wpmath0000001_0

13_(number).html

  1. 13 × x 13\times x
  2. 13 x 13^{x}\,
  3. x 13 x^{13}\,

1995.html

  1. M w M_{\mathrm{w}}

1_(number).html

  1. I \mathrm{I}
  2. ( { x | x < 1 } , { x | x 1 } ) (\left\{x|x<1\right\},\left\{x|x\geq 1\right\})
  3. 2 5 = 1 3 + 1 15 \frac{2}{5}=\frac{1}{3}+\frac{1}{15}
  4. 1 1 - x = 1 + x + x 2 + x 3 + \frac{1}{1-x}=1+x+x^{2}+x^{3}+\cdots
  5. | x | < 1 |x|<1
  6. 1 × x 1\times x
  7. 1 ÷ x 1\div x
  8. 3 ¯ \overline{3}
  9. 6 ¯ \overline{6}
  10. 142857 ¯ \overline{142857}
  11. 1 ¯ \overline{1}
  12. 09 ¯ \overline{09}
  13. 3 ¯ \overline{3}
  14. 076923 ¯ \overline{076923}
  15. 714285 ¯ \overline{714285}
  16. 6 ¯ \overline{6}
  17. x ÷ 1 x\div 1
  18. 1 x 1^{x}\,
  19. x 1 x^{1}\,

2_(number).html

  1. 3 n - 1 3n-1
  2. k = 0 1 2 k = 1 + 1 2 + 1 4 + 1 8 + 1 16 + = 2. \sum_{k=0}^{\infty}\frac{1}{2^{k}}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac% {1}{16}+\cdots=2.
  3. k = 0 1 n k = 1 + 1 n - 1 for all n > 1. \sum_{k=0}^{\infty}\frac{1}{n^{k}}=1+\frac{1}{n-1}\quad\mbox{for all}~{}\quad n% \in\mathbb{R}>1.
  4. { { } , } \{\{\emptyset\},\emptyset\}
  5. k = 0 n - 1 2 k = 2 n - 1 \sum_{k=0}^{n-1}2^{k}=2^{n}-1
  6. k = a n - 1 2 k = 2 n - k = 0 a - 1 2 k - 1 \sum_{k=a}^{n-1}2^{k}=2^{n}-\sum_{k=0}^{a-1}2^{k}-1
  7. χ = V - E + F = 2 , \chi=V-E+F=2,
  8. 2 × x 2\times x
  9. 2 ÷ x 2\div x
  10. 6 ¯ \overline{6}
  11. 3 ¯ \overline{3}
  12. 285714 ¯ \overline{285714}
  13. 2 ¯ \overline{2}
  14. 18 ¯ \overline{18}
  15. 6 ¯ \overline{6}
  16. 153846 ¯ \overline{153846}
  17. 142857 ¯ \overline{142857}
  18. 3 ¯ \overline{3}
  19. x ÷ 2 x\div 2
  20. 2 x 2^{x}\,
  21. x 2 x^{2}\,

2D_computer_graphics.html

  1. T δ T_{\mathbf{\delta}}
  2. T δ f ( 𝐯 ) = f ( 𝐯 + δ ) . T_{\mathbf{\delta}}f(\mathbf{v})=f(\mathbf{v}+\mathbf{\delta}).
  3. T 𝐯 = [ 1 0 0 v x 0 1 0 v y 0 0 1 v z 0 0 0 1 ] T_{\mathbf{v}}=\begin{bmatrix}1&0&0&v_{x}\\ 0&1&0&v_{y}\\ 0&0&1&v_{z}\\ 0&0&0&1\end{bmatrix}
  4. T 𝐯 𝐩 = [ 1 0 0 v x 0 1 0 v y 0 0 1 v z 0 0 0 1 ] [ p x p y p z 1 ] = [ p x + v x p y + v y p z + v z 1 ] = 𝐩 + 𝐯 T_{\mathbf{v}}\mathbf{p}=\begin{bmatrix}1&0&0&v_{x}\\ 0&1&0&v_{y}\\ 0&0&1&v_{z}\\ 0&0&0&1\end{bmatrix}\begin{bmatrix}p_{x}\\ p_{y}\\ p_{z}\\ 1\end{bmatrix}=\begin{bmatrix}p_{x}+v_{x}\\ p_{y}+v_{y}\\ p_{z}+v_{z}\\ 1\end{bmatrix}=\mathbf{p}+\mathbf{v}
  5. T 𝐯 - 1 = T - 𝐯 . T^{-1}_{\mathbf{v}}=T_{-\mathbf{v}}.\!
  6. T 𝐮 T 𝐯 = T 𝐮 + 𝐯 . T_{\mathbf{u}}T_{\mathbf{v}}=T_{\mathbf{u}+\mathbf{v}}.\!
  7. R = [ cos θ - sin θ sin θ cos θ ] R=\begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\\ \end{bmatrix}
  8. R T = R - 1 , det R = 1 R^{T}=R^{-1},\det R=1\,
  9. S O ( n ) SO(n)
  10. R ( θ ) = [ cos θ - sin θ sin θ cos θ ] R(\theta)=\begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\\ \end{bmatrix}
  11. [ x y ] = [ cos θ - sin θ sin θ cos θ ] [ x y ] \begin{bmatrix}x^{\prime}\\ y^{\prime}\\ \end{bmatrix}=\begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\\ \end{bmatrix}\begin{bmatrix}x\\ y\\ \end{bmatrix}
  12. x = x cos θ - y sin θ x^{\prime}=x\cos\theta-y\sin\theta\,
  13. y = x sin θ + y cos θ y^{\prime}=x\sin\theta+y\cos\theta\,
  14. R ( - θ ) = [ cos θ sin θ - sin θ cos θ ] R(-\theta)=\begin{bmatrix}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\\ \end{bmatrix}\,
  15. R ( 90 ) = [ 0 - 1 1 0 ] R(90^{\circ})=\begin{bmatrix}0&-1\\ 1&0\\ \end{bmatrix}
  16. R ( 180 ) = [ - 1 0 0 - 1 ] R(180^{\circ})=\begin{bmatrix}-1&0\\ 0&-1\\ \end{bmatrix}
  17. R ( 270 ) = [ 0 1 - 1 0 ] R(270^{\circ})=\begin{bmatrix}0&1\\ -1&0\\ \end{bmatrix}
  18. S v = [ v x 0 0 0 v y 0 0 0 v z ] . S_{v}=\begin{bmatrix}v_{x}&0&0\\ 0&v_{y}&0\\ 0&0&v_{z}\\ \end{bmatrix}.
  19. S v p = [ v x 0 0 0 v y 0 0 0 v z ] [ p x p y p z ] = [ v x p x v y p y v z p z ] . S_{v}p=\begin{bmatrix}v_{x}&0&0\\ 0&v_{y}&0\\ 0&0&v_{z}\\ \end{bmatrix}\begin{bmatrix}p_{x}\\ p_{y}\\ p_{z}\end{bmatrix}=\begin{bmatrix}v_{x}p_{x}\\ v_{y}p_{y}\\ v_{z}p_{z}\end{bmatrix}.
  20. S v = [ v x 0 0 0 0 v y 0 0 0 0 v z 0 0 0 0 1 ] . S_{v}=\begin{bmatrix}v_{x}&0&0&0\\ 0&v_{y}&0&0\\ 0&0&v_{z}&0\\ 0&0&0&1\end{bmatrix}.
  21. S v p = [ v x 0 0 0 0 v y 0 0 0 0 v z 0 0 0 0 1 ] [ p x p y p z 1 ] = [ v x p x v y p y v z p z 1 ] . S_{v}p=\begin{bmatrix}v_{x}&0&0&0\\ 0&v_{y}&0&0\\ 0&0&v_{z}&0\\ 0&0&0&1\end{bmatrix}\begin{bmatrix}p_{x}\\ p_{y}\\ p_{z}\\ 1\end{bmatrix}=\begin{bmatrix}v_{x}p_{x}\\ v_{y}p_{y}\\ v_{z}p_{z}\\ 1\end{bmatrix}.
  22. S v = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 s ] . S_{v}=\begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&\frac{1}{s}\end{bmatrix}.
  23. S v p = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 s ] [ p x p y p z 1 ] = [ p x p y p z 1 s ] S_{v}p=\begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&\frac{1}{s}\end{bmatrix}\begin{bmatrix}p_{x}\\ p_{y}\\ p_{z}\\ 1\end{bmatrix}=\begin{bmatrix}p_{x}\\ p_{y}\\ p_{z}\\ \frac{1}{s}\end{bmatrix}
  24. [ s p x s p y s p z 1 ] . \begin{bmatrix}sp_{x}\\ sp_{y}\\ sp_{z}\\ 1\end{bmatrix}.

3-sphere.html

  1. i = 0 3 ( x i - C i ) 2 = ( x 0 - C 0 ) 2 + ( x 1 - C 1 ) 2 + ( x 2 - C 2 ) 2 + ( x 3 - C 3 ) 2 = r 2 . \sum_{i=0}^{3}(x_{i}-C_{i})^{2}=(x_{0}-C_{0})^{2}+(x_{1}-C_{1})^{2}+(x_{2}-C_{% 2})^{2}+(x_{3}-C_{3})^{2}=r^{2}.
  2. S 3 = { ( x 0 , x 1 , x 2 , x 3 ) 4 : x 0 2 + x 1 2 + x 2 2 + x 3 2 = 1 } . S^{3}=\left\{(x_{0},x_{1},x_{2},x_{3})\in\mathbb{R}^{4}:x_{0}^{2}+x_{1}^{2}+x_% {2}^{2}+x_{3}^{2}=1\right\}.
  3. S 3 = { ( z 1 , z 2 ) 2 : | z 1 | 2 + | z 2 | 2 = 1 } S^{3}=\left\{(z_{1},z_{2})\in\mathbb{C}^{2}:|z_{1}|^{2}+|z_{2}|^{2}=1\right\}
  4. S 3 = { q : || q || = 1 } . S^{3}=\left\{q\in\mathbb{H}:||q||=1\right\}.
  5. 2 π 2 r 3 2\pi^{2}r^{3}\,
  6. 1 2 π 2 r 4 . \begin{matrix}\frac{1}{2}\end{matrix}\pi^{2}r^{4}.
  7. 3 \mathbb{R}^{3}
  8. ( z 1 , z 2 ) λ = ( z 1 λ , z 2 λ ) λ 𝕋 (z_{1},z_{2})\cdot\lambda=(z_{1}\lambda,z_{2}\lambda)\quad\forall\lambda\in% \mathbb{T}
  9. p p
  10. N N
  11. p p
  12. N p Np
  13. π \pi
  14. x 0 2 + x 1 2 + x 2 2 + x 3 2 = 1 {x_{0}}^{2}+{x_{1}}^{2}+{x_{2}}^{2}+{x_{3}}^{2}=1
  15. x 0 = r cos ψ x_{0}=r\cos\psi
  16. x 1 = r sin ψ cos θ x_{1}=r\sin\psi\cos\theta
  17. x 2 = r sin ψ sin θ cos ϕ x_{2}=r\sin\psi\sin\theta\cos\phi
  18. x 3 = r sin ψ sin θ sin ϕ x_{3}=r\sin\psi\sin\theta\sin\phi
  19. d s 2 = r 2 [ d ψ 2 + sin 2 ψ ( d θ 2 + sin 2 θ d ϕ 2 ) ] ds^{2}=r^{2}\left[d\psi^{2}+\sin^{2}\psi\left(d\theta^{2}+\sin^{2}\theta\,d% \phi^{2}\right)\right]
  20. d V = r 3 ( sin 2 ψ sin θ ) d r d ψ d θ d ϕ . dV=r^{3}\left(\sin^{2}\psi\,\sin\theta\right)\,dr\wedge d\psi\wedge d\theta% \wedge d\phi.
  21. z 1 = e i ξ 1 sin η z_{1}=e^{i\,\xi_{1}}\sin\eta
  22. z 2 = e i ξ 2 cos η . z_{2}=e^{i\,\xi_{2}}\cos\eta.
  23. x 0 = cos ξ 1 sin η x_{0}=\cos\xi_{1}\sin\eta
  24. x 1 = sin ξ 1 sin η x_{1}=\sin\xi_{1}\sin\eta
  25. x 2 = cos ξ 2 cos η x_{2}=\cos\xi_{2}\cos\eta
  26. x 3 = sin ξ 2 cos η x_{3}=\sin\xi_{2}\cos\eta
  27. S 1 S 3 S 2 . S^{1}\to S^{3}\to S^{2}.\,
  28. d s 2 = d η 2 + sin 2 η d ξ 1 2 + cos 2 η d ξ 2 2 ds^{2}=d\eta^{2}+\sin^{2}\eta\,d\xi_{1}^{2}+\cos^{2}\eta\,d\xi_{2}^{2}
  29. d V = sin η cos η d η d ξ 1 d ξ 2 . dV=\sin\eta\cos\eta\,d\eta\wedge d\xi_{1}\wedge d\xi_{2}.
  30. z 1 = e i ( ξ 1 + ξ 2 ) sin η z_{1}=e^{i\,(\xi_{1}+\xi_{2})}\sin\eta
  31. z 2 = e i ( ξ 1 - ξ 2 ) cos η . z_{2}=e^{i\,(\xi_{1}-\xi_{2})}\cos\eta.
  32. p = ( 1 - u 2 1 + u 2 , 2 𝐮 1 + u 2 ) = 1 + 𝐮 1 - 𝐮 p=\left(\frac{1-\|u\|^{2}}{1+\|u\|^{2}},\frac{2\mathbf{u}}{1+\|u\|^{2}}\right)% =\frac{1+\mathbf{u}}{1-\mathbf{u}}
  33. 𝐮 = 1 1 + x 0 ( x 1 , x 2 , x 3 ) . \mathbf{u}=\frac{1}{1+x_{0}}\left(x_{1},x_{2},x_{3}\right).
  34. p = ( - 1 + v 2 1 + v 2 , 2 𝐯 1 + v 2 ) = - 1 + 𝐯 1 + 𝐯 p=\left(\frac{-1+\|v\|^{2}}{1+\|v\|^{2}},\frac{2\mathbf{v}}{1+\|v\|^{2}}\right% )=\frac{-1+\mathbf{v}}{1+\mathbf{v}}
  35. 𝐯 = 1 1 - x 0 ( x 1 , x 2 , x 3 ) . \mathbf{v}=\frac{1}{1-x_{0}}\left(x_{1},x_{2},x_{3}\right).
  36. 𝐯 = 1 u 2 𝐮 \mathbf{v}=\frac{1}{\|u\|^{2}}\mathbf{u}
  37. x 1 + x 2 i + x 3 j + x 4 k ( x 1 + i x 2 x 3 + i x 4 - x 3 + i x 4 x 1 - i x 2 ) . x_{1}+x_{2}i+x_{3}j+x_{4}k\mapsto\begin{pmatrix}\;\;\,x_{1}+ix_{2}&x_{3}+ix_{4% }\\ -x_{3}+ix_{4}&x_{1}-ix_{2}\end{pmatrix}.
  38. ( e i ξ 1 sin η e i ξ 2 cos η - e - i ξ 2 cos η e - i ξ 1 sin η ) . \begin{pmatrix}e^{i\,\xi_{1}}\sin\eta&e^{i\,\xi_{2}}\cos\eta\\ -e^{-i\,\xi_{2}}\cos\eta&e^{-i\,\xi_{1}}\sin\eta\end{pmatrix}.
  39. U S U ( 2 ) U\in SU(2)
  40. U = α 0 I + i = 1 3 α i J i U=\alpha_{0}I+\sum_{i=1}^{3}\alpha_{i}J_{i}
  41. α i \alpha_{i}

61_Cygni.html

  1. 79.4 2 + 62 2 = 100 \begin{smallmatrix}\sqrt{79.4^{2}\ +\ 62^{2}}\ =\ 100\end{smallmatrix}
  2. 87 2 + 64 2 = 106 \begin{smallmatrix}\sqrt{87^{2}\ +\ 64^{2}}\ =\ 106\end{smallmatrix}
  3. α R = 138 D \begin{smallmatrix}\alpha_{R}\ =\ \frac{138}{D}\end{smallmatrix}
  4. r p e r = ( 1 - e ) a 44 \begin{smallmatrix}r_{per}\ =\ (1\ -\ e)\cdot a\ \approx\ 44\end{smallmatrix}
  5. r a p = ( 1 + e ) a 124 \begin{smallmatrix}r_{ap}\ =\ (1\ +\ e)\cdot a\ \approx\ 124\end{smallmatrix}

A-law_algorithm.html

  1. F ( x ) = sgn ( x ) { A | x | 1 + ln ( A ) , | x | < 1 A 1 + ln ( A | x | ) 1 + ln ( A ) , 1 A | x | 1 , F(x)=\operatorname{sgn}(x)\begin{cases}{A|x|\over 1+\ln(A)},&|x|<{1\over A}\\ \frac{1+\ln(A|x|)}{1+\ln(A)},&{1\over A}\leq|x|\leq 1,\end{cases}
  2. A = ′′′ 87.6 ′′′ A=^{\prime\prime\prime}87.6^{\prime\prime\prime}
  3. F - 1 ( y ) = sgn ( y ) { | y | ( 1 + ln ( A ) ) A , | y | < 1 1 + ln ( A ) exp ( | y | ( 1 + ln ( A ) ) - 1 ) A , 1 1 + ln ( A ) | y | < 1. F^{-1}(y)=\operatorname{sgn}(y)\begin{cases}{|y|(1+\ln(A))\over A},&|y|<{1% \over 1+\ln(A)}\\ {\exp(|y|(1+\ln(A))-1)\over A},&{1\over 1+\ln(A)}\leq|y|<1.\end{cases}

Abbe_number.html

  1. V D = n D - 1 n F - n C , V_{D}=\frac{n_{D}-1}{n_{F}-n_{C}},
  2. V d = n d - 1 n F - n C V_{d}=\frac{n_{d}-1}{n_{F}-n_{C}}
  3. V e = n e - 1 n F - n C V_{e}=\frac{n_{e}-1}{n_{F^{\prime}}-n_{C^{\prime}}}

Abductive_reasoning.html

  1. a a
  2. b b
  3. a a
  4. b b
  5. a a
  6. b b
  7. a a
  8. b b
  9. b b
  10. a a
  11. b b
  12. a a
  13. b b
  14. a a
  15. b b
  16. a a
  17. a a
  18. b b
  19. b b
  20. a a
  21. b b
  22. a a
  23. b b
  24. a a
  25. b b
  26. b b
  27. T T
  28. O O
  29. O O
  30. T T
  31. E E
  32. O O
  33. T T
  34. O O
  35. E E
  36. T T
  37. E E
  38. T T
  39. O O
  40. E E
  41. E E
  42. O O
  43. T T
  44. T E O T\cup E\models O
  45. T E T\cup E
  46. E E
  47. O O
  48. E E
  49. T T
  50. E E
  51. H H
  52. M M
  53. e e
  54. H H H^{\prime}\subseteq H
  55. e ( H ) e(H^{\prime})
  56. H H H^{\prime}\subseteq H
  57. M e ( H ) M\subseteq e(H^{\prime})
  58. H H^{\prime}
  59. e ( H ) e(H^{\prime})
  60. M M
  61. H H H^{\prime}\subseteq H
  62. e ( H ) = h H e ( { h } ) e(H^{\prime})=\bigcup_{h\in H^{\prime}}e(\{h\})
  63. x x
  64. x ¯ \overline{x}
  65. y y
  66. y ¯ \overline{y}
  67. p ( x | y ) p(x|y)
  68. p ( x | y ¯ ) p(x|\overline{y})
  69. p ( x | y ) p(x|y)
  70. p ( x | y ¯ ) p(x|\overline{y})
  71. p ( y | x ) p(y|x)
  72. p ( y | x ¯ ) p(y|\overline{x})
  73. p ( y x ) = p ( x ) p ( y | x ) + p ( x ¯ ) p ( y | x ¯ ) p(y\|x)=p(x)p(y|x)+p(\overline{x})p(y|\overline{x})
  74. \|
  75. { p ( x | y ) = p ( x y ) p ( y ) p ( y | x ) = p ( x y ) p ( x ) p ( y | x ) = p ( y ) p ( x | y ) p ( x ) . \begin{cases}p(x|y)=\frac{p(x\land y)}{p(y)}\\ p(y|x)=\frac{p(x\land y)}{p(x)}\end{cases}\;\;\Rightarrow\;\;\;\;p(y|x)=\frac{% p(y)p(x|y)}{p(x)}\;.
  76. p ( y ) p(y)
  77. p ( x ) p(x)
  78. a ( y ) a(y)
  79. a ( y ¯ ) = 1 - a ( y ) a(\overline{y})=1-a(y)
  80. y y
  81. y ¯ \overline{y}
  82. p ( x ) = a ( y ) p ( x | y ) + a ( y ¯ ) p ( x | y ¯ ) p(x)=a(y)p(x|y)+a(\overline{y})p(x|\overline{y})
  83. p ( y | x ) p(y|x)
  84. p ( y | x ¯ ) p(y|\overline{x})
  85. { p ( y | x ) = a ( y ) p ( x | y ) a ( y ) p ( x | y ) + a ( y ¯ ) p ( x | y ¯ ) p ( y | x ¯ ) = a ( y ) p ( x ¯ | y ) a ( y ) p ( x ¯ | y ) + a ( y ¯ ) p ( x ¯ | y ¯ ) \begin{cases}p(y|x)=\frac{a(y)p(x|y)}{a(y)p(x|y)+a(\overline{y})p(x|\overline{% y})}\\ p(y|\overline{x})=\frac{a(y)p(\overline{x}|y)}{a(y)p(\overline{x}|y)+a(% \overline{y})p(\overline{x}|\overline{y})}\end{cases}
  86. p ( y ¯ x ) p(y\overline{\|}x)
  87. p ( y ¯ x ) = p ( x ) ( a ( y ) p ( x | y ) a ( y ) p ( x | y ) + a ( y ¯ ) p ( x | y ¯ ) ) + p ( x ¯ ) ( a ( y ) p ( x ¯ | y ) a ( y ) p ( x ¯ | y ) + a ( y ¯ ) p ( x ¯ | y ¯ ) ) p(y\overline{\|}x)=p(x)\left(\frac{a(y)p(x|y)}{a(y)p(x|y)+a(\overline{y})p(x|% \overline{y})}\right)+p(\overline{x})\left(\frac{a(y)p(\overline{x}|y)}{a(y)p(% \overline{x}|y)+a(\overline{y})p(\overline{x}|\overline{y})}\right)
  88. p ( y ¯ x ) = a ( y ) ( p ( x | y ) + p ( x ¯ | y ) ) p(y\overline{\|}x)=a(y)\left(p(x|y)+p(\overline{x}|y)\right)
  89. p ( x ) = 1 p(x)=1
  90. p ( x ¯ ) = 1 p(\overline{x})=1
  91. p ( y ¯ x ) = p ( y | x ) p(y\overline{\|}x)=p(y|x)
  92. p ( y | x ) = p ( x | y ) p(y|x)=p(x|y)
  93. X X
  94. x i x_{i}
  95. Y Y
  96. y j y_{j}
  97. X X\,\!
  98. x i x_{i}\,\!
  99. ω X = ( b , u , a ) \omega_{X}=(\vec{b},u,\vec{a})\,\!
  100. b \vec{b}\,\!
  101. X X\,\!
  102. u u\,\!
  103. a \vec{a}\,\!
  104. X X\,\!
  105. u + b ( x i ) = 1 u+\sum\vec{b}(x_{i})=1\,\!
  106. a ( x i ) = 1 \sum\vec{a}(x_{i})=1\,\!
  107. b ( x i ) , u , a ( x i ) [ 0 , 1 ] \vec{b}(x_{i}),u,\vec{a}(x_{i})\in[0,1]\,\!
  108. X X
  109. Y Y
  110. ω X | Y \omega_{X|Y}
  111. ω X | Y ¯ \omega_{X|\overline{Y}}
  112. ω X \omega_{X}
  113. X X
  114. a Y a_{Y}
  115. Y Y
  116. ω Y | X \omega_{Y|X}
  117. ω Y | X ¯ \omega_{Y|\overline{X}}
  118. ¯ \overline{\|}
  119. ¯ \overline{\circledcirc}
  120. ω Y ¯ X = ω X ¯ ( ω X | Y , ω X | Y ¯ , a Y ) \omega_{Y\overline{\|}X}=\omega_{X}\;\overline{\circledcirc}\;(\omega_{X|Y},% \omega_{X|\overline{Y}},a_{Y})\,\!
  121. \therefore
  122. \therefore
  123. \therefore
  124. \therefore
  125. \therefore
  126. \therefore

Abelian_category.html

  1. 0 M M M ′′ 0 0\to M^{\prime}\to M\to M^{\prime\prime}\to 0
  2. M , M , M ′′ M^{\prime},M,M^{\prime\prime}
  3. 0 M M M ′′ 0 0\to M^{\prime}\to M\to M^{\prime\prime}\to 0
  4. M , M ′′ M^{\prime},M^{\prime\prime}
  5. Q : 𝐀 𝐀 / 𝐂 Q\colon\mathbf{A}\to\mathbf{A}/\mathbf{C}
  6. T n T_{n}
  7. n × n n\times n
  8. 𝐀 n \mathbf{A}_{n}
  9. T n T_{n}
  10. 𝐀 n \mathbf{A}_{n}
  11. I : 𝐀 2 𝐀 3 I\colon\mathbf{A}_{2}\to\mathbf{A}_{3}

Abelian_group.html

  1. i = 1 u 𝐙 k i \bigoplus_{i=1}^{u}\ \mathbf{Z}_{k_{i}}
  2. e 1 e 2 e n e_{1}\leq e_{2}\leq\cdots\leq e_{n}
  3. 𝐙 p e 1 𝐙 p e n . \mathbf{Z}_{p^{e_{1}}}\oplus\cdots\oplus\mathbf{Z}_{p^{e_{n}}}.
  4. 𝐙 p 𝐙 p , \mathbf{Z}_{p}\oplus\cdots\oplus\mathbf{Z}_{p},
  5. Aut ( P ) GL ( n , 𝐅 p ) , \mathrm{Aut}(P)\cong\mathrm{GL}(n,\mathbf{F}_{p}),
  6. | Aut ( P ) | = ( p n - 1 ) ( p n - p n - 1 ) . |\mathrm{Aut}(P)|=(p^{n}-1)\cdots(p^{n}-p^{n-1}).
  7. d k = max { r | e r = e k } d_{k}=\mathrm{max}\{r|e_{r}=e_{k}\}
  8. c k = min { r | e r = e k } c_{k}=\mathrm{min}\{r|e_{r}=e_{k}\}
  9. | Aut ( P ) | = k = 1 n ( p d k - p k - 1 ) j = 1 n ( p e j ) n - d j i = 1 n ( p e i - 1 ) n - c i + 1 . |\mathrm{Aut}(P)|=\prod_{k=1}^{n}{(p^{d_{k}}-p^{k-1})}\prod_{j=1}^{n}{(p^{e_{j% }})^{n-d_{j}}}\prod_{i=1}^{n}{(p^{e_{i}-1})^{n-c_{i}+1}}.

Aberration_of_light.html

  1. u x u_{x}
  2. u y u_{y}
  3. tan ( θ ) = u y / u x \tan(\theta)=u_{y}/u_{x}
  4. v v
  5. u x = u x - v u_{x}^{\prime}=u_{x}-v
  6. u y = u y u_{y}^{\prime}=u_{y}
  7. tan ( ϕ ) = u y u x = u y ( u x - v ) = sin ( θ ) ( cos ( θ ) - v / c ) \tan(\phi)=\frac{u_{y}^{\prime}}{u_{x}^{\prime}}=\frac{u_{y}}{(u_{x}-v)}=\frac% {\sin(\theta)}{(\cos(\theta)-v/c)}
  8. θ = 90 \theta=90^{\circ}
  9. tan ( θ - ϕ ) = - v / c \tan(\theta-\phi)=-v/c
  10. u x = ( u x - v ) / ( 1 - u x v / c 2 ) u_{x}^{\prime}=(u_{x}-v)/(1-u_{x}v/c^{2})
  11. u y = u y / γ ( 1 - u x v / c 2 ) u_{y}^{\prime}=u_{y}/\gamma(1-u_{x}v/c^{2})
  12. γ = 1 / 1 - v 2 / c 2 \gamma=1/\sqrt{1-v^{2}/c^{2}}
  13. tan ( ϕ ) = u y u x = u y γ ( u x - v ) = sin ( θ ) γ ( cos ( θ ) - v / c ) \tan(\phi)=\frac{u_{y}^{\prime}}{u_{x}^{\prime}}=\frac{u_{y}}{\gamma(u_{x}-v)}% =\frac{\sin(\theta)}{\gamma(\cos(\theta)-v/c)}
  14. θ = 90 \theta=90^{\circ}
  15. sin ( θ - ϕ ) = - v / c \sin(\theta-\phi)=-v/c
  16. v / c 1 v/c\ll 1
  17. θ - ϕ = - v / c \theta-\phi=-v/c
  18. u x 2 + u y 2 = c \sqrt{u_{x}^{2}+u_{y}^{2}}=c
  19. v v
  20. κ \kappa
  21. κ = θ - ϕ v / c \kappa=\theta-\phi\approx v/c
  22. v v
  23. c c
  24. κ \kappa
  25. κ \kappa
  26. κ \kappa
  27. κ \kappa
  28. κ \kappa
  29. Δ x = v t \Delta x=vt
  30. t = R / c t=R/c
  31. R R
  32. tan ( θ ) θ = Δ x / R \tan(\theta)\approx\theta=\Delta x/R
  33. θ = v / c = κ \theta=v/c=\kappa
  34. θ \theta
  35. c c
  36. h / c h/c
  37. v v
  38. v h / c vh/c
  39. ϕ \phi
  40. θ \theta
  41. ϕ \phi
  42. ϕ \phi
  43. tan ( ϕ ) = h sin ( θ ) h v / c + h cos ( θ ) = sin ( θ ) v / c + cos ( θ ) \tan(\phi)=\frac{h\sin(\theta)}{hv/c+h\cos(\theta)}=\frac{\sin(\theta)}{v/c+% \cos(\theta)}
  44. θ = 90 \theta=90^{\circ}
  45. tan ( θ - ϕ ) = v / c \tan(\theta-\phi)=v/c
  46. c / n c/n
  47. c c
  48. n n
  49. tan ( ϕ ) = sin ( θ ) v / ( c / n ) + cos ( θ ) \tan(\phi)=\frac{\sin(\theta)}{v/(c/n)+\cos(\theta)}
  50. n n
  51. v v
  52. ( 1 - 1 / n 2 ) v (1-1/n^{2})v
  53. 1 - v 2 / c 2 \sqrt{1-v^{2}/c^{2}}
  54. γ \gamma

Abraham_de_Moivre.html

  1. cos x = 1 2 ( cos ( n x ) + i sin ( n x ) ) 1 / n + 1 2 ( cos ( n x ) - i sin ( n x ) ) 1 / n \cos x=\tfrac{1}{2}(\cos(nx)+i\sin(nx))^{1/n}+\tfrac{1}{2}(\cos(nx)-i\sin(nx))% ^{1/n}
  2. ( cos x + i sin x ) n = cos ( n x ) + i sin ( n x ) . (\cos x+i\sin x)^{n}=\cos(nx)+i\sin(nx).\,

Abscissa.html

  1. ( x abscissa , y ordinate ) (\overbrace{x}\text{abscissa},\overbrace{y}\text{ordinate})

Absolute_convergence.html

  1. n = 0 a n \textstyle\sum_{n=0}^{\infty}a_{n}
  2. n = 0 | a n | = L \textstyle\sum_{n=0}^{\infty}\left|a_{n}\right|=L
  3. L \textstyle L
  4. 0 f ( x ) d x \textstyle\int_{0}^{\infty}f(x)\,dx
  5. 0 | f ( x ) | d x = L . \textstyle\int_{0}^{\infty}\left|f(x)\right|dx=L.
  6. n = 0 a n \sum_{n=0}^{\infty}a_{n}
  7. : G \|\cdot\|:G\to\mathbb{R}
  8. 0 = 0. \|0\|=0.
  9. x = 0 \|x\|=0
  10. x = 0. x=0.
  11. - x = x . \|-x\|=\|x\|.
  12. x + y x + y . \|x+y\|\leq\|x\|+\|y\|.
  13. d ( x , y ) = x - y d(x,y)=\|x-y\|
  14. n = 0 a n < . \sum_{n=0}^{\infty}\|a_{n}\|<\infty.
  15. a n a_{n}
  16. | a n | \sum|a_{n}|
  17. 2 | a n | 2\sum|a_{n}|
  18. 0 a n + | a n | 2 | a n | 0\leq a_{n}+|a_{n}|\leq 2|a_{n}|
  19. 0 n = 1 m ( a n + | a n | ) n = 1 m 2 | a n | n = 1 2 | a n | 0\leq\sum_{n=1}^{m}(a_{n}+|a_{n}|)\leq\sum_{n=1}^{m}2|a_{n}|\leq\sum_{n=1}^{% \infty}2|a_{n}|
  20. n = 1 m ( a n + | a n | ) \sum_{n=1}^{m}(a_{n}+|a_{n}|)
  21. a n = ( a n + | a n | ) - | a n | \sum a_{n}=\sum(a_{n}+|a_{n}|)-\sum|a_{n}|
  22. k = 1 n x k \scriptstyle\sum_{k=1}^{n}\|x_{k}\|
  23. | k = 1 m x k - k = 1 n x k | = k = n + 1 m x k < ε . \left|\sum_{k=1}^{m}\|x_{k}\|-\sum_{k=1}^{n}\|x_{k}\|\right|=\sum_{k=n+1}^{m}% \|x_{k}\|<\varepsilon.
  24. k = 1 m x k - k = 1 n x k = k = n + 1 m x k k = n + 1 m x k < ε , \left\|\sum_{k=1}^{m}x_{k}-\sum_{k=1}^{n}x_{k}\right\|=\left\|\sum_{k=n+1}^{m}% x_{k}\right\|\leq\sum_{k=n+1}^{m}\|x_{k}\|<\varepsilon,
  25. k = 1 n x k \scriptstyle\sum_{k=1}^{n}x_{k}
  26. n = 0 a n \sum_{n=0}^{\infty}a_{n}
  27. n = 0 a σ ( n ) \sum_{n=0}^{\infty}a_{\sigma(n)}
  28. i = 1 a i = A G , i = 1 a i < \sum_{i=1}^{\infty}a_{i}=A\in G,\quad\sum_{i=1}^{\infty}\|a_{i}\|<\infty
  29. σ : 𝐍 𝐍 σ:\mathbf{N}→\mathbf{N}
  30. i = 1 a σ ( i ) = A . \sum_{i=1}^{\infty}a_{\sigma(i)}=A.
  31. a n = 1 n e n , a_{n}=\tfrac{1}{n}e_{n},
  32. { e n } n = 1 \{e_{n}\}_{n=1}^{\infty}
  33. κ ε , λ ε 𝐍 \kappa_{\varepsilon},\lambda_{\varepsilon}\in\mathbf{N}
  34. N > κ ε \displaystyle\forall N>\kappa_{\varepsilon}
  35. N ε = max { κ ε , λ ε } M σ , ε = max { σ - 1 ( { 1 , , N ε } ) } \begin{aligned}\displaystyle N_{\varepsilon}&\displaystyle=\max\left\{\kappa_{% \varepsilon},\lambda_{\varepsilon}\right\}\\ \displaystyle M_{\sigma,\varepsilon}&\displaystyle=\max\left\{\sigma^{-1}\left% (\left\{1,\dots,N_{\varepsilon}\right\}\right)\right\}\end{aligned}
  36. N > M σ , ε N>M_{\sigma,\varepsilon}
  37. I σ , ε = { 1 , , N } σ - 1 ( { 1 , , N ε } ) S σ , ε = min { σ ( k ) : k I σ , ε } L σ , ε = max { σ ( k ) : k I σ , ε } \begin{aligned}\displaystyle I_{\sigma,\varepsilon}&\displaystyle=\left\{1,% \ldots,N\right\}\setminus\sigma^{-1}\left(\left\{1,\dots,N_{\varepsilon}\right% \}\right)\\ \displaystyle S_{\sigma,\varepsilon}&\displaystyle=\min\left\{\sigma(k)\ :\ k% \in I_{\sigma,\varepsilon}\right\}\\ \displaystyle L_{\sigma,\varepsilon}&\displaystyle=\max\left\{\sigma(k)\ :\ k% \in I_{\sigma,\varepsilon}\right\}\end{aligned}
  38. i = 1 N a σ ( i ) - A = i σ - 1 ( { 1 , , N ε } ) a σ ( i ) - A + i I σ , ε a σ ( i ) j = 1 N ε a j - A + i I σ , ε a σ ( i ) j = 1 N ε a j - A + i I σ , ε a σ ( i ) j = 1 N ε a j - A + j = S σ , ε L σ , ε a j j = 1 N ε a j - A + j = N ε + 1 a j S σ , ε N ε + 1 < ε \begin{aligned}\displaystyle\left\|\sum_{i=1}^{N}a_{\sigma(i)}-A\right\|&% \displaystyle=\left\|\sum_{i\in\sigma^{-1}\left(\{1,\dots,N_{\varepsilon}\}% \right)}a_{\sigma(i)}-A+\sum_{i\in I_{\sigma,\varepsilon}}a_{\sigma(i)}\right% \|\\ &\displaystyle\leq\left\|\sum_{j=1}^{N_{\varepsilon}}a_{j}-A\right\|+\left\|% \sum_{i\in I_{\sigma,\varepsilon}}a_{\sigma(i)}\right\|\\ &\displaystyle\leq\left\|\sum_{j=1}^{N_{\varepsilon}}a_{j}-A\right\|+\sum_{i% \in I_{\sigma,\varepsilon}}\left\|a_{\sigma(i)}\right\|\\ &\displaystyle\leq\left\|\sum_{j=1}^{N_{\varepsilon}}a_{j}-A\right\|+\sum_{j=S% _{\sigma,\varepsilon}}^{L_{\sigma,\varepsilon}}\left\|a_{j}\right\|\\ &\displaystyle\leq\left\|\sum_{j=1}^{N_{\varepsilon}}a_{j}-A\right\|+\sum_{j=N% _{\varepsilon}+1}^{\infty}\left\|a_{j}\right\|&&\displaystyle S_{\sigma,% \varepsilon}\geq N_{\varepsilon}+1\\ &\displaystyle<\varepsilon\end{aligned}
  39. ε > 0 , M σ , ε , N > M σ , ε i = 1 N a σ ( i ) - A < ε , \forall\varepsilon>0,\exists M_{\sigma,\varepsilon},\forall N>M_{\sigma,% \varepsilon}\quad\left\|\sum_{i=1}^{N}a_{\sigma(i)}-A\right\|<\varepsilon,
  40. i = 1 a σ ( i ) = A \sum_{i=1}^{\infty}a_{\sigma(i)}=A
  41. n = 0 a n = A \sum_{n=0}^{\infty}a_{n}=A
  42. n = 0 b n = B \sum_{n=0}^{\infty}b_{n}=B
  43. c n = k = 0 n a k b n - k . c_{n}=\sum_{k=0}^{n}a_{k}b_{n-k}.
  44. n = 0 c n = A B . \sum_{n=0}^{\infty}c_{n}=AB.
  45. A f ( x ) d x \int_{A}f(x)\,dx
  46. A | f ( x ) | d x < . \int_{A}\left|f(x)\right|\,dx<\infty.
  47. S [ a , b ] S\subset[a,b]
  48. f = χ S - 1 / 2 , f=\chi_{S}-1/2,
  49. χ S \chi_{S}
  50. n = 0 a n \sum_{n=0}^{\infty}a_{n}
  51. f a : [ 0 , ) 𝐑 f_{a}:[0,\infty)\rightarrow\mathbf{R}
  52. f a ( [ n , n + 1 ) ) = a n f_{a}([n,n+1))=a_{n}
  53. 0 f a d x \int_{0}^{\infty}f_{a}\,dx
  54. n = 0 a n . \sum_{n=0}^{\infty}a_{n}.
  55. 𝐑 sin x x d x \int_{\mathbf{R}}\frac{\sin x}{x}\,dx

Absolute_magnitude.html

  1. M b o l = M V + B C M_{bol}=M_{V}+BC
  2. M M\!\,
  3. m m\!\,
  4. D L D_{L}\!\,
  5. M = m - 5 ( ( log 10 D L ) - 1 ) M=m-5((\log_{10}{D_{L}})-1)\!\,
  6. D L D_{L}\!\,
  7. M M\!\,
  8. M = m + 5 ( 1 + log 10 p ) M=m+5(1+\log_{10}{p})\!\,
  9. M M\!\,
  10. m m\!\,
  11. μ \mu\!\,
  12. M = m - μ . M=m-{\mu}.\!\,
  13. m V = 0.12 m_{V}=0.12
  14. M V = 0.12 - 5 ( log 10 860 3.2616 - 1 ) = - 7.02. M_{V}=0.12-5\cdot(\log_{10}\frac{860}{3.2616}-1)=-7.02.
  15. M V = 0.03 + 5 ( 1 + log 10 0.129 ) = + 0.6. M_{V}=0.03+5\cdot(1+\log_{10}{0.129})=+0.6.
  16. M V = - 0.01 + 5 ( 1 + log 10 0.742 ) = + 4.3. M_{V}=-0.01+5\cdot(1+\log_{10}{0.742})=+4.3.
  17. M V = 9.36 - 31.06 = - 21.7. M_{V}=9.36-31.06=-21.7.
  18. M M\!\,
  19. m m\!\,
  20. d d\!\,
  21. m = M - 5 ( 1 - log 10 d ) . m=M-5(1-\log_{10}{d}).\!\,
  22. M M\!\,
  23. m m\!\,
  24. p p\!\,
  25. m = M - 5 ( 1 + log 10 p ) . m=M-5(1+\log_{10}p).\!\,
  26. M M\!\,
  27. μ \mu\!\,
  28. m = M + μ . m=M+{\mu}.\!\,
  29. M b o l star - M b o l Sun = - 2.5 log 10 L star L M_{bol_{\rm star}}-M_{bol_{\rm Sun}}=-2.5\log_{10}{\frac{L_{\rm star}}{L_{% \odot}}}
  30. L star L = 10 ( ( M b o l Sun - M b o l star ) / 2.5 ) \frac{L_{\rm star}}{L_{\odot}}=10^{((M_{bol_{\rm Sun}}-M_{bol_{\rm star}})/2.5)}
  31. L L_{\odot}
  32. L star L_{\rm star}
  33. M b o l Sun M_{bol_{\rm Sun}}
  34. M b o l star M_{bol_{\rm star}}
  35. M b o l = 0 M_{bol}=0
  36. M b o l Sun = 4.74 M_{bol_{\rm Sun}}=4.74
  37. m b o l = 0 m_{bol}=0
  38. f o = 2.518021002... e - 8 W / m 2 f_{o}=2.518021002...e-8W/m2
  39. m b o l Sun = - 26.832 m_{bol_{\rm Sun}}=-26.832
  40. m = H + 2.5 log 10 ( d B S 2 d B O 2 p ( χ ) d 0 4 ) m=H+2.5\log_{10}{\left(\frac{d_{BS}^{2}d_{BO}^{2}}{p(\chi)d_{0}^{4}}\right)}\!\,
  41. d 0 d_{0}\!\,
  42. χ \chi\!\,
  43. cos χ = d B O 2 + d B S 2 - d O S 2 2 d B O d B S . \cos{\chi}=\frac{d_{BO}^{2}+d_{BS}^{2}-d_{OS}^{2}}{2d_{BO}d_{BS}}.\!\,
  44. p ( χ ) p(\chi)\!\,
  45. p ( χ ) = 2 3 ( ( 1 - χ π ) cos χ + 1 π sin χ ) . p(\chi)=\frac{2}{3}\left(\left(1-\frac{\chi}{\pi}\right)\cos{\chi}+\frac{1}{% \pi}\sin{\chi}\right).\!\,
  46. 2 / 3 {2}/{3}
  47. d B O d_{BO}\!\,
  48. d B S d_{BS}\!\,
  49. d O S d_{OS}\!\,
  50. H M o o n H_{Moon}\!\,
  51. d O S d_{OS}\!\,
  52. d B S d_{BS}\!\,
  53. d B O d_{BO}\!\,
  54. χ \chi\!\,
  55. p ( χ ) p(\chi)\!\,
  56. m M o o n = 0.25 + 2.5 log 10 ( 3 2 0.00257 2 ) = - 12.26 m_{Moon}=0.25+2.5\log_{10}{\left(\frac{3}{2}0.00257^{2}\right)}=-12.26\!\,
  57. χ \chi\!\,
  58. p ( χ ) 2 3 π p(\chi)\approx\frac{2}{3\pi}\!\,
  59. m M o o n = 0.25 + 2.5 log 10 ( 3 π 2 0.00257 2 ) = - 11.02 m_{Moon}=0.25+2.5\log_{10}{\left(\frac{3\pi}{2}0.00257^{2}\right)}=-11.02\!\,

Absolute_value.html

  1. | x | |x|
  2. x x
  3. x x
  4. | x | = x |x|=x
  5. x x
  6. | x | = x |x|=−x
  7. x x
  8. x −x
  9. | 0 | = 0 |0|=0
  10. | x | |x|
  11. x x
  12. x x
  13. | x | |x|
  14. | x | = { x , if x 0 - x , if x < 0. |x|=\begin{cases}x,&\mbox{if }~{}x\geq 0\\ -x,&\mbox{if }~{}x<0.\end{cases}
  15. x x
  16. | a | = a 2 |a|=\sqrt{a^{2}}
  17. | a | 0 |a|\geq 0
  18. | a | = 0 a = 0 |a|=0\iff a=0
  19. | a b | = | a | | b | |ab|=|a||b|
  20. | a + b | | a | + | b | |a+b|\leq|a|+|b|
  21. | ( | a | ) | = | a | |(|a|)|=|a|
  22. | - a | = | a | |-a|=|a|
  23. | a - b | = 0 a = b |a-b|=0\iff a=b
  24. | a - b | | a - c | + | c - b | |a-b|\leq|a-c|+|c-b|
  25. | a b | = | a | | b | \left|\frac{a}{b}\right|=\frac{|a|}{|b|}
  26. b 0 b\neq 0
  27. | a - b | | ( | a | - | b | ) | |a-b|\geq|(|a|-|b|)|
  28. | a | b - b a b |a|\leq b\iff-b\leq a\leq b
  29. | a | b a - b |a|\geq b\iff a\leq-b
  30. b a b\leq a
  31. | x - 3 | 9 |x-3|\leq 9
  32. - 9 x - 3 9 \iff-9\leq x-3\leq 9
  33. - 6 x 12 \iff-6\leq x\leq 12
  34. z z
  35. r r
  36. z z
  37. z z
  38. z ¯ \overline{z}
  39. z = x + i y , z=x+iy,
  40. x x
  41. y y
  42. z z
  43. | z | |z|
  44. | z | = x 2 + y 2 . |z|=\sqrt{x^{2}+y^{2}}.
  45. y y
  46. x x
  47. z z
  48. z = r e i θ z=re^{i\theta}
  49. r 0 r≥0
  50. | z | = r |z|=r
  51. | z | = z z ¯ |z|=\sqrt{z\cdot\overline{z}}
  52. z ¯ \overline{z}
  53. z z
  54. | z | z 2 |z|\neq\sqrt{z^{2}}
  55. x x
  56. ( , 0 ] (−∞,0]
  57. [ 0 , + ) [0,+∞)
  58. | x | = x sgn ( x ) , |x|=x\operatorname{sgn}(x),
  59. | x | sgn ( x ) = x , |x|\operatorname{sgn}(x)=x,
  60. x 0 x≠0
  61. sgn ( x ) = | x | x . \operatorname{sgn}(x)=\frac{|x|}{x}.
  62. x 0 x≠0
  63. x = 0 x=0
  64. x 0 x≠0
  65. d | x | d x = x | x | = { - 1 x < 0 1 x > 0. \frac{d|x|}{dx}=\frac{x}{|x|}=\begin{cases}-1&x<0\\ 1&x>0.\end{cases}
  66. | x | |x|
  67. x = 0 x=0
  68. [ 1 , 1 ] [−1,1]
  69. | x | |x|
  70. x x
  71. | x | d x = x | x | 2 + C , \int|x|dx=\frac{x|x|}{2}+C,
  72. C C
  73. a = ( a 1 , a 2 , , a n ) a=(a_{1},a_{2},\dots,a_{n})
  74. b = ( b 1 , b 2 , , b n ) b=(b_{1},b_{2},\dots,b_{n})
  75. n n
  76. i = 1 n ( a i - b i ) 2 . \sqrt{\sum_{i=1}^{n}(a_{i}-b_{i})^{2}}.
  77. | a b | |a−b|
  78. a a
  79. b b
  80. | a - b | = ( a - b ) 2 . |a-b|=\sqrt{(a-b)^{2}}.
  81. a = a 1 + i a 2 a=a_{1}+ia_{2}
  82. b = b 1 + i b 2 b=b_{1}+ib_{2}
  83. | a - b | |a-b|
  84. = | ( a 1 + i a 2 ) - ( b 1 + i b 2 ) | =|(a_{1}+ia_{2})-(b_{1}+ib_{2})|
  85. = | ( a 1 - b 1 ) + i ( a 2 - b 2 ) | =|(a_{1}-b_{1})+i(a_{2}-b_{2})|
  86. = ( a 1 - b 1 ) 2 + ( a 2 - b 2 ) 2 . =\sqrt{(a_{1}-b_{1})^{2}+(a_{2}-b_{2})^{2}}.
  87. d d
  88. X × X X×X
  89. X X
  90. d ( a , b ) 0 d(a,b)\geq 0
  91. d ( a , b ) = 0 a = b d(a,b)=0\iff a=b
  92. d ( a , b ) = d ( b , a ) d(a,b)=d(b,a)
  93. d ( a , b ) d ( a , c ) + d ( c , b ) d(a,b)\leq d(a,c)+d(c,b)
  94. a a
  95. a a
  96. | a | |a|
  97. | a | = { a , if a 0 - a , if a 0 |a|=\begin{cases}a,&\mbox{if }~{}a\geq 0\\ -a,&\mbox{if }~{}a\leq 0\end{cases}\;
  98. a −a
  99. a a
  100. v v
  101. F F
  102. v ( a ) 0 v(a)\geq 0
  103. v ( a ) = 0 a = 𝟎 v(a)=0\iff a=\mathbf{0}
  104. v ( a b ) = v ( a ) v ( b ) v(ab)=v(a)v(b)
  105. v ( a + b ) v ( a ) + v ( b ) v(a+b)\leq v(a)+v(b)
  106. F F
  107. v ( 𝟏 ) = 1 v(\mathbf{1})=1
  108. F F
  109. v v
  110. F F
  111. d d
  112. F × F F×F
  113. d ( a , b ) = v ( a b ) d(a,b)=v(a−b)
  114. d d
  115. d ( x , y ) max ( d ( x , z ) , d ( y , z ) ) d(x,y)\leq\max(d(x,z),d(y,z))
  116. x x
  117. y y
  118. z z
  119. F F
  120. { v ( k = 1 n 𝟏 ) : n } \big\{v\Big({\textstyle\sum_{k=1}^{n}}\mathbf{1}\Big):n\in\mathbb{N}\big\}
  121. v ( k = 1 n 𝟏 ) 1 v\Big({\textstyle\sum_{k=1}^{n}}\mathbf{1}\Big)\leq 1
  122. n . n\in\mathbb{N}.
  123. v ( a ) 1 v ( 1 + a ) 1 v(a)\leq 1\Rightarrow v(1+a)\leq 1
  124. a F . a\in F.
  125. v ( a + b ) max { v ( a ) , v ( b ) } v(a+b)\leq\mathrm{max}\{v(a),v(b)\}
  126. a , b F . a,b\in F.
  127. V V
  128. F F
  129. · ‖·‖
  130. a a
  131. F F
  132. 𝐯 \mathbf{v}
  133. 𝐮 \mathbf{u}
  134. V V
  135. 𝐯 0 \|\mathbf{v}\|\geq 0
  136. 𝐯 = 0 𝐯 = 0 \|\mathbf{v}\|=0\iff\mathbf{v}=0
  137. a 𝐯 = | a | 𝐯 \|a\mathbf{v}\|=|a|\|\mathbf{v}\|
  138. 𝐯 + 𝐮 𝐯 + 𝐮 \|\mathbf{v}+\mathbf{u}\|\leq\|\mathbf{v}\|+\|\mathbf{u}\|
  139. ( x 1 , x 2 , , x n ) = i = 1 n x i 2 \|(x_{1},x_{2},\dots,x_{n})\|=\sqrt{\sum_{i=1}^{n}x_{i}^{2}}
  140. 𝐑 \mathbf{R}
  141. 𝐑 < s u p > 1 \mathbf{R}<sup>1
  142. 0 = d ( a , a ) d ( a , b ) + d ( b , a ) = 2 d ( a , b ) 0=d(a,a)≤d(a,b)+d(b,a)=2d(a,b)

Absolute_zero.html

  1. lim T 0 Δ S = 0 \lim_{T\to 0}\Delta S=0
  2. Δ G = Δ H - T Δ S \Delta G=\Delta H-T\Delta S\,

Accelerating_universe.html

  1. a ( t ) a(t)
  2. H 2 = ( a ˙ a ) 2 = 8 π G 3 ρ - K c 2 R 2 a 2 H^{2}={\left(\frac{\dot{a}}{a}\right)}^{2}=\frac{8{\pi}G}{3}\rho-\frac{{K}c^{2% }}{R^{2}a^{2}}
  3. K K
  4. a ( t ) a(t)
  5. ρ \rho
  6. H H
  7. ρ c = 3 H 2 8 π G \rho_{c}=\frac{3H^{2}}{8{\pi}G}
  8. Ω = ρ ρ c \Omega=\frac{\rho}{\rho_{c}}
  9. H ( a ) = H 0 Ω k a - 2 + Ω m a - 3 + Ω r a - 4 + Ω D E a - 3 ( 1 + w ) H(a)=H_{0}\sqrt{{\Omega_{k}a^{-2}+\Omega}_{m}a^{-3}+\Omega_{r}a^{-4}+\Omega_{% DE}a^{-3(1+w)}}
  10. a ¨ a = - 4 π G 3 ( ρ + 3 P c 2 ) \frac{\ddot{a}}{a}=-\frac{4{\pi}G}{3}\left(\rho+\frac{3P}{c^{2}}\right)
  11. q 0 q_{0}
  12. a ( t ) = 1 1 + z a(t)=\frac{1}{1+z}
  13. Ω M = 0.2 \Omega_{M}=0.2
  14. - 1 {}^{-1}
  15. w w
  16. Ω m \Omega_{m}
  17. t 0 = 0 1 d a a ˙ t_{0}=\int\limits_{0}^{1}\frac{da}{\dot{a}}
  18. P = w c 2 ρ P=wc^{2}\rho
  19. ρ \rho
  20. Ω λ \Omega_{\lambda}
  21. q 0 q_{0}
  22. w < - 1 w<-1
  23. H 0 H_{0}

Acceleration.html

  1. ( Δ 𝐯 ) (\Delta\mathbf{v})
  2. ( Δ t ) (\Delta t)
  3. 𝐚 ¯ = Δ 𝐯 Δ t . \mathbf{\bar{a}}=\frac{\Delta\mathbf{v}}{\Delta t}.
  4. 𝐚 = lim Δ t 0 Δ 𝐯 Δ t = d 𝐯 d t \mathbf{a}=\lim_{{\Delta t}\to 0}\frac{\Delta\mathbf{v}}{\Delta t}=\frac{d% \mathbf{v}}{dt}
  5. a ( t ) a(t)
  6. v ( t ) v(t)
  7. a a
  8. t t
  9. 𝐯 = 𝐚 d t \mathbf{v}=\int\mathbf{a}\ dt
  10. 𝐯 \mathbf{v}
  11. t t
  12. 𝐱 \mathbf{x}
  13. 𝐱 \mathbf{x}
  14. t t
  15. 𝐚 = d 𝐯 d t = d 2 𝐱 d t 2 \mathbf{a}=\frac{d\mathbf{v}}{dt}=\frac{d^{2}\mathbf{x}}{dt^{2}}
  16. 𝐅 = m 𝐚 𝐚 = 𝐅 / m \mathbf{F}=m\mathbf{a}\quad\to\quad\mathbf{a}=\mathbf{F}/m
  17. 𝐯 ( t ) = v ( t ) 𝐯 ( t ) v ( t ) = v ( t ) 𝐮 t ( t ) , \mathbf{v}(t)=v(t)\frac{\mathbf{v}(t)}{v(t)}=v(t)\mathbf{u}_{\mathrm{t}}(t),
  18. 𝐮 t = 𝐯 ( t ) v ( t ) , \mathbf{u}_{\mathrm{t}}=\frac{\mathbf{v}(t)}{v(t)}\ ,
  19. 𝐚 \displaystyle\mathbf{a}
  20. 𝐅 = m 𝐠 \mathbf{F}=m\mathbf{g}
  21. v = v 0 + a t v=v_{0}+at
  22. s = v 0 t + 1 2 a t 2 = v 0 + v 2 t s=v_{0}t+\frac{1}{2}at^{2}=\frac{v_{0}+v}{2}t
  23. | v | 2 = | v 0 | 2 + 2 a s |v|^{2}=|v_{0}|^{2}+2\,a\cdot s
  24. s s
  25. v 0 v_{0}
  26. v v
  27. a a
  28. t t
  29. a = v 2 r \textrm{a}={{v^{2}}\over{r}}
  30. v v
  31. 𝐚 \mathbf{a}
  32. ω \omega
  33. 𝐚 = - ω 2 𝐫 \mathbf{a}={-\omega^{2}}\mathbf{r}
  34. 𝐫 \mathbf{r}
  35. a = r α . a=r\alpha.
  36. α \alpha

Accuracy_and_precision.html

  1. accuracy = number of true positives + number of true negatives number of true positives + false positives + false negatives + true negatives \,\text{accuracy}=\frac{\,\text{number of true positives}+\,\text{number of % true negatives}}{\,\text{number of true positives}+\,\text{false positives}+\,% \text{false negatives}+\,\text{true negatives}}
  2. precision = number of true positives number of true positives + false positives \,\text{precision}=\frac{\,\text{number of true positives}}{\,\text{number of % true positives}+\,\text{false positives}}
  3. accuracy = ( sensitivity ) ( prevalence ) + ( specificity ) ( 1 - prevalence ) \,\text{accuracy}=(\,\text{sensitivity})(\,\text{prevalence})+(\,\text{% specificity})(1-\,\text{prevalence})
  4. balanced accuracy = sensitivity + specificity 2 \,\text{balanced accuracy}=\frac{\,\text{sensitivity}+\,\text{specificity}}{2}
  5. = 0.5 * true positives true positives + false negatives + 0.5 * true negatives true negatives + false positives =\frac{0.5*\,\text{true positives}}{\,\text{true positives}+\,\text{false % negatives}}+\frac{0.5*\,\text{true negatives}}{\,\text{true negatives}+\,\text% {false positives}}
  6. Informedness = sensitivity + specificity - 1 = 2 * balanced accuracy - 1 \,\text{Informedness}=\,\text{sensitivity}+\,\text{specificity}-1=2*\,\text{% balanced accuracy}-1

Acetylene.html

  1. Fe ( CO ) 5 + 4 C 2 H 2 + 2 H 2 O 50 - 80 C 20 - 25 atm basic conditions 2 C 6 H 4 ( OH ) 2 + FeCO 3 \mathrm{Fe(CO)_{5}+4C_{2}H_{2}+2H_{2}O\xrightarrow[basic\ conditions]{\begin{% array}[]{c}50-80\ ^{\circ}\mathrm{C}\\ 20-25\ \mathrm{atm}\end{array}}2C_{6}H_{4}(OH)_{2}+FeCO_{3}}

Acid.html

  1. K a = [ H ] + [ A ] - [ HA ] K_{a}=\frac{[\mbox{H}~{}^{+}][\mbox{A}~{}^{-}]}{[\mbox{HA}~{}]}
  2. α H 2 A = [ H + ] 2 [ H + ] 2 + [ H + ] K 1 + K 1 K 2 = [ H 2 A ] [ H 2 A ] + [ H A - ] + [ A 2 - ] \alpha_{H_{2}A}={{[H^{+}]^{2}}\over{[H^{+}]^{2}+[H^{+}]K_{1}+K_{1}K_{2}}}={{[H% _{2}A]}\over{[H_{2}A]+[HA^{-}]+[A^{2-}]}}
  3. α H A - = [ H + ] K 1 [ H + ] 2 + [ H + ] K 1 + K 1 K 2 = [ H A - ] [ H 2 A ] + [ H A - ] + [ A 2 - ] \alpha_{HA^{-}}={{[H^{+}]K_{1}}\over{[H^{+}]^{2}+[H^{+}]K_{1}+K_{1}K_{2}}}={{[% HA^{-}]}\over{[H_{2}A]+[HA^{-}]+[A^{2-}]}}
  4. α A 2 - = K 1 K 2 [ H + ] 2 + [ H + ] K 1 + K 1 K 2 = [ A 2 - ] [ H 2 A ] + [ H A - ] + [ A 2 - ] \alpha_{A^{2-}}={{K_{1}K_{2}}\over{[H^{+}]^{2}+[H^{+}]K_{1}+K_{1}K_{2}}}={{[A^% {2-}]}\over{[H_{2}A]+[HA^{-}]+[A^{2-}]}}
  5. α H n - i A i - = [ H + ] n - i j = 0 i K j i = 0 n [ [ H + ] n - i j = 0 i K j ] \alpha_{H_{n-i}A^{i-}}={{[H^{+}]^{n-i}\displaystyle\prod_{j=0}^{i}K_{j}}\over{% \displaystyle\sum_{i=0}^{n}\Big[[H^{+}]^{n-i}\displaystyle\prod_{j=0}^{i}K_{j}% }\Big]}

Acid_dissociation_constant.html

  1. HA + H 2 O A - + H 3 O + \mathrm{HA+H_{2}O\rightleftharpoons A^{-}+H_{3}O^{+}}
  2. K a = [ A - ] [ H 3 O + ] [ HA ] [ H 2 O ] K_{\mathrm{a}}=\mathrm{\frac{[A^{-}][H_{3}O^{+}]}{[HA][H_{2}O]}}
  3. HA A - + H + : K a = [ A - ] [ H + ] [ HA ] \mathrm{HA\rightleftharpoons A^{-}+H^{+}}:K_{\mathrm{a}}=\mathrm{\frac{[A^{-}]% [H^{+}]}{[HA]}}
  4. p K a = - log 10 K a \ \mathrm{p}K_{\mathrm{a}}=-\log_{10}K_{\mathrm{a}}
  5. HA + S A - + SH + ; K a = [ A - ] [ SH + ] [ HA ] [ S ] \mathrm{HA+S\rightleftharpoons A^{-}+SH^{+}};K_{\mathrm{a}}=\mathrm{\frac{[A^{% -}][SH^{+}]}{[HA][S]}}
  6. K a = [ A - ] [ H + ] [ HA ] K_{\mathrm{a}}=\mathrm{\frac{[A^{-}][H^{+}]}{[HA]}}
  7. K = { A - } { H 3 O + } { HA } { H 2 O } K^{\ominus}=\mathrm{\frac{\{A^{-}\}\{H_{3}O^{+}\}}{\{HA\}\{H_{2}O\}}}
  8. K = [ A - ] [ H 3 O + ] [ HA ] [ H 2 O ] × γ A - γ H 3 O + γ HA γ H 2 O = [ A - ] [ H 3 O + ] [ HA ] [ H 2 O ] × Γ K^{\ominus}=\mathrm{\frac{[A^{-}][H_{3}O^{+}]}{[HA][H_{2}O]}\times\frac{\gamma% _{A^{-}}\ \gamma_{H_{3}O^{+}}}{\gamma_{HA}\ \gamma_{H_{2}O}}=\mathrm{\frac{[A^% {-}][H_{3}O^{+}]}{[HA][H_{2}O]}}\times\Gamma}
  9. Γ \Gamma
  10. Γ \Gamma
  11. K a = [ A - ] [ H + ] [ HA ] K_{\mathrm{a}}=\mathrm{\frac{[A^{-}][H^{+}]}{[HA]}}
  12. pH = p K a + log [ A - ] [ HA ] \mathrm{pH}=\mathrm{p}K_{\mathrm{a}}+\log\mathrm{\frac{[A^{-}]}{[HA]}}
  13. p I = p K 1 + p K 2 2 \mathrm{p}I=\frac{\mathrm{p}K_{1}+\mathrm{p}K_{2}}{2}
  14. K a = [ H 3 O + ] [ OH - ] [ H 2 O ] K_{\mathrm{a}}=\mathrm{\frac{[H_{3}O^{+}][OH^{-}]}{[H_{2}O]}}
  15. K w = [ H 3 O + ] [ OH - ] K_{\mathrm{w}}=[\mathrm{H_{3}O}^{+}][\mathrm{OH}^{-}]\,
  16. K b = [ HB + ] [ OH - ] [ B ] K_{\mathrm{b}}=\mathrm{\frac{[HB^{+}][OH^{-}]}{[B]}}
  17. [ OH - ] = K w [ H + ] \mathrm{[OH^{-}]}=\frac{K_{\mathrm{w}}}{\mathrm{[H^{+}]}}
  18. K b = [ HB + ] K w [ B ] [ H + ] = K w K a K_{\mathrm{b}}=\frac{[\mathrm{HB^{+}}]K_{\mathrm{w}}}{\mathrm{[B][H^{+}]}}=% \frac{K_{\mathrm{w}}}{K_{\mathrm{a}}}
  19. p K b 14 - p K a pK_{\mathrm{b}}\approx 14-pK_{\mathrm{a}}
  20. d ln K d T = Δ H R T 2 \frac{\operatorname{d}\ln K}{\operatorname{d}T}=\frac{{\Delta H}^{\ominus}}{RT% ^{2}}

Ackermann_function.html

  1. A ( m , n ) = { n + 1 if m = 0 A ( m - 1 , 1 ) if m > 0 and n = 0 A ( m - 1 , A ( m , n - 1 ) ) if m > 0 and n > 0. A(m,n)=\begin{cases}n+1&\mbox{if }~{}m=0\\ A(m-1,1)&\mbox{if }~{}m>0\mbox{ and }~{}n=0\\ A(m-1,A(m,n-1))&\mbox{if }~{}m>0\mbox{ and }~{}n>0.\end{cases}
  2. φ \varphi\,\!
  3. φ ( m , n , p ) \varphi(m,n,p)\,\!
  4. φ ( m , n , 0 ) = m + n , \varphi(m,n,0)=m+n,\,\!
  5. φ ( m , n , 1 ) = m n , \varphi(m,n,1)=m\cdot n,\,\!
  6. φ ( m , n , 2 ) = m n , \varphi(m,n,2)=m^{n},\,\!
  7. φ ( m , n , p ) \varphi(m,n,p)\,\!
  8. φ ( m , n , p ) = { φ ( m , n , 0 ) = m + n φ ( m , 0 , 1 ) = 0 φ ( m , 0 , 2 ) = 1 φ ( m , 0 , p ) = m for p > 2 φ ( m , n , p ) = φ ( m , φ ( m , n - 1 , p ) , p - 1 ) for n > 0 and p > 0. \varphi(m,n,p)=\begin{cases}\varphi(m,n,0)=m+n\\ \varphi(m,0,1)=0\\ \varphi(m,0,2)=1\\ \varphi(m,0,p)=m&\,\text{ for }p>2\\ \varphi(m,n,p)=\varphi(m,\varphi(m,n-1,p),p-1)&\,\text{ for }n>0\,\text{ and }% p>0.\end{cases}\,\!
  9. A ( m , n ) = { n + 1 if m = 0 A ( m - 1 , 1 ) if m > 0 and n = 0 A ( m - 1 , A ( m , n - 1 ) ) if m > 0 and n > 0. A(m,n)=\begin{cases}n+1&\mbox{if }~{}m=0\\ A(m-1,1)&\mbox{if }~{}m>0\mbox{ and }~{}n=0\\ A(m-1,A(m,n-1))&\mbox{if }~{}m>0\mbox{ and }~{}n>0.\end{cases}
  10. A ( m , n ) A(m,n)
  11. A ( m , n ) = 2 m - 2 ( n + 3 ) - 3. A(m,n)=2\uparrow^{m-2}(n+3)-3.
  12. 2 m + 1 3 = 2 m 4. 2\uparrow^{m+1}3=2\uparrow^{m}4.
  13. A ( m , n ) = 2 [ m ] ( n + 3 ) - 3 A(m,n)=2[m](n+3)-3
  14. A ( m , n ) = ( 2 ( n + 3 ) ( m - 2 ) ) - 3 A(m,n)=(2\rightarrow(n+3)\rightarrow(m-2))-3
  15. m 3 m\geq 3
  16. 2 n m = A ( m + 2 , n - 3 ) + 3 2\rightarrow n\rightarrow m=A(m+2,n-3)+3
  17. n > 2 n>2
  18. × 10 1 9728 \times 10^{1}9728
  19. A × B C A ( B C ) A\times B\rightarrow C\cong A\rightarrow(B\rightarrow C)
  20. Ack ( 0 ) = Succ Ack ( m + 1 ) = Iter ( Ack ( m ) ) \begin{array}[]{lcl}\operatorname{Ack}(0)&=&\operatorname{Succ}\\ \operatorname{Ack}(m+1)&=&\operatorname{Iter}(\operatorname{Ack}(m))\end{array}
  21. Iter ( f ) ( 0 ) = f ( 1 ) Iter ( f ) ( n + 1 ) = f ( Iter ( f ) ( n ) ) . \begin{array}[]{lcl}\operatorname{Iter}(f)(0)&=&f(1)\\ \operatorname{Iter}(f)(n+1)&=&f(\operatorname{Iter}(f)(n)).\end{array}
  22. n + 1 n+1
  23. n + 2 = 2 + ( n + 3 ) - 3 n+2=2+(n+3)-3
  24. 2 n + 3 = 2 ( n + 3 ) - 3 2n+3=2\cdot(n+3)-3
  25. 2 ( n + 3 ) - 3 2^{(n+3)}-3
  26. 2 2 2 - 3 {2^{2^{2}}}-3
  27. 2 2 2 2 - 3 {2^{2^{2^{2}}}}-3
  28. 2 2 2 2 2 - 3 {2^{2^{2^{2^{2}}}}}-3
  29. 2 2 65536 - 3 {2^{2^{65536}}}-3
  30. 2 2 2 2 2 2 - 3 {2^{2^{2^{2^{2^{2}}}}}}-3
  31. 2 2 2 65536 - 3 {2^{2^{2^{65536}}}}-3
  32. 2 2 2 2 2 2 2 - 3 {2^{2^{2^{2^{2^{2^{2}}}}}}}-3
  33. 2 2 2 - 3 n + 3 \begin{matrix}\underbrace{{2^{2}}^{{\cdot}^{{\cdot}^{{\cdot}^{2}}}}}-3\\ n+3\end{matrix}
  34. 2 3 - 3 2\uparrow\uparrow\uparrow 3-3
  35. 2 4 - 3 2\uparrow\uparrow\uparrow 4-3
  36. 2 5 - 3 2\uparrow\uparrow\uparrow 5-3
  37. 2 6 - 3 2\uparrow\uparrow\uparrow 6-3
  38. 2 7 - 3 2\uparrow\uparrow\uparrow 7-3
  39. 2 ( n + 3 ) - 3 2\uparrow\uparrow\uparrow(n+3)-3
  40. 2 3 - 3 2\uparrow\uparrow\uparrow\uparrow 3-3
  41. 2 4 - 3 2\uparrow\uparrow\uparrow\uparrow 4-3
  42. 2 5 - 3 2\uparrow\uparrow\uparrow\uparrow 5-3
  43. 2 6 - 3 2\uparrow\uparrow\uparrow\uparrow 6-3
  44. 2 7 - 3 2\uparrow\uparrow\uparrow\uparrow 7-3
  45. 2 ( n + 3 ) - 3 2\uparrow\uparrow\uparrow\uparrow(n+3)-3
  46. n + 1 n+1
  47. A ( 1 , 2 ) A(1,2)
  48. A ( 1 , 2 ) \displaystyle A(1,2)
  49. A ( 4 , 3 ) A(4,3)
  50. A ( 4 , 3 ) \displaystyle A(4,3)
  51. × 10 1 9727 \times 10^{1}9727
  52. 2 2 2 2 16 2^{2^{2^{2^{16}}}}
  53. x \lfloor x\rfloor
  54. α ( m , n ) = min { i 1 : A ( i , m / n ) log 2 n } . \alpha(m,n)=\min\{i\geq 1:A(i,\lfloor m/n\rfloor)\geq\log_{2}n\}.
  55. 6 [ 6 ] 6 = 6 [ 5 ] ( 6 [ 5 ] ( 6 [ 5 ] ( 6 [ 5 ] ( 6 [ 5 ] 6 ) ) ) ) = 6[6]6=6[5](6[5](6[5](6[5](6[5]6))))=
  56. \quad
  57. 6 6 6 } 6 6 6 } 6 6 6 6 } 6 , \left.\begin{matrix}6^{6^{\cdot^{\cdot^{\cdot^{\cdot^{6}}}}}}\end{matrix}% \right\}\left.\begin{matrix}6^{6^{\cdot^{\cdot^{\cdot^{6}}}}}\end{matrix}% \right\}\dots\left.\begin{matrix}6^{6^{6^{6}}}\end{matrix}\right\}6,
  58. 6 6 6 } 6 6 6 } 6 6 6 6 } 6 , \left.\begin{matrix}6^{6^{\cdot^{\cdot^{\cdot^{\cdot^{6}}}}}}\end{matrix}% \right\}\left.\begin{matrix}6^{6^{\cdot^{\cdot^{\cdot^{6}}}}}\end{matrix}% \right\}\dots\left.\begin{matrix}6^{6^{6^{6}}}\end{matrix}\right\}6,
  59. 6 6 6 } 6 6 6 } 6 6 6 6 } 6 , \left.\begin{matrix}6^{6^{\cdot^{\cdot^{\cdot^{6}}}}}\end{matrix}\right\}\left% .\begin{matrix}6^{6^{\cdot^{\cdot^{\cdot^{6}}}}}\end{matrix}\right\}\left.% \begin{matrix}6^{6^{6^{6}}}\end{matrix}\right\}6,
  60. 6 6 6 } 6 6 6 } 6 6 6 6 } 6 , \left.\begin{matrix}6^{6^{\cdot^{\cdot^{\cdot^{6}}}}}\end{matrix}\right\}\left% .\begin{matrix}6^{6^{\cdot^{\cdot^{\cdot^{6}}}}}\end{matrix}\right\}\left.% \begin{matrix}6^{6^{6^{6}}}\end{matrix}\right\}6,
  61. 6 6 6 } 6 6 6 } 6 6 6 6 } 6 , \left.\begin{matrix}6^{6^{\cdot^{\cdot^{\cdot^{6}}}}}\end{matrix}\right\}\left% .\begin{matrix}6^{6^{\cdot^{\cdot^{\cdot^{6}}}}}\end{matrix}\right\}\left.% \begin{matrix}6^{6^{6^{6}}}\end{matrix}\right\}6,
  62. 6 [ 5 ] n = 6 [ 4 ] ( 6 [ 4 ] ( 6 [ 4 ] ( 6 [ 4 ] ( 6 [ 4 ] 6 ) ) ) ) n 6 s . 6[5]n=\underbrace{6[4](6[4](6[4](6[4](6[4]6))))}_{n\ 6\,\text{s}}.

Acoustic_theory.html

  1. p t + κ 𝐮 \displaystyle\frac{\partial p}{\partial t}+\kappa~{}\nabla\cdot\mathbf{u}
  2. p ( 𝐱 , t ) p(\mathbf{x},t)
  3. 𝐮 ( 𝐱 , t ) \mathbf{u}(\mathbf{x},t)
  4. 𝐱 \mathbf{x}
  5. x , y , z x,y,z
  6. t t
  7. ρ 0 \rho_{0}
  8. κ \kappa
  9. c 0 c_{0}
  10. κ = ρ 0 c 0 2 . \kappa=\rho_{0}c_{0}^{2}~{}.
  11. × 𝐮 = 𝟎 \nabla\times\mathbf{u}=\mathbf{0}
  12. 2 𝐮 t 2 - c 0 2 2 𝐮 = 0 or 2 p t 2 - c 0 2 2 p = 0 , \cfrac{\partial^{2}\mathbf{u}}{\partial t^{2}}-c_{0}^{2}~{}\nabla^{2}\mathbf{u% }=0\qquad\,\text{or}\qquad\cfrac{\partial^{2}p}{\partial t^{2}}-c_{0}^{2}~{}% \nabla^{2}p=0,
  13. 2 𝐮 = ( 𝐮 ) - × ( × 𝐮 ) \nabla^{2}\mathbf{u}=\nabla(\nabla\cdot\mathbf{u})-\nabla\times(\nabla\times% \mathbf{u})
  14. φ \varphi
  15. 𝐮 = φ \mathbf{u}=\nabla\varphi
  16. 2 φ t 2 - c 0 2 2 φ = 0 \cfrac{\partial^{2}\varphi}{\partial t^{2}}-c_{0}^{2}~{}\nabla^{2}\varphi=0
  17. p + ρ 0 φ t = 0 ; ρ + ρ 0 c 0 2 φ t = 0 . p+\rho_{0}~{}\cfrac{\partial\varphi}{\partial t}=0~{};~{}~{}\rho+\cfrac{\rho_{% 0}}{c_{0}^{2}}~{}\cfrac{\partial\varphi}{\partial t}=0~{}.
  18. ρ ( 𝐮 t + 𝐮 𝐮 ) = - p + \cdotsymbol τ + ρ 𝐠 \rho\left(\frac{\partial\mathbf{u}}{\partial t}+\mathbf{u}\cdot\nabla\mathbf{u% }\right)=-\nabla p+\nabla\cdotsymbol{\tau}+\rho\mathbf{g}
  19. 𝐠 \mathbf{g}
  20. p p
  21. s y m b o l τ symbol{\tau}
  22. s y m b o l τ symbol{\tau}
  23. p := - 1 3 tr ( s y m b o l τ ) ; s y m b o l τ := s y m b o l τ + p s y m b o l 1 p:=-\tfrac{1}{3}~{}\,\text{tr}(symbol{\tau})~{};~{}~{}symbol{\tau}:=symbol{% \tau}+p~{}symbol{\mathit{1}}
  24. s y m b o l 1 symbol{\mathit{1}}
  25. s y m b o l τ = μ [ 𝐮 + ( 𝐮 ) T ] + λ ( 𝐮 ) s y m b o l 1 symbol{\tau}=\mu~{}\left[\nabla\mathbf{u}+(\nabla\mathbf{u})^{T}\right]+% \lambda~{}(\nabla\cdot\mathbf{u})~{}symbol{\mathit{1}}
  26. μ \mu
  27. λ \lambda
  28. s y m b o l τ symbol{\tau}
  29. \cdotsymbol τ s i j x i = μ [ x i ( u i x j + u j x i ) ] + λ [ x i ( u k x k ) ] δ i j = μ 2 u i x i x j + μ 2 u j x i x i + λ 2 u k x k x j = ( μ + λ ) 2 u i x i x j + μ 2 u j x i 2 ( μ + λ ) ( 𝐮 ) + μ 2 𝐮 . \begin{aligned}\displaystyle\nabla\cdotsymbol{\tau}\equiv\cfrac{\partial s_{ij% }}{\partial x_{i}}&\displaystyle=\mu\left[\cfrac{\partial}{\partial x_{i}}% \left(\cfrac{\partial u_{i}}{\partial x_{j}}+\cfrac{\partial u_{j}}{\partial x% _{i}}\right)\right]+\lambda~{}\left[\cfrac{\partial}{\partial x_{i}}\left(% \cfrac{\partial u_{k}}{\partial x_{k}}\right)\right]\delta_{ij}\\ &\displaystyle=\mu~{}\cfrac{\partial^{2}u_{i}}{\partial x_{i}\partial x_{j}}+% \mu~{}\cfrac{\partial^{2}u_{j}}{\partial x_{i}\partial x_{i}}+\lambda~{}\cfrac% {\partial^{2}u_{k}}{\partial x_{k}\partial x_{j}}\\ &\displaystyle=(\mu+\lambda)~{}\cfrac{\partial^{2}u_{i}}{\partial x_{i}% \partial x_{j}}+\mu~{}\cfrac{\partial^{2}u_{j}}{\partial x_{i}^{2}}\\ &\displaystyle\equiv(\mu+\lambda)~{}\nabla(\nabla\cdot\mathbf{u})+\mu~{}\nabla% ^{2}\mathbf{u}~{}.\end{aligned}
  30. 2 𝐮 = ( 𝐮 ) - × × 𝐮 \nabla^{2}\mathbf{u}=\nabla(\nabla\cdot\mathbf{u})-\nabla\times\nabla\times% \mathbf{u}
  31. \cdotsymbol τ = ( 2 μ + λ ) ( 𝐮 ) - μ × × 𝐮 . \nabla\cdotsymbol{\tau}=(2\mu+\lambda)~{}\nabla(\nabla\cdot\mathbf{u})-\mu~{}% \nabla\times\nabla\times\mathbf{u}~{}.
  32. ρ ( 𝐮 t + 𝐮 𝐮 ) = - p + ( 2 μ + λ ) ( 𝐮 ) - μ × × 𝐮 + ρ 𝐠 \rho\left(\frac{\partial\mathbf{u}}{\partial t}+\mathbf{u}\cdot\nabla\mathbf{u% }\right)=-\nabla p+(2\mu+\lambda)~{}\nabla(\nabla\cdot\mathbf{u})-\mu~{}\nabla% \times\nabla\times\mathbf{u}+\rho\mathbf{g}
  33. × 𝐮 = 0 \nabla\times\mathbf{u}=0
  34. ρ ( 𝐮 t + 𝐮 𝐮 ) = - p + ( 2 μ + λ ) ( 𝐮 ) + ρ 𝐠 \rho\left(\frac{\partial\mathbf{u}}{\partial t}+\mathbf{u}\cdot\nabla\mathbf{u% }\right)=-\nabla p+(2\mu+\lambda)~{}\nabla(\nabla\cdot\mathbf{u})+\rho\mathbf{g}
  35. ρ ( 𝐮 t + 𝐮 𝐮 ) = - p + ( 2 μ + λ ) ( 𝐮 ) \rho\left(\frac{\partial\mathbf{u}}{\partial t}+\mathbf{u}\cdot\nabla\mathbf{u% }\right)=-\nabla p+(2\mu+\lambda)~{}\nabla(\nabla\cdot\mathbf{u})
  36. ρ ( 𝐮 t + 𝐮 𝐮 ) = - p \rho\left(\frac{\partial\mathbf{u}}{\partial t}+\mathbf{u}\cdot\nabla\mathbf{u% }\right)=-\nabla p
  37. \langle\cdot\rangle
  38. ~ \tilde{\cdot}
  39. p = p + p ~ ; ρ = ρ + ρ ~ ; 𝐮 = 𝐮 + 𝐮 ~ p=\langle p\rangle+\tilde{p}~{};~{}~{}\rho=\langle\rho\rangle+\tilde{\rho}~{};% ~{}~{}\mathbf{u}=\langle\mathbf{u}\rangle+\tilde{\mathbf{u}}
  40. p t = 0 ; ρ t = 0 ; 𝐮 t = 𝟎 . \cfrac{\partial\langle p\rangle}{\partial t}=0~{};~{}~{}\cfrac{\partial\langle% \rho\rangle}{\partial t}=0~{};~{}~{}\cfrac{\partial\langle\mathbf{u}\rangle}{% \partial t}=\mathbf{0}~{}.
  41. [ ρ + ρ ~ ] [ 𝐮 ~ t + [ 𝐮 + 𝐮 ~ ] [ 𝐮 + 𝐮 ~ ] ] = - [ p + p ~ ] \left[\langle\rho\rangle+\tilde{\rho}\right]\left[\frac{\partial\tilde{\mathbf% {u}}}{\partial t}+\left[\langle\mathbf{u}\rangle+\tilde{\mathbf{u}}\right]% \cdot\nabla\left[\langle\mathbf{u}\rangle+\tilde{\mathbf{u}}\right]\right]=-% \nabla\left[\langle p\rangle+\tilde{p}\right]
  42. ρ 𝐮 ~ t + [ ρ + ρ ~ ] [ 𝐮 𝐮 ] + ρ [ 𝐮 𝐮 ~ + 𝐮 ~ 𝐮 ] = - [ p + p ~ ] \begin{aligned}\displaystyle\langle\rho\rangle~{}\frac{\partial\tilde{\mathbf{% u}}}{\partial t}&\displaystyle+\left[\langle\rho\rangle+\tilde{\rho}\right]% \left[\langle\mathbf{u}\rangle\cdot\nabla\langle\mathbf{u}\rangle\right]+% \langle\rho\rangle\left[\langle\mathbf{u}\rangle\cdot\nabla\tilde{\mathbf{u}}+% \tilde{\mathbf{u}}\cdot\nabla\langle\mathbf{u}\rangle\right]\\ &\displaystyle=-\nabla\left[\langle p\rangle+\tilde{p}\right]\end{aligned}
  43. p \langle p\rangle
  44. ρ \langle\rho\rangle
  45. p = 0 ; ρ = 0 . \nabla\langle p\rangle=0~{};~{}~{}\nabla\langle\rho\rangle=0~{}.
  46. ρ 𝐮 ~ t + [ ρ + ρ ~ ] [ 𝐮 𝐮 ] + ρ [ 𝐮 𝐮 ~ + 𝐮 ~ 𝐮 ] = - p ~ \langle\rho\rangle~{}\frac{\partial\tilde{\mathbf{u}}}{\partial t}+\left[% \langle\rho\rangle+\tilde{\rho}\right]\left[\langle\mathbf{u}\rangle\cdot% \nabla\langle\mathbf{u}\rangle\right]+\langle\rho\rangle\left[\langle\mathbf{u% }\rangle\cdot\nabla\tilde{\mathbf{u}}+\tilde{\mathbf{u}}\cdot\nabla\langle% \mathbf{u}\rangle\right]=-\nabla\tilde{p}
  47. 𝐮 = 0 \langle\mathbf{u}\rangle=0
  48. ρ 𝐮 ~ t = - p ~ \langle\rho\rangle~{}\frac{\partial\tilde{\mathbf{u}}}{\partial t}=-\nabla% \tilde{p}
  49. ρ 0 := ρ \rho_{0}:=\langle\rho\rangle
  50. ρ 0 𝐮 t + p = 0 . \rho_{0}~{}\frac{\partial\mathbf{u}}{\partial t}+\nabla p=0~{}.
  51. ρ t + ( ρ 𝐮 ) = 0 \frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\mathbf{u})=0
  52. ρ ( 𝐱 , t ) \rho(\mathbf{x},t)
  53. 𝐮 ( 𝐱 , t ) \mathbf{u}(\mathbf{x},t)
  54. p = p + p ~ ; ρ = ρ + ρ ~ ; 𝐮 = 𝐮 + 𝐮 ~ p=\langle p\rangle+\tilde{p}~{};~{}~{}\rho=\langle\rho\rangle+\tilde{\rho}~{};% ~{}~{}\mathbf{u}=\langle\mathbf{u}\rangle+\tilde{\mathbf{u}}
  55. p t = 0 ; ρ t = 0 ; 𝐮 t = 𝟎 . \cfrac{\partial\langle p\rangle}{\partial t}=0~{};~{}~{}\cfrac{\partial\langle% \rho\rangle}{\partial t}=0~{};~{}~{}\cfrac{\partial\langle\mathbf{u}\rangle}{% \partial t}=\mathbf{0}~{}.
  56. ρ ~ t + [ ρ + ρ ~ ] [ 𝐮 + 𝐮 ~ ] + [ ρ + ρ ~ ] [ 𝐮 + 𝐮 ~ ] = 0 \frac{\partial\tilde{\rho}}{\partial t}+\left[\langle\rho\rangle+\tilde{\rho}% \right]\nabla\cdot\left[\langle\mathbf{u}\rangle+\tilde{\mathbf{u}}\right]+% \nabla\left[\langle\rho\rangle+\tilde{\rho}\right]\cdot\left[\langle\mathbf{u}% \rangle+\tilde{\mathbf{u}}\right]=0
  57. ρ ~ t + [ ρ + ρ ~ ] 𝐮 + ρ 𝐮 ~ + [ ρ + ρ ~ ] 𝐮 + ρ 𝐮 ~ = 0 \frac{\partial\tilde{\rho}}{\partial t}+\left[\langle\rho\rangle+\tilde{\rho}% \right]\nabla\cdot\langle\mathbf{u}\rangle+\langle\rho\rangle\nabla\cdot\tilde% {\mathbf{u}}+\nabla\left[\langle\rho\rangle+\tilde{\rho}\right]\cdot\langle% \mathbf{u}\rangle+\nabla\langle\rho\rangle\cdot\tilde{\mathbf{u}}=0
  58. ρ = 0 . \nabla\langle\rho\rangle=0~{}.
  59. ρ ~ t + [ ρ + ρ ~ ] 𝐮 + ρ 𝐮 ~ + ρ ~ 𝐮 = 0 \frac{\partial\tilde{\rho}}{\partial t}+\left[\langle\rho\rangle+\tilde{\rho}% \right]\nabla\cdot\langle\mathbf{u}\rangle+\langle\rho\rangle\nabla\cdot\tilde% {\mathbf{u}}+\nabla\tilde{\rho}\cdot\langle\mathbf{u}\rangle=0
  60. 𝐮 = 0 \langle\mathbf{u}\rangle=0
  61. ρ ~ t + ρ 𝐮 ~ = 0 \frac{\partial\tilde{\rho}}{\partial t}+\langle\rho\rangle\nabla\cdot\tilde{% \mathbf{u}}=0
  62. d p d ρ = γ p ρ ; γ := c p c v ; c 2 = γ p ρ . \cfrac{dp}{d\rho}=\cfrac{\gamma~{}p}{\rho}~{};~{}~{}\gamma:=\cfrac{c_{p}}{c_{v% }}~{};~{}~{}c^{2}=\cfrac{\gamma~{}p}{\rho}~{}.
  63. c p c_{p}
  64. c v c_{v}
  65. c c
  66. γ \gamma
  67. d p d ρ p ~ ρ ~ ; p ρ p ρ ; c 2 c 0 2 = γ p ρ . \cfrac{dp}{d\rho}\approx\cfrac{\tilde{p}}{\tilde{\rho}}~{};~{}~{}\cfrac{p}{% \rho}\approx\cfrac{\langle p\rangle}{\langle\rho\rangle}~{};~{}~{}c^{2}\approx c% _{0}^{2}=\cfrac{\gamma~{}\langle p\rangle}{\langle\rho\rangle}~{}.
  68. c 0 c_{0}
  69. p ~ ρ ~ = γ p ρ = c 0 2 p ~ t = c 0 2 ρ ~ t \cfrac{\tilde{p}}{\tilde{\rho}}=\gamma~{}\cfrac{\langle p\rangle}{\langle\rho% \rangle}=c_{0}^{2}\qquad\implies\qquad\cfrac{\partial\tilde{p}}{\partial t}=c_% {0}^{2}\cfrac{\partial\tilde{\rho}}{\partial t}
  70. 1 c 0 2 p ~ t + ρ 𝐮 ~ = 0 \cfrac{1}{c_{0}^{2}}\frac{\partial\tilde{p}}{\partial t}+\langle\rho\rangle% \nabla\cdot\tilde{\mathbf{u}}=0
  71. ρ 0 := ρ \rho_{0}:=\langle\rho\rangle
  72. p t + ρ 0 c 0 2 𝐮 = 0 . \frac{\partial p}{\partial t}+\rho_{0}~{}c_{0}^{2}~{}\nabla\cdot\mathbf{u}=0~{}.
  73. ( r , θ , z ) (r,\theta,z)
  74. 𝐞 r , 𝐞 θ , 𝐞 z \mathbf{e}_{r},\mathbf{e}_{\theta},\mathbf{e}_{z}
  75. p p
  76. 𝐮 \mathbf{u}
  77. p = p r 𝐞 r + 1 r p θ 𝐞 θ + p z 𝐞 z 𝐮 = u r r + 1 r ( u θ θ + u r ) + u z z \begin{aligned}\displaystyle\nabla p&\displaystyle=\cfrac{\partial p}{\partial r% }~{}\mathbf{e}_{r}+\cfrac{1}{r}~{}\cfrac{\partial p}{\partial\theta}~{}\mathbf% {e}_{\theta}+\cfrac{\partial p}{\partial z}~{}\mathbf{e}_{z}\\ \displaystyle\nabla\cdot\mathbf{u}&\displaystyle=\cfrac{\partial u_{r}}{% \partial r}+\cfrac{1}{r}\left(\cfrac{\partial u_{\theta}}{\partial\theta}+u_{r% }\right)+\cfrac{\partial u_{z}}{\partial z}\end{aligned}
  78. 𝐮 = u r 𝐞 r + u θ 𝐞 θ + u z 𝐞 z \mathbf{u}=u_{r}~{}\mathbf{e}_{r}+u_{\theta}~{}\mathbf{e}_{\theta}+u_{z}~{}% \mathbf{e}_{z}
  79. ρ 0 [ u r t 𝐞 r + u θ t 𝐞 θ + u z t 𝐞 z ] + p r 𝐞 r + 1 r p θ 𝐞 θ + p z 𝐞 z = 0 \rho_{0}~{}\left[\cfrac{\partial u_{r}}{\partial t}~{}\mathbf{e}_{r}+\cfrac{% \partial u_{\theta}}{\partial t}~{}\mathbf{e}_{\theta}+\cfrac{\partial u_{z}}{% \partial t}~{}\mathbf{e}_{z}\right]+\cfrac{\partial p}{\partial r}~{}\mathbf{e% }_{r}+\cfrac{1}{r}~{}\cfrac{\partial p}{\partial\theta}~{}\mathbf{e}_{\theta}+% \cfrac{\partial p}{\partial z}~{}\mathbf{e}_{z}=0
  80. ρ 0 u r t + p r = 0 ; ρ 0 u θ t + 1 r p θ = 0 ; ρ 0 u z t + p z = 0 . \rho_{0}~{}\cfrac{\partial u_{r}}{\partial t}+\cfrac{\partial p}{\partial r}=0% ~{};~{}~{}\rho_{0}~{}\cfrac{\partial u_{\theta}}{\partial t}+\cfrac{1}{r}~{}% \cfrac{\partial p}{\partial\theta}=0~{};~{}~{}\rho_{0}~{}\cfrac{\partial u_{z}% }{\partial t}+\cfrac{\partial p}{\partial z}=0~{}.
  81. p t + κ [ u r r + 1 r ( u θ θ + u r ) + u z z ] = 0 . \cfrac{\partial p}{\partial t}+\kappa\left[\cfrac{\partial u_{r}}{\partial r}+% \cfrac{1}{r}\left(\cfrac{\partial u_{\theta}}{\partial\theta}+u_{r}\right)+% \cfrac{\partial u_{z}}{\partial z}\right]=0~{}.
  82. p ( 𝐱 , t ) = p ^ ( 𝐱 ) e - i ω t ; 𝐮 ( 𝐱 , t ) = 𝐮 ^ ( 𝐱 ) e - i ω t ; i := - 1 p(\mathbf{x},t)=\hat{p}(\mathbf{x})~{}e^{-i\omega t}~{};~{}~{}\mathbf{u}(% \mathbf{x},t)=\hat{\mathbf{u}}(\mathbf{x})~{}e^{-i\omega t}~{};~{}~{}i:=\sqrt{% -1}
  83. ω \omega
  84. p ^ r = i ω ρ 0 u ^ r ; 1 r p ^ θ = i ω ρ 0 u ^ θ ; p ^ z = i ω ρ 0 u ^ z \cfrac{\partial\hat{p}}{\partial r}=i\omega~{}\rho_{0}~{}\hat{u}_{r}~{};~{}~{}% \cfrac{1}{r}~{}\cfrac{\partial\hat{p}}{\partial\theta}=i\omega~{}\rho_{0}~{}% \hat{u}_{\theta}~{};~{}~{}\cfrac{\partial\hat{p}}{\partial z}=i\omega~{}\rho_{% 0}~{}\hat{u}_{z}
  85. i ω p ^ κ = u ^ r r + 1 r ( u ^ θ θ + u ^ r ) + u ^ z z . \cfrac{i\omega\hat{p}}{\kappa}=\cfrac{\partial\hat{u}_{r}}{\partial r}+\cfrac{% 1}{r}\left(\cfrac{\partial\hat{u}_{\theta}}{\partial\theta}+\hat{u}_{r}\right)% +\cfrac{\partial\hat{u}_{z}}{\partial z}~{}.
  86. u r , u θ u_{r},u_{\theta}
  87. 2 p r 2 + 1 r p r + 1 r 2 2 p θ 2 + ω 2 ρ 0 κ p = 0 \frac{\partial^{2}p}{\partial r^{2}}+\frac{1}{r}\frac{\partial p}{\partial r}+% \frac{1}{r^{2}}~{}\frac{\partial^{2}p}{\partial\theta^{2}}+\frac{\omega^{2}% \rho_{0}}{\kappa}~{}p=0
  88. p ( r , θ ) = R ( r ) Q ( θ ) p(r,\theta)=R(r)~{}Q(\theta)
  89. r 2 R d 2 R d r 2 + r R d R d r + r 2 ω 2 ρ 0 κ = - 1 Q d 2 Q d θ 2 \cfrac{r^{2}}{R}~{}\cfrac{d^{2}R}{dr^{2}}+\cfrac{r}{R}~{}\cfrac{dR}{dr}+\cfrac% {r^{2}\omega^{2}\rho_{0}}{\kappa}=-\cfrac{1}{Q}~{}\cfrac{d^{2}Q}{d\theta^{2}}
  90. θ \theta
  91. r r
  92. r 2 d 2 R d r 2 + r d R d r + r 2 ω 2 ρ 0 κ R = α 2 R ; d 2 Q d θ 2 = - α 2 Q r^{2}~{}\cfrac{d^{2}R}{dr^{2}}+r~{}\cfrac{dR}{dr}+\cfrac{r^{2}\omega^{2}\rho_{% 0}}{\kappa}~{}R=\alpha^{2}~{}R~{};~{}~{}\cfrac{d^{2}Q}{d\theta^{2}}=-\alpha^{2% }~{}Q
  93. α 2 \alpha^{2}
  94. r ~ ( ω ρ 0 κ ) r = k r \tilde{r}\leftarrow\left(\omega\sqrt{\cfrac{\rho_{0}}{\kappa}}\right)r=k~{}r
  95. r ~ 2 d 2 R d r ~ 2 + r ~ d R d r ~ + ( r ~ 2 - α 2 ) R = 0 ; d 2 Q d θ 2 = - α 2 Q \tilde{r}^{2}~{}\cfrac{d^{2}R}{d\tilde{r}^{2}}+\tilde{r}~{}\cfrac{dR}{d\tilde{% r}}+(\tilde{r}^{2}-\alpha^{2})~{}R=0~{};~{}~{}\cfrac{d^{2}Q}{d\theta^{2}}=-% \alpha^{2}~{}Q
  96. R ( r ) = A α J α ( k r ) + B α J - α ( k r ) R(r)=A_{\alpha}~{}J_{\alpha}(k~{}r)+B_{\alpha}~{}J_{-\alpha}(k~{}r)
  97. J α J_{\alpha}
  98. A α , B α A_{\alpha},B_{\alpha}
  99. Q ( θ ) = C α e i α θ + D α e - i α θ Q(\theta)=C_{\alpha}~{}e^{i\alpha\theta}+D_{\alpha}~{}e^{-i\alpha\theta}
  100. C α , D α C_{\alpha},D_{\alpha}
  101. p ( r , θ ) = [ A α J α ( k r ) + B α J - α ( k r ) ] ( C α e i α θ + D α e - i α θ ) p(r,\theta)=\left[A_{\alpha}~{}J_{\alpha}(k~{}r)+B_{\alpha}~{}J_{-\alpha}(k~{}% r)\right]\left(C_{\alpha}~{}e^{i\alpha\theta}+D_{\alpha}~{}e^{-i\alpha\theta}\right)
  102. α \alpha

Actinide.html

  1. U 92 238 + n 0 1 U 92 239 β - 23.5 min Np 93 239 β - 2.3 days Pu 94 239 2.4 10 4 years 𝛼 \mathrm{{}^{238}_{92}U+{}^{1}_{0}n\ \xrightarrow{\ }\ {}^{239}_{92}U\ % \xrightarrow[23.5\ min]{\beta^{-}}\ {}^{239}_{93}Np\ \xrightarrow[2.3\ days]{% \beta^{-}}\ {}^{239}_{94}Pu\ \xrightarrow[2.4\cdot 10^{4}\ years]{\alpha}}
  2. U 92 238 + Ne 10 22 No 102 256 + 4 n 0 1 ~{}\mathrm{{}_{~{}92}^{238}{}U+{}_{10}^{22}{}Ne\longrightarrow{}_{102}^{256}{}% No+4{}_{0}^{1}{}n}
  3. Pu 94 239 + He 2 4 Cm 96 242 + n 0 1 ~{}\mathrm{{}_{~{}94}^{239}{}Pu+{}_{2}^{4}{}He\longrightarrow{}_{~{}96}^{242}{% }Cm+{}_{0}^{1}{}n}
  4. × 10 10 \times 10^{−}10
  5. × 10 1 4 \times 10^{1}4
  6. × 10 3 \times 10^{3}
  7. × 10 9 \times 10^{−}9
  8. × 10 1 2 \times 10^{1}2
  9. × 10 6 \times 10^{−}6
  10. × 10 3 \times 10^{−}3
  11. × 10 1 7 \times 10^{1}7
  12. × 10 9 \times 10^{9}
  13. × 10 4 \times 10^{−}4
  14. × 10 3 \times 10^{−}3
  15. × 10 1 5 \times 10^{1}5
  16. × 10 9 \times 10^{9}
  17. × 10 3 \times 10^{3}
  18. × 10 7 \times 10^{−}7
  19. × 10 1 2 \times 10^{1}2
  20. × 10 4 \times 10^{4}
  21. × 10 3 \times 10^{3}
  22. × 10 1 3 \times 10^{1}3
  23. × 10 9 \times 10^{9}
  24. × 10 7 \times 10^{7}
  25. × 10 9 \times 10^{9}
  26. × 10 5 \times 10^{5}
  27. × 10 1 1 \times 10^{1}1
  28. × 10 1 0 \times 10^{1}0
  29. × 10 8 \times 10^{−}8
  30. × 10 3 \times 10^{−}3
  31. × 10 4 \times 10^{−}4
  32. × 10 5 \times 10^{−}5
  33. × 10 1 6 \times 10^{1}6
  34. × 10 7 \times 10^{7}
  35. × 10 7 \times 10^{−}7
  36. × 10 1 4 \times 10^{1}4
  37. × 10 5 \times 10^{5}
  38. × 10 1 5 \times 10^{1}5
  39. × 10 1 2 \times 10^{1}2
  40. × 10 1 3 \times 10^{1}3
  41. × 10 2 \times 10^{−}2
  42. × 10 2 \times 10^{−}2
  43. × 10 1 6 \times 10^{1}6
  44. × 10 1 4 \times 10^{1}4
  45. × 10 2 \times 10^{−}2
  46. × 10 1 4 \times 10^{1}4
  47. × 10 1 7 \times 10^{1}7
  48. × 10 6 \times 10^{−}6
  49. × 10 3 \times 10^{−}3
  50. × 10 1 7 \times 10^{1}7
  51. × 10 1 0 \times 10^{1}0
  52. × 10 1 6 \times 10^{1}6
  53. × 10 1 7 \times 10^{1}7
  54. × 10 1 6 \times 10^{1}6
  55. × 10 1 3 \times 10^{1}3
  56. × 10 5 \times 10^{−}5
  57. × 10 1 9 \times 10^{1}9
  58. × 10 1 2 \times 10^{1}2
  59. × 10 2 0 \times 10^{2}0
  60. × 10 1 9 \times 10^{1}9
  61. × 10 1 7 \times 10^{1}7
  62. × 10 1 4 \times 10^{1}4
  63. × 10 2 0 \times 10^{2}0
  64. × 10 1 9 \times 10^{1}9
  65. × 10 1 7 \times 10^{1}7
  66. × 10 2 1 \times 10^{2}1
  67. × 10 2 0 \times 10^{2}0
  68. × 10 2 2 \times 10^{2}2
  69. × 10 2 4 \times 10^{2}4
  70. × 10 - 10 \times 10^{-}10
  71. × 10 - 8 \times 10^{-}8
  72. × 10 9 \times 10^{9}
  73. × 10 6 \times 10^{6}
  74. × 10 5 \times 10^{5}
  75. × 10 7 \times 10^{7}
  76. × 10 - 3 \times 10^{-}3
  77. × 10 7 \times 10^{7}
  78. × 10 - 3 \times 10^{-}3
  79. × 10 - 4 \times 10^{-}4
  80. × 10 9 \times 10^{9}
  81. × 10 1 2 \times 10^{1}2
  82. × 10 - 12 \times 10^{-}12
  83. × 10 - 12 \times 10^{-}12
  84. × 10 - 12 \times 10^{-}12
  85. × 10 - 14 \times 10^{-}14
  86. × 10 - 14 \times 10^{-}14
  87. × 10 - 15 \times 10^{-}15
  88. × 10 - 20 \times 10^{-}20
  89. 2 A m F 3 + 3 B a 1150 - 1350 C 3 BaF 2 + 2 A m ~{}\mathrm{2AmF_{3}+3Ba\ \xrightarrow{1150-1350^{\circ}C}\ 3BaF_{2}+2Am}
  90. PuF 4 + 2 B a 1200 C 2 BaF 2 + Pu ~{}\mathrm{PuF_{4}+2Ba\ \xrightarrow{1200^{\circ}C}\ 2BaF_{2}+Pu}
  91. UF 4 + 2 M g > 500 C U + 2 M g F 2 ~{}\mathrm{UF_{4}+2Mg\ \xrightarrow{>500^{\circ}C}\ U+2MgF_{2}}
  92. E M 4 + A n O 2 2 + ~{}E_{\frac{M^{4+}}{AnO^{2+}_{2}}}
  93. 3 ¯ \overline{3}
  94. Th + O 2 1000 C ThO 2 ~{}\mathrm{Th+O_{2}\ \xrightarrow{1000^{\circ}C}\ ThO_{2}}
  95. U 235 + 0 1 n 115 Rh + 118 Ag + 118 3 0 1 n ~{}\mathrm{{}^{235}U+\ ^{1}_{0}n\longrightarrow^{115}Rh+^{118}Ag+3^{1}_{0}n}
  96. × 10 8 \times 10^{8}

Actinium.html

  1. 3 ¯ \overline{3}
  2. 3 ¯ \overline{3}
  3. 3 ¯ \overline{3}
  4. 4 ¯ \overline{4}
  5. 3 ¯ \overline{3}
  6. 3 ¯ \overline{3}
  7. Ra 88 226 + 0 1 n 88 227 Ra β - 42.2 min 89 227 Ac \mathrm{{}^{226}_{\ 88}Ra\ +\ ^{1}_{0}n\ \longrightarrow\ ^{227}_{\ 88}Ra\ % \xrightarrow[42.2\ min]{\beta^{-}}\ ^{227}_{\ 89}Ac}
  8. Be 4 9 + 2 4 He 6 12 C + 0 1 n + 0 1 γ \mathrm{{}^{9}_{4}Be\ +\ ^{4}_{2}He\ \longrightarrow\ ^{12}_{\ 6}C\ +\ ^{1}_{0% }n\ +\ \gamma}

Activation_energy.html

  1. kJ mol ~{}\frac{\mathrm{kJ}}{\mathrm{mol}}
  2. k = A e - E a / R T k=Ae^{{-E_{a}}/{RT}}
  3. Δ G \ \Delta G^{\ddagger}

Active_laser_medium.html

  1. ω p ~{}\omega_{\rm p}~{}
  2. ω p ~{}\omega_{\rm p}~{}
  3. ω s ~{}\omega_{\rm s}~{}
  4. N ~{}N~{}
  5. N 1 ~{}N_{1}~{}
  6. N 2 ~{}N_{2}~{}
  7. N 1 + N 2 = N ~{}N_{1}+N_{2}=N~{}
  8. n 1 = N 1 / N ~{}n_{1}=N_{1}/N~{}
  9. n 2 = N 2 / N ~{}n_{2}=N_{2}/N~{}
  10. W u = I p σ ap ω p + I s σ as ω s ~{}W_{\rm u}=\frac{I_{\rm p}\sigma_{\rm ap}}{\hbar\omega_{\rm p}}+\frac{I_{\rm s% }\sigma_{\rm as}}{\hbar\omega_{\rm s}}~{}
  11. W d = I p σ ep ω p + I s σ es ω s + 1 τ ~{}W_{\rm d}=\frac{I_{\rm p}\sigma_{\rm ep}}{\hbar\omega_{\rm p}}+\frac{I_{\rm s% }\sigma_{\rm es}}{\hbar\omega_{\rm s}}+\frac{1}{\tau}~{}
  12. σ as ~{}\sigma_{\rm as}~{}
  13. σ ap ~{}\sigma_{\rm ap}~{}
  14. σ es ~{}\sigma_{\rm es}~{}
  15. σ ep ~{}\sigma_{\rm ep}~{}
  16. 1 τ ~{}\frac{1}{\tau}~{}
  17. d n 2 d t = W u n 1 - W d n 2 ~{}\frac{{\rm d}n_{2}}{{\rm d}t}=W_{\rm u}n_{1}-W_{\rm d}n_{2}~{}
  18. d n 1 d t = - W u n 1 + W d n 2 ~{}\frac{{\rm d}n_{1}}{{\rm d}t}=-W_{\rm u}n_{1}+W_{\rm d}n_{2}~{}
  19. n 1 + n 2 = 1 ~{}n_{1}+n_{2}=1~{}
  20. A ~{}A~{}
  21. G ~{}G~{}
  22. A = N 1 σ pa - N 2 σ pe ~{}A=N_{1}\sigma_{\rm pa}-N_{2}\sigma_{\rm pe}~{}
  23. G = N 2 σ se - N 1 σ sa ~{}G=N_{2}\sigma_{\rm se}-N_{1}\sigma_{\rm sa}~{}
  24. n 2 = W u W u + W d ~{}n_{2}=\frac{W_{\rm u}}{W_{\rm u}+W_{\rm d}}~{}
  25. n 1 = W d W u + W d . ~{}n_{1}=\frac{W_{\rm d}}{W_{\rm u}+W_{\rm d}}.
  26. I po = ω p ( σ ap + σ ep ) τ ~{}I_{\rm po}=\frac{\hbar\omega_{\rm p}}{(\sigma_{\rm ap}+\sigma_{\rm ep})\tau% }~{}
  27. I so = ω s ( σ as + σ es ) τ ~{}I_{\rm so}=\frac{\hbar\omega_{\rm s}}{(\sigma_{\rm as}+\sigma_{\rm es})\tau% }~{}
  28. A 0 = N D σ as + σ es ~{}A_{0}=\frac{ND}{\sigma_{\rm as}+\sigma_{\rm es}}~{}
  29. G 0 = N D σ ap + σ ep ~{}G_{0}=\frac{ND}{\sigma_{\rm ap}+\sigma_{\rm ep}}~{}
  30. D = σ pa σ se - σ pe σ sa ~{}D=\sigma_{\rm pa}\sigma_{\rm se}-\sigma_{\rm pe}\sigma_{\rm sa}~{}
  31. G 0 ~{}G_{0}~{}
  32. A 0 U ~{}A_{0}U~{}
  33. I p ~{}I_{\rm p}~{}
  34. I s ~{}I_{\rm s}~{}
  35. A = A 0 U + s 1 + p + s ~{}A=A_{0}\frac{U+s}{1+p+s}~{}
  36. G = G 0 p - V 1 + p + s ~{}G=G_{0}\frac{p-V}{1+p+s}~{}
  37. p = I p / I po ~{}p=I_{\rm p}/I_{\rm po}~{}
  38. s = I s / I so ~{}s=I_{\rm s}/I_{\rm so}~{}
  39. U = ( σ as + σ es ) σ ap D ~{}U=\frac{(\sigma_{\rm as}+\sigma_{\rm es})\sigma_{\rm ap}}{D}~{}
  40. V = ( σ ap + σ ep ) σ as D ~{}V=\frac{(\sigma_{\rm ap}+\sigma_{\rm ep})\sigma_{\rm as}}{D}~{}
  41. U - V = 1 U-V=1~{}
  42. A / A 0 + G / G 0 = 1 . ~{}A/A_{0}+G/G_{0}=1~{}.
  43. E = I s G I p A ~{}E=\frac{I_{\rm s}G}{I_{\rm p}A}~{}
  44. E = ω s ω p 1 - V / p 1 + U / s ~{}E=\frac{\omega_{\rm s}}{\omega_{\rm p}}\frac{1-V/p}{1+U/s}~{}
  45. p V 1 ~{}\frac{p}{V}\gg 1~{}
  46. s U 1 ~{}\frac{s}{U}\gg 1~{}

Adaptive_expectations.html

  1. p e p^{e}
  2. p - 1 e p^{e}_{-1}
  3. p p
  4. p e = p - 1 e + λ ( p - p - 1 e ) p^{e}=p^{e}_{-1}+\lambda(p-p^{e}_{-1})
  5. λ \lambda
  6. p e = λ j = 0 ( ( 1 - λ ) j p j ) p^{e}=\lambda\sum_{j=0}^{\infty}((1-\lambda)^{j}p_{j})
  7. p j p_{j}
  8. j j

Adder–subtractor.html

  1. A A
  2. D = 1 D=1
  3. S = B - A S=B-A
  4. A A
  5. B B
  6. S = A + B S=A+B
  7. B - A B-A
  8. S = B + A ¯ + 1 S=B+\overline{A}+1
  9. A A
  10. I 0 I_{0}
  11. A i A_{i}
  12. I 1 I_{1}
  13. A i ¯ \overline{A_{i}}
  14. D D
  15. D D
  16. D = 0 D=0
  17. D = 1 D=1
  18. D = 1 D=1
  19. A A
  20. A ¯ \overline{A}
  21. 1 1
  22. B B
  23. A ¯ \overline{A}
  24. 1 1
  25. B - A B-A
  26. A A
  27. D D
  28. D = 0 D=0
  29. D = 1 D=1
  30. D D
  31. B i B_{i}
  32. B i B_{i}
  33. D = 1 D=1
  34. A A
  35. - A = A ¯ + 1 -A=\overline{A}+1
  36. A A
  37. B i B_{i}
  38. 0
  39. A i A_{i}
  40. B i B_{i}
  41. D = 0 D=0
  42. 1 1
  43. A i A_{i}
  44. B i B_{i}
  45. D = 1 D=1
  46. A A
  47. B i B_{i}
  48. B B
  49. A i A_{i}
  50. A + 1 A+1
  51. B i B_{i}
  52. D = 1 D=1
  53. B + 1 B+1
  54. A i A_{i}
  55. D = 1 D=1
  56. A + B A+B
  57. A - B A-B
  58. B - A B-A
  59. A ¯ \overline{A}
  60. A i A_{i}
  61. B i B_{i}
  62. D = 0 D=0
  63. - A -A
  64. A i A_{i}
  65. B i B_{i}
  66. D = 1 D=1
  67. B ¯ \overline{B}
  68. B i B_{i}
  69. A i A_{i}
  70. D = 0 D=0
  71. - B -B
  72. B i B_{i}
  73. A i A_{i}
  74. D = 1 D=1

Addition.html

  1. 1 + 1 = 2 1+1=2
  2. 2 + 2 = 4 2+2=4
  3. 3 + 3 = 6 3+3=6
  4. 5 + 4 + 2 = 11 5+4+2=11
  5. 3 + 3 + 3 + 3 = 12 3+3+3+3=12
  6. k = 1 5 k 2 = 1 2 + 2 2 + 3 2 + 4 2 + 5 2 = 55. \sum_{k=1}^{5}k^{2}=1^{2}+2^{2}+3^{2}+4^{2}+5^{2}=55.
  7. b t h b^{th}
  8. x x
  9. x x
  10. x = a × 10 b x=a\times 10^{b}
  11. a a
  12. 10 b 10^{b}
  13. 2.34 × 10 - 5 + 5.67 × 10 - 6 = 2.34 × 10 - 5 + 0.567 × 10 - 5 = 2.907 × 10 - 5 2.34\times 10^{-5}+5.67\times 10^{-6}=2.34\times 10^{-5}+0.567\times 10^{-5}=2% .907\times 10^{-5}
  14. N ( A B ) N(A\cup B)
  15. a b + c d = a d + b c b d . \frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}.
  16. 3 4 + 1 8 = 3 × 8 + 4 × 1 4 × 8 = 24 + 4 32 = 28 32 = 7 8 \frac{3}{4}+\frac{1}{8}=\frac{3\times 8+4\times 1}{4\times 8}=\frac{24+4}{32}=% \frac{28}{32}=\frac{7}{8}
  17. a c + b c = a + b c \frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}
  18. 1 4 + 2 4 = 1 + 2 4 = 3 4 \frac{1}{4}+\frac{2}{4}=\frac{1+2}{4}=\frac{3}{4}
  19. a + b = { q + r q a , r b } . a+b=\{q+r\mid q\in a,r\in b\}.
  20. lim n a n + lim n b n = lim n ( a n + b n ) . \lim_{n}a_{n}+\lim_{n}b_{n}=\lim_{n}(a_{n}+b_{n}).
  21. ( a + b i ) + ( c + d i ) = ( a + c ) + ( b + d ) i . (a+bi)+(c+di)=(a+c)+(b+d)i.
  22. A + B = [ a 11 a 12 a 1 n a 21 a 22 a 2 n a m 1 a m 2 a m n ] + [ b 11 b 12 b 1 n b 21 b 22 b 2 n b m 1 b m 2 b m n ] = [ a 11 + b 11 a 12 + b 12 a 1 n + b 1 n a 21 + b 21 a 22 + b 22 a 2 n + b 2 n a m 1 + b m 1 a m 2 + b m 2 a m n + b m n ] \begin{aligned}\displaystyle{A}+{B}&\displaystyle=\begin{bmatrix}a_{11}&a_{12}% &\cdots&a_{1n}\\ a_{21}&a_{22}&\cdots&a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{m1}&a_{m2}&\cdots&a_{mn}\\ \end{bmatrix}+\begin{bmatrix}b_{11}&b_{12}&\cdots&b_{1n}\\ b_{21}&b_{22}&\cdots&b_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ b_{m1}&b_{m2}&\cdots&b_{mn}\\ \end{bmatrix}\\ &\displaystyle=\begin{bmatrix}a_{11}+b_{11}&a_{12}+b_{12}&\cdots&a_{1n}+b_{1n}% \\ a_{21}+b_{21}&a_{22}+b_{22}&\cdots&a_{2n}+b_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{m1}+b_{m1}&a_{m2}+b_{m2}&\cdots&a_{mn}+b_{mn}\\ \end{bmatrix}\\ \end{aligned}\,\!
  23. [ 1 3 1 0 1 2 ] + [ 0 0 7 5 2 1 ] = [ 1 + 0 3 + 0 1 + 7 0 + 5 1 + 2 2 + 1 ] = [ 1 3 8 5 3 3 ] \begin{bmatrix}1&3\\ 1&0\\ 1&2\end{bmatrix}+\begin{bmatrix}0&0\\ 7&5\\ 2&1\end{bmatrix}=\begin{bmatrix}1+0&3+0\\ 1+7&0+5\\ 1+2&2+1\end{bmatrix}=\begin{bmatrix}1&3\\ 8&5\\ 3&3\end{bmatrix}
  24. x x
  25. x x
  26. x x
  27. x x
  28. max ( a , b ) = lim h 0 h log ( e a / h + e b / h ) . \max(a,b)=\lim_{h\to 0}h\log(e^{a/h}+e^{b/h}).

Additive_category.html

  1. X Y X Y X\coprod Y\to X\prod Y
  2. H o m ( X , Y ) Hom(X,Y)
  3. R R
  4. R R
  5. R R
  6. R R
  7. K K
  8. A A
  9. : A A A ∆:A→A⊕A
  10. k = 1 , 2 k=1, 2
  11. : A A A ∇:A⊕A→A
  12. k = 1 , 2 k=1, 2
  13. k = l k=l
  14. A Δ A A A α 1 α 2 α 3 B B B B A\ \xrightarrow{\quad\Delta\quad}\ A\oplus A\oplus A\ \xrightarrow{\alpha_{1}% \,\oplus\,\alpha_{2}\,\oplus\,\alpha_{3}}\ B\oplus B\oplus B\ \xrightarrow{% \quad\nabla\quad}\ B
  15. α + 0 = α α+0=α
  16. β = ( β β ) ∆∘β=(β⊕β)∘∆
  17. A B A⊕B
  18. A B C D A⊕B→C⊕D
  19. m m
  20. n n
  21. ( f 11 f 12 f 1 n f 21 f 22 f 2 n f m 1 f m 2 f m n ) \begin{pmatrix}f_{11}&f_{12}&\cdots&f_{1n}\\ f_{21}&f_{22}&\cdots&f_{2n}\\ \vdots&\vdots&\cdots&\vdots\\ f_{m1}&f_{m2}&\cdots&f_{mn}\end{pmatrix}
  22. f k l := p k f i l : A l B k . f_{kl}:=p_{k}\circ f\circ i_{l}\colon A_{l}\to B_{k}.
  23. < s u b > k i k p k = 1 ∑<sub>ki_{k} ∘p_{k}=1

Additive_synthesis.html

  1. f k f_{k}\,
  2. r k r_{k}\,
  3. y ( t ) = y ( t + P ) y(t)=y(t+P)
  4. t t\,
  5. P P\,
  6. y ( t ) \displaystyle y(t)
  7. f 0 = 1 / P f_{0}=1/P\,
  8. a k = r k cos ( ϕ k ) = 2 f 0 0 P y ( t ) cos ( 2 π k f 0 t ) d t , k 0 a_{k}=r_{k}\cos(\phi_{k})=2f_{0}\int_{0}^{P}y(t)\cos(2\pi kf_{0}t)\,dt,\quad k% \geq 0\,
  9. b k = r k sin ( ϕ k ) = - 2 f 0 0 P y ( t ) sin ( 2 π k f 0 t ) d t , k 1 b_{k}=r_{k}\sin(\phi_{k})=-2f_{0}\int_{0}^{P}y(t)\sin(2\pi kf_{0}t)\,dt,\quad k% \geq 1\,
  10. r k = a k 2 + b k 2 r_{k}=\sqrt{a_{k}^{2}+b_{k}^{2}}\,
  11. k k\,
  12. ϕ k = atan2 ( b k , a k ) \phi_{k}=\operatorname{atan2}(b_{k},a_{k})\,
  13. k k\,
  14. a 0 / 2 a_{0}/2\,
  15. K f 0 Kf_{0}\,
  16. y ( t ) y(t)\,
  17. r k r_{k}\,
  18. k f 0 kf_{0}\,
  19. ϕ k \phi_{k}\,
  20. k k\,
  21. K K\,
  22. f 0 f_{0}\,
  23. r k ( t ) r_{k}(t)\,
  24. r k ( t ) r_{k}(t)\,
  25. r k ( t ) r_{k}(t)\,
  26. f 0 f_{0}\,
  27. y ( t ) = k = 1 K r k ( t ) cos ( 2 π f k t + ϕ k ) , y(t)=\sum_{k=1}^{K}r_{k}(t)\cos\left(2\pi f_{k}t+\phi_{k}\right),
  28. f k f_{k}\,
  29. k k\,
  30. 2 π 2\pi\,
  31. f k ( t ) f_{k}(t)\,
  32. y ( t ) = k = 1 K r k ( t ) cos ( 2 π 0 t f k ( u ) d u + ϕ k ) . y(t)=\sum_{k=1}^{K}r_{k}(t)\cos\left(2\pi\int_{0}^{t}f_{k}(u)\ du+\phi_{k}% \right).
  33. y [ n ] y[n]\,
  34. n n\,
  35. y ( t ) y(t)\,
  36. f s / 2 f_{\mathrm{s}}/2\,
  37. T = 1 / f s T=1/f_{\mathrm{s}}\,
  38. y ( t ) = k = 1 K r k ( t ) cos ( 2 π 0 t f k ( u ) d u + ϕ k ) y(t)=\sum_{k=1}^{K}r_{k}(t)\cos\left(2\pi\int_{0}^{t}f_{k}(u)\ du+\phi_{k}\right)
  39. t = n T = n / f s t=nT=n/f_{\mathrm{s}}\,
  40. y [ n ] = y ( n T ) = k = 1 K r k ( n T ) cos ( 2 π 0 n T f k ( u ) d u + ϕ k ) = k = 1 K r k ( n T ) cos ( 2 π i = 1 n ( i - 1 ) T i T f k ( u ) d u + ϕ k ) = k = 1 K r k ( n T ) cos ( 2 π i = 1 n ( T f k [ i ] ) + ϕ k ) = k = 1 K r k [ n ] cos ( 2 π f s i = 1 n f k [ i ] + ϕ k ) \begin{aligned}\displaystyle y[n]&\displaystyle=y(nT)=\sum_{k=1}^{K}r_{k}(nT)% \cos\left(2\pi\int_{0}^{nT}f_{k}(u)\ du+\phi_{k}\right)\\ &\displaystyle=\sum_{k=1}^{K}r_{k}(nT)\cos\left(2\pi\sum_{i=1}^{n}\int_{(i-1)T% }^{iT}f_{k}(u)\ du+\phi_{k}\right)\\ &\displaystyle=\sum_{k=1}^{K}r_{k}(nT)\cos\left(2\pi\sum_{i=1}^{n}(Tf_{k}[i])+% \phi_{k}\right)\\ &\displaystyle=\sum_{k=1}^{K}r_{k}[n]\cos\left(\frac{2\pi}{f_{\mathrm{s}}}\sum% _{i=1}^{n}f_{k}[i]+\phi_{k}\right)\\ \end{aligned}
  41. r k [ n ] = r k ( n T ) r_{k}[n]=r_{k}(nT)\,
  42. f k [ n ] = 1 T ( n - 1 ) T n T f k ( t ) d t f_{k}[n]=\frac{1}{T}\int_{(n-1)T}^{nT}f_{k}(t)\ dt\,
  43. y [ n ] = k = 1 K r k [ n ] cos ( θ k [ n ] ) y[n]=\sum_{k=1}^{K}r_{k}[n]\cos\left(\theta_{k}[n]\right)
  44. θ k [ n ] = 2 π f s i = 1 n f k [ i ] + ϕ k = θ k [ n - 1 ] + 2 π f s f k [ n ] \begin{aligned}\displaystyle\theta_{k}[n]&\displaystyle=\frac{2\pi}{f_{\mathrm% {s}}}\sum_{i=1}^{n}f_{k}[i]+\phi_{k}\\ &\displaystyle=\theta_{k}[n-1]+\frac{2\pi}{f_{\mathrm{s}}}f_{k}[n]\\ \end{aligned}
  45. n > 0 n>0\,
  46. θ k [ 0 ] = ϕ k . \theta_{k}[0]=\phi_{k}.\,

Adenosine_triphosphate.html

  1. \to
  2. 1 2 NADH + cyt c ox + ADP + P i 1 2 NAD + + cyt c red + ATP \frac{1}{2}\mathrm{NADH}+\mathrm{cyt~{}c_{ox}}+\mathrm{ADP}+P_{i}\iff\frac{1}{% 2}\mathrm{NAD^{+}}+\mathrm{cyt~{}c_{red}}+\mathrm{ATP}
  3. cyt c red cyt c ox = ( [ NADH ] [ NAD ] + ) 1 2 ( [ ADP ] [ P i ] [ ATP ] ) K e q \frac{\mathrm{cyt~{}c_{red}}}{\mathrm{cyt~{}c_{ox}}}=\left(\frac{[\mathrm{NADH% }]}{[\mathrm{NAD}]^{+}}\right)^{\frac{1}{2}}\left(\frac{[\mathrm{ADP}][P_{i}]}% {[\mathrm{ATP}]}\right)K_{eq}

Adiabatic_process.html

  1. W = 0 W=0
  2. Q > 0 Q>0
  3. W = 0 W=0
  4. Q = 0 Q=0
  5. Q = 0 Q=0
  6. W 0 W≠0
  7. Δ S = 0 ΔS=0
  8. Q > 0 Q>0
  9. Δ S > 0 ΔS>0
  10. P d V PdV
  11. d V = 0 dV=0
  12. Δ S > 0 ΔS>0
  13. P V γ = constant PV^{\gamma}=\operatorname{constant}\qquad
  14. γ = C P C V = f + 2 f , \gamma={C_{P}\over C_{V}}=\frac{f+2}{f},
  15. C P C_{P}
  16. C V C_{V}
  17. γ \gamma
  18. f f
  19. γ = 5 / 3 \gamma=5/3\,
  20. γ = 7 / 5 \gamma=7/5\,
  21. P 1 - γ T γ = constant P^{1-\gamma}T^{\gamma}=\operatorname{constant}
  22. V T f / 2 = constant VT^{f/2}=\operatorname{constant}
  23. T V γ - 1 = constant TV^{\gamma-1}=\operatorname{constant}
  24. γ \gamma
  25. P V γ = constant = 100 , 000 pa × 1000 7 / 5 = 100 × 10 3 × 15.8 × 10 3 = 1.58 × 10 9 PV^{\gamma}=\operatorname{constant}=100,000\operatorname{pa}\times 1000^{7/5}=% 100\times 10^{3}\times 15.8\times 10^{3}=1.58\times 10^{9}
  26. P V γ = constant = 1.58 × 10 9 = P × 100 7 / 5 PV^{\gamma}=\operatorname{constant}=1.58\times 10^{9}=P\times 100^{7/5}
  27. P = 1.58 × 10 9 / 100 7 / 5 = 1.58 × 10 9 / 630.9 = 2.50 × 10 6 Pa P=1.58\times 10^{9}/{100^{7/5}}=1.58\times 10^{9}/630.9=2.50\times 10^{6}% \operatorname{Pa}
  28. P V T = constant = 10 5 × 10 3 300 = 3.33 × 10 5 {PV\over T}=\operatorname{constant}={{10^{5}\times 10^{3}}\over{300}}=3.33% \times 10^{5}
  29. P V constant = T = 2.50 × 10 6 × 100 3.33 × 10 5 = 751 {PV\over{\operatorname{constant}}}=T={{2.50\times 10^{6}\times 100}\over{3.33% \times 10^{5}}}=751
  30. δ Q = 0 \delta Q=0
  31. (1) d U + δ W = δ Q = 0 , \,\text{(1)}\qquad dU+\delta W=\delta Q=0,
  32. d U dU
  33. δ W \delta W
  34. δ W \delta W
  35. U U
  36. δ Q \delta Q
  37. δ W \delta W
  38. (2) δ W = P d V . \,\text{(2)}\qquad\delta W=P\,dV.
  39. P P
  40. V V
  41. d P dP
  42. d V dV
  43. (3) U = α n R T , \,\text{(3)}\qquad U=\alpha nRT,
  44. α {\alpha}
  45. R R
  46. n n
  47. P V = n R T PV=nRT
  48. (4) d U = α n R d T = α d ( P V ) = α ( P d V + V d P ) . \,\text{(4)}\qquad dU=\alpha nR\,dT=\alpha\,d(PV)=\alpha(P\,dV+V\,dP).
  49. d U = n C V d T dU=nC_{V}\,dT
  50. C V = α R C_{V}=\alpha R
  51. - P d V = α P d V + α V d P , -P\,dV=\alpha P\,dV+\alpha V\,dP,
  52. - P d V , -P\,dV,
  53. - ( α + 1 ) P d V = α V d P , -(\alpha+1)P\,dV=\alpha V\,dP,
  54. - ( α + 1 ) d V V = α d P P . -(\alpha+1){dV\over V}=\alpha{dP\over P}.
  55. V 0 V_{0}
  56. V V
  57. P 0 P_{0}
  58. P P
  59. ln ( P P 0 ) = - α + 1 α ln ( V V 0 ) . \ln\left({P\over P_{0}}\right)={-{\alpha+1\over\alpha}}\ln\left({V\over V_{0}}% \right).
  60. α + 1 α {\alpha+1\over\alpha}
  61. γ \gamma
  62. ( P P 0 ) = ( V V 0 ) - γ , \left({P\over P_{0}}\right)=\left({V\over V_{0}}\right)^{-{\gamma}},
  63. ( P P 0 ) = ( V 0 V ) γ . \left({P\over P_{0}}\right)=\left({V_{0}\over V}\right)^{\gamma}.
  64. ( P P 0 ) ( V V 0 ) γ = 1 \left({P\over P_{0}}\right)\left({V\over V_{0}}\right)^{\gamma}=1
  65. P 0 V 0 γ = P V γ = constant . P_{0}V_{0}^{\gamma}=PV^{\gamma}=\operatorname{constant}.
  66. P ( n R T / P ) γ = constant . P(nRT/P)^{\gamma}=\operatorname{constant}.
  67. P ( 1 - γ ) T γ = constant . P^{(1-\gamma)}T^{\gamma}=\operatorname{constant}.
  68. (1) Δ U = α R n T 2 - α R n T 1 = α R n Δ T \,\text{(1)}\qquad\Delta U=\alpha RnT_{2}-\alpha RnT_{1}=\alpha Rn\Delta T
  69. (2) W = V 1 V 2 P d V \,\text{(2)}\qquad W=\int_{V_{1}}^{V_{2}}P\,dV
  70. (3) Δ U + W = 0 \,\text{(3)}\qquad\Delta U+W=0
  71. (4) P V γ = constant = P 1 V 1 γ \,\text{(4)}\qquad PV^{\gamma}=\,\text{constant}=P_{1}V_{1}^{\gamma}
  72. P = P 1 ( V 1 V ) γ P=P_{1}\left(\frac{V_{1}}{V}\right)^{\gamma}
  73. W = V 1 V 2 P 1 ( V 1 V ) γ d V W=\int_{V_{1}}^{V_{2}}P_{1}\left(\frac{V_{1}}{V}\right)^{\gamma}\,dV
  74. W = P 1 V 1 γ V 2 1 - γ - V 1 1 - γ 1 - γ W=P_{1}V_{1}^{\gamma}\frac{V_{2}^{1-\gamma}-V_{1}^{1-\gamma}}{1-\gamma}
  75. γ = α + 1 α \gamma=\frac{\alpha+1}{\alpha}
  76. W = - α P 1 V 1 γ ( V 2 1 - γ - V 1 1 - γ ) W=-\alpha P_{1}V_{1}^{\gamma}\left(V_{2}^{1-\gamma}-V_{1}^{1-\gamma}\right)
  77. W = - α P 1 V 1 ( ( V 2 V 1 ) 1 - γ - 1 ) W=-\alpha P_{1}V_{1}\left(\left(\frac{V_{2}}{V_{1}}\right)^{1-\gamma}-1\right)
  78. W = - α n R T 1 ( ( V 2 V 1 ) 1 - γ - 1 ) W=-\alpha nRT_{1}\left(\left(\frac{V_{2}}{V_{1}}\right)^{1-\gamma}-1\right)
  79. P 2 P 1 = ( V 2 V 1 ) - γ \frac{P_{2}}{P_{1}}=\left(\frac{V_{2}}{V_{1}}\right)^{-\gamma}
  80. ( P 2 P 1 ) - 1 γ = V 2 V 1 \left(\frac{P_{2}}{P_{1}}\right)^{-1\over\gamma}=\frac{V_{2}}{V_{1}}
  81. W W
  82. W = - α n R T 1 ( ( P 2 P 1 ) γ - 1 γ - 1 ) W=-\alpha nRT_{1}\left(\left(\frac{P_{2}}{P_{1}}\right)^{\frac{\gamma-1}{% \gamma}}-1\right)
  83. α n R ( T 2 - T 1 ) = α n R T 1 ( ( P 2 P 1 ) γ - 1 γ - 1 ) \alpha nR(T_{2}-T_{1})=\alpha nRT_{1}\left(\left(\frac{P_{2}}{P_{1}}\right)^{% \frac{\gamma-1}{\gamma}}-1\right)
  84. T 2 - T 1 = T 1 ( ( P 2 P 1 ) γ - 1 γ - 1 ) T_{2}-T_{1}=T_{1}\left(\left(\frac{P_{2}}{P_{1}}\right)^{\frac{\gamma-1}{% \gamma}}-1\right)
  85. T 2 T 1 - 1 = ( P 2 P 1 ) γ - 1 γ - 1 \frac{T_{2}}{T_{1}}-1=\left(\frac{P_{2}}{P_{1}}\right)^{\frac{\gamma-1}{\gamma% }}-1
  86. T 2 = T 1 ( P 2 P 1 ) γ - 1 γ T_{2}=T_{1}\left(\frac{P_{2}}{P_{1}}\right)^{\frac{\gamma-1}{\gamma}}
  87. c 1 2 = 1 , c 2 2 = 0 , c 3 2 = 0 , , c_{1}^{2}=1,\,\,c_{2}^{2}=0,\,\,c_{3}^{2}=0,\,...\,,

Adjoint_functors.html

  1. F : 𝒟 𝒞 F:\mathcal{D}\rightarrow\mathcal{C}
  2. G : 𝒞 𝒟 G:\mathcal{C}\rightarrow\mathcal{D}
  3. hom 𝒞 ( F Y , X ) hom 𝒟 ( Y , G X ) \mathrm{hom}_{\mathcal{C}}(FY,X)\cong\mathrm{hom}_{\mathcal{D}}(Y,GX)
  4. F G . F\dashv G.
  5. φ : hom 𝒞 ( F Y , X ) hom 𝒟 ( Y , G X ) \varphi:\mathrm{hom}_{\mathcal{C}}(FY,X)\cong\mathrm{hom}_{\mathcal{D}}(Y,GX)
  6. φ \varphi
  7. f f
  8. hom 𝒞 ( F Y , X ) \mathrm{hom}_{\mathcal{C}}(FY,X)
  9. φ f \varphi f
  10. f f
  11. F F
  12. G G
  13. ε : F G 1 𝒞 η : 1 𝒟 G F \begin{aligned}\displaystyle\varepsilon&\displaystyle:FG\to 1_{\mathcal{C}}\\ \displaystyle\eta&\displaystyle:1_{\mathcal{D}}\to GF\end{aligned}
  14. F F η F G F ε F F F\xrightarrow{\;F\eta\;}FGF\xrightarrow{\;\varepsilon F\,}F
  15. G η G G F G G ε G G\xrightarrow{\;\eta G\;}GFG\xrightarrow{\;G\varepsilon\,}G
  16. ( ε , η ) : F G (\varepsilon,\eta):F\dashv G
  17. F G F\dashv G
  18. 1 F = ε F F η 1 G = G ε η G \begin{aligned}\displaystyle 1_{F}&\displaystyle=\varepsilon F\circ F\eta\\ \displaystyle 1_{G}&\displaystyle=G\varepsilon\circ\eta G\end{aligned}
  19. 1 F Y = ε F Y F ( η Y ) 1 G X = G ( ε X ) η G X \begin{aligned}\displaystyle 1_{FY}&\displaystyle=\varepsilon_{FY}\circ F(\eta% _{Y})\\ \displaystyle 1_{GX}&\displaystyle=G(\varepsilon_{X})\circ\eta_{GX}\end{aligned}
  20. 1 1
  21. 1 = ε η 1=\varepsilon\circ\eta
  22. Φ : hom C ( F - , - ) hom D ( - , G - ) \Phi:\mathrm{hom}_{C}(F-,-)\to\mathrm{hom}_{D}(-,G-)
  23. Φ Y , X : hom C ( F Y , X ) hom D ( Y , G X ) \Phi_{Y,X}:\mathrm{hom}_{C}(FY,X)\to\mathrm{hom}_{D}(Y,GX)
  24. Φ : F G \Phi:F\dashv G
  25. F G F\dashv G
  26. f = Φ Y , X - 1 ( g ) = ε X F ( g ) hom C ( F ( Y ) , X ) g = Φ Y , X ( f ) = G ( f ) η Y hom D ( Y , G ( X ) ) Φ G X , X - 1 ( 1 G X ) = ε X hom C ( F G ( X ) , X ) Φ Y , F Y ( 1 F Y ) = η Y hom D ( Y , G F ( Y ) ) \begin{aligned}\displaystyle f=\Phi_{Y,X}^{-1}(g)&\displaystyle=\varepsilon_{X% }\circ F(g)&\displaystyle\in&\displaystyle\,\,\mathrm{hom}_{C}(F(Y),X)\\ \displaystyle g=\Phi_{Y,X}(f)&\displaystyle=G(f)\circ\eta_{Y}&\displaystyle\in% &\displaystyle\,\,\mathrm{hom}_{D}(Y,G(X))\\ \displaystyle\Phi_{GX,X}^{-1}(1_{GX})&\displaystyle=\varepsilon_{X}&% \displaystyle\in&\displaystyle\,\,\mathrm{hom}_{C}(FG(X),X)\\ \displaystyle\Phi_{Y,FY}(1_{FY})&\displaystyle=\eta_{Y}&\displaystyle\in&% \displaystyle\,\,\mathrm{hom}_{D}(Y,GF(Y))\\ \end{aligned}
  27. 1 F Y = ε F Y F ( η Y ) 1 G X = G ( ε X ) η G X \begin{aligned}\displaystyle 1_{FY}&\displaystyle=\varepsilon_{FY}\circ F(\eta% _{Y})\\ \displaystyle 1_{GX}&\displaystyle=G(\varepsilon_{X})\circ\eta_{GX}\end{aligned}
  28. \dashv
  29. Φ Y , X ( f ) = G ( f ) η Y Ψ Y , X ( g ) = ε X F ( g ) \begin{aligned}\displaystyle\Phi_{Y,X}(f)=G(f)\circ\eta_{Y}\\ \displaystyle\Psi_{Y,X}(g)=\varepsilon_{X}\circ F(g)\end{aligned}
  30. Ψ Φ f = ε X F G ( f ) F ( η Y ) = f ε F Y F ( η Y ) = f 1 F Y = f \begin{aligned}\displaystyle\Psi\Phi f&\displaystyle=\varepsilon_{X}\circ FG(f% )\circ F(\eta_{Y})\\ &\displaystyle=f\circ\varepsilon_{FY}\circ F(\eta_{Y})\\ &\displaystyle=f\circ 1_{FY}=f\end{aligned}
  31. Φ Ψ g = G ( ε X ) G F ( g ) η Y = G ( ε X ) η G X g = 1 G X g = g \begin{aligned}\displaystyle\Phi\Psi g&\displaystyle=G(\varepsilon_{X})\circ GF% (g)\circ\eta_{Y}\\ &\displaystyle=G(\varepsilon_{X})\circ\eta_{GX}\circ g\\ &\displaystyle=1_{GX}\circ g=g\end{aligned}
  32. ( ε , η ) : F G (\varepsilon,\eta):F\dashv G
  33. ε X = Φ G X , X - 1 ( 1 G X ) hom C ( F G X , X ) \varepsilon_{X}=\Phi_{GX,X}^{-1}(1_{GX})\in\mathrm{hom}_{C}(FGX,X)
  34. 1 G X hom D ( G X , G X ) 1_{GX}\in\mathrm{hom}_{D}(GX,GX)
  35. η Y = Φ Y , F Y ( 1 F Y ) hom D ( Y , G F Y ) \eta_{Y}=\Phi_{Y,FY}(1_{FY})\in\mathrm{hom}_{D}(Y,GFY)
  36. 1 F Y hom C ( F Y , F Y ) 1_{FY}\in\mathrm{hom}_{C}(FY,FY)
  37. Φ Y , X ( f ) = G ( f ) η Y Φ Y , X - 1 ( g ) = ε X F ( g ) \begin{aligned}\displaystyle\Phi_{Y,X}(f)=G(f)\circ\eta_{Y}\\ \displaystyle\Phi_{Y,X}^{-1}(g)=\varepsilon_{X}\circ F(g)\end{aligned}
  38. 1 F Y = ε F Y F ( η Y ) 1_{FY}=\varepsilon_{FY}\circ F(\eta_{Y})
  39. 1 G X = G ( ε X ) η G X 1_{GX}=G(\varepsilon_{X})\circ\eta_{GX}
  40. - A -\otimes A
  41. T x , y = x , U y \langle Tx,y\rangle=\langle x,Uy\rangle
  42. ε X : F G X X \varepsilon_{X}:FGX\to X
  43. ( G X , ε X ) (GX,\varepsilon_{X})
  44. ε X : F G X X \varepsilon_{X}:FGX\to X
  45. η Y : Y G F Y \eta_{Y}:Y\to GFY
  46. ( F Y , η Y ) (FY,\eta_{Y})
  47. η Y : Y G F Y \eta_{Y}:Y\to GFY
  48. ( ε , η ) : F G (\varepsilon,\eta):F\dashv G
  49. 1 F = ε F F η 1_{F}=\varepsilon F\circ F\eta
  50. F Y F ( η Y ) F G F Y ε F Y F Y FY\xrightarrow{\;F(\eta_{Y})\;}FGFY\xrightarrow{\;\varepsilon_{FY}\,}FY
  51. F ( η Y ) F(\eta_{Y})
  52. ε F Y \varepsilon_{FY}
  53. 1 G = G ε η G 1_{G}=G\varepsilon\circ\eta G
  54. G X η G X G F G X G ( ε X ) G X GX\xrightarrow{\;\eta_{GX}\;}GFGX\xrightarrow{\;G(\varepsilon_{X})\,}GX
  55. η G X \eta_{GX}
  56. G ( ε X ) G(\varepsilon_{X})
  57. a A a\in A
  58. ϕ Y \phi_{Y}
  59. Y = { y ϕ Y ( y ) } Y=\{y\mid\phi_{Y}(y)\}
  60. T Y T\subset Y
  61. T T
  62. Y Y
  63. ϕ T ( y ) = ϕ Y ( y ) φ ( y ) \phi_{T}(y)=\phi_{Y}(y)\land\varphi(y)
  64. ψ f \psi_{f}
  65. X X
  66. Y Y
  67. X X
  68. { y Y x . ψ f ( x , y ) ϕ S ( x ) } \{y\in Y\mid\exists x.\,\psi_{f}(x,y)\land\phi_{S}(x)\}
  69. y y
  70. Y Y
  71. x x
  72. ψ f \psi_{f}
  73. ϕ S \phi_{S}
  74. \cap
  75. \land
  76. Y Y
  77. f : X Y f:X\to Y
  78. f * : Sub ( Y ) Sub ( X ) f^{*}:\,\text{Sub}(Y)\longrightarrow\,\text{Sub}(X)
  79. T T
  80. Y Y
  81. T Y T\to Y
  82. X × Y T X\times_{Y}T
  83. f \exists_{f}
  84. f \forall_{f}
  85. Sub ( X ) \,\text{Sub}(X)
  86. Sub ( Y ) \,\text{Sub}(Y)
  87. S X S\subset X
  88. f f
  89. X X
  90. X × Y T X\times_{Y}T
  91. Y Y
  92. Set \operatorname{Set}
  93. f * T = X × Y T f^{*}T=X\times_{Y}T
  94. T T
  95. Y Y
  96. f f
  97. f f
  98. T T
  99. Y Y
  100. f - 1 [ T ] X f^{-1}[T]\subseteq X
  101. S X S\subseteq X
  102. Hom ( f S , T ) Hom ( S , f * T ) , {\operatorname{Hom}}(\exists_{f}S,T)\cong{\operatorname{Hom}}(S,f^{*}T),
  103. f S T S f - 1 [ T ] \exists_{f}S\subseteq T\leftrightarrow S\subseteq f^{-1}[T]
  104. f [ S ] T f[S]\subseteq T
  105. S f - 1 [ f [ S ] ] f - 1 [ T ] S\subseteq f^{-1}[f[S]]\subseteq f^{-1}[T]
  106. x S x\in S
  107. x f - 1 [ T ] x\in f^{-1}[T]
  108. f ( x ) T f(x)\in T
  109. S f - 1 [ T ] S\subseteq f^{-1}[T]
  110. f [ S ] T f[S]\subseteq T
  111. f * f^{*}
  112. S S
  113. f \exists_{f}
  114. y y
  115. f - 1 [ { y } ] S f^{-1}[\{y\}]\cap S
  116. y Y y\in Y
  117. f [ S ] f[S]
  118. f S = { y Y ( x f - 1 [ { y } ] ) . x S } = f [ S ] . \exists_{f}S=\{y\in Y\mid\exists(x\in f^{-1}[\{y\}]).\,x\in S\;\}=f[S].
  119. { y Y x . ψ f ( x , y ) ϕ S ( x ) } \{y\in Y\mid\exists x.\,\psi_{f}(x,y)\land\phi_{S}(x)\}
  120. f S = { y Y ( x f - 1 [ { y } ] ) . x S } . \forall_{f}S=\{y\in Y\mid\forall(x\in f^{-1}[\{y\}]).\,x\in S\;\}.
  121. f S \forall_{f}S
  122. Y Y
  123. y y
  124. { y } \{y\}
  125. f f
  126. S S
  127. \exists
  128. \forall
  129. η = ( τ σ ) η ε = ε ( σ - 1 τ - 1 ) . \begin{aligned}\displaystyle\eta^{\prime}&\displaystyle=(\tau\ast\sigma)\circ% \eta\\ \displaystyle\varepsilon^{\prime}&\displaystyle=\varepsilon\circ(\sigma^{-1}% \ast\tau^{-1}).\end{aligned}
  130. \circ
  131. \ast
  132. F F : 𝒞 F^{\prime}\circ F:\mathcal{C}\leftarrow\mathcal{E}
  133. G G : 𝒞 . G\circ G^{\prime}:\mathcal{C}\to\mathcal{E}.
  134. 1 𝜂 G F G η F G G F F \displaystyle 1_{\mathcal{E}}\xrightarrow{\eta}GF\xrightarrow{G\eta^{\prime}F}% GG^{\prime}F^{\prime}F
  135. Φ Y , X : hom 𝒞 ( F Y , X ) hom 𝒟 ( Y , G X ) \Phi_{Y,X}:\mathrm{hom}_{\mathcal{C}}(FY,X)\cong\mathrm{hom}_{\mathcal{D}}(Y,GX)
  136. T : 𝒟 𝒟 T:\mathcal{D}\to\mathcal{D}
  137. η : 1 𝒟 T \eta:1_{\mathcal{D}}\to T
  138. μ : T 2 T \mu:T^{2}\to T\,

Advanced_Encryption_Standard.html

  1. a i , j a_{i,j}
  2. S ( a i , j ) S(a_{i,j})
  3. S ( a i , j ) a i , j S(a_{i,j})\neq a_{i,j}
  4. S ( a i , j ) a i , j 0xFF S(a_{i,j})\oplus a_{i,j}\neq\,\text{0xFF}
  5. [ 2 3 1 1 1 2 3 1 1 1 2 3 3 1 1 2 ] \begin{bmatrix}2&3&1&1\\ 1&2&3&1\\ 1&1&2&3\\ 3&1&1&2\end{bmatrix}

Aerosol.html

  1. d f = f ( d p ) d d p \mathrm{d}f=f(d_{p})\,\mathrm{d}d_{p}
  2. d p d_{p}
  3. d f \,\mathrm{d}f
  4. d p d_{p}
  5. d p d_{p}
  6. d d p \mathrm{d}d_{p}
  7. f ( d p ) f(d_{p})
  8. f a b = a b f ( d p ) d d p f_{ab}=\int_{a}^{b}f(d_{p})\,\mathrm{d}d_{p}
  9. d N = N ( d p ) d d p dN=N(d_{p})\,\mathrm{d}d_{p}
  10. S = π / 2 0 N ( d p ) d p 2 d d p S=\pi/2\int_{0}^{\infty}N(d_{p})d_{p}^{2}\,\mathrm{d}d_{p}
  11. V = π / 6 0 N ( d p ) d p 3 d d p V=\pi/6\int_{0}^{\infty}N(d_{p})d_{p}^{3}\,\mathrm{d}d_{p}
  12. d f = 1 σ 2 π e - ( d p - d p ¯ ) 2 2 σ 2 d d p \mathrm{d}f=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(d_{p}-\bar{d_{p}})^{2}}{2% \sigma^{2}}}\mathrm{d}d_{p}
  13. σ \sigma
  14. d p ¯ \bar{d_{p}}
  15. F D = 3 π η V d C c F_{D}=\frac{3\pi\eta Vd}{C_{c}}
  16. F D F_{D}
  17. η \eta
  18. V V
  19. C c C_{c}
  20. V T S = ρ p d 2 g C c 18 η V_{TS}=\frac{\rho_{p}d^{2}gC_{c}}{18\eta}
  21. V T S V_{TS}
  22. B = V F D = C c 3 π η d B=\frac{V}{F_{D}}=\frac{C_{c}}{3\pi\eta d}
  23. V ( t ) = V f - ( V f - V 0 ) e - t τ V(t)=V_{f}-(V_{f}-V_{0})e^{-\frac{t}{\tau}}
  24. V ( t ) V(t)
  25. V f V_{f}
  26. V 0 V_{0}
  27. χ = F D 3 π η V d e \chi=\frac{F_{D}}{3\pi\eta Vd_{e}}
  28. χ \chi
  29. V T S = ρ 0 d a 2 g 18 η V_{TS}=\frac{\rho_{0}d_{a}^{2}g}{18\eta}
  30. ρ 0 \ \rho_{0}
  31. d a = d e ( ρ p ρ 0 χ ) 1 2 d_{a}=d_{e}\left(\frac{\rho_{p}}{\rho_{0}\chi}\right)^{\frac{1}{2}}
  32. n i t = - n i 𝐪 + D p i + ( n i t ) g r o w t h + ( n i t ) c o a g - 𝐪 F n i \frac{\partial{n_{i}}}{\partial{t}}=-\nabla\cdot n_{i}\mathbf{q}+\nabla\cdot D% _{p}\nabla_{i}+\left(\frac{\partial{n_{i}}}{\partial{t}}\right)_{growth}+\left% (\frac{\partial{n_{i}}}{\partial{t}}\right)_{coag}-\nabla\cdot\mathbf{q}_{F}n_% {i}
  33. n i n_{i}
  34. i i
  35. 𝐪 \mathbf{q}
  36. D p D_{p}
  37. 𝐪 F \mathbf{q}_{F}
  38. K n = 2 λ d K_{n}=\frac{2\lambda}{d}
  39. λ \lambda
  40. d d
  41. I = π a 2 k b ( P T - P A T A ) C A α I=\frac{\pi a^{2}}{k_{b}}\left(\frac{P_{\infty}}{T_{\infty}}-\frac{P_{A}}{T_{A% }}\right)\cdot C_{A}\alpha
  42. I c o n t 4 π a M A D A B R T ( P A - P A S ) I_{cont}\sim\frac{4\pi aM_{A}D_{AB}}{RT}\left(P_{A\infty}-P_{AS}\right)
  43. I = I c o n t 1 + K n 1 + 1.71 K n + 1.33 K n 2 I=I_{cont}\cdot\frac{1+K_{n}}{1+1.71K_{n}+1.33{K_{n}}^{2}}
  44. R H = p s p 0 × 100 % = S × 100 % RH=\frac{p_{s}}{p_{0}}\times 100\%=S\times 100\%
  45. p s p_{s}
  46. ln p s p 0 = 2 σ M R T ρ r p \ln{p_{s}\over p_{0}}=\frac{2\sigma M}{RT\rho\cdot r_{p}}

Affine_transformation.html

  1. X X
  2. Y Y
  3. f : X Y f:X\to Y
  4. x M x + b x\mapsto Mx+b
  5. M M
  6. X X
  7. b b
  8. Y Y
  9. f : 𝒜 f:\mathcal{A}\to\mathcal{B}
  10. f f
  11. φ \varphi
  12. P , Q 𝒜 P,Q\in\mathcal{A}
  13. f ( P ) f ( Q ) = φ ( P Q ) \overrightarrow{f(P)~{}f(Q)}=\varphi(\overrightarrow{PQ})
  14. f ( Q ) - f ( P ) = φ ( Q - P ) f(Q)-f(P)=\varphi(Q-P)
  15. O 𝒜 O\in\mathcal{A}
  16. B B
  17. f ( O ) f(O)\in\mathcal{B}
  18. x \vec{x}
  19. f : ( O + x ) ( B + φ ( x ) ) . f:(O+\vec{x})\mapsto(B+\varphi(\vec{x})).
  20. O O^{\prime}\in\mathcal{B}
  21. g : 𝒜 g:\mathcal{A}\to\mathcal{B}
  22. O O O\mapsto O^{\prime}
  23. g : ( O + x ) ( O + φ ( x ) ) , g:(O+\vec{x})\mapsto(O^{\prime}+\varphi(\vec{x})),
  24. b = O B \vec{b}=\overrightarrow{O^{\prime}B}
  25. f f
  26. 𝒜 \mathcal{A}
  27. \mathcal{B}
  28. f : 𝒜 f:\mathcal{A}\to\mathcal{B}
  29. { ( a i , λ i ) } i I \{(a_{i},\lambda_{i})\}_{i\in I}
  30. 𝒜 \mathcal{A}
  31. i I λ i = 1 , \sum_{i\in I}\lambda_{i}=1,
  32. f ( i I λ i a i ) = i I λ i f ( a i ) . f\left(\sum_{i\in I}\lambda_{i}a_{i}\right)=\sum_{i\in I}\lambda_{i}f(a_{i})\,.
  33. f f
  34. b \vec{b}
  35. f f
  36. x \vec{x}
  37. y = f ( x ) = A x + b . \vec{y}=f(\vec{x})=A\vec{x}+\vec{b}.
  38. [ y 1 ] = [ A b 0 0 1 ] [ x 1 ] \begin{bmatrix}\vec{y}\\ 1\end{bmatrix}=\left[\begin{array}[]{ccc|c}&A&&\vec{b}\\ 0&\ldots&0&1\end{array}\right]\begin{bmatrix}\vec{x}\\ 1\end{bmatrix}
  39. y = A x + b . \vec{y}=A\vec{x}+\vec{b}.
  40. x 1 , , x n + 1 \vec{x}_{1},\ldots,\vec{x}_{n+1}
  41. y 1 , , y n + 1 \vec{y}_{1},\ldots,\vec{y}_{n+1}
  42. [ y 1 ] = M [ x 1 ] \left[\begin{array}[]{c}\vec{y}\\ 1\end{array}\right]=M\left[\begin{array}[]{c}\vec{x}\\ 1\end{array}\right]
  43. M = [ y 1 y n + 1 1 1 ] [ x 1 x n + 1 1 1 ] - 1 M=\left[\begin{array}[]{ccc}\vec{y}_{1}&\ldots&\vec{y}_{n+1}\\ 1&\ldots&1\end{array}\right]\left[\begin{array}[]{ccc}\vec{x}_{1}&\ldots&\vec{% x}_{n+1}\\ 1&\ldots&1\end{array}\right]^{-1}
  44. p 1 , p 2 , p 3 , p_{1},\,p_{2},\,p_{3},
  45. p 1 p 2 \overrightarrow{p_{1}p_{2}}
  46. p 2 p 3 \overrightarrow{p_{2}p_{3}}
  47. f ( p 1 ) f ( p 2 ) \overrightarrow{f(p_{1})f(p_{2})}
  48. f ( p 2 ) f ( p 3 ) \overrightarrow{f(p_{2})f(p_{3})}
  49. [ A - 1 - A - 1 b 0 0 1 ] \left[\begin{array}[]{ccc|c}&A^{-1}&&-A^{-1}\vec{b}\\ 0&\ldots&0&1\end{array}\right]
  50. { a } = M { a } { v } , \{\,a^{\prime}\,\}=M\{\,a\,\}\oplus\{\,v\,\},
  51. M { a } = [ 1 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 ] M\{\,a\,\}=\begin{bmatrix}1&0&0&0&1&1&1&1\\ 1&1&0&0&0&1&1&1\\ 1&1&1&0&0&0&1&1\\ 1&1&1&1&0&0&0&1\\ 1&1&1&1&1&0&0&0\\ 0&1&1&1&1&1&0&0\\ 0&0&1&1&1&1&1&0\\ 0&0&0&1&1&1&1&1\end{bmatrix}
  52. { v } = [ 1 1 0 0 0 1 1 0 ] . \{\,v\,\}=\begin{bmatrix}1\\ 1\\ 0\\ 0\\ 0\\ 1\\ 1\\ 0\end{bmatrix}.
  53. a 0 = a 0 a 4 a 5 a 6 a 7 1 = 0 0 0 1 1 1 = 1 a_{0}^{\prime}=a_{0}\oplus a_{4}\oplus a_{5}\oplus a_{6}\oplus a_{7}\oplus 1=0% \oplus 0\oplus 0\oplus 1\oplus 1\oplus 1=1
  54. a 1 = a 0 a 1 a 5 a 6 a 7 1 = 0 1 0 1 1 1 = 0 a_{1}^{\prime}=a_{0}\oplus a_{1}\oplus a_{5}\oplus a_{6}\oplus a_{7}\oplus 1=0% \oplus 1\oplus 0\oplus 1\oplus 1\oplus 1=0
  55. a 2 = a 0 a 1 a 2 a 6 a 7 0 = 0 1 0 1 1 0 = 1 a_{2}^{\prime}=a_{0}\oplus a_{1}\oplus a_{2}\oplus a_{6}\oplus a_{7}\oplus 0=0% \oplus 1\oplus 0\oplus 1\oplus 1\oplus 0=1
  56. a 3 = a 0 a 1 a 2 a 3 a 7 0 = 0 1 0 1 1 0 = 1 a_{3}^{\prime}=a_{0}\oplus a_{1}\oplus a_{2}\oplus a_{3}\oplus a_{7}\oplus 0=0% \oplus 1\oplus 0\oplus 1\oplus 1\oplus 0=1
  57. a 4 = a 0 a 1 a 2 a 3 a 4 0 = 0 1 0 1 0 0 = 0 a_{4}^{\prime}=a_{0}\oplus a_{1}\oplus a_{2}\oplus a_{3}\oplus a_{4}\oplus 0=0% \oplus 1\oplus 0\oplus 1\oplus 0\oplus 0=0
  58. a 5 = a 1 a 2 a 3 a 4 a 5 1 = 1 0 1 0 0 1 = 1 a_{5}^{\prime}=a_{1}\oplus a_{2}\oplus a_{3}\oplus a_{4}\oplus a_{5}\oplus 1=1% \oplus 0\oplus 1\oplus 0\oplus 0\oplus 1=1
  59. a 6 = a 2 a 3 a 4 a 5 a 6 1 = 0 1 0 0 1 1 = 1 a_{6}^{\prime}=a_{2}\oplus a_{3}\oplus a_{4}\oplus a_{5}\oplus a_{6}\oplus 1=0% \oplus 1\oplus 0\oplus 0\oplus 1\oplus 1=1
  60. a 7 = a 3 a 4 a 5 a 6 a 7 0 = 1 0 0 1 1 0 = 1. a_{7}^{\prime}=a_{3}\oplus a_{4}\oplus a_{5}\oplus a_{6}\oplus a_{7}\oplus 0=1% \oplus 0\oplus 0\oplus 1\oplus 1\oplus 0=1.
  61. [ x y ] [ 0 1 2 1 ] [ x y ] + [ - 100 - 100 ] \begin{bmatrix}x\\ y\end{bmatrix}\mapsto\begin{bmatrix}0&1\\ 2&1\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix}+\begin{bmatrix}-100\\ -100\end{bmatrix}

Affirming_the_consequent.html

  1. ( P Q ) ( ¬ Q ¬ P ) (P\to Q)\leftrightarrow(\neg Q\to\neg P)

AI-complete.html

  1. Φ H , Φ M \langle\Phi_{H},\Phi_{M}\rangle
  2. O ( 1 ) , p o l y ( n ) \langle O(1),poly(n)\rangle
  3. n n
  4. O ( n ) , O ( n ) \langle O(n),O(n)\rangle
  5. n n
  6. O ( n ) , O ( n 2 ) \langle O(n),O(n^{2})\rangle
  7. n n
  8. O ( n 2 ) , O ( n 2 ) \langle O(n^{2}),O(n^{2})\rangle
  9. O ( n ) , O ( n ) \langle O(n),O(n)\rangle
  10. O ( n ) , O ( n ) \langle O(n),O(n)\rangle
  11. O ( n ) , O ( n ) \langle O(n),O(n)\rangle
  12. O ( log n ) , O ( n log n ) \langle O(\log n),O(n\log n)\rangle

Albedo.html

  1. α ¯ ( θ i ) {\bar{\alpha}(\theta_{i})}
  2. α ¯ ¯ \bar{\bar{\alpha}}
  3. D {D}
  4. α {\alpha}
  5. α = ( 1 - D ) α ¯ ( θ i ) + D α ¯ ¯ . {\alpha}=(1-D)\bar{\alpha}(\theta_{i})+D\bar{\bar{\alpha}}.
  6. A = ( 1329 × 10 - H / 5 D ) 2 A=\left(\frac{1329\times 10^{-H/5}}{D}\right)^{2}
  7. A A
  8. D D
  9. H H

Alcubierre_drive.html

  1. d s 2 = - ( α 2 - β i β i ) d t 2 + 2 β i d x i d t + γ i j d x i d x j ds^{2}=-\left(\alpha^{2}-\beta_{i}\beta^{i}\right)\,dt^{2}+2\beta_{i}\,dx^{i}% \,dt+\gamma_{ij}\,dx^{i}\,dx^{j}
  2. α \alpha
  3. β i \beta^{i}
  4. γ i j \gamma_{ij}
  5. α = 1 \alpha=1\,
  6. β x = - v s ( t ) f ( r s ( t ) ) \beta^{x}=-v_{s}(t)f\left(r_{s}(t)\right)
  7. β y = β z = 0 \beta^{y}=\beta^{z}=0\,\!
  8. γ i j = δ i j \gamma_{ij}=\delta_{ij}\,\!
  9. v s ( t ) = d x s ( t ) d t , v_{s}(t)=\frac{dx_{s}(t)}{dt},
  10. r s ( t ) = ( x - x s ( t ) ) 2 + y 2 + z 2 , r_{s}(t)=\sqrt{(x-x_{s}(t))^{2}+y^{2}+z^{2}},
  11. f ( r s ) = tanh ( σ ( r s + R ) ) - tanh ( σ ( r s - R ) ) 2 tanh ( σ R ) , f(r_{s})=\frac{\tanh(\sigma(r_{s}+R))-\tanh(\sigma(r_{s}-R))}{2\tanh(\sigma R)},
  12. R > 0 R>0
  13. σ > 0 \sigma>0
  14. d s 2 = ( v s ( t ) 2 f ( r s ( t ) ) 2 - 1 ) d t 2 - 2 v s ( t ) f ( r s ( t ) ) d x d t + d x 2 + d y 2 + d z 2 . ds^{2}=\left(v_{s}(t)^{2}f(r_{s}(t))^{2}-1\right)\,dt^{2}-2v_{s}(t)f(r_{s}(t))% \,dx\,dt+dx^{2}+dy^{2}+dz^{2}.
  15. - c 4 8 π G v s 2 ( y 2 + z 2 ) 4 g 2 r s 2 ( d f d r s ) 2 , -\frac{c^{4}}{8\pi G}\frac{v_{s}^{2}(y^{2}+z^{2})}{4g^{2}r_{s}^{2}}\left(\frac% {df}{dr_{s}}\right)^{2},
  16. g g\!

Alexandroff_extension.html

  1. S - 1 : 2 S 2 S^{-1}:\mathbb{R}^{2}\hookrightarrow S^{2}
  2. = ( 0 , 0 , 1 ) \infty=(0,0,1)
  3. z = c z=c
  4. r = ( 1 + c ) / ( 1 - c ) r=\sqrt{(1+c)/(1-c)}
  5. ( 1 , 0 , 0 ) (1,0,0)
  6. c z < 1 c\leq z<1
  7. r ( 1 + c ) / ( 1 - c ) r\geq\sqrt{(1+c)/(1-c)}
  8. \infty
  9. S - 1 ( 2 K ) { } S^{-1}(\mathbb{R}^{2}\setminus K)\cup\{\infty\}
  10. 2 \mathbb{R}^{2}
  11. c : X Y c:X\hookrightarrow Y
  12. { } = Y c ( X ) \{\infty\}=Y\setminus c(X)
  13. \infty
  14. \infty
  15. \infty
  16. X * = X { } X^{*}=X\cup\{\infty\}
  17. X * X^{*}
  18. \infty
  19. X V X\setminus V
  20. c : X X * c:X\rightarrow X^{*}
  21. X * X^{*}
  22. X * X^{*}
  23. X * X^{*}
  24. X * X^{*}
  25. c : X X * c:X\rightarrow X^{*}
  26. 𝒞 ( X ) \mathcal{C}(X)
  27. κ \kappa
  28. κ \kappa
  29. c : X Y c:X\rightarrow Y
  30. c 1 : X 1 Y 1 c_{1}:X_{1}\rightarrow Y_{1}
  31. c 2 : X 2 Y 2 c_{2}:X_{2}\rightarrow Y_{2}
  32. f X : X 1 X 2 , f Y : Y 1 Y 2 f_{X}:X_{1}\rightarrow X_{2},\ f_{Y}:Y_{1}\rightarrow Y_{2}
  33. f Y c 1 = c 2 f X f_{Y}\circ c_{1}=c_{2}\circ f_{X}
  34. { a n } \{a_{n}\}
  35. X X
  36. a a
  37. X X
  38. f : * X f\colon\mathbb{N}^{*}\to X
  39. f ( n ) = a n f(n)=a_{n}
  40. n n
  41. \mathbb{N}
  42. f ( ) = a f(\infty)=a
  43. \mathbb{N}
  44. C ( Ω ) C\left(\Omega\right)
  45. Ω \Omega
  46. f ( x ) = 1 f(x)=1
  47. x x

Algebraic_closure.html

  1. S = { f λ | λ Λ } S=\{f_{\lambda}|\lambda\in\Lambda\}
  2. f λ S f_{\lambda}\in S
  3. u λ , 1 , , u λ , d u_{\lambda,1},\ldots,u_{\lambda,d}
  4. d = degree ( f λ ) d={\rm degree}(f_{\lambda})
  5. u λ , i u_{\lambda,i}
  6. λ Λ \lambda\in\Lambda
  7. i degree ( f λ ) i\leq{\rm degree}(f_{\lambda})
  8. f λ - i = 1 d ( x - u λ , i ) = j = 0 d - 1 r λ , j x j R [ x ] f_{\lambda}-\prod_{i=1}^{d}(x-u_{\lambda,i})=\sum_{j=0}^{d-1}r_{\lambda,j}% \cdot x^{j}\in R[x]
  9. r λ , j R r_{\lambda,j}\in R
  10. r λ , j r_{\lambda,j}
  11. f λ f_{\lambda}
  12. x - ( u λ , i + M ) x-(u_{\lambda,i}+M)
  13. K ( X ) ( X p ) K ( X ) K(X)(\sqrt[p]{X})\supset K(X)

Algebraic_extension.html

  1. { y N | p ( y ) } \left\{y\in N\Big|p(y)\right\}

Algebraic_geometry.html

  1. x 2 + y 2 + z 2 - 1 = 0. x^{2}+y^{2}+z^{2}-1=0.\,
  2. x 2 + y 2 + z 2 - 1 = 0 , x^{2}+y^{2}+z^{2}-1=0,\,
  3. x + y + z = 0. x+y+z=0.\,
  4. V ( S ) = { ( t 1 , , t n ) | p S , p ( t 1 , , t n ) = 0 } . V(S)=\{(t_{1},\dots,t_{n})|\forall p\in S,p(t_{1},\dots,t_{n})=0\}.\,
  5. x 2 + y 2 - 1 = 0 x^{2}+y^{2}-1=0
  6. x = 2 t 1 + t 2 x=\frac{2\,t}{1+t^{2}}
  7. y = 1 - t 2 1 + t 2 , y=\frac{1-t^{2}}{1+t^{2}}\,,
  8. x 2 + y 2 - a = 0 x^{2}+y^{2}-a=0
  9. a > 0 a>0
  10. a < 0 a<0
  11. x y - 1 = 0 xy-1=0
  12. x y - 1 = 0 xy-1=0
  13. x > 0 x>0
  14. x y - 1 = 0 xy-1=0
  15. x + y > 0 x+y>0
  16. d 2 c n d^{2^{cn}}
  17. d 2 c n d^{2^{c^{\prime}n}}
  18. d O ( n 2 ) d^{O(n^{2})}
  19. d O ( n 2 ) d^{O(n^{2})}

Algebraic_number.html

  1. π \pi
  2. x = a / b x=a/b
  3. b x - a bx-a
  4. a x 2 + b x + c ax^{2}+bx+c
  5. a a
  6. b b
  7. c c
  8. ( a = 1 ) (a=1)
  9. i i
  10. - i -i
  11. 3 + 2 i 3+\sqrt{2}i
  12. x 5 - x + 1 x^{5}-x+1
  13. a + b i a+bi
  14. a a
  15. b b
  16. π \pi
  17. cos ( π / 7 ) \cos(\pi/7)
  18. cos ( 3 π / 7 ) \cos(3\pi/7)
  19. cos ( 5 π / 7 ) \cos(5\pi/7)
  20. 8 x 3 - 4 x 2 - 4 x + 1 = 0 8x^{3}-4x^{2}-4x+1=0
  21. tan ( 3 π / 16 ) \tan(3\pi/16)
  22. tan ( 7 π / 16 ) \tan(7\pi/16)
  23. tan ( 11 π / 16 ) \tan(11\pi/16)
  24. tan ( 15 π / 16 ) \tan(15\pi/16)
  25. x 4 - 4 x 3 - 6 x 2 + 4 x + 1 x^{4}-4x^{3}-6x^{2}+4x+1
  26. 2 \sqrt{2}
  27. 3 3 / 2 \sqrt[3]{3}/2
  28. x 2 - 2 x^{2}-2
  29. 8 x 3 - 3 8x^{3}-3
  30. ϕ \phi
  31. x 2 - x - 1 x^{2}-x-1
  32. π \pi
  33. e e
  34. n n
  35. n n

Algebraically_closed_field.html

  1. ( 0 0 0 - a 0 1 0 0 - a 1 0 1 0 - a 2 0 0 1 - a n - 1 ) . \begin{pmatrix}0&0&\cdots&0&-a_{0}\\ 1&0&\cdots&0&-a_{1}\\ 0&1&\cdots&0&-a_{2}\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&\cdots&1&-a_{n-1}\end{pmatrix}.
  2. 1 p ( x ) - q ( x ) = 1 - p ( x ) q ( x ) p ( x ) \frac{1}{p(x)}-q(x)=\frac{1-p(x)q(x)}{p(x)}

Algorithms_for_calculating_variance.html

  1. σ 2 = ( x 2 ) ¯ - x ¯ 2 = i = 1 N x i 2 - ( i = 1 N x i ) 2 / N N . \sigma^{2}=\bar{(x^{2})}-\bar{x}^{2}=\displaystyle\frac{\sum_{i=1}^{N}x_{i}^{2% }-(\sum_{i=1}^{N}x_{i})^{2}/N}{N}.\!
  2. s 2 = i = 1 n x i 2 - ( i = 1 n x i ) 2 / n n - 1 . s^{2}=\displaystyle\frac{\sum_{i=1}^{n}x_{i}^{2}-(\sum_{i=1}^{n}x_{i})^{2}/n}{% n-1}.\!
  3. n 0 , S u m 0 , S u m S q 0 n←0,Sum←0,SumSq←0
  4. x x
  5. n n + 1 n←n+1
  6. S u m S u m + x Sum←Sum+x
  7. S u m S q S u m S q + x × x SumSq←SumSq+x×x
  8. V a r = ( S u m S q ( S u m × S u m ) / n ) / ( n 1 ) Var=(SumSq−(Sum×Sum)/n)/(n−1)
  9. S u m S q SumSq
  10. ( S u m × S u m ) / n (Sum×Sum)/n
  11. Var ( X - k ) = Var ( X ) . \operatorname{Var}(X-k)=\operatorname{Var}(X).
  12. k k
  13. s 2 = i = 1 n ( x i - K ) 2 - ( i = 1 n ( x i - K ) ) 2 / n n - 1 . s^{2}=\displaystyle\frac{\sum_{i=1}^{n}(x_{i}-K)^{2}-(\sum_{i=1}^{n}(x_{i}-K))% ^{2}/n}{n-1}.\!
  14. K K
  15. ( x i - K ) (x_{i}-K)
  16. K K
  17. x ¯ = j = 1 n x j n \bar{x}=\displaystyle\frac{\sum_{j=1}^{n}x_{j}}{n}
  18. variance = s 2 = i = 1 n ( x i - x ¯ ) 2 n - 1 \mathrm{variance}=s^{2}=\displaystyle\frac{\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}}{% n-1}\!
  19. x i x_{i}
  20. x new x_{\mathrm{new}}
  21. x ¯ \overline{x}
  22. x ¯ n = ( n - 1 ) x ¯ n - 1 + x n n = x ¯ n - 1 + x n - x ¯ n - 1 n \bar{x}_{n}=\frac{(n-1)\,\bar{x}_{n-1}+x_{n}}{n}=\bar{x}_{n-1}+\frac{x_{n}-% \bar{x}_{n-1}}{n}\!
  23. s n 2 = ( n - 2 ) ( n - 1 ) s n - 1 2 + ( x n - x ¯ n - 1 ) 2 n , n > 1 s^{2}_{n}=\frac{(n-2)}{(n-1)}\,s^{2}_{n-1}+\frac{(x_{n}-\bar{x}_{n-1})^{2}}{n}% ,\quad n>1
  24. σ n 2 = ( n - 1 ) σ n - 1 2 + ( x n - x ¯ n - 1 ) ( x n - x ¯ n ) n . \sigma^{2}_{n}=\frac{(n-1)\,\sigma^{2}_{n-1}+(x_{n}-\bar{x}_{n-1})(x_{n}-\bar{% x}_{n})}{n}.
  25. i = 1 n ( x i - x ¯ n ) 2 \textstyle\sum_{i=1}^{n}(x_{i}-\bar{x}_{n})^{2}
  26. M 2 , n M_{2,n}
  27. M 2 , n = M 2 , n - 1 + ( x n - x ¯ n - 1 ) ( x n - x ¯ n ) M_{2,n}\!=M_{2,n-1}+(x_{n}-\bar{x}_{n-1})(x_{n}-\bar{x}_{n})
  28. s n 2 = M 2 , n n - 1 s^{2}_{n}=\frac{M_{2,n}}{n-1}
  29. σ n 2 = M 2 , n n \sigma^{2}_{n}=\frac{M_{2,n}}{n}
  30. M k = x ¯ k M_{k}=\bar{x}_{k}
  31. S k = M 2 , k S_{k}=M_{2,k}
  32. X X
  33. X A X_{A}
  34. X B X_{B}
  35. δ = x ¯ B - x ¯ A \delta\!=\bar{x}_{B}-\bar{x}_{A}
  36. x ¯ X = x ¯ A + δ n B n X \bar{x}_{X}=\bar{x}_{A}+\delta\cdot\frac{n_{B}}{n_{X}}
  37. M 2 , X = M 2 , A + M 2 , B + δ 2 n A n B n X M_{2,X}=M_{2,A}+M_{2,B}+\delta^{2}\cdot\frac{n_{A}n_{B}}{n_{X}}
  38. n A n B n_{A}\approx n_{B}
  39. x ¯ B - x ¯ A \bar{x}_{B}-\bar{x}_{A}
  40. n B = 1 n_{B}=1
  41. x ¯ X = n A x ¯ A + n B x ¯ B n A + n B \bar{x}_{X}=\frac{n_{A}\bar{x}_{A}+n_{B}\bar{x}_{B}}{n_{A}+n_{B}}
  42. M 3 , X = M 3 , A + M 3 , B + δ 3 n A n B ( n A - n B ) n X 2 + 3 δ n A M 2 , B - n B M 2 , A n X M_{3,X}=M_{3,A}+M_{3,B}+\delta^{3}\frac{n_{A}n_{B}(n_{A}-n_{B})}{n_{X}^{2}}+3% \delta\frac{n_{A}M_{2,B}-n_{B}M_{2,A}}{n_{X}}
  43. M 4 , X = M 4 , A + M 4 , B + δ 4 n A n B ( n A 2 - n A n B + n B 2 ) n X 3 + 6 δ 2 n A 2 M 2 , B + n B 2 M 2 , A n X 2 + 4 δ n A M 3 , B - n B M 3 , A n X \begin{aligned}\displaystyle M_{4,X}=M_{4,A}+M_{4,B}&\displaystyle+\delta^{4}% \frac{n_{A}n_{B}\left(n_{A}^{2}-n_{A}n_{B}+n_{B}^{2}\right)}{n_{X}^{3}}\\ &\displaystyle+6\delta^{2}\frac{n_{A}^{2}M_{2,B}+n_{B}^{2}M_{2,A}}{n_{X}^{2}}+% 4\delta\frac{n_{A}M_{3,B}-n_{B}M_{3,A}}{n_{X}}\\ \end{aligned}
  44. M k M_{k}
  45. Σ ( x - x ¯ ) k \Sigma(x-\overline{x})^{k}
  46. g 1 = n M 3 M 2 3 / 2 , g_{1}=\frac{\sqrt{n}M_{3}}{M_{2}^{3/2}},
  47. g 2 = n M 4 M 2 2 - 3. g_{2}=\frac{nM_{4}}{M_{2}^{2}}-3.
  48. B = { x } B=\{x\}
  49. δ = x - m \delta\!=x-m
  50. m = m + δ n m^{\prime}=m+\frac{\delta}{n}
  51. M 2 = M 2 + δ 2 n - 1 n M_{2}^{\prime}=M_{2}+\delta^{2}\frac{n-1}{n}
  52. M 3 = M 3 + δ 3 ( n - 1 ) ( n - 2 ) n 2 - 3 δ M 2 n M_{3}^{\prime}=M_{3}+\delta^{3}\frac{(n-1)(n-2)}{n^{2}}-\frac{3\delta M_{2}}{n}
  53. M 4 = M 4 + δ 4 ( n - 1 ) ( n 2 - 3 n + 3 ) n 3 + 6 δ 2 M 2 n 2 - 4 δ M 3 n M_{4}^{\prime}=M_{4}+\frac{\delta^{4}(n-1)(n^{2}-3n+3)}{n^{3}}+\frac{6\delta^{% 2}M_{2}}{n^{2}}-\frac{4\delta M_{3}}{n}
  54. δ / n \delta/n
  55. H ( x k ) = h ( x k ) A H(x_{k})=\frac{h(x_{k})}{A}
  56. h ( x k ) h(x_{k})
  57. H ( x k ) H(x_{k})
  58. x k x_{k}
  59. A = k = 1 K h ( x k ) Δ x k A=\sum_{k=1}^{K}h(x_{k})\,\Delta x_{k}
  60. n n
  61. x ( t ) x(t)
  62. m n ( h ) = k = 1 K x k n H ( x k ) Δ x k = 1 A k = 1 K x k n h ( x k ) Δ x k m_{n}^{(h)}=\sum_{k=1}^{K}x_{k}^{n}\,H(x_{k})\Delta x_{k}=\frac{1}{A}\sum_{k=1% }^{K}x_{k}^{n}\,h(x_{k})\Delta x_{k}
  63. θ n ( h ) = k = 1 K ( x k - m 1 ( h ) ) n H ( x k ) Δ x k = 1 A k = 1 K ( x k - m 1 ( h ) ) n h ( x k ) Δ x k \theta_{n}^{(h)}=\sum_{k=1}^{K}\Big(x_{k}-m_{1}^{(h)}\Big)^{n}\,H(x_{k})\Delta x% _{k}=\frac{1}{A}\sum_{k=1}^{K}\Big(x_{k}-m_{1}^{(h)}\Big)^{n}\,h(x_{k})\Delta x% _{k}
  64. ( h ) {}^{(h)}
  65. Δ x k = Δ x \Delta x_{k}=\Delta x
  66. I = A / Δ x I=A/\Delta x
  67. m n ( h ) = 1 I k = 1 K x k n h ( x k ) m_{n}^{(h)}=\frac{1}{I}{\sum_{k=1}^{K}x_{k}^{n}\,h(x_{k})}
  68. θ n ( h ) = 1 I k = 1 K ( x k - m 1 ( h ) ) n h ( x k ) \theta_{n}^{(h)}=\frac{1}{I}{\sum_{k=1}^{K}\Big(x_{k}-m_{1}^{(h)}\Big)^{n}\,h(% x_{k})}
  69. Q Q
  70. ( γ 0 , q , μ q , σ q 2 , α 3 , q , α 4 , q ) (\gamma_{0,q},\mu_{q},\sigma^{2}_{q},\alpha_{3,q},\alpha_{4,q})\quad
  71. q = 1 , 2 , , Q q=1,2,...,Q
  72. γ n \gamma_{n}
  73. n n
  74. γ n , q = m n , q γ 0 , q for n = 1 , 2 , 3 , 4 and q = 1 , 2 , , Q \gamma_{n,q}=m_{n,q}\gamma_{0,q}\qquad\quad\textrm{for}\quad n=1,2,3,4\quad\,% \text{ and }\quad q=1,2,\dots,Q
  75. γ 0 , q \gamma_{0,q}
  76. q t h q^{th}
  77. Δ t \Delta t
  78. γ \gamma
  79. Q Q
  80. Q Q
  81. γ n , c = q = 1 Q γ n , q for n = 0 , 1 , 2 , 3 , 4 \gamma_{n,c}=\sum_{q=1}^{Q}\gamma_{n,q}\quad\quad\textrm{for}\quad n=0,1,2,3,4
  82. c {}_{c}
  83. γ \gamma
  84. γ \gamma
  85. m n , c = γ n , c γ 0 , c for n = 1 , 2 , 3 , 4 m_{n,c}=\frac{\gamma_{n,c}}{\gamma_{0,c}}\quad\textrm{for}\quad n=1,2,3,4
  86. m n m_{n}
  87. θ n = E [ ( x - μ ) n ] ) \theta_{n}=E[(x-\mu)^{n}])
  88. μ c = m 1 , c σ c 2 = θ 2 , c α 3 , c = θ 3 , c σ c 3 α 4 , c = θ 4 , c σ c 4 - 3 \mu_{c}=m_{1,c}\ \ \ \ \ \sigma^{2}_{c}=\theta_{2,c}\ \ \ \ \ \alpha_{3,c}=% \frac{\theta_{3,c}}{\sigma_{c}^{3}}\ \ \ \ \ \alpha_{4,c}={\frac{\theta_{4,c}}% {\sigma_{c}^{4}}}-3
  89. Cov ( X , Y ) = i = 1 n x i y i - ( i = 1 n x i ) ( i = 1 n y i ) / n n . \operatorname{Cov}(X,Y)=\displaystyle\frac{\sum_{i=1}^{n}x_{i}y_{i}-(\sum_{i=1% }^{n}x_{i})(\sum_{i=1}^{n}y_{i})/n}{n}.\!
  90. K x K_{x}
  91. K y K_{y}
  92. Cov ( X , Y ) = Cov ( X - k x , Y - k y ) = i = 1 n ( x i - K x ) ( y i - K y ) - ( i = 1 n ( x i - K x ) ) ( i = 1 n ( y i - K y ) ) / n n . \operatorname{Cov}(X,Y)=\operatorname{Cov}(X-k_{x},Y-k_{y})=\displaystyle\frac% {\sum_{i=1}^{n}(x_{i}-K_{x})(y_{i}-K_{y})-(\sum_{i=1}^{n}(x_{i}-K_{x}))(\sum_{% i=1}^{n}(y_{i}-K_{y}))/n}{n}.\!
  93. x i \textstyle\sum x_{i}
  94. y i \textstyle\sum y_{i}
  95. C n = i = 1 n ( x i - x ¯ n ) ( y i - y ¯ n ) \textstyle C_{n}=\sum_{i=1}^{n}(x_{i}-\bar{x}_{n})(y_{i}-\bar{y}_{n})
  96. x ¯ n = x ¯ n - 1 + x n - x ¯ n - 1 n \bar{x}_{n}=\bar{x}_{n-1}+\frac{x_{n}-\bar{x}_{n-1}}{n}\!
  97. y ¯ n = y ¯ n - 1 + y n - y ¯ n - 1 n \bar{y}_{n}=\bar{y}_{n-1}+\frac{y_{n}-\bar{y}_{n-1}}{n}\!
  98. C n = C n - 1 + ( x n - x ¯ n ) ( y n - y ¯ n - 1 ) = C n - 1 + ( y n - y ¯ n ) ( x n - x ¯ n - 1 ) C_{n}=C_{n-1}+(x_{n}-\bar{x}_{n})(y_{n}-\bar{y}_{n-1})=C_{n-1}+(y_{n}-\bar{y}_% {n})(x_{n}-\bar{x}_{n-1})
  99. ( x n - x ¯ n ) = n - 1 n ( x n - x ¯ n - 1 ) \textstyle(x_{n}-\bar{x}_{n})=\frac{n-1}{n}(x_{n}-\bar{x}_{n-1})
  100. n - 1 n ( x n - x ¯ n - 1 ) ( y n - y ¯ n - 1 ) \textstyle\frac{n-1}{n}(x_{n}-\bar{x}_{n-1})(y_{n}-\bar{y}_{n-1})
  101. Cov N ( X , Y ) = C N N = Cov N - 1 ( X , Y ) ( N - 1 ) + ( x n - x ¯ n ) ( y n - y ¯ n - 1 ) N = Cov N - 1 ( X , Y ) ( N - 1 ) + ( y n - y ¯ n ) ( x n - x ¯ n - 1 ) N = Cov N - 1 ( X , Y ) ( N - 1 ) + N - 1 N ( x n - x ¯ n - 1 ) ( y n - y ¯ n - 1 ) N . \begin{aligned}\displaystyle\operatorname{Cov}_{N}(X,Y)=\frac{C_{N}}{N}&% \displaystyle=\frac{\operatorname{Cov}_{N-1}(X,Y)\cdot(N-1)+(x_{n}-\bar{x}_{n}% )(y_{n}-\bar{y}_{n-1})}{N}\\ &\displaystyle=\frac{\operatorname{Cov}_{N-1}(X,Y)\cdot(N-1)+(y_{n}-\bar{y}_{n% })(x_{n}-\bar{x}_{n-1})}{N}\\ &\displaystyle=\frac{\operatorname{Cov}_{N-1}(X,Y)\cdot(N-1)+\frac{N-1}{N}(x_{% n}-\bar{x}_{n-1})(y_{n}-\bar{y}_{n-1})}{N}.\end{aligned}
  102. C X = C A + C B + ( x ¯ A - x ¯ B ) ( y ¯ A - y ¯ B ) n A n B n X C_{X}=C_{A}+C_{B}+(\bar{x}_{A}-\bar{x}_{B})(\bar{y}_{A}-\bar{y}_{B})\cdot\frac% {n_{A}n_{B}}{n_{X}}

Alkyne.html

  1. \overrightarrow{\leftarrow}

Allan_variance.html

  1. σ y 2 ( τ ) . \sigma_{y}^{2}(\tau).\,
  2. σ y ( τ ) . \sigma_{y}(\tau).\,
  3. τ \tau
  4. σ y 2 ( M , T , τ ) . \sigma_{y}^{2}(M,T,\tau).\,
  5. T = τ T=\tau
  6. τ \tau
  7. τ \tau
  8. τ \tau
  9. n n
  10. τ 0 \tau_{0}
  11. τ = n τ 0 \tau=n\,\tau_{0}
  12. f - 2 f^{-2}
  13. f - 3 f^{-3}
  14. M M
  15. σ y 2 ( M , T , τ ) = 1 M - 1 { i = 0 M - 1 [ x ( i T + τ ) - x ( i T ) τ ] 2 - 1 M [ i = 0 M - 1 x ( i T + τ ) - x ( i T ) τ ] 2 } \sigma_{y}^{2}(M,T,\tau)=\frac{1}{M-1}\left\{\sum_{i=0}^{M-1}\left[\frac{x(iT+% \tau)-x(iT)}{\tau}\right]^{2}-\frac{1}{M}\left[\sum_{i=0}^{M-1}\frac{x(iT+\tau% )-x(iT)}{\tau}\right]^{2}\right\}
  16. x ( t ) x(t)
  17. t t
  18. σ y 2 ( M , T , τ ) = 1 M - 1 { i = 0 M - 1 y ¯ i 2 - 1 M [ i = 0 M - 1 y ¯ i ] 2 } \sigma_{y}^{2}(M,T,\tau)=\frac{1}{M-1}\left\{\sum_{i=0}^{M-1}\bar{y}_{i}^{2}-% \frac{1}{M}\left[\sum_{i=0}^{M-1}\bar{y}_{i}\right]^{2}\right\}
  19. M M
  20. T T
  21. τ \tau
  22. M M
  23. T T
  24. τ \tau
  25. σ y 2 ( τ ) = σ y 2 ( 2 , τ , τ ) \sigma_{y}^{2}(\tau)=\langle\sigma_{y}^{2}(2,\tau,\tau)\rangle
  26. σ y 2 ( τ ) = 1 2 ( y ¯ n + 1 - y ¯ n ) 2 = 1 2 τ 2 ( x n + 2 - 2 x n + 1 + x n ) 2 \sigma_{y}^{2}(\tau)=\frac{1}{2}\langle(\bar{y}_{n+1}-\bar{y}_{n})^{2}\rangle=% \frac{1}{2\tau^{2}}\langle(x_{n+2}-2x_{n+1}+x_{n})^{2}\rangle
  27. τ \tau
  28. y ¯ n \bar{y}_{n}
  29. τ \tau
  30. T = τ T=\tau\,
  31. σ y ( τ ) = σ y 2 ( τ ) \sigma_{y}(\tau)=\sqrt{\sigma_{y}^{2}(\tau)}\,
  32. V ( t ) = V 0 sin ( Φ ( t ) ) . V(t)=V_{0}\sin(\Phi(t)).
  33. ν n \nu_{\mathrm{n}}
  34. ω n \omega_{\mathrm{n}}
  35. ω n = 2 π ν n . \omega_{\mathrm{n}}=2\pi\nu_{\mathrm{n}}.
  36. ω n t \omega_{n}t
  37. ϕ ( t ) \phi(t)
  38. Φ ( t ) = ω n t + ϕ ( t ) = 2 π ν n t + ϕ ( t ) . \Phi(t)=\omega_{\mathrm{n}}t+\phi(t)=2\pi\nu_{\mathrm{n}}t+\phi(t).
  39. x ( t ) = ϕ ( t ) 2 π ν n = Φ ( t ) 2 π ν n - t = T ( t ) - t x(t)=\frac{\phi(t)}{2\pi\nu_{n}}=\frac{\Phi(t)}{2\pi\nu_{n}}-t=T(t)-t
  40. T E ( t ) = T ( t ) - T REF ( t ) . TE(t)=T(t)-T\text{REF}(t).\,
  41. ν ( t ) \nu(t)
  42. ν ( t ) = 1 2 π d Φ ( t ) d t . \nu(t)=\frac{1}{2\pi}\frac{d\Phi(t)}{dt}.
  43. ν ( t ) \nu(t)
  44. ν n \nu_{\mathrm{n}}
  45. y ( t ) = ν ( t ) - ν n ν n = ν ( t ) ν n - 1. y(t)=\frac{\nu(t)-\nu_{\mathrm{n}}}{\nu_{\mathrm{n}}}=\frac{\nu(t)}{\nu_{% \mathrm{n}}}-1.
  46. y ¯ ( t , τ ) = 1 τ 0 τ y ( t + t v ) d t v \bar{y}(t,\tau)=\frac{1}{\tau}\int\limits_{0}^{\tau}y(t+t_{v})\,dt_{v}
  47. y ¯ ( t , τ ) = x ( t + τ ) - x ( t ) τ . \bar{y}(t,\tau)=\frac{x(t+\tau)-x(t)}{\tau}.
  48. N = M + 1 N=M+1\,
  49. x i = x ( i T ) x_{i}=x(iT)\,
  50. y ¯ i \bar{y}_{i}
  51. y ¯ i = y ¯ ( T i , τ ) \bar{y}_{i}=\bar{y}(Ti,\tau)\,
  52. y ¯ i = 1 τ 0 τ y ( i T + t v ) d t v = x ( i T + τ ) - x ( i T ) τ \bar{y}_{i}=\frac{1}{\tau}\int\limits_{0}^{\tau}y(iT+t_{v})\,dt_{v}=\frac{x(iT% +\tau)-x(iT)}{\tau}
  53. y ¯ i = x i + 1 - x i τ . \bar{y}_{i}=\frac{x_{i+1}-x_{i}}{\tau}.
  54. y ¯ 0 y ¯ M - 1 \bar{y}_{0}\ldots\bar{y}_{M-1}
  55. σ y 2 ( τ , M ) = AVAR ( τ , M ) = 1 2 ( M - 1 ) i = 0 M - 2 ( y ¯ i + 1 - y ¯ i ) 2 \sigma_{y}^{2}(\tau,M)=\,\text{AVAR}(\tau,M)=\frac{1}{2(M-1)}\sum_{i=0}^{M-2}(% \bar{y}_{i+1}-\bar{y}_{i})^{2}
  56. σ y 2 ( τ , N ) = AVAR ( τ , N ) = 1 2 τ 2 ( N - 2 ) i = 0 N - 3 ( x i + 2 - 2 x i + 1 + x i ) 2 \sigma_{y}^{2}(\tau,N)=\,\text{AVAR}(\tau,N)=\frac{1}{2\tau^{2}(N-2)}\sum_{i=0% }^{N-3}(x_{i+2}-2x_{i+1}+x_{i})^{2}
  57. σ y 2 ( n τ 0 , M ) = AVAR ( n τ 0 , M ) = 1 2 n ( M - 1 ) i = 0 M - 1 n - 1 ( y ¯ n i + n - y ¯ n i ) 2 \sigma_{y}^{2}(n\tau_{0},M)=\,\text{AVAR}(n\tau_{0},M)=\frac{1}{2n(M-1)}\sum_{% i=0}^{\frac{M-1}{n}-1}(\bar{y}_{ni+n}-\bar{y}_{ni})^{2}
  58. n M - 1 n\leq M-1
  59. σ y 2 ( n τ 0 , N ) = AVAR ( n τ 0 , N ) = 1 2 n 2 τ 0 2 ( N - 1 n - 1 ) i = 0 N - 1 n - 2 ( x n i + 2 n - 2 x n i + n + x n i ) 2 \sigma_{y}^{2}(n\tau_{0},N)=\,\text{AVAR}(n\tau_{0},N)=\frac{1}{2n^{2}\tau_{0}% ^{2}(\frac{N-1}{n}-1)}\sum_{i=0}^{\frac{N-1}{n}-2}(x_{ni+2n}-2x_{ni+n}+x_{ni})% ^{2}
  60. n N - 1 2 n\leq\frac{N-1}{2}
  61. σ y 2 ( n τ 0 , M ) = AVAR ( n τ 0 , M ) = 1 2 n 2 ( M - 2 n + 1 ) j = 0 M - 2 n ( i = j j + n - 1 y ¯ i + n - y ¯ i ) 2 \sigma_{y}^{2}(n\tau_{0},M)=\,\text{AVAR}(n\tau_{0},M)=\frac{1}{2n^{2}(M-2n+1)% }\sum_{j=0}^{M-2n}\left(\sum_{i=j}^{j+n-1}\bar{y}_{i+n}-\bar{y}_{i}\right)^{2}
  62. σ y 2 ( n τ 0 , N ) = AVAR ( n τ 0 , N ) = 1 2 n 2 τ 0 2 ( N - 2 n ) i = 0 N - 2 n - 1 ( x i + 2 n - 2 x i + n + x i ) 2 \sigma_{y}^{2}(n\tau_{0},N)=\,\text{AVAR}(n\tau_{0},N)=\frac{1}{2n^{2}\tau_{0}% ^{2}(N-2n)}\sum_{i=0}^{N-2n-1}(x_{i+2n}-2x_{i+n}+x_{i})^{2}
  63. σ x 2 ( τ ) = τ 2 3 M o d . σ y 2 ( τ ) \sigma_{x}^{2}(\tau)=\frac{\tau^{2}}{3}Mod.\sigma_{y}^{2}(\tau)
  64. σ x ( τ ) = τ 3 M o d . σ y ( τ ) . \sigma_{x}(\tau)=\frac{\tau}{\sqrt{3}}Mod.\sigma_{y}(\tau).
  65. χ 2 = ( d . f . ) s 2 σ 2 \chi^{2}=\frac{(d.f.)s^{2}}{\sigma^{2}}
  66. χ 2 ( 0.05 ) ( d . f . ) s 2 σ 2 χ 2 ( 0.95 ) \chi^{2}(0.05)\leq\frac{(d.f.)s^{2}}{\sigma^{2}}\leq\chi^{2}(0.95)
  67. ( d . f . ) s 2 χ 2 ( 0.95 ) σ 2 ( d . f . ) s 2 χ 2 ( 0.05 ) \frac{(d.f.)s^{2}}{\chi^{2}(0.95)}\leq\sigma^{2}\leq\frac{(d.f.)s^{2}}{\chi^{2% }(0.05)}
  68. d . f . ( N + 1 ) ( N - 2 n ) 2 ( N - n ) d.f.\cong\frac{(N+1)(N-2n)}{2(N-n)}
  69. d . f . exp [ ( ln N - 1 2 n ln ( 2 n + 1 ) ( N - 1 ) 4 ) - 1 / 2 ] d.f.\cong\exp\left[\left(\ln\frac{N-1}{2n}\ln\frac{(2n+1)(N-1)}{4}\right)^{-1/% 2}\right]
  70. d . f . [ 3 ( N - 1 ) 2 n - 2 ( N - 2 ) N ] 4 n 2 4 n 2 + 5 d.f.\cong\left[\frac{3(N-1)}{2n}-\frac{2(N-2)}{N}\right]\frac{4n^{2}}{4n^{2}+5}
  71. d . f . { 2 ( N - 2 ) 2.3 N - 4.9 n = 1 5 N 2 4 n ( N + 3 n ) n 2 d.f.\cong\begin{cases}\frac{2(N-2)}{2.3N-4.9}&n=1\\ \frac{5N^{2}}{4n(N+3n)}&n\geq 2\end{cases}
  72. d . f . N - 2 n ( N - 1 ) 2 - 3 n ( N - 1 ) + 4 n 2 ( N - 3 ) 2 d.f.\cong\frac{N-2}{n}\frac{(N-1)^{2}-3n(N-1)+4n^{2}}{(N-3)^{2}}
  73. f 0 = 1 f^{0}=1
  74. f 2 f^{2}
  75. h 2 h_{2}
  76. S x ( f ) = 1 ( 2 π ) 2 h 2 S_{x}(f)=\frac{1}{(2\pi)^{2}}h_{2}
  77. σ y 2 ( τ ) = 3 f H 4 π 2 τ 2 h 2 \sigma_{y}^{2}(\tau)=\frac{3f_{H}}{4\pi^{2}\tau^{2}}h_{2}
  78. σ y ( τ ) = 3 f H 2 π τ h 2 \sigma_{y}(\tau)=\frac{\sqrt{3f_{H}}}{2\pi\tau}\sqrt{h_{2}}
  79. f - 1 f^{-1}
  80. f 1 = f f^{1}=f
  81. h 1 h_{1}
  82. S x ( f ) = 1 ( 2 π ) 2 f h 1 S_{x}(f)=\frac{1}{(2\pi)^{2}f}h_{1}
  83. σ y 2 ( τ ) = 3 [ γ + ln ( 2 π f H τ ) ] - ln 2 4 π 2 τ 2 h 1 \sigma_{y}^{2}(\tau)=\frac{3[\gamma+\ln(2\pi f_{H}\tau)]-\ln 2}{4\pi^{2}\tau^{% 2}}h_{1}
  84. σ t ( τ ) = 3 [ γ + ln ( 2 π f H τ ) ] - ln 2 2 π τ h 1 \sigma_{t}(\tau)=\frac{\sqrt{3[\gamma+\ln(2\pi f_{H}\tau)]-\ln 2}}{2\pi\tau}% \sqrt{h_{1}}
  85. f - 2 f^{-2}
  86. f 0 = 1 f^{0}=1
  87. h 0 h_{0}
  88. S x ( f ) = 1 ( 2 π ) 2 f 2 h 0 S_{x}(f)=\frac{1}{(2\pi)^{2}f^{2}}h_{0}
  89. σ y 2 ( τ ) = 1 2 τ h 0 \sigma_{y}^{2}(\tau)=\frac{1}{2\tau}h_{0}
  90. σ y ( τ ) = 1 2 τ h 0 \sigma_{y}(\tau)=\frac{1}{\sqrt{2\tau}}\sqrt{h_{0}}
  91. f - 3 f^{-3}
  92. f - 1 f^{-1}
  93. h - 1 h_{-1}
  94. S x ( f ) = 1 ( 2 π ) 2 f 3 h - 1 S_{x}(f)=\frac{1}{(2\pi)^{2}f^{3}}h_{-1}
  95. σ y 2 ( τ ) = 2 ln ( 2 ) h - 1 \sigma_{y}^{2}(\tau)=2\ln(2)h_{-1}
  96. σ y ( τ ) = 2 ln ( 2 ) h - 1 \sigma_{y}(\tau)=\sqrt{2\ln(2)}\sqrt{h_{-1}}
  97. f - 4 f^{-4}
  98. f - 2 f^{-2}
  99. h - 2 h_{-2}
  100. S x ( f ) = 1 ( 2 π ) 2 f 4 h - 2 S_{x}(f)=\frac{1}{(2\pi)^{2}f^{4}}h_{-2}
  101. σ y 2 ( τ ) = 2 π 2 τ 3 h - 2 \sigma_{y}^{2}(\tau)=\frac{2\pi^{2}\tau}{3}h_{-2}
  102. σ y ( τ ) = π 2 τ 3 h - 2 \sigma_{y}(\tau)=\frac{\pi\sqrt{2\tau}}{\sqrt{3}}\sqrt{h_{-2}}
  103. τ 1 2 π f H \tau\gg\frac{1}{2\pi f_{H}}
  104. S x ( f ) = 1 4 π 2 h α f α - 2 = 1 4 π 2 h α f β S_{x}(f)=\frac{1}{4\pi^{2}}h_{\alpha}f^{\alpha-2}=\frac{1}{4\pi^{2}}h_{\alpha}% f^{\beta}
  105. β α - 2 \beta\equiv\alpha-2
  106. S y ( f ) = h α f α S_{y}(f)=h_{\alpha}f^{\alpha}
  107. σ y 2 ( τ ) = K α h α τ μ \sigma_{y}^{2}(\tau)=K_{\alpha}h_{\alpha}\tau^{\mu}
  108. 2 π 2 3 \frac{2\pi^{2}}{3}
  109. 2 ln 2 2\ln{2}
  110. 1 2 \frac{1}{2}
  111. 3 [ γ + ln ( 2 π f H τ ) ] - ln 2 4 π 2 \frac{3[\gamma+\ln(2\pi f_{H}\tau)]-\ln 2}{4\pi^{2}}
  112. 3 f H 4 π 2 \frac{3f_{H}}{4\pi^{2}}
  113. S ϕ S_{\phi}
  114. σ y 2 ( τ ) = 2 ν 0 2 0 f b S ϕ ( f ) sin 4 ( π τ f ) ( π τ ) 2 d f \sigma^{2}_{y}(\tau)=\frac{2}{\nu_{0}^{2}}\int^{f_{b}}_{0}S_{\phi}(f)\frac{% \sin^{4}(\pi\tau f)}{(\pi\tau)^{2}}df
  115. x 0 x_{0}
  116. 0
  117. 0
  118. 0
  119. y 0 t y_{0}t
  120. y 0 y_{0}
  121. 0
  122. 0
  123. D t 2 2 \frac{Dt^{2}}{2}
  124. D t Dt
  125. D 2 τ 2 2 \frac{D^{2}\tau^{2}}{2}
  126. D τ 2 \frac{D\tau}{\sqrt{2}}
  127. σ y 2 ( τ ) = 1 2 ( y ¯ i + 1 - y ¯ i ) 2 \sigma_{y}^{2}(\tau)=\frac{1}{2}\langle(\bar{y}_{i+1}-\bar{y}_{i})^{2}\rangle
  128. y ¯ i = 1 τ 0 τ y ( i τ + t ) d t . \bar{y}_{i}=\frac{1}{\tau}\int\limits_{0}^{\tau}y(i\tau+t)\,dt.
  129. y i y_{i}
  130. S y ( f ) S_{y}(f)
  131. σ y 2 ( τ ) = 0 S y ( f ) 2 sin 4 π τ f ( π τ f ) 2 d f \sigma_{y}^{2}(\tau)=\int_{0}^{\infty}S_{y}(f)\frac{2\sin^{4}\pi\tau f}{(\pi% \tau f)^{2}}\,df
  132. | H A ( f ) | 2 = 2 sin 4 π τ f ( π τ f ) 2 . \left|H_{A}(f)\right|^{2}=\frac{2\sin^{4}\pi\tau f}{(\pi\tau f)^{2}}.
  133. B 1 ( N , r , μ ) = σ y 2 ( N , T , τ ) σ y 2 ( 2 , T , τ ) B_{1}(N,r,\mu)=\frac{\left\langle\sigma_{y}^{2}(N,T,\tau)\right\rangle}{\left% \langle\sigma_{y}^{2}(2,T,\tau)\right\rangle}
  134. r = T τ . r=\frac{T}{\tau}.
  135. B 1 ( N , r , μ ) = 1 + n = 1 N - 1 N - n N ( N - 1 ) [ 2 ( r n ) μ + 2 - ( r n + 1 ) μ + 2 - | r n - 1 | μ + 2 ] 1 + 1 2 [ 2 r μ + 2 - ( r + 1 ) μ + 2 - | r - 1 | μ + 2 ] . B_{1}(N,r,\mu)=\frac{1+\sum_{n=1}^{N-1}\frac{N-n}{N(N-1)}\left[2\left(rn\right% )^{\mu+2}-\left(rn+1\right)^{\mu+2}-\left|rn-1\right|^{\mu+2}\right]}{1+\frac{% 1}{2}\left[2r^{\mu+2}-\left(r+1\right)^{\mu+2}-\left|r-1\right|^{\mu+2}\right]}.
  136. B 2 ( r , μ ) = σ y 2 ( 2 , T , τ ) σ y 2 ( 2 , τ , τ ) B_{2}(r,\mu)=\frac{\left\langle\sigma_{y}^{2}(2,T,\tau)\right\rangle}{\left% \langle\sigma_{y}^{2}(2,\tau,\tau)\right\rangle}
  137. r = T τ . r=\frac{T}{\tau}.
  138. B 2 ( r , μ ) = 1 + 1 2 [ 2 r μ + 2 - ( r + 1 ) μ + 2 - | r - 1 | μ + 2 ] 2 ( 1 - 2 μ ) . B_{2}(r,\mu)=\frac{1+\frac{1}{2}\left[2r^{\mu+2}-\left(r+1\right)^{\mu+2}-% \left|r-1\right|^{\mu+2}\right]}{2\left(1-2^{\mu}\right)}.
  139. B 3 ( N , M , r , μ ) = σ y 2 ( N , M , T , τ ) σ y 2 ( N , T , τ ) B_{3}(N,M,r,\mu)=\frac{\left\langle\sigma_{y}^{2}(N,M,T,\tau)\right\rangle}{% \left\langle\sigma_{y}^{2}(N,T,\tau)\right\rangle}
  140. T = M T 0 T=MT_{0}\,
  141. τ = M τ 0 . \tau=M\tau_{0}.\,
  142. B 3 ( 2 , M , r , μ ) = 2 M + M F ( M r ) - n = 1 M - 1 ( M - n ) [ 2 F ( n r ) - F ( ( M + n ) r ) + F ( ( M - n ) r ) ] M μ + 2 [ F ( r ) + 2 ] B_{3}(2,M,r,\mu)=\frac{2M+MF(Mr)-\sum_{n=1}^{M-1}(M-n)\left[2F(nr)-F((M+n)r)+F% ((M-n)r)\right]}{M^{\mu+2}\left[F(r)+2\right]}
  143. F ( A ) = 2 A μ + 2 - ( A + 1 ) μ + 2 - | A - 1 | μ + 2 . F(A)=2A^{\mu+2}-(A+1)^{\mu+2}-|A-1|^{\mu+2}.\,
  144. B τ ( τ 1 , τ 2 , μ ) = σ y 2 ( 2 , τ 2 , τ 2 ) σ y 2 ( 2 , τ 1 , τ 1 ) . B_{\tau}(\tau_{1},\tau_{2},\mu)=\frac{\left\langle\sigma_{y}^{2}(2,\tau_{2},% \tau_{2})\right\rangle}{\left\langle\sigma_{y}^{2}(2,\tau_{1},\tau_{1})\right% \rangle}.\,
  145. B τ ( τ 1 , τ 2 , μ ) = ( τ 2 τ 1 ) μ . B_{\tau}(\tau_{1},\tau_{2},\mu)=\left(\frac{\tau_{2}}{\tau_{1}}\right)^{\mu}.
  146. σ y 2 ( N 2 , T 2 , τ 2 ) = ( τ 2 τ 1 ) μ [ B 1 ( N 2 , r 2 , μ ) B 2 ( r 2 , μ ) B 1 ( N 1 , r 1 , μ ) B 2 ( r 1 , μ ) ] σ y 2 ( N 1 , T 1 , τ 1 ) \left\langle\sigma_{y}^{2}(N_{2},T_{2},\tau_{2})\right\rangle=\left(\frac{\tau% _{2}}{\tau_{1}}\right)^{\mu}\left[\frac{B_{1}(N_{2},r_{2},\mu)B_{2}(r_{2},\mu)% }{B_{1}(N_{1},r_{1},\mu)B_{2}(r_{1},\mu)}\right]\left\langle\sigma_{y}^{2}(N_{% 1},T_{1},\tau_{1})\right\rangle
  147. r 1 = T 1 r 1 r_{1}=\frac{T_{1}}{r_{1}}
  148. r 2 = T 2 r 2 r_{2}=\frac{T_{2}}{r_{2}}
  149. σ y 2 ( N 2 , M 2 , T 2 , τ 2 ) = ( τ 2 τ 1 ) μ [ B 3 ( N 2 , M 2 , r 2 , μ ) B 1 ( N 2 , r 2 , μ ) B 2 ( r 2 , μ ) B 3 ( N 1 , M 1 , r 1 , μ ) B 1 ( N 1 , r 1 , μ ) B 2 ( r 1 , μ ) ] σ y 2 ( N 1 , M 1 , T 1 , τ 1 ) . \left\langle\sigma_{y}^{2}(N_{2},M_{2},T_{2},\tau_{2})\right\rangle=\left(% \frac{\tau_{2}}{\tau_{1}}\right)^{\mu}\left[\frac{B_{3}(N_{2},M_{2},r_{2},\mu)% B_{1}(N_{2},r_{2},\mu)B_{2}(r_{2},\mu)}{B_{3}(N_{1},M_{1},r_{1},\mu)B_{1}(N_{1% },r_{1},\mu)B_{2}(r_{1},\mu)}\right]\left\langle\sigma_{y}^{2}(N_{1},M_{1},T_{% 1},\tau_{1})\right\rangle.
  150. f H f_{H}
  151. τ \tau
  152. τ 1 2 π f H . \tau\gg\frac{1}{2\pi f_{H}}.
  153. f H f_{H}
  154. n τ 0 n\tau_{0}
  155. τ 0 \tau_{0}
  156. τ \tau

Allele.html

  1. p + q = 1 p+q=1\,
  2. p 2 + 2 p q + q 2 = 1 p^{2}+2pq+q^{2}=1\,
  3. p + q + r = 1 p+q+r=1\,
  4. p 2 + 2 p q + 2 p r + q 2 + 2 q r + r 2 = 1. p^{2}+2pq+2pr+q^{2}+2qr+r^{2}=1.\,
  5. G = a ( a + 1 ) 2 . G=\frac{a(a+1)}{2}.

Almost_all.html

  1. ( n ) P ( n ) . (\forall^{\infty}n)P(n).

Almost_everywhere.html

  1. a b f ( x ) d x 0 \int_{a}^{b}f(x)\,dx\geq 0
  2. a b | f ( x ) | d x < \int_{a}^{b}|f(x)|\,dx<\infty
  3. 1 2 ϵ x - ϵ x + ϵ f ( t ) d t \frac{1}{2\epsilon}\int_{x-\epsilon}^{x+\epsilon}f(t)\,dt
  4. ϵ \epsilon

Alpha_Centauri.html

  1. [ ( 11.2 + 35.6 ) / 2 ] 3 / 79.91 2 = 2.0 \begin{smallmatrix}[(11.2+35.6)/2]^{3}/79.91^{2}=2.0\end{smallmatrix}

Alpha_compositing.html

  1. C o = C a α a + C b α b ( 1 - α a ) α a + α b ( 1 - α a ) C_{o}=\frac{C_{a}\alpha_{a}+C_{b}\alpha_{b}\left(1-\alpha_{a}\right)}{\alpha_{% a}+\alpha_{b}\left(1-\alpha_{a}\right)}
  2. C o C_{o}
  3. C a C_{a}
  4. C b C_{b}
  5. α a \alpha_{a}
  6. α b \alpha_{b}
  7. c i = α i C i c_{i}=\alpha_{i}C_{i}
  8. c o = c a + c b ( 1 - α a ) c_{o}=c_{a}+c_{b}\left(1-\alpha_{a}\right)
  9. α o = c o C o = α a + α b ( 1 - α a ) \alpha_{o}=\frac{c_{o}}{C_{o}}=\alpha_{a}+\alpha_{b}\left(1-\alpha_{a}\right)
  10. a b a\odot b
  11. α b = 1 \alpha_{b}=1
  12. a a
  13. b b
  14. C o = α a C a + ( 1 - α a ) C b C_{o}=\alpha_{a}C_{a}+(1-\alpha_{a})C_{b}
  15. ( a b ) c = a ( b c ) (a\odot b)\odot c=a\odot(b\odot c)
  16. a a
  17. b b
  18. c c
  19. o = a b o=a\odot b
  20. o c = a ( b c ) o\odot c=a\odot(b\odot c)
  21. c c
  22. b c b\odot c
  23. \odot
  24. α o C o + ( 1 - α o ) C c \displaystyle\alpha_{o}C_{o}+(1-\alpha_{o})C_{c}
  25. X 0 + Y 0 C c = X 1 + Y 1 C c X_{0}+Y_{0}C_{c}=X_{1}+Y_{1}C_{c}
  26. X 0 = X 1 X_{0}=X_{1}
  27. Y 0 = Y 1 Y_{0}=Y_{1}
  28. α o = 1 - ( 1 - α a ) ( 1 - α b ) , C o = α a C a + ( 1 - α a ) α b C b α o , \begin{aligned}\displaystyle\alpha_{o}&\displaystyle=1-(1-\alpha_{a})(1-\alpha% _{b}),\\ \displaystyle C_{o}&\displaystyle=\frac{\alpha_{a}C_{a}+(1-\alpha_{a})\alpha_{% b}C_{b}}{\alpha_{o}},\end{aligned}
  29. a b a\odot b
  30. ( 1 - α a ) α b = α o - α a (1-\alpha_{a})\alpha_{b}=\alpha_{o}-\alpha_{a}
  31. C o = α a α o C a + ( 1 - α a α o ) C b C_{o}=\frac{\alpha_{a}}{\alpha_{o}}C_{a}+\left(1-\frac{\alpha_{a}}{\alpha_{o}}% \right)C_{b}
  32. \odot
  33. e e
  34. e a = a e = a e\odot a=a\odot e=a
  35. C , α \langle C,\alpha\rangle
  36. α = 0 \alpha=0
  37. { out A = src A + dst A ( 1 - src A ) out R G B = ( src R G B src A + dst R G B dst A ( 1 - src A ) ) ÷ out A out A = 0 out R G B = 0 \begin{cases}\mathrm{out}_{A}=\mathrm{src}_{A}+\mathrm{dst}_{A}(1-\mathrm{src}% _{A})\\ \mathrm{out}_{RGB}=\bigl(\mathrm{src}_{RGB}\mathrm{src}_{A}+\mathrm{dst}_{RGB}% \mathrm{dst}_{A}\left(1-\mathrm{src}_{A}\right)\bigr)\div\mathrm{out}_{A}\\ \mathrm{out}_{A}=0\Rightarrow\mathrm{out}_{RGB}=0\end{cases}
  38. d s t A = 1 dst_{A}=1
  39. { out A = 1 out R G B = src R G B src A + dst R G B ( 1 - src A ) \begin{cases}\mathrm{out}_{A}=1\\ \mathrm{out}_{RGB}=\mathrm{src}_{RGB}\mathrm{src}_{A}+\mathrm{dst}_{RGB}(1-% \mathrm{src}_{A})\end{cases}
  40. { out A = src A + dst A ( 1 - src A ) out R G B = src R G B + dst R G B ( 1 - src A ) \begin{cases}\mathrm{out}_{A}=\mathrm{src}_{A}+\mathrm{dst}_{A}(1-\mathrm{src}% _{A})\\ \mathrm{out}_{RGB}=\mathrm{src}_{RGB}+\mathrm{dst}_{RGB}\left(1-\mathrm{src}_{% A}\right)\end{cases}

Alpha_decay.html

  1. U 92 238 Th 90 234 + α \mathrm{~{}^{238}_{92}U}\rightarrow\mathrm{~{}^{234}_{90}Th}+{\alpha}
  2. U 92 238 Th 90 234 + He 2 4 \mathrm{~{}^{238}_{92}U}\rightarrow\mathrm{~{}^{234}_{90}Th}+\mathrm{~{}^{4}_{% 2}He}

Alpha_helix.html

  1. i + 4 i i+4\rightarrow i
  2. i + 4 i i+4\rightarrow i
  3. i + 3 i i+3\rightarrow i
  4. i + 5 i i+5\rightarrow i
  5. 3 cos Ω = 1 - 4 cos 2 [ ( ϕ + ψ ) / 2 ] 3\cos\Omega=1-4\cos^{2}\left[\left(\phi+\psi\right)/2\right]
  6. C α \mathrm{C^{\alpha}}
  7. C β \mathrm{C^{\beta}}
  8. C \mathrm{C^{\prime}}
  9. Δ ( Δ G ) \Delta(\Delta G)

Alternating_current.html

  1. P L P_{\rm L}
  2. P L = I 2 R . P_{\rm L}=I^{2}R\,.
  3. P T = I V . P_{\rm T}=IV\,.
  4. v ( t ) = V peak sin ( ω t ) v(t)=V_{\mathrm{peak}}\cdot\sin(\omega t)
  5. V peak \displaystyle V_{\rm peak}
  6. ω \displaystyle\omega
  7. f \displaystyle f
  8. ω = 2 π f \displaystyle\omega=2\pi f
  9. t \displaystyle t
  10. sin ( x ) \sin(x)
  11. + V peak +V_{\rm peak}
  12. - V peak -V_{\rm peak}
  13. V pp V_{\rm pp}
  14. V P - P V_{\rm P-P}
  15. V peak - ( - V peak ) = 2 V peak V_{\rm peak}-(-V_{\rm peak})=2V_{\rm peak}
  16. p ( t ) = v 2 ( t ) R p(t)=\frac{v^{2}(t)}{R}
  17. R R
  18. p ( t ) p(t)
  19. V rms V_{\rm rms}
  20. P time averaged = V 2 rms R . P_{\rm time~{}averaged}=\frac{{V^{2}}_{\rm rms}}{R}.
  21. v ( t ) = V peak sin ( ω t ) v(t)=V_{\mathrm{peak}}\sin(\omega t)
  22. i ( t ) = v ( t ) R = V peak R sin ( ω t ) i(t)=\frac{v(t)}{R}=\frac{V_{\mathrm{peak}}}{R}\sin(\omega t)
  23. P ( t ) = v ( t ) i ( t ) = ( V peak ) 2 R sin 2 ( ω t ) P(t)=v(t)\ i(t)=\frac{(V_{\mathrm{peak}})^{2}}{R}\sin^{2}(\omega t)
  24. sin 2 x = 1 - cos 2 x 2 \sin^{2}x=\frac{1-\cos 2x}{2}
  25. V rms \displaystyle V_{\mathrm{rms}}
  26. 2 \sqrt{2}
  27. V rms = V peak 3 . V_{\mathrm{rms}}=\frac{V_{\mathrm{peak}}}{\sqrt{3}}.
  28. V rms = V peak . \displaystyle V_{\mathrm{rms}}=V_{\mathrm{peak}}.
  29. v ( t ) v(t)
  30. T T
  31. V rms = 1 T 0 T [ v ( t ) ] 2 d t . V_{\mathrm{rms}}=\sqrt{\frac{1}{T}\int_{0}^{T}{[v(t)]^{2}dt}}.
  32. V peak = 2 V rms . V_{\mathrm{peak}}=\sqrt{2}\ V_{\mathrm{rms}}.
  33. V peak \scriptstyle V_{\mathrm{peak}}
  34. 230 V × 2 \scriptstyle 230V\times\sqrt{2}

Alternating_group.html

  1. n n
  2. Aut ( A n ) \mbox{Aut}~{}(A_{n})
  3. Out ( A n ) \mbox{Out}~{}(A_{n})
  4. n 4 , n 6 n\geq 4,n\neq 6
  5. S n \mathrm{S}_{n}\,
  6. C 2 \mathrm{C}_{2}\,
  7. n = 1 , 2 n=1,2\,
  8. 1 1\,
  9. 1 1\,
  10. n = 3 n=3\,
  11. C 2 \mathrm{C}_{2}\,
  12. C 2 \mathrm{C}_{2}\,
  13. n = 6 n=6\,
  14. S 6 C 2 \mathrm{S}_{6}\rtimes\mathrm{C}_{2}
  15. V = C 2 × C 2 \mathrm{V}=\mathrm{C}_{2}\times\mathrm{C}_{2}
  16. A n \mathrm{A}_{n}
  17. H 1 ( A n , 𝐙 ) = 0 H_{1}(\mathrm{A}_{n},\mathbf{Z})=0
  18. n = 0 , 1 , 2 n=0,1,2
  19. H 1 ( A 3 , 𝐙 ) = A 3 ab = A 3 = 𝐙 / 3 H_{1}(\mathrm{A}_{3},\mathbf{Z})=\mathrm{A}_{3}^{\,\text{ab}}=\mathrm{A}_{3}=% \mathbf{Z}/3
  20. H 1 ( A 4 , 𝐙 ) = A 4 ab = 𝐙 / 3 H_{1}(\mathrm{A}_{4},\mathbf{Z})=\mathrm{A}_{4}^{\,\text{ab}}=\mathbf{Z}/3
  21. H 1 ( A n , 𝐙 ) = 0 H_{1}(\mathrm{A}_{n},\mathbf{Z})=0
  22. n 5 n\geq 5
  23. A n \mathrm{A}_{n}
  24. A n C 3 , \mathrm{A}_{n}\to\mathrm{C}_{3},
  25. n 5 n\geq 5
  26. n < 3 n<3
  27. A n \mathrm{A}_{n}
  28. A 3 \mathrm{A}_{3}
  29. A 4 \mathrm{A}_{4}
  30. A 3 C 3 \mathrm{A}_{3}\twoheadrightarrow\mathrm{C}_{3}
  31. A 4 C 3 . \mathrm{A}_{4}\twoheadrightarrow\mathrm{C}_{3}.
  32. H 2 ( A n , 𝐙 ) = 0 H_{2}(\mathrm{A}_{n},\mathbf{Z})=0
  33. n = 1 , 2 , 3 n=1,2,3
  34. H 2 ( A n , 𝐙 ) = 𝐙 / 2 H_{2}(\mathrm{A}_{n},\mathbf{Z})=\mathbf{Z}/2
  35. n = 4 , 5 n=4,5
  36. H 2 ( A n , 𝐙 ) = 𝐙 / 6 H_{2}(\mathrm{A}_{n},\mathbf{Z})=\mathbf{Z}/6
  37. n = 6 , 7 n=6,7
  38. H 2 ( A n , 𝐙 ) = 𝐙 / 2 H_{2}(\mathrm{A}_{n},\mathbf{Z})=\mathbf{Z}/2
  39. n 8 n\geq 8

Alternative_algebra.html

  1. x ( x y ) = ( x x ) y x(xy)=(xx)y
  2. ( y x ) x = y ( x x ) (yx)x=y(xx)
  3. [ x , y , z ] = ( x y ) z - x ( y z ) [x,y,z]=(xy)z-x(yz)
  4. [ x , x , y ] = 0 [x,x,y]=0
  5. [ y , x , x ] = 0. [y,x,x]=0.
  6. [ x σ ( 1 ) , x σ ( 2 ) , x σ ( 3 ) ] = sgn ( σ ) [ x 1 , x 2 , x 3 ] [x_{\sigma(1)},x_{\sigma(2)},x_{\sigma(3)}]=\operatorname{sgn}(\sigma)[x_{1},x% _{2},x_{3}]
  7. [ x , y , x ] = 0 [x,y,x]=0
  8. ( x y ) x = x ( y x ) . (xy)x=x(yx).
  9. x ( x y ) = ( x x ) y x(xy)=(xx)y
  10. ( y x ) x = y ( x x ) (yx)x=y(xx)
  11. ( x y ) x = x ( y x ) . (xy)x=x(yx).
  12. x , y , z x,y,z
  13. [ x , y , z ] = 0 [x,y,z]=0
  14. a ( x ( a y ) ) = ( a x a ) y a(x(ay))=(axa)y
  15. ( ( x a ) y ) a = x ( a y a ) ((xa)y)a=x(aya)
  16. ( a x ) ( y a ) = a ( x y ) a (ax)(ya)=a(xy)a
  17. x x
  18. y y
  19. y = x - 1 ( x y ) . y=x^{-1}(xy).
  20. [ x - 1 , x , y ] [x^{-1},x,y]
  21. x x
  22. y y
  23. x x
  24. y y
  25. x y xy
  26. ( x y ) - 1 = y - 1 x - 1 (xy)^{-1}=y^{-1}x^{-1}
  27. a a * a\mapsto a^{*}

Altimeter.html

  1. z = c T log ( P o / P ) , z=c\;T\;\log(P_{o}/P),

Ambiguity.html

  1. f = f ( x ) f=f(x)
  2. f = f ( y + 1 ) f=f(y+1)
  3. f = f ( x ) f=f(x)
  4. ( y + 1 ) (y+1)
  5. f f
  6. ( y + 1 ) (y+1)
  7. a / b c a/bc
  8. a / ( b c ) a/(bc)
  9. s i n α sin\alpha
  10. s s
  11. i i
  12. n n
  13. α \alpha
  14. sin [ α ] \sin[\alpha]
  15. T m n k T_{mnk}
  16. m m
  17. n n
  18. k k
  19. T m n k T_{mnk}
  20. T m , n , k T_{m,n,k}
  21. F x F_{x}
  22. sin 2 α / 2 \sin^{2}\alpha/2\,
  23. ( sin ( α / 2 ) ) 2 (\sin(\alpha/2))^{2}\,
  24. ( sin ( α ) ) 2 / 2 (\sin(\alpha))^{2}/2\,
  25. sin 2 ( x ) \sin^{2}(x)
  26. sin ( sin ( x ) ) \sin(\sin(x))
  27. exp 2 ( x ) \exp^{2}(x)
  28. exp ( exp ( x ) ) \exp(\exp(x))
  29. sin - 1 α \sin^{-1}\alpha
  30. arcsin ( α ) \arcsin(\alpha)
  31. ( sin ( α ) ) - 1 (\sin(\alpha))^{-1}
  32. sin n α \sin^{n}\alpha
  33. ( sin ( α ) ) n (\sin(\alpha))^{n}\,
  34. a / 2 b a/2b\,
  35. ( a / 2 ) b (a/2)b\,
  36. a / ( 2 b ) a/(2b)\,
  37. | α ~{}|\alpha\rangle~{}
  38. | n ~{}|n\rangle~{}
  39. n ~{}n~{}
  40. | x ~{}|x\rangle~{}
  41. | p ~{}|p\rangle~{}
  42. | 1 |1\rangle
  43. X = Y X=Y
  44. X = 2 , X = 3 X=2,X=3
  45. f ( ω ) = f ( t ) exp ( i ω t ) d t ~{}f(\omega)=\int f(t)\exp(i\omega t){\rm d}t
  46. f = 0 ~{}f=0~{}
  47. f ~{}f~{}
  48. f ( ω ) = 1 2 π f ( t ) exp ( i ω t ) d t ~{}f(\omega)=\frac{1}{\sqrt{2\pi}}\int f(t)\exp(i\omega t){\rm d}t
  49. | f ( x ) | 2 d x \int|f(x)|^{2}{\rm d}x
  50. t ~{}t~{}
  51. τ ~{}\tau~{}
  52. ω ~{}\omega~{}
  53. f ( ω t ) ~{}f(\omega t)~{}
  54. f ( y ) ~{}f(y)~{}
  55. t ~{}t~{}
  56. ω ~{}\omega~{}

Amdahl's_law.html

  1. n n\in\mathbb{N}
  2. B [ 0 , 1 ] B\in[0,1]
  3. T ( n ) T\left(n\right)
  4. n n
  5. T ( n ) = T ( 1 ) ( B + 1 n ( 1 - B ) ) T(n)=T(1)\left(B+\frac{1}{n}\left(1-B\right)\right)
  6. S ( n ) S(n)
  7. n n
  8. S ( n ) = T ( 1 ) T ( n ) = T ( 1 ) T ( 1 ) ( B + 1 n ( 1 - B ) ) = 1 B + 1 n ( 1 - B ) S(n)=\frac{T\left(1\right)}{T\left(n\right)}=\frac{T\left(1\right)}{T\left(1% \right)\left(B+\frac{1}{n}\left(1-B\right)\right)}=\frac{1}{B+\frac{1}{n}\left% (1-B\right)}
  9. 1 ( 1 - P ) + P S = 1 ( 1 - 0.3 ) + 0.3 2 = 1.1765 \frac{1}{(1-P)+\frac{P}{S}}=\frac{1}{(1-0.3)+\frac{0.3}{2}}=1.1765
  10. 0.11 1 + 0.18 5 + 0.23 20 + 0.48 1.6 = 0.4575. \frac{0.11}{1}+\frac{0.18}{5}+\frac{0.23}{20}+\frac{0.48}{1.6}=0.4575.
  11. 1 / 2 {1}/{2}
  12. S ( N ) = 1 ( 1 - P ) + P N S(N)=\frac{1}{(1-P)+\frac{P}{N}}
  13. P estimated = 1 S U - 1 1 N P - 1 P\text{estimated}=\frac{\frac{1}{SU}-1}{\frac{1}{NP}-1}
  14. 1 / ( 1 - P ) \scriptstyle 1/(1\,-\,P)
  15. p p
  16. maximum speedup p 1 + f ( p - 1 ) \,\text{maximum speedup }\leq\frac{p}{1+f\cdot(p-1)}
  17. f \scriptstyle f
  18. 0 < f < 1 \scriptstyle 0\;<\;f\;<\;1
  19. p = 5 \scriptstyle p\;=\;5
  20. t A = 3 \scriptstyle t_{A}\;=\;3
  21. t B = 1 \scriptstyle t_{B}\;=\;1
  22. f = t A / ( t A + t B ) = 0.75 \scriptstyle f\;=\;t_{A}/(t_{A}\,+\,t_{B})\;=\;0.75
  23. maximum speedup 5 1 + 0.75 ( 5 - 1 ) = 1.25 \,\text{maximum speedup }\leq\frac{5}{1+0.75\cdot(5-1)}=1.25
  24. p = 2 \scriptstyle p\;=\;2
  25. t B = 1 \scriptstyle t_{B}\;=\;1
  26. t A = 3 \scriptstyle t_{A}\;=\;3
  27. f = t B / ( t A + t B ) = 0.25 \scriptstyle f\;=\;t_{B}/(t_{A}\,+\,t_{B})\;=\;0.25
  28. maximum speedup 2 1 + 0.25 ( 2 - 1 ) = 1.60 \,\text{maximum speedup }\leq\frac{2}{1+0.25\cdot(2-1)}=1.60
  29. percentage improvement = ( 1 - 1 speedup factor ) 100 \,\text{percentage improvement}=\left(1-\frac{1}{\,\text{speedup factor}}% \right)\cdot 100

Americium.html

  1. Pu 94 239 ( n , γ ) 94 240 Pu ( n , γ ) 94 241 Pu β - 14.35 yr 95 241 Am ( 432.2 yr 𝛼 93 237 Np ) \mathrm{{}^{239}_{\ 94}Pu\ \xrightarrow{(n,\gamma)}\ ^{240}_{\ 94}Pu\ % \xrightarrow{(n,\gamma)}\ ^{241}_{\ 94}Pu\ \xrightarrow[14.35\ yr]{\beta^{-}}% \ ^{241}_{\ 95}Am\ \left(\ \xrightarrow[432.2\ yr]{\alpha}\ ^{237}_{\ 93}Np% \right)}
  2. Am 95 241 ( n , γ ) 95 242 Am ( β - 16.02 h 96 242 Cm ) \mathrm{{}^{241}_{\ 95}Am\ \xrightarrow{(n,\gamma)}\ ^{242}_{\ 95}Am\ \left(\ % \xrightarrow[16.02\ h]{\beta^{-}}\ ^{242}_{\ 96}Cm\right)}
  3. U 92 238 ( n , γ ) 92 239 U β - 23.5 min 93 239 Np β - 2.3565 d 94 239 Pu \mathrm{{}^{238}_{\ 92}U\ \xrightarrow{(n,\gamma)}\ ^{239}_{\ 92}U\ % \xrightarrow[23.5\ min]{\beta^{-}}\ ^{239}_{\ 93}Np\ \xrightarrow[2.3565\ d]{% \beta^{-}}\ ^{239}_{\ 94}Pu}
  4. Pu 94 239 2 ( n , γ ) 94 241 Pu β - 14.35 yr 95 241 Am \mathrm{{}^{239}_{\ 94}Pu\ \xrightarrow{2(n,\gamma)}\ ^{241}_{\ 94}Pu\ % \xrightarrow[14.35\ yr]{\beta^{-}}\ ^{241}_{\ 95}Am}
  5. Am 95 241 ( n , γ ) 95 242 Am \mathrm{{}^{241}_{\ 95}Am\ \xrightarrow{(n,\gamma)}\ ^{242}_{\ 95}Am}
  6. Am 95 241 ( n , γ ) 95 242 m Am \mathrm{{}^{241}_{\ 95}Am\ \xrightarrow{(n,\gamma)}\ ^{242m}_{\ \ \ 95}Am}
  7. Pu 94 239 4 ( n , γ ) 94 243 Pu β - 4.956 h 95 243 Am \mathrm{{}^{239}_{\ 94}Pu\ \xrightarrow{4(n,\gamma)}\ ^{243}_{\ 94}Pu\ % \xrightarrow[4.956\ h]{\beta^{-}}\ ^{243}_{\ 95}Am}
  8. 2 AmF 3 + 3 Ba 2 Am + 3 BaF 2 \mathrm{2\ AmF_{3}\ +\ 3\ Ba\ \longrightarrow\ 2\ Am\ +\ 3\ BaF_{2}}
  9. 3 AmO 2 + 4 La 3 Am + 2 La 2 O 3 \mathrm{3\ AmO_{2}\ +\ 4\ La\ \longrightarrow\ 3\ Am\ +\ 2\ La_{2}O_{3}}
  10. 3 ¯ \overline{3}
  11. × 10 - 6 \times 10^{-}6
  12. × 10 - 6 \times 10^{-}6
  13. 3 AmO 2 + + 4 H + 2 AmO 2 2 + + Am 3 + + 2 H 2 O \mathrm{3\ AmO_{2}^{+}\ +\ 4\ H^{+}\ \longrightarrow\ 2\ AmO_{2}^{2+}\ +\ Am^{% 3+}\ +\ 2\ H_{2}O}
  14. 2 Am ( V ) Am ( VI ) + Am ( IV ) \mathrm{2\ Am(V)\ \longrightarrow\ Am(VI)\ +\ Am(IV)}
  15. Am + HgX 2 400 - 500 C AmX 2 + Hg \mathrm{\ Am\ +\ HgX_{2}\ \xrightarrow{400-500^{\circ}C}\ AmX_{2}\ +\ Hg\ }
  16. Am ( aq ) 3 + + 3 F ( aq ) - AmF 3 ( s ) \mathrm{Am^{3+}\ _{(aq)}+\ 3\ F^{-}\ _{(aq)}\longrightarrow\ AmF_{3}\ {}_{(s)}\downarrow}
  17. 2 AmF 3 + F 2 2 AmF 4 \mathrm{2\ AmF_{3}\ +\ F_{2}\ \longrightarrow\ 2\ AmF_{4}}
  18. 3 ¯ \overline{3}
  19. AmCl 3 + H 2 O AmOCl + 2 HCl \mathrm{AmCl_{3}\ +\ \ H_{2}O\ \longrightarrow\ AmOCl\ +\ 2\ HCl}
  20. Am 95 241 93 237 Np + 2 4 He + 2 4 γ \mathrm{{}^{241\!\,}_{\ 95}Am\ \longrightarrow\ ^{237}_{\ 93}Np\ +\ ^{4}_{2}He% \ +\ \gamma}
  21. Be 4 9 + 2 4 He 6 12 C + 0 1 n + 0 1 γ \mathrm{{}^{9}_{4}Be\ +\ ^{4}_{2}He\ \longrightarrow\ ^{12}_{\ 6}C\ +\ ^{1}_{0% }n\ +\ \gamma}
  22. Am 95 243 ( n , γ ) 95 244 Am β - 10.1 h 96 244 Cm \mathrm{{}^{243}_{\ 95}Am\ \xrightarrow{(n,\gamma)}\ ^{244}_{\ 95}Am\ % \xrightarrow[10.1\ h]{\beta^{-}}\ ^{244}_{\ 96}Cm}

Amicable_numbers.html

  1. ( m , n ) (m,n)
  2. σ ( m ) - m = n \sigma(m)-m=n
  3. σ ( n ) - n = m \sigma(n)-n=m
  4. σ ( m ) = σ ( n ) = m + n \sigma(m)=\sigma(n)=m+n
  5. ( n 1 , n 2 , , n k ) (n_{1},n_{2},\ldots,n_{k})
  6. σ ( n 1 ) = σ ( n 2 ) = = σ ( n k ) = n 1 + n 2 + + n k \sigma(n_{1})=\sigma(n_{2})=\dots=\sigma(n_{k})=n_{1}+n_{2}+\dots+n_{k}
  7. 1264460 1547860 1727636 1305184 1264460 1264460\mapsto 1547860\mapsto 1727636\mapsto 1305184\mapsto 1264460\mapsto\dots

Amide.html

  1. \overrightarrow{\leftarrow}

Ampere.html

  1. 1 A = 1 C s . \rm 1\ A=1\tfrac{C}{s}.
  2. P ( t ) = I ( t ) V ( t ) P(t)=I(t)\cdot V(t)\,

Amplitude.html

  1. U ^ \scriptstyle\hat{U}
  2. 2 U ^ \scriptstyle 2\hat{U}
  3. U ^ / 2 \scriptstyle\hat{U}/\sqrt{2}
  4. x = A sin ( ω ( t - K ) ) + b , x=A\sin(\omega(t-K))+b\ ,
  5. Peak-to-peak = 2 2 × RMS 2.8 × RMS \mbox{Peak-to-peak}~{}=2\sqrt{2}\times\mbox{RMS}~{}\approx 2.8\times\mbox{RMS}% ~{}\,

Amplitude_modulation.html

  1. c ( t ) = A sin ( 2 π f c t ) c(t)=A\cdot\sin(2\pi f_{c}t)\,
  2. m ( t ) = M cos ( 2 π f m t + ϕ ) m(t)=M\cdot\cos(2\pi f_{m}t+\phi)\,
  3. y ( t ) y(t)\,
  4. = [ 1 + m ( t ) ] c ( t ) =[1+m(t)]\cdot c(t)\,
  5. = [ 1 + M cos ( 2 π f m t + ϕ ) ] A sin ( 2 π f c t ) =[1+M\cdot\cos(2\pi f_{m}t+\phi)]\cdot A\cdot\sin(2\pi f_{c}t)
  6. y ( t ) = A sin ( 2 π f c t ) + A M 2 [ sin ( 2 π ( f c + f m ) t + ϕ ) + sin ( 2 π ( f c - f m ) t - ϕ ) ] . y(t)=A\cdot\sin(2\pi f_{c}t)+\begin{matrix}\frac{AM}{2}\end{matrix}\left[\sin(% 2\pi(f_{c}+f_{m})t+\phi)+\sin(2\pi(f_{c}-f_{m})t-\phi)\right].\,
  7. h = peak value of m ( t ) A = M A h=\frac{\mathrm{peak\ value\ of\ }m(t)}{A}=\frac{M}{A}
  8. M M\,
  9. A A\,
  10. h = 0.5 h=0.5
  11. h = 1.0 h=1.0

Analog-to-digital_converter.html

  1. Q = E FSR < m t p l > 2 M , Q=\dfrac{E_{\mathrm{FSR}}}{<}mtpl>{{2^{M}}},
  2. E FSR = V RefHi - V RefLow , E_{\mathrm{FSR}}=V_{\mathrm{RefHi}}-V_{\mathrm{RefLow}},\,
  3. N = 2 M , N=2^{M},\,
  4. SQNR = 20 log 10 ( 2 Q ) 6.02 Q dB \mathrm{SQNR}=20\log_{10}(2^{Q})\approx 6.02\cdot Q\ \mathrm{dB}\,\!
  5. x ( t ) = A sin ( 2 π f 0 t ) x(t)=A\sin{(2\pi f_{0}t)}
  6. Δ t \Delta t
  7. E a p | x ( t ) Δ t | 2 A π f 0 Δ t E_{ap}\leq|x^{\prime}(t)\Delta t|\leq 2A\pi f_{0}\Delta t
  8. Δ t < 1 2 q π f 0 \Delta t<\frac{1}{2^{q}\pi f_{0}}

Analogy_of_the_Divided_Line.html

  1. 1 \scriptstyle 1
  2. x \scriptstyle x
  3. 0 < x < 1 \scriptstyle 0<x<1
  4. 0 < x < 1 / 2 \scriptstyle 0<x<1/2
  5. 1 - x \scriptstyle 1-x
  6. x - x × x ( 1 - x ) × x \scriptstyle x-x\times x\equiv(1-x)\times x

Analysis_of_algorithms.html

  1. { t 1 , t 2 } \{t1,t2\}
  2. { n 1 , n 2 } \{n1,n2\}
  3. t 2 / t 1 = ( n 2 / n 1 ) a t_{2}/t_{1}=(n_{2}/n_{1})^{a}
  4. a = log ( t 2 / t 1 ) / log ( n 2 / n 1 ) a=\log(t_{2}/t_{1})/\log(n_{2}/n_{1})
  5. T 1 + T 2 + T 3 + T 7 . T_{1}+T_{2}+T_{3}+T_{7}.\,
  6. T 6 + 2 T 6 + 3 T 6 + + ( n - 1 ) T 6 + n T 6 T_{6}+2T_{6}+3T_{6}+\cdots+(n-1)T_{6}+nT_{6}
  7. T 6 [ 1 + 2 + 3 + + ( n - 1 ) + n ] = T 6 [ 1 2 ( n 2 + n ) ] T_{6}\left[1+2+3+\cdots+(n-1)+n\right]=T_{6}\left[\frac{1}{2}(n^{2}+n)\right]
  8. 2 T 5 + 3 T 5 + 4 T 5 + + ( n - 1 ) T 5 + n T 5 + ( n + 1 ) T 5 2T_{5}+3T_{5}+4T_{5}+\cdots+(n-1)T_{5}+nT_{5}+(n+1)T_{5}
  9. = T 5 + 2 T 5 + 3 T 5 + 4 T 5 + + ( n - 1 ) T 5 + n T 5 + ( n + 1 ) T 5 - T 5 =T_{5}+2T_{5}+3T_{5}+4T_{5}+\cdots+(n-1)T_{5}+nT_{5}+(n+1)T_{5}-T_{5}
  10. T 5 [ 1 + 2 + 3 + + ( n - 1 ) + n + ( n + 1 ) ] - T 5 = [ 1 2 ( n 2 + n ) ] T 5 + ( n + 1 ) T 5 - T 5 = T 5 [ 1 2 ( n 2 + n ) ] + n T 5 = [ 1 2 ( n 2 + 3 n ) ] T 5 T_{5}\left[1+2+3+\cdots+(n-1)+n+(n+1)\right]-T_{5}=\left[\frac{1}{2}(n^{2}+n)% \right]T_{5}+(n+1)T_{5}-T_{5}=T_{5}\left[\frac{1}{2}(n^{2}+n)\right]+nT_{5}=% \left[\frac{1}{2}(n^{2}+3n)\right]T_{5}
  11. f ( n ) = T 1 + T 2 + T 3 + T 7 + ( n + 1 ) T 4 + [ 1 2 ( n 2 + n ) ] T 6 + [ 1 2 ( n 2 + 3 n ) ] T 5 f(n)=T_{1}+T_{2}+T_{3}+T_{7}+(n+1)T_{4}+\left[\frac{1}{2}(n^{2}+n)\right]T_{6}% +\left[\frac{1}{2}(n^{2}+3n)\right]T_{5}
  12. f ( n ) = [ 1 2 ( n 2 + n ) ] T 6 + [ 1 2 ( n 2 + 3 n ) ] T 5 + ( n + 1 ) T 4 + T 1 + T 2 + T 3 + T 7 f(n)=\left[\frac{1}{2}(n^{2}+n)\right]T_{6}+\left[\frac{1}{2}(n^{2}+3n)\right]% T_{5}+(n+1)T_{4}+T_{1}+T_{2}+T_{3}+T_{7}
  13. [ 1 2 ( n 2 + n ) ] T 6 + [ 1 2 ( n 2 + 3 n ) ] T 5 + ( n + 1 ) T 4 + T 1 + T 2 + T 3 + T 7 c n 2 , n n 0 \left[\frac{1}{2}(n^{2}+n)\right]T_{6}+\left[\frac{1}{2}(n^{2}+3n)\right]T_{5}% +(n+1)T_{4}+T_{1}+T_{2}+T_{3}+T_{7}\leq cn^{2},\ n\geq n_{0}
  14. [ 1 2 ( n 2 + n ) ] T 6 + [ 1 2 ( n 2 + 3 n ) ] T 5 + ( n + 1 ) T 4 + T 1 + T 2 + T 3 + T 7 \left[\frac{1}{2}(n^{2}+n)\right]T_{6}+\left[\frac{1}{2}(n^{2}+3n)\right]T_{5}% +(n+1)T_{4}+T_{1}+T_{2}+T_{3}+T_{7}
  15. ( n 2 + n ) T 6 + ( n 2 + 3 n ) T 5 + ( n + 1 ) T 4 + T 1 + T 2 + T 3 + T 7 \leq(n^{2}+n)T_{6}+(n^{2}+3n)T_{5}+(n+1)T_{4}+T_{1}+T_{2}+T_{3}+T_{7}
  16. T 6 ( n 2 + n ) + T 5 ( n 2 + 3 n ) + ( n + 1 ) T 4 + T 1 + T 2 + T 3 + T 7 k ( n 2 + n ) + k ( n 2 + 3 n ) + k n + 5 k T_{6}(n^{2}+n)+T_{5}(n^{2}+3n)+(n+1)T_{4}+T_{1}+T_{2}+T_{3}+T_{7}\leq k(n^{2}+% n)+k(n^{2}+3n)+kn+5k
  17. = 2 k n 2 + 5 k n + 5 k 2 k n 2 + 5 k n 2 + 5 k n 2 =2kn^{2}+5kn+5k\leq 2kn^{2}+5kn^{2}+5kn^{2}
  18. = 12 k n 2 =12kn^{2}
  19. [ 1 2 ( n 2 + n ) ] T 6 + [ 1 2 ( n 2 + 3 n ) ] T 5 + ( n + 1 ) T 4 + T 1 + T 2 + T 3 + T 7 c n 2 , n n 0 \left[\frac{1}{2}(n^{2}+n)\right]T_{6}+\left[\frac{1}{2}(n^{2}+3n)\right]T_{5}% +(n+1)T_{4}+T_{1}+T_{2}+T_{3}+T_{7}\leq cn^{2},n\geq n_{0}
  20. c = 12 k , n 0 = 1 c=12k,n_{0}=1
  21. 4 + i = 1 n i 4 + i = 1 n n = 4 + n 2 5 n 2 4+\sum_{i=1}^{n}i\leq 4+\sum_{i=1}^{n}n=4+n^{2}\leq 5n^{2}
  22. = O ( n 2 ) . =O(n^{2}).
  23. K > k log log n K>k\log\log n
  24. K / k > 6 K/k>6
  25. n < 2 2 6 = 2 64 n<2^{2^{6}}=2^{64}
  26. n log n n\log n
  27. n 2 n^{2}
  28. 1 + 2 + 3 + + ( n - 1 ) + n = n ( n + 1 ) 2 1+2+3+\cdots+(n-1)+n=\frac{n(n+1)}{2}

Analysis_of_variance.html

  1. ε \varepsilon
  2. ε N ( 0 , σ 2 ) . \varepsilon\thicksim N(0,\sigma^{2}).\,
  3. y i , j y_{i,j}
  4. i i
  5. j j
  6. y i y_{i}
  7. t j t_{j}
  8. y i , j = y i + t j . y_{i,j}=y_{i}+t_{j}.
  9. j j
  10. j j
  11. t j t_{j}
  12. s 2 = 1 n - 1 ( y i - y ¯ ) 2 s^{2}=\textstyle\frac{1}{n-1}\sum(y_{i}-\bar{y})^{2}
  13. S S Total = S S Error + S S Treatments SS\text{Total}=SS\text{Error}+SS\text{Treatments}
  14. D F Total = D F Error + D F Treatments DF\text{Total}=DF\text{Error}+DF\text{Treatments}
  15. F = variance between treatments variance within treatments F=\frac{\,\text{variance between treatments}}{\,\text{variance within % treatments}}
  16. F = M S Treatments M S Error = S S Treatments / ( I - 1 ) S S Error / ( n T - I ) F=\frac{MS\text{Treatments}}{MS\text{Error}}={{SS\text{Treatments}/(I-1)}\over% {SS\text{Error}/(n_{T}-I)}}
  17. I I
  18. n T n_{T}
  19. I - 1 I-1
  20. n T - I n_{T}-I
  21. 1 + n σ 2 Treatment / σ 2 Error 1+{n\sigma^{2}\text{Treatment}}/{\sigma^{2}\text{Error}}

Analytic_function.html

  1. f ( x ) = n = 0 a n ( x - x 0 ) n = a 0 + a 1 ( x - x 0 ) + a 2 ( x - x 0 ) 2 + a 3 ( x - x 0 ) 3 + f(x)=\sum_{n=0}^{\infty}a_{n}\left(x-x_{0}\right)^{n}=a_{0}+a_{1}(x-x_{0})+a_{% 2}(x-x_{0})^{2}+a_{3}(x-x_{0})^{3}+\cdots
  2. T ( x ) = n = 0 f ( n ) ( x 0 ) n ! ( x - x 0 ) n T(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(x_{0})}{n!}(x-x_{0})^{n}
  3. | d k f d x k ( x ) | C k + 1 k ! \left|\frac{d^{k}f}{dx^{k}}(x)\right|\leq C^{k+1}k!
  4. A ( Ω ) \scriptstyle A_{\infty}(\Omega)
  5. f ( x ) = 1 x 2 + 1 . f(x)=\frac{1}{x^{2}+1}.

Analytic_geometry.html

  1. y = m x + b y=mx+b\,
  2. 𝐫 0 \mathbf{r}_{0}
  3. P 0 = ( x 0 , y 0 , z 0 ) P_{0}=(x_{0},y_{0},z_{0})
  4. 𝐧 = ( a , b , c ) \mathbf{n}=(a,b,c)
  5. P P
  6. 𝐫 \mathbf{r}
  7. P 0 P_{0}
  8. P P
  9. 𝐧 \mathbf{n}
  10. 𝐫 \mathbf{r}
  11. 𝐧 ( 𝐫 - 𝐫 0 ) = 0. \mathbf{n}\cdot(\mathbf{r}-\mathbf{r}_{0})=0.
  12. a ( x - x 0 ) + b ( y - y 0 ) + c ( z - z 0 ) = 0 , a(x-x_{0})+b(y-y_{0})+c(z-z_{0})=0,
  13. a x + b y + c z + d = 0 , where d = - ( a x 0 + b y 0 + c z 0 ) . ax+by+cz+d=0,\,\text{ where }d=-(ax_{0}+by_{0}+cz_{0}).
  14. a x + b y + c z + d = 0 , ax+by+cz+d=0,
  15. 𝐧 = ( a , b , c ) \mathbf{n}=(a,b,c)
  16. x = x 0 + a t x=x_{0}+at\,
  17. y = y 0 + b t y=y_{0}+bt\,
  18. z = z 0 + c t z=z_{0}+ct\,
  19. A x 2 + B x y + C y 2 + D x + E y + F = 0 with A , B , C not all zero. Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0\,\text{ with }A,B,C\,\text{ not all zero.}\,
  20. 𝐏 5 . \mathbf{P}^{5}.
  21. B 2 - 4 A C . B^{2}-4AC.\,
  22. B 2 - 4 A C < 0 B^{2}-4AC<0
  23. A = C A=C
  24. B = 0 B=0
  25. B 2 - 4 A C = 0 B^{2}-4AC=0
  26. B 2 - 4 A C > 0 B^{2}-4AC>0
  27. A + C = 0 A+C=0
  28. i , j = 1 3 x i Q i j x j + i = 1 3 P i x i + R = 0 \sum_{i,j=1}^{3}x_{i}Q_{ij}x_{j}+\sum_{i=1}^{3}P_{i}x_{i}+R=0
  29. d = ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 , d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}},\!
  30. θ = arctan ( m ) \theta=\arctan(m)\!
  31. d = ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 + ( z 2 - z 1 ) 2 , d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}},\!
  32. 𝐀 𝐁 = 𝐀 𝐁 cos θ , \mathbf{A}\cdot\mathbf{B}=\|\mathbf{A}\|\,\|\mathbf{B}\|\cos\theta,
  33. R ( x , y ) R(x,y)
  34. x x
  35. x - h x-h
  36. h h
  37. y y
  38. y - k y-k
  39. k k
  40. x x
  41. x / b x/b
  42. b b
  43. x x
  44. y y
  45. y / a y/a
  46. x x
  47. x cos A + y sin A x\cos A+y\sin A
  48. y y
  49. - x sin A + y cos A -x\sin A+y\cos A
  50. A A
  51. y = 1 / x y=1/x
  52. y = f ( x ) y=f(x)
  53. y = a f ( b ( x - k ) ) + h y=af(b(x-k))+h
  54. a a
  55. a a
  56. x x
  57. b b
  58. a a
  59. y y
  60. k k
  61. h h
  62. h h
  63. k k
  64. h h
  65. k k
  66. R ( x , y ) R(x,y)
  67. x y xy
  68. x 2 + y 2 - 1 = 0 x^{2}+y^{2}-1=0
  69. P ( x , y ) P(x,y)
  70. Q ( x , y ) Q(x,y)
  71. ( x , y ) (x,y)
  72. P P
  73. ( 0 , 0 ) (0,0)
  74. P = { ( x , y ) | x 2 + y 2 = 1 } P=\{(x,y)|x^{2}+y^{2}=1\}
  75. Q Q
  76. ( 1 , 0 ) : Q = { ( x , y ) | ( x - 1 ) 2 + y 2 = 1 } (1,0):Q=\{(x,y)|(x-1)^{2}+y^{2}=1\}
  77. ( 0 , 0 ) (0,0)
  78. ( 0 , 0 ) (0,0)
  79. ( x , y ) (x,y)
  80. Q Q
  81. ( 0 - 1 ) 2 + 0 2 = 1 (0-1)^{2}+0^{2}=1
  82. ( - 1 ) 2 = 1 (-1)^{2}=1
  83. ( 0 , 0 ) (0,0)
  84. Q Q
  85. ( 0 , 0 ) (0,0)
  86. ( x , y ) (x,y)
  87. P P
  88. 0 2 + 0 2 = 1 0^{2}+0^{2}=1
  89. 0 = 1 0=1
  90. ( 0 , 0 ) (0,0)
  91. P P
  92. P P
  93. Q Q
  94. x 2 + y 2 = 1 x^{2}+y^{2}=1
  95. ( x - 1 ) 2 + y 2 = 1 (x-1)^{2}+y^{2}=1
  96. y y
  97. x x
  98. y y
  99. x 2 + y 2 = 1 x^{2}+y^{2}=1
  100. y 2 = 1 - x 2 y^{2}=1-x^{2}
  101. y 2 y^{2}
  102. ( x - 1 ) 2 + ( 1 - x 2 ) = 1 (x-1)^{2}+(1-x^{2})=1
  103. x x
  104. x 2 - 2 x + 1 + 1 - x 2 = 1 x^{2}-2x+1+1-x^{2}=1
  105. - 2 x = - 1 -2x=-1
  106. x = 1 / 2 x=1/2
  107. x x
  108. y y
  109. ( 1 / 2 ) 2 + y 2 = 1 (1/2)^{2}+y^{2}=1
  110. y 2 = 3 / 4 y^{2}=3/4
  111. y = ± 3 2 y=\frac{\pm\sqrt{3}}{2}
  112. ( 1 / 2 , + 3 2 ) and ( 1 / 2 , - 3 2 ) \left(1/2,\frac{+\sqrt{3}}{2}\right)\;\;\mathrm{and}\;\;\left(1/2,\frac{-\sqrt% {3}}{2}\right)
  113. ( x - 1 ) 2 - x 2 = 0 (x-1)^{2}-x^{2}=0
  114. y 2 y^{2}
  115. y 2 y^{2}
  116. y y
  117. y y
  118. x x
  119. x 2 - 2 x + 1 + 1 - x 2 = 1 x^{2}-2x+1+1-x^{2}=1
  120. - 2 x = - 1 -2x=-1
  121. x = 1 / 2 x=1/2
  122. x x
  123. y y
  124. ( 1 / 2 ) 2 + y 2 = 1 (1/2)^{2}+y^{2}=1
  125. y 2 = 3 / 4 y^{2}=3/4
  126. y = ± 3 2 y=\frac{\pm\sqrt{3}}{2}
  127. ( 1 / 2 , + 3 2 ) and ( 1 / 2 , - 3 2 ) \left(1/2,\frac{+\sqrt{3}}{2}\right)\;\;\mathrm{and}\;\;\left(1/2,\frac{-\sqrt% {3}}{2}\right)
  128. x x
  129. y y
  130. y y
  131. y y
  132. x x
  133. x x
  134. y = m x + b y=mx+b
  135. b b
  136. y y
  137. b b
  138. ( 0 , b ) (0,b)
  139. y y

Analytical_chemistry.html

  1. E = O - T E=O-T
  2. E T \frac{E}{T}
  3. E T \frac{E}{T}
  4. v < m t p l > R M S = 4 k B T R Δ f , v_{<}mtpl>{{RMS}}=\sqrt{4k_{B}TR\Delta f},
  5. Δ f \Delta f
  6. f f
  7. i < m t p l > R M S = 2 e I Δ f i_{<}mtpl>{{RMS}}=\sqrt{2\,e\,I\,\Delta f}

Ancient_Greek_architecture.html

  1. 1 φ = φ - 1 ; φ = 1 + 5 2 1.618 \frac{1}{\varphi}=\varphi-1;\;\varphi=\frac{1+\sqrt{5}}{2}\approx 1.618

Andrey_Markov.html

  1. ( 1 + x 2 ) d y d x = n ( 1 + y 2 ) (1+x^{2})\frac{dy}{dx}=n(1+y^{2})

Angle.html

  1. BAC ^ . \widehat{\rm BAC}.
  2. ( 0 , 1 4 ) (0,\tfrac{1}{4})
  3. 1 4 \tfrac{1}{4}
  4. ( 1 4 , 1 2 ) (\tfrac{1}{4},\tfrac{1}{2})
  5. 1 2 \tfrac{1}{2}
  6. ( 1 2 , 1 ) (\tfrac{1}{2},1)
  7. 1 1
  8. ( 0 , 1 2 π ) (0,\tfrac{1}{2}\pi)
  9. 1 2 π \tfrac{1}{2}\pi
  10. ( 1 2 π , π ) (\tfrac{1}{2}\pi,\pi)
  11. π \pi
  12. ( π , 2 π ) (\pi,2\pi)
  13. 2 π 2\pi\,
  14. sin 2 A + sin 2 B = 1. cos 2 A + cos 2 B = 1. \sin^{2}A+\sin^{2}B=1.\quad\cos^{2}A+\cos^{2}B=1.
  15. tan A = cot B . sec A = csc B . \tan A=\cot B.\quad\quad\quad\sec A=\csc B.
  16. k / 2 π k/2\pi
  17. k k
  18. θ = k s 2 π r . \theta=k\frac{s}{2\pi r}.
  19. τ \tau
  20. π \pi
  21. τ \tau
  22. π \pi
  23. π \pi
  24. π \pi
  25. π \pi
  26. π \pi
  27. π \pi
  28. π \pi
  29. π \pi
  30. π \pi
  31. π \pi
  32. π \pi
  33. 𝐮 𝐯 = cos ( θ ) 𝐮 𝐯 . \mathbf{u}\cdot\mathbf{v}=\cos(\theta)\ \left\|\mathbf{u}\right\|\ \left\|% \mathbf{v}\right\|.
  34. , \langle\cdot,\cdot\rangle
  35. 𝐮 , 𝐯 = cos ( θ ) 𝐮 𝐯 . \langle\mathbf{u},\mathbf{v}\rangle=\cos(\theta)\ \left\|\mathbf{u}\right\|\ % \left\|\mathbf{v}\right\|.
  36. Re ( 𝐮 , 𝐯 ) = cos ( θ ) 𝐮 𝐯 . \operatorname{Re}\left(\langle\mathbf{u},\mathbf{v}\rangle\right)=\cos(\theta)% \ \left\|\mathbf{u}\right\|\ \left\|\mathbf{v}\right\|.
  37. | 𝐮 , 𝐯 | = | cos ( θ ) | 𝐮 𝐯 . \left|\langle\mathbf{u},\mathbf{v}\rangle\right|=|\cos(\theta)|\ \left\|% \mathbf{u}\right\|\ \left\|\mathbf{v}\right\|.
  38. span ( 𝐮 ) \operatorname{span}(\mathbf{u})
  39. span ( 𝐯 ) \operatorname{span}(\mathbf{v})
  40. 𝐮 \mathbf{u}
  41. 𝐯 \mathbf{v}
  42. span ( 𝐮 ) \operatorname{span}(\mathbf{u})
  43. span ( 𝐯 ) \operatorname{span}(\mathbf{v})
  44. | 𝐮 , 𝐯 | = | cos ( θ ) | 𝐮 𝐯 \left|\langle\mathbf{u},\mathbf{v}\rangle\right|=|\cos(\theta)|\ \left\|% \mathbf{u}\right\|\ \left\|\mathbf{v}\right\|
  45. 𝒰 \mathcal{U}
  46. 𝒲 \mathcal{W}
  47. dim ( 𝒰 ) := k dim ( 𝒲 ) := l \operatorname{dim}(\mathcal{U}):=k\leq\operatorname{dim}(\mathcal{W}):=l
  48. k k
  49. cos θ = g i j U i V j | g i j U i U j | | g i j V i V j | . \cos\theta=\frac{g_{ij}U^{i}V^{j}}{\sqrt{\left|g_{ij}U^{i}U^{j}\right|\left|g_% {ij}V^{i}V^{j}\right|}}.

Angular_acceleration.html

  1. 2 {}^{2}
  2. α = < m t p l > d ω d t = d 2 θ d t 2 {\alpha}=\frac{<}{m}tpl>{{d\omega}}{dt}=\frac{d^{2}{\theta}}{dt^{2}}
  3. α = a T r {\alpha}=\frac{a_{T}}{r}
  4. ω {\omega}
  5. a T a_{T}
  6. r r
  7. θ \theta
  8. ω \omega
  9. L ^ \hat{L}
  10. τ = I α {\tau}=I\ {\alpha}
  11. τ {\tau}
  12. I I
  13. τ {\tau}
  14. α = τ I . {\alpha}=\frac{\tau}{I}.

Angular_displacement.html

  1. s = r θ s=r\theta\,
  2. θ = s r \theta=\frac{s}{r}
  3. θ = 2 π r r \theta=\frac{2\pi r}{r}
  4. θ = 2 π \theta=2\pi
  5. 2 π 2\pi
  6. δ t \delta t
  7. Δ θ = Δ θ 2 - Δ θ 1 \Delta\theta=\Delta\theta_{2}-\Delta\theta_{1}
  8. A 0 A_{0}
  9. A f A_{f}
  10. d A = A f . A 0 - 1 dA=A_{f}.A_{0}^{-1}