wpmath0000002_15

Pedal_triangle.html

  1. A N 2 + B L 2 + C M 2 = N B 2 + L C 2 + M A 2 . AN^{2}+BL^{2}+CM^{2}=NB^{2}+LC^{2}+MA^{2}.

Penrose–Hawking_singularity_theorems.html

  1. θ \theta
  2. θ ˙ = - σ a b σ a b - 1 3 θ 2 - E [ X ] a a \dot{\theta}=-\sigma_{ab}\sigma^{ab}-\frac{1}{3}\theta^{2}-{E[\vec{X}]^{a}}_{a}
  3. σ a b \sigma_{ab}
  4. E [ X ] a a {E[\vec{X}]^{a}}_{a}
  5. p p
  6. q q
  7. p p
  8. q q
  9. γ \gamma
  10. γ \gamma
  11. p p
  12. γ \gamma
  13. p p
  14. 𝒯 \mathcal{T}
  15. 𝒯 \mathcal{T}
  16. J ˙ ( 𝒯 ) \dot{J}(\mathcal{T})
  17. 𝒯 \mathcal{T}
  18. 𝒯 \mathcal{T}
  19. J ˙ ( 𝒯 ) \dot{J}(\mathcal{T})
  20. 𝒯 \mathcal{T}
  21. 𝒯 \mathcal{T}
  22. J ˙ ( 𝒯 ) \dot{J}(\mathcal{T})
  23. J ˙ ( 𝒯 ) \dot{J}(\mathcal{T})
  24. J ˙ ( 𝒯 ) \dot{J}(\mathcal{T})

Pentagram.html

  1. φ = 1 + 2 sin ( π / 10 ) = 1 + 2 sin 18 \varphi=1+2\sin(\pi/10)=1+2\sin 18^{\circ}\,
  2. φ = 1 / ( 2 sin ( π / 10 ) ) = 1 / ( 2 sin 18 ) \varphi=1/(2\sin(\pi/10))=1/(2\sin 18^{\circ})\,
  3. φ = 2 cos ( π / 5 ) = 2 cos 36 \varphi=2\cos(\pi/5)=2\cos 36^{\circ}\,
  4. red green = green blue = blue magenta = φ . \frac{\mathrm{red}}{\mathrm{green}}=\frac{\mathrm{green}}{\mathrm{blue}}=\frac% {\mathrm{blue}}{\mathrm{magenta}}=\varphi.
  5. sin π 10 = sin 18 = 5 - 1 4 = φ - 1 2 = 1 2 φ \sin\frac{\pi}{10}=\sin 18^{\circ}=\frac{\sqrt{5}-1}{4}=\frac{\varphi-1}{2}=% \frac{1}{2\varphi}
  6. cos π 10 = cos 18 = 2 ( 5 + 5 ) 4 \cos\frac{\pi}{10}=\cos 18^{\circ}=\frac{\sqrt{2(5+\sqrt{5})}}{4}
  7. tan π 10 = tan 18 = 5 ( 5 - 2 5 ) 5 \tan\frac{\pi}{10}=\tan 18^{\circ}=\frac{\sqrt{5(5-2\sqrt{5})}}{5}
  8. cot π 10 = cot 18 = 5 + 2 5 \cot\frac{\pi}{10}=\cot 18^{\circ}=\sqrt{5+2\sqrt{5}}
  9. sin π 5 = sin 36 = 2 ( 5 - 5 ) 4 \sin\frac{\pi}{5}=\sin 36^{\circ}=\frac{\sqrt{2(5-\sqrt{5})}}{4}
  10. cos π 5 = cos 36 = 5 + 1 4 = φ 2 \cos\frac{\pi}{5}=\cos 36^{\circ}=\frac{\sqrt{5}+1}{4}=\frac{\varphi}{2}
  11. tan π 5 = tan 36 = 5 - 2 5 \tan\frac{\pi}{5}=\tan 36^{\circ}=\sqrt{5-2\sqrt{5}}
  12. cot π 5 = cot 36 = 5 ( 5 + 2 5 ) 5 \cot\frac{\pi}{5}=\cot 36^{\circ}=\frac{\sqrt{5(5+2\sqrt{5})}}{5}

Percentile_rank.html

  1. c + 0.5 f i N × 100 % \frac{\,\text{c}_{\ell}+0.5f_{i}}{N}\times 100\%

Perceptron.html

  1. x x
  2. f ( x ) f(x)
  3. f ( x ) = { 1 if w x + b > 0 0 otherwise f(x)=\begin{cases}1&\,\text{if }w\cdot x+b>0\\ 0&\,\text{otherwise}\end{cases}
  4. w w
  5. w x w\cdot x
  6. i w i x i \sum_{i}w_{i}x_{i}
  7. b b
  8. f ( x ) f(x)
  9. x x
  10. b b
  11. | b | |b|
  12. y = f ( 𝐳 ) y=f(\mathbf{z})\,
  13. 𝐳 \mathbf{z}
  14. b b\,
  15. D = { ( 𝐱 1 , d 1 ) , , ( 𝐱 s , d s ) } D=\{(\mathbf{x}_{1},d_{1}),\dots,(\mathbf{x}_{s},d_{s})\}\,
  16. s s
  17. 𝐱 j \mathbf{x}_{j}
  18. n n
  19. d j d_{j}\,
  20. x j , i x_{j,i}\,
  21. i i
  22. j j
  23. x j , 0 = 1 x_{j,0}=1\,
  24. w i w_{i}\,
  25. i i
  26. i i
  27. x j , 0 = 1 x_{j,0}=1\,
  28. w 0 w_{0}\,
  29. b b
  30. 𝐰 \mathbf{w}
  31. w i ( t ) w_{i}(t)\,
  32. i i
  33. t t
  34. α \alpha\,
  35. 0 < α 1 0<\alpha\leq 1
  36. j j\,
  37. D D\,
  38. 𝐱 j \mathbf{x}_{j}\,
  39. d j d_{j}\,
  40. y j ( t ) = f [ 𝐰 ( t ) 𝐱 j ] = f [ w 0 ( t ) + w 1 ( t ) x j , 1 + w 2 ( t ) x j , 2 + + w n ( t ) x j , n ] y_{j}(t)=f[\mathbf{w}(t)\cdot\mathbf{x}_{j}]=f[w_{0}(t)+w_{1}(t)x_{j,1}+w_{2}(% t)x_{j,2}+\cdots+w_{n}(t)x_{j,n}]
  41. w i ( t + 1 ) = w i ( t ) + α ( d j - y j ( t ) ) x j , i w_{i}(t+1)=w_{i}(t)+\alpha(d_{j}-y_{j}(t))x_{j,i}\,
  42. 0 i n 0\leq i\leq n
  43. 1 s j = 1 s | d j - y j ( t ) | \frac{1}{s}\sum_{j=1}^{s}|d_{j}-y_{j}(t)|\,
  44. γ \gamma\,
  45. D D
  46. γ \gamma
  47. 𝐰 , || 𝐰 || = 1 \mathbf{w},||\mathbf{w}||=1
  48. b b
  49. 𝐰 𝐱 j + b > γ \mathbf{w}\cdot\mathbf{x}_{j}+b>\gamma
  50. j : d j = 1 j:d_{j}=1
  51. 𝐰 𝐱 j + b < - γ \mathbf{w}\cdot\mathbf{x}_{j}+b<-\gamma
  52. j : d j = 0 j:d_{j}=0
  53. R R
  54. O ( R 2 / γ 2 ) O(R^{2}/\gamma^{2})
  55. O ( t ) O(\sqrt{t})
  56. O ( t ) O(t)
  57. 𝐰 \mathbf{w}
  58. α \alpha\,
  59. 𝐰 / α \mathbf{w}/\alpha\,
  60. α \alpha
  61. x 1 x_{1}\,
  62. x 2 x_{2}\,
  63. x 0 x_{0}\,
  64. x 1 x_{1}\,
  65. x 2 x_{2}\,
  66. x 0 x_{0}\,
  67. t t
  68. b b
  69. r r
  70. { ( ( 1 , 0 , 0 ) , 1 ) , ( ( 1 , 0 , 1 ) , 1 ) , ( ( 1 , 1 , 0 ) , 1 ) , ( ( 1 , 1 , 1 ) , 0 ) } \{((1,0,0),1),((1,0,1),1),((1,1,0),1),((1,1,1),0)\}\,
  71. x 0 x_{0}
  72. x 1 x_{1}
  73. x 2 x_{2}
  74. z z
  75. w 0 w_{0}
  76. w 1 w_{1}
  77. w 2 w_{2}
  78. c 0 c_{0}
  79. c 1 c_{1}
  80. c 2 c_{2}
  81. s s
  82. n n
  83. e e
  84. d d
  85. w 0 w_{0}
  86. w 1 w_{1}
  87. w 2 w_{2}
  88. x 0 * w 0 x_{0}*w_{0}
  89. x 1 * w 1 x_{1}*w_{1}
  90. x 2 * w 2 x_{2}*w_{2}
  91. c 0 + c 1 + c 2 c_{0}+c_{1}+c_{2}
  92. s > t s>t
  93. z - n z-n
  94. r * e r*e
  95. Δ ( w 0 + x 0 * d ) \Delta(w_{0}+x_{0}*d)
  96. Δ ( w 1 + x 1 * d ) \Delta(w_{1}+x_{1}*d)
  97. Δ ( w 2 + x 2 * d ) \Delta(w_{2}+x_{2}*d)
  98. x x
  99. y y
  100. f ( x , y ) f(x,y)
  101. w w
  102. y ^ = argmax y f ( x , y ) w . \hat{y}=\operatorname{argmax}_{y}f(x,y)\cdot w.
  103. w t + 1 = w t + f ( x , y ) - f ( x , y ^ ) . w_{t+1}=w_{t}+f(x,y)-f(x,\hat{y}).
  104. x x
  105. y y
  106. { 0 , 1 } \{0,1\}
  107. f ( x , y ) = y x f(x,y)=yx
  108. argmax y f ( x , y ) w \mathrm{argmax}_{y}f(x,y)\cdot w
  109. y y

Perfect_fifth.html

  1. ( 2 12 ) 7 (\sqrt[12]{2})^{7}

Perfect_gas.html

  1. C V C_{V}
  2. C P C_{P}
  3. p V = n R T pV=nRT
  4. C p - C V = R C_{p}-C_{V}=R
  5. e = e ( T ) e=e(T)
  6. h = h ( T ) h=h(T)
  7. d e = C v d T de=C_{v}dT
  8. d h = C p d T dh=C_{p}dT
  9. e = C v T e=C_{v}T
  10. h = C p T h=C_{p}T

Periodic_function.html

  1. P . P.
  2. f ( x + P ) = f ( x ) f(x+P)=f(x)\,\!
  3. sin ( x + 2 π ) = sin x \sin(x+2\pi)=\sin x\,\!
  4. e i k x = cos k x + i sin k x e^{ikx}=\cos kx+i\,\sin kx
  5. L = 2 π / k L=2\pi/k
  6. f ( x + P ) = e i k P f ( x ) f(x+P)=e^{ikP}f(x)\,\!
  7. / = { x + : x } = { { y : y y - x } : x } {\mathbb{R}/\mathbb{Z}}=\{x+\mathbb{Z}:x\in\mathbb{R}\}=\{\{y:y\in\mathbb{R}% \land y-x\in\mathbb{Z}\}:x\in\mathbb{R}\}
  8. / {\mathbb{R}/\mathbb{Z}}
  9. f : / f:{\mathbb{R}/\mathbb{Z}}\to\mathbb{R}

Permanent.html

  1. perm ( A ) = σ S n i = 1 n a i , σ ( i ) . \operatorname{perm}(A)=\sum_{\sigma\in S_{n}}\prod_{i=1}^{n}a_{i,\sigma(i)}.
  2. perm ( a b c d ) = a d + b c , \operatorname{perm}\begin{pmatrix}a&b\\ c&d\end{pmatrix}=ad+bc,
  3. perm ( a b c d e f g h i ) = a e i + b f g + c d h + c e g + b d i + a f h . \operatorname{perm}\begin{pmatrix}a&b&c\\ d&e&f\\ g&h&i\end{pmatrix}=aei+bfg+cdh+ceg+bdi+afh.
  4. | + | + \overset{+}{|}\quad\overset{+}{|}
  5. A = ( a i j ) A=(a_{ij})
  6. A = ( a i j ) A=(a_{ij})
  7. B = ( b i j ) B=(b_{ij})
  8. perm ( A + B ) = s , t perm ( a i j ) i s , j t perm ( b i j ) i s ¯ , j t ¯ , \operatorname{perm}(A+B)=\sum_{s,t}\operatorname{perm}(a_{ij})_{i\in s,j\in t}% \operatorname{perm}(b_{ij})_{i\in\bar{s},j\in\bar{t}},
  9. s ¯ , t ¯ \bar{s},\bar{t}
  10. 4 = perm ( 1 1 1 1 ) perm ( 1 1 1 1 ) perm ( ( 1 1 1 1 ) ( 1 1 1 1 ) ) = perm ( 2 2 2 2 ) = 8. 4=\operatorname{perm}\left(\begin{matrix}1&1\\ 1&1\end{matrix}\right)\operatorname{perm}\left(\begin{matrix}1&1\\ 1&1\end{matrix}\right)\neq\operatorname{perm}\left(\left(\begin{matrix}1&1\\ 1&1\end{matrix}\right)\left(\begin{matrix}1&1\\ 1&1\end{matrix}\right)\right)=\operatorname{perm}\left(\begin{matrix}2&2\\ 2&2\end{matrix}\right)=8.
  11. perm ( 1 1 1 1 2 1 0 0 3 0 1 0 4 0 0 1 ) = 1 perm ( 1 0 0 0 1 0 0 0 1 ) + 2 perm ( 1 1 1 0 1 0 0 0 1 ) + 3 perm ( 1 1 1 1 0 0 0 0 1 ) + 4 perm ( 1 1 1 1 0 0 0 1 0 ) = 1 ( 1 ) + 2 ( 1 ) + 3 ( 1 ) + 4 ( 1 ) = 10 , \operatorname{perm}\left(\begin{matrix}1&1&1&1\\ 2&1&0&0\\ 3&0&1&0\\ 4&0&0&1\end{matrix}\right)=1\cdot\operatorname{perm}\left(\begin{matrix}1&0&0% \\ 0&1&0\\ 0&0&1\end{matrix}\right)+2\cdot\operatorname{perm}\left(\begin{matrix}1&1&1\\ 0&1&0\\ 0&0&1\end{matrix}\right)+3\cdot\operatorname{perm}\left(\begin{matrix}1&1&1\\ 1&0&0\\ 0&0&1\end{matrix}\right)+4\cdot\operatorname{perm}\left(\begin{matrix}1&1&1\\ 1&0&0\\ 0&1&0\end{matrix}\right)=1(1)+2(1)+3(1)+4(1)=10,
  12. perm ( 1 1 1 1 2 1 0 0 3 0 1 0 4 0 0 1 ) = 4 perm ( 1 1 1 1 0 0 0 1 0 ) + 0 perm ( 1 1 1 2 0 0 3 1 0 ) + 0 perm ( 1 1 1 2 1 0 3 0 0 ) + 1 perm ( 1 1 1 2 1 0 3 0 1 ) = 4 ( 1 ) + 0 + 0 + 1 ( 6 ) = 10. \operatorname{perm}\left(\begin{matrix}1&1&1&1\\ 2&1&0&0\\ 3&0&1&0\\ 4&0&0&1\end{matrix}\right)=4\cdot\operatorname{perm}\left(\begin{matrix}1&1&1% \\ 1&0&0\\ 0&1&0\end{matrix}\right)+0\cdot\operatorname{perm}\left(\begin{matrix}1&1&1\\ 2&0&0\\ 3&1&0\end{matrix}\right)+0\cdot\operatorname{perm}\left(\begin{matrix}1&1&1\\ 2&1&0\\ 3&0&0\end{matrix}\right)+1\cdot\operatorname{perm}\left(\begin{matrix}1&1&1\\ 2&1&0\\ 3&0&1\end{matrix}\right)=4(1)+0+0+1(6)=10.
  13. H H
  14. k H \vee^{k}H
  15. k k
  16. H H
  17. k H \vee^{k}H
  18. H H
  19. x 1 , x 2 , , x k H x_{1},x_{2},\dots,x_{k}\in H
  20. x 1 x 2 x k = ( k ! ) - 1 / 2 σ S k x σ ( 1 ) x σ ( 2 ) x σ ( k ) x_{1}\vee x_{2}\vee\cdots\vee x_{k}=(k!)^{-1/2}\sum_{\sigma\in S_{k}}x_{\sigma% (1)}\otimes x_{\sigma(2)}\otimes\cdots\otimes x_{\sigma(k)}
  21. k H \vee^{k}H
  22. k H \otimes^{k}H
  23. H H
  24. k H \vee^{k}H
  25. x j , y j H x_{j},y_{j}\in H
  26. x 1 x 2 x k , y 1 y 2 y k = perm [ x i , y j ] i , j = 1 k \langle x_{1}\vee x_{2}\vee\cdots\vee x_{k},y_{1}\vee y_{2}\vee\cdots\vee y_{k% }\rangle=\operatorname{perm}\left[\langle x_{i},y_{j}\rangle\right]_{i,j=1}^{k}
  27. perm [ x i , x j ] i , j = 1 k 0 \operatorname{perm}\left[\langle x_{i},x_{j}\rangle\right]_{i,j=1}^{k}\geq 0
  28. | perm [ x i , y j ] i , j = 1 k | 2 perm [ x i , x j ] i , j = 1 k perm [ y i , y j ] i , j = 1 k \left|\operatorname{perm}\left[\langle x_{i},y_{j}\rangle\right]_{i,j=1}^{k}% \right|^{2}\leq\operatorname{perm}\left[\langle x_{i},x_{j}\rangle\right]_{i,j% =1}^{k}\cdot\operatorname{perm}\left[\langle y_{i},y_{j}\rangle\right]_{i,j=1}% ^{k}
  29. A = ( a i j ) A=(a_{ij})
  30. a i j a_{ij}
  31. σ ( i ) \sigma(i)
  32. σ \sigma
  33. { 1 , 2 , , n } \{1,2,\dots,n\}
  34. σ \sigma
  35. { 1 , 2 , , n } \{1,2,\dots,n\}
  36. σ ( i ) \sigma(i)
  37. Weight ( σ ) = i = 1 n a i , σ ( i ) . \operatorname{Weight}(\sigma)=\prod_{i=1}^{n}a_{i,\sigma(i)}.
  38. n × n n\times n
  39. perm ( A ) = σ i = 1 n a i , σ ( i ) \operatorname{perm}(A)=\sum_{\sigma}\prod_{i=1}^{n}a_{i,\sigma(i)}
  40. σ \sigma
  41. { 1 , 2 , , n } \{1,2,\dots,n\}
  42. A = ( a i j ) A=(a_{ij})
  43. x 1 , x 2 , , x n x_{1},x_{2},\dots,x_{n}
  44. y 1 , y 2 , , y n y_{1},y_{2},\dots,y_{n}
  45. a i j a_{ij}
  46. x i x_{i}
  47. y j y_{j}
  48. σ \sigma
  49. x i x_{i}
  50. y σ ( i ) y_{\sigma(i)}
  51. Weight ( σ ) = i = 1 n a i , σ ( i ) . \operatorname{Weight}(\sigma)=\prod_{i=1}^{n}a_{i,\sigma(i)}.
  52. a i j a_{ij}
  53. x i x_{i}
  54. y j y_{j}
  55. A = ( a i j ) A=(a_{ij})
  56. perm ( J - I ) = perm ( 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 ) = n ! i = 0 n ( - 1 ) i i ! , \operatorname{perm}(J-I)=\operatorname{perm}\left(\begin{matrix}0&1&1&\dots&1% \\ 1&0&1&\dots&1\\ 1&1&0&\dots&1\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 1&1&1&\dots&0\end{matrix}\right)=n!\sum_{i=0}^{n}\frac{(-1)^{i}}{i!},
  57. perm ( J - I - I ) = perm ( 0 0 1 1 1 0 0 1 1 1 0 1 0 1 1 0 ) = 2 n ! k = 0 n ( - 1 ) k 2 n 2 n - k ( 2 n - k k ) ( n - k ) ! , \operatorname{perm}(J-I-I^{\prime})=\operatorname{perm}\left(\begin{matrix}0&0% &1&\dots&1\\ 1&0&0&\dots&1\\ 1&1&0&\dots&1\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&1&1&\dots&0\end{matrix}\right)=2\cdot n!\sum_{k=0}^{n}(-1)^{k}\frac{2n}{2n-k% }{2n-k\choose k}(n-k)!,
  58. perm A i = 1 n ( r i ) ! 1 / r i . \operatorname{perm}A\leq\prod_{i=1}^{n}(r_{i})!^{1/r_{i}}.
  59. A k A_{k}
  60. P ( A k ) P(A_{k})
  61. A k A_{k}
  62. Σ k \Sigma_{k}
  63. P ( A k ) P(A_{k})
  64. A k A_{k}
  65. perm ( A ) = k = 0 n - 1 ( - 1 ) k Σ k . \operatorname{perm}(A)=\sum_{k=0}^{n-1}(-1)^{k}\Sigma_{k}.
  66. perm ( A ) = ( - 1 ) n S { 1 , , n } ( - 1 ) | S | i = 1 n j S a i j . \operatorname{perm}(A)=(-1)^{n}\sum_{S\subseteq\{1,\dots,n\}}(-1)^{|S|}\prod_{% i=1}^{n}\sum_{j\in S}a_{ij}.
  67. A = ( a i j ) A=(a_{ij})
  68. F ( x 1 , x 2 , , x n ) = i = 1 n ( j = 1 n a i j x j ) = ( j = 1 n a 1 j x j ) ( j = 1 n a 2 j x j ) ( j = 1 n a n j x j ) . F(x_{1},x_{2},\dots,x_{n})=\prod_{i=1}^{n}\left(\sum_{j=1}^{n}a_{ij}x_{j}% \right)=\left(\sum_{j=1}^{n}a_{1j}x_{j}\right)\left(\sum_{j=1}^{n}a_{2j}x_{j}% \right)\cdots\left(\sum_{j=1}^{n}a_{nj}x_{j}\right).
  69. x 1 x 2 x n x_{1}x_{2}\dots x_{n}
  70. F ( x 1 , x 2 , , x n ) F(x_{1},x_{2},\dots,x_{n})
  71. s 1 , s 2 , , s n s_{1},s_{2},\dots,s_{n}
  72. perm ( s 1 , s 2 , , s n ) ( A ) := coefficient of x 1 s 1 x 2 s 2 x n s n in ( j = 1 n a 1 j x j ) s 1 ( j = 1 n a 2 j x j ) s 2 ( j = 1 n a n j x j ) s n . \operatorname{perm}^{(s_{1},s_{2},\dots,s_{n})}(A):=\,\text{ coefficient of }x% _{1}^{s_{1}}x_{2}^{s_{2}}\cdots x_{n}^{s_{n}}\,\text{ in }\left(\sum_{j=1}^{n}% a_{1j}x_{j}\right)^{s_{1}}\left(\sum_{j=1}^{n}a_{2j}x_{j}\right)^{s_{2}}\cdots% \left(\sum_{j=1}^{n}a_{nj}x_{j}\right)^{s_{n}}.
  73. perm ( s 1 , s 2 , , s n ) ( A ) = coefficient of x 1 s 1 x 2 s 2 x n s n in 1 det ( I - X A ) , \operatorname{perm}^{(s_{1},s_{2},\dots,s_{n})}(A)=\,\text{ coefficient of }x_% {1}^{s_{1}}x_{2}^{s_{2}}\cdots x_{n}^{s_{n}}\,\text{ in }\frac{1}{\det(I-XA)},
  74. [ x 1 , x 2 , , x n ] . [x_{1},x_{2},\dots,x_{n}].
  75. A = ( a i j ) A=(a_{ij})
  76. perm ( A ) = σ P ( n , m ) a 1 σ ( 1 ) a 2 σ ( 2 ) a m σ ( m ) \operatorname{perm}(A)=\sum_{\sigma\in\operatorname{P}(n,m)}a_{1\sigma(1)}a_{2% \sigma(2)}\ldots a_{m\sigma(m)}
  77. A k A_{k}
  78. P ( A k ) P(A_{k})
  79. A k A_{k}
  80. σ k \sigma_{k}
  81. P ( A k ) P(A_{k})
  82. A k A_{k}
  83. perm ( A ) = k = 0 m - 1 ( - 1 ) k ( n - m + k k ) σ n - m + k . \operatorname{perm}(A)=\sum_{k=0}^{m-1}(-1)^{k}{\left({{n-m+k}\atop{k}}\right)% }\sigma_{n-m+k}.

Permutation_matrix.html

  1. π : { 1 , , m } { 1 , , m } \pi:\{1,\ldots,m\}\to\{1,\ldots,m\}
  2. ( 1 2 m π ( 1 ) π ( 2 ) π ( m ) ) , \begin{pmatrix}1&2&\cdots&m\\ \pi(1)&\pi(2)&\cdots&\pi(m)\end{pmatrix},
  3. P π = [ 𝐞 π ( 1 ) 𝐞 π ( 2 ) 𝐞 π ( m ) ] , P_{\pi}=\begin{bmatrix}\mathbf{e}_{\pi(1)}\\ \mathbf{e}_{\pi(2)}\\ \vdots\\ \mathbf{e}_{\pi(m)}\end{bmatrix},
  4. 𝐞 j \mathbf{e}_{j}
  5. P σ P π = P π σ P_{\sigma}P_{\pi}=P_{\pi\,\circ\,\sigma}
  6. 𝐞 π ( i ) \mathbf{e}_{\pi(i)}
  7. P σ π P_{\sigma\,\circ\,\pi}
  8. P π P π T = I P_{\pi}P_{\pi}^{T}=I
  9. P π - 1 = P π - 1 = P π T . P_{\pi}^{-1}=P_{\pi^{-1}}=P_{\pi}^{T}.
  10. P π P_{\pi}
  11. P π 𝐠 = [ 𝐞 π ( 1 ) 𝐞 π ( 2 ) 𝐞 π ( n ) ] [ g 1 g 2 g n ] = [ g π ( 1 ) g π ( 2 ) g π ( n ) ] . P_{\pi}\mathbf{g}=\begin{bmatrix}\mathbf{e}_{\pi(1)}\\ \mathbf{e}_{\pi(2)}\\ \vdots\\ \mathbf{e}_{\pi(n)}\end{bmatrix}\begin{bmatrix}g_{1}\\ g_{2}\\ \vdots\\ g_{n}\end{bmatrix}=\begin{bmatrix}g_{\pi(1)}\\ g_{\pi(2)}\\ \vdots\\ g_{\pi(n)}\end{bmatrix}.
  12. P σ P_{\sigma}
  13. P π P_{\pi}
  14. P π σ P_{\pi\circ\sigma}
  15. P π 𝐠 = 𝐠 P_{\pi}\mathbf{g}=\mathbf{g}^{\prime}
  16. g i = g π ( i ) g^{\prime}_{i}=g_{\pi(i)}\,
  17. P σ ( P π ( 𝐠 ) ) = P σ ( 𝐠 ) = [ g σ ( 1 ) g σ ( 2 ) g σ ( n ) ] = [ g π ( σ ( 1 ) ) g π ( σ ( 2 ) ) g π ( σ ( n ) ) ] . P_{\sigma}(P_{\pi}(\mathbf{g}))=P_{\sigma}(\mathbf{g}^{\prime})=\begin{bmatrix% }g^{\prime}_{\sigma(1)}\\ g^{\prime}_{\sigma(2)}\\ \vdots\\ g^{\prime}_{\sigma(n)}\end{bmatrix}=\begin{bmatrix}g_{\pi(\sigma(1))}\\ g_{\pi(\sigma(2))}\\ \vdots\\ g_{\pi(\sigma(n))}\end{bmatrix}.
  18. P π P_{\pi}
  19. P π P_{\pi}
  20. 𝐡 P π = [ h 1 h 2 h n ] [ 𝐞 π ( 1 ) 𝐞 π ( 2 ) 𝐞 π ( n ) ] = [ h π - 1 ( 1 ) h π - 1 ( 2 ) h π - 1 ( n ) ] \mathbf{h}P_{\pi}=\begin{bmatrix}h_{1}\;h_{2}\;\dots\;h_{n}\end{bmatrix}\begin% {bmatrix}\mathbf{e}_{\pi(1)}\\ \mathbf{e}_{\pi(2)}\\ \vdots\\ \mathbf{e}_{\pi(n)}\end{bmatrix}=\begin{bmatrix}h_{\pi^{-1}(1)}\;h_{\pi^{-1}(2% )}\;\dots\;h_{\pi^{-1}(n)}\end{bmatrix}
  21. ( 𝐡 P σ ) P π = 𝐡 P π σ (\mathbf{h}P_{\sigma})P_{\pi}=\mathbf{h}P_{\pi\circ\sigma}
  22. ( 𝐀𝐁 ) T = 𝐁 T 𝐀 T \left(\mathbf{AB}\right)^{\mathrm{T}}=\mathbf{B}^{\mathrm{T}}\mathbf{A}^{% \mathrm{T}}\,
  23. π = ( 1 2 3 4 5 1 4 2 5 3 ) , \pi=\begin{pmatrix}1&2&3&4&5\\ 1&4&2&5&3\end{pmatrix},
  24. P π = [ 𝐞 π ( 1 ) 𝐞 π ( 2 ) 𝐞 π ( 3 ) 𝐞 π ( 4 ) 𝐞 π ( 5 ) ] = [ 𝐞 1 𝐞 4 𝐞 2 𝐞 5 𝐞 3 ] = [ 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 ] . P_{\pi}=\begin{bmatrix}\mathbf{e}_{\pi(1)}\\ \mathbf{e}_{\pi(2)}\\ \mathbf{e}_{\pi(3)}\\ \mathbf{e}_{\pi(4)}\\ \mathbf{e}_{\pi(5)}\end{bmatrix}=\begin{bmatrix}\mathbf{e}_{1}\\ \mathbf{e}_{4}\\ \mathbf{e}_{2}\\ \mathbf{e}_{5}\\ \mathbf{e}_{3}\end{bmatrix}=\begin{bmatrix}1&0&0&0&0\\ 0&0&0&1&0\\ 0&1&0&0&0\\ 0&0&0&0&1\\ 0&0&1&0&0\end{bmatrix}.
  25. P π 𝐠 = [ 𝐞 π ( 1 ) 𝐞 π ( 2 ) 𝐞 π ( 3 ) 𝐞 π ( 4 ) 𝐞 π ( 5 ) ] [ g 1 g 2 g 3 g 4 g 5 ] = [ g 1 g 4 g 2 g 5 g 3 ] . P_{\pi}\mathbf{g}=\begin{bmatrix}\mathbf{e}_{\pi(1)}\\ \mathbf{e}_{\pi(2)}\\ \mathbf{e}_{\pi(3)}\\ \mathbf{e}_{\pi(4)}\\ \mathbf{e}_{\pi(5)}\end{bmatrix}\begin{bmatrix}g_{1}\\ g_{2}\\ g_{3}\\ g_{4}\\ g_{5}\end{bmatrix}=\begin{bmatrix}g_{1}\\ g_{4}\\ g_{2}\\ g_{5}\\ g_{3}\end{bmatrix}.
  26. [ 𝐞 a 1 𝐞 a 2 𝐞 a j ] \begin{bmatrix}\mathbf{e}_{a_{1}}\\ \mathbf{e}_{a_{2}}\\ \vdots\\ \mathbf{e}_{a_{j}}\\ \end{bmatrix}
  27. [ 1 2 j a 1 a 2 a j ] \begin{bmatrix}1&2&\ldots&j\\ a_{1}&a_{2}&\ldots&a_{j}\end{bmatrix}
  28. ( 1 2 j a 1 a 2 a j ) . \begin{pmatrix}1&2&\ldots&j\\ a_{1}&a_{2}&\ldots&a_{j}\end{pmatrix}.

Perpendicular.html

  1. A B ¯ \overline{AB}
  2. C D ¯ \overline{CD}
  3. A B ¯ C D ¯ \overline{AB}\perp\overline{CD}
  4. C D ¯ \overline{CD}
  5. C D ¯ \overline{CD}
  6. 2 . \sqrt{2}.

Perturbation_theory.html

  1. A A
  2. ε ε
  3. A = A 0 + ε 1 A 1 + ε 2 A 2 + A=A_{0}+\varepsilon^{1}A_{1}+\varepsilon^{2}A_{2}+\cdots
  4. ε ε
  5. A A 0 + ε A 1 . A\approx A_{0}+\varepsilon A_{1}~{}.
  6. F = m a . {F}=m{a}~{}.
  7. 𝐅 \mathbf{F}
  8. 𝐚 \mathbf{a}
  9. D g ( x ) = λ g ( x ) , Dg(x)=\lambda g(x)~{},
  10. D D
  11. λ λ
  12. D = D ( 0 ) + ε D ( 1 ) D=D^{(0)}+\varepsilon D^{(1)}
  13. ε ε
  14. f n ( 0 ) ( x ) f^{(0)}_{n}(x)
  15. n n
  16. D ( 0 ) f n ( 0 ) ( x ) = λ n ( 0 ) f n ( 0 ) ( x ) . D^{(0)}f^{(0)}_{n}(x)=\lambda^{(0)}_{n}f^{(0)}_{n}(x).
  17. { f n ( 0 ) ( x ) } \{f^{(0)}_{n}(x)\}
  18. f m ( 0 ) ( x ) f n ( 0 ) ( x ) d x = δ m n \int f^{(0)}_{m}(x)f^{(0)}_{n}(x)\,dx=\delta_{mn}
  19. g ( x ) g(x)
  20. f n ( 0 ) ( x ) f^{(0)}_{n}(x)
  21. g ( x ) = f n ( 0 ) ( x ) + 𝒪 ( ε ) g(x)=f^{(0)}_{n}(x)+\mathcal{O}(\varepsilon)
  22. λ = λ n ( 0 ) + 𝒪 ( ε ) . \lambda=\lambda^{(0)}_{n}+\mathcal{O}(\varepsilon).
  23. 𝒪 \mathcal{O}
  24. g ( x ) g(x)
  25. f n ( 0 ) ( x ) f^{(0)}_{n}(x)
  26. g ( x ) = m c m f m ( 0 ) ( x ) g(x)=\sum_{m}c_{m}f^{(0)}_{m}(x)
  27. c m = 𝒪 ( ε ) c_{m}=\mathcal{O}(\varepsilon)
  28. n n
  29. c n = 𝒪 ( 1 ) c_{n}=\mathcal{O}(1)
  30. f n ( 0 ) ( x ) f^{(0)}_{n}(x)
  31. c n λ n ( 0 ) + ε m c m f n ( 0 ) ( x ) D ( 1 ) f m ( 0 ) ( x ) d x = λ c n . c_{n}\lambda^{(0)}_{n}+\varepsilon\sum_{m}c_{m}\int f^{(0)}_{n}(x)D^{(1)}f^{(0% )}_{m}(x)\,dx=\lambda c_{n}~{}.
  32. m A n m c m = λ c n \sum_{m}A_{nm}c_{m}=\lambda c_{n}
  33. A n m = δ n m λ n ( 0 ) + ε f n ( 0 ) ( x ) D ( 1 ) f m ( 0 ) ( x ) d x . A_{nm}=\delta_{nm}\lambda^{(0)}_{n}+\varepsilon\int f^{(0)}_{n}(x)D^{(1)}f^{(0% )}_{m}(x)\,dx~{}.
  34. ε ε
  35. λ = λ n ( 0 ) + ε f n ( 0 ) ( x ) D ( 1 ) f n ( 0 ) ( x ) d x \lambda=\lambda^{(0)}_{n}+\varepsilon\int f^{(0)}_{n}(x)D^{(1)}f^{(0)}_{n}(x)% \,dx
  36. 𝒪 ( ε 2 ) \mathcal{O}(\varepsilon^{2})
  37. g ( x ) g(x)
  38. g ( x ) = f n ( 0 ) ( x ) + ε f n ( 1 ) ( x ) g(x)=f^{(0)}_{n}(x)+\varepsilon f^{(1)}_{n}(x)
  39. ( D ( 0 ) + ε D ( 1 ) ) ( f n ( 0 ) ( x ) + ε f n ( 1 ) ( x ) ) = ( λ n ( 0 ) + ε λ n ( 1 ) ) ( f n ( 0 ) ( x ) + ε f n ( 1 ) ( x ) ) \left(D^{(0)}+\varepsilon D^{(1)}\right)\left(f^{(0)}_{n}(x)+\varepsilon f^{(1% )}_{n}(x)\right)=\left(\lambda^{(0)}_{n}+\varepsilon\lambda^{(1)}_{n}\right)% \left(f^{(0)}_{n}(x)+\varepsilon f^{(1)}_{n}(x)\right)
  40. f n ( 1 ) ( x ) f^{(1)}_{n}(x)
  41. δ ( x - y ) = n f n ( 0 ) ( x ) f n ( 0 ) ( y ) \delta(x-y)=\sum_{n}f^{(0)}_{n}(x)f^{(0)}_{n}(y)
  42. f n ( 1 ) ( x ) = m ( n ) f m ( 0 ) ( x ) λ n ( 0 ) - λ m ( 0 ) f m ( 0 ) ( y ) D ( 1 ) f n ( 0 ) ( y ) d y f^{(1)}_{n}(x)=\sum_{m\,(\neq n)}\frac{f^{(0)}_{m}(x)}{\lambda^{(0)}_{n}-% \lambda^{(0)}_{m}}\int f^{(0)}_{m}(y)D^{(1)}f^{(0)}_{n}(y)\,dy
  43. ε ε
  44. λ n ( 0 ) - λ m ( 0 ) \lambda^{(0)}_{n}-\lambda^{(0)}_{m}
  45. D ( 0 ) f n , i ( 0 ) ( x ) = λ n ( 0 ) f n , i ( 0 ) ( x ) , D^{(0)}f^{(0)}_{n,i}(x)=\lambda^{(0)}_{n}f^{(0)}_{n,i}(x)~{},
  46. i i
  47. λ n ( 0 ) \lambda^{(0)}_{n}
  48. g ( x ) = k c n , k f n , k ( 0 ) ( x ) , g(x)=\sum_{k}c_{n,k}f^{(0)}_{n,k}(x)~{},
  49. f n , k ( 0 ) f^{(0)}_{n,k}
  50. c n , i λ n ( 0 ) + ε k c n , k f n , i ( 0 ) ( x ) D ( 1 ) f n , k ( 0 ) ( x ) d x = λ c n , i c_{n,i}\lambda^{(0)}_{n}+\varepsilon\sum_{k}c_{n,k}\int f^{(0)}_{n,i}(x)D^{(1)% }f^{(0)}_{n,k}(x)\,dx=\lambda c_{n,i}
  51. n n
  52. n n
  53. i i
  54. 4 × 4 4×4
  55. x x
  56. x = 1 + ε x 5 . x=1+\varepsilon x^{5}.
  57. ε = 0 ε=0
  58. ε ε
  59. x = x 0 + ε x 1 + x=x_{0}+\varepsilon x_{1}+\cdots
  60. ε ε
  61. ε ε
  62. ε 0 ε→0
  63. ε = 0 ε=0
  64. x x
  65. x = y ε - ν x=y\varepsilon^{-\nu}
  66. y y
  67. ν ν
  68. y y
  69. ε - ν y = 1 + ε 1 - 5 ν y 5 \varepsilon^{-\nu}y=1+\varepsilon^{1-5\nu}y^{5}
  70. ν ν
  71. ε ε
  72. y y
  73. ε ε
  74. ν = 1 4 ν=\frac{1}{4}
  75. ν ν
  76. y y
  77. ν ν
  78. ν = 1 4 ν=\frac{1}{4}
  79. y = ε 1 4 + y 5 y=\varepsilon^{\frac{1}{4}}+y^{5}
  80. x x
  81. y = y 0 + ε 1 4 y 1 + ε 1 2 y 2 + y=y_{0}+\varepsilon^{\frac{1}{4}}y_{1}+\varepsilon^{\frac{1}{2}}y_{2}+\cdots
  82. y = 0 y=0
  83. ε = 0 ε=0
  84. ε 0 ε→0
  85. x x
  86. x = ε - 1 4 [ y 0 - 1 4 ε 1 4 + ] x=\varepsilon^{-\frac{1}{4}}\left[y_{0}-\tfrac{1}{4}\varepsilon^{\frac{1}{4}}+% \cdots\right]
  87. ω ω
  88. H = H 0 + ε x H=H_{0}+\varepsilon x
  89. H 0 = 𝐩 2 2 - 1 r - ω L z , H_{0}=\tfrac{\mathbf{p}^{2}}{2}-\tfrac{1}{r}-\omega L_{z},
  90. z z
  91. ε x εx
  92. E n , m = - 1 2 n 2 - m ω E_{n,m}=-\tfrac{1}{2}n^{2}-m\omega
  93. | n , l , m , | 1 , 0 , 0 \left|n,l,m\right\rangle,\left|1,0,0\right\rangle
  94. | 2 , 1 , 1 \left|2,1,1\right\rangle
  95. [ E 1 , 0 ε d ε d E 1 , 0 ] [ a b ] = E [ a b ] \begin{bmatrix}E_{1,0}&{\varepsilon}d\\ {\varepsilon}d&E_{1,0}\end{bmatrix}\begin{bmatrix}a\\ b\end{bmatrix}=E\begin{bmatrix}a\\ b\end{bmatrix}
  96. d = 128 243 a 0 d=\tfrac{128}{243}a_{0}
  97. ( E 1 , 0 - E ) 2 - d 2 ε 2 = 0 \left(E_{1,0}-E\right)^{2}-d^{2}\varepsilon^{2}=0
  98. | χ 1 \displaystyle|\chi 1\rangle

Perturbation_theory_(quantum_mechanics).html

  1. α α
  2. α α
  3. n 1 / α n~{}1/α
  4. g g
  5. g g
  6. g g
  7. H 0 | n ( 0 ) = E n ( 0 ) | n ( 0 ) , n = 1 , 2 , 3 , H_{0}\left|n^{(0)}\right\rangle=E_{n}^{(0)}\left|n^{(0)}\right\rangle,\qquad n% =1,2,3,\cdots
  8. ( 0 ) (0)
  9. V V
  10. V V
  11. λ λ
  12. H = H 0 + λ V H=H_{0}+\lambda V
  13. ( H 0 + λ V ) | n = E n | n . \left(H_{0}+\lambda V\right)|n\rangle=E_{n}|n\rangle.
  14. | n |n\rangle
  15. λ λ
  16. E n \displaystyle E_{n}
  17. E n ( k ) = 1 k ! d k E n d λ k | n ( k ) = 1 k ! d k | n d λ k \begin{aligned}\displaystyle E_{n}^{(k)}&\displaystyle=\frac{1}{k!}\frac{d^{k}% E_{n}}{d\lambda^{k}}\\ \displaystyle\left|n^{(k)}\right\rangle&\displaystyle=\frac{1}{k!}\frac{d^{k}|% n\rangle}{d\lambda^{k}}\end{aligned}
  18. λ = 0 λ=0
  19. ( H 0 + λ V ) ( | n ( 0 ) + λ | n ( 1 ) + ) = ( E n ( 0 ) + λ E n ( 1 ) + λ 2 E n ( 2 ) + ) ( | n ( 0 ) + λ | n ( 1 ) + ) \left(H_{0}+\lambda V\right)\left(\left|n^{(0)}\right\rangle+\lambda\left|n^{(% 1)}\right\rangle+\cdots\right)=\left(E_{n}^{(0)}+\lambda E_{n}^{(1)}+\lambda^{% 2}E_{n}^{(2)}+\cdots\right)\left(\left|n^{(0)}\right\rangle+\lambda\left|n^{(1% )}\right\rangle+\cdots\right)
  20. H 0 | n ( 1 ) + V | n ( 0 ) = E n ( 0 ) | n ( 1 ) + E n ( 1 ) | n ( 0 ) H_{0}\left|n^{(1)}\right\rangle+V\left|n^{(0)}\right\rangle=E_{n}^{(0)}\left|n% ^{(1)}\right\rangle+E_{n}^{(1)}\left|n^{(0)}\right\rangle
  21. n ( 0 ) | \langle n^{(0)}|
  22. E n ( 1 ) = n ( 0 ) | V | n ( 0 ) E_{n}^{(1)}=\left\langle n^{(0)}\right|V\left|n^{(0)}\right\rangle
  23. | n ( 0 ) |n^{(0)}\rangle
  24. n ( 0 ) | V | n ( 0 ) \langle n^{(0)}|V|n^{(0)}\rangle
  25. | n ( 0 ) |n^{(0)}\rangle
  26. n ( 0 ) | n ( 0 ) = 1 , \left\langle n^{(0)}\right|\left.n^{(0)}\right\rangle=1,
  27. n | n = 1 \langle n|n\rangle=1
  28. λ λ
  29. ( n ( 0 ) | + λ n ( 1 ) | ) ( | n ( 0 ) + λ | n ( 1 ) ) = 1 \left(\left\langle n^{(0)}\right|+\lambda\left\langle n^{(1)}\right|\right)% \left(\left|n^{(0)}\right\rangle+\lambda\left|n^{(1)}\right\rangle\right)=1
  30. n ( 0 ) | n ( 0 ) + λ n ( 0 ) | n ( 1 ) + λ n ( 1 ) | n ( 0 ) + \cancel λ 2 n ( 1 ) | n ( 1 ) = 1 \left\langle n^{(0)}\right|\left.n^{(0)}\right\rangle+\lambda\left\langle n^{(% 0)}\right|\left.n^{(1)}\right\rangle+\lambda\left\langle n^{(1)}\right|\left.n% ^{(0)}\right\rangle+\cancel{\lambda^{2}\left\langle n^{(1)}\right|\left.n^{(1)% }\right\rangle}=1
  31. n ( 0 ) | n ( 1 ) + n ( 1 ) | n ( 0 ) = 0. \left\langle n^{(0)}\right|\left.n^{(1)}\right\rangle+\left\langle n^{(1)}% \right|\left.n^{(0)}\right\rangle=0.
  32. n ( 0 ) | n \langle n^{(0)}|n\rangle
  33. n ( 0 ) | n ( 1 ) = - n ( 0 ) | n ( 1 ) , \left\langle n^{(0)}\right|\left.n^{(1)}\right\rangle=-\left\langle n^{(0)}% \right|\left.n^{(1)}\right\rangle,
  34. n ( 0 ) | n ( 1 ) = 0. \left\langle n^{(0)}\right|\left.n^{(1)}\right\rangle=0.
  35. λ λ
  36. V | n ( 0 ) = ( k n | k ( 0 ) k ( 0 ) | ) V | n ( 0 ) + ( | n ( 0 ) n ( 0 ) | ) V | n ( 0 ) = k n | k ( 0 ) k ( 0 ) | V | n ( 0 ) + E n ( 1 ) | n ( 0 ) , \begin{aligned}\displaystyle V\left|n^{(0)}\right\rangle&\displaystyle=\left(% \sum_{k\neq n}\left|k^{(0)}\right\rangle\left\langle k^{(0)}\right|\right)V% \left|n^{(0)}\right\rangle+\left(\left|n^{(0)}\right\rangle\left\langle n^{(0)% }\right|\right)V\left|n^{(0)}\right\rangle\\ &\displaystyle=\sum_{k\neq n}\left|k^{(0)}\right\rangle\left\langle k^{(0)}% \right|V\left|n^{(0)}\right\rangle+E_{n}^{(1)}\left|n^{(0)}\right\rangle,\end{aligned}
  37. | k ( 0 ) |k^{(0)}\rangle
  38. | n ( 0 ) |n^{(0)}\rangle
  39. ( E n ( 0 ) - H 0 ) | n ( 1 ) = k n | k ( 0 ) k ( 0 ) | V | n ( 0 ) \left(E_{n}^{(0)}-H_{0}\right)\left|n^{(1)}\right\rangle=\sum_{k\neq n}\left|k% ^{(0)}\right\rangle\left\langle k^{(0)}\right|V\left|n^{(0)}\right\rangle
  40. | n ( 0 ) |n^{(0)}\rangle
  41. E n ( 0 ) E_{n}^{(0)}
  42. k ( 0 ) | \langle k^{(0)}|
  43. ( E n ( 0 ) - E k ( 0 ) ) k ( 0 ) | n ( 1 ) = k ( 0 ) | V | n ( 0 ) \left(E_{n}^{(0)}-E_{k}^{(0)}\right)\left\langle k^{(0)}\right.\left|n^{(1)}% \right\rangle=\left\langle k^{(0)}\right|V\left|n^{(0)}\right\rangle
  44. | k ( 0 ) |k^{(0)}\rangle
  45. E n ( 0 ) E k ( 0 ) E_{n}^{(0)}\neq E_{k}^{(0)}
  46. | n ( 1 ) = k n k ( 0 ) | V | n ( 0 ) E n ( 0 ) - E k ( 0 ) | k ( 0 ) \left|n^{(1)}\right\rangle=\sum_{k\neq n}\frac{\left\langle k^{(0)}\right|V% \left|n^{(0)}\right\rangle}{E_{n}^{(0)}-E_{k}^{(0)}}\left|k^{(0)}\right\rangle
  47. n n
  48. k n k≠n
  49. k ( 0 ) | V | n ( 0 ) \langle k^{(0)}|V|n^{(0)}\rangle
  50. n n
  51. k k
  52. k k
  53. n n
  54. n n
  55. 2 n ( 0 ) | n ( 2 ) + n ( 1 ) | n ( 1 ) = 0. 2\left\langle n^{(0)}\right|\left.n^{(2)}\right\rangle+\left\langle n^{(1)}% \right|\left.n^{(1)}\right\rangle=0.
  56. E n ( λ ) = E n ( 0 ) + λ n ( 0 ) | V | n ( 0 ) + λ 2 k n | k ( 0 ) | V | n ( 0 ) | 2 E n ( 0 ) - E k ( 0 ) + O ( λ 3 ) E_{n}(\lambda)=E_{n}^{(0)}+\lambda\left\langle n^{(0)}\right|V\left|n^{(0)}% \right\rangle+\lambda^{2}\sum_{k\neq n}\frac{\left|\left\langle k^{(0)}\right|% V\left|n^{(0)}\right\rangle\right|^{2}}{E_{n}^{(0)}-E_{k}^{(0)}}+O(\lambda^{3})
  57. | n ( λ ) = | n ( 0 ) + λ k n | k ( 0 ) k ( 0 ) | V | n ( 0 ) E n ( 0 ) - E k ( 0 ) + λ 2 k n n | k ( 0 ) k ( 0 ) | V | ( 0 ) ( 0 ) | V | n ( 0 ) ( E n ( 0 ) - E k ( 0 ) ) ( E n ( 0 ) - E ( 0 ) ) - λ 2 k n | k ( 0 ) n ( 0 ) | V | n ( 0 ) k ( 0 ) | V | n ( 0 ) ( E n ( 0 ) - E k ( 0 ) ) 2 - 1 2 λ 2 | n ( 0 ) k n n ( 0 ) | V | k ( 0 ) k ( 0 ) | V | n ( 0 ) ( E n ( 0 ) - E k ( 0 ) ) 2 + O ( λ 3 ) . \begin{aligned}\displaystyle|n(\lambda)\rangle=\left|n^{(0)}\right\rangle&% \displaystyle+\lambda\sum_{k\neq n}\left|k^{(0)}\right\rangle\frac{\left% \langle k^{(0)}\right|V\left|n^{(0)}\right\rangle}{E_{n}^{(0)}-E_{k}^{(0)}}+% \lambda^{2}\sum_{k\neq n}\sum_{\ell\neq n}\left|k^{(0)}\right\rangle\frac{% \left\langle k^{(0)}\right|V\left|\ell^{(0)}\right\rangle\left\langle\ell^{(0)% }\right|V\left|n^{(0)}\right\rangle}{\left(E_{n}^{(0)}-E_{k}^{(0)}\right)\left% (E_{n}^{(0)}-E_{\ell}^{(0)}\right)}\\ &\displaystyle-\lambda^{2}\sum_{k\neq n}\left|k^{(0)}\right\rangle\frac{\left% \langle n^{(0)}\right|V\left|n^{(0)}\right\rangle\left\langle k^{(0)}\right|V% \left|n^{(0)}\right\rangle}{\left(E_{n}^{(0)}-E_{k}^{(0)}\right)^{2}}-\frac{1}% {2}\lambda^{2}\left|n^{(0)}\right\rangle\sum_{k\neq n}\frac{\left\langle n^{(0% )}\right|V\left|k^{(0)}\right\rangle\left\langle k^{(0)}\right|V\left|n^{(0)}% \right\rangle}{\left(E_{n}^{(0)}-E_{k}^{(0)}\right)^{2}}+O(\lambda^{3}).\end{aligned}
  58. E n ( 3 ) = k n m n n ( 0 ) | V | m ( 0 ) m ( 0 ) | V | k ( 0 ) k ( 0 ) | V | n ( 0 ) ( E m ( 0 ) - E n ( 0 ) ) ( E k ( 0 ) - E n ( 0 ) ) - n ( 0 ) | V | n ( 0 ) m n | n ( 0 ) | V | m ( 0 ) | 2 ( E m ( 0 ) - E n ( 0 ) ) 2 . E_{n}^{(3)}=\sum_{k\neq n}\sum_{m\neq n}\frac{\langle n^{(0)}|V|m^{(0)}\rangle% \langle m^{(0)}|V|k^{(0)}\rangle\langle k^{(0)}|V|n^{(0)}\rangle}{\left(E_{m}^% {(0)}-E_{n}^{(0)}\right)\left(E_{k}^{(0)}-E_{n}^{(0)}\right)}-\langle n^{(0)}|% V|n^{(0)}\rangle\sum_{m\neq n}\frac{|\langle n^{(0)}|V|m^{(0)}\rangle|^{2}}{% \left(E_{m}^{(0)}-E_{n}^{(0)}\right)^{2}}.
  59. | n ( 1 ) = V k 1 n E n k 1 | k 1 ( 0 ) | n ( 2 ) = ( V k 1 k 2 V k 2 n E n k 1 E n k 2 - V n n V k 1 n E n k 1 2 ) | k 1 ( 0 ) - 1 2 V n k 1 V k 1 n E k 1 n 2 | n ( 0 ) | n ( 3 ) = [ - V k 1 k 2 V k 2 k 3 V k 3 n E k 1 n E n k 2 E n k 3 + V n n V k 1 k 2 V k 2 n E k 1 n E n k 2 ( 1 E n k 1 - 1 E n k 2 ) + | V n n | 2 V k 1 n E k 1 n 3 ] | k 1 ( 0 ) - [ V n k 2 V k 2 k 1 V k 1 n + V k 2 n V k 1 k 2 V n k 1 E n k 2 2 E n k 1 + | V n k 1 | 2 V n n E n k 1 3 ] | n ( 0 ) | n ( 4 ) = [ V k 1 k 2 V k 2 k 3 V k 3 k 4 V k 4 k 2 + V k 3 k 2 V k 1 k 2 V k 4 k 3 V k 2 k 4 2 E k 1 n E k 2 k 3 2 E k 2 k 4 - V k 2 k 3 V k 3 k 4 V k 4 n V k 1 k 2 E k 1 n E k 2 n E n k 3 E n k 4 + V k 1 k 2 E k 1 n ( | V k 2 k 3 | 2 V k 2 k 2 E k 2 k 3 3 - | V n k 3 | 2 V k 2 n E k 3 n 2 E k 2 n ) + V n n V k 1 k 2 V k 3 n V k 2 k 3 E k 1 n E n k 3 E k 2 n ( 1 E n k 3 + 1 E k 2 n + 1 E k 1 n ) + | V k 2 n | 2 V k 1 k 3 E n k 2 E k 1 n ( V k 3 n E n k 1 E n k 3 - V k 3 k 1 E k 3 k 1 2 ) - V n n ( V k 3 k 2 V k 1 k 3 V k 2 k 1 + V k 3 k 1 V k 2 k 3 V k 1 k 2 ) 2 E k 1 n E k 1 k 3 2 E k 1 k 2 + | V n n | 2 E k 1 n ( V k 1 n V n n E k 1 n 3 + V k 1 k 2 V k 2 n E k 2 n 3 ) - | V k 1 k 2 | 2 V n n V k 1 n E k 1 n E k 1 k 2 3 ] | k 1 ( 0 ) + 1 2 [ V n k 1 V k 1 k 2 E n k 1 E k 2 n 2 ( V k 2 n V n n E k 2 n - V k 2 k 3 V k 3 n E n k 3 ) - V k 1 n V k 2 k 1 E k 1 n 2 E n k 2 ( V k 3 k 2 V n k 3 E n k 3 + V n n V n k 2 E n k 2 ) + | V n k 1 | 2 E k 1 n 2 ( 3 | V n k 2 | 2 4 E k 2 n 2 - 2 | V n n | 2 E k 1 n 2 ) - V k 2 k 3 V k 3 k 1 | V n k 1 | 2 E n k 3 2 E n k 1 E n k 2 ] | n ( 0 ) \begin{aligned}\displaystyle|n^{(1)}\rangle&\displaystyle=\frac{V_{k_{1}n}}{E_% {nk_{1}}}|k_{1}^{(0)}\rangle\\ \displaystyle|n^{(2)}\rangle&\displaystyle=\left(\frac{V_{k_{1}k_{2}}V_{k_{2}n% }}{E_{nk_{1}}E_{nk_{2}}}-\frac{V_{nn}V_{k_{1}n}}{E_{nk_{1}}^{2}}\right)|k_{1}^% {(0)}\rangle-\frac{1}{2}\frac{V_{nk_{1}}V_{k_{1}n}}{E_{k_{1}n}^{2}}|n^{(0)}% \rangle\\ \displaystyle|n^{(3)}\rangle&\displaystyle=\Bigg[-\frac{V_{k_{1}k_{2}}V_{k_{2}% k_{3}}V_{k_{3}n}}{E_{k_{1}n}E_{nk_{2}}E_{nk_{3}}}+\frac{V_{nn}V_{k_{1}k_{2}}V_% {k_{2}n}}{E_{k_{1}n}E_{nk_{2}}}\left(\frac{1}{E_{nk_{1}}}-\frac{1}{E_{nk_{2}}}% \right)+\frac{|V_{nn}|^{2}V_{k_{1}n}}{E_{k_{1}n}^{3}}\Bigg]|k_{1}^{(0)}\rangle% -\Bigg[\frac{V_{nk_{2}}V_{k_{2}k_{1}}V_{k_{1}n}+V_{k_{2}n}V_{k_{1}k_{2}}V_{nk_% {1}}}{E_{nk_{2}}^{2}E_{nk_{1}}}+\frac{|V_{nk_{1}}|^{2}V_{nn}}{E_{nk_{1}}^{3}}% \Bigg]|n^{(0)}\rangle\\ \displaystyle|n^{(4)}\rangle&\displaystyle=\Bigg[\frac{V_{k_{1}k_{2}}V_{k_{2}k% _{3}}V_{k_{3}k_{4}}V_{k_{4}k_{2}}+V_{k_{3}k_{2}}V_{k_{1}k_{2}}V_{k_{4}k_{3}}V_% {k_{2}k_{4}}}{2E_{k_{1}n}E_{k_{2}k_{3}}^{2}E_{k_{2}k_{4}}}-\frac{V_{k_{2}k_{3}% }V_{k_{3}k_{4}}V_{k_{4}n}V_{k_{1}k_{2}}}{E_{k_{1}n}E_{k_{2}n}E_{nk_{3}}E_{nk_{% 4}}}+\frac{V_{k_{1}k_{2}}}{E_{k_{1}n}}\left(\frac{|V_{k_{2}k_{3}}|^{2}V_{k_{2}% k_{2}}}{E_{k_{2}k_{3}}^{3}}-\frac{|V_{nk_{3}}|^{2}V_{k_{2}n}}{E_{k_{3}n}^{2}E_% {k_{2}n}}\right)\\ &\displaystyle\quad+\frac{V_{nn}V_{k_{1}k_{2}}V_{k_{3}n}V_{k_{2}k_{3}}}{E_{k_{% 1}n}E_{nk_{3}}E_{k_{2}n}}\left(\frac{1}{E_{nk_{3}}}+\frac{1}{E_{k_{2}n}}+\frac% {1}{E_{k_{1}n}}\right)+\frac{|V_{k_{2}n}|^{2}V_{k_{1}k_{3}}}{E_{nk_{2}}E_{k_{1% }n}}\left(\frac{V_{k_{3}n}}{E_{nk_{1}}E_{nk_{3}}}-\frac{V_{k_{3}k_{1}}}{E_{k_{% 3}k_{1}}^{2}}\right)-\frac{V_{nn}\left(V_{k_{3}k_{2}}V_{k_{1}k_{3}}V_{k_{2}k_{% 1}}+V_{k_{3}k_{1}}V_{k_{2}k_{3}}V_{k_{1}k_{2}}\right)}{2E_{k_{1}n}E_{k_{1}k_{3% }}^{2}E_{k_{1}k_{2}}}\\ &\displaystyle\quad+\frac{|V_{nn}|^{2}}{E_{k_{1}n}}\left(\frac{V_{k_{1}n}V_{nn% }}{E_{k_{1}n}^{3}}+\frac{V_{k_{1}k_{2}}V_{k_{2}n}}{E_{k_{2}n}^{3}}\right)-% \frac{|V_{k_{1}k_{2}}|^{2}V_{nn}V_{k_{1}n}}{E_{k_{1}n}E_{k_{1}k_{2}}^{3}}\Bigg% ]|k_{1}^{(0)}\rangle+\frac{1}{2}\left[\frac{V_{nk_{1}}V_{k_{1}k_{2}}}{E_{nk_{1% }}E_{k_{2}n}^{2}}\left(\frac{V_{k_{2}n}V_{nn}}{E_{k_{2}n}}-\frac{V_{k_{2}k_{3}% }V_{k_{3}n}}{E_{nk_{3}}}\right)\right.\\ &\displaystyle\quad\left.-\frac{V_{k_{1}n}V_{k_{2}k_{1}}}{E_{k_{1}n}^{2}E_{nk_% {2}}}\left(\frac{V_{k_{3}k_{2}}V_{nk_{3}}}{E_{nk_{3}}}+\frac{V_{nn}V_{nk_{2}}}% {E_{nk_{2}}}\right)+\frac{|V_{nk_{1}}|^{2}}{E_{k_{1}n}^{2}}\left(\frac{3|V_{nk% _{2}}|^{2}}{4E_{k_{2}n}^{2}}-\frac{2|V_{nn}|^{2}}{E_{k_{1}n}^{2}}\right)-\frac% {V_{k_{2}k_{3}}V_{k_{3}k_{1}}|V_{nk_{1}}|^{2}}{E_{nk_{3}}^{2}E_{nk_{1}}E_{nk_{% 2}}}\right]|n^{(0)}\rangle\end{aligned}
  60. E n ( 0 ) - H 0 E_{n}^{(0)}-H_{0}
  61. D D
  62. D D
  63. D D
  64. D D
  65. | n = k D α n k | k ( 0 ) + λ | n ( 1 ) |n\rangle=\sum_{k\in D}\alpha_{nk}|k^{(0)}\rangle+\lambda|n^{(1)}\rangle
  66. D D
  67. D D
  68. V | k ( 0 ) = ϵ k | k ( 0 ) | k ( 0 ) D . V|k^{(0)}\rangle=\epsilon_{k}|k^{(0)}\rangle\qquad\forall\;|k^{(0)}\rangle\in D.
  69. ϵ k \epsilon_{k}
  70. k ( 0 ) | V | l ( 0 ) = V k l | k ( 0 ) , | l ( 0 ) D . \langle k^{(0)}|V|l^{(0)}\rangle=V_{kl}\qquad\forall\;|k^{(0)}\rangle,|l^{(0)}% \rangle\in D.
  71. D D
  72. ϵ k \epsilon_{k}
  73. ( E n ( 0 ) - H 0 ) | n ( 1 ) = k D ( k ( 0 ) | V | n ( 0 ) ) | k ( 0 ) . \left(E_{n}^{(0)}-H_{0}\right)|n^{(1)}\rangle=\sum_{k\not\in D}\left(\langle k% ^{(0)}|V|n^{(0)}\rangle\right)|k^{(0)}\rangle.
  74. D D
  75. | n ( 1 ) = k D k ( 0 ) | V | n ( 0 ) E n ( 0 ) - E k ( 0 ) | k ( 0 ) , |n^{(1)}\rangle=\sum_{k\not\in D}\frac{\langle k^{(0)}|V|n^{(0)}\rangle}{E_{n}% ^{(0)}-E_{k}^{(0)}}|k^{(0)}\rangle,
  76. ϵ k \epsilon_{k}
  77. x μ = ( x 1 , x 2 , ) x^{\mu}=(x^{1},x^{2},\cdots)
  78. ( x 1 , x 2 , ) (x^{1},x^{2},\cdots)
  79. | n ( x μ ) |n(x^{\mu})\rangle
  80. n n
  81. | n ( x μ ) |n(x^{\mu})\rangle
  82. E n ( x 0 μ ) E_{n}(x^{\mu}_{0})
  83. | n ( x 0 μ ) |n(x^{\mu}_{0})\rangle
  84. x 0 μ x^{\mu}_{0}
  85. | n ( x μ ) |n(x^{\mu})\rangle
  86. x 0 μ = 0 x^{\mu}_{0}=0
  87. H ( x μ ) = H ( 0 ) + x μ F μ . H(x^{\mu})=H(0)+x^{\mu}F_{\mu}.
  88. μ μ
  89. μ μ
  90. E n ( x μ ) \displaystyle E_{n}(x^{\mu})
  91. | μ n |\partial_{\mu}n\rangle
  92. | n | n ( 0 ) |n\rangle\equiv|n(0)\rangle
  93. μ E n = n | μ H | n \partial_{\mu}E_{n}=\langle n|\partial_{\mu}H|n\rangle
  94. m | μ n = m | μ H | n E n - E m , μ m | n = m | μ H | n E m - E n . \langle m|\partial_{\mu}n\rangle=\frac{\langle m|\partial_{\mu}H|n\rangle}{E_{% n}-E_{m}},\qquad\langle\partial_{\mu}m|n\rangle=\frac{\langle m|\partial_{\mu}% H|n\rangle}{E_{m}-E_{n}}.
  95. H | n = E n | n , H|n\rangle=E_{n}|n\rangle,
  96. μ H | n + H | μ n = μ E n | n + E n | μ n . \partial_{\mu}H|n\rangle+H|\partial_{\mu}n\rangle=\partial_{\mu}E_{n}|n\rangle% +E_{n}|\partial_{\mu}n\rangle.
  97. m | \langle m|
  98. m | H = m | E m \langle m|H=\langle m|E_{m}
  99. m | μ H | n + E m m | μ n = μ E n m | n + E n m | μ n . \langle m|\partial_{\mu}H|n\rangle+E_{m}\langle m|\partial_{\mu}n\rangle=% \partial_{\mu}E_{n}\langle m|n\rangle+E_{n}\langle m|\partial_{\mu}n\rangle.
  100. m | n = δ m n \langle m|n\rangle=\delta_{mn}
  101. m = n m=n
  102. m n m≠n
  103. E n ( x μ ) = n | H | n + n | μ H | n x μ + m n n | ν H | m m | μ H | n E n - E m x μ x ν + . E_{n}(x^{\mu})=\langle n|H|n\rangle+\langle n|\partial_{\mu}H|n\rangle x^{\mu}% +\sum_{m\neq n}\frac{\langle n|\partial_{\nu}H|m\rangle\langle m|\partial_{\mu% }H|n\rangle}{E_{n}-E_{m}}x^{\mu}x^{\nu}+\cdots.
  104. n | ν H | n \langle n|\partial_{\nu}H|n\rangle
  105. μ ν E n = μ n | ν H | n + n | μ ν H | n + n | ν H | μ n . \partial_{\mu}\partial_{\nu}E_{n}=\langle\partial_{\mu}n|\partial_{\nu}H|n% \rangle+\langle n|\partial_{\mu}\partial_{\nu}H|n\rangle+\langle n|\partial_{% \nu}H|\partial_{\mu}n\rangle.
  106. μ ν E n = m ( μ n | m m | ν H | n + n | ν H | m m | μ n ) , \partial_{\mu}\partial_{\nu}E_{n}=\sum_{m}\left(\langle\partial_{\mu}n|m% \rangle\langle m|\partial_{\nu}H|n\rangle+\langle n|\partial_{\nu}H|m\rangle% \langle m|\partial_{\mu}n\rangle\right),
  107. μ n | n = n | μ n = 0 \langle\partial_{\mu}n|n\rangle=\langle n|\partial_{\mu}n\rangle=0
  108. m = n m=n
  109. | n ( x μ ) = | n + m n m | μ H | n E n - E m | m x μ + ( m n l n m | μ H | l l | ν H | n ( E n - E m ) ( E n - E l ) | m - m n m | μ H | n n | ν H | n ( E n - E m ) 2 | m - 1 2 m n n | μ H | m m | ν H | n ( E n - E m ) 2 | m ) x μ x ν + . \begin{aligned}\displaystyle\left|n\left(x^{\mu}\right)\right\rangle=|n\rangle% &\displaystyle+\sum_{m\neq n}\frac{\langle m|\partial_{\mu}H|n\rangle}{E_{n}-E% _{m}}|m\rangle x^{\mu}\\ &\displaystyle+\left(\sum_{m\neq n}\sum_{l\neq n}\frac{\langle m|\partial_{\mu% }H|l\rangle\langle l|\partial_{\nu}H|n\rangle}{(E_{n}-E_{m})(E_{n}-E_{l})}|m% \rangle-\sum_{m\neq n}\frac{\langle m|\partial_{\mu}H|n\rangle\langle n|% \partial_{\nu}H|n\rangle}{(E_{n}-E_{m})^{2}}|m\rangle-\frac{1}{2}\sum_{m\neq n% }\frac{\langle n|\partial_{\mu}H|m\rangle\langle m|\partial_{\nu}H|n\rangle}{(% E_{n}-E_{m})^{2}}|m\rangle\right)x^{\mu}x^{\nu}+\cdots.\end{aligned}
  110. H ( 0 ) H(0)
  111. L \mathcal{H}_{L}
  112. H \mathcal{H}_{H}
  113. H ( 0 ) H(0)
  114. m | H ( 0 ) | l = 0 \langle m|H(0)|l\rangle=0
  115. m L , l H m\in\mathcal{H}_{L},l\in\mathcal{H}_{H}
  116. H m n eff ( x μ ) = m | H | n + m | μ H | n x μ + 1 2 ! l H ( m | μ H | l l | ν H | n E m - E l + m | ν H | l l | μ H | n E n - E l ) x μ x ν + . H_{mn}^{\,\text{eff}}\left(x^{\mu}\right)=\langle m|H|n\rangle+\langle m|% \partial_{\mu}H|n\rangle x^{\mu}+\frac{1}{2!}\sum_{l\in\mathcal{H}_{H}}\left(% \frac{\langle m|\partial_{\mu}H|l\rangle\langle l|\partial_{\nu}H|n\rangle}{E_% {m}-E_{l}}+\frac{\langle m|\partial_{\nu}H|l\rangle\langle l|\partial_{\mu}H|n% \rangle}{E_{n}-E_{l}}\right)x^{\mu}x^{\nu}+\cdots.
  117. m , n L m,n\in\mathcal{H}_{L}
  118. m | H ( x μ ) | n \langle m|H(x^{\mu})|n\rangle
  119. V ( t ) V(t)
  120. H H
  121. A A
  122. A A
  123. A A
  124. x x
  125. | n {|n\rangle}
  126. | j |j\rangle
  127. t t
  128. | j ( t ) = e - i E j t / | j . |j(t)\rangle=e^{-iE_{j}t/\hbar}|j\rangle~{}.
  129. V ( t ) V(t)
  130. H = H 0 + V ( t ) . H=H_{0}+V(t)~{}.
  131. | ψ ( t ) |\psi(t)\rangle
  132. t t
  133. H | ψ ( t ) = i t | ψ ( t ) . H|\psi(t)\rangle=i\hbar\frac{\partial}{\partial t}|\psi(t)\rangle~{}.
  134. | n |n\rangle
  135. | ψ ( t ) = n c n ( t ) e - i E n t / | n , |\psi(t)\rangle=\sum_{n}c_{n}(t)e^{-iE_{n}t/\hbar}|n\rangle~{},
  136. t t
  137. exp ( - i E n t / ) \exp(-iE_{n}t/\hbar)
  138. | j |j\rangle
  139. t t
  140. n j n≠j
  141. n n
  142. t t
  143. | c n ( t ) | 2 = | n | ψ ( t ) | 2 . \left|c_{n}(t)\right|^{2}=\left|\langle n|\psi(t)\rangle\right|^{2}~{}.
  144. n ( i c n t - c n ( t ) V ( t ) ) e - i E n t / | n = 0 . \sum_{n}\left(i\hbar\frac{\partial c_{n}}{\partial t}-c_{n}(t)V(t)\right)e^{-% iE_{n}t/\hbar}|n\rangle=0~{}.
  145. V V
  146. c n t = - i k n | V ( t ) | k c k ( t ) e - i ( E k - E n ) t / . \frac{\partial c_{n}}{\partial t}=\frac{-i}{\hbar}\sum_{k}\langle n|V(t)|k% \rangle\,c_{k}(t)\,e^{-i(E_{k}-E_{n})t/\hbar}~{}.
  147. V V
  148. V V
  149. n n
  150. c n ( t ) = c n ( 0 ) + - i k 0 t d t n | V ( t ) | k c k ( t ) e - i ( E k - E n ) t / . c_{n}(t)=c_{n}(0)+\frac{-i}{\hbar}\sum_{k}\int_{0}^{t}dt^{\prime}\;\langle n|V% (t^{\prime})|k\rangle\,c_{k}(t^{\prime})\,e^{-i(E_{k}-E_{n})t^{\prime}/\hbar}~% {}.
  151. c n ( t ) = c n ( 0 ) + c n ( 1 ) + c n ( 2 ) + c_{n}(t)=c_{n}^{(0)}+c_{n}^{(1)}+c_{n}^{(2)}+\cdots
  152. c n ( 1 ) ( t ) = - i k 0 t d t n | V ( t ) | k c k ( 0 ) e - i ( E k - E n ) t / . c_{n}^{(1)}(t)=\frac{-i}{\hbar}\sum_{k}\int_{0}^{t}dt^{\prime}\;\langle n|V(t^% {\prime})|k\rangle\,c_{k}(0)\,e^{-i(E_{k}-E_{n})t^{\prime}/\hbar}~{}.
  153. H ( t ) | ψ ( t ) = i | ψ ( t ) t H(t)|\psi(t)\rangle=i\hbar\frac{\partial|\psi(t)\rangle}{\partial t}
  154. | ψ ( t ) = T exp [ - i t 0 t d t H ( t ) ] | ψ ( t 0 ) , |\psi(t)\rangle=T\exp{\left[-\frac{i}{\hbar}\int_{t_{0}}^{t}dt^{\prime}H(t^{% \prime})\right]}|\psi(t_{0})\rangle~{},
  155. T T
  156. T A ( t 1 ) A ( t 2 ) = { A ( t 1 ) A ( t 2 ) t 1 > t 2 A ( t 2 ) A ( t 1 ) t 2 > t 1 . TA(t_{1})A(t_{2})=\begin{cases}A(t_{1})A(t_{2})&t_{1}>t_{2}\\ A(t_{2})A(t_{1})&t_{2}>t_{1}\end{cases}~{}.
  157. | ψ ( t ) = [ 1 - i t 0 t d t 1 H ( t 1 ) - 1 2 t 0 t d t 1 t 0 t 1 d t 2 H ( t 1 ) H ( t 2 ) + ] | ψ ( t 0 ) . |\psi(t)\rangle=\left[1-\frac{i}{\hbar}\int_{t_{0}}^{t}dt_{1}H(t_{1})-\frac{1}% {\hbar^{2}}\int_{t_{0}}^{t}dt_{1}\int_{t_{0}}^{t_{1}}dt_{2}H(t_{1})H(t_{2})+% \ldots\right]|\psi(t_{0})\rangle~{}.
  158. [ H 0 + λ V ( t ) ] | ψ ( t ) = i | ψ ( t ) t , [H_{0}+\lambda V(t)]|\psi(t)\rangle=i\hbar\frac{\partial|\psi(t)\rangle}{% \partial t}~{},
  159. λ λ
  160. H 0 | n = E n | n H_{0}|n\rangle=E_{n}|n\rangle
  161. | ψ ( t ) = e - i H 0 ( t - t 0 ) | ψ I ( t ) . |\psi(t)\rangle=e^{-\frac{i}{\hbar}H_{0}(t-t_{0})}|\psi_{I}(t)\rangle~{}.
  162. λ e i H 0 ( t - t 0 ) V ( t ) e - i H 0 ( t - t 0 ) | ψ I ( t ) = i | ψ I ( t ) t , \lambda e^{\frac{i}{\hbar}H_{0}(t-t_{0})}V(t)e^{-\frac{i}{\hbar}H_{0}(t-t_{0})% }|\psi_{I}(t)\rangle=i\hbar\frac{\partial|\psi_{I}(t)\rangle}{\partial t}~{},
  163. | ψ I ( t ) = [ 1 - i λ t 0 t d t 1 e i H 0 ( t 1 - t 0 ) V ( t 1 ) e - i H 0 ( t 1 - t 0 ) - λ 2 2 t 0 t d t 1 t 0 t 1 d t 2 e i H 0 ( t 1 - t 0 ) V ( t 1 ) e - i H 0 ( t 1 - t 0 ) e i H 0 ( t 2 - t 0 ) V ( t 2 ) e - i H 0 ( t 2 - t 0 ) + ] | ψ ( t 0 ) , |\psi_{I}(t)\rangle=\left[1-\frac{i\lambda}{\hbar}\int_{t_{0}}^{t}dt_{1}e^{% \frac{i}{\hbar}H_{0}(t_{1}-t_{0})}V(t_{1})e^{-\frac{i}{\hbar}H_{0}(t_{1}-t_{0}% )}-\frac{\lambda^{2}}{\hbar^{2}}\int_{t_{0}}^{t}dt_{1}\int_{t_{0}}^{t_{1}}dt_{% 2}e^{\frac{i}{\hbar}H_{0}(t_{1}-t_{0})}V(t_{1})e^{-\frac{i}{\hbar}H_{0}(t_{1}-% t_{0})}e^{\frac{i}{\hbar}H_{0}(t_{2}-t_{0})}V(t_{2})e^{-\frac{i}{\hbar}H_{0}(t% _{2}-t_{0})}+\ldots\right]|\psi(t_{0})\rangle~{},
  164. λ λ
  165. H 0 | n = E n | n H_{0}|n\rangle=E_{n}|n\rangle
  166. n | n n | = 1 \sum_{n}|n\rangle\langle n|=1
  167. | ψ I ( t ) = [ 1 - i λ m n t 0 t d t 1 m | V ( t 1 ) | n e - i ( E n - E m ) ( t 1 - t 0 ) | m n | + ] | ψ ( t 0 ) . |\psi_{I}(t)\rangle=\left[1-\frac{i\lambda}{\hbar}\sum_{m}\sum_{n}\int_{t_{0}}% ^{t}dt_{1}\langle m|V(t_{1})|n\rangle e^{-\frac{i}{\hbar}(E_{n}-E_{m})(t_{1}-t% _{0})}|m\rangle\langle n|+\ldots\right]|\psi(t_{0})\rangle~{}.
  168. | α = | ψ ( t 0 ) |\alpha\rangle=|\psi(t_{0})\rangle
  169. | β |\beta\rangle
  170. A α β = - i λ t 0 t d t 1 β | V ( t 1 ) | α e - i ( E α - E β ) ( t 1 - t 0 ) , A_{\alpha\beta}=-\frac{i\lambda}{\hbar}\int_{t_{0}}^{t}dt_{1}\langle\beta|V(t_% {1})|\alpha\rangle e^{-\frac{i}{\hbar}(E_{\alpha}-E_{\beta})(t_{1}-t_{0})}~{},
  171. U ( t ) = 1 - i λ t 0 t d t 1 e i H 0 ( t 1 - t 0 ) V ( t 1 ) e - i H 0 ( t 1 - t 0 ) - λ 2 2 t 0 t d t 1 t 0 t 1 d t 2 e i H 0 ( t 1 - t 0 ) V ( t 1 ) e - i H 0 ( t 1 - t 0 ) e i H 0 ( t 2 - t 0 ) V ( t 2 ) e - i H 0 ( t 2 - t 0 ) + U(t)=1-\frac{i\lambda}{\hbar}\int_{t_{0}}^{t}dt_{1}e^{\frac{i}{\hbar}H_{0}(t_{% 1}-t_{0})}V(t_{1})e^{-\frac{i}{\hbar}H_{0}(t_{1}-t_{0})}-\frac{\lambda^{2}}{% \hbar^{2}}\int_{t_{0}}^{t}dt_{1}\int_{t_{0}}^{t_{1}}dt_{2}e^{\frac{i}{\hbar}H_% {0}(t_{1}-t_{0})}V(t_{1})e^{-\frac{i}{\hbar}H_{0}(t_{1}-t_{0})}e^{\frac{i}{% \hbar}H_{0}(t_{2}-t_{0})}V(t_{2})e^{-\frac{i}{\hbar}H_{0}(t_{2}-t_{0})}+\cdots
  172. V V
  173. n | n n | = 1 \sum_{n}|n\rangle\langle n|=1
  174. H 0 | n = E n | n H_{0}|n\rangle=E_{n}|n\rangle
  175. U ( t ) = 1 - { i λ t 0 t d t 1 m n m | V | n e - i ( E n - E m ) ( t 1 - t 0 ) | m n | } - { λ 2 2 t 0 t d t 1 t 0 t 1 d t 2 m n q e - i ( E n - E m ) ( t 1 - t 0 ) m | V | n n | V | q e - i ( E q - E n ) ( t 2 - t 0 ) | m q | } + \begin{aligned}\displaystyle U(t)=1&\displaystyle-\left\{\frac{i\lambda}{\hbar% }\int_{t_{0}}^{t}dt_{1}\sum_{m}\sum_{n}\langle m|V|n\rangle e^{-\frac{i}{\hbar% }(E_{n}-E_{m})(t_{1}-t_{0})}|m\rangle\langle n|\right\}\\ &\displaystyle-\left\{\frac{\lambda^{2}}{\hbar^{2}}\int_{t_{0}}^{t}dt_{1}\int_% {t_{0}}^{t_{1}}dt_{2}\sum_{m}\sum_{n}\sum_{q}e^{-\frac{i}{\hbar}(E_{n}-E_{m})(% t_{1}-t_{0})}\langle m|V|n\rangle\langle n|V|q\rangle e^{-\frac{i}{\hbar}(E_{q% }-E_{n})(t_{2}-t_{0})}|m\rangle\langle q|\right\}+\cdots\end{aligned}
  176. t 0 = 0 t_{0}=0
  177. e - ϵ t e^{-\epsilon t}
  178. ε ε
  179. t t→∞
  180. U ( t ) = 1 - i λ n n | V | n t - i λ 2 m n n | V | m m | V | n E n - E m t - 1 2 λ 2 2 m , n n | V | m m | V | n t 2 + + λ m n m | V | n E n - E m | m n | + λ 2 m n q n n m | V | n n | V | q ( E n - E m ) ( E q - E n ) | m q | + \begin{aligned}\displaystyle U(t)=1&\displaystyle-\frac{i\lambda}{\hbar}\sum_{% n}\langle n|V|n\rangle t-\frac{i\lambda^{2}}{\hbar}\sum_{m\neq n}\frac{\langle n% |V|m\rangle\langle m|V|n\rangle}{E_{n}-E_{m}}t-\frac{1}{2}\frac{\lambda^{2}}{% \hbar^{2}}\sum_{m,n}\langle n|V|m\rangle\langle m|V|n\rangle t^{2}+\ldots\\ &\displaystyle+\lambda\sum_{m\neq n}\frac{\langle m|V|n\rangle}{E_{n}-E_{m}}|m% \rangle\langle n|+\lambda^{2}\sum_{m\neq n}\sum_{q\neq n}\sum_{n}\frac{\langle m% |V|n\rangle\langle n|V|q\rangle}{(E_{n}-E_{m})(E_{q}-E_{n})}|m\rangle\langle q% |+\ldots\end{aligned}
  181. | n ( λ ) = U ( 0 ; λ ) | n ) |n(\lambda)\rangle=U(0;\lambda)|n\rangle)
  182. H ( t ) | ψ ( t ) = i | ψ ( t ) t H(t)|\psi(t)\rangle=i\hbar\frac{\partial|\psi(t)\rangle}{\partial t}
  183. [ H 0 + λ V ( t ) ] | ψ ( t ) = i | ψ ( t ) t [H_{0}+\lambda V(t)]|\psi(t)\rangle=i\hbar\frac{\partial|\psi(t)\rangle}{% \partial t}
  184. λ λ→∞
  185. | ψ = | ψ 0 + 1 λ | ψ 1 + 1 λ 2 | ψ 2 + |\psi\rangle=|\psi_{0}\rangle+\frac{1}{\lambda}|\psi_{1}\rangle+\frac{1}{% \lambda^{2}}|\psi_{2}\rangle+\ldots
  186. τ = λ t \tau=\lambda t
  187. V ( t ) | ψ 0 = i | ψ 0 τ V(t)|\psi_{0}\rangle=i\hbar\frac{\partial|\psi_{0}\rangle}{\partial\tau}
  188. V ( t ) | ψ 1 + H 0 | ψ 0 = i | ψ 1 τ V(t)|\psi_{1}\rangle+H_{0}|\psi_{0}\rangle=i\hbar\frac{\partial|\psi_{1}% \rangle}{\partial\tau}
  189. \vdots
  190. V ( t ) V(t)
  191. | ψ ( t ) = e - i λ V ( t - t 0 ) | ψ F ( t ) |\psi(t)\rangle=e^{-\frac{i}{\hbar}\lambda V(t-t_{0})}|\psi_{F}(t)\rangle
  192. e i λ V ( t - t 0 ) H 0 e - i λ V ( t - t 0 ) | ψ F ( t ) = i | ψ F ( t ) t e^{\frac{i}{\hbar}\lambda V(t-t_{0})}H_{0}e^{-\frac{i}{\hbar}\lambda V(t-t_{0}% )}|\psi_{F}(t)\rangle=i\hbar\frac{\partial|\psi_{F}(t)\rangle}{\partial t}
  193. λ λ
  194. λ λ
  195. | ψ F ( t ) = [ 1 - i t 0 t d t 1 e i λ V ( t 1 - t 0 ) H 0 e - i λ V ( t 1 - t 0 ) - 1 2 t 0 t d t 1 t 0 t 1 d t 2 e i λ V ( t 1 - t 0 ) H 0 e - i λ V ( t 1 - t 0 ) e i λ V ( t 2 - t 0 ) H 0 e - i λ V ( t 2 - t 0 ) + ] | ψ ( t 0 ) . |\psi_{F}(t)\rangle=\left[1-\frac{i}{\hbar}\int_{t_{0}}^{t}dt_{1}e^{\frac{i}{% \hbar}\lambda V(t_{1}-t_{0})}H_{0}e^{-\frac{i}{\hbar}\lambda V(t_{1}-t_{0})}-% \frac{1}{\hbar^{2}}\int_{t_{0}}^{t}dt_{1}\int_{t_{0}}^{t_{1}}dt_{2}e^{\frac{i}% {\hbar}\lambda V(t_{1}-t_{0})}H_{0}e^{-\frac{i}{\hbar}\lambda V(t_{1}-t_{0})}e% ^{\frac{i}{\hbar}\lambda V(t_{2}-t_{0})}H_{0}e^{-\frac{i}{\hbar}\lambda V(t_{2% }-t_{0})}+\ldots\right]|\psi(t_{0})\rangle.
  196. τ = λ t \tau=\lambda t
  197. 1 / λ 1/\lambda
  198. V V
  199. H 0 = p 2 / 2 m H_{0}=p^{2}/2m
  200. H = - 2 2 m 2 x 2 + m ω 2 x 2 2 + λ x 4 H=-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial x^{2}}+\frac{m\omega^{2}x^% {2}}{2}+\lambda x^{4}
  201. ψ 0 = ( α π ) 1 4 e - α x 2 / 2 \psi_{0}=\left(\frac{\alpha}{\pi}\right)^{\frac{1}{4}}e^{-\alpha x^{2}/2}
  202. α = m ω / \alpha=m\omega/\hbar
  203. E 0 ( 0 ) = 1 2 ω . E_{0}^{(0)}=\frac{1}{2}\hbar\omega.\,
  204. E 0 ( 1 ) = λ ( α π ) 1 2 e - α x 2 / 2 x 4 e - α x 2 / 2 d x = λ ( α π ) 1 2 2 α 2 e - α x 2 d x E_{0}^{(1)}=\lambda\left(\frac{\alpha}{\pi}\right)^{\frac{1}{2}}\int e^{-% \alpha x^{2}/2}x^{4}e^{-\alpha x^{2}/2}dx=\lambda\left(\frac{\alpha}{\pi}% \right)^{\frac{1}{2}}\frac{\partial^{2}}{\partial\alpha^{2}}\int e^{-\alpha x^% {2}}dx
  205. E 0 ( 1 ) = λ ( α π ) 1 2 2 α 2 ( π α ) 1 2 = λ 3 4 1 α 2 = 3 4 2 λ m 2 ω 2 E_{0}^{(1)}=\lambda\left(\frac{\alpha}{\pi}\right)^{\frac{1}{2}}\frac{\partial% ^{2}}{\partial\alpha^{2}}\left(\frac{\pi}{\alpha}\right)^{\frac{1}{2}}=\lambda% \frac{3}{4}\frac{1}{\alpha^{2}}=\frac{3}{4}\frac{\hbar^{2}\lambda}{m^{2}\omega% ^{2}}
  206. H = - 2 2 m a 2 2 ϕ 2 - λ cos ϕ H=-\frac{\hbar^{2}}{2ma^{2}}\frac{\partial^{2}}{\partial\phi^{2}}-\lambda\cos\phi
  207. - λ cos ϕ -\lambda\cos\phi
  208. V = - cos ϕ \frac{}{}V=-\cos\phi
  209. ψ n ( ϕ ) = e i n ϕ 2 π \psi_{n}(\phi)=\frac{e^{in\phi}}{\sqrt{2\pi}}
  210. E n ( 0 ) = 2 n 2 2 m a 2 E_{n}^{(0)}=\frac{\hbar^{2}n^{2}}{2ma^{2}}
  211. E n ( 1 ) = - 1 2 π e - i n ϕ cos ϕ e i n ϕ = - 1 2 π cos ϕ = 0 E_{n}^{(1)}=-\frac{1}{2\pi}\int e^{-in\phi}\cos\phi e^{in\phi}=-\frac{1}{2\pi}% \int\cos\phi=0
  212. E n ( 2 ) = m a 2 2 π 2 2 k | e - i k ϕ cos ϕ e i n ϕ | 2 n 2 - k 2 E_{n}^{(2)}=\frac{ma^{2}}{2\pi^{2}\hbar^{2}}\sum_{k}\frac{\left|\int e^{-ik% \phi}\cos\phi e^{in\phi}\right|^{2}}{n^{2}-k^{2}}
  213. E n ( 2 ) = m a 2 2 2 k | ( δ n , 1 - k + δ n , - 1 - k ) | 2 n 2 - k 2 E_{n}^{(2)}=\frac{ma^{2}}{2\hbar^{2}}\sum_{k}\frac{\left|\left(\delta_{n,1-k}+% \delta_{n,-1-k}\right)\right|^{2}}{n^{2}-k^{2}}
  214. E n ( 2 ) = m a 2 2 2 ( 1 2 n - 1 + 1 - 2 n - 1 ) = m a 2 2 1 4 n 2 - 1 E_{n}^{(2)}=\frac{ma^{2}}{2\hbar^{2}}\left(\frac{1}{2n-1}+\frac{1}{-2n-1}% \right)=\frac{ma^{2}}{\hbar^{2}}\frac{1}{4n^{2}-1}

Peter_Gustav_Lejeune_Dirichlet.html

  1. [ i ] \mathbb{Z}[i]

Petersen_graph.html

  1. K 5 K_{5}
  2. K G 5 , 2 KG_{5,2}
  3. K G 2 n - 1 , n - 1 KG_{2n-1,n-1}
  4. K 5 K_{5}
  5. K 3 , 3 K_{3,3}
  6. K 5 K_{5}
  7. K 3 , 3 K_{3,3}
  8. S 5 S_{5}
  9. S 5 S_{5}
  10. ( 3 , 5 ) (3,5)
  11. ( 3 , 5 ) (3,5)
  12. t ( t - 1 ) ( t - 2 ) ( t 7 - 12 t 6 + 67 t 5 - 230 t 4 + 529 t 3 - 814 t 2 + 775 t - 352 ) . t(t-1)(t-2)\left(t^{7}-12t^{6}+67t^{5}-230t^{4}+529t^{3}-814t^{2}+775t-352% \right).
  13. ( t - 1 ) 5 ( t + 2 ) 4 ( t - 3 ) (t-1)^{5}(t+2)^{4}(t-3)
  14. \subseteq

Peter–Weyl_theorem.html

  1. φ = L π \varphi=L\circ\pi
  2. g v , g w = v , w \langle gv,gw\rangle=\langle v,w\rangle
  3. ρ ( h ) f ( g ) = f ( h - 1 g ) . \rho(h)f(g)=f(h^{-1}g).
  4. L 2 ( G ) = π Σ ^ E π dim E π L^{2}(G)=\underset{\pi\in\Sigma}{\widehat{\bigoplus}}E_{\pi}^{\oplus\dim E_{% \pi}}
  5. u i j ( π ) \scriptstyle{u_{ij}^{(\pi)}}
  6. u i j ( π ) ( g ) = π ( g ) e i , e j . u^{(\pi)}_{ij}(g)=\langle\pi(g)e_{i},e_{j}\rangle.
  7. { d ( π ) u i j ( π ) π Σ , 1 i , j d ( π ) } \left\{\sqrt{d^{(\pi)}}u^{(\pi)}_{ij}\mid\,\pi\in\Sigma,\,\,1\leq i,j\leq d^{(% \pi)}\right\}

Petri_net.html

  1. N = ( P , T , F ) N=(P,T,F)
  2. P P
  3. T T
  4. F ( P × T ) ( T × P ) F\subset(P\times T)\cup(T\times P)
  5. ( S , T , W ) (S,T,W)\!
  6. W : ( S × T ) ( T × S ) W:(S\times T)\cup(T\times S)\to\mathbb{N}
  7. F = { ( x , y ) W ( x , y ) > 0 } F=\{(x,y)\mid W(x,y)>0\}
  8. ( S T , F ) (S\cup T,F)
  9. t = { s S W ( s , t ) > 0 } {}^{\bullet}t=\{s\in S\mid W(s,t)>0\}
  10. t = { s S W ( t , s ) > 0 } t^{\bullet}=\{s\in S\mid W(t,s)>0\}
  11. M : S M:S\to\mathbb{N}
  12. ( S , T , W , M 0 ) (S,T,W,M_{0})\!
  13. ( S , T , W ) (S,T,W)
  14. M 0 M_{0}
  15. W ( s , t ) W(s,t)
  16. W ( t , s ) W(t,s)
  17. s : M ( s ) W ( s , t ) \forall s:M(s)\geq W(s,t)
  18. M M^{\prime}
  19. M G M M\to_{G}M^{\prime}
  20. M G * M M{\to_{G}}^{*}M^{\prime}
  21. G * {\to_{G}}^{*}
  22. G \to_{G}
  23. N = ( S , T , W , M 0 ) N=(S,T,W,M_{0})\!
  24. M 0 M_{0}
  25. R ( N ) = D { M M 0 ( S , T , W ) * M } R(N)\ \stackrel{D}{=}\ \{M^{\prime}\mid M_{0}{\to_{(S,T,W)}}^{*}M^{\prime}\}
  26. G \to_{G}
  27. R ( N ) R(N)
  28. M 0 M_{0}
  29. σ = t i 1 t i n \vec{\sigma}=\langle t_{i_{1}}\ldots t_{i_{n}}\rangle
  30. M 0 G , t i 1 M 1 M n - 1 G , t i n M n M_{0}\to_{G,t_{i_{1}}}M_{1}\wedge\ldots\wedge M_{n-1}\to_{G,t_{i_{n}}}M_{n}
  31. L ( N ) L(N)
  32. F ( S × T ) ( T × S ) F\subseteq(S\times T)\cup(T\times S)
  33. ( S , T , W , M 0 ) (S,T,W,M_{0})\!
  34. | S | |S|
  35. | S | |S|
  36. | T | |T|
  37. W - W^{-}
  38. s , t : W - [ s , t ] = W ( s , t ) \forall s,t:W^{-}[s,t]=W(s,t)
  39. W + W^{+}
  40. s , t : W + [ s , t ] = W ( t , s ) . \forall s,t:W^{+}[s,t]=W(t,s).
  41. W T = W + - W - W^{T}=W^{+}-W^{-}
  42. o ( w ) o(w)
  43. R ( N ) = { M w : M = M 0 + W T o ( w ) w R(N)=\{M\mid\exists w:M=M_{0}+W^{T}\cdot o(w)\wedge w\!
  44. N } N\}\!
  45. W + = [ * t 1 t 2 p 1 0 1 p 2 1 0 p 3 1 0 p 4 0 1 ] W^{+}=\begin{bmatrix}*&t1&t2\\ p1&0&1\\ p2&1&0\\ p3&1&0\\ p4&0&1\end{bmatrix}
  46. W - = [ * t 1 t 2 p 1 1 0 p 2 0 1 p 3 0 1 p 4 0 0 ] W^{-}=\begin{bmatrix}*&t1&t2\\ p1&1&0\\ p2&0&1\\ p3&0&1\\ p4&0&0\end{bmatrix}
  47. W T = [ * t 1 t 2 p 1 - 1 1 p 2 1 - 1 p 3 1 - 1 p 4 0 1 ] W^{T}=\begin{bmatrix}*&t1&t2\\ p1&-1&1\\ p2&1&-1\\ p3&1&-1\\ p4&0&1\end{bmatrix}
  48. M 0 = [ 1 0 2 1 ] M_{0}=\begin{bmatrix}1&0&2&1\end{bmatrix}
  49. M R ( N ) M\in R(N)
  50. t 0 t_{0}
  51. j > 0 , j>0,
  52. t j t_{j}
  53. L j L_{j}
  54. L 1 - L 4 L_{1}-L_{4}
  55. ( N , M 0 ) (N,M_{0})
  56. L k L_{k}
  57. L k L_{k}
  58. L ( N , M 0 ) L(N,M_{0})
  59. L 1 L_{1}
  60. L ( N , M 0 ) L(N,M_{0})
  61. L 2 L_{2}
  62. L ( N , M 0 ) L(N,M_{0})
  63. L 3 L_{3}
  64. V L ( N , M 0 ) {}_{V\subseteq L(N,M_{0})}
  65. L 4 L_{4}
  66. L 1 L_{1}
  67. R ( N , M 0 ) R(N,M_{0})
  68. L j + 1 L_{j+1}
  69. L j L_{j}
  70. j 1 , 2 , 3 {}_{j\in{1,2,3}}
  71. L 0 L_{0}
  72. ( S , T , W , C , M 0 ) (S,T,W,C,M_{0})
  73. ( S , T , W , M 0 ) (S,T,W,M_{0})
  74. C : P I N C:P\rightarrow\!\!\!\shortmid I\!N
  75. t T : | t | = | t | = 1 \forall t\in T:|t^{\bullet}|=|{}^{\bullet}t|=1
  76. s S : | s | = | s | = 1 \forall s\in S:|s^{\bullet}|=|{}^{\bullet}s|=1
  77. s S : ( | s | 1 ) ( ( s ) = { s } ) \forall s\in S:(|s^{\bullet}|\leq 1)\vee({}^{\bullet}(s^{\bullet})=\{s\})
  78. s 1 , s 2 S : ( s 1 s 2 ) [ ( s 1 s 2 ) ( s 2 s 1 ) ] \forall s_{1},s_{2}\in S:(s_{1}{}^{\bullet}\cap s_{2}{}^{\bullet}\neq\emptyset% )\to[(s_{1}{}^{\bullet}\subseteq s_{2}{}^{\bullet})\vee(s_{2}{}^{\bullet}% \subseteq s_{1}{}^{\bullet})]

PH_indicator.html

  1. HInd + H 2 O H 3 O + + Ind - \textrm{HInd}+\textrm{H}_{2}\textrm{O}\rightleftharpoons\textrm{H}_{3}\textrm{% O}^{+}+\textrm{Ind}^{-}
  2. pH = pK a + log [ Ind - ] [ HInd ] \textrm{pH}=\textrm{pK}_{a}+\log\frac{[\textrm{Ind}^{-}]}{[\textrm{HInd}]}
  3. A x = [ H A ] ϵ H A x + [ A - ] ϵ A - x A_{x}=[HA]\epsilon^{x}_{HA}+[A^{-}]\epsilon^{x}_{A^{-}}
  4. A y = [ H A ] ϵ H A y + [ A - ] ϵ A - y A_{y}=[HA]\epsilon^{y}_{HA}+[A^{-}]\epsilon^{y}_{A^{-}}
  5. pH = pK a + log [ A - ] [ HA ] \textrm{pH}=\textrm{pK}_{a}+\log\frac{[\textrm{A}^{-}]}{[\textrm{HA}]}

Phase_rule.html

  1. F = C - P + 2 F\;=\;C\;-\;P\;+\;2

Phase_space.html

  1. d y / d t = f ( y ) , dy/dt=f(y),

Phi.html

  1. 1 + 5 2 \tfrac{1+\sqrt{5}}{2}\approx
  2. Φ \Phi\,\!
  3. ϕ \phi\,\!
  4. φ \varphi
  5. Φ \Phi\,\!
  6. ϕ \phi\,\!
  7. φ \varphi\,\!

Phillips_curve.html

  1. g W = g W T - f ( U ) gW=gW^{T}-f(U)
  2. g W = g W T - f ( U ) + λ . g P e x . gW=gW^{T}-f(U)+\lambda.gP^{ex}.
  3. Y = Y n + a ( P - P e ) Y=Y_{n}+a(P-P_{e})\,
  4. P = P e + Y - Y n a P=P_{e}+\frac{Y-Y_{n}}{a}
  5. P = P e + Y - Y n a + v P=P_{e}+\frac{Y-Y_{n}}{a}+v
  6. P - P - 1 π P-P_{-1}\ \approx\pi
  7. P e - P - 1 π e P_{e}-P_{-1}\ \approx\pi_{e}
  8. Y - Y n a = - b ( U - U n ) \frac{Y-Y_{n}}{a}=-b(U-U_{n})
  9. π = π e - b ( U - U n ) + v \pi=\pi_{e}-b(U-U_{n})+v\,
  10. π t = β E t [ π t + 1 ] + κ y t \pi_{t}=\beta E_{t}[\pi_{t+1}]+\kappa y_{t}
  11. κ = α [ 1 - ( 1 - α ) β ] ϕ 1 - α \kappa=\frac{\alpha[1-(1-\alpha)\beta]\phi}{1-\alpha}
  12. β E t [ π t + 1 ] \beta E_{t}[\pi_{t+1}]

Phong_reflection_model.html

  1. i s i\text{s}
  2. i d i\text{d}
  3. i a i\text{a}
  4. k s k\text{s}
  5. k d k\text{d}
  6. k a k\text{a}
  7. α \alpha
  8. L ^ m \hat{L}_{m}
  9. m m
  10. N ^ \hat{N}
  11. R ^ m \hat{R}_{m}
  12. V ^ \hat{V}
  13. I p I\text{p}
  14. I p = k a i a + m lights ( k d ( L ^ m N ^ ) i m , d + k s ( R ^ m V ^ ) α i m , s ) . I\text{p}=k\text{a}i\text{a}+\sum_{m\;\in\;\,\text{lights}}(k\text{d}(\hat{L}_% {m}\cdot\hat{N})i_{m,\,\text{d}}+k\text{s}(\hat{R}_{m}\cdot\hat{V})^{\alpha}i_% {m,\,\text{s}}).
  15. R ^ m \hat{R}_{m}
  16. L ^ m \hat{L}_{m}
  17. N ^ \hat{N}
  18. R ^ m = 2 ( L ^ m N ^ ) N ^ - L ^ m \hat{R}_{m}=2(\hat{L}_{m}\cdot\hat{N})\hat{N}-\hat{L}_{m}
  19. V ^ \hat{V}
  20. V ^ \hat{V}
  21. R ^ m \hat{R}_{m}
  22. α \alpha
  23. R ^ m \hat{R}_{m}
  24. V ^ \hat{V}
  25. α \alpha
  26. k a , k\text{a},
  27. k d k\text{d}
  28. k s k\text{s}
  29. max ( 0 , R ^ m V ^ ) α = max ( 0 , 1 - λ ) β γ = ( max ( 0 , 1 - λ ) β ) γ max ( 0 , 1 - β λ ) γ \max(0,\hat{R}_{m}\cdot\hat{V})^{\alpha}=\max(0,1-\lambda)^{\beta\gamma}=\left% (\max(0,1-\lambda)^{\beta}\right)^{\gamma}\approx\max(0,1-\beta\lambda)^{\gamma}
  30. λ = 1 - R ^ m V ^ \lambda=1-\hat{R}_{m}\cdot\hat{V}
  31. γ \gamma
  32. β = α / γ \beta=\alpha/\gamma\,
  33. λ \lambda
  34. λ = ( R ^ m - V ^ ) ( R ^ m - V ^ ) / 2 \lambda=(\hat{R}_{m}-\hat{V})\cdot(\hat{R}_{m}-\hat{V})/2
  35. R ^ m \hat{R}_{m}
  36. V ^ \hat{V}
  37. λ = 1 - R ^ m V ^ \lambda=1-\hat{R}_{m}\cdot\hat{V}
  38. γ \gamma
  39. γ = 2 n \gamma=2^{n}
  40. n n
  41. ( 1 - β λ ) γ (1-\beta\lambda)^{\gamma}
  42. ( 1 - β λ ) (1-\beta\lambda)
  43. n n
  44. β \beta
  45. α \alpha
  46. N = [ N x , N z ] N=[N_{x},N_{z}]
  47. I p ( x ) = C a + C d ( L ( x ) N ( x ) ) I_{p}(x)=C_{a}+C_{d}(L(x)\cdot N(x))
  48. C a C_{a}
  49. C d C_{d}
  50. ( I p ( x ) - C a ) / C d = L ( x ) N ( x ) (I_{p}(x)-C_{a})/C_{d}=L(x)\cdot N(x)
  51. ( I p - C a ) / C d = L x N x + L z N z (I_{p}-C_{a})/C_{d}=L_{x}N_{x}+L_{z}N_{z}
  52. L = [ 0.71 , 0.71 ] L=[0.71,0.71]
  53. ( I p - C a ) / C d = 0.71 N x + 0.71 N z (I_{p}-C_{a})/C_{d}=0.71N_{x}+0.71N_{z}
  54. 1 = ( N x 2 + N z 2 ) 1=\sqrt{(N_{x}^{2}+N_{z}^{2})}
  55. N = [ N x , N y , N z ] N=[N_{x},N_{y},N_{z}]

Phonon.html

  1. V \,V
  2. i j V ( r i - r j ) \,\sum_{i\neq j}V(r_{i}-r_{j})
  3. r i \,r_{i}
  4. i \,i
  5. V \,V
  6. V \,V
  7. V \,V
  8. V \,V
  9. x 2 \,x^{2}
  10. x \,x
  11. x \,x
  12. { i j } ( n n ) 1 2 m ω 2 ( R i - R j ) 2 . \sum_{\{ij\}(nn)}{1\over 2}m\omega^{2}(R_{i}-R_{j})^{2}.
  13. ω \,\omega
  14. R i \,R_{i}
  15. i \,i
  16. λ \,\lambda
  17. \cdots
  18. \cdots
  19. u n - 1 u n u n + 1 u_{n-1}\qquad\quad u_{n}\qquad\quad u_{n+1}
  20. n n
  21. n n
  22. d d
  23. u n u_{n}
  24. n n
  25. C C
  26. m m
  27. n n
  28. - 2 C u n + C ( u n + 1 + u n - 1 ) = m d 2 u n d t 2 -2Cu_{n}+C(u_{n+1}+u_{n-1})=m{\operatorname{d^{2}}u_{n}\over\operatorname{d}t^% {2}}
  29. u n = k = 1 N U k e i k n d u_{n}=\sum_{k=1}^{N}U_{k}e^{iknd}
  30. n d nd
  31. x x
  32. U k U_{k}
  33. 2 C ( cos k d - 1 ) U k = m d 2 U k d t 2 2C(\cos\,kd-1)U_{k}=m{\operatorname{d^{2}}U_{k}\over\operatorname{d}t^{2}}
  34. U k = A k e i ω k t ; ω k = 2 C m ( 1 - cos k d ) U_{k}=A_{k}e^{i\omega_{k}t};\qquad\quad\omega_{k}=\sqrt{{2C\over m}(1-\cos{kd})}
  35. U k U_{k}
  36. k k
  37. ω k \omega_{k}
  38. x 1 , x 2 , x_{1},x_{2},...
  39. x i = 0 x_{i}=0
  40. i i
  41. x i x_{i}
  42. 𝐇 = i = 1 N p i 2 2 m + 1 2 m ω 2 { i j } ( n n ) ( x i - x j ) 2 \mathbf{H}=\sum_{i=1}^{N}{p_{i}^{2}\over 2m}+{1\over 2}m\omega^{2}\sum_{\{ij\}% (nn)}(x_{i}-x_{j})^{2}
  43. m \,m
  44. x i \,x_{i}
  45. p i \,p_{i}
  46. i \,i
  47. N \,N
  48. Q k \,Q_{k}
  49. x \,x
  50. N \,N
  51. Π \,\Pi
  52. p \,p
  53. Q k = 1 N l e i k a l x l Q_{k}={1\over\sqrt{N}}\sum_{l}e^{ikal}x_{l}
  54. Π k = 1 N l e - i k a l p l . \Pi_{k}={1\over\sqrt{N}}\sum_{l}e^{-ikal}p_{l}.
  55. k n \,k_{n}
  56. 2 π \,2\,\pi
  57. [ x l , p m ] \displaystyle\left[x_{l},p_{m}\right]
  58. l x l x l + m \displaystyle\sum_{l}x_{l}x_{l+m}
  59. 1 2 m ω 2 j ( x j - x j + 1 ) 2 = 1 2 m ω 2 k Q k Q - k ( 2 - e i k a - e - i k a ) = 1 2 k m ω k 2 Q k Q - k {1\over 2}m\omega^{2}\sum_{j}(x_{j}-x_{j+1})^{2}={1\over 2}m\omega^{2}\sum_{k}% Q_{k}Q_{-k}(2-e^{ika}-e^{-ika})={1\over 2}\sum_{k}m{\omega_{k}}^{2}Q_{k}Q_{-k}
  60. ω k = 2 ω 2 [ 1 - cos ( k a ) ] = 2 ω | sin ( k a 2 ) | \omega_{k}=\sqrt{2\omega^{2}\left[1-\cos(ka)\right]}=2\omega\left|\sin\left({{% ka}\over 2}\right)\right|
  61. 𝐇 = 1 2 m k ( Π k Π - k + m 2 ω k 2 Q k Q - k ) \mathbf{H}={1\over{2m}}\sum_{k}\left({\Pi_{k}\Pi_{-k}}+m^{2}\omega_{k}^{2}Q_{k% }Q_{-k}\right)
  62. Q \,Q
  63. Π \,\Pi
  64. N \,N
  65. ( N + 1 ) \,(N+1)
  66. k = k n = 2 π n N a for n = 0 , ± 1 , ± 2 , , ± N 2 . k=k_{n}={2\pi n\over Na}\quad\hbox{for}\ n=0,\pm 1,\pm 2,...,\pm{N\over 2}.
  67. n \,n
  68. a \,a
  69. ω k \omega_{k}
  70. E n = ( 1 2 + n ) ω k n = 0 , 1 , 2 , 3...... E_{n}=\left({1\over 2}+n\right)\hbar\omega_{k}\quad\quad\quad n=0,1,2,3......
  71. 1 2 ω , 3 2 ω , 5 2 ω {1\over 2}\hbar\omega,\quad{3\over 2}\hbar\omega,\quad{5\over 2}\hbar\omega% \quad......
  72. 1 2 ω {1\over 2}\hbar\omega
  73. ω \hbar\omega
  74. ω ± 2 = K ( 1 m 1 + 1 m 2 ) ± K ( 1 m 1 + 1 m 2 ) 2 - 4 sin 2 ( k a / 2 ) m 1 m 2 , \omega_{\pm}^{2}=K\left(\frac{1}{m_{1}}+\frac{1}{m_{2}}\right)\pm K\sqrt{\left% (\frac{1}{m_{1}}+\frac{1}{m_{2}}\right)^{2}-\frac{4\sin^{2}(ka/2)}{m_{1}m_{2}}% }\ ,
  75. ω k k \,\tfrac{\partial\omega_{k}}{\partial k}
  76. k \,k
  77. ω a \,\omega a
  78. k \,k
  79. b k = 1 2 ( Q k l k + i Π - k / l k ) , Q k = l k 1 2 ( b k + b - k ) b_{k}={1\over\sqrt{2}}\left({Q_{k}\over l_{k}}+i{\Pi_{-k}\over\hbar/l_{k}}% \right)\quad,\quad Q_{k}=l_{k}{1\over\sqrt{2}}({b_{k}}^{\dagger}+b_{-k})
  80. b k = 1 2 ( Q k l k - i Π - k / l k ) , Π k = l k i 2 ( b k - b - k ) {b_{k}}^{\dagger}={1\over\sqrt{2}}\left({Q_{k}\over l_{k}}-i{\Pi_{-k}\over% \hbar/l_{k}}\right)\quad,\quad\Pi_{k}={\hbar\over l_{k}}{i\over\sqrt{2}}({b_{k% }}^{\dagger}-b_{-k})
  81. l k = < m t p l > m ω k l_{k}=\sqrt{<}mtpl>{{\hbar\over m\omega_{k}}}
  82. H p h = k ω k ( b k b k + 1 2 ) H_{ph}=\sum_{k}\hbar\omega_{k}\left({b_{k}}^{\dagger}b_{k}+{1\over 2}\right)
  83. [ b k , b k ] = δ k , k , [ b k , b k ] = [ b k , b k ] = 0. [b_{k},b_{k^{\prime}}^{\dagger}]=\delta_{k,k^{\prime}},[b_{k},b_{k^{\prime}}]=% [b_{k}^{\dagger},b_{k^{\prime}}^{\dagger}]=0.
  84. b k \,b_{k}^{\dagger}
  85. b k \,b_{k}
  86. ω k \,\hbar\omega_{k}
  87. b k \,b_{k}^{\dagger}
  88. b \,b
  89. | k \,|k\rangle
  90. k \,k
  91. | k = b k | 0 . \begin{matrix}|k\rangle=b_{k}^{\dagger}|0\rangle.\end{matrix}
  92. j \,j
  93. \,\ell
  94. k | x j ( t ) x ( 0 ) | k = N m ω k cos [ k ( j - ) a - ω k t ] + 0 | x j ( t ) x ( 0 ) | 0 \langle k|x_{j}(t)x_{\ell}(0)|k\rangle=\frac{\hbar}{Nm\omega_{k}}\cos\left[k(j% -\ell)a-\omega_{k}t\right]+\langle 0|x_{j}(t)x_{\ell}(0)|0\rangle
  95. ω k \,\omega_{k}
  96. k \,k
  97. 𝐇 = k s = 1 3 ω k , s ( b k , s b k , s + 1 / 2 ) . \mathbf{H}=\sum_{k}\sum_{s=1}^{3}\hbar\,\omega_{k,s}\left(b_{k,s}^{\dagger}b_{% k,s}+1/2\right).
  98. ω \omega
  99. k \,k
  100. k \,\hbar k
  101. k \,\hbar k
  102. k \,k
  103. Q \,Q
  104. Π \,\Pi
  105. Q k = def Q k + K ; Π k = def Π k + K Q_{k}\ \stackrel{\mathrm{def}}{=}\ Q_{k+K}\quad;\quad\Pi_{k}\ \stackrel{% \mathrm{def}}{=}\ \Pi_{k+K}\quad
  106. K = 2 n π / a \,K=2n\pi/a
  107. n \,n
  108. k \,k
  109. k ± 2 π a \,k\pm\tfrac{2\,\pi}{a}
  110. k ± 4 π a \,k\pm\tfrac{4\,\pi}{a}
  111. k \,k
  112. ( | k | ) \,(|k|)
  113. n ( ω k , s ) = 1 exp ( ω k , s / k B T ) - 1 n(\omega_{k,s})=\frac{1}{\exp(\hbar\omega_{k,s}/k_{B}T)-1}
  114. ω k , s \,\omega_{k,s}
  115. k B \,k_{B}
  116. T \,T
  117. 𝐇 = 1 2 α ( p α 2 + ω α 2 q α 2 - 1 2 ω α ) \mathbf{H}=\frac{1}{2}\sum_{\alpha}(p_{\alpha}^{2}+\omega^{2}_{\alpha}q_{% \alpha}^{2}-\frac{1}{2}\hbar\omega_{\alpha})
  118. 𝐇 = α ω α a α a α \mathbf{H}=\sum_{\alpha}\hbar\omega_{\alpha}a_{\alpha}^{\dagger}a_{\alpha}
  119. 1 2 ω q \frac{1}{2}\hbar\omega_{q}
  120. 1 2 ω q \frac{1}{2}\hbar\omega_{q}
  121. 1 2 ω q \frac{1}{2}\hbar\omega_{q}
  122. 1 2 ω q \frac{1}{2}\hbar\omega_{q}
  123. | n 1 n 2 n 3 |n_{1}n_{2}n_{3}...\rangle
  124. n α n_{\alpha}
  125. α \alpha
  126. n α n_{\alpha}
  127. α \alpha
  128. ω q \hbar\omega_{q}
  129. n 1 ω 1 + n 2 ω 2 + n_{1}\hbar\omega_{1}+n_{2}\hbar\omega_{2}+...
  130. a α | n 1 n α - 1 n α n α + 1 = n α + 1 | n 1 , n α - 1 , n α + 1 , n α + 1 a^{\dagger}_{\alpha}|n_{1}...n_{\alpha-1}n_{\alpha}n_{\alpha+1}...\rangle=% \sqrt{n_{\alpha}+1}|n_{1}...,n_{\alpha-1},n_{\alpha}+1,n_{\alpha+1}...\rangle
  131. a α | n 1 n α - 1 n α n α + 1 = n α | n 1 , n α - 1 , ( n α - 1 ) , n α + 1 , a_{\alpha}|n_{1}...n_{\alpha-1}n_{\alpha}n_{\alpha+1}...\rangle=\sqrt{n_{% \alpha}}|n_{1}...,n_{\alpha-1},(n_{\alpha}-1),n_{\alpha+1},...\rangle
  132. a α a^{\dagger}_{\alpha}
  133. α \alpha
  134. a α a_{\alpha}
  135. N = α a α a α N=\sum_{\alpha}a_{\alpha}^{\dagger}a_{\alpha}
  136. a a
  137. a a^{\dagger}
  138. | α , β = | β , α |\alpha,\beta\rangle=|\beta,\alpha\rangle

Phosphagen.html

  1. \leftrightarrow

Phosphorescence.html

  1. S 0 + h ν S 1 T 1 S 0 + h ν S_{0}+h\nu\to S_{1}\to T_{1}\to S_{0}+h\nu^{\prime}

Phylogenetic_tree.html

  1. ( 2 n - 3 ) ! ! = ( 2 n - 3 ) ! 2 n - 2 ( n - 2 ) ! , for n 2 (2n-3)!!=\frac{(2n-3)!}{2^{n-2}(n-2)!}\,,\,\,\text{for}\,n\geq 2
  2. ( 2 n - 5 ) ! ! = ( 2 n - 5 ) ! 2 n - 3 ( n - 3 ) ! , for n 3 (2n-5)!!=\frac{(2n-5)!}{2^{n-3}(n-3)!}\,,\,\,\text{for}\,n\geq 3
  3. n n
  4. n n
  5. n - 1 n-1

Physical_geodesy.html

  1. 𝐠 = W = grad W = W X 𝐢 + W Y 𝐣 + W Z 𝐤 \mathbf{g}=\nabla W=\mathrm{grad}\ W=\frac{\partial W}{\partial X}\mathbf{i}+% \frac{\partial W}{\partial Y}\mathbf{j}+\frac{\partial W}{\partial Z}\mathbf{k}
  2. W W
  3. { 𝐢 , 𝐣 , 𝐤 } \{\mathbf{i},\mathbf{j},\mathbf{k}\}
  4. X , Y , Z X,Y,Z
  5. W = V + Φ W=V+\Phi\,
  6. V V
  7. W W
  8. Φ \Phi
  9. 𝐠 c = ω 2 𝐩 , \mathbf{g}_{c}=\omega^{2}\mathbf{p},
  10. 𝐩 = X 𝐢 + Y 𝐣 + 0 𝐤 \mathbf{p}=X\mathbf{i}+Y\mathbf{j}+0\cdot\mathbf{k}
  11. Φ = 1 2 ω 2 ( X 2 + Y 2 ) . \Phi=\frac{1}{2}\omega^{2}(X^{2}+Y^{2}).
  12. \nabla
  13. X X
  14. Y Y
  15. Z Z
  16. W G M R W\approx\frac{GM}{R}
  17. U U
  18. Ψ \Psi
  19. Φ \Phi
  20. U 0 = 62636860.850 m 2 s - 2 U_{0}=62636860.850\ \textrm{m}^{2}\,\textrm{s}^{-2}
  21. R R\rightarrow\infty
  22. U U
  23. W W
  24. T = W - U T=W-U
  25. N N
  26. N = T / γ , N=T/\gamma\,,
  27. γ \gamma
  28. U U
  29. N = R 4 π γ 0 σ Δ g S ( ψ ) d σ . N=\frac{R}{4\pi\gamma_{0}}\iint_{\sigma}\Delta g\,S(\psi)\,d\sigma.
  30. Δ g \Delta g
  31. N N
  32. Δ g \Delta g
  33. Δ g \Delta g
  34. g = g g=\|\vec{g}\|
  35. γ = γ = U \gamma=\|\vec{\gamma}\|=\|\nabla U\|
  36. Δ g = g - γ . \Delta g=g-\gamma.\,
  37. a B = 2 π G ρ H , a_{B}=2\pi G\rho H,\,
  38. Δ g B = Δ g F A - a B , \Delta g_{B}=\Delta g_{FA}-a_{B},\,
  39. Δ g F A \Delta g_{FA}
  40. Δ g \Delta g

Physics_of_firearms.html

  1. M M
  2. m m
  3. V V
  4. v v
  5. M V + m v = 0 ( 1 ) MV+mv=0\qquad(1)
  6. m m
  7. b b
  8. v v
  9. b b
  10. m m
  11. t t
  12. v v
  13. t t
  14. m m
  15. b b
  16. v v
  17. b b
  18. m m
  19. t t
  20. v v
  21. t t
  22. v v
  23. t t
  24. m m
  25. b b
  26. v v
  27. b b
  28. m m
  29. t t
  30. 1 2 M V 2 \begin{matrix}\frac{1}{2}\end{matrix}MV^{2}
  31. 1 2 m v 2 \begin{matrix}\frac{1}{2}\end{matrix}mv^{2}
  32. 1 2 M V 2 = 1 2 M ( m v M ) 2 = m M 1 2 m v 2 \frac{1}{2}MV^{2}=\frac{1}{2}M\left(\frac{mv}{M}\right)^{2}=\frac{m}{M}\frac{1% }{2}mv^{2}
  33. 1 2 M V 2 1 2 m v 2 = m M ( 2 ) \frac{\frac{1}{2}MV^{2}}{\frac{1}{2}mv^{2}}=\frac{m}{M}\qquad(2)

Pi_(letter).html

  1. ϖ \varpi\,\!
  2. ω \omega

Pinhole_camera.html

  1. d = 2 f λ d=\sqrt{2f\lambda}

Pitch_(music).html

  1. p = 69 + 12 × log 2 ( f 440 Hz ) p=69+12\times\log_{2}{\left(\frac{f}{440\;\mbox{Hz}~{}}\right)}

Planck's_law.html

  1. T T
  2. B ν ( ν , T ) = 2 h ν 3 c 2 1 e h ν k B T - 1 B_{\nu}(\nu,T)=\frac{2h\nu^{3}}{c^{2}}\frac{1}{e^{\frac{h\nu}{k_{\mathrm{B}}T}% }-1}
  3. h h
  4. c c
  5. B λ ( λ , T ) = 2 h c 2 λ 5 1 e h c λ k B T - 1 B_{\lambda}(\lambda,T)=\frac{2hc^{2}}{\lambda^{5}}\frac{1}{e^{\frac{hc}{% \lambda k_{\mathrm{B}}T}}-1}
  6. ε ε
  7. θ θ
  8. d A dA
  9. d Ω
  10. d ν
  11. ν ν
  12. ν \nu
  13. B ν ( ν , T ) = 2 h ν 3 c 2 1 e h ν / ( k B T ) - 1 B_{\nu}(\nu,T)=\frac{2h\nu^{3}}{c^{2}}\frac{1}{e^{h\nu/(k_{\mathrm{B}}T)}-1}
  14. ω \omega
  15. B ω ( ω , T ) = ω 3 4 π 3 c 2 1 e ω / ( k B T ) - 1 B_{\omega}(\omega,T)=\frac{\hbar\omega^{3}}{4\pi^{3}c^{2}}\frac{1}{e^{\hbar% \omega/(k_{\mathrm{B}}T)}-1}
  16. λ \lambda
  17. B λ ( λ , T ) = 2 h c 2 λ 5 1 e h c / ( λ k B T ) - 1 B_{\lambda}(\lambda,T)=\frac{2hc^{2}}{\lambda^{5}}\frac{1}{e^{hc/(\lambda k_{% \mathrm{B}}T)}-1}
  18. y y
  19. B y ( y , T ) = c 2 4 π 3 y 5 1 e c / ( y k B T ) - 1 B_{y}(y,T)=\frac{\hbar c^{2}}{4\pi^{3}y^{5}}\frac{1}{e^{\hbar c/(yk_{\mathrm{B% }}T)}-1}
  20. ν ~ \tilde{\nu}
  21. B ν ~ ( ν ~ , T ) = 2 h c 2 ν ~ 3 1 e h c ν ~ / ( k B T ) - 1 B_{\tilde{\nu}}(\tilde{\nu},T)=2hc^{2}\tilde{\nu}^{3}\frac{1}{e^{hc\tilde{\nu}% /(k_{\mathrm{B}}T)}-1}
  22. k k
  23. B k ( k , T ) = c 2 k 3 4 π 3 1 e c k / ( k B T ) - 1 B_{k}(k,T)=\frac{\hbar c^{2}k^{3}}{4\pi^{3}}\frac{1}{e^{\hbar ck/(k_{\mathrm{B% }}T)}-1}
  24. d ν
  25. d λ
  26. B λ ( λ , T ) d λ = - B ν ( ν ( λ ) , T ) d ν , B_{\lambda}(\lambda,\ T)\ \mathrm{d}\lambda=-B_{\nu}(\nu(\lambda),\ T)\ % \mathrm{d}\nu\ ,
  27. B λ ( λ , T ) = - d ν d λ B ν ( ν ( λ ) , T ) . B_{\lambda}(\lambda,\ T)\ =\ -\ \frac{\mathrm{d}\nu}{\mathrm{d}\lambda}B_{\nu}% (\nu(\lambda),\ T).
  28. ν ( λ ) = c / λ ν(λ)=c/λ
  29. B λ ( T ) B ν ( T ) = c λ 2 = ν 2 c . \frac{B_{\lambda}(T)}{B_{\nu}(T)}=\frac{c}{\lambda^{2}}=\frac{\nu^{2}}{c}.
  30. u i ( T ) = 4 π c B i ( T ) . u_{i}(T)=\frac{4\pi}{c}B_{i}(T).
  31. L ( λ , T ) = c 1 L λ 5 1 exp ( c 2 λ T ) - 1 L(\lambda,T)=\frac{c_{1L}}{\lambda^{5}}\frac{1}{\exp\left(\frac{c_{2}}{{% \lambda}T}\right)-1}
  32. π \pi
  33. M ( λ , T ) = c 1 λ 5 1 exp ( c 2 λ T ) - 1 M(\lambda,T)=\frac{c_{1}}{\lambda^{5}}\frac{1}{\exp\left(\frac{c_{2}}{{\lambda% }T}\right)-1}
  34. λ i = 2 L n i , \lambda_{i}=\frac{2L}{n_{i}},
  35. E n 1 , n 2 , n 3 ( r ) = ( r + 1 2 ) h c 2 L n 1 2 + n 2 2 + n 3 2 . (1) E_{n_{1},n_{2},n_{3}}\left(r\right)=\left(r+\frac{1}{2}\right)\frac{hc}{2L}% \sqrt{n_{1}^{2}+n_{2}^{2}+n_{3}^{2}}.\qquad\,\text{(1)}
  36. P r = exp ( - β E ( r ) ) Z ( β ) . P_{r}=\frac{\exp\left(-\beta E\left(r\right)\right)}{Z\left(\beta\right)}.
  37. β = def 1 / ( k B T ) . \beta\ \stackrel{\mathrm{def}}{=}\ 1/\left(k_{\mathrm{B}}T\right).
  38. Z ( β ) = r = 0 e - β E ( r ) = e - β ε / 2 1 - e - β ε . Z\left(\beta\right)=\sum_{r=0}^{\infty}e^{-\beta E\left(r\right)}=\frac{e^{-% \beta\varepsilon/2}}{1-e^{-\beta\varepsilon}}.
  39. ε = def h c 2 L n 1 2 + n 2 2 + n 3 2 , \varepsilon\ \stackrel{\mathrm{def}}{=}\ \frac{hc}{2L}\sqrt{n_{1}^{2}+n_{2}^{2% }+n_{3}^{2}},
  40. E = - d log ( Z ) d β = ε 2 + ε e β ε - 1 . \left\langle E\right\rangle=-\frac{d\log\left(Z\right)}{d\beta}=\frac{% \varepsilon}{2}+\frac{\varepsilon}{e^{\beta\varepsilon}-1}.
  41. E - ε 2 \scriptstyle{\left\langle E\right\rangle}-\frac{\varepsilon}{2}
  42. E - ε 2 \scriptstyle{\left\langle E\right\rangle}-\frac{\varepsilon}{2}
  43. U = 0 ε e β ε - 1 g ( ε ) d ε . (2) U=\int_{0}^{\infty}\frac{\varepsilon}{e^{\beta\varepsilon}-1}g(\varepsilon)\,d% \varepsilon.\qquad\mbox{(2)}~{}
  44. ε = def h c 2 L n , \varepsilon\ \stackrel{\mathrm{def}}{=}\ \frac{hc}{2L}n,
  45. n = n 1 2 + n 2 2 + n 3 2 . n=\sqrt{n_{1}^{2}+n_{2}^{2}+n_{3}^{2}}.
  46. g ( ε ) d ε = 2 1 8 4 π n 2 d n = 8 π L 3 h 3 c 3 ε 2 d ε . g(\varepsilon)\,d\varepsilon=2\frac{1}{8}4\pi n^{2}\,dn=\frac{8\pi L^{3}}{h^{3% }c^{3}}\varepsilon^{2}\,d\varepsilon.
  47. U = L 3 8 π h 3 c 3 0 ε 3 e β ε - 1 d ε . (3) U=L^{3}\frac{8\pi}{h^{3}c^{3}}\int_{0}^{\infty}\frac{\varepsilon^{3}}{e^{\beta% \varepsilon}-1}\,d\varepsilon.\qquad\,\text{(3)}
  48. u ν ( T ) u_{\nu}(T)
  49. U L 3 = 0 u ν ( T ) d ν , \frac{U}{L^{3}}=\int_{0}^{\infty}u_{\nu}(T)\,d\nu,
  50. u ν ( T ) = 8 π h ν 3 c 3 1 e h ν / k B T - 1 . u_{\nu}(T)={8\pi h\nu^{3}\over c^{3}}{1\over e^{h\nu/k_{\mathrm{B}}T}-1}.
  51. U L 3 = 0 u λ ( T ) d λ , \frac{U}{L^{3}}=\int_{0}^{\infty}u_{\lambda}(T)\,d\lambda,
  52. u λ ( T ) = 8 π h c λ 5 1 e h c / λ k B T - 1 . u_{\lambda}(T)={8\pi hc\over\lambda^{5}}{1\over e^{hc/\lambda k_{\mathrm{B}}T}% -1}.
  53. ε = k B T x , \varepsilon=k_{\mathrm{B}}Tx,
  54. u ( T ) = 8 π ( k B T ) 4 ( h c ) 3 J , u(T)=\frac{8\pi(k_{\mathrm{B}}T)^{4}}{(hc)^{3}}J,
  55. J = 0 x 3 e x - 1 d x = π 4 15 . J=\int_{0}^{\infty}\frac{x^{3}}{e^{x}-1}\,dx=\frac{\pi^{4}}{15}.
  56. U V = 8 π 5 ( k B T ) 4 15 ( h c ) 3 , {U\over V}=\frac{8\pi^{5}(k_{\mathrm{B}}T)^{4}}{15(hc)^{3}},
  57. U V = 4 σ T 4 c . {U\over V}=\frac{4\sigma T^{4}}{c}.
  58. B ν ( T ) = u ν ( T ) c 4 π , B_{\nu}(T)=\frac{u_{\nu}(T)\,c}{4\pi},
  59. B ν ( T ) = 2 h ν 3 c 2 1 e h ν / k B T - 1 . B_{\nu}(T)=\frac{2h\nu^{3}}{c^{2}}~{}\frac{1}{e^{h\nu/k_{\mathrm{B}}T}-1}.
  60. ν \nu
  61. B λ ( T ) = B ν ( T ) | d ν d λ | . B_{\lambda}(T)=B_{\nu}(T)\left|\frac{d\nu}{d\lambda}\right|.
  62. X X
  63. Y Y
  64. X X
  65. ν ν
  66. X X
  67. X X
  68. ν ν
  69. X X
  70. X X
  71. q ( ν , T X , T Y ) = α ν , X , Y ( T X , T Y ) I ν , Y ( T Y ) - I ν , X ( T X ) . q(\nu,T_{X},T_{Y})=\alpha_{\nu,X,Y}(T_{X},T_{Y})I_{\nu,Y}(T_{Y})-I_{\nu,X}(T_{% X}).
  72. T T
  73. X X
  74. Y Y
  75. T T
  76. X X
  77. in thermodynamic equilibrium , when T = T X = T Y , it is true that 0 = α ν , X , Y ( T , T ) B ν ( T ) - ϵ ν , X ( T ) B ν ( T ) . \mathrm{in\,thermodynamic\,equilibrium,\,when}\,\,T=T_{X}=T_{Y}\,\mathrm{,\,it% \,is\,true\,that}\,\,0=\alpha_{\nu,X,Y}(T,T)B_{\nu}(T)\,-\,\epsilon_{\nu,X}(T)% B_{\nu}(T).
  78. in thermodynamic equilibrium , when T = T X = T Y , it is true that α ν , X , Y ( T , T ) = ϵ ν , X ( T ) . \mathrm{in\,thermodynamic\,equilibrium,\,when}\,\,T=T_{X}=T_{Y}\,\mathrm{,\,it% \,is\,true\,that}\,\,\alpha_{\nu,X,Y}(T,T)=\epsilon_{\nu,X}(T).
  79. X X
  80. T T
  81. α ν , X ( T ) = ϵ ν , X ( T ) at thermodynamic equilibrium . \alpha_{\nu,X}(T)=\epsilon_{\nu,X}(T)\,\,\mathrm{at\,thermodynamic\,% equilibrium.}
  82. T T
  83. B B
  84. ν ν
  85. ν ν
  86. T T
  87. T T
  88. d Φ ( d A , θ , d Ω , d ν ) dΦ(dA,θ,dΩ,dν)
  89. d A dA
  90. θ θ
  91. d A dA
  92. d Ω
  93. θ θ
  94. d ν
  95. d Φ ( d A , θ , d Ω , d ν ) d Ω = L 0 ( d A , d ν ) d A d ν cos θ \frac{\mathrm{d}\Phi(\mathrm{d}A,\theta,\mathrm{d}\Omega,\mathrm{d}\nu)}{% \mathrm{d}\Omega}=L^{0}(\mathrm{d}A,\mathrm{d}\nu)\,\mathrm{d}A\,\mathrm{d}\nu% \,\cos\theta
  96. d A dA
  97. θ = 0 θ=0
  98. c o s θ cosθ
  99. θ θ
  100. d Φ ( d A , θ , d Ω , d ν ) d Ω = B ν ( T ) d A d ν cos θ . \frac{\mathrm{d}\Phi(\mathrm{d}A,\theta,\mathrm{d}\Omega,\mathrm{d}\nu)}{% \mathrm{d}{\Omega}}=B_{\nu}(T)\,\mathrm{d}A\,\mathrm{d}\nu\,\cos\theta.
  101. P = 0 d ν h d Ω B ν cos ( θ ) P=\int_{0}^{\infty}d\nu\int_{h}d\Omega\,B_{\nu}\cos(\theta)
  102. d Ω = sin ( θ ) d θ d ϕ . d\Omega=\sin(\theta)\,d\theta\,d\phi.
  103. P = 0 d ν 0 π / 2 d θ 0 2 π d ϕ B ν ( T ) cos ( θ ) sin ( θ ) = σ T 4 P=\int_{0}^{\infty}d\nu\int_{0}^{\pi/2}d\theta\int_{0}^{2\pi}d\phi\,B_{\nu}(T)% \cos(\theta)\sin(\theta)=\sigma\,T^{4}
  104. σ = 2 k B 4 π 5 15 c 2 h 3 5.670400 × 10 - 8 J s - 1 m - 2 K - 4 \sigma=\frac{2k_{\mathrm{B}}^{4}\pi^{5}}{15c^{2}h^{3}}\approx 5.670400\times 1% 0^{-8}\,\mathrm{J\,s^{-1}m^{-2}K^{-4}}
  105. I ν I_{\nu}
  106. α \alpha
  107. ρ \rho
  108. κ ν = α / ρ \kappa_{\nu}=\alpha/\rho
  109. d I ν = - κ ν ρ I ν d s dI_{\nu}=-\kappa_{\nu}\,\rho\,I_{\nu}\,ds
  110. j ν j_{\nu}
  111. d I ν = j ν ρ d s dI_{\nu}=j_{\nu}\,\rho\,ds
  112. d I ν d s = j ν ρ - κ ν ρ I ν . \frac{dI_{\nu}}{ds}=j_{\nu}\rho-\kappa_{\nu}\rho I_{\nu}.
  113. d I ν = 0 dI_{\nu}=0
  114. κ ν B ν = j ν \kappa_{\nu}B_{\nu}=j_{\nu}\,
  115. d I ν d s = κ ν ρ ( B ν - I ν ) . \frac{dI_{\nu}}{ds}=\kappa_{\nu}\rho(B_{\nu}-I_{\nu}).
  116. E 1 E_{1}
  117. E 2 E_{2}
  118. ν \nu
  119. E 2 - E 1 = h ν E_{2}-E_{1}=h\nu
  120. n 1 n_{1}
  121. n 2 n_{2}
  122. ( d n 1 d t ) spon = A 21 n 2 \left(\frac{dn_{1}}{dt}\right)_{\mathrm{spon}}=A_{21}n_{2}
  123. ( d n 1 d t ) stim = B 21 n 2 I ν ( T ) \left(\frac{dn_{1}}{dt}\right)_{\mathrm{stim}}=B_{21}n_{2}I_{\nu}(T)
  124. ( d n 2 d t ) abs = B 12 n 1 I ν ( T ) \left(\frac{dn_{2}}{dt}\right)_{\mathrm{abs}}=B_{12}n_{1}I_{\nu}(T)
  125. I ν ( T ) I_{\nu}(T)
  126. A 21 A_{21}
  127. B 21 B_{21}
  128. B 12 B_{12}
  129. ( ν ) (\nu)
  130. 0 = A 21 n 2 + B 21 n 2 B ν ( T ) - B 12 n 1 B ν ( T ) 0=A_{21}n_{2}+B_{21}n_{2}B_{\nu}(T)-B_{12}n_{1}B_{\nu}(T)\,
  131. n 2 n 1 = g 2 g 1 e - h ν / k B T \frac{n_{2}}{n_{1}}=\frac{g_{2}}{g_{1}}e^{-h\nu/k_{\mathrm{B}}T}
  132. g 1 g_{1}
  133. g 2 g_{2}
  134. A 21 B 21 = 2 h ν 3 c 2 \frac{A_{21}}{B_{21}}=\frac{2h\nu^{3}}{c^{2}}
  135. B 21 B 12 = g 1 g 2 \frac{B_{21}}{B_{12}}=\frac{g_{1}}{g_{2}}
  136. j ν j_{\nu}
  137. κ ν \kappa_{\nu}\,
  138. j ν = κ ν B ν . j_{\nu}=\kappa_{\nu}B_{\nu}.\,
  139. B ν , B ω , B ν ~ B_{\nu},B_{\omega},B_{\tilde{\nu}}
  140. B k B_{k}
  141. E = [ 3 + W ( - 3 e 3 ) ] k B T 2.821 k B T , E=\left[3+W\left(\frac{-3}{e^{3}}\right)\right]k_{\mathrm{B}}T\approx 2.821\ k% _{\mathrm{B}}T,
  142. B λ B_{\lambda}
  143. B y B_{y}
  144. E = [ 5 + W ( - 5 e 5 ) ] k B T 4.965 k B T , E=\left[5+W\left(\frac{-5}{e^{5}}\right)\right]k_{\mathrm{B}}T\approx 4.965\ k% _{\mathrm{B}}T,
  145. B ν B_{\nu}
  146. B λ B_{\lambda}
  147. ν \nu
  148. λ \lambda
  149. | d ν d λ | = c / λ 2 \left|\frac{d\nu}{d\lambda}\right|=c/\lambda^{2}
  150. 1 / λ 2 1/\lambda^{2}
  151. h c / k B = 14 387.770 μ m·K . hc/k_{\mathrm{B}}=\,\text{14 387.770 }\mu\,\text{m·K}.
  152. B ν , max ( T ) = 2 k B 3 T 3 ( 3 + W ( - 3 exp ( - 3 ) ) 3 h 2 c 2 1 e 3 + W ( - 3 exp ( - 3 ) ) - 1 ( 1.896 × 10 - 19 W m 2 Hz K 3 ) × T 3 B_{\nu,\,\text{max}}(T)=\frac{2k_{\mathrm{B}}^{3}T^{3}(3+W(-3\exp(-3))^{3}}{h^% {2}c^{2}}\frac{1}{e^{3+W(-3\exp(-3))}-1}\approx\left(1.896\times 10^{-19}\frac% {\,\text{W}}{\,\text{m}^{2}\cdot\,\text{Hz}\cdot\,\text{K}^{3}}\right)\times T% ^{3}
  153. B λ , max ( T ) = 2 k B 5 T 5 ( 5 + W ( - 5 exp ( - 5 ) ) 5 h 4 c 3 1 e 5 + W ( - 5 exp ( - 5 ) ) - 1 ( 4.096 × 10 - 6 W m 3 K 5 ) × T 5 B_{\lambda,\,\text{max}}(T)=\frac{2k_{\mathrm{B}}^{5}T^{5}(5+W(-5\exp(-5))^{5}% }{h^{4}c^{3}}\frac{1}{e^{5+W(-5\exp(-5))}-1}\approx\left(4.096\times 10^{-6}% \frac{\,\text{W}}{\,\text{m}^{3}\cdot\,\text{K}^{5}}\right)\times~{}T^{5}
  154. B ν ( T ) 2 ν 2 c 2 k B T B_{\nu}(T)\approx\frac{2\nu^{2}}{c^{2}}\,k_{\mathrm{B}}T
  155. B λ ( T ) 2 c λ 4 k B T . \qquad B_{\lambda}(T)\approx\frac{2c}{\lambda^{4}}\,k_{\mathrm{B}}T.
  156. B ν ( T ) 2 h ν 3 c 2 e - h ν k B T B_{\nu}(T)\approx\frac{2h\nu^{3}}{c^{2}}\,e^{-\frac{h\nu}{k_{\mathrm{B}}T}}
  157. B λ ( T ) 2 h c 2 λ 5 e - h c λ k B T . B_{\lambda}(T)\approx\frac{2hc^{2}}{\lambda^{5}}\,e^{-\frac{hc}{\lambda k_{% \mathrm{B}}T}}.
  158. 1 / λ 5 1/\lambda^{5}
  159. ν 3 \nu^{3}
  160. ν 2 / λ 2 \nu^{2}/\lambda^{2}
  161. ν = c / λ \nu=c/\lambda
  162. λ \lambda
  163. T T
  164. E ( T , i ) E(T,i)
  165. i i
  166. T T
  167. a ( T , i ) a(T,i)
  168. T T
  169. E ( T , i ) / a ( T , i ) E(T,i)/a(T,i)
  170. a ( T , i ) a(T,i)
  171. T T
  172. E ( λ , T , i ) E(λ,T,i)
  173. a ( λ , T , i ) a(λ,T,i)
  174. E ( λ , T , i ) / a ( λ , T , i ) E(λ,T,i)/a(λ,T,i)
  175. E ( λ , T , i ) / a ( λ , T , i ) E(λ,T,i)/a(λ,T,i)
  176. i i
  177. E ( T , i ) / a ( T , i ) E(T,i)/a(T,i)
  178. i i
  179. E ( λ , T , i ) / a ( λ , T , i ) E(λ,T,i)/a(λ,T,i)
  180. i i
  181. B B BB
  182. T T
  183. E ( λ , T , i ) / a ( λ , T , i ) E(λ,T,i)/a(λ,T,i)
  184. i i
  185. λ λ
  186. T T
  187. E ( λ , T , B B ) E(λ,T,BB)
  188. a ( λ , T , B B ) a(λ,T,BB)
  189. E ( λ , T , B B ) / a ( λ , T , B B ) E(λ,T,BB)/a(λ,T,BB)
  190. E ( λ , T , B B ) E(λ,T,BB)
  191. T T
  192. E ( λ , T , i ) / a ( λ , T , i ) E(λ,T,i)/a(λ,T,i)
  193. E ( λ , T , B B ) E(λ,T,BB)
  194. λ λ
  195. T T
  196. T T
  197. λ λ
  198. i i
  199. T T
  200. λ λ
  201. e e
  202. I λ = B 1 θ 3 2 exp ( - c λ 2 θ ) λ - 6 , I_{\lambda}=B_{1}\theta^{\frac{3}{2}}\mathrm{exp}(-\frac{c}{\lambda^{2}\theta}% )\lambda^{-6}\,,
  203. λ λ
  204. θ θ
  205. c c
  206. C C
  207. c c
  208. λ λ
  209. T T
  210. d 2 S / d U λ 2 = α U λ ( β + U λ ) . \mathrm{d}^{2}S/\mathrm{d}U_{\lambda}^{2}=\frac{\alpha}{U_{\lambda}(\beta+U_{% \lambda})}.
  211. B λ ( T ) = C λ - 5 e c λ T - 1 , B_{\lambda}(T)=\frac{C\lambda^{-5}}{e^{\frac{c}{\lambda T}}-1},
  212. C C
  213. c c
  214. h h
  215. h h
  216. ν ν
  217. h h
  218. ϵ = h ν . \epsilon=h\nu.
  219. h ν h\nu
  220. R β ν / N R\beta\nu/N
  221. U = n h ν U=nhν
  222. h ν
  223. W τ 2 - W τ 1 = h ν W_{\tau_{2}}-W_{\tau_{1}}=h\nu
  224. W τ 2 W_{\tau_{2}}
  225. W τ 1 W_{\tau_{1}}
  226. τ 2 \tau_{2}
  227. τ 1 \tau_{1}
  228. ν \nu
  229. ν \nu
  230. 2 π ν 2\pi\nu
  231. h ν h\nu
  232. h ν h\nu

Planck_mass.html

  1. m P = c G m\text{P}=\sqrt{\frac{\hbar c}{G}}
  2. c 8 π G \sqrt{\frac{\hbar{}c}{8\pi G}}
  3. 1 / 8 π 1/\sqrt{8\pi}
  4. m P = c n 1 G n 2 n 3 , m\text{P}=c^{n_{1}}G^{n_{2}}\hbar^{n_{3}},
  5. n 1 , n 2 , n 3 n_{1},n_{2},n_{3}
  6. [ c ] = 𝖫𝖳 - 1 [c]=\mathsf{LT}^{-1}
  7. [ G ] = 𝖬 - 1 𝖫 3 𝖳 - 2 [G]=\mathsf{M}^{-1}\mathsf{L}^{3}\mathsf{T}^{-2}
  8. [ ] = 𝖬 1 𝖫 2 𝖳 - 1 [\hbar]=\mathsf{M}^{1}\mathsf{L}^{2}\mathsf{T}^{-1}
  9. [ c n 1 G n 2 n 3 ] = 𝖬 - n 2 + n 3 𝖫 n 1 + 3 n 2 + 2 n 3 𝖳 - n 1 - 2 n 2 - n 3 [c^{n_{1}}G^{n_{2}}\hbar^{n_{3}}]=\mathsf{M}^{-n_{2}+n_{3}}\mathsf{L}^{n_{1}+3% n_{2}+2n_{3}}\mathsf{T}^{-n_{1}-2n_{2}-n_{3}}
  10. - n 2 + n 3 = 1 -n_{2}+n_{3}=1
  11. n 1 + 3 n 2 + 2 n 3 = 0 n_{1}+3n_{2}+2n_{3}=0
  12. - n 1 - 2 n 2 - n 3 = 0 -n_{1}-2n_{2}-n_{3}=0
  13. n 1 = 1 / 2 , n 2 = - 1 / 2 , n 3 = 1 / 2. n_{1}=1/2,n_{2}=-1/2,n_{3}=1/2.
  14. m P = c 1 / 2 G - 1 / 2 1 / 2 = c G . m\text{P}=c^{1/2}G^{-1/2}\hbar^{1/2}=\sqrt{\frac{c\hbar}{G}}.
  15. E = G m P 2 r = c r . E=\frac{Gm\text{P}^{2}}{r}=\frac{\hbar c}{r}.
  16. m P = c G m\text{P}=\sqrt{\frac{\hbar c}{G}}
  17. λ c = h m c \lambda_{c}=\frac{h}{mc}
  18. r s = 2 G m c 2 r_{s}=\frac{2Gm}{c^{2}}
  19. m = h c 2 G = π c G m=\sqrt{\frac{hc}{2G}}=\sqrt{\frac{\pi c\hbar}{G}}
  20. π \sqrt{\pi}

Planck_temperature.html

  1. T P = m P c 2 k = c 5 G k 2 T\text{P}=\frac{m\text{P}c^{2}}{k}=\sqrt{\frac{\hbar c^{5}}{Gk^{2}}}
  2. \hbar
  3. = h 2 π , \hbar\ =\frac{h}{2\pi},

Planck_time.html

  1. t P G c 5 5.39106 ( 32 ) × 10 - 44 s t_{\mathrm{P}}\equiv\sqrt{\frac{\hbar G}{c^{5}}}\approx 5.39106(32)\times 10^{% -44}\ \mathrm{s}
  2. < v a r > ħ = < v a r > h < / v a r > 2 < v a r > / π <var>ħ=<var>h</var>{2<var>}/{\pi}

Plane_(geometry).html

  1. 𝐫 0 \mathbf{r}_{0}
  2. P 0 = ( x 0 , y 0 , z 0 ) P_{0}=(x_{0},y_{0},z_{0})
  3. 𝐧 = ( a , b , c ) \mathbf{n}=(a,b,c)
  4. P P
  5. 𝐫 \mathbf{r}
  6. P 0 P_{0}
  7. P P
  8. 𝐧 \mathbf{n}
  9. 𝐫 \mathbf{r}
  10. 𝐧 ( 𝐫 - 𝐫 0 ) = 0. \mathbf{n}\cdot(\mathbf{r}-\mathbf{r}_{0})=0.
  11. 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,
  12. 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}).
  13. a x + b y + c z + d = 0 , ax+by+cz+d=0,
  14. 𝐧 = ( a , b , c ) \mathbf{n}=(a,b,c)
  15. r = r 0 + s v + t w , r={r}_{0}+s{v}+t{w},
  16. v v
  17. 𝐰 \mathbf{w}
  18. v v
  19. 𝐰 \mathbf{w}
  20. v v
  21. 𝐰 \mathbf{w}
  22. | x - x 1 y - y 1 z - z 1 x 2 - x 1 y 2 - y 1 z 2 - z 1 x 3 - x 1 y 3 - y 1 z 3 - z 1 | = | x - x 1 y - y 1 z - z 1 x - x 2 y - y 2 z - z 2 x - x 3 y - y 3 z - z 3 | = 0. \begin{vmatrix}x-x_{1}&y-y_{1}&z-z_{1}\\ x_{2}-x_{1}&y_{2}-y_{1}&z_{2}-z_{1}\\ x_{3}-x_{1}&y_{3}-y_{1}&z_{3}-z_{1}\end{vmatrix}=\begin{vmatrix}x-x_{1}&y-y_{1% }&z-z_{1}\\ x-x_{2}&y-y_{2}&z-z_{2}\\ x-x_{3}&y-y_{3}&z-z_{3}\end{vmatrix}=0.
  23. a x + b y + c z + d = 0 ax+by+cz+d=0
  24. a x 1 + b y 1 + c z 1 + d = 0 \,ax_{1}+by_{1}+cz_{1}+d=0
  25. a x 2 + b y 2 + c z 2 + d = 0 \,ax_{2}+by_{2}+cz_{2}+d=0
  26. a x 3 + b y 3 + c z 3 + d = 0. \,ax_{3}+by_{3}+cz_{3}+d=0.
  27. D = | x 1 y 1 z 1 x 2 y 2 z 2 x 3 y 3 z 3 | D=\begin{vmatrix}x_{1}&y_{1}&z_{1}\\ x_{2}&y_{2}&z_{2}\\ x_{3}&y_{3}&z_{3}\end{vmatrix}
  28. a = - d D | 1 y 1 z 1 1 y 2 z 2 1 y 3 z 3 | a=\frac{-d}{D}\begin{vmatrix}1&y_{1}&z_{1}\\ 1&y_{2}&z_{2}\\ 1&y_{3}&z_{3}\end{vmatrix}
  29. b = - d D | x 1 1 z 1 x 2 1 z 2 x 3 1 z 3 | b=\frac{-d}{D}\begin{vmatrix}x_{1}&1&z_{1}\\ x_{2}&1&z_{2}\\ x_{3}&1&z_{3}\end{vmatrix}
  30. c = - d D | x 1 y 1 1 x 2 y 2 1 x 3 y 3 1 | . c=\frac{-d}{D}\begin{vmatrix}x_{1}&y_{1}&1\\ x_{2}&y_{2}&1\\ x_{3}&y_{3}&1\end{vmatrix}.
  31. n = ( p 2 - p 1 ) × ( p 3 - p 1 ) , n=(p_{2}-p_{1})\times(p_{3}-p_{1}),
  32. Π : a x + b y + c z + d = 0 \Pi:ax+by+cz+d=0\,
  33. p 1 = ( x 1 , y 1 , z 1 ) p_{1}=(x_{1},y_{1},z_{1})
  34. p 1 p_{1}
  35. D = | a x 1 + b y 1 + c z 1 + d | a 2 + b 2 + c 2 . D=\frac{\left|ax_{1}+by_{1}+cz_{1}+d\right|}{\sqrt{a^{2}+b^{2}+c^{2}}}.
  36. p 1 p_{1}
  37. a 2 + b 2 + c 2 = 1 \sqrt{a^{2}+b^{2}+c^{2}}=1
  38. D = | a x 1 + b y 1 + c z 1 + d | . D=\ |ax_{1}+by_{1}+cz_{1}+d|.
  39. 𝐧 𝐫 - D 0 = 0 , \mathbf{n}\cdot\mathbf{r}-D_{0}=0,
  40. 𝐧 \mathbf{n}
  41. 𝐫 \mathbf{r}
  42. 𝐧 ( 𝐫 - 𝐫 0 ) = 0 \mathbf{n}\cdot(\mathbf{r}-\mathbf{r}_{0})=0
  43. 𝐧 \mathbf{n}
  44. 𝐫 0 = ( x 10 , x 20 , , x N 0 ) \mathbf{r}_{0}=(x_{10},x_{20},\dots,x_{N0})
  45. 𝐫 1 = ( x 11 , x 21 , , x N 1 ) \mathbf{r}_{1}=(x_{11},x_{21},\dots,x_{N1})
  46. i = 1 N a i x i = - a 0 \sum_{i=1}^{N}a_{i}x_{i}=-a_{0}
  47. { a i } \{a_{i}\}
  48. 𝐧 \mathbf{n}
  49. ( a 1 , a 2 , , a N ) (a_{1},a_{2},\dots,a_{N})
  50. 𝐫 1 - 𝐫 0 \mathbf{r}_{1}-\mathbf{r}_{0}
  51. 𝐧 \mathbf{n}
  52. 𝐧 𝐫 0 = 𝐫 0 𝐧 = - a 0 \mathbf{n}\cdot\mathbf{r}_{0}=\mathbf{r}_{0}\cdot\mathbf{n}=-a_{0}
  53. 𝐫 0 \mathbf{r}_{0}
  54. D = | ( 𝐫 1 - 𝐫 0 ) 𝐧 | | 𝐧 | = | 𝐫 1 𝐧 - 𝐫 0 𝐧 | | 𝐧 | = | 𝐫 1 𝐧 + a 0 | | 𝐧 | = | a 1 x 11 + a 2 x 21 + + a N x N 1 + a 0 | a 1 2 + a 2 2 + + a N 2 D=\frac{|(\mathbf{r}_{1}-\mathbf{r}_{0})\cdot\mathbf{n}|}{|\mathbf{n}|}=\frac{% |\mathbf{r}_{1}\cdot\mathbf{n}-\mathbf{r}_{0}\cdot\mathbf{n}|}{|\mathbf{n}|}=% \frac{|\mathbf{r}_{1}\cdot\mathbf{n}+a_{0}|}{|\mathbf{n}|}=\frac{|a_{1}x_{11}+% a_{2}x_{21}+\dots+a_{N}x_{N1}+a_{0}|}{\sqrt{a_{1}^{2}+a_{2}^{2}+\dots+a_{N}^{2% }}}
  55. Π 1 : n 1 r = h 1 \Pi_{1}:{n}_{1}\cdot r=h_{1}
  56. Π 2 : n 2 r = h 2 \Pi_{2}:{n}_{2}\cdot r=h_{2}
  57. n i {n}_{i}
  58. r = ( c 1 n 1 + c 2 n 2 ) + λ ( n 1 × n 2 ) {r}=(c_{1}{n}_{1}+c_{2}{n}_{2})+\lambda({n}_{1}\times{n}_{2})
  59. c 1 = h 1 - h 2 ( n 1 n 2 ) 1 - ( n 1 n 2 ) 2 c_{1}=\frac{h_{1}-h_{2}({n}_{1}\cdot{n}_{2})}{1-({n}_{1}\cdot{n}_{2})^{2}}
  60. c 2 = h 2 - h 1 ( n 1 n 2 ) 1 - ( n 1 n 2 ) 2 . c_{2}=\frac{h_{2}-h_{1}({n}_{1}\cdot{n}_{2})}{1-({n}_{1}\cdot{n}_{2})^{2}}.
  61. n 1 × n 2 {n}_{1}\times{n}_{2}
  62. r = c 1 n 1 + c 2 n 2 + λ ( n 1 × n 2 ) r=c_{1}{n}_{1}+c_{2}{n}_{2}+\lambda({n}_{1}\times{n}_{2})
  63. { n 1 , n 2 , ( n 1 × n 2 ) } \{{n}_{1},{n}_{2},({n}_{1}\times{n}_{2})\}
  64. c 1 c_{1}
  65. c 2 c_{2}
  66. n 1 {n}_{1}
  67. n 2 {n}_{2}
  68. r 0 = h 1 n 1 + h 2 n 2 r_{0}=h_{1}{n}_{1}+h_{2}{n}_{2}
  69. Π 1 : a 1 x + b 1 y + c 1 z + d 1 = 0 \Pi_{1}:a_{1}x+b_{1}y+c_{1}z+d_{1}=0\,
  70. Π 2 : a 2 x + b 2 y + c 2 z + d 2 = 0 \Pi_{2}:a_{2}x+b_{2}y+c_{2}z+d_{2}=0\,
  71. α \alpha
  72. cos α = n ^ 1 n ^ 2 | n ^ 1 | | n ^ 2 | = a 1 a 2 + b 1 b 2 + c 1 c 2 a 1 2 + b 1 2 + c 1 2 a 2 2 + b 2 2 + c 2 2 . \cos\alpha=\frac{\hat{n}_{1}\cdot\hat{n}_{2}}{|\hat{n}_{1}||\hat{n}_{2}|}=% \frac{a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}% \sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}.
  73. a 2 + b 2 + c 2 \sqrt{a^{2}+b^{2}+c^{2}}

Plasmon.html

  1. E p = E_{p}=
  2. \hbar
  3. n e 2 m ϵ 0 = \sqrt{\frac{ne^{2}}{m\epsilon_{0}}}=
  4. \hbar
  5. ω p , \omega_{p},
  6. n n
  7. e e
  8. m m
  9. ϵ 0 \epsilon_{0}
  10. \hbar
  11. ω p \omega_{p}

Plasticity_(physics).html

  1. σ 1 - σ 3 σ 0 \sigma_{1}-\sigma_{3}\geq\sigma_{0}
  2. σ v 2 = 1 2 [ ( σ 11 - σ 22 ) 2 + ( σ 22 - σ 33 ) 2 + ( σ 11 - σ 33 ) 2 + 6 ( σ 23 2 + σ 31 2 + σ 12 2 ) ] \sigma_{v}^{2}=\tfrac{1}{2}[(\sigma_{11}-\sigma_{22})^{2}+(\sigma_{22}-\sigma_% {33})^{2}+(\sigma_{11}-\sigma_{33})^{2}+6(\sigma_{23}^{2}+\sigma_{31}^{2}+% \sigma_{12}^{2})]

Pochhammer_symbol.html

  1. n n
  2. ( x n ) {\textstyle\left({{x}\atop{n}}\right)}
  3. ( x ) n = x ( x - 1 ) ( x - 2 ) ( x - n + 1 ) (x)_{n}=x(x-1)(x-2)\cdots(x-n+1)
  4. x ( n ) = x ( x + 1 ) ( x + 2 ) ( x + n - 1 ) . x^{(n)}=x(x+1)(x+2)\cdots(x+n-1).
  5. x x
  6. n n
  7. x x
  8. n n
  9. x x
  10. x x
  11. n n
  12. x x
  13. x ( n ) n ! = ( x + n - 1 n ) and ( x ) n n ! = ( x n ) . \frac{x^{(n)}}{n!}={x+n-1\choose n}\quad\mbox{and}~{}\quad\frac{(x)_{n}}{n!}={% x\choose n}.
  14. x ( n ) = ( x + n - 1 ) n , x^{(n)}={(x+n-1)}_{n},
  15. x ( n ) = ( - 1 ) n ( - x ) < m t p l > n x^{(n)}={(-1)}^{n}{(-x)}_{<}mtpl>{{n}}
  16. n n
  17. x x
  18. x + n x+n
  19. x ( n ) = Γ ( x + n ) Γ ( x ) , x^{(n)}=\frac{\Gamma(x+n)}{\Gamma(x)},
  20. ( x ) n = Γ ( x + 1 ) Γ ( x - n + 1 ) . (x)_{n}=\frac{\Gamma(x+1)}{\Gamma(x-n+1)}.
  21. D D
  22. x x
  23. D n ( x a ) = ( a ) n x a - n . D^{n}(x^{a})=(a)_{n}\,\,x^{a-n}.
  24. ( a ) < m t p l > n {(a)}_{<}mtpl>{{n}}
  25. Δ Δ
  26. Δ ( x ) k = k ( x ) k - 1 , \Delta(x)_{k}=k\ (x)_{k-1},
  27. D x k = k x k - 1 , Dx^{k}=k\ x^{k-1},
  28. ( a + b ) ( n ) = < m t p l > j = 0 (a+b)^{(n)}=\sum_{<}mtpl>{{j=0}}
  29. ( a + b ) n = < m t p l > j = 0 (a+b)_{n}=\sum_{<}mtpl>{{j=0}}
  30. n = 0 ( x ) n t n n ! = ( 1 + t ) x , \sum_{n=0}^{\infty}(x)_{n}~{}\frac{t^{n}}{n!}=(1+t)^{x}~{},
  31. ( x ) m ( x ) n = k = 0 m ( m k ) ( n k ) k ! ( x ) m + n - k . (x)_{m}(x)_{n}=\sum_{k=0}^{m}{m\choose k}{n\choose k}k!\,(x)_{m+n-k}.
  32. k k
  33. m m
  34. n n
  35. x m ¯ = x ( x + 1 ) ( x + m - 1 ) m factors for integer m 0 , x^{\overline{m}}=\overbrace{x(x+1)\ldots(x+m-1)}^{m~{}\mathrm{factors}}\qquad% \mbox{for integer }~{}m\geq 0,
  36. x m ¯ = x ( x - 1 ) ( x - m + 1 ) m factors for integer m 0 ; x^{\underline{m}}=\overbrace{x(x-1)\ldots(x-m+1)}^{m~{}\mathrm{factors}}\qquad% \mbox{for integer }~{}m\geq 0;
  37. x x
  38. m m
  39. x x
  40. m m
  41. P ( x , n ) P(x,n)
  42. [ f ( x ) ] k / - h = f ( x ) f ( x - h ) f ( x - 2 h ) f ( x - ( k - 1 ) h ) , [f(x)]^{k/-h}=f(x)\cdot f(x-h)\cdot f(x-2h)\cdots f(x-(k-1)h),
  43. h −h
  44. k k
  45. [ f ( x ) ] k / h = f ( x ) f ( x + h ) f ( x + 2 h ) f ( x + ( k - 1 ) h ) . [f(x)]^{k/h}=f(x)\cdot f(x+h)\cdot f(x+2h)\cdots f(x+(k-1)h).

Poincaré_group.html

  1. 𝐑 1 , 3 SO ( 1 , 3 ) . \mathbf{R}^{1,3}\rtimes\mathrm{SO}(1,3)\,.
  2. P P
  3. M M
  4. η η
  5. [ J m , P n ] = i ϵ m n k P k , [J_{m},P_{n}]=i\epsilon_{mnk}P_{k}~{},
  6. [ J i , P 0 ] = 0 , [J_{i},P_{0}]=0~{},
  7. [ K i , P k ] = i η i k P 0 , [K_{i},P_{k}]=i\eta_{ik}P_{0}~{},
  8. [ K i , P 0 ] = - i P i , [K_{i},P_{0}]=-iP_{i}~{},
  9. [ J m , J n ] = i ϵ m n k J k , [J_{m},J_{n}]=i\epsilon_{mnk}J_{k}~{},
  10. [ J m , K n ] = i ϵ m n k K k , [J_{m},K_{n}]=i\epsilon_{mnk}K_{k}~{},
  11. [ K m , K n ] = - i ϵ m n k J k , [K_{m},K_{n}]=-i\epsilon_{mnk}J_{k}~{},
  12. J < s u b > m + i K m , J n i K n u b 0 J<sub>m+iK_{m},J_{n}−iK_{n}ub>=0

Poisson's_equation.html

  1. Δ φ = f \Delta\varphi=f
  2. Δ \Delta
  3. 2 φ = f . \nabla^{2}\varphi=f.
  4. ( 2 x 2 + 2 y 2 + 2 z 2 ) φ ( x , y , z ) = f ( x , y , z ) . \left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+% \frac{\partial^{2}}{\partial z^{2}}\right)\varphi(x,y,z)=f(x,y,z).
  5. f = 0 f=0
  6. g = - 4 π G ρ \nabla\cdot{g}=-4\pi G\rho
  7. g = - Φ {g}=-\nabla\Phi
  8. ( - Φ ) = - 4 π G ρ \nabla\cdot(-\nabla\Phi)=-4\pi G\rho
  9. 2 Φ = 4 π G ρ . {\nabla}^{2}\Phi=4\pi G\rho.
  10. ρ f \rho_{f}
  11. 𝐃 = ρ f \mathbf{\nabla}\cdot\mathbf{D}=\rho_{f}
  12. \mathbf{\nabla}\cdot
  13. 𝐃 = ε 𝐄 \mathbf{D}=\varepsilon\mathbf{E}
  14. 𝐄 = ρ f ε \mathbf{\nabla}\cdot\mathbf{E}=\frac{\rho_{f}}{\varepsilon}
  15. × 𝐄 = - 𝐁 t = 0 \nabla\times\mathbf{E}=-\dfrac{\partial\mathbf{B}}{\partial t}=0
  16. × \nabla\times
  17. φ \varphi
  18. 𝐄 = - φ \mathbf{E}=-\nabla\varphi
  19. E = ( - φ ) = - 2 φ = ρ f ε , \nabla\cdot{E}=\nabla\cdot(-\nabla\varphi)=-{\nabla}^{2}\varphi=\frac{\rho_{f}% }{\varepsilon},
  20. 2 φ = - ρ f ε . {\nabla}^{2}\varphi=-\frac{\rho_{f}}{\varepsilon}.
  21. ρ f ( r ) = Q σ 3 2 π 3 e - r 2 / ( 2 σ 2 ) , \rho_{f}(r)=\frac{Q}{\sigma^{3}\sqrt{2\pi}^{3}}\,e^{-r^{2}/(2\sigma^{2})},
  22. 2 φ = - ρ f ε {\nabla}^{2}\varphi=-{\rho_{f}\over\varepsilon}
  23. φ ( r ) = 1 4 π ε Q r erf ( r 2 σ ) \varphi(r)={1\over 4\pi\varepsilon}\frac{Q}{r}\,\mbox{erf}~{}\left(\frac{r}{% \sqrt{2}\sigma}\right)
  24. 2 φ {\nabla}^{2}\varphi
  25. φ 1 4 π ε Q r \varphi\approx{1\over 4\pi\varepsilon}{Q\over r}

Poisson's_ratio.html

  1. ν \nu
  2. ν = - d ε trans d ε axial = - d ε y d ε x = - d ε z d ε x \nu=-\frac{d\varepsilon_{\mathrm{trans}}}{d\varepsilon_{\mathrm{axial}}}=-% \frac{d\varepsilon_{\mathrm{y}}}{d\varepsilon_{\mathrm{x}}}=-\frac{d% \varepsilon_{\mathrm{z}}}{d\varepsilon_{\mathrm{x}}}
  3. ν \nu
  4. ε trans \varepsilon_{\mathrm{trans}}
  5. ε axial \varepsilon_{\mathrm{axial}}
  6. Δ L \Delta L
  7. Δ L \Delta L^{\prime}
  8. d ε x = d x x d ε y = d y y d ε z = d z z . d\varepsilon_{x}=\frac{dx}{x}\qquad d\varepsilon_{y}=\frac{dy}{y}\qquad d% \varepsilon_{z}=\frac{dz}{z}.
  9. - ν L L + Δ L d x x = L L - Δ L d y y = L L - Δ L d z z . -\nu\int\limits_{L}^{L+\Delta L}\frac{dx}{x}=\int\limits_{L}^{L-\Delta L^{% \prime}}\frac{dy}{y}=\int\limits_{L}^{L-\Delta L^{\prime}}\frac{dz}{z}.
  10. Δ L \Delta L
  11. Δ L \Delta L^{\prime}
  12. ( 1 + Δ L L ) - ν = 1 - Δ L L . \left(1+\frac{\Delta L}{L}\right)^{-\nu}=1-\frac{\Delta L^{\prime}}{L}.
  13. Δ L \Delta L
  14. Δ L \Delta L^{\prime}
  15. ν Δ L Δ L . \nu\approx\frac{\Delta L^{\prime}}{\Delta L}.
  16. V = L 3 V=L^{3}
  17. V + Δ V = ( L + Δ L ) ( L - Δ L ) 2 V+\Delta V=(L+\Delta L)(L-\Delta L^{\prime})^{2}
  18. Δ V V = ( 1 + Δ L L ) ( 1 - Δ L L ) 2 - 1 \frac{\Delta V}{V}=\left(1+\frac{\Delta L}{L}\right)\left(1-\frac{\Delta L^{% \prime}}{L}\right)^{2}-1
  19. Δ L \Delta L
  20. Δ L \Delta L^{\prime}
  21. Δ V V = ( 1 + Δ L L ) 1 - 2 ν - 1 \frac{\Delta V}{V}=\left(1+\frac{\Delta L}{L}\right)^{1-2\nu}-1
  22. Δ L \Delta L
  23. Δ L \Delta L^{\prime}
  24. Δ V V ( 1 - 2 ν ) Δ L L \frac{\Delta V}{V}\approx(1-2\nu)\frac{\Delta L}{L}
  25. ν 1 2 - E 6 K \nu\approx\frac{1}{2}-\frac{E}{6K}
  26. K K
  27. - 1 < ν < 0.5 -1<\nu<0.5
  28. 0.2 < ν < 0.5 0.2<\nu<0.5
  29. Δ d = - d ν Δ L L \Delta d=-d\cdot\nu{{\Delta L}\over L}
  30. Δ d = - d ( 1 - ( 1 + Δ L L ) - ν ) \Delta d=-d\cdot\left(1-{\left(1+{{\Delta L}\over L}\right)}^{-\nu}\right)
  31. d d
  32. Δ d \Delta d
  33. ν \nu
  34. L L
  35. Δ L \Delta L
  36. ε x = 1 E [ σ x - ν ( σ y + σ z ) ] \varepsilon_{x}=\frac{1}{E}\left[\sigma_{x}-\nu\left(\sigma_{y}+\sigma_{z}% \right)\right]
  37. ε y = 1 E [ σ y - ν ( σ x + σ z ) ] \varepsilon_{y}=\frac{1}{E}\left[\sigma_{y}-\nu\left(\sigma_{x}+\sigma_{z}% \right)\right]
  38. ε z = 1 E [ σ z - ν ( σ x + σ y ) ] \varepsilon_{z}=\frac{1}{E}\left[\sigma_{z}-\nu\left(\sigma_{x}+\sigma_{y}% \right)\right]
  39. ε i = 1 E [ σ i ( 1 + ν ) - ν ( σ x + σ y + σ z ) ] \varepsilon_{i}=\frac{1}{E}\left[\sigma_{i}(1+\nu)-\nu\left(\sigma_{x}+\sigma_% {y}+\sigma_{z}\right)\right]
  40. ε x \varepsilon_{x}
  41. ε y \varepsilon_{y}
  42. ε z \varepsilon_{z}
  43. x x
  44. y y
  45. z z
  46. σ x \sigma_{x}
  47. σ y \sigma_{y}
  48. σ z \sigma_{z}
  49. x x
  50. y y
  51. z z
  52. E E
  53. x x
  54. y y
  55. z z
  56. ν \nu
  57. x x
  58. y y
  59. z z
  60. ε i j = 1 E [ σ i j ( 1 + ν ) - ν δ i j σ k k ] \varepsilon_{ij}=\frac{1}{E}\left[\sigma_{ij}(1+\nu)-\nu\delta_{ij}\sigma_{kk}\right]
  61. δ i j \delta_{ij}
  62. σ k k = σ x + σ y + σ z = σ 11 + σ 22 + σ 33 \sigma_{kk}=\sigma_{x}+\sigma_{y}+\sigma_{z}=\sigma_{11}+\sigma_{22}+\sigma_{3% 3}\,
  63. [ ϵ < m t p l > xx ϵ yy ϵ zz 2 ϵ yz 2 ϵ zx 2 ϵ xy ] = [ 1 E x - ν yx E y - ν zx E z 0 0 0 - ν xy E x 1 E y - ν zy E z 0 0 0 - ν xz E x - ν yz E y 1 E z 0 0 0 0 0 0 1 G yz 0 0 0 0 0 0 1 G zx 0 0 0 0 0 0 1 G xy ] [ σ xx σ yy σ zz σ yz σ zx σ xy ] \begin{bmatrix}\epsilon_{<}mtpl>{{\rm xx}}\\ \epsilon_{\rm yy}\\ \epsilon_{\rm zz}\\ 2\epsilon_{\rm yz}\\ 2\epsilon_{\rm zx}\\ 2\epsilon_{\rm xy}\end{bmatrix}=\begin{bmatrix}\tfrac{1}{E_{\rm x}}&-\tfrac{% \nu_{\rm yx}}{E_{\rm y}}&-\tfrac{\nu_{\rm zx}}{E_{\rm z}}&0&0&0\\ -\tfrac{\nu_{\rm xy}}{E_{\rm x}}&\tfrac{1}{E_{\rm y}}&-\tfrac{\nu_{\rm zy}}{E_% {\rm z}}&0&0&0\\ -\tfrac{\nu_{\rm xz}}{E_{\rm x}}&-\tfrac{\nu_{\rm yz}}{E_{\rm y}}&\tfrac{1}{E_% {\rm z}}&0&0&0\\ 0&0&0&\tfrac{1}{G_{\rm yz}}&0&0\\ 0&0&0&0&\tfrac{1}{G_{\rm zx}}&0\\ 0&0&0&0&0&\tfrac{1}{G_{\rm xy}}\\ \end{bmatrix}\begin{bmatrix}\sigma_{\rm xx}\\ \sigma_{\rm yy}\\ \sigma_{\rm zz}\\ \sigma_{\rm yz}\\ \sigma_{\rm zx}\\ \sigma_{\rm xy}\end{bmatrix}
  64. E i {E}_{\rm i}\,
  65. i i
  66. G ij G_{\rm ij}\,
  67. j j
  68. i i
  69. ν ij \nu_{\rm ij}\,
  70. j j
  71. i i
  72. ν yx E y = ν xy E x , ν zx E z = ν xz E x , ν yz E y = ν zy E z \frac{\nu_{\rm yx}}{E_{\rm y}}=\frac{\nu_{\rm xy}}{E_{\rm x}}~{},\qquad\frac{% \nu_{\rm zx}}{E_{\rm z}}=\frac{\nu_{\rm xz}}{E_{\rm x}}~{},\qquad\frac{\nu_{% \rm yz}}{E_{\rm y}}=\frac{\nu_{\rm zy}}{E_{\rm z}}
  73. E x > E y E_{\rm x}>E_{\rm y}
  74. ν xy > ν yx \nu_{\rm xy}>\nu_{\rm yx}
  75. ν xy \nu_{\rm xy}
  76. ν yx \nu_{\rm yx}
  77. y - z y-z
  78. [ ϵ < m t p l > xx ϵ yy ϵ zz 2 ϵ yz 2 ϵ zx 2 ϵ xy ] = [ 1 E x - ν yx E y - ν zx E z 0 0 0 - ν xy E x 1 E y - ν zy E z 0 0 0 - ν xy E x - ν yz E y 1 E z 0 0 0 0 0 0 1 G yz 0 0 0 0 0 0 1 G zx 0 0 0 0 0 0 1 G xy ] [ σ xx σ yy σ zz σ yz σ zx σ xy ] \begin{bmatrix}\epsilon_{<}mtpl>{{\rm xx}}\\ \epsilon_{\rm yy}\\ \epsilon_{\rm zz}\\ 2\epsilon_{\rm yz}\\ 2\epsilon_{\rm zx}\\ 2\epsilon_{\rm xy}\end{bmatrix}=\begin{bmatrix}\tfrac{1}{E_{\rm x}}&-\tfrac{% \nu_{\rm yx}}{E_{\rm y}}&-\tfrac{\nu_{\rm zx}}{E_{\rm z}}&0&0&0\\ -\tfrac{\nu_{\rm xy}}{E_{\rm x}}&\tfrac{1}{E_{\rm y}}&-\tfrac{\nu_{\rm zy}}{E_% {\rm z}}&0&0&0\\ -\tfrac{\nu_{\rm xy}}{E_{\rm x}}&-\tfrac{\nu_{\rm yz}}{E_{\rm y}}&\tfrac{1}{E_% {\rm z}}&0&0&0\\ 0&0&0&\tfrac{1}{G_{\rm yz}}&0&0\\ 0&0&0&0&\tfrac{1}{G_{\rm zx}}&0\\ 0&0&0&0&0&\tfrac{1}{G_{\rm xy}}\\ \end{bmatrix}\begin{bmatrix}\sigma_{\rm xx}\\ \sigma_{\rm yy}\\ \sigma_{\rm zz}\\ \sigma_{\rm yz}\\ \sigma_{\rm zx}\\ \sigma_{\rm xy}\end{bmatrix}
  79. y - z y-z
  80. E y = E z , ν x y = ν x z , ν y x = ν z x E_{y}=E_{z},~{}\nu_{xy}=\nu_{xz},~{}\nu_{yx}=\nu_{zx}
  81. ν xy E x = ν yx E y , ν yz = ν zy . \cfrac{\nu_{\rm xy}}{E_{\rm x}}=\cfrac{\nu_{\rm yx}}{E_{\rm y}}~{},~{}~{}\nu_{% \rm yz}=\nu_{\rm zy}~{}.
  82. E x , E y , G xy , G yz , ν xy , ν yz E_{\rm x},E_{\rm y},G_{\rm xy},G_{\rm yz},\nu_{\rm xy},\nu_{\rm yz}
  83. G yz G_{\rm yz}
  84. E y , ν yz E_{\rm y},\nu_{\rm yz}
  85. G yz = E y 2 ( 1 + ν yz ) . G_{\rm yz}=\cfrac{E_{\rm y}}{2(1+\nu_{\rm yz})}~{}.
  86. ν xy \nu_{\rm xy}
  87. ν yx \nu_{\rm yx}
  88. x - y x-y
  89. x x
  90. x - y x-y

Poisson_bracket.html

  1. f , g , h f,g,h
  2. { f , g } = - { g , f } \{f,g\}=-\{g,f\}
  3. { f + g , h } = { f , h } + { g , h } \{f+g,h\}=\{f,h\}+\{g,h\}
  4. { f g , h } = { f , h } g + { g , h } f \{fg,h\}=\{f,h\}g+\{g,h\}f
  5. { f , { g , h } } + { g , { h , f } } + { h , { f , g } } = 0 \{f,\{g,h\}\}+\{g,\{h,f\}\}+\{h,\{f,g\}\}=0
  6. k k
  7. { f , k } = 0 \{f,k\}=0
  8. f f
  9. ( q i , p i ) (q_{i},p_{i})
  10. f ( p i , q i , t ) f(p_{i},q_{i},t)
  11. g ( p i , q i , t ) g(p_{i},q_{i},t)
  12. { f , g } = i = 1 N ( f q i g p i - f p i g q i ) . \{f,g\}=\sum_{i=1}^{N}\left(\frac{\partial f}{\partial q_{i}}\frac{\partial g}% {\partial p_{i}}-\frac{\partial f}{\partial p_{i}}\frac{\partial g}{\partial q% _{i}}\right).
  13. { q i , q j } = 0 \{q_{i},q_{j}\}=0
  14. { p i , p j } = 0 \{p_{i},p_{j}\}=0
  15. { q i , p j } = δ i j \{q_{i},p_{j}\}=\delta_{ij}
  16. f ( p , q , t ) f(p,q,t)
  17. d d t f ( p , q , t ) = f q d q d t + f p d p d t + f t . \frac{\mathrm{d}}{\mathrm{d}t}f(p,q,t)=\frac{\partial f}{\partial q}\frac{% \mathrm{d}q}{\mathrm{d}t}+\frac{\partial f}{\partial p}\frac{\mathrm{d}p}{% \mathrm{d}t}+\frac{\partial f}{\partial t}.
  18. p = p ( t ) p=p(t)
  19. q = q ( t ) q=q(t)
  20. { q ˙ = H p = { q , H } p ˙ = - H q = { p , H } \begin{cases}\dot{q}=\frac{\partial H}{\partial p}=\{q,H\}\\ \dot{p}=-\frac{\partial H}{\partial q}=\{p,H\}\end{cases}
  21. d d t f ( p , q , t ) = f q H p - f p H q + f t = { f , H } + f t . \begin{aligned}\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}f(p,q,t)&% \displaystyle=\frac{\partial f}{\partial q}\frac{\partial H}{\partial p}-\frac% {\partial f}{\partial p}\frac{\partial H}{\partial q}+\frac{\partial f}{% \partial t}\\ &\displaystyle=\{f,H\}+\frac{\partial f}{\partial t}~{}.\end{aligned}
  22. f f
  23. d d t f = ( t - { H , } ) f . \frac{\mathrm{d}}{\mathrm{d}t}f=\left(\frac{\partial}{\partial t}-\{H,\cdot\}% \right)f~{}.
  24. i i
  25. 0 = d f d t 0=\frac{\mathrm{d}f}{\mathrm{d}t}
  26. 0 = d d t f ( p , q ) = { f , H } + f t 0=\frac{\mathrm{d}}{\mathrm{d}t}f(p,q)=\{f,H\}+\frac{\partial f}{\partial t}
  27. A A
  28. B B
  29. { A , B } \{A,B\}
  30. 2 n - 1 2n-1
  31. A A
  32. B B
  33. 2 n \mathbb{R}^{2n}
  34. ω = i = 1 n d q i d p i . \omega=\sum_{i=1}^{n}dq_{i}\wedge dp_{i}.
  35. ι v ω \iota_{v}\omega
  36. ( ι v ω ) ( w ) = ω ( v , w ) (\iota_{v}\omega)(w)=\omega(v,w)
  37. Ω α \Omega_{\alpha}
  38. ι Ω α ω = α \iota_{\Omega_{\alpha}}\omega=\alpha
  39. Ω d H = ω - 1 ( d H ) \Omega_{dH}=\omega^{-1}(dH)
  40. Ω d H \Omega_{dH}
  41. X p i = q i X_{p_{i}}=\frac{\partial}{\partial q_{i}}
  42. X q i = - p i . X_{q_{i}}=-\frac{\partial}{\partial p_{i}}.
  43. { , } \{\cdot,\cdot\}
  44. { f , g } = ω ( X f , X g ) \{f,g\}=\omega(X_{f},X_{g})
  45. { f , g } = ω ( X f , X g ) = - ω ( X g , X f ) = - { g , f } \{f,g\}=\omega(X_{f},X_{g})=-\omega(X_{g},X_{f})=-\{g,f\}
  46. X g f \mathcal{L}_{X_{g}}f
  47. ϕ x ( t ) \phi_{x}(t)
  48. ϕ x ( 0 ) = x \phi_{x}(0)=x
  49. d ϕ x d t = Ω α | ϕ x ( t ) . \frac{d\phi_{x}}{dt}=\Omega_{\alpha}|_{\phi_{x}(t)}.
  50. ϕ x ( t ) \phi_{x}(t)
  51. Ω α ω = 0 \mathcal{L}_{\Omega_{\alpha}}\omega=0
  52. X ω = d ( ι X ω ) + ι X d ω \mathcal{L}_{X}\omega=d(\iota_{X}\omega)+\iota_{X}d\omega
  53. Ω α ω = d ( ι Ω α ω ) = d α \mathcal{L}_{\Omega_{\alpha}}\omega=d(\iota_{\Omega_{\alpha}}\omega)=d\alpha
  54. d ( d f ) = d 2 f = 0 d(df)=d^{2}f=0
  55. d d t f ( ϕ x ( t ) ) = X H f = { f , H } . \frac{d}{dt}f(\phi_{x}(t))=X_{H}f=\{f,H\}.
  56. { p i , p j } = { q i , q j } = 0 \{p_{i},p_{j}\}=\{q_{i},q_{j}\}=0
  57. { q i , p j } = δ i j \{q_{i},p_{j}\}=\delta_{ij}
  58. { f g , h } = f { g , h } + g { f , h } \{fg,h\}=f\{g,h\}+g\{f,h\}
  59. { f , g h } = g { f , h } + h { f , g } \{f,gh\}=g\{f,h\}+h\{f,g\}
  60. v ι w ω = ι v w ω + ι w v ω = ι [ v , w ] ω + ι w v ω \mathcal{L}_{v}\iota_{w}\omega=\iota_{\mathcal{L}_{v}w}\omega+\iota_{w}% \mathcal{L}_{v}\omega=\iota_{[v,w]}\omega+\iota_{w}\mathcal{L}_{v}\omega
  61. v ω = 0 \mathcal{L}_{v}\omega=0
  62. ι w ω \iota_{w}\omega
  63. ι [ v , w ] ω = v ι w ω = d ( ι v ι w ω ) + ι v d ( ι w ω ) = d ( ι v ι w ω ) = d ( ω ( w , v ) ) . \iota_{[v,w]}\omega=\mathcal{L}_{v}\iota_{w}\omega=d(\iota_{v}\iota_{w}\omega)% +\iota_{v}d(\iota_{w}\omega)=d(\iota_{v}\iota_{w}\omega)=d(\omega(w,v)).
  64. [ v , w ] = X ω ( w , v ) [v,w]=X_{\omega(w,v)}
  65. { f , { g , h } } + { g , { h , f } } + { h , { f , g } } = 0 \ \{f,\{g,h\}\}+\{g,\{h,f\}\}+\{h,\{f,g\}\}=0
  66. ad { f , g } = [ ad f , ad g ] \operatorname{ad}_{\{f,g\}}=[\operatorname{ad}_{f},\operatorname{ad}_{g}]
  67. ad g \operatorname{ad}_{g}
  68. ad g ( ) = { , g } \operatorname{ad}_{g}(\cdot)=\{\cdot,g\}
  69. [ A , B ] = A B - B A [\operatorname{A},\operatorname{B}]=\operatorname{A}\operatorname{B}-% \operatorname{B}\operatorname{A}
  70. ad g \operatorname{ad}_{g}
  71. { P X , P Y } = - P [ X , Y ] . \{P_{X},P_{Y}\}=-P_{[X,Y]}.\,
  72. X q = i X i ( q ) q i X_{q}=\sum_{i}X^{i}(q)\frac{\partial}{\partial q^{i}}
  73. / q i \partial/\partial q^{i}
  74. P X ( q , p ) = i X i ( q ) p i P_{X}(q,p)=\sum_{i}X^{i}(q)\;p_{i}
  75. { P X , P Y } ( q , p ) = i j { X i ( q ) p i , Y j ( q ) p j } = i j p i Y j ( q ) X i q j - p j X i ( q ) Y j q i = - i p i [ X , Y ] i ( q ) = - P [ X , Y ] ( q , p ) . \begin{aligned}\displaystyle\{P_{X},P_{Y}\}(q,p)&\displaystyle=\sum_{i}\sum_{j% }\left\{X^{i}(q)\;p_{i},Y^{j}(q)\;p_{j}\right\}\\ &\displaystyle=\sum_{ij}p_{i}Y^{j}(q)\frac{\partial X^{i}}{\partial q^{j}}-p_{% j}X^{i}(q)\frac{\partial Y^{j}}{\partial q^{i}}\\ &\displaystyle=-\sum_{i}p_{i}\;[X,Y]^{i}(q)\\ &\displaystyle=-P_{[X,Y]}(q,p).\end{aligned}
  76. f ( p i , q i , t ) f(p_{i},q_{i},t)

Poisson_process.html

  1. t ( a , b t∈(a,b
  2. P [ N ( t + τ ) - N ( t ) = k ] = e - λ τ ( λ τ ) k k ! k = 0 , 1 , , P[N(t+\tau)-N(t)=k]=\frac{e^{-\lambda\tau}(\lambda\tau)^{k}}{k!}\qquad k=0,1,\ldots,
  3. N a , b = a b λ ( t ) d t . N_{a,b}=\int_{a}^{b}\lambda(t)\,dt.
  4. P [ N ( b ) - N ( a ) = k ] = e - N a , b ( N a , b ) k k ! k = 0 , 1 , . P[N(b)-N(a)=k]=\frac{e^{-N_{a,b}}(N_{a,b})^{k}}{k!}\qquad k=0,1,\ldots.
  5. λ ( x , t ) \lambda(x,t)
  6. x V x\in V
  7. S V S\subset V
  8. μ ( S ) \mu(S)
  9. λ S ( t ) = S λ ( x , t ) d μ ( x ) . \lambda_{S}(t)=\int_{S}\lambda(x,t)\,d\mu(x).
  10. λ ( x , t ) = f ( x ) λ ( t ) \lambda(x,t)=f(x)\lambda(t)\,
  11. f ( x ) f(x)
  12. V f ( x ) d μ ( x ) = 1. \int_{V}f(x)\,d\mu(x)=1.
  13. f ( x ) f(x)
  14. X X
  15. f ( x ) f(x)
  16. lim Δ t 0 P ( N ( t + Δ t ) - N ( t ) > 1 N ( t + Δ t ) - N ( t ) 1 ) = 0 \lim_{\Delta t\to 0}P(N(t+\Delta t)-N(t)>1\mid N(t+\Delta t)-N(t)\geq 1)=0
  17. P ( T k > t ) = P ( N ( t ) < k ) . P(T_{k}>t)=P(N(t)<k).\,
  18. P ( T 1 > t ) = P ( N ( t ) = 0 ) = P [ N ( t ) - N ( 0 ) = 0 ] = e - λ t ( λ t ) 0 0 ! = e - λ t . P(T_{1}>t)=P(N(t)=0)=P[N(t)-N(0)=0]=\frac{e^{-\lambda t}(\lambda t)^{0}}{0!}=e% ^{-\lambda t}.
  19. P ( T 1 t ) = P ( N ( t ) > 0 ) = 1 - e - λ t P(T_{1}\leq t)=P(N(t)>0)=1-e^{-\lambda t}

Poker_probability.html

  1. ( 52 5 ) = 2 , 598 , 960 \,\begin{matrix}{52\choose 5}=2,598,960\end{matrix}
  2. 4 2 , 598 , 960 \frac{4}{2,598,960}
  3. ( 52 5 ) = 2 , 598 , 960 \begin{matrix}{52\choose 5}=2,598,960\end{matrix}
  4. ( 4 1 ) {4\choose 1}
  5. ( 10 1 ) ( 4 1 ) - ( 4 1 ) {10\choose 1}{4\choose 1}-{4\choose 1}
  6. ( 13 1 ) ( 12 1 ) ( 4 1 ) {13\choose 1}{12\choose 1}{4\choose 1}
  7. ( 13 1 ) ( 4 3 ) ( 12 1 ) ( 4 2 ) {13\choose 1}{4\choose 3}{12\choose 1}{4\choose 2}
  8. ( 13 5 ) ( 4 1 ) - ( 10 1 ) ( 4 1 ) {13\choose 5}{4\choose 1}-{10\choose 1}{4\choose 1}
  9. ( 10 1 ) ( 4 1 ) 5 - ( 10 1 ) ( 4 1 ) {10\choose 1}{4\choose 1}^{5}-{10\choose 1}{4\choose 1}
  10. ( 13 1 ) ( 4 3 ) ( 12 2 ) ( 4 1 ) 2 {13\choose 1}{4\choose 3}{12\choose 2}{4\choose 1}^{2}
  11. ( 13 2 ) ( 4 2 ) 2 ( 11 1 ) ( 4 1 ) {13\choose 2}{4\choose 2}^{2}{11\choose 1}{4\choose 1}
  12. ( 13 1 ) ( 4 2 ) ( 12 3 ) ( 4 1 ) 3 {13\choose 1}{4\choose 2}{12\choose 3}{4\choose 1}^{3}
  13. [ ( 13 5 ) - 10 ] [ ( 4 1 ) 5 - 4 ] \left[{13\choose 5}-10\right]\left[{4\choose 1}^{5}-4\right]
  14. ( 52 5 ) {52\choose 5}
  15. ( 10 1 ) ( 4 1 ) = 40 {10\choose 1}{4\choose 1}=40
  16. ( 5 5 ) ( 8 0 ) ( 4 1 ) = 4 {5\choose 5}{8\choose 0}{4\choose 1}=4
  17. ( 4 1 ) = 4 {4\choose 1}=4
  18. ( 13 1 ) ( 4 4 ) ( 12 1 ) ( 4 1 ) = 624 {13\choose 1}{4\choose 4}{12\choose 1}{4\choose 1}=624
  19. ( 13 1 ) ( 4 3 ) ( 12 1 ) ( 4 2 ) = 3 , 744 {13\choose 1}{4\choose 3}{12\choose 1}{4\choose 2}=3,744
  20. ( 13 5 ) ( 4 1 ) - 40 = 5 , 108 {13\choose 5}{4\choose 1}-40=5,108
  21. ( 10 1 ) ( 4 1 ) 5 - 40 = 10 , 200 {10\choose 1}{4\choose 1}^{5}-40=10,200
  22. ( 13 1 ) ( 4 3 ) ( 12 2 ) ( 4 1 ) 2 = 54 , 912 {13\choose 1}{4\choose 3}{12\choose 2}{4\choose 1}^{2}=54,912
  23. ( 13 2 ) ( 4 2 ) 2 ( 11 1 ) ( 4 1 ) = 123 , 552 {13\choose 2}{4\choose 2}^{2}{11\choose 1}{4\choose 1}=123,552
  24. ( 13 1 ) ( 4 2 ) ( 12 3 ) ( 4 1 ) 3 = 1 , 098 , 240 {13\choose 1}{4\choose 2}{12\choose 3}{4\choose 1}^{3}=1,098,240
  25. [ ( 13 5 ) - 10 ] [ ( 4 1 ) 5 - 4 ] = ( 52 5 ) - 1 , 296 , 420 = 1 , 302 , 540 \left[{13\choose 5}-10\right]\left[{4\choose 1}^{5}-4\right]={52\choose 5}-1,2% 96,420=1,302,540
  26. ( n r ) = n ! r ! ( n - r ) ! = ( 52 5 ) = 52 ! 5 ! ( 52 - 5 ) ! = 2 , 598 , 960 {n\choose r}={{n!}\over{r!(n-r)!}}={52\choose 5}={{52!}\over{5!(52-5)!}}=2,598% ,960
  27. ( 52 7 ) = 133 , 784 , 560 \begin{matrix}{52\choose 7}=133,784,560\end{matrix}
  28. ( 4 1 ) [ ( 1 1 ) ( 47 2 ) + ( 9 1 ) ( 46 2 ) ] = 41 , 584 {4\choose 1}\left[{1\choose 1}{47\choose 2}+{9\choose 1}{46\choose 2}\right]=4% 1,584
  29. ( 13 1 ) ( 48 3 ) = 224 , 848 {13\choose 1}{48\choose 3}=224,848
  30. ( 13 1 ) ( 4 3 ) ( 12 1 ) ( 4 2 ) ( 11 2 ) ( 4 1 ) 2 = 3 , 294 , 720 {13\choose 1}{4\choose 3}{12\choose 1}{4\choose 2}{11\choose 2}{4\choose 1}^{2% }=3,294,720
  31. ( 13 1 ) ( 4 3 ) ( 12 2 ) ( 4 2 ) 2 = 123 , 552 {13\choose 1}{4\choose 3}{12\choose 2}{4\choose 2}^{2}=123,552
  32. ( 13 2 ) ( 4 3 ) 2 ( 11 1 ) ( 4 1 ) = 54 , 912 {13\choose 2}{4\choose 3}^{2}{11\choose 1}{4\choose 1}=54,912
  33. 3 , 294 , 720 + 123 , 552 + 54 , 912 = 3 , 473 , 184 3,294,720+123,552+54,912=3,473,184\,
  34. ( 4 1 ) [ ( 13 5 ) ( 39 2 ) + ( 13 6 ) ( 39 1 ) + ( 13 7 ) ] - 41 , 584 = 4 , 047 , 644 {4\choose 1}\left[{13\choose 5}{39\choose 2}+{13\choose 6}{39\choose 1}+{13% \choose 7}\right]-41,584=4,047,644
  35. ( 8 2 ) + ( 9 1 ) ( 7 2 ) = 217 {8\choose 2}+{9\choose 1}{7\choose 2}=217
  36. ( 4 1 ) 7 = 16 , 384 {4\choose 1}^{7}=16,384\,
  37. ( 4 1 ) [ 1 + ( 7 6 ) ( 3 1 ) + ( 7 5 ) ( 3 1 ) 2 ] = 844 {4\choose 1}\left[1+{7\choose 6}{3\choose 1}+{7\choose 5}{3\choose 1}^{2}% \right]=844
  38. 16 , 384 - 844 = 15 , 540 16,384-844=15,540\,
  39. 217 15 , 540 = 3 , 372 , 180 217\cdot 15,540=3,372,180\,
  40. ( 8 1 ) + ( 9 1 ) ( 7 1 ) = 71 {8\choose 1}+{9\choose 1}{7\choose 1}=71
  41. ( 6 1 ) ( 4 2 ) = 36 {6\choose 1}{4\choose 2}=36
  42. ( 4 1 ) 5 = 1 , 024 {4\choose 1}^{5}=1,024\,
  43. ( 4 1 ) + ( 5 4 ) ( 2 1 ) ( 3 1 ) = 34 {4\choose 1}+{5\choose 4}{2\choose 1}{3\choose 1}=34
  44. 1 , 024 - 34 = 990 1,024-34=990\,
  45. 71 36 990 = 2 , 530 , 440 71\cdot 36\cdot 990=2,530,440\,
  46. ( 5 1 ) ( 4 3 ) = 20 {5\choose 1}{4\choose 3}=20
  47. 4 4 4^{4}
  48. ( 4 1 ) 4 - ( 3 1 ) = 253 {4\choose 1}^{4}-{3\choose 1}=253\,
  49. 10 20 253 = 50 , 600 10\cdot 20\cdot 253=50,600\,
  50. ( 5 2 ) = 10 {5\choose 2}=10
  51. ( 4 2 ) 2 = 36 {4\choose 2}^{2}=36
  52. 4 3 4^{3}
  53. 6 [ 64 - ( 2 1 ) ] + 24 ( 64 - 1 ) + 6 64 = 2 , 268 6\cdot\left[64-{2\choose 1}\right]+24\cdot(64-1)+6\cdot 64=2,268\,
  54. 10 10 2 , 268 = 226 , 800 10\cdot 10\cdot 2,268=226,800\,
  55. 3 , 372 , 180 + 2 , 530 , 440 + 50 , 600 + 226 , 800 = 6 , 180 , 020 3,372,180+2,530,440+50,600+226,800=6,180,020\,
  56. ( 13 5 ) - 10 = 1 , 277 {13\choose 5}-10=1,277
  57. ( 5 1 ) ( 4 3 ) = 20 {5\choose 1}{4\choose 3}=20
  58. 4 4 4^{4}
  59. ( 4 1 ) 4 - ( 3 1 ) = 253 {4\choose 1}^{4}-{3\choose 1}=253
  60. 1 , 277 20 253 = 6 , 461 , 620 1,277\cdot 20\cdot 253=6,461,620\,
  61. ( 13 4 ) ( 4 3 ) ( 4 2 ) 3 ( 4 1 ) = 2 , 471 , 040 {13\choose 4}{4\choose 3}{4\choose 2}^{3}{4\choose 1}=2,471,040
  62. [ ( 13 5 ) - 10 ] ( 5 2 ) 2 , 268 = 28 , 962 , 360 \left[{13\choose 5}-10\right]{5\choose 2}\cdot 2,268=28,962,360
  63. 2 , 471 , 040 + 28 , 962 , 360 = 31 , 433 , 400 2,471,040+28,962,360=31,433,400\,
  64. ( 13 6 ) - 9 - [ 2 ( 7 1 ) + 8 ( 6 1 ) ] = 1 , 645 {13\choose 6}-9-\left[2\cdot{7\choose 1}+8\cdot{6\choose 1}\right]=1,645
  65. 4 5 4^{5}
  66. ( 4 1 ) 5 - 34 = 990 {4\choose 1}^{5}-34=990
  67. ( 6 1 ) ( 4 2 ) = 36 {6\choose 1}{4\choose 2}=36
  68. 1645 990 36 = 58 , 627 , 800 1645\cdot 990\cdot 36=58,627,800
  69. ( 13 7 ) - 8 - [ 2 ( 6 1 ) + 7 ( 5 1 ) ] - [ 2 ( 7 2 ) + 8 ( 6 2 ) ] = 1 , 499 {13\choose 7}-8-\left[2\cdot{6\choose 1}+7\cdot{5\choose 1}\right]-\left[2% \cdot{7\choose 2}+8\cdot{6\choose 2}\right]=1,499
  70. 4 7 4^{7}
  71. ( 4 1 ) 7 - 844 = 15 , 540 {4\choose 1}^{7}-844=15,540
  72. 1 , 499 15 , 540 = 23 , 294 , 460 1,499\cdot 15,540=23,294,460\,
  73. ( 52 5 ) = 2 , 598 , 960 \begin{matrix}{52\choose 5}=2,598,960\end{matrix}
  74. ( 4 1 ) 5 = 1 , 024 {4\choose 1}^{5}=1,024
  75. ( 5 5 ) = 1 {5\choose 5}=1
  76. ( 5 4 ) = 5 {5\choose 4}=5
  77. r r
  78. D = ( r - 1 4 ) D={r-1\choose 4}
  79. r r
  80. 1 , 024 D 1,024\cdot D
  81. r - 4 r-4
  82. r r
  83. [ ( 4 1 ) 5 - 4 ] [ ( r - 1 4 ) - ( r - 4 ) ] \left[{4\choose 1}^{5}-4\right]\cdot\left[{r-1\choose 4}-(r-4)\right]
  84. ( 52 7 ) = 133 , 784 , 560 \begin{matrix}{52\choose 7}=133,784,560\end{matrix}
  85. r r
  86. ( r - 1 4 ) {r-1\choose 4}
  87. r r
  88. r r
  89. ( 13 - r 2 ) ( 4 1 ) 7 = ( 13 - r 2 ) 16 , 384 {13-r\choose 2}{4\choose 1}^{7}={13-r\choose 2}\cdot 16,384
  90. r r
  91. r r
  92. ( 13 - r 1 ) ( 6 1 ) ( 4 2 ) ( 4 1 ) 5 = ( 13 - r 1 ) 36 , 864 {13-r\choose 1}{6\choose 1}{4\choose 2}{4\choose 1}^{5}={13-r\choose 1}\cdot 3% 6,864
  93. ( 5 2 ) ( 4 2 ) 2 ( 4 1 ) 3 = 23 , 040 {5\choose 2}{4\choose 2}^{2}{4\choose 1}^{3}=23,040
  94. ( 5 1 ) ( 4 3 ) ( 4 1 ) 4 = 5 , 120 {5\choose 1}{4\choose 3}{4\choose 1}^{4}=5,120
  95. 23 , 040 + 5 , 120 = 28 , 160 23,040+5,120=28,160
  96. r r
  97. r r
  98. ( r - 1 4 ) [ ( 13 - r 2 ) 16 , 384 + ( 13 - r 1 ) 36 , 864 + 28 , 160 ] {r-1\choose 4}\cdot\left[{13-r\choose 2}\cdot 16,384+{13-r\choose 1}\cdot 36,8% 64+28,160\right]
  99. ( 11 4 ) [ ( 1 1 ) 36 , 864 + 28 , 160 ] = ( 11 4 ) 65 , 024 = 21 , 457 , 920 {11\choose 4}\cdot\left[{1\choose 1}\cdot 36,864+28,160\right]={11\choose 4}% \cdot 65,024=21,457,920
  100. ( 12 4 ) 28 , 160 = 13 , 939 , 200 {12\choose 4}\cdot 28,160=13,939,200

Pole_(complex_analysis).html

  1. 1 z n \frac{1}{z^{n}}
  2. f ( z ) = g ( z ) ( z - a ) n f(z)=\frac{g(z)}{(z-a)^{n}}
  3. f ( z ) = 1 h ( z ) f(z)=\frac{1}{h(z)}
  4. f ( z ) = a - n ( z - a ) n + + a - 1 ( z - a ) + k 0 a k ( z - a ) k . f(z)=\frac{a_{-n}}{(z-a)^{n}}+\cdots+\frac{a_{-1}}{(z-a)}+\sum_{k\,\geq\,0}a_{% k}(z-a)^{k}.
  5. k 0 a k ( z - a ) k \scriptstyle\sum_{k\,\geq\,0}a_{k}(z\,-\,a)^{k}
  6. z 1 z \scriptstyle z\mapsto\frac{1}{z}
  7. f ( 1 z ) \scriptstyle f(\frac{1}{z})
  8. z = 0 \scriptstyle z=0
  9. z = 0 \scriptstyle z=0
  10. f : M \scriptstyle f:\;M\,\rightarrow\,\mathbb{C}
  11. U \scriptstyle U
  12. a \scriptstyle a
  13. ϕ : U \scriptstyle\phi:\;U\,\rightarrow\,\mathbb{C}
  14. f ϕ - 1 : \scriptstyle f\,\circ\,\phi^{-1}:\;\mathbb{C}\,\rightarrow\,\mathbb{C}
  15. ϕ ( a ) \scriptstyle\phi(a)
  16. ϕ ( z ) = 1 z \scriptstyle\phi(z)\,=\,\frac{1}{z}
  17. f ( z ) = 3 z f(z)=\frac{3}{z}
  18. z = 0 z=0
  19. f ( z ) = z + 2 ( z - 5 ) 2 ( z + 7 ) 3 f(z)=\frac{z+2}{(z-5)^{2}(z+7)^{3}}
  20. z = 5 z=5
  21. z = - 7 z=-7
  22. f ( z ) = z - 4 e z - 1 f(z)=\frac{z-4}{e^{z}-1}
  23. z = 2 π n i for n = , - 1 , 0 , 1 , . z\,=\,2\pi ni\,\text{ for }n\,=\,\dots,\,-1,\,0,\,1,\,\dots.
  24. e z e^{z}
  25. f ( z ) = z f(z)=z

Polygonal_number.html

  1. P ( s , n ) = n 2 ( s - 2 ) - n ( s - 4 ) 2 P(s,n)=\frac{n^{2}(s-2)-n(s-4)}{2}
  2. P ( s , n ) = ( s - 2 ) n ( n - 1 ) 2 + n P(s,n)=(s-2)\frac{n(n-1)}{2}+n
  3. P ( s , n ) = ( s - 2 ) T n - 1 + n = ( s - 3 ) T n - 1 + T n . P(s,n)=(s-2)T_{n-1}+n=(s-3)T_{n-1}+T_{n}\,.
  4. P ( s , n + 1 ) - P ( s , n ) = ( s - 2 ) n + 1 , P(s,n+1)-P(s,n)=(s-2)n+1\,,
  5. P ( s + 1 , n ) - P ( s , n ) = T n - 1 = n ( n - 1 ) 2 . P(s+1,n)-P(s,n)=T_{n-1}=\frac{n(n-1)}{2}\,.
  6. n = 8 ( s - 2 ) x + ( s - 4 ) 2 + ( s - 4 ) 2 ( s - 2 ) . n=\frac{\sqrt{8(s-2)x+(s-4)^{2}}+(s-4)}{2(s-2)}.
  7. P ( s , n ) = ( s - 2 ) T n - 1 + n P(s,n)=(s-2)T_{n-1}+n
  8. P ( 6 , n ) = 4 T n - 1 + n P(6,n)=4T_{n-1}+n
  9. T n - 1 = n ( n - 1 ) / 2 T_{n-1}=n(n-1)/2
  10. P ( 6 , n ) = 4 n ( n - 1 ) / 2 + n = 2 n ( 2 n - 1 ) / 2 = T 2 n - 1 P(6,n)=4n(n-1)/2+n=2n(2n-1)/2=T_{2n-1}
  11. n t h n^{th}
  12. P ( 6 , n ) P(6,n)
  13. ( 2 n - 1 ) s t (2n-1)^{st}
  14. T 2 n - 1 T_{2n-1}
  15. 2 {2}
  16. π 2 6 {\pi^{2}\over 6}
  17. 3 ln ( 3 ) - π 3 3 {3\ln\left(3\right)}-{\pi\sqrt{3}\over 3}
  18. 2 ln ( 2 ) {2\ln\left(2\right)}
  19. 2 3 ln ( 5 ) + 1 + 5 3 ln ( 1 2 10 - 2 5 ) + 1 - 5 3 ln ( 1 2 10 + 2 5 ) + 1 15 π 25 - 10 5 \begin{matrix}\frac{2}{3}\ln(5)\\ +\frac{{1}+\sqrt{5}}{3}\ln\left(\frac{1}{2}\sqrt{10-2\sqrt{5}}\right)\\ +\frac{{1}-\sqrt{5}}{3}\ln\left(\frac{1}{2}\sqrt{10+2\sqrt{5}}\right)\\ +\frac{1}{15}{\pi}{\sqrt{25-10\sqrt{5}}}\end{matrix}
  20. 3 ln ( 3 ) 4 + π 3 12 {{3\ln\left(3\right)\over 4}+{\pi\sqrt{3}\over 12}}
  21. ln ( 2 ) + π 6 {{\ln\left(2\right)}+{\pi\over 6}}
  22. 2 ln ( 2 ) 5 + 3 ln ( 3 ) 10 + π 3 10 {{2\ln\left(2\right)\over 5}+{3\ln\left(3\right)\over 10}+{\pi\sqrt{3}\over 10}}
  23. 4 7 ln ( 2 ) - 2 14 ln ( 3 - 2 2 ) + π ( 1 + 2 ) 14 \begin{matrix}\frac{4}{7}\ln{\left(2\right)}\\ -\frac{\sqrt{2}}{14}\ln\left(3-2\sqrt{2}\right)\\ +\pi\frac{\left(1+\sqrt{2}\right)}{14}\end{matrix}
  24. 2 P ( s , n ) = P ( s + k , n ) + P ( s - k , n ) , 2\,P(s,n)=P(s+k,n)+P(s-k,n),
  25. k = 0 , 1 , 2 , 3 , , s - 3. k=0,1,2,3,...,s-3.

Polymer_physics.html

  1. P ( θ ) exp ( - U ( θ ) / k T ) P(\theta)\propto{}\exp\left(-U(\theta)/kT\right)
  2. U ( θ ) U(\theta)
  3. θ \theta
  4. R g N ν R_{g}\sim N^{\nu}
  5. R g R_{g}
  6. N N
  7. ν \nu
  8. ν = 3 / 5 \nu=3/5
  9. ν = 1 / 3 \nu=1/3
  10. θ \theta
  11. ν = 1 / 2 \nu=1/2
  12. S i = 0 \langle S_{i}\rangle=0
  13. S i S j = b 2 δ i j . \langle S_{i}S_{j}\rangle=b^{2}\delta_{ij}.
  14. x = i = 1 N S i x=\sum_{i=1}^{N}S_{i}
  15. x = i = 1 N S i \langle x\rangle=\left\langle\sum_{i=1}^{N}S_{i}\right\rangle
  16. x = i = 1 N S i . \langle x\rangle=\sum_{i=1}^{N}\langle S_{i}\rangle.
  17. S i = 0 \langle S_{i}\rangle=0
  18. x rms = x 2 = b N . x_{\mathrm{rms}}=\sqrt{\langle x^{2}\rangle}=b\sqrt{N}.
  19. 𝐑 = i = 1 N 𝐫 i \mathbf{R}=\sum_{i=1}^{N}\mathbf{r}_{i}
  20. 𝐫 i = 0 \langle\mathbf{r}_{i}\rangle=0
  21. 𝐫 i 𝐫 j = 3 b 2 δ i j \langle\mathbf{r}_{i}\cdot\mathbf{r}_{j}\rangle=3b^{2}\delta_{ij}
  22. 𝐑 = 0 \langle\mathbf{R}\rangle=0
  23. 𝐑 𝐑 = 3 N b 2 \langle\mathbf{R}\cdot\mathbf{R}\rangle=3Nb^{2}
  24. P = 1 ( 2 π N b 2 3 ) 3 / 2 exp ( - 3 𝐑 𝐑 2 N b 2 ) . P=\frac{1}{\left(\frac{2\pi Nb^{2}}{3}\right)^{3/2}}\exp\left(\frac{-3\mathbf{% R}\cdot\mathbf{R}}{2Nb^{2}}\right).
  25. Ω ( 𝐑 ) = c P ( 𝐑 ) \Omega\left(\mathbf{R}\right)=cP\left(\mathbf{R}\right)
  26. S ( 𝐑 ) = k B ln Ω ( 𝐑 ) S\left(\mathbf{R}\right)=k_{B}\ln\Omega{\left(\mathbf{R}\right)}
  27. Δ S ( 𝐑 ) = S ( 𝐑 ) - S ( 0 ) \Delta S\left(\mathbf{R}\right)=S\left(\mathbf{R}\right)-S\left(0\right)
  28. Δ F = - T Δ S ( 𝐑 ) \Delta F=-T\Delta S\left(\mathbf{R}\right)
  29. Δ F = k B T 3 R 2 2 N b 2 = 1 2 K R 2 ; K = 3 k B T N b 2 . \Delta F=k_{B}T\frac{3R^{2}}{2Nb^{2}}=\frac{1}{2}KR^{2}\quad;K=\frac{3k_{B}T}{% Nb^{2}}.

Polynomial-time_reduction.html

  1. A m P B A\leq_{m}^{P}B
  2. A t t P B A\leq_{tt}^{P}B
  3. A T P B A\leq_{T}^{P}B
  4. A m P C A\leq_{m}^{P}C
  5. \exists\mathbb{R}
  6. \exists\mathbb{R}
  7. \exists\mathbb{R}
  8. \exists\mathbb{R}

Polynomial_interpolation.html

  1. n + 1 n+1
  2. p p
  3. n n
  4. p ( x i ) = y i , i = 0 , , n . p(x_{i})=y_{i},\qquad i=0,\ldots,n.
  5. n + 1 n+1
  6. L n : 𝕂 n + 1 Π n L_{n}:\mathbb{K}^{n+1}\to\Pi_{n}
  7. n n
  8. p ( x ) = a n x n + a n - 1 x n - 1 + + a 2 x 2 + a 1 x + a 0 . ( 1 ) p(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{2}x^{2}+a_{1}x+a_{0}.\qquad(1)
  9. p ( x i ) = y i for all i { 0 , 1 , , n } . p(x_{i})=y_{i}\qquad\mbox{for all }~{}i\in\left\{0,1,\dots,n\right\}.
  10. [ x 0 n x 0 n - 1 x 0 n - 2 x 0 1 x 1 n x 1 n - 1 x 1 n - 2 x 1 1 x n n x n n - 1 x n n - 2 x n 1 ] [ a n a n - 1 a 0 ] = [ y 0 y 1 y n ] . \begin{bmatrix}x_{0}^{n}&x_{0}^{n-1}&x_{0}^{n-2}&\ldots&x_{0}&1\\ x_{1}^{n}&x_{1}^{n-1}&x_{1}^{n-2}&\ldots&x_{1}&1\\ \vdots&\vdots&\vdots&&\vdots&\vdots\\ x_{n}^{n}&x_{n}^{n-1}&x_{n}^{n-2}&\ldots&x_{n}&1\end{bmatrix}\begin{bmatrix}a_% {n}\\ a_{n-1}\\ \vdots\\ a_{0}\end{bmatrix}=\begin{bmatrix}y_{0}\\ y_{1}\\ \vdots\\ y_{n}\end{bmatrix}.
  11. p ( x ) \displaystyle p(x)
  12. n + 1 n+1
  13. n n
  14. n + 1 n+1
  15. n n
  16. n + 1 n+1
  17. r ( x ) = p ( x ) - q ( x ) r(x)=p(x)-q(x)
  18. n n
  19. n + 1 n+1
  20. r ( x i ) = p ( x i ) - q ( x i ) = y i - y i = 0 r(x_{i})=p(x_{i})-q(x_{i})=y_{i}-y_{i}=0
  21. n + 1 n+1
  22. n ≤n
  23. r ( x ) = A ( x - x 0 ) ( x - x 1 ) ( x - x n ) r(x)=A(x-x_{0})(x-x_{1})\cdots(x-x_{n})
  24. n + 1 n+1
  25. A x n + 1 Ax^{n+1}
  26. A = 0 A=0
  27. r ( x ) = 0 r(x)=0
  28. r ( x ) = 0 = p ( x ) - q ( x ) p ( x ) = q ( x ) r(x)=0=p(x)-q(x)\implies p(x)=q(x)
  29. V a = y Va=y
  30. det ( V ) = i , j = 0 , i < j n ( x i - x j ) \det(V)=\prod_{i,j=0,i<j}^{n}(x_{i}-x_{j})
  31. n + 1 n+1
  32. x i - x j x_{i}-x_{j}
  33. n n
  34. n n
  35. f ( x ) - p n ( x ) = f [ x 0 , , x n , x ] i = 0 n ( x - x i ) f(x)-p_{n}(x)=f[x_{0},\ldots,x_{n},x]\prod_{i=0}^{n}(x-x_{i})
  36. f [ x 0 , , x n , x ] f[x_{0},\ldots,x_{n},x]
  37. n + 1 n+1
  38. p n ( x ) p_{n}(x)
  39. n n
  40. n + 1 n+1
  41. ξ ξ
  42. f ( x ) - p n ( x ) = f ( n + 1 ) ( ξ ) ( n + 1 ) ! i = 0 n ( x - x i ) f(x)-p_{n}(x)=\frac{f^{(n+1)}(\xi)}{(n+1)!}\prod_{i=0}^{n}(x-x_{i})
  43. R n ( x ) = f ( x ) - p n ( x ) R_{n}(x)=f(x)-p_{n}(x)
  44. Y ( t ) = R n ( t ) - R n ( x ) W ( x ) W ( t ) Y(t)=R_{n}(t)-\frac{R_{n}(x)}{W(x)}W(t)
  45. W ( u ) = i = 0 n ( u - x i ) W(u)=\prod_{i=0}^{n}(u-x_{i})
  46. f f
  47. p n p_{n}
  48. Y Y
  49. n + 2 n+2
  50. Y ( t ) Y^{\prime}(t)
  51. n + 1 n+1
  52. Y ( n + 1 ) ( t ) Y^{(n+1)}(t)
  53. ξ ξ
  54. ξ ξ
  55. I I
  56. Y ( n + 1 ) ( t ) = R n ( n + 1 ) ( t ) - R n ( x ) W ( x ) ( n + 1 ) ! Y^{(n+1)}(t)=R_{n}^{(n+1)}(t)-\frac{R_{n}(x)}{W(x)}\ (n+1)!
  57. p n ( x ) p_{n}(x)
  58. n n
  59. R n ( n + 1 ) ( t ) = f ( n + 1 ) ( t ) R_{n}^{(n+1)}(t)=f^{(n+1)}(t)
  60. Y ( n + 1 ) ( t ) = f ( n + 1 ) ( t ) - R n ( x ) W ( x ) ( n + 1 ) ! Y^{(n+1)}(t)=f^{(n+1)}(t)-\frac{R_{n}(x)}{W(x)}\ (n+1)!
  61. ξ ξ
  62. Y ( n + 1 ) ( t ) Y^{(n+1)}(t)
  63. Y ( n + 1 ) ( ξ ) = f ( n + 1 ) ( ξ ) - R n ( x ) W ( x ) ( n + 1 ) ! = 0 Y^{(n+1)}(\xi)=f^{(n+1)}(\xi)-\frac{R_{n}(x)}{W(x)}\ (n+1)!=0
  64. R n ( x ) = f ( x ) - p n ( x ) = f ( n + 1 ) ( ξ ) ( n + 1 ) ! i = 0 n ( x - x i ) R_{n}(x)=f(x)-p_{n}(x)=\frac{f^{(n+1)}(\xi)}{(n+1)!}\prod_{i=0}^{n}(x-x_{i})
  65. x i = x 0 + i h x_{i}=x_{0}+ih
  66. ( h n + 1 ) (h^{n+1})
  67. f ( n + 1 ) ( ξ ) f^{(n+1)}(\xi)
  68. h n + 1 h^{n+1}
  69. f ( n + 1 ) ( ξ ) h n + 1 1 f^{(n+1)}(\xi)h^{n+1}<<1
  70. n n→∞
  71. | ( x - x i ) | , \left|\prod(x-x_{i})\right|,
  72. f - X ( f ) ( L + 1 ) f - p * . \|f-X(f)\|\leq(L+1)\|f-p^{*}\|.
  73. L 2 π log ( n + 1 ) + 1. L\leq\frac{2}{\pi}\log(n+1)+1.
  74. n n→∞
  75. 5 , 55 −5,55
  76. n n→∞
  77. 1 , 11 −1,11
  78. p n ( x ) p_{n}(x)
  79. p n * ( x ) p^{*}_{n}(x)
  80. p n * ( x ) p^{*}_{n}(x)
  81. n + 1 n+1
  82. lim n X n f = f , for every f C ( [ a , b ] ) . \lim_{n\to\infty}X_{n}f=f,\,\text{ for every }f\in C([a,b]).
  83. X n 2 π log ( n + 1 ) + C . \|X_{n}\|\geq\tfrac{2}{\pi}\log(n+1)+C.
  84. 1 , 11 −1,11
  85. 1 , 11 −1,11
  86. n n

Polynomial_long_division.html

  1. x 3 - 2 x 2 - 4 , x^{3}-2x^{2}-4,
  2. x - 3 , x-3,
  3. x 3 - 2 x 2 + 0 x - 4. x^{3}-2x^{2}+0x-4.
  4. x 3 - 2 x 2 - 4 = ( x - 3 ) ( x 2 + x + 3 ) q ( x ) + 5 r ( x ) {x^{3}-2x^{2}-4}=(x-3)\,\underbrace{(x^{2}+x+3)}_{q(x)}+\underbrace{5}_{r(x)}
  5. A = B Q + R , A=BQ+R,
  6. x = 1 x=1
  7. y = x 3 - 12 x 2 - 42. y=x^{3}-12x^{2}-42.
  8. ( x - 1 ) 2 = x 2 - 2 x + 1 (x-1)^{2}=x^{2}-2x+1
  9. x - 10 x 2 - 2 x + 1 ) x 3 - 12 x 2 + 0 x - 42 ¯ x 3 - 2 x 2 + x ¯ - 10 x 2 - x - 42 - 10 x 2 + 20 x - 10 ¯ - 21 x - 32 \begin{matrix}\qquad\qquad\qquad\qquad\qquad\qquad x\;-10\\ \quad x^{2}-2x+1\overline{)x^{3}-12x^{2}+0x-42}\\ \qquad\qquad\underline{x^{3}-\;\;2x^{2}+\;\;x}\\ \qquad\qquad\qquad\qquad-10x^{2}-\;x-42\\ \qquad\qquad\qquad\;\;\;\underline{-10x^{2}+20x-10}\\ \qquad\qquad\qquad\qquad\qquad\;\;-21x-32\end{matrix}
  10. y = - 21 x - 32. y=-21x-32.

Polyomino.html

  1. × 10 3 1 \times 10^{3}1
  2. A n c λ n n A_{n}\sim\frac{c\lambda^{n}}{n}
  3. lim n ( A n ) 1 / n = λ \lim_{n\rightarrow\infty}(A_{n})^{1/n}=\lambda

Polyphosphate.html

  1. \approx
  2. \approx
  3. \approx

Polytrope.html

  1. P = K ρ ( n + 1 ) / n , P=K\rho^{(n+1)/n},
  2. < v a r > P <var>P

Population_genetics.html

  1. V t p q ( 1 - exp { - t 2 N e } ) . V_{t}\approx pq\left(1-\exp\left\{-\frac{t}{2N_{e}}\right\}\right).
  2. G 1 T 1 P 1 T 2 P 2 T 3 G 2 T 4 G 1 G_{1}\;\stackrel{T_{1}}{\rightarrow}\;P_{1}\;\stackrel{T_{2}}{\rightarrow}\;P_% {2}\;\stackrel{T_{3}}{\rightarrow}\;G_{2}\;\stackrel{T_{4}}{\rightarrow}\;G_{1% }^{\prime}\;\rightarrow\cdots

Positive_feedback.html

  1. G c = A / ( 1 - A B ) G_{c}=A/(1-AB)

Potentiometer.html

  1. V L = R 2 R L R 1 R L + R 2 R L + R 1 R 2 V s . V_{\mathrm{L}}={R_{2}R_{\mathrm{L}}\over R_{1}R_{\mathrm{L}}+R_{2}R_{\mathrm{L% }}+R_{1}R_{2}}\cdot V_{s}.
  2. V L = R 2 R 1 + R 2 V s . V_{\mathrm{L}}={R_{2}\over R_{1}+R_{2}}\cdot V_{s}.
  3. V S = 10 V V_{\mathrm{S}}=10\ \mathrm{V}
  4. R 1 = 1 k Ω R_{1}=1\ \mathrm{k\Omega}
  5. R 2 = 2 k Ω R_{2}=2\ \mathrm{k\Omega}
  6. R L = 100 k Ω . R_{\mathrm{L}}=100\ \mathrm{k\Omega}.
  7. 2 k Ω 1 k Ω + 2 k Ω 10 V = 2 3 10 V 6.667 V . {2\ \mathrm{k\Omega}\over 1\ \mathrm{k\Omega}+2\ \mathrm{k\Omega}}\cdot 10\ % \mathrm{V}={2\over 3}\cdot 10\ \mathrm{V}\approx 6.667\ \mathrm{V}.
  8. 6.623 V ≈6.623V
  9. V < s u b > S V<sub>S

Pound_(force).html

  1. 1 lbf = 1 lbm × g n = 1 lbm × 9.80665 m s 2 / 0.3048 m ft 1 lbm × 32.174049 ft s 2 = 32.174049 ft lbm s 2 1 lbf = 1 lbm × 0.45359237 kg lbm × g n = 0.45359237 kg × 9.80665 m s 2 = 4.4482216152605 N \begin{aligned}\displaystyle 1\,\,\text{lbf}&\displaystyle=1\,\,\text{lbm}% \times g\text{n}\\ &\displaystyle=1\,\,\text{lbm}\times 9.80665\,\tfrac{\,\text{m}}{\,\text{s}^{2% }}/0.3048\,\tfrac{\,\text{m}}{\,\text{ft}}\\ &\displaystyle\approx 1\,\,\text{lbm}\times 32.174049\,\mathrm{\tfrac{ft}{s^{2% }}}\\ &\displaystyle=32.174049\,\mathrm{\tfrac{ft{\cdot}lbm}{s^{2}}}\\ \displaystyle 1\,\,\text{lbf}&\displaystyle=1\,\,\text{lbm}\times 0.45359237\,% \tfrac{\,\text{kg}}{\,\text{lbm}}\times g\text{n}\\ &\displaystyle=0.45359237\,\,\text{kg}\times 9.80665\,\tfrac{\,\text{m}}{\,% \text{s}^{2}}\\ &\displaystyle=4.4482216152605\,\,\text{N}\end{aligned}
  2. 1 lbf \displaystyle 1\,\,\text{lbf}

Power-to-weight_ratio.html

  1. P = lim Δ t 0 Δ W ( t ) Δ t = lim Δ t 0 P avg P=\lim_{\Delta t\rightarrow 0}\tfrac{\Delta W(t)}{\Delta t}=\lim_{\Delta t% \rightarrow 0}P_{\mathrm{avg}}\,
  2. W k g \tfrac{W}{kg}\;
  3. m 2 s 3 \tfrac{m^{2}}{s^{3}}\;
  4. m m\;
  5. | 𝐯 ( t ) | |\mathbf{v}(t)|\;
  6. ϕ \phi\;
  7. E K = 1 2 m | 𝐯 ( t ) | 2 E_{K}=\tfrac{1}{2}m|\mathbf{v}(t)|^{2}
  8. m m\;
  9. | 𝐯 ( t ) | |\mathbf{v}(t)|\;
  10. P K = 1 2 m 2 | 𝐯 ( t ) | lim Δ t 0 Δ | 𝐯 ( t ) | Δ t = m 𝐚 ( t ) 𝐯 ( t ) = 𝐅 ( t ) 𝐯 ( t ) = τ ( t ) ω ( t ) P_{K}=\tfrac{1}{2}m2|\mathbf{v}(t)|\lim_{\Delta t\rightarrow 0}\tfrac{\Delta|% \mathbf{v}(t)|}{\Delta t}=m\mathbf{a}(t)\cdot\mathbf{v}(t)=\mathbf{F}(t)\cdot% \mathbf{v}(t)=\mathbf{\tau}(t)\cdot\mathbf{\omega}(t)
  11. 𝐚 ( t ) \mathbf{a}(t)\;
  12. 𝐅 ( t ) \mathbf{F}(t)\;
  13. 𝐯 ( t ) \mathbf{v}(t)\;
  14. τ ( t ) \mathbf{\tau}(t)\;
  15. ω ( t ) \mathbf{\omega}(t)\;
  16. P-to-W = | 𝐚 ( t ) | | 𝐯 ( t ) | | 𝐠 | \mbox{P-to-W}~{}=\frac{|\mathbf{a}(t)||\mathbf{v}(t)|}{|\mathbf{g}|}\;
  17. | 𝐯 ( t ) | |\mathbf{v}(t)|\;

Power_chord.html

  1. 3 1 {}_{1}^{3}

Power_inverter.html

  1. THD = V 2 2 + V 3 2 + V 4 2 + + V n 2 V 1 \mbox{THD}~{}={\sqrt{V_{2}^{2}+V_{3}^{2}+V_{4}^{2}+\cdots+V_{n}^{2}}\over V_{1}}

Power_rule.html

  1. d d x x n = n x n - 1 , n 0. \frac{d}{dx}x^{n}=nx^{n-1},\qquad n\neq 0.
  2. x 0 x^{0}
  3. 0
  4. 0 x - 1 0\cdot x^{-1}
  5. x = 0 x=0
  6. x x
  7. x - 1 x^{-1}
  8. n 0 n\geq 0
  9. x n d x = x n + 1 n + 1 + C , n - 1. \int\!x^{n}\,dx=\frac{x^{n+1}}{n+1}+C,\qquad n\neq-1.
  10. C C
  11. x - 1 x^{-1}
  12. x - 1 d x = ln | x | + C , \int\!x^{-1}\,dx=\ln|x|+C,
  13. x 100 x^{100}
  14. 100 x 99 100x^{99}
  15. x 100 x^{100}
  16. 1 101 x 101 + C \frac{1}{101}x^{101}+C
  17. x n x^{n}
  18. n 0 n\geq 0
  19. n 1 n\geq 1
  20. f ( x ) = x n f(x)=x^{n}\!
  21. f ( x ) = n x n - 1 , f^{\prime}(x)=nx^{n-1},\!
  22. ( x n ) = n x n - 1 . \left(x^{n}\right)^{\prime}=nx^{n-1}.
  23. x n d x = x n + 1 n + 1 + C \int\!x^{n}\,dx=\frac{x^{n+1}}{n+1}+C
  24. n 0 n\geq 0
  25. n 1 n\geq 1
  26. f ( x ) - f ( a ) = x n - a n = ( x - a ) ( x n - 1 + a x n - 2 + + a n - 2 x + a n - 1 ) f(x)-f(a)=x^{n}-a^{n}=(x-a)(x^{n-1}+ax^{n-2}+\cdots+a^{n-2}x+a^{n-1})
  27. f ( a ) = lim x a x n - a n x - a = lim x a x n - 1 + a x n - 2 + + a n - 2 x + a n - 1 f^{\prime}(a)=\lim_{x\rightarrow a}\frac{x^{n}-a^{n}}{x-a}=\lim_{x\rightarrow a% }x^{n-1}+ax^{n-2}+\cdots+a^{n-2}x+a^{n-1}
  28. f ( a ) = lim x a x n - 1 + a x n - 2 + + a n - 2 x + a n - 1 = a n - 1 + a n - 1 + + a n - 1 + a n - 1 = n a n - 1 f^{\prime}(a)=\lim_{x\rightarrow a}x^{n-1}+ax^{n-2}+\cdots+a^{n-2}x+a^{n-1}=a^% {n-1}+a^{n-1}+\cdots+a^{n-1}+a^{n-1}=n\cdot a^{n-1}
  29. n n
  30. n n
  31. ( r = 0 n a r x r ) = r = 0 n ( a r x r ) = r = 0 n a r ( x r ) = r = 0 n r a r x r - 1 . \left(\sum_{r=0}^{n}a_{r}x^{r}\right)^{\prime}=\sum_{r=0}^{n}\left(a_{r}x^{r}% \right)^{\prime}=\sum_{r=0}^{n}a_{r}\left(x^{r}\right)^{\prime}=\sum_{r=0}^{n}% ra_{r}x^{r-1}.
  32. ( k = 0 n a k x k ) d x = k = 0 n a k x k + 1 k + 1 + C . \int\!\left(\sum^{n}_{k=0}a_{k}x^{k}\right)\,dx=\sum^{n}_{k=0}\frac{a_{k}x^{k+% 1}}{k+1}+C.
  33. r r
  34. d d x x r = r x r - 1 \frac{d}{dx}x^{r}=rx^{r-1}
  35. ln x r = r ln x \ln x^{r}=r\ln x
  36. d d x ln x r = d d x ( r ln x ) \frac{d}{dx}\ln x^{r}=\frac{d}{dx}(r\ln x)
  37. 1 x r d d x x r = r d d x ln x \frac{1}{x^{r}}\cdot\frac{d}{dx}x^{r}=r\cdot\frac{d}{dx}\ln x
  38. 1 x r d d x x r = r x \frac{1}{x^{r}}\cdot\frac{d}{dx}x^{r}=\frac{r}{x}
  39. d d x x r = r x r x \frac{d}{dx}x^{r}=\frac{rx^{r}}{x}
  40. d d x x r = r x r - 1 \frac{d}{dx}x^{r}=rx^{r-1}
  41. x x
  42. r r
  43. x - 1 d x = ln | x | + C , \int\!x^{-1}\,dx=\ln|x|+C,
  44. x x

Prandtl–Glauert_singularity.html

  1. c p = c p 0 | 1 - M 2 | c_{p}=\frac{c_{p0}}{\sqrt{|1-{M_{\infty}}^{2}|}}

Presentation_of_a_group.html

  1. S R . \langle S\mid R\rangle.
  2. a a n = 1 . \langle a\mid a^{n}=1\rangle.
  3. a a n , \langle a\mid a^{n}\rangle,
  4. s 1 a 1 s 2 a 2 s n a n s_{1}^{a_{1}}s_{2}^{a_{2}}\ldots s_{n}^{a_{n}}
  5. r , f r 8 = f 2 = ( r f ) 2 = 1 . \langle r,f\mid r^{8}=f^{2}=(rf)^{2}=1\rangle.
  6. S R = F S / N . \langle S\mid R\rangle=F_{S}/N.
  7. g i g j g k - 1 g_{i}g_{j}g_{k}^{-1}
  8. g i g j = g k g_{i}g_{j}=g_{k}
  9. S \langle S\mid\varnothing\rangle
  10. a a n \langle a\mid a^{n}\rangle
  11. r , f r n , f 2 , ( r f ) 2 \langle r,f\mid r^{n},f^{2},(rf)^{2}\rangle
  12. r , f f 2 , ( r f ) 2 \langle r,f\mid f^{2},(rf)^{2}\rangle
  13. r , f r 2 n , r n = f 2 , f r f - 1 = r - 1 \langle r,f\mid r^{2n},r^{n}=f^{2},frf^{-1}=r^{-1}\rangle
  14. x , y x y = y x \langle x,y\mid xy=yx\rangle
  15. x , y x m , y n , x y = y x \langle x,y\mid x^{m},y^{n},xy=yx\rangle
  16. S R \langle S\mid R\rangle
  17. σ 1 , , σ n - 1 \sigma_{1},\ldots,\sigma_{n-1}
  18. σ i 2 = 1 \sigma_{i}^{2}=1
  19. σ i σ j = σ j σ i if j i ± 1 \sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i}\mbox{ if }~{}j\neq i\pm 1
  20. σ i σ i + 1 σ i = σ i + 1 σ i σ i + 1 \sigma_{i}\sigma_{i+1}\sigma_{i}=\sigma_{i+1}\sigma_{i}\sigma_{i+1}
  21. ( σ i σ i + 1 ) 3 = 1 {(\sigma_{i}\sigma_{i+1}})^{3}=1
  22. σ i 2 = 1 \sigma_{i}^{2}=1
  23. σ 1 , , σ n - 1 \sigma_{1},\ldots,\sigma_{n-1}
  24. σ i σ j = σ j σ i if j i ± 1 \sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i}\mbox{ if }~{}j\neq i\pm 1
  25. σ i σ i + 1 σ i = σ i + 1 σ i σ i + 1 \sigma_{i}\sigma_{i+1}\sigma_{i}=\sigma_{i+1}\sigma_{i}\sigma_{i+1}
  26. σ i 2 = 1 \sigma_{i}^{2}=1
  27. s , t s 2 , t 3 , ( s t ) 3 \langle s,t\mid s^{2},t^{3},(st)^{3}\rangle
  28. s , t s 2 , t 3 , ( s t ) 4 \langle s,t\mid s^{2},t^{3},(st)^{4}\rangle
  29. s , t s 2 , t 3 , ( s t ) 5 \langle s,t\mid s^{2},t^{3},(st)^{5}\rangle
  30. i , j j i j = i , i j i = j \langle i,j\mid jij=i,iji=j\rangle\,
  31. a , b a b a = b a b , ( a b a ) 4 \langle a,b\mid aba=bab,(aba)^{4}\rangle
  32. a , b , j a b a = b a b , ( a b a ) 4 , j 2 , ( j a ) 2 , ( j b ) 2 \langle a,b,j\mid aba=bab,(aba)^{4},j^{2},(ja)^{2},(jb)^{2}\rangle
  33. a , b a 2 , b 3 \langle a,b\mid a^{2},b^{3}\rangle
  34. x , y , z z = x y x - 1 y - 1 , x z = z x , y z = z y \langle x,y,z\mid z=xyx^{-1}y^{-1},xz=zx,yz=zy\rangle
  35. a , b a n = b a m b - 1 \langle a,b\mid a^{n}=ba^{m}b^{-1}\rangle
  36. a , b a 2 , b 3 , ( a b ) 13 , [ a , b ] 5 , [ a , b a b ] 4 , ( a b a b a b a b a b - 1 ) 6 \langle a,b\mid a^{2},b^{3},(ab)^{13},[a,b]^{5},[a,bab]^{4},(ababababab^{-1})^% {6}\rangle
  37. 𝐙 𝐙 \mathbf{Z}\wr\mathbf{Z}

Price_discrimination.html

  1. P 1 / P 2 = [ 1 + 1 / E 2 ] / [ 1 + 1 / E 1 ] P1/P2=[1+1/E2]/[1+1/E1]

Price_elasticity_of_demand.html

  1. e p = d Q / Q d P / P e_{\langle p\rangle}=\frac{\mathrm{d}Q/Q}{\mathrm{d}P/P}
  2. E d = P Q d × d Q d d P E_{d}=\frac{P}{Q_{d}}\times\frac{\mathrm{d}Q_{d}}{\mathrm{d}P}
  3. x ( p , w ) \displaystyle x(p,w)
  4. x 1 , x 2 , , x L x_{1},x_{2},\dots,x_{L}
  5. x l ( p , w ) \displaystyle x_{l}(p,w)
  6. l \displaystyle l
  7. x l ( p , w ) \displaystyle x_{l}(p,w)
  8. p k p_{k}
  9. E x l , p k = x l ( p , w ) p k p k x l ( p , w ) = log x l ( p , w ) log p k E_{x_{l},p_{k}}=\frac{\partial x_{l}(p,w)}{\partial p_{k}}\cdot\frac{p_{k}}{x_% {l}(p,w)}=\frac{\partial\log x_{l}(p,w)}{\partial\log p_{k}}
  10. Q d = f ( P ) Q_{d}=f(P)
  11. d Q d / d P {dQ_{d}/dP}
  12. E d = P 1 + P 2 2 Q d 1 + Q d 2 2 × Δ Q d Δ P = P 1 + P 2 Q d 1 + Q d 2 × Δ Q d Δ P E_{d}=\frac{\frac{P_{1}+P_{2}}{2}}{\frac{Q_{d_{1}}+Q_{d_{2}}}{2}}\times\frac{% \Delta Q_{d}}{\Delta P}=\frac{P_{1}+P_{2}}{Q_{d_{1}}+Q_{d_{2}}}\times\frac{% \Delta Q_{d}}{\Delta P}
  13. E d = 0 E_{d}=0
  14. - 1 < E d < 0 -1<E_{d}<0
  15. E d = - 1 E_{d}=-1
  16. - < E d < - 1 -\infty<E_{d}<-1
  17. E d = - E_{d}=-\infty
  18. R = P ( 1 - 1 E d ) R^{\prime}=P\,\left(1-\dfrac{1}{E_{d}}\right)
  19. R = T R Q = Q ( P Q ) = P + Q P Q R^{\prime}=\dfrac{\partial TR}{\partial Q}=\dfrac{\partial}{\partial Q}(P\,Q)=% P+Q\,\dfrac{\partial P}{\partial Q}
  20. E d = - Q P P Q - E d Q P = Q P - P E d Q = P Q E_{d}=-\dfrac{\partial Q}{\partial P}\cdot\dfrac{P}{Q}\Rightarrow-E_{d}\cdot% \dfrac{Q}{P}=\dfrac{\partial Q}{\partial P}\Rightarrow-\dfrac{P}{E_{d}\cdot Q}% =\dfrac{\partial P}{\partial Q}
  21. R = P + Q - P E d Q = P ( 1 - 1 E d ) R^{\prime}=P+Q\cdot-\dfrac{P}{E_{d}\cdot Q}=P\,\left(1-\dfrac{1}{E_{d}}\right)
  22. Revenue = P Q d \mbox{Revenue}~{}=PQ_{d}
  23. L Q = K + E × L P LQ=K+E\times LP
  24. L Q = ln ( Q ) , L P = ln ( P ) , E LQ=\ln(Q),LP=\ln(P),E
  25. K K
  26. n n
  27. n n
  28. L Q l = K l + E l , k × L P k LQ_{l}=K_{l}+E_{l,k}\times LP^{k}
  29. l l
  30. k = 1 , , n , L Q l = ln ( Q l ) , L P l = ln ( P l ) k=1,\ldots,n,LQ_{l}=\ln(Q_{l}),LP^{l}=\ln(P^{l})
  31. K l K_{l}
  32. ln ( Q ) \ln(Q)
  33. Q Q
  34. Q Q
  35. ln ( Q ) \ln(Q)
  36. Q Q
  37. L Q = K + E 1 × L P + E 2 × L P 2 LQ=K+E_{1}\times LP+E_{2}\times LP^{2}
  38. L Q l = K l + E 1 l , k × L P k + E 2 l , k × ( L P k ) 2 LQ_{l}=K_{l}+E1_{l,k}\times LP^{k}+E2_{l,k}\times(LP^{k})^{2}

Primality_test.html

  1. 100 \scriptstyle\sqrt{100}
  2. n \scriptstyle\sqrt{n}
  3. n \scriptstyle{}\leq\sqrt{n}
  4. \cdot
  5. ( p - 1 ) ! - 1 ( mod p ) (p-1)!\equiv-1\;\;(\mathop{{\rm mod}}p)\,
  6. 2 340 1 ( mod 341 ) 2^{340}\equiv 1\;\;(\mathop{{\rm mod}}341)
  7. a d 1 ( mod n ) a^{d}\not\equiv 1\;\;(\mathop{{\rm mod}}n)
  8. a 2 r d - 1 ( mod n ) a^{2^{r}d}\not\equiv-1\;\;(\mathop{{\rm mod}}n)
  9. 0 r s - 1 0\leq r\leq s-1
  10. ( a n ) \left(\frac{a}{n}\right)
  11. O ( log 3 n log log n log log log n ) O(\log^{3}n\log\log n\log\log\log n)

Primary_production.html

  1. \rightarrow
  2. \rightarrow

Prime-factor_FFT_algorithm.html

  1. X k = n = 0 N - 1 x n e - 2 π i N n k k = 0 , , N - 1. X_{k}=\sum_{n=0}^{N-1}x_{n}e^{-\frac{2\pi i}{N}nk}\qquad k=0,\dots,N-1.
  2. n = n 1 N 2 + n 2 N 1 mod N , n=n_{1}N_{2}+n_{2}N_{1}\mod N,
  3. k = k 1 N 2 - 1 N 2 + k 2 N 1 - 1 N 1 mod N , k=k_{1}N_{2}^{-1}N_{2}+k_{2}N_{1}^{-1}N_{1}\mod N,
  4. X k 1 N 2 - 1 N 2 + k 2 N 1 - 1 N 1 = n 1 = 0 N 1 - 1 ( n 2 = 0 N 2 - 1 x n 1 N 2 + n 2 N 1 e - 2 π i N 2 n 2 k 2 ) e - 2 π i N 1 n 1 k 1 . X_{k_{1}N_{2}^{-1}N_{2}+k_{2}N_{1}^{-1}N_{1}}=\sum_{n_{1}=0}^{N_{1}-1}\left(% \sum_{n_{2}=0}^{N_{2}-1}x_{n_{1}N_{2}+n_{2}N_{1}}e^{-\frac{2\pi i}{N_{2}}n_{2}% k_{2}}\right)e^{-\frac{2\pi i}{N_{1}}n_{1}k_{1}}.

Primitive_equations.html

  1. u u
  2. v v
  3. T T
  4. f f
  5. 2 Ω sin ( ϕ ) 2\Omega\sin(\phi)
  6. Ω \Omega
  7. 2 π / 24 2\pi/24
  8. ϕ \phi
  9. R R
  10. p p
  11. c p c_{p}
  12. J J
  13. W W
  14. θ \theta
  15. f m = 1 ρ d p d x . \frac{f}{m}=\frac{1}{\rho}\frac{dp}{dx}.
  16. f r = f a 1 ρ μ ( ( μ v ) + ( λ v ) ) . f_{r}={f\over a}{1\over\rho}\mu\left(\nabla\cdot(\mu\nabla v)+\nabla(\lambda% \nabla\cdot v)\right).
  17. d v d t = - ( 1 / ρ ) p - g ( r / r ) + f r \frac{dv}{dt}=-(1/\rho)\nabla p-g(r/r)+f_{r}
  18. g = g e . g=g_{e}.\,
  19. d v d t = - ( 1 / ρ ) p - g ( r / r ) + ( 1 / ρ ) [ ( μ v ) + ( λ v ) ] \frac{dv}{dt}=-(1/\rho)\nabla p-g(r/r)+(1/\rho)\left[\nabla\cdot(\mu\nabla v)+% \nabla(\lambda\nabla\cdot v)\right]
  20. c v d T d t + p d α d t = q + f c_{v}\frac{dT}{dt}+p\frac{d\alpha}{dt}=q+f
  21. d ρ d t + ρ v = 0 \frac{d\rho}{dt}+\rho\nabla\cdot v=0
  22. p = ρ R T . p=\rho RT.
  23. D u D t - f v = - ϕ x \frac{Du}{Dt}-fv=-\frac{\partial\phi}{\partial x}
  24. D v D t + f u = - ϕ y \frac{Dv}{Dt}+fu=-\frac{\partial\phi}{\partial y}
  25. 0 = - ϕ p - R T p 0=-\frac{\partial\phi}{\partial p}-\frac{RT}{p}
  26. d p = - ρ d ϕ dp=-\rho\,d\phi
  27. u x + v y + ω p = 0 \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial% \omega}{\partial p}=0
  28. T t + u T x + v T y + ω ( T p - R T p c p ) = J c p \frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T% }{\partial y}+\omega\left(\frac{\partial T}{\partial p}-\frac{RT}{pc_{p}}% \right)=\frac{J}{c_{p}}
  29. u t = η v - Φ x - c p θ π x - z u σ - ( u 2 + v 2 2 ) x \frac{\partial u}{\partial t}=\eta v-\frac{\partial\Phi}{\partial x}-c_{p}% \theta\frac{\partial\pi}{\partial x}-z\frac{\partial u}{\partial\sigma}-\frac{% \partial(\frac{u^{2}+v^{2}}{2})}{\partial x}
  30. v t = - η u v - Φ y - c p θ π y - z v σ - ( u 2 + v 2 2 ) y \frac{\partial v}{\partial t}=-\eta\frac{u}{v}-\frac{\partial\Phi}{\partial y}% -c_{p}\theta\frac{\partial\pi}{\partial y}-z\frac{\partial v}{\partial\sigma}-% \frac{\partial(\frac{u^{2}+v^{2}}{2})}{\partial y}
  31. δ T t = T t + u T x + v T y + w T z \frac{\delta T}{\partial t}=\frac{\partial T}{\partial t}+u\frac{\partial T}{% \partial x}+v\frac{\partial T}{\partial y}+w\frac{\partial T}{\partial z}
  32. δ W t = u W x + v W y + w W z \frac{\delta W}{\partial t}=u\frac{\partial W}{\partial x}+v\frac{\partial W}{% \partial y}+w\frac{\partial W}{\partial z}
  33. t p σ = u x x p σ + v y y p σ + w z z p σ \frac{\partial}{\partial t}\frac{\partial p}{\partial\sigma}=u\frac{\partial}{% \partial x}x\frac{\partial p}{\partial\sigma}+v\frac{\partial}{\partial y}y% \frac{\partial p}{\partial\sigma}+w\frac{\partial}{\partial z}z\frac{\partial p% }{\partial\sigma}
  34. { u , v , ϕ } = { u ^ , v ^ , ϕ ^ } e i ( s λ + σ t ) \begin{Bmatrix}u,v,\phi\end{Bmatrix}=\begin{Bmatrix}\hat{u},\hat{v},\hat{\phi}% \end{Bmatrix}e^{i(s\lambda+\sigma t)}
  35. σ \sigma

Primitive_root_modulo_n.html

  1. 3 1 = 3 = 3 0 × 3 1 × 3 = 3 3 ( mod 7 ) 3 2 = 9 = 3 1 × 3 3 × 3 = 9 2 ( mod 7 ) 3 3 = 27 = 3 2 × 3 2 × 3 = 6 6 ( mod 7 ) 3 4 = 81 = 3 3 × 3 6 × 3 = 18 4 ( mod 7 ) 3 5 = 243 = 3 4 × 3 4 × 3 = 12 5 ( mod 7 ) 3 6 = 729 = 3 5 × 3 5 × 3 = 15 1 ( mod 7 ) \begin{array}[]{rcrcrcrcrcr}3^{1}&=&3&=&3^{0}\times 3&\equiv&1\times 3&=&3&% \equiv&3\;\;(\mathop{{\rm mod}}7)\\ 3^{2}&=&9&=&3^{1}\times 3&\equiv&3\times 3&=&9&\equiv&2\;\;(\mathop{{\rm mod}}% 7)\\ 3^{3}&=&27&=&3^{2}\times 3&\equiv&2\times 3&=&6&\equiv&6\;\;(\mathop{{\rm mod}% }7)\\ 3^{4}&=&81&=&3^{3}\times 3&\equiv&6\times 3&=&18&\equiv&4\;\;(\mathop{{\rm mod% }}7)\\ 3^{5}&=&243&=&3^{4}\times 3&\equiv&4\times 3&=&12&\equiv&5\;\;(\mathop{{\rm mod% }}7)\\ 3^{6}&=&729&=&3^{5}\times 3&\equiv&5\times 3&=&15&\equiv&1\;\;(\mathop{{\rm mod% }}7)\\ \end{array}
  2. φ ( n ) \varphi\left(n\right)
  3. φ ( n ) \varphi\left(n\right)
  4. φ ( n ) \varphi\left(n\right)
  5. φ ( n ) \varphi\left(n\right)
  6. m φ ( n ) / p i mod n for i = 1 , , k m^{\varphi(n)/p_{i}}\mod n\qquad\mbox{ for }~{}i=1,\ldots,k
  7. φ ( φ ( n ) ) \varphi\left(\varphi\left(n\right)\right)
  8. φ ( r ) \varphi\left(r\right)
  9. g p C p 1 4 + ϵ . g_{p}\leq Cp^{\frac{1}{4}+\epsilon}.
  10. p > e e 24 p>e^{e^{24}}
  11. g p < p 0.499 g_{p}<p^{0.499}

Principal_bundle.html

  1. X × G X×G
  2. X X
  3. G G
  4. P P
  5. G G
  6. P P
  7. ( x , g ) h = ( x , g h ) (x,g)h=(x,gh)
  8. X X
  9. ( x , g ) x (x,g)↦x
  10. ( x , e ) (x,e)
  11. G G
  12. X × G G X×G→G
  13. F E FE
  14. E E
  15. G G
  16. G G
  17. G G
  18. G G
  19. π : P X π:P→X
  20. P × G P P×G→P
  21. G G
  22. P P
  23. g G g∈G
  24. G G
  25. X X
  26. π : P X π:P→X
  27. G G
  28. P / G P/G
  29. X X
  30. G G
  31. G G
  32. G G
  33. G G
  34. π : P X π:P→X
  35. G G
  36. G G
  37. G G
  38. P P
  39. π π
  40. G G
  41. G G
  42. π : P X π:P→X
  43. G G
  44. P P
  45. M M
  46. F M FM
  47. G L ( M ) GL(M)
  48. x M x∈M
  49. G L ( n , 𝐑 ) GL(n,\mathbf{R})
  50. G L ( n , 𝐑 ) GL(n,\mathbf{R})
  51. M M
  52. O ( n ) O(n)
  53. E E
  54. k k
  55. M M
  56. E E
  57. G L ( k , 𝐑 ) GL(k,\mathbf{R})
  58. F ( E ) F(E)
  59. p : C X p:C→X
  60. G = π 1 ( X ) / p * ( π 1 ( C ) ) G=\pi_{1}(X)/p_{*}(\pi_{1}(C))
  61. p p
  62. X X
  63. X X
  64. G G
  65. H H
  66. G G
  67. H H
  68. G / H G/H
  69. H H
  70. G G
  71. H H
  72. H H
  73. n n
  74. O ( 1 ) O(1)
  75. O ( 1 ) O(1)
  76. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  77. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  78. n n
  79. O ( 1 ) S ( n + 1 ) n \mbox{O}~{}(1)\to S(\mathbb{R}^{n+1})\to\mathbb{RP}^{n}
  80. U ( 1 ) S ( n + 1 ) n \mbox{U}~{}(1)\to S(\mathbb{C}^{n+1})\to\mathbb{CP}^{n}
  81. Sp ( 1 ) S ( n + 1 ) n . \mbox{Sp}~{}(1)\to S(\mathbb{H}^{n+1})\to\mathbb{HP}^{n}.
  82. S ( V ) S(V)
  83. V V
  84. n = 1 n=1
  85. π : P X π:P→X
  86. G G
  87. U U
  88. X X
  89. U U
  90. Φ : π - 1 ( U ) U × G \Phi:\pi^{-1}(U)\to U\times G
  91. s : U π - 1 ( U ) ; s ( x ) = Φ - 1 ( x , e ) s:U\to\pi^{-1}(U);s(x)=\Phi^{-1}(x,e)\,
  92. e e
  93. G G
  94. s s
  95. Φ Φ
  96. Φ - 1 ( x , g ) = s ( x ) g . \Phi^{-1}(x,g)=s(x)\cdot g.
  97. G G
  98. P P
  99. G G
  100. Φ : π - 1 ( U ) U × G \Phi:\pi^{-1}(U)\to U\times G
  101. Φ ( p ) = ( π ( p ) , φ ( p ) ) , \Phi(p)=(\pi(p),\varphi(p)),
  102. φ : P G \varphi:P\to G
  103. φ ( p g ) = φ ( p ) g . \varphi(p\cdot g)=\varphi(p)g.
  104. G G
  105. s s
  106. φ φ
  107. φ ( s ( x ) g ) = g . \varphi(s(x)\cdot g)=g.
  108. P P
  109. G G
  110. t i j = U i U j G t_{ij}=U_{i}\cap U_{j}\to G\,
  111. s j ( x ) = s i ( x ) t i j ( x ) . s_{j}(x)=s_{i}(x)\cdot t_{ij}(x).
  112. P / H P/H
  113. G / H G/H
  114. G L ( 2 n , ) GL(2n,\mathbb{R})
  115. GL ( n , ) GL ( 2 n , ) \mathrm{GL}(n,\mathbb{C})\subset\mathrm{GL}(2n,\mathbb{R})
  116. GL ( k , ) GL ( n , ) \mathrm{GL}(k,\mathbb{R})\subset\mathrm{GL}(n,\mathbb{R})
  117. SO ( n ) GL ( n , ) \mathrm{SO}(n)\subset\mathrm{GL}(n,\mathbb{R})
  118. SO ( n ) \mathrm{SO}(n)
  119. Spin ( n ) \mathrm{Spin}(n)
  120. SO ( n ) \mathrm{SO}(n)
  121. E = P × G V E=P\times_{G}V
  122. G G
  123. B G BG
  124. G G
  125. E G EG
  126. G G
  127. E G B G EG→BG
  128. G G
  129. B B
  130. B B G B→BG

Principal_component_analysis.html

  1. 𝐰 ( k ) = ( w 1 , , w p ) ( k ) \mathbf{w}_{(k)}=(w_{1},\dots,w_{p})_{(k)}
  2. 𝐱 ( i ) \mathbf{x}_{(i)}
  3. 𝐭 ( i ) = ( t 1 , , t p ) ( i ) \mathbf{t}_{(i)}=(t_{1},\dots,t_{p})_{(i)}
  4. t k ( i ) = 𝐱 ( i ) 𝐰 ( k ) {t_{k}}_{(i)}=\mathbf{x}_{(i)}\cdot\mathbf{w}_{(k)}
  5. 𝐰 ( 1 ) = arg max 𝐰 = 1 { i ( t 1 ) ( i ) 2 } = arg max 𝐰 = 1 { i ( 𝐱 ( i ) 𝐰 ) 2 } \mathbf{w}_{(1)}=\underset{\|\mathbf{w}\|=1}{\operatorname{\arg\,max}}\,\left% \{\sum_{i}\left(t_{1}\right)^{2}_{(i)}\right\}=\underset{\|\mathbf{w}\|=1}{% \operatorname{\arg\,max}}\,\left\{\sum_{i}\left(\mathbf{x}_{(i)}\cdot\mathbf{w% }\right)^{2}\right\}
  6. 𝐰 ( 1 ) = arg max 𝐰 = 1 { 𝐗𝐰 2 } = arg max 𝐰 = 1 { 𝐰 T 𝐗 T 𝐗𝐰 } \mathbf{w}_{(1)}=\underset{\|\mathbf{w}\|=1}{\operatorname{\arg\,max}}\,\{\|% \mathbf{Xw}\|^{2}\}=\underset{\|\mathbf{w}\|=1}{\operatorname{\arg\,max}}\,% \left\{\mathbf{w}^{T}\mathbf{X}^{T}\mathbf{Xw}\right\}
  7. 𝐰 ( 1 ) = arg max { 𝐰 T 𝐗 T 𝐗𝐰 𝐰 T 𝐰 } \mathbf{w}_{(1)}={\operatorname{\arg\,max}}\,\left\{\frac{\mathbf{w}^{T}% \mathbf{X}^{T}\mathbf{Xw}}{\mathbf{w}^{T}\mathbf{w}}\right\}
  8. 𝐗 ^ k = 𝐗 - s = 1 k - 1 𝐗𝐰 ( s ) 𝐰 ( s ) T \mathbf{\hat{X}}_{k}=\mathbf{X}-\sum_{s=1}^{k-1}\mathbf{X}\mathbf{w}_{(s)}% \mathbf{w}_{(s)}^{\rm T}
  9. 𝐰 ( k ) = arg max 𝐰 = 1 { 𝐗 ^ k 𝐰 2 } = arg max { 𝐰 T 𝐗 ^ k T 𝐗 ^ k 𝐰 𝐰 T 𝐰 } \mathbf{w}_{(k)}=\underset{\|\mathbf{w}\|=1}{\operatorname{arg\,max}}\left\{\|% \mathbf{\hat{X}}_{k}\mathbf{w}\|^{2}\right\}={\operatorname{\arg\,max}}\,\left% \{\tfrac{\mathbf{w}^{T}\mathbf{\hat{X}}_{k}^{T}\mathbf{\hat{X}}_{k}\mathbf{w}}% {\mathbf{w}^{T}\mathbf{w}}\right\}
  10. 𝐓 = 𝐗𝐖 \mathbf{T}=\mathbf{X}\mathbf{W}
  11. Q ( PC ( j ) , PC ( k ) ) \displaystyle Q(\mathrm{PC}_{(j)},\mathrm{PC}_{(k)})
  12. 𝐐 𝐗 T 𝐗 = 𝐖 𝚲 𝐖 T \mathbf{Q}\propto\mathbf{X}^{T}\mathbf{X}=\mathbf{W}\mathbf{\Lambda}\mathbf{W}% ^{T}
  13. 𝐖 T 𝐐𝐖 𝐖 T 𝐖 𝚲 𝐖 T 𝐖 = 𝚲 \mathbf{W}^{T}\mathbf{Q}\mathbf{W}\propto\mathbf{W}^{T}\mathbf{W}\,\mathbf{% \Lambda}\,\mathbf{W}^{T}\mathbf{W}=\mathbf{\Lambda}
  14. 𝐓 L = 𝐗𝐖 L \mathbf{T}_{L}=\mathbf{X}\mathbf{W}_{L}
  15. t = W T x , x R p , t R L , t=W^{T}x,x\in R^{p},t\in R^{L},
  16. p × L p×L
  17. 𝐓𝐖 T - 𝐓 L 𝐖 L T 2 2 \|\mathbf{T}\mathbf{W}^{T}-\mathbf{T}_{L}\mathbf{W}^{T}_{L}\|_{2}^{2}
  18. 𝐗 - 𝐗 L 2 2 \|\mathbf{X}-\mathbf{X}_{L}\|_{2}^{2}
  19. 𝐗 = 𝐔 𝚺 𝐖 T \mathbf{X}=\mathbf{U}\mathbf{\Sigma}\mathbf{W}^{T}
  20. 𝐗 T 𝐗 = 𝐖 𝚺 𝐔 T 𝐔 𝚺 𝐖 T = 𝐖 𝚺 2 𝐖 T \begin{aligned}\displaystyle\mathbf{X}^{T}\mathbf{X}&\displaystyle=\mathbf{W}% \mathbf{\Sigma}\mathbf{U}^{T}\mathbf{U}\mathbf{\Sigma}\mathbf{W}^{T}\\ &\displaystyle=\mathbf{W}\mathbf{\Sigma}^{2}\mathbf{W}^{T}\end{aligned}
  21. 𝐓 = 𝐗𝐖 = 𝐔 𝚺 𝐖 T 𝐖 = 𝐔 𝚺 \begin{aligned}\displaystyle\mathbf{T}&\displaystyle=\mathbf{X}\mathbf{W}\\ &\displaystyle=\mathbf{U}\mathbf{\Sigma}\mathbf{W}^{T}\mathbf{W}\\ &\displaystyle=\mathbf{U}\mathbf{\Sigma}\end{aligned}
  22. n × L n×L
  23. 𝐓 L = 𝐔 L 𝚺 L = 𝐗𝐖 L \mathbf{T}_{L}=\mathbf{U}_{L}\mathbf{\Sigma}_{L}=\mathbf{X}\mathbf{W}_{L}
  24. 𝐗 = { X [ i , j ] } \mathbf{X}=\{X[i,j]\}
  25. n × p n\times p
  26. i = 1 n i=1\ldots n
  27. j = 1 p j=1\ldots p
  28. n n\,
  29. 1 × 1 1\times 1
  30. p p\,
  31. 1 × 1 1\times 1
  32. L L\,
  33. 1 L p 1\leq L\leq p
  34. 1 × 1 1\times 1
  35. 𝐮 = { u [ j ] } \mathbf{u}=\{u[j]\}
  36. p × 1 p\times 1
  37. j = 1 p j=1\ldots p
  38. 𝐬 = { s [ j ] } \mathbf{s}=\{s[j]\}
  39. p × 1 p\times 1
  40. j = 1 p j=1\ldots p
  41. 𝐡 = { h [ i ] } \mathbf{h}=\{h[i]\}
  42. 1 × n 1\times n
  43. i = 1 n i=1\ldots n
  44. 𝐁 = { B [ i , j ] } \mathbf{B}=\{B[i,j]\}
  45. n × p n\times p
  46. i = 1 n i=1\ldots n
  47. j = 1 p j=1\ldots p
  48. 𝐙 = { Z [ m , n ] } \mathbf{Z}=\{Z[m,n]\}
  49. n × p n\times p
  50. i = 1 n i=1\ldots n
  51. j = 1 p j=1\ldots p
  52. 𝐂 = { C [ k , l ] } \mathbf{C}=\{C[k,l]\}
  53. p × p p\times p
  54. k = 1 p k=1\ldots p
  55. l = 1 p l=1\ldots p
  56. 𝐑 = { R [ k , l ] } \mathbf{R}=\{R[k,l]\}
  57. p × p p\times p
  58. k = 1 p k=1\ldots p
  59. l = 1 p l=1\ldots p
  60. 𝐕 = { V [ j , k ] } \mathbf{V}=\{V[j,k]\}
  61. p × p p\times p
  62. j = 1 p j=1\ldots p
  63. k = 1 p k=1\ldots p
  64. 𝐃 = { D [ k , l ] } \mathbf{D}=\{D[k,l]\}
  65. p × p p\times p
  66. k = 1 p k=1\ldots p
  67. l = 1 p l=1\ldots p
  68. 𝐖 = { W [ j , k ] } \mathbf{W}=\{W[j,k]\}
  69. p × L p\times L
  70. j = 1 p j=1\ldots p
  71. k = 1 L k=1\ldots L
  72. 𝐓 = { T [ i , k ] } \mathbf{T}=\{T[i,k]\}
  73. n × L n\times L
  74. i = 1 n i=1\ldots n
  75. k = 1 L k=1\ldots L
  76. y = 𝐁 x y=\mathbf{B^{\prime}}x
  77. y y
  78. 𝐁 \mathbf{B^{\prime}}
  79. < m t p l > Σ y = 𝐁 𝚺 𝐁 \mathbf{<}mtpl>{{\Sigma}}_{y}=\mathbf{B^{\prime}}\mathbf{\Sigma}\mathbf{B}
  80. y y
  81. 𝚺 y \mathbf{\Sigma}_{y}
  82. tr ( 𝚺 y ) \,\text{tr}(\mathbf{\Sigma}_{y})
  83. 𝐁 = 𝐀 q \mathbf{B}=\mathbf{A}_{q}
  84. 𝐀 q \mathbf{A}_{q}
  85. 𝐀 \mathbf{A}
  86. ( 𝐁 (\mathbf{B^{\prime}}
  87. 𝐁 ) \mathbf{B})
  88. y = 𝐁 x y=\mathbf{B^{\prime}}x
  89. x , 𝐁 , 𝐀 x,\mathbf{B},\mathbf{A}
  90. 𝚺 y \mathbf{\Sigma}_{y}
  91. tr ( 𝚺 y ) \,\text{tr}(\mathbf{\Sigma}_{y})
  92. 𝐁 = 𝐀 q * , \mathbf{B}=\mathbf{A}_{q}^{*},
  93. 𝐀 q * \mathbf{A}_{q}^{*}
  94. 𝐀 \mathbf{A}
  95. x x
  96. x x
  97. 𝚺 \mathbf{Σ}
  98. < m t p l > Σ = λ 1 α 1 α 1 + + λ p α p α p \mathbf{<}mtpl>{{\Sigma}}=\lambda_{1}\alpha_{1}\alpha_{1}^{\prime}+\cdots+% \lambda_{p}\alpha_{p}\alpha_{p}^{\prime}
  99. Var ( x j ) = k = 1 P λ k α k j 2 \,\text{Var}(x_{j})=\sum_{k=1}^{P}\lambda_{k}\alpha_{kj}^{2}
  100. x x
  101. λ k α k α k \lambda_{k}\alpha_{k}\alpha_{k}^{\prime}
  102. λ k α k α k \lambda_{k}\alpha_{k}\alpha_{k}^{\prime}
  103. k k
  104. λ k α k α k \lambda_{k}\alpha_{k}\alpha_{k}^{\prime}
  105. k k
  106. α k \alpha_{k}
  107. α k α k = 1 , k = 1 , , p \alpha_{k}^{\prime}\alpha_{k}=1,k=1,\cdots,p
  108. 𝐱 = 𝐬 + 𝐧 \mathbf{x}=\mathbf{s}+\mathbf{n}
  109. 𝐱 \mathbf{x}
  110. 𝐬 \mathbf{s}
  111. 𝐧 \mathbf{n}
  112. 𝐬 \mathbf{s}
  113. 𝐧 \mathbf{n}
  114. I ( 𝐲 ; 𝐬 ) I(\mathbf{y};\mathbf{s})
  115. 𝐬 \mathbf{s}
  116. 𝐲 = 𝐖 L T 𝐱 \mathbf{y}=\mathbf{W}_{L}^{T}\mathbf{x}
  117. 𝐧 \mathbf{n}
  118. 𝐬 \mathbf{s}
  119. I ( 𝐱 ; 𝐬 ) - I ( 𝐲 ; 𝐬 ) . I(\mathbf{x};\mathbf{s})-I(\mathbf{y};\mathbf{s}).
  120. 𝐧 \mathbf{n}
  121. 𝐬 \mathbf{s}
  122. 𝐧 \mathbf{n}
  123. 𝐘 = 𝕂 𝕃 𝕋 { 𝐗 } \mathbf{Y}=\mathbb{KLT}\{\mathbf{X}\}
  124. 𝐱 i \mathbf{x}_{i}
  125. 𝐱 1 𝐱 n \mathbf{x}_{1}\ldots\mathbf{x}_{n}
  126. u [ j ] = 1 n i = 1 n X [ i , j ] u[j]={1\over n}\sum_{i=1}^{n}X[i,j]
  127. 𝐁 = 𝐗 - 𝐡𝐮 T \mathbf{B}=\mathbf{X}-\mathbf{h}\mathbf{u}^{T}
  128. n × 1 n×1
  129. h [ i ] = 1 for i = 1 , , n h[i]=1\,\qquad\qquad\,\text{for }i=1,\ldots,n
  130. 𝐂 = 1 n - 1 𝐁 * 𝐁 \mathbf{C}={1\over{n-1}}\mathbf{B}^{*}\cdot\mathbf{B}
  131. * *
  132. N 1 N−1
  133. 𝐕 - 1 𝐂𝐕 = 𝐃 \mathbf{V}^{-1}\mathbf{C}\mathbf{V}=\mathbf{D}
  134. D [ k , l ] = λ k for k = l D[k,l]=\lambda_{k}\qquad\,\text{for }k=l
  135. D [ k , l ] = 0 for k l . D[k,l]=0\qquad\,\text{for }k\neq l.
  136. g [ j ] = k = 1 j D [ k , k ] for j = 1 , , p g[j]=\sum_{k=1}^{j}D[k,k]\qquad\mathrm{for}\qquad j=1,\dots,p
  137. W [ k , l ] = V [ k , l ] for k = 1 , , p l = 1 , , L W[k,l]=V[k,l]\qquad\mathrm{for}\qquad k=1,\dots,p\qquad l=1,\dots,L
  138. 1 L p . 1\leq L\leq p.
  139. g [ L ] g [ p ] 0.9 \frac{g[L]}{g[p]}\geq 0.9\,
  140. 𝐬 = { s [ j ] } = { C [ j , j ] } for j = 1 , , p \mathbf{s}=\{s[j]\}=\{\sqrt{C[j,j]}\}\qquad\,\text{for }j=1,\ldots,p
  141. 𝐙 = 𝐁 𝐡 𝐬 T \mathbf{Z}={\mathbf{B}\over\mathbf{h}\cdot\mathbf{s}^{T}}
  142. 𝐓 = 𝐙 𝐖 = 𝕂 𝕃 𝕋 { 𝐗 } . \mathbf{T}=\mathbf{Z}\cdot\mathbf{W}=\mathbb{KLT}\{\mathbf{X}\}.
  143. ( ) (\ast)\,
  144. d × d d×d
  145. P P
  146. var ( P X ) \displaystyle\operatorname{var}(PX)
  147. ( ) (\ast)\,
  148. var ( X ) \operatorname{var}(X)
  149. P P
  150. 𝐫 = \mathbf{r}=
  151. 𝐬 = 0 \mathbf{s}=0
  152. 𝐱 𝐗 \mathbf{x}\in\mathbf{X}
  153. 𝐬 = 𝐬 + ( 𝐱 𝐫 ) 𝐱 \mathbf{s}=\mathbf{s}+(\mathbf{x}\cdot\mathbf{r})\mathbf{x}
  154. 𝐫 = 𝐬 | 𝐬 | \mathbf{r}=\frac{\mathbf{s}}{|\mathbf{s}|}
  155. 𝐫 \mathbf{r}
  156. < v a r > k <var>k

Principle_of_indifference.html

  1. f ( L ) = | V L | f ( V ) = 3 L 2 f ( L 3 ) f(L)=|{\partial V\over\partial L}|f(V)=3L^{2}f(L^{3})
  2. f ( L ) = K L f(L)={K\over L}
  3. K - 1 = 3 5 d L L = log ( 5 3 ) K^{-1}=\int_{3}^{5}{dL\over L}=\log({5\over 3})
  4. P r ( L < 4 ) = 3 4 d L L log ( 5 3 ) = log ( 4 3 ) log ( 5 3 ) 0.56 Pr(L<4)=\int_{3}^{4}{dL\over L\log({5\over 3})}={\log({4\over 3})\over\log({5% \over 3})}\approx 0.56
  5. f ( V < m t p l > 1 3 ) 1 3 V - 2 3 = 1 3 V log ( 5 3 ) f(V^{<}mtpl>{{1\over 3}}){1\over 3}V^{-{2\over 3}}={1\over 3V\log({5\over 3})}
  6. P r ( V < 64 ) = 27 64 d V 3 V log ( 5 3 ) = log ( 64 27 ) 3 log ( 5 3 ) = 3 log ( 4 3 ) 3 log ( 5 3 ) = log ( 4 3 ) log ( 5 3 ) 0.56 Pr(V<64)=\int_{27}^{64}{dV\over 3V\log({5\over 3})}={\log({64\over 27})\over 3% \log({5\over 3})}={3\log({4\over 3})\over 3\log({5\over 3})}={\log({4\over 3})% \over\log({5\over 3})}\approx 0.56

Principle_of_least_action.html

  1. 𝒮 \mathcal{S}
  2. 𝒮 [ 𝐪 ( t ) ] = t 1 t 2 L ( 𝐪 ( t ) , 𝐪 ˙ ( t ) , t ) d t \mathcal{S}[\mathbf{q}(t)]=\int_{t_{1}}^{t_{2}}L(\mathbf{q}(t),\mathbf{\dot{q}% }(t),t)dt
  3. δ 𝒮 = 0 \delta\mathcal{S}=0
  4. δ t 1 t 2 L ( 𝐪 , 𝐪 ˙ , t ) d t = 0 \delta\int_{t_{1}}^{t_{2}}L(\mathbf{q},\mathbf{\dot{q}},t)dt=0
  5. L = T - V L=T-V
  6. 𝐅 = m 𝐚 \mathbf{F}=m\mathbf{a}

Principle_of_maximum_entropy.html

  1. p i = 1 n for all i { 1 , , n } . p_{i}=\frac{1}{n}\ {\rm for\ all}\ i\in\{\,1,\dots,n\,\}.
  2. i = 1 n Pr ( x i ) f k ( x i ) = F k k = 1 , , m . \sum_{i=1}^{n}\Pr(x_{i})f_{k}(x_{i})=F_{k}\qquad k=1,\ldots,m.
  3. i = 1 n Pr ( x i ) = 1. \sum_{i=1}^{n}\Pr(x_{i})=1.
  4. Pr ( x i ) = 1 Z ( λ 1 , , λ m ) exp [ λ 1 f 1 ( x i ) + + λ m f m ( x i ) ] . \Pr(x_{i})=\frac{1}{Z(\lambda_{1},\ldots,\lambda_{m})}\exp\left[\lambda_{1}f_{% 1}(x_{i})+\cdots+\lambda_{m}f_{m}(x_{i})\right].
  5. Z ( λ 1 , , λ m ) = i = 1 n exp [ λ 1 f 1 ( x i ) + + λ m f m ( x i ) ] , Z(\lambda_{1},\ldots,\lambda_{m})=\sum_{i=1}^{n}\exp\left[\lambda_{1}f_{1}(x_{% i})+\cdots+\lambda_{m}f_{m}(x_{i})\right],
  6. F k = λ k log Z ( λ 1 , , λ m ) . F_{k}=\frac{\partial}{\partial\lambda_{k}}\log Z(\lambda_{1},\ldots,\lambda_{m% }).
  7. H c = - p ( x ) log p ( x ) m ( x ) d x H_{c}=-\int p(x)\log\frac{p(x)}{m(x)}\,dx
  8. p ( x ) f k ( x ) d x = F k k = 1 , , m . \int p(x)f_{k}(x)dx=F_{k}\qquad k=1,\ldots,m.
  9. p ( x ) d x = 1. \int p(x)dx=1.
  10. p ( x ) = 1 Z ( λ 1 , , λ m ) m ( x ) exp [ λ 1 f 1 ( x ) + + λ m f m ( x ) ] p(x)=\frac{1}{Z(\lambda_{1},\ldots,\lambda_{m})}m(x)\exp\left[\lambda_{1}f_{1}% (x)+\cdots+\lambda_{m}f_{m}(x)\right]
  11. Z ( λ 1 , , λ m ) = m ( x ) exp [ λ 1 f 1 ( x ) + + λ m f m ( x ) ] d x . Z(\lambda_{1},\ldots,\lambda_{m})=\int m(x)\exp\left[\lambda_{1}f_{1}(x)+% \cdots+\lambda_{m}f_{m}(x)\right]dx.
  12. λ k \lambda_{k}
  13. F k = λ k log Z ( λ 1 , , λ m ) . F_{k}=\frac{\partial}{\partial\lambda_{k}}\log Z(\lambda_{1},\ldots,\lambda_{m% }).
  14. p ( x ) = A m ( x ) , a < x < b p(x)=A\cdot m(x),\qquad a<x<b
  15. p i = n i N p_{i}=\frac{n_{i}}{N}
  16. P r ( 𝐩 ) = W m - N Pr(\mathbf{p})=W\cdot m^{-N}
  17. W = N ! n 1 ! n 2 ! n m ! W=\frac{N!}{n_{1}!\,n_{2}!\,\cdots\,n_{m}!}
  18. 1 N log W = 1 N log N ! n 1 ! n 2 ! n m ! = 1 N log N ! ( N p 1 ) ! ( N p 2 ) ! ( N p m ) ! = 1 N ( log N ! - i = 1 m log ( ( N p i ) ! ) ) . \begin{array}[]{rcl}\frac{1}{N}\log W&=&\frac{1}{N}\log\frac{N!}{n_{1}!\,n_{2}% !\,\cdots\,n_{m}!}\\ \\ &=&\frac{1}{N}\log\frac{N!}{(Np_{1})!\,(Np_{2})!\,\cdots\,(Np_{m})!}\\ \\ &=&\frac{1}{N}\left(\log N!-\sum_{i=1}^{m}\log((Np_{i})!)\right).\end{array}
  19. N N\to\infty
  20. lim N ( 1 N log W ) = 1 N ( N log N - i = 1 m N p i log ( N p i ) ) = log N - i = 1 m p i log ( N p i ) = log N - log N i = 1 m p i - i = 1 m p i log p i = ( 1 - i = 1 m p i ) log N - i = 1 m p i log p i = - i = 1 m p i log p i = H ( 𝐩 ) . \begin{array}[]{rcl}\lim_{N\to\infty}\left(\frac{1}{N}\log W\right)&=&\frac{1}% {N}\left(N\log N-\sum_{i=1}^{m}Np_{i}\log(Np_{i})\right)\\ \\ &=&\log N-\sum_{i=1}^{m}p_{i}\log(Np_{i})\\ \\ &=&\log N-\log N\sum_{i=1}^{m}p_{i}-\sum_{i=1}^{m}p_{i}\log p_{i}\\ \\ &=&\left(1-\sum_{i=1}^{m}p_{i}\right)\log N-\sum_{i=1}^{m}p_{i}\log p_{i}\\ \\ &=&-\sum_{i=1}^{m}p_{i}\log p_{i}\\ \\ &=&H(\mathbf{p}).\end{array}
  21. N N

Prism.html

  1. θ 0 = arcsin ( n 0 n 1 sin θ 0 ) θ 1 = α - θ 0 θ 1 = arcsin ( n 1 n 2 sin θ 1 ) θ 2 = θ 1 - α \begin{aligned}\displaystyle\theta^{\prime}_{0}&\displaystyle=\,\,\text{arcsin% }\Big(\frac{n_{0}}{n_{1}}\,\sin\theta_{0}\Big)\\ \displaystyle\theta_{1}&\displaystyle=\alpha-\theta^{\prime}_{0}\\ \displaystyle\theta^{\prime}_{1}&\displaystyle=\,\,\text{arcsin}\Big(\frac{n_{% 1}}{n_{2}}\,\sin\theta_{1}\Big)\\ \displaystyle\theta_{2}&\displaystyle=\theta^{\prime}_{1}-\alpha\end{aligned}
  2. n 0 = n 2 1 n_{0}=n_{2}\simeq 1
  3. n = n 1 n=n_{1}
  4. δ \delta
  5. δ = θ 0 + θ 2 = θ 0 + arcsin ( n sin [ α - arcsin ( 1 n sin θ 0 ) ] ) - α \delta=\theta_{0}+\theta_{2}=\theta_{0}+\,\text{arcsin}\Big(n\,\sin\Big[\alpha% -\,\text{arcsin}\Big(\frac{1}{n}\,\sin\theta_{0}\Big)\Big]\Big)-\alpha
  6. θ 0 \theta_{0}
  7. α \alpha
  8. sin θ θ \sin\theta\approx\theta
  9. arcsin x x \,\text{arcsin}x\approx x
  10. δ \delta
  11. δ θ 0 - α + ( n [ ( α - 1 n θ 0 ) ] ) = θ 0 - α + n α - θ 0 = ( n - 1 ) α . \delta\approx\theta_{0}-\alpha+\Big(n\,\Big[\Big(\alpha-\frac{1}{n}\,\theta_{0% }\Big)\Big]\Big)=\theta_{0}-\alpha+n\alpha-\theta_{0}=(n-1)\alpha\ .
  12. δ ( λ ) [ n ( λ ) - 1 ] α \delta(\lambda)\approx[n(\lambda)-1]\alpha

Prism_(geometry).html

  1. V = B h V=B\cdot h
  2. V = n 4 h s 2 cot π n . V=\frac{n}{4}hs^{2}\cot\frac{\pi}{n}.
  3. A = n 2 s 2 cot π n + n s h . A=\frac{n}{2}s^{2}\cot{\frac{\pi}{n}}+nsh.

Probabilistic_method.html

  1. R ( r , r ) R(r,r)
  2. n n
  3. n n
  4. r r
  5. 1 / 2 1/2
  6. 1 / 2 1/2
  7. r r
  8. S S
  9. r r
  10. X ( S ) X(S)
  11. 1 1
  12. r r
  13. 0
  14. r r
  15. X ( S ) X(S)
  16. S S
  17. X ( S ) X(S)
  18. ( r 2 ) {r\choose 2}
  19. S S
  20. 2 2 - ( r 2 ) 2\cdot 2^{-{r\choose 2}}
  21. 2 2
  22. C ( n , r ) C(n,r)
  23. E X X ( S ) EXX(S)
  24. S S
  25. ( n r ) 2 1 - ( r 2 ) . {n\choose r}2^{1-{r\choose 2}}.
  26. r r
  27. ( n r ) 2 1 - ( r 2 ) . {n\choose r}2^{1-{r\choose 2}}.
  28. 1 1
  29. r r
  30. 0
  31. ( n r ) < 2 ( r 2 ) - 1 , {n\choose r}<2^{{r\choose 2}-1},
  32. n n
  33. r r
  34. ( n r ) {n\choose r}
  35. ( r 2 ) {r\choose 2}
  36. 2 ( r 2 ) 2^{r\choose 2}
  37. 2 ( n r ) 2{n\choose r}
  38. 2 ( r 2 ) > 2 ( n r ) 2^{r\choose 2}>2{n\choose r}
  39. R ( r , r ) R(r,r)
  40. n n
  41. R ( r , r ) R(r,r)
  42. r r
  43. r r
  44. g g
  45. k k
  46. G G
  47. g g
  48. G G
  49. k k
  50. g g
  51. k k
  52. n n
  53. G G
  54. n n
  55. G G
  56. G G
  57. n / 2 n/2
  58. g g
  59. X X
  60. g g
  61. i i
  62. n n
  63. n ! 2 i ( n - i ) ! n i 2 \frac{n!}{2\cdot i\cdot(n-i)!}\leq\frac{n^{i}}{2}
  64. G G
  65. Pr ( X > n 2 ) 2 n E [ X ] 1 n i = 3 g - 1 p i n i = 1 n i = 3 g - 1 n i g g n n g - 1 g g n - 1 g = o ( 1 ) . \Pr\left(X>\tfrac{n}{2}\right)\leq\frac{2}{n}E[X]\leq\frac{1}{n}\sum_{i=3}^{g-% 1}p^{i}n^{i}=\frac{1}{n}\sum_{i=3}^{g-1}n^{\frac{i}{g}}\leq\frac{g}{n}n^{\frac% {g-1}{g}}\leq gn^{-\frac{1}{g}}=o(1).
  66. G G
  67. n 2 k \lceil\tfrac{n}{2k}\rceil
  68. Y Y
  69. G G
  70. Pr ( Y y ) ( n y ) ( 1 - p ) y ( y - 1 ) 2 n y e - p y ( y - 1 ) 2 = e - y 2 ( p y - 2 ln n - p ) = o ( 1 ) , \Pr(Y\geq y)\leq{n\choose y}(1-p)^{\frac{y(y-1)}{2}}\leq n^{y}e^{-\frac{py(y-1% )}{2}}=e^{-\frac{y}{2}\cdot(py-2\ln n-p)}=o(1),
  71. y = n 2 k . y=\left\lceil\frac{n}{2k}\right\rceil.
  72. G G
  73. n / 2 n/2
  74. G G
  75. G G′
  76. n n′
  77. g g
  78. n k \lceil\frac{n^{\prime}}{k}\rceil
  79. G G′
  80. k k