wpmath0000006_14

Vector_autoregression.html

  1. y t = c + A 1 y t - 1 + A 2 y t - 2 + + A p y t - p + e t , y_{t}=c+A_{1}y_{t-1}+A_{2}y_{t-2}+\cdots+A_{p}y_{t-p}+e_{t},\,
  2. E ( e t ) = 0 \mathrm{E}(e_{t})=0\,
  3. E ( e t e t ) = Ω \mathrm{E}(e_{t}e_{t}^{\prime})=\Omega\,
  4. E ( e t e t - k ) = 0 \mathrm{E}(e_{t}e_{t-k}^{\prime})=0\,
  5. Y = B Z + U Y=BZ+U\,
  6. [ y 1 , t y 2 , t ] = [ c 1 c 2 ] + [ A 1 , 1 A 1 , 2 A 2 , 1 A 2 , 2 ] [ y 1 , t - 1 y 2 , t - 1 ] + [ e 1 , t e 2 , t ] , \begin{bmatrix}y_{1,t}\\ y_{2,t}\end{bmatrix}=\begin{bmatrix}c_{1}\\ c_{2}\end{bmatrix}+\begin{bmatrix}A_{1,1}&A_{1,2}\\ A_{2,1}&A_{2,2}\end{bmatrix}\begin{bmatrix}y_{1,t-1}\\ y_{2,t-1}\end{bmatrix}+\begin{bmatrix}e_{1,t}\\ e_{2,t}\end{bmatrix},
  7. y 1 , t = c 1 + A 1 , 1 y 1 , t - 1 + A 1 , 2 y 2 , t - 1 + e 1 , t y_{1,t}=c_{1}+A_{1,1}y_{1,t-1}+A_{1,2}y_{2,t-1}+e_{1,t}\,
  8. y 2 , t = c 2 + A 2 , 1 y 1 , t - 1 + A 2 , 2 y 2 , t - 1 + e 2 , t . y_{2,t}=c_{2}+A_{2,1}y_{1,t-1}+A_{2,2}y_{2,t-1}+e_{2,t}.\,
  9. y t = c + A 1 y t - 1 + A 2 y t - 2 + e t y_{t}=c+A_{1}y_{t-1}+A_{2}y_{t-2}+e_{t}
  10. [ y t y t - 1 ] = [ c 0 ] + [ A 1 A 2 I 0 ] [ y t - 1 y t - 2 ] + [ e t 0 ] , \begin{bmatrix}y_{t}\\ y_{t-1}\end{bmatrix}=\begin{bmatrix}c\\ 0\end{bmatrix}+\begin{bmatrix}A_{1}&A_{2}\\ I&0\end{bmatrix}\begin{bmatrix}y_{t-1}\\ y_{t-2}\end{bmatrix}+\begin{bmatrix}e_{t}\\ 0\end{bmatrix},
  11. B 0 y t = c 0 + B 1 y t - 1 + B 2 y t - 2 + + B p y t - p + ϵ t , B_{0}y_{t}=c_{0}+B_{1}y_{t-1}+B_{2}y_{t-2}+\cdots+B_{p}y_{t-p}+\epsilon_{t},
  12. E ( ϵ t ϵ t ) = Σ \mathrm{E}(\epsilon_{t}\epsilon_{t}^{\prime})=\Sigma
  13. [ 1 B 0 ; 1 , 2 B 0 ; 2 , 1 1 ] [ y 1 , t y 2 , t ] = [ c 0 ; 1 c 0 ; 2 ] + [ B 1 ; 1 , 1 B 1 ; 1 , 2 B 1 ; 2 , 1 B 1 ; 2 , 2 ] [ y 1 , t - 1 y 2 , t - 1 ] + [ ϵ 1 , t ϵ 2 , t ] , \begin{bmatrix}1&B_{0;1,2}\\ B_{0;2,1}&1\end{bmatrix}\begin{bmatrix}y_{1,t}\\ y_{2,t}\end{bmatrix}=\begin{bmatrix}c_{0;1}\\ c_{0;2}\end{bmatrix}+\begin{bmatrix}B_{1;1,1}&B_{1;1,2}\\ B_{1;2,1}&B_{1;2,2}\end{bmatrix}\begin{bmatrix}y_{1,t-1}\\ y_{2,t-1}\end{bmatrix}+\begin{bmatrix}\epsilon_{1,t}\\ \epsilon_{2,t}\end{bmatrix},
  14. Σ = E ( ϵ t ϵ t ) = [ σ 1 2 0 0 σ 2 2 ] ; \Sigma=\mathrm{E}(\epsilon_{t}\epsilon_{t}^{\prime})=\begin{bmatrix}\sigma_{1}% ^{2}&0\\ 0&\sigma_{2}^{2}\end{bmatrix};
  15. var ( ϵ i ) = σ i 2 \mathrm{var}(\epsilon_{i})=\sigma_{i}^{2}
  16. cov ( ϵ 1 , ϵ 2 ) = 0 \mathrm{cov}(\epsilon_{1},\epsilon_{2})=0
  17. y 1 , t = c 0 ; 1 - B 0 ; 1 , 2 y 2 , t + B 1 ; 1 , 1 y 1 , t - 1 + B 1 ; 1 , 2 y 2 , t - 1 + ϵ 1 , t y_{1,t}=c_{0;1}-B_{0;1,2}y_{2,t}+B_{1;1,1}y_{1,t-1}+B_{1;1,2}y_{2,t-1}+% \epsilon_{1,t}\,
  18. y t = B 0 - 1 c 0 + B 0 - 1 B 1 y t - 1 + B 0 - 1 B 2 y t - 2 + + B 0 - 1 B p y t - p + B 0 - 1 ϵ t , y_{t}=B_{0}^{-1}c_{0}+B_{0}^{-1}B_{1}y_{t-1}+B_{0}^{-1}B_{2}y_{t-2}+\cdots+B_{% 0}^{-1}B_{p}y_{t-p}+B_{0}^{-1}\epsilon_{t},
  19. B 0 - 1 c 0 = c , B 0 - 1 B i = A i for i = 1 , , p and B 0 - 1 ϵ t = e t B_{0}^{-1}c_{0}=c,\quad B_{0}^{-1}B_{i}=A_{i}\,\text{ for }i=1,\dots,p\,\text{% and }B_{0}^{-1}\epsilon_{t}=e_{t}
  20. y t = c + A 1 y t - 1 + A 2 y t - 2 + + A p y t - p + e t y_{t}=c+A_{1}y_{t-1}+A_{2}y_{t-2}+\cdots+A_{p}y_{t-p}+e_{t}
  21. Ω = E ( e t e t ) = E ( B 0 - 1 ϵ t ϵ t ( B 0 - 1 ) ) = B 0 - 1 Σ ( B 0 - 1 ) \Omega=\mathrm{E}(e_{t}e_{t}^{\prime})=\mathrm{E}(B_{0}^{-1}\epsilon_{t}% \epsilon_{t}^{\prime}(B_{0}^{-1})^{\prime})=B_{0}^{-1}\Sigma(B_{0}^{-1})^{% \prime}\,
  22. Y = B Z + U Y=BZ+U\,
  23. B ^ = Y Z ( Z Z ) - 1 \hat{B}=YZ^{{}^{\prime}}(ZZ^{{}^{\prime}})^{-1}
  24. Vec ( B ^ ) = ( ( Z Z ) - 1 Z I k ) Vec ( Y ) \operatorname{Vec}(\hat{B})=((ZZ^{{}^{\prime}})^{-1}Z\otimes I_{k})\ % \operatorname{Vec}(Y)
  25. \otimes
  26. Σ ^ = 1 T t = 1 T ϵ ^ t ϵ ^ t \hat{\Sigma}=\frac{1}{T}\sum_{t=1}^{T}\hat{\epsilon}_{t}\hat{\epsilon}_{t}^{\prime}
  27. Σ ^ = 1 T - k p - 1 t = 1 T ϵ ^ t ϵ ^ t \hat{\Sigma}=\frac{1}{T-kp-1}\sum_{t=1}^{T}\hat{\epsilon}_{t}\hat{\epsilon}_{t% }^{\prime}
  28. Σ ^ = 1 T - k p - 1 ( Y - B ^ Z ) ( Y - B ^ Z ) . \hat{\Sigma}=\frac{1}{T-kp-1}(Y-\hat{B}Z)(Y-\hat{B}Z)^{\prime}.
  29. Cov ^ ( Vec ( B ^ ) ) = ( Z Z ) - 1 Σ ^ . \widehat{\mbox{Cov}}~{}(\mbox{Vec}~{}(\hat{B}))=({ZZ^{\prime}})^{-1}\otimes% \hat{\Sigma}.\,

Vector_calculus_identities.html

  1. 𝐀 \mathbf{A}
  2. grad ( 𝐀 ) = 𝐀 \operatorname{grad}(\mathbf{A})=\nabla\mathbf{A}
  3. ψ \psi
  4. grad ( ψ ) = ψ \operatorname{grad}(\psi)=\nabla\psi
  5. 𝐀 \mathbf{A}
  6. div ( 𝐀 ) = 𝐀 \operatorname{div}(\mathbf{A})=\nabla\cdot\mathbf{A}
  7. ( 𝐁 𝐀 ^ ) = 𝐀 ^ ( 𝐁 ) + ( 𝐁 ) 𝐀 ^ \nabla\cdot(\mathbf{B}\otimes\hat{\mathbf{A}})=\hat{\mathbf{A}}(\nabla\cdot% \mathbf{B})+(\mathbf{B}\cdot\nabla)\hat{\mathbf{A}}
  8. 𝐁 \mathbf{B}\cdot\nabla
  9. 𝐁 \mathbf{B}
  10. ( 𝐚𝐛 T ) = 𝐛 ( 𝐚 ) + ( 𝐚 ) 𝐛 . \nabla\cdot(\mathbf{a}\mathbf{b}^{\mathrm{T}})=\mathbf{b}(\nabla\cdot\mathbf{a% })+(\mathbf{a}\cdot\nabla)\mathbf{b}\ .
  11. 𝐯 \mathbf{v}
  12. × 𝐯 \nabla\times\mathbf{v}
  13. ε i j k v k x j \varepsilon_{ijk}\frac{\partial v_{k}}{\partial x_{j}}
  14. 𝐀 \mathbf{A}
  15. Δ 𝐀 = 2 𝐀 = ( ) 𝐀 \Delta\mathbf{A}=\nabla^{2}\mathbf{A}=(\nabla\cdot\nabla)\mathbf{A}
  16. 𝐁 ( 𝐀 𝐁 ) = 𝐀 × ( × 𝐁 ) + ( 𝐀 ) 𝐁 \nabla_{\mathbf{B}}\left(\mathbf{A\cdot B}\right)=\mathbf{A}\times\left(\nabla% \times\mathbf{B}\right)+\left(\mathbf{A}\cdot\nabla\right)\mathbf{B}
  17. ˙ ( 𝐀 𝐁 ˙ ) = 𝐀 × ( × 𝐁 ) + ( 𝐀 ) 𝐁 \dot{\nabla}\left(\mathbf{A}\cdot\dot{\mathbf{B}}\right)=\mathbf{A}\times\left% (\nabla\times\mathbf{B}\right)+\left(\mathbf{A}\cdot\nabla\right)\mathbf{B}
  18. ( ψ + ϕ ) = ψ + ϕ \nabla(\psi+\phi)=\nabla\psi+\nabla\phi
  19. ( 𝐀 + 𝐁 ) = 𝐀 + 𝐁 \nabla\cdot(\mathbf{A}+\mathbf{B})=\nabla\cdot\mathbf{A}+\nabla\cdot\mathbf{B}
  20. × ( 𝐀 + 𝐁 ) = × 𝐀 + × 𝐁 \nabla\times(\mathbf{A}+\mathbf{B})=\nabla\times\mathbf{A}+\nabla\times\mathbf% {B}
  21. ψ \psi
  22. ϕ \phi
  23. ( ψ ϕ ) = ϕ ψ + ψ ϕ \nabla(\psi\,\phi)=\phi\,\nabla\psi+\psi\,\nabla\phi
  24. ( ψ 𝐀 ) = 𝐀 ψ + ψ ( 𝐀 ) \nabla\cdot(\psi\mathbf{A})=\mathbf{A}\cdot\nabla\psi+\psi(\nabla\cdot\mathbf{% A})
  25. × ( ψ 𝐀 ) = ψ ( × 𝐀 ) + ( ψ ) × 𝐀 \nabla\times(\psi\mathbf{A})=\psi(\nabla\times\mathbf{A})+(\nabla\psi)\times% \mathbf{A}
  26. ( f g ) = g f - f g g 2 \nabla\left(\frac{f}{g}\right)=\frac{g\nabla f-f\nabla g}{g^{2}}
  27. ( 𝐀 g ) = ( 𝐀 ) g - 𝐀 g g 2 \nabla\cdot\left(\frac{\mathbf{A}}{g}\right)=\frac{(\nabla\cdot\mathbf{A})g-% \mathbf{A}\cdot\nabla g}{g^{2}}
  28. × ( 𝐀 g ) = ( × 𝐀 ) g + 𝐀 × g g 2 \nabla\times\left(\frac{\mathbf{A}}{g}\right)=\frac{(\nabla\times\mathbf{A})g+% \mathbf{A}\times\nabla g}{g^{2}}
  29. ( f g ) = ( f g ) g \nabla(f\circ g)=(f^{\prime}\circ g)\nabla g
  30. ( f 𝐀 ) = ( f 𝐀 ) 𝐀 \nabla(f\circ\mathbf{A})=(\nabla f\circ\mathbf{A})\nabla\mathbf{A}
  31. ( 𝐀 f ) = ( 𝐀 f ) f \nabla\cdot(\mathbf{A}\circ f)=(\mathbf{A}^{\prime}\circ f)\cdot\nabla f
  32. × ( 𝐀 f ) = - ( 𝐀 f ) × f \nabla\times(\mathbf{A}\circ f)=-(\mathbf{A}^{\prime}\circ f)\times\nabla f
  33. ( 𝐀 𝐁 ) = 𝐉 𝐀 T 𝐁 + 𝐉 𝐁 T 𝐀 = ( 𝐀 ) 𝐁 + ( 𝐁 ) 𝐀 + 𝐀 × ( × 𝐁 ) + 𝐁 × ( × 𝐀 ) . \begin{aligned}\displaystyle\nabla(\mathbf{A}\cdot\mathbf{B})&\displaystyle=% \mathbf{J}^{\mathrm{T}}_{\mathbf{A}}\mathbf{B}+\mathbf{J}^{\mathrm{T}}_{% \mathbf{B}}\mathbf{A}\\ &\displaystyle=(\mathbf{A}\cdot\nabla)\mathbf{B}+(\mathbf{B}\cdot\nabla)% \mathbf{A}+\mathbf{A}\times(\nabla\times\mathbf{B})+\mathbf{B}\times(\nabla% \times\mathbf{A})\ .\end{aligned}
  34. 𝐀 \mathbf{A}
  35. ( 𝐀 𝐁 ) = 𝐀 ( 𝐀 𝐁 ) + 𝐁 ( 𝐀 𝐁 ) . \nabla(\mathbf{A}\cdot\mathbf{B})=\nabla_{\mathbf{A}}(\mathbf{A}\cdot\mathbf{B% })+\nabla_{\mathbf{B}}(\mathbf{A}\cdot\mathbf{B})\ .
  36. 𝐀 = 𝐁 \mathbf{A}=\mathbf{B}
  37. 1 2 ( 𝐀 𝐀 ) = 𝐉 𝐀 T 𝐀 = ( 𝐀 ) 𝐀 + 𝐀 × ( × 𝐀 ) . \begin{aligned}\displaystyle\frac{1}{2}\nabla\left(\mathbf{A}\cdot\mathbf{A}% \right)&\displaystyle=\mathbf{J}^{\mathrm{T}}_{\mathbf{A}}\mathbf{A}\\ &\displaystyle=(\mathbf{A}\cdot\nabla)\mathbf{A}+\mathbf{A}\times(\nabla\times% \mathbf{A})\ .\end{aligned}
  38. ( 𝐀 × 𝐁 ) = ( × 𝐀 ) 𝐁 - 𝐀 ( × 𝐁 ) \nabla\cdot(\mathbf{A}\times\mathbf{B})=(\nabla\times\mathbf{A})\cdot\mathbf{B% }-\mathbf{A}\cdot(\nabla\times\mathbf{B})
  39. × ( 𝐀 × 𝐁 ) = 𝐀 ( 𝐁 ) - 𝐁 ( 𝐀 ) + ( 𝐁 ) 𝐀 - ( 𝐀 ) 𝐁 = ( 𝐁 + 𝐁 ) 𝐀 - ( 𝐀 + 𝐀 ) 𝐁 = ( 𝐁𝐀 T ) - ( 𝐀𝐁 T ) = ( 𝐁𝐀 T - 𝐀𝐁 T ) \begin{aligned}\displaystyle\nabla\times(\mathbf{A}\times\mathbf{B})&% \displaystyle=\mathbf{A}(\nabla\cdot\mathbf{B})-\mathbf{B}(\nabla\cdot\mathbf{% A})+(\mathbf{B}\cdot\nabla)\mathbf{A}-(\mathbf{A}\cdot\nabla)\mathbf{B}\\ &\displaystyle=(\nabla\cdot\mathbf{B}+\mathbf{B}\cdot\nabla)\mathbf{A}-(\nabla% \cdot\mathbf{A}+\mathbf{A}\cdot\nabla)\mathbf{B}\\ &\displaystyle=\nabla\cdot(\mathbf{B}\mathbf{A}^{\mathrm{T}})-\nabla\cdot(% \mathbf{A}\mathbf{B}^{\mathrm{T}})\\ &\displaystyle=\nabla\cdot(\mathbf{B}\mathbf{A}^{\mathrm{T}}-\mathbf{A}\mathbf% {B}^{\mathrm{T}})\end{aligned}
  40. ϕ \ \phi
  41. × ( ϕ ) = 𝟎 \nabla\times(\nabla\phi)=\mathbf{0}
  42. ( × 𝐀 ) = 0 \nabla\cdot(\nabla\times\mathbf{A})=0
  43. 2 ψ = ( ψ ) \nabla^{2}\psi=\nabla\cdot(\nabla\psi)
  44. × ( × 𝐀 ) = ( 𝐀 ) - 2 𝐀 \nabla\times\left(\nabla\times\mathbf{A}\right)=\nabla(\nabla\cdot\mathbf{A})-% \nabla^{2}\mathbf{A}
  45. 𝐀 + 𝐁 = 𝐁 + 𝐀 \mathbf{A}+\mathbf{B}=\mathbf{B}+\mathbf{A}
  46. 𝐀 𝐁 = 𝐁 𝐀 \mathbf{A}\cdot\mathbf{B}=\mathbf{B}\cdot\mathbf{A}
  47. 𝐀 × 𝐁 = - 𝐁 × 𝐀 \mathbf{A}\times\mathbf{B}=\mathbf{-B}\times\mathbf{A}
  48. ( 𝐀 + 𝐁 ) 𝐂 = 𝐀 𝐂 + 𝐁 𝐂 \left(\mathbf{A}+\mathbf{B}\right)\cdot\mathbf{C}=\mathbf{A}\cdot\mathbf{C}+% \mathbf{B}\cdot\mathbf{C}
  49. ( 𝐀 + 𝐁 ) × 𝐂 = 𝐀 × 𝐂 + 𝐁 × 𝐂 \left(\mathbf{A}+\mathbf{B}\right)\times\mathbf{C}=\mathbf{A}\times\mathbf{C}+% \mathbf{B}\times\mathbf{C}
  50. 𝐀 ( 𝐁 × 𝐂 ) = 𝐁 ( 𝐂 × 𝐀 ) = 𝐂 ( 𝐀 × 𝐁 ) \mathbf{A}\cdot\left(\mathbf{B}\times\mathbf{C}\right)=\mathbf{B}\cdot\left(% \mathbf{C}\times\mathbf{A}\right)=\mathbf{C}\cdot\left(\mathbf{A}\times\mathbf% {B}\right)
  51. 𝐀 × ( 𝐁 × 𝐂 ) = ( 𝐀 𝐂 ) 𝐁 - ( 𝐀 𝐁 ) 𝐂 \mathbf{A}\times\left(\mathbf{B}\times\mathbf{C}\right)=\left(\mathbf{A}\cdot% \mathbf{C}\right)\mathbf{B}-\left(\mathbf{A}\cdot\mathbf{B}\right)\mathbf{C}
  52. ( 𝐀 × 𝐁 ) × 𝐂 = ( 𝐀 𝐂 ) 𝐁 - ( 𝐁 𝐂 ) 𝐀 \left(\mathbf{A}\times\mathbf{B}\right)\times\mathbf{C}=\left(\mathbf{A}\cdot% \mathbf{C}\right)\mathbf{B}-\left(\mathbf{B}\cdot\mathbf{C}\right)\mathbf{A}
  53. ( 𝐀 × 𝐁 ) ( 𝐂 × 𝐃 ) = ( 𝐀 𝐂 ) ( 𝐁 𝐃 ) - ( 𝐁 𝐂 ) ( 𝐀 𝐃 ) \left(\mathbf{A}\times\mathbf{B}\right)\cdot\left(\mathbf{C}\times\mathbf{D}% \right)=\left(\mathbf{A}\cdot\mathbf{C}\right)\left(\mathbf{B}\cdot\mathbf{D}% \right)-\left(\mathbf{B}\cdot\mathbf{C}\right)\left(\mathbf{A}\cdot\mathbf{D}\right)
  54. ( 𝐀 ( 𝐁 × 𝐂 ) ) 𝐃 = ( 𝐀 𝐃 ) ( 𝐁 × 𝐂 ) + ( 𝐁 𝐃 ) ( 𝐂 × 𝐀 ) + ( 𝐂 𝐃 ) ( 𝐀 × 𝐁 ) \left(\mathbf{A}\cdot\left(\mathbf{B}\times\mathbf{C}\right)\right)\mathbf{D}=% \left(\mathbf{A}\cdot\mathbf{D}\right)\left(\mathbf{B}\times\mathbf{C}\right)+% \left(\mathbf{B}\cdot\mathbf{D}\right)\left(\mathbf{C}\times\mathbf{A}\right)+% \left(\mathbf{C}\cdot\mathbf{D}\right)\left(\mathbf{A}\times\mathbf{B}\right)
  55. ( 𝐀 × 𝐁 ) × ( 𝐂 × 𝐃 ) = ( 𝐀 ( 𝐁 × 𝐃 ) ) 𝐂 - ( 𝐀 ( 𝐁 × 𝐂 ) ) 𝐃 \left(\mathbf{A}\times\mathbf{B}\right)\times\left(\mathbf{C}\times\mathbf{D}% \right)=\left(\mathbf{A}\cdot\left(\mathbf{B}\times\mathbf{D}\right)\right)% \mathbf{C}-\left(\mathbf{A}\cdot\left(\mathbf{B}\times\mathbf{C}\right)\right)% \mathbf{D}
  56. ( ψ + ϕ ) = ψ + ϕ \nabla(\psi+\phi)=\nabla\psi+\nabla\phi
  57. ( ψ ϕ ) = ϕ ψ + ψ ϕ \nabla(\psi\,\phi)=\phi\,\nabla\psi+\psi\,\nabla\phi
  58. ( 𝐀 𝐁 ) = ( 𝐀 ) 𝐁 + ( 𝐁 ) 𝐀 + 𝐀 × ( × 𝐁 ) + 𝐁 × ( × 𝐀 ) \nabla\left(\mathbf{A}\cdot\mathbf{B}\right)=\left(\mathbf{A}\cdot\nabla\right% )\mathbf{B}+\left(\mathbf{B}\cdot\nabla\right)\mathbf{A}+\mathbf{A}\times\left% (\nabla\times\mathbf{B}\right)+\mathbf{B}\times\left(\nabla\times\mathbf{A}\right)
  59. ( 𝐀 + 𝐁 ) = 𝐀 + 𝐁 \nabla\cdot(\mathbf{A}+\mathbf{B})=\nabla\cdot\mathbf{A}+\nabla\cdot\mathbf{B}
  60. ( ψ 𝐀 ) = ψ 𝐀 + 𝐀 ψ \nabla\cdot\left(\psi\mathbf{A}\right)=\psi\nabla\cdot\mathbf{A}+\mathbf{A}% \cdot\nabla\psi
  61. ( 𝐀 × 𝐁 ) = 𝐁 ( × 𝐀 ) - 𝐀 ( × 𝐁 ) \nabla\cdot\left(\mathbf{A}\times\mathbf{B}\right)=\mathbf{B}\cdot(\nabla% \times\mathbf{A})-\mathbf{A}\cdot(\nabla\times\mathbf{B})
  62. × ( 𝐀 + 𝐁 ) = × 𝐀 + × 𝐁 \nabla\times(\mathbf{A}+\mathbf{B})=\nabla\times\mathbf{A}+\nabla\times\mathbf% {B}
  63. × ( ψ 𝐀 ) = ψ × 𝐀 + ψ × 𝐀 \nabla\times\left(\psi\mathbf{A}\right)=\psi\nabla\times\mathbf{A}+\nabla\psi% \times\mathbf{A}
  64. × ( 𝐀 × 𝐁 ) = 𝐀 ( 𝐁 ) - 𝐁 ( 𝐀 ) + ( 𝐁 ) 𝐀 - ( 𝐀 ) 𝐁 \nabla\times\left(\mathbf{A}\times\mathbf{B}\right)=\mathbf{A}\left(\nabla% \cdot\mathbf{B}\right)-\mathbf{B}\left(\nabla\cdot\mathbf{A}\right)+\left(% \mathbf{B}\cdot\nabla\right)\mathbf{A}-\left(\mathbf{A}\cdot\nabla\right)% \mathbf{B}
  65. ( × 𝐀 ) = 0 \nabla\cdot(\nabla\times\mathbf{A})=0
  66. × ( ψ ) = 𝟎 \nabla\times(\nabla\psi)=\mathbf{0}
  67. ( ψ ) = 2 ψ \nabla\cdot(\nabla\psi)=\nabla^{2}\psi
  68. ( 𝐀 ) - × ( × 𝐀 ) = 2 𝐀 \nabla\left(\nabla\cdot\mathbf{A}\right)-\nabla\times\left(\nabla\times\mathbf% {A}\right)=\nabla^{2}\mathbf{A}
  69. ( ϕ ψ ) = ϕ 2 ψ + ϕ ψ \nabla\cdot(\phi\nabla\psi)=\phi\nabla^{2}\psi+\nabla\phi\cdot\nabla\psi
  70. ψ 2 ϕ - ϕ 2 ψ = ( ψ ϕ - ϕ ψ ) \psi\nabla^{2}\phi-\phi\nabla^{2}\psi=\nabla\cdot\left(\psi\nabla\phi-\phi% \nabla\psi\right)
  71. 2 ( ϕ ψ ) = ϕ 2 ψ + 2 ϕ ψ + ψ 2 ϕ \nabla^{2}(\phi\psi)=\phi\nabla^{2}\psi+2\nabla\phi\cdot\nabla\psi+\psi\nabla^% {2}\phi
  72. 2 ( ψ 𝐀 ) = 𝐀 2 ψ + 2 ( ψ ) 𝐀 + ψ 2 𝐀 \nabla^{2}(\psi\mathbf{A})=\mathbf{A}\nabla^{2}\psi+2(\nabla\psi\cdot\nabla)% \mathbf{A}+\psi\nabla^{2}\mathbf{A}
  73. 2 ( 𝐀 𝐁 ) = 𝐀 2 𝐁 - 𝐁 2 𝐀 + 2 ( ( 𝐁 ) 𝐀 + 𝐁 × × 𝐀 ) \nabla^{2}(\mathbf{A}\cdot\mathbf{B})=\mathbf{A}\cdot\nabla^{2}\mathbf{B}-% \mathbf{B}\cdot\nabla^{2}\mathbf{A}+2\nabla\cdot((\mathbf{B}\cdot\nabla)% \mathbf{A}+\mathbf{B}\times\nabla\times\mathbf{A})
  74. 2 ( ψ ) = ( ( ψ ) ) = ( 2 ψ ) \nabla^{2}(\nabla\psi)=\nabla(\nabla\cdot(\nabla\psi))=\nabla(\nabla^{2}\psi)
  75. 2 ( 𝐀 ) = ( ( 𝐀 ) ) = ( 2 𝐀 ) \nabla^{2}(\nabla\cdot\mathbf{A})=\nabla\cdot(\nabla(\nabla\cdot\mathbf{A}))=% \nabla\cdot(\nabla^{2}\mathbf{A})
  76. 2 ( × 𝐀 ) = - × ( × ( × 𝐀 ) ) = × ( 2 𝐀 ) \nabla^{2}(\nabla\times\mathbf{A})=-\nabla\times(\nabla\times(\nabla\times% \mathbf{A}))=\nabla\times(\nabla^{2}\mathbf{A})
  77. = V ( ψ 2 φ - φ 2 ψ ) d V \displaystyle=\iiint_{V}\left(\psi\nabla^{2}\varphi-\varphi\nabla^{2}\psi% \right)dV\,\!
  78. S 𝐀 d s y m b o l = S ( × 𝐀 ) d 𝐬 \oint_{\partial S}\mathbf{A}\cdot dsymbol{\ell}=\iint_{S}\left(\nabla\times% \mathbf{A}\right)\cdot d\mathbf{s}
  79. S ψ d s y m b o l = S ( 𝐧 ^ × ψ ) d S \oint_{\partial S}\psi dsymbol{\ell}=\iint_{S}\left(\hat{\mathbf{n}}\times% \nabla\psi\right)dS

Vector_decomposition.html

  1. x ^ \hat{x}
  2. i ^ \hat{i}
  3. y ^ \hat{y}
  4. j ^ \hat{j}

Vector_flow.html

  1. γ ˙ ( 0 ) = V . \dot{\gamma}(0)=V.
  2. \mapsto

Velocity_potential.html

  1. × 𝐮 = 0 , \nabla\times\mathbf{u}=0,
  2. 𝐮 \mathbf{u}
  3. 𝐮 \mathbf{u}
  4. Φ \Phi\;
  5. 𝐮 = Φ = Φ x 𝐢 + Φ y 𝐣 + Φ z 𝐤 . \mathbf{u}=\nabla\Phi\ =\frac{\partial\Phi}{\partial x}\mathbf{i}+\frac{% \partial\Phi}{\partial y}\mathbf{j}+\frac{\partial\Phi}{\partial z}\mathbf{k}.
  6. Φ \Phi\;
  7. 𝐮 \mathbf{u}
  8. a a\;
  9. Φ + a ( t ) \Phi+a(t)\;
  10. 𝐮 \mathbf{u}\;
  11. Ψ \Psi\;
  12. 𝐮 \mathbf{u}\;
  13. Ψ = Φ + b \Psi=\Phi+b\;
  14. b ( t ) b(t)\;
  15. × ( × 𝐮 ) \nabla\times(\nabla\times\mathbf{u})
  16. Φ \Phi\;
  17. p p\;
  18. 𝐮 \mathbf{u}\;
  19. 2 Φ - 1 c 2 2 Φ t 2 = 0 \nabla^{2}\Phi-{1\over c^{2}}{\partial^{2}\Phi\over\partial t^{2}}=0
  20. p p\;
  21. 𝐮 \mathbf{u}\;
  22. Φ \Phi\;
  23. 𝐮 \mathbf{u}\;
  24. p p\;
  25. p = - ρ t Φ p=-\rho{\partial\over\partial t}\Phi

Ventricular_hypertrophy.html

  1. 𝐅 \mathbf{F}
  2. 𝐅 e \mathbf{F}^{e}
  3. 𝐅 g \mathbf{F}^{g}
  4. 𝐅 = 𝐅 e 𝐅 g \mathbf{F}=\mathbf{F}^{e}\mathbf{F}^{g}
  5. 𝐅 g = ϑ f 𝐟 0 𝐟 0 + ϑ s 𝐬 0 𝐬 0 + ϑ n 𝐧 0 𝐧 0 \mathbf{F}^{g}=\vartheta^{f}\mathbf{f}_{0}\otimes\mathbf{f}_{0}+\vartheta^{s}% \mathbf{s}_{0}\otimes\mathbf{s}_{0}+\vartheta^{n}\mathbf{n}_{0}\otimes\mathbf{% n}_{0}
  6. 𝐟 0 , 𝐬 0 \mathbf{f}_{0},\mathbf{s}_{0}
  7. 𝐧 0 \mathbf{n}_{0}
  8. ϑ = [ ϑ f , ϑ s , ϑ n ] \mathbf{\vartheta}=[\vartheta^{f},\vartheta^{s},\vartheta^{n}]
  9. 𝐟 0 \mathbf{f}_{0}
  10. 𝐅 g = 𝐈 + [ ϑ - 1 ] 𝐟 0 𝐟 0 \mathbf{F}^{g}=\mathbf{I}+[\vartheta^{\parallel}-1]\mathbf{f}_{0}\otimes% \mathbf{f}_{0}
  11. 𝐈 \mathbf{I}
  12. 𝐅 g = 𝐈 + [ ϑ - 1 ] 𝐬 0 𝐬 0 \mathbf{F}^{g}=\mathbf{I}+[\vartheta^{\perp}-1]\mathbf{s}_{0}\otimes\mathbf{s}% _{0}
  13. 𝐬 0 \mathbf{s}_{0}
  14. ϑ \vartheta^{\parallel}
  15. ϑ \vartheta^{\perp}
  16. ϑ \vartheta^{\parallel}

Verifiable_random_function.html

  1. P K = g S K PK=g^{SK}
  2. F S K ( x ) = e ( g , g ) 1 / ( x + S K ) and p S K ( x ) = g 1 / ( x + S K ) , F_{SK}(x)=e(g,g)^{1/(x+SK)}\quad\mbox{and}~{}\quad p_{SK}(x)=g^{1/(x+SK)},
  3. F S K ( x ) F_{SK}(x)
  4. e ( g x P K , p S K ( x ) ) = e ( g , g ) e(g^{x}PK,p_{SK}(x))=e(g,g)
  5. ( g , g x , , g ( x q ) , R ) (g,g^{x},\ldots,g^{(x^{q})},R)
  6. R = e ( g , g ) 1 / x R=e(g,g)^{1/x}

Verma_module.html

  1. λ \lambda
  2. λ \lambda
  3. F F
  4. 𝔤 \mathfrak{g}
  5. F F
  6. 𝒰 ( 𝔤 ) \mathcal{U}(\mathfrak{g})
  7. 𝔟 \mathfrak{b}
  8. 𝔤 \mathfrak{g}
  9. 𝒰 ( 𝔟 ) \mathcal{U}(\mathfrak{b})
  10. 𝔥 \mathfrak{h}
  11. 𝔤 \mathfrak{g}
  12. λ 𝔥 * \lambda\in\mathfrak{h}^{*}
  13. F λ F_{\lambda}
  14. F F
  15. F F
  16. 𝔟 \mathfrak{b}
  17. 𝔥 \mathfrak{h}
  18. λ \lambda
  19. F λ F_{\lambda}
  20. 𝔟 \mathfrak{b}
  21. 𝒰 ( 𝔟 ) \mathcal{U}(\mathfrak{b})
  22. 𝒰 ( 𝔟 ) \mathcal{U}(\mathfrak{b})
  23. 𝒰 ( 𝔤 ) \mathcal{U}(\mathfrak{g})
  24. 𝒰 ( 𝔤 ) \mathcal{U}(\mathfrak{g})
  25. 𝔤 \mathfrak{g}
  26. ( 𝔤 , 𝒰 ( 𝔟 ) ) (\mathfrak{g},\mathcal{U}(\mathfrak{b}))
  27. λ \lambda
  28. M λ = 𝒰 ( 𝔤 ) 𝒰 ( 𝔟 ) F λ M_{\lambda}=\mathcal{U}(\mathfrak{g})\otimes_{\mathcal{U}(\mathfrak{b})}F_{\lambda}
  29. 𝔤 \mathfrak{g}
  30. 𝔤 \mathfrak{g}
  31. M λ M_{\lambda}
  32. 𝒰 ( 𝔤 - ) F F λ \mathcal{U}(\mathfrak{g}_{-})\otimes_{F}F_{\lambda}
  33. 𝔤 - \mathfrak{g}_{-}
  34. 𝔤 \mathfrak{g}
  35. 𝔤 \mathfrak{g}
  36. 1 1 1\otimes 1
  37. 1 1
  38. 𝒰 ( 𝔤 ) \mathcal{U}(\mathfrak{g})
  39. F F
  40. 𝔟 \mathfrak{b}
  41. F λ F_{\lambda}
  42. λ \lambda
  43. M λ M_{\lambda}
  44. M λ M_{\lambda}
  45. μ \mu
  46. M μ M_{\mu}
  47. λ - μ \lambda-\mu
  48. V V
  49. λ \lambda
  50. 𝔤 \mathfrak{g}
  51. M λ V . M_{\lambda}\to V.
  52. λ \lambda
  53. M λ . M_{\lambda}.
  54. M λ M_{\lambda}
  55. λ . \lambda.
  56. M λ M_{\lambda}
  57. λ \lambda
  58. { 0 , 1 , 2 , } \{0,1,2,\ldots\}
  59. M λ M_{\lambda}
  60. λ ~ \tilde{\lambda}
  61. λ = w λ ~ \lambda=w\cdot\tilde{\lambda}
  62. \cdot
  63. M λ M_{\lambda}
  64. λ ~ \tilde{\lambda}
  65. λ ~ + δ \tilde{\lambda}+\delta
  66. λ , μ \lambda,\mu
  67. M μ M λ M_{\mu}\rightarrow M_{\lambda}
  68. μ \mu
  69. λ \lambda
  70. W W
  71. 𝔤 \mathfrak{g}
  72. dim ( Hom ( M μ , M λ ) ) 1 \dim(\operatorname{Hom}(M_{\mu},M_{\lambda}))\leq 1
  73. μ , λ \mu,\lambda
  74. M μ M λ M_{\mu}\rightarrow M_{\lambda}
  75. M μ M_{\mu}
  76. M λ M_{\lambda}
  77. M μ M λ M_{\mu}\rightarrow M_{\lambda}
  78. μ = ν 0 ν 1 ν k = λ \mu=\nu_{0}\leq\nu_{1}\leq\ldots\leq\nu_{k}=\lambda
  79. ν i - 1 + δ = s γ i ( ν i + δ ) \nu_{i-1}+\delta=s_{\gamma_{i}}(\nu_{i}+\delta)
  80. γ i \gamma_{i}
  81. s γ i s_{\gamma_{i}}
  82. δ \delta
  83. 1 i k , ( ν i + δ ) ( H γ i ) 1\leq i\leq k,(\nu_{i}+\delta)(H_{\gamma_{i}})
  84. H γ i H_{\gamma_{i}}
  85. γ i \gamma_{i}
  86. M μ M_{\mu}
  87. M λ M_{\lambda}
  88. λ ~ \tilde{\lambda}
  89. μ = w λ ~ \mu=w^{\prime}\cdot\tilde{\lambda}
  90. λ = w λ ~ , \lambda=w\cdot\tilde{\lambda},
  91. \cdot
  92. M μ M λ M_{\mu}\to M_{\lambda}
  93. w w w\leq w^{\prime}
  94. 0 A B M λ 0\subset A\subset B\subset M_{\lambda}
  95. 𝔤 \mathfrak{g}
  96. M μ M λ M_{\mu}\to M_{\lambda}
  97. V μ , V λ V_{\mu},V_{\lambda}
  98. V μ V λ V_{\mu}\subset V_{\lambda}
  99. M μ M λ M_{\mu}\to M_{\lambda}
  100. V λ V_{\lambda}
  101. 𝔤 \mathfrak{g}
  102. M w λ M w λ M_{w^{\prime}\cdot\lambda}\to M_{w\cdot\lambda}
  103. w w w\leq w^{\prime}
  104. V λ V_{\lambda}
  105. 𝔤 \mathfrak{g}
  106. 0 w W , ( w ) = n M w λ w W , ( w ) = 2 M w λ w W , ( w ) = 1 M w λ M λ V λ 0 0\to\oplus_{w\in W,\,\,\ell(w)=n}M_{w\cdot\lambda}\to\cdots\to\oplus_{w\in W,% \,\,\ell(w)=2}M_{w\cdot\lambda}\to\oplus_{w\in W,\,\,\ell(w)=1}M_{w\cdot% \lambda}\to M_{\lambda}\to V_{\lambda}\to 0

Versor.html

  1. q = exp ( a 𝐫 ) = cos a + 𝐫 sin a , 𝐫 2 = - 1 , a [ 0 , π ] , q=\exp(a\mathbf{r})=\cos a+\mathbf{r}\sin a,\quad\mathbf{r}^{2}=-1,\quad a\in[% 0,\pi],
  2. q = exp ( a 𝐫 ) q=\exp(a\mathbf{r})
  3. v q v q - 1 v\mapsto qvq^{-1}
  4. ( π , π ] (−π, π]
  5. q = β : α = O B : O A q=\beta:\alpha=OB:OA
  6. q = γ : β = O C : O B q^{\prime}=\gamma:\beta=OC:OB
  7. q q = γ : α = O C : O A . q^{\prime}q=\gamma:\alpha=OC:OA.
  8. exp ( c 𝐫 ) exp ( a 𝐬 ) = exp ( b 𝐭 ) \exp(c\mathbf{r})\exp(a\mathbf{s})=\exp(b\mathbf{t})\!
  9. exp ( a r ) = cosh a + r sinh a \exp(ar)=\cosh a+r\sinh a
  10. r 2 = + 1. r^{2}=+1.
  11. a exp ( a r ) a\mapsto\exp(ar)

Vietoris–Begle_mapping_theorem.html

  1. X X
  2. Y Y
  3. f : X Y f:X\to Y
  4. f f
  5. H ~ r ( f - 1 ( y ) ) = 0 , \tilde{H}_{r}(f^{-1}(y))=0,
  6. 0 r n - 1 0\leq r\leq n-1
  7. y Y y\in Y
  8. H ~ r \tilde{H}_{r}
  9. r r
  10. f * : H ~ r ( X ) H ~ r ( Y ) f_{*}:\tilde{H}_{r}(X)\to\tilde{H}_{r}(Y)
  11. r n - 1 r\leq n-1
  12. r = n r=n

Virtual_displacement.html

  1. δ 𝐫 i \delta\mathbf{r}_{i}\,
  2. t t\,
  3. ϵ \epsilon\,
  4. δ \delta\,
  5. ϵ | ϵ = 0 \textstyle{\partial\over{\partial\epsilon}}\big|_{\epsilon=0}\,
  6. 𝐫 i \mathbf{r}_{i}\,
  7. { q 1 , q 2 , , q m } \{q_{1},q_{2},...,q_{m}\}\,
  8. t t\,
  9. d 𝐫 i = 𝐫 i t d t + j = 1 m 𝐫 i q j d q j d\mathbf{r}_{i}=\frac{\partial\mathbf{r}_{i}}{\partial t}dt+\sum_{j=1}^{m}% \frac{\partial\mathbf{r}_{i}}{\partial q_{j}}dq_{j}\,
  10. δ 𝐫 i = j = 1 m 𝐫 i q j δ q j \delta\mathbf{r}_{i}=\sum_{j=1}^{m}\frac{\partial\mathbf{r}_{i}}{\partial q_{j% }}\delta q_{j}\,
  11. q j q_{j}\,
  12. δ W \delta W\,
  13. Q j Q_{j}\,
  14. d 𝐫 d\mathbf{r}\,
  15. δ 𝐫 \delta\mathbf{r}\,
  16. θ \theta\,
  17. z z\,
  18. z + d z z+dz\,
  19. θ \theta\,
  20. θ + δ θ \theta+\delta\theta\,
  21. δ θ \delta\theta\,
  22. δ t = 0 \delta t=0\,

Virtual_retinal_display.html

  1. angular resolution = 1.22 λ D \mathrm{angular\ resolution}=\frac{1.22\lambda}{D}

Viscosity_index.html

  1. V = 100 < m t p l > ( L - U ) ( L - H ) V=100\cfrac{<}{m}tpl>{{(L-U)}}{{(L-H)}}

Viscosity_solution.html

  1. F ( x , u , D u , D 2 u ) = 0 F(x,u,Du,D^{2}u)=0
  2. x Ω x\in\Omega
  3. x x
  4. u u
  5. D u Du
  6. D 2 u D^{2}u
  7. D u Du
  8. D 2 u D^{2}u
  9. F ( x , u , D u , D 2 u ) = 0 F(x,u,Du,D^{2}u)=0
  10. Ω \Omega
  11. X X
  12. Y Y
  13. Y - X Y-X
  14. x Ω x\in\Omega
  15. u u\in\mathbb{R}
  16. p n p\in\mathbb{R}^{n}
  17. F ( x , u , p , X ) F ( x , u , p , Y ) F(x,u,p,X)\geq F(x,u,p,Y)
  18. - Δ u = 0 -\Delta u=0
  19. u u
  20. Ω \Omega
  21. x 0 Ω x_{0}\in\Omega
  22. C 2 C^{2}
  23. ϕ \phi
  24. ϕ ( x 0 ) = u ( x 0 ) \phi(x_{0})=u(x_{0})
  25. ϕ u \phi\geq u
  26. x 0 x_{0}
  27. F ( x 0 , ϕ ( x 0 ) , D ϕ ( x 0 ) , D 2 ϕ ( x 0 ) ) 0 F(x_{0},\phi(x_{0}),D\phi(x_{0}),D^{2}\phi(x_{0}))\leq 0
  28. u u
  29. Ω \Omega
  30. x 0 Ω x_{0}\in\Omega
  31. C 2 C^{2}
  32. ϕ \phi
  33. ϕ ( x 0 ) = u ( x 0 ) \phi(x_{0})=u(x_{0})
  34. ϕ u \phi\leq u
  35. x 0 x_{0}
  36. F ( x 0 , ϕ ( x 0 ) , D ϕ ( x 0 ) , D 2 ϕ ( x 0 ) ) 0 F(x_{0},\phi(x_{0}),D\phi(x_{0}),D^{2}\phi(x_{0}))\geq 0
  37. u + H ( x , u ) = 0 u+H(x,\nabla u)=0
  38. F ( D 2 u , D u , u ) = 0 F(D^{2}u,Du,u)=0
  39. F F
  40. r s r\leq s
  41. X Y X\geq Y
  42. F ( Y , p , s ) F ( X , p , r ) + λ || X - Y || F(Y,p,s)\geq F(X,p,r)+\lambda||X-Y||
  43. λ > 0 \lambda>0
  44. L L^{\infty}
  45. L L^{\infty}

Visual_angle.html

  1. S S
  2. D D
  3. O O
  4. O O
  5. A A
  6. A A
  7. a a
  8. B B
  9. b b
  10. V V
  11. A A
  12. B B
  13. V V
  14. O O
  15. V = 2 arctan ( S 2 D ) V=2\arctan\left(\frac{S}{2D}\right)
  16. tan ( V ) = S D . \tan\left(V\right)=\frac{S}{D}.
  17. a a
  18. b b
  19. R R
  20. R n = tan V , \frac{R}{n}=\tan V,
  21. n n
  22. R 0.17 mm R\approx 0.17\,\text{ mm}
  23. 0.15 mm 0.15\,\text{ mm}
  24. S = 2160 miles S=2160\,\text{ miles}
  25. D D
  26. 238 , 000 miles 238,000\,\text{ miles}
  27. V 0.009 rad V\approx 0.009\,\text{ rad}
  28. 0.52 deg \approx 0.52\,\text{ deg}
  29. S S
  30. V V
  31. V V
  32. V V^{\prime}
  33. V V
  34. S S^{\prime}
  35. S S
  36. R R
  37. R R
  38. V V
  39. R R
  40. V V^{\prime}

Viviani's_curve.html

  1. 2 a 2a
  2. x 2 + y 2 + z 2 = 4 a 2 x^{2}+y^{2}+z^{2}=4a^{2}\,
  3. ( a , 0 , 0 ) (a,0,0)
  4. a a
  5. ( x - a ) 2 + y 2 = a 2 . (x-a)^{2}+y^{2}=a^{2}.\,
  6. V V
  7. t t
  8. V ( t ) = a ( 1 + cos ( t ) ) , a sin ( t ) , 2 a sin ( t 2 ) . V(t)=\left\langle a(1+\cos(t)),a\sin(t),2a\sin\left(\frac{t}{2}\right)\right\rangle.
  9. m = 1 m=1
  10. θ = t - π 2 \theta=\frac{t-\pi}{2}

Volatility_arbitrage.html

  1. C = f ( S , σ , ) C=f(S,\sigma,\cdot)\,
  2. S S\,
  3. σ \sigma\,
  4. f ( ) f()\,
  5. σ \sigma\,
  6. g ( ) g()\,
  7. C ¯ \bar{C}\,
  8. σ C ¯ = g ( S , C ¯ , ) \sigma_{\bar{C}}=g(S,\bar{C},\cdot)\,
  9. S S\,
  10. σ C ¯ \sigma_{\bar{C}}\,
  11. C ¯ \bar{C}\,
  12. σ C ¯ \sigma_{\bar{C}}\,
  13. σ C ¯ \sigma_{\bar{C}}\,
  14. σ \sigma\,

Voltage_drop.html

  1. V = I R V=IR
  2. E = I Z E=IZ

Volterra_integral_equation.html

  1. f ( t ) = a t K ( t , s ) x ( s ) d s f(t)=\int_{a}^{t}K(t,s)\,x(s)\,ds
  2. x ( t ) = f ( t ) + a t K ( t , s ) x ( s ) d s . x(t)=f(t)+\int_{a}^{t}K(t,s)x(s)\,ds.
  3. x ( t ) = f ( t ) + t 0 t K ( t - s ) x ( s ) d s . x(t)=f(t)+\int_{t_{0}}^{t}K(t-s)x(s)\,ds.
  4. K K

Von_Neumann_entropy.html

  1. ρ ρ
  2. S = - tr ( ρ ln ρ ) , S=-\mathrm{tr}(\rho\ln\rho),
  3. ρ ρ
  4. ρ = j η j | j j | , \rho=\sum_{j}\eta_{j}\left|j\right\rangle\left\langle j\right|~{},
  5. S = - j η j ln η j . S=-\sum_{j}\eta_{j}\ln\eta_{j}.
  6. | Ψ = i a i | ψ i . \left|\Psi\right\rangle\,=\,\sum_{i}a_{i}\,\left|\psi_{i}\right\rangle.
  7. B = i , j a i * a j i | B | j . \left\langle B\right\rangle\,=\,\sum_{i,j}a_{i}^{*}a_{j}\,\left\langle i\right% |B\left|j\right\rangle.
  8. | a i | 2 \left|a_{i}\right|^{2}
  9. j | ρ | i = a j a i * . \left\langle j\right|\,\rho\,\left|i\right\rangle\,=\,a_{j}\,a_{i}^{*}.
  10. B = tr ( ρ B ) . \left\langle B\right\rangle\,=\,\mathrm{tr}(\rho\,B)~{}.
  11. S ( ρ ) = - tr ( ρ ln ρ ) , S(\rho)\,=\,-\mathrm{tr}(\rho\,{\rm\ln}\rho),
  12. ρ = j η j | j j | ~{}\rho=\sum_{j}\eta_{j}\left|j\right\rangle\left\langle j\right|
  13. S ( ρ ) = - j η j ln η j . S(\rho)\,=\,-\sum_{j}\eta_{j}\ln\eta_{j}~{}.
  14. Ψ = ( | 0 + | 1 ) / 2 \Psi=(\left|0\right\rangle+\left|1\right\rangle)/\sqrt{2}
  15. ρ = 1 2 ( 1 1 1 1 ) \rho={1\over 2}\begin{pmatrix}1&1\\ 1&1\end{pmatrix}
  16. ρ = 1 2 ( 1 0 0 1 ) \rho={1\over 2}\begin{pmatrix}1&0\\ 0&1\end{pmatrix}
  17. S ( ρ ) S(ρ)
  18. ρ ρ
  19. S ( ρ ) S(ρ)
  20. l n N lnN
  21. N N
  22. S ( ρ ) S(ρ)
  23. ρ ρ
  24. U U
  25. S ( ρ ) S(ρ)
  26. Σ i λ i = 1 \Sigma_{i}\lambda_{i}=1
  27. S ( i = 1 k λ i ρ i ) i = 1 k λ i S ( ρ i ) . S\bigg(\sum_{i=1}^{k}\lambda_{i}\,\rho_{i}\bigg)\,\geq\,\sum_{i=1}^{k}\lambda_% {i}\,S(\rho_{i}).
  28. S ( ρ ) S(ρ)
  29. S ( ρ A ρ B ) = S ( ρ A ) + S ( ρ B ) S(\rho_{A}\otimes\rho_{B})=S(\rho_{A})+S(\rho_{B})
  30. S ( ρ ) S(ρ)
  31. S ( ρ A B C ) + S ( ρ B ) S ( ρ A B ) + S ( ρ B C ) . S(\rho_{ABC})+S(\rho_{B})\leq S(\rho_{AB})+S(\rho_{BC}).
  32. S ( ρ ) S(ρ)
  33. S ( ρ A C ) S ( ρ A ) + S ( ρ C ) . S(\rho_{AC})\leq S(\rho_{A})+S(\rho_{C}).
  34. | S ( ρ A ) - S ( ρ B ) | S ( ρ A B ) S ( ρ A ) + S ( ρ B ) . \left|S(\rho_{A})\,-\,S(\rho_{B})\right|\,\leq\,S(\rho_{AB})\,\leq\,S(\rho_{A}% )\,+\,S(\rho_{B})~{}.
  35. | ψ = | + | , \left|\psi\right\rangle=\left|\uparrow\downarrow\right\rangle+\left|\downarrow% \uparrow\right\rangle,
  36. A A
  37. B B
  38. S ( ρ A B C ) + S ( ρ B ) S ( ρ A B ) + S ( ρ B C ) . S(\rho_{ABC})\,+\,S(\rho_{B})\,\leq\,S(\rho_{AB})\,+\,S(\rho_{BC}).
  39. S ( ρ A ) + S ( ρ C ) S ( ρ A B ) + S ( ρ B C ) S(\rho_{A})\,+\,S(\rho_{C})\,\leq\,S(\rho_{AB})\,+\,S(\rho_{BC})
  40. ρ < s u b > A B ρ<sub>AB

W_Ursae_Majoris_variable.html

  1. q = 0.72 q=0.72
  2. q q

Wandering_set.html

  1. f : X X f:X\to X
  2. x X x\in X
  3. n > N n>N
  4. f n ( U ) U = . f^{n}(U)\cap U=\varnothing.\,
  5. ( X , Σ , μ ) (X,\Sigma,\mu)
  6. Σ \Sigma
  7. μ \mu
  8. μ ( f n ( U ) U ) = 0. \mu\left(f^{n}(U)\cap U\right)=0.\,
  9. φ t : X X \varphi_{t}:X\to X
  10. φ \varphi
  11. φ t + s = φ t φ s . \varphi_{t+s}=\varphi_{t}\circ\varphi_{s}.\,
  12. x X x\in X
  13. t > T t>T
  14. μ ( φ t ( U ) U ) = 0. \mu\left(\varphi_{t}(U)\cap U\right)=0.\,
  15. Ω = ( X , Σ , μ ) \Omega=(X,\Sigma,\mu)
  16. Γ \Gamma
  17. x Ω x\in\Omega
  18. { γ x : γ Γ } \{\gamma\cdot x:\gamma\in\Gamma\}
  19. x Ω x\in\Omega
  20. Γ \Gamma
  21. μ ( γ U U ) = 0 \mu\left(\gamma\cdot U\cap U\right)=0
  22. γ Γ - V \gamma\in\Gamma-V
  23. x X x\in X
  24. μ ( f n ( U ) U ) > 0. \mu\left(f^{n}(U)\cap U\right)>0.
  25. Ω \Omega
  26. Γ \Gamma
  27. γ Γ - { e } \gamma\in\Gamma-\{e\}
  28. γ W W \gamma W\cap W\,
  29. Γ \Gamma
  30. ( Ω , Γ ) (\Omega,\Gamma)
  31. W * = γ Γ γ W . W^{*}=\cup_{\gamma\in\Gamma}\;\;\gamma W.
  32. Γ \Gamma
  33. W * W^{*}
  34. Ω \Omega
  35. Ω - W * \Omega-W^{*}\,

War_of_attrition_(game).html

  1. v i > 0 v_{i}>0
  2. v i / 2 v_{i}/2
  3. p ( t ) = 1 V e ( - t / V ) p(t)=\frac{1}{V}e^{(-t/V)}
  4. δ \delta

Water_(data_page).html

  1. p K w = - log ( [ H + ] [ OH - ] ) pK_{w}=-\log([\mathrm{H}^{+}][\mathrm{OH}^{-}])

Watt's_curve.html

  1. r 2 = b 2 - [ a sin θ ± c 2 - a 2 cos 2 θ ] 2 . r^{2}=b^{2}-\left[a\sin\theta\pm\sqrt{c^{2}-a^{2}\cos^{2}\theta}\right]^{2}.
  2. a + b e i ρ = r e i θ + c e i ψ . a+be^{i\rho}=re^{i\theta}+ce^{i\psi}.\,
  3. - a + b e i λ = r e i θ - c e i ψ -a+be^{i\lambda}=re^{i\theta}-ce^{i\psi}\,
  4. r e i θ = b 2 ( e i ρ + e i λ ) = b cos ( ρ - λ 2 ) e i ρ + λ 2 . re^{i\theta}=\tfrac{b}{2}(e^{i\rho}+e^{i\lambda})=b\cos(\tfrac{\rho-\lambda}{2% })e^{i\tfrac{\rho+\lambda}{2}}.
  5. r = b cos α , θ = ρ + λ 2 where α = ρ - λ 2 . r=b\cos\alpha,\ \theta=\tfrac{\rho+\lambda}{2}\ \mbox{where}~{}\ \alpha=\tfrac% {\rho-\lambda}{2}.
  6. c e i ψ - a = b 2 ( e i ρ - e i λ ) = i b sin α e i θ . ce^{i\psi}-a=\tfrac{b}{2}(e^{i\rho}-e^{i\lambda})=ib\sin\alpha e^{i\theta}.
  7. a = a cos θ e i θ - i a sin θ e i θ . a=a\cos\theta\ e^{i\theta}-ia\sin\theta\ e^{i\theta}.\,
  8. c e i ψ = i b sin α e i θ + a cos θ e i θ - i a sin θ e i θ = ( a cos θ + i ( b sin α - a sin θ ) ) e i θ , ce^{i\psi}=ib\sin\alpha e^{i\theta}+a\cos\theta\ e^{i\theta}-ia\sin\theta\ e^{% i\theta}=(a\cos\theta\ +i(b\sin\alpha-a\sin\theta))e^{i\theta},
  9. c 2 = a 2 cos 2 θ + ( b sin α - a sin θ ) 2 , c^{2}=a^{2}\cos^{2}\theta+(b\sin\alpha-a\sin\theta)^{2},\,
  10. b sin α = a sin θ ± c 2 - a 2 cos 2 θ , b\sin\alpha=a\sin\theta\pm\sqrt{c^{2}-a^{2}\cos^{2}\theta},\,
  11. r 2 = b 2 cos 2 α = b 2 - b 2 sin 2 α = b 2 - [ a sin θ ± c 2 - a 2 cos 2 θ ] 2 . , r^{2}=b^{2}\cos^{2}\alpha=b^{2}-b^{2}\sin^{2}\alpha=b^{2}-\left[a\sin\theta\pm% \sqrt{c^{2}-a^{2}\cos^{2}\theta}\right]^{2}.,\,
  12. r 2 = b 2 - ( a 2 sin 2 θ + c 2 - a 2 cos 2 θ ± 2 a sin θ c 2 - a 2 cos 2 θ ) , r^{2}=b^{2}-(a^{2}\sin^{2}\theta\ +c^{2}-a^{2}\cos^{2}\theta\pm 2a\sin\theta% \sqrt{c^{2}-a^{2}\cos^{2}\theta}),\,
  13. r 2 - a 2 - b 2 + c 2 + 2 a 2 sin 2 θ = ± 2 a sin θ c 2 - a 2 cos 2 θ ) , r^{2}-a^{2}-b^{2}+c^{2}+2a^{2}\sin^{2}\theta=\pm 2a\sin\theta\sqrt{c^{2}-a^{2}% \cos^{2}\theta}),\,
  14. ( r 2 - a 2 - b 2 + c 2 ) 2 + 4 a 2 ( r 2 - a 2 - b 2 + c 2 ) sin 2 θ + 4 a 4 sin 4 θ = 4 a 2 sin 2 θ ( c 2 - a 2 cos 2 θ ) , (r^{2}-a^{2}-b^{2}+c^{2})^{2}+4a^{2}(r^{2}-a^{2}-b^{2}+c^{2})\sin^{2}\theta+4a% ^{4}\sin^{4}\theta=4a^{2}\sin^{2}\theta(c^{2}-a^{2}\cos^{2}\theta),\,
  15. ( r 2 - a 2 - b 2 + c 2 ) 2 + 4 a 2 ( r 2 - b 2 ) sin 2 θ = 0 , (r^{2}-a^{2}-b^{2}+c^{2})^{2}+4a^{2}(r^{2}-b^{2})\sin^{2}\theta=0,\,
  16. ( x 2 + y 2 ) ( x 2 + y 2 - a 2 - b 2 + c 2 ) 2 + 4 a 2 y 2 ( x 2 + y 2 - b 2 ) = 0. (x^{2}+y^{2})(x^{2}+y^{2}-a^{2}-b^{2}+c^{2})^{2}+4a^{2}y^{2}(x^{2}+y^{2}-b^{2}% )=0.\,
  17. ( x 2 + y 2 ) ( x 2 + y 2 - d 2 ) 2 + 4 a 2 y 2 ( x 2 + y 2 - b 2 ) = 0. (x^{2}+y^{2})(x^{2}+y^{2}-d^{2})^{2}+4a^{2}y^{2}(x^{2}+y^{2}-b^{2})=0.\,

Weak_convergence_(Hilbert_space).html

  1. x n x_{n}
  2. x n x_{n}
  3. x lim inf n x n , \|x\|\leq\liminf_{n\to\infty}\|x_{n}\|,
  4. x n x_{n}
  5. x x
  6. x n x \lVert x_{n}\rVert\to\lVert x\rVert
  7. x n x_{n}
  8. x x
  9. x - x n , x - x n = x , x + x n , x n - x n , x - x , x n 0. \langle x-x_{n},x-x_{n}\rangle=\langle x,x\rangle+\langle x_{n},x_{n}\rangle-% \langle x_{n},x\rangle-\langle x,x_{n}\rangle\rightarrow 0.
  10. f n ( x ) = sin ( n x ) f_{n}(x)=\sin(nx)
  11. [ 0 , 2 π ] [0,2\pi]
  12. n n\rightarrow\infty
  13. f n f_{n}
  14. f = 0 f=0
  15. L 2 [ 0 , 2 π ] L^{2}[0,2\pi]
  16. [ 0 , 2 π ] [0,2\pi]
  17. f , g = 0 2 π f ( x ) g ( x ) d x , \langle f,g\rangle=\int_{0}^{2\pi}f(x)\cdot g(x)\,dx,
  18. f 1 , f 2 , f_{1},f_{2},\ldots
  19. f n ( x ) = sin ( n x ) f_{n}(x)=\sin(nx)
  20. L 2 [ 0 , 2 π ] L^{2}[0,2\pi]
  21. 0 2 π sin ( n x ) g ( x ) d x . \int_{0}^{2\pi}\sin(nx)\cdot g(x)\,dx.
  22. g g
  23. [ 0 , 2 π ] [0,2\pi]
  24. n n
  25. f n , g 0 , g = 0. \langle f_{n},g\rangle\to\langle 0,g\rangle=0.
  26. f n f_{n}
  27. [ 0 , 2 π ] [0,2\pi]
  28. n n
  29. n n
  30. f n f_{n}
  31. L L_{\infty}
  32. L 2 L_{2}
  33. e n e_{n}
  34. e n , e m = δ m n \langle e_{n},e_{m}\rangle=\delta_{mn}
  35. δ m n \delta_{mn}
  36. n | e n , x | 2 x 2 \sum_{n}|\langle e_{n},x\rangle|^{2}\leq\|x\|^{2}
  37. | e n , x | 2 0 |\langle e_{n},x\rangle|^{2}\rightarrow 0
  38. e n , x 0. \langle e_{n},x\rangle\rightarrow 0.
  39. x n x_{n}
  40. x n k x_{n_{k}}
  41. 1 N k = 1 N x n k \frac{1}{N}\sum_{k=1}^{N}x_{n_{k}}
  42. ( x n ) (x_{n})
  43. f ( x n ) f ( x ) f(x_{n})\to f(x)
  44. f f
  45. B B
  46. f f
  47. B . B^{\prime}.
  48. B B
  49. f f
  50. f ( ) = , y f(\cdot)=\langle\cdot,y\rangle
  51. y y
  52. B B

Wedge_(geometry).html

  1. V = b h ( a 3 + c 6 ) , V=bh\left(\frac{a}{3}+\frac{c}{6}\right),

Weierstrass_point.html

  1. P P
  2. C C
  3. C C
  4. P P
  5. L ( 0 ) , L ( P ) , L ( 2 P ) , L ( 3 P ) , L(0),\ L(P),\ L(2P),\ L(3P),\ldots
  6. L ( k P ) L(kP)
  7. C C
  8. P P
  9. k k
  10. C C
  11. g g
  12. C C
  13. k k
  14. l ( k P ) = k - g + 1 l(kP)=k-g+1
  15. k ; 2 g - 1 k\geq;2g-1
  16. f f
  17. g g
  18. P P
  19. f + c g f+cg
  20. c c
  21. 2 g - 2 2g-2
  22. g = 0 g=0
  23. 1 1
  24. g 2 g\geq 2
  25. g - 1 g-1
  26. g - 1 g-1
  27. C C
  28. P P
  29. k k
  30. C C
  31. k k
  32. P P
  33. 1 , 2 , , g 1,\ 2,\ \ldots,\ g
  34. g g
  35. F F
  36. P P
  37. 4 , 6 4,\ 6
  38. P P
  39. a , b , c , a,\ b,\ c,\ \ldots
  40. ( a - 1 ) + ( b - 2 ) + ( c - 3 ) + (a-1)+(b-2)+(c-3)+\ldots
  41. g ( g 2 - 1 ) g(g^{2}-1)
  42. C C
  43. C C

Weierstrass–Enneper_parameterization.html

  1. x k ( ζ ) \displaystyle x_{k}(\zeta)

Weil_cohomology_theory.html

  1. H i ( X ) H dR 2 n - i ( X ) 𝐂 H_{i}(X)\otimes H_{\,\text{dR}}^{2n-i}(X)\rightarrow\mathbf{C}
  2. H i ( X ) H dR 2 n - i ( X ) H i ( X ) . H_{i}(X)\cong H_{\,\text{dR}}^{2n-i}(X)^{\vee}\cong H^{i}(X).

Weinberg–Witten_theorem.html

  1. J μ J^{\mu}
  2. T μ ν T^{\mu\nu}
  3. d 3 x J 0 \int d^{3}x\,J^{0}
  4. J μ J^{\mu}
  5. | p |p\rangle
  6. | p |p^{\prime}\rangle
  7. ( p - p ) (p-p^{\prime})
  8. q δ 3 ( p - p ) = p | Q | p \displaystyle q\delta^{3}(\vec{p^{\prime}}-\vec{p})=\langle p^{\prime}|Q|p\rangle
  9. p | J 0 ( 0 ) | p = q ( 2 π ) 3 \langle p^{\prime}|J^{0}(0)|p\rangle=\frac{q}{(2\pi)^{3}}
  10. q 0 q\neq 0
  11. p | J 0 ( 0 ) | p \langle p^{\prime}|J^{0}(0)|p\rangle
  12. p | J 3 ( 0 ) | p \langle p^{\prime}|J^{3}(0)|p\rangle
  13. e i ( h - ( - h ) ) θ = e 2 i h θ e^{i(h-(-h))\theta}=e^{2ih\theta}
  14. p | J 1 ( 0 ) + i J 2 ( 0 ) | p \langle p^{\prime}|J^{1}(0)+iJ^{2}(0)|p\rangle
  15. p | J 1 ( 0 ) - i J 2 ( 0 ) | p \langle p^{\prime}|J^{1}(0)-iJ^{2}(0)|p\rangle
  16. e i ( 2 h + 1 ) θ e^{i(2h+1)\theta}
  17. e i ( 2 h - 1 ) θ e^{i(2h-1)\theta}
  18. p | J 0 ( 0 ) | p = lim p p p | J 0 ( 0 ) | p \langle p|J^{0}(0)|p\rangle=\lim_{p^{\prime}\rightarrow p}\langle p^{\prime}|J% ^{0}(0)|p\rangle
  19. δ 3 ( p - p ) \delta^{3}(\vec{p^{\prime}}-\vec{p})
  20. d 3 x d^{3}x
  21. T μ ν T^{\mu\nu}
  22. p μ = d 3 x T μ 0 ( x , 0 ) p^{\mu}=\int d^{3}x\,T^{\mu 0}(\vec{x},0)
  23. p | T 00 ( 0 ) | p = E ( 2 π ) 3 \langle p|T^{00}(0)|p\rangle=\frac{E}{(2\pi)^{3}}
  24. E 0 E\neq 0
  25. p | 𝐓 ( 0 ) | p \langle p^{\prime}|\mathbf{T}(0)|p\rangle
  26. e i ( 2 h - 2 ) θ e^{i(2h-2)\theta}
  27. e i ( 2 h - 1 ) θ e^{i(2h-1)\theta}
  28. e i ( 2 h ) θ e^{i(2h)\theta}
  29. e i ( 2 h + 1 ) θ e^{i(2h+1)\theta}
  30. e i ( 2 h + 2 ) θ e^{i(2h+2)\theta}
  31. | h | = 0 , 1 2 , 1 |h|=0,\frac{1}{2},1
  32. p | J | p \langle p^{\prime}|J|p\rangle
  33. J μ ( x ) δ δ A μ ( x ) S matter J^{\mu}(x)\equiv\frac{\delta}{\delta A_{\mu}(x)}S_{\mathrm{matter}}
  34. D μ J μ = 0 D_{\mu}J^{\mu}=0
  35. μ J μ = 0 \partial_{\mu}J^{\mu}=0
  36. N f - 2 N c > 2 3 N f N_{f}-2\geq N_{c}>\frac{2}{3}N_{f}
  37. S U ( N f - N c ) SU(N_{f}-N_{c})
  38. < p | J μ ( 0 ) | p Align g t ; <p^{\prime}|J^{\mu}(0)|p&gt;
  39. p | J 0 ( 0 ) | p = lim p p p | J 0 ( 0 ) | p \langle p|J^{0}(0)|p\rangle=\lim_{p^{\prime}\rightarrow p}\langle p^{\prime}|J% ^{0}(0)|p\rangle
  40. T M N ( x ) 1 - g δ δ g M N ( x ) Γ [ background ] . T^{MN}(x)\equiv\frac{1}{\sqrt{-g}}\frac{\delta}{\delta g_{MN}(x)}\Gamma[\,% \text{background}].

Well_control.html

  1. E C D = M W + P a 0.052 * T V D ECD=MW+{\frac{P_{a}}{0.052*TVD}}

Wess–Zumino_model.html

  1. diag ( - 1 , 1 , 1 , 1 ) \mathrm{diag}(-1,1,1,1)
  2. = - 1 2 ( S ) 2 - 1 2 ( P ) 2 - 1 2 ψ ¯ / ψ \mathcal{L}=-\frac{1}{2}(\partial S)^{2}-\frac{1}{2}(\partial P)^{2}-\frac{1}{% 2}\bar{\psi}\partial\!\!\!/\psi
  3. S S
  4. P P
  5. ψ \psi
  6. δ ϵ S = ϵ ¯ ψ \delta_{\epsilon}S=\bar{\epsilon}\psi
  7. δ ϵ P = ϵ ¯ γ 5 ψ \delta_{\epsilon}P=\bar{\epsilon}\gamma_{5}\psi
  8. δ ϵ ψ = / ( S + P γ 5 ) ϵ \delta_{\epsilon}\psi=\partial\!\!\!/(S+P\gamma_{5})\epsilon
  9. ϵ \epsilon
  10. γ 5 \gamma_{5}
  11. S S
  12. P P
  13. ψ \psi

Wet_gas.html

  1. χ = m m g ρ g ρ , \chi=\frac{m_{\ell}}{m_{g}}\sqrt{\frac{\rho_{g}}{\rho_{\ell}}},
  2. m m_{\ell}
  3. m g m_{g}
  4. ρ g \rho_{g}
  5. ρ \rho_{\ell}

Wheeler–Feynman_absorber_theory.html

  1. t 0 = 0 t_{0}=0
  2. x 0 = 0 x_{0}=0
  3. x 1 x_{1}
  4. t 1 = x 1 / c t_{1}=x_{1}/c
  5. c c
  6. t 2 = x 1 / c t_{2}=x_{1}/c
  7. E tot ( 𝐱 , t ) = n E n ret ( 𝐱 , t ) + E n adv ( 𝐱 , t ) 2 . E_{\mathrm{tot}}(\mathbf{x},t)=\sum_{n}\frac{E_{n}^{\mathrm{ret}}(\mathbf{x},t% )+E_{n}^{\mathrm{adv}}(\mathbf{x},t)}{2}.
  8. E free ( 𝐱 , t ) = n E n ret ( 𝐱 , t ) - E n adv ( 𝐱 , t ) 2 = 0 E_{\mathrm{free}}(\mathbf{x},t)=\sum_{n}\frac{E_{n}^{\mathrm{ret}}(\mathbf{x},% t)-E_{n}^{\mathrm{adv}}(\mathbf{x},t)}{2}=0
  9. E free E_{\mathrm{free}}
  10. E tot ( 𝐱 , t ) = n E n ret ( 𝐱 , t ) + E n adv ( 𝐱 , t ) 2 + n E n ret ( 𝐱 , t ) - E n adv ( 𝐱 , t ) 2 = n E n ret ( 𝐱 , t ) . E_{\mathrm{tot}}(\mathbf{x},t)=\sum_{n}\frac{E_{n}^{\mathrm{ret}}(\mathbf{x},t% )+E_{n}^{\mathrm{adv}}(\mathbf{x},t)}{2}+\sum_{n}\frac{E_{n}^{\mathrm{ret}}(% \mathbf{x},t)-E_{n}^{\mathrm{adv}}(\mathbf{x},t)}{2}=\sum_{n}E_{n}^{\mathrm{% ret}}(\mathbf{x},t).
  11. F = m a F=ma
  12. E damping ( 𝐱 j , t ) = E j ret ( 𝐱 j , t ) - E j adv ( 𝐱 j , t ) 2 E^{\mathrm{damping}}(\mathbf{x}_{j},t)=\frac{E_{j}^{\mathrm{ret}}(\mathbf{x}_{% j},t)-E_{j}^{\mathrm{adv}}(\mathbf{x}_{j},t)}{2}
  13. j j
  14. x j x_{j}
  15. E tot ( 𝐱 j , t ) = n j E n ret ( 𝐱 j , t ) + E n adv ( 𝐱 j , t ) 2 . E^{\mathrm{tot}}(\mathbf{x}_{j},t)=\sum_{n\neq j}\frac{E_{n}^{\mathrm{ret}}(% \mathbf{x}_{j},t)+E_{n}^{\mathrm{adv}}(\mathbf{x}_{j},t)}{2}\ \,\text{.}
  16. E tot ( 𝐱 j , t ) = n j E n ret ( 𝐱 j , t ) + E n adv ( 𝐱 j , t ) 2 + n E n ret ( 𝐱 j , t ) - E n adv ( 𝐱 j , t ) 2 E^{\mathrm{tot}}(\mathbf{x}_{j},t)=\sum_{n\neq j}\frac{E_{n}^{\mathrm{ret}}(% \mathbf{x}_{j},t)+E_{n}^{\mathrm{adv}}(\mathbf{x}_{j},t)}{2}+\sum_{n}\frac{E_{% n}^{\mathrm{ret}}(\mathbf{x}_{j},t)-E_{n}^{\mathrm{adv}}(\mathbf{x}_{j},t)}{2}
  17. E tot ( 𝐱 j , t ) = n j E n ret ( 𝐱 j , t ) + E damping ( 𝐱 j , t ) . E^{\mathrm{tot}}(\mathbf{x}_{j},t)=\sum_{n\neq j}E_{n}^{\mathrm{ret}}(\mathbf{% x}_{j},t)+E^{\mathrm{damping}}(\mathbf{x}_{j},t).
  18. E damping ( 𝐱 j , t ) = e 6 π c 3 d 3 d t 3 x E^{\mathrm{damping}}(\mathbf{x}_{j},t)=\frac{e}{6\pi c^{3}}\frac{\mathrm{d}^{3% }}{\mathrm{d}t^{3}}x
  19. p 1 p_{1}
  20. p 1 p_{1}
  21. L 1 = T 1 - 1 2 ( ( V R ) 1 2 + ( V A ) 1 2 ) L_{1}=T_{1}-\frac{1}{2}\left((V_{R})^{2}_{1}+(V_{A})^{2}_{1}\right)
  22. T i T_{i}
  23. p i p_{i}
  24. ( V R ) i j (V_{R})^{j}_{i}
  25. ( V A ) i j (V_{A})^{j}_{i}
  26. p i p_{i}
  27. p j p_{j}
  28. p 1 p_{1}
  29. L 2 = T 2 - 1 2 ( ( V R ) 2 1 + ( V A ) 2 1 ) . L_{2}=T_{2}-\frac{1}{2}\left((V_{R})^{1}_{2}+(V_{A})^{1}_{2}\right).
  30. ( V R ) j i - ( V A ) i j (V_{R})^{i}_{j}-(V_{A})^{j}_{i}
  31. L = i = 1 N T i - 1 2 i j N ( V R ) j i L=\sum_{i=1}^{N}T_{i}-\frac{1}{2}\sum_{i\neq j}^{N}(V_{R})^{i}_{j}
  32. p i p_{i}
  33. p j p_{j}
  34. N = 2 N=2
  35. L 1 L_{1}
  36. L 2 L_{2}

Whitehead_product.html

  1. f π k ( X ) , g π l ( X ) f\in\pi_{k}(X),g\in\pi_{l}(X)
  2. [ f , g ] π k + l - 1 ( X ) [f,g]\in\pi_{k+l-1}(X)\,
  3. S k × S l S^{k}\times S^{l}
  4. ( k + l ) (k+l)
  5. S k S l S^{k}\vee S^{l}
  6. S k + l - 1 S k S l . S^{k+l-1}\to S^{k}\vee S^{l}.\,
  7. f f
  8. g g
  9. f : S k X f\colon S^{k}\to X\,
  10. g : S l X , g\colon S^{l}\to X,\,
  11. S k + l - 1 S k S l X S^{k+l-1}\to S^{k}\vee S^{l}\to X\,
  12. π k + l - 1 ( X ) . \pi_{k+l-1}(X).\,
  13. π k ( X ) \pi_{k}(X)
  14. ( k - 1 ) (k-1)
  15. L k = π k + 1 ( X ) L_{k}=\pi_{k+1}(X)
  16. L 0 = π 1 ( X ) L_{0}=\pi_{1}(X)
  17. f π 1 ( X ) f\in\pi_{1}(X)
  18. π 1 \pi_{1}
  19. π k \pi_{k}
  20. [ f , g ] = g f - g , [f,g]=g^{f}-g,\,
  21. g f g^{f}
  22. g g
  23. f f
  24. k = 1 k=1
  25. [ f , g ] = f g f - 1 g - 1 , [f,g]=fgf^{-1}g^{-1},\,

Whittaker_function.html

  1. d 2 w d z 2 + ( - 1 4 + κ z + 1 / 4 - μ 2 z 2 ) w = 0. \frac{d^{2}w}{dz^{2}}+\left(-\frac{1}{4}+\frac{\kappa}{z}+\frac{1/4-\mu^{2}}{z% ^{2}}\right)w=0.
  2. M κ , μ ( z ) = exp ( - z / 2 ) z μ + 1 2 M ( μ - κ + 1 2 , 1 + 2 μ ; z ) M_{\kappa,\mu}\left(z\right)=\exp\left(-z/2\right)z^{\mu+\tfrac{1}{2}}M\left(% \mu-\kappa+\frac{1}{2},1+2\mu;z\right)
  3. W κ , μ ( z ) = exp ( - z / 2 ) z μ + 1 2 U ( μ - κ + 1 2 , 1 + 2 μ ; z ) . W_{\kappa,\mu}\left(z\right)=\exp\left(-z/2\right)z^{\mu+\tfrac{1}{2}}U\left(% \mu-\kappa+\frac{1}{2},1+2\mu;z\right).

Widlar_current_source.html

  1. V B = V B E 1 = V B E 2 + ( β 2 + 1 ) I B 2 R 2 , V_{B}=V_{BE1}=V_{BE2}+(\beta_{2}+1)I_{B2}R_{2}\ ,
  2. ( β 2 + 1 ) I B 2 = ( 1 + 1 / β 2 ) I C 2 = V B E 1 - V B E 2 R 2 = V T R 2 ln ( I C 1 I S 2 I C 2 I S 1 ) , (\beta_{2}+1)I_{B2}=\left(1+1/\beta_{2}\right)I_{C2}=\frac{V_{BE1}-V_{BE2}}{R_% {2}}=\frac{V_{T}}{R_{2}}\ln\left(\frac{I_{C1}I_{S2}}{I_{C2}I_{S1}}\right)\ ,
  3. I C 1 = β 1 β 1 + 1 ( I R 1 - I C 2 / β 2 ) . I_{C1}=\frac{\beta_{1}}{\beta_{1}+1}\left(I_{R1}-I_{C2}/\beta_{2}\right)\ .
  4. V B E 1 = V T ln ( I C 1 I S ) = V A . V_{BE1}=V_{T}\ln\left(\frac{I_{C1}}{I_{S}}\right)=V_{A}\ .
  5. R 1 = V C C - V A I R 1 . R_{1}=\frac{V_{CC}-V_{A}}{I_{R1}}\ .
  6. R 2 = V T ( 1 + 1 / β 2 ) I C 2 ln ( I C 1 I C 2 ) . R_{2}=\frac{V_{T}}{\left(1+1/\beta_{2}\right)I_{C2}}\ln\left(\frac{I_{C1}}{I_{% C2}}\right)\ .
  7. I R 1 = I C 1 + I B 1 + I B 2 I_{R1}=I_{C1}+I_{B1}+I_{B2}
  8. = I C 1 + I C 1 β 1 + I C 2 β 2 =I_{C1}+\frac{I_{C1}}{\beta_{1}}+\frac{I_{C2}}{\beta_{2}}
  9. = V C C - V B E 1 R 1 . =\frac{V_{CC}-V_{BE1}}{R_{1}}\ .
  10. I C 1 = β 1 β 1 + 1 ( V C C - V B E 1 R 1 - I C 2 β 2 ) I_{C1}=\frac{\beta_{1}}{\beta_{1}+1}\left(\frac{V_{CC}-V_{BE1}}{R_{1}}-\frac{I% _{C2}}{\beta_{2}}\right)
  11. V B E 1 = V T ln ( I C 1 I S 1 ) . V_{BE1}=V_{T}\ln\left(\frac{I_{C1}}{I_{S1}}\right)\ .
  12. I C 2 = V T ( 1 + 1 / β 2 ) R 2 ln ( I C 1 I C 2 ) . I_{C2}=\frac{V_{T}}{\left(1+1/\beta_{2}\right)R_{2}}\ln\left(\frac{I_{C1}}{I_{% C2}}\right)\ .
  13. V B E 1 = V T ln ( I C 1 I S 1 ) . V_{BE1}=V_{T}\ln\left(\frac{I_{C1}}{I_{S1}}\right)\ .
  14. I C 1 = β 1 β 1 + 1 ( V C C - V B E 1 R 1 - I C 2 β 2 ) I_{C1}=\frac{\beta_{1}}{\beta_{1}+1}\left(\frac{V_{CC}-V_{BE1}}{R_{1}}-\frac{I% _{C2}}{\beta_{2}}\right)
  15. I C 2 = V T ( 1 + 1 / β 2 ) R 2 ln ( I C 1 I C 2 ) . I_{C2}=\frac{V_{T}}{\left(1+1/\beta_{2}\right)R_{2}}\ln\left(\frac{I_{C1}}{I_{% C2}}\right)\ .
  16. Q 2 Q_{2}
  17. I b ( ( R 1 r E ) + r π ) + ( I x + I b ) R 2 = 0 . I_{b}\left((R_{1}\parallel r_{E})+r_{\pi}\right)+(I_{x}+I_{b})R_{2}=0\ .
  18. I b = - I x R 2 ( R 1 r E ) + r π + R 2 . I_{b}=-I_{x}\frac{R_{2}}{(R_{1}\parallel r_{E})+r_{\pi}+R_{2}}\ .
  19. V x = I x ( r O + R 2 ) + I b ( R 2 - β r O ) , V_{x}=I_{x}(r_{O}+R_{2})+I_{b}(R_{2}-\beta r_{O})\ ,
  20. R O = V x I x = r O ( 1 + β R 2 ( R 1 r E ) + r π + R 2 ) R_{O}=\frac{V_{x}}{I_{x}}=r_{O}\left(1+\frac{\beta R_{2}}{(R_{1}\parallel r_{E% })+r_{\pi}+R_{2}}\right)
  21. + R 2 ( ( R 1 r E ) + r π ( R 1 r E ) + r π + R 2 ) . +\ R_{2}\left(\frac{(R_{1}\parallel r_{E})+r_{\pi}}{(R_{1}\parallel r_{E})+r_{% \pi}+R_{2}}\right)\ .
  22. r π = v b e i b | v c e = 0 r_{\pi}=\frac{v_{be}}{i_{b}}\Bigg|_{v_{ce}=0}
  23. = V T I B2 = β 2 V T I C2 , =\ \frac{V_{\mathrm{T}}}{I_{\mathrm{B2}}}=\beta_{2}\frac{V_{\mathrm{T}}}{I_{% \mathrm{C2}}}\ ,
  24. r O = v c e i c | v b e = 0 r_{O}=\frac{v_{ce}}{i_{c}}\Bigg|_{v_{be}=0}
  25. = V A I C 2 =\ \frac{V_{A}}{I_{C2}}
  26. R 2 = V T ( 1 + 1 / β 2 ) I C 2 ln ( I C 1 I C 2 ) . R_{2}=\frac{V_{T}}{\left(1+1/\beta_{2}\right)I_{C2}}\ln\left(\frac{I_{C1}}{I_{% C2}}\right)\ .
  27. R O r O ( 1 + β 2 R 2 r π + R 2 ) R_{O}\approx r_{O}\left(1+\frac{\beta_{2}R_{2}}{r_{\pi}+R_{2}}\right)
  28. = r O ( 1 + β 2 ln ( I C 1 I C 2 ) ( β 2 + 1 ) + ln ( I C 1 I C 2 ) ) , =r_{O}\left(1+\frac{\beta_{2}\ln\left(\frac{I_{C1}}{I_{C2}}\right)}{\left(% \beta_{2}+1\right)+\ln\left(\frac{I_{C1}}{I_{C2}}\right)}\right)\ ,

Willmore_energy.html

  1. 𝒲 = S H 2 d A - S K d A \mathcal{W}=\int_{S}H^{2}\,dA-\int_{S}K\,dA
  2. H H
  3. K K
  4. χ ( S ) \chi(S)
  5. S K d A = 2 π χ ( S ) , \int_{S}K\,dA=2\pi\chi(S),
  6. 3 \mathbb{R}^{3}
  7. 𝒲 = S H 2 d A - 2 π χ ( S ) \mathcal{W}=\int_{S}H^{2}\,dA-2\pi\chi(S)
  8. 𝒲 = 1 4 S ( k 1 - k 2 ) 2 d A \mathcal{W}={1\over 4}\int_{S}(k_{1}-k_{2})^{2}\,dA
  9. k 1 k_{1}
  10. k 2 k_{2}
  11. S H 2 d A \int_{S}H^{2}\,dA
  12. π \pi
  13. L 2 L^{2}
  14. e [ ] = 1 2 H 2 d A e[{\mathcal{M}}]=\frac{1}{2}\int_{\mathcal{M}}H^{2}\,\mathrm{d}A
  15. \mathcal{M}
  16. t x ( t ) = - 𝒲 [ x ( t ) ] \partial_{t}x(t)=-\nabla\mathcal{W}[x(t)]\,
  17. x x
  18. \mathcal{M}

Winning_percentage.html

  1. Points percentage = Points Total possible points = Overtime Losses + ( 2 × Wins ) 2 × Games Played \mathrm{Points}\ \mathrm{percentage}=\frac{\mathrm{Points}}{\mathrm{Total\ % possible\ points}}=\frac{\mathrm{Overtime\ Losses+(2\times Wins)}}{\mathrm{2% \times Games\ Played}}

Wiswesser's_rule.html

  1. W ( n , l ) = n + l - ( l l + 1 ) W(n,l)=n+l-\left(\frac{l}{l+1}\right)
  2. n n
  3. l l
  4. W ( n , l ) W(n,l)

Witten_index.html

  1. T r [ ( - 1 ) F e - β H ] Tr[(-1)^{F}e^{-\beta H}]

Working_set.html

  1. W ( t , τ ) W(t,\tau)
  2. t t
  3. ( t - τ , t ) (t-\tau,t)
  4. τ \tau

Worm-like_chain.html

  1. T = 0 T=0
  2. l l
  3. s ( 0 , l ) s\in(0,l)
  4. t ^ ( s ) \hat{t}(s)
  5. s s
  6. r ( s ) \vec{r}(s)
  7. t ^ ( s ) r ( s ) s \hat{t}(s)\equiv\frac{\partial\vec{r}(s)}{\partial s}
  8. R = 0 l t ^ ( s ) d s \vec{R}=\int_{0}^{l}\hat{t}(s)ds
  9. t ^ ( s ) t ^ ( 0 ) = cos θ ( s ) = e - s / P \langle\hat{t}(s)\cdot\hat{t}(0)\rangle=\langle\cos\;\theta(s)\rangle=e^{-s/P}\,
  10. P P
  11. R 2 = R R = 0 l t ^ ( s ) d s 0 l t ^ ( s ) d s = 0 l d s 0 l t ^ ( s ) t ^ ( s ) d s = 0 l d s 0 l e - | s - s | / P d s \langle R^{2}\rangle=\langle\vec{R}\cdot\vec{R}\rangle=\left\langle\int_{0}^{l% }\hat{t}(s)ds\cdot\int_{0}^{l}\hat{t}(s^{\prime})ds^{\prime}\right\rangle=\int% _{0}^{l}ds\int_{0}^{l}\langle\hat{t}(s)\cdot\hat{t}(s^{\prime})\rangle ds^{% \prime}=\int_{0}^{l}ds\int_{0}^{l}e^{-\left|s-s^{\prime}\right|/P}ds^{\prime}
  12. R 2 = 2 P l [ 1 - P l ( 1 - e - l / P ) ] \langle R^{2}\rangle=2Pl\left[1-\frac{P}{l}\left(1-e^{-l/P}\right)\right]
  13. l P l\gg P
  14. R 2 = 2 P l \langle R^{2}\rangle=2Pl
  15. L 0 L_{0}
  16. H = H entropic + H external = 1 2 k B T 0 L 0 P ( 2 r ( s ) s 2 ) 2 d s - x F H=H_{\rm entropic}+H_{\rm external}=\frac{1}{2}k_{B}T\int_{0}^{L_{0}}P\cdot% \left(\frac{\partial^{2}\vec{r}(s)}{\partial s^{2}}\right)^{2}ds-xF
  17. L 0 L_{0}
  18. P P
  19. x F xF
  20. F P k B T = 1 4 ( 1 - x L 0 ) - 2 - 1 4 + x L 0 \frac{FP}{k_{B}T}=\frac{1}{4}\left(1-\frac{x}{L_{0}}\right)^{-2}-\frac{1}{4}+% \frac{x}{L_{0}}
  21. k B k_{B}
  22. T T
  23. K 0 K_{0}
  24. H = H entropic + H enthalpic + H external = 1 2 k B T 0 L 0 P ( r ( s ) s ) 2 d s + 1 2 K 0 L 0 x 2 - x F H=H_{\rm entropic}+H_{\rm enthalpic}+H_{\rm external}=\frac{1}{2}k_{B}T\int_{0% }^{L_{0}}P\cdot\left(\frac{\partial\vec{r}(s)}{\partial s}\right)^{2}ds+\frac{% 1}{2}\frac{K_{0}}{L_{0}}x^{2}-xF
  25. L 0 L_{0}
  26. P P
  27. x x
  28. F F
  29. F P k B T = 1 4 ( 1 - x L 0 + F K 0 ) - 2 - 1 4 + x L 0 - F K 0 \frac{FP}{k_{B}T}=\frac{1}{4}\left(1-\frac{x}{L_{0}}+\frac{F}{K_{0}}\right)^{-% 2}-\frac{1}{4}+\frac{x}{L_{0}}-\frac{F}{K_{0}}
  30. x = L 0 ( 1 - 1 2 ( k B T F P ) 1 / 2 + F K 0 ) x=L_{0}\left(1-\frac{1}{2}\left(\frac{k_{B}T}{FP}\right)^{1/2}+\frac{F}{K_{0}}\right)

Wöhler_synthesis.html

  1. Pb ( NCO ) 2 + 2 N H 3 + 2 H 2 O Pb ( OH ) 2 + 2 N H 4 ( NCO ) \mathrm{Pb(NCO)_{2}+2NH_{3}+2H_{2}O\rightarrow Pb(OH)_{2}+2NH_{4}(NCO)}
  2. NH 4 ( NCO ) NH 3 + HNCO ( NH 2 ) 2 CO \mathrm{NH_{4}(NCO)\rightarrow NH_{3}+HNCO\leftrightarrow(NH_{2})_{2}CO}

Xbar_and_R_chart.html

  1. x ¯ ± A 2 R ¯ \bar{x}\pm A_{2}\bar{R}
  2. x ¯ i = j = 1 n x i j n \bar{x}_{i}=\frac{\sum_{j=1}^{n}x_{ij}}{n}
  3. x ¯ \bar{x}
  4. x ¯ \bar{x}
  5. x ¯ \bar{x}
  6. x ¯ \bar{x}
  7. x ¯ i \bar{x}_{i}
  8. D 3 R ¯ D_{3}\bar{R}
  9. D 4 R ¯ D_{4}\bar{R}
  10. x ¯ ± A 2 R ¯ \bar{x}\pm A_{2}\bar{R}
  11. x ¯ \bar{x}
  12. R ¯ = i = 1 m ( R m a x - R m i n ) m \bar{R}=\frac{\sum_{i=1}^{m}\left(R_{max}-R_{min}\right)}{m}
  13. x ¯ \bar{x}
  14. x ¯ \bar{x}
  15. x ¯ \bar{x}
  16. x ¯ \bar{x}
  17. x ¯ \bar{x}
  18. x ¯ \bar{x}

Y-homeomorphism.html

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

Yale_shooting_problem.html

  1. a l i v e alive
  2. l o a d e d loaded
  3. 0
  4. 1 1
  5. 2 2
  6. 3 3
  7. a l i v e alive
  8. a l i v e ( t ) alive(t)
  9. a l i v e ( 0 ) alive(0)
  10. ¬ l o a d e d ( 0 ) \neg loaded(0)
  11. t r u e l o a d e d ( 1 ) true\rightarrow loaded(1)
  12. l o a d e d ( 2 ) ¬ a l i v e ( 3 ) loaded(2)\rightarrow\neg alive(3)
  13. 0
  14. 2 2
  15. ¬ a l i v e ( 1 ) \neg alive(1)
  16. a l i v e ( 0 ) a l i v e ( 1 ) alive(0)\equiv alive(1)
  17. l o a d e d loaded
  18. a l i v e alive
  19. a l i v e ( 0 ) alive(0)
  20. a l i v e ( 1 ) alive(1)
  21. a l i v e ( 2 ) alive(2)
  22. ¬ a l i v e ( 3 ) \neg alive(3)
  23. ¬ l o a d e d ( 0 ) \neg loaded(0)
  24. l o a d e d ( 1 ) loaded(1)
  25. l o a d e d ( 2 ) loaded(2)
  26. l o a d e d ( 3 ) loaded(3)
  27. l o a d e d loaded
  28. a l i v e alive
  29. a l i v e ( 0 ) alive(0)
  30. a l i v e ( 1 ) alive(1)
  31. a l i v e ( 2 ) alive(2)
  32. a l i v e ( 3 ) alive(3)
  33. ¬ l o a d e d ( 0 ) \neg loaded(0)
  34. l o a d e d ( 1 ) loaded(1)
  35. ¬ l o a d e d ( 2 ) \neg loaded(2)
  36. ¬ l o a d e d ( 3 ) \neg loaded(3)
  37. l o a d e d loaded
  38. l o a d e d loaded

Yang–Mills_existence_and_mass_gap.html

  1. 4 \mathbb{R}^{4}
  2. 4 \mathbb{R}^{4}
  3. P 0 , P j P_{0},P_{j}
  4. P 0 0 P_{0}\geq 0
  5. P 0 2 - P j P j 0. P_{0}^{2}-P_{j}P_{j}\geq 0.
  6. A 1 ( f ) , , A n ( f ) A_{1}(f),\ldots,A_{n}(f)
  7. U ( a , L ) A ( x ) U ( a , L ) = S ( L ) A ( L - 1 ( x - a ) ) . U(a,L)^{\dagger}A(x)U(a,L)=S(L)A(L^{-1}(x-a)).
  8. H i n H^{in}
  9. H o u t H^{out}
  10. ϕ ( x ) \phi(x)
  11. ϕ ( 0 , t ) ϕ ( 0 , 0 ) n A n exp ( - Δ n t ) \langle\phi(0,t)\phi(0,0)\rangle\sim\sum_{n}A_{n}\exp\left(-\Delta_{n}t\right)
  12. Δ 0 > 0 \Delta_{0}>0

Zariski_surface.html

  1. z p = f ( x , y ) . z^{p}=f(x,y).

Zaslavskii_map.html

  1. x n , y n x_{n},y_{n}
  2. x n + 1 = [ x n + ν ( 1 + μ y n ) + ϵ ν μ cos ( 2 π x n ) ] ( mod 1 ) x_{n+1}=[x_{n}+\nu(1+\mu y_{n})+\epsilon\nu\mu\cos(2\pi x_{n})]\,(\textrm{mod}% \,1)
  3. y n + 1 = e - r ( y n + ϵ cos ( 2 π x n ) ) y_{n+1}=e^{-r}(y_{n}+\epsilon\cos(2\pi x_{n}))\,
  4. μ = 1 - e - r r \mu=\frac{1-e^{-r}}{r}

Zero-sum_problem.html

  1. k k
  2. / n \mathbb{Z}/n\mathbb{Z}
  3. k = 2 n - 1. k=2n-1.

Zero_moment_point.html

  1. F g i = m g - m a G F^{gi}=mg-ma_{G}
  2. m m
  3. g g
  4. G G
  5. a G a_{G}
  6. X X
  7. M X g i = X G × m g - X G × m a G - H ˙ G M_{X}^{gi}=\overrightarrow{XG}\times mg-\overrightarrow{XG}\times ma_{G}-\dot{% H}_{G}
  8. H ˙ G \dot{H}_{G}
  9. F c + m g = m a G F^{c}+mg=ma_{G}
  10. M X c + X G × m g = H ˙ G + X G × m a G M_{X}^{c}+\overrightarrow{XG}\times mg=\dot{H}_{G}+\overrightarrow{XG}\times ma% _{G}
  11. F c F^{c}
  12. M X c M_{X}^{c}
  13. F c + ( m g - m a G ) = 0 F^{c}+(mg-ma_{G})=0
  14. M X c + ( X G × m g - X G × m a G - H ˙ G ) = 0 M_{X}^{c}+(\overrightarrow{XG}\times mg-\overrightarrow{XG}\times ma_{G}-\dot{% H}_{G})=0
  15. F c + F g i = 0 F^{c}+F^{gi}=0
  16. M X c + M X g i = 0 M_{X}^{c}+M_{X}^{gi}=0
  17. Δ g i \Delta^{gi}
  18. n n
  19. n n
  20. Δ g i \Delta^{gi}
  21. M Z g i = Z G × m g - Z G × m a G - H ˙ G M_{Z}^{gi}=\overrightarrow{ZG}\times mg-\overrightarrow{ZG}\times ma_{G}-\dot{% H}_{G}
  22. M Z g i × n = 0 M_{Z}^{gi}\times n=0
  23. Z Z
  24. Z Z
  25. P Z = n × M P g i F g i n \overrightarrow{PZ}=\frac{n\times M_{P}^{gi}}{F^{gi}\cdot n}
  26. P P

Zero_state_response.html

  1. f ( t ) f(t)\,
  2. y ( t ) = y ( t 0 ) + t 0 t f ( τ ) d τ y(t)=y(t_{0})+\int_{t_{0}}^{t}f(\tau)d\tau
  3. f ( t ) . f(t).
  4. y ( t ) . y(t).
  5. y ( t ) . y(t).
  6. y ( t ) = y ( t 0 ) Z e r o - i n p u t r e s p o n s e + t 0 t f ( τ ) d τ Z e r o - s t a t e r e s p o n s e . y(t)=\underbrace{y(t_{0})}_{Zero-input\ response}+\underbrace{\int_{t_{0}}^{t}% f(\tau)d\tau}_{Zero-state\ response}.
  7. y ( t 0 ) y(t_{0})\,
  8. f ( t ) f(t)\,
  9. y ( t ) y(t)\,
  10. y ( t 0 ) y(t_{0})\,
  11. t 0 t f ( τ ) d τ \int_{t_{0}}^{t}f(\tau)d\tau
  12. y ( t 0 ) y(t_{0})\,
  13. f ( t ) . f(t).\,
  14. t 0 t f ( τ ) d τ \int_{t_{0}}^{t}f(\tau)d\tau
  15. y ( t ) y(t)\,
  16. y ( t 0 ) = 0. y(t_{0})=0.\,
  17. y ( t ) = t 0 t f ( τ ) d τ . y(t)=\int_{t_{0}}^{t}f(\tau)d\tau.\,
  18. f 1 ( t ) f_{1}(t)\,
  19. y 1 ( t ) y_{1}(t)\,
  20. f 2 ( t ) f_{2}(t)\,
  21. y 2 ( t ) . y_{2}(t).\,
  22. K f 1 ( t ) + K f 2 ( t ) Kf_{1}(t)+Kf_{2}(t)\,
  23. K y 1 ( t ) + K y 2 ( t ) . Ky_{1}(t)+Ky_{2}(t).\,
  24. y ( t ) = t 0 t f ( τ ) d τ y(t)=\int_{t_{0}}^{t}f(\tau)d\tau\,
  25. i ( t ) = C d v d t i(t)=C\frac{dv}{dt}
  26. a b i ( t ) d t = a b C d v d t d t . \int_{a}^{b}i(t)dt=\int_{a}^{b}C\frac{dv}{dt}dt.
  27. a b i ( t ) d t = C [ v ( b ) - v ( a ) ] . \int_{a}^{b}i(t)dt=C[v(b)-v(a)].
  28. C v ( a ) Cv(a)\,
  29. C v ( b ) = C v ( a ) + a b i ( t ) d t . Cv(b)=Cv(a)+\int_{a}^{b}i(t)dt.
  30. C C\,
  31. v ( b ) = v ( a ) + 1 C a b i ( t ) d t . v(b)=v(a)+\frac{1}{C}\int_{a}^{b}i(t)dt.
  32. t t\,
  33. b b\,
  34. t o t_{o}\,
  35. a a\,
  36. τ \tau\,
  37. v ( t ) = v ( t 0 ) + 1 C t 0 t i ( τ ) d τ v(t)=v(t_{0})+\frac{1}{C}\int_{t_{0}}^{t}i(\tau)d\tau
  38. v ( t ) = v ( t 0 ) + t 0 t i ( τ ) d τ , v(t)=v(t_{0})+\int_{t_{0}}^{t}i(\tau)d\tau,
  39. v ( t 0 ) = 0 v(t_{0})=0\,
  40. v ( t ) = t 0 t i ( τ ) d τ . v(t)=\int_{t_{0}}^{t}i(\tau)d\tau.\,
  41. i 1 ( t ) i_{1}(t)\,
  42. i 2 ( t ) i_{2}(t)\,
  43. v 1 ( t ) = t 0 t i 1 ( τ ) d τ v_{1}(t)=\int_{t_{0}}^{t}i_{1}(\tau)d\tau\,
  44. v 2 ( t ) = t 0 t i 2 ( τ ) d τ v_{2}(t)=\int_{t_{0}}^{t}i_{2}(\tau)d\tau\,
  45. i 1 ( t ) i_{1}(t)\,
  46. i 2 ( t ) i_{2}(t)\,
  47. i 3 ( t ) = K 1 i 1 ( t ) + K 2 i 2 ( t ) i_{3}(t)=K_{1}i_{1}(t)+K_{2}i_{2}(t)\,
  48. v 3 ( t ) = t 0 t ( K 1 i 1 ( τ ) + K 2 i 2 ( τ ) ) d τ . v_{3}(t)=\int_{t_{0}}^{t}(K_{1}i_{1}(\tau)+K_{2}i_{2}(\tau))d\tau.
  49. v 3 ( t ) = K 1 t 0 t i 1 ( τ ) d τ + K 2 t 0 t i 2 ( τ ) d τ , v_{3}(t)=K_{1}\int_{t_{0}}^{t}i_{1}(\tau)d\tau+K_{2}\int_{t_{0}}^{t}i_{2}(\tau% )d\tau,
  50. v 3 ( t ) = K 1 v 1 ( t ) + K 2 v 2 ( t ) v_{3}(t)=K_{1}v_{1}(t)+K_{2}v_{2}(t)\,
  51. v ( t 0 ) = 0 v(t_{0})=0\,

Zeta_function_regularization.html

  1. ζ A ( s ) = 1 a 1 s + 1 a 2 s + \zeta_{A}(s)=\frac{1}{a_{1}^{s}}+\frac{1}{a_{2}^{s}}+\cdots
  2. ζ ( - s ) = n = 1 n s = 1 s + 2 s + 3 s + = - B s + 1 s + 1 \zeta(-s)=\sum_{n=1}^{\infty}n^{s}=1^{s}+2^{s}+3^{s}+\ldots=-\frac{B_{s+1}}{s+1}
  3. 0 | T 00 | 0 = n | ω n | 2 \langle 0|T_{00}|0\rangle=\sum_{n}\frac{\hbar|\omega_{n}|}{2}
  4. T 00 T_{00}
  5. ω n \omega_{n}
  6. ω n \omega_{n}
  7. 0 | T 00 ( s ) | 0 = n | ω n | 2 | ω n | - s \langle 0|T_{00}(s)|0\rangle=\sum_{n}\frac{\hbar|\omega_{n}|}{2}|\omega_{n}|^{% -s}
  8. ϵ i , j , k \epsilon_{i,j,k}
  9. f ~ ( s ) = n = 1 f ( n ) n - s \tilde{f}(s)=\sum_{n=1}^{\infty}f(n)n^{-s}
  10. F ( t ) = n = 1 f ( n ) e - t n . F(t)=\sum_{n=1}^{\infty}f(n)e^{-tn}.
  11. F ( t ) = a N t N + a N - 1 t N - 1 + F(t)=\frac{a_{N}}{t^{N}}+\frac{a_{N-1}}{t^{N-1}}+\cdots
  12. f ~ ( s ) = a N s - N + . \tilde{f}(s)=\frac{a_{N}}{s-N}+\cdots.\,
  13. Γ ( s + 1 ) = 0 x s e - x d x \Gamma(s+1)=\int_{0}^{\infty}x^{s}e^{-x}\,dx
  14. Γ ( s + 1 ) f ~ ( s + 1 ) = 0 t s F ( t ) d t \Gamma(s+1)\tilde{f}(s+1)=\int_{0}^{\infty}t^{s}F(t)\,dt
  15. f ( s ) = n a n e - s | ω n | f(s)=\sum_{n}a_{n}e^{-s|\omega_{n}|}
  16. ω n \omega_{n}
  17. f ( s ) = 0 e - s t d α ( t ) f(s)=\int_{0}^{\infty}e^{-st}\,d\alpha(t)
  18. α ( t ) \alpha(t)
  19. a n a_{n}
  20. t = | ω n | t=|\omega_{n}|
  21. L = lim sup n log | k = 1 n a k | | ω n | L=\limsup_{n\to\infty}\frac{\log|\sum_{k=1}^{n}a_{k}|}{|\omega_{n}|}
  22. f ( s ) f(s)
  23. ( s ) > L \Re(s)>L
  24. ( s ) > L \Re(s)>L
  25. L = 0 L=0
  26. a x m - s d x \int_{a}^{\infty}x^{m-s}dx
  27. x - s x^{-s}
  28. ζ ( s - m ) \zeta(s-m)
  29. s 0 s\to 0

Zeta_function_universality.html

  1. | z e t a ( s + i t ) - f ( s ) | < ε for all s U . \left|\ zeta(s+it)-f(s)\right|<\varepsilon\quad\mbox{for all}~{}\quad s\in U.
  2. 0 < lim inf T 1 T λ ( { t [ 0 , T ] max s U | ζ ( s + i t ) - f ( s ) | < ε } ) , 0<\liminf_{T\to\infty}\frac{1}{T}\,\lambda\!\left(\left\{t\in[0,T]\mid\max_{s% \in U}\left|\zeta(s+it)-f(s)\right|<\varepsilon\right\}\right),
  3. U = { s : | s - 3 / 4 | < r } with 0 < r < 1 / 4 U=\{s\in\mathbb{C}:|s-3/4|<r\}\quad\mbox{with}~{}\quad 0<r<1/4
  4. | ln ζ ( s + i t ) - g ( s ) | < ε for all s U . \left|\ln\zeta(s+it)-g(s)\right|<\varepsilon\quad\,\text{for all}\quad s\in U.
  5. ζ ( s ) = p ( 1 - 1 p s ) - 1 , \zeta(s)=\prod_{p\in\mathbb{P}}\left(1-\frac{1}{p^{s}}\right)^{-1},
  6. θ = ( θ p ) p \theta=(\theta_{p})_{p\in\mathbb{P}}
  7. ζ M ( s , θ ) = p M ( 1 - e - 2 π i θ p p s ) - 1 . \zeta_{M}(s,\theta)=\prod_{p\in M}\left(1-\frac{e^{-2\pi i\theta_{p}}}{p^{s}}% \right)^{-1}.
  8. θ ^ = ( 1 4 , 2 4 , 3 4 , 4 4 , 5 4 , ) \hat{\theta}=\left(\frac{1}{4},\frac{2}{4},\frac{3}{4},\frac{4}{4},\frac{5}{4}% ,\ldots\right)
  9. ln ( ζ M ( s , θ ^ ) ) \ln(\zeta_{M}(s,\hat{\theta}))
  10. u k ( s ) = ln ( 1 - e - π i k / 2 p k s ) u_{k}(s)=\ln\left(1-\frac{e^{-\pi ik/2}}{p_{k}^{s}}\right)
  11. k = 1 u k \sum_{k=1}^{\infty}u_{k}
  12. ln 2 2 π , ln 3 2 π , ln 5 2 π , , ln p N 2 π \frac{\ln 2}{2\pi},\frac{\ln 3}{2\pi},\frac{\ln 5}{2\pi},\ldots,\frac{\ln p_{N% }}{2\pi}
  13. ln ( ζ M ( s , θ ^ ) ) \ln(\zeta_{M}(s,\hat{\theta}))
  14. ln ( ζ M ( s + i t , 0 ) ) \ln(\zeta_{M}(s+it,0))
  15. ln ( ζ M ( s + i t , 0 ) ) \ln(\zeta_{M}(s+it,0))
  16. ln ( ζ ( s + i t ) ) \ln(\zeta(s+it))
  17. 1 2 < σ < 1 \tfrac{1}{2}<\sigma<1
  18. γ ( t ) = ( ζ ( σ + i t ) , ζ ( σ + i t ) , , ζ ( n - 1 ) ( σ + i t ) ) \gamma(t)=(\zeta(\sigma+it),\zeta^{\prime}(\sigma+it),\dots,\zeta^{(n-1)}(% \sigma+it))
  19. n . \mathbb{C}^{n}.
  20. Φ \Phi
  21. h 1 , h 2 , , h n h_{1},h_{2},\dots,h_{n}
  22. ζ ( s ) \zeta(s)
  23. Φ { ζ ( s + h 1 ) , ζ ( s + h 1 ) , , ζ ( n 1 ) ( s + h 1 ) , ζ ( s + h 2 ) , ζ ( s + h 2 ) , , ζ ( n 2 ) ( s + h 2 ) , } = 0 \Phi\{\zeta(s+h_{1}),\zeta^{\prime}(s+h_{1}),\dots,\zeta^{(n_{1})}(s+h_{1}),% \zeta(s+h_{2}),\zeta^{\prime}(s+h_{2}),\dots,\zeta^{(n_{2})}(s+h_{2}),\dots\}=0
  24. Φ \Phi

Zillmerisation.html

  1. S A x + t : n - t | - N P x : n | a x + t : n - t | S\cdot~{}A_{x+t:\begin{smallmatrix}\hline~{}n-t|\end{smallmatrix}}-NP_{x:% \begin{smallmatrix}\hline~{}n|\end{smallmatrix}}a_{x+t:\begin{smallmatrix}% \hline~{}n-t|\end{smallmatrix}}
  2. E / a x : n | E/a_{x:\begin{smallmatrix}\hline~{}n|\end{smallmatrix}}
  3. S A x + t : n - t | - N P x + 1 : n - 1 | a x + t : n - t | S\cdot~{}A_{x+t:\begin{smallmatrix}\hline~{}n-t|\end{smallmatrix}}-NP_{x+1:% \begin{smallmatrix}\hline~{}n-1|\end{smallmatrix}}a_{x+t:\begin{smallmatrix}% \hline~{}n-t|\end{smallmatrix}}

Zolotarev's_lemma.html

  1. ( a p ) \left(\frac{a}{p}\right)
  2. ( a p ) = ε ( π a ) \left(\frac{a}{p}\right)=\varepsilon(\pi_{a})
  3. ( 3 11 ) \left(\frac{3}{11}\right)
  4. U : x a x ( mod p ) U:x\mapsto ax\;\;(\mathop{{\rm mod}}p)
  5. ( a n ) , \left(\frac{a}{n}\right),

Zome.html

  1. R 3 R^{3}
  2. φ \varphi
  3. H 3 H_{3}
  4. ( 0 , ± φ , ± 1 ) (0,\pm\varphi,\pm 1)
  5. ( ± φ , ± 1 , 0 ) (\pm\varphi,\pm 1,0)
  6. ( ± 1 , 0 , ± φ ) (\pm 1,0,\pm\varphi)
  7. H 3 H_{3}
  8. H 3 H_{3}
  9. A 5 A_{5}
  10. A 5 A_{5}
  11. ( 2 , 0 , 0 ) (2,0,0)
  12. A 5 A_{5}
  13. ( 1 , 1 , 1 ) (1,1,1)
  14. A 5 A_{5}
  15. ( 0 , φ , 1 ) (0,\varphi,1)
  16. φ n \varphi^{n}
  17. n n
  18. R 3 R^{3}
  19. ( N , S ) (N,S)
  20. N N
  21. S S
  22. ( v , w ) (v,w)
  23. v v
  24. w w
  25. N N
  26. v - w v-w
  27. A 5 A_{5}
  28. ( 2 , 2 , 0 ) (2,2,0)
  29. φ n \varphi^{n}
  30. | A 5 | |A_{5}|

Γ-convergence.html

  1. X X
  2. F n : X [ 0 , + ) F_{n}:X\to[0,+\infty)
  3. X X
  4. F n F_{n}
  5. Γ \Gamma
  6. Γ \Gamma
  7. F : X [ 0 , + ) F:X\to[0,+\infty)
  8. x n X x_{n}\in X
  9. x n x x_{n}\to x
  10. n + n\to+\infty
  11. F ( x ) lim inf n F n ( x n ) . F(x)\leq\liminf_{n\to\infty}F_{n}(x_{n}).
  12. x X x\in X
  13. x n x_{n}
  14. x x
  15. F ( x ) lim sup n F n ( x n ) F(x)\geq\limsup_{n\to\infty}F_{n}(x_{n})
  16. F F
  17. F n F_{n}
  18. F n F_{n}
  19. Γ \Gamma
  20. F F
  21. x n x_{n}
  22. F n F_{n}
  23. x n x_{n}
  24. F F
  25. Γ \Gamma
  26. Γ \Gamma
  27. F n F_{n}
  28. Γ \Gamma
  29. F F
  30. G : X [ 0 , + ) G:X\to[0,+\infty)
  31. F n + G F_{n}+G
  32. Γ \Gamma
  33. F + G F+G
  34. F n = F F_{n}=F
  35. Γ \Gamma
  36. F F
  37. F F
  38. F F
  39. Γ \Gamma

Ε-net.html

  1. ε \varepsilon

Θ_(set_theory).html

  1. ( 2 0 ) + (2^{\aleph_{0}})^{+}