wpmath0000002_10

IS-95.html

  1. 2 15 {2}^{15}
  2. 80 3 \frac{80}{3}
  3. 80 3 \frac{80}{3}

ISAAC_(cipher).html

  1. 4.67 × 10 1240 4.67\times 10^{1240}
  2. 10 2466 10^{2466}

Isaac_Newton's_occult_studies.html

  1. π \pi
  2. V = ( 2 / 3 ) π r 3 V=(2/3)\pi r^{3}

Ising_model.html

  1. H ( σ ) = - < i j > J i j σ i σ j - μ j h j σ j H(\sigma)=-\sum_{<i~{}j>}J_{ij}\sigma_{i}\sigma_{j}-\mu\sum_{j}h_{j}\sigma_{j}
  2. P β ( σ ) = e - β H ( σ ) Z β , P_{\beta}(\sigma)={e^{-\beta H(\sigma)}\over Z_{\beta}},
  3. Z β = σ e - β H ( σ ) Z_{\beta}=\sum_{\sigma}e^{-\beta H(\sigma)}
  4. f β = σ f ( σ ) P β ( σ ) \langle f\rangle_{\beta}=\sum_{\sigma}f(\sigma)P_{\beta}(\sigma)\,
  5. J i j > 0 J_{ij}>0
  6. J i j < 0 J_{ij}<0
  7. J i j = 0 J_{ij}=0
  8. h j > 0 h_{j}>0
  9. h j < 0 h_{j}<0
  10. h j = 0 h_{j}=0
  11. H ( σ ) = - < i j > J i j σ i σ j . H(\sigma)=-\sum_{<i~{}j>}J_{ij}\sigma_{i}\sigma_{j}.
  12. H ( σ ) = - J < i j > σ i σ j . H(\sigma)=-J\sum_{<i~{}j>}\sigma_{i}\sigma_{j}.
  13. σ i σ j β C exp ( - c ( β ) | i - j | ) , \langle\sigma_{i}\sigma_{j}\rangle_{\beta}\leq C\exp(-c(\beta)|i-j|),\,
  14. σ i σ j β c ( β ) > 0. \langle\sigma_{i}\sigma_{j}\rangle_{\beta}\geq c(\beta)>0.\,
  15. M = 1 N i = 1 N S i . M={1\over N}\sum_{i=1}^{N}S_{i}.
  16. N ( L ) = ( 2 d ) L N(L)=(2d)^{L}
  17. L ( 2 d ) L ( ε ) L \sum_{L}(2d)^{L}(\varepsilon)^{L}
  18. N ( L ) < 4 2 L . N(L)<4^{2L}.
  19. L L 2 4 - 2 L e - 4 β L \sum_{L}L^{2}4^{-2L}e^{-4\beta L}
  20. H ( σ ) = - J < i j > σ i σ j - h j σ j . H(\sigma)=-J\sum_{<ij>}\sigma_{i}\sigma_{j}-h\sum_{j}\sigma_{j}.
  21. H ( σ ) = - J < i j > σ i σ j . H(\sigma)=-J\sum_{<ij>}\sigma_{i}\sigma_{j}.
  22. P ( μ , ν ) P ( ν , μ ) = g ( μ , ν ) A ( μ , ν ) g ( ν , μ ) A ( ν , μ ) = A ( μ , ν ) A ( ν , μ ) = P β ( ν ) P β ( μ ) = 1 Z e - β ( H ν ) 1 Z e - β ( H μ ) = e - β ( H ν - H μ ) . \frac{P(\mu,\nu)}{P(\nu,\mu)}=\frac{g(\mu,\nu)A(\mu,\nu)}{g(\nu,\mu)A(\nu,\mu)% }=\frac{A(\mu,\nu)}{A(\nu,\mu)}=\frac{P_{\beta}(\nu)}{P_{\beta}(\mu)}=\frac{% \frac{1}{Z}e^{-\beta(H_{\nu})}}{\frac{1}{Z}e^{-\beta(H_{\mu})}}=e^{-\beta(H_{% \nu}-H_{\mu})}.
  23. A ( μ , ν ) A ( ν , μ ) = e - β ( H ν - H μ ) . \frac{A(\mu,\nu)}{A(\nu,\mu)}=e^{-\beta(H_{\nu}-H_{\mu})}.
  24. A ( μ , ν ) = { e - β ( H ν - H μ ) , if H ν - H μ > 0 1 , otherwise . A(\mu,\nu)=\begin{cases}e^{-\beta(H_{\nu}-H_{\mu})},&\,\text{if }H_{\nu}-H_{% \mu}>0\\ 1,&\,\text{otherwise}.\end{cases}
  25. e - β ( H ν - H μ ) . e^{-\beta(H_{\nu}-H_{\mu})}.
  26. J i j | i - j | - α J_{ij}\sim|i-j|^{-\alpha}
  27. J i j | i - j | - α J_{ij}\sim|i-j|^{-\alpha}
  28. J i j | i - j | - 2 J_{ij}\sim|i-j|^{-2}
  29. J i j | i - j | - α J_{ij}\sim|i-j|^{-\alpha}
  30. H ( σ ) = - J i = 1 , , L σ i σ i + 1 - h i σ i H(\sigma)=-J\sum_{i=1,\ldots,L}\sigma_{i}\sigma_{i+1}-h\sum_{i}\sigma_{i}
  31. f ( β , h ) = - lim L 1 β L ln ( Z ( β ) ) = - 1 β ln ( e β J cosh β h + e 2 β J ( sinh β h ) 2 + e - 2 β J ) f(\beta,h)=-\lim_{L\to\infty}\frac{1}{\beta L}\ln(Z(\beta))=-\frac{1}{\beta}% \ln\left(e^{\beta J}\cosh\beta h+\sqrt{e^{2\beta J}(\sinh\beta h)^{2}+e^{-2% \beta J}}\right)
  32. σ i σ j - σ i σ j = C ( β ) e - c ( β ) | i - j | \langle\sigma_{i}\sigma_{j}\rangle-\langle\sigma_{i}\rangle\langle\sigma_{j}% \rangle=C(\beta)e^{-c(\beta)|i-j|}
  33. H ( σ ) = - J ( σ 1 σ 2 + + σ L - 1 σ L ) . H(\sigma)=-J(\sigma_{1}\sigma_{2}+\cdots+\sigma_{L-1}\sigma_{L}).
  34. σ j = σ j σ j - 1 j 2. \sigma^{\prime}_{j}=\sigma_{j}\sigma_{j-1}\qquad j\geq 2.
  35. Z ( β ) = σ 1 , , σ L e β J σ 1 σ 2 e β J σ 2 σ 3 e β J σ L - 1 σ L = 2 j = 2 L σ j e β J σ j = 2 [ e β J + e - β J ] L - 1 . Z(\beta)=\sum_{\sigma_{1},\ldots,\sigma_{L}}e^{\beta J\sigma_{1}\sigma_{2}}\;e% ^{\beta J\sigma_{2}\sigma_{3}}\;\cdots e^{\beta J\sigma_{L-1}\sigma_{L}}=2% \prod_{j=2}^{L}\sum_{\sigma^{\prime}_{j}}e^{\beta J\sigma^{\prime}_{j}}=2\left% [e^{\beta J}+e^{-\beta J}\right]^{L-1}.
  36. f ( β , 0 ) = - 1 β ln [ e β J + e - β J ] . f(\beta,0)=-\frac{1}{\beta}\ln\left[e^{\beta J}+e^{-\beta J}\right].
  37. σ j σ j + N = [ e β J - e - β J e β J + e - β J ] N \langle\sigma_{j}\sigma_{j+N}\rangle=\left[\frac{e^{\beta J}-e^{-\beta J}}{e^{% \beta J}+e^{-\beta J}}\right]^{N}
  38. Z ( β ) = σ 1 , , σ L e β h σ 1 e β J σ 1 σ 2 e β h σ 2 e β J σ 2 σ 3 e β h σ L e β J σ L σ 1 = σ 1 , , σ L V σ 1 , σ 2 V σ 2 , σ 3 V σ L , σ 1 . Z(\beta)=\sum_{\sigma_{1},\ldots,\sigma_{L}}e^{\beta h\sigma_{1}}e^{\beta J% \sigma_{1}\sigma_{2}}\;e^{\beta h\sigma_{2}}e^{\beta J\sigma_{2}\sigma_{3}}\;% \cdots e^{\beta h\sigma_{L}}e^{\beta J\sigma_{L}\sigma_{1}}=\sum_{\sigma_{1},% \ldots,\sigma_{L}}V_{\sigma_{1},\sigma_{2}}V_{\sigma_{2},\sigma_{3}}\cdots V_{% \sigma_{L},\sigma_{1}}.
  39. V σ , σ V_{\sigma,\sigma^{\prime}}
  40. V σ , σ = e β h 2 σ e β J σ σ e β h 2 σ V_{\sigma,\sigma^{\prime}}=e^{\frac{\beta h}{2}\sigma}e^{\beta J\sigma\sigma^{% \prime}}e^{\frac{\beta h}{2}\sigma^{\prime}}
  41. V = [ e β ( h + J ) e - β J e - β J e - β ( h - J ) ] . V=\begin{bmatrix}e^{\beta(h+J)}&e^{-\beta J}\\ e^{-\beta J}&e^{-\beta(h-J)}\end{bmatrix}.
  42. Z ( β ) = Tr V L = λ 1 L + λ 2 L = λ 1 L [ 1 + ( λ 2 λ 1 ) L ] Z(\beta)={\rm Tr}V^{L}=\lambda_{1}^{L}+\lambda_{2}^{L}=\lambda_{1}^{L}\left[1+% \left(\frac{\lambda_{2}}{\lambda_{1}}\right)^{L}\right]
  43. λ 1 = e β J cosh β h + e 2 β J ( sinh β h ) 2 + e - 2 β J \lambda_{1}=e^{\beta J}\cosh\beta h+\sqrt{e^{2\beta J}(\sinh\beta h)^{2}+e^{-2% \beta J}}
  44. p 1 - p = e - 2 β J . {p\over 1-p}=e^{-2\beta J}.
  45. S i S j e - p | i - j | . \langle S_{i}S_{j}\rangle\,\propto\,e^{-p|i-j|}.
  46. Z = configs e k S k = k ( 1 + p ) = ( 1 + p ) L . Z=\sum_{\mathrm{configs}}e^{\sum_{k}S_{k}}=\prod_{k}(1+p)=(1+p)^{L}.
  47. β f = log ( 1 + p ) = log ( 1 + e - 2 β J 1 + e - 2 β J ) , \beta f=\log(1+p)=\log\left(1+{e^{-2\beta J}\over 1+e^{-2\beta J}}\right),
  48. h = 0 h=0
  49. J 1 J_{1}
  50. J 2 J_{2}
  51. - β f = ln 2 + 1 8 π 2 0 2 π d θ 1 0 2 π d θ 2 ln [ cosh ( 2 β J 1 ) cosh ( 2 β J 2 ) - sinh ( 2 β J 1 ) cos ( θ 1 ) - sinh ( 2 β J 2 ) cos ( θ 2 ) ] . -\beta f=\ln 2+\frac{1}{8\pi^{2}}\int_{0}^{2\pi}d\theta_{1}\int_{0}^{2\pi}d% \theta_{2}\ln[\cosh(2\beta J_{1})\cosh(2\beta J_{2})-\sinh(2\beta J_{1})\cos(% \theta_{1})-\sinh(2\beta J_{2})\cos(\theta_{2})].
  52. T c T_{c}
  53. sinh ( 2 J 1 k T c ) sinh ( 2 J 2 k T c ) = 1 \sinh\left(\frac{2J_{1}}{kT_{c}}\right)\sinh\left(\frac{2J_{2}}{kT_{c}}\right)=1
  54. J 1 = J 2 = J J_{1}=J_{2}=J
  55. T c T_{c}
  56. T c = 2 J k ln ( 1 + 2 ) T_{c}=\frac{2J}{k\ln(1+\sqrt{2})}
  57. J 1 J_{1}
  58. J 2 J_{2}
  59. h = 0 h=0
  60. S exp ( i j S i , j S i , j + 1 + S i , j S i + 1 , j ) . \sum_{S}\exp\biggl(\sum_{ij}S_{i,j}S_{i,j+1}+S_{i,j}S_{i+1,j}\biggr).
  61. U = e i H Δ t U=e^{iH\Delta t}
  62. U N = ( e i H Δ t ) N = D X e i L U^{N}=(e^{iH\Delta t})^{N}=\int DXe^{iL}
  63. T C 1 C 2 . T_{C_{1}C_{2}}.
  64. | A = S A ( S ) | S |A\rangle=\sum_{S}A(S)|S\rangle
  65. | S |S\rangle
  66. Z = tr ( T N ) . Z=\mathrm{tr}(T^{N}).
  67. σ i x . \sigma^{x}_{i}.
  68. σ i z . \sigma^{z}_{i}.
  69. i A σ i x + B σ i z σ i + 1 z \sum_{i}A\sigma^{x}_{i}+B\sigma^{z}_{i}\sigma^{z}_{i+1}
  70. C ψ i ψ i . \sum C\psi^{\dagger}_{i}\psi_{i}.\,
  71. σ i x = D ψ i ψ i + 1 + D * ψ i ψ i - 1 + C ψ i ψ i + 1 + C * ψ i ψ i + 1 . \sigma^{x}_{i}=D{\psi^{\dagger}}_{i}\psi_{i+1}+D^{*}{\psi^{\dagger}}_{i}\psi_{% i-1}+C\psi_{i}\psi_{i+1}+C^{*}{\psi^{\dagger}}_{i}{\psi^{\dagger}}_{i+1}.
  72. M = ( 1 - [ sinh 2 β J 1 sinh 2 β J 2 ] - 2 ) 1 8 M=\left(1-\left[\sinh 2\beta J_{1}\sinh 2\beta J_{2}\right]^{-2}\right)^{\frac% {1}{8}}
  73. J 1 J_{1}
  74. J 2 J_{2}
  75. β F = d d x [ A H 2 + i = 1 d Z i ( i H ) 2 + λ H 4 ] . \beta F=\int d^{d}x\left[AH^{2}+\sum_{i=1}^{d}Z_{i}(\partial_{i}H)^{2}+\lambda H% ^{4}...\right].
  76. Z i j i H j H Z_{ij}\partial_{i}H\partial_{j}H
  77. P ( H ) e - d d x [ A H 2 + Z | H | 2 + λ H 4 ] . P(H)\propto e^{-\int d^{d}x\left[AH^{2}+Z|\nabla H|^{2}+\lambda H^{4}\right]}.
  78. H ( x 1 ) H ( x 2 ) H ( x n ) = D H P ( H ) H ( x 1 ) H ( x 2 ) H ( x n ) D H P ( H ) . \langle H(x_{1})H(x_{2})\cdots H(x_{n})\rangle={\int DHP(H)H(x_{1})H(x_{2})% \cdots H(x_{n})\over\int DHP(H)}.
  79. Z = D H e d d x [ A H 2 + Z | H | 2 + λ H 4 ] Z=\int DHe^{\int d^{d}x\left[AH^{2}+Z|\nabla H|^{2}+\lambda H^{4}\right]}
  80. F = d d x A H 2 . F=\int d^{d}xAH^{2}.
  81. F = d d x [ t H 2 + λ H 4 + Z ( H ) 2 ] , F=\int d^{d}x\left[tH^{2}+\lambda H^{4}+Z(\nabla H)^{2}\right],
  82. H ( x ) 4 = - H ( x ) 2 2 + 2 H ( x ) 2 H ( x ) 2 + ( H ( x ) 2 - H ( x ) 2 ) 2 H(x)^{4}=-\langle H(x)^{2}\rangle^{2}+2\langle H(x)^{2}\rangle H(x)^{2}+\left(% H(x)^{2}-\langle H(x)^{2}\rangle\right)^{2}
  83. H ( t H 2 + λ H 4 ) = 2 t H + 4 λ H 3 = 0 {\partial\over\partial H}\left(tH^{2}+\lambda H^{4}\right)=2tH+4\lambda H^{3}=0
  84. S ( x ) S ( y ) H ( x ) H ( y ) = G ( x - y ) = d k ( 2 π ) d e i k ( x - y ) k 2 + t \langle S(x)S(y)\rangle\propto\langle H(x)H(y)\rangle=G(x-y)=\int{dk\over(2\pi% )^{d}}{e^{ik(x-y)}\over k^{2}+t}
  85. S ( x 1 ) S ( x 2 ) S ( x 2 n ) = C n G ( x i 1 , x j 1 ) G ( x i 2 , X j 2 ) G ( x i n , x j n ) \langle S(x_{1})S(x_{2})...S(x_{2n})\rangle=C^{n}\sum G(x_{i1},x_{j1})G(x_{i2}% ,X_{j2})\ldots G(x_{in},x_{jn})
  86. ( - x 2 + t ) H ( x ) H ( y ) \displaystyle\left(-\nabla_{x}^{2}+t\right)\langle H(x)H(y)\rangle
  87. E = G E=\nabla G
  88. E = 0 \nabla\cdot E=0
  89. d d - 1 S E r = constant \int d^{d-1}SE_{r}=\mathrm{constant}
  90. E = C r d - 1 E={C\over r^{d-1}}
  91. G ( r ) = C r d - 2 G(r)={C\over r^{d-2}}
  92. 2 G + t G = 0 1 r d - 1 d d r ( r d - 1 d G d r ) + t G ( r ) = 0 \nabla^{2}G+tG=0\to{1\over r^{d-1}}{d\over dr}\left(r^{d-1}{dG\over dr}\right)% +tG(r)=0
  93. t \sqrt{t}
  94. G ( x ) = d τ 1 ( 2 π τ ) d e - x 2 4 τ - t τ G(x)=\int d\tau{1\over\left(\sqrt{2\pi\tau}\right)^{d}}e^{-{x^{2}\over 4\tau}-% t\tau}
  95. G ( k ) = d τ e - ( k 2 - t ) τ = 1 k 2 - t G(k)=\int d\tau e^{-(k^{2}-t)\tau}={1\over k^{2}-t}
  96. e - t τ = e - t r 2 e^{-t\tau}=e^{-tr^{2}}
  97. e - t r e^{-\sqrt{t}r}
  98. G ( r ) e - t r r d - 2 G(r)\approx{e^{-\sqrt{t}r}\over r^{d-2}}
  99. F = d 4 x [ Z 2 | H | 2 + t 2 H 2 + λ 4 ! H 4 ] F=\int d^{4}x\left[{Z\over 2}|\nabla H|^{2}+{t\over 2}H^{2}+{\lambda\over 4!}H% ^{4}\right]\,
  100. 2 H + t H = - λ 6 H 3 . \nabla^{2}H+tH=-{\lambda\over 6}H^{3}.
  101. δ H 3 = 3 H Λ < | k | < ( 1 + b ) Λ d 4 k ( 2 π ) 4 1 ( k 2 + t ) \delta H^{3}=3H\int_{\Lambda<|k|<(1+b)\Lambda}{d^{4}k\over(2\pi)^{4}}{1\over(k% ^{2}+t)}
  102. d k 1 k 2 - t d k 1 k 2 ( k 2 + t ) = A Λ 2 b + B b t \int dk{1\over k^{2}}-t\int dk{1\over k^{2}(k^{2}+t)}=A\Lambda^{2}b+Bbt
  103. δ t = ( 2 - B λ 2 ) b t \delta t=\left(2-{B\lambda\over 2}\right)bt
  104. δ λ = - 3 λ 2 2 k d k 1 ( k 2 + t ) 2 = - 3 λ 2 2 b \delta\lambda=-{3\lambda^{2}\over 2}\int_{k}dk{1\over(k^{2}+t)^{2}}=-{3\lambda% ^{2}\over 2}b
  105. δ λ = - 3 B λ 2 b \delta\lambda=-3B\lambda^{2}b
  106. d t t \displaystyle{dt\over t}
  107. B b = Λ < | k | < ( 1 + b ) Λ d 4 k ( 2 π ) 4 1 k 4 Bb=\int_{\Lambda<|k|<(1+b)\Lambda}{d^{4}k\over(2\pi)^{4}}{1\over k^{4}}
  108. B = ( 2 π 2 Λ 3 ) 1 ( 2 π ) 4 b Λ 1 b Λ 4 = 1 8 π 2 B=(2\pi^{2}\Lambda^{3}){1\over(2\pi)^{4}}{b\Lambda}{1\over b\Lambda^{4}}={1% \over 8\pi^{2}}
  109. G ( x - y ) = d τ 1 t d 2 e x 2 2 τ + t τ G(x-y)=\int d\tau{1\over t^{d\over 2}}e^{{x^{2}\over 2\tau}+t\tau}
  110. d λ λ = ε - 3 B λ d t t = 2 - λ B \begin{aligned}\displaystyle{d\lambda\over\lambda}&\displaystyle=\varepsilon-3% B\lambda\\ \displaystyle{dt\over t}&\displaystyle=2-\lambda B\end{aligned}
  111. λ = ε 3 B \lambda={\varepsilon\over 3B}
  112. 1 2 ( 1 - ε 3 ) \tfrac{1}{2}\left(1-{\varepsilon\over 3}\right)
  113. p 1 - p = e 2 β J H {p\over 1-p}=e^{2\beta JH}
  114. p = 1 1 + e - 2 β J H p={1\over 1+e^{-2\beta JH}}
  115. H = 2 p - 1 = 1 - e - 2 β J H 1 + e - 2 β J H = tanh ( β J H ) H=2p-1={1-e^{-2\beta JH}\over 1+e^{-2\beta JH}}=\tanh(\beta JH)
  116. H = 3 ε H=\sqrt{3\varepsilon}
  117. H = ( β F ) h = ( 1 + A ε ) H + B H 3 + H={\partial(\beta F)\over\partial h}=(1+A\varepsilon)H+BH^{3}+\cdots
  118. H ε 0.308 H\propto\varepsilon^{0.308}
  119. H ε 0.125 H\propto\varepsilon^{0.125}
  120. E = i j J i j S i S j + J i j k l S i S j S k S l . E=\sum_{ij}J_{ij}S_{i}S_{j}+\sum J_{ijkl}S_{i}S_{j}S_{k}S_{l}\ldots.
  121. 2 \scriptstyle\sqrt{2}
  122. e J ( N + - N - ) + e J ( N - - N + ) = 2 cosh ( J [ N + - N - ] ) e^{J(N_{+}-N_{-})}+e^{J(N_{-}-N_{+})}=2\cosh(J[N_{+}-N_{-}])
  123. F = log ( cosh [ J ( N + - N - ) ] ) . F=\log(\cosh[J(N_{+}-N_{-})]).
  124. Δ F = ln ( cosh [ 4 J ] ) . \Delta F=\ln(\cosh[4J]).
  125. 3 J = ln ( cosh [ 4 J ] ) . 3J^{\prime}=\ln(\cosh[4J]).
  126. E = - 1 2 i , j 4 J B i B j + i μ B i E=-\frac{1}{2}\sum_{\langle i,j\rangle}4JB_{i}B_{j}+\sum_{i}\mu B_{i}
  127. B i = ( S i + 1 ) / 2. B_{i}=(S_{i}+1)/2.
  128. E = - 1 2 i , j J S i S j - 1 2 i ( 4 J - μ ) S i E=-\frac{1}{2}\sum_{\langle i,j\rangle}JS_{i}S_{j}-\frac{1}{2}\sum_{i}(4J-\mu)% S_{i}
  129. E = - i h i S i E=-\sum_{i}h_{i}S_{i}
  130. E = - 1 2 i j J i j S i S j - i h i S i E=-\tfrac{1}{2}\sum_{ij}J_{ij}S_{i}S_{j}-\sum_{i}h_{i}S_{i}
  131. J i j J_{ij}
  132. H ^ = - 1 2 J i , k S i S k , \hat{H}=-\frac{1}{2}\,\sum J_{i,k}\,S_{i}\,S_{k},

Isometric_projection.html

  1. 1 / 3 {1}/{\sqrt{3}}
  2. 1 / 2 {1}/{\sqrt{2}}
  3. 2 \scriptstyle\sqrt{2}
  4. 2 \scriptstyle\sqrt{2}
  5. 1 / 2 \scriptstyle 1/\sqrt{2}
  6. a x , y , z a_{x,y,z}
  7. b x , y b_{x,y}
  8. [ 𝐜 x 𝐜 y 𝐜 z ] = [ 1 0 0 0 cos α sin α 0 - sin α cos α ] [ cos β 0 - sin β 0 1 0 sin β 0 cos β ] [ 𝐚 x 𝐚 y 𝐚 z ] = 1 6 [ 3 0 - 3 1 2 1 2 - 2 2 ] [ 𝐚 x 𝐚 y 𝐚 z ] \begin{bmatrix}\mathbf{c}_{x}\\ \mathbf{c}_{y}\\ \mathbf{c}_{z}\\ \end{bmatrix}=\begin{bmatrix}1&0&0\\ 0&{\cos\alpha}&{\sin\alpha}\\ 0&{-\sin\alpha}&{\cos\alpha}\\ \end{bmatrix}\begin{bmatrix}{\cos\beta}&0&{-\sin\beta}\\ 0&1&0\\ {\sin\beta}&0&{\cos\beta}\\ \end{bmatrix}\begin{bmatrix}\mathbf{a}_{x}\\ \mathbf{a}_{y}\\ \mathbf{a}_{z}\\ \end{bmatrix}=\frac{1}{\sqrt{6}}\begin{bmatrix}\sqrt{3}&0&-\sqrt{3}\\ 1&2&1\\ \sqrt{2}&-\sqrt{2}&\sqrt{2}\\ \end{bmatrix}\begin{bmatrix}\mathbf{a}_{x}\\ \mathbf{a}_{y}\\ \mathbf{a}_{z}\\ \end{bmatrix}
  9. α = arcsin ( tan 30 ) 35.264 \alpha=\arcsin(\tan 30^{\circ})\approx 35.264^{\circ}
  10. β = 45 \beta=45^{\circ}
  11. β \beta
  12. α \alpha
  13. [ 𝐛 x 𝐛 y 0 ] = [ 1 0 0 0 1 0 0 0 0 ] [ 𝐜 x 𝐜 y 𝐜 z ] \begin{bmatrix}\mathbf{b}_{x}\\ \mathbf{b}_{y}\\ 0\\ \end{bmatrix}=\begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&0\\ \end{bmatrix}\begin{bmatrix}\mathbf{c}_{x}\\ \mathbf{c}_{y}\\ \mathbf{c}_{z}\\ \end{bmatrix}

Isometry.html

  1. T T
  2. d Y ( f ( a ) , f ( b ) ) = d X ( a , b ) . d_{Y}\left(f(a),f(b)\right)=d_{X}(a,b).
  3. x | x | x\mapsto|x|
  4. {\mathbb{R}}\to{\mathbb{R}}
  5. f ( v ) = v \|f(v)\|=\|v\|
  6. f : X Y f:X\to Y
  7. a 𝔄 a\in\mathfrak{A}
  8. a * a = 1 a^{*}\cdot a=1

Isosceles_triangle.html

  1. 1 2 4 a 2 - b 2 . \tfrac{1}{2}\sqrt{4a^{2}-b^{2}}.
  2. 2 p b 3 - p 2 b 2 + 16 T 2 = 0. 2pb^{3}-p^{2}b^{2}+16T^{2}=0.
  3. T = b 4 4 a 2 - b 2 . T=\frac{b}{4}\sqrt{4a^{2}-b^{2}}.
  4. ( b / 2 ) 2 + h 2 = a 2 (b/2)^{2}+h^{2}=a^{2}
  5. h = 4 a 2 - b 2 2 h=\frac{\sqrt{4a^{2}-b^{2}}}{2}
  6. T = b 4 4 a 2 - b 2 . T=\frac{b}{4}\sqrt{4a^{2}-b^{2}}.
  7. T = 2 ( 1 2 a sin ( θ 2 ) a cos ( θ 2 ) ) T=2\left(\frac{1}{2}a\sin\left(\frac{\theta}{2}\right)a\cos\left(\frac{\theta}% {2}\right)\right)
  8. = a 2 sin ( θ 2 ) cos ( θ 2 ) =a^{2}\sin\left(\frac{\theta}{2}\right)\cos\left(\frac{\theta}{2}\right)
  9. T = 1 2 a 2 sin θ T=\frac{1}{2}a^{2}\sin\theta
  10. p 2 > 12 3 T . p^{2}>12\sqrt{3}T.
  11. 2 a c a + c > t > a c 2 a + c . \frac{2ac}{a+c}>t>\frac{ac\sqrt{2}}{a+c}.

Isostasy.html

  1. ( h 1 + c + b 1 ) ρ c = ( c ρ c ) + ( b 1 ρ m ) (h_{1}+c+b_{1})\rho_{c}=(c\rho_{c})+(b_{1}\rho_{m})
  2. b 1 ( ρ m - ρ c ) = h 1 ρ c {b_{1}(\rho_{m}-\rho_{c})}=h_{1}\rho_{c}
  3. b 1 = h 1 ρ c ρ m - ρ c b_{1}=\frac{h_{1}\rho_{c}}{\rho_{m}-\rho_{c}}
  4. ρ m \rho_{m}
  5. ρ c \rho_{c}
  6. c ρ c = ( h 2 ρ w ) + ( b 2 ρ m ) + [ ( c - h 2 - b 2 ) ρ c ] c\rho_{c}=(h_{2}\rho_{w})+(b_{2}\rho_{m})+[(c-h_{2}-b_{2})\rho_{c}]
  7. b 2 ( ρ m - ρ c ) = h 2 ( ρ c - ρ w ) {b_{2}(\rho_{m}-\rho_{c})}={h_{2}(\rho_{c}-\rho_{w})}
  8. b 2 = ( ρ c - ρ w ρ m - ρ c ) h 2 b_{2}=(\frac{\rho_{c}-\rho_{w}}{\rho_{m}-\rho_{c}}){h_{2}}
  9. ρ m \rho_{m}
  10. ρ c \rho_{c}
  11. ρ w \rho_{w}
  12. ρ 1 = ρ c c h 1 + c \rho_{1}=\rho_{c}\frac{c}{h_{1}+c}
  13. h 1 h_{1}

Israeli_Air_Force.html

  1. } \Bigg\}
  2. } \Big\}
  3. } \Big\}
  4. } \Big\}
  5. } \Big\}
  6. } \Big\}
  7. } \Big\}
  8. } \Big\}
  9. } \Big\}
  10. } \Big\}
  11. } \Big\}

Itemized_deduction.html

  1. 4 , 000 - .02 ( 50 , 000 ) = 3 , 000 4,000-.02(50,000)=3,000
  2. .03 ( 500 , 000 - 275 , 000 ) = $ 6 , 750 .03(500,000-275,000)=\$6,750
  3. .80 ( 20 , 000 ) = $ 16 , 000 .80(20,000)=\$16,000
  4. $ 20 , 000 - $ 6 , 750 = $ 13 , 250 \$20,000-\$6,750=\$13,250

Itō's_lemma.html

  1. d X t = μ t d t + σ t d B t , dX_{t}=\mu_{t}\,dt+\sigma_{t}\,dB_{t},
  2. f ( t , x ) f(t,x)
  3. d f = f t d t + f x d x + 1 2 2 f x 2 d x 2 + . df=\frac{\partial f}{\partial t}\,dt+\frac{\partial f}{\partial x}\,dx+\frac{1% }{2}\frac{\partial^{2}f}{\partial x^{2}}\,dx^{2}+\cdots.
  4. x x
  5. d f = f t d t + f x ( μ t d t + σ t d B t ) + 1 2 2 f x 2 ( μ t 2 d t 2 + 2 μ t σ t d t d B t + σ t 2 d B t 2 ) + . df=\frac{\partial f}{\partial t}\,dt+\frac{\partial f}{\partial x}(\mu_{t}\,dt% +\sigma_{t}\,dB_{t})+\frac{1}{2}\frac{\partial^{2}f}{\partial x^{2}}\left(\mu_% {t}^{2}\,dt^{2}+2\mu_{t}\sigma_{t}\,dt\,dB_{t}+\sigma_{t}^{2}\,dB_{t}^{2}% \right)+\cdots.
  6. d t 0 dt→0
  7. O ( d t ) O(dt)
  8. d t dt
  9. d t dt
  10. d B dB
  11. d f = ( f t + μ t f x + σ t 2 2 2 f x 2 ) d t + σ t f x d B t df=\left(\frac{\partial f}{\partial t}+\mu_{t}\frac{\partial f}{\partial x}+% \frac{\sigma_{t}^{2}}{2}\frac{\partial^{2}f}{\partial x^{2}}\right)dt+\sigma_{% t}\frac{\partial f}{\partial x}\,dB_{t}
  12. d X t = μ t d t + σ t d B t dX_{t}=\mu_{t}\,dt+\sigma_{t}\,dB_{t}
  13. f ( t , x ) f(t,x)
  14. t t
  15. x x
  16. d f ( t , X t ) = ( f t + μ t f x + σ t 2 2 2 f x 2 ) d t + σ t f x d B t . df(t,X_{t})=\left(\frac{\partial f}{\partial t}+\mu_{t}\frac{\partial f}{% \partial x}+\frac{\sigma_{t}^{2}}{2}\frac{\partial^{2}f}{\partial x^{2}}\right% )dt+\sigma_{t}\frac{\partial f}{\partial x}\,dB_{t}.
  17. 𝐗 t = ( X t 1 , X t 2 , , X t n ) T \mathbf{X}_{t}=(X^{1}_{t},X^{2}_{t},\ldots,X^{n}_{t})^{T}
  18. d 𝐗 t = s y m b o l μ t d t + 𝐆 t d 𝐁 t d\mathbf{X}_{t}=symbol{\mu}_{t}\,dt+\mathbf{G}_{t}\,d\mathbf{B}_{t}
  19. s y m b o l μ t symbol{\mu}_{t}
  20. 𝐆 t \mathbf{G}_{t}
  21. d f ( t , 𝐗 t ) = f t d t + ( 𝐗 f ) T d 𝐗 t + 1 2 ( d 𝐗 t ) T ( H 𝐗 f ) d 𝐗 t , = { f t + ( 𝐗 f ) T s y m b o l μ t + 1 2 Tr [ 𝐆 t T ( H 𝐗 f ) 𝐆 t ] } d t + ( 𝐗 f ) T 𝐆 t d 𝐁 t \begin{aligned}\displaystyle df(t,\mathbf{X}_{t})&\displaystyle=\frac{\partial f% }{\partial t}\,dt+\left(\nabla_{\mathbf{X}}f\right)^{T}\,d\mathbf{X}_{t}+\frac% {1}{2}\left(d\mathbf{X}_{t}\right)^{T}\left(H_{\mathbf{X}}f\right)\,d\mathbf{X% }_{t},\\ &\displaystyle=\left\{\frac{\partial f}{\partial t}+\left(\nabla_{\mathbf{X}}f% \right)^{T}symbol{\mu}_{t}+\frac{1}{2}\,\text{Tr}\left[\mathbf{G}_{t}^{T}\left% (H_{\mathbf{X}}f\right)\mathbf{G}_{t}\right]\right\}dt+\left(\nabla_{\mathbf{X% }}f\right)^{T}\mathbf{G}_{t}\,d\mathbf{B}_{t}\end{aligned}
  22. f f
  23. X X
  24. f f
  25. X X
  26. T r Tr
  27. h h
  28. t t , t + Δ t tt,t+Δt
  29. h Δ t hΔt
  30. h h
  31. 0 , t 0,t
  32. d p s ( t ) = - p s ( t ) h ( t ) d t . dp_{s}(t)=-p_{s}(t)h(t)\,dt.
  33. p s ( t ) = exp ( - 0 t h ( u ) d u ) . p_{s}(t)=\exp\left(-\int_{0}^{t}h(u)\,du\right).
  34. S ( t ) S(t)
  35. S ( t - ) S(t^{-})
  36. d j S ( t ) d_{j}S(t)
  37. S ( t ) S(t)
  38. d j S ( t ) = lim Δ t 0 ( S ( t + Δ t ) - S ( t - ) ) d_{j}S(t)=\lim_{\Delta t\to 0}(S(t+\Delta t)-S(t^{-}))
  39. η ( S ( t - ) , z ) \eta(S(t^{-}),z)
  40. E [ d j S ( t ) ] = h ( S ( t - ) ) d t z z η ( S ( t - ) , z ) d z . E[d_{j}S(t)]=h(S(t^{-}))\,dt\int_{z}z\eta(S(t^{-}),z)\,dz.
  41. d J S ( t ) dJ_{S}(t)
  42. d J S ( t ) = d j S ( t ) - E [ d j S ( t ) ] = S ( t ) - S ( t - ) - ( h ( S ( t - ) ) z z η ( S ( t - ) , z ) d z ) d t . dJ_{S}(t)=d_{j}S(t)-E[d_{j}S(t)]=S(t)-S(t^{-})-\left(h(S(t^{-}))\int_{z}z\eta% \left(S(t^{-}),z\right)\,dz\right)\,dt.
  43. d j S ( t ) = E [ d j S ( t ) ] + d J S ( t ) = h ( S ( t - ) ) ( z z η ( S ( t - ) , z ) d z ) d t + d J S ( t ) . d_{j}S(t)=E[d_{j}S(t)]+dJ_{S}(t)=h(S(t^{-}))\left(\int_{z}z\eta(S(t^{-}),z)\,% dz\right)dt+dJ_{S}(t).
  44. g ( S ( t ) , t ) g(S(t),t)
  45. d S ( t ) dS(t)
  46. S ( t ) S(t)
  47. Δ s Δs
  48. g ( t ) g(t)
  49. Δ g Δg
  50. Δ g Δg
  51. η g ( ) \eta_{g}()
  52. g ( t - ) g(t^{-})
  53. S ( t - ) S(t^{-})
  54. g g
  55. g ( t ) - g ( t - ) = h ( t ) d t Δ g Δ g η g ( ) d Δ g + d J g ( t ) . g(t)-g(t^{-})=h(t)\,dt\int_{\Delta g}\,\Delta g\eta_{g}(\cdot)\,d\Delta g+dJ_{% g}(t).
  56. S S
  57. g ( S ( t ) , t ) g(S(t),t)
  58. d g ( t ) = ( g t + μ g S + σ 2 2 2 g S 2 + h ( t ) Δ g ( Δ g η g ( ) d Δ g ) ) d t + g S σ d W ( t ) + d J g ( t ) . dg(t)=\left(\frac{\partial g}{\partial t}+\mu\frac{\partial g}{\partial S}+% \frac{\sigma^{2}}{2}\frac{\partial^{2}g}{\partial S^{2}}+h(t)\int_{\Delta g}% \left(\Delta g\eta_{g}(\cdot)\,d{\Delta}g\right)\,\right)dt+\frac{\partial g}{% \partial S}\sigma\,dW(t)+dJ_{g}(t).
  59. d d
  60. t t
  61. d d
  62. f ( X t ) = f ( X 0 ) + i = 1 d 0 t f i ( X s - ) d X s i + 1 2 i , j = 1 d 0 t f i , j ( X s - ) d [ X i , X j ] s + s t ( Δ f ( X s ) - i = 1 d f i ( X s - ) Δ X s i - 1 2 i , j = 1 d f i , j ( X s - ) Δ X s i Δ X s j ) . \begin{aligned}\displaystyle f(X_{t})&\displaystyle=f(X_{0})+\sum_{i=1}^{d}% \int_{0}^{t}f_{i}(X_{s-})\,dX^{i}_{s}+\frac{1}{2}\sum_{i,j=1}^{d}\int_{0}^{t}f% _{i,j}(X_{s-})\,d[X^{i},X^{j}]_{s}\\ &\displaystyle\qquad+\sum_{s\leq t}\left(\Delta f(X_{s})-\sum_{i=1}^{d}f_{i}(X% _{s-})\,\Delta X^{i}_{s}-\frac{1}{2}\sum_{i,j=1}^{d}f_{i,j}(X_{s-})\,\Delta X^% {i}_{s}\,\Delta X^{j}_{s}\right).\end{aligned}
  63. t t
  64. d S = S ( σ d B + μ d t ) dS=S(σdB+μdt)
  65. d log ( S ) = f ( S ) d S + 1 2 f ′′ ( S ) S 2 σ 2 d t = 1 S ( σ S d B + μ S d t ) - 1 2 σ 2 d t = σ d B + ( μ - σ 2 2 ) d t . \begin{aligned}\displaystyle d\log(S)&\displaystyle=f^{\prime}(S)\,dS+\frac{1}% {2}f^{\prime\prime}(S)S^{2}\sigma^{2}\,dt\\ &\displaystyle=\frac{1}{S}\left(\sigma S\,dB+\mu S\,dt\right)-\frac{1}{2}% \sigma^{2}\,dt\\ &\displaystyle=\sigma\,dB+\left(\mu-\tfrac{\sigma^{2}}{2}\right)\,dt.\end{aligned}
  66. log ( S t ) = log ( S 0 ) + σ B t + ( μ - σ 2 2 ) t , \log(S_{t})=\log(S_{0})+\sigma B_{t}+\left(\mu-\tfrac{\sigma^{2}}{2}\right)t,
  67. S t = S 0 exp ( σ B t + ( μ - σ 2 2 ) t ) . S_{t}=S_{0}\exp\left(\sigma B_{t}+\left(\mu-\tfrac{\sigma^{2}}{2}\right)t% \right).
  68. d Y = Y d X dY=YdX
  69. [ u u n i c o d e , u 190 ] ( X ) [u^{\prime}unicode^{\prime},u^{\prime}\u{0}190^{\prime}](X)
  70. d log ( Y ) = 1 Y d Y - 1 2 Y 2 d [ Y ] = d X - 1 2 d [ X ] . \begin{aligned}\displaystyle d\log(Y)&\displaystyle=\frac{1}{Y}\,dY-\frac{1}{2% Y^{2}}\,d[Y]\\ &\displaystyle=dX-\tfrac{1}{2}\,d[X].\end{aligned}
  71. Y t = exp ( X t - X 0 - 1 2 [ X ] t ) . Y_{t}=\exp\left(X_{t}-X_{0}-\tfrac{1}{2}[X]_{t}\right).
  72. d S = S ( σ d B + μ d t ) dS=S(σdB+μdt)
  73. t t
  74. d f ( t , S t ) = ( f t + 1 2 ( S t σ ) 2 2 f S 2 ) d t + f S d S t . df(t,S_{t})=\left(\frac{\partial f}{\partial t}+\frac{1}{2}\left(S_{t}\sigma% \right)^{2}\frac{\partial^{2}f}{\partial S^{2}}\right)\,dt+\frac{\partial f}{% \partial S}\,dS_{t}.
  75. f d S ∂\frac{f}{∂}dS
  76. f ∂\frac{f}{∂}
  77. d V t = r ( V t - f S S t ) d t + f S d S t . dV_{t}=r\left(V_{t}-\frac{\partial f}{\partial S}S_{t}\right)\,dt+\frac{% \partial f}{\partial S}\,dS_{t}.
  78. f t + σ 2 S 2 2 2 f S 2 + r S f S - r f = 0. \frac{\partial f}{\partial t}+\frac{\sigma^{2}S^{2}}{2}\frac{\partial^{2}f}{% \partial S^{2}}+rS\frac{\partial f}{\partial S}-rf=0.

Î.html

  1. 𝐬𝐲𝐦𝐛𝐨𝐥 ı ^ \mathbf{\hat{symbol{\imath}}}

Jacobi_identity.html

  1. a × ( b × c ) + c × ( a × b ) + b × ( c × a ) = 0 a , b , c S . a\times(b\times c)+c\times(a\times b)+b\times(c\times a)=0\quad\forall{a,b,c}% \in S.
  2. [ A , [ B , C ] ] + [ B , [ C , A ] ] + [ C , [ A , B ] ] = 0 [A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0\,
  3. [ [ A , B ] , C ] = [ A , [ B , C ] ] - [ B , [ A , C ] ] . [[A,B],C]=[A,[B,C]]-[B,[A,C]]~{}.
  4. B B
  5. A , ( A A , B B , A,(AA,BB,⋅
  6. A A
  7. B , ( B B , A A , B,(BB,AA,⋅
  8. A , B A , B A,BA,B
  9. A , B A,B
  10. C C
  11. [ x , [ y , z ] ] + [ z , [ x , y ] ] + [ y , [ z , x ] ] = 0. [x,[y,z]]+[z,[x,y]]+[y,[z,x]]=0.
  12. ad x : y [ x , y ] , \operatorname{ad}_{x}:y\mapsto[x,y],
  13. ad x [ y , z ] = [ ad x y , z ] + [ y , ad x z ] . \operatorname{ad}_{x}[y,z]=[\operatorname{ad}_{x}y,z]+[y,\operatorname{ad}_{x}% z].
  14. ad [ x , y ] = [ ad x , ad y ] . \operatorname{ad}_{[x,y]}=[\operatorname{ad}_{x},\operatorname{ad}_{y}].

Jacobi_symbol.html

  1. a a
  2. n n
  3. n n
  4. ( a n ) = ( a p 1 ) α 1 ( a p 2 ) α 2 ( a p k ) α k where n = p 1 α 1 p 2 α 2 p k α k . \Bigg(\frac{a}{n}\Bigg)=\left(\frac{a}{p_{1}}\right)^{\alpha_{1}}\left(\frac{a% }{p_{2}}\right)^{\alpha_{2}}\cdots\left(\frac{a}{p_{k}}\right)^{\alpha_{k}}% \mbox{ where }~{}n=p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}\cdots p_{k}^{\alpha_{k% }}.
  5. ( a p ) \left(\tfrac{a}{p}\right)
  6. a a
  7. p p
  8. ( a p ) = { 0 if a 0 ( mod p ) , 1 if a 0 ( mod p ) and for some integer x : a x 2 ( mod p ) , - 1 if a 0 ( mod p ) and there is no such x . \left(\frac{a}{p}\right)=\left\{\begin{array}[]{rl}0&\,\text{if }a\equiv 0\;\;% (\mathop{{\rm mod}}p),\\ 1&\,\text{if }a\not\equiv 0\;\;(\mathop{{\rm mod}}p)\,\text{ and for some % integer }x:\;a\equiv x^{2}\;\;(\mathop{{\rm mod}}p),\\ -1&\,\text{if }a\not\equiv 0\;\;(\mathop{{\rm mod}}p)\,\text{ and there is no % such }x.\end{array}\right.
  9. ( a 1 ) = 1. \left(\tfrac{a}{1}\right)=1.
  10. n n
  11. ( a n ) \left(\frac{a}{n}\right)
  12. a b ( mod n ) a\equiv b\;\;(\mathop{{\rm mod}}n)
  13. ( a n ) = ( b n ) . \left(\frac{a}{n}\right)=\left(\frac{b}{n}\right).
  14. ( a n ) = { 0 if gcd ( a , n ) 1 , ± 1 if gcd ( a , n ) = 1. \left(\frac{a}{n}\right)=\begin{cases}\;\;\,0\mbox{ if }~{}\gcd(a,n)\neq 1,\\ \pm 1\mbox{ if }~{}\gcd(a,n)=1.\end{cases}
  15. ( a b n ) = ( a n ) ( b n ) \left(\frac{ab}{n}\right)=\left(\frac{a}{n}\right)\left(\frac{b}{n}\right)
  16. ( a 2 n ) = 1 or 0. \left(\frac{a^{2}}{n}\right)=1\,\text{ or }0.
  17. ( a m n ) = ( a m ) ( a n ) \left(\frac{a}{mn}\right)=\left(\frac{a}{m}\right)\left(\frac{a}{n}\right)
  18. ( a n 2 ) = 1 or 0. \left(\frac{a}{n^{2}}\right)=1\,\text{ or }0.
  19. m m
  20. n n
  21. ( m n ) ( n m ) = ( - 1 ) m - 1 2 n - 1 2 = { 1 if n 1 ( mod 4 ) or m 1 ( mod 4 ) , - 1 if n m 3 ( mod 4 ) \left(\frac{m}{n}\right)\left(\frac{n}{m}\right)=(-1)^{\tfrac{m-1}{2}\tfrac{n-% 1}{2}}=\left\{\begin{array}[]{rl}1&\,\text{if }n\equiv 1\;\;(\mathop{{\rm mod}% }4)\,\text{ or }m\equiv 1\;\;(\mathop{{\rm mod}}4),\\ -1&\,\text{if }n\equiv m\equiv 3\;\;(\mathop{{\rm mod}}4)\end{array}\right.
  22. ( - 1 n ) = ( - 1 ) n - 1 2 = { 1 if n 1 ( mod 4 ) , - 1 if n 3 ( mod 4 ) , \left(\frac{-1}{n}\right)=(-1)^{\tfrac{n-1}{2}}=\left\{\begin{array}[]{rl}1&\,% \text{if }n\equiv 1\;\;(\mathop{{\rm mod}}4),\\ -1&\,\text{if }n\equiv 3\;\;(\mathop{{\rm mod}}4),\end{array}\right.
  23. ( 2 n ) = ( - 1 ) n 2 - 1 8 = { 1 if n 1 , 7 ( mod 8 ) , - 1 if n 3 , 5 ( mod 8 ) . \left(\frac{2}{n}\right)=(-1)^{\tfrac{n^{2}-1}{8}}=\left\{\begin{array}[]{rl}1% &\,\text{if }n\equiv 1,7\;\;(\mathop{{\rm mod}}8),\\ -1&\,\text{if }n\equiv 3,5\;\;(\mathop{{\rm mod}}8).\end{array}\right.
  24. ( a n ) = - 1 \left(\frac{a}{n}\right)=-1
  25. a a
  26. mod n \bmod{n}
  27. a a
  28. mod n \bmod{n}
  29. gcd ( a , n ) = 1 \gcd(a,n)=1
  30. ( a n ) = 1 \left(\frac{a}{n}\right)=1
  31. ( a n ) = 1 \left(\frac{a}{n}\right)=1
  32. a a
  33. mod n \bmod{n}
  34. a a
  35. mod n \bmod n
  36. n n
  37. a a
  38. n n
  39. ( a n ) (\tfrac{a}{n})
  40. n n
  41. O ( log a log b ) O(\log a\log b)
  42. 1 1
  43. 1 1
  44. 0
  45. ( a p ) (\tfrac{a}{p})
  46. p p
  47. ( - 1 p ) (\tfrac{-1}{p})
  48. ( 2 p ) (\tfrac{2}{p})
  49. 9907 9907
  50. ( 1001 9907 ) . \left(\frac{1001}{9907}\right).
  51. ( 1001 9907 ) = ( 7 9907 ) ( 11 9907 ) ( 13 9907 ) . \left(\frac{1001}{9907}\right)=\left(\frac{7}{9907}\right)\left(\frac{11}{9907% }\right)\left(\frac{13}{9907}\right).
  52. ( 7 9907 ) = - ( 9907 7 ) = - ( 2 7 ) = - 1. \left(\frac{7}{9907}\right)=-\left(\frac{9907}{7}\right)=-\left(\frac{2}{7}% \right)=-1.
  53. ( 11 9907 ) = - ( 9907 11 ) = - ( 7 11 ) = ( 11 7 ) = ( 4 7 ) = 1. \left(\frac{11}{9907}\right)=-\left(\frac{9907}{11}\right)=-\left(\frac{7}{11}% \right)=\left(\frac{11}{7}\right)=\left(\frac{4}{7}\right)=1.
  54. ( 13 9907 ) = ( 9907 13 ) = ( 1 13 ) = 1. \left(\frac{13}{9907}\right)=\left(\frac{9907}{13}\right)=\left(\frac{1}{13}% \right)=1.
  55. ( 1001 9907 ) = - 1. \left(\frac{1001}{9907}\right)=-1.
  56. ( 1001 9907 ) = ( 9907 1001 ) = ( 898 1001 ) = ( 2 1001 ) ( 449 1001 ) = ( 449 1001 ) \left(\frac{1001}{9907}\right)=\left(\frac{9907}{1001}\right)=\left(\frac{898}% {1001}\right)=\left(\frac{2}{1001}\right)\left(\frac{449}{1001}\right)=\left(% \frac{449}{1001}\right)
  57. = ( 1001 449 ) = ( 103 449 ) = ( 449 103 ) = ( 37 103 ) = ( 103 37 ) =\left(\frac{1001}{449}\right)=\left(\frac{103}{449}\right)=\left(\frac{449}{1% 03}\right)=\left(\frac{37}{103}\right)=\left(\frac{103}{37}\right)
  58. = ( 29 37 ) = ( 37 29 ) = ( 8 29 ) = ( 2 29 ) 3 = - 1. =\left(\frac{29}{37}\right)=\left(\frac{37}{29}\right)=\left(\frac{8}{29}% \right)=\left(\frac{2}{29}\right)^{3}=-1.
  59. - 1 -1
  60. 1 1
  61. ( 19 45 ) = 1 and 19 ( 45 - 1 ) / 2 1 ( mod 45 ) . \left(\frac{19}{45}\right)=1\,\,\text{ and }\,19^{(45-1)/2}\equiv 1\;\;(% \mathop{{\rm mod}}45).
  62. ( 8 21 ) = - 1 but 8 ( 21 - 1 ) / 2 1 ( mod 21 ) . \left(\frac{8}{21}\right)=-1\,\,\text{ but }\,8^{(21-1)/2}\equiv 1\;\;(\mathop% {{\rm mod}}21).
  63. ( 5 21 ) = 1 but 5 ( 21 - 1 ) / 2 16 ( mod 21 ) . \left(\frac{5}{21}\right)=1\,\,\text{ but }\,5^{(21-1)/2}\equiv 16\;\;(\mathop% {{\rm mod}}21).
  64. n n
  65. a a
  66. ( a n ) (\tfrac{a}{n})
  67. n n
  68. n n
  69. n n
  70. a a
  71. n n
  72. O ( e ( ln N ) 1 / 3 ( ln ln N ) 2 / 3 ( C + o ( 1 ) ) ) O\left(e^{(\ln N)^{1/3}(\ln\ln N)^{2/3}(C+o(1))}\right)
  73. N N

Jacobian_matrix_and_determinant.html

  1. 𝐉 \mathbf{J}
  2. 𝐟 \mathbf{f}
  3. m × n m×n
  4. 𝐉 = d 𝐟 d 𝐱 = [ 𝐟 x 1 𝐟 x n ] = [ f 1 x 1 f 1 x n f m x 1 f m x n ] \mathbf{J}=\frac{d\mathbf{f}}{d\mathbf{x}}=\begin{bmatrix}\dfrac{\partial% \mathbf{f}}{\partial x_{1}}&\cdots&\dfrac{\partial\mathbf{f}}{\partial x_{n}}% \end{bmatrix}=\begin{bmatrix}\dfrac{\partial f_{1}}{\partial x_{1}}&\cdots&% \dfrac{\partial f_{1}}{\partial x_{n}}\\ \vdots&\ddots&\vdots\\ \dfrac{\partial f_{m}}{\partial x_{1}}&\cdots&\dfrac{\partial f_{m}}{\partial x% _{n}}\end{bmatrix}
  5. 𝐉 i , j = f i x j . \mathbf{J}_{i,j}=\frac{\partial f_{i}}{\partial x_{j}}.
  6. 𝐱 \mathbf{x}
  7. D 𝐟 D\mathbf{f}
  8. 𝐟 \mathbf{f}
  9. 𝐱 \mathbf{x}
  10. 𝐟 \mathbf{f}
  11. 𝐱 \mathbf{x}
  12. 𝐟 \mathbf{f}
  13. 𝐱 \mathbf{x}
  14. m m
  15. n n
  16. 𝐟 \mathbf{f}
  17. 𝐟 \mathbf{f}
  18. 𝐟 \mathbf{f}
  19. 𝐱 \mathbf{x}
  20. 𝐱 \mathbf{x}
  21. m m
  22. 𝐟 \mathbf{f}
  23. 𝐟 \mathbf{f}
  24. 𝐟 \mathbf{f}
  25. ( x , y ) = 𝐟 ( x , y ) (x′,y′)=\mathbf{f}(x,y)
  26. ( x , y ) (x,y)
  27. 𝐩 \mathbf{p}
  28. 𝐟 \mathbf{f}
  29. 𝐩 \mathbf{p}
  30. 𝐟 \mathbf{f}
  31. 𝐩 \mathbf{p}
  32. 𝐟 ( 𝐱 ) = 𝐟 ( 𝐩 ) + 𝐉 𝐟 ( 𝐩 ) ( 𝐱 - 𝐩 ) + o ( 𝐱 - 𝐩 ) \mathbf{f}(\mathbf{x})=\mathbf{f}(\mathbf{p})+\mathbf{J}_{\mathbf{f}}(\mathbf{% p})(\mathbf{x}-\mathbf{p})+o(\|\mathbf{x}-\mathbf{p}\|)
  33. 𝐱 \mathbf{x}
  34. 𝐩 \mathbf{p}
  35. o o
  36. 𝐱 𝐩 \mathbf{x}→\mathbf{p}
  37. 𝐱 𝐩 ‖\mathbf{x}−\mathbf{p}‖
  38. 𝐱 \mathbf{x}
  39. 𝐩 \mathbf{p}
  40. f ( x ) = f ( p ) + f ( p ) ( x - p ) + o ( x - p ) . f(x)=f(p)+f^{\prime}(p)(x-p)+o(x-p).
  41. m m
  42. n n
  43. 𝐟 \mathbf{f}
  44. 𝐟 \mathbf{f}
  45. 𝐟 \mathbf{f}
  46. 𝐩 \mathbf{p}
  47. 𝐩 \mathbf{p}
  48. 𝐟 \mathbf{f}
  49. 𝐩 \mathbf{p}
  50. 𝐟 \mathbf{f}
  51. 𝐩 \mathbf{p}
  52. 𝐟 \mathbf{f}
  53. 𝐩 \mathbf{p}
  54. n n
  55. d V dV
  56. n n
  57. 𝐩 \mathbf{p}
  58. 𝐟 \mathbf{f}
  59. 𝐩 \mathbf{p}
  60. 𝐉 𝐟 - 1 𝐟 = 𝐉 𝐟 - 1 . \mathbf{J}_{\mathbf{f}^{-1}}\circ\mathbf{f}={\mathbf{J}_{\mathbf{f}}}^{-1}.
  61. 𝐟 \mathbf{f}
  62. k k
  63. 𝐟 \mathbf{f}
  64. k k
  65. 𝐟 \mathbf{f}
  66. m m
  67. n n
  68. k k
  69. 𝐟 ( x , y ) = [ x 2 y 5 x + sin y ] . \mathbf{f}(x,y)=\begin{bmatrix}x^{2}y\\ 5x+\sin y\end{bmatrix}.
  70. f 1 ( x , y ) = x 2 y f_{1}(x,y)=x^{2}y
  71. f 2 ( x , y ) = 5 x + sin y f_{2}(x,y)=5x+\sin y
  72. 𝐅 \mathbf{F}
  73. 𝐉 𝐟 ( x , y ) = [ f 1 x f 1 y f 2 x f 2 y ] = [ 2 x y x 2 5 cos y ] \mathbf{J}_{\mathbf{f}}(x,y)=\begin{bmatrix}\dfrac{\partial f_{1}}{\partial x}% &\dfrac{\partial f_{1}}{\partial y}\\ \dfrac{\partial f_{2}}{\partial x}&\dfrac{\partial f_{2}}{\partial y}\end{% bmatrix}=\begin{bmatrix}2xy&x^{2}\\ 5&\cos y\end{bmatrix}
  74. det ( 𝐉 𝐟 ( x , y ) ) = 2 x y cos y - 5 x 2 . \det(\mathbf{J}_{\mathbf{f}}(x,y))=2xy\cos y-5x^{2}.
  75. ( r , φ ) (r,φ)
  76. x \displaystyle x
  77. 𝐉 ( r , φ ) = [ x r x φ y r y φ ] = [ cos φ - r sin φ sin φ r cos φ ] \mathbf{J}(r,\varphi)=\begin{bmatrix}\dfrac{\partial x}{\partial r}&\dfrac{% \partial x}{\partial\varphi}\\ \dfrac{\partial y}{\partial r}&\dfrac{\partial y}{\partial\varphi}\end{bmatrix% }=\begin{bmatrix}\cos\varphi&-r\sin\varphi\\ \sin\varphi&r\cos\varphi\end{bmatrix}
  78. r r
  79. A f ( x , y ) d x d y = A f ( r cos ϕ , r sin ϕ ) r d r d ϕ . \iint_{A}f(x,y)\,dx\,dy=\iint_{A}f(r\cos\phi,r\sin\phi)\,r\,dr\,d\phi.
  80. ( r , θ , φ ) (r,θ,φ)
  81. x \displaystyle x
  82. 𝐉 𝐅 ( r , θ , φ ) = [ x r x θ x φ y r y θ y φ z r z θ z φ ] = [ sin θ cos φ r cos θ cos φ - r sin θ sin φ sin θ sin φ r cos θ sin φ r sin θ cos φ cos θ - r sin θ 0 ] . \mathbf{J}_{\mathbf{F}}(r,\theta,\varphi)=\begin{bmatrix}\dfrac{\partial x}{% \partial r}&\dfrac{\partial x}{\partial\theta}&\dfrac{\partial x}{\partial% \varphi}\\ \dfrac{\partial y}{\partial r}&\dfrac{\partial y}{\partial\theta}&\dfrac{% \partial y}{\partial\varphi}\\ \dfrac{\partial z}{\partial r}&\dfrac{\partial z}{\partial\theta}&\dfrac{% \partial z}{\partial\varphi}\end{bmatrix}=\begin{bmatrix}\sin\theta\cos\varphi% &r\cos\theta\cos\varphi&-r\sin\theta\sin\varphi\\ \sin\theta\sin\varphi&r\cos\theta\sin\varphi&r\sin\theta\cos\varphi\\ \cos\theta&-r\sin\theta&0\end{bmatrix}.
  83. y 1 = x 1 y 2 = 5 x 3 y 3 = 4 x 2 2 - 2 x 3 y 4 = x 3 sin x 1 \begin{aligned}\displaystyle y_{1}&\displaystyle=x_{1}\\ \displaystyle y_{2}&\displaystyle=5x_{3}\\ \displaystyle y_{3}&\displaystyle=4x_{2}^{2}-2x_{3}\\ \displaystyle y_{4}&\displaystyle=x_{3}\sin x_{1}\end{aligned}
  84. 𝐉 𝐅 ( x 1 , x 2 , x 3 ) = [ y 1 x 1 y 1 x 2 y 1 x 3 y 2 x 1 y 2 x 2 y 2 x 3 y 3 x 1 y 3 x 2 y 3 x 3 y 4 x 1 y 4 x 2 y 4 x 3 ] = [ 1 0 0 0 0 5 0 8 x 2 - 2 x 3 cos x 1 0 sin x 1 ] . \mathbf{J}_{\mathbf{F}}(x_{1},x_{2},x_{3})=\begin{bmatrix}\dfrac{\partial y_{1% }}{\partial x_{1}}&\dfrac{\partial y_{1}}{\partial x_{2}}&\dfrac{\partial y_{1% }}{\partial x_{3}}\\ \dfrac{\partial y_{2}}{\partial x_{1}}&\dfrac{\partial y_{2}}{\partial x_{2}}&% \dfrac{\partial y_{2}}{\partial x_{3}}\\ \dfrac{\partial y_{3}}{\partial x_{1}}&\dfrac{\partial y_{3}}{\partial x_{2}}&% \dfrac{\partial y_{3}}{\partial x_{3}}\\ \dfrac{\partial y_{4}}{\partial x_{1}}&\dfrac{\partial y_{4}}{\partial x_{2}}&% \dfrac{\partial y_{4}}{\partial x_{3}}\end{bmatrix}=\begin{bmatrix}1&0&0\\ 0&0&5\\ 0&8x_{2}&-2\\ x_{3}\cos x_{1}&0&\sin x_{1}\end{bmatrix}.
  85. y 1 = 5 x 2 y 2 = 4 x 1 2 - 2 sin ( x 2 x 3 ) y 3 = x 2 x 3 \begin{aligned}\displaystyle y_{1}&\displaystyle=5x_{2}\\ \displaystyle y_{2}&\displaystyle=4x_{1}^{2}-2\sin(x_{2}x_{3})\\ \displaystyle y_{3}&\displaystyle=x_{2}x_{3}\end{aligned}
  86. | 0 5 0 8 x 1 - 2 x 3 cos ( x 2 x 3 ) - 2 x 2 cos ( x 2 x 3 ) 0 x 3 x 2 | = - 8 x 1 | 5 0 x 3 x 2 | = - 40 x 1 x 2 . \begin{vmatrix}0&5&0\\ 8x_{1}&-2x_{3}\cos(x_{2}x_{3})&-2x_{2}\cos(x_{2}x_{3})\\ 0&x_{3}&x_{2}\end{vmatrix}=-8x_{1}\begin{vmatrix}5&0\\ x_{3}&x_{2}\end{vmatrix}=-40x_{1}x_{2}.
  87. 𝐅 \mathbf{F}
  88. x < s u b > 1 x<sub>1

Jacques_Cazotte.html

  1. 1 1 1 / < m a t h > < m a t h > ⊢\frac{1}{\frac{1}{{1}/{<math><math>}}}

Jacques_Hadamard.html

  1. ζ ( s ) \zeta(s)

James_Gregory_(mathematician).html

  1. θ = tan θ - ( 1 / 3 ) tan 3 θ + ( 1 / 5 ) tan 5 θ - \theta=\tan\theta-(1/3)\tan^{3}\theta+(1/5)\tan^{5}\theta-\ldots

Jansky.html

  1. S S
  2. B B
  3. S = source B ( θ , ϕ ) d Ω S=\iint_{\mathrm{source}}B(\theta,\phi)\mathrm{d}\Omega
  4. 1 Jy = 10 - 26 W m 2 Hz 1\ \mathrm{Jy}=10^{-26}\frac{\mathrm{W}}{\mathrm{m^{2}}\cdot\mathrm{Hz}}
  5. = 10 - 23 erg s cm 2 Hz =10^{-23}\frac{\mathrm{erg}}{\mathrm{s}\cdot\mathrm{cm^{2}}\cdot\mathrm{Hz}}
  6. S v [ μ Jy ] = 10 6 10 23 10 - ( AB + 48.6 ) / 2.5 = 10 ( 23.9 - AB ) / 2.5 S_{v}\ [\mathrm{\mu Jy}]=10^{6}\cdot 10^{23}\cdot 10^{-(\mathrm{AB}+48.6)/2.5}% =10^{(23.9-\mathrm{AB})/2.5}
  7. P d B W / m 2 / H z = 10 log 10 ( P J y ) - 260 P_{dBW/m^{2}/Hz}=10\log_{10}(P_{Jy})-260
  8. P d B m / m 2 / H z = 10 log 10 ( P J y ) - 230 P_{dBm/m^{2}/Hz}=10\log_{10}(P_{Jy})-230

Jensen's_inequality.html

  1. t f ( x 1 ) + ( 1 - t ) f ( x 2 ) , tf(x_{1})+(1-t)f(x_{2}),
  2. f ( t x 1 + ( 1 - t ) x 2 ) . f\left(tx_{1}+(1-t)x_{2}\right).
  3. φ φ
  4. φ ( 𝔼 [ X ] ) 𝔼 [ φ ( X ) ] . \varphi\left(\mathbb{E}[X]\right)\leq\mathbb{E}\left[\varphi(X)\right].
  5. φ φ
  6. φ ( a i x i a i ) a i φ ( x i ) a i ( 1 ) \varphi\left(\frac{\sum a_{i}x_{i}}{\sum a_{i}}\right)\leq\frac{\sum a_{i}% \varphi(x_{i})}{\sum a_{i}}\qquad\qquad(1)
  7. φ φ
  8. φ ( a i x i a i ) a i φ ( x i ) a i . ( 2 ) \varphi\left(\frac{\sum a_{i}x_{i}}{\sum a_{i}}\right)\geq\frac{\sum a_{i}% \varphi(x_{i})}{\sum a_{i}}.\qquad\qquad(2)
  9. x 1 = x 2 = = x n x_{1}=x_{2}=\cdots=x_{n}
  10. φ φ
  11. φ ( x i n ) φ ( x i ) n ( 3 ) \varphi\left(\frac{\sum x_{i}}{n}\right)\leq\frac{\sum\varphi(x_{i})}{n}\qquad% \qquad(3)
  12. φ ( x i n ) φ ( x i ) n ( 4 ) \varphi\left(\frac{\sum x_{i}}{n}\right)\geq\frac{\sum\varphi(x_{i})}{n}\qquad% \qquad(4)
  13. l o g ( x ) log(x)
  14. φ ( x ) = l o g ( x ) φ(x)=log(x)
  15. x 1 + x 2 + + x n n x 1 x 2 x n n . or log ( i = 1 n x i n ) i = 1 n log ( x i ) n \frac{x_{1}+x_{2}+\cdots+x_{n}}{n}\geq\sqrt[n]{x_{1}\cdot x_{2}\cdots x_{n}}.% \quad\,\text{or}\quad\log\!\left(\frac{\sum_{i=1}^{n}x_{i}}{n}\right)\geq\frac% {\sum_{i=1}^{n}\log\!\left(x_{i}\right)}{n}
  16. f ( x ) f(x)
  17. ( Ω , A , μ ) \scriptstyle(\Omega,A,\mu)
  18. μ ( Ω ) = 1 μ(Ω)=1
  19. φ φ
  20. φ ( Ω g d μ ) Ω φ g d μ . \varphi\left(\int_{\Omega}g\,d\mu\right)\leq\int_{\Omega}\varphi\circ g\,d\mu.
  21. φ ( a b f ( x ) d x ) , \varphi\left(\int_{a}^{b}f(x)\,dx\right),
  22. a , b 𝐑 a,b∈\mathbf{R}
  23. f : a a , b 𝐑 f:aa,b→\mathbf{R}
  24. a a , b aa,b
  25. φ ( a b f ( x ) d x ) 1 b - a a b φ ( ( b - a ) f ( x ) ) d x . \varphi\left(\int_{a}^{b}f(x)\,dx\right)\leq\frac{1}{b-a}\int_{a}^{b}\varphi((% b-a)f(x))\,dx.
  26. ( Ω , 𝔉 , ) \scriptstyle(\Omega,\mathfrak{F},\mathbb{P})
  27. φ φ
  28. φ ( 𝔼 [ X ] ) 𝔼 [ φ ( X ) ] . \varphi\left(\mathbb{E}[X]\right)\leq\mathbb{E}\left[\varphi(X)\right].
  29. μ μ
  30. \scriptstyle\mathbb{P}
  31. μ μ
  32. 𝔼 \scriptstyle\mathbb{E}
  33. φ φ
  34. 𝔼 [ X ] \scriptstyle\mathbb{E}[X]
  35. 𝔼 | z , X | < \scriptstyle\mathbb{E}|\langle z,X\rangle|<\infty
  36. z , 𝔼 [ X ] = 𝔼 [ z , X ] \scriptstyle\langle z,\mathbb{E}[X]\rangle=\mathbb{E}[\langle z,X\rangle]
  37. φ φ
  38. 𝔊 \scriptstyle\mathfrak{G}
  39. 𝔉 \scriptstyle\mathfrak{F}
  40. φ ( 𝔼 [ X | 𝔊 ] ) 𝔼 [ φ ( X ) | 𝔊 ] . \varphi\left(\mathbb{E}\left[X|\mathfrak{G}\right]\right)\leq\mathbb{E}\left[% \varphi(X)|\mathfrak{G}\right].
  41. 𝔼 [ | 𝔊 ] \scriptstyle\mathbb{E}[\cdot|\mathfrak{G}]
  42. 𝔊 \scriptstyle\mathfrak{G}
  43. T T
  44. 𝔊 \scriptstyle\mathfrak{G}
  45. σ σ
  46. X X
  47. X X
  48. Y Y
  49. Y Y
  50. Y ( X ) Y(X)
  51. X X
  52. X X
  53. X X
  54. 𝔼 [ X ] \scriptstyle\mathbb{E}[X]
  55. φ ( 𝔼 [ X ] ) \scriptstyle\varphi(\mathbb{E}[X])
  56. Y = φ ( X ) Y=φ(X)
  57. Y Y
  58. X X
  59. Y Y
  60. X 0 = 𝔼 [ X ] \scriptstyle X_{0}=\mathbb{E}[X]
  61. φ ( 𝔼 [ X ] ) \scriptstyle\varphi(\mathbb{E}[X])
  62. X X
  63. 𝔼 [ Y ] = 𝔼 [ φ ( X ) ] φ ( 𝔼 [ X ] ) , \mathbb{E}[Y]=\mathbb{E}[\varphi(X)]\geq\varphi(\mathbb{E}[X]),
  64. φ ( X ) φ(X)
  65. X X
  66. φ φ
  67. x 1 , x 2 : φ ( λ 1 x 1 + λ 2 x 2 ) λ 1 φ ( x 1 ) + λ 2 φ ( x 2 ) . \forall x_{1},x_{2}:\qquad\varphi\left(\lambda_{1}x_{1}+\lambda_{2}x_{2}\right% )\leq\lambda_{1}\,\varphi(x_{1})+\lambda_{2}\,\varphi(x_{2}).
  68. φ ( λ 1 x 1 + λ 2 x 2 + + λ n x n ) λ 1 φ ( x 1 ) + λ 2 φ ( x 2 ) + + λ n φ ( x n ) , \varphi(\lambda_{1}x_{1}+\lambda_{2}x_{2}+\cdots+\lambda_{n}x_{n})\leq\lambda_% {1}\,\varphi(x_{1})+\lambda_{2}\,\varphi(x_{2})+\cdots+\lambda_{n}\,\varphi(x_% {n}),
  69. n + 1 n+1
  70. φ ( i = 1 n + 1 λ i x i ) \displaystyle\varphi\left(\sum_{i=1}^{n+1}\lambda_{i}x_{i}\right)
  71. i = 2 n + 1 λ i 1 - λ 1 = 1 , \sum_{i=2}^{n+1}\frac{\lambda_{i}}{1-\lambda_{1}}=1,
  72. φ ( x d μ n ( x ) ) φ ( x ) d μ n ( x ) , \varphi\left(\int x\,d\mu_{n}(x)\right)\leq\int\varphi(x)\,d\mu_{n}(x),
  73. μ n = i = 1 n λ i δ x i . \mu_{n}=\sum_{i=1}^{n}\lambda_{i}\delta_{x_{i}}.
  74. φ φ
  75. φ φ
  76. x x
  77. φ φ
  78. x x
  79. φ φ
  80. x 0 := Ω g d μ , x_{0}:=\int_{\Omega}g\,d\mu,
  81. a x + b φ ( x ) , ax+b\leq\varphi(x),
  82. x x
  83. a x 0 + b = φ ( x 0 ) . ax_{0}+b=\varphi(x_{0}).
  84. φ g ( x ) a g ( x ) + b \varphi\circ g(x)\geq ag(x)+b
  85. x x
  86. μ ( Ω ) = 1 μ(Ω)=1
  87. Ω φ g d μ Ω ( a g + b ) d μ = a Ω g d μ + b Ω d μ = a x 0 + b = φ ( x 0 ) = φ ( Ω g d μ ) , \int_{\Omega}\varphi\circ g\,d\mu\geq\int_{\Omega}(ag+b)\,d\mu=a\int_{\Omega}g% \,d\mu+b\int_{\Omega}d\mu=ax_{0}+b=\varphi(x_{0})=\varphi\left(\int_{\Omega}g% \,d\mu\right),
  88. φ : T 𝐑 φ:T→\mathbf{R}
  89. x , y T x,y\in T
  90. φ ( x + θ y ) - φ ( x ) θ , \frac{\varphi(x+\theta\,y)-\varphi(x)}{\theta},
  91. θ θ
  92. φ φ
  93. x x
  94. y y
  95. ( D φ ) ( x ) y := lim θ 0 φ ( x + θ y ) - φ ( x ) θ = inf θ 0 φ ( x + θ y ) - φ ( x ) θ . (D\varphi)(x)\cdot y:=\lim_{\theta\downarrow 0}\frac{\varphi(x+\theta\,y)-% \varphi(x)}{\theta}=\inf_{\theta\neq 0}\frac{\varphi(x+\theta\,y)-\varphi(x)}{% \theta}.
  96. y y
  97. θ = 1 θ=1
  98. φ ( x ) φ ( x + y ) - ( D φ ) ( x ) y . \varphi(x)\leq\varphi(x+y)-(D\varphi)(x)\cdot y.
  99. σ σ
  100. 𝔊 \scriptstyle\mathfrak{G}
  101. x = 𝔼 [ X | 𝔊 ] , y = X - 𝔼 [ X | 𝔊 ] \scriptstyle x=\mathbb{E}[X|\mathfrak{G}],\,y=X-\mathbb{E}[X|\mathfrak{G}]
  102. φ ( 𝔼 [ X | 𝔊 ] ) φ ( X ) - ( D φ ) ( 𝔼 [ X | 𝔊 ] ) ( X - 𝔼 [ X | 𝔊 ] ) . \varphi(\mathbb{E}[X|\mathfrak{G}])\leq\varphi(X)-(D\varphi)(\mathbb{E}[X|% \mathfrak{G}])\cdot(X-\mathbb{E}[X|\mathfrak{G}]).
  103. 𝔊 \scriptstyle\mathfrak{G}
  104. 𝔼 [ [ ( D φ ) ( 𝔼 [ X | 𝔊 ] ) ( X - 𝔼 [ X | 𝔊 ] ) ] | 𝔊 ] = ( D φ ) ( 𝔼 [ X | 𝔊 ] ) 𝔼 [ ( X - 𝔼 [ X | 𝔊 ] ) | 𝔊 ] = 0 , \mathbb{E}\left[\left[(D\varphi)(\mathbb{E}[X|\mathfrak{G}])\cdot(X-\mathbb{E}% [X|\mathfrak{G}])\right]|\mathfrak{G}\right]=(D\varphi)(\mathbb{E}[X|\mathfrak% {G}])\cdot\mathbb{E}[\left(X-\mathbb{E}[X|\mathfrak{G}]\right)|\mathfrak{G}]=0,
  105. 𝔼 [ ( 𝔼 [ X | 𝔊 ] ) | 𝔊 ] = 𝔼 [ X | 𝔊 ] . \mathbb{E}\left[\left(\mathbb{E}[X|\mathfrak{G}]\right)|\mathfrak{G}\right]=% \mathbb{E}[X|\mathfrak{G}].
  106. Ω Ω
  107. - f ( x ) d x = 1. \int_{-\infty}^{\infty}f(x)\,dx=1.
  108. φ φ
  109. φ ( - g ( x ) f ( x ) d x ) - φ ( g ( x ) ) f ( x ) d x . \varphi\left(\int_{-\infty}^{\infty}g(x)f(x)\,dx\right)\leq\int_{-\infty}^{% \infty}\varphi(g(x))f(x)\,dx.
  110. φ ( - x f ( x ) d x ) - φ ( x ) f ( x ) d x . \varphi\left(\int_{-\infty}^{\infty}x\,f(x)\,dx\right)\leq\int_{-\infty}^{% \infty}\varphi(x)\,f(x)\,dx.
  111. μ μ
  112. Ω Ω
  113. φ ( i = 1 n g ( x i ) λ i ) i = 1 n φ ( g ( x i ) ) λ i , \varphi\left(\sum_{i=1}^{n}g(x_{i})\lambda_{i}\right)\leq\sum_{i=1}^{n}\varphi% (g(x_{i}))\lambda_{i},
  114. λ 1 + + λ n = 1. \lambda_{1}+\cdots+\lambda_{n}=1.
  115. e 𝔼 [ X ] 𝔼 [ e X ] , e^{\mathbb{E}[X]}\leq\mathbb{E}\left[e^{X}\right],
  116. X X
  117. 𝔼 [ e X ] = e 𝔼 [ X ] 𝔼 [ e X - 𝔼 [ X ] ] \mathbb{E}\left[e^{X}\right]=e^{\mathbb{E}[X]}\mathbb{E}\left[e^{X-\mathbb{E}[% X]}\right]
  118. p ( x ) p(x)
  119. x x
  120. q ( x ) q(x)
  121. φ ( y ) = l o g ( y ) φ(y)=−log(y)
  122. 𝔼 [ φ ( Y ) ] φ ( 𝔼 [ Y ] ) \mathbb{E}[\varphi(Y)]\geq\varphi(\mathbb{E}[Y])
  123. - D ( p ( x ) q ( x ) ) = p ( x ) log ( q ( x ) p ( x ) ) d x log ( p ( x ) q ( x ) p ( x ) d x ) = log ( q ( x ) d x ) = 0 -D(p(x)\|q(x))=\int p(x)\log\left(\frac{q(x)}{p(x)}\right)\,dx\leq\log\left(% \int p(x)\frac{q(x)}{p(x)}\,dx\right)=\log\left(\int q(x)\,dx\right)=0
  124. l o g ( x ) −log(x)
  125. x > 0 x>0
  126. p ( x ) p(x)
  127. q ( x ) q(x)
  128. L ( 𝔼 [ δ ( X ) ] ) 𝔼 [ L ( δ ( X ) ) ] 𝔼 [ L ( 𝔼 [ δ ( X ) ] ) ] 𝔼 [ L ( δ ( X ) ) ] . L(\mathbb{E}[\delta(X)])\leq\mathbb{E}[L(\delta(X))]\quad\Rightarrow\quad% \mathbb{E}[L(\mathbb{E}[\delta(X)])]\leq\mathbb{E}[L(\delta(X))].
  129. δ 1 ( X ) = 𝔼 θ [ δ ( X ) | T ( X ) = T ( X ) ] , \delta_{1}(X)=\mathbb{E}_{\theta}[\delta(X^{\prime})\,|\,T(X^{\prime})=T(X)],

Jet_aircraft.html

  1. η \eta
  2. η = η c η p \eta=\eta_{c}\eta_{p}
  3. η c \eta_{c}
  4. η p \eta_{p}
  5. η p = 2 1 + c v \eta_{p}=\frac{2}{1+\frac{c}{v}}
  6. V = a M V=aM
  7. M M
  8. a a
  9. R = a M c T C L C D l n W 1 W 2 R=\frac{aM}{c_{T}}\frac{C_{L}}{C_{D}}ln\frac{W_{1}}{W_{2}}

John_Machin.html

  1. π 4 = 4 arctan 1 5 - arctan 1 239 \frac{\pi}{4}=4\arctan\frac{1}{5}-\arctan\frac{1}{239}

John_Smeaton.html

  1. L = k V 2 A C l L=kV^{2}AC_{l}\,
  2. L L
  3. k k
  4. C l C_{l}
  5. A A

John_Wallis.html

  1. {\infty}
  2. 1 x \frac{1}{x}
  3. x 0 = 1 x^{0}=1
  4. x - 1 = 1 x x^{-1}=\frac{1}{x}
  5. x - n = 1 x n etc. x^{-n}=\frac{1}{x^{n}}\,\text{ etc.}
  6. x 1 / 2 = x x^{1/2}=\sqrt{x}
  7. x 2 / 3 = x 2 3 etc. x^{2/3}=\sqrt[3]{x^{2}}\,\text{ etc.}
  8. x 1 / n = x n x^{1/n}=\sqrt[n]{x}
  9. x p / q = x p q x^{p/q}=\sqrt[q]{x^{p}}
  10. y = m a x m y=\sum_{m}ax^{m}
  11. 0 1 x 1 / m d x . \int_{0}^{1}x^{1/m}\,dx.
  12. y = 1 - x 2 y=\sqrt{1-x^{2}}
  13. y = 1 - x 2 y=\sqrt{1-x^{2}}
  14. y = ( 1 - x 2 ) 0 y=(1-x^{2})^{0}
  15. y = ( 1 - x 2 ) 1 y=(1-x^{2})^{1}
  16. 0 1 1 - x 2 d x \int_{0}^{1}\sqrt{1-x^{2}}\,dx
  17. 1 4 π \begin{matrix}\frac{1}{4}\end{matrix}\pi
  18. 0 1 ( 1 - x 2 ) 0 d x and 0 1 ( 1 - x 2 ) 1 d x \int_{0}^{1}(1-x^{2})^{0}\,dx\,\text{ and }\int_{0}^{1}(1-x^{2})^{1}\,dx
  19. 2 3 \begin{matrix}\frac{2}{3}\end{matrix}
  20. 4 2 3 4\sqrt{\begin{matrix}\frac{2}{3}\end{matrix}}
  21. 1 , 1 6 , 1 30 , 1 140 , 1,\begin{matrix}\frac{1}{6}\end{matrix},\begin{matrix}\frac{1}{30}\end{matrix}% ,\begin{matrix}\frac{1}{140}\end{matrix},
  22. 1 6 \begin{matrix}\frac{1}{6}\end{matrix}
  23. π 2 = 2 1 2 3 4 3 4 5 6 5 6 7 \frac{\pi}{2}=\frac{2}{1}\cdot\frac{2}{3}\cdot\frac{4}{3}\cdot\frac{4}{5}\cdot% \frac{6}{5}\cdot\frac{6}{7}\cdots

Johnson–Nyquist_noise.html

  1. v n 2 ¯ = 4 k B T R \overline{v_{n}^{2}}=4k\text{B}TR
  2. v n 2 ¯ = 0.13 R nV / Hz . \sqrt{\overline{v_{n}^{2}}}=0.13\sqrt{R}~{}\mathrm{nV}/\sqrt{\mathrm{Hz}}.
  3. v n 2 ¯ = 4 1.38 10 - 23 J / K 300 K 1 k Ω = 4.07 nV / Hz . \sqrt{\overline{v_{n}^{2}}}=\sqrt{4\cdot 1.38\cdot 10^{-23}~{}\mathrm{J}/% \mathrm{K}\cdot 300~{}\mathrm{K}\cdot 1~{}\mathrm{k}\Omega}=4.07~{}\mathrm{nV}% /\sqrt{\mathrm{Hz}}.
  4. v n v_{n}
  5. v n = v n 2 ¯ Δ f = 4 k B T R Δ f v_{n}=\sqrt{\overline{v_{n}^{2}}}\sqrt{\Delta f}=\sqrt{4k\text{B}TR\Delta f}
  6. P = v n 2 / R = 4 k B T Δ f . P={v_{n}^{2}}/R=4k\text{B}\,T\Delta f.
  7. P = k B T Δ f P=k\text{B}\,T\Delta f
  8. i n = 4 k B T Δ f R . i_{n}=\sqrt{{4k\text{B}T\Delta f}\over R}.
  9. P dBm = 10 log 10 ( k B T Δ f × 1000 ) P_{\mathrm{dBm}}=10\ \log_{10}(k\text{B}T\Delta f\times 1000)
  10. P dBm = 10 log 10 ( k B T × 1000 ) + 10 log 10 ( Δ f ) P_{\mathrm{dBm}}=10\ \log_{10}(k\text{B}T\times 1000)+10\ \log_{10}(\Delta f)
  11. P dBm = - 174 + 10 log 10 ( Δ f ) P_{\mathrm{dBm}}=-174+10\ \log_{10}(\Delta f)
  12. Δ f \Delta f
  13. Δ f \Delta f
  14. ( Δ f ) (\Delta f)
  15. v n 2 ¯ = k B T / C \overline{v_{n}^{2}}=k\text{B}T/C
  16. v n = k B T / C . v_{n}=\sqrt{k\text{B}T/C}.
  17. Q n = k B T C . Q_{n}=\sqrt{k\text{B}TC}.
  18. k B T C k\text{B}TC
  19. k B T / C \sqrt{k\text{B}T/C}
  20. 4 k B T R 4k\text{B}TR
  21. Z ( f ) Z(f)
  22. S v n v n ( f ) = 4 k B T η ( f ) Re [ Z ( f ) ] . S_{v_{n}v_{n}}(f)=4k\text{B}T\eta(f)\operatorname{Re}[Z(f)].
  23. η ( f ) \eta(f)
  24. Re [ Z ( f ) ] \operatorname{Re}[Z(f)]
  25. f 1 f_{1}
  26. f 2 f_{2}
  27. v n 2 = f 1 f 2 S v n v n ( f ) d f \sqrt{\langle v_{n}^{2}\rangle}=\sqrt{\int_{f_{1}}^{f_{2}}S_{v_{n}v_{n}}(f)df}
  28. S i n i n ( f ) = 4 k B T η ( f ) Re [ Y ( f ) ] . S_{i_{n}i_{n}}(f)=4k\text{B}T\eta(f)\operatorname{Re}[Y(f)].
  29. Y ( f ) = 1 / Z ( f ) Y(f)=1/Z(f)
  30. Re [ Y ( f ) ] = Re [ Z ( f ) ] / | Z ( f ) | 2 \operatorname{Re}[Y(f)]=\operatorname{Re}[Z(f)]/|Z(f)|^{2}
  31. η ( f ) \eta(f)
  32. η ( f ) = h f / k B T e h f / k B T - 1 , \eta(f)=\frac{hf/k\text{B}T}{e^{hf/k\text{B}T}-1},
  33. h h
  34. f k B T / h f\gtrsim k\text{B}T/h
  35. η ( f ) \eta(f)
  36. η ( f ) = 1 \eta(f)=1
  37. S v m v n ( f ) = 2 k B T η ( f ) ( Z m n ( f ) + Z n m ( f ) * ) S_{v_{m}v_{n}}(f)=2k\text{B}T\eta(f)(Z_{mn}(f)+Z_{nm}(f)^{*})
  38. Z m n Z_{mn}
  39. 𝐙 \mathbf{Z}
  40. S i m i n ( f ) = 2 k B T η ( f ) ( Y m n ( f ) + Y n m ( f ) * ) S_{i_{m}i_{n}}(f)=2k\text{B}T\eta(f)(Y_{mn}(f)+Y_{nm}(f)^{*})
  41. 𝐘 = 𝐙 - 1 \mathbf{Y}=\mathbf{Z}^{-1}

Jordan_curve_theorem.html

  1. H ~ q ( Y ) = { , q = n - k 0 , otherwise . \tilde{H}_{q}(Y)=\begin{cases}\mathbb{Z},\quad q=n-k\\ 0,\quad\,\text{otherwise}.\end{cases}

Jordan_normal_form.html

  1. A = [ 5 4 2 1 0 1 - 1 - 1 - 1 - 1 3 0 1 1 - 1 2 ] . A=\left[\!\!\!\begin{array}[]{*{20}{r}}5&4&2&1\\ 0&1&-1&-1\\ -1&-1&3&0\\ 1&1&-1&2\end{array}\!\!\right].
  2. J = [ 1 0 0 0 0 2 0 0 0 0 4 1 0 0 0 4 ] . J=\begin{bmatrix}1&0&0&0\\ 0&2&0&0\\ 0&0&4&1\\ 0&0&0&4\end{bmatrix}.
  3. J = [ J 1 J p ] J=\begin{bmatrix}J_{1}&&\\ &\ddots&\\ &&J_{p}\end{bmatrix}
  4. J i = [ λ i 1 λ i 1 λ i ] . J_{i}=\begin{bmatrix}\lambda_{i}&1&&\\ &\lambda_{i}&\ddots&\\ &&\ddots&1\\ &&&\lambda_{i}\end{bmatrix}.
  5. 2 dim ker ( A - λ i I ) j - dim ker ( A - λ i I ) j + 1 - dim ker ( A - λ i I ) j - 1 2\dim\ker(A-\lambda_{i}I)^{j}-\dim\ker(A-\lambda_{i}I)^{j+1}-\dim\ker(A-% \lambda_{i}I)^{j-1}
  6. A P = P J . \;AP=PJ.
  7. A [ p 1 p 2 p 3 p 4 ] = [ p 1 p 2 p 3 p 4 ] [ 1 0 0 0 0 2 0 0 0 0 4 1 0 0 0 4 ] = [ p 1 2 p 2 4 p 3 p 3 + 4 p 4 ] . A\begin{bmatrix}p_{1}&p_{2}&p_{3}&p_{4}\end{bmatrix}=\begin{bmatrix}p_{1}&p_{2% }&p_{3}&p_{4}\end{bmatrix}\begin{bmatrix}1&0&0&0\\ 0&2&0&0\\ 0&0&4&1\\ 0&0&0&4\end{bmatrix}=\begin{bmatrix}p_{1}&2p_{2}&4p_{3}&p_{3}+4p_{4}\end{% bmatrix}.
  8. ( A - 1 I ) p 1 = 0 \;(A-1I)p_{1}=0
  9. ( A - 2 I ) p 2 = 0 \;(A-2I)p_{2}=0
  10. ( A - 4 I ) p 3 = 0 \;(A-4I)p_{3}=0
  11. ( A - 4 I ) p 4 = p 3 . \;(A-4I)p_{4}=p_{3}.
  12. p i Ker ( A - λ i I ) p_{i}\in\operatorname{Ker}(A-\lambda_{i}I)
  13. ( A - 4 I ) (A-4I)
  14. ( A - 4 I ) 2 p 4 = ( A - 4 I ) p 3 . \;(A-4I)^{2}p_{4}=(A-4I)p_{3}.
  15. ( A - 4 I ) p 3 = 0 (A-4I)p_{3}=0
  16. ( A - 4 I ) 2 p 4 = 0. \;(A-4I)^{2}p_{4}=0.
  17. p 4 Ker ( A - 4 I ) 2 . p_{4}\in\operatorname{Ker}(A-4I)^{2}.
  18. p 4 p_{4}
  19. Ran ( A - λ I ) Ker ( A - λ I ) = { 0 } , \mathrm{Ran}(A-\lambda I)\cap\mathrm{Ker}(A-\lambda I)=\{0\},
  20. Q = Ran ( A - λ I ) Ker ( A - λ I ) { 0 } , Q=\mathrm{Ran}(A-\lambda I)\cap\mathrm{Ker}(A-\lambda I)\neq\{0\},
  21. ( A - λ I ) q i = p i for i = r - s + 1 , , r . \;(A-\lambda I)q_{i}=p_{i}\mbox{ for }~{}i=r-s+1,\ldots,r.
  22. Ker ( A - λ I ) / Q . \;\mathrm{Ker}(A-\lambda I)/Q.
  23. ( A - λ I ) k 1 = 0 (A-\lambda I)^{k_{1}}=0
  24. ( A - λ I ) k 1 - 1 (A-\lambda I)^{k_{1}-1}
  25. ( A - λ I ) k 1 - 2 (A-\lambda I)^{k_{1}-2}
  26. λ i \lambda_{i}
  27. λ i = a i + i b i \lambda_{i}=a_{i}+ib_{i}
  28. C i = [ a i b i - b i a i ] C_{i}=\begin{bmatrix}a_{i}&b_{i}\\ -b_{i}&a_{i}\\ \end{bmatrix}
  29. λ i \lambda_{i}
  30. J i = [ C i I C i I C i ] . J_{i}=\begin{bmatrix}C_{i}&I&&\\ &C_{i}&\ddots&\\ &&\ddots&I\\ &&&C_{i}\\ \end{bmatrix}.
  31. p p
  32. A A
  33. p ( A ) = 0 p(A)=0
  34. λ λ
  35. m m
  36. λ λ
  37. p p
  38. λ λ
  39. p p
  40. p ( A ) p(A)
  41. Σ Σ
  42. n = i = 1 k X i \mathbb{C}^{n}=\bigoplus_{i=1}^{k}X_{i}
  43. Y i = Ker ( λ i I - A ) ν ( λ i ) . \;Y_{i}=\operatorname{Ker}(\lambda_{i}I-A)^{\nu(\lambda_{i})}.
  44. n = i = 1 l Y i \mathbb{C}^{n}=\bigoplus_{i=1}^{l}Y_{i}
  45. Ker ( λ - A ) ν ( λ ) = Ker ( λ - A ) m , m ν ( λ ) . \mathrm{Ker}(\lambda-A)^{\nu(\lambda)}=\operatorname{Ker}(\lambda-A)^{m},\;% \forall m\geq\nu(\lambda).
  46. f ( T ) = 1 2 π i Γ f ( z ) ( z - T ) - 1 d z . f(T)=\frac{1}{2\pi i}\int_{\Gamma}f(z)(z-T)^{-1}dz.
  47. Φ ( f ) = f ( T ) . \;\Phi(f)=f(T).
  48. e i ( T ) \;e_{i}(T)
  49. f ( z ) = ( z - λ i ) ν i . f(z)=(z-\lambda_{i})^{\nu_{i}}.
  50. f ( T ) e i ( T ) = ( T - λ i ) ν i e i ( T ) f(T)e_{i}(T)=(T-\lambda_{i})^{\nu_{i}}e_{i}(T)
  51. Ran e i ( T ) = Ker ( T - λ i ) ν i . \mathrm{Ran}\;e_{i}(T)=\mathrm{Ker}(T-\lambda_{i})^{\nu_{i}}.
  52. i e i = 1 \;\sum_{i}e_{i}=1
  53. n = i Ran e i ( T ) = i Ker ( T - λ i ) ν i \mathbb{C}^{n}=\bigoplus_{i}\;\mathrm{Ran}\;e_{i}(T)=\bigoplus_{i}\;\mathrm{% Ker}(T-\lambda_{i})^{\nu_{i}}
  54. n = i Y i \mathbb{C}^{n}=\bigoplus_{i}Y_{i}
  55. f ( T ) = λ i σ ( T ) k = 0 ν i - 1 f ( k ) k ! ( T - λ i ) k e i ( T ) . f(T)=\sum_{\lambda_{i}\in\sigma(T)}\sum_{k=0}^{\nu_{i}-1}\frac{f^{(k)}}{k!}(T-% \lambda_{i})^{k}e_{i}(T).
  56. R T ( λ ) = ( λ - T ) - 1 \;R_{T}(\lambda)=(\lambda-T)^{-1}
  57. R T ( z ) = - a m ( λ - z ) m R_{T}(z)=\sum_{-\infty}^{\infty}a_{m}(\lambda-z)^{m}
  58. a - m = - 1 2 π i C ( λ - z ) m - 1 ( z - T ) - 1 d z a_{-m}=-\frac{1}{2\pi i}\int_{C}(\lambda-z)^{m-1}(z-T)^{-1}dz
  59. a - m = - ( λ - T ) m - 1 e λ ( T ) \;a_{-m}=-(\lambda-T)^{m-1}e_{\lambda}(T)
  60. e λ \;e_{\lambda}
  61. B ϵ ( λ ) \;B_{\epsilon}(\lambda)
  62. a - m 0 a_{-m}\neq 0
  63. a - l = 0 l m a_{-l}=0\;\;\forall\;l\geq m
  64. A = [ 5 4 2 1 0 1 - 1 - 1 - 1 - 1 3 0 1 1 - 1 2 ] A=\begin{bmatrix}5&4&2&1\\ 0&1&-1&-1\\ -1&-1&3&0\\ 1&1&-1&2\end{bmatrix}
  65. χ ( λ ) = det ( λ I - A ) = λ 4 - 11 λ 3 + 42 λ 2 - 64 λ + 32 = ( λ - 1 ) ( λ - 2 ) ( λ - 4 ) 2 . \chi(\lambda)=\det(\lambda I-A)=\lambda^{4}-11\lambda^{3}+42\lambda^{2}-64% \lambda+32=(\lambda-1)(\lambda-2)(\lambda-4)^{2}.\,
  66. J = J 1 ( 1 ) J 1 ( 2 ) J 2 ( 4 ) = [ 1 0 0 0 0 2 0 0 0 0 4 1 0 0 0 4 ] . J=J_{1}(1)\oplus J_{1}(2)\oplus J_{2}(4)=\begin{bmatrix}1&0&0&0\\ 0&2&0&0\\ 0&0&4&1\\ 0&0&0&4\end{bmatrix}.
  67. ker ( A - 4 I ) 2 = span { [ 1 0 0 0 ] , [ 1 0 - 1 1 ] } . \ker{(A-4I)}^{2}=\operatorname{span}\,\left\{\begin{bmatrix}1\\ 0\\ 0\\ 0\end{bmatrix},\begin{bmatrix}1\\ 0\\ -1\\ 1\end{bmatrix}\right\}.
  68. P = [ v | w | x | y ] = [ - 1 1 1 1 1 - 1 0 0 0 0 - 1 0 0 1 1 0 ] . P=\Big[\,v\,\Big|\,w\,\Big|\,x\,\Big|\,y\,\Big]=\begin{bmatrix}-1&1&1&1\\ 1&-1&0&0\\ 0&0&-1&0\\ 0&1&1&0\end{bmatrix}.
  69. P - 1 A P = J = [ 1 0 0 0 0 2 0 0 0 0 4 1 0 0 0 4 ] . P^{-1}AP=J=\begin{bmatrix}1&0&0&0\\ 0&2&0&0\\ 0&0&4&1\\ 0&0&0&4\end{bmatrix}.
  70. A = [ 1 1 ε 1 ] . A=\begin{bmatrix}1&1\\ \varepsilon&1\end{bmatrix}.
  71. [ 1 1 0 1 ] . \begin{bmatrix}1&1\\ 0&1\end{bmatrix}.
  72. [ 1 + ε 0 0 1 - ε ] . \begin{bmatrix}1+\sqrt{\varepsilon}&0\\ 0&1-\sqrt{\varepsilon}\end{bmatrix}.
  73. [ 2 1 0 0 0 0 2 1 0 0 0 0 2 0 0 0 0 0 5 1 0 0 0 0 5 ] 4 = [ 16 32 24 0 0 0 16 32 0 0 0 0 16 0 0 0 0 0 625 500 0 0 0 0 625 ] . \begin{bmatrix}2&1&0&0&0\\ 0&2&1&0&0\\ 0&0&2&0&0\\ 0&0&0&5&1\\ 0&0&0&0&5\end{bmatrix}^{4}=\begin{bmatrix}16&32&24&0&0\\ 0&16&32&0&0\\ 0&0&16&0&0\\ 0&0&0&625&500\\ 0&0&0&0&625\end{bmatrix}.
  74. ( n 1 ) {\textstyle\left({{n}\atop{1}}\right)}
  75. ( n k ) ( n | n | ) k ( | n | k ) {\textstyle\left({{n}\atop{k}}\right)}\mapsto\left(\frac{n}{|n|}\right)^{k}{% \textstyle\left({{|n|}\atop{k}}\right)}
  76. [ λ 1 1 0 0 0 0 λ 1 1 0 0 0 0 λ 1 0 0 0 0 0 λ 2 1 0 0 0 0 λ 2 ] n = [ λ 1 n ( n 1 ) λ 1 n - 1 ( n 2 ) λ 1 n - 2 0 0 0 λ 1 n ( n 1 ) λ 1 n - 1 0 0 0 0 λ 1 n 0 0 0 0 0 λ 2 n ( n 1 ) λ 2 n - 1 0 0 0 0 λ 2 n ] . \begin{bmatrix}\lambda_{1}&1&0&0&0\\ 0&\lambda_{1}&1&0&0\\ 0&0&\lambda_{1}&0&0\\ 0&0&0&\lambda_{2}&1\\ 0&0&0&0&\lambda_{2}\end{bmatrix}^{n}=\begin{bmatrix}\lambda_{1}^{n}&{% \textstyle\left({{n}\atop{1}}\right)}\lambda_{1}^{n-1}&{\textstyle\left({{n}% \atop{2}}\right)}\lambda_{1}^{n-2}&0&0\\ 0&\lambda_{1}^{n}&{\textstyle\left({{n}\atop{1}}\right)}\lambda_{1}^{n-1}&0&0% \\ 0&0&\lambda_{1}^{n}&0&0\\ 0&0&0&\lambda_{2}^{n}&{\textstyle\left({{n}\atop{1}}\right)}\lambda_{2}^{n-1}% \\ 0&0&0&0&\lambda_{2}^{n}\end{bmatrix}.

Joseph-Louis_Lagrange.html

  1. y = a sin ( m x ) sin ( n t ) y=a\sin(mx)\sin(nt)\,
  2. x 2 - n y 2 = 1 x^{2}-ny^{2}=1
  3. a x 2 + b y 2 + c x y ax^{2}+by^{2}+cxy
  4. d d t T θ ˙ - T θ + V θ = 0 , \frac{d}{dt}\frac{\partial T}{\partial\dot{\theta}}-\frac{\partial T}{\partial% \theta}+\frac{\partial V}{\partial\theta}=0,
  5. a p - 1 - 1 = 0 ( mod p ) a^{p-1}-1=0\;\;(\mathop{{\rm mod}}p)

Joule_per_mole.html

  1. k B T k_{B}T

Joule–Thomson_effect.html

  1. T T
  2. P P
  3. H H
  4. μ JT \mu_{\mathrm{JT}}
  5. V V
  6. C p C_{\mathrm{p}}
  7. α \alpha
  8. μ JT = ( T P ) H = V C p ( α T - 1 ) \mu_{\mathrm{JT}}=\left({\partial T\over\partial P}\right)_{H}=\frac{V}{C_{% \mathrm{p}}}\left(\alpha T-1\right)\,
  9. μ JT \mu_{\mathrm{JT}}
  10. μ JT \mu_{\mathrm{JT}}
  11. P \partial P
  12. μ JT \mu_{\mathrm{JT}}
  13. P \partial P
  14. T \partial T
  15. μ JT \mu_{\mathrm{JT}}
  16. m P 1 v 1 - m P 2 v 2 . mP_{1}v_{1}-mP_{2}v_{2}.
  17. m u 2 - m u 1 - m P 1 v 1 + m P 2 v 2 = 0 mu_{2}-mu_{1}-mP_{1}v_{1}+mP_{2}v_{2}=0
  18. h 1 = h 2 h_{1}=h_{2}
  19. h d = x d h e + ( 1 - x d ) h f . h_{d}=x_{d}h_{e}+(1-x_{d})h_{f}.
  20. μ JT ( T P ) H = V C p ( α T - 1 ) \mu_{\mathrm{JT}}\equiv\left(\frac{\partial T}{\partial P}\right)_{H}=\frac{V}% {C_{\mathrm{p}}}\left(\alpha T-1\right)\,
  21. d H = T d S + V d P . \mathrm{d}H=T\mathrm{d}S+V\mathrm{d}P.
  22. d H = T ( S T ) P d T + [ V + T ( S P ) T ] d P . \mathrm{d}H=T\left(\frac{\partial S}{\partial T}\right)_{P}\mathrm{d}T+\left[V% +T\left(\frac{\partial S}{\partial P}\right)_{T}\right]\mathrm{d}P.
  23. C p = T ( S T ) P , C_{\mathrm{p}}=T\left(\frac{\partial S}{\partial T}\right)_{P},
  24. d H = C p d T + [ V + T ( S P ) T ] d P . \mathrm{d}H=C_{\mathrm{p}}\mathrm{d}T+\left[V+T\left(\frac{\partial S}{% \partial P}\right)_{T}\right]\mathrm{d}P.
  25. d G = - S d T + V d P . \mathrm{d}G=-S\mathrm{d}T+V\mathrm{d}P.
  26. ( S P ) T = - ( V T ) P = - V α \left(\frac{\partial S}{\partial P}\right)_{T}=-\left(\frac{\partial V}{% \partial T}\right)_{P}=-V\alpha\,
  27. d H = C p d T + V ( 1 - T α ) d P . \mathrm{d}H=C_{\mathrm{p}}\mathrm{d}T+V\left(1-T\alpha\right)\mathrm{d}P.
  28. ( T / P ) H (\partial T/\partial P)_{H}
  29. ( T P ) H = V C p ( α T - 1 ) . \left(\frac{\partial T}{\partial P}\right)_{H}=\frac{V}{C_{\mathrm{p}}}\left(% \alpha T-1\right).

Jules_Richard.html

  1. 0 \aleph_{0}
  2. 0 \aleph_{0}

Julian_Schwinger.html

  1. α 2 π \frac{\alpha}{2\pi}

Ĵ.html

  1. s y m b o l ȷ ^ symbol{\hat{\jmath}}

K-theory.html

  1. ch ( L ) = exp ( c 1 ( L ) ) := m = 0 c 1 ( L ) m m ! . \operatorname{ch}(L)=\exp(c_{1}(L)):=\sum_{m=0}^{\infty}\frac{c_{1}(L)^{m}}{m!}.
  2. V = L 1 L n V=L_{1}\oplus...\oplus L_{n}
  3. x i = c 1 ( L i ) , x_{i}=c_{1}(L_{i}),
  4. ch ( V ) = e x 1 + + e x n := m = 0 1 m ! ( x 1 m + + x n m ) . \operatorname{ch}(V)=e^{x_{1}}+\dots+e^{x_{n}}:=\sum_{m=0}^{\infty}\frac{1}{m!% }(x_{1}^{m}+...+x_{n}^{m}).
  5. Coh G ( X ) \operatorname{Coh}^{G}(X)
  6. K i G ( X ) = π i ( B + Coh G ( X ) ) . K_{i}^{G}(X)=\pi_{i}(B^{+}\operatorname{Coh}^{G}(X)).
  7. K 0 G ( C ) K_{0}^{G}(C)
  8. Coh G ( X ) \operatorname{Coh}^{G}(X)

Kaiser_window.html

  1. w [ n ] = { I 0 ( π α 1 - ( 2 n N - 1 - 1 ) 2 ) I 0 ( π α ) , 0 n N - 1 0 otherwise , w[n]=\left\{\begin{matrix}\frac{I_{0}\left(\pi\alpha\sqrt{1-\left(\frac{2n}{N-% 1}-1\right)^{2}}\right)}{I_{0}(\pi\alpha)},&0\leq n\leq N-1\\ \\ 0&\mbox{otherwise}~{},\\ \end{matrix}\right.
  2. w [ ( N - 1 ) / 2 ] = 1 , \scriptstyle w[(N-1)/2]=1,
  3. w [ N / 2 - 1 ] = w [ N / 2 ] < 1. \scriptstyle w[N/2-1]\ =\ w[N/2]\ <\ 1.
  4. I 0 ( π α 1 - ( 2 t ( N - 1 ) T ) 2 ) I 0 ( π α ) w 0 ( t ) ( N - 1 ) T sinh ( π α 2 - ( ( N - 1 ) T f ) 2 ) I 0 ( π α ) π α 2 - ( ( N - 1 ) T f ) 2 W 0 ( f ) . \underbrace{\frac{I_{0}\left(\pi\alpha\sqrt{1-\left(\frac{2t}{(N-1)T}\right)^{% 2}}\right)}{I_{0}(\pi\alpha)}}_{w_{0}(t)}\quad\stackrel{\mathcal{F}}{% \Longleftrightarrow}\quad\underbrace{\frac{(N-1)T\cdot\sinh\left(\pi\sqrt{% \alpha^{2}-\left((N-1)T\cdot f\right)^{2}}\right)}{I_{0}(\pi\alpha)\cdot\pi% \sqrt{\alpha^{2}-\left((N-1)T\cdot f\right)^{2}}}}_{W_{0}(f)}.
  5. w 0 ( t - ( N - 1 ) T 2 ) rect ( t - ( N - 1 ) T / 2 N T ) , w_{0}\left(t-\tfrac{(N-1)T}{2}\right)\cdot\operatorname{rect}\left(\tfrac{t-(N% -1)T/2}{NT}\right),
  6. f = 1 + α 2 N T , f=\frac{\sqrt{1+\alpha^{2}}}{NT},
  7. 1 + α 2 . \scriptstyle\sqrt{1+\alpha^{2}}.
  8. d n = { i = 0 n w [ i ] i = 0 M w [ i ] if 0 n < M i = 0 2 M - 1 - n w [ i ] i = 0 M w [ i ] if M n < 2 M 0 otherwise . d_{n}=\left\{\begin{matrix}\sqrt{\frac{\sum_{i=0}^{n}w[i]}{\sum_{i=0}^{M}w[i]}% }&\mbox{if }~{}0\leq n<M\\ \\ \sqrt{\frac{\sum_{i=0}^{2M-1-n}w[i]}{\sum_{i=0}^{M}w[i]}}&\mbox{if }~{}M\leq n% <2M\\ \\ 0&\mbox{otherwise}~{}.\\ \end{matrix}\right.

Kalman_filter.html

  1. x ^ k k - 1 \hat{x}_{k\mid k-1}
  2. P k k - 1 P_{k\mid k-1}
  3. 𝐱 k = 𝐅 k 𝐱 k - 1 + 𝐁 k 𝐮 k + 𝐰 k \mathbf{x}_{k}=\mathbf{F}_{k}\mathbf{x}_{k-1}+\mathbf{B}_{k}\mathbf{u}_{k}+% \mathbf{w}_{k}
  4. 𝐰 k N ( 0 , 𝐐 k ) \mathbf{w}_{k}\sim N(0,\mathbf{Q}_{k})
  5. 𝐳 k = 𝐇 k 𝐱 k + 𝐯 k \mathbf{z}_{k}=\mathbf{H}_{k}\mathbf{x}_{k}+\mathbf{v}_{k}
  6. 𝐯 k N ( 0 , 𝐑 k ) \mathbf{v}_{k}\sim N(0,\mathbf{R}_{k})
  7. 𝐱 ^ n m \hat{\mathbf{x}}_{n\mid m}
  8. 𝐱 \mathbf{x}
  9. 𝐱 ^ k k \hat{\mathbf{x}}_{k\mid k}
  10. 𝐏 k k \mathbf{P}_{k\mid k}
  11. 𝐱 ^ k k - 1 = 𝐅 k 𝐱 ^ k - 1 k - 1 + 𝐁 k 𝐮 k \hat{\mathbf{x}}_{k\mid k-1}=\mathbf{F}_{k}\hat{\mathbf{x}}_{k-1\mid k-1}+% \mathbf{B}_{k}\mathbf{u}_{k}
  12. 𝐏 k k - 1 = 𝐅 k 𝐏 k - 1 k - 1 𝐅 k T + 𝐐 k \mathbf{P}_{k\mid k-1}=\mathbf{F}_{k}\mathbf{P}_{k-1\mid k-1}\mathbf{F}_{k}^{% \,\text{T}}+\mathbf{Q}_{k}
  13. 𝐲 ~ k = 𝐳 k - 𝐇 k 𝐱 ^ k k - 1 \tilde{\mathbf{y}}_{k}=\mathbf{z}_{k}-\mathbf{H}_{k}\hat{\mathbf{x}}_{k\mid k-1}
  14. 𝐒 k = 𝐇 k 𝐏 k k - 1 𝐇 k T + 𝐑 k \mathbf{S}_{k}=\mathbf{H}_{k}\mathbf{P}_{k\mid k-1}\mathbf{H}_{k}^{T}+\mathbf{% R}_{k}
  15. 𝐊 k = 𝐏 k k - 1 𝐇 k T 𝐒 k - 1 \mathbf{K}_{k}=\mathbf{P}_{k\mid k-1}\mathbf{H}_{k}^{T}\mathbf{S}_{k}^{-1}
  16. 𝐱 ^ k k = 𝐱 ^ k k - 1 + 𝐊 k 𝐲 ~ k \hat{\mathbf{x}}_{k\mid k}=\hat{\mathbf{x}}_{k\mid k-1}+\mathbf{K}_{k}\tilde{% \mathbf{y}}_{k}
  17. 𝐏 k | k = ( I - 𝐊 k 𝐇 k ) 𝐏 k | k - 1 \mathbf{P}_{k|k}=(I-\mathbf{K}_{k}\mathbf{H}_{k})\mathbf{P}_{k|k-1}
  18. 𝐱 ^ 0 0 \hat{\mathbf{x}}_{0\mid 0}
  19. 𝐏 0 0 \mathbf{P}_{0\mid 0}
  20. 𝐄 [ 𝐱 k - 𝐱 ^ k k ] = E [ 𝐱 k - 𝐱 ^ k k - 1 ] = 0 \mathbf{E}[\mathbf{x}_{k}-\hat{\mathbf{x}}_{k\mid k}]=\textrm{E}[\mathbf{x}_{k% }-\hat{\mathbf{x}}_{k\mid k-1}]=0
  21. E [ 𝐲 ~ k ] = 0 \textrm{E}[\tilde{\mathbf{y}}_{k}]=0
  22. E [ ξ ] \textrm{E}[\xi]
  23. ξ \xi
  24. 𝐏 k k = cov ( 𝐱 k - 𝐱 ^ k k ) \mathbf{P}_{k\mid k}=\mathrm{cov}(\mathbf{x}_{k}-\hat{\mathbf{x}}_{k\mid k})
  25. 𝐏 k k - 1 = cov ( 𝐱 k - 𝐱 ^ k k - 1 ) \mathbf{P}_{k\mid k-1}=\mathrm{cov}(\mathbf{x}_{k}-\hat{\mathbf{x}}_{k\mid k-1})
  26. 𝐒 k = cov ( 𝐲 ~ k ) \mathbf{S}_{k}=\mathrm{cov}(\tilde{\mathbf{y}}_{k})
  27. 𝐅 , 𝐇 , 𝐑 , 𝐐 \mathbf{F},\mathbf{H},\mathbf{R},\mathbf{Q}
  28. 𝐱 k = [ x x ˙ ] \mathbf{x}_{k}=\begin{bmatrix}x\\ \dot{x}\end{bmatrix}
  29. x ˙ \dot{x}
  30. 𝐱 k = 𝐅𝐱 k - 1 + 𝐆 a k \mathbf{x}_{k}=\mathbf{F}\mathbf{x}_{k-1}+\mathbf{G}a_{k}
  31. 𝐁 u \mathbf{B}u
  32. 𝐆 \mathbf{G}
  33. 𝐅 = [ 1 Δ t 0 1 ] \mathbf{F}=\begin{bmatrix}1&\Delta t\\ 0&1\end{bmatrix}
  34. 𝐆 = [ Δ t 2 2 Δ t ] \mathbf{G}=\begin{bmatrix}\frac{\Delta t^{2}}{2}\\ \Delta t\end{bmatrix}
  35. 𝐱 k = 𝐅𝐱 k - 1 + 𝐰 k \mathbf{x}_{k}=\mathbf{F}\mathbf{x}_{k-1}+\mathbf{w}_{k}
  36. 𝐰 k N ( 0 , 𝐐 ) \mathbf{w}_{k}\sim N(0,\mathbf{Q})
  37. 𝐐 = 𝐆𝐆 T σ a 2 = [ Δ t 4 4 Δ t 3 2 Δ t 3 2 Δ t 2 ] σ a 2 . \mathbf{Q}=\mathbf{G}\mathbf{G}^{\,\text{T}}\sigma_{a}^{2}=\begin{bmatrix}% \frac{\Delta t^{4}}{4}&\frac{\Delta t^{3}}{2}\\ \frac{\Delta t^{3}}{2}&\Delta t^{2}\end{bmatrix}\sigma_{a}^{2}.
  38. 𝐳 k = 𝐇𝐱 k + 𝐯 k \mathbf{z}_{k}=\mathbf{Hx}_{k}+\mathbf{v}_{k}
  39. 𝐇 = [ 1 0 ] \mathbf{H}=\begin{bmatrix}1&0\end{bmatrix}
  40. 𝐑 = E [ 𝐯 k 𝐯 k T ] = [ σ z 2 ] \mathbf{R}=\textrm{E}[\mathbf{v}_{k}\mathbf{v}_{k}^{\,\text{T}}]=\begin{% bmatrix}\sigma_{z}^{2}\end{bmatrix}
  41. 𝐱 ^ 0 0 = [ 0 0 ] \hat{\mathbf{x}}_{0\mid 0}=\begin{bmatrix}0\\ 0\end{bmatrix}
  42. 𝐏 0 0 = [ 0 0 0 0 ] \mathbf{P}_{0\mid 0}=\begin{bmatrix}0&0\\ 0&0\end{bmatrix}
  43. 𝐏 0 0 = [ L 0 0 L ] \mathbf{P}_{0\mid 0}=\begin{bmatrix}L&0\\ 0&L\end{bmatrix}
  44. 𝐏 k k = cov ( 𝐱 k - 𝐱 ^ k k ) \mathbf{P}_{k\mid k}=\mathrm{cov}(\mathbf{x}_{k}-\hat{\mathbf{x}}_{k\mid k})
  45. 𝐱 ^ k k \hat{\mathbf{x}}_{k\mid k}
  46. 𝐏 k k = cov ( 𝐱 k - ( 𝐱 ^ k k - 1 + 𝐊 k 𝐲 ~ k ) ) \mathbf{P}_{k\mid k}=\textrm{cov}(\mathbf{x}_{k}-(\hat{\mathbf{x}}_{k\mid k-1}% +\mathbf{K}_{k}\tilde{\mathbf{y}}_{k}))
  47. 𝐲 ~ k \tilde{\mathbf{y}}_{k}
  48. 𝐏 k k = cov ( 𝐱 k - ( 𝐱 ^ k k - 1 + 𝐊 k ( 𝐳 k - 𝐇 k 𝐱 ^ k k - 1 ) ) ) \mathbf{P}_{k\mid k}=\textrm{cov}(\mathbf{x}_{k}-(\hat{\mathbf{x}}_{k\mid k-1}% +\mathbf{K}_{k}(\mathbf{z}_{k}-\mathbf{H}_{k}\hat{\mathbf{x}}_{k\mid k-1})))
  49. 𝐳 k \mathbf{z}_{k}
  50. 𝐏 k k = cov ( 𝐱 k - ( 𝐱 ^ k k - 1 + 𝐊 k ( 𝐇 k 𝐱 k + 𝐯 k - 𝐇 k 𝐱 ^ k k - 1 ) ) ) \mathbf{P}_{k\mid k}=\textrm{cov}(\mathbf{x}_{k}-(\hat{\mathbf{x}}_{k\mid k-1}% +\mathbf{K}_{k}(\mathbf{H}_{k}\mathbf{x}_{k}+\mathbf{v}_{k}-\mathbf{H}_{k}\hat% {\mathbf{x}}_{k\mid k-1})))
  51. 𝐏 k | k = cov ( ( I - 𝐊 k 𝐇 k ) ( 𝐱 k - 𝐱 ^ k k - 1 ) - 𝐊 k 𝐯 k ) \mathbf{P}_{k|k}=\textrm{cov}((I-\mathbf{K}_{k}\mathbf{H}_{k})(\mathbf{x}_{k}-% \hat{\mathbf{x}}_{k\mid k-1})-\mathbf{K}_{k}\mathbf{v}_{k})
  52. 𝐏 k | k = cov ( ( I - 𝐊 k 𝐇 k ) ( 𝐱 k - 𝐱 ^ k k - 1 ) ) + cov ( 𝐊 k 𝐯 k ) \mathbf{P}_{k|k}=\textrm{cov}((I-\mathbf{K}_{k}\mathbf{H}_{k})(\mathbf{x}_{k}-% \hat{\mathbf{x}}_{k\mid k-1}))+\textrm{cov}(\mathbf{K}_{k}\mathbf{v}_{k})
  53. 𝐏 k k = ( I - 𝐊 k 𝐇 k ) cov ( 𝐱 k - 𝐱 ^ k k - 1 ) ( I - 𝐊 k 𝐇 k ) T + 𝐊 k cov ( 𝐯 k ) 𝐊 k T \mathbf{P}_{k\mid k}=(I-\mathbf{K}_{k}\mathbf{H}_{k})\textrm{cov}(\mathbf{x}_{% k}-\hat{\mathbf{x}}_{k\mid k-1})(I-\mathbf{K}_{k}\mathbf{H}_{k})^{\,\text{T}}+% \mathbf{K}_{k}\textrm{cov}(\mathbf{v}_{k})\mathbf{K}_{k}^{\,\text{T}}
  54. 𝐏 k k = ( I - 𝐊 k 𝐇 k ) 𝐏 k k - 1 ( I - 𝐊 k 𝐇 k ) T + 𝐊 k 𝐑 k 𝐊 k T \mathbf{P}_{k\mid k}=(I-\mathbf{K}_{k}\mathbf{H}_{k})\mathbf{P}_{k\mid k-1}(I-% \mathbf{K}_{k}\mathbf{H}_{k})\text{T}+\mathbf{K}_{k}\mathbf{R}_{k}\mathbf{K}_{% k}\text{T}
  55. 𝐱 k - 𝐱 ^ k k \mathbf{x}_{k}-\hat{\mathbf{x}}_{k\mid k}
  56. E [ 𝐱 k - 𝐱 ^ k | k 2 ] \textrm{E}[\|\mathbf{x}_{k}-\hat{\mathbf{x}}_{k|k}\|^{2}]
  57. 𝐏 k | k \mathbf{P}_{k|k}
  58. 𝐏 k k \displaystyle\mathbf{P}_{k\mid k}
  59. tr ( 𝐏 k k ) 𝐊 k = - 2 ( 𝐇 k 𝐏 k k - 1 ) T + 2 𝐊 k 𝐒 k = 0. \frac{\partial\;\mathrm{tr}(\mathbf{P}_{k\mid k})}{\partial\;\mathbf{K}_{k}}=-% 2(\mathbf{H}_{k}\mathbf{P}_{k\mid k-1})\text{T}+2\mathbf{K}_{k}\mathbf{S}_{k}=0.
  60. 𝐊 k 𝐒 k = ( 𝐇 k 𝐏 k k - 1 ) T = 𝐏 k k - 1 𝐇 k T \mathbf{K}_{k}\mathbf{S}_{k}=(\mathbf{H}_{k}\mathbf{P}_{k\mid k-1})\text{T}=% \mathbf{P}_{k\mid k-1}\mathbf{H}_{k}\text{T}
  61. 𝐊 k = 𝐏 k k - 1 𝐇 k T 𝐒 k - 1 \mathbf{K}_{k}=\mathbf{P}_{k\mid k-1}\mathbf{H}_{k}\text{T}\mathbf{S}_{k}^{-1}
  62. 𝐊 k 𝐒 k 𝐊 k T = 𝐏 k k - 1 𝐇 k T 𝐊 k T \mathbf{K}_{k}\mathbf{S}_{k}\mathbf{K}_{k}^{\mathrm{T}}=\mathbf{P}_{k\mid k-1}% \mathbf{H}_{k}^{\mathrm{T}}\mathbf{K}_{k}^{\mathrm{T}}
  63. 𝐏 k k = 𝐏 k k - 1 - 𝐊 k 𝐇 k 𝐏 k k - 1 - 𝐏 k k - 1 𝐇 k T 𝐊 k T + 𝐊 k 𝐒 k 𝐊 k T \mathbf{P}_{k\mid k}=\mathbf{P}_{k\mid k-1}-\mathbf{K}_{k}\mathbf{H}_{k}% \mathbf{P}_{k\mid k-1}-\mathbf{P}_{k\mid k-1}\mathbf{H}_{k}^{\mathrm{T}}% \mathbf{K}_{k}^{\mathrm{T}}+\mathbf{K}_{k}\mathbf{S}_{k}\mathbf{K}_{k}^{% \mathrm{T}}
  64. 𝐏 k k = 𝐏 k k - 1 - 𝐊 k 𝐇 k 𝐏 k k - 1 = ( I - 𝐊 k 𝐇 k ) 𝐏 k k - 1 . \mathbf{P}_{k\mid k}=\mathbf{P}_{k\mid k-1}-\mathbf{K}_{k}\mathbf{H}_{k}% \mathbf{P}_{k\mid k-1}=(I-\mathbf{K}_{k}\mathbf{H}_{k})\mathbf{P}_{k\mid k-1}.
  65. 𝐱 ^ k k \hat{\mathbf{x}}_{k\mid k}
  66. 𝐏 k k \mathbf{P}_{k\mid k}
  67. 𝐐 k \mathbf{Q}_{k}
  68. 𝐑 k \mathbf{R}_{k}
  69. 𝐏 k k = ( 𝐈 - 𝐊 k 𝐇 k ) 𝐏 k k - 1 ( 𝐈 - 𝐊 k 𝐇 k ) T + 𝐊 k 𝐑 k 𝐊 k T \mathbf{P}_{k\mid k}=(\mathbf{I}-\mathbf{K}_{k}\mathbf{H}_{k})\mathbf{P}_{k% \mid k-1}(\mathbf{I}-\mathbf{K}_{k}\mathbf{H}_{k})^{\mathrm{T}}+\mathbf{K}_{k}% \mathbf{R}_{k}\mathbf{K}_{k}^{\mathrm{T}}
  70. 𝐏 k k E [ ( 𝐱 k - 𝐱 ^ k k ) ( 𝐱 k - 𝐱 ^ k k ) T ] \mathbf{P}_{k\mid k}\neq E[(\mathbf{x}_{k}-\hat{\mathbf{x}}_{k\mid k})(\mathbf% {x}_{k}-\hat{\mathbf{x}}_{k\mid k})^{\mathrm{T}}]
  71. 𝐅 k \mathbf{F}_{k}
  72. 𝐇 k \mathbf{H}_{k}
  73. 𝐐 k a \mathbf{Q}^{a}_{k}
  74. 𝐑 k a \mathbf{R}^{a}_{k}
  75. 𝐐 k \mathbf{Q}_{k}
  76. 𝐑 k \mathbf{R}_{k}
  77. 𝐏 k k a \mathbf{P}_{k\mid k}^{a}
  78. 𝐏 k k \mathbf{P}_{k\mid k}
  79. 𝐐 k 𝐐 k a \mathbf{Q}_{k}\equiv\mathbf{Q}^{a}_{k}
  80. 𝐑 k 𝐑 k a \mathbf{R}_{k}\equiv\mathbf{R}^{a}_{k}
  81. 𝐏 k k = 𝐏 k k a \mathbf{P}_{k\mid k}=\mathbf{P}_{k\mid k}^{a}
  82. 𝐏 k k a = E [ ( 𝐱 k - 𝐱 ^ k k ) ( 𝐱 k - 𝐱 ^ k k ) T ] \mathbf{P}_{k\mid k}^{a}=E[(\mathbf{x}_{k}-\hat{\mathbf{x}}_{k\mid k})(\mathbf% {x}_{k}-\hat{\mathbf{x}}_{k\mid k})^{\mathrm{T}}]
  83. 𝐱 ^ k k \widehat{\mathbf{x}}_{k\mid k}
  84. E [ 𝐰 k 𝐰 k T ] = 𝐐 k a E[\mathbf{w}_{k}\mathbf{w}_{k}^{\mathrm{T}}]=\mathbf{Q}_{k}^{a}
  85. E [ 𝐯 k 𝐯 k T ] = 𝐑 k a E[\mathbf{v}_{k}\mathbf{v}_{k}^{\mathrm{T}}]=\mathbf{R}_{k}^{a}
  86. 𝐏 k k a \mathbf{P}_{k\mid k}^{a}
  87. 𝐏 k k - 1 a = 𝐅 k 𝐏 k - 1 k - 1 a 𝐅 k T + 𝐐 k a \mathbf{P}_{k\mid k-1}^{a}=\mathbf{F}_{k}\mathbf{P}_{k-1\mid k-1}^{a}\mathbf{F% }_{k}^{\mathrm{T}}+\mathbf{Q}_{k}^{a}
  88. 𝐏 k k a = ( 𝐈 - 𝐊 k 𝐇 k ) 𝐏 k k - 1 a ( 𝐈 - 𝐊 k 𝐇 k ) T + 𝐊 k 𝐑 k a 𝐊 k T \mathbf{P}_{k\mid k}^{a}=(\mathbf{I}-\mathbf{K}_{k}\mathbf{H}_{k})\mathbf{P}_{% k\mid k-1}^{a}(\mathbf{I}-\mathbf{K}_{k}\mathbf{H}_{k})^{\mathrm{T}}+\mathbf{K% }_{k}\mathbf{R}_{k}^{a}\mathbf{K}_{k}^{\mathrm{T}}
  89. 𝐏 k k \mathbf{P}_{k\mid k}
  90. E [ 𝐰 k 𝐰 k T ] = 𝐐 k E[\mathbf{w}_{k}\mathbf{w}_{k}^{\mathrm{T}}]=\mathbf{Q}_{k}
  91. E [ 𝐯 k 𝐯 k T ] = 𝐑 k E[\mathbf{v}_{k}\mathbf{v}_{k}^{\mathrm{T}}]=\mathbf{R}_{k}
  92. 𝐏 k k a \mathbf{P}_{k\mid k}^{a}
  93. 𝐏 k k \mathbf{P}_{k\mid k}
  94. 𝐐 k a \mathbf{Q}_{k}^{a}
  95. 𝐑 k a \mathbf{R}_{k}^{a}
  96. 𝐐 k \mathbf{Q}_{k}
  97. 𝐑 k \mathbf{R}_{k}
  98. p ( 𝐱 k 𝐱 0 , , 𝐱 k - 1 ) = p ( 𝐱 k 𝐱 k - 1 ) p(\,\textbf{x}_{k}\mid\,\textbf{x}_{0},\dots,\,\textbf{x}_{k-1})=p(\,\textbf{x% }_{k}\mid\,\textbf{x}_{k-1})
  99. p ( 𝐳 k 𝐱 0 , , 𝐱 k ) = p ( 𝐳 k 𝐱 k ) p(\,\textbf{z}_{k}\mid\,\textbf{x}_{0},\dots,\,\textbf{x}_{k})=p(\,\textbf{z}_% {k}\mid\,\textbf{x}_{k})
  100. p ( 𝐱 0 , , 𝐱 k , 𝐳 1 , , 𝐳 k ) = p ( 𝐱 0 ) i = 1 k p ( 𝐳 i 𝐱 i ) p ( 𝐱 i 𝐱 i - 1 ) p(\,\textbf{x}_{0},\dots,\,\textbf{x}_{k},\,\textbf{z}_{1},\dots,\,\textbf{z}_% {k})=p(\,\textbf{x}_{0})\prod_{i=1}^{k}p(\,\textbf{z}_{i}\mid\,\textbf{x}_{i})% p(\,\textbf{x}_{i}\mid\,\textbf{x}_{i-1})
  101. x k - 1 x_{k-1}
  102. p ( 𝐱 k 𝐙 k - 1 ) = p ( 𝐱 k 𝐱 k - 1 ) p ( 𝐱 k - 1 𝐙 k - 1 ) d 𝐱 k - 1 p(\,\textbf{x}_{k}\mid\,\textbf{Z}_{k-1})=\int p(\,\textbf{x}_{k}\mid\,\textbf% {x}_{k-1})p(\,\textbf{x}_{k-1}\mid\,\textbf{Z}_{k-1})\,d\,\textbf{x}_{k-1}
  103. 𝐙 t = { 𝐳 1 , , 𝐳 t } \,\textbf{Z}_{t}=\left\{\,\textbf{z}_{1},\dots,\,\textbf{z}_{t}\right\}
  104. p ( 𝐱 k 𝐙 k ) = p ( 𝐳 k 𝐱 k ) p ( 𝐱 k 𝐙 k - 1 ) p ( 𝐳 k 𝐙 k - 1 ) p(\,\textbf{x}_{k}\mid\,\textbf{Z}_{k})=\frac{p(\,\textbf{z}_{k}\mid\,\textbf{% x}_{k})p(\,\textbf{x}_{k}\mid\,\textbf{Z}_{k-1})}{p(\,\textbf{z}_{k}\mid\,% \textbf{Z}_{k-1})}
  105. p ( 𝐳 k 𝐙 k - 1 ) = p ( 𝐳 k 𝐱 k ) p ( 𝐱 k 𝐙 k - 1 ) d 𝐱 k p(\,\textbf{z}_{k}\mid\,\textbf{Z}_{k-1})=\int p(\,\textbf{z}_{k}\mid\,\textbf% {x}_{k})p(\,\textbf{x}_{k}\mid\,\textbf{Z}_{k-1})d\,\textbf{x}_{k}
  106. p ( 𝐱 k 𝐱 k - 1 ) = 𝒩 ( 𝐅 k 𝐱 k - 1 , 𝐐 k ) p(\,\textbf{x}_{k}\mid\,\textbf{x}_{k-1})=\mathcal{N}(\,\textbf{F}_{k}\,% \textbf{x}_{k-1},\,\textbf{Q}_{k})
  107. p ( 𝐳 k 𝐱 k ) = 𝒩 ( 𝐇 k 𝐱 k , 𝐑 k ) p(\,\textbf{z}_{k}\mid\,\textbf{x}_{k})=\mathcal{N}(\,\textbf{H}_{k}\,\textbf{% x}_{k},\,\textbf{R}_{k})
  108. p ( 𝐱 k - 1 𝐙 k - 1 ) = 𝒩 ( 𝐱 ^ k - 1 , 𝐏 k - 1 ) p(\,\textbf{x}_{k-1}\mid\,\textbf{Z}_{k-1})=\mathcal{N}(\hat{\,\textbf{x}}_{k-% 1},\,\textbf{P}_{k-1})
  109. 𝐱 k \mathbf{x}_{k}
  110. 𝐙 k \mathbf{Z}_{k}
  111. 𝐱 0 \mathbf{x}_{0}
  112. p ( 𝐱 0 ) = 𝒩 ( 𝐱 ^ 0 0 , 𝐏 0 0 ) p(\mathbf{x}_{0})=\mathcal{N}(\hat{\mathbf{x}}_{0\mid 0},\mathbf{P}_{0\mid 0})
  113. 𝐳 0 \mathbf{z}_{0}
  114. p ( 𝐳 0 𝐱 0 ) = 𝒩 ( 𝐇 0 𝐱 0 , 𝐑 0 ) p(\mathbf{z}_{0}\mid\mathbf{x}_{0})=\mathcal{N}(\mathbf{H}_{0}\mathbf{x}_{0},% \mathbf{R}_{0})
  115. k = 1 , 2 , 3 , k=1,2,3,...\ldots
  116. 𝐱 k \mathbf{x}_{k}
  117. p ( 𝐱 k 𝐱 k - 1 ) = 𝒩 ( 𝐅 k 𝐱 k - 1 + 𝐁 k 𝐮 k , 𝐐 k ) p(\mathbf{x}_{k}\mid\mathbf{x}_{k-1})=\mathcal{N}(\mathbf{F}_{k}\mathbf{x}_{k-% 1}+\mathbf{B}_{k}\mathbf{u}_{k},\mathbf{Q}_{k})
  118. 𝐳 k \mathbf{z}_{k}
  119. p ( 𝐳 k 𝐱 k ) = 𝒩 ( 𝐇 k 𝐱 k , 𝐑 k ) p(\mathbf{z}_{k}\mid\mathbf{x}_{k})=\mathcal{N}(\mathbf{H}_{k}\mathbf{x}_{k},% \mathbf{R}_{k})
  120. p ( 𝐳 ) = k = 0 T p ( 𝐳 k 𝐳 k - 1 , , 𝐳 0 ) p(\mathbf{z})=\prod_{k=0}^{T}p(\mathbf{z}_{k}\mid\mathbf{z}_{k-1},\ldots,% \mathbf{z}_{0})
  121. 𝐱 ^ k k - 1 , 𝐏 k k - 1 \hat{\mathbf{x}}_{k\mid k-1},\mathbf{P}_{k\mid k-1}
  122. p ( 𝐳 ) = k = 0 T p ( 𝐳 k 𝐱 k ) p ( 𝐱 k 𝐳 k - 1 , , 𝐳 0 ) d 𝐱 k = k = 0 T 𝒩 ( 𝐳 k ; 𝐇 k 𝐱 k , 𝐑 k ) 𝒩 ( 𝐱 k ; 𝐱 ^ k k - 1 , 𝐏 k k - 1 ) d 𝐱 k = k = 0 T 𝒩 ( 𝐳 k ; 𝐇 k 𝐱 ^ k k - 1 , 𝐑 k + 𝐇 k 𝐏 k k - 1 𝐇 k T ) = k = 0 T 𝒩 ( 𝐳 k ; 𝐇 k 𝐱 ^ k k - 1 , 𝐒 k ) , \begin{aligned}\displaystyle p(\mathbf{z})&\displaystyle=\prod_{k=0}^{T}\int p% (\mathbf{z}_{k}\mid\mathbf{x}_{k})p(\mathbf{x}_{k}\mid\mathbf{z}_{k-1},\ldots,% \mathbf{z}_{0})d\mathbf{x}_{k}\\ &\displaystyle=\prod_{k=0}^{T}\int\mathcal{N}(\mathbf{z}_{k};\mathbf{H}_{k}% \mathbf{x}_{k},\mathbf{R}_{k})\mathcal{N}(\mathbf{x}_{k};\hat{\mathbf{x}}_{k% \mid k-1},\mathbf{P}_{k\mid k-1})d\mathbf{x}_{k}\\ &\displaystyle=\prod_{k=0}^{T}\mathcal{N}(\mathbf{z}_{k};\mathbf{H}_{k}\hat{% \mathbf{x}}_{k\mid k-1},\mathbf{R}_{k}+\mathbf{H}_{k}\mathbf{P}_{k\mid k-1}% \mathbf{H}_{k}^{T})\\ &\displaystyle=\prod_{k=0}^{T}\mathcal{N}(\mathbf{z}_{k};\mathbf{H}_{k}\hat{% \mathbf{x}}_{k\mid k-1},\mathbf{S}_{k}),\end{aligned}
  123. 𝐇 k 𝐱 ^ k k - 1 , 𝐒 k \mathbf{H}_{k}\hat{\mathbf{x}}_{k\mid k-1},\mathbf{S}_{k}
  124. = log p ( 𝐳 ) \ell=\log p(\mathbf{z})
  125. ( - 1 ) = 0 \ell^{(-1)}=0
  126. ( k ) = ( k - 1 ) - 1 2 ( 𝐲 ~ k T 𝐒 k - 1 𝐲 ~ k + log | 𝐒 k | + d y log 2 π ) \ell^{(k)}=\ell^{(k-1)}-\frac{1}{2}\left(\tilde{\mathbf{y}}_{k}^{T}\mathbf{S}^% {-1}_{k}\tilde{\mathbf{y}}_{k}+\log\left|\mathbf{S}_{k}\right|+d_{y}\log 2\pi\right)
  127. d y d_{y}
  128. 𝐘 k k = 𝐏 k k - 1 \,\textbf{Y}_{k\mid k}=\,\textbf{P}_{k\mid k}^{-1}
  129. 𝐲 ^ k k = 𝐏 k k - 1 𝐱 ^ k k \hat{\,\textbf{y}}_{k\mid k}=\,\textbf{P}_{k\mid k}^{-1}\hat{\,\textbf{x}}_{k% \mid k}
  130. 𝐘 k k - 1 = 𝐏 k k - 1 - 1 \,\textbf{Y}_{k\mid k-1}=\,\textbf{P}_{k\mid k-1}^{-1}
  131. 𝐲 ^ k k - 1 = 𝐏 k k - 1 - 1 𝐱 ^ k k - 1 \hat{\,\textbf{y}}_{k\mid k-1}=\,\textbf{P}_{k\mid k-1}^{-1}\hat{\,\textbf{x}}% _{k\mid k-1}
  132. 𝐈 k = 𝐇 k T 𝐑 k - 1 𝐇 k \,\textbf{I}_{k}=\,\textbf{H}_{k}^{\,\text{T}}\,\textbf{R}_{k}^{-1}\,\textbf{H% }_{k}
  133. 𝐢 k = 𝐇 k T 𝐑 k - 1 𝐳 k \,\textbf{i}_{k}=\,\textbf{H}_{k}^{\,\text{T}}\,\textbf{R}_{k}^{-1}\,\textbf{z% }_{k}
  134. 𝐘 k k = 𝐘 k k - 1 + 𝐈 k \,\textbf{Y}_{k\mid k}=\,\textbf{Y}_{k\mid k-1}+\,\textbf{I}_{k}
  135. 𝐲 ^ k k = 𝐲 ^ k k - 1 + 𝐢 k \hat{\,\textbf{y}}_{k\mid k}=\hat{\,\textbf{y}}_{k\mid k-1}+\,\textbf{i}_{k}
  136. 𝐘 k k = 𝐘 k k - 1 + j = 1 N 𝐈 k , j \,\textbf{Y}_{k\mid k}=\,\textbf{Y}_{k\mid k-1}+\sum_{j=1}^{N}\,\textbf{I}_{k,j}
  137. 𝐲 ^ k k = 𝐲 ^ k k - 1 + j = 1 N 𝐢 k , j \hat{\,\textbf{y}}_{k\mid k}=\hat{\,\textbf{y}}_{k\mid k-1}+\sum_{j=1}^{N}\,% \textbf{i}_{k,j}
  138. 𝐌 k = [ 𝐅 k - 1 ] T 𝐘 k - 1 k - 1 𝐅 k - 1 \,\textbf{M}_{k}=[\,\textbf{F}_{k}^{-1}]^{\,\text{T}}\,\textbf{Y}_{k-1\mid k-1% }\,\textbf{F}_{k}^{-1}
  139. 𝐂 k = 𝐌 k [ 𝐌 k + 𝐐 k - 1 ] - 1 \,\textbf{C}_{k}=\,\textbf{M}_{k}[\,\textbf{M}_{k}+\,\textbf{Q}_{k}^{-1}]^{-1}
  140. 𝐋 k = I - 𝐂 k \,\textbf{L}_{k}=I-\,\textbf{C}_{k}
  141. 𝐘 k k - 1 = 𝐋 k 𝐌 k 𝐋 k T + 𝐂 k 𝐐 k - 1 𝐂 k T \,\textbf{Y}_{k\mid k-1}=\,\textbf{L}_{k}\,\textbf{M}_{k}\,\textbf{L}_{k}^{\,% \text{T}}+\,\textbf{C}_{k}\,\textbf{Q}_{k}^{-1}\,\textbf{C}_{k}^{\,\text{T}}
  142. 𝐲 ^ k k - 1 = 𝐋 k [ 𝐅 k - 1 ] T 𝐲 ^ k - 1 k - 1 \hat{\,\textbf{y}}_{k\mid k-1}=\,\textbf{L}_{k}[\,\textbf{F}_{k}^{-1}]^{\,% \text{T}}\hat{\,\textbf{y}}_{k-1\mid k-1}
  143. 𝐱 ^ k - N k \hat{\,\textbf{x}}_{k-N\mid k}
  144. N N
  145. 𝐳 1 \,\textbf{z}_{1}
  146. 𝐳 k \,\textbf{z}_{k}
  147. [ 𝐱 ^ t t 𝐱 ^ t - 1 t 𝐱 ^ t - N + 1 t ] = [ 𝐈 0 0 ] 𝐱 ^ t t - 1 + [ 0 0 𝐈 0 0 I ] [ 𝐱 ^ t - 1 t - 1 𝐱 ^ t - 2 t - 1 𝐱 ^ t - N + 1 t - 1 ] + [ 𝐊 ( 0 ) 𝐊 ( 1 ) 𝐊 ( N - 1 ) ] 𝐲 t t - 1 \begin{bmatrix}\hat{\,\textbf{x}}_{t\mid t}\\ \hat{\,\textbf{x}}_{t-1\mid t}\\ \vdots\\ \hat{\,\textbf{x}}_{t-N+1\mid t}\\ \end{bmatrix}=\begin{bmatrix}\,\textbf{I}\\ 0\\ \vdots\\ 0\\ \end{bmatrix}\hat{\,\textbf{x}}_{t\mid t-1}+\begin{bmatrix}0&\ldots&0\\ \,\textbf{I}&0&\vdots\\ \vdots&\ddots&\vdots\\ 0&\ldots&I\\ \end{bmatrix}\begin{bmatrix}\hat{\,\textbf{x}}_{t-1\mid t-1}\\ \hat{\,\textbf{x}}_{t-2\mid t-1}\\ \vdots\\ \hat{\,\textbf{x}}_{t-N+1\mid t-1}\\ \end{bmatrix}+\begin{bmatrix}\,\textbf{K}^{(0)}\\ \,\textbf{K}^{(1)}\\ \vdots\\ \,\textbf{K}^{(N-1)}\\ \end{bmatrix}\,\textbf{y}_{t\mid t-1}
  148. 𝐱 ^ t t - 1 \hat{\,\textbf{x}}_{t\mid t-1}
  149. 𝐲 t t - 1 = 𝐳 ( t ) - 𝐇 𝐱 ^ t t - 1 \,\textbf{y}_{t\mid t-1}=\,\textbf{z}(t)-\,\textbf{H}\hat{\,\textbf{x}}_{t\mid t% -1}
  150. 𝐱 ^ t - i t \hat{\,\textbf{x}}_{t-i\mid t}
  151. i = 1 , , N - 1 i=1,\ldots,N-1
  152. 𝐊 ( i ) = 𝐏 ( i ) 𝐇 T [ 𝐇 𝐏 𝐇 T + 𝐑 ] - 1 \,\textbf{K}^{(i)}=\,\textbf{P}^{(i)}\,\textbf{H}^{T}\left[\,\textbf{H}\,% \textbf{P}\,\textbf{H}^{\mathrm{T}}+\,\textbf{R}\right]^{-1}
  153. 𝐏 ( i ) = 𝐏 [ [ 𝐅 - 𝐊 𝐇 ] T ] i \,\textbf{P}^{(i)}=\,\textbf{P}\left[\left[\,\textbf{F}-\,\textbf{K}\,\textbf{% H}\right]^{T}\right]^{i}
  154. 𝐏 \,\textbf{P}
  155. 𝐊 \,\textbf{K}
  156. 𝐏 t t - 1 \,\textbf{P}_{t\mid t-1}
  157. 𝐏 i := E [ ( 𝐱 t - i - 𝐱 ^ t - i t ) * ( 𝐱 t - i - 𝐱 ^ t - i t ) z 1 z t ] , \,\textbf{P}_{i}:=E\left[\left(\,\textbf{x}_{t-i}-\hat{\,\textbf{x}}_{t-i\mid t% }\right)^{*}\left(\,\textbf{x}_{t-i}-\hat{\,\textbf{x}}_{t-i\mid t}\right)\mid z% _{1}\ldots z_{t}\right],
  158. 𝐱 t - i \,\textbf{x}_{t-i}
  159. 𝐏 - 𝐏 i = j = 0 i [ 𝐏 ( j ) 𝐇 T [ 𝐇 𝐏 𝐇 T + 𝐑 ] - 1 𝐇 ( 𝐏 ( i ) ) T ] \,\textbf{P}-\,\textbf{P}_{i}=\sum_{j=0}^{i}\left[\,\textbf{P}^{(j)}\,\textbf{% H}^{T}\left[\,\textbf{H}\,\textbf{P}\,\textbf{H}^{\mathrm{T}}+\,\textbf{R}% \right]^{-1}\,\textbf{H}\left(\,\textbf{P}^{(i)}\right)^{\mathrm{T}}\right]
  160. 𝐱 ^ k n \hat{\,\textbf{x}}_{k\mid n}
  161. k < n k<n
  162. 𝐳 1 \,\textbf{z}_{1}
  163. 𝐳 n \,\textbf{z}_{n}
  164. 𝐱 ^ k k \hat{\,\textbf{x}}_{k\mid k}
  165. 𝐏 k k \,\textbf{P}_{k\mid k}
  166. 𝐱 ^ k n \hat{\,\textbf{x}}_{k\mid n}
  167. 𝐏 k n \,\textbf{P}_{k\mid n}
  168. 𝐱 ^ k n = 𝐱 ^ k k + 𝐂 k ( 𝐱 ^ k + 1 n - 𝐱 ^ k + 1 k ) \hat{\,\textbf{x}}_{k\mid n}=\hat{\,\textbf{x}}_{k\mid k}+\,\textbf{C}_{k}(% \hat{\,\textbf{x}}_{k+1\mid n}-\hat{\,\textbf{x}}_{k+1\mid k})
  169. 𝐏 k n = 𝐏 k k + 𝐂 k ( 𝐏 k + 1 n - 𝐏 k + 1 k ) 𝐂 k T \,\textbf{P}_{k\mid n}=\,\textbf{P}_{k\mid k}+\,\textbf{C}_{k}(\,\textbf{P}_{k% +1\mid n}-\,\textbf{P}_{k+1\mid k})\,\textbf{C}_{k}^{\mathrm{T}}
  170. 𝐂 k = 𝐏 k k 𝐅 k + 1 T 𝐏 k + 1 k - 1 \,\textbf{C}_{k}=\,\textbf{P}_{k\mid k}\,\textbf{F}_{k+1}^{\mathrm{T}}\,% \textbf{P}_{k+1\mid k}^{-1}
  171. Λ ~ k = 𝐇 k T 𝐒 k - 1 𝐇 k + 𝐂 ^ k T Λ ^ k 𝐂 ^ k \tilde{\Lambda}_{k}=\,\textbf{H}_{k}^{T}\,\textbf{S}_{k}^{-1}\,\textbf{H}_{k}+% \hat{\,\textbf{C}}_{k}^{T}\hat{\Lambda}_{k}\hat{\,\textbf{C}}_{k}
  172. Λ ^ k - 1 = 𝐅 k T Λ ~ k 𝐅 k \hat{\Lambda}_{k-1}=\,\textbf{F}_{k}^{T}\tilde{\Lambda}_{k}\,\textbf{F}_{k}
  173. Λ ^ n = 0 \hat{\Lambda}_{n}=0
  174. λ ~ k = - 𝐇 k T 𝐒 k - 1 𝐲 k + 𝐂 ^ k T λ ^ k \tilde{\lambda}_{k}=-\,\textbf{H}_{k}^{T}\,\textbf{S}_{k}^{-1}\,\textbf{y}_{k}% +\hat{\,\textbf{C}}_{k}^{T}\hat{\lambda}_{k}
  175. λ ^ k - 1 = 𝐅 k T λ ~ k \hat{\lambda}_{k-1}=\,\textbf{F}_{k}^{T}\tilde{\lambda}_{k}
  176. λ ^ n = 0 \hat{\lambda}_{n}=0
  177. 𝐒 k \,\textbf{S}_{k}
  178. 𝐂 ^ k = 𝐈 - 𝐊 k 𝐇 k \hat{\,\textbf{C}}_{k}=\,\textbf{I}-\,\textbf{K}_{k}\,\textbf{H}_{k}
  179. 𝐏 k n = 𝐏 k k - 𝐏 k k Λ ^ k 𝐏 k k \,\textbf{P}_{k\mid n}=\,\textbf{P}_{k\mid k}-\,\textbf{P}_{k\mid k}\hat{% \Lambda}_{k}\,\textbf{P}_{k\mid k}
  180. 𝐱 k n = 𝐱 k k - 𝐏 k k λ ^ k \,\textbf{x}_{k\mid n}=\,\textbf{x}_{k\mid k}-\,\textbf{P}_{k\mid k}\hat{% \lambda}_{k}
  181. 𝐏 k n = 𝐏 k k - 1 - 𝐏 k k - 1 Λ ~ k 𝐏 k k - 1 \,\textbf{P}_{k\mid n}=\,\textbf{P}_{k\mid k-1}-\,\textbf{P}_{k\mid k-1}\tilde% {\Lambda}_{k}\,\textbf{P}_{k\mid k-1}
  182. 𝐱 k n = 𝐱 k k - 1 - 𝐏 k k - 1 λ ~ k . \,\textbf{x}_{k\mid n}=\,\textbf{x}_{k\mid k-1}-\,\textbf{P}_{k\mid k-1}\tilde% {\lambda}_{k}.
  183. 𝐱 ^ k + 1 k = (F k - 𝐊 k 𝐇 k ) 𝐱 ^ k k - 1 + 𝐊 k 𝐳 k \hat{\,\textbf{x}}_{k+1\mid k}=\,\textbf{(F}_{k}-\,\textbf{K}_{k}\,\textbf{H}_% {k})\hat{\,\textbf{x}}_{k\mid k-1}+\,\textbf{K}_{k}\,\textbf{z}_{k}
  184. α k = - 𝐒 k - 1 / 2 𝐇 k 𝐱 ^ k k - 1 + 𝐒 k - 1 / 2 𝐳 k {\alpha}_{k}=-\,\textbf{S}_{k}^{-1/2}\,\textbf{H}_{k}\hat{\,\textbf{x}}_{k\mid k% -1}+\,\textbf{S}_{k}^{-1/2}\,\textbf{z}_{k}
  185. β k \beta_{k}
  186. α k \alpha_{k}
  187. 𝐲 ^ k N = 𝐳 k - 𝐑 k β k \hat{\,\textbf{y}}_{k\mid N}=\,\textbf{z}_{k}-\,\textbf{R}_{k}\beta_{k}
  188. 𝐲 ^ k k = 𝐳 k - 𝐑 k 𝐒 k - 1 / 2 α k \hat{\,\textbf{y}}_{k\mid k}=\,\textbf{z}_{k}-\,\textbf{R}_{k}\,\textbf{S}_{k}% ^{-1/2}\alpha_{k}
  189. 𝐲 \,\textbf{y}
  190. 𝐲 ^ \hat{\,\textbf{y}}
  191. 𝐖 \,\textbf{W}
  192. 𝐖 \,\textbf{W}
  193. 𝐲 \,\textbf{y}
  194. 𝐲 ^ \hat{\,\textbf{y}}
  195. 𝐖 - 1 𝐲 ^ \,\textbf{W}^{-1}\hat{\,\textbf{y}}
  196. 𝐖 \,\textbf{W}
  197. 𝐖 \,\textbf{W}
  198. 𝐱 k = f ( 𝐱 k - 1 , 𝐮 k ) + 𝐰 k \,\textbf{x}_{k}=f(\,\textbf{x}_{k-1},\,\textbf{u}_{k})+\,\textbf{w}_{k}
  199. 𝐳 k = h ( 𝐱 k ) + 𝐯 k \,\textbf{z}_{k}=h(\,\textbf{x}_{k})+\,\textbf{v}_{k}
  200. f f
  201. h h
  202. 𝐱 k - 1 k - 1 a = [ 𝐱 ^ k - 1 k - 1 T E [ 𝐰 k T ] ] T \,\textbf{x}_{k-1\mid k-1}^{a}=[\hat{\,\textbf{x}}_{k-1\mid k-1}^{\mathrm{T}}% \quad E[\,\textbf{w}_{k}^{\mathrm{T}}]\ ]^{\mathrm{T}}
  203. 𝐏 k - 1 k - 1 a = [ 𝐏 k - 1 k - 1 0 0 𝐐 k ] \,\textbf{P}_{k-1\mid k-1}^{a}=\begin{bmatrix}&\,\textbf{P}_{k-1\mid k-1}&&0&% \\ &0&&\,\textbf{Q}_{k}&\end{bmatrix}
  204. χ k - 1 k - 1 0 = 𝐱 k - 1 k - 1 a \chi_{k-1\mid k-1}^{0}=\,\textbf{x}_{k-1\mid k-1}^{a}
  205. χ k - 1 k - 1 i = 𝐱 k - 1 k - 1 a + ( ( L + λ ) 𝐏 k - 1 k - 1 a ) i , i = 1 , , L \chi_{k-1\mid k-1}^{i}=\,\textbf{x}_{k-1\mid k-1}^{a}+\left(\sqrt{(L+\lambda)% \,\textbf{P}_{k-1\mid k-1}^{a}}\right)_{i},\qquad i=1,\ldots,L
  206. χ k - 1 k - 1 i = 𝐱 k - 1 k - 1 a - ( ( L + λ ) 𝐏 k - 1 k - 1 a ) i - L , i = L + 1 , , 2 L \chi_{k-1\mid k-1}^{i}=\,\textbf{x}_{k-1\mid k-1}^{a}-\left(\sqrt{(L+\lambda)% \,\textbf{P}_{k-1\mid k-1}^{a}}\right)_{i-L},\qquad i=L+1,\dots{},2L
  207. ( ( L + λ ) 𝐏 k - 1 k - 1 a ) i \left(\sqrt{(L+\lambda)\,\textbf{P}_{k-1\mid k-1}^{a}}\right)_{i}
  208. ( L + λ ) 𝐏 k - 1 k - 1 a (L+\lambda)\,\textbf{P}_{k-1\mid k-1}^{a}
  209. 𝐀 \,\textbf{A}
  210. 𝐁 \,\textbf{B}
  211. 𝐁 𝐀 𝐀 T . \,\textbf{B}\triangleq\,\textbf{A}\,\textbf{A}^{\mathrm{T}}.\,
  212. χ k k - 1 i = f ( χ k - 1 k - 1 i ) i = 0 , , 2 L \chi_{k\mid k-1}^{i}=f(\chi_{k-1\mid k-1}^{i})\quad i=0,\dots,2L
  213. f : R L R | 𝐱 | f:R^{L}\rightarrow R^{|\,\textbf{x}|}
  214. 𝐱 ^ k k - 1 = i = 0 2 L W s i χ k k - 1 i \hat{\,\textbf{x}}_{k\mid k-1}=\sum_{i=0}^{2L}W_{s}^{i}\chi_{k\mid k-1}^{i}
  215. 𝐏 k k - 1 = i = 0 2 L W c i [ χ k k - 1 i - 𝐱 ^ k k - 1 ] [ χ k k - 1 i - 𝐱 ^ k k - 1 ] T \,\textbf{P}_{k\mid k-1}=\sum_{i=0}^{2L}W_{c}^{i}\ [\chi_{k\mid k-1}^{i}-\hat{% \,\textbf{x}}_{k\mid k-1}][\chi_{k\mid k-1}^{i}-\hat{\,\textbf{x}}_{k\mid k-1}% ]^{\mathrm{T}}
  216. W s 0 = λ L + λ W_{s}^{0}=\frac{\lambda}{L+\lambda}
  217. W c 0 = λ L + λ + ( 1 - α 2 + β ) W_{c}^{0}=\frac{\lambda}{L+\lambda}+(1-\alpha^{2}+\beta)
  218. W s i = W c i = 1 2 ( L + λ ) W_{s}^{i}=W_{c}^{i}=\frac{1}{2(L+\lambda)}
  219. λ = α 2 ( L + κ ) - L \lambda=\alpha^{2}(L+\kappa)-L\,\!
  220. α \alpha
  221. κ \kappa
  222. β \beta
  223. x x
  224. α = 10 - 3 \alpha=10^{-3}
  225. κ = 0 \kappa=0
  226. β = 2 \beta=2
  227. x x
  228. β = 2 \beta=2
  229. 𝐱 k k - 1 a = [ 𝐱 ^ k k - 1 T E [ 𝐯 k T ] ] T \,\textbf{x}_{k\mid k-1}^{a}=[\hat{\,\textbf{x}}_{k\mid k-1}^{\mathrm{T}}\quad E% [\,\textbf{v}_{k}^{\mathrm{T}}]\ ]^{\mathrm{T}}
  230. 𝐏 k k - 1 a = [ 𝐏 k k - 1 0 0 𝐑 k ] \,\textbf{P}_{k\mid k-1}^{a}=\begin{bmatrix}&\,\textbf{P}_{k\mid k-1}&&0&\\ &0&&\,\textbf{R}_{k}&\end{bmatrix}
  231. χ k k - 1 0 \displaystyle\chi_{k\mid k-1}^{0}
  232. χ k k - 1 := [ χ k k - 1 T E [ 𝐯 k T ] ] T ± ( L + λ ) 𝐑 k a \chi_{k\mid k-1}:=[\chi_{k\mid k-1}^{\mathrm{T}}\quad E[\,\textbf{v}_{k}^{% \mathrm{T}}]\ ]^{\mathrm{T}}\pm\sqrt{(L+\lambda)\,\textbf{R}_{k}^{a}}
  233. 𝐑 k a = [ 0 0 0 𝐑 k ] \,\textbf{R}_{k}^{a}=\begin{bmatrix}&0&&0&\\ &0&&\,\textbf{R}_{k}&\end{bmatrix}
  234. γ k i = h ( χ k k - 1 i ) i = 0..2 L \gamma_{k}^{i}=h(\chi_{k\mid k-1}^{i})\quad i=0..2L
  235. 𝐳 ^ k = i = 0 2 L W s i γ k i \hat{\,\textbf{z}}_{k}=\sum_{i=0}^{2L}W_{s}^{i}\gamma_{k}^{i}
  236. 𝐏 z k z k = i = 0 2 L W c i [ γ k i - 𝐳 ^ k ] [ γ k i - 𝐳 ^ k ] T \,\textbf{P}_{z_{k}z_{k}}=\sum_{i=0}^{2L}W_{c}^{i}\ [\gamma_{k}^{i}-\hat{\,% \textbf{z}}_{k}][\gamma_{k}^{i}-\hat{\,\textbf{z}}_{k}]^{\mathrm{T}}
  237. 𝐏 x k z k = i = 0 2 L W c i [ χ k k - 1 i - 𝐱 ^ k k - 1 ] [ γ k i - 𝐳 ^ k ] T \,\textbf{P}_{x_{k}z_{k}}=\sum_{i=0}^{2L}W_{c}^{i}\ [\chi_{k\mid k-1}^{i}-\hat% {\,\textbf{x}}_{k\mid k-1}][\gamma_{k}^{i}-\hat{\,\textbf{z}}_{k}]^{\mathrm{T}}
  238. K k = 𝐏 x k z k 𝐏 z k z k - 1 K_{k}=\,\textbf{P}_{x_{k}z_{k}}\,\textbf{P}_{z_{k}z_{k}}^{-1}
  239. 𝐱 ^ k k = 𝐱 ^ k k - 1 + K k ( 𝐳 k - 𝐳 ^ k ) \hat{\,\textbf{x}}_{k\mid k}=\hat{\,\textbf{x}}_{k\mid k-1}+K_{k}(\,\textbf{z}% _{k}-\hat{\,\textbf{z}}_{k})
  240. 𝐏 k k = 𝐏 k k - 1 - K k 𝐏 z k z k K k T \,\textbf{P}_{k\mid k}=\,\textbf{P}_{k\mid k-1}-K_{k}\,\textbf{P}_{z_{k}z_{k}}% K_{k}^{\mathrm{T}}
  241. d d t 𝐱 ( t ) = 𝐅 ( t ) 𝐱 ( t ) + 𝐁 ( t ) 𝐮 ( t ) + 𝐰 ( t ) \frac{d}{dt}\mathbf{x}(t)=\mathbf{F}(t)\mathbf{x}(t)+\mathbf{B}(t)\mathbf{u}(t% )+\mathbf{w}(t)
  242. 𝐳 ( t ) = 𝐇 ( t ) 𝐱 ( t ) + 𝐯 ( t ) \mathbf{z}(t)=\mathbf{H}(t)\mathbf{x}(t)+\mathbf{v}(t)
  243. 𝐐 ( t ) \mathbf{Q}(t)
  244. 𝐑 ( t ) \mathbf{R}(t)
  245. 𝐰 ( t ) \mathbf{w}(t)
  246. 𝐯 ( t ) \mathbf{v}(t)
  247. d d t 𝐱 ^ ( t ) = 𝐅 ( t ) 𝐱 ^ ( t ) + 𝐁 ( t ) 𝐮 ( t ) + 𝐊 ( t ) ( 𝐳 ( t ) - 𝐇 ( t ) 𝐱 ^ ( t ) ) \frac{d}{dt}\hat{\mathbf{x}}(t)=\mathbf{F}(t)\hat{\mathbf{x}}(t)+\mathbf{B}(t)% \mathbf{u}(t)+\mathbf{K}(t)(\mathbf{z}(t)-\mathbf{H}(t)\hat{\mathbf{x}}(t))
  248. d d t 𝐏 ( t ) = 𝐅 ( t ) 𝐏 ( t ) + 𝐏 ( t ) 𝐅 T ( t ) + 𝐐 ( t ) - 𝐊 ( t ) 𝐑 ( t ) 𝐊 T ( t ) \frac{d}{dt}\mathbf{P}(t)=\mathbf{F}(t)\mathbf{P}(t)+\mathbf{P}(t)\mathbf{F}^{% T}(t)+\mathbf{Q}(t)-\mathbf{K}(t)\mathbf{R}(t)\mathbf{K}^{T}(t)
  249. 𝐊 ( t ) = 𝐏 ( t ) 𝐇 T ( t ) 𝐑 - 1 ( t ) \mathbf{K}(t)=\mathbf{P}(t)\mathbf{H}^{T}(t)\mathbf{R}^{-1}(t)
  250. 𝐊 ( t ) \mathbf{K}(t)
  251. 𝐑 ( t ) \mathbf{R}(t)
  252. 𝐲 ~ ( t ) = 𝐳 ( t ) - 𝐇 ( t ) 𝐱 ^ ( t ) \tilde{\mathbf{y}}(t)=\mathbf{z}(t)-\mathbf{H}(t)\hat{\mathbf{x}}(t)
  253. 𝐱 ˙ ( t ) = 𝐅 ( t ) 𝐱 ( t ) + 𝐁 ( t ) 𝐮 ( t ) + 𝐰 ( t ) , 𝐰 ( t ) N ( 𝟎 , 𝐐 ( t ) ) 𝐳 k = 𝐇 k 𝐱 k + 𝐯 k , 𝐯 k N ( 𝟎 , 𝐑 k ) \begin{aligned}\displaystyle\dot{\mathbf{x}}(t)&\displaystyle=\mathbf{F}(t)% \mathbf{x}(t)+\mathbf{B}(t)\mathbf{u}(t)+\mathbf{w}(t),&\displaystyle\mathbf{w% }(t)&\displaystyle\sim N\bigl(\mathbf{0},\mathbf{Q}(t)\bigr)\\ \displaystyle\mathbf{z}_{k}&\displaystyle=\mathbf{H}_{k}\mathbf{x}_{k}+\mathbf% {v}_{k},&\displaystyle\mathbf{v}_{k}&\displaystyle\sim N(\mathbf{0},\mathbf{R}% _{k})\end{aligned}
  254. 𝐱 k = 𝐱 ( t k ) \mathbf{x}_{k}=\mathbf{x}(t_{k})
  255. 𝐱 ^ 0 0 = E [ 𝐱 ( t 0 ) ] , 𝐏 0 0 = V a r [ 𝐱 ( t 0 ) ] \hat{\mathbf{x}}_{0\mid 0}=E\bigl[\mathbf{x}(t_{0})\bigr],\mathbf{P}_{0\mid 0}% =Var\bigl[\mathbf{x}(t_{0})\bigr]
  256. 𝐱 ^ ˙ ( t ) = 𝐅 ( t ) 𝐱 ^ ( t ) + 𝐁 ( t ) 𝐮 ( t ) , with 𝐱 ^ ( t k - 1 ) = 𝐱 ^ k - 1 k - 1 𝐱 ^ k k - 1 = 𝐱 ^ ( t k ) 𝐏 ˙ ( t ) = 𝐅 ( t ) 𝐏 ( t ) + 𝐏 ( t ) 𝐅 ( t ) T + 𝐐 ( t ) , with 𝐏 ( t k - 1 ) = 𝐏 k - 1 k - 1 𝐏 k k - 1 = 𝐏 ( t k ) \begin{aligned}&\displaystyle\dot{\hat{\mathbf{x}}}(t)=\mathbf{F}(t)\hat{% \mathbf{x}}(t)+\mathbf{B}(t)\mathbf{u}(t)\,\text{, with }\hat{\mathbf{x}}(t_{k% -1})=\hat{\mathbf{x}}_{k-1\mid k-1}\\ \displaystyle\Rightarrow&\displaystyle\hat{\mathbf{x}}_{k\mid k-1}=\hat{% \mathbf{x}}(t_{k})\\ &\displaystyle\dot{\mathbf{P}}(t)=\mathbf{F}(t)\mathbf{P}(t)+\mathbf{P}(t)% \mathbf{F}(t)^{T}+\mathbf{Q}(t)\,\text{, with }\mathbf{P}(t_{k-1})=\mathbf{P}_% {k-1\mid k-1}\\ \displaystyle\Rightarrow&\displaystyle\mathbf{P}_{k\mid k-1}=\mathbf{P}(t_{k})% \end{aligned}
  257. 𝐊 ( t ) = 0 \mathbf{K}(t)=0
  258. 𝐊 k = 𝐏 k k - 1 𝐇 k T ( 𝐇 k 𝐏 k k - 1 𝐇 k T + 𝐑 k ) - 1 \mathbf{K}_{k}=\mathbf{P}_{k\mid k-1}\mathbf{H}_{k}^{T}\bigl(\mathbf{H}_{k}% \mathbf{P}_{k\mid k-1}\mathbf{H}_{k}^{T}+\mathbf{R}_{k}\bigr)^{-1}
  259. 𝐱 ^ k k = 𝐱 ^ k k - 1 + 𝐊 k ( 𝐳 k - 𝐇 k 𝐱 ^ k k - 1 ) \hat{\mathbf{x}}_{k\mid k}=\hat{\mathbf{x}}_{k\mid k-1}+\mathbf{K}_{k}(\mathbf% {z}_{k}-\mathbf{H}_{k}\hat{\mathbf{x}}_{k\mid k-1})
  260. 𝐏 k k = ( 𝐈 - 𝐊 k 𝐇 k ) 𝐏 k k - 1 \mathbf{P}_{k\mid k}=(\mathbf{I}-\mathbf{K}_{k}\mathbf{H}_{k})\mathbf{P}_{k% \mid k-1}

Kamal_(navigation).html

  1. D = S d s D=\frac{Sd}{s}
  2. D D
  3. S S
  4. d d
  5. s s

Kardashev_scale.html

  1. × 10 1 9 \times 10^{1}9
  2. × 10 3 3 \times 10^{3}3
  3. × 10 3 3 \times 10^{3}3
  4. × 10 4 4 \times 10^{4}4
  5. × 10 4 4 \times 10^{4}4
  6. K = log 10 P - 6 10 K=\frac{\log_{10}P-6}{10}

Karl_Schwarzschild.html

  1. p p
  2. i = f ( I t p ) i=f(I\cdot t^{p})
  3. i i
  4. I I
  5. t t
  6. p p
  7. S = ( 1 / 2 ) ( H 2 - E 2 ) d V + ρ ( ϕ - A u ) d V S=(1/2)\int(H^{2}-E^{2})dV+\int\rho(\phi-\vec{A}\vec{u})dV
  8. E , H \vec{E},\vec{H}
  9. A \vec{A}
  10. ϕ \phi
  11. S = i m i C i d s i + 1 2 i , j C i , C j q i q j δ ( P i P j ) d 𝐬 i d 𝐬 j S=\sum_{i}m_{i}\int_{C_{i}}ds_{i}+\frac{1}{2}\sum_{i,j}\iint_{C_{i},C_{j}}q_{i% }q_{j}\delta\left(\left\|P_{i}P_{j}\right\|\right)d\mathbf{s}_{i}d\mathbf{s}_{j}
  12. C α C_{\alpha}
  13. d 𝐬 α d\mathbf{s}_{\alpha}
  14. δ ( P i P j ) \delta\left(\left\|P_{i}P_{j}\right\|\right)
  15. R s = 2 G M c 2 R_{s}=\frac{2GM}{c^{2}}
  16. R s R_{s}

Karl_Weierstrass.html

  1. f ( x ) \displaystyle f(x)
  2. x = x 0 \displaystyle x=x_{0}
  3. ε > 0 δ > 0 \displaystyle\forall\ \varepsilon>0\ \exists\ \delta>0
  4. x x
  5. f f
  6. | x - x 0 | < δ | f ( x ) - f ( x 0 ) | < ε . \displaystyle\ |x-x_{0}|<\delta\Rightarrow|f(x)-f(x_{0})|<\varepsilon.

Kater's_pendulum.html

  1. T = 2 π L g ( 1 ) T=2\pi\sqrt{\frac{L}{g}}\qquad\qquad\qquad(1)\,
  2. g = π 2 L g=\pi^{2}L\,
  3. T 2 = T 1 2 + T 2 2 2 + T 1 2 - T 2 2 2 ( h 1 + h 2 h 1 - h 2 ) ( 2 ) T^{2}=\frac{T_{1}^{2}+T_{2}^{2}}{2}+\frac{T_{1}^{2}-T_{2}^{2}}{2}\left(\frac{h% _{1}+h_{2}}{h_{1}-h_{2}}\right)\,\qquad\qquad\qquad(2)
  4. h 1 h_{1}\,
  5. h 2 h_{2}\,
  6. h 1 + h 2 h_{1}+h_{2}\,
  7. h 1 h_{1}\,
  8. h 2 h_{2}\,
  9. h 1 - h 2 h_{1}-h_{2}\,
  10. T 1 2 - T 2 2 T_{1}^{2}-T_{2}^{2}\,
  11. T 1 2 + T 2 2 T_{1}^{2}+T_{2}^{2}\,
  12. h 1 - h 2 h_{1}-h_{2}\,

Kennedy–Thorndike_experiment.html

  1. L = L 0 1 - v 2 / c 2 = L 0 / γ ( v ) L=L_{0}\sqrt{1-v^{2}/c^{2}}=L_{0}/{\gamma(v)}
  2. L 0 L_{0}
  3. L L
  4. v v\,
  5. c c\,
  6. γ ( v ) 1 1 - v 2 / c 2 \gamma(v)\equiv\frac{1}{\sqrt{1-v^{2}/c^{2}}}
  7. T L - T T = 2 ( L L - L T ) c T_{L}-T_{T}=\frac{2(L_{L}-L_{T})}{c}
  8. T L = T 1 + T 2 = L L / γ ( v ) c - v + L L / γ ( v ) c + v T_{L}=T_{1}+T_{2}=\frac{L_{L}/\gamma(v)}{c-v}+\frac{L_{L}/\gamma(v)}{c+v}
  9. = 2 L L / γ ( v ) c 1 1 - v 2 c 2 =\frac{2L_{L}/\gamma(v)}{c}\frac{1}{1-\frac{v^{2}}{c^{2}}}
  10. = 2 L L γ ( v ) c =\frac{2L_{L}\gamma(v)}{c}
  11. T T = 2 L T c 2 - v 2 = 2 L T c 1 1 - v 2 c 2 T_{T}=\frac{2L_{T}}{\sqrt{c^{2}-v^{2}}}=\frac{2L_{T}}{c}\frac{1}{\sqrt{1-\frac% {v^{2}}{c^{2}}}}
  12. = 2 L T γ ( v ) c =\frac{2L_{T}\gamma(v)}{c}
  13. T L - T T = 2 ( L L - L T ) γ ( v ) c T_{L}-T_{T}=\frac{2(L_{L}-L_{T})\gamma(v)}{c}
  14. Δ L A = 2 ( L L - L T ) 1 - v A 2 / c 2 , Δ L B = 2 ( L L - L T ) 1 - v B 2 / c 2 \Delta L_{A}=\frac{2\left(L_{L}-L_{T}\right)}{\sqrt{1-v_{A}^{2}/c^{2}}},\qquad% \Delta L_{B}=\frac{2\left(L_{L}-L_{T}\right)}{\sqrt{1-v_{B}^{2}/c^{2}}}
  15. Δ N = Δ L A - Δ L B λ \Delta N=\frac{\Delta L_{A}-\Delta L_{B}}{\lambda}
  16. = 2 ( L L - L T ) λ ( 1 1 - v A 2 / c 2 - 1 1 - v B 2 / c 2 ) =\frac{2\left(L_{L}-L_{T}\right)}{\lambda}\left(\frac{1}{\sqrt{1-v_{A}^{2}/c^{% 2}}}-\frac{1}{\sqrt{1-v_{B}^{2}/c^{2}}}\right)
  17. L L - L T λ ( v A 2 - v B 2 c 2 ) \approx\frac{L_{L}-L_{T}}{\lambda}\left(\frac{v_{A}^{2}-v_{B}^{2}}{c^{2}}\right)
  18. 10 - 5 \lesssim 10^{-5}
  19. 10 - 7 \lesssim 10^{-7}
  20. 10 - 8 \lesssim 10^{-8}

Kilowatt_hour.html

  1. kW h = ( 3600 s ) [ kW ] = 3600 [ s ] [ kJ s ] = 3600 kJ = 3.6 MJ \mathrm{kW\cdot h}=(3600\,\mathrm{s})[\mathrm{kW}]=3600\,[\mathrm{s}]\Bigg[% \frac{\mathrm{kJ}}{\mathrm{s}}\Bigg]=3600\,\mathrm{kJ}=3.6\,\mathrm{MJ}

Kip_Thorne.html

  1. 4 π G M c 2 \begin{matrix}\frac{4\pi GM}{c^{2}}\end{matrix}

Kite_(geometry).html

  1. A = p q 2 . A=\frac{p\cdot q}{2}.
  2. A = a b sin θ . \displaystyle A=ab\cdot\sin{\theta}.
  3. r 1 + r 3 = r 2 + r 4 . r_{1}+r_{3}=r_{2}+r_{4}.
  4. R 1 + R 3 = R 2 + R 4 . R_{1}+R_{3}=R_{2}+R_{4}.

Kleene's_recursion_theorem.html

  1. φ \varphi
  2. e e
  3. φ e \varphi_{e}
  4. e e
  5. φ e \varphi_{e}
  6. F F
  7. F F
  8. e e
  9. φ e φ F ( e ) \varphi_{e}\simeq\varphi_{F(e)}
  10. e e
  11. F ( e ) F(e)
  12. F F
  13. h h
  14. x x
  15. h h
  16. y y
  17. φ x ( x ) \varphi_{x}(x)
  18. e e
  19. φ e ( y ) \varphi_{e}(y)
  20. x x
  21. φ x ( x ) \varphi_{x}(x)
  22. φ h ( x ) = φ φ x ( x ) \varphi_{h(x)}=\varphi_{\varphi_{x}(x)}
  23. φ x ( x ) \varphi_{x}(x)
  24. φ h ( x ) \varphi_{h(x)}\,
  25. φ h ( x ) φ φ x ( x ) \varphi_{h(x)}\simeq\varphi_{\varphi_{x}(x)}
  26. h h
  27. g ( x , y ) = φ φ x ( x ) ( y ) g(x,y)=\varphi_{\varphi_{x}(x)}(y)
  28. x x
  29. h ( x ) h(x)
  30. y g ( x , y ) y\mapsto g(x,y)
  31. F F
  32. h h
  33. e e
  34. F h F\circ h
  35. φ h ( e ) φ φ e ( e ) \varphi_{h(e)}\simeq\varphi_{\varphi_{e}(e)}
  36. h h
  37. e e
  38. F h F\circ h
  39. φ e ( e ) = ( F h ) ( e ) \varphi_{e}(e)=(F\circ h)(e)
  40. φ φ e ( e ) φ F ( h ( e ) ) \varphi_{\varphi_{e}(e)}\simeq\varphi_{F(h(e))}
  41. \simeq
  42. φ h ( e ) φ F ( h ( e ) ) \varphi_{h(e)}\simeq\varphi_{F(h(e))}
  43. φ n φ F ( n ) \varphi_{n}\simeq\varphi_{F(n)}
  44. n = h ( e ) n=h(e)
  45. F F
  46. φ e ≄ φ F ( e ) \varphi_{e}\not\simeq\varphi_{F(e)}
  47. e e
  48. Q ( x , y ) Q(x,y)
  49. p p
  50. φ p λ y . Q ( p , y ) \varphi_{p}\simeq\lambda y.Q(p,y)
  51. Q Q
  52. p p
  53. p p
  54. y y
  55. F ( p ) F(p)
  56. φ F ( p ) ( y ) = Q ( p , y ) \varphi_{F(p)}(y)=Q(p,y)
  57. F F
  58. p p
  59. g g
  60. h h
  61. f f
  62. f ( 0 , y ) g ( y ) , f(0,y)\simeq g(y),\,
  63. f ( x + 1 , y ) h ( f ( x , y ) , x , y ) , f(x+1,y)\simeq h(f(x,y),x,y),\,
  64. φ F ( e , x , y ) \varphi_{F}(e,x,y)
  65. e e
  66. φ F ( e , 0 , y ) g ( y ) , \varphi_{F}(e,0,y)\simeq g(y),\,
  67. φ F ( e , x + 1 , y ) h ( φ e ( x , y ) , x , y ) . \varphi_{F}(e,x+1,y)\simeq h(\varphi_{e}(x,y),x,y).\,
  68. φ f \varphi_{f}
  69. φ f ( x , y ) φ F ( f , x , y ) \varphi_{f}(x,y)\simeq\varphi_{F}(f,x,y)
  70. f f
  71. Q ( x , y ) = x Q(x,y)=x
  72. p p
  73. p p
  74. Q Q
  75. φ \varphi
  76. Φ ( X ) = { n A X [ ( A , n ) Φ ] } . \Phi(X)=\{n\mid\exists A\subseteq X[(A,n)\in\Phi]\}.
  77. ( , ( 0 , 1 ) ) (\varnothing,(0,1))
  78. ( { ( n , m ) } , ( n + 1 , ( n + 1 ) m ) ) (\{(n,m)\},(n+1,(n+1)\cdot m))
  79. k = 0 , 1 , k=0,1,\ldots
  80. F k Φ ( F k ) F_{k}\cup\Phi(F_{k})
  81. F k \bigcup F_{k}
  82. ν \nu
  83. f f
  84. t t
  85. n : ν f ( n , t ( n ) ) = ν t ( n ) . \forall n\in\mathbb{N}:\nu\circ f(n,t(n))=\nu\circ t(n).

Kleene_algebra.html

  1. V * = i = 0 V n V^{*}=\bigoplus_{i=0}^{\infty}V^{n}
  2. \lor
  3. \land
  4. \lor
  5. \land

Klein–Gordon_equation.html

  1. 1 c 2 2 t 2 ψ - 2 ψ + m 2 c 2 2 ψ = 0. \frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}\psi-\nabla^{2}\psi+\frac{m^% {2}c^{2}}{\hbar^{2}}\psi=0.
  2. ( + μ 2 ) ψ = 0 , (\Box+\mu^{2})\psi=0,
  3. μ = m c ħ μ=\frac{mc}{ħ}
  4. = - η μ ν μ ν = 1 c 2 2 t 2 - 2 . \Box=-\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}=\frac{1}{c^{2}}\frac{\partial^% {2}}{\partial t^{2}}-\nabla^{2}.
  5. - t 2 ψ + 2 ψ = m 2 ψ -\partial_{t}^{2}\psi+\nabla^{2}\psi=m^{2}\psi
  6. ψ = e - i ω t + i k x = e i k μ x μ \psi=e^{-i\omega t+ik\cdot x}=e^{ik_{\mu}x^{\mu}}
  7. - p μ p μ = E 2 - P 2 = ω 2 - k 2 = - k μ k μ = m 2 -p_{\mu}p^{\mu}=E^{2}-P^{2}=\omega^{2}-k^{2}=-k_{\mu}k^{\mu}=m^{2}
  8. ω ω
  9. k k
  10. [ 2 - m 2 c 2 2 ] ψ ( 𝐫 ) = 0 \left[\nabla^{2}-\frac{m^{2}c^{2}}{\hbar^{2}}\right]\psi(\mathbf{r})=0
  11. 4 n 2 4\frac{n}{2}
  12. n n
  13. j j
  14. j j
  15. 𝐩 2 2 m = E . \frac{\mathbf{p}^{2}}{2m}=E.
  16. 𝐩 ^ 2 2 m ψ = E ^ ψ \frac{\mathbf{\hat{p}}^{2}}{2m}\psi=\hat{E}\psi
  17. 𝐩 ^ = - i \mathbf{\hat{p}}=-i\hbar\mathbf{\nabla}
  18. E ^ = i t \hat{E}=i\hbar\dfrac{\partial}{\partial t}
  19. 𝐩 2 c 2 + m 2 c 4 = E \sqrt{\mathbf{p}^{2}c^{2}+m^{2}c^{4}}=E
  20. ( - i ) 2 c 2 + m 2 c 4 ψ = i t ψ . \sqrt{(-i\hbar\mathbf{\nabla})^{2}c^{2}+m^{2}c^{4}}\psi=i\hbar\frac{\partial}{% \partial t}\psi.
  21. 𝐩 2 c 2 + m 2 c 4 = E 2 \mathbf{p}^{2}c^{2}+m^{2}c^{4}=E^{2}
  22. ( ( - i ) 2 c 2 + m 2 c 4 ) ψ = ( i t ) 2 ψ \left((-i\hbar\mathbf{\nabla})^{2}c^{2}+m^{2}c^{4}\right)\psi=\left(i\hbar% \frac{\partial}{\partial t}\right)^{2}\psi
  23. - 2 c 2 2 ψ + m 2 c 4 ψ = - 2 2 t 2 ψ . -\hbar^{2}c^{2}\mathbf{\nabla}^{2}\psi+m^{2}c^{4}\psi=-\hbar^{2}\frac{\partial% ^{2}}{\partial t^{2}}\psi.
  24. 1 c 2 2 t 2 ψ - 2 ψ + m 2 c 2 2 ψ = 0. \frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}\psi-\mathbf{\nabla}^{2}\psi% +\frac{m^{2}c^{2}}{\hbar^{2}}\psi=0.
  25. - η μ ν μ ν ψ + m 2 c 2 2 ψ = 0 -\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}\psi+\frac{m^{2}c^{2}}{\hbar^{2}}% \psi=0
  26. ( + μ 2 ) ψ = 0 , (\Box+\mu^{2})\psi=0,
  27. μ = m c \mu=\frac{mc}{\hbar}
  28. = 1 c 2 2 t 2 - 2 . \Box=\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}-\nabla^{2}.
  29. V ( ψ ) V(ψ)
  30. ψ + V ψ = 0 \Box\psi+\frac{\partial{}V}{\partial\psi}=0
  31. ϕ ( x ) \phi(x)\in\mathbb{C}
  32. μ J μ ( x ) = 0 , J μ ( x ) ϕ * ( x ) μ ϕ ( x ) - ϕ ( x ) μ ϕ * ( x ) . \partial_{\mu}J^{\mu}(x)=0,\qquad J^{\mu}(x)\equiv\phi^{*}(x)\partial^{\mu}% \phi(x)-\phi(x)\partial^{\mu}\phi^{*}(x).
  33. ϕ ( x ) \phi(x)
  34. m m
  35. ( + m 2 ) ϕ ( x ) = 0 , (\square+m^{2})\phi(x)=0,
  36. ( + m 2 ) ϕ * ( x ) = 0 , (\square+m^{2})\phi^{*}(x)=0,
  37. ϕ * ( x ) \phi^{*}(x)
  38. ϕ ( x ) \phi(x)
  39. x x
  40. ϕ * ( + m 2 ) ϕ = 0 , \phi^{*}(\square+m^{2})\phi=0,
  41. ϕ ( + m 2 ) ϕ * = 0. \phi(\square+m^{2})\phi^{*}=0.
  42. ϕ * ϕ - ϕ ϕ * = 0 \phi^{*}\square\phi-\phi\square\phi^{*}=0
  43. μ J μ ( x ) = 0 , J μ ( x ) ϕ * ( x ) μ ϕ ( x ) - ϕ ( x ) μ ϕ * ( x ) . \partial_{\mu}J^{\mu}(x)=0,\qquad J^{\mu}(x)\equiv\phi^{*}(x)\partial^{\mu}% \phi(x)-\phi(x)\partial^{\mu}\phi^{*}(x).
  44. 2 ψ - 1 c 2 2 t 2 ψ = m 2 c 2 2 ψ \mathbf{\nabla}^{2}\psi-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}\psi% =\frac{m^{2}c^{2}}{\hbar^{2}}\psi
  45. ψ ( 𝐫 , t ) = e i ( 𝐤 𝐫 - ω t ) \psi(\mathbf{r},t)=e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}
  46. ω ω∈ℝ
  47. - | 𝐤 | 2 + ω 2 c 2 = m 2 c 2 2 . -|\mathbf{k}|^{2}+\frac{\omega^{2}}{c^{2}}=\frac{m^{2}c^{2}}{\hbar^{2}}.
  48. ω ω
  49. 𝐤 \mathbf{k}
  50. 𝐩 = ψ | - i | ψ = 𝐤 , \langle\mathbf{p}\rangle=\left\langle\psi\left|-i\hbar\mathbf{\nabla}\right|% \psi\right\rangle=\hbar\mathbf{k},
  51. E = ψ | i t | ψ = ω . \langle E\rangle=\left\langle\psi\left|i\hbar\frac{\partial}{\partial t}\right% |\psi\right\rangle=\hbar\omega.
  52. E 2 = m 2 c 4 + 𝐩 2 c 2 . \langle E\rangle^{2}=m^{2}c^{4}+\langle\mathbf{p}\rangle^{2}c^{2}.
  53. m = 0 m=0
  54. E = | 𝐩 | c . \langle E\rangle=\langle|\mathbf{p}|\rangle c.
  55. 𝒮 = ( - 2 m η μ ν μ ψ ¯ ν ψ - m c 2 ψ ¯ ψ ) d 4 x \mathcal{S}=\int\left(-\frac{\hbar^{2}}{m}\eta^{\mu\nu}\partial_{\mu}\bar{\psi% }\partial_{\nu}\psi-mc^{2}\bar{\psi}\psi\right)\mathrm{d}^{4}x
  56. ψ ψ
  57. m m
  58. ψ ψ
  59. [ u o v e r b a r , u 3 c 8 ] [u^{\prime}overbar^{\prime},u^{\prime}\u{0}3c8^{\prime}]
  60. [ u o v e r b a r , u 3 c 8 ] = ψ [u^{\prime}overbar^{\prime},u^{\prime}\u{0}3c8^{\prime}]=ψ
  61. T μ ν = 2 m ( η μ α η ν β + η μ β η ν α - η μ ν η α β ) α ψ ¯ β ψ - η μ ν m c 2 ψ ¯ ψ . T^{\mu\nu}=\frac{\hbar^{2}}{m}\left(\eta^{\mu\alpha}\eta^{\nu\beta}+\eta^{\mu% \beta}\eta^{\nu\alpha}-\eta^{\mu\nu}\eta^{\alpha\beta}\right)\partial_{\alpha}% \bar{\psi}\partial_{\beta}\psi-\eta^{\mu\nu}mc^{2}\bar{\psi}\psi.
  62. D μ D μ ϕ = - ( t - i e A 0 ) 2 ϕ + ( i - i e A i ) 2 ϕ = m 2 ϕ D_{\mu}D^{\mu}\phi=-(\partial_{t}-ieA_{0})^{2}\phi+(\partial_{i}-ieA_{i})^{2}% \phi=m^{2}\phi
  63. A A
  64. D μ D μ ϕ + A F μ ν D μ ϕ D ν ( D α D α ϕ ) = 0 D_{\mu}D^{\mu}\phi+AF^{\mu\nu}D_{\mu}\phi D_{\nu}(D_{\alpha}D^{\alpha}\phi)=0
  65. i i
  66. S = x ( μ ϕ * + i e A μ ϕ * ) ( ν ϕ - i e A ν ϕ ) η μ ν = x | D ϕ | 2 S=\int_{x}\left(\partial_{\mu}\phi^{*}+ieA_{\mu}\phi^{*}\right)\left(\partial_% {\nu}\phi-ieA_{\nu}\phi\right)\eta^{\mu\nu}=\int_{x}|D\phi|^{2}
  67. 0 = - g μ ν μ ν ψ + m 2 c 2 2 ψ = - g μ ν μ ( ν ψ ) + m 2 c 2 2 ψ = - g μ ν μ ν ψ + g μ ν Γ σ σ μ ν ψ + m 2 c 2 2 ψ \begin{aligned}\displaystyle 0&\displaystyle=-g^{\mu\nu}\nabla_{\mu}\nabla_{% \nu}\psi+\dfrac{m^{2}c^{2}}{\hbar^{2}}\psi=-g^{\mu\nu}\nabla_{\mu}(\partial_{% \nu}\psi)+\dfrac{m^{2}c^{2}}{\hbar^{2}}\psi\\ &\displaystyle=-g^{\mu\nu}\partial_{\mu}\partial_{\nu}\psi+g^{\mu\nu}\Gamma^{% \sigma}{}_{\mu\nu}\partial_{\sigma}\psi+\dfrac{m^{2}c^{2}}{\hbar^{2}}\psi\end{aligned}
  68. - 1 - g μ ( g μ ν - g ν ψ ) + m 2 c 2 2 ψ = 0 \frac{-1}{\sqrt{-g}}\partial_{\mu}\left(g^{\mu\nu}\sqrt{-g}\partial_{\nu}\psi% \right)+\frac{m^{2}c^{2}}{\hbar^{2}}\psi=0
  69. < s u p > α β <sup>αβ

KN.html

  1. n n
  2. K n K_{n}

Knaster–Tarski_theorem.html

  1. A F ( A ) A\subseteq F(A)
  2. A = F ( A ) A=F(A)
  3. L , \langle L,\leq\rangle
  4. f : L L f\colon L\rightarrow L
  5. P , \langle P,\leq\rangle
  6. P = { x L x f ( x ) } \bigvee P=\bigvee\{x\in L\mid x\leq f(x)\}
  7. P = { x L x f ( x ) } \bigwedge P=\bigwedge\{x\in L\mid x\geq f(x)\}
  8. u = D u=\bigvee D
  9. L o p , \langle L^{op},\geq\rangle
  10. 1 L = L 1_{L}=\bigvee L
  11. w = W w=\bigvee W

Knot_invariant.html

  1. 3 \mathbb{R}^{3}
  2. K κ d s 4 π , \oint_{K}\kappa\,ds\leq 4\pi,
  3. κ ( p ) \kappa(p)
  4. K κ d s > 4 π . \oint_{K}\kappa\,ds>4\pi.\,

Knot_polynomial.html

  1. Δ ( t ) \Delta(t)
  2. ( z ) \nabla(z)
  3. V ( q ) V(q)
  4. H ( a , z ) H(a,z)
  5. 0 1 0_{1}
  6. 1 1
  7. 1 1
  8. 1 1
  9. 1 1
  10. 3 1 3_{1}
  11. t - 1 + t - 1 t-1+t^{-1}
  12. z 2 + 1 z^{2}+1
  13. q - 1 + q - 3 - q - 4 q^{-1}+q^{-3}-q^{-4}
  14. - a 4 + a 2 z 2 + 2 a 2 -a^{4}+a^{2}z^{2}+2a^{2}
  15. 4 1 4_{1}
  16. - t + 3 - t - 1 -t+3-t^{-1}
  17. - z 2 + 1 -z^{2}+1
  18. q 2 - q + 1 - q - 1 + q - 2 q^{2}-q+1-q^{-1}+q^{-2}
  19. a 2 + a - 2 - z 2 - 1 a^{2}+a^{-2}-z^{2}-1
  20. 5 1 5_{1}
  21. t 2 - t + 1 - t - 1 + t - 2 t^{2}-t+1-t^{-1}+t^{-2}
  22. z 4 + 3 z 2 + 1 z^{4}+3z^{2}+1
  23. q - 2 + q - 4 - q - 5 + q - 6 - q - 7 q^{-2}+q^{-4}-q^{-5}+q^{-6}-q^{-7}
  24. - a 6 z 2 - 2 a 6 + a 4 z 4 + 4 a 4 z 2 + 3 a 4 -a^{6}z^{2}-2a^{6}+a^{4}z^{4}+4a^{4}z^{2}+3a^{4}
  25. - -
  26. ( t - 1 + t - 1 ) 2 \left(t-1+t^{-1}\right)^{2}
  27. ( z 2 + 1 ) 2 \left(z^{2}+1\right)^{2}
  28. ( q - 1 + q - 3 - q - 4 ) 2 \left(q^{-1}+q^{-3}-q^{-4}\right)^{2}
  29. ( - a 4 + a 2 z 2 + 2 a 2 ) 2 \left(-a^{4}+a^{2}z^{2}+2a^{2}\right)^{2}
  30. - -
  31. ( t - 1 + t - 1 ) 2 \left(t-1+t^{-1}\right)^{2}
  32. ( z 2 + 1 ) 2 \left(z^{2}+1\right)^{2}
  33. ( q - 1 + q - 3 - q - 4 ) ( q + q 3 - q 4 ) \left(q^{-1}+q^{-3}-q^{-4}\right)\left(q+q^{3}-q^{4}\right)
  34. ( - a 4 + a 2 z 2 + 2 a 2 ) × \left(-a^{4}+a^{2}z^{2}+2a^{2}\right)\times
  35. ( - a - 4 + a - 2 z - 2 + 2 a - 2 ) \left(-a^{-4}+a^{-2}z^{-2}+2a^{-2}\right)
  36. 3 \mathbb{R}^{3}

Knot_theory.html

  1. K K
  2. K : [ 0 , 1 ] 3 K:[0,1]\to\mathbb{R}^{3}
  3. K ( 0 ) = K ( 1 ) K(0)=K(1)
  4. K 1 , K 2 K_{1},K_{2}
  5. h : \R 3 \R 3 h\colon\R^{3}\to\R^{3}
  6. h ( K 1 ) = K 2 h(K_{1})=K_{2}
  7. L + , L - , L 0 L_{+},L_{-},L_{0}
  8. L + L_{+}
  9. L - L_{-}
  10. C ( L + ) = C ( L - ) + z C ( L 0 ) . C(L_{+})=C(L_{-})+zC(L_{0}).
  11. 1 2 {}^{2}_{1}

Knuth's_up-arrow_notation.html

  1. a × b = a + a + + a b copies of a \begin{matrix}a\times b&=&\underbrace{a+a+\dots+a}\\ &&b\mbox{ copies of }~{}a\end{matrix}
  2. 4 × 3 = 4 + 4 + 4 = 12 3 copies of 4 \begin{matrix}4\times 3&=&\underbrace{4+4+4}&=&12\\ &&3\mbox{ copies of }~{}4\end{matrix}
  3. b b
  4. a b = a b = a × a × × a b copies of a \begin{matrix}a\uparrow b=a^{b}=&\underbrace{a\times a\times\dots\times a}\\ &b\mbox{ copies of }~{}a\end{matrix}
  5. 4 3 = 4 3 = 4 × 4 × 4 = 64 3 copies of 4 \begin{matrix}4\uparrow 3=4^{3}=&\underbrace{4\times 4\times 4}&=&64\\ &3\mbox{ copies of }~{}4\end{matrix}
  6. a b = a b = a a . . . a = a ( a ( a ) ) b copies of a b copies of a \begin{matrix}a\uparrow\uparrow b&={\ {}^{b}a}=&\underbrace{a^{a^{{}^{.\,^{.\,% ^{.\,^{a}}}}}}}&=&\underbrace{a\uparrow(a\uparrow(\dots\uparrow a))}\\ &&b\mbox{ copies of }~{}a&&b\mbox{ copies of }~{}a\end{matrix}
  7. 4 3 = 4 3 = 4 4 4 = 4 ( 4 4 ) = 4 256 1.34078079 × 10 154 3 copies of 4 3 copies of 4 \begin{matrix}4\uparrow\uparrow 3&={\ {}^{3}4}=&\underbrace{4^{4^{4}}}&=&% \underbrace{4\uparrow(4\uparrow 4)}&=&4^{256}&\approx&1.34078079\times 10^{154% }&\\ &&3\mbox{ copies of }~{}4&&3\mbox{ copies of }~{}4\end{matrix}
  8. 3 2 = 3 3 = 27 3\uparrow\uparrow 2=3^{3}=27
  9. 3 3 = 3 3 3 = 3 27 = 7625597484987 3\uparrow\uparrow 3=3^{3^{3}}=3^{27}=7625597484987
  10. 3 4 = 3 3 3 3 = 3 3 27 = 3 7625597484987 1.2580143 × 10 3638334640024 3\uparrow\uparrow 4=3^{3^{3^{3}}}=3^{3^{27}}=3^{7625597484987}\approx 1.258014% 3\times 10^{3638334640024}
  11. 3 5 = 3 3 3 3 3 = 3 3 3 27 = 3 3 7625597484987 3\uparrow\uparrow 5=3^{3^{3^{3^{3}}}}=3^{3^{3^{27}}}=3^{3^{7625597484987}}
  12. a b = a ( a ( a ) ) b copies of a \begin{matrix}a\uparrow\uparrow\uparrow b=&\underbrace{a\uparrow\uparrow(a% \uparrow\uparrow(\dots\uparrow\uparrow a))}\\ &b\mbox{ copies of }~{}a\end{matrix}
  13. a b = a ( a ( a ) ) b copies of a \begin{matrix}a\uparrow\uparrow\uparrow\uparrow b=&\underbrace{a\uparrow% \uparrow\uparrow(a\uparrow\uparrow\uparrow(\dots\uparrow\uparrow\uparrow a))}% \\ &b\mbox{ copies of }~{}a\end{matrix}
  14. n n
  15. n - 1 n-1
  16. a n b = a n - 1 ( a n - 1 ( n - 1 a ) ) b copies of a \begin{matrix}a\ \underbrace{\uparrow\uparrow\!\!\dots\!\!\uparrow}_{n}\ b=% \underbrace{a\ \underbrace{\uparrow\!\!\dots\!\!\uparrow}_{n-1}\ (a\ % \underbrace{\uparrow\!\!\dots\!\!\uparrow}_{n-1}\ (\dots\ \underbrace{\uparrow% \!\!\dots\!\!\uparrow}_{n-1}\ a))}_{b\,\text{ copies of }a}\end{matrix}
  17. 3 2 = 3 3 = 3 3 3 = 3 27 = 7 , 625 , 597 , 484 , 987 3\uparrow\uparrow\uparrow 2=3\uparrow\uparrow 3=3^{3^{3}}=3^{27}=7,625,597,484% ,987
  18. 3 3 = 3 ( 3 3 ) = 3 ( 3 3 3 ) = 3 3 3 3 3 3 copies of 3 = 3 3 3 7,625,597,484,987 copies of 3 = 3 3 3 3 3 7,625,597,484,987 copies of 3 \begin{matrix}3\uparrow\uparrow\uparrow 3=3\uparrow\uparrow(3\uparrow\uparrow 3% )=3\uparrow\uparrow(3\uparrow 3\uparrow 3)=&\underbrace{3\uparrow 3\uparrow% \dots\uparrow 3}\\ &3\uparrow 3\uparrow 3\mbox{ copies of }~{}3\end{matrix}\begin{matrix}=&% \underbrace{3\uparrow 3\uparrow\dots\uparrow 3}\\ &\mbox{7,625,597,484,987 copies of 3}\end{matrix}\begin{matrix}=&\underbrace{3% ^{3^{3^{3^{\cdot^{\cdot^{\cdot^{\cdot^{3}}}}}}}}}\\ &\mbox{7,625,597,484,987 copies of 3}\end{matrix}
  19. a n b a\uparrow^{n}b
  20. a b a\uparrow\uparrow\dots\uparrow b
  21. a n b a\uparrow^{n}b
  22. 39 14 39\uparrow\uparrow 14
  23. 77 77 77 77\uparrow^{77}77
  24. a b a^{b}
  25. b b
  26. a a
  27. a b a\uparrow b
  28. a b a^{b}
  29. a b a\uparrow b
  30. a n b a\uparrow^{n}b
  31. a 4 b = a b a\uparrow^{4}b=a\uparrow\uparrow\uparrow\uparrow b
  32. a b a\uparrow\uparrow b
  33. a 4 = a ( a ( a a ) ) = a a a a a\uparrow\uparrow 4=a\uparrow(a\uparrow(a\uparrow a))=a^{a^{a^{a}}}
  34. a b = a a . . . a b a\uparrow\uparrow b=\underbrace{a^{a^{.^{.^{.{a}}}}}}_{b}
  35. a b a\uparrow\uparrow\uparrow b
  36. a 4 = a ( a ( a a ) ) = a a . . . a a a . . . a a a . . . a a a\uparrow\uparrow\uparrow 4=a\uparrow\uparrow(a\uparrow\uparrow(a\uparrow% \uparrow a))=\underbrace{a^{a^{.^{.^{.{a}}}}}}_{\underbrace{a^{a^{.^{.^{.{a}}}% }}}_{\underbrace{a^{a^{.^{.^{.{a}}}}}}_{a}}}
  37. a b = a a . . . a a a . . . a a } b a\uparrow\uparrow\uparrow b=\left.\underbrace{a^{a^{.^{.^{.{a}}}}}}_{% \underbrace{a^{a^{.^{.^{.{a}}}}}}_{\underbrace{\vdots}_{a}}}\right\}b
  38. a b a\uparrow\uparrow\uparrow\uparrow b
  39. a 4 = a ( a ( a a ) ) = a a . . . a a a . . . a a } a a . . . a a a . . . a a } a a . . . a a a . . . a a } a a\uparrow\uparrow\uparrow\uparrow 4=a\uparrow\uparrow\uparrow(a\uparrow% \uparrow\uparrow(a\uparrow\uparrow\uparrow a))=\left.\left.\left.\underbrace{a% ^{a^{.^{.^{.{a}}}}}}_{\underbrace{a^{a^{.^{.^{.{a}}}}}}_{\underbrace{\vdots}_{% a}}}\right\}\underbrace{a^{a^{.^{.^{.{a}}}}}}_{\underbrace{a^{a^{.^{.^{.{a}}}}% }}_{\underbrace{\vdots}_{a}}}\right\}\underbrace{a^{a^{.^{.^{.{a}}}}}}_{% \underbrace{a^{a^{.^{.^{.{a}}}}}}_{\underbrace{\vdots}_{a}}}\right\}a
  40. a b = a a . . . a a a . . . a a } a a . . . a a a . . . a a } } a b a\uparrow\uparrow\uparrow\uparrow b=\underbrace{\left.\left.\left.\underbrace{% a^{a^{.^{.^{.{a}}}}}}_{\underbrace{a^{a^{.^{.^{.{a}}}}}}_{\underbrace{\vdots}_% {a}}}\right\}\underbrace{a^{a^{.^{.^{.{a}}}}}}_{\underbrace{a^{a^{.^{.^{.{a}}}% }}}_{\underbrace{\vdots}_{a}}}\right\}\cdots\right\}a}_{b}
  41. a n b a\uparrow^{n}b
  42. a b {}^{b}a
  43. a b = a b a\uparrow\uparrow b={}^{b}a
  44. a b = a a . . . a b a\uparrow\uparrow\uparrow b=\underbrace{{}^{{}^{{}^{{}^{{}^{a}.}.}.}a}a}_{b}
  45. a b = a a . . . a a a . . . a a } b a\uparrow\uparrow\uparrow\uparrow b=\left.\underbrace{{}^{{}^{{}^{{}^{{}^{a}.}% .}.}a}a}_{\underbrace{{}^{{}^{{}^{{}^{{}^{a}.}.}.}a}a}_{\underbrace{\vdots}_{a% }}}\right\}b
  46. 4 4 4 4\uparrow^{4}4
  47. 4 4 . . . 4 4 4 . . . 4 4 4 . . . 4 4 = 4 4 . . . 4 4 4 . . . 4 4 4 4 4 \underbrace{{}^{{}^{{}^{{}^{{}^{4}.}.}.}4}4}_{\underbrace{{}^{{}^{{}^{{}^{{}^{% 4}.}.}.}4}4}_{\underbrace{{}^{{}^{{}^{{}^{{}^{4}.}.}.}4}4}_{4}}}=\underbrace{{% }^{{}^{{}^{{}^{{}^{4}.}.}.}4}4}_{\underbrace{{}^{{}^{{}^{{}^{{}^{4}.}.}.}4}4}_% {{}^{{}^{{}^{4}4}4}4}}
  48. n \uparrow^{n}
  49. a n b = a [ n + 2 ] b = a b n (Knuth) (hyperoperation) (Conway) \begin{matrix}a\uparrow^{n}b&=&a[n+2]b&=&a\to b\to n\\ \mbox{(Knuth)}&&\mbox{(hyperoperation)}&&\mbox{(Conway)}\end{matrix}
  50. a n b = { a b , if n = 0 ; 1 , if n 1 and b = 0 ; a n - 1 ( a n ( b - 1 ) ) , otherwise a\uparrow^{n}b=\left\{\begin{matrix}ab,&\mbox{if }~{}n=0;\\ 1,&\mbox{if }~{}n\geq 1\mbox{ and }~{}b=0;\\ a\uparrow^{n-1}(a\uparrow^{n}(b-1)),&\mbox{otherwise }\end{matrix}\right.
  51. a , b , n a,b,n
  52. b 0 , n 0 b\geq 0,n\geq 0
  53. ( a 0 b = a b ) (a\uparrow^{0}b=ab)
  54. ( a 1 b = a b = a b ) (a\uparrow^{1}b=a\uparrow b=a^{b})
  55. ( a 2 b = a b ) (a\uparrow^{2}b=a\uparrow\uparrow b)
  56. a b a\uparrow b
  57. a b c = a ( b c ) a\uparrow b\uparrow c=a\uparrow(b\uparrow c)
  58. ( a b ) c (a\uparrow b)\uparrow c
  59. 3 3 = 3 3 3 3\uparrow\uparrow 3=3^{3^{3}}
  60. 3 ( 3 3 ) = 3 27 = 7625597484987 3^{(3^{3})}=3^{27}=7625597484987
  61. ( 3 3 ) 3 = 27 3 = 19683. \left(3^{3}\right)^{3}=27^{3}=19683.
  62. b 1 , n 1 b\geq 1,n\geq 1
  63. a n b = a n - 1 a n - 1 a n - 1 a ( with b a ’s ) = a n - 1 a n - 1 a n - 1 a n - 1 1 ( with b a ’s ) = ( a n - 1 ) b 1 \begin{array}[]{lcl}a\uparrow^{n}b&=&a\uparrow^{n-1}a\uparrow^{n-1}\cdots a% \uparrow^{n-1}a\ \ (\,\text{with }b\ a\,\text{'s})\\ &=&a\uparrow^{n-1}a\uparrow^{n-1}\cdots a\uparrow^{n-1}a\uparrow^{n-1}1\ \ (\,% \text{with }b\ a\,\text{'s})\\ &=&(a\uparrow^{n-1})^{b}1\end{array}
  64. a a
  65. ( a m ) b (a\uparrow^{m})^{b}
  66. f ( x ) = a m x f(x)=a\uparrow^{m}x
  67. ( a m ) 0 n = n (a\uparrow^{m})^{0}n=n
  68. a n b = { a b , if n = 0 ; ( a n - 1 ) b 1 if n 1 a\uparrow^{n}b=\left\{\begin{matrix}ab,&\mbox{if }~{}n=0;\\ (a\uparrow^{n-1})^{b}1&\mbox{if }~{}n\geq 1\end{matrix}\right.
  69. a , b , n a,b,n
  70. b 0 , n 0 b\geq 0,n\geq 0
  71. 2 m n 2\uparrow^{m}n
  72. 2 n 2^{n}
  73. 2 m n 2\uparrow^{m}n
  74. 2 n 2^{n}
  75. 2 65 536 2.0 × 10 19 728 2^{65\,536}\approx 2.0\times 10^{19\,728}
  76. 2 2 65 536 10 6.0 × 10 19 727 2^{2^{65\,536}}\approx 10^{6.0\times 10^{19\,727}}
  77. 2 n 2\uparrow\uparrow n
  78. 2 2 . . . 2 65536 copies of 2 \begin{matrix}\underbrace{2^{2^{{}^{.\,^{.\,^{.\,^{2}}}}}}}\\ 65536\mbox{ copies of }~{}2\end{matrix}
  79. 2 2 . . . 2 2 2 . . . 2 65536 copies of 2 \begin{matrix}\underbrace{2^{2^{{}^{.\,^{.\,^{.\,^{2}}}}}}}\\ \underbrace{2^{2^{{}^{.\,^{.\,^{.\,^{2}}}}}}}\\ 65536\mbox{ copies of }~{}2\end{matrix}
  80. 2 2 . . . 2 2 2 . . . 2 2 2 . . . 2 65536 copies of 2 \begin{matrix}\underbrace{2^{2^{{}^{.\,^{.\,^{.\,^{2}}}}}}}\\ \underbrace{2^{2^{{}^{.\,^{.\,^{.\,^{2}}}}}}}\\ \underbrace{2^{2^{{}^{.\,^{.\,^{.\,^{2}}}}}}}\\ 65536\mbox{ copies of }~{}2\end{matrix}
  81. 2 n 2\uparrow\uparrow\uparrow n
  82. 2 2 . . . 2 65536 copies of 2 \begin{matrix}\underbrace{2^{2^{{}^{.\,^{.\,^{.\,^{2}}}}}}}\\ 65536\mbox{ copies of }~{}2\end{matrix}
  83. 2 n 2\uparrow\uparrow\uparrow\uparrow n
  84. m m
  85. n n
  86. 3 n 3^{n}
  87. 3 m n 3\uparrow^{m}n
  88. 3 n 3^{n}
  89. 3 7 , 625 , 597 , 484 , 987 3^{7{,}625{,}597{,}484{,}987}
  90. 3 n 3\uparrow\uparrow n
  91. 3 3 . . . 3 7 , 625 , 597 , 484 , 987 copies of 3 \begin{matrix}\underbrace{3^{3^{{}^{.\,^{.\,^{.\,^{3}}}}}}}\\ 7{,}625{,}597{,}484{,}987\mbox{ copies of }~{}3\end{matrix}
  92. 3 n 3\uparrow\uparrow\uparrow n
  93. 3 3 . . . 3 7 , 625 , 597 , 484 , 987 copies of 3 \begin{matrix}\underbrace{3^{3^{{}^{.\,^{.\,^{.\,^{3}}}}}}}\\ 7{,}625{,}597{,}484{,}987\mbox{ copies of }~{}3\end{matrix}
  94. 3 n 3\uparrow\uparrow\uparrow\uparrow n
  95. 10 n 10^{n}
  96. 10 m n 10\uparrow^{m}n
  97. 10 n 10^{n}
  98. 10 10 , 000 , 000 , 000 10^{10,000,000,000}
  99. 10 10 10 , 000 , 000 , 000 10^{10^{10,000,000,000}}
  100. 10 10 10 10 , 000 , 000 , 000 10^{10^{10^{10,000,000,000}}}
  101. 10 n 10\uparrow\uparrow n
  102. 10 10 . . . 10 10 copies of 10 \begin{matrix}\underbrace{10^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}}\\ 10\mbox{ copies of }~{}10\end{matrix}
  103. 10 10 . . . 10 10 10 . . . 10 10 copies of 10 \begin{matrix}\underbrace{10^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}}\\ \underbrace{10^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}}\\ 10\mbox{ copies of }~{}10\end{matrix}
  104. 10 10 . . . 10 10 10 . . . 10 10 10 . . . 10 10 copies of 10 \begin{matrix}\underbrace{10^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}}\\ \underbrace{10^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}}\\ \underbrace{10^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}}\\ 10\mbox{ copies of }~{}10\end{matrix}
  105. 10 n 10\uparrow\uparrow\uparrow n
  106. 10 10 . . . 10 10 copies of 10 \begin{matrix}\underbrace{{}^{{}^{{}^{{}^{{}^{10}.}.}.}10}10}\\ 10\mbox{ copies of }~{}10\end{matrix}
  107. 10 10 . . . 10 10 10 . . . 10 10 copies of 10 \begin{matrix}\underbrace{{}^{{}^{{}^{{}^{{}^{10}.}.}.}10}10}\\ \underbrace{{}^{{}^{{}^{{}^{{}^{10}.}.}.}10}10}\\ 10\mbox{ copies of }~{}10\end{matrix}
  108. 10 n 10\uparrow\uparrow\uparrow\uparrow n
  109. 10 m n 10\uparrow^{m}n
  110. ( + , × , , , ) (+,\ \times,\ \uparrow,\ \uparrow\uparrow,\ \dots)\,\!
  111. ( + , × , , , ) (+,\ \times,\ \uparrow,\ \uparrow\uparrow,\ \dots)\,\!

Knuth–Morris–Pratt_algorithm.html

  1. x x
  2. x x
  3. i i
  4. i i
  5. x x
  6. x x

Kolmogorov's_zero–one_law.html

  1. X 1 , X 2 , X 3 , X_{1},X_{2},X_{3},\dots\,
  2. \mathcal{F}
  3. X i X_{i}
  4. F F\in\mathcal{F}
  5. F F
  6. \mathcal{F}
  7. F F
  8. X i X_{i}
  9. G n = σ ( k = n F k ) G_{n}=\sigma\bigg(\bigcup_{k=n}^{\infty}F_{k}\bigg)
  10. F n = 1 G n F\in\bigcap_{n=1}^{\infty}G_{n}
  11. n = 1 G n \textstyle{\bigcap_{n=1}^{\infty}G_{n}}

König's_lemma.html

  1. ω < ω \omega^{<\omega}\,
  2. ω \omega
  3. ω < ω \omega^{<\omega}\,
  4. ω \omega
  5. ω < ω \omega^{<\omega}\,
  6. ω < ω \omega^{<\omega}\,
  7. ω < ω \omega^{<\omega}\,
  8. ω < ω \omega^{<\omega}\,
  9. ω < ω \omega^{<\omega}\,
  10. ω < ω \omega^{<\omega}\,
  11. Π 1 1 \Pi^{1}_{1}
  12. Σ 1 1 \Sigma^{1}_{1}
  13. ω < ω \omega^{<\omega}\,
  14. ω \omega
  15. ω \omega
  16. ω < ω \omega^{<\omega}\,
  17. 0 0^{\prime}
  18. ω \omega
  19. { 0 , 1 } < ω \{0,1\}^{<\omega}\,
  20. ω \omega
  21. { 0 , 1 } \{0,1\}
  22. ω \omega
  23. { 0 , 1 } \{0,1\}
  24. N N
  25. { 0 , 1 } ω \{0,1\}^{\omega}
  26. { 0 , , k } < ω \{0,\ldots,k\}^{<\omega}\,

König's_theorem_(set_theory).html

  1. m i < n i m_{i}<n_{i}\!
  2. i I m i < i I n i . \sum_{i\in I}m_{i}<\prod_{i\in I}n_{i}.
  3. A i < B i A_{i}<B_{i}\!
  4. i I A i < i I B i , \sum_{i\in I}A_{i}<\prod_{i\in I}B_{i},
  5. m i < n i m_{i}<n_{i}\!
  6. i I m i i I n i \sum_{i\in I}m_{i}\leq\sum_{i\in I}n_{i}
  7. m i = 1 m_{i}=1
  8. n i = 2 n_{i}=2
  9. 0 \aleph_{0}
  10. κ \kappa\,
  11. κ < 2 κ . \kappa<2^{\kappa}.\!
  12. i I ( { } < B i ) \forall i\in I(\{\}<B_{i})
  13. { } < i I B i . \{\}<\prod_{i\in I}B_{i}.
  14. κ 0 \kappa\geq\aleph_{0}
  15. κ < κ c f ( κ ) . \kappa<\kappa^{cf(\kappa)}.\!
  16. κ 0 \kappa\geq\aleph_{0}
  17. λ 2 \lambda\geq 2
  18. κ < c f ( λ κ ) . \kappa<cf(\lambda^{\kappa}).\!
  19. μ = λ κ \mu=\lambda^{\kappa}\!
  20. κ c f ( μ ) \kappa\geq cf(\mu)
  21. μ < μ c f ( μ ) μ κ = ( λ κ ) κ = λ κ κ = λ κ = μ \mu<\mu^{cf(\mu)}\leq\mu^{\kappa}=(\lambda^{\kappa})^{\kappa}=\lambda^{\kappa% \cdot\kappa}=\lambda^{\kappa}=\mu
  22. i I A i < B i \forall i\in I\quad A_{i}<B_{i}
  23. i I A i < i I B i . \sum_{i\in I}A_{i}<\prod_{i\in I}B_{i}.

Kronecker_delta.html

  1. δ i j = { 0 if i j , 1 if i = j . \delta_{ij}=\begin{cases}0&\,\text{if }i\neq j,\\ 1&\,\text{if }i=j.\end{cases}
  2. ( I ) i j = δ i j \left(I\right)_{ij}=\delta_{ij}\,
  3. s y m b o l a \cdotsymbol b = i j a i δ i j b j . \textstyle symbol{a}\cdotsymbol{b}=\sum_{ij}a_{i}\delta_{ij}b_{j}.
  4. j δ i j a j \displaystyle\sum_{j}\delta_{ij}a_{j}
  5. δ i j = [ i = j ] . \delta_{ij}=[i=j].\,
  6. δ i \delta_{i}
  7. δ i = { 0 , if i 0 1 , if i = 0 \delta_{i}=\begin{cases}0,&\mbox{if }~{}i\neq 0\\ 1,&\mbox{if }~{}i=0\end{cases}
  8. δ j i \delta^{i}_{j}
  9. \mathbb{Z}
  10. δ [ n ] = { 0 , n 0 1 , n = 0. \delta[n]=\begin{cases}0,&n\neq 0\\ 1,&n=0.\end{cases}
  11. j j\in\mathbb{Z}
  12. i = - a i δ i j = a j . \sum_{i=-\infty}^{\infty}a_{i}\delta_{ij}=a_{j}.
  13. - δ ( x - y ) f ( x ) d x = f ( y ) , \int_{-\infty}^{\infty}\delta(x-y)f(x)dx=f(y),
  14. δ ( t ) \delta(t)\,
  15. δ [ n ] \delta[n]\,
  16. 𝐱 = { x 1 , , x n } \mathbf{x}=\{x_{1},\dots,x_{n}\}
  17. p 1 , , p n p_{1},\dots,p_{n}\,
  18. p ( x ) p(x)\,
  19. 𝐱 \mathbf{x}
  20. p ( x ) = i = 1 n p i δ x x i . p(x)=\sum_{i=1}^{n}p_{i}\delta_{xx_{i}}.
  21. f ( x ) f(x)\,
  22. f ( x ) = i = 1 n p i δ ( x - x i ) . f(x)=\sum_{i=1}^{n}p_{i}\delta(x-x_{i}).
  23. δ j i \delta^{i}_{j}
  24. δ j i = { 0 ( i j ) , 1 ( i = j ) . \delta^{i}_{j}=\begin{cases}0&(i\neq j),\\ 1&(i=j).\end{cases}
  25. V V V\to V
  26. V * V * V^{*}\to V^{*}
  27. V * V K V^{*}\otimes V\to K
  28. K V * V K\to V^{*}\otimes V
  29. δ ν 1 ν p μ 1 μ p = { + 1 if ν 1 ν p are distinct integers and are an even permutation of μ 1 μ p - 1 if ν 1 ν p are distinct integers and are an odd permutation of μ 1 μ p 0 in all other cases . \delta^{\mu_{1}\dots\mu_{p}}_{\nu_{1}\dots\nu_{p}}=\begin{cases}+1&\quad\,% \text{if }\nu_{1}\dots\nu_{p}\,\text{ are distinct integers and are an even % permutation of }\mu_{1}\dots\mu_{p}\\ -1&\quad\,\text{if }\nu_{1}\dots\nu_{p}\,\text{ are distinct integers and are % an odd permutation of }\mu_{1}\dots\mu_{p}\\ \;\;0&\quad\,\text{in all other cases}.\end{cases}
  30. 𝔖 p \mathfrak{S}_{p}
  31. δ ν 1 ν p μ 1 μ p = σ 𝔖 p sgn ( σ ) δ ν σ ( 1 ) μ 1 δ ν σ ( p ) μ p = σ 𝔖 p sgn ( σ ) δ ν 1 μ σ ( 1 ) δ ν p μ σ ( p ) . \delta^{\mu_{1}\dots\mu_{p}}_{\nu_{1}\dots\nu_{p}}=\sum_{\sigma\in\mathfrak{S}% _{p}}\operatorname{sgn}(\sigma)\,\delta^{\mu_{1}}_{\nu_{\sigma(1)}}\cdots% \delta^{\mu_{p}}_{\nu_{\sigma(p)}}=\sum_{\sigma\in\mathfrak{S}_{p}}% \operatorname{sgn}(\sigma)\,\delta^{\mu_{\sigma(1)}}_{\nu_{1}}\cdots\delta^{% \mu_{\sigma(p)}}_{\nu_{p}}.
  32. δ ν 1 ν p μ 1 μ p = p ! δ [ ν 1 μ 1 δ ν p ] μ p = p ! δ ν 1 [ μ 1 δ ν p μ p ] . \delta^{\mu_{1}\dots\mu_{p}}_{\nu_{1}\dots\nu_{p}}=p!\delta^{\mu_{1}}_{[\nu_{1% }}\dots\delta^{\mu_{p}}_{\nu_{p}]}=p!\delta^{[\mu_{1}}_{\nu_{1}}\dots\delta^{% \mu_{p}]}_{\nu_{p}}.
  33. δ ν 1 ν p μ 1 μ p = | δ ν 1 μ 1 δ ν p μ 1 δ ν 1 μ p δ ν p μ p | . \delta^{\mu_{1}\dots\mu_{p}}_{\nu_{1}\dots\nu_{p}}=\begin{vmatrix}\delta^{\mu_% {1}}_{\nu_{1}}&\cdots&\delta^{\mu_{1}}_{\nu_{p}}\\ \vdots&\ddots&\vdots\\ \delta^{\mu_{p}}_{\nu_{1}}&\cdots&\delta^{\mu_{p}}_{\nu_{p}}\end{vmatrix}.
  34. δ ν 1 ν p μ 1 μ p = k = 1 p ( - 1 ) p + k δ ν k μ p δ ν 1 ν ˇ k ν p μ 1 μ k μ ˇ p = δ ν p μ p δ ν 1 ν p - 1 μ 1 μ p - 1 - k = 1 p - 1 δ ν k μ p δ ν 1 ν k - 1 ν p ν k + 1 ν p - 1 μ 1 μ k - 1 μ k μ k + 1 μ p - 1 , \begin{aligned}\displaystyle\delta^{\mu_{1}\dots\mu_{p}}_{\nu_{1}\dots\nu_{p}}% &\displaystyle=\sum_{k=1}^{p}(-1)^{p+k}\delta^{\mu_{p}}_{\nu_{k}}\delta^{\mu_{% 1}\dots\mu_{k}\dots\check{\mu}_{p}}_{\nu_{1}\dots\check{\nu}_{k}\dots\nu_{p}}% \\ &\displaystyle=\delta^{\mu_{p}}_{\nu_{p}}\delta^{\mu_{1}\dots\mu_{p-1}}_{\nu_{% 1}\dots\nu_{p-1}}-\sum_{k=1}^{p-1}\delta^{\mu_{p}}_{\nu_{k}}\delta^{\mu_{1}% \dots\mu_{k-1}\;\mu_{k}\;\mu_{k+1}\dots\mu_{p-1}}_{\nu_{1}\dots\;\nu_{k-1}\;% \nu_{p}\;\nu_{k+1}\;\dots\nu_{p-1}},\end{aligned}
  35. ˇ \check{~{}}
  36. δ ν 1 ν n μ 1 μ n = ε μ 1 μ n ε ν 1 ν n . \delta^{\mu_{1}\dots\mu_{n}}_{\nu_{1}\dots\nu_{n}}=\varepsilon^{\mu_{1}\dots% \mu_{n}}\varepsilon_{\nu_{1}\dots\nu_{n}}.
  37. 1 p ! δ ν 1 ν p μ 1 μ p a ν 1 ν p = a [ μ 1 μ p ] , \frac{1}{p!}\delta^{\mu_{1}\dots\mu_{p}}_{\nu_{1}\dots\nu_{p}}a^{\nu_{1}\dots% \nu_{p}}=a^{[\mu_{1}\dots\mu_{p}]},
  38. 1 p ! δ ν 1 ν p μ 1 μ p a μ 1 μ p = a [ ν 1 ν p ] . \frac{1}{p!}\delta^{\mu_{1}\dots\mu_{p}}_{\nu_{1}\dots\nu_{p}}a_{\mu_{1}\dots% \mu_{p}}=a_{[\nu_{1}\dots\nu_{p}]}.
  39. 1 p ! δ ν 1 ν p μ 1 μ p a [ ν 1 ν p ] = a [ μ 1 μ p ] , \frac{1}{p!}\delta^{\mu_{1}\dots\mu_{p}}_{\nu_{1}\dots\nu_{p}}a^{[\nu_{1}\dots% \nu_{p}]}=a^{[\mu_{1}\dots\mu_{p}]},
  40. 1 p ! δ ν 1 ν p μ 1 μ p a [ μ 1 μ p ] = a [ ν 1 ν p ] , \frac{1}{p!}\delta^{\mu_{1}\dots\mu_{p}}_{\nu_{1}\dots\nu_{p}}a_{[\mu_{1}\dots% \mu_{p}]}=a_{[\nu_{1}\dots\nu_{p}]},
  41. 1 p ! δ ν 1 ν p μ 1 μ p δ ρ 1 ρ p ν 1 ν p = δ ρ 1 ρ p μ 1 μ p , \frac{1}{p!}\delta^{\mu_{1}\dots\mu_{p}}_{\nu_{1}\dots\nu_{p}}\delta^{\nu_{1}% \dots\nu_{p}}_{\rho_{1}\dots\rho_{p}}=\delta^{\mu_{1}\dots\mu_{p}}_{\rho_{1}% \dots\rho_{p}},
  42. δ ν 1 ν s μ s + 1 μ p μ 1 μ s μ s + 1 μ p = ( n - s ) ! ( n - p ) ! δ ν 1 ν s μ 1 μ s . \delta^{\mu_{1}\dots\mu_{s}\,\mu_{s+1}\dots\mu_{p}}_{\nu_{1}\dots\nu_{s}\,\mu_% {s+1}\dots\mu_{p}}=\tfrac{(n-s)!}{(n-p)!}\delta^{\mu_{1}\dots\mu_{s}}_{\nu_{1}% \dots\nu_{s}}.
  43. δ ν 1 ν s μ 1 μ s = 1 ( n - s ) ! ε μ 1 μ s ρ s + 1 ρ n ε ν 1 ν s ρ s + 1 ρ n . \delta^{\mu_{1}\dots\mu_{s}}_{\nu_{1}\dots\nu_{s}}={1\over(n-s)!}\,\varepsilon% ^{\mu_{1}\dots\mu_{s}\,\rho_{s+1}\dots\rho_{n}}\varepsilon_{\nu_{1}\dots\nu_{s% }\,\rho_{s+1}\dots\rho_{n}}.
  44. δ x , n = 1 2 π i | z | = 1 z x - n - 1 d z = 1 2 π 0 2 π e i ( x - n ) φ d φ \delta_{x,n}=\frac{1}{2\pi i}\oint_{|z|=1}z^{x-n-1}dz=\frac{1}{2\pi}\int_{0}^{% 2\pi}e^{i(x-n)\varphi}d\varphi
  45. Δ N [ n ] = k = - δ [ n - k N ] , \Delta_{N}[n]=\sum_{k=-\infty}^{\infty}\delta[n-kN],
  46. S u v w S_{uvw}
  47. S x y z S_{xyz}
  48. R u v w R_{uvw}
  49. R x y z R_{xyz}
  50. S u v w S_{uvw}
  51. S u v w S_{uvw}
  52. S x y z S_{xyz}
  53. u = u ( s , t ) , v = v ( s , t ) , w = w ( s , t ) , u=u(s,t),v=v(s,t),w=w(s,t),
  54. ( u s i + v s j + w s k ) × ( u t i + v t j + w t k ) . (u_{s}i+v_{s}j+w_{s}k)\times(u_{t}i+v_{t}j+w_{t}k).
  55. S u v w S_{uvw}
  56. S u v w S_{uvw}
  57. S x y z S_{xyz}
  58. δ \delta
  59. 1 / 4 π 1/4\pi
  60. S u v w S_{uvw}
  61. S x y z S_{xyz}
  62. R x y z R_{xyz}
  63. δ \delta
  64. δ = 1 4 π R s t | x y z x s y s z s x t y t z t | ( x 2 + y 2 + z 2 ) x 2 + y 2 + z 2 d s d t . \delta=\frac{1}{4\pi}\iint_{R_{st}}\frac{\begin{vmatrix}x&y&z\\ \dfrac{\partial x}{\partial s}&\dfrac{\partial y}{\partial s}&\dfrac{\partial z% }{\partial s}\\ \dfrac{\partial x}{\partial t}&\dfrac{\partial y}{\partial t}&\dfrac{\partial z% }{\partial t}\end{vmatrix}}{(x^{2}+y^{2}+z^{2})\sqrt{x^{2}+y^{2}+z^{2}}}dsdt.

Lagrange_inversion_theorem.html

  1. f ( w ) = z f(w)=z\,
  2. w = g ( z ) w=g(z)\,
  3. g ( z ) = a + n = 1 ( lim w a ( ( z - f ( a ) ) n n ! d n - 1 d w n - 1 ( w - a f ( w ) - f ( a ) ) n ) ) . g(z)=a+\sum_{n=1}^{\infty}\left(\lim_{w\to a}\left({\frac{(z-f(a))^{n}}{n!}}% \frac{\mathrm{d}^{\,n-1}}{\mathrm{d}w^{\,n-1}}\left(\frac{w-a}{f(w)-f(a)}% \right)^{n}\right)\right).
  4. f ( w ) = w / ϕ ( w ) f(w)=w/\phi(w)
  5. ϕ ( 0 ) 0. \phi(0)\neq 0.
  6. a = 0 a=0
  7. b = f ( 0 ) = 0. b=f(0)=0.
  8. g ( z ) = n = 1 ( lim w 0 ( d n - 1 d w n - 1 ( w w / ϕ ( w ) ) n ) z n n ! ) g(z)=\sum_{n=1}^{\infty}\left(\lim_{w\to 0}\left(\frac{\mathrm{d}^{n-1}}{% \mathrm{d}w^{n-1}}\left(\frac{w}{w/\phi(w)}\right)^{n}\right)\frac{z^{n}}{n!}\right)
  9. = n = 1 1 n ( 1 ( n - 1 ) ! lim w 0 ( d n - 1 d w n - 1 ϕ ( w ) n ) ) z n , =\sum_{n=1}^{\infty}\frac{1}{n}\left(\frac{1}{(n-1)!}\lim_{w\to 0}\left(\frac{% \mathrm{d}^{n-1}}{\mathrm{d}w^{n-1}}\phi(w)^{n}\right)\right)z^{n},
  10. [ z n ] g ( z ) = 1 n [ w n - 1 ] ϕ ( w ) n , [z^{n}]g(z)=\frac{1}{n}[w^{n-1}]\phi(w)^{n},
  11. [ w r ] [w^{r}]
  12. w r w^{r}
  13. [ z n ] H ( g ( z ) ) = 1 n [ w n - 1 ] ( H ( w ) ϕ ( w ) n ) [z^{n}]H(g(z))=\frac{1}{n}[w^{n-1}](H^{\prime}(w)\phi(w)^{n})
  14. H H
  15. W ( z ) W(z)
  16. W ( z ) e W ( z ) = z . W(z)e^{W(z)}=z.\,
  17. W ( z ) W(z)
  18. z = 0. z=0.
  19. f ( w ) = w e w f(w)=w\mathrm{e}^{w}
  20. a = b = 0. a=b=0.
  21. d n d x n e α x = α n e α x \frac{\mathrm{d}^{n}}{\mathrm{d}x^{n}}\ \mathrm{e}^{\alpha\,x}\,=\,\alpha^{n}% \,\mathrm{e}^{\alpha\,x}
  22. W ( z ) = n = 1 lim w 0 ( d n - 1 d w n - 1 e - n w ) z n n ! = n = 1 ( - n ) n - 1 z n n ! = z - z 2 + 3 2 z 3 - 8 3 z 4 + O ( z 5 ) . W(z)=\sum_{n=1}^{\infty}\lim_{w\to 0}\left(\frac{\mathrm{d}^{\,n-1}}{\mathrm{d% }w^{\,n-1}}\ \mathrm{e}^{-nw}\right){\frac{z^{n}}{n!}}\,=\,\sum_{n=1}^{\infty}% (-n)^{n-1}\,\frac{z^{n}}{n!}=z-z^{2}+\frac{3}{2}z^{3}-\frac{8}{3}z^{4}+O(z^{5}).
  23. e - 1 e^{-1}
  24. f ( z ) = W ( e z ) - 1 f(z)=W(e^{z})-1\,
  25. 1 + f ( z ) + ln ( 1 + f ( z ) ) = z . 1+f(z)+\ln(1+f(z))=z.\,
  26. z + ln ( 1 + z ) z+\ln(1+z)\,
  27. f ( z + 1 ) = W ( e z + 1 ) - 1 f(z+1)=W(e^{z+1})-1\,
  28. W ( e 1 + z ) = 1 + z 2 + z 2 16 - z 3 192 - z 4 3072 + 13 z 5 61440 - 47 z 6 1474560 - 73 z 7 41287680 + 2447 z 8 1321205760 + O ( z 9 ) . W(e^{1+z})=1+\frac{z}{2}+\frac{z^{2}}{16}-\frac{z^{3}}{192}-\frac{z^{4}}{3072}% +\frac{13z^{5}}{61440}-\frac{47z^{6}}{1474560}-\frac{73z^{7}}{41287680}+\frac{% 2447z^{8}}{1321205760}+O(z^{9}).
  29. W ( x ) W(x)\,
  30. ln x - 1 \ln x-1\,
  31. W ( 1 ) = 0.567143 W(1)=0.567143\,
  32. \mathcal{B}
  33. \mathcal{B}
  34. B n B_{n}
  35. B ( z ) = n = 0 B n z n B(z)=\sum_{n=0}^{\infty}B_{n}z^{n}
  36. B ( z ) = 1 + z B ( z ) 2 . B(z)=1+zB(z)^{2}.
  37. C ( z ) = B ( z ) - 1 C(z)=B(z)-1
  38. z = C ( z ) ( C ( z ) + 1 ) 2 . z=\frac{C(z)}{(C(z)+1)^{2}}.
  39. ϕ ( w ) = ( w + 1 ) 2 : \phi(w)=(w+1)^{2}:
  40. B n = [ z n ] C ( z ) = 1 n [ w n - 1 ] ( w + 1 ) 2 n = 1 n ( 2 n n - 1 ) = 1 n + 1 ( 2 n n ) . B_{n}=[z^{n}]C(z)=\frac{1}{n}[w^{n-1}](w+1)^{2n}=\frac{1}{n}{2n\choose n-1}=% \frac{1}{n+1}{2n\choose n}.
  41. B n B_{n}

Lagrange_multiplier.html

  1. f ( x , y ) f(x,y)
  2. g ( x , y ) = c g(x,y)=c
  3. f f
  4. g g
  5. λ λ
  6. Λ ( x , y , λ ) = f ( x , y ) + λ ( g ( x , y ) - c ) , \Lambda(x,y,\lambda)=f(x,y)+\lambda\cdot\Big(g(x,y)-c\Big),
  7. λ λ
  8. f ( x , y ) f(x,y)
  9. Λ Λ
  10. f ( x , y ) f(x,y)
  11. g ( x , y ) = 0 g(x,y)=0
  12. f ( x , y ) f(x,y)
  13. g = 0 g=0
  14. g = 0 g=0
  15. f f
  16. f ( x , y ) = d f(x,y)=d
  17. d d
  18. g g
  19. g ( x , y ) = 0 g(x,y)=0
  20. g = 0 g=0
  21. f f
  22. f f
  23. f f
  24. f f
  25. g g
  26. f f
  27. f f
  28. f f
  29. g g
  30. f f
  31. g g
  32. ( x , y ) (x,y)
  33. g ( x , y ) = 0 g(x,y)=0
  34. x , y f = - λ x , y g \nabla_{x,y}f=-\lambda\nabla_{x,y}g
  35. λ λ
  36. x , y f = ( f x , f y ) , x , y g = ( g x , g y ) . \nabla_{x,y}f=\left(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y% }\right),\qquad\nabla_{x,y}g=\left(\frac{\partial g}{\partial x},\frac{% \partial g}{\partial y}\right).
  37. λ λ
  38. f f
  39. λ = 0 λ=0
  40. g g
  41. Λ ( x , y , λ ) = f ( x , y ) + λ g ( x , y ) , \Lambda(x,y,\lambda)=f(x,y)+\lambda\cdot g(x,y),
  42. x , y , λ Λ ( x , y , λ ) = 0. \nabla_{x,y,\lambda}\Lambda(x,y,\lambda)=0.
  43. λ Λ ( x , y , λ ) = 0 \nabla_{\lambda}\Lambda(x,y,\lambda)=0
  44. g ( x , y ) = 0 g(x,y)=0
  45. f f
  46. Λ Λ
  47. Λ Λ
  48. f ( p ) f(p)
  49. ( p , f ( p ) ) (p,f(p))
  50. f f
  51. f f
  52. f f
  53. ( p , f ( p ) ) (p,f(p))
  54. f f
  55. v v
  56. f x 1 v x 1 + f x 2 v x 2 + + f x N v x N = 0 \frac{\partial f}{\partial x_{1}}v_{x_{1}}+\frac{\partial f}{\partial x_{2}}v_% {x_{2}}+\cdots+\frac{\partial f}{\partial x_{N}}v_{x_{N}}=0
  57. [ f x 1 f x 2 f x N ] f T [ v x 1 v x 2 v x N ] v = 0 \begin{matrix}\underbrace{\begin{matrix}\left[\begin{matrix}\frac{\partial f}{% \partial x_{1}}&\frac{\partial f}{\partial x_{2}}&...&\frac{\partial f}{% \partial x_{N}}\end{matrix}\right]\\ \\ \end{matrix}}_{\nabla f^{T}}&\underbrace{\begin{matrix}\left[\begin{matrix}v_{% x_{1}}\\ v_{x_{2}}\\ \vdots\\ v_{x_{N}}\\ \end{matrix}\right]\\ \\ \end{matrix}}_{v}&=\,\,0\\ \end{matrix}
  58. f T v = 0. \nabla f^{T}\cdot v=0.
  59. p p
  60. f f
  61. f ( p ) ∇f(p)
  62. f f
  63. p p
  64. v v
  65. g x 1 v x 1 + g x 2 v x 2 + + g x N v x N = 0 g T v = 0. \frac{\partial g}{\partial x_{1}}v_{x_{1}}+\frac{\partial g}{\partial x_{2}}v_% {x_{2}}+\cdots+\frac{\partial g}{\partial x_{N}}v_{x_{N}}=0\quad\Rightarrow% \quad\nabla g^{T}\cdot v=0.
  66. p p
  67. g ( p ) ∇g(p)
  68. f f
  69. f f
  70. g ( p ) ∇g(p)
  71. f f
  72. f f
  73. f f
  74. g g
  75. f f
  76. f ( p ) = λ g ( p ) f ( p ) - λ g ( p ) = 0. \nabla f(p)=\lambda\,\nabla g(p)\qquad\Rightarrow\qquad\nabla f(p)-\lambda\,% \nabla g(p)=0.
  77. { g ( p ) = 0 p satisfies constraint f ( p ) - λ g ( p ) = 0 p is a stationary point . \begin{cases}g(p)=0&p\,\text{ satisfies constraint}\\ \nabla f(p)-\lambda\,\nabla g(p)=0&p\,\text{ is a stationary point}.\end{cases}
  78. N + 1 N+1
  79. N + 1 N+1
  80. λ λ
  81. g ( x 1 , x 2 , , x N ) \displaystyle g\left(x_{1},x_{2},\ldots,x_{N}\right)
  82. g g
  83. p \textstyle p
  84. g ( p ) \textstyle\nabla g(p)
  85. A \textstyle A
  86. S \textstyle S
  87. A = S \textstyle A=S^{\perp}
  88. S \textstyle S
  89. p \textstyle p
  90. f ( p ) \textstyle\nabla f(p)
  91. f ( p ) A = S \nabla f(p)\in A^{\perp}=S
  92. f ( p ) = k = 1 M λ k g k ( p ) f ( p ) - k = 1 M λ k g k ( p ) = 0. \nabla f(p)=\sum_{k=1}^{M}\lambda_{k}\nabla g_{k}(p)\quad\Rightarrow\quad% \nabla f(p)-\sum_{k=1}^{M}{\lambda_{k}\nabla g_{k}(p)}=0.
  93. { g 1 ( p ) = g 2 ( p ) = = g M ( p ) = 0 p satisfies all constraints f ( p ) - k = 1 M λ k g k ( p ) = 0 p is a stationary point . \begin{cases}g_{1}(p)=g_{2}(p)=\cdots=g_{M}(p)=0&p\,\text{ satisfies all % constraints}\\ \nabla f(p)-\sum_{k=1}^{M}{\lambda_{k}\,\nabla g_{k}(p)}=0&p\,\text{ is a % stationary point}.\end{cases}
  94. L L
  95. L L
  96. L ( x 1 , x 2 , , x N , λ 1 , λ 2 , , λ M ) = f ( x 1 , x 2 , , x N ) - k = 1 M λ k g k ( x 1 , x 2 , , x N ) . L\left(x_{1},x_{2},\ldots,x_{N},\lambda_{1},\lambda_{2},\ldots,\lambda_{M}% \right)=f\left(x_{1},x_{2},\ldots,x_{N}\right)-\sum\limits_{k=1}^{M}{\lambda_{% k}g_{k}\left(x_{1},x_{2},\ldots,x_{N}\right)}.
  97. f f
  98. h ( 𝐱 ) c h(\mathbf{x}) ≤c
  99. f : U f:U\to\mathbb{R}
  100. U U
  101. n \mathbb{R}^{n}
  102. x U x\in U
  103. D x f = 0 D_{x}f=0
  104. H x f H_{x}f
  105. D x f = 0 D_{x}f=0
  106. g : n g:\mathbb{R}^{n}\to\mathbb{R}
  107. D x g 0 D_{x}g\neq 0
  108. x x
  109. c c
  110. g - 1 ( c ) g^{-1}(c)
  111. n - 1 n-1
  112. M M
  113. M M
  114. φ : V n - 1 \varphi:V\to\mathbb{R}^{n-1}
  115. V M V\subseteq M
  116. U n - 1 U\subseteq\mathbb{R}^{n-1}
  117. f = f φ - 1 : U f^{\prime}=f\circ\varphi^{-1}:U\to\mathbb{R}
  118. φ - 1 \varphi^{-1}
  119. D x f D_{x}f^{\prime}
  120. D x φ - 1 D_{x}\varphi^{-1}
  121. ker D φ - 1 ( x ) g n \mathrm{ker}D_{\varphi^{-1}(x)}g\subseteq\mathbb{R}^{n}
  122. 0 = D x f = D x ( f φ - 1 ) = D φ - 1 ( x ) f D x φ - 1 0=D_{x}f^{\prime}=D_{x}(f\circ\varphi^{-1})=D_{\varphi^{-1}(x)}f\circ D_{x}% \varphi^{-1}
  123. ker D y g ker D y f \mathrm{ker}D_{y}g\subseteq\mathrm{ker}D_{y}f
  124. y y
  125. φ - 1 ( x ) \varphi^{-1}(x)
  126. L : L:\mathbb{R}\to\mathbb{R}
  127. L D y g = D y f L\circ D_{y}g=D_{y}f
  128. L ( x ) = λ x L(x)=\lambda x
  129. λ \lambda\in\mathbb{R}
  130. f f^{\prime}
  131. λ D y g = D y f \lambda D_{y}g=D_{y}f
  132. g ( y ) = c g(y)=c
  133. y n - 1 y\in\mathbb{R}^{n-1}
  134. λ \lambda\in\mathbb{R}
  135. n n
  136. n n
  137. g : n m g:\mathbb{R}^{n}\to\mathbb{R}^{m}
  138. D x g 0 D_{x}g\neq 0
  139. x g - 1 ( c ) x\in g^{-1}(c)
  140. D x g D_{x}g
  141. L L
  142. m \mathbb{R}^{m}\to\mathbb{R}
  143. m m
  144. L ( x 1 , x 2 , ; λ 1 , λ 2 , ) = f ( x 1 , x 2 , ) + λ 1 ( c 1 - g 1 ( x 1 , x 2 , ) ) + λ 2 ( c 2 - g 2 ( x 1 , x 2 , ) ) + L(x_{1},x_{2},\dots;\lambda_{1},\lambda_{2},\dots)=f(x_{1},x_{2},\dots)+% \lambda_{1}(c_{1}-g_{1}(x_{1},x_{2},\dots))+\lambda_{2}(c_{2}-g_{2}(x_{1},x_{2% },\dots))+\dots
  145. L c k = λ k . \frac{\partial L}{\partial{c_{k}}}=\lambda_{k}.
  146. F = V F=−∇V
  147. d f ( x 1 * ( c 1 , c 2 , ) , x 2 * ( c 1 , c 2 , ) , ) d c k = λ k * . \frac{\,\text{d}f(x_{1}^{*}(c_{1},c_{2},\dots),x_{2}^{*}(c_{1},c_{2},\dots),% \dots)}{\,\text{d}c_{k}}=\lambda_{k}^{*}.
  148. λ * λ*
  149. f ( x , y ) = x + y f(x,y)=x+y
  150. x 2 + y 2 = 1 x^{2}+y^{2}=1
  151. f f
  152. ( 2 2 , 2 2 ) \left(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2}\right)
  153. ( - 2 2 , - 2 2 ) \left(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2}\right)
  154. g ( x , y ) - c = x 2 + y 2 - 1 g(x,y)-c=x^{2}+y^{2}-1
  155. Λ ( x , y , λ ) = f ( x , y ) + λ ( g ( x , y ) - c ) = x + y + λ ( x 2 + y 2 - 1 ) . \Lambda(x,y,\lambda)=f(x,y)+\lambda(g(x,y)-c)=x+y+\lambda(x^{2}+y^{2}-1).
  156. x , y , λ Λ ( x , y , λ ) = 0 \nabla_{x,y,\lambda}\Lambda(x,y,\lambda)=0
  157. Λ x = 1 + 2 λ x = 0 , Λ y = 1 + 2 λ y = 0 , Λ λ = x 2 + y 2 - 1 = 0 , \begin{aligned}\displaystyle\frac{\partial\Lambda}{\partial x}&\displaystyle=1% +2\lambda x&&\displaystyle=0,\\ \displaystyle\frac{\partial\Lambda}{\partial y}&\displaystyle=1+2\lambda y&&% \displaystyle=0,\\ \displaystyle\frac{\partial\Lambda}{\partial\lambda}&\displaystyle=x^{2}+y^{2}% -1&&\displaystyle=0,\end{aligned}
  158. x = y = - 1 2 λ , λ 0. x=y=-\frac{1}{2\lambda},\qquad\lambda\neq 0.
  159. 1 / ( 4 λ 2 ) + 1 / ( 4 λ 2 ) = 1 1/(4\lambda^{2})+1/(4\lambda^{2})=1
  160. λ = 1 / 2 \lambda=\mp 1/\sqrt{2}
  161. ( 2 / 2 , 2 / 2 ) (\sqrt{2}/2,\sqrt{2}/2)
  162. ( - 2 / 2 , - 2 / 2 ) (-\sqrt{2}/2,-\sqrt{2}/2)
  163. f f
  164. f ( 2 / 2 , 2 / 2 ) = 2 and f ( - 2 / 2 , - 2 / 2 ) = - 2 , f(\sqrt{2}/2,\sqrt{2}/2)=\sqrt{2}\mbox{ and }~{}f(-\sqrt{2}/2,-\sqrt{2}/2)=-% \sqrt{2},
  165. 2 \sqrt{2}
  166. ( 2 / 2 , 2 / 2 ) (\sqrt{2}/2,\sqrt{2}/2)
  167. - 2 -\sqrt{2}
  168. ( - 2 / 2 , - 2 / 2 ) (-\sqrt{2}/2,-\sqrt{2}/2)
  169. f ( x , y ) = x 2 y f(x,y)=x^{2}y
  170. x x
  171. y y
  172. g ( x , y ) = x 2 + y 2 = 3. g(x,y)=x^{2}+y^{2}=3.
  173. λ λ
  174. g ( x , y ) - 3 g(x,y)-3
  175. g ( x , y ) - 3 g(x,y)-3
  176. f ( x , y ) f(x,y)
  177. f ( x , y ) f(x,y)
  178. Λ ( x , y , λ ) = f ( x , y ) + λ ( g ( x , y ) - 3 ) = x 2 y + λ ( x 2 + y 2 - 3 ) . \Lambda(x,y,\lambda)=f(x,y)+\lambda(g(x,y)-3)=x^{2}y+\lambda(x^{2}+y^{2}-3).
  179. Λ Λ
  180. Λ x = 2 x y + 2 λ x = 0 , (i) Λ y = x 2 + 2 λ y = 0 , (ii) Λ λ = x 2 + y 2 - 3 = 0. (iii) \begin{aligned}\displaystyle\frac{\partial\Lambda}{\partial x}&\displaystyle=2% xy+2\lambda x&&\displaystyle=0,\qquad\,\text{(i)}\\ \displaystyle\frac{\partial\Lambda}{\partial y}&\displaystyle=x^{2}+2\lambda y% &&\displaystyle=0,\qquad\,\text{(ii)}\\ \displaystyle\frac{\partial\Lambda}{\partial\lambda}&\displaystyle=x^{2}+y^{2}% -3&&\displaystyle=0.\qquad\,\text{(iii)}\end{aligned}
  181. x = 0 x=0
  182. λ = y λ=−y
  183. x = 0 x=0
  184. y = ± 3 y=\pm\sqrt{3}
  185. λ = y λ=−y
  186. x 2 - 2 y 2 = 0. x^{2}-2y^{2}=0.
  187. y y
  188. y = ± 1 y=±1
  189. ( 2 , 1 ) ; ( - 2 , 1 ) ; ( 2 , - 1 ) ; ( - 2 , - 1 ) ; ( 0 , 3 ) ; ( 0 , - 3 ) . (\sqrt{2},1);\quad(-\sqrt{2},1);\quad(\sqrt{2},-1);\quad(-\sqrt{2},-1);\quad(0% ,\sqrt{3});\quad(0,-\sqrt{3}).
  190. f ( ± 2 , 1 ) = 2 ; f ( ± 2 , - 1 ) = - 2 ; f ( 0 , ± 3 ) = 0. f(\pm\sqrt{2},1)=2;\quad f(\pm\sqrt{2},-1)=-2;\quad f(0,\pm\sqrt{3})=0.
  191. ( ± 2 , 1 ) (\pm\sqrt{2},1)
  192. ( ± 2 , - 1 ) . (\pm\sqrt{2},-1).
  193. ( 0 , 3 ) (0,\sqrt{3})
  194. ( 0 , - 3 ) (0,-\sqrt{3})
  195. Λ ( x , y , 0 ) \Lambda(x,y,0)
  196. ( 2 , 1 , - 1 ) (\sqrt{2},1,-1)
  197. Λ Λ
  198. Λ ( 2 + ϵ , 1 , - 1 + δ ) = 2 + δ ( ϵ 2 + ( 2 2 ) ϵ ) \Lambda(\sqrt{2}+\epsilon,1,-1+\delta)=2+\delta(\epsilon^{2}+(2\sqrt{2})\epsilon)
  199. ( 2 , 1 , - 1 ) (\sqrt{2},1,-1)
  200. ϵ \epsilon
  201. δ \delta
  202. Λ Λ
  203. 2 2
  204. { p 1 , p 2 , , p n } \{p_{1},p_{2},\ldots,p_{n}\}
  205. { p 1 , p 2 , , p n } \{p_{1},p_{2},\ldots,p_{n}\}
  206. f ( p 1 , p 2 , , p n ) = - j = 1 n p j log 2 p j . f(p_{1},p_{2},\ldots,p_{n})=-\sum_{j=1}^{n}p_{j}\log_{2}p_{j}.
  207. p i p_{i}
  208. x i x_{i}
  209. g ( p 1 , p 2 , , p n ) = j = 1 n p j = 1. g(p_{1},p_{2},\ldots,p_{n})=\sum_{j=1}^{n}p_{j}=1.
  210. p * \vec{p}^{\,*}
  211. p \vec{p}
  212. { x 1 , x 2 , , x n } \{x_{1},x_{2},\ldots,x_{n}\}
  213. p ( f + λ ( g - 1 ) ) | p = p * = 0 , \left.\frac{\partial}{\partial\vec{p}}(f+\lambda(g-1))\right|_{\vec{p}=\vec{p}% ^{\,*}}=0,
  214. n n
  215. k = 1 , , n k={1,\ldots,n}
  216. p k { - ( j = 1 n p j log 2 p j ) + λ ( j = 1 n p j - 1 ) } | p k = p k * = 0. \left.\frac{\partial}{\partial p_{k}}\left\{-\left(\sum_{j=1}^{n}p_{j}\log_{2}% p_{j}\right)+\lambda\left(\sum_{j=1}^{n}p_{j}-1\right)\right\}\right|_{p_{k}=p% ^{*}_{k}}=0.
  217. n n
  218. - ( 1 ln 2 + log 2 p k * ) + λ = 0. -\left(\frac{1}{\ln 2}+\log_{2}p^{*}_{k}\right)+\lambda=0.
  219. p k * p^{*}_{k}
  220. λ λ
  221. j p j = 1 , \sum_{j}p_{j}=1,
  222. p k * = 1 n . p^{*}_{k}=\frac{1}{n}.
  223. n n
  224. x x
  225. f ( x ) = x 2 f(x)=x^{2}
  226. x 2 = 1 x^{2}=1
  227. Λ ( x , λ ) = x 2 + λ ( x 2 - 1 ) . \Lambda(x,\lambda)=x^{2}+\lambda(x^{2}-1).
  228. x = 1 x=1
  229. x = 1 x=−1
  230. Λ x = 2 x + 2 x λ \frac{\partial\Lambda}{\partial x}=2x+2x\lambda
  231. Λ λ = x 2 - 1. \frac{\partial\Lambda}{\partial\lambda}=x^{2}-1.
  232. Λ x Λ ( x + ϵ , λ ) - Λ ( x , λ ) ϵ \frac{\partial\Lambda}{\partial x}\approx\frac{\Lambda(x+\epsilon,\lambda)-% \Lambda(x,\lambda)}{\epsilon}
  233. Λ λ Λ ( x , λ + ϵ ) - Λ ( x , λ ) ϵ \frac{\partial\Lambda}{\partial\lambda}\approx\frac{\Lambda(x,\lambda+\epsilon% )-\Lambda(x,\lambda)}{\epsilon}
  234. ϵ \epsilon
  235. h ( x , λ ) = ( 2 x + 2 x λ ) 2 + ( x 2 - 1 ) 2 ( Λ ( x + ϵ , λ ) - Λ ( x , λ ) ϵ ) 2 + ( Λ ( x , λ + ϵ ) - Λ ( x , λ ) ϵ ) 2 . h(x,\lambda)=\sqrt{(2x+2x\lambda)^{2}+(x^{2}-1)^{2}}\approx\sqrt{\left(\frac{% \Lambda(x+\epsilon,\lambda)-\Lambda(x,\lambda)}{\epsilon}\right)^{2}+\left(% \frac{\Lambda(x,\lambda+\epsilon)-\Lambda(x,\lambda)}{\epsilon}\right)^{2}}.
  236. h h
  237. x = 1 x=1
  238. x = 1 x=−1
  239. Λ Λ
  240. Λ Λ
  241. h h

Lagrange_polynomial.html

  1. x j x_{j}
  2. y j y_{j}
  3. x j x_{j}
  4. y j y_{j}
  5. ( x 0 , y 0 ) , , ( x j , y j ) , , ( x k , y k ) (x_{0},y_{0}),\ldots,(x_{j},y_{j}),\ldots,(x_{k},y_{k})
  6. x j x_{j}
  7. L ( x ) := j = 0 k y j j ( x ) L(x):=\sum_{j=0}^{k}y_{j}\ell_{j}(x)
  8. j ( x ) := 0 m k m j x - x m x j - x m = ( x - x 0 ) ( x j - x 0 ) ( x - x j - 1 ) ( x j - x j - 1 ) ( x - x j + 1 ) ( x j - x j + 1 ) ( x - x k ) ( x j - x k ) , \ell_{j}(x):=\prod_{\begin{smallmatrix}0\leq m\leq k\\ m\neq j\end{smallmatrix}}\frac{x-x_{m}}{x_{j}-x_{m}}=\frac{(x-x_{0})}{(x_{j}-x% _{0})}\cdots\frac{(x-x_{j-1})}{(x_{j}-x_{j-1})}\frac{(x-x_{j+1})}{(x_{j}-x_{j+% 1})}\cdots\frac{(x-x_{k})}{(x_{j}-x_{k})},
  9. 0 j k 0\leq j\leq k
  10. x i x_{i}
  11. x j - x m 0 x_{j}-x_{m}\neq 0
  12. x i = x j x_{i}=x_{j}
  13. y i y j y_{i}\neq y_{j}
  14. L L
  15. y i = L ( x i ) y_{i}=L(x_{i})
  16. x i x_{i}
  17. y i = y j y_{i}=y_{j}
  18. i j i\neq j
  19. j ( x ) \ell_{j}(x)
  20. ( x - x i ) (x-x_{i})
  21. x = x i x=x_{i}
  22. j i ( x i ) = m j x i - x m x j - x m = ( x i - x 0 ) ( x j - x 0 ) ( x i - x i ) ( x j - x i ) ( x i - x k ) ( x j - x k ) = 0. \ell_{j\neq i}(x_{i})=\prod_{m\neq j}\frac{x_{i}-x_{m}}{x_{j}-x_{m}}=\frac{(x_% {i}-x_{0})}{(x_{j}-x_{0})}\cdots\frac{(x_{i}-x_{i})}{(x_{j}-x_{i})}\cdots\frac% {(x_{i}-x_{k})}{(x_{j}-x_{k})}=0.
  23. i ( x i ) := m i x i - x m x i - x m = 1 \ell_{i}(x_{i}):=\prod_{m\neq i}\frac{x_{i}-x_{m}}{x_{i}-x_{m}}=1
  24. x = x i x=x_{i}
  25. i ( x ) \ell_{i}(x)
  26. i ( x i ) = 1 \ell_{i}(x_{i})=1
  27. ( x - x i ) (x-x_{i})
  28. y i i ( x i ) = y i y_{i}\ell_{i}(x_{i})=y_{i}
  29. x i x_{i}
  30. L ( x i ) = y i + 0 + 0 + + 0 = y i L(x_{i})=y_{i}+0+0+\dots+0=y_{i}
  31. L L
  32. x x
  33. y j y_{j}
  34. x j x_{j}
  35. j j
  36. L ( x j ) = y j j = 0 , , k L(x_{j})=y_{j}\qquad j=0,\ldots,k
  37. j ( x ) \ell_{j}(x)
  38. j ( x i ) = m = 0 , m j k x i - x m x j - x m \ell_{j}(x_{i})=\prod_{m=0,\,m\neq j}^{k}\frac{x_{i}-x_{m}}{x_{j}-x_{m}}
  39. m = j m=j
  40. i = j i=j
  41. x j - x m x j - x m = 1 \frac{x_{j}-x_{m}}{x_{j}-x_{m}}=1
  42. x j = x m x_{j}=x_{m}
  43. m = j m=j
  44. m j m\neq j
  45. i j i\neq j
  46. i = j i=j
  47. i j i\neq j
  48. m j m\neq j
  49. m = i m=i
  50. x i - x i x j - x i = 0 \frac{x_{i}-x_{i}}{x_{j}-x_{i}}=0
  51. j ( x i ) = δ j i = { 1 , if j = i 0 , if j i \ell_{j}(x_{i})=\delta_{ji}=\begin{cases}1,&\,\text{if }j=i\\ 0,&\,\text{if }j\neq i\end{cases}
  52. δ i j \delta_{ij}
  53. L ( x i ) = j = 0 k y j j ( x i ) = j = 0 k y j δ j i = y i . L(x_{i})=\sum_{j=0}^{k}y_{j}\ell_{j}(x_{i})=\sum_{j=0}^{k}y_{j}\delta_{ji}=y_{% i}.
  54. L ( x i ) = y i L(x_{i})=y_{i}
  55. L ( x ) L(x)
  56. x 0 \displaystyle x_{0}
  57. L ( x ) \displaystyle L(x)
  58. x 0 = 1 x_{0}=1\,
  59. f ( x 0 ) = 1 f(x_{0})=1\,
  60. x 1 = 2 x_{1}=2\,
  61. f ( x 1 ) = 8 f(x_{1})=8\,
  62. x 2 = 3 x_{2}=3\,
  63. f ( x 2 ) = 27 f(x_{2})=27\,
  64. L ( x ) \displaystyle L(x)
  65. ( x ) = ( x - x 0 ) ( x - x 1 ) ( x - x k ) \ell(x)=(x-x_{0})(x-x_{1})\cdots(x-x_{k})
  66. ( x j ) = d ( x ) d x | x = x j = i = 0 , i j k ( x j - x i ) \ell^{\prime}(x_{j})=\frac{\mathrm{d}\ell(x)}{\mathrm{d}x}\Big|_{x=x_{j}}=% \prod_{i=0,i\neq j}^{k}(x_{j}-x_{i})
  67. j ( x ) = ( x ) ( x j ) ( x - x j ) \ell_{j}(x)=\frac{\ell(x)}{\ell^{\prime}(x_{j})(x-x_{j})}
  68. w j = 1 ( x j ) w_{j}=\frac{1}{\ell^{\prime}(x_{j})}
  69. j ( x ) = ( x ) w j x - x j \ell_{j}(x)=\ell(x)\frac{w_{j}}{x-x_{j}}
  70. L ( x ) = ( x ) j = 0 k w j x - x j y j L(x)=\ell(x)\sum_{j=0}^{k}\frac{w_{j}}{x-x_{j}}y_{j}
  71. w j w_{j}
  72. 𝒪 ( n ) \mathcal{O}(n)
  73. ( x ) \ell(x)
  74. w j / ( x - x j ) w_{j}/(x-x_{j})
  75. 𝒪 ( n 2 ) \mathcal{O}(n^{2})
  76. j ( x ) \ell_{j}(x)
  77. x k + 1 x_{k+1}
  78. w j w_{j}
  79. j = 0 k j=0\dots k
  80. ( x j - x k + 1 ) (x_{j}-x_{k+1})
  81. w k + 1 w_{k+1}
  82. g ( x ) 1 g(x)\equiv 1
  83. g ( x ) = ( x ) j = 0 k w j x - x j . g(x)=\ell(x)\sum_{j=0}^{k}\frac{w_{j}}{x-x_{j}}.
  84. L ( x ) L(x)
  85. g ( x ) g(x)
  86. L ( x ) = j = 0 k w j x - x j y j j = 0 k w j x - x j L(x)=\frac{\sum_{j=0}^{k}\frac{w_{j}}{x-x_{j}}y_{j}}{\sum_{j=0}^{k}\frac{w_{j}% }{x-x_{j}}}
  87. ( x ) \ell(x)
  88. L ( x ) L(x)

Lah_number.html

  1. L ( n , k ) = ( n - 1 k - 1 ) n ! k ! . L(n,k)={n-1\choose k-1}\frac{n!}{k!}.
  2. L ( n , k ) = ( - 1 ) n ( n - 1 k - 1 ) n ! k ! . L^{\prime}(n,k)=(-1)^{n}{n-1\choose k-1}\frac{n!}{k!}.
  3. L ( n , k ) = n k . L(n,k)=\left\lfloor\begin{matrix}n\\ k\end{matrix}\right\rfloor.
  4. x ( n ) x^{(n)}
  5. x ( x + 1 ) ( x + 2 ) ( x + n - 1 ) x(x+1)(x+2)\cdots(x+n-1)
  6. ( x ) n (x)_{n}
  7. x ( x - 1 ) ( x - 2 ) ( x - n + 1 ) x(x-1)(x-2)\cdots(x-n+1)
  8. x ( n ) = k = 1 n L ( n , k ) ( x ) k x^{(n)}=\sum_{k=1}^{n}L(n,k)(x)_{k}
  9. ( x ) n = k = 1 n ( - 1 ) n - k L ( n , k ) x ( k ) . (x)_{n}=\sum_{k=1}^{n}(-1)^{n-k}L(n,k)x^{(k)}.
  10. x ( x + 1 ) ( x + 2 ) = \color r e d 6 x + \color r e d 6 x ( x - 1 ) + \color r e d 1 x ( x - 1 ) ( x - 2 ) . x(x+1)(x+2)={\color{red}6}x+{\color{red}6}x(x-1)+{\color{red}1}x(x-1)(x-2).
  11. L ( n , k ) = ( n - 1 k - 1 ) n ! k ! = ( n k ) ( n - 1 ) ! ( k - 1 ) ! L(n,k)={n-1\choose k-1}\frac{n!}{k!}={n\choose k}\frac{(n-1)!}{(k-1)!}
  12. L ( n , k ) = n ! ( n - 1 ) ! k ! ( k - 1 ) ! 1 ( n - k ) ! = ( n ! k ! ) 2 k n ( n - k ) ! L(n,k)=\frac{n!(n-1)!}{k!(k-1)!}\cdot\frac{1}{(n-k)!}=\left(\frac{n!}{k!}% \right)^{2}\frac{k}{n(n-k)!}
  13. L ( n , k + 1 ) = n - k k ( k + 1 ) L ( n , k ) . L(n,k+1)=\frac{n-k}{k(k+1)}L(n,k).
  14. L ( n , k ) = j [ n j ] { j k } , L(n,k)=\sum_{j}\left[{n\atop j}\right]\left\{{j\atop k}\right\},
  15. s ( n , j ) s(n,j)
  16. S ( j , k ) S(j,k)
  17. L ( 0 , 0 ) = 1 L(0,0)=1
  18. L ( n , k ) = 0 L(n,k)=0
  19. k > n k>n
  20. L ( n , 1 ) = n ! L(n,1)=n!
  21. L ( n , 2 ) = ( n - 1 ) n ! / 2 L(n,2)=(n-1)n!/2
  22. L ( n , 3 ) = ( n - 2 ) ( n - 1 ) n ! / 12 L(n,3)=(n-2)(n-1)n!/12
  23. L ( n , n - 1 ) = n ( n - 1 ) L(n,n-1)=n(n-1)
  24. L ( n , n ) = 1 L(n,n)=1
  25. k n {}_{n}\!\!\diagdown\!\!^{k}

Lambert_W_function.html

  1. z = f - 1 ( z e z ) = W ( z e z ) z=f^{-1}(ze^{z})=W(ze^{z})
  2. z = W ( z ) z=W(z)
  3. z = W ( z ) e W ( z ) z=W(z)e^{W(z)}
  4. ( ) \scriptstyle(\cdot)
  5. z ( 1 + W ) d W d z = W for z - 1 / e . z(1+W)\frac{{\rm d}W}{{\rm d}z}=W\quad\,\text{for }z\neq-1/e.
  6. d W d z = W ( z ) z ( 1 + W ( z ) ) for z { 0 , - 1 / e } . \frac{{\rm d}W}{{\rm d}z}=\frac{W(z)}{z(1+W(z))}\quad\,\text{for }z\not\in\{0,% -1/e\}.
  7. d W d z | z = 0 = 1. \left.\frac{{\rm d}W}{{\rm d}z}\right|_{z=0}=1.
  8. W ( x ) d x = x ( W ( x ) - 1 + 1 W ( x ) ) + C . \int W(x)\,{\rm d}x=x\left(W(x)-1+\frac{1}{W(x)}\right)+C.
  9. W ( e ) = 1 W(e)=1
  10. 0 e W ( x ) d x = e - 1 \int_{0}^{e}W(x)\,{\rm d}x=e-1
  11. W 0 W_{0}
  12. W 0 ( x ) = n = 1 ( - n ) n - 1 n ! x n = x - x 2 + 3 2 x 3 - 8 3 x 4 + 125 24 x 5 - W_{0}(x)=\sum_{n=1}^{\infty}\frac{(-n)^{n-1}}{n!}\ x^{n}=x-x^{2}+\frac{3}{2}x^% {3}-\frac{8}{3}x^{4}+\frac{125}{24}x^{5}-\cdots
  13. W 0 ( x ) = L 1 - L 2 + L 2 L 1 + L 2 ( - 2 + L 2 ) 2 L 1 2 + L 2 ( 6 - 9 L 2 + 2 L 2 2 ) 6 L 1 3 + L 2 ( - 12 + 36 L 2 - 22 L 2 2 + 3 L 2 3 ) 12 L 1 4 + W_{0}(x)=L_{1}-L_{2}+\frac{L_{2}}{L_{1}}+\frac{L_{2}(-2+L_{2})}{2L_{1}^{2}}+% \frac{L_{2}(6-9L_{2}+2L_{2}^{2})}{6L_{1}^{3}}+\frac{L_{2}(-12+36L_{2}-22L_{2}^% {2}+3L_{2}^{3})}{12L_{1}^{4}}+\cdots
  14. W 0 ( x ) = L 1 - L 2 + = 0 m = 1 ( - 1 ) [ + m + 1 ] m ! L 1 - - m L 2 m W_{0}(x)=L_{1}-L_{2}+\sum_{\ell=0}^{\infty}\sum_{m=1}^{\infty}\frac{(-1)^{\ell% }\left[\begin{matrix}\ell+m\\ \ell+1\end{matrix}\right]}{m!}L_{1}^{-\ell-m}L_{2}^{m}
  15. L 1 = ln x L_{1}=\ln x
  16. L 2 = ln ln x L_{2}=\ln\ln x
  17. [ + m + 1 ] \left[\begin{matrix}\ell+m\\ \ell+1\end{matrix}\right]
  18. W 0 ( x ) = ln x - ln ln x + o ( 1 ) . W_{0}(x)=\ln x-\ln\ln x+o(1).
  19. W - 1 W_{-1}
  20. L 1 = ln ( - x ) L_{1}=\ln(-x)
  21. L 2 = ln ( - ln ( - x ) ) L_{2}=\ln(-\ln(-x))
  22. W 0 W_{0}
  23. 0
  24. W 0 ( x ) 2 = n = 2 - 2 ( - n ) n - 3 ( n - 2 ) ! x n = x 2 - 2 x 3 + 4 x 4 - 25 3 x 5 + 18 x 6 - W_{0}(x)^{2}=\sum_{n=2}^{\infty}\frac{-2(-n)^{n-3}}{(n-2)!}\ x^{n}=x^{2}-2x^{3% }+4x^{4}-\frac{25}{3}x^{5}+18x^{6}-\cdots
  25. r \Z , r\in\Z,
  26. W 0 ( x ) r = n = r - r ( - n ) n - r - 1 ( n - r ) ! x n , W_{0}(x)^{r}=\sum_{n=r}^{\infty}\frac{-r(-n)^{n-r-1}}{(n-r)!}\ x^{n},
  27. W 0 ( x ) / x W_{0}(x)/x
  28. ( W 0 ( x ) x ) r = exp ( - r W 0 ( x ) ) = n = 0 r ( n + r ) n - 1 n ! ( - x ) n , \left(\frac{W_{0}(x)}{x}\right)^{r}=\exp(-rW_{0}(x))=\sum_{n=0}^{\infty}\frac{% r(n+r)^{n-1}}{n!}\ (-x)^{n},
  29. r \C r\in\C
  30. | x | < e - 1 |x|<e^{-1}
  31. W ( x e x ) = x W(x\cdot e^{x})=x
  32. W ( x ) e W ( x ) = x W(x)\cdot e^{W(x)}=x
  33. e W ( x ) = x W ( x ) e^{W(x)}=\frac{x}{W(x)}
  34. e W ( ln x ) = ln x W ( ln x ) e^{W(\ln x)}=\frac{\ln x}{W(\ln x)}
  35. e n W ( x ) = ( x W ( x ) ) n e^{n\cdot W(x)}=\left(\frac{x}{W(x)}\right)^{n}
  36. ln W ( x ) = ln ( x ) - W ( x ) \ln W(x)=\ln(x)-W(x)
  37. W ( x ) = ln ( x W ( x ) ) W(x)=\ln\left(\frac{x}{W(x)}\right)
  38. W ( x ln x ) = ln x W(x\cdot\ln x)=\ln x
  39. W ( x ln x ) = W ( x ) + ln W ( x ) W(x\cdot\ln x)=W(x)+\ln W(x)
  40. h ( x ) = W ( - ln ( x ) ) - ln ( x ) h(x)=\frac{W(-\ln(x))}{-\ln(x)}
  41. W ( - π 2 ) = π 2 i W\left(-\frac{\pi}{2}\right)=\frac{\pi}{2}{\rm{i}}
  42. W ( - ln a a ) = - ln a ( 1 e a e ) W\left(-\frac{\ln a}{a}\right)=-\ln a\quad\left(\frac{1}{e}\leq a\leq e\right)
  43. W ( - 1 e ) = - 1 W\left(-\frac{1}{e}\right)=-1
  44. W ( 0 ) = 0 W\left(0\right)=0\,
  45. W ( 1 ) = Ω = 1 - + d t ( e t - t ) 2 + π 2 - 1 0.56714329 W\left(1\right)=\Omega=\frac{1}{\displaystyle\int_{-\infty}^{+\infty}\frac{\,% dt}{(e^{t}-t)^{2}+\pi^{2}}}-1\approx 0.56714329\dots\,
  46. W ( 1 ) = e - W ( 1 ) = ln ( 1 W ( 1 ) ) = - ln W ( 1 ) W\left(1\right)=e^{-W(1)}=\ln\left(\frac{1}{W(1)}\right)=-\ln W(1)
  47. W ( e ) = 1 W\left(e\right)=1\,
  48. W ( - 1 ) - 0.31813 - 1.33723 i W\left(-1\right)\approx-0.31813-1.33723{\rm{i}}\,
  49. W ( 0 ) = 1 W^{\prime}\left(0\right)=1\,
  50. 0 π W ( 2 cot 2 ( x ) ) sec 2 ( x ) d x = 4 π \int_{0}^{\pi}W\bigl(2\cot^{2}(x)\bigr)\sec^{2}(x)\;\mathrm{d}x=4\sqrt{\pi}
  51. 0 W ( x ) x x d x = 2 2 π \int_{0}^{\infty}\frac{W(x)}{x\sqrt{x}}\mathrm{d}x=2\sqrt{2\pi}
  52. 0 W ( 1 x 2 ) d x = 2 π \int_{0}^{\infty}W\left(\frac{1}{x^{2}}\right)\;\mathrm{d}x=\sqrt{2\pi}
  53. u = W ( x ) u=W(x)
  54. x = u e u x=ue^{u}
  55. d x d u = ( u + 1 ) e u \frac{dx}{du}=(u+1)e^{u}
  56. 0 W ( x ) x x d x = 0 u u e u u e u ( u + 1 ) e u d u \int_{0}^{\infty}\frac{W(x)}{x\sqrt{x}}\mathrm{d}x=\int_{0}^{\infty}\frac{u}{% ue^{u}\sqrt{ue^{u}}}(u+1)e^{u}\mathrm{d}u
  57. = 0 u + 1 u e u d u =\int_{0}^{\infty}\frac{u+1}{\sqrt{ue^{u}}}\mathrm{d}u
  58. = 0 u + 1 u 1 e u d u =\int_{0}^{\infty}\frac{u+1}{\sqrt{u}}\frac{1}{\sqrt{e^{u}}}\mathrm{d}u
  59. = 0 u 1 2 e - u 2 d u + 0 u - 1 2 e - u 2 d u =\int_{0}^{\infty}u^{\frac{1}{2}}e^{-\frac{u}{2}}\mathrm{d}u+\int_{0}^{\infty}% u^{-\frac{1}{2}}e^{-\frac{u}{2}}\mathrm{d}u
  60. = 2 0 ( 2 w ) 1 2 e - w d w + 2 0 ( 2 w ) - 1 2 e - w d w =2\int_{0}^{\infty}(2w)^{\frac{1}{2}}e^{-w}\mathrm{d}w+2\int_{0}^{\infty}(2w)^% {-\frac{1}{2}}e^{-w}\mathrm{d}w
  61. u = 2 w u=2w
  62. = 2 2 0 w 1 2 e - w d w + 2 0 w - 1 2 e - w d w =2\sqrt{2}\int_{0}^{\infty}w^{\frac{1}{2}}e^{-w}\mathrm{d}w+\sqrt{2}\int_{0}^{% \infty}w^{-\frac{1}{2}}e^{-w}\mathrm{d}w
  63. = 2 2 Γ ( 3 2 ) + 2 Γ ( 1 2 ) =2\sqrt{2}\Gamma(\frac{3}{2})+\sqrt{2}\Gamma(\frac{1}{2})
  64. = 2 2 ( 1 2 π ) + 2 ( π ) =2\sqrt{2}(\frac{1}{2}\sqrt{\pi})+\sqrt{2}(\sqrt{\pi})
  65. = 2 2 π =2\sqrt{2\pi}
  66. u = 1 x 2 u=\frac{1}{x^{2}}
  67. Y = X e X X = W ( Y ) Y=Xe^{X}\;\Longleftrightarrow\;X=W(Y)
  68. 2 t = 5 t 1 = 5 t 2 t 1 = 5 t e - t ln 2 1 5 = t e - t ln 2 - ln 2 5 = ( - t ln 2 ) e ( - t ln 2 ) W ( - ln 2 5 ) = - t ln 2 t = - W ( - ln 2 5 ) ln 2 \begin{aligned}\displaystyle 2^{t}&\displaystyle=5t\\ \displaystyle 1&\displaystyle=\frac{5t}{2^{t}}\\ \displaystyle 1&\displaystyle=5t\,e^{-t\ln 2}\\ \displaystyle\frac{1}{5}&\displaystyle=t\,e^{-t\ln 2}\\ \displaystyle\frac{-\,\ln 2}{5}&\displaystyle=(-\,t\,\ln 2)\,e^{(-t\ln 2)}\\ \displaystyle W\left(\frac{-\ln 2}{5}\right)&\displaystyle=-t\ln 2\\ \displaystyle t&\displaystyle=-\frac{W\left(\frac{-\ln 2}{5}\right)}{\ln 2}% \end{aligned}
  69. p a x + b = c x + d ~{}p^{ax+b}=cx+d
  70. p > 0 and c , a 0 p>0\,\text{ and }c,a\neq 0
  71. - t = a x + a d c -t=ax+\frac{ad}{c}
  72. t p t = R = - a c p b - a d c tp^{t}=R=-\frac{a}{c}p^{b-\frac{ad}{c}}
  73. t = W ( R ln p ) ln p t=\frac{W(R\ln p)}{\ln p}
  74. x = - W ( - a ln p c p b - a d c ) a ln p - d c x=-\frac{W(-\frac{a\ln p}{c}\,p^{b-\frac{ad}{c}})}{a\ln p}-\frac{d}{c}
  75. x x = z x^{x}=z\,
  76. x ln x = ln z \Rightarrow x\ln x=\ln z\,
  77. e ln x ln x = ln z \Rightarrow e^{\ln x}\cdot\ln x=\ln z\,
  78. ln x = W ( ln z ) \Rightarrow\ln x=W(\ln z)\,
  79. x = e W ( ln z ) , \Rightarrow x=e^{W(\ln z)}\,,
  80. x = ln z W ( ln z ) , x=\frac{\ln z}{W(\ln z)},
  81. ln z = W ( ln z ) e W ( ln z ) \ln z=W(\ln z)e^{W(\ln z)}\,
  82. z z z z^{z^{z^{\cdot^{\cdot^{\cdot}}}}}\!
  83. c = W ( - ln ( z ) ) - ln ( z ) c=\frac{W(-\ln(z))}{-\ln(z)}
  84. z c = c z^{c}=c
  85. z = c 1 c z=c^{\frac{1}{c}}
  86. z - 1 = c - 1 c \Rightarrow z^{-1}=c^{-\frac{1}{c}}
  87. 1 z = ( 1 c ) ( 1 c ) \Rightarrow\frac{1}{z}=\left(\frac{1}{c}\right)^{\left(\frac{1}{c}\right)}
  88. - ln ( z ) = ( 1 c ) ln ( 1 c ) \Rightarrow-\ln(z)=\left(\frac{1}{c}\right)\ln\left(\frac{1}{c}\right)
  89. - ln ( z ) = e ln ( 1 c ) ln ( 1 c ) \Rightarrow-\ln(z)=e^{\ln\left(\frac{1}{c}\right)}\ln\left(\frac{1}{c}\right)
  90. ln ( 1 c ) = W ( - ln ( z ) ) \Rightarrow\ln\left(\frac{1}{c}\right)=W(-\ln(z))
  91. 1 c = e W ( - ln ( z ) ) \Rightarrow\frac{1}{c}=e^{W(-\ln(z))}
  92. 1 c = - ln ( z ) W ( - ln ( z ) ) \Rightarrow\frac{1}{c}=\frac{-\ln(z)}{W(-\ln(z))}
  93. c = W ( - ln ( z ) ) - ln ( z ) \Rightarrow c=\frac{W(-\ln(z))}{-\ln(z)}
  94. x log b ( x ) = a x\log_{b}\left(x\right)=a
  95. x = e W ( a ln b ) . x=e^{W(a\ln b)}.
  96. y ˙ ( t ) = a y ( t - 1 ) \dot{y}(t)=ay(t-1)
  97. λ = a e - λ \lambda=ae^{-\lambda}
  98. λ = W k ( a ) \lambda=W_{k}(a)
  99. y ( t ) = e W k ( a ) t y(t)=e^{W_{k}(a)t}
  100. k k
  101. a e - 1 a\geq e^{-1}
  102. W 0 ( a ) W_{0}(a)
  103. H ( x ) = 1 + W [ ( H ( 0 ) - 1 ) exp ( ( H ( 0 ) - 1 ) - x L ) ] , H(x)=1+W[(H(0)-1)\exp((H(0)-1)-\frac{x}{L})],
  104. Y = X 1 - e X Y=\frac{X}{1-e^{X}}
  105. W 0 W_{0}
  106. W - 1 W_{-1}
  107. X ( Y ) = W - 1 ( Y e Y ) - W 0 ( Y e Y ) = Y - W 0 ( Y e Y ) for Y < - 1. X(Y)=W_{-1}(Ye^{Y})-W_{0}(Ye^{Y})=Y-W_{0}(Ye^{Y})\,\text{for }Y<-1.
  108. X ( Y ) = W 0 ( Y e Y ) - W - 1 ( Y e Y ) = Y - W - 1 ( Y e Y ) for - 1 < Y < 0. X(Y)=W_{0}(Ye^{Y})-W_{-1}(Ye^{Y})=Y-W_{-1}(Ye^{Y})\,\text{for }-1<Y<0.
  109. e - c x = a o ( x - r ) ( 1 ) e^{-cx}=a_{o}(x-r)~{}~{}\quad\qquad\qquad\qquad\qquad(1)
  110. x = r + 1 c W ( c e - c r a o ) x=r+\frac{1}{c}W\!\left(\frac{c\,e^{-cr}}{a_{o}}\right)\,
  111. e - c x = a o ( x - r 1 ) ( x - r 2 ) ( 2 ) e^{-cx}=a_{o}(x-r_{1})(x-r_{2})~{}~{}\qquad\qquad(2)
  112. e - c x = a o i = 1 ( x - r i ) i = 1 ( x - s i ) ( 3 ) e^{-cx}=a_{o}\frac{\displaystyle\prod_{i=1}^{\infty}(x-r_{i})}{\displaystyle% \prod_{i=1}^{\infty}(x-s_{i})}\qquad\qquad\qquad(3)
  113. w = W ( z ) w=W(z)
  114. z = w e w z=we^{w}
  115. w j + 1 = w j - w j e w j - z e w j + w j e w j . w_{j+1}=w_{j}-\frac{w_{j}e^{w_{j}}-z}{e^{w_{j}}+w_{j}e^{w_{j}}}.
  116. w j + 1 = w j - w j e w j - z e w j ( w j + 1 ) - ( w j + 2 ) ( w j e w j - z ) 2 w j + 2 w_{j+1}=w_{j}-\frac{w_{j}e^{w_{j}}-z}{e^{w_{j}}(w_{j}+1)-\frac{(w_{j}+2)(w_{j}% e^{w_{j}}-z)}{2w_{j}+2}}

Landau's_function.html

  1. lim n ln ( g ( n ) ) n ln ( n ) = 1 \lim_{n\to\infty}\frac{\ln(g(n))}{\sqrt{n\ln(n)}}=1
  2. ln g ( n ) < Li - 1 ( n ) \ln g(n)<\sqrt{\mathrm{Li}^{-1}(n)}
  3. g ( n ) < e n / e g(n)<e^{n/e}
  4. g ( n ) exp ( 1.05314 n log n ) . g(n)\leq\exp\left(1.05314\sqrt{n\log n}\right).

Langevin_equation.html

  1. m d 2 𝐱 d t 2 = - λ d 𝐱 d t + s y m b o l η ( t ) . m\frac{d^{2}\mathbf{x}}{dt^{2}}=-\lambda\frac{d\mathbf{x}}{dt}+symbol{\eta}% \left(t\right).
  2. η i ( t ) η j ( t ) = 2 λ k B T δ i , j δ ( t - t ) , \left\langle\eta_{i}\left(t\right)\eta_{j}\left(t^{\prime}\right)\right\rangle% =2\lambda k_{B}T\delta_{i,j}\delta\left(t-t^{\prime}\right),
  3. d A i d t = k B T j [ A i , A j ] d d A j - j λ i , j ( A ) d d A j + j d λ i , j ( A ) d A j + η i ( t ) . \frac{dA_{i}}{dt}=k_{B}T\sum\limits_{j}{\left[{A_{i},A_{j}}\right]\frac{{d}% \mathcal{H}}{{dA_{j}}}}-\sum\limits_{j}{\lambda_{i,j}\left(A\right)\frac{d% \mathcal{H}}{{dA_{j}}}+}\sum\limits_{j}{\frac{d{\lambda_{i,j}\left(A\right)}}{% {dA_{j}}}}+\eta_{i}\left(t\right).
  4. η i ( t ) η j ( t ) = 2 λ i , j ( A ) δ ( t - t ) . \left\langle{\eta_{i}\left(t\right)\eta_{j}\left(t^{\prime}\right)}\right% \rangle=2\lambda_{i,j}\left(A\right)\delta\left(t-t^{\prime}\right).
  5. \mathcal{H}
  6. \mathcal{H}
  7. \mathcal{H}
  8. d U d t = - U R C + η ( t ) , η ( t ) η ( t ) = 2 k B T R C 2 δ ( t - t ) . \frac{dU}{dt}=-\frac{U}{RC}+\eta\left(t\right),\;\;\left\langle\eta\left(t% \right)\eta\left(t^{\prime}\right)\right\rangle=\frac{2k_{B}T}{RC^{2}}\delta% \left(t-t^{\prime}\right).
  9. U ( t ) U ( t ) = ( k B T / C ) exp ( - | t - t | / R C ) 2 R k B T δ ( t - t ) , \left\langle U\left(t\right)U\left(t^{\prime}\right)\right\rangle=\left(k_{B}T% /C\right)\exp\left(-\left|t-t^{\prime}\right|/RC\right)\approx 2Rk_{B}T\delta% \left(t-t^{\prime}\right),
  10. φ ( 𝐱 , t ) t = - λ δ δ φ + η ( 𝐱 , t ) , = d d x { 1 2 φ [ r 0 - 2 ] φ + u φ 4 } , η ( 𝐱 , t ) η ( 𝐱 , t ) = 2 λ δ ( 𝐱 - 𝐱 ) δ ( t - t ) . \begin{aligned}\displaystyle\frac{\partial\varphi\left(\mathbf{x},t\right)}{% \partial t}&\displaystyle=-\lambda\frac{\delta\mathcal{H}}{\delta\varphi}+\eta% \left(\mathbf{x},t\right),\\ \displaystyle\mathcal{H}&\displaystyle=\int d^{d}x\left\{\frac{1}{2}\varphi% \left[r_{0}-\nabla^{2}\right]\varphi+u\varphi^{4}\right\},\\ \displaystyle\left\langle\eta\left(\mathbf{x},t\right)\eta\left(\mathbf{x}^{% \prime},t^{\prime}\right)\right\rangle&\displaystyle=2\lambda\delta\left(% \mathbf{x}-\mathbf{x}^{\prime}\right)\delta\left(t-t^{\prime}\right).\end{aligned}
  11. P ( A , t ) t = i , j A i ( - k B T [ A i , A j ] A j + λ i , j A j + λ i , j A j ) P ( A , t ) . \frac{\partial P\left(A,t\right)}{\partial t}=\sum_{i,j}\frac{\partial}{% \partial A_{i}}\left(-k_{B}T\left[A_{i},A_{j}\right]\frac{\partial\mathcal{H}}% {\partial A_{j}}+\lambda_{i,j}\frac{\partial\mathcal{H}}{\partial A_{j}}+% \lambda_{i,j}\frac{\partial}{\partial A_{j}}\right)P\left(A,t\right).
  12. \mathcal{H}
  13. A ~ \tilde{A}
  14. P ( A , A ~ ) d A d A ~ = N exp ( L ( A , A ~ ) ) d A d A ~ , \int P(A,\tilde{A})\,dA\,d\tilde{A}=N\int\exp\left(L(A,\tilde{A})\right)dA\,d% \tilde{A},
  15. L ( A , A ~ ) = i , j { A ~ i λ i , j A ~ j - A ~ i { δ i , j d A j d t - k B T [ A i , A j ] d d A j + λ i , j d d A j - d λ i , j d A j } } d t . L(A,\tilde{A})=\int\sum_{i,j}\left\{\tilde{A}_{i}\lambda_{i,j}\tilde{A}_{j}-% \widetilde{A}_{i}\left\{\delta_{i,j}\frac{dA_{j}}{dt}-k_{B}T\left[A_{i},A_{j}% \right]\frac{d\mathcal{H}}{dA_{j}}+\lambda_{i,j}\frac{d\mathcal{H}}{dA_{j}}-% \frac{d\lambda_{i,j}}{dA_{j}}\right\}\right\}dt.

Laplace_operator.html

  1. = ( x 1 , , x n ) . \nabla=\left(\frac{\partial}{\partial x_{1}},\dots,\frac{\partial}{\partial x_% {n}}\right).
  2. x i x_{i}
  3. V u 𝐧 d S = 0 , \int_{\partial V}\nabla u\cdot\mathbf{n}\,dS=0,
  4. V div u d V = V u 𝐧 d S = 0. \int_{V}\operatorname{div}\nabla u\,dV=\int_{\partial V}\nabla u\cdot\mathbf{n% }\,dS=0.
  5. div u = Δ u = 0. \operatorname{div}\nabla u=\Delta u=0.
  6. V 𝐄 𝐧 d S = V φ 𝐧 d S = V q d V , \int_{\partial V}\mathbf{E}\cdot\mathbf{n}\,dS=\int_{\partial V}\nabla\varphi% \cdot\mathbf{n}\,dS=\int_{V}q\,dV,
  7. V Δ φ d V = V q d V , \int_{V}\Delta\varphi\,dV=\int_{V}q\,dV,
  8. Δ f = 0 \Delta f=0
  9. E ( f ) = 1 2 U f 2 d x . E(f)=\frac{1}{2}\int_{U}\|\nabla f\|^{2}\,dx.
  10. f : U f\colon U\to\mathbb{R}
  11. u : U u\colon U\to\mathbb{R}
  12. d d ε | ε = 0 E ( f + ε u ) = U f u d x = - U u Δ f d x \frac{d}{d\varepsilon}\Big|_{\varepsilon=0}E(f+\varepsilon u)=\int_{U}\nabla f% \cdot\nabla u\,dx=-\int_{U}u\Delta f\,dx
  13. Δ f = 0 \Delta f=0
  14. Δ f = 0 \Delta f=0
  15. Δ f = 2 f x 2 + 2 f y 2 \Delta f=\frac{\partial^{2}f}{\partial x^{2}}+\frac{\partial^{2}f}{\partial y^% {2}}
  16. Δ f = 1 r r ( r f r ) + 1 r 2 2 f θ 2 = 2 f r 2 + 1 r f r + 1 r 2 2 f θ 2 . \begin{aligned}\displaystyle\Delta f&\displaystyle={1\over r}{\partial\over% \partial r}\left(r{\partial f\over\partial r}\right)+{1\over r^{2}}{\partial^{% 2}f\over\partial\theta^{2}}\\ &\displaystyle={\partial^{2}f\over\partial r^{2}}+{1\over r}{\partial f\over% \partial r}+{1\over r^{2}}{\partial^{2}f\over\partial\theta^{2}}.\end{aligned}
  17. Δ f = 2 f x 2 + 2 f y 2 + 2 f z 2 . \Delta f=\frac{\partial^{2}f}{\partial x^{2}}+\frac{\partial^{2}f}{\partial y^% {2}}+\frac{\partial^{2}f}{\partial z^{2}}.
  18. Δ f = 1 ρ ρ ( ρ f ρ ) + 1 ρ 2 2 f φ 2 + 2 f z 2 . \Delta f={1\over\rho}{\partial\over\partial\rho}\left(\rho{\partial f\over% \partial\rho}\right)+{1\over\rho^{2}}{\partial^{2}f\over\partial\varphi^{2}}+{% \partial^{2}f\over\partial z^{2}}.
  19. Δ f = 1 r 2 r ( r 2 f r ) + 1 r 2 sin θ θ ( sin θ f θ ) + 1 r 2 sin 2 θ 2 f φ 2 . \Delta f={1\over r^{2}}{\partial\over\partial r}\left(r^{2}{\partial f\over% \partial r}\right)+{1\over r^{2}\sin\theta}{\partial\over\partial\theta}\left(% \sin\theta{\partial f\over\partial\theta}\right)+{1\over r^{2}\sin^{2}\theta}{% \partial^{2}f\over\partial\varphi^{2}}.
  20. ξ 1 , ξ 2 , ξ 3 \xi^{1},\xi^{2},\xi^{3}
  21. 2 = ξ m ξ n 2 ξ m ξ n + 2 ξ m ξ m , \nabla^{2}=\nabla\xi^{m}\cdot\nabla\xi^{n}{\partial^{2}\over\partial\xi^{m}% \partial\xi^{n}}+\nabla^{2}\xi^{m}{\partial\over\partial\xi^{m}},
  22. Δ f = 2 f r 2 + N - 1 r f r + 1 r 2 Δ S N - 1 f \Delta f=\frac{\partial^{2}f}{\partial r^{2}}+\frac{N-1}{r}\frac{\partial f}{% \partial r}+\frac{1}{r^{2}}\Delta_{S^{N-1}}f
  23. Δ S N - 1 \Delta_{S^{N-1}}
  24. 1 r N - 1 r ( r N - 1 f r ) . \frac{1}{r^{N-1}}\frac{\partial}{\partial r}\Bigl(r^{N-1}\frac{\partial f}{% \partial r}\Bigr).
  25. - Δ f = λ f . -\Delta f=\lambda f.
  26. Δ f = tr ( H ( f ) ) \Delta f=\mathrm{tr}(H(f))\,\!
  27. Δ f = d * d f \Delta f=d^{*}df\,
  28. Δ α = d * d α + d d * α . \Delta\alpha=d^{*}d\alpha+dd^{*}\alpha.\,
  29. = 1 c 2 2 t 2 - 2 x 2 - 2 y 2 - 2 z 2 . \square=\frac{1}{c^{2}}{\partial^{2}\over\partial t^{2}}-{\partial^{2}\over% \partial x^{2}}-{\partial^{2}\over\partial y^{2}}-{\partial^{2}\over\partial z% ^{2}}.

Lapse_rate.html

  1. γ = - d T d z \gamma=-\frac{dT}{dz}
  2. γ \gamma
  3. Γ \Gamma
  4. α \alpha
  5. γ \gamma
  6. P d V = - V d P / γ PdV=-VdP/\gamma
  7. m c v d T - V d p / γ = 0 mc_{v}dT-Vdp/\gamma=0
  8. α = V / m \alpha=V/m
  9. γ = c p / c v \gamma=c_{p}/c_{v}
  10. c p d T - α d P = 0 c_{p}dT-\alpha dP=0
  11. c p c_{p}
  12. α \alpha
  13. d P = - ρ g d z dP=-\rho gdz
  14. ρ \rho
  15. Γ d = - d T d z = g c p = 9.8 C / km \Gamma_{d}=-\frac{dT}{dz}=\frac{g}{c_{p}}=9.8\ ^{\circ}\mathrm{C}/\mathrm{km}
  16. Γ w = g 1 + H v r R s d T c p d + H v 2 r R s w T 2 = g 1 + H v r R s d T c p d + H v 2 r ϵ R s d T 2 \Gamma_{w}=g\,\frac{1+\dfrac{H_{v}\,r}{R_{sd}\,T}}{c_{pd}+\dfrac{H_{v}^{2}\,r}% {R_{sw}\,T^{2}}}=g\,\frac{1+\dfrac{H_{v}\,r}{R_{sd}\,T}}{c_{pd}+\dfrac{H_{v}^{% 2}\,r\,\epsilon}{R_{sd}\,T^{2}}}
  17. Γ w \Gamma_{w}
  18. g g
  19. H v H_{v}
  20. R s d R_{sd}
  21. R s w R_{sw}
  22. ϵ = R s d R s w \epsilon=\frac{R_{sd}}{R_{sw}}
  23. e e
  24. p p
  25. r = ϵ e / ( p - e ) r=\epsilon e/(p-e)
  26. T T
  27. c p d c_{pd}

Large_numbers.html

  1. 10 10 10 10 10 1.1 years 10^{10^{10^{10^{10^{1.1}}}}}\mbox{ years}~{}
  2. 10 10 10^{10}
  3. 10 99 10^{99}
  4. 10 100 . 10^{100}.
  5. 10 303 10^{303}
  6. 10 600 10^{600}
  7. 2 57 , 885 , 161 - 1 2^{57,885,161}-1
  8. 10 googol = 10 10 100 10^{\,\text{googol}}=10^{10^{100}}
  9. 10 10 10 34 10^{10^{10^{34}}}
  10. 10 10 10 964 10^{10^{10^{964}}}
  11. 2 1.6 × 10 18 10 4.8 × 10 17 2^{1.6\times 10^{18}}\approx 10^{4.8\times 10^{17}}
  12. 1.1 1.1 1.1 1000 10 10 1.02 × 10 40 1.1^{1.1^{1.1^{1000}}}\approx 10^{10^{1.02\times 10^{40}}}
  13. 1000 1000 1000 10 10 3000.48 1000^{1000^{1000}}\approx 10^{10^{3000.48}}
  14. 10 10 10 10^{10^{10}}
  15. f 0 ( n ) f_{0}(n)
  16. f 1 ( n ) = f 0 n ( n ) f_{1}(n)=f_{0}^{n}(n)
  17. f k ( n ) = f k - 1 n ( n ) f_{k}(n)=f_{k-1}^{n}(n)
  18. f ω ( n ) = f n ( n ) f_{\omega}(n)=f_{n}(n)
  19. f k f_{k}
  20. 10 n = 10 n 2 = ( 10 ) n 1 10\uparrow\uparrow n=10\to n\to 2=(10\uparrow)^{n}1
  21. ( 10 ) n (10\uparrow)^{n}
  22. f ( n ) = 10 n f(n)=10^{n}
  23. ( 10 ) n a (10\uparrow)^{n}a
  24. 10 10 10 10 10 4.829 10^{10^{10^{10^{10^{4.829}}}}}
  25. 10 5 10\uparrow\uparrow 5
  26. 10 6 10\uparrow\uparrow 6
  27. 10 n < ( 10 ) n a < 10 ( n + 1 ) 10\uparrow\uparrow n<(10\uparrow)^{n}a<10\uparrow\uparrow(n+1)
  28. 1 < a < 10 1<a<10
  29. 10 10 100 = ( 10 ) 2 100 = ( 10 ) 3 2 10^{10^{100}}=(10\uparrow)^{2}100=(10\uparrow)^{3}2
  30. 2 4 = 2 2 . . . 2 65 , 536 copies of 2 ( 10 ) 65 , 531 ( 6.0 × 10 19 , 728 ) ( 10 ) 65 , 533 4.3 2\uparrow\uparrow\uparrow 4=\begin{matrix}\underbrace{2^{2^{{}^{.\,^{.\,^{.\,^% {2}}}}}}}\\ \qquad\quad\ \ \ 65,536\mbox{ copies of }~{}2\end{matrix}\approx(10\uparrow)^{% 65,531}(6.0\times 10^{19,728})\approx(10\uparrow)^{65,533}4.3
  31. 10 65 , 533 10\uparrow\uparrow 65,533
  32. 10 65 , 534 10\uparrow\uparrow 65,534
  33. l o g 10 log_{10}
  34. 10 n 10\uparrow\uparrow n
  35. 10 ( n + 1 ) 10\uparrow\uparrow(n+1)
  36. 10 ( 10 ) n x = ( 10 ) n 10 x 10^{(10\uparrow)^{n}x}=(10\uparrow)^{n}10^{x}
  37. ( 10 ) n x (10\uparrow)^{n}x
  38. 10 ( 7.21 × 10 8 ) 10\uparrow\uparrow(7.21\times 10^{8})
  39. 10 10 10 10 3.81 × 10 17 10\uparrow\uparrow 10^{\,\!10^{10^{3.81\times 10^{17}}}}
  40. 10 2 10\uparrow\uparrow\uparrow 2
  41. 10 3 10\uparrow\uparrow\uparrow 3
  42. 10 10 ( 10 ) 497 ( 9.73 × 10 32 ) = ( 10 ) 2 ( 10 ) 497 ( 9.73 × 10 32 ) 10\uparrow\uparrow 10\uparrow\uparrow(10\uparrow)^{497}(9.73\times 10^{32})=(1% 0\uparrow\uparrow)^{2}(10\uparrow)^{497}(9.73\times 10^{32})
  43. 10 4 10\uparrow\uparrow\uparrow 4
  44. 10 5 10\uparrow\uparrow\uparrow 5
  45. ( 10 ) (10\uparrow)
  46. ( 10 ) (10\uparrow)
  47. ( 10 ) (10\uparrow\uparrow)
  48. ( 10 ) 3 ( 2.8 × 10 12 ) (10\uparrow\uparrow)^{3}(2.8\times 10^{12})
  49. ( 10 ) (10\uparrow\uparrow)
  50. ( 10 ) (10\uparrow\uparrow)
  51. 10 ( 7.3 × 10 6 ) 10\uparrow\uparrow\uparrow(7.3\times 10^{6})
  52. 10 ( 10 ) 2 ( 10 ) 497 ( 9.73 × 10 32 ) 10\uparrow\uparrow\uparrow(10\uparrow\uparrow)^{2}(10\uparrow)^{497}(9.73% \times 10^{32})
  53. 10 10 4 10\uparrow\uparrow\uparrow 10\uparrow\uparrow\uparrow 4
  54. 10 10 5 10\uparrow\uparrow\uparrow 10\uparrow\uparrow\uparrow 5
  55. n \uparrow^{n}
  56. a n b a\uparrow^{n}b
  57. ( a n ) k b (a\uparrow^{n})^{k}b
  58. ( 10 2 ) 3 b (10\uparrow^{2})^{3}b
  59. 10 3 3 = ( 10 2 ) 3 1 10\uparrow^{3}3=(10\uparrow^{2})^{3}1
  60. ( 10 n ) k n (10\uparrow^{n})^{k_{n}}
  61. k n {k_{n}}
  62. k n {k_{n}}
  63. k n + 1 {k_{n+1}}
  64. ( n + 1 ) k n + 1 ({n+1})^{k_{n+1}}
  65. 10 ( 10 ) 5 a = ( 10 ) 6 a 10\uparrow(10\uparrow\uparrow)^{5}a=(10\uparrow\uparrow)^{6}a
  66. 10 ( 10 3 ) = 10 ( 10 10 + 1 ) 10 3 10\uparrow(10\uparrow\uparrow\uparrow 3)=10\uparrow\uparrow(10\uparrow\uparrow 1% 0+1)\approx 10\uparrow\uparrow\uparrow 3
  67. 10 n 10 = ( 10 10 n ) 10\uparrow^{n}10=(10\to 10\to n)
  68. n \uparrow^{n}
  69. 3 × 10 5 3\times 10^{5}
  70. 10 10 3 × 10 5 10 10 10\uparrow^{10\uparrow^{3\times 10^{5}}10}10\!
  71. f ( n ) = 10 n 10 f(n)=10\uparrow^{n}10
  72. f m ( n ) f^{m}(n)
  73. f 2 ( 3 × 10 5 ) f^{2}(3\times 10^{5})
  74. ( 10 10 3 2 ) = 10 10 10 10 10 10 (10\to 10\to 3\to 2)=10\uparrow^{10\uparrow^{10^{10}}10}10\!
  75. G < 3 3 65 2 < ( 10 10 65 2 ) = f 65 ( 1 ) G<3\rightarrow 3\rightarrow 65\rightarrow 2<(10\to 10\to 65\to 2)=f^{65}(1)
  76. G < f 64 ( 4 ) < f 65 ( 1 ) G<f^{64}(4)<f^{65}(1)
  77. f m ( n ) f^{m}(n)
  78. g ( n ) = f n ( 1 ) g(n)=f^{n}(1)
  79. g m ( n ) g^{m}(n)
  80. f k m ( n ) f_{k}^{m}(n)
  81. f k ( n ) = f k - 1 n ( 1 ) f_{k}(n)=f_{k-1}^{n}(1)
  82. f k m k {f_{k}}^{m_{k}}
  83. ( 10 n ) p n (10\uparrow^{n})^{p_{n}}
  84. f n ( 10 ) {f_{n}}(10)
  85. 10 n 10^{n}
  86. 10 n 10 10\uparrow^{n}10
  87. f n ( 10 ) {f_{n}}(10)
  88. f q k m q k {f_{qk}}^{m_{qk}}
  89. ( 10 n ) p n (10\uparrow^{n})^{p_{n}}
  90. 2 2 2 2 2^{2^{2^{2}}}
  91. 2 2 2 2 2 = 2 5 = 2 65 , 536 2.0 × 10 19 , 728 ( 10 ) 2 4.3 2^{2^{2^{2^{2}}}}=2\uparrow\uparrow 5=2^{65,536}\approx 2.0\times 10^{19,728}% \approx(10\uparrow)^{2}4.3
  92. M 57 , 885 , 161 5.81 × 10 17 , 425 , 169 10 10 7.2 = ( 10 ) 2 7.2 M_{57,885,161}\approx 5.81\times 10^{17,425,169}\approx 10^{10^{7.2}}=(10% \uparrow)^{2}7.2
  93. 3 3 3 3 = 3 4 1.26 × 10 3 , 638 , 334 , 640 , 024 ( 10 ) 3 1.10 3^{3^{3^{3}}}=3\uparrow\uparrow 4\approx 1.26\times 10^{3,638,334,640,024}% \approx(10\uparrow)^{3}1.10
  94. 10 10 100 = ( 10 ) 3 2 10^{10^{100}}=(10\uparrow)^{3}2
  95. 2 2 2 2 2 2 = 2 6 = 2 2 65 , 536 2 ( 10 ) 2 4.3 10 ( 10 ) 2 4.3 = ( 10 ) 3 4.3 2^{2^{2^{2^{2^{2}}}}}=2\uparrow\uparrow 6=2^{2^{65,536}}\approx 2^{(10\uparrow% )^{2}4.3}\approx 10^{(10\uparrow)^{2}4.3}=(10\uparrow)^{3}4.3
  96. 10 10 10 10 = 10 4 = ( 10 ) 4 1 10^{10^{10^{10}}}=10\uparrow\uparrow 4=(10\uparrow)^{4}1
  97. 3 3 3 3 3 = 3 5 3 10 3.6 × 10 12 ( 10 ) 4 1.10 3^{3^{3^{3^{3}}}}=3\uparrow\uparrow 5\approx 3^{10^{3.6\times 10^{12}}}\approx% (10\uparrow)^{4}1.10
  98. 2 2 2 2 2 2 2 = 2 7 ( 10 ) 4 4.3 2^{2^{2^{2^{2^{2^{2}}}}}}=2\uparrow\uparrow 7\approx(10\uparrow)^{4}4.3
  99. 10 3 = ( 10 ) 3 1 10\uparrow\uparrow\uparrow 3=(10\uparrow\uparrow)^{3}1
  100. ( 10 ) 2 11 (10\uparrow\uparrow)^{2}11
  101. ( 10 ) 2 10 10 10 3.81 × 10 17 (10\uparrow\uparrow)^{2}10^{\,\!10^{10^{3.81\times 10^{17}}}}
  102. 10 4 = ( 10 ) 4 1 10\uparrow\uparrow\uparrow 4=(10\uparrow\uparrow)^{4}1
  103. ( 10 ) 2 ( 10 ) 497 ( 9.73 × 10 32 ) (10\uparrow\uparrow)^{2}(10\uparrow)^{497}(9.73\times 10^{32})
  104. 10 5 = ( 10 ) 5 1 10\uparrow\uparrow\uparrow 5=(10\uparrow\uparrow)^{5}1
  105. 10 6 = ( 10 ) 6 1 10\uparrow\uparrow\uparrow 6=(10\uparrow\uparrow)^{6}1
  106. 10 7 = ( 10 ) 7 1 10\uparrow\uparrow\uparrow 7=(10\uparrow\uparrow)^{7}1
  107. 10 8 = ( 10 ) 8 1 10\uparrow\uparrow\uparrow 8=(10\uparrow\uparrow)^{8}1
  108. 10 9 = ( 10 ) 9 1 10\uparrow\uparrow\uparrow 9=(10\uparrow\uparrow)^{9}1
  109. 10 2 = 10 10 = ( 10 ) 1 01 10\uparrow\uparrow\uparrow\uparrow 2=10\uparrow\uparrow\uparrow 10=(10\uparrow% \uparrow)^{1}01
  110. 10 3 = ( 10 ) 3 1 10\uparrow\uparrow\uparrow\uparrow 3=(10\uparrow\uparrow\uparrow)^{3}1
  111. 4 4 4\uparrow\uparrow\uparrow\uparrow 4
  112. ( 10 ) 2 ( 10 ) 3 154 \approx(10\uparrow\uparrow\uparrow)^{2}(10\uparrow\uparrow)^{3}154
  113. 10 4 = ( 10 ) 4 1 10\uparrow\uparrow\uparrow\uparrow 4=(10\uparrow\uparrow\uparrow)^{4}1
  114. 10 5 = ( 10 ) 5 1 10\uparrow\uparrow\uparrow\uparrow 5=(10\uparrow\uparrow\uparrow)^{5}1
  115. 10 6 = ( 10 ) 6 1 10\uparrow\uparrow\uparrow\uparrow 6=(10\uparrow\uparrow\uparrow)^{6}1
  116. 10 7 = ( 10 ) 7 1 = 10\uparrow\uparrow\uparrow\uparrow 7=(10\uparrow\uparrow\uparrow)^{7}1=
  117. 10 8 = ( 10 ) 8 1 = 10\uparrow\uparrow\uparrow\uparrow 8=(10\uparrow\uparrow\uparrow)^{8}1=
  118. 10 9 = ( 10 ) 9 1 = 10\uparrow\uparrow\uparrow\uparrow 9=(10\uparrow\uparrow\uparrow)^{9}1=
  119. 10 2 = 10 10 = ( 10 ) 10 1 10\uparrow\uparrow\uparrow\uparrow\uparrow 2=10\uparrow\uparrow\uparrow% \uparrow 10=(10\uparrow\uparrow\uparrow)^{10}1
  120. 10 10 10^{10}
  121. 10 10 10 10 10\uparrow^{10^{10}}10\!
  122. 10 10 10^{10}
  123. 10 10 10 10 10 10 10\uparrow^{10\uparrow^{10^{10}}10}10\!
  124. 100 12 = 10 24 100^{12}=10^{24}
  125. 100 100 12 = 10 2 * 10 24 100^{100^{12}}=10^{2*10^{24}}
  126. 100 100 100 12 10 10 2 * 10 24 + 0.30103 100^{100^{100^{12}}}\approx 10^{10^{2*10^{24}+0.30103}}
  127. 100 2 = 10 200 100\uparrow\uparrow 2=10^{200}
  128. 100 3 = 10 2 × 10 200 100\uparrow\uparrow 3=10^{2\times 10^{200}}
  129. 100 4 = ( 10 ) 2 ( 2 × 10 200 + 0.3 ) = ( 10 ) 2 ( 2 × 10 200 ) = ( 10 ) 3 200.3 = ( 10 ) 4 2.3 100\uparrow\uparrow 4=(10\uparrow)^{2}(2\times 10^{200}+0.3)=(10\uparrow)^{2}(% 2\times 10^{200})=(10\uparrow)^{3}200.3=(10\uparrow)^{4}2.3
  130. 100 n = ( 10 ) n - 2 ( 2 × 10 200 ) = ( 10 ) n - 1 200.3 = ( 10 ) n 2.3 < 10 ( n + 1 ) 100\uparrow\uparrow n=(10\uparrow)^{n-2}(2\times 10^{200})=(10\uparrow)^{n-1}2% 00.3=(10\uparrow)^{n}2.3<10\uparrow\uparrow(n+1)
  131. 100 n 100\uparrow\uparrow n
  132. 10 n 10\uparrow\uparrow n
  133. 100 2 = ( 10 ) 98 ( 2 × 10 200 ) = ( 10 ) 100 2.3 100\uparrow\uparrow\uparrow 2=(10\uparrow)^{98}(2\times 10^{200})=(10\uparrow)% ^{100}2.3
  134. 100 3 = 10 ( 10 ) 98 ( 2 × 10 200 ) = 10 ( 10 ) 100 2.3 100\uparrow\uparrow\uparrow 3=10\uparrow\uparrow(10\uparrow)^{98}(2\times 10^{% 200})=10\uparrow\uparrow(10\uparrow)^{100}2.3
  135. 100 n = ( 10 ) n - 2 ( 10 ) 98 ( 2 × 10 200 ) = ( 10 ) n - 2 ( 10 ) 100 2.3 < 10 ( n + 1 ) 100\uparrow\uparrow\uparrow n=(10\uparrow\uparrow)^{n-2}(10\uparrow)^{98}(2% \times 10^{200})=(10\uparrow\uparrow)^{n-2}(10\uparrow)^{100}2.3<10\uparrow% \uparrow\uparrow(n+1)
  136. 10 n = ( 10 ) n - 2 ( 10 ) 10 1 < 10 ( n + 1 ) 10\uparrow\uparrow\uparrow n=(10\uparrow\uparrow)^{n-2}(10\uparrow)^{10}1<10% \uparrow\uparrow\uparrow(n+1)
  137. 100 n 100\uparrow\uparrow\uparrow n
  138. 10 n 10\uparrow\uparrow\uparrow n
  139. 100 2 = ( 10 ) 98 ( 10 ) 100 2.3 100\uparrow\uparrow\uparrow\uparrow 2=(10\uparrow\uparrow)^{98}(10\uparrow)^{1% 00}2.3
  140. 10 2 = ( 10 ) 8 ( 10 ) 10 1 10\uparrow\uparrow\uparrow\uparrow 2=(10\uparrow\uparrow)^{8}(10\uparrow)^{10}1
  141. 100 3 = 10 ( 10 ) 98 ( 10 ) 100 2.3 100\uparrow\uparrow\uparrow\uparrow 3=10\uparrow\uparrow\uparrow(10\uparrow% \uparrow)^{98}(10\uparrow)^{100}2.3
  142. 10 3 = 10 ( 10 ) 8 ( 10 ) 10 1 10\uparrow\uparrow\uparrow\uparrow 3=10\uparrow\uparrow\uparrow(10\uparrow% \uparrow)^{8}(10\uparrow)^{10}1
  143. 100 n = ( 10 ) n - 2 ( 10 ) 98 ( 10 ) 100 2.3 100\uparrow\uparrow\uparrow\uparrow n=(10\uparrow\uparrow\uparrow)^{n-2}(10% \uparrow\uparrow)^{98}(10\uparrow)^{100}2.3
  144. 10 n = ( 10 ) n - 2 ( 10 ) 8 ( 10 ) 10 1 10\uparrow\uparrow\uparrow\uparrow n=(10\uparrow\uparrow\uparrow)^{n-2}(10% \uparrow\uparrow)^{8}(10\uparrow)^{10}1
  145. 10 n 10^{n}
  146. 10 6.2 × 10 3 10^{\,\!6.2\times 10^{3}}
  147. 10 50 10^{50}
  148. 10 10 10^{10}
  149. 10 9 10^{9}
  150. 1 - 10 9 10 10 = 1 - 1 10 = 90 % 1-\frac{10^{9}}{10^{10}}=1-\frac{1}{10}=90\%
  151. 10 a 10^{a}
  152. 10 b 10^{b}
  153. 10 10 a 10^{10^{a}}
  154. 10 10 b 10^{10^{b}}
  155. 10 10 10 10^{10^{10}}
  156. 10 10 9 10^{10^{9}}
  157. log 10 ( log 10 ( 10 10 10 ) ) = 10 \log_{10}(\log_{10}(10^{10^{10}}))=10
  158. log 10 ( log 10 ( 10 10 9 ) ) = 9 \log_{10}(\log_{10}(10^{10^{9}}))=9
  159. ( 10 a ) 10 b = 10 a 10 b = 10 10 b + log 10 a (10^{a})^{\,\!10^{b}}=10^{a10^{b}}=10^{10^{b+\log_{10}a}}
  160. n n 10 n n^{n}\approx 10^{n}
  161. 2 n 10 n 2^{n}\approx 10^{n}
  162. 2 65536 10 65533 2\uparrow\uparrow 65536\approx 10\uparrow\uparrow 65533
  163. 𝔠 \mathfrak{c}
  164. 𝔠 = 1 \mathfrak{c}=\aleph_{1}
  165. 10 n 10 < 3 n + 1 3 10\uparrow^{n}10<3\uparrow^{n+1}3

Latent_heat.html

  1. L = Q m . L=\frac{Q}{m}.
  2. Q = m L Q={m}{L}
  3. L water ( T ) = ( 2500.8 - 2.36 T + 0.0016 T 2 - 0.00006 T 3 ) J/g , L\text{water}(T)=(2500.8-2.36T+0.0016T^{2}-0.00006T^{3})~{}\,\text{J/g},
  4. T T
  5. L ice ( T ) = ( 2834.1 - 0.29 T - 0.004 T 2 ) J/g . L\text{ice}(T)=(2834.1-0.29T-0.004T^{2})~{}\,\text{J/g}.

Launch_window.html

  1. β \beta

Law_of_sines.html

  1. a sin A = b sin B = c sin C = D \frac{a}{\sin A}\,=\,\frac{b}{\sin B}\,=\,\frac{c}{\sin C}\,=\,D\!
  2. sin A a = sin B b = sin C c \frac{\sin A}{a}\,=\,\frac{\sin B}{b}\,=\,\frac{\sin C}{c}\!
  3. T T
  4. T = 1 2 b ( c sin A ) = 1 2 c ( a sin B ) = 1 2 a ( b sin C ) . T=\frac{1}{2}b(c\sin A)=\frac{1}{2}c(a\sin B)=\frac{1}{2}a(b\sin C)\,.
  5. 2 / a b c 2/abc
  6. 2 T a b c = sin A a = sin B b = sin C c . \frac{2T}{abc}=\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}\,.
  7. sin α = opposite hypotenuse \textstyle\sin\alpha=\frac{\,\text{opposite}}{\,\text{hypotenuse}}
  8. sin A = h 1 c ; sin C = h 1 a \sin A=\frac{h_{1}}{c}\,\text{; }\sin C=\frac{h_{1}}{a}
  9. h 1 = c sin A ; h 1 = a sin C h_{1}=c\sin A\,\text{; }h_{1}=a\sin C\,
  10. h 1 = c sin A = a sin C h_{1}=c\sin A=a\sin C\,
  11. a sin A = c sin C . \frac{a}{\sin A}=\frac{c}{\sin C}.
  12. sin B = h 2 c ; sin C = h 2 b \sin B=\frac{h_{2}}{c}\,\text{; }\sin C=\frac{h_{2}}{b}
  13. h 2 = c sin B ; h 2 = b sin C h_{2}=c\sin B\,\text{; }h_{2}=b\sin C\,
  14. b sin B = c sin C \frac{b}{\sin B}=\frac{c}{\sin C}
  15. c sin C \textstyle\frac{c}{\sin C}
  16. a sin A = b sin B = c sin C \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}
  17. sin A = a c ; sin B = b c \sin A=\frac{a}{c}\,\text{; }\sin B=\frac{b}{c}
  18. c = a sin A ; c = b sin B c=\frac{a}{\sin A}\,\text{; }c=\frac{b}{\sin B}
  19. a sin A = b sin B \frac{a}{\sin A}=\frac{b}{\sin B}
  20. c = c sin C c=\frac{c}{\sin C}
  21. a sin A = b sin B = c sin C \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}
  22. sin A = h 1 c ; sin C = h 1 a \sin A^{\prime}=\frac{h_{1}}{c}\,\text{; }\sin C=\frac{h_{1}}{a}
  23. sin π - θ = sin θ \textstyle\sin\pi-\theta=\sin\theta
  24. sin A \textstyle\sin A^{\prime}
  25. sin A \textstyle\sin A
  26. A + A = π A+A^{\prime}=\pi
  27. A = π - A A=\pi-A^{\prime}
  28. sin A = sin ( π - A ) = sin A \sin A=\sin(\pi-A^{\prime})=\sin A^{\prime}
  29. sin A = h 1 c ; sin C = h 1 a \sin A=\frac{h_{1}}{c}\,\text{; }\sin C=\frac{h_{1}}{a}
  30. a sin A = c sin C \frac{a}{\sin A}=\frac{c}{\sin C}
  31. sin B = h 2 c ; sin C = h 2 b \sin B=\frac{h_{2}}{c}\,\text{; }\sin C=\frac{h_{2}}{b}
  32. b sin B = c sin C \frac{b}{\sin B}=\frac{c}{\sin C}
  33. c sin C \textstyle\frac{c}{\sin C}
  34. a sin A = b sin B = c sin C \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}
  35. C = arcsin c sin A a or C = π - arcsin c sin A a C^{\prime}=\arcsin{c\sin A\over a}\,\text{ or }C=\pi-\arcsin{c\sin A\over a}
  36. sin A 20 = sin 40 24 . \frac{\sin A}{20}=\frac{\sin 40^{\circ}}{24}.
  37. A = arcsin ( 20 sin 40 24 ) 32.39 . A=\arcsin\left(\frac{20\sin 40^{\circ}}{24}\right)\approx 32.39^{\circ}.
  38. A = B = 180 - C 2 = 90 - C 2 \angle A=\angle B=\frac{180^{\circ}-\angle C}{2}=90-\frac{\angle C}{2}\!
  39. x sin A = chord sin C or x sin B = chord sin C {x\over\sin A}={\mbox{chord}~{}\over\sin C}\,\text{ or }{x\over\sin B}={\mbox{% chord}~{}\over\sin C}\,\!
  40. chord sin A sin C = x or chord sin B sin C = x . {\mbox{chord}~{}\,\sin A\over\sin C}=x\,\text{ or }{\mbox{chord}~{}\,\sin B% \over\sin C}=x.\!
  41. a sin A = b sin B = c sin C , \frac{a}{\sin A}\,=\,\frac{b}{\sin B}\,=\,\frac{c}{\sin C},\!
  42. a b c 2 T = a b c 2 s ( s - a ) ( s - b ) ( s - c ) = 2 a b c ( a 2 + b 2 + c 2 ) 2 - 2 ( a 4 + b 4 + c 4 ) , \begin{aligned}\displaystyle\frac{abc}{2T}&\displaystyle{}=\frac{abc}{2\sqrt{s% (s-a)(s-b)(s-c)}}\\ &\displaystyle{}=\frac{2abc}{\sqrt{(a^{2}+b^{2}+c^{2})^{2}-2(a^{4}+b^{4}+c^{4}% )}},\end{aligned}
  43. s = a + b + c 2 . s=\frac{a+b+c}{2}.
  44. sin A sin α = sin B sin β = sin C sin γ . \frac{\sin A}{\sin\alpha}=\frac{\sin B}{\sin\beta}=\frac{\sin C}{\sin\gamma}.
  45. lim α 0 sin α α = 1 \lim_{\alpha\rightarrow 0}\frac{\sin\alpha}{\alpha}=1
  46. sin β {\sin\beta}
  47. sin γ {\sin\gamma}
  48. sin A sinh a = sin B sinh b = sin C sinh c . \frac{\sin A}{\sinh a}=\frac{\sin B}{\sinh b}=\frac{\sin C}{\sinh c}\,.
  49. sin C = sinh c sinh b \sin C=\frac{\sinh c}{\sinh b}\,
  50. K K
  51. sin K x = x - K x 3 3 ! + K 2 x 5 5 ! - K 3 x 7 7 ! + . \sin_{K}x=x-\frac{Kx^{3}}{3!}+\frac{K^{2}x^{5}}{5!}-\frac{K^{3}x^{7}}{7!}+\cdots.
  52. K K
  53. sin A sin K a = sin B sin K b = sin C sin K c . \frac{\sin A}{\sin_{K}a}=\frac{\sin B}{\sin_{K}b}=\frac{\sin C}{\sin_{K}c}\,.
  54. K = 0 K=0
  55. K = 1 K=1
  56. K = - 1 K=-1
  57. p K ( r ) p_{K}(r)
  58. r r
  59. K K
  60. p K ( r ) = 2 π sin K r p_{K}(r)=2\pi\sin_{K}r
  61. sin A p K ( a ) = sin B p K ( b ) = sin C p K ( c ) . \frac{\sin A}{p_{K}(a)}=\frac{\sin B}{p_{K}(b)}=\frac{\sin C}{p_{K}(c)}\,.
  62. | psin ( 𝐧 𝟐 , 𝐧 𝟑 , 𝐧 𝟒 ) | Area 1 = | psin ( 𝐧 𝟏 , 𝐧 𝟑 , 𝐧 𝟒 ) | Area 2 = | psin ( 𝐧 𝟏 , 𝐧 𝟐 , 𝐧 𝟒 ) | Area 3 = | psin ( 𝐧 𝟏 , 𝐧 𝟐 , 𝐧 𝟑 ) | Area 4 . \frac{|\mathrm{psin}(\mathbf{n_{2}},\mathbf{n_{3}},\mathbf{n_{4}})|}{\mathrm{% Area}_{1}}=\frac{|\mathrm{psin}(\mathbf{n_{1}},\mathbf{n_{3}},\mathbf{n_{4}})|% }{\mathrm{Area}_{2}}=\frac{|\mathrm{psin}(\mathbf{n_{1}},\mathbf{n_{2}},% \mathbf{n_{4}})|}{\mathrm{Area}_{3}}=\frac{|\mathrm{psin}(\mathbf{n_{1}},% \mathbf{n_{2}},\mathbf{n_{3}})|}{\mathrm{Area}_{4}}\,.

Law_of_total_expectation.html

  1. A 1 , A 2 , , A n A_{1},A_{2},\ldots,A_{n}
  2. E ( X ) = i = 1 n E ( X A i ) P ( A i ) . \operatorname{E}(X)=\sum_{i=1}^{n}{\operatorname{E}(X\mid A_{i})\operatorname{% P}(A_{i})}.
  3. E ( L ) = E ( L X ) P ( X ) + E ( L Y ) P ( Y ) = 5000 ( .6 ) + 4000 ( .4 ) = 4600 \operatorname{E}(L)=\operatorname{E}(L\mid X)\operatorname{P}(X)+\operatorname% {E}(L\mid Y)\operatorname{P}(Y)=5000(.6)+4000(.4)=4600
  4. E ( L ) \operatorname{E}(L)
  5. Pr ( X ) = 6 10 \Pr(X)={6\over 10}
  6. Pr ( Y ) = 4 10 \Pr(Y)={4\over 10}
  7. E ( L X ) = 5000 \operatorname{E}(L\mid X)=5000
  8. E ( L Y ) = 4000 \operatorname{E}(L\mid Y)=4000
  9. E Y ( E X Y ( X Y ) ) = E Y [ x x P ( X = x Y ) ] = y [ x x P ( X = x Y = y ) ] P ( Y = y ) = y x x P ( X = x Y = y ) P ( Y = y ) = x x y P ( X = x Y = y ) P ( Y = y ) = x x y P ( X = x , Y = y ) = x x P ( X = x ) = E ( X ) . \begin{aligned}\displaystyle\operatorname{E}_{Y}\left(\operatorname{E}_{X\mid Y% }(X\mid Y)\right)&\displaystyle{}=\operatorname{E}_{Y}\Bigg[\sum_{x}x\cdot% \operatorname{P}(X=x\mid Y)\Bigg]\\ &\displaystyle{}=\sum_{y}\Bigg[\sum_{x}x\cdot\operatorname{P}(X=x\mid Y=y)% \Bigg]\cdot\operatorname{P}(Y=y)\\ &\displaystyle{}=\sum_{y}\sum_{x}x\cdot\operatorname{P}(X=x\mid Y=y)\cdot% \operatorname{P}(Y=y)\\ &\displaystyle{}=\sum_{x}x\sum_{y}\operatorname{P}(X=x\mid Y=y)\cdot% \operatorname{P}(Y=y)\\ &\displaystyle{}=\sum_{x}x\sum_{y}\operatorname{P}(X=x,Y=y)\\ &\displaystyle{}=\sum_{x}x\cdot\operatorname{P}(X=x)\\ &\displaystyle{}=\operatorname{E}(X).\end{aligned}
  10. ( Ω , , P ) (\Omega,\mathcal{F},P)
  11. σ \sigma
  12. 𝒢 1 𝒢 2 \mathcal{G}_{1}\subseteq\mathcal{G}_{2}\subseteq\mathcal{F}
  13. X X
  14. E [ E [ X 𝒢 2 ] 𝒢 1 ] = E [ X 𝒢 1 ] . \operatorname{E}[\operatorname{E}[X\mid\mathcal{G}_{2}]\mid\mathcal{G}_{1}]=% \operatorname{E}[X\mid\mathcal{G}_{1}].
  15. E [ E [ X 𝒢 2 ] 𝒢 1 ] is 𝒢 1 \operatorname{E}[\operatorname{E}[X\mid\mathcal{G}_{2}]\mid\mathcal{G}_{1}]% \mbox{ is }~{}\mathcal{G}_{1}
  16. G 1 E [ E [ X 𝒢 2 ] 𝒢 1 ] d P = G 1 X d P holds for all G 1 𝒢 2 \int_{G_{1}}\operatorname{E}[\operatorname{E}[X\mid\mathcal{G}_{2}]\mid% \mathcal{G}_{1}]dP=\int_{G_{1}}XdP\mbox{ holds for all }~{}G_{1}\in\mathcal{G}% _{2}
  17. G 1 𝒢 1 𝒢 2 G_{1}\in\mathcal{G}_{1}\subseteq\mathcal{G}_{2}
  18. G 1 E [ E [ X 𝒢 2 ] 𝒢 1 ] d P = G 1 E [ X 𝒢 2 ] d P = G 1 X d P . \int_{G_{1}}\operatorname{E}[\operatorname{E}[X\mid\mathcal{G}_{2}]\mid% \mathcal{G}_{1}]dP=\int_{G_{1}}\operatorname{E}[X\mid\mathcal{G}_{2}]dP=\int_{% G_{1}}XdP.
  19. 𝒢 1 = { , Ω } \mathcal{G}_{1}=\{,\Omega\}
  20. 𝒢 2 = σ ( Y ) \mathcal{G}_{2}=\sigma(Y)
  21. E [ E [ X Y ] ] = E [ X ] . \operatorname{E}[\operatorname{E}[X\mid Y]]=\operatorname{E}[X].
  22. E \operatorname{E}
  23. E ( E ( X Y ) ) \operatorname{E}\left(\operatorname{E}(X\mid Y)\right)
  24. E Y ( E X Y ( X Y ) ) \operatorname{E}_{Y}\left(\operatorname{E}_{X\mid Y}(X\mid Y)\right)
  25. X X
  26. Y Y
  27. Y Y
  28. E ( X I 1 ) = E ( E ( X I 2 ) I 1 ) , \operatorname{E}(X\mid I_{1})=\operatorname{E}(\operatorname{E}(X\mid I_{2})% \mid I_{1}),
  29. E t ( X ) = E t ( E t + 1 ( X ) ) . \operatorname{E}_{t}(X)=\operatorname{E}_{t}(\operatorname{E}_{t+1}(X)).

Law_of_total_probability.html

  1. { B n : n = 1 , 2 , 3 , } \left\{{B_{n}:n=1,2,3,\ldots}\right\}
  2. B n B_{n}
  3. A A
  4. Pr ( A ) = n Pr ( A B n ) \Pr(A)=\sum_{n}\Pr(A\cap B_{n})\,
  5. Pr ( A ) = n Pr ( A B n ) Pr ( B n ) , \Pr(A)=\sum_{n}\Pr(A\mid B_{n})\Pr(B_{n}),\,
  6. n n\,
  7. Pr ( B n ) = 0 \Pr(B_{n})=0\,
  8. Pr ( A B n ) \Pr(A\mid B_{n})\,
  9. Pr ( A ) \Pr(A)
  10. B n B_{n}
  11. C C
  12. B n B_{n}
  13. Pr ( A C ) = n Pr ( A C B n ) Pr ( B n C ) = n Pr ( A C B n ) Pr ( B n ) \Pr(A\mid C)=\sum_{n}\Pr(A\mid C\cap B_{n})\Pr(B_{n}\mid C)=\sum_{n}\Pr(A\mid C% \cap B_{n})\Pr(B_{n})
  14. A A
  15. B n B_{n}
  16. A A
  17. Pr ( A ) \Pr(A)
  18. Pr ( A ) = Pr ( A | B 1 ) Pr ( B 1 ) + Pr ( A | B 2 ) Pr ( B 2 ) = 99 100 6 10 + 95 100 4 10 = 594 + 380 1000 = 974 1000 {\Pr(A)=\Pr(A|B_{1})}\cdot{\Pr(B_{1})}+{\Pr(A|B_{2})}\cdot{\Pr(B_{2})}={99% \over 100}\cdot{6\over 10}+{95\over 100}\cdot{4\over 10}={{594+380}\over 1000}% ={974\over 1000}
  19. Pr ( B 1 ) = 6 10 \Pr(B_{1})={6\over 10}
  20. Pr ( B 2 ) = 4 10 \Pr(B_{2})={4\over 10}
  21. Pr ( A | B 1 ) = 99 100 \Pr(A|B_{1})={99\over 100}
  22. Pr ( A | B 2 ) = 95 100 \Pr(A|B_{2})={95\over 100}
  23. B n B_{n}
  24. X = x n X=x_{n}
  25. X = x n X=x_{n}
  26. Pr ( A ) = n Pr ( A X = x n ) Pr ( X = x n ) = E [ Pr ( A X ) ] , \Pr(A)=\sum_{n}\Pr(A\mid X=x_{n})\Pr(X=x_{n})=\operatorname{E}[\Pr(A\mid X)],
  27. Pr ( A ) = E [ Pr ( A X ) ] , \Pr(A)=\operatorname{E}[\Pr(A\mid\mathcal{F}_{X})],
  28. X \mathcal{F}_{X}

Law_of_total_variance.html

  1. Var [ Y ] = E X ( Var [ Y X ] ) + Var X ( E [ Y X ] ) . \operatorname{Var}[Y]=\operatorname{E}_{X}(\operatorname{Var}[Y\mid X])+% \operatorname{Var}_{X}(\operatorname{E}[Y\mid X]).\,
  2. Var [ Y ] = E ( Var [ Y X 1 , X 2 ] ) + E ( Var [ E [ Y X 1 , X 2 ] X 1 ] ) + Var ( E [ Y X 1 ] ) , \operatorname{Var}[Y]=\operatorname{E}(\operatorname{Var}[Y\mid X_{1},X_{2}])+% \operatorname{E}(\operatorname{Var}[\operatorname{E}[Y\mid X_{1},X_{2}]\mid X_% {1}])+\operatorname{Var}(\operatorname{E}[Y\mid X_{1}]),\,
  3. Var [ Y X 1 ] = E ( Var [ Y X 1 , X 2 ] X 1 ) + Var ( E [ Y X 1 , X 2 ] X 1 ) . \operatorname{Var}[Y\mid X_{1}]=\operatorname{E}(\operatorname{Var}[Y\mid X_{1% },X_{2}]\mid X_{1})+\operatorname{Var}(\operatorname{E}[Y\mid X_{1},X_{2}]\mid X% _{1}).\,
  4. A 1 , A 2 , , A n A_{1},A_{2},\ldots,A_{n}
  5. Var ( X ) = i = 1 n Var ( X A i ) P ( A i ) + i = 1 n E ( X A i ) 2 ( 1 - P ( A i ) ) P ( A i ) - 2 i = 1 n j = 1 i - 1 E ( X A i ) P ( A i ) E ( X A j ) P ( A j ) . \operatorname{Var}(X)=\sum_{i=1}^{n}{\operatorname{Var}(X\mid A_{i})% \operatorname{P}(A_{i})}+\sum_{i=1}^{n}{\operatorname{E}(X\mid A_{i})^{2}(1-% \operatorname{P}(A_{i}))\operatorname{P}(A_{i})}-2\sum_{i=1}^{n}\sum_{j=1}^{i-% 1}\operatorname{E}(X\mid A_{i})\operatorname{P}(A_{i})\operatorname{E}(X\mid A% _{j})\operatorname{P}(A_{j}).
  6. Var [ Y ] = E [ Y 2 ] - [ E [ Y ] ] 2 \operatorname{Var}[Y]=\operatorname{E}[Y^{2}]-[\operatorname{E}[Y]]^{2}
  7. = E X [ E [ Y 2 X ] ] - [ E X [ E [ Y X ] ] ] 2 =\operatorname{E}_{X}\!\left[\operatorname{E}[Y^{2}\mid X]\right]-[% \operatorname{E}_{X}[\operatorname{E}[Y\mid X]]]^{2}
  8. = E X [ Var [ Y X ] + [ E [ Y X ] ] 2 ] - [ E X [ E [ Y X ] ] ] 2 =\operatorname{E}_{X}\!\left[\operatorname{Var}[Y\mid X]+[\operatorname{E}[Y% \mid X]]^{2}\right]-[\operatorname{E}_{X}[\operatorname{E}[Y\mid X]]]^{2}
  9. = E X [ Var [ Y X ] ] + ( E X [ [ E [ Y X ] ] 2 ] - [ E X [ E [ Y X ] ] ] 2 ) =\operatorname{E}_{X}[\operatorname{Var}[Y\mid X]]+\left(\operatorname{E}_{X}[% [\operatorname{E}[Y\mid X]]^{2}]-[\operatorname{E}_{X}[\operatorname{E}[Y\mid X% ]]]^{2}\right)
  10. = E X [ Var [ Y X ] ] + Var X [ E [ Y X ] ] =\operatorname{E}_{X}[\operatorname{Var}[Y\mid X]]+\operatorname{Var}_{X}[% \operatorname{E}[Y\mid X]]
  11. H 1 t , H 2 t , , H c - 1 , t H_{1t},H_{2t},\ldots,H_{c-1,t}
  12. Var [ Y ( t ) ] = E ( Var [ Y ( t ) H 1 t , H 2 t , , H c - 1 , t ] ) + j = 2 c - 1 E ( Var [ E [ Y ( t ) H 1 t , H 2 t , , H j t ] H 1 t , H 2 t , , H j - 1 , t ] ) + Var ( E [ Y ( t ) H 1 t ] ) . \operatorname{Var}[Y(t)]=\operatorname{E}(\operatorname{Var}[Y(t)\mid H_{1t},H% _{2t},\ldots,H_{c-1,t}])+\sum_{j=2}^{c-1}\operatorname{E}(\operatorname{Var}[% \operatorname{E}[Y(t)\mid H_{1t},H_{2t},\ldots,H_{jt}]\mid H_{1t},H_{2t},% \ldots,H_{j-1,t}])+\operatorname{Var}(\operatorname{E}[Y(t)\mid H_{1t}]).\,
  13. E ( Y X ) = a X + b , \operatorname{E}(Y\mid X)=aX+b,\,
  14. a = Cov ( Y , X ) Var ( X ) a={\operatorname{Cov}(Y,X)\over\operatorname{Var}(X)}
  15. b = E ( Y ) - Cov ( Y , X ) Var ( X ) E ( X ) b=\operatorname{E}(Y)-{\operatorname{Cov}(Y,X)\over\operatorname{Var}(X)}% \operatorname{E}(X)
  16. Var ( E ( Y X ) ) Var ( Y ) = Corr ( X , Y ) 2 . {\operatorname{Var}(\operatorname{E}(Y\mid X))\over\operatorname{Var}(Y)}=% \operatorname{Corr}(X,Y)^{2}.\,
  17. ι Y X = Var ( E ( Y X ) ) Var ( Y ) = Corr ( E ( Y X ) , Y ) 2 , \iota_{Y\mid X}={\operatorname{Var}(\operatorname{E}(Y\mid X))\over% \operatorname{Var}(Y)}=\operatorname{Corr}(\operatorname{E}(Y\mid X),Y)^{2},\,
  18. I ( Y ; X ) ln ( [ 1 - ι Y X ] - 1 / 2 ) . \operatorname{I}(Y;X)\geq\ln([1-\iota_{Y\mid X}]^{-1/2}).\,
  19. μ 3 ( Y ) = E ( μ 3 ( Y X ) ) + μ 3 ( E ( Y X ) ) + 3 cov ( E ( Y X ) , var ( Y X ) ) . \mu_{3}(Y)=\operatorname{E}(\mu_{3}(Y\mid X))+\mu_{3}(\operatorname{E}(Y\mid X% ))+3\,\operatorname{cov}(\operatorname{E}(Y\mid X),\operatorname{var}(Y\mid X)% ).\,

Laws_of_science.html

  1. ρ t = - 𝐉 \frac{\partial\rho}{\partial t}=-\nabla\cdot\mathbf{J}
  2. ρ t = - ( ρ 𝐮 ) \frac{\partial\rho}{\partial t}=-\nabla\cdot(\rho\mathbf{u})
  3. ρ t = - 𝐉 \frac{\partial\rho}{\partial t}=-\nabla\cdot\mathbf{J}
  4. u t = - 𝐪 \frac{\partial u}{\partial t}=-\nabla\cdot\mathbf{q}
  5. | Ψ | 2 t = - 𝐣 \frac{\partial|\Psi|^{2}}{\partial t}=-\nabla\cdot\mathbf{j}
  6. δ 𝒮 = δ t 1 t 2 L ( 𝐪 , 𝐪 ˙ , t ) d t = 0 \delta\mathcal{S}=\delta\int_{t_{1}}^{t_{2}}L(\mathbf{q},\mathbf{\dot{q}},t)dt=0
  7. 𝒮 \mathcal{S}
  8. L ( 𝐪 , 𝐪 ˙ , t ) = T ( 𝐪 ˙ , t ) - V ( 𝐪 , 𝐪 ˙ , t ) L(\mathbf{q},\mathbf{\dot{q}},t)=T(\mathbf{\dot{q}},t)-V(\mathbf{q},\mathbf{% \dot{q}},t)
  9. p i = L q ˙ i p_{i}=\frac{\partial L}{\partial\dot{q}_{i}}
  10. E = T + V E=T+V
  11. 𝒮 = t 1 t 2 L d t \mathcal{S}=\int_{t_{1}}^{t_{2}}L\mathrm{d}t\,\!
  12. d d t ( L q ˙ i ) = L q i \frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial L}{\partial\dot{q}_{i}}% \right)=\frac{\partial L}{\partial q_{i}}
  13. p i = L q ˙ i p ˙ i = L q i p_{i}=\frac{\partial L}{\partial\dot{q}_{i}}\quad\dot{p}_{i}=\frac{\partial L}% {\partial{q}_{i}}
  14. 𝐩 t = - H 𝐪 \dfrac{\partial\mathbf{p}}{\partial t}=-\dfrac{\partial H}{\partial\mathbf{q}}
  15. 𝐪 t = H 𝐩 \dfrac{\partial\mathbf{q}}{\partial t}=\dfrac{\partial H}{\partial\mathbf{p}}
  16. H ( 𝐪 , 𝐩 , t ) = 𝐩 𝐪 ˙ - L H(\mathbf{q},\mathbf{p},t)=\mathbf{p}\cdot\mathbf{\dot{q}}-L
  17. H ( 𝐪 , S 𝐪 , t ) = - S t H\left(\mathbf{q},\frac{\partial S}{\partial\mathbf{q}},t\right)=-\frac{% \partial S}{\partial t}
  18. 𝐅 = d 𝐩 d t , 𝐅 i j = - 𝐅 j i \mathbf{F}=\frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t},\quad\mathbf{F}_{ij}=-% \mathbf{F}_{ji}
  19. d 𝐩 i d t = 𝐅 E + i j 𝐅 ij \frac{\mathrm{d}\mathbf{p}_{\mathrm{i}}}{\mathrm{d}t}=\mathbf{F}_{E}+\sum_{% \mathrm{i}\neq\mathrm{j}}\mathbf{F}_{\mathrm{ij}}\,\!
  20. A = Λ A A^{\prime}=\Lambda A
  21. E 2 = ( p c ) 2 + ( m c 2 ) 2 E^{2}=(pc)^{2}+(mc^{2})^{2}
  22. R μ ν + ( Λ - R 2 ) g μ ν = 8 π G c 4 T μ ν R_{\mu\nu}+\left(\Lambda-\frac{R}{2}\right)g_{\mu\nu}=\frac{8\pi G}{c^{4}}T_{% \mu\nu}\,\!
  23. d 2 x λ d t 2 + Γ μ ν λ d x μ d t d x ν d t = 0 , \frac{{\rm d}^{2}x^{\lambda}}{{\rm d}t^{2}}+\Gamma^{\lambda}_{\mu\nu}\frac{{% \rm d}x^{\mu}}{{\rm d}t}\frac{{\rm d}x^{\nu}}{{\rm d}t}=0\ ,
  24. 𝐠 = - 4 π G ρ \nabla\cdot\mathbf{g}=-4\pi G\rho\,\!
  25. 𝐇 = 𝟎 \nabla\cdot\mathbf{H}=\mathbf{0}\,\!
  26. × 𝐠 = - 𝐇 t \nabla\times\mathbf{g}=-\frac{\partial\mathbf{H}}{\partial t}\,\!
  27. × 𝐇 = 4 c 2 ( - 4 π G 𝐉 + 𝐠 t ) \nabla\times\mathbf{H}=\frac{4}{c^{2}}\left(-4\pi G\mathbf{J}+\frac{\partial% \mathbf{g}}{\partial t}\right)\,\!
  28. 𝐅 = γ ( 𝐯 ) m ( 𝐠 + 𝐯 × 𝐇 ) \mathbf{F}=\gamma(\mathbf{v})m\left(\mathbf{g}+\mathbf{v}\times\mathbf{H}\right)
  29. 𝐅 = G m 1 m 2 | 𝐫 | 2 𝐫 ^ \mathbf{F}=\frac{Gm_{1}m_{2}}{\left|\mathbf{r}\right|^{2}}\mathbf{\hat{r}}\,\!
  30. 𝐠 = G V 𝐫 ρ d V | 𝐫 | 3 \mathbf{g}=G\int_{V}\frac{\mathbf{r}\rho\mathrm{d}{V}}{\left|\mathbf{r}\right|% ^{3}}\,\!
  31. 𝐠 = 4 π G ρ \nabla\cdot\mathbf{g}=4\pi G\rho\,\!
  32. r = l 1 + e cos θ r=\frac{l}{1+e\cos\theta}\,\!
  33. e = 1 - ( b / a ) 2 e=\sqrt{1-(b/a)^{2}}
  34. d A d t = | 𝐋 | 2 m \frac{\mathrm{d}A}{\mathrm{d}t}=\frac{\left|\mathbf{L}\right|}{2m}\,\!
  35. T 2 = 4 π 2 G ( m + M ) a 3 T^{2}=\frac{4\pi^{2}}{G\left(m+M\right)}a^{3}\,\!
  36. d U = δ Q - δ W \mathrm{d}U=\delta Q-\delta W\,
  37. Δ S 0 \Delta S\geq 0
  38. T A = T B , T B = T C T A = T C T_{A}=T_{B}\,,T_{B}=T_{C}\Rightarrow T_{A}=T_{C}\,\!
  39. d U = T d S - P d V + i μ i d N i \mathrm{d}U=T\mathrm{d}S-P\mathrm{d}V+\sum_{i}\mu_{i}\mathrm{d}N_{i}\,\!
  40. 𝐉 u = L u u ( 1 / T ) - L u r ( m / T ) \mathbf{J}_{u}=L_{uu}\,\nabla(1/T)-L_{ur}\,\nabla(m/T)\!
  41. 𝐉 r = L r u ( 1 / T ) - L r r ( m / T ) \mathbf{J}_{r}=L_{ru}\,\nabla(1/T)-L_{rr}\,\nabla(m/T)\!
  42. 𝐄 = ρ ε 0 \nabla\cdot\mathbf{E}=\frac{\rho}{\varepsilon_{0}}
  43. 𝐁 = 0 \nabla\cdot\mathbf{B}=0
  44. × 𝐄 = - 𝐁 t \nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}
  45. × 𝐁 = μ 0 𝐉 + 1 c 2 𝐄 t \nabla\times\mathbf{B}=\mu_{0}\mathbf{J}+\frac{1}{c^{2}}\frac{\partial\mathbf{% E}}{\partial t}
  46. 𝐅 = q ( 𝐄 + 𝐯 × 𝐁 ) \mathbf{F}=q\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right)
  47. i d d t | ψ = H ^ | ψ i\hbar\frac{d}{dt}\left|\psi\right\rangle=\hat{H}\left|\psi\right\rangle
  48. | ψ |\psi\rangle
  49. E = h ν = ω E=h\nu=\hbar\omega
  50. 𝐩 = h λ 𝐤 ^ = 𝐤 \mathbf{p}=\frac{h}{\lambda}\mathbf{\hat{k}}=\hbar\mathbf{k}
  51. Δ x Δ p 2 , Δ E Δ t 2 \Delta x\Delta p\geq\frac{\hbar}{2},\,\Delta E\Delta t\geq\frac{\hbar}{2}
  52. i t ψ = - 2 2 m 2 ψ + V ψ i\hbar\frac{\partial}{\partial t}\psi=-\frac{\hbar^{2}}{2m}\nabla^{2}\psi+V\psi
  53. ψ ( 𝐫 i 𝐫 j ) = ( - 1 ) 2 s ψ ( 𝐫 j 𝐫 i ) \psi(\cdots\mathbf{r}_{i}\cdots\mathbf{r}_{j}\cdots)=(-1)^{2s}\psi(\cdots% \mathbf{r}_{j}\cdots\mathbf{r}_{i}\cdots)