wpmath0000005_9

Next-bit_test.html

  1. i i
  2. i i
  3. ( i + 1 ) (i+1)
  4. P P
  5. S = { S k } S=\{S_{k}\}
  6. S k S_{k}
  7. P ( k ) P(k)
  8. ฮผ k \mu_{k}
  9. S k S_{k}
  10. C = { C k i } C=\{C_{k}^{i}\}
  11. C k i C_{k}^{i}
  12. P C ( k ) P_{C}(k)
  13. i i
  14. p k , i C p_{k,i}^{C}
  15. i i
  16. s s
  17. S k S_{k}
  18. ฮผ k ( s ) \mu_{k}(s)
  19. s i + 1 s_{i+1}
  20. p k , i C = ๐’ซ [ C k ( s 1 โ€ฆ s i ) = s i + 1 | s โˆˆ S k with probability ฮผ k ( s ) ] p_{k,i}^{C}={\mathcal{P}}\left[C_{k}(s_{1}\ldots s_{i})=s_{i+1}\right|s\in S_{% k}\,\text{ with probability }\mu_{k}(s)]
  21. { S k } k \{S_{k}\}_{k}
  22. C C
  23. Q Q
  24. p k , i C < 1 2 + 1 Q ( k ) p_{k,i}^{C}<\frac{1}{2}+\frac{1}{Q(k)}
  25. โ„ณ \mathcal{M}
  26. p k , i โ„ณ p_{k,i}^{\mathcal{M}}
  27. โ„ณ \mathcal{M}
  28. ( i + 1 ) (i+1)
  29. p k , i โ„ณ = ๐’ซ [ M ( s 1 โ€ฆ s i ) = s i + 1 | s โˆˆ S k with probability ฮผ k ( s ) ] p_{k,i}^{\mathcal{M}}={\mathcal{P}}[M(s_{1}\ldots s_{i})=s_{i+1}|s\in S_{k}\,% \text{ with probability }\mu_{k}(s)]
  30. S = { S k } S=\{S_{k}\}
  31. Q Q
  32. k k
  33. 0 < i < k 0<i<k
  34. p k , i โ„ณ < 1 2 + 1 Q ( k ) p_{k,i}^{\mathcal{M}}<\frac{1}{2}+\frac{1}{Q(k)}
  35. โ„ณ \mathcal{M}
  36. Q Q
  37. k k
  38. | p k , S โ„ณ - p k , U โ„ณ | โ‰ฅ 1 Q ( k ) |p_{k,S}^{\mathcal{M}}-p_{k,U}^{\mathcal{M}}|\geq\frac{1}{Q(k)}
  39. R k , i = { s 1 โ€ฆ s i u i + 1 โ€ฆ u P ( k ) | s โˆˆ S k , u โˆˆ { 0 , 1 } P ( k ) } R_{k,i}=\{s_{1}\ldots s_{i}u_{i+1}\ldots u_{P(k)}|s\in S_{k},u\in\{0,1\}^{P(k)}\}
  40. R k , 0 = { 0 , 1 } P ( k ) R_{k,0}=\{0,1\}^{P(k)}
  41. R k , P ( k ) = S k R_{k,P(k)}=S_{k}
  42. โˆ‘ i = 0 P ( k ) | p k , R k , i + 1 โ„ณ - p k , R k , i โ„ณ | โ‰ฅ | p k , R k , P ( k ) โ„ณ - p k , R k , 0 โ„ณ | = | p k , S โ„ณ - p k , U โ„ณ | โ‰ฅ 1 Q ( k ) \sum_{i=0}^{P(k)}|p_{k,R_{k,i+1}}^{\mathcal{M}}-p_{k,R_{k,i}}^{\mathcal{M}}|% \geq|p^{\mathcal{M}}_{k,R_{k,P(k)}}-p^{\mathcal{M}}_{k,R_{k,0}}|=|p_{k,S}^{% \mathcal{M}}-p_{k,U}^{\mathcal{M}}|\geq\frac{1}{Q(k)}
  43. | p k , R k , i + 1 โ„ณ - p k , R k , i โ„ณ | |p_{k,R_{k,i+1}}^{\mathcal{M}}-p_{k,R_{k,i}}^{\mathcal{M}}|
  44. 1 Q ( k ) P ( k ) \frac{1}{Q(k)P(k)}
  45. ฮผ k , i \mu_{k,i}
  46. ฮผ k , i ยฏ \overline{\mu_{k,i}}
  47. R k , i R_{k,i}
  48. ฮผ k , i \mu_{k,i}
  49. i i
  50. S k S_{k}
  51. ฮผ k \mu_{k}
  52. P ( k ) - i P(k)-i
  53. ฮผ k , i ( w 1 โ€ฆ w P ( k ) ) = ( โˆ‘ s โˆˆ S k , s 1 โ€ฆ s i = w 1 โ€ฆ w i ฮผ k ( s ) ) ( 1 2 ) P ( k ) - i \mu_{k,i}(w_{1}\ldots w_{P(k)})=\left(\sum_{s\in S_{k},s_{1}\ldots s_{i}=w_{1}% \ldots w_{i}}\mu_{k}(s)\right)\left(\frac{1}{2}\right)^{P(k)-i}
  54. ฮผ k , i ยฏ ( w 1 โ€ฆ w P ( k ) ) = ( โˆ‘ s โˆˆ S k , s 1 โ€ฆ s i - 1 ( 1 - s i ) = w 1 โ€ฆ w i ฮผ k ( s ) ) ( 1 2 ) P ( k ) - i \overline{\mu_{k,i}}(w_{1}\ldots w_{P(k)})=\left(\sum_{s\in S_{k},s_{1}\ldots s% _{i-1}(1-s_{i})=w_{1}\ldots w_{i}}\mu_{k}(s)\right)\left(\frac{1}{2}\right)^{P% (k)-i}
  55. ฮผ k , i = 1 2 ( ฮผ k , i + 1 + ฮผ k , i + 1 ยฏ ) \mu_{k,i}=\frac{1}{2}(\mu_{k,i+1}+\overline{\mu_{k,i+1}})
  56. ฮผ k , i + 1 \mu_{k,i+1}
  57. ฮผ k , i + 1 ยฏ \overline{\mu_{k,i+1}}
  58. โ„ณ \mathcal{M}
  59. p ฮผ k , i + 1 โ„ณ - p ฮผ k , i + 1 ยฏ โ„ณ โ‰ฅ 1 2 + 1 R ( k ) p^{\mathcal{M}}_{\mu_{k,i+1}}-p^{\mathcal{M}}_{\overline{\mu_{k,i+1}}}\geq% \frac{1}{2}+\frac{1}{R(k)}
  60. R R
  61. i i
  62. ๐’ฉ \mathcal{N}
  63. l l
  64. P ( k ) - i - 1 P(k)-i-1
  65. โ„ณ \mathcal{M}
  66. l l
  67. 1 1
  68. 1 - l 1-l

Neฬronโ€“Severi_group.html

  1. 1 โ†’ Pic 0 ( V ) โ†’ Pic ( V ) โ†’ NS ( V ) โ†’ 0 1\to\mathrm{Pic}^{0}(V)\to\mathrm{Pic}(V)\to\mathrm{NS}(V)\to 0
  2. 0 โ†’ 2 ฯ€ i โ„ค โ†’ ๐’ช V โ†’ ๐’ช V * โ†’ 0 0\to 2\pi i\mathbb{Z}\to\mathcal{O}_{V}\to\mathcal{O}_{V}^{*}\to 0
  3. โ‹ฏ โ†’ H 1 ( V , ๐’ช V * ) โ†’ H 2 ( V , โ„ค ) โ†’ H 2 ( V , ๐’ช V ) โ†’ โ‹ฏ . \cdots\to H^{1}(V,\mathcal{O}_{V}^{*})\to H^{2}(V,\mathbb{Z})\to H^{2}(V,% \mathcal{O}_{V})\to\cdots.
  4. c 1 : Pic ( V ) โ†’ H 2 ( V , โ„ค ) , c_{1}:\mathrm{Pic}(V)\to H^{2}(V,\mathbb{Z}),
  5. exp * : H 2 ( V , 2 i ฯ€ โ„ค ) โ†’ H 2 ( V , ๐’ช V ) . \exp^{*}:H^{2}(V,2i\pi\mathbb{Z})\to H^{2}(V,\mathcal{O}_{V}).

Nicholas_Mercator.html

  1. ln ( 1 + x ) = x - 1 2 x 2 + 1 3 x 3 - 1 4 x 4 + โ‹ฏ . \ln(1+x)=x-\frac{1}{2}x^{2}+\frac{1}{3}x^{3}-\frac{1}{4}x^{4}+\cdots.

Nichols_plot.html

  1. G ( s ) = Y ( s ) X ( s ) G(s)=\frac{Y(s)}{X(s)}
  2. M ( s ) = G ( s ) ( 1 + G ( s ) ) M(s)=\frac{G(s)}{(1+G(s))}
  3. 20 log 10 ( | G ( s ) | ) 20\log_{10}(|G(s)|)
  4. arg ( G ( s ) ) \arg(G(s))
  5. 20 log 10 ( | M ( s ) | ) 20\log_{10}(|M(s)|)
  6. arg ( M ( s ) ) \arg(M(s))
  7. ฯ‰ \omega
  8. 20 log 10 ( | G ( s ) | ) 20\log_{10}(|G(s)|)
  9. log 10 ( ฯ‰ ) \log_{10}(\omega)
  10. arg ( G ( s ) ) \arg(G(s))
  11. log 10 ( ฯ‰ ) \log_{10}(\omega)
  12. arg ( G ( s ) ) \arg(G(s))

Nick_Katz.html

  1. L L

Niemeier_lattice.html

  1. 4 \sqrt{4}
  2. 6 \sqrt{6}
  3. 8 \sqrt{8}

No-communication_theorem.html

  1. H = H A โŠ— H B . H=H_{A}\otimes H_{B}.
  2. ฯƒ = โˆ‘ i T i โŠ— S i \sigma=\sum_{i}T_{i}\otimes S_{i}
  3. P ( ฯƒ ) = โˆ‘ k ( V k โŠ— I H B ) * ฯƒ ( V k โŠ— I H B ) , P(\sigma)=\sum_{k}(V_{k}\otimes I_{H_{B}})^{*}\ \sigma\ (V_{k}\otimes I_{H_{B}% }),
  4. โˆ‘ k V k V k * = I H A . \sum_{k}V_{k}V_{k}^{*}=I_{H_{A}}.
  5. I H B I_{H_{B}}
  6. ( V k โŠ— I H B ) (V_{k}\otimes I_{H_{B}})
  7. tr H A ( P ( ฯƒ ) ) \operatorname{tr}_{H_{A}}(P(\sigma))
  8. tr H A \operatorname{tr}_{H_{A}}
  9. tr H A ( P ( ฯƒ ) ) = tr H A ( โˆ‘ k ( V k โŠ— I H B ) * ฯƒ ( V k โŠ— I H B ) ) \operatorname{tr}_{H_{A}}(P(\sigma))=\operatorname{tr}_{H_{A}}\left(\sum_{k}(V% _{k}\otimes I_{H_{B}})^{*}\sigma(V_{k}\otimes I_{H_{B}})\right)
  10. = tr H A ( โˆ‘ k โˆ‘ i V k * T i V k โŠ— S i ) =\operatorname{tr}_{H_{A}}\left(\sum_{k}\sum_{i}V_{k}^{*}T_{i}V_{k}\otimes S_{% i}\right)
  11. = โˆ‘ i โˆ‘ k tr ( V k * T i V k ) S i =\sum_{i}\sum_{k}\operatorname{tr}(V_{k}^{*}T_{i}V_{k})S_{i}
  12. = โˆ‘ i โˆ‘ k tr ( T i V k V k * ) S i =\sum_{i}\sum_{k}\operatorname{tr}(T_{i}V_{k}V_{k}^{*})S_{i}
  13. = โˆ‘ i tr ( T i โˆ‘ k V k V k * ) S i =\sum_{i}\operatorname{tr}\left(T_{i}\sum_{k}V_{k}V_{k}^{*}\right)S_{i}
  14. = โˆ‘ i tr ( T i ) S i =\sum_{i}\operatorname{tr}(T_{i})S_{i}
  15. = tr H A ( ฯƒ ) . =\operatorname{tr}_{H_{A}}(\sigma).
  16. P ( ฯƒ ) P(\sigma)
  17. | z + โŸฉ B |z+\rangle_{B}
  18. | z - โŸฉ B |z-\rangle_{B}
  19. | z + โŸฉ B |z+\rangle_{B}
  20. | z - โŸฉ B |z-\rangle_{B}

No_free_lunch_in_search_and_optimization.html

  1. Y X Y^{X}
  2. X X
  3. Y Y
  4. Y X Y^{X}
  5. Y X Y^{X}
  6. Y X Y^{X}
  7. X X
  8. Y Y
  9. โˆ‘ f P ( d m y | f , m , a 1 ) = โˆ‘ f P ( d m y | f , m , a 2 ) , \sum_{f}P(d_{m}^{y}|f,m,a_{1})=\sum_{f}P(d_{m}^{y}|f,m,a_{2}),
  10. d m y d_{m}^{y}
  11. m m
  12. y โˆˆ Y y\in Y
  13. x โˆˆ X x\in X
  14. f : X โ†’ Y f:X\rightarrow Y
  15. P ( d m y | f , m , a ) P(d_{m}^{y}|f,m,a)
  16. a a
  17. m m
  18. f f

No_free_lunch_theorem.html

  1. โˆ‘ f P ( d m y | f , m , a 1 ) = โˆ‘ f P ( d m y | f , m , a 2 ) , \sum_{f}P(d_{m}^{y}|f,m,a_{1})=\sum_{f}P(d_{m}^{y}|f,m,a_{2}),
  2. d m y d_{m}^{y}
  3. m m
  4. y y
  5. x โˆˆ X x\in X
  6. f : X โ†’ Y f:X\rightarrow Y
  7. P ( d m y | f , m , a ) P(d_{m}^{y}|f,m,a)
  8. a a
  9. m m
  10. f f
  11. V V
  12. S S
  13. f : V โ†’ S f:V\to S
  14. V S V^{S}
  15. V V
  16. S S
  17. f f
  18. V V
  19. v โˆˆ V v\in V
  20. V V

Noise_shaping.html

  1. y [ n ] = x [ n ] + e [ n - 1 ] , \ y[n]=x[n]+e[n-1],
  2. y [ n ] = x [ n ] + A 1 e [ n - 1 ] \ y[n]=x[n]+A_{1}e[n-1]
  3. y [ n ] = x [ n ] + โˆ‘ i = 1 9 A i e [ n - i ] \ y[n]=x[n]+\sum_{i=1}^{9}A_{i}e[n-i]
  4. y [ n ] = x [ n ] + A 1 e [ n - 1 ] + dither , \ y[n]=x[n]+A_{1}e[n-1]+\mathrm{dither},

Nome_(mathematics).html

  1. q = e - ฯ€ K โ€ฒ K = e i ฯ€ ฯ‰ 2 ฯ‰ 1 = e i ฯ€ ฯ„ q=e^{-\frac{\pi K^{\prime}}{K}}=e^{\frac{{\rm{i}}\pi\omega_{2}}{\omega_{1}}}=e% ^{{\rm{i}}\pi\tau}\,
  2. q 1 = e - ฯ€ K K โ€ฒ . q_{1}=e^{-\frac{\pi K}{K^{\prime}}}.\,

Non-circular_gear.html

  1. r 1 ( ฮธ 1 ) r_{1}(\theta_{1})
  2. ฮธ 1 \theta_{1}
  3. r 2 ( ฮธ 2 ) r_{2}(\theta_{2})
  4. ฮธ 2 \theta_{2}
  5. r 1 ( ฮธ 1 ) + r 2 ( ฮธ 2 ) = a r_{1}(\theta_{1})+r_{2}(\theta_{2})=a\,
  6. r 1 d ฮธ 1 = r 2 d ฮธ 2 r_{1}\,d\theta_{1}=r_{2}\,d\theta_{2}
  7. z = e i ฮธ z=e^{i\theta}
  8. d z = i z d ฮธ dz=iz\,d\theta
  9. d z 2 z 2 = r 1 ( z 1 ) a - r 1 ( z 1 ) d z 1 z 1 \frac{dz_{2}}{z_{2}}=\frac{r_{1}(z_{1})}{a-r_{1}(z_{1})}\,\frac{dz_{1}}{z_{1}}
  10. z 1 z_{1}
  11. z 2 z_{2}
  12. ln ( z 2 ) = ln ( K ) + โˆซ r 1 ( z 1 ) a - r 1 ( z 1 ) d z 1 z 1 \ln(z_{2})=\ln(K)+\int\frac{r_{1}(z_{1})}{a-r_{1}(z_{1})}\,\frac{dz_{1}}{z_{1}}
  13. ln ( K ) \ln(K)

Non-linear_sigma_model.html

  1. โ„’ = 1 2 g ( โˆ‚ ฮผ ฮฃ , โˆ‚ ฮผ ฮฃ ) - V ( ฮฃ ) \mathcal{L}={1\over 2}g(\partial^{\mu}\Sigma,\partial_{\mu}\Sigma)-V(\Sigma)
  2. โˆ‚ ฮฃ \partial\Sigma
  3. โ„’ = 1 2 g a b ( ฮฃ ) ( โˆ‚ ฮผ ฮฃ a ) ( โˆ‚ ฮผ ฮฃ b ) - V ( ฮฃ ) . \mathcal{L}={1\over 2}g_{ab}(\Sigma)(\partial^{\mu}\Sigma^{a})(\partial_{\mu}% \Sigma^{b})-V(\Sigma).
  4. ฯƒ ฯƒ
  5. det g ๐’Ÿ ฮฃ . \sqrt{\det g}\mathcal{D}\Sigma.
  6. ฮป โˆ‚ g a b โˆ‚ ฮป = ฮฒ a b ( T - 1 g ) = R a b + O ( T 2 ) . \lambda\frac{\partial g_{ab}}{\partial\lambda}=\beta_{ab}(T^{-1}g)=R_{ab}+O(T^% {2}).
  7. R a b R_{ab}
  8. ฯƒ ฯƒ
  9. โ„’ = 1 2 โˆ‚ ฮผ n ^ โ‹… โˆ‚ ฮผ n ^ \mathcal{L}=\tfrac{1}{2}\ \partial^{\mu}\hat{n}\cdot\partial_{\mu}\hat{n}
  10. n ^ = ( n 1 , n 2 , n 3 ) \hat{n}=(n_{1},n_{2},n_{3})
  11. n ^ โ‹… n ^ = 1 \hat{n}\cdot\hat{n}=1
  12. ฮผ ฮผ
  13. n ^ = const. \hat{n}=\textrm{const.}
  14. n ^ \hat{n}
  15. S 2 โ†’ S 2 S^{2}\rightarrow S^{2}

Non-perturbative.html

  1. f ( x ) = e - 1 / x 2 f(x)=e^{-1/x^{2}}

Noncentral_F-distribution.html

  1. X X
  2. ฮป \lambda
  3. ฮฝ 1 \nu_{1}
  4. Y Y
  5. ฮฝ 2 \nu_{2}
  6. X X
  7. F = X / ฮฝ 1 Y / ฮฝ 2 F=\frac{X/\nu_{1}}{Y/\nu_{2}}
  8. p ( f ) = โˆ‘ k = 0 โˆž e - ฮป / 2 ( ฮป / 2 ) k B ( ฮฝ 2 2 , ฮฝ 1 2 + k ) k ! ( ฮฝ 1 ฮฝ 2 ) ฮฝ 1 2 + k ( ฮฝ 2 ฮฝ 2 + ฮฝ 1 f ) ฮฝ 1 + ฮฝ 2 2 + k f ฮฝ 1 / 2 - 1 + k p(f)=\sum\limits_{k=0}^{\infty}\frac{e^{-\lambda/2}(\lambda/2)^{k}}{B\left(% \frac{\nu_{2}}{2},\frac{\nu_{1}}{2}+k\right)k!}\left(\frac{\nu_{1}}{\nu_{2}}% \right)^{\frac{\nu_{1}}{2}+k}\left(\frac{\nu_{2}}{\nu_{2}+\nu_{1}f}\right)^{% \frac{\nu_{1}+\nu_{2}}{2}+k}f^{\nu_{1}/2-1+k}
  9. f โ‰ฅ 0 f\geq 0
  10. ฮฝ 1 \nu_{1}
  11. ฮฝ 2 \nu_{2}
  12. ฮป \lambda
  13. B ( x , y ) B(x,y)
  14. B ( x , y ) = ฮ“ ( x ) ฮ“ ( y ) ฮ“ ( x + y ) . B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}.
  15. F ( x | d 1 , d 2 , ฮป ) = โˆ‘ j = 0 โˆž ( ( 1 2 ฮป ) j j ! e - ฮป 2 ) I ( d 1 x d 2 + d 1 x | d 1 2 + j , d 2 2 ) F(x|d_{1},d_{2},\lambda)=\sum\limits_{j=0}^{\infty}\left(\frac{\left(\frac{1}{% 2}\lambda\right)^{j}}{j!}e^{-\frac{\lambda}{2}}\right)I\left(\frac{d_{1}x}{d_{% 2}+d_{1}x}\bigg|\frac{d_{1}}{2}+j,\frac{d_{2}}{2}\right)
  16. I I
  17. E [ F ] = { ฮฝ 2 ( ฮฝ 1 + ฮป ) ฮฝ 1 ( ฮฝ 2 - 2 ) ฮฝ 2 > 2 Does not exist ฮฝ 2 โ‰ค 2 \operatorname{E}\left[F\right]=\begin{cases}\frac{\nu_{2}(\nu_{1}+\lambda)}{% \nu_{1}(\nu_{2}-2)}&\nu_{2}>2\\ \,\text{Does not exist}&\nu_{2}\leq 2\\ \end{cases}
  18. Var [ F ] = { 2 ( ฮฝ 1 + ฮป ) 2 + ( ฮฝ 1 + 2 ฮป ) ( ฮฝ 2 - 2 ) ( ฮฝ 2 - 2 ) 2 ( ฮฝ 2 - 4 ) ( ฮฝ 2 ฮฝ 1 ) 2 ฮฝ 2 > 4 Does not exist ฮฝ 2 โ‰ค 4. \operatorname{Var}\left[F\right]=\begin{cases}2\frac{(\nu_{1}+\lambda)^{2}+(% \nu_{1}+2\lambda)(\nu_{2}-2)}{(\nu_{2}-2)^{2}(\nu_{2}-4)}\left(\frac{\nu_{2}}{% \nu_{1}}\right)^{2}&\nu_{2}>4\\ \,\text{Does not exist}&\nu_{2}\leq 4.\\ \end{cases}
  19. { 4 x ( ฮฝ 2 + ฮฝ 1 x ) f โ€ฒโ€ฒ 2 ( x ) + f โ€ฒ ( x ) ( - 2 ฮฝ 2 2 ฮฝ 1 + 8 ฮฝ 2 2 + 16 ฮฝ 1 2 x 2 + 4 ฮฝ 2 ฮฝ 1 2 x 2 - 2 ฮป ฮฝ 2 ฮฝ 1 x - 2 ฮฝ 2 ฮฝ 1 2 x + 4 ฮฝ 2 2 ฮฝ 1 x + 24 ฮฝ 2 ฮฝ 1 x ) + ฮฝ 1 ( ฮฝ 2 + 2 ) f ( x ) ( - ฮป ฮฝ 2 - ฮฝ 2 ฮฝ 1 + 4 ฮฝ 2 + 4 ฮฝ 1 x + ฮฝ 2 ฮฝ 1 x ) = 0 , f ( 1 ) = e - ฮป / 2 ฮฝ 1 ฮฝ 1 2 ฮฝ 2 ฮฝ 2 2 ( ฮฝ 1 + ฮฝ 2 ) F 1 1 1 2 ( - ฮฝ 1 - ฮฝ 2 ) ( 1 2 ( ฮฝ 1 + ฮฝ 2 ) ; ฮฝ 1 2 ; ฮป ฮฝ 1 2 ( ฮฝ 1 + ฮฝ 2 ) ) B ( ฮฝ 1 2 , ฮฝ 2 2 ) , f โ€ฒ ( 1 ) = e - ฮป / 2 ฮฝ 1 ฮฝ 1 2 ฮฝ 2 ฮฝ 2 2 ( ฮฝ 1 + ฮฝ 2 ) ( ฮฝ 2 ( ฮป 1 F 1 ( 1 2 ( ฮฝ 1 + ฮฝ 2 + 2 ) ; 1 2 ( ฮฝ 1 + 2 ) ; ฮป ฮฝ 1 2 ( ฮฝ 1 + ฮฝ 2 ) ) - 2 1 F 1 ( 1 2 ( ฮฝ 1 + ฮฝ 2 ) ; ฮฝ 1 2 ; ฮป ฮฝ 1 2 ( ฮฝ 1 + ฮฝ 2 ) ) ) - 2 ฮฝ 1 F 1 1 ( 1 2 ( ฮฝ 1 + ฮฝ 2 ) ; ฮฝ 1 2 ; ฮป ฮฝ 1 2 ( ฮฝ 1 + ฮฝ 2 ) ) ) 1 2 ( - ฮฝ 1 - ฮฝ 2 - 2 ) 2 B ( ฮฝ 1 2 , ฮฝ 2 2 ) } \left\{\begin{array}[]{l}4x\left(\nu_{2}+\nu_{1}x\right){}^{2}f^{\prime\prime}% (x)+f^{\prime}(x)\left(-2\nu_{2}^{2}\nu_{1}+8\nu_{2}^{2}+16\nu_{1}^{2}x^{2}+4% \nu_{2}\nu_{1}^{2}x^{2}-2\lambda\nu_{2}\nu_{1}x-2\nu_{2}\nu_{1}^{2}x+4\nu_{2}^% {2}\nu_{1}x+24\nu_{2}\nu_{1}x\right)+\nu_{1}\left(\nu_{2}+2\right)f(x)\left(-% \lambda\nu_{2}-\nu_{2}\nu_{1}+4\nu_{2}+4\nu_{1}x+\nu_{2}\nu_{1}x\right)=0,\\ f(1)=\frac{e^{-\lambda/2}\nu_{1}^{\frac{\nu_{1}}{2}}\nu_{2}^{\frac{\nu_{2}}{2}% }\left(\nu_{1}+\nu_{2}\right){}^{\frac{1}{2}\left(-\nu_{1}-\nu_{2}\right)}\,_{% 1}F_{1}\left(\frac{1}{2}\left(\nu_{1}+\nu_{2}\right);\frac{\nu_{1}}{2};\frac{% \lambda\nu_{1}}{2\left(\nu_{1}+\nu_{2}\right)}\right)}{B\left(\frac{\nu_{1}}{2% },\frac{\nu_{2}}{2}\right)},\\ f^{\prime}(1)=\frac{e^{-\lambda/2}\nu_{1}^{\frac{\nu_{1}}{2}}\nu_{2}^{\frac{% \nu_{2}}{2}}\left(\nu_{1}+\nu_{2}\right){}^{\frac{1}{2}\left(-\nu_{1}-\nu_{2}-% 2\right)}\left(\nu_{2}\left(\lambda\,_{1}F_{1}\left(\frac{1}{2}\left(\nu_{1}+% \nu_{2}+2\right);\frac{1}{2}\left(\nu_{1}+2\right);\frac{\lambda\nu_{1}}{2% \left(\nu_{1}+\nu_{2}\right)}\right)-2\,_{1}F_{1}\left(\frac{1}{2}\left(\nu_{1% }+\nu_{2}\right);\frac{\nu_{1}}{2};\frac{\lambda\nu_{1}}{2\left(\nu_{1}+\nu_{2% }\right)}\right)\right)-2\nu_{1}\,{}_{1}F_{1}\left(\frac{1}{2}\left(\nu_{1}+% \nu_{2}\right);\frac{\nu_{1}}{2};\frac{\lambda\nu_{1}}{2\left(\nu_{1}+\nu_{2}% \right)}\right)\right)}{2B\left(\frac{\nu_{1}}{2},\frac{\nu_{2}}{2}\right)}% \end{array}\right\}
  20. Z = lim ฮฝ 2 โ†’ โˆž ฮฝ 1 F Z=\lim_{\nu_{2}\to\infty}\nu_{1}F

Nondimensionalization.html

  1. a d x d t + b x = A f ( t ) . a\frac{dx}{dt}+bx=Af(t).
  2. x = ฯ‡ x c , t = ฯ„ t c x=\chi x_{c},\ t=\tau t_{c}
  3. a x c t c d ฯ‡ d ฯ„ + b x c ฯ‡ = A f ( ฯ„ t c ) = def A F ( ฯ„ ) . a\frac{x_{c}}{t_{c}}\frac{d\chi}{d\tau}+bx_{c}\chi=Af(\tau t_{c})\ \stackrel{% \mathrm{def}}{=}\ AF(\tau).
  4. d ฯ‡ d ฯ„ + b t c a ฯ‡ = A t c a x c F ( ฯ„ ) . \frac{d\chi}{d\tau}+\frac{bt_{c}}{a}\chi=\frac{At_{c}}{ax_{c}}F(\tau).
  5. b t c a = 1 โ‡’ t c = a b . \frac{bt_{c}}{a}=1\Rightarrow t_{c}=\frac{a}{b}.
  6. A t c a x c = A b x c = 1 โ‡’ x c = A b . \frac{At_{c}}{ax_{c}}=\frac{A}{bx_{c}}=1\Rightarrow x_{c}=\frac{A}{b}.
  7. d ฯ‡ d ฯ„ + ฯ‡ = F ( ฯ„ ) . \frac{d\chi}{d\tau}+\chi=F(\tau).
  8. ฯ„ = t t c โ‡’ t = ฯ„ t c \tau=\frac{t}{t_{c}}\Rightarrow t=\tau t_{c}
  9. ฯ‡ = x x c โ‡’ x = ฯ‡ x c . \chi=\frac{x}{x_{c}}\Rightarrow x=\chi x_{c}.
  10. t = ฯ„ t c โ‡’ d t = t c d ฯ„ โ‡’ d ฯ„ d t = 1 t c . \,\!t=\tau t_{c}\Rightarrow dt=t_{c}d\tau\Rightarrow\frac{d\tau}{dt}=\frac{1}{% t_{c}}.
  11. d d t = d ฯ„ d t d d ฯ„ = 1 t c d d ฯ„ โ‡’ d n d t n = ( d d t ) n = ( 1 t c d d ฯ„ ) n = 1 t c n d n d ฯ„ n . \frac{d}{dt}=\frac{d\tau}{dt}\frac{d}{d\tau}=\frac{1}{t_{c}}\frac{d}{d\tau}% \Rightarrow\frac{d^{n}}{dt^{n}}=\left(\frac{d}{dt}\right)^{n}=\left(\frac{1}{t% _{c}}\frac{d}{d\tau}\right)^{n}=\frac{1}{t_{c}^{n}}\frac{d^{n}}{d\tau^{n}}.
  12. f ( t ) = f ( ฯ„ t c ) = f ( t ( ฯ„ ) ) = F ( ฯ„ ) . \,\!f(t)=f(\tau t_{c})=f(t(\tau))=F(\tau).
  13. a d x d t + b x = A f ( t ) . a\frac{dx}{dt}+bx=Af(t).
  14. t c = a b , x c = A b . t_{c}=\frac{a}{b},\ x_{c}=\frac{A}{b}.
  15. a d 2 x d t 2 + b d x d t + c x = A f ( t ) . a\frac{d^{2}x}{dt^{2}}+b\frac{dx}{dt}+cx=Af(t).
  16. a x c t c 2 d 2 ฯ‡ d ฯ„ 2 + b x c t c d ฯ‡ d ฯ„ + c x c ฯ‡ = A f ( ฯ„ t c ) = A F ( ฯ„ ) . a\frac{x_{c}}{t_{c}^{2}}\frac{d^{2}\chi}{d\tau^{2}}+b\frac{x_{c}}{t_{c}}\frac{% d\chi}{d\tau}+cx_{c}\chi=Af(\tau t_{c})=AF(\tau).
  17. d 2 ฯ‡ d ฯ„ 2 + t c b a d ฯ‡ d ฯ„ + t c 2 c a ฯ‡ = A t c 2 a x c F ( ฯ„ ) . \frac{d^{2}\chi}{d\tau^{2}}+t_{c}\frac{b}{a}\frac{d\chi}{d\tau}+t_{c}^{2}\frac% {c}{a}\chi=\frac{At_{c}^{2}}{ax_{c}}F(\tau).
  18. t c = a b t_{c}=\frac{a}{b}
  19. t c = a c t_{c}=\sqrt{\frac{a}{c}}
  20. 1 = A t c 2 a x c = A c x c โ‡’ x c = A c . 1=\frac{At_{c}^{2}}{ax_{c}}=\frac{A}{cx_{c}}\Rightarrow x_{c}=\frac{A}{c}.
  21. d 2 ฯ‡ d ฯ„ 2 + b a c d ฯ‡ d ฯ„ + ฯ‡ = F ( ฯ„ ) . \frac{d^{2}\chi}{d\tau^{2}}+\frac{b}{\sqrt{ac}}\frac{d\chi}{d\tau}+\chi=F(\tau).
  22. 2 ฮถ = def b a c . 2\zeta\ \stackrel{\mathrm{def}}{=}\ \frac{b}{\sqrt{ac}}.
  23. d 2 ฯ‡ d ฯ„ 2 + 2 ฮถ d ฯ‡ d ฯ„ + ฯ‡ = F ( ฯ„ ) . \frac{d^{2}\chi}{d\tau^{2}}+2\zeta\frac{d\chi}{d\tau}+\chi=F(\tau).
  24. a n d n x ( t ) d t n + a n - 1 d n - 1 x ( t ) d t n - 1 + โ€ฆ + a 1 d x ( t ) d t + a 0 x ( t ) = โˆ‘ k = 0 n a k d k x ( t ) d t k = A f ( t ) . a_{n}\frac{d^{n}x(t)}{dt^{n}}+a_{n-1}\frac{d^{n-1}x(t)}{dt^{n-1}}+\ldots+a_{1}% \frac{dx(t)}{dt}+a_{0}x(t)=\sum_{k=0}^{n}a_{k}\frac{d^{k}x(t)}{dt^{k}}=Af(t).
  25. m d 2 x d t 2 + B d x d t + k x = F 0 cos ( ฯ‰ t ) m\frac{d^{2}x}{dt^{2}}+B\frac{dx}{dt}+kx=F_{0}\cos(\omega t)
  26. x c = F 0 k . x_{c}=\frac{F_{0}}{k}.
  27. t c = m k t_{c}=\sqrt{\frac{m}{k}}
  28. 2 ฮถ = B m k 2\zeta=\frac{B}{\sqrt{mk}}
  29. R d Q d t + Q C = V ( t ) โ‡’ d ฯ‡ d ฯ„ + ฯ‡ = F ( ฯ„ ) R\frac{dQ}{dt}+\frac{Q}{C}=V(t)\Rightarrow\frac{d\chi}{d\tau}+\chi=F(\tau)
  30. Q = ฯ‡ x c , t = ฯ„ t c , x c = C V 0 , t c = R C , F = V . Q=\chi x_{c},\ t=\tau t_{c},\ x_{c}=CV_{0},\ t_{c}=RC,\ F=V.
  31. L d 2 Q d t 2 + R d Q d t + Q C = V 0 cos ( ฯ‰ t ) โ‡’ d 2 ฯ‡ d ฯ„ 2 + 2 ฮถ d ฯ‡ d ฯ„ + ฯ‡ = cos ( ฮฉ ฯ„ ) L\frac{d^{2}Q}{dt^{2}}+R\frac{dQ}{dt}+\frac{Q}{C}=V_{0}\cos(\omega t)% \Rightarrow\frac{d^{2}\chi}{d\tau^{2}}+2\zeta\frac{d\chi}{d\tau}+\chi=\cos(% \Omega\tau)
  32. Q = ฯ‡ x c , t = ฯ„ t c , x c = C V 0 , t c = L C , 2 ฮถ = R C L , ฮฉ = t c ฯ‰ . Q=\chi x_{c},\ t=\tau t_{c},\ \ x_{c}=CV_{0},\ t_{c}=\sqrt{LC},\ 2\zeta=R\sqrt% {\frac{C}{L}},\ \Omega=t_{c}\omega.
  33. ( - โ„ 2 2 m d 2 d x 2 + 1 2 m ฯ‰ 2 x 2 ) ฯˆ ( x ) = E ฯˆ ( x ) . \left(-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}+\frac{1}{2}m\omega^{2}x^{2}% \right)\psi(x)=E\psi(x).
  34. x x
  35. x ~ โ‰ก x x c , \tilde{x}\equiv\frac{x}{x_{\,\text{c}}},
  36. ฯˆ ~ \tilde{\psi}
  37. ฯˆ ( x ) = ฯˆ ( x ~ x c ) โ‰ก ฯˆ ~ ( x ~ ) x c . \psi(x)=\psi(\tilde{x}x_{\,\text{c}})\equiv\frac{\tilde{\psi}(\tilde{x})}{x_{% \,\text{c}}}.
  38. ( - โ„ 2 2 m 1 x c 2 d 2 d x ~ 2 + 1 2 m ฯ‰ 2 x c 2 x ~ 2 ) ฯˆ ~ ( x ~ ) x ~ c = E ฯˆ ~ ( x ~ ) x c โ‡’ ( - d 2 d x ~ 2 + m 2 ฯ‰ 2 x c 4 โ„ 2 x ~ 2 ) ฯˆ ( x ~ ) = 2 m x c 2 E โ„ 2 ฯˆ ( x ~ ) . \left(-\frac{\hbar^{2}}{2m}\frac{1}{x_{\,\text{c}}^{2}}\frac{d^{2}}{d\tilde{x}% ^{2}}+\frac{1}{2}m\omega^{2}x_{\,\text{c}}^{2}\tilde{x}^{2}\right)\frac{\tilde% {\psi}(\tilde{x})}{\tilde{x}_{\,\text{c}}}=E\frac{\tilde{\psi}(\tilde{x})}{x_{% \,\text{c}}}\Rightarrow\left(-\frac{d^{2}}{d\tilde{x}^{2}}+\frac{m^{2}\omega^{% 2}x_{\,\text{c}}^{4}}{\hbar^{2}}\tilde{x}^{2}\right)\psi(\tilde{x})=\frac{2mx_% {\,\text{c}}^{2}E}{\hbar^{2}}\psi(\tilde{x}).
  39. x ~ 2 \tilde{x}^{2}
  40. m 2 ฯ‰ 2 x c 4 โ„ 2 = 1 โ‡’ x c = โ„ m ฯ‰ . \frac{m^{2}\omega^{2}x_{\,\text{c}}^{4}}{\hbar^{2}}=1\Rightarrow x_{\,\text{c}% }=\sqrt{\frac{\hbar}{m\omega}}.
  41. ( - d 2 d x ~ 2 + x ~ 2 ) ฯˆ ~ ( x ~ ) = E ~ ฯˆ ~ ( ฯ‡ ~ ) , \left(-\frac{d^{2}}{d\tilde{x}^{2}}+\tilde{x}^{2}\right)\tilde{\psi}(\tilde{x}% )=\tilde{E}\tilde{\psi}(\tilde{\chi}),
  42. E โ‰ก โ„ ฯ‰ 2 E ~ . E\equiv\frac{\hbar\omega}{2}\tilde{E}.
  43. E ~ \tilde{E}
  44. โ„ ฯ‰ 2 ( - d 2 d x ~ 2 + x ~ 2 ) ฯˆ ~ ( x ~ ) = E ฯˆ ~ ( x ~ ) . \frac{\hbar\omega}{2}\left(-\frac{d^{2}}{d\tilde{x}^{2}}+\tilde{x}^{2}\right)% \tilde{\psi}(\tilde{x})=E\tilde{\psi}(\tilde{x}).

Nonelementary_integral.html

  1. 1 - x 4 \sqrt{1-x^{4}}
  2. ln ( ln x ) \ln(\ln x)\,
  3. 1 ln x \frac{1}{\ln x}
  4. e e x e^{e^{x}}\,
  5. e - x 2 2 e^{-\frac{x^{2}}{2}}\,
  6. x x x^{x}
  7. sin ( sin ( x ) ) \sin(\sin(x))

Nonlinear_Schroฬˆdinger_equation.html

  1. H = โˆซ d x [ 1 2 | โˆ‚ x ฯˆ | 2 + ฮบ 2 | ฯˆ | 4 ] H=\int\mathrm{d}x\left[{1\over 2}|\partial_{x}\psi|^{2}+{\kappa\over 2}|\psi|^% {4}\right]
  2. { ฯˆ ( x ) , ฯˆ ( y ) } = { ฯˆ * ( x ) , ฯˆ * ( y ) } = 0 \{\psi(x),\psi(y)\}=\{\psi^{*}(x),\psi^{*}(y)\}=0\,
  3. { ฯˆ * ( x ) , ฯˆ ( y ) } = i ฮด ( x - y ) . \{\psi^{*}(x),\psi(y)\}=i\delta(x-y).\,
  4. [ ฯˆ ( x ) , ฯˆ ( y ) ] = [ ฯˆ * ( x ) , ฯˆ * ( y ) ] = 0 [ ฯˆ * ( x ) , ฯˆ ( y ) ] = - ฮด ( x - y ) \begin{aligned}\displaystyle{}[\psi(x),\psi(y)]&\displaystyle=[\psi^{*}(x),% \psi^{*}(y)]=0\\ \displaystyle{}[\psi^{*}(x),\psi(y)]&\displaystyle=-\delta(x-y)\end{aligned}
  5. H = โˆซ d x [ 1 2 โˆ‚ x ฯˆ โ€  โˆ‚ x ฯˆ + ฮบ 2 ฯˆ โ€  ฯˆ โ€  ฯˆ ฯˆ ] . H=\int dx\left[{1\over 2}\partial_{x}\psi^{\dagger}\partial_{x}\psi+{\kappa% \over 2}\psi^{\dagger}\psi^{\dagger}\psi\psi\right].
  6. ฯ• x \displaystyle\phi_{x}
  7. ฮ› = ( ฮป 1 0 0 ฮป 2 ) , J = i ฯƒ z = ( i 0 0 - i ) , U = i ( 0 q r 0 ) . \Lambda=\begin{pmatrix}\lambda_{1}&0\\ 0&\lambda_{2}\end{pmatrix},\quad J=i\sigma_{z}=\begin{pmatrix}i&0\\ 0&-i\end{pmatrix},\quad U=i\begin{pmatrix}0&q\\ r&0\end{pmatrix}.
  8. ฯ• x t = ฯ• t x โ‡’ U t = - J U x x + 2 J U 2 U โ‡” { i q t = q x x + 2 q r q i r t = - r x x - 2 q r r . \phi_{xt}=\phi_{tx}\quad\Rightarrow\quad U_{t}=-JU_{xx}+2JU^{2}U\quad% \Leftrightarrow\quad\begin{cases}iq_{t}=q_{xx}+2qrq\\ ir_{t}=-r_{xx}-2qrr.\end{cases}\,
  9. ฯ• โ†’ ฯ• [ 1 ] = ฯ• ฮ› - ฯƒ ฯ• \displaystyle\phi\to\phi[1]=\phi\Lambda-\sigma\phi
  10. ฯ† x \displaystyle\varphi_{x}
  11. e - i v ( x + v t / 2 ) e^{-iv(x+vt/2)}\,
  12. ฯˆ ( x , t ) โ†ฆ ฯˆ [ v ] ( x , t ) = ฯˆ ( x + v t , t ) e - i v ( x + v t / 2 ) . \psi(x,t)\mapsto\psi_{[v]}(x,t)=\psi(x+vt,t)\;e^{-iv(x+vt/2)}.
  13. ฮท = a ( x 0 , t 0 ) cos [ k 0 x 0 - ฯ‰ 0 t 0 - ฮธ ( x 0 , t 0 ) ] , \eta=a(x_{0},t_{0})\;\cos\left[k_{0}\,x_{0}-\omega_{0}\,t_{0}-\theta(x_{0},t_{% 0})\right],
  14. ฯˆ = a exp ( i ฮธ ) . \psi=a\;\exp\left(i\theta\right).
  15. x = k 0 [ x 0 - ฮฉ โ€ฒ ( k 0 ) t 0 ] , t = k 0 2 [ - ฮฉ โ€ฒโ€ฒ ( k 0 ) ] t 0 x=k_{0}\left[x_{0}-\Omega^{\prime}(k_{0})\;t_{0}\right],\quad t=k_{0}^{2}\left% [-\Omega^{\prime\prime}(k_{0})\right]\;t_{0}
  16. ฮบ = - 2 k 0 2 , ฮฉ ( k 0 ) = g k 0 = ฯ‰ 0 \kappa=-2k_{0}^{2},\quad\Omega(k_{0})=\sqrt{gk_{0}}=\omega_{0}\,\!
  17. ฮฉ โ€ฒ ( k 0 ) = 1 2 ฯ‰ 0 k 0 , ฮฉ โ€ฒโ€ฒ ( k 0 ) = - 1 4 ฯ‰ 0 3 k 0 3 \Omega^{\prime}(k_{0})=\frac{1}{2}\frac{\omega_{0}}{k_{0}},\quad\Omega^{\prime% \prime}(k_{0})=-\frac{1}{4}\frac{\omega_{0}^{3}}{k_{0}^{3}}\,\!
  18. S โ†’ t = S โ†’ โˆง S โ†’ x x . \vec{S}_{t}=\vec{S}\wedge\vec{S}_{xx}.\qquad

Normal-form_game.html

  1. S k = { 1 , 2 , โ€ฆ , n k } . S_{k}=\{1,2,\ldots,n_{k}\}.
  2. s โ†’ = ( s 1 , s 2 , โ€ฆ , s m ) \vec{s}=(s_{1},s_{2},\ldots,s_{m})
  3. s 1 โˆˆ S 1 , s 2 โˆˆ S 2 , โ€ฆ , s m โˆˆ S m s_{1}\in S_{1},s_{2}\in S_{2},\ldots,s_{m}\in S_{m}
  4. F : S 1 ร— S 2 ร— โ€ฆ ร— S m โ†’ โ„ . F:S_{1}\times S_{2}\times\ldots\times S_{m}\rightarrow\mathbb{R}.
  5. G = โŸจ P , ๐’ , ๐… โŸฉ G=\langle P,\mathbf{S},\mathbf{F}\rangle
  6. P = { 1 , 2 , โ€ฆ , m } P=\{1,2,\ldots,m\}
  7. ๐’ = { S 1 , S 2 , โ€ฆ , S m } \mathbf{S}=\{S_{1},S_{2},\ldots,S_{m}\}
  8. ๐… = { F 1 , F 2 , โ€ฆ , F m } \mathbf{F}=\{F_{1},F_{2},\ldots,F_{m}\}

Normal_family.html

  1. f n ( x ) f_{n}(x)
  2. f ( x ) f(x)
  3. lim n โ†’ โˆž sup x โˆˆ K d Y ( f n ( x ) , f ( x ) ) = 0 \lim_{n\rightarrow\infty}\sup_{x\in K}d_{Y}(f_{n}(x),f(x))=0
  4. d Y ( y 1 , y 2 ) d_{Y}(y_{1},y_{2})
  5. d Y ( y 1 , y 2 ) = | y 1 - y 2 | d_{Y}(y_{1},y_{2})=|y_{1}-y_{2}|
  6. f n ( z ) โ†’ f ( z ) f_{n}(z)\to f(z)
  7. d ( f n ( z ) , f ( z ) ) d\left(f_{n}(z),f(z)\right)\,

Normalization_property_(abstract_rewriting).html

  1. ฮป x . x x x \lambda x.xxx
  2. t t
  3. ( ฮป x . x x x ) t โ†’ t t t (\mathbf{\lambda}x.xxx)t\rightarrow ttt
  4. ฮป x . x x x \lambda x.xxx
  5. ( ฮป x . x x x ) ( ฮป x . x x x ) (\mathbf{\lambda}x.xxx)(\lambda x.xxx)
  6. โ†’ ( ฮป x . x x x ) ( ฮป x . x x x ) ( ฮป x . x x x ) \rightarrow(\mathbf{\lambda}x.xxx)(\lambda x.xxx)(\lambda x.xxx)
  7. โ†’ ( ฮป x . x x x ) ( ฮป x . x x x ) ( ฮป x . x x x ) ( ฮป x . x x x ) \rightarrow(\mathbf{\lambda}x.xxx)(\lambda x.xxx)(\lambda x.xxx)(\lambda x.xxx)
  8. โ†’ ( ฮป x . x x x ) ( ฮป x . x x x ) ( ฮป x . x x x ) ( ฮป x . x x x ) ( ฮป x . x x x ) \rightarrow(\mathbf{\lambda}x.xxx)(\lambda x.xxx)(\lambda x.xxx)(\lambda x.xxx% )(\lambda x.xxx)
  9. โ†’ โ€ฆ \rightarrow\ \ldots\,
  10. ( ฮป x . x x x ) ( ฮป x . x x x ) (\lambda x.xxx)(\lambda x.xxx)

Normalized_number.html

  1. ยฑ d 0 . d 1 d 2 d 3 โ€ฆ ร— 10 n \pm d_{0}.d_{1}d_{2}d_{3}\dots\times 10^{n}
  2. d 0 , d_{0},
  3. d 1 , d_{1},
  4. d 2 d_{2}
  5. d 3 d_{3}
  6. d 0 d_{0}
  7. x = 918.082 x=918.082
  8. 9.18082 ร— 10 2 9.18082\times 10^{2}
  9. - 5.74012 ร— 10 - 3 . -5.74012\times 10^{-3}.
  10. ยฑ d 0 . d 1 d 2 d 3 โ€ฆ ร— b n , \pm d_{0}.d_{1}d_{2}d_{3}\dots\times b^{n},
  11. d 0 โ‰  0 , d_{0}\not=0,
  12. d 0 , d_{0},
  13. d 1 , d_{1},
  14. d 2 d_{2}
  15. d 3 d_{3}
  16. 0
  17. b - 1 b-1

Normally_distributed_and_uncorrelated_does_not_imply_independent.html

  1. cov ( X , Y ) \displaystyle\operatorname{cov}(X,Y)
  2. Pr ( Y โ‰ค x ) \displaystyle\Pr(Y\leq x)
  3. ฮฆ ( x ) \Phi(x)
  4. Y = { X if | X | โ‰ค c - X if | X | > c Y=\left\{\begin{matrix}X&\,\text{if }\left|X\right|\leq c\\ -X&\,\text{if }\left|X\right|>c\end{matrix}\right.
  5. Pr ( Y โ‰ค x ) = Pr ( { | X | โ‰ค c and X โ‰ค x } or { | X | > c and - X โ‰ค x } ) = Pr ( | X | โ‰ค c and X โ‰ค x ) + Pr ( | X | > c and - X โ‰ค x ) = Pr ( | X | โ‰ค c and X โ‰ค x ) + Pr ( | X | > c and X โ‰ค x ) = Pr ( X โ‰ค x ) . \begin{aligned}\displaystyle\Pr(Y\leq x)&\displaystyle=\Pr(\{|X|\leq c\,\text{% and }X\leq x\}\,\text{ or }\{|X|>c\,\text{ and }-X\leq x\})\\ &\displaystyle=\Pr(|X|\leq c\,\text{ and }X\leq x)+\Pr(|X|>c\,\text{ and }-X% \leq x)\\ &\displaystyle=\Pr(|X|\leq c\,\text{ and }X\leq x)+\Pr(|X|>c\,\text{ and }X% \leq x)\\ &\displaystyle=\Pr(X\leq x).\end{aligned}\,

Normed_algebra.html

  1. โˆ€ x , y โˆˆ A โˆฅ x y โˆฅ โ‰ค โˆฅ x โˆฅ โˆฅ y โˆฅ \forall x,y\in A\qquad\|xy\|\leq\|x\|\|y\|
  2. A {}_{A}
  3. A {}_{A}

Nose_cone_design.html

  1. y = x R L y={xR\over L}
  2. ฯ• \phi\;
  3. ฯ• = arctan ( R L ) \phi=\arctan\left({R\over L}\right)
  4. y = x tan ( ฯ• ) y=x\tan(\phi)\;
  5. x t = L 2 R r n 2 R 2 + L 2 x_{t}=\frac{L^{2}}{R}\sqrt{\frac{r_{n}^{2}}{R^{2}+L^{2}}}
  6. y t = x t R L y_{t}=\frac{x_{t}R}{L}
  7. r n r_{n}
  8. x o = x t + r n 2 - y t 2 x_{o}=x_{t}+\sqrt{r_{n}^{2}-y_{t}^{2}}
  9. x a = x o - r n x_{a}=x_{o}-r_{n}
  10. 0 โ‰ค x โ‰ค L 1 0\leq x\leq L_{1}
  11. y = x R 1 L 1 y={xR_{1}\over L_{1}}
  12. ฯ• 1 = arctan ( R 1 L 1 ) \phi_{1}=\arctan\left({R_{1}\over L_{1}}\right)
  13. y = x tan ( ฯ• 1 ) y=x\tan(\phi_{1})\;
  14. L 1 โ‰ค x โ‰ค L L_{1}\leq x\leq L
  15. y = R 1 + ( x - L 1 ) ( R 2 - R 1 ) L 2 y=R_{1}+{(x-L_{1})(R_{2}-R_{1})\over L_{2}}
  16. ฯ• 2 = arctan ( R 2 - R 1 L 2 ) \phi_{2}=\arctan\left({R_{2}-R_{1}\over L_{2}}\right)
  17. y = R 1 + ( x - L 1 ) tan ( ฯ• 2 ) y=R_{1}+(x-L_{1})\tan(\phi_{2})\;
  18. ฯ \rho
  19. ฯ = R 2 + L 2 2 R \rho={R^{2}+L^{2}\over 2R}
  20. y = ฯ 2 - ( L - x ) 2 + R - ฯ y=\sqrt{\rho^{2}-(L-x)^{2}}+R-\rho
  21. ฯ \rho
  22. x o = L - ( ฯ - r n ) 2 - ( ฯ - R ) 2 x_{o}=L-\sqrt{(\rho-r_{n})^{2}-(\rho-R)^{2}}
  23. y t = r n ( ฯ - R ) ฯ - r n y_{t}=\frac{r_{n}(\rho-R)}{\rho-r_{n}}
  24. x t = x o - r n 2 - y t 2 x_{t}=x_{o}-\sqrt{r_{n}^{2}-y_{t}^{2}}
  25. r n r_{n}
  26. x o x_{o}
  27. x a = x o - r n x_{a}=x_{o}-r_{n}
  28. ฯ \rho
  29. ฯ > R 2 + L 2 2 R \rho>{R^{2}+L^{2}\over 2R}
  30. ฮฑ = arctan ( R L ) - arccos ( L 2 + R 2 2 ฯ ) \alpha=\arctan\left({R\over L}\right)-\arccos\left({\sqrt{L^{2}+R^{2}}\over 2% \rho}\right)
  31. y = ฯ 2 - ( ฯ cos ฮฑ - x ) 2 + ฯ sin ฮฑ y=\sqrt{\rho^{2}-(\rho\cos\alpha-x)^{2}}+\rho\sin\alpha
  32. ฯ \rho
  33. ฯ \rho
  34. L / 2 < ฯ < R 2 + L 2 2 R {L/2}<\rho<{R^{2}+L^{2}\over 2R}
  35. y = R 1 - x 2 L 2 y=R\sqrt{1-{x^{2}\over L^{2}}}
  36. 0 โ‰ค K โ€ฒ โ‰ค 1 0\leq K^{\prime}\leq 1
  37. y = R ( 2 ( x L ) - K โ€ฒ ( x L ) 2 2 - K โ€ฒ ) y=R\left({2({x\over L})-K^{\prime}({x\over L})^{2}\over 2-K^{\prime}}\right)
  38. 0 โ‰ค n โ‰ค 1 : y = R ( x L ) n 0\leq n\leq 1:y=R\left({x\over L}\right)^{n}
  39. ฮธ = arccos ( 1 - 2 x L ) \theta=\arccos\left(1-{2x\over L}\right)
  40. y = R ฯ€ ฮธ - sin ( 2 ฮธ ) 2 + C sin 3 ฮธ y={R\over\sqrt{\pi}}\sqrt{\theta-{\sin(2\theta)\over 2}+C\sin^{3}\theta}

Note_value.html

  1. 2 n - 1 2 n \tfrac{2^{n}-1}{2^{n}}

Nth_root_algorithm.html

  1. A n \sqrt[n]{A}
  2. x n = A x^{n}=A
  3. A > 0 A>0
  4. A n \sqrt[n]{A}
  5. x 0 x_{0}
  6. x k + 1 = 1 n [ ( n - 1 ) x k + A x k n - 1 ] x_{k+1}=\frac{1}{n}\left[{(n-1)x_{k}+\frac{A}{x_{k}^{n-1}}}\right]
  7. ฮ” x k = 1 n [ A x k n - 1 - x k ] ; x k + 1 = x k + ฮ” x k \Delta x_{k}=\frac{1}{n}\left[{\frac{A}{x_{k}^{n-1}}}-x_{k}\right];x_{k+1}=x_{% k}+\Delta x_{k}
  8. | ฮ” x k | < ฯต |\Delta x_{k}|<\epsilon
  9. x k + 1 = 1 2 ( x k + A x k ) x_{k+1}=\frac{1}{2}\left(x_{k}+\frac{A}{x_{k}}\right)
  10. f ( x ) f(x)
  11. x k n - 1 x_{k}^{n-1}
  12. x 0 x_{0}
  13. x k + 1 = x k - f ( x k ) f โ€ฒ ( x k ) x_{k+1}=x_{k}-\frac{f(x_{k})}{f^{\prime}(x_{k})}
  14. f ( x ) = x n - A f(x)=x^{n}-A
  15. f โ€ฒ ( x ) = n x n - 1 f^{\prime}(x)=nx^{n-1}
  16. x k + 1 = x k - f ( x k ) f โ€ฒ ( x k ) x_{k+1}=x_{k}-\frac{f(x_{k})}{f^{\prime}(x_{k})}
  17. = x k - x k n - A n x k n - 1 =x_{k}-\frac{x_{k}^{n}-A}{nx_{k}^{n-1}}
  18. = x k - x k n + A n x k n - 1 =x_{k}-\frac{x_{k}}{n}+\frac{A}{nx_{k}^{n-1}}
  19. = 1 n [ ( n - 1 ) x k + A x k n - 1 ] =\frac{1}{n}\left[{(n-1)x_{k}+\frac{A}{x_{k}^{n-1}}}\right]

Nuclear_density.html

  1. R = A 1 / 3 R 0 R=A^{1/3}R_{0}
  2. A A
  3. r 0 r_{0}
  4. n = A 4 3 ฯ€ R 3 n=\frac{A}{{4\over 3}\pi R^{3}}
  5. n = 3 4 ฯ€ R 0 3 = 0.122 fm - 3 = 1.22 โ‹… 10 44 m - 3 n={3\over 4\pi{R_{0}}^{3}}=0.122\ \mathrm{fm}^{-3}=1.22\cdot 10^{44}\ \mathrm{% m}^{-3}
  6. ( 1.67 โ‹… 10 - 27 kg ) ( 1.22 โ‹… 10 44 m - 3 ) = 2.04 โ‹… 10 17 kg โ‹… m - 3 (1.67\cdot 10^{-27}\ \mathrm{kg})(1.22\cdot 10^{44}\ \mathrm{m}^{-3})=2.04% \cdot 10^{17}\ \mathrm{kg}\cdot\mathrm{m}^{-3}

Nuclear_force.html

  1. V Yukawa ( r ) = - g 2 e - ฮผ r r , V\text{Yukawa}(r)=-g^{2}\frac{e^{-\mu r}}{r},
  2. ฮผ \mu

Nuclear_magnetic_moment.html

  1. ฮผ = โŸจ ( l , s ) , j , m j = j | ฮผ z | ( l , s ) , j , m j = j โŸฉ \mu=\langle(l,s),j,m_{j}=j|\mu_{z}|(l,s),j,m_{j}=j\rangle
  2. ฮผ = โŸจ ( l , s ) , j , m j = j | ฮผ โ†’ โ‹… j โ†’ | ( l , s ) , j , m j = j โŸฉ โŸจ ( l , s ) j , m j = j | j z | ( l , s ) j , m j = j โŸฉ โŸจ ( l , s ) j , m j = j | j โ†’ โ‹… j โ†’ | ( l , s ) j , m j = j โŸฉ = 1 j + 1 โŸจ ( l , s ) , j , m j = j | ฮผ โ†’ โ‹… j โ†’ | ( l , s ) , j , m j = j โŸฉ \begin{aligned}\displaystyle\mu&\displaystyle=\left\langle(l,s),j,m_{j}=j\left% |\vec{\mu}\cdot\vec{j}\right|(l,s),j,m_{j}=j\right\rangle\frac{\left\langle(l,% s)j,m_{j}=j\left|j_{z}\right|(l,s)j,m_{j}=j\right\rangle}{\left\langle(l,s)j,m% _{j}=j\left|\vec{j}\cdot\vec{j}\right|(l,s)j,m_{j}=j\right\rangle}\\ &\displaystyle=\frac{1}{j+1}\left\langle(l,s),j,m_{j}=j\left|\vec{\mu}\cdot% \vec{j}\right|(l,s),j,m_{j}=j\right\rangle\end{aligned}
  3. ฮผ โ†’ \vec{\mu}
  4. ฮผ โ†’ = g ( l ) l โ†’ + g ( s ) s โ†’ \vec{\mu}=g^{(l)}\vec{l}+g^{(s)}\vec{s}
  5. l โ†’ โ‹… j โ†’ = 1 2 ( j โ†’ โ‹… j โ†’ + l โ†’ โ‹… l โ†’ - s โ†’ โ‹… s โ†’ ) s โ†’ โ‹… j โ†’ = 1 2 ( j โ†’ โ‹… j โ†’ - l โ†’ โ‹… l โ†’ + s โ†’ โ‹… s โ†’ ) ฮผ = 1 2 โŸจ ( l , s ) , j , m j = j | g ( l ) 1 2 ( j โ†’ โ‹… j โ†’ + l โ†’ โ‹… l โ†’ - s โ†’ โ‹… s โ†’ ) + g ( s ) 1 2 ( j โ†’ โ‹… j โ†’ - l โ†’ โ‹… l โ†’ + s โ†’ โ‹… s โ†’ ) | ( l , s ) , j , m j = j โŸฉ = 1 j + 1 ( g ( l ) 1 2 [ j ( j + 1 ) + l ( l + 1 ) - s ( s + 1 ) ] + g ( s ) 1 2 [ j ( j + 1 ) - l ( l + 1 ) + s ( s + 1 ) ] ) \begin{aligned}\displaystyle\vec{l}\cdot\vec{j}&\displaystyle=\frac{1}{2}\left% (\vec{j}\cdot\vec{j}+\vec{l}\cdot\vec{l}-\vec{s}\cdot\vec{s}\right)\\ \displaystyle\vec{s}\cdot\vec{j}&\displaystyle=\frac{1}{2}\left(\vec{j}\cdot% \vec{j}-\vec{l}\cdot\vec{l}+\vec{s}\cdot\vec{s}\right)\\ \displaystyle\mu&\displaystyle=\frac{1}{2}\left\langle(l,s),j,m_{j}=j\left|g^{% (l)}\frac{1}{2}\left(\vec{j}\cdot\vec{j}+\vec{l}\cdot\vec{l}-\vec{s}\cdot\vec{% s}\right)+g^{(s)}\frac{1}{2}\left(\vec{j}\cdot\vec{j}-\vec{l}\cdot\vec{l}+\vec% {s}\cdot\vec{s}\right)\right|(l,s),j,m_{j}=j\right\rangle\\ &\displaystyle=\frac{1}{j+1}\left(g^{(l)}\frac{1}{2}\left[j(j+1)+l(l+1)-s(s+1)% \right]+g^{(s)}\frac{1}{2}\left[j(j+1)-l(l+1)+s(s+1)\right]\right)\end{aligned}
  6. s = 1 / 2 s=1/2
  7. j = l + 1 / 2 j=l+1/2
  8. ฮผ j = g ( l ) l + 1 2 g ( s ) \mu_{j}=g^{(l)}l+{1\over 2}g^{(s)}
  9. j = l - 1 / 2 j=l-1/2
  10. ฮผ j = j j + 1 ( g ( l ) ( l + 1 ) - 1 2 g ( s ) ) \mu_{j}={j\over j+1}\left(g^{(l)}(l+1)-\frac{1}{2}g^{(s)}\right)

Nucleation.html

  1. 4 ฯ€ r 2 4\pi r^{2}

Nucleotide_diversity.html

  1. ฯ€ = โˆ‘ i j x i x j ฯ€ i j = 2 * โˆ‘ i = 1 n โˆ‘ j = 1 i - 1 x i x j ฯ€ i j \pi=\sum_{ij}x_{i}x_{j}\pi_{ij}=2*\sum_{i=1}^{n}\sum_{j=1}^{i-1}x_{i}x_{j}\pi_% {ij}

Null_(SQL).html

  1. R โ‹ˆ R = R R\bowtie R=R
  2. q q
  3. q ยฏ \bar{q}
  4. q ยฏ \bar{q}
  5. Models ( q ยฏ ( T ) ) = { q ( R ) | R โˆˆ Models ( T ) } \mathop{\mathrm{Models}}(\bar{q}(T))=\{q(R)\,|R\in\mathop{\mathrm{Models}}(T)\}
  6. โ‹‚ Models ( q ยฏ ( T ) ) = โ‹‚ { q ( R ) | R โˆˆ Models ( T ) } \bigcap\mathop{\mathrm{Models}}(\bar{q}(T))=\bigcap\{q(R)\,|R\in\mathop{% \mathrm{Models}}(T)\}

Null_dust_solution.html

  1. G a b = 8 ฯ€ ฮฆ k a k b G^{ab}=8\pi\Phi\,k^{a}\,k^{b}
  2. k โ†’ \vec{k}
  3. T a b = ฮฆ k a k b T^{ab}=\Phi\,k^{a}\,k^{b}
  4. e โ†’ 0 , e โ†’ 1 , e โ†’ 2 , e โ†’ 3 \vec{e}_{0},\;\vec{e}_{1},\;\vec{e}_{2},\;\vec{e}_{3}
  5. G a ^ b ^ = 8 ฯ€ ฯต [ 1 0 0 ยฑ 1 0 0 0 0 0 0 0 0 ยฑ 1 0 0 1 ] G^{\hat{a}\hat{b}}=8\pi\epsilon\,\left[\begin{matrix}1&0&0&\pm 1\\ 0&0&0&0\\ 0&0&0&0\\ \pm 1&0&0&1\end{matrix}\right]
  6. e โ†’ 0 \vec{e}_{0}
  7. ฯต \epsilon
  8. k โ†’ \vec{k}
  9. e โ†’ 3 \vec{e}_{3}
  10. e โ†’ 3 \vec{e}_{3}
  11. E ( 2 ) E(2)

Null_graph.html

  1. K 0 K_{0}
  2. K 0 K_{0}
  3. K 0 K_{0}
  4. K 0 K_{0}
  5. K 0 K_{0}
  6. K 0 K_{0}
  7. K 0 K_{0}
  8. K 0 K_{0}
  9. K 1 K_{1}
  10. K 0 K_{0}
  11. K 0 ยฏ \overline{K_{0}}
  12. K 0 K_{0}
  13. K ยฏ n \overline{K}_{n}
  14. K ยฏ n \overline{K}_{n}
  15. K n K_{n}

Null_vector.html

  1. โ‹ƒ r > 0 { x = a + b : q ( a ) = - q ( b ) = r } . \bigcup_{r>0}\{x=a+b:q(a)=-q(b)=r\}.

Numerical_differentiation.html

  1. f ( x + h ) - f ( x ) h . {f(x+h)-f(x)\over h}.
  2. f โ€ฒ ( x ) = lim h โ†’ 0 f ( x + h ) - f ( x ) h . f^{\prime}(x)=\lim_{h\to 0}{f(x+h)-f(x)\over h}.
  3. f ( x + h ) - f ( x - h ) 2 h . {f(x+h)-f(x-h)\over 2h}.
  4. h 2 h^{2}
  5. R = - f ( 3 ) ( c ) 6 h 2 R={{-f^{(3)}(c)}\over{6}}h^{2}
  6. c c
  7. x - h x-h
  8. x + h x+h
  9. h h
  10. ฮต x \sqrt{\varepsilon}x
  11. f โ€ฒ ( x ) = - f ( x + 2 h ) + 8 f ( x + h ) - 8 f ( x - h ) + f ( x - 2 h ) 12 h + h 4 30 f ( 5 ) ( c ) f^{\prime}(x)=\frac{-f(x+2h)+8f(x+h)-8f(x-h)+f(x-2h)}{12h}+\frac{h^{4}}{30}f^{% (5)}(c)
  12. c โˆˆ [ x - 2 h , x + 2 h ] c\in[x-2h,x+2h]
  13. f f
  14. x x
  15. f โ€ฒ ( x ) โ‰ˆ โ„‘ ( f ( x + i h ) ) / h f^{\prime}(x)\approx\Im(f(x+ih))/h
  16. f ( n ) ( a ) = n ! 2 ฯ€ i โˆฎ ฮณ f ( z ) ( z - a ) n + 1 d z f^{(n)}(a)={n!\over 2\pi i}\oint_{\gamma}{f(z)\over(z-a)^{n+1}}\,\mathrm{d}z

Numerical_model_of_the_Solar_System.html

  1. a โ†’ j = โˆ‘ i โ‰  j n G M i | r โ†’ i - r โ†’ j | 3 ( r โ†’ i - r โ†’ j ) \vec{a}_{j}=\sum_{i\neq j}^{n}G\frac{M_{i}}{|\vec{r}_{i}-\vec{r}_{j}|^{3}}(% \vec{r}_{i}-\vec{r}_{j})
  2. ( a j ) x = โˆ‘ i โ‰  j n G M i ( ( x i - x j ) 2 + ( y i - y j ) 2 + ( z i - z j ) 2 ) 3 / 2 ( x i - x j ) (a_{j})_{x}=\sum_{i\neq j}^{n}G\frac{M_{i}}{((x_{i}-x_{j})^{2}+(y_{i}-y_{j})^{% 2}+(z_{i}-z_{j})^{2})^{3/2}}(x_{i}-x_{j})
  3. a x = d v x d t a_{x}=\frac{dv_{x}}{dt}
  4. v x = d x d t v_{x}=\frac{dx}{dt}
  5. ฮ” v x = a x ฮ” t \Delta v_{x}=a_{x}\Delta t
  6. ฮ” x = v x ฮ” t \Delta x=v_{x}\Delta t
  7. r = r 0 + r 0 โ€ฒ t + r 0 โ€ฒโ€ฒ t 2 2 ! + โ€ฆ r=r_{0}+r^{\prime}_{0}t+r^{\prime\prime}_{0}\frac{t^{2}}{2!}+...
  8. r = f r 0 + g r 0 โ€ฒ r=fr_{0}+gr^{\prime}_{0}
  9. a = G M r 2 + e a=\frac{GM}{r^{2}+e}

Numerical_weather_prediction.html

  1. z z
  2. z z
  3. ฯƒ \sigma

Numeฬraire.html

  1. M ( t ) = exp ( โˆซ 0 t r ( s ) d s ) M(t)=\exp\left(\int_{0}^{t}r(s)ds\right)
  2. t t
  3. S ( t ) S(t)
  4. Q Q
  5. S ( t ) M ( t ) = E Q [ S ( T ) M ( T ) | โ„ฑ ( t ) ] โˆ€ t โ‰ค T . \frac{S(t)}{M(t)}=E_{Q}\left[\left.\frac{S(T)}{M(T)}\right|\mathcal{F}(t)% \right]\qquad\forall\,t\leq T.
  6. N ( t ) > 0 N\left(t\right)>0
  7. Q N Q^{N}
  8. d Q N d Q = M ( 0 ) M ( T ) N ( T ) N ( 0 ) . \frac{dQ^{N}}{dQ}=\frac{M(0)}{M(T)}\frac{N(T)}{N(0)}.
  9. S ( t ) S(t)
  10. Q N Q^{N}
  11. N ( t ) N(t)
  12. E Q N [ S ( T ) N ( T ) | โ„ฑ ( t ) ] \displaystyle{}\quad E_{Q^{N}}\left[\left.\frac{S(T)}{N(T)}\right|\mathcal{F}(% t)\right]

Nyquist_stability_criterion.html

  1. G ( s ) = 1 s 2 + s + 1 G(s)=\frac{1}{s^{2}+s+1}
  2. G ( s ) G(s)
  3. H ( s ) H(s)
  4. G 1 + G H \frac{G}{1+GH}
  5. 1 + G H 1+GH
  6. ๐’ฏ ( s ) \mathcal{T}(s)
  7. ๐’ฏ ( s ) = N ( s ) D ( s ) . \mathcal{T}(s)=\frac{N(s)}{D(s)}.
  8. N ( s ) N(s)
  9. ๐’ฏ ( s ) \mathcal{T}(s)
  10. D ( s ) D(s)
  11. ๐’ฏ ( s ) \mathcal{T}(s)
  12. ๐’ฏ ( s ) \mathcal{T}(s)
  13. D ( s ) = 0 D(s)=0
  14. ๐’ฏ ( s ) \mathcal{T}(s)
  15. ๐’ฏ ( s ) \mathcal{T}(s)
  16. G ( s ) = A ( s ) B ( s ) G(s)=\frac{A(s)}{B(s)}
  17. 1 + G ( s ) 1+G(s)
  18. A ( s ) + B ( s ) = 0 A(s)+B(s)=0
  19. ฮ“ s \Gamma_{s}
  20. s s
  21. F ( s ) F(s)
  22. F ( s ) F(s)
  23. F ( s ) F(s)
  24. F ( s ) F(s)
  25. ฮ“ F ( s ) = F ( ฮ“ s ) \Gamma_{F(s)}=F(\Gamma_{s})
  26. s = - 1 / k s={-1/k}
  27. F ( s ) F(s)
  28. N N
  29. N = Z - P N=Z-P
  30. Z Z
  31. P P
  32. 1 + k F ( s ) 1+kF(s)
  33. F ( s ) F(s)
  34. ฮ“ s \Gamma_{s}
  35. F ( s ) F(s)
  36. ฮ“ s \Gamma_{s}
  37. j ฯ‰ j\omega
  38. 0 - j โˆž 0-j\infty
  39. 0 + j โˆž 0+j\infty
  40. r โ†’ โˆž r\to\infty
  41. 0 + j โˆž 0+j\infty
  42. 0 - j โˆž 0-j\infty
  43. 1 + G ( s ) 1+G(s)
  44. 1 + G ( s ) 1+G(s)
  45. 1 + G ( s ) 1+G(s)
  46. 1 + G ( s ) 1+G(s)
  47. G ( s ) G(s)
  48. G ( s ) G(s)
  49. 1 + G ( s ) 1+G(s)
  50. 1 + G ( s ) 1+G(s)
  51. 1 + G ( s ) 1+G(s)
  52. G ( s ) G(s)
  53. ฮ“ s \Gamma_{s}
  54. P P
  55. G ( s ) G(s)
  56. ฮ“ s \Gamma_{s}
  57. Z Z
  58. 1 + G ( s ) 1+G(s)
  59. ฮ“ s \Gamma_{s}
  60. Z Z
  61. G ( s ) G(s)
  62. ฮ“ G ( s ) \Gamma_{G(s)}
  63. ( - 1 + j 0 ) (-1+j0)
  64. N N
  65. N = Z - P N=Z-P
  66. - 1 + j 0 -1+j0
  67. G ( s ) G(s)
  68. 0 + j ฯ‰ 0+j\omega
  69. 0 + j ฯ‰ 0+j\omega
  70. r โ†’ 0 r\to 0
  71. 0 + j ฯ‰ 0+j\omega
  72. 0 + j ( ฯ‰ - r ) 0+j(\omega-r)
  73. 0 + j ( ฯ‰ + r ) 0+j(\omega+r)
  74. G ( s ) G(s)
  75. - l ฯ€ -l\pi
  76. l l
  77. T ( s ) = k G ( s ) 1 + k G ( s ) T(s)=\frac{kG(s)}{1+kG(s)}
  78. D ( s ) = 1 + k G ( s ) = 0 D(s)=1+kG(s)=0
  79. ฮ“ s \Gamma_{s}
  80. G ( s ) G(s)
  81. - 1 2 ฯ€ i โˆฎ ฮ“ s D โ€ฒ ( s ) D ( s ) d s = N = Z - P -{{1}\over{2\pi i}}\oint_{\Gamma_{s}}{D^{\prime}(s)\over D(s)}\,ds=N=Z-P
  82. Z Z
  83. D ( s ) D(s)
  84. P P
  85. D ( s ) D(s)
  86. Z = N + P Z=N+P
  87. Z = - 1 2 ฯ€ i โˆฎ ฮ“ s D โ€ฒ ( s ) D ( s ) d s + P Z=-{{1}\over{2\pi i}}\oint_{\Gamma_{s}}{D^{\prime}(s)\over D(s)}\,ds+P
  88. D ( s ) = 1 + k G ( s ) D(s)=1+kG(s)
  89. G ( s ) G(s)
  90. P P
  91. G ( s ) G(s)
  92. u ( s ) = D ( s ) u(s)=D(s)
  93. N = - 1 2 ฯ€ i โˆฎ ฮ“ s D โ€ฒ ( s ) D ( s ) d s = - 1 2 ฯ€ i โˆฎ u ( ฮ“ s ) 1 u d u N=-{{1}\over{2\pi i}}\oint_{\Gamma_{s}}{D^{\prime}(s)\over D(s)}\,ds=-{{1}% \over{2\pi i}}\oint_{u(\Gamma_{s})}{1\over u}\,du
  94. v ( u ) = u - 1 k v(u)={{u-1}\over{k}}
  95. N = - 1 2 ฯ€ i โˆฎ u ( ฮ“ s ) 1 u d u = - 1 2 ฯ€ i โˆฎ v ( u ( ฮ“ s ) ) 1 v + 1 / k d v N=-{{1}\over{2\pi i}}\oint_{u(\Gamma_{s})}{1\over u}\,du=-{{1}\over{2\pi i}}% \oint_{v(u(\Gamma_{s}))}{1\over{v+1/k}}\,dv
  96. v ( u ( ฮ“ s ) ) = D ( ฮ“ s ) - 1 k = G ( ฮ“ s ) v(u(\Gamma_{s}))={{D(\Gamma_{s})-1}\over{k}}=G(\Gamma_{s})
  97. G ( s ) G(s)
  98. N = - 1 2 ฯ€ i โˆฎ G ( ฮ“ s ) ) 1 v + 1 / k d v N=-{{1}\over{2\pi i}}\oint_{G(\Gamma_{s}))}{1\over{v+1/k}}\,dv
  99. - 1 / k -1/k
  100. Z = N + P = (number of times the Nyquist plot encircles -1/k clockwise) + (number of poles of G(s) in ORHP) Z=N+P=\,\text{(number of times the Nyquist plot encircles -1/k clockwise)}+\,% \text{(number of poles of G(s) in ORHP)}
  101. T ( s ) T(s)
  102. Z Z
  103. G ( s ) G(s)
  104. l l
  105. ฯ‰ = 0 \omega=0
  106. l l
  107. G ( s ) G(s)
  108. G ( s ) G(s)
  109. G ( s ) G(s)
  110. G ( s ) G(s)
  111. - 1 + j 0 -1+j0
  112. j ฯ‰ j\omega

Object_theory.html

  1. F [ X ] [ Y ] {{F}[X]}[Y]

Omitted-variable_bias.html

  1. y = a + b x + c z + u y=a+bx+cz+u
  2. z = d + f x + e z=d+fx+e
  3. y = ( a + c d ) + ( b + c f ) x + ( u + c e ) . y=(a+cd)+(b+cf)x+(u+ce).
  4. y i = x i ฮฒ + z i ฮด + u i , i = 1 , โ€ฆ , n y_{i}=x_{i}\beta+z_{i}\delta+u_{i},\qquad i=1,\dots,n
  5. X = [ x 1 โ‹ฎ x n ] โˆˆ โ„ n ร— p , X=\left[\begin{array}[]{c}x_{1}\\ \vdots\\ x_{n}\end{array}\right]\in\mathbb{R}^{n\times p},
  6. Y = [ y 1 โ‹ฎ y n ] , Z = [ z 1 โ‹ฎ z n ] , U = [ u 1 โ‹ฎ u n ] โˆˆ โ„ n ร— 1 . Y=\left[\begin{array}[]{c}y_{1}\\ \vdots\\ y_{n}\end{array}\right],\quad Z=\left[\begin{array}[]{c}z_{1}\\ \vdots\\ z_{n}\end{array}\right],\quad U=\left[\begin{array}[]{c}u_{1}\\ \vdots\\ u_{n}\end{array}\right]\in\mathbb{R}^{n\times 1}.
  7. ฮฒ ^ = ( X โ€ฒ X ) - 1 X โ€ฒ Y \hat{\beta}=(X^{\prime}X)^{-1}X^{\prime}Y\,
  8. ฮฒ ^ = ( X โ€ฒ X ) - 1 X โ€ฒ ( X ฮฒ + Z ฮด + U ) = ( X โ€ฒ X ) - 1 X โ€ฒ X ฮฒ + ( X โ€ฒ X ) - 1 X โ€ฒ Z ฮด + ( X โ€ฒ X ) - 1 X โ€ฒ U = ฮฒ + ( X โ€ฒ X ) - 1 X โ€ฒ Z ฮด + ( X โ€ฒ X ) - 1 X โ€ฒ U . \begin{aligned}\displaystyle\hat{\beta}&\displaystyle=(X^{\prime}X)^{-1}X^{% \prime}(X\beta+Z\delta+U)\\ &\displaystyle=(X^{\prime}X)^{-1}X^{\prime}X\beta+(X^{\prime}X)^{-1}X^{\prime}% Z\delta+(X^{\prime}X)^{-1}X^{\prime}U\\ &\displaystyle=\beta+(X^{\prime}X)^{-1}X^{\prime}Z\delta+(X^{\prime}X)^{-1}X^{% \prime}U.\end{aligned}
  9. E [ ฮฒ ^ | X ] \displaystyle E[\hat{\beta}|X]

One-_and_two-tailed_tests.html

  1. X ยฏ . \bar{X}.
  2. 1 / 32 = 0.03125 โ‰ˆ 0.03 1/32=0.03125\approx 0.03
  3. p โ‰ˆ 0.03 p\approx 0.03
  4. 2 / 32 = 0.0625 โ‰ˆ 0.06 2/32=0.0625\approx 0.06
  5. ฮผ \mu
  6. ฮผ 0 , \mu_{0},
  7. ฮผ > ฮผ 0 \mu>\mu_{0}
  8. ฮผ < ฮผ 0 \mu<\mu_{0}
  9. ฮผ โ‰  ฮผ 0 . \mu\neq\mu_{0}.

Open-loop_gain.html

  1. 10 5 10^{5}
  2. A OL = V out ( V + - V - ) , A_{\,\text{OL}}=\frac{V_{\,\text{out}}}{\left(V^{+}-V^{-}\right)},
  3. V + - V - V^{+}-V^{-}

Operating_leverage.html

  1. FC TC = FC FC + VC \frac{\,\text{FC}}{\,\text{TC}}=\frac{\,\text{FC}}{\,\text{FC}+\,\text{VC}}
  2. Debt Assets = Debt Debt + Equity \frac{\,\text{Debt}}{\,\text{Assets}}=\frac{\,\text{Debt}}{\,\text{Debt}+\,% \text{Equity}}
  3. FC VC \frac{\,\text{FC}}{\,\text{VC}}
  4. Debt Equity \frac{\,\text{Debt}}{\,\text{Equity}}
  5. DOL = % change in Operating Income % change in Sales \,\text{DOL}=\frac{\%\,\text{ change in Operating Income}}{\%\,\text{ change % in Sales}}
  6. DOL = Total Contribution Operating Income = Total Contribution Total Contribution - Fixed Costs = ( P - V ) ร— X ( P - V ) ร— X - FC \,\text{DOL}=\frac{\,\text{Total Contribution}}{\,\text{Operating Income}}=% \frac{\,\text{Total Contribution}}{\,\text{Total Contribution}-\,\text{Fixed % Costs}}=\frac{(\,\text{P}-\,\text{V})\times\,\text{X}}{(\,\text{P}-\,\text{V})% \times\,\text{X}-\,\text{FC}}
  7. DOL = Contribution Margin Ratio Operating Margin \,\text{DOL}=\frac{\,\text{Contribution Margin Ratio}}{\,\text{Operating % Margin}}
  8. $ 40m $ 30m = 1 1 3 โ‰ˆ 1.33 \frac{\$\,\text{40m}}{\$\,\text{30m}}=1\frac{1}{3}\approx 1.33
  9. ร— 1 1 3 = \times 1\frac{1}{3}=
  10. Operating Income Sales = Unit Price - Unit Variable Cost Unit Price \frac{\,\text{Operating Income}}{\,\text{Sales}}=\frac{\,\text{Unit Price}-\,% \text{Unit Variable Cost}}{\,\text{Unit Price}}

Operating_margin.html

  1. Operating margin = ( Operating income Revenue ) \mathrm{Operating\ margin}=\left(\frac{\mathrm{Operating\ income}}{\mathrm{% Revenue}}\right)
  2. Operating margin = ( 6 , 318 20 , 088 ) = 31.45 % ยฏ ยฏ \mathrm{Operating\ margin}=\left(\frac{6,318}{20,088}\right)=\underline{% \underline{31.45\%}}

Operative_temperature.html

  1. t o t_{o}
  2. t o = ( h r t m r + h c t a ) h r + h c t_{o}=\frac{(h_{r}t_{mr}+h_{c}t_{a})}{h_{r}+h_{c}}
  3. h c h_{c}
  4. h r h_{r}
  5. t a t_{a}
  6. t m r t_{mr}
  7. t o = ( t m r + ( t a ร— 10 v ) ) 1 + 10 v t_{o}=\frac{(t_{mr}+(t_{a}\times\sqrt{10v}))}{1+\sqrt{10v}}
  8. v v
  9. t a t_{a}
  10. t m r t_{mr}
  11. t o = ( t a + t m r ) 2 t_{o}=\frac{(t_{a}+t_{mr})}{2}
  12. t a t_{a}
  13. t m r t_{mr}

Operator_product_expansion.html

  1. z z
  2. w w
  3. 1 / ( z - w ) 1/(z-w)
  4. z z
  5. z ยฏ \bar{z}
  6. log z \log z
  7. y y
  8. A A
  9. B B
  10. O O
  11. y y
  12. x โˆˆ O โˆ– { y } x\in O\setminus\{y\}
  13. A ( x ) B ( y ) = โˆ‘ i c i ( x - y ) i C i ( y ) A(x)B(y)=\sum_{i}c_{i}(x-y)^{i}C_{i}(y)
  14. O โˆ– { y } O\setminus\{y\}
  15. O โˆ– { y } O\setminus\{y\}
  16. F ( x , y ) โˆผ G ( x , y ) F(x,y)\sim G(x,y)

Optical_autocorrelation.html

  1. E ( t ) E(t)
  2. A ( ฯ„ ) = โˆซ - โˆž + โˆž E ( t ) E * ( t - ฯ„ ) d t A(\tau)=\int_{-\infty}^{+\infty}E(t)E^{*}(t-\tau)dt
  3. E ( t ) E(t)
  4. E ( t ) E(t)
  5. E ( t ) E(t)
  6. E ( t - ฯ„ ) E(t-\tau)
  7. E ( t ) E(t)
  8. I M I_{M}
  9. ฯ„ \tau
  10. I M ( ฯ„ ) = โˆซ - โˆž + โˆž | E ( t ) + E ( t - ฯ„ ) | 2 d t I_{M}(\tau)=\int_{-\infty}^{+\infty}|E(t)+E(t-\tau)|^{2}dt
  11. I M ( ฯ„ ) I_{M}(\tau)
  12. A ( ฯ„ ) A(\tau)
  13. E ( t ) E(t)
  14. E ( t ) E(t)
  15. I ( t ) = | E ( t ) | 2 I(t)=|E(t)|^{2}
  16. A ( ฯ„ ) = โˆซ - โˆž + โˆž I ( t ) I ( t - ฯ„ ) d t A(\tau)=\int_{-\infty}^{+\infty}I(t)I(t-\tau)dt
  17. ( E ( t ) + E ( t - ฯ„ ) ) 2 (E(t)+E(t-\tau))^{2}
  18. E ( t ) E ( t - ฯ„ ) E(t)E(t-\tau)
  19. I M ( ฯ„ ) = โˆซ - โˆž + โˆž | E ( t ) E ( t - ฯ„ ) | 2 d t = โˆซ - โˆž + โˆž I ( t ) I ( t - ฯ„ ) d t I_{M}(\tau)=\int_{-\infty}^{+\infty}|E(t)E(t-\tau)|^{2}dt=\int_{-\infty}^{+% \infty}I(t)I(t-\tau)dt
  20. I M ( ฯ„ ) I_{M}(\tau)
  21. A ( ฯ„ ) A(\tau)
  22. 2 \sqrt{2}
  23. I M ( ฯ„ ) = โˆซ - โˆž + โˆž | ( E ( t ) + E ( t - ฯ„ ) ) 2 | 2 d t I_{M}(\tau)=\int_{-\infty}^{+\infty}|(E(t)+E(t-\tau))^{2}|^{2}dt
  24. I M ( ฯ„ ) I_{M}(\tau)
  25. T ( w ) = โˆซ w / 2 1 โˆซ 0 1 - x 2 f ( x , y ) f * ( x - w , y ) d y d x โˆซ 0 1 โˆซ 0 1 - x 2 f ( x , y ) 2 d y d x T(w)=\frac{\int_{w/2}^{1}\int_{0}^{\sqrt{1-x^{2}}}f(x,y)f^{*}(x-w,y)dydx}{\int% _{0}^{1}\int_{0}^{\sqrt{1-x^{2}}}f(x,y)^{2}dydx}

Optical_medium.html

  1. ฮท = E x H y \eta={E_{x}\over H_{y}}
  2. E x E_{x}
  3. H y H_{y}
  4. ฮท = ฮผ ฮต . \eta=\sqrt{\mu\over\varepsilon}\ .
  5. Z 0 = ฮผ 0 ฮต 0 . Z_{0}=\sqrt{\mu_{0}\over\varepsilon_{0}}\ .
  6. c w = ฮฝ ฮป c_{w}=\nu\lambda
  7. ฮฝ \nu
  8. ฮป \lambda
  9. c w = ฯ‰ k , c_{w}={\omega\over k}\ ,
  10. ฯ‰ \omega
  11. k k
  12. ฮฒ \beta
  13. k k
  14. c 0 = 1 ฮต 0 ฮผ 0 , c_{0}={1\over\sqrt{\varepsilon_{0}\mu_{0}}}\ ,
  15. ฮต 0 \varepsilon_{0}
  16. ฮผ 0 ~{}\mu_{0}

Optical_resolution.html

  1. r r
  2. r = 1.22 ฮป 2 n sin ฮธ = 0.61 ฮป NA r=\frac{1.22\lambda}{2n\sin{\theta}}=\frac{0.61\lambda}{\mathrm{NA}}
  3. r r
  4. ฮป \lambda
  5. ฮป \lambda
  6. n n
  7. ฮธ \theta
  8. NA \mathrm{NA}
  9. r = 0.4 ฮป NA r=\frac{0.4\lambda}{\mathrm{NA}}
  10. r = ฮป 2 n sin ฮธ = ฮป 2 N A r=\frac{\lambda}{2n\sin{\theta}}=\frac{\lambda}{2\mathrm{NA}}
  11. r = 1.22 ฮป NA o b j + NA c o n d r=\frac{1.22\lambda}{\mathrm{NA}_{obj}+\mathrm{NA}_{cond}}
  12. NA o b j + NA c o n d = 2 N A o b j \mathrm{NA}_{obj}+\mathrm{NA}_{cond}=2\mathrm{NA}_{obj}
  13. ฮธ = 1.22 ฮป D \theta=1.22\frac{\lambda}{D}
  14. ๐Ž๐“๐… ( ฮพ , ฮท ) = ๐Œ๐“๐… ( ฮพ , ฮท ) โ‹… ๐๐“๐… ( ฮพ , ฮท ) \mathbf{OTF(\xi,\eta)}=\mathbf{MTF(\xi,\eta)}\cdot\mathbf{PTF(\xi,\eta)}
  15. ๐Œ๐“๐… ( ฮพ , ฮท ) = | ๐Ž๐“๐… ( ฮพ , ฮท ) | \mathbf{MTF(\xi,\eta)}=|\mathbf{OTF(\xi,\eta)}|
  16. ๐๐“๐… ( ฮพ , ฮท ) = e - i 2 โ‹… ฯ€ โ‹… ฮป ( ฮพ , ฮท ) \mathbf{PTF(\xi,\eta)}=e^{-i2\cdot\pi\cdot\lambda(\xi,\eta)}
  17. ( ฮพ , ฮท ) (\xi,\eta)
  18. F F = a โ‹… b c โ‹… d FF=\frac{a\cdot b}{c\cdot d}
  19. comb ( ฮพ , ฮท ) \operatorname{comb}(\xi,\eta)
  20. sinc ( ฮพ , ฮท ) \operatorname{sinc}(\xi,\eta)
  21. sinc ( ฮพ , ฮท ) \operatorname{sinc}(\xi,\eta)
  22. ๐’ ( x , y ) = [ comb ( x c , y d ) * rect ( x a , y b ) ] โ‹… rect ( x M โ‹… c , y N โ‹… d ) \mathbf{S}(x,y)=\left[\operatorname{comb}\left(\frac{x}{c},\frac{y}{d}\right)*% \operatorname{rect}\left(\frac{x}{a},\frac{y}{b}\right)\right]\cdot% \operatorname{rect}\left(\frac{x}{M\cdot c},\frac{y}{N\cdot d}\right)
  23. ๐Œ๐“๐… ๐ฌ๐ž๐ง๐ฌ๐จ๐ซ ( ฮพ , ฮท ) \mathbf{MTF_{sensor}}(\xi,\eta)
  24. = โ„ฑ โ„ฑ ( ๐’ ( x , y ) ) =\mathcal{FF}(\mathbf{S}(x,y))
  25. = [ sinc ( ( M โ‹… c ) โ‹… ฮพ , ( N โ‹… d ) โ‹… ฮท ) * comb ( c โ‹… ฮพ , d โ‹… ฮท ) ] โ‹… sinc ( a โ‹… ฮพ , b โ‹… ฮท ) =[\operatorname{sinc}((M\cdot c)\cdot\xi,(N\cdot d)\cdot\eta)*\operatorname{% comb}(c\cdot\xi,d\cdot\eta)]\cdot\operatorname{sinc}(a\cdot\xi,b\cdot\eta)
  26. ๐ˆ๐ฆ๐š๐ ๐ž ( ๐ฑ , ๐ฒ ) = \mathbf{Image(x,y)=}
  27. ๐Ž๐›๐ฃ๐ž๐œ๐ญ ( ๐ฑ , ๐ฒ ) * ๐๐’๐… ๐š๐ญ๐ฆ๐จ๐ฌ๐ฉ๐ก๐ž๐ซ๐ž ( ๐ฑ , ๐ฒ ) * \mathbf{Object(x,y)*PSF_{atmosphere}(x,y)*}
  28. ๐๐’๐… ๐ฅ๐ž๐ง๐ฌ ( ๐ฑ , ๐ฒ ) * ๐๐’๐… ๐ฌ๐ž๐ง๐ฌ๐จ๐ซ ( ๐ฑ , ๐ฒ ) * \mathbf{PSF_{lens}(x,y)*PSF_{sensor}(x,y)*}
  29. ๐๐’๐… ๐ญ๐ซ๐š๐ง๐ฌ๐ฆ๐ข๐ฌ๐ฌ๐ข๐จ๐ง ( ๐ฑ , ๐ฒ ) * ๐๐’๐… ๐๐ข๐ฌ๐ฉ๐ฅ๐š๐ฒ ( ๐ฑ , ๐ฒ ) \mathbf{PSF_{transmission}(x,y)*PSF_{display}(x,y)}
  30. ๐Œ๐“๐… ๐ฌ๐ฒ๐ฌ ( ฮพ , ฮท ) = \mathbf{MTF_{sys}(\xi,\eta)=}
  31. ๐Œ๐“๐… ๐š๐ญ๐ฆ๐จ๐ฌ๐ฉ๐ก๐ž๐ซ๐ž ( ฮพ , ฮท ) โ‹… ๐Œ๐“๐… ๐ฅ๐ž๐ง๐ฌ ( ฮพ , ฮท ) โ‹… \mathbf{MTF_{atmosphere}(\xi,\eta)\cdot MTF_{lens}(\xi,\eta)\cdot}
  32. ๐Œ๐“๐… ๐ฌ๐ž๐ง๐ฌ๐จ๐ซ ( ฮพ , ฮท ) โ‹… ๐Œ๐“๐… ๐ญ๐ซ๐š๐ง๐ฌ๐ฆ๐ข๐ฌ๐ฌ๐ข๐จ๐ง ( ฮพ , ฮท ) โ‹… \mathbf{MTF_{sensor}(\xi,\eta)\cdot MTF_{transmission}(\xi,\eta)\cdot}
  33. ๐Œ๐“๐… ๐๐ข๐ฌ๐ฉ๐ฅ๐š๐ฒ ( ฮพ , ฮท ) \mathbf{MTF_{display}(\xi,\eta)}
  34. MTF s ( ฮฝ ) = e - 3.44 โ‹… ( ฮป f ฮฝ / r 0 ) 5 / 3 โ‹… [ 1 - b โ‹… ( ฮป f ฮฝ / D ) 1 / 3 ] \operatorname{MTF}_{s}(\nu)=e^{-3.44\cdot(\lambda f\nu/r_{0})^{5/3}\cdot[1-b% \cdot(\lambda f\nu/D)^{1/3}]}
  35. ฮฝ \nu
  36. ฮป \lambda
  37. r 0 r_{0}
  38. contrast = C max - C min C max + C min \,\text{contrast}=\frac{C_{\max}-C_{\min}}{C_{\max}+C_{\min}}
  39. C max C_{\max}
  40. C min C_{\min}
  41. R e s o l u t i o n = 2 g r o u p + e l e m e n t - 1 6 Resolution=2^{{group}+{\frac{element-1}{6}}}

Optimal_asymmetric_encryption_padding.html

  1. f f
  2. f f

Orbital_hybridisation.html

  1. 3 \sqrt{3}
  2. 3 \sqrt{3}

Ordinal_utility.html

  1. u ( x , y ) u(x,y)
  2. g ( x , y ) โ‰ก f ( u ( x , y ) ) , g(x,y)\equiv f(u(x,y)),

Ordinary_least_squares.html

  1. y i = x i T ฮฒ + ฮต i , y_{i}=x_{i}^{T}\beta+\varepsilon_{i},\,
  2. y = X ฮฒ + ฮต , y=X\beta+\varepsilon,\,
  3. E [ ฮต | X ] = 0. \operatorname{E}[\,\varepsilon|X\,]=0.
  4. Pr [ rank ( X ) = p ] = 1. \Pr\!\big[\,\operatorname{rank}(X)=p\,\big]=1.
  5. Var [ ฮต โˆฃ X ] = ฯƒ 2 I n , \operatorname{Var}[\,\varepsilon\mid X\,]=\sigma^{2}I_{n},
  6. ฮต โˆฃ X โˆผ ๐’ฉ ( 0 , ฯƒ 2 I n ) . \varepsilon\mid X\sim\mathcal{N}(0,\sigma^{2}I_{n}).
  7. S ( b ) = โˆ‘ i = 1 n ( y i - x i T b ) 2 = ( y - X b ) T ( y - X b ) , S(b)=\sum_{i=1}^{n}(y_{i}-x_{i}^{T}b)^{2}=(y-Xb)^{T}(y-Xb),
  8. b = ฮฒ ^ b=\hat{\beta}
  9. ฮฒ ^ = arg min b โˆˆ โ„ p S ( b ) = ( 1 n โˆ‘ i = 1 n x i x i T ) - 1 โ‹… 1 n โˆ‘ i = 1 n x i y i \hat{\beta}={\rm arg}\min_{b\in\mathbb{R}^{p}}S(b)=\bigg(\frac{1}{n}\sum_{i=1}% ^{n}x_{i}x_{i}^{T}\bigg)^{\!-1}\!\!\cdot\,\frac{1}{n}\sum_{i=1}^{n}x_{i}y_{i}
  10. ฮฒ ^ = ( X T X ) - 1 X T y . \hat{\beta}=(X^{T}X)^{-1}X^{T}y\ .
  11. y ^ = X ฮฒ ^ = P y , \hat{y}=X\hat{\beta}=Py,
  12. ฮต ^ = y - X ฮฒ ^ = M y = M ฮต . \hat{\varepsilon}=y-X\hat{\beta}=My=M\varepsilon.
  13. s 2 = ฮต ^ T ฮต ^ n - p = y T M y n - p = S ( ฮฒ ^ ) n - p , ฯƒ ^ 2 = n - p n s 2 s^{2}=\frac{\hat{\varepsilon}^{T}\hat{\varepsilon}}{n-p}=\frac{y^{T}My}{n-p}=% \frac{S(\hat{\beta})}{n-p},\qquad\hat{\sigma}^{2}=\frac{n-p}{n}\;s^{2}
  14. ฯƒ ^ 2 \scriptstyle\hat{\sigma}^{2}
  15. R 2 = โˆ‘ ( y ^ i - y ยฏ ) 2 โˆ‘ ( y i - y ยฏ ) 2 = y T P T L P y y T L y = 1 - y T M y y T L y = 1 - SSR TSS R^{2}=\frac{\sum(\hat{y}_{i}-\overline{y})^{2}}{\sum(y_{i}-\overline{y})^{2}}=% \frac{y^{T}P^{T}LPy}{y^{T}Ly}=1-\frac{y^{T}My}{y^{T}Ly}=1-\frac{\rm SSR}{\rm TSS}
  16. y i = ฮฑ + ฮฒ x i + ฮต i . y_{i}=\alpha+\beta x_{i}+\varepsilon_{i}.
  17. ฮฒ ^ = โˆ‘ x i y i - 1 n โˆ‘ x i โˆ‘ y i โˆ‘ x i 2 - 1 n ( โˆ‘ x i ) 2 = Cov [ x , y ] ฯƒ x 2 , ฮฑ ^ = y ยฏ - ฮฒ ^ x ยฏ . \hat{\beta}=\frac{\sum{x_{i}y_{i}}-\frac{1}{n}\sum{x_{i}}\sum{y_{i}}}{\sum{x_{% i}^{2}}-\frac{1}{n}(\sum{x_{i}})^{2}}=\frac{\mathrm{Cov}[x,y]}{\sigma^{2}_{x}}% ,\quad\hat{\alpha}=\overline{y}-\hat{\beta}\,\overline{x}\ .
  18. ฯƒ x 2 \sigma_{x}^{2}
  19. ฮฒ ^ \scriptstyle\hat{\beta}
  20. ฮฒ ^ = arg min ฮฒ โˆฅ y - X ฮฒ โˆฅ , \hat{\beta}={\rm arg}\min_{\beta}\,\lVert y-X\beta\rVert,
  21. ฮฒ ^ \scriptstyle\hat{\beta}
  22. E [ x i ( y i - x i T ฮฒ ) ] = 0. \mathrm{E}\big[\,x_{i}(y_{i}-x_{i}^{T}\beta)\,\big]=0.
  23. ฮฒ ^ \scriptstyle\hat{\beta}
  24. E [ ฮฒ ^ โˆฃ X ] = ฮฒ , E [ s 2 โˆฃ X ] = ฯƒ 2 . \operatorname{E}[\,\hat{\beta}\mid X\,]=\beta,\quad\operatorname{E}[\,s^{2}% \mid X\,]=\sigma^{2}.
  25. ฮฒ ^ \scriptstyle\hat{\beta}
  26. Var [ ฮฒ ^ โˆฃ X ] = ฯƒ 2 ( X T X ) - 1 . \operatorname{Var}[\,\hat{\beta}\mid X\,]=\sigma^{2}(X^{T}X)^{-1}.
  27. ฮฒ ^ j \scriptstyle\hat{\beta}_{j}
  28. s . e ^ ( ฮฒ ^ j ) = s 2 ( X T X ) j j - 1 \widehat{\operatorname{s.\!e}}(\hat{\beta}_{j})=\sqrt{s^{2}(X^{T}X)^{-1}_{jj}}
  29. ฮฒ ^ \scriptstyle\hat{\beta}
  30. Cov [ ฮฒ ^ , ฮต ^ โˆฃ X ] = 0. \operatorname{Cov}[\,\hat{\beta},\hat{\varepsilon}\mid X\,]=0.
  31. ฮฒ ^ \scriptstyle\hat{\beta}
  32. ฮฒ ~ \scriptstyle\tilde{\beta}
  33. Var [ ฮฒ ~ โˆฃ X ] - Var [ ฮฒ ^ โˆฃ X ] โ‰ฅ 0 \operatorname{Var}[\,\tilde{\beta}\mid X\,]-\operatorname{Var}[\,\hat{\beta}% \mid X\,]\geq 0
  34. ฮฒ ^ \scriptstyle\hat{\beta}
  35. ฮฒ ^ โˆผ ๐’ฉ ( ฮฒ , ฯƒ 2 ( X T X ) - 1 ) \hat{\beta}\ \sim\ \mathcal{N}\big(\beta,\ \sigma^{2}(X^{T}X)^{-1}\big)
  36. s 2 โˆผ ฯƒ 2 n - p โ‹… ฯ‡ n - p 2 s^{2}\ \sim\ \frac{\sigma^{2}}{n-p}\cdot\chi^{2}_{n-p}
  37. ฮฒ ^ \scriptstyle\hat{\beta}
  38. ฮฒ ^ \scriptstyle\hat{\beta}
  39. ฮฒ ^ ( j ) - ฮฒ ^ = - 1 1 - h j ( X T X ) - 1 x j T ฮต ^ j , \hat{\beta}^{(j)}-\hat{\beta}=-\frac{1}{1-h_{j}}(X^{T}X)^{-1}x_{j}^{T}\hat{% \varepsilon}_{j}\,,
  40. y ^ j ( j ) - y ^ j = x j T ฮฒ ^ ( j ) - x j T ฮฒ ^ = - h j 1 - h j ฮต ^ j \hat{y}_{j}^{(j)}-\hat{y}_{j}=x_{j}^{T}\hat{\beta}^{(j)}-x_{j}^{T}\hat{\beta}=% -\frac{h_{j}}{1-h_{j}}\,\hat{\varepsilon}_{j}
  41. y = X 1 ฮฒ 1 + X 2 ฮฒ 2 + ฮต , y=X_{1}\beta_{1}+X_{2}\beta_{2}+\varepsilon,
  42. ฮต ^ \hat{\varepsilon}
  43. ฮฒ ^ 2 \scriptstyle\hat{\beta}_{2}
  44. M 1 y = M 1 X 2 ฮฒ 2 + ฮท , M_{1}y=M_{1}X_{2}\beta_{2}+\eta\,,
  45. H 0 : Q T ฮฒ = c , H_{0}\colon\quad Q^{T}\beta=c,\,
  46. ฮฒ ^ c = ฮฒ ^ - ( X T X ) - 1 Q ( Q T ( X T X ) - 1 Q ) - 1 ( Q T ฮฒ ^ - c ) \hat{\beta}^{c}=\hat{\beta}-(X^{T}X)^{-1}Q\Big(Q^{T}(X^{T}X)^{-1}Q\Big)^{-1}(Q% ^{T}\hat{\beta}-c)
  47. ฮฒ ^ c = R ( R T X T X R ) - 1 R T X T y + ( I p - R ( R T X T X R ) - 1 R T X T X ) Q ( Q T Q ) - 1 c , \hat{\beta}^{c}=R(R^{T}X^{T}XR)^{-1}R^{T}X^{T}y+\Big(I_{p}-R(R^{T}X^{T}XR)^{-1% }R^{T}X^{T}X\Big)Q(Q^{T}Q)^{-1}c,
  48. ฮฒ ^ \hat{\beta}
  49. ฯƒ ^ 2 \hat{\sigma}^{2}
  50. ฮฒ ^ \hat{\beta}
  51. ( ฮฒ ^ - ฮฒ ) โ†’ ๐‘‘ ๐’ฉ ( 0 , ฯƒ 2 Q x x - 1 ) , (\hat{\beta}-\beta)\ \xrightarrow{d}\ \mathcal{N}\big(0,\;\sigma^{2}Q_{xx}^{-1% }\big),
  52. Q x x = X T X . Q_{xx}=X^{T}X.
  53. ฮฒ ^ \hat{\beta}
  54. ฮฒ j โˆˆ [ ฮฒ ^ j ยฑ q 1 - ฮฑ / 2 ๐’ฉ ( 0 , 1 ) ฯƒ ^ 2 [ Q x x - 1 ] j j ] \beta_{j}\in\bigg[\ \hat{\beta}_{j}\pm q^{\mathcal{N}(0,1)}_{1-\alpha/2}\!% \sqrt{\hat{\sigma}^{2}\big[Q_{xx}^{-1}\big]_{jj}}\ \bigg]
  55. ( ฯƒ ^ 2 - ฯƒ 2 ) โ†’ ๐‘‘ ๐’ฉ ( 0 , E [ ฮต i 4 ] - ฯƒ 4 ) . (\hat{\sigma}^{2}-\sigma^{2})\ \xrightarrow{d}\ \mathcal{N}\big(0,\;% \operatorname{E}[\varepsilon_{i}^{4}]-\sigma^{4}\big).
  56. x 0 x_{0}
  57. y 0 = x 0 T ฮฒ y_{0}=x_{0}^{T}\beta
  58. y ^ 0 = x 0 T ฮฒ ^ \hat{y}_{0}=x_{0}^{T}\hat{\beta}
  59. ฮฒ ^ \hat{\beta}
  60. ( y ^ 0 - y 0 ) โ†’ ๐‘‘ ๐’ฉ ( 0 , ฯƒ 2 x 0 T Q x x - 1 x 0 ) , (\hat{y}_{0}-y_{0})\ \xrightarrow{d}\ \mathcal{N}\big(0,\;\sigma^{2}x_{0}^{T}Q% _{xx}^{-1}x_{0}\big),
  61. y 0 y_{0}
  62. y 0 โˆˆ [ x 0 T ฮฒ ^ ยฑ q 1 - ฮฑ / 2 ๐’ฉ ( 0 , 1 ) ฯƒ ^ 2 x 0 T Q x x - 1 x 0 ] y_{0}\in\bigg[\ x_{0}^{T}\hat{\beta}\pm q^{\mathcal{N}(0,1)}_{1-\alpha/2}\!% \sqrt{\hat{\sigma}^{2}x_{0}^{T}Q_{xx}^{-1}x_{0}}\ \bigg]
  63. w i = ฮฒ 1 + ฮฒ 2 h i + ฮฒ 3 h i 2 + ฮต i . w_{i}=\beta_{1}+\beta_{2}h_{i}+\beta_{3}h_{i}^{2}+\varepsilon_{i}.
  64. ฮฒ \beta
  65. h h
  66. h 2 h^{2}
  67. ฯƒ ^ j = ( ฯƒ ^ 2 [ Q x x - 1 ] j j ) 1 / 2 \hat{\sigma}_{j}=\big(\hat{\sigma}^{2}[Q_{xx}^{-1}]_{jj}\big)^{1/2}
  68. t = ฮฒ ^ j / ฯƒ ^ j t=\hat{\beta}_{j}/\hat{\sigma}_{j}
  69. R 2 R^{2}
  70. R 2 R^{2}
  71. R ยฏ 2 = 1 - n - 1 n - p ( 1 - R 2 ) {\overline{R}}^{2}=1-\tfrac{n-1}{n-p}(1-R^{2})
  72. y ^ \hat{y}

Organotin_chemistry.html

  1. โ† โ†’ \overrightarrow{\leftarrow}
  2. โ† โ†’ \overrightarrow{\leftarrow}

Orientation_(vector_space).html

  1. A 1 = ( cos ฮฑ - sin ฮฑ 0 sin ฮฑ cos ฮฑ 0 0 0 1 ) {A}_{1}=\begin{pmatrix}\cos\alpha&-\sin\alpha&0\\ \sin\alpha&\cos\alpha&0\\ 0&0&1\end{pmatrix}
  2. A 2 = ( 1 0 0 0 1 0 0 0 - 1 ) {A}_{2}=\begin{pmatrix}1&0&0\\ 0&1&0\\ 0&0&-1\end{pmatrix}
  3. { ยฑ 1 } \{\pm 1\}
  4. GL n โ†’ ยฑ 1 \operatorname{GL}_{n}\to\pm 1
  5. { ยฑ 1 } \{\pm 1\}
  6. K K
  7. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  8. T : V โ†’ V T\colon V\to V
  9. ฯ€ 0 ( GL ( V ) ) = ( GL ( V ) / GL + ( V ) = { ยฑ 1 } \pi_{0}(\operatorname{GL}(V))=(\operatorname{GL}(V)/\operatorname{GL}^{+}(V)=% \{\pm 1\}
  10. V V
  11. GL ( V ) \operatorname{GL}(V)
  12. V n ( V ) / GL + ( V ) V_{n}(V)/\operatorname{GL}^{+}(V)
  13. { ยฑ 1 } \{\pm 1\}

Orifice_plate.html

  1. q v = q m ฯ 1 q_{v}=\frac{q_{m}}{\rho_{1}}
  2. q m = C 1 - ฮฒ 4 ฯต ฯ€ 4 d 2 2 ฯ 1 ฮ” p q_{m}=\frac{C}{\sqrt{1-\beta^{4}}}\;\epsilon\;\frac{\pi}{4}\;d^{2}\;\sqrt{2\;% \rho_{1}\Delta p\;}
  3. C = 0.5961 + 0.0261 ฮฒ 2 - 0.216 ฮฒ 8 + 0.000521 ( 10 6 ฮฒ R e D ) 0.7 + ( 0.0188 + 0.0063 A ) ฮฒ 3.5 ( 10 6 R e D ) 0.3 + ( 0.043 + 0.080 e - 10 L 1 - 0.123 e - 7 L 1 ) ( 1 - 0.11 A ) ฮฒ 4 1 - ฮฒ 4 - 0.031 ( M 2 โ€ฒ - 0.8 M 2 โ€ฒ 1.1 ) ฮฒ 1.3 C=0.5961+0.0261\beta^{2}-0.216\beta^{8}+0.000521\bigg(\frac{10^{6}\beta}{Re_{D% }}\bigg)^{0.7}+(0.0188+0.0063A)\beta^{3.5}\bigg(\frac{10^{6}}{Re_{D}}\bigg)^{0% .3}+(0.043+0.080e^{-10{L_{1}}}-0.123e^{-7{L_{1}}})(1-0.11A)\frac{\beta^{4}}{1-% \beta^{4}}-0.031(M^{\prime}_{2}-0.8{M^{\prime}_{2}}^{1.1})\beta^{1.3}
  4. + 0.011 ( 0.75 - ฮฒ ) ( 2.8 - D 0.025.4 ) +0.011(0.75-\beta)\bigg(2.8-\frac{D}{0.025.4}\bigg)
  5. A = ( 19000 ฮฒ R e D ) 0.8 A=\bigg(\frac{19000\beta}{Re_{D}}\bigg)^{0.8}
  6. M 2 โ€ฒ = 2 L 2 โ€ฒ 1 - ฮฒ M^{\prime}_{2}=\frac{2L^{\prime}_{2}}{1-\beta}
  7. L 1 = L 2 โ€ฒ = 0 L_{1}=L^{\prime}_{2}=0
  8. L 1 = L 2 โ€ฒ = 0.0254 D L_{1}=L^{\prime}_{2}=\frac{0.0254}{D}
  9. L 1 = 1 L_{1}=1
  10. L 2 โ€ฒ = 0.47 L^{\prime}_{2}=0.47
  11. p 2 / p 1 > 0.75 p_{2}/p_{1}>0.75
  12. ฯต = 1 - ( 0.351 + 0.256 ฮฒ 4 + 0.93 ฮฒ 8 ) [ 1 - ( p 2 p 1 ) 1 ฮบ ] \epsilon=1-(0.351+0.256\beta^{4}+0.93\beta^{8})\bigg[1-\bigg(\frac{p_{2}}{p_{1% }}\bigg)^{\frac{1}{\kappa}}\bigg]
  13. ฯต = 1 \epsilon=1
  14. C C
  15. d d
  16. D D
  17. p 1 p_{1}
  18. p 2 p_{2}
  19. q m q_{m}
  20. q v q_{v}
  21. R e D Re_{D}
  22. 4 q m ฯ€ ฮผ D \frac{4q_{m}}{\pi\mu D}
  23. ฮฒ \beta
  24. d D \frac{d}{D}
  25. ฮ” p \Delta p
  26. ฯต \epsilon
  27. ฮบ \kappa
  28. ฮผ \mu
  29. ฯ 1 \rho_{1}
  30. ฮ” ฯ‰ ยฏ ฮ” p = 1 - ฮฒ 1.9 \frac{\Delta\bar{\omega}}{\Delta p}=1-\beta^{1.9}
  31. ฮ” ฯ‰ ยฏ ฮ” p = 1 - ฮฒ 4 ( 1 - C 2 ) - C ฮฒ 2 1 - ฮฒ 4 ( 1 - C 2 ) + C ฮฒ 2 \frac{\Delta\bar{\omega}}{\Delta p}=\frac{\sqrt{1-\beta^{4}(1-C^{2})}-C\beta^{% 2}}{\sqrt{1-\beta^{4}(1-C^{2})}+C\beta^{2}}
  32. ฮ” ฯ‰ ยฏ \Delta\bar{\omega}
  33. P 1 + 1 2 โ‹… ฯ โ‹… V 1 2 = P 2 + 1 2 โ‹… ฯ โ‹… V 2 2 P_{1}+\frac{1}{2}\cdot\rho\cdot V_{1}^{2}=P_{2}+\frac{1}{2}\cdot\rho\cdot V_{2% }^{2}
  34. P 1 - P 2 = 1 2 โ‹… ฯ โ‹… V 2 2 - 1 2 โ‹… ฯ โ‹… V 1 2 P_{1}-P_{2}=\frac{1}{2}\cdot\rho\cdot V_{2}^{2}-\frac{1}{2}\cdot\rho\cdot V_{1% }^{2}
  35. Q = A 1 โ‹… V 1 = A 2 โ‹… V 2 Q=A_{1}\cdot V_{1}=A_{2}\cdot V_{2}
  36. V 1 = Q / A 1 V_{1}=Q/A_{1}
  37. V 2 = Q / A 2 V_{2}=Q/A_{2}
  38. P 1 - P 2 = 1 2 โ‹… ฯ โ‹… ( Q A 2 ) 2 - 1 2 โ‹… ฯ โ‹… ( Q A 1 ) 2 P_{1}-P_{2}=\frac{1}{2}\cdot\rho\cdot\bigg(\frac{Q}{A_{2}}\bigg)^{2}-\frac{1}{% 2}\cdot\rho\cdot\bigg(\frac{Q}{A_{1}}\bigg)^{2}
  39. Q Q
  40. Q = A 2 2 ( P 1 - P 2 ) / ฯ 1 - ( A 2 / A 1 ) 2 Q=A_{2}\;\sqrt{\frac{2\;(P_{1}-P_{2})/\rho}{1-(A_{2}/A_{1})^{2}}}
  41. Q = A 2 1 1 - ( d 2 / d 1 ) 4 2 ( P 1 - P 2 ) / ฯ Q=A_{2}\;\sqrt{\frac{1}{1-(d_{2}/d_{1})^{4}}}\;\sqrt{2\;(P_{1}-P_{2})/\rho}
  42. Q Q
  43. ฮฒ = d 2 / d 1 \beta=d_{2}/d_{1}
  44. C d C_{d}
  45. Q = C d A 2 1 1 - ฮฒ 4 2 ( P 1 - P 2 ) / ฯ Q=C_{d}\;A_{2}\;\sqrt{\frac{1}{1-\beta^{4}}}\;\sqrt{2\;(P_{1}-P_{2})/\rho}
  46. C C
  47. C = C d 1 - ฮฒ 4 C=\frac{C_{d}}{\sqrt{1-\beta^{4}}}
  48. ( 1 ) Q = C A 2 2 ( P 1 - P 2 ) / ฯ (1)\qquad Q=C\;A_{2}\;\sqrt{2\;(P_{1}-P_{2})/\rho}
  49. ( 2 ) m ห™ = ฯ Q = C A 2 2 ฯ ( P 1 - P 2 ) (2)\qquad\dot{m}=\rho\;Q=C\;A_{2}\;\sqrt{2\;\rho\;(P_{1}-P_{2})}
  50. Q Q
  51. m ห™ \dot{m}
  52. C d C_{d}
  53. C C
  54. A 1 A_{1}
  55. A 2 A_{2}
  56. d 1 d_{1}
  57. d 2 d_{2}
  58. ฮฒ \beta
  59. V 1 V_{1}
  60. V 2 V_{2}
  61. P 1 P_{1}
  62. P 2 P_{2}
  63. ฯ \rho
  64. C d C_{d}
  65. 1 1 - ฮฒ 4 \frac{1}{\sqrt{1-\beta^{4}}}
  66. C C
  67. ฮฒ \beta
  68. Y Y
  69. m ห™ = ฯ 1 Q 1 = C Y A 2 2 ฯ 1 ( P 1 - P 2 ) \dot{m}=\rho_{1}\;Q_{1}=C\;Y\;A_{2}\;\sqrt{2\;\rho_{1}\;(P_{1}-P_{2})}
  70. Y Y
  71. m ห™ = C A 2 2 ฯ 1 P 1 ( k k - 1 ) [ ( P 2 / P 1 ) 2 / k - ( P 2 / P 1 ) ( k + 1 ) / k ] \dot{m}=C\;A_{2}\;\sqrt{2\;\rho_{1}\;P_{1}\;\bigg(\frac{k}{k-1}\bigg)\bigg[(P_% {2}/P_{1})^{2/k}-(P_{2}/P_{1})^{(k+1)/k}\bigg]}
  72. k k
  73. c p / c v c_{p}/c_{v}
  74. Q 1 Q_{1}
  75. ฯ 1 \rho_{1}

Orthogonal_transformation.html

  1. โŸจ u , v โŸฉ = โŸจ T u , T v โŸฉ . \langle u,v\rangle=\langle Tu,Tv\rangle\,.

Osculating_circle.html

  1. T ( s ) = ฮณ โ€ฒ ( s ) , T โ€ฒ ( s ) = k ( s ) N ( s ) , R ( s ) = 1 | k ( s ) | . T(s)=\gamma^{\prime}(s),\quad T^{\prime}(s)=k(s)N(s),\quad R(s)=\frac{1}{\left% |k(s)\right|}.
  2. ฮณ ( t ) = ( x 1 ( t ) x 2 ( t ) ) \gamma(t)\,=\,\begin{pmatrix}x_{1}(t)\\ x_{2}(t)\end{pmatrix}\,
  3. ฮณ โ€ฒ ( t ) โ‰  0 \gamma^{\prime}(t)\neq 0
  4. t t
  5. k ( t ) = x 1 โ€ฒ ( t ) โ‹… x 2 โ€ฒโ€ฒ ( t ) - x 1 โ€ฒโ€ฒ ( t ) โ‹… x 2 โ€ฒ ( t ) ( x 1 โ€ฒ ( t ) 2 + x 2 โ€ฒ ( t ) 2 ) 3 2 N ( t ) = 1 || ฮณ โ€ฒ ( t ) || โ‹… ( - x 2 โ€ฒ ( t ) x 1 โ€ฒ ( t ) ) k(t)=\frac{x_{1}^{\prime}(t)\cdot x_{2}^{\prime\prime}(t)-x_{1}^{\prime\prime}% (t)\cdot x_{2}^{\prime}(t)}{\Big(x_{1}^{\prime}(t)^{2}+x_{2}^{\prime}(t)^{2}% \Big)^{\frac{3}{2}}}\qquad\qquad\qquad\qquad\qquad N(t)\,=\,\frac{1}{||\gamma^% {\prime}(t)||}\cdot\begin{pmatrix}-x_{2}^{\prime}(t)\\ x_{1}^{\prime}(t)\end{pmatrix}
  6. R ( t ) = | ( x 1 โ€ฒ ( t ) 2 + x 2 โ€ฒ ( t ) 2 ) 3 2 x 1 โ€ฒ ( t ) โ‹… x 2 โ€ฒโ€ฒ ( t ) - x 1 โ€ฒโ€ฒ ( t ) โ‹… x 2 โ€ฒ ( t ) | and Q ( t ) = ฮณ ( t ) + 1 k ( t ) โ‹… || ฮณ โ€ฒ ( t ) || โ‹… ( - x 2 โ€ฒ ( t ) x 1 โ€ฒ ( t ) ) . R(t)=\left|\frac{\Big(x_{1}^{\prime}(t)^{2}+x_{2}^{\prime}(t)^{2}\Big)^{\frac{% 3}{2}}}{x_{1}^{\prime}(t)\cdot x_{2}^{\prime\prime}(t)-x_{1}^{\prime\prime}(t)% \cdot x_{2}^{\prime}(t)}\right|\qquad\qquad\mathrm{and}\qquad\qquad Q(t)\,=\,% \gamma(t)\,+\,\frac{1}{k(t)\cdot||\gamma^{\prime}(t)||}\cdot\begin{pmatrix}-x_% {2}^{\prime}(t)\\ x_{1}^{\prime}(t)\end{pmatrix}\,.
  7. ฮณ ( t ) = ( t t 2 ) \gamma(t)=\begin{pmatrix}t\\ t^{2}\end{pmatrix}
  8. R ( t ) = | ( 1 + 4 โ‹… t 2 ) 3 2 2 | R(t)=\left|\frac{\left(1+4\cdot t^{2}\right)^{\frac{3}{2}}}{2}\right|
  9. ฮณ ( 0 ) = ( 0 0 ) \gamma(0)=\begin{pmatrix}0\\ 0\end{pmatrix}
  10. ฮณ ( t ) = ( cos ( 3 t ) sin ( 2 t ) ) . \gamma(t)\,=\,\begin{pmatrix}\cos(3t)\\ \sin(2t)\end{pmatrix}\,.
  11. k ( t ) = 6 cos ( t ) ( 8 cos ( t ) 4 - 10 cos ( t ) 2 + 5 ) ( 232 cos ( t ) 4 - 97 cos ( t ) 2 + 13 - 144 cos ( t ) 6 ) 3 / 2 , k(t)=\frac{6\cos(t)(8\cos(t)^{4}-10\cos(t)^{2}+5)}{(232\cos(t)^{4}-97\cos(t)^{% 2}+13-144\cos(t)^{6})^{3/2}}\,,
  12. N ( t ) = 1 || ฮณ โ€ฒ ( t ) || โ‹… ( - 2 cos ( 2 t ) - 3 sin ( 3 t ) ) N(t)\,=\,\frac{1}{||\gamma^{\prime}(t)||}\cdot\begin{pmatrix}-2\cos(2t)\\ -3\sin(3t)\end{pmatrix}
  13. R ( t ) = | ( 232 cos ( t ) 4 - 97 cos ( t ) 2 + 13 - 144 cos ( t ) 6 ) 3 / 2 6 cos ( t ) ( 8 cos ( t ) 4 - 10 cos ( t ) 2 + 5 ) | . R(t)=\left|\frac{(232\cos(t)^{4}-97\cos(t)^{2}+13-144\cos(t)^{6})^{3/2}}{6\cos% (t)(8\cos(t)^{4}-10\cos(t)^{2}+5)}\right|\,.
  14. d 2 ฮณ ( s ) d s 2 \frac{\mathrm{d}^{2}\gamma(s)}{\mathrm{d}s^{2}}
  15. s s

Otoacoustic_emission.html

  1. f 1 f_{1}
  2. f 2 f_{2}
  3. f 1 : f 2 f_{1}\mbox{ }~{}:\mbox{ }~{}f_{2}
  4. f d p f_{dp}
  5. f d p = 2 f 1 - f 2 f_{dp}=2f_{1}-f_{2}
  6. f d p = f 2 - f 1 f_{dp}=f_{2}-f_{1}

Otsu's_method.html

  1. ฯƒ w 2 ( t ) = ฯ‰ 1 ( t ) ฯƒ 1 2 ( t ) + ฯ‰ 2 ( t ) ฯƒ 2 2 ( t ) \sigma^{2}_{w}(t)=\omega_{1}(t)\sigma^{2}_{1}(t)+\omega_{2}(t)\sigma^{2}_{2}(t)
  2. ฯ‰ i \omega_{i}
  3. t t
  4. ฯƒ i 2 \sigma^{2}_{i}
  5. ฯƒ b 2 ( t ) = ฯƒ 2 - ฯƒ w 2 ( t ) = ฯ‰ 1 ( t ) ฯ‰ 2 ( t ) [ ฮผ 1 ( t ) - ฮผ 2 ( t ) ] 2 \sigma^{2}_{b}(t)=\sigma^{2}-\sigma^{2}_{w}(t)=\omega_{1}(t)\omega_{2}(t)\left% [\mu_{1}(t)-\mu_{2}(t)\right]^{2}
  6. ฯ‰ i \omega_{i}
  7. ฮผ i \mu_{i}
  8. ฯ‰ 1 ( t ) \omega_{1}(t)
  9. t t
  10. ฯ‰ 1 ( t ) = ฮฃ 0 t p ( i ) \omega_{1}(t)=\Sigma_{0}^{t}p(i)
  11. ฮผ 1 ( t ) \mu_{1}(t)
  12. ฮผ 1 ( t ) = [ ฮฃ 0 t p ( i ) x ( i ) ] / ฯ‰ 1 \mu_{1}(t)=\left[\Sigma_{0}^{t}p(i)\,x(i)\right]/\omega_{1}
  13. x ( i ) x(i)
  14. i i
  15. ฯ‰ 2 ( t ) \omega_{2}(t)
  16. ฮผ 2 \mu_{2}
  17. t t
  18. ฯ‰ i ( 0 ) \omega_{i}(0)
  19. ฮผ i ( 0 ) \mu_{i}(0)
  20. t = 1 โ€ฆ t=1\ldots
  21. ฯ‰ i \omega_{i}
  22. ฮผ i \mu_{i}
  23. ฯƒ b 2 ( t ) \sigma^{2}_{b}(t)
  24. ฯƒ b 2 ( t ) \sigma^{2}_{b}(t)
  25. ฯƒ b 1 2 ( t ) \sigma^{2}_{b1}(t)
  26. ฯƒ b 2 2 ( t ) \sigma^{2}_{b2}(t)
  27. threshold 1 + threshold 2 2 \frac{\,\text{threshold}_{1}+\,\text{threshold}_{2}}{2}

Oudin_coil.html

  1. f = 1 2 ฯ€ L 1 C / 2 f={1\over 2\pi\sqrt{L_{1}C/2}}\;

Output_elasticity.html

  1. E Q = โˆ‚ Q โˆ‚ ๐ฑ โ‹… ๐ฑ Q E_{Q}=\frac{\partial Q}{\partial\,\textbf{x}}\cdot\frac{\,\textbf{x}}{Q}

Overhead_power_line.html

  1. D = ( d . d .2 d ) 1 3 D=(d.d.2d)^{1\over 3}
  2. L = 2 โˆ™ 10 - 7 โˆ™ ln ( D โˆ™ e 1 4 r x ) L=2\bullet 10^{-7}\bullet\ln\left({D\bullet e^{1\over 4}\over r_{x}}\right)
  3. Two-Conductor Bundle Equation: D B E = r x . D B \,\text{Two-Conductor Bundle Equation: }D_{BE}=\sqrt{r_{x}.D_{B}}
  4. Three-Conductor Bundle Equation: D B E = r x . ( D B ) 2 3 \,\text{Three-Conductor Bundle Equation: }D_{BE}=\sqrt[3]{r_{x}.(D_{B})^{2}}
  5. Four-Conductor Bundle Equation: D B E = r x . ( D B ) 2 . D B 2 4 \,\text{Four-Conductor Bundle Equation: }D_{BE}=\sqrt[4]{r_{x}.(D_{B})^{2}.D_{% B}\sqrt{2}}
  6. n-Conductor Bundle Equation: D B E = D B 2. s i n ( ฯ€ ฯ€ ฯ€ n ) . n . r x .2. s i n ( ฯ€ n ) D B n \,\text{n-Conductor Bundle Equation: }D_{BE}={D_{B}\over{2.sin({\pi\pi\pi\over% {n}})}}.\sqrt[n]{n.r_{x}.2.sin({\pi\over{n}})\over{D_{B}}}
  7. L = 2 โˆ™ 10 - 7 โˆ™ ln ( D โˆ™ e 1 4 D B E ) L=2\bullet 10^{-7}\bullet\ln\left({D\bullet e^{1\over 4}\over D_{BE}}\right)
  8. Z = z l = ( R + j ฯ‰ L ) l Z=zl=(R+j\omega L)l
  9. ฯ‰ \omega
  10. Y = y l = j ฯ‰ C l Y=yl=j\omega Cl

Overlapping_generations_model.html

  1. N t t = ( 1 + n ) t N_{t}^{t}=(1+n)^{t}
  2. u ( c t t , c t t + 1 ) = U ( c t t ) + ฮฒ U ( c t t + 1 ) , u(c_{t}^{t},c_{t}^{t+1})=U(c_{t}^{t})+\beta U(c_{t}^{t+1}),
  3. ฮฒ \beta

Oversampling.html

  1. n n
  2. number of samples = ( 2 n ) 2 = 2 2 n . \mbox{number of samples}~{}=(2^{n})^{2}=2^{2n}.
  3. n n
  4. 2 2 n 2^{2n}
  5. 2 n 2^{n}
  6. scaled mean = โˆ‘ i = 0 2 2 n - 1 2 n d a t a i 2 2 n = โˆ‘ i = 0 2 2 n - 1 d a t a i 2 n . \mbox{scaled mean}~{}=\frac{\sum\limits^{2^{2n}-1}_{i=0}2^{n}data_{i}}{2^{2n}}% =\frac{\sum\limits^{2^{2n}-1}_{i=0}data_{i}}{2^{n}}.
  7. 2 n 2^{n}

P-wave.html

  1. v p = K + 4 3 ฮผ ฯ = ฮป + 2 ฮผ ฯ v_{p}=\sqrt{\frac{K+\frac{4}{3}\mu}{\rho}}=\sqrt{\frac{\lambda+2\mu}{\rho}}
  2. ฮผ \mu
  3. ฯ \rho
  4. ฮป \lambda
  5. M M
  6. M = K + 4 ฮผ / 3 M=K+4\mu/3
  7. v p = M / ฯ . v_{p}=\sqrt{M/\rho}.
  8. V p = a ( M ยฏ ) + b ฯ V_{p}=a(\bar{M})+b\rho

Pacific_decadal_oscillation.html

  1. d y d t = v ( t ) - ฮป t {\operatorname{d}y\over\operatorname{d}t}=v(t)-\lambda t
  2. G ( w ) = F w 2 + ฮป 2 {G(w)=\frac{F}{w^{2}+\lambda^{2}}}
  3. โˆ‚ h โˆ‚ t - c โˆ‚ h โˆ‚ t = - โˆ‡ ร— ฯ„ โ†’ ฯ 0 f 0 {\partial h\over\partial t}-c{\partial h\over\partial t}=\frac{-\nabla\times% \vec{\tau}}{\rho_{0}f_{0}}

Packed_storage_matrix.html

  1. m ร— n m\times n

Painleveฬ_transcendents.html

  1. y โ€ฒโ€ฒ = R ( y โ€ฒ , y , t ) y^{\prime\prime}=R(y^{\prime},y,t)
  2. ( z - z 0 ) - 2 - z 0 10 ( z - z 0 ) 2 - 1 6 ( z - z 0 ) 3 + h ( z - z 0 ) 4 + z 0 2 300 ( z - z 0 ) 6 + โ‹ฏ (z-z_{0})^{-2}-\frac{z_{0}}{10}(z-z_{0})^{2}-\frac{1}{6}(z-z_{0})^{3}+h(z-z_{0% })^{4}+\frac{z_{0}^{2}}{300}(z-z_{0})^{6}+\cdots
  3. โ†— \nearrow
  4. โ†˜ \searrow
  5. โ†˜ \searrow
  6. โ†— \nearrow
  7. q = y , p = y โ€ฒ + y 2 + t / 2 \displaystyle q=y,\quad p=y^{\prime}+y^{2}+t/2
  8. y โ€ฒโ€ฒ = 2 y 3 + t y + b - 1 / 2 \displaystyle y^{\prime\prime}=2y^{3}+ty+b-1/2
  9. q โ€ฒ = โˆ‚ H โˆ‚ p = p - q 2 - t / 2 \displaystyle q^{\prime}=\frac{\partial H}{\partial p}=p-q^{2}-t/2
  10. p โ€ฒ = - โˆ‚ H โˆ‚ q = 2 p q + b \displaystyle p^{\prime}=-\frac{\partial H}{\partial q}=2pq+b
  11. H = p ( p - 2 q 2 - t ) / 2 - b q . \displaystyle H=p(p-2q^{2}-t)/2-bq.
  12. y โ€ฒโ€ฒ = 6 y 2 + t y^{\prime\prime}=6y^{2}+t
  13. y โ€ฒโ€ฒ = 2 y 3 + t y + b - 1 / 2 \displaystyle y^{\prime\prime}=2y^{3}+ty+b-1/2
  14. q = y , p = y โ€ฒ + y 2 + t / 2 \displaystyle q=y,p=y^{\prime}+y^{2}+t/2
  15. ( q , p , b ) โ†’ ( q + b / p , p , - b ) \displaystyle(q,p,b)\rightarrow(q+b/p,p,-b)
  16. ( q , p , b ) โ†’ ( - q , - p + 2 q 2 + t , 1 - b ) . \displaystyle(q,p,b)\rightarrow(-q,-p+2q^{2}+t,1-b).

Palladium-hydrogen_electrode.html

  1. 1 2 H 2 = H a d s = H a b s \tfrac{1}{2}\mathrm{H}_{2}=\mathrm{H}_{ads}=\mathrm{H}_{abs}
  2. E = E 0 + R T F ln a H + ( p H2 p 0 ) 1 / 2 E=E^{0}+{RT\over F}\ln{a_{\mathrm{H}^{+}}\over(\frac{p_{\mathrm{H}2}}{p^{0}})^% {1/2}}

Pancreatic_lipase.html

  1. โ‡Œ \rightleftharpoons

Pappus's_hexagon_theorem.html

  1. | A B C a b c X Y Z | \left|\begin{matrix}A&B&C\\ a&b&c\\ X&Y&Z\end{matrix}\right|

Parabolic_cylinder_function.html

  1. d 2 f d z 2 + ( a ~ z 2 + b ~ z + c ~ ) f = 0. \frac{d^{2}f}{dz^{2}}+\left(\tilde{a}z^{2}+\tilde{b}z+\tilde{c}\right)f=0.
  2. d 2 f d z 2 - ( 1 4 z 2 + a ) f = 0 \frac{d^{2}f}{dz^{2}}-\left(\tfrac{1}{4}z^{2}+a\right)f=0
  3. d 2 f d z 2 + ( 1 4 z 2 - a ) f = 0. \frac{d^{2}f}{dz^{2}}+\left(\tfrac{1}{4}z^{2}-a\right)f=0.
  4. f ( a , z ) f(a,z)\,
  5. f ( a , - z ) , f ( - a , i z ) and f ( - a , - i z ) . f(a,-z),f(-a,iz)\,\text{ and }f(-a,-iz).\,
  6. f ( a , z ) f(a,z)\,
  7. f ( - i a , z e ( 1 / 4 ) ฯ€ i ) f(-ia,ze^{(1/4)\pi i})\,
  8. f ( - i a , - z e ( 1 / 4 ) ฯ€ i ) , f ( i a , - z e - ( 1 / 4 ) ฯ€ i ) and f ( i a , z e - ( 1 / 4 ) ฯ€ i ) f(-ia,-ze^{(1/4)\pi i}),f(ia,-ze^{-(1/4)\pi i})\,\text{ and }f(ia,ze^{-(1/4)% \pi i})\,
  9. y 1 ( a ; z ) = exp ( - z 2 / 4 ) 1 F 1 ( 1 2 a + 1 4 ; 1 2 ; z 2 2 ) ( even ) y_{1}(a;z)=\exp(-z^{2}/4)\;_{1}F_{1}\left(\tfrac{1}{2}a+\tfrac{1}{4};\;\tfrac{% 1}{2}\;;\;\frac{z^{2}}{2}\right)\,\,\,\,\,\,(\mathrm{even})
  10. y 2 ( a ; z ) = z exp ( - z 2 / 4 ) 1 F 1 ( 1 2 a + 3 4 ; 3 2 ; z 2 2 ) ( odd ) y_{2}(a;z)=z\exp(-z^{2}/4)\;_{1}F_{1}\left(\tfrac{1}{2}a+\tfrac{3}{4};\;\tfrac% {3}{2}\;;\;\frac{z^{2}}{2}\right)\,\,\,\,\,\,(\mathrm{odd})
  11. F 1 1 ( a ; b ; z ) = M ( a ; b ; z ) \;{}_{1}F_{1}(a;b;z)=M(a;b;z)
  12. U ( a , z ) = 1 2 ฮพ ฯ€ [ cos ( ฮพ ฯ€ ) ฮ“ ( 1 / 2 - ฮพ ) y 1 ( a , z ) - 2 sin ( ฮพ ฯ€ ) ฮ“ ( 1 - ฮพ ) y 2 ( a , z ) ] U(a,z)=\frac{1}{2^{\xi}\sqrt{\pi}}\left[\cos(\xi\pi)\Gamma(1/2-\xi)\,y_{1}(a,z% )-\sqrt{2}\sin(\xi\pi)\Gamma(1-\xi)\,y_{2}(a,z)\right]
  13. V ( a , z ) = 1 2 ฮพ ฯ€ ฮ“ [ 1 / 2 - a ] [ sin ( ฮพ ฯ€ ) ฮ“ ( 1 / 2 - ฮพ ) y 1 ( a , z ) + 2 cos ( ฮพ ฯ€ ) ฮ“ ( 1 - ฮพ ) y 2 ( a , z ) ] V(a,z)=\frac{1}{2^{\xi}\sqrt{\pi}\Gamma[1/2-a]}\left[\sin(\xi\pi)\Gamma(1/2-% \xi)\,y_{1}(a,z)+\sqrt{2}\cos(\xi\pi)\Gamma(1-\xi)\,y_{2}(a,z)\right]
  14. ฮพ = 1 2 a + 1 4 . \xi=\frac{1}{2}a+\frac{1}{4}.
  15. lim z โ†’ โˆž V ( a , z ) / 2 ฯ€ e z 2 / 4 z a - 1 / 2 = 1 ( for arg ( z ) = 0 ) . \lim_{z\rightarrow\infty}V(a,z)/\sqrt{\frac{2}{\pi}}e^{z^{2}/4}z^{a-1/2}=1\,\,% \,\,(\,\text{for}\,\arg(z)=0).
  16. U ( a , x ) = D - a - 1 2 ( x ) , U(a,x)=D_{-a-\tfrac{1}{2}}(x),
  17. V ( a , x ) = ฮ“ ( 1 2 + a ) ฯ€ [ sin ( ฯ€ a ) D - a - 1 2 ( x ) + D - a - 1 2 ( - x ) ] . V(a,x)=\frac{\Gamma(\tfrac{1}{2}+a)}{\pi}[\sin(\pi a)D_{-a-\tfrac{1}{2}}(x)+D_% {-a-\tfrac{1}{2}}(-x)].
  18. โˆ‚ 2 u / โˆ‚ x 2 + โˆ‚ 2 u / โˆ‚ y 2 + k 2 u = 0 \partial^{2}u/\partial x^{2}+\partial^{2}u/\partial y^{2}+k^{2}u=0

Parabolic_SAR.html

  1. S A R n + 1 = S A R n + ฮฑ ( E P - S A R n ) {SAR}_{n+1}={SAR}_{n}+\alpha(EP-{SAR}_{n})

Parallel_tempering.html

  1. p = min ( 1 , exp ( - E j k T i - E i k T j ) exp ( - E i k T i - E j k T j ) ) = min ( 1 , e ( E i - E j ) ( 1 k T i - 1 k T j ) ) , p=\min\left(1,\frac{\exp\left(-\frac{E_{j}}{kT_{i}}-\frac{E_{i}}{kT_{j}}\right% )}{\exp\left(-\frac{E_{i}}{kT_{i}}-\frac{E_{j}}{kT_{j}}\right)}\right)=\min% \left(1,e^{(E_{i}-E_{j})\left(\frac{1}{kT_{i}}-\frac{1}{kT_{j}}\right)}\right),

Parametric_derivative.html

  1. x ( t ) = 4 t 2 x(t)=4t^{2}\,
  2. y ( t ) = 3 t . y(t)=3t.\,
  3. d y d t d x d t = y ห™ ( t ) x ห™ ( t ) , \frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{\dot{y}(t)}{\dot{x}(t)},
  4. x ห™ ( t ) \dot{x}(t)
  5. d y d x = d y d t โ‹… d t d x , \frac{dy}{dx}=\frac{dy}{dt}\cdot\frac{dt}{dx},
  6. d y d x = d y d t d x d t . \frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}.
  7. d y d t = d y d x โ‹… d x d t \frac{dy}{dt}=\frac{dy}{dx}\cdot\frac{dx}{dt}
  8. d x d t \frac{dx}{dt}
  9. d x d t = 8 t \frac{dx}{dt}=8t
  10. d y d t = 3 , \frac{dy}{dt}=3,
  11. d y d x = y ห™ x ห™ = 3 8 t , \frac{dy}{dx}=\frac{\dot{y}}{\dot{x}}=\frac{3}{8t},
  12. x ห™ \dot{x}
  13. y ห™ \dot{y}
  14. d 2 y d x 2 \frac{d^{2}y}{dx^{2}}
  15. = d d x ( d y d x ) =\frac{d}{dx}\left(\frac{dy}{dx}\right)
  16. = d d t ( d y d x ) โ‹… d t d x =\frac{d}{dt}\left(\frac{dy}{dx}\right)\cdot\frac{dt}{dx}
  17. = d d t ( y ห™ x ห™ ) 1 x ห™ =\frac{d}{dt}\left(\frac{\dot{y}}{\dot{x}}\right)\frac{1}{\dot{x}}
  18. = x ห™ y ยจ - y ห™ x ยจ x ห™ 3 =\frac{\dot{x}\ddot{y}-\dot{y}\ddot{x}}{\dot{x}^{3}}

Parity_(physics).html

  1. ๐ : ( x y z ) โ†ฆ ( - x - y - z ) . \mathbf{P}:\begin{pmatrix}x\\ y\\ z\end{pmatrix}\mapsto\begin{pmatrix}-x\\ -y\\ -z\end{pmatrix}.
  2. V x : ( x y z ) โ†ฆ ( - x y z ) , V_{x}:\begin{pmatrix}x\\ y\\ z\end{pmatrix}\mapsto\begin{pmatrix}-x\\ y\\ z\end{pmatrix},
  3. t \ t
  4. m \ m
  5. E \ E
  6. P \ P
  7. ฯ \ \rho
  8. V \ V
  9. ฯ \ \rho
  10. ๐‹ \mathbf{L}
  11. ๐ \mathbf{B}
  12. ๐‡ \mathbf{H}
  13. ๐Œ \mathbf{M}
  14. T i j \ T_{ij}
  15. h \ h
  16. ฮฆ \ \Phi
  17. ๐ฑ \mathbf{x}
  18. ๐ฏ \mathbf{v}
  19. ๐š \mathbf{a}
  20. ๐ฉ \mathbf{p}
  21. ๐… \mathbf{F}
  22. ๐‰ \mathbf{J}
  23. ๐„ \mathbf{E}
  24. ๐ƒ \mathbf{D}
  25. ๐ \mathbf{P}
  26. ๐€ \mathbf{A}
  27. ๐’ \mathbf{S}
  28. โŸฉ \rangle
  29. โŸฉ \rangle
  30. โŸจ \langle
  31. โŸฉ \rangle
  32. โŸฉ \rangle
  33. โŸฉ \rangle
  34. โŸฉ \rangle
  35. โŸฉ \rangle
  36. โŸฉ \rangle

Parity_anomaly.html

  1. m ฯˆ ยฏ ฯˆ m\overline{\psi}\psi
  2. i ฯˆ ยฏ ( โˆ‚ ฮผ + A ฮผ ) ฮ“ ฮผ ฯˆ i\overline{\psi}(\partial_{\mu}+A_{\mu})\Gamma^{\mu}\psi
  3. M ร— S 1 M\times S^{1}
  4. M ร— S 1 M\times S^{1}
  5. M ร— S 1 M\times S^{1}
  6. M ร— S 1 M\times S^{1}
  7. M ร— S 1 M\times S^{1}
  8. ๐’ฉ = 1 \mathcal{N}=1
  9. ๐’ฉ = 2 \mathcal{N}=2

Partial_isometry.html

  1. โ„ W := โ„› W * W , โ„ฑ W := โ„› W W * \mathcal{I}W:=\mathcal{R}W^{*}W,\,\mathcal{F}W:=\mathcal{R}WW^{*}
  2. ( W * W ) 2 = W * W โ‡” W W * W = W โ‡” W * W W * = W * โ‡” ( W W * ) 2 = W W * (W^{*}W)^{2}=W^{*}W\iff WW^{*}W=W\iff W^{*}WW^{*}=W^{*}\iff(WW^{*})^{2}=WW^{*}
  3. P = W * W , Q = W W * P=W^{*}W,\,Q=WW^{*}
  4. P : โ„‹ โ†’ โ„‹ : โ„ P = โ„ฑ P P:\mathcal{H}\rightarrow\mathcal{H}:\quad\mathcal{I}P=\mathcal{F}P
  5. J : โ„‹ โ†ช ๐’ฆ : โ„ J = โ„‹ J:\mathcal{H}\hookrightarrow\mathcal{K}:\quad\mathcal{I}J=\mathcal{H}
  6. U : โ„‹ โ†” ๐’ฆ : โ„ U = โ„‹ , โ„ฑ U = ๐’ฆ U:\mathcal{H}\leftrightarrow\mathcal{K}:\quad\mathcal{I}U=\mathcal{H},\,% \mathcal{F}U=\mathcal{K}
  7. ( 0 1 0 0 ) \begin{pmatrix}0&1\\ 0&0\end{pmatrix}
  8. { 0 } โŠ• โ„‚ \{0\}\oplus\mathbb{C}
  9. โ„‚ โŠ• { 0 } . \mathbb{C}\oplus\{0\}.
  10. R : โ„“ 2 ( โ„• ) โ†’ โ„“ 2 ( โ„• ) : ( x 1 , x 2 , โ€ฆ ) โ†ฆ ( 0 , x 1 , x 2 , โ€ฆ ) R:\ell^{2}(\mathbb{N})\to\ell^{2}(\mathbb{N}):(x_{1},x_{2},\ldots)\mapsto(0,x_% {1},x_{2},\ldots)
  11. L : โ„“ 2 ( โ„• ) โ†’ โ„“ 2 ( โ„• ) : ( x 1 , x 2 , โ€ฆ ) โ†ฆ ( x 2 , x 3 , โ€ฆ ) L:\ell^{2}(\mathbb{N})\to\ell^{2}(\mathbb{N}):(x_{1},x_{2},\ldots)\mapsto(x_{2% },x_{3},\ldots)
  12. R * = L R^{*}=L
  13. L R ( x 1 , x 2 , โ€ฆ ) = ( x 1 , x 2 , โ€ฆ ) LR(x_{1},x_{2},\ldots)=(x_{1},x_{2},\ldots)
  14. R L ( x 1 , x 2 , โ€ฆ ) = ( 0 , x 2 , โ€ฆ ) RL(x_{1},x_{2},\ldots)=(0,x_{2},\ldots)

Particle_acceleration.html

  1. ๐š = d ๐ฏ d t \mathbf{a}={d\mathbf{v}\over dt}
  2. ๐š ยฏ = ๐ฏ - ๐ฎ t \mathbf{\bar{a}}={\mathbf{v}-\mathbf{u}\over t}
  3. ๐š ยฏ \mathbf{\bar{a}}
  4. ๐ฎ \mathbf{u}
  5. ๐ฏ \mathbf{v}
  6. t t
  7. ๐š = - v 2 r ๐ซ r = - ฯ‰ 2 ๐ซ \mathbf{a}=-\frac{v^{2}}{r}\frac{\mathbf{r}}{r}=-\omega^{2}\mathbf{r}
  8. a a
  9. F F
  10. m m
  11. F = m โ‹… a F=m\cdot a
  12. a = ฮด โ‹… ฯ‰ 2 = v โ‹… ฯ‰ = p โ‹… ฯ‰ Z = ฯ‰ J Z = ฯ‰ E ฯ = ฯ‰ P a c Z โ‹… A a=\delta\cdot\omega^{2}=v\cdot\omega=\frac{p\cdot\omega}{Z}=\omega\sqrt{\frac{% J}{Z}}=\omega\sqrt{\frac{E}{\rho}}=\omega\sqrt{\frac{P_{ac}}{Z\cdot A}}
  13. ฯ‰ \omega
  14. ฯ€ \pi

Particle_filter.html

  1. X 0 โ†’ X 1 โ†’ X 2 โ†’ X 3 โ†’ โ€ฆ signal โ†“ โ†“ โ†“ โ†“ โ€ฆ Y 0 Y 1 Y 2 Y 3 โ€ฆ observation \begin{array}[]{cccccccccc}X_{0}&\rightarrow&X_{1}&\rightarrow&X_{2}&% \rightarrow&X_{3}&\rightarrow&\ldots&\mbox{signal}\\ \downarrow&&\downarrow&&\downarrow&&\downarrow&&\ldots&\\ Y_{0}&&Y_{1}&&Y_{2}&&Y_{3}&&\ldots&\mbox{observation}\end{array}
  2. X k X_{k}
  3. Y 0 , โ€ฆ , Y k , Y_{0},\ldots,Y_{k}~{},
  4. X k X_{k}
  5. X n = ( X n ( 1 ) X n ( 2 ) X n ( 3 ) ) X_{n}=\left(\begin{array}[]{c}X^{(1)}_{n}\\ X^{(2)}_{n}\\ X^{(3)}_{n}\end{array}\right)
  6. X n ( 1 ) X^{(1)}_{n}
  7. n n
  8. X n ( 2 ) , X n ( 3 ) X^{(2)}_{n},~{}X^{(3)}_{n}
  9. { X n ( 1 ) = X n - 1 ( 1 ) + ฯต n W n X n ( 2 ) = ( 1 - ฮฑ ฮ” ) X n - 1 ( 2 ) + ฮฒ ฮ” X n ( 1 ) X n ( 3 ) = X n - 1 ( 3 ) + ฮ” X n ( 2 ) \left\{\begin{array}[]{rcl}X_{n}^{(1)}&=&X_{n-1}^{(1)}+\epsilon_{n}~{}W_{n}\\ X_{n}^{(2)}&=&(1-\alpha~{}\Delta)~{}X_{n-1}^{(2)}+\beta~{}\Delta~{}X_{n}^{(1)}% \\ X_{n}^{(3)}&=&X_{n-1}^{(3)}+\Delta~{}X_{n}^{(2)}\end{array}\right.
  10. ฮฑ , ฮฒ > 0 \alpha,~{}\beta~{}>0
  11. ฮ” > 0 \Delta>0
  12. { 0 , 1 } \{0,1\}
  13. ฯต n \epsilon_{n}
  14. W n W_{n}
  15. โˆ€ n โ‰ฅ 0 Y n = X n ( 3 ) + ฮ” V n \forall n\geq 0\,\qquad Y_{n}=X_{n}^{(3)}+\Delta~{}V_{n}
  16. V n V_{n}
  17. X n X_{n}~{}
  18. Y 0 = y 0 , โ€ฆ , Y n = y n Y_{0}=y_{0},\ldots,Y_{n}=y_{n}~{}
  19. n . n.
  20. ( X 0 , โ€ฆ , X n ) (X_{0},\ldots,X_{n})~{}
  21. Y 0 = y 0 , โ€ฆ , Y n = y n Y_{0}=y_{0},\ldots,Y_{n}=y_{n}~{}
  22. n . n.
  23. X n X_{n}~{}
  24. Y 0 = y 0 , โ€ฆ , Y n - 1 = y n - 1 , Y n = y n Y_{0}=y_{0},\ldots,Y_{n-1}=y_{n-1},Y_{n}=y_{n}~{}
  25. X n X_{n}~{}
  26. Y 0 = y 0 , โ€ฆ , Y n - 1 = y n - 1 Y_{0}=y_{0},\ldots,Y_{n-1}=y_{n-1}
  27. X n X_{n}~{}
  28. Y 0 = y 0 , โ€ฆ , Y n : y n , โ€ฆ , Y m = y m Y_{0}=y_{0},\ldots,Y_{n}:y_{n},\ldots,Y_{m}=y_{m}
  29. m > n m>n
  30. X k X_{k}
  31. Y k Y_{k}
  32. X 0 , X 1 , โ€ฆ X_{0},X_{1},\ldots
  33. โ„ d x \mathbb{R}^{d_{x}}
  34. d x โ‰ฅ 1 d_{x}\geq 1
  35. p ( x k | x k - 1 ) p(x_{k}|x_{k-1})
  36. X k | X k - 1 = x k - 1 โˆผ p ( x k | x k - 1 ) X_{k}|X_{k-1}=x_{k-1}\sim p(x_{k}|x_{k-1})
  37. p ( x 0 ) p(x_{0})
  38. Y 0 , Y 1 , โ€ฆ Y_{0},Y_{1},\ldots
  39. โ„ d y \mathbb{R}^{d_{y}}
  40. d y โ‰ฅ 1 d_{y}\geq 1
  41. X 0 , X 1 , โ€ฆ X_{0},X_{1},\ldots
  42. Y k Y_{k}
  43. X k X_{k}
  44. Y k Y_{k}
  45. X k = x k X_{k}=x_{k}
  46. Y k | X k = x k โˆผ p ( y k | x k ) Y_{k}|X_{k}=x_{k}\sim p(y_{k}|x_{k})
  47. X k = g ( X k - 1 ) + W k X_{k}=g(X_{k-1})+W_{k}\,
  48. Y k = h ( X k ) + V k Y_{k}=h(X_{k})+V_{k}\,
  49. W k W_{k}
  50. V k V_{k}
  51. g ( โ‹… ) g(\cdot)
  52. h ( โ‹… ) h(\cdot)
  53. g ( โ‹… ) g(\cdot)
  54. h ( โ‹… ) h(\cdot)
  55. W k W_{k}
  56. V k V_{k}
  57. X k - 1 โ†’ X k X_{k-1}~{}\rightarrow~{}X_{k}
  58. X k , X_{k},
  59. x k โ†ฆ p ( y k | x k ) x_{k}\mapsto p(y_{k}|x_{k})
  60. X k X_{k}
  61. X k X_{k}
  62. ๐’ณ k = ( X k , Y k ) \mathcal{X}_{k}=\left(X_{k},Y_{k}\right)
  63. ๐’ด k = Y k + ฯต ๐’ฑ k for some parameter ฯต โˆˆ [ 0 , 1 ] \mathcal{Y}_{k}=Y_{k}+\epsilon~{}\mathcal{V}_{k}\quad\mbox{for some parameter}% ~{}\quad\epsilon\in[0,1]
  64. Law ( X k | ๐’ด 0 = y 0 , โ€ฆ , ๐’ด k = y k ) โ‰ˆ ฯต โ†“ 0 Law ( X k | Y 0 = y 0 , โ€ฆ , Y k = y k ) \mbox{Law}~{}\left(X_{k}~{}|~{}\mathcal{Y}_{0}=y_{0},\ldots,\mathcal{Y}_{k}=y_% {k}\right)\approx_{\epsilon\downarrow 0}~{}\mbox{Law}~{}\left(X_{k}~{}|~{}Y_{0% }=y_{0},\ldots,Y_{k}=y_{k}\right)
  65. ๐’ณ k = ( X k , Y k ) \mathcal{X}_{k}=\left(X_{k},Y_{k}\right)
  66. ๐’ด 0 = y 0 , โ€ฆ , ๐’ด k = y k \mathcal{Y}_{0}=y_{0},\ldots,\mathcal{Y}_{k}=y_{k}
  67. โ„ d x + d y \mathbb{R}^{d_{x}+d_{y}}
  68. p ( ๐’ด k | ๐’ณ k ) p(\mathcal{Y}_{k}~{}|~{}\mathcal{X}_{k})
  69. p ( x 0 , โ€ฆ , x k | y 0 , โ€ฆ , y k ) = p ( y 0 , โ€ฆ , y k | x 0 , โ€ฆ , x k ) ร— p ( x 0 , โ€ฆ , x k ) p ( y 0 , โ€ฆ , y k ) with p ( y 0 , โ€ฆ , y k ) = โˆซ p ( y 0 , โ€ฆ , y k | x 0 , โ€ฆ , x k ) p ( x 0 , โ€ฆ , x k ) d x 0 โ€ฆ d x k p(x_{0},...,x_{k}|y_{0},\dots,y_{k})=\frac{p(y_{0},\dots,y_{k}|x_{0},...,x_{k}% )\times p(x_{0},...,x_{k})}{p(y_{0},\dots,y_{k})}\quad\mbox{with}~{}\quad p(y_% {0},\dots,y_{k})=\int p(y_{0},\dots,y_{k}|x_{0},...,x_{k})~{}p(x_{0},...,x_{k}% )~{}dx_{0}\ldots dx_{k}
  70. p ( y 0 , โ€ฆ , y k | x 0 , โ€ฆ , x k ) = โˆ 0 โ‰ค l โ‰ค k p ( y l | x l ) with p ( x 0 , โ€ฆ , x k ) = p 0 ( x 0 ) โˆ 1 โ‰ค l โ‰ค k p ( x l | x l - 1 ) p(y_{0},\dots,y_{k}|x_{0},...,x_{k})=\prod_{0\leq l\leq k}p(y_{l}|x_{l})\qquad% \mbox{with}~{}\qquad p(x_{0},...,x_{k})=p_{0}(x_{0})\prod_{1\leq l\leq k}p(x_{% l}|x_{l-1})
  71. p ( x 0 | y 0 , โ€ฆ , y - 1 ) = p ( x 0 ) p(x_{0}|y_{0},\ldots,y_{-1})=p(x_{0})
  72. Y 0 = y 0 , โ€ฆ , Y n = y n Y_{0}=y_{0},\ldots,Y_{n}=y_{n}
  73. G k ( x k ) = p ( y k | x k ) , for each k = 0 , โ€ฆ , n G_{k}(x_{k})=p(y_{k}|x_{k}),~{}\mbox{for each}~{}~{}k=0,\ldots,n
  74. X k X_{k}
  75. โˆซ F ( x 0 , โ€ฆ , x n ) p ( x 0 , โ€ฆ , x n | y 0 , โ€ฆ , y n ) d x 0 โ€ฆ d x n = โˆซ F ( x 0 , โ€ฆ , x n ) { โˆ 0 โ‰ค k โ‰ค n p ( y k | x k ) } p ( x 0 , โ€ฆ , x n ) d x 0 โ€ฆ d x n โˆซ { โˆ 0 โ‰ค k โ‰ค n p ( y k | x k ) } p ( x 0 , โ€ฆ , x n ) d x 0 โ€ฆ d x n = E ( F ( X 0 , โ€ฆ , X n ) โˆ 0 โ‰ค k โ‰ค n G k ( X k ) ) E ( โˆ 0 โ‰ค k โ‰ค n G k ( X k ) ) \begin{array}[]{rcl}\displaystyle\int F(x_{0},\ldots,x_{n})~{}p(x_{0},\ldots,x% _{n}|y_{0},\ldots,y_{n})~{}dx_{0}\ldots dx_{n}&=&\displaystyle\frac{\int F(x_{% 0},\ldots,x_{n})~{}\left\{\prod_{0\leq k\leq n}p(y_{k}|x_{k})\right\}p(x_{0},% \ldots,x_{n})~{}dx_{0}\ldots dx_{n}}{\int~{}\left\{\prod_{0\leq k\leq n}p(y_{k% }|x_{k})\right\}p(x_{0},\ldots,x_{n})~{}dx_{0}\ldots dx_{n}}\\ &&\\ &=&\displaystyle\frac{E\left(F(X_{0},\ldots,X_{n})\prod_{0\leq k\leq n}G_{k}(X% _{k})\right)}{E\left(\prod_{0\leq k\leq n}G_{k}(X_{k})\right)}\end{array}
  76. G n ( x n ) = 1 A ( x n ) G_{n}(x_{n})=1_{A}(x_{n})
  77. E ( F ( X 0 , โ€ฆ , X n ) | X 0 โˆˆ A , โ€ฆ , X n โˆˆ A ) = E ( F ( X 0 , โ€ฆ , X n ) โˆ 0 โ‰ค k โ‰ค n G k ( X k ) ) E ( โˆ 0 โ‰ค k โ‰ค n G k ( X k ) ) and P ( X 0 โˆˆ A , โ€ฆ , X n โˆˆ A ) = E ( โˆ 0 โ‰ค k โ‰ค n G k ( X k ) ) E\left(F(X_{0},\ldots,X_{n})~{}|~{}X_{0}\in A,~{}\ldots,X_{n}\in A\right)=% \displaystyle\frac{E\left(F(X_{0},\ldots,X_{n})\prod_{0\leq k\leq n}G_{k}(X_{k% })\right)}{E\left(\prod_{0\leq k\leq n}G_{k}(X_{k})\right)}\quad\mbox{and}~{}% \quad P\left(X_{0}\in A,~{}\ldots,X_{n}\in A\right)=E\left(\prod_{0\leq k\leq n% }G_{k}(X_{k})\right)
  78. N N
  79. ( ฮพ 0 i ) 1 โ‰ค i โ‰ค N \left(\xi^{i}_{0}\right)_{1\leq i\leq N}
  80. p ( x 0 ) p(x_{0})
  81. ฮพ k := ( ฮพ k i ) 1 โ‰ค i โ‰ค N - - - - - - - - - - โŸถ selection ฮพ ^ k := ( ฮพ ^ k i ) 1 โ‰ค i โ‰ค N - - - - - - - - - - โŸถ mutation ฮพ k + 1 := ( ฮพ k + 1 i ) 1 โ‰ค i โ‰ค N \xi_{k}:=\left(\xi^{i}_{k}\right)_{1\leq i\leq N}\stackrel{\mbox{selection}~{}% }{-\!\!\!\!-\!\!\!\!-\!\!\!\!-\!\!\!\!-\!\!\!\!-\!\!\!\!-\!\!\!\!-\!\!\!\!-\!% \!\!\!-\!\!\!\!\longrightarrow}~{}\widehat{\xi}_{k}:=\left(\widehat{\xi}^{i}_{% k}\right)_{1\leq i\leq N}\stackrel{\mbox{mutation}~{}}{-\!\!\!\!-\!\!\!\!-\!\!% \!\!-\!\!\!\!-\!\!\!\!-\!\!\!\!-\!\!\!\!-\!\!\!\!-\!\!\!\!-\!\!\!\!% \longrightarrow}~{}\xi_{k+1}:=\left(\xi^{i}_{k+1}\right)_{1\leq i\leq N}
  82. ฮพ ^ k := ( ฮพ ^ k i ) 1 โ‰ค i โ‰ค N \widehat{\xi}_{k}:=\left(\widehat{\xi}^{i}_{k}\right)_{1\leq i\leq N}
  83. โˆ‘ 1 โ‰ค i โ‰ค N p ( y k | ฮพ k i ) โˆ‘ 1 โ‰ค j โ‰ค N p ( y k | ฮพ k j ) ฮด ฮพ k i ( d x k ) \sum_{1\leq i\leq N}~{}\frac{p(y_{k}~{}|~{}\xi^{i}_{k})}{\sum_{1\leq j\leq N}p% (y_{k}~{}|~{}\xi^{j}_{k})}~{}\delta_{\xi^{i}_{k}}(dx_{k})
  84. ฮพ ^ k i \widehat{\xi}^{i}_{k}
  85. ฮพ ^ k i โŸถ ฮพ k + 1 i โˆผ p ( x k + 1 | ฮพ ^ k i ) \widehat{\xi}^{i}_{k}~{}\longrightarrow\xi^{i}_{k+1}~{}~{}\sim~{}~{}p(x_{k+1}|% \widehat{\xi}^{i}_{k})
  86. i = 1 , โ€ฆ , N . i=1,\ldots,N.
  87. p ( y k | ฮพ k i ) p(y_{k}~{}|~{}\xi^{i}_{k})
  88. x k โ†ฆ p ( y k | x k ) x_{k}\mapsto p(y_{k}~{}|~{}x_{k})
  89. x k = ฮพ k i x_{k}=\xi^{i}_{k}
  90. p ( x k + 1 | ฮพ ^ k i ) p(x_{k+1}~{}|~{}\widehat{\xi}^{i}_{k})
  91. p ( x k + 1 | x k ) p(x_{k+1}~{}|~{}x_{k})
  92. x k = ฮพ ^ k i x_{k}=\widehat{\xi}^{i}_{k}
  93. p ^ ( d x k | y 0 , โ€ฆ , y k ) := 1 N โˆ‘ 1 โ‰ค i โ‰ค N ฮด ฮพ ^ k i ( d x k ) โ‰ˆ N โ†‘ โˆž p ( d x k | y 0 , โ€ฆ , y k ) โ‰ˆ N โ†‘ โˆž โˆ‘ 1 โ‰ค i โ‰ค N p ( y k | ฮพ k i ) โˆ‘ 1 โ‰ค j โ‰ค N p ( y k | ฮพ k j ) ฮด ฮพ k i ( d x k ) \widehat{p}(dx_{k}~{}|~{}y_{0},\ldots,y_{k}):=\frac{1}{N}\sum_{1\leq i\leq N}% \delta_{\widehat{\xi}^{i}_{k}}(dx_{k})~{}~{}\approx_{N\uparrow\infty}~{}~{}p(% dx_{k}~{}|~{}y_{0},\ldots,y_{k})~{}~{}\approx_{N\uparrow\infty}~{}~{}\sum_{1% \leq i\leq N}~{}\frac{p(y_{k}~{}|~{}\xi^{i}_{k})}{\sum_{1\leq j\leq N}p(y_{k}~% {}|~{}\xi^{j}_{k})}~{}\delta_{\xi^{i}_{k}}(dx_{k})
  94. p ^ ( d x k | y 0 , โ€ฆ , y k - 1 ) := 1 N โˆ‘ 1 โ‰ค i โ‰ค N ฮด ฮพ k i ( d x k ) โ‰ˆ N โ†‘ โˆž p ( d x k | y 0 , โ€ฆ , y k - 1 ) \widehat{p}(dx_{k}~{}|~{}y_{0},\ldots,y_{k-1}):=\frac{1}{N}\sum_{1\leq i\leq N% }\delta_{\xi^{i}_{k}}(dx_{k})~{}~{}\approx_{N\uparrow\infty}~{}~{}p(dx_{k}~{}|% ~{}y_{0},\ldots,y_{k-1})
  95. p ( x k | y 0 , โ€ฆ , y k ) p(x_{k}|y_{0},\dots,y_{k})
  96. N N
  97. X k X_{k}
  98. ฮพ ^ k 1 , ฮพ ^ k 2 , โ€ฆ , ฮพ ^ k N \widehat{\xi}_{k}^{1},\widehat{\xi}_{k}^{2},\ldots,\widehat{\xi}_{k}^{N}
  99. ฮด a \delta_{a}
  100. f ( โ‹… ) f(\cdot)
  101. f ( โ‹… ) f(\cdot)
  102. p ( d x k | y 0 , โ€ฆ , y k ) := p ( x k | y 0 , โ€ฆ , y k ) d x k โ‰ˆ N โ†‘ โˆž p ^ ( d x k | y 0 , โ€ฆ , y k ) = 1 N โˆ‘ 1 โ‰ค i โ‰ค N ฮด ฮพ ^ k i ( d x k ) p(dx_{k}|y_{0},\dots,y_{k}):=p(x_{k}|y_{0},\dots,y_{k})~{}dx_{k}~{}\approx_{N% \uparrow\infty}\widehat{p}(dx_{k}|y_{0},\dots,y_{k})=\frac{1}{N}\sum_{1\leq i% \leq N}\delta_{\widehat{\xi}^{i}_{k}}(dx_{k})
  103. ( ฮพ ^ 0 , k i , ฮพ ^ 1 , k i , โ€ฆ , ฮพ ^ k - 1 , k i , ฮพ ^ k , k i ) \left(\widehat{\xi}^{i}_{0,k},\widehat{\xi}^{i}_{1,k},...,\widehat{\xi}^{i}_{k% -1,k},\widehat{\xi}^{i}_{k,k}\right)
  104. i = 1 , โ€ฆ , N i=1,...,N
  105. ฮพ ^ l , k i \widehat{\xi}^{i}_{l,k}
  106. ฮพ ^ k , k i = ฮพ ^ k i \widehat{\xi}^{i}_{k,k}=\widehat{\xi}^{i}_{k}
  107. p ( d ( x 0 , โ€ฆ , x k ) | y 0 , โ€ฆ , y k ) := p ( x 0 , โ€ฆ , x k | y 0 , โ€ฆ , y k ) d x 0 โ€ฆ d x k โ‰ˆ N โ†‘ โˆž p ^ ( d ( x 0 , โ€ฆ , x k ) | y 0 , โ€ฆ , y k ) := 1 N โˆ‘ 1 โ‰ค i โ‰ค N ฮด ( ฮพ ^ 0 , k i , ฮพ ^ 1 , k i , โ€ฆ , ฮพ ^ k , k i ) ( d ( x 0 , โ€ฆ , x k ) ) p(d(x_{0},...,x_{k})|y_{0},\dots,y_{k}):=p(x_{0},...,x_{k}|y_{0},\dots,y_{k})% \,dx_{0}...dx_{k}\approx_{N\uparrow\infty}\widehat{p}(d(x_{0},\ldots,x_{k})|y_% {0},\dots,y_{k}):=\frac{1}{N}\sum_{1\leq i\leq N}\delta_{\left(\widehat{\xi}^{% i}_{0,k},\widehat{\xi}^{i}_{1,k},\ldots,\widehat{\xi}^{i}_{k,k}\right)}(d(x_{0% },\ldots,x_{k}))
  108. ฮท n + 1 = ฮฆ n + 1 ( ฮท n ) \eta_{n+1}=\Phi_{n+1}\left(\eta_{n}\right)
  109. ฮฆ n + 1 \Phi_{n+1}
  110. ฮท n ( d x n ) = p ( x n | y 0 , โ€ฆ , y n - 1 ) d x n \eta_{n}(dx_{n})=p(x_{n}|y_{0},\ldots,y_{n-1})dx_{n}
  111. ฮท 0 ( d x 0 ) = p ( x 0 ) d x 0 \eta_{0}(dx_{0})=p(x_{0})dx_{0}
  112. N N
  113. ( ฮพ 0 i ) 1 โ‰ค i โ‰ค N \left(\xi^{i}_{0}\right)_{1\leq i\leq N}
  114. ฮท 0 ( d x 0 ) = p ( x 0 ) d x 0 \eta_{0}(dx_{0})=p(x_{0})dx_{0}
  115. N N
  116. ( ฮพ n i ) 1 โ‰ค i โ‰ค N \left(\xi^{i}_{n}\right)_{1\leq i\leq N}
  117. 1 N โˆ‘ 1 โ‰ค i โ‰ค N ฮด ฮพ n i ( d x n ) โ‰ˆ N โ†‘ โˆž ฮท n ( d x n ) \frac{1}{N}\sum_{1\leq i\leq N}\delta_{\xi^{i}_{n}}(dx_{n})~{}\approx_{N% \uparrow\infty}~{}\eta_{n}(dx_{n})
  118. N N
  119. ฮพ n + 1 := ( ฮพ n + 1 i ) 1 โ‰ค i โ‰ค N \xi_{n+1}:=\left(\xi^{i}_{n+1}\right)_{1\leq i\leq N}
  120. ฮฆ n + 1 ( 1 N โˆ‘ 1 โ‰ค i โ‰ค N ฮด ฮพ n i ) โ‰ˆ N โ†‘ โˆž ฮฆ n + 1 ( ฮท n ) = ฮท n + 1 \Phi_{n+1}\left(\frac{1}{N}\sum_{1\leq i\leq N}\delta_{\xi^{i}_{n}}\right)~{}% \approx_{N\uparrow\infty}~{}\Phi_{n+1}\left(\eta_{n}\right)=\eta_{n+1}
  121. p ( x 0 | y 0 , โ€ฆ , y - 1 ) := p ( x 0 ) p(x_{0}|y_{0},\dots,y_{-1}):=p(x_{0})
  122. p ^ ( d x 0 ) = 1 N โˆ‘ 1 โ‰ค i โ‰ค N ฮด ฮพ 0 i ( d x 0 ) โ‰ˆ N โ†‘ โˆž p ( x 0 ) d x 0 in the sense that โˆซ f ( x 0 ) p ^ ( d x 0 ) = 1 N โˆ‘ 1 โ‰ค i โ‰ค N f ( ฮพ 0 i ) โ‰ˆ N โ†‘ โˆž โˆซ f ( x 0 ) p ( d x 0 ) d x 0 \widehat{p}(dx_{0})=\frac{1}{N}\sum_{1\leq i\leq N}\delta_{\xi^{i}_{0}}(dx_{0}% )\approx_{N\uparrow\infty}~{}p(x_{0})dx_{0}\quad\mbox{in the sense that}~{}% \quad\int f(x_{0})\widehat{p}(dx_{0})=\frac{1}{N}\sum_{1\leq i\leq N}f(\xi^{i}% _{0})\approx_{N\uparrow\infty}\int f(x_{0})p(dx_{0})dx_{0}
  123. f f
  124. ( ฮพ k i ) 1 โ‰ค i โ‰ค N \left(\xi^{i}_{k}\right)_{1\leq i\leq N}
  125. p ^ ( d x k | y 0 , โ€ฆ , y k - 1 ) := 1 N โˆ‘ 1 โ‰ค i โ‰ค N ฮด ฮพ k i ( d x k ) โ‰ˆ N โ†‘ โˆž p ( x k | y 0 , โ€ฆ , y k - 1 ) d x k \widehat{p}(dx_{k}|y_{0},\ldots,y_{k-1}):=\frac{1}{N}\sum_{1\leq i\leq N}% \delta_{\xi^{i}_{k}}(dx_{k})\approx_{N\uparrow\infty}~{}p(x_{k}~{}|~{}y_{0},% \ldots,y_{k-1})dx_{k}
  126. f f
  127. โˆซ f ( x k ) p ^ ( d x k | y 0 , โ€ฆ , y k - 1 ) = 1 N โˆ‘ 1 โ‰ค i โ‰ค N f ( ฮพ k i ) โ‰ˆ N โ†‘ โˆž โˆซ f ( x k ) p ( d x k | y 0 , โ€ฆ , y k - 1 ) \int f(x_{k})\widehat{p}(dx_{k}~{}|~{}y_{0},\ldots,y_{k-1})=\frac{1}{N}\sum_{1% \leq i\leq N}f(\xi^{i}_{k})\approx_{N\uparrow\infty}\int f(x_{k})p(dx_{k}|y_{0% },\ldots,y_{k-1})
  128. 1 " > p ( x k | y 0 , โ€ฆ , y k - 1 ) d x k 1">p(x_{k}~{}|~{}y_{0},\ldots,y_{k-1})~{}dx_{k}
  129. 1 " > p ^ ( d x k | y 0 , โ€ฆ , y k - 1 ) 1">\widehat{p}(dx_{k}~{}|~{}y_{0},\ldots,y_{k-1})
  130. p ( x k + 1 | y 0 , โ€ฆ , y k ) โ‰ˆ N โ†‘ โˆž โˆซ p ( x k + 1 | x k โ€ฒ ) p ( y k | x k โ€ฒ ) p ^ ( d x k โ€ฒ | y 0 , โ€ฆ , y k - 1 ) โˆซ p ( y k | x k โ€ฒโ€ฒ ) p ^ ( d x k โ€ฒโ€ฒ | y 0 , โ€ฆ , y k - 1 ) p(x_{k+1}|y_{0},\ldots,y_{k})\approx_{N\uparrow\infty}\int~{}p(x_{k+1}|x^{% \prime}_{k})~{}\displaystyle\frac{p(y_{k}|x_{k}^{\prime})~{}\widehat{p}(dx^{% \prime}_{k}|y_{0},\dots,y_{k-1})}{\displaystyle\int~{}p(y_{k}|x^{\prime\prime}% _{k})~{}\widehat{p}(dx^{\prime\prime}_{k}|y_{0},\ldots,y_{k-1})}
  131. โˆซ p ( x k + 1 | x k โ€ฒ ) p ( y k | x k โ€ฒ ) p ^ ( d x k โ€ฒ | y 0 , โ€ฆ , y k - 1 ) โˆซ p ( y k | x k โ€ฒโ€ฒ ) p ^ ( d x k โ€ฒโ€ฒ | y 0 , โ€ฆ , y k - 1 ) = โˆ‘ 1 โ‰ค i โ‰ค N p ( y k | ฮพ k i ) โˆ‘ 1 โ‰ค j โ‰ค N p ( y k | ฮพ k j ) p ( x k + 1 | ฮพ k i ) = : q ^ ( x k + 1 | y 0 , โ€ฆ , y k ) \int~{}p(x_{k+1}|x^{\prime}_{k})~{}\displaystyle\frac{p(y_{k}|x_{k}^{\prime})~% {}\widehat{p}(dx^{\prime}_{k}|y_{0},\dots,y_{k-1})}{\displaystyle\int~{}p(y_{k% }|x^{\prime\prime}_{k})~{}\widehat{p}(dx^{\prime\prime}_{k}|y_{0},\ldots,y_{k-% 1})}=\sum_{1\leq i\leq N}~{}\frac{p(y_{k}~{}|~{}\xi^{i}_{k})}{\sum_{1\leq j% \leq N}p(y_{k}~{}|~{}\xi^{j}_{k})}~{}~{}p(x_{k+1}~{}|~{}\xi^{i}_{k})=:\widehat% {q}(x_{k+1}|y_{0},\ldots,y_{k})
  132. p ( y k | ฮพ k i ) p(y_{k}~{}|~{}\xi^{i}_{k})
  133. p ( y k | x k ) p(y_{k}~{}|~{}x_{k})
  134. x k = ฮพ k i x_{k}=\xi^{i}_{k}
  135. p ( x k + 1 | ฮพ k i ) p(x_{k+1}~{}|~{}\xi^{i}_{k})
  136. p ( x k + 1 | x k ) p(x_{k+1}~{}|~{}x_{k})
  137. x k = ฮพ k i x_{k}=\xi^{i}_{k}
  138. i = 1 , โ€ฆ , N . i=1,\ldots,N.
  139. N N
  140. ( ฮพ k + 1 i ) 1 โ‰ค i โ‰ค N \left(\xi^{i}_{k+1}\right)_{1\leq i\leq N}
  141. q ^ ( x k + 1 | y 0 , โ€ฆ , y k ) \widehat{q}(x_{k+1}|y_{0},\ldots,y_{k})
  142. p ^ ( d x k + 1 | y 0 , โ€ฆ , y k ) := 1 N โˆ‘ 1 โ‰ค i โ‰ค N ฮด ฮพ k + 1 i ( d x k + 1 ) โ‰ˆ N โ†‘ โˆž q ^ ( x k + 1 | y 0 , โ€ฆ , y k ) d x k + 1 โ‰ˆ N โ†‘ โˆž p ( x k + 1 | y 0 , โ€ฆ , y k ) d x k + 1 \widehat{p}(dx_{k+1}|y_{0},\ldots,y_{k}):=\frac{1}{N}\sum_{1\leq i\leq N}% \delta_{\xi^{i}_{k+1}}(dx_{k+1})\approx_{N\uparrow\infty}~{}\widehat{q}(x_{k+1% }~{}|~{}y_{0},\ldots,y_{k})dx_{k+1}\approx_{N\uparrow\infty}~{}p(x_{k+1}~{}|~{% }y_{0},\ldots,y_{k})dx_{k+1}
  143. p ^ ( d x k | y 0 , โ€ฆ , y k - 1 ) := 1 N โˆ‘ 1 โ‰ค i โ‰ค N ฮด ฮพ k i ( d x k ) โ‰ˆ N โ†‘ โˆž p ( d x k | y 0 , โ€ฆ , y k - 1 ) := p ( x k | y 0 , โ€ฆ , y k - 1 ) d x k \widehat{p}(dx_{k}|y_{0},\ldots,y_{k-1}):=\frac{1}{N}\sum_{1\leq i\leq N}% \delta_{\xi^{i}_{k}}(dx_{k})\approx_{N\uparrow\infty}~{}p(dx_{k}~{}|~{}y_{0},% \ldots,y_{k-1}):=p(x_{k}~{}|~{}y_{0},\ldots,y_{k-1})dx_{k}
  144. p ( d x k | y 0 , โ€ฆ , y k ) โ‰ˆ N โ†‘ โˆž p ( y k | x k ) p ^ ( d x k | y 0 , โ€ฆ , y k - 1 ) โˆซ p ( y k | x k โ€ฒ ) p ^ ( d x k โ€ฒ | y 0 , โ€ฆ , y k - 1 ) = โˆ‘ 1 โ‰ค i โ‰ค N p ( y k | ฮพ k i ) โˆ‘ 1 โ‰ค j โ‰ค N p ( y k | ฮพ k j ) ฮด ฮพ k i ( d x k ) p(dx_{k}|y_{0},\ldots,y_{k})~{}\approx_{N\uparrow\infty}~{}\displaystyle\frac{% p(y_{k}|x_{k})~{}\widehat{p}(dx_{k}|y_{0},\ldots,y_{k-1})}{\displaystyle\int p% (y_{k}|x^{\prime}_{k})\widehat{p}(dx^{\prime}_{k}|y_{0},\ldots,y_{k-1})}=\sum_% {1\leq i\leq N}~{}\frac{p(y_{k}~{}|~{}\xi^{i}_{k})}{\sum_{1\leq j\leq N}p(y_{k% }~{}|~{}\xi^{j}_{k})}~{}\delta_{\xi^{i}_{k}}(dx_{k})
  145. p ( d x k | y 0 , โ€ฆ , y k - 1 ) p(dx_{k}~{}|~{}y_{0},\ldots,y_{k-1})
  146. p ^ ( d x k | y 0 , โ€ฆ , y k - 1 ) \widehat{p}(dx_{k}~{}|~{}y_{0},\ldots,y_{k-1})
  147. I k ( f ) := โˆซ f ( x k ) p ( d x k | y 0 , โ€ฆ , y k - 1 ) โ‰ˆ N โ†‘ โˆž I ^ k ( f ) := โˆซ f ( x k ) p ^ ( d x k | y 0 , โ€ฆ , y k - 1 ) I_{k}(f):=\int f(x_{k})~{}p(dx_{k}|y_{0},\ldots,y_{k-1})~{}\approx_{N\uparrow% \infty}~{}\widehat{I}_{k}(f):=\int f(x_{k})~{}\widehat{p}(dx_{k}|y_{0},\ldots,% y_{k-1})~{}
  148. sup k โ‰ฅ 0 | E ( I ^ k ( f ) ) - I k ( f ) | โ‰ค c 1 / N and sup k โ‰ฅ 0 E ( [ I ^ k ( f ) - I k ( f ) ] 2 ) โ‰ค c 2 / N \sup_{k\geq 0}\left|E\left(\widehat{I}_{k}(f)\right)-I_{k}(f)\right|\leq{c_{1}% }/{N}\quad\mbox{and}~{}\quad\sup_{k\geq 0}E\left(\left[\widehat{I}_{k}(f)-I_{k% }(f)\right]^{2}\right)\leq{c_{2}}/{N}
  149. c 1 , c 2 < โˆž . c_{1},c_{2}<\infty~{}.
  150. x โ‰ฅ 0 x\geq 0
  151. | I ^ k ( f ) - I k ( f ) | โ‰ค c 1 x N + c 2 x N and sup 0 โ‰ค k โ‰ค n | I ^ k ( f ) - I k ( f ) | โ‰ค c x log ( n ) N \left|\widehat{I}_{k}(f)-I_{k}(f)\right|\leq c_{1}~{}\frac{x}{N}+c_{2}~{}\sqrt% {\frac{x}{N}}\quad\mbox{and}~{}\quad\sup_{0\leq k\leq n}\left|\widehat{I}_{k}(% f)-I_{k}(f)\right|\leq c~{}\sqrt{\frac{x\log(n)}{N}}
  152. 1 - e - x 1-e^{-x}
  153. c 1 and c 2 < โˆž c_{1}~{}\mbox{and}~{}~{}c_{2}<\infty~{}
  154. ( ฮพ ^ 0 , k i , ฮพ ^ 1 , k i , โ€ฆ , ฮพ ^ k - 1 , k i , ฮพ ^ k , k i ) \left(\widehat{\xi}^{i}_{0,k},\widehat{\xi}^{i}_{1,k},\ldots,\widehat{\xi}^{i}% _{k-1,k},\widehat{\xi}^{i}_{k,k}\right)
  155. ( ฮพ 0 , k i , ฮพ 1 , k i , โ€ฆ , ฮพ k - 1 , k i , ฮพ k , k ) \left(\xi^{i}_{0,k},\xi^{i}_{1,k},\ldots,\xi^{i}_{k-1,k},\xi_{k,k}\right)
  156. ฮพ ^ k i ( = ฮพ ^ k , k i ) \widehat{\xi}^{i}_{k}\left(=\widehat{\xi}^{i}_{k,k}\right)
  157. ฮพ k i ( = ฮพ k , k i ) {\xi}^{i}_{k}\left(={\xi}^{i}_{k,k}\right)
  158. p ^ ( d ( x 0 , โ€ฆ , x k ) | y 0 , โ€ฆ , y k ) := 1 N โˆ‘ 1 โ‰ค i โ‰ค N ฮด ( ฮพ ^ 0 , k i , โ€ฆ , ฮพ ^ 0 , k i ) ( d ( x 0 , โ€ฆ , x k ) ) โ‰ˆ N โ†‘ โˆž p ( d ( x 0 , โ€ฆ , x k ) | y 0 , โ€ฆ , y k ) \widehat{p}(d(x_{0},\ldots,x_{k})~{}|~{}y_{0},\ldots,y_{k}):=\frac{1}{N}\sum_{% 1\leq i\leq N}\delta_{\left(\widehat{\xi}^{i}_{0,k},\ldots,\widehat{\xi}^{i}_{% 0,k}\right)}(d(x_{0},\ldots,x_{k}))~{}~{}\approx_{N\uparrow\infty}~{}~{}p(d(x_% {0},\ldots,x_{k})~{}|~{}y_{0},\ldots,y_{k})
  159. p ( d ( x 0 , โ€ฆ , x k ) | y 0 , โ€ฆ , y k ) โ‰ˆ N โ†‘ โˆž โˆ‘ 1 โ‰ค i โ‰ค N p ( y k | ฮพ k , k i ) โˆ‘ 1 โ‰ค j โ‰ค N p ( y k | ฮพ k , k j ) ฮด ( ฮพ 0 , k i , โ€ฆ , ฮพ 0 , k i ) ( d ( x 0 , โ€ฆ , x k ) ) p(d(x_{0},\ldots,x_{k})~{}|~{}y_{0},\ldots,y_{k})~{}\approx_{N\uparrow\infty}~% {}~{}\sum_{1\leq i\leq N}~{}\frac{p(y_{k}~{}|~{}\xi^{i}_{k,k})}{\sum_{1\leq j% \leq N}p(y_{k}~{}|~{}\xi^{j}_{k,k})}~{}\delta_{\left(\xi^{i}_{0,k},\ldots,\xi^% {i}_{0,k}\right)}(d(x_{0},\ldots,x_{k}))
  160. p ^ ( d ( x 0 , โ€ฆ , x k ) | y 0 , โ€ฆ , y k - 1 ) := 1 N โˆ‘ 1 โ‰ค i โ‰ค N ฮด ( ฮพ 0 , k i , โ€ฆ , ฮพ k , k i ) ( d ( x 0 , โ€ฆ , x k ) ) โ‰ˆ N โ†‘ โˆž p ( d ( x 0 , โ€ฆ , x k ) | y 0 , โ€ฆ , y k - 1 ) := p ( x 0 , โ€ฆ , x k | y 0 , โ€ฆ , y k - 1 ) d x 0 , โ€ฆ , d x k \widehat{p}(d(x_{0},\ldots,x_{k})~{}|~{}y_{0},\ldots,y_{k-1}):=\frac{1}{N}\sum% _{1\leq i\leq N}\delta_{\left(\xi^{i}_{0,k},\ldots,\xi^{i}_{k,k}\right)}(d(x_{% 0},\ldots,x_{k}))~{}~{}\approx_{N\uparrow\infty}~{}~{}p(d(x_{0},\ldots,x_{k})~% {}|~{}y_{0},\ldots,y_{k-1}):=p(x_{0},\ldots,x_{k}~{}|~{}y_{0},\ldots,y_{k-1})~% {}dx_{0},\ldots,dx_{k}
  161. โˆซ F ( x 0 , โ€ฆ , x n ) p ^ ( d ( x 0 , โ€ฆ , x k ) | y 0 , โ€ฆ , y k ) := 1 N โˆ‘ 1 โ‰ค i โ‰ค N F ( ฮพ ^ 0 , k i , โ€ฆ , ฮพ ^ 0 , k i ) โ‰ˆ N โ†‘ โˆž โˆซ F ( x 0 , โ€ฆ , x n ) p ( d ( x 0 , โ€ฆ , x k ) | y 0 , โ€ฆ , y k ) \int F(x_{0},\ldots,x_{n})~{}\widehat{p}(d(x_{0},\ldots,x_{k})~{}|~{}y_{0},% \ldots,y_{k}):=\frac{1}{N}\sum_{1\leq i\leq N}F\left(\widehat{\xi}^{i}_{0,k},% \ldots,\widehat{\xi}^{i}_{0,k}\right)~{}~{}\approx_{N\uparrow\infty}~{}~{}\int% ~{}F(x_{0},\ldots,x_{n})~{}p(d(x_{0},\ldots,x_{k})~{}|~{}y_{0},\ldots,y_{k})
  162. โˆซ F ( x 0 , โ€ฆ , x n ) p ( d ( x 0 , โ€ฆ , x k ) | y 0 , โ€ฆ , y k ) โ‰ˆ N โ†‘ โˆž โˆ‘ 1 โ‰ค i โ‰ค N p ( y k | ฮพ k , k i ) โˆ‘ 1 โ‰ค j โ‰ค N p ( y k | ฮพ k , k j ) F ( ฮพ 0 , k i , ฮพ 1 , k i , โ€ฆ , ฮพ k , k i ) \int F(x_{0},\ldots,x_{n})~{}p(d(x_{0},\ldots,x_{k})~{}|~{}y_{0},\ldots,y_{k})% ~{}\approx_{N\uparrow\infty}~{}~{}\sum_{1\leq i\leq N}~{}\frac{p(y_{k}~{}|~{}% \xi^{i}_{k,k})}{\sum_{1\leq j\leq N}p(y_{k}~{}|~{}\xi^{j}_{k,k})}~{}F\left(\xi% ^{i}_{0,k},\xi^{i}_{1,k},\ldots,\xi^{i}_{k,k}\right)
  163. โˆซ F ( x 0 , โ€ฆ , x n ) p ^ ( d ( x 0 , โ€ฆ , x k ) | y 0 , โ€ฆ , y k - 1 ) := 1 N โˆ‘ 1 โ‰ค i โ‰ค N F ( ฮพ 0 , k i , โ€ฆ , ฮพ k , k i ) โ‰ˆ N โ†‘ โˆž โˆซ F ( x 0 , โ€ฆ , x n ) p ( d ( x 0 , โ€ฆ , x k ) | y 0 , โ€ฆ , y k - 1 ) \int~{}F(x_{0},\ldots,x_{n})~{}\widehat{p}(d(x_{0},\ldots,x_{k})~{}|~{}y_{0},% \ldots,y_{k-1}):=\frac{1}{N}\sum_{1\leq i\leq N}F\left(\xi^{i}_{0,k},\ldots,% \xi^{i}_{k,k}\right)~{}~{}\approx_{N\uparrow\infty}~{}~{}\int~{}F(x_{0},\ldots% ,x_{n})~{}p(d(x_{0},\ldots,x_{k})~{}|~{}y_{0},\ldots,y_{k-1})
  164. p ( y 0 , โ€ฆ , y n ) = โˆ 0 โ‰ค k โ‰ค n p ( y k | y 0 , โ€ฆ , y k - 1 ) with p ( y k | y 0 , โ€ฆ , y k - 1 ) = โˆซ p ( y k | x k ) p ( d x k | y 0 , โ€ฆ , y k - 1 ) p(y_{0},\ldots,y_{n})=\prod_{0\leq k\leq n}p(y_{k}|y_{0},\ldots,y_{k-1})\quad% \mbox{with}~{}\quad p(y_{k}|y_{0},\ldots,y_{k-1})=\int~{}p(y_{k}|x_{k})~{}p(dx% _{k}|y_{0},\ldots,y_{k-1})
  165. p ( y 0 | y 0 , โ€ฆ , y - 1 ) = p ( y 0 ) p(y_{0}|y_{0},\ldots,y_{-1})=p(y_{0})
  166. p ( x 0 | y 0 , โ€ฆ , y - 1 ) = p ( x 0 ) p(x_{0}|y_{0},\ldots,y_{-1})=p(x_{0})
  167. p ( x k | y 0 , โ€ฆ , y k - 1 ) d x k p(x_{k}|y_{0},\ldots,y_{k-1})dx_{k}
  168. p ^ ( d x k | y 0 , โ€ฆ , y k - 1 ) := 1 N โˆ‘ 1 โ‰ค i โ‰ค N ฮด ฮพ k i ( d x k ) โ‰ˆ N โ†‘ โˆž p ( d x k | y 0 , โ€ฆ , y k - 1 ) \widehat{p}(dx_{k}~{}|~{}y_{0},\ldots,y_{k-1}):=\frac{1}{N}\sum_{1\leq i\leq N% }\delta_{\xi^{i}_{k}}(dx_{k})~{}~{}\approx_{N\uparrow\infty}~{}~{}p(dx_{k}~{}|% ~{}y_{0},\ldots,y_{k-1})
  169. p ( y 0 , โ€ฆ , y n ) โ‰ˆ N โ†‘ โˆž p ^ ( y 0 , โ€ฆ , y n ) = โˆ 0 โ‰ค k โ‰ค n p ^ ( y k | y 0 , โ€ฆ , y k - 1 ) with p ^ ( y k | y 0 , โ€ฆ , y k - 1 ) = โˆซ p ( y k | x k ) p ^ ( d x k | y 0 , โ€ฆ , y k - 1 ) = 1 N โˆ‘ 1 โ‰ค i โ‰ค N p ( y k | ฮพ k i ) p(y_{0},\ldots,y_{n})~{}~{}\approx_{N\uparrow\infty}~{}~{}\widehat{p}(y_{0},% \ldots,y_{n})=\prod_{0\leq k\leq n}\widehat{p}(y_{k}|y_{0},\ldots,y_{k-1})% \quad\mbox{with}~{}\quad\widehat{p}(y_{k}|y_{0},\ldots,y_{k-1})=\int~{}p(y_{k}% |x_{k})~{}\widehat{p}(dx_{k}|y_{0},\ldots,y_{k-1})=\frac{1}{N}\sum_{1\leq i% \leq N}~{}p(y_{k}|\xi^{i}_{k})
  170. p ( y k | ฮพ k i ) p(y_{k}~{}|~{}\xi^{i}_{k})
  171. p ( y k | x k ) p(y_{k}~{}|~{}x_{k})
  172. x k = ฮพ k i x_{k}=\xi^{i}_{k}
  173. p ( ( x 0 , โ€ฆ , x n ) | ( y 0 , โ€ฆ , y n - 1 ) ) = p ( x n | ( y 0 , โ€ฆ , y n - 1 ) ) p ( x n - 1 | x n , ( y 0 , โ€ฆ , y n - 1 ) ) ร— p ( x n - 2 | x n - 1 , ( y 0 , โ€ฆ , y n - 2 ) ) โ€ฆ p ( x 1 | x 2 , ( y 0 , y 1 ) ) p ( x 0 | x 1 , y 0 ) \begin{array}[]{l}p((x_{0},\ldots,x_{n})~{}|~{}(y_{0},\ldots,y_{n-1}))\\ \\ =p(x_{n}~{}|~{}(y_{0},\ldots,y_{n-1}))~{}p(x_{n-1}~{}|~{}x_{n},(y_{0},\ldots,y% _{n-1}))~{}\times p(x_{n-2}~{}|~{}x_{n-1},(y_{0},\ldots,y_{n-2}))~{}~{}\ldots~% {}~{}p(x_{1}~{}|~{}x_{2},(y_{0},y_{1}))~{}p(x_{0}~{}|~{}x_{1},y_{0})\end{array}
  174. p ( x k - 1 | x k , ( y 0 , โ€ฆ , y k - 1 ) ) โˆ p ( x k | x k - 1 ) p ( x k - 1 | ( y 0 , โ€ฆ , y k - 1 ) ) and p ( x k - 1 | ( y 0 , โ€ฆ , y k - 1 ) โˆ p ( y k - 1 | x k - 1 ) p ( x k - 1 | ( y 0 , โ€ฆ , y k - 2 ) p(x_{k-1}~{}|~{}x_{k},(y_{0},\ldots,y_{k-1}))\propto p(x_{k}~{}|~{}x_{k-1})~{}% p(x_{k-1}~{}|~{}(y_{0},\ldots,y_{k-1}))\quad\mbox{and}~{}\quad p(x_{k-1}~{}|~{% }(y_{0},\ldots,y_{k-1})\propto p(y_{k-1}|x_{k-1})~{}p(x_{k-1}~{}|~{}(y_{0},% \ldots,y_{k-2})
  175. p ( x k - 1 | x k , ( y 0 , โ€ฆ , y k - 1 ) ) = p ( y k - 1 | x k - 1 ) p ( x k | x k - 1 ) p ( x k - 1 | y 0 , โ€ฆ , y k - 2 ) โˆซ p ( y k - 1 | x k - 1 โ€ฒ ) p ( x k | x k - 1 โ€ฒ ) p ( x k - 1 โ€ฒ | y 0 , โ€ฆ , y k - 2 ) d x k - 1 โ€ฒ p(x_{k-1}~{}|~{}x_{k},(y_{0},\ldots,y_{k-1}))=\frac{p(y_{k-1}|x_{k-1})~{}p(x_{% k}~{}|~{}x_{k-1})~{}p(x_{k-1}|y_{0},\ldots,y_{k-2})}{\int~{}p(y_{k-1}|x^{% \prime}_{k-1})~{}p(x_{k}~{}|~{}x^{\prime}_{k-1})~{}p(x^{\prime}_{k-1}|y_{0},% \ldots,y_{k-2})~{}dx^{\prime}_{k-1}}
  176. p ( x k - 1 | ( y 0 , โ€ฆ , y k - 2 ) ) d x k - 1 p(x_{k-1}~{}|~{}(y_{0},\ldots,y_{k-2}))dx_{k-1}
  177. p ^ ( d x k - 1 | ( y 0 , โ€ฆ , y k - 2 ) ) = 1 N โˆ‘ 1 โ‰ค i โ‰ค N ฮด ฮพ k - 1 i ( d x k - 1 ) ( โ‰ˆ N โ†‘ โˆž p ( d x k - 1 | ( y 0 , โ€ฆ , y k - 2 ) ) := p ( x k - 1 | ( y 0 , โ€ฆ , y k - 2 ) ) d x k - 1 ) \widehat{p}(dx_{k-1}~{}|~{}(y_{0},\ldots,y_{k-2}))=\frac{1}{N}\sum_{1\leq i% \leq N}\delta_{\xi^{i}_{k-1}}(dx_{k-1})~{}\left(\approx_{N\uparrow\infty}~{}{p% }(dx_{k-1}~{}|~{}(y_{0},\ldots,y_{k-2})):={p}(x_{k-1}~{}|~{}(y_{0},\ldots,y_{k% -2}))~{}dx_{k-1}\right)
  178. p ( d x k - 1 | x k , ( y 0 , โ€ฆ , y k - 1 ) ) โ‰ˆ N โ†‘ โˆž p ^ ( d x k - 1 | x k , ( y 0 , โ€ฆ , y k - 1 ) ) : = p ( y k - 1 | x k - 1 ) p ( x k | x k - 1 ) p ^ ( d x k - 1 | y 0 , โ€ฆ , y k - 2 ) โˆซ p ( y k - 1 | x k - 1 โ€ฒ ) p ( x k | x k - 1 โ€ฒ ) p ^ ( d x k - 1 โ€ฒ | y 0 , โ€ฆ , y k - 2 ) = โˆ‘ 1 โ‰ค i โ‰ค N p ( y k - 1 | ฮพ k - 1 i ) p ( x k | ฮพ k - 1 i ) โˆ‘ 1 โ‰ค j โ‰ค N p ( y k - 1 | ฮพ k - 1 j ) p ( x k | ฮพ k - 1 j ) ฮด ฮพ k - 1 i ( d x k - 1 ) p(dx_{k-1}~{}|~{}x_{k},(y_{0},\ldots,y_{k-1}))\approx_{N\uparrow\infty}~{}% \begin{array}[t]{rcl}\widehat{p}(dx_{k-1}~{}|~{}x_{k},(y_{0},\ldots,y_{k-1}))&% :=&\displaystyle\frac{p(y_{k-1}|x_{k-1})~{}p(x_{k}~{}|~{}x_{k-1})~{}\widehat{p% }(dx_{k-1}|y_{0},\ldots,y_{k-2})}{\int~{}p(y_{k-1}|x^{\prime}_{k-1})~{}p(x_{k}% ~{}|~{}x^{\prime}_{k-1})~{}\widehat{p}(dx^{\prime}_{k-1}|y_{0},\ldots,y_{k-2})% }\\ &&\\ &=&\displaystyle\sum_{1\leq i\leq N}\frac{p(y_{k-1}|\xi^{i}_{k-1})~{}p(x_{k}~{% }|~{}\xi^{i}_{k-1})}{\sum_{1\leq j\leq N}p(y_{k-1}|\xi^{j}_{k-1})~{}p(x_{k}~{}% |~{}\xi^{j}_{k-1})}~{}\delta_{\xi^{i}_{k-1}}(dx_{k-1})\end{array}
  179. p ( d ( x 0 , โ€ฆ , x n ) | ( y 0 , โ€ฆ , y n - 1 ) ) โ‰ˆ N โ†‘ โˆž p ^ b a c k w a r d ( d ( x 0 , โ€ฆ , x n ) | ( y 0 , โ€ฆ , y n - 1 ) ) p(d(x_{0},\ldots,x_{n})~{}|~{}(y_{0},\ldots,y_{n-1}))\approx_{N\uparrow\infty}% \widehat{p}_{backward}(d(x_{0},\ldots,x_{n})~{}|~{}(y_{0},\ldots,y_{n-1}))
  180. p ^ b a c k w a r d ( d ( x 0 , โ€ฆ , x n ) | ( y 0 , โ€ฆ , y n - 1 ) ) = p ^ ( d x n | ( y 0 , โ€ฆ , y n - 1 ) ) p ^ ( d x n - 1 | x n , ( y 0 , โ€ฆ , y n - 1 ) ) ร— p ^ ( d x n - 2 | x n - 1 , ( y 0 , โ€ฆ , y n - 2 ) ) โ€ฆ p ^ ( d x 1 | x 2 , ( y 0 , y 1 ) ) p ^ ( d x 0 | x 1 , y 0 ) \begin{array}[]{l}\widehat{p}_{backward}(d(x_{0},\ldots,x_{n})~{}|~{}(y_{0},% \ldots,y_{n-1}))\\ \\ =\widehat{p}(dx_{n}~{}|~{}(y_{0},\ldots,y_{n-1}))~{}\widehat{p}(dx_{n-1}~{}|~{% }x_{n},(y_{0},\ldots,y_{n-1}))~{}\times\widehat{p}(dx_{n-2}~{}|~{}x_{n-1},(y_{% 0},\ldots,y_{n-2}))~{}~{}\ldots~{}~{}\widehat{p}(dx_{1}~{}|~{}x_{2},(y_{0},y_{% 1}))~{}\widehat{p}(dx_{0}~{}|~{}x_{1},y_{0})\end{array}
  181. p ^ b a c k w a r d ( d ( x 0 , โ€ฆ , x n ) | ( y 0 , โ€ฆ , y n - 1 ) ) \widehat{p}_{backward}(d(x_{0},\ldots,x_{n})~{}|~{}(y_{0},\ldots,y_{n-1}))
  182. ( ๐• k , n โ™ญ ) 0 โ‰ค k โ‰ค n \left(\mathbb{X}^{\flat}_{k,n}\right)_{0\leq k\leq n}
  183. ฮพ k i , with i = 1 , โ€ฆ , N . \xi^{i}_{k},~{}\mbox{with}~{}~{}i=1,\ldots,N.
  184. ๐• n , n โ™ญ \mathbb{X}^{\flat}_{n,n}
  185. p ^ ( d x n | ( y 0 , โ€ฆ , y n - 1 ) ) = 1 N โˆ‘ 1 โ‰ค i โ‰ค N ฮด ฮพ n i ( d x n ) \widehat{p}(dx_{n}~{}|~{}(y_{0},\ldots,y_{n-1}))=\frac{1}{N}\sum_{1\leq i\leq N% }\delta_{\xi^{i}_{n}}(dx_{n})
  186. ๐• k , n โ™ญ = ฮพ k i ( for some i = 1 , โ€ฆ , N ) \mathbb{X}^{\flat}_{k,n}=\xi^{i}_{k}~{}(\mbox{for some}~{}~{}i=1,\ldots,N)
  187. ๐• k - 1 , n โ™ญ \mathbb{X}^{\flat}_{k-1,n}
  188. p ^ ( d x k - 1 | ฮพ k i , ( y 0 , โ€ฆ , y k - 1 ) ) = โˆ‘ 1 โ‰ค j โ‰ค N p ( y k - 1 | ฮพ k - 1 j ) p ( ฮพ k i | ฮพ k - 1 j ) โˆ‘ 1 โ‰ค l โ‰ค N p ( y k - 1 | ฮพ k - 1 l ) p ( ฮพ k i | ฮพ k - 1 l ) ฮด ฮพ k - 1 j ( d x k - 1 ) \widehat{p}(dx_{k-1}~{}|~{}\xi^{i}_{k},(y_{0},\ldots,y_{k-1}))=\displaystyle% \sum_{1\leq j\leq N}\frac{p(y_{k-1}|\xi^{j}_{k-1})~{}p(\xi^{i}_{k}~{}|~{}\xi^{% j}_{k-1})}{\displaystyle\sum_{1\leq l\leq N}p(y_{k-1}|\xi^{l}_{k-1})~{}p(\xi^{% i}_{k}~{}|~{}\xi^{l}_{k-1})}~{}\delta_{\xi^{j}_{k-1}}(dx_{k-1})
  189. p ^ ( d x k - 1 | ฮพ k i , ( y 0 , โ€ฆ , y k - 1 ) ) \widehat{p}(dx_{k-1}~{}|~{}\xi^{i}_{k},(y_{0},\ldots,y_{k-1}))
  190. p ^ ( d x k - 1 | x k , ( y 0 , โ€ฆ , y k - 1 ) ) \widehat{p}(dx_{k-1}~{}|~{}x_{k},(y_{0},\ldots,y_{k-1}))
  191. x k = ฮพ k i x_{k}=\xi^{i}_{k}
  192. p ( y k - 1 | ฮพ k - 1 j ) p(y_{k-1}|\xi^{j}_{k-1})
  193. p ( ฮพ k i | ฮพ k - 1 j ) p(\xi^{i}_{k}~{}|~{}\xi^{j}_{k-1})
  194. p ( y k - 1 | x k - 1 ) p(y_{k-1}|x_{k-1})
  195. p ( x k | x k - 1 ) p(x_{k}~{}|~{}x_{k-1})
  196. x k = ฮพ k i x_{k}=\xi^{i}_{k}
  197. x k - 1 = ฮพ k - 1 j . x_{k-1}=\xi^{j}_{k-1}.
  198. p ( ( x 0 , โ€ฆ , x n ) | ( y 0 , โ€ฆ , y n - 1 ) ) p((x_{0},\ldots,x_{n})~{}|~{}(y_{0},\ldots,y_{n-1}))
  199. f k ( . ) f_{k}(.)
  200. โˆซ p ( d ( x 0 , โ€ฆ , x n ) | ( y 0 , โ€ฆ , y n - 1 ) ) f k ( x k ) โ‰ˆ N โ†‘ โˆž โˆซ p ^ b a c k w a r d ( d ( x 0 , โ€ฆ , x n ) | ( y 0 , โ€ฆ , y n - 1 ) ) f k ( x k ) = โˆซ p ^ ( d x n | ( y 0 , โ€ฆ , y n - 1 ) ) p ^ ( d x n - 1 | x n , ( y 0 , โ€ฆ , y n - 1 ) ) ร— p ^ ( d x n - 2 | x n - 1 , ( y 0 , โ€ฆ , y n - 2 ) ) โ€ฆ p ^ ( d x k | x k + 1 , ( y 0 , โ€ฆ , y k ) ) f k ( x k ) = [ 1 N , โ€ฆ , 1 N ] โŸ N times ๐•„ n - 1 ๐•„ n - 2 โ€ฆ ๐•„ k [ f k ( ฮพ k 1 ) โ‹ฎ f k ( ฮพ k N ) ] \begin{array}[]{l}\displaystyle\int~{}p(d(x_{0},\ldots,x_{n})~{}|~{}(y_{0},% \ldots,y_{n-1}))~{}f_{k}(x_{k})\\ \\ ~{}\approx_{N\uparrow\infty}~{}\displaystyle\int\widehat{p}_{backward}(d(x_{0}% ,\ldots,x_{n})~{}|~{}(y_{0},\ldots,y_{n-1}))~{}f_{k}(x_{k})\\ \\ =\displaystyle\int~{}\widehat{p}(dx_{n}~{}|~{}(y_{0},\ldots,y_{n-1}))~{}% \widehat{p}(dx_{n-1}~{}|~{}x_{n},(y_{0},\ldots,y_{n-1}))~{}\times\widehat{p}(% dx_{n-2}~{}|~{}x_{n-1},(y_{0},\ldots,y_{n-2}))~{}~{}\ldots~{}~{}\widehat{p}(dx% _{k}~{}|~{}x_{k+1},(y_{0},\ldots,y_{k}))~{}f_{k}(x_{k})\\ \\ =\underbrace{\left[\frac{1}{N},\ldots,\frac{1}{N}\right]}_{\mbox{N times}~{}}% \mathbb{M}_{n-1}\mathbb{M}_{n-2}\ldots\mathbb{M}_{k}\left[\begin{array}[]{c}f_% {k}(\xi^{1}_{k})\\ \vdots\\ f_{k}(\xi^{N}_{k})\end{array}\right]\end{array}
  201. ( N ร— N ) (N\times N)
  202. ๐•„ k = ( ๐•„ k ( i , j ) ) 1 โ‰ค i , j โ‰ค N \mathbb{M}_{k}=\left(\mathbb{M}_{k}(i,j)\right)_{1\leq i,j\leq N}
  203. ๐•„ k ( i , j ) = p ( ฮพ k i | ฮพ k - 1 j ) p ( y k - 1 | ฮพ k - 1 j ) โˆ‘ 1 โ‰ค l โ‰ค N p ( ฮพ k i | ฮพ k - 1 l ) p ( y k - 1 | ฮพ k - 1 l ) \mathbb{M}_{k}(i,j)=\frac{p(\xi^{i}_{k}~{}|~{}\xi^{j}_{k-1})~{}p(y_{k-1}|\xi^{% j}_{k-1})~{}}{\displaystyle\sum_{1\leq l\leq N}~{}p(\xi^{i}_{k}~{}|~{}\xi^{l}_% {k-1})~{}p(y_{k-1}|\xi^{l}_{k-1})}
  204. F ยฏ ( x 0 , โ€ฆ , x n ) := 1 n + 1 โˆ‘ 0 โ‰ค k โ‰ค n f k ( x k ) โŸน โˆซ F ยฏ ( x 0 , โ€ฆ , x n ) p ( d ( x 0 , โ€ฆ , x n ) | ( y 0 , โ€ฆ , y n - 1 ) ) โ‰ˆ N โ†‘ โˆž โˆซ F ยฏ ( x 0 , โ€ฆ , x n ) p ^ b a c k w a r d ( d ( x 0 , โ€ฆ , x n ) | ( y 0 , โ€ฆ , y n - 1 ) ) = 1 n + 1 โˆ‘ 0 โ‰ค k โ‰ค n [ 1 N , โ€ฆ , 1 N ] โŸ N times ๐•„ n - 1 ๐•„ n - 2 โ€ฆ ๐•„ k [ f k ( ฮพ k 1 ) โ‹ฎ f k ( ฮพ k N ) ] \overline{F}(x_{0},\ldots,x_{n}):=\frac{1}{n+1}\sum_{0\leq k\leq n}f_{k}(x_{k}% )\quad\Longrightarrow\quad\begin{array}[t]{l}\displaystyle\int~{}\overline{F}(% x_{0},\ldots,x_{n})~{}p(d(x_{0},\ldots,x_{n})~{}|~{}(y_{0},\ldots,y_{n-1}))\\ \\ ~{}\approx_{N\uparrow\infty}~{}\displaystyle\int\overline{F}(x_{0},\ldots,x_{n% })~{}\widehat{p}_{backward}(d(x_{0},\ldots,x_{n})~{}|~{}(y_{0},\ldots,y_{n-1})% )\\ \\ \displaystyle=\frac{1}{n+1}\sum_{0\leq k\leq n}\underbrace{\left[\frac{1}{N},% \ldots,\frac{1}{N}\right]}_{\mbox{N times}~{}}\mathbb{M}_{n-1}\mathbb{M}_{n-2}% \ldots\mathbb{M}_{k}\left[\begin{array}[]{c}f_{k}(\xi^{1}_{k})\\ \vdots\\ f_{k}(\xi^{N}_{k})\end{array}\right]\end{array}
  205. E ( p ^ ( y 0 , โ€ฆ , y n ) ) = p ( y 0 , โ€ฆ , y n ) and E ( [ p ^ ( y 0 , โ€ฆ , y n ) p ( y 0 , โ€ฆ , y n ) - 1 ] 2 ) โ‰ค c n / N E\left(\widehat{p}(y_{0},\ldots,y_{n})\right)=p(y_{0},\ldots,y_{n})\quad\mbox{% and}~{}\quad E\left(\left[\frac{\widehat{p}(y_{0},\ldots,y_{n})}{p(y_{0},% \ldots,y_{n})}-1\right]^{2}\right)\leq{c}~{}n/{N}
  206. x โ‰ฅ 0 x\geq 0
  207. | 1 n log p ^ ( y 0 , โ€ฆ , y n ) - 1 n log p ^ ( y 0 , โ€ฆ , y n ) | โ‰ค c 1 x N + c 2 x N \left|\frac{1}{n}\log{\widehat{p}(y_{0},\ldots,y_{n})}-\frac{1}{n}\log{% \widehat{p}(y_{0},\ldots,y_{n})}\right|\leq c_{1}~{}\frac{x}{N}+c_{2}~{}\sqrt{% \frac{x}{N}}
  208. 1 - e - x 1-e^{-x}
  209. c 1 and c 2 < โˆž c_{1}~{}\mbox{and}~{}~{}c_{2}<\infty~{}
  210. I k p a t h ( F ) := โˆซ F ( x 0 , โ€ฆ , x k ) p ( d ( x 0 , โ€ฆ , x k ) | y 0 , โ€ฆ , y k - 1 ) โ‰ˆ N โ†‘ โˆž I ^ k p a t h ( F ) := โˆซ F ( x 0 , โ€ฆ , x k ) p ^ ( d ( x 0 , โ€ฆ , x k ) | y 0 , โ€ฆ , y k - 1 ) = 1 N โˆ‘ 1 โ‰ค i โ‰ค N F ( ฮพ 0 , k i , โ€ฆ , ฮพ k , k i ) I^{path}_{k}(F):=\int F(x_{0},\ldots,x_{k})~{}p(d(x_{0},\ldots,x_{k})|y_{0},% \ldots,y_{k-1})~{}\approx_{N\uparrow\infty}~{}\widehat{I}^{path}_{k}(F):=\int F% (x_{0},\ldots,x_{k})~{}\widehat{p}(d(x_{0},\ldots,x_{k})|y_{0},\ldots,y_{k-1})% ~{}=\frac{1}{N}\sum_{1\leq i\leq N}F\left(\xi^{i}_{0,k},\ldots,\xi^{i}_{k,k}\right)
  211. | E ( I ^ k p a t h ( F ) ) - I k p a t h ( F ) | โ‰ค c 1 k / N and E ( [ I ^ k p a t h ( F ) - I k p a t h ( F ) ] 2 ) โ‰ค c 2 k / N \left|E\left(\widehat{I}^{path}_{k}(F)\right)-I_{k}^{path}(F)\right|\leq{c_{1}% }~{}k/{N}\quad\mbox{and}~{}\quad E\left(\left[\widehat{I}^{path}_{k}(F)-I_{k}^% {path}(F)\right]^{2}\right)\leq{c_{2}}~{}k/{N}
  212. c 1 , c 2 < โˆž . c_{1},c_{2}<\infty~{}.
  213. x โ‰ฅ 0 x\geq 0
  214. | I ^ k p a t h ( F ) - I k p a t h ( F ) | โ‰ค c 1 k x N + c 2 k x N and sup 0 โ‰ค k โ‰ค n | I ^ k p a t h ( F ) - I k p a t h ( F ) | โ‰ค c x n log ( n ) N \left|\widehat{I}^{path}_{k}(F)-I_{k}^{path}(F)\right|\leq c_{1}~{}\frac{k~{}x% }{N}+c_{2}~{}\sqrt{\frac{k~{}x}{N}}\quad\mbox{and}~{}\quad\sup_{0\leq k\leq n}% \left|\widehat{I}_{k}^{path}(F)-I^{path}_{k}(F)\right|\leq c~{}\sqrt{\frac{x~{% }n~{}\log(n)}{N}}
  215. 1 - e - x 1-e^{-x}
  216. c 1 and c 2 < โˆž c_{1}~{}\mbox{and}~{}~{}c_{2}<\infty~{}
  217. F ยฏ ( x 0 , โ€ฆ , x n ) := 1 n + 1 โˆ‘ 0 โ‰ค k โ‰ค n f k ( x k ) with I n p a t h ( F ยฏ ) โ‰ˆ N โ†‘ โˆž I n โ™ญ , p a t h ( F ยฏ ) := โˆซ F ยฏ ( x 0 , โ€ฆ , x n ) p ^ b a c k w a r d ( d ( x 0 , โ€ฆ , x n ) | ( y 0 , โ€ฆ , y n - 1 ) ) \overline{F}(x_{0},\ldots,x_{n}):=\frac{1}{n+1}\sum_{0\leq k\leq n}f_{k}(x_{k}% )\quad\mbox{with}~{}\quad I^{path}_{n}(\overline{F})~{}\approx_{N\uparrow% \infty}~{}I^{\flat,path}_{n}(\overline{F}):=\displaystyle\int\overline{F}(x_{0% },\ldots,x_{n})~{}\widehat{p}_{backward}(d(x_{0},\ldots,x_{n})~{}|~{}(y_{0},% \ldots,y_{n-1}))
  218. f k ( . ) f_{k}(.)
  219. sup n โ‰ฅ 0 | E ( I ^ n โ™ญ , p a t h ( F ยฏ ) ) - I n p a t h ( F ยฏ ) | โ‰ค c 1 / N and E ( [ I ^ n โ™ญ , p a t h ( F ) - I n p a t h ( F ) ] 2 ) โ‰ค c 2 / ( n N ) + c 3 / N 2 \sup_{n\geq 0}{\left|E\left(\widehat{I}^{\flat,path}_{n}(\overline{F})\right)-% I_{n}^{path}(\overline{F})\right|}~{}\leq{c_{1}}/{N}\quad\mbox{and}~{}\quad E% \left(\left[\widehat{I}^{\flat,path}_{n}(F)-I_{n}^{path}(F)\right]^{2}\right)% \leq{c_{2}}/{(nN)}+{c_{3}}/{N^{2}}
  220. c 1 , c 2 , c 3 < โˆž c_{1},c_{2},c_{3}<\infty
  221. p ( x k | y 0 , โ€ฆ , y k ) p(x_{k}|y_{0},\ldots,y_{k})
  222. { ( w k ( i ) , x k ( i ) ) : i โˆˆ { 1 , โ€ฆ , N } } . \{(w^{(i)}_{k},x^{(i)}_{k})~{}:~{}i\in\{1,\ldots,N\}\}.
  223. w k ( i ) w^{(i)}_{k}
  224. โˆ‘ i = 1 N w k ( i ) = 1 \sum_{i=1}^{N}w^{(i)}_{k}=1
  225. f ( โ‹… ) f(\cdot)
  226. โˆซ f ( x k ) p ( x k | y 0 , โ€ฆ , y k ) d x k โ‰ˆ โˆ‘ i = 1 N w k ( i ) f ( x k ( i ) ) . \int f(x_{k})p(x_{k}|y_{0},\dots,y_{k})dx_{k}\approx\sum_{i=1}^{N}w_{k}^{(i)}f% (x_{k}^{(i)}).
  227. ฯ€ ( x k | x 0 : k - 1 , y 0 : k ) \pi(x_{k}|x_{0:k-1},y_{0:k})\,
  228. ฯ€ ( x k | x 0 : k - 1 , y 0 : k ) = p ( x k | x k - 1 , y k ) = p ( y k | x k ) โˆซ p ( y k | x k ) p ( x k | x k - 1 ) d x k p ( x k | x k - 1 ) . \pi(x_{k}|x_{0:k-1},y_{0:k})=p(x_{k}|x_{k-1},y_{k})=\frac{p(y_{k}|x_{k})}{\int% ~{}p(y_{k}|x_{k})p(x_{k}|x_{k-1})dx_{k}}~{}p(x_{k}|x_{k-1}).\,
  229. p ( x k | x k - 1 , y k ) p(x_{k}|x_{k-1},y_{k})
  230. p ( y k | x k ) โˆซ p ( y k | x k ) p ( x k | x k - 1 ) d x k p ( x k | x k - 1 ) d x k โ‰ƒ N โ†‘ โˆž p ( y k | x k ) โˆซ p ( y k | x k ) p ^ ( d x k | x k - 1 ) p ^ ( d x k | x k - 1 ) = โˆ‘ 1 โ‰ค i โ‰ค N p ( y k | X k i ( x k - 1 ) ) โˆ‘ 1 โ‰ค j โ‰ค N p ( y k | X k j ( x k - 1 ) ) ฮด X k i ( x k - 1 ) ( d x k ) \frac{p(y_{k}|x_{k})}{\int~{}p(y_{k}|x_{k})p(x_{k}|x_{k-1})dx_{k}}~{}p(x_{k}|x% _{k-1})dx_{k}\simeq_{N\uparrow\infty}\frac{p(y_{k}|x_{k})}{\int~{}p(y_{k}|x_{k% })\widehat{p}(dx_{k}|x_{k-1})}~{}\widehat{p}(dx_{k}|x_{k-1})=\sum_{1\leq i\leq N% }\frac{p(y_{k}|X^{i}_{k}(x_{k-1}))}{\sum_{1\leq j\leq N}p(y_{k}|X^{j}_{k}(x_{k% -1}))}~{}\delta_{X^{i}_{k}(x_{k-1})}(dx_{k})
  231. p ^ ( d x k | x k - 1 ) = 1 N โˆ‘ i = 1 N ฮด X k i ( x k - 1 ) ( d x k ) โ‰ƒ N โ†‘ โˆž p ( x k | x k - 1 ) d x k \widehat{p}(dx_{k}|x_{k-1})=\frac{1}{N}\sum_{i=1}^{N}\delta_{X^{i}_{k}(x_{k-1}% )}(dx_{k})~{}\simeq_{N\uparrow\infty}~{}p(x_{k}|x_{k-1})dx_{k}
  232. N N
  233. X k i ( x k - 1 ) , i = 1 , โ€ฆ , N X^{i}_{k}(x_{k-1}),~{}i=1,...,N
  234. X k X_{k}
  235. X k - 1 = x k - 1 X_{k-1}=x_{k-1}
  236. ฮด a \delta_{a}
  237. ฯ€ ( x k | x 0 : k - 1 , y 0 : k ) = p ( x k | x k - 1 ) . \pi(x_{k}|x_{0:k-1},y_{0:k})=p(x_{k}|x_{k-1}).\,
  238. i = 1 , โ€ฆ , N i=1,\ldots,N
  239. x k ( i ) โˆผ ฯ€ ( x k | x 0 : k - 1 ( i ) , y 0 : k ) x^{(i)}_{k}\sim\pi(x_{k}|x^{(i)}_{0:k-1},y_{0:k})
  240. i = 1 , โ€ฆ , N i=1,\ldots,N
  241. w ^ k ( i ) = w k - 1 ( i ) p ( y k | x k ( i ) ) p ( x k ( i ) | x k - 1 ( i ) ) ฯ€ ( x k ( i ) | x 0 : k - 1 ( i ) , y 0 : k ) . \hat{w}^{(i)}_{k}=w^{(i)}_{k-1}\frac{p(y_{k}|x^{(i)}_{k})p(x^{(i)}_{k}|x^{(i)}% _{k-1})}{\pi(x_{k}^{(i)}|x^{(i)}_{0:k-1},y_{0:k})}.
  242. ฯ€ ( x k ( i ) | x 0 : k - 1 ( i ) , y 0 : k ) = p ( x k ( i ) | x k - 1 ( i ) ) \pi(x_{k}^{(i)}|x^{(i)}_{0:k-1},y_{0:k})=p(x^{(i)}_{k}|x^{(i)}_{k-1})
  243. w ^ k ( i ) = w k - 1 ( i ) p ( y k | x k ( i ) ) , \hat{w}^{(i)}_{k}=w^{(i)}_{k-1}p(y_{k}|x^{(i)}_{k}),
  244. i = 1 , โ€ฆ , N i=1,\ldots,N
  245. w k ( i ) = w ^ k ( i ) โˆ‘ j = 1 N w ^ k ( j ) w^{(i)}_{k}=\frac{\hat{w}^{(i)}_{k}}{\sum_{j=1}^{N}\hat{w}^{(j)}_{k}}
  246. N ^ ๐‘’๐‘“๐‘“ = 1 โˆ‘ i = 1 N ( w k ( i ) ) 2 \hat{N}_{\mathit{eff}}=\frac{1}{\sum_{i=1}^{N}\left(w^{(i)}_{k}\right)^{2}}
  247. N ^ ๐‘’๐‘“๐‘“ < N t h r \hat{N}_{\mathit{eff}}<N_{thr}
  248. N N
  249. i = 1 , โ€ฆ , N i=1,\ldots,N
  250. w k ( N ) = 1 / N . w^{(N)}_{k}=1/N.
  251. x x
  252. k k
  253. p x k | y 1 : k ( x | y 1 : k ) p_{x_{k}|y_{1:k}}(x|y_{1:k})
  254. { 1 , โ€ฆ , N } \{1,...,N\}
  255. x ^ \hat{x}
  256. p ( x k | x k - 1 ) , with x k - 1 = x k - 1 | k - 1 ( i ) p(x_{k}|x_{k-1}),~{}\mbox{with}~{}~{}x_{k-1}=x_{k-1|k-1}^{(i)}
  257. y ^ \hat{y}
  258. x ^ \hat{x}
  259. p ( y k | x k ) , with x k = x ^ p(y_{k}|x_{k}),~{}\mbox{with}~{}~{}x_{k}=\hat{x}
  260. y k y_{k}
  261. [ 0 , m k ] [0,m_{k}]
  262. m k = sup x k p ( y k | x k ) m_{k}=\sup_{x_{k}}p(y_{k}|x_{k})
  263. p ( y ^ ) p\left(\hat{y}\right)
  264. x ^ \hat{x}
  265. x k | k ( i ) x_{k|k}^{(i)}
  266. k k
  267. k - 1 k-1
  268. x k x_{k}
  269. x k - 1 x_{k-1}
  270. k - 1 k-1
  271. k k
  272. k k
  273. x x
  274. k k
  275. x ( k , i ) x(k,i)
  276. k k
  277. x k ( i ) x_{k}^{(i)}
  278. x k x_{k}
  279. x k - 1 ( i ) x_{k-1}^{(i)}
  280. k - 1 k-1
  281. x k x_{k}
  282. x k - 1 x_{k-1}

Pauliโ€“Villars_regularization.html

  1. 1 k 2 + i ฯต \frac{1}{k^{2}+i\epsilon}
  2. 1 k 2 + i ฯต - 1 k 2 - ฮ› 2 + i ฯต \frac{1}{k^{2}+i\epsilon}-\frac{1}{k^{2}-\Lambda^{2}+i\epsilon}
  3. ฮ› \Lambda

Pbar.html

  1. p ยฏ \bar{p}

PCF_theory.html

  1. c f ( โˆ A / D ) cf(\prod A/D)
  2. โˆ A \prod A
  3. f < g f<g
  4. { x โˆˆ A : f ( x ) < g ( x ) } โˆˆ D \{x\in A:f(x)<g(x)\}\in D
  5. pcf ( A ) = { c f ( โˆ A / D ) : D is an ultrafilter on A } . {\rm pcf}(A)=\{cf(\prod A/D):D\,\,\mbox{is an ultrafilter on}~{}\,\,A\}.
  6. A โІ pcf ( A ) A\subseteq{\rm pcf}(A)
  7. | A | < min ( A ) |A|<\min(A)
  8. { B ฮธ : ฮธ โˆˆ pcf ( A ) } \{B_{\theta}:\theta\in{\rm pcf}(A)\}
  9. c f ( โˆ A / D ) cf(\prod A/D)
  10. B ฮธ โˆˆ D B_{\theta}\in D
  11. | pcf ( A ) | โ‰ค 2 | A | |{\rm pcf}(A)|\leq 2^{|A|}
  12. 2 โ„ต ฯ‰ < โ„ต ฯ‰ 4 2^{\aleph_{\omega}}<\aleph_{\omega_{4}}
  13. c f ( โˆ A / D ) < ฮป cf(\prod A/D)<\lambda
  14. { B ฮธ : ฮธ โˆˆ pcf ( A ) , ฮธ < ฮป } \{B_{\theta}:\theta\in{\rm pcf}(A),\theta<\lambda\}
  15. โˆ B ฮป \prod B_{\lambda}
  16. โˆ A \prod A
  17. 2 โ„ต ฯ‰ < โ„ต ฯ‰ 1 2^{\aleph_{\omega}}<\aleph_{\omega_{1}}
  18. 2 โ„ต ฯ‰ 1 < โ„ต ฯ‰ 2 2^{\aleph_{\omega_{1}}}<\aleph_{\omega_{2}}

Peak_expiratory_flow.html

  1. EU < m t p l โ‰ฅ 50.356 + ( 0.4 ร— Wright ) + ( 0.0008814 ร— Wright 2 ) - ( 0.0000001116 ร— Wright 3 ) \,\text{EU}<mtpl>{{=}}50.356+(0.4\times\,\text{Wright})+(0.0008814\times\,% \text{Wright}^{2})-(0.0000001116\times\,\text{Wright}^{3})
  2. Wright < m t p l โ‰ฅ - 61.1 + ( 1.798 ร— EU ) - ( 0.001594 ร— EU 2 ) + ( 0.0000007713 ร— EU 3 ) \,\text{Wright}<mtpl>{{=}}-61.1+(1.798\times\,\text{EU})-(0.001594\times\,% \text{EU}^{2})+(0.0000007713\times\,\text{EU}^{3})

Peak_signal-to-noise_ratio.html

  1. ๐‘€๐‘†๐ธ = 1 m n โˆ‘ i = 0 m - 1 โˆ‘ j = 0 n - 1 [ I ( i , j ) - K ( i , j ) ] 2 \mathit{MSE}=\frac{1}{m\,n}\sum_{i=0}^{m-1}\sum_{j=0}^{n-1}[I(i,j)-K(i,j)]^{2}
  2. ๐‘ƒ๐‘†๐‘๐‘… \displaystyle\mathit{PSNR}

Pearson_distribution.html

  1. ฮฒ 1 = ฮณ 1 2 \beta_{1}=\gamma_{1}^{2}
  2. p โ€ฒ ( x ) p ( x ) + a + x - ฮป b 2 ( x - ฮป ) 2 + b 1 ( x - ฮป ) + b 0 = 0. ( 1 ) \frac{p^{\prime}(x)}{p(x)}+\frac{a+x-\lambda}{b_{2}(x-\lambda)^{2}+b_{1}(x-% \lambda)+b_{0}}=0.\qquad(1)\!
  3. b 0 = 4 ฮฒ 2 - 3 ฮฒ 1 10 ฮฒ 2 - 12 ฮฒ 1 - 18 ฮผ 2 , b_{0}=\frac{4\beta_{2}-3\beta_{1}}{10\beta_{2}-12\beta_{1}-18}\mu_{2},
  4. a = b 1 = ฮผ 2 ฮฒ 1 ฮฒ 2 + 3 10 ฮฒ 2 - 12 ฮฒ 1 - 18 , a=b_{1}=\sqrt{\mu_{2}}\sqrt{\beta_{1}}\frac{\beta_{2}+3}{10\beta_{2}-12\beta_{% 1}-18},
  5. b 2 = 2 ฮฒ 2 - 3 ฮฒ 1 - 6 10 ฮฒ 2 - 12 ฮฒ 1 - 18 . b_{2}=\frac{2\beta_{2}-3\beta_{1}-6}{10\beta_{2}-12\beta_{1}-18}.
  6. p โ€ฒ ( ฮป - a ) = 0 p^{\prime}(\lambda-a)=0\!
  7. p ( x ) โˆ exp ( - โˆซ x - a b 2 x 2 + b 1 x + b 0 d x ) . p(x)\propto\exp\left(-\!\int\!\!\frac{x-a}{b_{2}x^{2}+b_{1}x+b_{0}}\,\mathrm{d% }x\right).
  8. f ( x ) = b 2 x 2 + b 1 x + b 0 . ( 2 ) f(x)=b_{2}\,x^{2}+b_{1}\,x+b_{0}.\qquad(2)\!
  9. b 1 2 - 4 b 2 b 0 < 0 b_{1}^{2}-4b_{2}b_{0}<0
  10. y = x + b 1 2 b 2 y=x+\frac{b_{1}}{2\,b_{2}}\!
  11. ฮฑ = 4 b 2 b 0 - b 1 2 2 b 2 . \alpha=\frac{\sqrt{4\,b_{2}\,b_{0}-b_{1}^{2}\,}}{2\,b_{2}}.\!
  12. 4 b 2 b 0 - b 1 2 > 0 4b_{2}b_{0}-b_{1}^{2}>0
  13. f ( x ) = b 2 ( y 2 + ฮฑ 2 ) . f(x)=b_{2}\,(y^{2}+\alpha^{2}).\!
  14. p ( y ) โˆ exp ( - 1 b 2 โˆซ y - b 1 2 b 2 - a y 2 + ฮฑ 2 d y ) . p(y)\propto\exp\left(-\frac{1}{b_{2}}\,\int\frac{y-\frac{b_{1}}{2\,b_{2}}-a}{y% ^{2}+\alpha^{2}}\,\mathrm{d}y\right).\!
  15. โˆซ y - 2 b 2 a + b 1 2 b 2 y 2 + ฮฑ 2 d y = 1 2 ln ( y 2 + ฮฑ 2 ) - 2 b 2 a + b 1 2 b 2 ฮฑ arctan ( y ฮฑ ) + C 0 \int\frac{y-\frac{2\,b_{2}\,a+b_{1}}{2\,b_{2}}}{y^{2}+\alpha^{2}}\,\mathrm{d}y% =\frac{1}{2}\ln(y^{2}+\alpha^{2})-\frac{2\,b_{2}\,a+b_{1}}{2\,b_{2}\,\alpha}% \arctan\left(\frac{y}{\alpha}\right)+C_{0}
  16. p ( y ) โˆ exp [ - 1 2 b 2 ln ( 1 + y 2 ฮฑ 2 ) - ln ฮฑ b 2 + 2 b 2 a + b 1 2 b 2 2 ฮฑ arctan ( y ฮฑ ) + C 1 ] p(y)\propto\exp\left[-\frac{1}{2\,b_{2}}\ln\!\left(1+\frac{y^{2}}{\alpha^{2}}% \right)-\frac{\ln\alpha}{b_{2}}+\frac{2\,b_{2}\,a+b_{1}}{2\,b_{2}^{2}\,\alpha}% \arctan\left(\frac{y}{\alpha}\right)+C_{1}\right]
  17. m = 1 2 b 2 m=\frac{1}{2\,b_{2}}\!
  18. ฮฝ = - 2 b 2 a + b 1 2 b 2 2 ฮฑ \nu=-\frac{2\,b_{2}\,a+b_{1}}{2\,b_{2}^{2}\,\alpha}\!
  19. p ( y ) โˆ [ 1 + y 2 ฮฑ 2 ] - m exp [ - ฮฝ arctan ( y ฮฑ ) ] p(y)\propto\left[1+\frac{y^{2}}{\alpha^{2}}\right]^{-m}\exp\left[-\nu\arctan% \left(\frac{y}{\alpha}\right)\right]
  20. p ( x ) = | ฮ“ ( m + ฮฝ 2 i ) ฮ“ ( m ) | 2 ฮฑ B ( m - 1 2 , 1 2 ) [ 1 + ( x - ฮป ฮฑ ) 2 ] - m exp [ - ฮฝ arctan ( x - ฮป ฮฑ ) ] . p(x)=\frac{\left|\frac{\Gamma\!\left(m+\frac{\nu}{2}i\right)}{\Gamma(m)}\right% |^{2}}{\alpha\,\mathrm{B}\!\left(m-\frac{1}{2},\frac{1}{2}\right)}\left[1+% \left(\frac{x-\lambda}{\alpha}\right)^{\!2\,}\right]^{-m}\exp\left[-\nu\arctan% \left(\frac{x-\lambda}{\alpha}\right)\right].
  21. p ( x ) = 1 ฮฑ B ( m - 1 2 , 1 2 ) [ 1 + ( x - ฮป ฮฑ ) 2 ] - m , p(x)=\frac{1}{\alpha\,\mathrm{B}\!\left(m-\frac{1}{2},\frac{1}{2}\right)}\left% [1+\left(\frac{x-\lambda}{\alpha}\right)^{\!2\,}\right]^{-m},
  22. ฮฑ = ฯƒ 2 m - 3 , \alpha=\sigma\,\sqrt{2\,m-3},\!
  23. lim m โ†’ โˆž 1 ฯƒ 2 m - 3 B ( m - 1 2 , 1 2 ) [ 1 + ( x - ฮป ฯƒ 2 m - 3 ) 2 ] - m \lim_{m\to\infty}\frac{1}{\sigma\,\sqrt{2\,m-3}\,\mathrm{B}\!\left(m-\frac{1}{% 2},\frac{1}{2}\right)}\left[1+\left(\frac{x-\lambda}{\sigma\,\sqrt{2\,m-3}}% \right)^{\!2\,}\right]^{-m}
  24. = 1 ฯƒ 2 ฮ“ ( 1 2 ) ร— lim m โ†’ โˆž ฮ“ ( m ) ฮ“ ( m - 1 2 ) m - 3 2 ร— lim m โ†’ โˆž [ 1 + ( x - ฮป ฯƒ ) 2 2 m - 3 ] - m =\frac{1}{\sigma\,\sqrt{2}\,\Gamma\!\left(\frac{1}{2}\right)}\times\lim_{m\to% \infty}\frac{\Gamma(m)}{\Gamma\!\left(m-\frac{1}{2}\right)\sqrt{m-\frac{3}{2}}% }\times\lim_{m\to\infty}\left[1+\frac{\left(\frac{x-\lambda}{\sigma}\right)^{2% }}{2\,m-3}\right]^{-m}
  25. = 1 ฯƒ 2 ฯ€ ร— 1 ร— exp [ - 1 2 ( x - ฮป ฯƒ ) 2 ] =\frac{1}{\sigma\sqrt{2\,\pi}}\times 1\times\exp\!\left[-\frac{1}{2}\left(% \frac{x-\lambda}{\sigma}\right)^{\!2\,}\right]
  26. m = 5 2 + 3 ฮณ 2 . m=\frac{5}{2}+\frac{3}{\gamma_{2}}.\!
  27. ฮป = ฮผ , \lambda=\mu,\!
  28. ฮฑ = ฮฝ ฯƒ 2 , \alpha=\sqrt{\nu\sigma^{2}},\!
  29. m = ฮฝ + 1 2 , m=\frac{\nu+1}{2},\!
  30. p ( x | ฮผ , ฯƒ 2 , ฮฝ ) = 1 ฮฝ ฯƒ 2 B ( ฮฝ 2 , 1 2 ) ( 1 + 1 ฮฝ ( x - ฮผ ) 2 ฯƒ 2 ) - ฮฝ + 1 2 , p(x|\mu,\sigma^{2},\nu)=\frac{1}{\sqrt{\nu\sigma^{2}}\,\mathrm{B}\!\left(\frac% {\nu}{2},\frac{1}{2}\right)}\left(1+\frac{1}{\nu}\frac{(x-\mu)^{2}}{\sigma^{2}% }\right)^{-\frac{\nu+1}{2}},
  31. ฮป = 0 , \lambda=0,\!
  32. ฮฑ = ฮฝ , \alpha=\sqrt{\nu},\!
  33. m = ฮฝ + 1 2 , m=\frac{\nu+1}{2},\!
  34. p ( x ) = 1 ฮฝ B ( ฮฝ 2 , 1 2 ) ( 1 + x 2 ฮฝ ) - ฮฝ + 1 2 , p(x)=\frac{1}{\sqrt{\nu}\,\mathrm{B}\!\left(\frac{\nu}{2},\frac{1}{2}\right)}% \left(1+\frac{x^{2}}{\nu}\right)^{-\frac{\nu+1}{2}},
  35. b 1 2 - 4 b 2 b 0 โ‰ฅ 0 b_{1}^{2}-4b_{2}b_{0}\geq 0
  36. a 1 = - b 1 - b 1 2 - 4 b 2 b 0 2 b 2 , a_{1}=\frac{-b_{1}-\sqrt{b_{1}^{2}-4b_{2}b_{0}}}{2b_{2}},\!
  37. a 2 = - b 1 + b 1 2 - 4 b 2 b 0 2 b 2 , a_{2}=\frac{-b_{1}+\sqrt{b_{1}^{2}-4b_{2}b_{0}}}{2b_{2}},\!
  38. f ( x ) = b 2 ( x - a 1 ) ( x - a 2 ) , f(x)=b_{2}\,(x-a_{1})(x-a_{2}),\!
  39. p ( x ) โˆ exp ( - 1 b 2 โˆซ x - a ( x - a 1 ) ( x - a 2 ) d x ) . p(x)\propto\exp\left(-\frac{1}{b_{2}}\int\!\!\frac{x-a}{(x-a_{1})(x-a_{2})}\,% \mathrm{d}x\right).\!
  40. โˆซ x - a ( x - a 1 ) ( x - a 2 ) d x = ( a 1 - a ) ln ( x - a 1 ) - ( a 2 - a ) ln ( x - a 2 ) a 1 - a 2 + C \int\!\!\frac{x-a}{(x-a_{1})(x-a_{2})}\,\mathrm{d}x=\frac{(a_{1}-a)\ln(x-a_{1}% )-(a_{2}-a)\ln(x-a_{2})}{a_{1}-a_{2}}+C
  41. ฮฝ = 1 b 2 ( a 1 - a 2 ) \nu=\frac{1}{b_{2}\,(a_{1}-a_{2})}\!
  42. p ( x ) โˆ ( x - a 1 ) - ฮฝ ( a 1 - a ) ( x - a 2 ) ฮฝ ( a 2 - a ) . p(x)\propto(x-a_{1})^{-\nu(a_{1}-a)}(x-a_{2})^{\nu(a_{2}-a)}.
  43. p ( x ) โˆ ( 1 - x a 1 ) - ฮฝ ( a 1 - a ) ( 1 - x a 2 ) ฮฝ ( a 2 - a ) p(x)\propto\left(1-\frac{x}{a_{1}}\right)^{-\nu(a_{1}-a)}\left(1-\frac{x}{a_{2% }}\right)^{\nu(a_{2}-a)}
  44. a 1 < 0 < a 2 a_{1}<0<a_{2}
  45. ( a 1 , a 2 ) (a_{1},a_{2})
  46. x = a 1 + y ( a 2 - a 1 ) where 0 < y < 1 , x=a_{1}+y(a_{2}-a_{1})\qquad\mbox{where}~{}\ 0<y<1,\!
  47. p ( y ) โˆ ( a 1 - a 2 a 1 y ) ( - a 1 + a ) ฮฝ ( a 2 - a 1 a 2 ( 1 - y ) ) ( a 2 - a ) ฮฝ . p(y)\propto\left(\frac{a_{1}-a_{2}}{a_{1}}\;y\right)^{(-a_{1}+a)\nu}\left(% \frac{a_{2}-a_{1}}{a_{2}}\;(1-y)\right)^{(a_{2}-a)\nu}.
  48. m 1 = a - a 1 b 2 ( a 1 - a 2 ) m_{1}=\frac{a-a_{1}}{b_{2}(a_{1}-a_{2})}\!
  49. m 2 = a - a 2 b 2 ( a 2 - a 1 ) m_{2}=\frac{a-a_{2}}{b_{2}(a_{2}-a_{1})}\!
  50. p ( y ) โˆ y m 1 ( 1 - y ) m 2 , p(y)\propto y^{m_{1}}(1-y)^{m_{2}},\!
  51. x - ฮป - a 1 a 2 - a 1 \frac{x-\lambda-a_{1}}{a_{2}-a_{1}}
  52. B ( m 1 + 1 , m 2 + 1 ) B(m_{1}+1,m_{2}+1)
  53. ฮป = ฮผ 1 - ( a 2 - a 1 ) m 1 + 1 m 1 + m 2 + 2 - a 1 \lambda=\mu_{1}-(a_{2}-a_{1})\frac{m_{1}+1}{m_{1}+m_{2}+2}-a_{1}
  54. y = y 0 ( 1 - x 2 a 2 ) m y=y_{0}\left(1-\frac{x^{2}}{a^{2}}\right)^{m}
  55. x = โˆ‘ d 2 / 2 - ( n 3 - n ) / 12 x=\sum d^{2}/2-(n^{3}-n)/12
  56. โˆ‘ d 2 \sum d^{2}
  57. m = 5 ฮฒ 2 - 9 2 ( 3 - ฮฒ 2 ) m=\frac{5\beta_{2}-9}{2(3-\beta_{2})}
  58. a 2 = 2 ฮผ 2 ฮฒ 2 3 - ฮฒ 2 a^{2}=\frac{2\mu_{2}\beta_{2}}{3-\beta_{2}}
  59. y 0 = N [ ฮ“ ( 2 m + 2 ) ] a [ 2 2 m + 1 ] [ ฮ“ ( m + 1 ) ] y_{0}=\frac{N[\Gamma(2m+2)]}{a[2^{2m+1}][\Gamma(m+1)]}
  60. ฮผ 2 = ( n - 1 ) [ ( n 2 + n ) / 12 ] 2 \mu_{2}=(n-1)[(n^{2}+n)/12]^{2}
  61. ฮฒ 2 = 3 ( 25 n 4 - 13 n 3 - 73 n 2 + 37 n + 72 ) 25 n ( n + 1 ) 2 ( n - 1 ) \beta_{2}=\frac{3(25n^{4}-13n^{3}-73n^{2}+37n+72)}{25n(n+1)^{2}(n-1)}
  62. ฮป = ฮผ 1 + b 0 b 1 - ( m + 1 ) b 1 \lambda=\mu_{1}+\frac{b_{0}}{b_{1}}-(m+1)b_{1}\!
  63. b 0 + b 1 ( x - ฮป ) b_{0}+b_{1}(x-\lambda)\!
  64. Gamma ( m + 1 , b 1 2 ) \mathrm{Gamma}(m+1,b_{1}^{2})\!
  65. C 1 = b 1 2 b 2 C_{1}=\frac{b_{1}}{2b_{2}}\!
  66. ฮป = ฮผ 1 - a - C 1 1 - 2 b 2 \lambda=\mu_{1}-\frac{a-C_{1}}{1-2b_{2}}\!
  67. x - ฮป x-\lambda\!
  68. InverseGamma ( 1 b 2 - 1 , a - C 1 b 2 ) \operatorname{InverseGamma}(\frac{1}{b_{2}}-1,\frac{a-C_{1}}{b_{2}})\!
  69. ฮป = ฮผ 1 + ( a 2 - a 1 ) m 2 + 1 m 2 + m 1 + 2 - a 2 \lambda=\mu_{1}+(a_{2}-a_{1})\frac{m_{2}+1}{m_{2}+m_{1}+2}-a_{2}\!
  70. x - ฮป - a 2 a 2 - a 1 \frac{x-\lambda-a_{2}}{a_{2}-a_{1}}\!
  71. ฮฒ โ€ฒ ( m 2 + 1 , - m 2 - m 1 - 1 ) \beta^{\prime}(m_{2}+1,-m_{2}-m_{1}-1)\!

Pedosphere.html

  1. H 2 O + CO 2 โŸถ H + + HCO 3 - โŸถ H 2 CO 3 \mathrm{H_{2}O+CO_{2}\longrightarrow H^{+}+HCO_{3}^{-}\longrightarrow H_{2}CO_% {3}}
  2. 2 NaAlSi 3 O 8 + 2 H 2 CO 3 + 9 H 2 O โŸถ 2 Na + + 2 HCO 3 - + 4 H 4 SiO 4 + Al 2 Si 2 O 5 ( OH ) 4 \mathrm{2\ NaAlSi_{3}O_{8}+2\ H_{2}CO_{3}+9\ H_{2}O\longrightarrow 2\ Na^{+}+2% \ HCO_{3}^{-}+4\ H_{4}SiO_{4}+Al_{2}Si_{2}O_{5}(OH)_{4}}
  3. CaCO 3 + H 2 CO 3 โŸถ Ca 2 + + 2 HCO 3 - \mathrm{CaCO_{3}+H_{2}CO_{3}\longrightarrow Ca^{2+}+2\ HCO_{3}^{-}}
  4. CaCO 3 โŸถ Ca 2 + + CO 3 2 - \mathrm{CaCO_{3}\longrightarrow Ca^{2+}+CO_{3}^{2-}}
  5. O 2 + 4 e - + 4 H + โŸถ H 2 O \mathrm{O_{2}+4\ e^{-}+4\ H^{+}\longrightarrow H_{2}O}
  6. 2 H + + SO 4 2 - + 2 ( CH 2 O ) โŸถ 2 CO 2 + H 2 S + 2 H 2 O \mathrm{2\ H^{+}+SO_{4}^{2-}+2(CH_{2}O)\longrightarrow 2\ CO_{2}+H_{2}S+2\ H_{% 2}O}

PEG_ratio.html

  1. PEG Ratio = Price/Earnings Annual EPS Growth \mbox{PEG Ratio}~{}\,=\,\frac{\mbox{Price/Earnings}~{}}{\mbox{Annual EPS % Growth}~{}}

Pell_number.html

  1. P n = { 0 if n = 0 ; 1 if n = 1 ; 2 P n - 1 + P n - 2 otherwise. P_{n}=\begin{cases}0&\mbox{if }~{}n=0;\\ 1&\mbox{if }~{}n=1;\\ 2P_{n-1}+P_{n-2}&\mbox{otherwise.}\end{cases}
  2. P n = ( 1 + 2 ) n - ( 1 - 2 ) n 2 2 . P_{n}=\frac{(1+\sqrt{2})^{n}-(1-\sqrt{2})^{n}}{2\sqrt{2}}.
  3. ( 1 + 2 ) n \scriptstyle(1+\sqrt{2})^{n}
  4. ( 1 + 2 ) \scriptstyle(1+\sqrt{2})
  5. ( P n + 1 P n P n P n - 1 ) = ( 2 1 1 0 ) n . \begin{pmatrix}P_{n+1}&P_{n}\\ P_{n}&P_{n-1}\end{pmatrix}=\begin{pmatrix}2&1\\ 1&0\end{pmatrix}^{n}.
  6. P n + 1 P n - 1 - P n 2 = ( - 1 ) n , P_{n+1}P_{n-1}-P_{n}^{2}=(-1)^{n},
  7. x 2 - 2 y 2 = ยฑ 1 , \displaystyle x^{2}-2y^{2}=\pm 1,
  8. x y \tfrac{x}{y}
  9. 2 \scriptstyle\sqrt{2}
  10. 1 , 3 2 , 7 5 , 17 12 , 41 29 , 99 70 , โ€ฆ 1,\frac{3}{2},\frac{7}{5},\frac{17}{12},\frac{41}{29},\frac{99}{70},\dots
  11. P n - 1 + P n P n \tfrac{P_{n-1}+P_{n}}{P_{n}}
  12. 2 โ‰ˆ 577 408 \sqrt{2}\approx\frac{577}{408}
  13. 2 \scriptstyle\sqrt{2}
  14. 2 = 1 + 1 2 + 1 2 + 1 2 + 1 2 + 1 2 + โ‹ฑ . \sqrt{2}=1+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\ddots% \,}}}}}.
  15. 577 408 = 1 + 1 2 + 1 2 + 1 2 + 1 2 + 1 2 + 1 2 + 1 2 . \frac{577}{408}=1+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+% \cfrac{1}{2+\cfrac{1}{2}}}}}}}.
  16. 2 \scriptstyle\sqrt{2}
  17. ( ยฑ P i , ยฑ P i + 1 ) (\pm P_{i},\pm P_{i+1})
  18. ( ยฑ P i + 1 , ยฑ P i ) (\pm P_{i+1},\pm P_{i})
  19. ( ยฑ ( P i + P i - 1 ) , 0 ) (\pm(P_{i}+P_{i-1}),0)
  20. ( 0 , ยฑ ( P i + P i - 1 ) ) (0,\pm(P_{i}+P_{i-1}))
  21. ( ยฑ P i , ยฑ P i ) (\pm P_{i},\pm P_{i})
  22. P n P_{n}
  23. P a P_{a}
  24. P b P_{b}
  25. ( ( P k - 1 + P k ) โ‹… P k ) 2 = ( P k - 1 + P k ) 2 โ‹… ( ( P k - 1 + P k ) 2 - ( - 1 ) k ) 2 . \bigl((P_{k-1}+P_{k})\cdot P_{k}\bigr)^{2}=\frac{(P_{k-1}+P_{k})^{2}\cdot\left% ((P_{k-1}+P_{k})^{2}-(-1)^{k}\right)}{2}.
  26. P 4 n + 1 P_{4n+1}
  27. โˆ‘ i = 0 4 n + 1 P i = ( โˆ‘ r = 0 n 2 r ( 2 n + 1 2 r ) ) 2 = ( P 2 n + P 2 n + 1 ) 2 . \sum_{i=0}^{4n+1}P_{i}=\left(\sum_{r=0}^{n}2^{r}{2n+1\choose 2r}\right)^{2}=(P% _{2n}+P_{2n+1})^{2}.
  28. P 5 P_{5}
  29. 0 + 1 + 2 + 5 + 12 + 29 = 49 0+1+2+5+12+29=49
  30. P 2 + P 3 = 2 + 5 = 7 P_{2}+P_{3}=2+5=7
  31. P 2 n + P 2 n + 1 P_{2n}+P_{2n+1}
  32. ( 2 P n P n + 1 , P n + 1 2 - P n 2 , P n + 1 2 + P n 2 = P 2 n + 1 ) . (2P_{n}P_{n+1},P_{n+1}^{2}-P_{n}^{2},P_{n+1}^{2}+P_{n}^{2}=P_{2n+1}).
  33. Q n = { 2 if n = 0 ; 2 if n = 1 ; 2 Q n - 1 + Q n - 2 otherwise. Q_{n}=\begin{cases}2&\mbox{if }~{}n=0;\\ 2&\mbox{if }~{}n=1;\\ 2Q_{n-1}+Q_{n-2}&\mbox{otherwise.}\end{cases}
  34. Q n = P 2 n P n Q_{n}=\frac{P_{2n}}{P_{n}}
  35. Q n = ( 1 + 2 ) n + ( 1 - 2 ) n . Q_{n}=(1+\sqrt{2})^{n}+(1-\sqrt{2})^{n}.
  36. 2 \scriptstyle\sqrt{2}
  37. Q n 2 \frac{Q_{n}}{2}
  38. ฮด = ฮด S = 1 + 2 \delta=\delta_{S}=1+\sqrt{2}
  39. ฮด ยฏ = 1 - 2 \bar{\delta}=1-\sqrt{2}
  40. n n
  41. ( 1 + 2 ) n (1+\sqrt{2})^{n}
  42. ( 1 - 2 ) n (1-\sqrt{2})^{n}
  43. 1 + 0 2 = 1.0 1+0\sqrt{2}=1.0
  44. 1 - 0 2 = 1.0 1-0\sqrt{2}=1.0
  45. 1 + 1 2 = 2.41421 โ€ฆ 1+1\sqrt{2}=2.41421\ldots
  46. 1 - 1 2 = - 0.41421 โ€ฆ 1-1\sqrt{2}=-0.41421\ldots
  47. 3 + 2 2 = 5.82842 โ€ฆ 3+2\sqrt{2}=5.82842\ldots
  48. 3 - 2 2 = 0.17157 โ€ฆ 3-2\sqrt{2}=0.17157\ldots
  49. 7 + 5 2 = 14.07106 โ€ฆ 7+5\sqrt{2}=14.07106\ldots
  50. 7 - 5 2 = - 0.07106 โ€ฆ 7-5\sqrt{2}=-0.07106\ldots
  51. 17 + 12 2 = 33.97056 โ€ฆ 17+12\sqrt{2}=33.97056\ldots
  52. 17 - 12 2 = 0.02943 โ€ฆ 17-12\sqrt{2}=0.02943\ldots
  53. 41 + 29 2 = 82.01219 โ€ฆ 41+29\sqrt{2}=82.01219\ldots
  54. 41 - 29 2 = - 0.01219 โ€ฆ 41-29\sqrt{2}=-0.01219\ldots
  55. 99 + 70 2 = 197.9949 โ€ฆ 99+70\sqrt{2}=197.9949\ldots
  56. 99 - 70 2 = 0.0050 โ€ฆ 99-70\sqrt{2}=0.0050\ldots
  57. 239 + 169 2 = 478.00209 โ€ฆ 239+169\sqrt{2}=478.00209\ldots
  58. 239 - 169 2 = - 0.00209 โ€ฆ 239-169\sqrt{2}=-0.00209\ldots
  59. 577 + 408 2 = 1153.99913 โ€ฆ 577+408\sqrt{2}=1153.99913\ldots
  60. 577 - 408 2 = 0.00086 โ€ฆ 577-408\sqrt{2}=0.00086\ldots
  61. 1393 + 985 2 = 2786.00035 โ€ฆ 1393+985\sqrt{2}=2786.00035\ldots
  62. 1393 - 985 2 = - 0.00035 โ€ฆ 1393-985\sqrt{2}=-0.00035\ldots
  63. 3363 + 2378 2 = 6725.99985 โ€ฆ 3363+2378\sqrt{2}=6725.99985\ldots
  64. 3363 - 2378 2 = 0.00014 โ€ฆ 3363-2378\sqrt{2}=0.00014\ldots
  65. 8119 + 5741 2 = 16238.00006 โ€ฆ 8119+5741\sqrt{2}=16238.00006\ldots
  66. 8119 - 5741 2 = - 0.00006 โ€ฆ 8119-5741\sqrt{2}=-0.00006\ldots
  67. 19601 + 13860 2 = 39201.99997 โ€ฆ 19601+13860\sqrt{2}=39201.99997\ldots
  68. 19601 - 13860 2 = 0.00002 โ€ฆ 19601-13860\sqrt{2}=0.00002\ldots
  69. H n H_{n}
  70. P n P_{n}
  71. H 2 - 2 P 2 = ยฑ 1 H^{2}-2P^{2}=\pm 1
  72. N = t ( t + 1 ) 2 = s 2 N=\frac{t(t+1)}{2}=s^{2}
  73. t t\,
  74. s s\,
  75. a 2 + b 2 = c 2 a^{2}+b^{2}=c^{2}
  76. a + 1 = b a+1=b
  77. H n H_{n}
  78. n n
  79. H n H_{n}
  80. P n P_{n}
  81. H n H_{n}
  82. P n P_{n}
  83. ( 1 + 2 ) n = H n + P n 2 (1+\sqrt{2})^{n}=H_{n}+P_{n}\sqrt{2}
  84. ( 1 - 2 ) n = H n - P n 2 . (1-\sqrt{2})^{n}=H_{n}-P_{n}\sqrt{2}.
  85. H n = ( 1 + 2 ) n + ( 1 - 2 ) n 2 . H_{n}=\frac{(1+\sqrt{2})^{n}+(1-\sqrt{2})^{n}}{2}.
  86. P n 2 = ( 1 + 2 ) n - ( 1 - 2 ) n 2 . P_{n}\sqrt{2}=\frac{(1+\sqrt{2})^{n}-(1-\sqrt{2})^{n}}{2}.
  87. H n = { 1 if n = 0 ; H n - 1 + 2 P n - 1 otherwise. H_{n}=\begin{cases}1&\mbox{if }~{}n=0;\\ H_{n-1}+2P_{n-1}&\mbox{otherwise.}\end{cases}
  88. P n = { 0 if n = 0 ; H n - 1 + P n - 1 otherwise. P_{n}=\begin{cases}0&\mbox{if }~{}n=0;\\ H_{n-1}+P_{n-1}&\mbox{otherwise.}\end{cases}
  89. ( H n P n ) = ( 1 2 1 1 ) ( H n - 1 P n - 1 ) = ( 1 2 1 1 ) n ( 1 0 ) . \begin{pmatrix}H_{n}\\ P_{n}\end{pmatrix}=\begin{pmatrix}1&2\\ 1&1\end{pmatrix}\begin{pmatrix}H_{n-1}\\ P_{n-1}\end{pmatrix}=\begin{pmatrix}1&2\\ 1&1\end{pmatrix}^{n}\begin{pmatrix}1\\ 0\end{pmatrix}.
  90. ( H n 2 P n P n H n ) = ( 1 2 1 1 ) n . \begin{pmatrix}H_{n}&2P_{n}\\ P_{n}&H_{n}\end{pmatrix}=\begin{pmatrix}1&2\\ 1&1\end{pmatrix}^{n}.
  91. H n H_{n}\,
  92. P n 2 P_{n}\sqrt{2}
  93. ( 1 - 2 ) n โ‰ˆ ( - 0.41421 ) n (1-\sqrt{2})^{n}\approx(-0.41421)^{n}
  94. ( 1 + 2 ) n = H n + P n 2 (1+\sqrt{2})^{n}=H_{n}+P_{n}\sqrt{2}
  95. 2 H n 2H_{n}\,
  96. H n P n \frac{H_{n}}{P_{n}}
  97. 2 \sqrt{2}\,
  98. H n H n - 1 \frac{H_{n}}{H_{n-1}}\,
  99. P n P n - 1 \frac{P_{n}}{P_{n-1}}\,
  100. 1 + 2 1+\sqrt{2}\,
  101. 2 \sqrt{2}
  102. H P = 2 \frac{H}{P}=\sqrt{2}\,
  103. H 2 P 2 = 2 P 2 P 2 \frac{H^{2}}{P^{2}}=\frac{2P^{2}}{P^{2}}\,
  104. H 2 P 2 = 2 P 2 - 1 P 2 \frac{H^{2}}{P^{2}}=\frac{2P^{2}-1}{P^{2}}\,
  105. H 2 P 2 = 2 P 2 + 1 P 2 \frac{H^{2}}{P^{2}}=\frac{2P^{2}+1}{P^{2}}
  106. H 2 - 2 P 2 = 1 H^{2}-2P^{2}=1\,
  107. H n , P n with n H_{n},P_{n}\mbox{ with }~{}n\,
  108. H 2 - 2 P 2 = - 1 H^{2}-2P^{2}=-1\,
  109. H n , P n with n H_{n},P_{n}\mbox{ with }~{}n\,
  110. H n + 1 2 - 2 P n + 1 2 = ( H n + 2 P n ) 2 - 2 ( H n + P n ) 2 = - ( H n 2 - 2 P n 2 ) H_{n+1}^{2}-2P_{n+1}^{2}=(H_{n}+2P_{n})^{2}-2(H_{n}+P_{n})^{2}=-(H_{n}^{2}-2P_% {n}^{2})\,
  111. H 0 2 - 2 P 0 2 = 1 H_{0}^{2}-2P_{0}^{2}=1\,
  112. 1 and - 1 1\mbox{ and }~{}-1\,
  113. ( 2 P - H ) 2 - 2 ( H - P ) 2 = - ( H 2 - 2 P 2 ) (2P-H)^{2}-2(H-P)^{2}=-(H^{2}-2P^{2})\,
  114. H = P = 1 H=P=1\,
  115. H 0 = 1 and P 0 = 0 H_{0}=1\mbox{ and }~{}P_{0}=0\,
  116. t ( t + 1 ) 2 = s 2 \frac{t(t+1)}{2}=s^{2}\,
  117. 4 t 2 + 4 t + 1 = 8 s 2 + 1 4t^{2}+4t+1=8s^{2}+1\,
  118. H 2 = 2 P 2 + 1 H^{2}=2P^{2}+1
  119. H = 2 t + 1 and P = 2 s H=2t+1\mbox{ and }~{}P=2s
  120. t n = H 2 n - 1 2 t_{n}=\frac{H_{2n}-1}{2}
  121. s n = P 2 n 2 . s_{n}=\frac{P_{2n}}{2}.
  122. t t
  123. t + 1 t+1
  124. t ( t + 1 ) 2 = s 2 \frac{t(t+1)}{2}=s^{2}\,
  125. H 2 H^{2}
  126. 2 P 2 2P^{2}
  127. t n = { 2 P n 2 if n is even ; H n 2 if n is odd. t_{n}=\begin{cases}2P_{n}^{2}&\mbox{if }~{}n\mbox{ is even}~{};\\ H_{n}^{2}&\mbox{if }~{}n\mbox{ is odd.}\end{cases}
  128. s n = H n P n s_{n}=H_{n}P_{n}\,
  129. n n
  130. H n H_{n}
  131. P n P_{n}
  132. c 2 = a 2 + ( a + 1 ) 2 = 2 a 2 + 2 a + 1 c^{2}=a^{2}+(a+1)^{2}=2a^{2}+2a+1
  133. 2 c 2 = 4 a 2 + 4 a + 2 2c^{2}=4a^{2}+4a+2
  134. 2 P 2 = H 2 + 1 2P^{2}=H^{2}+1
  135. H = 2 a + 1 and P = c H=2a+1\mbox{ and }~{}P=c
  136. a n = H 2 n + 1 - 1 2 a_{n}=\frac{H_{2n+1}-1}{2}
  137. c n = P 2 n + 1 . c_{n}={P_{2n+1}}.\,
  138. a n and b n = a n + 1 a_{n}\mbox{ and }~{}b_{n}=a_{n}+1
  139. H n H n + 1 and 2 P n P n + 1 H_{n}H_{n+1}\mbox{ and }~{}2P_{n}P_{n+1}
  140. c n = H n + 1 P n + P n + 1 H n . c_{n}=H_{n+1}P_{n}+P_{n+1}H_{n}.

Pen_tilt.html

  1. { X , Y } \{X,Y\}
  2. { Z } \{Z\}

Pentadecagon.html

  1. A = 15 4 a 2 cot ฯ€ 15 = 15 a 2 8 ( 3 + 15 + 2 5 + 5 ) โ‰ƒ 17.6424 a 2 . \begin{aligned}\displaystyle A&\displaystyle=\frac{15}{4}a^{2}\cot\frac{\pi}{1% 5}\\ &\displaystyle=\frac{15a^{2}}{8}\left(\sqrt{3}+\sqrt{15}+\sqrt{2}\sqrt{5+\sqrt% {5}}\right)\\ &\displaystyle\simeq 17.6424\,a^{2}.\end{aligned}

PEPA.html

  1. P : := ( a , ฮป ) . P | P + Q | P L โ–ท โ— Q | P / L | A P::=(a,\lambda).P\,\,\,|\,\,\,P+Q\,\,\,|\,\,\,P\stackrel{\triangleright\!\!% \triangleleft}{\scriptstyle{L}}Q\,\,\,|\,\,\,P/L\,\,\,|\,\,\,A
  2. ( a , ฮป ) . P (a,\lambda).P
  3. ฮป \lambda
  4. ฯ„ \tau
  5. A = def P A\overset{\underset{\mathrm{def}}{}}{=}P

Permeance.html

  1. ๐’ซ \mathcal{P}
  2. ๐’ซ \mathcal{P}
  3. โ„› \mathcal{R}
  4. ๐’ซ = 1 โ„› \mathcal{P}=\frac{1}{\mathcal{R}}
  5. ๐’ซ = ฮฆ B N I \mathcal{P}=\frac{\Phi_{B}}{NI}
  6. โ„ฑ = ฮฆ B โ„› = N I \mathcal{F}=\Phi_{B}\mathcal{R}=NI
  7. ๐’ซ = ฮผ A โ„“ \mathcal{P}=\frac{\mu A}{\ell}
  8. โ„“ \ell

Permutation_automaton.html

  1. A A
  2. A A
  3. x x
  4. O ( n 2 / 5 ( log n ) 3 / 5 ) O(n^{2/5}(\log n)^{3/5})
  5. O ( n 1 / 2 ) O(n^{1/2})

Peroxy_acid.html

  1. โ† โ†’ \overrightarrow{\leftarrow}

Perronโ€“Frobenius_theorem.html

  1. A = ( a i j ) A=(a_{ij})
  2. n ร— n n\times n
  3. a i j > 0 a_{ij}>0
  4. 1 โ‰ค i , j โ‰ค n 1\leq i,j\leq n
  5. lim k โ†’ โˆž A k / r k = v w T \lim_{k\rightarrow\infty}A^{k}/r^{k}=vw^{T}
  6. min i โˆ‘ j a i j โ‰ค r โ‰ค max i โˆ‘ j a i j . \min_{i}\sum_{j}a_{ij}\leq r\leq\max_{i}\sum_{j}a_{ij}.
  7. ( 0 1 1 0 ) , ( 0 1 0 0 ) \begin{pmatrix}0&1\\ 1&0\end{pmatrix},\begin{pmatrix}0&1\\ 0&0\end{pmatrix}
  8. { exp ( t A ) , t โˆˆ โ„ } \left\{\exp(tA),t\in\mathbb{R}\right\}
  9. P A P - 1 = ( 0 A 1 0 0 โ€ฆ 0 0 0 A 2 0 โ€ฆ 0 โ‹ฎ โ‹ฎ โ‹ฎ โ‹ฎ โ‹ฎ 0 0 0 0 โ€ฆ A h - 1 A h 0 0 0 โ€ฆ 0 ) , PAP^{-1}=\begin{pmatrix}0&A_{1}&0&0&\ldots&0\\ 0&0&A_{2}&0&\ldots&0\\ \vdots&\vdots&\vdots&\vdots&&\vdots\\ 0&0&0&0&\ldots&A_{h-1}\\ A_{h}&0&0&0&\ldots&0\end{pmatrix},
  10. min i โˆ‘ j a i j โ‰ค r โ‰ค max i โˆ‘ j a i j . \min_{i}\sum_{j}a_{ij}\leq r\leq\max_{i}\sum_{j}a_{ij}.
  11. ( 0 0 1 0 0 1 1 1 0 ) \begin{pmatrix}0&0&1\\ 0&0&1\\ 1&1&0\\ \end{pmatrix}
  12. P A P - 1 = ( A 1 0 0 โ€ฆ 0 0 A 2 0 โ€ฆ 0 โ‹ฎ โ‹ฎ โ‹ฎ โ‹ฎ 0 0 0 โ€ฆ A d ) PAP^{-1}=\begin{pmatrix}A_{1}&0&0&\dots&0\\ 0&A_{2}&0&\dots&0\\ \vdots&\vdots&\vdots&&\vdots\\ 0&0&0&\dots&A_{d}\\ \end{pmatrix}
  13. lim k โ†’ โˆž 1 / k โˆ‘ i = 0 , โ€ฆ , k A i / r i = ( v w t ) , \lim_{k\rightarrow\infty}1/k\sum_{i=0,...,k}A^{i}/r^{i}=(vw^{t}),
  14. M = ( 0 1 0 0 โ€ฆ 0 0 0 1 0 โ€ฆ 0 0 0 0 1 โ€ฆ 0 โ‹ฎ โ‹ฎ โ‹ฎ โ‹ฎ โ‹ฎ 0 0 0 0 โ€ฆ 1 1 1 0 0 โ€ฆ 0 ) M=\begin{pmatrix}0&1&0&0&...&0\\ 0&0&1&0&...&0\\ 0&0&0&1&...&0\\ \vdots&\vdots&\vdots&\vdots&&\vdots\\ 0&0&0&0&...&1\\ 1&1&0&0&...&0\\ \end{pmatrix}
  15. ( B 1 * * โ‹ฏ * 0 B 2 * โ‹ฏ * โ‹ฎ โ‹ฎ โ‹ฎ โ‹ฎ 0 0 0 โ‹ฏ * 0 0 0 โ‹ฏ B h ) \left(\begin{smallmatrix}B_{1}&*&*&\cdots&*\\ 0&B_{2}&*&\cdots&*\\ \vdots&\vdots&\vdots&&\vdots\\ 0&0&0&\cdots&*\\ 0&0&0&\cdots&B_{h}\end{smallmatrix}\right)
  16. โˆฅ v โˆฅ โˆž = โˆฅ A k v โˆฅ โˆž โ‰ฅ โˆฅ A k โˆฅ โˆž min i ( v i ) , โ‡’ โˆฅ A k โˆฅ โˆž โ‰ค โˆฅ v โˆฅ / min i ( v i ) \|v\|_{\infty}=\|A^{k}v\|_{\infty}\geq\|A^{k}\|_{\infty}\min_{i}(v_{i}),~{}~{}% \Rightarrow~{}~{}\|A^{k}\|_{\infty}\leq\|v\|/\min_{i}(v_{i})
  17. J k = ( ฮป 1 0 ฮป ) k = ( ฮป k k ฮป k - 1 0 ฮป k ) , J^{k}=\begin{pmatrix}\lambda&1\\ 0&\lambda\end{pmatrix}^{k}=\begin{pmatrix}\lambda^{k}&k\lambda^{k-1}\\ 0&\lambda^{k}\end{pmatrix},
  18. r โ‰ค max i โˆ‘ j A i j . r\;\leq\;\max_{i}\sum_{j}A_{ij}.
  19. ฮป \scriptstyle\lambda
  20. | ฮป | โ‰ค max i โˆ‘ j | A i j | . \scriptstyle|\lambda|\;\leq\;\max_{i}\sum_{j}|A_{ij}|.
  21. โˆฅ A โˆฅ โ‰ฅ | ฮป | \scriptstyle\|A\|\geq|\lambda|
  22. ฮป \scriptstyle\lambda
  23. x \scriptstyle x
  24. โˆฅ A โˆฅ โ‰ฅ | A x | / | x | = | ฮป x | / | x | = | ฮป | \scriptstyle\|A\|\geq|Ax|/|x|=|\lambda x|/|x|=|\lambda|
  25. โˆฅ A โˆฅ โˆž = max 1 โ‰ค i โ‰ค m โˆ‘ j = 1 n | A i j | . \scriptstyle\left\|A\right\|_{\infty}=\max\limits_{1\leq i\leq m}\sum_{j=1}^{n% }|A_{ij}|.
  26. โˆฅ A โˆฅ โˆž โ‰ฅ | ฮป | \scriptstyle\|A\|_{\infty}\geq|\lambda|
  27. min i โˆ‘ j A i j โ‰ค r . \min_{i}\sum_{j}A_{ij}\;\leq\;r.
  28. h - 1 โˆ‘ 1 h ฮป - k R k \scriptstyle h^{-1}\sum^{h}_{1}\lambda^{-k}R^{k}
  29. ( 1 0 0 1 0 0 1 1 1 ) \left(\begin{smallmatrix}1&0&0\\ 1&0&0\\ 1&1&1\end{smallmatrix}\right)
  30. ( 1 0 0 1 0 0 - 1 1 1 ) \left(\begin{smallmatrix}\;\;\;1&0&0\\ \;\;\;1&0&0\\ -1&1&1\end{smallmatrix}\right)
  31. ( 0 1 1 1 0 1 1 1 0 ) \left(\begin{smallmatrix}0&1&1\\ 1&0&1\\ 1&1&0\end{smallmatrix}\right)
  32. ( 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 ) \left(\begin{smallmatrix}0&1&0&0&0\\ 1&0&0&0&0\\ 0&0&0&1&0\\ 0&0&0&0&1\\ 0&0&1&0&0\end{smallmatrix}\right)

Personal_armor.html

  1. V 50 = ( U * ) 1 / 3 f ( A d A p ) . V_{50}=(U^{*})^{1/3}f\left(\frac{A_{d}}{A_{p}}\right).
  2. U * = ฯƒ ฯต 2 ฯ E ฯ U^{*}=\frac{\sigma\epsilon}{2\rho}\sqrt{\frac{E}{\rho}}
  3. ฯƒ , ฯต , ฯ , E \sigma,\epsilon,\rho,E
  4. A d A_{d}
  5. A p A_{p}

Perturbation_(astronomy).html

  1. n n
  2. i i
  3. j j
  4. ๐ซ ยจ i = โˆ‘ j = 1 j โ‰  i n G m j ( ๐ซ j - ๐ซ i ) r i j 3 \mathbf{\ddot{r}}_{i}=\sum_{\underset{j\neq i}{j=1}}^{n}{Gm_{j}(\mathbf{r}_{j}% -\mathbf{r}_{i})\over r_{ij}^{3}}
  5. ๐ซ ยจ i \mathbf{\ddot{r}}_{i}
  6. i i
  7. G G
  8. m j m_{j}
  9. j j
  10. ๐ซ i \mathbf{r}_{i}
  11. ๐ซ j \mathbf{r}_{j}
  12. i i
  13. j j
  14. r i j r_{ij}
  15. i i
  16. j j
  17. x x
  18. y y
  19. z z
  20. s y m b o l ฯ symbol{\rho}
  21. ๐ซ \mathbf{r}
  22. ฮด ๐ซ \delta\mathbf{r}
  23. ๐ซ ยจ \mathbf{\ddot{r}}
  24. s y m b o l ฯ ยจ symbol{\ddot{\rho}}
  25. ๐ซ \mathbf{r}
  26. s y m b o l ฯ symbol{\rho}
  27. ฮผ = G ( M + m ) \mu=G(M+m)
  28. M M
  29. m m
  30. ๐š per \mathbf{a}_{\,\text{per}}
  31. r r
  32. ฯ \rho
  33. ๐ซ \mathbf{r}
  34. s y m b o l ฯ symbol{\rho}
  35. ฮด ๐ซ \delta\mathbf{r}
  36. s y m b o l ฯ symbol{\rho}
  37. ฮด ๐ซ \delta\mathbf{r}
  38. ๐ซ \mathbf{r}
  39. s y m b o l ฯ ฯ 3 - ๐ซ r 3 {symbol{\rho}\over\rho^{3}}-{\mathbf{r}\over r^{3}}

Perverse_sheaf.html

  1. H - i ( j x * C ) โ‰  0 H^{-i}(j_{x}^{*}C)\neq 0
  2. H i ( j x ! C ) โ‰  0 H^{i}(j_{x}^{!}C)\neq 0

PGL2.html

  1. PGL 2 \mathrm{PGL}_{2}

Phantom_energy.html

  1. w < - 1 w<-1

Phase-change_material.html

  1. V H C = ฯ c p VHC=\rho c_{p}
  2. I = k ฯ c p = e = ( k ฯ c p ) 1 / 2 I=\sqrt{k\rho c_{p}}=e={(k\rho c_{p})}^{1/2}

Phase_correlation.html

  1. g a \ g_{a}
  2. g b \ g_{b}
  3. ๐† a = โ„ฑ { g a } , ๐† b = โ„ฑ { g b } \ \mathbf{G}_{a}=\mathcal{F}\{g_{a}\},\;\mathbf{G}_{b}=\mathcal{F}\{g_{b}\}
  4. R = ๐† a โˆ˜ ๐† b * | ๐† a โˆ˜ ๐† b * | \ R=\frac{\mathbf{G}_{a}\circ\mathbf{G}_{b}^{*}}{|\mathbf{G}_{a}\circ\mathbf{G% }_{b}^{*}|}
  5. โˆ˜ \circ
  6. r = โ„ฑ - 1 { R } \ r=\mathcal{F}^{-1}\{R\}
  7. r \ r
  8. ( ฮ” x , ฮ” y ) = arg max ( x , y ) { r } \ (\Delta x,\Delta y)=\arg\max_{(x,y)}\{r\}
  9. g a \ g_{a}
  10. g b \ g_{b}
  11. g b ( x , y ) = def g a ( ( x - ฮ” x ) mod M , ( y - ฮ” y ) mod N ) \ g_{b}(x,y)\ \stackrel{\mathrm{def}}{=}\ g_{a}((x-\Delta x)\bmod M,(y-\Delta y% )\bmod N)
  12. M ร— N \ M\times N
  13. ๐† b ( u , v ) = ๐† a ( u , v ) e - 2 ฯ€ i ( u ฮ” x M + v ฮ” y N ) \mathbf{G}_{b}(u,v)=\mathbf{G}_{a}(u,v)e^{-2\pi i(\frac{u\Delta x}{M}+\frac{v% \Delta y}{N})}
  14. R ( u , v ) \displaystyle R(u,v)
  15. ๐† a ๐† a * \ \mathbf{G}_{a}\mathbf{G}_{a}^{*}
  16. r ( x , y ) = ฮด ( x + ฮ” x , y + ฮ” y ) \ r(x,y)=\delta(x+\Delta x,y+\Delta y)
  17. g b \ g_{b}
  18. g a \ g_{a}
  19. r \ r

Phasor.html

  1. A โ‹… e i ( ฯ‰ t + ฮธ ) \scriptstyle A\cdot e^{i(\omega t+\theta)}
  2. A โ‹… cos ( ฯ‰ t + ฮธ ) = A โ‹… e i ( ฯ‰ t + ฮธ ) + e - i ( ฯ‰ t + ฮธ ) 2 , A\cdot\cos(\omega t+\theta)=A\cdot\frac{e^{i(\omega t+\theta)}+e^{-i(\omega t+% \theta)}}{2},
  3. A โ‹… cos ( ฯ‰ t + ฮธ ) = Re { A โ‹… e i ( ฯ‰ t + ฮธ ) } = Re { A e i ฮธ โ‹… e i ฯ‰ t } . \begin{aligned}\displaystyle A\cdot\cos(\omega t+\theta)=\operatorname{Re}\{A% \cdot e^{i(\omega t+\theta)}\}=\operatorname{Re}\{Ae^{i\theta}\cdot e^{i\omega t% }\}.\end{aligned}
  4. A โ‹… e i ( ฯ‰ t + ฮธ ) A\cdot e^{i(\omega t+\theta)}
  5. A โ‹… cos ( ฯ‰ t + ฮธ ) . A\cdot\cos(\omega t+\theta).
  6. , A e i ฮธ . ,Ae^{i\theta}.
  7. A โˆ  ฮธ . A\angle\theta.\,
  8. A e i ฮธ e i ฯ‰ t Ae^{i\theta}e^{i\omega t}\,
  9. B e i ฯ• Be^{i\phi}\,
  10. Re { ( A e i ฮธ โ‹… B e i ฯ• ) โ‹… e i ฯ‰ t } = Re { ( A B e i ( ฮธ + ฯ• ) ) โ‹… e i ฯ‰ t } = A B cos ( ฯ‰ t + ( ฮธ + ฯ• ) ) \begin{aligned}\displaystyle\operatorname{Re}\{(Ae^{i\theta}\cdot Be^{i\phi})% \cdot e^{i\omega t}\}&\displaystyle=\operatorname{Re}\{(ABe^{i(\theta+\phi)})% \cdot e^{i\omega t}\}\\ &\displaystyle=AB\cos(\omega t+(\theta+\phi))\end{aligned}
  11. B e i ฯ• Be^{i\phi}\,
  12. Re { d d t ( A e i ฮธ โ‹… e i ฯ‰ t ) } = Re { A e i ฮธ โ‹… i ฯ‰ e i ฯ‰ t } = Re { A e i ฮธ โ‹… e i ฯ€ / 2 ฯ‰ e i ฯ‰ t } = Re { ฯ‰ A e i ( ฮธ + ฯ€ / 2 ) โ‹… e i ฯ‰ t } = ฯ‰ A โ‹… cos ( ฯ‰ t + ฮธ + ฯ€ / 2 ) \begin{aligned}\displaystyle\operatorname{Re}\{\frac{d}{dt}(Ae^{i\theta}\cdot e% ^{i\omega t})\}=\operatorname{Re}\{Ae^{i\theta}\cdot i\omega e^{i\omega t}\}=% \operatorname{Re}\{Ae^{i\theta}\cdot e^{i\pi/2}\omega e^{i\omega t}\}=% \operatorname{Re}\{\omega Ae^{i(\theta+\pi/2)}\cdot e^{i\omega t}\}=\omega A% \cdot\cos(\omega t+\theta+\pi/2)\end{aligned}
  13. i ฯ‰ = ( e i ฯ€ / 2 โ‹… ฯ‰ ) . i\omega=(e^{i\pi/2}\cdot\omega).\,
  14. 1 i ฯ‰ = e - i ฯ€ / 2 ฯ‰ . \frac{1}{i\omega}=\frac{e^{-i\pi/2}}{\omega}.\,
  15. e i ฯ‰ t e^{i\omega t}\,
  16. e i ฯ‰ t e^{i\omega t}\,
  17. d v C ( t ) d t + 1 R C v C ( t ) = 1 R C v S ( t ) \frac{d\ v_{C}(t)}{dt}+\frac{1}{RC}v_{C}(t)=\frac{1}{RC}v_{S}(t)
  18. v S ( t ) = V P โ‹… cos ( ฯ‰ t + ฮธ ) , v_{S}(t)=V_{P}\cdot\cos(\omega t+\theta),\,
  19. v S ( t ) = Re { V s โ‹… e i ฯ‰ t } \begin{aligned}\displaystyle v_{S}(t)&\displaystyle=\operatorname{Re}\{V_{s}% \cdot e^{i\omega t}\}\\ \end{aligned}
  20. v C ( t ) = Re { V c โ‹… e i ฯ‰ t } , v_{C}(t)=\operatorname{Re}\{V_{c}\cdot e^{i\omega t}\},
  21. V s = V P e i ฮธ , V_{s}=V_{P}e^{i\theta},\,
  22. V c V_{c}\,
  23. t t\,
  24. t - ฯ€ 2 ฯ‰ , t-\frac{\pi}{2\omega},\,
  25. d Re { V c โ‹… e i ฯ‰ t } d t = Re { d ( V c โ‹… e i ฯ‰ t ) d t } = Re { i ฯ‰ V c โ‹… e i ฯ‰ t } \frac{d\ \operatorname{Re}\{V_{c}\cdot e^{i\omega t}\}}{dt}=\operatorname{Re}% \left\{\frac{d\left(V_{c}\cdot e^{i\omega t}\right)}{dt}\right\}=\operatorname% {Re}\left\{i\omega V_{c}\cdot e^{i\omega t}\right\}
  26. d Im { V c โ‹… e i ฯ‰ t } d t = Im { d ( V c โ‹… e i ฯ‰ t ) d t } = Im { i ฯ‰ V c โ‹… e i ฯ‰ t } \frac{d\ \operatorname{Im}\{V_{c}\cdot e^{i\omega t}\}}{dt}=\operatorname{Im}% \left\{\frac{d\left(V_{c}\cdot e^{i\omega t}\right)}{dt}\right\}=\operatorname% {Im}\left\{i\omega V_{c}\cdot e^{i\omega t}\right\}
  27. i , i,\,
  28. i ฯ‰ V c โ‹… e i ฯ‰ t + 1 R C V c โ‹… e i ฯ‰ t = 1 R C V s โ‹… e i ฯ‰ t i\omega V_{c}\cdot e^{i\omega t}+\frac{1}{RC}V_{c}\cdot e^{i\omega t}=\frac{1}% {RC}V_{s}\cdot e^{i\omega t}
  29. ( i ฯ‰ V c + 1 R C V c ) โ‹… e i ฯ‰ t = ( 1 R C V s ) โ‹… e i ฯ‰ t \left(i\omega V_{c}+\frac{1}{RC}V_{c}\right)\cdot e^{i\omega t}=\left(\frac{1}% {RC}V_{s}\right)\cdot e^{i\omega t}
  30. i ฯ‰ V c + 1 R C V c = 1 R C V s ( QED ) i\omega V_{c}+\frac{1}{RC}V_{c}=\frac{1}{RC}V_{s}\quad\quad(\mathrm{QED})
  31. i ฯ‰ V c + 1 R C V c = 1 R C V s i\omega V_{c}+\frac{1}{RC}V_{c}=\frac{1}{RC}V_{s}
  32. V c = 1 1 + i ฯ‰ R C โ‹… ( V s ) = 1 - i ฯ‰ R C 1 + ( ฯ‰ R C ) 2 โ‹… ( V P e i ฮธ ) V_{c}=\frac{1}{1+i\omega RC}\cdot(V_{s})=\frac{1-i\omega RC}{1+(\omega RC)^{2}% }\cdot(V_{P}e^{i\theta})\,
  33. V s V_{s}\,
  34. v C ( t ) v_{C}(t)\,
  35. V P V_{P}\,
  36. ฮธ . \theta.\,
  37. 1 1 + ( ฯ‰ R C ) 2 โ‹… e - i ฯ• ( ฯ‰ ) , where ฯ• ( ฯ‰ ) = arctan ( ฯ‰ R C ) . \frac{1}{\sqrt{1+(\omega RC)^{2}}}\cdot e^{-i\phi(\omega)},\,\text{ where }% \phi(\omega)=\arctan(\omega RC).\,
  38. v C ( t ) = 1 1 + ( ฯ‰ R C ) 2 โ‹… V P cos ( ฯ‰ t + ฮธ - ฯ• ( ฯ‰ ) ) v_{C}(t)=\frac{1}{\sqrt{1+(\omega RC)^{2}}}\cdot V_{P}\cos(\omega t+\theta-% \phi(\omega))
  39. A 1 cos ( ฯ‰ t + ฮธ 1 ) + A 2 cos ( ฯ‰ t + ฮธ 2 ) \displaystyle A_{1}\cos(\omega t+\theta_{1})+A_{2}\cos(\omega t+\theta_{2})
  40. A 3 2 = ( A 1 cos ฮธ 1 + A 2 cos ฮธ 2 ) 2 + ( A 1 sin ฮธ 1 + A 2 sin ฮธ 2 ) 2 , A_{3}^{2}=(A_{1}\cos\theta_{1}+A_{2}\cos\theta_{2})^{2}+(A_{1}\sin\theta_{1}+A% _{2}\sin\theta_{2})^{2},
  41. ฮธ 3 = arctan ( A 1 sin ฮธ 1 + A 2 sin ฮธ 2 A 1 cos ฮธ 1 + A 2 cos ฮธ 2 ) \theta_{3}=\arctan\left(\frac{A_{1}\sin\theta_{1}+A_{2}\sin\theta_{2}}{A_{1}% \cos\theta_{1}+A_{2}\cos\theta_{2}}\right)
  42. A 3 2 = A 1 2 + A 2 2 - 2 A 1 A 2 cos ( 180 โˆ˜ - ฮ” ฮธ ) , = A 1 2 + A 2 2 + 2 A 1 A 2 cos ( ฮ” ฮธ ) , A_{3}^{2}=A_{1}^{2}+A_{2}^{2}-2A_{1}A_{2}\cos(180^{\circ}-\Delta\theta),=A_{1}% ^{2}+A_{2}^{2}+2A_{1}A_{2}\cos(\Delta\theta),
  43. ฮ” ฮธ = ฮธ 1 - ฮธ 2 \Delta\theta=\theta_{1}-\theta_{2}
  44. A 1 โˆ  ฮธ 1 + A 2 โˆ  ฮธ 2 = A 3 โˆ  ฮธ 3 . A_{1}\angle\theta_{1}+A_{2}\angle\theta_{2}=A_{3}\angle\theta_{3}.\,
  45. ฮป ฮป
  46. cos ( ฯ‰ t ) + cos ( ฯ‰ t + 2 ฯ€ / 3 ) + cos ( ฯ‰ t - 2 ฯ€ / 3 ) = 0. \cos(\omega t)+\cos(\omega t+2\pi/3)+\cos(\omega t-2\pi/3)=0.\,
  47. ฮป \lambda

Photoelasticity.html

  1. ฮ” = 2 ฯ€ t ฮป C ( ฯƒ 1 - ฯƒ 2 ) \Delta=\frac{2\pi t}{\lambda}C(\sigma_{1}-\sigma_{2})
  2. N = ฮ” 2 ฯ€ N=\frac{\Delta}{2\pi}

Photosynthetically_active_radiation.html

  1. ฮท v ( T ) = โˆซ ฮป 1 ฮป 2 B ( ฮป , T ) 683 [ lm / W ] y ( ฮป ) d ฮป โˆซ ฮป 1 ฮป 2 B ( ฮป , T ) d ฮป , \eta_{v}(T)=\frac{\int_{\lambda_{1}}^{\lambda_{2}}B(\lambda,T)\,683\mathrm{~{}% [lm/W]}\,y(\lambda)\,d\lambda}{\int_{\lambda_{1}}^{\lambda_{2}}B(\lambda,T)\,d% \lambda},
  2. ฮท photon ( T ) = โˆซ ฮป 1 ฮป 2 B ( ฮป , T ) ฮป h c N A d ฮป โˆซ ฮป 1 ฮป 2 B ( ฮป , T ) d ฮป , \eta_{\mathrm{photon}}(T)=\frac{\int_{\lambda_{1}}^{\lambda_{2}}B(\lambda,T)\,% \frac{\lambda}{hcN_{A}}\,d\lambda}{\int_{\lambda_{1}}^{\lambda_{2}}B(\lambda,T% )\,d\lambda},
  3. ฮท PAR ( T ) = โˆซ ฮป 1 ฮป 2 B ( ฮป , T ) d ฮป โˆซ 0 โˆž B ( ฮป , T ) d ฮป , \eta_{\mathrm{PAR}}(T)=\frac{\int_{\lambda_{1}}^{\lambda_{2}}B(\lambda,T)\,d% \lambda}{\int_{0}^{\infty}B(\lambda,T)\,d\lambda},
  4. B ( ฮป , T ) B(\lambda,T)
  5. y y
  6. ฮป 1 , ฮป 2 \lambda_{1},\lambda_{2}
  7. N A N_{A}

Phragmeฬnโ€“Lindeloฬˆf_principle.html

  1. g ( z ) = exp ( exp ( z ) ) g(z)=\exp(\exp(z))
  2. - ฯ€ / 2 < Im { z } < ฯ€ / 2. -\pi/2<\mbox{Im}~{}\{z\}<\pi/2.
  3. S = { z | ฮฑ < arg z < ฮฒ } S=\left\{z\,\big|\,\alpha<\arg z<\beta\right\}
  4. | F ( z ) | โ‰ค e C | z | ฯ |F(z)|\leq e^{C|z|^{\rho}}
  5. lim inf r โ†’ โˆž sup ฮฑ < ฮธ < ฮฒ log | F ( r e i ฮธ ) | r ฯ = 0 for some 0 โ‰ค ฯ < ฮป , \liminf_{r\to\infty}\sup_{\alpha<\theta<\beta}\frac{\log|F(re^{i\theta})|}{r^{% \rho}}=0\quad\,\text{for some}\quad 0\leq\rho<\lambda~{},

Physiologically_based_pharmacokinetic_modelling.html

  1. d Q i d t = F i ( C a r t - Q i P i V i ) {dQ_{i}\over dt}=F_{i}(C_{art}-{{Q_{i}}\over{P_{i}V_{i}}})
  2. d Q g d t = F g ( C a r t - Q g P g V g ) {dQ_{g}\over dt}=F_{g}(C_{art}-{{Q_{g}}\over{P_{g}V_{g}}})
  3. d Q k d t = F k ( C a r t - Q k P k V k ) {dQ_{k}\over dt}=F_{k}(C_{art}-{{Q_{k}}\over{P_{k}V_{k}}})
  4. d Q p d t = F p ( C a r t - Q p P p V p ) {dQ_{p}\over dt}=F_{p}(C_{art}-{{Q_{p}}\over{P_{p}V_{p}}})
  5. d Q b d t = F b ( C a r t - Q b P b V b ) {dQ_{b}\over dt}=F_{b}(C_{art}-{{Q_{b}}\over{P_{b}V_{b}}})
  6. d Q h d t = F h ( C a r t - Q h P h V h ) {dQ_{h}\over dt}=F_{h}(C_{art}-{{Q_{h}}\over{P_{h}V_{h}}})
  7. d Q p n d t = F p n ( C a r t - Q p n P p n V p n ) {dQ_{pn}\over dt}=F_{pn}(C_{art}-{{Q_{pn}}\over{P_{pn}V_{pn}}})
  8. d Q l d t = F a C a r t + F g ( Q g P g V g ) + F p n ( Q p n P p n V p n ) - ( F a + F g + F p n ) ( Q l P l V l ) {dQ_{l}\over dt}=F_{a}C_{art}+F_{g}({{Q_{g}}\over{P_{g}V_{g}}})+F_{pn}({{Q_{pn% }}\over{P_{pn}V_{pn}}})-(F_{a}+F_{g}+F_{pn})({{Q_{l}}\over{P_{l}V_{l}}})
  9. d Q g d t = F g ( C a r t - Q g P g V g ) + K a Q i n g {dQ_{g}\over dt}=F_{g}(C_{art}-{{Q_{g}}\over{P_{g}V_{g}}})+K_{a}Q_{ing}
  10. d Q i n g d t = - K a Q i n g {dQ_{ing}\over dt}=-K_{a}Q_{ing}
  11. d Q g d t = F g ( C a r t - Q g P g V g ) + R i n g {dQ_{g}\over dt}=F_{g}(C_{art}-{{Q_{g}}\over{P_{g}V_{g}}})+R_{ing}
  12. v = d [ P ] d t = V max [ S ] K m + [ S ] v=\frac{d[P]}{dt}=\frac{V_{\max}{[S]}}{K_{m}+[S]}

Piano_key_frequencies.html

  1. f f
  2. n n
  3. f ( n ) = ( 2 12 ) n - 49 ร— 440 Hz f(n)=(\sqrt[12]{2}\,)^{n-49}\times 440\,\,\text{Hz}\,
  4. f ( n ) = 2 n - 49 12 ร— 440 Hz f(n)=2^{\frac{n-49}{12}}\times 440\,\,\text{Hz}\,
  5. n = 12 log 2 ( f 440 Hz ) + 49 n=12\,\log_{2}\left({\frac{f}{440\,\,\text{Hz}}}\right)+49

Picard_group.html

  1. H 1 ( X , ๐’ช X * ) . H^{1}(X,\mathcal{O}_{X}^{*}).\,
  2. ๐’ช ( m ) , \mathcal{O}(m),\,
  3. 1 โ†’ Pic 0 ( V ) โ†’ Pic ( V ) โ†’ NS ( V ) โ†’ 0. 1\to\mathrm{Pic}^{0}(V)\to\mathrm{Pic}(V)\to\mathrm{NS}(V)\to 0.\,
  4. Pic X / S ( T ) = Pic ( X T ) / f T * ( Pic ( T ) ) \operatorname{Pic}_{X/S}(T)=\operatorname{Pic}(X_{T})/f_{T}^{*}(\operatorname{% Pic}(T))
  5. f T : X T โ†’ T f_{T}:X_{T}\to T
  6. Pic X / S ( T ) \operatorname{Pic}_{X/S}(T)
  7. s * L s^{*}L

Pin_group.html

  1. v 2 = Q ( v ) โˆˆ C โ„“ ( V , Q ) v^{2}=Q(v)\in C\ell(V,Q)
  2. Pin ( n ) + := Pin ( n , 0 ) Pin ( n ) - := Pin ( 0 , n ) \mbox{Pin}~{}_{+}(n):=\mbox{Pin}~{}(n,0)\qquad\mbox{Pin}~{}_{-}(n):=\mbox{Pin}% ~{}(0,n)
  3. 1 โ†’ { ยฑ 1 } โ†’ Pin ( V ) ยฑ โ†’ O ( V ) โ†’ 1. 1\to\{\pm 1\}\to\mbox{Pin}~{}_{\pm}(V)\to O(V)\to 1.
  4. r ~ 2 = ยฑ 1 \tilde{r}^{2}=\pm 1
  5. r ~ \tilde{r}
  6. r ~ \tilde{r}
  7. Dic < n Pin ( 2 ) - \mbox{Dic}~{}_{n}<\mbox{Pin}~{}_{-}(2)
  8. Pin ( 1 ) + \displaystyle\mbox{Pin}~{}_{+}(1)
  9. C โ„“ ( n , 0 ) โ‰…ฬธ C โ„“ ( 0 , n ) C\ell(n,0)\not\cong C\ell(0,n)
  10. C โ„“ ( n , 0 ) = C โ„“ ( 0 , n ) = ฮ› * ๐‘ n C\ell(n,0)=C\ell(0,n)=\Lambda^{*}\mathbf{R}^{n}

Pivot_language.html

  1. n - 1 n-1
  2. ( ( n 2 ) = n 2 - n 2 ) \left(\textstyle{{\left({{n}\atop{2}}\right)}}=\frac{n^{2}-n}{2}\right)

Planar_algebra.html

  1. I I
  2. W W
  3. V ฯ‰ V_{\omega}
  4. ฯ‰ \omega
  5. W W
  6. I I
  7. w 1 , w 2 , โ€ฆ , w k w_{1},w_{2},\ldots,w_{k}
  8. w 0 w_{0}
  9. ( w 1 , w 2 , โ€ฆ , w k ) (w_{1},w_{2},\dots,w_{k})
  10. w 0 w_{0}
  11. w 0 w_{0}
  12. k k
  13. w i w_{i}
  14. I I
  15. W W
  16. V ฯ‰ V_{\omega}
  17. T T
  18. ( w 1 , w 2 , โ€ฆ , w k ) (w_{1},w_{2},\ldots,w_{k})
  19. w 0 w_{0}
  20. Z T : V w 1 โŠ— V w 2 โŠ— โ‹ฏ โŠ— V w k โŸถ V w 0 Z_{T}:V_{w_{1}}\otimes V_{w_{2}}\otimes\cdots\otimes V_{w_{k}}\longrightarrow V% _{w_{0}}
  21. ฮด โˆˆ R \delta\in R
  22. 2 n 2n
  23. V 2 n V_{2n}
  24. ฮด \delta
  25. i โˆˆ I i\in I
  26. ฮด i โˆˆ R \delta_{i}\in R
  27. ฯ‰ โˆˆ W \omega\in W
  28. V ฯ‰ V_{\omega}
  29. I I
  30. ฯ‰ \omega
  31. i i
  32. ฮด i \delta_{i}
  33. V โˆ… V_{\emptyset}

Planck_(spacecraft).html

  1. k m / M p c ยท s {km}/{Mpcยทs}
  2. t 0 t_{0}
  3. H 0 H_{0}
  4. ฮฉ b h 2 \Omega_{b}h^{2}
  5. ฮฉ c h 2 \Omega_{c}h^{2}
  6. ฮฉ ฮ› \Omega_{\Lambda}
  7. ฯƒ 8 \sigma_{8}
  8. n s n_{s}
  9. ฯ„ \tau

Planck_particle.html

  1. ฮป = h m c = 2 G m c 2 \lambda=\frac{h}{mc}=\frac{2Gm}{c^{2}}
  2. m = h c 2 G m=\sqrt{\frac{hc}{2G}}
  3. ฯ€ \sqrt{\pi}
  4. r = h m c = 2 G h c 3 r=\frac{h}{mc}=\sqrt{\frac{2Gh}{c^{3}}}

Planckian_locus.html

  1. X T = โˆซ 0 โˆž X ( ฮป ) M ( ฮป , T ) d ฮป X_{T}=\int_{0}^{\infty}X(\lambda)M(\lambda,T)\,d\lambda
  2. Y T = โˆซ 0 โˆž Y ( ฮป ) M ( ฮป , T ) d ฮป Y_{T}=\int_{0}^{\infty}Y(\lambda)M(\lambda,T)\,d\lambda
  3. Z T = โˆซ 0 โˆž Z ( ฮป ) M ( ฮป , T ) d ฮป Z_{T}=\int_{0}^{\infty}Z(\lambda)M(\lambda,T)\,d\lambda
  4. M ( ฮป , T ) = c 1 ฮป 5 1 exp ( c 2 ฮป T ) - 1 M(\lambda,T)=\frac{c_{1}}{\lambda^{5}}\frac{1}{\exp\left(\frac{c_{2}}{{\lambda% }T}\right)-1}
  5. ฯ€ \pi
  6. x T = X T X T + Y T + Z T x_{T}=\frac{X_{T}}{X_{T}+Y_{T}+Z_{T}}
  7. y T = Y T X T + Y T + Z T y_{T}=\frac{Y_{T}}{X_{T}+Y_{T}+Z_{T}}
  8. x c = { - 0.2661239 10 9 T 3 - 0.2343580 10 6 T 2 + 0.8776956 10 3 T + 0.179910 1667 K โ‰ค T โ‰ค 4000 K - 3.0258469 10 9 T 3 + 2.1070379 10 6 T 2 + 0.2226347 10 3 T + 0.240390 4000 K โ‰ค T โ‰ค 25000 K x_{c}=\begin{cases}-0.2661239\frac{10^{9}}{T^{3}}-0.2343580\frac{10^{6}}{T^{2}% }+0.8776956\frac{10^{3}}{T}+0.179910&1667\,\text{K}\leq T\leq 4000\,\text{K}\\ -3.0258469\frac{10^{9}}{T^{3}}+2.1070379\frac{10^{6}}{T^{2}}+0.2226347\frac{10% ^{3}}{T}+0.240390&4000\,\text{K}\leq T\leq 25000\,\text{K}\end{cases}
  9. y c = { - 1.1063814 x c 3 - 1.34811020 x c 2 + 2.18555832 x c - 0.20219683 1667 K โ‰ค T โ‰ค 2222 K - 0.9549476 x c 3 - 1.37418593 x c 2 + 2.09137015 x c - 0.16748867 2222 K โ‰ค T โ‰ค 4000 K + 3.0817580 x c 3 - 5.87338670 x c 2 + 3.75112997 x c - 0.37001483 4000 K โ‰ค T โ‰ค 25000 K y_{c}=\begin{cases}-1.1063814x_{c}^{3}-1.34811020x_{c}^{2}+2.18555832x_{c}-0.2% 0219683&1667\,\text{K}\leq T\leq 2222\,\text{K}\\ -0.9549476x_{c}^{3}-1.37418593x_{c}^{2}+2.09137015x_{c}-0.16748867&2222\,\text% {K}\leq T\leq 4000\,\text{K}\\ +3.0817580x_{c}^{3}-5.87338670x_{c}^{2}+3.75112997x_{c}-0.37001483&4000\,\text% {K}\leq T\leq 25000\,\text{K}\end{cases}
  10. u ยฏ ( T ) = 0.860117757 + 1.54118254 ร— 10 - 4 T + 1.28641212 ร— 10 - 7 T 2 1 + 8.42420235 ร— 10 - 4 T + 7.08145163 ร— 10 - 7 T 2 \bar{u}(T)=\frac{0.860117757+1.54118254\times 10^{-4}T+1.28641212\times 10^{-7% }T^{2}}{1+8.42420235\times 10^{-4}T+7.08145163\times 10^{-7}T^{2}}
  11. v ยฏ ( T ) = 0.317398726 + 4.22806245 ร— 10 - 5 T + 4.20481691 ร— 10 - 8 T 2 1 - 2.89741816 ร— 10 - 5 T + 1.61456053 ร— 10 - 7 T 2 \bar{v}(T)=\frac{0.317398726+4.22806245\times 10^{-5}T+4.20481691\times 10^{-8% }T^{2}}{1-2.89741816\times 10^{-5}T+1.61456053\times 10^{-7}T^{2}}
  12. | u - u ยฏ | < 8 ร— 10 - 5 \left|u-\bar{u}\right|<8\times 10^{-5}
  13. | v - v ยฏ | < 9 ร— 10 - 5 \left|v-\bar{v}\right|<9\times 10^{-5}
  14. 1000 K < T < 15 , 000 K 1000K<T<15,000K
  15. ฮ” u v = ยฑ 0.05 \Delta_{uv}=\pm 0.05
  16. c 2 = h c / k c_{2}=hc/k
  17. c 2 = 1.432 ร— 10 - 2 mยทK c_{2}=1.432\times 10^{-2}\,\text{mยทK}
  18. c 2 = 1.4380 ร— 10 - 2 mยทK c_{2}=1.4380\times 10^{-2}\,\text{mยทK}
  19. c 2 = 1.4388 ร— 10 - 2 mยทK c_{2}=1.4388\times 10^{-2}\,\text{mยทK}
  20. c 2 = 1.4387770 ( 13 ) ร— 10 - 2 mยทK c_{2}=1.4387770(13)\times 10^{-2}\,\text{mยทK}

Planetary_protection.html

  1. P c = N 0 R P S P t P R P g P_{c}=N_{0}RP_{S}P_{t}P_{R}P_{g}
  2. N 0 N_{0}
  3. R R
  4. P S P_{S}
  5. P t P_{t}
  6. P R P_{R}
  7. P g P_{g}
  8. P c < 10 - 4 P_{c}<10^{-4}
  9. 10 - 4 10^{-4}

Plasma_acceleration.html

  1. E = c โ‹… m e โ‹… ฯ ฮต 0 . E=c\cdot\sqrt{\frac{m_{e}\cdot\rho}{\varepsilon_{0}}}.
  2. E E
  3. c c
  4. m e m_{e}
  5. ฯ \rho
  6. ฮต 0 \varepsilon_{0}

Plasma_stealth.html

  1. ฯ‰ p e = ( 4 ฯ€ n e e 2 / m e ) 1 / 2 = 5.64 ร— 10 4 n e 1 / 2 rad/s \omega_{pe}=(4\pi n_{e}e^{2}/m_{e})^{1/2}=5.64\times 10^{4}n_{e}^{1/2}\mbox{% rad/s}~{}

Player_efficiency_rating.html

  1. u P E R = 1 m i n ร— ( 3 P + [ 2 3 ร— A S T ] + [ ( 2 - f a c t o r ร— t m A S T t m F G ) ร— F G ] + [ 0.5 ร— F T ร— ( 2 - t m A S T t m F G + 2 3 ร— t m A S T t m F G ) ] - [ V O P ร— T O ] - [ V O P ร— D R B P ร— ( F G A - F G ) ] - [ V O P ร— 0.44 ร— ( 0.44 + ( 0.56 ร— D R B P ) ) ร— ( F T A - F T ) ] + [ V O P ร— ( 1 - D R B P ) ร— ( T R B - O R B ) ] + [ V O P ร— D R B P ร— O R B ] + [ V O P ร— S T L ] + [ V O P ร— D R B P ร— B L K ] - [ P F ร— ( l g F T l g P F - 0.44 ร— l g F T A l g P F ร— V O P ) ] ) uPER=\frac{1}{min}\times\left(3P+\left[\frac{2}{3}\times AST\right]+\left[% \left(2-factor\times\frac{tmAST}{tmFG}\right)\times FG\right]+\left[0.5\times FT% \times\left(2-\frac{tmAST}{tmFG}+\frac{2}{3}\times\frac{tmAST}{tmFG}\right)% \right]-\left[VOP\times TO\right]-\left[VOP\times DRBP\times\left(FGA-FG\right% )\right]-\left[VOP\times 0.44\times\left(0.44+\left(0.56\times DRBP\right)% \right)\times\left(FTA-FT\right)\right]+\left[VOP\times\left(1-DRBP\right)% \times\left(TRB-ORB\right)\right]+\left[VOP\times DRBP\times ORB\right]+\left[% VOP\times STL\right]+\left[VOP\times DRBP\times BLK\right]-\left[PF\times\left% (\frac{lgFT}{lgPF}-0.44\times\frac{lgFTA}{lgPF}\times VOP\right)\right]\right)
  2. u P E R = 1 m i n ร— ( 3 P - P F ร— l g F T l g P F + [ F T 2 ร— ( 2 - t m A S T 3 ร— t m F G ) ] + [ F G ร— ( 2 - f a c t o r ร— t m A S T t m F G ) ] + 2 ร— A S T 3 + V O P ร— [ D R B P ร— ( 2 ร— O R B + B L K - 0.2464 ร— [ F T A - F T ] - [ F G A - F G ] - T R B ) + 0.44 ร— l g F T A ร— P F l g P F - ( T O + O R B ) + S T L + T R B - 0.1936 ( F T A - F T ) ] ) uPER=\frac{1}{min}\times\left(3P-\frac{PF\times lgFT}{lgPF}+\left[\frac{FT}{2}% \times\left(2-\frac{tmAST}{3\times tmFG}\right)\right]+\left[FG\times\left(2-% \frac{factor\times tmAST}{tmFG}\right)\right]+\frac{2\times AST}{3}+VOP\times% \left[DRBP\times\left(2\times ORB+BLK-0.2464\times\left[FTA-FT\right]-\left[% FGA-FG\right]-TRB\right)+\frac{0.44\times lgFTA\times PF}{lgPF}-\left(TO+ORB% \right)+STL+TRB-0.1936\left(FTA-FT\right)\right]\right)
  3. f a c t o r = 2 3 - [ ( 0.5 ร— l g A S T l g F G ) รท ( 2 ร— l g F G l g F T ) ] \ factor=\frac{2}{3}-\left[\left(0.5\times\frac{lgAST}{lgFG}\right)\div\left(2% \times\frac{lgFG}{lgFT}\right)\right]
  4. V O P = l g P T S l g F G A - l g O R B + l g T O + 0.44 ร— l g F T A \ VOP=\frac{lgPTS}{lgFGA-lgORB+lgTO+0.44\times lgFTA}
  5. D R B P = l g T R B - l g O R B l g T R B \ DRBP=\frac{lgTRB-lgORB}{lgTRB}
  6. P E R = ( u P E R ร— l g P a c e t m P a c e ) ร— 15 l g u P E R \ PER=\left(uPER\times\frac{lgPace}{tmPace}\right)\times\frac{15}{lguPER}

Plebanski_action.html

  1. A a i A^{i}_{a}
  2. i i
  3. a a
  4. S P l e b a n s k i = โˆซ ฮฃ ร— R ฯต i j k l B i j โˆง F k l ( A a i ) + ฯ• i j k l B i j โˆง B k l S_{Plebanski}=\int_{\Sigma\times R}\epsilon_{ijkl}B^{ij}\wedge F^{kl}(A^{i}_{a% })+\phi_{ijkl}B^{ij}\wedge B^{kl}
  5. i , j , l , k i,j,l,k
  6. F F
  7. S O ( 3 , 1 ) SO(3,1)
  8. A a i A^{i}_{a}
  9. ฯ• i j k l \phi_{ijkl}
  10. ฯต i j k l \epsilon_{ijkl}
  11. S O ( 3 , 1 ) SO(3,1)
  12. B i j = e i โˆง e j B^{ij}=e^{i}\wedge e^{j}

Plug_flow.html

  1. ฮด s \delta_{s}
  2. u * = ( ฯ„ w ฯ ) 1 / 2 u^{*}=\left(\frac{\tau_{w}}{\rho}\right)^{1/2}
  3. ฯ„ w = D ฮ” P 4 L \tau_{w}=\frac{D\Delta P}{4L}
  4. ฮ” P L = f ฯ V 2 2 D \frac{\Delta P}{L}=\frac{f\rho V^{2}}{2D}
  5. 1 f = - 2.0 log 10 ( ฯต D 3.7 + 2.51 R e f ) , (turbulent flow) {1\over\sqrt{\mathit{f}}}=-2.0\log_{10}\left(\frac{\frac{\epsilon}{D}}{3.7}+{% \frac{2.51}{Re\sqrt{\mathit{f}}}}\right),\,\text{(turbulent flow)}
  6. ฮด s \delta_{s}
  7. ฯ \rho
  8. ฯ„ w \tau_{w}
  9. ฮ” P \Delta P
  10. ฯต \epsilon

PN.html

  1. n n
  2. P n P_{n}
  3. โ„™ n \mathbb{P}_{n}

Pohligโ€“Hellman_algorithm.html

  1. ฯ† ( p ) = p 1 โ‹… p 2 โ‹ฏ p n \varphi(p)=p_{1}\cdot p_{2}\cdots p_{n}
  2. e ฯ† ( p ) / p 1 โ‰ก ( g x ) ฯ† ( p ) / p 1 ( mod p ) โ‰ก ( g ฯ† ( p ) ) a 1 g b 1 ฯ† ( p ) / p 1 ( mod p ) โ‰ก ( g ฯ† ( p ) / p 1 ) b 1 ( mod p ) \begin{aligned}\displaystyle e^{\varphi(p)/p_{1}}&\displaystyle\equiv(g^{x})^{% \varphi(p)/p_{1}}\;\;(\mathop{{\rm mod}}p)\\ &\displaystyle\equiv(g^{\varphi(p)})^{a_{1}}g^{b_{1}\varphi(p)/p_{1}}\;\;(% \mathop{{\rm mod}}p)\\ &\displaystyle\equiv(g^{\varphi(p)/p_{1}})^{b_{1}}\;\;(\mathop{{\rm mod}}p)% \end{aligned}
  3. g ฯ† ( p ) / p 1 โ‰ข 1 ( mod p ) g^{\varphi(p)/p_{1}}\not\equiv 1\;\;(\mathop{{\rm mod}}p)
  4. g ฯ† ( p ) / p 1 โ‰ก 1 ( mod p ) g^{\varphi(p)/p_{1}}\equiv 1\;\;(\mathop{{\rm mod}}p)
  5. e ฯ† ( p ) / p 1 mod p e^{\varphi(p)/p_{1}}\mod p
  6. g ฯ† ( p ) / p i โ‰ก 1 ( mod p ) g^{\varphi(p)/p_{i}}\equiv 1\;\;(\mathop{{\rm mod}}p)
  7. O ( n ) O(\sqrt{n})
  8. โˆ i p i e i \prod_{i}p_{i}^{e_{i}}
  9. O ( โˆ‘ i e i ( log n + p i ) ) O\left(\sum_{i}{e_{i}(\log n+\sqrt{p}_{i})}\right)

Poincareฬ_metric.html

  1. d s 2 = ฮป 2 ( z , z ยฏ ) d z d z ยฏ ds^{2}=\lambda^{2}(z,\overline{z})\,dz\,d\overline{z}
  2. z z
  3. z ยฏ \overline{z}
  4. l ( ฮณ ) = โˆซ ฮณ ฮป ( z , z ยฏ ) | d z | l(\gamma)=\int_{\gamma}\lambda(z,\overline{z})\,|dz|
  5. Area ( M ) = โˆซ M ฮป 2 ( z , z ยฏ ) i 2 d z โˆง d z ยฏ \,\text{Area}(M)=\int_{M}\lambda^{2}(z,\overline{z})\,\frac{i}{2}\,dz\wedge d% \overline{z}
  6. โˆง \wedge
  7. ฮป 4 \lambda^{4}
  8. ฮป 2 \lambda^{2}
  9. d x โˆง d y dx\wedge dy
  10. d z โˆง d z ยฏ = ( d x + i d y ) โˆง ( d x - i d y ) = - 2 i d x โˆง d y . dz\wedge d\overline{z}=(dx+i\,dy)\wedge(dx-i\,dy)=-2i\,dx\wedge dy.
  11. ฮฆ ( z , z ยฏ ) \Phi(z,\overline{z})
  12. 4 โˆ‚ โˆ‚ z โˆ‚ โˆ‚ z ยฏ ฮฆ ( z , z ยฏ ) = ฮป 2 ( z , z ยฏ ) . 4\frac{\partial}{\partial z}\frac{\partial}{\partial\overline{z}}\Phi(z,% \overline{z})=\lambda^{2}(z,\overline{z}).
  13. ฮ” = 4 ฮป 2 โˆ‚ โˆ‚ z โˆ‚ โˆ‚ z ยฏ = 1 ฮป 2 ( โˆ‚ 2 โˆ‚ x 2 + โˆ‚ 2 โˆ‚ y 2 ) . \Delta=\frac{4}{\lambda^{2}}\frac{\partial}{\partial z}\frac{\partial}{% \partial\overline{z}}=\frac{1}{\lambda^{2}}\left(\frac{\partial^{2}}{\partial x% ^{2}}+\frac{\partial^{2}}{\partial y^{2}}\right).
  14. K = - ฮ” log ฮป . K=-\Delta\log\lambda.\,
  15. ฮป 2 ( z , z ยฏ ) d z d z ยฏ \lambda^{2}(z,\overline{z})\,dz\,d\overline{z}
  16. ฮผ 2 ( w , w ยฏ ) d w d w ยฏ \mu^{2}(w,\overline{w})\,dw\,d\overline{w}
  17. f : S โ†’ T f:S\to T\,
  18. f = w ( z ) f=w(z)
  19. ฮผ 2 ( w , w ยฏ ) โˆ‚ w โˆ‚ z โˆ‚ w ยฏ โˆ‚ z ยฏ = ฮป 2 ( z , z ยฏ ) \mu^{2}(w,\overline{w})\;\frac{\partial w}{\partial z}\frac{\partial\overline{% w}}{\partial\overline{z}}=\lambda^{2}(z,\overline{z})
  20. w ( z , z ยฏ ) = w ( z ) , w(z,\overline{z})=w(z),
  21. โˆ‚ โˆ‚ z ยฏ w ( z ) = 0. \frac{\partial}{\partial\overline{z}}w(z)=0.
  22. d s 2 = d x 2 + d y 2 y 2 = d z d z ยฏ y 2 ds^{2}=\frac{dx^{2}+dy^{2}}{y^{2}}=\frac{dz\,d\overline{z}}{y^{2}}
  23. d z = d x + i d y . dz=dx+i\,dy.
  24. z โ€ฒ = x โ€ฒ + i y โ€ฒ = a z + b c z + d z^{\prime}=x^{\prime}+iy^{\prime}=\frac{az+b}{cz+d}
  25. a d - b c = 1 ad-bc=1
  26. x โ€ฒ = a c ( x 2 + y 2 ) + x ( a d + b c ) + b d | c z + d | 2 x^{\prime}=\frac{ac(x^{2}+y^{2})+x(ad+bc)+bd}{|cz+d|^{2}}
  27. y โ€ฒ = y | c z + d | 2 . y^{\prime}=\frac{y}{|cz+d|^{2}}.
  28. d z โ€ฒ = d z | c z + d | 2 dz^{\prime}=\frac{dz}{|cz+d|^{2}}
  29. d z โ€ฒ d z ยฏ โ€ฒ = d z d z ยฏ | c z + d | 4 dz^{\prime}d\overline{z}^{\prime}=\frac{dz\,d\overline{z}}{|cz+d|^{4}}
  30. d ฮผ = d x d y y 2 . d\mu=\frac{dx\,dy}{y^{2}}.
  31. ฯ ( z 1 , z 2 ) = 2 tanh - 1 | z 1 - z 2 | | z 1 - z 2 ยฏ | \rho(z_{1},z_{2})=2\tanh^{-1}\frac{|z_{1}-z_{2}|}{|z_{1}-\overline{z_{2}}|}
  32. ฯ ( z 1 , z 2 ) = log | z 1 - z 2 ยฏ | + | z 1 - z 2 | | z 1 - z 2 ยฏ | - | z 1 - z 2 | \rho(z_{1},z_{2})=\log\frac{|z_{1}-\overline{z_{2}}|+|z_{1}-z_{2}|}{|z_{1}-% \overline{z_{2}}|-|z_{1}-z_{2}|}
  33. z 1 , z 2 โˆˆ โ„ z_{1},z_{2}\in\mathbb{H}
  34. z 1 , z 2 , z 3 z_{1},z_{2},z_{3}
  35. z 4 z_{4}
  36. โ„‚ ^ = โ„‚ โˆช โˆž \hat{\mathbb{C}}=\mathbb{C}\cup\infty
  37. ( z 1 , z 2 ; z 3 , z 4 ) = ( z 1 - z 3 ) ( z 2 - z 4 ) ( z 1 - z 4 ) ( z 2 - z 3 ) . (z_{1},z_{2};z_{3},z_{4})=\frac{(z_{1}-z_{3})(z_{2}-z_{4})}{(z_{1}-z_{4})(z_{2% }-z_{3})}.
  38. ฯ ( z 1 , z 2 ) = log ( z 1 , z 2 ; z 1 ร— , z 2 ร— ) . \rho(z_{1},z_{2})=\log(z_{1},z_{2};z_{1}^{\times},z_{2}^{\times}).
  39. z 1 ร— z_{1}^{\times}
  40. z 2 ร— z_{2}^{\times}
  41. z 1 z_{1}
  42. z 2 z_{2}
  43. z 1 z_{1}
  44. z 1 ร— z_{1}^{\times}
  45. z 2 z_{2}
  46. w = e i ฯ• z - z 0 z - z 0 ยฏ w=e^{i\phi}\frac{z-z_{0}}{z-\overline{z_{0}}}
  47. โ„‘ z = 0 \Im z=0
  48. | w | = 1. |w|=1.
  49. ฯ• \phi
  50. w = i z + 1 z + i w=\frac{iz+1}{z+i}
  51. U = { z = x + i y : | z | = x 2 + y 2 < 1 } U=\{z=x+iy:|z|=\sqrt{x^{2}+y^{2}}<1\}
  52. d s 2 = 4 ( d x 2 + d y 2 ) ( 1 - ( x 2 + y 2 ) ) 2 = 4 d z d z ยฏ ( 1 - | z | 2 ) 2 . ds^{2}=\frac{4(dx^{2}+dy^{2})}{(1-(x^{2}+y^{2}))^{2}}=\frac{4dz\,d\overline{z}% }{(1-|z|^{2})^{2}}.
  53. d ฮผ = 4 d x d y ( 1 - ( x 2 + y 2 ) ) 2 = 4 d x d y ( 1 - | z | 2 ) 2 . d\mu=\frac{4dx\,dy}{(1-(x^{2}+y^{2}))^{2}}=\frac{4dx\,dy}{(1-|z|^{2})^{2}}.
  54. ฯ ( z 1 , z 2 ) = 2 tanh - 1 | z 1 - z 2 1 - z 1 z 2 ยฏ | \rho(z_{1},z_{2})=2\tanh^{-1}\left|\frac{z_{1}-z_{2}}{1-z_{1}\overline{z_{2}}}\right|
  55. z 1 , z 2 โˆˆ U . z_{1},z_{2}\in U.
  56. q = exp ( i ฯ€ ฯ„ ) q=\exp(i\pi\tau)
  57. โ„‘ ฯ„ > 0 \Im\tau>0
  58. d s 2 = 4 | q | 2 ( log | q | 2 ) 2 d q d q ยฏ ds^{2}=\frac{4}{|q|^{2}(\log|q|^{2})^{2}}dq\,d\overline{q}
  59. ฮฆ ( q , q ยฏ ) = 4 log log | q | - 2 \Phi(q,\overline{q})=4\log\log|q|^{-2}

Poincareฬโ€“Hopf_theorem.html

  1. โˆ‘ i index x i ( v ) = ฯ‡ ( M ) \sum_{i}\operatorname{index}_{x_{i}}(v)=\chi(M)\,
  2. ฯ‡ ( M ) \chi(M)

Pointed_set.html

  1. ( X , x 0 ) (X,x_{0})
  2. X X
  3. x 0 x_{0}
  4. X X
  5. ( X , x 0 ) (X,x_{0})
  6. ( Y , y 0 ) (Y,y_{0})
  7. X X
  8. Y Y
  9. f : X โ†’ Y f:X\to Y
  10. f ( x 0 ) = y 0 f(x_{0})=y_{0}
  11. f : ( X , x 0 ) โ†’ ( Y , y 0 ) f:(X,x_{0})\to(Y,y_{0})
  12. ( { a } , a ) (\{a\},a)
  13. ๐Ÿ โ†“ ๐’๐ž๐ญ \mathbf{1}\downarrow\mathbf{Set}
  14. ๐Ÿ \mathbf{1}
  15. ๐”ฝ \mathbb{F}
  16. X X

Pointed_space.html

  1. ฮฉ \Omega
  2. X X
  3. ฮฉ X \Omega X

Polar_decomposition.html

  1. z = r e i ฮธ z=re^{i\theta}\,
  2. e i ฮธ e^{i\theta}
  3. A = U P A=UP\,
  4. A A
  5. A ยฏ = U ยฏ \overline{A}=\overline{U}
  6. P ยฏ \overline{P}
  7. det A = det U det P = r e i ฮธ \det A=\det U\,\det P=re^{i\theta}
  8. det P = r = | det A | \det P=r=|\det A|
  9. det U = e i ฮธ \det U=e^{i\theta}
  10. P = A * A P=\sqrt{A^{*}A}
  11. U = A P - 1 . U=AP^{-1}.\,
  12. P = V ฮฃ V * P=V\Sigma V^{*}\,
  13. U = W V * U=WV^{*}\,
  14. A = P โ€ฒ U A=P^{\prime}U\,
  15. P โ€ฒ = U P U - 1 = A A * = W ฮฃ W * . P^{\prime}=UPU^{-1}=\sqrt{AA^{*}}=W\Sigma W^{*}.
  16. A = ( Q ฮฆ Q * ) ( Q ฮฃ Q * ) , A=(Q\Phi Q^{*})(Q\Sigma Q^{*}),\,
  17. A = Q ฮ› Q * A=Q\Lambda Q^{*}\,
  18. A * A = ( A * A ) 1 2 ( A * A ) 1 2 , A^{*}A=(A^{*}A)^{\frac{1}{2}}(A^{*}A)^{\frac{1}{2}},
  19. A = U ( A * A ) 1 2 A=U(A^{*}A)^{\frac{1}{2}}
  20. A = U | A | A=U|A|\,
  21. { x i + y j + z k โˆˆ H : x 2 + y 2 + z 2 = 1 } \{xi+yj+zk\in H:x^{2}+y^{2}+z^{2}=1\}
  22. q = t e a r . q=te^{ar}\!.
  23. cosh ( a ) + j sinh ( a ) = exp ( a j ) = e a j \cosh(a)+j\ \sinh(a)=\exp(aj)=e^{aj}
  24. r e a j , - r e a j , r j e a j , - r j e a j , r > 0 re^{aj},-re^{aj},rje^{aj},-rje^{aj},\quad r>0
  25. U 0 = A U_{0}=A
  26. U k + 1 = 1 2 ( U k + ( U k * ) - 1 ) U_{k+1}=\frac{1}{2}\left(U_{k}+(U_{k}^{*})^{-1}\right)
  27. U k U_{k}
  28. ฮณ k U k \gamma_{k}U_{k}
  29. ฮณ k \gamma_{k}
  30. ฮณ k = โˆฅ U k - 1 โˆฅ 1 โˆฅ U k - 1 โˆฅ โˆž โˆฅ U k โˆฅ 1 โˆฅ U k โˆฅ โˆž 4 \gamma_{k}=\sqrt[4\;]{\frac{\|U_{k}^{-1}\|_{1}\,\|U_{k}^{-1}\|_{\infty}}{\|U_{% k}\|_{1}\,\|U_{k}\|_{\infty}}}
  31. ฮณ k = โˆฅ U k - 1 โˆฅ F โˆฅ U k โˆฅ F \gamma_{k}=\sqrt{\frac{\|U_{k}^{-1}\|_{F}}{\|U_{k}\|_{F}}}
  32. U k + 1 = 1 2 ( ฮณ k U k + 1 ฮณ k ( U k * ) - 1 ) U_{k+1}=\frac{1}{2}\left(\gamma_{k}U_{k}+\frac{1}{\gamma_{k}}(U_{k}^{*})^{-1}\right)
  33. x 2 - 1 = 0 x^{2}-1=0
  34. U k + 1 = U k ( I + 3 U k * U k ) - 1 ( 3 I + U k * U k ) U_{k+1}=U_{k}\left(I+3\,U_{k}^{*}U_{k}\right)^{-1}\left(3\,I+U_{k}^{*}U_{k}\right)

Polar_set.html

  1. ( X , Y ) (X,Y)
  2. A A
  3. X X
  4. A โˆ˜ A^{\circ}
  5. Y Y
  6. A โˆ˜ := { y โˆˆ Y : sup x โˆˆ A | โŸจ x , y โŸฉ | โ‰ค 1 } A^{\circ}:=\{y\in Y:\sup_{x\in A}|\langle x,y\rangle|\leq 1\}
  7. A A
  8. X X
  9. A โˆ˜ A^{\circ}
  10. A โˆ˜ โˆ˜ A^{\circ\circ}
  11. X X
  12. A โˆ˜ A^{\circ}
  13. A โІ B A\subseteq B
  14. B โˆ˜ โІ A โˆ˜ B^{\circ}\subseteq A^{\circ}
  15. โ‹ƒ i โˆˆ I A i โˆ˜ โІ ( โ‹‚ i โˆˆ I A i ) โˆ˜ \bigcup_{i\in I}A_{i}^{\circ}\subseteq(\bigcap_{i\in I}A_{i})^{\circ}
  16. ฮณ โ‰  0 \gamma\neq 0
  17. ( ฮณ A ) โˆ˜ = 1 โˆฃ ฮณ โˆฃ A โˆ˜ (\gamma A)^{\circ}=\frac{1}{\mid\gamma\mid}A^{\circ}
  18. ( โ‹ƒ i โˆˆ I A i ) โˆ˜ = โ‹‚ i โˆˆ I A i โˆ˜ (\bigcup_{i\in I}A_{i})^{\circ}=\bigcap_{i\in I}A_{i}^{\circ}
  19. ( X , Y ) (X,Y)
  20. A โˆ˜ A^{\circ}
  21. Y Y
  22. Y Y
  23. A โˆ˜ โˆ˜ A^{\circ\circ}
  24. A A
  25. A A
  26. A A
  27. A A
  28. A โˆ˜ โˆ˜ = A A^{\circ\circ}=A
  29. C C
  30. X X
  31. C C
  32. C โˆ˜ = { y โˆˆ Y : sup { โŸจ x , y โŸฉ : x โˆˆ C } โ‰ค 1 } C^{\circ}=\{y\in Y:\sup\{\langle x,y\rangle:x\in C\}\leq 1\}
  33. x 0 x_{0}
  34. x x
  35. โŸจ x , x 0 โŸฉ = 0 \langle x,x_{0}\rangle=0

Polar_topology.html

  1. ๐’œ \mathcal{A}
  2. ๐’œ \mathcal{A}
  3. ( X , Y , โŸจ , โŸฉ ) (X,Y,\langle,\rangle)
  4. ( X , Y , โŸจ , โŸฉ ) (X,Y,\langle,\rangle)
  5. X X
  6. Y Y
  7. ๐”ฝ \mathbb{F}
  8. A โІ X A\subseteq X
  9. X X
  10. Y Y
  11. y โˆˆ Y y\in Y
  12. { โŸจ x , y โŸฉ ; x โˆˆ A } \{\langle x,y\rangle;x\in A\}
  13. ๐”ฝ \mathbb{F}
  14. โˆ€ y โˆˆ Y sup x โˆˆ A | โŸจ x , y โŸฉ | < โˆž . \forall y\in Y\qquad\sup_{x\in A}|\langle x,y\rangle|<\infty.
  15. A โˆ˜ A^{\circ}
  16. A A
  17. Y Y
  18. A โˆ˜ = { y โˆˆ Y : sup x โˆˆ A | โŸจ x , y โŸฉ | โ‰ค 1 } A^{\circ}=\{y\in Y:\quad\sup_{x\in A}|\langle x,y\rangle|\leq 1\}
  19. Y Y
  20. โ‹ƒ ฮป โˆˆ ๐”ฝ ฮป โ‹… A โˆ˜ = Y . \bigcup_{\lambda\in{\mathbb{F}}}\lambda\cdot A^{\circ}=Y.
  21. ๐’œ \mathcal{A}
  22. X X
  23. Y Y
  24. x x
  25. X X
  26. A โˆˆ ๐’œ A\in{\mathcal{A}}
  27. โˆ€ x โˆˆ X โˆƒ A โˆˆ ๐’œ x โˆˆ A , \forall x\in X\qquad\exists A\in{\mathcal{A}}\qquad x\in A,
  28. A โˆˆ ๐’œ A\in{\mathcal{A}}
  29. B โˆˆ ๐’œ B\in{\mathcal{A}}
  30. C โˆˆ ๐’œ C\in{\mathcal{A}}
  31. โˆ€ A , B โˆˆ ๐’œ โˆƒ C โˆˆ ๐’œ A โˆช B โІ C , \forall A,B\in{\mathcal{A}}\qquad\exists C\in{\mathcal{A}}\qquad A\cup B% \subseteq C,
  32. ๐’œ {\mathcal{A}}
  33. โˆ€ A โˆˆ ๐’œ โˆ€ ฮป โˆˆ ๐”ฝ ฮป โ‹… A โˆˆ ๐’œ . \forall A\in{\mathcal{A}}\qquad\forall\lambda\in{\mathbb{F}}\qquad\lambda\cdot A% \in{\mathcal{A}}.
  34. โˆฅ y โˆฅ A = sup x โˆˆ A | โŸจ x , y โŸฉ | , A โˆˆ ๐’œ , \|y\|_{A}=\sup_{x\in A}|\langle x,y\rangle|,\qquad A\in{\mathcal{A}},
  35. Y Y
  36. Y Y
  37. ๐’œ {\mathcal{A}}
  38. U B = { x โˆˆ V : โˆฅ ฯ† โˆฅ B < 1 } , B โˆˆ โ„ฌ , U_{B}=\{x\in V:\quad\|\varphi\|_{B}<1\},\qquad B\in{\mathcal{B}},
  39. y i โˆˆ Y y_{i}\in Y
  40. y โˆˆ Y y\in Y
  41. โˆ€ A โˆˆ ๐’œ โˆฅ y i - y โˆฅ A = sup x โˆˆ A | โŸจ x , y i โŸฉ - โŸจ x , y โŸฉ | โŸถ i โ†’ โˆž 0. \forall A\in{\mathcal{A}}\qquad\|y_{i}-y\|_{A}=\sup_{x\in A}|\langle x,y_{i}% \rangle-\langle x,y\rangle|\underset{i\to\infty}{\longrightarrow}0.
  42. ๐’œ \mathcal{A}
  43. โˆฅ y โˆฅ A \|y\|_{A}
  44. A โˆ˜ A^{\circ}
  45. ๐’œ \mathcal{A}
  46. X X
  47. Y Y
  48. ๐’œ \mathcal{A}
  49. X X
  50. Y Y
  51. X X
  52. X X
  53. ๐’œ \mathcal{A}
  54. A โІ X โ€ฒ A\subseteq X^{\prime}
  55. X โ€ฒ X^{\prime}
  56. A โˆˆ ๐’œ A\in{\mathcal{A}}
  57. A โІ X โ€ฒ A\subseteq X^{\prime}
  58. U โІ X U\subseteq X
  59. sup x โˆˆ U , f โˆˆ A | f ( x ) | < โˆž \sup_{x\in U,f\in A}|f(x)|<\infty

Polarimetry.html

  1. [ ฮฑ ] ฮป T = 100 ฮฑ / l ฯ [\alpha]_{\lambda}^{T}=100\alpha/l\rho\,\!
  2. ฯ \rho

Polarizability.html

  1. ฮฑ \alpha
  2. s y m b o l p symbol{p}
  3. s y m b o l E symbol{E}
  4. s y m b o l p = ฮฑ s y m b o l E symbol{p}=\alpha symbol{E}
  5. ฮฑ ( cm 3 ) = 10 6 4 ฯ€ ฮต 0 ฮฑ ( C โ‹… m 2 โ‹… V - 1 ) = 10 6 4 ฯ€ ฮต 0 ฮฑ ( F โ‹… m 2 ) \alpha(\mathrm{cm}^{3})=\frac{10^{6}}{4\pi\varepsilon_{0}}\alpha(\mathrm{C}% \cdot\mathrm{m}^{2}\cdot\mathrm{V}^{-1})=\frac{10^{6}}{4\pi\varepsilon_{0}}% \alpha(\mathrm{F}\cdot\mathrm{m}^{2})
  6. ฮฑ ( F โ‹… m 2 ) \alpha(\mathrm{F}\cdot\mathrm{m}^{2})
  7. ฮต 0 \varepsilon_{0}
  8. ฮฑ โ€ฒ \alpha^{\prime}
  9. 4 ฯ€ ฮต 0 ฮฑ โ€ฒ = ฮฑ 4\pi\varepsilon_{0}\alpha^{\prime}=\alpha
  10. ฮฑ \alpha
  11. x x
  12. x x
  13. s y m b o l p symbol{p}
  14. x x
  15. y y
  16. z z
  17. s y m b o l p symbol{p}
  18. ฮฑ \alpha
  19. 3 ร— 3 3\times 3
  20. s y m b o l p symbol{p}
  21. < b i g > S โ‰ฅ 3 / 2 <big>Sโ‰ฅ{3}/{2}

Pollard's_rho_algorithm_for_logarithms.html

  1. ฮณ \gamma
  2. ฮฑ ฮณ = ฮฒ \alpha^{\gamma}=\beta
  3. ฮฒ \beta
  4. G G
  5. ฮฑ \alpha
  6. a a
  7. b b
  8. A A
  9. B B
  10. ฮฑ a ฮฒ b = ฮฑ A ฮฒ B \alpha^{a}\beta^{b}=\alpha^{A}\beta^{B}
  11. n n
  12. ฮณ \gamma
  13. ( B - b ) ฮณ = ( a - A ) ( mod n ) (B-b)\gamma=(a-A)\;\;(\mathop{{\rm mod}}n)
  14. a a
  15. b b
  16. A A
  17. B B
  18. x i = ฮฑ a i ฮฒ b i x_{i}=\alpha^{a_{i}}\beta^{b_{i}}
  19. f : x i โ†ฆ x i + 1 f:x_{i}\mapsto x_{i+1}
  20. ฯ€ n 2 \sqrt{\frac{\pi n}{2}}
  21. G G
  22. S 0 S_{0}
  23. S 1 S_{1}
  24. S 2 S_{2}
  25. x i x_{i}
  26. S 0 S_{0}
  27. a a
  28. b b
  29. x i โˆˆ S 1 x_{i}\in S_{1}
  30. a a
  31. x i โˆˆ S 2 x_{i}\in S_{2}
  32. b b
  33. G G
  34. p p
  35. ฮฑ , ฮฒ โˆˆ G \alpha,\beta\in G
  36. G = S 0 โˆช S 1 โˆช S 2 G=S_{0}\cup S_{1}\cup S_{2}
  37. f : G โ†’ G f:G\to G
  38. f ( x ) = { ฮฒ x x โˆˆ S 0 x 2 x โˆˆ S 1 ฮฑ x x โˆˆ S 2 f(x)=\left\{\begin{matrix}\beta x&x\in S_{0}\\ x^{2}&x\in S_{1}\\ \alpha x&x\in S_{2}\end{matrix}\right.
  39. g : G ร— โ„ค โ†’ โ„ค g:G\times\mathbb{Z}\to\mathbb{Z}
  40. h : G ร— โ„ค โ†’ โ„ค h:G\times\mathbb{Z}\to\mathbb{Z}
  41. g ( x , n ) = { n x โˆˆ S 0 2 n mod p x โˆˆ S 1 n + 1 mod p x โˆˆ S 2 g(x,n)=\left\{\begin{matrix}n&x\in S_{0}\\ 2n\ (\bmod\ p)&x\in S_{1}\\ n+1\ (\bmod\ p)&x\in S_{2}\end{matrix}\right.
  42. h ( x , n ) = { n + 1 mod p x โˆˆ S 0 2 n mod p x โˆˆ S 1 n x โˆˆ S 2 h(x,n)=\left\{\begin{matrix}n+1\ (\bmod\ p)&x\in S_{0}\\ 2n\ (\bmod\ p)&x\in S_{1}\\ n&x\in S_{2}\end{matrix}\right.
  43. N = 1019 N=1019
  44. n = 1018 n=1018

Polydisc.html

  1. D ( z , r ) D(z,r)
  2. D ( z 1 , r 1 ) ร— โ€ฆ ร— D ( z n , r n ) . D(z_{1},r_{1})\times\dots\times D(z_{n},r_{n}).
  3. { w = ( w 1 , w 2 , โ€ฆ , w n ) โˆˆ ๐‚ n โˆฃ | z k - w k | < r k , for all k = 1 , โ€ฆ , n } . \{w=(w_{1},w_{2},\dots,w_{n})\in{\mathbf{C}}^{n}\mid|z_{k}-w_{k}|<r_{k},\mbox{% for all }~{}k=1,\dots,n\}.
  4. { w โˆˆ ๐‚ n โˆฃ โˆฅ z - w โˆฅ < r } . \{w\in\mathbf{C}^{n}\mid\lVert z-w\rVert<r\}.
  5. n > 1 n>1
  6. n = 2 n=2

Polymer_solar_cell.html

  1. ฮท = V O C ร— J S C ร— F F P i n \eta=\frac{V_{OC}\times J_{SC}\times FF}{P_{in}}

Position_(vector).html

  1. r = O P โ†’ . {r}=\overrightarrow{OP}.
  2. r ( t ) โ‰ก r ( x , y , z ) โ‰ก x ( t ) e ^ x + y ( t ) e ^ y + z ( t ) e ^ z โ‰ก r ( r , ฮธ , ฯ• ) โ‰ก r ( t ) e ^ r ( ฮธ ( t ) , ฯ• ( t ) ) โ‰ก r ( r , ฮธ , z ) โ‰ก r ( t ) e ^ r ( ฮธ ( t ) ) + z ( t ) e ^ z โ‹ฏ \begin{aligned}\displaystyle{r}(t)&\displaystyle\equiv{r}\left(x,y,z\right)% \equiv x(t){\hat{e}}_{x}+y(t){\hat{e}}_{y}+z(t){\hat{e}}_{z}\\ &\displaystyle\equiv{r}\left(r,\theta,\phi\right)\equiv r(t){\hat{e}}_{r}(% \theta(t),\phi(t))\\ &\displaystyle\equiv{r}\left(r,\theta,z\right)\equiv r(t){\hat{e}}_{r}(\theta(% t))+z(t){\hat{e}}_{z}\\ &\displaystyle\,\!\cdots\\ \end{aligned}
  3. r = โˆ‘ i = 1 n x i e i = x 1 e 1 + x 2 e 2 + โ‹ฏ x n e n {r}=\sum_{i=1}^{n}x_{i}{e}_{i}=x_{1}{e}_{1}+x_{2}{e}_{2}+\cdots x_{n}{e}_{n}
  4. x โ‰ก x โ‰ก x ( t ) , r โ‰ก r ( t ) , s โ‰ก s ( t ) โ‹ฏ {x}\equiv x\equiv x(t),\quad r\equiv r(t),\quad s\equiv s(t)\cdots\,\!
  5. v = d r d t {v}=\frac{\mathrm{d}{r}}{\mathrm{d}t}
  6. a = d v d t = d 2 r d t 2 {a}=\frac{\mathrm{d}{v}}{\mathrm{d}t}=\frac{\mathrm{d}^{2}{r}}{\mathrm{d}t^{2}}
  7. j = d a d t = d 2 v d t 2 = d 3 r d t 3 {j}=\frac{\mathrm{d}{a}}{\mathrm{d}t}=\frac{\mathrm{d}^{2}{v}}{\mathrm{d}t^{2}% }=\frac{\mathrm{d}^{3}{r}}{\mathrm{d}t^{3}}

Positive_and_negative_predictive_values.html

  1. PPV = number of true positives number of true positives + number of false positives = number of true positives number of positive calls \,\text{PPV}=\frac{\,\text{number of true positives}}{\,\text{number of true % positives}+\,\text{number of false positives}}=\frac{\,\text{number of true % positives}}{\,\text{number of positive calls}}
  2. PPV = sensitivity ร— prevalence sensitivity ร— prevalence + ( 1 - specificity ) ร— ( 1 - prevalence ) \,\text{PPV}=\frac{\,\text{sensitivity}\times\,\text{prevalence}}{\,\text{% sensitivity}\times\,\text{prevalence}+(1-\,\text{specificity})\times(1-\,\text% {prevalence})}
  3. FDR = 1 - PPV = number of false positives number of true positives + number of false positives = number of false positives number of positive calls \,\text{FDR}=1-\,\text{PPV}=\frac{\,\text{number of false positives}}{\,\text{% number of true positives}+\,\text{number of false positives}}=\frac{\,\text{% number of false positives}}{\,\text{number of positive calls}}
  4. NPV = number of true negatives number of true negatives + number of false negatives = number of true negatives number of negative calls \,\text{NPV}=\frac{\,\text{number of true negatives}}{\,\text{number of true % negatives}+\,\text{number of false negatives}}=\frac{\,\text{number of true % negatives}}{\,\text{number of negative calls}}
  5. NPV = specificity ร— ( 1 - prevalence ) ( 1 - sensitivity ) ร— prevalence + specificity ร— ( 1 - prevalence ) \,\text{NPV}=\frac{\,\text{specificity}\times(1-\,\text{prevalence})}{(1-\,% \text{sensitivity})\times\,\text{prevalence}+\,\text{specificity}\times(1-\,% \text{prevalence})}
  6. FOR = 1 - NPV = number of false negatives number of true negatives + number of false negatives = number of false negatives number of negative calls \,\text{FOR}=1-\,\text{NPV}=\frac{\,\text{number of false negatives}}{\,\text{% number of true negatives}+\,\text{number of false negatives}}=\frac{\,\text{% number of false negatives}}{\,\text{number of negative calls}}

Possibility_theory.html

  1. pos \operatorname{pos}
  2. 2 ฮฉ 2^{\Omega}
  3. pos ( โˆ… ) = 0 \operatorname{pos}(\varnothing)=0
  4. pos ( ฮฉ ) = 1 \operatorname{pos}(\Omega)=1
  5. pos ( U โˆช V ) = max ( pos ( U ) , pos ( V ) ) \operatorname{pos}(U\cup V)=\max\left(\operatorname{pos}(U),\operatorname{pos}% (V)\right)
  6. U U
  7. V V
  8. pos ( U ) = max ฯ‰ โˆˆ U pos ( { ฯ‰ } ) \operatorname{pos}(U)=\max_{\omega\in U}\operatorname{pos}(\{\omega\})
  9. pos \operatorname{pos}
  10. pos ( U โˆช V ) = max ( pos ( U ) , pos ( V ) ) \operatorname{pos}(U\cup V)=\max\left(\operatorname{pos}(U),\operatorname{pos}% (V)\right)
  11. U U
  12. V V
  13. pos ( U โˆฉ V ) โ‰ค min ( pos ( U ) , pos ( V ) ) \operatorname{pos}(U\cap V)\leq\min\left(\operatorname{pos}(U),\operatorname{% pos}(V)\right)
  14. I I
  15. U i , i โˆˆ I U_{i,\,i\in I}
  16. pos ( โˆช i โˆˆ I U i ) = sup i โˆˆ I pos ( U i ) . \operatorname{pos}\left(\cup_{i\in I}U_{i}\right)=\sup_{i\in I}\operatorname{% pos}(U_{i}).
  17. U U
  18. nec ( U ) = 1 - pos ( U ยฏ ) \operatorname{nec}(U)=1-\operatorname{pos}(\overline{U})
  19. U ยฏ \overline{U}
  20. U U
  21. ฮฉ \Omega
  22. U U
  23. nec ( U ) โ‰ค pos ( U ) \operatorname{nec}(U)\leq\operatorname{pos}(U)
  24. U U
  25. nec ( U โˆฉ V ) = min ( nec ( U ) , nec ( V ) ) \operatorname{nec}(U\cap V)=\min(\operatorname{nec}(U),\operatorname{nec}(V))
  26. U U
  27. pos ( U ) + pos ( U ยฏ ) โ‰ฅ 1 \operatorname{pos}(U)+\operatorname{pos}(\overline{U})\geq 1
  28. U U
  29. pos ( U ) = 1 \operatorname{pos}(U)=1
  30. nec ( U ) = 0 \operatorname{nec}(U)=0
  31. nec ( U ) = 1 \operatorname{nec}(U)=1
  32. U U
  33. U U
  34. pos ( U ) = 1 \operatorname{pos}(U)=1
  35. pos ( U ) = 0 \operatorname{pos}(U)=0
  36. U U
  37. U U
  38. nec ( U ) = 0 \operatorname{nec}(U)=0
  39. pos ( U ) = 1 \operatorname{pos}(U)=1
  40. U U
  41. U U
  42. nec ( U ) \operatorname{nec}(U)
  43. nec ( U ) = 0 \operatorname{nec}(U)=0
  44. U U
  45. U U
  46. pos ( U ) \operatorname{pos}(U)
  47. nec ( U ) = 0 \operatorname{nec}(U)=0
  48. pos ( U ) = 1 \operatorname{pos}(U)=1
  49. U U
  50. { p : โˆ€ S p ( S ) โ‰ค pos ( S ) } . \left\{\,p:\forall S\ p(S)\leq\operatorname{pos}(S)\,\right\}.

Posterior_ischemic_optic_neuropathy.html

  1. O p t i c n e r v e p e r f u s i o n = H c t ร— ( B P - T P ) Opticnerveperfusion=Hct\times(BP-TP)

Postulates_of_special_relativity.html

  1. ( x 1 , x 2 , x 3 , t ) (x_{1},x_{2},x_{3},t)
  2. ( p 1 , p 2 , p 3 , E ) (p_{1},p_{2},p_{3},E)
  3. ( E 1 , E 2 , E 3 , B 1 , B 2 , B 3 ) (E_{1},E_{2},E_{3},B_{1},B_{2},B_{3})
  4. ( x 1 , x 2 , x 3 , t ) (x_{1},x_{2},x_{3},t)
  5. ( p 1 , p 2 , p 3 , E ) (p_{1},p_{2},p_{3},E)
  6. 0 < c < โˆž 0<c<\infty
  7. ( x 1 , x 2 , x 3 , t ) (x_{1},x_{2},x_{3},t)
  8. ( y 1 , y 2 , y 3 , s ) (y_{1},y_{2},y_{3},s)
  9. F F
  10. ( x 1 โ€ฒ , x 2 โ€ฒ , x 3 โ€ฒ , t โ€ฒ ) (x^{\prime}_{1},x^{\prime}_{2},x^{\prime}_{3},t^{\prime})
  11. ( y 1 โ€ฒ , y 2 โ€ฒ , y 3 โ€ฒ , s โ€ฒ ) (y^{\prime}_{1},y^{\prime}_{2},y^{\prime}_{3},s^{\prime})
  12. F โ€ฒ F^{\prime}
  13. ( x 1 - y 1 ) 2 + ( x 2 - y 2 ) 2 + ( x 3 - y 3 ) 2 = c ( s - t ) \sqrt{(x_{1}-y_{1})^{2}+(x_{2}-y_{2})^{2}+(x_{3}-y_{3})^{2}}=c(s-t)\quad
  14. ( x 1 โ€ฒ - y 1 โ€ฒ ) 2 + ( x 2 โ€ฒ - y 2 โ€ฒ ) 2 + ( x 3 โ€ฒ - y 3 โ€ฒ ) 2 = c ( s โ€ฒ - t โ€ฒ ) \quad\sqrt{(x^{\prime}_{1}-y^{\prime}_{1})^{2}+(x^{\prime}_{2}-y^{\prime}_{2})% ^{2}+(x^{\prime}_{3}-y^{\prime}_{3})^{2}}=c(s^{\prime}-t^{\prime})
  15. c 2 ( s - t ) 2 - ( x 1 - y 1 ) 2 - ( x 2 - y 2 ) 2 - ( x 3 - y 3 ) 2 c^{2}(s-t)^{2}-(x_{1}-y_{1})^{2}-(x_{2}-y_{2})^{2}-(x_{3}-y_{3})^{2}
  16. = c 2 ( s โ€ฒ - t โ€ฒ ) 2 - ( x 1 โ€ฒ - y 1 โ€ฒ ) 2 - ( x 2 โ€ฒ - y 2 โ€ฒ ) 2 - ( x 3 โ€ฒ - y 3 โ€ฒ ) 2 =c^{2}(s^{\prime}-t^{\prime})^{2}-(x^{\prime}_{1}-y^{\prime}_{1})^{2}-(x^{% \prime}_{2}-y^{\prime}_{2})^{2}-(x^{\prime}_{3}-y^{\prime}_{3})^{2}
  17. c โ†’ โˆž c\to\infty
  18. ( x 1 , x 2 , x 3 , t ) (x_{1},x_{2},x_{3},t)
  19. ( y 1 , y 2 , y 3 , s ) (y_{1},y_{2},y_{3},s)
  20. F F
  21. ( x 1 โ€ฒ , x 2 โ€ฒ , x 3 โ€ฒ , t โ€ฒ ) (x^{\prime}_{1},x^{\prime}_{2},x^{\prime}_{3},t^{\prime})
  22. ( y 1 โ€ฒ , y 2 โ€ฒ , y 3 โ€ฒ , s โ€ฒ ) (y^{\prime}_{1},y^{\prime}_{2},y^{\prime}_{3},s^{\prime})
  23. F โ€ฒ F^{\prime}
  24. s - t = s โ€ฒ - t โ€ฒ s-t=s^{\prime}-t^{\prime}
  25. s - t = s โ€ฒ - t โ€ฒ = 0 s-t=s^{\prime}-t^{\prime}=0
  26. ( x 1 - y 1 ) 2 + ( x 2 - y 2 ) 2 + ( x 3 - y 3 ) 2 \quad\sqrt{(x_{1}-y_{1})^{2}+(x_{2}-y_{2})^{2}+(x_{3}-y_{3})^{2}}
  27. = ( x 1 โ€ฒ - y 1 โ€ฒ ) 2 + ( x 2 โ€ฒ - y 2 โ€ฒ ) 2 + ( x 3 โ€ฒ - y 3 โ€ฒ ) 2 =\sqrt{(x^{\prime}_{1}-y^{\prime}_{1})^{2}+(x^{\prime}_{2}-y^{\prime}_{2})^{2}% +(x^{\prime}_{3}-y^{\prime}_{3})^{2}}
  28. E = m c 2 E=mc^{2}

Potential_theory.html

  1. n n
  2. n n
  3. n n

Power_closed.html

  1. G G
  2. H H
  3. G G
  4. p k p^{k}
  5. p k p^{k}
  6. p k p^{k}
  7. p k p^{k}

Powerful_p-group.html

  1. G G
  2. [ G , G ] [G,G]
  3. G p = โŸจ g p | g โˆˆ G โŸฉ G^{p}=\langle g^{p}|g\in G\rangle
  4. p p
  5. [ G , G ] [G,G]
  6. G 4 G^{4}
  7. G G
  8. ฮฆ ( G ) \Phi(G)
  9. G G
  10. ฮฆ ( G ) = G p . \Phi(G)=G^{p}.
  11. G p k = { g p k | g โˆˆ G } G^{p^{k}}=\{g^{p^{k}}|g\in G\}
  12. k โ‰ฅ 1. k\geq 1.
  13. p p
  14. p p
  15. G = โŸจ g 1 , โ€ฆ , g d โŸฉ G=\langle g_{1},\ldots,g_{d}\rangle
  16. G p k = โŸจ g 1 p k , โ€ฆ , g d p k โŸฉ G^{p^{k}}=\langle g_{1}^{p^{k}},\ldots,g_{d}^{p^{k}}\rangle
  17. k โ‰ฅ 1. k\geq 1.
  18. k k
  19. G G
  20. ฮณ k ( G ) โ‰ค G p k - 1 \gamma_{k}(G)\leq G^{p^{k-1}}
  21. k โ‰ฅ 1. k\geq 1.
  22. G G
  23. G . G.
  24. G G
  25. G p k G^{p^{k}}
  26. G G

Pp-wave_spacetime.html

  1. d s 2 = H ( u , x , y ) d u 2 + 2 d u d v + d x 2 + d y 2 ds^{2}=H(u,x,y)\,du^{2}+2\,du\,dv+dx^{2}+dy^{2}
  2. H H
  3. k k
  4. k k
  5. โˆ‡ k = 0. \nabla k=0.
  6. k = โˆ‚ v k=\partial_{v}
  7. v = v 0 v=v_{0}
  8. k k
  9. k a ; b = 0 k_{a;b}=0
  10. ฮฆ i j \Phi_{ij}
  11. ฮจ i \Psi_{i}
  12. โ„“ โ†’ = โˆ‚ u - H / 2 โˆ‚ v \vec{\ell}=\partial_{u}-H/2\,\partial_{v}
  13. n โ†’ = โˆ‚ v \vec{n}=\partial_{v}
  14. m โ†’ = 1 2 ( โˆ‚ x + i โˆ‚ y ) \vec{m}=\frac{1}{\sqrt{2}}\,\left(\partial_{x}+i\,\partial_{y}\right)
  15. ฮฆ 00 = 1 4 ( H x x + H y y ) \Phi_{00}=\frac{1}{4}\,\left(H_{xx}+H_{yy}\right)
  16. ฮจ 0 = 1 4 ( ( H x x - H y y ) + 2 i H x y ) . \Psi_{0}=\frac{1}{4}\,\left(\left(H_{xx}-H_{yy}\right)+2i\,H_{xy}\right).
  17. k = โˆ‚ v k=\partial_{v}
  18. k k
  19. H H
  20. x , y x,y
  21. โˆ‚ v \partial_{v}
  22. k = โˆ‚ v k=\partial_{v}
  23. H H
  24. H H
  25. H ( u , x , y ) = a ( u ) ( x 2 - y 2 ) + 2 b ( u ) x y + c ( u ) ( x 2 + y 2 ) H(u,x,y)=a(u)\,(x^{2}-y^{2})+2\,b(u)\,xy+c(u)\,(x^{2}+y^{2})
  26. a , b , c a,b,c
  27. u u
  28. a , b a,b
  29. c c
  30. c = 0 c=0
  31. X X
  32. X = โˆ‚ v X=\partial_{v}
  33. X = โˆ‚ โˆ‚ u ( p x + q y ) โˆ‚ v + p โˆ‚ x + q โˆ‚ y X=\frac{\partial}{\partial u}(px+qy)\,\partial_{v}+p\,\partial_{x}+q\,\partial% _{y}
  34. p ยจ = - a p + b q - c p \ddot{p}=-ap+bq-cp
  35. q ยจ = a q - b p - c q . \ddot{q}=aq-bp-cq.
  36. u 1 < u < u 2 u_{1}<u<u_{2}
  37. โˆ‚ v \partial_{v}
  38. H 1 , H 2 H_{1},H_{2}
  39. H 1 + H 2 H_{1}+H_{2}
  40. S 3 S^{3}

Practical_number.html

  1. n = p 1 ฮฑ 1 โ€ฆ p k ฮฑ k n=p_{1}^{\alpha_{1}}...p_{k}^{\alpha_{k}}
  2. n > 1 n>1
  3. p 1 < p 2 < โ€ฆ < p k p_{1}<p_{2}<\dots<p_{k}
  4. p 1 = 2 p_{1}=2
  5. p i โ‰ค 1 + ฯƒ ( p 1 ฮฑ 1 โ€ฆ p i - 1 ฮฑ i - 1 ) = 1 + โˆ j = 1 i - 1 p j ฮฑ j + 1 - 1 p j - 1 , p_{i}\leq 1+\sigma(p_{1}^{\alpha_{1}}\dots p_{i-1}^{\alpha_{i-1}})=1+\prod_{j=% 1}^{i-1}\frac{p_{j}^{\alpha_{j}+1}-1}{p_{j}-1},
  6. ฯƒ ( x ) \sigma(x)
  7. p i - 1 p_{i}-1
  8. m โ‰ค ฯƒ ( n ) m\leq\sigma(n)
  9. q = min { โŒŠ m / p k ฮฑ k โŒ‹ , ฯƒ ( n / p k ฮฑ k ) } q=\min\{\lfloor m/p_{k}^{\alpha_{k}}\rfloor,\sigma(n/p_{k}^{\alpha_{k}})\}
  10. r = m - q p k ฯƒ k r=m-qp_{k}^{\sigma_{k}}
  11. q โ‰ค ฯƒ ( n / p k ฮฑ k ) q\leq\sigma(n/p_{k}^{\alpha_{k}})
  12. n / p k ฮฑ k n/p_{k}^{\alpha_{k}}
  13. n / p k ฮฑ k n/p_{k}^{\alpha_{k}}
  14. r โ‰ค ฯƒ ( n ) - p k ฮฑ k ฯƒ ( n / p k ฮฑ k ) = ฯƒ ( n / p k ) r\leq\sigma(n)-p_{k}^{\alpha_{k}}\sigma(n/p_{k}^{\alpha_{k}})=\sigma(n/p_{k})
  15. n / p k n/p_{k}
  16. n / p k n/p_{k}
  17. p k ฮฑ k p_{k}^{\alpha_{k}}
  18. 13 20 = 10 20 + 2 20 + 1 20 = 1 2 + 1 10 + 1 20 . \frac{13}{20}=\frac{10}{20}+\frac{2}{20}+\frac{1}{20}=\frac{1}{2}+\frac{1}{10}% +\frac{1}{20}.
  19. O ( log y ) \scriptstyle O(\sqrt{\log y})
  20. O ( log n i - 1 ) \scriptstyle O(\sqrt{\log n_{i-1}})
  21. x y = q n i + r y n i \scriptstyle\frac{x}{y}=\frac{q}{n_{i}}+\frac{r}{yn_{i}}
  22. O ( y log 2 y log log y ) \scriptstyle O(\frac{y\log^{2}y}{\log\log y})
  23. c 1 x log x < p ( x ) < c 2 x log x , c_{1}\frac{x}{\log x}<p(x)<c_{2}\frac{x}{\log x},

Precipitation_hardening.html

  1. A = 2 ฯ€ r b A=2\pi rb\,\!
  2. E = 2 ฯ€ r b ฮณ s E=2\pi rb\gamma_{s}\,\!
  3. ฮณ s \gamma_{s}\,\!
  4. ฯ„ = ฯ€ r ฮณ b L \tau=\frac{\pi r\gamma}{bL}\,\!
  5. ฯ„ \tau
  6. r r
  7. ฮณ \gamma
  8. b b
  9. L L
  10. r r
  11. ฯ„ = G b L - 2 r \tau=\frac{Gb}{L-2r}\,\!
  12. ฯ„ \tau
  13. G G
  14. b b
  15. L L
  16. r r

Predicate_transformer_semantics.html

  1. w p ( S , R ) wp(S,R)
  2. { P } S { Q } \{P\}S\{Q\}
  3. โˆ€ x , P โ‡’ w p ( S , Q ) \forall x,P\Rightarrow wp(S,Q)
  4. w p ( ๐ฌ๐ค๐ข๐ฉ , R ) = R wp(\,\textbf{skip},R)\ =\ R
  5. w p ( ๐š๐›๐จ๐ซ๐ญ , R ) = ๐Ÿ๐š๐ฅ๐ฌ๐ž wp(\,\textbf{abort},R)\ =\ \,\textbf{false}
  6. R [ x โ† E ] R[x\leftarrow E]
  7. w p ( x := E , R ) = โˆ€ y , y = E โ‡’ R [ x โ† y ] wp(x:=E,R)\ =\ \forall y,y=E\Rightarrow R[x\leftarrow y]
  8. w p ( x := E , R ) = R [ x โ† E ] wp(x:=E,R)\ =\ R[x\leftarrow E]
  9. w p ( x := x - 5 , x > 10 ) = x - 5 > 10 โ‡” x > 15 \begin{array}[]{rcl}wp(x:=x-5,x>10)&=&x-5>10\\ &\Leftrightarrow&x>15\end{array}
  10. w p ( S 1 ; S 2 , R ) = w p ( S 1 , w p ( S 2 , R ) ) wp(S_{1};S_{2},R)\ =\ wp(S_{1},wp(S_{2},R))
  11. w p ( x := x - 5 ; x := x * 2 , x > 20 ) = w p ( x := x - 5 , w p ( x := x * 2 , x > 20 ) ) = w p ( x := x - 5 , x * 2 > 20 ) = ( x - 5 ) * 2 > 20 = x > 15 \begin{array}[t]{rcl}wp(x:=x-5;x:=x*2\ ,\ x>20)&=&wp(x:=x-5,wp(x:=x*2,x>20))\\ &=&wp(x:=x-5,x*2>20)\\ &=&(x-5)*2>20\\ &=&x>15\end{array}
  12. w p ( ๐ข๐Ÿ E ๐ญ๐ก๐ž๐ง S 1 ๐ž๐ฅ๐ฌ๐ž S 2 ๐ž๐ง๐ , R ) = ( E โ‡’ w p ( S 1 , R ) ) โˆง ( ยฌ E โ‡’ w p ( S 2 , R ) ) wp(\,\textbf{if}\ E\ \,\textbf{then}\ S_{1}\ \,\textbf{else}\ S_{2}\ \,\textbf% {end},R)\ =\ (E\Rightarrow wp(S_{1},R))\wedge(\neg E\Rightarrow wp(S_{2},R))
  13. w p ( ๐ข๐Ÿ x < y ๐ญ๐ก๐ž๐ง x := y ๐ž๐ฅ๐ฌ๐ž ๐ฌ๐ค๐ข๐ฉ ๐ž๐ง๐ , x โ‰ฅ y ) = ( x < y โ‡’ w p ( x := y , x โ‰ฅ y ) ) โˆง ( ยฌ ( x < y ) โ‡’ w p ( ๐ฌ๐ค๐ข๐ฉ , x โ‰ฅ y ) ) = ( x < y โ‡’ y โ‰ฅ y ) โˆง ( ยฌ ( x < y ) โ‡’ x โ‰ฅ y ) โ‡” ๐ญ๐ซ๐ฎ๐ž \begin{array}[t]{rcl}wp(\,\textbf{if}\ x<y\ \,\textbf{then}\ x:=y\ \,\textbf{% else}\;\;\,\textbf{skip}\;\;\,\textbf{end},\ x\geq y)&=&(x<y\Rightarrow wp(x:=% y,x\geq y))\ \wedge\ (\neg(x<y)\Rightarrow wp(\,\textbf{skip},x\geq y))\\ &=&(x<y\Rightarrow y\geq y)\ \wedge\ (\neg(x<y)\Rightarrow x\geq y)\\ &\Leftrightarrow&\,\textbf{true}\end{array}
  14. w l p ( ๐ฐ๐ก๐ข๐ฅ๐ž E ๐๐จ S ๐๐จ๐ง๐ž , R ) = I โˆง ( ( E โˆง I ) โ‡’ w p ( S , I ) ) โˆง ( ( ยฌ E โˆง I ) โ‡’ R ) wlp(\,\textbf{while}\ E\ \,\textbf{do}\ S\ \,\textbf{done},R)\ =\ I\wedge\ ((E% \wedge I)\Rightarrow wp(S,I))\wedge\ ((\neg E\wedge I)\Rightarrow R)
  15. w l p ( ๐ฐ๐ก๐ข๐ฅ๐ž E ๐๐จ S ๐๐จ๐ง๐ž , R ) = I โˆง ( E โ‡’ w p ( S , I ) ) โˆง ( ยฌ E โ‡’ R ) wlp(\,\textbf{while}\ E\ \,\textbf{do}\ S\ \,\textbf{done},R)\ =\ I\wedge\ (E% \Rightarrow wp(S,I))\wedge\ (\neg E\Rightarrow R)
  16. w p ( ๐ฐ๐ก๐ข๐ฅ๐ž E ๐๐จ S ๐๐จ๐ง๐ž , R ) = I โˆง โˆ€ y , ( ( E โˆง I ) โ‡’ w p ( S , I โˆง x < y ) ) [ x โ† y ] โˆง โˆ€ y , ( ( ยฌ E โˆง I ) โ‡’ R ) [ x โ† y ] wp(\,\textbf{while}\ E\ \,\textbf{do}\ S\ \,\textbf{done},R)\ =\ \begin{array}% [t]{l}I\\ \wedge\ \forall y,((E\wedge I)\Rightarrow wp(S,I\wedge x<y))[x\leftarrow y]\\ \wedge\ \forall y,((\neg E\wedge I)\Rightarrow R)[x\leftarrow y]\end{array}
  17. ( A k ) k โˆˆ โ„• (A_{k})_{k\in\mathbb{N}}
  18. A k A_{k}
  19. A 0 = โˆ… A k + 1 = { y โˆˆ U | ( ( E โ‡’ w p ( S , x โˆˆ A k ) ) โˆง ( ยฌ E โ‡’ R ) ) [ x โ† y ] } \begin{array}[]{rcl}A_{0}&=&\emptyset\\ A_{k+1}&=&\left\{\ y\in U\ |\ ((E\Rightarrow wp(S,x\in A_{k}))\wedge(\neg E% \Rightarrow R))[x\leftarrow y]\ \right\}\\ \end{array}
  20. โˆƒ k , x โˆˆ A k \exists k,x\in A_{k}
  21. y < z y<z
  22. โˆƒ i , y โˆˆ A i โˆง ( โˆ€ j , z โˆˆ A j โ‡’ i < j ) \exists i,y\in A_{i}\wedge(\forall j,z\in A_{j}\Rightarrow i<j)
  23. w p ( ๐ฐ๐ก๐ข๐ฅ๐ž E ๐๐จ S ๐๐จ๐ง๐ž , R ) wp(\,\textbf{while}\ E\ \,\textbf{do}\ S\ \,\textbf{done},R)
  24. โˆƒ k , x โˆˆ A k \exists k,x\in A_{k}
  25. w p ( ๐ข๐Ÿ E 1 โ†’ S 1 [ ] โ€ฆ [ ] E n โ†’ S n ๐Ÿ๐ข , R ) = ( E 1 โˆจ โ€ฆ โˆจ E n ) โˆง ( E 1 โ‡’ w p ( S 1 , R ) ) โ€ฆ โˆง ( E n โ‡’ w p ( S n , R ) ) wp(\mathbf{if}\ E_{1}\rightarrow S_{1}\ [\!]\ \ldots\ [\!]\ E_{n}\rightarrow S% _{n}\ \mathbf{fi},R)\ =\begin{array}[t]{l}(E_{1}\vee\ldots\vee E_{n})\\ \wedge\ (E_{1}\Rightarrow wp(S_{1},R))\\ \ldots\\ \wedge\ (E_{n}\Rightarrow wp(S_{n},R))\\ \end{array}
  26. E i E_{i}
  27. E j E_{j}
  28. S i S_{i}
  29. S j S_{j}
  30. P | P\ |
  31. y . Q โ†’ x := y y.\ Q\rightarrow x:=y
  32. w p ( P | wp(P\ |
  33. y . Q โ†’ x := y , R ) = P โˆง โˆ€ z , Q [ y โ† z ] โ‡’ R [ x โ† z ] y.\ Q\rightarrow x:=y\ ,\ R)\ =\ P\wedge\forall z,Q[y\leftarrow z]\Rightarrow R% [x\leftarrow z]
  34. โˆ€ x 0 , ( P โ‡’ w p ( S , Q [ x โ† x 0 ] [ y โ† x ] ) ) [ x โ† x 0 ] \forall x_{0},(P\Rightarrow wp(S,Q[x\leftarrow x_{0}][y\leftarrow x]))[x% \leftarrow x_{0}]
  35. x 0 x_{0}
  36. w p ( P | wp(P\ |
  37. y . Q โ†’ x := y , R ) โ‡’ w p ( S , R ) y.\ Q\rightarrow x:=y\ ,\ R)\Rightarrow wp(S,R)
  38. w l p ( S , R ) wlp(S,R)
  39. s p ( S , R ) sp(S,R)
  40. { P } S { Q } \{P\}S\{Q\}
  41. โˆ€ x , s p ( S , P ) โ‡’ Q \forall x,sp(S,P)\Rightarrow Q
  42. ( โˆ€ x , P โ‡’ w l p ( S , Q ) ) โ‡” ( โˆ€ x , s p ( S , P ) โ‡’ Q ) (\forall x,P\Rightarrow wlp(S,Q))\ \Leftrightarrow\ (\forall x,sp(S,P)% \Rightarrow Q)
  43. s p ( x := E , R ) = โˆƒ y , x = E [ x โ† y ] โˆง R [ x โ† y ] sp(x:=E,R)\ =\ \exists y,x=E[x\leftarrow y]\wedge R[x\leftarrow y]
  44. s p ( x := x - 5 , x > 15 ) = โˆƒ y , x = y - 5 โˆง y > 15 โ‡” x > 10 sp(x:=x-5,x>15)\ =\ \exists y,x=y-5\wedge y>15\ \Leftrightarrow\ x>10
  45. s p ( S 1 ; S 2 , R ) = s p ( S 2 , s p ( S 1 , R ) ) sp(S_{1};S_{2}\ ,\ R)\ =\ sp(S_{2},sp(S_{1},R))
  46. ( โˆ€ x , P โ‡’ Q ) โ‡’ ( โˆ€ x , T ( P ) โ‡’ T ( Q ) ) (\forall x,P\Rightarrow Q)\Rightarrow(\forall x,T(P)\Rightarrow T(Q))
  47. T ( ๐Ÿ๐š๐ฅ๐ฌ๐ž ) โ‡” ๐Ÿ๐š๐ฅ๐ฌ๐ž T(\mathbf{false})\ \Leftrightarrow\ \mathbf{false}
  48. w l p ( S , ๐Ÿ๐š๐ฅ๐ฌ๐ž ) wlp(S,\mathbf{false})
  49. w l p ( ๐ฐ๐ก๐ข๐ฅ๐ž ๐ญ๐ซ๐ฎ๐ž ๐๐จ ๐ฌ๐ค๐ข๐ฉ ๐๐จ๐ง๐ž , ๐Ÿ๐š๐ฅ๐ฌ๐ž ) โ‡” ๐ญ๐ซ๐ฎ๐ž wlp(\mathbf{while}\ \mathbf{true}\ \mathbf{do}\ \mathbf{skip}\ \mathbf{done},% \mathbf{false})\ \Leftrightarrow\mathbf{true}
  50. T ( ๐ญ๐ซ๐ฎ๐ž ) โ‡” ๐ญ๐ซ๐ฎ๐ž T(\mathbf{true})\ \Leftrightarrow\ \mathbf{true}
  51. w p ( S , ๐ญ๐ซ๐ฎ๐ž ) wp(S,\mathbf{true})
  52. T ( P โˆง Q ) โ‡” ( T ( P ) โˆง T ( Q ) ) T(P\wedge Q)\ \Leftrightarrow\ (T(P)\wedge T(Q))
  53. w p ( S , . ) wp(S,.)
  54. T ( P โˆจ Q ) โ‡” ( T ( P ) โˆจ T ( Q ) ) T(P\vee Q)\ \Leftrightarrow\ (T(P)\vee T(Q))
  55. w p ( S , . ) wp(S,.)
  56. S = ๐ข๐Ÿ ๐ญ๐ซ๐ฎ๐ž โ†’ x := 0 [ ] ๐ญ๐ซ๐ฎ๐ž โ†’ x := 1 ๐Ÿ๐ข S\ =\ \mathbf{if}\ \mathbf{true}\rightarrow x:=0\ [\!]\ \mathbf{true}% \rightarrow x:=1\ \mathbf{fi}
  57. w p ( S , R ) wp(S,R)
  58. R [ x โ† 0 ] โˆง R [ x โ† 1 ] R[x\leftarrow 0]\wedge R[x\leftarrow 1]
  59. w p ( S , x = 0 โˆจ x = 1 ) wp(S,\ x=0\vee x=1)
  60. ( 0 = 0 โˆจ 0 = 1 ) โˆง ( 1 = 0 โˆจ 1 = 1 ) (0=0\vee 0=1)\wedge(1=0\vee 1=1)
  61. w p ( S , x = 0 ) โˆจ w p ( S , x = 1 ) wp(S,x=0)\vee wp(S,x=1)
  62. ( 0 = 0 โˆง 1 = 0 ) โˆจ ( 1 = 0 โˆง 1 = 1 ) (0=0\wedge 1=0)\vee(1=0\wedge 1=1)
  63. S = ๐ญ๐ซ๐ฎ๐ž | S\ =\ \mathbf{true}\ |
  64. y . y โˆˆ { 0 , 1 } โ†’ x := y y.\ y\in\{0,1\}\rightarrow x:=y