wpmath0000007_6

Hyperelastic_material.html

  1. s y m b o l S = λ tr ( s y m b o l E ) s y m b o l 1 + 2 \musymbol E symbol{S}=\lambda~{}\,\text{tr}(symbol{E})symbol{\mathit{1}}+2\musymbol{E}
  2. s y m b o l S symbol{S}
  3. s y m b o l E symbol{E}
  4. λ \lambda
  5. μ \mu
  6. W ( s y m b o l E ) = λ 2 [ tr ( s y m b o l E ) ] 2 + μ tr ( s y m b o l E 2 ) W(symbol{E})=\frac{\lambda}{2}[\,\text{tr}(symbol{E})]^{2}+\mu\,\text{tr}(% symbol{E}^{2})
  7. s y m b o l S = W s y m b o l E . symbol{S}=\cfrac{\partial W}{\partial symbol{E}}~{}.
  8. ( λ 1 , λ 2 , λ 3 ) (\lambda_{1},\lambda_{2},\lambda_{3})
  9. W = f ( λ 1 ) + f ( λ 2 ) + f ( λ 3 ) . W=f(\lambda_{1})+f(\lambda_{2})+f(\lambda_{3})\,.
  10. W ( s y m b o l F ) W(symbol{F})
  11. s y m b o l P = W s y m b o l F or P i K = W F i K . symbol{P}=\frac{\partial W}{\partial symbol{F}}\qquad\,\text{or}\qquad P_{iK}=% \frac{\partial W}{\partial F_{iK}}.
  12. s y m b o l F symbol{F}
  13. s y m b o l E symbol{E}
  14. s y m b o l P = s y m b o l F W s y m b o l E or P i K = F i L W E L K . symbol{P}=symbol{F}\cdot\frac{\partial W}{\partial symbol{E}}\qquad\,\text{or}% \qquad P_{iK}=F_{iL}~{}\frac{\partial W}{\partial E_{LK}}~{}.
  15. s y m b o l C symbol{C}
  16. s y m b o l P = 2 s y m b o l F W s y m b o l C or P i K = 2 F i L W C L K . symbol{P}=2~{}symbol{F}\cdot\frac{\partial W}{\partial symbol{C}}\qquad\,\text% {or}\qquad P_{iK}=2~{}F_{iL}~{}\frac{\partial W}{\partial C_{LK}}~{}.
  17. s y m b o l S symbol{S}
  18. s y m b o l S = s y m b o l F - 1 W s y m b o l F or S I J = F I k - 1 W F k J . symbol{S}=symbol{F}^{-1}\cdot\frac{\partial W}{\partial symbol{F}}\qquad\,% \text{or}\qquad S_{IJ}=F^{-1}_{Ik}\frac{\partial W}{\partial F_{kJ}}~{}.
  19. s y m b o l S = W s y m b o l E or S I J = W E I J . symbol{S}=\frac{\partial W}{\partial symbol{E}}\qquad\,\text{or}\qquad S_{IJ}=% \frac{\partial W}{\partial E_{IJ}}~{}.
  20. s y m b o l S = 2 W s y m b o l C or S I J = 2 W C I J . symbol{S}=2~{}\frac{\partial W}{\partial symbol{C}}\qquad\,\text{or}\qquad S_{% IJ}=2~{}\frac{\partial W}{\partial C_{IJ}}~{}.
  21. s y m b o l σ = 1 J W s y m b o l F \cdotsymbol F T ; J := \detsymbol F or σ i j = 1 J W F i K F j K . symbol{\sigma}=\cfrac{1}{J}~{}\cfrac{\partial W}{\partial symbol{F}}% \cdotsymbol{F}^{T}~{};~{}~{}J:=\detsymbol{F}\qquad\,\text{or}\qquad\sigma_{ij}% =\cfrac{1}{J}~{}\cfrac{\partial W}{\partial F_{iK}}~{}F_{jK}~{}.
  22. s y m b o l σ = 1 J s y m b o l F W s y m b o l E \cdotsymbol F T or σ i j = 1 J F i K W E K L F j L . symbol{\sigma}=\cfrac{1}{J}~{}symbol{F}\cdot\cfrac{\partial W}{\partial symbol% {E}}\cdotsymbol{F}^{T}\qquad\,\text{or}\qquad\sigma_{ij}=\cfrac{1}{J}~{}F_{iK}% ~{}\cfrac{\partial W}{\partial E_{KL}}~{}F_{jL}~{}.
  23. s y m b o l σ = 2 J s y m b o l F W s y m b o l C \cdotsymbol F T or σ i j = 2 J F i K W C K L F j L . symbol{\sigma}=\cfrac{2}{J}~{}symbol{F}\cdot\cfrac{\partial W}{\partial symbol% {C}}\cdotsymbol{F}^{T}\qquad\,\text{or}\qquad\sigma_{ij}=\cfrac{2}{J}~{}F_{iK}% ~{}\cfrac{\partial W}{\partial C_{KL}}~{}F_{jL}~{}.
  24. s y m b o l σ = 2 J s y m b o l B W s y m b o l B or σ i j = 2 J B i k W B k j . symbol{\sigma}=\cfrac{2}{J}~{}symbol{B}\cdot\cfrac{\partial W}{\partial symbol% {B}}\qquad\,\text{or}\qquad\sigma_{ij}=\cfrac{2}{J}~{}B_{ik}~{}\cfrac{\partial W% }{\partial B_{kj}}~{}.
  25. J := \detsymbol F = 1 J:=\detsymbol{F}=1
  26. J - 1 = 0 J-1=0
  27. W = W ( s y m b o l F ) - p ( J - 1 ) W=W(symbol{F})-p~{}(J-1)
  28. p p
  29. s y m b o l P = - p s y m b o l F - T + W s y m b o l F = - p s y m b o l F - T + s y m b o l F W s y m b o l E = - p s y m b o l F - T + 2 s y m b o l F W s y m b o l C . symbol{P}=-p~{}symbol{F}^{-T}+\frac{\partial W}{\partial symbol{F}}=-p~{}% symbol{F}^{-T}+symbol{F}\cdot\frac{\partial W}{\partial symbol{E}}=-p~{}symbol% {F}^{-T}+2~{}symbol{F}\cdot\frac{\partial W}{\partial symbol{C}}~{}.
  30. s y m b o l σ = s y m b o l P \cdotsymbol F T = - p s y m b o l 1 + W s y m b o l F \cdotsymbol F T = - p s y m b o l 1 + s y m b o l F W s y m b o l E \cdotsymbol F T = - p s y m b o l 1 + 2 s y m b o l F W s y m b o l C \cdotsymbol F T . symbol{\sigma}=symbol{P}\cdotsymbol{F}^{T}=-p~{}symbol{\mathit{1}}+\frac{% \partial W}{\partial symbol{F}}\cdotsymbol{F}^{T}=-p~{}symbol{\mathit{1}}+% symbol{F}\cdot\frac{\partial W}{\partial symbol{E}}\cdotsymbol{F}^{T}=-p~{}% symbol{\mathit{1}}+2~{}symbol{F}\cdot\frac{\partial W}{\partial symbol{C}}% \cdotsymbol{F}^{T}~{}.
  31. W ( s y m b o l F ) = W ^ ( I 1 , I 2 , I 3 ) = W ¯ ( I ¯ 1 , I ¯ 2 , J ) = W ~ ( λ 1 , λ 2 , λ 3 ) W(symbol{F})=\hat{W}(I_{1},I_{2},I_{3})=\bar{W}(\bar{I}_{1},\bar{I}_{2},J)=% \tilde{W}(\lambda_{1},\lambda_{2},\lambda_{3})
  32. s y m b o l σ = 2 I 3 [ ( W ^ I 1 + I 1 W ^ I 2 ) s y m b o l B - W ^ I 2 s y m b o l B \cdotsymbol B ] + 2 I 3 W ^ I 3 s y m b o l 1 = 2 J [ 1 J 2 / 3 ( W ¯ I ¯ 1 + I ¯ 1 W ¯ I ¯ 2 ) s y m b o l B - 1 J 4 / 3 W ¯ I ¯ 2 s y m b o l B \cdotsymbol B ] + [ W ¯ J - 2 3 J ( I ¯ 1 W ¯ I ¯ 1 + 2 I ¯ 2 W ¯ I ¯ 2 ) ] s y m b o l 1 = 2 J [ ( W ¯ I ¯ 1 + I ¯ 1 W ¯ I ¯ 2 ) s y m b o l B ¯ - W ¯ I ¯ 2 s y m b o l B ¯ s y m b o l B ¯ ] + [ W ¯ J - 2 3 J ( I ¯ 1 W ¯ I ¯ 1 + 2 I ¯ 2 W ¯ I ¯ 2 ) ] s y m b o l 1 = λ 1 λ 1 λ 2 λ 3 W ~ λ 1 𝐧 1 𝐧 1 + λ 2 λ 1 λ 2 λ 3 W ~ λ 2 𝐧 2 𝐧 2 + λ 3 λ 1 λ 2 λ 3 W ~ λ 3 𝐧 3 𝐧 3 \begin{aligned}\displaystyle symbol{\sigma}&\displaystyle=\cfrac{2}{\sqrt{I_{3% }}}\left[\left(\cfrac{\partial\hat{W}}{\partial I_{1}}+I_{1}~{}\cfrac{\partial% \hat{W}}{\partial I_{2}}\right)symbol{B}-\cfrac{\partial\hat{W}}{\partial I_{2% }}~{}symbol{B}\cdotsymbol{B}\right]+2\sqrt{I_{3}}~{}\cfrac{\partial\hat{W}}{% \partial I_{3}}~{}symbol{\mathit{1}}\\ &\displaystyle=\cfrac{2}{J}\left[\cfrac{1}{J^{2/3}}\left(\cfrac{\partial\bar{W% }}{\partial\bar{I}_{1}}+\bar{I}_{1}~{}\cfrac{\partial\bar{W}}{\partial\bar{I}_% {2}}\right)symbol{B}-\cfrac{1}{J^{4/3}}~{}\cfrac{\partial\bar{W}}{\partial\bar% {I}_{2}}~{}symbol{B}\cdotsymbol{B}\right]\\ &\displaystyle\qquad\qquad+\left[\cfrac{\partial\bar{W}}{\partial J}-\cfrac{2}% {3J}\left(\bar{I}_{1}~{}\cfrac{\partial\bar{W}}{\partial\bar{I}_{1}}+2~{}\bar{% I}_{2}~{}\cfrac{\partial\bar{W}}{\partial\bar{I}_{2}}\right)\right]~{}symbol{% \mathit{1}}\\ &\displaystyle=\cfrac{2}{J}\left[\left(\cfrac{\partial\bar{W}}{\partial\bar{I}% _{1}}+\bar{I}_{1}~{}\cfrac{\partial\bar{W}}{\partial\bar{I}_{2}}\right)\bar{% symbol{B}}-\cfrac{\partial\bar{W}}{\partial\bar{I}_{2}}~{}\bar{symbol{B}}\cdot% \bar{symbol{B}}\right]+\left[\cfrac{\partial\bar{W}}{\partial J}-\cfrac{2}{3J}% \left(\bar{I}_{1}~{}\cfrac{\partial\bar{W}}{\partial\bar{I}_{1}}+2~{}\bar{I}_{% 2}~{}\cfrac{\partial\bar{W}}{\partial\bar{I}_{2}}\right)\right]~{}symbol{% \mathit{1}}\\ &\displaystyle=\cfrac{\lambda_{1}}{\lambda_{1}\lambda_{2}\lambda_{3}}~{}\cfrac% {\partial\tilde{W}}{\partial\lambda_{1}}~{}\mathbf{n}_{1}\otimes\mathbf{n}_{1}% +\cfrac{\lambda_{2}}{\lambda_{1}\lambda_{2}\lambda_{3}}~{}\cfrac{\partial% \tilde{W}}{\partial\lambda_{2}}~{}\mathbf{n}_{2}\otimes\mathbf{n}_{2}+\cfrac{% \lambda_{3}}{\lambda_{1}\lambda_{2}\lambda_{3}}~{}\cfrac{\partial\tilde{W}}{% \partial\lambda_{3}}~{}\mathbf{n}_{3}\otimes\mathbf{n}_{3}\end{aligned}
  33. s y m b o l S = 2 W s y m b o l C symbol{S}=2~{}\cfrac{\partial W}{\partial symbol{C}}
  34. s y m b o l C = s y m b o l F T \cdotsymbol F symbol{C}=symbol{F}^{T}\cdotsymbol{F}
  35. s y m b o l F symbol{F}
  36. s y m b o l σ = 1 J s y m b o l F \cdotsymbol S \cdotsymbol F T = 2 J s y m b o l F W s y m b o l C \cdotsymbol F T symbol{\sigma}=\cfrac{1}{J}~{}symbol{F}\cdotsymbol{S}\cdotsymbol{F}^{T}=\cfrac% {2}{J}~{}symbol{F}\cdot\cfrac{\partial W}{\partial symbol{C}}\cdotsymbol{F}^{T}
  37. J = \detsymbol F J=\detsymbol{F}
  38. I 1 , I 2 , I 3 I_{1},I_{2},I_{3}
  39. s y m b o l C symbol{C}
  40. W s y m b o l C = W I 1 I 1 s y m b o l C + W I 2 I 2 s y m b o l C + W I 3 I 3 s y m b o l C . \cfrac{\partial W}{\partial symbol{C}}=\cfrac{\partial W}{\partial I_{1}}~{}% \cfrac{\partial I_{1}}{\partial symbol{C}}+\cfrac{\partial W}{\partial I_{2}}~% {}\cfrac{\partial I_{2}}{\partial symbol{C}}+\cfrac{\partial W}{\partial I_{3}% }~{}\cfrac{\partial I_{3}}{\partial symbol{C}}~{}.
  41. s y m b o l C symbol{C}
  42. I 1 s y m b o l C = s y m b o l 1 ; I 2 s y m b o l C = I 1 s y m b o l 1 - s y m b o l C ; I 3 s y m b o l C = det ( s y m b o l C ) s y m b o l C - 1 \frac{\partial I_{1}}{\partial symbol{C}}=symbol{\mathit{1}}~{};~{}~{}\frac{% \partial I_{2}}{\partial symbol{C}}=I_{1}~{}symbol{\mathit{1}}-symbol{C}~{};~{% }~{}\frac{\partial I_{3}}{\partial symbol{C}}=\det(symbol{C})~{}symbol{C}^{-1}
  43. W s y m b o l C = W I 1 s y m b o l 1 + W I 2 ( I 1 s y m b o l 1 - s y m b o l F T \cdotsymbol F ) + W I 3 I 3 s y m b o l F - 1 \cdotsymbol F - T . \cfrac{\partial W}{\partial symbol{C}}=\cfrac{\partial W}{\partial I_{1}}~{}% symbol{\mathit{1}}+\cfrac{\partial W}{\partial I_{2}}~{}(I_{1}~{}symbol{% \mathit{1}}-symbol{F}^{T}\cdotsymbol{F})+\cfrac{\partial W}{\partial I_{3}}~{}% I_{3}~{}symbol{F}^{-1}\cdotsymbol{F}^{-T}~{}.
  44. s y m b o l σ = 2 J [ W I 1 s y m b o l F \cdotsymbol F T + W I 2 ( I 1 s y m b o l F \cdotsymbol F T - s y m b o l F \cdotsymbol F T \cdotsymbol F \cdotsymbol F T ) + W I 3 I 3 s y m b o l 1 ] symbol{\sigma}=\cfrac{2}{J}~{}\left[\cfrac{\partial W}{\partial I_{1}}~{}% symbol{F}\cdotsymbol{F}^{T}+\cfrac{\partial W}{\partial I_{2}}~{}(I_{1}~{}% symbol{F}\cdotsymbol{F}^{T}-symbol{F}\cdotsymbol{F}^{T}\cdotsymbol{F}% \cdotsymbol{F}^{T})+\cfrac{\partial W}{\partial I_{3}}~{}I_{3}~{}symbol{% \mathit{1}}\right]
  45. s y m b o l B = s y m b o l F \cdotsymbol F T symbol{B}=symbol{F}\cdotsymbol{F}^{T}
  46. I 3 = J 2 I_{3}=J^{2}
  47. s y m b o l σ = 2 I 3 [ ( W I 1 + I 1 W I 2 ) s y m b o l B - W I 2 s y m b o l B \cdotsymbol B ] + 2 I 3 W I 3 s y m b o l 1 . symbol{\sigma}=\cfrac{2}{\sqrt{I_{3}}}~{}\left[\left(\cfrac{\partial W}{% \partial I_{1}}+I_{1}~{}\cfrac{\partial W}{\partial I_{2}}\right)~{}symbol{B}-% \cfrac{\partial W}{\partial I_{2}}~{}symbol{B}\cdotsymbol{B}\right]+2~{}\sqrt{% I_{3}}~{}\cfrac{\partial W}{\partial I_{3}}~{}symbol{\mathit{1}}~{}.
  48. I 3 = 1 I_{3}=1
  49. W = W ( I 1 , I 2 ) W=W(I_{1},I_{2})
  50. W s y m b o l C = W I 1 I 1 s y m b o l C + W I 2 I 2 s y m b o l C = W I 1 s y m b o l 1 + W I 2 ( I 1 s y m b o l 1 - s y m b o l F T \cdotsymbol F ) \cfrac{\partial W}{\partial symbol{C}}=\cfrac{\partial W}{\partial I_{1}}~{}% \cfrac{\partial I_{1}}{\partial symbol{C}}+\cfrac{\partial W}{\partial I_{2}}~% {}\cfrac{\partial I_{2}}{\partial symbol{C}}=\cfrac{\partial W}{\partial I_{1}% }~{}symbol{\mathit{1}}+\cfrac{\partial W}{\partial I_{2}}~{}(I_{1}~{}symbol{% \mathit{1}}-symbol{F}^{T}\cdotsymbol{F})
  51. s y m b o l σ = 2 [ ( W I 1 + I 1 W I 2 ) s y m b o l B - W I 2 s y m b o l B \cdotsymbol B ] - p s y m b o l 1 . symbol{\sigma}=2\left[\left(\cfrac{\partial W}{\partial I_{1}}+I_{1}~{}\cfrac{% \partial W}{\partial I_{2}}\right)~{}symbol{B}-\cfrac{\partial W}{\partial I_{% 2}}~{}symbol{B}\cdotsymbol{B}\right]-p~{}symbol{\mathit{1}}~{}.
  52. p p
  53. I 1 = I 2 I_{1}=I_{2}
  54. W = W ( I 1 ) W=W(I_{1})
  55. W s y m b o l C = W I 1 I 1 s y m b o l C = W I 1 s y m b o l 1 \cfrac{\partial W}{\partial symbol{C}}=\cfrac{\partial W}{\partial I_{1}}~{}% \cfrac{\partial I_{1}}{\partial symbol{C}}=\cfrac{\partial W}{\partial I_{1}}~% {}symbol{\mathit{1}}
  56. s y m b o l σ = 2 W I 1 s y m b o l B - p s y m b o l 1 . symbol{\sigma}=2\cfrac{\partial W}{\partial I_{1}}~{}symbol{B}-p~{}symbol{% \mathit{1}}~{}.
  57. s y m b o l F ¯ := J - 1 / 3 s y m b o l F \bar{symbol{F}}:=J^{-1/3}symbol{F}
  58. s y m b o l B ¯ := s y m b o l F ¯ s y m b o l F ¯ T = J - 2 / 3 s y m b o l B \bar{symbol{B}}:=\bar{symbol{F}}\cdot\bar{symbol{F}}^{T}=J^{-2/3}symbol{B}
  59. s y m b o l B ¯ \bar{symbol{B}}
  60. I ¯ 1 = tr ( s y m b o l B ¯ ) = J - 2 / 3 tr ( s y m b o l B ) = J - 2 / 3 I 1 I ¯ 2 = 1 2 ( tr ( s y m b o l B ¯ ) 2 - tr ( s y m b o l B ¯ 2 ) ) = 1 2 ( ( J - 2 / 3 tr ( s y m b o l B ) ) 2 - tr ( J - 4 / 3 s y m b o l B 2 ) ) = J - 4 / 3 I 2 I ¯ 3 = det ( s y m b o l B ¯ ) = J - 6 / 3 det ( s y m b o l B ) = J - 2 I 3 = J - 2 J 2 = 1 \begin{aligned}\displaystyle\bar{I}_{1}&\displaystyle=\,\text{tr}(\bar{symbol{% B}})=J^{-2/3}\,\text{tr}(symbol{B})=J^{-2/3}I_{1}\\ \displaystyle\bar{I}_{2}&\displaystyle=\frac{1}{2}\left(\,\text{tr}(\bar{% symbol{B}})^{2}-\,\text{tr}(\bar{symbol{B}}^{2})\right)=\frac{1}{2}\left(\left% (J^{-2/3}\,\text{tr}(symbol{B})\right)^{2}-\,\text{tr}(J^{-4/3}symbol{B}^{2})% \right)=J^{-4/3}I_{2}\\ \displaystyle\bar{I}_{3}&\displaystyle=\det(\bar{symbol{B}})=J^{-6/3}\det(% symbol{B})=J^{-2}I_{3}=J^{-2}J^{2}=1\end{aligned}
  61. J J
  62. I ¯ 1 , I ¯ 2 , J \bar{I}_{1},\bar{I}_{2},J
  63. I ¯ 1 = J - 2 / 3 I 1 = I 3 - 1 / 3 I 1 ; I ¯ 2 = J - 4 / 3 I 2 = I 3 - 2 / 3 I 2 ; J = I 3 1 / 2 . \bar{I}_{1}=J^{-2/3}~{}I_{1}=I_{3}^{-1/3}~{}I_{1}~{};~{}~{}\bar{I}_{2}=J^{-4/3% }~{}I_{2}=I_{3}^{-2/3}~{}I_{2}~{};~{}~{}J=I_{3}^{1/2}~{}.
  64. W I 1 = W I ¯ 1 I ¯ 1 I 1 + W I ¯ 2 I ¯ 2 I 1 + W J J I 1 = I 3 - 1 / 3 W I ¯ 1 = J - 2 / 3 W I ¯ 1 W I 2 = W I ¯ 1 I ¯ 1 I 2 + W I ¯ 2 I ¯ 2 I 2 + W J J I 2 = I 3 - 2 / 3 W I ¯ 2 = J - 4 / 3 W I ¯ 2 W I 3 = W I ¯ 1 I ¯ 1 I 3 + W I ¯ 2 I ¯ 2 I 3 + W J J I 3 = - 1 3 I 3 - 4 / 3 I 1 W I ¯ 1 - 2 3 I 3 - 5 / 3 I 2 W I ¯ 2 + 1 2 I 3 - 1 / 2 W J = - 1 3 J - 8 / 3 J 2 / 3 I ¯ 1 W I ¯ 1 - 2 3 J - 10 / 3 J 4 / 3 I ¯ 2 W I ¯ 2 + 1 2 J - 1 W J = - 1 3 J - 2 ( I ¯ 1 W I ¯ 1 + 2 I ¯ 2 W I ¯ 2 ) + 1 2 J - 1 W J \begin{aligned}\displaystyle\cfrac{\partial W}{\partial I_{1}}&\displaystyle=% \cfrac{\partial W}{\partial\bar{I}_{1}}~{}\cfrac{\partial\bar{I}_{1}}{\partial I% _{1}}+\cfrac{\partial W}{\partial\bar{I}_{2}}~{}\cfrac{\partial\bar{I}_{2}}{% \partial I_{1}}+\cfrac{\partial W}{\partial J}~{}\cfrac{\partial J}{\partial I% _{1}}\\ &\displaystyle=I_{3}^{-1/3}~{}\cfrac{\partial W}{\partial\bar{I}_{1}}=J^{-2/3}% ~{}\cfrac{\partial W}{\partial\bar{I}_{1}}\\ \displaystyle\cfrac{\partial W}{\partial I_{2}}&\displaystyle=\cfrac{\partial W% }{\partial\bar{I}_{1}}~{}\cfrac{\partial\bar{I}_{1}}{\partial I_{2}}+\cfrac{% \partial W}{\partial\bar{I}_{2}}~{}\cfrac{\partial\bar{I}_{2}}{\partial I_{2}}% +\cfrac{\partial W}{\partial J}~{}\cfrac{\partial J}{\partial I_{2}}\\ &\displaystyle=I_{3}^{-2/3}~{}\cfrac{\partial W}{\partial\bar{I}_{2}}=J^{-4/3}% ~{}\cfrac{\partial W}{\partial\bar{I}_{2}}\\ \displaystyle\cfrac{\partial W}{\partial I_{3}}&\displaystyle=\cfrac{\partial W% }{\partial\bar{I}_{1}}~{}\cfrac{\partial\bar{I}_{1}}{\partial I_{3}}+\cfrac{% \partial W}{\partial\bar{I}_{2}}~{}\cfrac{\partial\bar{I}_{2}}{\partial I_{3}}% +\cfrac{\partial W}{\partial J}~{}\cfrac{\partial J}{\partial I_{3}}\\ &\displaystyle=-\cfrac{1}{3}~{}I_{3}^{-4/3}~{}I_{1}~{}\cfrac{\partial W}{% \partial\bar{I}_{1}}-\cfrac{2}{3}~{}I_{3}^{-5/3}~{}I_{2}~{}\cfrac{\partial W}{% \partial\bar{I}_{2}}+\cfrac{1}{2}~{}I_{3}^{-1/2}~{}\cfrac{\partial W}{\partial J% }\\ &\displaystyle=-\cfrac{1}{3}~{}J^{-8/3}~{}J^{2/3}~{}\bar{I}_{1}~{}\cfrac{% \partial W}{\partial\bar{I}_{1}}-\cfrac{2}{3}~{}J^{-10/3}~{}J^{4/3}~{}\bar{I}_% {2}~{}\cfrac{\partial W}{\partial\bar{I}_{2}}+\cfrac{1}{2}~{}J^{-1}~{}\cfrac{% \partial W}{\partial J}\\ &\displaystyle=-\cfrac{1}{3}~{}J^{-2}~{}\left(\bar{I}_{1}~{}\cfrac{\partial W}% {\partial\bar{I}_{1}}+2~{}\bar{I}_{2}~{}\cfrac{\partial W}{\partial\bar{I}_{2}% }\right)+\cfrac{1}{2}~{}J^{-1}~{}\cfrac{\partial W}{\partial J}\end{aligned}
  65. s y m b o l σ = 2 I 3 [ ( W I 1 + I 1 W I 2 ) s y m b o l B - W I 2 s y m b o l B \cdotsymbol B ] + 2 I 3 W I 3 s y m b o l 1 . symbol{\sigma}=\cfrac{2}{\sqrt{I_{3}}}~{}\left[\left(\cfrac{\partial W}{% \partial I_{1}}+I_{1}~{}\cfrac{\partial W}{\partial I_{2}}\right)~{}symbol{B}-% \cfrac{\partial W}{\partial I_{2}}~{}symbol{B}\cdotsymbol{B}\right]+2~{}\sqrt{% I_{3}}~{}\cfrac{\partial W}{\partial I_{3}}~{}symbol{\mathit{1}}~{}.
  66. I ¯ 1 , I ¯ 2 , J \bar{I}_{1},\bar{I}_{2},J
  67. s y m b o l σ = 2 J [ ( W I 1 + J 2 / 3 I ¯ 1 W I 2 ) s y m b o l B - W I 2 s y m b o l B \cdotsymbol B ] + 2 J W I 3 s y m b o l 1 . symbol{\sigma}=\cfrac{2}{J}~{}\left[\left(\cfrac{\partial W}{\partial I_{1}}+J% ^{2/3}~{}\bar{I}_{1}~{}\cfrac{\partial W}{\partial I_{2}}\right)~{}symbol{B}-% \cfrac{\partial W}{\partial I_{2}}~{}symbol{B}\cdotsymbol{B}\right]+2~{}J~{}% \cfrac{\partial W}{\partial I_{3}}~{}symbol{\mathit{1}}~{}.
  68. W W
  69. I ¯ 1 , I ¯ 2 , J \bar{I}_{1},\bar{I}_{2},J
  70. s y m b o l σ = 2 J [ ( J - 2 / 3 W I ¯ 1 + J - 2 / 3 I ¯ 1 W I ¯ 2 ) s y m b o l B - J - 4 / 3 W I ¯ 2 s y m b o l B \cdotsymbol B ] + 2 J [ - 1 3 J - 2 ( I ¯ 1 W I ¯ 1 + 2 I ¯ 2 W I ¯ 2 ) + 1 2 J - 1 W J ] s y m b o l 1 \begin{aligned}\displaystyle symbol{\sigma}&\displaystyle=\cfrac{2}{J}~{}\left% [\left(J^{-2/3}~{}\cfrac{\partial W}{\partial\bar{I}_{1}}+J^{-2/3}~{}\bar{I}_{% 1}~{}\cfrac{\partial W}{\partial\bar{I}_{2}}\right)~{}symbol{B}-J^{-4/3}~{}% \cfrac{\partial W}{\partial\bar{I}_{2}}~{}symbol{B}\cdotsymbol{B}\right]+\\ &\displaystyle\qquad 2~{}J~{}\left[-\cfrac{1}{3}~{}J^{-2}~{}\left(\bar{I}_{1}~% {}\cfrac{\partial W}{\partial\bar{I}_{1}}+2~{}\bar{I}_{2}~{}\cfrac{\partial W}% {\partial\bar{I}_{2}}\right)+\cfrac{1}{2}~{}J^{-1}~{}\cfrac{\partial W}{% \partial J}\right]~{}symbol{\mathit{1}}\end{aligned}
  71. s y m b o l σ = 2 J [ 1 J 2 / 3 ( W I ¯ 1 + I ¯ 1 W I ¯ 2 ) s y m b o l B - 1 J 4 / 3 W I ¯ 2 s y m b o l B \cdotsymbol B ] + [ W J - 2 3 J ( I ¯ 1 W I ¯ 1 + 2 I ¯ 2 W I ¯ 2 ) ] s y m b o l 1 \begin{aligned}\displaystyle symbol{\sigma}&\displaystyle=\cfrac{2}{J}~{}\left% [\cfrac{1}{J^{2/3}}~{}\left(\cfrac{\partial W}{\partial\bar{I}_{1}}+\bar{I}_{1% }~{}\cfrac{\partial W}{\partial\bar{I}_{2}}\right)~{}symbol{B}-\cfrac{1}{J^{4/% 3}}~{}\cfrac{\partial W}{\partial\bar{I}_{2}}~{}symbol{B}\cdotsymbol{B}\right]% \\ &\displaystyle\qquad+\left[\cfrac{\partial W}{\partial J}-\cfrac{2}{3J}\left(% \bar{I}_{1}~{}\cfrac{\partial W}{\partial\bar{I}_{1}}+2~{}\bar{I}_{2}~{}\cfrac% {\partial W}{\partial\bar{I}_{2}}\right)\right]symbol{\mathit{1}}\end{aligned}
  72. s y m b o l B symbol{B}
  73. s y m b o l σ = 2 J [ ( W I ¯ 1 + I ¯ 1 W I ¯ 2 ) s y m b o l B ¯ - W I ¯ 2 s y m b o l B ¯ s y m b o l B ¯ ] + [ W J - 2 3 J ( I ¯ 1 W I ¯ 1 + 2 I ¯ 2 W I ¯ 2 ) ] s y m b o l 1 \begin{aligned}\displaystyle symbol{\sigma}&\displaystyle=\cfrac{2}{J}~{}\left% [\left(\cfrac{\partial W}{\partial\bar{I}_{1}}+\bar{I}_{1}~{}\cfrac{\partial W% }{\partial\bar{I}_{2}}\right)~{}\bar{symbol{B}}-\cfrac{\partial W}{\partial% \bar{I}_{2}}~{}\bar{symbol{B}}\cdot\bar{symbol{B}}\right]\\ &\displaystyle\qquad+\left[\cfrac{\partial W}{\partial J}-\cfrac{2}{3J}\left(% \bar{I}_{1}~{}\cfrac{\partial W}{\partial\bar{I}_{1}}+2~{}\bar{I}_{2}~{}\cfrac% {\partial W}{\partial\bar{I}_{2}}\right)\right]symbol{\mathit{1}}\end{aligned}
  74. J = 1 J=1
  75. W = W ( I ¯ 1 , I ¯ 2 ) W=W(\bar{I}_{1},\bar{I}_{2})
  76. s y m b o l σ = 2 [ ( W I ¯ 1 + I 1 W I ¯ 2 ) s y m b o l B ¯ - W I ¯ 2 s y m b o l B ¯ s y m b o l B ¯ ] - p s y m b o l 1 . symbol{\sigma}=2\left[\left(\cfrac{\partial W}{\partial\bar{I}_{1}}+I_{1}~{}% \cfrac{\partial W}{\partial\bar{I}_{2}}\right)~{}\bar{symbol{B}}-\cfrac{% \partial W}{\partial\bar{I}_{2}}~{}\bar{symbol{B}}\cdot\bar{symbol{B}}\right]-% p~{}symbol{\mathit{1}}~{}.
  77. p p
  78. I ¯ 1 = I ¯ 2 \bar{I}_{1}=\bar{I}_{2}
  79. W = W ( I ¯ 1 ) W=W(\bar{I}_{1})
  80. s y m b o l σ = 2 W I ¯ 1 s y m b o l B ¯ - p s y m b o l 1 . symbol{\sigma}=2\cfrac{\partial W}{\partial\bar{I}_{1}}~{}\bar{symbol{B}}-p~{}% symbol{\mathit{1}}~{}.
  81. λ 1 , λ 2 , λ 3 \lambda_{1},\lambda_{2},\lambda_{3}
  82. λ i \partialsymbol C = 1 2 λ i s y m b o l R T ( 𝐧 i 𝐧 i ) \cdotsymbol R ; i = 1 , 2 , 3 . \cfrac{\partial\lambda_{i}}{\partialsymbol{C}}=\cfrac{1}{2\lambda_{i}}~{}% symbol{R}^{T}\cdot(\mathbf{n}_{i}\otimes\mathbf{n}_{i})\cdotsymbol{R}~{};~{}~{% }i=1,2,3~{}.
  83. W \partialsymbol C = W λ 1 λ 1 \partialsymbol C + W λ 2 λ 2 \partialsymbol C + W λ 3 λ 3 \partialsymbol C = s y m b o l R T [ 1 2 λ 1 W λ 1 𝐧 1 𝐧 1 + 1 2 λ 2 W λ 2 𝐧 2 𝐧 2 + 1 2 λ 3 W λ 3 𝐧 3 𝐧 3 ] \cdotsymbol R \begin{aligned}\displaystyle\cfrac{\partial W}{\partialsymbol{C}}&% \displaystyle=\cfrac{\partial W}{\partial\lambda_{1}}~{}\cfrac{\partial\lambda% _{1}}{\partialsymbol{C}}+\cfrac{\partial W}{\partial\lambda_{2}}~{}\cfrac{% \partial\lambda_{2}}{\partialsymbol{C}}+\cfrac{\partial W}{\partial\lambda_{3}% }~{}\cfrac{\partial\lambda_{3}}{\partialsymbol{C}}\\ &\displaystyle=symbol{R}^{T}\cdot\left[\cfrac{1}{2\lambda_{1}}~{}\cfrac{% \partial W}{\partial\lambda_{1}}~{}\mathbf{n}_{1}\otimes\mathbf{n}_{1}+\cfrac{% 1}{2\lambda_{2}}~{}\cfrac{\partial W}{\partial\lambda_{2}}~{}\mathbf{n}_{2}% \otimes\mathbf{n}_{2}+\cfrac{1}{2\lambda_{3}}~{}\cfrac{\partial W}{\partial% \lambda_{3}}~{}\mathbf{n}_{3}\otimes\mathbf{n}_{3}\right]\cdotsymbol{R}\end{aligned}
  84. s y m b o l σ = 2 J s y m b o l F W s y m b o l C \cdotsymbol F T = 2 J ( s y m b o l V \cdotsymbol R ) W s y m b o l C ( s y m b o l R T \cdotsymbol V ) symbol{\sigma}=\cfrac{2}{J}~{}symbol{F}\cdot\cfrac{\partial W}{\partial symbol% {C}}\cdotsymbol{F}^{T}=\cfrac{2}{J}~{}(symbol{V}\cdotsymbol{R})\cdot\cfrac{% \partial W}{\partial symbol{C}}\cdot(symbol{R}^{T}\cdotsymbol{V})
  85. W W
  86. s y m b o l σ = 2 J s y m b o l V [ 1 2 λ 1 W λ 1 𝐧 1 𝐧 1 + 1 2 λ 2 W λ 2 𝐧 2 𝐧 2 + 1 2 λ 3 W λ 3 𝐧 3 𝐧 3 ] \cdotsymbol V symbol{\sigma}=\cfrac{2}{J}~{}symbol{V}\cdot\left[\cfrac{1}{2\lambda_{1}}~{}% \cfrac{\partial W}{\partial\lambda_{1}}~{}\mathbf{n}_{1}\otimes\mathbf{n}_{1}+% \cfrac{1}{2\lambda_{2}}~{}\cfrac{\partial W}{\partial\lambda_{2}}~{}\mathbf{n}% _{2}\otimes\mathbf{n}_{2}+\cfrac{1}{2\lambda_{3}}~{}\cfrac{\partial W}{% \partial\lambda_{3}}~{}\mathbf{n}_{3}\otimes\mathbf{n}_{3}\right]\cdotsymbol{V}
  87. s y m b o l V symbol{V}
  88. s y m b o l V ( 𝐧 i 𝐧 i ) \cdotsymbol V = λ i 2 𝐧 i 𝐧 i ; i = 1 , 2 , 3. symbol{V}\cdot(\mathbf{n}_{i}\otimes\mathbf{n}_{i})\cdotsymbol{V}=\lambda_{i}^% {2}~{}\mathbf{n}_{i}\otimes\mathbf{n}_{i}~{};~{}~{}i=1,2,3.
  89. J = det ( s y m b o l F ) = det ( s y m b o l V ) det ( s y m b o l R ) = det ( s y m b o l V ) = λ 1 λ 2 λ 3 . J=\det(symbol{F})=\det(symbol{V})\det(symbol{R})=\det(symbol{V})=\lambda_{1}% \lambda_{2}\lambda_{3}~{}.
  90. s y m b o l σ = 1 λ 1 λ 2 λ 3 [ λ 1 W λ 1 𝐧 1 𝐧 1 + λ 2 W λ 2 𝐧 2 𝐧 2 + λ 3 W λ 3 𝐧 3 𝐧 3 ] symbol{\sigma}=\cfrac{1}{\lambda_{1}\lambda_{2}\lambda_{3}}~{}\left[\lambda_{1% }~{}\cfrac{\partial W}{\partial\lambda_{1}}~{}\mathbf{n}_{1}\otimes\mathbf{n}_% {1}+\lambda_{2}~{}\cfrac{\partial W}{\partial\lambda_{2}}~{}\mathbf{n}_{2}% \otimes\mathbf{n}_{2}+\lambda_{3}~{}\cfrac{\partial W}{\partial\lambda_{3}}~{}% \mathbf{n}_{3}\otimes\mathbf{n}_{3}\right]
  91. λ 1 λ 2 λ 3 = 1 \lambda_{1}\lambda_{2}\lambda_{3}=1
  92. W = W ( λ 1 , λ 2 ) W=W(\lambda_{1},\lambda_{2})
  93. s y m b o l σ = λ 1 W λ 1 𝐧 1 𝐧 1 + λ 2 W λ 2 𝐧 2 𝐧 2 + λ 3 W λ 3 𝐧 3 𝐧 3 - p s y m b o l 1 symbol{\sigma}=\lambda_{1}~{}\cfrac{\partial W}{\partial\lambda_{1}}~{}\mathbf% {n}_{1}\otimes\mathbf{n}_{1}+\lambda_{2}~{}\cfrac{\partial W}{\partial\lambda_% {2}}~{}\mathbf{n}_{2}\otimes\mathbf{n}_{2}+\lambda_{3}~{}\cfrac{\partial W}{% \partial\lambda_{3}}~{}\mathbf{n}_{3}\otimes\mathbf{n}_{3}-p~{}symbol{\mathit{% 1}}~{}
  94. σ 11 - σ 33 = λ 1 W λ 1 - λ 3 W λ 3 ; σ 22 - σ 33 = λ 2 W λ 2 - λ 3 W λ 3 \sigma_{11}-\sigma_{33}=\lambda_{1}~{}\cfrac{\partial W}{\partial\lambda_{1}}-% \lambda_{3}~{}\cfrac{\partial W}{\partial\lambda_{3}}~{};~{}~{}\sigma_{22}-% \sigma_{33}=\lambda_{2}~{}\cfrac{\partial W}{\partial\lambda_{2}}-\lambda_{3}~% {}\cfrac{\partial W}{\partial\lambda_{3}}
  95. λ 1 = λ 2 \lambda_{1}=\lambda_{2}
  96. σ 11 = σ 22 \sigma_{11}=\sigma_{22}
  97. σ 11 - σ 33 = σ 22 - σ 33 = λ 1 W λ 1 - λ 3 W λ 3 \sigma_{11}-\sigma_{33}=\sigma_{22}-\sigma_{33}=\lambda_{1}~{}\cfrac{\partial W% }{\partial\lambda_{1}}-\lambda_{3}~{}\cfrac{\partial W}{\partial\lambda_{3}}
  98. W ( s y m b o l F ) = W ^ ( I 1 , I 2 ) W(symbol{F})=\hat{W}(I_{1},I_{2})
  99. s y m b o l σ = - p s y m b o l 1 + 2 [ ( W ^ I 1 + I 1 W ^ I 2 ) s y m b o l B - W ^ I 2 s y m b o l B \cdotsymbol B ] = - p s y m b o l 1 + 2 [ ( W I ¯ 1 + I 1 W I ¯ 2 ) s y m b o l B ¯ - W I ¯ 2 s y m b o l B ¯ s y m b o l B ¯ ] = - p s y m b o l 1 + λ 1 W λ 1 𝐧 1 𝐧 1 + λ 2 W λ 2 𝐧 2 𝐧 2 + λ 3 W λ 3 𝐧 3 𝐧 3 \begin{aligned}\displaystyle symbol{\sigma}&\displaystyle=-p~{}symbol{\mathit{% 1}}+2\left[\left(\cfrac{\partial\hat{W}}{\partial I_{1}}+I_{1}~{}\cfrac{% \partial\hat{W}}{\partial I_{2}}\right)symbol{B}-\cfrac{\partial\hat{W}}{% \partial I_{2}}~{}symbol{B}\cdotsymbol{B}\right]\\ &\displaystyle=-p~{}symbol{\mathit{1}}+2\left[\left(\cfrac{\partial W}{% \partial\bar{I}_{1}}+I_{1}~{}\cfrac{\partial W}{\partial\bar{I}_{2}}\right)~{}% \bar{symbol{B}}-\cfrac{\partial W}{\partial\bar{I}_{2}}~{}\bar{symbol{B}}\cdot% \bar{symbol{B}}\right]\\ &\displaystyle=-p~{}symbol{\mathit{1}}+\lambda_{1}~{}\cfrac{\partial W}{% \partial\lambda_{1}}~{}\mathbf{n}_{1}\otimes\mathbf{n}_{1}+\lambda_{2}~{}% \cfrac{\partial W}{\partial\lambda_{2}}~{}\mathbf{n}_{2}\otimes\mathbf{n}_{2}+% \lambda_{3}~{}\cfrac{\partial W}{\partial\lambda_{3}}~{}\mathbf{n}_{3}\otimes% \mathbf{n}_{3}\end{aligned}
  100. p p
  101. σ 11 - σ 33 = λ 1 W λ 1 - λ 3 W λ 3 ; σ 22 - σ 33 = λ 2 W λ 2 - λ 3 W λ 3 \sigma_{11}-\sigma_{33}=\lambda_{1}~{}\cfrac{\partial W}{\partial\lambda_{1}}-% \lambda_{3}~{}\cfrac{\partial W}{\partial\lambda_{3}}~{};~{}~{}\sigma_{22}-% \sigma_{33}=\lambda_{2}~{}\cfrac{\partial W}{\partial\lambda_{2}}-\lambda_{3}~% {}\cfrac{\partial W}{\partial\lambda_{3}}
  102. I 1 = I 2 I_{1}=I_{2}
  103. s y m b o l σ = 2 W I 1 s y m b o l B - p s y m b o l 1 . symbol{\sigma}=2\cfrac{\partial W}{\partial I_{1}}~{}symbol{B}-p~{}symbol{% \mathit{1}}~{}.
  104. λ 1 = λ 2 \lambda_{1}=\lambda_{2}
  105. σ 11 - σ 33 = σ 22 - σ 33 = λ 1 W λ 1 - λ 3 W λ 3 \sigma_{11}-\sigma_{33}=\sigma_{22}-\sigma_{33}=\lambda_{1}~{}\cfrac{\partial W% }{\partial\lambda_{1}}-\lambda_{3}~{}\cfrac{\partial W}{\partial\lambda_{3}}
  106. s y m b o l σ = λ tr ( s y m b o l ε ) s y m b o l 1 + 2 \musymbol ε symbol{\sigma}=\lambda~{}\mathrm{tr}(symbol{\varepsilon})~{}symbol{\mathit{1}}% +2\musymbol{\varepsilon}
  107. λ , μ \lambda,\mu
  108. W = 1 2 λ [ tr ( s y m b o l ε ) ] 2 + μ tr ( s y m b o l ε 2 ) W=\tfrac{1}{2}\lambda~{}[\mathrm{tr}(symbol{\varepsilon})]^{2}+\mu~{}\mathrm{% tr}(symbol{\varepsilon}^{2})
  109. tr ( s y m b o l ε ) = 0 \mathrm{tr}(symbol{\varepsilon})=0
  110. W = μ tr ( s y m b o l ε 2 ) W=\mu~{}\mathrm{tr}(symbol{\varepsilon}^{2})
  111. W ( λ 1 , λ 2 , λ 3 ) W(\lambda_{1},\lambda_{2},\lambda_{3})
  112. W ( 1 , 1 , 1 ) = 0 ; W λ i ( 1 , 1 , 1 ) = 0 2 W λ i λ j ( 1 , 1 , 1 ) = λ + 2 μ δ i j \begin{aligned}&\displaystyle W(1,1,1)=0~{};~{}~{}\cfrac{\partial W}{\partial% \lambda_{i}}(1,1,1)=0\\ &\displaystyle\cfrac{\partial^{2}W}{\partial\lambda_{i}\partial\lambda_{j}}(1,% 1,1)=\lambda+2\mu\delta_{ij}\end{aligned}
  113. W ( 1 , 1 , 1 ) = 0 \displaystyle W(1,1,1)=0
  114. I 1 I_{1}
  115. I 1 I_{1}
  116. W = W ( I 1 ) W=W(I_{1})
  117. I 1 = 3 , λ i = λ j = 1 I_{1}=3,\lambda_{i}=\lambda_{j}=1
  118. W ( I 1 ) | I 1 = 3 = 0 and W I 1 | I 1 = 3 = μ 2 . W(I_{1})\biggr|_{I_{1}=3}=0\quad\,\text{and}\quad\cfrac{\partial W}{\partial I% _{1}}\biggr|_{I_{1}=3}=\frac{\mu}{2}\,.
  119. W λ i = W I 1 I 1 λ i = 2 λ i W I 1 and 2 W λ i λ j = 2 δ i j W I 1 + 4 λ i λ j 2 W I 1 2 . \cfrac{\partial W}{\partial\lambda_{i}}=\cfrac{\partial W}{\partial I_{1}}% \cfrac{\partial I_{1}}{\partial\lambda_{i}}=2\lambda_{i}\cfrac{\partial W}{% \partial I_{1}}\quad\,\text{and}\quad\cfrac{\partial^{2}W}{\partial\lambda_{i}% \partial\lambda_{j}}=2\delta_{ij}\cfrac{\partial W}{\partial I_{1}}+4\lambda_{% i}\lambda_{j}\cfrac{\partial^{2}W}{\partial I_{1}^{2}}\,.

Hyperexponential_distribution.html

  1. f X ( x ) = i = 1 n f Y i ( x ) p i , f_{X}(x)=\sum_{i=1}^{n}f_{Y_{i}}(x)\;p_{i},
  2. E [ X ] = - x f ( x ) d x = i = 1 n p i 0 x λ i e - λ i x d x = i = 1 n p i λ i E[X]=\int_{-\infty}^{\infty}xf(x)\,dx=\sum_{i=1}^{n}p_{i}\int_{0}^{\infty}x% \lambda_{i}e^{-\lambda_{i}x}\,dx=\sum_{i=1}^{n}\frac{p_{i}}{\lambda_{i}}
  3. E [ X 2 ] = - x 2 f ( x ) d x = i = 1 n p i 0 x 2 λ i e - λ i x d x = i = 1 n 2 λ i 2 p i , E\!\left[X^{2}\right]=\int_{-\infty}^{\infty}x^{2}f(x)\,dx=\sum_{i=1}^{n}p_{i}% \int_{0}^{\infty}x^{2}\lambda_{i}e^{-\lambda_{i}x}\,dx=\sum_{i=1}^{n}\frac{2}{% \lambda_{i}^{2}}p_{i},
  4. Var [ X ] = E [ X 2 ] - E [ X ] 2 = i = 1 n 2 λ i 2 p i - [ i = 1 n p i λ i ] 2 = [ i = 1 n p i λ i ] 2 + i = 1 n j = 1 n p i p j ( 1 λ i - 1 λ j ) 2 . \operatorname{Var}[X]=E\!\left[X^{2}\right]-E\!\left[X\right]^{2}=\sum_{i=1}^{% n}\frac{2}{\lambda_{i}^{2}}p_{i}-\left[\sum_{i=1}^{n}\frac{p_{i}}{\lambda_{i}}% \right]^{2}=\left[\sum_{i=1}^{n}\frac{p_{i}}{\lambda_{i}}\right]^{2}+\sum_{i=1% }^{n}\sum_{j=1}^{n}p_{i}p_{j}\left(\frac{1}{\lambda_{i}}-\frac{1}{\lambda_{j}}% \right)^{2}.
  5. E [ e t x ] = - e t x f ( x ) d x = i = 1 n p i 0 e t x λ i e - λ i x d x = i = 1 n λ i λ i - t p i . E\!\left[e^{tx}\right]=\int_{-\infty}^{\infty}e^{tx}f(x)\,dx=\sum_{i=1}^{n}p_{% i}\int_{0}^{\infty}e^{tx}\lambda_{i}e^{-\lambda_{i}x}\,dx=\sum_{i=1}^{n}\frac{% \lambda_{i}}{\lambda_{i}-t}p_{i}.

Hyperplane_separation_theorem.html

  1. δ = inf { | x | x K } . \delta=\inf\{|x|\mid x\in K\}.
  2. x j x_{j}
  3. | x j | δ |x_{j}|\to\delta
  4. ( x i + x j ) / 2 (x_{i}+x_{j})/2
  5. | x i + x j | 2 4 δ 2 |x_{i}+x_{j}|^{2}\geq 4\delta^{2}
  6. | x i - x j | 2 = 2 | x i | 2 + 2 | x j | 2 - | x i + x j | 2 2 | x i | 2 + 2 | x j | 2 - 4 δ 2 0 |x_{i}-x_{j}|^{2}=2|x_{i}|^{2}+2|x_{j}|^{2}-|x_{i}+x_{j}|^{2}\leq 2|x_{i}|^{2}% +2|x_{j}|^{2}-4\delta^{2}\to 0
  7. i , j i,j\to\infty
  8. x i x_{i}
  9. | x - y | 2 2 | x | 2 + 2 | y | 2 - 4 δ 2 = 0 |x-y|^{2}\leq 2|x|^{2}+2|y|^{2}-4\delta^{2}=0
  10. \square
  11. K = A + ( - B ) = { x - y x A , y B } . K=A+(-B)=\{x-y\mid x\in A,y\in B\}.
  12. - B -B
  13. K ¯ \overline{K}
  14. K ¯ \overline{K}
  15. v + t ( x - v ) , 0 t 1 v+t(x-v),\,0\leq t\leq 1
  16. K ¯ \overline{K}
  17. | v | 2 | v + t ( x - v ) | 2 = | v | 2 + 2 t v , x - v + t 2 | x - v | 2 |v|^{2}\leq|v+t(x-v)|^{2}=|v|^{2}+2t\langle v,x-v\rangle+t^{2}|x-v|^{2}
  18. 0 < t 1 0<t\leq 1
  19. 0 2 v , x - 2 | v | 2 + t | x - v | 2 0\leq 2\langle v,x\rangle-2|v|^{2}+t|x-v|^{2}
  20. t 0 t\to 0
  21. x , v | v | 2 \langle x,v\rangle\geq|v|^{2}
  22. x - y , v | v | 2 \langle x-y,v\rangle\geq|v|^{2}
  23. inf x A x , v | v | 2 + sup y B y , v . \inf_{x\in A}\langle x,v\rangle\geq|v|^{2}+\sup_{y\in B}\langle y,v\rangle.
  24. K n K_{n}
  25. K n K_{n}
  26. v n v_{n}
  27. x , v n 0 \langle x,v_{n}\rangle\geq 0
  28. x K n x\in K_{n}
  29. v n v_{n}
  30. x , v 0 \langle x,v\rangle\geq 0
  31. , v = c \langle\cdot,v\rangle=c
  32. x , v c \langle x,v\rangle\geq c
  33. \square
  34. A = { ( x , y ) : x 0 } A=\{(x,y):x\leq 0\}
  35. B = { ( x , y ) : x > 0 , y 1 / x } . B=\{(x,y):x>0,y\geq 1/x\}.

Ideal_point.html

  1. π / - K \pi/-K
  2. d ( p , q ) = 1 2 log | q a | | b p | | p a | | b q | , d(p,q)=\frac{1}{2}\log\frac{\left|qa\right|\left|bp\right|}{\left|pa\right|% \left|bq\right|},
  3. d ( p , q ) = log | q a | | b p | | p a | | b q | , d(p,q)=\log\frac{\left|qa\right|\left|bp\right|}{\left|pa\right|\left|bq\right% |},
  4. y = y=\infty

Identity_theorem.html

  1. S = { z D | f ( k ) ( z ) = g ( k ) ( z ) for all k 0 } . S=\{z\in D\;|\;f^{(k)}(z)=g^{(k)}(z)\quad\mbox{for all}~{}\;k\geq 0\}.
  2. ( f - g ) ( z ) \displaystyle(f-g)(z)

Image_gradient.html

  1. f = f x x ^ + f y y ^ \nabla f=\frac{\partial f}{\partial x}\hat{x}+\frac{\partial f}{\partial y}% \hat{y}
  2. f x \textstyle\frac{\partial f}{\partial x}
  3. f y \textstyle\frac{\partial f}{\partial y}
  4. θ = atan2 ( f y , f x ) \theta=\operatorname{atan2}\left(\frac{\partial f}{\partial y},\frac{\partial f% }{\partial x}\right)

Imaging_spectroscopy.html

  1. λ / Δ λ \lambda/\Delta\lambda
  2. p = A * x p=A*x\,
  3. p p
  4. A A
  5. x x
  6. x x

Implementation_of_mathematics_in_set_theory.html

  1. ϕ \phi
  2. ϕ \phi
  3. ϕ \phi
  4. ϕ \phi
  5. ϕ \phi
  6. { x ϕ } \{x\mid\phi\}
  7. x A ϕ x\in A\leftrightarrow\phi
  8. ϕ \phi
  9. { x B ϕ } \{x\in B\mid\phi\}
  10. { x x B ϕ } \{x\mid x\in B\wedge\phi\}
  11. { f ( x 1 , , x n ) ϕ } \{f(x_{1},\ldots,x_{n})\mid\phi\}
  12. { z x 1 , , x n ( z = f ( x 1 , , x n ) ϕ ) } \{z\mid\exists x_{1},\ldots,x_{n}\,(z=f(x_{1},\dots,x_{n})\wedge\phi)\}
  13. f ( x 1 , , x n ) f(x_{1},\ldots,x_{n})
  14. { x x x } \{x\mid x\not\in x\}
  15. ω \omega
  16. ω \omega
  17. = def . { x : x x } \left.\varnothing\right.\overset{\mathrm{def.}}{=}\left\{x:x\neq x\right\}
  18. x x
  19. { x } \{x\}
  20. x x
  21. { x } = def . { y : y = x } \left\{x\right\}\overset{\mathrm{def.}}{=}\left\{y:y=x\right\}
  22. x x
  23. y y
  24. { x , y } \{x,y\}
  25. x x
  26. y y
  27. { x , y } = def . { z : z = x z = y } \left\{x,y\right\}\overset{\mathrm{def.}}{=}\left\{z:z=x\vee z=y\right\}
  28. x y = def . { z : z x z y } \left.x\cup y\right.\overset{\mathrm{def.}}{=}\left\{z:z\in x\vee z\in y\right\}
  29. n n
  30. n n
  31. { x 1 , , x n , x n + 1 } = def . { x 1 , , x n } { x n + 1 } \left\{x_{1},\ldots,x_{n},x_{n+1}\right\}\overset{\mathrm{def.}}{=}\left\{x_{1% },\ldots,x_{n}\right\}\cup\left\{x_{n+1}\right\}
  32. x y = { x , y } x\cup y=\bigcup\{x,y\}
  33. ( x , y ) = def { { { x } , } , { { y } } } (x,y)\overset{\mathrm{def}}{=}\{\{\{x\},\emptyset\},\{\{y\}\}\}
  34. n > 1 n>1
  35. ( x , y ) = def . { { x } , { x , y } } (x,y)\overset{\mathrm{def.}}{=}\{\{x\},\{x,y\}\}
  36. ( x , y ) (x,y)
  37. ( x , y ) (x,y)
  38. ( x , y ) = ( z , w ) x = z y = w (x,y)=(z,w)\equiv x=z\wedge y=w
  39. R R
  40. { ( x , y ) x R y } \{(x,y)\mid xRy\}
  41. { z π 1 ( z ) R π 2 ( z ) } \{z\mid\pi_{1}(z)R\pi_{2}(z)\}
  42. R R
  43. x R y xRy
  44. ( x , y ) R \left(x,y\right)\in R
  45. { ( x , y ) x y } \{(x,y)\mid x\in y\}
  46. x x
  47. y y
  48. x x
  49. y y
  50. R R
  51. S S
  52. R R
  53. { ( y , x ) : x R y } \left\{\left(y,x\right):xRy\right\}
  54. R R
  55. { x : y ( x R y ) } \left\{x:\exists y\left(xRy\right)\right\}
  56. R R
  57. R R
  58. R R
  59. R R
  60. x x
  61. R R
  62. { y : y R x } \left\{y:yRx\right\}
  63. x x
  64. R R
  65. D D
  66. x x
  67. z R y zRy
  68. y D y\in D
  69. R R
  70. R | S R|S
  71. R R
  72. S S
  73. { ( x , z ) : y ( x R y y S z ) } \left\{\left(x,z\right):\exists y\,\left(xRy\wedge ySz\right)\right\}
  74. R R
  75. B B
  76. ( R , B ) \left(R,B\right)
  77. A A
  78. B B
  79. A × B = { ( a , b ) : a A b B } A\times B=\left\{\left(a,b\right):a\in A\wedge b\in B\right\}
  80. 𝒫 ( A B ) \mathcal{P}\!\left(A\cup B\right)
  81. { ( x , y ) A × B : x R y } \left\{\left(x,y\right)\in A\times B:xRy\right\}
  82. x x
  83. y y
  84. R R
  85. x R y xRy
  86. R R
  87. R R
  88. x R x xRx
  89. x x
  90. R R
  91. x , y ( x R y y R x ) \forall x,y\,(xRy\to yRx)
  92. x , y , z ( x R y y R z x R z ) \forall x,y,z\,(xRy\wedge yRz\rightarrow xRz)
  93. x , y ( x R y y R x x = y ) \forall x,y\,(xRy\wedge yRx\rightarrow x=y)
  94. S S
  95. R R
  96. x S \ \exists x\in S
  97. R R
  98. S S
  99. x , y x,y
  100. R R
  101. x = y x=y
  102. x x
  103. y y
  104. R R
  105. R R
  106. R R
  107. R R
  108. R R
  109. x , y x,y
  110. R R
  111. x R y xRy
  112. y R x yRx
  113. R R
  114. R R
  115. R R
  116. F F
  117. x , y , z ( x F y x F z y = z ) \forall x,y,z\,\left(xFy\wedge xFz\to y=z\right)
  118. F F
  119. { ( x , y ) : x F y } \left\{\left(x,y\right):xFy\right\}
  120. F F
  121. x , y , z ( ( x , y ) F ( x , z ) F y = z ) \forall x,y,z\,\left(\left(x,y\right)\in F\wedge\left(x,z\right)\in F\to y=z\right)
  122. F ( x ) F\!\left(x\right)
  123. y y
  124. x F y xFy
  125. x x
  126. F F
  127. y y
  128. f f
  129. x x
  130. y y
  131. y y
  132. ( x , y ) F \left(x,y\right)\in F
  133. F ( x ) F\!\left(x\right)
  134. F F
  135. x x
  136. F ( x ) F\!\left(x\right)
  137. F F
  138. F ( x ) F\!\left(x\right)
  139. F [ A ] F\left[A\right]
  140. { y : x ( x A y = F ( x ) ) } \left\{y:\exists x\,\left(x\in A\wedge y=F\!\left(x\right)\right)\right\}
  141. A A
  142. { F ( x ) : x A } \left\{F\!\left(x\right):x\in A\right\}
  143. A A
  144. F F
  145. F [ A ] F\left[A\right]
  146. F [ A ] F\left[A\right]
  147. A A
  148. F F
  149. F [ A ] F\left[A\right]
  150. I ( x ) = x I\!\left(x\right)=x
  151. I ( x ) I\!\left(x\right)
  152. S ( x ) = { x } S\!\left(x\right)=\left\{x\right\}
  153. S ( x ) S\!\left(x\right)
  154. f f
  155. g g
  156. f f
  157. g g
  158. g f g\circ f
  159. f g f\mid g
  160. g f g\circ f
  161. ( g f ) ( x ) = g ( f ( x ) ) \left(g\circ f\right)\!\left(x\right)=g\!\left(f\!\left(x\right)\right)
  162. f f
  163. g g
  164. f f
  165. f ( - 1 ) f^{\left(}-1\right)
  166. f f
  167. A A
  168. i A i_{A}
  169. { ( x , x ) x A } \left\{\left(x,x\right)\mid x\in A\right\}
  170. A A
  171. B B
  172. f f
  173. A A
  174. B B
  175. f f
  176. A A
  177. B B
  178. f f
  179. A A
  180. B B
  181. f f
  182. A A
  183. B B
  184. f f
  185. A A
  186. B B
  187. A A
  188. B B
  189. f f
  190. B B
  191. A A
  192. B B
  193. f f
  194. | A | = | B | |A|=|B|
  195. | A | |A|
  196. | B | |B|
  197. A B A\sim B
  198. | A | | B | |A|\leq|B|
  199. i A i_{A}
  200. | A | = | B | |A|=|B|
  201. f - 1 f^{-1}
  202. | B | = | A | |B|=|A|
  203. | A | = | B | |A|=|B|
  204. | B | = | C | |B|=|C|
  205. g f g\circ f
  206. | A | = | C | |A|=|C|
  207. | A | | B | |A|\leq|B|
  208. | A | | B | | B | | A | | A | = | B | |A|\leq|B|\wedge|B|\leq|A|\rightarrow|A|=|B|
  209. | A | | B | | B | | A | |A|\leq|B|\vee|B|\leq|A|
  210. \emptyset
  211. y { y } y\cup\{y\}
  212. y A y\in A
  213. { x A B ( B y ( y B y { y } B ) x B ) } \{x\in A\mid\forall B\,(\emptyset\in B\wedge\forall y\,(y\in B\rightarrow y% \cup\{y\}\in B)\rightarrow x\in B)\}
  214. y y { y } y\mapsto y\cup\{y\}
  215. A A
  216. n N n\in N
  217. | n | = | A | |n|=|A|
  218. | A | |A|
  219. ( ( x , ) , x ) ((x,\emptyset),x)
  220. x x
  221. ( ( x , y { y } ) , z { z } ) ((x,y\cup\{y\}),z\cup\{z\})
  222. ( ( x , y ) , z ) ((x,y),z)
  223. y { y } y\cup\{y\}
  224. { A F ( F x , y ( x F x { y } F ) A F ) } \{A\mid\forall F\,(\emptyset\in F\wedge\forall x,y\,(x\in F\rightarrow x\cup\{% y\}\in F)\rightarrow A\in F)\}
  225. A F i n A\in Fin
  226. | A | |A|
  227. { B A B } \{B\mid A\sim B\}
  228. { | A | A F i n } \{|A|\mid A\in Fin\}
  229. V F i n V\not\in Fin
  230. | A | |A|
  231. | A { x } | |A\cup\{x\}|
  232. x A x\not\in A
  233. | A B | |A\cup B|
  234. { A B , C ( B m C n B C = A = B C ) } \{A\mid\exists B,C\,(B\in m\wedge C\in n\wedge B\cap C=\emptyset\wedge A=B\cup C)\}
  235. [ x ] R = { y A x R y } [x]_{R}=\{y\in A\mid xRy\}
  236. P P
  237. A = P A=\bigcup P
  238. { [ x ] R x A } \{[x]_{R}\mid x\in A\}
  239. { ( x , y ) A P ( x A y A ) } \{(x,y)\mid\exists A\in P\,(x\in A\wedge y\in A)\}
  240. [ x ] R [x]_{R}
  241. y R x yRx
  242. z R x zRx
  243. [ x ] R [x]_{R}
  244. [ x ] R [x]_{R}
  245. x [ x ] R x\mapsto[x]_{R}
  246. { x } [ x ] R \{x\}\mapsto[x]_{R}
  247. [ x ] R [x]_{R}
  248. W 1 W_{1}
  249. W 2 W_{2}
  250. W 1 W 2 W_{1}\sim W_{2}
  251. W 1 W_{1}
  252. W 2 W_{2}
  253. x W 1 y f ( x ) W 2 f ( y ) xW_{1}y\leftrightarrow f(x)W_{2}f(y)
  254. \leq
  255. { ( x , y ) x y x y } \{(x,y)\mid x\leq y\wedge x\neq y\}
  256. A A \bigcup A\subseteq A
  257. α β \alpha\leq\beta
  258. W 1 α W_{1}\in\alpha
  259. W 2 β W_{2}\in\beta
  260. Ω \Omega
  261. Ω \Omega
  262. Ω \Omega
  263. Ω \Omega
  264. α \alpha
  265. α \alpha
  266. α \alpha
  267. α \alpha
  268. α \alpha
  269. T 4 ( α ) T^{4}(\alpha)
  270. α \alpha
  271. T ( α ) T(\alpha)
  272. W ι = { ( { x } , { y } ) x W y } W^{\iota}=\{(\{x\},\{y\})\mid xWy\}
  273. W α W\in\alpha
  274. Ω \Omega
  275. T 4 ( Ω ) T^{4}(\Omega)
  276. T 4 ( Ω ) < Ω T^{4}(\Omega)<\Omega
  277. T 4 T^{4}
  278. T 2 T^{2}
  279. x { x } x\mapsto\{x\}
  280. W ι W^{\iota}
  281. T 4 ( Ω ) < Ω T^{4}(\Omega)<\Omega
  282. Ω > T ( Ω ) > T 2 ( Ω ) \Omega>T(\Omega)>T^{2}(\Omega)\ldots
  283. A \in\lceil A
  284. α \alpha
  285. α \alpha
  286. T ( α ) T(\alpha)
  287. | A | = def { B B A } |A|=_{\mathrm{def}}\{B\mid B\sim A\}
  288. | A | |A|
  289. | A | < | P ( A ) | . |A|<|P(A)|.
  290. | P 1 ( A ) | < | P ( A ) | |P_{1}(A)|<|P(A)|
  291. P 1 ( A ) P_{1}(A)
  292. | P 1 ( V ) | < | P ( V ) | |P_{1}(V)|<|P(V)|
  293. x { x } x\mapsto\{x\}
  294. P 1 ( V ) P_{1}(V)
  295. | P 1 ( V ) | < | P ( V ) | | V | |P_{1}(V)|<|P(V)|\ll|V|
  296. \ll
  297. T ( | A | ) = | P 1 ( A ) | T(|A|)=|P_{1}(A)|
  298. | A | = | P 1 ( A ) | = T ( | A | ) |A|=|P_{1}(A)|=T(|A|)
  299. | A | |A|
  300. ( x { x } ) A (x\mapsto\{x\})\lceil A
  301. | A | + | B | = { C D C A D B C D = } |A|+|B|=\{C\cup D\mid C\sim A\wedge D\sim B\wedge C\cap D=\emptyset\}
  302. | A | | B | |A|\cdot|B|
  303. | A × B | |A\times B|
  304. | A | | B | |A|\cdot|B|
  305. T - 2 ( | A × B | ) T^{-2}(|A\times B|)
  306. B A B^{A}
  307. | B | | A | |B|^{|A|}
  308. T - 3 ( | B A | ) T^{-3}(|B^{A}|)
  309. T - 1 T^{-1}
  310. T - 3 T^{-3}
  311. 2 | V | 2^{|V|}
  312. | B | | A | |B|^{|A|}
  313. | B A | |B^{A}|
  314. κ κ = κ \kappa\cdot\kappa=\kappa
  315. | V | | V | = | V | |V|\cdot|V|=|V|
  316. | V | | V | = T - 2 ( | V × V | ) |V|\cdot|V|=T^{-2}(|V\times V|)
  317. | V | |V|
  318. | V × V | = T 2 ( | V | ) = | P 1 2 ( V ) | |V\times V|=T^{2}(|V|)=|P_{1}^{2}(V)|
  319. ( a , b ) (a,b)
  320. { { c } } \{\{c\}\}
  321. ( a , b ) (a,b)
  322. { { c } } \{\{c\}\}
  323. ( a , b ) (a,b)
  324. T ( n ) T(n)
  325. | N | |N|
  326. ω \omega
  327. T ( n ) = n T(n)=n
  328. T ( n ) = n T(n)=n
  329. | { 1 , , n } | = n |\{1,\ldots,n\}|=n
  330. | { 1 , , n } | = T 2 ( n ) |\{1,\ldots,n\}|=T^{2}(n)
  331. a A a\in A
  332. f ( a ) \bigcup f(a)
  333. f ( a ) f(a)
  334. f - 1 ( { a } ) f^{-1}(\{a\})
  335. f ( a ) = { a } f(a)=\{a\}
  336. a A a\in A
  337. p q \frac{p}{q}
  338. ( p , q ) (p,q)
  339. p q = r s p s = q r \frac{p}{q}=\frac{r}{s}\leftrightarrow ps=qr
  340. \sim
  341. ( p , q ) ( r , s ) p s = q r (p,q)\sim(r,s)\leftrightarrow ps=qr
  342. m - n m-n
  343. ( m , n ) (m,n)
  344. \sim
  345. ( m , n ) ( r , s ) m + s = n + r (m,n)\sim(r,s)\leftrightarrow m+s=n+r
  346. ( x 1 , x 2 , , x n ) (x_{1},x_{2},\ldots,x_{n})
  347. ( x 1 , ( x 2 , , x n ) ) (x_{1},(x_{2},\ldots,x_{n}))
  348. A 1 × A 2 × × A n = A 1 × ( A 2 × × A n ) A_{1}\times A_{2}\times\ldots\times A_{n}=A_{1}\times(A_{2}\times\ldots\times A% _{n})
  349. Π i I A i \Pi_{i\in I}A_{i}
  350. f ( i ) A i f(i)\in A_{i}
  351. A i A_{i}
  352. { i } \{i\}
  353. P 1 ( I ) P_{1}(I)
  354. A i A_{i}
  355. Π i I A i \Pi_{i\in I}A_{i}
  356. f ( i ) A i f(i)\in A_{i}
  357. f ( i ) A i = A ( { i } ) f(i)\in A_{i}=A(\{i\})
  358. A i A_{i}
  359. Π i I | A i | \Pi_{i\in I}|A_{i}|
  360. Π i I A i \Pi_{i\in I}A_{i}
  361. | A i | |A_{i}|
  362. T - 1 ( | Π i I A i | ) T^{-1}(|\Pi_{i\in I}A_{i}|)
  363. P 1 ( I ) P_{1}(I)
  364. A i A_{i}
  365. A ( { i } ) A(\{i\})
  366. Σ i I A i \Sigma_{i\in I}A_{i}
  367. { ( i , a ) a A i } \{(i,a)\mid a\in A_{i}\}
  368. A i A_{i}
  369. Σ i I | A i | \Sigma_{i\in I}|A_{i}|
  370. | Σ i I A i | |\Sigma_{i\in I}A_{i}|
  371. Σ i I A i \Sigma_{i\in I}A_{i}
  372. { ( i , a ) a A i } \{(i,a)\mid a\in A_{i}\}
  373. A i A_{i}
  374. A ( i ) A(i)
  375. i = { i } i=\{i\}
  376. V 0 = V_{0}=\emptyset
  377. V α + 1 = P ( V α ) V_{\alpha+1}=P(V_{\alpha})
  378. V λ = { V β β < λ } V_{\lambda}=\bigcup\{V_{\beta}\mid\beta<\lambda\}
  379. λ \lambda
  380. α \alpha
  381. A V α + 1 - V α A\in V_{\alpha+1}-V_{\alpha}
  382. | P ( V ω + α ) | |P(V_{\omega+\alpha})|
  383. α \beth_{\alpha}
  384. P ( A ) P(A)
  385. α \beth_{\alpha}
  386. 2 | A | 2^{|A|}
  387. T - 1 ( | { 0 , 1 } A | ) T^{-1}(|\{0,1\}^{A}|)
  388. { 0 , 1 } \{0,1\}
  389. | { 0 , 1 } A | = | P ( A ) | |\{0,1\}^{A}|=|P(A)|
  390. \beth
  391. | N | |N|
  392. 2 | A | 2^{|A|}
  393. | A | |A|
  394. W α W_{\alpha}
  395. W W
  396. W W
  397. { y y W x } \{y\mid yWx\}
  398. α \alpha
  399. α \beth_{\alpha}
  400. α \alpha
  401. \beth
  402. α \aleph_{\alpha}
  403. α \alpha
  404. 0 = | N | \aleph_{0}=|N|
  405. α \alpha
  406. W α W_{\alpha}
  407. T C ( A ) TC(A)
  408. T C ( A ) \in\lceil TC(A)
  409. T C ( A ) \in\lceil TC(A)
  410. x x
  411. [ x ] [x]
  412. [ x ] E [ y ] [x]E[y]
  413. [ x ] [x]
  414. A B A\in B
  415. x ι = { ( { a } , { b } ) ( a , b ) x } x^{\iota}=\{(\{a\},\{b\})\mid(a,b)\in x\}
  416. T ( [ x ] ) T([x])
  417. [ x ι ] [x^{\iota}]
  418. [ x ] E [ y ] T ( [ x ] ) = T ( [ y ] ) [x]E[y]\leftrightarrow T([x])=T([y])
  419. { x ι x S } \{x^{\iota}\mid x\in S\}
  420. x ι x^{\iota}
  421. { a } \{a\}
  422. x ι x^{\iota}
  423. ( x , { a } ) (x,\{a\})
  424. [ x ι ] = T ( [ x ] ) [x^{\iota}]=T([x])
  425. [ y ] [y]
  426. T [ v ] v T[v]\neq v
  427. α \alpha
  428. R α R_{\alpha}
  429. | R α | = α |R_{\alpha}|=\beth_{\alpha}
  430. R α R_{\alpha}
  431. R α R_{\alpha}
  432. R T ( α ) R_{T(\alpha)}
  433. T ( α ) < α T(\alpha)<\alpha
  434. [ x ] N F U [ y ] [x]\in_{NFU}[y]
  435. T ( [ x ] ) E [ y ] [ y ] R T ( α ) + 1 T([x])E[y]\wedge[y]\in R_{T(\alpha)+1}
  436. E N F U E_{NFU}
  437. \in

Impossible_world.html

  1. A A \vdash A\Rightarrow\ \vdash\Box A
  2. A \vdash A
  3. A A
  4. A A
  5. A \Box A
  6. \Box
  7. A A
  8. A A
  9. A \Box A
  10. A \Box A
  11. A A
  12. A \Box A
  13. A \vdash A
  14. A A
  15. ( A ( A B ) ) ( A B ) (A\rightarrow(A\rightarrow B))\rightarrow(A\rightarrow B)

Imprecise_probability.html

  1. P ¯ ( A ) \underline{P}(A)
  2. P ¯ ( A ) \overline{P}(A)
  3. P ¯ ( A ) = P ¯ ( A ) \underline{P}(A)=\overline{P}(A)
  4. A A
  5. P ¯ ( A ) = 0 \underline{P}(A)=0
  6. P ¯ ( A ) = 1 \overline{P}(A)=1
  7. P ( A ) P(A)
  8. P ¯ ( A c ) = 1 - P ¯ ( A ) \underline{P}(A^{c})=1-\overline{P}(A)
  9. A c A^{c}
  10. A A
  11. [ P ¯ ( A ) , P ¯ ( A ) ] [\underline{P}(A),\overline{P}(A)]

In-phase_and_quadrature_components.html

  1. f , \scriptstyle f,
  2. sin ( 2 π f t + ϕ ) , \scriptstyle\sin(2\pi ft+\phi),
  3. sin ( 2 π f t ) cos ( ϕ ) \scriptstyle\sin(2\pi ft)\cos(\phi)
  4. sin ( 2 π f t + π / 2 ) sin ( ϕ ) , \scriptstyle\sin(2\pi ft+\pi/2)\sin(\phi),
  5. 1 / f \scriptstyle 1/f
  6. f , \scriptstyle f,
  7. sin [ 2 π f t + ϕ ( t ) ] = sin ( 2 π f t ) cos [ ϕ ( t ) ] in-phase + sin ( 2 π f t + π 2 ) cos ( 2 π f t ) sin [ ϕ ( t ) ] quadrature . \sin[2\pi ft+\phi(t)]\ =\ \underbrace{\sin(2\pi ft)\cdot\cos[\phi(t)]}_{\text{% in-phase}}\ +\ \underbrace{\overbrace{\sin\left(2\pi ft+\tfrac{\pi}{2}\right)}% ^{\cos(2\pi ft)}\cdot\sin[\phi(t)]}_{\text{quadrature}}.
  8. 2 π f t , \scriptstyle 2\pi ft,

Inada_conditions.html

  1. f ( x ) f(x)
  2. f ( x ) f(x)
  3. f ( 0 ) = 0 f(0)=0
  4. x i x_{i}
  5. f ( x ) / x i > 0 \partial f(x)/\partial x_{i}>0
  6. x i x_{i}
  7. 2 f ( x ) / x i 2 < 0 \partial^{2}f(x)/\partial x_{i}^{2}<0
  8. x i x_{i}
  9. lim x i 0 f ( x ) / x i = + \lim_{x_{i}\to 0}\partial f(x)/\partial x_{i}=+\infty
  10. x i x_{i}
  11. lim x i + f ( x ) / x i = 0 \lim_{x_{i}\to+\infty}\partial f(x)/\partial x_{i}=0

Incomplete_markets.html

  1. q 1 q_{1}
  2. q 2 q_{2}
  3. q 2 q_{2}
  4. q 1 q_{1}
  5. q 1 q_{1}
  6. q 2 q_{2}

Increasing_process.html

  1. ( X t ) t M (X_{t})_{t\in M}
  2. X t X_{t}
  3. 0 = X 0 X t 1 . 0=X_{0}\leq X_{t_{1}}\leq\cdots.
  4. M M

Indeterminate_equation.html

  1. a n x n + a n - 1 x n - 1 + + a 2 x 2 + a 1 x + a 0 = 0 , a_{n}x^{n}+a_{n-1}x^{n-1}+\dots+a_{2}x^{2}+a_{1}x+a_{0}=0,
  2. a n ( x - b ) n = 0 a_{n}(x-b)^{n}=0
  3. A x 2 + B x y + C y 2 + D x + E y + F = 0 , Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0,
  4. x 2 - P y 2 = 1 , \ x^{2}-Py^{2}=1,
  5. x 2 + y 2 = z 2 , x^{2}+y^{2}=z^{2},
  6. a m + b n = c k , a^{m}+b^{n}=c^{k},

Indiana_Pi_Bill.html

  1. π \pi
  2. π \pi
  3. π \pi
  4. π = 4 1.25 = 3.2 \pi=\frac{4}{1.25}=3.2
  5. 2 = 10 7 1.429 \sqrt{2}=\frac{10}{7}\approx 1.429
  6. 2 \sqrt{2}
  7. 1 / 5 {1}/{5}
  8. 1 / 5 {1}/{5}
  9. 1 / 5 {1}/{5}
  10. π / 4 {\pi}/{4}
  11. π \pi
  12. π / 4 {\pi}/{4}

Indicator_value.html

  1. I n d V a l i j = A i j * B i j * 100 IndVal_{ij}=A_{ij}*B_{ij}*100

Indiscernibles.html

  1. [ φ ( a , b ) and φ ( b , a ) and φ ( a , c ) and φ ( c , a ) and φ ( b , c ) and φ ( c , b ) ] [ ¬ φ ( a , b ) and ¬ φ ( b , a ) and ¬ φ ( a , c ) and ¬ φ ( c , a ) and ¬ φ ( b , c ) and ¬ φ ( c , b ) ] . [\varphi(a,b)\and\varphi(b,a)\and\varphi(a,c)\and\varphi(c,a)\and\varphi(b,c)% \and\varphi(c,b)][\lnot\varphi(a,b)\and\lnot\varphi(b,a)\and\lnot\varphi(a,c)% \and\lnot\varphi(c,a)\and\lnot\varphi(b,c)\and\lnot\varphi(c,b)]\,.
  2. ( [ φ ( a , b ) and φ ( a , c ) and φ ( b , c ) ] [ ¬ φ ( a , b ) and ¬ φ ( a , c ) and ¬ φ ( b , c ) ] ) and ( [ φ ( b , a ) and φ ( c , a ) and φ ( c , b ) ] [ ¬ φ ( b , a ) and ¬ φ ( c , a ) and ¬ φ ( c , b ) ] ) . ([\varphi(a,b)\and\varphi(a,c)\and\varphi(b,c)][\lnot\varphi(a,b)\and\lnot% \varphi(a,c)\and\lnot\varphi(b,c)])\and([\varphi(b,a)\and\varphi(c,a)\and% \varphi(c,b)][\lnot\varphi(b,a)\and\lnot\varphi(c,a)\and\lnot\varphi(c,b)])\,.

Induced_topology.html

  1. X 0 , X 1 X_{0},X_{1}
  2. f : X 0 X 1 f:X_{0}\to X_{1}
  3. τ 0 \tau_{0}
  4. X 0 X_{0}
  5. X 1 X_{1}
  6. f f
  7. { U 1 X 1 | f - 1 ( U 1 ) τ 0 } \{U_{1}\subseteq X_{1}|f^{-1}(U_{1})\in\tau_{0}\}
  8. τ 1 \tau_{1}
  9. X 1 X_{1}
  10. X 0 X_{0}
  11. f f
  12. { f - 1 ( U 1 ) | U 1 τ 1 } \{f^{-1}(U_{1})|U_{1}\in\tau_{1}\}
  13. X 0 = { - 2 , - 1 , 1 , 2 } X_{0}=\{-2,-1,1,2\}
  14. { { - 2 , - 1 } , { 1 , 2 } } \{\{-2,-1\},\{1,2\}\}
  15. X 1 = { - 1 , 0 , 1 } X_{1}=\{-1,0,1\}
  16. f : X 0 X 1 f:X_{0}\to X_{1}
  17. f ( - 2 ) = - 1 , f ( - 1 ) = 0 , f ( 1 ) = 0 , f ( 2 ) = 1 f(-2)=-1,f(-1)=0,f(1)=0,f(2)=1
  18. τ 1 = { f ( U 0 ) | U 0 τ 0 } \tau_{1}=\{f(U_{0})|U_{0}\in\tau_{0}\}
  19. { { - 1 , 0 } , { 0 , 1 } } τ 1 \{\{-1,0\},\{0,1\}\}\subseteq\tau_{1}
  20. { - 1 , 0 } { 0 , 1 } τ 1 \{-1,0\}\cap\{0,1\}\notin\tau_{1}
  21. τ 1 \tau_{1}
  22. X 1 X_{1}
  23. f f
  24. f f
  25. ( X 0 , τ 0 ) ( X 1 , τ 1 ) (X_{0},\tau_{0})\to(X_{1},\tau_{1})
  26. X 1 X_{1}
  27. τ 0 \tau_{0}
  28. X 0 X_{0}
  29. f f
  30. f f
  31. ( X 0 , τ 0 ) ( X 1 , τ 1 ) (X_{0},\tau_{0})\to(X_{1},\tau_{1})
  32. X 0 X_{0}
  33. f f
  34. f f
  35. X 0 X_{0}

Inelastic_neutron_scattering.html

  1. S ( 𝐐 , ω ) S(\mathbf{Q},\omega)
  2. χ ′′ ( 𝐐 , ω ) \chi^{\prime\prime}(\mathbf{Q},\omega)
  3. 𝐐 \mathbf{Q}
  4. ω \hbar\omega
  5. ω \omega

Infinite_dihedral_group.html

  1. r , s s 2 = 1 , s r s = r - 1 \langle r,s\mid s^{2}=1,srs=r^{-1}\rangle\,\!
  2. x , y x 2 = y 2 = 1 \langle x,y\mid x^{2}=y^{2}=1\rangle\,\!

Info-gap_decision_theory.html

  1. α \alpha
  2. 𝒰 ( α , u ~ ) \mathcal{U}(\alpha,\tilde{u})
  3. u ~ \tilde{u}
  4. 𝔘 := α 𝒰 ( α , u ~ ) . \mathfrak{U}:=\bigcup_{\alpha}\mathcal{U}(\alpha,\tilde{u}).
  5. u ~ \tilde{u}
  6. 𝔘 , \mathfrak{U},
  7. 𝒰 ( α , u ~ ) = { u : | u - u ~ | α } , α 0 \mathcal{U}(\alpha,{\tilde{u}})=\left\{u:\ |u-{\tilde{u}}|\leq\alpha\right\},% \qquad\alpha\geq 0
  8. u ~ = $ 100 , \tilde{u}=\$100,
  9. α ^ \hat{\alpha}
  10. β ^ \hat{\beta}
  11. 𝒰 ( α , u ~ ) = { u : | u - u ~ | α u ~ } , α 0 \mathcal{U}(\alpha,{\tilde{u}})=\left\{u:\ |u-{\tilde{u}}|\leq\alpha\tilde{u}% \right\},\qquad\alpha\geq 0
  12. u ( x ) , u(x),
  13. u ~ ( x ) , {\tilde{u}}(x),
  14. u ~ {\tilde{u}}
  15. 𝒰 ( α , u ~ ) = { u ( x ) : | u ( x ) - u ~ ( x ) | α u ~ ( x ) , for all x X } , α 0. \mathcal{U}(\alpha,{\tilde{u}})=\left\{u(x):\ |u(x)-{\tilde{u}}(x)|\leq\alpha{% \tilde{u}}(x),\ \mbox{for all}~{}\ x\in X\right\},\ \ \ \alpha\geq 0.
  16. U ( α , y ) U(\alpha,y)
  17. 𝒰 ( α , u ~ ) = { u ( x ) : u ( x ) U ( α , u ~ ( x ) ) , for all x X } , α 0. \mathcal{U}(\alpha,{\tilde{u}})=\left\{u(x):\ u(x)\in U(\alpha,{\tilde{u}}(x))% ,\ \mbox{for all}~{}\ x\in X\right\},\ \ \ \alpha\geq 0.
  18. q q
  19. q q
  20. q q
  21. q q^{\prime}
  22. q q^{\prime}
  23. q q
  24. q q
  25. q q
  26. u ( x ) u(x)\!\,
  27. u ~ ( x ) {\tilde{u}}(x)
  28. 𝒰 ( α , u ~ ) = { u ( x ) : | u ( x ) - u ~ ( x ) | α u ~ ( x ) , for all x } , α 0 \mathcal{U}(\alpha,{\tilde{u}})=\left\{u(x):\ |u(x)-{\tilde{u}}(x)|\leq\alpha{% \tilde{u}}(x),\ \mbox{for all}~{}\ x\right\},\ \ \ \alpha\geq 0
  29. α \alpha
  30. 𝒰 ( α , u ~ ) \mathcal{U}(\alpha,{\tilde{u}})
  31. u ( x ) u(x)\!\,
  32. u ~ ( x ) {\tilde{u}}(x)
  33. α \alpha
  34. 𝒰 ( α , u ~ ) \mathcal{U}(\alpha,{\tilde{u}})
  35. α < α \alpha<\alpha^{\prime}
  36. 𝒰 ( α , u ~ ) 𝒰 ( α , u ~ ) \mathcal{U}(\alpha,{\tilde{u}})\ \subseteq\ \mathcal{U}(\alpha^{\prime},{% \tilde{u}})
  37. 𝒰 ( 0 , u ~ ) \mathcal{U}(0,{\tilde{u}})
  38. 𝒰 ( 0 , u ~ ) = { u ~ } \mathcal{U}(0,{\tilde{u}})=\{{\tilde{u}}\}
  39. 𝒰 ( α , u ) \mathcal{U}(\alpha,u)
  40. α \alpha
  41. α \alpha
  42. u ~ {\tilde{u}}
  43. u u
  44. u ~ , \tilde{u},
  45. ϕ : 𝔘 [ 0 , + ) \phi\colon\mathfrak{U}\to[0,+\infty)
  46. ϕ ( u ) := min { α u 𝒰 ( α , u ~ ) } \phi(u):=\min\{\alpha\mid u\in\mathcal{U}(\alpha,{\tilde{u}})\}
  47. 𝒰 ( α , u ~ ) \mathcal{U}(\alpha,\tilde{u})
  48. ϕ \phi
  49. 𝒰 ( α , u ~ ) := ϕ - 1 ( [ 0 , α ] ) \mathcal{U}(\alpha,\tilde{u}):=\phi^{-1}([0,\alpha])
  50. α \alpha
  51. α \alpha
  52. ϕ : 𝔘 [ 0 , + ) , \phi\colon\mathfrak{U}\to[0,+\infty),
  53. ϕ - 1 ( 0 ) = { u ~ } \phi^{-1}(0)=\{\tilde{u}\}
  54. ϕ ( u ) = 0 \phi(u)=0
  55. u = u ~ u=\tilde{u}
  56. ϕ ( u ) := d ( u ~ , u ) , \phi(u):=d(\tilde{u},u),
  57. 𝒰 ( α , u ~ ) = { u d ( u ~ , u ) α } . \mathcal{U}(\alpha,\tilde{u})=\{u\mid d(\tilde{u},u)\leq\alpha\}.
  58. ϕ - 1 ( 0 ) = { u ~ } \phi^{-1}(0)=\{\tilde{u}\}
  59. u 1 𝒰 ( α , u ~ ) u_{1}\in\mathcal{U}(\alpha,\tilde{u})
  60. α > 1 , \alpha>1,
  61. α = 1 \alpha=1
  62. u 1 ϕ - 1 ( [ 0 , 1 ] ) , u_{1}\in\phi^{-1}([0,1]),
  63. u 1 𝒰 ( 1 , u ~ ) . u_{1}\not\in\mathcal{U}(1,\tilde{u}).
  64. ϵ , \epsilon,
  65. q q
  66. α \alpha
  67. α ^ ( q ) = max { α : minimal requirements are always satisfied } {\hat{\alpha}}(q)=\max\{\alpha:\ \mbox{minimal requirements are always % satisfied}~{}\}
  68. β ^ ( q ) = min { α : sweeping success is possible } {\hat{\beta}}(q)=\min\{\alpha:\ \mbox{sweeping success is possible}~{}\}
  69. α ^ ( q ) = max { α : minimal requirements are satisfied for all u 𝒰 ( α , u ~ ) } {\hat{\alpha}}(q)=\max\{\alpha:\ \mbox{minimal requirements are satisfied for % all }~{}u\in\mathcal{U}(\alpha,\tilde{u})\}
  70. β ^ ( q ) = min { α : windfall is achieved for at least one u 𝒰 ( α , u ~ ) } {\hat{\beta}}(q)=\min\{\alpha:\ \mbox{windfall is achieved for at least one }~% {}u\in\mathcal{U}(\alpha,\tilde{u})\}
  71. α ^ ( q ) {\hat{\alpha}}(q)
  72. q q
  73. α \alpha
  74. α ^ ( q ) {\hat{\alpha}}(q)
  75. α ^ ( q ) {\hat{\alpha}}(q)
  76. β ^ ( q ) {\hat{\beta}}(q)
  77. α \alpha
  78. q q
  79. β ^ ( q ) {\hat{\beta}}(q)
  80. β ^ ( q ) {\hat{\beta}}(q)
  81. β ^ ( q ) {\hat{\beta}}(q)
  82. α ^ ( q ) {\hat{\alpha}}(q)
  83. β ^ ( q ) {\hat{\beta}}(q)
  84. α ^ ( q ) {\hat{\alpha}}(q)
  85. β ^ ( q ) {\hat{\beta}}(q)
  86. q q
  87. q , q , q,\,q^{\prime},\,\ldots
  88. α ^ ( q ) , α ^ ( q ) , {\hat{\alpha}}(q),\,{\hat{\alpha}}(q^{\prime}),\,\ldots
  89. q q
  90. β ^ ( q ) {\hat{\beta}}(q)
  91. q q
  92. R ( q , u ) R(q,u)
  93. q q
  94. u u
  95. R ( q , u ) R(q,u)
  96. r c {r_{\rm c}}
  97. r w {r_{\rm w}}
  98. r c {r_{\rm c}}
  99. r c {r_{\rm c}}
  100. r w {r_{\rm w}}
  101. r w {r_{\rm w}}
  102. r c {r_{\rm c}}
  103. r w > r c {r_{\rm w}}>{r_{\rm c}}
  104. α ^ ( q , r c ) = max { α : r c min u 𝒰 ( α , u ~ ) R ( q , u ) } {\hat{\alpha}}(q,{r_{\rm c}})=\max\left\{\alpha:r_{\rm c}\leq\min_{u\in% \mathcal{U}(\alpha,\tilde{u})}R(q,u)\right\}
  105. β ^ ( q , r w ) = min { α : r w max u 𝒰 ( α , u ~ ) R ( q , u ) } {\hat{\beta}}(q,{r_{\rm w}})=\min\left\{\alpha:r_{\rm w}\leq\max_{u\in\mathcal% {U}(\alpha,\tilde{u})}R(q,u)\right\}
  106. α ^ ( q , r c ) {\hat{\alpha}}(q,{r_{\rm c}})
  107. r c {r_{\rm c}}
  108. β ^ ( q , r w ) {\hat{\beta}}(q,{r_{\rm w}})
  109. r w {r_{\rm w}}
  110. R ( q , u ) R(q,u)
  111. 𝒬 \mathcal{Q}
  112. q q
  113. α ^ ( q , r c ) {\hat{\alpha}}(q,{r_{\rm c}})
  114. q r q q\succ_{\rm r}q^{\prime}
  115. α ^ ( q , r c ) > α ^ ( q , r c ) . {\hat{\alpha}}(q,{r_{\rm c}})>{\hat{\alpha}}(q^{\prime},{r_{\rm c}}).
  116. r c {r_{\rm c}}
  117. α ^ , \hat{\alpha},
  118. α ^ ( r c ) , \hat{\alpha}({r_{\rm c}}),
  119. q ^ < m t p l > c \hat{q}_{<}mtpl>{{\rm c}}
  120. q ^ < m t p l > c ( r c ) , {\hat{q}_{<}mtpl>{{\rm c}}}({r_{\rm c}}),
  121. α ^ ( r c ) \displaystyle\hat{\alpha}({r_{\rm c}})
  122. q ^ < m t p l > c ( r c ) {\hat{q}_{<}mtpl>{{\rm c}}}({r_{\rm c}})
  123. r c {r_{\rm c}}
  124. β ^ ( q , r c ) {\hat{\beta}}(q,{r_{\rm c}})
  125. q w q q\succ_{\rm w}q^{\prime}
  126. β ^ ( q , r w ) < β ^ ( q , r w ) . {\hat{\beta}}(q,{r_{\rm w}})<{\hat{\beta}}(q^{\prime},{r_{\rm w}}).
  127. q ^ < m t p l > w ( r w ) {\hat{q}_{<}mtpl>{{\rm w}}}({r_{\rm w}})
  128. β ^ , \hat{\beta},
  129. β ^ ( r w ) , \hat{\beta}({r_{\rm w}}),
  130. q ^ < m t p l > w \hat{q}_{<}mtpl>{{\rm w}}
  131. q ^ < m t p l > w ( r w ) , {\hat{q}_{<}mtpl>{{\rm w}}}({r_{\rm w}}),
  132. β ^ ( r w ) \displaystyle\hat{\beta}({r_{\rm w}})
  133. q ^ < m t p l > c ( r c ) {\hat{q}_{<}mtpl>{{\rm c}}}({r_{\rm c}})
  134. q ^ < m t p l > w ( r w ) {\hat{q}_{<}mtpl>{{\rm w}}}({r_{\rm w}})
  135. r c {r_{\rm c}}
  136. r w . {r_{\rm w}}.
  137. 𝔘 \mathfrak{U}
  138. [ - 100 % , + % ) [-100\%,+\infty\%)
  139. α = 1 \alpha=1
  140. u ~ \displaystyle\tilde{u}
  141. u \displaystyle u
  142. u ~ \displaystyle\tilde{u}
  143. u ~ \displaystyle\tilde{u}
  144. α \alpha
  145. α \alpha
  146. ρ ^ ( p ~ ) := max { ρ 0 : p P ( s ) , p B ( ρ , p ~ ) } \hat{\rho}(\tilde{p}):=\max\ \{\rho\geq 0:p\in P(s),\forall p\in B(\rho,\tilde% {p})\}
  147. B ( ρ , p ~ ) B(\rho,\tilde{p})
  148. ρ \rho
  149. p ~ \tilde{p}
  150. P ( s ) P(s)
  151. p p
  152. r c R ( q , p ) r_{c}\leq R(q,p)
  153. A A
  154. B B
  155. A A
  156. B B
  157. u ~ \displaystyle\tilde{u}
  158. u \displaystyle u
  159. u ~ \displaystyle\tilde{u}
  160. u ~ \displaystyle\tilde{u}
  161. u \displaystyle u
  162. 𝔘 \displaystyle\mathfrak{U}
  163. u \displaystyle u
  164. u ~ \ \displaystyle\tilde{u}
  165. u \displaystyle u
  166. 𝔘 \displaystyle\mathfrak{U}
  167. u ~ \displaystyle\tilde{u}
  168. u \displaystyle\ u
  169. u \displaystyle\ u
  170. u \displaystyle u^{\circ}
  171. u \ \displaystyle u
  172. u ~ \displaystyle\tilde{u}
  173. u \displaystyle u^{\circ}
  174. u ~ \displaystyle\tilde{u}
  175. u ~ \displaystyle\tilde{u}
  176. u ~ \ \displaystyle\tilde{u}
  177. u \ \displaystyle u
  178. 𝔘 \ \displaystyle\mathfrak{U}
  179. 𝔘 \ \displaystyle\mathfrak{U}
  180. 𝔘 \ \displaystyle\mathfrak{U}
  181. 𝔘 \ \displaystyle\mathfrak{U}
  182. α ^ ( q , r c ) \ \displaystyle\hat{\alpha}(q,r_{c})
  183. q 𝒬 \ \displaystyle q\in\mathcal{Q}
  184. 𝒰 ( α * , u ~ ) 𝔘 \ \displaystyle\ \mathcal{U}(\alpha^{*},\tilde{u})\subseteq\mathfrak{U}
  185. α * = α ^ ( q , r c ) + ε \ \displaystyle\alpha^{*}=\hat{\alpha}(q,r_{c})+\varepsilon
  186. ε > 0 \ \displaystyle\varepsilon>0
  187. q \ \displaystyle q
  188. α ^ ( q , r c ) \ \displaystyle\hat{\alpha}(q,r_{c})
  189. 𝔘 \ \displaystyle\mathfrak{U}
  190. 𝔘 \ \displaystyle\mathfrak{U}
  191. q \ \displaystyle q
  192. 𝔘 \ \displaystyle\mathfrak{U}
  193. 𝔘 \ \displaystyle\mathfrak{U}
  194. 𝒰 ( α ^ ( q , r c ) + ε , u ~ ) 𝔘 \mathcal{U}(\hat{\alpha}(q,r_{c})+\varepsilon,\tilde{u})\subseteq\mathfrak{U}
  195. ε > 0 . \ \displaystyle\varepsilon>0\ .
  196. \Box
  197. 𝒰 ( α * , u ~ ) \ \displaystyle\ \mathcal{U}(\alpha^{*},\tilde{u})
  198. 𝔘 \ \displaystyle\mathfrak{U}
  199. 𝔘 ′′′ \ \displaystyle\mathfrak{U}^{\prime\prime\prime}
  200. u ~ \ \displaystyle\tilde{u}
  201. α ^ ( q , r c ) \ \displaystyle\hat{\alpha}(q,r_{c})
  202. β ^ ( q , r c ) \ \displaystyle\hat{\beta}(q,r_{c})
  203. 𝒰 ( α ^ ( q , r c ) , u ~ ) \mathcal{U}(\hat{\alpha}(q,r_{c}),\tilde{u})
  204. 𝒰 ( β ^ ( q , r c ) , u ~ ) \mathcal{U}(\hat{\beta}(q,r_{c}),\tilde{u})
  205. u ~ \ \displaystyle\tilde{u}
  206. 𝔘 = ( - , ) , \mathfrak{U}=(-\infty,\infty),
  207. u ~ = 0 \ \displaystyle\tilde{u}=0
  208. q ^ \ \displaystyle\hat{q}
  209. 𝒰 ( α ^ ( q ^ , r c ) , u ~ ) = ( - 2 , 2 ) \mathcal{U}(\hat{\alpha}(\hat{q},r_{c}),\tilde{u})=(-2,2)
  210. 𝒰 ( α ^ ( q , r c ) + ε , u ~ ) \ \displaystyle\mathcal{U}(\hat{\alpha}(q,r_{c})+\varepsilon,\tilde{u})
  211. q ^ \ \displaystyle\hat{q}
  212. u ~ \ \displaystyle\tilde{u}
  213. 𝒰 ( α , u ~ ) \ \displaystyle\mathcal{U}(\alpha,\tilde{u})
  214. α 0 \ \displaystyle\alpha\geq 0
  215. α \ \displaystyle\alpha
  216. α ^ ( q , r c ) \ \displaystyle\hat{\alpha}(q,r_{c})
  217. R \ \displaystyle R
  218. r c \ \displaystyle r_{c}
  219. α ^ \ \displaystyle\hat{\alpha}
  220. α \ \displaystyle\alpha
  221. r c R ( q , u ) , u 𝒰 ( α , u ~ ) r_{c}\leq R(q,u),\forall u\in\mathcal{U}(\alpha,\tilde{u})
  222. α 0 \ \displaystyle\alpha\geq 0
  223. 𝒰 ( α , u ~ ) , α 0 \displaystyle\ \mathcal{U}(\alpha,\tilde{u}),\alpha\geq 0
  224. D \ \displaystyle D
  225. { S ( d ) : d D } \ \displaystyle\{S(d):d\in D\}
  226. D \ \displaystyle D
  227. g = g ( d , s ) \ \displaystyle g=g(d,s)
  228. ( d , s ) \ \displaystyle(d,s)
  229. g \ \displaystyle g
  230. d D \ \displaystyle d\in D
  231. d \ \displaystyle d
  232. s S ( d ) \ \displaystyle s\in S(d)
  233. g ( d , s ) \ \displaystyle g(d,s)
  234. z * = O p t D M o p t N a t u r e g ( d , s ) d D s S ( d ) \begin{array}[]{cccc}z^{*}=&\stackrel{DM}{\mathop{Opt}}&\stackrel{Nature}{% \mathop{opt}}&g(d,s)\\ &d\in D&s\in S(d)&\end{array}
  235. O p t \ \displaystyle\mathop{Opt}
  236. o p t \displaystyle\mathop{opt}
  237. max \ \displaystyle\max
  238. min \ \displaystyle\min
  239. O p t = o p t \ \displaystyle\mathop{Opt}=\mathop{opt}
  240. O p t o p t \ \displaystyle\mathop{Opt}\neq\mathop{opt}
  241. Worst-Case Pessimism Best-Case Optimism M a x i m i n M i n i m a x M i n i m i n M a x i m a x max d D min s S ( d ) g ( d , s ) min d D max s S ( d ) g ( d , s ) min d D min s S ( d ) g ( d , s ) max d D max s S ( d ) g ( d , s ) \begin{array}[]{c| |c}\,\textit{Worst-Case\ Pessimism}&\,\textit{Best-Case\ % Optimism}\\ \hline Maximin\ \ \ \ \ \ \ \ \ \ \ Minimax&Minimin\ \ \ \ \ \ \ \ \ \ \ \ \ % Maximax\\ \displaystyle\max_{d\in D}\,\min_{s\in S(d)}\,g(d,s)\ \ \ \displaystyle\min_{d% \in D}\,\max_{s\in S(d)}\,g(d,s)&\displaystyle\min_{d\in D}\,\min_{s\in S(d)}% \,g(d,s)\ \ \ \displaystyle\max_{d\in D}\,\max_{s\in S(d)}\,g(d,s)\end{array}
  242. max d D min s S ( d ) g ( d , s ) \ \displaystyle\max_{d\in D}\,\min_{s\in S(d)}\,g(d,s)
  243. d D \ \displaystyle d\in D
  244. g ( d , s ) \ \displaystyle g(d,s)
  245. d \ \displaystyle d
  246. S ( d ) \ \displaystyle S(d)
  247. g ( d , s ) \ \displaystyle g(d,s)
  248. S ( d ) \ \displaystyle S(d)
  249. g ( d , s ) \ \displaystyle g(d,s)
  250. min d D min s S ( d ) g ( d , s ) \ \displaystyle\min_{d\in D}\,\min_{s\in S(d)}\,g(d,s)
  251. d D \ \displaystyle d\in D
  252. g ( d , s ) \ \displaystyle g(d,s)
  253. d \ \displaystyle d
  254. S ( d ) \ \displaystyle S(d)
  255. g ( d , s ) \ \displaystyle g(d,s)
  256. S ( d ) \ \displaystyle S(d)
  257. g ( d , s ) \ \displaystyle g(d,s)
  258. α \ \displaystyle\alpha
  259. u \ \displaystyle u
  260. 𝒰 ( α , u ~ ) \ \displaystyle\mathcal{U}(\alpha,\tilde{u})
  261. u \ \displaystyle u
  262. ( q , α ) \ \displaystyle(q,\alpha)
  263. u 𝒰 ( α , u ~ ) \ \displaystyle u\in\mathcal{U}(\alpha,\tilde{u})
  264. r c R ( q , u ) \ \displaystyle r_{c}\leq R(q,u)
  265. R ( q , u ) \ \displaystyle R(q,u)
  266. 𝒰 ( α , u ~ ) \ \displaystyle\mathcal{U}(\alpha,\tilde{u})
  267. φ ( q , α , u ) := { α , r c R ( q , u ) - , r c > R ( q , u ) , q 𝒬 , α 0 , u 𝒰 ( α , u ~ ) \varphi(q,\alpha,u):=\begin{cases}\quad\alpha&,\ \ r_{c}\leq R(q,u)\\ -\infty&,\ \ r_{c}>R(q,u)\end{cases}\ ,\ q\in\mathcal{Q},\alpha\geq 0,u\in% \mathcal{U}(\alpha,\tilde{u})
  268. ( q , α , u ) \ \ (q,\alpha,u)
  269. r c R ( q , u ) α φ ( q , α , u ) r_{c}\leq R(q,u)\ \ \ \longleftrightarrow\ \ \ \alpha\leq\varphi(q,\alpha,u)
  270. φ ( q , α , u ) \ \displaystyle\varphi(q,\alpha,u)
  271. q \ \displaystyle q
  272. α ^ ( q , r c ) := max α 0 min u 𝒰 ( α , u ~ ) φ ( q , α , u ) \ \displaystyle\hat{\alpha}(q,r_{c}):=\max_{\alpha\geq 0}\,\min_{u\in\mathcal{% U}(\alpha,\tilde{u})}\,\varphi(q,\alpha,u)
  273. α 0 \ \displaystyle\alpha\geq 0
  274. φ ( q , α , u ) \ \displaystyle\varphi(q,\alpha,u)
  275. α \ \displaystyle\alpha
  276. u 𝒰 ( α , u ~ ) \ \displaystyle u\in\mathcal{U}(\alpha,\tilde{u})
  277. φ ( q , α , u ) \ \displaystyle\varphi(q,\alpha,u)
  278. 𝒰 ( α , u ~ ) \ \displaystyle\mathcal{U}(\alpha,\tilde{u})
  279. φ ( q , α , u ) \ \displaystyle\varphi(q,\alpha,u)
  280. α \ \displaystyle\alpha
  281. u 𝒰 ( α , u ~ ) \ \displaystyle u\in\mathcal{U}(\alpha,\tilde{u})
  282. α ^ ( q , r c ) = max { α : r c min u 𝒰 ( α , u ~ ) R ( q , u ) } = max α 0 min u 𝒰 ( α , u ~ ) φ ( q , α , u ) {\hat{\alpha}}(q,{r_{c}})=\max\left\{\alpha:\ {r_{\rm c}}\leq\min_{u\in% \mathcal{U}(\alpha,\tilde{u})}R(q,u)\right\}=\max_{\alpha\geq 0}\min_{u\in% \mathcal{U}(\alpha,\tilde{u})}\varphi(q,\alpha,u)\quad\quad\Box
  283. β ^ ( q , r c ) = min { α : r c max u 𝒰 ( α , u ~ ) R ( q , u ) } = min α 0 min u 𝒰 ( α , u ~ ) ψ ( q , α , u ) {\hat{\beta}}(q,{r_{c}})=\min\left\{\alpha:\ {r_{c}}\leq\max_{u\in\mathcal{U}(% \alpha,\tilde{u})}R(q,u)\right\}=\min_{\alpha\geq 0}\min_{u\in\mathcal{U}(% \alpha,\tilde{u})}\psi(q,\alpha,u)
  284. ψ ( q , α , u ) = { α , r c R ( q , u ) , r c > R ( q , u ) , α 0 , u 𝒰 ( α , u ~ ) \psi(q,\alpha,u)=\left\{\begin{matrix}\alpha&,&{r_{c}}\leq R(q,u)\\ \infty&,&{r_{c}}>R(q,u)\end{matrix}\right.\ ,\ \alpha\geq 0,u\in\mathcal{U}(% \alpha,\tilde{u})
  285. ( q , α , u ) \ \ (q,\alpha,u)
  286. r w R ( q , u ) α ψ ( q , α , u ) r_{w}\leq R(q,u)\ \ \ \longleftrightarrow\ \ \ \alpha\geq\psi(q,\alpha,u)
  287. ( q , α ) \ \displaystyle(q,\alpha)
  288. ψ ( q , α , u ) \ \displaystyle\psi(q,\alpha,u)
  289. 𝒰 ( α , u ~ ) \ \displaystyle\mathcal{U}(\alpha,\tilde{u})
  290. 𝑀𝑜𝑑𝑒𝑙 Classical Format MP Format Maximin: max d D min s S ( d ) g ( d , s ) max d D , α { α : α min s S ( d ) g ( d , s ) } Minimin: min d D min s S ( d ) g ( d , s ) min d D , α { α : α min s S ( d ) g ( d , s ) } \begin{array}[]{c|c|c}\,\textit{Model}&\,\textit{Classical\ Format}&\,\textit{% MP\ Format}\\ \hline\,\textit{Maximin:}&\displaystyle\max_{d\in D}\ \min_{s\in S(d)}\ g(d,s)% &\displaystyle\max_{d\in D,\alpha\in\mathbb{R}}\{\alpha:\alpha\leq\min_{s\in S% (d)}g(d,s)\}\\ \,\textit{Minimin:}&\displaystyle\min_{d\in D}\ \min_{s\in S(d)}\ g(d,s)&% \displaystyle\min_{d\in D,\alpha\in\mathbb{R}}\{\alpha:\alpha\geq\min_{s\in S(% d)}g(d,s)\}\end{array}
  291. 𝑀𝑜𝑑𝑒𝑙 Info-Gap Format MP Format Classical Format 𝑅𝑜𝑏𝑢𝑠𝑡𝑛𝑒𝑠𝑠 max { α : r c min u 𝒰 ( α , u ~ ) R ( q , u ) } max { α : α min u 𝒰 ( α , u ~ ) φ ( q , α , u ) } max α 0 min u 𝒰 ( α , u ~ ) φ ( q , α , u ) 𝑂𝑝𝑝𝑜𝑟𝑡𝑢𝑛𝑒𝑛𝑒𝑠𝑠 min { α : r c max u 𝒰 ( α , u ~ ) R ( q , u ) } min { α : α min u 𝒰 ( α , u ~ ) ψ ( q , α , u ) } min α 0 min u 𝒰 ( α , u ~ ) ψ ( q , α , u ) \begin{array}[]{c|c|c|c}\,\textit{Model}&\,\textit{Info-Gap\ Format}&\,\textit% {MP\ Format}&\,\textit{Classical\ Format}\\ \hline\,\textit{Robustness}&\displaystyle\max\{\alpha:r_{c}\leq\min_{u\in% \mathcal{U}(\alpha,\tilde{u})}R(q,u)\}&\displaystyle\displaystyle\max\{\alpha:% \alpha\leq\min_{u\in\mathcal{U}(\alpha,\tilde{u})}\varphi(q,\alpha,u)\}&% \displaystyle\max_{\alpha\geq 0}\ \min_{u\in\mathcal{U}(\alpha,\tilde{u})}\ % \varphi(q,\alpha,u)\\ \,\textit{Opportuneness}&\displaystyle\min\{\alpha:r_{c}\leq\max_{u\in\mathcal% {U}(\alpha,\tilde{u})}R(q,u)\}&\displaystyle\displaystyle\min\{\alpha:\alpha% \geq\min_{u\in\mathcal{U}(\alpha,\tilde{u})}\psi(q,\alpha,u)\}&\displaystyle% \min_{\alpha\geq 0}\ \min_{u\in\mathcal{U}(\alpha,\tilde{u})}\ \psi(q,\alpha,u% )\end{array}
  292. ( q , α , u ) \ \displaystyle(q,\alpha,u)
  293. α φ ( q , α , u ) r c R ( q , u ) \alpha\leq\varphi(q,\alpha,u)\ \ \ \longleftrightarrow\ \ \ r_{c}\leq R(q,u)
  294. α ψ ( q , α , u ) r w R ( q , u ) \alpha\geq\psi(q,\alpha,u)\ \ \ \longleftrightarrow\ \ \ r_{w}\leq R(q,u)
  295. R ( q , u ) \ \displaystyle R(q,u)
  296. φ ( q , α , u ) \ \displaystyle\varphi(q,\alpha,u)
  297. R ( q , u ) \ \displaystyle R(q,u)
  298. ψ ( q , α , u ) \ \displaystyle\psi(q,\alpha,u)
  299. ( q , α ) \ \displaystyle(q,\alpha)
  300. 𝒰 ( α , u ~ ) \ \displaystyle\mathcal{U}(\alpha,\tilde{u})
  301. q \ \displaystyle q
  302. α \ \displaystyle\alpha
  303. u \ \displaystyle u
  304. 𝒰 ( α , u ~ ) \ \displaystyle\mathcal{U}(\alpha,\tilde{u})
  305. r c R ( q , u ) \ \displaystyle r_{c}\leq R(q,u)
  306. ( q , α ) \ \displaystyle(q,\alpha)
  307. 𝒰 ( α , u ~ ) \ \displaystyle\mathcal{U}(\alpha,\tilde{u})
  308. q \ \displaystyle q
  309. α \ \displaystyle\alpha
  310. u \ \displaystyle u
  311. 𝒰 ( α , u ~ ) \ \displaystyle\mathcal{U}(\alpha,\tilde{u})
  312. r w R ( q , u ) \ \displaystyle r_{w}\leq R(q,u)
  313. u \ \displaystyle u
  314. \ \clubsuit
  315. r f ( x ) 1 I ( x ) r\leq f(x)\ \ \longleftrightarrow\ \ 1\leq I(x)
  316. I ( x ) := { 1 , r f ( x ) 0 , r > f ( x ) , x X I(x):=\begin{cases}1&,\ \ r\leq f(x)\\ 0&,\ \ r>f(x)\end{cases}\ ,\ x\in X
  317. x X , r f ( x ) x = arg max x X I ( x ) x\in X,r\leq f(x)\ \ \ \longleftrightarrow\ \ \ x=\arg\,\max_{x\in X}I(x)
  318. min \ \displaystyle\min
  319. max \ \displaystyle\max
  320. z ( q ) := max { α : R ( q , u ) C , u 𝒰 ( α , u ~ ) } z(q):=\max\{\alpha:R(q,u)\in C,\forall u\in\mathcal{U}(\alpha,\tilde{u})\}
  321. C \ \displaystyle C
  322. R \ R
  323. 𝒬 × 𝔘 \ \displaystyle\mathcal{Q}\times\mathfrak{U}
  324. R \ \displaystyle R
  325. R ( q , u ) C , u 𝒰 ( α , u ~ ) R(q,u)\in C\ ,\ \forall u\in\mathcal{U}(\alpha,\tilde{u})
  326. ( q , α . u ) \ \displaystyle(q,\alpha.u)
  327. R ( q , u ) C α I ( q , α , u ) R(q,u)\in C\ \ \ \longleftrightarrow\ \ \ \alpha\leq I(q,\alpha,u)
  328. I ( q , α , u ) := { α , R ( q , u ) C - , R ( q , u ) C , q 𝒬 , u 𝒰 ( α , u ~ ) I(q,\alpha,u):=\begin{cases}\quad\alpha&,\ \ R(q,u)\in C\\ -\infty&,\ \ R(q,u)\notin C\end{cases}\ ,\ q\in\mathcal{Q},u\in\mathcal{U}(% \alpha,\tilde{u})
  329. max { α : R ( q , u ) C , u 𝒰 ( α , u ~ ) } = max { α : α I ( q , α , u ) , u 𝒰 ( α , u ~ ) } = max { α : α min u 𝒰 ( α , u ~ ) I ( q , α , u ) } \begin{array}[]{ccl}\max\{\alpha:R(q,u)\in C,\forall u\in\mathcal{U}(\alpha,% \tilde{u})\}&=&\max\{\alpha:\alpha\leq I(q,\alpha,u),\forall u\in\mathcal{U}(% \alpha,\tilde{u})\}\\ &=&\max\{\alpha:\alpha\leq\displaystyle\min_{u\in\mathcal{U}(\alpha,\tilde{u})% }I(q,\alpha,u)\}\end{array}
  330. max { α : R ( q , u ) C , u 𝒰 ( α , u ~ ) } = max α 0 min u 𝒰 ( α , u ~ ) I ( q , α , u ) } \max\{\alpha:R(q,u)\in C,\forall u\in\mathcal{U}(\alpha,\tilde{u})\}=\max_{% \alpha\geq 0}\ \min_{u\in\mathcal{U}(\alpha,\tilde{u})}I(q,\alpha,u)\}
  331. \ \displaystyle\forall
  332. max { α : r c max u 𝒰 ( α , u ~ ) R ( q , u ) } \max\{\alpha:r_{c}\geq\max_{u\in\mathcal{U}(\alpha,\tilde{u})}R(q,u)\}
  333. max α 0 min u 𝒰 ( α , u ~ ) ϑ ( q , α , u ) \max_{\alpha\geq 0}\min_{u\in\mathcal{U}(\alpha,\tilde{u})}\vartheta(q,\alpha,u)
  334. ϑ ( q , α , u ) := { α , r c R ( q , α ) - , r c < R ( q , α ) \vartheta(q,\alpha,u):=\begin{cases}\quad\alpha&,\ \ r_{c}\geq R(q,\alpha)\\ -\infty&,\ \ r_{c}<R(q,\alpha)\end{cases}
  335. min x X f ( x ) = max y Y min x X g ( y , x ) \min_{x\in X}f(x)=\max_{y\in Y}\min_{x\in X}g(y,x)
  336. g ( y , x ) = f ( x ) , x X , y Y g(y,x)=f(x)\ ,\ \forall x\in X,y\in Y
  337. Y \ \displaystyle Y
  338. Decision Space D = ( 0 , ) State Spaces S ( d ) = 𝒰 ( d , u ~ ) Outcomes g ( d , s ) = φ ( q , d , s ) \begin{array}[]{rccl}\,\text{Decision Space}&D&=&(0,\infty)\\ \,\text{State Spaces}&S(d)&=&\mathcal{U}(d,\tilde{u})\\ \,\text{Outcomes}&g(d,s)&=&\varphi(q,d,s)\end{array}
  339. Classical Maximin Format MP Maximin Format max d D min s S ( d ) g ( d , s ) = max d D , α { α : α min s S ( d ) g ( d , s ) } \begin{array}[]{ccc}\,\text{Classical Maximin Format}&&\,\text{MP Maximin % Format}\\ \displaystyle\max_{d\in D}\ \min_{s\in S(d)}\ g(d,s)&=&\displaystyle\max_{d\in D% ,\alpha\in\mathbb{R}}\{\alpha:\alpha\leq\min_{s\in S(d)}g(d,s)\}\end{array}
  340. \ \mathbb{R}
  341. Robustness Model \begin{array}[]{c}\,\textit{Robustness\ Model}\end{array}
  342. Info-gap Format MP Maximin Format Classical Maximin Format max { α : r c min u 𝒰 ( α , u ~ ) R ( q , u ) } max { α : α min u 𝒰 ( α , u ~ ) φ ( q , α , u ) } max α 0 min u 𝒰 ( α , u ~ ) φ ( q , α , u ) \begin{array}[]{c|c|c}\,\text{Info-gap Format}&\,\text{MP Maximin Format}&\,% \text{Classical Maximin Format}\\ \hline\\ \displaystyle\max\{\alpha:r_{c}\leq\min_{u\in\mathcal{U}(\alpha,\tilde{u})}R(q% ,u)\}&\displaystyle\max\{\alpha:\alpha\leq\min_{u\in\mathcal{U}(\alpha,\tilde{% u})}\ \varphi(q,\alpha,u)\}&\displaystyle\max_{\alpha\geq 0}\ \min_{u\in% \mathcal{U}(\alpha,\tilde{u})}\varphi(q,\alpha,u)\end{array}
  343. r c R ( q , u ) α φ ( q , α , u ) r_{c}\leq R(q,u)\longleftrightarrow\alpha\leq\varphi(q,\alpha,u)
  344. φ \ \displaystyle\varphi
  345. Opportuneness Model \begin{array}[]{c}\,\textit{Opportuneness\ Model}\end{array}
  346. Info-gap Format MP Minimin Format Classical Minimin Format min { α : r w max u 𝒰 ( α , u ~ ) R ( q , u ) } min { α : α min u 𝒰 ( α , u ~ ) ψ ( q , α , u ) } min α 0 min u 𝒰 ( α , u ~ ) ψ ( q , α , u ) \begin{array}[]{c|c|c}\,\text{Info-gap Format}&\,\text{MP Minimin Format}&\,% \text{Classical Minimin Format}\\ \hline\\ \displaystyle\min\{\alpha:r_{w}\leq\max_{u\in\mathcal{U}(\alpha,\tilde{u})}R(q% ,u)\}&\displaystyle\min\{\alpha:\alpha\geq\min_{u\in\mathcal{U}(\alpha,\tilde{% u})}\ \psi(q,\alpha,u)\}&\displaystyle\min_{\alpha\geq 0}\ \min_{u\in\mathcal{% U}(\alpha,\tilde{u})}\psi(q,\alpha,u)\end{array}
  347. r c R ( q , u ) α ψ ( q , α , u ) r_{c}\leq R(q,u)\longleftrightarrow\alpha\geq\psi(q,\alpha,u)
  348. ψ \ \displaystyle\psi
  349. α \ \displaystyle\alpha
  350. u 𝒰 ( α , u ~ ) u\in\mathcal{U}(\alpha,\tilde{u})
  351. ψ ( q , α , u ) \ \psi(q,\alpha,u)
  352. 𝒰 ( α , u ~ ) \ \displaystyle\mathcal{U}(\alpha,\tilde{u})
  353. g ( α , u ) := α ( r c R ( q , u ) ) g(\alpha,u):=\alpha\cdot\left(r_{c}\preceq R(q,u)\right)
  354. \ \ \preceq\
  355. a b := { 1 , a b 0 , a > b a\preceq b:=\begin{cases}1&,\ \ a\leq b\\ 0&,\ \ a>b\end{cases}
  356. max { α : α min u 𝒰 ( α , u ~ ) α ( r c R ( q , u ) ) } = max { α : 1 min u 𝒰 ( α , u ~ ) ( r c R ( q , u ) ) } \max\{\alpha:\alpha\leq\min_{u\in\mathcal{U}(\alpha,\tilde{u})}\alpha\cdot% \left(r_{c}\preceq R(q,u)\right)\}=\max\{\alpha:1\leq\min_{u\in\mathcal{U}(% \alpha,\tilde{u})}\left(r_{c}\preceq R(q,u)\right)\}
  357. α \ \alpha
  358. r c R ( q , u ) \ r_{c}\leq R(q,u)
  359. u 𝒰 ( α , u ~ ) \ u\in\mathcal{U}(\alpha,\tilde{u})
  360. α \ \displaystyle\alpha
  361. α \ \displaystyle\alpha

Information_ratio.html

  1. I R IR
  2. I R = E [ R p - R b ] σ = α ω = E [ R p - R b ] var [ R p - R b ] IR=\frac{E[R_{p}-R_{b}]}{\sigma}=\frac{\alpha}{\omega}=\frac{E[R_{p}-R_{b}]}{% \sqrt{\mathrm{var}[R_{p}-R_{b}]}}
  3. R p R_{p}
  4. R b R_{b}
  5. α = E [ R p - R b ] \alpha=E[R_{p}-R_{b}]
  6. ω = σ \omega=\sigma
  7. α \alpha

Inhibition_theory.html

  1. λ ( t ) = lim Δ t 0 P ( t T < t + Δ t | T t ) Δ t \lambda(t)=\lim_{\Delta t\to 0}\frac{P(t\leq T<t+\Delta t|T\geq t)}{\Delta t}
  2. λ ( t ) = f ( t ) 1 - F ( t ) \lambda(t)=\frac{f(t)}{1-F(t)}
  3. l 1 ( y ) = c 1 M M - y l_{1}(y)=\frac{c_{1}M}{M-y}
  4. l 0 ( y ) = c 0 y l_{0}(y)=\frac{c_{0}}{y}
  5. λ 1 ( t ) = c 1 M M - ( y 0 + a 1 t ) \lambda_{1}(t)=\frac{c_{1}M}{M-(y_{0}+a_{1}t)}
  6. λ 0 ( t ) = c 0 y 0 - a 0 t \lambda_{0}(t)=\frac{c_{0}}{y_{0}-a_{0}t}
  7. E ( T ) = A + a 1 a 0 A E(T)=A+\frac{a_{1}}{a_{0}}A

Inhomogeneous_electromagnetic_wave_equation.html

  1. 𝐄 = ρ ε 0 \nabla\cdot\mathbf{E}=\frac{\rho}{\varepsilon_{0}}
  2. 𝐄 = 4 π ρ \nabla\cdot\mathbf{E}=4\pi\rho
  3. 𝐁 = 0 \nabla\cdot\mathbf{B}=0
  4. 𝐁 = 0 \nabla\cdot\mathbf{B}=0
  5. × 𝐄 = - 𝐁 t \nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}
  6. × 𝐄 = - 1 c 𝐁 t \nabla\times\mathbf{E}=-\frac{1}{c}\frac{\partial\mathbf{B}}{\partial t}
  7. × 𝐁 = μ 0 ( 𝐉 + ε 0 𝐄 t ) \nabla\times\mathbf{B}=\mu_{0}\left(\mathbf{J}+\varepsilon_{0}\frac{\partial% \mathbf{E}}{\partial t}\right)
  8. × 𝐁 = 1 c ( 4 π 𝐉 + 𝐄 t ) \nabla\times\mathbf{B}=\frac{1}{c}\left(4\pi\mathbf{J}+\frac{\partial\mathbf{E% }}{\partial t}\right)
  9. ε 0 μ 0 = 1 c 2 . \varepsilon_{0}\mu_{0}=\dfrac{1}{c^{2}}\,.
  10. 1 c 2 2 𝐄 t 2 - 2 𝐄 = - ( 1 ε 0 ρ + μ 0 𝐉 t ) . \dfrac{1}{c^{2}}\dfrac{\partial^{2}\mathbf{E}}{\partial t^{2}}-\nabla^{2}% \mathbf{E}=-\left(\dfrac{1}{\varepsilon_{0}}\nabla\rho+\mu_{0}\dfrac{\partial% \mathbf{J}}{\partial t}\right)\,.
  11. 1 c 2 2 𝐁 t 2 - 2 𝐁 = μ 0 × 𝐉 . \dfrac{1}{c^{2}}\dfrac{\partial^{2}\mathbf{B}}{\partial t^{2}}-\nabla^{2}% \mathbf{B}=\mu_{0}\nabla\times\mathbf{J}\,.
  12. 𝐄 = - φ - 𝐀 t , 𝐁 = × 𝐀 , \mathbf{E}=-\nabla\varphi-{\partial\mathbf{A}\over\partial t}\,,\quad\mathbf{B% }=\nabla\times\mathbf{A}\,,
  13. 2 φ + t ( 𝐀 ) = - ρ ε 0 , \nabla^{2}\varphi+{{\partial}\over\partial t}\left(\nabla\cdot\mathbf{A}\right% )=-{\rho\over\varepsilon_{0}}\,,
  14. 2 𝐀 - 1 c 2 2 𝐀 t 2 - ( 1 c 2 φ t + 𝐀 ) = - μ 0 𝐉 . \nabla^{2}\mathbf{A}-{1\over c^{2}}{\partial^{2}\mathbf{A}\over\partial t^{2}}% -\nabla\left({1\over c^{2}}{{\partial\varphi}\over{\partial t}}+\nabla\cdot% \mathbf{A}\right)=-\mu_{0}\mathbf{J}\,.
  15. 1 c 2 φ t + 𝐀 = 0 {1\over c^{2}}{{\partial\varphi}\over{\partial t}}+\nabla\cdot\mathbf{A}=0
  16. 2 φ - 1 c 2 2 φ t 2 = - ρ ε 0 , \nabla^{2}\varphi-{1\over c^{2}}{\partial^{2}\varphi\over\partial t^{2}}=-{% \rho\over\varepsilon_{0}}\,,
  17. 2 𝐀 - 1 c 2 2 𝐀 t 2 = - μ 0 𝐉 . \nabla^{2}\mathbf{A}-{1\over c^{2}}{\partial^{2}\mathbf{A}\over\partial t^{2}}% =-\mu_{0}\mathbf{J}\,.
  18. 2 φ - 1 c 2 2 φ t 2 = - 4 π ρ \nabla^{2}\varphi-{1\over c^{2}}{\partial^{2}\varphi\over\partial t^{2}}=-{4% \pi\rho}
  19. 2 𝐀 - 1 c 2 2 𝐀 t 2 = - 4 π c 𝐉 \nabla^{2}\mathbf{A}-{1\over c^{2}}{\partial^{2}\mathbf{A}\over\partial t^{2}}% =-{4\pi\over c}\mathbf{J}
  20. 1 c φ t + 𝐀 = 0 . {1\over c}{{\partial\varphi}\over{\partial t}}+\nabla\cdot\mathbf{A}=0\,.
  21. A μ = def β β A μ = def A μ , β β = - μ 0 J μ SI \Box A^{\mu}\ \stackrel{\mathrm{def}}{=}\ \partial_{\beta}\partial^{\beta}A^{% \mu}\ \stackrel{\mathrm{def}}{=}\ {A^{\mu,\beta}}_{\beta}=-\mu_{0}J^{\mu}\quad% \,\text{SI}
  22. A μ = def β β A μ = def A μ , β β = - 4 π c J μ cgs \Box A^{\mu}\ \stackrel{\mathrm{def}}{=}\ \partial_{\beta}\partial^{\beta}A^{% \mu}\ \stackrel{\mathrm{def}}{=}\ {A^{\mu,\beta}}_{\beta}=-\frac{4\pi}{c}J^{% \mu}\quad\,\text{cgs}
  23. = β β = 2 - 1 c 2 2 t 2 \Box=\partial_{\beta}\partial^{\beta}=\nabla^{2}-{1\over c^{2}}\frac{\partial^% {2}}{\partial t^{2}}
  24. J μ = ( c ρ , 𝐉 ) J^{\mu}=\left(c\rho,\mathbf{J}\right)
  25. x a = def a = def = def , a ( / c t , ) {\partial\over{\partial x^{a}}}\ \stackrel{\mathrm{def}}{=}\ \partial_{a}\ % \stackrel{\mathrm{def}}{=}\ {}_{,a}\ \stackrel{\mathrm{def}}{=}\ (\partial/% \partial ct,\nabla)
  26. A μ = ( φ , 𝐀 c ) SI A^{\mu}=(\varphi,\mathbf{A}c)\quad\,\text{SI}
  27. A μ = ( φ , 𝐀 ) cgs A^{\mu}=(\varphi,\mathbf{A})\quad\,\text{cgs}
  28. μ A μ = 0 . \partial_{\mu}A^{\mu}=0\,.
  29. - A α ; β β + R α β A β = μ 0 J α -{A^{\alpha;\beta}}_{\beta}+{R^{\alpha}}_{\beta}A^{\beta}=\mu_{0}J^{\alpha}
  30. R α β {R^{\alpha}}_{\beta}
  31. A μ ; μ = 0 . {A^{\mu}}_{;\mu}=0\,.
  32. 𝐀 ( 𝐫 , t ) = δ ( t + | 𝐫 - 𝐫 | c - t ) | 𝐫 - 𝐫 | 𝐉 ( 𝐫 , t ) c d 3 r d t \mathbf{A}(\mathbf{r},t)=\int{{\delta\left(t^{\prime}+{{\left|\mathbf{r}-% \mathbf{r}^{\prime}\right|}\over c}-t\right)}\over{{\left|\mathbf{r}-\mathbf{r% }^{\prime}\right|}}}{\mathbf{J}(\mathbf{r}^{\prime},t^{\prime})\over c}d^{3}r^% {\prime}dt^{\prime}
  33. δ ( t + | 𝐫 - 𝐫 | c - t ) {\delta\left(t^{\prime}+{{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}\over c}% -t\right)}
  34. φ ( 𝐫 , t ) = δ ( t - | 𝐫 - 𝐫 | c - t ) | 𝐫 - 𝐫 | ρ ( 𝐫 , t ) d 3 r d t \varphi(\mathbf{r},t)=\int{{\delta\left(t^{\prime}-{{\left|\mathbf{r}-\mathbf{% r}^{\prime}\right|}\over c}-t\right)}\over{{\left|\mathbf{r}-\mathbf{r}^{% \prime}\right|}}}\rho(\mathbf{r}^{\prime},t^{\prime})d^{3}r^{\prime}dt^{\prime}
  35. 𝐀 ( 𝐫 , t ) = δ ( t - | 𝐫 - 𝐫 | c - t ) | 𝐫 - 𝐫 | 𝐉 ( 𝐫 , t ) c d 3 r d t . \mathbf{A}(\mathbf{r},t)=\int{{\delta\left(t^{\prime}-{{\left|\mathbf{r}-% \mathbf{r}^{\prime}\right|}\over c}-t\right)}\over{{\left|\mathbf{r}-\mathbf{r% }^{\prime}\right|}}}{\mathbf{J}(\mathbf{r}^{\prime},t^{\prime})\over c}d^{3}r^% {\prime}dt^{\prime}\,.

Inhomogeneous_Poisson_process.html

  1. λ ( t ) \lambda(t)
  2. N ( t ) N(t)
  3. t t
  4. h h
  5. N ( 0 ) = 0 N(0)=0
  6. P ( N ( t + h ) - N ( t ) = 1 ) = λ ( t ) h + o ( h ) P(N(t+h)-N(t)=1)=\lambda(t)h+o(h)
  7. P ( N ( t + h ) - N ( t ) > 1 ) = o ( h ) P(N(t+h)-N(t)>1)=o(h)
  8. o ( h ) h 0 as h 0 \scriptstyle\frac{o(h)}{h}\rightarrow 0\;\mathrm{as}\,h\,\rightarrow 0
  9. P ( N ( t + h ) - N ( t ) > 1 ) = o ( h 2 ) P(N(t+h)-N(t)>1)=o(h^{2})
  10. m ( t ) = 0 t λ ( u ) d u \scriptstyle m(t)=\int_{0}^{t}\lambda(u)\,\text{d}u
  11. ( N ( t ) = k ) = m ( t ) k k ! e - m ( t ) . \mathbb{P}(N(t)=k)=\frac{m(t)^{k}}{k!}e^{-m(t)}.

Initial_algebra.html

  1. X X
  2. p : X 2 p\colon X\to 2
  3. f : X X f\colon X\to X
  4. x X x\in X
  5. p ( x ) = 0 p(x)=0
  6. { ω } \mathbb{N}\cup\{\omega\}
  7. ω \omega
  8. p p
  9. p ( 0 ) = 1 p(0)=1
  10. p ( n + 1 ) = p ( ω ) = 0 p(n+1)=p(\omega)=0
  11. f f
  12. ω \omega
  13. f ( n + 1 ) = n f(n+1)=n
  14. f ( ω ) = ω f(\omega)=\omega
  15. 1 + × ( - ) 1+\mathbb{N}\times(\mathord{-})
  16. F : 𝐒𝐞𝐭 𝐒𝐞𝐭 F:\mathbf{Set}\to\mathbf{Set}
  17. X X
  18. 1 + X 1+X
  19. N N
  20. [ 𝑧𝑒𝑟𝑜 , 𝑠𝑢𝑐𝑐 ] : 1 + N N [\mathit{zero},\mathit{succ}]\colon 1+N\to N
  21. 𝑧𝑒𝑟𝑜 : 1 N \mathit{zero}\colon 1\to N
  22. 𝑠𝑢𝑐𝑐 : N N \mathit{succ}\colon N\to N
  23. F F
  24. F F
  25. ( A , [ e , f ] ) (A,[e,f])
  26. e : 1 A e\colon 1\to A
  27. A A
  28. f : A A f\colon A\to A
  29. A A
  30. n n
  31. f n ( e ) f^{n}(e)
  32. f ( f ( ( f ( e ) ) ) ) f(f(\dots(f(e))\dots))
  33. n n
  34. f f
  35. e e
  36. 𝐿𝑖𝑠𝑡 ( A ) \mathit{List}(A)
  37. A A
  38. 𝑛𝑖𝑙 : 1 𝐿𝑖𝑠𝑡 ( A ) \mathit{nil}\colon 1\to\mathit{List}(A)
  39. 𝑐𝑜𝑛𝑠 : A × 𝐿𝑖𝑠𝑡 ( A ) 𝐿𝑖𝑠𝑡 ( A ) \mathit{cons}\colon A\times\mathit{List}(A)\to\mathit{List}(A)
  40. [ 𝑛𝑖𝑙 , 𝑐𝑜𝑛𝑠 ] : 1 + ( A × 𝐿𝑖𝑠𝑡 ( A ) ) 𝐿𝑖𝑠𝑡 ( A ) [\mathit{nil},\mathit{cons}]\colon 1+(A\times\mathit{List}(A))\to\mathit{List}% (A)
  41. F F
  42. F F
  43. X X
  44. 1 + ( A × X ) 1+(A\times X)
  45. F F
  46. [ 𝑡𝑖𝑝 , 𝑗𝑜𝑖𝑛 ] : A + ( 𝑇𝑟𝑒𝑒 ( A ) × 𝑇𝑟𝑒𝑒 ( A ) ) 𝑇𝑟𝑒𝑒 ( A ) [\mathit{tip},\mathit{join}]\colon A+(\mathit{Tree}(A)\times\mathit{Tree}(A))% \to\mathit{Tree}(A)
  47. F F

Initial_mass_function.html

  1. N ( m ) d m N(m)dm
  2. ξ ( m ) Δ m \xi(m)\Delta m
  3. m m
  4. m + d m m+dm
  5. m - α m^{-\alpha}
  6. α \alpha
  7. α = 2.35 \alpha=2.35
  8. ξ ( m ) Δ m = ξ 0 ( m M sun ) - 2.35 ( Δ m M sun ) . \xi(m)\Delta m=\xi_{0}\left(\frac{m}{M_{\mathrm{sun}}}\right)^{-2.35}\left(% \frac{\Delta m}{M_{\mathrm{sun}}}\right).
  9. α = 0 \alpha=0
  10. ξ ( m ) Δ m = 0.158 ( 1 / m ) exp [ - ( log ( m ) - log ( 0.08 ) ) 2 / ( 2 × 0.69 2 ) ] \xi(m)\Delta m=0.158(1/m)\exp[-(\log(m)-\log(0.08))^{2}/(2\times 0.69^{2})]
  11. m < 1 , m<1,
  12. ξ ( m ) = k m - α \xi(m)=km^{-\alpha}
  13. m > 1 , α = 2.3 ± 0.3 m>1,\alpha=2.3\pm 0.3
  14. ξ ( m ) Δ m = 0.086 ( 1 / m ) exp [ - ( log ( m ) - log ( 0.22 ) ) 2 / ( 2 × 0.57 2 ) ] \xi(m)\Delta m=0.086(1/m)\exp[-(\log(m)-\log(0.22))^{2}/(2\times 0.57^{2})]
  15. m < 1 , m<1,
  16. ξ ( m ) = k m - α \xi(m)=km^{-\alpha}
  17. m > 1 , α = 2.3 ± 0.3 m>1,\alpha=2.3\pm 0.3
  18. α = 2.3 \alpha=2.3
  19. α = 1.3 \alpha=1.3
  20. α = 0.3 \alpha=0.3
  21. ξ ( m ) = m - α , \xi(m)=m^{-\alpha},
  22. α = 0.3 \alpha=0.3
  23. m < 0.08 , m<0.08,
  24. α = 1.3 \alpha=1.3
  25. 0.08 < m < 0.5 , 0.08<m<0.5,
  26. α = 2.3 \alpha=2.3
  27. 0.5 < m 0.5<m

Initial_volume_of_distribution.html

  1. V D = total amount of drug in the body per body weight unit ( Kg ) ( i . e . Dose ) drug blood concentration {V_{D}}=\frac{\mathrm{total\ amount\ of\ drug\ in\ the\ body\ per\ body\ % weight\ unit\ (Kg)\ (i.e.Dose)}}{\mathrm{drug\ blood\ concentration}}
  2. V D = V P + V T ( f u f u t ) {V_{D}}={V_{P}}+{V_{T}}\left(\frac{fu}{fu_{t}}\right)

Inquiry.html

  1. X Y X\Rightarrow Y
  2. Y Z Y\Rightarrow Z
  3. X Z . X\Rightarrow Z.
  4. X Y X\Rightarrow Y
  5. X Z X\Rightarrow Z
  6. Y Z . Y\Rightarrow Z.
  7. X Z X\Rightarrow Z
  8. Y Z Y\Rightarrow Z
  9. X Y . X\Rightarrow Y.

Insect_wing.html

  1. C L ( α ) = 2 L ρ U 2 S and C D ( α ) = 2 D ρ U 2 S . C\text{L}(\alpha)=\frac{2L}{\rho U^{2}S}\quad\,\text{and}\quad C\text{D}(% \alpha)=\frac{2D}{\rho U^{2}S}.
  2. C L = 2 π sin α and C D = 0. C\text{L}=2\pi\sin\alpha\quad\,\text{and}\quad C\text{D}=0.
  3. 𝐮 t + ( 𝐮 ) 𝐮 \displaystyle\frac{\partial\mathbf{u}}{\partial t}+\left(\mathbf{u}\cdot\nabla% \right)\mathbf{u}
  4. h = g ( Δ t 2 ) 2 h=\frac{g(\Delta t^{2})}{2}
  5. Δ t = ( 2 h g ) 1 / 2 = 2 × 10 - 2 cm 980 cm / sec 2 4.5 × 10 - 3 sec \Delta t=\left(\frac{2h}{g}\right)^{1/2}=\sqrt{\frac{2\times 10^{-2}\,\text{ % cm}}{980\,\text{ cm}/\,\text{sec}^{2}}}\approx 4.5\times 10^{-3}\,\text{ sec}
  6. T = 2 Δ t = 9 × 10 - 3 sec T=2\,\Delta t=9\times 10^{-3}\,\text{ sec}
  7. f = 1 T 110 sec - 1 f=\frac{1}{T}\approx 110\,\text{ sec}^{-1}
  8. Work = F a v × d = 2Wd \text{Work}=F_{av}\times d=\,\text{2Wd}\,\!
  9. Work = 2 × 0.1 × 980 × 0.57 = 112 erg \text{Work}=2\times 0.1\times 980\times 0.57=112\,\text{erg}\,\!
  10. E = mgh = 0.1 × 980 × 10 - 2 = 0.98 erg \text{E}=\text{mgh}=0.1\times 980\times 10^{-2}=0.98\text{erg}\,\!
  11. P = 112 erg × 110 /s = 1.23 × 10 4 erg/s = 1.23 × 10 - 3 W \text{P}=112\text{erg}\times 110\text{/s}=1.23\times 10^{4}\text{erg/s}=1.23% \times 10^{-3}\text{W}
  12. K E = 1 2 I ω max 2 KE=\frac{1}{2}I\omega_{\text{max}}^{2}
  13. I = m 3 3 I=\frac{m\ell^{3}}{3}
  14. ω max = v max / 2 \omega_{\text{max}}=\frac{v_{\text{max}}}{\ell/2}
  15. v a v = d Δ t = 0.57 4.5 × 10 - 3 = 127 cm/s v_{av}=\frac{d}{\Delta t}=\frac{0.57}{4.5\times 10^{-3}}=127\text{cm/s}
  16. ω max = 254 / 2 \omega_{\text{max}}=\frac{254}{\ell/2}
  17. K E = 1 2 I ω m a x 2 = ( 10 - 3 2 3 ) ( 254 / 2 ) 2 = 43 erg KE=\frac{1}{2}I\omega_{max}^{2}=\left(10^{-3}\frac{\ell^{2}}{3}\right)\left(% \frac{254}{\ell/2}\right)^{2}=43\text{ erg}
  18. E = 1 2 Y A Δ 2 E=\frac{1}{2}\frac{YA\Delta\ell^{2}}{\ell}
  19. E = 1 2 1.8 × 10 7 × 4 × 10 - 4 × 10 - 4 2 × 10 - 2 = 18 erg E=\frac{1}{2}\frac{1.8\times 10^{7}\times 4\times 10^{-4}\times 10^{-4}}{2% \times 10^{-2}}=18\ \,\text{erg}

Inside–outside_algorithm.html

  1. β j ( p , q ) \beta_{j}(p,q)
  2. w p w q w_{p}\cdots w_{q}
  3. N j N^{j}
  4. G G
  5. β j ( p , q ) = P ( w p q | N p q j , G ) \beta_{j}(p,q)=P(w_{pq}|N^{j}_{pq},G)
  6. α j ( p , q ) \alpha_{j}(p,q)
  7. N 1 N^{1}
  8. N p q j N^{j}_{pq}
  9. w p w q w_{p}\cdots w_{q}
  10. G G
  11. α j ( p , q ) = P ( w 1 ( p - 1 ) , N p q j , w ( q + 1 ) m | G ) \alpha_{j}(p,q)=P(w_{1(p-1)},N^{j}_{pq},w_{(q+1)m}|G)
  12. β j ( p , p ) = P ( w p | N j , G ) \beta_{j}(p,p)=P(w_{p}|N^{j},G)
  13. N j N r N s N_{j}\rightarrow N_{r}N_{s}
  14. w p w q w_{p}\cdots w_{q}
  15. N j N_{j}
  16. k = p k = q - 1 P ( N j N r N s ) β r ( p , k ) β s ( k + 1 , q ) \sum_{k=p}^{k=q-1}P(N_{j}\rightarrow N_{r}N_{s})\beta_{r}(p,k)\beta_{s}(k+1,q)
  17. β j ( p , q ) \beta_{j}(p,q)
  18. β j ( p , q ) = N r , N s k = p k = q - 1 P ( N j N r N s ) β r ( p , k ) β s ( k + 1 , q ) \beta_{j}(p,q)=\sum_{N_{r},N_{s}}\sum_{k=p}^{k=q-1}P(N_{j}\rightarrow N_{r}N_{% s})\beta_{r}(p,k)\beta_{s}(k+1,q)
  19. α j ( 1 , n ) = { 1 if j = 1 0 otherwise \alpha_{j}(1,n)=\begin{cases}1&\mbox{if }~{}j=1\\ 0&\mbox{otherwise}\end{cases}
  20. N 1 N_{1}

Integer_lattice.html

  1. ( 2 ) n S n (\mathbb{Z}_{2})^{n}\rtimes S_{n}
  2. × \scriptstyle\mathbb{Z}\times\mathbb{Z}
  3. \scriptstyle\mathbb{Z}

Integer_matrix.html

  1. ( 5 2 6 0 4 7 3 8 5 9 0 4 3 1 0 - 3 9 0 2 1 ) \left(\begin{array}[]{cccc}5&2&6&0\\ 4&7&3&8\\ 5&9&0&4\\ 3&1&0&-3\\ 9&0&2&1\end{array}\right)
  2. ( 1 5 0 0 9 2 1 7 3 ) \left(\begin{array}[]{ccc}1&5&0\\ 0&9&2\\ 1&7&3\end{array}\right)
  3. M M
  4. M M
  5. 1 1
  6. - 1 -1
  7. 1 1
  8. SL n ( 𝐙 ) \mathrm{SL}_{n}(\mathbf{Z})
  9. n = 2 n=2

Integral_energy.html

  1. θ i \theta_{i}
  2. θ f \theta_{f}
  3. θ i > θ f \theta_{i}>\theta_{f}
  4. ψ ( θ ) \psi(\theta)
  5. θ \theta
  6. E i = θ i θ f 1 θ i - θ f ψ ( θ ) d θ E_{i}=\int_{\theta_{i}}^{\theta_{f}}\frac{1}{\theta_{i}-\theta_{f}}\psi(\theta% )\,d\theta

Integrated_Encryption_Scheme.html

  1. E E
  2. ( p , a , b , G , n , h ) (p,a,b,G,n,h)
  3. ( m , f ( x ) , a , b , G , n , h ) (m,f(x),a,b,G,n,h)
  4. K B K_{B}
  5. K B = k B G K_{B}=k_{B}G
  6. k B k_{B}
  7. k B [ 1 , n - 1 ] k_{B}\in[1,n-1]
  8. S 1 S_{1}
  9. S 2 S_{2}
  10. m m
  11. r [ 1 , n - 1 ] r\in[1,n-1]
  12. R = r G R=rG
  13. S = P x S=P_{x}
  14. P = ( P x , P y ) = r K B P=(P_{x},P_{y})=rK_{B}
  15. P O P\neq O
  16. k E k M = KDF ( S S 1 ) k_{E}\|k_{M}=\textrm{KDF}(S\|S_{1})
  17. c = E ( k E ; m ) c=E(k_{E};m)
  18. S 2 S_{2}
  19. d = MAC ( k M ; c S 2 ) d=\textrm{MAC}(k_{M};c\|S_{2})
  20. R c d R\|c\|d
  21. R c d R\|c\|d
  22. S = P x S=P_{x}
  23. P = ( P x , P y ) = k B R P=(P_{x},P_{y})=k_{B}R
  24. P = k B R = k B r G = r k B G = r K B P=k_{B}R=k_{B}rG=rk_{B}G=rK_{B}
  25. P = O P=O
  26. k E k M = KDF ( S S 1 ) k_{E}\|k_{M}=\textrm{KDF}(S\|S_{1})
  27. d MAC ( k M ; c S 2 ) d\neq\textrm{MAC}(k_{M};c\|S_{2})
  28. m = E - 1 ( k E ; c ) m=E^{-1}(k_{E};c)

Interlingual_machine_translation.html

  1. n ( n - 1 ) n(n-1)
  2. n n
  3. 2 n 2n
  4. n n

Interpolation_attack.html

  1. c i = ( c i - 1 k i ) 3 , c_{i}=(c_{i-1}\oplus k_{i})^{3},
  2. c 0 c_{0}
  3. k i K k_{i}\in K
  4. r r
  5. c r c_{r}
  6. x x
  7. c c
  8. c 1 = ( x + k 1 ) 3 = ( x 2 + k 1 2 ) ( x + k 1 ) = x 3 + k 1 2 x + x 2 k 1 + k 1 3 , c_{1}=(x+k_{1})^{3}=(x^{2}+k_{1}^{2})(x+k_{1})=x^{3}+k_{1}^{2}x+x^{2}k_{1}+k_{% 1}^{3},
  9. c 2 = c = ( c 1 + k 2 ) 3 = ( x 3 + k 1 2 x + x 2 k 1 + k 1 3 + k 2 ) 3 c_{2}=c=(c_{1}+k_{2})^{3}=(x^{3}+k_{1}^{2}x+x^{2}k_{1}+k_{1}^{3}+k_{2})^{3}
  10. = x 9 + x 8 k 1 + x 6 k 2 + x 4 k 1 2 k 2 + x 3 k 2 2 + x 2 ( k 1 k 2 2 + k 1 4 k 2 ) + x ( k 1 2 k 2 2 + k 1 8 ) + k 1 3 k 2 2 + k 1 9 + k 2 3 , =x^{9}+x^{8}k_{1}+x^{6}k_{2}+x^{4}k_{1}^{2}k_{2}+x^{3}k_{2}^{2}+x^{2}(k_{1}k_{% 2}^{2}+k_{1}^{4}k_{2})+x(k_{1}^{2}k_{2}^{2}+k_{1}^{8})+k_{1}^{3}k_{2}^{2}+k_{1% }^{9}+k_{2}^{3},
  11. p ( x ) = a 1 x 9 + a 2 x 8 + a 3 x 6 + a 4 x 4 + a 5 x 3 + a 6 x 2 + a 7 x + a 8 , p(x)=a_{1}x^{9}+a_{2}x^{8}+a_{3}x^{6}+a_{4}x^{4}+a_{5}x^{3}+a_{6}x^{2}+a_{7}x+% a_{8},
  12. a i a_{i}
  13. p ( x ) p(x)
  14. p ( x ) p(x)
  15. K K
  16. m m
  17. 2 m 2^{m}
  18. 2 m 2^{m}
  19. p / c p/c
  20. n n
  21. p ( x ) p(x)
  22. p / c p/c
  23. n 2 m n\leq 2^{m}
  24. p ( x ) p(x)
  25. p / c p/c
  26. n n
  27. p ( x ) p(x)
  28. n n
  29. n n
  30. p / c p/c
  31. r r
  32. m m
  33. z z
  34. s s
  35. s < r s<r
  36. z z
  37. x x
  38. c c
  39. g ( x ) G F ( 2 m ) [ x ] g(x)\in GF(2^{m})[x]
  40. z z
  41. x x
  42. h ( c ) G F ( 2 m ) [ c ] h(c)\in GF(2^{m})[c]
  43. z z
  44. c c
  45. g ( x ) g(x)
  46. s s
  47. h ( c ) h(c)
  48. r r
  49. s + 1 s+1
  50. g ( x ) = h ( c ) , g(x)=h(c),
  51. g g
  52. h h
  53. g ( x ) g(x)
  54. p p
  55. h ( c ) h(c)
  56. q q
  57. p + q p+q
  58. p / c p/c
  59. p + q - 2 p+q-2
  60. p / c p/c
  61. p + q - 2 p+q-2
  62. p + q - 2 p+q-2
  63. p / c p/c
  64. p / c p/c
  65. K K
  66. r r
  67. m m
  68. y ~ = c r - 1 \tilde{y}=c_{r-1}
  69. k r k_{r}
  70. y ~ \tilde{y}
  71. y ~ \tilde{y}
  72. x x
  73. p ( x ) G F ( 2 m ) [ x ] p(x)\in GF(2^{m})[x]
  74. p ( x ) p(x)
  75. n n
  76. n n
  77. p / c p/c
  78. p / c p/c
  79. p ( x ) = y ~ . p(x)=\tilde{y}.
  80. z z
  81. s < r s<r
  82. x x
  83. y ~ \tilde{y}
  84. g ( x ) g(x)
  85. h ( y ~ ) h(\tilde{y})
  86. p p
  87. q q
  88. q + p - 2 q+p-2
  89. p / c p/c
  90. p / c p/c
  91. g ( x ) = h ( y ~ ) . g(x)=h(\tilde{y}).
  92. m m
  93. 2 m 2^{m}
  94. 1 / 2 m 1/2^{m}
  95. 1 / 2 2 m 1/2\cdot 2^{m}
  96. 2 m - 1 ( n + 1 ) 2^{m-1}(n+1)
  97. n + 1 n+1
  98. c / p c/p
  99. 2 m - 1 ( p + q - 1 ) 2^{m-1}(p+q-1)
  100. p + q - 1 p+q-1
  101. c / p c/p
  102. m m
  103. S : f ( x ) = x - 1 = x 2 m - 2 S:f(x)=x^{-1}=x^{2^{m}-2}
  104. G F ( 2 m ) GF(2^{m})
  105. S : f ( x ) = x - 1 S:f(x)=x^{-1}
  106. ( n , m , r ) (n,m,r)
  107. n m nm
  108. n n
  109. m m
  110. r r
  111. ( 8 , 8 , 4 ) (8,8,4)
  112. 2 21 2^{21}
  113. ( 8 , 16 , 7 ) (8,16,7)
  114. 2 61 2^{61}

Invariant_basis_number.html

  1. f : A n A p f\colon A^{n}\to A^{p}
  2. ( e 1 , , e n ) (e_{1},\dots,e_{n})
  3. e i A n e_{i}\in A^{n}
  4. ( i 1 , , i n ) I n (i_{1},\dots,i_{n})\in I^{n}
  5. f ( i 1 , , i n ) = k = 0 n i k f ( e k ) I p f(i_{1},\dots,i_{n})=\sum_{k=0}^{n}i_{k}f(e_{k})\in I^{p}
  6. f : ( A I ) n ( A I ) p f^{\prime}\colon\left(\frac{A}{I}\right)^{n}\to\left(\frac{A}{I}\right)^{p}
  7. 𝔽 𝕄 ( R ) \mathbb{CFM}_{\mathbb{N}}(R)
  8. × \mathbb{N}\times\mathbb{N}
  9. 𝔽 𝕄 ( R ) 𝔽 𝕄 ( R ) 2 \mathbb{CFM}_{\mathbb{N}}(R)\cong\mathbb{CFM}_{\mathbb{N}}(R)^{2}
  10. ψ : 𝔽 𝕄 ( R ) 𝔽 𝕄 ( R ) 2 M ( odd columns of M , even columns of M ) \begin{array}[]{rcl}\psi:\mathbb{CFM}_{\mathbb{N}}(R)&\to&\mathbb{CFM}_{% \mathbb{N}}(R)^{2}\\ M&\mapsto&(\,\text{odd columns of }M,\,\text{ even columns of }M)\end{array}
  11. 𝔽 𝕄 ( R ) = S \mathbb{CFM}_{\mathbb{N}}(R)=S

Inventory_turnover.html

  1. Inventory Turnover = Cost of Goods Sold Average Inventory \mbox{Inventory Turnover}~{}=\frac{\mbox{Cost of Goods Sold}~{}}{\mbox{Average% Inventory}~{}}
  2. Inventory Turnover = Cost of Material − Change in inventories (of 1/2 and 1/1 goods) Inventories \mbox{Inventory Turnover}~{}=\frac{\mbox{Cost of Material − Change in % inventories (of 1/2 and 1/1 goods)}~{}}{\mbox{Inventories}~{}}
  3. Average Inventory = Beginning inventory+Ending inventory 2 \mbox{Average Inventory}~{}=\frac{\mbox{Beginning inventory+Ending inventory}~% {}}{\mbox{2}~{}}
  4. Average days to sell the inventory = 365 days Inventory Turnover Ratio \mbox{Average days to sell the inventory}~{}=\frac{\mbox{365 days}~{}}{\mbox{% Inventory Turnover Ratio}~{}}
  5. Inventory Turn = Number of Units Sold (Over a given period) Average Number of Units (For the period) \mbox{Inventory Turn}~{}=\frac{\,\text{Number of Units Sold (Over a given % period)}}{\,\text{Average Number of Units (For the period)}}

Inverse_beta_decay.html

  1. ν ¯ e + p e + + n \bar{\nu}_{e}+p\to e^{+}+n
  2. e - + p ν e + n e^{-}+p\to\nu_{e}+n

Inverse_Faraday_effect.html

  1. M ( 0 ) \vec{M}(0)
  2. ω \omega
  3. E \vec{E}
  4. E * \vec{E}^{*}
  5. M ( 0 ) [ E ( ω ) × E * ( ω ) ] \vec{M}(0)\propto[\vec{E}(\omega)\times\vec{E}^{*}(\omega)]
  6. ω \omega
  7. k \vec{k}
  8. E \vec{E}

Inverse_hyperbolic_function.html

  1. arsinh z \displaystyle\operatorname{arsinh}\,z
  2. Log \operatorname{Log}{}
  3. x + 1 x - 1 = x 2 - 1 \sqrt{x+1}\sqrt{x-1}=\sqrt{x^{2}-1}
  4. ln ( 1 + x ) - ln ( 1 - x ) = ln ( 1 + x 1 - x ) \ln\left({1+x}\right)-\ln\left({1-x}\right)=\ln\left(\frac{1+x}{1-x}\right)
  5. arsinh x \displaystyle\operatorname{arsinh}\,x
  6. arcosh x = ln 2 x - ( ( 1 2 ) x - 2 2 + ( 1 3 2 4 ) x - 4 4 + ( 1 3 5 2 4 6 ) x - 6 6 + ) = ln 2 x - n = 1 ( ( 2 n ) ! 2 2 n ( n ! ) 2 ) x - 2 n ( 2 n ) , x > 1 \begin{aligned}\displaystyle\operatorname{arcosh}\,x&\displaystyle=\ln 2x-% \left(\left(\frac{1}{2}\right)\frac{x^{-2}}{2}+\left(\frac{1\cdot 3}{2\cdot 4}% \right)\frac{x^{-4}}{4}+\left(\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\right)% \frac{x^{-6}}{6}+\cdots\right)\\ &\displaystyle=\ln 2x-\sum_{n=1}^{\infty}\left(\frac{(2n)!}{2^{2n}(n!)^{2}}% \right)\frac{x^{-2n}}{(2n)},\qquad x>1\end{aligned}
  7. artanh x = x + x 3 3 + x 5 5 + x 7 7 + = n = 0 x 2 n + 1 ( 2 n + 1 ) , | x | < 1 \begin{aligned}\displaystyle\operatorname{artanh}\,x&\displaystyle=x+\frac{x^{% 3}}{3}+\frac{x^{5}}{5}+\frac{x^{7}}{7}+\cdots\\ &\displaystyle=\sum_{n=0}^{\infty}\frac{x^{2n+1}}{(2n+1)},\qquad\left|x\right|% <1\end{aligned}
  8. arcsch x = arsinh 1 x = x - 1 - ( 1 2 ) x - 3 3 + ( 1 3 2 4 ) x - 5 5 - ( 1 3 5 2 4 6 ) x - 7 7 + = n = 0 ( ( - 1 ) n ( 2 n ) ! 2 2 n ( n ! ) 2 ) x - ( 2 n + 1 ) ( 2 n + 1 ) , | x | > 1 \begin{aligned}\displaystyle\operatorname{arcsch}\,x=\operatorname{arsinh}% \frac{1}{x}&\displaystyle=x^{-1}-\left(\frac{1}{2}\right)\frac{x^{-3}}{3}+% \left(\frac{1\cdot 3}{2\cdot 4}\right)\frac{x^{-5}}{5}-\left(\frac{1\cdot 3% \cdot 5}{2\cdot 4\cdot 6}\right)\frac{x^{-7}}{7}+\cdots\\ &\displaystyle=\sum_{n=0}^{\infty}\left(\frac{(-1)^{n}(2n)!}{2^{2n}(n!)^{2}}% \right)\frac{x^{-(2n+1)}}{(2n+1)},\qquad\left|x\right|>1\end{aligned}
  9. arsech x = arcosh 1 x = ln 2 x - ( ( 1 2 ) x 2 2 + ( 1 3 2 4 ) x 4 4 + ( 1 3 5 2 4 6 ) x 6 6 + ) = ln 2 x - n = 1 ( ( 2 n ) ! 2 2 n ( n ! ) 2 ) x 2 n 2 n , 0 < x 1 \begin{aligned}\displaystyle\operatorname{arsech}\,x=\operatorname{arcosh}% \frac{1}{x}&\displaystyle=\ln\frac{2}{x}-\left(\left(\frac{1}{2}\right)\frac{x% ^{2}}{2}+\left(\frac{1\cdot 3}{2\cdot 4}\right)\frac{x^{4}}{4}+\left(\frac{1% \cdot 3\cdot 5}{2\cdot 4\cdot 6}\right)\frac{x^{6}}{6}+\cdots\right)\\ &\displaystyle=\ln\frac{2}{x}-\sum_{n=1}^{\infty}\left(\frac{(2n)!}{2^{2n}(n!)% ^{2}}\right)\frac{x^{2n}}{2n},\qquad 0<x\leq 1\end{aligned}
  10. arcoth x = artanh 1 x = x - 1 + x - 3 3 + x - 5 5 + x - 7 7 + = n = 0 x - ( 2 n + 1 ) ( 2 n + 1 ) , | x | > 1 \begin{aligned}\displaystyle\operatorname{arcoth}\,x=\operatorname{artanh}% \frac{1}{x}&\displaystyle=x^{-1}+\frac{x^{-3}}{3}+\frac{x^{-5}}{5}+\frac{x^{-7% }}{7}+\cdots\\ &\displaystyle=\sum_{n=0}^{\infty}\frac{x^{-(2n+1)}}{(2n+1)},\qquad\left|x% \right|>1\end{aligned}
  11. arsinh x = ln 2 x + n = 1 ( - 1 ) n - 1 ( 2 n - 1 ) ! ! 2 n ( 2 n ) ! ! 1 x 2 n \operatorname{arsinh}\,x=\ln 2x+\sum\limits_{n=1}^{\infty}{\left({-1}\right)^{% n-1}\frac{{\left({2n-1}\right)!!}}{{2n\left({2n}\right)!!}}}\frac{1}{{x^{2n}}}
  12. d d x arsinh x = 1 1 + x 2 , for all real x d d x arcosh x = 1 x 2 - 1 , for all real x > 1 d d x artanh x = 1 1 - x 2 , for all real | x | < 1 d d x arcoth x = 1 1 - x 2 , for all real | x | > 1 d d x arsech x = - 1 x 1 - x 2 , for all real x ( 0 , 1 ) d d x arcsch x = - 1 | x | 1 + x 2 , for all real x , except 0 \begin{aligned}\displaystyle\frac{d}{dx}\operatorname{arsinh}\,x&\displaystyle% {}=\frac{1}{\sqrt{1+x^{2}}},\,\text{ for all real }x\\ \displaystyle\frac{d}{dx}\operatorname{arcosh}\,x&\displaystyle{}=\frac{1}{% \sqrt{x^{2}-1}},\,\text{ for all real }x>1\\ \displaystyle\frac{d}{dx}\operatorname{artanh}\,x&\displaystyle{}=\frac{1}{1-x% ^{2}},\,\text{ for all real }|x|<1\\ \displaystyle\frac{d}{dx}\operatorname{arcoth}\,x&\displaystyle{}=\frac{1}{1-x% ^{2}},\,\text{ for all real }|x|>1\\ \displaystyle\frac{d}{dx}\operatorname{arsech}\,x&\displaystyle{}=\frac{-1}{x% \sqrt{1-x^{2}}},\,\text{ for all real }x\in(0,1)\\ \displaystyle\frac{d}{dx}\operatorname{arcsch}\,x&\displaystyle{}=\frac{-1}{|x% |\sqrt{1+x^{2}}},\,\text{ for all real }x\,\text{, except }0\\ \end{aligned}
  13. d arsinh x d x = d θ d sinh θ = 1 cosh θ = 1 1 + sinh 2 θ = 1 1 + x 2 \frac{d\,\operatorname{arsinh}\,x}{dx}=\frac{d\theta}{d\sinh\theta}=\frac{1}{% \cosh\theta}=\frac{1}{\sqrt{1+\sinh^{2}\theta}}=\frac{1}{\sqrt{1+x^{2}}}
  14. sinh ( arcosh x ) = x 2 - 1 for | x | > 1 sinh ( artanh x ) = x 1 - x 2 for - 1 < x < 1 cosh ( arsinh x ) = 1 + x 2 cosh ( artanh x ) = 1 1 - x 2 for - 1 < x < 1 tanh ( arsinh x ) = x 1 + x 2 tanh ( arcosh x ) = x 2 - 1 x for | x | > 1 \begin{aligned}&\displaystyle\sinh(\operatorname{arcosh}\,x)=\sqrt{x^{2}-1}% \quad\,\text{for}\quad|x|>1\\ &\displaystyle\sinh(\operatorname{artanh}\,x)=\frac{x}{\sqrt{1-x^{2}}}\quad\,% \text{for}\quad-1<x<1\\ &\displaystyle\cosh(\operatorname{arsinh}\,x)=\sqrt{1+x^{2}}\\ &\displaystyle\cosh(\operatorname{artanh}\,x)=\frac{1}{\sqrt{1-x^{2}}}\quad\,% \text{for}\quad-1<x<1\\ &\displaystyle\tanh(\operatorname{arsinh}\,x)=\frac{x}{\sqrt{1+x^{2}}}\\ &\displaystyle\tanh(\operatorname{arcosh}\,x)=\frac{\sqrt{x^{2}-1}}{x}\quad\,% \text{for}\quad|x|>1\end{aligned}
  15. arsinh u ± arsinh v = arsinh ( u 1 + v 2 ± v 1 + u 2 ) \operatorname{arsinh}\;u\pm\operatorname{arsinh}\;v=\operatorname{arsinh}\left% (u\sqrt{1+v^{2}}\pm v\sqrt{1+u^{2}}\right)
  16. arcosh u ± arcosh v = arcosh ( u v ± ( u 2 - 1 ) ( v 2 - 1 ) ) \operatorname{arcosh}\;u\pm\operatorname{arcosh}\;v=\operatorname{arcosh}\left% (uv\pm\sqrt{(u^{2}-1)(v^{2}-1)}\right)
  17. artanh u ± artanh v = artanh ( u ± v 1 ± u v ) \operatorname{artanh}\;u\pm\operatorname{artanh}\;v=\operatorname{artanh}\left% (\frac{u\pm v}{1\pm uv}\right)
  18. arsinh u + arcosh v = arsinh ( u v + ( 1 + u 2 ) ( v 2 - 1 ) ) = arcosh ( v 1 + u 2 + u v 2 - 1 ) \begin{aligned}\displaystyle\operatorname{arsinh}\;u+\operatorname{arcosh}\;v&% \displaystyle=\operatorname{arsinh}\left(uv+\sqrt{(1+u^{2})(v^{2}-1)}\right)\\ &\displaystyle=\operatorname{arcosh}\left(v\sqrt{1+u^{2}}+u\sqrt{v^{2}-1}% \right)\end{aligned}
  19. 2 arcosh x = arcosh ( 2 x 2 - 1 ) for x 1 4 arcosh x = arcosh ( 8 x 4 - 8 x 2 + 1 ) for x 1 2 arsinh x = arcosh ( 2 x 2 + 1 ) for x 0 4 arsinh x = arcosh ( 8 x 4 + 8 x 2 + 1 ) for x 0 \begin{aligned}\displaystyle 2\operatorname{arcosh}x&\displaystyle=% \operatorname{arcosh}(2x^{2}-1)&\displaystyle\quad\hbox{ for }x\geq 1\\ \displaystyle 4\operatorname{arcosh}x&\displaystyle=\operatorname{arcosh}(8x^{% 4}-8x^{2}+1)&\displaystyle\quad\hbox{ for }x\geq 1\\ \displaystyle 2\operatorname{arsinh}x&\displaystyle=\operatorname{arcosh}(2x^{% 2}+1)&\displaystyle\quad\hbox{ for }x\geq 0\\ \displaystyle 4\operatorname{arsinh}x&\displaystyle=\operatorname{arcosh}(8x^{% 4}+8x^{2}+1)&\displaystyle\quad\hbox{ for }x\geq 0\end{aligned}

Inverse_Mills_ratio.html

  1. E [ X | X > α ] = μ + σ ϕ ( α - μ σ ) 1 - Φ ( α - μ σ ) , \displaystyle\operatorname{E}[\,X\,|\ X>\alpha\,]=\mu+\sigma\frac{\phi\big(% \tfrac{\alpha-\mu}{\sigma}\big)}{1-\Phi\big(\tfrac{\alpha-\mu}{\sigma}\big)},

Inverse_relation.html

  1. X and Y X\,\text{ and }Y
  2. L X × Y L\subseteq X\times Y
  3. L - 1 L^{-1}
  4. y L - 1 x y\,L^{-1}\,x
  5. x L y x\,L\,y
  6. L - 1 = { ( y , x ) Y × X ( x , y ) L } L^{-1}=\{(y,x)\in Y\times X\mid(x,y)\in L\}
  7. L ˘ \breve{L}
  8. - 1 = , < - 1 = Align g t ; \leq^{-1}=\ \geq,~{}<^{-1}=\ &gt;
  9. f : X Y f:X\to Y
  10. f - 1 : Y X f^{-1}:Y\to X
  11. graph f - 1 = { ( y , x ) y = f ( x ) } \operatorname{graph}\,f^{-1}=\{(y,x)\mid y=f(x)\}
  12. f - 1 f^{-1}
  13. f - 1 f^{-1}
  14. f - 1 f^{-1}
  15. f - 1 f^{-1}
  16. L L - 1 L\circ L^{-1}
  17. ( L - 1 ) - 1 = L (L^{-1})^{-1}=L
  18. ( L R ) - 1 = R - 1 L - 1 (L\circ R)^{-1}=R^{-1}\circ L^{-1}

Inverse_scattering_transform.html

  1. L 2 L^{2}
  2. u t - 6 u u x + 3 u x 3 = 0 , \frac{\partial u}{\partial t}-6\,u\,\frac{\partial u}{\partial x}+\frac{% \partial^{3}u}{\partial x^{3}}=0,\,
  3. u t \frac{\partial u}{\partial t}
  4. u x \frac{\partial u}{\partial x}
  5. u ( x , 0 ) u(x,0)
  6. 2 ψ x 2 - u ( x , t ) ψ = λ ψ . \frac{\partial^{2}\psi}{\partial x^{2}}-u(x,t)\psi=\lambda\psi.
  7. ψ \psi
  8. t = 0 t=0
  9. λ \lambda
  10. u = 1 ψ 2 ψ x 2 - λ . u=\frac{1}{\psi}\frac{\partial^{2}\psi}{\partial x^{2}}-\lambda.
  11. ψ t + 3 ψ x 3 - 3 ( u - λ ) ψ x = C ψ + D ψ d x ψ 2 \frac{\partial\psi}{\partial t}+\frac{\partial^{3}\psi}{\partial x^{3}}-3(u-% \lambda)\frac{\partial\psi}{\partial x}=C\psi+D\psi\int\frac{dx}{\psi^{2}}
  12. L L
  13. M M
  14. L v = λ v Lv=\lambda v
  15. d v d t = M v \frac{dv}{dt}=Mv
  16. λ \lambda
  17. d λ d t = 0. \frac{d\lambda}{dt}=0.
  18. L v = λ v Lv=\lambda v
  19. d L d t v + L d v d t = d λ d t v + λ d v d t . \frac{dL}{dt}v+L\frac{dv}{dt}=\frac{d\lambda}{dt}v+\lambda\frac{dv}{dt}.
  20. M v Mv
  21. d v d t \frac{dv}{dt}
  22. d L d t v + L M v = d λ d t v + λ M v . \frac{dL}{dt}v+LMv=\frac{d\lambda}{dt}v+\lambda Mv.
  23. d L d t v + L M v = d λ d t v + M L v . \frac{dL}{dt}v+LMv=\frac{d\lambda}{dt}v+MLv.
  24. d L d t v + L M v - M L v = d λ d t v . \frac{dL}{dt}v+LMv-MLv=\frac{d\lambda}{dt}v.
  25. v 0 v\not=0
  26. d λ d t = 0 \frac{d\lambda}{dt}=0
  27. d L d t + L M - M L = 0. \frac{dL}{dt}+LM-ML=0.\,
  28. d L d t \frac{dL}{dt}
  29. L L
  30. t t
  31. L L
  32. L = d 2 d x 2 + u , L=\frac{d^{2}}{dx^{2}}+u,
  33. d L d t v + L d v d t \frac{dL}{dt}v+L\frac{dv}{dt}
  34. t ( d 2 v d x 2 + u v ) \frac{\partial}{\partial t}\left(\frac{d^{2}v}{dx^{2}}+uv\right)
  35. L t = d u d t , \frac{\partial L}{\partial t}=\frac{du}{dt},
  36. λ \lambda
  37. L L

Ion_Barbu.html

  1. d ( a , b ) = log max p J ( p a / p b ) + log max q J ( q b / q a ) . d(a,b)=\log\underset{p\in J}{\max}(pa/pb)+\log\underset{q\in J}{\max}(qb/qa).

IP_set.html

  1. S S\,
  2. S = C 1 C 2 C n S=C_{1}\cup C_{2}\cup...\cup C_{n}
  3. C i C_{i}\,

Irregular_matrix.html

  1. [ 1 31 12 - 3 7 2 1 2 2 ] \begin{bmatrix}1&31&12&-3\\ 7&2\\ 1&2&2\end{bmatrix}

Isenthalpic_process.html

  1. h 1 = h 2 h_{1}=h_{2}
  2. d h = 0 dh=0
  3. d h = 0 = c p d T {d}h=0=c_{p}{d}T

ISO_31-11.html

  1. i = 1 n A i \bigcup_{i=1}^{n}A_{i}
  2. i = 1 n A i = A 1 A 2 A n \bigcup_{i=1}^{n}A_{i}=A_{1}\cup A_{2}\cup\ldots\cup A_{n}
  3. i = 1 n \bigcup{}_{i=1}^{n}
  4. i I \bigcup_{i\in I}
  5. i I \bigcup{}_{i\in I}
  6. i = 1 n A i \bigcap_{i=1}^{n}A_{i}
  7. i = 1 n A i = A 1 A 2 A n \bigcap_{i=1}^{n}A_{i}=A_{1}\cap A_{2}\cap\ldots\cap A_{n}
  8. i = 1 n \bigcap{}_{i=1}^{n}
  9. i I \bigcap_{i\in I}
  10. = def \ \stackrel{\mathrm{def}}{=}
  11. a b a\not\equiv b
  12. \langle\rangle
  13. \langle
  14. \rangle
  15. \langle\rangle
  16. \langle\rangle
  17. \perp
  18. \perp
  19. f : D C f:D\rightarrow C
  20. f ( S ) f\left(S\right)
  21. { f ( x ) x S } \left\{f\left(x\right)\mid x\in S\right\}
  22. a \vec{a}

ISO_31-3.html

  1. s y m b o l τ symbol{\tau}
  2. σ , τ \sigma,\tau

ISO_31-4.html

  1. α l = 1 l l T \alpha_{l}=\frac{1}{l}\frac{l}{T}

ISO_31-5.html

  1. 1 μ 0 c 0 2 \begin{matrix}\frac{1}{\mu_{0}{c_{0}}^{2}}\end{matrix}

Isogenous_series.html

  1. T e 2 Te_{2}
  2. T e 2 Te_{2}

Isothermal–isobaric_ensemble.html

  1. T T\,
  2. P P\,
  3. N p T NpT
  4. N N\,
  5. Z ( N , V , T ) Z(N,V,T)\,
  6. Δ ( N , P , T ) = Z ( N , V , T ) exp ( - β P V ) C d V . \Delta(N,P,T)=\int Z(N,V,T)\exp(-\beta PV)CdV.\,\;
  7. β = 1 / k B T \beta=1/k_{B}T\,
  8. k B k_{B}\,
  9. V V\,
  10. C C\,
  11. C = N / V C=N/V\,
  12. C = β P C=\beta P\,
  13. G ( N , P , T ) = - k B T ln Δ ( N , P , T ) G(N,P,T)=-k_{B}T\ln\Delta(N,P,T)\;\,
  14. F F\,
  15. G = F + P V . G=F+PV.\;\,

Isotope_dilution.html

  1. n A = n B × 10 1 = 50 n_{\mathrm{A}}=n_{\mathrm{B}}\times\frac{10}{1}=50
  2. n A = n B × R B - R AB R AB - R A × 1 + R A 1 + R B n_{\mathrm{A}}=n_{\mathrm{B}}\times\frac{R_{\mathrm{B}}-R_{\mathrm{AB}}}{R_{% \mathrm{AB}}-R_{\mathrm{A}}}\times\frac{1+R_{\mathrm{A}}}{1+R_{\mathrm{B}}}
  3. n A = n B R B - R AB R AB - R A × x ( j A ) B x ( j A ) A n_{\mathrm{A}}=n_{\mathrm{B}}\frac{R_{\mathrm{B}}-R_{\mathrm{AB}}}{R_{\mathrm{% AB}}-R_{\mathrm{A}}}\times\frac{x(^{j}\mathrm{A})_{\mathrm{B}}}{x(^{j}\mathrm{% A})_{\mathrm{A}}}
  4. n A = n B R B - R AB R AB - R A × 1 + R A 1 + R B n_{\mathrm{A}}=n_{\mathrm{B}}\frac{R_{\mathrm{B}}-R_{\mathrm{AB}}}{R_{\mathrm{% AB}}-R_{\mathrm{A}}}\times\frac{1+R_{\mathrm{A}}}{1+R_{\mathrm{B}}}
  5. u ( n A ) 2 ( n A R AB ) 2 u ( R AB ) 2 = n A 2 ( R A - R B ) 2 ( R A - R AB ) 2 ( R AB - R B ) 2 u ( R AB ) 2 u(n_{\mathrm{A}})^{2}\propto\left({\frac{\partial{n_{\mathrm{A}}}}{\partial R_% {\mathrm{AB}}}}\right)^{2}u(R_{\mathrm{AB}})^{2}=n_{\mathrm{A}}^{2}\frac{(R_{% \mathrm{A}}-R_{\mathrm{B}})^{2}}{(R_{\mathrm{A}}-R_{\mathrm{AB}})^{2}(R_{% \mathrm{AB}}-R_{\mathrm{B}})^{2}}u(R_{\mathrm{AB}})^{2}
  6. u r ( n A ) 2 ( R A - R B ) 2 ( R A - R AB ) 2 ( R AB - R B ) 2 u ( R AB ) 2 u_{\mathrm{r}}(n_{\mathrm{A}})^{2}\propto\frac{(R_{\mathrm{A}}-R_{\mathrm{B}})% ^{2}}{(R_{\mathrm{A}}-R_{\mathrm{AB}})^{2}(R_{\mathrm{AB}}-R_{\mathrm{B}})^{2}% }u(R_{\mathrm{AB}})^{2}
  7. u r ( n A ) min ( ( R A - R B ) ( R A - R AB ) ( R AB - R B ) R AB ) / R AB = 0 u_{\mathrm{r}}(n_{\mathrm{A}})_{\mathrm{min}}\mapsto\partial\left(\frac{(R_{% \mathrm{A}}-R_{\mathrm{B}})}{(R_{\mathrm{A}}-R_{\mathrm{AB}})(R_{\mathrm{AB}}-% R_{\mathrm{B}})}R_{\mathrm{AB}}\right)/\partial R_{\mathrm{AB}}=0
  8. R AB = R A R B R_{\mathrm{AB}}=\sqrt{R_{\mathrm{A}}R_{\mathrm{B}}}
  9. n B = n A * R A * - R A * B R A * B - R B × x ( j A ) A * x ( j A ) B n_{\mathrm{B}}=n_{\mathrm{A*}}\frac{R_{\mathrm{A*}}-R_{\mathrm{A*B}}}{R_{% \mathrm{A*B}}-R_{\mathrm{B}}}\times\frac{x(^{j}\mathrm{A})_{\mathrm{A*}}}{x(^{% j}\mathrm{A})_{\mathrm{B}}}
  10. n A = n B R B - R AB R AB - R A × x ( j A ) B x ( j A ) A n_{\mathrm{A}}=n_{\mathrm{B}}\frac{R_{\mathrm{B}}-R_{\mathrm{AB}}}{R_{\mathrm{% AB}}-R_{\mathrm{A}}}\times\frac{x(^{j}\mathrm{A})_{\mathrm{B}}}{x(^{j}\mathrm{% A})_{\mathrm{A}}}
  11. n A = n A * R A * - R A * B R A * B - R B × R B - R AB R AB - R A n_{\mathrm{A}}=n_{\mathrm{A*}}\frac{R_{\mathrm{A*}}-R_{\mathrm{A*B}}}{R_{% \mathrm{A*B}}-R_{\mathrm{B}}}\times\frac{R_{\mathrm{B}}-R_{\mathrm{AB}}}{R_{% \mathrm{AB}}-R_{\mathrm{A}}}
  12. n A = n A * ( R A * B = R AB and R A * = R A ) n_{\mathrm{A}}=n_{\mathrm{A*}}\;(R_{\mathrm{A*B}}=R_{\mathrm{AB}}\and R_{% \mathrm{A*}}=R_{\mathrm{A}})

Iterated_monodromy_group.html

  1. IMG f := π 1 ( X , t ) n Ker ϝ n \mathrm{IMG}f:=\frac{\pi_{1}(X,t)}{\bigcap_{n\in\mathbb{N}}\mathrm{Ker}\,% \digamma^{n}}
  2. f : X 1 X f:X_{1}\rightarrow X
  3. X 1 X_{1}
  4. π 1 ( X , t ) \pi_{1}(X,t)
  5. ϝ : π 1 ( X , t ) Sym f - 1 ( t ) \digamma:\pi_{1}(X,t)\rightarrow\mathrm{Sym}\,f^{-1}(t)
  6. ϝ n : π 1 ( X , t ) Sym f - n ( t ) \digamma^{n}:\pi_{1}(X,t)\rightarrow\mathrm{Sym}\,f^{-n}(t)
  7. n th n^{\mathrm{th}}
  8. n 0 \forall n\in\mathbb{N}_{0}
  9. T f := n 0 f - n ( t ) , T_{f}:=\bigsqcup_{n\geq 0}f^{-n}(t),
  10. z f - n ( t ) z\in f^{-n}(t)
  11. f ( z ) f - ( n - 1 ) ( t ) f(z)\in f^{-(n-1)}(t)
  12. P f P_{f}
  13. P f P_{f}
  14. f : C ^ f - 1 ( P f ) C ^ P f f:\hat{C}\setminus f^{-1}(P_{f})\rightarrow\hat{C}\setminus P_{f}
  15. C ^ \hat{C}
  16. z 2 - 1 z^{2}-1

Ives–Stilwell_experiment.html

  1. f o f s = c c ± v , \frac{f_{o}}{f_{s}}=\frac{c}{c\pm v},
  2. f o f s = 1 - v 2 c 2 \frac{f_{o}}{f_{s}}=\sqrt{1-\frac{v^{2}}{c^{2}}}
  3. 1 + v / c 1+v/c
  4. 1 v / c 1−v/c
  5. | | | |\cdot\cdot\cdot\cdot\cdot|\cdot\cdot\cdot\cdot\cdot|\,
  6. | | | |\cdot\cdot\cdot\cdot|\cdot\cdot\cdot\cdot\cdot\cdot|\,
  7. ν a ν p ν 1 ν 2 = 1 + 2 α ^ β 2 \frac{\nu_{a}\nu_{p}}{\nu_{1}\nu_{2}}=1+2\hat{\alpha}\beta^{2}
  8. ν a \nu_{a}
  9. ν p \nu_{p}
  10. ν 1 \nu_{1}
  11. ν 2 \nu_{2}
  12. β = v / c \beta=v/c
  13. v v
  14. c c
  15. ν a ν p ν 0 2 = 1 + 2 α ^ β 2 \frac{\nu_{a}\nu_{p}}{\nu_{0}^{2}}=1+2\hat{\alpha}\beta^{2}
  16. ν 0 \nu_{0}
  17. α ^ \hat{\alpha}
  18. | α ^ | |\hat{\alpha}|

J_integral.html

  1. J := Γ ( W d x 2 - 𝐭 𝐮 x 1 d s ) = Γ ( W d x 2 - t i u i x 1 d s ) J:=\int_{\Gamma}\left(W~{}dx_{2}-\mathbf{t}\cdot\cfrac{\partial\mathbf{u}}{% \partial x_{1}}~{}ds\right)=\int_{\Gamma}\left(W~{}dx_{2}-t_{i}\,\cfrac{% \partial u_{i}}{\partial x_{1}}~{}ds\right)
  2. W = 0 ϵ s y m b o l σ : d s y m b o l ϵ ; s y m b o l ϵ = 1 2 [ s y m b o l 𝐮 + ( s y m b o l 𝐮 ) T ] . W=\int_{0}^{\epsilon}symbol{\sigma}:dsymbol{\epsilon}~{};~{}~{}symbol{\epsilon% }=\tfrac{1}{2}\left[symbol{\nabla}\mathbf{u}+(symbol{\nabla}\mathbf{u})^{T}% \right]~{}.
  3. J i := lim ϵ 0 Γ ϵ ( W n i - n j σ j k u k x i ) d Γ J_{i}:=\lim_{\epsilon\rightarrow 0}\int_{\Gamma_{\epsilon}}\left(Wn_{i}-n_{j}% \sigma_{jk}~{}\cfrac{\partial u_{k}}{\partial x_{i}}\right)d\Gamma
  4. J i J_{i}
  5. x i x_{i}
  6. ϵ \epsilon
  7. Γ \Gamma
  8. J J
  9. J 1 J_{1}
  10. J 1 := Γ ( W n 1 - n j σ j k u k x 1 ) d Γ J_{1}:=\int_{\Gamma}\left(Wn_{1}-n_{j}\sigma_{jk}~{}\cfrac{\partial u_{k}}{% \partial x_{1}}\right)d\Gamma
  11. J 1 = Γ ( W δ 1 j - σ j k u k x 1 ) n j d Γ J_{1}=\int_{\Gamma}\left(W\delta_{1j}-\sigma_{jk}~{}\cfrac{\partial u_{k}}{% \partial x_{1}}\right)n_{j}d\Gamma
  12. Γ f j n j d Γ = A f j x j d A \int_{\Gamma}f_{j}~{}n_{j}~{}d\Gamma=\int_{A}\cfrac{\partial f_{j}}{\partial x% _{j}}~{}dA
  13. J 1 J_{1}
  14. J 1 = A x j ( W δ 1 j - σ j k u k x 1 ) d A = A [ W x 1 - σ j k x j u k x 1 - σ j k 2 u k x 1 x j ] d A \begin{aligned}\displaystyle J_{1}&\displaystyle=\int_{A}\cfrac{\partial}{% \partial x_{j}}\left(W\delta_{1j}-\sigma_{jk}~{}\cfrac{\partial u_{k}}{% \partial x_{1}}\right)dA\\ &\displaystyle=\int_{A}\left[\cfrac{\partial W}{\partial x_{1}}-\cfrac{% \partial\sigma_{jk}}{\partial x_{j}}~{}\cfrac{\partial u_{k}}{\partial x_{1}}-% \sigma_{jk}~{}\cfrac{\partial^{2}u_{k}}{\partial x_{1}\partial x_{j}}\right]~{% }dA\end{aligned}
  15. A A
  16. Γ \Gamma
  17. s y m b o l \cdotsymbol σ = 𝟎 σ j k x j = 0 . symbol{\nabla}\cdotsymbol{\sigma}=\mathbf{0}\qquad\implies\qquad\cfrac{% \partial\sigma_{jk}}{\partial x_{j}}=0~{}.
  18. s y m b o l ϵ = 1 2 [ s y m b o l 𝐮 + ( s y m b o l 𝐮 ) T ] ϵ j k = 1 2 ( u k x j + u j x k ) . symbol{\epsilon}=\tfrac{1}{2}\left[symbol{\nabla}\mathbf{u}+(symbol{\nabla}% \mathbf{u})^{T}\right]\qquad\implies\qquad\epsilon_{jk}=\tfrac{1}{2}\left(% \cfrac{\partial u_{k}}{\partial x_{j}}+\cfrac{\partial u_{j}}{\partial x_{k}}% \right)~{}.
  19. σ j k ϵ j k x 1 = 1 2 ( σ j k 2 u k x 1 x j + σ j k 2 u j x 1 x k ) \sigma_{jk}\cfrac{\partial\epsilon_{jk}}{\partial x_{1}}=\tfrac{1}{2}\left(% \sigma_{jk}\cfrac{\partial^{2}u_{k}}{\partial x_{1}\partial x_{j}}+\sigma_{jk}% \cfrac{\partial^{2}u_{j}}{\partial x_{1}\partial x_{k}}\right)
  20. σ j k = σ k j \sigma_{jk}=\sigma_{kj}
  21. σ j k ϵ j k x 1 = σ j k 2 u j x 1 x k \sigma_{jk}\cfrac{\partial\epsilon_{jk}}{\partial x_{1}}=\sigma_{jk}\cfrac{% \partial^{2}u_{j}}{\partial x_{1}\partial x_{k}}
  22. J 1 = A [ W x 1 - σ j k ϵ j k x 1 ] d A J_{1}=\int_{A}\left[\cfrac{\partial W}{\partial x_{1}}-\sigma_{jk}~{}\cfrac{% \partial\epsilon_{jk}}{\partial x_{1}}\right]~{}dA
  23. W W
  24. σ j k = W ϵ j k \sigma_{jk}=\cfrac{\partial W}{\partial\epsilon_{jk}}
  25. σ j k ϵ j k x 1 = W ϵ j k ϵ j k x 1 = W x 1 \sigma_{jk}~{}\cfrac{\partial\epsilon_{jk}}{\partial x_{1}}=\cfrac{\partial W}% {\partial\epsilon_{jk}}~{}\cfrac{\partial\epsilon_{jk}}{\partial x_{1}}=\cfrac% {\partial W}{\partial x_{1}}
  26. J 1 = 0 J_{1}=0
  27. Γ = Γ 1 + Γ + + Γ 2 + Γ - \Gamma=\Gamma_{1}+\Gamma^{+}+\Gamma_{2}+\Gamma^{-}
  28. J = J ( 1 ) + J + - J ( 2 ) - J - = 0 J=J_{(1)}+J^{+}-J_{(2)}-J^{-}=0
  29. x 1 x_{1}
  30. n 1 = 0 n_{1}=0
  31. t k = 0 t_{k}=0
  32. J + = J - = Γ ( W n 1 - t k u k x 1 ) d Γ = 0 J^{+}=J^{-}=\int_{\Gamma}\left(Wn_{1}-t_{k}~{}\cfrac{\partial u_{k}}{\partial x% _{1}}\right)d\Gamma=0
  33. J ( 1 ) = J ( 2 ) J_{(1)}=J_{(2)}
  34. J Ic = G Ic = K Ic 2 ( 1 - ν 2 E ) J_{\rm Ic}=G_{\rm Ic}=K_{\rm Ic}^{2}\left(\frac{1-\nu^{2}}{E}\right)
  35. G Ic G_{\rm Ic}
  36. K Ic K_{\rm Ic}
  37. ν \nu
  38. K IIc K_{\rm IIc}
  39. J IIc = G IIc = K IIc 2 [ 1 - ν 2 E ] J_{\rm IIc}=G_{\rm IIc}=K_{\rm IIc}^{2}\left[\frac{1-\nu^{2}}{E}\right]
  40. J IIIc = G IIIc = K IIIc 2 ( 1 + ν E ) J_{\rm IIIc}=G_{\rm IIIc}=K_{\rm IIIc}^{2}\left(\frac{1+\nu}{E}\right)
  41. ε ε y = σ σ y + α ( σ σ y ) n \frac{\varepsilon}{\varepsilon_{y}}=\frac{\sigma}{\sigma_{y}}+\alpha\left(% \frac{\sigma}{\sigma_{y}}\right)^{n}
  42. J Γ 1 = π ( σ far ) 2 . J_{\Gamma_{1}}=\pi\,(\sigma_{\,\text{far}})^{2}\,.
  43. J Γ 1 = - J Γ 2 . J_{\Gamma_{1}}=-J_{\Gamma_{2}}\,.
  44. J Γ 2 = - α K n + 1 r ( n + 1 ) ( s - 2 ) + 1 I J_{\Gamma_{2}}=-\alpha\,K^{n+1}\,r^{(n+1)(s-2)+1}\,I
  45. s = 2 n + 1 n + 1 s=\frac{2n+1}{n+1}
  46. K = ( β π α I ) 1 n + 1 ( σ far ) 2 n + 1 K=\left(\frac{\beta\,\pi}{\alpha\,I}\right)^{\frac{1}{n+1}}\,(\sigma_{\,\text{% far}})^{\frac{2}{n+1}}
  47. σ i j = σ y ( E J r α σ y 2 I ) 1 n + 1 σ ~ i j ( n , θ ) \sigma_{ij}=\sigma_{y}\left(\frac{EJ}{r\,\alpha\sigma_{y}^{2}I}\right)^{{1}% \over{n+1}}\tilde{\sigma}_{ij}(n,\theta)
  48. ε i j = α ε y E ( E J r α σ y 2 I ) n n + 1 ε ~ i j ( n , θ ) \varepsilon_{ij}=\frac{\alpha\varepsilon_{y}}{E}\left(\frac{EJ}{r\,\alpha% \sigma_{y}^{2}I}\right)^{{n}\over{n+1}}\tilde{\varepsilon}_{ij}(n,\theta)
  49. σ ~ i j \tilde{\sigma}_{ij}
  50. ε ~ i j \tilde{\varepsilon}_{ij}

Jackson_network.html

  1. α > 0 \alpha>0
  2. p 0 j 0 p_{0j}\geq 0
  3. j = 1 J p 0 j = 1 \sum_{j=1}^{J}p_{0j}=1
  4. p i j p_{ij}
  5. p i 0 = 1 - j = 1 J p i j p_{i0}=1-\sum_{j=1}^{J}p_{ij}
  6. λ i \lambda_{i}
  7. λ i = α p 0 i + j = 1 J λ j p j i , i = 1 , , J . ( 1 ) \lambda_{i}=\alpha p_{0i}+\sum_{j=1}^{J}\lambda_{j}p_{ji},i=1,\ldots,J.\qquad(1)
  8. a = ( α p 0 i ) i = 1 J a=(\alpha p_{0i})_{i=1}^{J}
  9. λ = ( I - P ) - 1 a \lambda=(I-P^{\prime})^{-1}a
  10. μ i ( x i ) \mu_{i}(x_{i})
  11. x i x_{i}
  12. X i ( t ) X_{i}(t)
  13. 𝐗 = ( X i ) i = 1 J \mathbf{X}=(X_{i})_{i=1}^{J}
  14. 𝐗 \mathbf{X}
  15. π ( 𝐱 ) = P ( 𝐗 = 𝐱 ) \pi(\mathbf{x})=P(\mathbf{X}=\mathbf{x})
  16. π ( 𝐱 ) i = 1 J [ α p 0 i + μ i ( x i ) ( 1 - p i i ) ] \pi(\mathbf{x})\sum_{i=1}^{J}[\alpha p_{0i}+\mu_{i}(x_{i})(1-p_{ii})]
  17. = i = 1 J [ π ( 𝐱 - 𝐞 i ) α p 0 i + π ( 𝐱 + 𝐞 i ) μ i ( x i + 1 ) p i 0 ] + i = 1 J j i π ( 𝐱 + 𝐞 i - 𝐞 j ) μ i ( x i + 1 ) p i j . ( 2 ) =\sum_{i=1}^{J}[\pi(\mathbf{x}-\mathbf{e}_{i})\alpha p_{0i}+\pi(\mathbf{x}+% \mathbf{e}_{i})\mu_{i}(x_{i}+1)p_{i0}]+\sum_{i=1}^{J}\sum_{j\neq i}\pi(\mathbf% {x}+\mathbf{e}_{i}-\mathbf{e}_{j})\mu_{i}(x_{i}+1)p_{ij}.\qquad(2)
  18. 𝐞 i \mathbf{e}_{i}
  19. i t h i^{th}
  20. ( Y 1 , , Y J ) (Y_{1},\ldots,Y_{J})
  21. Y i Y_{i}
  22. P ( Y i = n ) = p ( Y i = 0 ) λ i n M i ( n ) , ( 3 ) P(Y_{i}=n)=p(Y_{i}=0)\cdot\frac{\lambda_{i}^{n}}{M_{i}(n)},\quad(3)
  23. M i ( n ) = j = 1 n μ i ( j ) M_{i}(n)=\prod_{j=1}^{n}\mu_{i}(j)
  24. n = 1 λ i n M i ( n ) < \sum_{n=1}^{\infty}\frac{\lambda_{i}^{n}}{M_{i}(n)}<\infty
  25. P ( Y i = 0 ) = ( 1 + n = 1 λ i n M i ( n ) ) - 1 P(Y_{i}=0)=\left(1+\sum_{n=1}^{\infty}\frac{\lambda_{i}^{n}}{M_{i}(n)}\right)^% {-1}
  26. π ( 𝐱 ) = i = 1 J P ( Y i = x i ) . \pi(\mathbf{x})=\prod_{i=1}^{J}P(Y_{i}=x_{i}).
  27. 𝐱 𝒵 + J \mathbf{x}\in\mathcal{Z}_{+}^{J}
  28. ( 2 ) (2)
  29. π ( 𝐱 ) = π ( 𝐱 + 𝐞 i ) μ i ( x i + 1 ) / λ i = π ( 𝐱 + 𝐞 i - 𝐞 j ) μ i ( x i + 1 ) λ j / [ λ i μ j ( x j ) ] \pi(\mathbf{x})=\pi(\mathbf{x}+\mathbf{e}_{i})\mu_{i}(x_{i}+1)/\lambda_{i}=\pi% (\mathbf{x}+\mathbf{e}_{i}-\mathbf{e}_{j})\mu_{i}(x_{i}+1)\lambda_{j}/[\lambda% _{i}\mu_{j}(x_{j})]
  30. ( 2 ) (2)
  31. i = 1 J [ α p 0 i + μ i ( x i ) ( 1 - p i i ) ] = i = 1 J [ α p 0 i λ i μ i ( x i ) + λ i p i 0 ] + i = 1 J j i λ i λ j p i j μ j ( x j ) . ( 4 ) \sum_{i=1}^{J}[\alpha p_{0i}+\mu_{i}(x_{i})(1-p_{ii})]=\sum_{i=1}^{J}[\frac{% \alpha p_{0i}}{\lambda_{i}}\mu_{i}(x_{i})+\lambda_{i}p_{i0}]+\sum_{i=1}^{J}% \sum_{j\neq i}\frac{\lambda_{i}}{\lambda_{j}}p_{ij}\mu_{j}(x_{j}).\qquad(4)
  32. ( 1 ) (1)
  33. i = 1 J j i λ i λ j p i j μ j ( x j ) = j = 1 J [ i j λ i λ j p i j ] μ j ( x j ) = j = 1 J [ 1 - p j j - α p 0 j λ j ] μ j ( x j ) . \sum_{i=1}^{J}\sum_{j\neq i}\frac{\lambda_{i}}{\lambda_{j}}p_{ij}\mu_{j}(x_{j}% )=\sum_{j=1}^{J}[\sum_{i\neq j}\frac{\lambda_{i}}{\lambda_{j}}p_{ij}]\mu_{j}(x% _{j})=\sum_{j=1}^{J}[1-p_{jj}-\frac{\alpha p_{0j}}{\lambda_{j}}]\mu_{j}(x_{j}).
  34. ( 4 ) (4)
  35. i = 1 J α p 0 i = i = 1 J λ i p i 0 \sum_{i=1}^{J}\alpha p_{0i}=\sum_{i=1}^{J}\lambda_{i}p_{i0}
  36. i = 1 J α p 0 i = i = 1 J λ i - i = 1 J j = 1 J λ j p j i = i = 1 J λ i - j = 1 J λ j ( 1 - p j 0 ) = i = 1 J λ i p i 0 \sum_{i=1}^{J}\alpha p_{0i}=\sum_{i=1}^{J}\lambda_{i}-\sum_{i=1}^{J}\sum_{j=1}% ^{J}\lambda_{j}p_{ji}=\sum_{i=1}^{J}\lambda_{i}-\sum_{j=1}^{J}\lambda_{j}(1-p_% {j0})=\sum_{i=1}^{J}\lambda_{i}p_{i0}
  37. ( 2 ) (2)
  38. 𝐗 \mathbf{X}
  39. 𝐘 \mathbf{Y}
  40. α = 5 , p 01 = p 02 = 0.5 , p 03 = 0 , \alpha=5,\quad p_{01}=p_{02}=0.5,\quad p_{03}=0,\quad
  41. P = [ 0 0.5 0.5 0 0 0 0 0 0 ] , μ = [ μ 1 ( x 1 ) μ 2 ( x 2 ) μ 3 ( x 3 ) ] = [ 15 12 10 ] for all x i > 0 P=\begin{bmatrix}0&0.5&0.5\\ 0&0&0\\ 0&0&0\end{bmatrix},\quad\mu=\begin{bmatrix}\mu_{1}(x_{1})\\ \mu_{2}(x_{2})\\ \mu_{3}(x_{3})\end{bmatrix}=\begin{bmatrix}15\\ 12\\ 10\end{bmatrix}\,\text{ for all }x_{i}>0
  42. λ = ( I - P ) - 1 a = [ 1 0 0 - 0.5 1 0 - 0.5 0 1 ] - 1 [ 0.5 × 5 0.5 × 5 0 ] = [ 1 0 0 0.5 1 0 0.5 0 1 ] [ 2.5 2.5 0 ] = [ 2.5 3.75 1.25 ] \lambda=(I-P^{\prime})^{-1}a=\begin{bmatrix}1&0&0\\ -0.5&1&0\\ -0.5&0&1\end{bmatrix}^{-1}\begin{bmatrix}0.5\times 5\\ 0.5\times 5\\ 0\end{bmatrix}=\begin{bmatrix}1&0&0\\ 0.5&1&0\\ 0.5&0&1\end{bmatrix}\begin{bmatrix}2.5\\ 2.5\\ 0\end{bmatrix}=\begin{bmatrix}2.5\\ 3.75\\ 1.25\end{bmatrix}
  43. 𝐘 \mathbf{Y}
  44. P ( Y 1 = 0 ) = ( 1 + n = 1 ( 2.5 15 ) n ) - 1 = 5 6 P(Y_{1}=0)=\left(1+\sum_{n=1}^{\infty}\left(\frac{2.5}{15}\right)^{n}\right)^{% -1}=\frac{5}{6}
  45. P ( Y 2 = 0 ) = ( 1 + n = 1 ( 3.75 12 ) n ) - 1 = 11 16 P(Y_{2}=0)=\left(1+\sum_{n=1}^{\infty}\left(\frac{3.75}{12}\right)^{n}\right)^% {-1}=\frac{11}{16}
  46. P ( Y 3 = 0 ) = ( 1 + n = 1 ( 1.25 10 ) n ) - 1 = 7 8 P(Y_{3}=0)=\left(1+\sum_{n=1}^{\infty}\left(\frac{1.25}{10}\right)^{n}\right)^% {-1}=\frac{7}{8}
  47. π ( 1 , 1 , 1 ) = 5 6 2.5 15 11 16 3.75 12 7 8 1.25 10 0.00326 \pi(1,1,1)=\frac{5}{6}\cdot\frac{2.5}{15}\cdot\frac{11}{16}\cdot\frac{3.75}{12% }\cdot\frac{7}{8}\cdot\frac{1.25}{10}\approx 0.00326
  48. Y i Y_{i}
  49. Q ( t ) Q(t)
  50. R B M Q ( 0 ) ( θ , Γ ; R ) . RBM_{Q(0)}(\theta,\Gamma;R).
  51. θ \theta
  52. Γ \Gamma
  53. R R
  54. θ = α - ( I - P ) μ \theta=\alpha-(I-P^{\prime})\mu
  55. Γ = ( Γ k l ) \Gamma=(\Gamma_{kl})
  56. Γ k l = j = 1 J ( λ j μ j ) [ p j k ( δ k l - p j l ) + c j 2 ( p j k - δ j k ) ( p j l - δ j l ) ] + α k c 0 , k 2 δ k l \Gamma_{kl}=\sum_{j=1}^{J}(\lambda_{j}\wedge\mu_{j})[p_{jk}(\delta_{kl}-p_{jl}% )+c_{j}^{2}(p_{jk}-\delta_{jk})(p_{jl}-\delta_{jl})]+\alpha_{k}c_{0,k}^{2}% \delta_{kl}
  57. R = I - P R=I-P^{\prime}
  58. α = ( α j ) j = 1 J \alpha=(\alpha_{j})_{j=1}^{J}
  59. μ = ( μ ) j = 1 J \mu=(\mu)_{j=1}^{J}
  60. P P
  61. λ j \lambda_{j}
  62. j t h j^{th}
  63. c j c_{j}
  64. j t h j^{th}
  65. c 0 , j c_{0,j}
  66. j t h j^{th}
  67. δ i j \delta_{ij}
  68. A ( t ) A(t)
  69. A ( t ) - α t A ^ ( t ) A(t)-\alpha t{}\approx\hat{A}(t)
  70. A ^ ( t ) \hat{A}(t)
  71. Γ 0 = ( Γ i j 0 ) \Gamma^{0}=(\Gamma^{0}_{ij})
  72. Γ i j 0 = α i c 0 , i 2 δ i j \Gamma^{0}_{ij}=\alpha_{i}c_{0,i}^{2}\delta_{ij}
  73. i , j { 1 , , J } i,j\in\{1,\dots,J\}

Jacobi_eigenvalue_algorithm.html

  1. S = G S G S^{\prime}=GSG^{\top}\,
  2. S i i \displaystyle S^{\prime}_{ii}
  3. S i j = cos ( 2 θ ) S i j + 1 2 sin ( 2 θ ) ( S i i - S j j ) S^{\prime}_{ij}=\cos(2\theta)S_{ij}+\tfrac{1}{2}\sin(2\theta)(S_{ii}-S_{jj})
  4. tan ( 2 θ ) = 2 S i j S j j - S i i \tan(2\theta)=\frac{2S_{ij}}{S_{jj}-S_{ii}}
  5. S j j = S i i S_{jj}=S_{ii}
  6. θ = π 4 \theta=\frac{\pi}{4}
  7. p = S k l p=S_{kl}
  8. | S i j | | p | |S_{ij}|\leq|p|
  9. 1 i , j n , i j 1\leq i,j\leq n,i\neq j
  10. p 2 Γ ( S ) 2 2 N p 2 p^{2}\leq\Gamma(S)^{2}\leq 2Np^{2}
  11. 2 p 2 Γ ( S ) 2 / N 2p^{2}\geq\Gamma(S)^{2}/N
  12. Γ ( S J ) 2 ( 1 - 1 / N ) Γ ( S ) 2 \Gamma(S^{J})^{2}\leq(1-1/N)\Gamma(S)^{2}
  13. Γ ( S J ) ( 1 - 1 / N ) 1 / 2 Γ ( S ) \Gamma(S^{J})\leq(1-1/N)^{1/2}\Gamma(S)
  14. ( 1 - 1 / N ) 1 / 2 (1-1/N)^{1/2}
  15. S σ S^{\sigma}
  16. Γ ( S σ ) ( 1 - 1 / N ) N / 2 Γ ( S ) \Gamma(S^{\sigma})\leq(1-1/N)^{N/2}\Gamma(S)
  17. e 1 / 2 e^{1/2}
  18. λ 1 , , λ m \lambda_{1},...,\lambda_{m}
  19. ν 1 , , ν m \nu_{1},...,\nu_{m}
  20. N S := 1 2 n ( n - 1 ) - μ = 1 m 1 2 ν μ ( ν μ - 1 ) N N_{S}:=\frac{1}{2}n(n-1)-\sum_{\mu=1}^{m}\frac{1}{2}\nu_{\mu}(\nu_{\mu}-1)\leq N
  21. S s S^{s}
  22. Γ ( S s ) n 2 - 1 γ 2 d - 2 γ , γ := Γ ( S ) \Gamma(S^{s})\leq\sqrt{\frac{n}{2}-1}\frac{\gamma^{2}}{d-2\gamma},\quad\gamma:% =\Gamma(S)
  23. Γ ( S ) < d / ( 2 + n 2 - 1 ) \Gamma(S)<d/(2+\sqrt{\frac{n}{2}-1})
  24. m 1 , , m n - 1 m_{1},\,\dots\,,\,m_{n-1}
  25. m i m_{i}
  26. ( i , m i ) (i,m_{i})
  27. m k and m l m_{k}\mbox{ and }~{}m_{l}
  28. m i m_{i}
  29. N S < N N_{S}<N
  30. e i e_{i}
  31. E i E_{i}
  32. e i e_{i}
  33. e k e_{k}
  34. e l e_{l}
  35. e 1 , , e n e_{1},\,...\,,e_{n}
  36. S = ( 4 - 30 60 - 35 - 30 300 - 675 420 60 - 675 1620 - 1050 - 35 420 - 1050 700 ) S=\begin{pmatrix}4&-30&60&-35\\ -30&300&-675&420\\ 60&-675&1620&-1050\\ -35&420&-1050&700\end{pmatrix}
  37. e 1 = 2585.25381092892231 e_{1}=2585.25381092892231
  38. E 1 = ( 0.0291933231647860588 - 0.328712055763188997 0.791411145833126331 - 0.514552749997152907 ) E_{1}=\begin{pmatrix}0.0291933231647860588\\ -0.328712055763188997\\ 0.791411145833126331\\ -0.514552749997152907\end{pmatrix}
  39. e 2 = 37.1014913651276582 e_{2}=37.1014913651276582
  40. E 2 = ( - 0.179186290535454826 0.741917790628453435 - 0.100228136947192199 - 0.638282528193614892 ) E_{2}=\begin{pmatrix}-0.179186290535454826\\ 0.741917790628453435\\ -0.100228136947192199\\ -0.638282528193614892\end{pmatrix}
  41. e 3 = 1.4780548447781369 e_{3}=1.4780548447781369
  42. E 3 = ( - 0.582075699497237650 0.370502185067093058 0.509578634501799626 0.514048272222164294 ) E_{3}=\begin{pmatrix}-0.582075699497237650\\ 0.370502185067093058\\ 0.509578634501799626\\ 0.514048272222164294\end{pmatrix}
  43. e 4 = 0.1666428611718905 e_{4}=0.1666428611718905
  44. E 4 = ( 0.792608291163763585 0.451923120901599794 0.322416398581824992 0.252161169688241933 ) E_{4}=\begin{pmatrix}0.792608291163763585\\ 0.451923120901599794\\ 0.322416398581824992\\ 0.252161169688241933\end{pmatrix}
  45. A T A A^{T}A
  46. S T S = S 2 S^{T}S=S^{2}
  47. A x 2 \|Ax\|_{2}
  48. x 2 = 1 \|x\|_{2}=1
  49. cond ( A ) = A 2 A - 1 2 \mbox{cond}~{}(A)=\|A\|_{2}\|A^{-1}\|_{2}
  50. X = A + X=A^{+}
  51. A + = A - 1 A^{+}=A^{-1}
  52. S = E T Diag ( e ) E S=E^{T}\mbox{Diag}~{}(e)E
  53. e + e^{+}
  54. e i e_{i}
  55. 1 / e i 1/e_{i}
  56. e i 0 e_{i}\leq 0
  57. e i e_{i}
  58. S + = E T Diag ( e + ) E S^{+}=E^{T}\mbox{Diag}~{}(e^{+})E
  59. A x - b 2 \|Ax-b\|_{2}
  60. x = A + b x=A^{+}b
  61. x = S + b = E T Diag ( e + ) E b x=S^{+}b=E^{T}\mbox{Diag}~{}(e^{+})Eb
  62. S = E T Diag ( e ) E S=E^{T}\mbox{Diag}~{}(e)E
  63. exp S = E T Diag ( exp e ) E \exp S=E^{T}\mbox{Diag}~{}(\exp e)E
  64. e i e_{i}
  65. exp e i \exp e_{i}
  66. x ( t ) = E T Diag ( exp t e ) E a x(t)=E^{T}\mbox{Diag}~{}(\exp te)Ea
  67. a = i = 1 n a i E i a=\sum_{i=1}^{n}a_{i}E_{i}
  68. x ( t ) = i = 1 n a i exp ( t e i ) E i x(t)=\sum_{i=1}^{n}a_{i}\exp(te_{i})E_{i}
  69. W s W^{s}
  70. W u W^{u}
  71. a W s a\in W^{s}
  72. lim x t ( t ) = 0 \mbox{lim}~{}_{t\ \infty}x(t)=0
  73. a W u a\in W^{u}
  74. lim x t ( t ) = \mbox{lim}~{}_{t\ \infty}x(t)=\infty
  75. W s W^{s}
  76. W u W^{u}
  77. W u W^{u}
  78. t t\ \infty
  79. S = A T A S=A^{T}A
  80. J S J T = J A T A J T = J A T J T J A J T = B T B JSJ^{T}=JA^{T}AJ^{T}=JA^{T}J^{T}JAJ^{T}=B^{T}B
  81. B := J A J T B\,:=JAJ^{T}

James–Stein_estimator.html

  1. m m
  2. m m
  3. 𝐲 N ( s y m b o l θ , σ 2 I ) . {\mathbf{y}}\sim N({symbol\theta},\sigma^{2}I).\,
  4. s y m b o l θ ^ = s y m b o l θ ^ ( 𝐲 ) \widehat{symbol\theta}=\widehat{symbol\theta}({\mathbf{y}})
  5. s y m b o l θ ^ L S = 𝐲 \widehat{symbol\theta}_{LS}={\mathbf{y}}
  6. E { s y m b o l θ - s y m b o l θ ^ 2 } E\{\|{symbol\theta}-\widehat{symbol\theta}\|^{2}\}
  7. σ 2 \sigma^{2}
  8. s y m b o l θ ^ J S = ( 1 - ( m - 2 ) σ 2 𝐲 2 ) 𝐲 . \widehat{symbol\theta}_{JS}=\left(1-\frac{(m-2)\sigma^{2}}{\|{\mathbf{y}}\|^{2% }}\right){\mathbf{y}}.
  9. s y m b o l θ ^ L S \widehat{symbol\theta}_{LS}
  10. m 3 m\geq 3
  11. m 3 m\geq 3
  12. ( m - 2 ) σ 2 < 𝐲 2 (m-2)\sigma^{2}<\|{\mathbf{y}}\|^{2}
  13. 𝐲 \mathbf{y}
  14. m m
  15. s y m b o l θ ^ J S = ( 1 - ( m - 2 ) σ 2 𝐲 - s y m b o l ν 2 ) ( 𝐲 - s y m b o l ν ) + s y m b o l ν . \widehat{symbol\theta}_{JS}=\left(1-\frac{(m-2)\sigma^{2}}{\|{\mathbf{y}}-{% symbol\nu}\|^{2}}\right)({\mathbf{y}}-{symbol\nu})+{symbol\nu}.
  16. s y m b o l θ - s y m b o l ν \|{symbol\theta-symbol\nu}\|
  17. N ( 0 , A ) \sim N(0,A)
  18. m m
  19. m 2 m\leq 2
  20. 𝐲 - s y m b o l ν , \|{\mathbf{y}}-{symbol\nu}\|,
  21. 𝐲 - s y m b o l ν {\mathbf{y}}-{symbol\nu}
  22. s y m b o l θ ^ J S + = ( 1 - ( m - 2 ) σ 2 𝐲 - s y m b o l ν 2 ) + ( 𝐲 - s y m b o l ν ) + s y m b o l ν . \widehat{symbol\theta}_{JS+}=\left(1-\frac{(m-2)\sigma^{2}}{\|{\mathbf{y}}-{% symbol\nu}\|^{2}}\right)^{+}({\mathbf{y}}-{symbol\nu})+{symbol\nu}.
  23. σ 2 \sigma^{2}
  24. σ ^ 2 = 1 n ( y i - y ¯ ) 2 \widehat{\sigma}^{2}=\frac{1}{n}\sum(y_{i}-\overline{y})^{2}
  25. m > 2 m>2
  26. n n
  27. s y m b o l θ ^ J S = ( 1 - ( m - 2 ) σ 2 n 𝐲 ¯ 2 ) 𝐲 ¯ , \widehat{symbol\theta}_{JS}=\left(1-\frac{(m-2)\frac{\sigma^{2}}{n}}{\|{% \overline{\mathbf{y}}}\|^{2}}\right){\overline{\mathbf{y}}},
  28. 𝐲 ¯ {\overline{\mathbf{y}}}
  29. m m
  30. n n

Janus_(programming_language).html

  1. λ v . ( v , v ) \lambda v^{\prime}.\oplus\left(v^{\prime},v\right)
  2. \oplus

Jeans_instability.html

  1. d p d r = - G ρ ( r ) M e n c ( r ) r 2 \frac{dp}{dr}=-\frac{G\rho(r)M_{enc}(r)}{r^{2}}
  2. M e n c ( r ) M_{enc}(r)
  3. p p
  4. ρ ( r ) \rho(r)
  5. r r
  6. G G
  7. r r
  8. R R
  9. M M
  10. c s c_{s}
  11. t s o u n d = R c s ( 5 × 10 5 yr ) ( R 0.1 pc ) ( c s 0.2 km s - 1 ) - 1 t_{sound}=\frac{R}{c_{s}}\simeq(5\times 10^{5}\mbox{ yr}~{})\left(\frac{R}{0.1% \mbox{ pc}~{}}\right)\left(\frac{c_{s}}{0.2\mbox{ km s}~{}^{-1}}\right)^{-1}
  12. t ff = 1 G ρ ( 2 Myr ) ( n 10 3 cm - 3 ) - 1 / 2 t_{\rm ff}=\frac{1}{\sqrt{G\rho}}\simeq(2\mbox{ Myr}~{})\left(\frac{n}{10^{3}% \mbox{ cm}~{}^{-3}}\right)^{-1/2}
  13. G G
  14. ρ \rho
  15. n = ρ / μ n=\rho/\mu
  16. μ = 3.9 × 10 - 24 \mu=3.9\times 10^{-24}
  17. t ff < t s o u n d . t_{\rm ff}<t_{sound}.
  18. λ J \lambda_{J}
  19. λ J = c s G ρ ( 0.4 pc ) ( c s 0.2 km s - 1 ) ( n 10 3 cm - 3 ) - 1 / 2 . \lambda_{J}=\frac{c_{s}}{\sqrt{G\rho}}\simeq(0.4\mbox{ pc}~{})\left(\frac{c_{s% }}{0.2\mbox{ km s}~{}^{-1}}\right)\left(\frac{n}{10^{3}\mbox{ cm}~{}^{-3}}% \right)^{-1/2}.
  20. M J M_{J}
  21. R J R_{J}
  22. R J R_{J}
  23. M J = ( 4 π 3 ) ρ R J 3 = ( π 6 ) c s 3 G 3 / 2 ρ 1 / 2 ( 2 M ) ( c s 0.2 km s - 1 ) 3 ( n 10 3 cm - 3 ) - 1 / 2 . M_{J}=\left(\frac{4\pi}{3}\right)\rho R_{J}^{3}=\left(\frac{\pi}{6}\right)% \frac{c_{s}^{3}}{G^{3/2}\rho^{1/2}}\simeq(2\mbox{ M}~{}_{\odot})\left(\frac{c_% {s}}{0.2\mbox{ km s}~{}^{-1}}\right)^{3}\left(\frac{n}{10^{3}\mbox{ cm}~{}^{-3% }}\right)^{-1/2}.
  24. λ J = 15 k B T 4 π G μ ρ , \lambda_{J}=\sqrt{\frac{15k_{B}T}{4\pi G\mu\rho}},
  25. k B k_{B}
  26. T T
  27. r r
  28. μ \mu
  29. G G
  30. ρ \rho
  31. 15 15
  32. 4 π 4\pi
  33. ρ \rho
  34. M r 3 \frac{M}{r^{3}}
  35. λ J k B T r 3 G M μ . \lambda_{J}\approx\sqrt{\frac{k_{B}Tr^{3}}{GM\mu}}.
  36. λ J = r \lambda_{J}=r
  37. k B T = G M μ r k_{B}T=\frac{GM\mu}{r}
  38. λ J = 2 π k J = c s ( π G ρ ) 1 / 2 , \lambda_{J}=\frac{2\pi}{k_{J}}=c_{s}\left(\frac{\pi}{G\rho}\right)^{1/2},
  39. c s c_{s}
  40. ρ \rho
  41. P V γ = constant V P - 1 / γ . PV^{\gamma}=\,\text{constant}\rightarrow V\sim P^{-1/\gamma}.
  42. P V = n R T P . P - 1 / γ T P T γ γ - 1 . PV=nRT\rightarrow P.P^{-1/\gamma}\sim T\rightarrow P\sim T^{\frac{\gamma}{% \gamma-1}}.
  43. P = K ρ γ T ρ γ - 1 . P=K\rho^{\gamma}\rightarrow T\sim\rho^{\gamma-1}.
  44. M J T 3 / 2 ρ - 1 / 2 ρ 3 2 ( γ - 1 ) ρ - 1 / 2 . M_{J}\sim T^{3/2}\rho^{-1/2}\sim\rho^{\frac{3}{2}(\gamma-1)}\rho^{-1/2}.
  45. M J ρ 3 2 ( γ - 4 3 ) . M_{J}\sim\rho^{\frac{3}{2}\left(\gamma-\frac{4}{3}\right)}.
  46. γ > 4 3 \gamma>\frac{4}{3}
  47. γ < 4 3 \gamma<\frac{4}{3}

Jeffrey_Lagarias.html

  1. σ ( n ) H n + e H n ln H n \sigma(n)\leq H_{n}+e^{H_{n}}\ln H_{n}\,
  2. n n

Jefimenko's_equations.html

  1. 𝐄 ( 𝐫 , t ) = 1 4 π ϵ 0 [ ( ρ ( 𝐫 , t r ) | 𝐫 - 𝐫 | 3 + 1 | 𝐫 - 𝐫 | 2 c ρ ( 𝐫 , t r ) t ) ( 𝐫 - 𝐫 ) - 1 | 𝐫 - 𝐫 | c 2 𝐉 ( 𝐫 , t r ) t ] d 3 𝐫 \mathbf{E}(\mathbf{r},t)=\frac{1}{4\pi\epsilon_{0}}\int\left[\left(\frac{\rho(% \mathbf{r}^{\prime},t_{r})}{|\mathbf{r}-\mathbf{r}^{\prime}|^{3}}+\frac{1}{|% \mathbf{r}-\mathbf{r}^{\prime}|^{2}c}\frac{\partial\rho(\mathbf{r}^{\prime},t_% {r})}{\partial t}\right)(\mathbf{r}-\mathbf{r}^{\prime})-\frac{1}{|\mathbf{r}-% \mathbf{r}^{\prime}|c^{2}}\frac{\partial\mathbf{J}(\mathbf{r}^{\prime},t_{r})}% {\partial t}\right]\mathrm{d}^{3}\mathbf{r}^{\prime}
  2. 𝐁 ( 𝐫 , t ) = μ 0 4 π [ 𝐉 ( 𝐫 , t r ) | 𝐫 - 𝐫 | 3 + 1 | 𝐫 - 𝐫 | 2 c 𝐉 ( 𝐫 , t r ) t ] × ( 𝐫 - 𝐫 ) d 3 𝐫 \mathbf{B}(\mathbf{r},t)=\frac{\mu_{0}}{4\pi}\int\left[\frac{\mathbf{J}(% \mathbf{r}^{\prime},t_{r})}{|\mathbf{r}-\mathbf{r}^{\prime}|^{3}}+\frac{1}{|% \mathbf{r}-\mathbf{r}^{\prime}|^{2}c}\frac{\partial\mathbf{J}(\mathbf{r}^{% \prime},t_{r})}{\partial t}\right]\times(\mathbf{r}-\mathbf{r}^{\prime})% \mathrm{d}^{3}\mathbf{r}^{\prime}
  3. t r = t - | 𝐫 - 𝐫 | c t_{r}=t-\frac{|\mathbf{r}-\mathbf{r}^{\prime}|}{c}
  4. φ ( 𝐫 , t ) = 1 4 π ϵ 0 ρ ( 𝐫 , t r ) | 𝐫 - 𝐫 | d 3 𝐫 \displaystyle\varphi(\mathbf{r},t)=\dfrac{1}{4\pi\epsilon_{0}}\int\dfrac{\rho(% \mathbf{r}^{\prime},t_{r})}{|\mathbf{r}-\mathbf{r}^{\prime}|}\mathrm{d}^{3}% \mathbf{r}^{\prime}
  5. - 𝐄 = φ + 𝐀 t , 𝐁 = × 𝐀 -\mathbf{E}=\nabla\varphi+\dfrac{\partial\mathbf{A}}{\partial t}\,,\quad% \mathbf{B}=\nabla\times\mathbf{A}
  6. c 2 = 1 ϵ 0 μ 0 c^{2}=\frac{1}{\epsilon_{0}\mu_{0}}

Jenny_Harrison.html

  1. C 2 C^{2}

Jensen–Shannon_divergence.html

  1. M + 1 ( A ) M_{+}^{1}(A)
  2. M + 1 ( A ) × M + 1 ( A ) [ 0 , ) M_{+}^{1}(A)\times M_{+}^{1}(A)\rightarrow[0,\infty{})
  3. D ( P Q ) D(P\parallel Q)
  4. JSD ( P Q ) = 1 2 D ( P M ) + 1 2 D ( Q M ) {\rm JSD}(P\parallel Q)=\frac{1}{2}D(P\parallel M)+\frac{1}{2}D(Q\parallel M)
  5. M = 1 2 ( P + Q ) M=\frac{1}{2}(P+Q)
  6. JSD π 1 , , π n ( P 1 , P 2 , , P n ) = H ( i = 1 n π i P i ) - i = 1 n π i H ( P i ) {\rm JSD}_{\pi_{1},\ldots,\pi_{n}}(P_{1},P_{2},\ldots,P_{n})=H\left(\sum_{i=1}% ^{n}\pi_{i}P_{i}\right)-\sum_{i=1}^{n}\pi_{i}H(P_{i})
  7. π 1 , , π n \pi_{1},\ldots,\pi_{n}
  8. P 1 , P 2 , , P n P_{1},P_{2},\ldots,P_{n}
  9. H ( P ) H(P)
  10. P P
  11. P 1 = P , P 2 = Q , π 1 = π 2 = 1 2 . P_{1}=P,P_{2}=Q,\pi_{1}=\pi_{2}=\frac{1}{2}.
  12. 0 JSD ( P Q ) 1 0\leq{\rm JSD}(P\parallel Q)\leq 1
  13. 0 JSD ( P Q ) ln ( 2 ) 0\leq{\rm JSD}(P\parallel Q)\leq\ln(2)
  14. X X
  15. P P
  16. Q Q
  17. Z Z
  18. P P
  19. Q Q
  20. X X
  21. X X
  22. P P
  23. Z = 0 Z=0
  24. Q Q
  25. Z = 1 Z=1
  26. X X
  27. M = ( P + Q ) / 2 M=(P+Q)/2
  28. I ( X ; Z ) = H ( X ) - H ( X | Z ) = - M log M + 1 2 [ P log P + Q log Q ] = - P 2 log M - Q 2 log M + 1 2 [ P log P + Q log Q ] = 1 2 P ( log P - log M ) + 1 2 Q ( log Q - log M ) = JSD ( P Q ) \begin{aligned}\displaystyle I(X;Z)&\displaystyle=H(X)-H(X|Z)\\ &\displaystyle=-\sum M\log M+\frac{1}{2}\left[\sum P\log P+\sum Q\log Q\right]% \\ &\displaystyle=-\sum\frac{P}{2}\log M-\sum\frac{Q}{2}\log M+\frac{1}{2}\left[% \sum P\log P+\sum Q\log Q\right]\\ &\displaystyle=\frac{1}{2}\sum P\left(\log P-\log M\right)+\frac{1}{2}\sum Q% \left(\log Q-\log M\right)\\ &\displaystyle={\rm JSD}(P\parallel Q)\end{aligned}
  29. H ( Z ) = 1 H(Z)=1
  30. Z Z
  31. ( ρ 1 , , ρ n ) (\rho_{1},\ldots,\rho_{n})
  32. π = ( π 1 , , π n ) \pi=(\pi_{1},\ldots,\pi_{n})
  33. QJSD ( ρ 1 , , ρ n ) = S ( i = 1 n π i ρ i ) - i = 1 n π i S ( ρ i ) {\rm QJSD}(\rho_{1},\ldots,\rho_{n})=S\left(\sum_{i=1}^{n}\pi_{i}\rho_{i}% \right)-\sum_{i=1}^{n}\pi_{i}S(\rho_{i})
  34. S ( π i ) S(\pi_{i})
  35. ( ρ 1 , , ρ n ) (\rho_{1},\ldots,\rho_{n})
  36. π \pi
  37. π = ( 1 2 , 1 2 ) \pi=\left(\frac{1}{2},\frac{1}{2}\right)

Jet_engine_performance.html

  1. T 1 = t 0 ( 1 + ( γ c - 1 ) M 2 / 2 ) T_{1}=t_{0}\cdot(1+({\gamma}_{c}-1)\cdot M^{2}/2)
  2. P 1 = p 0 ( T 1 / t 0 ) γ c / ( γ c - 1 ) P_{1}=p_{0}\cdot(T_{1}/t_{0})^{{\gamma}_{c}/({\gamma}_{c}-1)}
  3. T 2 = T 1 T_{2}=T_{1}\,
  4. P 2 = P 1 prf P_{2}=P_{1}\cdot\mathrm{prf}
  5. T 3 = T 2 ( P 3 / P 2 ) ( γ c - 1 ) / ( γ c η p c ) T_{3}=T_{2}\cdot(P_{3}/P_{2})^{{(\gamma}_{c}-1)/({\gamma}_{c}\cdot{\eta}pc)}
  6. P 3 = P 2 ( P 3 / P 2 ) P_{3}=P_{2}\cdot(P_{3}/P_{2})
  7. T 4 = RIT T_{4}=\mathrm{RIT}\,
  8. P 4 = P 3 ( P 4 / P 3 ) P_{4}=P_{3}\cdot(P_{4}/P_{3})
  9. w 4 C pt ( T 4 - T 5 ) = w 2 C pc ( T 3 - T 2 ) w_{4}\cdot C_{\mathrm{pt}}(T_{4}-T_{5})=w_{2}\cdot C_{\mathrm{pc}}(T_{3}-T_{2})
  10. P 4 / P 5 = ( T 4 / T 5 ) γ t / ( ( γ t - 1 ) . η pt ) P_{4}/P_{5}=(T_{4}/T_{5})^{{\gamma}_{t}/(({\gamma}_{t}-1).{\eta}_{\mathrm{pt}})}
  11. P 5 = P 4 / ( P 4 / P 5 ) P_{5}=P_{4}/(P_{4}/P_{5})\,
  12. T 8 = T 5 T_{8}=T_{5}\,
  13. P 8 = P 5 ( P 8 / P 5 ) P_{8}=P_{5}\cdot(P_{8}/P_{5})\,
  14. ( P 8 / p 8 s ) c r i t = ( ( γ t + 1 ) / 2 ) ) γ t / ( γ t - 1 ) (P_{8}/p_{\mathrm{8s}})crit=(({\gamma}_{t}+1)/2))^{{\gamma}_{t}/({\gamma}_{t}-% 1)}\,
  15. ( P 8 / p 0 ) ( P 8 / p 8 s ) c r i t (P_{8}/p_{0})>=(P_{8}/p_{\mathrm{8s}})crit\,
  16. ( P 8 / p 0 ) < ( P 8 / p 8 s ) c r i t (P_{8}/p_{0})<(P_{8}/p_{\mathrm{8s}})crit\,
  17. t 8 s = T 8 / ( ( γ t + 1 ) / 2 ) t_{\mathrm{8s}}=T_{8}/(({\gamma}_{t}+1)/2)\,
  18. p 8 s = P 8 / ( T 8 / t 8 s ) γ t / ( γ t - 1 ) p_{\mathrm{8s}}=P_{8}/(T_{8}/t_{\mathrm{8s}})^{{\gamma}_{t}/({\gamma}_{t}-1)}
  19. V 8 2 = 2 g J C p t ( T 8 - t 8 s ) V_{8}^{2}=2gJC_{pt}(T_{8}-t_{\mathrm{8s}})
  20. ρ 8 s = p 8 s / ( R t 8 s ) {\rho}_{\mathrm{8s}}=p_{\mathrm{8s}}/(R\cdot t_{\mathrm{8s}})
  21. A 8 = w 8 / ( ρ 8 s V 8 ) A_{8}=w_{8}/({\rho}_{\mathrm{8s}}\cdot V_{8})
  22. F g = C x ( ( w 8 V 8 / g ) + A 8 ( p 8 s - p 0 ) ) F_{g}=C_{\mathrm{x}}((w_{8}\cdot V_{8}/g)+A_{8}(p_{\mathrm{8s}}-p_{0}))\,
  23. p 8 s = p 0 p_{\mathrm{8s}}=p_{0}\,
  24. t 8 s = T 8 / ( P 8 / p 8 s ) ( γ t - 1 ) / γ t t_{\mathrm{8s}}=T_{8}/(P_{8}/p_{\mathrm{8s}})^{{(\gamma}_{t}-1)/{\gamma}_{t}}
  25. V 8 2 = 2 g J C p t ( T 8 - t 8 s ) V_{8}^{2}=2gJC_{pt}(T_{8}-t_{\mathrm{8s}})
  26. F g = C x ( ( w 8 V 8 / g ) F_{g}=C_{\mathrm{x}}((w_{8}\cdot V_{8}/g)\,
  27. F r = w 0 V 0 / g F_{r}=w_{0}\cdot V_{0}/g
  28. F n = F g - F r F_{n}=F_{g}-F_{r}\,
  29. w 2 = 100 lb / s w_{2}=100\ \mathrm{lb/s}\,
  30. P 3 / P 2 = 10.0 P_{3}/P_{2}=10.0\,
  31. T 4 = RIT = 1400 K T_{4}=\mathrm{RIT}=1400\ \mathrm{K}\,
  32. prf = 1.0 \mathrm{prf}=1.0\,
  33. η p c = 0.89 ( i . e .89 % ) {\eta}pc=0.89\ (i.e.89\%)\,
  34. η p t = 0.90 ( i . e .90 % ) {\eta}pt=0.90\ (i.e.90\%)\,
  35. P 4 / P 3 = 0.95 P_{4}/P_{3}=0.95\,
  36. P 8 / P 5 = 0.99 P_{8}/P_{5}=0.99\,
  37. C x = 0.995 C_{\mathrm{x}}=0.995\,
  38. γ c = 1.4 {\gamma}_{c}=1.4\,
  39. γ t = 1.333 {\gamma}_{t}=1.333\,
  40. C pc = 0.6111 hp s lb K C_{\mathrm{pc}}=0.6111\ \frac{\mathrm{hp}\cdot\mathrm{s}}{\mathrm{lb}\cdot% \mathrm{K}}\,
  41. C pt = 0.697255 hp s lb K C_{\mathrm{pt}}=0.697255\ \frac{\mathrm{hp}\cdot\mathrm{s}}{\mathrm{lb}\cdot% \mathrm{K}}\,
  42. g = 32.174 ft / s 2 g=32.174\ \mathrm{ft}/\mathrm{s}^{2}\,
  43. J = 550 ft lb / ( s hp ) J=550\ \mathrm{ft}\cdot\mathrm{lb}/(\mathrm{s}\cdot\mathrm{hp})\,
  44. R = 96.034 ft lbf / ( lb K ) R=96.034\ \mathrm{ft}\cdot\mathrm{lbf}/(\mathrm{lb}\cdot\mathrm{K})\,
  45. p 0 = 14.696 psia p_{0}=14.696\ \mathrm{psia}\,
  46. t 0 = 288.15 K t_{0}=288.15\ \mathrm{K}\,
  47. 15 C + 273.15 C 15\ ^{\circ}\mathrm{C}+273.15\ ^{\circ}\mathrm{C}\,
  48. V 0 V_{0}\,
  49. M M\,
  50. T 1 = t 0 = 288.15 K T_{1}=t_{0}=288.15\ \mathrm{K}\,
  51. P 1 = p 0 = 14.696 psia P_{1}=p_{0}=14.696\ \mathrm{psia}\,
  52. T 2 = T 1 = 288.15 K T_{2}=T_{1}=288.15\ \mathrm{K}\,
  53. P 2 = P 1 prf P_{2}=P_{1}\cdot\mathrm{prf}\,
  54. P 2 = 14.696 * 1.0 = 14.696 psia P_{2}=14.696*1.0=14.696\ \mathrm{psia}\,
  55. T 3 = T 2 ( ( P 3 / P 2 ) ( γ c - 1 ) / ( γ c η p c ) = 288.15 * 10 ( 1.4 - 1 ) / ( 1.4 * 0.89 ) = 603.456 K T_{3}=T_{2}\cdot((P_{3}/P_{2})^{({\gamma}_{c}-1)/({\gamma}_{c}\cdot{\eta}pc)}=% 288.15*10^{(1.4-1)/(1.4*0.89)}=603.456\ \mathrm{K}
  56. P 3 = P 2 ( P 3 / P 2 ) P_{3}=P_{2}\cdot(P_{3}/P_{2})\,
  57. P 3 = 14.696 * 10 = 146.96 psia P_{3}=14.696*10=146.96\ \mathrm{psia}\,
  58. T 4 = RIT = 1400 K T_{4}=\mathrm{RIT}=1400\ \mathrm{K}\,
  59. P 4 = P 3 ( P 4 / P 3 ) = 146.96 * 0.95 = 139.612 psia P_{4}=P_{3}\cdot(P_{4}/P_{3})=146.96*0.95=139.612\ \mathrm{psia}\,
  60. w 4 C pt ( T 4 - T 5 ) = w 2 C pc ( T 3 - T 2 ) w_{4}\cdot C_{\mathrm{pt}}(T_{4}-T_{5})=w_{2}\cdot C_{\mathrm{pc}}(T_{3}-T_{2})\,
  61. 100 * 0.697255 * ( 1400 - T 5 ) = 100 * 0.6111 * ( 603.456 - 288.15 ) 100*0.697255*(1400-T_{5})=100*0.6111*(603.456-288.15)\,
  62. T 5 = 1123.65419 K T_{5}=1123.65419\ \mathrm{K}\,
  63. P 4 / P 5 = ( T 4 / T 5 ) γ t / ( ( γ t - 1 ) . η pt ) P4/P5=(T4/T5)^{{\gamma}_{t}/(({\gamma}_{t}-1).{\eta}_{\mathrm{pt}})}\,
  64. P 4 / P 5 = ( 1400 / 1123.65419 ) 1.333 / ( ( 1.333 - 1 ) * 0.9 ) P4/P5=(1400/1123.65419)^{1.333/((1.333-1)*0.9)}\,
  65. P 4 / P 5 = 2.65914769 P4/P5=2.65914769\,
  66. T 8 = T 5 = 1123.65419 K T_{8}=T_{5}=1123.65419\ \mathrm{K}\,
  67. P 5 = P 4 / ( P 4 / P 5 ) P_{5}=P_{4}/(P_{4}/P_{5})\,
  68. P 5 = 139.612 / 2.65914769 = 52.502537 psia P_{5}=139.612/2.65914769=52.502537\ \mathrm{psia}\,
  69. P 8 = P 5 ( P 8 / P 5 ) P_{8}=P_{5}\cdot(P_{8}/P_{5})\,
  70. P 8 = 52.502537 * 0.99 = 51.9775116 psia P_{8}=52.502537*0.99=51.9775116\ \mathrm{psia}\,
  71. P 8 / p 0 = 51.9775116 / 14.696 = 3.53684755 P_{8}/p_{0}=51.9775116/14.696=3.53684755\,
  72. ( P 8 / p 8 s ) c r i t = ( ( γ t + 1 ) / 2 ) ) γ t / ( ( γ t - 1 ) (P_{8}/p_{\mathrm{8s}})crit=(({\gamma}_{t}+1)/2))^{{\gamma}_{t}/(({\gamma}_{t}% -1)}\,
  73. ( P 8 / p 8 s ) c r i t = ( ( 1.333 + 1 ) / 2 ) 1.333 / ( 1.333 - 1 ) = 1.85242156 (P_{8}/p_{\mathrm{8s}})crit=((1.333+1)/2)^{1.333/(1.333-1)}=1.85242156\,
  74. P 8 / p 0 > P 8 / p 8 s P_{8}/p_{0}>P_{8}/p_{\mathrm{8s}}\,
  75. t 8 s = T 8 / ( ( γ t + 1 ) / 2 ) t_{\mathrm{8s}}=T_{8}/(({\gamma}_{t}+1)/2)\,
  76. t 8 s = 1123.65419 / ( ( 1.333 + 1 ) / 2 ) t_{\mathrm{8s}}=1123.65419/((1.333+1)/2)\,
  77. t 8 s = 963.269773 K t_{\mathrm{8s}}=963.269773\ \mathrm{K}\,
  78. p 8 s = P 8 / ( ( T 8 / t 8 s ) γ t / ( γ t - 1 ) ) p_{\mathrm{8s}}=P_{8}/((T_{8}/t_{\mathrm{8s}})^{{\gamma}_{t}/({\gamma}_{t}-1)})
  79. p 8 s = 51.9775116 / ( 1123.65419 / 963.269773 ) ) 1.333 / ( ( 1.333 - 1 ) ) p_{\mathrm{8s}}=51.9775116/(1123.65419/963.269773))^{1.333/((1.333-1))}\,
  80. p 8 s = 28.059224 psia p_{\mathrm{8s}}=28.059224\ \mathrm{psia}\,
  81. V 8 2 = 2 g J C p t ( T 8 - t 8 s ) V_{8}^{2}=2gJC_{pt}(T_{8}-t_{\mathrm{8s}})\,
  82. V 8 2 = 2 * 32.174 * 550 * 0.697255 * ( 1123.65419 - 963.269773 ) = 3957779.09 V_{8}^{2}=2*32.174*550*0.697255*(1123.65419-963.269773)=3957779.09\,
  83. V 8 = 3957779.09 0.5 = 1989.41677 ft / s V_{8}=3957779.09^{0.5}=1989.41677\ \mathrm{ft}/\mathrm{s}\,
  84. ρ 8 s = p 8 s / ( R t 8 s ) {\rho}_{\mathrm{8s}}=p_{\mathrm{8s}}/(R\cdot t_{\mathrm{8s}})\,
  85. ρ 8 s = ( 28.059224 * 144 ) / ( 96.034 * 963.269773 ) = 0.0436782467 lb / ft 3 {\rho}_{\mathrm{8s}}=(28.059224*144)/(96.034*963.269773)=0.0436782467\ \mathrm% {lb}/\mathrm{ft}^{3}\,
  86. A 8 = w 8 / ( ρ 8 s V 8 ) A_{8}=w_{8}/({\rho}_{\mathrm{8s}}\cdot V_{8})\,
  87. A 8 = ( 100 * 144 ) / ( 0.0436782467 * 1989.41677 ) = 165.718701 i n 2 A_{8}=(100*144)/(0.0436782467*1989.41677)=165.718701in^{2}\,
  88. F g = C x ( ( w 8 V 8 / g ) + A 8 ( p 8 s - p 0 ) ) F_{g}=C_{\mathrm{x}}((w_{8}\cdot V_{8}/g)+A_{8}(p_{\mathrm{8s}}-p_{0}))\,
  89. F g = 0.995 ( ( ( 100 * 1989.41677 ) / 32.174 ) + ( 165.718701 * ( 28.059224 - 14.696 ) ) ) F_{g}=0.995(((100*1989.41677)/32.174)+(165.718701*(28.059224-14.696)))\,
  90. F g = 6152.38915 + 2203.46344 F_{g}=6152.38915+2203.46344\,
  91. F g = 8355.85259 lbf F_{g}=8355.85259\ \mathrm{lbf}\,
  92. F r = w 0 V 0 / g F_{r}=w_{0}\cdot V_{0}/g\,
  93. F r = ( 100 * 0 ) / 32.174 = 0 F_{r}=(100*0)/32.174=0\,
  94. F n = F g - F r F_{n}=F_{g}-F_{r}\,
  95. F n = 8355.85259 - 0 = 8356 lbf F_{n}=8355.85259-0=8356\ \mathrm{lbf}\,
  96. w rotorexit C pt T rotorexit = w rotorbleed C pc T rotorbleed + w rotorentry C pt T rotorentry w_{\mathrm{rotorexit}}\cdot C_{\mathrm{pt}}\cdot T_{\mathrm{rotorexit}}=w_{% \mathrm{rotorbleed}}\cdot C_{\mathrm{pc}}\cdot T_{\mathrm{rotorbleed}}+w_{% \mathrm{rotorentry}}\cdot C_{\mathrm{pt}}\cdot T_{\mathrm{rotorentry}}\,
  97. w rotorexit = w rotorbleed + w rotorentry w_{\mathrm{rotorexit}}=w_{\mathrm{rotorbleed}}+w_{\mathrm{rotorentry}}\,
  98. R I T RIT\,
  99. w 2 c o r w_{\mathrm{2cor}}\,
  100. P 3 / P 2 P_{3}/P_{2}\,
  101. F n Fn\,
  102. w fe w_{\mathrm{fe}}\,
  103. T 3 T_{3}\,
  104. A 8 c a l c A_{\mathrm{8calc}}\,
  105. A 8 d e s p t A_{\mathrm{8despt}}\,
  106. w 4 c o r c a l c w_{\mathrm{4corcalc}}\,
  107. w 4 c o r d e s p t w_{\mathrm{4cordespt}}\,
  108. T turb / R I T \triangle T_{\mathrm{turb}}/RIT\,
  109. R I T RIT\,
  110. T turb \triangle T_{\mathrm{turb}}\,
  111. T comp \triangle T_{\mathrm{comp}}\,
  112. T turb \triangle T_{\mathrm{turb}}\,
  113. T comp / T 1 \triangle T_{\mathrm{comp}}/T_{1}\,
  114. R I T / T 1 RIT/T_{1}\,
  115. R I T RIT\,
  116. R I T RIT\,
  117. N cor N_{\mathrm{cor}}\,
  118. β {\beta}\,
  119. F n F_{n}\,
  120. w fe w_{\mathrm{fe}}\,
  121. T 3 T_{\mathrm{3}}\,
  122. A 8 g e o m e t r i c d e s i g n A_{\mathrm{8geometricdesign}}\,
  123. A 8 c a l c / C dcalc A_{\mathrm{8calc}}/C_{\mathrm{dcalc}}\,
  124. w 4 c o r c a l c w_{\mathrm{4corcalc}}\,
  125. w 4 c o r t u r b c h a r w_{\mathrm{4corturbchar}}\,
  126. N cor N_{\mathrm{cor}}\,
  127. β {\beta}\,
  128. N turbcor N_{\mathrm{turbcor}}\,
  129. ( δ H / T ) turb ({\delta}H/T)_{\mathrm{turb}}\,
  130. w 4 c o r t u r b c h a r w_{\mathrm{4corturbchar}}\,
  131. η pt {\eta}_{\mathrm{pt}}\,
  132. N N\,
  133. δ P w {\delta}P_{w}\,
  134. δ τ {\delta}\,{\tau}\,
  135. I I\,
  136. K K\,
  137. d N dN\,
  138. d t dt\,
  139. d N dN\,
  140. δ τ {\delta}\,{\tau}\,
  141. I I\,
  142. K K\,
  143. d t dt\,
  144. δ P w {\delta}P_{w}\,
  145. 2 2\,
  146. π {\pi}\,
  147. N N\,
  148. δ τ {\delta}\,{\tau}\,
  149. K 1 K_{1}\,
  150. d N dN\,
  151. K 1 K_{1}\,
  152. δ P w {\delta}P_{w}\,
  153. 2 2\,
  154. π {\pi}\,
  155. I I\,
  156. N N\,
  157. K K\,
  158. d t dt\,
  159. δ N {\delta}N\,
  160. K 2 K_{2}\,
  161. δ P w {\delta}P_{w}\,
  162. I I\,
  163. N N\,
  164. δ t {\delta}t\,
  165. N new N_{\mathrm{new}}\,
  166. N old N_{\mathrm{old}}\,
  167. δ N {\delta}N\,
  168. t new t_{\mathrm{new}}\,
  169. t old t_{\mathrm{old}}\,
  170. δ t {\delta}t\,
  171. t n e w tnew\,
  172. R I T RIT\,
  173. T turb / R I T \triangle T_{\mathrm{turb}}/RIT\,
  174. R I T RIT\,
  175. F n / δ Fn/{\delta}\,
  176. δ {\delta}\,
  177. S F C / θ SFC/\sqrt{\theta}\,
  178. θ {\theta}\,
  179. F n = ( F n / δ ) δ Fn=(Fn/{\delta})\cdot{\delta}
  180. S F C = ( S F C / θ ) θ SFC=(SFC/\sqrt{\theta})\cdot\sqrt{\theta}
  181. S O T / θ SOT/{\theta}\,
  182. N F / θ N_{F}/\sqrt{\theta}\,
  183. N F / θ T N_{F}/\sqrt{\theta}_{T}\,
  184. S O T = ( S O T / θ ) θ SOT=(SOT/{\theta})\cdot{\theta}\,
  185. N F = ( N F / θ ) θ N_{F}=(N_{F}/\sqrt{\theta})\cdot\sqrt{\theta}\,
  186. N F / θ T N_{F}/\sqrt{\theta}_{T}\,
  187. A A\,
  188. A 8 c a l c A_{\mathrm{8calc}}\,
  189. A 8 d e s p t A_{\mathrm{8despt}}\,
  190. A 8 g e o m e t r i c d e s i g n A_{\mathrm{8geometricdesign}}\,
  191. α {\alpha}\,
  192. β {\beta}\,
  193. C pc C_{\mathrm{pc}}\,
  194. C pt C_{\mathrm{pt}}\,
  195. C dcalc C_{\mathrm{dcalc}}\,
  196. C x C_{x}\,
  197. δ {\delta}\,
  198. ( δ H / T ) turb ({\delta}H/T)_{\mathrm{turb}}\,
  199. δ N {\delta}N\,
  200. δ P w {\delta}P_{w}\,
  201. δ τ {\delta}\,{\tau}\,
  202. η pc {\eta}_{\mathrm{pc}}\,
  203. η pt {\eta}_{\mathrm{pt}}\,
  204. g g\,
  205. F g F_{g}\,
  206. F n F_{n}\,
  207. F r F_{r}\,
  208. γ c {\gamma}_{\mathrm{c}}\,
  209. γ t {\gamma}_{\mathrm{t}}\,
  210. I I\,
  211. J J\,
  212. K K\,
  213. K 1 K_{1}\,
  214. K 2 K_{2}\,
  215. M M\,
  216. N N\,
  217. N cor N_{\mathrm{cor}}\,
  218. N turbcor N_{\mathrm{turbcor}}\,
  219. p p\,
  220. P P\,
  221. P 3 / P 2 P_{3}/P_{2}\,
  222. p r f prf\,
  223. R R\,
  224. ρ {\rho}\,
  225. S F C SFC\,
  226. R I T RIT\,
  227. t t\,
  228. T T\,
  229. T 1 T_{1}\,
  230. T 3 T_{3}\,
  231. θ {\theta}\,
  232. θ T {\theta}_{T}\,
  233. V V\,
  234. w w\,
  235. w 4 c o r c a l c w_{\mathrm{4corcalc}}\,
  236. w 2 c o r w_{\mathrm{2cor}}\,
  237. w 4 c o r d e s p t w_{\mathrm{4cordespt}}\,
  238. w 4 c o r t u r b c h a r w_{\mathrm{4corturbchar}}\,
  239. w fe w_{\mathrm{fe}}\,

Jing_Fang.html

  1. 3 11 = 177147 3^{11}=177147
  2. 177147 / 3 = 59049 177147/3=59049
  3. 177147 + 59049 = 226196 177147+59049=226196
  4. 4 / 3 4/3
  5. 1 / 3 1/3
  6. 177147 / 176776 177147/176776

Joel_Henry_Hildebrand.html

  1. δ = ( Δ H v - R T ) / V m \delta=\sqrt{(\Delta H_{v}-RT)/V_{m}}
  2. δ \delta

John's_equation.html

  1. f : n f\colon\mathbb{R}^{n}\rightarrow\mathbb{R}
  2. n \mathbb{R}^{n}
  3. x , y n x,y\in\mathbb{R}^{n}
  4. x y x\neq y
  5. u u
  6. u ( x , y ) = - f ( x + t ( y - x ) ) d t . u(x,y)=\int\limits_{-\infty}^{\infty}f(x+t(y-x))dt.
  7. u u
  8. 2 u x i y j - 2 u y i x j = 0 \frac{\partial^{2}u}{\partial x_{i}\partial y_{j}}-\frac{\partial^{2}u}{% \partial y_{i}\partial x_{j}}=0
  9. i , j = 1 2 n a i j 2 u x i x j + i = 1 2 n b i u x i + c u = 0 \sum\limits_{i,j=1}^{2n}a_{ij}\frac{\partial^{2}u}{\partial x_{i}\partial x_{j% }}+\sum\limits_{i=1}^{2n}b_{i}\frac{\partial u}{\partial x_{i}}+cu=0
  10. n 2 n\geq 2
  11. i , j = 1 2 n a i j ξ i ξ j \sum\limits_{i,j=1}^{2n}a_{ij}\xi_{i}\xi_{j}
  12. i = 1 n ξ i 2 - i = n + 1 2 n ξ i 2 . \sum\limits_{i=1}^{n}\xi_{i}^{2}-\sum\limits_{i=n+1}^{2n}\xi_{i}^{2}.

John_R._Philip.html

  1. I = S t + A t I=S\sqrt{t}\ +At

Johnson_bound.html

  1. C C
  2. n n
  3. 𝔽 q n \mathbb{F}_{q}^{n}
  4. d d
  5. C C
  6. d = min x , y C , x y d ( x , y ) d=\min_{x,y\in C,x\neq y}d(x,y)
  7. d ( x , y ) d(x,y)
  8. x x
  9. y y
  10. C q ( n , d ) C_{q}(n,d)
  11. n n
  12. d d
  13. C q ( n , d , w ) C_{q}(n,d,w)
  14. C q ( n , d ) C_{q}(n,d)
  15. w w
  16. | C | |C|
  17. C C
  18. A q ( n , d ) A_{q}(n,d)
  19. n n
  20. d d
  21. A q ( n , d ) = max C C q ( n , d ) | C | . A_{q}(n,d)=\max_{C\in C_{q}(n,d)}|C|.
  22. A q ( n , d , w ) A_{q}(n,d,w)
  23. C q ( n , d , w ) C_{q}(n,d,w)
  24. A q ( n , d , w ) = max C C q ( n , d , w ) | C | . A_{q}(n,d,w)=\max_{C\in C_{q}(n,d,w)}|C|.
  25. A q ( n , d ) A_{q}(n,d)
  26. d = 2 t + 1 d=2t+1
  27. A q ( n , d ) q n i = 0 t ( n i ) ( q - 1 ) i + ( n t + 1 ) ( q - 1 ) t + 1 - ( d t ) A q ( n , d , d ) A q ( n , d , t + 1 ) . A_{q}(n,d)\leq\frac{q^{n}}{\sum_{i=0}^{t}{n\choose i}(q-1)^{i}+\frac{{n\choose t% +1}(q-1)^{t+1}-{d\choose t}A_{q}(n,d,d)}{A_{q}(n,d,t+1)}}.
  28. d = 2 t d=2t
  29. A q ( n , d ) q n i = 0 t ( n i ) ( q - 1 ) i + ( n t + 1 ) ( q - 1 ) t + 1 A q ( n , d , t + 1 ) . A_{q}(n,d)\leq\frac{q^{n}}{\sum_{i=0}^{t}{n\choose i}(q-1)^{i}+\frac{{n\choose t% +1}(q-1)^{t+1}}{A_{q}(n,d,t+1)}}.
  30. A q ( n , d , w ) A_{q}(n,d,w)
  31. d > 2 w d>2w
  32. A q ( n , d , w ) = 1. A_{q}(n,d,w)=1.
  33. d 2 w d\leq 2w
  34. e e
  35. d d
  36. e e
  37. d = 2 e d=2e
  38. d d
  39. e e
  40. d = 2 e - 1 d=2e-1
  41. q * = q - 1 q^{*}=q-1
  42. A q ( n , d , w ) n q * w ( n - 1 ) q * w - 1 ( n - w + e ) q * e A_{q}(n,d,w)\leq\lfloor\frac{nq^{*}}{w}\lfloor\frac{(n-1)q^{*}}{w-1}\lfloor% \cdots\lfloor\frac{(n-w+e)q^{*}}{e}\rfloor\cdots\rfloor\rfloor
  43. \lfloor~{}~{}\rfloor
  44. A q ( n , d ) A_{q}(n,d)

Join-calculus.html

  1. π \pi
  2. π \pi
  3. π \pi
  4. π \pi
  5. π \pi

Jordan's_lemma.html

  1. f f
  2. C R = { R e i θ θ [ 0 , π ] } C_{R}=\{Re^{i\theta}\mid\theta\in[0,\pi]\}
  3. R R
  4. f f
  5. f ( z ) = e i a z g ( z ) , z C R , f(z)=e^{iaz}g(z),\quad z\in C_{R},
  6. a a
  7. | C R f ( z ) d z | π a M R where M R := max θ [ 0 , π ] | g ( R e i θ ) | . \left|\int_{C_{R}}f(z)\,dz\right|\leq\frac{\pi}{a}M_{R}\quad\,\text{where}% \quad M_{R}:=\max_{\theta\in[0,\pi]}\left|g\left(Re^{i\theta}\right)\right|.
  8. g g
  9. f f
  10. R R
  11. lim R C R f ( z ) d z = 0. \lim_{R\to\infty}\int_{C_{R}}f(z)\,dz=0.
  12. a = 0 a=0
  13. C C
  14. C C
  15. C f ( z ) d z = C 1 f ( z ) d z + C 2 f ( z ) d z . \oint_{C}f(z)\,dz=\int_{C_{1}}f(z)\,dz+\int_{C_{2}}f(z)\,dz\,.
  16. z z
  17. C 2 f ( z ) d z = - R R f ( x ) d x . \int_{C_{2}}f(z)\,dz=\int_{-R}^{R}f(x)\,dx\,.
  18. R R
  19. C f ( z ) d z = 2 π i k = 1 n Res ( f , z k ) , \oint_{C}f(z)\,dz=2\pi i\sum_{k=1}^{n}\operatorname{Res}(f,z_{k})\,,
  20. f f
  21. f f
  22. R R
  23. - f ( x ) d x = 2 π i k = 1 n Res ( f , z k ) . \int_{-\infty}^{\infty}f(x)\,dx=2\pi i\sum_{k=1}^{n}\operatorname{Res}(f,z_{k}% )\,.
  24. f ( z ) = e i z 1 + z 2 , z { i , - i } , f(z)=\frac{e^{iz}}{1+z^{2}},\qquad z\in{\mathbb{C}}\setminus\{i,-i\},
  25. a = 1 a=1
  26. R > 0 R>0
  27. R 1 R≠1
  28. R > 1 R>1
  29. M R = max θ [ 0 , π ] 1 | 1 + R 2 e 2 i θ | = 1 R 2 - 1 , M_{R}=\max_{\theta\in[0,\pi]}\frac{1}{|1+R^{2}e^{2i\theta}|}=\frac{1}{R^{2}-1}\,,
  30. f f
  31. z = i z=i
  32. - e i x 1 + x 2 d x = 2 π i Res ( f , i ) . \int_{-\infty}^{\infty}\frac{e^{ix}}{1+x^{2}}\,dx=2\pi i\,\operatorname{Res}(f% ,i)\,.
  33. z = i z=i
  34. f f
  35. Res ( f , i ) = lim z i ( z - i ) f ( z ) = lim z i e i z z + i = e - 1 2 i \operatorname{Res}(f,i)=\lim_{z\to i}(z-i)f(z)=\lim_{z\to i}\frac{e^{iz}}{z+i}% =\frac{e^{-1}}{2i}
  36. - cos x 1 + x 2 d x = Re - e i x 1 + x 2 d x = π e . \int_{-\infty}^{\infty}\frac{\cos x}{1+x^{2}}\,dx=\operatorname{Re}\int_{-% \infty}^{\infty}\frac{e^{ix}}{1+x^{2}}\,dx=\frac{\pi}{e}\,.
  37. C R f ( z ) d z = 0 π g ( R e i θ ) e i a R ( cos θ + i sin θ ) i R e i θ d θ = R 0 π g ( R e i θ ) e a R ( i cos θ - sin θ ) i e i θ d θ . \int_{C_{R}}f(z)\,dz=\int_{0}^{\pi}g(Re^{i\theta})\,e^{iaR(\cos\theta+i\sin% \theta)}\,iRe^{i\theta}\,d\theta=R\int_{0}^{\pi}g(Re^{i\theta})\,e^{aR(i\cos% \theta-\sin\theta)}\,ie^{i\theta}\,d\theta\,.
  38. | a b f ( x ) d x | a b | f ( x ) | d x \biggl|\int_{a}^{b}f(x)\,dx\biggr|\leq\int_{a}^{b}\left|f(x)\right|\,dx
  39. I R := | C R f ( z ) d z | R 0 π | g ( R e i θ ) e a R ( i cos θ - sin θ ) i e i θ | d θ = R 0 π | g ( R e i θ ) | e - a R sin θ d θ . I_{R}:=\biggl|\int_{C_{R}}f(z)\,dz\biggr|\leq R\int_{0}^{\pi}\bigl|g(Re^{i% \theta})\,e^{aR(i\cos\theta-\sin\theta)}\,ie^{i\theta}\bigr|\,d\theta=R\int_{0% }^{\pi}\bigl|g(Re^{i\theta})\bigr|\,e^{-aR\sin\theta}\,d\theta\,.
  40. s i n θ = s i n ( π θ ) sinθ=sin(π–θ)
  41. I R R M R 0 π e - a R sin θ d θ = 2 R M R 0 π / 2 e - a R sin θ d θ . I_{R}\leq RM_{R}\int_{0}^{\pi}e^{-aR\sin\theta}\,d\theta=2RM_{R}\int_{0}^{\pi/% 2}e^{-aR\sin\theta}\,d\theta\,.
  42. s i n θ sinθ
  43. θ 0 , π 2 θ∈0,π⁄2
  44. s i n θ sinθ
  45. sin θ 2 θ π \sin\theta\geq\frac{2\theta}{\pi}\quad
  46. θ 0 , π 2 θ∈0,π⁄2
  47. I R 2 R M R 0 π / 2 e - 2 a R θ / π d θ = π a ( 1 - e - a R ) M R π a M R . I_{R}\leq 2RM_{R}\int_{0}^{\pi/2}e^{-2aR\theta/\pi}\,d\theta=\frac{\pi}{a}(1-e% ^{-aR})M_{R}\leq\frac{\pi}{a}M_{R}\,.

Jordan_matrix.html

  1. R R
  2. λ R \lambda\in R
  3. ( λ 1 0 0 0 λ 1 0 0 0 0 λ 1 0 0 0 0 λ ) \begin{pmatrix}\lambda&1&0&\cdots&0\\ 0&\lambda&1&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\lambda&1\\ 0&0&0&0&\lambda\end{pmatrix}
  4. λ \lambda
  5. J λ , n J_{\lambda,n}
  6. \oplus
  7. diag \mathrm{diag}
  8. ( l + m + n ) × ( l + m + n ) (l+m+n)\times(l+m+n)
  9. J α , l J_{\alpha,l}
  10. J β , m J_{\beta,m}
  11. J γ , n J_{\gamma,n}
  12. J α , l J β , m J γ , n J_{\alpha,l}\oplus J_{\beta,m}\oplus J_{\gamma,n}
  13. diag ( J α , l , J β , m , J γ , n ) \mathrm{diag}\left(J_{\alpha,l},J_{\beta,m},J_{\gamma,n}\right)
  14. J = ( 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 i 1 0 0 0 0 0 0 0 0 0 i 0 0 0 0 0 0 0 0 0 0 i 1 0 0 0 0 0 0 0 0 0 i 0 0 0 0 0 0 0 0 0 0 7 1 0 0 0 0 0 0 0 0 0 7 1 0 0 0 0 0 0 0 0 0 7 ) J=\left(\begin{matrix}0&1&0&0&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&i&1&0&0&0&0&0\\ 0&0&0&0&i&0&0&0&0&0\\ 0&0&0&0&0&i&1&0&0&0\\ 0&0&0&0&0&0&i&0&0&0\\ 0&0&0&0&0&0&0&7&1&0\\ 0&0&0&0&0&0&0&0&7&1\\ 0&0&0&0&0&0&0&0&0&7\end{matrix}\right)
  15. 10 × 10 10\times 10
  16. 3 × 3 3\times 3
  17. 0
  18. 2 × 2 2\times 2
  19. 3 × 3 3\times 3
  20. J 0 , 3 J i , 2 J i , 2 J 7 , 3 J_{0,3}\oplus J_{i,2}\oplus J_{i,2}\oplus J_{7,3}
  21. diag ( J 0 , 3 , J i , 2 , J i , 2 , J 7 , 3 ) \mathrm{diag}\left(J_{0,3},J_{i,2},J_{i,2},J_{7,3}\right)
  22. n × n n\times n
  23. A A
  24. K K
  25. J J
  26. 𝕄 n ( K ) \mathbb{M}_{n}(K)
  27. J J
  28. A A
  29. 1 × 1 1\times 1
  30. J = J λ 1 , m 1 J λ 2 , m 2 J λ N , m N J=J_{\lambda_{1},m_{1}}\oplus J_{\lambda_{2},m_{2}}\oplus\ldots\oplus J_{% \lambda_{N},m_{N}}
  31. k th k\text{th}
  32. 1 k N 1\leq k\leq N
  33. J λ k , m k J_{\lambda_{k},m_{k}}
  34. λ k \lambda_{k}
  35. λ K \lambda\in K
  36. J J
  37. gmul J λ \mathrm{gmul}_{J}\lambda\,
  38. λ \lambda
  39. λ \lambda
  40. J J
  41. idx J λ \mathrm{idx}_{J}\lambda\,
  42. A A
  43. J J
  44. idx A λ \mathrm{idx}_{A}\lambda\,
  45. A A
  46. λ spec A \lambda\in\mathrm{spec}A
  47. λ \lambda
  48. A A
  49. A A
  50. A A
  51. mul A λ \mathrm{mul}_{A}\lambda\,
  52. A A
  53. det ( A - x I ) K [ x ] \det(A-xI)\in K[x]
  54. A A
  55. K K
  56. 1 1
  57. A 𝕄 n ( ) A\in\mathbb{M}_{n}(\mathbb{C})
  58. n × n n\times n
  59. C GL n ( ) C\in\mathrm{GL}_{n}(\mathbb{C})
  60. A A
  61. A = C - 1 J C A=C^{-1}JC
  62. f ( z ) f(z)
  63. Ω \mathit{\Omega}
  64. spec A Ω \mathrm{spec}A\subset\mathit{\Omega}\subseteq\mathbb{C}
  65. f f
  66. f ( z ) = h = 0 a h ( z - z 0 ) h f(z)=\sum_{h=0}^{\infty}a_{h}(z-z_{0})^{h}
  67. f f
  68. z 0 Ω \ spec A z_{0}\in\mathit{\Omega}\backslash\mathrm{spec}A
  69. f ( A ) f(A)
  70. f ( A ) = h = 0 a h A h f(A)=\sum_{h=0}^{\infty}a_{h}A^{h}
  71. 𝕄 n ( ) \mathbb{M}_{n}(\mathbb{C})
  72. f ( A ) f(A)\,
  73. f f
  74. 0
  75. 𝕄 n ( ) \mathbb{M}_{n}(\mathbb{C})
  76. k th k^{\mathrm{th}}
  77. k 0 k\in\mathbb{N}_{0}
  78. k th k^{\mathrm{th}}
  79. ( A 1 A 2 A 3 ) k = A 1 k A 2 k A 3 k \left(A_{1}\oplus A_{2}\oplus A_{3}\oplus\ldots\right)^{k}=A^{k}_{1}\oplus A_{% 2}^{k}\oplus A_{3}^{k}\oplus\ldots
  80. A k = C - 1 J k C A^{k}=C^{-1}J^{k}C\,
  81. f ( A ) = C - 1 f ( J ) C = C - 1 ( k = 1 N f ( J λ k , m k ) ) C f(A)=C^{-1}f(J)C=C^{-1}\left(\bigoplus_{k=1}^{N}f\left(J_{\lambda_{k},m_{k}}% \right)\right)C
  82. λ Ω \lambda\in\mathit{\Omega}
  83. f ( J λ , n ) f(J_{\lambda,n})\,
  84. f ( J λ , n ) = ( f ( λ ) f ( λ ) f ′′ ( λ ) 2 f ( n - 2 ) ( λ ) ( n - 2 ) ! f ( n - 1 ) ( λ ) ( n - 1 ) ! 0 f ( λ ) f ( λ ) f ( n - 3 ) ( λ ) ( n - 3 ) ! f ( n - 2 ) ( λ ) ( n - 2 ) ! 0 0 f ( λ ) f ( n - 4 ) ( λ ) ( n - 4 ) ! f ( n - 3 ) ( λ ) ( n - 3 ) ! 0 0 0 f ( λ ) f ( λ ) 0 0 0 0 f ( λ ) ) = ( a 0 a 1 a 2 a n - 1 0 a 0 a 1 a n - 2 0 0 a 0 a n - 3 0 0 0 a 1 0 0 0 a 0 ) . f(J_{\lambda,n})=\left(\begin{matrix}f(\lambda)&f^{\prime}(\lambda)&\frac{f^{% \prime\prime}(\lambda)}{2}&\cdots&\frac{f^{(n-2)}(\lambda)}{(n-2)!}&\frac{f^{(% n-1)}(\lambda)}{(n-1)!}\\ 0&f(\lambda)&f^{\prime}(\lambda)&\cdots&\frac{f^{(n-3)}(\lambda)}{(n-3)!}&% \frac{f^{(n-2)}(\lambda)}{(n-2)!}\\ 0&0&f(\lambda)&\cdots&\frac{f^{(n-4)}(\lambda)}{(n-4)!}&\frac{f^{(n-3)}(% \lambda)}{(n-3)!}\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&0&\cdots&f(\lambda)&f^{\prime}(\lambda)\\ 0&0&0&\cdots&0&f(\lambda)\\ \end{matrix}\right)=\left(\begin{matrix}a_{0}&a_{1}&a_{2}&\cdots&a_{n-1}\\ 0&a_{0}&a_{1}&\cdots&a_{n-2}\\ 0&0&a_{0}&\cdots&a_{n-3}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\cdots&a_{1}\\ 0&0&0&\cdots&a_{0}\end{matrix}\right).
  85. spec f ( A ) = f ( spec A ) \mathrm{spec}f(A)=f(\mathrm{spec}A)
  86. λ spec A \lambda\in\mathrm{spec}A
  87. f ( λ ) spec f ( A ) f(\lambda)\in\mathrm{spec}f(A)
  88. mul f ( A ) f ( λ ) = μ spec A f - 1 ( f ( λ ) ) mul A μ . \,\text{mul}_{f(A)}f(\lambda)=\sum_{\mu\in\,\text{spec}A\cap f^{-1}(f(\lambda)% )}~{}\,\text{mul}_{A}\mu.\,
  89. f ( T ) f(T)
  90. T T
  91. 𝐳 ˙ ( t ) = A ( 𝐜 ) 𝐳 ( t ) , \dot{\mathbf{z}}(t)=A(\mathbf{c})\mathbf{z}(t),
  92. 𝐳 ( 0 ) = 𝐳 0 n , \mathbf{z}(0)=\mathbf{z}_{0}\in\mathbb{C}^{n},
  93. 𝐳 : + \mathbf{z}:\mathbb{R_{+}}\rightarrow\mathcal{R}
  94. n n
  95. \mathcal{R}
  96. A ( 𝐜 ) A(\mathbf{c})
  97. n × n n\times n
  98. d d
  99. 𝐜 d \mathbf{c}\in\mathbb{C}^{d}
  100. A 𝕄 n ( C 0 ( d ) ) A\in\mathbb{M}_{n}\left(\mathrm{C}^{0}(\mathbb{C}^{d})\right)
  101. A A
  102. 𝐜 \mathbf{c}
  103. d \mathbb{C}^{d}
  104. d \mathbb{C}^{d}
  105. A ( 𝐜 ) A(\mathbf{c})
  106. A 𝕄 n ( ) A\in\mathbb{M}_{n}(\mathbb{C})
  107. 𝐳 0 n \mathbf{z}_{0}\in\mathbb{C}^{n}
  108. 𝐳 ˙ ( t ) = A 𝐳 ( t ) , \dot{\mathbf{z}}(t)=A\mathbf{z}(t),
  109. 𝐳 ( 0 ) = 𝐳 0 , \mathbf{z}(0)=\mathbf{z}_{0},
  110. 𝐳 ( t ) = e t A 𝐳 0 . \mathbf{z}(t)=e^{tA}\mathbf{z}_{0}.
  111. n n
  112. 𝐳 L loc 1 ( + ) n \mathbf{z}\in\mathrm{L}_{\mathrm{loc}}^{1}(\mathbb{R}_{+})^{n}
  113. 𝐙 ( s ) = [ 𝐳 ] ( s ) \mathbf{Z}(s)=\mathcal{L}[\mathbf{z}](s)
  114. 𝐙 ( s ) = ( s I - A ) - 1 𝐳 0 . \mathbf{Z}(s)=\left(sI-A\right)^{-1}\mathbf{z}_{0}.
  115. ( A - s I ) - 1 \left(A-sI\right)^{-1}
  116. d d t - A \frac{\mathrm{d}}{\mathrm{d}t}-A
  117. s s\in\mathbb{C}
  118. det ( A - s I ) \det(A-sI)
  119. A A
  120. ord ( A - s I ) - 1 λ = idx A λ \mathrm{ord}_{(A-sI)^{-1}}\lambda=\mathrm{idx}_{A}\lambda

Jordan_measure.html

  1. C = [ a 1 , b 1 ) × [ a 2 , b 2 ) × × [ a n , b n ) C=[a_{1},b_{1})\times[a_{2},b_{2})\times\cdots\times[a_{n},b_{n})
  2. m ( C ) = ( b 1 - a 1 ) ( b 2 - a 2 ) ( b n - a n ) . m(C)=(b_{1}-a_{1})(b_{2}-a_{2})\cdots(b_{n}-a_{n}).
  3. S = C 1 C 2 C k S=C_{1}\cup C_{2}\cup\cdots\cup C_{k}
  4. [ a 1 , b 1 ] × [ a 2 , b 2 ] × × [ a n , b n ] [a_{1},b_{1}]\times[a_{2},b_{2}]\times\cdots\times[a_{n},b_{n}]
  5. m * ( B ) = sup S B m ( S ) m_{*}(B)=\sup_{S\subset B}m(S)
  6. m * ( B ) = inf S B m ( S ) m^{*}(B)=\inf_{S\supset B}m(S)

Josephson_penetration_depth.html

  1. λ J \lambda_{J}
  2. λ J = Φ 0 2 π μ 0 d j c , \lambda_{J}=\sqrt{\frac{\Phi_{0}}{2\pi\mu_{0}d^{\prime}j_{c}}},
  3. Φ 0 \Phi_{0}
  4. j c j_{c}
  5. ( A / m 2 ) \mathrm{(A/m^{2})}
  6. d d^{\prime}
  7. d = d I + λ 1 coth ( d 1 λ 1 ) + λ 2 coth ( d 2 λ 2 ) , d^{\prime}=d_{I}+\lambda_{1}\coth\left(\frac{d_{1}}{\lambda_{1}}\right)+% \lambda_{2}\coth\left(\frac{d_{2}}{\lambda_{2}}\right),
  8. d I d_{I}
  9. d 1 , 2 d_{1,2}
  10. λ 1 , 2 \lambda_{1,2}

Josephson_phase.html

  1. Ψ 1 \Psi_{1}
  2. Ψ 2 \Psi_{2}
  3. Ψ 1 = n s e i θ 1 \Psi_{1}=\sqrt{n_{s}}e^{i\theta_{1}}
  4. Ψ 2 = n s e i θ 2 \Psi_{2}=\sqrt{n_{s}}e^{i\theta_{2}}
  5. ϕ = def θ 2 - θ 1 \phi\ \stackrel{\mathrm{def}}{=}\ \theta_{2}-\theta_{1}

Joukowsky_transform.html

  1. z = ζ + 1 ζ z=\zeta+\frac{1}{\zeta}
  2. z = x + i y z=x+iy
  3. ζ = χ + i η \zeta=\chi+i\eta
  4. ζ \zeta
  5. ζ \zeta
  6. ζ \zeta
  7. μ x + i μ y \mu_{x}+i\mu_{y}
  8. ζ \zeta
  9. z z
  10. z = x + i y = ζ + 1 ζ = χ + i η + 1 χ + i η = χ + i η + ( χ - i η ) χ 2 + η 2 = χ ( χ 2 + η 2 + 1 ) χ 2 + η 2 + i η ( χ 2 + η 2 - 1 ) χ 2 + η 2 . \begin{aligned}\displaystyle z&\displaystyle=x+iy=\zeta+\frac{1}{\zeta}\\ &\displaystyle=\chi+i\eta+\frac{1}{\chi+i\eta}\\ &\displaystyle=\chi+i\eta+\frac{(\chi-i\eta)}{\chi^{2}+\eta^{2}}\\ &\displaystyle=\frac{\chi(\chi^{2}+\eta^{2}+1)}{\chi^{2}+\eta^{2}}+i\frac{\eta% (\chi^{2}+\eta^{2}-1)}{\chi^{2}+\eta^{2}}.\end{aligned}
  11. x \displaystyle x
  12. | ζ | = χ 2 + η 2 = 1 which gives χ 2 + η 2 = 1. |\zeta|=\sqrt{\chi^{2}+\eta^{2}}=1\quad\,\text{which gives}\quad\chi^{2}+\eta^% {2}=1.
  13. x = χ ( 1 + 1 ) 1 = 2 χ x=\frac{\chi(1+1)}{1}=2\chi
  14. y = η ( 1 - 1 ) 1 = 0. y=\frac{\eta(1-1)}{1}=0.
  15. W ~ \tilde{W}
  16. ζ \zeta
  17. W ~ = V e - i α + i Γ 2 π ( ζ - μ ) - V R 2 e i α ( ζ - μ ) 2 \tilde{W}=V_{\infty}e^{-i\alpha}+\frac{i\Gamma}{2\pi(\zeta-\mu)}-\frac{V_{% \infty}R^{2}e^{i\alpha}}{(\zeta-\mu)^{2}}
  18. μ = μ x + i μ y \mu=\mu_{x}+i\mu_{y}
  19. V V_{\infty}
  20. α \alpha
  21. R = ( 1 - μ x ) 2 + μ y 2 R=\sqrt{(1-\mu_{x})^{2}+\mu_{y}^{2}}
  22. Γ \Gamma
  23. Γ = 4 π V R sin ( α + sin - 1 ( μ y R ) ) . \Gamma=4\pi V_{\infty}R\sin\left(\ \alpha+\sin^{-1}\left(\frac{\mu_{y}}{R}% \right)\right).
  24. W = W ~ d z d ζ = W ~ 1 - 1 ζ 2 . W=\frac{\tilde{W}}{\frac{dz}{d\zeta}}=\frac{\tilde{W}}{1-\frac{1}{\zeta^{2}}}.
  25. W = u x - i u y , W=u_{x}-iu_{y},
  26. u x u_{x}
  27. u y u_{y}
  28. x x
  29. y y
  30. z = x + i y , z=x+iy,
  31. x x
  32. y y
  33. z = n ( 1 + 1 ζ ) n + ( 1 - 1 ζ ) n ( 1 + 1 ζ ) n - ( 1 - 1 ζ ) n , z=n\frac{\left(1+\frac{1}{\zeta}\right)^{n}+\left(1-\frac{1}{\zeta}\right)^{n}% }{\left(1+\frac{1}{\zeta}\right)^{n}-\left(1-\frac{1}{\zeta}\right)^{n}},
  34. α = 2 π - n π and n = 2 - α π . \alpha=2\pi\,-\,n\pi\quad\,\text{ and }\quad n=2-\frac{\alpha}{\pi}.
  35. d z / d ζ dz/d\zeta
  36. d z d ζ = 4 n 2 ζ 2 - 1 ( 1 + 1 ζ ) n ( 1 - 1 ζ ) n [ ( 1 + 1 ζ ) n - ( 1 - 1 ζ ) n ] 2 . \frac{dz}{d\zeta}=\frac{4n^{2}}{\zeta^{2}-1}\frac{\left(1+\frac{1}{\zeta}% \right)^{n}\left(1-\frac{1}{\zeta}\right)^{n}}{\left[\left(1+\frac{1}{\zeta}% \right)^{n}-\left(1-\frac{1}{\zeta}\right)^{n}\right]^{2}}.
  37. z + 2 \displaystyle z+2
  38. z - 2 z + 2 = ( ζ - 1 ζ + 1 ) 2 . \frac{z-2}{z+2}=\left(\frac{\zeta-1}{\zeta+1}\right)^{2}.
  39. ζ = + 1. \zeta=+1.
  40. ζ \zeta
  41. z - n z + n = ( ζ - 1 ζ + 1 ) n , \frac{z-n}{z+n}=\left(\frac{\zeta-1}{\zeta+1}\right)^{n},