wpmath0000012_14

Vicsek_fractal.html

  1. log ( 5 ) log ( 3 ) \textstyle{\frac{\log(5)}{\log(3)}}
  2. ( 5 9 ) n \textstyle{(\frac{5}{9})^{n}}
  3. 4 ( 5 3 ) n \textstyle{4(\frac{5}{3})^{n}}
  4. log ( 7 ) log ( 3 ) \textstyle{\frac{\log(7)}{\log(3)}}
  5. ( 7 27 ) n \textstyle{(\frac{7}{27})^{n}}

Victor_Henri.html

  1. d x d t = K Φ ( a - x ) 1 + m ( a - x ) + n x \frac{dx}{dt}=\frac{K\cdot\Phi\cdot(a-x)}{1+m\cdot(a-x)+n\cdot x}
  2. v = const. S 1 + S K 1 + P K 2 ( ) v=\frac{\,\text{const.}\cdot S}{1+\frac{S}{K_{1}}+\frac{P}{K_{2}}}\ (\ast)

Vincenty's_formulae.html

  1. sin σ = ( cos U 2 sin λ ) 2 + ( cos U 1 sin U 2 - sin U 1 cos U 2 cos λ ) 2 \sin\sigma=\sqrt{(\cos U_{2}\sin\lambda)^{2}+(\cos U_{1}\sin U_{2}-\sin U_{1}% \cos U_{2}\cos\lambda)^{2}}
  2. cos σ = sin U 1 sin U 2 + cos U 1 cos U 2 cos λ \cos\sigma=\sin U_{1}\sin U_{2}+\cos U_{1}\cos U_{2}\cos\lambda\,
  3. σ = arctan sin σ cos σ \sigma=\arctan\frac{\sin\sigma}{\cos\sigma}\,
  4. sin α = cos U 1 cos U 2 sin λ sin σ \sin\alpha=\frac{\cos U_{1}\cos U_{2}\sin\lambda}{\sin\sigma}\,
  5. cos 2 α = 1 - sin 2 α \cos^{2}\alpha=1-\sin^{2}\alpha\,
  6. cos ( 2 σ m ) = cos σ - 2 sin U 1 sin U 2 cos 2 α \cos(2\sigma_{m})=\cos\sigma-\frac{2\sin U_{1}\sin U_{2}}{\cos^{2}\alpha}\,
  7. cos ( 2 σ m ) \cos(2\sigma_{m})
  8. cos ( 2 σ m ) = - 1 \cos(2\sigma_{m})=-1
  9. C = f 16 cos 2 α [ 4 + f ( 4 - 3 cos 2 α ) ] C=\frac{f}{16}\cos^{2}\alpha\big[4+f(4-3\cos^{2}\alpha)\big]\,
  10. λ = L + ( 1 - C ) f sin α { σ + C sin σ [ cos ( 2 σ m ) + C cos σ ( - 1 + 2 cos 2 ( 2 σ m ) ) ] } \lambda=L+(1-C)f\sin\alpha\left\{\sigma+C\sin\sigma\left[\cos(2\sigma_{m})+C% \cos\sigma(-1+2\cos^{2}(2\sigma_{m}))\right]\right\}\,
  11. u 2 = cos 2 α a 2 - b 2 b 2 u^{2}=\cos^{2}\alpha\frac{a^{2}-b^{2}}{b^{2}}\,
  12. A = 1 + u 2 16384 { 4096 + u 2 [ - 768 + u 2 ( 320 - 175 u 2 ) ] } A=1+\frac{u^{2}}{16384}\left\{4096+u^{2}\left[-768+u^{2}(320-175u^{2})\right]\right\}
  13. B = u 2 1024 { 256 + u 2 [ - 128 + u 2 ( 74 - 47 u 2 ) ] } B=\frac{u^{2}}{1024}\left\{256+u^{2}\left[-128+u^{2}(74-47u^{2})\right]\right\}
  14. Δ σ = B sin σ { cos ( 2 σ m ) + 1 4 B [ cos σ ( - 1 + 2 cos 2 ( 2 σ m ) ) - 1 6 B cos ( 2 σ m ) ( - 3 + 4 sin 2 σ ) ( - 3 + 4 cos 2 ( 2 σ m ) ) ] } \Delta\sigma=B\sin\sigma\Big\{\cos(2\sigma_{m})+\tfrac{1}{4}B\big[\cos\sigma% \big(-1+2\cos^{2}(2\sigma_{m})\big)-\tfrac{1}{6}B\cos(2\sigma_{m})(-3+4\sin^{2% }\sigma)\big(-3+4\cos^{2}(2\sigma_{m})\big)\big]\Big\}
  15. s = b A ( σ - Δ σ ) s=bA(\sigma-\Delta\sigma)\,
  16. α 1 = arctan ( cos U 2 sin λ cos U 1 sin U 2 - sin U 1 cos U 2 cos λ ) \alpha_{1}=\arctan\left(\frac{\cos U_{2}\sin\lambda}{\cos U_{1}\sin U_{2}-\sin U% _{1}\cos U_{2}\cos\lambda}\right)
  17. α 2 = arctan ( cos U 1 sin λ - sin U 1 cos U 2 + cos U 1 sin U 2 cos λ ) \alpha_{2}=\arctan\left(\frac{\cos U_{1}\sin\lambda}{-\sin U_{1}\cos U_{2}+% \cos U_{1}\sin U_{2}\cos\lambda}\right)
  18. U 1 = arctan ( ( 1 - f ) tan ϕ 1 ) U_{1}=\arctan\left((1-f)\tan\phi_{1}\right)\,
  19. σ 1 = arctan ( tan U 1 cos α 1 ) \sigma_{1}=\arctan\left(\frac{\tan U_{1}}{\cos\alpha_{1}}\right)\,
  20. sin α = cos U 1 sin α 1 \sin\alpha=\cos U_{1}\sin\alpha_{1}\,
  21. cos 2 α = 1 - sin 2 α \cos^{2}\alpha=1-\sin^{2}\alpha\,
  22. u 2 = cos 2 α a 2 - b 2 b 2 u^{2}=\cos^{2}\alpha\frac{a^{2}-b^{2}}{b^{2}}\,
  23. A = 1 + u 2 16384 { 4096 + u 2 [ - 768 + u 2 ( 320 - 175 u 2 ) ] } A=1+\frac{u^{2}}{16384}\left\{4096+u^{2}\left[-768+u^{2}(320-175u^{2})\right]\right\}
  24. B = u 2 1024 { 256 + u 2 [ - 128 + u 2 ( 74 - 47 u 2 ) ] } B=\frac{u^{2}}{1024}\left\{256+u^{2}\left[-128+u^{2}(74-47u^{2})\right]\right\}
  25. σ = s b A \sigma=\tfrac{s}{bA}
  26. 2 σ m = 2 σ 1 + σ 2\sigma_{m}=2\sigma_{1}+\sigma\,
  27. Δ σ = B sin σ { cos ( 2 σ m ) + 1 4 B [ cos σ ( - 1 + 2 cos 2 ( 2 σ m ) ) - 1 6 B cos ( 2 σ m ) ( - 3 + 4 sin 2 σ ) ( - 3 + 4 cos 2 ( 2 σ m ) ) ] } \Delta\sigma=B\sin\sigma\Big\{\cos(2\sigma_{m})+\tfrac{1}{4}B\big[\cos\sigma% \big(-1+2\cos^{2}(2\sigma_{m})\big)-\tfrac{1}{6}B\cos(2\sigma_{m})(-3+4\sin^{2% }\sigma)\big(-3+4\cos^{2}(2\sigma_{m})\big)\big]\Big\}
  28. σ = s b A + Δ σ \sigma=\frac{s}{bA}+\Delta\sigma\,
  29. ϕ 2 = arctan ( sin U 1 cos σ + cos U 1 sin σ cos α 1 ( 1 - f ) sin 2 α + ( sin U 1 sin σ - cos U 1 cos σ cos α 1 ) 2 ) \phi_{2}=\arctan\left(\frac{\sin U_{1}\cos\sigma+\cos U_{1}\sin\sigma\cos% \alpha_{1}}{(1-f)\sqrt{\sin^{2}\alpha+(\sin U_{1}\sin\sigma-\cos U_{1}\cos% \sigma\cos\alpha_{1})^{2}}}\right)\,
  30. λ = arctan ( sin σ sin α 1 cos U 1 cos σ - sin U 1 sin σ cos α 1 ) \lambda=\arctan\left(\frac{\sin\sigma\sin\alpha_{1}}{\cos U_{1}\cos\sigma-\sin U% _{1}\sin\sigma\cos\alpha_{1}}\right)\,
  31. C = f 16 cos 2 α [ 4 + f ( 4 - 3 cos 2 α ) ] C=\frac{f}{16}\cos^{2}\alpha\big[4+f(4-3\cos^{2}\alpha)\big]\,
  32. L = λ - ( 1 - C ) f sin α { σ + C sin σ [ cos ( 2 σ m ) + C cos σ ( - 1 + 2 cos 2 ( 2 σ m ) ) ] } L=\lambda-(1-C)f\sin\alpha\left\{\sigma+C\sin\sigma\left[\cos(2\sigma_{m})+C% \cos\sigma(-1+2\cos^{2}(2\sigma_{m}))\right]\right\}\,
  33. L 2 = L + L 1 L_{2}=L+L_{1}\,
  34. α 2 = arctan ( sin α - sin U 1 sin σ + cos U 1 cos σ cos α 1 ) \alpha_{2}=\arctan\left(\frac{\sin\alpha}{-\sin U_{1}\sin\sigma+\cos U_{1}\cos% \sigma\cos\alpha_{1}}\right)\,
  35. A = 1 + 1 4 ( k 1 ) 2 1 - k 1 A=\frac{1+\frac{1}{4}(k_{1})^{2}}{1-k_{1}}
  36. B = k 1 ( 1 - 3 8 ( k 1 ) 2 ) B=k_{1}(1-\tfrac{3}{8}(k_{1})^{2})
  37. k 1 = ( 1 + u 2 ) - 1 ( 1 + u 2 ) + 1 k_{1}=\frac{\sqrt{(1+u^{2})}-1}{\sqrt{(1+u^{2})}+1}

Viper_telescope.html

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Viscoplasticity.html

  1. s y m b o l ε = s y m b o l ε e + s y m b o l ε vp symbol{\varepsilon}=symbol{\varepsilon}_{\mathrm{e}}+symbol{\varepsilon}_{% \mathrm{vp}}
  2. s y m b o l ε e symbol{\varepsilon}_{\mathrm{e}}
  3. s y m b o l ε vp symbol{\varepsilon}_{\mathrm{vp}}
  4. ( 0 s y m b o l ε s y m b o l ε 1 ) (0\leq symbol{\varepsilon}\leq symbol{\varepsilon}_{1})
  5. ( s y m b o l ε 1 s y m b o l ε s y m b o l ε 2 ) (symbol{\varepsilon}_{1}\leq symbol{\varepsilon}\leq symbol{\varepsilon}_{2})
  6. ( s y m b o l ε 2 s y m b o l ε s y m b o l ε R ) (symbol{\varepsilon}_{2}\leq symbol{\varepsilon}\leq symbol{\varepsilon}_{R})
  7. d s y m b o l ε d t = d s y m b o l ε e d t + d s y m b o l ε vp d t . \cfrac{\mathrm{d}symbol{\varepsilon}}{\mathrm{d}t}=\cfrac{\mathrm{d}symbol{% \varepsilon}_{\mathrm{e}}}{\mathrm{d}t}+\cfrac{\mathrm{d}symbol{\varepsilon}_{% \mathrm{vp}}}{\mathrm{d}t}~{}.
  8. d s y m b o l ε e d t = 𝖤 - 1 d s y m b o l σ d t \cfrac{\mathrm{d}symbol{\varepsilon}_{\mathrm{e}}}{\mathrm{d}t}=\mathsf{E}^{-1% }~{}\cfrac{\mathrm{d}symbol{\sigma}}{\mathrm{d}t}
  9. d s y m b o l ε vp d t = - 𝖤 - 1 d s y m b o l σ d t \cfrac{\mathrm{d}symbol{\varepsilon}_{\mathrm{vp}}}{\mathrm{d}t}=-\mathsf{E}^{% -1}~{}\cfrac{\mathrm{d}symbol{\sigma}}{\mathrm{d}t}
  10. d s y m b o l ε d t = 0 \cfrac{\mathrm{d}symbol{\varepsilon}}{\mathrm{d}t}=0
  11. s y m b o l ε ˙ \dot{symbol{\varepsilon}}
  12. s y m b o l σ ˙ \dot{symbol{\sigma}}
  13. s y m b o l ε e = 0 symbol{\varepsilon}_{e}=0
  14. σ y = 0 \sigma_{y}=0
  15. s y m b o l σ = η s y m b o l ε ˙ vp s y m b o l ε ˙ vp = s y m b o l σ η symbol{\sigma}=\eta~{}\dot{symbol{\varepsilon}}_{\mathrm{vp}}\implies\dot{% symbol{\varepsilon}}_{\mathrm{vp}}=\cfrac{symbol{\sigma}}{\eta}
  16. η \eta
  17. η \eta
  18. η = λ [ λ || s y m b o l σ || ] N - 1 \eta=\lambda\left[\cfrac{\lambda}{||symbol{\sigma}||}\right]^{N-1}
  19. N N
  20. || s y m b o l σ || = s y m b o l σ : s y m b o l σ = σ i j σ i j ||symbol{\sigma}||=\sqrt{symbol{\sigma}:symbol{\sigma}}=\sqrt{\sigma_{ij}% \sigma_{ij}}
  21. s y m b o l ε ˙ vp = s y m b o l σ λ [ || s y m b o l σ || λ ] N - 1 \dot{symbol{\varepsilon}}_{\mathrm{vp}}=\cfrac{symbol{\sigma}}{\lambda}\left[% \cfrac{||symbol{\sigma}||}{\lambda}\right]^{N-1}
  22. σ = λ ( ε ˙ vp ) 1 / N \sigma=\lambda~{}\left(\dot{\varepsilon}_{\mathrm{vp}}\right)^{1/N}
  23. N = 1.0 N=1.0
  24. s y m b o l s = 2 K ( 3 ε ˙ eq ) m - 1 s y m b o l ε ˙ vp symbol{s}=2K~{}\left(\sqrt{3}\dot{\varepsilon}_{\mathrm{eq}}\right)^{m-1}~{}% \dot{symbol{\varepsilon}}_{\mathrm{vp}}
  25. s y m b o l s symbol{s}
  26. ε ˙ eq \dot{\varepsilon}_{\mathrm{eq}}
  27. K , m K,m
  28. ϵ ¯ ˙ = ( 2 ¯ 3 ϵ ¯ ¯ ˙ : ϵ ¯ ¯ ˙ ) \dot{\bar{\epsilon}}=\sqrt{(\begin{array}[]{c}\underline{2}\\ 3\end{array}\dot{\bar{\bar{\epsilon}}}:\dot{\bar{\bar{\epsilon}}}~{})}
  29. s y m b o l σ = 𝖤 s y m b o l ε for s y m b o l σ < σ y s y m b o l ε ˙ = s y m b o l ε ˙ e + s y m b o l ε ˙ vp = 𝖤 - 1 s y m b o l σ ˙ + s y m b o l σ η [ 1 - σ y s y m b o l σ ] for s y m b o l σ σ y \begin{aligned}&\displaystyle symbol{\sigma}=\mathsf{E}~{}symbol{\varepsilon}&% &\displaystyle\mathrm{for}~{}\|symbol{\sigma}\|<\sigma_{y}\\ &\displaystyle\dot{symbol{\varepsilon}}=\dot{symbol{\varepsilon}}_{\mathrm{e}}% +\dot{symbol{\varepsilon}}_{\mathrm{vp}}=\mathsf{E}^{-1}~{}\dot{symbol{\sigma}% }+\cfrac{symbol{\sigma}}{\eta}\left[1-\cfrac{\sigma_{y}}{\|symbol{\sigma}\|}% \right]&&\displaystyle\mathrm{for}~{}\|symbol{\sigma}\|\geq\sigma_{y}\end{aligned}
  30. η \eta
  31. s y m b o l σ η = s y m b o l σ λ [ s y m b o l σ λ ] N - 1 \cfrac{symbol{\sigma}}{\eta}=\cfrac{symbol{\sigma}}{\lambda}\left[\cfrac{\|% symbol{\sigma}\|}{\lambda}\right]^{N-1}
  32. s y m b o l ε ˙ = 𝖤 - 1 s y m b o l σ ˙ + s y m b o l σ λ [ s y m b o l σ λ ] N - 1 [ 1 - σ y s y m b o l σ ] for s y m b o l σ σ y \dot{symbol{\varepsilon}}=\mathsf{E}^{-1}~{}\dot{symbol{\sigma}}+\cfrac{symbol% {\sigma}}{\lambda}\left[\cfrac{\|symbol{\sigma}\|}{\lambda}\right]^{N-1}\left[% 1-\cfrac{\sigma_{y}}{\|symbol{\sigma}\|}\right]\quad\mathrm{for}~{}\|symbol{% \sigma}\|\geq\sigma_{y}
  33. s y m b o l ε ˙ = 𝖤 - 1 s y m b o l σ ˙ + f ( s y m b o l σ , σ y ) s y m b o l σ for s y m b o l σ σ y \dot{symbol{\varepsilon}}=\mathsf{E}^{-1}~{}\dot{symbol{\sigma}}+f(symbol{% \sigma},\sigma_{y})~{}symbol{\sigma}\quad\mathrm{for}~{}\|symbol{\sigma}\|\geq% \sigma_{y}
  34. s y m b o l ε = s y m b o l ε e = 𝖤 - 1 s y m b o l σ = s y m b o l ε for || s y m b o l σ || < σ y s y m b o l ε ˙ = s y m b o l ε ˙ e + s y m b o l ε ˙ vp = 𝖤 - 1 s y m b o l σ ˙ + f ( s y m b o l σ , σ y , s y m b o l ε vp ) s y m b o l σ for || s y m b o l σ || σ y \begin{aligned}&\displaystyle symbol{\varepsilon}=symbol{\varepsilon}_{\mathrm% {e}}=\mathsf{E}^{-1}~{}symbol{\sigma}=~{}symbol{\varepsilon}&&\displaystyle% \mathrm{for}~{}||symbol{\sigma}||<\sigma_{y}\\ &\displaystyle\dot{symbol{\varepsilon}}=\dot{symbol{\varepsilon}}_{\mathrm{e}}% +\dot{symbol{\varepsilon}}_{\mathrm{vp}}=\mathsf{E}^{-1}~{}\dot{symbol{\sigma}% }+f(symbol{\sigma},\sigma_{y},symbol{\varepsilon}_{\mathrm{vp}})~{}symbol{% \sigma}&&\displaystyle\mathrm{for}~{}||symbol{\sigma}||\geq\sigma_{y}\end{aligned}
  35. ε ˙ vp = f ( s y m b o l σ , s y m b o l q ) τ = { f ( s y m b o l σ , s y m b o l q ) τ if f ( symbol σ , symbolq ) > 0 0 otherwise \dot{\varepsilon}_{\mathrm{vp}}=\left\langle\cfrac{f(symbol{\sigma},symbol{q})% }{\tau}\right\rangle=\begin{cases}\cfrac{f(symbol{\sigma},symbol{q})}{\tau}&% \rm{if}~{}f(symbol{\sigma},symbol{q})>0\\ 0&\rm{otherwise}\\ \end{cases}
  36. f ( . , . ) f(.,.)
  37. s y m b o l σ symbol{\sigma}
  38. s y m b o l q symbol{q}
  39. s y m b o l ε vp symbol{\varepsilon}_{\mathrm{vp}}
  40. τ \tau
  41. ε ˙ vp = f f 0 n ( s y m b o l σ - s y m b o l χ ) \dot{\varepsilon}_{\mathrm{vp}}=\left\langle\frac{f}{f_{0}}\right\rangle^{n}(% symbol{\sigma}-symbol{\chi})
  42. f 0 f_{0}
  43. f f
  44. s y m b o l χ symbol{\chi}
  45. ε ˙ vp = { s y m b o l σ - 𝒫 s y m b o l σ τ if f ( symbol σ , symbolq ) > 0 0 otherwise \dot{\varepsilon}_{\mathrm{vp}}=\begin{cases}\cfrac{symbol{\sigma}-\mathcal{P}% symbol{\sigma}}{\tau}&\rm{if}~{}f(symbol{\sigma},symbol{q})>0\\ 0&\rm{otherwise}\end{cases}
  46. 𝒫 s y m b o l σ \mathcal{P}symbol{\sigma}
  47. f ( s y m b o l σ , s y m b o l q ) f(symbol{\sigma},symbol{q})
  48. f f
  49. J 2 J_{2}
  50. σ y \sigma_{y}
  51. (1) σ y ( ε p , ε p ˙ , T ) = [ A + B ( ε p ) n ] [ 1 + C ln ( ε p ˙ * ) ] [ 1 - ( T * ) m ] \,\text{(1)}\qquad\sigma_{y}(\varepsilon_{\rm{p}},\dot{\varepsilon_{\rm{p}}},T% )=\left[A+B(\varepsilon_{\rm{p}})^{n}\right]\left[1+C\ln(\dot{\varepsilon_{\rm% {p}}}^{*})\right]\left[1-(T^{*})^{m}\right]
  52. ε p \varepsilon_{\rm{p}}
  53. ε p ˙ \dot{\varepsilon_{\rm{p}}}
  54. A , B , C , n , m A,B,C,n,m
  55. ε p ˙ * := ε p ˙ ε p0 ˙ and T * := ( T - T 0 ) ( T m - T 0 ) \dot{\varepsilon_{\rm{p}}}^{*}:=\cfrac{\dot{\varepsilon_{\rm{p}}}}{\dot{% \varepsilon_{\rm{p0}}}}\qquad\,\text{and}\qquad T^{*}:=\cfrac{(T-T_{0})}{(T_{m% }-T_{0})}
  56. ε p0 ˙ \dot{\varepsilon_{\rm{p0}}}
  57. ε p ˙ * \dot{\varepsilon_{\rm{p}}}^{*}
  58. T 0 T_{0}
  59. T m T_{m}
  60. T * < 0 T^{*}<0
  61. m = 1 m=1
  62. (2) σ y ( ε p , ε p ˙ , T ) = [ σ a f ( ε p ) + σ t ( ε p ˙ , T ) ] μ ( p , T ) μ 0 ; σ a f σ max and σ t σ p \,\text{(2)}\qquad\sigma_{y}(\varepsilon_{\rm{p}},\dot{\varepsilon_{\rm{p}}},T% )=\left[\sigma_{a}f(\varepsilon_{\rm{p}})+\sigma_{t}(\dot{\varepsilon_{\rm{p}}% },T)\right]\frac{\mu(p,T)}{\mu_{0}};\quad\sigma_{a}f\leq\sigma_{\,\text{max}}~% {}~{}\,\text{and}~{}~{}\sigma_{t}\leq\sigma_{p}
  63. σ a \sigma_{a}
  64. f ( ε p ) f(\varepsilon_{\rm{p}})
  65. σ t \sigma_{t}
  66. μ ( p , T ) \mu(p,T)
  67. μ 0 \mu_{0}
  68. σ max \sigma_{\,\text{max}}
  69. σ p \sigma_{p}
  70. f f
  71. f ( ε p ) = [ 1 + β ( ε p + ε p i ) ] n f(\varepsilon_{\rm{p}})=[1+\beta(\varepsilon_{\rm{p}}+\varepsilon_{\rm{p}}i)]^% {n}
  72. β , n \beta,n
  73. ε p i \varepsilon_{\rm{p}}i
  74. σ t \sigma_{t}
  75. ε p ˙ = [ 1 C 1 exp [ 2 U k k b T ( 1 - σ t σ p ) 2 ] + C 2 σ t ] - 1 ; σ t σ p \dot{\varepsilon_{\rm{p}}}=\left[\frac{1}{C_{1}}\exp\left[\frac{2U_{k}}{k_{b}~% {}T}\left(1-\frac{\sigma_{t}}{\sigma_{p}}\right)^{2}\right]+\frac{C_{2}}{% \sigma_{t}}\right]^{-1};\quad\sigma_{t}\leq\sigma_{p}
  76. 2 U k 2U_{k}
  77. L d L_{d}
  78. k b k_{b}
  79. σ p \sigma_{p}
  80. C 1 , C 2 C_{1},C_{2}
  81. C 1 := ρ d L d a b 2 ν 2 w 2 ; C 2 := D ρ d b 2 C_{1}:=\frac{\rho_{d}L_{d}ab^{2}\nu}{2w^{2}};\quad C_{2}:=\frac{D}{\rho_{d}b^{% 2}}
  82. ρ d \rho_{d}
  83. L d L_{d}
  84. a a
  85. b b
  86. ν \nu
  87. w w
  88. D D
  89. (3) σ y ( ε p , ε p ˙ , T ) = σ a + B exp ( - β T ) + B 0 ε p exp ( - α T ) . \,\text{(3)}\qquad\sigma_{y}(\varepsilon_{\rm{p}},\dot{\varepsilon_{\rm{p}}},T% )=\sigma_{a}+B\exp(-\beta T)+B_{0}\sqrt{\varepsilon_{\rm{p}}}\exp(-\alpha T)~{}.
  90. σ a \sigma_{a}
  91. σ a := σ g + k h l + K ε p n , \sigma_{a}:=\sigma_{g}+\frac{k_{h}}{\sqrt{l}}+K\varepsilon_{\rm{p}}^{n},
  92. σ g \sigma_{g}
  93. k h k_{h}
  94. l l
  95. K K
  96. B , B 0 B,B_{0}
  97. α \alpha
  98. β \beta
  99. α = α 0 - α 1 ln ( ε p ˙ ) ; β = β 0 - β 1 ln ( ε p ˙ ) ; \alpha=\alpha_{0}-\alpha_{1}\ln(\dot{\varepsilon_{\rm{p}}});\quad\beta=\beta_{% 0}-\beta_{1}\ln(\dot{\varepsilon_{\rm{p}}});
  100. α 0 , α 1 , β 0 , β 1 \alpha_{0},\alpha_{1},\beta_{0},\beta_{1}
  101. (4) σ y ( ε p , ε ˙ , T ) = σ a + ( S i σ i + S e σ e ) μ ( p , T ) μ 0 \,\text{(4)}\qquad\sigma_{y}(\varepsilon_{\rm{p}},\dot{\varepsilon},T)=\sigma_% {a}+(S_{i}\sigma_{i}+S_{e}\sigma_{e})\frac{\mu(p,T)}{\mu_{0}}
  102. σ a \sigma_{a}
  103. σ i \sigma_{i}
  104. σ e \sigma_{e}
  105. S i , S e S_{i},S_{e}
  106. μ 0 \mu_{0}
  107. S i = [ 1 - ( k b T g 0 i b 3 μ ( p , T ) ln ε 0 ˙ ε ˙ ) 1 / q i ] 1 / p i S e = [ 1 - ( k b T g 0 e b 3 μ ( p , T ) ln ε 0 ˙ ε ˙ ) 1 / q e ] 1 / p e \begin{aligned}\displaystyle S_{i}&\displaystyle=\left[1-\left(\frac{k_{b}~{}T% }{g_{0i}b^{3}\mu(p,T)}\ln\frac{\dot{\varepsilon_{\rm{0}}}}{\dot{\varepsilon}}% \right)^{1/q_{i}}\right]^{1/p_{i}}\\ \displaystyle S_{e}&\displaystyle=\left[1-\left(\frac{k_{b}~{}T}{g_{0e}b^{3}% \mu(p,T)}\ln\frac{\dot{\varepsilon_{\rm{0}}}}{\dot{\varepsilon}}\right)^{1/q_{% e}}\right]^{1/p_{e}}\end{aligned}
  108. k b k_{b}
  109. b b
  110. g 0 i , g 0 e g_{0i},g_{0e}
  111. ε ˙ , ε 0 ˙ \dot{\varepsilon},\dot{\varepsilon_{\rm{0}}}
  112. q i , p i , q e , p e q_{i},p_{i},q_{e},p_{e}
  113. σ e \sigma_{e}
  114. (5) d σ e d ε p = θ ( σ e ) \,\text{(5)}\qquad\frac{d\sigma_{e}}{d\varepsilon_{\rm{p}}}=\theta(\sigma_{e})
  115. θ ( σ e ) = θ 0 [ 1 - F ( σ e ) ] + θ I V F ( σ e ) θ 0 = a 0 + a 1 ln ε p ˙ + a 2 ε p ˙ - a 3 T F ( σ e ) = tanh ( α σ e σ e s ) tanh ( α ) ln ( σ e s σ 0 e s ) = ( k T g 0 e s b 3 μ ( p , T ) ) ln ( ε p ˙ ε p ˙ ) \begin{aligned}\displaystyle\theta(\sigma_{e})&\displaystyle=\theta_{0}[1-F(% \sigma_{e})]+\theta_{IV}F(\sigma_{e})\\ \displaystyle\theta_{0}&\displaystyle=a_{0}+a_{1}\ln\dot{\varepsilon_{\rm{p}}}% +a_{2}\sqrt{\dot{\varepsilon_{\rm{p}}}}-a_{3}T\\ \displaystyle F(\sigma_{e})&\displaystyle=\cfrac{\tanh\left(\alpha\cfrac{% \sigma_{e}}{\sigma_{es}}\right)}{\tanh(\alpha)}\\ \displaystyle\ln(\cfrac{\sigma_{es}}{\sigma_{0es}})&\displaystyle=\left(\frac{% kT}{g_{0es}b^{3}\mu(p,T)}\right)\ln\left(\cfrac{\dot{\varepsilon_{\rm{p}}}}{% \dot{\varepsilon_{\rm{p}}}}\right)\end{aligned}
  116. θ 0 \theta_{0}
  117. θ I V \theta_{IV}
  118. a 0 , a 1 , a 2 , a 3 , α a_{0},a_{1},a_{2},a_{3},\alpha
  119. σ e s \sigma_{es}
  120. σ 0 e s \sigma_{0es}
  121. g 0 e s g_{0es}
  122. ε p ˙ \dot{\varepsilon_{\rm{p}}}
  123. 10 7 10^{7}
  124. (6) σ y ( ε p , ε p ˙ , T ) = { 2 [ τ s + α ln [ 1 - φ exp ( - β - θ ε p α φ ) ] ] μ ( p , T ) thermal regime 2 τ s μ ( p , T ) shock regime \,\text{(6)}\qquad\sigma_{y}(\varepsilon_{\rm{p}},\dot{\varepsilon_{\rm{p}}},T% )=\begin{cases}2\left[\tau_{s}+\alpha\ln\left[1-\varphi\exp\left(-\beta-\cfrac% {\theta\varepsilon_{\rm{p}}}{\alpha\varphi}\right)\right]\right]\mu(p,T)&\,% \text{thermal regime}\\ 2\tau_{s}\mu(p,T)&\,\text{shock regime}\end{cases}
  125. α := s 0 - τ y d ; β := τ s - τ y α ; φ := exp ( β ) - 1 \alpha:=\frac{s_{0}-\tau_{y}}{d};\quad\beta:=\frac{\tau_{s}-\tau_{y}}{\alpha};% \quad\varphi:=\exp(\beta)-1
  126. τ s \tau_{s}
  127. s 0 s_{0}
  128. τ s \tau_{s}
  129. τ y \tau_{y}
  130. θ \theta
  131. d d
  132. τ s = max { s 0 - ( s 0 - s ) erf [ κ T ^ ln ( γ ξ ˙ ε p ˙ ) ] , s 0 ( ε p ˙ γ ξ ˙ ) s 1 } τ y = max { y 0 - ( y 0 - y ) erf [ κ T ^ ln ( γ ξ ˙ ε p ˙ ) ] , min { y 1 ( ε p ˙ γ ξ ˙ ) y 2 , s 0 ( ε p ˙ γ ξ ˙ ) s 1 } } \begin{aligned}\displaystyle\tau_{s}&\displaystyle=\max\left\{s_{0}-(s_{0}-s_{% \infty})\rm{erf}\left[\kappa\hat{T}\ln\left(\cfrac{\gamma\dot{\xi}}{\dot{% \varepsilon_{\rm{p}}}}\right)\right],s_{0}\left(\cfrac{\dot{\varepsilon_{\rm{p% }}}}{\gamma\dot{\xi}}\right)^{s_{1}}\right\}\\ \displaystyle\tau_{y}&\displaystyle=\max\left\{y_{0}-(y_{0}-y_{\infty})\rm{erf% }\left[\kappa\hat{T}\ln\left(\cfrac{\gamma\dot{\xi}}{\dot{\varepsilon_{\rm{p}}% }}\right)\right],\min\left\{y_{1}\left(\cfrac{\dot{\varepsilon_{\rm{p}}}}{% \gamma\dot{\xi}}\right)^{y_{2}},s_{0}\left(\cfrac{\dot{\varepsilon_{\rm{p}}}}{% \gamma\dot{\xi}}\right)^{s_{1}}\right\}\right\}\end{aligned}
  133. s s_{\infty}
  134. τ s \tau_{s}
  135. y 0 , y y_{0},y_{\infty}
  136. τ y \tau_{y}
  137. ( κ , γ ) (\kappa,\gamma)
  138. T ^ = T / T m \hat{T}=T/T_{m}
  139. s 1 , y 1 , y 2 s_{1},y_{1},y_{2}
  140. ξ ˙ = 1 2 ( 4 π ρ 3 M ) 1 / 3 ( μ ( p , T ) ρ ) 1 / 2 \dot{\xi}=\frac{1}{2}\left(\cfrac{4\pi\rho}{3M}\right)^{1/3}\left(\cfrac{\mu(p% ,T)}{\rho}\right)^{1/2}
  141. ρ \rho
  142. M M

Viscosity.html

  1. u u
  2. u u
  3. F F
  4. u u
  5. A A
  6. y y
  7. F = μ A u y . F=\mu A\frac{u}{y}.
  8. u / y u/y
  9. τ = μ u y , \tau=\mu\frac{\partial u}{\partial y},
  10. τ = F / A \tau=F/A
  11. u / y {\partial u}/{\partial y}
  12. y y
  13. y y
  14. ν = μ ρ \nu=\frac{\mu}{\rho}
  15. R e = ρ u L μ = u L ν , Re=\frac{\rho uL}{\mu}=\frac{uL}{\nu}\;,
  16. L L
  17. a a
  18. b b
  19. F χ a F a + χ b F b , F\approx\chi_{a}F_{a}+\chi_{b}F_{b},
  20. μ 1 χ a / μ a + χ b / μ b , \mu\approx\frac{1}{\chi_{a}/\mu_{a}+\chi_{b}/\mu_{b}},
  21. τ = μ d u x d y \tau=\mu\frac{\mathrm{d}u_{x}}{\mathrm{d}y}
  22. τ = p ˙ A = m ˙ u x A . \tau=\frac{\dot{p}}{A}=\frac{\dot{m}\langle u_{x}\rangle}{A}.
  23. u x \langle u_{x}\rangle
  24. m ˙ = ρ u ¯ A \dot{m}=\rho\bar{u}A
  25. u x = 1 2 λ d u x d y \langle u_{x}\rangle=\frac{1}{2}\,\lambda\frac{\mathrm{d}u_{x}}{\mathrm{d}y}
  26. τ = 1 2 ρ u ¯ λ μ d u x d y ν = μ ρ = 1 2 u ¯ λ , \tau=\underbrace{\frac{1}{2}\,\rho\bar{u}\lambda}_{\mu}\cdot\frac{\mathrm{d}u_% {x}}{\mathrm{d}y}\;\;\Rightarrow\;\;\nu=\frac{\mu}{\rho}=\tfrac{1}{2}\,\bar{u}\lambda,
  27. m ˙ \dot{m}
  28. u ¯ = u 2 \bar{u}=\sqrt{\langle u^{2}\rangle}
  29. μ = μ 0 T 0 + C T + C ( T T 0 ) 3 / 2 . {\mu}={\mu}_{0}\frac{T_{0}+C}{T+C}\left(\frac{T}{T_{0}}\right)^{3/2}.
  30. λ T 3 / 2 T + C , \lambda\,\frac{T^{3/2}}{T+C}\,,
  31. λ = μ 0 ( T 0 + C ) T 0 3 / 2 \lambda=\frac{\mu_{0}(T_{0}+C)}{T_{0}^{3/2}}\,
  32. μ 0 × 10 6 = 2.6693 ( M T ) 1 / 2 σ 2 ω ( T * ) , {\mu}_{0}\times 10^{6}={2.6693}\frac{(MT)^{1/2}}{\sigma^{2}\omega(T^{*})},
  33. VBN = 14.534 × ln [ ln ( ν + 0.8 ) ] + 10.975 \mbox{VBN}~{}=14.534\times\ln\left[\ln(\nu+0.8)\right]+10.975\,
  34. VBN Blend = [ x A × VBN ] A + [ x B × VBN ] B + + [ x N × VBN ] N \mbox{VBN}~\text{Blend}=\left[x_{A}\times\mbox{VBN}~{}_{A}\right]+\left[x_{B}% \times\mbox{VBN}~{}_{B}\right]+\cdots+\left[x_{N}\times\mbox{VBN}~{}_{N}\right]\,
  35. ν = exp ( exp ( VBNBlend - 10.975 14.534 ) ) - 0.8 , \nu=\exp\left(\exp\left(\frac{\,\text{VBN}\text{Blend}-10.975}{14.534}\right)% \right)-0.8,
  36. × 10 5 \times 10^{−}5
  37. × 10 5 \times 10^{−}5
  38. × 10 - 5 \times 10^{-}5
  39. × 10 - 3 \times 10^{-}3
  40. × 10 2 1 \times 10^{2}1
  41. × 10 2 1 \times 10^{2}1
  42. μ s = μ r μ l , \mu_{s}=\mu_{r}\cdot\mu_{l},
  43. μ r = 1 + 2.5 ϕ \mu_{r}=1+2.5\cdot\phi
  44. μ r = 1 + 2.5 ϕ + 14.1 ϕ 2 \mu_{r}=1+2.5\cdot\phi+14.1\cdot\phi^{2}
  45. μ r = 1 + 2.5 ϕ + 10.05 ϕ 2 + A e B ϕ , \mu_{r}=1+2.5\cdot\phi+10.05\cdot\phi^{2}+A\cdot e^{B\cdot\phi},
  46. μ r = ( 1 - ϕ A ) - 2 , \mu_{r}=(1-\frac{\phi}{A})^{-2},
  47. μ = A e Q / R T , \mu=A\cdot e^{Q/RT},
  48. R D = Q H Q L R_{D}=\frac{Q_{H}}{Q_{L}}
  49. μ = A 1 T [ 1 + A 2 e B / R T ] [ 1 + C e D / R T ] , \mu=A_{1}\cdot T\cdot\left[1+A_{2}\cdot e^{B/RT}]\cdot[1+C\cdot e^{D/RT}\right],
  50. μ = A L T e Q H / R T \mu=A_{L}T\cdot e^{Q_{H}/RT}
  51. Q H = H d + H m , Q_{H}=H_{d}+H_{m},\,
  52. μ = A H T e Q L / R T , \mu=A_{H}T\cdot e^{Q_{L}/RT},
  53. Q L = H m . Q_{L}=H_{m}.\,

Viscous_remanent_magnetization.html

  1. τ τ
  2. M ( t ) = M 0 + ( M eq - M 0 ) exp ( - t / τ ) . M(t)=M_{0}+\left(M\text{eq}-M_{0}\right)\exp(-t/\tau).
  3. τ τ
  4. S = M log t . S=\frac{\partial M}{\partial\log t}.

VisualRank.html

  1. V R = S * × V R VR=S^{*}\times VR
  2. S * S^{*}
  3. V R VR
  4. S * S^{*}
  5. l 2 l_{2}

Volume_entropy.html

  1. M ~ . \tilde{M}.
  2. x ~ 0 M ~ \tilde{x}_{0}\in\tilde{M}
  3. h = h ( M , g ) h=h(M,g)
  4. h ( M , g ) = lim R + log ( vol B ( R ) ) R , h(M,g)=\lim_{R\rightarrow+\infty}\frac{\log\left(\operatorname{vol}B(R)\right)% }{R},
  5. M ~ \tilde{M}
  6. x ~ 0 \tilde{x}_{0}
  7. h n ( Y , g ) vol ( Y , g ) | deg ( f ) | h n ( X , g 0 ) vol ( X , g 0 ) , h^{n}(Y,g)\operatorname{vol}(Y,g)\geq|\operatorname{deg}(f)|h^{n}(X,g_{0})% \operatorname{vol}(X,g_{0}),

Volume_of_an_n-ball.html

  1. V n ( R ) = π n / 2 Γ ( n 2 + 1 ) R n , V_{n}(R)=\frac{\pi^{n/2}}{\Gamma(\frac{n}{2}+1)}R^{n},
  2. V 2 k ( R ) = π k k ! R 2 k , V_{2k}(R)=\frac{\pi^{k}}{k!}R^{2k},
  3. V 2 k + 1 ( R ) = 2 k + 1 π k ( 2 k + 1 ) ! ! R 2 k + 1 = 2 ( k ! ) ( 4 π ) k ( 2 k + 1 ) ! R 2 k + 1 . V_{2k+1}(R)=\frac{2^{k+1}\pi^{k}}{(2k+1)!!}R^{2k+1}=\frac{2(k!)(4\pi)^{k}}{(2k% +1)!}R^{2k+1}.
  4. R n ( V ) = Γ ( n 2 + 1 ) 1 / n π V 1 / n . R_{n}(V)=\frac{\Gamma(\frac{n}{2}+1)^{1/n}}{\sqrt{\pi}}V^{1/n}.
  5. R 2 k ( V ) = ( k ! V ) 1 / 2 k π , R_{2k}(V)=\frac{(k!V)^{1/2k}}{\sqrt{\pi}},
  6. R 2 k + 1 ( V ) = ( ( 2 k + 1 ) ! ! V 2 k + 1 π k ) 1 / ( 2 k + 1 ) . R_{2k+1}(V)=\left(\frac{(2k+1)!!V}{2^{k+1}\pi^{k}}\right)^{1/(2k+1)}.
  7. V n ( R ) = 2 π R 2 n V n - 2 ( R ) . V_{n}(R)=\frac{2\pi R^{2}}{n}V_{n-2}(R).
  8. V n ( R ) = R π Γ ( n + 1 2 ) Γ ( n 2 + 1 ) V n - 1 ( R ) . V_{n}(R)=R\sqrt{\pi}\frac{\Gamma(\frac{n+1}{2})}{\Gamma(\frac{n}{2}+1)}V_{n-1}% (R).
  9. V 2 k ( R ) \displaystyle V_{2k}(R)
  10. 1 1
  11. 2 R 2R
  12. V / 2 V/2
  13. π R 2 \pi R^{2}
  14. V 1 / 2 π \frac{V^{1/2}}{\sqrt{\pi}}
  15. 4 3 π R 3 \frac{4}{3}\pi R^{3}
  16. ( 3 V 4 π ) 1 / 3 \left(\frac{3V}{4\pi}\right)^{1/3}
  17. π 2 2 R 4 \frac{\pi^{2}}{2}R^{4}
  18. ( 2 V ) 1 / 4 π \frac{(2V)^{1/4}}{\sqrt{\pi}}
  19. 8 π 2 15 R 5 \frac{8\pi^{2}}{15}R^{5}
  20. ( 15 V 8 π 2 ) 1 / 5 \left(\frac{15V}{8\pi^{2}}\right)^{1/5}
  21. π 3 6 R 6 \frac{\pi^{3}}{6}R^{6}
  22. ( 6 V ) 1 / 6 π \frac{(6V)^{1/6}}{\sqrt{\pi}}
  23. 16 π 3 105 R 7 \frac{16\pi^{3}}{105}R^{7}
  24. ( 105 V 16 π 3 ) 1 / 7 \left(\frac{105V}{16\pi^{3}}\right)^{1/7}
  25. π 4 24 R 8 \frac{\pi^{4}}{24}R^{8}
  26. ( 24 V ) 1 / 8 π \frac{(24V)^{1/8}}{\sqrt{\pi}}
  27. 32 π 4 945 R 9 \frac{32\pi^{4}}{945}R^{9}
  28. ( 945 V 32 π 4 ) 1 / 9 \left(\frac{945V}{32\pi^{4}}\right)^{1/9}
  29. π 5 120 R 10 \frac{\pi^{5}}{120}R^{10}
  30. ( 120 V ) 1 / 10 π \frac{(120V)^{1/10}}{\sqrt{\pi}}
  31. V n ( R ) 1 n π ( 2 π e n ) n / 2 R n . V_{n}(R)\sim\frac{1}{\sqrt{n\pi}}\left(\frac{2\pi e}{n}\right)^{n/2}R^{n}.
  32. R 2 π e / n R\sqrt{2\pi e/n}
  33. A n ( R ) A_{n}(R)
  34. A n ( R ) = d d R V n + 1 ( R ) . A_{n}(R)=\frac{d}{dR}V_{n+1}(R).
  35. V 0 ( R ) = 1 , V_{0}(R)=1,
  36. A 0 ( R ) = 2 , A_{0}(R)=2,
  37. V n + 1 ( R ) = R n + 1 A n ( R ) , V_{n+1}(R)=\frac{R}{n+1}A_{n}(R),
  38. A n + 1 ( R ) = ( 2 π R ) V n ( R ) . A_{n+1}(R)=(2\pi R)V_{n}(R).
  39. V n ( R ) R n . V_{n}(R)\propto R^{n}.
  40. V n ( R ) = - R R V n - 1 ( R 2 - x 2 ) d x , V_{n}(R)=\int_{-R}^{R}V_{n-1}(\sqrt{R^{2}-x^{2}})\,dx,
  41. V n ( R ) = R n - 1 - R R V n - 1 ( 1 - ( x / R ) 2 ) d x . V_{n}(R)=R^{n-1}\int_{-R}^{R}V_{n-1}\left(\sqrt{1-(x/R)^{2}}\right)\,dx.
  42. V n ( R ) = R n - 1 1 V n - 1 ( 1 - t 2 ) d t = R n V n ( 1 ) , V_{n}(R)=R^{n}\int_{-1}^{1}V_{n-1}(\sqrt{1-t^{2}})\,dt=R^{n}V_{n}(1),
  43. R 2 - r 2 \sqrt{R^{2}-r^{2}}
  44. V n ( R ) = 0 2 π 0 R V n - 2 ( R 2 - r 2 ) r d r d θ , V_{n}(R)=\int_{0}^{2\pi}\int_{0}^{R}V_{n-2}(\sqrt{R^{2}-r^{2}})\,r\,dr\,d\theta,
  45. V n ( R ) = ( 2 π ) V n - 2 ( R ) 0 R ( 1 - ( r / R ) 2 ) ( n - 2 ) / 2 r d r . V_{n}(R)=(2\pi)V_{n-2}(R)\int_{0}^{R}(1-(r/R)^{2})^{(n-2)/2}\,r\,dr.
  46. V n ( R ) \displaystyle V_{n}(R)
  47. Γ ( 1 ) = 1 \Gamma(1)=1
  48. Γ ( 3 / 2 ) = ( 1 / 2 ) Γ ( 1 / 2 ) = π / 2 \Gamma(3/2)=(1/2)\cdot\Gamma(1/2)=\sqrt{\pi}/2
  49. V n ( R ) = V n - 1 ( R ) - R R ( 1 - ( x / R ) 2 ) ( n - 1 ) / 2 d x . V_{n}(R)=V_{n-1}(R)\int_{-R}^{R}(1-(x/R)^{2})^{(n-1)/2}\,dx.
  50. V n - 1 ( R ) R 0 1 ( 1 - u ) ( n - 1 ) / 2 u - 1 / 2 d u V_{n-1}(R)\cdot R\cdot\int_{0}^{1}(1-u)^{(n-1)/2}u^{-1/2}\,du
  51. V n ( R ) = V n - 1 ( R ) R B ( n + 1 2 , 1 2 ) . V_{n}(R)=V_{n-1}(R)\cdot R\cdot B(\textstyle\frac{n+1}{2},\textstyle\frac{1}{2% }).
  52. V n ( R ) = V n - 1 ( R ) R Γ ( n + 1 2 ) Γ ( 1 2 ) Γ ( n 2 + 1 ) . V_{n}(R)=V_{n-1}(R)\cdot R\cdot\frac{\Gamma(\frac{n+1}{2})\Gamma(\frac{1}{2})}% {\Gamma(\frac{n}{2}+1)}.
  53. Γ ( 1 / 2 ) = π \Gamma(1/2)=\sqrt{\pi}
  54. V n ( R ) = R π Γ ( n + 1 2 ) Γ ( n 2 + 1 ) V n - 1 ( R ) . V_{n}(R)=R\sqrt{\pi}\frac{\Gamma(\frac{n+1}{2})}{\Gamma(\frac{n}{2}+1)}V_{n-1}% (R).
  55. d V = r n - 1 sin n - 2 ( ϕ 1 ) sin n - 3 ( ϕ 2 ) sin ( ϕ n - 2 ) d r d ϕ 1 d ϕ 2 d ϕ n - 1 , dV=r^{n-1}\sin^{n-2}(\phi_{1})\sin^{n-3}(\phi_{2})\cdots\sin(\phi_{n-2})\,dr\,% d\phi_{1}\,d\phi_{2}\cdots d\phi_{n-1},
  56. V n ( R ) = 0 R 0 π 0 2 π r n - 1 sin n - 2 ( ϕ 1 ) sin ( ϕ n - 2 ) d ϕ n - 1 d ϕ 1 d r . V_{n}(R)=\int_{0}^{R}\int_{0}^{\pi}\cdots\int_{0}^{2\pi}r^{n-1}\sin^{n-2}(\phi% _{1})\cdots\sin(\phi_{n-2})\,d\phi_{n-1}\cdots d\phi_{1}\,dr.
  57. V n ( R ) = ( 0 R r n - 1 d r ) ( 0 π sin n - 2 ( ϕ 1 ) d ϕ 1 ) ( 0 2 π d ϕ n - 1 ) . V_{n}(R)=\bigg(\int_{0}^{R}r^{n-1}\,dr\bigg)\bigg(\int_{0}^{\pi}\sin^{n-2}(% \phi_{1})\,d\phi_{1}\bigg)\cdots\bigg(\int_{0}^{2\pi}d\phi_{n-1}\bigg).
  58. V n ( R ) = R n n ( 2 0 π / 2 sin n - 2 ( ϕ 1 ) d ϕ 1 ) ( 4 0 π / 2 d ϕ n - 1 ) . V_{n}(R)=\frac{R^{n}}{n}\bigg(2\int_{0}^{\pi/2}\sin^{n-2}(\phi_{1})\,d\phi_{1}% \bigg)\cdots\bigg(4\int_{0}^{\pi/2}d\phi_{n-1}\bigg).
  59. V n ( R ) = R n n B ( n - 1 2 , 1 2 ) B ( n - 2 2 , 1 2 ) B ( 2 2 , 1 2 ) 2 B ( 1 2 , 1 2 ) . V_{n}(R)=\frac{R^{n}}{n}\textstyle B(\frac{n-1}{2},\frac{1}{2})B(\frac{n-2}{2}% ,\frac{1}{2})\cdots B(\frac{2}{2},\frac{1}{2})\cdot 2B(\frac{1}{2},\frac{1}{2}).
  60. V n ( R ) = R n n Γ ( n - 1 2 ) Γ ( 1 2 ) Γ ( n 2 ) Γ ( n - 2 2 ) Γ ( 1 2 ) Γ ( n - 1 2 ) Γ ( 2 2 ) Γ ( 1 2 ) Γ ( 3 2 ) 2 Γ ( 1 2 ) Γ ( 1 2 ) Γ ( 2 2 ) . V_{n}(R)=\frac{R^{n}}{n}\frac{\Gamma(\frac{n-1}{2})\Gamma(\frac{1}{2})}{\Gamma% (\frac{n}{2})}\frac{\Gamma(\frac{n-2}{2})\Gamma(\frac{1}{2})}{\Gamma(\frac{n-1% }{2})}\cdots\frac{\Gamma(\frac{2}{2})\Gamma(\frac{1}{2})}{\Gamma(\frac{3}{2})}% \cdot 2\frac{\Gamma(\frac{1}{2})\Gamma(\frac{1}{2})}{\Gamma(\frac{2}{2})}.
  61. Γ ( 1 / 2 ) = π \Gamma(1/2)=\sqrt{\pi}
  62. Γ ( 1 ) = 1 \Gamma(1)=1
  63. V n ( R ) = 2 π n / 2 R n n Γ ( n 2 ) = π n / 2 R n Γ ( n 2 + 1 ) . V_{n}(R)=\frac{2\pi^{n/2}R^{n}}{n\Gamma(\frac{n}{2})}=\frac{\pi^{n/2}R^{n}}{% \Gamma(\frac{n}{2}+1)}.
  64. f ( x 1 , , x n ) = exp ( - 1 2 i = 1 n x i 2 ) . f(x_{1},\ldots,x_{n})=\exp\Big(\mathord{-}\textstyle\frac{1}{2}\displaystyle% \sum_{i=1}^{n}x_{i}^{2}\Big).
  65. 𝐑 n f d V = i = 1 n ( - exp ( - x i 2 / 2 ) d x i ) = ( 2 π ) n / 2 , \int_{\mathbf{R}^{n}}f\,dV=\prod_{i=1}^{n}\Big(\int_{-\infty}^{\infty}\exp% \left(-x_{i}^{2}/2\right)\,dx_{i}\Big)=(2\pi)^{n/2},
  66. 𝐑 n f d V = 0 S n - 1 ( r ) exp ( - r 2 / 2 ) d A d r , \int_{\mathbf{R}^{n}}f\,dV=\int_{0}^{\infty}\int_{S^{n-1}(r)}\exp\left(-r^{2}/% 2\right)\,dA\,dr,
  67. A n - 1 ( r ) = r n - 1 A n - 1 ( 1 ) . A_{n-1}(r)=r^{n-1}A_{n-1}(1).
  68. A n - 1 ( 1 ) 0 exp ( - r 2 / 2 ) r n - 1 d r . A_{n-1}(1)\int_{0}^{\infty}\exp\left(-r^{2}/2\right)\,r^{n-1}\,dr.
  69. A n - 1 ( 1 ) 2 n / 2 - 1 0 e - t t n / 2 - 1 d t . A_{n-1}(1)2^{n/2-1}\int_{0}^{\infty}e^{-t}t^{n/2-1}\,dt.
  70. A n - 1 ( 1 ) = 2 π n / 2 Γ ( n 2 ) . A_{n-1}(1)=\frac{2\pi^{n/2}}{\Gamma(\frac{n}{2})}.
  71. V n ( R ) = 0 R 2 π n / 2 Γ ( n 2 ) r n - 1 d r = 2 π n / 2 n Γ ( n 2 ) R n = π n / 2 Γ ( n 2 + 1 ) R n . V_{n}(R)=\int_{0}^{R}\frac{2\pi^{n/2}}{\Gamma(\frac{n}{2})}\,r^{n-1}\,dr=\frac% {2\pi^{n/2}}{n\Gamma(\frac{n}{2})}R^{n}=\frac{\pi^{n/2}}{\Gamma(\frac{n}{2}+1)% }R^{n}.
  72. ( | x i | p ) 1 / p \textstyle(\sum|x_{i}|^{p})^{1/p}
  73. V n p ( R ) = ( 2 Γ ( 1 p + 1 ) R ) n Γ ( n p + 1 ) . V^{p}_{n}(R)=\frac{(2\Gamma(\frac{1}{p}+1)R)^{n}}{\Gamma(\frac{n}{p}+1)}.
  74. V n p ( R ) = ( 2 Γ ( 1 p + 1 ) R ) Γ ( n - 1 p + 1 ) Γ ( n p + 1 ) V n - 1 p ( R ) . V^{p}_{n}(R)=(2\Gamma(\textstyle\frac{1}{p}+1)R)\displaystyle\frac{\Gamma(% \frac{n-1}{p}+1)}{\Gamma(\frac{n}{p}+1)}V^{p}_{n-1}(R).
  75. 2 Γ ( 3 / 2 ) = π 2\Gamma(3/2)=\sqrt{\pi}
  76. V n 1 ( R ) = 2 n n ! R n , V^{1}_{n}(R)=\frac{2^{n}}{n!}R^{n},
  77. V n ( R ) = ( 2 R ) n . V^{\infty}_{n}(R)=(2R)^{n}.
  78. n \sqrt{n}
  79. n \sqrt{n}
  80. B p 1 , , p n = { x = ( x 1 , , x n ) 𝐑 n : | x 1 | p 1 + + | x n | p n 1 } . B_{p_{1},\ldots,p_{n}}=\{x=(x_{1},\ldots,x_{n})\in\mathbf{R}^{n}:|x_{1}|^{p_{1% }}+\cdots+|x_{n}|^{p_{n}}\leq 1\}.
  81. Vol ( B p 1 , , p n ) = 2 n Γ ( 1 + p 1 - 1 ) Γ ( 1 + p n - 1 ) Γ ( 1 + p 1 - 1 + + p n - 1 ) . \operatorname{Vol}(B_{p_{1},\ldots,p_{n}})=2^{n}\frac{\Gamma(1+p_{1}^{-1})% \cdots\Gamma(1+p_{n}^{-1})}{\Gamma(1+p_{1}^{-1}+\cdots+p_{n}^{-1})}.

Volume_of_fluid_method.html

  1. C C
  2. C C
  3. C C
  4. C = 1 C=1
  5. 0 < C < 1 0<C<1
  6. C C
  7. C C
  8. 0 < C < 1 0<C<1
  9. m m
  10. n n
  11. ϕ \phi
  12. C m t + 𝐯 C m = 0 , \frac{\partial C_{m}}{\partial t}+\mathbf{v}\cdot\nabla C_{m}=0,
  13. m = 1 n C m = 1 \sum_{m=1}^{n}C_{m}=1
  14. ρ \rho
  15. ρ = ρ m ϕ m . \rho=\sum\rho_{m}\phi_{m}.
  16. C C
  17. C C
  18. C C
  19. 𝐧 x + 𝐧 y + 𝐧 z = α , \mathbf{n}_{x}+\mathbf{n}_{y}+\mathbf{n}_{z}=\alpha,
  20. 𝐧 \mathbf{n}
  21. α \alpha
  22. C C
  23. C C

Von_Neumann's_inequality.html

  1. L p L^{p}
  2. || P ( T ) || L p || P ( S ) || p ||P(T)||_{L^{p}}\leq||P(S)||_{\ell^{p}}
  3. p = 2 p=2
  4. p = 1 p=1
  5. p = p=\infty

Von_Neumann_paradox.html

  1. σ u 1 τ v 1 σ u 2 τ v 2 σ u n τ v n \sigma^{u_{1}}\tau^{v_{1}}\sigma^{u_{2}}\tau^{v_{2}}\cdots\sigma^{u_{n}}\tau^{% v_{n}}
  2. u u
  3. v v
  4. u u
  5. v v
  6. u 1 u_{1}
  7. C 1 , C 2 , , C m C_{1},C_{2},\dots,C_{m}
  8. C 1 C_{1}
  9. C 1 C_{1}

Von_Neumann_stability_analysis.html

  1. u t = α 2 u x 2 \frac{\partial u}{\partial t}=\alpha\frac{\partial^{2}u}{\partial x^{2}}
  2. L L
  3. ( 1 ) u j n + 1 = u j n + r ( u j + 1 n - 2 u j n + u j - 1 n ) \quad(1)\qquad u_{j}^{n+1}=u_{j}^{n}+r\left(u_{j+1}^{n}-2u_{j}^{n}+u_{j-1}^{n}\right)
  4. r = α Δ t Δ x 2 r=\frac{\alpha\,\Delta t}{\Delta x^{2}}
  5. u j n u_{j}^{n}
  6. u ( x , t ) u(x,t)
  7. ϵ j n \epsilon_{j}^{n}
  8. ϵ j n = N j n - u j n \epsilon_{j}^{n}=N_{j}^{n}-u_{j}^{n}
  9. u j n u_{j}^{n}
  10. N j n N_{j}^{n}
  11. u j n u_{j}^{n}
  12. ϵ j n \epsilon_{j}^{n}
  13. ( 2 ) ϵ j n + 1 = ϵ j n + r ( ϵ j + 1 n - 2 ϵ j n + ϵ j - 1 n ) \quad(2)\qquad\epsilon_{j}^{n+1}=\epsilon_{j}^{n}+r\left(\epsilon_{j+1}^{n}-2% \epsilon_{j}^{n}+\epsilon_{j-1}^{n}\right)
  14. L L
  15. ( 3 ) ϵ ( x ) = m = 1 M A m e i k m x \quad(3)\qquad\epsilon(x)=\sum_{m=1}^{M}A_{m}e^{ik_{m}x}
  16. k m = π m L k_{m}=\frac{\pi m}{L}
  17. m = 1 , 2 , , M m=1,2,\ldots,M
  18. M = L / Δ x M=L/\Delta x
  19. A m A_{m}
  20. ( 4 ) ϵ ( x , t ) = m = 1 M e a t e i k m x \quad(4)\qquad\epsilon(x,t)=\sum_{m=1}^{M}e^{at}e^{ik_{m}x}
  21. a a
  22. ( 5 ) ϵ m ( x , t ) = e a t e i k m x \quad(5)\qquad\epsilon_{m}(x,t)=e^{at}e^{ik_{m}x}
  23. ϵ j n = e a t e i k m x ϵ j n + 1 = e a ( t + Δ t ) e i k m x ϵ j + 1 n = e a t e i k m ( x + Δ x ) ϵ j - 1 n = e a t e i k m ( x - Δ x ) , \begin{aligned}\displaystyle\epsilon_{j}^{n}&\displaystyle=e^{at}e^{ik_{m}x}\\ \displaystyle\epsilon_{j}^{n+1}&\displaystyle=e^{a(t+\Delta t)}e^{ik_{m}x}\\ \displaystyle\epsilon_{j+1}^{n}&\displaystyle=e^{at}e^{ik_{m}(x+\Delta x)}\\ \displaystyle\epsilon_{j-1}^{n}&\displaystyle=e^{at}e^{ik_{m}(x-\Delta x)},% \end{aligned}
  24. ( 6 ) e a Δ t = 1 + α Δ t Δ x 2 ( e i k m Δ x + e - i k m Δ x - 2 ) . \quad(6)\qquad e^{a\Delta t}=1+\frac{\alpha\Delta t}{\Delta x^{2}}\left(e^{ik_% {m}\Delta x}+e^{-ik_{m}\Delta x}-2\right).
  25. cos ( k m Δ x ) = e i k m Δ x + e - i k m Δ x 2 and sin 2 k m Δ x 2 = 1 - cos ( k m Δ x ) 2 \qquad\cos(k_{m}\Delta x)=\frac{e^{ik_{m}\Delta x}+e^{-ik_{m}\Delta x}}{2}% \qquad\,\text{and}\qquad\sin^{2}\frac{k_{m}\Delta x}{2}=\frac{1-\cos(k_{m}% \Delta x)}{2}
  26. ( 7 ) e a Δ t = 1 - 4 α Δ t Δ x 2 sin 2 ( k m Δ x / 2 ) \quad(7)\qquad e^{a\Delta t}=1-\frac{4\alpha\Delta t}{\Delta x^{2}}\sin^{2}(k_% {m}\Delta x/2)
  27. G ϵ j n + 1 ϵ j n G\equiv\frac{\epsilon_{j}^{n+1}}{\epsilon_{j}^{n}}
  28. | G | 1. |G|\leq 1.
  29. ( 8 ) G = e a ( t + Δ t ) e i k m x e a t e i k m x = e a Δ t \quad(8)\qquad G=\frac{e^{a(t+\Delta t)}e^{ik_{m}x}}{e^{at}e^{ik_{m}x}}=e^{a% \Delta t}
  30. ( 9 ) | 1 - 4 α Δ t Δ x 2 sin 2 ( k m Δ x / 2 ) | 1 \quad(9)\qquad\left|1-\frac{4\alpha\Delta t}{\Delta x^{2}}\sin^{2}(k_{m}\Delta x% /2)\right|\leq 1
  31. 4 α Δ t Δ x 2 sin 2 ( k m Δ x / 2 ) \frac{4\alpha\Delta t}{\Delta x^{2}}\sin^{2}(k_{m}\Delta x/2)
  32. ( 10 ) 4 α Δ t Δ x 2 sin 2 ( k m Δ x / 2 ) 2 \quad(10)\qquad\frac{4\alpha\Delta t}{\Delta x^{2}}\sin^{2}(k_{m}\Delta x/2)\leq 2
  33. sin 2 ( k m Δ x / 2 ) \sin^{2}(k_{m}\Delta x/2)
  34. ( 11 ) α Δ t Δ x 2 1 2 \quad(11)\qquad\frac{\alpha\Delta t}{\Delta x^{2}}\leq\frac{1}{2}
  35. Δ x \Delta x
  36. Δ t \Delta t

Voronoi_deformation_density.html

  1. Q A = - Voronoi cell A ( ρ ( 𝐫 ) - B ρ B ( 𝐫 ) ) d 𝐫 Q_{A}=-\int\limits_{\,\text{Voronoi cell }A}\Big(\rho(\mathbf{r})-\sum_{B}\rho% _{B}(\mathbf{r})\Big)d\mathbf{r}

Vortex_lattice_method.html

  1. φ \varphi
  2. 𝐕 \mathbf{V}
  3. 𝐕 = 𝐕 + φ \mathbf{V}=\mathbf{V}_{\infty}+\nabla\varphi
  4. φ \varphi
  5. φ 1 \varphi_{1}
  6. φ 2 \varphi_{2}
  7. c 1 φ 1 + c 2 φ 2 c_{1}\varphi_{1}+c_{2}\varphi_{2}
  8. c 1 c_{1}
  9. c 2 c_{2}
  10. Γ \Gamma
  11. Γ \Gamma
  12. 𝐧 \mathbf{n}
  13. N N
  14. i i
  15. 𝐰 i j \mathbf{w}_{ij}
  16. φ i = j = 1 N 𝐰 i j Γ j \nabla\varphi_{i}=\sum_{j=1}^{N}\mathbf{w}_{ij}\Gamma_{j}
  17. V V_{\infty}
  18. α , β \alpha,\beta
  19. 𝐕 = V [ cos α cos β - sin β sin α cos β ] \mathbf{V}_{\infty}=V_{\infty}\begin{bmatrix}\cos\alpha\cos\beta\\ -\sin\beta\\ \sin\alpha\cos\beta\end{bmatrix}
  20. 𝐕 i 𝐧 i = ( 𝐕 + j = 1 N 𝐰 i j Γ j ) 𝐧 i = 0 \mathbf{V}_{i}\cdot\mathbf{n}_{i}=\left(\mathbf{V}_{\infty}+\sum_{j=1}^{N}% \mathbf{w}_{ij}\Gamma_{j}\right)\cdot\mathbf{n}_{i}=0
  21. a i j = 𝐰 i j 𝐧 i a_{ij}=\mathbf{w}_{ij}\cdot\mathbf{n}_{i}
  22. b i = V [ - cos α cos β , sin β , - sin α cos β ] 𝐧 i b_{i}=V_{\infty}[-\cos\alpha\cos\beta,\sin\beta,-\sin\alpha\cos\beta]\cdot% \mathbf{n}_{i}
  23. [ a 11 a 12 a 1 N a 21 a N 1 a N N ] [ Γ 1 Γ 2 Γ N ] = [ b 1 b 2 b N ] \begin{bmatrix}a_{11}&a_{12}&\cdots&a_{1N}\\ a_{21}&\ddots&&\vdots\\ \vdots&&\ddots&\vdots\\ a_{N1}&\cdots&\cdots&a_{NN}\end{bmatrix}\begin{bmatrix}\Gamma_{1}\\ \Gamma_{2}\\ \vdots\\ \Gamma_{N}\end{bmatrix}=\begin{bmatrix}b_{1}\\ b_{2}\\ \vdots\\ b_{N}\end{bmatrix}
  24. Γ i \Gamma_{i}
  25. 𝐅 \mathbf{F}
  26. 𝐌 \mathbf{M}
  27. 𝐅 i \mathbf{F}_{i}
  28. ρ \rho
  29. 𝐅 i = ρ Γ i ( 𝐕 + 𝐯 i ) × 𝐥 i \mathbf{F}_{i}=\rho\Gamma_{i}(\mathbf{V}_{\infty}+\mathbf{v}_{i})\times\mathbf% {l}_{i}
  30. 𝐅 = i = 1 N 𝐅 i \mathbf{F}=\sum_{i=1}^{N}\mathbf{F}_{i}
  31. 𝐌 = i = 1 N 𝐅 i × 𝐫 i \mathbf{M}=\sum_{i=1}^{N}\mathbf{F}_{i}\times\mathbf{r}_{i}
  32. 𝐥 i \mathbf{l}_{i}
  33. 𝐯 i \mathbf{v}_{i}
  34. 𝐫 i \mathbf{r}_{i}
  35. x , y , z x,y,z
  36. 𝐅 \mathbf{F}
  37. D i = F x cos α + F z sin α L = - F x sin α + F z cos α \begin{array}[]{rcl}D_{i}&=&\;\;F_{x}\cos\alpha+F_{z}\sin\alpha\\ L&=&\!-F_{x}\sin\alpha+F_{z}\cos\alpha\end{array}

VRPM.html

  1. P C I O P S ( m e a n ) = 1 / [ 0.5 / S S D I O P S ( I w r i t e ) ] + [ 0.5 / S S D I O P S ( I r e a d ) ] PCIOPS(mean)=1/[0.5/SSDIOPS(Iwrite)]+[0.5/SSDIOPS(Iread)]
  2. v R P M = 50 / [ 0.5 / S S D I O P S ( I w r i t e ) ] + [ 0.5 / S S D I O P S ( I r e a d ) ] vRPM=50/[0.5/SSDIOPS(Iwrite)]+[0.5/SSDIOPS(Iread)]

Wagner_graph.html

  1. ( x - 3 ) ( x - 1 ) 2 ( x + 1 ) ( x 2 + 2 x - 1 ) 2 (x-3)(x-1)^{2}(x+1)(x^{2}+2x-1)^{2}

Wang_and_Landau_algorithm.html

  1. Ω \Omega
  2. E Γ = [ E min , E max ] E\in\Gamma=[E_{\min},E_{\max}]
  3. ρ ( E ) exp ( S ( E ) ) \rho(E)\equiv\exp(S(E))
  4. Γ \Gamma
  5. Δ \Delta
  6. N = E max - E min Δ , N=\frac{E_{\max}-E_{\min}}{\Delta},
  7. S ( E i ) = 0 i = 1 , 2 , , N S(E_{i})=0\ \ i=1,2,...,N
  8. f = 1 f=1
  9. s y m b o l r Ω symbol{r}\in\Omega
  10. P ( s y m b o l r ) = 1 / ρ ( E ( s y m b o l r ) ) = exp ( - S ( E ( s y m b o l r ) ) ) P(symbol{r})=1/\rho(E(symbol{r}))=\exp(-S(E(symbol{r})))
  11. g ( s y m b o l r s y m b o l r ) g(symbol{r}\rightarrow symbol{r}^{\prime})
  12. H ( E ) H(E)
  13. s y m b o l r Ω symbol{r}^{\prime}\in\Omega
  14. g ( s y m b o l r s y m b o l r ) g(symbol{r}\rightarrow symbol{r}^{\prime})
  15. A ( s y m b o l r s y m b o l r ) = min ( 1 , e S - S g ( s y m b o l r s y m b o l r ) g ( s y m b o l r s y m b o l r ) ) A(symbol{r}\rightarrow symbol{r}^{\prime})=\min\left(1,e^{S-S^{\prime}}\frac{g% (symbol{r}^{\prime}\rightarrow symbol{r})}{g(symbol{r}\rightarrow symbol{r}^{% \prime})}\right)
  16. S = S ( E ( s y m b o l r ) ) S=S(E(symbol{r}))
  17. S = S ( E ( s y m b o l r ) ) S^{\prime}=S(E(symbol{r}^{\prime}))
  18. E i E_{i}
  19. H ( E i ) H(E_{i})
  20. S ( E i ) S ( E i ) + f S(E_{i})\leftarrow S(E_{i})+f
  21. E i E_{i}
  22. H ( E ) H(E)
  23. f f / 2 f\leftarrow f/2
  24. 1 / t 1/t
  25. t t
  26. E ( x ) = x 2 , E(x)=x^{2},\,
  27. g ( E ) = δ ( E ( x ) - E 0 ) d x = δ ( x 2 - E 0 ) d x , g(E)=\int\delta(E(x)-E_{0})\,dx=\int\delta(x^{2}-E_{0})\,dx,
  28. g ( E ) E - 1 / 2 , g(E)\propto E^{-1/2},
  29. ρ ( E ) \rho(E)
  30. g ( E ) g(E)
  31. g ( s y m b o l x s y m b o l x ) g ( s y m b o l x s y m b o l x ) = 1 \frac{g(symbol{x}^{\prime}\rightarrow symbol{x})}{g(symbol{x}\rightarrow symbol% {x}^{\prime})}=1

Wang_Xiaotong.html

  1. x 3 + p x 2 + q x = N x^{3}+px^{2}+qx=N

WASP-11b::HAT-P-10b.html

  1. Θ = 1 2 ( Planetary escape velocity Orbital velocity ) 2 \textstyle\Theta=\frac{1}{2}\left(\frac{\mathrm{Planetary\ escape\ velocity}}{% \mathrm{Orbital\ velocity}}\right)^{2}

Water_use.html

  1. W F p r o c , g r e y = L c m a x - c n a t WF_{proc,grey}=\frac{L}{c_{max}-c_{nat}}

Watt.html

  1. W = J s = N m s = kg m 2 s 3 \mathrm{W=\frac{J}{s}=\frac{N\cdot m}{s}=\frac{kg\cdot m^{2}}{s^{3}}}
  2. W = V A \mathrm{W=V\cdot A}
  3. W = V 2 Ω = A 2 Ω \mathrm{W=\frac{V^{2}}{\Omega}=A^{2}\cdot\Omega}
  4. Ω \Omega

Wave_height.html

  1. H = 2 a . H=2a.\,
  2. H = max { η ( x - c p t ) } - min { η ( x - c p t ) } , H=\max\left\{\eta(x\,-\,c_{p}\,t)\right\}-\min\left\{\eta(x-c_{p}\,t)\right\},\,
  3. H 1 / 3 = 1 1 3 N m = 1 1 3 N H m , H_{1/3}=\frac{1}{\frac{1}{3}\,N}\,\sum_{m=1}^{\frac{1}{3}\,N}\,H_{m},
  4. H m 0 = 4 m 0 = 4 σ η , H_{m_{0}}=4\sqrt{m_{0}}=4\sigma_{\eta},\,
  5. H rms = 1 N m = 1 N H m 2 , H\text{rms}=\sqrt{\frac{1}{N}\sum_{m=1}^{N}H_{m}^{2}},\,

Wave_shoaling.html

  1. d d s ( c g E ) = 0 , \frac{d}{ds}(c_{g}E)=0,
  2. c g E c_{g}E
  3. c g c_{g}
  4. S = S ( 𝐱 , t ) , 0 S < 2 π S=S(\mathbf{x},t),0\leq S<2\pi
  5. 𝐤 = S \mathbf{k}=\nabla S
  6. ω = - S / t \omega=-\partial S/\partial t
  7. k t + ω x = 0 \frac{\partial k}{\partial t}+\frac{\partial\omega}{\partial x}=0
  8. / t = 0 \partial/\partial t=0
  9. ω / x = 0 \partial\omega/\partial x=0
  10. λ = 2 π / k \lambda=2\pi/k
  11. ω / k c = g h \omega/k\equiv c=\sqrt{gh}
  12. k = ω / g h k=\omega/\sqrt{gh}
  13. λ \lambda
  14. ω \omega

Weak_gravitational_lensing.html

  1. κ κ = λ κ + ( 1 - λ ) \kappa\rightarrow\kappa^{\prime}=\lambda\kappa+(1-\lambda)
  2. β \vec{\beta}
  3. θ \vec{\theta}
  4. Φ \Phi
  5. β i θ j = δ i j + 0 r d r g ( r ) 2 Φ ( x ( r ) ) x i x j \frac{\partial\beta_{i}}{\partial\theta_{j}}=\delta_{ij}+\int_{0}^{r_{\infty}}% drg(r)\frac{\partial^{2}\Phi(\vec{x}(r))}{\partial x^{i}\partial x^{j}}
  6. r r
  7. x i x^{i}
  8. g ( r ) = 2 r r r ( 1 - r r ) W ( r ) g(r)=2r\int^{r_{\infty}}_{r}\left(1-\frac{r^{\prime}}{r}\right)W(r^{\prime})
  9. W ( r ) W(r)
  10. ξ + + ( Δ θ ) = γ + ( θ ) γ + ( θ + Δ θ ) \xi_{++}(\Delta\theta)=\langle\gamma_{+}(\vec{\theta})\gamma_{+}(\vec{\theta}+% \vec{\Delta\theta})\rangle
  11. ξ × × ( Δ θ ) = γ × ( θ ) γ × ( θ + Δ θ ) \xi_{\times\times}(\Delta\theta)=\langle\gamma_{\times}(\vec{\theta})\gamma_{% \times}(\vec{\theta}+\vec{\Delta\theta})\rangle
  12. ξ × + ( Δ θ ) = ξ + × ( Δ θ ) = γ + ( θ ) γ × ( θ + Δ θ ) \xi_{\times+}(\Delta\theta)=\xi_{+\times}(\Delta\theta)=\langle\gamma_{+}(\vec% {\theta})\gamma_{\times}(\vec{\theta}+\vec{\Delta\theta})\rangle
  13. γ + \gamma_{+~{}}
  14. Δ θ \vec{\Delta\theta}
  15. γ × \gamma_{\times}
  16. ξ × + \xi_{\times+}
  17. ξ + + \xi_{++~{}}
  18. ξ × × \xi_{\times\times}
  19. ξ + + \xi_{++~{}}
  20. ξ × × \xi_{\times\times}
  21. M a p 2 ( θ ) = 0 2 θ ϕ d ϕ θ 2 [ ξ + + ( ϕ ) + ξ × × ( ϕ ) ] T + ( ϕ θ ) = 0 2 θ ϕ d ϕ θ 2 [ ξ + + ( ϕ ) - ξ × × ( ϕ ) ] T - ( ϕ θ ) \langle M_{ap}^{2}\rangle(\theta)=\int_{0}^{2\theta}\frac{\phi d\phi}{\theta^{% 2}}\left[\xi_{++}(\phi)+\xi_{\times\times}(\phi)\right]T_{+}\left(\frac{\phi}{% \theta}\right)=\int_{0}^{2\theta}\frac{\phi d\phi}{\theta^{2}}\left[\xi_{++}(% \phi)-\xi_{\times\times}(\phi)\right]T_{-}\left(\frac{\phi}{\theta}\right)
  22. T + ( x ) = 576 0 d t t 3 J 0 ( x t ) [ J 4 ( t ) ] 2 T_{+}(x)=576\int^{\infty}_{0}\frac{dt}{t^{3}}J_{0}(xt)[J_{4}(t)]^{2}
  23. T - ( x ) = 576 0 d t t 3 J 4 ( x t ) [ J 4 ( t ) ] 2 T_{-}(x)=576\int^{\infty}_{0}\frac{dt}{t^{3}}J_{4}(xt)[J_{4}(t)]^{2}
  24. J 0 J_{0}~{}
  25. J 4 J_{4}~{}
  26. T + T_{+}~{}
  27. T - T_{-}~{}
  28. θ = 0 \theta=0~{}
  29. Ω m \Omega_{m}~{}
  30. σ 8 \sigma_{8}~{}

Web_Services_Modeling_Language.html

  1. 𝒮 𝒬 ( 𝒟 ) \mathcal{SHIQ}^{\mathcal{(D)}}

Weber_function.html

  1. f , f 1 , f 2 f,f_{1},f_{2}

Wedge_prism.html

  1. δ ( n - 1 ) α , \delta\approx(n-1)\alpha\ ,

Weierstrass_transform.html

  1. F ( x ) = 1 4 π - f ( y ) e - ( x - y ) 2 4 d y = 1 4 π - f ( x - y ) e - y 2 4 d y , F(x)=\frac{1}{\sqrt{4\pi}}\int_{-\infty}^{\infty}f(y)\;e^{-\frac{(x-y)^{2}}{4}% }\;dy=\frac{1}{\sqrt{4\pi}}\int_{-\infty}^{\infty}f(x-y)\;e^{-\frac{y^{2}}{4}}% \;dy,
  2. 1 4 π e - x 2 / 4 \frac{1}{\sqrt{4\pi}}e^{-x^{2}/4}
  3. 1 1 - 4 a e a x 2 1 - 4 a \frac{1}{\sqrt{1-4a}}e^{\frac{ax^{2}}{1-4a}}
  4. g ( x ) = e - x 2 4 f ( x ) g(x)=e^{-\frac{x^{2}}{4}}f(x)
  5. W [ f ] ( x ) = 1 4 π e - x 2 / 4 L [ g ] ( - x 2 ) . W[f](x)=\frac{1}{\sqrt{4\pi}}e^{-x^{2}/4}L[g]\left(-\frac{x}{2}\right).
  6. e u 2 = 1 4 π - e - u y e - y 2 / 4 d y . e^{u^{2}}=\frac{1}{\sqrt{4\pi}}\int_{-\infty}^{\infty}e^{-uy}e^{-y^{2}/4}\;dy.
  7. e - y D f ( x ) = f ( x - y ) e^{-yD}f(x)=f(x-y)
  8. e D 2 f ( x ) = 1 4 π - e - y D f ( x ) e - y 2 / 4 d y = 1 4 π - f ( x - y ) e - y 2 / 4 d y = W [ f ] ( x ) \begin{aligned}\displaystyle e^{D^{2}}f(x)&\displaystyle=\frac{1}{\sqrt{4\pi}}% \int_{-\infty}^{\infty}e^{-yD}f(x)e^{-y^{2}/4}\;dy\\ &\displaystyle=\frac{1}{\sqrt{4\pi}}\int_{-\infty}^{\infty}f(x-y)e^{-y^{2}/4}% \;dy=W[f](x)\end{aligned}
  9. W = e D 2 W=e^{D^{2}}\,
  10. e D 2 f ( x ) = k = 0 D 2 k f ( x ) k ! . e^{D^{2}}f(x)=\sum_{k=0}^{\infty}\frac{D^{2k}f(x)}{k!}.
  11. W - 1 = e - D 2 . W^{-1}=e^{-D^{2}}.\,
  12. F ( x ) = n = 0 a n x n F(x)=\sum_{n=0}^{\infty}a_{n}x^{n}
  13. f ( x ) = W - 1 [ F ( x ) ] = n = 0 a n W - 1 [ x n ] = n = 0 a n H n ( x / 2 ) f(x)=W^{-1}[F(x)]=\sum_{n=0}^{\infty}a_{n}W^{-1}[x^{n}]=\sum_{n=0}^{\infty}a_{% n}H_{n}(x/2)
  14. 1 4 π t e - x 2 4 t \frac{1}{\sqrt{4\pi t}}e^{-\frac{x^{2}}{4t}}
  15. 1 4 π e - x 2 4 \frac{1}{\sqrt{4\pi}}e^{-\frac{x^{2}}{4}}
  16. W s W t = W s + t , W_{s}\circ W_{t}=W_{s+t},
  17. 1 4 π t e - x 2 4 t \frac{1}{\sqrt{4\pi t}}e^{-\frac{x^{2}}{4t}}
  18. 1 4 π e - x 2 / 4 \frac{1}{\sqrt{4\pi}}e^{-x^{2}/4}
  19. f ( x ) = lim r 1 i 4 π x 0 - i r x 0 + i r F ( z ) e ( x - z ) 2 4 d z f(x)=\lim_{r\to\infty}\frac{1}{i\sqrt{4\pi}}\int_{x_{0}-ir}^{x_{0}+ir}F(z)e^{% \frac{(x-z)^{2}}{4}}\;dz

Weingarten_equations.html

  1. r u = r u , r v = r v {r}_{u}=\frac{\partial{r}}{\partial u},\quad{r}_{v}=\frac{\partial{r}}{% \partial v}
  2. n u = F M - G L E G - F 2 r u + F L - E M E G - F 2 r v {n}_{u}=\frac{FM-GL}{EG-F^{2}}{r}_{u}+\frac{FL-EM}{EG-F^{2}}{r}_{v}
  3. n v = F N - G M E G - F 2 r u + F M - E N E G - F 2 r v {n}_{v}=\frac{FN-GM}{EG-F^{2}}{r}_{u}+\frac{FM-EN}{EG-F^{2}}{r}_{v}

Well-behaved_statistic.html

  1. 𝔘 , 𝔛 \mathfrak{U},\mathfrak{X}
  2. s y m b o l x = { x 1 , , x m } symbolx=\{x_{1},\ldots,x_{m}\}
  3. ( g θ , Z ) (g_{\theta},Z)
  4. θ \theta
  5. s y m b o l x = { g θ ( z 1 ) , , g θ ( z m ) } . symbolx=\{g_{\theta}(z_{1}),\ldots,g_{\theta}(z_{m})\}.
  6. ( g θ , s y m b o l z ) (g_{\theta},symbolz)
  7. { x 1 , , x m } \{x_{1},\ldots,x_{m}\}
  8. 𝔖 \mathfrak{S}
  9. s = ρ ( x 1 , , x m ) = ρ ( g θ ( z 1 ) , , g θ ( z m ) ) = h ( θ , z 1 , , z m ) , ( 1 ) s=\rho(x_{1},\ldots,x_{m})=\rho(g_{\theta}(z_{1}),\ldots,g_{\theta}(z_{m}))=h(% \theta,z_{1},\ldots,z_{m}),\qquad\qquad\qquad(1)
  10. s y m b o l z = { z 1 , , z m } symbolz=\{z_{1},\ldots,z_{m}\}
  11. s y m b o l x symbolx
  12. { z 1 , , z m } \{z_{1},\ldots,z_{m}\}
  13. { x 1 , , x m } 𝔛 m \{x_{1},\ldots,x_{m}\}\in\mathfrak{X}^{m}
  14. ρ ( x 1 , , x m ) = s \rho(x_{1},\ldots,x_{m})=s
  15. 𝔛 m \mathfrak{X}^{m}
  16. 𝔖 \mathfrak{S}
  17. s s
  18. { x 1 , , x m } \{x_{1},\ldots,x_{m}\}
  19. { θ ˘ 1 , , θ ˘ N } \{\breve{\theta}_{1},\ldots,\breve{\theta}_{N}\}
  20. θ ˘ j = h - 1 ( s , z ˘ 1 j , , z ˘ m j ) \breve{\theta}_{j}=h^{-1}(s,\breve{z}_{1}^{j},\ldots,\breve{z}_{m}^{j})
  21. { z ˘ 1 j , , z ˘ m j } \{\breve{z}_{1}^{j},\ldots,\breve{z}_{m}^{j}\}
  22. { Z 1 , , Z m | S = s } \{Z_{1},\ldots,Z_{m}|S=s\}
  23. x i x_{i}
  24. { X 1 , , X m | S = s } \{X_{1},\ldots,X_{m}|S=s\}
  25. i = 1 m x i \sum_{i=1}^{m}x_{i}
  26. g p ( u ) = 1 g_{p}(u)=1
  27. u p u\leq p
  28. g λ ( u ) = - log u / λ g_{\lambda}(u)=-\log u/\lambda
  29. s p = i = 1 m I [ 0 , p ] ( u i ) s_{p}=\sum_{i=1}^{m}I_{[0,p]}(u_{i})
  30. s λ = - 1 λ i = 1 m log u i . s_{\lambda}=-\frac{1}{\lambda}\sum_{i=1}^{m}\log u_{i}.
  31. [ 0 , A ] [0,A]
  32. { c , c / 2 , c / 3 } \{c,c/2,c/3\}
  33. s A = 11 / 6 c s^{\prime}_{A}=11/6c
  34. g a ( u ) = u a g_{a}(u)=ua
  35. s A = i = 1 m u i a s_{A}=\sum_{i=1}^{m}u_{i}a
  36. { 0.8 , 0.8 , 0.8 } \{0.8,0.8,0.8\}
  37. a ˘ = 0.76 c \breve{a}=0.76c
  38. s A = max { x 1 , , x m } s_{A}=\max\{x_{1},\ldots,x_{m}\}
  39. s 1 = i = 1 m log x i s_{1}=\sum_{i=1}^{m}\log x_{i}
  40. s 2 = min i = 1 , , m { x i } s_{2}=\min_{i=1,\ldots,m}\{x_{i}\}
  41. f ( x ; n ) = 1 / n I { 1 , 2 , , n } ( x ) f(x;n)=1/nI_{\{1,2,\ldots,n\}}(x)
  42. s n = max i x i s_{n}=\max_{i}x_{i}
  43. f ( x ; p ) = p x ( 1 - p ) 1 - x I { 0 , 1 } ( x ) f(x;p)=p^{x}(1-p)^{1-x}I_{\{0,1\}}(x)
  44. s P = i = 1 m x i s_{P}=\sum_{i=1}^{m}x_{i}
  45. f ( x ; n , p ) = ( n x ) p x ( 1 - p ) n - x I 0 , 1 , , n ( x ) f(x;n,p)={\left({{n}\atop{x}}\right)}p^{x}(1-p)^{n-x}I_{0,1,\ldots,n}(x)
  46. s P = i = 1 m x i s_{P}=\sum_{i=1}^{m}x_{i}
  47. f ( x ; p ) = p ( 1 - p ) x I { 0 , 1 , } ( x ) f(x;p)=p(1-p)^{x}I_{\{0,1,\ldots\}}(x)
  48. s P = i = 1 m x i s_{P}=\sum_{i=1}^{m}x_{i}
  49. f ( x ; μ ) = e - μ x μ x / x ! I { 0 , 1 , } ( x ) f(x;\mu)=\mathrm{e}^{-\mu x}\mu^{x}/x!I_{\{0,1,\ldots\}}(x)
  50. s M = i = 1 m x i s_{M}=\sum_{i=1}^{m}x_{i}
  51. f ( x ; a , b ) = 1 / ( b - a ) I [ a , b ] ( x ) f(x;a,b)=1/(b-a)I_{[a,b]}(x)
  52. s A = min i x i ; s B = max i x i s_{A}=\min_{i}x_{i};s_{B}=\max_{i}x_{i}
  53. f ( x ; λ ) = λ e - λ x I [ 0 , ] ( x ) f(x;\lambda)=\lambda\mathrm{e}^{-\lambda x}I_{[0,\infty]}(x)
  54. s Λ = i = 1 m x i s_{\Lambda}=\sum_{i=1}^{m}x_{i}
  55. f ( x ; a , k ) = a k ( x k ) - a - 1 I [ k , ] ( x ) f(x;a,k)=\frac{a}{k}\left(\frac{x}{k}\right)^{-a-1}I_{[k,\infty]}(x)
  56. s A = i = 1 m log x i ; s K = min i x i s_{A}=\sum_{i=1}^{m}\log x_{i};s_{K}=\min_{i}x_{i}
  57. f ( x , μ , σ ) = 1 / ( 2 π σ ) e - ( x - μ 2 ) / ( 2 σ 2 ) f(x,\mu,\sigma)=1/(\sqrt{2\pi}\sigma)\mathrm{e}^{-(x-\mu^{2})/(2\sigma^{2})}
  58. s M = i = 1 m x i ; s Σ = i = 1 m ( x i - x ¯ ) 2 s_{M}=\sum_{i=1}^{m}x_{i};s_{\Sigma}=\sqrt{\sum_{i=1}^{m}(x_{i}-\bar{x})^{2}}
  59. f ( x ; r , λ ) = λ / Γ ( r ) ( λ x ) r - 1 e - λ x I [ 0 , ] ( x ) f(x;r,\lambda)=\lambda/\Gamma(r)(\lambda x)^{r-1}\mathrm{e}^{-\lambda x}I_{[0,% \infty]}(x)
  60. s Λ = i = 1 m x i ; s K = i = 1 m x i s_{\Lambda}=\sum_{i=1}^{m}x_{i};s_{K}=\prod_{i=1}^{m}x_{i}

Western_non-interpolations.html

  1. 𝔓 \mathfrak{P}

Wheelie.html

  1. Δ W e i g h t f r o n t = a h w m \Delta Weight_{front}=a\frac{h}{w}m
  2. Δ W e i g h t f r o n t \Delta Weight_{front}
  3. a a
  4. h h
  5. w w
  6. m m
  7. a m i n = g b h a_{min}=g\frac{b}{h}
  8. g g
  9. b b
  10. h h
  11. P m i n = m v a m i n P_{min}=mva_{min}

White_and_Middleton.html

  1. b.hp. i.hp. × 100 \dfrac{\mbox{b.hp.}~{}}{\mbox{i.hp.}~{}}\times 100
  2. b.hp. b.hp. + avg. f.hp. × 100 \dfrac{\mbox{b.hp.}~{}}{\mbox{b.hp.}~{}+\mbox{avg. f.hp.}~{}}\times 100

Whitney_covering_lemma.html

  1. diam ( Q j ) dist ( Q j , A ) 4 diam ( Q j ) . \mathrm{diam}(Q_{j})\leq\,\text{dist}(Q_{j},\partial A)\leq 4\,\,\text{diam}(Q% _{j}).\,

Whittaker_model.html

  1. f ( ( 1 b 0 1 ) g ) = τ ( b ) f ( g ) . f\left(\begin{pmatrix}1&b\\ 0&1\end{pmatrix}g\right)=\tau(b)f(g).
  2. G G
  3. GL n {\rm GL}_{n}
  4. ψ \psi
  5. F F
  6. U U
  7. GL n {\rm GL}_{n}
  8. U U
  9. χ ( u ) = ψ ( α 1 x 12 + α 2 x 23 + + α n - 1 x n - 1 n ) , \chi(u)=\psi(\alpha_{1}x_{12}+\alpha_{2}x_{23}+\cdots+\alpha_{n-1}x_{n-1n}),
  10. u = ( x i j ) u=(x_{ij})
  11. U U
  12. α 1 \alpha_{1}
  13. α n - 1 \alpha_{n-1}
  14. F F
  15. ( π , V ) (\pi,V)
  16. G ( F ) G(F)
  17. λ \lambda
  18. V V
  19. λ ( π ( u ) v ) = χ ( u ) λ ( v ) \lambda(\pi(u)v)=\chi(u)\lambda(v)
  20. u u
  21. U U
  22. v v
  23. V V
  24. π \pi
  25. G L n GL_{n}

Wien_filter.html

  1. F = q E \vec{F}=q\vec{E}
  2. F = q v × B \vec{F}=q\vec{v}\times\vec{B}
  3. × \times
  4. F = q v B F=qvB
  5. E B = v \frac{E}{B}=v
  6. E \vec{E}
  7. B \vec{B}
  8. v \vec{v}

Wiener_sausage.html

  1. W δ ( t ) ( b ) W_{\delta}(t)({b})
  2. E ( | W δ ( t ) | ) = 2 π δ t + 4 δ 2 2 π t + 4 π δ 3 / 3. E(|W_{\delta}(t)|)=2\pi\delta t+4\delta^{2}\sqrt{2\pi t}+4\pi\delta^{3}/3.
  3. δ d - 2 π d / 2 2 t / Γ ( ( d - 2 ) / 2 ) \delta^{d-2}\pi^{d/2}2t/\Gamma((d-2)/2)
  4. 8 t / π \sqrt{8t/\pi}
  5. 2 π t / log ( t ) 2{\pi}t/\log(t)

Wilf–Zeilberger_pair.html

  1. F ( n + 1 , k ) - F ( n , k ) = G ( n , k + 1 ) - G ( n , k ) F(n+1,k)-F(n,k)=G(n,k+1)-G(n,k)\,
  2. lim M ± G ( n , M ) = 0. \lim_{M\to\pm\infty}G(n,M)=0.\,
  3. k = - [ F ( n + 1 , k ) - F ( n , k ) ] = 0 \sum_{k=-\infty}^{\infty}[F(n+1,k)-F(n,k)]=0
  4. k = - [ F ( n + 1 , k ) - F ( n , k ) ] \displaystyle\sum_{k=-\infty}^{\infty}[F(n+1,k)-F(n,k)]
  5. G ( n , k ) = R ( n , k ) F ( n , k ) , G(n,k)=R(n,k)F(n,k),
  6. R ( n , k ) R(n,k)
  7. k = 0 ( - 1 ) k ( n k ) ( 2 k k ) 4 n - k = ( 2 n n ) \sum_{k=0}^{\infty}(-1)^{k}{n\choose k}{2k\choose k}4^{n-k}={2n\choose n}
  8. R ( n , k ) = 2 k - 1 2 n + 1 . R(n,k)=\frac{2k-1}{2n+1}.
  9. F ( n , k ) \displaystyle F(n,k)

Willam-Warnke_yield_criterion.html

  1. f ( I 1 , J 2 , J 3 ) = 0 f(I_{1},J_{2},J_{3})=0\,
  2. I 1 I_{1}
  3. J 2 , J 3 J_{2},J_{3}
  4. σ c \sigma_{c}
  5. σ t \sigma_{t}
  6. σ b \sigma_{b}
  7. I 1 , J 2 , J 3 I_{1},J_{2},J_{3}
  8. f := J 2 + λ ( J 2 , J 3 ) ( I 1 3 - B ) = 0 f:=\sqrt{J_{2}}+\lambda(J_{2},J_{3})~{}(\tfrac{I_{1}}{3}-B)=0
  9. λ \lambda
  10. J 2 , J 3 J_{2},J_{3}
  11. B B
  12. λ \lambda
  13. θ \theta
  14. B B
  15. f := 1 3 z I 1 σ c + 2 5 1 r ( θ ) J 2 σ c - 1 0 f:=\cfrac{1}{3z}~{}\cfrac{I_{1}}{\sigma_{c}}+\sqrt{\cfrac{2}{5}}~{}\cfrac{1}{r% (\theta)}\cfrac{\sqrt{J_{2}}}{\sigma_{c}}-1\leq 0
  16. I 1 I_{1}
  17. J 2 J_{2}
  18. σ c \sigma_{c}
  19. θ \theta
  20. θ = 1 3 cos - 1 ( 3 3 2 J 3 J 2 3 / 2 ) . \theta=\tfrac{1}{3}\cos^{-1}\left(\cfrac{3\sqrt{3}}{2}~{}\cfrac{J_{3}}{J_{2}^{% 3/2}}\right)~{}.
  21. r ( θ ) r(\theta)
  22. r ( θ ) := u ( θ ) + v ( θ ) w ( θ ) r(\theta):=\cfrac{u(\theta)+v(\theta)}{w(\theta)}
  23. u ( θ ) := 2 r c ( r c 2 - r t 2 ) cos θ v ( θ ) := r c ( 2 r t - r c ) 4 ( r c 2 - r t 2 ) cos 2 θ + 5 r t 2 - 4 r t r c w ( θ ) := 4 ( r c 2 - r t 2 ) cos 2 θ + ( r c - 2 r t ) 2 \begin{aligned}\displaystyle u(\theta):=&\displaystyle 2~{}r_{c}~{}(r_{c}^{2}-% r_{t}^{2})~{}\cos\theta\\ \displaystyle v(\theta):=&\displaystyle r_{c}~{}(2~{}r_{t}-r_{c})\sqrt{4~{}(r_% {c}^{2}-r_{t}^{2})~{}\cos^{2}\theta+5~{}r_{t}^{2}-4~{}r_{t}~{}r_{c}}\\ \displaystyle w(\theta):=&\displaystyle 4(r_{c}^{2}-r_{t}^{2})\cos^{2}\theta+(% r_{c}-2~{}r_{t})^{2}\end{aligned}
  24. r t r_{t}
  25. r c r_{c}
  26. θ = 0 , 60 \theta=0^{\circ},60^{\circ}
  27. σ c , σ b , σ t \sigma_{c},\sigma_{b},\sigma_{t}
  28. r c := 6 5 [ σ b σ t 3 σ b σ t + σ c ( σ b - σ t ) ] ; r t := 6 5 [ σ b σ t σ c ( 2 σ b + σ t ) ] r_{c}:=\sqrt{\cfrac{6}{5}}\left[\cfrac{\sigma_{b}\sigma_{t}}{3\sigma_{b}\sigma% _{t}+\sigma_{c}(\sigma_{b}-\sigma_{t})}\right]~{};~{}~{}r_{t}:=\sqrt{\cfrac{6}% {5}}\left[\cfrac{\sigma_{b}\sigma_{t}}{\sigma_{c}(2\sigma_{b}+\sigma_{t})}\right]
  29. z z
  30. z := σ b σ t σ c ( σ b - σ t ) . z:=\cfrac{\sigma_{b}\sigma_{t}}{\sigma_{c}(\sigma_{b}-\sigma_{t})}~{}.
  31. f ( ξ , ρ , θ ) = 0 f := λ ¯ ( θ ) ρ + B ¯ ξ - σ c 0 f(\xi,\rho,\theta)=0\,\quad\equiv\quad f:=\bar{\lambda}(\theta)~{}\rho+\bar{B}% ~{}\xi-\sigma_{c}\leq 0
  32. B ¯ := 1 3 z ; λ ¯ := 1 5 r ( θ ) . \bar{B}:=\cfrac{1}{\sqrt{3}~{}z}~{};~{}~{}\bar{\lambda}:=\cfrac{1}{\sqrt{5}~{}% r(\theta)}~{}.
  33. f ( ξ , ρ , θ ) = 0 or f := ρ + λ ¯ ( θ ) ( ξ - B ¯ ) = 0 f(\xi,\rho,\theta)=0\,\quad\,\text{or}\quad f:=\rho+\bar{\lambda}(\theta)~{}% \left(\xi-\bar{B}\right)=0
  34. λ ¯ := 2 3 u ( θ ) + v ( θ ) w ( θ ) ; B ¯ := 1 3 [ σ b σ t σ b - σ t ] \bar{\lambda}:=\sqrt{\tfrac{2}{3}}~{}\cfrac{u(\theta)+v(\theta)}{w(\theta)}~{}% ;~{}~{}\bar{B}:=\tfrac{1}{\sqrt{3}}~{}\left[\cfrac{\sigma_{b}\sigma_{t}}{% \sigma_{b}-\sigma_{t}}\right]
  35. r t := 3 ( σ b - σ t ) 2 σ b - σ t r c := 3 σ c ( σ b - σ t ) ( σ c + σ t ) σ b - σ c σ t \begin{aligned}\displaystyle r_{t}:=&\displaystyle\cfrac{\sqrt{3}~{}(\sigma_{b% }-\sigma_{t})}{2\sigma_{b}-\sigma_{t}}\\ \displaystyle r_{c}:=&\displaystyle\cfrac{\sqrt{3}~{}\sigma_{c}~{}(\sigma_{b}-% \sigma_{t})}{(\sigma_{c}+\sigma_{t})\sigma_{b}-\sigma_{c}\sigma_{t}}\end{aligned}
  36. r c , r t r_{c},r_{t}
  37. 2 r t r c r t / 2 2~{}r_{t}\geq r_{c}\geq r_{t}/2
  38. 0 θ π 3 0\leq\theta\leq\cfrac{\pi}{3}
  39. σ c = 1 , σ t = 0.3 , σ b = 1.7 \sigma_{c}=1,\sigma_{t}=0.3,\sigma_{b}=1.7
  40. σ 1 - σ 2 \sigma_{1}-\sigma_{2}
  41. σ c = 1 , σ t = 0.3 , σ b = 1.7 \sigma_{c}=1,\sigma_{t}=0.3,\sigma_{b}=1.7

Wilson_polynomials.html

  1. p n ( t 2 ) = ( a + b ) n ( a + c ) n ( a + d ) n F 3 4 ( - n a + b + c + d + n - 1 a - t a + t a + b a + c a + d ; 1 ) . p_{n}(t^{2})=(a+b)_{n}(a+c)_{n}(a+d)_{n}{}_{4}F_{3}\left(\begin{matrix}-n&a+b+% c+d+n-1&a-t&a+t\\ a+b&a+c&a+d\end{matrix};1\right).

Wind_turbine_aerodynamics.html

  1. P = F v P=\vec{F}\cdot\vec{v}
  2. C P = P 1 2 ρ A V 3 C_{P}=\frac{P}{\frac{1}{2}\rho AV^{3}}
  3. C P C_{P}
  4. ρ \rho
  5. C T = T 1 2 ρ A V 2 C_{T}=\frac{T}{\frac{1}{2}\rho AV^{2}}
  6. λ = U V \lambda=\frac{U}{V}
  7. C L = L 1 2 ρ A W 2 C_{L}=\frac{L}{\frac{1}{2}\rho AW^{2}}
  8. C D = D 1 2 ρ A W 2 C_{D}=\frac{D}{\frac{1}{2}\rho AW^{2}}
  9. C L C_{L}
  10. C D C_{D}
  11. W W
  12. W = V - U \vec{W}=\vec{V}-\vec{U}
  13. P = 1 2 ρ A C D ( U V 2 - 2 V U 2 + U 3 ) P=\frac{1}{2}\rho AC_{D}\left(UV^{2}-2VU^{2}+U^{3}\right)
  14. λ = 1 / 3 \lambda=1/3
  15. λ > 1 \lambda>1
  16. C P = 4 27 C D C_{P}=\frac{4}{27}C_{D}
  17. C D C_{D}
  18. C P C_{P}
  19. P = 1 2 ρ A U 2 + V 2 ( C L U V - C D U 2 ) P=\frac{1}{2}\rho A\sqrt{U^{2}+V^{2}}\left(C_{L}UV-C_{D}U^{2}\right)
  20. γ = C D / C L \gamma=C_{D}/C_{L}
  21. C P = C L 1 + λ 2 ( λ - γ λ 2 ) C_{P}=C_{L}\sqrt{1+\lambda^{2}}\left(\lambda-\gamma\lambda^{2}\right)
  22. γ \gamma
  23. C P C_{P}
  24. γ \gamma
  25. γ \gamma
  26. λ \lambda
  27. C P C_{P}
  28. C L C_{L}
  29. C L C_{L}
  30. C L C_{L}
  31. C L C_{L}
  32. C L C_{L}
  33. C L C_{L}
  34. C L C_{L}
  35. C L C_{L}
  36. C L C_{L}
  37. γ \gamma
  38. C P C_{P}
  39. C P C_{P}
  40. C P C_{P}
  41. C L C_{L}
  42. C D C_{D}
  43. C P C_{P}
  44. C P C_{P}
  45. a U 1 - U 2 U 1 a\equiv\frac{U_{1}-U_{2}}{U_{1}}
  46. U 2 \displaystyle U_{2}
  47. C p C_{p}
  48. P \displaystyle P
  49. C p C_{p}
  50. C p = 4 a ( 1 - a ) 2 C_{p}=4a(1-a)^{2}
  51. σ \sigma
  52. ϕ \phi
  53. C n C_{n}
  54. C t C_{t}
  55. a \displaystyle a
  56. F \displaystyle F
  57. a c = 0.2 a_{c}=0.2
  58. C T = 4 [ a c 2 + ( 1 - 2 a c ) a ] C_{T}=4\left[a_{c}^{2}+(1-2a_{c})a\right]
  59. a > a c a>a_{c}
  60. C T C_{T}
  61. C t C_{t}
  62. a a
  63. c t c_{t}

World_population.html

  1. N = C τ \arccot T 0 - T τ N=\frac{C}{\tau}\arccot\frac{T_{0}-T}{\tau}
  2. τ \tau

Wrapped_normal_distribution.html

  1. f W N ( θ ; μ , σ ) = 1 σ 2 π k = - exp [ - ( θ - μ + 2 π k ) 2 2 σ 2 ] f_{WN}(\theta;\mu,\sigma)=\frac{1}{\sigma\sqrt{2\pi}}\sum^{\infty}_{k=-\infty}% \exp\left[\frac{-(\theta-\mu+2\pi k)^{2}}{2\sigma^{2}}\right]
  2. f W N ( θ ; μ , σ ) = 1 2 π n = - e - σ 2 n 2 / 2 + i n ( θ - μ ) = 1 2 π ϑ ( θ - μ 2 π , i σ 2 2 π ) , f_{WN}(\theta;\mu,\sigma)=\frac{1}{2\pi}\sum_{n=-\infty}^{\infty}e^{-\sigma^{2% }n^{2}/2+in(\theta-\mu)}=\frac{1}{2\pi}\vartheta\left(\frac{\theta-\mu}{2\pi},% \frac{i\sigma^{2}}{2\pi}\right),
  3. ϑ ( θ , τ ) \vartheta(\theta,\tau)
  4. ϑ ( θ , τ ) = n = - ( w 2 ) n q n 2 where w e i π θ \vartheta(\theta,\tau)=\sum_{n=-\infty}^{\infty}(w^{2})^{n}q^{n^{2}}\,\text{ % where }w\equiv e^{i\pi\theta}
  5. q e i π τ . q\equiv e^{i\pi\tau}.
  6. f W N ( θ ; μ , σ ) = 1 2 π n = 1 ( 1 - q n ) ( 1 + q n - 1 / 2 z ) ( 1 + q n - 1 / 2 / z ) . f_{WN}(\theta;\mu,\sigma)=\frac{1}{2\pi}\prod_{n=1}^{\infty}(1-q^{n})(1+q^{n-1% /2}z)(1+q^{n-1/2}/z).
  7. z = e i ( θ - μ ) z=e^{i(\theta-\mu)}\,
  8. q = e - σ 2 . q=e^{-\sigma^{2}}.
  9. z = e i θ z=e^{i\theta}
  10. z n = Γ e i n θ f W N ( θ ; μ , σ ) d θ = e i n μ - n 2 σ 2 / 2 . \langle z^{n}\rangle=\int_{\Gamma}e^{in\theta}\,f_{WN}(\theta;\mu,\sigma)\,d% \theta=e^{in\mu-n^{2}\sigma^{2}/2}.
  11. Γ \Gamma\,
  12. 2 π 2\pi
  13. z = e i μ - σ 2 / 2 \langle z\rangle=e^{i\mu-\sigma^{2}/2}
  14. θ μ = Arg z = μ \theta_{\mu}=\mathrm{Arg}\langle z\rangle=\mu
  15. R = | z | = e - σ 2 / 2 R=|\langle z\rangle|=e^{-\sigma^{2}/2}
  16. s = ln ( 1 / R 2 ) = σ s=\sqrt{\ln(1/R^{2})}=\sigma
  17. z ¯ = 1 N n = 1 N z n \overline{z}=\frac{1}{N}\sum_{n=1}^{N}z_{n}
  18. z ¯ = e i μ - σ 2 / 2 . \langle\overline{z}\rangle=e^{i\mu-\sigma^{2}/2}.\,
  19. R ¯ 2 = z ¯ z * ¯ = ( 1 N n = 1 N cos θ n ) 2 + ( 1 N n = 1 N sin θ n ) 2 \overline{R}^{2}=\overline{z}\,\overline{z^{*}}=\left(\frac{1}{N}\sum_{n=1}^{N% }\cos\theta_{n}\right)^{2}+\left(\frac{1}{N}\sum_{n=1}^{N}\sin\theta_{n}\right% )^{2}\,
  20. R ¯ 2 = 1 N + N - 1 N e - σ 2 \left\langle\overline{R}^{2}\right\rangle=\frac{1}{N}+\frac{N-1}{N}\,e^{-% \sigma^{2}}\,
  21. R e 2 = N N - 1 ( R ¯ 2 - 1 N ) R_{e}^{2}=\frac{N}{N-1}\left(\overline{R}^{2}-\frac{1}{N}\right)
  22. H = - Γ f W N ( θ ; μ , σ ) ln ( f W N ( θ ; μ , σ ) ) d θ H=-\int_{\Gamma}f_{WN}(\theta;\mu,\sigma)\,\ln(f_{WN}(\theta;\mu,\sigma))\,d\theta
  23. Γ \Gamma
  24. 2 π 2\pi
  25. z = e i ( θ - μ ) z=e^{i(\theta-\mu)}
  26. q = e - σ 2 q=e^{-\sigma^{2}}
  27. f W N ( θ ; μ , σ ) = ϕ ( q ) 2 π m = 1 ( 1 + q m - 1 / 2 z ) ( 1 + q m - 1 / 2 z - 1 ) f_{WN}(\theta;\mu,\sigma)=\frac{\phi(q)}{2\pi}\prod_{m=1}^{\infty}(1+q^{m-1/2}% z)(1+q^{m-1/2}z^{-1})
  28. ϕ ( q ) \phi(q)\,
  29. ln ( f W N ( θ ; μ , σ ) ) = ln ( ϕ ( q ) 2 π ) + m = 1 ln ( 1 + q m - 1 / 2 z ) + m = 1 ln ( 1 + q m - 1 / 2 z - 1 ) \ln(f_{WN}(\theta;\mu,\sigma))=\ln\left(\frac{\phi(q)}{2\pi}\right)+\sum_{m=1}% ^{\infty}\ln(1+q^{m-1/2}z)+\sum_{m=1}^{\infty}\ln(1+q^{m-1/2}z^{-1})
  30. ln ( 1 + x ) = - k = 1 ( - 1 ) k k x k \ln(1+x)=-\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}\,x^{k}
  31. m = 1 ln ( 1 + q m - 1 / 2 z ± 1 ) = - m = 1 k = 1 ( - 1 ) k k q m k - k / 2 z ± k = - k = 1 ( - 1 ) k k q k / 2 1 - q k z ± k \sum_{m=1}^{\infty}\ln(1+q^{m-1/2}z^{\pm 1})=-\sum_{m=1}^{\infty}\sum_{k=1}^{% \infty}\frac{(-1)^{k}}{k}\,q^{mk-k/2}z^{\pm k}=-\sum_{k=1}^{\infty}\frac{(-1)^% {k}}{k}\,\frac{q^{k/2}}{1-q^{k}}\,z^{\pm k}
  32. ln ( f W N ( θ ; μ , σ ) ) = ln ( ϕ ( q ) 2 π ) - k = 1 ( - 1 ) k k q k / 2 1 - q k ( z k + z - k ) \ln(f_{WN}(\theta;\mu,\sigma))=\ln\left(\frac{\phi(q)}{2\pi}\right)-\sum_{k=1}% ^{\infty}\frac{(-1)^{k}}{k}\frac{q^{k/2}}{1-q^{k}}\,(z^{k}+z^{-k})
  33. θ \theta\,
  34. f W N ( θ ; μ , σ ) = 1 2 π n = - q n 2 / 2 z n f_{WN}(\theta;\mu,\sigma)=\frac{1}{2\pi}\sum_{n=-\infty}^{\infty}q^{n^{2}/2}\,% z^{n}
  35. H = - ln ( ϕ ( q ) 2 π ) + 1 2 π Γ ( n = - k = 1 ( - 1 ) k k q ( n 2 + k ) / 2 1 - q k ( z n + k + z n - k ) ) d θ H=-\ln\left(\frac{\phi(q)}{2\pi}\right)+\frac{1}{2\pi}\int_{\Gamma}\left(\sum_% {n=-\infty}^{\infty}\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}\frac{q^{(n^{2}+k)/2}% }{1-q^{k}}\left(z^{n+k}+z^{n-k}\right)\right)\,d\theta
  36. H = - ln ( ϕ ( q ) 2 π ) + 2 k = 1 ( - 1 ) k k q ( k 2 + k ) / 2 1 - q k H=-\ln\left(\frac{\phi(q)}{2\pi}\right)+2\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}% \,\frac{q^{(k^{2}+k)/2}}{1-q^{k}}

Wright_Omega_function.html

  1. ω ( z ) = W Im ( z ) - π 2 π ( e z ) . \omega(z)=W_{\big\lceil\frac{\mathrm{Im}(z)-\pi}{2\pi}\big\rceil}(e^{z}).
  2. z x ± i π z\neq x\pm i\pi
  3. W k ( z ) = ω ( ln ( z ) + 2 π i k ) W_{k}(z)=\omega(\ln(z)+2\pi ik)
  4. d ω d z = ω 1 + ω \frac{d\omega}{dz}=\frac{\omega}{1+\omega}
  5. ln ( ω ) + ω = z \ln(\omega)+\omega=z
  6. w n d z = { ω n + 1 - 1 n + 1 + ω n n if n - 1 , ln ( ω ) - 1 ω if n = - 1. \int w^{n}\,dz=\begin{cases}\frac{\omega^{n+1}-1}{n+1}+\frac{\omega^{n}}{n}&% \mbox{if }~{}n\neq-1,\\ \ln(\omega)-\frac{1}{\omega}&\mbox{if }~{}n=-1.\end{cases}
  7. a = ω a + ln ( ω a ) a=\omega_{a}+\ln(\omega_{a})
  8. ω ( z ) = n = 0 + q n ( ω a ) ( 1 + ω a ) 2 n - 1 ( z - a ) n n ! \omega(z)=\sum_{n=0}^{+\infty}\frac{q_{n}(\omega_{a})}{(1+\omega_{a})^{2n-1}}% \frac{(z-a)^{n}}{n!}
  9. q n ( w ) = k = 0 n - 1 n + 1 k ( - 1 ) k w k + 1 q_{n}(w)=\sum_{k=0}^{n-1}\bigg\langle\!\!\bigg\langle\begin{matrix}n+1\\ k\end{matrix}\bigg\rangle\!\!\bigg\rangle(-1)^{k}w^{k+1}
  10. n k \bigg\langle\!\!\bigg\langle\begin{matrix}n\\ k\end{matrix}\bigg\rangle\!\!\bigg\rangle
  11. ω ( 0 ) = W 0 ( 1 ) 0.56714 ω ( 1 ) = 1 ω ( - 1 ± i π ) = - 1 ω ( - 1 3 + ln ( 1 3 ) + i π ) = - 1 3 ω ( - 1 3 + ln ( 1 3 ) - i π ) = W - 1 ( - 1 3 e - 1 3 ) - 2.237147028 \begin{array}[]{lll}\omega(0)&=W_{0}(1)&\approx 0.56714\\ \omega(1)&=1&\\ \omega(-1\pm i\pi)&=-1&\\ \omega(-\frac{1}{3}+\ln\left(\frac{1}{3}\right)+i\pi)&=-\frac{1}{3}&\\ \omega(-\frac{1}{3}+\ln\left(\frac{1}{3}\right)-i\pi)&=W_{-1}\left(-\frac{1}{3% }e^{-\frac{1}{3}}\right)&\approx-2.237147028\\ \end{array}

Wulff_construction.html

  1. Δ G i = j γ j O j \Delta G_{i}=\sum_{j}\gamma_{j}O_{j}~{}
  2. γ j \gamma_{j}
  3. j j
  4. O j O_{j}
  5. Δ G i \Delta G_{i}
  6. i i
  7. i i
  8. Δ G i \Delta G_{i}
  9. h j h_{j}
  10. γ j \gamma_{j}
  11. h j = λ γ j h_{j}=\lambda\gamma_{j}
  12. h j h_{j}
  13. j j
  14. γ ( n ^ ) \gamma(\hat{n})
  15. n ^ \hat{n}
  16. n ^ \hat{n}
  17. Δ G i = j γ j O j \Delta G_{i}=\sum_{j}\gamma_{j}O_{j}\,\!
  18. δ j γ j O j = j γ j δ O j = 0 \delta\sum_{j}\gamma_{j}O_{j}=\sum_{j}\gamma_{j}\delta O_{j}=0\,\!
  19. δ V c = 1 3 δ j h j O j = 0 \delta V_{c}=\frac{1}{3}\delta\sum_{j}h_{j}O_{j}=0
  20. j h j δ O j + j O j δ h j = 0 \sum_{j}h_{j}\delta O_{j}+\sum_{j}O_{j}\delta h_{j}=0\,\!
  21. j h j δ O j = 0 \sum_{j}h_{j}\delta O_{j}=0\,\!
  22. j γ j δ O j = 0 \sum_{j}\gamma_{j}\delta O_{j}=0\,\!
  23. j ( h i - λ γ j ) δ O j = 0 \sum_{j}(h_{i}-\lambda\gamma_{j})\delta O_{j}=0\,\!
  24. ( δ O j ) (\delta O_{j})
  25. h j = λ γ j h_{j}=\lambda\gamma_{j}

X-ray_Raman_scattering.html

  1. d 2 σ d Ω d E = ( d σ d Ω ) Th × S ( q , E ) {d^{2}\sigma\over d\Omega dE}=({d\sigma\over d\Omega})_{\rm Th}\times S(q,E)
  2. ( d σ / d Ω ) Th (d\sigma/d\Omega)_{\rm Th}
  3. S ( q , E ) S(q,E)
  4. q q
  5. E E
  6. q q

X-ray_transform.html

  1. X f ( L ) = L f = 𝐑 f ( x 0 + t θ ) d t Xf(L)=\int_{L}f=\int_{\mathbf{R}}f(x_{0}+t\theta)dt

Xbar_and_s_chart.html

  1. B 4 S ¯ B_{4}\bar{S}
  2. B 3 S ¯ B_{3}\bar{S}
  3. s ¯ i = j = 1 n ( x i j - x ¯ i ) 2 n - 1 \bar{s}_{i}=\sqrt{\frac{\sum_{j=1}^{n}\left(x_{ij}-\bar{x}_{i}\right)^{2}}{n-1}}
  4. x ¯ = i = 1 m j = 1 n x i j m n \bar{x}=\frac{\sum_{i=1}^{m}\sum_{j=1}^{n}x_{ij}}{mn}
  5. x ¯ ± A 3 s ¯ \bar{x}\pm A_{3}\bar{s}
  6. x ¯ i = j = 1 n x i j n \bar{x}_{i}=\frac{\sum_{j=1}^{n}x_{ij}}{n}
  7. x ¯ \bar{x}
  8. x ¯ \bar{x}
  9. x ¯ \bar{x}
  10. x ¯ \bar{x}
  11. x ¯ i \bar{x}_{i}
  12. s = i = 1 n ( x i - x ¯ ) 2 n - 1 s=\sqrt{\frac{\sum_{i=1}^{n}{\left(x_{i}-\bar{x}\right)}^{2}}{n-1}}
  13. B 3 s ¯ B_{3}\bar{s}
  14. B 4 s ¯ B_{4}\bar{s}
  15. x ¯ ± A 3 s ¯ \bar{x}\pm A_{3}\bar{s}
  16. x ¯ \bar{x}
  17. s ¯ = i = 1 m s i m \bar{s}=\frac{\sum_{i=1}^{m}s_{i}}{m}
  18. x ¯ \bar{x}
  19. x ¯ \bar{x}
  20. x ¯ \bar{x}
  21. x ¯ \bar{x}
  22. x ¯ \bar{x}
  23. x ¯ \bar{x}

Yarowsky_algorithm.html

  1. log ( Pr ( Sense A | Collocation i ) Pr ( Sense B | Collocation i ) ) \log\left(\frac{\Pr(\,\text{Sense}_{A}|\,\text{Collocation}_{i})}{\Pr(\,\text{% Sense}_{B}|\,\text{Collocation}_{i})}\right)

Yeoh_(hyperelastic_model).html

  1. I 1 , I 2 , I 3 I_{1},I_{2},I_{3}
  2. I 1 I_{1}
  3. I 3 I_{3}
  4. I 1 I_{1}
  5. W = i = 1 3 C i ( I 1 - 3 ) i W=\sum_{i=1}^{3}C_{i}~{}(I_{1}-3)^{i}
  6. C i C_{i}
  7. 2 C 1 2C_{1}
  8. n n
  9. W = i = 1 n C i ( I 1 - 3 ) i . W=\sum_{i=1}^{n}C_{i}~{}(I_{1}-3)^{i}~{}.
  10. n = 1 n=1
  11. 2 W I 1 ( 3 ) = μ ( i j ) 2\cfrac{\partial W}{\partial I_{1}}(3)=\mu~{}~{}(i\neq j)
  12. μ \mu
  13. I 1 = 3 ( λ i = λ j = 1 ) I_{1}=3(\lambda_{i}=\lambda_{j}=1)
  14. W I 1 = C 1 \cfrac{\partial W}{\partial I_{1}}=C_{1}
  15. 2 C 1 = μ 2C_{1}=\mu\,
  16. s y m b o l σ = - p s y m b o l 1 + 2 W I 1 s y m b o l B ; W I 1 = i = 1 n i C i ( I 1 - 3 ) i - 1 . symbol{\sigma}=-p~{}symbol{\mathit{1}}+2~{}\cfrac{\partial W}{\partial I_{1}}~% {}symbol{B}~{};~{}~{}\cfrac{\partial W}{\partial I_{1}}=\sum_{i=1}^{n}i~{}C_{i% }~{}(I_{1}-3)^{i-1}~{}.
  17. 𝐧 1 \mathbf{n}_{1}
  18. λ 1 = λ , λ 2 = λ 3 \lambda_{1}=\lambda,~{}\lambda_{2}=\lambda_{3}
  19. λ 1 λ 2 λ 3 = 1 \lambda_{1}~{}\lambda_{2}~{}\lambda_{3}=1
  20. λ 2 2 = λ 3 2 = 1 / λ \lambda_{2}^{2}=\lambda_{3}^{2}=1/\lambda
  21. I 1 = λ 1 2 + λ 2 2 + λ 3 2 = λ 2 + 2 λ . I_{1}=\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}=\lambda^{2}+\cfrac{2}{% \lambda}~{}.
  22. s y m b o l B = λ 2 𝐧 1 𝐧 1 + 1 λ ( 𝐧 2 𝐧 2 + 𝐧 3 𝐧 3 ) . symbol{B}=\lambda^{2}~{}\mathbf{n}_{1}\otimes\mathbf{n}_{1}+\cfrac{1}{\lambda}% ~{}(\mathbf{n}_{2}\otimes\mathbf{n}_{2}+\mathbf{n}_{3}\otimes\mathbf{n}_{3})~{}.
  23. σ 11 = - p + 2 λ 2 W I 1 ; σ 22 = - p + 2 λ W I 1 = σ 33 . \sigma_{11}=-p+2~{}\lambda^{2}~{}\cfrac{\partial W}{\partial I_{1}}~{};~{}~{}% \sigma_{22}=-p+\cfrac{2}{\lambda}~{}\cfrac{\partial W}{\partial I_{1}}=\sigma_% {33}~{}.
  24. σ 22 = σ 33 = 0 \sigma_{22}=\sigma_{33}=0
  25. p = 2 λ W I 1 . p=\cfrac{2}{\lambda}~{}\cfrac{\partial W}{\partial I_{1}}~{}.
  26. σ 11 = 2 ( λ 2 - 1 λ ) W I 1 . \sigma_{11}=2~{}\left(\lambda^{2}-\cfrac{1}{\lambda}\right)~{}\cfrac{\partial W% }{\partial I_{1}}~{}.
  27. λ - 1 \lambda-1\,
  28. T 11 = σ 11 / λ = 2 ( λ - 1 λ 2 ) W I 1 . T_{11}=\sigma_{11}/\lambda=2~{}\left(\lambda-\cfrac{1}{\lambda^{2}}\right)~{}% \cfrac{\partial W}{\partial I_{1}}~{}.
  29. 𝐧 1 \mathbf{n}_{1}
  30. 𝐧 2 \mathbf{n}_{2}
  31. λ 1 = λ 2 = λ \lambda_{1}=\lambda_{2}=\lambda\,
  32. λ 1 λ 2 λ 3 = 1 \lambda_{1}~{}\lambda_{2}~{}\lambda_{3}=1
  33. λ 3 = 1 / λ 2 \lambda_{3}=1/\lambda^{2}\,
  34. I 1 = λ 1 2 + λ 2 2 + λ 3 2 = 2 λ 2 + 1 λ 4 . I_{1}=\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}=2~{}\lambda^{2}+\cfrac{1% }{\lambda^{4}}~{}.
  35. s y m b o l B = λ 2 𝐧 1 𝐧 1 + λ 2 𝐧 2 𝐧 2 + 1 λ 4 𝐧 3 𝐧 3 . symbol{B}=\lambda^{2}~{}\mathbf{n}_{1}\otimes\mathbf{n}_{1}+\lambda^{2}~{}% \mathbf{n}_{2}\otimes\mathbf{n}_{2}+\cfrac{1}{\lambda^{4}}~{}\mathbf{n}_{3}% \otimes\mathbf{n}_{3}~{}.
  36. σ 11 = - p + 2 λ 2 W I 1 = σ 22 ; σ 33 = - p + 2 λ 4 W I 1 . \sigma_{11}=-p+2~{}\lambda^{2}~{}\cfrac{\partial W}{\partial I_{1}}=\sigma_{22% }~{};~{}~{}\sigma_{33}=-p+\cfrac{2}{\lambda^{4}}~{}\cfrac{\partial W}{\partial I% _{1}}~{}.
  37. σ 33 = 0 \sigma_{33}=0
  38. p = 2 λ 4 W I 1 . p=\cfrac{2}{\lambda^{4}}~{}\cfrac{\partial W}{\partial I_{1}}~{}.
  39. σ 11 = 2 ( λ 2 - 1 λ 4 ) W I 1 = σ 22 . \sigma_{11}=2~{}\left(\lambda^{2}-\cfrac{1}{\lambda^{4}}\right)~{}\cfrac{% \partial W}{\partial I_{1}}=\sigma_{22}~{}.
  40. λ - 1 \lambda-1\,
  41. T 11 = σ 11 λ = 2 ( λ - 1 λ 5 ) W I 1 = T 22 . T_{11}=\cfrac{\sigma_{11}}{\lambda}=2~{}\left(\lambda-\cfrac{1}{\lambda^{5}}% \right)~{}\cfrac{\partial W}{\partial I_{1}}=T_{22}~{}.
  42. 𝐧 1 \mathbf{n}_{1}
  43. 𝐧 3 \mathbf{n}_{3}
  44. λ 1 = λ , λ 3 = 1 \lambda_{1}=\lambda,~{}\lambda_{3}=1
  45. λ 1 λ 2 λ 3 = 1 \lambda_{1}~{}\lambda_{2}~{}\lambda_{3}=1
  46. λ 2 = 1 / λ \lambda_{2}=1/\lambda\,
  47. I 1 = λ 1 2 + λ 2 2 + λ 3 2 = λ 2 + 1 λ 2 + 1 . I_{1}=\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}=\lambda^{2}+\cfrac{1}{% \lambda^{2}}+1~{}.
  48. s y m b o l B = λ 2 𝐧 1 𝐧 1 + 1 λ 2 𝐧 2 𝐧 2 + 𝐧 3 𝐧 3 . symbol{B}=\lambda^{2}~{}\mathbf{n}_{1}\otimes\mathbf{n}_{1}+\cfrac{1}{\lambda^% {2}}~{}\mathbf{n}_{2}\otimes\mathbf{n}_{2}+\mathbf{n}_{3}\otimes\mathbf{n}_{3}% ~{}.
  49. σ 11 = - p + 2 λ 2 W I 1 ; σ 22 = - p + 2 λ 2 W I 1 ; σ 33 = - p + 2 W I 1 . \sigma_{11}=-p+2~{}\lambda^{2}~{}\cfrac{\partial W}{\partial I_{1}}~{};~{}~{}% \sigma_{22}=-p+\cfrac{2}{\lambda^{2}}~{}\cfrac{\partial W}{\partial I_{1}}~{};% ~{}~{}\sigma_{33}=-p+2~{}\cfrac{\partial W}{\partial I_{1}}~{}.
  50. σ 22 = 0 \sigma_{22}=0
  51. p = 2 λ 2 W I 1 . p=\cfrac{2}{\lambda^{2}}~{}\cfrac{\partial W}{\partial I_{1}}~{}.
  52. σ 11 = 2 ( λ 2 - 1 λ 2 ) W I 1 ; σ 22 = 0 ; σ 33 = 2 ( 1 - 1 λ 2 ) W I 1 . \sigma_{11}=2~{}\left(\lambda^{2}-\cfrac{1}{\lambda^{2}}\right)~{}\cfrac{% \partial W}{\partial I_{1}}~{};~{}~{}\sigma_{22}=0~{};~{}~{}\sigma_{33}=2~{}% \left(1-\cfrac{1}{\lambda^{2}}\right)~{}\cfrac{\partial W}{\partial I_{1}}~{}.
  53. λ - 1 \lambda-1\,
  54. T 11 = σ 11 λ = 2 ( λ - 1 λ 3 ) W I 1 . T_{11}=\cfrac{\sigma_{11}}{\lambda}=2~{}\left(\lambda-\cfrac{1}{\lambda^{3}}% \right)~{}\cfrac{\partial W}{\partial I_{1}}~{}.
  55. I 3 = J 2 I_{3}=J^{2}
  56. W = i = 1 n C i 0 ( I ¯ 1 - 3 ) i + k = 1 n C k 1 ( J - 1 ) 2 k W=\sum_{i=1}^{n}C_{i0}~{}(\bar{I}_{1}-3)^{i}+\sum_{k=1}^{n}C_{k1}~{}(J-1)^{2k}
  57. I ¯ 1 = J - 2 / 3 I 1 \bar{I}_{1}=J^{-2/3}~{}I_{1}
  58. C i 0 , C k 1 C_{i0},C_{k1}
  59. C 10 C_{10}
  60. C 11 C_{11}
  61. n = 1 n=1

Z2.html

  1. 2 \mathbb{Z}_{2}
  2. / 2 \mathbb{Z}/2\mathbb{Z}

Zadoff–Chu_sequence.html

  1. x u ( n ) = e - j π u n ( n + 1 + 2 q ) N ZC , x_{u}(n)=e^{-j\frac{\pi un(n+1+2q)}{N\text{ZC}}},\,
  2. 0 n < N ZC , 0\leq n<N\text{ZC},\,
  3. 0 < u < N ZC and gcd ( N ZC , u ) = 1 , 0<u<N\text{ZC}\,\and\,\,\text{gcd}(N\text{ZC},u)=1,
  4. q \Z , q\in\Z,
  5. N ZC = length of sequence. N\text{ZC}=\,\text{length of sequence.}\,
  6. q = 0 q=0
  7. N ZC N\text{ZC}
  8. N ZC N\text{ZC}
  9. x u ( n + N ZC ) = x u ( n ) x_{u}(n+N_{\,\text{ZC}})=x_{u}(n)
  10. N ZC N\text{ZC}
  11. X u [ k ] = x u * ( u ~ k ) X u [ 0 ] X_{u}[k]=x_{u}^{*}(\tilde{u}k)X_{u}[0]
  12. u ~ \tilde{u}
  13. N ZC N\text{ZC}
  14. u , u = u 1 , u = u 2 u,u=u_{1},u=u_{2}
  15. N ZC \sqrt{N\text{ZC}}
  16. u 1 - u 2 u_{1}-u_{2}
  17. N ZC N\text{ZC}

Zakharov_system.html

  1. i t u + 2 u = u n i\partial_{t}u+\nabla^{2}u=un
  2. n = - 2 ( | u | 2 ) \Box n=-\nabla^{2}(|u|^{2})

Zakharov–Schulman_system.html

  1. i t u + L 1 u = ϕ u i\partial_{t}u+L_{1}u=\phi u
  2. L 2 ϕ = L 3 ( | u | 2 ) L_{2}\phi=L_{3}(|u|^{2})

Zero-forcing_precoding.html

  1. N t N_{t}
  2. K K
  3. k k
  4. y k = 𝐡 k T 𝐱 + n k , k = 1 , 2 , , K y_{k}=\mathbf{h}_{k}^{T}\mathbf{x}+n_{k},\quad k=1,2,\ldots,K
  5. 𝐱 = i = 1 K s i P i 𝐰 i \mathbf{x}=\sum_{i=1}^{K}s_{i}P_{i}\mathbf{w}_{i}
  6. N t × 1 N_{t}\times 1
  7. n k n_{k}
  8. 𝐡 k \mathbf{h}_{k}
  9. N t × 1 N_{t}\times 1
  10. 𝐰 i \mathbf{w}_{i}
  11. N t × 1 N_{t}\times 1
  12. y k = 𝐡 k T i = 1 K s i P i 𝐰 i + n k = 𝐡 k T s k P k 𝐰 k + n k , k = 1 , 2 , , K y_{k}=\mathbf{h}_{k}^{T}\sum_{i=1}^{K}s_{i}P_{i}\mathbf{w}_{i}+n_{k}=\mathbf{h% }_{k}^{T}s_{k}P_{k}\mathbf{w}_{k}+n_{k},\quad k=1,2,\ldots,K
  13. N r N_{r}
  14. K K
  15. 𝐲 = i = 1 K s i 𝐡 i + 𝐧 \mathbf{y}=\sum_{i=1}^{K}s_{i}\mathbf{h}_{i}+\mathbf{n}
  16. s i s_{i}
  17. i i
  18. 𝐧 \mathbf{n}
  19. N r × 1 N_{r}\times 1
  20. 𝐡 k \mathbf{h}_{k}
  21. N r × 1 N_{r}\times 1
  22. Δ R = R Z F - R F B l o g 2 g \Delta R=R_{ZF}-R_{FB}\leq log_{2}g
  23. B = ( M - 1 ) log 2 ρ b , m - ( M - 1 ) log 2 ( g - 1 ) B=(M-1)\log_{2}\rho_{b,m}-(M-1)\log_{2}(g-1)
  24. ρ b , m \rho_{b,m}
  25. b F B log 2 ( 1 + ρ F B ) B b_{FB}\log_{2}(1+\rho_{FB})\geq B
  26. b = Ω F B T F B b=\Omega_{FB}T_{FB}
  27. ρ F B \rho_{FB}
  28. Δ R log 2 g \Delta R\leq\log_{2}g
  29. b F B B log 2 ( 1 + ρ F B ) = ( M - 1 ) log 2 ρ b , m - ( M - 1 ) log 2 ( g - 1 ) log 2 ( 1 + ρ F B ) b_{FB}\geq\frac{B}{\log_{2}(1+\rho_{FB})}=\frac{(M-1)\log_{2}\rho_{b,m}-(M-1)% \log_{2}(g-1)}{\log_{2}(1+\rho_{FB})}
  30. ρ b , m / ρ F B ) = C u p , d n \rho_{b,m}/\rho_{FB})=C_{up,dn}
  31. b F B , m i n * = lim ρ F B ( M - 1 ) log 2 ρ b , m - ( M - 1 ) log 2 ( g - 1 ) log 2 ( 1 + ρ F B ) = M - 1 b_{FB,min}^{*}=\lim_{\rho_{FB}\to\infty}\frac{(M-1)\log_{2}\rho_{b,m}-(M-1)% \log_{2}(g-1)}{\log_{2}(1+\rho_{FB})}=M-1
  32. b F B b_{FB}

Ziegler–Nichols_method.html

  1. K p K_{p}
  2. K u K_{u}
  3. K u K_{u}
  4. T u T_{u}
  5. K p K_{p}
  6. K i K_{i}
  7. K d K_{d}
  8. 0.5 K u 0.5K_{u}
  9. 0.45 K u 0.45K_{u}
  10. 1.2 K p / T u 1.2K_{p}/T_{u}
  11. 0.8 K u 0.8K_{u}
  12. K p T u / 8 K_{p}T_{u}/8
  13. 0.60 K u 0.60K_{u}
  14. 2 K p / T u 2K_{p}/T_{u}
  15. K p T u / 8 K_{p}T_{u}/8
  16. 0.7 K u 0.7K_{u}
  17. 2.5 K p / T u 2.5K_{p}/T_{u}
  18. 3 K p T u / 20 3K_{p}T_{u}/20
  19. 0.33 K u 0.33K_{u}
  20. 2 K p / T u 2K_{p}/T_{u}
  21. K p T u / 3 K_{p}T_{u}/3
  22. 0.2 K u 0.2K_{u}
  23. 2 K p / T u 2K_{p}/T_{u}
  24. K p T u / 3 K_{p}T_{u}/3

Zig-zag_lemma.html

  1. ( 𝒜 , ) , ( , ) (\mathcal{A},\partial_{\bullet}),(\mathcal{B},\partial_{\bullet}^{\prime})
  2. ( 𝒞 , ′′ ) (\mathcal{C},\partial_{\bullet}^{\prime\prime})
  3. 0 𝒜 α β 𝒞 0 0\longrightarrow\mathcal{A}\stackrel{\alpha}{\longrightarrow}\mathcal{B}% \stackrel{\beta}{\longrightarrow}\mathcal{C}\longrightarrow 0
  4. δ n : H n ( 𝒞 ) H n - 1 ( 𝒜 ) , \delta_{n}:H_{n}(\mathcal{C})\longrightarrow H_{n-1}(\mathcal{A}),
  5. α * \alpha_{*}
  6. β * \beta_{*}
  7. δ n \delta_{n}
  8. δ n \delta_{n}
  9. c C n c\in C_{n}
  10. H n ( 𝒞 ) H_{n}(\mathcal{C})
  11. n ′′ ( c ) = 0 \partial_{n}^{\prime\prime}(c)=0
  12. β n \beta_{n}
  13. b B n b\in B_{n}
  14. β n ( b ) = c \beta_{n}(b)=c
  15. β n - 1 n ( b ) = n ′′ β n ( b ) = n ′′ ( c ) = 0. \beta_{n-1}\partial_{n}^{\prime}(b)=\partial_{n}^{\prime\prime}\beta_{n}(b)=% \partial_{n}^{\prime\prime}(c)=0.
  16. n ( b ) ker β n - 1 = im α n - 1 . \partial_{n}^{\prime}(b)\in\ker\beta_{n-1}=\mathrm{im}\alpha_{n-1}.
  17. α n - 1 \alpha_{n-1}
  18. a A n - 1 a\in A_{n-1}
  19. α n - 1 ( a ) = n ( b ) \alpha_{n-1}(a)=\partial_{n}^{\prime}(b)
  20. α n - 2 \alpha_{n-2}
  21. α n - 2 n - 1 ( a ) = n - 1 α n - 1 ( a ) = n - 1 n ( b ) = 0 , \alpha_{n-2}\partial_{n-1}(a)=\partial_{n-1}^{\prime}\alpha_{n-1}(a)=\partial_% {n-1}^{\prime}\partial_{n}^{\prime}(b)=0,
  22. 2 = 0 \partial^{2}=0
  23. n - 1 ( a ) ker α n - 2 = { 0 } \partial_{n-1}(a)\in\ker\alpha_{n-2}=\{0\}
  24. a a
  25. H n - 1 ( 𝒜 ) H_{n-1}(\mathcal{A})
  26. δ [ c ] = [ a ] . \delta[c]=[a].\,

Zinc-copper_couple.html

  1. Zn n ( Cu ) + CH 2 I 2 IZnCH 2 I + Zn n - 1 ( Cu ) \mathrm{Zn_{n}(Cu)+CH_{2}I_{2}\longrightarrow IZnCH_{2}I+Zn_{n-1}(Cu)}

Zinc_smelting.html

  1. 2 ZnS + 3 O 2 2 ZnO + 2 SO 2 \mathrm{2\,ZnS+3\,O_{2}\rightarrow 2\,ZnO+2\,SO_{2}}
  2. 2 SO 2 + O 2 2 SO 3 \mathrm{2\,SO_{2}+O_{2}\rightarrow 2\,SO_{3}}
  3. ZnO + SO 3 ZnSO 4 \mathrm{ZnO+SO_{3}\rightarrow ZnSO_{4}}
  4. ZnO + CO Zn ( v a p o u r ) + CO 2 \mathrm{ZnO+CO\rightarrow Zn}(vapour)+\mathrm{CO}_{2}

Zodi.html

  1. 10 - 7 10^{-7}

Zonal_spherical_function.html

  1. P = K ρ ( k ) d k P=\int_{K}\rho(k)\,dk
  2. λ ( G ) = ρ ( G ) ′′ , \lambda(G)^{\prime}=\rho(G)^{\prime\prime},
  3. π ( G ) = P ρ ( G ) ′′ P . \pi(G)^{\prime}=P\rho(G)^{\prime\prime}P.
  4. P ρ ( f ) P = G f ( g ) ( P ρ ( g ) P ) d g P\rho(f)P=\int_{G}f(g)(P\rho(g)P)\,dg
  5. F ( g ) = K K f ( k g k ) d k d k F(g)=\int_{K}\int_{K}f(kgk^{\prime})\,dk\,dk^{\prime}
  6. F * ( g ) = F ( g - 1 ) ¯ , F^{*}(g)=\overline{F(g^{-1})},
  7. π ( G ) \pi(G)^{\prime}
  8. C c ( K \ G / K ) C_{c}(K\backslash G/K)
  9. A ( K \ G / K ) . A(K\backslash G/K).
  10. π ( F ) G | F ( g ) | d g F 1 , \|\pi(F)\|\leq\int_{G}|F(g)|\,dg\equiv\|F\|_{1},
  11. χ ( π ( F ) ) = G F ( g ) h ( g ) d g , \chi(\pi(F))=\int_{G}F(g)h(g)\,dg,
  12. h ( x ) h ( y ) = K h ( x k y ) d k ( x , y G ) . h(x)h(y)=\int_{K}h(xky)\,dk\,\,(x,y\in G).
  13. h ( g ) = ( σ ( g ) v , v ) . h(g)=(\sigma(g)v,v).
  14. | G f ( g ) h ( g ) d g | π ( f ) \left|\int_{G}f(g)h(g)\,dg\right|\leq\|\pi(f)\|
  15. G | f ( g ) | 2 d g = X | χ ( π ( f ) ) | 2 d μ ( χ ) . \int_{G}|f(g)|^{2}\,dg=\int_{X}|\chi(\pi(f))|^{2}\,d\mu(\chi).
  16. G = P K , G=P\cdot K,
  17. P = { g G | τ ( g ) = g - 1 } . P=\{g\in G|\tau(g)=g^{-1}\}.
  18. A K A\rtimes K
  19. \cap
  20. S = K T = T τ . S=K\cap T=T^{\tau}.
  21. 𝔤 = 𝔨 𝔞 𝔫 , \mathfrak{g}=\mathfrak{k}\oplus\mathfrak{a}\oplus\mathfrak{n},
  22. 𝔤 \mathfrak{g}
  23. 𝔨 \mathfrak{k}
  24. 𝔞 \mathfrak{a}
  25. \cap
  26. 𝔫 = 𝔤 α , \mathfrak{n}=\bigoplus\mathfrak{g}_{\alpha},
  27. 𝔤 \mathfrak{g}
  28. 𝔞 𝔭 \mathfrak{a}\oplus\mathfrak{p}
  29. v K = c Q v 0 \displaystyle{v_{K}=cQv_{0}}
  30. f ( g ) = ( g v 0 , v 0 ) \displaystyle{f(g)=(gv_{0},v_{0})}
  31. G f ( g ) d g = S K f ( x k ) d x d k = S K f ( k x ) Δ S ( x ) d x d k . \int_{G}f(g)\,dg=\int_{S}\int_{K}f(x\cdot k)\,dx\,dk=\int_{S}\int_{K}f(k\cdot x% )\Delta_{S}(x)\,dx\,dk.
  32. ( σ ( g ) ξ ) ( k ) = α ( g - 1 k ) - 1 ξ ( U ( g - 1 k ) ) , (\sigma(g)\xi)(k)=\alpha^{\prime}(g^{-1}k)^{-1}\,\xi(U(g^{-1}k)),
  33. g = U ( g ) X ( g ) g=U(g)\cdot X(g)
  34. α ( k x ) = Δ S ( x ) 1 / 2 α ( x ) \alpha^{\prime}(kx)=\Delta_{S}(x)^{1/2}\alpha(x)
  35. φ α ( g ) = ( σ ( g ) v , v ) \varphi_{\alpha}(g)=(\sigma(g)v,v)
  36. φ α ( g ) = K α ( g k ) - 1 d k \varphi_{\alpha}(g)=\int_{K}\alpha^{\prime}(gk)^{-1}\,dk
  37. W ( A ) = N K ( A ) / C K ( A ) , W(A)=N_{K}(A)/C_{K}(A),
  38. π ( D ) f = λ D f , \displaystyle\pi(D)f=\lambda_{D}f,
  39. a = ( e r / 2 0 0 e - r / 2 ) , a=\begin{pmatrix}e^{r/2}&0\\ 0&e^{-r/2}\end{pmatrix},
  40. L = - r 2 - 2 coth r r . L=-\partial_{r}^{2}-2\coth r\partial_{r}.
  41. r 2 \partial_{r}^{2}
  42. φ ( r ) = sin ( r ) sinh r \varphi(r)={\sin(\ell r)\over\ell\sinh r}
  43. \ell
  44. 𝔤 {\mathfrak{g}}
  45. 𝔨 \mathfrak{k}
  46. 𝔤 = 𝔨 i 𝔨 . \mathfrak{g}=\mathfrak{k}\oplus i\mathfrak{k}.
  47. 𝔱 \mathfrak{t}
  48. A = exp i 𝔱 , P = exp i 𝔨 . A=\exp i\mathfrak{t},\,\,P=\exp i\mathfrak{k}.
  49. W = N K ( T ) / T W=N_{K}(T)/T
  50. 𝔱 \mathfrak{t}
  51. 𝔱 * \mathfrak{t}^{*}
  52. 𝔨 𝔱 \mathfrak{k}\ominus\mathfrak{t}
  53. χ λ ( e X ) Tr π ( z ) = A λ + ρ ( e X ) / A ρ ( e X ) , \displaystyle\chi_{\lambda}(e^{X})\equiv{\rm Tr}\,\pi(z)=A_{\lambda+\rho}(e^{X% })/A_{\rho}(e^{X}),
  54. 𝔱 * \mathfrak{t}^{*}
  55. A μ ( e X ) = s W ε ( s ) e i μ ( s X ) , \displaystyle A_{\mu}(e^{X})=\sum_{s\in W}\varepsilon(s)e^{i\mu(sX)},
  56. A ρ ( e X ) = e i ρ ( X ) α > 0 ( 1 - e - i α ( X ) ) , \displaystyle A_{\rho}(e^{X})=e^{i\rho(X)}\prod_{\alpha>0}(1-e^{-i\alpha(X)}),
  57. χ λ ( 1 ) dim V = α > 0 ( λ + ρ , α ) α > 0 ( ρ , α ) . \displaystyle\chi_{\lambda}(1)\equiv{\rm dim}\,V={\prod_{\alpha>0}(\lambda+% \rho,\alpha)\over\prod_{\alpha>0}(\rho,\alpha)}.
  58. 𝔱 * \mathfrak{t}^{*}
  59. 𝔨 \mathfrak{k}
  60. 𝔱 * \mathfrak{t}^{*}
  61. i 𝔱 * i\mathfrak{t}^{*}
  62. i 𝔱 i\mathfrak{t}
  63. φ λ ( e X ) = χ λ ( e X ) χ λ ( 1 ) . \varphi_{\lambda}(e^{X})={\chi_{\lambda}(e^{X})\over\chi_{\lambda}(1)}.
  64. Δ K = h - 1 Δ T h + ρ 2 , \displaystyle\Delta_{K}=h^{-1}\circ\Delta_{T}\circ h+\|\rho\|^{2},
  65. h ( e X ) = A ρ ( e X ) \displaystyle h(e^{X})=A_{\rho}(e^{X})
  66. 𝔱 \mathfrak{t}
  67. Δ T φ = ( λ + ρ 2 - ρ 2 ) φ . \Delta_{T}\varphi=(\|\lambda+\rho\|^{2}-\|\rho\|^{2})\varphi.
  68. λ + ρ 2 = μ + ρ 2 . \displaystyle\|\lambda+\rho\|^{2}=\|\mu+\rho\|^{2}.
  69. Δ G = H - 1 Δ A H - ρ 2 , \displaystyle\Delta_{G}=H^{-1}\circ\Delta_{A}\circ H-\|\rho\|^{2},
  70. H ( e X ) = A ρ ( e X ) \displaystyle H(e^{X})=A_{\rho}(e^{X})
  71. i 𝔱 i\mathfrak{t}
  72. f = H φ , \displaystyle f=H\cdot\varphi,
  73. i 𝔱 i\mathfrak{t}
  74. Δ = - 4 y 2 ( x 2 + y 2 ) . \displaystyle\Delta=-4y^{2}(\partial_{x}^{2}+\partial_{y}^{2}).
  75. f s ( z ) = y s = exp ( s log y ) , \displaystyle f_{s}(z)=y^{s}=\exp({s}\cdot\log y),
  76. Δ f s = 4 s ( 1 - s ) f s . \displaystyle\Delta f_{s}=4s(1-s)f_{s}.
  77. φ s ( z ) = K f s ( k z ) d k \varphi_{s}(z)=\int_{K}f_{s}(k\cdot z)\,dk
  78. s = 1 2 + i τ , \displaystyle s={1\over 2}+i\tau,
  79. ( σ ( e X ) v , v ) = n = 0 ( σ ( X ) n v , v ) / n ! \displaystyle(\sigma(e^{X})v,v)=\sum_{n=0}^{\infty}(\sigma(X)^{n}v,v)/n!
  80. i 𝔱 i\mathfrak{t}
  81. Δ = - r 2 - coth ( r ) r , \displaystyle\Delta=-\partial_{r}^{2}-\coth(r)\cdot\partial_{r},
  82. φ ′′ + coth r φ = α φ \displaystyle\varphi^{\prime\prime}+\coth r\,\varphi^{\prime}=\alpha\,\varphi
  83. φ ( r ) = P ρ ( cosh r ) = 1 2 π 0 2 π ( cosh r + sinh r cos θ ) ρ d θ , \varphi(r)=P_{\rho}(\cosh r)={1\over 2\pi}\int_{0}^{2\pi}(\cosh r+\sinh r\,% \cos\theta)^{\rho}\,d\theta,
  84. σ ( f ) = G f ( g ) σ ( g ) d g \sigma(f)=\int_{G}f(g)\sigma(g)\,dg

Δ13C.html

  1. δ 13 C = ( ( C 13 C 12 ) sample ( C 13 C 12 ) standard - 1 ) * 1000 o / o o \mathrm{\delta^{13}C}=\Biggl(\mathrm{\frac{\bigl(\frac{{}^{13}C}{{}^{12}C}% \bigr)_{sample}}{\bigl(\frac{{}^{13}C}{{}^{12}C}\bigr)_{standard}}}-1\Biggr)\ % *1000\ ^{o}\!/\!_{oo}

Bispinor.html

  1. ψ = ( ψ L ψ R ) \psi=\left(\begin{array}[]{c}\psi_{L}\\ \psi_{R}\end{array}\right)
  2. ψ L \psi_{L}
  3. ψ R \psi_{R}
  4. S O ( 1 , 3 ) SO(1,3)
  5. ψ 1 2 [ 1 1 - 1 1 ] ψ = 1 2 ( ψ R + ψ L ψ R - ψ L ) . \psi\rightarrow{1\over\sqrt{2}}\left[\begin{array}[]{cc}1&1\\ -1&1\end{array}\right]\psi={1\over\sqrt{2}}\left(\begin{array}[]{c}\psi_{R}+% \psi_{L}\\ \psi_{R}-\psi_{L}\end{array}\right).
  6. ψ ( x ) \psi(x)
  7. ψ a ( x ) ψ a ( x ) = S [ Λ ] b a ψ b ( Λ - 1 x ) \psi^{a}(x)\to{\psi^{\prime}}^{a}(x)=S[\Lambda]^{a}_{b}\psi^{b}(\Lambda^{-1}x)
  8. Λ \Lambda
  9. Λ \Lambda
  10. Λ - 1 x \Lambda^{-1}x
  11. x x
  12. Λ b o o s t \Lambda_{boost}
  13. Λ r o t \Lambda_{rot}
  14. S [ Λ b o o s t ] = ( e + χ σ / 2 0 0 e - χ σ / 2 ) S[\Lambda_{boost}]=\left(\begin{array}[]{cc}e^{+\chi\cdot\sigma/2}&0\\ 0&e^{-\chi\cdot\sigma/2}\end{array}\right)
  15. S [ Λ r o t ] = ( e + i ϕ σ / 2 0 0 e + i ϕ σ / 2 ) S[\Lambda_{rot}]=\left(\begin{array}[]{cc}e^{+i\phi\cdot\sigma/2}&0\\ 0&e^{+i\phi\cdot\sigma/2}\end{array}\right)
  16. χ \chi
  17. ϕ i \phi^{i}
  18. x i x^{i}
  19. σ i \sigma_{i}
  20. ψ ¯ ψ \bar{\psi}\psi
  21. ψ ¯ γ 5 ψ \bar{\psi}\gamma_{5}\psi
  22. ψ ¯ γ μ ψ \bar{\psi}\gamma^{\mu}\psi
  23. ψ ¯ γ μ γ 5 ψ \bar{\psi}\gamma^{\mu}\gamma_{5}\psi
  24. ψ ¯ ( γ μ γ ν - γ ν γ μ ) ψ \bar{\psi}(\gamma^{\mu}\gamma^{\nu}-\gamma^{\nu}\gamma^{\mu})\psi
  25. ψ ¯ ψ γ 0 \bar{\psi}\equiv\psi^{\dagger}\gamma_{0}
  26. { γ μ , γ 5 } \{\gamma^{\mu},\gamma_{5}\}
  27. = i 2 ( ψ ¯ γ μ μ ψ - μ ψ ¯ γ μ ψ ) - m ψ ¯ ψ . \mathcal{L}={i\over 2}\left(\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi-\partial_% {\mu}\bar{\psi}\gamma^{\mu}\psi\right)-m\bar{\psi}\psi\;.
  28. 𝐬𝐨 ( 3 ; 1 ) \mathbf{so}(3;1)
  29. 𝐬𝐨 ( 3 ; 1 ) \mathbf{so}(3;1)
  30. 𝐎 ( 3 ; 1 ) \mathbf{O}(3;1)
  31. 4 × 4 4×4
  32. ( 1 2 , 0 ) ( 0 , 1 2 ) (\frac{1}{2},0)⊕(0,\frac{1}{2})
  33. 𝐬𝐨 ( 3 ; 1 ) \mathbf{so}(3;1)
  34. [ M μ ν , M ρ σ ] = i ( η σ μ M ρ ν + η ν σ M μ ρ - η ρ μ M σ ν - η ν ρ M μ σ ) [M^{\mu\nu},M^{\rho\sigma}]=i(\eta^{\sigma\mu}M^{\rho\nu}+\eta^{\nu\sigma}M^{% \mu\rho}-\eta^{\rho\mu}M^{\sigma\nu}-\eta^{\nu\rho}M^{\mu\sigma})
  35. η = d i a g ( 1 , 1 , 1 , 1 ) η=diag(−1,1,1,1)
  36. 4 × 4 4×4
  37. 𝐬𝐨 ( 3 ; 1 ) \mathbf{so}(3;1)
  38. 𝐬𝐨 ( 3 ; 1 ) \mathbf{so}(3;1)
  39. 𝐬𝐨 ( 3 ; 1 ) \mathbf{so}(3;1)
  40. 𝐬𝐨 ( 3 ; 1 ) \mathbf{so}(3;1)
  41. π π
  42. u S ( Λ ) u = e i π ( ω μ ν M μ ν ) u ; u α [ e ω μ ν σ < m t p l > μ ν 1.43 42 𝐬𝐨 ( 3 ; 1 ) 1.44 43 S : S O ( 3 ; 1 ) + GL ( U ) ; Λ e i π ( X ) ; e i X = Λ , X = ω μ ν M μ ν 𝔰 𝔬 ( 3 ; 1 ) 1.45 44 S 1.46 45 ( 1 2 , 0 ) ( 0 , 1 2 ) 1.47 46 S ( Λ B ) = e i π ( ϕ 𝐉 ) = ( e - 1 2 χ σ 0 0 e 1 2 χ σ ) , S ( Λ R ) = e i π ( χ 𝐊 ) = ( e i 2 ϕ σ 0 0 e i 2 ϕ σ ) , 1.48 47 φ 1.49 48 χ 1.50 49 ψ = ( ψ R ψ L ) , 1.51 50 P = d i a g ( 1 , 1 , 1 , 1 ) 1.52 51 X = 2 π M < s u p > 12 u\rightarrow S(\Lambda)u=e^{i\pi(\omega_{\mu\nu}M^{\mu\nu})}u;\quad u^{\alpha}% \rightarrow[e^{\omega_{\mu\nu}\sigma^{<}mtpl>{{\mu\nu}}$\par \par \@@section{% subsection}{S1.SS43}{1.43}{1.43}{{\@tag[][]{1.43}42}}{{\@tag[][]{1.43}42}}% \par $\mathbf{so}(3;1)$\par \par \@@section{subsection}{S1.SS44}{1.44}{1.44}{{% \@tag[][]{1.44}43}}{{\@tag[][]{1.44}43}}\par $S:SO(3;1)^{+}\rightarrow\mathrm{% GL}(U);\quad\Lambda\rightarrow e^{i\pi(X)};\quad e^{iX}=\Lambda,\quad X=\omega% _{\mu\nu}M^{\mu\nu}\in\mathfrak{so}(3;1)$\par \par \@@section{subsection}{S1.% SS45}{1.45}{1.45}{{\@tag[][]{1.45}44}}{{\@tag[][]{1.45}44}}\par $S$\par \par % \@@section{subsection}{S1.SS46}{1.46}{1.46}{{\@tag[][]{1.46}45}}{{\@tag[][]{1.% 46}45}}\par $(\frac{1}{2},0)⊕(0,\frac{1}{2})$\par \par \@@section{subsection}{% S1.SS47}{1.47}{1.47}{{\@tag[][]{1.47}46}}{{\@tag[][]{1.47}46}}\par $\begin{% aligned}\displaystyle S(\Lambda_{B})&\displaystyle=e^{i\pi(\phi\cdot\mathbf{J}% )}=\biggl(\begin{matrix}e^{-\frac{1}{2}\chi\cdot\sigma}&0\\ 0&e^{\frac{1}{2}\chi\cdot\sigma}\\ \end{matrix}\biggr),\\ \displaystyle S(\Lambda_{R})&\displaystyle=e^{i\pi(\chi\cdot\mathbf{K})}=% \biggl(\begin{matrix}e^{\frac{i}{2}\phi\cdot\sigma}&0\\ 0&e^{\frac{i}{2}\phi\cdot\sigma}\\ \end{matrix}\biggr)\\ \end{aligned},$\par \par \@@section{subsection}{S1.SS48}{1.48}{1.48}{{\@tag[][% ]{1.48}47}}{{\@tag[][]{1.48}47}}\par $φ$\par \par \@@section{subsection}{S1.SS% 49}{1.49}{1.49}{{\@tag[][]{1.49}48}}{{\@tag[][]{1.49}48}}\par $χ$\par \par % \@@section{subsection}{S1.SS50}{1.50}{1.50}{{\@tag[][]{1.50}49}}{{\@tag[][]{1.% 50}49}}\par $\psi=\begin{pmatrix}\psi_{R}\\ \psi_{L}\end{pmatrix},$\par \par \@@section{subsection}{S1.SS51}{1.51}{1.51}{{% \@tag[][]{1.51}50}}{{\@tag[][]{1.51}50}}\par $P=diag(1,−1,−1,−1)$\par \par % \@@section{subsection}{S1.SS52}{1.52}{1.52}{{\@tag[][]{1.52}51}}{{\@tag[][]{1.% 52}51}}\par $X=2πM<sup>12$\end{document}}