+0.3*300+0.2\times(-800)=680
  10. Exp = m u t u a l f u n d 0.5 × 900 + 0.3 * 600 + 0.2 × ( - 200 ) = 590 \mbox{Exp}~{}_{mutualfund}=0.5\times 900+0.3*600+0.2\times(-200)=590
  11. Exp = c e r t i f i c a t e o f d e p o s i t 0.5 × 500 + 0.3 × 500 + 0.2 × 500 = 500 \mbox{Exp}~{}_{certificateofdeposit}=0.5\times 500+0.3\times 500+0.2\times 500% =500
  12. EMV = 680 \mbox{EMV}~{}=680
  13. EV | PI = 0.5 × 1500 + 0.3 × 600 + 0.2 × 500 = 1030 \mbox{EV}~{}|\mbox{PI}~{}=0.5\times 1500+0.3\times 600+0.2\times 500=1030
  14. EVPI = EV | PI - EMV = 1030 - 680 = 350. \mbox{EVPI}~{}=\mbox{EV}~{}|\mbox{PI}~{}-\mbox{EMV}~{}=1030-680=350.\,

Explained_sum_of_squares.html

  1. a ^ \hat{a}
  2. b ^ i \hat{b}_{i}
  3. y ^ i = a ^ + b 1 ^ x 1 i + b 2 ^ x 2 i + \hat{y}_{i}=\hat{a}+\hat{b_{1}}x_{1i}+\hat{b_{2}}x_{2i}+\cdots\,
  4. ESS = i = 1 n ( y ^ i - y ¯ ) 2 . \,\text{ESS}=\sum_{i=1}^{n}\left(\hat{y}_{i}-\bar{y}\right)^{2}.
  5. i = 1 n ( y i - y ¯ ) 2 = i = 1 n ( y i - y ^ i ) 2 + i = 1 n ( y ^ i - y ¯ ) 2 . \sum_{i=1}^{n}\left(y_{i}-\bar{y}\right)^{2}=\sum_{i=1}^{n}\left(y_{i}-\hat{y}% _{i}\right)^{2}+\sum_{i=1}^{n}\left(\hat{y}_{i}-\bar{y}\right)^{2}.
  6. ( y i - y ¯ ) = ( y i - y ^ i ) + ( y ^ i - y ¯ ) . \begin{aligned}\displaystyle(y_{i}-\bar{y})=(y_{i}-\hat{y}_{i})+(\hat{y}_{i}-% \bar{y}).\end{aligned}
  7. i = 1 n ( y i - y ¯ ) 2 = i = 1 n ( y i - y ^ i ) 2 + i = 1 n ( y ^ i - y ¯ ) 2 + i = 1 n 2 ( y ^ i - y ¯ ) ( y i - y ^ i ) . \sum_{i=1}^{n}(y_{i}-\bar{y})^{2}=\sum_{i=1}^{n}(y_{i}-\hat{y}_{i})^{2}+\sum_{% i=1}^{n}(\hat{y}_{i}-\bar{y})^{2}+\sum_{i=1}^{n}2(\hat{y}_{i}-\bar{y})(y_{i}-% \hat{y}_{i}).
  8. a ^ = y ¯ - b ^ x ¯ \hat{a}=\bar{y}-\hat{b}\bar{x}
  9. i = 1 n 2 ( y ^ i - y ¯ ) ( y i - y ^ i ) = i = 1 n 2 ( ( y ¯ - b ^ x ¯ + b ^ x i ) - y ¯ ) ( y i - y ^ i ) = i = 1 n 2 ( ( y ¯ + b ^ ( x i - x ¯ ) ) - y ¯ ) ( y i - y ^ i ) = i = 1 n 2 ( b ^ ( x i - x ¯ ) ) ( y i - y ^ i ) = i = 1 n 2 b ^ ( x i - x ¯ ) ( y i - ( y ¯ + b ^ ( x i - x ¯ ) ) ) = i = 1 n 2 b ^ ( ( y i - y ¯ ) ( x i - x ¯ ) - b ^ ( x i - x ¯ ) 2 ) . \begin{aligned}\displaystyle\sum_{i=1}^{n}2(\hat{y}_{i}-\bar{y})(y_{i}-\hat{y}% _{i})&\displaystyle=\sum_{i=1}^{n}2((\bar{y}-\hat{b}\bar{x}+\hat{b}x_{i})-\bar% {y})(y_{i}-\hat{y}_{i})\\ &\displaystyle=\sum_{i=1}^{n}2((\bar{y}+\hat{b}(x_{i}-\bar{x}))-\bar{y})(y_{i}% -\hat{y}_{i})\\ &\displaystyle=\sum_{i=1}^{n}2(\hat{b}(x_{i}-\bar{x}))(y_{i}-\hat{y}_{i})\\ &\displaystyle=\sum_{i=1}^{n}2\hat{b}(x_{i}-\bar{x})(y_{i}-(\bar{y}+\hat{b}(x_% {i}-\bar{x})))\\ &\displaystyle=\sum_{i=1}^{n}2\hat{b}((y_{i}-\bar{y})(x_{i}-\bar{x})-\hat{b}(x% _{i}-\bar{x})^{2}).\end{aligned}
  10. b ^ = i = 1 n ( x i - x ¯ ) ( y i - y ¯ ) i = 1 n ( x i - x ¯ ) 2 , \hat{b}=\frac{\sum_{i=1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i=1}^{n}(x_{% i}-\bar{x})^{2}},
  11. i = 1 n 2 ( y ^ i - y ¯ ) ( y i - y ^ i ) = i = 1 n 2 b ^ ( ( y i - y ¯ ) ( x i - x ¯ ) - b ^ ( x i - x ¯ ) 2 ) = 2 b ^ ( i = 1 n ( y i - y ¯ ) ( x i - x ¯ ) - b ^ i = 1 n ( x i - x ¯ ) 2 ) = 2 b ^ i = 1 n ( ( y i - y ¯ ) ( x i - x ¯ ) - ( y i - y ¯ ) ( x i - x ¯ ) ) = 2 b ^ 0 = 0. \begin{aligned}\displaystyle\sum_{i=1}^{n}2(\hat{y}_{i}-\bar{y})(y_{i}-\hat{y}% _{i})&\displaystyle=\sum_{i=1}^{n}2\hat{b}\left((y_{i}-\bar{y})(x_{i}-\bar{x})% -\hat{b}(x_{i}-\bar{x})^{2}\right)\\ &\displaystyle=2\hat{b}\left(\sum_{i=1}^{n}(y_{i}-\bar{y})(x_{i}-\bar{x})-\hat% {b}\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}\right)\\ &\displaystyle=2\hat{b}\sum_{i=1}^{n}\left((y_{i}-\bar{y})(x_{i}-\bar{x})-(y_{% i}-\bar{y})(x_{i}-\bar{x})\right)\\ &\displaystyle=2\hat{b}\cdot 0=0.\end{aligned}
  12. y = X β + e y=X\beta+e
  13. β \beta
  14. β \beta
  15. β ^ = ( X T X ) - 1 X T y . \hat{\beta}=(X^{T}X)^{-1}X^{T}y.
  16. e ^ \hat{e}
  17. y - X β ^ = y - X ( X T X ) - 1 X T y y-X\hat{\beta}=y-X(X^{T}X)^{-1}X^{T}y
  18. e ^ T e ^ \hat{e}^{T}\hat{e}
  19. R S S = y T y - y T X ( X T X ) - 1 X T y . RSS=y^{T}y-y^{T}X(X^{T}X)^{-1}X^{T}y.
  20. y ¯ \bar{y}
  21. y m y_{m}
  22. T S S = ( y - y ¯ ) T ( y - y ¯ ) = y T y - 2 y T y ¯ + y ¯ T y ¯ . TSS=(y-\bar{y})^{T}(y-\bar{y})=y^{T}y-2y^{T}\bar{y}+\bar{y}^{T}\bar{y}.
  23. E S S = ( y ^ - y ¯ ) T ( y ^ - y ¯ ) = y ^ T y ^ - 2 y ^ T y ¯ + y ¯ T y ¯ . ESS=(\hat{y}-\bar{y})^{T}(\hat{y}-\bar{y})=\hat{y}^{T}\hat{y}-2\hat{y}^{T}\bar% {y}+\bar{y}^{T}\bar{y}.
  24. y ^ = X β ^ \hat{y}=X\hat{\beta}
  25. y ^ T y ^ = y T X ( X T X ) - 1 X T y \hat{y}^{T}\hat{y}=y^{T}X(X^{T}X)^{-1}X^{T}y
  26. y T y ¯ = y ^ T y ¯ y^{T}\bar{y}=\hat{y}^{T}\bar{y}
  27. y m y_{m}
  28. y m y_{m}
  29. y ^ \hat{y}
  30. y ^ \hat{y}
  31. y i - y ^ i y_{i}-\hat{y}_{i}
  32. X T e ^ = X T [ I - X ( X T X ) - 1 X T ] y = 0 X^{T}\hat{e}=X^{T}[I-X(X^{T}X)^{-1}X^{T}]y=0
  33. X T e ^ X^{T}\hat{e}
  34. R S S = y - y ^ 2 2 RSS=\|y-{\hat{y}}\|_{2}^{2}
  35. T S S = y - y ¯ 2 2 TSS=\|y-\bar{y}\|_{2}^{2}
  36. E S S = y ^ - y ¯ 2 2 ESS=\|{\hat{y}}-\bar{y}\|_{2}^{2}
  37. y T y ^ = y ^ T y ^ y^{T}{\hat{y}}={\hat{y}}^{T}{\hat{y}}
  38. y ^ T y ^ = y T X ( X T X ) - 1 X T X ( X T X ) - 1 X T y = y T X ( X T X ) - 1 X T y = y T y ^ , {\hat{y}}^{T}{\hat{y}}=y^{T}X(X^{T}X)^{-1}X^{T}X(X^{T}X)^{-1}X^{T}y=y^{T}X(X^{% T}X)^{-1}X^{T}y=y^{T}{\hat{y}},
  39. T S S = y - y ¯ 2 2 = y - y ^ + y ^ - y ¯ 2 2 TSS=\|y-\bar{y}\|_{2}^{2}=\|y-{\hat{y}}+{\hat{y}}-\bar{y}\|_{2}^{2}
  40. T S S = y - y ^ 2 2 + y ^ - y ¯ 2 2 + 2 < y - y ^ , y ^ - y ¯ Align g t ; TSS=\|y-{\hat{y}}\|_{2}^{2}+\|{\hat{y}}-\bar{y}\|_{2}^{2}+2<y-{\hat{y}},{\hat{% y}}-{\bar{y}}&gt;
  41. T S S = R S S + E S S + 2 y T y ^ - 2 y ^ T y ^ - 2 y T y ¯ + 2 y ^ T y ¯ TSS=RSS+ESS+2y^{T}{\hat{y}}-2{\hat{y}}^{T}{\hat{y}}-2y^{T}{\bar{y}}+2{\hat{y}}% ^{T}{\bar{y}}
  42. T S S = R S S + E S S - 2 y T y ¯ + 2 y ^ T y ¯ TSS=RSS+ESS-2y^{T}{\bar{y}}+2{\hat{y}}^{T}{\bar{y}}
  43. y T y ¯ = y ^ T y ¯ y^{T}\bar{y}=\hat{y}^{T}\bar{y}

Explicit_formulae_(L-function).html

  1. π 0 ( x ) = 1 2 lim h 0 ( π ( x + h ) + π ( x - h ) ) . \pi_{0}(x)=\frac{1}{2}\lim_{h\to 0}(\pi(x+h)+\pi(x-h)).
  2. f ( x ) = π ( x ) + 1 2 π ( x 1 / 2 ) + 1 3 π ( x 1 / 3 ) + f(x)=\pi(x)+\frac{1}{2}\pi(x^{1/2})+\frac{1}{3}\pi(x^{1/3})+\cdots
  3. π 0 ( x ) = n μ ( n ) f ( x 1 / n ) / n = f ( x ) - 1 2 f ( x 1 / 2 ) - 1 3 f ( x 1 / 3 ) - . \pi_{0}(x)=\sum_{n}\mu(n)f(x^{1/n})/n=f(x)-\frac{1}{2}f(x^{1/2})-\frac{1}{3}f(% x^{1/3})-\cdots.
  4. f ( x ) = li ( x ) - ρ li ( x ρ ) - log ( 2 ) + x d t t ( t 2 - 1 ) log ( t ) f(x)=\operatorname{li}(x)-\sum_{\rho}\operatorname{li}(x^{\rho})-\log(2)+\int_% {x}^{\infty}\frac{dt}{t(t^{2}-1)\log(t)}
  5. li ( x ) = 0 x d t log ( t ) . \operatorname{li}(x)=\int_{0}^{x}\frac{dt}{\log(t)}.
  6. ψ 0 \psi_{0}
  7. ψ 0 ( x ) = 1 2 π i 0 ( - ζ ( s ) ζ ( s ) ) x s s d s = x - ρ x ρ ρ - log ( 2 π ) - log ( 1 - x - 2 ) / 2 \psi_{0}(x)=\dfrac{1}{2\pi i}\int_{0}^{\infty}\left(-\dfrac{\zeta^{\prime}(s)}% {\zeta(s)}\right)\dfrac{x^{s}}{s}ds=x-\sum_{\rho}\frac{x^{\rho}}{\rho}-\log(2% \pi)-\log(1-x^{-2})/2
  8. ρ x ρ ρ = lim T S ( x , T ) \sum_{\rho}\frac{x^{\rho}}{\rho}=\lim_{T\rightarrow\infty}S(x,T)
  9. S ( x , T ) = ρ : | ρ | T x ρ ρ . S(x,T)=\sum_{\rho:|\Im\rho|\leq T}\frac{x^{\rho}}{\rho}\ .
  10. x 2 log 2 T T + log x . x^{2}\frac{\log^{2}T}{T}+\log x\ .
  11. Φ ( 1 ) + Φ ( 0 ) - ρ Φ ( ρ ) = p , m log ( p ) p m / 2 ( F ( log ( p m ) ) + F ( - log ( p m ) ) ) - 1 2 π - φ ( t ) Ψ ( t ) d t \begin{aligned}&\displaystyle{}\quad\Phi(1)+\Phi(0)-\sum_{\rho}\Phi(\rho)\\ &\displaystyle=\sum_{p,m}\frac{\log(p)}{p^{m/2}}(F(\log(p^{m}))+F(-\log(p^{m})% ))-\frac{1}{2\pi}\int_{-\infty}^{\infty}\varphi(t)\Psi(t)\,dt\end{aligned}
  12. φ \varphi
  13. φ ( t ) = - F ( x ) e i t x d x \varphi(t)=\int_{-\infty}^{\infty}F(x)e^{itx}\,dx
  14. Φ ( 1 / 2 + i t ) = φ ( t ) \Phi(1/2+it)=\varphi(t)
  15. Ψ ( t ) = - log ( π ) + R e ( ψ ( 1 / 4 + i t / 2 ) ) \Psi(t)=-\log(\pi)+Re(\psi(1/4+it/2))
  16. ψ \psi
  17. ζ * ( s ) = Γ ( s / 2 ) π - s / 2 p 1 1 - p - s \zeta^{*}(s)=\Gamma(s/2)\pi^{-s/2}\prod_{p}\frac{1}{1-p^{-s}}
  18. d d u [ n e | u | Λ ( n ) + 1 2 ln ( 1 - e - 2 | u | ) ] \frac{d}{du}\left[\sum_{n\leq e^{|u|}}\Lambda(n)+\frac{1}{2}\ln(1-e^{-2|u|})\right]
  19. = n = 1 Λ ( n ) [ δ ( u - ln n ) + δ ( u - ln n ) ] + d ln ( 1 - e - 2 | u | ) d u = e u - ρ e ρ u =\sum_{n=1}^{\infty}\Lambda(n)\left[\delta(u-\ln n)+\delta(u-\ln n)\right]+% \frac{d\ln(1-e^{-2|u|})}{du}=e^{u}-\sum{\rho}e^{\rho u}
  20. - f ( u ) g * ( u ) d u = - F ( t ) G * ( t ) d t \int_{-\infty}^{\infty}f(u)g^{*}(u)du=\int_{-\infty}^{\infty}F(t)G^{*}(t)dt
  21. F , G F,G
  22. f , g f,g
  23. g g
  24. g ( u ) = n = 1 Λ ( n ) [ δ ( u - ln n ) + δ ( u - ln n ) ] g(u)=\sum_{n=1}^{\infty}\Lambda(n)\left[\delta(u-\ln n)+\delta(u-\ln n)\right]
  25. δ ( u ) \delta(u)
  26. f f
  27. ρ F ( ρ ) = Tr ( F ( T ^ ) ) . \sum_{\rho}F(\rho)=\operatorname{Tr}(F(\widehat{T})).\!

Exponential_object.html

  1. eval : ( Z Y × Y ) Z \mathrm{eval}\colon(Z^{Y}\times Y)\rightarrow Z
  2. λ g : X Z Y \lambda g\colon X\to Z^{Y}
  3. Hom ( X × Y , Z ) Hom ( X , Z Y ) . \mathrm{Hom}(X\times Y,Z)\cong\mathrm{Hom}(X,Z^{Y}).
  4. λ g \lambda g
  5. g g
  6. λ g \lambda g
  7. A , B A,B
  8. A B = Hom ( B , A ) A^{B}=\mathrm{Hom}(B,A)
  9. Z Y Z^{Y}
  10. Y Y
  11. Z Z
  12. eval : ( Z Y × Y ) Z \mathrm{eval}\colon(Z^{Y}\times Y)\to Z
  13. g : ( X × Y ) Z g\colon(X\times Y)\rightarrow Z
  14. λ g : X Z Y \lambda g\colon X\to Z^{Y}
  15. g g
  16. λ g ( x ) ( y ) = g ( x , y ) . \lambda g(x)(y)=g(x,y).\,

Extension_and_contraction_of_ideals.html

  1. 𝔞 \mathfrak{a}
  2. f ( 𝔞 ) f(\mathfrak{a})
  3. 𝔞 e \mathfrak{a}^{e}
  4. 𝔞 \mathfrak{a}
  5. f ( 𝔞 ) f(\mathfrak{a})
  6. 𝔞 e = { y i f ( x i ) : x i 𝔞 , y i B } \mathfrak{a}^{e}=\Big\{\sum y_{i}f(x_{i}):x_{i}\in\mathfrak{a},y_{i}\in B\Big\}
  7. 𝔟 \mathfrak{b}
  8. f - 1 ( 𝔟 ) f^{-1}(\mathfrak{b})
  9. 𝔟 c \mathfrak{b}^{c}
  10. 𝔟 \mathfrak{b}
  11. 𝔞 \mathfrak{a}
  12. 𝔟 \mathfrak{b}
  13. 𝔟 \mathfrak{b}
  14. \Rightarrow
  15. 𝔟 c \mathfrak{b}^{c}
  16. 𝔞 e c 𝔞 \mathfrak{a}^{ec}\supseteq\mathfrak{a}
  17. 𝔟 c e 𝔟 \mathfrak{b}^{ce}\subseteq\mathfrak{b}
  18. 𝔞 \mathfrak{a}
  19. 𝔞 e \mathfrak{a}^{e}
  20. [ i ] \mathbb{Z}\to\mathbb{Z}\left[i\right]
  21. B = [ i ] B=\mathbb{Z}\left[i\right]
  22. 2 = ( 1 + i ) ( 1 - i ) 2=(1+i)(1-i)
  23. 1 + i , 1 - i 1+i,1-i
  24. ( 2 ) e (2)^{e}
  25. ( 1 ± i ) 2 = ± 2 i (1\pm i)^{2}=\pm 2i
  26. ( 1 + i ) = ( ( 1 - i ) - ( 1 - i ) 2 ) (1+i)=((1-i)-(1-i)^{2})
  27. ( 1 - i ) = ( ( 1 + i ) - ( 1 + i ) 2 ) (1-i)=((1+i)-(1+i)^{2})
  28. ( 2 ) e = ( 1 + i ) 2 (2)^{e}=(1+i)^{2}
  29. 𝔞 ker f \mathfrak{a}\supseteq\mathop{\mathrm{ker}}f
  30. 𝔞 e c = 𝔞 \mathfrak{a}^{ec}=\mathfrak{a}
  31. 𝔟 c e = 𝔟 \mathfrak{b}^{ce}=\mathfrak{b}
  32. 𝔞 \mathfrak{a}
  33. \Leftrightarrow
  34. 𝔞 e \mathfrak{a}^{e}
  35. 𝔞 \mathfrak{a}
  36. \Leftrightarrow
  37. 𝔞 e \mathfrak{a}^{e}
  38. 𝔞 = 𝔭 \mathfrak{a}=\mathfrak{p}

Étale_fundamental_group.html

  1. X X
  2. x x
  3. X , X,
  4. C C
  5. ( Y , f ) (Y,f)
  6. f : Y X f\colon Y\to X
  7. Y . Y.
  8. ( Y , f ) ( Y , f ) (Y,f)\to(Y^{\prime},f^{\prime})
  9. Y Y Y\to Y^{\prime}
  10. X . X.
  11. F ( Y ) = Hom X ( x , Y ) ; F(Y)=\operatorname{Hom}_{X}(x,Y);
  12. Y X Y\to X
  13. x , x,
  14. x . x.
  15. F F
  16. X X
  17. { X j X i i < j I } \{X_{j}\to X_{i}\mid i<j\in I\}
  18. C C
  19. I , I,
  20. X i X_{i}
  21. X X
  22. X , X,
  23. # Aut X ( X i ) = deg ( X i / X ) \#\operatorname{Aut}_{X}(X_{i})=\operatorname{deg}(X_{i}/X)
  24. F ( Y ) = lim i I Hom C ( X i , Y ) F(Y)=\underrightarrow{\lim}_{i\in I}\operatorname{Hom}_{C}(X_{i},Y)
  25. P lim i I F ( X i ) P\in\underleftarrow{\lim}_{i\in I}F(X_{i})
  26. X i , X j X_{i},X_{j}
  27. X j X i X_{j}\to X_{i}
  28. Aut X ( X j ) Aut X ( X i ) \operatorname{Aut}_{X}(X_{j})\to\operatorname{Aut}_{X}(X_{i})
  29. { X i } \{X_{i}\}
  30. π 1 ( X , x ) \pi_{1}(X,x)
  31. X X
  32. x x
  33. π 1 ( X , x ) = lim i I Aut X ( X i ) , \pi_{1}(X,x)=\underleftarrow{\lim}_{i\in I}{\operatorname{Aut}}_{X}(X_{i}),
  34. F F
  35. C C
  36. π 1 ( X , x ) \pi_{1}(X,x)
  37. C C
  38. π 1 ( X , x ) \pi_{1}(X,x)
  39. x ¯ \overline{x}
  40. x ¯ \overline{x}
  41. 𝐀 k 1 \mathbf{A}^{1}_{k}

Étale_morphism.html

  1. ϕ : R S \phi:R\to S
  2. S S
  3. R R
  4. f f
  5. R [ x ] R[x]
  6. g g
  7. R [ x ] R[x]
  8. f f^{\prime}
  9. f f
  10. ( R [ x ] / f R [ x ] ) g (R[x]/fR[x])_{g}
  11. ϕ \phi
  12. f f
  13. g g
  14. S S
  15. R R
  16. ( R [ x ] / f R [ x ] ) g (R[x]/fR[x])_{g}
  17. ϕ \phi
  18. f : X Y f:X\to Y
  19. f f
  20. f f
  21. f f
  22. f f
  23. y y
  24. Y Y
  25. f - 1 ( y ) f^{-1}(y)
  26. κ ( y ) \kappa(y)
  27. f f
  28. y y
  29. Y Y
  30. k k^{\prime}
  31. κ ( y ) \kappa(y)
  32. f - 1 ( y ) κ ( y ) k f^{-1}(y)\otimes_{\kappa(y)}k^{\prime}
  33. Spec k \mbox{Spec }~{}k^{\prime}
  34. f f
  35. f f
  36. f f
  37. x x
  38. X X
  39. y = f ( x ) y=f(x)
  40. y y
  41. x x
  42. f f
  43. f f
  44. f f
  45. Y Y
  46. x x
  47. X X
  48. y = f ( x ) y=f(x)
  49. 𝒪 ^ Y , y 𝒪 ^ X , x \hat{\mathcal{O}}_{Y,y}\to\hat{\mathcal{O}}_{X,x}
  50. f f
  51. x x
  52. X X
  53. x x
  54. X X
  55. 𝒪 ^ X , x \hat{\mathcal{O}}_{X,x}
  56. 𝒪 ^ Y , y \hat{\mathcal{O}}_{Y,y}
  57. 𝒪 ^ X , x / m y 𝒪 ^ X , x \hat{\mathcal{O}}_{X,x}/m_{y}\hat{\mathcal{O}}_{X,x}
  58. κ ( y ) \kappa(y)
  59. m y m_{y}
  60. 𝒪 ^ Y , y \hat{\mathcal{O}}_{Y,y}
  61. κ ( y ) κ ( x ) \kappa(y)\to\kappa(x)
  62. κ ( y ) \kappa(y)
  63. f f
  64. x x
  65. X X
  66. R S = R [ x 1 , , x n ] g / ( f 1 , , f n ) R\to S=R[x_{1},\ldots,x_{n}]_{g}/(f_{1},\ldots,f_{n})
  67. f i f_{i}
  68. det ( f i / x j ) \det(\partial f_{i}/\partial x_{j})
  69. S S
  70. f f
  71. f f
  72. f f
  73. f f
  74. f : X Y f:X\to Y
  75. A K K ¯ K ¯ K ¯ , A\otimes_{K}\bar{K}\cong\bar{K}\oplus...\oplus\bar{K},
  76. K ¯ \bar{K}
  77. K ¯ \bar{K}
  78. f : X Y f:X\to Y
  79. X X
  80. f f
  81. Y Y
  82. f ( U ) : X × Y U U f_{(U)}:X\times_{Y}U\to U
  83. U U
  84. V = Spec ( B ) U = Spec ( A ) V=\operatorname{Spec}(B)\to U=\operatorname{Spec}(A)
  85. { f α : X α Y } \{f_{\alpha}:X_{\alpha}\to Y\}
  86. f α : X α Y \coprod f_{\alpha}:\coprod X_{\alpha}\to Y
  87. f α f_{\alpha}
  88. f : X Y f:X\to Y
  89. g : Y Z g:Y\to Z
  90. g g
  91. g f gf
  92. f f
  93. X X
  94. X X^{\prime}
  95. Y Y
  96. Y Y
  97. X X
  98. X X^{\prime}
  99. f : X Y f:X\to Y
  100. f : X Y f:X\to Y
  101. dim X = dim Y \dim X=\dim Y
  102. f : X Y f:X\to Y
  103. X × Y V V X\times_{Y}V\to V
  104. f : X Y f:X\to Y
  105. 𝔸 Y n \mathbb{A}^{n}_{Y}

F-algebra.html

  1. α : 1 + G + G × G G 1 1 x x - 1 ( x , y ) x y \begin{matrix}\alpha:{1}+G+G\times G&\to&G\\ 1&\mapsto&1\\ x&\mapsto&x^{-1}\\ (x,y)&\mapsto&x\cdot y\end{matrix}
  2. F : 𝐒𝐞𝐭 𝐒𝐞𝐭 F:\mathbf{Set}\to\mathbf{Set}
  3. X X
  4. 1 + X 1+X
  5. + +
  6. \mathbb{N}
  7. [ zero , succ ] : 1 + [\mathrm{zero},\mathrm{succ}]:1+\mathbb{N}\to\mathbb{N}
  8. zero : 1 \mathrm{zero}:1\to\mathbb{N}
  9. succ : \mathrm{succ}:\mathbb{N}\to\mathbb{N}
  10. F F
  11. ( , [ zero , succ ] ) (\mathbb{N},[\mathrm{zero},\mathrm{succ}])

F-term.html

  1. θ 1 , θ 2 , θ ¯ 1 , θ ¯ 2 \theta^{1},\theta^{2},\bar{\theta}^{1},\bar{\theta}^{2}
  2. θ \theta
  3. F θ 1 θ 2 F\theta^{1}\theta^{2}
  4. d 4 x F ( x ) \int{d^{4}x\,F(x)}
  5. θ \theta

Factored_language_model.html

  1. w i = { f i 1 , , f i k } . w_{i}=\{f_{i}^{1},...,f_{i}^{k}\}.
  2. P ( f | f 1 , , f N ) P(f|f_{1},...,f_{N})
  3. f f
  4. N N
  5. { f 1 , , f N } \{f_{1},...,f_{N}\}
  6. w w
  7. t t
  8. P ( w i | w i - 2 , w i - 1 , t i - 1 ) P(w_{i}|w_{i-2},w_{i-1},t_{i-1})

Faithful_representation.html

  1. K [ G ] K[G]
  2. K [ G ] K[G]
  3. K [ G ] K[G]
  4. V n = V V V n times V^{\otimes n}=\underbrace{V\otimes V\otimes\cdots\otimes V}_{n\,\text{ times}}

Fanning_friction_factor.html

  1. τ = f ρ v 2 2 \tau=\frac{f\rho v^{2}}{2}
  2. τ \tau
  3. f f
  4. v v
  5. ρ \rho
  6. 2 π R L 2\pi RL
  7. π R 2 \pi R^{2}
  8. f = Δ P L R ρ v 2 f=\frac{\Delta P}{L}\frac{R}{\rho v^{2}}
  9. h f = 4 f v 2 L 2 g D h_{f}=\frac{4fv^{2}L}{2gD}
  10. h f h_{f}
  11. f f
  12. v v
  13. L L
  14. g g
  15. D D
  16. f = 16 R e f=\frac{16}{Re}
  17. f = 14.227 R e f=\frac{14.227}{Re}
  18. f = 8 τ w ρ V a v g 2 f=\frac{8\tau_{w}}{\rho V_{avg}^{2}}
  19. τ w \tau_{w}
  20. ρ \rho
  21. V a v g V_{avg}
  22. f f
  23. 1 f = - 4.0 log 10 ( ϵ d 3.7 + 1.256 R e f ) , turbulent flow {1\over\sqrt{\mathit{f}}}=-4.0\log_{10}\left(\frac{\frac{\epsilon}{d}}{3.7}+{% \frac{1.256}{Re\sqrt{\mathit{f}}}}\right),\,\text{turbulent flow}
  24. f D f_{D}
  25. f f
  26. 1 4 \frac{1}{4}
  27. f = 2 ( ( 8 R e ) 12 + ( A + B ) - 1.5 ) 1 12 f=2\left(\left(\frac{8}{Re}\right)^{12}+\left(A+B\right)^{-1.5}\right)^{\frac{% 1}{12}}
  28. A = ( 2.457 ln ( ( ( 7 R e ) 0.9 + 0.27 ϵ D ) - 1 ) ) 16 A=\left(2.457\ln\left(\left(\left(\frac{7}{Re}\right)^{0.9}+0.27\frac{\epsilon% }{D}\right)^{-1}\right)\right)^{16}
  29. B = ( 37530 R e ) 16 B=\left(\frac{37530}{Re}\right)^{16}

Fano_resonance.html

  1. 2 s 2 p 2s2p
  2. 2 Γ res 2\Gamma_{\mathrm{res}}
  3. π \pi
  4. σ \sigma
  5. σ ( q Γ res / 2 + E - E res ) 2 ( Γ res / 2 ) 2 + ( E - E res ) 2 \sigma\approx\frac{\left(q\Gamma_{\mathrm{res}}/2+E-E_{\mathrm{res}}\right)^{2% }}{\left(\Gamma_{\mathrm{res}}/2\right)^{2}+\left(E-E_{\mathrm{res}}\right)^{2}}
  6. Γ res \Gamma_{\mathrm{res}}
  7. ( Γ res / 2 ) 2 ( Γ res / 2 ) 2 + ( E - E res ) 2 \frac{\left(\Gamma_{\mathrm{res}}/2\right)^{2}}{\left(\Gamma_{\mathrm{res}}/2% \right)^{2}+\left(E-E_{\mathrm{res}}\right)^{2}}

Faraday_paradox.html

  1. = - d Φ B d t = - d d t Σ ( t ) d s y m b o l A 𝐁 ( 𝐫 , t ) , \mathcal{E}=-\frac{d\Phi_{B}}{dt}=-\frac{d}{dt}\iint_{\Sigma(t)}dsymbol{A}% \cdot\mathbf{B}(\mathbf{r},\ t)\ ,
  2. A = θ 2 π π R 2 , A=\frac{\theta}{2\pi}\pi R^{2}\ ,
  3. = - d Φ B d t = B d A d t = B R 2 2 d θ d t = B R 2 2 ω , \mathcal{E}=-\frac{d\Phi_{B}}{dt}=B\frac{dA}{dt}=B\ \frac{R^{2}}{2}\ \frac{d% \theta}{dt}=B\ \frac{R^{2}}{2}\omega\ ,
  4. | 𝐯 × 𝐁 | |\mathbf{v}×\mathbf{B}|
  5. = 0 R d r B r ω = R 2 2 B ω , \mathcal{E}=\int_{0}^{R}drBr\omega=\frac{R^{2}}{2}B\omega\ ,
  6. 𝐅 21 = μ 0 4 π I 1 I 2 C 1 C 2 d 𝐥 𝟏 × ( d 𝐥 𝟐 × 𝐫 ^ 21 ) r 21 2 \mathbf{F}_{21}=\frac{\mu_{0}}{4\pi}I_{1}I_{2}\oint_{C_{1}}\oint_{C_{2}}\frac{% d\mathbf{l_{1}}\ \mathbf{\times}\ (d\mathbf{l_{2}}\ \mathbf{\times}\ \hat{% \mathbf{r}}_{21})}{r_{21}^{2}}
  7. 𝐁 2 = μ 0 4 π I 2 C 2 ( d 𝐥 𝟐 × 𝐫 ^ 21 ) r 21 2 \mathbf{B}_{2}=\frac{\mu_{0}}{4\pi}I_{2}\oint_{C_{2}}\frac{(d\mathbf{l_{2}}\ % \mathbf{\times}\ \hat{\mathbf{r}}_{21})}{r_{21}^{2}}
  8. 𝐅 21 = I 1 C 1 d 𝐥 𝟏 × 𝐁 2 \mathbf{F}_{21}=I_{1}\oint_{C_{1}}d\mathbf{l_{1}}\ \mathbf{\times}\mathbf{B}_{2}
  9. d 𝐥 d\mathbf{l}
  10. d 𝐫 d\mathbf{r}
  11. d W = d 𝐅 d 𝐫 dW=d\mathbf{F}\cdot d\mathbf{r}
  12. d 𝐅 d\mathbf{F}
  13. d W = ( I d 𝐥 × 𝐁 ) d 𝐫 dW=(Id\mathbf{l}\mathbf{\times}\mathbf{B})\cdot d\mathbf{r}
  14. d 𝐒 = d 𝐫 × d 𝐥 d\mathbf{S}=d\mathbf{r}\mathbf{\times}d\mathbf{l}
  15. d W = I 𝐁 d 𝐒 = I d Φ B dW=I\mathbf{B}\cdot d\mathbf{S}=Id\Phi_{B}
  16. d q dq
  17. V V
  18. d W = V d q = V I d t dW=Vdq=VIdt
  19. d Φ B = V d t d\Phi_{B}=Vdt
  20. d W dW
  21. E M F = ( 𝐄 + 𝐯 × 𝐁 ) d s y m b o l EMF=\oint\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right)\cdot\,\text{d}% symbol{\ell}
  22. 𝐯 \mathbf{v}

Farkas'_lemma.html

  1. A x < 0 Ax<0
  2. A T y = 0 A^{T}y=0
  3. A x Ax
  4. b b
  5. y T A = 0 y^{T}A=0
  6. y T b = - 1 y^{T}b=-1
  7. y y
  8. 0
  9. y T A y^{T}A
  10. y T b y^{T}b

Farnesyltransferase.html

  1. \rightleftharpoons

Fary–Milnor_theorem.html

  1. If K κ ( s ) d s 4 π then K is an unknot . \,\text{If}\,\oint_{K}\!\kappa(s)\,\operatorname{d}s\leq 4\pi\ \,\text{then}\ % K\ \,\text{is an unknot}.

Fed_model.html

  1. ( Y 10 ) (Y_{\,\text{10}})
  2. E P = Y 10 \frac{E}{P}=Y_{\,\text{10}}
  3. P = D ( 1 + G ) R f + R P - G P=\frac{D(1+G)}{R_{\,\text{f}}+RP-G}
  4. R f R_{\,\text{f}}
  5. R f R_{\,\text{f}}
  6. E x P x = R x [ 1 - T ] \frac{E_{\,\text{x}}}{P_{\,\text{x}}}=R_{\,\text{x}}\ [1-T]

Feigenbaum_function.html

  1. g ( x ) = 1 - λ g ( g ( λ x ) ) g(x)=\frac{1}{-\lambda}g(g(\lambda x))

Fenchel's_duality_theorem.html

  1. min x ( f ( x ) - g ( x ) ) = max p ( g ( p ) - f ( p ) ) . \min_{x}(f(x)-g(x))=\max_{p}(g_{\star}(p)-f^{\star}(p)).\,
  2. f ( x * ) := sup { x * , x - f ( x ) | x n } f^{\star}\left(x^{*}\right):=\sup\left\{\left.\left\langle x^{*},x\right% \rangle-f\left(x\right)\right|x\in\mathbb{R}^{n}\right\}
  3. g ( x * ) := inf { x * , x - g ( x ) | x n } g_{\star}\left(x^{*}\right):=\inf\left\{\left.\left\langle x^{*},x\right% \rangle-g\left(x\right)\right|x\in\mathbb{R}^{n}\right\}
  4. f : X { + } f:X\to\mathbb{R}\cup\{+\infty\}
  5. g : Y { + } g:Y\to\mathbb{R}\cup\{+\infty\}
  6. A : X Y A:X\to Y
  7. p * = inf x X { f ( x ) + g ( A x ) } p^{*}=\inf_{x\in X}\{f(x)+g(Ax)\}
  8. d * = sup y * Y * { - f * ( A * y * ) - g * ( - y * ) } d^{*}=\sup_{y^{*}\in Y^{*}}\{-f^{*}(A^{*}y^{*})-g^{*}(-y^{*})\}
  9. p * d * p^{*}\geq d^{*}
  10. f * , g * f^{*},g^{*}
  11. A * A^{*}
  12. F ( x , y ) = f ( x ) + g ( A x - y ) F(x,y)=f(x)+g(Ax-y)
  13. 0 core ( dom g - A dom f ) 0\in\operatorname{core}(\operatorname{dom}g-A\operatorname{dom}f)
  14. core \operatorname{core}
  15. dom h \operatorname{dom}h
  16. { z : h ( z ) < + } \{z:h(z)<+\infty\}
  17. A dom f cont g A\operatorname{dom}f\cap\operatorname{cont}g\neq\emptyset
  18. cont \operatorname{cont}
  19. p * = d * p^{*}=d^{*}
  20. d * d^{*}\in\mathbb{R}

Fenchel's_theorem.html

  1. 2 π P , \frac{2\pi}{P},
  2. 2 π P \geq\frac{2\pi}{P}

Fermat's_factorization_method.html

  1. N = a 2 - b 2 . N=a^{2}-b^{2}.
  2. ( a + b ) ( a - b ) (a+b)(a-b)
  3. N = c d N=cd
  4. N = ( c + d 2 ) 2 - ( c - d 2 ) 2 N=\left(\frac{c+d}{2}\right)^{2}-\left(\frac{c-d}{2}\right)^{2}
  5. a 2 - N = b 2 a^{2}-N=b^{2}
  6. N = 5959 N=5959
  7. 5959 5959
  8. 78 78
  9. b 2 = 78 2 - 5959 = 125 b^{2}=78^{2}-5959=125
  10. a = 80 a=80
  11. b = 21 b=21
  12. 5959 5959
  13. a - b = 59 a-b=59
  14. a + b = 101 a+b=101
  15. a + b a+b
  16. a - b = N / ( a + b ) a-b=N/(a+b)
  17. N = 1 N N=1\cdot N
  18. N = c d N=cd
  19. a = ( c + d ) / 2 a=(c+d)/2
  20. ( c + d ) / 2 - N = ( d - c ) 2 / 2 = ( N - c ) 2 / 2 c (c+d)/2-\sqrt{N}=(\sqrt{d}-\sqrt{c})^{2}/2=(\sqrt{N}-c)^{2}/2c
  21. d = 1 d=1
  22. O ( N ) O(N)
  23. ( 4 N ) 1 / 4 {\left(4N\right)}^{1/4}
  24. N \sqrt{N}
  25. N \sqrt{N}
  26. a - b = 47830.1 a-b=47830.1
  27. c > N c>\sqrt{N}
  28. N \sqrt{N}
  29. c c
  30. c - c 2 - N c-\sqrt{c^{2}-N}
  31. c = 48436 c=48436
  32. c = 55000 c=55000
  33. a 2 - N a^{2}-N
  34. a a
  35. N = 2345678917 N=2345678917
  36. a a
  37. a 2 - N a^{2}-N
  38. a 2 a^{2}
  39. a a
  40. b 2 b^{2}
  41. b b
  42. N = 2345678917 N=2345678917
  43. a 2 a^{2}
  44. a a
  45. a 2 a^{2}
  46. a a
  47. d / c d/c
  48. v / u v/u
  49. N u v = c v d u Nuv=cv\cdot du
  50. gcd ( N , c v ) = c \gcd(N,cv)=c
  51. gcd ( N , d u ) = d \gcd(N,du)=d
  52. u / v u/v
  53. O ( N 1 / 3 ) O(N^{1/3})
  54. a 2 - n a^{2}-n

Fermat_point.html

  1. OA , OB , OC , OX \overrightarrow{\mathrm{OA}},\overrightarrow{\mathrm{OB}},\overrightarrow{% \mathrm{OC}},\overrightarrow{\mathrm{OX}}

Fermi_acceleration.html

  1. d N ( ϵ ) d ϵ ϵ - p \frac{dN(\epsilon)}{d\epsilon}\propto\epsilon^{-p}
  2. p 2 p\gtrsim 2
  3. β s \beta_{s}
  4. β m 2 \beta_{m}^{2}

Fermionic_field.html

  1. ψ ( x ) \psi(x)
  2. ( i γ μ μ - m ) ψ ( x ) = 0. (i\gamma^{\mu}\partial_{\mu}-m)\psi(x)=0.\,
  3. γ μ \gamma^{\mu}
  4. m m
  5. ψ 1 ( x ) = u ( p ) e - i p . x \psi_{1}(x)=u(p)e^{-ip.x}\,
  6. ψ 2 ( x ) = v ( p ) e i p . x \psi_{2}(x)=v(p)e^{ip.x}\,
  7. ψ ( x ) \psi(x)
  8. ψ ( x ) = d 3 p ( 2 π ) 3 1 2 E p s ( a 𝐩 s u s ( p ) e - i p x + b 𝐩 s v s ( p ) e i p x ) . \psi(x)=\int\frac{d^{3}p}{(2\pi)^{3}}\frac{1}{\sqrt{2E_{p}}}\sum_{s}\left(a^{s% }_{\mathbf{p}}u^{s}(p)e^{-ip\cdot x}+b^{s\dagger}_{\mathbf{p}}v^{s}(p)e^{ip% \cdot x}\right).\,
  9. ψ ( x ) \psi(x)
  10. a 𝐩 s a^{s}_{\mathbf{p}}
  11. b 𝐩 s b^{s\dagger}_{\mathbf{p}}
  12. ψ ( x ) \psi(x)
  13. ψ ( y ) \psi(y)^{\dagger}
  14. { ψ a ( 𝐱 ) , ψ b ( 𝐲 ) } = δ ( 3 ) ( 𝐱 - 𝐲 ) δ a b , \{\psi_{a}(\mathbf{x}),\psi_{b}^{\dagger}(\mathbf{y})\}=\delta^{(3)}(\mathbf{x% }-\mathbf{y})\delta_{ab},
  15. ψ ( x ) \psi(x)
  16. ψ ( y ) \psi(y)
  17. { a 𝐩 r , a 𝐪 s } = { b 𝐩 r , b 𝐪 s } = ( 2 π ) 3 δ 3 ( 𝐩 - 𝐪 ) δ r s , \{a^{r}_{\mathbf{p}},a^{s\dagger}_{\mathbf{q}}\}=\{b^{r}_{\mathbf{p}},b^{s% \dagger}_{\mathbf{q}}\}=(2\pi)^{3}\delta^{3}(\mathbf{p}-\mathbf{q})\delta^{rs},\,
  18. a 𝐩 s a^{s\dagger}_{\mathbf{p}}
  19. b 𝐪 r b^{r\dagger}_{\mathbf{q}}
  20. ψ ( x ) \psi(x)
  21. ψ ¯ = def ψ γ 0 \bar{\psi}\ \stackrel{\mathrm{def}}{=}\ \psi^{\dagger}\gamma^{0}
  22. ψ ¯ ψ \overline{\psi}\psi\,
  23. ψ ¯ = ψ γ 0 \bar{\psi}=\psi^{\dagger}\gamma^{0}
  24. ψ ψ \psi^{\dagger}\psi
  25. γ 0 \gamma^{0}\,
  26. ψ ¯ γ μ μ ψ \overline{\psi}\gamma^{\mu}\partial_{\mu}\psi
  27. D = ψ ¯ ( i γ μ μ - m ) ψ \mathcal{L}_{D}=\bar{\psi}(i\gamma^{\mu}\partial_{\mu}-m)\psi\,
  28. D = ψ ¯ a ( i γ a b μ μ - m 𝕀 a b ) ψ b \mathcal{L}_{D}=\bar{\psi}_{a}(i\gamma^{\mu}_{ab}\partial_{\mu}-m\mathbb{I}_{% ab})\psi_{b}\,
  29. ψ ( x ) \psi(x)
  30. D F ( x - y ) = 0 | T ( ψ ( x ) ψ ¯ ( y ) ) | 0 D_{F}(x-y)=\langle 0|T(\psi(x)\bar{\psi}(y))|0\rangle
  31. T ( ψ ( x ) ψ ¯ ( y ) ) = def θ ( x 0 - y 0 ) ψ ( x ) ψ ¯ ( y ) - θ ( y 0 - x 0 ) ψ ¯ ( y ) ψ ( x ) . T(\psi(x)\bar{\psi}(y))\ \stackrel{\mathrm{def}}{=}\ \theta(x^{0}-y^{0})\psi(x% )\bar{\psi}(y)-\theta(y^{0}-x^{0})\bar{\psi}(y)\psi(x).
  32. D F ( x - y ) = d 4 p ( 2 π ) 4 i ( p / + m ) p 2 - m 2 + i ϵ e - i p ( x - y ) D_{F}(x-y)=\int\frac{d^{4}p}{(2\pi)^{4}}\frac{i(p\!\!\!/+m)}{p^{2}-m^{2}+i% \epsilon}e^{-ip\cdot(x-y)}
  33. i ( p / + m ) p 2 - m 2 \frac{i(p\!\!\!/+m)}{p^{2}-m^{2}}
  34. ψ ( x ) \psi(x)

Fermi–Pasta–Ulam_problem.html

  1. p j = j h , j = 0 , , N - 1 p_{j}=jh,j=0,\dots,N-1
  2. h = l / ( N - 1 ) h=l/(N-1)
  3. X j ( t ) = p j + x j ( t ) X_{j}(t)=p_{j}+x_{j}(t)
  4. x j ( t ) x_{j}(t)
  5. m x ¨ j = k ( x j + 1 + x j - 1 - 2 x j ) [ 1 + α ( x j + 1 - x j - 1 ) ] m\ddot{x}_{j}=k(x_{j+1}+x_{j-1}-2x_{j})[1+\alpha(x_{j+1}-x_{j-1})]
  6. k ( x j + 1 + x j - 1 - 2 x j ) k(x_{j+1}+x_{j-1}-2x_{j})
  7. α \alpha
  8. c = κ / ρ c=\sqrt{\kappa/\rho}
  9. κ = k / h \kappa=k/h
  10. ρ = m / h 3 \rho=m/h^{3}
  11. x ¨ j = c 2 h 2 ( x j + 1 + x j - 1 - 2 x j ) [ 1 + α ( x j + 1 - x j - 1 ) ] \ddot{x}_{j}=\frac{c^{2}}{h^{2}}(x_{j+1}+x_{j-1}-2x_{j})[1+\alpha(x_{j+1}-x_{j% -1})]
  12. u ( p j , t ) u(p_{j},t)
  13. x j ( t ) x_{j}(t)
  14. x ¨ j = c 2 ( x j + 1 + x j - 1 - 2 x j h 2 ) [ 1 + α ( x j + 1 - x j - 1 ) ] \ddot{x}_{j}=c^{2}\left(\frac{x_{j+1}+x_{j-1}-2x_{j}}{h^{2}}\right)[1+\alpha(x% _{j+1}-x_{j-1})]
  15. h h
  16. ( x j + 1 + x j - 1 - 2 x j h 2 ) \displaystyle\left(\frac{x_{j+1}+x_{j-1}-2x_{j}}{h^{2}}\right)
  17. α ( x j + 1 - x j - 1 ) = 2 α h u x ( x , t ) + ( α h 3 3 ) u x x x ( x , t ) + O ( h 5 ) . \alpha(x_{j+1}-x_{j-1})=2\alpha hu_{x}(x,t)+\left(\frac{\alpha h^{3}}{3}\right% )u_{xxx}(x,t)+O(h^{5}).
  18. 1 c 2 u t t - u x x = ( 2 α h ) u x u x x + ( h 2 12 ) u x x x x + O ( α h 2 , h 4 ) . \frac{1}{c^{2}}u_{tt}-u_{xx}=(2\alpha h)u_{x}u_{xx}+\left(\frac{h^{2}}{12}% \right)u_{xxxx}+O(\alpha h^{2},h^{4}).
  19. 2 α h 2\alpha h
  20. 1 c 2 u t t - u x x = ( 2 α h ) u x u x x + ( h 2 12 ) u x x x x . \frac{1}{c^{2}}u_{tt}-u_{xx}=(2\alpha h)u_{x}u_{xx}+\left(\frac{h^{2}}{12}% \right)u_{xxxx}.
  21. α , h \alpha,h
  22. ξ = x - c t , τ = ( α h ) c t , y ( ξ , τ ) = u ( x , t ) \xi=x-ct,\tau=(\alpha h)ct,y(\xi,\tau)=u(x,t)
  23. y ξ τ - ( α h 2 ) y τ τ = - y ξ y ξ ξ - ( h 24 α ) y ξ ξ ξ ξ . y_{\xi\tau}-\left(\frac{\alpha h}{2}\right)y_{\tau\tau}=-y_{\xi}y_{\xi\xi}-% \left(\frac{h}{24\alpha}\right)y_{\xi\xi\xi\xi}.
  24. α / h \alpha/h
  25. α , h \alpha,h
  26. δ = lim h 0 h / ( 24 α ) \delta=\lim_{h\rightarrow 0}\sqrt{h/(24\alpha)}
  27. y ξ τ = - y ξ y ξ ξ - δ 2 y ξ ξ ξ ξ . y_{\xi\tau}=-y_{\xi}y_{\xi\xi}-\delta^{2}y_{\xi\xi\xi\xi}.
  28. v = y ξ v=y_{\xi}
  29. v τ + v v ξ + δ 2 v ξ ξ ξ = 0. v_{\tau}+vv_{\xi}+\delta^{2}v_{\xi\xi\xi}=0.

Feshbach_resonance.html

  1. A + B A + B A+B\rightarrow A^{\prime}+B^{\prime}
  2. A ( B , B ) A A(B,B^{\prime})A^{\prime}
  3. r A \vec{r}_{A}
  4. r B \vec{r}_{B}
  5. R | r A - r B | R\equiv|\vec{r}_{A}-\vec{r}_{B}|
  6. V c ( R ) V_{c}(R)
  7. E = T + V C ( R ) + Δ ( P ) E=T+V_{C}(R)+\Delta(\vec{P})
  8. T T
  9. Δ \Delta
  10. P \vec{P}
  11. V D ( R ) V_{D}(R)
  12. E D . E_{D}.
  13. E D T + V C ( R ) + Δ ( P 0 ) E_{D}\approx T+V_{C}(R)+\Delta(\vec{P}_{0})
  14. { P 0 } \{\vec{P}_{0}\}

Feshbach–Fano_partitioning.html

  1. H eff ( E ) = Q H Q + lim ε 0 Q H P 1 E + i ε - P H P P H Q = Q H Q + Δ ( E ) - i Γ ( E ) / 2 , H_{\mathrm{eff}}(E)=QHQ+\lim_{\varepsilon\to 0}QHP{1\over E+i\varepsilon-PHP}% PHQ=QHQ+\Delta(E)-i\Gamma(E)/2,\,
  2. E res E_{\mathrm{res}}
  3. Γ res \Gamma_{\mathrm{res}}
  4. det [ H eff ( z ) - z ] = 0 \det[H_{\mathrm{eff}}(z)-z]=0\,
  5. z res = E res - i Γ res z_{\mathrm{res}}=E_{\mathrm{res}}-i\Gamma_{\mathrm{res}}\,
  6. z res z_{\mathrm{res}}
  7. T tot = T background + T resonances . T_{\mathrm{tot}}=T_{\mathrm{background}}+T_{\mathrm{resonances}}.\,

Feynman_parametrization.html

  1. 1 A B = 0 1 d u [ u A + ( 1 - u ) B ] 2 \frac{1}{AB}=\int^{1}_{0}\frac{du}{\left[uA+(1-u)B\right]^{2}}
  2. d p A ( p ) B ( p ) = d p 0 1 d u [ u A ( p ) + ( 1 - u ) B ( p ) ] 2 = 0 1 d u d p [ u A ( p ) + ( 1 - u ) B ( p ) ] 2 . \int\frac{dp}{A(p)B(p)}=\int dp\int^{1}_{0}\frac{du}{\left[uA(p)+(1-u)B(p)% \right]^{2}}=\int^{1}_{0}du\int\frac{dp}{\left[uA(p)+(1-u)B(p)\right]^{2}}.
  3. 1 A 1 A n \displaystyle\frac{1}{A_{1}\cdots A_{n}}
  4. Re ( α j ) > 0 \,\text{Re}(\alpha_{j})>0
  5. 1 j n 1\leq j\leq n
  6. 1 A 1 α 1 A n α n = Γ ( α 1 + + α n ) Γ ( α 1 ) Γ ( α n ) 0 1 d u 1 0 1 d u n δ ( k = 1 n u k - 1 ) u 1 α 1 - 1 u n α n - 1 [ u 1 A 1 + + u n A n ] k = 1 n α k . \frac{1}{A_{1}^{\alpha_{1}}\cdots A_{n}^{\alpha_{n}}}=\frac{\Gamma(\alpha_{1}+% \dots+\alpha_{n})}{\Gamma(\alpha_{1})\cdots\Gamma(\alpha_{n})}\int_{0}^{1}du_{% 1}\cdots\int_{0}^{1}du_{n}\frac{\delta(\sum_{k=1}^{n}u_{k}-1)u_{1}^{\alpha_{1}% -1}\cdots u_{n}^{\alpha_{n}-1}}{\left[u_{1}A_{1}+\cdots+u_{n}A_{n}\right]^{% \sum_{k=1}^{n}\alpha_{k}}}.
  7. 1 A B = 1 A - B ( 1 B - 1 A ) = 1 A - B B A d z z 2 . \frac{1}{AB}=\frac{1}{A-B}\left(\frac{1}{B}-\frac{1}{A}\right)=\frac{1}{A-B}% \int_{B}^{A}\frac{dz}{z^{2}}.
  8. u = ( z - B ) / ( A - B ) u=(z-B)/(A-B)
  9. d u = d z / ( A - B ) du=dz/(A-B)
  10. z = u A + ( 1 - u ) B z=uA+(1-u)B
  11. 1 A B = 0 1 d u [ u A + ( 1 - u ) B ] 2 . \frac{1}{AB}=\int_{0}^{1}\frac{du}{\left[uA+(1-u)B\right]^{2}}.
  12. 1 A 1 A n \frac{1}{A_{1}...A_{n}}
  13. 1 A i = 0 d s i e - s i A i , i = 1 , , n \frac{1}{A_{i}}=\int^{\infty}_{0}ds_{i}\,e^{-s_{i}A_{i}},\forall i=1,...,n
  14. 1 A 1 A n = 0 d s 1 d s n exp [ - ( s 1 A 1 + s n A n ) ] . \frac{1}{A_{1}...A_{n}}=\int_{0}^{\infty}ds_{1}...ds_{n}\exp\left[-\left(s_{1}% A_{1}+...s_{n}A_{n}\right)\right].
  15. α = s 1 + + s n , \alpha=s_{1}+...+s_{n},
  16. α i = s i s 1 + s n ; i = 1 , , n - 1 , \alpha_{i}=\frac{s_{i}}{s_{1}+...s_{n}};i=1,\ldots,n-1,
  17. 1 A 1 A n = 0 1 d α 1 d α n - 1 0 d α α N - 1 exp ( - α { α 1 A 1 + α n - 1 A n - 1 + ( 1 - α 1 - α n - 1 ) A n } ) . \frac{1}{A_{1}...A_{n}}=\int_{0}^{1}d\alpha_{1}...d\alpha_{n-1}\int_{0}^{% \infty}d\alpha\alpha^{N-1}\exp\left(-\alpha\left\{\alpha_{1}A_{1}+...\alpha_{n% -1}A_{n-1}+\left(1-\alpha_{1}-...\alpha_{n-1}\right)A_{n}\right\}\right).
  18. α \alpha
  19. 0 d α α n - 1 exp ( - α x ) = n - 1 ( - x ) n - 1 ( 0 d α exp ( - α x ) ) = ( n - 1 ) ! x n . \int_{0}^{\infty}d\alpha\alpha^{n-1}\exp(-\alpha x)=\frac{\partial^{n-1}}{% \partial(-x)^{n-1}}\left(\int_{0}^{\infty}d\alpha\exp(-\alpha x)\right)=\frac{% \left(n-1\right)!}{x^{n}}.
  20. x = α 1 A 1 + α n - 1 A n - 1 + ( 1 - α 1 - α n - 1 ) A n . x=\alpha_{1}A_{1}+...\alpha_{n-1}A_{n-1}+\left(1-\alpha_{1}-...\alpha_{n-1}% \right)A_{n}.
  21. 1 A 1 A n = ( n - 1 ) ! 0 1 d α 1 d α n - 1 1 [ α 1 A 1 + + α n - 1 A n - 1 + ( 1 - α 1 - α n - 1 ) A n ] n , \frac{1}{A_{1}...A_{n}}=\left(n-1\right)!\int_{0}^{1}d\alpha_{1}...d\alpha_{n-% 1}\frac{1}{[\alpha_{1}A_{1}+...+\alpha_{n-1}A_{n-1}+\left(1-\alpha_{1}-...% \alpha_{n-1}\right)A_{n}]^{n}},
  22. 1 A 1 A n = ( n - 1 ) ! 0 1 d α 1 d α n δ ( 1 - α 1 - - α n ) [ α 1 A 1 + + α n A n ] n . \frac{1}{A_{1}...A_{n}}=\left(n-1\right)!\int_{0}^{1}d\alpha_{1}...d\alpha_{n}% \frac{\delta\left(1-\alpha_{1}-...-\alpha_{n}\right)}{[\alpha_{1}A_{1}+...+% \alpha_{n}A_{n}]^{n}}.
  23. 1 A 1 α 1 A n α n \frac{1}{A_{1}^{\alpha_{1}}...A_{n}^{\alpha_{n}}}
  24. 1 A 1 α 1 = 1 ( α 1 - 1 ) ! 0 d s 1 s 1 α 1 - 1 e - s 1 A 1 = 1 Γ ( α 1 ) α 1 - 1 ( - A 1 ) α 1 - 1 ( 0 d s 1 e - s 1 A 1 ) \frac{1}{A_{1}^{\alpha_{1}}}=\frac{1}{\left(\alpha_{1}-1\right)!}\int^{\infty}% _{0}ds_{1}\,s_{1}^{\alpha_{1}-1}e^{-s_{1}A_{1}}=\frac{1}{\Gamma(\alpha_{1})}% \frac{\partial^{\alpha_{1}-1}}{\partial(-A_{1})^{\alpha_{1}-1}}\left(\int_{0}^% {\infty}ds_{1}e^{-s_{1}A_{1}}\right)
  25. [ - 1 , 1 ] [-1,1]
  26. 1 A B = 2 - 1 1 d u [ ( 1 + u ) A + ( 1 - u ) B ] 2 . \frac{1}{AB}=2\int_{-1}^{1}\frac{du}{\left[(1+u)A+(1-u)B\right]^{2}}.

Fiber_Bragg_grating.html

  1. λ B \scriptstyle\lambda_{B}
  2. λ B = 2 n e Λ \lambda_{B}=2n_{e}\Lambda\,
  3. n e \scriptstyle n_{e}
  4. Λ \scriptstyle\Lambda
  5. n e \scriptstyle n_{e}
  6. Δ λ \scriptstyle\Delta\lambda
  7. Δ λ = [ 2 δ n 0 η π ] λ B \Delta\lambda=\left[\frac{2\delta n_{0}\eta}{\pi}\right]\lambda_{B}
  8. δ n 0 \scriptstyle\delta n_{0}
  9. n 3 - n 2 \scriptstyle n_{3}\,-\,n_{2}
  10. η \scriptstyle\eta
  11. L g \scriptstyle L_{g}
  12. λ B \scriptstyle\lambda_{B}
  13. δ n 0 \scriptstyle\delta n_{0}
  14. P B ( λ B ) \scriptstyle P_{B}(\lambda_{B})
  15. P B ( λ B ) tanh 2 [ N η ( V ) δ n 0 n ] P_{B}(\lambda_{B})\approx\tanh^{2}\left[\frac{N\eta(V)\delta n_{0}}{n}\right]
  16. N \scriptstyle N
  17. P B ( λ ) \scriptstyle P_{B}(\lambda)
  18. P B ( λ ) = sinh 2 [ η ( V ) δ n 0 1 - Γ 2 N Λ λ ] cosh 2 [ η ( V ) δ n 0 1 - Γ 2 N Λ λ ] - Γ 2 P_{B}(\lambda)=\frac{\sinh^{2}\left[\eta(V)\delta n_{0}\sqrt{1-\Gamma^{2}}% \frac{N\Lambda}{\lambda}\right]}{\cosh^{2}\left[\eta(V)\delta n_{0}\sqrt{1-% \Gamma^{2}}\frac{N\Lambda}{\lambda}\right]-\Gamma^{2}}
  19. Γ ( λ ) = 1 η ( V ) δ n 0 [ λ λ B - 1 ] \Gamma(\lambda)=\frac{1}{\eta(V)\delta n_{0}}\left[\frac{\lambda}{\lambda_{B}}% -1\right]
  20. L g \scriptstyle L_{g}
  21. L g = N Λ L_{g}=N\Lambda\,
  22. δ n 0 η \scriptstyle\delta n_{0}\eta
  23. δ n 0 \scriptstyle\delta n_{0}
  24. N \scriptstyle N
  25. Δ λ B \scriptstyle\Delta\lambda_{B}
  26. Δ λ B / λ B \scriptstyle\Delta\lambda_{B}/\lambda_{B}
  27. ϵ \scriptstyle\epsilon
  28. Δ T \scriptstyle\Delta T
  29. [ Δ λ B λ B ] = C S ϵ + C T Δ T \left[\frac{\Delta\lambda_{B}}{\lambda_{B}}\right]=C_{S}\epsilon+C_{T}\Delta T
  30. [ Δ λ B λ B ] = ( 1 - p e ) ϵ + ( α Λ + α n ) Δ T \left[\frac{\Delta\lambda_{B}}{\lambda_{B}}\right]=(1-p_{e})\epsilon+(\alpha_{% \Lambda}+\alpha_{n})\Delta T
  31. C S \scriptstyle C_{S}
  32. p e \scriptstyle p_{e}
  33. C T \scriptstyle C_{T}
  34. α Λ \scriptstyle\alpha_{\Lambda}
  35. α n \scriptstyle\alpha_{n}