wpmath0000009_3

Dihydroorotate_dehydrogenase.html

  1. \rightleftharpoons

Dihydropteroate_synthase.html

  1. \rightleftharpoons

Dilution_assay.html

  1. U U
  2. f : U f:U\rightarrow\mathbb{R}
  3. u U u\in U
  4. F F
  5. z [ 0 , ) z\in[0,\infty)
  6. f ( u ) = F ( z ) + e f(u)=F(z)+e
  7. e e
  8. F F
  9. T T
  10. S S
  11. F T ( z ) = F S ( ρ z ) F_{T}(z)=F_{S}(\rho z)
  12. z z
  13. ρ > 0 \rho>0
  14. T T
  15. x = z λ x=z^{\lambda}
  16. λ > 0 \lambda>0
  17. x = log ( z ) x=\log(z)
  18. λ 0 \lambda\downarrow 0
  19. λ = 0 \lambda=0
  20. F T ( x ) = F S ( ρ λ x ) F_{T}(x)=F_{S}(\rho^{\lambda}x)
  21. x x
  22. F ^ \hat{F}
  23. F F
  24. F F
  25. F F
  26. F F
  27. F F
  28. ρ \rho
  29. F F
  30. λ > 0 \lambda>0
  31. F F
  32. λ = 0 \lambda=0
  33. F F
  34. λ = 0 \lambda=0
  35. e e
  36. u u
  37. f ( u ) f(u)
  38. z z
  39. x = log ( z ) x=\log(z)
  40. log ( ρ ^ ) \log(\hat{\rho})
  41. ρ \rho

Dimension_function.html

  1. μ s ( E ) = lim δ 0 μ δ s ( E ) , \mu^{s}(E)=\lim_{\delta\to 0}\mu_{\delta}^{s}(E),
  2. μ δ s ( E ) = inf { i = 1 diam ( C i ) s | diam ( C i ) δ , i = 1 C i E } . \mu_{\delta}^{s}(E)=\inf\left\{\left.\sum_{i=1}^{\infty}\mathrm{diam}(C_{i})^{% s}\right|\mathrm{diam}(C_{i})\leq\delta,\bigcup_{i=1}^{\infty}C_{i}\supseteq E% \right\}.
  3. μ h ( E ) = lim δ 0 μ δ h ( E ) , \mu^{h}(E)=\lim_{\delta\to 0}\mu_{\delta}^{h}(E),
  4. μ δ h ( E ) = inf { i = 1 h ( diam ( C i ) ) | diam ( C i ) δ , i = 1 C i E } . \mu_{\delta}^{h}(E)=\inf\left\{\left.\sum_{i=1}^{\infty}h\left(\mathrm{diam}(C% _{i})\right)\right|\mathrm{diam}(C_{i})\leq\delta,\bigcup_{i=1}^{\infty}C_{i}% \supseteq E\right\}.
  5. h ( r ) = r 2 log 1 r log log log 1 r . h(r)=r^{2}\cdot\log\frac{1}{r}\cdot\log\log\log\frac{1}{r}.
  6. h ( r ) = r 2 log log 1 r . h(r)=r^{2}\cdot\log\log\frac{1}{r}.

Dirac_algebra.html

  1. { γ μ , γ ν } = γ μ γ ν + γ ν γ μ = 2 η μ ν 1 \displaystyle\{\gamma^{\mu},\gamma^{\nu}\}=\gamma^{\mu}\gamma^{\nu}+\gamma^{% \nu}\gamma^{\mu}=2\eta^{\mu\nu}{1}
  2. η μ ν \eta^{\mu\nu}\,
  3. 1 {1}
  4. a , b = μ ν η μ ν a μ b ν \displaystyle\langle a,b\rangle=\sum_{\mu\nu}\eta^{\mu\nu}a_{\mu}b^{\dagger}_{\nu}
  5. a = μ a μ γ μ \,a=\sum_{\mu}a_{\mu}\gamma^{\mu}
  6. b = ν b ν γ ν \,b=\sum_{\nu}b_{\nu}\gamma^{\nu}
  7. - i γ μ μ ψ + m c ψ = 0 . -i\hbar\gamma^{\mu}\partial_{\mu}\psi+mc\psi=0\,.
  8. - t 2 ψ + 2 ψ = m 2 ψ -\partial_{t}^{2}\psi+\nabla^{2}\psi=m^{2}\psi
  9. ψ ( i γ μ μ ψ + m c ) ( - i γ ν ν ψ + m c ) ψ = 0 . \psi^{\dagger}(i\hbar\gamma^{\mu}\partial_{\mu}\psi+mc)(-i\hbar\gamma^{\nu}% \partial_{\nu}\psi+mc)\psi=0\,.
  10. { γ μ , γ ν } = γ μ γ ν + γ ν γ μ = 2 η μ ν I 4 \displaystyle\{\gamma^{\mu},\gamma^{\nu}\}=\gamma^{\mu}\gamma^{\nu}+\gamma^{% \nu}\gamma^{\mu}=2\eta^{\mu\nu}I_{4}
  11. { , } \{,\}
  12. η μ ν \eta^{\mu\nu}\,
  13. I 4 \ I_{4}\,
  14. C l 1 , 3 ( ) = C l 1 , 3 ( ) . Cl_{1,3}(\mathbb{C})=Cl_{1,3}(\mathbb{R})\otimes\mathbb{C}.

Direct_linear_transformation.html

  1. 𝐱 k 𝐀 𝐲 k \mathbf{x}_{k}\propto\mathbf{A}\,\mathbf{y}_{k}
  2. k = 1 , , N \,k=1,\ldots,N
  3. 𝐱 k \mathbf{x}_{k}
  4. 𝐲 k \mathbf{y}_{k}
  5. \,\propto
  6. 𝐀 \mathbf{A}
  7. 𝐱 k = 𝐀 𝐲 k \mathbf{x}_{k}=\mathbf{A}\,\mathbf{y}_{k}
  8. k = 1 , , N \,k=1,\ldots,N
  9. 𝐗 = 𝐀 𝐘 \mathbf{X}=\mathbf{A}\,\mathbf{Y}
  10. 𝐗 \mathbf{X}
  11. 𝐘 \mathbf{Y}
  12. 𝐱 k \mathbf{x}_{k}
  13. 𝐲 k \mathbf{y}_{k}
  14. 𝐀 = 𝐗 𝐘 T ( 𝐘 𝐘 T ) - 1 . \mathbf{A}=\mathbf{X}\,\mathbf{Y}^{T}\,(\mathbf{Y}\,\mathbf{Y}^{T})^{-1}.
  15. 𝐀 \mathbf{A}
  16. 𝐱 k 2 \mathbf{x}_{k}\in\mathbb{R}^{2}
  17. 𝐲 k 3 \mathbf{y}_{k}\in\mathbb{R}^{3}
  18. 2 × 3 2\times 3
  19. 𝐀 \mathbf{A}
  20. α k 𝐱 k = 𝐀 𝐲 k \alpha_{k}\,\mathbf{x}_{k}=\mathbf{A}\,\mathbf{y}_{k}
  21. k = 1 , , N \,k=1,\ldots,N
  22. α k 0 \alpha_{k}\neq 0
  23. 𝐇 = ( 0 - 1 1 0 ) \mathbf{H}=\begin{pmatrix}0&-1\\ 1&0\end{pmatrix}
  24. 𝐱 k T 𝐇 \mathbf{x}_{k}^{T}\,\mathbf{H}
  25. α k 𝐱 k T 𝐇 𝐱 k = 𝐱 k T 𝐇 𝐀 𝐲 k \alpha_{k}\,\mathbf{x}_{k}^{T}\,\mathbf{H}\,\mathbf{x}_{k}=\mathbf{x}_{k}^{T}% \,\mathbf{H}\,\mathbf{A}\,\mathbf{y}_{k}
  26. k = 1 , , N . \,k=1,\ldots,N.
  27. 𝐱 k T 𝐇 𝐱 k = 0 , \mathbf{x}_{k}^{T}\,\mathbf{H}\,\mathbf{x}_{k}=0,
  28. 0 = 𝐱 k T 𝐇 𝐀 𝐲 k 0=\mathbf{x}_{k}^{T}\,\mathbf{H}\,\mathbf{A}\,\mathbf{y}_{k}
  29. k = 1 , , N . \,k=1,\ldots,N.
  30. 𝐀 \mathbf{A}
  31. 𝐱 k \mathbf{x}_{k}
  32. 𝐲 k \mathbf{y}_{k}
  33. 𝐀 \mathbf{A}
  34. 𝐱 k = ( x 1 k x 2 k ) \mathbf{x}_{k}=\begin{pmatrix}x_{1k}\\ x_{2k}\end{pmatrix}
  35. 𝐲 k = ( y 1 k y 2 k y 3 k ) \mathbf{y}_{k}=\begin{pmatrix}y_{1k}\\ y_{2k}\\ y_{3k}\end{pmatrix}
  36. 𝐀 = ( a 11 a 12 a 13 a 21 a 22 a 23 ) \mathbf{A}=\begin{pmatrix}a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\end{pmatrix}
  37. 0 = a 11 x 2 k y 1 k - a 21 x 1 k y 1 k + a 12 x 2 k y 2 k - a 22 x 1 k y 2 k + a 13 x 2 k y 3 k - a 23 x 1 k y 3 k 0=a_{11}\,x_{2k}\,y_{1k}-a_{21}\,x_{1k}\,y_{1k}+a_{12}\,x_{2k}\,y_{2k}-a_{22}% \,x_{1k}\,y_{2k}+a_{13}\,x_{2k}\,y_{3k}-a_{23}\,x_{1k}\,y_{3k}
  38. k = 1 , , N . \,k=1,\ldots,N.
  39. 0 = 𝐛 k T 𝐚 0=\mathbf{b}_{k}^{T}\,\mathbf{a}
  40. k = 1 , , N \,k=1,\ldots,N
  41. 𝐛 k \mathbf{b}_{k}
  42. 𝐚 \mathbf{a}
  43. 𝐛 k = ( x 2 k y 1 k - x 1 k y 1 k x 2 k y 2 k - x 1 k y 2 k x 2 k y 3 k - x 1 k y 3 k ) \mathbf{b}_{k}=\begin{pmatrix}x_{2k}\,y_{1k}\\ -x_{1k}\,y_{1k}\\ x_{2k}\,y_{2k}\\ -x_{1k}\,y_{2k}\\ x_{2k}\,y_{3k}\\ -x_{1k}\,y_{3k}\end{pmatrix}
  44. 𝐚 = ( a 11 a 21 a 12 a 22 a 13 a 23 ) . \mathbf{a}=\begin{pmatrix}a_{11}\\ a_{21}\\ a_{12}\\ a_{22}\\ a_{13}\\ a_{23}\end{pmatrix}.
  45. 𝟎 = 𝐁 𝐚 \mathbf{0}=\mathbf{B}\,\mathbf{a}
  46. 𝐁 \mathbf{B}
  47. N × 6 N\times 6
  48. 𝐛 k \mathbf{b}_{k}
  49. 𝐚 \mathbf{a}
  50. 𝐁 \mathbf{B}
  51. 𝐁 \mathbf{B}
  52. 𝐚 \mathbf{a}
  53. 𝐁 \mathbf{B}
  54. 𝐚 \mathbf{a}
  55. 𝐀 \mathbf{A}
  56. 2 × 3 2\times 3
  57. 𝐚 \mathbf{a}
  58. 𝐀 \mathbf{A}
  59. 𝐱 k \mathbf{x}_{k}
  60. 𝐲 k \mathbf{y}_{k}
  61. 𝐚 \mathbf{a}
  62. 𝟎 = 𝐁 𝐚 \mathbf{0}=\mathbf{B}\,\mathbf{a}
  63. 𝐚 \mathbf{a}
  64. 𝐁 . \mathbf{B}.
  65. 𝐱 k 2 \mathbf{x}_{k}\in\mathbb{R}^{2}
  66. 𝐲 k 3 \mathbf{y}_{k}\in\mathbb{R}^{3}
  67. 𝐱 k \mathbf{x}_{k}
  68. 𝐲 k . \mathbf{y}_{k}.
  69. 𝐱 k 2 \mathbf{x}_{k}\in\mathbb{R}^{2}
  70. 𝐲 k q \mathbf{y}_{k}\in\mathbb{R}^{q}
  71. 0 = 𝐱 k T 𝐇 𝐀 𝐲 k 0=\mathbf{x}_{k}^{T}\,\mathbf{H}\,\mathbf{A}\,\mathbf{y}_{k}
  72. k = 1 , , N \,k=1,\ldots,N
  73. 𝐀 \mathbf{A}
  74. 2 × q . 2\times q.
  75. 2 q 2q
  76. 𝐀 \mathbf{A}
  77. 𝐁 𝐚 = 𝟎 \mathbf{B}\,\mathbf{a}=\mathbf{0}
  78. N × 2 q N\times 2\,q
  79. 𝐁 \mathbf{B}
  80. 𝐚 . \mathbf{a}.
  81. 𝐱 k p \mathbf{x}_{k}\in\mathbb{R}^{p}
  82. 𝐲 k q \mathbf{y}_{k}\in\mathbb{R}^{q}
  83. 𝐇 \mathbf{H}
  84. p × p p\times p
  85. p > 0 p>0
  86. M = p ( p - 1 ) 2 . M=\frac{p\,(p-1)}{2}.
  87. 0 = 𝐱 k T 𝐇 m 𝐀 𝐲 k 0=\mathbf{x}_{k}^{T}\,\mathbf{H}_{m}\,\mathbf{A}\,\mathbf{y}_{k}
  88. m = 1 , , M \,m=1,\ldots,M
  89. k = 1 , , N \,k=1,\ldots,N
  90. 𝐇 m \mathbf{H}_{m}
  91. p × p p\times p
  92. 𝐇 m \mathbf{H}_{m}
  93. 𝐇 1 = ( 0 0 0 0 0 - 1 0 1 0 ) \mathbf{H}_{1}=\begin{pmatrix}0&0&0\\ 0&0&-1\\ 0&1&0\end{pmatrix}
  94. 𝐇 2 = ( 0 0 1 0 0 0 - 1 0 0 ) \mathbf{H}_{2}=\begin{pmatrix}0&0&1\\ 0&0&0\\ -1&0&0\end{pmatrix}
  95. 𝐇 3 = ( 0 - 1 0 1 0 0 0 0 0 ) . \mathbf{H}_{3}=\begin{pmatrix}0&-1&0\\ 1&0&0\\ 0&0&0\end{pmatrix}.
  96. 𝟎 = [ 𝐱 k ] × 𝐀 𝐲 k \mathbf{0}=[\mathbf{x}_{k}]_{\times}\,\mathbf{A}\,\mathbf{y}_{k}
  97. k = 1 , , N \,k=1,\ldots,N
  98. [ 𝐱 k ] × [\mathbf{x}_{k}]_{\times}
  99. 3 \mathbb{R}^{3}
  100. 𝐀 \mathbf{A}
  101. [ 𝐱 k ] × [\mathbf{x}_{k}]_{\times}
  102. 𝐇 m \mathbf{H}_{m}
  103. 𝐱 k \mathbf{x}_{k}
  104. 𝐁 \mathbf{B}
  105. 𝐇 m \mathbf{H}_{m}
  106. 𝐁 \mathbf{B}
  107. 𝐚 . \mathbf{a}.

Directed_infinity.html

  1. w + z = ( w + z ) w\infty+z\infty=(w+z)\infty
  2. z = z | z | . z\infty=\frac{z}{\left|z\right|}\infty.
  3. 0 is undefined, as is z w 0\infty\,\text{ is undefined, as is }\frac{z\infty}{w\infty}
  4. a z = { z if a > 0 , - z if a < 0. az\infty=\begin{cases}z\infty&\,\text{if }a>0,\\ -z\infty&\,\text{if }a<0.\end{cases}
  5. w z = ( w z ) w\infty z\infty=(wz)\infty

Dirichlet-multinomial_distribution.html

  1. s y m b o l α symbol{\alpha}
  2. z n z_{n}
  3. n = 1 N n=1\dots N
  4. k k
  5. k = 1 K k=1\dots K
  6. n k n_{k}
  7. k n k = N \sum_{k}n_{k}=N
  8. N N
  9. z 1 , , z N z_{1},\dots,z_{N}
  10. 𝐱 = ( n 1 , , n K ) \mathbf{x}=(n_{1},\dots,n_{K})
  11. 𝐩 = ( p 1 , p 2 , , p K ) , \mathbf{p}=(p_{1},p_{2},\dots,p_{K}),
  12. p k p_{k}
  13. k k
  14. 𝐩 \mathbf{p}
  15. P ( 𝐱 | 𝐩 ) P(\mathbf{x}|\mathbf{p})
  16. 𝐩 \mathbf{p}
  17. s y m b o l α = ( α 1 , α 2 , , α K ) symbol\alpha=(\alpha_{1},\alpha_{2},\ldots,\alpha_{K})
  18. 𝐩 \mathbf{p}
  19. = z 1 , , z N \mathbb{Z}=z_{1},\dots,z_{N}
  20. 𝐩 \mathbf{p}
  21. Pr ( \midsymbol α ) = 𝐩 Pr ( 𝐩 ) Pr ( 𝐩 \midsymbol α ) d 𝐩 \Pr(\mathbb{Z}\midsymbol{\alpha})=\int_{\mathbf{p}}\Pr(\mathbb{Z}\mid\mathbf{p% })\Pr(\mathbf{p}\midsymbol{\alpha})\textrm{d}\mathbf{p}
  22. Pr ( \midsymbol α ) = Γ ( A ) Γ ( N + A ) k = 1 K Γ ( n k + α k ) Γ ( α k ) \Pr(\mathbb{Z}\midsymbol{\alpha})=\frac{\Gamma\left(A\right)}{\Gamma\left(N+A% \right)}\prod_{k=1}^{K}\frac{\Gamma(n_{k}+\alpha_{k})}{\Gamma(\alpha_{k})}
  23. Γ \Gamma
  24. A = k α k and N = k n k , and where n k = number of z n ’s with the value k . A=\sum_{k}\alpha_{k}\,\text{ and }N=\sum_{k}n_{k}\,\text{, and where }n_{k}=\,% \text{number of }z_{n}\,\text{'s with the value }k\,\text{.}
  25. z 1 , , z N z_{1},\dots,z_{N}
  26. n k n_{k}
  27. z n z_{n}
  28. ( - n ) \mathbb{Z}^{(-n)}
  29. Pr ( z n = k ( - n ) , s y m b o l α ) n k ( - n ) + α k \Pr(z_{n}=k\mid\mathbb{Z}^{(-n)},symbol{\alpha})\propto n_{k}^{(-n)}+\alpha_{k}
  30. n k ( - n ) n_{k}^{(-n)}
  31. k k
  32. z n z_{n}
  33. z 1 , , z N z_{1},\dots,z_{N}
  34. z n z_{n}
  35. n k ( - n ) n_{k}^{(-n)}
  36. n j = { n j ( - n ) , if j k n j ( - n ) + 1 , if j = k n_{j}=\begin{cases}n_{j}^{(-n)},&\,\text{if }j\not=k\\ n_{j}^{(-n)}+1,&\,\text{if }j=k\end{cases}
  37. Γ ( n + 1 ) = n Γ ( n ) \Gamma(n+1)=n\Gamma(n)
  38. Pr ( z n = k ( - n ) , s y m b o l α ) \displaystyle\Pr(z_{n}=k\mid\mathbb{Z}^{(-n)},symbol{\alpha})
  39. k ( n k ( - n ) + α k ) = A + k n k ( - n ) = A + N - 1 \sum_{k}\left(n_{k}^{(-n)}+\alpha_{k}\right)=A+\sum_{k}n_{k}^{(-n)}=A+N-1
  40. Pr ( z n = k ( - n ) , s y m b o l α ) = n k ( - n ) + α k A + N - 1 \Pr(z_{n}=k\mid\mathbb{Z}^{(-n)},symbol{\alpha})=\frac{n_{k}^{(-n)}+\alpha_{k}% }{A+N-1}
  41. K K\to\infty
  42. DirMult ( \midsymbol α ) \operatorname{DirMult}(\mathbb{Z}\midsymbol{\alpha})
  43. P r ( \midsymbol α ) = DirMult ( \midsymbol α ) = Γ ( k α k ) Γ ( k n k + α k ) k = 1 K Γ ( n k + α k ) Γ ( α k ) Pr(\mathbb{Z}\midsymbol{\alpha})=\operatorname{DirMult}(\mathbb{Z}\midsymbol{% \alpha})=\frac{\Gamma\left(\sum_{k}\alpha_{k}\right)}{\Gamma\left(\sum_{k}n_{k% }+\alpha_{k}\right)}\prod_{k=1}^{K}\frac{\Gamma(n_{k}+\alpha_{k})}{\Gamma(% \alpha_{k})}
  44. s y m b o l α some distribution s y m b o l θ d = 1 M Dirichlet K ( s y m b o l α ) z d = 1 M , n = 1 N d Categorical K ( s y m b o l θ d ) \begin{array}[]{lcl}symbol\alpha&\sim&\,\text{some distribution}\\ symbol\theta_{d=1\dots M}&\sim&\operatorname{Dirichlet}_{K}(symbol\alpha)\\ z_{d=1\dots M,n=1\dots N_{d}}&\sim&\operatorname{Categorical}_{K}(symbol\theta% _{d})\end{array}
  45. Pr ( \midsymbol α ) = d DirMult ( d \midsymbol α ) \Pr(\mathbb{Z}\midsymbol\alpha)=\prod_{d}\operatorname{DirMult}(\mathbb{Z}_{d}% \midsymbol\alpha)
  46. d \mathbb{Z}_{d}
  47. Pr ( z d n = k ( - d n ) , s y m b o l α ) n k , d ( - n ) + α k \Pr(z_{dn}=k\mid\mathbb{Z}^{(-dn)},symbol\alpha)\ \propto\ n_{k,d}^{(-n)}+% \alpha_{k}
  48. n k , d ( - n ) n_{k,d}^{(-n)}
  49. d \mathbb{Z}_{d}
  50. z d n z_{dn}
  51. k k
  52. s y m b o l α some distribution s y m b o l θ d = 1 M Dirichlet K ( s y m b o l α ) z d = 1 M , n = 1 N d Categorical K ( s y m b o l θ d ) s y m b o l ϕ some other distribution w d = 1 M , n = 1 N d F ( w d n z d n , s y m b o l ϕ ) \begin{array}[]{lcl}symbol\alpha&\sim&\,\text{some distribution}\\ symbol\theta_{d=1\dots M}&\sim&\operatorname{Dirichlet}_{K}(symbol\alpha)\\ z_{d=1\dots M,n=1\dots N_{d}}&\sim&\operatorname{Categorical}_{K}(symbol\theta% _{d})\\ symbol\phi&\sim&\,\text{some other distribution}\\ w_{d=1\dots M,n=1\dots N_{d}}&\sim&\operatorname{F}(w_{dn}\mid z_{dn},symbol% \phi)\end{array}
  53. Pr ( , 𝕎 \midsymbol α , s y m b o l ϕ ) = d DirMult ( d \midsymbol α ) d = 1 M n = 1 N d F ( w d n z d n , s y m b o l ϕ ) \Pr(\mathbb{Z},\mathbb{W}\midsymbol\alpha,symbol\phi)=\prod_{d}\operatorname{% DirMult}(\mathbb{Z}_{d}\midsymbol\alpha)\prod_{d=1}^{M}\prod_{n=1}^{N_{d}}% \operatorname{F}(w_{dn}\mid z_{dn},symbol\phi)
  54. z d n z_{dn}
  55. ( - d n ) \mathbb{Z}^{(-dn)}
  56. α \alpha
  57. Pr ( z d n = k ( - d n ) , 𝕎 , s y m b o l α , s y m b o l ϕ ) ( n k , d ( - n ) + α k ) F ( w d n z d n , s y m b o l ϕ ) \Pr(z_{dn}=k\mid\mathbb{Z}^{(-dn)},\mathbb{W},symbol\alpha,symbol\phi)\ % \propto\ (n_{k,d}^{(-n)}+\alpha_{k})\operatorname{F}(w_{dn}\mid z_{dn},symbol\phi)
  58. F \operatorname{F}
  59. z d n z_{dn}
  60. z d n z_{dn}
  61. s y m b o l θ some distribution z n = 1 N Categorical K ( s y m b o l θ ) s y m b o l α some distribution s y m b o l ϕ k = 1 K Dirichlet V ( s y m b o l α ) w n = 1 N Categorical V ( s y m b o l ϕ z n ) \begin{array}[]{lcl}symbol\theta&\sim&\,\text{some distribution}\\ z_{n=1\dots N}&\sim&\operatorname{Categorical}_{K}(symbol\theta)\\ symbol\alpha&\sim&\,\text{some distribution}\\ symbol\phi_{k=1\dots K}&\sim&\operatorname{Dirichlet}_{V}(symbol\alpha)\\ w_{n=1\dots N}&\sim&\operatorname{Categorical}_{V}(symbol\phi_{z_{n}})\\ \end{array}
  62. 𝕎 \mathbb{W}
  63. K K
  64. V V
  65. \mathbb{Z}
  66. K K
  67. Pr ( 𝕎 \midsymbol α , ) = k = 1 K DirMult ( 𝕎 k , s y m b o l α ) = k = 1 K [ Γ ( v α v ) Γ ( v n v k + α v ) v = 1 V Γ ( n v k + α v ) Γ ( α v ) ] \Pr(\mathbb{W}\midsymbol\alpha,\mathbb{Z})=\prod_{k=1}^{K}\operatorname{% DirMult}(\mathbb{W}_{k}\mid\mathbb{Z},symbol\alpha)=\prod_{k=1}^{K}\left[\frac% {\Gamma\left(\sum_{v}\alpha_{v}\right)}{\Gamma\left(\sum_{v}n_{v}^{k}+\alpha_{% v}\right)}\prod_{v=1}^{V}\frac{\Gamma(n_{v}^{k}+\alpha_{v})}{\Gamma(\alpha_{v}% )}\right]
  68. n v k n_{v}^{k}
  69. Pr ( w n = v 𝕎 ( - n ) , , s y m b o l α ) n v k , ( - n ) + α v \Pr(w_{n}=v\mid\mathbb{W}^{(-n)},\mathbb{Z},symbol\alpha)\ \propto\ n_{v}^{k,(% -n)}+\alpha_{v}
  70. n v k , ( - n ) n_{v}^{k,(-n)}
  71. s y m b o l α A Dirichlet hyperprior, either a constant or a random variable s y m b o l β A Dirichlet hyperprior, either a constant or a random variable s y m b o l θ d = 1 M Dirichlet K ( s y m b o l α ) s y m b o l ϕ k = 1 K Dirichlet V ( s y m b o l β ) z d = 1 M , n = 1 N d Categorical K ( s y m b o l θ d ) w d = 1 M , n = 1 N d Categorical V ( s y m b o l ϕ z d n ) \begin{array}[]{lcl}symbol\alpha&\sim&\,\text{A Dirichlet hyperprior, either a% constant or a random variable}\\ symbol\beta&\sim&\,\text{A Dirichlet hyperprior, either a constant or a random% variable}\\ symbol\theta_{d=1\dots M}&\sim&\operatorname{Dirichlet}_{K}(symbol\alpha)\\ symbol\phi_{k=1\dots K}&\sim&\operatorname{Dirichlet}_{V}(symbol\beta)\\ z_{d=1\dots M,n=1\dots N_{d}}&\sim&\operatorname{Categorical}_{K}(symbol\theta% _{d})\\ w_{d=1\dots M,n=1\dots N_{d}}&\sim&\operatorname{Categorical}_{V}(symbol\phi_{% z_{dn}})\\ \end{array}
  72. Pr ( w d n = v 𝕎 ( - d n ) , , s y m b o l β ) # 𝕎 v k , ( - d n ) + β v Pr ( z d n = k ( - d n ) , w d n = v , 𝕎 ( - d n ) , s y m b o l α ) ( # k d , ( - d n ) + α k ) Pr ( w d n = v 𝕎 ( - d n ) , , s y m b o l β ) \begin{array}[]{lcl}\Pr(w_{dn}=v\mid\mathbb{W}^{(-dn)},\mathbb{Z},symbol\beta)% &\propto&\#\mathbb{W}_{v}^{k,(-dn)}+\beta_{v}\\ \Pr(z_{dn}=k\mid\mathbb{Z}^{(-dn)},w_{dn}=v,\mathbb{W}^{(-dn)},symbol\alpha)&% \propto&(\#\mathbb{Z}_{k}^{d,(-dn)}+\alpha_{k})\Pr(w_{dn}=v\mid\mathbb{W}^{(-% dn)},\mathbb{Z},symbol\beta)\\ \end{array}
  73. # 𝕎 v k , ( - d n ) = number of words having value v among topic k excluding w d n # k d , ( - d n ) = number of topics having value k among document d excluding z d n \begin{array}[]{lcl}\#\mathbb{W}_{v}^{k,(-dn)}&=&\,\text{number of words % having value }v\,\text{ among topic }k\,\text{ excluding }w_{dn}\\ \#\mathbb{Z}_{k}^{d,(-dn)}&=&\,\text{number of topics having value }k\,\text{ % among document }d\,\text{ excluding }z_{dn}\\ \end{array}
  74. Pr ( z d n = k ( - d n ) , w d n = v , 𝕎 ( - d n ) , s y m b o l α ) ( # k d , ( - d n ) + α k ) # 𝕎 v k , ( - d n ) + β v v = 1 V ( # 𝕎 v k , ( - d n ) + β v ) = ( # k d , ( - d n ) + α k ) # 𝕎 v k , ( - d n ) + β v # 𝕎 k + B - 1 \begin{array}[]{rcl}\Pr(z_{dn}=k\mid\mathbb{Z}^{(-dn)},w_{dn}=v,\mathbb{W}^{(-% dn)},symbol\alpha)&\propto&\bigl(\#\mathbb{Z}_{k}^{d,(-dn)}+\alpha_{k}\bigr)% \dfrac{\#\mathbb{W}_{v}^{k,(-dn)}+\beta_{v}}{\sum_{v^{\prime}=1}^{V}(\#\mathbb% {W}_{v^{\prime}}^{k,(-dn)}+\beta_{v^{\prime}})}\\ &&\\ &=&\bigl(\#\mathbb{Z}_{k}^{d,(-dn)}+\alpha_{k}\bigr)\dfrac{\#\mathbb{W}_{v}^{k% ,(-dn)}+\beta_{v}}{\#\mathbb{W}^{k}+B-1}\end{array}
  75. # 𝕎 k = number of words generated by topic k B = v = 1 V β v \begin{array}[]{lcl}\#\mathbb{W}^{k}&=&\,\text{number of words generated by % topic }k\\ B&=&\sum_{v=1}^{V}\beta_{v}\\ \end{array}
  76. z z
  77. F ( z ) \operatorname{F}(\dots\mid z)
  78. z z
  79. z d n z_{dn}
  80. w d n w_{dn}
  81. 𝕎 k \mathbb{W}^{k}
  82. z d n z_{dn}
  83. w d n w_{dn}
  84. p ( 𝕎 k z d n ) = p ( w d n 𝕎 k , ( - d n ) , z d n ) p ( 𝕎 k , ( - d n ) z d n ) = p ( w d n 𝕎 k , ( - d n ) , z d n ) p ( 𝕎 k , ( - d n ) ) p ( w d n 𝕎 k , ( - d n ) , z d n ) \begin{array}[]{lcl}p(\mathbb{W}^{k}\mid z_{dn})&=&p(w_{dn}\mid\mathbb{W}^{k,(% -dn)},z_{dn})\,p(\mathbb{W}^{k,(-dn)}\mid z_{dn})\\ &=&p(w_{dn}\mid\mathbb{W}^{k,(-dn)},z_{dn})\,p(\mathbb{W}^{k,(-dn)})\\ &\sim&p(w_{dn}\mid\mathbb{W}^{k,(-dn)},z_{dn})\end{array}
  85. 𝕎 k , ( - d n ) \mathbb{W}^{k,(-dn)}
  86. 𝕎 k \mathbb{W}^{k}
  87. w d n w_{dn}
  88. z d n z_{dn}
  89. s y m b o l α A Dirichlet hyperprior, either a constant or a random variable s y m b o l β A Dirichlet hyperprior, either a constant or a random variable s y m b o l θ d = 1 M Dirichlet K ( s y m b o l α ) s y m b o l ϕ k = 1 K Dirichlet V ( s y m b o l β ) z d = 1 M Categorical K ( s y m b o l θ d ) w d = 1 M , n = 1 N d Categorical V ( s y m b o l ϕ z d ) \begin{array}[]{lcl}symbol\alpha&\sim&\,\text{A Dirichlet hyperprior, either a% constant or a random variable}\\ symbol\beta&\sim&\,\text{A Dirichlet hyperprior, either a constant or a random% variable}\\ symbol\theta_{d=1\dots M}&\sim&\operatorname{Dirichlet}_{K}(symbol\alpha)\\ symbol\phi_{k=1\dots K}&\sim&\operatorname{Dirichlet}_{V}(symbol\beta)\\ z_{d=1\dots M}&\sim&\operatorname{Categorical}_{K}(symbol\theta_{d})\\ w_{d=1\dots M,n=1\dots N_{d}}&\sim&\operatorname{Categorical}_{V}(symbol\phi_{% z_{d}})\\ \end{array}
  90. Pr ( w d n = v 𝕎 ( - d n ) , , s y m b o l β ) # 𝕎 v k , ( - d n ) + β v \begin{array}[]{lcl}\Pr(w_{dn}=v\mid\mathbb{W}^{(-dn)},\mathbb{Z},symbol\beta)% &\propto&\#\mathbb{W}_{v}^{k,(-dn)}+\beta_{v}\\ \end{array}
  91. # 𝕎 v k , ( - d n ) = number of words having value v among documents with label k excluding w d n \begin{array}[]{lcl}\#\mathbb{W}_{v}^{k,(-dn)}&=&\,\text{number of words % having value }v\,\text{ among documents with label }k\,\text{ excluding }w_{dn% }\\ \end{array}
  92. F ( z d ) \operatorname{F}(\dots\mid z_{d})
  93. z d z_{d}
  94. 𝐱 = ( n 1 , , n K ) \mathbf{x}=(n_{1},\dots,n_{K})
  95. Pr ( 𝐱 \midsymbol α ) = 𝐩 Pr ( 𝐱 𝐩 ) Pr ( 𝐩 \midsymbol α ) d 𝐩 \Pr(\mathbf{x}\midsymbol{\alpha})=\int_{\mathbf{p}}\Pr(\mathbf{x}\mid\mathbf{p% })\Pr(\mathbf{p}\midsymbol{\alpha})\textrm{d}\mathbf{p}
  96. Pr ( 𝐱 \midsymbol α ) = N ! k ( n k ! ) Γ ( A ) Γ ( N + A ) k Γ ( n k + α k ) Γ ( α k ) \Pr(\mathbf{x}\midsymbol{\alpha})=\frac{N!}{\prod_{k}\left(n_{k}!\right)}\frac% {\Gamma\left(A\right)}{\Gamma\left(N+A\right)}\prod_{k}\frac{\Gamma(n_{k}+% \alpha_{k})}{\Gamma(\alpha_{k})}
  97. A = α k A=\sum\alpha_{k}
  98. Pr ( 𝐱 \midsymbol α ) = N B ( A , N ) k : n k > 0 n k B ( α k , n k ) . \Pr(\mathbf{x}\midsymbol{\alpha})=\frac{NB\left(A,N\right)}{\prod_{k:n_{k}>0}n% _{k}B\left(\alpha_{k},n_{k}\right)}.

Dirichlet_algebra.html

  1. ( X ) \mathcal{R}(X)
  2. X X
  3. X X
  4. 𝒮 = ( X ) + ( X ) ¯ \mathcal{S}=\mathcal{R}(X)+\bar{\mathcal{R}(X)}
  5. C ( X ) C(X)
  6. C ( X ) C\left(\partial X\right)
  7. 𝒮 \mathcal{S}
  8. C ( X ) C\left(\partial X\right)
  9. ( X ) \mathcal{R}(X)
  10. T T
  11. X X
  12. ( X ) \mathcal{R}(X)
  13. T T
  14. X = 𝔻 . X=\mathbb{D}.

Dirichlet_process.html

  1. H H
  2. α \alpha
  3. α 0 \alpha\rightarrow 0
  4. α \alpha\rightarrow\infty
  5. α \alpha
  6. X 1 , X 2 , X_{1},X_{2},\dots
  7. H H
  8. α \alpha
  9. X 1 X_{1}
  10. H H
  11. n > 1 n>1
  12. α α + n - 1 \frac{\alpha}{\alpha+n-1}
  13. X n X_{n}
  14. H H
  15. n x α + n - 1 \frac{n_{x}}{\alpha+n-1}
  16. X n = x X_{n}=x
  17. n x n_{x}
  18. X j , j < n X_{j},j<n
  19. X j = x X_{j}=x
  20. X 1 , X 2 , X_{1},X_{2},\dots
  21. P P
  22. X 1 , X 2 , X_{1},X_{2},\dots
  23. x x
  24. X 1 , X 2 , X_{1},X_{2},\dots
  25. P P
  26. P P
  27. DP \mathrm{DP}
  28. P P
  29. DP ( H , α ) \mathrm{DP}\left(H,\alpha\right)
  30. X 1 , X 2 X_{1},X_{2}\dots
  31. P P
  32. P P
  33. α \alpha
  34. DP ( H , α ) \mathrm{DP}(H,\alpha)
  35. { B i } i = 1 n \left\{B_{i}\right\}_{i=1}^{n}
  36. if X DP ( H , α ) \,\text{if }X\sim\mathrm{DP}\left(H,\alpha\right)
  37. then ( X ( B 1 ) , , X ( B n ) ) Dir ( α H ( B 1 ) , , α H ( B n ) ) \,\text{then }\left(X\left(B_{1}\right),\dots,X\left(B_{n}\right)\right)\sim% \mathrm{Dir}\left(\alpha H\left(B_{1}\right),\dots,\alpha H\left(B_{n}\right)\right)
  38. Dir \mathrm{Dir}
  39. X DP ( H , α ) X\sim\mathrm{DP}\left(H,\alpha\right)
  40. X X
  41. DP ( H , α ) \mathrm{DP}\left(H,\alpha\right)
  42. H H
  43. α \alpha
  44. f ( x ) = k = 1 β k δ x k ( x ) f\left(x\right)=\sum_{k=1}^{\infty}\beta_{k}\delta_{x_{k}}\left(x\right)
  45. { x k } k = 1 \left\{x_{k}\right\}_{k=1}^{\infty}
  46. H H
  47. δ x k \delta_{x_{k}}
  48. x k x_{k}
  49. δ x k ( x k ) = 1 \delta_{x_{k}}(x_{k})=1
  50. β k \beta_{k}
  51. Beta ( 1 , α ) \mathrm{Beta}\left(1,\alpha\right)
  52. N ( μ k , 1 / 4 ) N(\mu_{k},1/4)
  53. μ k \mu_{k}
  54. α = 0.5 \alpha=0.5
  55. H = N ( 2 , 16 ) H=N(2,16)
  56. v i N ( μ k , σ 2 ) v_{i}\sim N(\mu_{k},\sigma^{2})
  57. i i
  58. k k
  59. K K
  60. σ 2 \sigma^{2}
  61. i i
  62. k k
  63. z i = k z_{i}=k
  64. ( v i z i = k , μ k ) \displaystyle(v_{i}\mid z_{i}=k,\mu_{k})
  65. K K
  66. μ k \mu_{k}
  67. π k \pi_{k}
  68. k k
  69. Dir ( α / K 𝟏 K ) \mathrm{Dir}\left(\alpha/K\cdot\mathbf{1}_{K}\right)
  70. Dir \mathrm{Dir}
  71. 𝟏 K \mathbf{1}_{K}
  72. K K
  73. H ( λ ) H(\lambda)
  74. H H
  75. λ \lambda
  76. α \alpha
  77. λ \lambda
  78. ( v i μ ~ i ) \displaystyle(v_{i}\mid\tilde{\mu}_{i})
  79. μ ~ i \tilde{\mu}_{i}
  80. G G
  81. K K
  82. μ ~ i \tilde{\mu}_{i}
  83. G G
  84. G G
  85. K K
  86. G ( μ ~ i ) = k = 1 π k δ μ k ( μ ~ i ) G(\tilde{\mu}_{i})=\sum_{k=1}^{\infty}\pi_{k}\delta_{\mu_{k}}(\tilde{\mu}_{i})
  87. μ k \mu_{k}
  88. H ( λ ) H\left(\lambda\right)
  89. π k \pi_{k}
  90. ( v i μ ~ i ) \displaystyle(v_{i}\mid\tilde{\mu}_{i})
  91. n n
  92. K K
  93. K K
  94. μ k \mu_{k}
  95. H ( λ ) H(\lambda)
  96. π \pi
  97. Dir ( α / K 𝟏 K ) \mathrm{Dir}\left(\alpha/K\cdot\mathbf{1}_{K}\right)
  98. k k
  99. π k \pi_{k}
  100. N ( μ k , σ 2 ) N\left(\mu_{k},\sigma^{2}\right)
  101. s y m b o l π symbol{\pi}
  102. v i v_{i}
  103. s y m b o l π symbol{\pi}
  104. D D
  105. p ( s y m b o l π , s y m b o l μ D ) p\left(symbol{\pi},symbol{\mu}\mid D\right)
  106. K K
  107. K K
  108. J J
  109. { θ j } j = 1 J \left\{\theta_{j}\right\}_{j=1}^{J}
  110. ( J + 1 ) th \left(J+1\right)^{\mathrm{th}}
  111. θ J + 1 1 H ( S ) + J ( H + j = 1 J δ θ j ) \theta_{J+1}\sim\frac{1}{H\left(S\right)+J}\left(H+\sum_{j=1}^{J}\delta_{% \theta_{j}}\right)
  112. δ θ \delta_{\theta}
  113. θ \theta
  114. S S
  115. J J
  116. S S
  117. f ( θ ) = k = 1 β k δ θ k ( θ ) f(\theta)=\sum_{k=1}^{\infty}\beta_{k}\cdot\delta_{\theta_{k}}(\theta)
  118. δ θ k \delta_{\theta_{k}}
  119. δ θ k ( θ k ) = 1 \delta_{\theta_{k}}(\theta_{k})=1
  120. { θ k } k = 1 \left\{\theta_{k}\right\}_{k=1}^{\infty}
  121. { β k } k = 1 \left\{\beta_{k}\right\}_{k=1}^{\infty}
  122. θ k \theta_{k}
  123. H H
  124. β k \beta_{k}
  125. β k = β k i = 1 k - 1 ( 1 - β i ) \beta_{k}=\beta^{\prime}_{k}\cdot\prod_{i=1}^{k-1}\left(1-\beta^{\prime}_{i}\right)
  126. β k \beta^{\prime}_{k}
  127. Beta ( 1 , α ) \mathrm{Beta}\left(1,\alpha\right)
  128. β k \beta_{k}
  129. β k \beta^{\prime}_{k}
  130. β k \beta_{k}
  131. i = 1 k - 1 ( 1 - β i ) \prod_{i=1}^{k-1}\left(1-\beta^{\prime}_{i}\right)
  132. β k \beta^{\prime}_{k}
  133. β k \beta_{k}
  134. α \alpha
  135. α \alpha
  136. H H
  137. S S

Discharging_method_(discrete_mathematics).html

  1. V V
  2. F F
  3. E E
  4. 6 - d ( v ) 6-d(v)
  5. v v
  6. 6 - 2 d ( f ) 6-2d(f)
  7. f f
  8. d ( x ) d(x)
  9. f F 6 - 2 d ( f ) + v V 6 - d ( v ) = \displaystyle\sum_{f\in F}6-2d(f)+\sum_{v\in V}6-d(v)=
  10. d ( v ) / 5 d(v)/5
  11. 6 - d ( v ) 6-d(v)
  12. 6 - 4 d ( v ) / 5 6-4d(v)/5
  13. v v
  14. u u
  15. u v uv
  16. v v
  17. v v
  18. v v

Discounted_maximum_loss.html

  1. S S
  2. X X
  3. X s X_{s}
  4. s S s\in S
  5. X 1 : S , , X S : S X_{1:S},...,X_{S:S}
  6. - δ X 1 : S -\delta X_{1:S}
  7. δ \delta
  8. ( Ω , , ) (\Omega,\mathcal{F},\mathbb{P})
  9. X X
  10. δ X ( ω ) \delta X(\omega)
  11. ω Ω \omega\in\Omega
  12. - ess . inf δ X = - sup δ { x : ( X x ) = 1 } -\operatorname{ess.inf}\delta X=-\sup\delta\{x\in\mathbb{R}:\mathbb{P}(X\geq x% )=1\}
  13. ess . inf \operatorname{ess.inf}
  14. α = 0 \alpha=0
  15. ρ max \rho_{\max}
  16. ρ \rho
  17. X X
  18. ρ ( X ) ρ max ( X ) \rho(X)\leq\rho_{\max}(X)
  19. 80 × 0.8 = 64 80\times 0.8=64

Discrepancy_of_hypergraphs.html

  1. χ { - 1 , + 1 } \chi\rightarrow\{-1,+1\}
  2. χ - 1 ( - 1 ) \chi^{-1}(-1)
  3. χ - 1 ( + 1 ) \chi^{-1}(+1)
  4. E E\in\mathcal{E}
  5. χ ( E ) := v E χ ( v ) . \chi(E):=\sum_{v\in E}\chi(v).
  6. \mathcal{H}
  7. χ \chi
  8. \mathcal{H}
  9. d i s c ( , χ ) := max E | χ ( E ) | , disc(\mathcal{H},\chi):=\max_{E\in\mathcal{E}}|\chi(E)|,
  10. d i s c ( ) := min χ : V { - 1 , + 1 } d i s c ( , χ ) . disc(\mathcal{H}):=\min_{\chi:V\rightarrow\{-1,+1\}}disc(\mathcal{H},\chi).
  11. \mathcal{H}
  12. E 1 E 2 = E_{1}\cap E_{2}=\emptyset
  13. E 1 , E 2 E_{1},E_{2}\in\mathcal{E}
  14. ( V , 2 V ) (V,2^{V})
  15. 1 2 | V | \lceil\frac{1}{2}|V|\rceil
  16. χ \chi
  17. 1 2 | V | \lceil\frac{1}{2}|V|\rceil
  18. 1 2 | V | \lfloor\frac{1}{2}|V|\rfloor
  19. 1 2 | V | \lceil\frac{1}{2}|V|\rceil
  20. \mathcal{H}
  21. n = 4 k n=4k
  22. k 𝒩 k\in\mathcal{N}
  23. n = ( [ n ] , { E [ n ] | E [ 2 k ] | = | E [ 2 k ] | } ) \mathcal{H}_{n}=([n],\{E\subseteq[n]\mid|E\cap[2k]|=|E\setminus[2k]|\})
  24. n \mathcal{H}_{n}
  25. ( n / 2 n / 4 ) 2 = Θ ( 1 n 2 n ) {\left({{n/2}\atop{n/4}}\right)}^{2}=\Theta(\frac{1}{n}2^{n})
  26. d i s c ( ) 2 n ln ( 2 m ) . disc(\mathcal{H})\leq\sqrt{2n\ln(2m)}.
  27. χ : V { - 1 , 1 } \chi:V\rightarrow\{-1,1\}
  28. Pr ( χ ( v ) = - 1 ) = Pr ( χ ( v ) = 1 ) = 1 2 \Pr(\chi(v)=-1)=\Pr(\chi(v)=1)=\frac{1}{2}
  29. v V v\in V
  30. χ ( E ) = v E χ ( v ) \chi(E)=\sum_{v\in E}\chi(v)
  31. Pr ( | χ ( E ) | > λ ) < 2 exp ( - λ 2 / ( 2 n ) ) \Pr(|\chi(E)|>\lambda)<2\exp(-\lambda^{2}/(2n))
  32. E V E\subseteq V
  33. λ 0 \lambda\geq 0
  34. λ = 2 n ln ( 2 m ) \lambda=\sqrt{2n\ln(2m)}
  35. Pr ( d i s c ( , χ ) > λ ) E Pr ( | χ ( E ) | > λ ) < 1. \Pr(disc(\mathcal{H},\chi)>\lambda)\leq\sum_{E\in\mathcal{E}}\Pr(|\chi(E)|>% \lambda)<1.
  36. λ \lambda
  37. λ \lambda
  38. d i s c ( ) λ . disc(\mathcal{H})\leq\lambda.\ \Box
  39. \mathcal{H}
  40. m n m\geq n
  41. d i s c ( ) = O ( n ) . disc(\mathcal{H})=O(\sqrt{n}).
  42. m = O ( n ) m=O(n)
  43. m = n m=n
  44. d i s c ( ) 6 n disc(\mathcal{H})\leq 6\sqrt{n}
  45. \mathcal{H}
  46. d i s c ( ) < 2 t disc(\mathcal{H})<2t
  47. \mathcal{H}
  48. d i s c ( ) = O ( t ) disc(\mathcal{H})=O(\sqrt{t})
  49. d i s c ( ) 2 t - 3 disc(\mathcal{H})\leq 2t-3
  50. t 3 t\geq 3
  51. d i s c ( ) C t log m log n disc(\mathcal{H})\leq C\sqrt{t\log m}\log n
  52. d i s c ( ) = O ( t log n ) disc(\mathcal{H})=O(\sqrt{t\log n})

Discrete_exterior_calculus.html

  1. M d ω = M ω . \int_{M}\mathrm{d}\omega=\int_{\partial M}\omega.
  2. d ω M = ω M . \langle\mathrm{d}\omega\mid M\rangle=\langle\omega\mid\partial M\rangle.
  3. d ω S = ω S . \langle\mathrm{d}\omega\mid S\rangle=\langle\omega\mid\partial S\rangle.

Discrete_measure.html

  1. μ \mu
  2. [ 0 , ] [0,\infty]
  3. s 1 , s 2 , s_{1},s_{2},\dots\,
  4. μ ( \ { s 1 , s 2 , } ) = 0. \mu(\mathbb{R}\backslash\{s_{1},s_{2},\dots\})=0.
  5. δ . \delta.
  6. δ ( \ { 0 } ) = 0 \delta(\mathbb{R}\backslash\{0\})=0
  7. δ ( { 0 } ) = 1. \delta(\{0\})=1.
  8. s 1 , s 2 , s_{1},s_{2},\dots
  9. a 1 , a 2 , a_{1},a_{2},\dots
  10. [ 0 , ] [0,\infty]
  11. δ s i \delta_{s_{i}}
  12. δ s i ( X ) = { 1 if s i X 0 if s i X \delta_{s_{i}}(X)=\begin{cases}1&\mbox{ if }s_{i}\in X\\ 0&\mbox{ if }s_{i}\not\in X\\ \end{cases}
  13. X . X.
  14. μ = i a i δ s i \mu=\sum_{i}a_{i}\delta_{s_{i}}
  15. s 1 , s 2 , s_{1},s_{2},\dots
  16. a 1 , a 2 , a_{1},a_{2},\dots
  17. ( X , Σ ) , (X,\Sigma),
  18. μ \mu
  19. ν \nu
  20. μ \mu
  21. ν \nu
  22. S S
  23. X X
  24. { s } \{s\}
  25. s s
  26. S S
  27. S S
  28. ν ( S ) = 0 \nu(S)=0\,
  29. μ ( X \ S ) = 0. \mu(X\backslash S)=0.\,
  30. ν \nu
  31. μ \mu
  32. ( X , Σ ) (X,\Sigma)
  33. ν \nu
  34. μ \mu
  35. μ = i a i δ s i \mu=\sum_{i}a_{i}\delta_{s_{i}}
  36. S = { s 1 , s 2 , } , S=\{s_{1},s_{2},\dots\},
  37. { s i } \{s_{i}\}
  38. Σ , \Sigma,
  39. ν \nu
  40. ν \nu
  41. S S
  42. μ \mu
  43. X \ S . X\backslash S.

Discrete_phase-type_distribution.html

  1. m m
  2. P = [ T 𝐓 0 𝟎 1 ] , {P}=\left[\begin{matrix}{T}&\mathbf{T}^{0}\\ \mathbf{0}&1\end{matrix}\right],
  3. T {T}
  4. m × m m\times m
  5. 𝐓 0 + T 𝟏 = 𝟏 \mathbf{T}^{0}+{T}\mathbf{1}=\mathbf{1}
  6. T {T}
  7. { 0 , 1 , 2 , } \{0,1,2,...\}
  8. T {T}
  9. τ \tau
  10. PH d ( s y m b o l τ , T ) \mathrm{PH}_{d}(symbol{\tau},{T})
  11. DPH ( s y m b o l τ , T ) \mathrm{DPH}(symbol{\tau},{T})
  12. F ( k ) = 1 - s y m b o l τ T k 𝟏 , F(k)=1-symbol{\tau}{T}^{k}\mathbf{1},
  13. k = 0 , 1 , 2 , k=0,1,2,...
  14. f ( k ) = s y m b o l τ T k - 1 𝐓 𝟎 , f(k)=symbol{\tau}{T}^{k-1}\mathbf{T^{0}},
  15. k = 1 , 2 , k=1,2,...
  16. E [ K ( K - 1 ) ( K - n + 1 ) ] = n ! s y m b o l τ ( I - T ) - n T n - 1 𝟏 , E[K(K-1)...(K-n+1)]=n!symbol{\tau}(I-{T})^{-n}{T}^{n-1}\mathbf{1},
  17. I I

Discriminative_model.html

  1. y y
  2. x x
  3. P ( y | x ) P(y|x)
  4. y y
  5. x x
  6. x x
  7. y y

Disk_covering_problem.html

  1. r ( n ) r(n)
  2. n n
  3. r ( n ) r(n)
  4. 3 / 2 \sqrt{3}/2
  5. 2 / 2 \sqrt{2}/2
  6. 1 / 2 1/2

Disk_laser.html

  1. L ~{}L~{}
  2. L ~{}L~{}
  3. u = G L ~{}u=GL~{}
  4. G G~{}
  5. h ~{}h
  6. g = 2 G h ~{}g=2Gh~{}
  7. β \beta~{}
  8. g - β g\!-\!\beta~{}
  9. G G~{}
  10. β ~{}\beta~{}
  11. h h
  12. Q Q~{}
  13. η 0 = ω s / ω p \eta_{0}=\omega_{\rm s}/\omega_{\rm p}~{}~{}
  14. R R~{}
  15. P k = η 0 R 2 Q β 3 P_{\rm k}=\eta_{0}\frac{R^{2}}{Q\beta^{3}}~{}
  16. h R Q β h\sim\frac{R}{Q\beta}
  17. L R Q β 2 L\sim\frac{R}{Q\beta^{2}}
  18. β \beta
  19. P s P_{\rm s}
  20. β \beta
  21. β \beta
  22. β \beta
  23. P s P_{\rm s}
  24. s = P s / P d s=P_{\rm s}/P_{\rm d}
  25. P s P_{\rm s}
  26. P d = R 2 / Q P_{\rm d}=R^{2}/Q
  27. P k = P d / β 3 P_{\rm k}=P_{\rm d}/\beta^{3}
  28. β = s 1 / 3 \beta=s^{1/3}
  29. β \beta
  30. ( β , s ) (\beta,s)
  31. β \beta

Disk_loading.html

  1. A A
  2. v v
  3. ρ \rho
  4. m ˙ {}^{\dot{m}}
  5. m ˙ = ρ A v . \dot{m}=\rho\,A\,v.
  6. w w
  7. T T
  8. T = m ˙ w . T=\dot{m}\,w.
  9. T v = 1 2 m ˙ w 2 . T\,v=\tfrac{1}{2}\,\dot{m}\,{w^{2}}.
  10. T T
  11. v = 1 2 w . v=\tfrac{1}{2}\,w.
  12. p 0 p_{0}
  13. p 0 = p 1 + 1 2 ρ v 2 . p_{0}=\,p_{1}+\ \tfrac{1}{2}\,\rho\,v^{2}.
  14. p 2 + 1 2 ρ v 2 = p 0 + 1 2 ρ w 2 . p_{2}+\ \tfrac{1}{2}\,\rho\,v^{2}=\,p_{0}+\ \tfrac{1}{2}\,\rho\,w^{2}.
  15. T / A T/\,A
  16. T A = p 2 - p 1 = 1 2 ρ w 2 \frac{T}{A}=p_{2}-\,p_{1}=\tfrac{1}{2}\,\rho\,w^{2}
  17. p 0 + 1 2 ρ w 2 = p 0 + T A . p_{0}+\tfrac{1}{2}\,\rho\,w^{2}=\,p_{0}+\frac{T}{A}.
  18. p 0 - 1 2 ρ v 2 = p 0 - 1 4 T A . p_{0}-\tfrac{1}{2}\,\rho\,v^{2}=\,p_{0}-\,\tfrac{1}{4}\frac{T}{A}.
  19. p 0 + 3 2 ρ v 2 = p 0 + 3 4 T A . p_{0}+\tfrac{3}{2}\,\rho\,v^{2}=\,p_{0}+\,\tfrac{3}{4}\frac{T}{A}.
  20. T = m ˙ w = m ˙ ( 2 v ) = 2 ρ A v 2 . T=\dot{m}\,w=\dot{m}\,(2v)=2\rho\,A\,v^{2}.
  21. v = T A 1 2 ρ . v=\sqrt{\frac{T}{A}\cdot\frac{1}{2\rho}}.
  22. T / A T/A
  23. P P
  24. P = T v = T T A 1 2 ρ . P=Tv=T\sqrt{\frac{T}{A}\cdot\frac{1}{2\rho}}.
  25. v = P T = [ T P ] - 1 . v=\frac{P}{T}=\left[\frac{T}{P}\right]^{-1}.
  26. T / P T/P

Disodium_phosphate.html

  1. \overrightarrow{\leftarrow}

Disphenoid.html

  1. V = ( l 2 + m 2 - n 2 ) ( l 2 - m 2 + n 2 ) ( - l 2 + m 2 + n 2 ) 72 . V=\sqrt{\frac{(l^{2}+m^{2}-n^{2})(l^{2}-m^{2}+n^{2})(-l^{2}+m^{2}+n^{2})}{72}}.
  2. R = l 2 + m 2 + n 2 8 R=\sqrt{\frac{l^{2}+m^{2}+n^{2}}{8}}
  3. r = 3 V 4 T r=\frac{3V}{4T}
  4. 16 T 2 R 2 = l 2 m 2 n 2 + 9 V 2 . \displaystyle 16T^{2}R^{2}=l^{2}m^{2}n^{2}+9V^{2}.
  5. 1 2 ( l 2 + m 2 - n 2 ) , 1 2 ( l 2 - m 2 + n 2 ) , 1 2 ( - l 2 + m 2 + n 2 ) . \tfrac{1}{2}(l^{2}+m^{2}-n^{2}),\quad\tfrac{1}{2}(l^{2}-m^{2}+n^{2}),\quad% \tfrac{1}{2}(-l^{2}+m^{2}+n^{2}).

Displacement–length_ratio.html

  1. 𝐷𝐿𝑅 = 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 ( lb ) / 2240 ( 0.01 × 𝐿𝑊𝐿 ( ft ) ) 3 . \mathit{DLR}=\frac{\mathit{displacement}(\mathrm{lb})~{}/~{}2240}{(0.01\times% \mathit{LWL}(\mathrm{ft}))^{3}}.

Dissipative_operator.html

  1. ( λ I - A ) x λ x . \|(\lambda I-A)x\|\geq\lambda\|x\|.
  2. ( λ I - A ) x 0 , \|(\lambda I-A)x\|\neq 0,
  3. ( λ I - A ) x > λ x \|(\lambda I-A)x\|>\lambda\|x\|
  4. λ x + A x ( λ I - A ) x > λ x , \|\lambda x\|+\|Ax\|\geq\|(\lambda I-A)x\|>\lambda\|x\|,
  5. ( λ I - A ) - 1 z 1 λ z \|(\lambda I-A)^{-1}z\|\leq\frac{1}{\lambda}\|z\|
  6. z = ( λ I - A ) x . z=(\lambda I-A)x.
  7. ( I - κ A ) - 1 z z or ( I - κ A ) x x \|(I-\kappa A)^{-1}z\|\leq\|z\|\,\text{ or }\|(I-\kappa A)x\|\geq\|x\|
  8. J ( x ) := { x X : x X 2 = x X 2 = x , x } . J(x):=\left\{x^{\prime}\in X^{\prime}:\|x^{\prime}\|_{X^{\prime}}^{2}=\|x\|_{X% }^{2}=\langle x^{\prime},x\rangle\right\}.
  9. Re A x , x 0. {\rm Re}\langle Ax,x^{\prime}\rangle\leq 0.
  10. Re A x , x 0 {\rm Re}\langle Ax,x\rangle\leq 0
  11. x - A x 2 = x 2 + A x 2 - 2 R e A x , x x 2 + A x 2 + 2 R e A x , x = x + A x 2 \|x-Ax\|^{2}=\|x\|^{2}+\|Ax\|^{2}-2{\rm Re}\langle Ax,x\rangle\geq\|x\|^{2}+\|% Ax\|^{2}+2{\rm Re}\langle Ax,x\rangle=\|x+Ax\|^{2}
  12. x - A x x + A x \therefore\|x-Ax\|\geq\|x+Ax\|
  13. ( I + A ) ( I - A ) - 1 (I+A)(I-A)^{-1}
  14. ( λ I + A ) ( λ I - A ) - 1 (\lambda I+A)(\lambda I-A)^{-1}
  15. x A x = x ( - x ) = - x 2 0 , x\cdot Ax=x\cdot(-x)=-\|x\|^{2}\leq 0,
  16. x * A x , x^{*}Ax,
  17. x * A + A * 2 x . x^{*}\frac{A+A^{*}}{2}x.
  18. λ , λ - A \lambda,\lambda-A
  19. ( λ + A ) ( λ - A ) - 1 (\lambda+A)(\lambda-A)^{-1}
  20. u , A u = 0 1 u ( x ) u ( x ) d x = - 1 2 u ( 0 ) 2 0. \langle u,Au\rangle=\int_{0}^{1}u(x)u^{\prime}(x)\,\mathrm{d}x=-\frac{1}{2}u(0% )^{2}\leq 0.
  21. u - λ u = f u-\lambda u^{\prime}=f
  22. u , Δ u = Ω u ( x ) Δ u ( x ) d x = - Ω | u ( x ) | 2 d x = - u L 2 ( Ω ; 𝐑 ) 0 , \langle u,\Delta u\rangle=\int_{\Omega}u(x)\Delta u(x)\,\mathrm{d}x=-\int_{% \Omega}\big|\nabla u(x)\big|^{2}\,\mathrm{d}x=-\|\nabla u\|_{L^{2}(\Omega;% \mathbf{R})}\leq 0,

Dissipative_particle_dynamics.html

  1. f i = j i ( F i j C + F i j D + F i j R ) f_{i}=\sum_{j\neq i}(F^{C}_{ij}+F^{D}_{ij}+F^{R}_{ij})

Dissipative_soliton.html

  1. t s y m b o l q = s y m b o l D ¯ Δ s y m b o l q + s y m b o l R ( s y m b o l q ) . \partial_{t}symbol{q}=\underline{symbol{D}}\Delta symbol{q}+symbol{R}(symbol{q% }).
  2. ( τ u t u τ v t v ) = ( d u 2 0 0 d v 2 ) ( Δ u Δ v ) + ( λ u - u 3 - κ 3 v + κ 1 u - v ) . \left(\begin{array}[]{c}\tau_{u}\partial_{t}u\\ \tau_{v}\partial_{t}v\end{array}\right)=\left(\begin{array}[]{cc}d_{u}^{2}&0\\ 0&d_{v}^{2}\end{array}\right)\left(\begin{array}[]{c}\Delta u\\ \Delta v\end{array}\right)+\left(\begin{array}[]{c}\lambda u-u^{3}-\kappa_{3}v% +\kappa_{1}\\ u-v\end{array}\right).
  3. t q = ( d r + i d i ) Δ q + l r q + ( c r + i c i ) | q | 2 q + ( q r + i q i ) | q | 4 q . \partial_{t}q=(d_{r}+id_{i})\Delta q+l_{r}q+(c_{r}+ic_{i})|q|^{2}q+(q_{r}+iq_{% i})|q|^{4}q.
  4. t ρ + s y m b o l m = S = d r ( q Δ q + q Δ q ) + 2 l r ρ + 2 c r ρ 2 + 2 q r ρ 3 with \quadsymbol m = 2 d i Im ( q q ) . \partial_{t}\rho+\nabla\cdot symbol{m}=S=d_{r}(q\Delta q^{\ast}+q^{\ast}\Delta q% )+2l_{r}\rho+2c_{r}\rho^{2}+2q_{r}\rho^{3}\quad\,\text{with}\quadsymbol{m}=2d_% {i}\,\text{Im}(q^{\ast}\nabla q).
  5. t q = ( s r + i s i ) Δ 2 q + ( d r + i d i ) Δ q + l r q + ( c r + i c i ) | q | 2 q + ( q r + i q i ) | q | 4 q . \partial_{t}q=(s_{r}+is_{i})\Delta^{2}q+(d_{r}+id_{i})\Delta q+l_{r}q+(c_{r}+% ic_{i})|q|^{2}q+(q_{r}+iq_{i})|q|^{4}q.
  6. s y m b o l v ˙ = ( σ - σ 0 ) s y m b o l v - | s y m b o l v | 2 s y m b o l v \dot{symbol{v}}=(\sigma-\sigma_{0})symbol{v}-|symbol{v}|^{2}symbol{v}
  7. A ˙ = ( σ - σ 0 ) A - | A | 2 A \dot{A}=(\sigma-\sigma_{0})A-|A|^{2}A

Distance_from_a_point_to_a_plane.html

  1. a x + b y + c z = d ax+by+cz=d
  2. ( x , y , z ) (x,y,z)
  3. x = a d < m t p l > a 2 + b 2 + c 2 , y = b d a 2 + b 2 + c 2 , z = c d a 2 + b 2 + c 2 . \displaystyle x=\frac{ad}{<}mtpl>{{a^{2}+b^{2}+c^{2}}},\quad\quad\displaystyle y% =\frac{bd}{{a^{2}+b^{2}+c^{2}}},\quad\quad\displaystyle z=\frac{cd}{{a^{2}+b^{% 2}+c^{2}}}.
  4. a x + b y + c z ax+by+cz
  5. ( a , b , c ) ( x , y , z ) (a,b,c)\cdot(x,y,z)
  6. a 2 + b 2 + c 2 a^{2}+b^{2}+c^{2}
  7. | ( a , b , c ) | 2 |(a,b,c)|^{2}
  8. 𝐯 = ( a , b , c ) \mathbf{v}=(a,b,c)
  9. 𝐰 \mathbf{w}
  10. 𝐯 𝐰 = d \mathbf{v}\cdot\mathbf{w}=d
  11. 𝐩 = 𝐯 d | 𝐯 | 2 \mathbf{p}=\frac{\mathbf{v}d}{|\mathbf{v}|^{2}}
  12. d | 𝐯 | = d a 2 + b 2 + c 2 \frac{d}{|\mathbf{v}|}=\frac{d}{\sqrt{a^{2}+b^{2}+c^{2}}}
  13. 𝐩 \mathbf{p}
  14. 𝐯 \mathbf{v}
  15. 𝐪 \mathbf{q}
  16. 𝐩 \mathbf{p}
  17. 𝐩 \mathbf{p}
  18. 𝐩 \mathbf{p}
  19. 𝐪 \mathbf{q}
  20. q q
  21. | 𝐩 | 2 + | 𝐩 - 𝐪 | 2 \sqrt{|\mathbf{p}|^{2}+|\mathbf{p}-\mathbf{q}|^{2}}
  22. | 𝐩 - 𝐪 | 2 |\mathbf{p}-\mathbf{q}|^{2}
  23. | 𝐩 | |\mathbf{p}|
  24. 𝐩 \mathbf{p}
  25. 𝐩 \mathbf{p}
  26. 𝐯 \mathbf{v}
  27. 𝐩 \mathbf{p}

Divergent_geometric_series.html

  1. k = 0 a r k = a + a r + a r 2 + a r 3 + \sum_{k=0}^{\infty}ar^{k}=a+ar+ar^{2}+ar^{3}+\cdots
  2. k = 0 a r k = a 1 - r . \sum_{k=0}^{\infty}ar^{k}=\frac{a}{1-r}.

Diversity_factor.html

  1. k s k\text{s}
  2. f D i v e r s i t y = i = 1 n Max ( Load i ) i = 1 n Load i f_{Diversity}=\frac{\sum\limits_{i=1}^{n}\,\text{Max}(\,\text{Load}_{i})}{\sum% \limits_{i=1}^{n}\,\text{Load}_{i}}
  3. ( i = 1 n Load i ) \left(\sum\limits_{i=1}^{n}\,\text{Load}_{i}\right)
  4. f D i v e r s i f i c a t i o n = Diversified Load Maximum system load f_{Diversification}=\frac{\,\text{Diversified Load}}{\,\text{Maximum system % load}}

Diversity_gain.html

  1. k = 1 N 1 k \sum_{k=1}^{N}\frac{1}{k}

DJIA_divisor.html

  1. DJIA = p d \,\text{DJIA}={\sum p\over d}
  2. DJIA = p old d old = p new d new . \,\text{DJIA}={\sum p\text{old}\over d\text{old}}={\sum p\text{new}\over d% \text{new}}.

Doctor_sweetening_process.html

  1. PbO + 2 NaOH Na 2 PbO 2 + H 2 O \mathrm{PbO+2\ NaOH\longrightarrow Na_{2}PbO_{2}+H_{2}O}
  2. 2 RSH + Na 2 PbO 2 ( RS ) 2 Pb + 2 NaOH \mathrm{2\ RSH+Na_{2}PbO_{2}\longrightarrow(RS)_{2}Pb+2\ NaOH}
  3. - ( RS ) 2 Pb + S RS - SR + PbS \mathrm{-(RS)_{2}Pb+S\longrightarrow RS-SR+PbS}
  4. 2 ( RS ) 2 Pb + 4 NaOH + O 2 2 RS - SR + 2 Na 2 PbO 2 + 2 H 2 O \mathrm{2\ (RS)_{2}Pb+4\ NaOH+O_{2}\longrightarrow 2\ RS-SR+2\ Na_{2}PbO_{2}+2% \ H_{2}O}

Dom_Bédos_de_Celles.html

  1. sin ϕ \sin\phi
  2. tan h sin ϕ \tan h\sin\phi
  3. sin ϕ \sin\phi
  4. tan h sin ϕ \tan h\sin\phi

Domino_tiling.html

  1. A 0 A_{0}
  2. A 0 A_{0}
  3. A n + 1 A_{n+1}
  4. A n A_{n}
  5. A n A_{n}
  6. A n + 1 A_{n+1}
  7. 2 m × 2 n 2m\times 2n
  8. 2 m n 2mn
  9. j = 1 m k = 1 n ( 4 cos 2 π j m + 1 + 4 cos 2 π k n + 1 ) 1 4 . \prod_{j=1}^{m}\prod_{k=1}^{n}\left(4\cos^{2}\frac{\pi j}{m+1}+4\cos^{2}\frac{% \pi k}{n+1}\right)^{\frac{1}{4}}.
  10. 2 × n 2\times n
  11. m n × m n mn\times mn

Doob's_martingale_convergence_theorems.html

  1. N s 𝐄 [ N t | F s ] . N_{s}\geq\mathbf{E}\big[N_{t}\big|F_{s}\big].
  2. sup t > 0 𝐄 [ N t - ] < + . \sup_{t>0}\mathbf{E}\big[N_{t}^{-}\big]<+\infty.
  3. N ( ω ) = lim t + N t ( ω ) N(\omega)=\lim_{t\to+\infty}N_{t}(\omega)
  4. lim C sup t > 0 { ω Ω | N t ( ω ) > C } | N t ( ω ) | d 𝐏 ( ω ) = 0 ; \lim_{C\to\infty}\sup_{t>0}\int_{\{\omega\in\Omega|N_{t}(\omega)>C\}}\big|N_{t% }(\omega)\big|\,\mathrm{d}\mathbf{P}(\omega)=0;
  5. 𝐄 [ | N t - N | ] = Ω | N t ( ω ) - N ( ω ) | d 𝐏 ( ω ) 0 as t + . \mathbf{E}\big[\big|N_{t}-N\big|\big]=\int_{\Omega}\big|N_{t}(\omega)-N(\omega% )\big|\,\mathrm{d}\mathbf{P}(\omega)\to 0\mbox{ as }~{}t\to+\infty.
  6. sup t > 0 𝐄 [ | M t | p ] < + \sup_{t>0}\mathbf{E}\big[\big|M_{t}\big|^{p}\big]<+\infty
  7. sup k 𝐍 𝐄 [ | M k | p ] < + \sup_{k\in\mathbf{N}}\mathbf{E}\big[\big|M_{k}\big|^{p}\big]<+\infty
  8. 𝐄 [ X | F k ] 𝐄 [ X | F ] as k \mathbf{E}\big[X\big|F_{k}\big]\to\mathbf{E}\big[X\big|F_{\infty}\big]\mbox{ % as }~{}k\to\infty
  9. 𝐏 [ A | F k ] 𝟏 A \mathbf{P}[A|F_{k}]\to\mathbf{1}_{A}
  10. 𝐏 [ A ] = 𝟏 A \mathbf{P}[A]=\mathbf{1}_{A}
  11. 𝐏 [ A ] { 0 , 1 } \mathbf{P}[A]\in\{0,1\}

Doob's_martingale_inequality.html

  1. 𝐏 [ sup 0 t T X t C ] 𝐄 [ X T p ] C p . \mathbf{P}\left[\sup_{0\leq t\leq T}X_{t}\geq C\right]\leq\frac{\mathbf{E}% \left[X_{T}^{p}\right]}{C^{p}}.
  2. X : [ 0 , T ] × Ω [ 0 , + ) X:[0,T]\times\Omega\to[0,+\infty)
  3. 𝐄 [ X T ] = Ω X T ( ω ) d 𝐏 ( ω ) \mathbf{E}[X_{T}]=\int_{\Omega}X_{T}(\omega)\,\mathrm{d}\mathbf{P}(\omega)
  4. s \mathcal{F}_{s}
  5. S t = sup 0 s t X s , S_{t}=\sup_{0\leq s\leq t}X_{s},
  6. X t p = X t L p ( Ω , , 𝐏 ) = ( 𝐄 [ | X t | p ] ) 1 p . \|X_{t}\|_{p}=\|X_{t}\|_{L^{p}(\Omega,\mathcal{F},\mathbf{P})}=\left(\mathbf{E% }\left[|X_{t}|^{p}\right]\right)^{\frac{1}{p}}.
  7. 𝐏 [ S T C ] X T p p C p . \mathbf{P}\left[S_{T}\geq C\right]\leq\frac{\|X_{T}\|_{p}^{p}}{C^{p}}.
  8. S T p e e - 1 ( 1 + X T log X T p ) \|S_{T}\|_{p}\leq\frac{e}{e-1}\left(1+\|X_{T}\log X_{T}\|_{p}\right)
  9. X T p S T p p p - 1 X T p . \|X_{T}\|_{p}\leq\|S_{T}\|_{p}\leq\frac{p}{p-1}\|X_{T}\|_{p}.
  10. 𝐄 [ X 1 + + X n + X n + 1 | X 1 , , X n ] = X 1 + + X n + 𝐄 [ X n + 1 | X 1 , , X n ] = X 1 + + X n , \begin{aligned}\displaystyle\mathbf{E}\left[X_{1}+\dots+X_{n}+X_{n+1}\big|X_{1% },\dots,X_{n}\right]&\displaystyle=X_{1}+\dots+X_{n}+\mathbf{E}\left[X_{n+1}% \big|X_{1},\dots,X_{n}\right]\\ &\displaystyle=X_{1}+\cdots+X_{n},\end{aligned}
  11. 𝐏 [ max 1 i n | M i | λ ] 𝐄 [ M n 2 ] λ 2 , \mathbf{P}\left[\max_{1\leq i\leq n}\left|M_{i}\right|\geq\lambda\right]\leq% \frac{\mathbf{E}\left[M_{n}^{2}\right]}{\lambda^{2}},
  12. 𝐏 [ sup 0 t T B t C ] exp ( - C 2 2 T ) . \mathbf{P}\left[\sup_{0\leq t\leq T}B_{t}\geq C\right]\leq\exp\left(-\frac{C^{% 2}}{2T}\right).
  13. { sup 0 t T B t C } = { sup 0 t T exp ( λ B t ) exp ( λ C ) } . \left\{\sup_{0\leq t\leq T}B_{t}\geq C\right\}=\left\{\sup_{0\leq t\leq T}\exp% (\lambda B_{t})\geq\exp(\lambda C)\right\}.
  14. 𝐏 [ sup 0 t T B t C ] = 𝐏 [ sup 0 t T exp ( λ B t ) exp ( λ C ) ] 𝐄 [ exp ( λ B T ) ] exp ( λ C ) = exp ( 1 2 λ 2 T - λ C ) 𝐄 [ exp ( λ B t ) ] = exp ( 1 2 λ 2 t ) \begin{aligned}\displaystyle\mathbf{P}\left[\sup_{0\leq t\leq T}B_{t}\geq C% \right]&\displaystyle=\mathbf{P}\left[\sup_{0\leq t\leq T}\exp(\lambda B_{t})% \geq\exp(\lambda C)\right]\\ &\displaystyle\leq\frac{\mathbf{E}\left[\exp(\lambda B_{T})\right]}{\exp(% \lambda C)}\\ &\displaystyle=\exp\left(\tfrac{1}{2}\lambda^{2}T-\lambda C\right)&&% \displaystyle\mathbf{E}\left[\exp(\lambda B_{t})\right]=\exp\left(\tfrac{1}{2}% \lambda^{2}t\right)\end{aligned}

Doppler_spectroscopy.html

  1. r r
  2. r 3 = G M s t a r 4 π 2 P s t a r 2 r^{3}=\frac{GM_{star}}{4\pi^{2}}P_{star}^{2}\,
  3. r r
  4. V P L = G M s t a r / r V_{PL}=\sqrt{GM_{star}/r}\,
  5. V P L V_{PL}
  6. M P L = M s t a r V s t a r V P L M_{PL}=\frac{M_{star}V_{star}}{V_{PL}}\,
  7. V s t a r V_{star}
  8. K = V s t a r sin ( i ) K=V_{star}\sin(i)
  9. × 10 - 4 \times 10^{-}4
  10. × 10 - 3 \times 10^{-}3
  11. × 10 - 2 \times 10^{-}2
  12. × 10 - 1 \times 10^{-}1
  13. × 10 - 1 \times 10^{-}1

Double_exponential_function.html

  1. f ( x ) = a b x = a ( b x ) f(x)=a^{b^{x}}=a^{(b^{x})}
  2. F ( m ) = 2 2 m + 1 F(m)=2^{2^{m}}+1
  3. M M ( p ) = 2 2 p - 1 - 1 MM(p)=2^{2^{p}-1}-1
  4. s n = E 2 n + 1 + 1 2 s_{n}=\left\lfloor E^{2^{n+1}}+\frac{1}{2}\right\rfloor
  5. 2 2 k 2^{2^{k}}
  6. a ( n ) = A 3 n a(n)=\left\lfloor A^{3^{n}}\right\rfloor
  7. 2 4 n 2^{4^{n}}
  8. ( 8 d ) d 15 d 2 2 d + 1 (8d)^{d}\cdot 15^{d\cdot 2^{2d+1}}
  9. N ( y ) = 375.6 1.00185 1.00737 y - 1000 N(y)=375.6\cdot 1.00185^{1.00737^{y-1000}}\,

Dowker_space.html

  1. ω 0 \aleph_{\omega}^{\aleph_{0}}
  2. ω + 1 \aleph_{\omega+1}

Doxastic_logic.html

  1. 𝔹 \mathbb{B}
  2. 𝔹 \mathbb{B}
  3. b 1 , b 2 , , b n b_{1},b_{2},...,b_{n}
  4. p : p p \forall p:\mathcal{B}p\to p
  5. p : ¬ p p \exists p:\neg p\wedge\mathcal{B}p
  6. [ ¬ p : ¬ p p ] \mathcal{B}[\neg\exists p:\neg p\wedge\mathcal{B}p]
  7. [ p : p p ] \mathcal{B}[\forall p:\mathcal{B}p\to p]
  8. ¬ p : p ¬ p \neg\exists p:\mathcal{B}p\wedge\mathcal{B}\neg p
  9. p : p ¬ ¬ p \forall p:\mathcal{B}p\to\neg\mathcal{B}\neg p
  10. p : p p \forall p:\mathcal{B}p\to\mathcal{BB}p
  11. p : p ¬ p \exists p:\mathcal{B}p\wedge\mathcal{B\neg B}p
  12. p q p\to q
  13. p q \mathcal{B}p\to\mathcal{B}q
  14. p : q : ( p q ) ( p q ) \forall p:\forall q:\mathcal{B}(p\to q)\to\mathcal{B}(\mathcal{B}p\to\mathcal{% B}q)
  15. q ( q p ) q\equiv(\mathcal{B}q\to p)
  16. p : q : ( q ( q p ) ) \forall p:\exists q:\mathcal{B}(q\equiv(\mathcal{B}q\to p))
  17. p p \mathcal{B}p\to p
  18. p : p ¬ p \exists p:\mathcal{B}\mathcal{B}p\wedge\neg\mathcal{B}p
  19. p : p p \forall p:\mathcal{BB}p\to\mathcal{B}p
  20. p p \mathcal{B}p\to p
  21. p : ( p p ) p \forall p:\mathcal{B}(\mathcal{B}p\to p)\to\mathcal{B}p
  22. p \mathcal{B}p\to\mathcal{B}\bot
  23. P C p p \vdash_{PC}p\Rightarrow\vdash\mathcal{B}p
  24. p : q : ( p q ) ( p q ) \forall p:\forall q:\mathcal{B}(p\to q)\to(\mathcal{B}p\to\mathcal{B}q)
  25. p : q : ( p q ) ( p q ) \forall p:\forall q:\mathcal{B}(p\to q)\to\mathcal{B}(\mathcal{B}p\to\mathcal{% B}q)
  26. p : q : ( ( p ( p q ) ) q ) \forall p:\forall q:\mathcal{B}((\mathcal{B}p\wedge\mathcal{B}(p\to q))\to% \mathcal{B}q)

Drag_divergence_Mach_number.html

  1. M d d + c l , d e s i g n 10 + t c = K M_{dd}+\frac{c_{l,design}}{10}+\frac{t}{c}=K
  2. M d d M_{dd}
  3. c l , d e s i g n c_{l,design}
  4. t t
  5. c c
  6. K K

Drain-induced_barrier_lowering.html

  1. DIBL = - V T h D D - V T h low V D D - V D low , \mathrm{DIBL}=-\frac{V_{Th}^{DD}-V_{Th}^{\mathrm{low}}}{V_{DD}-V_{D}^{\mathrm{% low}}},
  2. V T h D D V_{Th}^{DD}
  3. V T h low V_{Th}^{\mathrm{low}}
  4. V D D V_{DD}
  5. V D low V_{D}^{\mathrm{low}}
  6. V T h D D V_{Th}^{DD}
  7. V T h low V_{Th}^{\mathrm{low}}
  8. Δ f f = - 2 D I B L V D D - V T h , \frac{\Delta f}{f}=-\frac{2\mathrm{DIBL}}{V_{DD}-V_{Th}},
  9. V D D V_{DD}
  10. V T h V_{Th}

Drude_particle.html

  1. ω \omega
  2. q q
  3. μ \mu
  4. σ = 1 / 2 μ ω \sigma=1/\sqrt{2\mu\omega}
  5. E ( 2 ) = l = 1 E l ( 2 ) = l = 1 - Q 2 α l 2 R 2 l + 2 E^{(2)}=\sum_{l=1}^{\infty}E_{l}^{(2)}=\sum_{l=1}^{\infty}-\frac{Q^{2}\alpha_{% l}}{2R^{2l+2}}
  6. α l \alpha_{l}
  7. α l = [ q 2 μ ω 2 ] [ ( 2 l - 1 ) ! ! l ] ( 2 μ ω ) l - 1 \alpha_{l}=\left[\frac{q^{2}}{\mu\omega^{2}}\right]\left[\frac{(2l-1)!!}{l}% \right]\left(\frac{\hbar}{2\mu\omega}\right)^{l-1}
  8. E ( 2 ) = l = 3 C 2 l R - 2 l E^{(2)}=\sum_{l=3}^{\infty}C_{2l}R^{-2l}
  9. C 6 = 3 4 α 1 α 1 ω C_{6}=\frac{3}{4}\alpha_{1}\alpha_{1}\hbar\omega
  10. C 8 = 5 α 1 α 2 ω C_{8}=5\alpha_{1}\alpha_{2}\hbar\omega
  11. C 10 = ( 21 2 α 1 α 3 + 70 4 α 2 α 2 ) ω C_{10}=\left(\frac{21}{2}\alpha_{1}\alpha_{3}+\frac{70}{4}\alpha_{2}\alpha_{2}% \right)\hbar\omega
  12. 20 9 α 2 α 1 α 3 = 1 \sqrt{\frac{20}{9}}\frac{\alpha_{2}}{\sqrt{\alpha_{1}\alpha_{3}}}=1
  13. 49 40 C 8 C 6 C 10 = 1 \sqrt{\frac{49}{40}}\frac{C_{8}}{\sqrt{C_{6}C_{10}}}=1
  14. C 6 α 1 4 C 9 = 1 \frac{C_{6}\alpha_{1}}{4C_{9}}=1

DSSP_(hydrogen_bond_estimation_algorithm).html

  1. E = 0.084 { 1 r O N + 1 r C H - 1 r O H - 1 r C N } 332 kcal / mol E=0.084\left\{\frac{1}{r_{ON}}+\frac{1}{r_{CH}}-\frac{1}{r_{OH}}-\frac{1}{r_{% CN}}\right\}\cdot 332\,\mathrm{kcal/mol}
  2. C i α C i + 2 α \overrightarrow{C_{i}^{\alpha}C_{i+2}^{\alpha}}
  3. C i - 2 α C i α \overrightarrow{C_{i-2}^{\alpha}C_{i}^{\alpha}}

Dual_cone_and_polar_cone.html

  1. C * = { y X * : y , x 0 x C } , C^{*}=\left\{y\in X^{*}:\langle y,x\rangle\geq 0\quad\forall x\in C\right\},
  2. C * internal := { y X : y , x 0 x C } . C^{*}\text{internal}:=\left\{y\in X:\langle y,x\rangle\geq 0\quad\forall x\in C% \right\}.
  3. C 2 * C 1 * C_{2}^{*}\subseteq C_{1}^{*}
  4. C o = { y X * : y , x 0 x C } . C^{o}=\left\{y\in X^{*}:\langle y,x\rangle\leq 0\quad\forall x\in C\right\}.

Dual_currency_deposit.html

  1. a = 1 + r d e l i v e r y t e r m d a y s c u r r e n c y b a s i s a=1+r\frac{deliverytermdays}{currencybasis}
  2. r = y i e l d o f o p t i o n r=yieldofoption
  3. m u l t i p l i e r = 1 + r d e l i v e r y t e r m d a y s c u r r e n c y b a s i s 1 - y multiplier=\frac{1+r\frac{deliverytermdays}{currencybasis}}{1-y}

Dual_quaternion.html

  1. i 2 = j 2 = k 2 = i j k = - 1. i^{2}=j^{2}=k^{2}=ijk=-1.\!
  2. G = A C = ( a 0 + 𝐀 ) ( c 0 + 𝐂 ) = ( a 0 c 0 - 𝐀 𝐂 ) + ( c 0 𝐀 + a 0 𝐂 + 𝐀 × 𝐂 ) . G=AC=(a_{0}+\mathbf{A})(c_{0}+\mathbf{C})=(a_{0}c_{0}-\mathbf{A}\cdot\mathbf{C% })+(c_{0}\mathbf{A}+a_{0}\mathbf{C}+\mathbf{A}\times\mathbf{C}).
  3. A ^ C ^ = ( A , B ) ( C , D ) = ( A C , A D + B C ) . \hat{A}\hat{C}=(A,B)(C,D)=(AC,AD+BC).\!
  4. G ^ = A ^ C ^ = ( a ^ 0 + 𝖠 ) ( c ^ 0 + 𝖢 ) = ( a ^ 0 c ^ 0 - 𝖠 𝖢 ) + ( c ^ 0 𝖠 + a ^ 0 𝖢 + 𝖠 × 𝖢 ) . \hat{G}=\hat{A}\hat{C}=(\hat{a}_{0}+\mathsf{A})(\hat{c}_{0}+\mathsf{C})=(\hat{% a}_{0}\hat{c}_{0}-\mathsf{A}\cdot\mathsf{C})+(\hat{c}_{0}\mathsf{A}+\hat{a}_{0% }\mathsf{C}+\mathsf{A}\times\mathsf{C}).
  5. A ^ = ( A , B ) = a 0 + a 1 i + a 2 j + a 3 k + b 0 ϵ + b 1 ϵ i + b 2 ϵ j + b 3 ϵ k , \hat{A}=(A,B)=a_{0}+a_{1}i+a_{2}j+a_{3}k+b_{0}\epsilon+b_{1}\epsilon i+b_{2}% \epsilon j+b_{3}\epsilon k,
  6. C ^ = ( C , D ) = c 0 + c 1 i + c 2 j + c 3 k + d 0 ϵ + d 1 ϵ i + d 2 ϵ j + d 3 ϵ k , \hat{C}=(C,D)=c_{0}+c_{1}i+c_{2}j+c_{3}k+d_{0}\epsilon+d_{1}\epsilon i+d_{2}% \epsilon j+d_{3}\epsilon k,
  7. A ^ + C ^ = ( A + C , B + D ) = ( a 0 + c 0 ) + ( a 1 + c 1 ) i + ( a 2 + c 2 ) j + ( a 3 + c 3 ) k + ( b 0 + d 0 ) ϵ + ( b 1 + d 1 ) ϵ i + ( b 2 + d 2 ) ϵ j + ( b 3 + d 3 ) ϵ k , \hat{A}+\hat{C}=(A+C,B+D)=(a_{0}+c_{0})+(a_{1}+c_{1})i+(a_{2}+c_{2})j+(a_{3}+c% _{3})k+(b_{0}+d_{0})\epsilon+(b_{1}+d_{1})\epsilon i+(b_{2}+d_{2})\epsilon j+(% b_{3}+d_{3})\epsilon k,
  8. A ^ = ( A , B ) = A + ϵ B , \hat{A}=(A,B)=A+\epsilon B,
  9. C ^ = ( C , D ) = C + ϵ D , \hat{C}=(C,D)=C+\epsilon D,
  10. A ^ C ^ = ( A + ϵ B ) ( C + ϵ D ) = A C + ϵ ( A D + B C ) . \hat{A}\hat{C}=(A+\epsilon B)(C+\epsilon D)=AC+\epsilon(AD+BC).\!
  11. A ^ * = ( A * , B * ) = A * + ϵ B * . \hat{A}^{*}=(A^{*},B^{*})=A^{*}+\epsilon B^{*}.\!
  12. G ^ * = ( A ^ C ^ ) * = C ^ * A ^ * . \hat{G}^{*}=(\hat{A}\hat{C})^{*}=\hat{C}^{*}\hat{A}^{*}.\!
  13. Sc ( A ^ ) = a ^ 0 , Vec ( A ^ ) = 𝖠 . \mbox{Sc}~{}(\hat{A})=\hat{a}_{0},\mbox{Vec}~{}(\hat{A})=\mathsf{A}.\!
  14. A ^ * = Sc ( A ^ ) - Vec ( A ^ ) . \hat{A}^{*}=\mbox{Sc}~{}(\hat{A})-\mbox{Vec}~{}(\hat{A}).\!
  15. ( a ^ 0 + 𝖠 ) * = a ^ 0 - 𝖠 . (\hat{a}_{0}+\mathsf{A})^{*}=\hat{a}_{0}-\mathsf{A}.\!
  16. A ^ A ^ * = ( a ^ 0 + 𝖠 ) ( a ^ 0 - 𝖠 ) = a ^ 0 2 + 𝖠 𝖠 . \hat{A}\hat{A}^{*}=(\hat{a}_{0}+\mathsf{A})(\hat{a}_{0}-\mathsf{A})=\hat{a}_{0% }^{2}+\mathsf{A}\cdot\mathsf{A}.\!
  17. = =
  18. A ^ A ^ * = ( A , B ) ( A * , B * ) = ( A A * , A B * + B A * ) = ( 1 , 0 ) . \hat{A}\hat{A}^{*}=(A,B)(A^{*},B^{*})=(AA^{*},AB^{*}+BA^{*})=(1,0).\!
  19. S ^ = cos ϕ ^ 2 + sin ϕ ^ 2 𝖲 . \hat{S}=\cos\frac{\hat{\phi}}{2}+\sin\frac{\hat{\phi}}{2}\mathsf{S}.
  20. A ^ = cos ( α ^ / 2 ) + sin ( α ^ / 2 ) 𝖠 and B ^ = cos ( β ^ / 2 ) + sin ( β ^ / 2 ) 𝖡 . \hat{A}=\cos(\hat{\alpha}/2)+\sin(\hat{\alpha}/2)\mathsf{A}\quad\,\text{and}% \quad\hat{B}=\cos(\hat{\beta}/2)+\sin(\hat{\beta}/2)\mathsf{B}.
  21. C ^ = cos γ ^ 2 + sin γ ^ 2 𝖢 = ( cos β ^ 2 + sin β ^ 2 𝖡 ) ( cos α ^ 2 + sin α ^ 2 𝖠 ) . \hat{C}=\cos\frac{\hat{\gamma}}{2}+\sin\frac{\hat{\gamma}}{2}\mathsf{C}=\Big(% \cos\frac{\hat{\beta}}{2}+\sin\frac{\hat{\beta}}{2}\mathsf{B}\Big)\Big(\cos% \frac{\hat{\alpha}}{2}+\sin\frac{\hat{\alpha}}{2}\mathsf{A}\Big).
  22. cos γ ^ 2 + sin γ ^ 2 𝖢 = ( cos β ^ 2 cos α ^ 2 - sin β ^ 2 sin α ^ 2 𝖡 𝖠 ) + ( sin β ^ 2 cos α ^ 2 𝖡 + sin α ^ 2 cos β ^ 2 𝖠 + sin β ^ 2 sin α ^ 2 𝖡 × 𝖠 ) . \cos\frac{\hat{\gamma}}{2}+\sin\frac{\hat{\gamma}}{2}\mathsf{C}=\Big(\cos\frac% {\hat{\beta}}{2}\cos\frac{\hat{\alpha}}{2}-\sin\frac{\hat{\beta}}{2}\sin\frac{% \hat{\alpha}}{2}\mathsf{B}\cdot\mathsf{A}\Big)+\Big(\sin\frac{\hat{\beta}}{2}% \cos\frac{\hat{\alpha}}{2}\mathsf{B}+\sin\frac{\hat{\alpha}}{2}\cos\frac{\hat{% \beta}}{2}\mathsf{A}+\sin\frac{\hat{\beta}}{2}\sin\frac{\hat{\alpha}}{2}% \mathsf{B}\times\mathsf{A}\Big).
  23. cos γ ^ 2 = cos β ^ 2 cos α ^ 2 - sin β ^ 2 sin α ^ 2 𝖡 𝖠 \cos\frac{\hat{\gamma}}{2}=\cos\frac{\hat{\beta}}{2}\cos\frac{\hat{\alpha}}{2}% -\sin\frac{\hat{\beta}}{2}\sin\frac{\hat{\alpha}}{2}\mathsf{B}\cdot\mathsf{A}
  24. tan γ ^ 2 𝖢 = tan β ^ 2 𝖡 + tan α ^ 2 𝖠 + tan β ^ 2 tan α ^ 2 𝖡 × 𝖠 1 - tan β ^ 2 tan α ^ 2 𝖡 𝖠 . \tan\frac{\hat{\gamma}}{2}\mathsf{C}=\frac{\tan\frac{\hat{\beta}}{2}\mathsf{B}% +\tan\frac{\hat{\alpha}}{2}\mathsf{A}+\tan\frac{\hat{\beta}}{2}\tan\frac{\hat{% \alpha}}{2}\mathsf{B}\times\mathsf{A}}{1-\tan\frac{\hat{\beta}}{2}\tan\frac{% \hat{\alpha}}{2}\mathsf{B}\cdot\mathsf{A}}.
  25. A C = [ A + ] C = [ a 0 - A 3 A 2 A 1 A 3 a 0 - A 1 A 2 - A 2 A 1 a 0 A 3 - A 1 - A 2 - A 3 a 0 ] { C 1 C 2 C 3 c 0 } . AC=[A^{+}]C=\begin{bmatrix}a_{0}&-A_{3}&A_{2}&A_{1}\\ A_{3}&a_{0}&-A_{1}&A_{2}\\ -A_{2}&A_{1}&a_{0}&A_{3}\\ -A_{1}&-A_{2}&-A_{3}&a_{0}\end{bmatrix}\begin{Bmatrix}C_{1}\\ C_{2}\\ C_{3}\\ c_{0}\end{Bmatrix}.
  26. A C = [ C - ] A = [ c 0 C 3 - C 2 C 1 - C 3 c 0 C 1 C 2 C 2 - C 1 c 0 C 3 - C 1 - C 2 - C 3 c 0 ] { A 1 A 2 A 3 a 0 } . AC=[C^{-}]A=\begin{bmatrix}c_{0}&C_{3}&-C_{2}&C_{1}\\ -C_{3}&c_{0}&C_{1}&C_{2}\\ C_{2}&-C_{1}&c_{0}&C_{3}\\ -C_{1}&-C_{2}&-C_{3}&c_{0}\end{bmatrix}\begin{Bmatrix}A_{1}\\ A_{2}\\ A_{3}\\ a_{0}\end{Bmatrix}.
  27. A ^ C ^ = [ A ^ + ] C ^ = [ A + 0 B + A + ] { C D } . \hat{A}\hat{C}=[\hat{A}^{+}]\hat{C}=\begin{bmatrix}A^{+}&0\\ B^{+}&A^{+}\end{bmatrix}\begin{Bmatrix}C\\ D\end{Bmatrix}.
  28. A ^ C ^ = [ C ^ - ] A ^ = [ C - 0 D - C - ] { A B } . \hat{A}\hat{C}=[\hat{C}^{-}]\hat{A}=\begin{bmatrix}C^{-}&0\\ D^{-}&C^{-}\end{bmatrix}\begin{Bmatrix}A\\ B\end{Bmatrix}.
  29. S ^ = S + ε 1 2 D S . \hat{S}=S+\varepsilon\frac{1}{2}DS.
  30. x ^ = 𝐱 + ε 𝐩 × 𝐱 and X ^ = 𝐗 + ε 𝐏 × 𝐗 . \hat{x}=\mathbf{x}+\varepsilon\mathbf{p}\times\mathbf{x}\quad\,\text{and}\quad% \hat{X}=\mathbf{X}+\varepsilon\mathbf{P}\times\mathbf{X}.
  31. X ^ = S ^ x ^ S ^ * . \hat{X}=\hat{S}\hat{x}\hat{S}^{*}.
  32. X ^ = [ S ^ + ] [ S ^ - ] * x ^ . \hat{X}=[\hat{S}^{+}][\hat{S}^{-}]^{*}\hat{x}.
  33. q ^ = r + d ε \hat{q}=r+d\varepsilon
  34. d d
  35. v = ( v 0 , v 1 , v 2 ) \vec{v}=(v_{0},v_{1},v_{2})
  36. v ^ := 1 + ε ( v 0 i + v 1 j + v 2 k ) \hat{v}:=1+\varepsilon(v_{0}i+v_{1}j+v_{2}k)
  37. q ^ \hat{q}
  38. v = q ^ v ^ q ^ - 1 \vec{v}^{\prime}=\hat{q}\cdot\hat{v}\cdot\hat{q}^{-1}
  39. r = r w + r x i + r y j + r z k = cos ( θ 2 ) + sin ( θ 2 ) a r=r_{w}+r_{x}i+r_{y}j+r_{z}k=\cos\left(\frac{\theta}{2}\right)+\sin\left(\frac% {\theta}{2}\right)\cdot\vec{a}
  40. θ \theta
  41. a \vec{a}
  42. R = ( r w 2 + r x 2 - r y 2 - r z 2 2 r x r y - 2 r w r z 2 r x r z + 2 r w r y 2 r x r y + 2 r w r z r w 2 - r x 2 + r y 2 - r z 2 2 r y r z - 2 r w r x 2 r x r z - 2 r w r y 2 r y r z + 2 r w r x r w 2 - r x 2 - r y 2 + r z 2 ) . R=\begin{pmatrix}r_{w}^{2}+r_{x}^{2}-r_{y}^{2}-r_{z}^{2}&2r_{x}r_{y}-2r_{w}r_{% z}&2r_{x}r_{z}+2r_{w}r_{y}\\ 2r_{x}r_{y}+2r_{w}r_{z}&r_{w}^{2}-r_{x}^{2}+r_{y}^{2}-r_{z}^{2}&2r_{y}r_{z}-2r% _{w}r_{x}\\ 2r_{x}r_{z}-2r_{w}r_{y}&2r_{y}r_{z}+2r_{w}r_{x}&r_{w}^{2}-r_{x}^{2}-r_{y}^{2}+% r_{z}^{2}\\ \end{pmatrix}.
  43. d r * = 0 + Δ x 2 i + Δ y 2 j + Δ z 2 k dr^{*}=0+\frac{\Delta x}{2}i+\frac{\Delta y}{2}j+\frac{\Delta z}{2}k
  44. [ R | t ] = ( Δ x R Δ y Δ z 0 0 0 1 ) [R|t]=\begin{pmatrix}&&&\Delta x\\ &R&&\Delta y\\ &&&\Delta z\\ 0&0&0&1\\ \end{pmatrix}

Ducci_sequence.html

  1. ( a 1 , a 2 , , a n ) (a_{1},a_{2},...,a_{n})
  2. ( a 1 , a 2 , , a n ) ( | a 1 - a 2 | , | a 2 - a 3 | , , | a n - a 1 | ) . (a_{1},a_{2},...,a_{n})\rightarrow(|a_{1}-a_{2}|,|a_{2}-a_{3}|,...,|a_{n}-a_{1% }|)\,.
  3. x 3 - x 2 - x - 1 = 0 x^{3}-x^{2}-x-1=0
  4. ( 0 , 653 , 1854 , 4063 ) ( 653 , 1201 , 2209 , 4063 ) ( 548 , 1008 , 1854 , 3410 ) (0,653,1854,4063)\rightarrow(653,1201,2209,4063)\rightarrow(548,1008,1854,3410)\rightarrow
  5. ( 0 , 0 , 128 , 128 ) ( 0 , 128 , 0 , 128 ) ( 128 , 128 , 128 , 128 ) ( 0 , 0 , 0 , 0 ) \cdots\rightarrow(0,0,128,128)\rightarrow(0,128,0,128)\rightarrow(128,128,128,% 128)\rightarrow(0,0,0,0)
  6. 15799 42208 20284 22642 04220 42020 22224 00022 00202 02222 20002 20020 20222 22000 02002 22022 02200 20200 22202 00220 02020 22220 00022 \begin{matrix}15799\rightarrow&42208\rightarrow&20284\rightarrow&22642% \rightarrow&04220\rightarrow&42020\rightarrow\\ 22224\rightarrow&00022\rightarrow&00202\rightarrow&02222\rightarrow&20002% \rightarrow&20020\rightarrow\\ 20222\rightarrow&22000\rightarrow&02002\rightarrow&22022\rightarrow&02200% \rightarrow&20200\rightarrow\\ 22202\rightarrow&00220\rightarrow&02020\rightarrow&22220\rightarrow&00022% \rightarrow&\cdots\\ \end{matrix}
  7. 121210 111111 000000 \begin{matrix}121210\rightarrow&111111\rightarrow&000000\\ \end{matrix}
  8. ( | a 1 - a 2 | , | a 2 - a 3 | , , | a n - a 1 | ) = ( a 1 + a 2 , a 2 + a 3 , , a n + a 1 ) m o d 2 (|a_{1}-a_{2}|,|a_{2}-a_{3}|,...,|a_{n}-a_{1}|)\ =(a_{1}+a_{2},a_{2}+a_{3},...% ,a_{n}+a_{1})\ mod2

Dunkerley's_method.html

  1. N = 94.251 E I m L 3 RPM N=94.251\sqrt{EI\over mL^{3}}\ \,\text{RPM}
  2. 1 N N 2 = 1 N A 2 + 1 N B 2 + + 1 N n 2 \frac{1}{N_{N}^{2}}=\frac{1}{N_{A}^{2}}+\frac{1}{N_{B}^{2}}+\cdots+\frac{1}{N_% {n}^{2}}

Dupuit–Forchheimer_assumption.html

  1. P z \displaystyle\frac{\partial P}{\partial z}
  2. P / z \partial P/\partial z
  3. γ \gamma
  4. ρ \rho
  5. g g
  6. h / z \partial h/\partial z

DVB-T2.html

  1. K b c h K_{bch}
  2. 1 + x 14 + x 15 1+x^{14}+x^{15}
  3. N l d p c N_{ldpc}
  4. a i a_{i}
  5. e i ( 1 ) e^{(1)}_{i}
  6. e i ( 2 ) e^{(2)}_{i}
  7. e i ( 1 ) = a i e^{(1)}_{i}=a_{i}
  8. e i + 1 ( 1 ) = a i + 1 e^{(1)}_{i+1}=a_{i+1}
  9. e i ( 2 ) = - a i + 1 * e^{(2)}_{i}=-a^{*}_{i+1}
  10. e i + 1 ( 2 ) = a i * e^{(2)}_{i+1}=a^{*}_{i}

Dvoretzky's_theorem.html

  1. | | = Q ( ) |\cdot|=\sqrt{Q(\cdot)}
  2. | x | x ( 1 + ϵ ) | x | for every x E . |x|\leq\|x\|\leq(1+\epsilon)|x|\quad\,\text{for every}\quad x\in E.
  3. N ( k , ϵ ) exp ( C ( ϵ ) k ) . N(k,\epsilon)\leq\exp(C(\epsilon)k).
  4. k = dim E c ( ϵ ) ( S n - 1 ξ d σ ( ξ ) max ξ S n - 1 ξ ) 2 N . k=\dim E\geq c(\epsilon)\,\left(\frac{\int_{S^{n-1}}\|\xi\|\,d\sigma(\xi)}{% \max_{\xi\in S^{n-1}}\|\xi\|}\right)^{2}\,N.
  5. c 1 log N N . c_{1}\sqrt{\frac{\log N}{N}}.

Dynamic_convex_hull.html

  1. Ω ( N ) \Omega(N)

Dynein_ATPase.html

  1. \rightleftharpoons

Dynkin's_formula.html

  1. d X t = b ( X t ) d t + σ ( X t ) d B t . \mathrm{d}X_{t}=b(X_{t})\,\mathrm{d}t+\sigma(X_{t})\,\mathrm{d}B_{t}.
  2. A f ( x ) = lim t 0 𝐄 x [ f ( X t ) ] - f ( x ) t Af(x)=\lim_{t\downarrow 0}\frac{\mathbf{E}^{x}[f(X_{t})]-f(x)}{t}
  3. A f ( x ) = i b i ( x ) f x i ( x ) + 1 2 i , j ( σ σ ) i , j ( x ) 2 f x i x j ( x ) . Af(x)=\sum_{i}b_{i}(x)\frac{\partial f}{\partial x_{i}}(x)+\frac{1}{2}\sum_{i,% j}\big(\sigma\sigma^{\top}\big)_{i,j}(x)\frac{\partial^{2}f}{\partial x_{i}\,% \partial x_{j}}(x).
  4. 𝐄 x [ f ( X τ ) ] = f ( x ) + 𝐄 x [ 0 τ A f ( X s ) d s ] . \mathbf{E}^{x}[f(X_{\tau})]=f(x)+\mathbf{E}^{x}\left[\int_{0}^{\tau}Af(X_{s})% \,\mathrm{d}s\right].
  5. K = K R = { x 𝐑 n | | x | R } , K=K_{R}=\{x\in\mathbf{R}^{n}|\,|x|\leq R\},
  6. 𝐄 a [ τ K ] = 1 n ( R 2 - | a | 2 ) . \mathbf{E}^{a}[\tau_{K}]=\frac{1}{n}\big(R^{2}-|a|^{2}\big).
  7. 𝐄 a [ f ( B σ j ) ] \mathbf{E}^{a}\left[f\big(B_{\sigma_{j}}\big)\right]
  8. = f ( a ) + 𝐄 a [ 0 σ j 1 2 Δ f ( B s ) d s ] =f(a)+\mathbf{E}^{a}\left[\int_{0}^{\sigma_{j}}\frac{1}{2}\Delta f(B_{s})\,% \mathrm{d}s\right]
  9. = | a | 2 + 𝐄 a [ 0 σ j n d s ] =|a|^{2}+\mathbf{E}^{a}\left[\int_{0}^{\sigma_{j}}n\,\mathrm{d}s\right]
  10. = | a | 2 + n 𝐄 a [ σ j ] . =|a|^{2}+n\mathbf{E}^{a}[\sigma_{j}].
  11. 𝐄 a [ σ j ] 1 n ( R 2 - | a | 2 ) . \mathbf{E}^{a}[\sigma_{j}]\leq\frac{1}{n}\big(R^{2}-|a|^{2}\big).
  12. 𝐄 a [ τ K ] = 1 n ( R 2 - | a | 2 ) , \mathbf{E}^{a}[\tau_{K}]=\frac{1}{n}\big(R^{2}-|a|^{2}\big),

Dynkin_index.html

  1. x λ x_{\lambda}
  2. | λ | |\lambda|
  3. λ \lambda
  4. tr ( t a t b ) = 2 x λ g a b {\rm tr}(t_{a}t_{b})=2x_{\lambda}g_{ab}
  5. | λ | |\lambda|
  6. t a t_{a}
  7. g a b g_{ab}
  8. tr ( t a t b ) = 2 g a b {\rm tr}(t_{a}t_{b})=2g_{ab}
  9. x λ = dim ( | λ | ) 2 dim ( g ) ( λ , λ + 2 ρ ) x_{\lambda}=\frac{\dim(|\lambda|)}{2\dim(g)}(\lambda,\lambda+2\rho)
  10. ρ = 1 2 α Δ + α \rho=\frac{1}{2}\sum_{\alpha\in\Delta^{+}}\alpha
  11. ( λ , λ + 2 ρ ) (\lambda,\lambda+2\rho)
  12. | λ | |\lambda|
  13. x λ x_{\lambda}
  14. λ \lambda
  15. | λ | |\lambda|
  16. x λ x_{\lambda}

Earth_mover's_distance.html

  1. π : P Q \pi:P\to Q
  2. P P P^{\prime}\subset P
  3. Q Q Q^{\prime}\subset Q
  4. | P - Q | π = | P P | + | Q Q | |P-Q|_{\pi}=|P\setminus P^{\prime}|+|Q\setminus Q^{\prime}|
  5. P P P\setminus P^{\prime}
  6. P P^{\prime}
  7. P P P\setminus P^{\prime}
  8. | P P | |P\setminus P^{\prime}|
  9. Q Q Q^{\prime}\subset Q

Earth_potential_rise.html

  1. 𝐕 g r i d = 𝐈 f × 𝐙 g r i d \mathbf{V}_{grid}=\mathbf{I}_{f}\times\mathbf{Z}_{grid}
  2. 𝐕 r = ρ 𝐈 𝟐 π 𝐫 𝐱 \mathbf{V}_{r}=\frac{\mathbf{\rho I}}{\mathbf{2\pi r_{x}}}
  3. 𝐫 𝐱 \mathbf{r_{x}}
  4. 𝐕 r \mathbf{V}_{r}
  5. 𝐫 𝐱 \mathbf{r_{x}}
  6. ρ \mathbf{\rho}
  7. 𝐈 \mathbf{I}

ECF_grading_system.html

  1. E = F - 700 7.5 E=\frac{F-700}{7.5}
  2. F = 7.5 E + 700 F=7.5E+700
  3. E = F - 650 8 E=\frac{F-650}{8}
  4. E = N - 600 8 E=\frac{N-600}{8}
  5. F = 8 E + 650 F=8E+650
  6. N = 8 E + 600 N=8E+600

Eckart_conditions.html

  1. A = 1 , , N A=1,\ldots,N
  2. A = 1 N M A ( δ i j | 𝐑 A 0 | 2 - R A i 0 R A j 0 ) = λ i 0 δ i j and A = 1 N M A 𝐑 A 0 = 0. \sum_{A=1}^{N}M_{A}\,\big(\delta_{ij}|\mathbf{R}_{A}^{0}|^{2}-R^{0}_{Ai}R^{0}_% {Aj}\big)=\lambda^{0}_{i}\delta_{ij}\quad\mathrm{and}\quad\sum_{A=1}^{N}M_{A}% \mathbf{R}_{A}^{0}=\mathbf{0}.
  3. 𝐅 = { f 1 , f 2 , f 3 } \vec{\mathbf{F}}=\{\vec{f}_{1},\vec{f}_{2},\vec{f}_{3}\}
  4. R A 0 𝐅 𝐑 A 0 = i = 1 3 f i R A i 0 , A = 1 , , N \vec{R}_{A}^{0}\equiv\vec{\mathbf{F}}\cdot\mathbf{R}_{A}^{0}=\sum_{i=1}^{3}% \vec{f}_{i}\,R^{0}_{Ai},\quad A=1,\ldots,N
  5. A = 1 , , N A=1,\ldots,N
  6. A M A 𝐑 A = 𝟎 \sum_{A}M_{A}\mathbf{R}_{A}=\mathbf{0}
  7. 𝐝 A 𝐑 A - 𝐑 A 0 \mathbf{d}_{A}\equiv\mathbf{R}_{A}-\mathbf{R}^{0}_{A}
  8. A = 1 N M A 𝐝 A = 0. \sum_{A=1}^{N}M_{A}\mathbf{d}_{A}=0.
  9. A = 1 N M A 𝐑 A 0 × 𝐝 A = 0 , \sum_{A=1}^{N}M_{A}\mathbf{R}^{0}_{A}\times\mathbf{d}_{A}=0,
  10. × \times
  11. R A \vec{R}_{A}
  12. 𝐑 3 N = 𝐑 ext 𝐑 int . \mathbf{R}^{3N}=\mathbf{R}_{\textrm{ext}}\oplus\mathbf{R}_{\textrm{int}}.
  13. s i A \displaystyle\vec{s}^{A}_{i}
  14. S t row ( M 1 s t 1 , , M N s t N ) for t = 1 , , 6. \vec{S}_{t}\equiv\operatorname{row}(\sqrt{M_{1}}\;\vec{s}^{\,1}_{t},\ldots,% \sqrt{M_{N}}\;\vec{s}^{\,N}_{t})\quad\mathrm{for}\quad t=1,\ldots,6.
  15. D col ( M 1 d 1 , , M N d N ) with d A 𝐅 𝐝 A . \vec{D}\equiv\operatorname{col}(\sqrt{M_{1}}\;\vec{d}^{\,1},\ldots,\sqrt{M_{N}% }\;\vec{d}^{\,N})\quad\mathrm{with}\quad\vec{d}^{\,A}\equiv\vec{\mathbf{F}}% \cdot\mathbf{d}_{A}.
  16. S i D = A = 1 N M A s i A d A = A = 1 N M A d A i = 0 , \vec{S}_{i}\cdot\vec{D}=\sum_{A=1}^{N}\;M_{A}\vec{s}^{\,A}_{i}\cdot\vec{d}^{\,% A}=\sum_{A=1}^{N}M_{A}d_{Ai}=0,
  17. S i D = A = 1 N M A ( f i × R A 0 ) d A = f i A = 1 N M A R A 0 × d A = A = 1 N M A ( 𝐑 A 0 × 𝐝 A ) i = 0 , \,\vec{S}_{i}\cdot\vec{D}=\sum_{A=1}^{N}\;M_{A}\big(\vec{f}_{i}\times\vec{R}_{% A}^{0}\big)\cdot\vec{d}^{\,A}=\vec{f}_{i}\cdot\sum_{A=1}^{N}M_{A}\vec{R}_{A}^{% 0}\times\vec{d}^{A}=\sum_{A=1}^{N}M_{A}\big(\mathbf{R}_{A}^{0}\times\mathbf{d}% _{A}\big)_{i}=0,
  18. D \vec{D}
  19. Q r row ( 1 M 1 q r 1 , , 1 M N q r N ) , for r = 1 , , 3 N - 6. \vec{Q}_{r}\equiv\operatorname{row}(\frac{1}{\sqrt{M_{1}}}\;\vec{q}_{r}^{\,1},% \ldots,\frac{1}{\sqrt{M_{N}}}\;\vec{q}_{r}^{\,N}),\quad\mathrm{for}\quad r=1,% \ldots,3N-6.
  20. q r A \vec{q}^{A}_{r}
  21. q r Q r D = A = 1 N q r A d A for r = 1 , , 3 N - 6. q_{r}\equiv\vec{Q}_{r}\cdot\vec{D}=\sum_{A=1}^{N}\vec{q}^{A}_{r}\cdot\vec{d}^{% \,A}\quad\mathrm{for}\quad r=1,\ldots,3N-6.
  22. q r A \vec{q}^{A}_{r}
  23. s t S t D = A = 1 N M A s t A d A = 0 for t = 1 , , 6. s_{t}\equiv\vec{S}_{t}\cdot\vec{D}=\sum_{A=1}^{N}M_{A}\;\vec{s}^{\,A}_{t}\cdot% \vec{d}^{\,A}=0\quad\mathrm{for}\quad t=1,\ldots,6.
  24. R A 0 R A 0 + t \vec{R}_{A}^{0}\mapsto\vec{R}_{A}^{0}+\vec{t}
  25. t \vec{t}
  26. R A 0 R A 0 + Δ φ ( n × R A 0 ) \vec{R}_{A}^{0}\mapsto\vec{R}_{A}^{0}+\Delta\varphi\;(\vec{n}\times\vec{R}_{A}% ^{0})
  27. n \vec{n}
  28. Q r \vec{Q}_{r}
  29. q r A \vec{q}^{A}_{r}
  30. A = 1 N q r A = 0 and A = 1 N R A 0 × q r A = 0 . \sum_{A=1}^{N}\vec{q}^{\,A}_{r}=\vec{0}\quad\mathrm{and}\quad\sum_{A=1}^{N}% \vec{R}^{0}_{A}\times\vec{q}^{A}_{r}=\vec{0}.
  31. q r A q r A ( d A - t ) = q r - t A q r A = q r . q_{r}\mapsto\sum_{A}\vec{q}^{\,A}_{r}\cdot(\vec{d}^{A}-\vec{t})=q_{r}-\vec{t}% \cdot\sum_{A}\vec{q}^{\,A}_{r}=q_{r}.
  32. q r A \vec{q}^{A}_{r}
  33. A q r A = 0 , \sum_{A}\vec{q}^{\,A}_{r}=0,
  34. t \vec{t}
  35. q r A q r A ( d A - Δ φ ( n × R A 0 ) ) = q r - Δ φ n A R A 0 × q r A = q r . q_{r}\mapsto\sum_{A}\vec{q}^{\,A}_{r}\cdot\big(\vec{d}^{A}-\Delta\varphi\;(% \vec{n}\times\vec{R}_{A}^{0})\big)=q_{r}-\Delta\varphi\;\vec{n}\cdot\sum_{A}% \vec{R}^{0}_{A}\times\vec{q}^{\,A}_{r}=q_{r}.
  36. A R A 0 × q r A = 0 . \sum_{A}\vec{R}^{0}_{A}\times\vec{q}^{\,A}_{r}=\vec{0}.
  37. s i \displaystyle s_{i}
  38. s i s i for i = 1 , 2 , 3 s i s i + Δ ϕ f i 𝐈 0 n for i = 4 , 5 , 6 , \begin{aligned}\displaystyle s_{i}&\displaystyle\mapsto s_{i}\quad\mathrm{for}% \quad i=1,2,3\\ \displaystyle s_{i}&\displaystyle\mapsto s_{i}+\Delta\phi\vec{f}_{i}\cdot% \mathbf{I}^{0}\cdot\vec{n}\quad\mathrm{for}\quad i=4,5,6,\\ \end{aligned}
  39. 2 T vib = A = 1 N M A 𝐑 ˙ A 𝐑 ˙ A = A = 1 N M A 𝐝 ˙ A 𝐝 ˙ A . 2T_{\mathrm{vib}}=\sum_{A=1}^{N}M_{A}\dot{\mathbf{R}}_{A}\cdot\dot{\mathbf{R}}% _{A}=\sum_{A=1}^{N}M_{A}\dot{\mathbf{d}}_{A}\cdot\dot{\mathbf{d}}_{A}.
  40. q r = A j d A j ( q r j A ) s i = A j d A j ( M A δ i j ) = 0 s i + 3 = A j d A j ( M A k ϵ i k j R A k 0 ) = 0 \begin{aligned}\displaystyle q_{r}=\sum_{Aj}d_{Aj}&\displaystyle\big(q^{A}_{rj% }\big)\\ \displaystyle s_{i}=\sum_{Aj}d_{Aj}&\displaystyle\big(M_{A}\delta_{ij}\big)=0% \\ \displaystyle s_{i+3}=\sum_{Aj}d_{Aj}&\displaystyle\big(M_{A}\sum_{k}\epsilon_% {ikj}R^{0}_{Ak}\big)=0\\ \end{aligned}
  41. 𝐯 ( q 1 q 3 N - 6 0 0 ) = ( 𝐁 int 𝐁 ext ) 𝐝 𝐁𝐝 . \mathbf{v}\equiv\begin{pmatrix}q_{1}\\ \vdots\\ \vdots\\ q_{3N-6}\\ 0\\ \vdots\\ 0\\ \end{pmatrix}=\begin{pmatrix}\mathbf{B}^{\mathrm{int}}\\ \cdots\\ \mathbf{B}^{\mathrm{ext}}\\ \end{pmatrix}\mathbf{d}\equiv\mathbf{B}\mathbf{d}.
  42. 𝐌 diag ( 𝐌 1 , 𝐌 2 , , 𝐌 N ) and 𝐌 A diag ( M A , M A , M A ) \mathbf{M}\equiv\operatorname{diag}(\mathbf{M}_{1},\mathbf{M}_{2},\ldots,% \mathbf{M}_{N})\quad\textrm{and}\quad\mathbf{M}_{A}\equiv\operatorname{diag}(M% _{A},M_{A},M_{A})
  43. 𝐁 ext 𝐌 - 1 ( 𝐁 ext ) T = diag ( N 1 , , N 6 ) 𝐍 , \mathbf{B}^{\mathrm{ext}}\mathbf{M}^{-1}(\mathbf{B}^{\mathrm{ext}})^{\mathrm{T% }}=\operatorname{diag}(N_{1},\ldots,N_{6})\equiv\mathbf{N},
  44. 𝐁 int 𝐌 - 1 ( 𝐁 ext ) T = 0. \mathbf{B}^{\mathrm{int}}\mathbf{M}^{-1}(\mathbf{B}^{\mathrm{ext}})^{\mathrm{T% }}=\mathbf{0}.
  45. 𝐆 𝐁 int 𝐌 - 1 ( 𝐁 int ) T . \mathbf{G}\equiv\mathbf{B}^{\mathrm{int}}\mathbf{M}^{-1}(\mathbf{B}^{\mathrm{% int}})^{\mathrm{T}}.
  46. ( 𝐁 T ) - 1 𝐌𝐁 - 1 = ( 𝐆 - 1 𝟎 𝟎 𝐍 - 1 ) , (\mathbf{B}^{\mathrm{T}})^{-1}\mathbf{M}\mathbf{B}^{-1}=\begin{pmatrix}\mathbf% {G}^{-1}&&\mathbf{0}\\ \mathbf{0}&&\mathbf{N}^{-1}\end{pmatrix},
  47. 2 T vib = 𝐝 ˙ T 𝐌 𝐝 ˙ = 𝐯 ˙ T ( 𝐁 T ) - 1 𝐌𝐁 - 1 𝐯 ˙ = r , r = 1 3 N - 6 ( G - 1 ) r r q ˙ r q ˙ r 2T_{\mathrm{vib}}=\dot{\mathbf{d}}^{\mathrm{T}}\mathbf{M}\dot{\mathbf{d}}=\dot% {\mathbf{v}}^{\mathrm{T}}\;(\mathbf{B}^{\mathrm{T}})^{-1}\mathbf{M}\mathbf{B}^% {-1}\;\dot{\mathbf{v}}=\sum_{r,r^{\prime}=1}^{3N-6}(G^{-1})_{rr^{\prime}}\dot{% q}_{r}\dot{q}_{r^{\prime}}
  48. 2 V harm = 𝐝 T 𝐇𝐝 = 𝐯 T ( 𝐁 T ) - 1 𝐇𝐁 - 1 𝐯 = r , r = 1 3 N - 6 F r r q r q r , 2V_{\mathrm{harm}}=\mathbf{d}^{\mathrm{T}}\mathbf{H}\mathbf{d}=\mathbf{v}^{% \mathrm{T}}(\mathbf{B}^{\mathrm{T}})^{-1}\mathbf{H}\mathbf{B}^{-1}\mathbf{v}=% \sum_{r,r^{\prime}=1}^{3N-6}F_{rr^{\prime}}q_{r}q_{r^{\prime}},
  49. 𝐇𝐂 = 𝐌𝐂 s y m b o l Φ , \mathbf{H}\mathbf{C}=\mathbf{M}\mathbf{C}symbol{\Phi},
  50. V ( 𝐑 1 , 𝐑 2 , , 𝐑 N ) V(\mathbf{R}_{1},\mathbf{R}_{2},\ldots,\mathbf{R}_{N})
  51. 𝐑 1 0 , , 𝐑 N 0 \mathbf{R}_{1}^{0},\ldots,\mathbf{R}_{N}^{0}
  52. s y m b o l Φ symbol{\Phi}
  53. 𝐇 ( 𝐭 𝐭 ) = ( 𝟎 𝟎 ) and 𝐇 ( 𝐬 × 𝐑 1 0 𝐬 × 𝐑 N 0 ) = ( 𝟎 𝟎 ) \mathbf{H}\begin{pmatrix}\mathbf{t}\\ \vdots\\ \mathbf{t}\end{pmatrix}=\begin{pmatrix}\mathbf{0}\\ \vdots\\ \mathbf{0}\end{pmatrix}\quad\mathrm{and}\quad\mathbf{H}\begin{pmatrix}\mathbf{% s}\times\mathbf{R}_{1}^{0}\\ \vdots\\ \mathbf{s}\times\mathbf{R}_{N}^{0}\end{pmatrix}=\begin{pmatrix}\mathbf{0}\\ \vdots\\ \mathbf{0}\end{pmatrix}
  54. 𝐂 T 𝐌𝐂 = 𝐈 \mathbf{C}^{\mathrm{T}}\mathbf{M}\mathbf{C}=\mathbf{I}

EcosimPro.html

  1. d y d t = ( x - y ) / t a u \frac{dy}{dt}=(x-y)/tau
  2. y = ( x - y ) / t a u y^{\prime}=(x-y)/tau
  3. x 2 + y 2 = L 2 x^{2}+y^{2}=L^{2}
  4. F x = - T x L F_{x}=-T\frac{x}{L}
  5. F y = - T y L - M g F_{y}=-T\frac{y}{L}-M\;g
  6. F x = M a x = M x ¨ F_{x}=M\;a_{x}=M\;\ddot{x}
  7. F y = M a y = M y ¨ F_{y}=M\;a_{y}=M\;\ddot{y}
  8. x ˙ \dot{x}
  9. x ¨ \ddot{x}

Edge_(geometry).html

  1. V - E + F = 2 , V-E+F=2,

EDGE_species.html

  1. E D G E = ln ( 1 + E D ) + G E * ln ( 2 ) = ln [ ( 1 + E D ) * 2 G E ] EDGE=\ln(1+ED)+GE*\ln(2)=\ln[(1+ED)*2^{GE}]

Edmonds'_algorithm.html

  1. D = V , E D=\langle V,E\rangle
  2. V V
  3. E E
  4. r V r\in V
  5. w ( e ) w(e)
  6. e E e\in E
  7. A A
  8. r r
  9. w ( A ) = e A w ( e ) w(A)=\sum_{e\in A}{w(e)}
  10. f ( D , r , w ) f(D,r,w)
  11. r r
  12. E E
  13. r r
  14. v v
  15. v v
  16. π ( v ) \pi(v)
  17. P = { ( π ( v ) , v ) v V { r } } P=\{(\pi(v),v)\mid v\in V\setminus\{r\}\}
  18. f ( D , r , w ) = P f(D,r,w)=P
  19. P P
  20. C C
  21. D = V , E D^{\prime}=\langle V^{\prime},E^{\prime}\rangle
  22. C C
  23. V V^{\prime}
  24. V V
  25. C C
  26. v C v_{C}
  27. ( u , v ) (u,v)
  28. E E
  29. u C u\notin C
  30. v C v\in C
  31. E E^{\prime}
  32. e = ( u , v C ) e=(u,v_{C})
  33. w ( e ) = w ( u , v ) - w ( π ( v ) , v ) w^{\prime}(e)=w(u,v)-w(\pi(v),v)
  34. ( u , v ) (u,v)
  35. E E
  36. u C u\in C
  37. v C v\notin C
  38. E E^{\prime}
  39. e = ( v C , v ) e=(v_{C},v)
  40. w ( e ) = w ( u , v ) w^{\prime}(e)=w(u,v)
  41. ( u , v ) (u,v)
  42. E E
  43. u C u\notin C
  44. v C v\notin C
  45. E E^{\prime}
  46. e = ( u , v ) e=(u,v)
  47. w ( e ) = w ( u , v ) w^{\prime}(e)=w(u,v)
  48. E E^{\prime}
  49. E E
  50. A A^{\prime}
  51. D D^{\prime}
  52. f ( D , r , w ) f(D^{\prime},r,w^{\prime})
  53. A A^{\prime}
  54. ( u , v C ) (u,v_{C})
  55. v C v_{C}
  56. A A^{\prime}
  57. ( u , v ) E (u,v)\in E
  58. v C v\in C
  59. ( π ( v ) , v ) (\pi(v),v)
  60. C C
  61. C C
  62. A A^{\prime}
  63. E E
  64. f ( D , r , w ) f(D,r,w)
  65. f ( D , r , w ) f(D,r,w)
  66. f ( D , r , w ) f(D^{\prime},r,w^{\prime})
  67. D D^{\prime}
  68. D D
  69. f ( D , r , w ) f(D,r,w)
  70. D D
  71. O ( E V ) O(EV)
  72. O ( E log V ) O(E\log V)
  73. O ( V 2 ) O(V^{2})
  74. O ( E + V log V ) O(E+V\log V)

Edmonds_matrix.html

  1. A A
  2. G ( U , V , E ) G(U,V,E)
  3. U = { u 1 , u 2 , , u n } U=\{u_{1},u_{2},\dots,u_{n}\}
  4. V = { v 1 , v 2 , , v n } V=\{v_{1},v_{2},\dots,v_{n}\}
  5. A i j = { x i j ( u i , v j ) E 0 ( u i , v j ) E A_{ij}=\left\{\begin{array}[]{ll}x_{ij}&(u_{i},v_{j})\in E\\ 0&(u_{i},v_{j})\notin E\end{array}\right.
  6. A A
  7. A A
  8. G G

Effective_dimension.html

  1. χ X \chi_{X}
  2. d ( σ ) = 1 2 ( d ( σ 0 ) + d ( σ 1 ) ) d(\sigma)=\frac{1}{2}(d(\sigma 0)+d(\sigma 1))
  3. d ( σ ) d(\sigma)
  4. d ( σ ) 1 2 ( d ( σ 0 ) + d ( σ 1 ) ) d(\sigma)\geq\frac{1}{2}(d(\sigma 0)+d(\sigma 1))
  5. d ( σ ) = e ( σ ) 2 ( 1 - s ) | σ | d(\sigma)=\frac{e(\sigma)}{2^{(1-s)|\sigma|}}
  6. d ( σ ) = e ( σ ) 2 ( 1 - s ) | σ | d(\sigma)=\frac{e(\sigma)}{2^{(1-s)|\sigma|}}
  7. lim sup n d ( X | n ) = \limsup_{n}d(X|n)=\infty
  8. X | n X|n
  9. lim inf n d ( X | n ) = \liminf_{n}d(X|n)=\infty
  10. d ( σ ) d(\sigma)
  11. inf { s : some c . e . s - gale succeeds on X } \inf\{s:\mathrm{some\ c.e.}\ s\mathrm{-gale\ succeeds\ on\ }X\}
  12. inf { s : some c . e . s - gale succeeds strongly on X } \inf\{s:\mathrm{some\ c.e.}\ s\mathrm{-gale\ succeeds\ strongly\ on\ }X\}
  13. lim inf n K ( X | n ) n \liminf_{n}\frac{K(X|n)}{n}
  14. lim sup n K ( X | n ) n \limsup_{n}\frac{K(X|n)}{n}
  15. inf { s : some s - gale succeeds on all elements of Z } \inf\{s:\mathrm{some}\ s\mathrm{-gale\ succeeds\ on\ all\ elements\ of\ }Z\}
  16. inf { s : some s - gale succeeds strongly on all elements of Z } \inf\{s:\mathrm{some}\ s\mathrm{-gale\ succeeds\ strongly\ on\ all\ elements\ % of\ }Z\}
  17. X X
  18. { X } \{X\}
  19. H β := { X 2 ω : X has effective Hausdorff dimension β } H_{\beta}:=\{X\in 2^{\omega}:X\ \mathrm{has\ effective\ Hausdorff\ dimension\ % }\beta\}
  20. H β := { X 2 ω : X has effective Hausdorff dimension β } H_{\leq\beta}:=\{X\in 2^{\omega}:X\ \mathrm{has\ effective\ Hausdorff\ % dimension\ }\leq\beta\}
  21. H < β := { X 2 ω : X has effective Hausdorff dimension < β } H_{<\beta}:=\{X\in 2^{\omega}:X\ \mathrm{has\ effective\ Hausdorff\ dimension% \ }<\beta\}
  22. P β := { X 2 ω : X has effective packing dimension β } P_{\beta}:=\{X\in 2^{\omega}:X\ \mathrm{has\ effective\ packing\ dimension\ }\beta\}
  23. P β := { X 2 ω : X has effective packing dimension β } P_{\leq\beta}:=\{X\in 2^{\omega}:X\ \mathrm{has\ effective\ packing\ dimension% \ }\leq\beta\}
  24. P < β := { X 2 ω : X has effective packing dimension < β } P_{<\beta}:=\{X\in 2^{\omega}:X\ \mathrm{has\ effective\ packing\ dimension\ }% <\beta\}
  25. β \beta
  26. H β , H β H_{\beta},H_{\leq\beta}
  27. H < β H_{<\beta}
  28. P β , P β P_{\beta},P_{\leq\beta}
  29. P < β P_{<\beta}
  30. β \beta

Effective_medium_approximations.html

  1. σ \sigma
  2. ϵ \epsilon
  3. i δ i σ i - σ e σ i + ( n - 1 ) σ e = 0 ( 1 ) \sum_{i}\,\delta_{i}\,\frac{\sigma_{i}-\sigma_{e}}{\sigma_{i}+(n-1)\sigma_{e}}% \,=\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)
  4. n n
  5. δ i \delta_{i}
  6. σ i \sigma_{i}
  7. σ e \sigma_{e}
  8. δ i \delta_{i}
  9. 1 n δ α + ( 1 - δ ) ( σ m - σ e ) σ m + ( n - 1 ) σ e = 0 ( 2 ) \frac{1}{n}\,\delta\alpha+\frac{(1-\delta)(\sigma_{m}-\sigma_{e})}{\sigma_{m}+% (n-1)\sigma_{e}}\,=\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2)
  10. σ \sigma
  11. σ m \sigma_{m}
  12. δ \delta
  13. n n
  14. α = 1 n j = 1 n σ - σ e σ e + L j ( σ - σ e ) ( 3 ) \alpha\,=\,\frac{1}{n}\sum_{j=1}^{n}\,\frac{\sigma-\sigma_{e}}{\sigma_{e}+L_{j% }(\sigma-\sigma_{e})}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(3)
  15. L j L_{j}
  16. L 1 = 1 / 2 L_{1}=1/2
  17. L 2 = 1 / 2 L_{2}=1/2
  18. L 1 = 1 / 3 L_{1}=1/3
  19. L 2 = 1 / 3 L_{2}=1/3
  20. L 3 = 1 / 3 L_{3}=1/3
  21. L j L_{j}
  22. σ 1 \sigma_{1}
  23. V V
  24. σ e \sigma_{e}
  25. E 0 ¯ \overline{E_{0}}
  26. p ¯ V σ 1 - σ e σ 1 + 2 σ e E 0 ¯ ( 4 ) . \overline{p}\,\propto\,V\,\frac{\sigma_{1}-\sigma_{e}}{\sigma_{1}+2\sigma_{e}}% \,\overline{E_{0}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(4)\,\,\,\,\,\,\,\,% \,\,\,\,\,\,\,\,\,\,\,\,.
  27. E 0 ¯ \overline{E_{0}}
  28. δ 1 σ 1 - σ e σ 1 + 2 σ e + δ 2 σ 2 - σ e σ 2 + 2 σ e = 0 ( 5 ) \delta_{1}\frac{\sigma_{1}-\sigma_{e}}{\sigma_{1}+2\sigma_{e}}\,+\,\delta_{2}% \frac{\sigma_{2}-\sigma_{e}}{\sigma_{2}+2\sigma_{e}}\,=\,0\,\,\,\,\,\,\,\,\,\,% \,\,\,\,\,\,\,\,\,\,(5)
  29. δ 1 \delta_{1}
  30. δ 2 \delta_{2}
  31. n n
  32. ε m \varepsilon_{m}
  33. ε i \varepsilon_{i}
  34. ( ε eff - ε m ε eff + 2 ε m ) = δ i ( ε i - ε m ε i + 2 ε m ) , ( 6 ) \left(\frac{\varepsilon_{\mathrm{eff}}-\varepsilon_{m}}{\varepsilon_{\mathrm{% eff}}+2\varepsilon_{m}}\right)=\delta_{i}\left(\frac{\varepsilon_{i}-% \varepsilon_{m}}{\varepsilon_{i}+2\varepsilon_{m}}\right),\,\,\,\,\,\,\,\,\,\,% \,\,\,\,\,\,\,\,\,\,(6)
  35. ε eff \varepsilon_{\mathrm{eff}}
  36. ε i \varepsilon_{i}
  37. ε m \varepsilon_{m}
  38. δ i \delta_{i}
  39. ε eff = ε m 2 δ i ( ε i - ε m ) + ε i + 2 ε m 2 ε m + ε i + δ i ( ε m - ε i ) , ( 7 ) \varepsilon_{\mathrm{eff}}\,=\,\varepsilon_{m}\,\frac{2\delta_{i}(\varepsilon_% {i}-\varepsilon_{m})+\varepsilon_{i}+2\varepsilon_{m}}{2\varepsilon_{m}+% \varepsilon_{i}+\delta_{i}(\varepsilon_{m}-\varepsilon_{i})},\,\,\,\,\,\,\,\,(7)

Egyptian_Mathematical_Leather_Roll.html

  1. 1 10 + 1 40 = 1 8 \frac{1}{10}+\frac{1}{40}=\frac{1}{8}
  2. 1 30 + 1 45 + 1 90 = 1 15 \frac{1}{30}+\frac{1}{45}+\frac{1}{90}=\frac{1}{15}
  3. 1 10 + 1 40 = 1 8 \frac{1}{10}+\frac{1}{40}=\frac{1}{8}
  4. 1 18 + 1 36 = 1 12 \frac{1}{18}+\frac{1}{36}=\frac{1}{12}
  5. 1 5 + 1 20 = 1 4 \frac{1}{5}+\frac{1}{20}=\frac{1}{4}
  6. 1 24 + 1 48 = 1 16 \frac{1}{24}+\frac{1}{48}=\frac{1}{16}
  7. 1 5 + 1 20 = 1 4 \frac{1}{5}+\frac{1}{20}=\frac{1}{4}
  8. 1 21 + 1 42 = 1 14 \frac{1}{21}+\frac{1}{42}=\frac{1}{14}
  9. 1 4 + 1 12 = 1 3 \frac{1}{4}+\frac{1}{12}=\frac{1}{3}
  10. 1 18 + 1 36 = 1 12 \frac{1}{18}+\frac{1}{36}=\frac{1}{12}
  11. 1 4 + 1 12 = 1 3 \frac{1}{4}+\frac{1}{12}=\frac{1}{3}
  12. 1 45 + 1 90 = 1 30 \frac{1}{45}+\frac{1}{90}=\frac{1}{30}
  13. 1 10 + 1 10 = 1 5 \frac{1}{10}+\frac{1}{10}=\frac{1}{5}
  14. 1 21 + 1 42 = 1 14 \frac{1}{21}+\frac{1}{42}=\frac{1}{14}
  15. 1 10 + 1 10 = 1 5 \frac{1}{10}+\frac{1}{10}=\frac{1}{5}
  16. 1 30 + 1 60 = 1 20 \frac{1}{30}+\frac{1}{60}=\frac{1}{20}
  17. 1 6 + 1 6 = 1 3 \frac{1}{6}+\frac{1}{6}=\frac{1}{3}
  18. 1 45 + 1 90 = 1 30 \frac{1}{45}+\frac{1}{90}=\frac{1}{30}
  19. 1 6 + 1 6 = 1 3 \frac{1}{6}+\frac{1}{6}=\frac{1}{3}
  20. 1 15 + 1 30 = 1 10 \frac{1}{15}+\frac{1}{30}=\frac{1}{10}
  21. 1 6 + 1 6 + 1 6 = 1 2 \frac{1}{6}+\frac{1}{6}+\frac{1}{6}=\frac{1}{2}
  22. 1 30 + 1 60 = 1 20 \frac{1}{30}+\frac{1}{60}=\frac{1}{20}
  23. 1 6 + 1 6 + 1 6 = 1 2 \frac{1}{6}+\frac{1}{6}+\frac{1}{6}=\frac{1}{2}
  24. 1 48 + 1 96 = 1 32 \frac{1}{48}+\frac{1}{96}=\frac{1}{32}
  25. 1 3 + 1 3 = 2 3 \frac{1}{3}+\frac{1}{3}=\frac{2}{3}
  26. 1 15 + 1 30 = 1 10 \frac{1}{15}+\frac{1}{30}=\frac{1}{10}
  27. 1 3 + 1 3 = 2 3 \frac{1}{3}+\frac{1}{3}=\frac{2}{3}
  28. 1 96 + 1 192 = 1 64 \frac{1}{96}+\frac{1}{192}=\frac{1}{64}
  29. 1 25 + 1 15 + 1 75 + 1 200 = 1 8 \frac{1}{25}+\frac{1}{15}+\frac{1}{75}+\frac{1}{200}=\frac{1}{8}
  30. 1 48 + 1 96 = 1 32 \frac{1}{48}+\frac{1}{96}=\frac{1}{32}
  31. 1 25 + 1 15 + 1 75 + 1 200 = 1 8 \frac{1}{25}+\frac{1}{15}+\frac{1}{75}+\frac{1}{200}=\frac{1}{8}
  32. 1 50 + 1 30 + 1 150 + 1 400 = 1 16 \frac{1}{50}+\frac{1}{30}+\frac{1}{150}+\frac{1}{400}=\frac{1}{16}
  33. 1 96 + 1 192 = 1 64 \frac{1}{96}+\frac{1}{192}=\frac{1}{64}
  34. 1 50 + 1 30 + 1 150 + 1 400 = 1 16 \frac{1}{50}+\frac{1}{30}+\frac{1}{150}+\frac{1}{400}=\frac{1}{16}
  35. 1 25 + 1 50 + 1 150 = 1 6 \frac{1}{25}+\frac{1}{50}+\frac{1}{150}=\frac{1}{6}
  36. 1 25 + 1 50 + 1 150 = 1 6 \frac{1}{25}+\frac{1}{50}+\frac{1}{150}=\frac{1}{6}
  37. 1 9 + 1 18 = 1 6 \frac{1}{9}+\frac{1}{18}=\frac{1}{6}
  38. 1 9 + 1 18 = 1 6 \frac{1}{9}+\frac{1}{18}=\frac{1}{6}
  39. 1 7 + 1 14 + 1 28 = 1 4 \frac{1}{7}+\frac{1}{14}+\frac{1}{28}=\frac{1}{4}
  40. 1 7 + 1 14 + 1 28 = 1 4 \frac{1}{7}+\frac{1}{14}+\frac{1}{28}=\frac{1}{4}
  41. 1 12 + 1 24 = 1 8 \frac{1}{12}+\frac{1}{24}=\frac{1}{8}
  42. 1 12 + 1 24 = 1 8 \frac{1}{12}+\frac{1}{24}=\frac{1}{8}
  43. 1 14 + 1 21 + 1 42 = 1 7 \frac{1}{14}+\frac{1}{21}+\frac{1}{42}=\frac{1}{7}
  44. 1 14 + 1 21 + 1 42 = 1 7 \frac{1}{14}+\frac{1}{21}+\frac{1}{42}=\frac{1}{7}
  45. 1 18 + 1 27 + 1 54 = 1 9 \frac{1}{18}+\frac{1}{27}+\frac{1}{54}=\frac{1}{9}
  46. 1 18 + 1 27 + 1 54 = 1 9 \frac{1}{18}+\frac{1}{27}+\frac{1}{54}=\frac{1}{9}
  47. 1 22 + 1 33 + 1 66 = 1 11 \frac{1}{22}+\frac{1}{33}+\frac{1}{66}=\frac{1}{11}
  48. 1 22 + 1 33 + 1 66 = 1 11 \frac{1}{22}+\frac{1}{33}+\frac{1}{66}=\frac{1}{11}
  49. 1 28 + 1 49 + 1 196 = 1 13 \frac{1}{28}+\frac{1}{49}+\frac{1}{196}=\frac{1}{13}
  50. 1 28 + 1 49 + 1 196 = 1 13 \frac{1}{28}+\frac{1}{49}+\frac{1}{196}=\frac{1}{13}
  51. 1 30 + 1 45 + 1 90 = 1 15 \frac{1}{30}+\frac{1}{45}+\frac{1}{90}=\frac{1}{15}
  52. 1 24 + 1 48 = 1 16 \frac{1}{24}+\frac{1}{48}=\frac{1}{16}
  53. 1 p q = 1 N × N p q \frac{1}{pq}=\frac{1}{N}\times\frac{N}{pq}
  54. 1 / 8 = 1 / 25 × 25 / 8 = 1 / 5 × 25 / 40 = 1 / 5 × ( 3 / 5 + 1 / 40 ) 1/8=1/25\times 25/8=1/5\times 25/40=1/5\times(3/5+1/40)
  55. = 1 / 5 × ( 1 / 5 + 2 / 5 + 1 / 40 ) = 1 / 5 × ( 1 / 5 + 1 / 3 + 1 / 15 + 1 / 40 ) = 1 / 25 + 1 / 15 + 1 / 75 + 1 / 200 =1/5\times(1/5+2/5+1/40)=1/5\times(1/5+1/3+1/15+1/40)=1/25+1/15+1/75+1/200

Eigenvalue_perturbation.html

  1. 𝐊 0 𝐱 0 i = λ 0 i 𝐌 0 𝐱 0 i . ( 0 ) \mathbf{K}_{0}\mathbf{x}_{0i}=\lambda_{0i}\mathbf{M}_{0}\mathbf{x}_{0i}.\qquad% (0)
  2. 𝐊 0 \mathbf{K}_{0}
  3. 𝐌 0 \mathbf{M}_{0}
  4. i = 1 , , N i=1,...,N
  5. 𝐊𝐱 i = λ i 𝐌𝐱 i ( 1 ) \mathbf{K}\mathbf{x}_{i}=\lambda_{i}\mathbf{M}\mathbf{x}_{i}\qquad(1)
  6. 𝐊 = 𝐊 0 + δ 𝐊 𝐌 = 𝐌 0 + δ 𝐌 \begin{aligned}\displaystyle\mathbf{K}&\displaystyle=\mathbf{K}_{0}+\delta% \mathbf{K}\\ \displaystyle\mathbf{M}&\displaystyle=\mathbf{M}_{0}+\delta\mathbf{M}\end{aligned}
  7. δ 𝐊 \delta\mathbf{K}
  8. δ 𝐌 \delta\mathbf{M}
  9. 𝐊 \mathbf{K}
  10. 𝐌 \mathbf{M}
  11. λ i \displaystyle\lambda_{i}
  12. 𝐱 0 j 𝐌 0 𝐱 0 i = δ i j ( 2 ) \mathbf{x}_{0j}^{\top}\mathbf{M}_{0}\mathbf{x}_{0i}=\delta_{ij}\qquad(2)
  13. 𝐊𝐱 i = λ i 𝐌𝐱 i . \mathbf{K}\mathbf{x}_{i}=\lambda_{i}\mathbf{M}\mathbf{x}_{i}.
  14. ( 𝐊 0 + δ 𝐊 ) ( 𝐱 0 i + δ 𝐱 i ) = ( λ 0 i + δ λ i ) ( 𝐌 0 + δ 𝐌 ) ( 𝐱 0 i + δ 𝐱 i ) , (\mathbf{K}_{0}+\delta\mathbf{K})(\mathbf{x}_{0i}+\delta\mathbf{x}_{i})=\left(% \lambda_{0i}+\delta\lambda_{i}\right)\left(\mathbf{M}_{0}+\delta\mathbf{M}% \right)\left(\mathbf{x}_{0i}+\delta\mathbf{x}_{i}\right),
  15. 𝐊 0 𝐱 0 i + δ 𝐊𝐱 0 i + 𝐊 0 δ 𝐱 i + δ 𝐊 δ 𝐱 i = = λ 0 i 𝐌 0 𝐱 0 i + λ 0 i 𝐌 0 δ 𝐱 i + λ 0 i δ 𝐌𝐱 0 i + δ λ i 𝐌 0 𝐱 0 i + λ 0 i δ 𝐌 δ 𝐱 i + δ λ i δ 𝐌𝐱 0 i + δ λ i 𝐌 0 δ 𝐱 i + δ λ i δ 𝐌 δ 𝐱 i . \begin{aligned}\displaystyle\mathbf{K}_{0}\mathbf{x}_{0i}&\displaystyle+\delta% \mathbf{K}\mathbf{x}_{0i}+\mathbf{K}_{0}\delta\mathbf{x}_{i}+\delta\mathbf{K}% \delta\mathbf{x}_{i}=\\ &\displaystyle=\lambda_{0i}\mathbf{M}_{0}\mathbf{x}_{0i}+\lambda_{0i}\mathbf{M% }_{0}\delta\mathbf{x}_{i}+\lambda_{0i}\delta\mathbf{M}\mathbf{x}_{0i}+\delta% \lambda_{i}\mathbf{M}_{0}\mathbf{x}_{0i}+\lambda_{0i}\delta\mathbf{M}\delta% \mathbf{x}_{i}+\delta\lambda_{i}\delta\mathbf{M}\mathbf{x}_{0i}+\delta\lambda_% {i}\mathbf{M}_{0}\delta\mathbf{x}_{i}+\delta\lambda_{i}\delta\mathbf{M}\delta% \mathbf{x}_{i}.\end{aligned}
  16. δ 𝐊𝐱 0 i + 𝐊 0 δ 𝐱 i + δ 𝐊 δ 𝐱 i = λ 0 i 𝐌 0 δ 𝐱 i + λ 0 i δ 𝐌𝐱 0 i + δ λ i 𝐌 0 𝐱 0 i + λ 0 i δ 𝐌 δ 𝐱 i + δ λ i δ 𝐌𝐱 0 i + δ λ i 𝐌 0 δ 𝐱 i + δ λ i δ 𝐌 δ 𝐱 i . \begin{aligned}\displaystyle\delta\mathbf{K}\mathbf{x}_{0i}+\mathbf{K}_{0}% \delta\mathbf{x}_{i}+\delta\mathbf{K}\delta\mathbf{x}_{i}=\lambda_{0i}\mathbf{% M}_{0}\delta\mathbf{x}_{i}+\lambda_{0i}\delta\mathbf{M}\mathbf{x}_{0i}+\delta% \lambda_{i}\mathbf{M}_{0}\mathbf{x}_{0i}+\lambda_{0i}\delta\mathbf{M}\delta% \mathbf{x}_{i}+\delta\lambda_{i}\delta\mathbf{M}\mathbf{x}_{0i}+\delta\lambda_% {i}\mathbf{M}_{0}\delta\mathbf{x}_{i}+\delta\lambda_{i}\delta\mathbf{M}\delta% \mathbf{x}_{i}.\end{aligned}
  17. 𝐊 0 δ 𝐱 i + δ 𝐊𝐱 0 i = λ 0 i 𝐌 0 δ 𝐱 i + λ 0 i δ 𝐌 x 0 i + δ λ i 𝐌 0 𝐱 0 i . ( 3 ) \mathbf{K}_{0}\delta\mathbf{x}_{i}+\delta\mathbf{K}\mathbf{x}_{0i}=\lambda_{0i% }\mathbf{M}_{0}\delta\mathbf{x}_{i}+\lambda_{0i}\delta\mathbf{M}\mathrm{x}_{0i% }+\delta\lambda_{i}\mathbf{M}_{0}\mathbf{x}_{0i}.\qquad(3)
  18. δ 𝐱 i = j = 1 N ε i j 𝐱 0 j ( 4 ) \delta\mathbf{x}_{i}=\sum_{j=1}^{N}\varepsilon_{ij}\mathbf{x}_{0j}\qquad(4)
  19. 𝐊 0 j = 1 N ε i j 𝐱 0 j + δ 𝐊𝐱 0 i = λ 0 i 𝐌 0 j = 1 N ε i j 𝐱 0 j + λ 0 i δ 𝐌𝐱 0 i + δ λ i 𝐌 0 𝐱 0 i ( 5 ) j = 1 N ε i j 𝐊 0 𝐱 0 j + δ 𝐊𝐱 0 i = λ 0 i 𝐌 0 j = 1 N ε i j 𝐱 0 j + λ 0 i δ 𝐌𝐱 0 i + δ λ i 𝐌 0 𝐱 0 i Applying 𝐊 0 to the sum j = 1 N ε i j λ 0 j 𝐌 0 𝐱 0 j + δ 𝐊𝐱 0 i = λ 0 i 𝐌 0 j = 1 N ε i j 𝐱 0 j + λ 0 i δ 𝐌𝐱 0 i + δ λ i 𝐌 0 𝐱 0 i Using Eq. ( 1 ) \begin{aligned}\displaystyle\mathbf{K}_{0}\sum_{j=1}^{N}\varepsilon_{ij}% \mathbf{x}_{0j}+\delta\mathbf{K}\mathbf{x}_{0i}&\displaystyle=\lambda_{0i}% \mathbf{M}_{0}\sum_{j=1}^{N}\varepsilon_{ij}\mathbf{x}_{0j}+\lambda_{0i}\delta% \mathbf{M}\mathbf{x}_{0i}+\delta\lambda_{i}\mathbf{M}_{0}\mathbf{x}_{0i}&&% \displaystyle(5)\\ \displaystyle\sum_{j=1}^{N}\varepsilon_{ij}\mathbf{K}_{0}\mathbf{x}_{0j}+% \delta\mathbf{K}\mathbf{x}_{0i}&\displaystyle=\lambda_{0i}\mathbf{M}_{0}\sum_{% j=1}^{N}\varepsilon_{ij}\mathbf{x}_{0j}+\lambda_{0i}\delta\mathbf{M}\mathbf{x}% _{0i}+\delta\lambda_{i}\mathbf{M}_{0}\mathbf{x}_{0i}&&\displaystyle\,\text{% Applying }\mathbf{K}_{0}\,\text{ to the sum}\\ \displaystyle\sum_{j=1}^{N}\varepsilon_{ij}\lambda_{0j}\mathbf{M}_{0}\mathbf{x% }_{0j}+\delta\mathbf{K}\mathbf{x}_{0i}&\displaystyle=\lambda_{0i}\mathbf{M}_{0% }\sum_{j=1}^{N}\varepsilon_{ij}\mathbf{x}_{0j}+\lambda_{0i}\delta\mathbf{M}% \mathbf{x}_{0i}+\delta\lambda_{i}\mathbf{M}_{0}\mathbf{x}_{0i}&&\displaystyle% \,\text{Using Eq. }(1)\end{aligned}
  20. 𝐱 0 i \mathbf{x}_{0i}^{\top}
  21. 𝐱 0 i ε i i λ 0 i 𝐌 0 𝐱 0 i + 𝐱 0 i δ 𝐊𝐱 0 i = λ 0 i 𝐱 0 i 𝐌 0 ε i i 𝐱 0 i + λ 0 i 𝐱 0 i δ 𝐌𝐱 0 i + δ λ i 𝐱 0 i 𝐌 0 𝐱 0 i . \mathbf{x}_{0i}^{\top}\varepsilon_{ii}\lambda_{0i}\mathbf{M}_{0}\mathbf{x}_{0i% }+\mathbf{x}_{0i}^{\top}\delta\mathbf{K}\mathbf{x}_{0i}=\lambda_{0i}\mathbf{x}% _{0i}^{\top}\mathbf{M}_{0}\varepsilon_{ii}\mathbf{x}_{0i}+\lambda_{0i}\mathbf{% x}_{0i}^{\top}\delta\mathbf{M}\mathbf{x}_{0i}+\delta\lambda_{i}\mathbf{x}_{0i}% ^{\top}\mathbf{M}_{0}\mathbf{x}_{0i}.
  22. 𝐱 0 i 𝐊 0 ε i i 𝐱 0 i + 𝐱 0 i δ 𝐊𝐱 0 i = λ 0 i 𝐱 0 i 𝐌 0 ε i i 𝐱 0 i + λ 0 i 𝐱 0 i δ 𝐌𝐱 0 i + δ λ i 𝐱 0 i 𝐌 0 𝐱 0 i . ( 6 ) \mathbf{x}_{0i}^{\top}\mathbf{K}_{0}\varepsilon_{ii}\mathbf{x}_{0i}+\mathbf{x}% _{0i}^{\top}\delta\mathbf{K}\mathbf{x}_{0i}=\lambda_{0i}\mathbf{x}_{0i}^{\top}% \mathbf{M}_{0}\varepsilon_{ii}\mathbf{x}_{0i}+\lambda_{0i}\mathbf{x}_{0i}^{% \top}\delta\mathbf{M}\mathbf{x}_{0i}+\delta\lambda_{i}\mathbf{x}_{0i}^{\top}% \mathbf{M}_{0}\mathbf{x}_{0i}.\qquad(6)
  23. 𝐱 0 i \mathbf{x}_{0i}^{\top}
  24. 𝐱 0 i 𝐊 0 𝐱 0 i = λ 0 i 𝐱 0 i 𝐌 0 𝐱 0 i . \mathbf{x}_{0i}^{\top}\mathbf{K}_{0}\mathbf{x}_{0i}=\lambda_{0i}\mathbf{x}_{0i% }^{\top}\mathbf{M}_{0}\mathbf{x}_{0i}.
  25. 𝐱 0 i δ 𝐊𝐱 0 i = λ 0 i 𝐱 0 i δ 𝐌𝐱 0 i + δ λ i 𝐱 0 i 𝐌 0 𝐱 0 i . \mathbf{x}_{0i}^{\top}\delta\mathbf{K}\mathbf{x}_{0i}=\lambda_{0i}\mathbf{x}_{% 0i}^{\top}\delta\mathbf{M}\mathbf{x}_{0i}+\delta\lambda_{i}\mathbf{x}_{0i}^{% \top}\mathbf{M}_{0}\mathbf{x}_{0i}.
  26. δ λ i = 𝐱 0 i ( δ 𝐊 - λ 0 i δ 𝐌 ) 𝐱 0 i 𝐱 0 i 𝐌 0 𝐱 0 i \delta\lambda_{i}=\frac{\mathbf{x}^{\top}_{0i}\left(\delta\mathbf{K}-\lambda_{% 0i}\delta\mathbf{M}\right)\mathbf{x}_{0i}}{\mathbf{x}_{0i}^{\top}\mathbf{M}_{0% }\mathbf{x}_{0i}}
  27. δ λ i = 𝐱 0 i ( δ 𝐊 - λ 0 i δ 𝐌 ) 𝐱 0 i . \delta\lambda_{i}=\mathbf{x}^{\top}_{0i}\left(\delta\mathbf{K}-\lambda_{0i}% \delta\mathbf{M}\right)\mathbf{x}_{0i}.
  28. ε i k = 𝐱 0 k ( δ 𝐊 - λ 0 i δ 𝐌 ) 𝐱 0 i λ 0 i - λ 0 k , i k . \varepsilon_{ik}=\frac{\mathbf{x}^{\top}_{0k}\left(\delta\mathbf{K}-\lambda_{0% i}\delta\mathbf{M}\right)\mathbf{x}_{0i}}{\lambda_{0i}-\lambda_{0k}},\qquad i% \neq k.
  29. ε i j = 𝐱 0 j ( δ 𝐊 - λ 0 i δ 𝐌 ) 𝐱 0 i λ 0 i - λ 0 j , i j . \varepsilon_{ij}=\frac{\mathbf{x}^{\top}_{0j}\left(\delta\mathbf{K}-\lambda_{0% i}\delta\mathbf{M}\right)\mathbf{x}_{0i}}{\lambda_{0i}-\lambda_{0j}},\qquad i% \neq j.
  30. 𝐱 i 𝐌𝐱 i = 1 \mathbf{x}^{\top}_{i}\mathbf{M}\mathbf{x}_{i}=1
  31. ε i i = - 1 2 𝐱 0 i δ 𝐌𝐱 0 i . \varepsilon_{ii}=-\tfrac{1}{2}\mathbf{x}^{\top}_{0i}\delta\mathbf{M}\mathbf{x}% _{0i}.
  32. λ i = λ 0 i + 𝐱 0 i ( δ 𝐊 - λ 0 i δ 𝐌 ) 𝐱 0 i 𝐱 i = 𝐱 0 i ( 1 - 1 2 𝐱 0 i δ 𝐌𝐱 0 i ) + j = 1 j i N 𝐱 0 j ( δ 𝐊 - λ 0 i δ 𝐌 ) 𝐱 0 i λ 0 i - λ 0 j 𝐱 0 j \begin{aligned}\displaystyle\lambda_{i}&\displaystyle=\lambda_{0i}+\mathbf{x}^% {\top}_{0i}\left(\delta\mathbf{K}-\lambda_{0i}\delta\mathbf{M}\right)\mathbf{x% }_{0i}\\ \displaystyle\mathbf{x}_{i}&\displaystyle=\mathbf{x}_{0i}\left(1-\tfrac{1}{2}% \mathbf{x}^{\top}_{0i}\delta\mathbf{M}\mathbf{x}_{0i}\right)+\sum_{j=1\atop j% \neq i}^{N}\frac{\mathbf{x}^{\top}_{0j}\left(\delta\mathbf{K}-\lambda_{0i}% \delta\mathbf{M}\right)\mathbf{x}_{0i}}{\lambda_{0i}-\lambda_{0j}}\mathbf{x}_{% 0j}\end{aligned}
  33. δ K δK
  34. δ M δM
  35. λ i 𝐊 ( k ) = 𝐊 ( k ) ( λ 0 i + 𝐱 0 i ( δ 𝐊 - λ 0 i δ 𝐌 ) 𝐱 0 i ) = x 0 i ( k ) x 0 i ( ) ( 2 - δ k ) λ i 𝐌 ( k ) = 𝐌 ( k ) ( λ 0 i + 𝐱 0 i ( δ 𝐊 - λ 0 i δ 𝐌 ) 𝐱 0 i ) = λ i x 0 i ( k ) x 0 i ( ) ( 2 - δ k ) . \begin{aligned}\displaystyle\frac{\partial\lambda_{i}}{\partial\mathbf{K}_{(k% \ell)}}&\displaystyle=\frac{\partial}{\partial\mathbf{K}_{(k\ell)}}\left(% \lambda_{0i}+\mathbf{x}^{\top}_{0i}\left(\delta\mathbf{K}-\lambda_{0i}\delta% \mathbf{M}\right)\mathbf{x}_{0i}\right)=x_{0i(k)}x_{0i(\ell)}\left(2-\delta_{k% \ell}\right)\\ \displaystyle\frac{\partial\lambda_{i}}{\partial\mathbf{M}_{(k\ell)}}&% \displaystyle=\frac{\partial}{\partial\mathbf{M}_{(k\ell)}}\left(\lambda_{0i}+% \mathbf{x}^{\top}_{0i}\left(\delta\mathbf{K}-\lambda_{0i}\delta\mathbf{M}% \right)\mathbf{x}_{0i}\right)=\lambda_{i}x_{0i(k)}x_{0i(\ell)}\left(2-\delta_{% k\ell}\right).\end{aligned}
  36. 𝐱 i 𝐊 ( k ) = j = 1 j i N x 0 j ( k ) x 0 i ( ) ( 2 - δ k ) λ 0 i - λ 0 j 𝐱 0 j 𝐱 i 𝐌 ( k ) = - 𝐱 0 i x 0 i ( k ) x 0 i ( ) 2 ( 2 - δ k ) - j = 1 j i N λ 0 i x 0 j ( k ) x 0 i ( ) λ 0 i - λ 0 j 𝐱 0 j ( 2 - δ k ) . \begin{aligned}\displaystyle\frac{\partial\mathbf{x}_{i}}{\partial\mathbf{K}_{% (k\ell)}}&\displaystyle=\sum_{j=1\atop j\neq i}^{N}\frac{x_{0j(k)}x_{0i(\ell)}% \left(2-\delta_{k\ell}\right)}{\lambda_{0i}-\lambda_{0j}}\mathbf{x}_{0j}\\ \displaystyle\frac{\partial\mathbf{x}_{i}}{\partial\mathbf{M}_{(k\ell)}}&% \displaystyle=-\mathbf{x}_{0i}\frac{x_{0i(k)}x_{0i(\ell)}}{2}(2-\delta_{k\ell}% )-\sum_{j=1\atop j\neq i}^{N}\frac{\lambda_{0i}x_{0j(k)}x_{0i(\ell)}}{\lambda_% {0i}-\lambda_{0j}}\mathbf{x}_{0j}\left(2-\delta_{k\ell}\right).\end{aligned}
  37. N N
  38. N N
  39. 𝐊 \mathbf{K}
  40. 𝐌 \mathbf{M}

Eikonal_approximation.html

  1. Ψ = e i S / \Psi=e^{iS/{\hbar}}
  2. - 2 2 m 2 Ψ = ( E - V ) Ψ -\frac{{\hbar}^{2}}{2m}{\nabla}^{2}\Psi=(E-V)\Psi
  3. - 2 2 m 2 e i S / = ( E - V ) e i S / -\frac{{\hbar}^{2}}{2m}{\nabla}^{2}{e^{iS/{\hbar}}}=(E-V)e^{iS/{\hbar}}
  4. 1 2 m ( S ) 2 - i 2 m 2 S = E - V \frac{1}{2m}{(\nabla S)}^{2}-\frac{i\hbar}{2m}{\nabla}^{2}S=E-V
  5. S = S 0 + i S 1 + S=S_{0}+\frac{\hbar}{i}S_{1}+...
  6. 1 2 m ( S 0 ) 2 = E - V \frac{1}{2m}{(\nabla S_{0})}^{2}=E-V
  7. 2 z 2 {\nabla}^{2}\rightarrow{\partial_{z}}^{2}
  8. S ( z = z 0 ) = k z 0 \frac{S(z=z_{0})}{\hbar}=kz_{0}
  9. d d z S 0 = k 2 - 2 m V / 2 \frac{d}{dz}\frac{S_{0}}{\hbar}=\sqrt{k^{2}-2mV/{\hbar}^{2}}
  10. S 0 ( z ) = k z - m 2 k - Z V d z \frac{S_{0}(z)}{\hbar}=kz-\frac{m}{{\hbar}^{2}k}\int_{-\infty}^{Z}{Vdz^{\prime}}

Eilenberg's_inequality.html

  1. Y * H m - n ( A f - 1 ( y ) ) d H n ( y ) v m - n v n v m ( Lip f ) n H m ( A ) , \int_{Y}^{*}H_{m-n}(A\cap f^{-1}(y))\,dH_{n}(y)\leq\frac{v_{m-n}v_{n}}{v_{m}}(% \,\text{Lip }f)^{n}H_{m}(A),

Eilenberg–Moore_spectral_sequence.html

  1. k k
  2. H ( - ) = H ( - , k ) , H ( - ) = H ( - , k ) H_{\ast}(-)=H_{\ast}(-,k),H^{\ast}(-)=H^{\ast}(-,k)
  3. E f E p X f B \begin{array}[]{c c c}E_{f}&\rightarrow&E\\ \downarrow&&\downarrow{p}\\ X&\rightarrow_{f}&B\\ \end{array}
  4. E 2 , = Tor H ( B ) , ( H ( X ) , H ( E ) ) H ( E f ) . E_{2}^{\ast,\ast}=\,\text{Tor}_{H^{\ast}(B)}^{\ast,\ast}(H^{\ast}(X),H^{\ast}(% E))\Rightarrow H^{\ast}(E_{f}).
  5. E , 2 = Cotor , H ( B ) ( H ( X ) , H ( E ) ) H ( E f ) . E^{2}_{\ast,\ast}=\,\text{Cotor}^{H_{\ast}(B)}_{\ast,\ast}(H_{\ast}(X),H_{\ast% }(E))\Rightarrow H_{\ast}(E_{f}).
  6. S ( - ) = S ( - , k ) S_{\ast}(-)=S_{\ast}(-,k)
  7. k k
  8. S ( B ) S_{\ast}(B)
  9. k k
  10. S ( B ) S ( B × B ) S ( B ) S ( B ) . S_{\ast}(B)\xrightarrow{\triangle}S_{\ast}(B\times B)\xrightarrow{\simeq}S_{% \ast}(B)\otimes S_{\ast}(B).
  11. f f
  12. p p
  13. f : S ( X ) S ( B ) f_{\ast}\colon S_{\ast}(X)\rightarrow S_{\ast}(B)
  14. p : S ( E ) S ( B ) p_{\ast}\colon S_{\ast}(E)\rightarrow S_{\ast}(B)
  15. S ( E ) S_{\ast}(E)
  16. S ( X ) S_{\ast}(X)
  17. S ( B ) S_{\ast}(B)
  18. S ( X ) S ( X ) S ( X ) f 1 S ( B ) S ( X ) S_{\ast}(X)\xrightarrow{\triangle}S_{\ast}(X)\otimes S_{\ast}(X)\xrightarrow{f% _{\ast}\otimes 1}S_{\ast}(B)\otimes S_{\ast}(X)
  19. S ( X ) S_{\ast}(X)
  20. S ( B ) S_{\ast}(B)
  21. 𝒞 ( S ( X ) , S ( B ) ) = δ 2 𝒞 - 2 ( S ( X ) , S ( B ) ) δ 1 𝒞 - 1 ( S ( X ) , S ( B ) ) δ 0 S ( X ) S ( B ) , \mathcal{C}(S_{\ast}(X),S_{\ast}(B))=\cdots\xleftarrow{\delta_{2}}\mathcal{C}_% {-2}(S_{\ast}(X),S_{\ast}(B))\xleftarrow{\delta_{1}}\mathcal{C}_{-1}(S_{\ast}(% X),S\ast(B))\xleftarrow{\delta_{0}}S_{\ast}(X)\otimes S_{\ast}(B),
  22. 𝒞 - n \mathcal{C}_{-n}
  23. 𝒞 - n ( S ( X ) , S ( B ) ) = S ( X ) S ( B ) S ( B ) n S ( B ) . \mathcal{C}_{-n}(S_{\ast}(X),S_{\ast}(B))=S_{\ast}(X)\otimes\underbrace{S_{% \ast}(B)\otimes\cdots\otimes S_{\ast}(B)}_{n}\otimes S_{\ast}(B).
  24. δ n \delta_{n}
  25. λ f 1 + i = 2 n 1 i 1 , \lambda_{f}\otimes\cdots\otimes 1+\sum_{i=2}^{n}1\otimes\cdots\otimes\triangle% _{i}\otimes\cdots\otimes 1,
  26. λ f \lambda_{f}
  27. S ( X ) S_{\ast}(X)
  28. S ( B ) S_{\ast}(B)
  29. 𝒞 \mathbf{\mathcal{C}}_{\bullet}
  30. Θ : 𝒞 S ( B ) S ( E ) S ( E f , k ) \Theta\colon\mathbf{\mathcal{C}}_{\bullet{\,\text{ }\Box_{S_{\ast}(B)}}}S_{% \ast}(E)\rightarrow S_{\ast}(E_{f},k)
  31. Θ : Cotor S ( B ) ( S ( X ) S ( E ) ) H ( E f ) , \Theta_{\ast}\colon\operatorname{Cotor}^{S_{\ast}(B)}(S_{\ast}(X)S_{\ast}(E))% \rightarrow H_{\ast}(E_{f}),
  32. S ( B ) \Box_{S_{\ast}(B)}
  33. H ( 𝒞 S ( B ) S ( E ) ) H_{\ast}(\mathbf{\mathcal{C}}_{\bullet{\,\text{ }\Box_{S_{\ast}(B)}}}S_{\ast}(% E))
  34. 𝒞 S ( B ) S ( E ) \mathbf{\mathcal{C}}_{\bullet{\,\text{ }\Box_{S_{\ast}(B)}}}S_{\ast}(E)
  35. E 2 = Cotor H ( B ) ( H ( X ) , H ( E ) ) . E^{2}=\operatorname{Cotor}^{H_{\ast}(B)}(H_{\ast}(X),H_{\ast}(E)).
  36. π 1 ( B ) \pi_{1}(B)
  37. H i ( E f ) H_{i}(E_{f})
  38. i 0 i\geq 0

Eilenberg–Zilber_theorem.html

  1. X × Y X\times Y
  2. X X
  3. Y Y
  4. X X
  5. Y Y
  6. C * ( X ) C_{*}(X)
  7. C * ( Y ) C_{*}(Y)
  8. C * ( X × Y ) C_{*}(X\times Y)
  9. C * ( X ) C * ( Y ) C_{*}(X)\otimes C_{*}(Y)
  10. δ ( σ τ ) = δ X σ τ + ( - 1 ) p σ δ Y τ \delta(\sigma\otimes\tau)=\delta_{X}\sigma\otimes\tau+(-1)^{p}\sigma\otimes% \delta_{Y}\tau
  11. σ C p ( X ) \sigma\in C_{p}(X)
  12. δ X \delta_{X}
  13. δ Y \delta_{Y}
  14. C * ( X ) C_{*}(X)
  15. C * ( Y ) C_{*}(Y)
  16. F : C * ( X × Y ) C * ( X ) C * ( Y ) , G : C * ( X ) C * ( Y ) C * ( X × Y ) F:C_{*}(X\times Y)\rightarrow C_{*}(X)\otimes C_{*}(Y),\quad G:C_{*}(X)\otimes C% _{*}(Y)\rightarrow C_{*}(X\times Y)
  17. F G FG
  18. G F GF
  19. X X
  20. Y Y
  21. H * ( C * ( X × Y ) ) H * ( C * ( X ) C * ( Y ) ) . H_{*}(C_{*}(X\times Y))\cong H_{*}(C_{*}(X)\otimes C_{*}(Y)).
  22. H * ( X × Y ) H_{*}(X\times Y)
  23. H * ( X ) H_{*}(X)
  24. H * ( Y ) H_{*}(Y)

Elasticity_of_cell_membranes.html

  1. k c , k ¯ k_{c},\bar{k}
  2. c 0 c_{0}
  3. H H
  4. K K
  5. Δ p \Delta p
  6. R R
  7. 2 \sqrt{2}
  8. C C
  9. d s ds
  10. γ \gamma
  11. k n k_{n}
  12. k g k_{g}
  13. τ g \tau_{g}
  14. 𝐞 2 \mathbf{e}_{2}
  15. 2 H 2H
  16. K K
  17. tr ( ε ) \mathrm{tr}(\varepsilon)
  18. det ( ε ) \det(\varepsilon)
  19. ε \varepsilon
  20. tr ε \mathrm{tr}\varepsilon
  21. - tr ε -\mathrm{tr}\varepsilon
  22. k d k_{d}
  23. μ \mu

Electrical_conductivity_meter.html

  1. σ T = σ T c a l [ 1 + α ( T - T c a l ) ] \sigma_{T}={\sigma_{T_{cal}}[1+\alpha(T-T_{cal})]}

Electro-gyration.html

  1. 4 ¯ 3 m \overline{4}3m\,
  2. E i = B i j 0 D j + δ ~ i j k D j x k = B i j 0 D j + ( i e i j l g ~ l k k k ) D j E_{i}=B_{ij}^{0}D_{j}+\tilde{\delta}_{ijk}\frac{\partial D_{j}}{\partial x_{k}% }=B_{ij}^{0}D_{j}+(ie_{ijl}\tilde{g}_{lk}k_{k})D_{j}\,
  3. D i = ϵ i j 0 E j + δ i j k E j x k = ϵ i j 0 E j + ( i e i j l g l k k k ) E j D_{i}=\epsilon_{ij}^{0}E_{j}+\delta_{ijk}\frac{\partial E_{j}}{\partial x_{k}}% =\epsilon_{ij}^{0}E_{j}+(ie_{ijl}{g}_{lk}k_{k})E_{j}\,
  4. B i j 0 B_{ij}^{0}
  5. ϵ i j 0 \epsilon_{ij}^{0}
  6. g ~ l k n ¯ = g k l \tilde{g}_{lk}\overline{n}=g_{kl}
  7. n ¯ \overline{n}
  8. D j D_{j}\,
  9. δ i j k \delta_{ijk}\,
  10. δ ~ i j k \tilde{\delta}_{ijk}
  11. e i j l e_{ijl}\,
  12. k k k_{k}\,
  13. g l k g_{lk}\,
  14. g ~ l k \tilde{g}_{lk}
  15. ρ \rho\,
  16. ρ = π λ n g l k l l l k = π λ n G \rho=\frac{\pi}{\lambda n}g_{lk}l_{l}l_{k}=\frac{\pi}{\lambda n}G\,
  17. n n\,
  18. λ \lambda\,
  19. l l l_{l}\,
  20. l k l_{k}\,
  21. l 1 = sin Θ cos φ l_{1}=\sin\Theta\cos\varphi\,
  22. l 2 = sin Θ sin φ , l 3 = cos Θ l_{2}=\sin\Theta\sin\varphi,l_{3}=\cos\Theta
  23. G G\,
  24. E m E_{m}\,
  25. E n E_{n}\,
  26. Δ g l k = γ l k m E m + β l k m n E m E n \Delta g_{lk}=\gamma_{lkm}E_{m}+\beta_{lkmn}E_{m}E_{n}\,
  27. γ l k m \gamma_{lkm}\,
  28. β l k m n \beta_{lkmn}\,
  29. Δ ρ = π λ n g l k l l l k = π λ n Δ G = π λ n ( γ l k m E m + β l k m n E m E n ) l l l k \Delta\rho=\frac{\pi}{\lambda n}g_{lk}l_{l}l_{k}=\frac{\pi}{\lambda n}\Delta G% =\frac{\pi}{\lambda n}(\gamma_{lkm}E_{m}+\beta_{lkmn}E_{m}E_{n})l_{l}l_{k}
  30. P m s P n s P_{m}^{s}P_{n}^{s}\,
  31. Δ ρ = π λ n g l k l l l k = π λ n Δ G = π λ n ( γ ~ l k m P m s + β ~ l k m n P m s P n s ) l l l k \Delta\rho=\frac{\pi}{\lambda n}g_{lk}l_{l}l_{k}=\frac{\pi}{\lambda n}\Delta G% =\frac{\pi}{\lambda n}(\tilde{\gamma}_{lkm}P_{m}^{s}+\tilde{\beta}_{lkmn}P_{m}% ^{s}P_{n}^{s})l_{l}l_{k}
  32. 2 \infty 2\,
  33. m m \infty mm\,
  34. E m E n E_{m}E_{n}\,
  35. / m m m \infty/mmm\,
  36. κ \kappa\,
  37. χ \chi\,
  38. κ = Δ G 2 Δ n n ¯ \kappa=\frac{\Delta G}{2\Delta n\overline{n}}\,
  39. tan 2 ( α - χ ) = 2 κ 1 + κ 2 tan s y m b o l Γ ( 1 + P tan 2 α + ( 1 - R ) R + tan 2 2 α ) \tan 2(\alpha-\chi)=\frac{2\kappa}{1+\kappa^{2}}\tan symbol{\Gamma}\left(1+% \frac{P\tan 2\alpha+(1-R)}{R+\tan^{2}2\alpha}\right)\,
  40. α \alpha\,
  41. Δ n \Delta n\,
  42. s y m b o l Γ symbol\Gamma\,
  43. P = ( 1 - κ 2 ) 2 2 κ ( 1 + κ 2 ) P=\frac{(1-\kappa^{2})^{2}}{2\kappa(1+\kappa^{2})}\,
  44. R = ( 2 κ 1 + κ 2 ) 2 + ( 1 - κ 2 1 + κ 2 ) 2 R=\left(\frac{2\kappa}{1+\kappa^{2}}\right)^{2}+\left(\frac{1-\kappa^{2}}{1+% \kappa^{2}}\right)^{2}\,
  45. κ = 1 \kappa=1\,
  46. 2 ( α - χ ) = s y m b o l Γ 2(\alpha-\chi)=symbol\Gamma\,
  47. ρ d = α - s y m b o l Γ 2 \rho d=\alpha-\frac{symbol\Gamma}{2}\,
  48. d d\,
  49. κ \kappa\,
  50. κ 2 \kappa^{2}\,
  51. α = 0 \alpha=0\,
  52. tan 2 χ = - 2 κ sin s y m b o l Γ \tan 2\chi=-2\kappa\sin symbol\Gamma\,
  53. g k l = 2 χ Δ n n ¯ g_{kl}=2\chi\Delta n\overline{n}\,
  54. s y m b o l Γ symbol\Gamma\,
  55. α - \alpha-\,
  56. α \alpha\,

Electrode_array.html

  1. ρ = 2 π a V I \rho=2\pi a\frac{V}{I}

Electron-transfer_dissociation.html

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

Electron_tomography.html

  1. 2 \scriptstyle\sqrt{2}

Element_(category_theory).html

  1. p : T A p\colon T\to A
  2. f g = f h f\circ g=f\circ h
  3. g = h g=h
  4. f : B C f\colon B\to C
  5. g , h : A B g,h\colon A\to B
  6. g f = h f g\circ f=h\circ f
  7. g = h g=h
  8. p : A × B A , p\colon A\times B\to A,
  9. q : A × B B q\colon A\times B\to B
  10. f : T A , g : T B f\colon T\to A,g\colon T\to B
  11. h : T A × B h\colon T\to A\times B
  12. f = p h f=p\circ h
  13. g = q h g=q\circ h
  14. p : A × B A p\colon A\times B\to A
  15. q : A × B B q\colon A\times B\to B
  16. ( Spec ) (\operatorname{Spec}\mathbb{C})
  17. ( Spec ) (\operatorname{Spec}\mathbb{Q})
  18. ( Spec 𝔽 p ) (\operatorname{Spec}\mathbb{F}_{p})
  19. x 2 + 1 = 0 x^{2}+1=0
  20. ± i \pm i

Element_distinctness_problem.html

  1. Θ ( n 2 / 3 ) \Theta\left(n^{2/3}\right)

Elitserien_(speedway).html

  1. N = C * M + ( ( p * h / 4 ) - I ) M N=\frac{C*M+((p*h/4)-I)}{M}
  2. 8 , 3966667 = 8 , 11 * 12 + ( ( 14 * 5 / 4 ) - 7 , 76 ) 12 8,3966667=\frac{8,11*12+((14*5/4)-7,76)}{12}

Elliptic_boundary_value_problem.html

  1. x , y x,y
  2. u x , u x x u_{x},u_{xx}
  3. u u
  4. x x
  5. y y
  6. D x D_{x}
  7. D y D_{y}
  8. x x
  9. y y
  10. D x 2 D_{x}^{2}
  11. D y 2 D_{y}^{2}
  12. u = ( u x , u y ) \nabla u=(u_{x},u_{y})
  13. Δ u = u x x + u y y \Delta u=u_{xx}+u_{yy}
  14. ( u , v ) = u x + v y \nabla\cdot(u,v)=u_{x}+v_{y}
  15. Δ u = ( u ) \Delta u=\nabla\cdot(\nabla u)
  16. Δ u = f in Ω , \Delta u=f\,\text{ in }\Omega,
  17. u = 0 on Ω ; u=0\text{ on }\partial\Omega;
  18. Ω \Omega
  19. Ω \partial\Omega
  20. f f
  21. u u
  22. u u
  23. Ω \Omega
  24. f f
  25. f ( x ) f(x)
  26. u u
  27. L u = a u x x + b u y y Lu=au_{xx}+bu_{yy}
  28. a a
  29. b b
  30. L = a D x 2 + b D y 2 L=aD_{x}^{2}+bD_{y}^{2}
  31. D x D_{x}
  32. x x
  33. D y D_{y}
  34. y y
  35. a x 2 + b y 2 ax^{2}+by^{2}
  36. k k
  37. a , b , k a,b,k
  38. a a
  39. b b
  40. L L
  41. a b > 0 ab>0
  42. a b < 0 ab<0
  43. L = D x + D y 2 L=D_{x}+D_{y}^{2}
  44. L L
  45. x 1 , , x n x_{1},...,x_{n}
  46. a i j ( x ) , b i ( x ) , c ( x ) a_{ij}(x),b_{i}(x),c(x)
  47. x = ( x 1 , , x n ) x=(x_{1},...,x_{n})
  48. L L
  49. L u ( x ) = i , j = 1 n ( a i j ( x ) u x i ) x j + i = 1 n b i ( x ) u x i ( x ) + c ( x ) u ( x ) Lu(x)=\sum_{i,j=1}^{n}(a_{ij}(x)u_{x_{i}})_{x_{j}}+\sum_{i=1}^{n}b_{i}(x)u_{x_% {i}}(x)+c(x)u(x)
  50. L u ( x ) = i , j = 1 n a i j ( x ) u x i x j + i = 1 n b ~ i u x i ( x ) + c ( x ) u ( x ) Lu(x)=\sum_{i,j=1}^{n}a_{ij}(x)u_{x_{i}x_{j}}+\sum_{i=1}^{n}\tilde{b}_{i}u_{x_% {i}}(x)+c(x)u(x)
  51. x i \cdot_{x_{i}}
  52. x i x_{i}
  53. b ~ i ( x ) = b i ( x ) + j a i j , x j ( x ) \tilde{b}_{i}(x)=b_{i}(x)+\sum_{j}a_{ij,x_{j}}(x)
  54. a ( x ) a(x)
  55. n × n n\times n
  56. x x
  57. b ( x ) b(x)
  58. n n
  59. x x
  60. L u = ( a u ) + b T u + c u Lu=\nabla\cdot(a\nabla u)+b^{T}\nabla u+cu
  61. a a
  62. i , j , x i,j,x
  63. a i j ( x ) = a j i ( x ) a_{ij}(x)=a_{ji}(x)
  64. L L
  65. α > 0 \alpha>0
  66. λ min ( a ( x ) ) > α x \lambda_{\min}(a(x))>\alpha\;\;\;\forall x
  67. u T a ( x ) u > α u T u u n u^{T}a(x)u>\alpha u^{T}u\;\;\;\forall u\in\mathbb{R}^{n}
  68. i , j = 1 n a i j u i u j > α i = 1 n u i 2 u n \sum_{i,j=1}^{n}a_{ij}u_{i}u_{j}>\alpha\sum_{i=1}^{n}u_{i}^{2}\;\;\;\forall u% \in\mathbb{R}^{n}
  69. L u = f in Ω Lu=f\,\text{ in }\Omega
  70. u = 0 on Ω u=0\,\text{ on }\partial\Omega
  71. L u = f in Ω Lu=f\,\text{ in }\Omega
  72. u ν = g on Ω u_{\nu}=g\,\text{ on }\partial\Omega
  73. u ν u_{\nu}
  74. u u
  75. Ω \partial\Omega
  76. B B
  77. L u = f in Ω Lu=f\,\text{ in }\Omega
  78. B u = g on Ω Bu=g\,\text{ on }\partial\Omega
  79. L L
  80. u = 0 on Ω u=0\,\text{ on }\partial\Omega
  81. H 1 ( Ω ) H^{1}(\Omega)
  82. Ω \Omega
  83. u u
  84. u x i u_{x_{i}}
  85. i = 1 , , n i=1,\dots,n
  86. H 1 H^{1}
  87. C k C^{k}
  88. k = 0 , 1 , k=0,1,\dots
  89. k k
  90. k k
  91. Δ u = f \Delta u=f
  92. φ \varphi
  93. - Ω u φ + Ω u ν φ = Ω f φ -\int_{\Omega}\nabla u\cdot\nabla\varphi+\int_{\partial\Omega}u_{\nu}\varphi=% \int_{\Omega}f\varphi
  94. u = 0 on Ω u=0\,\text{ on }\partial\Omega
  95. φ \varphi
  96. u u
  97. φ = 0 on Ω \varphi=0\,\text{ on }\partial\Omega
  98. Ω \int_{\partial\Omega}
  99. A ( u , φ ) = F ( φ ) A(u,\varphi)=F(\varphi)
  100. A ( u , φ ) = Ω u φ A(u,\varphi)=\int_{\Omega}\nabla u\cdot\nabla\varphi
  101. F ( φ ) = - Ω f φ F(\varphi)=-\int_{\Omega}f\varphi
  102. L L
  103. A ( u , φ ) = Ω u T a φ - Ω b T u φ - Ω c u φ A(u,\varphi)=\int_{\Omega}\nabla u^{T}a\nabla\varphi-\int_{\Omega}b^{T}\nabla u% \varphi-\int_{\Omega}cu\varphi
  104. A ( u , φ ) A(u,\varphi)
  105. H 0 1 H 1 H^{1}_{0}\subset H^{1}
  106. Ω \partial\Omega
  107. a , b , c a,b,c
  108. Ω \Omega
  109. a i j ( x ) a_{ij}(x)
  110. Ω ¯ \bar{\Omega}
  111. i , j = 1 , , n , i,j=1,\dots,n,
  112. b i ( x ) b_{i}(x)
  113. Ω ¯ \bar{\Omega}
  114. i = 1 , , n , i=1,\dots,n,
  115. c ( x ) c(x)
  116. Ω ¯ \bar{\Omega}
  117. Ω \Omega
  118. A ( u , φ ) A(u,\varphi)
  119. F ( φ ) F(\varphi)
  120. φ \varphi
  121. f f
  122. A A
  123. α > 0 \alpha>0
  124. u , φ H 0 1 ( Ω ) u,\varphi\in H_{0}^{1}(\Omega)
  125. A ( u , φ ) α Ω u φ . A(u,\varphi)\geq\alpha\int_{\Omega}\nabla u\cdot\nabla\varphi.
  126. α = 1 \alpha=1
  127. b = 0 b=0
  128. c 0 c\leq 0
  129. u T a u > α u T u u^{T}au>\alpha u^{T}u
  130. L L
  131. A ( u , φ ) A(u,\varphi)
  132. F ( φ ) F(\varphi)
  133. u H 0 1 ( Ω ) u\in H_{0}^{1}(\Omega)
  134. A ( u , φ ) A(u,\varphi)
  135. b = 0 b=0
  136. A ( u , φ ) A(u,\varphi)
  137. H 0 1 ( Ω ) H_{0}^{1}(\Omega)
  138. u H 0 1 ( Ω ) u\in H_{0}^{1}(\Omega)
  139. u u
  140. L u = f in Ω , Lu=f\,\text{ in }\Omega,
  141. u = 0 on Ω , u=0\,\text{ on }\partial\Omega,
  142. u u
  143. u x i x j u_{x_{i}x_{j}}
  144. L u Lu
  145. u u
  146. H 2 ( Ω ) H^{2}(\Omega)
  147. Ω \Omega
  148. C 2 C^{2}
  149. Ω \Omega
  150. Ω \partial\Omega
  151. C 2 C^{2}
  152. u u
  153. H 2 H^{2}
  154. u H 2 ( Ω ) u\in H^{2}(\Omega)
  155. u u
  156. L u = f Lu=f
  157. Ω n \Omega\subset\mathbb{R}^{n}
  158. f f
  159. u u
  160. L u = f Lu=f
  161. Ω \partial\Omega
  162. C k C^{k}
  163. f H k - 2 ( Ω ) f\in H^{k-2}(\Omega)
  164. k 2 k\geq 2
  165. u H k ( Ω ) u\in H^{k}(\Omega)
  166. H k ( Ω ) H^{k}(\Omega)
  167. C m ( Ω ¯ ) C^{m}(\bar{\Omega})
  168. 0 m < k - n / 2 0\leq m<k-n/2
  169. H 1 L 2 H^{1}\subset L^{2}
  170. A ( u , φ ) A(u,\varphi)
  171. S : f u S:f\rightarrow u
  172. L 2 ( Ω ) L^{2}(\Omega)
  173. L 2 ( Ω ) L^{2}(\Omega)
  174. u 1 , u 2 , H 1 ( Ω ) u_{1},u_{2},\dots\in H^{1}(\Omega)
  175. λ 1 , λ 2 , \lambda_{1},\lambda_{2},\dots\in\mathbb{R}
  176. S u k = λ k u k , k = 1 , 2 , , Su_{k}=\lambda_{k}u_{k},k=1,2,\dots,
  177. λ k 0 \lambda_{k}\rightarrow 0
  178. k k\rightarrow\infty
  179. λ k 0 k \lambda_{k}\gneqq 0\;\;\forall k
  180. Ω u j u k = 0 \int_{\Omega}u_{j}u_{k}=0
  181. j k j\neq k
  182. Ω u j u j = 1 \int_{\Omega}u_{j}u_{j}=1
  183. j = 1 , 2 , . j=1,2,\dots\,.
  184. L u = f Lu=f
  185. u = k = 1 u ^ ( k ) u k u=\sum_{k=1}^{\infty}\hat{u}(k)u_{k}
  186. u ^ ( k ) = λ k f ^ ( k ) , k = 1 , 2 , \hat{u}(k)=\lambda_{k}\hat{f}(k),\;\;k=1,2,\dots
  187. f ^ ( k ) = Ω f ( x ) u k ( x ) d x . \hat{f}(k)=\int_{\Omega}f(x)u_{k}(x)\,dx.
  188. L 2 L^{2}
  189. u - u x x - u y y = f ( x , y ) = x y u-u_{xx}-u_{yy}=f(x,y)=xy
  190. ( 0 , 1 ) × ( 0 , 1 ) , (0,1)\times(0,1),
  191. u ( x , 0 ) = u ( x , 1 ) = u ( 0 , y ) = u ( 1 , y ) = 0 ( x , y ) ( 0 , 1 ) × ( 0 , 1 ) u(x,0)=u(x,1)=u(0,y)=u(1,y)=0\;\;\forall(x,y)\in(0,1)\times(0,1)
  192. u j k ( x , y ) = sin ( π j x ) sin ( π k y ) u_{jk}(x,y)=\sin(\pi jx)\sin(\pi ky)
  193. j , k j,k\in\mathbb{N}
  194. λ j k = 1 1 + π 2 j 2 + π 2 k 2 . \lambda_{jk}={1\over 1+\pi^{2}j^{2}+\pi^{2}k^{2}}.
  195. g ( x ) = x g(x)=x
  196. g ^ ( n ) = ( - 1 ) n + 1 π n \hat{g}(n)={(-1)^{n+1}\over\pi n}
  197. f ^ ( j , k ) = ( - 1 ) j + k + 1 π 2 j k \hat{f}(j,k)={(-1)^{j+k+1}\over\pi^{2}jk}
  198. u ( x , y ) = j , k = 1 ( - 1 ) j + k + 1 π 2 j k ( 1 + π 2 j 2 + π 2 k 2 ) sin ( π j x ) sin ( π k y ) . u(x,y)=\sum_{j,k=1}^{\infty}{(-1)^{j+k+1}\over\pi^{2}jk(1+\pi^{2}j^{2}+\pi^{2}% k^{2})}\sin(\pi jx)\sin(\pi ky).
  199. u C 2 ( Ω ) C 1 ( Ω ¯ ) u\in C^{2}(\Omega)\cap C^{1}(\bar{\Omega})
  200. c ( x ) = 0 x Ω c(x)=0\;\forall x\in\Omega
  201. L u 0 Lu\leq 0
  202. Ω \Omega
  203. max x Ω ¯ u ( x ) = max x Ω u ( x ) \max_{x\in\bar{\Omega}}u(x)=\max_{x\in\partial\Omega}u(x)
  204. u ( x ) max y Ω u ( y ) u(x)\lneqq\max_{y\in\partial\Omega}u(y)
  205. x Ω x\in\Omega
  206. u u

Elliptic_cohomology.html

  1. S 1 S^{1}
  2. S 1 S^{1}
  3. A * A^{*}
  4. A i = 0 A^{i}=0
  5. u A 2 u\in A^{2}
  6. A 0 = R A^{0}=R
  7. A * ( X ) = M U * ( X ) M U * R [ u , u - 1 ] . A^{*}(X)=MU^{*}(X)\otimes_{MU^{*}}R[u,u^{-1}].\,
  8. \mathbb{Z}
  9. Spec R / p R \,\text{Spec }R/pR
  10. E x E_{x}
  11. x X x\in X
  12. 1 , 1 f g \mathcal{M}_{1,1}\to\mathcal{M}_{fg}

Elongatedness.html

  1. e l o n g a t e d n e s s = l e n g t h w i d t h = a r e a ( 2 d ) 2 elongatedness=\frac{length}{width}=\frac{area}{(2d)^{2}}
  2. d d

Energetic_space.html

  1. X X
  2. ( | ) (\cdot|\cdot)
  3. \|\cdot\|
  4. Y Y
  5. X X
  6. B : Y X B:Y\to X
  7. ( B u | v ) = ( u | B v ) (Bu|v)=(u|Bv)\,
  8. u , v u,v
  9. Y Y
  10. ( B u | u ) c u 2 (Bu|u)\geq c\|u\|^{2}
  11. c > 0 c>0
  12. u u
  13. Y . Y.
  14. ( u | v ) E = ( B u | v ) (u|v)_{E}=(Bu|v)\,
  15. u , v u,v
  16. Y Y
  17. u E = ( u | u ) E 1 2 \|u\|_{E}=(u|u)^{\frac{1}{2}}_{E}\,
  18. u u
  19. Y . Y.
  20. Y Y
  21. X E X_{E}
  22. Y Y
  23. X E X_{E}
  24. X , X,
  25. X X
  26. B B
  27. Y Y
  28. X E X_{E}
  29. ( u | v ) E = lim n ( u n | v n ) E (u|v)_{E}=\lim_{n\to\infty}(u_{n}|v_{n})_{E}
  30. ( u n ) (u_{n})
  31. ( v n ) (v_{n})
  32. X E X_{E}
  33. B B
  34. B E B_{E}
  35. B E : X E X E * B_{E}:X_{E}\to X^{*}_{E}
  36. X E X_{E}
  37. X E * X^{*}_{E}
  38. B E u | v E = ( u | v ) E \langle B_{E}u|v\rangle_{E}=(u|v)_{E}
  39. u , v u,v
  40. X E . X_{E}.
  41. | E \langle\cdot|\cdot\rangle_{E}
  42. X E * X^{*}_{E}
  43. X E , X_{E},
  44. B E u | v E \langle B_{E}u|v\rangle_{E}
  45. ( B E u ) ( v ) . (B_{E}u)(v).
  46. u u
  47. v v
  48. Y , Y,
  49. B E u | v E = ( u | v ) E = ( B u | v ) = u | B | v \langle B_{E}u|v\rangle_{E}=(u|v)_{E}=(Bu|v)=\langle u|B|v\rangle
  50. B u , Bu,
  51. X , X,
  52. X * X^{*}
  53. B u Bu
  54. X E * X_{E}^{*}
  55. B B
  56. B E u = B u . B_{E}u=Bu.
  57. B : Y X B:Y\to X
  58. B : Y X E * , B:Y\to X_{E}^{*},
  59. B E : X E X E * B_{E}:X_{E}\to X^{*}_{E}
  60. B B
  61. Y Y
  62. X E . X_{E}.
  63. a < b a<b
  64. x x
  65. ( a x b ) (a\leq x\leq b)
  66. f ( x ) 𝐞 f(x)\mathbf{e}
  67. 𝐞 \mathbf{e}
  68. f : [ a , b ] . f:[a,b]\to\mathbb{R}.
  69. u ( x ) u(x)
  70. x x
  71. 1 2 a b u ( x ) 2 d x \frac{1}{2}\int_{a}^{b}\!u^{\prime}(x)^{2}\,dx
  72. F ( u ) = 1 2 a b u ( x ) 2 d x - a b u ( x ) f ( x ) d x . F(u)=\frac{1}{2}\int_{a}^{b}\!u^{\prime}(x)^{2}\,dx-\int_{a}^{b}\!u(x)f(x)\,dx.
  73. u ( x ) u(x)
  74. - u ′′ = f -u^{\prime\prime}=f\,
  75. u ( a ) = u ( b ) = 0. u(a)=u(b)=0.\,
  76. X = L 2 ( a , b ) , X=L^{2}(a,b),
  77. u : [ a , b ] u:[a,b]\to\mathbb{R}
  78. ( u | v ) = a b u ( x ) v ( x ) d x , (u|v)=\int_{a}^{b}\!u(x)v(x)\,dx,
  79. u = ( u | u ) . \|u\|=\sqrt{(u|u)}.
  80. Y Y
  81. u : [ a , b ] u:[a,b]\to\mathbb{R}
  82. u ( a ) = u ( b ) = 0. u(a)=u(b)=0.
  83. Y Y
  84. X . X.
  85. B : Y X B:Y\to X
  86. B u = - u ′′ , Bu=-u^{\prime\prime},\,
  87. B u = f . Bu=f.
  88. ( B u | v ) = - a b u ′′ ( x ) v ( x ) d x = a b u ( x ) v ( x ) = ( u | B v ) (Bu|v)=-\int_{a}^{b}\!u^{\prime\prime}(x)v(x)\,dx=\int_{a}^{b}u^{\prime}(x)v^{% \prime}(x)=(u|Bv)
  89. u u
  90. v v
  91. Y . Y.
  92. B B
  93. B B
  94. u 2 = a b u 2 ( x ) d x C a b u ( x ) 2 d x = C ( B u | u ) \|u\|^{2}=\int_{a}^{b}u^{2}(x)\,dx\leq C\int_{a}^{b}u^{\prime}(x)^{2}\,dx=C\,(% Bu|u)
  95. C > 0. C>0.
  96. B B
  97. H 0 1 ( a , b ) . H^{1}_{0}(a,b).
  98. 1 2 a b u ( x ) 2 d x = 1 2 ( u | u ) E , \frac{1}{2}\int_{a}^{b}\!u^{\prime}(x)^{2}\,dx=\frac{1}{2}(u|u)_{E},
  99. u u
  100. u u
  101. F ( u ) F(u)
  102. ( u | v ) E = ( f | v ) (u|v)_{E}=(f|v)\,
  103. v v
  104. X E X_{E}
  105. u u
  106. u h u_{h}
  107. u h u_{h}
  108. u h u_{h}
  109. u u
  110. u h u_{h}

Energy_drift.html

  1. E = m 𝐯 2 = m 𝐯 t r u e 2 + m δ 𝐯 2 E=\sum m\mathbf{v}^{2}=\sum m\mathbf{v}_{true}^{2}+\sum m\ \delta\mathbf{v}^{2}
  2. 𝒪 ( Δ t p ) \mathcal{O}\left(\Delta t^{p}\right)
  3. 2 π Δ t \frac{2\pi}{\Delta t}
  4. n m ω = 2 π Δ t \frac{n}{m}\omega=\frac{2\pi}{\Delta t}
  5. ( n m = 4 ) \left(\frac{n}{m}=4\right)
  6. Δ t < 2 ω 0.225 p \Delta t<\frac{\sqrt{2}}{\omega}\approx 0.225p

Energy_minimization.html

  1. 𝐫 \mathbf{r}
  2. E ( 𝐫 ) E(\mathbf{r})
  3. 𝐫 \mathbf{r}
  4. E ( 𝐫 ) E(\mathbf{r})
  5. E / 𝐫 ∂E/∂\mathbf{r}
  6. 𝐫 \mathbf{r}
  7. E ( 𝐫 ) E(\mathbf{r})
  8. - E / 𝐫 -∂E/∂\mathbf{r}
  9. 𝐫 ∆\mathbf{r}
  10. E ( 𝐫 ) E(\mathbf{r})
  11. E / 𝐫 ∂E/∂\mathbf{r}
  12. 𝐫 \mathbf{r}
  13. E ( 𝐫 ) E(\mathbf{r})
  14. E / 𝐫 ∂E/∂\mathbf{r}
  15. N N
  16. 3 N 6 3N–6
  17. 3 N 3N
  18. 1 = E / 𝐫 = 𝟎 1=∂E/∂\mathbf{r}=\mathbf{0}
  19. N + 1 N+1
  20. i i
  21. 𝐫 i = i N 𝐫 product + ( 1 - i N ) 𝐫 reactant \mathbf{r}_{i}=\frac{i}{N}\mathbf{r}_{\mathrm{product}}+\left(1-\frac{i}{N}% \right)\mathbf{r}_{\mathrm{reactant}}
  22. i 0 , 1 , , N i∈0,1,...,N
  23. 𝐠 i = 𝐠 i - τ i ( τ i 𝐠 i ) = ( I - τ i τ i T ) 𝐠 i \mathbf{g}_{i}^{\perp}=\mathbf{g}_{i}-\mathbf{\tau}_{i}(\mathbf{\tau}_{i}\cdot% \mathbf{g}_{i})=\left(I-\mathbf{\tau}_{i}\mathbf{\tau}_{i}^{T}\right)\mathbf{g% }_{i}
  24. I I
  25. 𝐟 i = 𝐟 i - 𝐠 i \mathbf{f}_{i}=\mathbf{f}_{i}^{\parallel}-\mathbf{g}_{i}^{\perp}
  26. 𝐟 i = k [ ( ( 𝐫 i + 1 - 𝐫 i ) - ( 𝐫 i - 𝐫 i - 1 ) ) τ i ] τ i \mathbf{f}_{i}^{\parallel}=k\left[\left(\left(\mathbf{r}_{i+1}-\mathbf{r}_{i}% \right)-\left(\mathbf{r}_{i}-\mathbf{r}_{i-1}\right)\right)\cdot\tau_{i}\right% ]\tau_{i}
  27. 𝐫 < s u b > i \mathbf{r}<sub>i

Enhanced_vegetation_index.html

  1. E V I = G × ( N I R - R E D ) ( N I R + C 1 × R E D - C 2 × B l u e + L ) EVI=G\times\frac{(NIR-RED)}{(NIR+C1\times RED-C2\times Blue+L)}

Ensemble_Kalman_filter.html

  1. 𝐱 \mathbf{x}
  2. n n
  3. μ \mathbf{\mu}
  4. Q Q
  5. p ( 𝐱 ) exp ( - 1 2 ( 𝐱 - μ ) T Q - 1 ( 𝐱 - μ ) ) . p(\mathbf{x})\propto\exp\left(-\frac{1}{2}(\mathbf{x}-\mathbf{\mu})^{\mathrm{T% }}Q^{-1}(\mathbf{x}-\mathbf{\mu})\right).
  6. \propto
  7. p ( 𝐱 ) p(\mathbf{x})
  8. 𝐝 \mathbf{d}
  9. R R
  10. H 𝐱 H\mathbf{x}
  11. H H
  12. R R
  13. 𝐝 \mathbf{d}
  14. R R
  15. 𝐝 \mathbf{d}
  16. H 𝐱 H\mathbf{x}
  17. 𝐱 \mathbf{x}
  18. p ( 𝐝 | 𝐱 ) p(\mathbf{d}|\mathbf{x})
  19. 𝐝 \mathbf{d}
  20. 𝐱 \mathbf{x}
  21. p ( 𝐝 | 𝐱 ) exp ( - 1 2 ( 𝐝 - H 𝐱 ) T R - 1 ( 𝐝 - H 𝐱 ) ) . p\left(\mathbf{d}|\mathbf{x}\right)\propto\exp\left(-\frac{1}{2}(\mathbf{d}-H% \mathbf{x})^{\mathrm{T}}R^{-1}(\mathbf{d}-H\mathbf{x})\right).
  22. 𝐱 \mathbf{x}
  23. 𝐝 \mathbf{d}
  24. p ( 𝐱 | 𝐝 ) p ( 𝐝 | 𝐱 ) p ( 𝐱 ) . p\left(\mathbf{x}|\mathbf{d}\right)\propto p\left(\mathbf{d}|\mathbf{x}\right)% p(\mathbf{x}).
  25. 𝐝 \mathbf{d}
  26. 𝐱 ^ \mathbf{\hat{x}}
  27. 𝐱 | 𝐝 \mathbf{x}|\mathbf{d}
  28. p ( 𝐱 ^ ) p\left(\mathbf{\hat{x}}\right)
  29. p ( 𝐱 ^ ) exp ( - 1 2 ( 𝐱 ^ - μ ^ ) T Q ^ - 1 ( 𝐱 ^ - μ ^ ) ) , p\left(\mathbf{\hat{x}}\right)\propto\exp\left(-\frac{1}{2}(\mathbf{\hat{x}}-% \mathbf{\hat{\mu}})^{\mathrm{T}}\hat{Q}^{-1}(\mathbf{\hat{x}}-\mathbf{\hat{\mu% }})\right),
  30. μ ^ \mathbf{\hat{\mu}}
  31. Q ^ \hat{Q}
  32. μ ^ = μ + K ( 𝐝 - H μ ) , Q ^ = ( I - K H ) Q , \mathbf{\hat{\mu}}=\mathbf{\mu}+K\left(\mathbf{d}-H\mathbf{\mu}\right),\quad% \hat{Q}=\left(I-KH\right)Q,
  33. K = Q H T ( H Q H T + R ) - 1 K=QH^{\mathrm{T}}\left(HQH^{\mathrm{T}}+R\right)^{-1}
  34. 𝐱 \mathbf{x}
  35. X = [ 𝐱 1 , , 𝐱 N ] = [ 𝐱 i ] . X=\left[\mathbf{x}_{1},\ldots,\mathbf{x}_{N}\right]=\left[\mathbf{x}_{i}\right].
  36. X X
  37. n × N n\times N
  38. 𝐝 \mathbf{d}
  39. m × N m\times N
  40. D = [ 𝐝 1 , , 𝐝 N ] = [ 𝐝 i ] , 𝐝 i = 𝐝 + ϵ 𝐢 , ϵ 𝐢 = N ( 0 , R ) , D=\left[\mathbf{d}_{1},\ldots,\mathbf{d}_{N}\right]=\left[\mathbf{d}_{i}\right% ],\quad\mathbf{d}_{i}=\mathbf{d}+\mathbf{\epsilon_{i}},\quad\mathbf{\epsilon_{% i}}=N(0,R),
  41. 𝐝 i \mathbf{d}_{i}
  42. 𝐝 \mathbf{d}
  43. m m
  44. N ( 0 , R ) N(0,R)
  45. X X
  46. X ^ = X + K ( D - H X ) \hat{X}=X+K(D-HX)
  47. H = 1 H=1
  48. x i = μ + ξ i , ξ i N ( 0 , σ x 2 ) x_{i}=\mu+\xi_{i},\;\xi_{i}\sim N(0,\sigma_{x}^{2})
  49. d i = d + ϵ i , ϵ i N ( 0 , σ d 2 ) . d_{i}=d+\epsilon_{i},\;\epsilon_{i}\sim N(0,\sigma_{d}^{2}).
  50. x ^ i = ( 1 / σ x 2 1 / σ x 2 + 1 / σ d 2 μ + 1 / σ d 2 1 / σ x 2 + 1 / σ d 2 d ) + ( 1 / σ x 2 1 / σ x 2 + 1 / σ d 2 ξ i + 1 / σ d 2 1 / σ x 2 + 1 / σ d 2 ϵ i ) \hat{x}_{i}=\left(\frac{1/\sigma_{x}^{2}}{1/\sigma_{x}^{2}+1/\sigma_{d}^{2}}% \mu+\frac{1/\sigma_{d}^{2}}{1/\sigma_{x}^{2}+1/\sigma_{d}^{2}}d\right)+\left(% \frac{1/\sigma_{x}^{2}}{1/\sigma_{x}^{2}+1/\sigma_{d}^{2}}\xi_{i}+\frac{1/% \sigma_{d}^{2}}{1/\sigma_{x}^{2}+1/\sigma_{d}^{2}}\epsilon_{i}\right)
  51. ( 1 / σ x 2 1 / σ x 2 + 1 / σ d 2 ) 2 σ x 2 + ( 1 / σ d 2 1 / σ x 2 + 1 / σ d 2 ) 2 σ d 2 = 1 1 / σ x 2 + 1 / σ d 2 \left(\frac{1/\sigma_{x}^{2}}{1/\sigma_{x}^{2}+1/\sigma_{d}^{2}}\right)^{2}% \sigma_{x}^{2}+\left(\frac{1/\sigma_{d}^{2}}{1/\sigma_{x}^{2}+1/\sigma_{d}^{2}% }\right)^{2}\sigma_{d}^{2}=\frac{1}{1/\sigma_{x}^{2}+1/\sigma_{d}^{2}}
  52. Q Q
  53. K K
  54. C C
  55. K = C H T ( H C H T + R ) - 1 K=CH^{\mathrm{T}}\left(HCH^{\mathrm{T}}+R\right)^{-1}
  56. X X
  57. D D
  58. E ( X ) = 1 N k = 1 N 𝐱 k , C = A A T N - 1 , E\left(X\right)=\frac{1}{N}\sum_{k=1}^{N}\mathbf{x}_{k},\quad C=\frac{AA^{T}}{% N-1},
  59. A = X - E ( X ) 𝐞 1 × N = X - 1 N ( X 𝐞 N × 1 ) 𝐞 1 × N , A=X-E\left(X\right)\mathbf{e}_{1\times N}=X-\frac{1}{N}\left(X\mathbf{e}_{N% \times 1}\right)\mathbf{e}_{1\times N},
  60. 𝐞 \mathbf{e}
  61. X p X^{p}
  62. X p = X + C H T ( H C H T + R ) - 1 ( D - H X ) , X^{p}=X+CH^{T}\left(HCH^{T}+R\right)^{-1}(D-HX),
  63. D D
  64. R R
  65. R R
  66. D ~ D ~ T / ( N - 1 ) \tilde{D}\tilde{D}^{T}/\left(N-1\right)
  67. D ~ = D - 1 N d 𝐞 1 × N \tilde{D}=D-\frac{1}{N}d\,\mathbf{e}_{1\times N}
  68. H H
  69. h ( 𝐱 ) h(\mathbf{x})
  70. h ( 𝐱 ) = H 𝐱 . h(\mathbf{x})=H\mathbf{x}.
  71. h h
  72. h ( 𝐱 ) h(\mathbf{x})
  73. 𝐱 \mathbf{x}
  74. X p = X + 1 N - 1 A ( H A ) T P - 1 ( D - H X ) X^{p}=X+\frac{1}{N-1}A\left(HA\right)^{T}P^{-1}(D-HX)
  75. H A = H X - 1 N ( ( H X ) 𝐞 N × 1 ) 𝐞 1 × N , HA=HX-\frac{1}{N}\left(\left(HX\right)\mathbf{e}_{N\times 1}\right)\mathbf{e}_% {1\times N},
  76. P = 1 N - 1 H A ( H A ) T + R , P=\frac{1}{N-1}HA\left(HA\right)^{T}+R,
  77. [ H A ] i = H 𝐱 i - H 1 N j = 1 N 𝐱 j = h ( 𝐱 i ) - 1 N j = 1 N h ( 𝐱 j ) . \left[HA\right]_{i}=H\mathbf{x}_{i}-H\frac{1}{N}\sum_{j=1}^{N}\mathbf{x}_{j}\ % =h\left(\mathbf{x}_{i}\right)-\frac{1}{N}\sum_{j=1}^{N}h\left(\mathbf{x}_{j}% \right).
  78. h h
  79. H H
  80. h ( 𝐱 ) = H 𝐱 + 𝐟 h(\mathbf{x})=H\mathbf{x+f}
  81. 𝐟 \mathbf{f}
  82. h h
  83. m m
  84. P - 1 P^{-1}
  85. m m
  86. R R
  87. ( R + U V T ) - 1 = R - 1 - R - 1 U ( I + V T R - 1 U ) - 1 V T R - 1 , (R+UV^{T})^{-1}=R^{-1}-R^{-1}U(I+V^{T}R^{-1}U)^{-1}V^{T}R^{-1},
  88. U = 1 N - 1 H A , V = H A , U=\frac{1}{N-1}HA,\quad V=HA,
  89. P - 1 = ( R + 1 N - 1 H A ( H A ) T ) - 1 = = R - 1 [ I - 1 N - 1 ( H A ) ( I + ( H A ) T R - 1 1 N - 1 ( H A ) ) - 1 ( H A ) T R - 1 ] , \begin{aligned}\displaystyle P^{-1}&\displaystyle=\left(R+\frac{1}{N-1}HA\left% (HA\right)^{T}\right)^{-1}\ =\\ &\displaystyle=R^{-1}\left[I-\frac{1}{N-1}\left(HA\right)\left(I+\left(HA% \right)^{T}R^{-1}\frac{1}{N-1}\left(HA\right)\right)^{-1}\left(HA\right)^{T}R^% {-1}\right],\end{aligned}
  90. R R
  91. N N
  92. m m

Entropy_maximization.html

  1. f 0 ( x ) = - i = 1 n x i log x i f_{0}(\vec{x})=-\sum_{i=1}^{n}x_{i}\log x_{i}
  2. A x b , 𝟏 T x = | x | 1 = 1 A\vec{x}\leq b,\quad\mathbf{1}^{T}\vec{x}=|\vec{x}|_{1}=1
  3. x + + n \vec{x}\in\mathbb{R}^{n}_{++}
  4. A m × n A\in\mathbb{R}^{m\times n}
  5. b m b\in\mathbb{R}^{m}
  6. 𝟏 \mathbf{1}

Entropy_power_inequality.html

  1. h ( X ) = - n f ( x ) log f ( x ) d x h(X)=-\int_{\mathbb{R}^{n}}f(x)\log f(x)\,dx
  2. N ( X ) = 1 2 π e e 2 n h ( X ) . N(X)=\frac{1}{2\pi e}e^{\frac{2}{n}h(X)}.
  3. N ( X + Y ) N ( X ) + N ( Y ) . N(X+Y)\geq N(X)+N(Y).\,

Entropy_rate.html

  1. H ( X ) = lim n 1 n H ( X 1 , X 2 , X n ) H(X)=\lim_{n\to\infty}\frac{1}{n}H(X_{1},X_{2},\dots X_{n})
  2. H ( X ) = lim n H ( X n | X n - 1 , X n - 2 , X 1 ) H^{\prime}(X)=\lim_{n\to\infty}H(X_{n}|X_{n-1},X_{n-2},\dots X_{1})
  3. H ( X ) = H ( X ) H(X)=H^{\prime}(X)
  4. H ( Y ) = - i j μ i P i j log P i j \displaystyle H(Y)=-\sum_{ij}\mu_{i}P_{ij}\log P_{ij}

Eötvös_rule.html

  1. γ V 2 / 3 = k ( T c - T ) \gamma V^{2/3}=k(T_{c}-T)\,
  2. γ V 2 / 3 = k ( T c - 6 K - T ) \gamma V^{2/3}=k(T_{c}-6\ \mathrm{K}-T)\,
  3. V = M / ρ V=M/\rho\,
  4. γ V 2 / 3 \gamma V^{2/3}
  5. γ m o l = γ V 2 / 3 \gamma_{mol}=\gamma V^{2/3}\,
  6. γ = k ( M ρ N A ) - 2 / 3 ( T c - 6 K - T ) = k ( N A V ) 2 / 3 ( T c - 6 K - T ) \gamma=k^{\prime}\left(\frac{M}{\rho N_{A}}\right)^{-2/3}(T_{c}-6\ \mathrm{K}-% T)=k^{\prime}\left(\frac{N_{A}}{V}\right)^{2/3}(T_{c}-6\ \mathrm{K}-T)
  7. γ = 0.07275 N / m ( 1 - 0.002 ( T - 291 K ) ) \gamma=0.07275\ \mathrm{N/m}\cdot(1-0.002\cdot(T-291\ \mathrm{K}))

Epidemic_model.html

  1. M M
  2. S S
  3. E E
  4. I I
  5. R R
  6. β \beta
  7. μ \mu
  8. B B
  9. 1 / ε 1/\varepsilon
  10. 1 / γ 1/\gamma
  11. R 0 R_{0}
  12. N N
  13. f f
  14. δ \delta
  15. S ( t ) S(t)
  16. I ( t ) I(t)
  17. R ( t ) R(t)
  18. S ( t ) S(t)
  19. I ( t ) I(t)
  20. R ( t ) R(t)
  21. \color b l u e 𝒮 {\color{blue}{\mathcal{S}\rightarrow\mathcal{I}\rightarrow\mathcal{R}}}
  22. N = S ( t ) + I ( t ) + R ( t ) N=S(t)+I(t)+R(t)
  23. d S d t = - β S I N \frac{dS}{dt}=-\frac{\beta SI}{N}
  24. d I d t = β S I N - γ I \frac{dI}{dt}=\frac{\beta SI}{N}-\gamma I
  25. d R d t = γ I \frac{dR}{dt}=\gamma I
  26. β \beta
  27. β N \beta N
  28. S / N S/N
  29. β N ( S / N ) \beta N(S/N)
  30. β N ( S / N ) I = β S I \beta N(S/N)I=\beta SI
  31. γ \gamma
  32. 1 / γ 1/\gamma
  33. μ \mu
  34. d S d t = - β S I N + μ ( N - S ) \frac{dS}{dt}=-\frac{\beta SI}{N}+\mu(N-S)
  35. d I d t = β S I N - γ I - μ I \frac{dI}{dt}=\frac{\beta SI}{N}-\gamma I-\mu I
  36. d R d t = γ I - μ R \frac{dR}{dt}=\gamma I-\mu R
  37. \color b l u e 𝒮 𝒮 {\color{blue}{\mathcal{S}\rightarrow\mathcal{I}\rightarrow\mathcal{S}}}
  38. d S d t = - β S I N + μ ( N - S ) + γ I \frac{dS}{dt}=-\frac{\beta SI}{N}+\mu(N-S)+\gamma I
  39. d I d t = β S I N - γ I - μ I \frac{dI}{dt}=\frac{\beta SI}{N}-\gamma I-\mu I
  40. \color b l u e 𝒮 𝒮 {\color{blue}{\mathcal{S}\rightarrow\mathcal{I}\rightarrow\mathcal{R}% \rightarrow\mathcal{S}}}
  41. d S d t = - β S I N + μ ( N - S ) + f R \frac{dS}{dt}=-\frac{\beta SI}{N}+\mu(N-S)+fR
  42. d I d t = β S I N - γ I - μ I \frac{dI}{dt}=\frac{\beta SI}{N}-\gamma I-\mu I
  43. d R d t = γ I - μ R - f R \frac{dR}{dt}=\gamma I-\mu R-fR
  44. \color b l u e 𝒮 𝒮 {\color{blue}{\mathcal{S}\rightarrow\mathcal{E}\rightarrow\mathcal{I}% \rightarrow\mathcal{S}}}
  45. d S d T = B - β S I - μ S + γ I \tfrac{dS}{dT}=B-\beta SI-\mu S+\gamma I
  46. d E d T = β S I - ( ϵ + μ ) E \tfrac{dE}{dT}=\beta SI-(\epsilon+\mu)E
  47. d I d T = ε E - ( γ + μ ) I \tfrac{dI}{dT}=\varepsilon E-(\gamma+\mu)I
  48. \color r e d 𝒮 {\color{red}{\mathcal{S}\rightarrow\mathcal{E}\rightarrow\mathcal{I}% \rightarrow\mathcal{R}}}
  49. d S d T = B - β S I - μ S \tfrac{dS}{dT}=B-\beta SI-\mu S
  50. d E d T = β S I - ( ε + μ ) E \tfrac{dE}{dT}=\beta SI-(\varepsilon+\mu)E
  51. d I d T = ε E - ( γ + μ ) I \tfrac{dI}{dT}=\varepsilon E-(\gamma+\mu)I
  52. d R d T = γ I - μ R \tfrac{dR}{dT}=\gamma I-\mu R
  53. \color b l u e 𝒮 {\color{blue}{\mathcal{M}\rightarrow\mathcal{S}\rightarrow\mathcal{I}% \rightarrow\mathcal{R}}}
  54. d M d T = B - δ M - μ M \tfrac{dM}{dT}=B-\delta M-\mu M
  55. d S d T = δ M - β S I - μ S \tfrac{dS}{dT}=\delta M-\beta SI-\mu S
  56. d I d T = β S I - γ I - μ I \tfrac{dI}{dT}=\beta SI-\gamma I-\mu I
  57. d R d T = γ I - μ R \tfrac{dR}{dT}=\gamma I-\mu R
  58. \color b l u e 𝒮 {\color{blue}{\mathcal{M}\rightarrow\mathcal{S}\rightarrow\mathcal{E}% \rightarrow\mathcal{I}\rightarrow\mathcal{R}}}
  59. d M d T = B - δ M - μ M \tfrac{dM}{dT}=B-\delta M-\mu M
  60. d S d T = δ M - β S I - μ S \tfrac{dS}{dT}=\delta M-\beta SI-\mu S
  61. d E d T = β S I - ( ε + μ ) E \tfrac{dE}{dT}=\beta SI-(\varepsilon+\mu)E
  62. d I d T = ε E - ( γ + μ ) I \tfrac{dI}{dT}=\varepsilon E-(\gamma+\mu)I
  63. d R d T = γ I - μ R \tfrac{dR}{dT}=\gamma I-\mu R
  64. \color b l u e 𝒮 𝒮 {\color{blue}{\mathcal{M}\rightarrow\mathcal{S}\rightarrow\mathcal{E}% \rightarrow\mathcal{I}\rightarrow\mathcal{R}\rightarrow\mathcal{S}}}

Epistemic_closure.html

  1. S S
  2. p p
  3. S S
  4. p p
  5. q q
  6. S S
  7. q q

Epsilon-equilibrium.html

  1. ε \varepsilon
  2. ε \varepsilon
  3. ε \varepsilon
  4. ε \varepsilon
  5. ε = 0 \varepsilon=0
  6. G = ( N , A = A 1 × × A N , u : A R N ) G=(N,A=A_{1}\times\cdots\times A_{N},u\colon A\to R^{N})
  7. N N
  8. A i A_{i}
  9. i i
  10. u u
  11. u i ( s ) u_{i}(s)
  12. i i
  13. s s
  14. Δ i \Delta_{i}
  15. A i A_{i}
  16. σ Δ = Δ 1 × × Δ N \sigma\in\Delta=\Delta_{1}\times\cdots\times\Delta_{N}
  17. ε \varepsilon
  18. G G
  19. u i ( σ ) u i ( σ i , σ - i ) - ε u_{i}(\sigma)\geq u_{i}(\sigma_{i}^{^{\prime}},\sigma_{-i})-\varepsilon
  20. σ i Δ i , i N . \sigma_{i}^{^{\prime}}\in\Delta_{i},i\in N.
  21. a a
  22. a a
  23. ε \varepsilon
  24. x s x_{s}
  25. s s
  26. p p
  27. S - p S_{-p}
  28. p p
  29. s S - p s\in S_{-p}
  30. j j
  31. p p
  32. j s js
  33. p p
  34. j j
  35. s s
  36. u p ( s ) u_{p}(s)
  37. p p
  38. s s
  39. s S - p u p ( j s ) x s > ε + s S - p u p ( j s ) x s x j p = 0. \sum_{s\in S_{-p}}u_{p}(js)x_{s}>\varepsilon+\sum_{s\in S_{-p}}u_{p}(j^{\prime% }s)x_{s}\Longrightarrow x^{p}_{j^{\prime}}=0.
  40. ϵ \epsilon
  41. ϵ \epsilon
  42. ϵ \epsilon

Epsilon_calculus.html

  1. ( x ) A ( x ) A ( ϵ x A ) (\exists x)A(x)\ \equiv\ A(\epsilon x\ A)
  2. ( x ) A ( x ) A ( ϵ x ( ¬ A ) ) (\forall x)A(x)\ \equiv\ A(\epsilon x\ (\neg A))
  3. ( x ) A ( x ) ( τ x ( A ) | x ) A (\exists x)A(x)\ \equiv\ (\tau_{x}(A)|x)A
  4. ( x ) A ( x ) ¬ ( τ x ( ¬ A ) | x ) ¬ A ( τ x ( ¬ A ) | x ) A (\forall x)A(x)\ \equiv\ \neg(\tau_{x}(\neg A)|x)\neg A\ \equiv\ (\tau_{x}(% \neg A)|x)A
  5. τ x ( A ) \tau_{x}(A)
  6. τ \tau
  7. \square
  8. τ \tau

Equable_shape.html

  1. x 2 = 4 x . \displaystyle x^{2}=4x.

Equiareal_map.html

  1. | d f p ( v ) × d f p ( w ) | = | v × w | |df_{p}(v)\times df_{p}(w)|=|v\times w|\,
  2. f ( x , y , z ) = ( x x 2 + y 2 , y x 2 + y 2 , z ) f(x,y,z)=\left(\frac{x}{\sqrt{x^{2}+y^{2}}},\frac{y}{\sqrt{x^{2}+y^{2}}},z\right)
  3. | d f p ( v ) × d f p ( w ) | = κ | v × w | |df_{p}(v)\times df_{p}(w)|=\kappa|v\times w|\,
  4. v v
  5. w w

Equilibrium_fractionation.html

  1. A l X + B h X A h X + B l X A^{l}X+B^{h}X\rightleftharpoons A^{h}X+B^{l}X
  2. α = ( h X / l X ) A X ( h X / l X ) B X \alpha=\frac{(^{h}X/^{l}X)_{AX}}{(^{h}X/^{l}X)_{BX}}
  3. α = 1 \alpha=1
  4. α > 1 \alpha>1
  5. α < 1 \alpha<1
  6. α \alpha
  7. α = ( K e q Π σ P r o d u c t s / Π σ R e a c t a n t s ) 1 / n \alpha=(K_{eq}\cdot\Pi\sigma_{Products}/\Pi\sigma_{Reactants})^{1/n}
  8. Π σ P r o d u c t s \Pi\sigma_{Products}
  9. Π σ R e a c t a n t s \Pi\sigma_{Reactants}
  10. n n
  11. H 2 16 O ( l ) + H 2 18 O ( g ) H 2 18 O ( l ) + H 2 16 O ( g ) H_{2}^{16}O_{(l)}+H_{2}^{18}O_{(g)}\rightleftharpoons H_{2}^{18}O_{(l)}+H_{2}^% {16}O_{(g)}
  12. α = ( 18 O / 16 O ) L i q u i d ( 18 O / 16 O ) V a p o r = 1.0098 \alpha=\frac{(^{18}O/^{16}O)_{Liquid}}{(^{18}O/^{16}O)_{Vapor}}=1.0098

Equivalent_concentration.html

  1. c i c_{i}
  2. f eq f_{\mathrm{eq}}
  3. = c i f eq =\frac{c_{i}}{f_{\mathrm{eq}}}
  4. 1 / f eq 1/f_{\mathrm{eq}}
  5. 1 / f eq 1/f_{\mathrm{eq}}
  6. 1 / f eq 1/f_{\mathrm{eq}}
  7. f eq f_{\mathrm{eq}}
  8. f eq f_{\mathrm{eq}}

Erdős–Anning_theorem.html

  1. ( cos θ , sin θ ) (\cos\theta,\sin\theta)
  2. tan θ 4 \tan\frac{\theta}{4}
  3. sin θ 2 \sin\frac{\theta}{2}
  4. cos θ 2 \cos\frac{\theta}{2}
  5. θ \theta
  6. ϕ \phi
  7. | 2 sin θ 2 cos ϕ 2 - 2 sin ϕ 2 cos θ 2 | |2\sin\frac{\theta}{2}\cos\frac{\phi}{2}-2\sin\frac{\phi}{2}\cos\frac{\theta}{% 2}|
  8. ρ \rho
  9. ρ 2 \rho^{2}
  10. δ \delta
  11. 4 ( δ + 1 ) 2 4(\delta+1)^{2}
  12. δ \delta
  13. d ( A , B ) d(A,B)
  14. d ( A , C ) d(A,C)
  15. d ( B , C ) d(B,C)
  16. | d ( A , X ) - d ( B , X ) | |d(A,X)-d(B,X)|
  17. δ \delta
  18. δ + 1 \delta+1
  19. | d ( A , X ) - d ( B , X ) | = i |d(A,X)-d(B,X)|=i
  20. δ + 1 \delta+1
  21. δ + 1 \delta+1
  22. 4 ( δ + 1 ) 2 4(\delta+1)^{2}
  23. 4 ( δ + 1 ) 2 4(\delta+1)^{2}

Erdős–Diophantine_graph.html

  1. 2 \scriptstyle\mathbb{Z}^{2}

Erdős–Rényi_model.html

  1. p M ( 1 - p ) ( n 2 ) - M . p^{M}(1-p)^{{n\choose 2}-M}.
  2. 2 ( n 2 ) 2^{\left({{n}\atop{2}}\right)}
  3. ( n 2 ) p {\textstyle\left({{n}\atop{2}}\right)}p
  4. M = ( n 2 ) p M={\textstyle\left({{n}\atop{2}}\right)}p
  5. G ( n , ( n 2 ) p ) G(n,{\textstyle\left({{n}\atop{2}}\right)}p)
  6. ( n 2 ) p {\textstyle\left({{n}\atop{2}}\right)}p
  7. P ( deg ( v ) = k ) = ( n - 1 k ) p k ( 1 - p ) n - 1 - k , P(\operatorname{deg}(v)=k)={n-1\choose k}p^{k}(1-p)^{n-1-k},
  8. P ( deg ( v ) = k ) ( n p ) k e - n p k ! as n and n p = const , P(\operatorname{deg}(v)=k)\to\frac{(np)^{k}\mathrm{e}^{-np}}{k!}\quad\mbox{ as% }~{}n\to\infty\mbox{ and }~{}np=\mathrm{const},
  9. p < ( 1 - ϵ ) ln n n p<\tfrac{(1-\epsilon)\ln n}{n}
  10. p > ( 1 + ϵ ) ln n n p>\tfrac{(1+\epsilon)\ln n}{n}
  11. ln n n \tfrac{\ln n}{n}
  12. p c = 1 k p^{\prime}_{c}=\tfrac{1}{\langle k\rangle}
  13. p c p^{\prime}_{c}
  14. P = p [ 1 - exp ( - k P ) ] . P_{\infty}=p^{\prime}[1-\exp(-\langle k\rangle P_{\infty})].\,
  15. N ( n ) = ( n 2 ) p N(n)={n\choose 2}p

Erdős–Stone_theorem.html

  1. ex ( n ; K r ( t ) ) = ( r - 2 r - 1 + o ( 1 ) ) ( n 2 ) . \mbox{ex}~{}(n;K_{r}(t))=\left(\frac{r-2}{r-1}+o(1)\right){n\choose 2}.
  2. ex ( n ; H ) = ( r - 2 r - 1 + o ( 1 ) ) ( n 2 ) . \mbox{ex}~{}(n;H)=\left(\frac{r-2}{r-1}+o(1)\right){n\choose 2}.
  3. s r , ε ( n ) ( log log r - 1 n ) 1 / 2 s_{r,\varepsilon}(n)\geq\left(\underbrace{\log\cdots\log}_{r-1}n\right)^{1/2}
  4. 1 500 log ( 1 / ε ) log n < s r , ε ( n ) < 5 log ( 1 / ε ) log n \frac{1}{500\log(1/\varepsilon)}\log n<s_{r,\varepsilon}(n)<\frac{5}{\log(1/% \varepsilon)}\log n

Eric_Bach.html

  1. ( / n ) * \left(\mathbb{Z}/n\mathbb{Z}\right)^{*}

Error_diffusion.html

  1. 1 16 [ - # 7 3 5 1 ] \frac{1}{16}\left[\begin{array}[]{ccccc}-&\#&7\\ 3&5&1\end{array}\right]
  2. - -
  3. 1 48 [ - - # 7 5 3 5 7 5 3 1 3 5 3 1 ] \frac{1}{48}\left[\begin{array}[]{ccccc}-&-&\#&7&5\\ 3&5&7&5&3\\ 1&3&5&3&1\end{array}\right]
  4. 1 4 [ # 2 1 1 ] \frac{1}{4}\left[\begin{array}[]{cc}\#&2\\ 1&1\end{array}\right]

Essential_extension.html

  1. H N = { 0 } H\cap N=\{0\}\,
  2. H = { 0 } H=\{0\}\,
  3. N e M N\subseteq_{e}M\,
  4. N M N\trianglelefteq M
  5. N + H = M N+H=M\,
  6. H = M H=M\,
  7. N s M N\subseteq_{s}M\,
  8. N M N\ll M
  9. \subset
  10. K e M K\subseteq_{e}M
  11. K e N K\subseteq_{e}N
  12. N e M N\subseteq_{e}M
  13. K H e M K\cap H\subseteq_{e}M
  14. K e M K\subseteq_{e}M
  15. H e M H\subseteq_{e}M
  16. N C e M N\oplus C\subseteq_{e}M
  17. N s M N\subseteq_{s}M
  18. K s M K\subseteq_{s}M
  19. N / K s M / K N/K\subseteq_{s}M/K
  20. K + H s M K+H\subseteq_{s}M
  21. K s M K\subseteq_{s}M
  22. H s M H\subseteq_{s}M

Etching_(microfabrication).html

  1. arctan 2 = 54.7 \arctan\sqrt{2}=54.7^{\circ}
  2. δ = 6 D S = 6 R 100 T R 100 / R 111 = 6 T R 111 \delta=\frac{\sqrt{6}D}{S}=\frac{\sqrt{6}R_{100}T}{R_{100}/R_{111}}=\sqrt{6}TR% _{111}

Etemadi's_inequality.html

  1. S k = X 1 + + X k . S_{k}=X_{1}+\cdots+X_{k}.\,
  2. ( max 1 k n | S k | 3 α ) 3 max 1 k n ( | S k | α ) . \mathbb{P}\left(\max_{1\leq k\leq n}|S_{k}|\geq 3\alpha\right)\leq 3\max_{1% \leq k\leq n}\mathbb{P}\left(|S_{k}|\geq\alpha\right).
  3. ( max 1 k n | S k | α ) 27 α 2 Var ( S n ) . \mathbb{P}\left(\max_{1\leq k\leq n}|S_{k}|\geq\alpha\right)\leq\frac{27}{% \alpha^{2}}\mathrm{Var}(S_{n}).

Ethernet_frame.html

  1. Protocol overhead = Packet size - Payload size Packet size \,\text{Protocol overhead}=\frac{\,\text{Packet size}-\,\text{Payload size}}{% \,\text{Packet size}}
  2. Protocol efficiency = Payload size Packet size \,\text{Protocol efficiency}=\frac{\,\text{Payload size}}{\,\text{Packet size}}
  3. 1500 1538 = 97.53 % \frac{1500}{1538}=97.53\%
  4. 1500 1542 = 97.28 % \frac{1500}{1542}=97.28\%
  5. Throughput = Efficiency × Net bit rate \,\text{Throughput}=\,\text{Efficiency}\times\,\text{Net bit rate}\,\!
  6. Channel utilization = Time spent transmitting data Total time \,\text{Channel utilization}=\frac{\,\text{Time spent transmitting data}}{\,% \text{Total time}}

Euclidean_relation.html

  1. a , b , c X ( a R b a R c b R c ) . \forall a,b,c\in X\,(a\,R\,b\land a\,R\,c\to b\,R\,c).
  2. a , b , c X ( b R a c R a b R c ) . \forall a,b,c\in X\,(b\,R\,a\land c\,R\,a\to b\,R\,c).

Euclid–Mullin_sequence.html

  1. ( i < n a i ) + 1 . \left(\prod_{i<n}a_{i}\right)+1\,.

Euler's_continued_fraction_formula.html

  1. a 0 + a 0 a 1 + a 0 a 1 a 2 + + a 0 a 1 a 2 a n = a 0 1 - a 1 1 + a 1 - a 2 1 + a 2 - a n - 1 1 + a n - 1 - a n 1 + a n a_{0}+a_{0}a_{1}+a_{0}a_{1}a_{2}+\cdots+a_{0}a_{1}a_{2}\cdots a_{n}=\cfrac{a_{% 0}}{1-\cfrac{a_{1}}{1+a_{1}-\cfrac{a_{2}}{1+a_{2}-\cfrac{\ddots}{\ddots\cfrac{% a_{n-1}}{1+a_{n-1}-\cfrac{a_{n}}{1+a_{n}}}}}}}\,
  2. x = 1 1 + a 2 b 2 + a 3 b 3 + a 4 b 4 + x=\cfrac{1}{1+\cfrac{a_{2}}{b_{2}+\cfrac{a_{3}}{b_{3}+\cfrac{a_{4}}{b_{4}+% \ddots}}}}\,
  3. r i = - a i + 1 b i - 1 b i + 1 . r_{i}=-\frac{a_{i+1}b_{i-1}}{b_{i+1}}.\,
  4. x = 1 1 + a 2 b 2 + a 3 b 3 + a 4 b 4 + = 1 1 - r 1 1 + r 1 - r 2 1 + r 2 - r 3 1 + r 3 - x=\cfrac{1}{1+\cfrac{a_{2}}{b_{2}+\cfrac{a_{3}}{b_{3}+\cfrac{a_{4}}{b_{4}+% \ddots}}}}=\cfrac{1}{1-\cfrac{r_{1}}{1+r_{1}-\cfrac{r_{2}}{1+r_{2}-\cfrac{r_{3% }}{1+r_{3}-\ddots}}}}\,
  5. x = 1 + i = 1 r 1 r 2 r i = 1 + i = 1 ( j = 1 i r j ) x=1+\sum_{i=1}^{\infty}r_{1}r_{2}\cdots r_{i}=1+\sum_{i=1}^{\infty}\left(\prod% _{j=1}^{i}r_{j}\right)\,
  6. e z = 1 + n = 1 z n n ! = 1 + n = 1 ( j = 1 n z j ) e^{z}=1+\sum_{n=1}^{\infty}\frac{z^{n}}{n!}=1+\sum_{n=1}^{\infty}\left(\prod_{% j=1}^{n}\frac{z}{j}\right)\,
  7. e z = 1 1 - z 1 + z - 1 2 z 1 + 1 2 z - 1 3 z 1 + 1 3 z - 1 4 z 1 + 1 4 z - . e^{z}=\cfrac{1}{1-\cfrac{z}{1+z-\cfrac{\frac{1}{2}z}{1+\frac{1}{2}z-\cfrac{% \frac{1}{3}z}{1+\frac{1}{3}z-\cfrac{\frac{1}{4}z}{1+\frac{1}{4}z-\ddots}}}}}.\,
  8. e z = 1 1 - z 1 + z - z 2 + z - 2 z 3 + z - 3 z 4 + z - e^{z}=\cfrac{1}{1-\cfrac{z}{1+z-\cfrac{z}{2+z-\cfrac{2z}{3+z-\cfrac{3z}{4+z-% \ddots}}}}}\,
  9. log 1 + z 1 - z = 2 ( z + z 3 3 + z 5 5 + ) = 2 n = 0 z 2 n + 1 2 n + 1 . \log\frac{1+z}{1-z}=2\left(z+\frac{z^{3}}{3}+\frac{z^{5}}{5}+\cdots\right)=2% \sum_{n=0}^{\infty}\frac{z^{2n+1}}{2n+1}.\,
  10. log 1 + z 1 - z = 2 z [ 1 + z 2 3 + z 4 5 + ] = 2 z [ 1 + z 2 3 + ( z 2 3 ) z 2 5 / 3 + ( z 2 3 ) ( z 2 5 / 3 ) z 2 7 / 5 + ] \begin{aligned}\displaystyle\log\frac{1+z}{1-z}&\displaystyle=2z\left[1+\frac{% z^{2}}{3}+\frac{z^{4}}{5}+\cdots\right]\\ &\displaystyle=2z\left[1+\frac{z^{2}}{3}+\left(\frac{z^{2}}{3}\right)\frac{z^{% 2}}{5/3}+\left(\frac{z^{2}}{3}\right)\left(\frac{z^{2}}{5/3}\right)\frac{z^{2}% }{7/5}+\cdots\right]\end{aligned}
  11. log 1 + z 1 - z = 2 z 1 - 1 3 z 2 1 + 1 3 z 2 - 3 5 z 2 1 + 3 5 z 2 - 5 7 z 2 1 + 5 7 z 2 - 7 9 z 2 1 + 7 9 z 2 - \log\frac{1+z}{1-z}=\cfrac{2z}{1-\cfrac{\frac{1}{3}z^{2}}{1+\frac{1}{3}z^{2}-% \cfrac{\frac{3}{5}z^{2}}{1+\frac{3}{5}z^{2}-\cfrac{\frac{5}{7}z^{2}}{1+\frac{5% }{7}z^{2}-\cfrac{\frac{7}{9}z^{2}}{1+\frac{7}{9}z^{2}-\ddots}}}}}\,
  12. log 1 + z 1 - z = 2 z 1 - z 2 z 2 + 3 - ( 3 z ) 2 3 z 2 + 5 - ( 5 z ) 2 5 z 2 + 7 - ( 7 z ) 2 7 z 2 + 9 - . \log\frac{1+z}{1-z}=\cfrac{2z}{1-\cfrac{z^{2}}{z^{2}+3-\cfrac{(3z)^{2}}{3z^{2}% +5-\cfrac{(5z)^{2}}{5z^{2}+7-\cfrac{(7z)^{2}}{7z^{2}+9-\ddots}}}}}.\,
  13. 1 + i 1 - i = i log 1 + i 1 - i = i π 2 . \frac{1+i}{1-i}=i\quad\Rightarrow\quad\log\frac{1+i}{1-i}=\frac{i\pi}{2}.\,
  14. π = 4 1 + 1 2 2 + 3 2 2 + 5 2 2 + 7 2 2 + . \pi=\cfrac{4}{1+\cfrac{1^{2}}{2+\cfrac{3^{2}}{2+\cfrac{5^{2}}{2+\cfrac{7^{2}}{% 2+\ddots}}}}}.\,

Euler_summation.html

  1. j = 0 E y a j := i = 0 1 ( 1 + y ) i + 1 j = 0 i ( i j ) y j + 1 a j . {}_{E_{y}}\,\sum_{j=0}^{\infty}a_{j}:=\sum_{i=0}^{\infty}\frac{1}{(1+y)^{i+1}}% \sum_{j=0}^{i}{i\choose j}y^{j+1}a_{j}.
  2. y j + 1 i = j ( i j ) 1 ( 1 + y ) i + 1 = 1. y^{j+1}\sum_{i=j}^{\infty}{i\choose j}\frac{1}{(1+y)^{i+1}}=1.
  3. E y 1 E y 2 = E y 1 y 2 1 + y 1 + y 2 . {}_{E_{y_{1}}}{}_{E_{y_{2}}}\sum=\,_{E_{\frac{y_{1}y_{2}}{1+y_{1}+y_{2}}}}\sum.
  4. j = 0 x j P k ( j ) = i = 0 k x i ( 1 - x ) i + 1 j = 0 i ( i j ) ( - 1 ) i - j P k ( j ) \sum_{j=0}^{\infty}x^{j}P_{k}(j)=\sum_{i=0}^{k}\frac{x^{i}}{(1-x)^{i+1}}\sum_{% j=0}^{i}{i\choose j}(-1)^{i-j}P_{k}(j)
  5. P k P_{k}
  6. P k ( j ) := ( j + 1 ) k P_{k}(j):=(j+1)^{k}
  7. ζ ( - k ) = - B k + 1 k + 1 \zeta(-k)=-\frac{B_{k+1}}{k+1}
  8. 1 1 - 2 k + 1 i = 0 k 1 2 i + 1 j = 0 i ( i j ) ( - 1 ) j ( j + 1 ) k \frac{1}{1-2^{k+1}}\sum_{i=0}^{k}\frac{1}{2^{i+1}}\sum_{j=0}^{i}{i\choose j}(-% 1)^{j}(j+1)^{k}
  9. k k
  10. j = 0 z j = i = 0 1 ( 1 + y ) i + 1 j = 0 i ( i j ) y j + 1 z j = y 1 + y i = 0 ( 1 + y z 1 + y ) i \sum_{j=0}^{\infty}z^{j}=\sum_{i=0}^{\infty}\frac{1}{(1+y)^{i+1}}\sum_{j=0}^{i% }{i\choose j}y^{j+1}z^{j}=\frac{y}{1+y}\sum_{i=0}\left(\frac{1+yz}{1+y}\right)% ^{i}
  11. y y
  12. 1 1 - z \frac{1}{1-z}

Evolvability_(computer_science).html

  1. F n F_{n}\,
  2. R n R_{n}\,
  3. n n\,
  4. f F n f\in F_{n}
  5. r R n r\in R_{n}
  6. f f\,
  7. Perf ( f , r ) \operatorname{Perf}(f,r)
  8. r r\,
  9. f f\,
  10. r , r R n r,r^{\prime}\in R_{n}
  11. r r r\neq r^{\prime}\,
  12. r ( x ) = r ( x ) r(x)=r^{\prime}(x)\,
  13. x X n x\in X_{n}
  14. N ( r ) N(r)\,
  15. r r\,
  16. r r\,
  17. X n = { - 1 , 1 } n X_{n}=\{-1,1\}^{n}\,
  18. D n D_{n}\,
  19. X n X_{n}\,
  20. Perf ( f , r ) = x X n f ( x ) r ( x ) D n ( x ) . \operatorname{Perf}(f,r)=\sum_{x\in X_{n}}f(x)r(x)D_{n}(x).
  21. Perf ( f , r ) = Prob ( f ( x ) = r ( x ) ) - Prob ( f ( x ) r ( x ) ) . \operatorname{Perf}(f,r)=\operatorname{Prob}(f(x)=r(x))-\operatorname{Prob}(f(% x)\neq r(x)).
  22. Perf s ( f , r ) = 1 s x S f ( x ) r ( x ) , \operatorname{Perf}_{s}(f,r)=\frac{1}{s}\sum_{x\in S}f(x)r(x),
  23. S S\,
  24. s s\,
  25. X n X_{n}\,
  26. D n D_{n}\,
  27. s s\,
  28. Perf s ( f , r ) \operatorname{Perf}_{s}(f,r)
  29. Perf ( f , r ) \operatorname{Perf}(f,r)
  30. f F n f\in F_{n}
  31. r R n r\in R_{n}
  32. s s\,
  33. t t\,
  34. Mut ( f , r , s , t ) \operatorname{Mut}(f,r,s,t)
  35. r N ( r ) r^{\prime}\in N(r)
  36. r r^{\prime}\,
  37. Perf s ( f , r ) - Perf s ( f , r ) t \operatorname{Perf}_{s}(f,r^{\prime})-\operatorname{Perf}_{s}(f,r)\geq t
  38. r r^{\prime}\,
  39. - t < Perf s ( f , r ) - Perf s ( f , r ) < t -t<\operatorname{Perf}_{s}(f,r^{\prime})-\operatorname{Perf}_{s}(f,r)<t
  40. r r^{\prime}\,
  41. Perf s ( f , r ) - Perf s ( f , r ) - t \operatorname{Perf}_{s}(f,r^{\prime})-\operatorname{Perf}_{s}(f,r)\leq-t
  42. Mut ( f , r , s , t ) \operatorname{Mut}(f,r,s,t)
  43. Mut ( f , r , s , t ) \operatorname{Mut}(f,r,s,t)
  44. r r\,
  45. r 0 R n r_{0}\in R_{n}
  46. r 0 , r 1 , r 2 , r_{0},r_{1},r_{2},\ldots
  47. r i + 1 = Mut ( f , r i , s , t ) r_{i+1}=\operatorname{Mut}(f,r_{i},s,t)
  48. r g r_{g}\,
  49. r 0 r_{0}\,
  50. g g\,
  51. F F\,
  52. R R\,
  53. D D\,
  54. X X\,
  55. F F\,
  56. R R\,
  57. D D\,
  58. p ( , ) p(\cdot,\cdot)
  59. s ( , ) s(\cdot,\cdot)
  60. t ( , ) t(\cdot,\cdot)
  61. g ( , ) g(\cdot,\cdot)
  62. n n\,
  63. ϵ > 0 \epsilon>0\,
  64. f F n f\in F_{n}
  65. r 0 R n r_{0}\in R_{n}
  66. 1 - ϵ 1-\epsilon\,
  67. Perf ( f , r g ( n , 1 / ϵ ) ) 1 - ϵ , \operatorname{Perf}(f,r_{g(n,1/\epsilon)})\geq 1-\epsilon,
  68. N ( r ) N(r)\,
  69. r R n r\in R_{n}\,
  70. p ( n , 1 / ϵ ) p(n,1/\epsilon)\,
  71. s ( n , 1 / ϵ ) s(n,1/\epsilon)\,
  72. t ( 1 / n , ϵ ) t(1/n,\epsilon)\,
  73. g ( n , 1 / ϵ ) g(n,1/\epsilon)\,
  74. F F\,
  75. D D\,
  76. R R\,
  77. D D\,
  78. F F\,
  79. D D\,

Exact_category.html

  1. M M M ′′ M^{\prime}\to M\to M^{\prime\prime}
  2. M M M ′′ M ′′ ; M^{\prime}\rightarrow M^{\prime}\oplus M^{\prime\prime}\rightarrow M^{\prime% \prime};
  3. M M ′′ M\to M^{\prime\prime}
  4. N M ′′ N\to M^{\prime\prime}
  5. N N
  6. M M M^{\prime}\to M
  7. M N M^{\prime}\to N
  8. N N
  9. M M ′′ M\to M^{\prime\prime}
  10. N M N\to M
  11. N M M ′′ N\to M\to M^{\prime\prime}
  12. M M ′′ . M\to M^{\prime\prime}.
  13. M M M^{\prime}\to M
  14. M N M\to N
  15. M M N M^{\prime}\to M\to N
  16. M M . M^{\prime}\to M.
  17. \rightarrowtail
  18. . \twoheadrightarrow.
  19. F F
  20. M M M ′′ M^{\prime}\rightarrowtail M\twoheadrightarrow M^{\prime\prime}
  21. F ( M ) F ( M ) F ( M ′′ ) F(M^{\prime})\rightarrowtail F(M)\twoheadrightarrow F(M^{\prime\prime})
  22. 0 M M M ′′ 0 0\to M^{\prime}\to M\to M^{\prime\prime}\to 0
  23. M , M ′′ M^{\prime},M^{\prime\prime}
  24. M M
  25. M M M ′′ M^{\prime}\to M\to M^{\prime\prime}
  26. 0 M M M ′′ 0 0\to M^{\prime}\to M\to M^{\prime\prime}\to 0
  27. 0 M 𝑓 M M ′′ 0 , 0\to M^{\prime}\xrightarrow{f}M\to M^{\prime\prime}\to 0,
  28. N M ′′ N\to M^{\prime\prime}
  29. N N
  30. M × M ′′ N M\times_{M^{\prime\prime}}N
  31. 0 M ( f , 0 ) M × M ′′ N N 0. 0\to M^{\prime}\xrightarrow{(f,0)}M\times_{M^{\prime\prime}}N\to N\to 0.
  32. M M ′′ M\to M^{\prime\prime}
  33. N M N\to M
  34. N M M ′′ N\to M\to M^{\prime\prime}
  35. M M ′′ M\to M^{\prime\prime}
  36. 0 A B C 0 0\to A\to B\to C\to 0
  37. A , C A,C
  38. B B
  39. b b
  40. C C
  41. C C
  42. b b
  43. C C
  44. A A
  45. b = 0 b=0
  46. 0 ( 1 2 ) 2 ( - 2 , 1 ) 0 , 0\to\mathbb{Z}\xrightarrow{\left(\begin{smallmatrix}1\\ 2\end{smallmatrix}\right)}\mathbb{Z}^{2}\xrightarrow{(-2,1)}\mathbb{Z}\to 0,
  47. 0 / 0 , 0\to\mathbb{Q}\to\mathbb{R}\to\mathbb{R}/\mathbb{Q}\to 0,
  48. 0 d Ω 0 ( S 1 ) Ω c 1 ( S 1 ) H dR 1 ( S 1 ) 0 , 0\to d\Omega^{0}(S^{1})\to\Omega^{1}_{c}(S^{1})\to H^{1}_{\,\text{dR}}(S^{1})% \to 0,
  49. Ω c 1 ( S 1 ) \Omega^{1}_{c}(S^{1})
  50. d Ω 0 ( S 1 ) d\Omega^{0}(S^{1})
  51. 0 A B C 0 0\to A\to B\to C\to 0
  52. A , C A,C
  53. B B
  54. A A
  55. 0 / 2 / 4 / 2 0 , 0\to\mathbb{Z}/2\mathbb{Z}\to\mathbb{Z}/4\mathbb{Z}\to\mathbb{Z}/2\mathbb{Z}% \to 0,
  56. 0 / 2 ( 1 , 0 , 0 ) ( / 2 ) 2 ( / 2 ) 0 , 0\to\mathbb{Z}/2\mathbb{Z}\xrightarrow{(1,0,0)}(\mathbb{Z}/2\mathbb{Z})^{2}% \oplus\mathbb{Z}\to(\mathbb{Z}/2\mathbb{Z})\oplus\mathbb{Z}\to 0,
  57. 0 ( / 2 ) ( / 2 ) 2 ( 0 , 1 , 0 ) / 2 0 , 0\to(\mathbb{Z}/2\mathbb{Z})\oplus\mathbb{Z}\to(\mathbb{Z}/2\mathbb{Z})^{2}% \oplus\mathbb{Z}\xrightarrow{(0,1,0)}\mathbb{Z}/2\mathbb{Z}\to 0,
  58. ( 1 , 0 , 0 ) (1,0,0)
  59. ( 0 , 1 , 0 ) (0,1,0)
  60. / 2 ( 1 , 0 , 0 ) ( / 2 ) 2 ( 0 , 1 , 0 ) / 2 \mathbb{Z}/2\mathbb{Z}\xrightarrow{(1,0,0)}(\mathbb{Z}/2\mathbb{Z})^{2}\oplus% \mathbb{Z}\xrightarrow{(0,1,0)}\mathbb{Z}/2\mathbb{Z}
  61. ( / 2 ) (\mathbb{Z}/2\mathbb{Z})\oplus\mathbb{Z}
  62. \mathbb{Z}

Examples_of_generating_functions.html

  1. G ( 1 ; x ) = n = 0 x n = 1 1 - x G(1;x)=\sum_{n=0}^{\infty}x^{n}=\frac{1}{1-x}
  2. x x
  3. a x ax
  4. G ( 1 ; a x ) = 1 1 - a x = 1 + ( a x ) + ( a x ) 2 + + ( a x ) n + = n = 0 a n x n = G ( a n ; x ) . G(1;ax)=\frac{1}{1-ax}=1+(ax)+(ax)^{2}+\cdots+(ax)^{n}+\cdots=\sum_{n=0}^{% \infty}a^{n}x^{n}=G(a^{n};x).
  5. ( 1 + x ) n (1+x)^{n}
  6. ( n k ) {\left({{n}\atop{k}}\right)}
  7. ( 1 + x ) n (1+x)^{n}
  8. a n a^{n}
  9. 1 / ( 1 - a y ) 1/(1-ay)
  10. 1 1 - ( 1 + x ) y = 1 + ( 1 + x ) y + ( 1 + x ) 2 y 2 + , \frac{1}{1-(1+x)y}=1+(1+x)y+(1+x)^{2}y^{2}+\dots,
  11. x k y n x^{k}y^{n}
  12. ( n k ) {\left({{n}\atop{k}}\right)}
  13. f = n 0 F n x n f=\sum_{n\geq 0}F_{n}x^{n}
  14. f = F 0 x 0 + F 1 x 1 + F 2 x 2 + + F i x i + x f = F 0 x 1 + F 1 x 2 + + F i - 1 x i + x 2 f = F 0 x 2 + + F i - 2 x i + ( x + x 2 ) f = F 0 x 1 + ( F 0 + F 1 ) x 2 + + ( F i - 1 + F i - 2 ) x i + = F 2 x 2 + + F i x i + \begin{array}[]{rcrcrcrcrcrcr}f&=&F_{0}x^{0}&+&F_{1}x^{1}&+&F_{2}x^{2}&+&% \cdots&+&F_{i}x^{i}&+&\cdots\\ xf&=&&&F_{0}x^{1}&+&F_{1}x^{2}&+&\cdots&+&F_{i-1}x^{i}&+&\cdots\\ x^{2}f&=&&&&&F_{0}x^{2}&+&\cdots&+&F_{i-2}x^{i}&+&\cdots\\ (x+x^{2})f&=&&&F_{0}x^{1}&+&(F_{0}+F_{1})x^{2}&+&\cdots&+&(F_{i-1}+F_{i-2})x^{% i}&+&\cdots\\ &=&&&&&F_{2}x^{2}&+&\cdots&+&F_{i}x^{i}&+&\cdots\\ \end{array}
  15. f = x f + x 2 f + x . f=xf+x^{2}f+x.\,\!
  16. f = x 1 - x - x 2 . f=\frac{x}{1-x-x^{2}}.
  17. f = 1 5 ( 1 1 - φ 1 x - 1 1 - φ 2 x ) . f=\frac{1}{\sqrt{5}}\left(\frac{1}{1-\varphi_{1}x}-\frac{1}{1-\varphi_{2}x}% \right).
  18. F n = 1 5 ( φ 1 n - φ 2 n ) . F_{n}=\frac{1}{\sqrt{5}}(\varphi_{1}^{n}-\varphi_{2}^{n}).

Existentially_closed_model.html

  1. N N
  2. M 1 N M\prec_{1}N

Exner_function.html

  1. Π = ( p p 0 ) R d / c p = T θ \Pi=\left(\frac{p}{p_{0}}\right)^{R_{d}/c_{p}}=\frac{T}{\theta}
  2. p 0 p_{0}
  3. R d R_{d}
  4. c p c_{p}
  5. T T
  6. θ \theta

Exothermic_welding.html

  1. Fe 2 O 3 + 2 Al 2 Fe + Al 2 O 3 \mathrm{Fe_{2}O_{3}+2\ Al\longrightarrow 2\ Fe+Al_{2}O_{3}}
  2. 3 CuO + 2 A l 3 Cu + Al 2 O 3 \mathrm{3\ CuO+2Al\longrightarrow 3\ Cu+Al_{2}O_{3}}

Expectation_value_(quantum_mechanics).html

  1. A A
  2. σ \sigma
  3. A A
  4. σ \sigma
  5. A σ \langle A\rangle_{\sigma}
  6. A A
  7. σ \sigma
  8. ψ \psi
  9. A A
  10. ψ \psi
  11. A ψ = ψ | A | ψ \langle A\rangle_{\psi}=\langle\psi|A|\psi\rangle
  12. ψ \psi
  13. A A
  14. A A
  15. ϕ j \phi_{j}
  16. a j a_{j}
  17. A ψ = j a j | ψ | ϕ j | 2 \langle A\rangle_{\psi}=\sum_{j}a_{j}|\langle\psi|\phi_{j}\rangle|^{2}
  18. a j a_{j}
  19. | ψ | ϕ j | 2 |\langle\psi|\phi_{j}\rangle|^{2}
  20. A A
  21. A ψ = A ψ 2 \langle A\rangle_{\psi}=\|A\psi\|^{2}
  22. Q Q
  23. ψ \psi
  24. ψ ( x ) \psi(x)
  25. Q Q
  26. Q ψ = - x | ψ ( x ) | 2 d x \langle Q\rangle_{\psi}=\int_{-\infty}^{\infty}\,x\,|\psi(x)|^{2}\,dx
  27. P P
  28. σ \sigma
  29. ρ = i ρ i | ψ i ψ i | \rho=\sum_{i}\rho_{i}|\psi_{i}\rangle\langle\psi_{i}|
  30. A ρ = Trace ( ρ A ) = i ρ i ψ i | A | ψ i = i ρ i A ψ i \langle A\rangle_{\rho}=\mathrm{Trace}(\rho A)=\sum_{i}\rho_{i}\langle\psi_{i}% |A|\psi_{i}\rangle=\sum_{i}\rho_{i}\langle A\rangle_{\psi_{i}}
  31. σ \sigma
  32. A A
  33. A σ = σ ( A ) \langle A\rangle_{\sigma}=\sigma(A)
  34. σ \sigma
  35. σ ( ) = Trace ( ρ ) \sigma(\cdot)=\mathrm{Trace}(\rho\;\cdot)
  36. ρ \rho
  37. ρ = | ψ ψ | \rho=|\psi\rangle\langle\psi|
  38. ψ \psi
  39. σ = ψ | ψ \sigma=\langle\psi|\cdot\;\psi\rangle
  40. A A
  41. A A
  42. A = a d P ( a ) A=\int a\,\mathrm{d}P(a)
  43. P P
  44. A A
  45. σ = ψ | ψ \sigma=\langle\psi|\cdot\,\psi\rangle
  46. A σ = a d ψ | P ( a ) ψ \langle A\rangle_{\sigma}=\int a\;\mathrm{d}\langle\psi|P(a)\psi\rangle
  47. = L 2 ( ) \mathcal{H}=L^{2}(\mathbb{R})
  48. ψ \psi\in\mathcal{H}
  49. ψ ( x ) \psi(x)
  50. ψ 1 | ψ 2 = ψ 1 ( x ) ψ 2 ( x ) d x \langle\psi_{1}|\psi_{2}\rangle=\int\psi_{1}^{\ast}(x)\psi_{2}(x)\,\mathrm{d}x
  51. p ( x ) d x = ψ * ( x ) ψ ( x ) d x p(x)dx=\psi^{*}(x)\psi(x)dx
  52. d x dx
  53. x x
  54. Q Q
  55. ψ \psi
  56. ( Q ψ ) ( x ) = x ψ ( x ) (Q\psi)(x)=x\psi(x)
  57. Q Q
  58. Q ψ = ψ | Q ψ = - ψ ( x ) x ψ ( x ) d x = - x p ( x ) d x \langle Q\rangle_{\psi}=\langle\psi|Q\psi\rangle=\int_{-\infty}^{\infty}\psi^{% \ast}(x)\,x\,\psi(x)\,\mathrm{d}x=\int_{-\infty}^{\infty}x\,p(x)\,\mathrm{d}x
  59. ψ \psi
  60. ψ \psi
  61. Q Q
  62. P = - i d / d x P=-i\hbar\,d/dx
  63. P ψ = - i - ψ ( x ) d ψ ( x ) d x d x \langle P\rangle_{\psi}=-i\hbar\int_{-\infty}^{\infty}\psi^{\ast}(x)\,\frac{d% \psi(x)}{dx}\,\mathrm{d}x
  64. ψ \psi
  65. ψ \psi
  66. ψ / ψ \psi/\|\psi\|

Expected_shortfall.html

  1. q q
  2. q q
  3. q q
  4. q q
  5. q q
  6. q q
  7. q q
  8. X L p ( ) X\in L^{p}(\mathcal{F})
  9. 0 < α < 1 0<\alpha<1
  10. E S α = 1 α 0 α V a R γ ( X ) d γ ES_{\alpha}=\frac{1}{\alpha}\int_{0}^{\alpha}VaR_{\gamma}(X)d\gamma
  11. V a R γ VaR_{\gamma}
  12. E S α = - 1 α ( E [ X 1 { X x α } ] + x α ( α - P [ X x α ] ) ) ES_{\alpha}=-\frac{1}{\alpha}\left(E[X\ 1_{\{X\leq x_{\alpha}\}}]+x_{\alpha}(% \alpha-P[X\leq x_{\alpha}])\right)
  13. x α = inf { x : P ( X x ) α } x_{\alpha}=\inf\{x\in\mathbb{R}:P(X\leq x)\geq\alpha\}
  14. α \alpha
  15. 1 A ( x ) = { 1 if x A 0 else 1_{A}(x)=\begin{cases}1&\,\text{if }x\in A\\ 0&\,\text{else}\end{cases}
  16. E S α = inf Q 𝒬 α E Q [ X ] ES_{\alpha}=\inf_{Q\in\mathcal{Q}_{\alpha}}E^{Q}[X]
  17. 𝒬 α \mathcal{Q}_{\alpha}
  18. P P
  19. d Q d P α - 1 \frac{dQ}{dP}\leq\alpha^{-1}
  20. d Q d P \frac{dQ}{dP}
  21. Q Q
  22. P P
  23. X X
  24. T C E α ( X ) = E [ - X X - V a R α ( X ) ] TCE_{\alpha}(X)=E[-X\mid X\leq-VaR_{\alpha}(X)]
  25. g ( x ) = { x 1 - α if 0 x < 1 - α , 1 if 1 - α x 1. g(x)=\begin{cases}\frac{x}{1-\alpha}&\,\text{if }0\leq x<1-\alpha,\\ 1&\,\text{if }1-\alpha\leq x\leq 1.\end{cases}
  26. E S q ES_{q}
  27. q q
  28. q q
  29. E S q ES_{q}
  30. E S 0.05 ES_{0.05}
  31. E S 0.20 ES_{0.20}
  32. 10 100 ( - 100 ) + 10 100 ( - 20 ) 20 100 = - 60. \frac{\frac{10}{100}(-100)+\frac{10}{100}(-20)}{\frac{20}{100}}=-60.
  33. q q
  34. q q
  35. E S 0.20 ES_{0.20}
  36. E S 1 ES_{1}
  37. 0.1 ( - 100 ) + 0.3 ( - 20 ) + 0.4 0 + 0.2 50 = - 6. 0.1(-100)+0.3(-20)+0.4\cdot 0+0.2\cdot 50=-6.\,
  38. q q
  39. VaR q \operatorname{VaR}_{q}
  40. q q
  41. q q
  42. q q
  43. q q
  44. E S q ES_{q}
  45. q q
  46. E S 1.0 ES_{1.0}
  47. E S q ES_{q}
  48. VaR q \operatorname{VaR}_{q}
  49. q q
  50. E S α t ( X ) = ess sup Q 𝒬 α t E Q [ - X t ] ES_{\alpha}^{t}(X)=\operatorname*{ess\sup}_{Q\in\mathcal{Q}_{\alpha}^{t}}E^{Q}% [-X\mid\mathcal{F}_{t}]
  51. 𝒬 α t = { Q = P | t : d Q d P α t - 1 a . s . } \mathcal{Q}_{\alpha}^{t}=\{Q=P\,|_{\mathcal{F}_{t}}:\frac{dQ}{dP}\leq\alpha_{t% }^{-1}\mathrm{a.s.}\}
  52. ρ α t ( X ) = ess sup Q 𝒬 ~ α t E Q [ - X t ] \rho_{\alpha}^{t}(X)=\operatorname*{ess\sup}_{Q\in\tilde{\mathcal{Q}}_{\alpha}% ^{t}}E^{Q}[-X\mid\mathcal{F}_{t}]
  53. 𝒬 ~ α t = { Q P : 𝔼 [ d Q d P τ + 1 ] α t - 1 𝔼 [ d Q d P τ ] τ t a . s . } . \tilde{\mathcal{Q}}_{\alpha}^{t}=\left\{Q\ll P:\mathbb{E}\left[\frac{dQ}{dP}% \mid\mathcal{F}_{\tau+1}\right]\leq\alpha_{t}^{-1}\mathbb{E}\left[\frac{dQ}{dP% }\mid\mathcal{F}_{\tau}\right]\;\forall\tau\geq t\;\mathrm{a.s.}\right\}.

Expected_value_of_sample_information.html

  1. d D the decision being made, chosen from space D x X an uncertain state, with true value in space X z Z an observed sample composed of n observations z 1 , z 2 , . . , z n U ( d , x ) the utility of selecting decision d from x p ( x ) your prior subjective probability distribution (density function) on x p ( z | x ) the conditional prior probability of observing the sample z \begin{array}[]{ll}d\in D&\mbox{the decision being made, chosen from space }~{% }D\\ x\in X&\mbox{an uncertain state, with true value in space }~{}X\\ z\in Z&\mbox{an observed sample composed of }~{}n\mbox{ observations }~{}% \langle z_{1},z_{2},..,z_{n}\rangle\\ U(d,x)&\mbox{the utility of selecting decision }~{}d\mbox{ from }~{}x\\ p(x)&\mbox{your prior subjective probability distribution (density function) % on }~{}x\\ p(z|x)&\mbox{the conditional prior probability of observing the sample }~{}z% \end{array}
  2. Z i = X Z_{i}=X
  3. p ( z | x ) = p ( z i | x ) p(z|x)=\prod p(z_{i}|x)
  4. z p ( z | x ) d z = x \int zp(z|x)dz=x
  5. x x
  6. E [ U ] = max d D X U ( d , x ) p ( x ) d x E[U]=\max_{d\in D}~{}\int_{X}U(d,x)p(x)~{}dx
  7. z z
  8. E [ U | z ] = max d D X U ( d , x ) p ( x | z ) d x E[U|z]=\max_{d\in D}~{}\int_{X}U(d,x)p(x|z)~{}dx
  9. p ( x | z ) p(x|z)
  10. p ( x | z ) = p ( z | x ) p ( x ) p ( z ) p(x|z)={{p(z|x)p(x)}\over{p(z)}}
  11. p ( z ) = p ( z | x ) p ( x ) d x p(z)=\int p(z|x)p(x)~{}dx
  12. E [ U | S I ] = Z E [ U | z ] p ( z ) d z = Z max d D X U ( d , x ) p ( z | x ) p ( x ) d x d z E[U|SI]=\int_{Z}E[U|z]p(z)dz=\int_{Z}\max_{d\in D}~{}\int_{X}U(d,x)p(z|x)p(x)~% {}dx~{}dz
  13. E V S I = E [ U | S I ] - E [ U ] = ( Z max d D X U ( d , x ) p ( z | x ) p ( x ) d x d z ) - ( max d D X U ( d , x ) p ( x ) d x ) \begin{array}[]{rl}EVSI&=E[U|SI]-E[U]\\ &=\left(\int_{Z}\max_{d\in D}~{}\int_{X}U(d,x)p(z|x)p(x)~{}dx~{}dz\right)-% \left(\max_{d\in D}~{}\int_{X}U(d,x)p(x)~{}dx\right)\end{array}
  14. z i = z 1 i , z 2 i , . . , z n i z^{i}=\langle z^{i}_{1},z^{i}_{2},..,z^{i}_{n}\rangle
  15. p ( x | z i ) p(x|z^{i})
  16. p ( x | z i ) p(x|z^{i})
  17. i = 1 , . . , M i=1,..,M
  18. n n
  19. Z i = Z_{i}=
  20. x x
  21. x = [ 5 % , 60 % , 20 % , 10 % , 5 % ] x=[5\%,60\%,20\%,10\%,5\%]
  22. p ( x ) p(x)
  23. x x