wpmath0000002_6

Exterior_derivative.html

  1. k k
  2. k k
  3. ( k + 1 ) (k+1)
  4. k k
  5. k + 1. k+1.
  6. f f
  7. 0
  8. f f
  9. f f
  10. d f df
  11. 1 1
  12. X X
  13. f f
  14. X X
  15. k k
  16. 𝐑 \mathbf{R}
  17. k k
  18. ( k + 1 ) (k+1)
  19. d f df
  20. f f
  21. f f
  22. d ( d f ) = 0 d(df)=0
  23. f f
  24. α α
  25. p p
  26. d d
  27. 1 1
  28. d ( d α ) = 0 d(dα)=0
  29. k k
  30. α α
  31. f f
  32. α α
  33. k k
  34. d ( f α ) = d ( f α ) = d f α + f d α d(fα)=d(f∧α)=df∧α+f∧dα
  35. 0
  36. 1 p k 1≤p≤k
  37. k k
  38. ω = f I d x I = f i 1 , i 2 i k d x i 1 d x i 2 d x i k \omega=f_{I}\mathrm{d}x^{I}=f_{i_{1},i_{2}\cdots i_{k}}\mathrm{d}x^{i_{1}}% \wedge\mathrm{d}x^{i_{2}}\wedge\cdots\wedge\mathrm{d}x^{i_{k}}
  39. d ω = i = 1 n f I x i d x i d x I . \mathrm{d}{\omega}=\sum_{i=1}^{n}\frac{\partial f_{I}}{\partial x^{i}}\mathrm{% d}x^{i}\wedge\mathrm{d}x^{I}.
  40. k k
  41. ω = I f I d x I , \omega=\sum_{I}f_{I}\mathrm{d}x^{I},
  42. I I
  43. i i
  44. I I
  45. d ω = d ( f I d x i 1 d x i k ) = d f I ( d x i 1 d x i k ) + f I d ( d x i 1 d x i k ) = d f I d x i 1 d x i k + p = 1 k ( - 1 ) ( p - 1 ) f I d x i 1 d x i p - 1 d 2 x i p d x i p + 1 d x i k = d f I d x i 1 d x i k = i = 1 n f I x i d x i d x i 1 d x i k \begin{aligned}\displaystyle\mathrm{d}{\omega}&\displaystyle=\mathrm{d}\left(f% _{I}\mathrm{d}x^{i_{1}}\wedge\cdots\wedge\mathrm{d}x^{i_{k}}\right)\\ &\displaystyle=\mathrm{d}f_{I}\wedge\left(\mathrm{d}x^{i_{1}}\wedge\cdots% \wedge\mathrm{d}x^{i_{k}}\right)+f_{I}\mathrm{d}\left(\mathrm{d}x^{i_{1}}% \wedge\cdots\wedge\mathrm{d}x^{i_{k}}\right)\\ &\displaystyle=\mathrm{d}f_{I}\wedge\mathrm{d}x^{i_{1}}\wedge\cdots\wedge% \mathrm{d}x^{i_{k}}+\sum_{p=1}^{k}(-1)^{(p-1)}f_{I}\mathrm{d}x^{i_{1}}\wedge% \cdots\wedge\mathrm{d}x^{i_{p-1}}\wedge\mathrm{d}^{2}x^{i_{p}}\wedge\mathrm{d}% x^{i_{p+1}}\wedge\cdots\wedge\mathrm{d}x^{i_{k}}\\ &\displaystyle=\mathrm{d}f_{I}\wedge\mathrm{d}x^{i_{1}}\wedge\cdots\wedge% \mathrm{d}x^{i_{k}}\\ &\displaystyle=\sum_{i=1}^{n}\frac{\partial f_{I}}{\partial x^{i}}\mathrm{d}x^% {i}\wedge\mathrm{d}x^{i_{1}}\wedge\cdots\wedge\mathrm{d}x^{i_{k}}\\ \end{aligned}
  46. 0
  47. k k
  48. ω ω
  49. k + 1 k+1
  50. d ω ( V 0 , , V k ) = i ( - 1 ) i V i ( ω ( V 0 , , V ^ i , , V k ) ) + i < j ( - 1 ) i + j ω ( [ V i , V j ] , V 0 , , V ^ i , , V ^ j , , V k ) \mathrm{d}\omega(V_{0},...,V_{k})=\sum_{i}(-1)^{i}V_{i}\left(\omega\left(V_{0}% ,\ldots,\hat{V}_{i},\ldots,V_{k}\right)\right)+\sum_{i<j}(-1)^{i+j}\omega\left% (\left[V_{i},V_{j}\right],V_{0},\ldots,\hat{V}_{i},\ldots,\hat{V}_{j},\ldots,V% _{k}\right)
  51. ω ( V 0 , , V ^ i , , V k ) = ω ( V 0 , , V i - 1 , V i + 1 , , V k ) . \omega\left(V_{0},\ldots,\hat{V}_{i},\ldots,V_{k}\right)=\omega\left(V_{0},% \ldots,V_{i-1},V_{i+1},\ldots,V_{k}\right).
  52. 1 1
  53. d ω ( X , Y ) = X ω ( Y ) Y ω ( X ) ω ( X X , Y ) ) dω(X,Y)=Xω(Y)−Yω(X)−ω(XX,Y))
  54. X X
  55. Y Y
  56. M M
  57. n n
  58. ω ω
  59. ( n 1 ) (n−1)
  60. M M
  61. M d ω = M ω \int_{M}\mathrm{d}\omega=\int_{\partial{M}}\omega
  62. M M
  63. M M
  64. 1 1
  65. d σ \displaystyle\mathrm{d}\sigma
  66. σ = u d x + v d y σ=udx+vdy
  67. 1 1
  68. d σ = ( i = 1 2 u x i d x i d x ) + ( i = 1 2 v x i d x i d y ) = ( u x d x d x + u y d y d x ) + ( v x d x d y + v y d y d y ) = 0 - u y d x d y + v x d x d y + 0 = ( v x - u y ) d x d y \begin{aligned}\displaystyle\mathrm{d}\sigma&\displaystyle=\left(\sum_{i=1}^{2% }\frac{\partial u}{\partial x^{i}}\mathrm{d}x^{i}\wedge\mathrm{d}x\right)+% \left(\sum_{i=1}^{2}\frac{\partial v}{\partial x^{i}}\mathrm{d}x^{i}\wedge% \mathrm{d}y\right)\\ &\displaystyle=\left(\frac{\partial{u}}{\partial{x}}\mathrm{d}x\wedge\mathrm{d% }x+\frac{\partial{u}}{\partial{y}}\mathrm{d}y\wedge\mathrm{d}x\right)+\left(% \frac{\partial{v}}{\partial{x}}\mathrm{d}x\wedge\mathrm{d}y+\frac{\partial{v}}% {\partial{y}}\mathrm{d}y\wedge\mathrm{d}y\right)\\ &\displaystyle=0-\frac{\partial{u}}{\partial{y}}\mathrm{d}x\wedge\mathrm{d}y+% \frac{\partial{v}}{\partial{x}}\mathrm{d}x\wedge\mathrm{d}y+0\\ &\displaystyle=\left(\frac{\partial{v}}{\partial{x}}-\frac{\partial{u}}{% \partial{y}}\right)\mathrm{d}x\wedge\mathrm{d}y\end{aligned}
  69. k k
  70. ω ω
  71. d ω = 0 dω=0
  72. ω ω
  73. ω = d α ω=dα
  74. ( k 1 ) (k−1)
  75. α α
  76. d d
  77. k k
  78. k k
  79. k k
  80. k > 0 k>0
  81. 𝐑 \mathbf{R}
  82. f : M N f:M→N
  83. k k
  84. f f
  85. f f
  86. d d
  87. 0
  88. 0
  89. 1 1
  90. d f = i = 1 n f x i d x i = f , . \mathrm{d}f=\sum_{i=1}^{n}\frac{\partial f}{\partial x^{i}}\,\mathrm{d}x^{i}=% \langle\nabla f,\cdot\rangle.
  91. d f df
  92. V V
  93. V V
  94. f ∇f
  95. f f
  96. 1 1
  97. d f df
  98. f f
  99. ( n 1 ) (n−1)
  100. ω V = v 1 ( d x 2 d x 3 d x n ) - v 2 ( d x 1 d x 3 d x n ) + + ( - 1 ) n - 1 v n ( d x 1 d x n - 1 ) = p = 1 n ( - 1 ) ( p - 1 ) v p ( d x 1 d x p - 1 d x p ^ d x p + 1 d x n ) \begin{aligned}\displaystyle\omega_{V}&\displaystyle=v_{1}\left(\mathrm{d}x^{2% }\wedge\mathrm{d}x^{3}\wedge\cdots\wedge\mathrm{d}x^{n}\right)-v_{2}\left(% \mathrm{d}x^{1}\wedge\mathrm{d}x^{3}\cdots\wedge\mathrm{d}x^{n}\right)+\cdots+% (-1)^{n-1}v_{n}\left(\mathrm{d}x^{1}\wedge\cdots\wedge\mathrm{d}x^{n-1}\right)% \\ &\displaystyle=\sum_{p=1}^{n}(-1)^{(p-1)}v_{p}\left(\mathrm{d}x^{1}\wedge% \cdots\wedge\mathrm{d}x^{p-1}\wedge\widehat{\mathrm{d}x^{p}}\wedge\mathrm{d}x^% {p+1}\wedge\cdots\wedge\mathrm{d}x^{n}\right)\end{aligned}
  101. d x p ^ \widehat{\mathrm{d}x^{p}}
  102. n = 3 n=3
  103. 2 2
  104. V V
  105. V V
  106. ( n 1 ) (n−1)
  107. n n
  108. d ω V = div ( V ) ( d x 1 d x 2 d x n ) . \mathrm{d}\omega_{V}=\operatorname{div}(V)\left(\mathrm{d}x^{1}\wedge\mathrm{d% }x^{2}\wedge\cdots\wedge\mathrm{d}x^{n}\right).
  109. V V
  110. 1 1
  111. η V = v 1 d x 1 + v 2 d x 2 + + v n d x n . \eta_{V}=v_{1}\mathrm{d}x^{1}+v_{2}\mathrm{d}x^{2}+\cdots+v_{n}\mathrm{d}x^{n}.
  112. V V
  113. V −V
  114. n = 3 n=3
  115. 1 1
  116. 2 2
  117. d η V = ω curl ( V ) . \mathrm{d}\eta_{V}=\omega_{\operatorname{curl}(V)}.
  118. grad ( f ) = f = ( d f ) div ( F ) = F = d ( F ) curl ( F ) = × F = [ ( d F ) ] , Δ f = 2 f = d ( d f ) \begin{array}[]{rcccl}\operatorname{grad}(f)&=&\nabla f&=&\left(\mathrm{d}f% \right)^{\sharp}\\ \operatorname{div}(F)&=&\nabla\cdot F&=&\star\mathrm{d}\left(\star F^{\flat}% \right)\\ \operatorname{curl}(F)&=&\nabla\times F&=&\left[\star\left(\mathrm{d}F^{\flat}% \right)\right]^{\sharp},\\ \Delta f&=&\nabla^{2}f&=&\star\mathrm{d}\left(\star\mathrm{d}f\right)\\ \end{array}
  119. \star
  120. \flat
  121. \sharp

Extreme_value_theorem.html

  1. f ( c ) f ( x ) f ( d ) for all x [ a , b ] . f(c)\geq f(x)\geq f(d)\quad\,\text{for all }x\in[a,b].\,
  2. m f ( x ) M for all x [ a , b ] . m\leq f(x)\leq M\quad\,\text{for all }x\in[a,b].\,
  3. x n k x_{n_{k}}
  4. x n k x_{n_{k}}
  5. d n k d_{n_{k}}
  6. d n k d_{n_{k}}
  7. f * ( x i 0 ) f * ( x i ) f^{*}(x_{i_{0}})\geq f^{*}(x_{i})
  8. c = 𝐬𝐭 ( x i 0 ) c=\mathbf{st}(x_{i_{0}})
  9. x [ x i , x i + 1 ] x\in[x_{i},x_{i+1}]
  10. f * ( x i 0 ) f * ( x i ) f^{*}(x_{i_{0}})\geq f^{*}(x_{i})
  11. 𝐬𝐭 ( f * ( x i 0 ) ) 𝐬𝐭 ( f * ( x i ) ) \mathbf{st}(f^{*}(x_{i_{0}}))\geq\mathbf{st}(f^{*}(x_{i}))
  12. 𝐬𝐭 ( f * ( x i 0 ) ) = f ( 𝐬𝐭 ( x i 0 ) ) = f ( c ) \mathbf{st}(f^{*}(x_{i_{0}}))=f(\mathbf{st}(x_{i_{0}}))=f(c)
  13. lim sup y x f ( y ) f ( x ) \limsup_{y\to x}f(y)\leq f(x)\,
  14. lim inf y x f ( y ) f ( x ) \liminf_{y\to x}f(y)\geq f(x)\,

Édouard_Lucas.html

  1. n = 1 N n 2 = M 2 \sum_{n=1}^{N}n^{2}=M^{2}\;

F-number.html

  1. N = f D N=\frac{f}{D}
  2. f f
  3. D D
  4. 2 \scriptstyle\sqrt{2}
  5. f / 1 ( 2 ) 0 \frac{f/1}{(\sqrt{2})^{0}}
  6. f / 1 ( 2 ) 1 \frac{f/1}{(\sqrt{2})^{1}}
  7. f / 1 ( 2 ) 2 \frac{f/1}{(\sqrt{2})^{2}}
  8. f / 1 ( 2 ) 3 \frac{f/1}{(\sqrt{2})^{3}}
  9. 2 \scriptstyle\sqrt{2}
  10. 2 \scriptstyle\sqrt{2}
  11. 2 \scriptstyle\sqrt{2}
  12. 2 A V \sqrt{2^{AV}}
  13. T = f transmittance . T=\frac{f}{\sqrt{\,\text{transmittance}}}.
  14. T = 2.0 0.75 = 2.309... T=\frac{2.0}{\sqrt{0.75}}=2.309...
  15. f f
  16. D D
  17. N N
  18. f f
  19. D D
  20. N = f D × D f = N D N=\frac{f}{D}\quad\xrightarrow{\times D}\quad f=ND
  21. N w 1 2 N A i ( 1 + | m | ) N N_{w}\equiv{1\over 2\mathrm{NA}_{i}}\approx(1+|m|)N
  22. | m | |m|
  23. 1 / N 1/N

F-space.html

  1. L 1 2 [ 0 , 1 ] \scriptstyle L^{\frac{1}{2}}[0,\,1]
  2. W p ( 𝔻 ) \scriptstyle W_{p}(\mathbb{D})
  3. f ( z ) = n 0 a n z n f(z)=\sum_{n\geq 0}a_{n}z^{n}
  4. 𝔻 \scriptstyle\mathbb{D}
  5. n | a n | p < \sum_{n}|a_{n}|^{p}<\infty
  6. W p ( 𝔻 ) \scriptstyle W_{p}(\mathbb{D})
  7. f p = n | a n | p ( 0 < p < 1 ) \|f\|_{p}=\sum_{n}|a_{n}|^{p}\qquad(0<p<1)
  8. W p \scriptstyle W_{p}
  9. ζ \scriptstyle\zeta
  10. | ζ | 1 \scriptstyle|\zeta|\;\leq\;1
  11. f f ( ζ ) \scriptstyle f\,\mapsto\,f(\zeta)
  12. W p ( 𝔻 ) \scriptstyle W_{p}(\mathbb{D})

F4_(mathematics).html

  1. [ 2 - 1 0 0 - 1 2 - 2 0 0 - 1 2 - 1 0 0 - 1 2 ] \left[\begin{array}[]{rrrr}2&-1&0&0\\ -1&2&-2&0\\ 0&-1&2&-1\\ 0&0&-1&2\end{array}\right]
  2. [ 0 1 - 1 0 0 0 1 - 1 0 0 0 1 1 2 - 1 2 - 1 2 - 1 2 ] \begin{bmatrix}0&1&-1&0\\ 0&0&1&-1\\ 0&0&0&1\\ \frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}\\ \end{bmatrix}
  3. C 1 = x + y + z C_{1}=x+y+z
  4. C 2 = x 2 + y 2 + z 2 + 2 X X ¯ + 2 Y Y ¯ + 2 Z Z ¯ C_{2}=x^{2}+y^{2}+z^{2}+2X\overline{X}+2Y\overline{Y}+2Z\overline{Z}
  5. C 3 = x y z - x X X ¯ - y Y Y ¯ - z Z Z ¯ + X Y Z + X Y Z ¯ C_{3}=xyz-xX\overline{X}-yY\overline{Y}-zZ\overline{Z}+XYZ+\overline{XYZ}
  6. M = [ x Z ¯ Y Z y X ¯ Y ¯ X z ] M=\begin{bmatrix}x&\overline{Z}&Y\\ Z&y&\overline{X}\\ \overline{Y}&X&z\end{bmatrix}

Fabry–Pérot_interferometer.html

  1. δ = ( 2 π λ ) 2 n cos θ . \delta=\left(\frac{2\pi}{\lambda}\right)2n\ell\cos\theta.
  2. T e = ( 1 - R ) 2 1 + R 2 - 2 R cos δ = 1 1 + F sin 2 ( δ / 2 ) , T_{e}=\frac{(1-R)^{2}}{1+R^{2}-2R\cos\delta}=\frac{1}{1+F\sin^{2}(\delta/2)},
  3. F = 4 R < m t p l > ( 1 - R ) 2 F=\frac{4R}{<}mtpl>{{(1-R)^{2}}}
  4. T e = 1 T_{e}=1
  5. 2 n l cos θ 2nl\cos\theta
  6. T e + R e = 1 T_{e}+R_{e}=1
  7. R max = 1 - 1 1 + F = 4 R ( 1 + R ) 2 R_{\max}=1-\frac{1}{1+F}=\frac{4R}{(1+R)^{2}}
  8. Δ λ = λ 0 2 2 n g cos θ + λ 0 λ 0 2 2 n g cos θ \Delta\lambda=\frac{\lambda_{0}^{2}}{2n_{\mathrm{g}}\ell\cos\theta+\lambda_{0}% }\approx\frac{\lambda_{0}^{2}}{2n_{\mathrm{g}}\ell\cos\theta}
  9. n g n_{\mathrm{g}}
  10. = Δ λ δ λ = π 2 arcsin ( 1 / F ) \mathcal{F}=\frac{\Delta\lambda}{\delta\lambda}=\frac{\pi}{2\arcsin(1/\sqrt{F})}
  11. π F 2 = π R 1 / 2 1 - R \mathcal{F}\approx\frac{\pi\sqrt{F}}{2}=\frac{\pi R^{1/2}}{1-R}
  12. π ( R 1 R 2 ) 1 / 4 1 - ( R 1 R 2 ) 1 / 2 \mathcal{F}\approx\frac{\pi(R_{1}R_{2})^{1/4}}{1-(R_{1}R_{2})^{1/2}}
  13. R \sqrt{R}
  14. T \sqrt{T}
  15. t 0 = T e i k / cos θ t_{0}=T\,e^{ik\ell/\cos\theta}
  16. k = 2 π n / λ k=2\pi n/\lambda
  17. t 1 = T R e 3 i k / cos θ t^{\prime}_{1}=TR\,e^{3ik\ell/\cos\theta}
  18. k 0 0 k_{0}\ell_{0}
  19. k 0 = 2 π n 0 / λ k_{0}=2\pi n_{0}/\lambda
  20. t 1 = T R e 3 i k / cos θ - i k 0 0 t_{1}=TR\,e^{3ik\ell/\cos\theta-ik_{0}\ell_{0}}
  21. 0 = 2 tan θ sin θ 0 \ell_{0}=2\ell\tan\theta\sin\theta_{0}\,
  22. δ = 2 k cos θ - k 0 0 \delta={2k\ell\over\cos\theta}-k_{0}\ell_{0}\,
  23. n sin θ = n 0 sin θ 0 n\sin\theta=n_{0}\sin\theta_{0}\,
  24. δ = 2 k cos θ \delta=2k\ell\,\cos\theta\,
  25. t m = T R m e i m δ t_{m}=TR^{m}e^{im\delta}\,
  26. t = m = 0 t m = T m = 0 R m e i m δ t=\sum_{m=0}^{\infty}t_{m}=T\sum_{m=0}^{\infty}R^{m}\,e^{im\delta}
  27. t = T 1 - R e i δ t=\frac{T}{1-Re^{i\delta}}
  28. T e = t t * = T 2 1 + R 2 - 2 R cos δ T_{e}=tt^{*}=\frac{T^{2}}{1+R^{2}-2R\cos\delta}
  29. γ = ln ( 1 / R ) \gamma=\ln(1/R)
  30. T e = T 2 1 - R 2 ( sinh γ cosh γ - cos δ ) T_{e}=\frac{T^{2}}{1-R^{2}}\left(\frac{\sinh\gamma}{\cosh\gamma-\cos\delta}\right)
  31. T e = 2 π T 2 1 - R 2 = - L ( δ - 2 π ; γ ) T_{e}=\frac{2\pi\,T^{2}}{1-R^{2}}\,\sum_{\ell=-\infty}^{\infty}L(\delta-2\pi% \ell;\gamma)
  32. L ( x ; γ ) = γ π ( x 2 + γ 2 ) L(x;\gamma)=\frac{\gamma}{\pi(x^{2}+\gamma^{2})}

Face_(geometry).html

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

Factor_analysis.html

  1. p p
  2. x 1 , , x p x_{1},\dots,x_{p}
  3. μ 1 , , μ p \mu_{1},\dots,\mu_{p}
  4. l i j l_{ij}
  5. k k
  6. F j F_{j}
  7. i 1 , , p i\in{1,\dots,p}
  8. j 1 , , k j\in{1,\dots,k}
  9. k < p k<p
  10. x i - μ i = l i 1 F 1 + + l i k F k + ε i . x_{i}-\mu_{i}=l_{i1}F_{1}+\cdots+l_{ik}F_{k}+\varepsilon_{i}.\,
  11. ε i \varepsilon_{i}
  12. i i
  13. Var ( ε i ) = ψ i \mathrm{Var}(\varepsilon_{i})=\psi_{i}
  14. Cov ( ε ) = Diag ( ψ 1 , , ψ p ) = Ψ and E ( ε ) = 0. \mathrm{Cov}(\varepsilon)=\mathrm{Diag}(\psi_{1},\dots,\psi_{p})=\Psi\,\text{ % and }\mathrm{E}(\varepsilon)=0.\,
  15. x - μ = L F + ε . x-\mu=LF+\varepsilon.\,
  16. n n
  17. x p × n x_{p\times n}
  18. L p × k L_{p\times k}
  19. F k × n F_{k\times n}
  20. x x
  21. F F
  22. L L
  23. F F
  24. F F
  25. ε \varepsilon
  26. E ( F ) = 0. \mathrm{E}(F)=0.
  27. Cov ( F ) = I \mathrm{Cov}(F)=I
  28. F F
  29. L L
  30. Cov ( x - μ ) = Σ \mathrm{Cov}(x-\mu)=\Sigma
  31. F F
  32. Cov ( x - μ ) = Cov ( L F + ε ) , \mathrm{Cov}(x-\mu)=\mathrm{Cov}(LF+\varepsilon),\,
  33. Σ = L Cov ( F ) L T + Cov ( ε ) , \Sigma=L\mathrm{Cov}(F)L^{T}+\mathrm{Cov}(\varepsilon),\,
  34. Σ = L L T + Ψ . \Sigma=LL^{T}+\Psi.\,
  35. Q Q
  36. L = L Q L=LQ
  37. F = Q T F F=Q^{T}F
  38. N a N_{a}
  39. N p N_{p}
  40. N i N_{i}
  41. N i = 1000 N_{i}=1000
  42. N a = 10 N_{a}=10
  43. x a i x_{ai}
  44. x a x_{a}
  45. x a i x_{ai}
  46. z a i = x a i - μ a σ a z_{ai}=\frac{x_{ai}-\mu_{a}}{\sigma_{a}}
  47. μ a = 1 N i i x a i \mu_{a}=\tfrac{1}{N_{i}}\sum_{i}x_{ai}
  48. σ a 2 = 1 N i i ( x a i - μ a ) 2 \sigma_{a}^{2}=\tfrac{1}{N_{i}}\sum_{i}(x_{ai}-\mu_{a})^{2}
  49. z 1 , i = 1 , 1 F 1 , i + 1 , 2 F 2 , i + ε 1 , i z 10 , i = 10 , 1 F 1 , i + 10 , 2 F 2 , i + ε 10 , i \begin{matrix}z_{1,i}&=&\ell_{1,1}F_{1,i}&+&\ell_{1,2}F_{2,i}&+&\varepsilon_{1% ,i}\\ \vdots&&\vdots&&\vdots&&\vdots\\ z_{10,i}&=&\ell_{10,1}F_{1,i}&+&\ell_{10,2}F_{2,i}&+&\varepsilon_{10,i}\end{matrix}
  50. z a i = p a p F p i + ε a i z_{ai}=\sum_{p}\ell_{ap}F_{pi}+\varepsilon_{ai}
  51. F 1 , i F_{1,i}
  52. F 2 , i F_{2,i}
  53. a p \ell_{ap}
  54. Z = L F + ϵ Z=LF+\epsilon
  55. i F p i F q i = δ p q \sum_{i}F_{pi}F_{qi}=\delta_{pq}
  56. δ p q \delta_{pq}
  57. p q p\neq q
  58. p = q p=q
  59. i F p i ε a i = 0 \sum_{i}F_{pi}\varepsilon_{ai}=0
  60. i z a i z b i = p a p b p + i ε a i ε b i \sum_{i}z_{ai}z_{bi}=\sum_{p}\ell_{ap}\ell_{bp}+\sum_{i}\varepsilon_{ai}% \varepsilon_{bi}
  61. N a N_{a}
  62. h a 2 = 1 - ψ a = p a p a p h_{a}^{2}=1-\psi_{a}=\sum_{p}\ell_{ap}\ell_{ap}
  63. z a i z_{ai}
  64. F p i F_{pi}
  65. a p \ell_{ap}
  66. ε 2 = a b , a b [ i z a i z b i - p a p b p ] 2 \varepsilon^{2}=\sum_{ab,a\neq b}\left[\sum_{i}z_{ai}z_{bi}-\sum_{p}\ell_{ap}% \ell_{bp}\right]^{2}
  67. 𝐳 a \mathbf{z}_{a}
  68. 𝐅 1 \mathbf{F}_{1}
  69. 𝐅 2 \mathbf{F}_{2}
  70. 𝐳 ^ a \hat{\mathbf{z}}_{a}
  71. s y m b o l ε a symbol{\varepsilon}_{a}
  72. 𝐳 a = 𝐳 ^ a + s y m b o l ε a \mathbf{z}_{a}=\hat{\mathbf{z}}_{a}+symbol{\varepsilon}_{a}
  73. 𝐳 ^ a \hat{\mathbf{z}}_{a}
  74. 𝐳 ^ a = a 1 𝐅 1 + a 2 𝐅 2 \hat{\mathbf{z}}_{a}=\ell_{a1}\mathbf{F}_{1}+\ell_{a2}\mathbf{F}_{2}
  75. | 𝐳 ^ a | 2 = h a 2 |\hat{\mathbf{z}}_{a}|^{2}=h^{2}_{a}
  76. 𝐳 b \mathbf{z}_{b}
  77. 𝐳 a \mathbf{z}_{a}
  78. 𝐳 b \mathbf{z}_{b}
  79. r a b r_{ab}
  80. z a i z_{ai}
  81. F p i F_{pi}
  82. ε a i \varepsilon_{ai}
  83. N i N_{i}
  84. 𝐳 a \mathbf{z}_{a}
  85. 𝐅 p \mathbf{F}_{p}
  86. s y m b o l ε a symbol{\varepsilon}_{a}
  87. 𝐳 a 𝐳 a = 1 \mathbf{z}_{a}\cdot\mathbf{z}_{a}=1
  88. N p N_{p}
  89. 𝐳 a = p a p 𝐅 p + s y m b o l ε a \mathbf{z}_{a}=\sum_{p}\ell_{ap}\mathbf{F}_{p}+symbol{\varepsilon}_{a}
  90. 𝐅 p \cdotsymbol ε a = 0 \mathbf{F}_{p}\cdotsymbol{\varepsilon}_{a}=0
  91. 𝐳 ^ a = p a p 𝐅 p \hat{\mathbf{z}}_{a}=\sum_{p}\ell_{ap}\mathbf{F}_{p}
  92. 𝐅 p 𝐅 q = δ p q \mathbf{F}_{p}\cdot\mathbf{F}_{q}=\delta_{pq}
  93. 𝐳 a \mathbf{z}_{a}
  94. r a b = 𝐳 a 𝐳 b r_{ab}=\mathbf{z}_{a}\cdot\mathbf{z}_{b}
  95. 𝐳 a \mathbf{z}_{a}
  96. 𝐳 b \mathbf{z}_{b}
  97. r ^ a b = 𝐳 ^ a 𝐳 ^ b \hat{r}_{ab}=\hat{\mathbf{z}}_{a}\cdot\hat{\mathbf{z}}_{b}
  98. ε 2 = a , b a ( r a b - r ^ a b ) 2 \varepsilon^{2}=\sum_{a,b\neq a}\left(r_{ab}-\hat{r}_{ab}\right)^{2}
  99. r a b - r ^ a b = s y m b o l ε a \cdotsymbol ε b r_{ab}-\hat{r}_{ab}=symbol{\varepsilon}_{a}\cdotsymbol{\varepsilon}_{b}
  100. z ^ a \hat{z}_{a}
  101. h a 2 = 𝐳 ^ a 𝐳 ^ a = p a p a p h_{a}^{2}=\hat{\mathbf{z}}_{a}\cdot\hat{\mathbf{z}}_{a}=\sum_{p}\ell_{ap}\ell_% {ap}

Factor_of_safety.html

  1. Factor of Safety = Material Strength Design Load \,\text{Factor of Safety}=\frac{\,\text{Material Strength}}{\,\text{Design % Load}}
  2. Margin of Safety = Failure Load Design Load - 1 \,\text{Margin of Safety}=\frac{\,\text{Failure Load}}{\,\text{Design Load}}-1
  3. Margin of Safety = Factor of Safety - 1 \,\text{Margin of Safety}={\,\text{Factor of Safety}}-1
  4. Margin of Safety = Failure Load Design Load × Design Safety Factor - 1 \,\text{Margin of Safety}=\frac{\,\text{Failure Load}}{\,\text{Design Load × % Design Safety Factor}}-1
  5. Margin of Safety = Realized Factor of Safety Design Safety Factor - 1 \,\text{Margin of Safety}=\frac{\,\text{Realized Factor of Safety}}{\,\text{% Design Safety Factor}}-1

Factorization.html

  1. x 2 - y = ( x + y ) ( x - y ) x^{2}-y=(x+\sqrt{y})(x-\sqrt{y})
  2. 3 x 2 - 6 x + 12 = 3 ( x 2 - 2 x + 4 ) 3x^{2}-6x+12=3(x^{2}-2x+4)
  3. x 2 - 2 = ( x + 2 ) ( x - 2 ) x^{2}-2=(x+\sqrt{2})(x-\sqrt{2})
  4. x 2 + 4 = ( x + 2 i ) ( x - 2 i ) x^{2}+4=(x+2i)(x-2i)
  5. 6 x 3 y 2 + 8 x 4 y 3 - 10 x 5 y 3 = ( 2 x 3 y 2 ) ( 3 + 4 x y - 5 x 2 y ) . 6x^{3}y^{2}+8x^{4}y^{3}-10x^{5}y^{3}=(2x^{3}y^{2})(3+4xy-5x^{2}y).
  6. 4 x 2 + 20 x + 3 y x + 15 y 4x^{2}+20x+3yx+15y\,
  7. ( 4 x 2 + 20 x ) + ( 3 y x + 15 y ) , (4x^{2}+20x)+(3yx+15y),\,
  8. 4 x ( x + 5 ) + 3 y ( x + 5 ) , 4x(x+5)+3y(x+5),\,
  9. ( x + 5 ) ( 4 x + 3 y ) . (x+5)(4x+3y).\,
  10. x 3 - 3 x + 2. x^{3}-3x+2.
  11. x 3 - 3 x + 2 = ( x - 1 ) ( x 2 + x - 2 ) . x^{3}-3x+2=(x-1)(x^{2}+x-2).
  12. x 3 - 5 x 2 - 16 x + 80 x^{3}-5x^{2}-16x+80
  13. r 1 , r 2 r_{1},r_{2}
  14. r 3 r_{3}
  15. r 1 + r 2 + r 3 \displaystyle r_{1}+r_{2}+r_{3}
  16. r 2 + r 3 = 0 r_{2}+r_{3}=0
  17. r 1 = 5 r_{1}=5
  18. r 2 2 = 16. r_{2}^{2}=16.
  19. x 3 - 5 x 2 - 16 x + 80 = ( x - 5 ) ( x - 4 ) ( x + 4 ) . x^{3}-5x^{2}-16x+80=(x-5)(x-4)(x+4).
  20. ( x - p q ) = 1 q ( q x - p ) . \left(x-\frac{p}{q}\right)=\frac{1}{q}(qx-p).
  21. a 0 x n + a 1 x n - 1 + + a n - 1 x + a n a_{0}x^{n}+a_{1}x^{n-1}+\ldots+a_{n-1}x+a_{n}
  22. 2 x 3 - 7 x 2 + 10 x - 6 2x^{3}-7x^{2}+10x-6
  23. p 3 - 7 p 2 + 20 p - 24 = 0 p^{3}-7p^{2}+20p-24=0
  24. 2 x 3 - 7 x 2 + 10 x - 6 = ( 2 x - 3 ) ( x 2 - 2 x + 2 ) . 2x^{3}-7x^{2}+10x-6=(2x-3)(x^{2}-2x+2).
  25. a x 2 + b x + c ax^{2}+bx+c
  26. b 2 - 4 a c b^{2}-4ac
  27. a x 2 + b x + c = a ( x 2 + b a x + c a ) = a ( 1 a ( a x + r ) 1 a ( a x + s ) ) = ( a x + r ) ( a x + s ) a . ax^{2}+bx+c=a\left(x^{2}+\frac{b}{a}x+\frac{c}{a}\right)=a\left(\frac{1}{a}(ax% +r)\frac{1}{a}(ax+s)\right)=\frac{(ax+r)(ax+s)}{a}.
  28. a 2 x 2 + a b x + a c = ( a x + r ) ( a x + s ) , a^{2}x^{2}+abx+ac=(ax+r)(ax+s),
  29. 6 x 2 + 13 x + 6. 6x^{2}+13x+6.
  30. 6 x 2 + 13 x + 6 = ( 6 x + 4 ) ( 6 x + 9 ) 6 = 2 ( 3 x + 2 ) ( 3 ) ( 2 x + 3 ) 6 = ( 3 x + 2 ) ( 2 x + 3 ) \begin{aligned}\displaystyle 6x^{2}+13x+6&\displaystyle=\frac{(6x+4)(6x+9)}{6}% \\ &\displaystyle=\frac{2(3x+2)(3)(2x+3)}{6}\\ &\displaystyle=(3x+2)(2x+3)\end{aligned}
  31. a 2 - b 2 = ( a + b ) ( a - b ) , a^{2}-b^{2}=(a+b)(a-b),\,\!
  32. a 2 + 2 a b + b 2 - x 2 + 2 x y - y 2 = ( a 2 + 2 a b + b 2 ) - ( x 2 - 2 x y + y 2 ) = ( a + b ) 2 - ( x - y ) 2 = ( a + b + x - y ) ( a + b - x + y ) . \begin{aligned}\displaystyle a^{2}+2ab+b^{2}-x^{2}+2xy-y^{2}&\displaystyle=(a^% {2}+2ab+b^{2})-(x^{2}-2xy+y^{2})\\ &\displaystyle=(a+b)^{2}-(x-y)^{2}\\ &\displaystyle=(a+b+x-y)(a+b-x+y).\end{aligned}
  33. a 3 + b 3 = ( a + b ) ( a 2 - a b + b 2 ) , a^{3}+b^{3}=(a+b)(a^{2}-ab+b^{2}),\,\!
  34. a 3 - b 3 = ( a - b ) ( a 2 + a b + b 2 ) . a^{3}-b^{3}=(a-b)(a^{2}+ab+b^{2}).\,\!
  35. a 4 - b 4 = ( a 2 + b 2 ) ( a 2 - b 2 ) = ( a 2 + b 2 ) ( a + b ) ( a - b ) . a^{4}-b^{4}=(a^{2}+b^{2})(a^{2}-b^{2})=(a^{2}+b^{2})(a+b)(a-b).\,\!
  36. a n - b n = ( a - b ) ( a n - 1 + b a n - 2 + b 2 a n - 3 + + b n - 2 a + b n - 1 ) . a^{n}-b^{n}=(a-b)(a^{n-1}+ba^{n-2}+b^{2}a^{n-3}+\ldots+b^{n-2}a+b^{n-1}).\!
  37. a n + b n = ( a + b ) ( a n - 1 - b a n - 2 + b 2 a n - 3 - - b n - 2 a + b n - 1 ) . a^{n}+b^{n}=(a+b)(a^{n-1}-ba^{n-2}+b^{2}a^{n-3}-\ldots-b^{n-2}a+b^{n-1}).\!
  38. a n + b n a^{n}+b^{n}\!
  39. n = m 2 k , m > 1 , k > 0 n=m\cdot 2^{k},m>1,k>0
  40. a n + b n = ( a 2 k + b 2 k ) ( a n - 2 k - a n - 2 2 k b 2 k + a n - 3 2 k b 2 2 k - - a 2 k b n - 2 2 k + b n - 2 k ) = ( a 2 k + b 2 k ) i = 1 m a ( m - i ) 2 k ( - b 2 k ) i - 1 . a^{n}+b^{n}=(a^{2^{k}}+b^{2^{k}})(a^{n-2^{k}}-a^{n-2\cdot 2^{k}}b^{2^{k}}+a^{n% -3\cdot 2^{k}}b^{2\cdot 2^{k}}-\ldots-a^{2^{k}}b^{n-2\cdot 2^{k}}+b^{n-2^{k}})% =(a^{2^{k}}+b^{2^{k}})\sum_{i=1}^{m}a^{(m-i)2^{k}}(-b^{2^{k}})^{i-1}.\!
  41. a 5 + b 5 = ( a + b ) ( a 4 - a 3 b + a 2 b 2 - a b 3 + b 4 ) , a^{5}+b^{5}=(a+b)(a^{4}-a^{3}b+a^{2}b^{2}-ab^{3}+b^{4}),\,\!
  42. a 5 - b 5 = ( a - b ) ( a 4 + a 3 b + a 2 b 2 + a b 3 + b 4 ) . a^{5}-b^{5}=(a-b)(a^{4}+a^{3}b+a^{2}b^{2}+ab^{3}+b^{4}).\,\!
  43. a 6 + b 6 = ( a 2 + b 2 ) ( a 4 - a 2 b 2 + b 4 ) , a^{6}+b^{6}=(a^{2}+b^{2})(a^{4}-a^{2}b^{2}+b^{4}),\,\!
  44. a 6 - b 6 = ( a 3 + b 3 ) ( a 3 - b 3 ) = ( a + b ) ( a - b ) ( a 2 - a b + b 2 ) ( a 2 + a b + b 2 ) . a^{6}-b^{6}=(a^{3}+b^{3})(a^{3}-b^{3})=(a+b)(a-b)(a^{2}-ab+b^{2})(a^{2}+ab+b^{% 2}).\,\!
  45. a 7 + b 7 = ( a + b ) ( a 6 - a 5 b + a 4 b 2 - a 3 b 3 + a 2 b 4 - a b 5 + b 6 ) , a^{7}+b^{7}=(a+b)(a^{6}-a^{5}b+a^{4}b^{2}-a^{3}b^{3}+a^{2}b^{4}-ab^{5}+b^{6}),\,\!
  46. a 7 - b 7 = ( a - b ) ( a 6 + a 5 b + a 4 b 2 + a 3 b 3 + a 2 b 4 + a b 5 + b 6 ) . a^{7}-b^{7}=(a-b)(a^{6}+a^{5}b+a^{4}b^{2}+a^{3}b^{3}+a^{2}b^{4}+ab^{5}+b^{6}).\,\!
  47. f ( a ) = a n ± b n : f(a)=a^{n}\pm b^{n}:
  48. a 2 n + b 2 n = k = 1 n ( a 2 ± 2 a b cos ( 2 k - 1 ) π 2 n + b 2 ) . a^{2n}+\ b^{2n}=\prod_{k=1}^{n}\Bigl(a^{2}\ \pm\ 2ab\cos\frac{(2k-1)\pi}{2n}\ % +\ b^{2}\Bigl).\!
  49. a 2 n - b 2 n = ( a - b ) ( a + b ) k = 1 n - 1 ( a 2 ± 2 a b cos k π n + b 2 ) . a^{2n}-\ b^{2n}=(a\ -\ b)(a\ +\ b)\prod_{k=1}^{n-1}\Bigl(a^{2}\ \pm\ 2ab\cos% \frac{k\pi}{n}\ +\ b^{2}\Bigl).\!
  50. a 2 n + 1 ± b 2 n + 1 = ( a ± b ) k = 1 n ( a 2 ± 2 a b cos 2 k π 2 n + 1 + b 2 ) = ( a ± b ) k = 1 n ( a 2 ± 2 a b ( - 1 ) k cos k π 2 n + 1 + b 2 ) . a^{2n+1}\ \pm\ b^{2n+1}=(a\ \pm\ b)\prod_{k=1}^{n}\Bigl(a^{2}\ \pm\ 2ab\cos% \frac{2k\pi}{2n+1}\ +\ b^{2}\Bigl)=(a\ \pm\ b)\prod_{k=1}^{n}\Bigl(a^{2}\ \pm% \ 2ab(-1)^{k}\cos\frac{k\pi}{2n+1}\ +\ b^{2}\Bigl).\!
  51. a 5 ± b 5 = ( a ± b ) ( a 2 1 - 5 2 a b + b 2 ) ( a 2 1 + 5 2 a b + b 2 ) , a^{5}\pm b^{5}=(a\pm b)\Biggl(a^{2}\mp\frac{1-\sqrt{5}}{2}ab+b^{2}\Biggl)% \Biggl(a^{2}\mp\frac{1+\sqrt{5}}{2}ab+b^{2}\Biggl),\!
  52. a 4 + b 4 = ( a 2 - 2 a b + b 2 ) ( a 2 + 2 a b + b 2 ) . a^{4}+b^{4}=(a^{2}-\sqrt{2}ab+b^{2})(a^{2}+\sqrt{2}ab+b^{2}).\!
  53. a 2 + 2 a b + b 2 = ( a + b ) 2 , a^{2}+2ab+b^{2}=(a+b)^{2},\,\!
  54. a 2 - 2 a b + b 2 = ( a - b ) 2 . a^{2}-2ab+b^{2}=(a-b)^{2}.\,\!
  55. a 3 + 3 a 2 b + 3 a b 2 + b 3 = ( a + b ) 3 , a^{3}+3a^{2}b+3ab^{2}+b^{3}=(a+b)^{3},
  56. a 3 - 3 a 2 b + 3 a b 2 - b 3 = ( a - b ) 3 . a^{3}-3a^{2}b+3ab^{2}-b^{3}=(a-b)^{3}.
  57. ( a + b ) n (a+b)^{n}
  58. ( a - b ) n (a-b)^{n}
  59. x 2 + y 2 + z 2 + 2 ( x y + y z + x z ) = ( x + y + z ) 2 x 3 + y 3 + z 3 - 3 x y z = ( x + y + z ) ( x 2 + y 2 + z 2 - x y - x z - y z ) x 3 + y 3 + z 3 + 3 x 2 ( y + z ) + 3 y 2 ( x + z ) + 3 z 2 ( x + y ) + 6 x y z = ( x + y + z ) 3 x 4 + x 2 y 2 + y 4 = ( x 2 + x y + y 2 ) ( x 2 - x y + y 2 ) . \begin{aligned}\displaystyle x^{2}+y^{2}+z^{2}+2(xy+yz+xz)&\displaystyle=(x+y+% z)^{2}\\ \displaystyle x^{3}+y^{3}+z^{3}-3xyz&\displaystyle=(x+y+z)(x^{2}+y^{2}+z^{2}-% xy-xz-yz)\\ \displaystyle x^{3}+y^{3}+z^{3}+3x^{2}(y+z)+3y^{2}(x+z)+3z^{2}(x+y)+6xyz&% \displaystyle=(x+y+z)^{3}\\ \displaystyle x^{4}+x^{2}y^{2}+y^{4}&\displaystyle=(x^{2}+xy+y^{2})(x^{2}-xy+y% ^{2}).\end{aligned}
  60. a x 2 + b x + c ax^{2}+bx+c
  61. a x 2 + b x + c = a ( x - α ) ( x - β ) = a ( x - - b + b 2 - 4 a c 2 a ) ( x - - b - b 2 - 4 a c 2 a ) , ax^{2}+bx+c=a(x-\alpha)(x-\beta)=a\left(x-\frac{-b+\sqrt{b^{2}-4ac}}{2a}\right% )\left(x-\frac{-b-\sqrt{b^{2}-4ac}}{2a}\right),
  62. α \alpha
  63. β \beta
  64. a 2 + b 2 = ( a + b i ) ( a - b i ) . a^{2}+b^{2}=(a+bi)(a-bi).\,\!
  65. 4 x 2 + 49 4x^{2}+49
  66. ( 2 x + 7 i ) ( 2 x - 7 i ) (2x+7i)(2x-7i)

Fading.html

  1. T c 1 D s T_{c}\approx\frac{1}{D_{s}}
  2. T c T_{c}
  3. D s D_{s}

Fan-out.html

  1. DC Fan-out = min ( I out high I in high , I out low I in low ) \,\text{DC Fan-out}=\operatorname{min}\left(\left\lfloor\frac{I_{\,\text{out % high}}}{I_{\,\text{in high}}}\right\rfloor,\left\lfloor\frac{I_{\,\text{out % low}}}{I_{\,\text{in low}}}\right\rfloor\right)
  2. \lfloor\;\rfloor

Farad.html

  1. 10 3 10^{−}3
  2. 10 6 10^{−}6
  3. 10 9 10^{−}9
  4. 10 12 10^{−}12
  5. 4 {}^{4}
  6. 2 {}^{2}
  7. 2 {}^{−2}
  8. 1 {}^{−1}
  9. F = A s V = J V 2 = W s V 2 = C V = C 2 J = C 2 N m = s 2 C 2 m 2 kg = s 4 A 2 m 2 kg = s Ω = s 2 H \mbox{F}~{}=\,\mathrm{\frac{A\cdot s}{V}=\dfrac{\mbox{J}~{}}{\mbox{V}~{}^{2}}=% \dfrac{\mbox{W}~{}\cdot\mbox{s}~{}}{\mbox{V}~{}^{2}}=\dfrac{\mbox{C}~{}}{\mbox% {V}~{}}=\dfrac{\mbox{C}~{}^{2}}{\mbox{J}~{}}=\dfrac{\mbox{C}~{}^{2}}{\mbox{N}~% {}\cdot\mbox{m}~{}}=\dfrac{\mbox{s}~{}^{2}\cdot\mbox{C}~{}^{2}}{\mbox{m}~{}^{2% }\cdot\mbox{kg}~{}}=\dfrac{\mbox{s}~{}^{4}\cdot\mbox{A}~{}^{2}}{\mbox{m}~{}^{2% }\cdot\mbox{kg}~{}}=\dfrac{\mbox{s}~{}}{\Omega}}=\dfrac{\mbox{s}~{}^{2}}{\mbox% {H}~{}}
  10. 10 - 15 10^{-}15
  11. 10 18 10^{−}18
  12. 10 9 10^{9}

Fatou's_lemma.html

  1. f ( s ) = lim inf n f n ( s ) , s S . f(s)=\liminf_{n\to\infty}f_{n}(s),\qquad s\in S.
  2. S f d μ lim inf n S f n d μ . \int_{S}f\,d\mu\leq\liminf_{n\to\infty}\int_{S}f_{n}\,d\mu\,.
  3. E f d μ lim inf n E f n d μ . \int_{E}f\,d\mu\leq\liminf_{n\to\infty}\int_{E}f_{n}\,d\mu\,.
  4. K = { x E | f n ( x ) f ( x ) } K=\{x\in E|f_{n}(x)\rightarrow f(x)\}
  5. E f d μ = K f d μ , E f n d μ = K f n d μ n 𝒩 . \int_{E}f\,d\mu=\int_{K}f\,d\mu,~{}~{}~{}\int_{E}f_{n}\,d\mu=\int_{K}f_{n}\,d% \mu~{}\forall n\in\mathcal{N}.
  6. E φ d μ lim inf n E f n d μ \int_{E}\varphi\,d\mu\leq\liminf_{n\rightarrow\infty}\int_{E}f_{n}\,d\mu
  7. E φ = \int_{E}\varphi=\infty
  8. A = { x E | φ ( x ) > a / 2 } A=\{x\in E|\varphi(x)>a/2\}
  9. E φ M μ ( A ) , \int_{E}\varphi\leq M\mu(A),
  10. A n = { x E | f k ( x ) > a / 2 k n } . A_{n}=\{x\in E|f_{k}(x)>a/2~{}\forall k\geq n\}.
  11. A n A n μ ( n A n ) = . A\subseteq\bigcup_{n}A_{n}\Rightarrow\mu(\bigcup_{n}A_{n})=\infty.
  12. lim n μ ( A n ) = . \lim_{n\rightarrow\infty}\mu(A_{n})=\infty.
  13. E f n d μ A n f n d μ a 2 μ ( A n ) lim inf n E f n d μ = = E φ d μ , \int_{E}f_{n}\,d\mu\geq\int_{A_{n}}f_{n}\,d\mu\geq\frac{a}{2}\mu(A_{n})% \Rightarrow\liminf_{n\to\infty}\int_{E}f_{n}\,d\mu=\infty=\int_{E}\varphi\,d\mu,
  14. E φ < \int_{E}\varphi<\infty
  15. A n = { x E | f k ( x ) > ( 1 - ϵ ) φ ( x ) k n } . A_{n}=\{x\in E|f_{k}(x)>(1-\epsilon)\varphi(x)~{}\forall k\geq n\}.
  16. lim n μ ( A - A n ) = 0. \lim_{n\rightarrow\infty}\mu(A-A_{n})=0.
  17. μ ( A - A k ) < ϵ , k n . \mu(A-A_{k})<\epsilon,~{}\forall k\geq n.
  18. k n k\geq n
  19. E f k d μ A k f k d μ ( 1 - ϵ ) A k φ d μ . \int_{E}f_{k}\,d\mu\geq\int_{A_{k}}f_{k}\,d\mu\geq(1-\epsilon)\int_{A_{k}}% \varphi\,d\mu.
  20. E φ d μ = A φ d μ = A k φ d μ + A - A k φ d μ . \int_{E}\varphi\,d\mu=\int_{A}\varphi\,d\mu=\int_{A_{k}}\varphi\,d\mu+\int_{A-% A_{k}}\varphi\,d\mu.
  21. ( 1 - ϵ ) A k φ d μ ( 1 - ϵ ) E φ d μ - A - A k φ d μ . (1-\epsilon)\int_{A_{k}}\varphi\,d\mu\geq(1-\epsilon)\int_{E}\varphi\,d\mu-% \int_{A-A_{k}}\varphi\,d\mu.
  22. E f k d μ ( 1 - ϵ ) E φ d μ - A - A k φ d μ E φ d μ - ϵ ( E φ d μ + M ) . \int_{E}f_{k}\,d\mu\geq(1-\epsilon)\int_{E}\varphi\,d\mu-\int_{A-A_{k}}\varphi% \,d\mu\geq\int_{E}\varphi\,d\mu-\epsilon\left(\int_{E}\varphi\,d\mu+M\right).
  23. lim inf n E f n d μ E φ d μ , \liminf_{n\rightarrow\infty}\int_{E}f_{n}\,d\mu\geq\int_{E}\varphi\,d\mu,
  24. g k = inf n k f n . g_{k}=\inf_{n\geq k}f_{n}.
  25. E g k d μ E f n d μ , \int_{E}g_{k}\,d\mu\leq\int_{E}f_{n}\,d\mu,
  26. E g k d μ inf n k E f n d μ . \int_{E}g_{k}\,d\mu\leq\inf_{n\geq k}\int_{E}f_{n}\,d\mu.
  27. E f d μ = lim k E g k d μ lim k inf n k E f n d μ = lim inf n E f n d μ . \int_{E}f\,d\mu=\lim_{k\to\infty}\int_{E}g_{k}\,d\mu\leq\lim_{k\to\infty}\inf_% {n\geq k}\int_{E}f_{n}\,d\mu=\liminf_{n\to\infty}\int_{E}f_{n}\,d\mu\,.
  28. S S
  29. S = [ 0 , 1 ] S=[0,1]
  30. n n
  31. f n ( x ) = { n for x ( 0 , 1 / n ) , 0 otherwise. f_{n}(x)=\begin{cases}n&\,\text{for }x\in(0,1/n),\\ 0&\,\text{otherwise.}\end{cases}
  32. S S
  33. f n ( x ) = { 1 n for x [ 0 , n ] , 0 otherwise. f_{n}(x)=\begin{cases}\frac{1}{n}&\,\text{for }x\in[0,n],\\ 0&\,\text{otherwise.}\end{cases}
  34. ( f n ) n 𝒩 (f_{n})_{n\in\mathcal{N}}
  35. S S
  36. f n f_{n}
  37. f n ( x ) = { - 1 n for x [ n , 2 n ] , 0 otherwise. f_{n}(x)=\begin{cases}-\frac{1}{n}&\,\text{for }x\in[n,2n],\\ 0&\,\text{otherwise.}\end{cases}
  38. lim sup n S f n d μ S lim sup n f n d μ . \limsup_{n\to\infty}\int_{S}f_{n}\,d\mu\leq\int_{S}\limsup_{n\to\infty}f_{n}\,% d\mu.
  39. S g d μ < \textstyle\int_{S}g\,d\mu<\infty
  40. S lim inf n f n d μ lim inf n S f n d μ . \int_{S}\liminf_{n\to\infty}f_{n}\,d\mu\leq\liminf_{n\to\infty}\int_{S}f_{n}\,% d\mu.
  41. S f d μ lim inf n S f n d μ . \int_{S}f\,d\mu\leq\liminf_{n\to\infty}\int_{S}f_{n}\,d\mu\,.
  42. lim k S f n k d μ = lim inf n S f n d μ . \lim_{k\to\infty}\int_{S}f_{n_{k}}\,d\mu=\liminf_{n\to\infty}\int_{S}f_{n}\,d\mu.
  43. μ n ( E ) μ ( E ) , E Σ . \mu_{n}(E)\to\mu(E),~{}\forall E\in\Sigma.
  44. S f d μ lim inf n S f n d μ n . \int_{S}f\,d\mu\leq\liminf_{n\to\infty}\int_{S}f_{n}\,d\mu_{n}.
  45. E f d μ lim inf n E f n d μ n . \int_{E}f\,d\mu\leq\liminf_{n\to\infty}\int_{E}f_{n}\,d\mu_{n}\,.
  46. K = { x E | f n ( x ) f ( x ) } K=\{x\in E|f_{n}(x)\rightarrow f(x)\}
  47. E f d μ = E - K f d μ , E f n d μ = E - K f n d μ n 𝒩 . \int_{E}f\,d\mu=\int_{E-K}f\,d\mu,~{}~{}~{}\int_{E}f_{n}\,d\mu=\int_{E-K}f_{n}% \,d\mu~{}\forall n\in\mathcal{N}.
  48. E ϕ d μ = lim n E ϕ d μ n . \int_{E}\phi\,d\mu=\lim_{n\to\infty}\int_{E}\phi\,d\mu_{n}.
  49. E ϕ d μ lim inf n E f n d μ n \int_{E}\phi\,d\mu\leq\liminf_{n\rightarrow\infty}\int_{E}f_{n}\,d\mu_{n}
  50. A = { x E | ϕ ( x ) > a } A=\{x\in E|\phi(x)>a\}
  51. E ϕ d μ = \int_{E}\phi\,d\mu=\infty
  52. E ϕ d μ M μ ( A ) , \int_{E}\phi\,d\mu\leq M\mu(A),
  53. A n = { x E | f k ( x ) > a k n } . A_{n}=\{x\in E|f_{k}(x)>a~{}\forall k\geq n\}.
  54. A n A n μ ( n A n ) = . A\subseteq\bigcup_{n}A_{n}\Rightarrow\mu(\bigcup_{n}A_{n})=\infty.
  55. lim n μ ( A n ) = . \lim_{n\rightarrow\infty}\mu(A_{n})=\infty.
  56. lim n μ n ( A n ) = μ ( A n ) = . \lim_{n\to\infty}\mu_{n}(A_{n})=\mu(A_{n})=\infty.
  57. E f n d μ n a μ n ( A n ) lim inf n E f n d μ n = = E ϕ d μ , \int_{E}f_{n}\,d\mu_{n}\geq a\mu_{n}(A_{n})\Rightarrow\liminf_{n\to\infty}\int% _{E}f_{n}\,d\mu_{n}=\infty=\int_{E}\phi\,d\mu,
  58. E ϕ d μ < \int_{E}\phi\,d\mu<\infty
  59. A n = { x E | f k ( x ) > ( 1 - ϵ ) ϕ ( x ) k n } . A_{n}=\{x\in E|f_{k}(x)>(1-\epsilon)\phi(x)~{}\forall k\geq n\}.
  60. lim n μ ( A - A n ) = 0. \lim_{n\rightarrow\infty}\mu(A-A_{n})=0.
  61. μ ( A - A k ) < ϵ , k n . \mu(A-A_{k})<\epsilon,~{}\forall k\geq n.
  62. lim n μ n ( A - A k ) = μ ( A - A k ) , \lim_{n\to\infty}\mu_{n}(A-A_{k})=\mu(A-A_{k}),
  63. μ k ( A - A k ) < ϵ , k N . \mu_{k}(A-A_{k})<\epsilon,~{}\forall k\geq N.
  64. k N k\geq N
  65. E f k d μ k A k f k d μ k ( 1 - ϵ ) A k ϕ d μ k . \int_{E}f_{k}\,d\mu_{k}\geq\int_{A_{k}}f_{k}\,d\mu_{k}\geq(1-\epsilon)\int_{A_% {k}}\phi\,d\mu_{k}.
  66. E ϕ d μ k = A ϕ d μ k = A k ϕ d μ k + A - A k ϕ d μ k . \int_{E}\phi\,d\mu_{k}=\int_{A}\phi\,d\mu_{k}=\int_{A_{k}}\phi\,d\mu_{k}+\int_% {A-A_{k}}\phi\,d\mu_{k}.
  67. ( 1 - ϵ ) A k ϕ d μ k ( 1 - ϵ ) E ϕ d μ k - A - A k ϕ d μ k . (1-\epsilon)\int_{A_{k}}\phi\,d\mu_{k}\geq(1-\epsilon)\int_{E}\phi\,d\mu_{k}-% \int_{A-A_{k}}\phi\,d\mu_{k}.
  68. E f k d μ k ( 1 - ϵ ) E ϕ d μ k - A - A k ϕ d μ k E ϕ d μ k - ϵ ( E ϕ d μ k + M ) . \int_{E}f_{k}\,d\mu_{k}\geq(1-\epsilon)\int_{E}\phi\,d\mu_{k}-\int_{A-A_{k}}% \phi\,d\mu_{k}\geq\int_{E}\phi\,d\mu_{k}-\epsilon\left(\int_{E}\phi\,d\mu_{k}+% M\right).
  69. lim inf n E f n d μ k E ϕ d μ , \liminf_{n\rightarrow\infty}\int_{E}f_{n}\,d\mu_{k}\geq\int_{E}\phi\,d\mu,
  70. ( Ω , , ) \scriptstyle(\Omega,\,\mathcal{F},\,\mathbb{P})
  71. ( Ω , , ) \scriptstyle(\Omega,\mathcal{F},\mathbb{P})
  72. 𝒢 \scriptstyle\mathcal{G}\,\subset\,\mathcal{F}
  73. 𝔼 [ lim inf n X n | 𝒢 ] lim inf n 𝔼 [ X n | 𝒢 ] \mathbb{E}\Bigl[\liminf_{n\to\infty}X_{n}\,\Big|\,\mathcal{G}\Bigr]\leq\liminf% _{n\to\infty}\,\mathbb{E}[X_{n}|\mathcal{G}]
  74. Y k = inf n k X n . Y_{k}=\inf_{n\geq k}X_{n}.
  75. 𝔼 [ Y k | 𝒢 ] 𝔼 [ X n | 𝒢 ] \mathbb{E}[Y_{k}|\mathcal{G}]\leq\mathbb{E}[X_{n}|\mathcal{G}]
  76. 𝔼 [ Y k | 𝒢 ] inf n k 𝔼 [ X n | 𝒢 ] \mathbb{E}[Y_{k}|\mathcal{G}]\leq\inf_{n\geq k}\mathbb{E}[X_{n}|\mathcal{G}]
  77. 𝔼 [ lim inf n X n | 𝒢 ] = 𝔼 [ X | 𝒢 ] = 𝔼 [ lim k Y k | 𝒢 ] = lim k 𝔼 [ Y k | 𝒢 ] lim k inf n k 𝔼 [ X n | 𝒢 ] = lim inf n 𝔼 [ X n | 𝒢 ] . \begin{aligned}\displaystyle\mathbb{E}\Bigl[\liminf_{n\to\infty}X_{n}\,\Big|\,% \mathcal{G}\Bigr]&\displaystyle=\mathbb{E}[X|\mathcal{G}]=\mathbb{E}\Bigl[\lim% _{k\to\infty}Y_{k}\,\Big|\,\mathcal{G}\Bigr]=\lim_{k\to\infty}\mathbb{E}[Y_{k}% |\mathcal{G}]\\ &\displaystyle\leq\lim_{k\to\infty}\inf_{n\geq k}\mathbb{E}[X_{n}|\mathcal{G}]% =\liminf_{n\to\infty}\,\mathbb{E}[X_{n}|\mathcal{G}].\end{aligned}
  78. ( Ω , , ) \scriptstyle(\Omega,\mathcal{F},\mathbb{P})
  79. 𝒢 \scriptstyle\mathcal{G}\,\subset\,\mathcal{F}
  80. X n - := max { - X n , 0 } , n , X_{n}^{-}:=\max\{-X_{n},0\},\qquad n\in{\mathbb{N}},
  81. 𝔼 [ X n - 1 { X n - > c } | 𝒢 ] < ε , for all n , almost surely \mathbb{E}\bigl[X_{n}^{-}1_{\{X_{n}^{-}>c\}}\,|\,\mathcal{G}\bigr]<\varepsilon% ,\qquad\,\text{for all }n\in\mathbb{N},\,\,\text{almost surely}
  82. 𝔼 [ lim inf n X n | 𝒢 ] lim inf n 𝔼 [ X n | 𝒢 ] \mathbb{E}\Bigl[\liminf_{n\to\infty}X_{n}\,\Big|\,\mathcal{G}\Bigr]\leq\liminf% _{n\to\infty}\,\mathbb{E}[X_{n}|\mathcal{G}]
  83. X := lim inf n X n X:=\liminf_{n\to\infty}X_{n}
  84. 𝔼 [ max { X , 0 } | 𝒢 ] = , \mathbb{E}[\max\{X,0\}\,|\,\mathcal{G}]=\infty,
  85. 𝔼 [ X n - 1 { X n - > c } | 𝒢 ] < ε for all n , almost surely . \mathbb{E}\bigl[X_{n}^{-}1_{\{X_{n}^{-}>c\}}\,|\,\mathcal{G}\bigr]<\varepsilon% \qquad\,\text{for all }n\in\mathbb{N},\,\,\text{almost surely}.
  86. X + c lim inf n ( X n + c ) + , X+c\leq\liminf_{n\to\infty}(X_{n}+c)^{+},
  87. 𝔼 [ X | 𝒢 ] + c 𝔼 [ lim inf n ( X n + c ) + | 𝒢 ] lim inf n 𝔼 [ ( X n + c ) + | 𝒢 ] \mathbb{E}[X\,|\,\mathcal{G}]+c\leq\mathbb{E}\Bigl[\liminf_{n\to\infty}(X_{n}+% c)^{+}\,\Big|\,\mathcal{G}\Bigr]\leq\liminf_{n\to\infty}\mathbb{E}[(X_{n}+c)^{% +}\,|\,\mathcal{G}]
  88. ( X n + c ) + = ( X n + c ) + ( X n + c ) - X n + c + X n - 1 { X n - > c } , (X_{n}+c)^{+}=(X_{n}+c)+(X_{n}+c)^{-}\leq X_{n}+c+X_{n}^{-}1_{\{X_{n}^{-}>c\}},
  89. 𝔼 [ ( X n + c ) + | 𝒢 ] 𝔼 [ X n | 𝒢 ] + c + ε \mathbb{E}[(X_{n}+c)^{+}\,|\,\mathcal{G}]\leq\mathbb{E}[X_{n}\,|\,\mathcal{G}]% +c+\varepsilon
  90. 𝔼 [ X | 𝒢 ] lim inf n 𝔼 [ X n | 𝒢 ] + ε \mathbb{E}[X\,|\,\mathcal{G}]\leq\liminf_{n\to\infty}\mathbb{E}[X_{n}\,|\,% \mathcal{G}]+\varepsilon

Feigenbaum_constants.html

  1. x i + 1 = f ( x i ) x_{i+1}=f(x_{i})
  2. δ = lim n a n - 1 - a n - 2 a n - a n - 1 = 4.669 201 609 \delta=\lim_{n\rightarrow\infty}\dfrac{a_{n-1}-a_{n-2}}{a_{n}-a_{n-1}}=4.669\,% 201\,609\,\cdots
  3. f ( x ) = a - x 2 . f(x)=a-x^{2}.
  4. a n - 1 - a n - 2 a n - a n - 1 \dfrac{a_{n-1}-a_{n-2}}{a_{n}-a_{n-1}}
  5. f ( x ) = a x ( 1 - x ) f(x)=ax(1-x)
  6. a n - 1 - a n - 2 a n - a n - 1 \dfrac{a_{n-1}-a_{n-2}}{a_{n}-a_{n-1}}
  7. f ( z ) = z 2 + c f(z)=z^{2}+c
  8. = c n - 1 - c n - 2 c n - c n - 1 =\dfrac{c_{n-1}-c_{n-2}}{c_{n}-c_{n-1}}
  9. α = \alpha=
  10. α \alpha
  11. π + tan - 1 ( e π ) \pi+\tan^{-1}(e^{\pi})
  12. δ \delta
  13. α \alpha
  14. 2 φ log 2 4.669 \frac{2\varphi}{\log 2}\approx 4.669
  15. 2 φ + 1 log 2 + 1 2.502 \frac{2\varphi+1}{\log 2+1}\approx 2.502
  16. φ \varphi
  17. log 2 \log 2

Femtosecond.html

  1. λ c = 600 × 10 - 9 m 3 × 10 8 m s - 1 = 2.0 × 10 - 15 s {\lambda\over{c}}={600\times 10^{-9}~{}{\rm m}\over 3\times 10^{8}~{}{\rm m}~{% }{\rm s}^{-1}}=2.0\times 10^{-15}~{}{\rm s}

Fermat's_spiral.html

  1. r = ± θ 1 / 2 r=\pm\theta^{1/2}\,
  2. r = c θ , r=c\sqrt{\theta},
  3. θ = n × 137.508 , \theta=n\times 137.508^{\circ},

Fermat_number.html

  1. F n = 2 ( 2 n ) + 1 F_{n}=2^{(2^{n})}+1
  2. F n = ( F n - 1 - 1 ) 2 + 1 F_{n}=(F_{n-1}-1)^{2}+1\!
  3. F n = F n - 1 + 2 2 n - 1 F 0 F n - 2 F_{n}=F_{n-1}+2^{2^{n-1}}F_{0}\cdots F_{n-2}\!
  4. F n = F n - 1 2 - 2 ( F n - 2 - 1 ) 2 F_{n}=F_{n-1}^{2}-2(F_{n-2}-1)^{2}\!
  5. F n = F 0 F n - 1 + 2 F_{n}=F_{0}\cdots F_{n-1}+2\!
  6. F 0 F j - 1 F_{0}\cdots F_{j-1}
  7. F 5 = 2 2 5 + 1 = 2 32 + 1 = 4294967297 = 641 × 6700417. F_{5}=2^{2^{5}}+1=2^{32}+1=4294967297=641\times 6700417.\;
  8. A n = 0 1 ln F n \displaystyle A\sum_{n=0}^{\infty}\frac{1}{\ln F_{n}}
  9. A n = 0 ln 2 n + 1 ln F n \displaystyle A\sum_{n=0}^{\infty}\frac{\ln 2^{n+1}}{\ln F_{n}}
  10. C n = 0 2 n + 1 ln F n \displaystyle C\sum_{n=0}^{\infty}\frac{2^{n+1}}{\ln F_{n}}
  11. F n = 2 2 n + 1 F_{n}=2^{2^{n}}+1
  12. F n F_{n}
  13. 3 ( F n - 1 ) / 2 - 1 ( mod F n ) . 3^{(F_{n}-1)/2}\equiv-1\;\;(\mathop{{\rm mod}}F_{n}).
  14. 3 ( F n - 1 ) / 2 3^{(F_{n}-1)/2}
  15. F n F_{n}
  16. a ( N - 1 ) / 2 - 1 ( mod N ) a^{(N-1)/2}\equiv-1\;\;(\mathop{{\rm mod}}N)\!
  17. ( a N ) = - 1 \left(\frac{a}{N}\right)=-1\!
  18. F n F_{n}
  19. k × 2 n + 2 + 1 k\times 2^{n+2}+1
  20. F 5 F_{5}
  21. 5 2 7 + 1 5\cdot 2^{7}+1
  22. F 5 F_{5}
  23. 52347 2 7 + 1 52347\cdot 2^{7}+1
  24. F 6 F_{6}
  25. 1071 2 8 + 1 1071\cdot 2^{8}+1
  26. F 6 F_{6}
  27. 262814145745 2 8 + 1 262814145745\cdot 2^{8}+1
  28. F 12 F_{12}
  29. 7 2 14 + 1 7\cdot 2^{14}+1
  30. F 23 F_{23}
  31. 5 2 25 + 1 5\cdot 2^{25}+1
  32. F 36 F_{36}
  33. 5 2 39 + 1 5\cdot 2^{39}+1
  34. F 11 F_{11}
  35. 39 2 13 + 1 39\cdot 2^{13}+1
  36. F 11 F_{11}
  37. 119 2 13 + 1 119\cdot 2^{13}+1
  38. F 9 F_{9}
  39. 37 2 16 + 1 37\cdot 2^{16}+1
  40. F 12 F_{12}
  41. 397 2 16 + 1 397\cdot 2^{16}+1
  42. F 12 F_{12}
  43. 973 2 16 + 1 973\cdot 2^{16}+1
  44. F 18 F_{18}
  45. 13 2 20 + 1 13\cdot 2^{20}+1
  46. F 38 F_{38}
  47. 3 2 41 + 1 3\cdot 2^{41}+1
  48. F 73 F_{73}
  49. 5 2 75 + 1 5\cdot 2^{75}+1
  50. F 15 F_{15}
  51. 579 2 21 + 1 579\cdot 2^{21}+1
  52. 2 F n - 1 1 ( mod F n ) 2^{F_{n}-1}\equiv 1\;\;(\mathop{{\rm mod}}F_{n})\,\!
  53. a n - b n = ( a - b ) k = 0 n - 1 a k b n - 1 - k . a^{n}-b^{n}=(a-b)\sum_{k=0}^{n-1}a^{k}b^{n-1-k}.
  54. ( a - b ) k = 0 n - 1 a k b n - 1 - k (a-b)\sum_{k=0}^{n-1}a^{k}b^{n-1-k}
  55. = k = 0 n - 1 a k + 1 b n - 1 - k - k = 0 n - 1 a k b n - k =\sum_{k=0}^{n-1}a^{k+1}b^{n-1-k}-\sum_{k=0}^{n-1}a^{k}b^{n-k}
  56. = a n + k = 1 n - 1 a k b n - k - k = 1 n - 1 a k b n - k - b n =a^{n}+\sum_{k=1}^{n-1}a^{k}b^{n-k}-\sum_{k=1}^{n-1}a^{k}b^{n-k}-b^{n}
  57. = a n - b n . =a^{n}-b^{n}.
  58. 2 k + 1 2^{k}+1
  59. k k
  60. k k
  61. k = r s k=rs
  62. 1 r < k 1\leq r<k
  63. 1 < s k 1<s\leq k
  64. s s
  65. m m
  66. ( a - b ) ( a m - b m ) (a-b)\mid(a^{m}-b^{m})
  67. \mid
  68. a = 2 r a=2^{r}
  69. b = - 1 b=-1
  70. m = s m=s
  71. s s
  72. ( 2 r + 1 ) ( 2 r s + 1 ) , (2^{r}+1)\mid(2^{rs}+1),
  73. ( 2 r + 1 ) ( 2 k + 1 ) . (2^{r}+1)\mid(2^{k}+1).
  74. 1 < 2 r + 1 < 2 k + 1 1<2^{r}+1<2^{k}+1
  75. 2 k + 1 2^{k}+1
  76. k k
  77. p = 2 m + 1 p=2^{m}+1
  78. 2 p - 1 1 ( mod p 2 ) 2^{p-1}\equiv 1\;\;(\mathop{{\rm mod}}p^{2})
  79. 2 m | p - 1 2m|p-1
  80. p - 1 = 2 m λ p-1=2m\lambda
  81. p 2 | 2 2 m λ - 1 p^{2}|2^{2m\lambda}-1
  82. 0 ( 2 2 m λ - 1 ) / ( 2 m + 1 ) = ( 2 m - 1 ) ( 1 + 2 2 m + 2 4 m + + 2 2 ( λ - 1 ) m ) - 2 λ ( mod 2 m + 1 ) . 0\equiv(2^{2m\lambda}-1)/(2^{m}+1)=(2^{m}-1)(1+2^{2m}+2^{4m}+...+2^{2(\lambda-% 1)m})\equiv-2\lambda\;\;(\mathop{{\rm mod}}2^{m}+1).
  83. 2 m + 1 | 2 λ 2^{m}+1|2\lambda
  84. 2 λ 2 m + 1 2\lambda\geq 2^{m}+1
  85. p - 1 m ( 2 m + 1 ) p-1\geq m(2^{m}+1)
  86. m 2 m\geq 2
  87. 2 2 𝑛 + 1 2^{2^{\overset{n}{}}}+1
  88. k 2 n + 2 + 1 k2^{n+2}+1
  89. 2 n + 1 2^{n+1}
  90. 2 2 n + 1 2^{2^{\overset{n+1}{}}}
  91. 2 2 𝑛 2^{2^{\overset{n}{}}}
  92. 2 n + 1 2^{n+1}
  93. k 2 n + 1 + 1 k2^{n+1}+1
  94. 2 n + 2 2^{n+2}
  95. 2 n + 2 2^{n+2}
  96. s 2 n + 2 + 1 s2^{n+2}+1
  97. ( 1 + 2 2 n - 1 ) 2 2 1 + 2 n - 1 (1+2^{2^{n-1}})^{2}\equiv 2^{1+2^{n-1}}
  98. V j + 1 = ( A × V j ) mod P V_{j+1}=\left(A\times V_{j}\right)\bmod P
  99. P ( F n ) 2 n + 2 ( 4 n + 9 ) + 1. P(F_{n})\geq 2^{n+2}(4n+9)+1.
  100. a 2 𝑛 + b 2 𝑛 a^{2^{\overset{n}{}}}+b^{2^{\overset{n}{}}}
  101. 2 2 0 + 1 2^{2^{0}}+1
  102. a 2 𝑛 + 1 a^{2^{\overset{n}{}}}+1
  103. a 2 𝑛 + 1 a^{2^{\overset{n}{}}}+1
  104. a 2 n + 1 2 \frac{a^{2^{n}}+1}{2}
  105. a 2 n + 1 a^{2^{n}}+1
  106. a 2 n + 1 2 \frac{a^{2^{n}}+1}{2}

Fermat_primality_test.html

  1. 0 < a < p 0<a<p
  2. a p - 1 1 ( mod p ) . a^{p-1}\equiv 1\;\;(\mathop{{\rm mod}}p).
  3. a n - 1 1 ( mod n ) a^{n-1}\equiv 1\;\;(\mathop{{\rm mod}}n)
  4. a n - 1 1 ( mod n ) a^{n-1}\not\equiv 1\;\;(\mathop{{\rm mod}}n)
  5. a n - 1 = 24 220 81 1 ( mod 221 ) . a^{n-1}=24^{220}\equiv 81\not\equiv 1\;\;(\mathop{{\rm mod}}221).
  6. a n - 1 1 ( mod n ) a^{n-1}\not\equiv 1\;\;(\mathop{{\rm mod}}n)
  7. n n
  8. a a
  9. g c d ( a , n ) = 1 gcd(a,n)=1
  10. n n
  11. a ( / n ) * a\in(\mathbb{Z}/n\mathbb{Z})^{*}
  12. a a
  13. a 1 a_{1}
  14. a 2 a_{2}
  15. a s a_{s}
  16. ( a a i ) n - 1 a n - 1 a i n - 1 a n - 1 1 ( mod n ) (a\cdot a_{i})^{n-1}\equiv a^{n-1}\cdot a_{i}^{n-1}\equiv a^{n-1}\not\equiv 1% \;\;(\mathop{{\rm mod}}n)
  17. a × a i a\times a_{i}
  18. i = 1 , 2 , , s i=1,2,...,s

Fermi_energy.html

  1. E n = E 0 + 2 π 2 2 m L 2 n 2 . E_{n}=E_{0}+\frac{\hbar^{2}\pi^{2}}{2mL^{2}}n^{2}.\,
  2. E 0 E_{0}
  3. E 1 E_{1}
  4. E 2 E_{2}
  5. E 0 E_{0}
  6. E F = E N / 2 - E 0 = 2 π 2 2 m L 2 ( N / 2 ) 2 , E_{F}=E_{N/2}-E_{0}=\frac{\hbar^{2}\pi^{2}}{2mL^{2}}(N/2)^{2},
  7. E n x , n y , n z = E 0 + 2 π 2 2 m L 2 ( n x 2 + n y 2 + n z 2 ) E_{n_{x},n_{y},n_{z}}=E_{0}+\frac{\hbar^{2}\pi^{2}}{2mL^{2}}\left(n_{x}^{2}+n_% {y}^{2}+n_{z}^{2}\right)\,
  8. E 211 = E 121 = E 112 E_{211}=E_{121}=E_{112}
  9. n = { n x , n y , n z } \vec{n}=\{n_{x},n_{y},n_{z}\}
  10. E n = E 0 + 2 π 2 2 m L 2 | n | 2 E_{\vec{n}}=E_{0}+\frac{\hbar^{2}\pi^{2}}{2mL^{2}}|\vec{n}|^{2}\,
  11. | n | 2 |\vec{n}|^{2}
  12. ( n x 2 + n y 2 + n z 2 ) 2 (\sqrt{n_{x}^{2}+n_{y}^{2}+n_{z}^{2}})^{2}
  13. | n F | |\vec{n}_{F}|
  14. N = 2 × 1 8 × 4 3 π n F 3 N=2\times\frac{1}{8}\times\frac{4}{3}\pi n_{F}^{3}\,
  15. n F = ( 3 N π ) 1 / 3 n_{F}=\left(\frac{3N}{\pi}\right)^{1/3}
  16. E F = 2 π 2 2 m L 2 n F 2 = 2 π 2 2 m L 2 ( 3 N π ) 2 / 3 E_{F}=\frac{\hbar^{2}\pi^{2}}{2mL^{2}}n_{F}^{2}=\frac{\hbar^{2}\pi^{2}}{2mL^{2% }}\left(\frac{3N}{\pi}\right)^{2/3}
  17. E F = 2 2 m ( 3 π 2 N V ) 2 / 3 E_{F}=\frac{\hbar^{2}}{2m}\left(\frac{3\pi^{2}N}{V}\right)^{2/3}\,
  18. N N
  19. E t = N E 0 + 0 N E F d N = ( 3 5 E F + E 0 ) N E_{t}=NE_{0}+\int_{0}^{N}E_{F}\,dN^{\prime}=\left(\frac{3}{5}E_{F}+E_{0}\right)N
  20. E av = E 0 + 3 5 E F E_{\mathrm{av}}=E_{0}+\frac{3}{5}E_{F}
  21. T F = E F k B T_{F}=\frac{E_{F}}{k_{B}}
  22. k B k_{B}
  23. E F E_{F}
  24. p F = 2 m e E F p_{F}=\sqrt{2m_{e}E_{F}}
  25. v F = p F m e v_{F}=\frac{p_{F}}{m_{e}}
  26. m e m_{e}
  27. p F = k F p_{F}=\hbar k_{F}
  28. k F k_{F}
  29. d d
  30. g ( E ) = 2 d d k ( 2 π ) d / V δ ( E - E 0 - 2 k 2 2 m ) = V d m d / 2 ( E - E 0 ) d / 2 - 1 ( 2 π ) d / 2 Γ ( d / 2 + 1 ) d g(E)=2\int\frac{d^{d}\vec{k}}{(2\pi)^{d}/V}\delta\left(E-E_{0}-\frac{\hbar^{2}% \vec{k}^{2}}{2m}\right)=V\frac{d\,m^{d/2}(E-E_{0})^{d/2-1}}{(2\pi)^{d/2}\ % \Gamma(d/2+1)\hbar^{d}}
  31. n = E 0 E 0 + E F g ( E ) d E n=\int_{E_{0}}^{E_{0}+E_{F}}g(E)\,dE
  32. E F = 2 π 2 m ( 1 2 Γ ( d 2 + 1 ) n ) 2 / d E_{F}=\frac{2\pi\hbar^{2}}{m}\left(\tfrac{1}{2}\Gamma\left(\tfrac{d}{2}+1% \right)n\right)^{2/d}
  33. N / V N/V
  34. E F = 2 2 m e ( 3 π 2 10 28 29 m - 3 ) 2 / 3 2 10 eV E_{F}=\frac{\hbar^{2}}{2m_{e}}\left(3\pi^{2}\ 10^{28\ \sim\ 29}\ \mathrm{m}^{-% 3}\right)^{2/3}\approx 2\ \sim\ 10\ \mathrm{eV}
  35. E F = 2 2 m e ( 3 π 2 ( 10 36 ) 1 m 3 ) 2 / 3 3 × 10 5 eV = 0.3 MeV E_{F}=\frac{\hbar^{2}}{2m_{e}}\left(\frac{3\pi^{2}(10^{36})}{1\ \mathrm{m}^{3}% }\right)^{2/3}\approx 3\times 10^{5}\ \mathrm{eV}=0.3\ \mathrm{MeV}
  36. R = ( 1.25 × 10 - 15 m ) × A 1 / 3 R=\left(1.25\times 10^{-15}\mathrm{m}\right)\times A^{1/3}
  37. n = A 4 3 π R 3 1.2 × 10 44 m - 3 n=\frac{A}{\begin{matrix}\frac{4}{3}\end{matrix}\pi R^{3}}\approx 1.2\times 10% ^{44}\ \mathrm{m}^{-3}
  38. E F = 2 2 m p ( 3 π 2 ( 6 × 10 43 ) 1 m 3 ) 2 / 3 3 × 10 7 eV = 30 MeV E_{F}=\frac{\hbar^{2}}{2m_{p}}\left(\frac{3\pi^{2}(6\times 10^{43})}{1\ % \mathrm{m}^{3}}\right)^{2/3}\approx 3\times 10^{7}\ \mathrm{eV}=30\ \mathrm{MeV}

Fermi_gas.html

  1. k T E F kT\ll E_{F}
  2. μ = E 0 + E F [ 1 - π 2 12 ( k T E F ) 2 - π 4 80 ( k T E F ) 4 + ] \mu=E_{0}+E_{F}\left[1-\frac{\pi^{2}}{12}\left(\frac{kT}{E_{F}}\right)^{2}-% \frac{\pi^{4}}{80}\left(\frac{kT}{E_{F}}\right)^{4}+\cdots\right]

Fermi_level.html

  1. ( V A - V B ) = - ( μ A - μ B ) / e (V_{\mathrm{A}}-V_{\mathrm{B}})=-(\mu_{\mathrm{A}}-\mu_{\mathrm{B}})/e
  2. f ( ϵ ) f(\epsilon)
  3. f ( ϵ ) = 1 e ( ϵ - μ ) / ( k T ) + 1 f(\epsilon)=\frac{1}{e^{(\epsilon-\mu)/(kT)}+1}
  4. ζ = μ - ϵ C . \zeta=\mu-\epsilon_{\rm C}.
  5. f ( ) = 1 e ( - ζ ) / ( k T ) + 1 . f(\mathcal{E})=\frac{1}{e^{(\mathcal{E}-\zeta)/(kT)}+1}.
  6. [ - F o r m u l a E r r o r - ] [-FormulaError-]

Fermi_liquid_theory.html

  1. Ψ 0 \Psi_{0}
  2. p < p F p<p_{F}
  3. τ \tau
  4. 1 τ ϵ p \frac{1}{\tau}\ll\epsilon_{p}
  5. ϵ p \epsilon_{p}
  6. G ( ω , p ) Z ω + μ - ϵ ( p ) G(\omega,p)\approx\frac{Z}{\omega+\mu-\epsilon(p)}
  7. μ \mu
  8. ϵ ( p ) \epsilon(p)
  9. Z Z
  10. A ( k , ω ) = Z δ ( ω - v F k ) A(\vec{k},\omega)=Z\delta(\omega-v_{F}k_{\|})
  11. v F v_{F}
  12. ϵ 1 \epsilon_{1}
  13. ϵ 2 \epsilon_{2}
  14. ϵ 3 , ϵ 4 > ϵ F \epsilon_{3},\epsilon_{4}>\epsilon_{F}
  15. ϵ ϵ F \epsilon\approx\epsilon_{F}
  16. ϵ 2 , ϵ 3 , ϵ 4 \epsilon_{2},\epsilon_{3},\epsilon_{4}
  17. δ n k \delta n_{k}
  18. k k
  19. δ n k \delta n_{k}
  20. δ n k ϵ k \delta n_{k}\epsilon_{k}
  21. ϵ k \epsilon_{k}
  22. 0 < Z < 1 0<Z<1
  23. Z Z
  24. T 2 T^{2}
  25. T 2 T^{2}
  26. T c T_{c}
  27. Z 0 Z\to 0
  28. Z > 0 Z>0

Fermi–Dirac_statistics.html

  1. i i
  2. n ¯ i = 1 e ( ϵ i - μ ) / k T + 1 \bar{n}_{i}=\frac{1}{e^{(\epsilon_{i}-\mu)/kT}+1}
  3. k k
  4. T T
  5. ϵ i \epsilon_{i}
  6. i i
  7. μ μ
  8. μ μ
  9. μ μ
  10. μ μ
  11. 0 < n ¯ i < 1 0<\bar{n}_{i}<1
  12. μ \mu
  13. ϵ > μ \epsilon>\mu
  14. ϵ \epsilon
  15. ϵ \epsilon
  16. ϵ i \epsilon_{i}
  17. n ¯ i \bar{n}_{i}
  18. g i g_{i}
  19. ϵ i \epsilon_{i}
  20. n ¯ ( ϵ i ) = g i n ¯ i = g i e ( ϵ i - μ ) / k T + 1 \begin{aligned}\displaystyle\bar{n}(\epsilon_{i})&\displaystyle=g_{i}\ \bar{n}% _{i}\\ &\displaystyle=\frac{g_{i}}{e^{(\epsilon_{i}-\mu)/kT}+1}\\ \end{aligned}
  21. g i 2 g_{i}\geq 2
  22. n ¯ ( ϵ i ) > 1 \ \bar{n}(\epsilon_{i})>1
  23. ϵ i \epsilon_{i}
  24. ϵ \epsilon
  25. g ( ϵ ) g(\epsilon)
  26. 𝒩 ¯ ( ϵ ) = g ( ϵ ) F ( ϵ ) \bar{\mathcal{N}}(\epsilon)=g(\epsilon)\ F(\epsilon)
  27. F ( ϵ ) F(\epsilon)
  28. n ¯ i \bar{n}_{i}
  29. F ( ϵ ) = 1 e ( ϵ - μ ) / k T + 1 F(\epsilon)=\frac{1}{e^{(\epsilon-\mu)/kT}+1}
  30. 𝒩 ¯ ( ϵ ) = g ( ϵ ) e ( ϵ - μ ) / k T + 1 \bar{\mathcal{N}}(\epsilon)=\frac{g(\epsilon)}{e^{(\epsilon-\mu)/kT}+1}
  31. R ¯ \bar{R}
  32. λ ¯ \bar{\lambda}
  33. R ¯ λ ¯ h 3 m k T \bar{R}\ \gg\ \bar{\lambda}\ \approx\ \frac{h}{\sqrt{3mkT}}
  34. h h
  35. m m
  36. R ¯ λ ¯ / 25 \bar{R}\approx\bar{\lambda}/25
  37. R ¯ \bar{R}
  38. 𝒵 \displaystyle\mathcal{Z}
  39. N = k B T 1 𝒵 ( 𝒵 μ ) V , T = 1 exp ( ( ϵ - μ ) / k B T ) + 1 \langle N\rangle=k_{B}T\frac{1}{\mathcal{Z}}\left(\frac{\partial\mathcal{Z}}{% \partial\mu}\right)_{V,T}=\frac{1}{\exp((\epsilon-\mu)/k_{B}T)+1}
  40. ( Δ N ) 2 = k B T ( d N d μ ) V , T = N ( 1 - N ) \langle(\Delta N)^{2}\rangle=k_{B}T\left(\frac{d\langle N\rangle}{d\mu}\right)% _{V,T}=\langle N\rangle(1-\langle N\rangle)
  41. ( Δ N ) 2 \langle(\Delta N)^{2}\rangle
  42. E R E_{R}
  43. R R
  44. E R = r n r ϵ r E_{R}=\sum_{r}n_{r}\epsilon_{r}\;
  45. n r n_{r}
  46. r r
  47. ϵ r \epsilon_{r}\;
  48. r r
  49. R R
  50. P R = e - β E R R e - β E R P_{R}=\frac{e^{-\beta E_{R}}}{\displaystyle\sum_{R^{\prime}}e^{-\beta E_{R^{% \prime}}}}
  51. β \beta\;
  52. = 1 / k T =1/kT
  53. k k
  54. T T
  55. - β E R \scriptstyle-\beta E_{R}
  56. R R^{\prime}
  57. n i n_{i}\;
  58. n ¯ i = R n i P R \bar{n}_{i}\ =\ \sum_{R}n_{i}\ P_{R}
  59. R R
  60. n 1 , n 2 , , n_{1},\,n_{2},\,...\;,
  61. P R = P n 1 , n 2 , = e - β ( n 1 ϵ 1 + n 2 ϵ 2 + ) n 1 , n 2 , e - β ( n 1 ϵ 1 + n 2 ϵ 2 + ) P_{R}=P_{n_{1},n_{2},...}=\frac{e^{-\beta(n_{1}\epsilon_{1}+n_{2}\epsilon_{2}+% ...)}}{\displaystyle\sum_{{n_{1}}^{\prime},{n_{2}}^{\prime},...}e^{-\beta({n_{% 1}}^{\prime}\epsilon_{1}+{n_{2}}^{\prime}\epsilon_{2}+...)}}
  62. n ¯ i \bar{n}_{i}
  63. n ¯ i = n 1 , n 2 , n i P n 1 , n 2 , = n 1 , n 2 , n i e - β ( n 1 ϵ 1 + n 2 ϵ 2 + + n i ϵ i + ) n 1 , n 2 , e - β ( n 1 ϵ 1 + n 2 ϵ 2 + + n i ϵ i + ) \begin{aligned}\displaystyle\bar{n}_{i}&\displaystyle=\sum_{n_{1},n_{2},\dots}% n_{i}\ P_{n_{1},n_{2},\dots}\\ \\ &\displaystyle=\frac{\displaystyle\sum_{n_{1},n_{2},\dots}n_{i}\ e^{-\beta(n_{% 1}\epsilon_{1}+n_{2}\epsilon_{2}+\cdots+n_{i}\epsilon_{i}+\cdots)}}{% \displaystyle\sum_{n_{1},n_{2},\dots}e^{-\beta(n_{1}\epsilon_{1}+n_{2}\epsilon% _{2}+\cdots+n_{i}\epsilon_{i}+\cdots)}}\\ \end{aligned}
  64. n 1 , n 2 , n_{1},n_{2},...\;
  65. r r
  66. n 1 , n 2 , n_{1},n_{2},...\;
  67. N N
  68. r n r = N \sum_{r}n_{r}=N\;
  69. n ¯ i = n i = 0 1 n i e - β ( n i ϵ i ) ( i ) n 1 , n 2 , ( i ) e - β ( n 1 ϵ 1 + n 2 ϵ 2 + ) n i = 0 1 e - β ( n i ϵ i ) ( i ) n 1 , n 2 , ( i ) e - β ( n 1 ϵ 1 + n 2 ϵ 2 + ) \bar{n}_{i}=\frac{\displaystyle\sum_{n_{i}=0}^{1}n_{i}\ e^{-\beta(n_{i}% \epsilon_{i})}\quad\sideset{}{{}^{(i)}}{\sum}_{n_{1},n_{2},\dots}e^{-\beta(n_{% 1}\epsilon_{1}+n_{2}\epsilon_{2}+\cdots)}}{\displaystyle\sum_{n_{i}=0}^{1}e^{-% \beta(n_{i}\epsilon_{i})}\qquad\sideset{}{{}^{(i)}}{\sum}_{n_{1},n_{2},\dots}e% ^{-\beta(n_{1}\epsilon_{1}+n_{2}\epsilon_{2}+\cdots)}}
  70. ( i ) {}^{(i)}
  71. n i n_{i}
  72. N i = N - n i N_{i}=N-n_{i}
  73. Σ ( i ) \Sigma^{(i)}
  74. n i n_{i}
  75. N i N_{i}
  76. n i = 0 n_{i}=0
  77. Σ ( i ) \Sigma^{(i)}
  78. N i = N , N_{i}=N,
  79. n i = 1 n_{i}=1
  80. Σ ( i ) \Sigma^{(i)}
  81. N i = N - 1. N_{i}=N-1.
  82. Σ ( i ) \Sigma^{(i)}
  83. n i n_{i}
  84. N - n i N-n_{i}
  85. Z i ( N - n i ) ( i ) n 1 , n 2 , e - β ( n 1 ϵ 1 + n 2 ϵ 2 + ) Z_{i}(N-n_{i})\equiv\ \sideset{}{{}^{(i)}}{\sum}_{n_{1},n_{2},...}e^{-\beta(n_% {1}\epsilon_{1}+n_{2}\epsilon_{2}+\cdots)}\;
  86. n ¯ i \bar{n}_{i}
  87. Z i Z_{i}
  88. n ¯ i = n i = 0 1 n i e - β ( n i ϵ i ) Z i ( N - n i ) n i = 0 1 e - β ( n i ϵ i ) Z i ( N - n i ) = 0 + e - β ϵ i Z i ( N - 1 ) Z i ( N ) + e - β ϵ i Z i ( N - 1 ) = 1 [ Z i ( N ) / Z i ( N - 1 ) ] e β ϵ i + 1 . \begin{aligned}\displaystyle\bar{n}_{i}&\displaystyle=\frac{\displaystyle\sum_% {n_{i}=0}^{1}n_{i}\ e^{-\beta(n_{i}\epsilon_{i})}\ \ Z_{i}(N-n_{i})}{% \displaystyle\sum_{n_{i}=0}^{1}e^{-\beta(n_{i}\epsilon_{i})}\qquad Z_{i}(N-n_{% i})}\\ \\ &\displaystyle=\ \frac{\quad 0\quad\;+e^{-\beta\epsilon_{i}}\;Z_{i}(N-1)}{Z_{i% }(N)+e^{-\beta\epsilon_{i}}\;Z_{i}(N-1)}\\ &\displaystyle=\ \frac{1}{[Z_{i}(N)/Z_{i}(N-1)]\;e^{\beta\epsilon_{i}}+1}\quad% .\end{aligned}
  89. Z i ( N ) / Z i ( N - 1 ) Z_{i}(N)/Z_{i}(N-1)
  90. ln Z i ( N - 1 ) ln Z i ( N ) - ln Z i ( N ) N = ln Z i ( N ) - α i \begin{aligned}\displaystyle\ln Z_{i}(N-1)&\displaystyle\simeq\ln Z_{i}(N)-% \frac{\partial\ln Z_{i}(N)}{\partial N}\\ &\displaystyle=\ln Z_{i}(N)-\alpha_{i}\end{aligned}
  91. α i ln Z i ( N ) N . \alpha_{i}\equiv\frac{\partial\ln Z_{i}(N)}{\partial N}\ .
  92. N N
  93. μ \mu\;
  94. α i - μ / k T . \alpha_{i}\simeq-\mu/kT\ .
  95. α i \alpha_{i}\,
  96. Z i ( N ) / Z i ( N - 1 ) = e - μ / k T Z_{i}(N)/Z_{i}(N-1)=e^{-\mu/kT}\,
  97. n ¯ i \bar{n}_{i}
  98. β \beta\;
  99. 1 / k T 1/kT
  100. β \beta\;
  101. n ¯ i = 1 e ( ϵ i - μ ) / k T + 1 \bar{n}_{i}=\ \frac{1}{e^{(\epsilon_{i}-\mu)/kT}+1}
  102. w ( n i , g i ) = g i ! n i ! ( g i - n i ) ! . w(n_{i},g_{i})=\frac{g_{i}!}{n_{i}!(g_{i}-n_{i})!}\ .
  103. W = i w ( n i , g i ) = i g i ! n i ! ( g i - n i ) ! . W=\prod_{i}w(n_{i},g_{i})=\prod_{i}\frac{g_{i}!}{n_{i}!(g_{i}-n_{i})!}.
  104. f ( n i ) = ln ( W ) + α ( N - n i ) + β ( E - n i ϵ i ) . f(n_{i})=\ln(W)+\alpha(N-\sum n_{i})+\beta(E-\sum n_{i}\epsilon_{i}).
  105. n i = g i e α + β ϵ i + 1 . n_{i}=\frac{g_{i}}{e^{\alpha+\beta\epsilon_{i}}+1}.
  106. β = 1 k T \beta=\frac{1}{kT}
  107. α = - μ k T \alpha=-\frac{\mu}{kT}
  108. μ \mu
  109. n ¯ i = n i g i = 1 e ( ϵ i - μ ) / k T + 1 . \bar{n}_{i}=\frac{n_{i}}{g_{i}}=\frac{1}{e^{(\epsilon_{i}-\mu)/kT}+1}.
  110. n ¯ i \bar{n}_{i}
  111. i i
  112. 0 < n ¯ i < 1 0<\bar{n}_{i}<1
  113. n ( ϵ ) n(\epsilon)\,
  114. n s n_{s}\,
  115. n ¯ i \bar{n}_{i}
  116. n ¯ ( ϵ i ) \bar{n}(\epsilon_{i})

Fiber_bundle.html

  1. π : E B \pi\colon E\to B
  2. F E 𝜋 B F\longrightarrow E\ \xrightarrow{\,\ \pi\ }\ B
  3. π - 1 ( U ) \pi^{-1}(U)
  4. φ i φ j - 1 : ( U i U j ) × F ( U i U j ) × F \varphi_{i}\varphi_{j}^{-1}\colon(U_{i}\cap U_{j})\times F\to(U_{i}\cap U_{j})\times F
  5. φ i φ j - 1 ( x , ξ ) = ( x , t i j ( x ) ξ ) \varphi_{i}\varphi_{j}^{-1}(x,\xi)=(x,t_{ij}(x)\xi)
  6. t i i ( x ) = 1 t_{ii}(x)=1\,
  7. t i j ( x ) = t j i ( x ) - 1 t_{ij}(x)=t_{ji}(x)^{-1}\,
  8. t i k ( x ) = t i j ( x ) t j k ( x ) . t_{ik}(x)=t_{ij}(x)t_{jk}(x).\,
  9. φ : E F , f : M N \varphi\colon E\to F,\quad f\colon M\to N
  10. π F φ = f π E \pi_{F}\circ\varphi=f\circ\pi_{E}
  11. φ : E F \varphi\colon E\to F
  12. φ ( x s ) = φ ( x ) s \varphi(xs)=\varphi(x)s
  13. x E x\in E
  14. s G s\in G
  15. π E = π F φ \pi_{E}=\pi_{F}\circ\varphi
  16. f id M f\equiv{\mathrm{id}}_{M}
  17. ( φ , f ) (\varphi,f)
  18. f id M f\equiv{\mathrm{id}}_{M}

Fibonacci_heap.html

  1. F d + 2 φ d F_{d+2}\geq\varphi^{d}
  2. d 0 d\geq 0
  3. φ = ( 1 + 5 ) / 2 1.618 \varphi=(1+\sqrt{5})/2\doteq 1.618
  4. n F d + 2 φ d n\geq F_{d+2}\geq\varphi^{d}
  5. φ \varphi
  6. d log φ n d\leq\log_{\varphi}n
  7. 𝐬𝐢𝐳𝐞 ( x ) 2 + i = 2 d 𝐬𝐢𝐳𝐞 ( y i ) 2 + i = 2 d F i = 1 + i = 0 d F i . \,\textbf{size}(x)\geq 2+\sum_{i=2}^{d}\,\textbf{size}(y_{i})\geq 2+\sum_{i=2}% ^{d}F_{i}=1+\sum_{i=0}^{d}F_{i}.
  8. 1 + i = 0 d F i = F d + 2 1+\sum_{i=0}^{d}F_{i}=F_{d+2}
  9. d 0 d\geq 0

Field_electron_emission.html

  1. F = γ F M . F=\gamma F_{\mathrm{M}}.
  2. 2 2 m d 2 Ψ ( x ) d x 2 = [ U ( x ) - E n ] Ψ ( x ) = M ( x ) Ψ ( x ) , ( 1 ) \frac{\hbar^{2}}{2m}\frac{\mathrm{d}^{2}\Psi(x)}{\mathrm{d}x^{2}}=\left[U(x)-E% _{\mathrm{n}}\right]\Psi(x)=M(x)\Psi(x),\qquad\qquad(1)
  3. M ET ( x ) = h - e F x ( 2 ) M^{\mathrm{ET}}(x)=h-eFx\qquad\qquad\qquad\qquad\qquad\;\;\;(2)
  4. M SN ( x ) = h - e F x - e 2 / ( 16 π ε 0 x ) , ( 3 ) M^{\rm{SN}}(x)=h-eFx-e^{2}/(16\pi\varepsilon_{0}x),\qquad\qquad(3)
  5. G ( h , F ) = g M 1 / 2 d x , ( 4 ) G(h,F)=g\int M^{1/2}\mbox{d}~{}x,\qquad\qquad(4)
  6. g = 2 2 m / 10.24624 eV - 1 / 2 nm - 1 . ( 5 ) g\,=2\sqrt{2m}/\hbar\approx 10.24624\;{\rm{eV}}^{-1/2}\;{\rm{nm}}^{-1}.\qquad% \qquad(5)
  7. D = P e - G 1 + P e - G , ( 6 ) \,D=\frac{P\mathrm{e}^{-G}}{1+P\mathrm{e}^{-G}},\qquad\qquad(6)
  8. D P e - G e - G . ( 7 ) D\approx P\mathrm{e}^{-G}\approx\mathrm{e}^{-G}.\qquad\qquad(7)
  9. b = 2 g 3 e = 4 2 m 3 e 6.830890 eV - 3 / 2 V nm - 1 . ( 8 ) b=\frac{2g}{3e}=\frac{4\sqrt{2m}}{3e\hbar}\approx 6.830890\;{\mathrm{eV}}^{-3/% 2}\;\mathrm{V}\;{\mathrm{nm}}^{-1}.\qquad\qquad(8)
  10. G ( h , F ) = ν ( h , F ) G ET = ν ( h , F ) b h 3 / 2 / F , ( 9 ) G(h,F)=\nu(h,F)G^{\mathrm{ET}}=\nu(h,F)bh^{3/2}/F,\qquad\qquad(9)
  11. ν \it{\nu}
  12. F h = ( 4 π ϵ 0 / e 3 ) h 2 = ( 0.6944617 V nm - 1 ) ( h / eV ) 2 . ( 10 ) \,F_{h}=(4\pi\epsilon_{0}/e^{3})h^{2}=(0.6944617\;\mathrm{V}\;{\mathrm{nm}}^{-% 1})(h/{\rm{eV}})^{2}.\qquad\qquad(10)
  13. v ( f h ) 1 - f h + 1 6 f h ln f h . . ( 11 ) v(f_{h})\approx 1-f_{h}+\tfrac{1}{6}f_{h}\ln f_{h}...........\rm{(11)}
  14. 1 d h = - d ( ln D ) d h . ( 12 ) \frac{1}{d_{h}}=-\frac{\mathrm{d}(\ln D)}{\mathrm{d}h}.\qquad\qquad(12)
  15. d h ( el ) = 2 F 3 b h = e F g h . d_{h}^{\mathrm{(el)}}=\frac{2F}{3b\sqrt{h}}=\frac{eF}{g\sqrt{h}}.
  16. d h = λ d d h ( el ) = λ d e F g h . ( 13 ) d_{h}=\lambda_{d}d_{h}^{\mathrm{(el)}}=\frac{\lambda_{d}eF}{g\sqrt{h}}.\qquad% \qquad(13)
  17. d F = λ d e F g ϕ e F g ϕ 0.09759678 eV 1 eV ϕ F 1 V nm - 1 . ( 14 ) d_{\mathrm{F}}=\frac{\lambda_{d}eF}{g\sqrt{\phi}}\approx\frac{eF}{g\sqrt{\phi}% }\approx 0.09759678\;\mathrm{eV}\,\cdot\sqrt{\frac{1\ \mathrm{eV}}{\phi}}\cdot% \frac{F}{1\ \mathrm{V}\ \mathrm{nm}^{-1}}.\qquad\qquad(14)
  18. D ( h , F ) P exp [ - ν ( h , F ) b h 3 / 2 F ] , ( 15 ) D(h,F)\approx P\exp\left[-\frac{\nu(h,F)bh^{3/2}}{F}\right],\qquad\qquad(15)
  19. ϵ n = ϵ - K p . . ( 16 ) \;\epsilon_{\mathrm{n}}=\epsilon-K_{\mathrm{p}}...........(16)
  20. z S f FD d ϵ d K p z_{\mathrm{S}}f_{\mathrm{FD}}\mathrm{d}{\it{\epsilon}}\mathrm{d}K_{\mathrm{p}}
  21. z S = 4 π e m / h P 3 = 1.618311 × 10 14 A m - 2 eV - 2 , . . ( 17 ) z_{\mathrm{S}}=4\mathrm{\pi}em/h_{\mathrm{P}}^{3}=1.618311\times 10^{14}\,\rm{% A}\,m^{-2}\,eV^{-2},...........(17)
  22. f FD f_{\mathrm{FD}}
  23. f FD ( ϵ ) = 1 / [ 1 + exp ( ϵ / k B T ) ] , . . ( 18 ) \,f_{\mathrm{FD}}(\epsilon)=1/[1+\mathrm{exp}(\epsilon/k_{\mathrm{B}}T)],.....% ......(18)
  24. h = ϕ - ϵ + K p . . ( 19 a ) \,h=\phi-\epsilon+K_{\mathrm{p}}...........(19a)
  25. D ( h , F ) D F exp ( ϵ / d F ) exp ( - K p / d F ) . ( 19 b ) , D(h,F)\approx D_{\mathrm{F}}\;\mathrm{exp}(\epsilon/d_{\mathrm{F}})\;\mathrm{% exp}(-K_{\mathrm{p}}/d_{\mathrm{F}})..........(19b),
  26. z S f FD D d ϵ d K p z_{\mathrm{S}}f_{\mathrm{FD}}D\mathrm{d}{\it{\epsilon}}\mathrm{d}K_{\mathrm{p}}
  27. j ( ϵ ) d ϵ = z S f FD [ D d K p ] d ϵ = z S f FD D F exp ( ϵ / d F ) [ 0 exp ( - K p / d F ) d K p ] d ϵ . . ( 20 ) j(\epsilon)\mathrm{d}\epsilon=z_{\mathrm{S}}f_{\mathrm{FD}}\left[\int D\mathrm% {d}K_{\mathrm{p}}\right]\mathrm{d}\epsilon=z_{\mathrm{S}}f_{\mathrm{FD}}D_{% \mathrm{F}}\mathrm{exp}(\epsilon/d_{\mathrm{F}})\left[\int_{0}^{\infty}\mathrm% {exp}(-K_{\mathrm{p}}/d_{\mathrm{F}})\;\mathrm{d}K_{\mathrm{p}}\right]\mathrm{% d}\epsilon...........(20)
  28. j ( ϵ ) = z S d F D F f FD ( ϵ ) exp ( ϵ / d F ) = j F f FD ( ϵ ) exp ( ϵ / d F ) , . . ( 21 ) \,j(\epsilon)=z_{\mathrm{S}}d_{\mathrm{F}}D_{\mathrm{F}}f_{\mathrm{FD}}(% \epsilon)\mathrm{exp}(\epsilon/d_{\mathrm{F}})=j_{\mathrm{F}}f_{\mathrm{FD}}(% \epsilon)\mathrm{exp}(\epsilon/d_{\mathrm{F}}),...........(21)
  29. d F d_{\mathrm{F}}
  30. D F D_{\mathrm{F}}
  31. j F [ = z S d F D F ] j_{\mathrm{F}}[\,=z_{\mathrm{S}}d_{\mathrm{F}}D_{\mathrm{F}}]
  32. j F j_{\mathrm{F}}
  33. f FD ( ϵ ) f_{\mathrm{FD}}(\epsilon)
  34. FWHM = d F ln ( 2 ) 0.693 d F . . ( 22 ) \mathrm{FWHM}\,=d_{\mathrm{F}}\mathrm{ln}(2)\approx 0.693\,d_{\mathrm{F}}.....% ......(22)
  35. J 0 = z S d F D F - 0 exp ( ϵ / d F ) d ϵ = z S d F 2 D F = Z F D F , . ( 23 ) J_{0}=z_{\mathrm{S}}d_{\mathrm{F}}D_{\mathrm{F}}\int_{-\infty}^{0}\mathrm{exp}% (\epsilon/d_{\mathrm{F}})\;\mathrm{d}\epsilon\;=\;z_{\mathrm{S}}{d_{\mathrm{F}% }}^{2}D_{\mathrm{F}}\;=\;Z_{\mathrm{F}}D_{\mathrm{F}},..........(23)
  36. Z F [ = z S d F 2 ] Z_{\mathrm{F}}\;[=z_{\mathrm{S}}{d_{\mathrm{F}}}^{2}]
  37. Z F = z S d F 2 = λ d 2 ( z S e 2 g - 2 ) ϕ - 1 F 2 = λ d 2 a ϕ - 1 F 2 , . ( 24 ) Z_{\mathrm{F}}=z_{\mathrm{S}}{d_{\mathrm{F}}}^{2}={\lambda_{d}}^{2}(z_{\mathrm% {S}}e^{2}g^{-2})\phi^{-1}F^{2}={\lambda_{d}}^{2}a\phi^{-1}F^{2},..........(24)
  38. a = z S e 2 g - 2 = e 3 / 8 π h P 1.541434 × 10 - 6 A eV V - 2 . . ( 25 ) a=z_{\mathrm{S}}e^{2}g^{-2}=e^{3}/8\pi h_{\mathrm{P}}\approx\;1.541434\times 1% 0^{-6}\;\mathrm{A\;eV}\;{\mathrm{V}}^{-2}...........(25)
  39. J = J 0 - exp ( ϵ / d F ) [ 1 + exp [ ( ϵ / d F ) ( d F / k B T ) ] ] d ( ϵ / d F ) = λ T J 0 , . ( 26 ) J=J_{0}\int_{-\infty}^{\infty}\frac{\mathrm{exp}(\epsilon/d_{\mathrm{F}})}{[1+% \mathrm{exp}[(\epsilon/d_{\mathrm{F}})(d_{\mathrm{F}}/k_{\mathrm{B}}T)]]}% \mathrm{d}(\epsilon/d_{\mathrm{F}})=\lambda_{T}J_{0},..........(26)
  40. w = d F / k B T w=d_{\mathrm{F}}/k_{\mathrm{B}}T
  41. x = ϵ / d F x=\epsilon/d_{\mathrm{F}}
  42. u = exp ( x ) u=\mathrm{exp}(x)
  43. - [ e x / ( 1 + e w x ) ] d x = 0 [ u / ( 1 + w u ) ] d u = ( π / w ) / sin ( π / w ) . . ( 27 ) \int_{-\infty}^{\infty}[{\mathrm{e}}^{x}/(1+{\mathrm{e}}^{wx})]\mathrm{d}x=% \int_{0}^{\infty}[u/(1+wu)]\mathrm{d}u=(\pi/w)/\mathrm{sin}(\pi/w)...........(% 27)
  44. λ T = ( π k B T / d F ) / sin ( π k B T / d F ) 1 + ( 1 / 6 ) ( π k B T / d F ) 2 , . ( 28 ) \lambda_{T}=(\pi k_{\mathrm{B}}T/d_{\mathrm{F}})/\mathrm{sin}(\pi k_{\mathrm{B% }}T/d_{\mathrm{F}})\approx 1+(1/6){(\pi k_{\mathrm{B}}T/d_{\mathrm{F}})}^{2},.% .........(28)
  45. ν \nu
  46. J = λ Z a ϕ - 1 F 2 P F exp [ - ν F b ϕ 3 / 2 / F ] , . ( 29 ) J\;=\lambda_{Z}a\phi^{-1}F^{2}P_{\mathrm{F}}\mathrm{exp}[-\nu_{\mathrm{F}}b% \phi^{3/2}/F],..........(29)
  47. ν F {\nu}_{\mathrm{F}}
  48. ν F {\nu}_{\mathrm{F}}
  49. ν F {\nu}_{\mathrm{F}}
  50. ν F {\nu}_{\mathrm{F}}
  51. ν F {\nu}_{\mathrm{F}}
  52. ν F {\nu}_{\mathrm{F}}
  53. J = a ϕ - 1 F 2 exp [ - v ( f ) b ϕ 3 / 2 / F ] , . ( 30 a ) J=\;a{\phi^{-1}}F^{2}\mathrm{exp}[-v(f)\;b\phi^{3/2}/F],..........(30a)
  54. a 1.541434 × 10 - 6 A eV V - 2 ; b 6.830890 eV - 3 / 2 V nm - 1 , . ( 30 b ) a\approx\;1.541434\times 10^{-6}\;\mathrm{A\;eV}\;{\mathrm{V}}^{-2};\;\;\;\;\;% b\approx 6.830890\;{\mathrm{eV}}^{-3/2}\;\mathrm{V}\;{\mathrm{nm}}^{-1},......% ....(30b)
  55. v ( f ) 1 - f + ( 1 / 6 ) f ln f . ( 30 c ) v(f)\approx 1-f+(1/6)f\mathrm{ln}f..........(30c)
  56. f = F / F ϕ = ( e 3 / 4 π ϵ 0 ) ( F / ϕ 2 ) = ( 1.439964 eV 2 V - 1 nm ) ( F / ϕ 2 ) . . ( 30 d ) f=\;F/F_{\phi}=(e^{3}/4\pi\epsilon_{0})(F/{\phi}^{2})=(1.439964\;{\mathrm{eV}}% ^{2}\;{\mathrm{V}}^{-1}\;\mathrm{nm})(F/{\phi}^{2})...........(30d)
  57. F = β V , . ( 31 ) F=\;\beta V,..........(31)
  58. i = A r J r = J d A , . ( 32 ) i=A_{\mathrm{r}}J_{\mathrm{r}}=\int J\mathrm{d}A,..........(32)
  59. i = A r a ϕ - 1 β 2 V 2 exp [ - v ( f ) b ϕ 3 / 2 / β V ] , . ( 33 ) i=\;A_{\mathrm{r}}a{\phi^{-1}}{\beta}^{2}V^{2}\mathrm{exp}[-v(f)\;b\phi^{3/2}/% \beta V],..........(33)
  60. A r = α r A M . . ( 34 ) A_{\mathrm{r}}=\;\alpha_{\mathrm{r}}A_{\mathrm{M}}...........(34)
  61. J M = i / A M = α r ( i / A r ) = α r J r . . ( 35 ) J_{\mathrm{M}}=\;i/A_{\mathrm{M}}=\alpha_{\mathrm{r}}(i/A_{\mathrm{r}})=\alpha% _{\mathrm{r}}J_{\mathrm{r}}...........(35)
  62. J M = α r a ϕ - 1 F 2 exp [ - v ( f ) b ϕ 3 / 2 / F ] , . ( 36 ) J_{\mathrm{M}}=\alpha_{\mathrm{r}}a{\phi^{-1}}F^{2}\mathrm{exp}[-v(f)\;b\phi^{% 3/2}/F],..........(36)
  63. i = α r A M a ϕ - 1 β 2 V 2 exp [ - v ( f ) b ϕ 3 / 2 / β V ] , . ( 37 ) i=\;\alpha_{\mathrm{r}}A_{\mathrm{M}}a{\phi^{-1}}{\beta}^{2}V^{2}\mathrm{exp}[% -v(f)\;b\phi^{3/2}/\beta V],..........(37)
  64. F M = β M V . . ( 38 ) F_{\mathrm{M}}=\;\beta_{\mathrm{M}}V...........(38)
  65. γ = F r / F M = β r / β M . . ( 39 ) \gamma=\;F_{\mathrm{r}}/F_{\mathrm{M}}=\beta_{\mathrm{r}}/\beta_{\mathrm{M}}..% .........(39)
  66. F = γ F M = β V . ( 40 ) ; β = β M γ . ( 41 ) ; F=\;\gamma F_{\mathrm{M}}=\beta V..........(40);\;\;\;\;\;\;\;\;\;\beta=\;% \beta_{\mathrm{M}}\gamma..........(41);
  67. γ F M \gamma F_{\mathrm{M}}
  68. i = C V κ exp [ - B / V ] , . ( 42 ) i=\;CV^{\kappa}\mathrm{exp}[-B/V],..........(42)
  69. - dln i / d ( 1 / V ) = κ V + B , . ( 43 ) -\mathrm{dln}i/\mathrm{d}(1/V)=\;\kappa V+B,..........(43)
  70. η = b ϕ 3 / 2 / F ϕ = ( b e 3 / 4 π ϵ 0 ) ϕ - 1 / 2 9.836239 ( eV / ϕ ) 1 / 2 . . ( 34 ) \eta=b\phi^{3/2}/F_{\phi}=\;(be^{3}/4\pi\epsilon_{0}){\phi}^{-1/2}\approx 9.83% 6239\;\;(\mathrm{eV}/\phi)^{1/2}...........(34)
  71. exp [ - v ( f ) b ϕ 3 / 2 / F ] = exp [ - v ( f ) η / f ] e η f - η / 6 exp [ - η / f ] = e η f - η / 6 exp [ - b ϕ 3 / 2 / F ] . . ( 45 ) \mathrm{exp}[-v(f)\;b{\phi}^{3/2}/F]\;=\;\mathrm{exp}[-v(f)\;\eta/f]\;\approx% \;{\mathrm{e}}^{\eta}f^{-\eta/6}\mathrm{exp}[-\eta/f]\;=\;{\mathrm{e}}^{\eta}f% ^{-\eta/6}\mathrm{exp}[-b{\phi}^{3/2}/F]...........(45)
  72. κ 2 - η / 6 = 2 - 0.77 = 1.23........... ( 46 ) \kappa\approx 2-\eta/6=2-0.77=1.23...........(46)
  73. S FN = - b ϕ 3 / 2 / β . . ( 47 ) S_{\mathrm{FN}}=\;-b{\phi}^{3/2}/\beta...........(47)
  74. γ = β W . . ( 48 ) \gamma=\;\beta W...........(48)
  75. S FN = - σ FN b ϕ 3 / 2 / β . . ( 49 ) S_{\mathrm{FN}}=\;-\sigma_{\mathrm{FN}}b{\phi}^{3/2}/\beta...........(49)
  76. ν F \nu_{\mathrm{F}}
  77. B = b H 3 / 2 / β . . ( 50 ) B=\;bH^{3/2}/\beta...........(50)
  78. B = - dln ( i ) / d ( 1 / V ) - κ ( 1 / V ) . . ( 51 ) B=\;-\mathrm{dln}(i)/\mathrm{d}(1/V)-\kappa(1/V)...........(51)

Field_of_view.html

  1. A A
  2. L L
  3. M M
  4. A 360 2 π L 3000 0.0191 × L A\approx{360^{\circ}\over 2\pi}\cdot{L\over 3000}\approx 0.0191\times L
  5. A 360 2 π M 1000 0.0573 × M A\approx{360^{\circ}\over 2\pi}\cdot{M\over 1000}\approx 0.0573\times M
  6. L 2 π 3000 360 A 52.36 × A L\approx{2\pi\cdot 3000\over 360^{\circ}}\cdot A\approx 52.36\times A
  7. M 2 π 1000 360 A 17.45 × A M\approx{2\pi\cdot 1000\over 360^{\circ}}\cdot A\approx 17.45\times A

Figure_of_the_Earth.html

  1. a a
  2. f f
  3. r p r_{p}
  4. r p = a 2 b , r_{p}=\frac{a^{2}}{b},
  5. r e r_{e}
  6. r e = b 2 a r_{e}=\frac{b^{2}}{a}
  7. a a
  8. b b
  9. C 22 , S 22 C_{22},S_{22}
  10. C 30 C_{30}

Film_score.html

  1. 60 b p m ( x ) = W \frac{60}{bpm}(x)=W
  2. 60 88 ( 4 ) = 2.72 \frac{60}{88}(4)=2.72
  3. b p m ( s p ) 60 + 1 = B \frac{bpm(sp)}{60}+1=B

Film_speed.html

  1. 100 19 3 = 2.06914... 2 \sqrt[19]{100}^{3}=2.06914...\approx 2
  2. log 10 ( 2 ) = 0.30103... 3 / 10 \log_{10}{(2)}=0.30103...\approx 3/10
  3. S = 10 log S + 1 S^{\circ}=10\log S+1
  4. S = 10 ( S - 1 ) / 10 S=10^{\left({S^{\circ}-1}\right)/10}
  5. S = 0.8 lx⋅s H m S=\frac{0.8\;\,\text{lx⋅s}}{H_{\mathrm{m}}}
  6. H = q L t N 2 , H=\frac{qLt}{N^{2}},
  7. q = π 4 T v ( θ ) cos 4 θ q=\frac{\pi}{4}T\,v(\theta)\,\cos^{4}\theta
  8. S sat = 78 lx⋅s H sat , S_{\mathrm{sat}}=\frac{78\;\,\text{lx⋅s}}{H_{\mathrm{sat}}},
  9. H sat H_{\mathrm{sat}}
  10. S sos = 10 lx⋅s H sos , S_{\mathrm{sos}}=\frac{10\;\,\text{lx⋅s}}{H_{\mathrm{sos}}},
  11. H sos H_{\mathrm{sos}}
  12. S 40 : 1 = 107 S_{40:1}=107
  13. S 10 : 1 = 1688 S_{10:1}=1688
  14. S sat = 49 S_{\mathrm{sat}}=49
  15. S 40 : 1 = 40 S_{40:1}=40
  16. S 10 : 1 = 800 S_{10:1}=800
  17. S sat = 200 S_{\mathrm{sat}}=200

Filter_design.html

  1. σ s 2 \sigma^{2}_{s}
  2. σ f 2 \sigma^{2}_{f}
  3. σ r 2 \sigma^{2}_{r}
  4. σ r 2 \sigma^{2}_{r}
  5. σ s 2 \sigma^{2}_{s}
  6. σ f 2 \sigma^{2}_{f}
  7. σ r > σ f \sigma_{r}>\sigma_{f}
  8. F ( ω ) F(\omega)
  9. n 0 n\geq 0
  10. t - n - 1 t^{-n-1}
  11. F I ( ω ) F_{I}(\omega)
  12. W ( ω ) W(\omega)
  13. x k x_{k}
  14. ε \varepsilon
  15. ε = W ( F I - { f } ) 2 \varepsilon=\|W\cdot(F_{I}-\mathcal{F}\{f\})\|^{2}
  16. f ( x ) f(x)
  17. \mathcal{F}
  18. L 2 L^{2}
  19. ε \varepsilon
  20. F I F_{I}
  21. { f } \mathcal{F}\{f\}
  22. W W
  23. f ( x ) f(x)
  24. ε \varepsilon
  25. L 2 L^{2}

Finite_geometry.html

  1. \ell
  2. p p
  3. \ell
  4. \ell^{\prime}
  5. p p
  6. = . \ell\cap\ell^{\prime}=\varnothing.
  7. = X - 1 X 0 X n = P . \varnothing=X_{-1}\subset X_{0}\subset\cdots\subset X_{n}=P.
  8. ( n + 1 k + 1 ) q = i = 0 k q n + 1 - i - 1 q i + 1 - 1 , {{n+1}\choose{k+1}}_{q}=\prod_{i=0}^{k}\frac{q^{n+1-i}-1}{q^{i+1}-1},

First_law_of_thermodynamics.html

  1. Δ U \Delta U
  2. Δ U = Q + W \Delta U=Q\,+\,W
  3. Q Q
  4. W W
  5. - P d V -PdV
  6. P P
  7. d V dV
  8. d V dV
  9. d U = δ Q + P d V \mathrm{d}U=\delta Q+P\,\mathrm{d}V
  10. δ Q δQ
  11. U U
  12. Q Q
  13. W W
  14. Δ U ΔU
  15. E tot = E kin + E pot + U . E^{\mathrm{tot}}=E^{\mathrm{kin}}+E^{\mathrm{pot}}+U\,\,.
  16. Δ E tot = Δ E kin + Δ E pot + Δ U . \Delta E^{\mathrm{tot}}=\Delta E^{\mathrm{kin}}+\Delta E^{\mathrm{pot}}+\Delta U% \,\,.
  17. Δ E tot = Q + W . \Delta E^{\mathrm{tot}}=Q+W\,\,.
  18. Q Q
  19. W W
  20. W W
  21. U U
  22. U U
  23. U U
  24. U U
  25. O O
  26. U ( O ) U(O)
  27. A A
  28. U ( A ) U(A)
  29. A A
  30. O O
  31. U ( A ) = U ( O ) - W O A adiabatic or U ( O ) = U ( A ) - W A O adiabatic . U(A)=U(O)-W^{\mathrm{adiabatic}}_{O\to A}\,\,\mathrm{or}\,\,U(O)=U(A)-W^{% \mathrm{adiabatic}}_{A\to O}\,.
  32. adiabatic , O A \mathrm{adiabatic},\,O\to A
  33. adiabatic , A O \mathrm{adiabatic},\,{A\to O}\,
  34. ( 1 ) W adiabatic, quasi-static A O = - W adiabatic, quasi-static O A . (1)\,\,\,\,\,\,\,W\text{adiabatic, quasi-static}_{A\to O}=-W\text{adiabatic, % quasi-static}_{O\to A}\,.
  35. A A
  36. B B
  37. O O
  38. - W A B adiabatic , quasi - static = - W A O adiabatic , quasi - static - W O B adiabatic , quasi - static = W O A adiabatic , quasi - static - W O B adiabatic , quasi - static = - U ( A ) + U ( B ) = Δ U -W^{\mathrm{adiabatic,\,quasi-static}}_{A\to B}=-W^{\mathrm{adiabatic,\,quasi-% static}}_{A\to O}-W^{\mathrm{adiabatic,\,quasi-static}}_{O\to B}=W^{\mathrm{% adiabatic,\,quasi-static}}_{O\to A}-W^{\mathrm{adiabatic,\,quasi-static}}_{O% \to B}=-U(A)+U(B)=\Delta U
  39. U U
  40. Δ U \Delta U
  41. Q A B adynamic = Δ U . Q^{\mathrm{adynamic}}_{A\to B}=\Delta U\,.
  42. W A B path P 0 , reversible W^{\mathrm{path}\,P_{0},\,\mathrm{reversible}}_{A\to B}
  43. Q A B path P 0 , reversible Q^{\mathrm{path}\,P_{0},\,\mathrm{reversible}}_{A\to B}
  44. P 0 P_{0}
  45. - W A B path P 0 , reversible + Q A B path P 0 , reversible = Δ U . -W^{\mathrm{path}\,P_{0},\,\mathrm{reversible}}_{A\to B}+Q^{\mathrm{path}\,P_{% 0},\,\mathrm{reversible}}_{A\to B}=\Delta U\,.
  46. Δ U = 0 \Delta U=0\,
  47. W A B path P 1 , irreversible W^{\mathrm{path}\,P_{1},\,\mathrm{irreversible}}_{A\to B}
  48. Q A B path P 1 , irreversible Q^{\mathrm{path}\,P_{1},\,\mathrm{irreversible}}_{A\to B}
  49. P 1 P_{1}
  50. - W A B path P 1 , irreversible + Q A B path P 1 , irreversible = Δ U . -W^{\mathrm{path}\,P_{1},\,\mathrm{irreversible}}_{A\to B}+Q^{\mathrm{path}\,P% _{1},\,\mathrm{irreversible}}_{A\to B}=\Delta U\,.
  51. U U
  52. Δ U \Delta U
  53. d U = T d S - P d V . dU=TdS-PdV.\,
  54. d U = T d S - P d V + i μ i d N i . dU=TdS-PdV+\sum_{i}\mu_{i}dN_{i}.\,
  55. d U = T d S - i X i d x i + j μ j d N j . dU=TdS-\sum_{i}X_{i}dx_{i}+\sum_{j}\mu_{j}dN_{j}.\,
  56. E E
  57. E = E kin + E pot + U E=E^{\mathrm{kin}}+E^{\mathrm{pot}}+U
  58. E kin E^{\mathrm{kin}}
  59. E pot E^{\mathrm{pot}}
  60. U U
  61. E 12 pot E^{\mathrm{pot}}_{12}
  62. E = E 1 kin + E 1 pot + U 1 + E 2 kin + E 2 pot + U 2 + E 12 pot E=E^{\mathrm{kin}}_{1}+E^{\mathrm{pot}}_{1}+U_{1}+E^{\mathrm{kin}}_{2}+E^{% \mathrm{pot}}_{2}+U_{2}+E^{\mathrm{pot}}_{12}
  63. E 12 pot E^{\mathrm{pot}}_{12}
  64. Δ U s + Δ U o = 0 , \Delta U_{s}+\Delta U_{o}=0\,,
  65. U U
  66. U U
  67. Δ N s + Δ N o = 0 , \Delta N_{s}+\Delta N_{o}=0\,,
  68. ( 2 ) Δ U 0 = Q - W - i = 1 m Δ U i (suitably defined surrounding subsystems, general process, quasi-static or irreversible), (2)\,\,\,\,\,\,\,\Delta U_{0}\,=\,Q\,-\,W\,-\,\sum_{i=1}^{m}\Delta U_{i}\,\,\,% \,\,\text{(suitably defined surrounding subsystems, general process, quasi-% static or irreversible),}
  69. i t h ith
  70. m m
  71. i t h ith
  72. Q Q
  73. W W
  74. ( 3 ) d U 0 = T d S - P d V + j = 1 n μ j d N j (3)\,\,\,\,\,\,\,\mathrm{d}U_{0}\,=\,T\,\mathrm{d}S\,-\,P\,\mathrm{d}V\,+\,% \sum_{j=1}^{n}\mu_{j}\,\mathrm{d}N_{j}
  75. δ Q = T d S and δ W = P d V (suitably defined surrounding subsystems, quasi-static transfers of energy) . \delta Q\,=\,T\,\mathrm{d}S\,\,\text{ and }\delta W\,=\,P\,\mathrm{d}V\,\,\,\,% \,\,\text{(suitably defined surrounding subsystems, quasi-static transfers of % energy)}\,.
  76. ( 4 ) d U 0 = δ Q - δ W + j = 1 n μ j d N j (suitably defined surrounding subsystems, quasi-static transfers) . (4)\,\,\,\,\,\,\,\mathrm{d}U_{0}\,=\,\delta Q\,-\,\delta W\,+\,\sum_{j=1}^{n}% \mu_{j}\,\mathrm{d}N_{j}\,\,\,\,\,\,\,\text{(suitably defined surrounding % subsystems, quasi-static transfers)}\,.
  77. ρ u 𝐯 ρu\mathbf{v}
  78. 𝐖 \mathbf{W}
  79. 𝐣 U U = ρ u 𝐯 + 𝐖 \mathbf{j}UU=ρu\mathbf{v}+\mathbf{W}
  80. u u
  81. E E
  82. e e
  83. U U
  84. I . I.

Fiscal_multiplier.html

  1. b C b_{C}
  2. b T b_{T}
  3. b M b_{M}
  4. Δ y \Delta y
  5. Δ a T \Delta a_{T}
  6. Δ b T \Delta b_{T}
  7. Δ G \Delta G
  8. Δ T \Delta T
  9. Δ I \Delta I
  10. Δ X \Delta X
  11. Δ y = Δ T * - b C 1 - b C ( 1 - b T ) + b M \Delta y=\Delta T*\frac{-b_{C}}{1-b_{C}(1-b_{T})+b_{M}}
  12. Δ b T \Delta b_{T}
  13. Δ a T \Delta a_{T}
  14. Δ y = Δ G * 1 1 - b C ( 1 - b T ) + b M \Delta y=\Delta G*\frac{1}{1-b_{C}(1-b_{T})+b_{M}}
  15. Δ y = Δ I * 1 1 - b C ( 1 - b T ) + b M \Delta y=\Delta I*\frac{1}{1-b_{C}(1-b_{T})+b_{M}}
  16. Δ y = Δ X * 1 1 - b C ( 1 - b T ) + b M \Delta y=\Delta X*\frac{1}{1-b_{C}(1-b_{T})+b_{M}}
  17. Δ y = Δ G * 1 \Delta y=\Delta G*1
  18. Δ y = Δ T * 1 \Delta y=\Delta T*1

Fitness_(biology).html

  1. w w
  2. w abs w_{\mathrm{abs}}
  3. w abs = N after N before {w_{\mathrm{abs}}}={{N_{\mathrm{after}}}\over{N_{\mathrm{before}}}}
  4. w = 1 w=1
  5. w a b s w ¯ a b s = w r e l w ¯ r e l {\frac{w_{abs}}{\overline{w}_{abs}}=\frac{w_{rel}}{\overline{w}_{rel}}}

Fixed-point_combinator.html

  1. y f = f ( y f ) y\ f=f\ (y\ f)
  2. x = y f x=y\ f
  3. x = f x x=f\ x
  4. f x = x 2 f\ x=x^{2}
  5. 0 = 0 2 0=0^{2}
  6. 1 = 1 2 1=1^{2}
  7. y f = f ( f ( y f ) ) y\ f=f\ (\ldots f\ (y\ f)\ldots)
  8. λ f . ( λ x . f ( x x ) ) ( λ x . f ( x x ) ) \lambda f.(\lambda x.f\ (x\ x))\ (\lambda x.f\ (x\ x))
  9. x 2 = - 1 x = - 1 x f x = - 1 x and Y f = x x^{2}=-1\Rightarrow x=\frac{-1}{x}\Rightarrow f\ x=\frac{-1}{x}\and Y\ f=x
  10. Y f Y\ f
  11. y f = f ( y f ) y\ f=f\ (y\ f)
  12. y f x = f ( y f ) x y\ f\ x=f\ (y\ f)\ x
  13. ( x f x = g x ) f = g (\forall xf\ x=g\ x)\equiv f=g
  14. F f n = ( IsZero n ) 1 ( multiply n ( f ( pred n ) ) ) F\ f\ n=(\operatorname{IsZero}\ n)\ 1\ (\operatorname{multiply}\ n\ (f\ (% \operatorname{pred}\ n)))
  15. y F n = F ( y F ) n y\ F\ n=F\ (y\ F)\ n
  16. y F n = ( IsZero n ) 1 ( multiply n ( ( y F ) ( pred n ) ) ) y\ F\ n=(\operatorname{IsZero}\ n)\ 1\ (\operatorname{multiply}\ n\ ((y\ F)\ (% \operatorname{pred}\ n)))
  17. y F = fact y\ F=\operatorname{fact}
  18. fact n = ( IsZero n ) 1 ( multiply n ( fact ( pred n ) ) ) \operatorname{fact}\ n=(\operatorname{IsZero}\ n)\ 1\ (\operatorname{multiply}% \ n\ (\operatorname{fact}\ (\operatorname{pred}\ n)))
  19. fact n = if n = 0 then 1 else n * fact ( n - 1 ) \operatorname{fact}\ n=\operatorname{if}n=0\operatorname{then}1\operatorname{% else}n*\operatorname{fact}\ (n-1)
  20. Y = λ f . ( λ x . f ( x x ) ) ( λ x . f ( x x ) ) Y=\lambda f.(\lambda x.f\ (x\ x))\ (\lambda x.f\ (x\ x))
  21. Y g Y\ g
  22. = ( λ f . ( λ x . f ( x x ) ) ( λ x . f ( x x ) ) ) g =(\lambda f.(\lambda x.f\ (x\ x))\ (\lambda x.f\ (x\ x)))\ g
  23. = ( λ x . g ( x x ) ) ( λ x . g ( x x ) ) =(\lambda x.g\ (x\ x))\ (\lambda x.g\ (x\ x))
  24. = g ( ( λ x . g ( x x ) ) ( λ x . g ( x x ) ) ) =g((\lambda x.g\ (x\ x))\ (\lambda x.g\ (x\ x)))
  25. = g ( Y g ) =g\ (Y\ g)
  26. Y g = g ( Y g ) = g ( g ( Y g ) ) = g ( g ( Y g ) ) Y\ g=g\ (Y\ g)=g\ (g\ (Y\ g))=g\ (\ldots g\ (Y\ g)\ldots)
  27. x = f x and y f = x x=f\ x\and y\ f=x
  28. ( x E and F ) let x : E in F (\exists xE\and F)\iff\operatorname{let}x:E\operatorname{in}F
  29. let x = f x in y f = x \operatorname{let}x=f\ x\operatorname{in}y\ f=x
  30. x FV ( E ) and x FV ( F ) let x : G in E F = E ( let x : G in F ) x\not\in\operatorname{FV}(E)\and x\in\operatorname{FV}(F)\to\operatorname{let}% x:G\operatorname{in}E\ F=E\ (\operatorname{let}x:G\operatorname{in}F)
  31. y f = let x = f x in x y\ f=\operatorname{let}x=f\ x\operatorname{in}x
  32. f x = y f = λ x . y f\ x=y\iff f=\lambda x.y
  33. y = λ f . let x = f x in x y=\lambda f.\operatorname{let}x=f\ x\operatorname{in}x
  34. λ f . let x = f x in x \lambda f.\operatorname{let}x=f\ x\operatorname{in}x
  35. z , y z = x \forall z,y\ z=x
  36. λ f . let y z = f ( y z ) in y z \lambda f.\operatorname{let}y\ z=f\ (y\ z)\operatorname{in}y\ z
  37. λ f . let y z = f ( y z ) in y y \lambda f.\operatorname{let}y\ z=f\ (y\ z)\operatorname{in}y\ y
  38. λ f . let y z = f ( z z ) in y y \lambda f.\operatorname{let}y\ z=f\ (z\ z)\operatorname{in}y\ y
  39. f x = y f = λ x . y f\ x=y\equiv f=\lambda x.y
  40. λ f . let y = λ z . f ( z z ) in y y \lambda f.\operatorname{let}y=\lambda z.f\ (z\ z)\operatorname{in}y\ y
  41. n F V ( E ) ( let n = E in L ( λ n . L ) E ) n\not\in FV(E)\to(\operatorname{let}n=E\operatorname{in}L\equiv(\lambda n.L)\ E)
  42. λ f . ( λ y . y y ) ( λ z . f ( z z ) ) \lambda f.(\lambda y.y\ y)\ (\lambda z.f\ (z\ z))
  43. λ f . ( λ z . f ( z z ) ) ( λ z . f ( z z ) ) \lambda f.(\lambda z.f\ (z\ z))\ (\lambda z.f\ (z\ z))
  44. f ( f f ( fix f ) ) ) x f\ (f...f\ (\operatorname{fix}\ f)...))\ x
  45. x x x~{}x
  46. Γ x : t 1 t 2 Γ x : t 1 Γ x x : t 2 {\Gamma\vdash x\!:\!t_{1}\to t_{2}\quad\Gamma\vdash x\!:\!t_{1}}\over{\Gamma% \vdash x~{}x\!:\!t_{2}}
  47. x x
  48. t 1 = t 1 t 2 t_{1}=t_{1}\to t_{2}
  49. x f x = x \forall xf\ x=x
  50. x f f x = f x \forall xf\ f\ x=f\ x

Fizeau–Foucault_apparatus.html

  1. θ = 2 h ω c = ω t . \theta=\tfrac{2h\omega}{c}=\omega t\ .
  2. c = 2 ω h θ . c=\tfrac{2\omega h}{\theta}\ .

Flash_(photography).html

  1. [ u r a d i c a l , u l e s s t h a n v a r > s / < v a r > t < / v a r > ] [u^{\prime}radical^{\prime},u^{\prime}\\ lessthanvar>s/<var>t</var>^{\prime}]

Flash_evaporation.html

  1. X = H u L - H d L H d V - H d L X=\frac{H_{u}^{L}-H_{d}^{L}}{H_{d}^{V}-H_{d}^{L}}
  2. X X
  3. H u L H_{u}^{L}
  4. H d V H_{d}^{V}
  5. H d L H_{d}^{L}
  6. X = c p ( T u - T d ) H v X=\frac{c_{p}(T_{u}-T_{d})}{H_{v}}
  7. X X
  8. c p c_{p}
  9. T u T_{u}
  10. T d T_{d}
  11. H v H_{v}
  12. i z i ( K i - 1 ) 1 + β ( K i - 1 ) = 0 \sum_{i}\frac{z_{i}\,(K_{i}-1)}{1+\beta\,(K_{i}-1)}=0
  13. y i = K i x i y_{i}=K_{i}\,x_{i}
  14. x i \displaystyle x_{i}
  15. 1 1 - K max = β min < β < β max = 1 1 - K min \frac{1}{1-K\text{max}}=\beta\text{min}<\beta<\beta\text{max}=\frac{1}{1-K% \text{min}}

Flexible_AC_transmission_system.html

  1. P = ( E V X ) sin ( δ ) P=\left(\frac{EV}{X}\right)\sin(\delta)
  2. δ = \delta=
  3. δ \delta
  4. V s ¯ = V cos ( δ 2 ) + j V sin ( δ 2 ) V r ¯ = V cos ( δ 2 ) - j V sin ( δ 2 ) I ¯ = V s ¯ - V r ¯ j X = 2 V sin ( δ 2 ) X \begin{aligned}\displaystyle\underline{V_{s}}&\displaystyle=V\cos\left(\frac{% \delta}{2}\right)+jV\sin\left(\frac{\delta}{2}\right)\\ \displaystyle\underline{V_{r}}&\displaystyle=V\cos\left(\frac{\delta}{2}\right% )-jV\sin\left(\frac{\delta}{2}\right)\\ \displaystyle\underline{I}&\displaystyle=\frac{\underline{V_{s}}-\underline{V_% {r}}}{jX}=\frac{2V\sin{\left(\frac{\delta}{2}\right)}}{X}\end{aligned}
  5. P s = P r = P = V cos ( δ 2 ) 2 V sin ( δ 2 ) X = V 2 X sin ( δ ) P_{s}=P_{r}=P=V\cos\left(\frac{\delta}{2}\right)\cdot\frac{2V\sin{\left(\frac{% \delta}{2}\right)}}{X}=\frac{V^{2}}{X}\sin(\delta)
  6. Q s = - Q r = Q = V sin ( δ 2 ) 2 V sin ( δ 2 ) X = V 2 X ( 1 - cos δ ) Q_{s}=-Q_{r}=Q=V\sin\left(\frac{\delta}{2}\right)\cdot\frac{2V\sin\left(\frac{% \delta}{2}\right)}{X}=\frac{V^{2}}{X}(1-\cos\delta)
  7. δ \delta
  8. δ \delta
  9. P = V 2 X - X c sin ( δ ) Q = V 2 X - X c ( 1 - cos δ ) \begin{aligned}\displaystyle P&\displaystyle=\frac{V^{2}}{X-Xc}\sin(\delta)\\ \displaystyle Q&\displaystyle=\frac{V^{2}}{X-Xc}(1-\cos\delta)\end{aligned}
  10. P = 2 V 2 X sin ( δ 2 ) Q = 4 V 2 X [ 1 - cos ( δ / 2 2 ) ] \begin{aligned}\displaystyle P&\displaystyle=\frac{2V^{2}}{X}\sin\left(\frac{% \delta}{2}\right)\\ \displaystyle Q&\displaystyle=\frac{4V^{2}}{X}\left[1-\cos\left(\frac{\delta/2% }{2}\right)\right]\end{aligned}

Flight.html

  1. g 0 g_{0}

Flight_dynamics_(fixed-wing_aircraft).html

  1. 𝐅 A = Σ ( - Δ p 𝐧 + 𝐟 ) d σ \mathbf{F}_{A}=\int_{\Sigma}(-\Delta p\mathbf{n}+\mathbf{f})\,d\sigma
  2. Δ p \Delta p\equiv
  3. 𝐧 \mathbf{n}\equiv
  4. 𝐟 \mathbf{f}\equiv
  5. Σ \Sigma\equiv
  6. 𝐅 A = - ( 𝐢 w D + 𝐣 w Q + 𝐤 w L ) \mathbf{F}_{A}=-(\mathbf{i}_{w}D+\mathbf{j}_{w}Q+\mathbf{k}_{w}L)
  7. D D\equiv
  8. Q Q\equiv
  9. L L\equiv
  10. q = 1 2 ρ V 2 \equiv q=\tfrac{1}{2}\,\rho\,V^{2}
  11. S \equiv S
  12. C p = p - p q \equiv C_{p}=\dfrac{p-p_{\infty}}{q}
  13. C f = f q \equiv C_{f}=\dfrac{f}{q}
  14. C d = D q S = - 1 S Σ [ ( - C p ) 𝐧 𝐢 𝐰 + C f 𝐭 𝐢 𝐰 ] d σ \equiv C_{d}=\dfrac{D}{qS}=-\dfrac{1}{S}\int_{\Sigma}[(-C_{p})\mathbf{n}% \bullet\mathbf{i_{w}}+C_{f}\mathbf{t}\bullet\mathbf{i_{w}}]\,d\sigma
  15. C Q = Q q S = - 1 S Σ [ ( - C p ) 𝐧 𝐣 𝐰 + C f 𝐭 𝐣 𝐰 ] d σ \equiv C_{Q}=\dfrac{Q}{qS}=-\dfrac{1}{S}\int_{\Sigma}[(-C_{p})\mathbf{n}% \bullet\mathbf{j_{w}}+C_{f}\mathbf{t}\bullet\mathbf{j_{w}}]\,d\sigma
  16. C L = L q S = - 1 S Σ [ ( - C p ) 𝐧 𝐤 𝐰 + C f 𝐭 𝐤 𝐰 ] d σ \equiv C_{L}=\dfrac{L}{qS}=-\dfrac{1}{S}\int_{\Sigma}[(-C_{p})\mathbf{n}% \bullet\mathbf{k_{w}}+C_{f}\mathbf{t}\bullet\mathbf{k_{w}}]\,d\sigma
  17. M = V a \equiv M=\dfrac{V}{a}
  18. R e = ρ V l μ \equiv Re=\dfrac{\rho Vl}{\mu}
  19. K n = λ l \equiv Kn=\dfrac{\lambda}{l}
  20. a = k R θ a=\sqrt{kR\theta}\equiv
  21. R R\equiv
  22. θ \theta\equiv
  23. λ = μ ρ π 2 R θ = M R e k π 2 \lambda=\dfrac{\mu}{\rho}\sqrt{\dfrac{\pi}{2R\theta}}=\dfrac{M}{Re}\sqrt{% \dfrac{k\pi}{2}}\equiv
  24. M R e 1 \dfrac{M}{Re}\ll 1
  25. M R e 1 \dfrac{M}{Re}\approx 1
  26. M R e 1 \dfrac{M}{Re}\gg 1
  27. 0 < M < 0.3 0<M<0.3
  28. 0.3 < M < 0.8 0.3<M<0.8
  29. 0.8 < M < 1.2 0.8<M<1.2
  30. 1.2 < M < 5 1.2<M<5
  31. 5 < M 5<M
  32. C p = C p ( α , M , R e , P ) C_{p}=C_{p}(\alpha,M,Re,P)
  33. C f = C f ( α , M , R e , P ) C_{f}=C_{f}(\alpha,M,Re,P)
  34. α \alpha\equiv
  35. P P\equiv
  36. { C D = C D ( α , M , R e ) C L = C L ( α , M , R e ) E = E ( α , M , R e ) = C L C D \begin{cases}C_{D}=C_{D}(\alpha,M,Re)\\ C_{L}=C_{L}(\alpha,M,Re)\\ E=E(\alpha,M,Re)=\dfrac{C_{L}}{C_{D}}\\ \end{cases}
  37. C D = C D ( C L , M , R e ) C_{D}=C_{D}(C_{L},M,Re)\equiv
  38. C D = C D f + C D p { C D f = D q S = - 1 S Σ C f 𝐭 𝐢 𝐰 d σ C D p = D q S = - 1 S Σ ( - C p ) 𝐧 𝐢 𝐰 d σ C_{D}=C_{Df}+C_{Dp}\begin{cases}C_{Df}=\dfrac{D}{qS}=-\dfrac{1}{S}\int_{\Sigma% }C_{f}\mathbf{t}\bullet\mathbf{i_{w}}\,d\sigma\\ C_{Dp}=\dfrac{D}{qS}=-\dfrac{1}{S}\int_{\Sigma}(-C_{p})\mathbf{n}\bullet% \mathbf{i_{w}}\,d\sigma\end{cases}
  39. C D = C D 0 + C D i { C D 0 = ( C D ) C L = 0 C D i C_{D}=C_{D0}+C_{Di}\begin{cases}C_{D0}=(C_{D})_{C_{L}=0}\\ C_{Di}\end{cases}
  40. C D i = k C L 2 C D = C D 0 + k C L 2 C_{Di}=kC_{L}^{2}\Rightarrow C_{D}=C_{D0}+kC_{L}^{2}
  41. E = C L C D 0 + k C L 2 { E m a x = 1 2 k C D 0 ( C L ) E m a x = C D 0 k ( C D i ) E m a x = C D 0 E=\dfrac{C_{L}}{C_{D0}+kC_{L}^{2}}\Rightarrow\begin{cases}E_{max}=\dfrac{1}{2% \sqrt{kC_{D0}}}\\ (C_{L})_{Emax}=\sqrt{\dfrac{C_{D0}}{k}}\\ (C_{Di})_{Emax}=C_{D0}\end{cases}
  42. C D m i n = ( C D ) C L = 0 = C D 0 C_{Dmin}=(C_{D})_{CL=0}=C_{D0}
  43. C D m i n = C D M ( C D ) C L = 0 C_{Dmin}=C_{DM}\neq(C_{D})_{CL=0}
  44. C D = C D M + k ( C L - C L M ) 2 C_{D}=C_{DM}+k(C_{L}-C_{LM})^{2}
  45. C p = C p 0 | 1 - M 2 | C_{p}=\frac{C_{p0}}{\sqrt{|1-{M_{\infty}}^{2}|}}
  46. θ \theta
  47. α \alpha
  48. θ - α \theta-\alpha
  49. u f = U cos ( θ - α ) u_{f}=U\cos(\theta-\alpha)
  50. w f = U sin ( θ - α ) w_{f}=U\sin(\theta-\alpha)
  51. u f u_{f}
  52. w f w_{f}
  53. X f = m d u f d t = m d U d t cos ( θ - α ) - m U d ( θ - α ) d t sin ( θ - α ) X_{f}=m\frac{du_{f}}{dt}=m\frac{dU}{dt}\cos(\theta-\alpha)-mU\frac{d(\theta-% \alpha)}{dt}\sin(\theta-\alpha)
  54. Z f = m d w f d t = m d U d t sin ( θ - α ) + m U d ( θ - α ) d t cos ( θ - α ) Z_{f}=m\frac{dw_{f}}{dt}=m\frac{dU}{dt}\sin(\theta-\alpha)+mU\frac{d(\theta-% \alpha)}{dt}\cos(\theta-\alpha)
  55. m d U d t m\frac{dU}{dt}
  56. X f = - m U d ( θ - α ) d t sin ( θ - α ) X_{f}=-mU\frac{d(\theta-\alpha)}{dt}\sin(\theta-\alpha)
  57. Z f = m U d ( θ - α ) d t cos ( θ - α ) Z_{f}=mU\frac{d(\theta-\alpha)}{dt}\cos(\theta-\alpha)
  58. Z = - Z f cos ( θ - α ) + X f sin ( θ - α ) Z=-Z_{f}\cos(\theta-\alpha)+X_{f}\sin(\theta-\alpha)
  59. Z = - m U d ( θ - α ) d t Z=-mU\frac{d(\theta-\alpha)}{dt}
  60. M = B d 2 θ d t 2 M=B\frac{d^{2}\theta}{dt^{2}}
  61. d θ d t = q \frac{d\theta}{dt}=q
  62. d α d t = q + Z m U \frac{d\alpha}{dt}=q+\frac{Z}{mU}
  63. d q d t = M B \frac{dq}{dt}=\frac{M}{B}
  64. α \alpha
  65. Z α Z_{\alpha}
  66. Z q Z_{q}
  67. Z α Z_{\alpha}
  68. M α M_{\alpha}
  69. M q M_{q}
  70. M α ˙ M_{\dot{\alpha}}
  71. M α ˙ M_{\dot{\alpha}}
  72. d α d t = ( 1 + Z q m U ) q + Z α m U α \frac{d\alpha}{dt}=\left(1+\frac{Z_{q}}{mU}\right)q+\frac{Z_{\alpha}}{mU}\alpha
  73. d q d t = M q B q + M α B α + M α ˙ B α ˙ \frac{dq}{dt}=\frac{M_{q}}{B}q+\frac{M_{\alpha}}{B}\alpha+\frac{M_{\dot{\alpha% }}}{B}\dot{\alpha}
  74. α \alpha
  75. d 2 α d t 2 - ( Z α m U + M q B + ( 1 + Z q m U ) M α ˙ B ) d α d t + ( Z α m U M q B - M α B ( 1 + Z q m U ) ) α = 0 \frac{d^{2}\alpha}{dt^{2}}-\left(\frac{Z_{\alpha}}{mU}+\frac{M_{q}}{B}+(1+% \frac{Z_{q}}{mU})\frac{M_{\dot{\alpha}}}{B}\right)\frac{d\alpha}{dt}+\left(% \frac{Z_{\alpha}}{mU}\frac{M_{q}}{B}-\frac{M_{\alpha}}{B}(1+\frac{Z_{q}}{mU})% \right)\alpha=0
  76. Z q m U \frac{Z_{q}}{mU}
  77. α \alpha
  78. M α < Z α m U M q M_{\alpha}<\frac{Z_{\alpha}}{mU}M_{q}
  79. M α M_{\alpha}
  80. γ \gamma
  81. m U d γ d t = - Z mU\frac{d\gamma}{dt}=-Z
  82. m d u d t = X - m g γ m\frac{du}{dt}=X-mg\gamma
  83. X u X_{u}
  84. Z u Z_{u}
  85. X u X_{u}
  86. Z u Z_{u}
  87. m U d γ d t = - Z u u mU\frac{d\gamma}{dt}=-Z_{u}u
  88. m d u d t = X u u - m g γ m\frac{du}{dt}=X_{u}u-mg\gamma
  89. d 2 u d t 2 - X u m d u d t - Z u g m U u = 0 \frac{d^{2}u}{dt^{2}}-\frac{X_{u}}{m}\frac{du}{dt}-\frac{Z_{u}g}{mU}u=0
  90. Z = 1 2 ρ U 2 c L S w = W Z=\frac{1}{2}\rho U^{2}c_{L}S_{w}=W
  91. ρ \rho
  92. S w S_{w}
  93. c L c_{L}
  94. Z u = 2 W U = 2 m g U Z_{u}=\frac{2W}{U}=\frac{2mg}{U}
  95. 2 π T = 2 g 2 U 2 \frac{2\pi}{T}=\sqrt{\frac{2g^{2}}{U^{2}}}
  96. T = 2 π U 2 g T=\frac{2\pi U}{\sqrt{2}g}
  97. d β d t = Y m U - r \frac{d\beta}{dt}=\frac{Y}{mU}-r
  98. β \beta
  99. v = - p z + x r v=-pz+xr
  100. d v d t = - d p d t z + d r d t x \frac{dv}{dt}=-\frac{dp}{dt}z+\frac{dr}{dt}x
  101. δ m x d v d t = - d p d t x z δ m + d r d t x 2 δ m \delta mx\frac{dv}{dt}=-\frac{dp}{dt}xz\delta m+\frac{dr}{dt}x^{2}\delta m
  102. d r d t y 2 δ m \frac{dr}{dt}y^{2}\delta m
  103. N = - d p d t x z d m + d r d t x 2 + y 2 d m = - E d p d t + C d r d t N=-\frac{dp}{dt}\int xzdm+\frac{dr}{dt}\int x^{2}+y^{2}dm=-E\frac{dp}{dt}+C% \frac{dr}{dt}
  104. L = A d p d t - E d r d t L=A\frac{dp}{dt}-E\frac{dr}{dt}
  105. β \beta
  106. Y β Y_{\beta}
  107. Y β Y_{\beta}
  108. Y β Y_{\beta}
  109. Y p Y_{p}
  110. Y r Y_{r}
  111. N β N_{\beta}
  112. N β N_{\beta}
  113. N p N_{p}
  114. N p N_{p}
  115. N p N_{p}
  116. N r N_{r}
  117. N r N_{r}
  118. L β L_{\beta}
  119. L r L_{r}
  120. L p L_{p}
  121. L β L_{\beta}
  122. L r L_{r}
  123. d β d t = - r \frac{d\beta}{dt}=-r
  124. C d r d t - E d p d t = N β β - N r d β d t + N p p C\frac{dr}{dt}-E\frac{dp}{dt}=N_{\beta}\beta-N_{r}\frac{d\beta}{dt}+N_{p}p
  125. A d p d t - E d r d t = L β β - L r d β d t + L p p A\frac{dp}{dt}-E\frac{dr}{dt}=L_{\beta}\beta-L_{r}\frac{d\beta}{dt}+L_{p}p
  126. - C d 2 β d t 2 = N β β - N r d β d t + N p p -C\frac{d^{2}\beta}{dt^{2}}=N_{\beta}\beta-N_{r}\frac{d\beta}{dt}+N_{p}p
  127. E d 2 β d t 2 = L β β - L r d β d t + L p p E\frac{d^{2}\beta}{dt^{2}}=L_{\beta}\beta-L_{r}\frac{d\beta}{dt}+L_{p}p
  128. ( N p C E A - L p A ) d 2 β d t 2 + ( L p A N r C - N p C L r A ) d β d t - ( L p A N β C - L β A N p C ) β = 0 \left(\frac{N_{p}}{C}\frac{E}{A}-\frac{L_{p}}{A}\right)\frac{d^{2}\beta}{dt^{2% }}+\left(\frac{L_{p}}{A}\frac{N_{r}}{C}-\frac{N_{p}}{C}\frac{L_{r}}{A}\right)% \frac{d\beta}{dt}-\left(\frac{L_{p}}{A}\frac{N_{\beta}}{C}-\frac{L_{\beta}}{A}% \frac{N_{p}}{C}\right)\beta=0
  129. ϕ \phi
  130. d ϕ d t = p \frac{d\phi}{dt}=p
  131. ϕ \phi
  132. L p A N r C - N p C L r A N p C E A - L p A \frac{\frac{L_{p}}{A}\frac{N_{r}}{C}-\frac{N_{p}}{C}\frac{L_{r}}{A}}{\frac{N_{% p}}{C}\frac{E}{A}-\frac{L_{p}}{A}}
  133. L β A N p C - L p A N β C N p C E A - L p A \frac{\frac{L_{\beta}}{A}\frac{N_{p}}{C}-\frac{L_{p}}{A}\frac{N_{\beta}}{C}}{% \frac{N_{p}}{C}\frac{E}{A}-\frac{L_{p}}{A}}
  134. L p L_{p}
  135. - L p N β -L_{p}N_{\beta}
  136. L p L_{p}
  137. N β N_{\beta}
  138. L β L_{\beta}
  139. N p N_{p}
  140. L β L_{\beta}
  141. Y β Y_{\beta}
  142. A d p d t = L p p . A\frac{dp}{dt}=L_{p}p.
  143. L p L_{p}
  144. μ \mu
  145. ψ \psi
  146. d μ d t = Y m U + g U ϕ \frac{d\mu}{dt}=\frac{Y}{mU}+\frac{g}{U}\phi
  147. d μ d t = Y β m U β + Y r m U r + Y p m U p + g U ϕ \frac{d\mu}{dt}=\frac{Y_{\beta}}{mU}\beta+\frac{Y_{r}}{mU}r+\frac{Y_{p}}{mU}p+% \frac{g}{U}\phi
  148. Y r Y_{r}
  149. Y p Y_{p}
  150. N β β + N r d μ d t + N p p = 0 N_{\beta}\beta+N_{r}\frac{d\mu}{dt}+N_{p}p=0
  151. L β β + L r d μ d t + L p p = 0 L_{\beta}\beta+L_{r}\frac{d\mu}{dt}+L_{p}p=0
  152. β \beta
  153. β = ( L r N p - L p N r ) ( L p N β - N p L β ) d μ d t \beta=\frac{(L_{r}N_{p}-L_{p}N_{r})}{(L_{p}N_{\beta}-N_{p}L_{\beta})}\frac{d% \mu}{dt}
  154. p = ( L β N r - L r N β ) ( L p N β - N p L β ) d μ d t p=\frac{(L_{\beta}N_{r}-L_{r}N_{\beta})}{(L_{p}N_{\beta}-N_{p}L_{\beta})}\frac% {d\mu}{dt}
  155. d ϕ d t = m g ( L β N r - N β L r ) m U ( L p N β - N p L β ) - Y β ( L r N p - L p N r ) ϕ \frac{d\phi}{dt}=mg\frac{(L_{\beta}N_{r}-N_{\beta}L_{r})}{mU(L_{p}N_{\beta}-N_% {p}L_{\beta})-Y_{\beta}(L_{r}N_{p}-L_{p}N_{r})}\phi
  156. ϕ \phi
  157. L β N r > N β L r L_{\beta}N_{r}>N_{\beta}L_{r}

Flipped_SU(5).html

  1. S U ( 5 ) SU(5)
  2. S U ( 5 ) SU(5)
  3. U ( 1 ) U(1)
  4. S U ( 5 ) SU(5)
  5. N N
  6. S U ( 5 ) SU(5)
  7. 5 ¯ - 3 ( 3 ¯ , 1 ) - 2 3 ( 1 , 2 ) - 1 2 \bar{5}_{-3}\to(\bar{3},1)_{-\frac{2}{3}}\oplus(1,2)_{-\frac{1}{2}}
  8. 10 1 ( 3 , 2 ) 1 6 ( 3 ¯ , 1 ) 1 3 ( 1 , 1 ) 0 10_{1}\to(3,2)_{\frac{1}{6}}\oplus(\bar{3},1)_{\frac{1}{3}}\oplus(1,1)_{0}
  9. 1 5 ( 1 , 1 ) 1 1_{5}\to(1,1)_{1}
  10. 24 0 ( 8 , 1 ) 0 ( 1 , 3 ) 0 ( 1 , 1 ) 0 ( 3 , 2 ) 1 6 ( 3 ¯ , 2 ) - 1 6 24_{0}\to(8,1)_{0}\oplus(1,3)_{0}\oplus(1,1)_{0}\oplus(3,2)_{\frac{1}{6}}% \oplus(\bar{3},2)_{-\frac{1}{6}}
  11. S U ( 5 ) SU(5)
  12. S U ( 5 ) SU(5)
  13. 𝟏𝟎 \mathbf{10}
  14. 𝟓 \mathbf{5}
  15. S U ( 5 ) SU(5)
  16. S U ( 5 ) SU(5)
  17. S U ( 5 ) SU(5)
  18. ( 1 15 0 0 0 0 0 1 15 0 0 0 0 0 1 15 0 0 0 0 0 - 1 10 0 0 0 0 0 - 1 10 ) SU ( 5 ) , χ / 5. \begin{pmatrix}{1\over 15}&0&0&0&0\\ 0&{1\over 15}&0&0&0\\ 0&0&{1\over 15}&0&0\\ 0&0&0&-{1\over 10}&0\\ 0&0&0&0&-{1\over 10}\end{pmatrix}\in\,\text{SU}(5),\qquad\chi/5.
  19. π 2 ( [ S U ( 5 ) × U ( 1 ) χ ] / 𝐙 5 [ S U ( 3 ) × S U ( 2 ) × U ( 1 ) Y ] / 𝐙 6 ) = 0 \pi_{2}\left(\frac{[SU(5)\times U(1)_{\chi}]/\mathbf{Z}_{5}}{[SU(3)\times SU(2% )\times U(1)_{Y}]/\mathbf{Z}_{6}}\right)=0
  20. X X
  21. ( 3 , 2 ) 1 6 (3,2)_{\frac{1}{6}}
  22. S U ( 5 ) SU(5)
  23. N = 1 N=1
  24. 3 + 1 3+1
  25. N = 1 N=1
  26. 3 + 1 3+1
  27. S U ( 5 ) × U ( 1 ) < s u b > χ SU(5)×U(1)<sub>χ
  28. R R
  29. S S S 10 H 10 ¯ H S 10 H α β 10 ¯ H α β 10 H 10 H H d ϵ α β γ δ ϵ 10 H α β 10 H γ δ H d ϵ 10 ¯ H 10 ¯ H H u ϵ α β γ δ ϵ 10 ¯ H α β 10 ¯ H γ δ H u ϵ H d 1010 ϵ α β γ δ ϵ H d α 10 i β γ 10 j δ ϵ H d 5 ¯ 1 H d α 5 ¯ i α 1 j H u 10 5 ¯ H u α 10 i α β 5 ¯ j β 10 ¯ H 10 ϕ 10 ¯ H α β 10 i α β ϕ j \begin{matrix}S&S\\ S10_{H}\overline{10}_{H}&S10_{H}^{\alpha\beta}\overline{10}_{H\alpha\beta}\\ 10_{H}10_{H}H_{d}&\epsilon_{\alpha\beta\gamma\delta\epsilon}10_{H}^{\alpha% \beta}10_{H}^{\gamma\delta}H_{d}^{\epsilon}\\ \overline{10}_{H}\overline{10}_{H}H_{u}&\epsilon^{\alpha\beta\gamma\delta% \epsilon}\overline{10}_{H\alpha\beta}\overline{10}_{H\gamma\delta}H_{u\epsilon% }\\ H_{d}1010&\epsilon_{\alpha\beta\gamma\delta\epsilon}H_{d}^{\alpha}10_{i}^{% \beta\gamma}10_{j}^{\delta\epsilon}\\ H_{d}\bar{5}1&H_{d}^{\alpha}\bar{5}_{i\alpha}1_{j}\\ H_{u}10\bar{5}&H_{u\alpha}10_{i}^{\alpha\beta}\bar{5}_{j\beta}\\ \overline{10}_{H}10\phi&\overline{10}_{H\alpha\beta}10_{i}^{\alpha\beta}\phi_{% j}\\ \end{matrix}
  30. i i
  31. j j
  32. i i
  33. j j
  34. φ φ
  35. ( 10 ¯ H 10 ) ( 10 ¯ H 10 ) 10 ¯ H α β 10 i α β 10 ¯ H γ δ 10 j γ δ 10 ¯ H 10 10 ¯ H 10 10 ¯ H α β 10 i β γ 10 ¯ H γ δ 10 j δ α \begin{matrix}(\overline{10}_{H}10)(\overline{10}_{H}10)&\overline{10}_{H% \alpha\beta}10^{\alpha\beta}_{i}\overline{10}_{H\gamma\delta}10^{\gamma\delta}% _{j}\\ \overline{10}_{H}10\overline{10}_{H}10&\overline{10}_{H\alpha\beta}10^{\beta% \gamma}_{i}\overline{10}_{H\gamma\delta}10^{\delta\alpha}_{j}\end{matrix}
  36. S U ( 5 ) SU(5)

Flocking_(behavior).html

  1. O ( n 2 ) O(n^{2})
  2. O ( n k ) O(nk)
  3. O ( 1 ) O(1)

FLOPS.html

  1. FLOPS = sockets × cores socket × clock × FLOPs cycle \,\text{FLOPS}=\,\text{sockets}\times\frac{\,\text{cores}}{\,\text{socket}}% \times\,\text{clock}\times\frac{\,\text{FLOPs}}{\,\text{cycle}}

Flow_measurement.html

  1. Q Q
  2. m ˙ \dot{m}
  3. ρ \rho
  4. m ˙ = ρ * Q \dot{m}=\rho*Q
  5. 11 / 2 1{1}/{2}
  6. 11 / 2 1{1}/{2}
  7. V = D / t V=D/t
  8. D D
  9. t t
  10. Q = K H X Q=KH^{X}
  11. Q Q
  12. K K
  13. H H
  14. X X
  15. f = S V / L f=SV/L
  16. f f
  17. L L
  18. V V
  19. S S
  20. t u p t_{up}
  21. t d o w n t_{down}
  22. L L
  23. α \alpha
  24. v = L < m t p l > 2 sin ( α ) t u p - t d o w n t u p t d o w n v=\frac{L}{<}mtpl>{{2\;\sin\left(\alpha\right)}}\;\frac{{t_{up}-t_{down}}}{{t_% {up}\;t_{down}}}
  25. c = L 2 t u p + t d o w n t u p t d o w n c=\frac{L}{2}\;\frac{{t_{up}+t_{down}}}{{t_{up}\;t_{down}}}
  26. v v
  27. c c

Floyd–Warshall_algorithm.html

  1. R R
  2. w ( i , j ) w(i,j)
  3. shortestPath ( i , j , 0 ) = w ( i , j ) \textrm{shortestPath}(i,j,0)=w(i,j)
  4. shortestPath ( i , j , k + 1 ) = min ( shortestPath ( i , j , k ) , shortestPath ( i , k + 1 , k ) + shortestPath ( k + 1 , j , k ) ) \textrm{shortestPath}(i,j,k+1)=\min(\textrm{shortestPath}(i,j,k),\,\textrm{% shortestPath}(i,k+1,k)+\textrm{shortestPath}(k+1,j,k))
  5. O ( | V | | E | log | V | ) O(|V|\cdot|E|\log|V|)
  6. O ( | V | 3 ) O(|V|^{3})
  7. | E | |E|
  8. | V | 2 |V|^{2}

Flywheel.html

  1. E k = 1 2 I ω 2 E_{k}=\frac{1}{2}I\omega^{2}
  2. I I
  3. I = 1 2 m r 2 I=\frac{1}{2}mr^{2}
  4. I = m r 2 I=mr^{2}
  5. I = 1 2 m ( r external 2 + r internal 2 ) I=\frac{1}{2}m({r_{\mathrm{external}}}^{2}+{r_{\mathrm{internal}}}^{2})
  6. σ t = ρ r 2 ω 2 \sigma_{t}=\rho r^{2}\omega^{2}
  7. σ t \sigma_{t}
  8. ρ \rho
  9. r r
  10. ω \omega
  11. σ t ρ = v 2 \frac{\sigma_{t}}{\rho}=v^{2}
  12. σ t ρ \frac{\sigma_{t}}{\rho}
  13. v v
  14. E k σ t ρ m E_{k}\varpropto\frac{\sigma_{t}}{\rho}m
  15. σ t ρ \frac{\sigma_{t}}{\rho}
  16. E k σ t V E_{k}\varpropto\sigma_{t}V
  17. I rim = K I flywheel I_{\mathrm{rim}}=KI_{\mathrm{flywheel}}
  18. R R
  19. I rim = M rim R 2 I_{\mathrm{rim}}=M_{\mathrm{rim}}R^{2}

Foam.html

  1. W = γ Δ A W=\gamma\Delta A\,\!
  2. F b = V g ( ρ 2 - ρ 1 ) F_{b}=Vg(\rho_{2}-\rho_{1})\!
  3. V V
  4. g g
  5. F s = 2 r π γ F_{s}=2r\pi\gamma\!
  6. r r
  7. V g ( ρ 2 - ρ 1 ) > 2 r π γ Vg(\rho_{2}-\rho_{1})>2r\pi\gamma\!
  8. R R
  9. V V
  10. R 3 = 3 r γ 2 g ( ρ 2 - ρ 1 ) R^{3}=\frac{3r\gamma}{2g(\rho_{2}-\rho_{1})}\!
  11. p p
  12. p 0 p_{0}
  13. p - p 0 = γ ( 1 R 1 + 1 R 2 ) p-p_{0}=\gamma\left(\frac{1}{R_{1}}+\frac{1}{R_{2}}\right)\!
  14. p - p 0 - ( ρ 2 - ρ 1 ) g z = γ ( 1 R 3 + 1 R 4 ) p-p_{0}-(\rho_{2}-\rho_{1})gz=\gamma\left(\frac{1}{R_{3}}+\frac{1}{R_{4}}% \right)\!
  15. ( ρ 2 - ρ 1 ) g z = γ ( - 1 R 3 + - 1 R 4 + 1 R 1 + 1 R 2 ) (\rho_{2}-\rho_{1})gz=\gamma\left(\frac{-1}{R_{3}}+\frac{-1}{R_{4}}+\frac{1}{R% _{1}}+\frac{1}{R_{2}}\right)\!
  16. r r
  17. u = 2 g r 2 9 η 2 ( ρ 2 - ρ 1 ) ( 3 η 1 + 3 η 2 3 η 1 + 2 η 2 ) u=\frac{2gr^{2}}{9\eta_{2}}(\rho_{2}-\rho_{1})\left(\frac{3\eta_{1}+3\eta_{2}}% {3\eta_{1}+2\eta_{2}}\right)\!
  18. u = g r 2 3 η 2 ( ρ 2 ) u=\frac{gr^{2}}{3\eta_{2}}(\rho_{2})\!
  19. u = 2 g r 2 9 η 2 ( ρ 2 - ρ 1 ) u=\frac{2gr^{2}}{9\eta_{2}}(\rho_{2}-\rho_{1})\!
  20. P c = γ ( 1 R 1 + 1 R 2 ) P_{c}=\gamma\left(\frac{1}{R_{1}}+\frac{1}{R_{2}}\right)\!
  21. γ \gamma
  22. P B , 1 = p 0 + g ρ 1 z P_{B},1=p_{0}+g\rho_{1}z\!
  23. P B , 2 = p 0 + g ρ 2 z P_{B},2=p_{0}+g\rho_{2}z\!
  24. P c = 2 γ ( 1 R A - 1 R B ) P_{c}=2\gamma\left(\frac{1}{R_{A}}-\frac{1}{R_{B}}\right)\!
  25. g z ( ρ 2 - ρ 1 ) = 2 γ ( 1 R A - 1 R B ) gz(\rho_{2}-\rho_{1})=2\gamma\left(\frac{1}{R_{A}}-\frac{1}{R_{B}}\right)\!

Focal_length.html

  1. 1 f = 1 u + 1 v , \frac{1}{f}=\frac{1}{u}+\frac{1}{v}\ ,
  2. 1 f = ( n - 1 ) [ 1 R 1 - 1 R 2 + ( n - 1 ) d n R 1 R 2 ] , \frac{1}{f}=(n-1)\left[\frac{1}{R_{1}}-\frac{1}{R_{2}}+\frac{(n-1)d}{nR_{1}R_{% 2}}\right],
  3. FFD = f ( 1 + ( n - 1 ) d n R 2 ) , \mbox{FFD}~{}=f\left(1+\frac{(n-1)d}{nR_{2}}\right),
  4. BFD = f ( 1 - ( n - 1 ) d n R 1 ) . \mbox{BFD}~{}=f\left(1-\frac{(n-1)d}{nR_{1}}\right).
  5. f = - R 2 f=-{R\over 2}
  6. R R
  7. f f
  8. s 1 s_{1}
  9. s 2 s_{2}
  10. 1 s 1 + 1 s 2 = 1 f \frac{1}{s_{1}}+\frac{1}{s_{2}}=\frac{1}{f}
  11. s 1 s_{1}
  12. s 2 s_{2}
  13. f = 50 mm f=50\,\text{ mm}
  14. s 1 s_{1}\approx\infty
  15. s 2 = 50 mm s_{2}=50\,\text{ mm}
  16. s 1 = 1000 mm s_{1}=1000\,\text{ mm}
  17. s 2 = 52.6 mm s_{2}=52.6\,\text{ mm}

Fock_space.html

  1. H H
  2. n n
  3. n n
  4. H H
  5. n n
  6. n n
  7. H H
  8. H H
  9. F ν ( H ) = n = 0 S ν H n ¯ . F_{\nu}(H)=\overline{\bigoplus_{n=0}^{\infty}S_{\nu}H^{\otimes n}}~{}.
  10. S ν S_{\nu}
  11. ( ν = + ) (\nu=+)
  12. ( ν = - ) (\nu=-)
  13. F + ( H ) = S * H ¯ F_{+}(H)=\overline{S^{*}H}
  14. F - ( H ) = * H ¯ F_{-}(H)=\overline{{\bigwedge}^{*}H}
  15. H H
  16. H H
  17. F ν ( H ) = n = 0 S ν H n = H ( S ν ( H H ) ) ( S ν ( H H H ) ) F_{\nu}(H)=\bigoplus_{n=0}^{\infty}S_{\nu}H^{\otimes n}=\mathbb{C}\oplus H% \oplus\left(S_{\nu}\left(H\otimes H\right)\right)\oplus\left(S_{\nu}\left(H% \otimes H\otimes H\right)\right)\oplus\ldots
  18. \mathbb{C}
  19. H H
  20. S ν ( H H ) S_{\nu}(H\otimes H)
  21. F ν ( H ) F_{\nu}(H)
  22. | Ψ ν = | Ψ 0 ν | Ψ 1 ν | Ψ 2 ν = a 0 | 0 | ψ 1 i j a i j | ψ 2 i , ψ 2 j ν |\Psi\rangle_{\nu}=|\Psi_{0}\rangle_{\nu}\oplus|\Psi_{1}\rangle_{\nu}\oplus|% \Psi_{2}\rangle_{\nu}\oplus\ldots=a_{0}|0\rangle\oplus|\psi_{1}\rangle\oplus% \sum_{ij}a_{ij}|\psi_{2i},\psi_{2j}\rangle_{\nu}\oplus\ldots
  23. | 0 |0\rangle
  24. a 0 \,a_{0}\in\mathbb{C}
  25. | ψ 1 H |\psi_{1}\rangle\in H
  26. | ψ 2 i ψ 2 j ν = 1 2 ( | ψ 2 i | ψ 2 j + ( - 1 ) ν | ψ 2 j | ψ 2 i ) S ν ( H H ) |\psi_{2i}\psi_{2j}\rangle_{\nu}=\frac{1}{2}(|\psi_{2i}\rangle\otimes|\psi_{2j% }\rangle+(-1)^{\nu}|\psi_{2j}\rangle\otimes|\psi_{2i}\rangle)\in S_{\nu}(H% \otimes H)
  27. a i j = ν a j i a_{ij}=\nu a_{ji}\in\mathbb{C}
  28. F ν ( H ) F_{\nu}(H)
  29. F ν ( H ) F_{\nu}(H)
  30. | Ψ ν = ( | Ψ 0 ν , | Ψ 1 ν , | Ψ 2 ν , ) |\Psi\rangle_{\nu}=(|\Psi_{0}\rangle_{\nu},|\Psi_{1}\rangle_{\nu},|\Psi_{2}% \rangle_{\nu},\ldots)
  31. | Ψ ν ν 2 = n = 1 Ψ n | Ψ n ν < \||\Psi\rangle_{\nu}\|_{\nu}^{2}=\sum_{n=1}^{\infty}\langle\Psi_{n}|\Psi_{n}% \rangle_{\nu}<\infty
  32. n n
  33. Ψ n | Ψ n ν = i 1 , i n , j 1 , j n a i 1 , , i n * a j 1 , , j n ψ i 1 | ψ j 1 ψ i n | ψ j n \langle\Psi_{n}|\Psi_{n}\rangle_{\nu}=\sum_{i_{1},\ldots i_{n},j_{1},\ldots j_% {n}}a_{i_{1},\ldots,i_{n}}^{*}a_{j_{1},\ldots,j_{n}}\langle\psi_{i_{1}}|\psi_{% j_{1}}\rangle\cdots\langle\psi_{i_{n}}|\psi_{j_{n}}\rangle
  34. H n H^{\otimes n}
  35. | Ψ ν = | Ψ 0 ν | Ψ 1 ν | Ψ 2 ν = a 0 | 0 | ψ 1 i j a i j | ψ 2 i , ψ 2 j ν |\Psi\rangle_{\nu}=|\Psi_{0}\rangle_{\nu}\oplus|\Psi_{1}\rangle_{\nu}\oplus|% \Psi_{2}\rangle_{\nu}\oplus\ldots=a_{0}|0\rangle\oplus|\psi_{1}\rangle\oplus% \sum_{ij}a_{ij}|\psi_{2i},\psi_{2j}\rangle_{\nu}\oplus\ldots
  36. | Φ ν = | Φ 0 ν | Φ 1 ν | Φ 2 ν = b 0 | 0 | ϕ 1 i j b i j | ϕ 2 i , ϕ 2 j ν |\Phi\rangle_{\nu}=|\Phi_{0}\rangle_{\nu}\oplus|\Phi_{1}\rangle_{\nu}\oplus|% \Phi_{2}\rangle_{\nu}\oplus\ldots=b_{0}|0\rangle\oplus|\phi_{1}\rangle\oplus% \sum_{ij}b_{ij}|\phi_{2i},\phi_{2j}\rangle_{\nu}\oplus\ldots
  37. F ν ( H ) F_{\nu}(H)
  38. Ψ | Φ ν := n Ψ n | Φ n ν = a 0 * b 0 + ψ 1 | ϕ 1 + i j k l a i j * b k l ϕ 2 i | ψ 2 k ψ 2 j | ϕ 2 l ν + \langle\Psi|\Phi\rangle_{\nu}:=\sum_{n}\langle\Psi_{n}|\Phi_{n}\rangle_{\nu}=a% _{0}^{*}b_{0}+\langle\psi_{1}|\phi_{1}\rangle+\sum_{ijkl}a_{ij}^{*}b_{kl}% \langle\phi_{2i}|\psi_{2k}\rangle\langle\psi_{2j}|\phi_{2l}\rangle_{\nu}+\ldots
  39. n n
  40. n n
  41. n n
  42. | Ψ ν = | ϕ 1 , ϕ 2 , , ϕ n ν = | ϕ 1 | ϕ 2 | ϕ n |\Psi\rangle_{\nu}=|\phi_{1},\phi_{2},\cdots,\phi_{n}\rangle_{\nu}=|\phi_{1}% \rangle|\phi_{2}\rangle\cdots|\phi_{n}\rangle
  43. n n
  44. ϕ 1 \phi_{1}\,
  45. ϕ 2 \phi_{2}\,
  46. n n
  47. ϕ i \phi_{i}\,
  48. H H
  49. ϕ i \phi_{i}\,
  50. | Ψ - |\Psi\rangle_{-}
  51. ϕ i \phi_{i}\,
  52. | ϕ i | ϕ i = 0 |\phi_{i}\rangle|\phi_{i}\rangle=0
  53. { | ψ i } i = 0 , 1 , 2 , \{|\psi_{i}\rangle\}_{i=0,1,2,\dots}
  54. H H
  55. n 0 n_{0}
  56. | ψ 0 |\psi_{0}\rangle
  57. n 1 n_{1}
  58. | ψ 1 |\psi_{1}\rangle
  59. n k n_{k}
  60. | ψ k |\psi_{k}\rangle
  61. | n 0 , n 1 , , n k ν = | ψ 0 n 0 | ψ 1 n 1 | ψ k n k , |n_{0},n_{1},\ldots,n_{k}\rangle_{\nu}=|\psi_{0}\rangle^{n_{0}}|\psi_{1}% \rangle^{n_{1}}\cdots|\psi_{k}\rangle^{n_{k}},
  62. n i n_{i}
  63. | ψ i |\psi_{i}\rangle
  64. a ( ϕ ) a^{\dagger}(\phi)\,
  65. a ( ϕ ) a(\phi)\,
  66. | ϕ |\phi\rangle
  67. | ϕ |\phi\rangle
  68. ϕ | \langle\phi|
  69. a ( ϕ ) a^{\dagger}(\phi)\,
  70. H H
  71. | ϕ i |\phi_{i}\rangle
  72. a ( ϕ i ) a ( ϕ i ) a^{\dagger}(\phi_{i})a(\phi_{i})\,
  73. H H
  74. L 2 ( X , μ ) L_{2}(X,\mu)
  75. X X
  76. μ \mu
  77. H = L 2 ( 3 , d 3 x ) H=L_{2}(\mathbb{R}^{3},d^{3}x)
  78. X 0 = { * } X^{0}=\{*\}
  79. X 1 = X X^{1}=X
  80. X 2 = X × X X^{2}=X\times X
  81. X 3 = X × X × X X^{3}=X\times X\times X
  82. X * = X 0 X 1 X 2 X 3 X^{*}=X^{0}\bigsqcup X^{1}\bigsqcup X^{2}\bigsqcup X^{3}\bigsqcup\ldots
  83. μ * \mu^{*}
  84. μ * ( X 0 ) = 1 \mu^{*}(X^{0})=1
  85. μ * \mu^{*}
  86. X n X^{n}
  87. μ n \mu^{n}
  88. F + ( L 2 ( X , μ ) ) F_{+}(L_{2}(X,\mu))\,
  89. L 2 ( X * , μ * ) L_{2}(X^{*},\mu^{*})
  90. F - ( L 2 ( X , μ ) ) F_{-}(L_{2}(X,\mu))\,
  91. L 2 ( X , μ ) n L 2 ( X n , μ n ) L_{2}(X,\mu)^{\otimes n}\to L_{2}(X^{n},\mu^{n})
  92. ψ 1 ( x ) ψ n ( x ) ψ 1 ( x 1 ) ψ n ( x n ) \psi_{1}(x)\otimes\cdots\otimes\psi_{n}(x)\mapsto\psi_{1}(x_{1})\cdots\psi_{n}% (x_{n})
  93. ψ 1 = ψ 1 ( x ) , , ψ n = ψ n ( x ) \psi_{1}=\psi_{1}(x),\ldots,\psi_{n}=\psi_{n}(x)
  94. Ψ ( x 1 , x n ) = 1 n ! | ψ 1 ( x 1 ) ψ n ( x 1 ) ψ 1 ( x n ) ψ n ( x n ) | \Psi(x_{1},\ldots x_{n})=\frac{1}{\sqrt{n!}}\left|\begin{matrix}\psi_{1}(x_{1}% )&\ldots&\psi_{n}(x_{1})\\ \vdots&&\vdots\\ \psi_{1}(x_{n})&\dots&\psi_{n}(x_{n})\\ \end{matrix}\right|
  95. X n X^{n}
  96. n n
  97. Ψ = 1 \|\Psi\|=1
  98. ψ 1 , , ψ n \psi_{1},\ldots,\psi_{n}
  99. n n
  100. B n B_{n}
  101. 2 ( n ) = { f : n f 2 ( n ) < } \mathcal{F}^{2}(\mathbb{C}^{n})=\{f\colon\mathbb{C}^{n}\to\mathbb{C}\mid\|f\|_% {\mathcal{F}^{2}(\mathbb{C}^{n})}<\infty\}
  102. f 2 ( n ) := n | f ( 𝐳 ) | 2 e - π | 𝐳 | 2 d 𝐳 \|f\|_{\mathcal{F}^{2}(\mathbb{C}^{n})}:=\int_{\mathbb{C}^{n}}|f(\mathbf{z})|^% {2}e^{-\pi|\mathbf{z}|^{2}}\,d\mathbf{z}
  103. B B_{\infty}
  104. B n B_{n}
  105. n 0 n\geq 0
  106. B B_{\infty}

Fock_state.html

  1. N 𝐤 i ^ \widehat{N_{{\mathbf{k}}_{i}}}
  2. | n 𝐤 1 , n 𝐤 2 , . . n 𝐤 i |n_{{\mathbf{k}}_{1}},n_{{\mathbf{k}}_{2}},..n_{{\mathbf{k}}_{i}}...\rangle
  3. n 𝐤 i n_{{\mathbf{k}}_{i}}
  4. N 𝐤 i ^ \widehat{N_{{\mathbf{k}}_{i}}}
  5. N 𝐤 i ^ | n 𝐤 1 , n 𝐤 2 , . . n 𝐤 i = n 𝐤 i | n 𝐤 1 , n 𝐤 2 , . . n 𝐤 i \widehat{N_{{\mathbf{k}}_{i}}}|n_{{\mathbf{k}}_{1}},n_{{\mathbf{k}}_{2}},..n_{% {\mathbf{k}}_{i}}...\rangle=n_{{\mathbf{k}}_{i}}|n_{{\mathbf{k}}_{1}},n_{{% \mathbf{k}}_{2}},..n_{{\mathbf{k}}_{i}}...\rangle
  6. n 𝐤 i n_{{\mathbf{k}}_{i}}
  7. Var ( N ^ ) = 0 {\rm Var}(\widehat{N})=0
  8. N ^ = i N 𝐤 i ^ \widehat{N}=\sum_{i}\widehat{N_{{\mathbf{k}}_{i}}}
  9. | f |f\rangle
  10. | 1 < m t p l > 𝐤 1 , 1 𝐤 2 |1_{<}mtpl>{{\mathbf{{k}}_{1}}},1_{{\mathbf{{k}}_{2}}}\rangle
  11. 𝕆 ^ \widehat{\mathbb{O}}
  12. | f | 𝕆 ^ | 1 < m t p l > 𝐤 1 , 1 𝐤 2 | 2 = | f | 𝕆 ^ | 1 𝐤 2 , 1 𝐤 1 | 2 |\langle f|\widehat{\mathbb{O}}|1_{<}mtpl>{{\mathbf{{k}}_{1}}},1_{{\mathbf{{k}% }_{2}}}\rangle|^{2}=|\langle f|\widehat{\mathbb{O}}|1_{{\mathbf{{k}}_{2}}},1_{% {\mathbf{{k}}_{1}}}\rangle|^{2}
  13. f | 𝕆 ^ | 1 < m t p l > 𝐤 1 , 1 𝐤 2 = e i δ f | 𝕆 ^ | 1 𝐤 2 , 1 𝐤 1 \langle f|\widehat{\mathbb{O}}|1_{<}mtpl>{{\mathbf{{k}}_{1}}},1_{{\mathbf{{k}}% _{2}}}\rangle=e^{i\delta}\langle f|\widehat{\mathbb{O}}|1_{{\mathbf{{k}}_{2}}}% ,1_{{\mathbf{{k}}_{1}}}\rangle
  14. e i δ = + 1 e^{i\delta}=+1
  15. - 1 -1
  16. f | \langle f|
  17. 𝕆 ^ \widehat{\mathbb{O}}
  18. | 1 < m t p l > 𝐤 1 , 1 𝐤 2 = + | 1 𝐤 2 , 1 𝐤 1 |1_{<}mtpl>{{\mathbf{{k}}_{1}}},1_{{\mathbf{{k}}_{2}}}\rangle=+|1_{{\mathbf{{k% }}_{2}}},1_{{\mathbf{{k}}_{1}}}\rangle
  19. | 1 < m t p l > 𝐤 1 , 1 𝐤 2 = - | 1 𝐤 2 , 1 𝐤 1 |1_{<}mtpl>{{\mathbf{{k}}_{1}}},1_{{\mathbf{{k}}_{2}}}\rangle=-|1_{{\mathbf{{k% }}_{2}}},1_{{\mathbf{{k}}_{1}}}\rangle
  20. P ^ | x 1 , x 2 = | x 2 , x 1 \hat{P}\left|x_{1},x_{2}\right\rangle=\left|x_{2},x_{1}\right\rangle
  21. b b^{\dagger}
  22. b b
  23. b 𝐤 l | n 𝐤 1 , n 𝐤 2 , n 𝐤 3 n 𝐤 l , = n 𝐤 l + 1 | n 𝐤 1 , n 𝐤 2 , n 𝐤 3 n 𝐤 l + 1 , b^{\dagger}_{{\mathbf{k}}_{l}}|n_{{\mathbf{k}}_{1}},n_{{\mathbf{k}}_{2}},n_{{% \mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}},...\rangle=\sqrt{n_{{\mathbf{k}}_{l}}+% 1}|n_{{\mathbf{k}}_{1}},n_{{\mathbf{k}}_{2}},n_{{\mathbf{k}}_{3}}...n_{{% \mathbf{k}}_{l}}+1,...\rangle
  24. b 𝐤 l | n 𝐤 1 , n 𝐤 2 , n 𝐤 3 n 𝐤 l , = n 𝐤 l | n 𝐤 1 , n 𝐤 2 , n 𝐤 3 n 𝐤 l - 1 , b_{{\mathbf{k}}_{l}}|n_{{\mathbf{k}}_{1}},n_{{\mathbf{k}}_{2}},n_{{\mathbf{k}}% _{3}}...n_{{\mathbf{k}}_{l}},...\rangle=\sqrt{n_{{\mathbf{k}}_{l}}}|n_{{% \mathbf{k}}_{1}},n_{{\mathbf{k}}_{2}},n_{{\mathbf{k}}_{3}}...n_{{\mathbf{k}}_{% l}}-1,...\rangle
  25. | n 𝐤 1 , n 𝐤 2 , n 𝐤 3 n 𝐤 l , |n_{{\mathbf{k}}_{1}},n_{{\mathbf{k}}_{2}},n_{{\mathbf{k}}_{3}}...n_{{\mathbf{% k}}_{l}},...\rangle
  26. n 𝐤 1 , n 𝐤 2 , n 𝐤 3 n 𝐤 l - 1 , | b 𝐤 l | n 𝐤 1 , n 𝐤 2 , n 𝐤 3 n 𝐤 l , = n 𝐤 l n 𝐤 1 , n 𝐤 2 , n 𝐤 3 n 𝐤 l - 1 , | n 𝐤 1 , n 𝐤 2 , n 𝐤 3 n 𝐤 l - 1 , \langle n_{{\mathbf{k}}_{1}},n_{{\mathbf{k}}_{2}},n_{{\mathbf{k}}_{3}}...n_{{% \mathbf{k}}_{l}}-1,...|b_{{\mathbf{k}}_{l}}|n_{{\mathbf{k}}_{1}},n_{{\mathbf{k% }}_{2}},n_{{\mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}},...\rangle=\sqrt{n_{{% \mathbf{k}}_{l}}}\langle n_{{\mathbf{k}}_{1}},n_{{\mathbf{k}}_{2}},n_{{\mathbf% {k}}_{3}}...n_{{\mathbf{k}}_{l}}-1,...|n_{{\mathbf{k}}_{1}},n_{{\mathbf{k}}_{2% }},n_{{\mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}}-1,...\rangle
  27. ( n 𝐤 1 , n 𝐤 2 , n 𝐤 3 n 𝐤 l , | b 𝐤 l | n 𝐤 1 , n 𝐤 2 , n 𝐤 3 n 𝐤 l - 1 , ) * (\langle n_{{\mathbf{k}}_{1}},n_{{\mathbf{k}}_{2}},n_{{\mathbf{k}}_{3}}...n_{{% \mathbf{k}}_{l}},...|b_{{\mathbf{k}}_{l}}|n_{{\mathbf{k}}_{1}},n_{{\mathbf{k}}% _{2}},n_{{\mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}}-1,...\rangle)^{*}
  28. = n 𝐤 1 , n 𝐤 2 , n 𝐤 3 n 𝐤 l - 1... | b 𝐤 l | n 𝐤 1 , n 𝐤 2 , n 𝐤 3 n 𝐤 l , = n 𝐤 l + 1 n 𝐤 1 , n 𝐤 2 , n 𝐤 3 n 𝐤 l - 1... | n 𝐤 1 , n 𝐤 2 , n 𝐤 3 n 𝐤 l + 1... =\langle n_{{\mathbf{k}}_{1}},n_{{\mathbf{k}}_{2}},n_{{\mathbf{k}}_{3}}...n_{{% \mathbf{k}}_{l}}-1...|b_{{\mathbf{k}}_{l}}^{\dagger}|n_{{\mathbf{k}}_{1}},n_{{% \mathbf{k}}_{2}},n_{{\mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}},...\rangle=\sqrt{% n_{{\mathbf{k}}_{l}}+1}\langle n_{{\mathbf{k}}_{1}},n_{{\mathbf{k}}_{2}},n_{{% \mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}}-1...|n_{{\mathbf{k}}_{1}},n_{{\mathbf{% k}}_{2}},n_{{\mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}}+1...\rangle
  29. [ b i , b j ] b i b j - b j b i = δ i j , [b_{i},b^{\dagger}_{j}]\equiv b_{i}b^{\dagger}_{j}-b^{\dagger}_{j}b_{i}=\delta% _{ij},
  30. [ b i , b j ] = [ b i , b j ] = 0 , [b^{\dagger}_{i},b^{\dagger}_{j}]=[b_{i},b_{j}]=0,
  31. [ , ] [\ \ ,\ \ ]
  32. δ i j \delta_{ij}
  33. | n 𝐤 1 , n 𝐤 2 , n 𝐤 3 n 𝐤 l , |n_{{\mathbf{k}}_{1}},n_{{\mathbf{k}}_{2}},n_{{\mathbf{k}}_{3}}...n_{{\mathbf{% k}}_{l}},...\rangle
  34. | 0 , 0 , 0... |0,0,0...\rangle
  35. | 1 , 0 , 0... |1,0,0...\rangle
  36. | 0 , 1 , 0... |0,1,0...\rangle
  37. | 0 , 0 , 1... |0,0,1...\rangle
  38. | 2 , 0 , 0... |2,0,0...\rangle
  39. | 1 , 1 , 0... |1,1,0...\rangle
  40. | 0 , 2 , 0... |0,2,0...\rangle
  41. | 0 𝐤 1 , 0 𝐤 2 , 0 𝐤 3 0 𝐤 l , |0_{{\mathbf{k}}_{1}},0_{{\mathbf{k}}_{2}},0_{{\mathbf{k}}_{3}}...0_{{\mathbf{% k}}_{l}},...\rangle
  42. b 𝐤 l | 0 𝐤 1 , 0 𝐤 2 , 0 𝐤 3 0 𝐤 l , = | 0 𝐤 1 , 0 𝐤 2 , 0 𝐤 3 1 𝐤 l , b^{\dagger}_{{\mathbf{k}}_{l}}|0_{{\mathbf{k}}_{1}},0_{{\mathbf{k}}_{2}},0_{{% \mathbf{k}}_{3}}...0_{{\mathbf{k}}_{l}},...\rangle=|0_{{\mathbf{k}}_{1}},0_{{% \mathbf{k}}_{2}},0_{{\mathbf{k}}_{3}}...1_{{\mathbf{k}}_{l}},...\rangle
  43. b 𝐤 l | 0 𝐤 1 , 0 𝐤 2 , 0 𝐤 3 0 𝐤 l , = 0 b_{{\mathbf{k}}_{l}}|0_{{\mathbf{k}}_{1}},0_{{\mathbf{k}}_{2}},0_{{\mathbf{k}}% _{3}}...0_{{\mathbf{k}}_{l}},...\rangle=0
  44. | n 𝐤 1 , n 𝐤 2 = ( b 𝐤 1 ) 𝐤 1 𝐤 1 ! ( b 𝐤 2 ) 𝐤 2 𝐤 2 ! | 0 𝐤 1 , 0 𝐤 2 , |n_{{\mathbf{k}}_{1}},n_{{\mathbf{k}}_{2}}...\rangle=\frac{(b^{\dagger}_{{% \mathbf{k}}_{1}})^{{\mathbf{k}}_{1}}}{\sqrt{{{\mathbf{k}}_{1}}!}}\frac{(b^{% \dagger}_{{\mathbf{k}}_{2}})^{{\mathbf{k}}_{2}}}{\sqrt{{{\mathbf{k}}_{2}}!}}..% .|0_{{\mathbf{k}}_{1}},0_{{\mathbf{k}}_{2}},...\rangle
  45. | n 𝐤 |n_{{\mathbf{k}}}\rangle
  46. b 𝐤 | n 𝐤 = n 𝐤 + 1 | n 𝐤 + 1 b^{\dagger}_{\mathbf{k}}|n_{{\mathbf{k}}}\rangle=\sqrt{n_{\mathbf{k}}+1}|n_{{% \mathbf{k}}}+1\rangle
  47. b 𝐤 | n 𝐤 = n 𝐤 | n 𝐤 - 1 b_{\mathbf{k}}|n_{{\mathbf{k}}}\rangle=\sqrt{n_{\mathbf{k}}}|n_{{\mathbf{k}}}-1\rangle
  48. N 𝐤 l ^ = b 𝐤 l b 𝐤 l \widehat{N_{{\mathbf{k}}_{l}}}=b^{\dagger}_{{\mathbf{k}}_{l}}b_{{\mathbf{k}}_{% l}}
  49. N 𝐤 l ^ | n 𝐤 1 , n 𝐤 2 , n 𝐤 3 n 𝐤 l = n 𝐤 l | n 𝐤 1 , n 𝐤 2 , n 𝐤 3 n 𝐤 l \widehat{N_{{\mathbf{k}}_{l}}}|n_{{\mathbf{k}}_{1}},n_{{\mathbf{k}}_{2}},n_{{% \mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}}...\rangle=n_{{\mathbf{k}}_{l}}|n_{{% \mathbf{k}}_{1}},n_{{\mathbf{k}}_{2}},n_{{\mathbf{k}}_{3}}...n_{{\mathbf{k}}_{% l}}...\rangle
  50. | n 𝐤 1 , n 𝐤 2 , . n 𝐤 m n 𝐤 l |n_{{\mathbf{k}}_{1}},n_{{\mathbf{k}}_{2}},....n_{{\mathbf{k}}_{m}}...n_{{% \mathbf{k}}_{l}}...\rangle
  51. k l k_{l}
  52. k m k_{m}
  53. b 𝐤 m b 𝐤 l b_{{\mathbf{k}}_{m}}^{\dagger}b_{{\mathbf{k}}_{l}}
  54. b 𝐤 m . b 𝐤 l = b 𝐤 l . b 𝐤 m b_{{\mathbf{k}}_{m}}^{\dagger}.b_{{\mathbf{k}}_{l}}=b_{{\mathbf{k}}_{l}}.b_{{% \mathbf{k}}_{m}}^{\dagger}
  55. b 𝐤 m . b 𝐤 l | n 𝐤 1 , n 𝐤 2 , . n 𝐤 m n 𝐤 l = b 𝐤 l . b 𝐤 m | n 𝐤 1 , n 𝐤 2 , . n 𝐤 m n 𝐤 l = n 𝐤 m + 1 n 𝐤 l | n 𝐤 1 , n 𝐤 2 , . n 𝐤 m + 1... n 𝐤 l - 1... b_{{\mathbf{k}}_{m}}^{\dagger}.b_{{\mathbf{k}}_{l}}|n_{{\mathbf{k}}_{1}},n_{{% \mathbf{k}}_{2}},....n_{{\mathbf{k}}_{m}}...n_{{\mathbf{k}}_{l}}...\rangle=b_{% {\mathbf{k}}_{l}}.b_{{\mathbf{k}}_{m}}^{\dagger}|n_{{\mathbf{k}}_{1}},n_{{% \mathbf{k}}_{2}},....n_{{\mathbf{k}}_{m}}...n_{{\mathbf{k}}_{l}}...\rangle=% \sqrt{n_{{\mathbf{k}}_{m}}+1}\sqrt{n_{{\mathbf{k}}_{l}}}|n_{{\mathbf{k}}_{1}},% n_{{\mathbf{k}}_{2}},....n_{{\mathbf{k}}_{m}}+1...n_{{\mathbf{k}}_{l}}-1...\rangle
  56. | n 𝐤 1 , n 𝐤 2 , n 𝐤 3 n 𝐤 l , |n_{{\mathbf{k}}_{1}},n_{{\mathbf{k}}_{2}},n_{{\mathbf{k}}_{3}}...n_{{\mathbf{% k}}_{l}},...\rangle
  57. c 𝐤 l | n 𝐤 1 , n 𝐤 2 , n 𝐤 3 n 𝐤 l , = n 𝐤 l + 1 | n 𝐤 1 , n 𝐤 2 , n 𝐤 3 n 𝐤 l + 1 , c^{\dagger}_{{\mathbf{k}}_{l}}|n_{{\mathbf{k}}_{1}},n_{{\mathbf{k}}_{2}},n_{{% \mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}},...\rangle=\sqrt{n_{{\mathbf{k}}_{l}}+% 1}|n_{{\mathbf{k}}_{1}},n_{{\mathbf{k}}_{2}},n_{{\mathbf{k}}_{3}}...n_{{% \mathbf{k}}_{l}}+1,...\rangle
  58. c 𝐤 l | n 𝐤 1 , n 𝐤 2 , n 𝐤 3 n 𝐤 l , = n 𝐤 l | n 𝐤 1 , n 𝐤 2 , n 𝐤 3 n 𝐤 l - 1 , c_{{\mathbf{k}}_{l}}|n_{{\mathbf{k}}_{1}},n_{{\mathbf{k}}_{2}},n_{{\mathbf{k}}% _{3}}...n_{{\mathbf{k}}_{l}},...\rangle=\sqrt{n_{{\mathbf{k}}_{l}}}|n_{{% \mathbf{k}}_{1}},n_{{\mathbf{k}}_{2}},n_{{\mathbf{k}}_{3}}...n_{{\mathbf{k}}_{% l}}-1,...\rangle
  59. { c i , c j } c i c j + c j c i = δ i j , \{c_{i},c^{\dagger}_{j}\}\equiv c_{i}c^{\dagger}_{j}+c^{\dagger}_{j}c_{i}=% \delta_{ij},
  60. { c i , c j } = { c i , c j } = 0 , \{c^{\dagger}_{i},c^{\dagger}_{j}\}=\{c_{i},c_{j}\}=0,
  61. , {\ \ ,\ \ }
  62. δ i j \delta_{ij}
  63. N 𝐤 l ^ = c 𝐤 l . c 𝐤 l \widehat{N_{{\mathbf{k}}_{l}}}=c^{\dagger}_{{\mathbf{k}}_{l}}.c_{{\mathbf{k}}_% {l}}
  64. N 𝐤 l ^ | n 𝐤 1 , n 𝐤 2 , n 𝐤 3 n 𝐤 l = n 𝐤 l | n 𝐤 1 , n 𝐤 2 , n 𝐤 3 n 𝐤 l \widehat{N_{{\mathbf{k}}_{l}}}|n_{{\mathbf{k}}_{1}},n_{{\mathbf{k}}_{2}},n_{{% \mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}}...\rangle=n_{{\mathbf{k}}_{l}}|n_{{% \mathbf{k}}_{1}},n_{{\mathbf{k}}_{2}},n_{{\mathbf{k}}_{3}}...n_{{\mathbf{k}}_{% l}}...\rangle
  65. | n 𝐤 1 , n 𝐤 2 , n 𝐤 3 n 𝐤 l |n_{{\mathbf{k}}_{1}},n_{{\mathbf{k}}_{2}},n_{{\mathbf{k}}_{3}}...n_{{\mathbf{% k}}_{l}}...\rangle
  66. | n 𝐤 1 , n 𝐤 2 , n 𝐤 3 n 𝐤 l = S - | i 1 , i 2 , i 3 i l = 1 N ! | | i 1 1 | i 1 N | i N 1 | i N N | |n_{{\mathbf{k}}_{1}},n_{{\mathbf{k}}_{2}},n_{{\mathbf{k}}_{3}}...n_{{\mathbf{% k}}_{l}}...\rangle=S_{-}|i_{1},i_{2},i_{3}...i_{l}...\rangle=\frac{1}{\sqrt{N!% }}\begin{vmatrix}|i_{1}\rangle_{1}&\cdots&|i_{1}\rangle_{N}\\ \vdots&\ddots&\vdots\\ |i_{N}\rangle_{1}&\cdots&|i_{N}\rangle_{N}\end{vmatrix}
  67. N 𝐤 l ^ \widehat{N_{{\mathbf{k}}_{l}}}
  68. | n 𝐤 1 , n 𝐤 2 , n 𝐤 3 n 𝐤 l , |n_{{\mathbf{k}}_{1}},n_{{\mathbf{k}}_{2}},n_{{\mathbf{k}}_{3}}...n_{{\mathbf{% k}}_{l}},...\rangle
  69. | 0 , 0 , 0... |0,0,0...\rangle
  70. | 1 , 0 , 0... |1,0,0...\rangle
  71. | 0 , 1 , 0... |0,1,0...\rangle
  72. | 0 , 0 , 1... |0,0,1...\rangle
  73. | 1 , 1 , 0... |1,1,0...\rangle
  74. | 0 , 1 , 1... |0,1,1...\rangle
  75. | 0 , 1 , 0 , 1... |0,1,0,1...\rangle
  76. | 1 , 0 , 1 , 0... |1,0,1,0...\rangle
  77. | 0 𝐤 |0_{{\mathbf{k}}}\rangle
  78. c 𝐤 | 0 𝐤 = | 1 𝐤 c^{\dagger}_{\mathbf{k}}|0_{{\mathbf{k}}}\rangle=|1_{{\mathbf{k}}}\rangle
  79. c 𝐤 | 1 𝐤 = 0 c^{\dagger}_{\mathbf{k}}|1_{{\mathbf{k}}}\rangle=0
  80. | 1 𝐤 |1_{{\mathbf{k}}}\rangle
  81. c 𝐤 | 1 𝐤 = | 0 𝐤 c_{\mathbf{k}}|1_{{\mathbf{k}}}\rangle=|0_{{\mathbf{k}}}\rangle
  82. c 𝐤 | 0 𝐤 = 0 c_{\mathbf{k}}|0_{{\mathbf{k}}}\rangle=0
  83. | n 𝐤 1 , n 𝐤 2 , n 𝐤 β , n 𝐤 α , |n_{{\mathbf{k}}_{1}},n_{{\mathbf{k}}_{2}},...n_{{\mathbf{k}}_{\beta}},n_{{% \mathbf{k}}_{\alpha}},...\rangle
  84. c 𝐤 α | n 𝐤 1 , n 𝐤 2 , n 𝐤 β , n 𝐤 α , = ( - 1 ) β < α n β | n 𝐤 1 , n 𝐤 2 , n 𝐤 β , 1 - n 𝐤 α , c_{{\mathbf{k}}_{\alpha}}|n_{{\mathbf{k}}_{1}},n_{{\mathbf{k}}_{2}},...n_{{% \mathbf{k}}_{\beta}},n_{{\mathbf{k}}_{\alpha}},...\rangle=(-1)^{\sum_{\beta<% \alpha}n\beta}|n_{{\mathbf{k}}_{1}},n_{{\mathbf{k}}_{2}},...n_{{\mathbf{k}}_{% \beta}},1-n_{{\mathbf{k}}_{\alpha}},...\rangle
  85. ( - 1 ) β < α n β (-1)^{\sum_{\beta<\alpha}n\beta}
  86. | n 𝐤 1 , n 𝐤 2 , . n 𝐤 m n 𝐤 l |n_{{\mathbf{k}}_{1}},n_{{\mathbf{k}}_{2}},....n_{{\mathbf{k}}_{m}}...n_{{% \mathbf{k}}_{l}}...\rangle
  87. k l k_{l}
  88. k m k_{m}
  89. c 𝐤 m . c 𝐤 l c_{{\mathbf{k}}_{m}}^{\dagger}.c_{{\mathbf{k}}_{l}}
  90. c 𝐤 m . c 𝐤 l = - c 𝐤 l . c 𝐤 m c_{{\mathbf{k}}_{m}}^{\dagger}.c_{{\mathbf{k}}_{l}}=-c_{{\mathbf{k}}_{l}}.c_{{% \mathbf{k}}_{m}}^{\dagger}
  91. c 𝐤 m . c 𝐤 l | n 𝐤 1 , n 𝐤 2 , . n 𝐤 m n 𝐤 l = n 𝐤 m + 1 n 𝐤 l | n 𝐤 1 , n 𝐤 2 , . n 𝐤 m + 1... n 𝐤 l - 1... c_{{\mathbf{k}}_{m}}^{\dagger}.c_{{\mathbf{k}}_{l}}|n_{{\mathbf{k}}_{1}},n_{{% \mathbf{k}}_{2}},....n_{{\mathbf{k}}_{m}}...n_{{\mathbf{k}}_{l}}...\rangle=% \sqrt{n_{{\mathbf{k}}_{m}}+1}\sqrt{n_{{\mathbf{k}}_{l}}}|n_{{\mathbf{k}}_{1}},% n_{{\mathbf{k}}_{2}},....n_{{\mathbf{k}}_{m}}+1...n_{{\mathbf{k}}_{l}}-1...\rangle
  92. c 𝐤 l . c 𝐤 m | n 𝐤 1 , n 𝐤 2 , . n 𝐤 m n 𝐤 l = - c 𝐤 m . c 𝐤 l | n 𝐤 1 , n 𝐤 2 , . n 𝐤 m n 𝐤 l = - n 𝐤 m + 1 n 𝐤 l | n 𝐤 1 , n 𝐤 2 , . n 𝐤 m + 1... n 𝐤 l - 1... c_{{\mathbf{k}}_{l}}.c_{{\mathbf{k}}_{m}}^{\dagger}|n_{{\mathbf{k}}_{1}},n_{{% \mathbf{k}}_{2}},....n_{{\mathbf{k}}_{m}}...n_{{\mathbf{k}}_{l}}...\rangle=-c_% {{\mathbf{k}}_{m}}^{\dagger}.c_{{\mathbf{k}}_{l}}|n_{{\mathbf{k}}_{1}},n_{{% \mathbf{k}}_{2}},....n_{{\mathbf{k}}_{m}}...n_{{\mathbf{k}}_{l}}...\rangle=-% \sqrt{n_{{\mathbf{k}}_{m}}+1}\sqrt{n_{{\mathbf{k}}_{l}}}|n_{{\mathbf{k}}_{1}},% n_{{\mathbf{k}}_{2}},....n_{{\mathbf{k}}_{m}}+1...n_{{\mathbf{k}}_{l}}-1...\rangle
  93. = 1 2 m i ψ * ( x ) i ψ ( x ) \mathfrak{H}=\frac{1}{2m}\nabla_{i}\psi^{*}(x)\nabla_{i}\psi(x)
  94. = d 3 x = d 3 x ψ * ( x ) ( - 2 2 m ) ψ ( x ) \mathcal{H}=\int d^{3}x\,\mathfrak{H}=\int d^{3}x\psi^{*}(x)\left(-\tfrac{% \nabla^{2}}{2m}\right)\psi(x)
  95. = - 2 2 m \therefore\mathfrak{H}=-\tfrac{\nabla^{2}}{2m}
  96. ψ n ( + ) ( x ) = - 2 2 m ψ n ( + ) ( x ) = E n 0 ψ n ( + ) ( x ) \mathfrak{H}\psi_{n}^{(+)}(x)=-\tfrac{\nabla^{2}}{2m}\psi_{n}^{(+)}(x)=E_{n}^{% 0}\psi_{n}^{(+)}(x)
  97. d 3 x ψ n ( + ) * ( x ) ψ n ( + ) ( x ) = δ n n \int d^{3}x\,\psi_{n}^{(+)^{*}}(x)\psi_{n^{\prime}}^{(+)}(x)=\delta_{nn^{% \prime}}
  98. ψ ( x ) = n a n ψ n ( + ) ( x ) \psi(x)=\sum_{n}a_{n}\psi_{n}^{(+)}(x)
  99. a n a_{n}
  100. = n , n d 3 x a n ψ n ( + ) * ( x ) a n ψ n ( + ) ( x ) \therefore\mathcal{H}=\sum_{n,n^{\prime}}\int d^{3}x\,a^{\dagger}_{n^{\prime}}% \psi_{n^{\prime}}^{(+)^{*}}(x)\mathfrak{H}a_{n}\psi_{n}^{(+)}(x)
  101. \mathfrak{H}
  102. a n a_{n}
  103. = n , n d 3 x a n ψ n ( + ) * ( x ) E n 0 ψ n ( + ) ( x ) a n = n , n E n 0 a n a n δ n n = n E n 0 a n a n = n E n 0 N ^ \mathcal{H}=\sum_{n,n^{\prime}}\int d^{3}x\,a^{\dagger}_{n^{\prime}}\psi_{n^{% \prime}}^{(+)^{*}}(x)E^{0}_{n}\psi_{n}^{(+)}(x)a_{n}=\sum_{n,n^{\prime}}E^{0}_% {n}a^{\dagger}_{n^{\prime}}a_{n}\delta_{nn^{\prime}}=\sum_{n}E^{0}_{n}a^{% \dagger}_{n}a_{n}=\sum_{n}E^{0}_{n}\widehat{N}
  104. | 0 |0\rangle
  105. a a
  106. a a^{\dagger}
  107. a | 0 = 0 = 0 | a a|0\rangle=0=\langle 0|a^{\dagger}
  108. F ( r , t ) = ε a e i k x - ω t + h . c F(\vec{r},t)=\varepsilon ae^{i\vec{k}x-\omega t}+h.c
  109. 0 | F | 0 = 0 \langle 0|F|0\rangle=0
  110. a 𝐤 l a_{{\mathbf{k}}_{l}}
  111. a 𝐤 l a^{\dagger}_{{\mathbf{k}}_{l}}
  112. | n 𝐤 l |n_{{\mathbf{k}}_{l}}\rangle
  113. | n 𝐤 l |n_{{\mathbf{k}}_{l}}\rangle
  114. | n 𝐤 1 | n 𝐤 2 | n 𝐤 3 | n 𝐤 1 , n 𝐤 2 , n 𝐤 3 n 𝐤 l | { n 𝐤 } |n_{{\mathbf{k}}_{1}}\rangle|n_{{\mathbf{k}}_{2}}\rangle|n_{{\mathbf{k}}_{3}}% \rangle...\equiv|n_{{\mathbf{k}}_{1}},n_{{\mathbf{k}}_{2}},n_{{\mathbf{k}}_{3}% }...n_{{\mathbf{k}}_{l}}...\rangle\equiv|\{n_{\mathbf{k}}\}\rangle
  115. a 𝐤 l | n 𝐤 1 , n 𝐤 2 , n 𝐤 3 n 𝐤 l , = n 𝐤 l | n 𝐤 1 , n 𝐤 2 , n 𝐤 3 n 𝐤 l - 1 , a_{{\mathbf{k}}_{l}}|n_{{\mathbf{k}}_{1}},n_{{\mathbf{k}}_{2}},n_{{\mathbf{k}}% _{3}}...n_{{\mathbf{k}}_{l}},...\rangle=\sqrt{n_{{\mathbf{k}}_{l}}}|n_{{% \mathbf{k}}_{1}},n_{{\mathbf{k}}_{2}},n_{{\mathbf{k}}_{3}}...n_{{\mathbf{k}}_{% l}}-1,...\rangle
  116. a 𝐤 l | n 𝐤 1 , n 𝐤 2 , n 𝐤 3 n 𝐤 l , = n 𝐤 l + 1 | n 𝐤 1 , n 𝐤 2 , n 𝐤 3 n 𝐤 l + 1 , a^{\dagger}_{{\mathbf{k}}_{l}}|n_{{\mathbf{k}}_{1}},n_{{\mathbf{k}}_{2}},n_{{% \mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}},...\rangle=\sqrt{n_{{\mathbf{k}}_{l}}+% 1}|n_{{\mathbf{k}}_{1}},n_{{\mathbf{k}}_{2}},n_{{\mathbf{k}}_{3}}...n_{{% \mathbf{k}}_{l}}+1,...\rangle
  117. n ^ 𝐤 = n ^ 𝐤 l \hat{n}_{\mathbf{k}}=\sum\hat{n}_{\mathbf{k}_{l}}
  118. n ^ 𝐤 | { n 𝐤 } = ( n 𝐤 l ) | { n 𝐤 } \hat{n}_{\mathbf{k}}|\{n_{\mathbf{k}}\}\rangle=\bigg(\sum n_{\mathbf{k}_{l}}% \bigg)|\{n_{\mathbf{k}}\}\rangle
  119. H ^ | { n 𝐤 } = ( ω ( n 𝐤 l + 1 2 ) ) | { n 𝐤 } \hat{H}|\{n_{\mathbf{k}}\}\rangle=\bigg(\sum\hbar\omega\big(n_{\mathbf{k}_{l}}% +\frac{1}{2}\big)\bigg)|\{n_{\mathbf{k}}\}\rangle
  120. φ ( α ) \scriptstyle\varphi(\alpha)\,
  121. 2 n 2n

Fokker–Planck_equation.html

  1. W t W_{t}
  2. d X t = μ ( X t , t ) d t + σ ( X t , t ) d W t dX_{t}=\mu(X_{t},t)dt+\sigma(X_{t},t)dW_{t}
  3. μ ( X t , t ) \mu(X_{t},t)
  4. D ( X t , t ) = σ 2 ( X t , t ) / 2 D(X_{t},t)=\sigma^{2}(X_{t},t)/2
  5. p ( x , t ) p(x,t)
  6. X t X_{t}
  7. t p ( x , t ) = - x [ μ ( x , t ) p ( x , t ) ] + 2 x 2 [ D ( x , t ) p ( x , t ) ] . \frac{\partial}{\partial t}p(x,t)=-\frac{\partial}{\partial x}\left[\mu(x,t)p(% x,t)\right]+\frac{\partial^{2}}{\partial x^{2}}\left[D(x,t)p(x,t)\right].
  8. σ = 2 D \sigma=\sqrt{2D}
  9. \mathcal{L}
  10. p ( X t ) = lim Δ t 0 1 Δ t ( 𝔼 [ p ( X t + Δ t ) | X t = x ] - p ( x ) ) \mathcal{L}p(X_{t})=\lim_{\Delta t\rightarrow 0}\frac{1}{\Delta t}\left(% \mathbb{E}\big[p(X_{t+\Delta t})|X_{t}=x\big]-p(x)\right)
  11. t , t ( x | x ) \mathbb{P}_{t,t^{\prime}}(x|x^{\prime})
  12. ( t , x ) (t^{\prime},x^{\prime})
  13. ( t , x ) (t,x)
  14. 𝔼 ( p ( X t + Δ t ) | X t = x ) = p ( y ) t + Δ t ( y | x ) d y \mathbb{E}(p(X_{t+\Delta t})|X_{t}=x)=\int p(y)\,\mathbb{P}_{t+\Delta t}(y|x)dy
  15. \mathcal{L}
  16. t , t ( x | x ) \mathbb{P}_{t,t^{\prime}}(x|x^{\prime})
  17. d x dx
  18. p ( y ) t + Δ t , t ( y | x ) t , t ( x | x ) d x d y - p ( x ) t , t ( x | x ) d x \begin{aligned}&\displaystyle\int p(y)\int\mathbb{P}_{t+\Delta t,t}(y|x)\,% \mathbb{P}_{t,t^{\prime}}(x|x^{\prime})dxdy-\int p(x)\,\mathbb{P}_{t,t^{\prime% }}(x|x^{\prime})\,dx\end{aligned}
  19. t + Δ t , t ( y | x ) t , t ( x | x ) d x = t + Δ t , t ( y | x ) \int\mathbb{P}_{t+\Delta t,t}(y|x)\,\mathbb{P}_{t,t^{\prime}}(x|x^{\prime})\,% dx=\mathbb{P}_{t+\Delta t,t^{\prime}}(y|x^{\prime})
  20. y y
  21. x x
  22. = p ( x ) lim Δ t 0 1 Δ t ( t + Δ t , t ( x | x ) - t , t ( x | x ) ) d x \begin{aligned}&\displaystyle=\int p(x)\lim_{\Delta t\rightarrow 0}\frac{1}{% \Delta t}\left(\mathbb{P}_{t+\Delta t,t^{\prime}}(x|x^{\prime})-\mathbb{P}_{t,% t^{\prime}}(x|x^{\prime})\right)dx\end{aligned}
  23. [ p ( x ) ] t , t ( x | x ) d x = p ( x ) t t , t ( x | x ) d x \int\left[\mathcal{L}p(x)\right]\mathbb{P}_{t,t^{\prime}}(x|x^{\prime})\,dx=% \int p(x)\,\partial_{t}\mathbb{P}_{t,t^{\prime}}(x|x^{\prime})\,dx
  24. \mathcal{L}
  25. \mathcal{L}^{\dagger}
  26. [ p ( x ) ] t , t ( x | x ) d x = p ( x ) [ t , t ( x | x ) ] d x \int\left[\mathcal{L}p(x)\right]\mathbb{P}_{t,t^{\prime}}(x|x^{\prime})dx=\int p% (x)\left[\mathcal{L}^{\dagger}\mathbb{P}_{t,t^{\prime}}(x|x^{\prime})\right]dx
  27. p ( x , t ) = t , t ( x | x ) p(x,t)=\mathbb{P}_{t,t^{\prime}}(x|x^{\prime})
  28. p ( x , t ) = t p ( x , t ) \mathcal{L}^{\dagger}p(x,t)=\partial_{t}p(x,t)
  29. \mathcal{L}
  30. 𝔼 ( p ( X t ) ) = p ( X 0 ) + 𝔼 ( 0 t ( t + μ x + σ 2 2 x 2 ) p ( X t ) d t ) \mathbb{E}(p(X_{t}))=p(X_{0})+\mathbb{E}(\int_{0}^{t}\left(\partial_{t}+\mu% \partial_{x}+\frac{\sigma^{2}}{2}\partial_{x}^{2}\right)p(X_{t^{\prime}})dt^{% \prime})
  31. d W t dW_{t}
  32. = μ x + σ 2 2 x 2 \mathcal{L}=\mu\partial_{x}+\frac{\sigma^{2}}{2}\partial_{x}^{2}
  33. = - x ( μ ) + 1 2 x 2 ( σ 2 ) \mathcal{L}^{\dagger}=-\partial_{x}(\mu\cdot)+\frac{1}{2}\partial_{x}^{2}(% \sigma^{2}\cdot)
  34. t p ( x , t ) = - x ( μ ( x , t ) p ( x , t ) ) + x 2 ( σ ( x , t ) 2 2 p ( x , t ) ) . \partial_{t}p(x,t)=-\partial_{x}\left(\mu(x,t)\cdot p(x,t)\right)+\partial_{x}% ^{2}\left(\frac{\sigma(x,t)^{2}}{2}\,p(x,t)\right).
  35. d X t = [ μ ( X t , t ) - 1 2 X t D ( X t , t ) ] d t + 2 D ( X t , t ) d W t . dX_{t}=\left[\mu(X_{t},t)-\frac{1}{2}\frac{\partial}{\partial X_{t}}D(X_{t},t)% \right]dt+\sqrt{2D(X_{t},t)}\circ dW_{t}.
  36. t p ( x , t ) = D 0 2 x 2 [ p ( x , t ) ] \frac{\partial}{\partial t}p(x,t)=D_{0}\frac{\partial^{2}}{\partial x^{2}}% \left[p(x,t)\right]
  37. { 0 x L } \{0\leqslant x\leqslant L\}
  38. p ( 0 , t ) = p ( L , t ) = 0 p(0,t)=p(L,t)=0
  39. p ( x , 0 ) = p 0 ( x ) p(x,0)=p_{0}(x)
  40. Δ x Δ v D 0 \Delta x\Delta v\geqslant D_{0}
  41. D 0 D_{0}
  42. D j {D_{j}}
  43. Δ x \Delta x
  44. Δ v \Delta v
  45. d 𝐗 t = s y m b o l μ ( 𝐗 t , t ) d t + s y m b o l σ ( 𝐗 t , t ) d 𝐖 t , d\mathbf{X}_{t}=symbol{\mu}(\mathbf{X}_{t},t)\,dt+symbol{\sigma}(\mathbf{X}_{t% },t)\,d\mathbf{W}_{t},
  46. 𝐗 t \mathbf{X}_{t}
  47. s y m b o l μ ( 𝐗 t , t ) symbol{\mu}(\mathbf{X}_{t},t)
  48. s y m b o l σ ( 𝐗 t , t ) symbol{\sigma}(\mathbf{X}_{t},t)
  49. × \times
  50. 𝐖 t \mathbf{W}_{t}
  51. p ( 𝐱 , t ) p(\mathbf{x},t)
  52. 𝐗 t \mathbf{X}_{t}
  53. p ( 𝐱 , t ) t = - i = 1 N x i [ μ i ( 𝐱 , t ) p ( 𝐱 , t ) ] + 1 2 i = 1 N j = 1 N 2 x i x j [ D i j ( 𝐱 , t ) p ( 𝐱 , t ) ] , \frac{\partial p(\mathbf{x},t)}{\partial t}=-\sum_{i=1}^{N}\frac{\partial}{% \partial x_{i}}\left[\mu_{i}(\mathbf{x},t)p(\mathbf{x},t)\right]+\frac{1}{2}% \sum_{i=1}^{N}\sum_{j=1}^{N}\frac{\partial^{2}}{\partial x_{i}\,\partial x_{j}% }\left[D_{ij}(\mathbf{x},t)p(\mathbf{x},t)\right],
  54. s y m b o l μ = ( μ 1 , , μ N ) symbol{\mu}=(\mu_{1},\ldots,\mu_{N})
  55. D i j ( 𝐱 , t ) = k = 1 M σ i k ( 𝐱 , t ) σ j k ( 𝐱 , t ) . D_{ij}(\mathbf{x},t)=\sum_{k=1}^{M}\sigma_{ik}(\mathbf{x},t)\sigma_{jk}(% \mathbf{x},t).
  56. d 𝐗 t = s y m b o l μ ( 𝐗 t , t ) d t + s y m b o l σ ( 𝐗 t , t ) d 𝐖 t , d\mathbf{X}_{t}=symbol{\mu}(\mathbf{X}_{t},t)\,dt+symbol{\sigma}(\mathbf{X}_{t% },t)\circ d\mathbf{W}_{t},
  57. p ( 𝐱 , t ) t = - i = 1 N x i [ μ i ( 𝐱 , t ) p ( 𝐱 , t ) ] + 1 2 k = 1 M i = 1 N x i { σ i k ( 𝐱 , t ) j = 1 N x j [ σ j k ( 𝐱 , t ) p ( 𝐱 , t ) ] } \frac{\partial p(\mathbf{x},t)}{\partial t}=-\sum_{i=1}^{N}\frac{\partial}{% \partial x_{i}}\left[\mu_{i}(\mathbf{x},t)\,p(\mathbf{x},t)\right]+\frac{1}{2}% \sum_{k=1}^{M}\sum_{i=1}^{N}\frac{\partial}{\partial x_{i}}\left\{\sigma_{ik}(% \mathbf{x},t)\sum_{j=1}^{N}\frac{\partial}{\partial x_{j}}\left[\sigma_{jk}(% \mathbf{x},t)\,p(\mathbf{x},t)\right]\right\}
  58. d X t = d W t . dX_{t}=dW_{t}.
  59. p ( x , t ) t = 1 2 2 p ( x , t ) x 2 , \frac{\partial p(x,t)}{\partial t}=\frac{1}{2}\frac{\partial^{2}p(x,t)}{% \partial x^{2}},
  60. p ( x , 0 ) = δ ( x ) p(x,0)=\delta(x)
  61. p ( x , t ) = 1 2 π t e - x 2 / ( 2 t ) . p(x,t)=\frac{1}{\sqrt{2\pi t}}e^{-{x^{2}}/({2t})}.
  62. d X t = - a X t d t + σ d W t dX_{t}=-aX_{t}dt+\sigma dW_{t}
  63. 0 < a < 1 0<a<1
  64. p ( x , t ) t = a x ( x p ( x , t ) ) + σ 2 2 2 p ( x , t ) x 2 , \frac{\partial p(x,t)}{\partial t}=a\frac{\partial}{\partial x}\left(x\,p(x,t)% \right)+\frac{\sigma^{2}}{2}\frac{\partial^{2}p(x,t)}{\partial x^{2}},
  65. t p = 0 \partial_{t}p=0
  66. p ( x , t ) = a π σ 2 e - a x 2 σ 2 . p(x,t)=\sqrt{\frac{a}{\pi\sigma^{2}}}e^{-\frac{ax^{2}}{\sigma^{2}}}.
  67. s s
  68. p s ( x , v , t ) p_{s}\left(\vec{x},\vec{v},t\right)
  69. p s t + v p s + Z s e m s ( E + v × B ) v p s = - v i ( p s Δ v i ) + 1 2 2 v i v j ( p s Δ v i Δ v j ) \frac{\partial p_{s}}{\partial t}+\vec{v}\cdot\vec{\nabla}p_{s}+\frac{Z_{s}e}{% m_{s}}\left(\vec{E}+\vec{v}\times\vec{B}\right)\cdot\vec{\nabla}_{v}p_{s}=-% \frac{\partial}{\partial v_{i}}\left(p_{s}\langle\Delta v_{i}\rangle\right)+% \frac{1}{2}\frac{\partial^{2}}{\partial v_{i}\partial v_{j}}\left(p_{s}\langle% \Delta v_{i}\Delta v_{j}\rangle\right)
  70. Δ v i \langle\Delta v_{i}\rangle
  71. Δ v i Δ v j \langle\Delta v_{i}\Delta v_{j}\rangle
  72. s s
  73. f ( 𝐯 , t ) d 𝐯 f(\mathbf{v},t)d\mathbf{v}
  74. ( 𝐯 , 𝐯 + d 𝐯 ) (\mathbf{v},\mathbf{v}+d\mathbf{v})
  75. 𝐯 0 \mathbf{v}_{0}
  76. f 0 ( x ) f_{0}(x)
  77. f ˙ 0 ( x ) = 0 \dot{f}_{0}(x)=0
  78. σ ( 𝐗 t , t ) {\sigma}(\mathbf{X}_{t},t)
  79. σ ( 𝐗 t , t ) {\sigma}(\mathbf{X}_{t},t)
  80. σ ( 𝐗 t , t ) {\sigma}(\mathbf{X}_{t},t)
  81. t p ( x , t ) = - d x ( [ D 1 ( x , t ) x + D 2 ( x , t ) 2 x 2 ] δ ( x - x ) ) p ( x , t ) . \frac{\partial}{\partial t}p\!\left(x^{\prime},t\right)=\int_{-\infty}^{\infty% }dx\left(\left[D_{1}\left(x,t\right)\frac{\partial}{\partial x}+D_{2}\left(x,t% \right)\frac{\partial^{2}}{\partial x^{2}}\right]\delta\left(x^{\prime}-x% \right)\right)p\!\left(x,t\right).
  82. δ \delta
  83. p ( x , t ) p(x,t)
  84. ε \varepsilon
  85. p ( x , t + ε ) = - d x ( ( 1 + ε [ D 1 ( x , t ) x + D 2 ( x , t ) 2 x 2 ] ) δ ( x - x ) ) p ( x , t ) + O ( ε 2 ) . p\!\left(x^{\prime},t+\varepsilon\right)=\int_{-\infty}^{\infty}\,dx\left(% \left(1+\varepsilon\left[D_{1}\left(x,t\right)\frac{\partial}{\partial x}+D_{2% }\left(x,t\right)\frac{\partial^{2}}{\partial x^{2}}\right]\right)\delta\left(% x^{\prime}-x\right)\right)p\!\left(x,t\right)+O\left(\varepsilon^{2}\right).
  86. δ ( x - x ) = - i i d x ~ 2 π i e x ~ ( x - x ) \delta\left(x^{\prime}-x\right)=\int_{-i\infty}^{i\infty}\frac{d\tilde{x}}{2% \pi i}e^{\tilde{x}\left(x-x^{\prime}\right)}
  87. δ \delta
  88. p ( x , t + ε ) = - d x - i i d x ~ 2 π i ( 1 + ε [ x ~ D 1 ( x , t ) + x ~ 2 D 2 ( x , t ) ] ) e x ~ ( x - x ) p ( x , t ) + O ( ε 2 ) = - d x - i i d x ~ 2 π i exp ( ε [ - x ~ ( x - x ) ε + x ~ D 1 ( x , t ) + x ~ 2 D 2 ( x , t ) ] ) p ( x , t ) + O ( ε 2 ) . \begin{aligned}\displaystyle p\!\left(x^{\prime},t+\varepsilon\right)&% \displaystyle=\int_{-\infty}^{\infty}dx\int_{-i\infty}^{i\infty}\frac{d\tilde{% x}}{2\pi i}\left(1+\varepsilon\left[\tilde{x}D_{1}\left(x,t\right)+\tilde{x}^{% 2}D_{2}\left(x,t\right)\right]\right)e^{\tilde{x}\left(x-x^{\prime}\right)}p\!% \left(x,t\right)+O\left(\varepsilon^{2}\right)\\ &\displaystyle=\int_{-\infty}^{\infty}dx\int_{-i\infty}^{i\infty}\frac{d\tilde% {x}}{2\pi i}\exp\left(\varepsilon\left[-\tilde{x}\frac{\left(x^{\prime}-x% \right)}{\varepsilon}+\tilde{x}D_{1}\left(x,t\right)+\tilde{x}^{2}D_{2}\left(x% ,t\right)\right]\right)p\!\left(x,t\right)+O\left(\varepsilon^{2}\right).\end{aligned}
  89. p ( x , t + ε ) p\!\left(x^{\prime},t+\varepsilon\right)
  90. p ( x , t ) p\!\left(x,t\right)
  91. ( t - t ) / ε \left(t^{\prime}-t\right)/\varepsilon
  92. ε 0 \varepsilon\longrightarrow 0
  93. L = d t [ x ~ D 1 ( x , t ) + x ~ 2 D 2 ( x , t ) - x ~ x t ] . L=\int dt\left[\tilde{x}D_{1}\left(x,t\right)+\tilde{x}^{2}D_{2}\left(x,t% \right)-\tilde{x}\frac{\partial x}{\partial t}\right].
  94. x ~ \tilde{x}
  95. x x

Food_web.html

  1. C = t L S ( S - 1 ) / 2 C=\cfrac{t_{L}}{S(S-1)/2}

Force_spectroscopy.html

  1. k B T x β \frac{k_{B}T}{x_{\beta}}
  2. x β x_{\beta}

Forced_perspective.html

  1. θ = 2 tan - 1 ( h 2 D ) \theta=2\tan^{-1}\left(\frac{h}{2D}\right)

Forcing_(mathematics).html

  1. 0 , 1 , 2 , {0,1,2,…}
  2. V * = V × { 0 , 1 } , V^{*}=V\times\{0,1\},\,
  3. x V x\in V
  4. ( x , 0 ) (x,0)
  5. ( x , 1 ) (x,1)
  6. ( P , , 1 ) (P,≤,1)
  7. P P
  8. p P p∈P
  9. q , r P q,r∈P
  10. q , r p q,r≤p
  11. s P s∈P
  12. s q , r s≤q,r
  13. P P
  14. 1 1
  15. p 1 p≤1
  16. p P p∈P
  17. P P
  18. p q p≤q
  19. p p
  20. q q
  21. P P
  22. P P
  23. P P
  24. [ u ( ] ( u , p ) : u [u^{\prime}(^{\prime}](u,p):u
  25. P P
  26. p P p∈P
  27. u u
  28. p p
  29. [ u ) ] [u^{\prime})^{\prime}]
  30. N a m e ( 0 ) = Name(0)={}
  31. N a m e ( α + 1 ) Name(α+1)
  32. ( N a m e ( α ) × P ) (Name(α)×P)
  33. λ λ
  34. [ u ) ] [u^{\prime})^{\prime}]
  35. P P
  36. [ u ) ] [u^{\prime})^{\prime}]
  37. P P
  38. x V x∈V
  39. x ˇ
  40. P P
  41. x ˇ = ( y ˇ , 1 ) : y x xˇ={(yˇ,1):y∈x}
  42. G G
  43. P P
  44. P P
  45. v a l ( u , G ) = v a l ( v , G ) : p G , ( v , p ) u val(u,G)={val(v,G):∃p∈G,(v,p)∈u}
  46. 1 1
  47. G G
  48. v a l ( x ˇ , G ) = x val(xˇ,G)=x
  49. ( B o r ( 𝐈 ) , , 𝐈 ) (Bor(\mathbf{I}),⊆,\mathbf{I})
  50. I = 0 , 11 I=0,11
  51. B o r ( I ) Bor(I)
  52. I I
  53. B o r ( I ) Bor(I)
  54. M , P \Vdash_{M,P}
  55. M , P \Vdash_{M,P}
  56. M , P \Vdash_{M,P}
  57. M , P \Vdash_{M,P}
  58. M , P \Vdash_{M,P}
  59. M , P \Vdash_{M,P}
  60. M , P \Vdash_{M,P}
  61. M , P \Vdash_{M,P}
  62. M , P \Vdash_{M,P}
  63. V V
  64. p P a b p\Vdash_{P}a\in b
  65. ( q p ) ( r q ) ( s , c ) ( ( s , c ) b r s r P a = c ) (\forall q\leq p)(\exists r\leq q)(\exists s,c)((s,c)\in b\land r\leq s\land r% \Vdash_{P}a=c)
  66. p P a = b p\Vdash_{P}a=b
  67. ( q p ) ( c V P ) ( q P c a q P c b ) (\forall q\leq p)(\forall c\in V^{P})(q\Vdash_{P}c\in a\,\Leftrightarrow q% \Vdash_{P}c\in b)
  68. p P ¬ f p\Vdash_{P}\lnot f
  69. ¬ ( q p ) q P f \lnot(\exists q\leq p)q\Vdash_{P}f
  70. p P ( f g ) p\Vdash_{P}(f\land g)
  71. ( p P f p P g ) (p\Vdash_{P}f\,\land\,p\Vdash_{P}g)
  72. p P ( x ) f p\Vdash_{P}(\forall x)f
  73. ( x V P ) p P f (\forall x\in V^{P})p\Vdash_{P}f
  74. p p
  75. a a
  76. a a
  77. f f
  78. g g
  79. f ( x 1 , , x n ) f(x_{1},\dots,x_{n})
  80. p P f ( x 1 , , x n ) p\vDash_{P}f(x_{1},\dots,x_{n})
  81. p , P , x 1 , , x n p,P,x_{1},\dots,x_{n}
  82. f ( x 1 , , x n ) f(x_{1},\dots,x_{n})
  83. ( p , P , x 1 , , x n ) ( p P f ( x 1 , , x n ) ( p o ( P ) p d o m ( P ) x 1 , , x n V P ) (\forall p,P,x_{1},\dots,x_{n})(p\Vdash_{P}f(x_{1},\dots,x_{n})\Rightarrow(po(% P)\land p\in dom(P)\land x_{1},\dots,x_{n}\in V^{P})
  84. p o ( P ) po(P)
  85. P P
  86. V V
  87. M M
  88. ( M , P , x 1 , , x n ) ( c t m ( M ) p o ( P ) P M p d o m ( P ) x 1 , , x n M P ( p M , P f ( x 1 , , x n ) M p P f ( x 1 , , x n ) ) ) (\forall M,P,x_{1},\dots,x_{n})(ctm(M)\land po(P)\land P\in M\land p\in dom(P)% \land x_{1},\dots,x_{n}\in M^{P}\Rightarrow(p\Vdash_{M,P}f(x_{1},\dots,x_{n})% \,\Leftrightarrow\,M\models p\Vdash_{P}f(x_{1},\dots,x_{n})))
  89. c t m ( M ) ctm(M)
  90. M M
  91. Z F ZF
  92. f f
  93. \Vdash
  94. 0 , 1 {0,1}
  95. p : p ( n ) i s d e f i n e d {p:p(n)isdefined}
  96. ( n ˇ , p ) : p ( n ) = 1 {(nˇ,p):p(n)=1}
  97. p : n , n d o m ( p ) a n d p ( n ) = 1 i f a n d o n l y i f n A {p:∃n,n∈dom(p)andp(n)=1ifandonlyifn∉A}
  98. p : n , p ( n , α ) p ( n , β ) {p:∃n,p(n,α)≠p(n,β)}
  99. p : p q , s o m e q A {p:p≤q,someq∈A}
  100. b : q p , q f o r c e s u ( a ˇ ) = b ˇ {b:∃q≤p,qforcesu(aˇ)=bˇ}
  101. B ( i n V ) : r B * ( i n V G G ) {B(inV):r∈B*(inVGG)}
  102. Z F C Con ( Z F C ) Con ( Z F C + H ) . ZFC\vdash\operatorname{Con}(ZFC)\rightarrow\operatorname{Con}(ZFC+H).
  103. Z F C + ¬ Con ( Z F C + H ) T ( Fin ( T ) T Z F C ( T ¬ H ) ) . ZFC+\lnot\operatorname{Con}(ZFC+H)\vdash\exists T(\operatorname{Fin}(T)\land T% \subset ZFC\land(T\vdash\lnot H)).
  104. Z F C T ( ( T ¬ H ) ( Z F C ( T ¬ H ) ) ) . ZFC\vdash\forall T((T\vdash\lnot H)\rightarrow(ZFC\vdash(T\vdash\lnot H))).
  105. Z F C + ¬ Con ( Z F C + H ) T ( Fin ( T ) T Z F C ( Z F C ( T ¬ H ) ) ) . ZFC+\lnot\operatorname{Con}(ZFC+H)\vdash\exists T(\operatorname{Fin}(T)\land T% \subset ZFC\land(ZFC\vdash(T\vdash\lnot H))).
  106. Z F C T ( Fin ( T ) T Z F C ( Z F C Con ( T + H ) ) ) ZFC\vdash\forall T(\operatorname{Fin}(T)\land T\subset ZFC\rightarrow(ZFC% \vdash\operatorname{Con}(T+H)))
  107. Z F C + ¬ Con ( Z F C + H ) T ( Fin ( T ) T Z F C ( Z F C ( T ¬ H ) ) ( Z F C Con ( T + H ) ) ) ZFC+\lnot\operatorname{Con}(ZFC+H)\vdash\exists T(\operatorname{Fin}(T)\land T% \subset ZFC\land(ZFC\vdash(T\vdash\lnot H))\land(ZFC\vdash\operatorname{Con}(T% +H)))
  108. Z F C + ¬ Con ( Z F C + H ) ¬ Con ( Z F C ) ZFC+\lnot\operatorname{Con}(ZFC+H)\vdash\lnot\operatorname{Con}(ZFC)
  109. Z F C Con ( Z F C ) Con ( Z F L ) ZFC\vdash\operatorname{Con}(ZFC)\leftrightarrow\operatorname{Con}(ZFL)

Forecasting.html

  1. y ^ T + h | T = y ¯ = ( y 1 + + y T ) / T \hat{y}_{T+h|T}=\bar{y}=(y_{1}+...+y_{T})/T
  2. y 1 , , y T y_{1},...,y_{T}
  3. y ^ T + h | T = y T \hat{y}_{T+h|T}=y_{T}
  4. y ^ T + h | T = 1 T - 1 t = 2 N ( y t - y t - 1 ) = y T + h ( y T - y T - 1 ) T - 1 . \hat{y}_{T+h|T}=\frac{1}{T-1}\sum_{t=2}^{N}(y_{t}-y_{t-1})=y_{T}+\frac{h(y_{T}% -y_{T-1})}{T-1}.
  5. T + h T+h
  6. y ^ T + h | T = y T + h - k m \hat{y}_{T+h|T}=y_{T+h-km}
  7. m m
  8. k k
  9. ( h - 1 ) / m (h-1)/m
  10. E t = Y t - F t \ E_{t}=Y_{t}-F_{t}
  11. M A E = t = 1 N | E t | N \ MAE=\frac{\sum_{t=1}^{N}|E_{t}|}{N}
  12. M A P E = 100 * t = 1 N | E t Y t | N \ MAPE=100*\frac{\sum_{t=1}^{N}|\frac{E_{t}}{Y_{t}}|}{N}
  13. M A D = t = 1 N | E t | N \ MAD=\frac{\sum_{t=1}^{N}|E_{t}|}{N}
  14. P M A D = t = 1 N | E t | t = 1 N | Y t | \ PMAD=\frac{\sum_{t=1}^{N}|E_{t}|}{\sum_{t=1}^{N}|Y_{t}|}
  15. M S E = t = 1 N E t 2 N \ MSE=\frac{\sum_{t=1}^{N}{E_{t}^{2}}}{N}
  16. R M S E = t = 1 N E t 2 N \ RMSE=\sqrt{\frac{\sum_{t=1}^{N}{E_{t}^{2}}}{N}}
  17. S S = 1 - M S E f o r e c a s t M S E r e f \ SS=1-\frac{MSE_{forecast}}{MSE_{ref}}
  18. E ¯ = i = 1 N E i N \ \bar{E}=\frac{\sum_{i=1}^{N}{E_{i}}}{N}
  19. P t = 100 E t / Y t P_{t}=100E_{t}/Y_{t}
  20. Y t Y_{t}

Formal_concept_analysis.html

  1. A ( o ) = o A(o)=o^{\prime}
  2. O ( a ) = a O(a)=a^{\prime}
  3. Y X X Y , X Y X Z Y Z , X Y , Y Z X Z \frac{Y\subseteq X}{X\rightarrow Y},\frac{X\rightarrow Y}{XZ\rightarrow YZ},% \frac{X\rightarrow Y,Y\rightarrow Z}{X\rightarrow Z}

Formula.html

  1. b i g = 1 V = 4 3 π r < s u p > 3 big=1V=\frac{4}{3}πr<sup>3
  2. 3 {}^{−}_{3}
  3. 𝐅 = m 𝐚 \mathbf{F}=m\mathbf{a}
  4. V = 4 3 π ( 2.0 cm ) 3 33.51 cm . 3 V=\frac{4}{3}\pi(2.0\mbox{ cm}~{})^{3}\approx 33.51\mbox{ cm}~{}^{3}.
  5. V VOL t b s p V\equiv\mathrm{VOL}~{}{tbsp}
  6. r RAD c m r\equiv\mathrm{RAD}~{}{cm}
  7. VOL t b s p = 4 3 π RAD 3 c m 3 . \mathrm{VOL}~{}{tbsp}=\frac{4}{3}\pi\mathrm{RAD}^{3}~{}{cm}^{3}.
  8. 1 t b s p = 14.787 c m 3 1~{}{tbsp}=14.787~{}{cm}^{3}
  9. VOL 0.2833 RAD 3 . \mathrm{VOL}\approx 0.2833~{}\mathrm{RAD}^{3}.
  10. V V
  11. r r
  12. V = 0.2833 r 3 . V=0.2833~{}r^{3}.
  13. S = k log W S=k\cdot\log W\!

Forward_contract.html

  1. K K
  2. S T S_{T}
  3. f T = S T - K f_{T}=S_{T}-K
  4. f T = K - S T f_{T}=K-S_{T}
  5. F 0 F_{0}
  6. S 0 S_{0}
  7. F 0 = S 0 e r T F_{0}=S_{0}e^{rT}
  8. r r
  9. F 0 = ( S 0 - I ) e r T F_{0}=(S_{0}-I)e^{rT}
  10. F 0 = S 0 e ( r - q ) T F_{0}=S_{0}e^{(r-q)T}
  11. I = I=
  12. t 0 < T t_{0}<T
  13. q % p . a . q\%p.a.
  14. I I
  15. q q
  16. F 0 = ( S 0 + U ) e r T F_{0}=(S_{0}+U)e^{rT}
  17. F 0 = S 0 e ( r + u ) T F_{0}=S_{0}e^{(r+u)T}
  18. U = U=
  19. t 0 < T t_{0}<T
  20. u % p . a . u\%p.a.
  21. F 0 = ( S 0 + U ) e ( r - y ) T F_{0}=(S_{0}+U)e^{(r-y)T}
  22. F 0 = S 0 e ( r + u - y ) T F_{0}=S_{0}e^{(r+u-y)T}
  23. y % p . a . y\%p.a.
  24. c c
  25. F 0 = ( S 0 + U - I ) e ( r - y ) T F_{0}=(S_{0}+U-I)e^{(r-y)T}
  26. F 0 = S 0 e c T , where c = r - q + u - y . F_{0}=S_{0}e^{cT},\,\text{ where }c=r-q+u-y.
  27. F 0 F_{0}
  28. E ( S T ) E(S_{T})
  29. E ( S T - K ) = E ( S T ) - K E(S_{T}-K)=E(S_{T})-K
  30. K K
  31. E ( S T ) - K > 0 E(S_{T})-K>0
  32. E ( S T ) > K E(S_{T})>K
  33. E ( S T ) > F 0 E(S_{T})>F_{0}
  34. K = F 0 K=F_{0}
  35. E ( S T ) > F 0 E(S_{T})>F_{0}
  36. E ( S T ) < F 0 E(S_{T})<F_{0}
  37. S t S_{t}
  38. t t
  39. r r
  40. T T
  41. F t , T = S t e r ( T - t ) F_{t,T}=S_{t}e^{r(T-t)}
  42. F t , T > S t e r ( T - t ) F_{t,T}>S_{t}e^{r(T-t)}
  43. t t
  44. S t S_{t}
  45. S t S_{t}
  46. T T
  47. T T
  48. t t
  49. - S t e r ( T - t ) -S_{t}e^{r(T-t)}
  50. F t , T F_{t,T}
  51. F t , T F_{t,T}
  52. S T S_{T}
  53. F t , T - S t e r ( T - t ) F_{t,T}-S_{t}e^{r(T-t)}
  54. F t , T < S t e r ( T - t ) F_{t,T}<S_{t}e^{r(T-t)}
  55. F V T ( X ) FV_{T}(X)
  56. T T
  57. F t , T = S t e r ( T - t ) - F V T ( all cash flows over the life of the contract ) F_{t,T}=S_{t}e^{r(T-t)}-FV_{T}(\,\text{all cash flows over the life of the % contract})
  58. F t , T = ( S t - I t ) e r ( T - t ) F_{t,T}=(S_{t}-I_{t})e^{r(T-t)}\,
  59. I t I_{t}

Forward_rate_agreement.html

  1. Payment = Notional Amount * ( ( Reference Rate - Fixed Rate ) * α ( 1 + Reference Rate * α ) ) \mbox{Payment}~{}=\mbox{Notional Amount}~{}*\left(\frac{(\mbox{Reference Rate}% ~{}-\mbox{Fixed Rate}~{})*\alpha}{(1+\mbox{Reference Rate}~{}*\alpha)}\right)
  2. α \alpha

Foucault_pendulum.html

  1. ω = 360 sin φ / day \omega=360\sin\varphi\ ^{\circ}/\mathrm{day}
  2. F c , x = 2 m Ω d y d t sin ( φ ) F c , y = - 2 m Ω d x d t sin ( φ ) \begin{aligned}\displaystyle F_{c,x}&\displaystyle=2m\Omega\dfrac{dy}{dt}\sin(% \varphi)\\ \displaystyle F_{c,y}&\displaystyle=-2m\Omega\dfrac{dx}{dt}\sin(\varphi)\end{aligned}
  3. F g , x = - m ω 2 x F g , y = - m ω 2 y . \begin{aligned}\displaystyle F_{g,x}&\displaystyle=-m\omega^{2}x\\ \displaystyle F_{g,y}&\displaystyle=-m\omega^{2}y.\end{aligned}
  4. d 2 x d t 2 = - ω 2 x + 2 Ω d y d t sin ( φ ) d 2 y d t 2 = - ω 2 y - 2 Ω d x d t sin ( φ ) . \begin{aligned}\displaystyle\dfrac{d^{2}x}{dt^{2}}&\displaystyle=-\omega^{2}x+% 2\Omega\dfrac{dy}{dt}\sin(\varphi)\\ \displaystyle\dfrac{d^{2}y}{dt^{2}}&\displaystyle=-\omega^{2}y-2\Omega\dfrac{% dx}{dt}\sin(\varphi)\,.\end{aligned}
  5. d 2 z d t 2 + 2 i Ω d z d t sin ( φ ) + ω 2 z = 0 . \frac{d^{2}z}{dt^{2}}+2i\Omega\frac{dz}{dt}\sin(\varphi)+\omega^{2}z=0\,.
  6. z = e - i Ω sin ( φ ) t ( c 1 e i ω t + c 2 e - i ω t ) . z=e^{-i\Omega\sin(\varphi)t}\left(c_{1}e^{i\omega t}+c_{2}e^{-i\omega t}\right% )\,.
  7. ϕ \phi
  8. ϕ \phi
  9. ϕ \phi
  10. 2 π sin ( ϕ ) 2\pi\,\sin(\phi)

Four-acceleration.html

  1. 𝐀 = d 𝐔 d τ = ( γ u γ ˙ u c , γ u 2 𝐚 + γ u γ ˙ u 𝐮 ) = ( γ u 4 𝐚 𝐮 c , γ u 2 𝐚 + γ u 4 ( 𝐚 𝐮 ) c 2 𝐮 ) \mathbf{A}=\frac{d\mathbf{U}}{d\tau}=\left(\gamma_{u}\dot{\gamma}_{u}c,\gamma_% {u}^{2}\mathbf{a}+\gamma_{u}\dot{\gamma}_{u}\mathbf{u}\right)=\left(\gamma_{u}% ^{4}\frac{\mathbf{a}\cdot\mathbf{u}}{c},\gamma_{u}^{2}\mathbf{a}+\gamma_{u}^{4% }\frac{\left(\mathbf{a}\cdot\mathbf{u}\right)}{c^{2}}\mathbf{u}\right)
  2. 𝐚 = d 𝐮 d t \mathbf{a}={d\mathbf{u}\over dt}
  3. γ ˙ u = 𝐚 𝐮 c 2 γ u 3 = 𝐚 𝐮 c 2 1 ( 1 - u 2 c 2 ) 3 / 2 \dot{\gamma}_{u}=\frac{\mathbf{a\cdot u}}{c^{2}}\gamma_{u}^{3}=\frac{\mathbf{a% \cdot u}}{c^{2}}\frac{1}{\left(1-\frac{u^{2}}{c^{2}}\right)^{3/2}}
  4. γ u \gamma_{u}
  5. u u
  6. τ \tau
  7. 𝐮 = 0 \mathbf{u}=0
  8. γ u = 1 \gamma_{u}=1
  9. γ ˙ u = 0 \dot{\gamma}_{u}=0
  10. 𝐀 = ( 0 , 𝐚 ) \mathbf{A}=\left(0,\mathbf{a}\right)
  11. F μ = m A μ F^{\mu}=mA^{\mu}
  12. A λ := D U λ d τ = d U λ d τ + Γ λ U μ μ ν U ν A^{\lambda}:=\frac{DU^{\lambda}}{d\tau}=\frac{dU^{\lambda}}{d\tau}+\Gamma^{% \lambda}{}_{\mu\nu}U^{\mu}U^{\nu}
  13. Γ λ μ ν \Gamma^{\lambda}{}_{\mu\nu}

Four-force.html

  1. 𝐅 = d 𝐏 d τ \mathbf{F}={d\mathbf{P}\over d\tau}
  2. 𝐏 = m 𝐔 \mathbf{P}=m\mathbf{U}\,
  3. 𝐔 = γ ( c , 𝐮 ) \mathbf{U}=\gamma(c,\mathbf{u})\,
  4. 𝐀 \mathbf{A}
  5. 𝐅 = m 𝐀 = ( γ 𝐟 𝐮 c , γ 𝐟 ) \mathbf{F}=m\mathbf{A}=\left(\gamma{\mathbf{f}\cdot\mathbf{u}\over c},\gamma{% \mathbf{f}}\right)
  6. 𝐟 = d d t ( γ m 𝐮 ) = d 𝐩 d t {\mathbf{f}}={d\over dt}\left(\gamma m{\mathbf{u}}\right)={d\mathbf{p}\over dt}
  7. 𝐟 𝐮 = d d t ( γ m c 2 ) = d E d t {\mathbf{f}\cdot\mathbf{u}}={d\over dt}\left(\gamma mc^{2}\right)={dE\over dt}
  8. 𝐮 \mathbf{u}
  9. 𝐩 \mathbf{p}
  10. 𝐟 \mathbf{f}
  11. F λ := D P λ d τ = d P λ d τ + Γ λ U μ μ ν P ν F^{\lambda}:=\frac{DP^{\lambda}}{d\tau}=\frac{dP^{\lambda}}{d\tau}+\Gamma^{% \lambda}{}_{\mu\nu}U^{\mu}P^{\nu}
  12. F μ = ( F 0 , 𝐅 ) F^{\mu}=(F^{0},\mathbf{F})
  13. m m
  14. f μ f^{\mu}
  15. v v
  16. 𝐟 = 𝐅 + ( γ - 1 ) 𝐯 𝐯 𝐅 v 2 , {\mathbf{f}}={\mathbf{F}}+(\gamma-1){\mathbf{v}}{{\mathbf{v}}\cdot{\mathbf{F}}% \over v^{2}},
  17. f 0 = γ s y m b o l β 𝐅 = s y m b o l β 𝐟 . f^{0}=\gamma symbol{\beta}\cdot\mathbf{F}=symbol{\beta}\cdot\mathbf{f}.
  18. s y m b o l β = 𝐯 / c symbol{\beta}=\mathbf{v}/c
  19. f μ = m D U μ d τ f^{\mu}=m{DU^{\mu}\over d\tau}
  20. D / d τ D/d\tau
  21. m d 2 x μ d τ 2 = f μ - m Γ ν λ μ d x ν d τ d x λ d τ , m{d^{2}x^{\mu}\over d\tau^{2}}=f^{\mu}-m\Gamma^{\mu}_{\nu\lambda}{dx^{\nu}% \over d\tau}{dx^{\lambda}\over d\tau},
  22. Γ ν λ μ \Gamma^{\mu}_{\nu\lambda}
  23. f f α f^{\alpha}_{f}
  24. ξ α \xi^{\alpha}
  25. x μ x^{\mu}
  26. f μ = x μ ξ α f f α . f^{\mu}={\partial x^{\mu}\over\partial\xi^{\alpha}}f^{\alpha}_{f}.
  27. F μ = q F μ ν U ν F_{\mu}=qF_{\mu\nu}U^{\nu}
  28. F μ ν F_{\mu\nu}
  29. U ν U^{\nu}
  30. q q

Four-momentum.html

  1. E E
  2. 𝐯 \mathbf{v}
  3. γ γ
  4. p = ( p 0 , p 1 , p 2 , p 3 ) = ( E c , p x , p y , p z ) . p=(p^{0},p^{1},p^{2},p^{3})=\left({E\over c},p_{x},p_{y},p_{z}\right).
  5. m 𝐯 m\mathbf{v}
  6. m m
  7. c c
  8. p p = p μ p μ = η μ ν p μ p ν = - E 2 c 2 + | 𝐩 | 2 = - m 2 c 2 p\cdot p=p^{\mu}p_{\mu}=\eta_{\mu\nu}p^{\mu}p^{\nu}=-{E^{2}\over c^{2}}+|% \mathbf{p}|^{2}=-m^{2}c^{2}
  9. η μ ν = [ - 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] \eta_{\mu\nu}=\left[\begin{smallmatrix}-1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{smallmatrix}\right]
  10. p p
  11. q q
  12. p q p⋅q
  13. p μ = m u μ , p^{\mu}=mu^{\mu},
  14. u u
  15. u = ( u 0 , u 1 , u 2 , u 3 ) = γ ( c , v x , v y , v z ) , u=(u^{0},u^{1},u^{2},u^{3})=\gamma(c,v_{x},v_{y},v_{z}),
  16. γ = 1 1 - ( v / c ) 2 \gamma=\frac{1}{\sqrt{1-\left(v/c\right)^{2}}}
  17. c c
  18. M M
  19. M M
  20. p μ A μ = η μ ν p μ A ν = η μ ν p μ d d τ p ν m = 1 2 m d d τ p p = 1 2 m d d τ ( - m 2 c 2 ) = 0. p^{\mu}A_{\mu}=\eta_{\mu\nu}p^{\mu}A^{\nu}=\eta_{\mu\nu}p^{\mu}\frac{d}{d\tau}% \frac{p^{\nu}}{m}=\frac{1}{2m}\frac{d}{d\tau}p\cdot p=\frac{1}{2m}\frac{d}{d% \tau}(-m^{2}c^{2})=0.
  21. q q
  22. A = ( A 0 , A 1 , A 2 , A 3 ) = ( ϕ c , A x , A y , A z ) A=(A^{0},A^{1},A^{2},A^{3})=\left({\phi\over c},A_{x},A_{y},A_{z}\right)
  23. Φ Φ
  24. P P
  25. P μ = p μ + q A μ . P^{\mu}=p^{\mu}+qA^{\mu}.\!

Four-vector.html

  1. 𝐀 = ( A 0 , A 1 , A 2 , A 3 ) = A 0 𝐄 0 + A 1 𝐄 1 + A 2 𝐄 2 + A 3 𝐄 3 = A 0 𝐄 0 + A i 𝐄 i = A α 𝐄 α \begin{aligned}\displaystyle\mathbf{A}&\displaystyle=(A^{0},\,A^{1},\,A^{2},\,% A^{3})\\ &\displaystyle=A^{0}\mathbf{E}_{0}+A^{1}\mathbf{E}_{1}+A^{2}\mathbf{E}_{2}+A^{% 3}\mathbf{E}_{3}\\ &\displaystyle=A^{0}\mathbf{E}_{0}+A^{i}\mathbf{E}_{i}\\ &\displaystyle=A^{\alpha}\mathbf{E}_{\alpha}\\ \end{aligned}
  2. 𝐀 \displaystyle\mathbf{A}
  3. 𝐀 = ( A t , A r , A θ , A ϕ ) = A t 𝐄 t + A r 𝐄 r + A θ 𝐄 θ + A ϕ 𝐄 ϕ \begin{aligned}\displaystyle\mathbf{A}&\displaystyle=(A_{t},\,A_{r},\,A_{% \theta},\,A_{\phi})\\ &\displaystyle=A_{t}\mathbf{E}_{t}+A_{r}\mathbf{E}_{r}+A_{\theta}\mathbf{E}_{% \theta}+A_{\phi}\mathbf{E}_{\phi}\\ \end{aligned}
  4. 𝐀 = ( A t , A r , A θ , A z ) = A t 𝐄 t + A r 𝐄 r + A θ 𝐄 θ + A z 𝐄 z \begin{aligned}\displaystyle\mathbf{A}&\displaystyle=(A_{t},\,A_{r},\,A_{% \theta},\,A_{z})\\ &\displaystyle=A_{t}\mathbf{E}_{t}+A_{r}\mathbf{E}_{r}+A_{\theta}\mathbf{E}_{% \theta}+A_{z}\mathbf{E}_{z}\\ \end{aligned}
  5. 𝐄 0 = ( 1 0 0 0 ) , 𝐄 1 = ( 0 1 0 0 ) , 𝐄 2 = ( 0 0 1 0 ) , 𝐄 3 = ( 0 0 0 1 ) \mathbf{E}_{0}=\begin{pmatrix}1\\ 0\\ 0\\ 0\end{pmatrix}\,,\quad\mathbf{E}_{1}=\begin{pmatrix}0\\ 1\\ 0\\ 0\end{pmatrix}\,,\quad\mathbf{E}_{2}=\begin{pmatrix}0\\ 0\\ 1\\ 0\end{pmatrix}\,,\quad\mathbf{E}_{3}=\begin{pmatrix}0\\ 0\\ 0\\ 1\end{pmatrix}
  6. 𝐀 = ( A 0 A 1 A 2 A 3 ) \mathbf{A}=\begin{pmatrix}A^{0}\\ A^{1}\\ A^{2}\\ A^{3}\end{pmatrix}
  7. A μ = η μ ν A ν , A_{\mu}=\eta_{\mu\nu}A^{\nu}\,,
  8. 𝐀 \displaystyle\mathbf{A}
  9. 𝐄 0 = ( 1 0 0 0 ) , 𝐄 1 = ( 0 1 0 0 ) , 𝐄 2 = ( 0 0 1 0 ) , 𝐄 3 = ( 0 0 0 1 ) \mathbf{E}^{0}=\begin{pmatrix}1&0&0&0\end{pmatrix}\,,\quad\mathbf{E}^{1}=% \begin{pmatrix}0&1&0&0\end{pmatrix}\,,\quad\mathbf{E}^{2}=\begin{pmatrix}0&0&1% &0\end{pmatrix}\,,\quad\mathbf{E}^{3}=\begin{pmatrix}0&0&0&1\end{pmatrix}
  10. 𝐀 = ( A 0 A 1 A 2 A 3 ) \mathbf{A}=\begin{pmatrix}A_{0}&A_{1}&A_{2}&A_{3}\end{pmatrix}
  11. 𝐀 = s y m b o l Λ 𝐀 \mathbf{A}^{\prime}=symbol{\Lambda}\mathbf{A}
  12. A μ = Λ μ A ν ν , A μ = Λ μ A ν ν {A^{\prime}}^{\mu}=\Lambda^{\mu}{}_{\nu}A^{\nu}\,,\quad{A^{\prime}}_{\mu}=% \Lambda_{\mu}{}^{\nu}A_{\nu}
  13. 𝐧 ^ = ( n ^ 1 , n ^ 2 , n ^ 3 ) , \hat{\mathbf{n}}=(\hat{n}_{1},\hat{n}_{2},\hat{n}_{3})\,,
  14. Λ 00 = 1 \Lambda_{00}=1
  15. Λ 0 i = Λ i 0 = 0 \Lambda_{0i}=\Lambda_{i0}=0
  16. Λ i j = ( δ i j - n ^ i n ^ j ) cos θ - ε i j k n ^ k sin θ + n ^ i n ^ j \Lambda_{ij}=(\delta_{ij}-\hat{n}_{i}\hat{n}_{j})\cos\theta-\varepsilon_{ijk}% \hat{n}_{k}\sin\theta+\hat{n}_{i}\hat{n}_{j}
  17. ( A 0 A 1 A 2 A 3 ) = ( 1 0 0 0 0 cos θ - sin θ 0 0 sin θ cos θ 0 0 0 0 1 ) ( A 0 A 1 A 2 A 3 ) . \begin{pmatrix}{A^{\prime}}^{0}\\ {A^{\prime}}^{1}\\ {A^{\prime}}^{2}\\ {A^{\prime}}^{3}\end{pmatrix}=\begin{pmatrix}1&0&0&0\\ 0&\cos\theta&-\sin\theta&0\\ 0&\sin\theta&\cos\theta&0\\ 0&0&0&1\\ \end{pmatrix}\begin{pmatrix}A^{0}\\ A^{1}\\ A^{2}\\ A^{3}\end{pmatrix}\ .
  18. s y m b o l β = ( β 1 , β 2 , β 3 ) = 1 c ( v 1 , v 2 , v 3 ) = 1 c 𝐯 . symbol{\beta}=(\beta_{1},\,\beta_{2},\,\beta_{3})=\frac{1}{c}(v_{1},\,v_{2},\,% v_{3})=\frac{1}{c}\mathbf{v}\,.
  19. Λ 00 = γ , Λ 0 i = Λ i 0 = - γ β i , Λ i j = Λ j i = ( γ - 1 ) β i β j β 2 + δ i j = ( γ - 1 ) v i v j v 2 + δ i j , \begin{aligned}\displaystyle\Lambda_{00}&\displaystyle=\gamma,\\ \displaystyle\Lambda_{0i}&\displaystyle=\Lambda_{i0}=-\gamma\beta_{i},\\ \displaystyle\Lambda_{ij}&\displaystyle=\Lambda_{ji}=(\gamma-1)\dfrac{\beta_{i% }\beta_{j}}{\beta^{2}}+\delta_{ij}=(\gamma-1)\dfrac{v_{i}v_{j}}{v^{2}}+\delta_% {ij},\\ \end{aligned}\,\!
  20. γ = 1 1 - s y m b o l β \cdotsymbol β , \gamma=\frac{1}{\sqrt{1-symbol{\beta}\cdotsymbol{\beta}}}\,,
  21. ( A 0 A 1 A 2 A 3 ) = ( cosh ϕ - sinh ϕ 0 0 - sinh ϕ cosh ϕ 0 0 0 0 1 0 0 0 0 1 ) ( A 0 A 1 A 2 A 3 ) \begin{pmatrix}A^{\prime 0}\\ A^{\prime 1}\\ A^{\prime 2}\\ A^{\prime 3}\end{pmatrix}=\begin{pmatrix}\cosh\phi&-\sinh\phi&0&0\\ -\sinh\phi&\cosh\phi&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{pmatrix}\begin{pmatrix}A^{0}\\ A^{1}\\ A^{2}\\ A^{3}\end{pmatrix}
  22. γ = cosh ϕ \gamma=\cosh\phi
  23. 𝐀 + 𝐁 = ( A 0 , A 1 , A 2 , A 3 ) + ( B 0 , B 1 , B 2 , B 3 ) = ( A 0 + B 0 , A 1 + B 1 , A 2 + B 2 , A 3 + B 3 ) \mathbf{A}+\mathbf{B}=(A^{0},A^{1},A^{2},A^{3})+(B^{0},B^{1},B^{2},B^{3})=(A^{% 0}+B^{0},A^{1}+B^{1},A^{2}+B^{2},A^{3}+B^{3})
  24. λ 𝐀 = λ ( A 0 , A 1 , A 2 , A 3 ) = ( λ A 0 , λ A 1 , λ A 2 , λ A 3 ) \lambda\mathbf{A}=\lambda(A^{0},A^{1},A^{2},A^{3})=(\lambda A^{0},\lambda A^{1% },\lambda A^{2},\lambda A^{3})
  25. 𝐀 + ( - 1 ) 𝐁 = ( A 0 , A 1 , A 2 , A 3 ) + ( - 1 ) ( B 0 , B 1 , B 2 , B 3 ) = ( A 0 - B 0 , A 1 - B 1 , A 2 - B 2 , A 3 - B 3 ) \mathbf{A}+(-1)\mathbf{B}=(A^{0},A^{1},A^{2},A^{3})+(-1)(B^{0},B^{1},B^{2},B^{% 3})=(A^{0}-B^{0},A^{1}-B^{1},A^{2}-B^{2},A^{3}-B^{3})
  26. 𝐀 𝐁 = A μ η μ ν B ν \mathbf{A}\cdot\mathbf{B}=A^{\mu}\eta_{\mu\nu}B^{\nu}
  27. 𝐀 𝐁 = ( A 0 A 1 A 2 A 3 ) ( η 00 η 01 η 02 η 03 η 10 η 11 η 12 η 13 η 20 η 21 η 22 η 23 η 30 η 31 η 32 η 33 ) ( B 0 B 1 B 2 B 3 ) \mathbf{A\cdot B}=\begin{pmatrix}A^{0}&A^{1}&A^{2}&A^{3}\end{pmatrix}\begin{% pmatrix}\eta_{00}&\eta_{01}&\eta_{02}&\eta_{03}\\ \eta_{10}&\eta_{11}&\eta_{12}&\eta_{13}\\ \eta_{20}&\eta_{21}&\eta_{22}&\eta_{23}\\ \eta_{30}&\eta_{31}&\eta_{32}&\eta_{33}\end{pmatrix}\begin{pmatrix}B^{0}\\ B^{1}\\ B^{2}\\ B^{3}\end{pmatrix}
  28. 𝐀 𝐁 = A ν B ν = A μ B μ \mathbf{A}\cdot\mathbf{B}=A_{\nu}B^{\nu}=A^{\mu}B_{\mu}
  29. 𝐀 𝐁 = ( A 0 A 1 A 2 A 3 ) ( B 0 B 1 B 2 B 3 ) = ( B 0 B 1 B 2 B 3 ) ( A 0 A 1 A 2 A 3 ) \mathbf{A\cdot B}=\begin{pmatrix}A_{0}&A_{1}&A_{2}&A_{3}\end{pmatrix}\begin{% pmatrix}B^{0}\\ B^{1}\\ B^{2}\\ B^{3}\end{pmatrix}=\begin{pmatrix}B_{0}&B_{1}&B_{2}&B_{3}\end{pmatrix}\begin{% pmatrix}A^{0}\\ A^{1}\\ A^{2}\\ A^{3}\end{pmatrix}
  30. 𝐀 𝐁 = A μ η μ ν B ν \mathbf{A}\cdot\mathbf{B}=A_{\mu}\eta^{\mu\nu}B_{\nu}
  31. 𝐀 𝐀 = A μ η μ ν A ν \mathbf{A\cdot A}=A^{\mu}\eta_{\mu\nu}A^{\nu}
  32. 𝐀 𝐁 = A 0 B 0 - A 1 B 1 - A 2 B 2 - A 3 B 3 \mathbf{A}\cdot\mathbf{B}=A^{0}B^{0}-A^{1}B^{1}-A^{2}B^{2}-A^{3}B^{3}
  33. 𝐀 𝐁 = ( A 0 A 1 A 2 A 3 ) ( 1 0 0 0 0 - 1 0 0 0 0 - 1 0 0 0 0 - 1 ) ( B 0 B 1 B 2 B 3 ) \mathbf{A\cdot B}=\begin{pmatrix}A^{0}&A^{1}&A^{2}&A^{3}\end{pmatrix}\begin{% pmatrix}1&0&0&0\\ 0&-1&0&0\\ 0&0&-1&0\\ 0&0&0&-1\end{pmatrix}\begin{pmatrix}B^{0}\\ B^{1}\\ B^{2}\\ B^{3}\end{pmatrix}
  34. 𝐀 𝐁 = A 0 B 0 - A 1 B 1 - A 2 B 2 - A 3 B 3 = C \mathbf{A}\cdot\mathbf{B}=A^{0}B^{0}-A^{1}B^{1}-A^{2}B^{2}-A^{3}B^{3}=C
  35. 𝐀 𝐁 = A 0 B 0 - A 1 B 1 - A 2 B 2 - A 3 B 3 = C \mathbf{A}^{\prime}\cdot\mathbf{B}^{\prime}={A^{\prime}}^{0}{B^{\prime}}^{0}-{% A^{\prime}}^{1}{B^{\prime}}^{1}-{A^{\prime}}^{2}{B^{\prime}}^{2}-{A^{\prime}}^% {3}{B^{\prime}}^{3}=C^{\prime}
  36. 𝐀 𝐁 = 𝐀 𝐁 \mathbf{A}\cdot\mathbf{B}=\mathbf{A}^{\prime}\cdot\mathbf{B}^{\prime}
  37. C = A 0 B 0 - A 1 B 1 - A 2 B 2 - A 3 B 3 = A 0 B 0 - A 1 B 1 - A 2 B 2 - A 3 B 3 C=A^{0}B^{0}-A^{1}B^{1}-A^{2}B^{2}-A^{3}B^{3}={A^{\prime}}^{0}{B^{\prime}}^{0}% -{A^{\prime}}^{1}{B^{\prime}}^{1}-{A^{\prime}}^{2}{B^{\prime}}^{2}-{A^{\prime}% }^{3}{B^{\prime}}^{3}
  38. 𝐀 𝐀 = ( A 0 ) 2 - ( A 1 ) 2 - ( A 2 ) 2 - ( A 3 ) 2 \mathbf{A\cdot A}=(A^{0})^{2}-(A^{1})^{2}-(A^{2})^{2}-(A^{3})^{2}
  39. 𝐀 𝐀 < 0 \mathbf{A\cdot A}<0
  40. 𝐀 𝐀 > 0 \mathbf{A\cdot A}>0
  41. 𝐀 𝐀 = 0 \mathbf{A\cdot A}=0
  42. 𝐀 𝐁 = - A 0 B 0 + A 1 B 1 + A 2 B 2 + A 3 B 3 \mathbf{A\cdot B}=-A^{0}B^{0}+A^{1}B^{1}+A^{2}B^{2}+A^{3}B^{3}
  43. 𝐀 𝐁 = ( A 0 A 1 A 2 A 3 ) ( - 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) ( B 0 B 1 B 2 B 3 ) \mathbf{A\cdot B}=\left(\begin{matrix}A^{0}&A^{1}&A^{2}&A^{3}\end{matrix}% \right)\left(\begin{matrix}-1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{matrix}\right)\left(\begin{matrix}B^{0}\\ B^{1}\\ B^{2}\\ B^{3}\end{matrix}\right)
  44. 𝐀 𝐁 = - A 0 B 0 + A 1 B 1 + A 2 B 2 + A 3 B 3 = - C \mathbf{A}\cdot\mathbf{B}=-A^{0}B^{0}+A^{1}B^{1}+A^{2}B^{2}+A^{3}B^{3}=-C
  45. 𝐀 𝐁 = - A 0 B 0 + A 1 B 1 + A 2 B 2 + A 3 B 3 = - C \mathbf{A}^{\prime}\cdot\mathbf{B}^{\prime}=-{A^{\prime}}^{0}{B^{\prime}}^{0}+% {A^{\prime}}^{1}{B^{\prime}}^{1}+{A^{\prime}}^{2}{B^{\prime}}^{2}+{A^{\prime}}% ^{3}{B^{\prime}}^{3}=-C^{\prime}
  46. - C = - A 0 B 0 + A 1 B 1 + A 2 B 2 + A 3 B 3 = - A 0 B 0 + A 1 B 1 + A 2 B 2 + A 3 B 3 -C=-A^{0}B^{0}+A^{1}B^{1}+A^{2}B^{2}+A^{3}B^{3}=-{A^{\prime}}^{0}{B^{\prime}}^% {0}+{A^{\prime}}^{1}{B^{\prime}}^{1}+{A^{\prime}}^{2}{B^{\prime}}^{2}+{A^{% \prime}}^{3}{B^{\prime}}^{3}
  47. 𝐀 𝐀 = - ( A 0 ) 2 + ( A 1 ) 2 + ( A 2 ) 2 + ( A 3 ) 2 \mathbf{A\cdot A}=-(A^{0})^{2}+(A^{1})^{2}+(A^{2})^{2}+(A^{3})^{2}
  48. 𝐀 𝐀 > 0 \mathbf{A\cdot A}>0
  49. 𝐀 𝐀 < 0 \mathbf{A\cdot A}<0
  50. 𝐀 𝐀 = 0 \mathbf{A\cdot A}=0
  51. 𝐀 𝐁 = A * ( 𝐁 ) = A B ν ν . \mathbf{A\cdot B}=A^{*}(\mathbf{B})=A{{}_{\nu}}B^{\nu}.
  52. d 𝐀 differential = d 𝐀 d λ derivative d λ differential \underset{\,\text{differential}}{d\mathbf{A}}=\underset{\,\text{derivative}}{% \frac{d\mathbf{A}}{d\lambda}}\underset{\,\text{differential}}{d\lambda}
  53. d 𝐀 = ( d A 0 , d A 1 , d A 2 , d A 3 ) d\mathbf{A}=(dA^{0},dA^{1},dA^{2},dA^{3})
  54. d 𝐀 = ( d A 0 , d A 1 , d A 2 , d A 3 ) d\mathbf{A}=(dA_{0},dA_{1},dA_{2},dA_{3})
  55. 𝐑 = ( c t , 𝐫 ) \mathbf{R}=\left(ct,\mathbf{r}\right)
  56. Δ 𝐑 = ( c Δ t , Δ 𝐫 ) \Delta\mathbf{R}=\left(c\Delta t,\Delta\mathbf{r}\right)
  57. d 𝐑 2 = 𝐝𝐑 𝐝𝐑 = d R μ d R μ = c 2 d τ 2 = d s 2 , \|d\mathbf{R}\|^{2}=\mathbf{dR\cdot dR}=dR^{\mu}dR_{\mu}=c^{2}d\tau^{2}=ds^{2}\,,
  58. d 𝐑 2 = ( c d t ) 2 - d 𝐫 d 𝐫 , \|d\mathbf{R}\|^{2}=(cdt)^{2}-d\mathbf{r}\cdot d\mathbf{r}\,,
  59. ( c d τ ) 2 = ( c d t ) 2 - d 𝐫 d 𝐫 . (cd\tau)^{2}=(cdt)^{2}-d\mathbf{r}\cdot d\mathbf{r}\,.
  60. ( c d τ c d t ) 2 = 1 - ( d 𝐫 c d t d 𝐫 c d t ) = 1 - 𝐮 𝐮 c 2 = 1 γ ( 𝐮 ) 2 , \left(\frac{cd\tau}{cdt}\right)^{2}=1-\left(\frac{d\mathbf{r}}{cdt}\cdot\frac{% d\mathbf{r}}{cdt}\right)=1-\frac{\mathbf{u}\cdot\mathbf{u}}{c^{2}}=\frac{1}{% \gamma(\mathbf{u})^{2}}\,,
  61. γ ( 𝐮 ) = 1 1 - 𝐮 𝐮 c 2 \gamma(\mathbf{u})=\frac{1}{\sqrt{1-\frac{\mathbf{u}\cdot\mathbf{u}}{c^{2}}}}
  62. d t = γ ( 𝐮 ) d τ . dt=\gamma(\mathbf{u})d\tau\,.
  63. s y m b o l \displaystyle symbol{\partial}
  64. s y m b o l \displaystyle symbol{\partial}
  65. μ μ = 1 c 2 2 t 2 - 2 \partial^{\mu}\partial_{\mu}=\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}% }-\nabla^{2}
  66. 𝐔 = d 𝐗 d τ = d 𝐗 d t d t d τ = γ ( 𝐮 ) ( c , 𝐮 ) , \mathbf{U}=\frac{d\mathbf{X}}{d\tau}=\frac{d\mathbf{X}}{dt}\frac{dt}{d\tau}=% \gamma(\mathbf{u})\left(c,\mathbf{u}\right),
  67. 𝐔 2 = U μ U μ = d X μ d τ d X μ d τ = d X μ d X μ d τ 2 = c 2 , \|\mathbf{U}\|^{2}=U^{\mu}U_{\mu}=\frac{dX^{\mu}}{d\tau}\frac{dX_{\mu}}{d\tau}% =\frac{dX^{\mu}dX_{\mu}}{d\tau^{2}}=c^{2}\,,
  68. 𝐔 2 = c 2 \|\mathbf{U}\|^{2}=c^{2}\,
  69. 𝐔 2 = γ ( 𝐮 ) 2 ( c 2 - 𝐮 𝐮 ) , \|\mathbf{U}\|^{2}={\gamma(\mathbf{u})}^{2}\left(c^{2}-\mathbf{u}\cdot\mathbf{% u}\right)\,,
  70. c 2 = γ ( 𝐮 ) 2 ( c 2 - 𝐮 𝐮 ) , c^{2}={\gamma(\mathbf{u})}^{2}\left(c^{2}-\mathbf{u}\cdot\mathbf{u}\right)\,,
  71. 𝐀 = d 𝐔 d τ = γ ( 𝐮 ) ( d γ ( 𝐮 ) d t c , d γ ( 𝐮 ) d t 𝐮 + γ ( 𝐮 ) 𝐚 ) . \mathbf{A}=\frac{d\mathbf{U}}{d\tau}=\gamma(\mathbf{u})\left(\frac{d{\gamma}(% \mathbf{u})}{dt}c,\frac{d{\gamma}(\mathbf{u})}{dt}\mathbf{u}+\gamma(\mathbf{u}% )\mathbf{a}\right).
  72. 𝐀 𝐔 = A μ U μ = d U μ d τ U μ = 1 2 d d τ ( U μ U μ ) = 0 \mathbf{A}\cdot\mathbf{U}=A^{\mu}U_{\mu}=\frac{dU^{\mu}}{d\tau}U_{\mu}=\frac{1% }{2}\,\frac{d}{d\tau}(U^{\mu}U_{\mu})=0\,
  73. 𝐏 = m 0 𝐔 = m 0 γ ( 𝐮 ) ( c , 𝐮 ) = ( E / c , 𝐩 ) \mathbf{P}=m_{0}\mathbf{U}=m_{0}\gamma(\mathbf{u})(c,\mathbf{u})=(E/c,\mathbf{% p})
  74. E = γ ( 𝐮 ) m 0 c 2 E=\gamma(\mathbf{u})m_{0}c^{2}
  75. 𝐩 = γ ( 𝐮 ) m 0 𝐮 \mathbf{p}=\gamma(\mathbf{u})m_{0}\mathbf{u}
  76. 𝐏 2 = P μ P μ = m 0 2 U μ U μ = m 0 2 c 2 \|\mathbf{P}\|^{2}=P^{\mu}P_{\mu}=m_{0}^{2}U^{\mu}U_{\mu}=m_{0}^{2}c^{2}
  77. 𝐏 2 = E 2 c 2 - 𝐩 𝐩 \|\mathbf{P}\|^{2}=\frac{E^{2}}{c^{2}}-\mathbf{p}\cdot\mathbf{p}
  78. E 2 = c 2 𝐩 𝐩 + ( m 0 c 2 ) 2 . E^{2}=c^{2}\mathbf{p}\cdot\mathbf{p}+(m_{0}c^{2})^{2}\,.
  79. 𝐅 = d 𝐏 d τ = γ ( 𝐮 ) ( 1 c d E d t , d 𝐩 d t ) = γ ( 𝐮 ) ( P / c , 𝐟 ) \mathbf{F}=\frac{d\mathbf{P}}{d\tau}=\gamma(\mathbf{u})\left(\frac{1}{c}\frac{% dE}{dt},\frac{d\mathbf{p}}{dt}\right)=\gamma(\mathbf{u})(P/c,\mathbf{f})
  80. 𝐅 = m 0 𝐀 = m 0 γ ( 𝐮 ) ( d γ ( 𝐮 ) d t c , ( d γ ( 𝐮 ) d t 𝐮 + γ ( 𝐮 ) 𝐚 ) ) \mathbf{F}=m_{0}\mathbf{A}=m_{0}\gamma(\mathbf{u})\left(\frac{d{\gamma}(% \mathbf{u})}{dt}c,\left(\frac{d{\gamma}(\mathbf{u})}{dt}\mathbf{u}+\gamma(% \mathbf{u})\mathbf{a}\right)\right)
  81. 𝐅 𝐔 = F μ U μ = m 0 A μ U μ = 0 \mathbf{F}\cdot\mathbf{U}=F^{\mu}U_{\mu}=m_{0}A^{\mu}U_{\mu}=0
  82. 𝐐 = - k s y m b o l T = - k ( 1 c T t , T ) \mathbf{Q}=-ksymbol{\partial}T=-k\left(\frac{1}{c}\frac{\partial T}{\partial t% },\nabla T\right)
  83. 𝐒 = n 𝐔 \mathbf{S}=n\mathbf{U}
  84. 𝐬 = s 𝐒 + 𝐐 T \mathbf{s}=s\mathbf{S}+\frac{\mathbf{Q}}{T}
  85. 𝐉 = ( ρ c , 𝐣 ) \mathbf{J}=\left(\rho c,\mathbf{j}\right)
  86. 𝐀 = ( ϕ / c , 𝐚 ) \mathbf{A}=\left(\phi/c,\mathbf{a}\right)
  87. 𝐍 = ν ( 1 , 𝐧 ^ ) \mathbf{N}=\nu\left(1,\hat{\mathbf{n}}\right)
  88. 𝐧 ^ \hat{\mathbf{n}}
  89. 𝐍 = N μ N μ = ν 2 ( 1 - 𝐧 ^ 𝐧 ^ ) = 0 \|\mathbf{N}\|=N^{\mu}N_{\mu}=\nu^{2}\left(1-\hat{\mathbf{n}}\cdot\hat{\mathbf% {n}}\right)=0
  90. 𝐊 = ( ω c , 𝐤 ) . \mathbf{K}=\left(\frac{\omega}{c},\mathbf{k}\right)\,.
  91. 𝐊 = 2 π c 𝐍 = 2 π c ν ( 1 , 𝐧 ^ ) = ω c ( 1 , 𝐧 ^ ) . \mathbf{K}=\frac{2\pi}{c}\mathbf{N}=\frac{2\pi}{c}\nu(1,\hat{\mathbf{n}})=% \frac{\omega}{c}\left(1,\hat{\mathbf{n}}\right)\,.
  92. 𝐏 = 𝐊 . \mathbf{P}=\hbar\mathbf{K}\,.
  93. 𝐊 2 = K μ K μ = ( ω c ) 2 - 𝐤 𝐤 , \|\mathbf{K}\|^{2}=K^{\mu}K_{\mu}=\left(\frac{\omega}{c}\right)^{2}-\mathbf{k}% \cdot\mathbf{k}\,,
  94. 𝐊 2 = 1 2 𝐏 2 = ( m 0 c ) 2 , \|\mathbf{K}\|^{2}=\frac{1}{\hbar^{2}}\|\mathbf{P}\|^{2}=\left(\frac{m_{0}c}{% \hbar}\right)^{2}\,,
  95. ( ω c ) 2 - 𝐤 𝐤 = ( m 0 c ) 2 . \left(\frac{\omega}{c}\right)^{2}-\mathbf{k}\cdot\mathbf{k}=\left(\frac{m_{0}c% }{\hbar}\right)^{2}\,.
  96. ( ω c ) 2 = 𝐤 𝐤 , \left(\frac{\omega}{c}\right)^{2}=\mathbf{k}\cdot\mathbf{k}\,,
  97. 𝐧 ^ \hat{\mathbf{n}}
  98. 𝐉 = ( ρ c , 𝐣 ) \mathbf{J}=(\rho c,\mathbf{j})
  99. 𝐀 = ( A 0 , A 1 , A 2 , A 3 ) = A 0 s y m b o l σ 0 + A 1 s y m b o l σ 1 + A 2 s y m b o l σ 2 + A 3 s y m b o l σ 3 = A 0 s y m b o l σ 0 + A i s y m b o l σ i = A \alphasymbol σ α \begin{aligned}\displaystyle\mathbf{A}&\displaystyle=(A^{0},\,A^{1},\,A^{2},\,% A^{3})\\ &\displaystyle=A^{0}symbol{\sigma}_{0}+A^{1}symbol{\sigma}_{1}+A^{2}symbol{% \sigma}_{2}+A^{3}symbol{\sigma}_{3}\\ &\displaystyle=A^{0}symbol{\sigma}_{0}+A^{i}symbol{\sigma}_{i}\\ &\displaystyle=A^{\alphasymbol}{\sigma}_{\alpha}\\ \end{aligned}
  100. 𝐀 \displaystyle\mathbf{A}
  101. | 𝐀 | \displaystyle|\mathbf{A}|
  102. 𝐀 / = A α γ α = A 0 γ 0 + A 1 γ 1 + A 2 γ 2 + A 3 γ 3 \mathbf{A}\!\!\!\!/=A_{\alpha}\gamma^{\alpha}=A_{0}\gamma^{0}+A_{1}\gamma^{1}+% A_{2}\gamma^{2}+A_{3}\gamma^{3}
  103. 𝐏 / = P α γ α = P 0 γ 0 + P 1 γ 1 + P 2 γ 2 + P 3 γ 3 = E c γ 0 - p x γ 1 - p y γ 2 - p z γ 3 \mathbf{P}\!\!\!\!/=P_{\alpha}\gamma^{\alpha}=P_{0}\gamma^{0}+P_{1}\gamma^{1}+% P_{2}\gamma^{2}+P_{3}\gamma^{3}=\dfrac{E}{c}\gamma^{0}-p_{x}\gamma^{1}-p_{y}% \gamma^{2}-p_{z}\gamma^{3}

Four-velocity.html

  1. x ( t ) = [ x 1 ( t ) x 2 ( t ) x 3 ( t ) ] . \vec{x}(t)=\begin{bmatrix}x^{1}(t)\\ x^{2}(t)\\ x^{3}(t)\end{bmatrix}\,.
  2. u {\vec{u}}
  3. u = [ u 1 u 2 u 3 ] = d x d t = [ d x 1 d t d x 2 d t d x 3 d t ] . {\vec{u}}=\begin{bmatrix}u^{1}\\ u^{2}\\ u^{3}\end{bmatrix}={d\vec{x}\over dt}=\begin{bmatrix}\tfrac{dx^{1}}{dt}\\ \tfrac{dx^{2}}{dt}\\ \tfrac{dx^{3}}{dt}\end{bmatrix}.
  4. u i = d x i d t u^{i}={dx^{i}\over dt}
  5. x 0 = c t , x^{0}=ct\,,
  6. 𝐱 = [ x 0 ( τ ) x 1 ( τ ) x 2 ( τ ) x 3 ( τ ) ] . \mathbf{x}=\begin{bmatrix}x^{0}(\tau)\\ x^{1}(\tau)\\ x^{2}(\tau)\\ x^{3}(\tau)\\ \end{bmatrix}\,.
  7. d t = γ ( u ) d τ dt=\gamma(u)d\tau
  8. γ ( u ) = 1 1 - u 2 c 2 , \gamma(u)=\frac{1}{\sqrt{1-\frac{u^{2}}{c^{2}}}}\,,
  9. u \vec{u}
  10. u = || u || = ( u 1 ) 2 + ( u 2 ) 2 + ( u 3 ) 2 . u=||\ \vec{u}\ ||=\sqrt{(u^{1})^{2}+(u^{2})^{2}+(u^{3})^{2}}\,.
  11. 𝐱 ( τ ) \mathbf{x}(\tau)
  12. 𝐔 = d 𝐱 d τ \mathbf{U}=\frac{d\mathbf{x}}{d\tau}
  13. 𝐱 \mathbf{x}
  14. τ \tau
  15. x 0 = c t . x^{0}=ct.
  16. U 0 = d x 0 d τ = d ( c t ) d τ = c d t d τ = c γ ( u ) U^{0}=\frac{dx^{0}}{d\tau}=\frac{d(ct)}{d\tau}=c\frac{dt}{d\tau}=c\gamma(u)
  17. U i = d x i d τ = d x i d t d t d τ = d x i d t γ ( u ) = γ ( u ) u i U^{i}=\frac{dx^{i}}{d\tau}=\frac{dx^{i}}{dt}\frac{dt}{d\tau}=\frac{dx^{i}}{dt}% \gamma(u)=\gamma(u)u^{i}
  18. u i = d x i d t , d t d τ = γ ( u ) u^{i}={dx^{i}\over dt}\,,\quad\frac{dt}{d\tau}=\gamma(u)
  19. 𝐔 \mathbf{U}
  20. 𝐔 = γ [ c u ] . \mathbf{U}=\gamma\begin{bmatrix}c\\ \vec{u}\\ \end{bmatrix}.
  21. γ u = d x / d τ \gamma\vec{u}=d\vec{x}/d\tau

Four_fours.html

  1. . 4 ¯ = .4444... = 4 / 9 .\overline{4}=.4444...=4/9
  2. n = - 4 ln [ ( ln 4 n ) / ln 4 ] ln 4 n=-\sqrt{4}\frac{\ln\left[\left(\ln\underbrace{\sqrt{\sqrt{\cdots\sqrt{4}}}}_{% n}\right)/\ln 4\right]}{\ln{4}}
  3. Γ ( Γ ( 4 ) ) - 4 ! + 4 4 \Gamma(\Gamma(4))-\frac{4!+4}{4}

Fourier.html

  1. 𝐹𝑜 \mathit{Fo}
  2. α t / d 2 \alpha t/d^{2}
  3. α t \alpha t
  4. d 2 d^{2}

Fourier_optics.html

  1. u = u ( 𝐫 , t ) u=u(\mathbf{r},t)
  2. 𝐫 = ( x , y , z ) \mathbf{r}=(x,y,z)
  3. ( 2 - 1 c 2 2 t 2 ) u ( 𝐫 , t ) = 0. \left(\nabla^{2}-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial{t}^{2}}\right)u(% \mathbf{r},t)=0.
  4. u ( 𝐫 , t ) = Re { ψ ( 𝐫 ) e j ω t } u(\mathbf{r},t)=\mathrm{Re}\left\{\psi(\mathbf{r})e^{j\omega t}\right\}
  5. ω = 2 π f \omega=2\pi f
  6. ψ ( 𝐫 ) = a ( 𝐫 ) e j ϕ ( 𝐫 ) \psi(\mathbf{r})=a(\mathbf{r})e^{j\phi(\mathbf{r})}
  7. ( 2 + k 2 ) ψ ( 𝐫 ) = 0 \left(\nabla^{2}+k^{2}\right)\psi(\mathbf{r})=0
  8. k = ω c = 2 π λ k={\omega\over c}={2\pi\over\lambda}
  9. ω \omega
  10. ψ ( x , y , z ) = f x ( x ) × f y ( y ) × f z ( z ) \psi(x,y,z)=f_{x}(x)\times f_{y}(y)\times f_{z}(z)
  11. 2 ψ = 2 ψ x 2 + 2 ψ y 2 + 2 ψ z 2 \nabla^{2}\psi=\frac{\partial^{2}\psi}{\partial x^{2}}+\frac{\partial^{2}\psi}% {\partial y^{2}}+\frac{\partial^{2}\psi}{\partial z^{2}}
  12. f x ′′ ( x ) f y ( y ) f z ( z ) + f x ( x ) f y ′′ ( y ) f z ( z ) + f x ( x ) f y ( y ) f z ′′ ( z ) + k 2 f x ( x ) f y ( y ) f z ( z ) = 0 f^{\prime\prime}_{x}(x)f_{y}(y)f_{z}(z)+f_{x}(x)f^{\prime\prime}_{y}(y)f_{z}(z% )+f_{x}(x)f_{y}(y)f^{\prime\prime}_{z}(z)+k^{2}f_{x}(x)f_{y}(y)f_{z}(z)=0\,
  13. f x ′′ ( x ) f x ( x ) + f y ′′ ( y ) f y ( y ) + f z ′′ ( z ) f z ( z ) + k 2 = 0 \frac{f^{\prime\prime}_{x}(x)}{f_{x}(x)}+\frac{f^{\prime\prime}_{y}(y)}{f_{y}(% y)}+\frac{f^{\prime\prime}_{z}(z)}{f_{z}(z)}+k^{2}=0
  14. d 2 d x 2 f x ( x ) + k x 2 f x ( x ) = 0 \frac{d^{2}}{dx^{2}}f_{x}(x)+k_{x}^{2}f_{x}(x)=0
  15. d 2 d y 2 f y ( y ) + k y 2 f y ( y ) = 0 \frac{d^{2}}{dy^{2}}f_{y}(y)+k_{y}^{2}f_{y}(y)=0
  16. d 2 d z 2 f z ( z ) + k z 2 f z ( z ) = 0 \frac{d^{2}}{dz^{2}}f_{z}(z)+k_{z}^{2}f_{z}(z)=0
  17. k x 2 + k y 2 + k z 2 = k 2 k_{x}^{2}+k_{y}^{2}+k_{z}^{2}=k^{2}
  18. ψ ( x , y , z ) = e j ( k x x + k y y + k z z ) \psi(x,y,z)=e^{j(k_{x}x+k_{y}y+k_{z}z)}
  19. = e j ( k x x + k y y ) e j k z z =e^{j(k_{x}x+k_{y}y)}e^{jk_{z}z}
  20. = e j ( k x x + k y y ) e ± j z k 2 - k x 2 - k y 2 =e^{j(k_{x}x+k_{y}y)}e^{\pm jz\sqrt{k^{2}-k_{x}^{2}-k_{y}^{2}}}
  21. ψ ( x , y , z ) = - + - + Ψ 0 ( k x , k y ) e j ( k x x + k y y ) e ± j z k 2 - k x 2 - k y 2 d k x d k y ( 2.1 ) \psi(x,y,z)=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\Psi_{0}(k_{x},k_{% y})~{}e^{j(k_{x}x+k_{y}y)}~{}e^{\pm jz\sqrt{k^{2}-k_{x}^{2}-k_{y}^{2}}}~{}dk_{% x}dk_{y}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(2.1)
  22. ψ 0 ( x , y ) = ψ ( x , y , z ) | z = 0 \psi_{0}(x,y)=\psi(x,y,z)|_{z=0}
  23. ψ 0 ( x , y ) = - + - + Ψ 0 ( k x , k y ) e j ( k x x + k y y ) d k x d k y \psi_{0}(x,y)=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\Psi_{0}(k_{x},k% _{y})~{}e^{j(k_{x}x+k_{y}y)}~{}dk_{x}dk_{y}
  24. Ψ 0 ( k x , k y ) = { ψ 0 ( x , y ) } \Psi_{0}(k_{x},k_{y})=\mathcal{F}\{\psi_{0}(x,y)\}
  25. ψ 0 ( x , y ) = - 1 { Ψ 0 ( k x , k y ) } \psi_{0}(x,y)=\mathcal{F}^{-1}\{\Psi_{0}(k_{x},k_{y})\}
  26. k x 2 + k y 2 > k 2 k_{x}^{2}+k_{y}^{2}>k^{2}
  27. ψ ( 𝐫 ) = A ( 𝐫 ) e - j 𝐤 𝐫 \psi(\mathbf{r})=A(\mathbf{r})e^{-j\mathbf{k}\cdot\mathbf{r}}
  28. 𝐤 = k x 𝐱 + k y 𝐲 + k z 𝐳 \mathbf{k}=k_{x}\mathbf{x}+k_{y}\mathbf{y}+k_{z}\mathbf{z}
  29. k = 𝐤 = k x 2 + k y 2 + k z 2 = ω c k=\|\mathbf{k}\|=\sqrt{k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}={\omega\over c}
  30. k x 2 + k y 2 k z 2 k_{x}^{2}+k_{y}^{2}\ll k_{z}^{2}
  31. sin θ θ \sin\theta\approx\theta
  32. k z = k cos θ k ( 1 - θ 2 / 2 ) k_{z}=k\cos\theta\approx k(1-\theta^{2}/2)
  33. ψ ( 𝐫 ) A ( 𝐫 ) e - j ( k x x + k y y ) e j k z θ 2 / 2 e - j k z \psi(\mathbf{r})\approx A(\mathbf{r})e^{-j(k_{x}x+k_{y}y)}e^{jkz\theta^{2}/2}e% ^{-jkz}
  34. T 2 A - 2 j k A z = 0 \nabla_{T}^{2}A-2jk{\partial A\over\partial z}=0
  35. T 2 = 2 - 2 z 2 = 2 x 2 + 2 y 2 \nabla_{T}^{2}=\nabla^{2}-{\partial^{2}\over\partial z^{2}}={\partial^{2}\over% \partial x^{2}}+{\partial^{2}\over\partial y^{2}}
  36. E u ( r , θ , ϕ ) = 2 π j ( k cos θ ) e - j k r r E u ( k sin θ cos ϕ , k sin θ sin ϕ ) ( 2.2 ) E_{u}(r,\theta,\phi)~{}=~{}2\pi j~{}(k~{}\cos\theta)~{}\frac{e^{-jkr}}{r}~{}E_% {u}(k~{}\sin\theta~{}\cos\phi,k~{}\sin\theta~{}\sin\phi)~{}~{}~{}~{}~{}~{}~{}~% {}~{}~{}~{}~{}(2.2)
  37. x = r sin θ cos ϕ x=r~{}\sin\theta~{}\cos\phi
  38. y = r sin θ sin ϕ y=r~{}\sin\theta~{}\sin\phi
  39. z = r cos θ z=r~{}\cos\theta~{}
  40. k x = k sin θ cos ϕ k_{x}=k~{}\sin\theta~{}\cos\phi
  41. k y = k sin θ sin ϕ k_{y}=k~{}\sin\theta~{}\sin\phi
  42. k z = k cos θ k_{z}=k~{}\cos\theta~{}
  43. e - j k r r \frac{e^{-jkr}}{r}
  44. M = 2 E a p e r x z ^ ~{}~{}{M}~{}=~{}2{E}^{aper}~{}{x}~{}{\hat{z}}
  45. e j ω t e^{j\omega t}
  46. 2 E u + k 2 E u = 0 ( 2.0 ) \nabla^{2}E_{u}+k^{2}E_{u}=0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(2.0)
  47. ( 2 + k 2 ) f = 0 , (\nabla^{2}+k^{2})~{}f=0,
  48. ( A - λ I ) x = 0 (A-\lambda I)~{}x=0
  49. 2 \nabla^{2}
  50. λ \lambda
  51. k x 2 + k y 2 + k z 2 = k 2 k_{x}^{2}+k_{y}^{2}+k_{z}^{2}=k^{2}
  52. k = k x x ^ + k y y ^ + k z z ^ {k}~{}=~{}k_{x}{\hat{x}}+k_{y}{\hat{y}}+k_{z}{\hat{z}}
  53. k x = k sin θ cos ϕ k_{x}=k~{}\sin\theta~{}\cos\phi
  54. k y = k sin θ sin ϕ k_{y}=k~{}\sin\theta~{}\sin\phi
  55. k z = k cos θ k_{z}=k~{}\cos\theta~{}
  56. U ( k x , k y ) = - - u ( x , y ) e - j ( k x x + k y y ) d x d y U(k_{x},k_{y})=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}u(x,y)e^{-j(k_{x}% x+k_{y}y)}dxdy
  57. u ( x , y ) = 1 ( 2 π ) 2 - - U ( k x , k y ) e j ( k x x + k y y ) d k x d k y u(x,y)=\frac{1}{(2\pi)^{2}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}U(k_{% x},k_{y})e^{j(k_{x}x+k_{y}y)}dk_{x}dk_{y}
  58. 1 ( 2 π ) 2 \frac{1}{(2\pi)^{2}}
  59. g ( x , y ) = h ( x , y ) * f ( x , y ) g(x,y)=h(x,y)*f(x,y)
  60. f ( x , y ) = U ( x , y , z ) | z = 0 f(x,y)=U(x,y,z)\big|_{z=0}
  61. g ( x , y ) = U ( x , y , z ) | z = d g(x,y)=U(x,y,z)\big|_{z=d}
  62. g ( x , y ) = h ( x , y ) * f ( x , y ) g(x,y)~{}=~{}h(x,y)*f(x,y)
  63. g ( x , y ) = - - h ( x - x , y - y ) f ( x , y ) d x d y ( 4.1 ) g(x,y)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}h(x-x^{\prime},y-y^{% \prime})~{}f(x^{\prime},y^{\prime})~{}dx^{\prime}dy^{\prime}~{}~{}~{}~{}~{}~{}% (4.1)~{}
  64. g ( x , y ) = - - h M ( x - M x , y - M y ) f ( x , y ) d x d y ( 4.2 ) g(x,y)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}h_{M}(x-Mx^{\prime},y-My^% {\prime})~{}f(x^{\prime},y^{\prime})~{}dx^{\prime}dy^{\prime}~{}~{}~{}~{}~{}~{% }(4.2)~{}
  65. g ( t ) = - h ( t - t ) f ( t ) d t g(t)=\int_{-\infty}^{\infty}h(t-t^{\prime})f(t^{\prime})dt^{\prime}
  66. G ( ω ) = H ( ω ) F ( ω ) G(\omega)~{}=~{}H(\omega)\cdot F(\omega)
  67. G ( ω ) G(\omega)~{}
  68. H ( ω ) H(\omega)~{}
  69. F ( ω ) F(\omega)~{}
  70. G ( k x , k y ) = H ( k x , k y ) F ( k x , k y ) G(k_{x},k_{y})~{}=~{}H(k_{x},k_{y})\cdot F(k_{x},k_{y})
  71. H ( ω ) H(\omega)
  72. G ( k x , k y ) ~{}G(k_{x},k_{y})
  73. ( k x , k y ) ~{}(k_{x},k_{y})
  74. k x k = sin θ θ \frac{k_{x}}{k}=\sin\theta\simeq\theta
  75. k z k = cos θ 1 - θ 2 2 \frac{k_{z}}{k}=\cos\theta\simeq 1-\frac{\theta^{2}}{2}
  76. 1 cos θ 1 1 - θ 2 2 1 + θ 2 2 \frac{1}{\cos\theta}\simeq\frac{1}{1-\frac{\theta^{2}}{2}}\simeq 1+\frac{% \theta^{2}}{2}
  77. e j k f cos θ e^{jkf\cos\theta}\,
  78. e j k f / cos θ e^{jkf/\cos\theta}\,
  79. k 2 k x 2 + k y 2 k^{2}\gg k_{x}^{2}+k_{y}^{2}

Fourth_normal_form.html

  1. \twoheadrightarrow
  2. \twoheadrightarrow
  3. \twoheadrightarrow
  4. \twoheadrightarrow
  5. \twoheadrightarrow
  6. \twoheadrightarrow
  7. \twoheadrightarrow

Fractal_dimension.html

  1. N N
  2. ϵ \epsilon
  3. D D
  4. N ϵ - D {{N\propto\epsilon^{-D}}}
  5. \propto
  6. N N
  7. ϵ \epsilon
  8. D D
  9. N N
  10. ϵ \epsilon
  11. D D
  12. N N
  13. ϵ \epsilon
  14. D D
  15. log ϵ N = - D = log N log ϵ {{\log_{\epsilon}{N}={-D}=\frac{\log{N}}{\log{\epsilon}}}}
  16. N N
  17. ϵ \epsilon
  18. D D
  19. D D
  20. D 0 = lim ϵ 0 log N ( ϵ ) log 1 ϵ . D_{0}=\lim_{\epsilon\rightarrow 0}\frac{\log N(\epsilon)}{\log\frac{1}{% \epsilon}}.
  21. p p
  22. D 1 = lim ϵ 0 - log p ϵ log 1 ϵ D_{1}=\lim_{\epsilon\rightarrow 0}\frac{-\langle\log p_{\epsilon}\rangle}{\log% \frac{1}{\epsilon}}
  23. M M
  24. D 2 = lim ϵ 0 , M log ( g ϵ / M 2 ) log ϵ D_{2}=\lim_{\epsilon\rightarrow 0,M\rightarrow\infty}\frac{\log(g_{\epsilon}/M% ^{2})}{\log\epsilon}
  25. D α = lim ϵ 0 1 1 - α log ( i p i α ) log 1 ϵ D_{\alpha}=\lim_{\epsilon\rightarrow 0}\frac{\frac{1}{1-\alpha}\log(\sum_{i}p_% {i}^{\alpha})}{\log\frac{1}{\epsilon}}
  26. D = d log ( L ( k ) ) d log ( k ) D=\frac{d\ \log(L(k))}{d\ \log(k)}

Fractional_calculus.html

  1. D = d d x , D=\dfrac{d}{dx},
  2. D = D 1 2 \sqrt{D}=D^{\frac{1}{2}}
  3. D a D^{a}
  4. H 2 f ( x ) = D f ( x ) = d d x f ( x ) = f ( x ) H^{2}f(x)=Df(x)=\dfrac{d}{dx}f(x)=f^{\prime}(x)
  5. ( P a f ) ( x ) = f ( x ) , (P^{a}f)(x)=f^{\prime}(x),
  6. ( J f ) ( x ) = 0 x f ( t ) d t (Jf)(x)=\int_{0}^{x}f(t)\;dt
  7. ( J 2 f ) ( x ) = 0 x ( J f ) ( t ) d t = 0 x ( 0 t f ( s ) d s ) d t , (J^{2}f)(x)=\int_{0}^{x}(Jf)(t)dt=\int_{0}^{x}\left(\int_{0}^{t}f(s)\;ds\right% )\;dt,
  8. ( J n f ) ( x ) = 1 ( n - 1 ) ! 0 x ( x - t ) n - 1 f ( t ) d t , (J^{n}f)(x)={1\over(n-1)!}\int_{0}^{x}(x-t)^{n-1}f(t)\;dt,
  9. ( J α f ) ( x ) = 1 Γ ( α ) 0 x ( x - t ) α - 1 f ( t ) d t (J^{\alpha}f)(x)={1\over\Gamma(\alpha)}\int_{0}^{x}(x-t)^{\alpha-1}f(t)\;dt
  10. ( J α ) ( J β f ) ( x ) = ( J β ) ( J α f ) ( x ) = ( J α + β f ) ( x ) = 1 Γ ( α + β ) 0 x ( x - t ) α + β - 1 f ( t ) d t (J^{\alpha})(J^{\beta}f)(x)=(J^{\beta})(J^{\alpha}f)(x)=(J^{\alpha+\beta}f)(x)% ={1\over\Gamma(\alpha+\beta)}\int_{0}^{x}(x-t)^{\alpha+\beta-1}f(t)\;dt
  11. ( J α ) ( J β f ) ( x ) = 1 Γ ( α ) 0 x ( x - t ) α - 1 ( J β f ) ( t ) d t = 1 Γ ( α ) Γ ( β ) 0 x 0 t ( x - t ) α - 1 ( t - s ) β - 1 f ( s ) d s d t = 1 Γ ( α ) Γ ( β ) 0 x f ( s ) ( s x ( x - t ) α - 1 ( t - s ) β - 1 d t ) d s \begin{aligned}\displaystyle(J^{\alpha})(J^{\beta}f)(x)&\displaystyle=\frac{1}% {\Gamma(\alpha)}\int_{0}^{x}(x-t)^{\alpha-1}(J^{\beta}f)(t)\;dt\\ &\displaystyle=\frac{1}{\Gamma(\alpha)\Gamma(\beta)}\int_{0}^{x}\int_{0}^{t}(x% -t)^{\alpha-1}(t-s)^{\beta-1}f(s)\;ds\;dt\\ &\displaystyle=\frac{1}{\Gamma(\alpha)\Gamma(\beta)}\int_{0}^{x}f(s)\left(\int% _{s}^{x}(x-t)^{\alpha-1}(t-s)^{\beta-1}\;dt\right)ds\end{aligned}
  12. ( J α ) ( J β f ) ( x ) = 1 Γ ( α ) Γ ( β ) 0 x ( x - s ) α + β - 1 f ( s ) ( 0 1 ( 1 - r ) α - 1 r β - 1 d r ) d s (J^{\alpha})(J^{\beta}f)(x)=\frac{1}{\Gamma(\alpha)\Gamma(\beta)}\int_{0}^{x}(% x-s)^{\alpha+\beta-1}f(s)\left(\int_{0}^{1}(1-r)^{\alpha-1}r^{\beta-1}\;dr% \right)ds
  13. 0 1 ( 1 - r ) α - 1 r β - 1 d r = B ( α , β ) = Γ ( α ) Γ ( β ) Γ ( α + β ) \int_{0}^{1}(1-r)^{\alpha-1}r^{\beta-1}\;dr=B(\alpha,\beta)=\dfrac{\Gamma(% \alpha)\,\Gamma(\beta)}{\Gamma(\alpha+\beta)}
  14. ( J α ) ( J β f ) ( x ) = 1 Γ ( α + β ) 0 x ( x - s ) α + β - 1 f ( s ) d s = ( J α + β f ) ( x ) (J^{\alpha})(J^{\beta}f)(x)=\frac{1}{\Gamma(\alpha+\beta)}\int_{0}^{x}(x-s)^{% \alpha+\beta-1}f(s)\;ds=(J^{\alpha+\beta}f)(x)
  15. 1 / 2 {1}/{2}
  16. f ( x ) = x k . f(x)=x^{k}\;.
  17. f ( x ) = d d x f ( x ) = k x k - 1 . f^{\prime}(x)=\dfrac{d}{dx}f(x)=kx^{k-1}\;.
  18. d a d x a x k = k ! ( k - a ) ! x k - a , \dfrac{d^{a}}{dx^{a}}x^{k}=\dfrac{k!}{(k-a)!}x^{k-a}\;,
  19. d a d x a x k = Γ ( k + 1 ) Γ ( k - a + 1 ) x k - a , k 0 \dfrac{d^{a}}{dx^{a}}x^{k}=\dfrac{\Gamma(k+1)}{\Gamma(k-a+1)}x^{k-a},\qquad k\geq 0
  20. k = 1 k=1
  21. a = 1 2 \textstyle a=\frac{1}{2}
  22. x x
  23. d 1 2 d x 1 2 x = Γ ( 1 + 1 ) Γ ( 1 - 1 2 + 1 ) x 1 - 1 2 = Γ ( 2 ) Γ ( 3 2 ) x 1 2 = 1 π 2 x 1 2 . \dfrac{d^{\frac{1}{2}}}{dx^{\frac{1}{2}}}x=\dfrac{\Gamma(1+1)}{\Gamma(1-\frac{% 1}{2}+1)}x^{1-\frac{1}{2}}=\dfrac{\Gamma(2)}{\Gamma(\frac{3}{2})}x^{\frac{1}{2% }}=\dfrac{1}{\frac{\sqrt{\pi}}{2}}x^{\frac{1}{2}}.
  24. d 1 2 d x 1 2 2 x 1 2 π = 2 π Γ ( 1 + 1 2 ) Γ ( 1 2 - 1 2 + 1 ) x 1 2 - 1 2 = 2 π Γ ( 3 2 ) Γ ( 1 ) x 0 = 2 π 2 x 0 π 0 ! = 1 , \dfrac{d^{\frac{1}{2}}}{dx^{\frac{1}{2}}}\dfrac{2x^{\frac{1}{2}}}{\sqrt{\pi}}=% \frac{2}{\sqrt{\pi}}\dfrac{\Gamma(1+\frac{1}{2})}{\Gamma(\frac{1}{2}-\frac{1}{% 2}+1)}x^{\frac{1}{2}-\frac{1}{2}}=\frac{2}{\sqrt{\pi}}\dfrac{\Gamma(\frac{3}{2% })}{\Gamma(1)}x^{0}=\dfrac{2\frac{\sqrt{\pi}}{2}x^{0}}{\sqrt{\pi}0!}=1,
  25. ( d 1 2 d x 1 2 d 1 2 d x 1 2 ) x = d d x x = 1. \left(\dfrac{d^{\frac{1}{2}}}{dx^{\frac{1}{2}}}\dfrac{d^{\frac{1}{2}}}{dx^{% \frac{1}{2}}}\right)x=\dfrac{d}{dx}x=1.
  26. d a d x a x - k = ( - 1 ) a Γ ( k + a ) Γ ( k ) x - ( k + a ) \dfrac{d^{a}}{dx^{a}}x^{-k}=(-1)^{a}\dfrac{\Gamma(k+a)}{\Gamma(k)}x^{-(k+a)}
  27. k 0 k\geq 0
  28. D 3 2 f ( x ) = D 1 2 D 1 f ( x ) = D 1 2 d d x f ( x ) D^{\frac{3}{2}}f(x)=D^{\frac{1}{2}}D^{1}f(x)=D^{\frac{1}{2}}\frac{d}{dx}f(x)
  29. { J f } ( s ) = { 0 t f ( τ ) d τ } ( s ) = 1 s ( { f } ) ( s ) \mathcal{L}\left\{Jf\right\}(s)=\mathcal{L}\left\{\int_{0}^{t}f(\tau)\,d\tau% \right\}(s)=\frac{1}{s}(\mathcal{L}\left\{f\right\})(s)
  30. { J 2 f } = 1 s ( { J f } ) ( s ) = 1 s 2 ( { f } ) ( s ) \mathcal{L}\left\{J^{2}f\right\}=\frac{1}{s}(\mathcal{L}\left\{Jf\right\})(s)=% \frac{1}{s^{2}}(\mathcal{L}\left\{f\right\})(s)
  31. J α f = - 1 { s - α ( { f } ) ( s ) } J^{\alpha}f=\mathcal{L}^{-1}\left\{s^{-\alpha}(\mathcal{L}\{f\})(s)\right\}
  32. J α ( t k ) = - 1 { Γ ( k + 1 ) s α + k + 1 } = Γ ( k + 1 ) Γ ( α + k + 1 ) t α + k J^{\alpha}\left(t^{k}\right)=\mathcal{L}^{-1}\left\{\dfrac{\Gamma(k+1)}{s^{% \alpha+k+1}}\right\}=\dfrac{\Gamma(k+1)}{\Gamma(\alpha+k+1)}t^{\alpha+k}
  33. { f * g } = ( { f } ) ( { g } ) \mathcal{L}\{f*g\}=(\mathcal{L}\{f\})(\mathcal{L}\{g\})
  34. ( J α f ) ( t ) = 1 Γ ( α ) - 1 { ( { p } ) ( { f } ) } = 1 Γ ( α ) ( p * f ) = 1 Γ ( α ) 0 t p ( t - τ ) f ( τ ) d τ = 1 Γ ( α ) 0 t ( t - τ ) α - 1 f ( τ ) d τ \begin{aligned}\displaystyle(J^{\alpha}f)(t)&\displaystyle=\frac{1}{\Gamma(% \alpha)}\mathcal{L}^{-1}\left\{\left(\mathcal{L}\{p\}\right)(\mathcal{L}\{f\})% \right\}\\ &\displaystyle=\frac{1}{\Gamma(\alpha)}(p*f)\\ &\displaystyle=\frac{1}{\Gamma(\alpha)}\int_{0}^{t}p(t-\tau)f(\tau)\,d\tau\\ &\displaystyle=\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-\tau)^{\alpha-1}f(\tau)% \,d\tau\\ \end{aligned}
  35. D t - α a f ( t ) = I t α a f ( t ) = 1 Γ ( α ) a t ( t - τ ) α - 1 f ( τ ) d τ {}_{a}D_{t}^{-\alpha}f(t)={}_{a}I_{t}^{\alpha}f(t)=\frac{1}{\Gamma(\alpha)}% \int_{a}^{t}(t-\tau)^{\alpha-1}f(\tau)d\tau
  36. 𝐃 t - α a f ( t ) = 1 Γ ( α ) a t ( log t τ ) α - 1 f ( τ ) d τ τ , t > a . {}_{a}\mathbf{D}_{t}^{-\alpha}f(t)=\frac{1}{\Gamma(\alpha)}\int_{a}^{t}\left(% \log\frac{t}{\tau}\right)^{\alpha-1}f(\tau)\frac{d\tau}{\tau},\qquad t>a.
  37. D t α a f ( t ) = d n d t n D t - ( n - α ) a f ( t ) = d n d t n I t n - α a f ( t ) {}_{a}D_{t}^{\alpha}f(t)=\frac{d^{n}}{dt^{n}}{}_{a}D_{t}^{-(n-\alpha)}f(t)=% \frac{d^{n}}{dt^{n}}{}_{a}I_{t}^{n-\alpha}f(t)
  38. D t α a C f ( t ) = 1 Γ ( n - α ) a t f ( n ) ( τ ) d τ ( t - τ ) α + 1 - n . {}_{a}^{C}D_{t}^{\alpha}f(t)=\frac{1}{\Gamma(n-\alpha)}\int_{a}^{t}\frac{f^{(n% )}(\tau)d\tau}{(t-\tau)^{\alpha+1-n}}.
  39. x - ν - α + 1 Γ ( α ) 0 x ( t - x ) α - 1 t - α - ν f ( t ) d t , \frac{x^{-\nu-\alpha+1}}{\Gamma(\alpha)}\int_{0}^{x}(t-x)^{\alpha-1}t^{-\alpha% -\nu}f(t)dt,
  40. ( a + α ρ f ) ( x ) = ρ 1 - α Γ ( α ) a x τ ρ - 1 f ( τ ) ( x ρ - τ ρ ) 1 - α d τ , x > a . \left({}^{\rho}\mathcal{I}^{\alpha}_{a+}f\right)(x)=\frac{\rho^{1-\alpha}}{% \Gamma({\alpha})}\int^{x}_{a}\frac{\tau^{\rho-1}f(\tau)}{(x^{\rho}-\tau^{\rho}% )^{1-\alpha}}\,d\tau,\qquad x>a.
  41. - ρ ( α u ) = Γ ( α + 1 ) Δ x 1 - α ρ ( β s + ϕ β w ) p t -\rho\left(\nabla^{\alpha}\cdot\vec{u}\right)=\Gamma(\alpha+1)\Delta x^{1-% \alpha}\rho\left(\beta_{s}+\phi\beta_{w}\right)\frac{\partial p}{\partial t}
  42. α u t α = - K ( - ) β u . \frac{\partial^{\alpha}u}{\partial t^{\alpha}}=-K(-\triangle)^{\beta}u.
  43. 2 u - 1 c 0 2 2 u t 2 + τ σ α α t α 2 u - τ ϵ β c 0 2 β + 2 u t β + 2 = 0. \nabla^{2}u-\dfrac{1}{c_{0}^{2}}\frac{\partial^{2}u}{\partial t^{2}}+\tau_{% \sigma}^{\alpha}\dfrac{\partial^{\alpha}}{\partial t^{\alpha}}\nabla^{2}u-% \dfrac{\tau_{\epsilon}^{\beta}}{c_{0}^{2}}\dfrac{\partial^{\beta+2}u}{\partial t% ^{\beta+2}}=0.
  44. i ψ ( 𝐫 , t ) t = D α ( - 2 Δ ) α 2 ψ ( 𝐫 , t ) + V ( 𝐫 , t ) ψ ( 𝐫 , t ) . i\hbar\frac{\partial\psi(\mathbf{r},t)}{\partial t}=D_{\alpha}(-\hbar^{2}% \Delta)^{\frac{\alpha}{2}}\psi(\mathbf{r},t)+V(\mathbf{r},t)\psi(\mathbf{r},t).
  45. ( - 2 Δ ) α 2 ψ ( 𝐫 , t ) = 1 ( 2 π ) 3 d 3 p e i 𝐩 𝐫 | 𝐩 | α φ ( 𝐩 , t ) . \left(-\hbar^{2}\Delta\right)^{\frac{\alpha}{2}}\psi(\mathbf{r},t)=\frac{1}{(2% \pi\hbar)^{3}}\int d^{3}pe^{\frac{i}{\hbar}\mathbf{p}\cdot\mathbf{r}}|\mathbf{% p}|^{\alpha}\varphi(\mathbf{p},t).

Fracture.html

  1. σ theoretical = E γ r o \sigma_{\mathrm{theoretical}}=\sqrt{\frac{E\gamma}{r_{o}}}
  2. E E
  3. γ \gamma
  4. r o r_{o}
  5. σ elliptical crack = σ applied ( 1 + 2 a ρ ) = 2 σ applied a ρ \sigma_{\mathrm{elliptical\ crack}}=\sigma_{\mathrm{applied}}\left(1+2\sqrt{% \frac{a}{\rho}}\right)=2\sigma_{\mathrm{applied}}\sqrt{\frac{a}{\rho}}
  6. σ applied \sigma_{\mathrm{applied}}
  7. a a
  8. ρ \rho
  9. σ fracture = E γ ρ 4 a r o . \sigma_{\mathrm{fracture}}=\sqrt{\frac{E\gamma\rho}{4ar_{o}}}.
  10. ρ \rho
  11. a a

Fraktur.html

  1. 𝔄 𝔅 𝔇 𝔈 𝔉 𝔊 𝔍 𝔎 𝔏 𝔐 𝔑 𝔒 𝔓 𝔔 𝔖 𝔗 𝔘 𝔙 𝔚 𝔛 𝔜 \mathfrak{A}~{}\mathfrak{B}~{}\mathfrak{C}~{}\mathfrak{D}~{}\mathfrak{E}~{}% \mathfrak{F}~{}\mathfrak{G}~{}\mathfrak{H}~{}\mathfrak{I}~{}\mathfrak{J}~{}% \mathfrak{K}~{}\mathfrak{L}~{}\mathfrak{M}~{}\mathfrak{N}~{}\mathfrak{O}~{}% \mathfrak{P}~{}\mathfrak{Q}~{}\mathfrak{R}~{}\mathfrak{S}~{}\mathfrak{T}~{}% \mathfrak{U}~{}\mathfrak{V}~{}\mathfrak{W}~{}\mathfrak{X}~{}\mathfrak{Y}~{}% \mathfrak{Z}
  2. 𝔞 𝔟 𝔠 𝔡 𝔢 𝔣 𝔤 𝔥 𝔦 𝔧 𝔨 𝔩 𝔪 𝔫 𝔬 𝔭 𝔮 𝔯 𝔰 𝔱 𝔲 𝔳 𝔴 𝔵 𝔶 𝔷 \mathfrak{a}~{}\mathfrak{b}~{}\mathfrak{c}~{}\mathfrak{d}~{}\mathfrak{e}~{}% \mathfrak{f}~{}\mathfrak{g}~{}\mathfrak{h}~{}\mathfrak{i}~{}\mathfrak{j}~{}% \mathfrak{k}~{}\mathfrak{l}~{}\mathfrak{m}~{}\mathfrak{n}~{}\mathfrak{o}~{}% \mathfrak{p}~{}\mathfrak{q}~{}\mathfrak{r}~{}\mathfrak{s}~{}\mathfrak{t}~{}% \mathfrak{u}~{}\mathfrak{v}~{}\mathfrak{w}~{}\mathfrak{x}~{}\mathfrak{y}~{}% \mathfrak{z}
  3. 𝔤 \mathfrak{g}
  4. 𝔞 \mathfrak{a}
  5. a 𝔞 a\in\mathfrak{a}
  6. 𝔠 \mathfrak{c}
  7. 𝔄 \mathfrak{A}

Frame_of_reference.html

  1. \mathfrak{R}
  2. \mathfrak{R}
  3. \mathfrak{R}
  4. 𝐫 = [ x 1 , x 2 , , x n ] . \mathbf{r}=[x^{1},\ x^{2},\ \dots\ ,x^{n}]\ .
  5. x j = x j ( x , y , z , ) , x^{j}=x^{j}(x,\ y,\ z,\ \dots)\ ,
  6. j = 1 , , n j=1,\ \dots\ ,\ n
  7. x j ( x , y , z , ) = constant , x^{j}(x,y,z,\dots)=\mathrm{constant}\ ,
  8. j = 1 , , n . j=1,\ \dots\ ,\ n\ .
  9. 𝐞 i ( 𝐫 ) = lim ϵ 0 𝐫 ( x 1 , , x i + ϵ , , x n ) - 𝐫 ( x 1 , , x i , , x n ) ϵ , \mathbf{e}_{i}(\mathbf{r})=\lim_{\epsilon\rightarrow 0}\frac{\mathbf{r}\left(x% ^{1},\ \dots,\ x^{i}+\epsilon,\ \dots,\ x^{n}\right)-\mathbf{r}\left(x^{1},\ % \dots,\ x^{i},\ \dots,\ x^{n}\right)}{\epsilon}\ ,
  10. i = 1 , , n i=1,\ \dots\ ,\ n
  11. ( d s ) 2 = g i k d x i d x k , (ds)^{2}=g_{ik}\ dx^{i}\ dx^{k}\ ,
  12. x 1 ( t ) x_{1}(t)
  13. x 2 ( t ) x_{2}(t)
  14. x 1 ( t ) = d + v 1 t = 200 + 22 t ; x 2 ( t ) = v 2 t = 30 t x_{1}(t)=d+v_{1}t=200\ +\ 22t\ ;\quad x_{2}(t)=v_{2}t=30t
  15. x 1 = x 2 x_{1}=x_{2}
  16. x 1 = x 2 x_{1}=x_{2}
  17. t t
  18. 200 + 22 t = 30 t 200+22t=30t\quad
  19. 8 t = 200 8t=200\quad
  20. t = 25 seconds t=25\quad\mathrm{seconds}
  21. 𝐫 = 𝐑 + 𝐫 \mathbf{r}=\mathbf{R}+\mathbf{r}^{\prime}
  22. 𝐯 = 𝐕 + 𝐯 \mathbf{v}=\mathbf{V}+\mathbf{v}^{\prime}
  23. 𝐚 = 𝐀 + 𝐚 \mathbf{a}=\mathbf{A}+\mathbf{a}^{\prime}
  24. 𝐅 = m 𝐚 = m 𝐀 + m 𝐚 \mathbf{F}=m\mathbf{a}=m\mathbf{A}+m\mathbf{a}^{\prime}
  25. 𝐚 = 𝐚 + s y m b o l ω ˙ × 𝐫 + 2 s y m b o l ω × 𝐯 + s y m b o l ω × ( s y m b o l ω × 𝐫 ) + 𝐀 0 \mathbf{a}=\mathbf{a}^{\prime}+\dot{symbol\omega}\times\mathbf{r}^{\prime}+2% symbol\omega\times\mathbf{v}^{\prime}+symbol\omega\times(symbol\omega\times% \mathbf{r}^{\prime})+\mathbf{A}_{0}
  26. 𝐚 = 𝐚 - s y m b o l ω ˙ × 𝐫 - 2 s y m b o l ω × 𝐯 - s y m b o l ω × ( s y m b o l ω × 𝐫 ) - 𝐀 0 \mathbf{a}^{\prime}=\mathbf{a}-\dot{symbol\omega}\times\mathbf{r}^{\prime}-2% symbol\omega\times\mathbf{v}^{\prime}-symbol\omega\times(symbol\omega\times% \mathbf{r}^{\prime})-\mathbf{A}_{0}
  27. 𝐅 = 𝐅 physical + 𝐅 Euler + 𝐅 Coriolis + 𝐅 centripetal - m 𝐀 0 \mathbf{F}^{\prime}=\mathbf{F}_{\mathrm{physical}}+\mathbf{F}^{\prime}_{% \mathrm{Euler}}+\mathbf{F}^{\prime}_{\mathrm{Coriolis}}+\mathbf{F}^{\prime}_{% \mathrm{centripetal}}-m\mathbf{A}_{0}
  28. 𝐅 Euler = - m s y m b o l ω ˙ × 𝐫 \mathbf{F}^{\prime}_{\mathrm{Euler}}=-m\dot{symbol\omega}\times\mathbf{r}^{\prime}
  29. 𝐅 Coriolis = - 2 m s y m b o l ω × 𝐯 \mathbf{F}^{\prime}_{\mathrm{Coriolis}}=-2msymbol\omega\times\mathbf{v}^{\prime}
  30. 𝐅 centrifugal = - m s y m b o l ω × ( s y m b o l ω × 𝐫 ) = m ( ω 2 𝐫 - ( s y m b o l ω 𝐫 ) s y m b o l ω ) \mathbf{F}^{\prime}_{\mathrm{centrifugal}}=-msymbol\omega\times(symbol\omega% \times\mathbf{r}^{\prime})=m(\omega^{2}\mathbf{r}^{\prime}-(symbol\omega\cdot% \mathbf{r}^{\prime})symbol\omega)

François_Viète.html

  1. X 2 + X b = c X^{2}+Xb=c
  2. X 3 + a X = b X^{3}+aX=b
  3. 2 π 45 \frac{2\pi}{45}
  4. π = 2 × 2 2 × 2 2 + 2 × 2 2 + 2 + 2 × 2 2 + 2 + 2 + 2 × \pi=2\times\frac{2}{\sqrt{2}}\times\frac{2}{\sqrt{2+\sqrt{2}}}\times\frac{2}{% \sqrt{2+\sqrt{2+\sqrt{2}}}}\times\frac{2}{\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}\times\cdots

Fredholm_operator.html

  1. S : Y X S:Y\to X
  2. Id X - S T and Id Y - T S \mathrm{Id}_{X}-ST\quad\,\text{and}\quad\mathrm{Id}_{Y}-TS
  3. ind T := dim ker T - codim ran T \mathrm{ind}\,T:=\dim\ker T-\mathrm{codim}\,\mathrm{ran}\,T
  4. ind T := dim ker T - dim coker T ; \mathrm{ind}\,T:=\dim\ker T-\mathrm{dim}\,\mathrm{coker}\,T;
  5. U T U\circ T
  6. ind ( U T ) = ind ( U ) + ind ( T ) . \mathrm{ind}(U\circ T)=\mathrm{ind}(U)+\mathrm{ind}(T).
  7. inf { S x : x X 0 , x = 1 } = 0. \inf\{\|Sx\|:x\in X_{0},\,\|x\|=1\}=0.\,
  8. S ( e n ) = e n + 1 , n 0. S(e_{n})=e_{n+1},\quad n\geq 0.\,
  9. S * ( e 0 ) = 0 , S * ( e n ) = e n - 1 , n 1. S^{*}(e_{0})=0,\ \ S^{*}(e_{n})=e_{n-1},\quad n\geq 1.\,
  10. e n : e i t 𝐓 e i n t , n 0 , e_{n}:\mathrm{e}^{\mathrm{i}t}\in\mathbf{T}\rightarrow\mathrm{e}^{\mathrm{i}nt% },\quad n\geq 0,\,
  11. T φ : f H 2 ( T ) P ( f φ ) H 2 ( T ) . T_{\varphi}:f\in H^{2}(\mathrm{T})\rightarrow P(f\varphi)\in H^{2}(\mathrm{T}).\,
  12. n n
  13. T n T_{n}
  14. T T
  15. R ( T n ) R(T^{n})
  16. R ( T n ) R(T^{n})
  17. R ( T n ) R(T^{n})
  18. T 0 = T T_{0}=T
  19. n n
  20. R ( T n ) R(T^{n})
  21. T n T_{n}
  22. T T
  23. T T
  24. T n T_{n}
  25. n n

Free_abelian_group.html

  1. e 1 = ( 1 , 0 ) \ e_{1}=(1,0)
  2. e 2 = ( 0 , 1 ) \ e_{2}=(0,1)
  3. ( 4 , 3 ) = 4 e 1 + 3 e 2 \ (4,3)=4e_{1}+3e_{2}
  4. 4 e 1 := e 1 + e 1 + e 1 + e 1 . \ 4e_{1}:=e_{1}+e_{1}+e_{1}+e_{1}.
  5. f 1 = ( 1 , 0 ) \ f_{1}=(1,0)
  6. f 2 = ( 1 , 1 ) \ f_{2}=(1,1)
  7. ( 4 , 3 ) = f 1 + 3 f 2 . \ (4,3)=f_{1}+3f_{2}.
  8. \mathbb{Z}^{\mathbb{N}}
  9. \mathbb{Z}
  10. \mathbb{Z}^{\mathbb{N}}
  11. \mathbb{Z}
  12. ( B ) \mathbb{Z}^{(B)}
  13. ( B ) \mathbb{Z}^{(B)}
  14. ( B ) \mathbb{Z}^{(B)}
  15. ( B ) \mathbb{Z}^{(B)}
  16. f = { x f ( x ) 0 } f ( x ) e x f=\sum_{\{x\mid f(x)\neq 0\}}f(x)e_{x}
  17. ( B ) \mathbb{Z}^{(B)}
  18. ( B ) \mathbb{Z}^{(B)}
  19. 0 x = 0 1 x = x n x = x + ( n - 1 ) x if n > 1 ( - n ) x = - ( n x ) if n < 0 \begin{aligned}\displaystyle 0\,x&\displaystyle=0\\ \displaystyle 1\,x&\displaystyle=x\\ \displaystyle n\,x&\displaystyle=x+(n-1)\,x\qquad\,\text{if}\quad n>1\\ \displaystyle(-n)\,x&\displaystyle=-(n\,x)\qquad\,\text{if}\quad n<0\end{aligned}
  20. \Z \Z
  21. \Z \Z
  22. \Z \Z
  23. \Z \Z
  24. 0 = 0 b = n b 0=0\,b=n\,b
  25. n \mathbb{Z}^{n}
  26. F F
  27. G F G\subset F
  28. G G
  29. G G
  30. F F
  31. G G
  32. ( e 1 , , e n ) (e_{1},\ldots,e_{n})
  33. F F
  34. d 1 | d 2 | | d k d_{1}|d_{2}|\ldots|d_{k}
  35. ( d 1 e 1 , , d k e k ) (d_{1}e_{1},\ldots,d_{k}e_{k})
  36. G . G.
  37. d 1 , d 2 , , d k d_{1},d_{2},\ldots,d_{k}
  38. F F
  39. G G
  40. ( e 1 , , e n ) (e_{1},\ldots,e_{n})
  41. d 1 d r d_{1}\cdots d_{r}
  42. F = ( A ) F=\mathbb{Z}^{(A)}
  43. f = { x f ( x ) 0 } f ( x ) e x { x f ( x ) 0 } f ( x ) x , f=\sum_{\{x\mid f(x)\neq 0\}}f(x)e_{x}\mapsto\sum_{\{x\mid f(x)\neq 0\}}f(x)x,
  44. e x x e_{x}\mapsto x
  45. \Z \Z
  46. ( ( A ) ) n \left(\mathbb{Z}^{(A)}\right)^{n}