wpmath0000003_5

Experience_curve_effects.html

  1. P x = x Y x - ( x - 1 ) Y x - 1 P_{x}=xY_{x}-(x-1)Y_{x-1}
  2. Y x = P 1 + P 2 + + P x x . Y_{x}=\frac{P_{1}+P_{2}+\cdots+P_{x}}{x}.
  3. Y x = K x log 2 ( b ) Y_{x}=Kx^{\log_{2}(b)}
  4. Y x ¯ = K 1 1 + log 2 ( b ) x 1 + log 2 ( b ) x \overline{Y_{x}}=K\frac{\frac{1}{1+\log_{2}(b)}x^{1+\log_{2}(b)}}{x}
  5. C n = C 1 n - a C_{n}=C_{1}n^{-a}

Experimental_psychology.html

  1. Δ I I = k , \frac{\Delta I}{I}=k,
  2. I I\!
  3. Δ I \Delta I\!
  4. Δ I \Delta I\!

Exponential_decay.html

  1. d N d t = - λ N . \frac{dN}{dt}=-\lambda N.
  2. N ( t ) = N 0 e - λ t . N(t)=N_{0}e^{-\lambda t}.\,
  3. τ = 1 λ . \tau=\frac{1}{\lambda}.
  4. N ( t ) = N 0 e - t / τ . N(t)=N_{0}e^{-t/\tau}.\,
  5. t 1 / 2 = ln ( 2 ) λ = τ ln ( 2 ) . t_{1/2}=\frac{\ln(2)}{\lambda}=\tau\ln(2).
  6. τ \tau
  7. N ( t ) = N 0 2 - t / t 1 / 2 . N(t)=N_{0}2^{-t/t_{1/2}}.\,
  8. τ \tau
  9. τ = t 1 / 2 ln 2 1.44 t 1 / 2 . \tau=\frac{t_{1/2}}{\ln 2}\approx 1.44\cdot t_{1/2}.
  10. d N d t = - λ N \frac{dN}{dt}=-\lambda N
  11. d N N = - λ d t . \frac{dN}{N}=-\lambda dt.
  12. ln N = - λ t + C \ln N=-\lambda t+C\,
  13. N ( t ) = e C e - λ t = N 0 e - λ t N(t)=e^{C}e^{-\lambda t}=N_{0}e^{-\lambda t}\,
  14. τ \tau
  15. N = N 0 e - λ t , N=N_{0}e^{-\lambda t},\,
  16. 1 = 0 c N 0 e - λ t d t = c N 0 λ 1=\int_{0}^{\infty}c\cdot N_{0}e^{-\lambda t}\,dt=c\cdot\frac{N_{0}}{\lambda}
  17. c = λ N 0 . c=\frac{\lambda}{N_{0}}.
  18. τ = t = 0 t c N 0 e - λ t d t = 0 λ t e - λ t d t = 1 λ . \tau=\langle t\rangle=\int_{0}^{\infty}t\cdot c\cdot N_{0}e^{-\lambda t}\,dt=% \int_{0}^{\infty}\lambda te^{-\lambda t}\,dt=\frac{1}{\lambda}.
  19. - d N ( t ) d t = N λ 1 + N λ 2 = ( λ 1 + λ 2 ) N . -\frac{dN(t)}{dt}=N\lambda_{1}+N\lambda_{2}=(\lambda_{1}+\lambda_{2})N.
  20. λ 1 + λ 2 \lambda_{1}+\lambda_{2}\,
  21. λ c \lambda_{c}
  22. N ( t ) = N 0 e - ( λ 1 + λ 2 ) t = N 0 e - ( λ c ) t . N(t)=N_{0}e^{-(\lambda_{1}+\lambda_{2})t}=N_{0}e^{-(\lambda_{c})t}.
  23. τ = 1 / λ \tau=1/\lambda
  24. τ c \tau_{c}
  25. λ \lambda
  26. 1 τ c = λ c = λ 1 + λ 2 = 1 τ 1 + 1 τ 2 \frac{1}{\tau_{c}}=\lambda_{c}=\lambda_{1}+\lambda_{2}=\frac{1}{\tau_{1}}+% \frac{1}{\tau_{2}}
  27. τ c = τ 1 τ 2 τ 1 + τ 2 . \tau_{c}=\frac{\tau_{1}\tau_{2}}{\tau_{1}+\tau_{2}}.
  28. τ \tau
  29. T 1 / 2 = t 1 t 2 t 1 + t 2 T_{1/2}=\frac{t_{1}t_{2}}{t_{1}+t_{2}}
  30. T 1 / 2 T_{1/2}
  31. t 1 t_{1}
  32. t 2 t_{2}
  33. T 1 / 2 T_{1/2}
  34. T 1 / 2 = ln 2 λ c = ln 2 λ 1 + λ 2 . T_{1/2}=\frac{\ln 2}{\lambda_{c}}=\frac{\ln 2}{\lambda_{1}+\lambda_{2}}.
  35. T 1 / 2 = ln 2 λ c = ln 2 λ 1 + λ 2 + λ 3 = t 1 t 2 t 3 ( t 1 t 2 ) + ( t 1 t 3 ) + ( t 2 t 3 ) . T_{1/2}=\frac{\ln 2}{\lambda_{c}}=\frac{\ln 2}{\lambda_{1}+\lambda_{2}+\lambda% _{3}}=\frac{t_{1}t_{2}t_{3}}{(t_{1}t_{2})+(t_{1}t_{3})+(t_{2}t_{3})}.

Exponential_family.html

  1. f X ( x | θ ) = h ( x ) exp ( η ( θ ) T ( x ) - A ( θ ) ) f_{X}(x|\theta)=h(x)\exp\left(\eta(\theta)\cdot T(x)-A(\theta)\right)
  2. f X ( x | θ ) = h ( x ) g ( θ ) exp ( η ( θ ) T ( x ) ) f_{X}(x|\theta)=h(x)g(\theta)\exp\left(\eta(\theta)\cdot T(x)\right)
  3. f X ( x | θ ) = exp ( η ( θ ) T ( x ) - A ( θ ) + B ( x ) ) f_{X}(x|\theta)=\exp\left(\eta(\theta)\cdot T(x)-A(\theta)+B(x)\right)
  4. η ( θ ) T ( x ) \eta(\theta)^{\prime}\cdot T(x)
  5. exp ( - c T ( x ) ) \exp(-c\cdot T(x))
  6. f ( x ) , g ( θ ) , c f ( x ) , c g ( θ ) , [ f ( x ) ] c , [ g ( θ ) ] c , [ f ( x ) ] g ( θ ) , [ g ( θ ) ] f ( x ) , [ f ( x ) ] h ( x ) g ( θ ) , or [ g ( θ ) ] h ( x ) j ( θ ) , f(x),g(\theta),c^{f(x)},c^{g(\theta)},{[f(x)]}^{c},{[g(\theta)]}^{c},{[f(x)]}^% {g(\theta)},{[g(\theta)]}^{f(x)},{[f(x)]}^{h(x)g(\theta)},\,\text{ or }{[g(% \theta)]}^{h(x)j(\theta)},
  7. [ f ( x ) g ( θ ) ] h ( x ) j ( θ ) , [ f ( x ) ] h ( x ) j ( θ ) [ g ( θ ) ] h ( x ) j ( θ ) , {[f(x)g(\theta)]}^{h(x)j(\theta)},\qquad{[f(x)]}^{h(x)j(\theta)}[g(\theta)]^{h% (x)j(\theta)},
  8. [ f ( x ) g ( θ ) ] h ( x ) j ( θ ) = [ f ( x ) ] h ( x ) j ( θ ) [ g ( θ ) ] h ( x ) j ( θ ) = e [ h ( x ) ln f ( x ) ] j ( θ ) + h ( x ) [ j ( θ ) ln g ( θ ) ] , {[f(x)g(\theta)]}^{h(x)j(\theta)}={[f(x)]}^{h(x)j(\theta)}[g(\theta)]^{h(x)j(% \theta)}=e^{[h(x)\ln f(x)]j(\theta)+h(x)[j(\theta)\ln g(\theta)]},
  9. [ f ( x ) ] g ( θ ) {[f(x)]}^{g(\theta)}
  10. [ f ( x ) ] g ( θ ) = e g ( θ ) ln f ( x ) {[f(x)]}^{g(\theta)}=e^{g(\theta)\ln f(x)}
  11. [ f ( x ) ] h ( x ) g ( θ ) = e h ( x ) g ( θ ) ln f ( x ) = e [ h ( x ) ln f ( x ) ] g ( θ ) {[f(x)]}^{h(x)g(\theta)}=e^{h(x)g(\theta)\ln f(x)}=e^{[h(x)\ln f(x)]g(\theta)}
  12. 1 + f ( x ) g ( θ ) 1+f(x)g(\theta)
  13. s y m b o l θ = ( θ 1 , θ 2 , , θ s ) T . {symbol\theta}=\left(\theta_{1},\theta_{2},\cdots,\theta_{s}\right)^{T}.
  14. f X ( x | s y m b o l θ ) = h ( x ) exp ( i = 1 s η i ( s y m b o l θ ) T i ( x ) - A ( s y m b o l θ ) ) f_{X}(x|symbol\theta)=h(x)\exp\left(\sum_{i=1}^{s}\eta_{i}({symbol\theta})T_{i% }(x)-A({symbol\theta})\right)
  15. f X ( x | s y m b o l θ ) = h ( x ) exp ( s y m b o l η ( s y m b o l θ ) 𝐓 ( x ) - A ( s y m b o l θ ) ) f_{X}(x|symbol\theta)=h(x)\exp\Big(symbol\eta({symbol\theta})\cdot\mathbf{T}(x% )-A({symbol\theta})\Big)
  16. s y m b o l η ( s y m b o l θ ) symbol\eta({symbol\theta})
  17. 𝐓 ( x ) \mathbf{T}(x)
  18. f X ( x | s y m b o l θ ) = h ( x ) g ( s y m b o l θ ) exp ( s y m b o l η ( s y m b o l θ ) 𝐓 ( x ) ) f_{X}(x|symbol\theta)=h(x)g(symbol\theta)\exp\Big(symbol\eta({symbol\theta})% \cdot\mathbf{T}(x)\Big)
  19. i : η i ( s y m b o l θ ) = θ i . \forall i:\quad\eta_{i}({symbol\theta})=\theta_{i}.
  20. s y m b o l θ = ( θ 1 , θ 2 , , θ d ) T {symbol\theta}=\left(\theta_{1},\theta_{2},\ldots,\theta_{d}\right)^{T}
  21. s y m b o l η ( s y m b o l θ ) = ( η 1 ( s y m b o l θ ) , η 2 ( s y m b o l θ ) , , η s ( s y m b o l θ ) ) T . {symbol\eta}(symbol\theta)=\left(\eta_{1}(symbol\theta),\eta_{2}(symbol\theta)% ,\ldots,\eta_{s}(symbol\theta)\right)^{T}.
  22. A ( s y m b o l θ ) A(symbol\theta)
  23. g ( s y m b o l θ ) g(symbol\theta)
  24. s y m b o l η symbol\eta
  25. s y m b o l η symbol\eta
  26. s y m b o l θ symbol\theta
  27. f X ( x | s y m b o l η ) = h ( x ) exp ( s y m b o l η 𝐓 ( x ) - A ( s y m b o l η ) ) f_{X}(x|symbol\eta)=h(x)\exp\Big(symbol\eta\cdot\mathbf{T}(x)-A({symbol\eta})\Big)
  28. f X ( x | s y m b o l η ) = h ( x ) g ( s y m b o l η ) exp ( s y m b o l η 𝐓 ( x ) ) f_{X}(x|symbol\eta)=h(x)g(symbol\eta)\exp\Big(symbol\eta\cdot\mathbf{T}(x)\Big)
  29. s y m b o l η T 𝐓 ( x ) symbol\eta^{T}\mathbf{T}(x)
  30. s y m b o l η 𝐓 ( x ) symbol\eta\cdot\mathbf{T}(x)
  31. 𝐱 = ( x 1 , x 2 , , x k ) . \mathbf{x}=\left(x_{1},x_{2},\cdots,x_{k}\right).
  32. s y m b o l η symbol\eta
  33. f X ( 𝐱 | s y m b o l θ ) = h ( 𝐱 ) exp ( i = 1 s η i ( s y m b o l θ ) T i ( 𝐱 ) - A ( s y m b o l θ ) ) f_{X}(\mathbf{x}|symbol\theta)=h(\mathbf{x})\exp\left(\sum_{i=1}^{s}\eta_{i}({% symbol\theta})T_{i}(\mathbf{x})-A({symbol\theta})\right)
  34. f X ( 𝐱 | s y m b o l θ ) = h ( 𝐱 ) exp ( s y m b o l η ( s y m b o l θ ) 𝐓 ( 𝐱 ) - A ( s y m b o l θ ) ) f_{X}(\mathbf{x}|symbol\theta)=h(\mathbf{x})\exp\Big(symbol\eta({symbol\theta}% )\cdot\mathbf{T}(\mathbf{x})-A({symbol\theta})\Big)
  35. f X ( 𝐱 | s y m b o l θ ) = h ( 𝐱 ) g ( s y m b o l θ ) exp ( s y m b o l η ( s y m b o l θ ) 𝐓 ( 𝐱 ) ) f_{X}(\mathbf{x}|symbol\theta)=h(\mathbf{x})\ g(symbol\theta)\ \exp\Big(symbol% \eta({symbol\theta})\cdot\mathbf{T}(\mathbf{x})\Big)
  36. d F ( 𝐱 | s y m b o l η ) = e s y m b o l η T 𝐓 ( 𝐱 ) - A ( s y m b o l η ) d H ( 𝐱 ) . dF(\mathbf{x}|symbol\eta)=e^{symbol\eta^{\rm T}\mathbf{T}(\mathbf{x})-A(symbol% \eta)}dH(\mathbf{x}).
  37. d ( x , y ) > 0 d(x,y)>0
  38. f X ( x ; θ ) f_{X}(x;\theta)
  39. f X ( x ; θ ) f_{X}(x;\theta)
  40. A ( η ) = ln ( x h ( x ) exp ( η ( θ ) T ( x ) ) d x ) A(\eta)=\ln\left(\int_{x}h(x)\exp(\eta(\theta)\cdot T(x))\operatorname{d}x\right)
  41. 𝔼 [ ln x ] \mathbb{E}[\ln x]
  42. K ( u | η ) = A ( η + u ) - A ( η ) , K(u|\eta)=A(\eta+u)-A(\eta),
  43. f σ ( x ; μ ) = 1 2 π σ 2 e - ( x - μ ) 2 2 σ 2 . f_{\sigma}(x;\mu)=\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{(x-\mu)^{2}}{2% \sigma^{2}}}.
  44. h σ ( x ) = 1 2 π σ 2 e - x 2 2 σ 2 T σ ( x ) = x σ A σ ( μ ) = μ 2 2 σ 2 η σ ( μ ) = μ σ . \begin{aligned}\displaystyle h_{\sigma}(x)&\displaystyle=\frac{1}{\sqrt{2\pi% \sigma^{2}}}e^{-\frac{x^{2}}{2\sigma^{2}}}\\ \displaystyle T_{\sigma}(x)&\displaystyle=\frac{x}{\sigma}\\ \displaystyle A_{\sigma}(\mu)&\displaystyle=\frac{\mu^{2}}{2\sigma^{2}}\\ \displaystyle\eta_{\sigma}(\mu)&\displaystyle=\frac{\mu}{\sigma}.\end{aligned}
  45. f ( x ; μ , σ ) = 1 2 π σ 2 e - ( x - μ ) 2 2 σ 2 . f(x;\mu,\sigma)=\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{(x-\mu)^{2}}{2\sigma^% {2}}}.
  46. s y m b o l η = ( μ σ 2 , - 1 2 σ 2 ) T h ( x ) = 1 2 π T ( x ) = ( x , x 2 ) T A ( s y m b o l η ) = μ 2 2 σ 2 + ln | σ | = - η 1 2 4 η 2 + 1 2 ln | 1 2 η 2 | \begin{aligned}\displaystyle symbol{\eta}&\displaystyle=\left(\frac{\mu}{% \sigma^{2}},-\frac{1}{2\sigma^{2}}\right)^{\rm T}\\ \displaystyle h(x)&\displaystyle=\frac{1}{\sqrt{2\pi}}\\ \displaystyle T(x)&\displaystyle=\left(x,x^{2}\right)^{\rm T}\\ \displaystyle A({symbol\eta})&\displaystyle=\frac{\mu^{2}}{2\sigma^{2}}+\ln|% \sigma|=-\frac{\eta_{1}^{2}}{4\eta_{2}}+\frac{1}{2}\ln\left|\frac{1}{2\eta_{2}% }\right|\end{aligned}
  47. f ( x ) = ( n x ) p x ( 1 - p ) n - x , x { 0 , 1 , 2 , , n } . f(x)={n\choose x}p^{x}(1-p)^{n-x},\quad x\in\{0,1,2,\ldots,n\}.
  48. f ( x ) = ( n x ) exp ( x log ( p 1 - p ) + n log ( 1 - p ) ) , f(x)={n\choose x}\exp\left(x\log\left(\frac{p}{1-p}\right)+n\log(1-p)\right),
  49. η = log p 1 - p . \eta=\log\frac{p}{1-p}.
  50. f X ( 𝐱 | s y m b o l θ ) = h ( 𝐱 ) exp ( s y m b o l η ( s y m b o l θ ) 𝐓 ( 𝐱 ) - A ( s y m b o l η ) ) f_{X}(\mathbf{x}|symbol\theta)=h(\mathbf{x})\exp\Big(symbol\eta({symbol\theta}% )\cdot\mathbf{T}(\mathbf{x})-A({symbol\eta})\Big)
  51. f X ( x | s y m b o l θ ) = h ( x ) exp ( s y m b o l η ( s y m b o l θ ) 𝐓 ( x ) - A ( s y m b o l η ) ) f_{X}(x|symbol\theta)=h(x)\exp\Big(symbol\eta({symbol\theta})\cdot\mathbf{T}(x% )-A({symbol\eta})\Big)
  52. f X ( x | s y m b o l θ ) = h ( x ) g ( s y m b o l θ ) exp ( s y m b o l η ( s y m b o l θ ) 𝐓 ( x ) ) f_{X}(x|symbol\theta)=h(x)g(symbol\theta)\exp\Big(symbol\eta({symbol\theta})% \cdot\mathbf{T}(x)\Big)
  53. f X ( 𝐱 | s y m b o l θ ) = h ( 𝐱 ) exp ( s y m b o l η ( s y m b o l θ ) 𝐓 ( 𝐱 ) - A ( s y m b o l η ) ) f_{X}(\mathbf{x}|symbol\theta)=h(\mathbf{x})\exp\Big(symbol\eta({symbol\theta}% )\cdot\mathbf{T}(\mathbf{x})-A({symbol\eta})\Big)
  54. A ( s y m b o l η ) A({symbol\eta})
  55. s y m b o l θ symbol\theta
  56. g ( s y m b o l η ) g(symbol\eta)
  57. A ( s y m b o l η ) = - ln g ( s y m b o l η ) A(symbol\eta)=-\ln g(symbol\eta)
  58. g ( s y m b o l η ) = e - A ( s y m b o l η ) g(symbol\eta)=e^{-A(symbol\eta)}
  59. h ( x ) h(x)
  60. T ( x ) T(x)
  61. A ( s y m b o l η ) A(symbol\eta)
  62. A ( s y m b o l θ ) A(symbol\theta)
  63. ln p 1 - p \ln\frac{p}{1-p}
  64. 1 1 + e - η = e η 1 + e η \frac{1}{1+e^{-\eta}}=\frac{e^{\eta}}{1+e^{\eta}}
  65. 1 1
  66. x x
  67. ln ( 1 + e η ) \ln(1+e^{\eta})
  68. - ln ( 1 - p ) -\ln(1-p)
  69. ln p 1 - p \ln\frac{p}{1-p}
  70. 1 1 + e - η = e η 1 + e η \frac{1}{1+e^{-\eta}}=\frac{e^{\eta}}{1+e^{\eta}}
  71. ( n x ) {n\choose x}
  72. x x
  73. n ln ( 1 + e η ) n\ln(1+e^{\eta})
  74. - n ln ( 1 - p ) -n\ln(1-p)
  75. ln λ \ln\lambda
  76. e η e^{\eta}
  77. 1 x ! \frac{1}{x!}
  78. x x
  79. e η e^{\eta}
  80. λ \lambda
  81. ln p \ln p
  82. e η e^{\eta}
  83. ( x + r - 1 x ) {x+r-1\choose x}
  84. x x
  85. - r ln ( 1 - e η ) -r\ln(1-e^{\eta})
  86. - r ln ( 1 - p ) -r\ln(1-p)
  87. - λ -\lambda
  88. - η -\eta
  89. 1 1
  90. x x
  91. - ln ( - η ) -\ln(-\eta)
  92. - ln λ -\ln\lambda
  93. - α - 1 -\alpha-1
  94. - 1 - η -1-\eta
  95. 1 1
  96. ln x \ln x
  97. - ln ( - 1 - η ) + ( 1 + η ) ln x m -\ln(-1-\eta)+(1+\eta)\ln x_{\mathrm{m}}
  98. - ln α - α ln x m -\ln\alpha-\alpha\ln x_{\mathrm{m}}
  99. - 1 λ k -\frac{1}{\lambda^{k}}
  100. ( - η ) 1 k (-\eta)^{\frac{1}{k}}
  101. x k - 1 x^{k-1}
  102. x k x^{k}
  103. ln ( - η ) - ln k \ln(-\eta)-\ln k
  104. k ln λ - ln k k\ln\lambda-\ln k
  105. - 1 b -\frac{1}{b}
  106. - 1 η -\frac{1}{\eta}
  107. 1 1
  108. | x - μ | |x-\mu|
  109. ln ( - 2 η ) \ln\left(-\frac{2}{\eta}\right)
  110. ln 2 b \ln 2b
  111. ν 2 - 1 \frac{\nu}{2}-1
  112. 2 ( η + 1 ) 2(\eta+1)
  113. e - x 2 e^{-\frac{x}{2}}
  114. ln x \ln x
  115. ln Γ ( η + 1 ) + ( η + 1 ) ln 2 \ln\Gamma(\eta+1)+(\eta+1)\ln 2
  116. ln Γ ( ν 2 ) + ν 2 ln 2 \ln\Gamma\left(\frac{\nu}{2}\right)+\frac{\nu}{2}\ln 2
  117. μ σ \frac{\mu}{\sigma}
  118. σ η \sigma\eta
  119. e - x 2 2 σ 2 2 π σ \frac{e^{-\frac{x^{2}}{2\sigma^{2}}}}{\sqrt{2\pi}\sigma}
  120. x σ \frac{x}{\sigma}
  121. η 2 2 \frac{\eta^{2}}{2}
  122. μ 2 2 σ 2 \frac{\mu^{2}}{2\sigma^{2}}
  123. [ μ σ 2 - 1 2 σ 2 ] \begin{bmatrix}\dfrac{\mu}{\sigma^{2}}\\ -\dfrac{1}{2\sigma^{2}}\end{bmatrix}
  124. [ - η 1 2 η 2 - 1 2 η 2 ] \begin{bmatrix}-\dfrac{\eta_{1}}{2\eta_{2}}\\ -\dfrac{1}{2\eta_{2}}\end{bmatrix}
  125. 1 2 π \frac{1}{\sqrt{2\pi}}
  126. [ x x 2 ] \begin{bmatrix}x\\ x^{2}\end{bmatrix}
  127. - η 1 2 4 η 2 - 1 2 ln ( - 2 η 2 ) -\frac{\eta_{1}^{2}}{4\eta_{2}}-\frac{1}{2}\ln(-2\eta_{2})
  128. μ 2 2 σ 2 + ln σ \frac{\mu^{2}}{2\sigma^{2}}+\ln\sigma
  129. [ μ σ 2 - 1 2 σ 2 ] \begin{bmatrix}\dfrac{\mu}{\sigma^{2}}\\ -\dfrac{1}{2\sigma^{2}}\end{bmatrix}
  130. [ - η 1 2 η 2 - 1 2 η 2 ] \begin{bmatrix}-\dfrac{\eta_{1}}{2\eta_{2}}\\ -\dfrac{1}{2\eta_{2}}\end{bmatrix}
  131. 1 2 π x \frac{1}{\sqrt{2\pi}x}
  132. [ ln x ( ln x ) 2 ] \begin{bmatrix}\ln x\\ (\ln x)^{2}\end{bmatrix}
  133. - η 1 2 4 η 2 - 1 2 ln ( - 2 η 2 ) -\frac{\eta_{1}^{2}}{4\eta_{2}}-\frac{1}{2}\ln(-2\eta_{2})
  134. μ 2 2 σ 2 + ln σ \frac{\mu^{2}}{2\sigma^{2}}+\ln\sigma
  135. [ - λ 2 μ 2 - λ 2 ] \begin{bmatrix}-\dfrac{\lambda}{2\mu^{2}}\\ -\dfrac{\lambda}{2}\end{bmatrix}
  136. [ η 2 η 1 - 2 η 2 ] \begin{bmatrix}\sqrt{\dfrac{\eta_{2}}{\eta_{1}}}\\ -2\eta_{2}\end{bmatrix}
  137. 1 2 π x 3 2 \frac{1}{\sqrt{2\pi}x^{\frac{3}{2}}}
  138. [ x 1 x ] \begin{bmatrix}x\\ \dfrac{1}{x}\end{bmatrix}
  139. - 2 η 1 η 2 - 1 2 ln ( - 2 η 2 ) -2\sqrt{\eta_{1}\eta_{2}}-\frac{1}{2}\ln(-2\eta_{2})
  140. - λ μ - 1 2 ln λ -\frac{\lambda}{\mu}-\frac{1}{2}\ln\lambda
  141. [ α - 1 - β ] \begin{bmatrix}\alpha-1\\ -\beta\end{bmatrix}
  142. [ η 1 + 1 - η 2 ] \begin{bmatrix}\eta_{1}+1\\ -\eta_{2}\end{bmatrix}
  143. 1 1
  144. [ ln x x ] \begin{bmatrix}\ln x\\ x\end{bmatrix}
  145. ln Γ ( η 1 + 1 ) - ( η 1 + 1 ) ln ( - η 2 ) \ln\Gamma(\eta_{1}+1)-(\eta_{1}+1)\ln(-\eta_{2})
  146. ln Γ ( α ) - α ln β \ln\Gamma(\alpha)-\alpha\ln\beta
  147. [ k - 1 - 1 θ ] \begin{bmatrix}k-1\\ -\dfrac{1}{\theta}\end{bmatrix}
  148. [ η 1 + 1 - 1 η 2 ] \begin{bmatrix}\eta_{1}+1\\ -\dfrac{1}{\eta_{2}}\end{bmatrix}
  149. ln Γ ( k ) + k ln θ \ln\Gamma(k)+k\ln\theta
  150. [ - α - 1 - β ] \begin{bmatrix}-\alpha-1\\ -\beta\end{bmatrix}
  151. [ - η 1 - 1 - η 2 ] \begin{bmatrix}-\eta_{1}-1\\ -\eta_{2}\end{bmatrix}
  152. 1 1
  153. [ ln x 1 x ] \begin{bmatrix}\ln x\\ \frac{1}{x}\end{bmatrix}
  154. ln Γ ( - η 1 - 1 ) - ( - η 1 - 1 ) ln ( - η 2 ) \ln\Gamma(-\eta_{1}-1)-(-\eta_{1}-1)\ln(-\eta_{2})
  155. ln Γ ( α ) - α ln β \ln\Gamma(\alpha)-\alpha\ln\beta
  156. [ - ν 2 - 1 - ν σ 2 2 ] \begin{bmatrix}-\dfrac{\nu}{2}-1\\ -\dfrac{\nu\sigma^{2}}{2}\end{bmatrix}
  157. [ - 2 ( η 1 + 1 ) η 2 η 1 + 1 ] \begin{bmatrix}-2(\eta_{1}+1)\\ \dfrac{\eta_{2}}{\eta_{1}+1}\end{bmatrix}
  158. 1 1
  159. [ ln x 1 x ] \begin{bmatrix}\ln x\\ \frac{1}{x}\end{bmatrix}
  160. ln Γ ( - η 1 - 1 ) - ( - η 1 - 1 ) ln ( - η 2 ) \ln\Gamma(-\eta_{1}-1)-(-\eta_{1}-1)\ln(-\eta_{2})
  161. ln Γ ( ν 2 ) - ν 2 ln ν σ 2 2 \ln\Gamma\left(\frac{\nu}{2}\right)-\frac{\nu}{2}\ln\frac{\nu\sigma^{2}}{2}
  162. [ α - 1 β - 1 ] \begin{bmatrix}\alpha-1\\ \beta-1\end{bmatrix}
  163. [ η 1 + 1 η 2 + 1 ] \begin{bmatrix}\eta_{1}+1\\ \eta_{2}+1\end{bmatrix}
  164. 1 1
  165. [ ln x ln ( 1 - x ) ] \begin{bmatrix}\ln x\\ \ln(1-x)\end{bmatrix}
  166. ln Γ ( η 1 ) + ln Γ ( η 2 ) - ln Γ ( η 1 + η 2 ) \ln\Gamma(\eta_{1})+\ln\Gamma(\eta_{2})-\ln\Gamma(\eta_{1}+\eta_{2})
  167. ln Γ ( α ) + ln Γ ( β ) - ln Γ ( α + β ) \ln\Gamma(\alpha)+\ln\Gamma(\beta)-\ln\Gamma(\alpha+\beta)
  168. [ s y m b o l Σ - 1 s y m b o l μ - 1 2 s y m b o l Σ - 1 ] \begin{bmatrix}symbol\Sigma^{-1}symbol\mu\\ -\frac{1}{2}symbol\Sigma^{-1}\end{bmatrix}
  169. [ - 1 2 s y m b o l η 2 - 1 s y m b o l η 1 - 1 2 s y m b o l η 2 - 1 ] \begin{bmatrix}-\frac{1}{2}symbol\eta_{2}^{-1}symbol\eta_{1}\\ -\frac{1}{2}symbol\eta_{2}^{-1}\end{bmatrix}
  170. ( 2 π ) - k 2 (2\pi)^{-\frac{k}{2}}
  171. [ 𝐱 𝐱𝐱 T ] \begin{bmatrix}\mathbf{x}\\ \mathbf{x}\mathbf{x}^{\mathrm{T}}\end{bmatrix}
  172. - 1 4 s y m b o l η 1 T s y m b o l η 2 - 1 s y m b o l η 1 - 1 2 ln | - 2 s y m b o l η 2 | -\frac{1}{4}symbol\eta_{1}^{\rm T}symbol\eta_{2}^{-1}symbol\eta_{1}-\frac{1}{2% }\ln\left|-2symbol\eta_{2}\right|
  173. 1 2 s y m b o l μ T s y m b o l Σ - 1 s y m b o l μ + 1 2 ln | s y m b o l Σ | \frac{1}{2}symbol\mu^{\rm T}symbol\Sigma^{-1}symbol\mu+\frac{1}{2}\ln|symbol\Sigma|
  174. i = 1 k p i = 1 \textstyle\sum_{i=1}^{k}p_{i}=1
  175. [ ln p 1 ln p k ] \begin{bmatrix}\ln p_{1}\\ \vdots\\ \ln p_{k}\end{bmatrix}
  176. [ e η 1 e η k ] \begin{bmatrix}e^{\eta_{1}}\\ \vdots\\ e^{\eta_{k}}\end{bmatrix}
  177. i = 1 k e η i = 1 \textstyle\sum_{i=1}^{k}e^{\eta_{i}}=1
  178. 1 1
  179. [ [ x = 1 ] [ x = k ] ] \begin{bmatrix}[x=1]\\ \vdots\\ {[x=k]}\end{bmatrix}
  180. [ x = i ] [x=i]
  181. x = i x=i
  182. 0
  183. 0
  184. i = 1 k p i = 1 \textstyle\sum_{i=1}^{k}p_{i}=1
  185. [ ln p 1 + C ln p k + C ] \begin{bmatrix}\ln p_{1}+C\\ \vdots\\ \ln p_{k}+C\end{bmatrix}
  186. [ 1 C e η 1 1 C e η k ] = \begin{bmatrix}\dfrac{1}{C}e^{\eta_{1}}\\ \vdots\\ \dfrac{1}{C}e^{\eta_{k}}\end{bmatrix}=
  187. [ e η 1 i = 1 k e η i e η k i = 1 k e η i ] \begin{bmatrix}\dfrac{e^{\eta_{1}}}{\sum_{i=1}^{k}e^{\eta_{i}}}\\ \vdots\\ \dfrac{e^{\eta_{k}}}{\sum_{i=1}^{k}e^{\eta_{i}}}\end{bmatrix}
  188. i = 1 k e η i = C \textstyle\sum_{i=1}^{k}e^{\eta_{i}}=C
  189. 1 1
  190. [ [ x = 1 ] [ x = k ] ] \begin{bmatrix}[x=1]\\ \vdots\\ {[x=k]}\end{bmatrix}
  191. [ x = i ] [x=i]
  192. x = i x=i
  193. 0
  194. 0
  195. p k = 1 - i = 1 k - 1 p i p_{k}=1-\textstyle\sum_{i=1}^{k-1}p_{i}
  196. [ ln p 1 p k ln p k - 1 p k 0 ] = \begin{bmatrix}\ln\dfrac{p_{1}}{p_{k}}\\ \vdots\\ \ln\dfrac{p_{k-1}}{p_{k}}\\ 0\end{bmatrix}=
  197. [ ln p 1 1 - i = 1 k - 1 p i ln p k - 1 1 - i = 1 k - 1 p i 0 ] \begin{bmatrix}\ln\dfrac{p_{1}}{1-\sum_{i=1}^{k-1}p_{i}}\\ \vdots\\ \ln\dfrac{p_{k-1}}{1-\sum_{i=1}^{k-1}p_{i}}\\ 0\end{bmatrix}
  198. [ e η 1 i = 1 k e η i e η k i = 1 k e η i ] = \begin{bmatrix}\dfrac{e^{\eta_{1}}}{\sum_{i=1}^{k}e^{\eta_{i}}}\\ \vdots\\ \dfrac{e^{\eta_{k}}}{\sum_{i=1}^{k}e^{\eta_{i}}}\end{bmatrix}=
  199. [ e η 1 1 + i = 1 k - 1 e η i e η k - 1 1 + i = 1 k - 1 e η i 1 1 + i = 1 k - 1 e η i ] \begin{bmatrix}\dfrac{e^{\eta_{1}}}{1+\sum_{i=1}^{k-1}e^{\eta_{i}}}\\ \vdots\\ \dfrac{e^{\eta_{k-1}}}{1+\sum_{i=1}^{k-1}e^{\eta_{i}}}\\ \dfrac{1}{1+\sum_{i=1}^{k-1}e^{\eta_{i}}}\end{bmatrix}
  200. 1 1
  201. [ [ x = 1 ] [ x = k ] ] \begin{bmatrix}[x=1]\\ \vdots\\ {[x=k]}\end{bmatrix}
  202. [ x = i ] [x=i]
  203. x = i x=i
  204. ln ( i = 1 k e η i ) = ln ( 1 + i = 1 k - 1 e η i ) \ln\left(\sum_{i=1}^{k}e^{\eta_{i}}\right)=\ln\left(1+\sum_{i=1}^{k-1}e^{\eta_% {i}}\right)
  205. - ln p k = - ln ( 1 - i = 1 k - 1 p i ) -\ln p_{k}=-\ln\left(1-\sum_{i=1}^{k-1}p_{i}\right)
  206. i = 1 k p i = 1 \textstyle\sum_{i=1}^{k}p_{i}=1
  207. [ ln p 1 ln p k ] \begin{bmatrix}\ln p_{1}\\ \vdots\\ \ln p_{k}\end{bmatrix}
  208. [ e η 1 e η k ] \begin{bmatrix}e^{\eta_{1}}\\ \vdots\\ e^{\eta_{k}}\end{bmatrix}
  209. i = 1 k e η i = 1 \textstyle\sum_{i=1}^{k}e^{\eta_{i}}=1
  210. n ! i = 1 k x i ! \frac{n!}{\prod_{i=1}^{k}x_{i}!}
  211. [ x 1 x k ] \begin{bmatrix}x_{1}\\ \vdots\\ x_{k}\end{bmatrix}
  212. 0
  213. 0
  214. i = 1 k p i = 1 \textstyle\sum_{i=1}^{k}p_{i}=1
  215. [ ln p 1 + C ln p k + C ] \begin{bmatrix}\ln p_{1}+C\\ \vdots\\ \ln p_{k}+C\end{bmatrix}
  216. [ 1 C e η 1 1 C e η k ] = \begin{bmatrix}\dfrac{1}{C}e^{\eta_{1}}\\ \vdots\\ \dfrac{1}{C}e^{\eta_{k}}\end{bmatrix}=
  217. [ e η 1 i = 1 k e η i e η k i = 1 k e η i ] \begin{bmatrix}\dfrac{e^{\eta_{1}}}{\sum_{i=1}^{k}e^{\eta_{i}}}\\ \vdots\\ \dfrac{e^{\eta_{k}}}{\sum_{i=1}^{k}e^{\eta_{i}}}\end{bmatrix}
  218. i = 1 k e η i = C \textstyle\sum_{i=1}^{k}e^{\eta_{i}}=C
  219. n ! i = 1 k x i ! \frac{n!}{\prod_{i=1}^{k}x_{i}!}
  220. [ x 1 x k ] \begin{bmatrix}x_{1}\\ \vdots\\ x_{k}\end{bmatrix}
  221. 0
  222. 0
  223. p k = 1 - i = 1 k - 1 p i p_{k}=1-\textstyle\sum_{i=1}^{k-1}p_{i}
  224. [ ln p 1 p k ln p k - 1 p k 0 ] = \begin{bmatrix}\ln\dfrac{p_{1}}{p_{k}}\\ \vdots\\ \ln\dfrac{p_{k-1}}{p_{k}}\\ 0\end{bmatrix}=
  225. [ ln p 1 1 - i = 1 k - 1 p i ln p k - 1 1 - i = 1 k - 1 p i 0 ] \begin{bmatrix}\ln\dfrac{p_{1}}{1-\sum_{i=1}^{k-1}p_{i}}\\ \vdots\\ \ln\dfrac{p_{k-1}}{1-\sum_{i=1}^{k-1}p_{i}}\\ 0\end{bmatrix}
  226. [ e η 1 i = 1 k e η i e η k i = 1 k e η i ] = \begin{bmatrix}\dfrac{e^{\eta_{1}}}{\sum_{i=1}^{k}e^{\eta_{i}}}\\ \vdots\\ \dfrac{e^{\eta_{k}}}{\sum_{i=1}^{k}e^{\eta_{i}}}\end{bmatrix}=
  227. [ e η 1 1 + i = 1 k - 1 e η i e η k - 1 1 + i = 1 k - 1 e η i 1 1 + i = 1 k - 1 e η i ] \begin{bmatrix}\dfrac{e^{\eta_{1}}}{1+\sum_{i=1}^{k-1}e^{\eta_{i}}}\\ \vdots\\ \dfrac{e^{\eta_{k-1}}}{1+\sum_{i=1}^{k-1}e^{\eta_{i}}}\\ \dfrac{1}{1+\sum_{i=1}^{k-1}e^{\eta_{i}}}\end{bmatrix}
  228. n ! i = 1 k x i ! \frac{n!}{\prod_{i=1}^{k}x_{i}!}
  229. [ x 1 x k ] \begin{bmatrix}x_{1}\\ \vdots\\ x_{k}\end{bmatrix}
  230. n ln ( i = 1 k e η i ) = n ln ( 1 + i = 1 k - 1 e η i ) n\ln\left(\sum_{i=1}^{k}e^{\eta_{i}}\right)=n\ln\left(1+\sum_{i=1}^{k-1}e^{% \eta_{i}}\right)
  231. - n ln p k = - n ln ( 1 - i = 1 k - 1 p i ) -n\ln p_{k}=-n\ln\left(1-\sum_{i=1}^{k-1}p_{i}\right)
  232. [ α 1 - 1 α k - 1 ] \begin{bmatrix}\alpha_{1}-1\\ \vdots\\ \alpha_{k}-1\end{bmatrix}
  233. [ η 1 + 1 η k + 1 ] \begin{bmatrix}\eta_{1}+1\\ \vdots\\ \eta_{k}+1\end{bmatrix}
  234. 1 1
  235. [ ln x 1 ln x k ] \begin{bmatrix}\ln x_{1}\\ \vdots\\ \ln x_{k}\end{bmatrix}
  236. i = 1 k ln Γ ( η i + 1 ) - ln Γ ( i = 1 k ( η i + 1 ) ) \sum_{i=1}^{k}\ln\Gamma(\eta_{i}+1)-\ln\Gamma\left(\sum_{i=1}^{k}\Big(\eta_{i}% +1\Big)\right)
  237. i = 1 k ln Γ ( α i ) - ln Γ ( i = 1 k α i ) \sum_{i=1}^{k}\ln\Gamma(\alpha_{i})-\ln\Gamma\left(\sum_{i=1}^{k}\alpha_{i}\right)
  238. [ - 1 2 𝐕 - 1 n - p - 1 2 ] \begin{bmatrix}-\frac{1}{2}\mathbf{V}^{-1}\\ \dfrac{n-p-1}{2}\end{bmatrix}
  239. [ - 1 2 s y m b o l η 1 - 1 2 η 2 + p + 1 ] \begin{bmatrix}-\frac{1}{2}{symbol\eta_{1}}^{-1}\\ 2\eta_{2}+p+1\end{bmatrix}
  240. 1 1
  241. [ 𝐗 ln | 𝐗 | ] \begin{bmatrix}\mathbf{X}\\ \ln|\mathbf{X}|\end{bmatrix}
  242. - ( η 2 + p + 1 2 ) ln | - s y m b o l η 1 | -\left(\eta_{2}+\frac{p+1}{2}\right)\ln|-symbol\eta_{1}|
  243. + ln Γ p ( η 2 + p + 1 2 ) = +\ln\Gamma_{p}\left(\eta_{2}+\frac{p+1}{2}\right)=
  244. - n 2 ln | - s y m b o l η 1 | + ln Γ p ( n 2 ) = -\frac{n}{2}\ln|-symbol\eta_{1}|+\ln\Gamma_{p}\left(\frac{n}{2}\right)=
  245. ( η 2 + p + 1 2 ) ( p ln 2 + ln | 𝐕 | ) \left(\eta_{2}+\frac{p+1}{2}\right)(p\ln 2+\ln|\mathbf{V}|)
  246. + ln Γ p ( η 2 + p + 1 2 ) +\ln\Gamma_{p}\left(\eta_{2}+\frac{p+1}{2}\right)
  247. n 2 ( p ln 2 + ln | 𝐕 | ) + ln Γ p ( n 2 ) \frac{n}{2}(p\ln 2+\ln|\mathbf{V}|)+\ln\Gamma_{p}\left(\frac{n}{2}\right)
  248. tr ( 𝐀 T 𝐁 ) = vec ( 𝐀 ) vec ( 𝐁 ) , {\rm tr}(\mathbf{A}^{\rm T}\mathbf{B})=\operatorname{vec}(\mathbf{A})\cdot% \operatorname{vec}(\mathbf{B}),
  249. 𝐕 T = 𝐕 . \mathbf{V}^{\rm T}=\mathbf{V}.
  250. [ - 1 2 s y m b o l Ψ - m + p + 1 2 ] \begin{bmatrix}-\frac{1}{2}symbol\Psi\\ -\dfrac{m+p+1}{2}\end{bmatrix}
  251. [ - 2 s y m b o l η 1 - ( 2 η 2 + p + 1 ) ] \begin{bmatrix}-2symbol\eta_{1}\\ -(2\eta_{2}+p+1)\end{bmatrix}
  252. 1 1
  253. [ 𝐗 - 1 ln | 𝐗 | ] \begin{bmatrix}\mathbf{X}^{-1}\\ \ln|\mathbf{X}|\end{bmatrix}
  254. ( η 2 + p + 1 2 ) ln | - s y m b o l η 1 | \left(\eta_{2}+\frac{p+1}{2}\right)\ln|-symbol\eta_{1}|
  255. + ln Γ p ( - ( η 2 + p + 1 2 ) ) = +\ln\Gamma_{p}\left(-\Big(\eta_{2}+\frac{p+1}{2}\Big)\right)=
  256. - m 2 ln | - s y m b o l η 1 | + ln Γ p ( m 2 ) = -\frac{m}{2}\ln|-symbol\eta_{1}|+\ln\Gamma_{p}\left(\frac{m}{2}\right)=
  257. - ( η 2 + p + 1 2 ) ( p ln 2 - ln | s y m b o l Ψ | ) -\left(\eta_{2}+\frac{p+1}{2}\right)(p\ln 2-\ln|symbol\Psi|)
  258. + ln Γ p ( - ( η 2 + p + 1 2 ) ) +\ln\Gamma_{p}\left(-\Big(\eta_{2}+\frac{p+1}{2}\Big)\right)
  259. m 2 ( p ln 2 - ln | s y m b o l Ψ | ) + ln Γ p ( m 2 ) \frac{m}{2}(p\ln 2-\ln|symbol\Psi|)+\ln\Gamma_{p}\left(\frac{m}{2}\right)
  260. [ α - 1 2 - β - λ μ 2 2 λ μ - λ 2 ] \begin{bmatrix}\alpha-\frac{1}{2}\\ -\beta-\dfrac{\lambda\mu^{2}}{2}\\ \lambda\mu\\ -\dfrac{\lambda}{2}\end{bmatrix}
  261. [ η 1 + 1 2 - η 2 + η 3 2 4 η 4 - η 3 2 η 4 - 2 η 4 ] \begin{bmatrix}\eta_{1}+\frac{1}{2}\\ -\eta_{2}+\dfrac{\eta_{3}^{2}}{4\eta_{4}}\\ -\dfrac{\eta_{3}}{2\eta_{4}}\\ -2\eta_{4}\end{bmatrix}
  262. 1 2 π \dfrac{1}{\sqrt{2\pi}}
  263. [ ln τ τ τ x τ x 2 ] \begin{bmatrix}\ln\tau\\ \tau\\ \tau x\\ \tau x^{2}\end{bmatrix}
  264. ln Γ ( η 1 + 1 2 ) - 1 2 ln ( - 2 η 4 ) - \ln\Gamma\left(\eta_{1}+\frac{1}{2}\right)-\frac{1}{2}\ln\left(-2\eta_{4}% \right)-
  265. - ( η 1 + 1 2 ) ln ( - η 2 + η 3 2 4 η 4 ) -\left(\eta_{1}+\frac{1}{2}\right)\ln\left(-\eta_{2}+\dfrac{\eta_{3}^{2}}{4% \eta_{4}}\right)
  266. ln Γ ( α ) - α ln β - 1 2 ln λ \ln\Gamma\left(\alpha\right)-\alpha\ln\beta-\frac{1}{2}\ln\lambda
  267. p i p_{i}
  268. i = 1 k p i = 1. \sum_{i=1}^{k}p_{i}=1.
  269. C = - ln p k . C=-\ln p_{k}.
  270. p ( x ) = 1 Z f ( x ) p(x)=\frac{1}{Z}f(x)
  271. Z = x f ( x ) d x . Z=\int_{x}f(x)dx.
  272. p ( x ; s y m b o l η ) = g ( s y m b o l η ) h ( x ) e s y m b o l η 𝐓 ( x ) , p(x;symbol\eta)=g(symbol\eta)h(x)e^{symbol\eta\cdot\mathbf{T}(x)},
  273. K ( x ) = h ( x ) e s y m b o l η 𝐓 ( x ) K(x)=h(x)e^{symbol\eta\cdot\mathbf{T}(x)}
  274. Z = x h ( x ) e s y m b o l η 𝐓 ( x ) d x . Z=\int_{x}h(x)e^{symbol\eta\cdot\mathbf{T}(x)}dx.
  275. 1 = x g ( s y m b o l η ) h ( x ) e s y m b o l η 𝐓 ( x ) d x = g ( s y m b o l η ) x h ( x ) e s y m b o l η 𝐓 ( x ) d x = g ( s y m b o l η ) Z . 1=\int_{x}g(symbol\eta)h(x)e^{symbol\eta\cdot\mathbf{T}(x)}dx=g(symbol\eta)% \int_{x}h(x)e^{symbol\eta\cdot\mathbf{T}(x)}dx=g(symbol\eta)Z.
  276. g ( s y m b o l η ) = 1 Z g(symbol\eta)=\frac{1}{Z}
  277. A ( s y m b o l η ) = - ln g ( s y m b o l η ) = ln Z . A(symbol\eta)=-\ln g(symbol\eta)=\ln Z.
  278. M T ( u ) E [ e u T T ( x ) | η ] = x h ( x ) e ( η + u ) T T ( x ) - A ( η ) d x = e A ( η + u ) - A ( η ) M_{T}(u)\equiv E[e^{u^{\rm T}T(x)}|\eta]=\int_{x}h(x)e^{(\eta+u)^{\rm T}T(x)-A% (\eta)}dx=e^{A(\eta+u)-A(\eta)}
  279. K ( u | η ) = A ( η + u ) - A ( η ) K(u|\eta)=A(\eta+u)-A(\eta)
  280. E ( T j ) = A ( η ) η j E(T_{j})=\frac{\partial A(\eta)}{\partial\eta_{j}}
  281. cov ( T i , T j ) = 2 A ( η ) η i η j . \mathrm{cov}\left(T_{i},T_{j}\right)=\frac{\partial^{2}A(\eta)}{\partial\eta_{% i}\partial\eta_{j}}.
  282. p ( x ) = g ( η ) h ( x ) e η T ( x ) . p(x)=g(\eta)h(x)e^{\eta T(x)}.
  283. 1 = x p ( x ) d x = x g ( η ) h ( x ) e η T ( x ) d x = g ( η ) x h ( x ) e η T ( x ) d x . 1=\int_{x}p(x)dx=\int_{x}g(\eta)h(x)e^{\eta T(x)}dx=g(\eta)\int_{x}h(x)e^{\eta T% (x)}dx.
  284. 0 \displaystyle 0
  285. 𝔼 [ T ( x ) ] = - d d η ln g ( η ) = d d η A ( η ) . \mathbb{E}[T(x)]=-\frac{d}{d\eta}\ln g(\eta)=\frac{d}{d\eta}A(\eta).
  286. p ( x ) = β α Γ ( α ) x α - 1 e - β x . p(x)=\frac{\beta^{\alpha}}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}.
  287. η 1 = α - 1 , \eta_{1}=\alpha-1,
  288. η 2 = - β , \eta_{2}=-\beta,
  289. α = η 1 + 1 , \alpha=\eta_{1}+1,
  290. β = - η 2 , \beta=-\eta_{2},
  291. ( ln x , x ) , (\ln x,x),
  292. A ( η 1 , η 2 ) = ln Γ ( η 1 + 1 ) - ( η 1 + 1 ) ln ( - η 2 ) . A(\eta_{1},\eta_{2})=\ln\Gamma(\eta_{1}+1)-(\eta_{1}+1)\ln(-\eta_{2}).
  293. 𝔼 [ ln x ] \displaystyle\mathbb{E}[\ln x]
  294. ψ ( x ) \psi(x)
  295. 𝔼 [ x ] \displaystyle\mathbb{E}[x]
  296. Var ( x ) \displaystyle\operatorname{Var}(x)
  297. p θ ( x ) = θ e - x ( 1 + e - x ) θ + 1 p_{\theta}(x)=\frac{\theta e^{-x}}{\left(1+e^{-x}\right)^{\theta+1}}
  298. θ ( 0 , ) \theta\in(0,\infty)
  299. e - x 1 + e - x exp ( - θ log ( 1 + e - x ) + log ( θ ) ) \frac{e^{-x}}{1+e^{-x}}\exp\left(-\theta\log\left(1+e^{-x}\right)+\log(\theta)\right)
  300. η = - θ , \eta=-\theta,
  301. T = log ( 1 + e - x ) , T=\log\left(1+e^{-x}\right),
  302. A ( η ) = - log ( θ ) = - log ( - η ) A(\eta)=-\log(\theta)=-\log(-\eta)
  303. E ( log ( 1 + e - X ) ) = E ( T ) = A ( η ) η = η [ - log ( - η ) ] = 1 - η = 1 θ , E(\log(1+e^{-X}))=E(T)=\frac{\partial A(\eta)}{\partial\eta}=\frac{\partial}{% \partial\eta}[-\log(-\eta)]=\frac{1}{-\eta}=\frac{1}{\theta},
  304. var ( log ( 1 + e - X ) ) = 2 A ( η ) η 2 = η [ 1 - η ] = 1 ( - η ) 2 = 1 θ 2 . \mathrm{var}(\log\left(1+e^{-X}\right))=\frac{\partial^{2}A(\eta)}{\partial% \eta^{2}}=\frac{\partial}{\partial\eta}\left[\frac{1}{-\eta}\right]=\frac{1}{(% -\eta)^{2}}=\frac{1}{\theta^{2}}.
  305. s y m b o l η 1 = - 1 2 𝐕 - 1 , symbol\eta_{1}=-\frac{1}{2}\mathbf{V}^{-1},
  306. η 2 = n - p - 1 2 , \eta_{2}=\frac{n-p-1}{2},
  307. 𝐕 = - 1 2 s y m b o l η 1 - 1 , \mathbf{V}=-\frac{1}{2}{symbol\eta_{1}}^{-1},
  308. n = 2 η 2 + p + 1 , n=2\eta_{2}+p+1,
  309. ( 𝐗 , ln | 𝐗 | ) . (\mathbf{X},\ln|\mathbf{X}|).
  310. A ( s y m b o l η 1 , n ) = - n 2 ln | - s y m b o l η 1 | + ln Γ p ( n 2 ) , A(symbol\eta_{1},n)=-\frac{n}{2}\ln|-symbol\eta_{1}|+\ln\Gamma_{p}\left(\frac{% n}{2}\right),
  311. A ( 𝐕 , η 2 ) = ( η 2 + p + 1 2 ) ( p ln 2 + ln | 𝐕 | ) + ln Γ p ( η 2 + p + 1 2 ) . A(\mathbf{V},\eta_{2})=\left(\eta_{2}+\frac{p+1}{2}\right)(p\ln 2+\ln|\mathbf{% V}|)+\ln\Gamma_{p}\left(\eta_{2}+\frac{p+1}{2}\right).
  312. ln | a 𝐗 | 𝐗 = ( 𝐗 - 1 ) T \frac{\partial\ln|a\mathbf{X}|}{\partial\mathbf{X}}=(\mathbf{X}^{-1})^{\rm T}
  313. 𝔼 [ 𝐗 ] \displaystyle\mathbb{E}[\mathbf{X}]
  314. ln Γ p ( a ) = ln ( π p ( p - 1 ) 4 j = 1 p Γ ( a + 1 - j 2 ) ) = p ( p - 1 ) 4 ln π + j = 1 p ln Γ [ a + 1 - j 2 ] \ln\Gamma_{p}(a)=\ln\left(\pi^{\frac{p(p-1)}{4}}\prod_{j=1}^{p}\Gamma\left(a+% \frac{1-j}{2}\right)\right)=\frac{p(p-1)}{4}\ln\pi+\sum_{j=1}^{p}\ln\Gamma% \left[a+\frac{1-j}{2}\right]
  315. ψ ( x ) = d d x ln Γ ( x ) . \psi(x)=\frac{d}{dx}\ln\Gamma(x).
  316. 𝔼 [ ln | 𝐗 | ] \displaystyle\mathbb{E}[\ln|\mathbf{X}|]
  317. S [ d F | d H ] = - d F d H ln d F d H d H S[dF|dH]=-\int\frac{dF}{dH}\ln\frac{dF}{dH}\,dH
  318. S [ d F | d H ] = ln d H d F d F S[dF|dH]=\int\ln\frac{dH}{dF}\,dF
  319. S = - i I p i ln p i S=-\sum_{i\in I}p_{i}\ln p_{i}
  320. s y m b o l η symbol\eta
  321. f ( x | s y m b o l η ) = h ( x ) exp ( s y m b o l η T 𝐓 ( x ) - A ( s y m b o l η ) ) f(x|symbol\eta)=h(x)\exp\left({symbol\eta}^{\rm T}\mathbf{T}(x)-A(symbol\eta)\right)
  322. p π ( s y m b o l η | s y m b o l χ , ν ) = f ( s y m b o l χ , ν ) exp ( s y m b o l η T s y m b o l χ - ν A ( s y m b o l η ) ) , p_{\pi}(symbol\eta|symbol\chi,\nu)=f(symbol\chi,\nu)\exp\left(symbol\eta^{\rm T% }symbol\chi-\nu A(symbol\eta)\right),
  323. p π ( s y m b o l η | s y m b o l χ , ν ) = f ( s y m b o l χ , ν ) g ( s y m b o l η ) ν exp ( s y m b o l η T s y m b o l χ ) , s y m b o l χ s p_{\pi}(symbol\eta|symbol\chi,\nu)=f(symbol\chi,\nu)g(symbol\eta)^{\nu}\exp% \left(symbol\eta^{\rm T}symbol\chi\right),\qquad symbol\chi\in\mathbb{R}^{s}
  324. s y m b o l η symbol\eta
  325. ν > 0 \nu>0
  326. s y m b o l χ symbol\chi
  327. s y m b o l χ symbol\chi
  328. f ( s y m b o l χ , ν ) f(symbol\chi,\nu)
  329. A ( s y m b o l η ) A(symbol\eta)
  330. g ( s y m b o l η ) g(symbol\eta)
  331. p F ( x | s y m b o l η ) = h ( x ) g ( s y m b o l η ) exp ( s y m b o l η T 𝐓 ( x ) ) p_{F}(x|symbol\eta)=h(x)g(symbol\eta)\exp\left(symbol\eta^{\rm T}\mathbf{T}(x)\right)
  332. 𝐗 = ( x 1 , , x n ) \mathbf{X}=(x_{1},\ldots,x_{n})
  333. p ( 𝐗 | s y m b o l η ) = ( i = 1 n h ( x i ) ) g ( s y m b o l η ) n exp ( s y m b o l η T i = 1 n 𝐓 ( x i ) ) p(\mathbf{X}|symbol\eta)=\left(\prod_{i=1}^{n}h(x_{i})\right)g(symbol\eta)^{n}% \exp\left(symbol\eta^{\rm T}\sum_{i=1}^{n}\mathbf{T}(x_{i})\right)
  334. p π ( s y m b o l η | s y m b o l χ , ν ) \displaystyle p_{\pi}(symbol\eta|symbol\chi,\nu)
  335. p ( s y m b o l η | 𝐗 , s y m b o l χ , ν ) \displaystyle p(symbol\eta|\mathbf{X},symbol\chi,\nu)
  336. p ( s y m b o l η | 𝐗 , s y m b o l χ , ν ) = p π ( s y m b o l η | s y m b o l χ + i = 1 n 𝐓 ( x i ) , ν + n ) p(symbol\eta|\mathbf{X},symbol\chi,\nu)=p_{\pi}\left(symbol\eta|symbol\chi+% \sum_{i=1}^{n}\mathbf{T}(x_{i}),\nu+n\right)
  337. 𝐓 ( 𝐗 ) = i = 1 n 𝐓 ( x i ) , \mathbf{T}(\mathbf{X})=\sum_{i=1}^{n}\mathbf{T}(x_{i}),
  338. s y m b o l η symbol\eta
  339. s y m b o l χ \displaystyle symbol\chi^{\prime}
  340. s y m b o l η symbol\eta
  341. s y m b o l θ . symbol\theta.

Exponential_sum.html

  1. e ( x ) = exp ( 2 π i x ) . e(x)=\exp(2\pi ix).\,
  2. n e ( x n ) , \sum_{n}e(x_{n}),
  3. n a n e ( x n ) \sum_{n}a_{n}e(x_{n})
  4. S = n e ( x n ) S=\sum_{n}e(x_{n})
  5. | S | N |S|\leq N\,
  6. | S | = O ( N ) |S|=O(\sqrt{N})\,
  7. | S | = o ( N ) |S|=o(N)\,
  8. G ( τ ) = n e i a f ( x ) + i a τ n G(\tau)=\sum_{n}e^{iaf(x)+ia\tau n}
  9. G ( 0 ) G(0)
  10. S ( x ) = e i a f ( x ) S(x)=e^{iaf(x)}
  11. ξ = e 2 π i / p \xi=e^{2\pi i/p}
  12. n = 0 p - 1 ξ n 2 = { p , p = 1 mod 4 i p , p = 3 mod 4 \sum_{n=0}^{p-1}\xi^{n^{2}}=\begin{cases}\sqrt{p},&p=1\mod 4\\ i\sqrt{p},&p=3\mod 4\end{cases}

Expression_(mathematics).html

  1. 0 + 0 0+0
  2. 8 x - 5 8x-5
  3. 7 x 2 + 4 x - 10 7{{x}^{2}}+4x-10
  4. x - 1 x 2 + 12 \frac{x-1}{{{x}^{2}}+12}
  5. f ( a ) + k = 1 n 1 k ! d k d t k | t = 0 f ( u ( t ) ) + 0 1 ( 1 - t ) n n ! d n + 1 d t n + 1 f ( u ( t ) ) d t . f(a)+\sum_{k=1}^{n}\left.\frac{1}{k!}\frac{d^{k}}{dt^{k}}\right|_{t=0}f(u(t))+% \int_{0}^{1}\frac{(1-t)^{n}}{n!}\frac{d^{n+1}}{dt^{n+1}}f(u(t))\,dt.
  6. × 4 ) x + , / y \times 4)x+,/y
  7. \oplus
  8. x / y x/y
  9. n = 1 3 ( 2 n x ) \sum_{n=1}^{3}(2nx)

Extrapolation.html

  1. x = 7 x=7
  2. x * x_{*}
  3. ( x k - 1 , y k - 1 ) (x_{k-1},y_{k-1})
  4. ( x k , y k ) (x_{k},y_{k})
  5. y ( x * ) = y k - 1 + x * - x k - 1 x k - x k - 1 ( y k - y k - 1 ) . y(x_{*})=y_{k-1}+\frac{x_{*}-x_{k-1}}{x_{k}-x_{k-1}}(y_{k}-y_{k-1}).
  6. x k - 1 < x * < x k x_{k-1}<x_{*}<x_{k}
  7. z ^ = 1 / z \hat{z}=1/z

Extraterrestrial_skies.html

  1. m 1 m_{1}
  2. d 1 d_{1}
  3. m 2 = m 1 - 2.5 log ( d 1 2 / d 2 2 ) = m 1 + 5 log ( d 2 / d 1 ) m_{2}=m_{1}-2.5\log(d_{1}^{2}/d_{2}^{2})=m_{1}+5\log(d_{2}/d_{1})
  4. d 2 d_{2}

Extremal_graph_theory.html

  1. n 2 4 \left\lfloor\frac{n^{2}}{4}\right\rfloor
  2. ( k - 2 ) n 2 2 ( k - 1 ) = ( 1 - 1 k - 1 ) n 2 2 \left\lfloor\frac{(k-2)n^{2}}{2(k-1)}\right\rfloor=\left\lfloor\left(1-\frac{1% }{k-1}\right)\frac{n^{2}}{2}\right\rfloor
  3. ( 1 2 + o ( 1 ) ) n 3 / 2 \left(\frac{1}{2}+o(1)\right)n^{3/2}
  4. ( n - 1 2 ) {\left({{n-1}\atop{2}}\right)}
  5. δ ( G ) = min v G d ( v ) . \delta(G)=\min_{v\in G}d(v).

Étale_cohomology.html

  1. F Γ ( F ) , F\to\Gamma(F),\,\!
  2. H i ( V , / k ) H^{i}(V,\mathbb{Z}/\ell^{k}\mathbb{Z})
  3. H i ( V , ) = lim H i ( V , / k ) H^{i}(V,\mathbb{Z}_{\ell})=\underleftarrow{\lim}H^{i}(V,\mathbb{Z}/\ell^{k}% \mathbb{Z})
  4. H q ( X , F ) = lim H q ( X , F i ) H^{q}(X,F)=\underleftarrow{\lim}H^{q}(X,F_{i})
  5. H q ( X , lim F i ) lim H q ( X , F i ) H^{q}(X,\underleftarrow{\lim}F_{i})\to\underleftarrow{\lim}H^{q}(X,F_{i})
  6. H i ( V , ) = H i ( V , ) H^{i}(V,\mathbb{Q}_{\ell})=H^{i}(V,\mathbb{Z}_{\ell})\otimes\mathbb{Q}_{\ell}
  7. 1 𝐆 m j * 𝐆 m , K x | X | i x * 𝐙 1 1\rightarrow\mathbf{G}_{m}\rightarrow j_{*}\mathbf{G}_{m,K}\rightarrow% \bigoplus_{x\in|X|}i_{x*}\mathbf{Z}\rightarrow 1
  8. 0 H 0 ( 𝐆 m ) H 0 ( j * 𝐆 m , K ) x | X | H 0 ( i x * 𝐙 ) H 1 ( 𝐆 m ) H 1 ( j * 𝐆 m , K ) 0\rightarrow H^{0}(\mathbf{G}_{m})\rightarrow H^{0}(j_{*}\mathbf{G}_{m,K})% \rightarrow\bigoplus_{x\in|X|}H^{0}(i_{x*}\mathbf{Z})\rightarrow H^{1}(\mathbf% {G}_{m})\rightarrow H^{1}(j_{*}\mathbf{G}_{m,K})\rightarrow\cdots
  9. K Div ( X ) H 1 ( 𝐆 m ) 1 K\longrightarrow\mathrm{Div}(X)\longrightarrow H^{1}(\mathbf{G}_{m})\longrightarrow 1
  10. 0 H 0 ( X , μ n ) H 0 ( X , 𝐆 m ) H 0 ( X , 𝐆 m ) 0\rightarrow H^{0}(X,\mu_{n})\rightarrow H^{0}(X,\mathbf{G}_{m})\rightarrow H^% {0}(X,\mathbf{G}_{m})\rightarrow
  11. H 1 ( X , μ n ) H 1 ( X , 𝐆 m ) H 1 ( X , 𝐆 m ) H 2 ( X , μ n ) H 2 ( X , 𝐆 m ) \rightarrow H^{1}(X,\mu_{n})\rightarrow H^{1}(X,\mathbf{G}_{m})\rightarrow H^{% 1}(X,\mathbf{G}_{m})\rightarrow H^{2}(X,\mu_{n})\rightarrow H^{2}(X,\mathbf{G}% _{m})
  12. 1 μ n 𝐆 m 𝑛 𝐆 m 1. 1\rightarrow\mu_{n}\rightarrow\mathbf{G}_{m}\xrightarrow{n}\mathbf{G}_{m}% \rightarrow 1.
  13. 1 H 1 ( X , μ n ) Pic ( X ) × n Pic ( X ) H 2 ( X , μ n ) 1. 1\rightarrow H^{1}(X,\mu_{n})\rightarrow\mathrm{Pic}(X)\xrightarrow{\times n}% \mathrm{Pic}(X)\rightarrow H^{2}(X,\mu_{n})\rightarrow 1.
  14. 0 𝐙 / p 𝐙 K x x p - x K 0 0\rightarrow\mathbf{Z}/p\mathbf{Z}\rightarrow K\xrightarrow{x\mapsto x^{p}-x}K\rightarrow 0
  15. H c q ( X , F ) = H q ( Y , j ! F ) H_{c}^{q}(X,F)=H^{q}(Y,j_{!}F)
  16. H c q ( X , F ) H_{c}^{q}(X,F)
  17. H q ( X , F ) H^{q}(X,F)
  18. R q f ! ( F ) = R q g * ( j ! F ) R^{q}f_{!}(F)=R^{q}g_{*}(j_{!}F)
  19. Tr : H c 2 N ( X , μ n N ) 𝐙 / n 𝐙 \mathrm{Tr}:H_{c}^{2N}(X,\mu_{n}^{N})\mapsto\mathbf{Z}/n\mathbf{Z}
  20. H c i ( X , μ n N ) H^{i}_{c}(X,\mu_{n}^{N})
  21. H 2 N - i ( X , 𝐙 / n 𝐙 ) H^{2N-i}(X,\mathbf{Z}/n\mathbf{Z})
  22. X X
  23. g g
  24. p p
  25. n 1 n≥1
  26. # X ( 𝐅 p n ) = p n + 1 - i = 1 2 g α i n , \#X\left(\mathbf{F}_{p^{n}}\right)=p^{n}+1-\sum_{i=1}^{2g}\alpha_{i}^{n},
  27. 0
  28. f : X X f:X→X
  29. i = 0 2 dim ( X ) ( - 1 ) i Tr ( f | H i ( X ) ) . \sum_{i=0}^{2\dim(X)}(-1)^{i}\mathrm{Tr}\left(f|_{H^{i}(X)}\right).
  30. X X
  31. p p
  32. X X
  33. 0 , 1 , 2 0,1,2
  34. 1 , 2 g 1,2g
  35. 1 1
  36. # X ( 𝐅 p n ) = Tr ( F n | H 0 ( X ) ) - Tr ( F n | H 1 ( X ) ) + Tr ( F n | H 2 ( X ) ) . \#X\left(\mathbf{F}_{p^{n}}\right)=\mathrm{Tr}\left(F^{n}|_{H^{0}(X)}\right)-% \mathrm{Tr}\left(F^{n}|_{H^{1}(X)}\right)+\mathrm{Tr}\left(F^{n}|_{H^{2}(X)}% \right).
  37. α < s u b > i α<sub>i

F-distribution.html

  1. I d 1 x d 1 x + d 2 ( d 1 2 , d 2 2 ) I_{\frac{d_{1}x}{d_{1}x+d_{2}}}\left(\tfrac{d_{1}}{2},\tfrac{d_{2}}{2}\right)
  2. d 2 d 2 - 2 \frac{d_{2}}{d_{2}-2}\!
  3. d 1 - 2 d 1 d 2 d 2 + 2 \frac{d_{1}-2}{d_{1}}\;\frac{d_{2}}{d_{2}+2}
  4. 2 d 2 2 ( d 1 + d 2 - 2 ) d 1 ( d 2 - 2 ) 2 ( d 2 - 4 ) \frac{2\,d_{2}^{2}\,(d_{1}+d_{2}-2)}{d_{1}(d_{2}-2)^{2}(d_{2}-4)}\!
  5. ( 2 d 1 + d 2 - 2 ) 8 ( d 2 - 4 ) ( d 2 - 6 ) d 1 ( d 1 + d 2 - 2 ) \frac{(2d_{1}+d_{2}-2)\sqrt{8(d_{2}-4)}}{(d_{2}-6)\sqrt{d_{1}(d_{1}+d_{2}-2)}}\!
  6. f ( x ; d 1 , d 2 ) = ( d 1 x ) d 1 d 2 d 2 ( d 1 x + d 2 ) d 1 + d 2 x B ( d 1 2 , d 2 2 ) = 1 B ( d 1 2 , d 2 2 ) ( d 1 d 2 ) d 1 2 x d 1 2 - 1 ( 1 + d 1 d 2 x ) - d 1 + d 2 2 \begin{aligned}\displaystyle f(x;d_{1},d_{2})&\displaystyle=\frac{\sqrt{\frac{% (d_{1}\,x)^{d_{1}}\,\,d_{2}^{d_{2}}}{(d_{1}\,x+d_{2})^{d_{1}+d_{2}}}}}{x\,% \mathrm{B}\!\left(\frac{d_{1}}{2},\frac{d_{2}}{2}\right)}\\ &\displaystyle=\frac{1}{\mathrm{B}\!\left(\frac{d_{1}}{2},\frac{d_{2}}{2}% \right)}\left(\frac{d_{1}}{d_{2}}\right)^{\frac{d_{1}}{2}}x^{\frac{d_{1}}{2}-1% }\left(1+\frac{d_{1}}{d_{2}}\,x\right)^{-\frac{d_{1}+d_{2}}{2}}\end{aligned}
  7. B \mathrm{B}
  8. F ( x ; d 1 , d 2 ) = I d 1 x d 1 x + d 2 ( d 1 2 , d 2 2 ) , F(x;d_{1},d_{2})=I_{\frac{d_{1}x}{d_{1}x+d_{2}}}\left(\tfrac{d_{1}}{2},\tfrac{% d_{2}}{2}\right),
  9. γ 2 = 12 d 1 ( 5 d 2 - 22 ) ( d 1 + d 2 - 2 ) + ( d 2 - 4 ) ( d 2 - 2 ) 2 d 1 ( d 2 - 6 ) ( d 2 - 8 ) ( d 1 + d 2 - 2 ) \gamma_{2}=12\frac{d_{1}(5d_{2}-22)(d_{1}+d_{2}-2)+(d_{2}-4)(d_{2}-2)^{2}}{d_{% 1}(d_{2}-6)(d_{2}-8)(d_{1}+d_{2}-2)}
  10. μ X ( k ) = ( d 2 d 1 ) k Γ ( d 1 2 + k ) Γ ( d 1 2 ) Γ ( d 2 2 - k ) Γ ( d 2 2 ) \mu_{X}(k)=\left(\frac{d_{2}}{d_{1}}\right)^{k}\frac{\Gamma\left(\tfrac{d_{1}}% {2}+k\right)}{\Gamma\left(\tfrac{d_{1}}{2}\right)}\frac{\Gamma\left(\tfrac{d_{% 2}}{2}-k\right)}{\Gamma\left(\tfrac{d_{2}}{2}\right)}
  11. φ d 1 , d 2 F ( s ) = Γ ( d 1 + d 2 2 ) Γ ( d 2 2 ) U ( d 1 2 , 1 - d 2 2 , - d 2 d 1 ı s ) \varphi^{F}_{d_{1},d_{2}}(s)=\frac{\Gamma(\frac{d_{1}+d_{2}}{2})}{\Gamma(% \tfrac{d_{2}}{2})}U\!\left(\frac{d_{1}}{2},1-\frac{d_{2}}{2},-\frac{d_{2}}{d_{% 1}}\imath s\right)
  12. X = U 1 / d 1 U 2 / d 2 X=\frac{U_{1}/d_{1}}{U_{2}/d_{2}}
  13. X = s 1 2 σ 1 2 / s 2 2 σ 2 2 X=\frac{s_{1}^{2}}{\sigma_{1}^{2}}\;/\;\frac{s_{2}^{2}}{\sigma_{2}^{2}}
  14. { 2 x ( d 1 x + d 2 ) f ( x ) + ( 2 d 1 x + d 2 d 1 x - d 2 d 1 + 2 d 2 ) f ( x ) = 0 , f ( 1 ) = d 1 d 1 2 d 2 d 2 2 ( d 1 + d 2 ) 1 2 ( - d 1 - d 2 ) B ( d 1 2 , d 2 2 ) } \left\{\begin{array}[]{l}2x\left(d_{1}x+d_{2}\right)f^{\prime}(x)+\left(2d_{1}% x+d_{2}d_{1}x-d_{2}d_{1}+2d_{2}\right)f(x)=0,\\ f(1)=\frac{d_{1}^{\frac{d_{1}}{2}}d_{2}^{\frac{d_{2}}{2}}\left(d_{1}+d_{2}% \right){}^{\frac{1}{2}\left(-d_{1}-d_{2}\right)}}{B\left(\frac{d_{1}}{2},\frac% {d_{2}}{2}\right)}\end{array}\right\}
  15. X χ d 1 2 X\sim\chi^{2}_{d_{1}}
  16. Y χ d 2 2 Y\sim\chi^{2}_{d_{2}}
  17. X / d 1 Y / d 2 F ( d 1 , d 2 ) \frac{X/d_{1}}{Y/d_{2}}\sim\mathrm{F}(d_{1},d_{2})
  18. X Beta ( d 1 / 2 , d 2 / 2 ) X\sim\operatorname{Beta}(d_{1}/2,d_{2}/2)
  19. d 2 X d 1 ( 1 - X ) F ( d 1 , d 2 ) \frac{d_{2}X}{d_{1}(1-X)}\sim\operatorname{F}(d_{1},d_{2})
  20. d 1 X / d 2 1 + d 1 X / d 2 Beta ( d 1 / 2 , d 2 / 2 ) \frac{d_{1}X/d_{2}}{1+d_{1}X/d_{2}}\sim\operatorname{Beta}(d_{1}/2,d_{2}/2)
  21. Y = lim d 2 d 1 X Y=\lim_{d_{2}\to\infty}d_{1}X
  22. χ d 1 2 \chi^{2}_{d_{1}}
  23. d 2 d 1 ( d 1 + d 2 - 1 ) T 2 ( d 1 , d 1 + d 2 - 1 ) \frac{d_{2}}{d_{1}(d_{1}+d_{2}-1)}\operatorname{T}^{2}(d_{1},d_{1}+d_{2}-1)
  24. X 2 F ( 1 , n ) X^{2}\sim\operatorname{F}(1,n)
  25. X - 2 F ( n , 1 ) X^{-2}\sim\operatorname{F}(n,1)
  26. | X - μ | | Y - μ | F ( 2 , 2 ) \tfrac{|X-\mu|}{|Y-\mu|}\sim\operatorname{F}(2,2)
  27. log X 2 FisherZ ( n , m ) \tfrac{\log{X}}{2}\sim\operatorname{FisherZ}(n,m)
  28. λ 1 = λ 2 = 0 \lambda_{1}=\lambda_{2}=0
  29. Q X ( p ) \operatorname{Q}_{X}(p)
  30. Q Y ( 1 - p ) \operatorname{Q}_{Y}(1-p)
  31. Q X ( p ) = 1 Q Y ( 1 - p ) \operatorname{Q}_{X}(p)=\frac{1}{\operatorname{Q}_{Y}(1-p)}

F-test.html

  1. F = explained variance unexplained variance , F=\frac{\,\text{explained variance}}{\,\text{unexplained variance}},
  2. F = between-group variability within-group variability . F=\frac{\,\text{between-group variability}}{\,\text{within-group variability}}.
  3. i n i ( Y ¯ i - Y ¯ ) 2 / ( K - 1 ) \sum_{i}n_{i}(\bar{Y}_{i\cdot}-\bar{Y})^{2}/(K-1)
  4. Y ¯ i \bar{Y}_{i\cdot}
  5. Y ¯ \bar{Y}
  6. i j ( Y i j - Y ¯ i ) 2 / ( N - K ) , \sum_{ij}(Y_{ij}-\bar{Y}_{i\cdot})^{2}/(N-K),
  7. F = ( RSS 1 - RSS 2 p 2 - p 1 ) ( RSS 2 n - p 2 ) , F=\frac{\left(\frac{\,\text{RSS}_{1}-\,\text{RSS}_{2}}{p_{2}-p_{1}}\right)}{% \left(\frac{\,\text{RSS}_{2}}{n-p_{2}}\right)},
  8. Y ¯ 1 \displaystyle\overline{Y}_{1}
  9. Y ¯ = i Y ¯ i a = Y ¯ 1 + Y ¯ 2 + Y ¯ 3 a = 5 + 9 + 10 3 = 8 \overline{Y}=\frac{\sum_{i}\overline{Y}_{i}}{a}=\frac{\overline{Y}_{1}+% \overline{Y}_{2}+\overline{Y}_{3}}{a}=\frac{5+9+10}{3}=8
  10. S B \displaystyle S_{B}
  11. f b = 3 - 1 = 2 f_{b}=3-1=2
  12. M S B = 84 / 2 = 42 MS_{B}=84/2=42
  13. S W = \displaystyle S_{W}=
  14. f W = a ( n - 1 ) = 3 ( 6 - 1 ) = 15 f_{W}=a(n-1)=3(6-1)=15
  15. M S W = S W / f W = 68 / 15 4.5 MS_{W}=S_{W}/f_{W}=68/15\approx 4.5
  16. F = M S B M S W 42 / 4.5 9.3 F=\frac{MS_{B}}{MS_{W}}\approx 42/4.5\approx 9.3
  17. 4.5 / 6 + 4.5 / 6 = 1.2 \sqrt{4.5/6+4.5/6}=1.2

Faà_di_Bruno's_formula.html

  1. d n d x n f ( g ( x ) ) = n ! m 1 ! 1 ! m 1 m 2 ! 2 ! m 2 m n ! n ! m n f ( m 1 + + m n ) ( g ( x ) ) j = 1 n ( g ( j ) ( x ) ) m j , {d^{n}\over dx^{n}}f(g(x))=\sum\frac{n!}{m_{1}!\,1!^{m_{1}}\,m_{2}!\,2!^{m_{2}% }\,\cdots\,m_{n}!\,n!^{m_{n}}}\cdot f^{(m_{1}+\cdots+m_{n})}(g(x))\cdot\prod_{% j=1}^{n}\left(g^{(j)}(x)\right)^{m_{j}},
  2. 1 m 1 + 2 m 2 + 3 m 3 + + n m n = n . 1\cdot m_{1}+2\cdot m_{2}+3\cdot m_{3}+\cdots+n\cdot m_{n}=n.\,
  3. d n d x n f ( g ( x ) ) = n ! m 1 ! m 2 ! m n ! f ( m 1 + + m n ) ( g ( x ) ) j = 1 n ( g ( j ) ( x ) j ! ) m j . {d^{n}\over dx^{n}}f(g(x))=\sum\frac{n!}{m_{1}!\,m_{2}!\,\cdots\,m_{n}!}\cdot f% ^{(m_{1}+\cdots+m_{n})}(g(x))\cdot\prod_{j=1}^{n}\left(\frac{g^{(j)}(x)}{j!}% \right)^{m_{j}}.
  4. d n d x n f ( g ( x ) ) = k = 1 n f ( k ) ( g ( x ) ) B n , k ( g ( x ) , g ′′ ( x ) , , g ( n - k + 1 ) ( x ) ) . {d^{n}\over dx^{n}}f(g(x))=\sum_{k=1}^{n}f^{(k)}(g(x))\cdot B_{n,k}\left(g^{% \prime}(x),g^{\prime\prime}(x),\dots,g^{(n-k+1)}(x)\right).
  5. d n d x n f ( g ( x ) ) = ( f g ) ( n ) ( x ) = π Π f ( | π | ) ( g ( x ) ) B π g ( | B | ) ( x ) {d^{n}\over dx^{n}}f(g(x))=(f\circ g)^{(n)}(x)=\sum_{\pi\in\Pi}f^{(\left|\pi% \right|)}(g(x))\cdot\prod_{B\in\pi}g^{(\left|B\right|)}(x)
  6. ( f g ) ′′′′ ( x ) \displaystyle(f\circ g)^{\prime\prime\prime\prime}(x)
  7. g ( x ) 4 1 + 1 + 1 + 1 f ′′′′ ( g ( x ) ) 1 g ′′ ( x ) g ( x ) 2 2 + 1 + 1 f ′′′ ( g ( x ) ) 6 g ′′ ( x ) 2 2 + 2 f ′′ ( g ( x ) ) 3 g ′′′ ( x ) g ( x ) 3 + 1 f ′′ ( g ( x ) ) 4 g ′′′′ ( x ) 4 f ( g ( x ) ) 1. \begin{aligned}\displaystyle g^{\prime}(x)^{4}&&\displaystyle\leftrightarrow&&% \displaystyle 1+1+1+1&&\displaystyle\leftrightarrow&&\displaystyle f^{\prime% \prime\prime\prime}(g(x))&&\displaystyle\leftrightarrow&&\displaystyle 1\\ \displaystyle g^{\prime\prime}(x)g^{\prime}(x)^{2}&&\displaystyle% \leftrightarrow&&\displaystyle 2+1+1&&\displaystyle\leftrightarrow&&% \displaystyle f^{\prime\prime\prime}(g(x))&&\displaystyle\leftrightarrow&&% \displaystyle 6\\ \displaystyle g^{\prime\prime}(x)^{2}&&\displaystyle\leftrightarrow&&% \displaystyle 2+2&&\displaystyle\leftrightarrow&&\displaystyle f^{\prime\prime% }(g(x))&&\displaystyle\leftrightarrow&&\displaystyle 3\\ \displaystyle g^{\prime\prime\prime}(x)g^{\prime}(x)&&\displaystyle% \leftrightarrow&&\displaystyle 3+1&&\displaystyle\leftrightarrow&&% \displaystyle f^{\prime\prime}(g(x))&&\displaystyle\leftrightarrow&&% \displaystyle 4\\ \displaystyle g^{\prime\prime\prime\prime}(x)&&\displaystyle\leftrightarrow&&% \displaystyle 4&&\displaystyle\leftrightarrow&&\displaystyle f^{\prime}(g(x))&% &\displaystyle\leftrightarrow&&\displaystyle 1.\end{aligned}
  8. g ′′ ( x ) g ( x ) 2 \scriptstyle g^{\prime\prime}(x)g^{\prime}(x)^{2}\;
  9. f ′′′ ( g ( x ) ) \scriptstyle f^{\prime\prime\prime}(g(x))\;
  10. g ′′ ( x ) 2 \scriptstyle g^{\prime\prime}(x)^{2}\;
  11. f ′′ ( g ( x ) ) \scriptstyle f^{\prime\prime}(g(x))\,\!
  12. 1 2 ( 4 2 ) = 3 \tfrac{1}{2}{\textstyle\left({{4}\atop{2}}\right)}=3
  13. D 1 ( f g ) 1 ! \displaystyle\frac{D^{1}(f\circ{}g)}{1!}
  14. n = 1 + + 1 m 1 + 2 + + 2 m 2 + 3 + + 3 m 3 + \displaystyle n=\underbrace{1+\cdots+1}_{m_{1}}\,+\,\underbrace{2+\cdots+2}_{m% _{2}}\,+\,\underbrace{3+\cdots+3}_{m_{3}}+\cdots
  15. n ! m 1 ! m 2 ! m 3 ! 1 ! m 1 2 ! m 2 3 ! m 3 . \frac{n!}{m_{1}!\,m_{2}!\,m_{3}!\,\cdots 1!^{m_{1}}\,2!^{m_{2}}\,3!^{m_{3}}\,% \cdots}.
  16. n x 1 x n f ( y ) = π Π f ( | π | ) ( y ) B π | B | y j B x j {\partial^{n}\over\partial x_{1}\cdots\partial x_{n}}f(y)=\sum_{\pi\in\Pi}f^{(% \left|\pi\right|)}(y)\cdot\prod_{B\in\pi}{\partial^{\left|B\right|}y\over\prod% _{j\in B}\partial x_{j}}
  17. 3 x 1 x 2 x 3 f ( y ) \displaystyle{\partial^{3}\over\partial x_{1}\,\partial x_{2}\,\partial x_{3}}% f(y)
  18. f ( x ) = n = 0 a n x n f(x)=\sum_{n=0}^{\infty}{a_{n}}x^{n}
  19. g ( x ) = n = 0 b n x n g(x)=\sum_{n=0}^{\infty}{b_{n}}x^{n}
  20. b 0 = 0 b_{0}=0
  21. f g f\circ g
  22. f ( g ( x ) ) = n = 0 c n x n , f(g(x))=\sum_{n=0}^{\infty}{c_{n}}x^{n},
  23. c n = 𝐢 𝒞 n a k b i 1 b i 2 b i k , c_{n}=\sum_{\mathbf{i}\in\mathcal{C}_{n}}a_{k}b_{i_{1}}b_{i_{2}}\cdots b_{i_{k% }},
  24. 𝒞 n = { ( i 1 , i 2 , , i k ) : 1 k n , i 1 + i 2 + + i k = n } \mathcal{C}_{n}=\{(i_{1},i_{2},\dots,i_{k})\,:\ 1\leq k\leq n,\ i_{1}+i_{2}+% \cdots+i_{k}=n\}
  25. c n = k = 1 n a k π 𝒫 n , k ( k π 1 , π 2 , , π n ) b 1 π 1 b 2 π 2 b n π n , c_{n}=\sum_{k=1}^{n}a_{k}\sum_{\mathbf{\pi}\in\mathcal{P}_{n,k}}{\left({{k}% \atop{\pi_{1},\pi_{2},...,\pi_{n}}}\right)}b_{1}^{\pi_{1}}b_{2}^{\pi_{2}}% \cdots b_{n}^{\pi_{n}},
  26. 𝒫 n , k = { ( π 1 , π 2 , , π n ) : π 1 + π 2 + + π n = k , π 1 1 + π 2 2 + + π n n = n } \mathcal{P}_{n,k}=\{(\pi_{1},\pi_{2},\dots,\pi_{n})\,:\ \pi_{1}+\pi_{2}+\cdots% +\pi_{n}=k,\ \pi_{1}\cdot 1+\pi_{2}\cdot 2+\cdots+\pi_{n}\cdot n=n\}
  27. ( b 1 x + b 2 x 2 + ) k (b_{1}x+b_{2}x^{2}+\cdots)^{k}
  28. f ( x ) = n a n n ! x n , f(x)=\sum_{n}{a_{n}\over n!}x^{n},
  29. f ( n ) ( 0 ) = a n . f^{(n)}(0)=a_{n}.\;
  30. g ( x ) = n = 0 b n n ! x n g(x)=\sum_{n=0}^{\infty}{b_{n}\over n!}x^{n}
  31. f ( x ) = n = 1 a n n ! x n f(x)=\sum_{n=1}^{\infty}{a_{n}\over n!}x^{n}
  32. g ( f ( x ) ) = h ( x ) = n = 0 c n n ! x n , g(f(x))=h(x)=\sum_{n=0}^{\infty}{c_{n}\over n!}x^{n},
  33. c n = π = { B 1 , , B k } a | B 1 | a | B k | b k c_{n}=\sum_{\pi=\left\{\,B_{1},\,\dots,\,B_{k}\,\right\}}a_{\left|B_{1}\right|% }\cdots a_{\left|B_{k}\right|}b_{k}
  34. g ( f ( x ) ) = b 0 + n = 1 k = 1 n b k B n , k ( a 1 , , a n - k + 1 ) n ! x n , g(f(x))=b_{0}+\sum_{n=1}^{\infty}{\sum_{k=1}^{n}b_{k}B_{n,k}(a_{1},\dots,a_{n-% k+1})\over n!}x^{n},

Factoring_(finance).html

  1. C B = i * n C F ( 2 * r ) CB=\sqrt{\frac{i*nCF}{(2*r)}}
  2. C B CB
  3. n C F nCF
  4. i i
  5. r r

FairTax.html

  1. .23 / .77 = .30 .23/.77=.30

Fano_plane.html

  1. 1 168 ( n 7 + 21 n 5 + 98 n 3 + 48 n ) . {1\over 168}\left(n^{7}+21n^{5}+98n^{3}+48n\right).

Faraday_effect.html

  1. 𝐁 ( ω ) = | μ 1 - i μ 2 0 i μ 2 μ 1 0 0 0 μ z | 𝐇 ( ω ) \mathbf{B}(\omega)=\begin{vmatrix}\mu_{1}&-i\mu_{2}&0\\ i\mu_{2}&\mu_{1}&0\\ 0&0&\mu_{z}\\ \end{vmatrix}\mathbf{H}(\omega)
  2. β = 𝒱 B d \beta=\mathcal{V}Bd
  3. 𝒱 \scriptstyle\mathcal{V}
  4. β = RM λ 2 \beta=\mathrm{RM}\lambda^{2}
  5. RM = e 3 2 π m 2 c 4 0 d n e ( s ) B | | ( s ) d s \mathrm{RM}=\frac{e^{3}}{2\pi m^{2}c^{4}}\int_{0}^{d}n_{e}(s)B_{||}(s)\;% \mathrm{d}s
  6. RM = e 3 8 π 2 ε 0 m 2 c 3 0 d n e ( s ) B | | ( s ) d s ( 2.62 × 10 - 13 T - 1 ) 0 d n e ( s ) B | | ( s ) d s \mathrm{RM}=\frac{e^{3}}{8\pi^{2}\varepsilon_{0}m^{2}c^{3}}\int_{0}^{d}n_{e}(s% )B_{||}(s)\;\mathrm{d}s\approx(2.62\times 10^{-13}\,T^{-1})\,\int_{0}^{d}n_{e}% (s)B_{||}(s)\;\mathrm{d}s
  7. ϵ 0 \scriptstyle\epsilon_{0}
  8. λ - 2 \lambda^{-2}

Faraday_rotator.html

  1. β = V B d \beta=VBd\!
  2. β \beta
  3. B B
  4. d d
  5. V V

Farey_sequence.html

  1. 0 1 \frac{0}{1}
  2. 1 1 \frac{1}{1}
  3. 0 1 \frac{0}{1}
  4. 1 2 \frac{1}{2}
  5. 1 1 \frac{1}{1}
  6. 0 1 \frac{0}{1}
  7. 1 3 \frac{1}{3}
  8. 1 2 \frac{1}{2}
  9. 2 3 \frac{2}{3}
  10. 1 1 \frac{1}{1}
  11. 0 1 \frac{0}{1}
  12. 1 4 \frac{1}{4}
  13. 1 3 \frac{1}{3}
  14. 1 2 \frac{1}{2}
  15. 2 3 \frac{2}{3}
  16. 3 4 \frac{3}{4}
  17. 1 1 \frac{1}{1}
  18. 0 1 \frac{0}{1}
  19. 1 5 \frac{1}{5}
  20. 1 4 \frac{1}{4}
  21. 1 3 \frac{1}{3}
  22. 2 5 \frac{2}{5}
  23. 1 2 \frac{1}{2}
  24. 3 5 \frac{3}{5}
  25. 2 3 \frac{2}{3}
  26. 3 4 \frac{3}{4}
  27. 4 5 \frac{4}{5}
  28. 1 1 \frac{1}{1}
  29. 0 1 \frac{0}{1}
  30. 1 6 \frac{1}{6}
  31. 1 5 \frac{1}{5}
  32. 1 4 \frac{1}{4}
  33. 1 3 \frac{1}{3}
  34. 2 5 \frac{2}{5}
  35. 1 2 \frac{1}{2}
  36. 3 5 \frac{3}{5}
  37. 2 3 \frac{2}{3}
  38. 3 4 \frac{3}{4}
  39. 4 5 \frac{4}{5}
  40. 5 6 \frac{5}{6}
  41. 1 1 \frac{1}{1}
  42. 0 1 \frac{0}{1}
  43. 1 7 \frac{1}{7}
  44. 1 6 \frac{1}{6}
  45. 1 5 \frac{1}{5}
  46. 1 4 \frac{1}{4}
  47. 2 7 \frac{2}{7}
  48. 1 3 \frac{1}{3}
  49. 2 5 \frac{2}{5}
  50. 3 7 \frac{3}{7}
  51. 1 2 \frac{1}{2}
  52. 4 7 \frac{4}{7}
  53. 3 5 \frac{3}{5}
  54. 2 3 \frac{2}{3}
  55. 5 7 \frac{5}{7}
  56. 3 4 \frac{3}{4}
  57. 4 5 \frac{4}{5}
  58. 5 6 \frac{5}{6}
  59. 6 7 \frac{6}{7}
  60. 1 1 \frac{1}{1}
  61. 0 1 \frac{0}{1}
  62. 1 8 \frac{1}{8}
  63. 1 7 \frac{1}{7}
  64. 1 6 \frac{1}{6}
  65. 1 5 \frac{1}{5}
  66. 1 4 \frac{1}{4}
  67. 2 7 \frac{2}{7}
  68. 1 3 \frac{1}{3}
  69. 3 8 \frac{3}{8}
  70. 2 5 \frac{2}{5}
  71. 3 7 \frac{3}{7}
  72. 1 2 \frac{1}{2}
  73. 4 7 \frac{4}{7}
  74. 3 5 \frac{3}{5}
  75. 5 8 \frac{5}{8}
  76. 2 3 \frac{2}{3}
  77. 5 7 \frac{5}{7}
  78. 3 4 \frac{3}{4}
  79. 4 5 \frac{4}{5}
  80. 5 6 \frac{5}{6}
  81. 6 7 \frac{6}{7}
  82. 7 8 \frac{7}{8}
  83. 1 1 \frac{1}{1}
  84. 0 1 \frac{0}{1}
  85. 1 1 \frac{1}{1}
  86. 0 1 \frac{0}{1}
  87. 1 2 \frac{1}{2}
  88. 1 1 \frac{1}{1}
  89. 0 1 \frac{0}{1}
  90. 1 3 \frac{1}{3}
  91. 1 2 \frac{1}{2}
  92. 2 3 \frac{2}{3}
  93. 1 1 \frac{1}{1}
  94. 0 1 \frac{0}{1}
  95. 1 4 \frac{1}{4}
  96. 1 3 \frac{1}{3}
  97. 1 2 \frac{1}{2}
  98. 2 3 \frac{2}{3}
  99. 3 4 \frac{3}{4}
  100. 1 1 \frac{1}{1}
  101. 0 1 \frac{0}{1}
  102. 1 5 \frac{1}{5}
  103. 1 4 \frac{1}{4}
  104. 1 3 \frac{1}{3}
  105. 2 5 \frac{2}{5}
  106. 1 2 \frac{1}{2}
  107. 3 5 \frac{3}{5}
  108. 2 3 \frac{2}{3}
  109. 3 4 \frac{3}{4}
  110. 4 5 \frac{4}{5}
  111. 1 1 \frac{1}{1}
  112. 0 1 \frac{0}{1}
  113. 1 6 \frac{1}{6}
  114. 1 5 \frac{1}{5}
  115. 1 4 \frac{1}{4}
  116. 1 3 \frac{1}{3}
  117. 2 5 \frac{2}{5}
  118. 1 2 \frac{1}{2}
  119. 3 5 \frac{3}{5}
  120. 2 3 \frac{2}{3}
  121. 3 4 \frac{3}{4}
  122. 4 5 \frac{4}{5}
  123. 5 6 \frac{5}{6}
  124. 1 1 \frac{1}{1}
  125. 0 1 \frac{0}{1}
  126. 1 7 \frac{1}{7}
  127. 1 6 \frac{1}{6}
  128. 1 5 \frac{1}{5}
  129. 1 4 \frac{1}{4}
  130. 2 7 \frac{2}{7}
  131. 1 3 \frac{1}{3}
  132. 2 5 \frac{2}{5}
  133. 3 7 \frac{3}{7}
  134. 1 2 \frac{1}{2}
  135. 4 7 \frac{4}{7}
  136. 3 5 \frac{3}{5}
  137. 2 3 \frac{2}{3}
  138. 5 7 \frac{5}{7}
  139. 3 4 \frac{3}{4}
  140. 4 5 \frac{4}{5}
  141. 5 6 \frac{5}{6}
  142. 6 7 \frac{6}{7}
  143. 1 1 \frac{1}{1}
  144. 0 1 \frac{0}{1}
  145. 1 8 \frac{1}{8}
  146. 1 7 \frac{1}{7}
  147. 1 6 \frac{1}{6}
  148. 1 5 \frac{1}{5}
  149. 1 4 \frac{1}{4}
  150. 2 7 \frac{2}{7}
  151. 1 3 \frac{1}{3}
  152. 3 8 \frac{3}{8}
  153. 2 5 \frac{2}{5}
  154. 3 7 \frac{3}{7}
  155. 1 2 \frac{1}{2}
  156. 4 7 \frac{4}{7}
  157. 3 5 \frac{3}{5}
  158. 5 8 \frac{5}{8}
  159. 2 3 \frac{2}{3}
  160. 5 7 \frac{5}{7}
  161. 3 4 \frac{3}{4}
  162. 4 5 \frac{4}{5}
  163. 5 6 \frac{5}{6}
  164. 6 7 \frac{6}{7}
  165. 7 8 \frac{7}{8}
  166. 1 1 \frac{1}{1}
  167. 1 6 \frac{1}{6}
  168. 5 6 \frac{5}{6}
  169. 1 2 \frac{1}{2}
  170. φ ( n ) \varphi(n)
  171. | F n | = | F n - 1 | + φ ( n ) . |F_{n}|=|F_{n-1}|+\varphi(n).
  172. | F n | = 1 + m = 1 n φ ( m ) . |F_{n}|=1+\sum_{m=1}^{n}\varphi(m).
  173. | F n | = 1 2 ( 3 + d = 1 n μ ( d ) n d 2 ) , |F_{n}|=\frac{1}{2}\left(3+\sum_{d=1}^{n}\mu(d)\left\lfloor\tfrac{n}{d}\right% \rfloor^{2}\right),
  174. | F n | = 1 2 ( n + 3 ) n - d = 2 n | F n / d | , |F_{n}|=\frac{1}{2}(n+3)n-\sum_{d=2}^{n}|F_{\lfloor n/d\rfloor}|,
  175. n d \lfloor\tfrac{n}{d}\rfloor
  176. | F n | 3 n 2 π 2 . |F_{n}|\sim\frac{3n^{2}}{\pi^{2}}.
  177. I n ( a k , n ) = k I_{n}(a_{k,n})=k
  178. a k , n a_{k,n}
  179. F n = { a k , n : k = 0 , 1 , , m n } F_{n}=\{a_{k,n}:k=0,1,\ldots,m_{n}\}
  180. a k , n a_{k,n}
  181. I n ( 0 / 1 ) = 0 , I_{n}(0/1)=0,
  182. I n ( 1 / n ) = 1 , I_{n}(1/n)=1,
  183. I n ( 1 / 2 ) = ( | F n | - 1 ) / 2 , I_{n}(1/2)=(|F_{n}|-1)/2,
  184. I n ( 1 / 1 ) = | F n | - 1 , I_{n}(1/1)=|F_{n}|-1,
  185. I n ( h / k ) = | F n | - 1 - I n ( ( k - h ) / k ) . I_{n}(h/k)=|F_{n}|-1-I_{n}((k-h)/k).
  186. a b \frac{a}{b}
  187. c d \frac{c}{d}
  188. a b \frac{a}{b}
  189. c d \frac{c}{d}
  190. a b \frac{a}{b}
  191. 1 b d \frac{1}{bd}
  192. c d - a b = b c - a d b d , \frac{c}{d}-\frac{a}{b}=\frac{bc-ad}{bd},
  193. 1 3 \frac{1}{3}
  194. 2 5 \frac{2}{5}
  195. 1 15 \frac{1}{15}
  196. c d \frac{c}{d}
  197. p q \frac{p}{q}
  198. a b \frac{a}{b}
  199. c d \frac{c}{d}
  200. a b \frac{a}{b}
  201. p q \frac{p}{q}
  202. a b \frac{a}{b}
  203. c d \frac{c}{d}
  204. p q = a + c b + d . \frac{p}{q}=\frac{a+c}{b+d}.
  205. p q \frac{p}{q}
  206. a + c a\frac{+}{c}
  207. a b \frac{a}{b}
  208. c d \frac{c}{d}
  209. a + c b + d , \frac{a+c}{b+d},
  210. 1 3 \frac{1}{3}
  211. 2 5 \frac{2}{5}
  212. 3 8 \frac{3}{8}
  213. 0 1 \frac{0}{1}
  214. 1 1 \frac{1}{1}
  215. p q \frac{p}{q}
  216. p q \frac{p}{q}
  217. 3 8 \frac{3}{8}
  218. 2 5 \frac{2}{5}
  219. 1 3 \frac{1}{3}
  220. { a k , n : k = 0 , 1 , , m n } \{a_{k,n}:k=0,1,\ldots,m_{n}\}
  221. d k , n = a k , n - k / m n d_{k,n}=a_{k,n}-k/m_{n}
  222. d k , n d_{k,n}
  223. k = 1 m n d k , n 2 = 𝒪 ( n r ) r > - 1 \sum_{k=1}^{m_{n}}d_{k,n}^{2}=\mathcal{O}(n^{r})\quad\forall r>-1
  224. k = 1 m n | d k , n | = 𝒪 ( n r ) r > 1 / 2 \sum_{k=1}^{m_{n}}|d_{k,n}|=\mathcal{O}(n^{r})\quad\forall r>1/2
  225. a b \frac{a}{b}
  226. c d \frac{c}{d}
  227. p q \frac{p}{q}
  228. c d \frac{c}{d}
  229. a + p a\frac{+}{p}
  230. c d \frac{c}{d}
  231. p ( k ) q ( k ) - c d = c b - d a d ( k d - b ) \frac{p(k)}{q(k)}-\frac{c}{d}=\frac{cb-da}{d(kd-b)}
  232. p q \frac{p}{q}
  233. c d \frac{c}{d}
  234. n + b n\frac{+}{b}
  235. p = n + b d c - a p=\left\lfloor\frac{n+b}{d}\right\rfloor c-a
  236. q = n + b d d - b q=\left\lfloor\frac{n+b}{d}\right\rfloor d-b

Fashionable_Nonsense.html

  1. ( L ) - 1 \scriptstyle(L)\sqrt{-1}

Fatigue_(material).html

  1. i = 1 k n i N i = C \sum_{i=1}^{k}\frac{n_{i}}{N_{i}}=C
  2. d a d N = C ( Δ K ) m \frac{\mathrm{d}a}{\mathrm{d}N}=C(\Delta K)^{m}
  3. Δ ε p 2 = ε f ( 2 N ) c \frac{\Delta\varepsilon_{p}}{2}=\varepsilon_{f}^{\prime}(2N)^{c}

Federation_Square.html

  1. 5 \sqrt{5}

Feistel_cipher.html

  1. F {\rm F}
  2. K 0 , K 1 , , K n K_{0},K_{1},\ldots,K_{n}
  3. 0 , 1 , , n 0,1,\ldots,n
  4. L 0 L_{0}
  5. R 0 R_{0}
  6. i = 0 , 1 , , n i=0,1,\dots,n
  7. L i + 1 = R i L_{i+1}=R_{i}\,
  8. R i + 1 = L i F ( R i , K i ) R_{i+1}=L_{i}\oplus{\rm F}(R_{i},K_{i})
  9. ( R n + 1 , L n + 1 ) (R_{n+1},L_{n+1})
  10. ( R n + 1 , L n + 1 ) (R_{n+1},L_{n+1})
  11. i = n , n - 1 , , 0 i=n,n-1,\ldots,0
  12. R i = L i + 1 R_{i}=L_{i+1}\,
  13. L i = R i + 1 F ( L i + 1 , K i ) L_{i}=R_{i+1}\oplus{\rm F}(L_{i+1},K_{i})
  14. ( L 0 , R 0 ) (L_{0},R_{0})
  15. F {\rm F}
  16. L 0 L_{0}
  17. R 0 R_{0}

Ferdinand_Georg_Frobenius.html

  1. x G - H x\in G-H
  2. N = G - x G - H H x N=G\,-\!\!\bigcup_{x\in G-H}\!\!H^{x}
  3. P S L ( 2 , p ) PSL(2,p)

Fermi_problem.html

  1. n \sqrt{n}
  2. σ \sigma
  3. σ n \sigma^{\sqrt{n}}
  4. n \sqrt{n}
  5. 9 = 3 \sqrt{9}=3
  6. 2 3 = 8 2^{3}=8
  7. 2 9 = 512 2^{9}=512

Ferroelectric_capacitor.html

  1. V = E f d + E e ( 2 λ ) V=E_{f}d+E_{e}\left(2\lambda\right)
  2. d d
  3. E f = V + 8 π P s a d + ϵ f ( 2 a ) E_{f}=\frac{V+8\pi P_{s}a}{d+\epsilon_{f}\left(2a\right)}
  4. E e = ϵ f ϵ e E f - 4 π ϵ e P s E_{e}=\frac{\epsilon_{f}}{\epsilon_{e}}E_{f}-\frac{4\pi}{\epsilon_{e}}P_{s}
  5. P s P_{s}
  6. a = λ ϵ e a=\frac{\lambda}{\epsilon_{e}}
  7. ϵ f \epsilon_{f}
  8. ϵ e \epsilon_{e}
  9. λ = 0 \lambda=0
  10. d a d\gg a
  11. V = E f d E f = V d V=E_{f}d\Rightarrow E_{f}=\frac{V}{d}

Feuerbach_point.html

  1. x x
  2. y y
  3. z z
  4. x + y + z = 2 max ( x , y , z ) , x+y+z=2\max(x,y,z),
  5. 1 - cos ( B - C ) : 1 - cos ( C - A ) : 1 - cos ( A - B ) . 1-\cos(B-C):1-\cos(C-A):1-\cos(A-B).
  6. 1 + cos ( B - C ) : 1 + cos ( C - A ) : 1 + cos ( A - B ) . 1+\cos(B-C):1+\cos(C-A):1+\cos(A-B).

Feynman–Kac_formula.html

  1. u t ( x , t ) + μ ( x , t ) u x ( x , t ) + 1 2 σ 2 ( x , t ) 2 u x 2 ( x , t ) - V ( x , t ) u ( x , t ) + f ( x , t ) = 0 , \frac{\partial u}{\partial t}(x,t)+\mu(x,t)\frac{\partial u}{\partial x}(x,t)+% \tfrac{1}{2}\sigma^{2}(x,t)\frac{\partial^{2}u}{\partial x^{2}}(x,t)-V(x,t)u(x% ,t)+f(x,t)=0,
  2. u ( x , T ) = ψ ( x ) , u(x,T)=\psi(x),
  3. u : × [ 0 , T ] u:\mathbb{R}\times[0,T]\to\mathbb{R}
  4. u ( x , t ) = E Q [ t T e - t r V ( X τ , τ ) d τ f ( X r , r ) d r + e - t T V ( X τ , τ ) d τ ψ ( X T ) | X t = x ] u(x,t)=E^{Q}\left[\int_{t}^{T}e^{-\int_{t}^{r}V(X_{\tau},\tau)\,d\tau}f(X_{r},% r)dr+e^{-\int_{t}^{T}V(X_{\tau},\tau)\,d\tau}\psi(X_{T})\Bigg|X_{t}=x\right]
  5. d X = μ ( X , t ) d t + σ ( X , t ) d W Q , dX=\mu(X,t)\,dt+\sigma(X,t)\,dW^{Q},
  6. Y ( s ) = e - t s V ( X τ , τ ) d τ u ( X s , s ) + t s e - t r V ( X τ , τ ) d τ f ( X r , r ) d r Y(s)=e^{-\int_{t}^{s}V(X_{\tau},\tau)\,d\tau}u(X_{s},s)+\int_{t}^{s}e^{-\int_{% t}^{r}V(X_{\tau},\tau)\,d\tau}f(X_{r},r)\,dr
  7. d Y = d ( e - t s V ( X τ , τ ) d τ ) u ( X s , s ) + e - t s V ( X τ , τ ) d τ d u ( X s , s ) + d ( e - t s V ( X τ , τ ) d τ ) d u ( X s , s ) + d ( t s e - t r V ( X τ , τ ) d τ f ( X r , r ) d r ) \begin{aligned}\displaystyle dY=&\displaystyle d\left(e^{-\int_{t}^{s}V(X_{% \tau},\tau)\,d\tau}\right)u(X_{s},s)+e^{-\int_{t}^{s}V(X_{\tau},\tau)\,d\tau}% \,du(X_{s},s)\\ &\displaystyle{}+d\left(e^{-\int_{t}^{s}V(X_{\tau},\tau)\,d\tau}\right)du(X_{s% },s)+d\left(\int_{t}^{s}e^{-\int_{t}^{r}V(X_{\tau},\tau)\,d\tau}f(X_{r},r)\,dr% \right)\end{aligned}
  8. d ( e - t s V ( X τ , τ ) d τ ) = - V ( X s , s ) e - t s V ( X τ , τ ) d τ d s , d\left(e^{-\int_{t}^{s}V(X_{\tau},\tau)\,d\tau}\right)=-V(X_{s},s)e^{-\int_{t}% ^{s}V(X_{\tau},\tau)\,d\tau}\,ds,
  9. O ( d t d u ) O(dt\,du)
  10. d ( t s e - t r V ( X τ , τ ) d τ f ( X r , r ) d r ) = e - t s V ( X τ , τ ) d τ f ( X s , s ) d s . d\left(\int_{t}^{s}e^{-\int_{t}^{r}V(X_{\tau},\tau)\,d\tau}f(X_{r},r)dr\right)% =e^{-\int_{t}^{s}V(X_{\tau},\tau)\,d\tau}f(X_{s},s)ds.
  11. d u ( X s , s ) du(X_{s},s)
  12. d Y = e - t s V ( X τ , τ ) d τ ( - V ( X s , s ) u ( X s , s ) + f ( X s , s ) + μ ( X s , s ) u X + u s + 1 2 σ 2 ( X s , s ) 2 u X 2 ) d s + e - t s V ( X τ , τ ) d τ σ ( X , s ) u X d W . \begin{aligned}\displaystyle dY=&\displaystyle e^{-\int_{t}^{s}V(X_{\tau},\tau% )\,d\tau}\,\left(-V(X_{s},s)u(X_{s},s)+f(X_{s},s)+\mu(X_{s},s)\frac{\partial u% }{\partial X}+\frac{\partial u}{\partial s}+\tfrac{1}{2}\sigma^{2}(X_{s},s)% \frac{\partial^{2}u}{\partial X^{2}}\right)\,ds\\ &\displaystyle{}+e^{-\int_{t}^{s}V(X_{\tau},\tau)\,d\tau}\sigma(X,s)\frac{% \partial u}{\partial X}\,dW.\end{aligned}
  13. d Y = e - t s V ( X τ , τ ) d τ σ ( X , s ) u X d W . dY=e^{-\int_{t}^{s}V(X_{\tau},\tau)\,d\tau}\sigma(X,s)\frac{\partial u}{% \partial X}\,dW.
  14. Y ( T ) - Y ( t ) = t T e - t s V ( X τ , τ ) d τ σ ( X , s ) u X d W . Y(T)-Y(t)=\int_{t}^{T}e^{-\int_{t}^{s}V(X_{\tau},\tau)\,d\tau}\sigma(X,s)\frac% {\partial u}{\partial X}\,dW.
  15. E [ Y ( T ) X t = x ] = E [ Y ( t ) X t = x ] = u ( x , t ) . E[Y(T)\mid X_{t}=x]=E[Y(t)\mid X_{t}=x]=u(x,t).
  16. E [ Y ( T ) X t = x ] = E [ e - t T V ( X τ , τ ) d τ u ( X T , T ) + t T e - t r V ( X τ , τ ) d τ f ( X r , r ) d r | X t = x ] E[Y(T)\mid X_{t}=x]=E\left[e^{-\int_{t}^{T}V(X_{\tau},\tau)\,d\tau}u(X_{T},T)+% \int_{t}^{T}e^{-\int_{t}^{r}V(X_{\tau},\tau)\,d\tau}f(X_{r},r)\,dr\,\Bigg|\,X_% {t}=x\right]
  17. u ( x , t ) = E [ e - t T V ( X τ , τ ) d τ ψ ( X T ) + t T e - t s V ( X τ , τ ) d τ f ( X s , s ) d s | X t = x ] u(x,t)=E\left[e^{-\int_{t}^{T}V(X_{\tau},\tau)\,d\tau}\psi(X_{T})+\int_{t}^{T}% e^{-\int_{t}^{s}V(X_{\tau},\tau)\,d\tau}f(X_{s},s)\,ds\,\Bigg|\,X_{t}=x\right]
  18. f ( x , t ) f(x,t)
  19. u : N × [ 0 , T ] u:\mathbb{R}^{N}\times[0,T]\to\mathbb{R}
  20. u t + i = 1 N μ i ( x , t ) u x i + 1 2 i = 1 N j = 1 N γ i j ( x , t ) 2 u x i x j - r ( x , t ) u = f ( x , t ) , \frac{\partial u}{\partial t}+\sum_{i=1}^{N}\mu_{i}(x,t)\frac{\partial u}{% \partial x_{i}}+\frac{1}{2}\sum_{i=1}^{N}\sum_{j=1}^{N}\gamma_{ij}(x,t)\frac{% \partial^{2}u}{\partial x_{i}x_{j}}-r(x,t)u=f(x,t),
  21. γ i j ( x , t ) = k = 1 N σ i k ( x , t ) σ j k ( x , t ) , \gamma_{ij}(x,t)=\sum_{k=1}^{N}\sigma_{ik}(x,t)\sigma_{jk}(x,t),
  22. e - 0 t V ( x ( τ ) ) d τ e^{-\int_{0}^{t}V(x(\tau))\,d\tau}
  23. u V ( x ) 0 uV(x)\geq 0
  24. E [ e - u 0 t V ( x ( τ ) ) d τ ] = - w ( x , t ) d x E\left[e^{-u\int_{0}^{t}V(x(\tau))\,d\tau}\right]=\int_{-\infty}^{\infty}w(x,t% )\,dx
  25. w t = 1 2 2 w x 2 - u V ( x ) w . \frac{\partial w}{\partial t}=\frac{1}{2}\frac{\partial^{2}w}{\partial x^{2}}-% uV(x)w.
  26. I = f ( x ( 0 ) ) e - u 0 t V ( x ( t ) ) d t g ( x ( t ) ) D x I=\int f(x(0))e^{-u\int_{0}^{t}V(x(t))\,dt}g(x(t))\,Dx
  27. I = w ( x , t ) g ( x ) d x I=\int w(x,t)g(x)\,dx
  28. w t = 1 2 2 w x 2 - u V ( x ) w \frac{\partial w}{\partial t}=\frac{1}{2}\frac{\partial^{2}w}{\partial x^{2}}-% uV(x)w

Fibonacci_polynomials.html

  1. F n ( x ) = { 0 , if n = 0 1 , if n = 1 x F n - 1 ( x ) + F n - 2 ( x ) , if n 2 F_{n}(x)=\begin{cases}0,&\mbox{if }~{}n=0\\ 1,&\mbox{if }~{}n=1\\ xF_{n-1}(x)+F_{n-2}(x),&\mbox{if }~{}n\geq 2\end{cases}
  2. F 0 ( x ) = 0 F_{0}(x)=0\,
  3. F 1 ( x ) = 1 F_{1}(x)=1\,
  4. F 2 ( x ) = x F_{2}(x)=x\,
  5. F 3 ( x ) = x 2 + 1 F_{3}(x)=x^{2}+1\,
  6. F 4 ( x ) = x 3 + 2 x F_{4}(x)=x^{3}+2x\,
  7. F 5 ( x ) = x 4 + 3 x 2 + 1 F_{5}(x)=x^{4}+3x^{2}+1\,
  8. F 6 ( x ) = x 5 + 4 x 3 + 3 x F_{6}(x)=x^{5}+4x^{3}+3x\,
  9. L n ( x ) = { 2 , if n = 0 x , if n = 1 x L n - 1 ( x ) + L n - 2 ( x ) , if n 2. L_{n}(x)=\begin{cases}2,&\mbox{if }~{}n=0\\ x,&\mbox{if }~{}n=1\\ xL_{n-1}(x)+L_{n-2}(x),&\mbox{if }~{}n\geq 2.\end{cases}
  10. L 0 ( x ) = 2 L_{0}(x)=2\,
  11. L 1 ( x ) = x L_{1}(x)=x\,
  12. L 2 ( x ) = x 2 + 2 L_{2}(x)=x^{2}+2\,
  13. L 3 ( x ) = x 3 + 3 x L_{3}(x)=x^{3}+3x\,
  14. L 4 ( x ) = x 4 + 4 x 2 + 2 L_{4}(x)=x^{4}+4x^{2}+2\,
  15. L 5 ( x ) = x 5 + 5 x 3 + 5 x L_{5}(x)=x^{5}+5x^{3}+5x\,
  16. L 6 ( x ) = x 6 + 6 x 4 + 9 x 2 + 2. L_{6}(x)=x^{6}+6x^{4}+9x^{2}+2.\,
  17. n = 0 F n ( x ) t n = t 1 - x t - t 2 \sum_{n=0}^{\infty}F_{n}(x)t^{n}=\frac{t}{1-xt-t^{2}}
  18. n = 0 L n ( x ) t n = 2 - x t 1 - x t - t 2 . \sum_{n=0}^{\infty}L_{n}(x)t^{n}=\frac{2-xt}{1-xt-t^{2}}.
  19. F n ( x ) = U n ( x , - 1 ) , F_{n}(x)=U_{n}(x,-1),\,
  20. L n ( x ) = V n ( x , - 1 ) . L_{n}(x)=V_{n}(x,-1).\,
  21. F - n ( x ) = ( - 1 ) n - 1 F n ( x ) , L - n ( x ) = ( - 1 ) n L n ( x ) . F_{-n}(x)=(-1)^{n-1}F_{n}(x),\,L_{-n}(x)=(-1)^{n}L_{n}(x).
  22. F m + n ( x ) = F m + 1 ( x ) F n ( x ) + F m ( x ) F n - 1 ( x ) F_{m+n}(x)=F_{m+1}(x)F_{n}(x)+F_{m}(x)F_{n-1}(x)\,
  23. L m + n ( x ) = L m ( x ) L n ( x ) - ( - 1 ) n L m - n ( x ) L_{m+n}(x)=L_{m}(x)L_{n}(x)-(-1)^{n}L_{m-n}(x)\,
  24. F n + 1 ( x ) F n - 1 ( x ) - F n ( x ) 2 = ( - 1 ) n F_{n+1}(x)F_{n-1}(x)-F_{n}(x)^{2}=(-1)^{n}\,
  25. F 2 n ( x ) = F n ( x ) L n ( x ) . F_{2n}(x)=F_{n}(x)L_{n}(x).\,
  26. F n ( x ) = α ( x ) n - β ( x ) n α ( x ) - β ( x ) , L n ( x ) = α ( x ) n + β ( x ) n , F_{n}(x)=\frac{\alpha(x)^{n}-\beta(x)^{n}}{\alpha(x)-\beta(x)},\,L_{n}(x)=% \alpha(x)^{n}+\beta(x)^{n},
  27. α ( x ) = x + x 2 + 4 2 , β ( x ) = x - x 2 + 4 2 \alpha(x)=\frac{x+\sqrt{x^{2}+4}}{2},\,\beta(x)=\frac{x-\sqrt{x^{2}+4}}{2}
  28. t 2 - x t - 1 = 0. t^{2}-xt-1=0.\,
  29. F n ( x ) = k = 0 n F ( n , k ) x k , F_{n}(x)=\sum_{k=0}^{n}F(n,k)x^{k},\,
  30. F ( n , k ) = ( n + k - 1 2 k ) F(n,k)={\left({{\tfrac{n+k-1}{2}}\atop{k}}\right)}

Fibration.html

  1. p : E B p:E→B
  2. E E
  3. b b
  4. B B
  5. B B
  6. B B
  7. F F
  8. B B
  9. E E
  10. F E B F→E→B
  11. F F
  12. E E
  13. B B
  14. B B
  15. B B
  16. B B
  17. S O ( 3 ) SO(3)
  18. n n
  19. n > 1 n>1
  20. S O ( n + 1 ) SO(n+1)
  21. n n
  22. F F
  23. i i
  24. F E F→E
  25. π n ( F ) π n ( E ) π n ( B ) π n - 1 ( F ) π 0 ( F ) π 0 ( E ) . \cdots\to\pi_{n}(F)\to\pi_{n}(E)\to\pi_{n}(B)\to\pi_{n-1}(F)\to\cdots\to\pi_{0% }(F)\to\pi_{0}(E).
  26. F F
  27. E E
  28. B B
  29. π < s u b > n ( F ) π n ( E ) π<sub>n(F)→π_{n}(E)

Field_norm.html

  1. m α : L L given by m α ( x ) = α x m_{\alpha}:L\to L\,\text{ given by }m_{\alpha}(x)=\alpha x
  2. 1 {}_{1}
  3. n {}_{n}
  4. N L / K ( α ) = ( j = 1 n σ j ( α ) ) [ L : K ( α ) ] \operatorname{N}_{L/K}(\alpha)=\left(\prod_{j=1}^{n}\sigma_{j}(\alpha)\right)^% {[L:K(\alpha)]}
  5. N L / K ( α ) = g Gal ( L / K ) g ( α ) \operatorname{N}_{L/K}(\alpha)=\prod_{g\in\operatorname{Gal}(L/K)}g(\alpha)
  6. \C \C
  7. \R \R
  8. \C \C
  9. K = \Q ( 2 ) K=\Q(\sqrt{2})
  10. K K
  11. \Q \Q
  12. 2 \sqrt{2}
  13. - 2 -\sqrt{2}
  14. 1 + 2 1+\sqrt{2}
  15. ( 1 + 2 ) ( 1 - 2 ) = - 1. (1+\sqrt{2})(1-\sqrt{2})=-1.
  16. \Q \Q
  17. \Q ( 2 ) \Q(\sqrt{2})
  18. { 1 , 2 } \{1,\sqrt{2}\}
  19. 1 + 2 1+\sqrt{2}
  20. 1 + 2 1+\sqrt{2}
  21. 2 \sqrt{2}
  22. 2 + 2 2+\sqrt{2}
  23. 1 + 2 1+\sqrt{2}
  24. ( 1 , 0 ) T (1,0)^{\mathrm{T}}
  25. ( 1 , 1 ) T (1,1)^{\mathrm{T}}
  26. ( 0 , 1 ) T (0,1)^{\mathrm{T}}
  27. 2 \sqrt{2}
  28. ( 2 , 1 ) T (2,1)^{\mathrm{T}}
  29. [ 1 2 1 1 ] . \begin{bmatrix}1&2\\ 1&1\end{bmatrix}.
  30. L / K {}_{L/K}
  31. N L / K ( α β ) = N L / K ( α ) N L / K ( β ) for all α , β L * \operatorname{N}_{L/K}(\alpha\beta)=\operatorname{N}_{L/K}(\alpha)% \operatorname{N}_{L/K}(\beta)\,\text{ for all }\alpha,\beta\in L^{*}
  32. N L / K ( a α ) = a [ L : K ] N L / K ( α ) for all α L \operatorname{N}_{L/K}(a\alpha)=a^{[L:K]}\operatorname{N}_{L/K}(\alpha)\,\text% { for all }\alpha\in L
  33. N L / K ( a ) = a [ L : K ] . \operatorname{N}_{L/K}(a)=a^{[L:K]}.
  34. N M / K = N L / K N M / L \operatorname{N}_{M/K}=\operatorname{N}_{L/K}\circ\operatorname{N}_{M/L}
  35. N L / K ( α ) = α α q α q n - 1 = α ( q n - 1 ) / ( q - 1 ) \operatorname{N}_{L/K}(\alpha)=\alpha\bullet\alpha^{q}\bullet\cdots\bullet% \alpha^{q^{n-1}}=\alpha^{(q^{n}-1)/(q-1)}
  36. N L / K ( α q ) = N L / K ( α ) for all α L \operatorname{N}_{L/K}(\alpha^{q})=\operatorname{N}_{L/K}(\alpha)\,\text{ for % all }\alpha\in L
  37. for any a K , we have N L / K ( a ) = a n \,\text{for any }a\in K,\,\text{ we have }\operatorname{N}_{L/K}(a)=a^{n}
  38. O K / I O_{K}/I

Field_trace.html

  1. m α : L L given by m α ( x ) = α x m_{\alpha}:L\to L\,\text{ given by }m_{\alpha}(x)=\alpha x
  2. 1 {}_{1}
  3. n {}_{n}
  4. Tr L / K ( α ) = [ L : K ( α ) ] j = 1 n σ j ( α ) \operatorname{Tr}_{L/K}(\alpha)=[L:K(\alpha)]\sum_{j=1}^{n}\sigma_{j}(\alpha)
  5. Tr L / K ( α ) = g Gal ( L / K ) g ( α ) \operatorname{Tr}_{L/K}(\alpha)=\sum_{g\in\operatorname{Gal}(L/K)}g(\alpha)
  6. L = ( d ) L=\mathbb{Q}(\sqrt{d})
  7. \mathbb{Q}
  8. L / is { 1 , d } . L/\mathbb{Q}\,\text{ is }\{1,\sqrt{d}\}.
  9. α = a + b d \alpha=a+b\sqrt{d}
  10. m α m_{\alpha}
  11. [ a b d b a ] \left[\begin{matrix}a&bd\\ b&a\end{matrix}\right]
  12. Tr L / ( α ) = 2 a \operatorname{Tr}_{L/\mathbb{Q}}(\alpha)=2a
  13. Tr L / K ( α a + β b ) = α Tr L / K ( a ) + β Tr L / K ( b ) for all α , β K \operatorname{Tr}_{L/K}(\alpha a+\beta b)=\alpha\operatorname{Tr}_{L/K}(a)+% \beta\operatorname{Tr}_{L/K}(b)\,\text{ for all }\alpha,\beta\in K
  14. Tr L / K ( α ) = [ L : K ] α . \operatorname{Tr}_{L/K}(\alpha)=[L:K]\alpha.
  15. Tr M / K = Tr L / K Tr M / L \operatorname{Tr}_{M/K}=\operatorname{Tr}_{L/K}\circ\operatorname{Tr}_{M/L}
  16. Tr L / K ( α ) = α + α q + + α q n - 1 \operatorname{Tr}_{L/K}(\alpha)=\alpha+\alpha^{q}+\cdots+\alpha^{q^{n-1}}
  17. Tr L / K ( a q ) = Tr L / K ( a ) for a L \operatorname{Tr}_{L/K}(a^{q})=\operatorname{Tr}_{L/K}(a)\,\text{ for }a\in L
  18. for any α K , we have | { b L : Tr L / K ( b ) = α } | = q n - 1 \,\text{for any }\alpha\in K,\,\text{ we have }|\{b\in L\colon\operatorname{Tr% }_{L/K}(b)=\alpha\}|=q^{n-1}
  19. a Tr L / K ( b a ) . a\mapsto\operatorname{Tr}_{L/K}(ba).
  20. a x 2 + b x + c = 0 , with a 0 , ax^{2}+bx+c=0,\,\text{ with }a\neq 0,
  21. GF ( q ) = 𝔽 q \operatorname{GF}(q)=\mathbb{F}_{q}
  22. x = c a x=\sqrt{\frac{c}{a}}
  23. y 2 + y + δ = 0 , where δ = a c b 2 y^{2}+y+\delta=0,\text{ where }\delta=\frac{ac}{b^{2}}
  24. Tr G F ( q ) / G F ( 2 ) ( δ ) = 0. \operatorname{Tr}_{GF(q)/GF(2)}(\delta)=0.
  25. Tr G F ( q ) / G F ( 2 ) ( k ) = 1. \operatorname{Tr}_{GF(q)/GF(2)}(k)=1.
  26. y = s = k δ 2 + ( k + k 2 ) δ 4 + + ( k + k 2 + + k 2 h - 2 ) δ 2 h - 1 y=s=k\delta^{2}+(k+k^{2})\delta^{4}+\ldots+(k+k^{2}+\ldots+k^{2^{h-2}})\delta^% {2^{h-1}}
  27. y = s = δ + δ 2 2 + δ 2 4 + + δ 2 2 m y=s=\delta+\delta^{2^{2}}+\delta^{2^{4}}+\ldots+\delta^{2^{2m}}
  28. L / K {}_{L/K}

Figurate_number.html

  1. ( n + 1 2 ) n+1\choose 2
  2. r 0 r\geq 0
  3. P 1 ( n ) = n 1 = ( n + 0 1 ) P_{1}(n)=\frac{n}{1}={n+0\choose 1}
  4. P 2 ( n ) = n ( n + 1 ) 2 = ( n + 1 2 ) P_{2}(n)=\frac{n(n+1)}{2}={n+1\choose 2}
  5. P 3 ( n ) = n ( n + 1 ) ( n + 2 ) 6 = ( n + 2 3 ) P_{3}(n)=\frac{n(n+1)(n+2)}{6}={n+2\choose 3}
  6. P 4 ( n ) = n ( n + 1 ) ( n + 2 ) ( n + 3 ) 24 = ( n + 3 4 ) P_{4}(n)=\frac{n(n+1)(n+2)(n+3)}{24}={n+3\choose 4}
  7. ...
  8. P r ( n ) = n ( n + 1 ) ( n + 2 ) ( n + r - 1 ) r ! = ( n + r - 1 r ) P_{r}(n)=\frac{n(n+1)(n+2)...(n+r-1)}{r!}={n+r-1\choose r}

Fine_structure.html

  1. 1 / 137 1/137
  2. H = H 0 + H kinetic + H so + H Darwinian . H=H_{0}+H_{\mathrm{kinetic}}+H_{\mathrm{so}}+H_{\mathrm{Darwinian}}.\!
  3. T = p 2 2 m , T=\frac{p^{2}}{2m},
  4. p p
  5. m m
  6. T = p 2 c 2 + m 2 c 4 - m c 2 , T=\sqrt{p^{2}c^{2}+m^{2}c^{4}}-mc^{2},
  7. c c
  8. T = p 2 2 m - p 4 8 m 3 c 2 + . T=\frac{p^{2}}{2m}-\frac{p^{4}}{8m^{3}c^{2}}+\cdots.
  9. H kinetic = - p 4 8 m 3 c 2 . H_{\mathrm{kinetic}}=-\frac{p^{4}}{8m^{3}c^{2}}.
  10. E n ( 1 ) = ψ 0 | H | ψ 0 = - 1 8 m 3 c 2 ψ 0 | p 4 | ψ 0 = - 1 8 m 3 c 2 ψ 0 | p 2 p 2 | ψ 0 E_{n}^{(1)}=\left\langle\psi^{0}\right|H^{\prime}\left|\psi^{0}\right\rangle=-% \frac{1}{8m^{3}c^{2}}\left\langle\psi^{0}\right|p^{4}\left|\psi^{0}\right% \rangle=-\frac{1}{8m^{3}c^{2}}\left\langle\psi^{0}\right|p^{2}p^{2}\left|\psi^% {0}\right\rangle
  11. ψ 0 \psi^{0}
  12. H 0 | ψ 0 \displaystyle H^{0}\left|\psi^{0}\right\rangle
  13. E n ( 1 ) \displaystyle E_{n}^{(1)}
  14. V ( r ) = - e 2 4 π ϵ 0 r V(r)=\frac{-e^{2}}{4\pi\epsilon_{0}r}
  15. 1 r = 1 a 0 n 2 \left\langle\frac{1}{r}\right\rangle=\frac{1}{a_{0}n^{2}}
  16. 1 r 2 = 1 ( l + 1 / 2 ) n 3 a 0 2 \left\langle\frac{1}{r^{2}}\right\rangle=\frac{1}{(l+1/2)n^{3}a_{0}^{2}}
  17. a 0 a_{0}
  18. n n
  19. l l
  20. E n ( 1 ) \displaystyle E_{n}^{(1)}
  21. E n = - e 2 2 a 0 n 2 E_{n}=-\frac{e^{2}}{2a_{0}n^{2}}
  22. - 9.056 × 10 - 4 eV -9.056\times 10^{-4}\ \,\text{eV}
  23. Z Z
  24. L \vec{L}
  25. S \vec{S}
  26. H s o = 1 2 ( Z e 2 4 π ϵ 0 ) ( g s 2 m e 2 c 2 ) L S r 3 H_{so}=\frac{1}{2}\left(\frac{Ze^{2}}{4\pi\epsilon_{0}}\right)\left(\frac{g_{s% }}{2m_{e}^{2}c^{2}}\right)\frac{\vec{L}\cdot\vec{S}}{r^{3}}
  27. m e m_{e}
  28. ϵ 0 \epsilon_{0}
  29. g s g_{s}
  30. r r
  31. B \vec{B}
  32. μ s \vec{\mu}_{s}
  33. Δ E S O = ξ ( r ) L S \Delta E_{SO}=\xi(r)\vec{L}\cdot\vec{S}
  34. 1 r 3 \displaystyle\left\langle\frac{1}{r^{3}}\right\rangle
  35. H S O = E n 2 m e c 2 n j ( j + 1 ) - l ( l + 1 ) - 3 4 l ( l + 1 2 ) ( l + 1 ) \left\langle H_{SO}\right\rangle=\frac{E_{n}{}^{2}}{m_{e}c^{2}}~{}n~{}\frac{j(% j+1)-l(l+1)-\frac{3}{4}}{l\left(l+\frac{1}{2}\right)(l+1)}
  36. Z 4 n 3 ( j + 1 / 2 ) 10 - 5 eV \frac{Z^{4}}{n^{3}(j+1/2)}10^{-5}\,\text{ eV}
  37. H Darwinian \displaystyle H_{\mathrm{Darwinian}}
  38. Δ t / Δ E / m c 2 \Delta t\approx\hbar/\Delta E\approx\hbar/mc^{2}
  39. ξ c Δ t / m c = λ c \xi\approx c\Delta t\approx\hbar/mc=\lambda_{c}
  40. r + ξ \vec{r}+\vec{\xi}
  41. U U
  42. U ( r + ξ ) U ( r ) + ξ U ( r ) + 1 2 i j ξ i ξ j i j U ( r ) U(\vec{r}+\vec{\xi})\approx U(\vec{r})+\xi\cdot\nabla U(\vec{r})+\frac{1}{2}% \sum_{ij}\xi_{i}\xi_{j}\partial_{i}\partial_{j}U(\vec{r})
  43. ξ \vec{\xi}
  44. ξ ¯ = 0 , ξ i ξ j ¯ = 1 3 ξ 2 ¯ δ i j , \overline{\xi}=0,\quad\overline{\xi_{i}\xi_{j}}=\frac{1}{3}\overline{\vec{\xi}% ^{2}}\delta_{ij},
  45. U ( r + ξ ) ¯ = U ( r ) + 1 6 ξ 2 ¯ 2 U ( r ) . \overline{U\left(\vec{r}+\vec{\xi}\right)}=U\left(\vec{r}\right)+\frac{1}{6}% \overline{\vec{\xi}^{2}}\nabla^{2}U\left(\vec{r}\right).
  46. ξ 2 ¯ λ c 2 \overline{\vec{\xi}^{2}}\approx\lambda_{c}^{2}
  47. δ U 1 6 λ c 2 2 U = 2 6 m 2 c 2 2 U \delta U\approx\frac{1}{6}\lambda_{c}^{2}\nabla^{2}U=\frac{\hbar^{2}}{6m^{2}c^% {2}}\nabla^{2}U
  48. 2 U = - 2 Z e 2 4 π ϵ 0 r = 4 π ( Z e 2 4 π ϵ 0 ) δ ( r ) δ U 2 6 m 2 c 2 4 π ( Z e 2 4 π ϵ 0 ) δ ( r ) \nabla^{2}U=-\nabla^{2}\frac{Ze^{2}}{4\pi\epsilon_{0}r}=4\pi\left(\frac{Ze^{2}% }{4\pi\epsilon_{0}}\right)\delta(\vec{r})\quad\Rightarrow\quad\delta U\approx% \frac{\hbar^{2}}{6m^{2}c^{2}}4\pi\left(\frac{Ze^{2}}{4\pi\epsilon_{0}}\right)% \delta(\vec{r})
  49. Δ E = E n ( Z α ) 2 n ( 1 j + 1 2 - 3 4 n ) , \Delta E=\frac{E_{n}(Z\alpha)^{2}}{n}\left(\frac{1}{j+\frac{1}{2}}-\frac{3}{4n% }\right)\,,
  50. j j
  51. j = 1 / 2 j=1/2
  52. l = 0 l=0
  53. j = l ± 1 / 2 j=l\pm 1/2

Fingerboard.html

  1. r = r=\infty
  2. r = r 1 = r 2 = c o n s t r=r_{1}=r_{2}=const
  3. r 1 r_{1}
  4. r 2 r_{2}
  5. r ( x ) = r 1 + x l ( r 2 - r 1 ) r(x)=r_{1}+\frac{x}{l}(r_{2}-r_{1})
  6. r ( x ) = f ( x ) r(x)=f(x)
  7. r ( l ) = r(l)=\infty
  8. l l
  9. x x
  10. l l
  11. r ( x ) r(x)
  12. f ( x ) f(x)

Finite_field_arithmetic.html

  1. a b = g log g ( a b ) = g log g ( a ) + log g ( b ) ab=g^{\log_{g}(ab)}=g^{\log_{g}(a)+\log_{g}(b)}
  2. a - 1 = g log g ( a - 1 ) = g - log g ( a ) = g | g | - log g ( a ) a^{-1}=g^{\log_{g}(a^{-1})}=g^{-\log_{g}(a)}=g^{|g|-\log_{g}(a)}
  3. a n = g log g ( a n ) = g n log g ( a ) = g n log g ( a ) ( m o d | g | ) a^{n}=g^{\log_{g}(a^{n})}=g^{n\log_{g}(a)}=g^{n\log_{g}(a)(mod|g|)}

Finite_Fourier_transform.html

  1. x ( t ) x(t)
  2. [ 0 , T ] [0,T]
  3. x ( t ) x(t)
  4. X ( ω ) X(\omega)
  5. x ( t ) x(t)
  6. [ 0 , T ] [0,T]
  7. X ( ω ) = 1 2 π 0 T x ( t ) e - i ω t d t X(\omega)=\frac{1}{\sqrt{2\pi}}\int_{0}^{T}x(t)e^{-i\omega t}\,dt

Finite_group.html

  1. p a q b p^{a}q^{b}

Finite_impulse_response.html

  1. y [ n ] = b 0 x [ n ] + b 1 x [ n - 1 ] + + b N x [ n - N ] = i = 0 N b i x [ n - i ] , \begin{aligned}\displaystyle y[n]&\displaystyle=b_{0}x[n]+b_{1}x[n-1]+\cdots+b% _{N}x[n-N]\\ &\displaystyle=\sum_{i=0}^{N}b_{i}\cdot x[n-i],\end{aligned}
  2. x [ n ] \scriptstyle x[n]
  3. y [ n ] \scriptstyle y[n]
  4. N \scriptstyle N
  5. N \scriptstyle N
  6. ( N + 1 ) \scriptstyle(N\,+\,1)
  7. b i \scriptstyle b_{i}
  8. 0 i N \scriptstyle\ 0\ \leq\ i\ \leq\ N
  9. N \scriptstyle N
  10. b i \scriptstyle b_{i}
  11. x [ n - i ] \scriptstyle x[n-i]
  12. h [ n ] = i = 0 N b i δ [ n - i ] = { b n 0 n N 0 otherwise . h[n]=\sum_{i=0}^{N}b_{i}\cdot\delta[n-i]=\begin{cases}b_{n}&\scriptstyle 0\leq n% \leq N\\ 0&\scriptstyle\,\text{otherwise}.\end{cases}
  13. | b i | \scriptstyle\sum|b_{i}|
  14. { x * h } Y ( ω ) = { x } X ( ω ) { h } H ( ω ) \underbrace{\mathcal{F}\{x*h\}}_{Y(\omega)}=\underbrace{\mathcal{F}\{x\}}_{X(% \omega)}\cdot\underbrace{\mathcal{F}\{h\}}_{H(\omega)}
  15. y [ n ] = x [ n ] * h [ n ] = - 1 { X ( ω ) H ( ω ) } , y[n]=x[n]*h[n]=\mathcal{F}^{-1}\big\{X(\omega)\cdot H(\omega)\big\},
  16. \mathcal{F}
  17. - 1 \mathcal{F}^{-1}
  18. H ( ω ) H(\omega)
  19. H 2 π ( ω ) = def n = - h [ n ] ( e i ω ) - n = n = 0 N b n ( e i ω ) - n , H_{2\pi}(\omega)\ \stackrel{\mathrm{def}}{=}\sum_{n=-\infty}^{\infty}h[n]\cdot% \left({e^{i\omega}}\right)^{-n}=\sum_{n=0}^{N}b_{n}\cdot\left({e^{i\omega}}% \right)^{-n},
  20. ω \omega
  21. ω = 2 π f , \omega=2\pi f,
  22. ( f ) \scriptstyle(f)
  23. f s f_{s}
  24. ω = 2 π f / f s \omega=2\pi f/f_{s}
  25. ( f ) \scriptstyle(f)
  26. f s . f_{s}.
  27. ω = π \omega=\pi
  28. f = f s 2 f=\tfrac{f_{s}}{2}
  29. = 1 2 =\tfrac{1}{2}
  30. , H 2 π ( ω ) , ,\ H_{2\pi}(\omega),
  31. H ( e i ω ) , H(e^{i\omega}),
  32. H H
  33. H ( z ) = def n = - h [ n ] z - n . H(z)\ \stackrel{\mathrm{def}}{=}\sum_{n=-\infty}^{\infty}h[n]\cdot z^{-n}.
  34. z = e i ω , - π ω π , z=e^{i\omega},\ \scriptstyle-\pi\leq\omega\leq\pi,
  35. ( N + 1 ) \scriptstyle(N\,+\,1)
  36. ( N + 1 ) \scriptstyle(N\,+\,1)
  37. b 0 , , b N \scriptstyle b_{0},\,\dots,\,b_{N}
  38. b i = 1 N + 1 b_{i}=\frac{1}{N+1}
  39. N = 2 N=2
  40. h [ n ] = 1 3 δ [ n ] + 1 3 δ [ n - 1 ] + 1 3 δ [ n - 2 ] h[n]=\frac{1}{3}\delta[n]+\frac{1}{3}\delta[n-1]+\frac{1}{3}\delta[n-2]
  41. H ( z ) = 1 3 + 1 3 z - 1 + 1 3 z - 2 = 1 3 z 2 + z + 1 z 2 . H(z)=\frac{1}{3}+\frac{1}{3}z^{-1}+\frac{1}{3}z^{-2}=\frac{1}{3}\frac{z^{2}+z+% 1}{z^{2}}.
  42. z 1 = - 1 2 + j 3 2 \scriptstyle z_{1}\;=\;-\frac{1}{2}\,+\,j\frac{\sqrt{3}}{2}
  43. z 2 = - 1 2 - j 3 2 \scriptstyle z_{2}\;=\;-\frac{1}{2}\,-\,j\frac{\sqrt{3}}{2}
  44. H ( e j ω ) = 1 3 + 1 3 e - j ω + 1 3 e - j 2 ω . H\left(e^{j\omega}\right)=\frac{1}{3}+\frac{1}{3}e^{-j\omega}+\frac{1}{3}e^{-j% 2\omega}.
  45. H ( e j ω ) . \scriptstyle H\left(e^{j\omega}\right).

Finite_intersection_property.html

  1. X X
  2. A = { A i } i I A=\{A_{i}\}_{i\in I}
  3. X X
  4. A A
  5. J I J\subseteq I
  6. i J A i . \bigcap_{i\in J}A_{i}.

Finite_volume_method.html

  1. ( 1 ) ρ t + f x = 0 , t 0. \quad(1)\qquad\qquad\frac{\partial\rho}{\partial t}+\frac{\partial f}{\partial x% }=0,\quad t\geq 0.
  2. ρ = ρ ( x , t ) \rho=\rho\left(x,t\right)
  3. f = f ( ρ ( x , t ) ) f=f\left(\rho\left(x,t\right)\right)
  4. ρ \rho
  5. f f
  6. f f
  7. x x
  8. i i
  9. i i
  10. ρ i ( t ) = ρ ( x , t ) {\rho}_{i}\left(t\right)=\rho\left(x,t\right)
  11. t = t 1 {t=t_{1}}
  12. x [ x i - 1 2 , x i + 1 2 ] {x\in\left[x_{i-\frac{1}{2}},x_{i+\frac{1}{2}}\right]}
  13. ( 2 ) ρ ¯ i ( t 1 ) = 1 x i + 1 2 - x i - 1 2 x i - 1 2 x i + 1 2 ρ ( x , t 1 ) d x , \quad(2)\qquad\qquad\bar{\rho}_{i}\left(t_{1}\right)=\frac{1}{x_{i+\frac{1}{2}% }-x_{i-\frac{1}{2}}}\int_{x_{i-\frac{1}{2}}}^{x_{i+\frac{1}{2}}}\rho\left(x,t_% {1}\right)\,dx,
  14. t = t 2 {t=t_{2}}
  15. ( 3 ) ρ ¯ i ( t 2 ) = 1 x i + 1 2 - x i - 1 2 x i - 1 2 x i + 1 2 ρ ( x , t 2 ) d x , \quad(3)\qquad\qquad\bar{\rho}_{i}\left(t_{2}\right)=\frac{1}{x_{i+\frac{1}{2}% }-x_{i-\frac{1}{2}}}\int_{x_{i-\frac{1}{2}}}^{x_{i+\frac{1}{2}}}\rho\left(x,t_% {2}\right)\,dx,
  16. x i - 1 2 x_{i-\frac{1}{2}}
  17. x i + 1 2 x_{i+\frac{1}{2}}
  18. i t h i^{th}
  19. ( 4 ) ρ ( x , t 2 ) = ρ ( x , t 1 ) - t 1 t 2 f x ( x , t ) d t , \quad(4)\qquad\qquad\rho\left(x,t_{2}\right)=\rho\left(x,t_{1}\right)-\int_{t_% {1}}^{t_{2}}f_{x}\left(x,t\right)\,dt,
  20. f x = f x f_{x}=\frac{\partial f}{\partial x}
  21. ρ ( x , t ) \rho\left(x,t\right)
  22. t = t 2 t=t_{2}
  23. ρ ( x , t 2 ) \rho\left(x,t_{2}\right)
  24. [ x i - 1 2 , x i + 1 2 ] \left[x_{i-\frac{1}{2}},x_{i+\frac{1}{2}}\right]
  25. Δ x i = x i + 1 2 - x i - 1 2 \Delta x_{i}=x_{i+\frac{1}{2}}-x_{i-\frac{1}{2}}
  26. ( 5 ) ρ ¯ i ( t 2 ) = 1 Δ x i x i - 1 2 x i + 1 2 { ρ ( x , t 1 ) - t 1 t 2 f x ( x , t ) d t } d x . \quad(5)\qquad\qquad\bar{\rho}_{i}\left(t_{2}\right)=\frac{1}{\Delta x_{i}}% \int_{x_{i-\frac{1}{2}}}^{x_{i+\frac{1}{2}}}\left\{\rho\left(x,t_{1}\right)-% \int_{t_{1}}^{t_{2}}f_{x}\left(x,t\right)dt\right\}dx.
  27. f f
  28. f x f f_{x}\triangleq\nabla f
  29. v f d v = S f d S \oint_{v}\nabla\cdot fdv=\oint_{S}f\,dS
  30. f ( x ) f(x)
  31. x i - 1 2 x_{i-\frac{1}{2}}
  32. x i + 1 2 x_{i+\frac{1}{2}}
  33. ( 6 ) ρ ¯ i ( t 2 ) = ρ ¯ i ( t 1 ) - 1 Δ x i ( t 1 t 2 f i + 1 2 d t - t 1 t 2 f i - 1 2 d t ) . \quad(6)\qquad\qquad\bar{\rho}_{i}\left(t_{2}\right)=\bar{\rho}_{i}\left(t_{1}% \right)-\frac{1}{\Delta x_{i}}\left(\int_{t_{1}}^{t_{2}}f_{i+\frac{1}{2}}dt-% \int_{t_{1}}^{t_{2}}f_{i-\frac{1}{2}}dt\right).
  34. f i ± 1 2 = f ( x i ± 1 2 , t ) f_{i\pm\frac{1}{2}}=f\left(x_{i\pm\frac{1}{2}},t\right)
  35. i i
  36. i ± 1 2 i\pm\frac{1}{2}
  37. ( 7 ) d ρ ¯ i d t + 1 Δ x i [ f i + 1 2 - f i - 1 2 ] = 0 , \quad(7)\qquad\qquad\frac{d\bar{\rho}_{i}}{dt}+\frac{1}{\Delta x_{i}}\left[f_{% i+\frac{1}{2}}-f_{i-\frac{1}{2}}\right]=0,
  38. f i ± 1 2 f_{i\pm\frac{1}{2}}
  39. ( 8 ) 𝐮 t + 𝐟 ( 𝐮 ) = 0. \quad(8)\qquad\qquad{{\partial{\mathbf{u}}}\over{\partial t}}+\nabla\cdot{% \mathbf{f}}\left({\mathbf{u}}\right)={\mathbf{0}}.
  40. 𝐮 {\mathbf{u}}
  41. 𝐟 \mathbf{f}
  42. i i
  43. v i v_{i}
  44. ( 9 ) v i 𝐮 t d v + v i 𝐟 ( 𝐮 ) d v = 0. \quad(9)\qquad\qquad\int_{v_{i}}{{\partial{\mathbf{u}}}\over{\partial t}}\,dv+% \int_{v_{i}}\nabla\cdot{\mathbf{f}}\left({\mathbf{u}}\right)\,dv={\mathbf{0}}.
  45. ( 10 ) v i d 𝐮 ¯ i d t + S i 𝐟 ( 𝐮 ) 𝐧 d S = 𝟎 , \quad(10)\qquad\qquad v_{i}{{d{\mathbf{\bar{u}}}_{i}}\over{dt}}+\oint_{S_{i}}{% \mathbf{f}}\left({\mathbf{u}}\right)\cdot{\mathbf{n}}\ dS={\mathbf{0}},
  46. S i S_{i}
  47. 𝐧 {\mathbf{n}}
  48. ( 11 ) d 𝐮 ¯ i d t + 1 v i S i 𝐟 ( 𝐮 ) 𝐧 d S = 0. \quad(11)\qquad\qquad{{d{\mathbf{\bar{u}}}_{i}}\over{dt}}+{{1}\over{v_{i}}}% \oint_{S_{i}}{\mathbf{f}}\left({\mathbf{u}}\right)\cdot{\mathbf{n}}\ dS={% \mathbf{0}}.

Finitely_generated_module.html

  1. R n M R^{n}\to M
  2. dim K ( M A K ) \operatorname{dim}_{K}(M\otimes_{A}K)
  3. ( M / F ) ( 0 ) = M ( 0 ) / F ( 0 ) = 0 (M/F)_{(0)}=M_{(0)}/F_{(0)}=0
  4. M / F M/F
  5. M [ f - 1 ] M[f^{-1}]
  6. A [ f - 1 ] A[f^{-1}]
  7. d i d_{i}
  8. P M ( t ) = dim k ( M n ) t n P_{M}(t)=\sum\operatorname{dim}_{k}(M_{n})t^{n}
  9. P M ( t ) = F ( t ) ( 1 - t d i ) - 1 P_{M}(t)=F(t)\prod(1-t^{d_{i}})^{-1}
  10. F ( 1 ) F(1)
  11. f M F fM\subset F
  12. f M fM
  13. f : M f M f:M\to fM
  14. i I N i = M \sum_{i\in I}N_{i}=M\,
  15. i F N i = M \sum_{i\in F}N_{i}=M\,
  16. i I N i = M \bigcup_{i\in I}N_{i}=M\,
  17. ϕ : i I R M \phi:\bigoplus_{i\in I}R\to M\,
  18. ϕ : i F R M \phi:\bigoplus_{i\in F}R\to M\,
  19. i I N i = { 0 } \bigcap_{i\in I}N_{i}=\{0\}\,
  20. i F N i = { 0 } \bigcap_{i\in F}N_{i}=\{0\}\,
  21. i I N i = { 0 } \bigcap_{i\in I}N_{i}=\{0\}\,
  22. ϕ : M i I R \phi:M\to\prod_{i\in I}R\,
  23. ϕ : M i F R \phi:M\to\prod_{i\in F}R\,

Finnish_verb_conjugation.html

  1. \quad
  2. \quad
  3. \rightarrow
  4. \rightarrow
  5. \rightarrow
  6. \rightarrow
  7. \rightarrow
  8. \rightarrow
  9. \rightarrow
  10. \rightarrow
  11. \rightarrow
  12. \rightarrow
  13. \rightarrow
  14. \rightarrow
  15. \rightarrow
  16. \rightarrow
  17. \rightarrow
  18. \rightarrow
  19. \rightarrow

Finsler_manifold.html

  1. L [ γ ] = a b F ( γ ( t ) , γ ˙ ( t ) ) d t , L[\gamma]=\int_{a}^{b}F(\gamma(t),\dot{\gamma}(t))\,dt,
  2. 𝐠 v ( X , Y ) := 1 2 2 s t [ F ( v + s X + t Y ) 2 ] | s = t = 0 , \mathbf{g}_{v}(X,Y):=\frac{1}{2}\left.\frac{\partial^{2}}{\partial s\partial t% }\left[F(v+sX+tY)^{2}\right]\right|_{s=t=0},
  3. b a := a i j b i b j < 1 , \|b\|_{a}:=\sqrt{a^{ij}b_{i}b_{j}}<1,
  4. ( a i j ) (a^{ij})
  5. ( a i j ) (a_{ij})
  6. F ( x , v ) := a i j ( x ) v i v j + b i ( x ) v i F(x,v):=\sqrt{a_{ij}(x)v^{i}v^{j}}+b_{i}(x)v^{i}
  7. 1 C φ ( y ) - φ ( x ) d ( x , y ) C φ ( y ) - φ ( x ) . \frac{1}{C}\|\varphi(y)-\varphi(x)\|\leq d(x,y)\leq C\|\varphi(y)-\varphi(x)\|.
  8. F ( x , v ) := lim t 0 + d ( γ ( 0 ) , γ ( t ) ) t , F(x,v):=\lim_{t\to 0+}\frac{d(\gamma(0),\gamma(t))}{t},
  9. d L ( x , y ) := inf { 0 1 F ( γ ( t ) , γ ˙ ( t ) ) d t | γ C 1 ( [ 0 , 1 ] , M ) , γ ( 0 ) = x , γ ( 1 ) = y } , d_{L}(x,y):=\inf\left\{\ \left.\int_{0}^{1}F(\gamma(t),\dot{\gamma}(t))\,dt\ % \right|\ \gamma\in C^{1}([0,1],M)\ ,\ \gamma(0)=x\ ,\ \gamma(1)=y\ \right\},
  10. L [ γ ] := a b F ( γ ( t ) , γ ˙ ( t ) ) d t L[\gamma]:=\int_{a}^{b}F(\gamma(t),\dot{\gamma}(t))\,dt
  11. E [ γ ] := 1 2 a b F 2 ( γ ( t ) , γ ˙ ( t ) ) d t E[\gamma]:=\frac{1}{2}\int_{a}^{b}F^{2}(\gamma(t),\dot{\gamma}(t))\,dt
  12. g i k ( γ ( t ) , γ ˙ ( t ) ) γ ¨ i ( t ) + ( g i k x j ( γ ( t ) , γ ˙ ( t ) ) - 1 2 g i j x k ( γ ( t ) , γ ˙ ( t ) ) ) γ ˙ i ( t ) γ ˙ j ( t ) = 0 , g_{ik}\Big(\gamma(t),\dot{\gamma}(t)\Big)\ddot{\gamma}^{i}(t)+\left(\frac{% \partial g_{ik}}{\partial x^{j}}\Big(\gamma(t),\dot{\gamma}(t)\Big)-\frac{1}{2% }\frac{\partial g_{ij}}{\partial x^{k}}\Big(\gamma(t),\dot{\gamma}(t)\Big)% \right)\dot{\gamma}^{i}(t)\dot{\gamma}^{j}(t)=0,
  13. g i j ( x , v ) := g v ( x i | x , x j | x ) . g_{ij}(x,v):=g_{v}\left(\tfrac{\partial}{\partial x^{i}}\big|_{x},\tfrac{% \partial}{\partial x^{j}}\big|_{x}\right).
  14. H | ( x , v ) := v i x i | ( x , v ) - 2 G i ( x , v ) v i | ( x , v ) , H|_{(x,v)}:=v^{i}\tfrac{\partial}{\partial x^{i}}\big|_{(x,v)}-\ 2G^{i}(x,v)% \tfrac{\partial}{\partial v^{i}}\big|_{(x,v)},
  15. G i ( x , v ) := g i j ( x , v ) 4 ( 2 g j k x ( x , v ) - g k x j ( x , v ) ) v k v . G^{i}(x,v):=\frac{g^{ij}(x,v)}{4}\left(2\frac{\partial g_{jk}}{\partial x^{% \ell}}(x,v)-\frac{\partial g_{k\ell}}{\partial x^{j}}(x,v)\right)v^{k}v^{\ell}.
  16. v : T ( T M 0 ) T ( T M 0 ) ; v := 1 2 ( I + H J ) . v:T(TM\setminus 0)\to T(TM\setminus 0)\quad;\quad v:=\tfrac{1}{2}\big(I+% \mathcal{L}_{H}J\big).
  17. D γ ˙ D γ ˙ X ( t ) + R γ ˙ ( γ ˙ ( t ) , X ( t ) ) = 0 D_{\dot{\gamma}}D_{\dot{\gamma}}X(t)+R_{\dot{\gamma}}(\dot{\gamma}(t),X(t))=0

First_fundamental_form.html

  1. I ( x , y ) = x , y . \!\mathrm{I}(x,y)=\langle x,y\rangle.
  2. I ( a X u + b X v , c X u + d X v ) = a c X u , X u + ( a d + b c ) X u , X v + b d X v , X v = E a c + F ( a d + b c ) + G b d , \begin{aligned}&\displaystyle{}\quad\mathrm{I}(aX_{u}+bX_{v},cX_{u}+dX_{v})\\ &\displaystyle=ac\langle X_{u},X_{u}\rangle+(ad+bc)\langle X_{u},X_{v}\rangle+% bd\langle X_{v},X_{v}\rangle\\ &\displaystyle=Eac+F(ad+bc)+Gbd,\end{aligned}
  3. I ( x , y ) = x T ( E F F G ) y \!\mathrm{I}(x,y)=x^{T}\begin{pmatrix}E&F\\ F&G\end{pmatrix}y
  4. I ( v ) = v , v = | v | 2 \!\mathrm{I}(v)=\langle v,v\rangle=|v|^{2}
  5. g i j g_{ij}
  6. ( g i j ) = ( g 11 g 12 g 21 g 22 ) = ( E F F G ) \left(g_{ij}\right)=\begin{pmatrix}g_{11}&g_{12}\\ g_{21}&g_{22}\end{pmatrix}=\begin{pmatrix}E&F\\ F&G\end{pmatrix}
  7. g i j = X i X j g_{ij}=X_{i}\cdot X_{j}
  8. d s 2 = E d u 2 + 2 F d u d v + G d v 2 ds^{2}=Edu^{2}+2Fdudv+Gdv^{2}\,
  9. d A = | X u × X v | d u d v dA=|X_{u}\times X_{v}|\ du\,dv
  10. d A = | X u × X v | d u d v = X u , X u X v , X v - X u , X v 2 d u d v = E G - F 2 d u d v . dA=|X_{u}\times X_{v}|\ du\,dv=\sqrt{\langle X_{u},X_{u}\rangle\langle X_{v},X% _{v}\rangle-\langle X_{u},X_{v}\rangle^{2}}\ du\,dv=\sqrt{EG-F^{2}}\,du\,dv.
  11. X ( u , v ) = ( cos u sin v sin u sin v cos v ) , ( u , v ) [ 0 , 2 π ) × [ 0 , π ] . X(u,v)=\begin{pmatrix}\cos u\sin v\\ \sin u\sin v\\ \cos v\end{pmatrix},\ (u,v)\in[0,2\pi)\times[0,\pi].
  12. X ( u , v ) X(u,v)
  13. X u = ( - sin u sin v cos u sin v 0 ) , X v = ( cos u cos v sin u cos v - sin v ) . X_{u}=\begin{pmatrix}-\sin u\sin v\\ \cos u\sin v\\ 0\end{pmatrix},\ X_{v}=\begin{pmatrix}\cos u\cos v\\ \sin u\cos v\\ -\sin v\end{pmatrix}.
  14. E = X u X u = sin 2 v E=X_{u}\cdot X_{u}=\sin^{2}v
  15. F = X u X v = 0 F=X_{u}\cdot X_{v}=0
  16. G = X v X v = 1 G=X_{v}\cdot X_{v}=1
  17. ( u ( t ) , v ( t ) ) = ( t , π 2 ) (u(t),v(t))=(t,\tfrac{\pi}{2})
  18. 2 π 2\pi
  19. 0 2 π E ( d u d t ) 2 + 2 F d u d t d v d t + G ( d v d t ) 2 d t = 0 2 π | sin v | d t = 2 π sin π 2 = 2 π \int_{0}^{2\pi}\sqrt{E\left(\frac{du}{dt}\right)^{2}+2F\frac{du}{dt}\frac{dv}{% dt}+G\left(\frac{dv}{dt}\right)^{2}}\,dt=\int_{0}^{2\pi}|\sin v|\,dt=2\pi\sin% \tfrac{\pi}{2}=2\pi
  20. 0 π 0 2 π E G - F 2 d u d v = 0 π 0 2 π sin v d u d v = 2 π [ - cos v ] 0 π = 4 π \int_{0}^{\pi}\int_{0}^{2\pi}\sqrt{EG-F^{2}}\ du\,dv=\int_{0}^{\pi}\int_{0}^{2% \pi}\sin v\,du\,dv=2\pi\left[-\cos v\right]_{0}^{\pi}=4\pi
  21. K = det I I det I = L N - M 2 E G - F 2 , K=\frac{\det\mathrm{I\!I}}{\det\mathrm{I}}=\frac{LN-M^{2}}{EG-F^{2}},

Fisher_equation.html

  1. r r
  2. i i
  3. π π
  4. i r + π i\approx r+\pi
  5. i = r + π i=r+\pi
  6. r = i - π r=i-\pi
  7. e e
  8. i = r + π e i=r+\pi^{e}
  9. 1 + i = ( 1 + r ) ( 1 + π ) . 1+i=(1+r)(1+\pi).
  10. t t
  11. t + 1 t+1
  12. t t
  13. t + 1 t+1
  14. ( 1 + r t + 1 ) = 1 + i t 1 + π t + 1 (1+r_{t+1})=\frac{1+i_{t}}{1+\pi_{t+1}}
  15. 1 + i t = ( 1 + r t + 1 ) ( 1 + π t + 1 ) = 1 + r t + 1 + π t + 1 + r t + 1 π t + 1 \begin{aligned}\displaystyle 1+i_{t}&\displaystyle=\left(1+r_{t+1}\right)\left% (1+\pi_{t+1}\right)\\ &\displaystyle=1+r_{t+1}+\pi_{t+1}+r_{t+1}\pi_{t+1}\end{aligned}
  16. i t = r t + 1 + π t + 1 + r t + 1 π t + 1 r t + 1 + π t + 1 \begin{aligned}\displaystyle i_{t}&\displaystyle=r_{t+1}+\pi_{t+1}+r_{t+1}\pi_% {t+1}\\ &\displaystyle\approx r_{t+1}+\pi_{t+1}\end{aligned}
  17. 1 1 + x \displaystyle\frac{1}{1+x}
  18. 1 + r = 1 + i 1 + π ( 1 + i ) ( 1 - π ) 1 + i - π , 1+r=\frac{1+i}{1+\pi}\approx(1+i)(1-\pi)\approx 1+i-\pi,
  19. r i - π . r\approx i-\pi.
  20. 1.02 × 1.01775 = ( 1 + 0.02 ) × ( 1 + 0.01775 ) = 1.0381. 1.02×1.01775=(1+0.02)×(1+0.01775)=1.0381.
  21. 0.02 × 0.01775 = 0.00035 0.02×0.01775=0.00035
  22. 0.035 % 0.035\%
  23. t t
  24. t , t = 1 , , n t,t=1,...,n
  25. P V N B PVNB
  26. PVNB = Z 1 1 + R 1 + Z 2 ( 1 + R 1 ) ( 1 + R 2 ) + + Z n ( 1 + R 1 ) ( 1 + R n ) \,\text{PVNB}=\frac{Z_{1}}{1+R_{1}}+\frac{Z_{2}}{(1+R_{1})(1+R_{2})}+\cdots+% \frac{Z_{n}}{(1+R_{1})\cdots(1+R_{n})}
  27. PVNB = Z 1 ( 1 + I 1 ) 1 + r 1 + Z 2 ( 1 + I 1 ) ( 1 + I 2 ) ( 1 + r 1 ) ( 1 + r 2 ) + + Z n ( 1 + I 1 ) ( 1 + I n ) ( 1 + r 1 ) ( 1 + r n ) \,\text{PVNB}=\frac{Z_{1}(1+I_{1})}{1+r_{1}}+\frac{Z_{2}(1+I_{1})(1+I_{2})}{(1% +r_{1})(1+r_{2})}+\cdots+\frac{Z_{n}(1+I_{1})\cdots(1+I_{n})}{(1+r_{1})\cdots(% 1+r_{n})}
  28. PVNB = Z 1 ( 1 + I 1 ) ( 1 + R 1 ) ( 1 + I 1 ) + Z 2 ( 1 + I 1 ) ( 1 + I 2 ) ( 1 + R 1 ) ( 1 + R 2 ) ( 1 + I 1 ) ( 1 + I 2 ) + + Z n ( 1 + I 1 ) ( 1 + I n ) ( 1 + R 1 ) ( 1 + R n ) ( 1 + I 1 ) ( 1 + I n ) \begin{aligned}\displaystyle\,\text{PVNB}&\displaystyle=\frac{Z_{1}(1+I_{1})}{% (1+R_{1})(1+I_{1})}+\frac{Z_{2}(1+I_{1})(1+I_{2})}{(1+R_{1})(1+R_{2})(1+I_{1})% (1+I_{2})}+\cdots\\ &\displaystyle{}\qquad\cdots+\frac{Z_{n}(1+I_{1})\cdots(1+I_{n})}{(1+R_{1})% \cdots(1+R_{n})(1+I_{1})\cdots(1+I_{n})}\end{aligned}
  29. i i
  30. r r

Fisher_information.html

  1. E [ θ log f ( X ; θ ) | θ ] = E [ θ f ( X ; θ ) f ( X ; θ ) | θ ] = θ f ( x ; θ ) f ( x ; θ ) f ( x ; θ ) d x = \operatorname{E}\left[\left.\frac{\partial}{\partial\theta}\log f(X;\theta)% \right|\theta\right]=\operatorname{E}\left[\left.\frac{\frac{\partial}{% \partial\theta}f(X;\theta)}{f(X;\theta)}\right|\theta\right]=\int\frac{\frac{% \partial}{\partial\theta}f(x;\theta)}{f(x;\theta)}f(x;\theta)\;\mathrm{d}x=
  2. = θ f ( x ; θ ) d x = θ f ( x ; θ ) d x = θ 1 = 0. =\int\frac{\partial}{\partial\theta}f(x;\theta)\;\mathrm{d}x=\frac{\partial}{% \partial\theta}\int f(x;\theta)\;\mathrm{d}x=\frac{\partial}{\partial\theta}\;% 1=0.
  3. ( θ ) = E [ ( θ log f ( X ; θ ) ) 2 | θ ] = ( θ log f ( x ; θ ) ) 2 f ( x ; θ ) d x , \mathcal{I}(\theta)=\operatorname{E}\left[\left.\left(\frac{\partial}{\partial% \theta}\log f(X;\theta)\right)^{2}\right|\theta\right]=\int\left(\frac{% \partial}{\partial\theta}\log f(x;\theta)\right)^{2}f(x;\theta)\;\mathrm{d}x\,,
  4. 0 ( θ ) < 0\leq\mathcal{I}(\theta)<\infty
  5. ( θ ) = - E [ 2 θ 2 log f ( X ; θ ) | θ ] , \mathcal{I}(\theta)=-\operatorname{E}\left[\left.\frac{\partial^{2}}{\partial% \theta^{2}}\log f(X;\theta)\right|\theta\right]\,,
  6. 2 θ 2 log f ( X ; θ ) = 2 θ 2 f ( X ; θ ) f ( X ; θ ) - ( θ f ( X ; θ ) f ( X ; θ ) ) 2 = 2 θ 2 f ( X ; θ ) f ( X ; θ ) - ( θ log f ( X ; θ ) ) 2 \frac{\partial^{2}}{\partial\theta^{2}}\log f(X;\theta)=\frac{\frac{\partial^{% 2}}{\partial\theta^{2}}f(X;\theta)}{f(X;\theta)}\;-\;\left(\frac{\frac{% \partial}{\partial\theta}f(X;\theta)}{f(X;\theta)}\right)^{2}=\frac{\frac{% \partial^{2}}{\partial\theta^{2}}f(X;\theta)}{f(X;\theta)}\;-\;\left(\frac{% \partial}{\partial\theta}\log f(X;\theta)\right)^{2}
  7. E [ 2 θ 2 f ( X ; θ ) f ( X ; θ ) | θ ] = = 2 θ 2 f ( x ; θ ) d x = 2 θ 2 1 = 0. \operatorname{E}\left[\left.\frac{\frac{\partial^{2}}{\partial\theta^{2}}f(X;% \theta)}{f(X;\theta)}\right|\theta\right]=\cdots=\frac{\partial^{2}}{\partial% \theta^{2}}\int f(x;\theta)\;\mathrm{d}x=\frac{\partial^{2}}{\partial\theta^{2% }}\;1=0.
  8. X , Y ( θ ) = X ( θ ) + Y ( θ ) . \mathcal{I}_{X,Y}(\theta)=\mathcal{I}_{X}(\theta)+\mathcal{I}_{Y}(\theta).
  9. f ( X ; θ ) = g ( T ( X ) , θ ) h ( X ) f(X;\theta)=g(T(X),\theta)h(X)\!
  10. θ log [ f ( X ; θ ) ] = θ log [ g ( T ( X ) ; θ ) ] \frac{\partial}{\partial\theta}\log\left[f(X;\theta)\right]=\frac{\partial}{% \partial\theta}\log\left[g(T(X);\theta)\right]
  11. T ( θ ) X ( θ ) \mathcal{I}_{T}(\theta)\leq\mathcal{I}_{X}(\theta)
  12. θ ^ ( X ) \hat{\theta}(X)
  13. E [ θ ^ ( X ) - θ | θ ] = [ θ ^ ( x ) - θ ] f ( x ; θ ) d x = 0. \operatorname{E}\left[\left.\hat{\theta}(X)-\theta\right|\theta\right]=\int% \left[\hat{\theta}(x)-\theta\right]\cdot f(x;\theta)\,\mathrm{d}x=0.
  14. θ [ θ ^ ( x ) - θ ] f ( x ; θ ) d x = ( θ ^ ( x ) - θ ) f θ d x - f d x = 0. \frac{\partial}{\partial\theta}\int\left[\hat{\theta}(x)-\theta\right]\cdot f(% x;\theta)\,\mathrm{d}x=\int\left(\hat{\theta}(x)-\theta\right)\frac{\partial f% }{\partial\theta}\,\mathrm{d}x-\int f\,\mathrm{d}x=0.
  15. f d x = 1. \int f\,\mathrm{d}x=1.
  16. f θ = f log f θ . \frac{\partial f}{\partial\theta}=f\,\frac{\partial\log f}{\partial\theta}.
  17. ( θ ^ - θ ) f log f θ d x = 1. \int\left(\hat{\theta}-\theta\right)f\,\frac{\partial\log f}{\partial\theta}\,% \mathrm{d}x=1.
  18. ( ( θ ^ - θ ) f ) ( f log f θ ) d x = 1. \int\left(\left(\hat{\theta}-\theta\right)\sqrt{f}\right)\left(\sqrt{f}\,\frac% {\partial\log f}{\partial\theta}\right)\,\mathrm{d}x=1.
  19. [ ( θ ^ - θ ) 2 f d x ] [ ( log f θ ) 2 f d x ] 1. \left[\int\left(\hat{\theta}-\theta\right)^{2}f\,\mathrm{d}x\right]\cdot\left[% \int\left(\frac{\partial\log f}{\partial\theta}\right)^{2}f\,\mathrm{d}x\right% ]\geq 1.
  20. ( θ ) = ( log f θ ) 2 f d x . \mathcal{I}\left(\theta\right)=\int\left(\frac{\partial\log f}{\partial\theta}% \right)^{2}f\,\mathrm{d}x.
  21. E [ ( θ ^ ( X ) - θ ) 2 | θ ] = ( θ ^ - θ ) 2 f d x . \operatorname{E}\left[\left.\left(\hat{\theta}\left(X\right)-\theta\right)^{2}% \right|\theta\right]=\int\left(\hat{\theta}-\theta\right)^{2}f\,\mathrm{d}x.
  22. Var ( θ ^ ) 1 ( θ ) . \operatorname{Var}\left(\hat{\theta}\right)\,\geq\,\frac{1}{\mathcal{I}\left(% \theta\right)}.
  23. ( θ ) = - E [ 2 θ 2 log ( f ( A ; θ ) ) | θ ] ( 1 ) = - E [ 2 θ 2 log ( θ A ( 1 - θ ) B ( A + B ) ! A ! B ! ) | θ ] ( 2 ) = - E [ 2 θ 2 ( A log ( θ ) + B log ( 1 - θ ) ) | θ ] ( 3 ) = - E [ θ ( A θ - B 1 - θ ) | θ ] ( 4 ) = + E [ A θ 2 + B ( 1 - θ ) 2 | θ ] ( 5 ) = n θ θ 2 + n ( 1 - θ ) ( 1 - θ ) 2 ( 6 ) since the expected value of A given θ is n θ , etc. = n θ ( 1 - θ ) ( 7 ) \begin{aligned}\displaystyle\mathcal{I}(\theta)&\displaystyle=-\operatorname{E% }\left[\left.\frac{\partial^{2}}{\partial\theta^{2}}\log(f(A;\theta))\right|% \theta\right]\qquad(1)\\ &\displaystyle=-\operatorname{E}\left[\left.\frac{\partial^{2}}{\partial\theta% ^{2}}\log\left(\theta^{A}(1-\theta)^{B}\frac{(A+B)!}{A!B!}\right)\right|\theta% \right]\qquad(2)\\ &\displaystyle=-\operatorname{E}\left[\left.\frac{\partial^{2}}{\partial\theta% ^{2}}\left(A\log(\theta)+B\log(1-\theta)\right)\right|\theta\right]\qquad(3)\\ &\displaystyle=-\operatorname{E}\left[\left.\frac{\partial}{\partial\theta}% \left(\frac{A}{\theta}-\frac{B}{1-\theta}\right)\right|\theta\right]\qquad(4)% \\ &\displaystyle=+\operatorname{E}\left[\left.\frac{A}{\theta^{2}}+\frac{B}{(1-% \theta)^{2}}\right|\theta\right]\qquad(5)\\ &\displaystyle=\frac{n\theta}{\theta^{2}}+\frac{n(1-\theta)}{(1-\theta)^{2}}% \qquad(6)\\ &\displaystyle\,\text{since the expected value of }A\,\text{ given }\theta\,% \text{ is }n\theta,\,\text{ etc.}\\ &\displaystyle=\frac{n}{\theta(1-\theta)}\qquad(7)\end{aligned}
  24. ( θ ) = n θ ( 1 - θ ) , \mathcal{I}(\theta)=\frac{n}{\theta(1-\theta)},
  25. θ = [ θ 1 , θ 2 , , θ N ] T , \theta=\begin{bmatrix}\theta_{1},\theta_{2},\dots,\theta_{N}\end{bmatrix}^{% \mathrm{T}},
  26. ( ( θ ) ) i , j = E [ ( θ i log f ( X ; θ ) ) ( θ j log f ( X ; θ ) ) | θ ] . {\left(\mathcal{I}\left(\theta\right)\right)}_{i,j}=\operatorname{E}\left[% \left.\left(\frac{\partial}{\partial\theta_{i}}\log f(X;\theta)\right)\left(% \frac{\partial}{\partial\theta_{j}}\log f(X;\theta)\right)\right|\theta\right].
  27. ( ( θ ) ) i , j = - E [ 2 θ i θ j log f ( X ; θ ) | θ ] . {\left(\mathcal{I}\left(\theta\right)\right)}_{i,j}=-\operatorname{E}\left[% \left.\frac{\partial^{2}}{\partial\theta_{i}\,\partial\theta_{j}}\log f(X;% \theta)\right|\theta\right]\,.
  28. μ ( θ ) = [ μ 1 ( θ ) , μ 2 ( θ ) , , μ N ( θ ) ] T , \mu(\theta)=\begin{bmatrix}\mu_{1}(\theta),\mu_{2}(\theta),\dots,\mu_{N}(% \theta)\end{bmatrix}^{\mathrm{T}},
  29. m , n \mathcal{I}_{m,n}
  30. m , n = μ T θ m Σ - 1 μ θ n + 1 2 tr ( Σ - 1 Σ θ m Σ - 1 Σ θ n ) , \mathcal{I}_{m,n}=\frac{\partial\mu^{\mathrm{T}}}{\partial\theta_{m}}\Sigma^{-% 1}\frac{\partial\mu}{\partial\theta_{n}}+\frac{1}{2}\operatorname{tr}\left(% \Sigma^{-1}\frac{\partial\Sigma}{\partial\theta_{m}}\Sigma^{-1}\frac{\partial% \Sigma}{\partial\theta_{n}}\right),
  31. ( . . ) T (..)^{\mathrm{T}}
  32. μ θ m = [ μ 1 θ m μ 2 θ m μ N θ m ] T ; \frac{\partial\mu}{\partial\theta_{m}}=\begin{bmatrix}\frac{\partial\mu_{1}}{% \partial\theta_{m}}&\frac{\partial\mu_{2}}{\partial\theta_{m}}&\cdots&\frac{% \partial\mu_{N}}{\partial\theta_{m}}\end{bmatrix}^{\mathrm{T}};
  33. Σ θ m = [ Σ 1 , 1 θ m Σ 1 , 2 θ m Σ 1 , N θ m Σ 2 , 1 θ m Σ 2 , 2 θ m Σ 2 , N θ m Σ N , 1 θ m Σ N , 2 θ m Σ N , N θ m ] . \frac{\partial\Sigma}{\partial\theta_{m}}=\begin{bmatrix}\frac{\partial\Sigma_% {1,1}}{\partial\theta_{m}}&\frac{\partial\Sigma_{1,2}}{\partial\theta_{m}}&% \cdots&\frac{\partial\Sigma_{1,N}}{\partial\theta_{m}}\\ \\ \frac{\partial\Sigma_{2,1}}{\partial\theta_{m}}&\frac{\partial\Sigma_{2,2}}{% \partial\theta_{m}}&\cdots&\frac{\partial\Sigma_{2,N}}{\partial\theta_{m}}\\ \\ \vdots&\vdots&\ddots&\vdots\\ \\ \frac{\partial\Sigma_{N,1}}{\partial\theta_{m}}&\frac{\partial\Sigma_{N,2}}{% \partial\theta_{m}}&\cdots&\frac{\partial\Sigma_{N,N}}{\partial\theta_{m}}\end% {bmatrix}.
  34. m , n = μ T θ m Σ - 1 μ θ n . \mathcal{I}_{m,n}=\frac{\partial\mu^{\mathrm{T}}}{\partial\theta_{m}}\Sigma^{-% 1}\frac{\partial\mu}{\partial\theta_{n}}.
  35. ( β , θ ) = diag ( ( β ) , ( θ ) ) \mathcal{I}\left(\beta,\theta\right)=\,\text{diag}\left(\mathcal{I}\left(\beta% \right),\mathcal{I}\left(\theta\right)\right)
  36. ( β ) m , n = μ T β m Σ - 1 μ β n \mathcal{I}{{\left(\beta\right)}_{m,n}}=\frac{\partial{{\mu}^{\,\text{T}}}}{% \partial{{\beta}_{m}}}{{\Sigma}^{-1}}\frac{\partial\mu}{\partial{{\beta}_{n}}}
  37. ( θ ) m , n = 1 2 tr ( Σ - 1 Σ θ m Σ - 1 Σ θ n ) \mathcal{I}{{\left(\theta\right)}_{m,n}}=\frac{1}{2}\operatorname{tr}\left({{% \Sigma}^{-1}}\frac{\partial\Sigma}{\partial{{\theta}_{m}}}{{\Sigma}^{-1}}\frac% {\partial\Sigma}{\partial{{\theta}_{n}}}\right)
  38. η ( η ) = θ ( θ ( η ) ) ( d θ d η ) 2 {\mathcal{I}}_{\eta}(\eta)={\mathcal{I}}_{\theta}(\theta(\eta))\left(\frac{{% \mathrm{d}}\theta}{{\mathrm{d}}\eta}\right)^{2}
  39. η {\mathcal{I}}_{\eta}
  40. θ {\mathcal{I}}_{\theta}
  41. s y m b o l θ {symbol\theta}
  42. s y m b o l η {symbol\eta}
  43. s y m b o l θ {symbol\theta}
  44. s y m b o l η {symbol\eta}
  45. s y m b o l η ( s y m b o l η ) = s y m b o l J T s y m b o l θ ( s y m b o l θ ( s y m b o l η ) ) s y m b o l J {\mathcal{I}}_{symbol\eta}({symbol\eta})={symbolJ}^{\mathrm{T}}{\mathcal{I}}_{% symbol\theta}({symbol\theta}({symbol\eta})){symbolJ}
  46. s y m b o l J symbolJ
  47. J i j = θ i η j , J_{ij}=\frac{\partial\theta_{i}}{\partial\eta_{j}}\,,
  48. s y m b o l J T {symbolJ}^{\mathrm{T}}
  49. s y m b o l J {symbolJ}
  50. f ( x ; θ ) f(x;\theta)
  51. θ \theta
  52. D ( θ | | θ ) = f ( x ; θ ) log f ( x ; θ ) f ( x ; θ ) d x D(\theta^{\prime}||\theta)=\int f(x;\theta^{\prime})\log\frac{f(x;\theta^{% \prime})}{f(x;\theta)}\mathrm{d}x
  53. ( θ ) = ( d 2 d θ i d θ j D ( θ | | θ ) ) θ = θ \mathcal{I}(\theta)=\left(\frac{\mathrm{d}^{2}}{\mathrm{d}\theta^{\prime}_{i}% \mathrm{d}\theta^{\prime}_{j}}D(\theta^{\prime}||\theta)\right)_{\theta=\theta% ^{\prime}}
  54. θ \theta
  55. θ = θ \theta^{\prime}=\theta
  56. θ \theta^{\prime}
  57. θ \theta
  58. D ( θ | | θ ) = 1 2 ( θ - θ ) ( d 2 d θ i d θ j D ( θ | | θ ) ) θ = θ Fisher info. ( θ - θ ) + D(\theta^{\prime}||\theta)=\frac{1}{2}(\theta^{\prime}-\theta)^{\top}% \underbrace{\left(\frac{\mathrm{d}^{2}}{\mathrm{d}\theta^{\prime}_{i}\mathrm{d% }\theta^{\prime}_{j}}D(\theta^{\prime}||\theta)\right)_{\theta=\theta^{\prime}% }}_{\,\text{Fisher info.}}(\theta^{\prime}-\theta)+\cdots

Fisher_information_metric.html

  1. θ = ( θ 1 , θ 2 , , θ n ) \theta=(\theta_{1},\theta_{2},\ldots,\theta_{n})
  2. p ( x , θ ) p(x,\theta)
  3. θ \theta
  4. x x
  5. R p ( x , θ ) d x = 1 \int_{R}p(x,\theta)\,dx=1
  6. g j k ( θ ) = R log p ( x , θ ) θ j log p ( x , θ ) θ k p ( x , θ ) d x . g_{jk}(\theta)=\int_{R}\frac{\partial\log p(x,\theta)}{\partial\theta_{j}}% \frac{\partial\log p(x,\theta)}{\partial\theta_{k}}p(x,\theta)\,dx.
  7. θ \theta
  8. θ \theta
  9. i = - ln ( p ) i=-\ln(p)
  10. g j k ( θ ) = X 2 i ( x , θ ) θ j θ k p ( x , θ ) d x = E [ 2 i ( x , θ ) θ j θ k ] . g_{jk}(\theta)=\int_{X}\frac{\partial^{2}i(x,\theta)}{\partial\theta_{j}\,% \partial\theta_{k}}p(x,\theta)\,dx=\mathrm{E}\left[\frac{\partial^{2}i(x,% \theta)}{\partial\theta_{j}\,\partial\theta_{k}}\right].
  11. P = P ( θ ) P=P(\theta)
  12. Q = P ( θ 0 ) Q=P(\theta_{0})
  13. P = Q + j Δ θ j Q j P=Q+\sum_{j}\Delta\theta^{j}Q_{j}
  14. Δ θ j \Delta\theta^{j}
  15. θ \theta
  16. Q j = P θ j | θ = θ 0 Q_{j}=\left.\frac{\partial P}{\partial\theta^{j}}\right|_{\theta=\theta_{0}}
  17. D KL ( P Q ) D_{\mathrm{KL}}(P\|Q)
  18. θ = θ 0 \theta=\theta_{0}
  19. f θ 0 ( θ ) := D KL ( P Q ) = 1 2 j k Δ θ j Δ θ k g j k ( θ 0 ) f_{\theta_{0}}(\theta):=D_{\mathrm{KL}}(P\|Q)=\frac{1}{2}\sum_{jk}\Delta\theta% ^{j}\Delta\theta^{k}g_{jk}(\theta_{0})
  20. g j k g_{jk}
  21. f θ 0 f_{\theta_{0}}
  22. θ 0 \theta_{0}
  23. A = 1 2 a b θ j t g j k ( θ ) θ k t d t A=\frac{1}{2}\int_{a}^{b}\frac{\partial\theta^{j}}{\partial t}g_{jk}(\theta)% \frac{\partial\theta^{k}}{\partial t}dt
  24. Δ S = ( b - a ) A \Delta S=(b-a)A\,
  25. ( b - a ) a b θ j t g j k θ k t d t = 8 a b d J S D (b-a)\int_{a}^{b}\frac{\partial\theta^{j}}{\partial t}g_{jk}\frac{\partial% \theta^{k}}{\partial t}\,dt=8\int_{a}^{b}dJSD
  26. a b θ j t g j k θ k t d t = 8 a b d J S D \int_{a}^{b}\sqrt{\frac{\partial\theta^{j}}{\partial t}g_{jk}\frac{\partial% \theta^{k}}{\partial t}}\,dt=\sqrt{8}\int_{a}^{b}\sqrt{dJSD}
  27. i y i 2 = 1 \sum_{i}y_{i}^{2}=1
  28. h = i d y i d y i h=\sum_{i}dy_{i}\;dy_{i}
  29. d y i \textstyle dy_{i}
  30. y j \textstyle\frac{\partial}{\partial y_{j}}
  31. d y j ( y k ) = δ j k dy_{j}\left(\frac{\partial}{\partial y_{k}}\right)=\delta_{jk}
  32. h j k flat = h ( y j , y k ) = δ j k h^{\mathrm{flat}}_{jk}=h\left(\frac{\partial}{\partial y_{j}},\frac{\partial}{% \partial y_{k}}\right)=\delta_{jk}
  33. y y
  34. p i = y i 2 p_{i}=y_{i}^{2}
  35. i p i = 1 \sum_{i}p_{i}=1
  36. h = i d y i d y i = i d p i d p i = 1 4 i d p i d p i p i = 1 4 i p i d ( log p i ) d ( log p i ) \begin{aligned}\displaystyle h&\displaystyle=\sum_{i}dy_{i}\;dy_{i}=\sum_{i}d% \sqrt{p_{i}}\;d\sqrt{p_{i}}\\ &\displaystyle=\frac{1}{4}\sum_{i}\frac{dp_{i}\;dp_{i}}{p_{i}}=\frac{1}{4}\sum% _{i}p_{i}\;d(\log p_{i})\;d(\log p_{i})\end{aligned}
  37. θ \theta
  38. p i = p i ( θ ) p_{i}=p_{i}(\theta)
  39. h \displaystyle h
  40. g j k ( θ ) = 4 h j k fisher \displaystyle g_{jk}(\theta)=4h_{jk}^{\mathrm{fisher}}
  41. d θ j ( θ k ) = δ j k . d\theta_{j}\left(\frac{\partial}{\partial\theta_{k}}\right)=\delta_{jk}.
  42. θ \theta
  43. p p
  44. ψ ( x ; θ ) = p ( x ; θ ) e i α ( x ; θ ) \psi(x;\theta)=\sqrt{p(x;\theta)}\;e^{i\alpha(x;\theta)}
  45. ψ ( x ; θ ) \psi(x;\theta)
  46. p ( x ; θ ) p(x;\theta)
  47. α ( x ; θ ) \alpha(x;\theta)
  48. α ( x ; θ ) = 0 \alpha(x;\theta)=0
  49. X p ( x ; θ ) d x = 1 \int_{X}p(x;\theta)\,dx=1
  50. X | ψ ( x ; θ ) | 2 d x = 1 \int_{X}|\psi(x;\theta)|^{2}\,dx=1
  51. ψ ( x ; θ ) \psi(x;\theta)
  52. d s 2 = δ ψ δ ψ ψ ψ - δ ψ ψ ψ δ ψ ψ ψ 2 . ds^{2}=\frac{\langle\delta\psi\mid\delta\psi\rangle}{\langle\psi\mid\psi% \rangle}-\frac{\langle\delta\psi\mid\psi\rangle\;\langle\psi\mid\delta\psi% \rangle}{{\langle\psi\mid\psi\rangle}^{2}}.
  53. x ψ = ψ ( x ; θ ) \langle x\mid\psi\rangle=\psi(x;\theta)
  54. ϕ ψ = X ϕ * ( x ; θ ) ψ ( x ; θ ) d x . \langle\phi\mid\psi\rangle=\int_{X}\phi^{*}(x;\theta)\psi(x;\theta)\,dx.
  55. | δ ψ |\delta\psi\rangle
  56. δ ψ = ( δ p 2 p + i δ α ) ψ \delta\psi=\left(\frac{\delta p}{2p}+i\delta\alpha\right)\psi
  57. d s 2 = \displaystyle ds^{2}=
  58. δ α = 0 \delta\alpha=0
  59. δ d \delta\to d
  60. d s 2 h ds^{2}\to h
  61. h = 1 4 E [ ( d log p ) 2 ] + E [ ( d α ) 2 ] - ( E [ d α ] ) 2 - i 2 E [ d log p d α ] h=\frac{1}{4}\mathrm{E}\left[(d\log p)^{2}\right]+\mathrm{E}\left[(d\alpha)^{2% }\right]-\left(\mathrm{E}\left[d\alpha\right]\right)^{2}-\frac{i}{2}\mathrm{E}% \left[d\log p\wedge d\alpha\right]
  62. h j k = \displaystyle h_{jk}=
  63. α = 0 \alpha=0
  64. ( X , Σ , μ ) (X,\Sigma,\mu)
  65. ( Ω , , P ) (\Omega,\mathcal{F},P)
  66. Ω = X \Omega=X
  67. = Σ \mathcal{F}=\Sigma
  68. P = μ P=\mu
  69. μ \mu
  70. Σ \Sigma
  71. μ S ( X ) \mu\in S(X)
  72. T μ S T_{\mu}S
  73. g ( σ 1 , σ 2 ) = X d σ 1 d μ d σ 2 d μ d μ g(\sigma_{1},\sigma_{2})=\int_{X}\frac{d\sigma_{1}}{d\mu}\frac{d\sigma_{2}}{d% \mu}d\mu
  74. σ 1 \sigma_{1}
  75. σ 2 \sigma_{2}
  76. σ 1 , σ 2 T μ S \sigma_{1},\sigma_{2}\in T_{\mu}S
  77. μ \mu
  78. μ \mu
  79. θ \theta
  80. θ \theta
  81. θ \theta
  82. σ T μ S \sigma\in T_{\mu}S
  83. p = exp ( σ ) p=\exp(\sigma)
  84. p S ( X ) p\in S(X)
  85. μ \mu
  86. p S ( X ) p\in S(X)
  87. μ \mu

Fitness_proportionate_selection.html

  1. f i f_{i}
  2. i i
  3. p i = f i Σ j = 1 N f j p_{i}=\frac{f_{i}}{\Sigma_{j=1}^{N}f_{j}}
  4. N N
  5. i i
  6. f i / f M f_{i}/f_{M}
  7. f M f_{M}

Fixed-point_arithmetic.html

  1. I I
  2. Q Q
  3. 2 I 2I
  4. 2 Q 2Q

Fixed-rate_mortgage.html

  1. c c
  2. r r
  3. N N
  4. P 0 P_{0}
  5. c c
  6. c = r 1 - ( 1 + r ) - N P 0 c={r\over{1-(1+r)^{-N}}}P_{0}
  7. P 0 = 200000 P_{0}=200000
  8. r = 6.5 / 100 / 12 r=6.5/100/12
  9. N = 30 * 12 = 360 N=30*12=360
  10. c = $ 1264.14 c=\$1264.14
  11. = 1264.14 {}=1264.14
  12. P 0 P_{0}
  13. P 1 = P 0 + P 0 * r - c P_{1}=P_{0}+P_{0}*r-c
  14. P 1 = P 0 ( 1 + r ) - c P_{1}=P_{0}(1+r)-c
  15. P 2 = P 1 ( 1 + r ) - c P_{2}=P_{1}(1+r)-c
  16. P 1 P_{1}
  17. P 2 = ( P 0 ( 1 + r ) - c ) ( 1 + r ) - c P_{2}=(P_{0}(1+r)-c)(1+r)-c
  18. P 2 = P 0 ( 1 + r ) 2 - c ( 1 + r ) - c P_{2}=P_{0}(1+r)^{2}-c(1+r)-c
  19. P 3 = P 2 ( 1 + r ) - c P_{3}=P_{2}(1+r)-c
  20. P 2 P_{2}
  21. P 3 = ( P 0 ( 1 + r ) 2 - c ( 1 + r ) - c ) ( 1 + r ) - c P_{3}=(P_{0}(1+r)^{2}-c(1+r)-c)(1+r)-c
  22. P 3 = P 0 ( 1 + r ) 3 - c ( 1 + r ) 2 - c ( 1 + r ) - c P_{3}=P_{0}(1+r)^{3}-c(1+r)^{2}-c(1+r)-c
  23. P N = P N - 1 ( 1 + r ) - c P_{N}=P_{N-1}(1+r)-c
  24. P N = P 0 ( 1 + r ) N - c ( 1 + r ) N - 1 - c ( 1 + r ) N - 2 . - c P_{N}=P_{0}(1+r)^{N}-c(1+r)^{N-1}-c(1+r)^{N-2}....-c
  25. P N = P 0 ( 1 + r ) N - c ( ( 1 + r ) N - 1 + ( 1 + r ) N - 2 . + 1 ) P_{N}=P_{0}(1+r)^{N}-c((1+r)^{N-1}+(1+r)^{N-2}....+1)
  26. P N = P 0 ( 1 + r ) N - c ( S ) P_{N}=P_{0}(1+r)^{N}-c(S)
  27. S = ( 1 + r ) N - 1 + ( 1 + r ) N - 2 . + 1 S=(1+r)^{N-1}+(1+r)^{N-2}....+1
  28. S ( 1 + r ) = ( 1 + r ) N + ( 1 + r ) N - 1 . + ( 1 + r ) S(1+r)=(1+r)^{N}+(1+r)^{N-1}....+(1+r)
  29. S S
  30. S ( 1 + r ) S(1+r)
  31. S ( 1 + r ) - S = ( 1 + r ) N - 1 S(1+r)-S=(1+r)^{N}-1
  32. S ( ( 1 + r ) - 1 ) = ( 1 + r ) N - 1 S((1+r)-1)=(1+r)^{N}-1
  33. S ( r ) = ( 1 + r ) N - 1 S(r)=(1+r)^{N}-1
  34. S = ( 1 + r ) N - 1 r S={{(1+r)^{N}-1}\over r}
  35. P N = P 0 ( 1 + r ) N - c ( 1 + r ) N - 1 r P_{N}=P_{0}(1+r)^{N}-c{{(1+r)^{N}-1}\over r}
  36. P N P_{N}
  37. 0 = P 0 ( 1 + r ) N - c ( 1 + r ) N - 1 r 0=P_{0}(1+r)^{N}-c{{(1+r)^{N}-1}\over r}
  38. c c
  39. c = r ( 1 + r ) N ( 1 + r ) N - 1 P 0 c={{r(1+r)^{N}}\over{(1+r)^{N}-1}}P_{0}
  40. ( 1 + r ) N (1+r)^{N}
  41. c = r 1 - ( 1 + r ) - N P 0 c={r\over{1-(1+r)^{-N}}}P_{0}

Fixed_point_(mathematics).html

  1. f ( x ) = x 2 - 3 x + 4 , f(x)=x^{2}-3x+4,
  2. x , f ( x ) , f ( f ( x ) ) , f ( f ( f ( x ) ) ) , x,\ f(x),\ f(f(x)),\ f(f(f(x))),\dots
  3. y = x y=x
  4. | f ( x 0 ) | < 1 |f\,^{\prime}(x_{0})|<1
  5. X X
  6. f : X X f\colon X\to X
  7. x X x\in X
  8. f ( x ) = x f(x)=x

Flag_(linear_algebra).html

  1. { 0 } = V 0 \sub V 1 \sub V 2 \sub \sub V k = V . \{0\}=V_{0}\sub V_{1}\sub V_{2}\sub\cdots\sub V_{k}=V.
  2. 0 = d 0 < d 1 < d 2 < < d k = n , 0=d_{0}<d_{1}<d_{2}<\cdots<d_{k}=n,
  3. 0 < e 1 < e 1 , e 2 < < e 1 , , e n = K n . 0<\left\langle e_{1}\right\rangle<\left\langle e_{1},e_{2}\right\rangle<\cdots% <\left\langle e_{1},\ldots,e_{n}\right\rangle=K^{n}.
  4. v i V i - 1 < V i v_{i}\in V_{i-1}^{\perp}<V_{i}
  5. T ( V i ) < V i T(V_{i})<V_{i}
  6. d i - d i - 1 d_{i}-d_{i-1}
  7. 𝐅 2 \mathbf{F}_{2}
  8. { 0 } { 0 , 1 } { 0 , 1 , 2 } \{0\}\subset\{0,1\}\subset\{0,1,2\}
  9. ( 0 , 1 , 2 ) (0,1,2)

Flag_of_Iran.html

  1. 1 : 75 28 ( 7 5 - 15 ) 1:\frac{75}{28}(7\sqrt{5}-15)

Flat_module.html

  1. F M : M o d ( R ) M o d ( R ) , N M R N F_{M}:Mod(R)\to Mod(R),\quad N\mapsto M\otimes_{R}N
  2. M o d ( R ) Mod(R)
  3. R R
  4. ϕ : K L \phi:K\to L
  5. R R
  6. K K
  7. L L
  8. F M ( ϕ ) : M R K M R L F_{M}(\phi):M\otimes_{R}K\to M\otimes_{R}L
  9. I R I\hookrightarrow R
  10. I R M R R M M I\otimes_{R}M\to R\otimes_{R}M\cong M
  11. R R
  12. { F α } α \{F_{\alpha}\}_{\alpha}
  13. α \alpha
  14. F α F_{\alpha}
  15. R R
  16. M M
  17. lim α F α = M \underrightarrow{\lim}_{\alpha}F_{\alpha}=M
  18. M M
  19. r T x = i = 1 k r i x i = 0 r^{T}x=\sum_{i=1}^{k}r_{i}x_{i}=0
  20. r i R , x i M r_{i}\in R,x_{i}\in M
  21. A R k × j A\in R^{k\times j}
  22. A y = x Ay=x
  23. y M j y\in M^{j}
  24. r T A = 0 r^{T}A=0
  25. R R
  26. N N
  27. Tor 1 R ( N , M ) = 0 \mathrm{Tor}_{1}^{R}(N,M)=0
  28. I R I\subset R
  29. Tor 1 R ( R / I , M ) = 0 \mathrm{Tor}_{1}^{R}(R/I,M)=0
  30. f : F M f:F\to M
  31. F F
  32. R R
  33. R R
  34. K ker f K\leq\ker f
  35. f f
  36. R R
  37. G G
  38. K K
  39. \otimes
  40. \otimes
  41. S - 1 R S^{-1}R
  42. \mathbb{Q}
  43. \mathbb{Z}
  44. / n \mathbb{Z}/n\mathbb{Z}
  45. \mathbb{Z}
  46. n : , x n x n:\mathbb{Z}\to\mathbb{Z},\,x\mapsto nx
  47. / n \mathbb{Z}/n\mathbb{Z}
  48. / \mathbb{Q}/\mathbb{Z}
  49. \mathbb{Z}
  50. R = k [ t ] , k R=k[t],k
  51. S = R [ x ] / ( t x - 1 ) S=R[x]/(tx-1)
  52. R [ t - 1 ] R[t^{-1}]
  53. R [ x ] / ( t x - t ) R[x]/(tx-t)
  54. A A ^ A\to\widehat{A}
  55. i I M i \bigoplus_{i\in I}M_{i}
  56. M i M_{i}
  57. M P M_{P}
  58. R P R_{P}
  59. 𝔪 R S 𝔪 S \mathfrak{m}_{R}S\subset\mathfrak{m}_{S}
  60. Tor 1 R ( M , R / 𝔪 R ) = 0. \operatorname{Tor}_{1}^{R}(M,R/\mathfrak{m}_{R})=0.
  61. dim S = dim R + dim S / 𝔪 R S \operatorname{dim}S=\operatorname{dim}R+\operatorname{dim}S/\mathfrak{m}_{R}S
  62. f : R S f\colon R\to S
  63. f * : Spec ( S ) Spec ( R ) f^{*}\colon\mathrm{Spec}(S)\to\mathrm{Spec}(R)
  64. n 1 n\geq 1
  65. n 1 n\geq 1
  66. n 1 n\geq 1

Flat_morphism.html

  1. 𝒪 Y \mathcal{O}_{Y^{\prime}}
  2. 𝒪 X \mathcal{O}_{X^{\prime}}
  3. f * J 𝒪 X f^{*}J\to\mathcal{O}_{X}
  4. 𝒪 X \mathcal{O}_{X}
  5. 𝒪 Y \mathcal{O}_{Y}
  6. Γ ( Y , G ) Γ ( X , f * G ) \Gamma(Y,G)\to\Gamma(X,f^{*}G)
  7. f ( Spec 𝒪 X , x ) = Spec 𝒪 Y , f ( x ) f(\operatorname{Spec}\,\mathcal{O}_{X,x})=\operatorname{Spec}\,\mathcal{O}_{Y,% f(x)}
  8. f - 1 ( Z ¯ ) = f - 1 ( Z ) ¯ f^{-1}(\bar{Z})=\overline{f^{-1}(Z)}
  9. [ x 2 , y 2 , x y ] [ x , y ] \mathbb{C}[x^{2},y^{2},xy]\subset\mathbb{C}[x,y]

Flex_lexical_analyser.html

  1. O ( n ) O(n)

Fluctuation-dissipation_theorem.html

  1. D = μ k B T D={\mu\,k_{B}T}
  2. k B T k_{B}T
  3. Δ ν \Delta\nu
  4. V 2 = 4 R k B T Δ ν . \langle V^{2}\rangle=4Rk_{B}T\,\Delta\nu.
  5. x ( t ) x(t)
  6. H 0 ( x ) H_{0}(x)
  7. x ( t ) x(t)
  8. x 0 \langle x\rangle_{0}
  9. S x ( ω ) = x ^ ( ω ) x ^ * ( ω ) S_{x}(\omega)=\hat{x}(\omega)\hat{x}^{*}(\omega)
  10. f ( t ) f(t)
  11. H ( x ) = H 0 ( x ) + f ( t ) x H(x)=H_{0}(x)+f(t)x
  12. x ( t ) x(t)
  13. f ( t ) f(t)
  14. χ ( t ) \chi(t)
  15. x ( t ) = x 0 + - t f ( τ ) χ ( t - τ ) d τ , \langle x(t)\rangle=\langle x\rangle_{0}+\int\limits_{-\infty}^{t}\!f(\tau)% \chi(t-\tau)\,d\tau,
  16. τ = - \tau=-\infty
  17. x x
  18. χ ^ ( ω ) \hat{\chi}(\omega)
  19. χ ( t ) \chi(t)
  20. S x ( ω ) = 2 k B T ω Im χ ^ ( ω ) . S_{x}(\omega)=\frac{2k_{\mathrm{B}}T}{\omega}\mathrm{Im}\,\hat{\chi}(\omega).
  21. x x
  22. f ( t ) = F sin ( ω t + ϕ ) f(t)=F\sin(\omega t+\phi)
  23. 2 k B T / ω 2k_{\mathrm{B}}T/\omega
  24. coth ( ω / 2 k B T ) {\hbar}\,\coth(\hbar\omega/2k_{\mathrm{B}}T)
  25. 0 \hbar\to 0
  26. 2 k B T / ω 2k_{\mathrm{B}}T/\omega
  27. f ( t ) = f 0 θ ( - t ) . f(t)=f_{0}\theta(-t).
  28. P ( x , t | x , 0 ) P(x^{\prime},t|x,0)
  29. x ( t ) = d x d x x P ( x , t | x , 0 ) W ( x , 0 ) . \langle x(t)\rangle=\int dx^{\prime}\int dx\,x^{\prime}P(x^{\prime},t|x,0)W(x,% 0).
  30. H ( x ) = H 0 ( x ) + x f 0 H(x)=H_{0}(x)+xf_{0}
  31. W ( x , 0 ) = exp ( - β H ( x ) ) d x exp ( - β H ( x ) ) , W(x,0)=\frac{\exp(-\beta H(x))}{\int dx^{\prime}\,\exp(-\beta H(x^{\prime}))}\;,
  32. β - 1 = k B T \beta^{-1}=k_{\rm B}T
  33. β x f 0 1 \beta xf_{0}\ll 1
  34. W ( x , 0 ) W 0 ( x ) ( 1 - β f 0 x ) , W(x,0)\approx W_{0}(x)(1-\beta f_{0}x),
  35. W 0 ( x ) W_{0}(x)
  36. x ( t ) \langle x(t)\rangle
  37. A ( t ) = x ( t ) x ( 0 ) 0 . A(t)=\langle x(t)x(0)\rangle_{0}.
  38. x ( t ) - x 0 \langle x(t)\rangle-\langle x\rangle_{0}
  39. f 0 0 d τ χ ( τ ) θ ( τ - t ) = β f 0 A ( t ) f_{0}\int_{0}^{\infty}d\tau\,\chi(\tau)\theta(\tau-t)=\beta f_{0}A(t)
  40. - χ ^ ( ω ) = i ω β 0 e - i ω t A ( t ) d t - β A 0 . -\hat{\chi}(\omega)=i\omega\beta\int\limits_{0}^{\infty}\mathrm{e}^{-i\omega t% }A(t)\,dt-\beta A_{0}.
  41. A ( t ) A(t)
  42. 2 Im [ χ ^ ( ω ) ] = ω β A ^ ( ω ) . 2\,\mathrm{Im}[\hat{\chi}(\omega)]=\omega\beta\hat{A}(\omega).
  43. S x ( ω ) = A ^ ( ω ) . S_{x}(\omega)=\hat{A}(\omega).
  44. S x ( ω ) = 2 k B T ω Im [ χ ^ ( ω ) ] . S_{x}(\omega)=\frac{2k\text{B}T}{\omega}\,\mathrm{Im}[\hat{\chi}(\omega)].

Fluctuation_theorem.html

  1. Σ ¯ t \overline{\Sigma}_{t}
  2. Σ ¯ t \overline{\Sigma}_{t}
  3. Pr ( Σ ¯ t = A ) Pr ( Σ ¯ t = - A ) = e A t . \frac{\Pr(\overline{\Sigma}_{t}=A)}{\Pr(\overline{\Sigma}_{t}=-A)}=e^{At}.
  4. Σ \Sigma
  5. Σ ¯ t 0 , t . \left\langle{\overline{\Sigma}_{t}}\right\rangle\geq 0,\quad\forall t.
  6. exp [ - Σ ¯ t t ] = 1 , for all t . \left\langle{\exp[-\overline{\Sigma}_{t}\;t]}\right\rangle=1,\quad\,\text{ for% all }t.
  7. Ω t ( Γ ) = 0 t d s Ω ( Γ ; s ) ln [ < m t p l > f ( Γ , 0 ) f ( Γ ( t ) , 0 ) ] + Δ Q ( Γ ; t ) k T \Omega_{t}(\Gamma)=\int_{0}^{t}{ds\;\Omega(\Gamma;s)}\equiv\ln\left[{\frac{<}{% m}tpl>{{f(\Gamma,0)}}{{f(\Gamma(t),0)}}}\right]+\frac{{\Delta Q(\Gamma;t)}}{kT}
  8. f ( Γ , 0 ) f(\Gamma,0)
  9. Γ \Gamma
  10. Γ ( t ) \Gamma(t)
  11. f ( Γ ( t ) , 0 ) f(\Gamma(t),0)
  12. f ( Γ ( t ) , 0 ) 0 , Γ ( 0 ) f(\Gamma(t),0)\neq 0,\;\forall\Gamma(0)
  13. Δ Q ( t ) \Delta Q(t)
  14. Ω = - J F e V / k T \Omega=-JF_{e}V/{kT}
  15. F e F_{e}

Fluorescence_recovery_after_photobleaching.html

  1. D = w 2 4 t D D=\frac{w^{2}}{4t_{D}}
  2. w w
  3. I 0 I_{0}
  4. I 1 I_{1}
  5. f ( t ) = e - 2 τ D / t ( I 0 ( 2 τ D / t ) + I 1 ( 2 τ D / t ) ) f(t)=e^{-2\tau_{D}/t}\left(I_{0}(2\tau_{D}/t)+I_{1}(2\tau_{D}/t)\right)
  6. τ D \tau_{D}
  7. t t
  8. f ( t ) f(t)
  9. t t
  10. w w
  11. τ D = w 2 / ( 4 D ) \tau_{D}=w^{2}/(4D)
  12. f b f_{b}
  13. t = 0 t=0
  14. f b ( r ) = b , r < w f_{b}(r)=b,~{}~{}r<w
  15. f b ( r ) = 0 , r > w f_{b}(r)=0,~{}~{}r>w
  16. r r
  17. τ D \tau_{D}
  18. τ D \tau_{D}
  19. f ( t ) = 1 - e - k off t f(t)=1-e^{-k_{\,\text{off}}t}
  20. 1 / k off r 2 / D 1/k_{\,\text{off}}>>r^{2}/D
  21. r - 2 r^{-2}

Folk_psychology.html

  1. s ( P , E i ) = k s ( P , E i k ) s(P,E_{i})=\prod_{k}s(P,E_{ik})

Ford_circle.html

  1. p q \frac{p}{q}
  2. ( p / q , 1 / ( 2 q 2 ) ) (p/q,1/(2q^{2}))
  3. 1 / ( 2 q 2 ) , 1/(2q^{2}),
  4. p / q p/q
  5. p p
  6. q q
  7. y = 0 , y=0,
  8. 1 r middle = 1 r left + 1 r right . \frac{1}{\sqrt{r\text{middle}}}=\frac{1}{\sqrt{r\text{left}}}+\frac{1}{\sqrt{r% \text{right}}}.
  9. p / q p/q
  10. C [ p / q ] C[p/q]
  11. C [ p , q ] . C[p,q].
  12. y = 1 y=1
  13. p = 1 , q = 0. p=1,q=0.
  14. p / q p/q
  15. C [ p / q ] C[p/q]
  16. C [ r / s ] C[r/s]
  17. | p s - q r | = 1 , |ps-qr|=1,
  18. r / s r/s
  19. p / q p/q
  20. C [ r / s ] C[r/s]
  21. r / s r/s
  22. p / q p/q
  23. p / q p/q
  24. r / s r/s
  25. C [ p / q ] C[p/q]
  26. C [ r / s ] C[r/s]
  27. ( p / q , 0 ) (p/q,0)
  28. ( r / s , 0 ) (r/s,0)
  29. x x
  30. y = 0 y=0
  31. y = 1 y=1
  32. C [ 0 / 1 ] . C[0/1].
  33. φ , \varphi,
  34. ζ , \zeta,
  35. ζ ( 3 ) . \zeta(3).
  36. { C [ p , q ] : 0 p q 1 } \left\{C[p,q]:0\leq\frac{p}{q}\leq 1\right\}
  37. A = q 1 ( p , q ) = 1 1 p < q π ( 1 2 q 2 ) 2 . A=\sum_{q\geq 1}\sum_{(p,q)=1\atop 1\leq p<q}\pi\left(\frac{1}{2q^{2}}\right)^% {2}.
  38. A = π 4 q 1 1 q 4 ( p , q ) = 1 1 p < q 1 = π 4 q 1 φ ( q ) q 4 = π 4 ζ ( 3 ) ζ ( 4 ) , A=\frac{\pi}{4}\sum_{q\geq 1}\frac{1}{q^{4}}\sum_{(p,q)=1\atop 1\leq p<q}1=% \frac{\pi}{4}\sum_{q\geq 1}\frac{\varphi(q)}{q^{4}}=\frac{\pi}{4}\frac{\zeta(3% )}{\zeta(4)},
  39. φ ( q ) . \varphi(q).
  40. ζ ( 4 ) = π 4 / 90 , \zeta(4)=\pi^{4}/90,
  41. A = 45 2 ζ ( 3 ) π 3 0.872284041. A=\frac{45}{2}\frac{\zeta(3)}{\pi^{3}}\approx 0.872284041.

Formal_group.html

  1. \mathbb{Z}
  2. F ( x , y ) = x + y . F(x,y)=x+y.
  3. F ( x , y ) = x + y + x y . F(x,y)=x+y+xy.
  4. F ( x , y ) = ( x 1 - y 4 + y 1 - x 4 ) / ( 1 + x 2 y 2 ) F(x,y)=(x\sqrt{1-y^{4}}+y\sqrt{1-x^{4}})/(1+x^{2}y^{2})
  5. 0 x d t 1 - t 4 + 0 y d t 1 - t 4 = 0 F ( x , y ) d t 1 - t 4 . \int_{0}^{x}{dt\over\sqrt{1-t^{4}}}+\int_{0}^{y}{dt\over\sqrt{1-t^{4}}}=\int_{% 0}^{F(x,y)}{dt\over\sqrt{1-t^{4}}}.
  6. a x p h ax^{p^{h}}
  7. E p - 1 E_{p-1}
  8. G G
  9. G G
  10. G ^ \widehat{G}
  11. Spf ( R [ [ T 1 , , T n ] ] ) \mathrm{Spf}(R[[T_{1},\ldots,T_{n}]])
  12. e ( F ( x , y ) ) = F ( e ( x ) , e ( y ) ) . e(F(x,y))=F(e(x),e(y)).

Formula_for_primes.html

  1. P ( 1 ) 0 ( mod p ) P(1)\equiv 0\;\;(\mathop{{\rm mod}}p)
  2. P ( 1 + k p ) 0 ( mod p ) P(1+kp)\equiv 0\;\;(\mathop{{\rm mod}}p)
  3. P ( 1 + k p ) P(1+kp)
  4. P ( 1 + k p ) = P ( 1 ) P(1+kp)=P(1)
  5. 163 = 4 41 - 1 163=4\cdot 41-1
  6. p = 2 , 3 , 5 , 11 , and 17 p=2,3,5,11,\,\text{ and }17
  7. L ( n ) = a n + b L(n)=an+b
  8. L ( n ) = a n + b L(n)=an+b
  9. α 0 = w z + h + j - q = 0 \alpha_{0}=wz+h+j-q=0
  10. α 1 = ( g k + 2 g + k + 1 ) ( h + j ) + h - z = 0 \alpha_{1}=(gk+2g+k+1)(h+j)+h-z=0
  11. α 2 = 16 ( k + 1 ) 3 ( k + 2 ) ( n + 1 ) 2 + 1 - f 2 = 0 \alpha_{2}=16(k+1)^{3}(k+2)(n+1)^{2}+1-f^{2}=0
  12. α 3 = 2 n + p + q + z - e = 0 \alpha_{3}=2n+p+q+z-e=0
  13. α 4 = e 3 ( e + 2 ) ( a + 1 ) 2 + 1 - o 2 = 0 \alpha_{4}=e^{3}(e+2)(a+1)^{2}+1-o^{2}=0
  14. α 5 = ( a 2 - 1 ) y 2 + 1 - x 2 = 0 \alpha_{5}=(a^{2}-1)y^{2}+1-x^{2}=0
  15. α 6 = 16 r 2 y 4 ( a 2 - 1 ) + 1 - u 2 = 0 \alpha_{6}=16r^{2}y^{4}(a^{2}-1)+1-u^{2}=0
  16. α 7 = n + l + v - y = 0 \alpha_{7}=n+l+v-y=0
  17. α 8 = ( a 2 - 1 ) l 2 + 1 - m 2 = 0 \alpha_{8}=(a^{2}-1)l^{2}+1-m^{2}=0
  18. α 9 = a i + k + 1 - l - i = 0 \alpha_{9}=ai+k+1-l-i=0
  19. α 10 = ( ( a + u 2 ( u 2 - a ) ) 2 - 1 ) ( n + 4 d y ) 2 + 1 - ( x + c u ) 2 = 0 \alpha_{10}=((a+u^{2}(u^{2}-a))^{2}-1)(n+4dy)^{2}+1-(x+cu)^{2}=0
  20. α 11 = p + l ( a - n - 1 ) + b ( 2 a n + 2 a - n 2 - 2 n - 2 ) - m = 0 \alpha_{11}=p+l(a-n-1)+b(2an+2a-n^{2}-2n-2)-m=0
  21. α 12 = q + y ( a - p - 1 ) + s ( 2 a p + 2 a - p 2 - 2 p - 2 ) - x = 0 \alpha_{12}=q+y(a-p-1)+s(2ap+2a-p^{2}-2p-2)-x=0
  22. α 13 = z + p l ( a - p ) + t ( 2 a p - p 2 - 1 ) - p m = 0 \alpha_{13}=z+pl(a-p)+t(2ap-p^{2}-1)-pm=0
  23. ( k + 2 ) ( 1 - α 0 2 - α 1 2 - - α 13 2 ) > 0 (k+2)(1-\alpha_{0}^{2}-\alpha_{1}^{2}-\cdots-\alpha_{13}^{2})>0
  24. ( k + 2 ) ( 1 - (k+2)(1-
  25. [ w z + h + j - q ] 2 - [wz+h+j-q]^{2}-
  26. [ ( g k + 2 g + k + 1 ) ( h + j ) + h - z ] 2 - [(gk+2g+k+1)(h+j)+h-z]^{2}-
  27. [ 16 ( k + 1 ) 3 ( k + 2 ) ( n + 1 ) 2 + 1 - f 2 ] 2 - [16(k+1)^{3}(k+2)(n+1)^{2}+1-f^{2}]^{2}-
  28. [ 2 n + p + q + z - e ] 2 - [2n+p+q+z-e]^{2}-
  29. [ e 3 ( e + 2 ) ( a + 1 ) 2 + 1 - o 2 ] 2 - [e^{3}(e+2)(a+1)^{2}+1-o^{2}]^{2}-
  30. [ ( a 2 - 1 ) y 2 + 1 - x 2 ] 2 - [(a^{2}-1)y^{2}+1-x^{2}]^{2}-
  31. [ 16 r 2 y 4 ( a 2 - 1 ) + 1 - u 2 ] 2 - [16r^{2}y^{4}(a^{2}-1)+1-u^{2}]^{2}-
  32. [ n + l + v - y ] 2 - [n+l+v-y]^{2}-
  33. [ ( a 2 - 1 ) l 2 + 1 - m 2 ] 2 - [(a^{2}-1)l^{2}+1-m^{2}]^{2}-
  34. [ a i + k + 1 - l - i ] 2 - [ai+k+1-l-i]^{2}-
  35. [ ( ( a + u 2 ( u 2 - a ) ) 2 - 1 ) ( n + 4 d y ) 2 + 1 - ( x + c u ) 2 ] 2 - [((a+u^{2}(u^{2}-a))^{2}-1)(n+4dy)^{2}+1-(x+cu)^{2}]^{2}-
  36. [ p + l ( a - n - 1 ) + b ( 2 a n + 2 a - n 2 - 2 n - 2 ) - m ] 2 - [p+l(a-n-1)+b(2an+2a-n^{2}-2n-2)-m]^{2}-
  37. [ q + y ( a - p - 1 ) + s ( 2 a p + 2 a - p 2 - 2 p - 2 ) - x ] 2 - [q+y(a-p-1)+s(2ap+2a-p^{2}-2p-2)-x]^{2}-
  38. [ z + p l ( a - p ) + t ( 2 a p - p 2 - 1 ) - p m ] 2 ) [z+pl(a-p)+t(2ap-p^{2}-1)-pm]^{2})
  39. > 0 >0
  40. A 3 n \lfloor A^{3^{n}}\;\rfloor
  41. a n = a n - 1 + gcd ( n , a n - 1 ) , a 1 = 7 , a_{n}=a_{n-1}+\operatorname{gcd}(n,a_{n-1}),\quad a_{1}=7,

Fourier_inversion_theorem.html

  1. f : f:\mathbb{R}\rightarrow\mathbb{C}
  2. ( f ) ( ξ ) := n e - 2 π i y ξ f ( y ) d y , (\mathcal{F}f)(\xi):=\int_{\mathbb{R}^{n}}e^{-2\pi iy\cdot\xi}\,f(y)\,dy,
  3. f ( x ) = n e 2 π i x ξ ( f ) ( ξ ) d ξ . f(x)=\int_{\mathbb{R}^{n}}e^{2\pi ix\cdot\xi}\,(\mathcal{F}f)(\xi)\,d\xi.
  4. f ( x ) = n n e 2 π i ( x - y ) ξ f ( y ) d y d ξ . f(x)=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}e^{2\pi i(x-y)\cdot\xi}\,f(y)\,% dy\,d\xi.
  5. R R
  6. R f ( x ) := f ( x ) Rf(x):=f(−x)
  7. - 1 = R = R . \mathcal{F}^{-1}=\mathcal{F}R=R\mathcal{F}.
  8. f f
  9. f f
  10. x x
  11. x x
  12. x x
  13. f f
  14. ( f ) ( ξ ) := n e - 2 π i y ξ f ( y ) d y . (\mathcal{F}f)(\xi):=\int_{\mathbb{R}^{n}}e^{-2\pi iy\cdot\xi}\,f(y)\,dy.
  15. g g
  16. - 1 g ( x ) := n e 2 π i x ξ g ( ξ ) d ξ . \mathcal{F}^{-1}g(x):=\int_{\mathbb{R}^{n}}e^{2\pi ix\cdot\xi}\,g(\xi)\,d\xi.
  17. - 1 ( g ) ( x ) = g ( x ) . \mathcal{F}^{-1}(\mathcal{F}g)(x)=g(x).
  18. f ( x ) = n n e 2 π i ( x - y ) ξ f ( y ) d y d ξ . f(x)=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}e^{2\pi i(x-y)\cdot\xi}\,f(y)\,% dy\,d\xi.
  19. f f
  20. f ( x ) = n n cos ( 2 π ( x - y ) ξ ) f ( y ) d y d ξ . f(x)=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\cos(2\pi(x-y)\cdot\xi)\,f(y)\,% dy\,d\xi.
  21. g g
  22. R R
  23. R g ( x ) := g ( - x ) . Rg(x):=g(-x).
  24. - 1 f := R f = R f . \mathcal{F}^{-1}f:=R\mathcal{F}f=\mathcal{F}Rf.
  25. R f R\mathcal{F}f
  26. R f \mathcal{F}Rf
  27. - 1 f \mathcal{F}^{-1}f
  28. - 1 ( f ) ( x ) = f ( x ) \mathcal{F}^{-1}(\mathcal{F}f)(x)=f(x)
  29. - 1 ( f ) ( x ) = f ( x ) . \mathcal{F}^{-1}(\mathcal{F}f)(x)=f(x).
  30. - 1 \mathcal{F}^{-1}
  31. ( - 1 f ) ( ξ ) = f ( ξ ) . \mathcal{F}(\mathcal{F}^{-1}f)(\xi)=f(\xi).
  32. - 1 \mathcal{F}^{-1}
  33. \mathcal{F}
  34. ζ := ξ ζ:=−ξ
  35. f \displaystyle f
  36. - 1 f \mathcal{F}^{-1}f
  37. f = - 1 ( f ) = R f = ( - 1 f ) . f=\mathcal{F}^{-1}(\mathcal{F}f)=\mathcal{F}R\mathcal{F}f=\mathcal{F}(\mathcal% {F}^{-1}f).
  38. f f
  39. f = g f=g
  40. g g
  41. - 1 ( f ) ( x ) = g ( x ) \mathcal{F}^{-1}(\mathcal{F}f)(x)=g(x)
  42. - 1 g ( x ) := lim R - R R e 2 π i x ξ g ( ξ ) d ξ . \mathcal{F}^{-1}g(x):=\lim_{R\to\infty}\int_{-R}^{R}e^{2\pi ix\xi}\,g(\xi)\,d\xi.
  43. x x∈ℝ
  44. - 1 ( f ) ( x ) = 1 2 ( f ( x - ) + f ( x + ) ) , \mathcal{F}^{-1}(\mathcal{F}f)(x)=\frac{1}{2}(f(x_{-})+f(x_{+})),
  45. - 1 ( f ) ( x ) \mathcal{F}^{-1}(\mathcal{F}f)(x)
  46. f f
  47. x x
  48. f f
  49. f ( x ) f(x)
  50. - 1 g ( x ) := lim R φ ( ξ / R ) e 2 π i x ξ g ( ξ ) d ξ , φ ( ξ ) := e - ξ 2 . \mathcal{F}^{-1}g(x):=\lim_{R\to\infty}\int_{\mathbb{R}}\varphi(\xi/R)\,e^{2% \pi ix\xi}\,g(\xi)\,d\xi,\qquad\varphi(\xi):=e^{-\xi^{2}}.
  51. f f
  52. - 1 g ( x ) := lim R n φ ( ξ / R ) e 2 π i x ξ g ( ξ ) d ξ , φ ( ξ ) := e - | ξ | 2 . \mathcal{F}^{-1}g(x):=\lim_{R\to\infty}\int_{\mathbb{R}^{n}}\varphi(\xi/R)\,e^% {2\pi ix\cdot\xi}\,g(\xi)\,d\xi,\qquad\varphi(\xi):=e^{-|\xi|^{2}}.
  53. - 1 ( f ) ( x ) = f ( x ) . \mathcal{F}^{-1}(\mathcal{F}f)(x)=f(x).
  54. f f
  55. - 1 ( f ) ( x ) = f ( x ) \mathcal{F}^{-1}(\mathcal{F}f)(x)=f(x)
  56. x x∈ℝ
  57. g k ( ξ ) := { y n : | y | k } e - 2 π i y ξ f ( y ) d y , k , g_{k}(\xi):=\int_{\{y\in\mathbb{R}^{n}:\left|y\right|\leq k\}}e^{-2\pi iy\cdot% \xi}\,f(y)\,dy,\qquad k\in\mathbb{N},
  58. f := lim k g k \textstyle\mathcal{F}f:=\lim_{k\to\infty}g_{k}
  59. f ( x ) = ( - 1 f ) ( x ) = - 1 ( f ) ( x ) f(x)=\mathcal{F}(\mathcal{F}^{-1}f)(x)=\mathcal{F}^{-1}(\mathcal{F}f)(x)
  60. x x∈ℝ
  61. 𝒮 ( n ) \mathcal{S}^{\prime}(\mathbb{R}^{n})
  62. f 𝒮 ( n ) f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})
  63. φ 𝒮 ( n ) \varphi\in\mathcal{S}(\mathbb{R}^{n})
  64. f , φ := f , φ , \langle\mathcal{F}f,\varphi\rangle:=\langle f,\mathcal{F}\varphi\rangle,
  65. φ \mathcal{F}\varphi
  66. f f
  67. - 1 : 𝒮 ( n ) 𝒮 ( n ) \mathcal{F}^{-1}\colon\mathcal{S}^{\prime}(\mathbb{R}^{n})\to\mathcal{S}^{% \prime}(\mathbb{R}^{n})
  68. - 1 = - 1 = Id 𝒮 ( n ) . \mathcal{F}\mathcal{F}^{-1}=\mathcal{F}^{-1}\mathcal{F}=\operatorname{Id}_{% \mathcal{S}^{\prime}(\mathbb{R}^{n})}.
  69. 0 , 22 π 0,22π
  70. 2 π
  71. f f
  72. 0 , 11 0,11
  73. f : n , f ^ : n , f\colon\mathbb{R}^{n}\to\mathbb{C},\quad\hat{f}\colon\mathbb{R}^{n}\to\mathbb{% C},
  74. f ^ ( ξ ) := n e - 2 π i y ξ f ( y ) d y , \hat{f}(\xi):=\int_{\mathbb{R}^{n}}e^{-2\pi iy\cdot\xi}\,f(y)\,dy,
  75. f ( x ) = n e 2 π i x ξ f ^ ( ξ ) d ξ . f(x)=\int_{\mathbb{R}^{n}}e^{2\pi ix\cdot\xi}\,\hat{f}(\xi)\,d\xi.
  76. f : [ 0 , 1 ] n , f ^ : n , f\colon[0,1]^{n}\to\mathbb{C},\quad\hat{f}\colon\mathbb{Z}^{n}\to\mathbb{C},
  77. f ^ ( k ) := [ 0 , 1 ] n e - 2 π i y k f ( y ) d y , \hat{f}(k):=\int_{[0,1]^{n}}e^{-2\pi iy\cdot k}\,f(y)\,dy,
  78. f ( x ) = k n e 2 π i x k f ^ ( k ) . f(x)=\sum_{k\in\mathbb{Z}^{n}}e^{2\pi ix\cdot k}\,\hat{f}(k).
  79. k k
  80. −∞
  81. f ( x ) = rect ( a x ) ( f ) ( ξ ) = 1 | a | sinc ( ξ a ) , f(x)=\operatorname{rect}(ax)\quad\Rightarrow\quad(\mathcal{F}f)(\xi)=\frac{1}{% |a|}\operatorname{sinc}\left(\frac{\xi}{a}\right)\!,
  82. g ( ξ ) = rect ( a ξ ) ( - 1 g ) ( x ) = 1 | a | sinc ( - x a ) . g(\xi)=\operatorname{rect}(a\xi)\quad\Rightarrow\quad(\mathcal{F}^{-1}g)(x)=% \frac{1}{|a|}\operatorname{sinc}\left(-\frac{x}{a}\right)\!.
  83. ( g ) ( ξ ) = ( f ) ( ξ - η ) (\mathcal{F}g)(\xi)=(\mathcal{F}f)(\xi-\eta)
  84. a a∈ℝ
  85. g ( x ) = f ( a x ) g(x)=f(ax)
  86. ( g ) ( ξ ) = ( f ) ( ξ / a ) / a n (\mathcal{F}g)(\xi)=(\mathcal{F}f)(\xi/a)/a^{n}
  87. f f
  88. g g
  89. f ( g ) = ( f ) g \textstyle\int f(\mathcal{F}g)=\int(\mathcal{F}f)g
  90. φ = φ . \mathcal{F}\varphi=\varphi.
  91. f L 1 ( n ) \mathcal{F}f\in L^{1}(\mathbb{R}^{n})
  92. n e 2 π i x ξ ( f ) ( ξ ) d ξ = lim ε 0 n e - π ε 2 | ξ | 2 + 2 π i x ξ ( f ) ( ξ ) d ξ . \int_{\mathbb{R}^{n}}e^{2\pi ix\cdot\xi}(\mathcal{F}f)(\xi)\,d\xi=\lim_{% \varepsilon\to 0}\int_{\mathbb{R}^{n}}e^{-\pi\varepsilon^{2}|\xi|^{2}+2\pi ix% \cdot\xi}(\mathcal{F}f)(\xi)\,d\xi.
  93. ( g ) ( y ) = 1 ε n e - π ε 2 | x - y | 2 . (\mathcal{F}g)(y)=\frac{1}{\varepsilon^{n}}e^{-\frac{\pi}{\varepsilon^{2}}|x-y% |^{2}}.
  94. f f
  95. g g
  96. n e - π ε 2 | ξ | 2 + 2 π i x ξ ( f ) ( ξ ) d ξ = n 1 ε n e - π ε 2 | x - y | 2 f ( y ) d y = ( ϕ ε * f ) ( x ) , \int_{\mathbb{R}^{n}}e^{-\pi\varepsilon^{2}|\xi|^{2}+2\pi ix\cdot\xi}(\mathcal% {F}f)(\xi)\,d\xi=\int_{\mathbb{R}^{n}}\frac{1}{\varepsilon^{n}}e^{-\frac{\pi}{% \varepsilon^{2}}|x-y|^{2}}f(y)\,dy=(\phi_{\varepsilon}*f)(x),
  97. f f
  98. lim ε 0 ϕ ε * f ( x ) = f ( x ) . \lim_{\varepsilon\to 0}\phi_{\varepsilon}*f(x)=f(x).
  99. n e 2 π i x ξ ( f ) ( ξ ) d ξ = f ( x ) . \int_{\mathbb{R}^{n}}e^{2\pi ix\cdot\xi}(\mathcal{F}f)(\xi)\,d\xi=f(x).\qquad\square

Fractal_flame.html

  1. { F 1 ( x , y ) , p 1 F 2 ( x , y ) , p 2 F n ( x , y ) , p n \begin{cases}F_{1}(x,y),\quad p_{1}\\ F_{2}(x,y),\quad p_{2}\\ \dots\\ F_{n}(x,y),\quad p_{n}\end{cases}
  2. F j ( x , y ) = V k V a r i a t i o n s w k V k ( a j x + b j y + c j , d j x + e j y + f j ) F_{j}(x,y)=\sum_{V_{k}\in Variations}w_{k}\cdot V_{k}(a_{j}x+b_{j}y+c_{j},d_{j% }x+e_{j}y+f_{j})
  3. w k w_{k}

Fractional-reserve_banking.html

  1. m = 1 R m=\frac{1}{R}
  2. R = 1 5 R=\tfrac{1}{5}
  3. m = 1 1 / 5 = 5 m=\frac{1}{1/5}=5

Fractional_ideal.html

  1. ( R : I ) = { x K : x I R } . (R:I)=\{x\in K:xI\subseteq R\}.
  2. I ~ \tilde{I}
  3. I ~ = ( R : ( R : I ) ) , \tilde{I}=(R:(R:I)),
  4. ( R : I ) = { x K : x I R } . (R:I)=\{x\in K:xI\subseteq R\}.
  5. I ~ = I \tilde{I}=I

Fractional_part.html

  1. x x
  2. frac ( x ) = x - x , x > 0 \operatorname{frac}(x)=x-\lfloor x\rfloor,\;x>0
  3. frac ( x ) = x - x \operatorname{frac}(x)=x-\lfloor x\rfloor
  4. frac ( x ) = | x | - | x | \operatorname{frac}(x)=|x|-\lfloor|x|\rfloor
  5. frac ( x ) = { x - x x 0 x - x x < 0 \operatorname{frac}(x)=\begin{cases}x-\lfloor x\rfloor&x\geq 0\\ x-\lceil x\rceil&x<0\end{cases}
  6. n + r n+r
  7. n n
  8. r r
  9. x x
  10. x x
  11. p / q p/q
  12. p p
  13. q q
  14. 0 p < q 0\leq p<q

Frame_bundle.html

  1. p : 𝐑 k E x . p:\mathbf{R}^{k}\to E_{x}.
  2. p g : 𝐑 k E x . p\circ g:\mathbf{R}^{k}\to E_{x}.
  3. F ( E ) = x X F x . \mathrm{F}(E)=\coprod_{x\in X}F_{x}.
  4. ψ i : π - 1 ( U i ) U i × GL ( k , 𝐑 ) \psi_{i}:\pi^{-1}(U_{i})\to U_{i}\times\mathrm{GL}(k,\mathbf{R})
  5. ψ i ( x , p ) = ( x , φ i , x p ) . \psi_{i}(x,p)=(x,\varphi_{i,x}\circ p).
  6. F ( E ) × ρ V \mathrm{F}(E)\times_{\rho}V
  7. [ p , v ] p ( v ) [p,v]\mapsto p(v)
  8. ψ ( p ) = ( x , s ( x ) - 1 p ) \psi(p)=(x,s(x)^{-1}\circ p)
  9. ( x 1 , , x n ) \left(\frac{\partial}{\partial x^{1}},\cdots,\frac{\partial}{\partial x^{n}}\right)
  10. p : 𝐑 n T x M p:\mathbf{R}^{n}\to T_{x}M
  11. θ p ( ξ ) = p - 1 d π ( ξ ) \theta_{p}(\xi)=p^{-1}\mathrm{d}\pi(\xi)
  12. R g * θ = g - 1 θ R_{g}^{*}\theta=g^{-1}\theta
  13. p : 𝐑 k E x p:\mathbf{R}^{k}\to E_{x}
  14. i : F O ( E ) F GL ( E ) i:{\mathrm{F}}_{\mathrm{O}}(E)\to{\mathrm{F}}_{\mathrm{GL}}(E)
  15. F G ( M ) F GL ( M ) {\mathrm{F}}_{G}(M)\to{\mathrm{F}}_{\mathrm{GL}}(M)

Frattini_subgroup.html

  1. Φ ( G ) = a p \Phi(G)=\left\langle a^{p}\right\rangle

Fredholm_integral_equation.html

  1. g ( t ) = a b K ( t , s ) f ( s ) d s g(t)=\int_{a}^{b}K(t,s)f(s)\,\mathrm{d}s
  2. K ( t , s ) K(t,s)
  3. g ( t ) g(t)
  4. f ( s ) f(s)
  5. K ( t , s ) = K ( t - s ) K(t,s)=K(t-s)
  6. ± \pm\infty
  7. f ( t ) = ω - 1 [ t [ g ( t ) ] ( ω ) t [ K ( t ) ] ( ω ) ] = - t [ g ( t ) ] ( ω ) t [ K ( t ) ] ( ω ) e 2 π i ω t d ω f(t)=\mathcal{F}_{\omega}^{-1}\left[{\mathcal{F}_{t}[g(t)](\omega)\over% \mathcal{F}_{t}[K(t)](\omega)}\right]=\int_{-\infty}^{\infty}{\mathcal{F}_{t}[% g(t)](\omega)\over\mathcal{F}_{t}[K(t)](\omega)}e^{2\pi i\omega t}\mathrm{d}\omega
  8. t \mathcal{F}_{t}
  9. ω - 1 \mathcal{F}_{\omega}^{-1}
  10. ϕ ( t ) = f ( t ) + λ a b K ( t , s ) ϕ ( s ) d s . \phi(t)=f(t)+\lambda\int_{a}^{b}K(t,s)\phi(s)\,\mathrm{d}s.
  11. K ( t , s ) K(t,s)
  12. f ( t ) f(t)
  13. ϕ ( t ) \phi(t)

Free-air_gravity_anomaly.html

  1. g F = g o b s - g λ + δ g F g_{F}=g_{obs}-g_{\lambda}+\delta g_{F}
  2. g F g_{F}
  3. g o b s g_{obs}
  4. g λ g_{\lambda}
  5. δ g F \delta g_{F}
  6. g = G M R 2 d g d R = - 2 G M R 3 = - 2 g R \begin{aligned}\displaystyle g&\displaystyle=\frac{GM}{R^{2}}\\ \displaystyle\frac{dg}{dR}&\displaystyle=-\frac{2GM}{R^{3}}=-\frac{2g}{R}\end{aligned}
  7. 2 g / R = 0.3086 2g/R=0.3086
  8. h h
  9. δ g F = 2 g R × h \delta g_{F}=\frac{2g}{R}\times h

Free_algebra.html

  1. ( X i 1 X i 2 X i m ) ( X j 1 X j 2 X j n ) = X i 1 X i 2 X i m X j 1 X j 2 X j n , \left(X_{i_{1}}X_{i_{2}}\cdots X_{i_{m}}\right)\cdot\left(X_{j_{1}}X_{j_{2}}% \cdots X_{j_{n}}\right)=X_{i_{1}}X_{i_{2}}\cdots X_{i_{m}}X_{j_{1}}X_{j_{2}}% \cdots X_{j_{n}},
  2. X = { X i ; i I } X=\{X_{i}\,;\;i\in I\}
  3. R X := w X R w R\langle X\rangle:=\bigoplus_{w\in X^{\ast}}Rw
  4. \oplus
  5. ( α X 1 X 2 2 + β X 2 X 3 ) ( γ X 2 X 1 + δ X 1 4 X 4 ) = α γ X 1 X 2 3 X 1 + α δ X 1 X 2 2 X 1 4 X 4 + β γ X 2 X 3 X 2 X 1 + β δ X 2 X 3 X 1 4 X 4 (\alpha X_{1}X_{2}^{2}+\beta X_{2}X_{3})\cdot(\gamma X_{2}X_{1}+\delta X_{1}^{% 4}X_{4})=\alpha\gamma X_{1}X_{2}^{3}X_{1}+\alpha\delta X_{1}X_{2}^{2}X_{1}^{4}% X_{4}+\beta\gamma X_{2}X_{3}X_{2}X_{1}+\beta\delta X_{2}X_{3}X_{1}^{4}X_{4}
  6. i 1 , i 2 , , i k { 1 , 2 , , n } a i 1 , i 2 , , i k X i 1 X i 2 X i k , \sum\limits_{i_{1},i_{2},\cdots,i_{k}\in\left\{1,2,\cdots,n\right\}}a_{i_{1},i% _{2},\cdots,i_{k}}X_{i_{1}}X_{i_{2}}\cdots X_{i_{k}},
  7. a i 1 , i 2 , , i k a_{i_{1},i_{2},...,i_{k}}
  8. a i 1 , i 2 , , i k a_{i_{1},i_{2},...,i_{k}}

Free_module.html

  1. S S
  2. S S
  3. S S
  4. S S
  5. R R
  6. M M
  7. E M E\subseteq M
  8. M M
  9. E E
  10. M M
  11. M M
  12. E E
  13. R R
  14. E E
  15. r 1 e 1 + r 2 e 2 + + r n e n = 0 M r_{1}e_{1}+r_{2}e_{2}+\cdots+r_{n}e_{n}=0_{M}
  16. e 1 , e 2 , , e n e_{1},e_{2},\ldots,e_{n}
  17. E E
  18. r 1 = r 2 = = r n = 0 R r_{1}=r_{2}=\cdots=r_{n}=0_{R}
  19. 0 M 0_{M}
  20. M M
  21. 0 R 0_{R}
  22. R R
  23. R R
  24. M M
  25. x M x\in M
  26. E E
  27. R R
  28. E E
  29. E E
  30. E E
  31. E E
  32. R R
  33. E E
  34. E E
  35. X X
  36. a X aX
  37. a a
  38. R R
  39. R R
  40. E E
  41. | E | |E|
  42. R R
  43. C ( E ) C(E)
  44. f : E R f:E\to R
  45. f ( x ) = 0 f(x)=0
  46. x x
  47. E E
  48. ( f + g ) ( x ) = f ( x ) + g ( x ) , x E , (f+g)(x)=f(x)+g(x),\quad\forall x\in E,
  49. ( a f ) ( x ) = a ( f ( x ) ) , x E . (af)(x)=a(f(x)),\quad\forall x\in E.
  50. C ( E ) C(E)
  51. δ a \delta_{a}
  52. δ a ( x ) = { 1 if x = a 0 if x a \delta_{a}(x)=\begin{cases}1\quad\mbox{if }~{}x=a\\ 0\quad\mbox{if }~{}x\neq a\end{cases}
  53. a δ a a\mapsto\delta_{a}
  54. E E
  55. C ( E ) C(E)
  56. ι : E R ( E ) \iota:E\to R^{(E)}
  57. φ : E M \varphi:E\to M
  58. E E
  59. R R
  60. M M
  61. ψ : R ( E ) M \psi:R^{(E)}\to M
  62. φ = ψ ι \varphi=\psi\circ\iota
  63. ι : E R ( E ) \iota:E\to R^{(E)}
  64. R R

Free_monoid.html

  1. p a ( s ) = { ε if s = ε , the empty string p a ( t ) if s = t a p a ( t ) b if s = t b and b a . p_{a}(s)=\begin{cases}\varepsilon&\,\text{if }s=\varepsilon,\,\text{ the empty% string}\\ p_{a}(t)&\,\text{if }s=ta\\ p_{a}(t)b&\,\text{if }s=tb\,\text{ and }b\neq a.\end{cases}
  2. p a ( Σ * ) = ( Σ - a ) * p_{a}\left(\Sigma^{*}\right)=\left(\Sigma-a\right)^{*}
  3. p a ( Σ * ) p_{a}\left(\Sigma^{*}\right)
  4. p ε p_{\varepsilon}
  5. p ε ( s ) = s p_{\varepsilon}(s)=s
  6. p a ( s t ) = p a ( s ) p a ( t ) p_{a}(st)=p_{a}(s)p_{a}(t)
  7. p a ( p b ( s ) ) = p b ( p a ( s ) ) . p_{a}(p_{b}(s))=p_{b}(p_{a}(s)).
  8. p a ( p a ( s ) ) = p a ( s ) p_{a}(p_{a}(s))=p_{a}(s)

Free_object.html

  1. X i F ( A ) f F ( g ) F ( B ) \begin{array}[]{c}X\xrightarrow{\quad i\quad}F(A)\\ {}_{f}\searrow\quad\swarrow{}_{F(g)}\\ F(B)\\ \end{array}
  2. { e , a , b , a - 1 , b - 1 } \{e,a,b,a^{-1},b^{-1}\}
  3. a - 1 a^{-1}
  4. b - 1 b^{-1}
  5. S = { a , b , c , d , e } S=\{a,b,c,d,e\}
  6. W ( S ) W(S)
  7. g e = e g = g ge=eg=g
  8. g g - 1 = g - 1 g = e gg^{-1}=g^{-1}g=e
  9. a e b e c e d e = a b a - 1 b - 1 aebecede=aba^{-1}b^{-1}
  10. a - 1 a^{-1}
  11. b - 1 b^{-1}
  12. a b d c = a b b - 1 a - 1 = e abdc=abb^{-1}a^{-1}=e
  13. \sim
  14. F 2 = W ( S ) / F_{2}=W(S)/\sim
  15. F 2 = W ( S ) / E F_{2}=W(S)/E
  16. W ( S ) = { a 1 a 2 a n | a k S ; n finite } W(S)=\{a_{1}a_{2}\ldots a_{n}\,|\;a_{k}\in S\,;\,n\mbox{ finite }~{}\}
  17. E = { a 1 a 2 a n | e = a 1 a 2 a n ; a k S ; n finite } E=\{a_{1}a_{2}\ldots a_{n}\,|\;e=a_{1}a_{2}\ldots a_{n}\,;\,a_{k}\in S\,;\,n% \mbox{ finite }~{}\}
  18. S S
  19. 𝐀 \mathbf{A}
  20. ρ \rho
  21. S S
  22. 𝐀 \mathbf{A}
  23. A A
  24. ψ : S A \psi:S\longrightarrow A
  25. ( (
  26. A A
  27. ψ ) \psi)
  28. 𝐀 \mathbf{A}
  29. ρ \rho
  30. S S
  31. 𝐁 \mathbf{B}
  32. ρ \rho
  33. τ : S B \tau:S\longrightarrow B
  34. B B
  35. 𝐁 \mathbf{B}
  36. σ : A B \sigma:A\longrightarrow B
  37. σ ψ = τ \sigma\psi=\tau
  38. U : 𝐂 𝐒𝐞𝐭 U:\mathbf{C}\to\mathbf{Set}
  39. F : 𝐒𝐞𝐭 𝐂 F:\mathbf{Set}\to\mathbf{C}
  40. η : X U ( F ( X ) ) \eta:X\to U(F(X))\,\!
  41. X F ( X ) X\to F(X)
  42. X U ( F ( X ) ) X\to U(F(X))
  43. η : id 𝐒𝐞𝐭 U F \eta:\operatorname{id}_{\mathbf{Set}}\to UF
  44. ε : F U id 𝐂 \varepsilon:FU\to\operatorname{id}_{\mathbf{C}}

Free_product.html

  1. G G
  2. U ( G ) U(G)
  3. s 1 s 2 s n , s_{1}s_{2}\cdots s_{n},
  4. g 1 h 1 g 2 h 2 g k h k . g_{1}h_{1}g_{2}h_{2}\cdots g_{k}h_{k}.
  5. G = S G R G G=\langle S_{G}\mid R_{G}\rangle
  6. H = S H R H H=\langle S_{H}\mid R_{H}\rangle
  7. G * H = S G S H R G R H . G*H=\langle S_{G}\cup S_{H}\mid R_{G}\cup R_{H}\rangle.
  8. G = x x 4 = 1 , G=\langle x\mid x^{4}=1\rangle,
  9. H = y y 5 = 1 . H=\langle y\mid y^{5}=1\rangle.
  10. G * H = x , y x 4 = y 5 = 1 . G*H=\langle x,y\mid x^{4}=y^{5}=1\rangle.
  11. F m * F n F m + n , F_{m}*F_{n}\cong F_{m+n},
  12. φ : F G and ψ : F H , \varphi:F\rightarrow G\mbox{ and }~{}\psi:F\rightarrow H,
  13. φ ( f ) ψ ( f ) - 1 = 1 \varphi(f)\psi(f)^{-1}=1
  14. ( G * H ) / N . (G*H)/N.\,

Freezing-point_depression.html

  1. Δ T F = Δ H T F f u s - 2 R T F ln ( a l i q ) - 2 Δ C p f u s T F 2 R ln ( a l i q ) + ( Δ H T F f u s ) 2 2 ( Δ H T F f u s T F + Δ C p f u s 2 - R ln ( a l i q ) ) {\Delta}T_{F}=\frac{{\Delta}H^{fus}_{T_{F}}-2RT_{F}{\cdot}\ln(a_{liq})-\sqrt{2% {\Delta}C^{fus}_{p}T^{2}_{F}R{\cdot}\ln(a_{liq})+({\Delta}H^{fus}_{T_{F}})^{2}% }}{2\left(\frac{{\Delta}H^{fus}_{T_{F}}}{T_{F}}+\frac{{\Delta}C^{fus}_{p}}{2}-% R{\cdot}\ln(a_{liq})\right)}

Frequency_mixer.html

  1. I = I S ( e q V D n k T - 1 ) I=I_{\mathrm{S}}\left(e^{qV_{\mathrm{D}}\over nkT}-1\right)
  2. e x = n = 0 x n n ! e^{x}=\sum_{n=0}^{\infty}\frac{x^{n}}{n!}
  3. e x - 1 x + x 2 2 e^{x}-1\approx x+\frac{x^{2}}{2}
  4. v 1 + v 2 v_{1}+v_{2}
  5. v o = ( v 1 + v 2 ) + 1 2 ( v 1 + v 2 ) 2 + v_{\mathrm{o}}=(v_{1}+v_{2})+\frac{1}{2}(v_{1}+v_{2})^{2}+\dots
  6. ( v 1 + v 2 ) 2 = v 1 2 + 2 v 1 v 2 + v 2 2 (v_{1}+v_{2})^{2}=v_{1}^{2}+2v_{1}v_{2}+v_{2}^{2}
  7. v 1 = sin a t v_{1}=\sin at
  8. v 2 = sin b t v_{2}=\sin bt
  9. V 0 V_{0}
  10. v o = ( sin a t + sin b t ) + 1 2 ( sin a t + sin b t ) 2 + v_{\mathrm{o}}=(\sin at+\sin bt)+\frac{1}{2}(\sin at+\sin bt)^{2}+\dots
  11. v o = ( sin a t + sin b t ) + 1 2 ( sin 2 a t + 2 sin a t sin b t + sin 2 b t ) + v_{\mathrm{o}}=(\sin at+\sin bt)+\frac{1}{2}(\sin^{2}at+2\sin at\sin bt+\sin^{% 2}bt)+\dots
  12. sin a t sin b t \sin at\sin bt
  13. sin a sin b = cos ( a - b ) - cos ( a + b ) 2 \sin a\sin b=\frac{\cos(a-b)-\cos(a+b)}{2}
  14. v o = cos ( ( a - b ) t ) - cos ( ( a + b ) t ) + v_{\mathrm{o}}=\cos((a-b)t)-\cos((a+b)t)+\dots

Friedmann–Lemaître–Robertson–Walker_metric.html

  1. - c 2 d τ 2 = - c 2 d t 2 + a ( t ) 2 d 𝚺 2 -c^{2}\mathrm{d}\tau^{2}=-c^{2}\mathrm{d}t^{2}+{a(t)}^{2}\mathrm{d}\mathbf{% \Sigma}^{2}
  2. 𝚺 \mathbf{\Sigma}
  3. d 𝚺 \mathrm{d}\mathbf{\Sigma}
  4. d 𝚺 2 = d r 2 1 - k r 2 + r 2 d 𝛀 2 , where d 𝛀 2 = d θ 2 + sin 2 θ d ϕ 2 . \mathrm{d}\mathbf{\Sigma}^{2}=\frac{\mathrm{d}r^{2}}{1-kr^{2}}+r^{2}\mathrm{d}% \mathbf{\Omega}^{2},\quad\,\text{where }\mathrm{d}\mathbf{\Omega}^{2}=\mathrm{% d}\theta^{2}+\sin^{2}\theta\,\mathrm{d}\phi^{2}.
  5. π \pi
  6. d 𝚺 \mathrm{d}\mathbf{\Sigma}
  7. d 𝚺 2 = d r 2 + S k ( r ) 2 d 𝛀 2 \mathrm{d}\mathbf{\Sigma}^{2}=\mathrm{d}r^{2}+S_{k}(r)^{2}\,\mathrm{d}\mathbf{% \Omega}^{2}
  8. d 𝛀 \mathrm{d}\mathbf{\Omega}
  9. S k ( r ) = { k - 1 sin ( r k ) , k > 0 r , k = 0 | k | - 1 sinh ( r | k | ) , k < 0. S_{k}(r)=\begin{cases}\sqrt{k}^{\,-1}\sin(r\sqrt{k}),&k>0\\ r,&k=0\\ \sqrt{|k|}^{\,-1}\sinh(r\sqrt{|k|}),&k<0.\end{cases}
  10. d 𝚺 \mathrm{d}\mathbf{\Sigma}
  11. S k ( r ) = n = 0 ( - 1 ) n k n r 2 n + 1 ( 2 n + 1 ) ! = r - k r 3 6 + k 2 r 5 120 - S_{k}(r)=\sum_{n=0}^{\infty}\frac{(-1)^{n}k^{n}r^{2n+1}}{(2n+1)!}=r-\frac{kr^{% 3}}{6}+\frac{k^{2}r^{5}}{120}-\cdots
  12. S k ( r ) = r sinc ( r k ) S_{k}(r)=r\;\mathrm{sinc}\,(r\sqrt{k})
  13. k \sqrt{k}
  14. d 𝚺 2 = d x 2 + d y 2 + d z 2 . \mathrm{d}\mathbf{\Sigma}^{2}=\mathrm{d}x^{2}+\mathrm{d}y^{2}+\mathrm{d}z^{2}.
  15. x = r cos θ x=r\cos\theta\,
  16. y = r sin θ cos ϕ y=r\sin\theta\cos\phi\,
  17. z = r sin θ sin ϕ z=r\sin\theta\sin\phi\,
  18. a ( t ) a(t)
  19. ρ ( t ) , \rho(t),
  20. G μ ν + Λ g μ ν = 8 π G c 4 T μ ν G_{\mu\nu}+\Lambda g_{\mu\nu}=\frac{8\pi G}{c^{4}}T_{\mu\nu}
  21. ( a ˙ a ) 2 + k c 2 a 2 - Λ c 2 3 = 8 π G 3 ρ \left(\frac{\dot{a}}{a}\right)^{2}+\frac{kc^{2}}{a^{2}}-\frac{\Lambda c^{2}}{3% }=\frac{8\pi G}{3}\rho
  22. 2 a ¨ a + ( a ˙ a ) 2 + k c 2 a 2 - Λ c 2 = - 8 π G c 2 p . 2\frac{\ddot{a}}{a}+\left(\frac{\dot{a}}{a}\right)^{2}+\frac{kc^{2}}{a^{2}}-% \Lambda c^{2}=-\frac{8\pi G}{c^{2}}p.
  23. ρ ˙ = - 3 a ˙ a ( ρ + p c 2 ) {\dot{\rho}}=-3\frac{\dot{a}}{a}\left(\rho+\frac{p}{c^{2}}\right)
  24. a ¨ a = - 4 π G 3 ( ρ + 3 p c 2 ) + Λ c 2 3 \frac{\ddot{a}}{a}=-\frac{4\pi G}{3}\left(\rho+\frac{3p}{c^{2}}\right)+\frac{% \Lambda c^{2}}{3}
  25. k k
  26. a ˙ {\dot{a}}
  27. ρ ρ + Λ c 2 8 π G \rho\rightarrow\rho+\frac{\Lambda c^{2}}{8\pi G}
  28. p p - Λ c 4 8 π G . p\rightarrow p-\frac{\Lambda c^{4}}{8\pi G}.
  29. p = - ρ c 2 . p=-\rho c^{2}.\,
  30. p < - ρ c 2 3 . p<-\frac{\rho c^{2}}{3}.\,
  31. - a 3 ρ ˙ = 3 a 2 a ˙ ρ + 3 a 2 p a ˙ c 2 -a^{3}{\dot{\rho}}=3a^{2}{\dot{a}}\rho+\frac{3a^{2}p{\dot{a}}}{c^{2}}\,
  32. a ˙ 2 2 - G 4 π a 3 3 ρ a = - k c 2 2 . \frac{{\dot{a}}^{2}}{2}-\frac{G\frac{4\pi a^{3}}{3}\rho}{a}=-\frac{kc^{2}}{2}\,.
  33. a ˙ = a ¨ = 0 \dot{a}=\ddot{a}=0
  34. R E = c / 4 π G ρ R_{E}=c/\sqrt{4\pi G\rho}
  35. c c
  36. G G
  37. ρ \rho

Frobenius_theorem_(differential_topology).html

  1. { f k i : 𝐑 n 𝐑 : 1 i n , 1 k r } \left\{f_{k}^{i}:\mathbf{R}^{n}\to\mathbf{R}\ :\ 1\leq i\leq n,1\leq k\leq r\right\}
  2. ( f [ u s u , u b = , u k , u p = , u i ] ) (f[u^{\prime}su^{\prime},u^{\prime}b=^{\prime},u^{\prime}k^{\prime},u^{\prime}% p=^{\prime},u^{\prime}i^{\prime}])
  3. ( 1 ) { L 1 u = def i f 1 i ( x ) u x i = 0 L 2 u = def i f 2 i ( x ) u x i = 0 L r u = def i f r i ( x ) u x i = 0 (1)\quad\begin{cases}L_{1}u\ \stackrel{\mathrm{def}}{=}\ \sum_{i}f_{1}^{i}(x)% \frac{\partial u}{\partial x^{i}}=0\\ L_{2}u\ \stackrel{\mathrm{def}}{=}\ \sum_{i}f_{2}^{i}(x)\frac{\partial u}{% \partial x^{i}}=0\\ \qquad\cdots\\ L_{r}u\ \stackrel{\mathrm{def}}{=}\ \sum_{i}f_{r}^{i}(x)\frac{\partial u}{% \partial x^{i}}=0\end{cases}
  4. L i L j u ( x ) - L j L i u ( x ) = k c i j k ( x ) L k u ( x ) L_{i}L_{j}u(x)-L_{j}L_{i}u(x)=\sum_{k}c_{ij}^{k}(x)L_{k}u(x)
  5. 1 i , j r 1≤i,j≤r
  6. { f x + f y = 0 f y + f z = 0 \begin{cases}\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}=0\\ \frac{\partial f}{\partial y}+\frac{\partial f}{\partial z}=0\end{cases}
  7. x y + z = C x−y+z=C
  8. C C
  9. f ( x , y , z ) = C ( t ) whenever x - y + z = t . f(x,y,z)=C(t)\,\text{ whenever }x-y+z=t.
  10. C ( t ) C(t)
  11. X X
  12. M M
  13. u : I M u:I\to M
  14. I I
  15. u ˙ ( t ) = X u ( t ) \dot{u}(t)=X_{u(t)}
  16. X X
  17. M M
  18. M M
  19. E T M E⊂TM
  20. X X
  21. E E
  22. X X , Y XX,Y
  23. E E
  24. X X
  25. M M
  26. E T M E⊂TM
  27. p M p\in M
  28. N M N⊂M
  29. E p = T p N E_{p}=T_{p}N
  30. E E
  31. M M
  32. M M
  33. p p
  34. U U
  35. 1 1
  36. Ω ( M ) Ω(M)
  37. M M
  38. α Ω k ( M ) \alpha\in\Omega^{k}(M)
  39. k { 1 , , dim M } k\in\{1,\dots,\operatorname{dim}M\}
  40. α ( v 1 , , v k ) = 0 \alpha(v_{1},\dots,v_{k})=0
  41. v 1 , , v k D v_{1},\dots,v_{k}\in D
  42. Ω ( M ) Ω(M)
  43. I ( D ) I(D)
  44. X X
  45. Y Y
  46. A X , B Y A⊂X,B⊂Y
  47. F : A × B L ( X , Y ) F:A\times B\to L(X,Y)
  48. L ( X , Y ) L(X,Y)
  49. X X
  50. ( 1 ) y = F ( x , y ) (1)\quad y^{\prime}=F(x,y)
  51. x A : u ( x ) = F ( x , u ( x ) ) . \forall x\in A:\quad u^{\prime}(x)=F(x,u(x)).
  52. ( x 0 , y 0 ) A × B (x_{0},y_{0})\in A\times B
  53. u ( x ) u(x)
  54. 𝐑 \mathbf{R}
  55. 𝐂 \mathbf{C}
  56. 𝐂 \mathbf{C}
  57. A × B A×B
  58. D 1 F ( x , y ) ( s 1 , s 2 ) + D 2 F ( x , y ) ( F ( x , y ) s 1 , s 2 ) = D 1 F ( x , y ) ( s 2 , s 1 ) + D 2 F ( x , y ) ( F ( x , y ) s 2 , s 1 ) D_{1}F(x,y)\cdot(s_{1},s_{2})+D_{2}F(x,y)\cdot(F(x,y)\cdot s_{1},s_{2})=D_{1}F% (x,y)\cdot(s_{2},s_{1})+D_{2}F(x,y)\cdot(F(x,y)\cdot s_{2},s_{1})
  59. F ( x , y ) L ( X , Y ) F(x,y)∈L(X,Y)
  60. M M
  61. E E
  62. M M
  63. E E
  64. p M p∈M
  65. X X
  66. E E
  67. X X
  68. [ X , Y ] p E p [X,Y]_{p}\in E_{p}
  69. E E
  70. p M p∈M
  71. φ : N M φ:N→M
  72. φ φ
  73. E E
  74. 𝐂 \mathbf{C}
  75. ω 1 , , ω r \omega^{1},\dots,\omega^{r}
  76. d ω j = i = 1 r ψ i j ω i d\omega^{j}=\sum_{i=1}^{r}\psi_{i}^{j}\wedge\omega^{i}
  77. ψ [ u s u , u b = , u i , u p = , u j ] , 1 i , j r ψ[u^{\prime}su^{\prime},u^{\prime}b=^{\prime},u^{\prime}i^{\prime},u^{\prime}p% =^{\prime},u^{\prime}j^{\prime}],1≤i,j≤r
  78. ω j = i = 1 r f i j d g i . \omega^{j}=\sum_{i=1}^{r}f_{i}^{j}dg^{i}.
  79. 𝐂 < s u p > n \mathbf{C}<sup>n
  80. 𝐑 < s u p > n \mathbf{R}<sup>n
  81. 𝐑 < s u p > n \mathbf{R}<sup>n

Froude_number.html

  1. Fr = u 0 g 0 l 0 \mathrm{Fr}=\frac{u_{0}}{\sqrt{g_{0}l_{0}}}
  2. Speed Length Ratio = u LWL \,\text{Speed Length Ratio}=\frac{u}{\sqrt{\,\text{LWL}}}
  3. ρ * ρ ρ 0 , \rho^{*}\equiv\frac{\rho}{\rho_{0}},
  4. u * u u 0 , u^{*}\equiv\frac{u}{u_{0}},
  5. r * r r 0 , r^{*}\equiv\frac{r}{r_{0}},
  6. t * u 0 r 0 t , t^{*}\equiv\frac{u_{0}}{r_{0}}t,
  7. * r 0 \nabla^{*}\equiv r_{0}\nabla
  8. 𝐠 * 𝐠 g 0 , \mathbf{g}^{*}\equiv\frac{\mathbf{g}}{g_{0}},
  9. s y m b o l σ * s y m b o l σ p 0 , symbol\sigma^{*}\equiv\frac{symbol\sigma}{p_{0}},
  10. Fr = u 0 g 0 r 0 , \mathrm{Fr}=\frac{u_{0}}{\sqrt{g_{0}r_{0}}},
  11. Eu = p 0 ρ 0 u 0 2 , \mathrm{Eu}=\frac{p_{0}}{\rho_{0}u_{0}^{2}},
  12. s y m b o l σ = p s y m b o l I symbol\sigma=psymbolI
  13. D u D t + Eu p ρ = 1 Fr 2 g ^ {Du\over Dt}+\mathrm{Eu}\,\frac{\nabla p}{\rho}=\frac{1}{\mathrm{Fr}^{2}}\hat{g}
  14. s y m b o l σ = p 𝐈 + μ ( 𝐮 + ( 𝐮 ) T ) symbol\sigma=p\mathbf{I}+\mu(\nabla\mathbf{u}+(\nabla\mathbf{u})\text{T})
  15. D u D t + Eu p ρ = - 1 Re 2 u + 1 Fr 2 g ^ {Du\over Dt}+\mathrm{Eu}\,\frac{\nabla p}{\rho}=-\frac{1}{\mathrm{Re}}\,\nabla% ^{2}u+\frac{1}{\mathrm{Fr}^{2}}\hat{g}
  16. Fn L = u g L , \mathrm{Fn_{L}}=\frac{u}{\sqrt{gL}},
  17. Fn V = u g V 1 / 3 . \mathrm{Fn_{V}}=\frac{u}{\sqrt{gV^{1/3}}}.
  18. c = g A B , c=\sqrt{g\frac{A}{B}},
  19. Fr = U g A B . \mathrm{Fr}=\frac{U}{\sqrt{\displaystyle g\frac{A}{B}}}.
  20. Fr = U g d . \mathrm{Fr}=\frac{U}{\sqrt{gd}}.
  21. Fr = u β h + s g ( x d - x ) , \mathrm{Fr}=\frac{u}{\sqrt{\beta h+s_{g}(x_{d}-x)}},
  22. u u
  23. β = g K cos ζ \beta=gK\cos\zeta
  24. K K
  25. ζ \zeta
  26. s g = g sin ζ s_{g}=g\sin\zeta
  27. x x
  28. x d x_{d}
  29. E p o t p = β h E_{pot}^{p}=\beta h
  30. E p o t g = s g ( x d - x ) E_{pot}^{g}=s_{g}(x_{d}-x)
  31. E p o t g = s g ( x d - x ) E_{pot}^{g}=s_{g}(x_{d}-x)
  32. β h \beta h
  33. β h \beta h
  34. 1 \ll 1
  35. u u
  36. s g ( x d - x ) s_{g}(x_{d}-x)
  37. β h \beta h
  38. Fr = ω r g . \mathrm{Fr}=\omega\sqrt{\frac{r}{g}}.
  39. Fr = u g h \mathrm{Fr}=\frac{u}{\sqrt{g^{\prime}h}}
  40. g g^{\prime}
  41. g = g ρ 1 - ρ 2 ρ 1 g^{\prime}=g{\rho_{1}-\rho_{2}\over{\rho_{1}}}
  42. Fr = centripetal force gravitational force = m v 2 / l m g = v 2 g l \mathrm{Fr}=\frac{\,\text{centripetal force}}{\,\text{gravitational force}}=% \frac{mv^{2}/l}{mg}=\frac{v^{2}}{gl}
  43. m m
  44. l l
  45. g g
  46. v v
  47. l l
  48. Fr = v 2 g l = ( l f ) 2 g l = l f 2 g . \mathrm{Fr}=\frac{v^{2}}{gl}=\frac{(lf)^{2}}{gl}=\frac{lf^{2}}{g}.
  49. Fr 0.5 \mathrm{Fr}\approx 0.5

FS.html

  1. f s f_{s}

FTSE_100_Index.html

  1. Index level = i Price of stock i × Number of shares i × Free float adjustment factor i Index divisor \,\text{Index level}=\frac{\sum_{i}\,\text{Price of stock}_{i}\times\,\text{% Number of shares}_{i}\times\,\text{Free float adjustment factor}_{i}}{\,\text{% Index divisor}}

Fuchsian_group.html

  1. d s = 1 y d x 2 + d y 2 . ds=\frac{1}{y}\sqrt{dx^{2}+dy^{2}}.
  2. ( a b c d ) z = a z + b c z + d . \begin{pmatrix}a&b\\ c&d\end{pmatrix}\cdot z=\frac{az+b}{cz+d}.
  3. ( a b c d ) z = a z + b c z + d \begin{pmatrix}a&b\\ c&d\end{pmatrix}\cdot z=\frac{az+b}{cz+d}
  4. ( a b c d ) \begin{pmatrix}a&b\\ c&d\end{pmatrix}
  5. | tr h | = 2 cosh L 2 . |\mathrm{tr}\;h|=2\cosh\frac{L}{2}.

Fuel_efficiency.html

  1. 235.215 x \textstyle\frac{235.215}{x}
  2. x x
  3. 282.481 x \textstyle\frac{282.481}{x}

Function_space.html

  1. y y
  2. 𝒞 ( a , b ) \mathcal{C}(a,b)
  3. y \|y\|_{\infty}
  4. 𝒞 ( a , b ) \mathcal{C}(a,b)
  5. y ( x ) y(x)
  6. a x b a≤x≤b
  7. y max a x b | y ( x ) | where y 𝒞 ( a , b ) . \|y\|\equiv\max_{a\leq x\leq b}|y(x)|\qquad\,\text{where}\ \ y\in\mathcal{C}(a% ,b)\,.

Functional_equation.html

  1. f ( s ) = 2 s π s - 1 sin ( π s 2 ) Γ ( 1 - s ) f ( 1 - s ) f(s)=2^{s}\pi^{s-1}\sin\left(\frac{\pi s}{2}\right)\Gamma(1-s)f(1-s)
  2. Γ Γ
  3. f ( x ) = f ( x + 1 ) x f(x)={f(x+1)\over x}\,\!
  4. f ( y ) f ( y + 1 2 ) = π 2 2 y - 1 f ( 2 y ) f(y)f\left(y+\frac{1}{2}\right)=\frac{\sqrt{\pi}}{2^{2y-1}}f(2y)
  5. f ( z ) f ( 1 - z ) = π sin ( π z ) f(z)f(1-z)={\pi\over\sin(\pi z)}\,\!\,\,\,
  6. f ( a z + b c z + d ) = ( c z + d ) k f ( z ) f\left({az+b\over cz+d}\right)=(cz+d)^{k}f(z)\,\!
  7. a , b , c , d a,b,c,d
  8. a d b c ad−bc
  9. | a b c d | \begin{vmatrix}a&b\\ c&d\end{vmatrix}
  10. f f
  11. k k
  12. f ( x + y ) = f ( x ) + f ( y ) f(x+y)=f(x)+f(y)\,\!
  13. f ( x + y ) = f ( x ) f ( y ) , f(x+y)=f(x)f(y),\,\!
  14. f ( x y ) = f ( x ) + f ( y ) f(xy)=f(x)+f(y)\,\!
  15. f ( x y ) = f ( x ) f ( y ) f(xy)=f(x)f(y)\,\!
  16. f ( x + y ) + f ( x - y ) = 2 [ f ( x ) + f ( y ) ] f(x+y)+f(x-y)=2[f(x)+f(y)]\,\!
  17. f ( ( x + y ) / 2 ) = ( f ( x ) + f ( y ) ) / 2 f((x+y)/2)=(f(x)+f(y))/2\,\!
  18. g ( x + y ) + g ( x - y ) = 2 [ g ( x ) g ( y ) ] g(x+y)+g(x-y)=2[g(x)g(y)]\,\!
  19. f ( h ( x ) ) = h ( x + 1 ) f(h(x))=h(x+1)\,\!
  20. f ( h ( x ) ) = c f ( x ) f(h(x))=cf(x)\,\!
  21. f ( h ( x ) ) = ( f ( x ) ) c f(h(x))=(f(x))^{c}\,\!
  22. f ( h ( x ) ) = h ( x ) f ( x ) f(h(x))=h(x)f(x)
  23. ω ( ω ( x , u ) , v ) = ω ( x , u + v ) \omega(\omega(x,u),v)=\omega(x,u+v)
  24. f ( x + y ) = f ( x ) g ( y ) + f ( y ) g ( x ) f(x+y)=f(x)g(y)+f(y)g(x)\,\!
  25. g ( x + y ) = g ( x ) g ( y ) - f ( y ) f ( x ) g(x+y)=g(x)g(y)-f(y)f(x)\,\!
  26. f ( x y ) = g l ( x ) h l ( y ) f(xy)=\sum g_{l}(x)h_{l}(y)\,\!
  27. a ( n ) = 3 a ( n - 1 ) + 4 a ( n - 2 ) a(n)=3a(n-1)+4a(n-2)\,\!
  28. ( a b ) c = a ( b c ) . (a\circ b)\circ c=a\circ(b\circ c)~{}.
  29. f ( f ( a , b ) , c ) = f ( a , f ( b , c ) ) . f(f(a,b),c)=f(a,f(b,c)).\,\!
  30. f ( x ) = 1 - x . f(x)=1-x\,.
  31. f f
  32. f ( f ( x ) ) = 1 - ( 1 - x ) = x . f(f(x))=1-(1-x)=x\,.
  33. f ( f ( x ) ) = x , f(f(x))=x~{},
  34. f ( x ) = x f(x)=−x
  35. f ( x ) = a x , f(x)=\frac{a}{x}\,,
  36. f ( x ) = b - x 1 + c x , f(x)=\frac{b-x}{1+cx}~{},
  37. f f
  38. f ( x + y ) 2 = f ( x ) 2 + f ( y ) 2 f(x+y)^{2}=f(x)^{2}+f(y)^{2}\,
  39. x , y x,y∈ℝ
  40. x x
  41. y y
  42. f ( 0 ) 2 = f ( 0 ) 2 + f ( 0 ) 2 . f(0)^{2}=f(0)^{2}+f(0)^{2}.\,
  43. y y
  44. x x
  45. f ( x - x ) 2 = f ( x ) 2 + f ( - x ) 2 f(x-x)^{2}=f(x)^{2}+f(-x)^{2}\,
  46. f ( 0 ) 2 = f ( x ) 2 + f ( - x ) 2 f(0)^{2}=f(x)^{2}+f(-x)^{2}\,
  47. 0 = f ( x ) 2 + f ( - x ) 2 . 0=f(x)^{2}+f(-x)^{2}~{}.
  48. x x
  49. ƒ ( x ) = 0 ƒ(x)=0

Functional_equation_(L-function).html

  1. Z ( s ) = Z ( 1 - s ) Z(s)=Z(1-s)\,
  2. Λ ( s , χ ) = ε Λ ( 1 - s , χ * ) \Lambda(s,\chi)=\varepsilon\Lambda(1-s,\chi^{*})
  3. G ( χ ) | G ( χ ) | G(\chi)\over{\left|G(\chi)\right|}

Functor_category.html

  1. 𝒞 \mathcal{C}^{\rightarrow}
  2. 𝒞 \mathcal{C}
  3. 𝒞 \mathcal{C}
  4. 𝒞 𝟐 \mathcal{C}^{\mathbf{2}}
  5. X Hom ( - , X ) X\mapsto\operatorname{Hom}(-,X)

Fundamental_domain.html

  1. U = { z H : | z | > 1 , | Re ( z ) | < 1 2 } . U=\left\{z\in H:\left|z\right|>1,\,\left|\,\mbox{Re}~{}(z)\,\right|<\frac{1}{2% }\right\}.
  2. D = U { z H : | z | 1 , Re ( z ) = - 1 2 } { z H : | z | = 1 , - 1 2 < Re ( z ) 0 } . D=U\cup\left\{z\in H:\left|z\right|\geq 1,\,\mbox{Re}~{}(z)=\frac{-1}{2}\right% \}\cup\left\{z\in H:\left|z\right|=1,\,\frac{-1}{2}<\mbox{Re}~{}(z)\leq 0% \right\}.

Fundamental_theorem_of_asset_pricing.html

  1. \scriptstyle\mathcal{F}
  2. 𝒫 \scriptstyle\mathcal{P}

Fused_quartz.html

  1. ε = n 2 = 1 + 0.6961663 λ 2 λ 2 - 0.0684043 2 + 0.4079426 λ 2 λ 2 - 0.1162414 2 + 0.8974794 λ 2 λ 2 - 9.896161 2 , \varepsilon=n^{2}=1+\frac{0.6961663\lambda^{2}}{\lambda^{2}-0.0684043^{2}}+% \frac{0.4079426\lambda^{2}}{\lambda^{2}-0.1162414^{2}}+\frac{0.8974794\lambda^% {2}}{\lambda^{2}-9.896161^{2}},
  2. λ \lambda

Future_value.html

  1. F V = P V ( 1 + r t ) FV=PV(1+rt)
  2. F V = P V ( 1 + i ) t FV=PV(1+i)^{t}
  3. i 2 = [ ( 1 + i 1 n 1 ) n 1 n 2 - 1 ] × n 2 i_{2}=\left[\left(1+\frac{i_{1}}{n_{1}}\right)^{\frac{n_{1}}{n_{2}}}-1\right]{% \times}n_{2}
  4. r = ( 1 + i n ) n - 1 r=\left(1+{i\over n}\right)^{n}-1
  5. F V annuity = ( 1 + r ) n - 1 r ( payment amount ) FV_{\mathrm{annuity}}={(1+r)^{n}-1\over r}\cdot\mathrm{(payment\ amount)}

Fσ_set.html

  1. 𝚺 2 0 \mathbf{\Sigma}^{0}_{2}
  2. \mathbb{Q}
  3. \mathbb{R}\setminus\mathbb{Q}
  4. x {x}
  5. A A
  6. ( x , y ) (x,y)
  7. x / y x/y
  8. A = r { ( r y , y ) y } , A=\bigcup_{r\in\mathbb{Q}}\{(ry,y)\mid y\in\mathbb{R}\},
  9. \mathbb{Q}

G-force.html

  1. h h
  2. h / d h/d
  3. d d

Gabriel's_Horn.html

  1. x 1 x x\mapsto\frac{1}{x}
  2. x 1 x\geq 1
  3. V V
  4. A A
  5. V = π 1 a ( 1 x ) 2 d x = π ( 1 - 1 a ) V=\pi\int_{1}^{a}\left({1\over x}\right)^{2}\,\mathrm{d}x=\pi\left(1-{1\over a% }\right)
  6. A = 2 π 1 a 1 x 1 + f ( x ) 2 d x > 2 π 1 a 1 x d x = 2 π ln a . A=2\pi\int_{1}^{a}{1\over x}\sqrt{1+f^{\prime}(x)^{2}}\,\mathrm{d}x>2\pi\int_{% 1}^{a}{1\over x}\,\mathrm{d}x=2\pi\ln a.
  7. a a
  8. x = 1 x=1
  9. x = a x=a
  10. π \pi
  11. π \pi
  12. a a
  13. π \pi
  14. a a
  15. lim a V = lim a π ( 1 - 1 a ) = π . \lim_{a\to\infty}V=\lim_{a\to\infty}\pi\left(1-{1\over a}\right)=\pi.
  16. 2 π 2\pi
  17. a a
  18. a a
  19. lim a A > lim a 2 π ln a = . \lim_{a\to\infty}A>\lim_{a\to\infty}2\pi\ln a=\infty.
  20. f : [ 1 , ) [ 0 , ) f:[1,\infty)\to[0,\infty)
  21. S S
  22. y = f ( x ) y=f(x)
  23. x x
  24. S S
  25. A A
  26. lim t sup x t f ( x ) 2 - f ( 1 ) 2 = lim sup t 1 t ( f ( x ) 2 ) d x \lim_{t\to\infty}\sup_{x\geq t}f(x)^{2}~{}-~{}f(1)^{2}=\limsup_{t\to\infty}% \int_{1}^{t}(f(x)^{2})^{\prime}\,\mathrm{d}x
  27. 1 | ( f ( x ) 2 ) | d x = 1 2 f ( x ) | f ( x ) | d x \leqslant\int_{1}^{\infty}|(f(x)^{2})^{\prime}|\,\mathrm{d}x=\int_{1}^{\infty}% 2f(x)|f^{\prime}(x)|\,\mathrm{d}x
  28. 1 2 f ( x ) 1 + f ( x ) 2 d x \leqslant\int_{1}^{\infty}2f(x)\sqrt{1+f^{\prime}(x)^{2}}\,\mathrm{d}x
  29. = A π < . ={A\over\pi}<\infty.
  30. t 0 t_{0}
  31. sup { f ( x ) x t 0 } \sup\{f(x)\mid x\geq t_{0}\}
  32. M = sup { f ( x ) x 1 } M=\sup\{f(x)\mid x\geq 1\}
  33. f f
  34. f f
  35. [ 1 , ) [1,\infty)
  36. V = 1 f ( x ) π f ( x ) d x 1 M 2 2 π f ( x ) d x M 2 1 2 π f ( x ) 1 + f ( x ) 2 d x V=\int_{1}^{\infty}f(x)\cdot\pi f(x)\,\mathrm{d}x\leqslant\int_{1}^{\infty}{M% \over 2}\cdot 2\pi f(x)\,\mathrm{d}x\leqslant{M\over 2}\cdot\int_{1}^{\infty}2% \pi f(x)\sqrt{1+f^{\prime}(x)^{2}}\,\mathrm{d}x
  37. = M 2 A . ={M\over 2}\cdot A.
  38. A A
  39. V V

Galois_extension.html

  1. K ¯ \bar{K}
  2. K K
  3. K K
  4. K K

Galois_module.html

  1. 𝐙 × \mathbf{Z}_{\ell}^{\times}
  2. r K : C K ~ W K ab r_{K}:C_{K}\tilde{\rightarrow}W_{K}^{\,\text{ab}}

Galton–Watson_process.html

  1. X n + 1 = j = 1 X n ξ j ( n ) X_{n+1}=\sum_{j=1}^{X_{n}}\xi_{j}^{(n)}
  2. { ξ j ( n ) : n , j } \{\xi_{j}^{(n)}:n,j\in\mathbb{N}\}
  3. ξ j ( n ) \xi_{j}^{(n)}
  4. lim n Pr ( X n = 0 ) . \lim_{n\to\infty}\Pr(X_{n}=0).\,
  5. x n + 1 = e λ ( x n - 1 ) , x_{n+1}=e^{\lambda(x_{n}-1)},\,

Gambler's_ruin.html

  1. Bankroll N \frac{\mbox{Bankroll}~{}}{N}
  2. P 1 = n 2 n 1 + n 2 P_{1}=\frac{n_{2}}{n_{1}+n_{2}}
  3. P 2 = n 1 n 1 + n 2 P_{2}=\frac{n_{1}}{n_{1}+n_{2}}
  4. ( P 1 ) (P_{1})
  5. P 2 P_{2}
  6. P 1 = 5 8 + 5 P_{1}=\frac{5}{8+5}
  7. = 5 13 =\frac{5}{13}
  8. P 2 = 8 8 + 5 P_{2}=\frac{8}{8+5}
  9. = 8 13 =\frac{8}{13}
  10. P 1 = 6 6 + 6 P_{1}=\frac{6}{6+6}
  11. 6 12 \frac{6}{12}
  12. 1 2 \frac{1}{2}
  13. P 2 = 6 6 + 6 P_{2}=\frac{6}{6+6}
  14. 6 12 \frac{6}{12}
  15. 1 2 \frac{1}{2}
  16. P 1 = 1 - ( p q ) n 2 1 - ( p q ) n 1 + n 2 P_{1}=\frac{1-(\frac{p}{q})^{n_{2}}}{1-(\frac{p}{q})^{n_{1}+n_{2}}}
  17. P 2 = 1 - ( q p ) n 1 1 - ( q p ) n 1 + n 2 P_{2}=\frac{1-(\frac{q}{p})^{n_{1}}}{1-(\frac{q}{p})^{n_{1}+n_{2}}}
  18. n > 1 n>1
  19. P ( R n ) P(R_{n})
  20. P ( R n ) = P ( R n | W ) P ( W ) + P ( R n | W ¯ ) P ( W ¯ ) P(R_{n})=P(R_{n}|W)P(W)+P(R_{n}|\bar{W})P(\bar{W})
  21. P ( W ) = p P(W)=p
  22. P ( W ¯ ) = 1 - p = q P(\bar{W})=1-p=q
  23. P ( R n | W ) P(R_{n}|W)
  24. n + 1 n+1
  25. P ( R n + 1 ) P(R_{n+1})
  26. P ( R n | W ¯ ) P(R_{n}|\bar{W})
  27. n - 1 n-1
  28. P ( R n - 1 ) P(R_{n-1})
  29. q n = P ( R n ) q_{n}=P(R_{n})
  30. q n = q n + 1 p + q n - 1 q q_{n}=q_{n+1}p+q_{n-1}q
  31. q 0 = 1 q_{0}=1
  32. q n 1 + n 2 = 0 q_{n_{1}+n_{2}}=0
  33. N 2 N\geq 2\,\,
  34. x 1 , x 2 , , x N x_{1},x_{2},\cdots,x_{N}\,\,
  35. N = 2 N=2\,
  36. x 1 , x 2 x_{1},x_{2}\,
  37. N 3 N\geq 3
  38. N 3 N\geq 3

Game_complexity.html

  1. G T C b d GTC\geq b^{d}

Game_semantics.html

  1. x y ϕ ( x , y ) \forall x\exists y\,\phi(x,y)
  2. f x ϕ ( x , f ( x ) ) \exists f\forall x\,\phi(x,f(x))

Gamma-glutamyl_transpeptidase.html

  1. \rightleftharpoons

Gas_compressor.html

  1. η C = I s e n t r o p i c C o m p r e s s o r W o r k A c t u a l C o m p r e s s o r W o r k = W s W a h 2 s - h 1 h 2 a - h 1 \eta_{C}=\frac{Isentropic\;Compressor\;Work}{Actual\;Compressor\;Work}=\frac{W% _{s}}{W_{a}}\cong\frac{h_{2s}-h_{1}}{h_{2a}-h_{1}}
  2. h 1 h_{1}
  3. h 2 a h_{2a}
  4. h 2 s h_{2s}
  5. q q
  6. w w
  7. k e ke
  8. p e pe
  9. δ q a c t - δ w a c t = d h + d k e + d p e \delta q_{act}-\delta w_{act}=dh+dke+dpe
  10. δ q r e v - δ w r e v = d h + d k e + d p e \delta q_{rev}-\delta w_{rev}=dh+dke+dpe
  11. δ q a c t - δ w a c t = δ q r e v - δ w r e v \delta q_{act}-\delta w_{act}=\delta q_{rev}-\delta w_{rev}
  12. δ w r e v - δ w a c t = δ q r e v - δ q a c t \delta w_{rev}-\delta w_{act}=\delta q_{rev}-\delta q_{act}
  13. δ q r e v = T d s \delta q_{rev}=Tds
  14. δ w r e v - δ w a c t T = d s - δ q a c t T 0 \frac{\delta w_{rev}-\delta w_{act}}{T}=ds-\frac{\delta q_{act}}{T}\geq 0
  15. d s δ q a c t T ds\geq\frac{\delta q_{act}}{T}
  16. T 0 T\geq 0
  17. δ w r e v δ w a c t \delta w_{rev}\geq\delta w_{act}
  18. w r e v w a c t w_{rev}\geq w_{act}
  19. P 1 P_{1}
  20. P 2 P_{2}
  21. P 1 P_{1}
  22. P 2 P_{2}
  23. P v k = c o n s t a n t Pv^{k}=constant
  24. k = C p / C v k=C_{p}/C_{v}
  25. W c o m p , i n = k R ( T 2 - T 1 ) k - 1 = k R T 1 k - 1 [ ( P 2 P 1 ) ( k - 1 ) / k - 1 ] W_{comp,in}=\frac{kR(T_{2}-T_{1})}{k-1}=\frac{kRT_{1}}{k-1}\left[\left(\frac{P% _{2}}{P_{1}}\right)^{(k-1)/k}-1\right]
  26. P v n = c o n s t a n t Pv^{n}=constant
  27. W c o m p , i n = n R ( T 2 - T 1 ) n - 1 = n R T 1 n - 1 [ ( P 2 P 1 ) ( n - 1 ) / n - 1 ] W_{comp,in}=\frac{nR(T_{2}-T_{1})}{n-1}=\frac{nRT_{1}}{n-1}\left[\left(\frac{P% _{2}}{P_{1}}\right)^{(n-1)/n}-1\right]
  28. T = c o n s t a n t T=constant
  29. P v = c o n s t a n t Pv=constant
  30. W c o m p , i n = R T l n ( P 2 P 1 ) W_{comp,in}=RTln\left(\frac{P_{2}}{P_{1}}\right)
  31. P 1 P_{1}
  32. P 2 P_{2}
  33. P v k = c o n s t a n t Pv^{k}=constant
  34. T = c o n s t a n t T=constant
  35. P v = c o n s t a n t Pv=constant
  36. P v n = c o n s t a n t Pv^{n}=constant
  37. W = V 1 V 2 p d V = p 1 V 1 n V 1 V 2 V - n d V W=\int_{V_{1}}^{V_{2}}pdV=p_{1}V_{1}^{n}\int_{V_{1}}^{V_{2}}V^{-n}dV
  38. p 2 p 1 = ( V 1 V 2 ) n \frac{p_{2}}{p_{1}}\ =\left(\frac{V_{1}}{V_{2}}\ \right)^{n}
  39. p 1 V 1 n = p 2 V 2 n = p V n p_{1}V_{1}^{n}=p_{2}V_{2}^{n}=pV^{n}
  40. p = p 1 V 1 n V n p=\frac{p_{1}V_{1}^{n}}{V^{n}}
  41. W = p 1 V 1 n 1 - n ( V 2 1 - n - V 1 1 - n ) W=\frac{{p_{1}}{V_{1}^{n}}}{1-n}\ ({V_{2}^{1-n}}-{V_{1}^{1-n}})
  42. T 2 = T 1 ( p 2 p 1 ) ( k - 1 ) / k T_{2}=T_{1}\left(\frac{p_{2}}{p_{1}}\right)^{(k-1)/k}
  43. W = - p 1 V 1 ln ( p 2 p 1 ) W=-{p_{1}}{V_{1}}\ln\left(\frac{p_{2}}{p_{1}}\ \right)

Gauss_map.html

  1. G ~ k , n \tilde{G}_{k,n}
  2. G ~ k , n G ~ n - k , n \tilde{G}_{k,n}\cong\tilde{G}_{n-k,n}
  3. G ~ 1 , n S n - 1 \tilde{G}_{1,n}\cong S^{n-1}
  4. M = 𝐑 n M=\mathbf{R}^{n}
  5. R | N u × N v | d u d v = R K | X u × X v | d u d v = R K d A \iint_{R}|N_{u}\times N_{v}|\ du\,dv=\iint_{R}K|X_{u}\times X_{v}|\ du\,dv=% \iint_{R}K\ dA

Gaussian_integral.html

  1. - + e - x 2 d x = π \int_{-\infty}^{+\infty}e^{-x^{2}}\,\mathrm{d}x=\sqrt{\pi}
  2. e - x 2 d x , \int e^{-x^{2}}\,dx,
  3. - + e - x 2 d x \int_{-\infty}^{+\infty}e^{-x^{2}}\,\mathrm{d}x
  4. ( - e - x 2 d x ) 2 = - e - x 2 d x - e - y 2 d y = - - e - ( x 2 + y 2 ) d x d y \left(\int_{-\infty}^{\infty}e^{-x^{2}}\,dx\right)^{2}=\int_{-\infty}^{\infty}% e^{-x^{2}}\,dx\int_{-\infty}^{\infty}e^{-y^{2}}\,dy=\int_{-\infty}^{\infty}% \int_{-\infty}^{\infty}e^{-(x^{2}+y^{2})}\,dx\,dy
  5. ( e - x 2 d x ) 2 ; \left(\int e^{-x^{2}}\,dx\right)^{2};
  6. 𝐑 2 e - ( x 2 + y 2 ) d ( x , y ) = 0 2 π 0 e - r 2 r d r d θ = 2 π 0 r e - r 2 d r = 2 π - 0 1 2 e s d s s = - r 2 = π - 0 e s d s = π ( e 0 - e - ) = π , \begin{aligned}\displaystyle\iint_{\mathbf{R}^{2}}e^{-(x^{2}+y^{2})}\,d(x,y)&% \displaystyle=\int_{0}^{2\pi}\int_{0}^{\infty}e^{-r^{2}}r\,dr\,d\theta\\ &\displaystyle=2\pi\int_{0}^{\infty}re^{-r^{2}}\,dr\\ &\displaystyle=2\pi\int_{-\infty}^{0}\tfrac{1}{2}e^{s}\,ds&&\displaystyle s=-r% ^{2}\\ &\displaystyle=\pi\int_{-\infty}^{0}e^{s}\,ds\\ &\displaystyle=\pi(e^{0}-e^{-\infty})\\ &\displaystyle=\pi,\end{aligned}
  7. ( - e - x 2 d x ) 2 = π , \left(\int_{-\infty}^{\infty}e^{-x^{2}}\,dx\right)^{2}=\pi,
  8. - e - x 2 d x = π \int_{-\infty}^{\infty}e^{-x^{2}}\,dx=\sqrt{\pi}
  9. I ( a ) = - a a e - x 2 d x . I(a)=\int_{-a}^{a}e^{-x^{2}}dx.
  10. - e - x 2 d x \int_{-\infty}^{\infty}e^{-x^{2}}\,dx
  11. lim a I ( a ) \lim_{a\to\infty}I(a)
  12. - e - x 2 d x . \int_{-\infty}^{\infty}e^{-x^{2}}\,dx.
  13. - | e - x 2 | d x < - - 1 - x e - x 2 d x + - 1 1 e - x 2 d x + 1 x e - x 2 d x < . \int_{-\infty}^{\infty}|e^{-x^{2}}|\,dx<\int_{-\infty}^{-1}-xe^{-x^{2}}\,dx+% \int_{-1}^{1}e^{-x^{2}}\,dx+\int_{1}^{\infty}xe^{-x^{2}}\,dx<\infty.
  14. - e - x 2 d x \int_{-\infty}^{\infty}e^{-x^{2}}\,dx
  15. lim a I ( a ) \lim_{a\to\infty}I(a)
  16. I ( a ) 2 = ( - a a e - x 2 d x ) ( - a a e - y 2 d y ) = - a a ( - a a e - y 2 d y ) e - x 2 d x = - a a - a a e - ( x 2 + y 2 ) d y d x . \begin{aligned}\displaystyle I(a)^{2}&\displaystyle=\left(\int_{-a}^{a}e^{-x^{% 2}}\,dx\right)\left(\int_{-a}^{a}e^{-y^{2}}\,dy\right)\\ &\displaystyle=\int_{-a}^{a}\left(\int_{-a}^{a}e^{-y^{2}}\,dy\right)\,e^{-x^{2% }}\,dx\\ &\displaystyle=\int_{-a}^{a}\int_{-a}^{a}e^{-(x^{2}+y^{2})}\,dy\,dx.\end{aligned}
  17. [ - a , a ] × [ - a , a ] e - ( x 2 + y 2 ) d ( x , y ) , \iint_{[-a,a]\times[-a,a]}e^{-(x^{2}+y^{2})}\,d(x,y),
  18. I ( a ) 2 I(a)^{2}
  19. I ( a ) 2 I(a)^{2}
  20. x \displaystyle x
  21. 0 2 π 0 a r e - r 2 d r d θ < I 2 ( a ) < 0 2 π 0 a 2 r e - r 2 d r d θ . \int_{0}^{2\pi}\int_{0}^{a}re^{-r^{2}}\,dr\,d\theta<I^{2}(a)<\int_{0}^{2\pi}% \int_{0}^{a\sqrt{2}}re^{-r^{2}}\,dr\,d\theta.
  22. π ( 1 - e - a 2 ) < I 2 ( a ) < π ( 1 - e - 2 a 2 ) . \pi(1-e^{-a^{2}})<I^{2}(a)<\pi(1-e^{-2a^{2}}).
  23. - e - x 2 d x = π . \int_{-\infty}^{\infty}e^{-x^{2}}\,dx=\sqrt{\pi}.
  24. y \displaystyle y
  25. - e - x 2 d x = 2 0 e - x 2 d x . \int_{-\infty}^{\infty}e^{-x^{2}}\,dx=2\int_{0}^{\infty}e^{-x^{2}}\,dx.
  26. I 2 \displaystyle I^{2}
  27. I = π I=\sqrt{\pi}
  28. - e - x 2 d x = 2 0 e - x 2 d x \int_{-\infty}^{\infty}e^{-x^{2}}dx=2\int_{0}^{\infty}e^{-x^{2}}dx
  29. x = t x=\sqrt{t}
  30. 2 0 e - x 2 d x = 2 0 1 2 e - t t - 1 2 d t = Γ ( 1 2 ) = π 2\int_{0}^{\infty}e^{-x^{2}}dx=2\int_{0}^{\infty}\frac{1}{2}\ e^{-t}\ t^{-% \frac{1}{2}}dt=\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}
  31. π \sqrt{\pi}
  32. 0 e - a x b d x = Γ ( 1 b ) b a 1 b \int_{0}^{\infty}e^{-ax^{b}}dx=\frac{\Gamma\left(\frac{1}{b}\right)}{ba^{\frac% {1}{b}}}
  33. - e - a ( x + b ) 2 d x = π a . \int_{-\infty}^{\infty}e^{-a(x+b)^{2}}\,dx=\sqrt{\frac{\pi}{a}}.
  34. - e - a x 2 + b x + c d x = π a e b 2 4 a + c , \int_{-\infty}^{\infty}e^{-ax^{2}+bx+c}\,dx=\sqrt{\frac{\pi}{a}}\,e^{\frac{b^{% 2}}{4a}+c},
  35. - exp ( - 1 2 i , j = 1 n A i j x i x j ) d n x = - exp ( - 1 2 x T A x ) d n x = ( 2 π ) n det A \int_{-\infty}^{\infty}\exp\left(-\frac{1}{2}\sum_{i,j=1}^{n}A_{ij}x_{i}x_{j}% \right)\,d^{n}x=\int_{-\infty}^{\infty}\exp\left(-\frac{1}{2}x^{T}Ax\right)\,d% ^{n}x=\sqrt{\frac{(2\pi)^{n}}{\det A}}
  36. x k 1 x k 2 N exp ( - 1 2 i , j = 1 n A i j x i x j ) d n x = ( 2 π ) n det A 1 2 N N ! σ S 2 N ( A - 1 ) k σ ( 1 ) k σ ( 2 ) ( A - 1 ) k σ ( 2 N - 1 ) k σ ( 2 N ) \int x^{k_{1}}\cdots x^{k_{2N}}\,\exp\left(-\frac{1}{2}\sum_{i,j=1}^{n}A_{ij}x% _{i}x_{j}\right)\,d^{n}x=\sqrt{\frac{(2\pi)^{n}}{\det A}}\,\frac{1}{2^{N}N!}\,% \sum_{\sigma\in S_{2N}}(A^{-1})^{k_{\sigma(1)}k_{\sigma(2)}}\cdots(A^{-1})^{k_% {\sigma(2N-1)}k_{\sigma(2N)}}
  37. f ( x ) exp ( - 1 2 i , j = 1 n A i j x i x j ) d n x = ( 2 π ) n det A exp ( 1 2 i , j = 1 n ( A - 1 ) i j x i x j ) f ( x ) | x = 0 \int f(\vec{x})\exp\left(-\frac{1}{2}\sum_{i,j=1}^{n}A_{ij}x_{i}x_{j}\right)d^% {n}x=\sqrt{(2\pi)^{n}\over\det A}\,\left.\exp\left({1\over 2}\sum_{i,j=1}^{n}(% A^{-1})_{ij}{\partial\over\partial x_{i}}{\partial\over\partial x_{j}}\right)f% (\vec{x})\right|_{\vec{x}=0}
  38. ( 2 π ) (2\pi)^{\infty}
  39. f ( x 1 ) f ( x 2 N ) e - 1 2 A ( x 2 N + 1 , x 2 N + 2 ) f ( x 2 N + 1 ) f ( x 2 N + 2 ) d d x 2 N + 1 d d x 2 N + 2 𝒟 f e - 1 2 A ( x 2 N + 1 , x 2 N + 2 ) f ( x 2 N + 1 ) f ( x 2 N + 2 ) d d x 2 N + 1 d d x 2 N + 2 𝒟 f = 1 2 N N ! σ S 2 N A - 1 ( x σ ( 1 ) , x σ ( 2 ) ) A - 1 ( x σ ( 2 N - 1 ) , x σ ( 2 N ) ) . \frac{\int f(x_{1})\cdots f(x_{2N})e^{-\iint\frac{1}{2}A(x_{2N+1},x_{2N+2})f(x% _{2N+1})f(x_{2N+2})d^{d}x_{2N+1}d^{d}x_{2N+2}}\mathcal{D}f}{\int e^{-\iint% \frac{1}{2}A(x_{2N+1},x_{2N+2})f(x_{2N+1})f(x_{2N+2})d^{d}x_{2N+1}d^{d}x_{2N+2% }}\mathcal{D}f}=\frac{1}{2^{N}N!}\sum_{\sigma\in S_{2N}}A^{-1}(x_{\sigma(1)},x% _{\sigma(2)})\cdots A^{-1}(x_{\sigma(2N-1)},x_{\sigma(2N)}).
  40. e - 1 2 i , j = 1 n A i j x i x j + i = 1 n B i x i d n x = e - 1 2 x T A x + B T x d n x = ( 2 π ) n det A e 1 2 B T A - 1 B . \int e^{-\frac{1}{2}\sum_{i,j=1}^{n}A_{ij}x_{i}x_{j}+\sum_{i=1}^{n}B_{i}x_{i}}% d^{n}x=\int e^{-\frac{1}{2}\vec{x}^{T}{A}\vec{x}+\vec{B}^{T}\vec{x}}d^{n}x=% \sqrt{\frac{(2\pi)^{n}}{\det{A}}}e^{\frac{1}{2}\vec{B}^{T}A^{-1}\vec{B}}.
  41. 0 x 2 n e - x 2 a 2 d x = π a 2 n + 1 ( 2 n - 1 ) ! ! 2 n + 1 \int_{0}^{\infty}x^{2n}e^{-\frac{x^{2}}{a^{2}}}\,dx=\sqrt{\pi}\frac{a^{2n+1}(2% n-1)!!}{2^{n+1}}
  42. 0 x 2 n + 1 e - x 2 a 2 d x = n ! 2 a 2 n + 1 \int_{0}^{\infty}x^{2n+1}e^{-\frac{x^{2}}{a^{2}}}\,dx=\frac{n!}{2}a^{2n+1}
  43. 0 x n e - a x 2 d x = Γ ( ( n + 1 ) 2 ) 2 a ( n + 1 ) 2 \int_{0}^{\infty}x^{n}e^{-a\,x^{2}}\,dx=\frac{\Gamma(\frac{(n+1)}{2})}{2\,a^{% \frac{(n+1)}{2}}}
  44. 0 x 2 n e - a x 2 d x = ( 2 n - 1 ) ! ! a n 2 n + 1 π a \int_{0}^{\infty}x^{2n}e^{-ax^{2}}\,dx=\frac{(2n-1)!!}{a^{n}2^{n+1}}\sqrt{% \frac{\pi}{a}}
  45. - x 2 n e - α x 2 d x = ( - 1 ) n - n α n e - α x 2 d x = ( - 1 ) n n α n - e - α x 2 d x = π ( - 1 ) n n α n α - 1 2 = π α ( 2 n - 1 ) ! ! ( 2 α ) n \int_{-\infty}^{\infty}x^{2n}e^{-\alpha x^{2}}\,dx=\left(-1\right)^{n}\int_{-% \infty}^{\infty}\frac{\partial^{n}}{\partial\alpha^{n}}e^{-\alpha x^{2}}\,dx=% \left(-1\right)^{n}\frac{\partial^{n}}{\partial\alpha^{n}}\int_{-\infty}^{% \infty}e^{-\alpha x^{2}}\,dx=\sqrt{\pi}\left(-1\right)^{n}\frac{\partial^{n}}{% \partial\alpha^{n}}\alpha^{-\frac{1}{2}}=\sqrt{\frac{\pi}{\alpha}}\frac{(2n-1)% !!}{\left(2\alpha\right)^{n}}
  46. - e a x 4 + b x 3 + c x 2 + d x + f d x = 1 2 e f n , m , p = 0 n + p = 0 mod 2 b n n ! c m m ! d p p ! Γ ( 3 n + 2 m + p + 1 4 ) ( - a ) 3 n + 2 m + p + 1 4 . \int_{-\infty}^{\infty}e^{ax^{4}+bx^{3}+cx^{2}+dx+f}\,dx=\frac{1}{2}e^{f}\ % \sum_{\begin{smallmatrix}n,m,p=0\\ n+p=0\mod 2\end{smallmatrix}}^{\infty}\ \frac{b^{n}}{n!}\frac{c^{m}}{m!}\frac{% d^{p}}{p!}\frac{\Gamma\left(\frac{3n+2m+p+1}{4}\right)}{(-a)^{\frac{3n+2m+p+1}% {4}}}.