wpmath0000004_1

Automatic_programming.html

  1. n n
  2. O ( n 2 ) O(n^{2})
  3. O ( n 3 ) O(n^{3})
  4. O ( n ) O(n)

Automatic_stabilizer.html

  1. M u l t i p l i e r = 1 1 - [ M P C ( 1 - T ) - M P I ] Multiplier=\frac{1}{1-[MPC(1-T)-MPI]}

Automobile_handling.html

  1. I = M ( h e i g h t 2 + w i d t h 2 ) / 12 I=M(height^{2}+width^{2})/12

Automorphic_form.html

  1. Γ G \Gamma\subset G
  2. γ Γ \gamma\in\Gamma
  3. γ Γ \gamma\in\Gamma

Autoregressive–moving-average_model.html

  1. X t = c + i = 1 p φ i X t - i + ε t . X_{t}=c+\sum_{i=1}^{p}\varphi_{i}X_{t-i}+\varepsilon_{t}.\,
  2. φ 1 , , φ p \varphi_{1},\ldots,\varphi_{p}
  3. c c
  4. ε t \varepsilon_{t}
  5. X t = μ + ε t + i = 1 q θ i ε t - i X_{t}=\mu+\varepsilon_{t}+\sum_{i=1}^{q}\theta_{i}\varepsilon_{t-i}\,
  6. X t X_{t}
  7. ε t \varepsilon_{t}
  8. ε t - 1 \varepsilon_{t-1}
  9. X t = c + ε t + i = 1 p φ i X t - i + i = 1 q θ i ε t - i . X_{t}=c+\varepsilon_{t}+\sum_{i=1}^{p}\varphi_{i}X_{t-i}+\sum_{i=1}^{q}\theta_% {i}\varepsilon_{t-i}.\,
  10. ε t \varepsilon_{t}
  11. ε t \varepsilon_{t}
  12. ε t = ( 1 - i = 1 p φ i L i ) X t = φ ( L ) X t \varepsilon_{t}=\left(1-\sum_{i=1}^{p}\varphi_{i}L^{i}\right)X_{t}=\varphi(L)X% _{t}\,
  13. φ \varphi
  14. φ ( L ) = 1 - i = 1 p φ i L i . \varphi(L)=1-\sum_{i=1}^{p}\varphi_{i}L^{i}.\,
  15. L L
  16. L d X t = X t - d . L^{d}X_{t}=X_{t-d}.
  17. X t = ( 1 + i = 1 q θ i L i ) ε t = θ ( L ) ε t , X_{t}=\left(1+\sum_{i=1}^{q}\theta_{i}L^{i}\right)\varepsilon_{t}=\theta(L)% \varepsilon_{t},\,
  18. θ ( L ) = 1 + i = 1 q θ i L i . \theta(L)=1+\sum_{i=1}^{q}\theta_{i}L^{i}.\,
  19. ( 1 - i = 1 p φ i L i ) X t = ( 1 + i = 1 q θ i L i ) ε t , \left(1-\sum_{i=1}^{p}\varphi_{i}L^{i}\right)X_{t}=\left(1+\sum_{i=1}^{q}% \theta_{i}L^{i}\right)\varepsilon_{t}\,,
  20. φ ( L ) X t = θ ( L ) ε t \varphi(L)X_{t}=\theta(L)\varepsilon_{t}\,
  21. φ ( L ) θ ( L ) X t = ε t . \frac{\varphi(L)}{\theta(L)}X_{t}=\varepsilon_{t}\,.
  22. ( 1 + i = 1 p ϕ i L i ) X t = ( 1 + i = 1 q θ i L i ) ε t . \left(1+\sum_{i=1}^{p}\phi_{i}L^{i}\right)X_{t}=\left(1+\sum_{i=1}^{q}\theta_{% i}L^{i}\right)\varepsilon_{t}\,.
  23. ϕ 0 = θ 0 = 1 \phi_{0}=\theta_{0}=1
  24. i = 0 p ϕ i L i X t = i = 0 q θ i L i ε t . \sum_{i=0}^{p}\phi_{i}L^{i}\;X_{t}=\sum_{i=0}^{q}\theta_{i}L^{i}\;\varepsilon_% {t}\,.
  25. d t d_{t}
  26. X t = ε t + i = 1 p φ i X t - i + i = 1 q θ i ε t - i + i = 0 b η i d t - i . X_{t}=\varepsilon_{t}+\sum_{i=1}^{p}\varphi_{i}X_{t-i}+\sum_{i=1}^{q}\theta_{i% }\varepsilon_{t-i}+\sum_{i=0}^{b}\eta_{i}d_{t-i}.\,
  27. η 1 , , η b \eta_{1},\ldots,\eta_{b}
  28. d t d_{t}
  29. X t - m t = ε t + i = 1 p φ i ( X t - i - m t - i ) + i = 1 q θ i ε t - i . X_{t}-m_{t}=\varepsilon_{t}+\sum_{i=1}^{p}\varphi_{i}(X_{t-i}-m_{t-i})+\sum_{i% =1}^{q}\theta_{i}\varepsilon_{t-i}.\,
  30. m t = c + i = 0 b η i d t - i . m_{t}=c+\sum_{i=0}^{b}\eta_{i}d_{t-i}.\,

Autotransformer.html

  1. V 1 V 2 = N 1 N 2 = a \frac{V_{1}}{V_{2}}=\frac{N_{1}}{N_{2}}=a
  2. F U = ( N 1 - N 2 ) I 1 = ( 1 - 1 a ) N 1 I 1 F_{U}=(N_{1}-N_{2})I_{1}=(1-\frac{1}{a})N_{1}I_{1}
  3. F L = N 2 ( I 2 - I 1 ) = N 1 a ( I 2 - I 1 ) F_{L}=N_{2}(I_{2}-I_{1})=\frac{N_{1}}{a}(I_{2}-I_{1})
  4. ( 1 - 1 a ) N 1 I 1 = N 1 a ( I 2 - I 1 ) (1-\frac{1}{a})N_{1}I_{1}=\frac{N_{1}}{a}(I_{2}-I_{1})
  5. I 1 I 2 = 1 a \frac{I_{1}}{I_{2}}=\frac{1}{a}

Axiom_of_constructibility.html

  1. Δ 2 1 \Delta^{1}_{2}

Axiom_of_determinacy.html

  1. G S e q ( S ) : \forall G\subseteq Seq(S):
  2. a S : a S : b S : b S : c S : c S : ( a , a , b , b , c , c ) G \forall a\in S:\exists a^{\prime}\in S:\forall b\in S:\exists b^{\prime}\in S:% \forall c\in S:\exists c^{\prime}\in S...:(a,a^{\prime},b,b^{\prime},c,c^{% \prime}...)\in G
  3. a S : a S : b S : b S : c S : c S : ( a , a , b , b , c , c ) G \exists a\in S:\forall a^{\prime}\in S:\exists b\in S:\forall b^{\prime}\in S:% \exists c\in S:\forall c^{\prime}\in S...:(a,a^{\prime},b,b^{\prime},c,c^{% \prime}...)\notin G
  4. ω \omega
  5. a : b : c : d : R ( a , b , c , d ) \forall a:\exists b:\forall c:\exists d:R(a,b,c,d)
  6. a : b : c : d : ¬ R ( a , b , c , d ) \exists a:\forall b:\exists c:\forall d:\lnot R(a,b,c,d)

Azimuthal_equidistant_projection.html

  1. x = ρ sin θ x=\rho\sin\theta
  2. y = - ρ cos θ y=-\rho\cos\theta
  3. cos ρ = sin φ 1 sin φ + cos φ 1 cos φ cos ( λ - λ 0 ) \cos\rho=\sin\varphi_{1}\sin\varphi+\cos\varphi_{1}\cos\varphi\cos\left(% \lambda-\lambda_{0}\right)
  4. tan θ = cos φ sin ( λ - λ 0 ) cos φ 1 sin φ - sin φ 1 cos φ cos ( λ - λ 0 ) \tan\theta=\frac{\cos\varphi\sin\left(\lambda-\lambda_{0}\right)}{\cos\varphi_% {1}\sin\varphi-\sin\varphi_{1}\cos\varphi\cos\left(\lambda-\lambda_{0}\right)}
  5. ρ = π 2 - φ \rho=\frac{\pi}{2}-\varphi
  6. θ = λ \theta=\lambda

Bad_Kreuznach.html

  1. 𝔛 \mathfrak{X}

Balassa–Samuelson_effect.html

  1. M P L n t , 1 = M P L n t , 2 = 1 MPL_{nt,1}=MPL_{nt,2}=1
  2. w 1 = p n t , 1 * M P L n t , 1 = p n t , 1 = p t * M P L t , 1 w_{1}=p_{nt,1}*MPL_{nt,1}=p_{nt,1}=p_{t}*MPL_{t,1}
  3. w 2 = p n t , 2 * M P L n t , 2 = p n t , 2 = p t * M P L t , 2 w_{2}=p_{nt,2}*MPL_{nt,2}=p_{nt,2}=p_{t}*MPL_{t,2}
  4. M P L t , 1 < M P L t , 2 MPL_{t,1}<MPL_{t,2}
  5. p n t , 1 < p n t , 2 p_{nt,1}<p_{nt,2}

Bargmann's_limit.html

  1. N l 1 2 l + 1 2 m 2 0 r | V ( r ) | V < 0 d r N_{l}\leq\frac{1}{2l+1}\frac{2m}{\hbar^{2}}\int_{0}^{\infty}r|V(r)|_{V<0}\,dr

Barometric_formula.html

  1. P = P b [ T b T b + L b ( h - h b ) ] g 0 M R * L b {P}=P_{b}\cdot\left[\frac{T_{b}}{T_{b}+L_{b}\cdot(h-h_{b})}\right]^{\textstyle% \frac{g_{0}\cdot M}{R^{*}\cdot L_{b}}}
  2. P = P b exp [ - g 0 M ( h - h b ) R * T b ] P=P_{b}\cdot\exp\left[\frac{-g_{0}\cdot M\cdot(h-h_{b})}{R^{*}\cdot T_{b}}\right]
  3. P b P_{b}
  4. T b T_{b}
  5. L b L_{b}
  6. h h
  7. h b h_{b}
  8. R * R^{*}
  9. g 0 g_{0}
  10. M M
  11. P b P_{b}
  12. T b T_{b}
  13. L b L_{b}
  14. h h
  15. h b h_{b}
  16. R * R^{*}
  17. g 0 g_{0}
  18. M M
  19. ρ = ρ b [ 1 - L b ( h - h b ) T b ] ( 1 + g 0 M R * L b ) {\rho}=\rho_{b}\cdot\left[1-\frac{L_{b}\cdot(h-h_{b})}{T_{b}}\right]^{\left(1+% \frac{g_{0}\cdot M}{R^{*}\cdot L_{b}}\right)}
  20. ρ = ρ b exp [ - g 0 M ( h - h b ) R * T b ] {\rho}=\rho_{b}\cdot\exp\left[\frac{-g_{0}\cdot M\cdot(h-h_{b})}{R^{*}\cdot T_% {b}}\right]
  21. ρ {\rho}
  22. T T
  23. L L
  24. h h
  25. R * R^{*}
  26. g 0 g_{0}
  27. M M
  28. ρ {\rho}
  29. T {T}
  30. L {L}
  31. h {h}
  32. R * {R^{*}}
  33. g 0 {g_{0}}
  34. M {M}
  35. ρ \rho
  36. ρ = M P R * T \rho=\frac{M\cdot P}{R^{*}\cdot T}
  37. P = ρ R * T M P=\frac{\rho\cdot{R^{*}}\cdot T}{M}
  38. d P = - ρ g d z dP=-\rho g\,dz\,
  39. d P dP
  40. P P
  41. d P P = - M g d z R * T \frac{dP}{P}=-\frac{Mg\,dz}{R^{*}T}
  42. P = P 0 e - 0 z M g d z / R * T P=P_{0}e^{-\int_{0}^{z}{Mgdz/R^{*}T}}\,
  43. P = P 0 e - M g z / R * T P=P_{0}e^{-Mgz/R^{*}T}\,

Barycentric_coordinate_system.html

  1. 𝐱 1 , , 𝐱 n \mathbf{x}_{1},\ldots,\mathbf{x}_{n}
  2. 𝐩 \mathbf{p}
  3. ( a 1 + + a n ) 𝐩 = a 1 𝐱 1 + + a n 𝐱 n (a_{1}+\cdots+a_{n})\mathbf{p}=a_{1}\,\mathbf{x}_{1}+\cdots+a_{n}\,\mathbf{x}_% {n}
  4. a 1 , , a n a_{1},\ldots,a_{n}
  5. a 1 , , a n a_{1},\ldots,a_{n}
  6. 𝐩 \mathbf{p}
  7. 𝐱 1 , , 𝐱 n \mathbf{x}_{1},\ldots,\mathbf{x}_{n}
  8. 𝐱 1 = ( 1 , 0 , 0 , , 0 ) , 𝐱 2 = ( 0 , 1 , 0 , , 0 ) , , 𝐱 n = ( 0 , 0 , 0 , , 1 ) \mathbf{x}_{1}=(1,0,0,\ldots,0),\mathbf{x}_{2}=(0,1,0,\ldots,0),\ldots,\mathbf% {x}_{n}=(0,0,0,\ldots,1)
  9. b a 1 , , b a n ba_{1},\ldots,ba_{n}
  10. 𝐩 \mathbf{p}
  11. 𝐱 1 , , 𝐱 n \mathbf{x}_{1},\ldots,\mathbf{x}_{n}
  12. a i = 1 \sum a_{i}=1
  13. T T
  14. 𝐫 1 \mathbf{r}_{1}\,
  15. 𝐫 2 \mathbf{r}_{2}\,
  16. 𝐫 3 \mathbf{r}_{3}\,
  17. 𝐫 \mathbf{r}
  18. 𝐫 \mathbf{r}
  19. λ 1 , λ 2 , λ 3 0 \lambda_{1},\lambda_{2},\lambda_{3}\geq 0
  20. λ 1 + λ 2 + λ 3 = 1 \lambda_{1}+\lambda_{2}+\lambda_{3}=1
  21. 𝐫 = λ 1 𝐫 1 + λ 2 𝐫 2 + λ 3 𝐫 3 , \mathbf{r}=\lambda_{1}\mathbf{r}_{1}+\lambda_{2}\mathbf{r}_{2}+\lambda_{3}% \mathbf{r}_{3},
  22. λ 1 , λ 2 , λ 3 \lambda_{1},\lambda_{2},\lambda_{3}
  23. 𝐫 \mathbf{r}
  24. α , β , γ \alpha,\beta,\gamma
  25. λ 1 , λ 2 , λ 3 \lambda_{1},\lambda_{2},\lambda_{3}
  26. λ 1 + λ 2 + λ 3 = 1 \lambda_{1}+\lambda_{2}+\lambda_{3}=1
  27. 𝐫 \mathbf{r}\,
  28. λ 1 \lambda_{1}\,
  29. λ 2 \lambda_{2}\,
  30. λ 3 \lambda_{3}\,
  31. ( x , y ) (x,y)\,
  32. 𝐫 \mathbf{r}
  33. 𝐫 1 \mathbf{r}_{1}
  34. 𝐫 2 \mathbf{r}_{2}
  35. 𝐫 3 \mathbf{r}_{3}
  36. 𝐫 i = ( x i , y i ) \mathbf{r}_{i}=(x_{i},y_{i})
  37. 𝐫 \mathbf{r}\,
  38. x = λ 1 x 1 + λ 2 x 2 + λ 3 x 3 y = λ 1 y 1 + λ 2 y 2 + λ 3 y 3 \begin{matrix}x=\lambda_{1}x_{1}+\lambda_{2}x_{2}+\lambda_{3}x_{3}\\ y=\lambda_{1}y_{1}+\lambda_{2}y_{2}+\lambda_{3}y_{3}\\ \end{matrix}\,
  39. λ 3 = 1 - λ 1 - λ 2 \lambda_{3}=1-\lambda_{1}-\lambda_{2}\,
  40. x = λ 1 x 1 + λ 2 x 2 + ( 1 - λ 1 - λ 2 ) x 3 y = λ 1 y 1 + λ 2 y 2 + ( 1 - λ 1 - λ 2 ) y 3 \begin{matrix}x=\lambda_{1}x_{1}+\lambda_{2}x_{2}+(1-\lambda_{1}-\lambda_{2})x% _{3}\\ y=\lambda_{1}y_{1}+\lambda_{2}y_{2}+(1-\lambda_{1}-\lambda_{2})y_{3}\\ \end{matrix}\,
  41. λ 1 ( x 1 - x 3 ) + λ 2 ( x 2 - x 3 ) + x 3 - x = 0 λ 1 ( y 1 - y 3 ) + λ 2 ( y 2 - y 3 ) + y 3 - y = 0 \begin{matrix}\lambda_{1}(x_{1}-x_{3})+\lambda_{2}(x_{2}-x_{3})+x_{3}-x=0\\ \lambda_{1}(y_{1}-y_{3})+\lambda_{2}(y_{2}-y_{3})+y_{3}-y=0\\ \end{matrix}\,
  42. 𝐓 λ = 𝐫 - 𝐫 3 \mathbf{T}\cdot\lambda=\mathbf{r}-\mathbf{r}_{3}\,
  43. λ \lambda
  44. 𝐫 \mathbf{r}
  45. 𝐓 \mathbf{T}
  46. 𝐓 = ( x 1 - x 3 x 2 - x 3 y 1 - y 3 y 2 - y 3 ) \mathbf{T}=\left(\begin{matrix}x_{1}-x_{3}&x_{2}-x_{3}\\ y_{1}-y_{3}&y_{2}-y_{3}\\ \end{matrix}\right)
  47. 𝐓 \mathbf{T}
  48. 𝐫 1 - 𝐫 3 \mathbf{r}_{1}-\mathbf{r}_{3}
  49. 𝐫 2 - 𝐫 3 \mathbf{r}_{2}-\mathbf{r}_{3}
  50. 𝐫 1 \mathbf{r}_{1}
  51. 𝐫 2 \mathbf{r}_{2}
  52. 𝐫 3 \mathbf{r}_{3}
  53. ( λ 1 λ 2 ) = 𝐓 - 1 ( 𝐫 - 𝐫 3 ) \left(\begin{matrix}\lambda_{1}\\ \lambda_{2}\end{matrix}\right)=\mathbf{T}^{-1}(\mathbf{r}-\mathbf{r}_{3})\,
  54. 𝐓 \mathbf{T}
  55. 𝐫 \mathbf{r}
  56. λ 1 = ( y 2 - y 3 ) ( x - x 3 ) + ( x 3 - x 2 ) ( y - y 3 ) det ( T ) = ( y 2 - y 3 ) ( x - x 3 ) + ( x 3 - x 2 ) ( y - y 3 ) ( y 2 - y 3 ) ( x 1 - x 3 ) + ( x 3 - x 2 ) ( y 1 - y 3 ) , \lambda_{1}=\frac{(y_{2}-y_{3})(x-x_{3})+(x_{3}-x_{2})(y-y_{3})}{\det(T)}=% \frac{(y_{2}-y_{3})(x-x_{3})+(x_{3}-x_{2})(y-y_{3})}{(y_{2}-y_{3})(x_{1}-x_{3}% )+(x_{3}-x_{2})(y_{1}-y_{3})}\,,
  57. λ 2 = ( y 3 - y 1 ) ( x - x 3 ) + ( x 1 - x 3 ) ( y - y 3 ) det ( T ) = ( y 3 - y 1 ) ( x - x 3 ) + ( x 1 - x 3 ) ( y - y 3 ) ( y 2 - y 3 ) ( x 1 - x 3 ) + ( x 3 - x 2 ) ( y 1 - y 3 ) , \lambda_{2}=\frac{(y_{3}-y_{1})(x-x_{3})+(x_{1}-x_{3})(y-y_{3})}{\det(T)}=% \frac{(y_{3}-y_{1})(x-x_{3})+(x_{1}-x_{3})(y-y_{3})}{(y_{2}-y_{3})(x_{1}-x_{3}% )+(x_{3}-x_{2})(y_{1}-y_{3})}\,,
  58. λ 3 = 1 - λ 1 - λ 2 . \lambda_{3}=1-\lambda_{1}-\lambda_{2}\,.
  59. 𝐫 = 𝐑 s y m b o l λ \mathbf{r}=\mathbf{R}symbol{\lambda}
  60. 𝐑 = ( 𝐫 1 | 𝐫 2 | 𝐫 3 ) \mathbf{R}=\left(\begin{matrix}\mathbf{r}_{1}|\mathbf{r}_{2}|\mathbf{r}_{3}% \end{matrix}\right)
  61. s y m b o l λ = ( λ 1 , λ 2 , λ 3 ) symbol{\lambda}=\left(\lambda_{1},\lambda_{2},\lambda_{3}\right)^{\top}
  62. λ 1 + λ 2 + λ 3 = 1 \lambda_{1}+\lambda_{2}+\lambda_{3}=1
  63. ( 1 , 1 , 1 ) s y m b o l λ = 1 \left(1,1,1\right)symbol{\lambda}=1
  64. ( x 1 x 2 x 3 y 1 y 2 y 3 1 1 1 ) s y m b o l λ = ( x y 1 ) \left(\begin{matrix}x_{1}&x_{2}&x_{3}\\ y_{1}&y_{2}&y_{3}\\ 1&1&1\end{matrix}\right)symbol{\lambda}=\left(\begin{matrix}x\\ y\\ 1\end{matrix}\right)
  65. λ 1...3 0 \lambda_{1...3}\geq 0
  66. ( 0 , 1 ) . (0,1).
  67. λ 1...3 \lambda_{1...3}
  68. ( 0 , 1 ) . (0,1).
  69. 𝐫 \mathbf{r}
  70. 0 < λ i < 1 i in 1 , 2 , 3 0<\lambda_{i}<1\;\forall\;i\,\text{ in }1,2,3
  71. 𝐫 \mathbf{r}
  72. 0 λ i 1 i in 1 , 2 , 3 0\leq\lambda_{i}\leq 1\;\forall\;i\,\text{ in }1,2,3
  73. 𝐫 \mathbf{r}
  74. f ( 𝐫 1 ) , f ( 𝐫 2 ) , f ( 𝐫 3 ) f(\mathbf{r}_{1}),f(\mathbf{r}_{2}),f(\mathbf{r}_{3})
  75. f f
  76. 𝐫 1 , 𝐫 2 , 𝐫 3 \mathbf{r}_{1},\mathbf{r}_{2},\mathbf{r}_{3}
  77. 𝐫 \mathbf{r}
  78. λ 1 \lambda_{1}\,
  79. λ 2 \lambda_{2}\,
  80. λ 3 \lambda_{3}\,
  81. f ( 𝐫 ) λ 1 f ( 𝐫 1 ) + λ 2 f ( 𝐫 2 ) + λ 3 f ( 𝐫 3 ) f(\mathbf{r})\approx\lambda_{1}f(\mathbf{r}_{1})+\lambda_{2}f(\mathbf{r}_{2})+% \lambda_{3}f(\mathbf{r}_{3})
  82. f f
  83. f f
  84. 𝐫 \mathbf{r}
  85. 𝐫 \mathbf{r}
  86. 0 λ i 1 i in 1 , 2 , 3 0\leq\lambda_{i}\leq 1\;\forall\;i\,\text{ in }1,2,3
  87. f ( 𝐫 ) f(\mathbf{r})
  88. λ 1 , λ 2 \lambda_{1},\lambda_{2}
  89. T f ( 𝐫 ) d 𝐫 = 2 A 0 1 0 1 - λ 2 f ( λ 1 𝐫 1 + λ 2 𝐫 2 + ( 1 - λ 1 - λ 2 ) 𝐫 3 ) d λ 1 d λ 2 \int_{T}f(\mathbf{r})\ d\mathbf{r}=2A\int_{0}^{1}\int_{0}^{1-\lambda_{2}}f(% \lambda_{1}\mathbf{r}_{1}+\lambda_{2}\mathbf{r}_{2}+(1-\lambda_{1}-\lambda_{2}% )\mathbf{r}_{3})\ d\lambda_{1}\ d\lambda_{2}\,
  90. A A
  91. 2 A 2A
  92. a 2 ( - a 2 + b 2 + c 2 ) : b 2 ( a 2 - b 2 + c 2 ) : c 2 ( a 2 + b 2 - c 2 ) a^{2}(-a^{2}+b^{2}+c^{2}):\;b^{2}(a^{2}-b^{2}+c^{2}):\;c^{2}(a^{2}+b^{2}-c^{2})\,
  93. = sin 2 A : sin 2 B : sin 2 C , =\sin 2A:\sin 2B:\sin 2C,
  94. tan A : tan B : tan C . \tan A:\tan B:\tan C.
  95. a : b : c = sin A : sin B : sin C . a:b:c=\sin A:\sin B:\sin C.
  96. a cos ( B - C ) : b cos ( C - A ) : c cos ( A - B ) a\cos(B-C):b\cos(C-A):c\cos(A-B)
  97. = a 2 ( b 2 + c 2 ) - ( b 2 - c 2 ) 2 : b 2 ( c 2 + a 2 ) - ( c 2 - a 2 ) 2 : c 2 ( a 2 + b 2 ) - ( a 2 - b 2 ) 2 . =a^{2}(b^{2}+c^{2})-(b^{2}-c^{2})^{2}:b^{2}(c^{2}+a^{2})-(c^{2}-a^{2})^{2}:c^{% 2}(a^{2}+b^{2})-(a^{2}-b^{2})^{2}.
  98. 𝐫 1 \mathbf{r}_{1}
  99. λ = ( 1 , 0 , 0 , 0 ) \lambda=(1,0,0,0)
  100. 𝐫 2 ( 0 , 1 , 0 , 0 ) \mathbf{r}_{2}\to(0,1,0,0)
  101. 𝐫 \mathbf{r}
  102. ( λ 1 λ 2 λ 3 ) = 𝐓 - 1 ( 𝐫 - 𝐫 4 ) \left(\begin{matrix}\lambda_{1}\\ \lambda_{2}\\ \lambda_{3}\end{matrix}\right)=\mathbf{T}^{-1}(\mathbf{r}-\mathbf{r}_{4})\,
  103. 𝐓 \mathbf{T}
  104. 𝐓 = ( x 1 - x 4 x 2 - x 4 x 3 - x 4 y 1 - y 4 y 2 - y 4 y 3 - y 4 z 1 - z 4 z 2 - z 4 z 3 - z 4 ) \mathbf{T}=\left(\begin{matrix}x_{1}-x_{4}&x_{2}-x_{4}&x_{3}-x_{4}\\ y_{1}-y_{4}&y_{2}-y_{4}&y_{3}-y_{4}\\ z_{1}-z_{4}&z_{2}-z_{4}&z_{3}-z_{4}\end{matrix}\right)
  105. ( a 1 + + a n ) p = a 1 x 1 + + a n x n (a_{1}+\cdots+a_{n})p=a_{1}x_{1}+\cdots+a_{n}x_{n}
  106. ( n - 1 ) (n-1)
  107. Δ n - 1 P . \Delta^{n-1}\twoheadrightarrow P.
  108. P ( 𝐑 0 ) f P\hookrightarrow(\mathbf{R}_{\geq 0})^{f}
  109. ( n - 1 ) (n-1)
  110. K n K^{n}
  111. { ( x 0 , , x n ) x i = 1 } K n + 1 \{(x_{0},\ldots,x_{n})\mid\sum x_{i}=1\}\subset K^{n+1}

Base_rate_fallacy.html

  1. p ( d r u n k | D ) p(drunk|D)
  2. p ( d r u n k | D ) = p ( D | d r u n k ) p ( d r u n k ) p ( D ) p(drunk|D)=\frac{p(D|drunk)\,p(drunk)}{p(D)}
  3. p ( d r u n k ) = 0.001 p(drunk)=0.001
  4. p ( s o b e r ) = 0.999 p(sober)=0.999
  5. p ( D | d r u n k ) = 1.00 p(D|drunk)=1.00
  6. p ( D | s o b e r ) = 0.05 p(D|sober)=0.05
  7. p ( D ) = p ( D | d r u n k ) p ( d r u n k ) + p ( D | s o b e r ) p ( s o b e r ) p(D)=p(D|drunk)\,p(drunk)+p(D|sober)\,p(sober)
  8. p ( D ) = 0.05095 p(D)=0.05095
  9. p ( d r u n k | D ) = 0.019627 p(drunk|D)=0.019627\cdot
  10. p ( d r u n k | D ) = 1 / 50.95 0.019627 p(drunk|D)=1/50.95\approx 0.019627
  11. p ( d r u n k | D ) = N ( d r u n k D ) N ( D ) = 1 51 = 0.0196 p(drunk|D)=\frac{N(drunk\cap D)}{N(D)}=\frac{1}{51}=0.0196

Basic_Linear_Algebra_Subprograms.html

  1. O ( n ) O(n)
  2. s y m b o l y α s y m b o l x + s y m b o l y symbol{y}\leftarrow\alpha symbol{x}+symbol{y}
  3. s y m b o l y α s y m b o l A s y m b o l x + β s y m b o l y symbol{y}\leftarrow\alpha symbol{A}symbol{x}+\beta symbol{y}
  4. 𝐱 \mathbf{x}
  5. s y m b o l T s y m b o l x = s y m b o l y symbol{T}symbol{x}=symbol{y}
  6. 𝐓 \mathbf{T}
  7. s y m b o l C α s y m b o l A s y m b o l B + β s y m b o l C symbol{C}\leftarrow\alpha symbol{A}symbol{B}+\beta symbol{C}
  8. 𝐀 \mathbf{A}
  9. 𝐁 \mathbf{B}
  10. 𝐀𝐁 \mathbf{AB}
  11. α α
  12. 𝐂 \mathbf{C}
  13. s y m b o l B α s y m b o l T - 1 s y m b o l B symbol{B}\leftarrow\alpha symbol{T}^{-1}symbol{B}
  14. 𝐓 \mathbf{T}
  15. 𝐀 \mathbf{A}
  16. 𝐁 \mathbf{B}
  17. β β
  18. β = 1 β=1
  19. 𝐂 \mathbf{C}

Batalin–Vilkovisky_formalism.html

  1. ( a , b ) := ( - 1 ) | a | Δ ( a b ) - ( - 1 ) | a | Δ ( a ) b - a Δ ( b ) + a Δ ( 1 ) b . (a,b):=(-1)^{\left|a\right|}\Delta(ab)-(-1)^{\left|a\right|}\Delta(a)b-a\Delta% (b)+a\Delta(1)b.
  2. Δ ρ := Δ - Δ ( 1 ) . {\Delta}_{\rho}:=\Delta-\Delta(1).
  3. Δ ρ ( a , b ) = ( Δ ρ ( a ) , b ) - ( - 1 ) | a | ( a , Δ ρ ( b ) ) {\Delta}_{\rho}(a,b)=({\Delta}_{\rho}(a),b)-(-1)^{\left|a\right|}(a,{\Delta}_{% \rho}(b))
  4. Δ ρ {\Delta}_{\rho}
  5. Δ ρ 2 = ( Δ ( 1 ) , ) {\Delta}_{\rho}^{2}=(\Delta(1),\cdot)
  6. Δ ρ {\Delta}_{\rho}
  7. Δ ρ 2 ( a b ) = Δ ρ 2 ( a ) b + a Δ ρ 2 ( b ) {\Delta}_{\rho}^{2}(ab)={\Delta}_{\rho}^{2}(a)b+a{\Delta}_{\rho}^{2}(b)
  8. Δ ρ {\Delta}_{\rho}
  9. Δ ρ 2 {\Delta}_{\rho}^{2}
  10. L a L_{a}
  11. L a ( b ) := a b , L_{a}(b):=ab,
  12. [ S , T ] := S T - ( - 1 ) | S | | T | T S [S,T]:=ST-(-1)^{\left|S\right|\left|T\right|}TS
  13. ( a , b ) := ( - 1 ) | a | [ [ Δ , L a ] , L b ] 1 , (a,b):=(-1)^{\left|a\right|}[[\Delta,L_{a}],L_{b}]1,
  14. [ [ [ Δ , L a ] , L b ] , L c ] 1 = 0 [[[\Delta,L_{a}],L_{b}],L_{c}]1=0
  15. [ Δ , L a ] [\Delta,L_{a}]
  16. [ [ Δ , L a ] , L b ] [[\Delta,L_{a}],L_{b}]
  17. ( S , S ) = 0. (S,S)=0.
  18. Δ exp [ i W ] = 0 , \Delta\exp\left[\frac{i}{\hbar}W\right]=0,
  19. 1 2 ( W , W ) = i Δ ρ ( W ) + 2 Δ ( 1 ) . \frac{1}{2}(W,W)=i\hbar{\Delta}_{\rho}(W)+\hbar^{2}\Delta(1).
  20. 1 2 ( W , W ) = i Δ ( W ) . \frac{1}{2}(W,W)=i\hbar\Delta(W).
  21. Φ n ( a 1 , , a n ) := [ [ [ Δ , L a 1 ] , ] , L a n ] n nested commutators 1. \Phi^{n}(a_{1},\ldots,a_{n}):=\underbrace{[[\ldots[\Delta,L_{a_{1}}],\ldots],L% _{a_{n}}]}_{n~{}{\rm nested~{}commutators}}1.
  22. Φ n ( a π ( 1 ) , , a π ( n ) ) = ( - 1 ) | a π | Φ n ( a 1 , , a n ) \Phi^{n}(a_{\pi(1)},\ldots,a_{\pi(n)})=(-1)^{\left|a_{\pi}\right|}\Phi^{n}(a_{% 1},\ldots,a_{n})
  23. π S n \pi\in S_{n}
  24. ( - 1 ) | a π | (-1)^{\left|a_{\pi}\right|}
  25. a π ( 1 ) a π ( n ) = ( - 1 ) | a π | a 1 a n a_{\pi(1)}\ldots a_{\pi(n)}=(-1)^{\left|a_{\pi}\right|}a_{1}\ldots a_{n}
  26. L L_{\infty}
  27. k = 0 n 1 k ! ( n - k ) ! π S n ( - 1 ) | a π | Φ n - k + 1 ( Φ k ( a π ( 1 ) , , a π ( k ) ) , a π ( k + 1 ) , , a π ( n ) ) = 0. \sum_{k=0}^{n}\frac{1}{k!(n\!-\!k)!}\sum_{\pi\in S_{n}}(-1)^{\left|a_{\pi}% \right|}\Phi^{n-k+1}\left(\Phi^{k}(a_{\pi(1)},\ldots,a_{\pi(k)}),a_{\pi(k+1)},% \ldots,a_{\pi(n)}\right)=0.
  28. Φ 0 := Δ ( 1 ) \Phi^{0}:=\Delta(1)
  29. Φ 1 ( a ) := [ Δ , L a ] 1 = Δ ( a ) - Δ ( 1 ) a = : Δ ρ ( a ) \Phi^{1}(a):=[\Delta,L_{a}]1=\Delta(a)-\Delta(1)a=:{\Delta}_{\rho}(a)
  30. Φ 2 ( a , b ) := [ [ Δ , L a ] , L b ] 1 = : ( - 1 ) | a | ( a , b ) \Phi^{2}(a,b):=[[\Delta,L_{a}],L_{b}]1=:(-1)^{\left|a\right|}(a,b)
  31. Φ 3 ( a , b , c ) := [ [ [ Δ , L a ] , L b ] , L c ] 1 \Phi^{3}(a,b,c):=[[[\Delta,L_{a}],L_{b}],L_{c}]1
  32. \vdots
  33. Φ 1 = Δ ρ \Phi^{1}={\Delta}_{\rho}
  34. Φ 2 \Phi^{2}
  35. Φ 1 ( Φ 0 ) = 0 \Phi^{1}(\Phi^{0})=0
  36. Δ ( 1 ) \Delta(1)
  37. Δ ρ \Delta_{\rho}
  38. Φ 2 ( Φ 0 , a ) + Φ 1 ( Φ 1 ( a ) ) \Phi^{2}(\Phi^{0},a)+\Phi^{1}\left(\Phi^{1}(a)\right)
  39. Δ ( 1 ) \Delta(1)
  40. Δ ρ 2 {\Delta}_{\rho}^{2}
  41. Φ 3 ( Φ 0 , a , b ) + Φ 2 ( Φ 1 ( a ) , b ) + ( - 1 ) | a | Φ 2 ( a , Φ 1 ( b ) ) + Φ 1 ( Φ 2 ( a , b ) ) = 0 \Phi^{3}(\Phi^{0},a,b)+\Phi^{2}\left(\Phi^{1}(a),b\right)+(-1)^{|a|}\Phi^{2}% \left(a,\Phi^{1}(b)\right)+\Phi^{1}\left(\Phi^{2}(a,b)\right)=0
  42. Δ ρ {\Delta}_{\rho}
  43. Φ 4 ( Φ 0 , a , b , c ) + Jac ( a , b , c ) + Φ 1 ( Φ 3 ( a , b , c ) ) + Φ 3 ( Φ 1 ( a ) , b , c ) + ( - 1 ) | a | Φ 3 ( a , Φ 1 ( b ) , c ) + ( - 1 ) | a | + | b | Φ 3 ( a , b , Φ 1 ( c ) ) = 0 \Phi^{4}(\Phi^{0},a,b,c)+{\rm Jac}(a,b,c)+\Phi^{1}\left(\Phi^{3}(a,b,c)\right)% +\Phi^{3}\left(\Phi^{1}(a),b,c\right)+(-1)^{\left|a\right|}\Phi^{3}\left(a,% \Phi^{1}(b),c\right)+(-1)^{\left|a\right|+\left|b\right|}\Phi^{3}\left(a,b,% \Phi^{1}(c)\right)=0
  44. \vdots
  45. Φ 2 \Phi^{2}
  46. Jac ( a 1 , a 2 , a 3 ) := 1 2 π S 3 ( - 1 ) | a π | Φ 2 ( Φ 2 ( a π ( 1 ) , a π ( 2 ) ) , a π ( 3 ) ) . {\rm Jac}(a_{1},a_{2},a_{3}):=\frac{1}{2}\sum_{\pi\in S_{3}}(-1)^{\left|a_{\pi% }\right|}\Phi^{2}\left(\Phi^{2}(a_{\pi(1)},a_{\pi(2)}),a_{\pi(3)}\right).
  47. Φ n + 1 \Phi^{n+1}
  48. Jac ( a , b , c ) = 0 {\rm Jac}(a,b,c)=0
  49. π i j \pi^{ij}
  50. ρ \rho
  51. x i x^{i}
  52. i f \partial_{i}f
  53. f i := ( - 1 ) | x i | ( | f | + 1 ) i f f\stackrel{\leftarrow}{\partial}_{i}:=(-1)^{\left|x^{i}\right|(|f|+1)}\partial% _{i}f
  54. x i x^{i}
  55. π i j \pi^{ij}
  56. | π i j | = | x i | + | x j | - 1 \left|\pi^{ij}\right|=\left|x^{i}\right|+\left|x^{j}\right|-1
  57. π j i = - ( - 1 ) ( | x i | + 1 ) ( | x j | + 1 ) π i j \pi^{ji}=-(-1)^{(\left|x^{i}\right|+1)(\left|x^{j}\right|+1)}\pi^{ij}
  58. ( - 1 ) ( | x i | + 1 ) ( | x k | + 1 ) π i π j k + cyclic ( i , j , k ) = 0 (-1)^{(\left|x^{i}\right|+1)(\left|x^{k}\right|+1)}\pi^{i\ell}\partial_{\ell}% \pi^{jk}+{\rm cyclic}(i,j,k)=0
  59. x i x i x^{i}\to x^{\prime i}
  60. π i j \pi^{ij}
  61. ρ \rho
  62. π k = x k i π i j j x \pi^{\prime k\ell}=x^{\prime k}\stackrel{\leftarrow}{\partial}_{i}\pi^{ij}% \partial_{j}x^{\prime\ell}
  63. ρ = ρ / sdet ( i x j ) \rho^{\prime}=\rho/{\rm sdet}(\partial_{i}x^{\prime j})
  64. ( f , g ) := f i π i j j g . (f,g):=f\stackrel{\leftarrow}{\partial}_{i}\pi^{ij}\partial_{j}g.
  65. X f X_{f}
  66. X f [ g ] := ( f , g ) . X_{f}[g]:=(f,g).
  67. X = X i i X=X^{i}\partial_{i}
  68. div ρ X := ( - 1 ) | x i | ( | X | + 1 ) ρ i ( ρ X i ) {\rm div}_{\rho}X:=\frac{(-1)^{\left|x^{i}\right|(|X|+1)}}{\rho}\partial_{i}(% \rho X^{i})
  69. Δ ρ {\Delta}_{\rho}
  70. Δ ρ ( f ) := ( - 1 ) | f | 2 div ρ X f = ( - 1 ) | x i | 2 ρ i ρ π i j j f . {\Delta}_{\rho}(f):=\frac{(-1)^{\left|f\right|}}{2}{\rm div}_{\rho}X_{f}=\frac% {(-1)^{\left|x^{i}\right|}}{2\rho}\partial_{i}\rho\pi^{ij}\partial_{j}f.
  71. π i j \pi^{ij}
  72. ρ \rho
  73. Δ ρ 2 {\Delta}_{\rho}^{2}
  74. Δ ρ {\Delta}_{\rho}
  75. π i j \pi^{ij}
  76. q 1 , , q n q^{1},\ldots,q^{n}
  77. p 1 , , p n p_{1},\ldots,p_{n}
  78. | q i | + | p i | = 1 , \left|q^{i}\right|+\left|p_{i}\right|=1,
  79. ( q i , p j ) = δ j i . (q^{i},p_{j})=\delta^{i}_{j}.
  80. q i q^{i}
  81. p j p_{j}
  82. ϕ i \phi^{i}
  83. ϕ j * \phi^{*}_{j}
  84. Δ π := ( - 1 ) | q i | q i p i \Delta_{\pi}:=(-1)^{\left|q^{i}\right|}\frac{\partial}{\partial q^{i}}\frac{% \partial}{\partial p_{i}}
  85. Δ π \Delta_{\pi}
  86. Δ π 2 = 0 \Delta_{\pi}^{2}=0
  87. ρ \rho
  88. Δ ( f ) := 1 ρ Δ π ( ρ f ) , \Delta(f):=\frac{1}{\sqrt{\rho}}\Delta_{\pi}(\sqrt{\rho}f),
  89. π i j \pi^{ij}
  90. ρ \rho

Bateman–Horn_conjecture.html

  1. P ( x ) C D 2 x d t ( log t ) m , P(x)\sim\frac{C}{D}\int_{2}^{x}\frac{dt}{(\log t)^{m}},\,
  2. C = p 1 - N ( p ) / p ( 1 - 1 / p ) m C=\prod_{p}\frac{1-N(p)/p}{(1-1/p)^{m}}
  3. N ( p ) N(p)
  4. f ( n ) 0 ( mod p ) . f(n)\equiv 0\;\;(\mathop{{\rm mod}}p).
  5. N ( p ) < p N(p)<p
  6. π 2 ( x ) 2 p 3 p ( p - 2 ) ( p - 1 ) 2 x ( log x ) 2 1.32 x ( log x ) 2 . \pi_{2}(x)\sim 2\prod_{p\geq 3}\frac{p(p-2)}{(p-1)^{2}}\frac{x}{(\log x)^{2}}% \approx 1.32\frac{x}{(\log x)^{2}}.
  7. x 3 + u x^{3}+u\,

Batting_average_against.html

  1. B A A = H B F - B B - H B P - S H - S F - C I N T BAA=\frac{H}{BF-BB-HBP-SH-SF-CINT}

Baudhayana_sutras.html

  1. x = a 2 2 - a 2 x={a\over 2}\sqrt{2}-{a\over 2}
  2. a 2 + x 3 {a\over 2}+{x\over 3}
  3. a 2 + a 6 ( 2 - 1 ) {a\over 2}+{a\over 6}(\sqrt{2}-1)
  4. a 6 ( 2 + 2 ) {a\over 6}(2+\sqrt{2})
  5. ( 2 + 2 ) 2 11.66 36.6 π (2+\sqrt{2})^{2}\approx 11.66\approx{36.6\over\pi}
  6. π r 2 π × a 2 6 2 × 36.6 π a 2 {\pi}r^{2}\approx\pi\times{a^{2}\over 6^{2}}\times{36.6\over\pi}\approx a^{2}
  7. 2 1 + 1 3 + 1 3 4 - 1 3 4 34 = 577 408 1.414216 , \sqrt{2}\approx 1+\frac{1}{3}+\frac{1}{3\cdot 4}-\frac{1}{3\cdot 4\cdot 34}=% \frac{577}{408}\approx 1.414216,

Baum–Welch_algorithm.html

  1. i t h i^{th}
  2. ( i - 1 ) t h (i-1)^{th}
  3. X t X_{t}
  4. N N
  5. P ( X t | X t - 1 ) P(X_{t}|X_{t-1})
  6. t t
  7. A = { a i j } = P ( X t = j | X t - 1 = i ) A=\{a_{ij}\}=P(X_{t}=j|X_{t-1}=i)
  8. t = 1 t=1
  9. π i = P ( X 1 = i ) \pi_{i}=P(X_{1}=i)
  10. Y t Y_{t}
  11. K K
  12. t t
  13. j j
  14. b j ( y t ) = P ( Y t = y t | X t = j ) b_{j}(y_{t})=P(Y_{t}=y_{t}|X_{t}=j)
  15. Y t Y_{t}
  16. X t X_{t}
  17. K K
  18. N N
  19. B = { b j ( y i ) } B=\{b_{j}(y_{i})\}
  20. Y = ( Y 1 = y 1 , Y 2 = y 2 , , Y T = y T ) Y=(Y_{1}=y_{1},Y_{2}=y_{2},...,Y_{T}=y_{T})
  21. θ = ( A , B , π ) \theta=(A,B,\pi)
  22. θ * = arg max θ P ( Y | θ ) \theta^{*}=\operatorname*{arg\,max}_{\theta}P(Y|\theta)
  23. θ \theta
  24. θ = ( A , B , π ) \theta=(A,B,\pi)
  25. α i ( t ) = P ( Y 1 = y 1 , , Y t = y t , X t = i | θ ) \alpha_{i}(t)=P(Y_{1}=y_{1},...,Y_{t}=y_{t},X_{t}=i|\theta)
  26. y 1 , y 2 , , y t y_{1},y_{2},...,y_{t}
  27. i i
  28. t t
  29. α i ( 1 ) = π i b i ( y 1 ) \alpha_{i}(1)=\pi_{i}b_{i}(y_{1})
  30. α j ( t + 1 ) = b j ( y t + 1 ) i = 1 N α i ( t ) a i j \alpha_{j}(t+1)=b_{j}(y_{t+1})\sum_{i=1}^{N}\alpha_{i}(t)a_{ij}
  31. β i ( t ) = P ( Y t + 1 = y t + 1 , , Y T = y T | X t = i , θ ) \beta_{i}(t)=P(Y_{t+1}=y_{t+1},...,Y_{T}=y_{T}|X_{t}=i,\theta)
  32. y t + 1 , , y T y_{t+1},...,y_{T}
  33. i i
  34. t t
  35. β i ( t ) \beta_{i}(t)
  36. β i ( T ) = 1 \beta_{i}(T)=1
  37. β i ( t ) = j = 1 N β j ( t + 1 ) a i j b j ( y t + 1 ) \beta_{i}(t)=\sum_{j=1}^{N}\beta_{j}(t+1)a_{ij}b_{j}(y_{t+1})
  38. γ i ( t ) = P ( X t = i | Y , θ ) = α i ( t ) β i ( t ) j = 1 N α j ( t ) β j ( t ) \gamma_{i}(t)=P(X_{t}=i|Y,\theta)=\frac{\alpha_{i}(t)\beta_{i}(t)}{\sum_{j=1}^% {N}\alpha_{j}(t)\beta_{j}(t)}
  39. i i
  40. t t
  41. Y Y
  42. θ \theta
  43. ξ i j ( t ) = P ( X t = i , X t + 1 = j | Y , θ ) = α i ( t ) a i j β j ( t + 1 ) b j ( y t + 1 ) k = 1 N l = 1 N α k ( t ) a k l β l ( t + 1 ) b l ( y t + 1 ) = α i ( t ) a i j β j ( t + 1 ) b j ( y t + 1 ) k = 1 N α k ( t ) β k ( t ) \xi_{ij}(t)=P(X_{t}=i,X_{t+1}=j|Y,\theta)=\frac{\alpha_{i}(t)a_{ij}\beta_{j}(t% +1)b_{j}(y_{t+1})}{\sum_{k=1}^{N}\sum_{l=1}^{N}\alpha_{k}(t)a_{kl}\beta_{l}(t+% 1)b_{l}(y_{t+1})}=\frac{\alpha_{i}(t)a_{ij}\beta_{j}(t+1)b_{j}(y_{t+1})}{\sum_% {k=1}^{N}\alpha_{k}(t)\beta_{k}(t)}
  44. i i
  45. j j
  46. t t
  47. t + 1 t+1
  48. Y Y
  49. θ \theta
  50. θ \theta
  51. π i * = γ i ( 1 ) \pi_{i}^{*}=\gamma_{i}(1)
  52. i i
  53. 1 1
  54. a i j * = t = 1 T - 1 ξ i j ( t ) t = 1 T - 1 γ i ( t ) a_{ij}^{*}=\frac{\sum^{T-1}_{t=1}\xi_{ij}(t)}{\sum^{T-1}_{t=1}\gamma_{i}(t)}
  55. b i * ( v k ) = t = 1 T 1 y t = v k γ i ( t ) t = 1 T γ i ( t ) b_{i}^{*}(v_{k})=\frac{\sum^{T}_{t=1}1_{y_{t}=v_{k}}\gamma_{i}(t)}{\sum^{T}_{t% =1}\gamma_{i}(t)}
  56. 1 y t = v k = { 1 , if y t = v k 0 , otherwise 1_{y_{t}=v_{k}}=\begin{cases}1,&\,\text{if }y_{t}=v_{k}\\ 0,&\,\text{otherwise}\\ \end{cases}
  57. b i * ( v k ) b_{i}^{*}(v_{k})
  58. v k v_{k}
  59. i i
  60. i i
  61. P ( Y | θ f i n a l ) > P ( Y | θ t r u e ) P(Y|\theta_{final})>P(Y|\theta_{true})
  62. 0.22 2.4234 = 0.0908 \frac{0.22}{2.4234}=0.0908
  63. 0.2394 0.2730 = 0.8769 \frac{0.2394}{0.2730}=0.8769

Bayes_factor.html

  1. Pr ( M | D ) = Pr ( D | M ) Pr ( M ) Pr ( D ) . \Pr(M|D)=\frac{\Pr(D|M)\Pr(M)}{\Pr(D)}.
  2. θ 1 \theta_{1}
  3. θ 2 \theta_{2}
  4. K = Pr ( D | M 1 ) Pr ( D | M 2 ) = Pr ( θ 1 | M 1 ) Pr ( D | θ 1 , M 1 ) d θ 1 Pr ( θ 2 | M 2 ) Pr ( D | θ 2 , M 2 ) d θ 2 . K=\frac{\Pr(D|M_{1})}{\Pr(D|M_{2})}=\frac{\int\Pr(\theta_{1}|M_{1})\Pr(D|% \theta_{1},M_{1})\,d\theta_{1}}{\int\Pr(\theta_{2}|M_{2})\Pr(D|\theta_{2},M_{2% })\,d\theta_{2}}.
  5. ( 200 115 ) q 115 ( 1 - q ) 85 . {{200\choose 115}q^{115}(1-q)^{85}}.
  6. P ( X = 115 M 1 ) = ( 200 115 ) ( 1 2 ) 200 = 0.005956... , P(X=115\mid M_{1})={200\choose 115}\left({1\over 2}\right)^{200}=0.005956...,\,
  7. P ( X = 115 M 2 ) = 0 1 ( 200 115 ) q 115 ( 1 - q ) 85 d q = 1 201 = 0.004975.... P(X=115\mid M_{2})=\int_{0}^{1}{200\choose 115}q^{115}(1-q)^{85}dq={1\over 201% }=0.004975....
  8. P ( X = 115 M 2 ) = ( 200 115 ) q 115 ( 1 - q ) 85 = 0.056991 \textstyle P(X=115\mid M_{2})={{200\choose 115}q^{115}(1-q)^{85}}=0.056991

Bäcklund_transform.html

  1. u x = v y , u y = - v x , u_{x}=v_{y},\quad u_{y}=-v_{x},\,
  2. u x x + u y y = 0 u_{xx}+u_{yy}=0
  3. u x y = u y x , v x y = v y x , . u_{xy}=u_{yx},\quad v_{xy}=v_{yx},.\,
  4. u x y = sin u . u_{xy}=\sin u.\,
  5. v x = u x + 2 a sin ( u + v 2 ) v y = - u y + 2 a sin ( v - u 2 ) \begin{aligned}\displaystyle v_{x}&\displaystyle=u_{x}+2a\sin\Bigl(\frac{u+v}{% 2}\Bigr)\\ \displaystyle v_{y}&\displaystyle=-u_{y}+\frac{2}{a}\sin\Bigl(\frac{v-u}{2}% \Bigr)\end{aligned}\,\!
  6. v x = u x + 2 a exp ( u + v 2 ) v y = - u y - 1 a exp ( u - v 2 ) \begin{aligned}\displaystyle v_{x}&\displaystyle=u_{x}+2a\exp\Bigl(\frac{u+v}{% 2}\Bigr)\\ \displaystyle v_{y}&\displaystyle=-u_{y}-\frac{1}{a}\exp\Bigl(\frac{u-v}{2}% \Bigr)\end{aligned}\,\!
  7. u x y = exp u u_{xy}=\exp u\,\!
  8. v x y = 0 v_{xy}=0

Begriffsschrift.html

  1. B A B\rightarrow A
  2. \equiv
  3. A , A \vdash A,\Vdash A
  4. p ( A ) = 1 p(A)=1
  5. p ( A ) = i p(A)=i
  6. ¬ A , A \neg A,\sim A
  7. B A B\rightarrow A
  8. B A B\supset A
  9. x : F ( x ) \forall x\colon F(x)
  10. x : F ( x ) \sim\forall x\colon\sim F(x)
  11. x : F ( x ) \exists x\colon F(x)
  12. A B A\equiv B
  13. A B A\equiv B
  14. A = B A=B
  15. A ( B A ) \vdash\ \ A\rightarrow\left(B\rightarrow A\right)
  16. [ A ( B C ) ] [ ( A B ) ( A C ) ] \vdash\ \ \left[\ A\rightarrow\left(B\rightarrow C\right)\ \right]\ % \rightarrow\ \left[\ \left(A\rightarrow B\right)\rightarrow\left(A\rightarrow C% \right)\ \right]
  17. [ D ( B A ) ] [ B ( D A ) ] \vdash\ \ \left[\ D\rightarrow\left(B\rightarrow A\right)\ \right]\ % \rightarrow\ \left[\ B\rightarrow\left(D\rightarrow A\right)\ \right]
  18. ( B A ) ( ¬ A ¬ B ) \vdash\ \ \left(B\rightarrow A\right)\ \rightarrow\ \left(\lnot A\rightarrow% \lnot B\right)
  19. ¬ ¬ A A \vdash\ \ \lnot\lnot A\rightarrow A
  20. A ¬ ¬ A \vdash\ \ A\rightarrow\lnot\lnot A
  21. ( c = d ) ( f ( c ) f ( d ) ) \vdash\ \ \left(c=d\right)\rightarrow\left(f(c)\rightarrow f(d)\right)
  22. c = c \vdash\ \ c=c
  23. a f ( a ) f ( c ) \vdash\ \ \forall af(a)\rightarrow\ f(c)
  24. B \vdash B
  25. A B \vdash A\to B
  26. A \vdash A
  27. P x A ( x ) \vdash P\to\forall xA(x)
  28. P A ( x ) \vdash P\to A(x)
  29. ¬ \neg

Belief_propagation.html

  1. p X i ( x i ) = 𝐱 : x i = x i p ( 𝐱 ) . p_{X_{i}}(x_{i})=\sum_{\mathbf{x}^{\prime}:x^{\prime}_{i}=x_{i}}p(\mathbf{x}^{% \prime}).
  2. p ( 𝐱 ) = a F f a ( 𝐱 a ) p(\mathbf{x})=\prod_{a\in F}f_{a}(\mathbf{x}_{a})
  3. μ v a \mu_{v\to a}
  4. μ a v \mu_{a\to v}
  5. x v D o m ( v ) , μ v a ( x v ) = a * N ( v ) { a } μ a * v ( x v ) . \forall x_{v}\in Dom(v),\;\mu_{v\to a}(x_{v})=\prod_{a^{*}\in N(v)\setminus\{a% \}}\mu_{a^{*}\to v}(x_{v}).
  6. N ( v ) { a } N(v)\setminus\{a\}
  7. μ v a ( x v ) \mu_{v\to a}(x_{v})
  8. x v D o m ( v ) , μ a v ( x v ) = 𝐱 a : x v = x v f a ( 𝐱 a ) v * N ( a ) { v } μ v * a ( x v * ) . \forall x_{v}\in Dom(v),\;\mu_{a\to v}(x_{v})=\sum_{\mathbf{x}^{\prime}_{a}:x^% {\prime}_{v}=x_{v}}f_{a}(\mathbf{x}^{\prime}_{a})\prod_{v^{*}\in N(a)\setminus% \{v\}}\mu_{v^{*}\to a}(x^{\prime}_{v^{*}}).
  9. N ( a ) { v } N(a)\setminus\{v\}
  10. μ a v ( x v ) = f a ( x v ) \mu_{a\to v}(x_{v})=f_{a}(x_{v})
  11. x v = x a x_{v}=x_{a}
  12. p X v ( x v ) a N ( v ) μ a v ( x v ) . p_{X_{v}}(x_{v})\propto\prod_{a\in N(v)}\mu_{a\to v}(x_{v}).
  13. p X a ( 𝐱 a ) f a ( 𝐱 a ) v N ( a ) μ v a ( x v ) . p_{X_{a}}(\mathbf{x}_{a})\propto f_{a}(\mathbf{x}_{a})\prod_{v\in N(a)}\mu_{v% \to a}(x_{v}).
  14. 𝐱 \mathbf{x}
  15. arg max 𝐱 g ( 𝐱 ) . \operatorname*{arg\,max}_{\mathbf{x}}g(\mathbf{x}).
  16. P ( 𝐗 ) = 1 Z f j f j ( x j ) P(\mathbf{X})=\frac{1}{Z}\prod_{f_{j}}f_{j}(x_{j})
  17. E ( 𝐗 ) = log f j f j ( x j ) . E(\mathbf{X})=\log\prod_{f_{j}}f_{j}(x_{j}).
  18. F = U - H = 𝐗 P ( 𝐗 ) E ( 𝐗 ) + 𝐗 P ( 𝐗 ) log P ( 𝐗 ) . F=U-H=\sum_{\mathbf{X}}P(\mathbf{X})E(\mathbf{X})+\sum_{\mathbf{X}}P(\mathbf{X% })\log P(\mathbf{X}).
  19. P ( x i ) = 1 Z j i exp ( - 1 / 2 x T A x + b T x ) d x j P(x_{i})=\frac{1}{Z}\int_{j\neq i}\exp(-1/2x^{T}Ax+b^{T}x)\,dx_{j}
  20. argmax 𝑥 P ( x ) = 1 Z exp ( - 1 / 2 x T A x + b T x ) . \underset{x}{\operatorname{argmax}}\ P(x)=\frac{1}{Z}\exp(-1/2x^{T}Ax+b^{T}x).
  21. min 𝑥 1 / 2 x T A x - b T x . \underset{x}{\operatorname{min}}\ 1/2x^{T}Ax-b^{T}x.
  22. A x = b . Ax=b.
  23. ρ ( I - | D - 1 / 2 A D - 1 / 2 | ) < 1 \rho(I-|D^{-1/2}AD^{-1/2}|)<1\,

Belief_revision.html

  1. A A
  2. B B
  3. A A
  4. B B
  5. C C
  6. C C
  7. { a b } \{a\vee b\}
  8. ¬ a \neg a
  9. a a
  10. b b
  11. { a b } \{a\vee b\}
  12. ¬ a \neg a
  13. ¬ a \neg a
  14. { a b } \{a\vee b\}
  15. a a
  16. b b
  17. ¬ a \neg a
  18. ¬ a b \neg a\wedge b
  19. a b a\vee b
  20. ¬ a \neg a
  21. ¬ a \neg a
  22. ¬ a b \neg a\wedge b
  23. K K
  24. P P
  25. * *
  26. + +
  27. K + P K+P
  28. K { P } K\cup\{P\}
  29. K * P K*P
  30. P K * P P\in K*P
  31. K * P K + P K*P\subseteq K+P
  32. If ( ¬ P ) K , then K * P = K + P \,\text{If }(\neg P)\not\in K,\,\text{ then }K*P=K+P
  33. K * P K*P
  34. P P
  35. K K
  36. If P and Q are logically equivalent, then K * P = K * Q \,\text{If }P\,\text{ and }Q\,\text{ are logically equivalent, then }K*P=K*Q
  37. K * ( P Q ) ( K * P ) + Q K*(P\wedge Q)\subseteq(K*P)+Q
  38. If ( ¬ Q ) K * P then ( K * P ) + Q K * ( P Q ) \,\text{If }(\neg Q)\not\in K*P\,\text{ then }(K*P)+Q\subseteq K*(P\wedge Q)
  39. K * P K*P
  40. K + P K+P
  41. P P
  42. * *
  43. K \leq_{K}
  44. K K
  45. K K
  46. K \leq_{K}
  47. K * P K*P
  48. P P
  49. K \leq_{K}
  50. K * P K*P
  51. P P
  52. K \leq_{K}
  53. K \leq_{K}
  54. I < K J I<_{K}J
  55. I I
  56. J J
  57. I I
  58. J J
  59. I I
  60. P P
  61. P P
  62. K K
  63. K - P K-P
  64. K * P = ( K - ¬ P ) + P K*P=(K-\neg P)+P
  65. K - P = K ( K * ¬ P ) K-P=K\cap(K*\neg P)
  66. K = ( K - P ) + P K=(K-P)+P
  67. P P
  68. a b a\vee b
  69. a a
  70. a b a\vee b
  71. a a
  72. a > b a>b
  73. a a
  74. b b
  75. K K
  76. a > b a>b
  77. b K * a b\in K*a
  78. > >
  79. a > b a>b
  80. K K
  81. * *
  82. P Q P\vdash Q
  83. K * P Q K*P\models Q
  84. P P
  85. Q Q
  86. Q Q
  87. { a , b } \{a,b\}
  88. { a b } \{a\wedge b\}
  89. ¬ a \neg a
  90. a a
  91. b b
  92. ¬ a \neg a
  93. b b
  94. { ¬ a , b } \{\neg a,b\}
  95. a b a\wedge b
  96. a a
  97. { ¬ a } \{\neg a\}
  98. K K
  99. P P
  100. P P
  101. K K
  102. P P
  103. P P
  104. P P
  105. K K
  106. K K
  107. P P
  108. K { P } K\cup\{P\}
  109. P P
  110. K K
  111. P P
  112. P P
  113. K K
  114. P P
  115. K K
  116. K K
  117. P P
  118. K K
  119. P P
  120. K P K\wedge P
  121. K K
  122. P P
  123. K K^{\prime}
  124. P P
  125. K K
  126. P P
  127. K K
  128. P P
  129. K K
  130. K K
  131. K * P K*P
  132. K * P K*P
  133. K * P * Q K*P*Q
  134. K K
  135. K * P K*P
  136. K K
  137. K K
  138. K * P K*P
  139. P P
  140. α μ \alpha\models\mu
  141. ( ψ * μ ) * α ψ * α (\psi*\mu)*\alpha\equiv\psi*\alpha
  142. α ¬ μ \alpha\models\neg\mu
  143. ( ψ * μ ) * α ψ * α (\psi*\mu)*\alpha\equiv\psi*\alpha
  144. ψ * α μ \psi*\alpha\models\mu
  145. ( ψ * μ ) * α μ (\psi*\mu)*\alpha\models\mu
  146. ψ * α ⊧̸ ¬ μ \psi*\alpha\not\models\neg\mu
  147. ( ψ * μ ) * α ⊧̸ ¬ μ (\psi*\mu)*\alpha\not\models\neg\mu
  148. P P
  149. P P
  150. P P
  151. P P
  152. P P
  153. P P
  154. P P
  155. P P
  156. P P
  157. P P
  158. K K
  159. P P
  160. K K
  161. P P
  162. K K
  163. P P
  164. K K
  165. P P
  166. K K
  167. T T
  168. K K
  169. T T
  170. K T K\vee T
  171. K K
  172. K K
  173. K K
  174. K * P Q K*P\models Q
  175. K K
  176. P P
  177. Q Q
  178. K K
  179. P P
  180. K * P K*P
  181. K K
  182. P P

Bell_state.html

  1. | Φ + |\Phi^{+}\rangle
  2. | Φ + = 1 2 ( | 0 A | 0 B + | 1 A | 1 B ) . |\Phi^{+}\rangle=\frac{1}{\sqrt{2}}(|0\rangle_{A}\otimes|0\rangle_{B}+|1% \rangle_{A}\otimes|1\rangle_{B}).
  3. | + = 1 2 ( | 0 + | 1 ) |+\rangle=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)
  4. | - = 1 2 ( | 0 - | 1 ) |-\rangle=\frac{1}{\sqrt{2}}(|0\rangle-|1\rangle)
  5. | + |+\rangle
  6. | - |-\rangle
  7. | Φ + = 1 2 ( | + A | + B + | - A | - B ) . |\Phi^{+}\rangle=\frac{1}{\sqrt{2}}(|+\rangle_{A}\otimes|+\rangle_{B}+|-% \rangle_{A}\otimes|-\rangle_{B}).
  8. 2 2 2\sqrt{2}
  9. 2 2 2\sqrt{2}
  10. | Φ + = 1 2 ( | 0 A | 0 B + | 1 A | 1 B ) |\Phi^{+}\rangle=\frac{1}{\sqrt{2}}(|0\rangle_{A}\otimes|0\rangle_{B}+|1% \rangle_{A}\otimes|1\rangle_{B})
  11. | Φ - = 1 2 ( | 0 A | 0 B - | 1 A | 1 B ) |\Phi^{-}\rangle=\frac{1}{\sqrt{2}}(|0\rangle_{A}\otimes|0\rangle_{B}-|1% \rangle_{A}\otimes|1\rangle_{B})
  12. | Ψ + = 1 2 ( | 0 A | 1 B + | 1 A | 0 B ) |\Psi^{+}\rangle=\frac{1}{\sqrt{2}}(|0\rangle_{A}\otimes|1\rangle_{B}+|1% \rangle_{A}\otimes|0\rangle_{B})
  13. | Ψ - = 1 2 ( | 0 A | 1 B - | 1 A | 0 B ) . |\Psi^{-}\rangle=\frac{1}{\sqrt{2}}(|0\rangle_{A}\otimes|1\rangle_{B}-|1% \rangle_{A}\otimes|0\rangle_{B}).
  14. n n
  15. 2 n + 1 - 1 2^{n+1}-1
  16. 4 n 4^{n}
  17. | Φ + |\Phi^{+}\rangle
  18. | Φ - |\Phi^{-}\rangle
  19. { | 0 , | 1 } \{|0\rangle,|1\rangle\}
  20. { | + , | - } \{|+\rangle,|-\rangle\}
  21. = 1 2 2 ( ( | + A + | - A ) ( | + B + | - B ) - ( | + A - | - A ) ( | + B - | - B ) ) =\frac{1}{2\sqrt{2}}((|+\rangle_{A}+|-\rangle_{A})(|+\rangle_{B}+|-\rangle_{B}% )-(|+\rangle_{A}-|-\rangle_{A})(|+\rangle_{B}-|-\rangle_{B}))
  22. = 1 2 2 ( | + + + | + - + | - + + | - - - | + + + | + - + | - + - | - - ) =\frac{1}{2\sqrt{2}}(|++\rangle+|+-\rangle+|-+\rangle+|--\rangle-|++\rangle+|+% -\rangle+|-+\rangle-|--\rangle)
  23. = 1 2 ( | + - + | - + ) =\frac{1}{\sqrt{2}}(|+-\rangle+|-+\rangle)
  24. | Ψ + |\Psi^{+}\rangle
  25. | Ψ + |\Psi^{+}\rangle
  26. b 1 b_{1}
  27. b 2 = X . b 1 b_{2}=X.b_{1}
  28. | Φ - |\Phi^{-}\rangle
  29. b 2 b_{2}
  30. | Φ + |\Phi^{+}\rangle
  31. b 1 b_{1}
  32. | Φ - |\Phi^{-}\rangle
  33. Z . b 1 Z.b_{1}
  34. | Ψ + |\Psi^{+}\rangle
  35. X . b 1 X.b_{1}
  36. | Ψ - |\Psi^{-}\rangle
  37. X . Z . b 1 X.Z.b_{1}

Belleville_washer.html

  1. K = k i = 1 g 1 n i K=\frac{k}{\sum_{i=1}^{g}\frac{1}{n_{i}}}
  2. K = k 1 2 + 1 3 + 1 1 + 1 2 K=\frac{k}{\frac{1}{2}+\frac{1}{3}+\frac{1}{1}+\frac{1}{2}}
  3. K = 3 7 k K=\frac{3}{7}k
  4. n i n_{i}

Bendixson–Dulac_theorem.html

  1. C 1 C^{1}
  2. φ ( x , y ) \varphi(x,y)
  3. ( φ f ) x + ( φ g ) y \frac{\partial(\varphi f)}{\partial x}+\frac{\partial(\varphi g)}{\partial y}
  4. 0 \neq 0
  5. d x d t = f ( x , y ) , \frac{dx}{dt}=f(x,y),
  6. d y d t = g ( x , y ) \frac{dy}{dt}=g(x,y)
  7. φ ( x , y ) \varphi(x,y)
  8. ( φ f ) x + ( φ g ) y > 0 \frac{\partial(\varphi f)}{\partial x}+\frac{\partial(\varphi g)}{\partial y}>0
  9. R R
  10. C C
  11. R R
  12. D D
  13. C C
  14. D ( ( φ f ) x + ( φ g ) y ) d x d y = C - φ g d x + φ f d y \iint_{D}{\left(\frac{\partial(\varphi f)}{\partial x}+\frac{\partial(\varphi g% )}{\partial y}\right)dxdy}=\oint_{C}{-\varphi gdx+\varphi fdy}
  15. = C φ ( - y ˙ d x + x ˙ d y ) . =\oint_{C}{\varphi\left(-\dot{y}dx+\dot{x}dy\right)}.
  16. C C
  17. d x = x ˙ d t dx=\dot{x}dt
  18. d y = y ˙ d t dy=\dot{y}dt
  19. C C

Bernstein's_inequality_(mathematical_analysis).html

  1. n n
  2. max | z | 1 ( | P ( z ) | ) n max | z | 1 ( | P ( z ) | ) \max_{|z|\leq 1}(|P^{\prime}(z)|)\leq n\cdot\max_{|z|\leq 1}(|P(z)|)
  3. max | z | 1 ( | P ( k ) ( z ) | ) n ! ( n - k ) ! max | z | 1 ( | P ( z ) | ) . \max_{|z|\leq 1}(|P^{(k)}(z)|)\leq\frac{n!}{(n-k)!}\cdot\max_{|z|\leq 1}(|P(z)% |).

Bertrand's_ballot_theorem.html

  1. p - q p + q . \frac{p-q}{p+q}.
  2. 2 10 = 1 5 , \frac{2}{10}=\frac{1}{5},
  3. 3 - 2 3 + 2 \frac{3-2}{3+2}
  4. ( p + q p ) {\textstyle\left({{p+q}\atop{p}}\right)}
  5. ( p + q - 1 p - 1 ) - ( p + q - 1 p ) {\textstyle\left({{p+q-1}\atop{p-1}}\right)}-{\textstyle\left({{p+q-1}\atop{p}% }\right)}
  6. p p + q - q p + q = p - q p + q \tfrac{p}{p+q}-\tfrac{q}{p+q}=\tfrac{p-q}{p+q}
  7. ( n n + m 2 ) - ( n n + m 2 + 1 ) = m + 1 n + m 2 + 1 ( n n + m 2 ) . {\left({{n}\atop{\tfrac{n+m}{2}}}\right)}-{\left({{n}\atop{\tfrac{n+m}{2}+1}}% \right)}=\frac{m+1}{\tfrac{n+m}{2}+1}{\left({{n}\atop{\tfrac{n+m}{2}}}\right)}.
  8. 1 n 2 + 1 ( n n 2 ) \frac{1}{\tfrac{n}{2}+1}{\left({{n}\atop{\tfrac{n}{2}}}\right)}
  9. q / ( p + q ) q/(p+q)
  10. = 1 - =1-
  11. = 1 - =1-
  12. = 1 - 2 q p + q = p - q p + q =1-2\frac{q}{p+q}=\frac{p-q}{p+q}
  13. a ( a + b ) ( a - 1 ) - b ( a + b - 1 ) + b ( a + b ) a - ( b - 1 ) ( a + b - 1 ) = a - b a + b . {a\over(a+b)}{(a-1)-b\over(a+b-1)}+{b\over(a+b)}{a-(b-1)\over(a+b-1)}={a-b% \over a+b}.
  14. 2 m - μ μ \frac{2m-\mu}{\mu}
  15. μ = p + q \mu=p+q
  16. m = p m=p
  17. P m + 1 , μ + 1 = P m , μ + P m + 1 , μ , P_{m+1,\mu+1}=P_{m,\mu}+P_{m+1,\mu},
  18. P m , μ P_{m,\mu}
  19. ( p + q - 1 q - 1 ) {\textstyle\left({{p+q-1}\atop{q-1}}\right)}
  20. ( p + q q ) - 2 ( p + q - 1 q - 1 ) = ( p + q q ) p - q p + q {\left({{p+q}\atop{q}}\right)}-2{\left({{p+q-1}\atop{q-1}}\right)}={\left({{p+% q}\atop{q}}\right)}\frac{p-q}{p+q}
  21. p - q p + q \frac{p-q}{p+q}
  22. p + 1 - q p + 1 . \frac{p+1-q}{p+1}.
  23. ( p + q q ) {\textstyle\left({{p+q}\atop{q}}\right)}
  24. ( p + q q - 1 ) {\textstyle\left({{p+q}\atop{q-1}}\right)}
  25. ( p + q q ) - ( p + q q - 1 ) = ( p + q q ) p + 1 - q p + 1 . {\left({{p+q}\atop{q}}\right)}-{\left({{p+q}\atop{q-1}}\right)}={\left({{p+q}% \atop{q}}\right)}\frac{p+1-q}{p+1}.
  26. ( p + q q ) {\textstyle\left({{p+q}\atop{q}}\right)}
  27. p + 1 - q p + 1 \tfrac{p+1-q}{p+1}
  28. p p + q p - 1 + 1 - q p - 1 + 1 = p - q p + q \frac{p}{p+q}\frac{p-1+1-q}{p-1+1}=\frac{p-q}{p+q}
  29. p + 1 - q p + 1 + q ( p + 1 + q q ) \tfrac{p+1-q}{p+1+q}{\textstyle\left({{p+1+q}\atop{q}}\right)}
  30. p + 1 - q p + 1 ( p + q q ) \tfrac{p+1-q}{p+1}{\textstyle\left({{p+q}\atop{q}}\right)}
  31. p + 1 - q p + 1 \tfrac{p+1-q}{p+1}

Best_response.html

  1. b ( ) b(\cdot)
  2. σ - i \sigma_{-i}
  3. b i ( σ - i ) b_{i}(\sigma_{-i})
  4. σ - i \sigma_{-i}
  5. e E ( 1 ) / γ e E ( 1 ) / γ + e E ( 2 ) / γ \frac{e^{E(1)/\gamma}}{e^{E(1)/\gamma}+e^{E(2)/\gamma}}
  6. E ( x ) E(x)
  7. x x
  8. γ \gamma
  9. γ \gamma

Beta-peptide.html

  1. C α \mathrm{C}^{\alpha}
  2. C β \mathrm{C}^{\beta}
  3. C β \mathrm{C}^{\beta}

Beta_(finance).html

  1. r a α + β r b r_{a}\approx\alpha+\beta r_{b}
  2. r a , t = α + β r b , t + ε t r_{a,t}=\alpha+\beta r_{b,t}+\varepsilon_{t}
  3. β = Cov ( r a , r b ) Var ( r b ) \beta=\frac{\mathrm{Cov}(r_{a},r_{b})}{\mathrm{Var}(r_{b})}
  4. β = ρ a , b ( σ a / σ b ) \beta=\rho_{a,b}(\sigma_{a}/\sigma_{b})
  5. SCL : r a , t = α a + β a r m , t + ε a , t \mathrm{SCL}:r_{a,t}=\alpha_{a}+\beta_{a}r_{m,t}+\varepsilon_{a,t}
  6. α a \alpha_{a}
  7. β a \beta_{a}
  8. SML : E ( R i ) - R f = β i ( E ( R M ) - R f ) . \mathrm{SML}:E(R_{i})-R_{f}=\beta_{i}(E(R_{M})-R_{f}).~{}
  9. Z = ( 1 - δ ) X + δ Y . Z=(1-\delta)X+\delta Y.
  10. Var ( Z ) = ( 1 - δ ) 2 Var ( X ) + 2 δ ( 1 - δ ) Cov ( X , Y ) + δ 2 Var ( Y ) \mathrm{Var}(Z)=(1-\delta)^{2}\mathrm{Var}(X)+2\delta(1-\delta)\mathrm{Cov}(X,% Y)+\delta^{2}\mathrm{Var}(Y)
  11. Var ( Z ) ( 1 - 2 δ ) Var ( X ) + 2 δ Cov ( X , Y ) . \mathrm{Var}(Z)\approx(1-2\delta)\mathrm{Var}(X)+2\delta\mathrm{Cov}(X,Y).
  12. β = Cov ( X , Y ) / Var ( X ) , \beta=\mathrm{Cov}(X,Y)/\mathrm{Var}(X),
  13. Var ( Z ) / Var ( X ) 1 + 2 δ ( β - 1 ) . \mathrm{Var}(Z)/\mathrm{Var}(X)\approx 1+2\delta(\beta-1).
  14. K E = R F + β E ( R M - R F ) K_{E}=R_{F}+\beta_{E}(R_{M}-R_{F})
  15. β E = β = [ β A - β D ( D V ) ] V E \beta_{E}=\beta=\left[\beta_{A}-\beta_{D}\left(\frac{D}{V}\right)\right]\frac{% V}{E}
  16. β A = β D ( D V ) + β E ( E V ) \beta_{A}=\beta_{D}\left(\frac{D}{V}\right)+\beta_{E}\left(\frac{E}{V}\right)

Beta_function_(physics).html

  1. β ( g ) = g log ( μ ) , \beta(g)=\frac{\partial g}{\partial\log(\mu)}~{},
  2. β ( e ) = e 3 12 π 2 , \beta(e)=\frac{e^{3}}{12\pi^{2}}~{},
  3. β ( α ) = 2 α 2 3 π , \beta(\alpha)=\frac{2\alpha^{2}}{3\pi}~{},
  4. n f n_{f}
  5. β ( g ) = - ( 11 - 2 n f 3 ) g 3 16 π 2 , \beta(g)=-\left(11-\frac{2n_{f}}{3}\right)\frac{g^{3}}{16\pi^{2}}~{},
  6. β ( α s ) = - ( 11 - 2 n f 3 ) α s 2 2 π , \beta(\alpha_{s})=-\left(11-\frac{2n_{f}}{3}\right)\frac{\alpha_{s}^{2}}{2\pi}% ~{},
  7. g 2 4 π \frac{g^{2}}{4\pi}
  8. S U ( 3 ) SU(3)
  9. N c N_{c}
  10. G = S U ( N c ) G=SU(N_{c})
  11. R R
  12. G G
  13. β ( g ) = - ( 11 3 C 2 ( G ) - 4 3 n f C ( R ) ) g 3 16 π 2 , \beta(g)=-\left(\frac{11}{3}C_{2}(G)-\frac{4}{3}n_{f}C(R)\right)\frac{g^{3}}{1% 6\pi^{2}}~{},
  14. C 2 ( G ) C_{2}(G)
  15. G G
  16. C ( R ) C(R)
  17. T r ( T R a T R b ) = C ( R ) δ a b Tr(T^{a}_{R}T^{b}_{R})=C(R)\delta^{ab}
  18. T R a , b T^{a,b}_{R}
  19. G G
  20. C 2 ( G ) = N c C_{2}(G)=N_{c}
  21. G G
  22. C ( R ) = 1 / 2 C(R)=1/2
  23. N c = 3 N_{c}=3

Beta_Pictoris.html

  1. L V L V = 10 0.4 ( M V - M V ) \scriptstyle\frac{L_{V_{\ast}}}{L_{V_{\odot}}}=10^{0.4(M_{V_{\odot}}-M_{V_{% \ast}})}
  2. Distance in parsecs = 1 parallax in arcseconds \scriptstyle\mathrm{Distance\ in\ parsecs}=\frac{1}{\mathrm{parallax\ in\ % arcseconds}}
  3. P rot = 2 π r v rot \scriptstyle P_{\mathrm{rot}}=\frac{2\pi r}{v_{\mathrm{rot}}}

Bhattacharyya_distance.html

  1. D B ( p , q ) = - ln ( B C ( p , q ) ) D_{B}(p,q)=-\ln\left(BC(p,q)\right)
  2. B C ( p , q ) = x X p ( x ) q ( x ) BC(p,q)=\sum_{x\in X}\sqrt{p(x)q(x)}
  3. B C ( p , q ) = p ( x ) q ( x ) d x BC(p,q)=\int\sqrt{p(x)q(x)}\,dx
  4. 0 B C 1 0\leq BC\leq 1
  5. 0 D B 0\leq D_{B}\leq\infty
  6. D B D_{B}
  7. 1 - B C \sqrt{1-BC}
  8. D B ( p , q ) = 1 4 ln ( 1 4 ( σ p 2 σ q 2 + σ q 2 σ p 2 + 2 ) ) + 1 4 ( ( μ p - μ q ) 2 σ p 2 + σ q 2 ) D_{B}(p,q)=\frac{1}{4}\ln\left(\frac{1}{4}\left(\frac{\sigma_{p}^{2}}{\sigma_{% q}^{2}}+\frac{\sigma_{q}^{2}}{\sigma_{p}^{2}}+2\right)\right)+\frac{1}{4}\left% (\frac{(\mu_{p}-\mu_{q})^{2}}{\sigma_{p}^{2}+\sigma_{q}^{2}}\right)
  9. D B ( p , q ) D_{B}(p,q)
  10. σ p \sigma_{p}
  11. μ p \mu_{p}
  12. p , q p,q
  13. p i = 𝒩 ( s y m b o l μ i , s y m b o l Σ i ) p_{i}=\mathcal{N}(symbol\mu_{i},\,symbol\Sigma_{i})
  14. D B = 1 8 ( s y m b o l μ 1 - s y m b o l μ 2 ) T s y m b o l Σ - 1 ( s y m b o l μ 1 - s y m b o l μ 2 ) + 1 2 ln ( det s y m b o l Σ det s y m b o l Σ 1 det s y m b o l Σ 2 ) , D_{B}={1\over 8}(symbol\mu_{1}-symbol\mu_{2})^{T}symbol\Sigma^{-1}(symbol\mu_{% 1}-symbol\mu_{2})+{1\over 2}\ln\,\left({\det symbol\Sigma\over\sqrt{\det symbol% \Sigma_{1}\,\det symbol\Sigma_{2}}}\right),
  15. s y m b o l μ i symbol\mu_{i}
  16. s y m b o l Σ i symbol\Sigma_{i}
  17. s y m b o l Σ = s y m b o l Σ 1 + s y m b o l Σ 2 2 . symbol\Sigma={symbol\Sigma_{1}+symbol\Sigma_{2}\over 2}.
  18. B C ( 𝐩 , 𝐪 ) = i = 1 n p i q i , BC(\mathbf{p},\mathbf{q})=\sum_{i=1}^{n}\sqrt{p_{i}q_{i}},
  19. p i p_{i}
  20. q i q_{i}

Bhāskara_I.html

  1. sin x 16 x ( π - x ) 5 π 2 - 4 x ( π - x ) , ( 0 x π 2 ) \sin x\approx\frac{16x(\pi-x)}{5\pi^{2}-4x(\pi-x)},\qquad(0\leq x\leq\frac{\pi% }{2})
  2. 16 5 π - 1 1.859 % \frac{16}{5\pi}-1\approx 1.859\%
  3. x = 0 x=0
  4. 8 x 2 + 1 = y 2 8x^{2}+1=y^{2}

Biaxial_nematic.html

  1. D 2 h D_{2h}
  2. C 2 C_{2}
  3. Q = ( - 1 2 S + T 0 0 0 - 1 2 S - T 0 0 0 S ) Q=\begin{pmatrix}-\frac{1}{2}S+T&0&0\\ 0&-\frac{1}{2}S-T&0\\ 0&0&S\\ \end{pmatrix}
  4. S S
  5. T T

BIBO_stability.html

  1. B > 0 B>0
  2. B B
  3. | y [ n ] | B n \ |y[n]|\leq B\quad\forall n\in\mathbb{Z}
  4. | y ( t ) | B t \ |y(t)|\leq B\quad\forall t\in\mathbb{R}
  5. - | h ( t ) | d t = h 1 < \int_{-\infty}^{\infty}\left|h(t)\right|\,\mathord{\operatorname{d}}t=\|h\|_{1% }<\infty
  6. 1 \ell^{1}
  7. n = - | h [ n ] | = h 1 < \ \sum_{n=-\infty}^{\infty}\left|h[n]\right|=\|h\|_{1}<\infty
  8. h [ n ] \ h[n]
  9. x [ n ] \ x[n]
  10. y [ n ] \ y[n]
  11. y [ n ] = h [ n ] * x [ n ] \ y[n]=h[n]*x[n]
  12. * *
  13. y [ n ] = k = - h [ k ] x [ n - k ] \ y[n]=\sum_{k=-\infty}^{\infty}h[k]x[n-k]
  14. x \|x\|_{\infty}
  15. | x [ n ] | \ |x[n]|
  16. L L_{\infty}
  17. | y [ n ] | = | k = - h [ n - k ] x [ k ] | \left|y[n]\right|=\left|\sum_{k=-\infty}^{\infty}h[n-k]x[k]\right|
  18. k = - | h [ n - k ] | | x [ k ] | \leq\sum_{k=-\infty}^{\infty}\left|h[n-k]\right|\left|x[k]\right|
  19. k = - | h [ n - k ] | x \displaystyle\leq\sum_{k=-\infty}^{\infty}\left|h[n-k]\right|\|x\|_{\infty}
  20. h [ n ] h[n]
  21. k = - | h [ k ] | = h 1 < \sum_{k=-\infty}^{\infty}{\left|h[k]\right|}=\|h\|_{1}<\infty
  22. x k = - | h [ k ] | = x h 1 \|x\|_{\infty}\sum_{k=-\infty}^{\infty}\left|h[k]\right|=\|x\|_{\infty}\|h\|_{1}
  23. h [ n ] h[n]
  24. | x [ n ] | \left|x[n]\right|
  25. | y [ n ] | \left|y[n]\right|
  26. x h 1 < \|x\|_{\infty}\|h\|_{1}<\infty
  27. - | h ( t ) | dt \displaystyle\int_{-\infty}^{\infty}\left|h(t)\right|\,\operatorname{dt}
  28. s = σ + j ω s=\sigma+j\omega
  29. Re ( s ) = σ = 0 \operatorname{Re}(s)=\sigma=0
  30. n = - | h [ n ] | \displaystyle\sum_{n=-\infty}^{\infty}\left|h[n]\right|
  31. z = r e j ω z=re^{j\omega}
  32. r = | z | = 1 r=|z|=1

Bicuspid_aortic_valve.html

  1. h 2 h^{2}

Bid–ask_spread.html

  1. offer - bid offer × 100 % \frac{\hbox{offer}-\hbox{bid}}{\hbox{offer}}\times 100\%

Bifundamental_representation.html

  1. S U ( M ) × S U ( N ) SU(M)\times SU(N)

Bigram.html

  1. P ( W n | W n - 1 ) = P ( W n - 1 , W n ) P ( W n - 1 ) P(W_{n}|W_{n-1})={P(W_{n-1},W_{n})\over P(W_{n-1})}
  2. P ( ) P()
  3. W n W_{n}
  4. W n - 1 W_{n-1}
  5. P ( W n - 1 , W n ) P(W_{n-1},W_{n})

Bijective_proof.html

  1. ( n k ) = ( n n - k ) . {n\choose k}={n\choose n-k}.
  2. ( n k ) = ( n - 1 k - 1 ) + ( n - 1 k ) for 1 k n - 1. {n\choose k}={n-1\choose k-1}+{n-1\choose k}\,\text{ for }1\leq k\leq n-1.
  3. ( n - 1 k - 1 ) {n-1\choose k-1}
  4. ( n - 1 k ) {n-1\choose k}
  5. ( n - 1 k - 1 ) + ( n - 1 k ) {n-1\choose k-1}+{n-1\choose k}
  6. \Box

Bilinear_filtering.html

  1. f ( x , y ) = c 11 x y + c 10 x + c 01 y + c 00 f(x,y)=c_{11}xy+c_{10}x+c_{01}y+c_{00}
  2. f ( x 1 , y 1 ) = z 11 f ( x 1 , y 2 ) = z 12 f ( x 2 , y 1 ) = z 21 f ( x 2 , y 2 ) = z 22 \begin{array}[]{lcl}f(x_{1},y_{1})=z_{11}\\ f(x_{1},y_{2})=z_{12}\\ f(x_{2},y_{1})=z_{21}\\ f(x_{2},y_{2})=z_{22}\\ \end{array}
  3. f f
  4. f 1 f_{1}
  5. f 2 f_{2}
  6. f 1 ( y 1 ) = z 11 f 1 ( y 2 ) = z 12 f 2 ( y 1 ) = z 21 f 2 ( y 2 ) = z 22 \begin{array}[]{lcl}f_{1}(y_{1})=z_{11}\\ f_{1}(y_{2})=z_{12}\\ f_{2}(y_{1})=z_{21}\\ f_{2}(y_{2})=z_{22}\\ \end{array}
  7. y y
  8. f f
  9. f ( x 1 , y ) = f 1 ( y ) f ( x 2 , y ) = f 2 ( y ) \begin{array}[]{lcl}f(x_{1},y)=f_{1}(y)\\ f(x_{2},y)=f_{2}(y)\\ \end{array}
  10. z i j z_{ij}
  11. f f
  12. f ( x , y ) f(x,y)
  13. [ x ] [x]
  14. x x
  15. { x } \{x\}
  16. x = [ x ] + { x } x=[x]+\{x\}
  17. { x } < 1 \{x\}<1
  18. x 1 = [ x ] x_{1}=[x]
  19. x 2 = [ x ] + 1 x_{2}=[x]+1
  20. y 1 = [ y ] y_{1}=[y]
  21. y 2 = [ y ] + 1 y_{2}=[y]+1
  22. z 11 z_{11}
  23. z 12 z_{12}
  24. z 21 z_{21}
  25. z 22 z_{22}
  26. f 1 ( y 1 ) = z 11 f_{1}(y_{1})=z_{11}
  27. f 1 ( y 2 ) = z 12 f_{1}(y_{2})=z_{12}
  28. f 1 f_{1}
  29. f 1 ( y 2 ) - f 1 ( y 1 ) = z 12 - z 11 f_{1}(y_{2})-f_{1}(y_{1})=z_{12}-z_{11}
  30. f 1 f_{1}
  31. ( z 12 - z 11 ) / ( y 2 - y 1 ) = z 12 - z 11 (z_{12}-z_{11})/(y_{2}-y_{1})=z_{12}-z_{11}
  32. f 1 ( y 1 ) = z 11 f_{1}(y_{1})=z_{11}
  33. f 1 ( y 1 + { y } ) = z 11 + { y } ( z 12 - z 11 ) f_{1}(y_{1}+\{y\})=z_{11}+\{y\}(z_{12}-z_{11})
  34. f 2 ( y 1 + { y } ) = z 21 + { y } ( z 22 - z 21 ) f_{2}(y_{1}+\{y\})=z_{21}+\{y\}(z_{22}-z_{21})
  35. y 1 + { y } = y y_{1}+\{y\}=y
  36. f 1 ( y ) f_{1}(y)
  37. f 2 ( y ) f_{2}(y)
  38. f ( x , y ) f(x,y)
  39. f ( x , y ) = f 1 ( y ) + { x } ( f 2 ( y ) - f 1 ( y ) ) f(x,y)=f_{1}(y)+\{x\}(f_{2}(y)-f_{1}(y))
  40. f ( x , y ) = ( 1 - { x } ) ( ( 1 - { y } ) z 11 + { y } z 12 ) + { x } ( ( 1 - { y } ) z 21 + { y } z 22 ) f(x,y)=(1-\{x\})((1-\{y\})z_{11}+\{y\}z_{12})+\{x\}((1-\{y\})z_{21}+\{y\}z_{22})

Bilinear_interpolation.html

  1. f ( x , y 1 ) x 2 - x x 2 - x 1 f ( Q 11 ) + x - x 1 x 2 - x 1 f ( Q 21 ) f ( x , y 2 ) x 2 - x x 2 - x 1 f ( Q 12 ) + x - x 1 x 2 - x 1 f ( Q 22 ) \begin{aligned}\displaystyle f(x,y_{1})&\displaystyle\approx\frac{x_{2}-x}{x_{% 2}-x_{1}}f(Q_{11})+\frac{x-x_{1}}{x_{2}-x_{1}}f(Q_{21})\\ \displaystyle f(x,y_{2})&\displaystyle\approx\frac{x_{2}-x}{x_{2}-x_{1}}f(Q_{1% 2})+\frac{x-x_{1}}{x_{2}-x_{1}}f(Q_{22})\end{aligned}
  2. f ( x , y ) \displaystyle f(x,y)
  3. f ( x , y ) a 0 + a 1 x + a 2 y + a 3 x y f(x,y)\approx a_{0}+a_{1}x+a_{2}y+a_{3}xy
  4. [ 1 x 1 y 1 x 1 y 1 1 x 1 y 2 x 1 y 2 1 x 2 y 1 x 2 y 1 1 x 2 y 2 x 2 y 2 ] [ a 0 a 1 a 2 a 3 ] = [ f ( Q 11 ) f ( Q 12 ) f ( Q 21 ) f ( Q 22 ) ] \begin{aligned}\displaystyle\begin{bmatrix}1&x_{1}&y_{1}&x_{1}y_{1}\\ 1&x_{1}&y_{2}&x_{1}y_{2}\\ 1&x_{2}&y_{1}&x_{2}y_{1}\\ 1&x_{2}&y_{2}&x_{2}y_{2}\end{bmatrix}\begin{bmatrix}a_{0}\\ a_{1}\\ a_{2}\\ a_{3}\end{bmatrix}=\begin{bmatrix}f(Q_{11})\\ f(Q_{12})\\ f(Q_{21})\\ f(Q_{22})\end{bmatrix}\end{aligned}
  5. f ( x , y ) b 11 f ( Q 11 ) + b 12 f ( Q 12 ) + b 21 f ( Q 21 ) + b 22 f ( Q 22 ) f(x,y)\approx b_{11}f(Q_{11})+b_{12}f(Q_{12})+b_{21}f(Q_{21})+b_{22}f(Q_{22})
  6. [ b 11 b 12 b 21 b 22 ] = ( [ 1 x 1 y 1 x 1 y 1 1 x 1 y 2 x 1 y 2 1 x 2 y 1 x 2 y 1 1 x 2 y 2 x 2 y 2 ] - 1 ) T [ 1 x y x y ] \begin{bmatrix}b_{11}\\ b_{12}\\ b_{21}\\ b_{22}\end{bmatrix}=\left(\begin{bmatrix}1&x_{1}&y_{1}&x_{1}y_{1}\\ 1&x_{1}&y_{2}&x_{1}y_{2}\\ 1&x_{2}&y_{1}&x_{2}y_{1}\\ 1&x_{2}&y_{2}&x_{2}y_{2}\end{bmatrix}^{-1}\right)^{T}\begin{bmatrix}1\\ x\\ y\\ xy\end{bmatrix}
  7. ( 0 , 0 ) , ( 0 , 1 ) , ( 1 , 0 ) (0,0),(0,1),(1,0)
  8. f ( x , y ) f ( 0 , 0 ) ( 1 - x ) ( 1 - y ) + f ( 1 , 0 ) x ( 1 - y ) + f ( 0 , 1 ) ( 1 - x ) y + f ( 1 , 1 ) x y . f(x,y)\approx f(0,0)(1-x)(1-y)+f(1,0)x(1-y)+f(0,1)(1-x)y+f(1,1)xy.
  9. f ( x , y ) [ 1 - x x ] [ f ( 0 , 0 ) f ( 0 , 1 ) f ( 1 , 0 ) f ( 1 , 1 ) ] [ 1 - y y ] . f(x,y)\approx\begin{bmatrix}1-x&x\end{bmatrix}\begin{bmatrix}f(0,0)&f(0,1)\\ f(1,0)&f(1,1)\end{bmatrix}\begin{bmatrix}1-y\\ y\end{bmatrix}.
  10. f ( x , y ) = i = 0 1 j = 0 1 a i j x i y j = a 00 + a 10 x + a 01 y + a 11 x y f(x,y)=\sum_{i=0}^{1}\sum_{j=0}^{1}a_{ij}x^{i}y^{j}=a_{00}+a_{10}x+a_{01}y+a_{% 11}xy
  11. a 00 = f ( 0 , 0 ) a_{00}=f(0,0)
  12. a 10 = f ( 1 , 0 ) - f ( 0 , 0 ) a_{10}=f(1,0)-f(0,0)
  13. a 01 = f ( 0 , 1 ) - f ( 0 , 0 ) a_{01}=f(0,1)-f(0,0)
  14. a 11 = f ( 1 , 1 ) + f ( 0 , 0 ) - ( f ( 1 , 0 ) + f ( 0 , 1 ) ) a_{11}=f(1,1)+f(0,0)-(f(1,0)+f(0,1))
  15. I 20 , 14.5 = 15 - 14.5 15 - 14 91 + 14.5 - 14 15 - 14 210 = 150.5 I 21 , 14.5 = 15 - 14.5 15 - 14 162 + 14.5 - 14 15 - 14 95 = 128.5 \begin{aligned}\displaystyle I_{20,14.5}&\displaystyle=\tfrac{15-14.5}{15-14}% \cdot 91+\tfrac{14.5-14}{15-14}\cdot 210=150.5\\ \displaystyle I_{21,14.5}&\displaystyle=\tfrac{15-14.5}{15-14}\cdot 162+\tfrac% {14.5-14}{15-14}\cdot 95=128.5\end{aligned}
  16. I 20.2 , 14.5 = 21 - 20.2 21 - 20 150.5 + 20.2 - 20 21 - 20 128.5 = 146.1 I_{20.2,14.5}=\tfrac{21-20.2}{21-20}\cdot 150.5+\tfrac{20.2-20}{21-20}\cdot 12% 8.5=146.1

Bilunabirotunda.html

  1. ( 0 , 0 , ± φ 2 ) \left(0,0,\pm\frac{\varphi}{2}\right)
  2. ( ± ( φ + 1 ) 2 , ± 1 2 , 0 ) \left(\pm\frac{(\varphi+1)}{2},\pm\frac{1}{2},0\right)
  3. ( ± 1 2 , ± φ 2 , ± 1 2 ) \left(\pm\frac{1}{2},\pm\frac{\varphi}{2},\pm\frac{1}{2}\right)
  4. φ = 1 + 5 2 \varphi=\frac{1+\sqrt{5}}{2}

Binomial_series.html

  1. x = 0 x=0
  2. f f
  3. f ( x ) = ( 1 + x ) α f(x)=(1+x)^{\alpha}
  4. α \alpha\in\mathbb{C}
  5. ( 1 + x ) α = k = 0 ( α k ) x k ( 1 ) = 1 + α x + α ( α - 1 ) 2 ! x 2 + , \begin{aligned}\displaystyle(1+x)^{\alpha}&\displaystyle=\sum_{k=0}^{\infty}\;% {\alpha\choose k}\;x^{k}\qquad\qquad\qquad(1)\\ &\displaystyle=1+\alpha x+\frac{\alpha(\alpha-1)}{2!}x^{2}+\cdots,\end{aligned}
  6. ( α k ) := α ( α - 1 ) ( α - 2 ) ( α - k + 1 ) k ! . {\alpha\choose k}:=\frac{\alpha(\alpha-1)(\alpha-2)\cdots(\alpha-k+1)}{k!}.
  7. 1 ( 1 - z ) β + 1 = k = 0 ( k + β k ) z k . \frac{1}{(1-z)^{\beta+1}}=\sum_{k=0}^{\infty}{k+\beta\choose k}z^{k}.
  8. α α
  9. x x
  10. α \alpha
  11. | x | = 1 |x|=1
  12. R e ( α ) > 0 Re(α)>0
  13. - 1 Align l t ; R e ( α ) 0 -1&lt;Re(α)≤0
  14. x 1 x≠−1
  15. x = 1 x=−1
  16. R e ( α ) - 1 Re(α)≤-1
  17. ( α 0 ) \displaystyle{\alpha\choose 0}
  18. ( α k ) = ( - 1 ) k Γ ( - α ) k 1 + α ( 1 + o ( 1 ) ) , as k . ( 4 ) {\alpha\choose k}=\frac{(-1)^{k}}{\Gamma(-\alpha)k^{1+\alpha}}\,(1+o(1)),\quad% \,\text{as }k\to\infty.\qquad\qquad(4)
  19. Γ ( z ) = lim k k ! k z z ( z + 1 ) ( z + k ) , \Gamma(z)=\lim_{k\to\infty}\frac{k!\;k^{z}}{z\;(z+1)\cdots(z+k)},\qquad
  20. m k 1 + Re α | ( α k ) | M k 1 + Re α , ( 5 ) \frac{m}{k^{1+\operatorname{Re}\,\alpha}}\leq\left|{\alpha\choose k}\right|% \leq\frac{M}{k^{1+\operatorname{Re}\,\alpha}},\qquad\qquad(5)
  21. k = 1 1 k p , \sum_{k=1}^{\infty}\;\frac{1}{k^{p}},\qquad
  22. ( 1 + x ) k = 0 n ( α k ) x k = k = 0 n ( α + 1 k ) x k + ( α n ) x n + 1 , (1+x)\sum_{k=0}^{n}\;{\alpha\choose k}\;x^{k}=\sum_{k=0}^{n}\;{\alpha+1\choose k% }\;x^{k}+{\alpha\choose n}\;x^{n+1},
  23. | ( α k ) x k | 1 , \left|{\alpha\choose k}\;x^{k}\right|\geq 1,
  24. k = 0 n ( α k ) ( - 1 ) k = ( α - 1 n ) ( - 1 ) n , \sum_{k=0}^{n}\;{\alpha\choose k}\;(-1)^{k}={\alpha-1\choose n}\;(-1)^{n},
  25. m - ( k - 1 ) k \tfrac{m-(k-1)}{k}
  26. ( 1 - x 2 ) 1 / 2 = 1 - x 2 2 - x 4 8 - x 6 16 (1-x^{2})^{1/2}=1-\frac{x^{2}}{2}-\frac{x^{4}}{8}-\frac{x^{6}}{16}\cdots
  27. ( 1 - x 2 ) 3 / 2 = 1 - 3 x 2 2 + 3 x 4 8 + x 6 16 (1-x^{2})^{3/2}=1-\frac{3x^{2}}{2}+\frac{3x^{4}}{8}+\frac{x^{6}}{16}\cdots
  28. ( 1 - x 2 ) 1 / 3 = 1 - x 2 3 - x 4 9 - 5 x 6 81 (1-x^{2})^{1/3}=1-\frac{x^{2}}{3}-\frac{x^{4}}{9}-\frac{5x^{6}}{81}\cdots
  29. | ( α k ) | M k 1 + Re α , k 1 \left|{\alpha\choose k}\right|\leq\frac{M}{k^{1+\mathrm{Re}\,\alpha}},\qquad% \forall k\geq 1
  30. M := exp ( | α | 2 + Re α ) M:=\exp\left(|\alpha|^{2}+\mathrm{Re}\,\alpha\right)
  31. | ( α k ) | 2 = j = 1 k | 1 - 1 + α j | 2 ( 1 k j = 1 k | 1 - 1 + α j | 2 ) k . \left|{\alpha\choose k}\right|^{2}=\prod_{j=1}^{k}\left|1-\frac{1+\alpha}{j}% \right|^{2}\leq\left(\frac{1}{k}\sum_{j=1}^{k}\left|1-\frac{1+\alpha}{j}\right% |^{2}\right)^{k}.
  32. | 1 - ζ | 2 = 1 - 2 R e ζ + | ζ | 2 \textstyle|1-\zeta|^{2}=1-2\mathrm{Re}\,\zeta+|\zeta|^{2}
  33. 1 k j = 1 k | 1 - 1 + α j | 2 = 1 + 1 k ( - 2 ( 1 + Re α ) j = 1 k 1 j + | 1 + α | 2 j = 1 k 1 j 2 ) . \frac{1}{k}\sum_{j=1}^{k}\left|1-\frac{1+\alpha}{j}\right|^{2}=1+\frac{1}{k}% \left(-2(1+\mathrm{Re}\,\alpha)\sum_{j=1}^{k}\frac{1}{j}+|1+\alpha|^{2}\sum_{j% =1}^{k}\frac{1}{j^{2}}\right)\ .
  34. ( 1 + r k ) k e r , \left(1+\frac{r}{k}\right)^{k}\leq\mathrm{e}^{r},
  35. j = 1 k 1 j 1 + log k ; j = 1 k 1 j 2 2. \sum_{j=1}^{k}\frac{1}{j}\leq 1+\log k;\qquad\sum_{j=1}^{k}\frac{1}{j^{2}}\leq 2.
  36. | ( α k ) | 2 exp ( - 2 ( 1 + Re α ) ( 1 + log k ) + 2 | 1 + α | 2 ) = M 2 k 2 ( 1 + Re α ) \left|{\alpha\choose k}\right|^{2}\leq\exp\left(-2(1+\mathrm{Re}\,\alpha)(1+% \log k)+2|1+\alpha|^{2}\right)=\frac{M^{2}}{k^{2(1+\mathrm{Re}\,\alpha)}}
  37. M := exp ( | α | 2 + Re α ) , M:=\exp\left(|\alpha|^{2}+\mathrm{Re}\,\alpha\right),\,

Bioavailability.html

  1. F a b s = 100 A U C p o D i v A U C i v D p o F_{abs}=100\cdot\frac{AUC_{po}\cdot D_{iv}}{AUC_{iv}\cdot D_{po}}
  2. F r e l = 100 A U C A D B A U C B D A F_{rel}=100\cdot\frac{AUC_{A}\cdot D_{B}}{AUC_{B}\cdot D_{A}}

Biorthogonal_system.html

  1. v ~ i \tilde{v}_{i}
  2. E E
  3. u ~ i \tilde{u}_{i}
  4. F F
  5. v ~ i , u ~ j = δ i , j , \langle\tilde{v}_{i},\tilde{u}_{j}\rangle=\delta_{i,j},
  6. , \langle,\rangle
  7. δ i , j \delta_{i,j}
  8. E = F E=F
  9. v ~ i = u ~ i \tilde{v}_{i}=\tilde{u}_{i}
  10. P := i I u ~ i v ~ i P:=\sum_{i\in I}\tilde{u}_{i}\otimes\tilde{v}_{i}
  11. ( u v ) ( x ) := u v , x \left(u\otimes v\right)(x):=u\langle v,x\rangle
  12. { u ~ i : i I } \{\tilde{u}_{i}:i\in I\}
  13. { v ~ i , = 0 : i I } \{\langle\tilde{v}_{i},\cdot\rangle=0:i\in I\}
  14. 𝐮 = ( u i ) \mathbf{u}=(u_{i})
  15. 𝐯 = ( v i ) \mathbf{v}=(v_{i})
  16. P = i , j u i ( 𝐯 , 𝐮 - 1 ) j , i v j P=\sum_{i,j}u_{i}\left(\langle\mathbf{v},\mathbf{u}\rangle^{-1}\right)_{j,i}% \otimes v_{j}
  17. 𝐯 , 𝐮 \langle\mathbf{v},\mathbf{u}\rangle
  18. ( 𝐯 , 𝐮 ) i , j = v i , u j \left(\langle\mathbf{v},\mathbf{u}\rangle\right)_{i,j}=\langle v_{i},u_{j}\rangle
  19. u ~ i := ( I - P ) u i \tilde{u}_{i}:=(I-P)u_{i}
  20. v ~ i := ( I - P ) * v i \tilde{v}_{i}:=\left(I-P\right)^{*}v_{i}

Biosynthesis.html

  1. R e a c t a n t e n z y m e P r o d u c t Reactant\xrightarrow[enzyme]{}Product
  2. P r e c u r s o r m o l e c u l e + A T P p r o d u c t A M P + P P i Precursor~{}molecule+ATP\rightleftharpoons{}product~{}AMP+PP_{i}
  3. P r e c u r s o r m o l e c u l e + C o f a c t o r e n z y m e m a c r o m o l e c u l e Precursor~{}molecule+Cofactor\xrightarrow[enzyme]{}macromolecule
  4. M o l e c u l e 1 + M o l e c u l e 2 m a c r o m o l e c u l e Molecule~{}1+Molecule~{}2\xrightarrow{}macromolecule
  5. A m i n o a c i d + A T P a m i n o a c y l A M P + P P i Amino~{}acid+ATP\rightleftharpoons{}aminoacyl~{}AMP+PP_{i}
  6. A m i n o a c y l A M P + t R N A a m i n o a c y l t R N A + A M P Aminoacyl~{}AMP+tRNA\rightleftharpoons{}aminoacyl~{}tRNA+AMP

Bipolar_coordinates.html

  1. x = a sinh τ cosh τ - cos σ x=a\ \frac{\sinh\tau}{\cosh\tau-\cos\sigma}
  2. y = a sin σ cosh τ - cos σ y=a\ \frac{\sin\sigma}{\cosh\tau-\cos\sigma}
  3. τ = ln d 1 d 2 \tau=\ln\frac{d_{1}}{d_{2}}
  4. x + i y = a i cot ( σ + i τ 2 ) x+iy=ai\cot\left(\frac{\sigma+i\tau}{2}\right)
  5. x 2 + ( y - a cot σ ) 2 = a 2 sin 2 σ x^{2}+\left(y-a\cot\sigma\right)^{2}=\frac{a^{2}}{\sin^{2}\sigma}
  6. τ \tau
  7. y 2 + ( x - a coth τ ) 2 = a 2 sinh 2 τ y^{2}+\left(x-a\coth\tau\right)^{2}=\frac{a^{2}}{\sinh^{2}\tau}
  8. π - σ = 2 arctan 2 a y a 2 - x 2 - y 2 + ( a 2 - x 2 - y 2 ) 2 + 4 a 2 y 2 . \pi-\sigma=2\arctan\frac{2ay}{a^{2}-x^{2}-y^{2}+\sqrt{(a^{2}-x^{2}-y^{2})^{2}+% 4a^{2}y^{2}}}.
  9. tanh τ = 2 a x x 2 + y 2 + a 2 \tanh\tau=\frac{2ax}{x^{2}+y^{2}+a^{2}}
  10. tan σ = 2 a y x 2 + y 2 - a 2 . \tan\sigma=\frac{2ay}{x^{2}+y^{2}-a^{2}}.
  11. h σ = h τ = a cosh τ - cos σ h_{\sigma}=h_{\tau}=\frac{a}{\cosh\tau-\cos\sigma}
  12. d A = a 2 ( cosh τ - cos σ ) 2 d σ d τ dA=\frac{a^{2}}{\left(\cosh\tau-\cos\sigma\right)^{2}}\,d\sigma\,d\tau
  13. 2 Φ = 1 a 2 ( cosh τ - cos σ ) 2 ( 2 Φ σ 2 + 2 Φ τ 2 ) \nabla^{2}\Phi=\frac{1}{a^{2}}\left(\cosh\tau-\cos\sigma\right)^{2}\left(\frac% {\partial^{2}\Phi}{\partial\sigma^{2}}+\frac{\partial^{2}\Phi}{\partial\tau^{2% }}\right)
  14. 𝐅 \nabla\cdot\mathbf{F}
  15. × 𝐅 \nabla\times\mathbf{F}
  16. x x

Birch–Murnaghan_equation_of_state.html

  1. P ( V ) = 3 B 0 2 [ ( V 0 V ) 7 3 - ( V 0 V ) 5 3 ] { 1 + 3 4 ( B 0 - 4 ) [ ( V 0 V ) 2 3 - 1 ] } . P(V)=\frac{3B_{0}}{2}\left[\left(\frac{V_{0}}{V}\right)^{\frac{7}{3}}-\left(% \frac{V_{0}}{V}\right)^{\frac{5}{3}}\right]\left\{1+\frac{3}{4}\left(B_{0}^{% \prime}-4\right)\left[\left(\frac{V_{0}}{V}\right)^{\frac{2}{3}}-1\right]% \right\}.
  2. B 0 = - V ( P V ) P = 0 B_{0}=-V\left(\frac{\partial P}{\partial V}\right)_{P=0}
  3. B 0 = ( B P ) P = 0 B_{0}^{\prime}=\left(\frac{\partial B}{\partial P}\right)_{P=0}
  4. f = 1 2 [ ( V V 0 ) - 2 3 - 1 ] . f=\frac{1}{2}\left[\left(\frac{V}{V_{0}}\right)^{-\frac{2}{3}}-1\right]\,.
  5. E ( V ) = E 0 + 9 V 0 B 0 16 { [ ( V 0 V ) 2 3 - 1 ] 3 B 0 + [ ( V 0 V ) 2 3 - 1 ] 2 [ 6 - 4 ( V 0 V ) 2 3 ] } . E(V)=E_{0}+\frac{9V_{0}B_{0}}{16}\left\{\left[\left(\frac{V_{0}}{V}\right)^{% \frac{2}{3}}-1\right]^{3}B_{0}^{\prime}+\left[\left(\frac{V_{0}}{V}\right)^{% \frac{2}{3}}-1\right]^{2}\left[6-4\left(\frac{V_{0}}{V}\right)^{\frac{2}{3}}% \right]\right\}.

Bird_strike.html

  1. E k E_{k}
  2. E k = 1 2 m v 2 E_{k}=\frac{1}{2}mv^{2}
  3. m m
  4. v v

Birkhoff's_axioms.html

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

Bisection_method.html

  1. f ( x ) = x π f(x)=x−π
  2. a a
  3. b b
  4. f ( a ) f(a)
  5. f ( b ) f(b)
  6. a = 1 a=1
  7. b = 2 b=2
  8. f ( 1 ) = ( 1 ) 3 - ( 1 ) - 2 = - 2 f(1)=(1)^{3}-(1)-2=-2
  9. f ( 2 ) = ( 2 ) 3 - ( 2 ) - 2 = + 4 . f(2)=(2)^{3}-(2)-2=+4\,.
  10. a 1 = 1 a_{1}=1
  11. b 1 = 2 b_{1}=2
  12. c 1 = 2 + 1 2 = 1.5 c_{1}=\frac{2+1}{2}=1.5
  13. f ( c 1 ) = ( 1.5 ) 3 - ( 1.5 ) - 2 = - 0.125 f(c_{1})=(1.5)^{3}-(1.5)-2=-0.125
  14. f ( c 1 ) f(c_{1})
  15. a = 1 a=1
  16. a = 1.5 a=1.5
  17. f ( a ) f(a)
  18. f ( b ) f(b)
  19. a a
  20. b b
  21. a n a_{n}
  22. b n b_{n}
  23. c n c_{n}
  24. f ( c n ) f(c_{n})
  25. | c n - c | | b - a | 2 n . |c_{n}-c|\leq\frac{|b-a|}{2^{n}}.
  26. n = log 2 ( ϵ 0 ϵ ) = log ϵ 0 - log ϵ log 2 , n=\log_{2}\left(\frac{\epsilon_{0}}{\epsilon}\right)=\frac{\log\epsilon_{0}-% \log\epsilon}{\log 2},
  27. ϵ 0 = initial bracket size = b - a . \epsilon_{0}=\,\text{initial bracket size}=b-a.
  28. ϵ n + 1 = constant × ϵ n m , m = 1. \epsilon_{n+1}=\,\text{constant}\times\epsilon_{n}^{m},\ m=1.

Bishop–Gromov_inequality.html

  1. M M
  2. Ric ( n - 1 ) K \mathrm{Ric}\geq(n-1)K\,
  3. K K\in\mathbb{R}
  4. M K n M_{K}^{n}
  5. K K
  6. ( n - 1 ) K (n-1)K
  7. M K n M_{K}^{n}
  8. 1 / K 1/\sqrt{K}
  9. K = 0 K=0
  10. K < 0 K<0
  11. p M p\in M
  12. p K M K n p_{K}\in M_{K}^{n}
  13. ϕ ( r ) = Vol B ( p , r ) Vol B ( p K , r ) \phi(r)=\frac{\mathrm{Vol}\,B(p,r)}{\mathrm{Vol}\,B(p_{K},r)}
  14. Vol B ( p , r ) Vol B ( p K , r ) . \mathrm{Vol}\,B(p,r)\leq\mathrm{Vol}\,B(p_{K},r).
  15. r r
  16. p p

Bivector.html

  1. ( 𝐚𝐛 ) 𝐜 = 𝐚 ( 𝐛𝐜 ) (\mathbf{ab})\mathbf{c}=\mathbf{a}(\mathbf{bc})
  2. 𝐚 ( 𝐛 + 𝐜 ) = 𝐚𝐛 + 𝐚𝐜 \mathbf{a}(\mathbf{b}+\mathbf{c})=\mathbf{ab}+\mathbf{ac}
  3. ( 𝐛 + 𝐜 ) 𝐚 = 𝐛𝐚 + 𝐜𝐚 (\mathbf{b}+\mathbf{c})\mathbf{a}=\mathbf{ba}+\mathbf{ca}
  4. 𝐚 2 = Q ( 𝐚 ) = ϵ 𝐚 | 𝐚 | 2 {\mathbf{a}}^{2}=Q(\mathbf{a})=\epsilon_{\mathbf{a}}{\left|\mathbf{a}\right|}^% {2}
  5. 𝐚 2 = | 𝐚 | 2 {\mathbf{a}}^{2}={\left|\mathbf{a}\right|}^{2}
  6. 1 2 ( 𝐚𝐛 + 𝐛𝐚 ) = 1 2 ( ( 𝐚 + 𝐛 ) 2 - 𝐚 2 - 𝐛 2 ) \frac{1}{2}(\mathbf{ab}+\mathbf{ba})=\frac{1}{2}((\mathbf{a}+\mathbf{b})^{2}-% \mathbf{a}^{2}-\mathbf{b}^{2})
  7. 𝐚 𝐛 = 1 2 ( 𝐚𝐛 + 𝐛𝐚 ) . \mathbf{a}\cdot\mathbf{b}=\frac{1}{2}(\mathbf{ab}+\mathbf{ba}).
  8. 𝐚 𝐛 = 1 2 ( 𝐚𝐛 - 𝐛𝐚 ) \mathbf{a}\wedge\mathbf{b}=\frac{1}{2}(\mathbf{ab}-\mathbf{ba})
  9. 𝐛 𝐚 = 1 2 ( 𝐛𝐚 - 𝐚𝐛 ) = - 1 2 ( 𝐚𝐛 - 𝐛𝐚 ) = - 𝐚 𝐛 \mathbf{b}\wedge\mathbf{a}=\frac{1}{2}(\mathbf{ba}-\mathbf{ab})=-\frac{1}{2}(% \mathbf{ab}-\mathbf{ba})=-\mathbf{a}\wedge\mathbf{b}
  10. 𝐚 𝐛 + 𝐚 𝐛 = 1 2 ( 𝐚𝐛 + 𝐛𝐚 ) + 1 2 ( 𝐚𝐛 - 𝐛𝐚 ) = 𝐚𝐛 \mathbf{a}\cdot\mathbf{b}+\mathbf{a}\wedge\mathbf{b}=\frac{1}{2}(\mathbf{ab}+% \mathbf{ba})+\frac{1}{2}(\mathbf{ab}-\mathbf{ba})=\mathbf{ab}
  11. ( 𝐚 𝐛 ) 2 - ( 𝐚 𝐛 ) 2 = 𝐚 2 𝐛 2 , (\mathbf{a}\cdot\mathbf{b})^{2}-(\mathbf{a}\wedge\mathbf{b})^{2}=\mathbf{a}^{2% }\mathbf{b}^{2},
  12. ( 𝐚 𝐛 ) 2 = ( 𝐚 𝐛 ) 2 - 𝐚 2 𝐛 2 = | 𝐚 | 2 | 𝐛 | 2 ( cos 2 θ - 1 ) = - | 𝐚 | 2 | 𝐛 | 2 sin 2 θ (\mathbf{a}\wedge\mathbf{b})^{2}=(\mathbf{a}\cdot\mathbf{b})^{2}-\mathbf{a}^{2% }\mathbf{b}^{2}=\left|\mathbf{a}\right|^{2}\left|\mathbf{b}\right|^{2}(\cos^{2% }\theta-1)=-\left|\mathbf{a}\right|^{2}\left|\mathbf{b}\right|^{2}\sin^{2}\theta
  13. 𝐀 = 𝐚 𝐛 = - 𝐛 𝐚 , \mathbf{A}=\mathbf{a}\wedge\mathbf{b}=-\mathbf{b}\wedge\mathbf{a}\ ,
  14. 𝒢 n \mathcal{G}_{n}
  15. 𝐁 | 𝐁 | . \frac{\mathbf{B}}{\left|\mathbf{B}\right|}.
  16. 𝐁 = 𝐞 1 𝐞 2 + 𝐞 3 𝐞 4 = 𝐞 1 𝐞 2 + 𝐞 3 𝐞 4 = 𝐞 12 + 𝐞 34 \mathbf{B}=\mathbf{e}_{1}\wedge\mathbf{e}_{2}+\mathbf{e}_{3}\wedge\mathbf{e}_{% 4}=\mathbf{e}_{1}\mathbf{e}_{2}+\mathbf{e}_{3}\mathbf{e}_{4}=\mathbf{e}_{12}+% \mathbf{e}_{34}
  17. 𝐀𝐁 = 𝐀 𝐁 + 𝐀 × 𝐁 + 𝐀 𝐁 . \mathbf{A}\mathbf{B}=\mathbf{A}\cdot\mathbf{B}+\mathbf{A}\times\mathbf{B}+% \mathbf{A}\wedge\mathbf{B}.
  18. 𝐀 × 𝐁 = 1 2 ( 𝐀𝐁 - 𝐁𝐀 ) , \mathbf{A}\times\mathbf{B}=\frac{1}{2}(\mathbf{AB}-\mathbf{BA}),
  19. 𝐀𝐀 = 𝐀 𝐀 + 𝐀 𝐀 . \mathbf{A}\mathbf{A}=\mathbf{A}\cdot\mathbf{A}+\mathbf{A}\wedge\mathbf{A}.
  20. 𝐚𝐛 \displaystyle\mathbf{a}\mathbf{b}
  21. 𝐚 𝐛 \displaystyle\mathbf{a}\cdot\mathbf{b}
  22. 𝐞 12 2 = - 1 , \mathbf{e}_{12}^{2}=-1,
  23. θ 𝐞 12 = i θ , \theta\mathbf{e}_{12}=i\theta,
  24. e θ 𝐞 12 = e i θ = cos θ + i sin θ , e^{\theta\mathbf{e}_{12}}=e^{i\theta}=\cos{\theta}+i\sin{\theta},
  25. ( x 𝐞 1 + y 𝐞 2 ) = ( x 𝐞 1 + y 𝐞 2 ) e i θ . (x^{\prime}\mathbf{e}_{1}+y^{\prime}\mathbf{e}_{2})=(x\mathbf{e}_{1}+y\mathbf{% e}_{2})e^{i\theta}.
  26. 𝐯 = 𝐯 e i θ = e - i θ 𝐯 = e - i θ 2 𝐯 e i θ 2 . \mathbf{v}^{\prime}=\mathbf{v}e^{i\theta}=e^{-i\theta}\mathbf{v}=e^{\frac{-i% \theta}{2}}\mathbf{v}e^{\frac{i\theta}{2}}.
  27. R = e - i θ 2 = e - θ 𝐞 12 2 , R=e^{\frac{-i\theta}{2}}=e^{\frac{-\theta\mathbf{e}_{12}}{2}},
  28. 𝐯 = R 𝐯 R - 1 . \mathbf{v}^{\prime}=R\mathbf{v}R^{-1}.\,
  29. 𝐚𝐛 \displaystyle\mathbf{ab}
  30. 𝐚 𝐛 \displaystyle\mathbf{a}\cdot\mathbf{b}
  31. 𝐀 = A 23 𝐞 23 + A 31 𝐞 31 + A 12 𝐞 12 , \mathbf{A}=A_{23}\mathbf{e}_{23}+A_{31}\mathbf{e}_{31}+A_{12}\mathbf{e}_{12},
  32. 𝐀 + 𝐁 = ( A 23 + B 23 ) 𝐞 23 + ( A 31 + B 31 ) 𝐞 31 + ( A 12 + B 12 ) 𝐞 12 . \mathbf{A}+\mathbf{B}=(A_{23}+B_{23})\mathbf{e}_{23}+(A_{31}+B_{31})\mathbf{e}% _{31}+(A_{12}+B_{12})\mathbf{e}_{12}.
  33. 𝐀𝐁 = - A 23 B 23 - A 31 B 31 - A 12 B 12 + ( A 12 B 31 - A 31 B 12 ) 𝐞 23 + ( A 23 B 12 - A 12 B 23 ) 𝐞 31 + ( A 31 B 23 - A 23 B 31 ) 𝐞 12 \mathbf{A}\mathbf{B}=-A_{23}B_{23}-A_{31}B_{31}-A_{12}B_{12}+(A_{12}B_{31}-A_{% 31}B_{12})\mathbf{e}_{23}+(A_{23}B_{12}-A_{12}B_{23})\mathbf{e}_{31}+(A_{31}B_% {23}-A_{23}B_{31})\mathbf{e}_{12}
  34. 𝐀 𝐁 \displaystyle\mathbf{A}\cdot\mathbf{B}
  35. e 𝐁 = e β 𝐁 β = cos β + 𝐁 β sin β . e^{\mathbf{B}}=e^{\beta\frac{\mathbf{B}}{\beta}}=\cos{\beta}+\frac{\mathbf{B}}% {\beta}\sin{\beta}.
  36. ( a + 𝐀 ) ( b + 𝐁 ) = a b + a 𝐁 + b 𝐀 + 𝐀 𝐁 + 𝐀 × 𝐁 . (a+\mathbf{A})(b+\mathbf{B})=ab+a\mathbf{B}+b\mathbf{A}+\mathbf{A}\cdot\mathbf% {B}+\mathbf{A}\times\mathbf{B}.
  37. q = ( cos ( θ 2 ) , ω sin ( θ 2 ) ) q=\left(\cos\left(\frac{\theta}{2}\right),\omega\sin\left(\frac{\theta}{2}% \right)\right)
  38. e s y m b o l Ω θ 2 = cos ( θ 2 ) + s y m b o l Ω sin ( θ 2 ) e^{\frac{symbol{\Omega}\theta}{2}}=\cos\left(\frac{\theta}{2}\right)+symbol{% \Omega}\sin\left(\frac{\theta}{2}\right)
  39. 𝐯 = e s y m b o l Ω θ 2 𝐯 e - s y m b o l Ω θ 2 . \mathbf{v}^{\prime}=e^{\frac{symbol{\Omega}\theta}{2}}\mathbf{v}e^{-\frac{% symbol{\Omega}\theta}{2}}.
  40. 𝐯 = R 𝐯 R - 1 . \mathbf{v}^{\prime}=R\mathbf{v}R^{-1}.\,
  41. M B = ( 0 - B 12 B 31 B 12 0 - B 23 - B 31 B 23 0 ) . M_{B}=\begin{pmatrix}0&-B_{12}&B_{31}\\ B_{12}&0&-B_{23}\\ -B_{31}&B_{23}&0\end{pmatrix}.
  42. M R = e M B . M_{R}=e^{M_{B}}.
  43. R = e B 2 , R=e^{\frac{B}{2}},
  44. M R = ( ( R 𝐞 1 R - 1 ) 𝐞 1 ( R 𝐞 2 R - 1 ) 𝐞 1 ( R 𝐞 3 R - 1 ) 𝐞 1 ( R 𝐞 1 R - 1 ) 𝐞 2 ( R 𝐞 2 R - 1 ) 𝐞 2 ( R 𝐞 3 R - 1 ) 𝐞 2 ( R 𝐞 1 R - 1 ) 𝐞 3 ( R 𝐞 2 R - 1 ) 𝐞 3 ( R 𝐞 3 R - 1 ) 𝐞 3 ) . M_{R}=\begin{pmatrix}(R\mathbf{e}_{1}R^{-1})\cdot\mathbf{e}_{1}&(R\mathbf{e}_{% 2}R^{-1})\cdot\mathbf{e}_{1}&(R\mathbf{e}_{3}R^{-1})\cdot\mathbf{e}_{1}\\ (R\mathbf{e}_{1}R^{-1})\cdot\mathbf{e}_{2}&(R\mathbf{e}_{2}R^{-1})\cdot\mathbf% {e}_{2}&(R\mathbf{e}_{3}R^{-1})\cdot\mathbf{e}_{2}\\ (R\mathbf{e}_{1}R^{-1})\cdot\mathbf{e}_{3}&(R\mathbf{e}_{2}R^{-1})\cdot\mathbf% {e}_{3}&(R\mathbf{e}_{3}R^{-1})\cdot\mathbf{e}_{3}\end{pmatrix}.
  45. 𝐀 = * 𝐚 , 𝐚 = * 𝐀 \mathbf{A}=*\mathbf{a}\,,\quad\mathbf{a}=*\mathbf{A}
  46. 𝐀 = 𝐚 i , 𝐚 = - 𝐀 i . \mathbf{A}=\mathbf{a}i\,,\quad\mathbf{a}=-\mathbf{A}i.
  47. 𝐚 × 𝐛 = | 𝐞 1 𝐞 2 𝐞 3 a 1 a 2 a 3 b 1 b 2 b 3 | , 𝐚 𝐛 = | 𝐞 23 𝐞 31 𝐞 12 a 1 a 2 a 3 b 1 b 2 b 3 | , \mathbf{a}\times\mathbf{b}=\begin{vmatrix}\mathbf{e}_{1}&\mathbf{e}_{2}&% \mathbf{e}_{3}\\ a_{1}&a_{2}&a_{3}\\ b_{1}&b_{2}&b_{3}\end{vmatrix}\,,\quad\mathbf{a}\wedge\mathbf{b}=\begin{% vmatrix}\mathbf{e}_{23}&\mathbf{e}_{31}&\mathbf{e}_{12}\\ a_{1}&a_{2}&a_{3}\\ b_{1}&b_{2}&b_{3}\end{vmatrix}\ ,
  48. * ( 𝐚 𝐛 ) = 𝐚 × 𝐛 , * ( 𝐚 × 𝐛 ) = 𝐚 𝐛 . {*(\mathbf{a}\wedge\mathbf{b})}=\mathbf{a\times b}\,,\quad{*(\mathbf{a\times b% })}=\mathbf{a}\wedge\mathbf{b}.
  49. a x b y - b x a y a_{x}b_{y}-b_{x}a_{y}
  50. | 𝐁 | = | 𝐚 | | 𝐛 | sin θ , \left|\mathbf{B}\right|=\left|\mathbf{a}\right|\left|\mathbf{b}\right|\sin{% \theta},
  51. 𝐀𝐁 = 𝐀 𝐁 + 𝐀 × 𝐁 . \mathbf{AB}=\mathbf{A}\cdot\mathbf{B}+\mathbf{A}\times\mathbf{B}.
  52. 𝐁 \displaystyle\mathbf{B}
  53. 𝐁 + 𝐂 \displaystyle\mathbf{B}+\mathbf{C}
  54. 𝐀 = a 12 𝐞 12 + a 13 𝐞 13 + a 14 𝐞 14 + a 23 𝐞 23 + a 24 𝐞 24 + a 34 𝐞 34 . \mathbf{A}=a_{12}\mathbf{e}_{12}+a_{13}\mathbf{e}_{13}+a_{14}\mathbf{e}_{14}+a% _{23}\mathbf{e}_{23}+a_{24}\mathbf{e}_{24}+a_{34}\mathbf{e}_{34}.
  55. ( 𝐞 12 + 𝐞 34 ) 2 = 𝐞 12 𝐞 12 + 𝐞 12 𝐞 34 + 𝐞 34 𝐞 12 + 𝐞 34 𝐞 34 = - 2 + 2 𝐞 1234 . (\mathbf{e}_{12}+\mathbf{e}_{34})^{2}=\mathbf{e}_{12}\mathbf{e}_{12}+\mathbf{e% }_{12}\mathbf{e}_{34}+\mathbf{e}_{34}\mathbf{e}_{12}+\mathbf{e}_{34}\mathbf{e}% _{34}=-2+2\mathbf{e}_{1234}.
  56. 𝐞 12 + 𝐞 34 = 𝐞 1 𝐞 2 + 𝐞 3 𝐞 4 . \mathbf{e}_{12}+\mathbf{e}_{34}=\mathbf{e}_{1}\wedge\mathbf{e}_{2}+\mathbf{e}_% {3}\wedge\mathbf{e}_{4}.
  57. v = R v R - 1 . v^{\prime}=RvR^{-1}.\,
  58. R = e 𝐁 1 + 𝐁 2 2 = e 𝐁 1 2 e 𝐁 2 2 = e 𝐁 2 2 e 𝐁 1 2 R=e^{\frac{\mathbf{B}_{1}+\mathbf{B}_{2}}{2}}=e^{\frac{\mathbf{B}_{1}}{2}}e^{% \frac{\mathbf{B}_{2}}{2}}=e^{\frac{\mathbf{B}_{2}}{2}}e^{\frac{\mathbf{B}_{1}}% {2}}
  59. 𝐞 i 2 = { 1 , i = 1 , 2 , 3 - 1 , i = 4 {\mathbf{e}_{i}}^{2}=\begin{cases}1,&i=1,2,3\\ -1,&i=4\end{cases}
  60. e s y m b o l Ω θ 2 = cosh ( θ 2 ) + s y m b o l Ω sinh ( θ 2 ) , e^{\frac{symbol{\Omega}\theta}{2}}=\cosh\left(\frac{\theta}{2}\right)+symbol{% \Omega}\sinh\left(\frac{\theta}{2}\right),
  61. R = e 𝐀 2 . R=e^{\frac{\mathbf{A}}{2}}.
  62. 𝐅 = 1 c E ¯ 𝐞 4 + B ¯ 𝐞 123 , \mathbf{F}=\frac{1}{c}\overline{E}\mathbf{e}_{4}+\overline{B}\mathbf{e}_{123},
  63. 𝐉 = j ¯ + c ρ 𝐞 4 , \mathbf{J}=\overline{j}+c\rho\mathbf{e}_{4},
  64. = - 𝐞 4 1 c t . \partial=\nabla-\mathbf{e}_{4}\frac{1}{c}\frac{\partial}{\partial t}.
  65. 𝐅 = 𝐉 . \partial\mathbf{F}=\mathbf{J}.
  66. 𝐀 = A ¯ + 1 c V 𝐞 4 , \mathbf{A}=\overline{A}+\frac{1}{c}V\mathbf{e}_{4},
  67. 𝐀 = - 𝐅 , \partial\mathbf{A}=-\mathbf{F},
  68. 𝐞 12 + 𝐞 34 + 𝐞 56 \mathbf{e}_{12}+\mathbf{e}_{34}+\mathbf{e}_{56}
  69. e 𝐁 2 e^{\frac{\mathbf{B}}{2}}
  70. 𝐩 = 𝐀𝐁 = ( 𝐀 × 𝐁 ) J - 1 . \mathbf{p}=\mathbf{A}\mathbf{B}=(\mathbf{A}\times\mathbf{B})J^{-1}.
  71. 𝐩 and 𝐂 = 0. \mathbf{p}\and\mathbf{C}=0.
  72. ( 𝐀𝐁 ) and 𝐂 = 0 , (\mathbf{A}\mathbf{B})\and\mathbf{C}=0,
  73. 𝐀𝐁𝐂 = 0 , \langle\mathbf{ABC}\rangle=0,

Black-body_radiation.html

  1. i w , t = 2 h c 2 w 5 ( exp ( h c / w k t ) - 1 ) i_{w,t}=\frac{2hc^{2}}{w^{5}(\exp(hc/wkt)-1)}
  2. I ( ν , T ) = 2 h ν 3 c 2 1 e h ν k T - 1 I(\nu,T)=\frac{2h\nu^{3}}{c^{2}}\frac{1}{e^{\frac{h\nu}{kT}}-1}
  3. λ max \lambda_{\max}
  4. λ max = b T \lambda_{\max}=\frac{b}{T}
  5. ν max = T × 58.8 GHz K - 1 \nu_{\max}=T\times 58.8\ \mathrm{GHz}\ \mathrm{K}^{-1}
  6. j = σ T 4 , j^{\star}=\sigma T^{4},
  7. I ( ν , T ) I(\nu,T)
  8. j = 0 d ν d Ω cos θ I ( ν , T ) j^{\star}=\int_{0}^{\infty}d\nu\int d\Omega\cos\theta\cdot I(\nu,T)
  9. cos θ \cos\theta
  10. 2 π 2\pi
  11. ϕ \phi
  12. θ \theta
  13. d Ω cos θ = 0 2 π d ϕ 0 π / 2 d θ sin θ cos θ = π \int d\Omega\cos\theta=\int_{0}^{2\pi}d\phi\int_{0}^{\pi/2}d\theta\sin\theta% \cos\theta=\pi
  14. I ( ν , T ) I(\nu,T)
  15. I ( ν , T ) I(\nu,T)
  16. j = 2 π ( k T ) 4 c 2 h 3 0 d x x 3 e x - 1 j^{\star}={2\pi(kT)^{4}\over c^{2}h^{3}}\int_{0}^{\infty}dx{x^{3}\over e^{x}-1}
  17. x h ν / k T x\equiv h\nu/kT
  18. x x
  19. π 4 / 15 \pi^{4}/15
  20. j = σ T 4 , σ 2 π 5 15 k 4 c 2 h 3 j^{\star}=\sigma T^{4},\sigma\equiv{2\pi^{5}\over 15}{k^{4}\over c^{2}h^{3}}
  21. P net = P emit - P absorb . P_{\mathrm{net}}=P_{\mathrm{emit}}-P_{\mathrm{absorb}}.\,
  22. P net = A σ ε ( T 4 - T 0 4 ) . P_{\rm net}=A\sigma\varepsilon\left(T^{4}-T_{0}^{4}\right).
  23. P net = 100 W . P_{\rm net}=100\ \mathrm{W}.
  24. λ peak = 2.898 × 10 - 3 K m 305 K = 9.50 μ m . \lambda_{\rm peak}=\frac{2.898\times 10^{-3}\ \mathrm{K}\cdot\mathrm{m}}{305\ % \mathrm{K}}=9.50\ \mu\mathrm{m}.
  25. P S emt = 4 π R S 2 σ T S 4 ( 1 ) P_{\rm S\ emt}=4\pi R_{\rm S}^{2}\sigma T_{\rm S}^{4}\qquad\qquad(1)
  26. σ \sigma\,
  27. T S T_{\rm S}\,
  28. R S R_{\rm S}\,
  29. P SE = P S emt ( π R E 2 4 π D 2 ) ( 2 ) P_{\rm SE}=P_{\rm S\ emt}\left(\frac{\pi R_{\rm E}^{2}}{4\pi D^{2}}\right)% \qquad\qquad(2)
  30. R E R_{\rm E}\,
  31. D D\,
  32. α \alpha
  33. α \alpha
  34. 1 - α 1-\alpha
  35. P abs = ( 1 - α ) P SE ( 3 ) P_{\rm abs}=(1-\alpha)\,P_{\rm SE}\qquad\qquad(3)
  36. π R 2 \pi R^{2}
  37. P emt bb = 4 π R E 2 σ T E 4 ( 4 ) P_{\rm emt\,bb}=4\pi R_{\rm E}^{2}\sigma T_{\rm E}^{4}\qquad\qquad(4)
  38. T E T_{\rm E}
  39. P abs = P emt bb P_{\rm abs}=P_{\rm emt\,bb}
  40. ϵ ¯ \overline{\epsilon}
  41. ϵ ¯ \overline{\epsilon}
  42. P emt = ϵ ¯ P emt bb ( 5 ) P_{\rm emt}=\overline{\epsilon}\,P_{\rm emt\,bb}\qquad\qquad(5)
  43. P abs = P emt ( 6 ) P_{\rm abs}=P_{\rm emt}\qquad\qquad(6)
  44. T P = T S R S 1 - α ε ¯ 2 D ( 7 ) T_{P}=T_{S}\sqrt{\frac{R_{S}\sqrt{\frac{1-\alpha}{\overline{\varepsilon}}}}{2D% }}\qquad\qquad(7)
  45. ( 1 - α ) = ε ¯ ({1-\alpha})={\overline{\varepsilon}}
  46. T S = 5778 K , T_{\rm S}=5778\ \mathrm{K},
  47. R S = 6.96 × 10 8 m , R_{\rm S}=6.96\times 10^{8}\ \mathrm{m},
  48. D = 1.496 × 10 11 m , D=1.496\times 10^{11}\ \mathrm{m},
  49. α = 0.306 \alpha=0.306
  50. ε ¯ \overline{\varepsilon}
  51. T E = 254.356 K T_{\rm E}=254.356\ \mathrm{K}
  52. f = f 1 - v c cos θ 1 - v 2 / c 2 , f^{\prime}=f\frac{1-\frac{v}{c}\cos\theta}{\sqrt{1-v^{2}/c^{2}}},
  53. T = T c - v c + v . T^{\prime}=T\sqrt{\frac{c-v}{c+v}}.
  54. T T
  55. E ( T , i ) E(T,i)
  56. i i
  57. T T
  58. a ( T , i ) a(T,i)
  59. T T
  60. E ( T , i ) / a ( T , i ) E(T,i)/a(T,i)
  61. a ( T , i ) a(T,i)
  62. T T
  63. E ( λ , T , i ) E(λ,T,i)
  64. a ( λ , T , i ) a(λ,T,i)
  65. E ( λ , T , i ) / a ( λ , T , i ) E(λ,T,i)/a(λ,T,i)
  66. E ( λ , T , i ) / a ( λ , T , i ) E(λ,T,i)/a(λ,T,i)
  67. i i
  68. E ( T , i ) / a ( T , i ) E(T,i)/a(T,i)
  69. i i
  70. E ( λ , T , i ) / a ( λ , T , i ) E(λ,T,i)/a(λ,T,i)
  71. i i
  72. B B BB
  73. T T
  74. E ( λ , T , i ) / a ( λ , T , i ) E(λ,T,i)/a(λ,T,i)
  75. i i
  76. λ λ
  77. T T
  78. E ( λ , T , B B ) E(λ,T,BB)
  79. a ( λ , T , B B ) a(λ,T,BB)
  80. E ( λ , T , B B ) / a ( λ , T , B B ) E(λ,T,BB)/a(λ,T,BB)
  81. E ( λ , T , B B ) E(λ,T,BB)
  82. T T
  83. E ( λ , T , i ) / a ( λ , T , i ) E(λ,T,i)/a(λ,T,i)
  84. E ( λ , T , B B ) E(λ,T,BB)
  85. B < s u b > λ ( λ , T ) B<sub>λ(λ,T)

Blaschke_product.html

  1. ( a n ) (a_{n})
  2. n ( 1 - | a n | ) < . \sum_{n}(1-|a_{n}|)<\infty.
  3. B ( z ) = n B ( a n , z ) B(z)=\prod_{n}B(a_{n},z)
  4. B ( a , z ) = | a | a a - z 1 - a ¯ z B(a,z)=\frac{|a|}{a}\;\frac{a-z}{1-\overline{a}z}
  5. a ¯ \overline{a}
  6. H H^{\infty}
  7. H 1 H^{1}
  8. Δ ¯ = { z | | z | 1 } \overline{\Delta}=\{z\in\mathbb{C}\,|\,|z|\leq 1\}
  9. B ( z ) = ζ i = 1 n ( z - a i 1 - a i ¯ z ) m i B(z)=\zeta\prod_{i=1}^{n}\left({{z-a_{i}}\over{1-\overline{a_{i}}z}}\right)^{m% _{i}}

Bloch_sphere.html

  1. | 0 |0\rangle
  2. | 1 |1\rangle
  3. | ψ |\psi\rangle
  4. | 0 |0\rangle
  5. | 1 |1\rangle
  6. | 0 |0\rangle
  7. ψ | ψ = 1 \langle\psi|\psi\rangle=1
  8. | | ψ | 2 = 1 ||\psi\rangle|^{2}=1
  9. | ψ |\psi\rangle
  10. | ψ = cos ( θ 2 ) | 0 + e i ϕ sin ( θ 2 ) | 1 = cos ( θ 2 ) | 0 + ( cos ϕ + i sin ϕ ) sin ( θ 2 ) | 1 |\psi\rangle=\cos\left(\tfrac{\theta}{2}\right)|0\rangle\,+\,e^{i\phi}\sin% \left(\tfrac{\theta}{2}\right)|1\rangle=\cos\left(\tfrac{\theta}{2}\right)|0% \rangle\,+\,(\cos\phi+i\sin\phi)\,\sin\left(\tfrac{\theta}{2}\right)|1\rangle
  11. 0 θ π 0\leq\theta\leq\pi
  12. 0 ϕ < 2 π 0\leq\phi<2\pi
  13. | ψ |\psi\rangle
  14. | 0 |0\rangle
  15. | 1 |1\rangle
  16. θ \theta\,
  17. ϕ \phi\,
  18. a = ( sin θ cos ϕ , sin θ sin ϕ , cos θ ) \vec{a}=(\sin\theta\cos\phi,\;\sin\theta\sin\phi,\;\cos\theta)
  19. 3 \mathbb{R}^{3}
  20. ρ \rho
  21. I I
  22. σ \vec{\sigma}
  23. ρ = 1 2 ( I + a σ ) \rho=\frac{1}{2}\left(I+\vec{a}\cdot\vec{\sigma}\right)
  24. a 3 \vec{a}\in\mathbb{R}^{3}
  25. ρ \rho
  26. 1 2 ( 1 ± | a | ) \frac{1}{2}\left(1\pm|\vec{a}|\right)
  27. | a | 1 |\vec{a}|\leq 1
  28. tr ( ρ 2 ) = 1 2 ( 1 + | a | 2 ) = 1 | a | = 1 \mathrm{tr}(\rho^{2})=\frac{1}{2}\left(1+|\vec{a}|^{2}\right)=1\quad% \Leftrightarrow\quad|\vec{a}|=1
  29. a \vec{a}
  30. ρ \rho
  31. u = ρ 10 + ρ 01 = 2 [ ρ 01 ] u=\rho_{10}+\rho_{01}=2\cdot\Re[\rho_{01}]
  32. v = i ( ρ 10 - ρ 01 ) = 2 [ ρ 01 ] v=i(\rho_{10}-\rho_{01})=2\cdot\Im[\rho_{01}]
  33. w = ρ 11 - ρ 00 w=\rho_{11}-\rho_{00}
  34. a = ( u , v , w ) \vec{a}=(u,v,w)
  35. w w
  36. U ( n ) / ( U ( n - 1 ) × U ( 1 ) ) . \operatorname{U}(n)/(\operatorname{U}(n-1)\times\operatorname{U}(1)).
  37. | ψ |\psi\rangle
  38. | ψ |\psi\rangle
  39. g g
  40. g | ψ = | ψ g|\psi\rangle=|\psi\rangle
  41. U ( n - 1 ) × U ( 1 ) . \operatorname{U}(n-1)\times\operatorname{U}(1).
  42. g g
  43. | ψ |\psi\rangle
  44. | ψ |\psi\rangle
  45. | ψ |\psi\rangle
  46. A e i A A\mapsto e^{iA}
  47. n 2 - ( ( n - 1 ) 2 + 1 ) = 2 n - 2. n^{2}-((n-1)^{2}+1)=2n-2.\quad
  48. U ( n 1 ) × × U ( n k ) . \operatorname{U}(n_{1})\times\cdots\times\operatorname{U}(n_{k}).
  49. U ( n ) / ( U ( n 1 ) × × U ( n k ) ) . \operatorname{U}(n)/(\operatorname{U}(n_{1})\times\cdots\times\operatorname{U}% (n_{k})).

Block_code.html

  1. C : Σ k Σ n C:\Sigma^{k}\to\Sigma^{n}
  2. Σ \Sigma
  3. k k
  4. n n
  5. Σ \Sigma
  6. | Σ | |\Sigma|
  7. q q
  8. q = 2 q=2
  9. q q
  10. Σ \Sigma
  11. 𝔽 q \mathbb{F}_{q}
  12. m m
  13. Σ k \Sigma^{k}
  14. k k
  15. k k
  16. n n
  17. c c
  18. Σ n \Sigma^{n}
  19. n n
  20. c = C ( m ) c=C(m)
  21. m m
  22. c c
  23. m m
  24. R = k / n R=k/n
  25. 1 - R 1-R
  26. 1 1
  27. C C
  28. d d
  29. δ \delta
  30. d / n d/n
  31. c 1 , c 2 Σ n c_{1},c_{2}\in\Sigma^{n}
  32. Δ ( c 1 , c 2 ) \Delta(c_{1},c_{2})
  33. c 1 c_{1}
  34. c 2 c_{2}
  35. c 1 c_{1}
  36. c 2 c_{2}
  37. d d
  38. C C
  39. d := min m 1 , m 2 Σ k ; m 1 m 2 Δ [ C ( m 1 ) , C ( m 2 ) ] d:=\min_{m_{1},m_{2}\in\Sigma^{k};m_{1}\neq m_{2}}\Delta[C(m_{1}),C(m_{2})]
  40. 1 1
  41. min m 1 , m 2 Σ k ; m 1 m 2 Δ [ C ( m 1 ) , C ( m 2 ) ] = min m 1 , m 2 Σ k ; m 1 m 2 Δ [ 𝟎 , C ( m 1 ) + C ( m 2 ) ] = min m Σ k ; m 𝟎 w [ C ( m ) ] = w m i n \min_{m_{1},m_{2}\in\Sigma^{k};m_{1}\neq m_{2}}\Delta[C(m_{1}),C(m_{2})]=\min_% {m_{1},m_{2}\in\Sigma^{k};m_{1}\neq m_{2}}\Delta[\mathbf{0},C(m_{1})+C(m_{2})]% =\min_{m\in\Sigma^{k};m\neq\mathbf{0}}w[C(m)]=w_{min}
  42. d d
  43. d - 1 d-1
  44. d - 1 d-1
  45. ( d - 1 ) / 2 (d-1)/2
  46. ( d - 1 ) / 2 (d-1)/2
  47. ( d - 1 ) / 2 (d-1)/2
  48. ( n , k , d ) q (n,k,d)_{q}
  49. Σ \Sigma
  50. q q
  51. n n
  52. k k
  53. d d
  54. [ n , k , d ] q [n,k,d]_{q}
  55. q = 2 q=2
  56. d = n - k + 1 d=n-k+1
  57. d d
  58. ( n , M , d ) q (n,M,d)_{q}
  59. M M
  60. n n
  61. k k
  62. q q
  63. M = q k M=q^{k}
  64. [ 7 , 4 , 3 ] 2 [7,4,3]_{2}
  65. [ n , k , d ] q [n,k,d]_{q}
  66. d = n - k + 1 d=n-k+1
  67. q q
  68. [ n , k , d ] q [n,k,d]_{q}
  69. d n - k + 1 d\leq n-k+1
  70. [ n , k , d ] 2 [n,k,d]_{2}
  71. n = 2 k - 1 n=2^{k-1}
  72. d = 2 k - 2 d=2^{k-2}
  73. c Σ n c\in\Sigma^{n}
  74. n n
  75. Σ n \Sigma^{n}
  76. 𝒞 \mathcal{C}
  77. Σ n \Sigma^{n}
  78. 𝒞 \mathcal{C}
  79. d d
  80. c 𝒞 \forall c\in\mathcal{C}
  81. c c
  82. d - 1 d-1
  83. n n
  84. c c
  85. d - 1 d-1
  86. 𝒞 \mathcal{C}
  87. d d
  88. 𝒞 \mathcal{C}
  89. d - 1 d-1
  90. c c
  91. d - 1 d-1
  92. d - 1 d-1
  93. 𝒞 \mathcal{C}
  94. 𝒞 \mathcal{C}
  95. d - 1 2 \textstyle\left\lfloor{{d-1}\over 2}\right\rfloor
  96. c c
  97. d - 1 d-1
  98. d - 1 2 \textstyle\left\lfloor{{d-1}\over 2}\right\rfloor
  99. y y
  100. d - 1 2 \textstyle\left\lfloor{{d-1}\over 2}\right\rfloor
  101. y y
  102. d - 1 2 \textstyle\left\lfloor{{d-1}\over 2}\right\rfloor
  103. ( d - 1 ) / 2 (d-1)/2
  104. 𝒞 \mathcal{C}
  105. d - 1 d-1
  106. q q
  107. i t h i^{th}
  108. i t h i^{th}
  109. d - 1 2 \textstyle\left\lfloor{{d-1}\over 2}\right\rfloor
  110. C = { C i } i 1 C=\{C_{i}\}_{i\geq 1}
  111. C i C_{i}
  112. ( n i , k i , d i ) q (n_{i},k_{i},d_{i})_{q}
  113. n i n_{i}
  114. C C
  115. R ( C ) = lim i k i n i R(C)=\lim_{i\to\infty}{k_{i}\over n_{i}}
  116. C C
  117. δ ( C ) = lim i d i n i \delta(C)=\lim_{i\to\infty}{d_{i}\over n_{i}}
  118. R ( C ) R(C)
  119. δ ( C ) \delta(C)
  120. R 1 - 1 n log q [ i = 0 δ n - 1 2 ( n i ) ( q - 1 ) i ] R\leq 1-{1\over n}\cdot\log_{q}\cdot\left[\sum_{i=0}^{\lfloor{{\delta\cdot n-1% }\over 2}\rfloor}{\left({{n}\atop{i}}\right)}(q-1)^{i}\right]
  121. R + δ 1 + 1 n R+\delta\leq 1+\frac{1}{n}
  122. k + d n + 1 k+d\leq n+1
  123. q = 2 q=2
  124. R + 2 δ 1 R+2\delta\leq 1
  125. C 𝔽 q n C\subseteq\mathbb{F}_{q}^{n}
  126. d d
  127. d = ( 1 - 1 q ) n , | C | 2 q n d=(1-{1\over q})n,|C|\leq 2qn
  128. d d
  129. ( 1 - 1 q ) n , | C | q d q d - ( q - 1 ) n (1-{1\over q})n,|C|\leq{qd\over{qd-(q-1)n}}
  130. q q
  131. δ \delta
  132. R 1 - ( q q - 1 ) δ + o ( 1 ) R\leq 1-({q\over{q-1}})\delta+o(1)
  133. R 1 - H q ( δ ) - ϵ R\geq 1-H_{q}(\delta)-\epsilon
  134. 0 δ 1 - 1 q , 0 ϵ 1 - H q ( δ ) 0\leq\delta\leq 1-{1\over q},0\leq\epsilon\leq 1-H_{q}(\delta)
  135. H q ( x ) d e f - x log q x q - 1 - ( 1 - x ) log q ( 1 - x ) H_{q}(x)\equiv_{def}-x\cdot\log_{q}{x\over{q-1}}-(1-x)\cdot\log_{q}{(1-x)}
  136. q q
  137. J q ( δ ) d e f ( 1 - 1 q ) ( 1 - 1 - q δ q - 1 ) J_{q}(\delta)\equiv_{def}(1-{1\over q})(1-\sqrt{1-{q\delta\over{q-1}}})
  138. J q ( n , d , e ) J_{q}(n,d,e)
  139. e e
  140. C 𝔽 q n C\subseteq\mathbb{F}_{q}^{n}
  141. d d
  142. J q ( n , d , e ) q n d J_{q}(n,d,e)\leq qnd
  143. e n q - 1 q ( 1 - 1 - q q - 1 d n ) = J q ( d n ) {e\over n}\leq{{q-1}\over q}\left({1-\sqrt{1-{q\over{q-1}}\cdot{d\over n}}}\,% \right)=J_{q}({d\over n})
  144. R = log q | C | n 1 - H q ( J q ( δ ) ) + o ( 1 ) R={\log_{q}{|C|}\over n}\leq 1-H_{q}(J_{q}(\delta))+o(1)

Block_design.html

  1. b k = v r , bk=vr,\,
  2. λ ( v - 1 ) = r ( k - 1 ) . \lambda(v-1)=r(k-1).\,
  3. λ ( v - 1 ) = k ( k - 1 ) . \lambda(v-1)=k(k-1).
  4. v - 1 = k ( k - 1 ) . v-1=k(k-1).\,
  5. λ i = λ ( v - i t - i ) / ( k - i t - i ) for i = 0 , 1 , , t , \lambda_{i}=\lambda\left.{\left({{v-i}\atop{t-i}}\right)}\right/{\left({{k-i}% \atop{t-i}}\right)}\,\text{ for }i=0,1,\ldots,t,
  6. t > 5 t>5
  7. R 0 = { ( x , x ) : x X } R_{0}=\{(x,x):x\in X\}
  8. R * := { ( x , y ) | ( y , x ) R } R^{*}:=\{(x,y)|(y,x)\in R\}
  9. ( x , y ) R k (x,y)\in R_{k}
  10. z X z\in X
  11. ( x , z ) R i (x,z)\in R_{i}
  12. ( z , y ) R j (z,y)\in R_{j}
  13. p i j k p^{k}_{ij}
  14. p i j k = p j i k p_{ij}^{k}=p_{ji}^{k}
  15. v r = b k vr=bk
  16. i = 1 m n i = v - 1 \sum_{i=1}^{m}n_{i}=v-1
  17. i = 1 m n i λ i = r ( k - 1 ) \sum_{i=1}^{m}n_{i}\lambda_{i}=r(k-1)
  18. u = 0 m p j u h = n j \sum_{u=0}^{m}p_{ju}^{h}=n_{j}
  19. n i p j h i = n j p i h j n_{i}p_{jh}^{i}=n_{j}p_{ih}^{j}

Body_surface_area.html

  1. B S A = 0.007184 × W 0.425 × H 0.725 {BSA}=0.007184\times W^{0.425}\times H^{0.725}
  2. B S A = W × H 3600 = 0.016667 × W 0.5 × H 0.5 {BSA}=\sqrt{\frac{W\times H}{3600}}=0.016667\times W^{0.5}\times H^{0.5}
  3. B S A = W × H / 60 {BSA}=\sqrt{W\times H}/{60}
  4. B S A = W × H t / 6 {BSA}=\sqrt{W\times Ht}/{6}
  5. 0.024265 × W 0.5378 × H 0.3964 0.024265\times W^{0.5378}\times H^{0.3964}
  6. 0.0235 × W 0.51456 × H 0.42246 0.0235\times W^{0.51456}\times H^{0.42246}
  7. 0.0003207 × weight ( g ) ( 0.7285 - 0.0188 log 10 weight ( g ) ) × H 0.3 0.0003207\times\mathrm{weight}\mathrm{(g)}^{(0.7285-0.0188\log_{10}{\mathrm{% weight}\mathrm{(g)}})}\times H^{0.3}
  8. 0.03330 × W ( 0.6157 - 0.0188 log 10 W ) × H 0.3 0.03330\times W^{(0.6157-0.0188\log_{10}{W})}\times H^{0.3}
  9. 0.008883 × W 0.444 × H 0.663 0.008883\times W^{0.444}\times H^{0.663}
  10. 0.007241 × W 0.425 × H 0.725 0.007241\times W^{0.425}\times H^{0.725}
  11. 0.000975482 × W 0.46 × H 1.08 0.000975482\times W^{0.46}\times H^{1.08}
  12. 0.000579479 × W 0.38 × H 1.24 0.000579479\times W^{0.38}\times H^{1.24}

Bogoliubov_transformation.html

  1. [ a ^ , a ^ ] = 1 . \left[\hat{a},\hat{a}^{\dagger}\right]=1~{}.
  2. b ^ = u a ^ + v a ^ \hat{b}=u\hat{a}+v\hat{a}^{\dagger}
  3. b ^ = u * a ^ + v * a ^ , \hat{b}^{\dagger}=u^{*}\hat{a}^{\dagger}+v^{*}\hat{a}~{},
  4. [ b ^ , b ^ ] = [ u a ^ + v a ^ , u * a ^ + v * a ^ ] = = ( | u | 2 - | v | 2 ) [ a ^ , a ^ ] . \left[\hat{b},\hat{b}^{\dagger}\right]=\left[u\hat{a}+v\hat{a}^{\dagger},u^{*}% \hat{a}^{\dagger}+v^{*}\hat{a}\right]=\cdots=\left(|u|^{2}-|v|^{2}\right)\left% [\hat{a},\hat{a}^{\dagger}\right].
  5. | u | 2 - | v | 2 = 1 \,|u|^{2}-|v|^{2}=1
  6. cosh 2 x - sinh 2 x = 1 \cosh^{2}x-\sinh^{2}x=1
  7. u u
  8. v v
  9. u = e i θ 1 cosh r u=e^{i\theta_{1}}\cosh r
  10. v = e i θ 2 sinh r . v=e^{i\theta_{2}}\sinh r~{}.
  11. { a ^ , a ^ } = 1 \left\{\hat{a},\hat{a}^{\dagger}\right\}=1
  12. { b ^ , b ^ } = ( | u | 2 + | v | 2 ) { a ^ , a ^ } \left\{\hat{b},\hat{b}^{\dagger}\right\}=(|u|^{2}+|v|^{2})\left\{\hat{a},\hat{% a}^{\dagger}\right\}
  13. u = e i θ 1 cos r u=e^{i\theta_{1}}\cos r\,\!
  14. v = e i θ 2 sin r . v=e^{i\theta_{2}}\sin r\,\!.
  15. a i + a j + \,\langle a_{i}^{+}a_{j}^{+}\rangle
  16. i a i | 0 = 0 \forall i\qquad a_{i}|0\rangle=0
  17. k = 1 n a i k | 0 \prod_{k=1}^{n}a_{i_{k}}^{\dagger}|0\rangle
  18. a i = j ( u i j a j + v i j a j ) a^{\prime}_{i}=\sum_{j}(u_{ij}a_{j}+v_{ij}a^{\dagger}_{j})
  19. u i j , v i j \,u_{ij},v_{ij}
  20. a i a^{\prime\dagger}_{i}
  21. a i a^{\prime}_{i}
  22. | 0 |0\rangle

Bohr_compactification.html

  1. 𝐛 : G 𝐁𝐨𝐡𝐫 ( G ) . \mathbf{b}:G\rightarrow\mathbf{Bohr}(G).
  2. [ f g ] ( x ) = f ( g - 1 x ) [{}_{g}f](x)=f(g^{-1}\cdot x)
  3. f = f 1 𝐛 . f=f_{1}\circ\mathbf{b}.

Bohr–Einstein_debates.html

  1. Δ t \Delta t
  2. ν 0 \nu_{0}
  3. Δ x \Delta x
  4. Δ t = Δ x / v \Delta t=\Delta x/v
  5. Δ ν \Delta\nu
  6. Δ ν 1 Δ t . \Delta\nu\geq\frac{1}{\Delta t}.
  7. E = h ν E=h\nu\,
  8. Δ E = h Δ ν h Δ t . \Delta E=h\,\Delta\nu\geq\frac{h}{\Delta t}.
  9. Δ E Δ t h \Delta E\,\Delta t\geq h
  10. Δ t \Delta t
  11. E = m c 2 E=mc^{2}
  12. c 2 c^{2}
  13. Δ E \Delta E
  14. Δ E Δ t \Delta E\Delta t
  15. Δ t \Delta t
  16. Δ E Δ t h \Delta E\Delta t\geq h
  17. | Ψ \left|\Psi\right\rangle
  18. | Ψ , t = 1 2 | 1 , V | 2 , V + 1 2 | 1 , H | 2 , H . \left|\Psi,t\right\rangle=\frac{1}{\sqrt{2}}\left|1,V\right\rangle\left|2,V% \right\rangle+\frac{1}{\sqrt{2}}\left|1,H\right\rangle\left|2,H\right\rangle.
  19. | Ψ , t + d t = | 1 , V | 2 , V . \left|\Psi,t+dt\right\rangle=\left|1,V\right\rangle\left|2,V\right\rangle.

Bohr–Mollerup_theorem.html

  1. x > 0 x>0
  2. Γ ( x ) = 0 t x - 1 e - t d t \Gamma(x)=\int_{0}^{\infty}t^{x-1}e^{-t}\,dt
  3. f f
  4. x > 0 x>0
  5. f ( 1 ) = 1 f(1)=1
  6. f ( x + 1 ) = x f ( x ) f(x+1)=xf(x)
  7. x > 0 x>0
  8. f f
  9. Γ ( x ) Γ(x)
  10. f ( x + 1 ) = x f ( x ) f(x+1)=xf(x)
  11. l o g ( f ( x ) ) log(f(x))
  12. f ( 1 ) = 1 f(1)=1
  13. Γ ( x ) Γ(x)
  14. Γ ( x + 1 ) = x Γ ( x ) Γ(x+1)=xΓ(x)
  15. l o g ( Γ ( x ) ) log(Γ(x))
  16. Γ ( 1 ) = 1 Γ(1)=1
  17. Γ ( x + 1 ) = x Γ ( x ) Γ(x+1)=xΓ(x)
  18. Γ ( x + n ) = ( x + n - 1 ) ( x + n - 2 ) ( x + n - 3 ) ( x + 1 ) x Γ ( x ) \Gamma(x+n)=(x+n-1)(x+n-2)(x+n-3)\cdots(x+1)x\Gamma(x)
  19. Γ ( 1 ) = 1 Γ(1)=1
  20. Γ ( x + 1 ) = x Γ ( x ) Γ(x+1)=xΓ(x)
  21. Γ ( n ) = ( n 1 ) ! Γ(n)=(n−1)!
  22. n 𝐍 n∈\mathbf{N}
  23. Γ ( x ) Γ(x)
  24. Γ ( x + n ) Γ(x+n)
  25. Γ ( x ) Γ(x)
  26. Γ ( x ) Γ(x)
  27. x x
  28. l o g ( Γ ( x ) ) log(Γ(x))
  29. S ( n - 1 , n ) S ( n , n + x ) S ( n , n + 1 ) 0 < x 1 [ 6 p t ] log ( Γ ( n ) ) - log ( Γ ( n - 1 ) ) n - ( n - 1 ) log ( Γ ( n ) ) - log ( Γ ( n + x ) ) n - ( n + x ) log ( Γ ( n ) ) - log ( Γ ( n + 1 ) ) n - ( n + 1 ) [ 6 p t ] log ( ( n - 1 ) ! ) - log ( ( n - 2 ) ! ) 1 log ( Γ ( n + x ) ) - log ( ( n - 1 ) ! ) x log ( n ! ) - log ( ( n - 1 ) ! ) 1 [ 6 p t ] log ( ( n - 1 ) ! ( n - 2 ) ! ) log ( Γ ( n + x ) ) - log ( ( n - 1 ) ! ) x log ( n ! ( n - 1 ) ! ) [ 6 p t ] log ( n - 1 ) log ( Γ ( n + x ) ) - log ( ( n - 1 ) ! ) x log ( n ) x log ( n - 1 ) log ( Γ ( n + x ) ) - log ( ( n - 1 ) ! ) x log ( n ) log ( ( n - 1 ) x ) + log ( ( n - 1 ) ! ) log ( Γ ( n + x ) ) log ( n x ) + log ( ( n - 1 ) ! ) log ( ( n - 1 ) x ( n - 1 ) ! ) log ( Γ ( n + x ) ) log ( n x ( n - 1 ) ! ) ( n - 1 ) x ( n - 1 ) ! Γ ( n + x ) n x ( n - 1 ) ! log is monotonically increasing [ 6 p t ] ( n - 1 ) x ( n - 1 ) ! ( x + n - 1 ) ( x + n - 2 ) ( x + 1 ) x Γ ( x ) n x ( n - 1 ) ! [ 6 p t ] ( n - 1 ) x ( n - 1 ) ! ( x + n - 1 ) ( x + n - 2 ) ( x + 1 ) x Γ ( x ) n x ( n - 1 ) ! ( x + n - 1 ) ( x + n - 2 ) ( x + 1 ) x [ 6 p t ] ( n - 1 ) x ( n - 1 ) ! ( x + n - 1 ) ( x + n - 2 ) ( x + 1 ) x Γ ( x ) n x n ! ( x + n ) ( x + n - 1 ) ( x + 1 ) x ( n + x n ) [ 6 p t ] \begin{aligned}\displaystyle S(n-1,n)&\displaystyle\leq S(n,n+x)\leq S(n,n+1)&% &\displaystyle 0<x\leq 1\\ \displaystyle[6pt]\frac{\log(\Gamma(n))-\log(\Gamma(n-1))}{n-(n-1)}&% \displaystyle\leq\frac{\log(\Gamma(n))-\log(\Gamma(n+x))}{n-(n+x)}\leq\frac{% \log(\Gamma(n))-\log(\Gamma(n+1))}{n-(n+1)}\\ \displaystyle[6pt]\frac{\log((n-1)!)-\log((n-2)!)}{1}&\displaystyle\leq\frac{% \log(\Gamma(n+x))-\log((n-1)!)}{x}\leq\frac{\log(n!)-\log((n-1)!)}{1}\\ \displaystyle[6pt]\log\left(\frac{(n-1)!}{(n-2)!}\right)&\displaystyle\leq% \frac{\log(\Gamma(n+x))-\log((n-1)!)}{x}\leq\log\left(\frac{n!}{(n-1)!}\right)% \\ \displaystyle[6pt]\log(n-1)&\displaystyle\leq\frac{\log(\Gamma(n+x))-\log((n-1% )!)}{x}\leq\log(n)\\ \displaystyle x\log(n-1)&\displaystyle\leq\log(\Gamma(n+x))-\log((n-1)!)\leq x% \log(n)\\ \displaystyle\log\left((n-1)^{x}\right)+\log((n-1)!)&\displaystyle\leq\log(% \Gamma(n+x))\leq\log\left(n^{x}\right)+\log((n-1)!)\\ \displaystyle\log\left((n-1)^{x}(n-1)!\right)&\displaystyle\leq\log(\Gamma(n+x% ))\leq\log\left(n^{x}(n-1)!\right)\\ \displaystyle(n-1)^{x}(n-1)!&\displaystyle\leq\Gamma(n+x)\leq n^{x}(n-1)!&&% \displaystyle\log\,\text{is monotonically increasing}\\ \displaystyle[6pt](n-1)^{x}(n-1)!&\displaystyle\leq(x+n-1)(x+n-2)\cdots(x+1)x% \Gamma(x)\leq n^{x}(n-1)!\\ \displaystyle[6pt]\frac{(n-1)^{x}(n-1)!}{(x+n-1)(x+n-2)\cdots(x+1)x}&% \displaystyle\leq\Gamma(x)\leq\frac{n^{x}(n-1)!}{(x+n-1)(x+n-2)\cdots(x+1)x}\\ \displaystyle[6pt]\frac{(n-1)^{x}(n-1)!}{(x+n-1)(x+n-2)\cdots(x+1)x}&% \displaystyle\leq\Gamma(x)\leq\frac{n^{x}n!}{(x+n)(x+n-1)\cdots(x+1)x}\left(% \frac{n+x}{n}\right)\\ \displaystyle[6pt]\end{aligned}
  30. n n
  31. Γ ( x ) Γ(x)
  32. n n
  33. Γ ( x ) Γ(x)
  34. n n
  35. n n
  36. n n
  37. n + 1 n+1
  38. ( ( n + 1 ) - 1 ) x ( ( n + 1 ) - 1 ) ! ( x + ( n + 1 ) - 1 ) ( x + ( n + 1 ) - 2 ) ( x + 1 ) x \displaystyle\frac{((n+1)-1)^{x}((n+1)-1)!}{(x+(n+1)-1)(x+(n+1)-2)\cdots(x+1)x}
  39. n n→∞
  40. lim n n + x n = 1 \lim_{n\to\infty}\frac{n+x}{n}=1
  41. n x n ! ( x + n ) ( x + n - 1 ) ( x + 1 ) x \frac{n^{x}n!}{(x+n)(x+n-1)\cdots(x+1)x}
  42. lim n n x n ! ( x + n ) ( x + n - 1 ) ( x + 1 ) x = Γ ( x ) . \lim_{n\to\infty}\frac{n^{x}n!}{(x+n)(x+n-1)\cdots(x+1)x}=\Gamma(x).
  43. lim n n x n ! ( x + n ) ( x + n - 1 ) ( x + 1 ) x \lim_{n\to\infty}\frac{n^{x}n!}{(x+n)(x+n-1)\cdots(x+1)x}
  44. Γ ( x ) Γ(x)
  45. Γ ( x ) Γ(x)
  46. Γ ( x ) Γ(x)
  47. Γ ( x ) Γ(x)
  48. Γ ( x ) Γ(x)
  49. x x
  50. lim n n x n ! ( x + n ) ( x + n - 1 ) ( x + 1 ) x \lim_{n\to\infty}\frac{n^{x}n!}{(x+n)(x+n-1)\cdots(x+1)x}
  51. S ( n - 1 , n ) S ( n + x , n ) S ( n + 1 , n ) S(n-1,n)\leq S(n+x,n)\leq S(n+1,n)
  52. x > 1 x>1
  53. S S
  54. Γ ( x + 1 ) = lim n x ( n x n ! ( x + n ) ( x + n - 1 ) ( x + 1 ) x ) n n + x + 1 Γ ( x ) = ( 1 x ) Γ ( x + 1 ) \begin{aligned}\displaystyle\Gamma(x+1)&\displaystyle=\lim_{n\to\infty}x\cdot% \left(\frac{n^{x}n!}{(x+n)(x+n-1)\cdots(x+1)x}\right)\frac{n}{n+x+1}\\ \displaystyle\Gamma(x)&\displaystyle=\left(\frac{1}{x}\right)\Gamma(x+1)\end{aligned}
  55. Γ ( x ) Γ(x)
  56. x x

Bolted_joint.html

  1. T = K P p r e d T=KP_{pre}d
  2. T T
  3. K K
  4. P p r e P_{pre}
  5. d d
  6. K = d m 2 d ( tan ψ + μ sec α 1 - μ tan ψ sec α ) + 0.625 μ c K=\frac{d_{m}}{2d}\,\left(\frac{\tan\psi+\mu\sec\alpha}{1-\mu\tan\psi\sec% \alpha}\right)+0.625\mu_{c}
  7. d m d_{m}
  8. d d
  9. tan ψ \tan\psi
  10. μ \mu
  11. α \alpha
  12. μ c \mu_{c}
  13. μ \mu
  14. μ c \mu_{c}
  15. T = 0.20 P p r e d T=0.20P_{pre}d
  16. L e = 2 × A t 0.5 π ( D - 0.64952 p ) L_{e}=\frac{2\times A_{t}}{0.5\pi\left(D-0.64952p\right)}
  17. J = tensile strength of external thread material tensile strength of internal thread material J=\frac{\,\text{tensile strength of external thread material}}{\,\text{tensile% strength of internal thread material}}
  18. L e 2 = J × L e L_{e2}=J\times L_{e}

Boltzmann_equation.html

  1. d 3 𝐫 d 3 𝐩 = d x d y d z d p x d p y d p z . d^{3}\mathbf{r}\,d^{3}\mathbf{p}=dx\,dy\,dz\,dp_{x}\,dp_{y}\,dp_{z}.
  2. \,
  3. d N = f ( 𝐫 , 𝐩 , t ) d 3 𝐫 d 3 𝐩 dN=f(\mathbf{r},\mathbf{p},t)\,d^{3}\mathbf{r}\,d^{3}\mathbf{p}
  4. N = positions d 3 𝐫 momenta d 3 𝐩 f ( 𝐫 , 𝐩 , t ) = positions momenta f ( x , y , z , p x , p y , p z , t ) d x d y d z d p x d p y d p z N=\int\limits_{\mathrm{positions}}d^{3}\mathbf{r}\int\limits_{\mathrm{momenta}% }d^{3}\mathbf{p}\,f(\mathbf{r},\mathbf{p},t)=\iiint\limits_{\mathrm{positions}% }\quad\iiint\limits_{\mathrm{momenta}}f(x,y,z,p_{x},p_{y},p_{z},t)\,dx\,dy\,dz% \,dp_{x}\,dp_{y}\,dp_{z}
  5. f t = ( f t ) force + ( f t ) diff + ( f t ) coll \frac{\partial f}{\partial t}=\left(\frac{\partial f}{\partial t}\right)_{% \mathrm{force}}+\left(\frac{\partial f}{\partial t}\right)_{\mathrm{diff}}+% \left(\frac{\partial f}{\partial t}\right)_{\mathrm{coll}}
  6. f ( 𝐫 + 𝐩 m Δ t , 𝐩 + 𝐅 Δ t , t + Δ t ) d 3 𝐫 d 3 𝐩 = f ( 𝐫 , 𝐩 , t ) d 3 𝐫 d 3 𝐩 f\left(\mathbf{r}+\frac{\mathbf{p}}{m}\Delta t,\mathbf{p}+\mathbf{F}\Delta t,t% +\Delta t\right)\,d^{3}\mathbf{r}\,d^{3}\mathbf{p}=f(\mathbf{r},\mathbf{p},t)% \,d^{3}\mathbf{r}\,d^{3}\mathbf{p}
  7. f 𝐩 = 𝐞 ^ x f p x + 𝐞 ^ y f p y + 𝐞 ^ z f p z = 𝐩 f \frac{\partial f}{\partial\mathbf{p}}=\mathbf{\hat{e}}_{x}\frac{\partial f}{% \partial p_{x}}+\mathbf{\hat{e}}_{y}\frac{\partial f}{\partial p_{y}}+\mathbf{% \hat{e}}_{z}\frac{\partial f}{\partial p_{z}}=\nabla_{\mathbf{p}}f
  8. f t + 𝐩 m f + 𝐅 f 𝐩 = ( f t ) coll \frac{\partial f}{\partial t}+\frac{\mathbf{p}}{m}\cdot\nabla f+\mathbf{F}% \cdot\frac{\partial f}{\partial\mathbf{p}}=\left(\frac{\partial f}{\partial t}% \right)_{\mathrm{coll}}
  9. ( f t ) coll = g I ( g , Ω ) [ f ( 𝐩 A , t ) f ( 𝐩 B , t ) - f ( 𝐩 A , t ) f ( 𝐩 B , t ) ] d Ω d 3 𝐩 A d 3 𝐩 B . \left(\frac{\partial f}{\partial t}\right)_{\mathrm{coll}}=\iint gI(g,\Omega)[% f(\mathbf{p^{\prime}}_{A},t)f(\mathbf{p^{\prime}}_{B},t)-f(\mathbf{p}_{A},t)f(% \mathbf{p}_{B},t)]\,d\Omega\,d^{3}\mathbf{p}_{A}\,d^{3}\mathbf{p}_{B}.
  10. g = | 𝐩 B - 𝐩 A | = | 𝐩 B - 𝐩 A | g=|\mathbf{p}_{B}-\mathbf{p}_{A}|=|\mathbf{p^{\prime}}_{B}-\mathbf{p^{\prime}}% _{A}|
  11. f i t + 𝐩 i m i f i + 𝐅 f i 𝐩 i = ( f i t ) coll \frac{\partial f_{i}}{\partial t}+\frac{\mathbf{p}_{i}}{m_{i}}\cdot\nabla f_{i% }+\mathbf{F}\cdot\frac{\partial f_{i}}{\partial\mathbf{p}_{i}}=\left(\frac{% \partial f_{i}}{\partial t}\right)_{\mathrm{coll}}
  12. ( f i t ) coll = j = 1 n g i j I i j ( g i j , Ω ) [ f i f j - f i f j ] d Ω d 3 𝐩 . \left(\frac{\partial f_{i}}{\partial t}\right)_{\mathrm{coll}}=\sum_{j=1}^{n}% \iint g_{ij}I_{ij}(g_{ij},\Omega)[f^{\prime}_{i}f^{\prime}_{j}-f_{i}f_{j}]\,d% \Omega\,d^{3}\mathbf{p^{\prime}}.
  13. g i j = | 𝐩 i - 𝐩 j | = | 𝐩 i - 𝐩 j | g_{ij}=|\mathbf{p}_{i}-\mathbf{p}_{j}|=|\mathbf{p^{\prime}}_{i}-\mathbf{p^{% \prime}}_{j}|
  14. n = f d 3 p n=\int f\,d^{3}p
  15. A = 1 n A f d 3 p \langle A\rangle=\frac{1}{n}\int Af\,d^{3}p
  16. 𝐱 x i \mathbf{x}\rightarrow x_{i}
  17. 𝐩 p i = m w i \mathbf{p}\rightarrow p_{i}=mw_{i}
  18. w i w_{i}
  19. g ( p i ) g(p_{i})
  20. p i p_{i}
  21. F i F_{i}
  22. p i ± p_{i}\rightarrow\pm\infty
  23. g f t d 3 p = t ( n g ) \int g\frac{\partial f}{\partial t}\,d^{3}p=\frac{\partial}{\partial t}(n% \langle g\rangle)
  24. p j g m f x j d 3 p = 1 m x j ( n g p j ) \int\frac{p_{j}g}{m}\frac{\partial f}{\partial x_{j}}\,d^{3}p=\frac{1}{m}\frac% {\partial}{\partial x_{j}}(n\langle gp_{j}\rangle)
  25. g F j f p j d 3 p = - n F j g p j \int gF_{j}\frac{\partial f}{\partial p_{j}}\,d^{3}p=-nF_{j}\left\langle\frac{% \partial g}{\partial p_{j}}\right\rangle
  26. g ( f t ) coll d 3 p = 0 \int g\left(\frac{\partial f}{\partial t}\right)_{\mathrm{coll}}\,d^{3}p=0
  27. g = m g=m
  28. t ρ + x j ( ρ V j ) = 0 \frac{\partial}{\partial t}\rho+\frac{\partial}{\partial x_{j}}(\rho V_{j})=0
  29. ρ = m n \rho=mn
  30. V i = w i V_{i}=\langle w_{i}\rangle
  31. g = m w i g=mw_{i}
  32. t ( ρ V i ) + x j ( ρ V i V j + P i j ) - n F i = 0 \frac{\partial}{\partial t}(\rho V_{i})+\frac{\partial}{\partial x_{j}}(\rho V% _{i}V_{j}+P_{ij})-nF_{i}=0
  33. P i j = ρ ( w i - V i ) ( w j - V j ) P_{ij}=\rho\langle(w_{i}-V_{i})(w_{j}-V_{j})\rangle
  34. g = 1 2 m w i w i g=\tfrac{1}{2}mw_{i}w_{i}
  35. t ( u + 1 2 ρ V i V i ) + x j ( u V j + 1 2 ρ V i V i V j + J q j + P i j V i ) - n F i V i = 0 \frac{\partial}{\partial t}(u+\tfrac{1}{2}\rho V_{i}V_{i})+\frac{\partial}{% \partial x_{j}}(uV_{j}+\tfrac{1}{2}\rho V_{i}V_{i}V_{j}+J_{qj}+P_{ij}V_{i})-nF% _{i}V_{i}=0
  36. u = 1 2 ρ ( w i - V i ) ( w i - V i ) u=\tfrac{1}{2}\rho\langle(w_{i}-V_{i})(w_{i}-V_{i})\rangle
  37. J q i = 1 2 ρ ( w i - V i ) ( w k - V k ) ( w k - V k ) J_{qi}=\tfrac{1}{2}\rho\langle(w_{i}-V_{i})(w_{k}-V_{k})(w_{k}-V_{k})\rangle
  38. 𝐋 ^ [ f ] = 𝐂 [ f ] , \hat{\mathbf{L}}[f]=\mathbf{C}[f],\,
  39. 𝐋 ^ NR = t + 𝐩 m + 𝐅 𝐩 . \hat{\mathbf{L}}_{\mathrm{NR}}=\frac{\partial}{\partial t}+\frac{\mathbf{p}}{m% }\cdot\nabla+\mathbf{F}\cdot\frac{\partial}{\partial\mathbf{p}}\,.
  40. 𝐋 ^ GR = p α x α - Γ α p β β γ p γ p α , \hat{\mathbf{L}}_{\mathrm{GR}}=p^{\alpha}\frac{\partial}{\partial x^{\alpha}}-% \Gamma^{\alpha}{}_{\beta\gamma}p^{\beta}p^{\gamma}\frac{\partial}{\partial p^{% \alpha}},

Boltzmann_machine.html

  1. w i j w_{ij}
  2. E E
  3. E = - ( i < j w i j s i s j + i θ i s i ) E=-\left(\sum_{i<j}w_{ij}\,s_{i}\,s_{j}+\sum_{i}\theta_{i}\,s_{i}\right)
  4. w i j w_{ij}
  5. j j
  6. i i
  7. s i s_{i}
  8. s i { 0 , 1 } s_{i}\in\{0,1\}
  9. i i
  10. θ i \theta_{i}
  11. i i
  12. - θ i -\theta_{i}
  13. w i i = 0 i w_{ii}=0\qquad\forall i
  14. w i j = w j i i , j w_{ij}=w_{ji}\qquad\forall i,j
  15. W W
  16. i i
  17. Δ E i \Delta E_{i}
  18. Δ E i = j w i j s j + θ i \Delta E_{i}=\sum_{j}w_{ij}\,s_{j}+\theta_{i}
  19. Δ E i = E i=off - E i=on \Delta E_{i}=E\text{i=off}-E\text{i=on}
  20. Δ E i = - k B T ln ( p i=off ) - ( - k B T ln ( p i=on ) ) \Delta E_{i}=-k_{B}\,T\ln(p\text{i=off})-(-k_{B}\,T\ln(p\text{i=on}))
  21. k B k_{B}
  22. T T
  23. Δ E i T = ln ( p i=on ) - ln ( p i=off ) \frac{\Delta E_{i}}{T}=\ln(p\text{i=on})-\ln(p\text{i=off})
  24. Δ E i T = ln ( p i=on ) - ln ( 1 - p i=on ) \frac{\Delta E_{i}}{T}=\ln(p\text{i=on})-\ln(1-p\text{i=on})
  25. Δ E i T = ln ( p i=on 1 - p i=on ) \frac{\Delta E_{i}}{T}=\ln\left(\frac{p\text{i=on}}{1-p\text{i=on}}\right)
  26. - Δ E i T = ln ( 1 - p i=on p i=on ) -\frac{\Delta E_{i}}{T}=\ln\left(\frac{1-p\text{i=on}}{p\text{i=on}}\right)
  27. - Δ E i T = ln ( 1 p i=on - 1 ) -\frac{\Delta E_{i}}{T}=\ln\left(\frac{1}{p\text{i=on}}-1\right)
  28. exp ( - Δ E i T ) = 1 p i=on - 1 \exp\left(-\frac{\Delta E_{i}}{T}\right)=\frac{1}{p\text{i=on}}-1
  29. p i=on p\text{i=on}
  30. i i
  31. p i=on = 1 1 + exp ( - Δ E i T ) p\text{i=on}=\frac{1}{1+\exp(-\frac{\Delta E_{i}}{T})}
  32. T T
  33. P + ( V ) P^{+}(V)
  34. P - ( V ) P^{-}(V)
  35. P + ( V ) P^{+}(V)
  36. P - ( V ) P^{-}(V)
  37. G G
  38. G = v P + ( v ) ln ( P + ( v ) P - ( v ) ) G=\sum_{v}{P^{+}(v)\ln\left({\frac{P^{+}(v)}{P^{-}(v)}}\right)}
  39. V V
  40. G G
  41. P - ( v ) P^{-}(v)
  42. G G
  43. w i j w_{ij}
  44. G G
  45. P + P^{+}
  46. w i j w_{ij}
  47. G w i j = - 1 R [ p i j + - p i j - ] \frac{\partial{G}}{\partial{w_{ij}}}=-\frac{1}{R}[p_{ij}^{+}-p_{ij}^{-}]
  48. p i j + p_{ij}^{+}
  49. p i j - p_{ij}^{-}
  50. R R
  51. P - ( s ) P^{-}(s)
  52. s s
  53. G θ i = - 1 R [ p i + - p i - ] \frac{\partial{G}}{\partial{\theta_{i}}}=-\frac{1}{R}[p_{i}^{+}-p_{i}^{-}]
  54. G G
  55. G = v P - ( v ) ln ( P - ( v ) P + ( v ) ) G^{\prime}=\sum_{v}{P^{-}(v)\ln\left({\frac{P^{-}(v)}{P^{+}(v)}}\right)}

Bond_convexity.html

  1. C = 1 B d 2 ( B ( r ) ) d r 2 . C=\frac{1}{B}\frac{d^{2}\left(B(r)\right)}{dr^{2}}.
  2. d d r B ( r ) = - D B . \frac{d}{dr}B(r)=-DB.
  3. C B = d ( - D B ) d r = ( - D ) ( - D B ) + ( - d D d r ) ( B ) , CB=\frac{d(-DB)}{dr}=(-D)(-DB)+\left(-\frac{dD}{dr}\right)(B),
  4. C = D 2 - d D d r . C=D^{2}-\frac{dD}{dr}.
  5. D = 1 1 + r i = 1 n P ( i ) t ( i ) B D=\frac{1}{1+r}\sum_{i=1}^{n}\frac{P(i)t(i)}{B}
  6. d D d r 0. \frac{dD}{dr}\leq 0.
  7. B ( r ) = i = 1 n c i e - r t i \scriptstyle B(r)\ =\ \sum_{i=1}^{n}c_{i}e^{-rt_{i}}
  8. d 2 B d r 2 = i = 1 n c i e - r t i t i 2 0. \frac{d^{2}B}{dr^{2}}=\sum_{i=1}^{n}c_{i}e^{-rt_{i}}t_{i}^{2}\geq 0.
  9. d B / d r = - D B \scriptstyle dB/dr\ =\ -DB
  10. Δ ( B ) = B [ C 2 ( Δ ( r ) ) 2 - D Δ ( r ) ] . \Delta(B)=B\left[\frac{C}{2}(\Delta(r))^{2}-D\Delta(r)\right].

Bond_duration.html

  1. V = i = 1 n P V i V=\sum_{i=1}^{n}PV_{i}
  2. M a c D = i = 1 n t i P V i V = i = 1 n t i < m t p l > P V i V MacD=\frac{\sum_{i=1}^{n}{t_{i}PV_{i}}}{V}=\sum_{i=1}^{n}t_{i}\frac{<}{m}tpl>{% {PV_{i}}}{V}
  3. i i
  4. P V i PV_{i}
  5. i i
  6. t i t_{i}
  7. i i
  8. V V
  9. P V i PV_{i}
  10. P V i V \frac{PV_{i}}{V}
  11. i i
  12. t 1 t_{1}
  13. ( t 1 , , t n ) (t_{1},...,t_{n})
  14. t 1 M a c D t n , t_{1}\leq MacD\leq t_{n},
  15. P V ( i ) PV(i)
  16. V = i = 1 n P V i = i = 1 n C F i e - y t i V=\sum_{i=1}^{n}PV_{i}=\sum_{i=1}^{n}CF_{i}\cdot e^{-y\cdot t_{i}}
  17. M a c D = i = 1 n t i C F i e - y t i V MacD=\sum_{i=1}^{n}t_{i}\frac{{CF_{i}\cdot e^{-y\cdot t_{i}}}}{V}
  18. i i
  19. P V i PV_{i}
  20. i i
  21. C F i CF_{i}
  22. i i
  23. y y
  24. t i t_{i}
  25. i i
  26. V V
  27. y y
  28. M o d D ( y ) - 1 V V y = - ln ( V ) y ModD(y)\equiv-\frac{1}{V}\cdot\frac{\partial V}{\partial y}=-\frac{\partial\ln% (V)}{\partial y}
  29. y y
  30. V y = - i = 1 n t i C F i e - y t i = - M a c D V , \frac{\partial V}{\partial y}=-\sum_{i=1}^{n}t_{i}\cdot CF_{i}\cdot e^{-y\cdot t% _{i}}=-MacD\cdot V,
  31. M o d D = M a c D ModD=MacD
  32. i i
  33. t i t_{i}
  34. i i
  35. V V
  36. V ( y k ) = i = 1 n P V i = i = 1 n C F i ( 1 + y k / k ) k t i V(y_{k})=\sum_{i=1}^{n}PV_{i}=\sum_{i=1}^{n}\frac{CF_{i}}{(1+y_{k}/k)^{k\cdot t% _{i}}}
  37. M a c D = i = 1 n t i V ( y k ) C F i ( 1 + y k / k ) k t i MacD=\sum_{i=1}^{n}\frac{t_{i}}{V(y_{k})}\cdot\frac{CF_{i}}{(1+y_{k}/k)^{k% \cdot t_{i}}}
  38. V V
  39. V y k = - 1 ( 1 + y k / k ) i = 1 n t i C F i ( 1 + y k / k ) k t i = - M a c D V ( y k ) ( 1 + y k / k ) \frac{\partial V}{\partial y_{k}}=-\frac{1}{(1+y_{k}/k)}\cdot\sum_{i=1}^{n}t_{% i}\cdot\frac{CF_{i}}{(1+y_{k}/k)^{k\cdot t_{i}}}=-\frac{MacD\cdot V(y_{k})}{(1% +y_{k}/k)}
  40. M a c D ( 1 + y k / k ) = - 1 V ( y k ) V y k M o d D \frac{MacD}{(1+y_{k}/k)}=-\frac{1}{V(y_{k})}\cdot\frac{\partial V}{\partial y_% {k}}\equiv ModD
  41. M o d D = M a c D ( 1 + y k / k ) ModD=\frac{MacD}{(1+y_{k}/k)}
  42. i i
  43. k k
  44. C F i CF_{i}
  45. i i
  46. t i t_{i}
  47. i i
  48. t i t_{i}
  49. y k y_{k}
  50. V V
  51. Δ y \Delta y
  52. M o d D - 1 V Δ V Δ y \rArr Δ V - V M o d D Δ y ModD\approx-\frac{1}{V}\frac{\Delta V}{\Delta y}\rArr\Delta V\approx-V\cdot ModD% \cdot\Delta y
  53. Dur = 1 P ( C ( 1 + a i ) ( 1 + i ) m - ( 1 + i ) - ( m - 1 + a ) i i 2 ( 1 + i ) ( m - 1 + a ) + F V ( m - 1 + a ) ( 1 + i ) ( m - 1 + a ) ) \,\text{Dur}=\frac{1}{P}\left(C\frac{(1+ai)(1+i)^{m}-(1+i)-(m-1+a)i}{i^{2}(1+i% )^{(m-1+a)}}+\frac{FV(m-1+a)}{(1+i)^{(m-1+a)}}\right)
  54. k k
  55. M a c D = [ ( 1 + y / k ) y / k - 100 ( 1 + y / k ) + m ( c / k - 100 y / k ) ( c / k ) [ ( 1 + y / k ) m - 1 ] + 100 y / k ] / k MacD=\left[\frac{(1+y/k)}{y/k}-\frac{100(1+y/k)+m(c/k-100y/k)}{(c/k)[(1+y/k)^{% m}-1]+100y/k}\right]/k
  56. V = i = 1 n P V i = i = 1 n C F i ( 1 + y / k ) k t i = i = 1 4 10 ( 1 + .04 / 2 ) i + 100 ( 1 + .04 / 2 ) 4 V=\sum_{i=1}^{n}PV_{i}=\sum_{i=1}^{n}\frac{CF_{i}}{(1+y/k)^{k\cdot t_{i}}}=% \sum_{i=1}^{4}\frac{10}{(1+.04/2)^{i}}+\frac{100}{(1+.04/2)^{4}}
  57. = 9.804 + 9.612 + 9.423 + 9.238 + 92.385 = 130.462 =9.804+9.612+9.423+9.238+92.385=130.462
  58. M a c D = 0.5 9.804 130.462 + 1.0 9.612 130.462 + 1.5 9.423 130.462 + 2.0 9.238 130.462 + 2.0 92.385 130.462 = 1.777 y e a r s MacD=0.5\cdot\frac{9.804}{130.462}+1.0\cdot\frac{9.612}{130.462}+1.5\cdot\frac% {9.423}{130.462}+2.0\cdot\frac{9.238}{130.462}+2.0\cdot\frac{92.385}{130.462}=% 1.777years
  59. M a c D = [ ( 1.02 ) 0.02 - 100 ( 1.02 ) + 4 ( 10 - 2 ) 10 [ ( 1.02 ) 4 - 1 ] + 2 ] / 2 = 1.777 y e a r s MacD=\left[\frac{(1.02)}{0.02}-\frac{100(1.02)+4(10-2)}{10[(1.02)^{4}-1]+2}% \right]/2=1.777years
  60. M o d D = M a c D ( 1 + y / k ) = 1.777 ( 1 + .04 / 2 ) = 1.742 % ModD=\frac{MacD}{(1+y/k)}=\frac{1.777}{(1+.04/2)}=1.742\%
  61. D V 01 = M o d D 130.462 100 = 2.27 DV01=\frac{ModD\cdot 130.462}{100}=2.27
  62. D $ = D V 01 = - V y . D_{\$}=DV01=-\frac{\partial V}{\partial y}.
  63. D $ = D V 01 = B P V = V M o d D / 100 D_{\$}=DV01=BPV=V\cdot ModD/100
  64. D $ = D V 01 = V M o d D / 10000 D_{\$}=DV01=V\cdot ModD/10000
  65. D $ D_{\$}
  66. r 1 , , r n r_{1},\ldots,r_{n}
  67. V = V ( r 1 , , r n ) V=V(r_{1},\ldots,r_{n})\,
  68. s y m b o l ω = ( ω 1 , , ω n ) symbol{\omega}=(\omega_{1},\ldots,\omega_{n})
  69. ω i = - D $ , i := V r i . \omega_{i}=-D_{\$,i}:=\frac{\partial V}{\partial r_{i}}.
  70. Δ V = i = 1 n ω i Δ r i + 1 i , j n O ( Δ r i Δ r j ) , \Delta V=\sum_{i=1}^{n}\omega_{i}\,\Delta r_{i}+\sum_{1\leq i,j\leq n}O(\Delta r% _{i}\,\Delta r_{j}),
  71. Effective duration = V - Δ y - V + Δ y 2 ( V 0 ) Δ y \,\text{Effective duration}=\frac{V_{-\Delta y}-V_{+\Delta y}}{2(V_{0})\Delta y}
  72. V - Δ y and V + Δ y V_{-\Delta y}\,\text{ and }V_{+\Delta y}

Bond_valuation.html

  1. P = ( C 1 + i + C ( 1 + i ) 2 + + C ( 1 + i ) N ) + M ( 1 + i ) N = ( n = 1 N C ( 1 + i ) n ) + M ( 1 + i ) N = C ( 1 - ( 1 + i ) - N i ) + M ( 1 + i ) - N \begin{aligned}\displaystyle P&\displaystyle=\begin{matrix}\left(\frac{C}{1+i}% +\frac{C}{(1+i)^{2}}+...+\frac{C}{(1+i)^{N}}\right)+\frac{M}{(1+i)^{N}}\end{% matrix}\\ &\displaystyle=\begin{matrix}\left(\sum_{n=1}^{N}\frac{C}{(1+i)^{n}}\right)+% \frac{M}{(1+i)^{N}}\end{matrix}\\ &\displaystyle=\begin{matrix}C\left(\frac{1-(1+i)^{-N}}{i}\right)+M(1+i)^{-N}% \end{matrix}\end{aligned}
  2. i i
  3. 1 2 σ ( r ) 2 2 P r 2 + [ a ( r ) + σ ( r ) + φ ( r , t ) ] P r + P t - r P = 0 \frac{1}{2}\sigma(r)^{2}\frac{\partial^{2}P}{\partial r^{2}}+[a(r)+\sigma(r)+% \varphi(r,t)]\frac{\partial P}{\partial r}+\frac{\partial P}{\partial t}-rP=0
  4. P [ t , T , r ( t ) ] = E t [ e - R ( t , T ) ] P[t,T,r(t)]=E_{t}^{\ast}[e^{-R(t,T)}]
  5. E t E_{t}^{\ast}
  6. R ( t , T ) R(t,T)
  7. i i
  8. P 0 P_{0}
  9. C C
  10. F F
  11. Coupon rate = C F \,\text{Coupon rate}=\frac{C}{F}
  12. C C
  13. P P
  14. Current yield = C P 0 . \,\text{Current yield}=\frac{C}{P_{0}}.
  15. n { 0 , 1 , , N - 1 } n\in\{0,1,...,N-1\}
  16. a n + 1 a_{n+1}
  17. a n + 1 = | i P - C | ( 1 + i ) n a_{n+1}=|iP-C|{(1+i)}^{n}
  18. | F - P | |F-P|
  19. a 1 + a 2 + + a N a_{1}+a_{2}+...+a_{N}
  20. F | i - i F | ( 1 - ( 1 + i ) - N i ) F|i-i_{F}|(\frac{1-(1+i)^{-N}}{i})

Boolean_data_type.html

  1. \wedge
  2. \vee
  3. \supset
  4. \equiv
  5. ¬ \neg

Borel_functional_calculus.html

  1. Δ −Δ
  2. e i t Δ . e^{it\Delta}.
  3. T e k = λ k e k , 1 k . Te_{k}=\lambda_{k}e_{k},\qquad 1\leq k\leq\ell.
  4. T n e k = λ k n e k . T^{n}e_{k}=\lambda_{k}^{n}e_{k}.
  5. h ( T ) e k = h ( λ k ) e k . h(T)e_{k}=h(\lambda_{k})e_{k}.
  6. [ T ψ ] ( x ) = f ( x ) ψ ( x ) [T\psi](x)=f(x)\psi(x)
  7. [ h ( T ) ψ ] ( x ) = [ h f ] ( x ) ψ ( x ) . [h(T)\psi](x)=[h\circ f](x)\psi(x).
  8. { π T : L ( , ) ( ) f f ( T ) \begin{cases}\pi_{T}:L^{\infty}(\mathbb{R},\mathbb{C})\to\mathcal{B}(\mathcal{% H})\\ f\mapsto f(T)\end{cases}
  9. ν ξ : E π T ( 𝟏 E ) ξ , ξ \nu_{\xi}:E\mapsto\langle\pi_{T}(\mathbf{1}_{E})\xi,\xi\rangle
  10. η η
  11. π T ( [ η + i ] - 1 ) = [ T + i ] - 1 . \pi_{T}\left([\eta+i]^{-1}\right)=[T+i]^{-1}.
  12. U t = e i t A , t U_{t}=e^{itA},\qquad t\in\mathbb{R}
  13. p ( T ) = sup λ σ ( T ) | p ( λ ) | . \|p(T)\|=\sup_{\lambda\in\sigma(T)}|p(\lambda)|.
  14. p p ( T ) p\mapsto p(T)
  15. dom S = { ξ H : h L ν ξ 2 ( ) } \operatorname{dom}S=\left\{\xi\in H:h\in L^{2}_{\nu_{\xi}}(\mathbb{R})\right\}
  16. S ξ , ξ = h ( t ) d ν ξ ( t ) , for ξ dom S \langle S\xi,\xi\rangle=\int_{\mathbb{R}}h(t)\ d\nu_{\xi}(t),\quad\mbox{for}~{% }\quad\xi\in\operatorname{dom}S
  17. Ω : E 𝟏 E ( T ) \Omega:E\mapsto\mathbf{1}_{E}(T)
  18. I = 1 d Ω \textstyle{I=\int 1\,d\Omega}
  19. I = 1 d Ω \textstyle{I=\int 1\,d\Omega}
  20. I = i | i i | I=\sum_{i}\left|i\right\rangle\left\langle i\right|
  21. | i |i\rangle
  22. { | i } \{|i\rangle\}
  23. I = d | i i | I=\int d|i\rangle\langle i|
  24. { | i } \{|i\rangle\}

Borel–Weil–Bott_theorem.html

  1. G G
  2. \mathbb{C}
  3. T T
  4. B B
  5. T T
  6. λ λ
  7. T T
  8. λ λ
  9. B B
  10. T = B / U T=B/U
  11. U U
  12. B B
  13. G G / B G→G/B
  14. B B
  15. G / B G/B
  16. H i ( G / B , L λ ) H^{i}(G/B,\,L_{\lambda})
  17. G G
  18. L λ L_{\lambda}
  19. G G
  20. G G
  21. ρ ρ
  22. λ λ
  23. w w
  24. W W
  25. w * λ := w ( λ + ρ ) - ρ w*\lambda:=w(\lambda+\rho)-\rho\,
  26. ρ ρ
  27. G G
  28. μ μ
  29. μ ( α ) 0 \mu(\alpha^{\vee})\geq 0
  30. α α
  31. [ u e l l ] [u^{\prime}ell^{\prime}]
  32. W W
  33. λ λ
  34. w W w\in W
  35. w * λ w*\lambda
  36. w W w\in W
  37. w * λ = λ w*\lambda=\lambda
  38. w W w\in W
  39. w * λ w*\lambda
  40. H i ( G / B , L λ ) = 0 H^{i}(G/B,\,L_{\lambda})=0
  41. i i
  42. H i ( G / B , L λ ) = 0 H^{i}(G/B,\,L_{\lambda})=0
  43. i ( w ) i\neq\ell(w)
  44. H ( w ) ( G / B , L λ ) H^{\ell(w)}(G/B,\,L_{\lambda})
  45. G G
  46. w * λ w*\lambda
  47. λ ( β ) = 0 \lambda(\beta^{\vee})=0
  48. β β
  49. λ λ
  50. w w
  51. e W e\in W
  52. G / B G/B
  53. n n
  54. ρ = 1 ρ=1
  55. 𝒪 ( n ) {\mathcal{O}}(n)
  56. n n
  57. G G
  58. 𝔰 𝔩 2 ( 𝐂 ) \mathfrak{sl}_{2}(\mathbf{C})
  59. Γ ( 𝒪 ( 1 ) ) \Gamma({\mathcal{O}}(1))
  60. Γ ( 𝒪 ( n ) ) \Gamma({\mathcal{O}}(n))
  61. n n
  62. H H
  63. X X
  64. Y Y
  65. 𝔰 𝔩 2 ( 𝐂 ) \mathfrak{sl}_{2}(\mathbf{C})
  66. H = x d d x - y d d y H=x\frac{d}{dx}-y\frac{d}{dy}
  67. X = x d d y X=x\frac{d}{dy}
  68. Y = y d d x . Y=y\frac{d}{dx}.
  69. G G
  70. p > 0 p>0
  71. H i ( G / B , L λ ) = 0 H^{i}(G/B,\,L_{\lambda})=0
  72. i i
  73. λ λ
  74. w * λ w*\lambda
  75. w W w\in W
  76. λ λ
  77. λ λ
  78. H i ( G / B , L λ ) = 0 H^{i}(G/B,\,L_{\lambda})=0
  79. i > 0 i>0
  80. G G
  81. λ λ
  82. G G
  83. λ λ
  84. H i ( G / B , L λ ) H^{i}(G/B,\,L_{\lambda})
  85. \mathbb{C}
  86. λ λ
  87. i i

Born–Haber_cycle.html

  1. Δ H f \Delta\,\text{H}_{\,\text{f}}
  2. Δ H f = V + 1 2 B + IE M - EAX + UL \Delta\,\text{H}_{\,\text{f}}=\,\text{V}+\frac{1}{2}\,\text{B}+\,\text{IE}_{\,% \text{M}}-\,\text{EA}\text{X}+\,\text{U}\text{L}
  3. IE M \,\text{IE}_{\,\text{M}}
  4. M + IEM M + + e - \,\text{M}+\,\text{IE}\text{M}\to\,\text{M}^{+}+\,\text{e}^{-}
  5. EAX \,\text{EA}\text{X}
  6. UL \,\text{U}\text{L}
  7. 0 = - Δ H f + V + 1 2 B + IE M - EAX + UL 0=-\Delta\,\text{H}_{\,\text{f}}+\,\text{V}+\frac{1}{2}\,\text{B}+\,\text{IE}_% {\,\text{M}}-\,\text{EA}\text{X}+\,\text{U}\text{L}

Borrowed_chord.html

  1. 5 6 {}^{6}_{5}

Bose_gas.html

  1. 𝒵 ( z , β , V ) = i ( 1 - z e - β ϵ i ) - g i \mathcal{Z}(z,\beta,V)=\prod_{i}\left(1-ze^{-\beta\epsilon_{i}}\right)^{-g_{i}}
  2. z ( β , μ ) = e β μ z(\beta,\mu)=e^{\beta\mu}
  3. β = 1 k T \beta=\frac{1}{kT}
  4. Ω = - ln ( 𝒵 ) = i g i ln ( 1 - z e - β ϵ i ) . \Omega=-\ln(\mathcal{Z})=\sum_{i}g_{i}\ln\left(1-ze^{-\beta\epsilon_{i}}\right).
  5. Ω 0 ln ( 1 - z e - β E ) d g . \Omega\approx\int_{0}^{\infty}\ln\left(1-ze^{-\beta E}\right)\,dg.
  6. d g = 1 Γ ( α ) E α - 1 E c α d E dg=\frac{1}{\Gamma(\alpha)}\,\frac{E^{\,\alpha-1}}{E_{c}^{\alpha}}~{}dE
  7. E c E_{c}
  8. 1 ( β E c ) α = V f Λ 3 \frac{1}{(\beta E_{c})^{\alpha}}=\frac{Vf}{\Lambda^{3}}
  9. 1 ( β E c ) α = f ( ω β ) 3 \frac{1}{(\beta E_{c})^{\alpha}}=\frac{f}{(\hbar\omega\beta)^{3}}
  10. Ω - Li α + 1 ( z ) ( β E c ) α . \Omega\approx-\frac{\textrm{Li}_{\alpha+1}(z)}{\left(\beta E_{c}\right)^{% \alpha}}.
  11. N = - z Ω z Li α ( z ) ( β E c ) α N=-z\frac{\partial\Omega}{\partial z}\approx\frac{\textrm{Li}_{\alpha}(z)}{(% \beta E_{c})^{\alpha}}
  12. N = ζ ( α ) ( β c E c ) α N=\frac{\zeta(\alpha)}{(\beta_{c}E_{c})^{\alpha}}
  13. T c = ( N ζ ( α ) ) 1 / α E c k T_{c}=\left(\frac{N}{\zeta(\alpha)}\right)^{1/\alpha}\frac{E_{c}}{k}
  14. α = 3 / 2 \alpha=3/2
  15. E c E_{c}
  16. T c = ( N V f ζ ( 3 / 2 ) ) 2 / 3 h 2 2 π m k T_{c}=\left(\frac{N}{Vf\zeta(3/2)}\right)^{2/3}\frac{h^{2}}{2\pi mk}
  17. N = N 0 + Li α ( z ) ( β E c ) α N=N_{0}+\frac{\textrm{Li}_{\alpha}(z)}{(\beta E_{c})^{\alpha}}
  18. N 0 = g 0 z 1 - z N_{0}=\frac{g_{0}\,z}{1-z}
  19. τ = T T c \tau=\frac{T}{T_{c}}
  20. N = g 0 z 1 - z + N Li α ( z ) ζ ( α ) τ α N=\frac{g_{0}\,z}{1-z}+N~{}\frac{\textrm{Li}_{\alpha}(z)}{\zeta(\alpha)}~{}% \tau^{\alpha}
  21. Ω = g 0 ln ( 1 - z ) - Li α + 1 ( z ) ( β E c ) α \Omega=g_{0}\ln(1-z)-\frac{\textrm{Li}_{\alpha+1}(z)}{\left(\beta E_{c}\right)% ^{\alpha}}
  22. τ α \tau^{\alpha}
  23. T T c T\ll T_{c}\,
  24. T T c T\gg T_{c}\,
  25. = 1 =1\,
  26. ζ ( α ) τ α - ζ 2 ( α ) 2 α τ 2 α \approx\frac{\zeta(\alpha)}{\tau^{\alpha}}-\frac{\zeta^{2}(\alpha)}{2^{\alpha}% \tau^{2\alpha}}
  27. 1 - N 0 N 1-\frac{N_{0}}{N}\,
  28. = Li α ( z ) ζ ( α ) τ α =\frac{\textrm{Li}_{\alpha}(z)}{\zeta(\alpha)}\,\tau^{\alpha}
  29. = τ α =\tau^{\alpha}\,
  30. = 1 =1\,
  31. P V β N = - Ω N \frac{PV\beta}{N}=-\frac{\Omega}{N}\,
  32. = Li α + 1 ( z ) ζ ( α ) τ α =\frac{\textrm{Li}_{\alpha\!+\!1}(z)}{\zeta(\alpha)}\,\tau^{\alpha}
  33. = ζ ( α + 1 ) ζ ( α ) τ α =\frac{\zeta(\alpha\!+\!1)}{\zeta(\alpha)}\,\tau^{\alpha}
  34. 1 - ζ ( α ) 2 α + 1 τ α \approx 1-\frac{\zeta(\alpha)}{2^{\alpha\!+\!1}\tau^{\alpha}}
  35. G = ln ( z ) G=\ln(z)\,
  36. = ln ( z ) =\ln(z)\,
  37. = 0 =0\,
  38. ln ( ζ ( α ) τ α ) - ζ ( α ) 2 α τ α \approx\ln\left(\frac{\zeta(\alpha)}{\tau^{\alpha}}\right)-\frac{\zeta(\alpha)% }{2^{\alpha}\tau^{\alpha}}
  39. U = Ω β = α P V U=\frac{\partial\Omega}{\partial\beta}=\alpha PV
  40. C v = U T = k ( α + 1 ) U β C_{v}=\frac{\partial U}{\partial T}=k(\alpha+1)\,U\beta
  41. T S = U + P V - G TS=U+PV-G\,
  42. T S = ( α + 1 ) + ln ( τ α ζ ( α ) ) TS=(\alpha+1)+\ln\left(\frac{\tau^{\alpha}}{\zeta(\alpha)}\right)

Bosonic_string_theory.html

  1. M 2 M^{2}
  2. M 2 = - 1 α M^{2}=-\frac{1}{\alpha^{\prime}}
  3. I 0 [ g , X ] = T 8 π M d 2 ξ g g m n m x μ n x ν G μ ν ( x ) I_{0}[g,X]=\frac{T}{8\pi}\int_{M}d^{2}\xi\sqrt{g}g^{mn}\partial_{m}x^{\mu}% \partial_{n}x^{\nu}G_{\mu\nu}(x)
  4. X μ ( ξ ) X^{\mu}(\xi)
  5. g g
  6. G G
  7. G μ ν = δ μ ν G_{\mu\nu}=\delta_{\mu\nu}
  8. ξ \xi
  9. T T
  10. T = 1 2 π α T=\frac{1}{2\pi\alpha^{\prime}}
  11. I 0 I_{0}
  12. I = I 0 + λ χ ( M ) + μ 0 2 M d 2 ξ g I=I_{0}+\lambda\chi(M)+\mu_{0}^{2}\int_{M}d^{2}\xi\sqrt{g}
  13. Z = h = 0 𝒟 g m n 𝒟 X μ 𝒩 exp ( - I [ g , X ] ) Z=\sum_{h=0}^{\infty}\int\frac{\mathcal{D}g_{mn}\mathcal{D}X^{\mu}}{\mathcal{N% }}\exp(-I[g,X])
  14. V i 1 ( k 1 μ ) V i p ( k p μ ) = h = 0 𝒟 g m n 𝒟 X μ 𝒩 exp ( - I [ g , X ] ) V i 1 ( k 1 μ ) V i p ( k p μ ) \left\langle V_{i_{1}}(k^{\mu}_{1})\cdots V_{i_{p}}(k_{p}^{\mu})\right\rangle=% \sum_{h=0}^{\infty}\int\frac{\mathcal{D}g_{mn}\mathcal{D}X^{\mu}}{\mathcal{N}}% \exp(-I[g,X])V_{i_{1}}(k_{1}^{\mu})\cdots V_{i_{p}}(k^{\mu}_{p})
  15. h h
  16. 𝒩 \mathcal{N}
  17. p p
  18. g g
  19. g ( ξ ) = e σ ( ξ ) g ( ξ ) g^{\prime}(\xi)=e^{\sigma(\xi)}g(\xi)
  20. h 4 h\geq 4
  21. Z 0 = 0 Z_{0}=0
  22. A 4 ( 2 π ) 26 δ 26 ( k ) Γ ( - 1 - s / 2 ) Γ ( - 1 - t / 2 ) Γ ( - 1 - u / 2 ) Γ ( 2 + s / 2 ) Γ ( 2 + t / 2 ) Γ ( 2 + u / 2 ) A_{4}\propto(2\pi)^{26}\delta^{26}(k)\frac{\Gamma(-1-s/2)\Gamma(-1-t/2)\Gamma(% -1-u/2)}{\Gamma(2+s/2)\Gamma(2+t/2)\Gamma(2+u/2)}
  23. k k
  24. s s
  25. t t
  26. u u
  27. Z 1 = 1 d 2 τ 8 π 2 τ 2 2 1 ( 4 π 2 τ 2 ) 12 | η ( τ ) | - 48 Z_{1}=\int_{\mathcal{M}_{1}}\frac{d^{2}\tau}{8\pi^{2}\tau_{2}^{2}}\frac{1}{(4% \pi^{2}\tau_{2})^{12}}\left|\eta(\tau)\right|^{-48}
  28. τ \tau
  29. τ 2 \tau_{2}
  30. 1 \mathcal{M}_{1}
  31. P S L ( 2 , ) PSL(2,\mathbb{Z})
  32. { τ 2 > 0 , | τ | 2 > 1 , - 1 2 < τ 1 < 1 2 } \left\{\tau_{2}>0,|\tau|^{2}>1,-\frac{1}{2}<\tau_{1}<\frac{1}{2}\right\}
  33. η ( τ ) \eta(\tau)
  34. d 2 τ τ 2 2 \frac{d^{2}\tau}{\tau_{2}^{2}}
  35. τ 2 | c τ + d | 2 τ 2 \tau_{2}\rightarrow|c\tau+d|^{2}\tau_{2}
  36. η ( τ ) \eta(\tau)

Bott_periodicity_theorem.html

  1. U ( 1 ) U ( 2 ) U = k = 1 U ( k ) U(1)\subset U(2)\subset\cdots\subset U=\bigcup_{k=1}^{\infty}U(k)
  2. π n S \pi_{n}^{S}
  3. π n S \pi_{n}^{S}
  4. π k ( U ) = π k + 2 ( U ) \pi_{k}(U)=\pi_{k+2}(U)\,\!
  5. π k ( O ) = π k + 4 ( Sp ) \pi_{k}(O)=\pi_{k+4}(\operatorname{Sp})\,\!
  6. π k ( Sp ) = π k + 4 ( O ) , k = 0 , 1 , . \pi_{k}(\operatorname{Sp})=\pi_{k+4}(O),\ \ k=0,1,\dots.\,\!
  7. π k ( O ) = π k + 8 ( O ) \pi_{k}(O)=\pi_{k+8}(O)\,\!
  8. π k ( Sp ) = π k + 8 ( Sp ) , k = 0 , 1 , . \pi_{k}(\operatorname{Sp})=\pi_{k+8}(\operatorname{Sp}),\ \ k=0,1,\dots.\,\!
  9. Ω 2 B U 𝐙 × B U \Omega^{2}BU\simeq\mathbf{Z}\times BU\,
  10. Ω 2 U U . \Omega^{2}U\simeq U.\,
  11. Ω 8 B O 𝐙 × B O ; \Omega^{8}BO\simeq\mathbf{Z}\times BO;\,
  12. Ω 8 O O , \Omega^{8}O\simeq O,\,
  13. Ω 8 BSp 𝐙 × BSp ; \Omega^{8}\operatorname{BSp}\simeq\mathbf{Z}\times\operatorname{BSp};\,
  14. Ω 8 Sp Sp . \Omega^{8}\operatorname{Sp}\simeq\operatorname{Sp}.\,
  15. U × U U U × U . U\times U\subset U\subset U\times U.\,
  16. O × O O U Sp Sp × Sp Sp U O O × O . O\times O\subset O\subset U\subset\operatorname{Sp}\subset\operatorname{Sp}% \times\operatorname{Sp}\subset\operatorname{Sp}\subset U\subset O\subset O% \times O.\,
  17. 𝐂 𝐂 𝐂 𝐂 𝐂 . \mathbf{C}\oplus\mathbf{C}\subset\mathbf{C}\subset\mathbf{C}\oplus\mathbf{C}.\,
  18. 𝐑 𝐑 𝐑 𝐂 𝐇 𝐇 × 𝐇 𝐇 𝐂 𝐑 𝐑 𝐑 \mathbf{R}\oplus\mathbf{R}\subset\mathbf{R}\subset\mathbf{C}\subset\mathbf{H}% \subset\mathbf{H}\times\mathbf{H}\subset\mathbf{H}\subset\mathbf{C}\subset% \mathbf{R}\subset\mathbf{R}\oplus\mathbf{R}\,
  19. Ω U \displaystyle\Omega U
  20. Ω ( 𝐙 × B O ) \displaystyle\Omega(\mathbf{Z}\times BO)
  21. Ω 0 \Omega^{0}
  22. 𝐙 × O / ( O × O ) \mathbf{Z}\times O/(O\times O)
  23. Ω 1 \Omega^{1}
  24. O = ( O × O ) / O O=(O\times O)/O
  25. Ω 2 \Omega^{2}
  26. O / U O/U
  27. Ω 3 \Omega^{3}
  28. U / Sp U/\mathrm{Sp}
  29. Ω 4 \Omega^{4}
  30. 𝐙 × Sp / ( Sp × Sp ) \mathbf{Z}\times\mathrm{Sp}/(\mathrm{Sp}\times\mathrm{Sp})
  31. Ω 5 \Omega^{5}
  32. Sp = ( Sp × Sp ) / Sp \mathrm{Sp}=(\mathrm{Sp}\times\mathrm{Sp})/\mathrm{Sp}
  33. Ω 6 \Omega^{6}
  34. Sp / U \mathrm{Sp}/U
  35. Ω 7 \Omega^{7}
  36. U / O U/O

Bouguer_anomaly.html

  1. g o b s g_{obs}
  2. g B = g o b s - g λ + δ g F - δ g B g_{B}=g_{obs}-g_{\lambda}+\delta g_{F}-\delta g_{B}
  3. g B = g F - δ g B g_{B}=g_{F}-\delta g_{B}
  4. g B g_{B}
  5. g o b s g_{obs}
  6. g λ g_{\lambda}
  7. δ g F \delta g_{F}
  8. δ g B \delta g_{B}
  9. g F g_{F}
  10. g g
  11. G G
  12. G G
  13. g g
  14. H \scriptstyle H
  15. δ g B = 2 π ρ G H \delta g_{B}=2\pi\rho GH
  16. ρ \rho
  17. G G

Bound_state.html

  1. n n
  2. H H
  3. U = { U ( t ) t } U=\{U(t)\mid t\in\mathbb{R}\}
  4. H H
  5. ρ = ρ ( t 0 ) \rho=\rho(t_{0})
  6. H H
  7. A A
  8. H H
  9. μ ( A , ρ ) \mu(A,\rho)
  10. A A
  11. ρ ρ
  12. \mathbb{R}
  13. ρ ρ
  14. U U
  15. A A
  16. lim R t t 0 μ ( A , ρ ( t ) ) ( > R ) = 0 \lim_{R\rightarrow\infty}\sum_{t\geq t_{0}}\mu(A,\rho(t))(\mathbb{R}_{>R})=0
  17. > R = { x x > R } \mathbb{R}_{>R}=\{x\in\mathbb{R}\mid x>R\}
  18. H = L 2 ( ) H=L^{2}(\mathbb{R})
  19. A A
  20. ρ = ρ ( 0 ) H \rho=\rho(0)\in H
  21. [ - 1 , 1 ] Supp ( ρ ) [-1,1]\subseteq\mathrm{Supp}(\rho)
  22. ρ ρ
  23. [ t - 1 , t + 1 ] Supp ( ρ ( t ) ) [t-1,t+1]\in\mathrm{Supp}(\rho(t))
  24. t 0 t\geq 0
  25. ρ ρ
  26. ρ \rho
  27. ρ ( t ) = ρ \rho(t)=\rho
  28. t 0 t\geq 0
  29. ρ \rho
  30. ρ ρ
  31. ρ ρ
  32. ρ ρ

Bounding_sphere.html

  1. O ( n d ϵ + 1 ϵ 4.5 log 1 ϵ ) O(\frac{nd}{\epsilon}+\frac{1}{\epsilon^{4.5}}\log{\frac{1}{\epsilon}})
  2. O ( n d ϵ 2 ) O(\frac{nd}{\epsilon^{2}})
  3. O ( n d ) + O ( d ϵ 2 ) O(nd)+O(\frac{d}{\epsilon^{2}})

Bounding_volume.html

  1. r = 0.5 L x | N x | + 0.5 L y | N y | + 0.5 L z | N z | r=0.5L_{x}|N_{x}|+0.5L_{y}|N_{y}|+0.5L_{z}|N_{z}|\,
  2. b = C * N b=C*N\,
  3. b = C x N x + C y N y + C z N z b=C_{x}N_{x}+C_{y}N_{y}+C_{z}N_{z}\,
  4. m = b - r , n = b + r m=b-r,n=b+r\,
  5. r = 0.5 L x | N * I | + 0.5 L y | N * J | + 0.5 L z | N * K | r=0.5L_{x}|N*I|+0.5L_{y}|N*J|+0.5L_{z}|N*K|\,
  6. m = C * N - r and n = C * N + r m=C*N-r\mbox{ and }~{}n=C*N+r\,

Bourbaki–Witt_theorem.html

  1. f : X X f:X\to X
  2. f ( x ) x f(x)\geq x
  3. x , x,
  4. x n + 1 = f ( x n ) , n = 0 , 1 , 2 , , x_{n+1}=f(x_{n}),n=0,1,2,\ldots,
  5. x n = x , x_{n}=x_{\infty},\quad
  6. y X y\in X
  7. K ( 0 ) = y \,K(0)=y
  8. K ( α + 1 ) = f ( K ( α ) ) . \,K(\alpha+1)=f(K(\alpha)).
  9. β \beta
  10. { K ( α ) : α < β } \{K(\alpha)\ :\ \alpha<\beta\}
  11. K ( β ) = sup { K ( α ) : α < β } . K(\beta)=\sup\{K(\alpha)\ :\ \alpha<\beta\}.
  12. α , K ( α + 1 ) = K ( α ) ; \alpha,\ \ K(\alpha+1)=K(\alpha);
  13. f ( K ( α ) ) = K ( α ) . \,f(K(\alpha))=K(\alpha).
  14. x = K ( α ) , \,x=K(\alpha),
  15. P ( X ) - { } . P(X)-\{\varnothing\}.
  16. f : X X f:X\to X
  17. f ( x ) = g ( { y : y > x } ) . f(x)=g(\{y\ :\ y>x\}).

Bow_shocks_in_astrophysics.html

  1. c s 2 = γ p / ρ c_{s}^{2}=\gamma p/\rho
  2. γ \gamma
  3. ρ \rho
  4. v v
  5. V A V_{A}
  6. M A 1 M_{A}\gg 1
  7. ρ 0 v 2 = B 0 2 2 μ 0 , \rho_{0}v^{2}={B_{0}^{2}\over 2\mu_{0}},
  8. ρ 0 \rho_{0}
  9. B 0 B_{0}
  10. v v

Bowling_average.html

  1. Bowling average = Runs conceded Wickets taken \mathrm{Bowling~{}average}=\frac{\mathrm{Runs~{}conceded}}{\mathrm{Wickets~{}% taken}}

Box_office.html

  1. T W G 2 - P B {TWG\over 2}-{PB}

Box_topology.html

  1. X X
  2. X := i I X i , X:=\prod_{i\in I}X_{i},
  3. X i X_{i}
  4. i I i\in I
  5. X X
  6. B = { i I U i | U i open in X i } . B=\left\{\prod_{i\in I}U_{i}\Big|U_{i}\,\text{ open in }X_{i}\right\}.
  7. X = X i X=\prod X_{i}
  8. X i = { 0 , 1 } X_{i}=\{0,1\}
  9. X X
  10. { x n } n = 1 \{x_{n}\}_{n=1}^{\infty}
  11. ( x n ) m = { 0 m < n 1 m n (x_{n})_{m}=\begin{cases}0&m<n\\ 1&m\geq n\end{cases}
  12. X X

Braess'_paradox.html

  1. A 100 + 45 \tfrac{A}{100}+45
  2. B 100 + 45 \tfrac{B}{100}+45
  3. A + B = 4000 A+B=4000
  4. A = B = 2000 A=B=2000
  5. 2000 100 + 45 = 65 \tfrac{2000}{100}+45=65
  6. T 100 = 4000 100 = 40 \tfrac{T}{100}=\tfrac{4000}{100}=40
  7. 0 + 4000 100 = 40 0+\tfrac{4000}{100}=40
  8. 4000 100 + 4000 100 = 80 \tfrac{4000}{100}+\tfrac{4000}{100}=80
  9. L e ( x ) L_{e}(x)
  10. x x
  11. e e
  12. L e ( x ) = a e x + b e > 0 L_{e}(x)=a_{e}x+b_{e}>0
  13. a e a_{e}
  14. b e b_{e}
  15. e x e_{x}
  16. e e
  17. E ( e ) E(e)
  18. i = 1 e x L e ( i ) = L e ( 1 ) + L e ( 2 ) + + L e ( e x ) \sum_{i=1}^{e_{x}}L_{e}(i)=L_{e}(1)+L_{e}(2)+\cdots+L_{e}(e_{x})
  19. x e = 0 x_{e}=0
  20. E ( e ) = 0 E(e)=0
  21. x 0 , x 1 , , x n x_{0},x_{1},\ldots,x_{n}
  22. y 0 , y 1 , , y m y_{0},y_{1},\ldots,y_{m}
  23. E E
  24. x 0 , x 1 , x n x_{0},x_{1},...x_{n}
  25. e e
  26. L e ( e x ) L_{e}(e_{x})
  27. E E
  28. i = 0 n L e ( e x ) \sum_{i=0}^{n}L_{e}(e_{x})
  29. y 0 , y 1 , , y m y_{0},y_{1},\ldots,y_{m}
  30. E E
  31. E E
  32. E E
  33. E E
  34. S j = starting point for car j S_{j}=\,\text{starting point for car }j\,
  35. T j = target for car j T_{j}=\,\text{target for car }j\,
  36. S j S_{j}
  37. T j T_{j}
  38. L e ( x ) = a e x + b e L_{e}(x)=a_{e}x+b_{e}
  39. a e , b e 0 a_{e},b_{e}\geq 0
  40. E ( e ) = L e ( 1 ) + L e ( 2 ) + + L e ( x ) E(e)=L_{e}(1)+L_{e}(2)+\cdots+L_{e}(x)
  41. T ( e ) = x L e ( x ) T(e)=xL_{e}(x)
  42. L e ( 1 ) + + L e ( x ) \displaystyle L_{e}(1)+\cdots+L_{e}(x)
  43. 1 2 T ( e ) E ( e ) T ( e ) \tfrac{1}{2}T(e)\leq E(e)\leq T(e)
  44. 1 2 social cost ( Z ) E ( Z ) social cost ( Z ) \tfrac{1}{2}\,\text{social cost}(Z)\leq E(Z)\leq\,\text{social cost}(Z)
  45. social cost ( Z ) \displaystyle\,\text{social cost}(Z^{\prime})

Branching_fraction.html

  1. t 1 / 2 = ln 2 λ . t_{1/2}=\frac{\ln 2}{\lambda}.

Brans–Dicke_theory.html

  1. ϕ \phi
  2. g a b g_{ab}
  3. R a b c d R_{abcd}
  4. ϕ \phi
  5. ω \omega
  6. ϕ = 1 \phi=1
  7. ω \omega
  8. ω \omega
  9. ω \omega
  10. ω > 5 \omega>5
  11. ω > 30 \omega>30
  12. ω \omega
  13. ω \omega\rightarrow\infty
  14. T μ μ = 0 T^{\mu}_{\mu}=0
  15. ϕ = 8 π 3 + 2 ω T \Box\phi=\frac{8\pi}{3+2\omega}T
  16. G a b = 8 π ϕ T a b + ω ϕ 2 ( a ϕ b ϕ - 1 2 g a b c ϕ c ϕ ) + 1 ϕ ( a b ϕ - g a b ϕ ) G_{ab}=\frac{8\pi}{\phi}T_{ab}+\frac{\omega}{\phi^{2}}(\partial_{a}\phi% \partial_{b}\phi-\frac{1}{2}g_{ab}\partial_{c}\phi\partial^{c}\phi)+\frac{1}{% \phi}(\nabla_{a}\nabla_{b}\phi-g_{ab}\Box\phi)
  17. ω \omega
  18. g a b g_{ab}
  19. G a b = R a b - 1 2 R g a b G_{ab}=R_{ab}-\tfrac{1}{2}Rg_{ab}
  20. R a b = R m a m b R_{ab}={R^{m}}_{amb}
  21. R = R m m R={R^{m}}_{m}
  22. T a b T_{ab}
  23. T = T a a T=T_{a}^{a}
  24. ϕ \phi
  25. \Box
  26. ϕ = ϕ ; a ; a \Box\phi=\phi^{;a}_{\;\;;a}
  27. ϕ \phi
  28. ϕ \phi
  29. ϕ \phi
  30. ϕ \phi
  31. ϕ \phi
  32. G a b = 8 π T a b . G_{ab}=8\pi T_{ab}.
  33. ϕ \phi\,
  34. S = d 4 x - g ( ϕ R - ω a ϕ a ϕ ϕ 16 π + M ) S=\int d^{4}x\sqrt{-g}\;\left(\frac{\phi R-\omega\frac{\partial_{a}\phi% \partial^{a}\phi}{\phi}}{16\pi}+\mathcal{L}_{\mathrm{M}}\right)
  35. g g
  36. - g d 4 x \sqrt{-g}\,d^{4}x
  37. M \mathcal{L}_{\mathrm{M}}
  38. g a b g_{ab}
  39. ϕ \phi
  40. δ R a b / δ g c d \delta R_{ab}/\delta g_{cd}
  41. δ ( ϕ R ) δ g a b = ϕ R a b + g a b g c d ϕ ; c d - ϕ ; a b \frac{\delta(\phi R)}{\delta g^{ab}}=\phi R_{ab}+g_{ab}g^{cd}\phi_{;cd}-\phi_{% ;ab}
  42. δ ( ϕ R ) = R δ ϕ + ϕ R m n δ g m n + ϕ s ( g m n δ Γ n m s - g m s δ Γ r m r ) \delta(\phi R)=R\delta\phi+\phi R_{mn}\delta g^{mn}+\phi\nabla_{s}(g^{mn}% \delta\Gamma^{s}_{nm}-g^{ms}\delta\Gamma^{r}_{rm})
  43. δ Γ \delta\Gamma
  44. ( g a b g c d ϕ ; c d - ϕ ; a b ) δ g a b (g_{ab}g^{cd}\phi_{;cd}-\phi_{;ab})\delta g^{ab}
  45. S = d 4 x - g ( R 16 π G + M ) S=\int d^{4}x\sqrt{-g}\;\left(\frac{R}{16\pi G}+\mathcal{L}_{\mathrm{M}}\right)
  46. g a b g_{ab}

Bravais_lattice.html

  1. 𝐑 = n 1 𝐚 1 + n 2 𝐚 2 + n 3 𝐚 3 \mathbf{R}=n_{1}\mathbf{a}_{1}+n_{2}\mathbf{a}_{2}+n_{3}\mathbf{a}_{3}
  2. a b c 1 - cos 2 α - cos 2 β - cos 2 γ + 2 cos α cos β cos γ abc\sqrt{1-\cos^{2}\alpha-\cos^{2}\beta-\cos^{2}\gamma+2\cos\alpha\cos\beta% \cos\gamma}
  3. a b c sin β abc~{}\sin\beta
  4. a b c abc
  5. a 2 c a^{2}c
  6. a 3 1 - 3 cos 2 α + 2 cos 3 α a^{3}\sqrt{1-3\cos^{2}\alpha+2\cos^{3}\alpha}
  7. 3 a 2 c 2 \frac{\sqrt{3\,}\,a^{2}c}{2}
  8. a 3 a^{3}

Brazilian_cruzado.html

  1. CzS \mathrm{CzS}\!\!\!\|

Brazilian_cruzeiro_real.html

  1. CRS \mathrm{CRS}\!\!\!\|

Breidbart_Index.html

  1. BI = k = 1 m n k \mbox{BI}~{}=\sum_{k=1}^{m}\sqrt{n_{k}}
  2. 9 + 16 = 3 + 4 = 7 \sqrt{9}+\sqrt{16}=3+4=7
  3. BI2 = k = 1 m n k + n k 2 \mbox{BI2}~{}=\sum_{k=1}^{m}\frac{n_{k}+\sqrt{n_{k}}}{2}
  4. 9 + 16 + 9 + 16 2 = 3 + 4 + 9 + 16 2 = 32 2 = 16 \frac{\sqrt{9}+\sqrt{16}+9+16}{2}=\frac{3+4+9+16}{2}=\frac{32}{2}=16
  5. 9 + 16 + 9 + 4 2 = 3 + 4 + 9 + 4 2 = 20 2 = 10 \frac{\sqrt{9}+\sqrt{16}+9+4}{2}=\frac{3+4+9+4}{2}=\frac{20}{2}=10

Broaching_(metalworking).html

  1. P 0.35 L w P\cong 0.35\sqrt{L_{\mathrm{w}}}

Brown's_representability_theorem.html

  1. F ( α X α ) α F ( X α ) , F(\vee_{\alpha}X_{\alpha})\cong\prod_{\alpha}F(X_{\alpha}),

Bruhat_decomposition.html

  1. G = B W B = w W B w B G=BWB=\coprod_{w\in W}BwB
  2. n × n n\times n
  3. n × n n\times n

Buchberger's_algorithm.html

  1. 2 ( d 2 2 + d ) 2 n - 1 2\left(\frac{d^{2}}{2}+d\right)^{2^{n-1}}
  2. n n
  3. d d
  4. d 2 n + o ( 1 ) d^{2^{n+o(1)}}
  5. d 2 Ω ( n ) d^{2^{\Omega(n)}}

Buckling.html

  1. F = π 2 E I ( K L ) 2 F=\frac{\pi^{2}EI}{(KL)^{2}}
  2. F F
  3. E E
  4. I I
  5. L L
  6. K K
  7. K K
  8. K K
  9. K K
  10. K K
  11. K L KL
  12. σ = F A = π 2 E ( / r ) 2 \sigma=\frac{F}{A}=\frac{\pi^{2}E}{(\ell/r)^{2}}
  13. F / A F/A
  14. l / r l/r
  15. 1 F m a x = 1 F e + 1 F c \frac{1}{F_{max}}=\frac{1}{F_{e}}+\frac{1}{F_{c}}
  16. ρ \rho
  17. E E
  18. A A
  19. h c r i t = ( 9 B 2 4 E I ρ g A ) 1 / 3 h_{crit}=\left(\frac{9B^{2}}{4}\,\frac{EI}{\rho gA}\right)^{1/3}
  20. { 1 3 ρ l 1 2 ( l 1 + 3 l 2 ) α ¨ 1 + 1 2 ρ l 1 l 2 2 cos ( α 1 - α 2 ) α ¨ 2 + 1 2 ρ l 1 l 2 2 sin ( α 1 - α 2 ) α ˙ 2 2 + ( k 1 + k 2 ) α 1 - k 2 α 2 + + ( β 1 + β 2 ) α ˙ 1 - β 2 α ˙ 2 - l 1 P sin ( α 1 - α 2 ) = 0 , 1 2 ρ l 1 l 2 2 cos ( α 1 - α 2 ) α ¨ 1 + 1 3 ρ l 2 3 α ¨ 2 - 1 2 ρ l 1 l 2 2 sin ( α 1 - α 2 ) α ˙ 1 2 - k 2 ( α 1 - α 2 ) - β 2 ( α ˙ 1 - α ˙ 2 ) = 0 , \ \left\{\begin{array}[]{l}\frac{1}{3}\rho l_{1}^{2}\left(l_{1}+3l_{2}\right)% \ddot{\alpha}_{1}+\frac{1}{2}\rho l_{1}l_{2}^{2}\cos(\alpha_{1}-\alpha_{2})% \ddot{\alpha}_{2}+\frac{1}{2}\rho l_{1}l_{2}^{2}\sin(\alpha_{1}-\alpha_{2})% \dot{\alpha}_{2}^{2}+(k_{1}+k_{2})\alpha_{1}-k_{2}\alpha_{2}\,+\\ +(\beta_{1}+\beta_{2})\dot{\alpha}_{1}-\beta_{2}\dot{\alpha}_{2}-l_{1}P\sin(% \alpha_{1}-\alpha_{2})=0,\\ \frac{1}{2}\rho l_{1}l_{2}^{2}\cos(\alpha_{1}-\alpha_{2})\ddot{\alpha}_{1}+% \frac{1}{3}\rho l_{2}^{3}\ddot{\alpha}_{2}-\frac{1}{2}\rho l_{1}l_{2}^{2}\sin(% \alpha_{1}-\alpha_{2})\dot{\alpha}_{1}^{2}-k_{2}(\alpha_{1}-\alpha_{2})-\beta_% {2}(\dot{\alpha}_{1}-\dot{\alpha}_{2})=0,\end{array}\right.
  21. { 1 3 ρ l 1 2 ( l 1 + 3 l 2 ) α ¨ 1 + 1 2 ρ l 1 l 2 2 α ¨ 2 + ( k 1 + k 2 ) α 1 - k 2 α 2 - l 1 P ( α 1 - α 2 ) = 0 , 1 2 ρ l 1 l 2 2 α ¨ 1 + 1 3 ρ l 2 3 α ¨ 2 - k 2 ( α 1 - α 2 ) = 0. \ \left\{\begin{array}[]{l}\frac{1}{3}\rho l_{1}^{2}\left(l_{1}+3l_{2}\right)% \ddot{\alpha}_{1}+\frac{1}{2}\rho l_{1}l_{2}^{2}\ddot{\alpha}_{2}+(k_{1}+k_{2}% )\alpha_{1}-k_{2}\alpha_{2}-l_{1}P(\alpha_{1}-\alpha_{2})=0,\\ \frac{1}{2}\rho l_{1}l_{2}^{2}\ddot{\alpha}_{1}+\frac{1}{3}\rho l_{2}^{3}\ddot% {\alpha}_{2}-k_{2}(\alpha_{1}-\alpha_{2})=0.\end{array}\right.
  22. α j = A j e - i Ω t , j = 1 , 2 , \ \alpha_{j}=A_{j}\,e^{-i\Omega\,t},~{}~{}~{}j=1,2,
  23. P f \ P_{f}
  24. P d \ P_{d}
  25. P f , d = k 2 l 1 k + ( 1 + λ ) 3 λ k ( 3 + 4 λ ) 1 + 3 λ / 2 \ P_{f,d}=\frac{k_{2}}{l_{1}}\cdot\frac{k+(1+\lambda)^{3}\mp\lambda\,\sqrt{k(3% +4\lambda)}}{1+3\lambda/2}
  26. λ = l 1 / l 2 \ \lambda=l_{1}/l_{2}
  27. k = k 1 / k 2 \ k=k_{1}/k_{2}
  28. F = E A α L Δ T F=EA\alpha_{L}\Delta T
  29. Δ L = α L Δ T L {\Delta L}=\alpha_{L}\Delta T\ L
  30. Δ L = F E A L \Delta L=\frac{F}{EA}L
  31. F E A L = α L Δ T L {\frac{F}{EA}L}=\alpha_{L}\Delta T\ L
  32. F = E A α L Δ T F=EA\alpha_{L}\Delta T
  33. U inner = E 2 I ( x ) ( w x x ( x ) ) 2 d x U\text{inner}=\frac{E}{2}\int I(x)(w_{xx}(x))^{2}\,dx
  34. U outer = P Crit 2 ( w x ( x ) ) 2 d x U\text{outer}=\frac{P\text{Crit}}{2}\int(w_{x}(x))^{2}\,dx
  35. w ( x ) w(x)
  36. x x
  37. x x xx
  38. U Inner = U Outer U\text{Inner}=U\text{Outer}\,
  39. C b = 12.5 M max 2.5 M max + 3 M A + 4 M B + 3 M C \ C_{b}=\frac{12.5M_{\max}}{2.5M_{\max}+3M_{A}+4M_{B}+3M_{C}}
  40. M max M_{\max}
  41. M A M_{A}
  42. M B M_{B}
  43. M C M_{C}
  44. σ L = ρ c 2 h \sigma L=\rho c^{2}h\,
  45. σ \sigma
  46. L L
  47. c c
  48. h h
  49. σ \sigma
  50. h h
  51. h h

Buddhabrot.html

  1. c c
  2. z n + 1 = z n 2 + c z_{n+1}={z_{n}}^{2}+c
  3. n n
  4. z 0 = 0 z_{0}=0
  5. ( i , j ) (i,j)
  6. i = 1 , , m i=1,\ldots,m
  7. j = 1 , , n j=1,\ldots,n
  8. Δ x \Delta x
  9. Δ y \Delta y
  10. Δ x = w / m \Delta x=w/m
  11. Δ y = h / n \Delta y=h/n
  12. w w
  13. h h
  14. c c
  15. c c
  16. z n z_{n}
  17. ( Re ( z n ) , Im ( z n ) ) (\,\text{Re}(z_{n}),\,\text{Im}(z_{n}))
  18. c c
  19. z n z_{n}
  20. z 2 + c z^{2}+c
  21. λ x ( 1 - x ) \lambda x(1-x)
  22. c r \displaystyle c_{r}
  23. c r c_{r}
  24. λ \lambda
  25. z 2 + c z^{2}+c
  26. c = ( random , 0 ) c=(\,\text{random},0)
  27. z 0 = ( 0 , 0 ) z_{0}=(0,0)
  28. { c r , z r } \{c_{r},z_{r}\}
  29. z 0 = ( random , 0 ) z_{0}=(\,\text{random},0)
  30. z r 0 z_{r0}
  31. { c r , c i , z r } \{c_{r},c_{i},z_{r}\}
  32. c = ( random , random ) c=(\,\text{random},\,\text{random})
  33. z 0 = ( 0 , 0 ) z_{0}=(0,0)
  34. { c r , c i } \{c_{r},c_{i}\}
  35. c = ( random , 0 ) c=(\,\text{random},0)
  36. z 0 = ( 0 , 0 ) z_{0}=(0,0)
  37. { c r , z r } \{c_{r},z_{r}\}
  38. { c i , z r } \{c_{i},z_{r}\}
  39. c r c_{r}
  40. { c r , c i } \{c_{r},c_{i}\}
  41. { c r , z r } \{c_{r},z_{r}\}
  42. { c r , - c i } \{c_{r},-c_{i}\}
  43. c r c_{r}
  44. { c r , z r } \{c_{r},z_{r}\}
  45. c i = 0 c_{i}=0
  46. { c r , c i , z r } \{c_{r},c_{i},z_{r}\}
  47. { c r , c i } \{c_{r},c_{i}\}
  48. c i = 0 c_{i}=0
  49. c i c_{i}

Buffon's_needle.html

  1. l l
  2. { 2 t : 0 x t 2 0 : elsewhere. \begin{cases}\frac{2}{t}&:\ 0\leq x\leq\frac{t}{2}\\ 0&:\,\text{elsewhere.}\end{cases}
  3. { 2 π : 0 θ π 2 0 : elsewhere. \begin{cases}\frac{2}{\pi}&:\ 0\leq\theta\leq\frac{\pi}{2}\\ 0&:\,\text{elsewhere.}\end{cases}
  4. { 4 t π : 0 x t 2 , 0 θ π 2 0 : elsewhere. \begin{cases}\frac{4}{t\pi}&:\ 0\leq x\leq\frac{t}{2},\ 0\leq\theta\leq\frac{% \pi}{2}\\ 0&:\,\text{elsewhere.}\end{cases}
  5. x l 2 sin θ . x\leq\frac{l}{2}\sin\theta.
  6. P = θ = 0 π 2 x = 0 ( l / 2 ) sin θ 4 t π d x d θ = 2 l t π . P=\int_{\theta=0}^{\frac{\pi}{2}}\int_{x=0}^{(l/2)\sin\theta}\frac{4}{t\pi}\,% dx\,d\theta=\frac{2l}{t\pi}.
  7. l > t l>t
  8. θ = 0 π 2 x = 0 m ( θ ) 4 t π d x d θ , \int_{\theta=0}^{\frac{\pi}{2}}\int_{x=0}^{m(\theta)}\frac{4}{t\pi}\,dx\,d\theta,
  9. m ( θ ) m(\theta)
  10. ( l / 2 ) sin θ (l/2)\sin\theta
  11. t / 2 t/2
  12. t < l t<l
  13. 2 l t π - 2 t π { l 2 - t 2 + t sin - 1 ( t l ) } + 1 \frac{2l}{t\pi}-\frac{2}{t\pi}\left\{\sqrt{l^{2}-t^{2}}+t\sin^{-1}\left(\frac{% t}{l}\right)\right\}+1
  14. 2 π cos - 1 t l + 2 π l t { 1 - 1 - ( t l ) 2 } . \frac{2}{\pi}\cos^{-1}\frac{t}{l}+\frac{2}{\pi}\frac{l}{t}\left\{1-\sqrt{1-% \left(\frac{t}{l}\right)^{2}}\right\}.
  15. P P
  16. P = P 1 P 2 P=P_{1}\cdot P_{2}
  17. P 1 P_{1}
  18. P 2 P_{2}
  19. l / 2 l/2
  20. l 2 + l 2 \frac{l}{2}+\frac{l}{2}
  21. t t
  22. P 1 = l t . P_{1}=\frac{l}{t}.
  23. P 2 P_{2}
  24. l = 2 l=2
  25. θ ( x ) = cos - 1 ( x ) \theta\left(x\right)=\cos^{-1}\left(x\right)
  26. P 2 = 0 1 2 θ ( x ) π d x = 2 π 0 1 cos - 1 ( x ) d x = 2 π 1 = 2 π . P_{2}=\int_{0}^{1}\frac{2\theta(x)}{\pi}\,dx=\frac{2}{\pi}\int_{0}^{1}\cos^{-1% }(x)\,dx=\frac{2}{\pi}\cdot 1=\frac{2}{\pi}.
  27. P = P 1 P 2 = l t 2 π = 2 l t π P=P_{1}\cdot P_{2}=\frac{l}{t}\frac{2}{\pi}=\frac{2l}{t\pi}
  28. l cos θ l\cos\theta
  29. θ \theta
  30. t t
  31. l cos θ l\cos\theta
  32. t t
  33. 0 θ π / 2 0\leq\theta\leq\pi/2
  34. P = 0 π 2 l cos θ d θ 0 π 2 t d θ = l t 0 π 2 cos θ d θ 0 π 2 d θ = l t 1 π 2 = 2 l t π P=\frac{\int_{0}^{\frac{\pi}{2}}l\cos\theta d\theta}{\int_{0}^{\frac{\pi}{2}}% td\theta}=\frac{l}{t}\frac{\int_{0}^{\frac{\pi}{2}}\cos\theta d\theta}{\int_{0% }^{\frac{\pi}{2}}d\theta}=\frac{l}{t}\frac{1}{\frac{\pi}{2}}=\frac{2l}{t\pi}
  35. P P
  36. π = 2 l t P \pi=\frac{2l}{tP}
  37. P P
  38. P P
  39. h / n h/n
  40. π 2 l n t h . \pi\approx\frac{2l\cdot n}{th}.
  41. 5 3 π \frac{5}{3\pi}

Bulk_modulus.html

  1. K K
  2. B B
  3. K > 0 K>0
  4. K = - V d P d V K=-V\frac{\mathrm{d}P}{\mathrm{d}V}
  5. P P
  6. V V
  7. d P / d V dP/dV
  8. K = ρ d P d ρ K=\rho\frac{\mathrm{d}P}{\mathrm{d}\rho}
  9. K T K_{T}
  10. K S K_{S}
  11. K S K_{S}
  12. K S = γ p K_{S}=\gamma\,p
  13. K T K_{T}
  14. K T = p K_{T}=p
  15. c = K ρ . c=\sqrt{\frac{K}{\rho}}.
  16. K S K_{S}
  17. K T K_{T}

Bump_function.html

  1. C 0 ( 𝐑 n ) C^{\infty}_{0}(\mathbf{R}^{n})
  2. C c ( 𝐑 n ) C^{\infty}_{c}(\mathbf{R}^{n})
  3. Ψ ( x ) = { e - 1 1 - x 2 for | x | < 1 0 otherwise \Psi(x)=\begin{cases}e^{-\frac{1}{1-x^{2}}}&\mbox{ for }~{}|x|<1\\ 0&\mbox{ otherwise}\end{cases}
  4. e - y 2 e^{-y^{2}}
  5. y 2 = 1 / ( 1 - x 2 ) y^{2}=1/(1-x^{2})
  6. Φ ( x 1 , x 2 , , x n ) = Ψ ( x 1 ) Ψ ( x 2 ) Ψ ( x n ) . \Phi(x_{1},x_{2},\dots,x_{n})=\Psi(x_{1})\Psi(x_{2})\cdots\Psi(x_{n}).
  7. χ V \chi_{V}
  8. χ V \chi_{V}
  9. χ V \chi_{V}
  10. Φ \Phi
  11. Ψ ( x ) = e - 1 1 - x 2 𝟏 { | x | < 1 } \Psi(x)=e^{-\frac{1}{1-x^{2}}}\mathbf{1}_{\{|x|<1\}}
  12. | k | - 3 4 e - | k | |k|^{-\frac{3}{4}}e^{-\sqrt{|k|}}

Burgers'_equation.html

  1. y ( x , t ) y(x,t)
  2. d d
  3. y t + y y x = d 2 y x 2 \frac{\partial y}{\partial t}+y\frac{\partial y}{\partial x}=d\frac{\partial^{% 2}y}{\partial x^{2}}
  4. η ( x , t ) \eta(x,t)
  5. y t + y y x = d 2 y x 2 - λ η x \frac{\partial y}{\partial t}+y\frac{\partial y}{\partial x}=d\frac{\partial^{% 2}y}{\partial x^{2}}-\lambda\frac{\partial\eta}{\partial x}
  6. h ( x , t ) h(x,t)
  7. y ( x , t ) = - λ h / x y(x,t)=-\lambda\partial h/\partial x
  8. y t + y y x = 0 , \frac{\partial y}{\partial t}+y\frac{\partial y}{\partial x}=0,
  9. y t + 1 2 x ( y 2 ) = 0. \frac{\partial y}{\partial t}+\frac{1}{2}\frac{\partial}{\partial x}\big(y^{2}% \big)=0.
  10. j ( y ) = 1 2 y 2 j(y)=\frac{1}{2}y^{2}
  11. y t + j x ( y ) = 0 y_{t}+j_{x}(y)=0
  12. X ( t ) X(t)
  13. d X ( t ) d t = y [ X ( t ) , t ] \frac{dX(t)}{dt}=y[X(t),t]
  14. Y ( t ) := y [ X ( t ) , t ] Y(t):=y[X(t),t]
  15. t t
  16. [ X ( t ) , Y ( t ) ] [X(t),Y(t)]
  17. d X d t = Y , and d Y d t = 0. \frac{dX}{dt}=Y,\quad\mbox{and}~{}\quad\frac{dY}{dt}=0.
  18. X ( t ) = X ( 0 ) + t Y ( 0 ) , and Y ( t ) = Y ( 0 ) . X(t)=X(0)+tY(0),\quad\mbox{and}~{}\quad Y(t)=Y(0).
  19. X ( 0 ) = η X(0)=\eta
  20. Y ( 0 ) = y [ X ( 0 ) , 0 ] = y ( η , 0 ) Y(0)=y[X(0),0]=y(\eta,0)
  21. X ( t ) = η + t y ( η , 0 ) and Y ( t ) = Y ( 0 ) . X(t)=\eta+ty(\eta,0)\quad\mbox{and}~{}\quad Y(t)=Y(0).
  22. y ( η , 0 ) = Y ( 0 ) = Y ( t ) = y [ X ( t ) , t ] = y [ η + t y ( η , 0 ) , t ] . y(\eta,0)=Y(0)=Y(t)=y[X(t),t]=y[\eta+ty(\eta,0),t].
  23. y = - 2 d 1 ϕ ϕ x , y=-2d\frac{1}{\phi}\frac{\partial\phi}{\partial x},
  24. x ( 1 ϕ ϕ t ) = d x ( 1 ϕ 2 ϕ x 2 ) \frac{\partial}{\partial x}\Bigl(\frac{1}{\phi}\frac{\partial\phi}{\partial t}% \Bigr)=d\frac{\partial}{\partial x}\Bigl(\frac{1}{\phi}\frac{\partial^{2}\phi}% {\partial x^{2}}\Bigr)
  25. ϕ t = d 2 ϕ x 2 + f ( t ) ϕ \frac{\partial\phi}{\partial t}=d\frac{\partial^{2}\phi}{\partial x^{2}}+f(t)\phi
  26. ϕ t = d 2 ϕ x 2 . \frac{\partial\phi}{\partial t}=d\frac{\partial^{2}\phi}{\partial x^{2}}.
  27. y ( x , t ) = - 2 d x ln { ( 4 π d t ) - 1 / 2 - exp [ - ( x - x ) 2 4 d t - 1 2 d 0 x y ( x ′′ , 0 ) d x ′′ ] d x } . y(x,t)=-2d\frac{\partial}{\partial x}\ln\Bigl\{(4\pi dt)^{-1/2}\int_{-\infty}^% {\infty}\exp\Bigl[-\frac{(x-x^{\prime})^{2}}{4dt}-\frac{1}{2d}\int_{0}^{x^{% \prime}}y(x^{\prime\prime},0)dx^{\prime\prime}\Bigr]dx^{\prime}\Bigr\}.
  28. y ( x , t ) - 2 α x ln y ( x , t ) y(x,t)\rightarrow-2\alpha\frac{\partial}{\partial x}\ln y(x,t)
  29. 2 α x [ 1 y ( - y t + α 2 y 2 x ) ] = 0 2\alpha\frac{\partial}{\partial x}\left[\frac{1}{y}\left(-\frac{\partial y}{% \partial t}+\alpha\frac{\partial^{2}y}{\partial^{2}x}\right)\right]=0
  30. y t - α 2 y 2 x = y d f ( t ) d t \frac{\partial y}{\partial t}-\alpha\frac{\partial^{2}y}{\partial^{2}x}=y\frac% {df(t)}{dt}
  31. y e f y y\rightarrow e^{f}y
  32. y t - α 2 y 2 x = 0 \frac{\partial y}{\partial t}-\alpha\frac{\partial^{2}y}{\partial^{2}x}=0
  33. y 0 ( x ) - 1 2 α 0 x y 0 ( x ) d x y_{0}(x)\rightarrow-\frac{1}{2\alpha}\int_{0}^{x}y_{0}(x^{\prime})dx^{\prime}

C1.html

  1. C 1 C^{1}

Cabibbo–Kobayashi–Maskawa_matrix.html

  1. d = V u d d + V u s s , d^{\prime}=V_{ud}d+V_{us}s,
  2. d = cos θ c d + sin θ c s . d^{\prime}=\cos\theta_{\mathrm{c}}d+\sin\theta_{\mathrm{c}}s.
  3. tan θ c = | V u s | | V u d | = 0.22534 0.97427 θ c = 13.02 . \tan\theta_{\mathrm{c}}=\frac{|V_{us}|}{|V_{ud}|}=\frac{0.22534}{0.97427}% \rightarrow\theta_{\mathrm{c}}=~{}13.02^{\circ}.
  4. d = V u d d + V u s s ; d^{\prime}=V_{ud}d+V_{us}s;
  5. s = V c d d + V c s s , s^{\prime}=V_{cd}d+V_{cs}s,
  6. d = cos θ c d + sin θ c s ; d^{\prime}=\cos{\theta_{\mathrm{c}}}d+\sin{\theta_{\mathrm{c}}}s;
  7. s = - sin θ c d + cos θ c s . s^{\prime}=-\sin{\theta_{\mathrm{c}}}d+\cos{\theta_{\mathrm{c}}}s.
  8. [ d s ] = [ V u d V u s V c d V c s ] [ d s ] , \begin{bmatrix}d^{\prime}\\ s^{\prime}\end{bmatrix}=\begin{bmatrix}V_{ud}&V_{us}\\ V_{cd}&V_{cs}\\ \end{bmatrix}\begin{bmatrix}d\\ s\end{bmatrix},
  9. [ d s ] = [ cos θ c sin θ c - sin θ c cos θ c ] [ d s ] , \begin{bmatrix}d^{\prime}\\ s^{\prime}\end{bmatrix}=\begin{bmatrix}\cos{\theta_{\mathrm{c}}}&\sin{\theta_{% \mathrm{c}}}\\ -\sin{\theta_{\mathrm{c}}}&\cos{\theta_{\mathrm{c}}}\\ \end{bmatrix}\begin{bmatrix}d\\ s\end{bmatrix},
  10. [ d s b ] = [ V u d V u s V u b V c d V c s V c b V t d V t s V t b ] [ d s b ] . \begin{bmatrix}d^{\prime}\\ s^{\prime}\\ b^{\prime}\end{bmatrix}=\begin{bmatrix}V_{ud}&V_{us}&V_{ub}\\ V_{cd}&V_{cs}&V_{cb}\\ V_{td}&V_{ts}&V_{tb}\end{bmatrix}\begin{bmatrix}d\\ s\\ b\end{bmatrix}.
  11. [ | V u d | | V u s | | V u b | | V c d | | V c s | | V c b | | V t d | | V t s | | V t b | ] = [ 0.97427 ± 0.00015 0.22534 ± 0.00065 0.00351 - 0.00014 + 0.00015 0.22520 ± 0.00065 0.97344 ± 0.00016 0.0412 - 0.0005 + 0.0011 0.00867 - 0.00031 + 0.00029 0.0404 - 0.0005 + 0.0011 0.999146 - 0.000046 + 0.000021 ] . \begin{bmatrix}|V_{ud}|&|V_{us}|&|V_{ub}|\\ |V_{cd}|&|V_{cs}|&|V_{cb}|\\ |V_{td}|&|V_{ts}|&|V_{tb}|\end{bmatrix}=\begin{bmatrix}0.97427\pm 0.00015&0.22% 534\pm 0.00065&0.00351^{+0.00015}_{-0.00014}\\ 0.22520\pm 0.00065&0.97344\pm 0.00016&0.0412^{+0.0011}_{-0.0005}\\ 0.00867^{+0.00029}_{-0.00031}&0.0404^{+0.0011}_{-0.0005}&0.999146^{+0.000021}_% {-0.000046}\end{bmatrix}.
  12. k | V i k | 2 = i | V i k | 2 = 1 \sum_{k}|V_{ik}|^{2}=\sum_{i}|V_{ik}|^{2}=1
  13. k V i k V j k * = 0. \sum_{k}V_{ik}V^{*}_{jk}=0.
  14. [ c 1 - s 1 c 3 - s 1 s 3 s 1 c 2 c 1 c 2 c 3 - s 2 s 3 e i δ c 1 c 2 s 3 + s 2 c 3 e i δ s 1 s 2 c 1 s 2 c 3 + c 2 s 3 e i δ c 1 s 2 s 3 - c 2 c 3 e i δ ] . \begin{bmatrix}c_{1}&-s_{1}c_{3}&-s_{1}s_{3}\\ s_{1}c_{2}&c_{1}c_{2}c_{3}-s_{2}s_{3}e^{i\delta}&c_{1}c_{2}s_{3}+s_{2}c_{3}e^{% i\delta}\\ s_{1}s_{2}&c_{1}s_{2}c_{3}+c_{2}s_{3}e^{i\delta}&c_{1}s_{2}s_{3}-c_{2}c_{3}e^{% i\delta}\end{bmatrix}.
  15. [ 1 0 0 0 c 23 s 23 0 - s 23 c 23 ] [ c 13 0 s 13 e - i δ 13 0 1 0 - s 13 e i δ 13 0 c 13 ] [ c 12 s 12 0 - s 12 c 12 0 0 0 1 ] \displaystyle\begin{bmatrix}1&0&0\\ 0&c_{23}&s_{23}\\ 0&-s_{23}&c_{23}\end{bmatrix}\begin{bmatrix}c_{13}&0&s_{13}e^{-i\delta_{13}}\\ 0&1&0\\ -s_{13}e^{i\delta_{13}}&0&c_{13}\end{bmatrix}\begin{bmatrix}c_{12}&s_{12}&0\\ -s_{12}&c_{12}&0\\ 0&0&1\end{bmatrix}
  16. [ 1 - λ 2 / 2 λ A λ 3 ( ρ - i η ) - λ 1 - λ 2 / 2 A λ 2 A λ 3 ( 1 - ρ - i η ) - A λ 2 1 ] . \begin{bmatrix}1-\lambda^{2}/2&\lambda&A\lambda^{3}(\rho-i\eta)\\ -\lambda&1-\lambda^{2}/2&A\lambda^{2}\\ A\lambda^{3}(1-\rho-i\eta)&-A\lambda^{2}&1\end{bmatrix}.

Cabtaxi_number.html

  1. Cabtaxi ( 1 ) = 1 = 1 3 ± 0 3 \begin{matrix}\mathrm{Cabtaxi}(1)&=&1&=&1^{3}\pm 0^{3}\end{matrix}
  2. Cabtaxi ( 2 ) = 91 = 3 3 + 4 3 = 6 3 - 5 3 \begin{matrix}\mathrm{Cabtaxi}(2)&=&91&=&3^{3}+4^{3}\\ &&&=&6^{3}-5^{3}\end{matrix}
  3. Cabtaxi ( 3 ) = 728 = 6 3 + 8 3 = 9 3 - 1 3 = 12 3 - 10 3 \begin{matrix}\mathrm{Cabtaxi}(3)&=&728&=&6^{3}+8^{3}\\ &&&=&9^{3}-1^{3}\\ &&&=&12^{3}-10^{3}\end{matrix}
  4. Cabtaxi ( 4 ) = 2741256 = 108 3 + 114 3 = 140 3 - 14 3 = 168 3 - 126 3 = 207 3 - 183 3 \begin{matrix}\mathrm{Cabtaxi}(4)&=&2741256&=&108^{3}+114^{3}\\ &&&=&140^{3}-14^{3}\\ &&&=&168^{3}-126^{3}\\ &&&=&207^{3}-183^{3}\end{matrix}
  5. Cabtaxi ( 5 ) = 6017193 = 166 3 + 113 3 = 180 3 + 57 3 = 185 3 - 68 3 = 209 3 - 146 3 = 246 3 - 207 3 \begin{matrix}\mathrm{Cabtaxi}(5)&=&6017193&=&166^{3}+113^{3}\\ &&&=&180^{3}+57^{3}\\ &&&=&185^{3}-68^{3}\\ &&&=&209^{3}-146^{3}\\ &&&=&246^{3}-207^{3}\end{matrix}
  6. Cabtaxi ( 6 ) = 1412774811 = 963 3 + 804 3 = 1134 3 - 357 3 = 1155 3 - 504 3 = 1246 3 - 805 3 = 2115 3 - 2004 3 = 4746 3 - 4725 3 \begin{matrix}\mathrm{Cabtaxi}(6)&=&1412774811&=&963^{3}+804^{3}\\ &&&=&1134^{3}-357^{3}\\ &&&=&1155^{3}-504^{3}\\ &&&=&1246^{3}-805^{3}\\ &&&=&2115^{3}-2004^{3}\\ &&&=&4746^{3}-4725^{3}\end{matrix}
  7. Cabtaxi ( 7 ) = 11302198488 = 1926 3 + 1608 3 = 1939 3 + 1589 3 = 2268 3 - 714 3 = 2310 3 - 1008 3 = 2492 3 - 1610 3 = 4230 3 - 4008 3 = 9492 3 - 9450 3 \begin{matrix}\mathrm{Cabtaxi}(7)&=&11302198488&=&1926^{3}+1608^{3}\\ &&&=&1939^{3}+1589^{3}\\ &&&=&2268^{3}-714^{3}\\ &&&=&2310^{3}-1008^{3}\\ &&&=&2492^{3}-1610^{3}\\ &&&=&4230^{3}-4008^{3}\\ &&&=&9492^{3}-9450^{3}\end{matrix}
  8. Cabtaxi ( 8 ) = 137513849003496 = 22944 3 + 50058 3 = 36547 3 + 44597 3 = 36984 3 + 44298 3 = 52164 3 - 16422 3 = 53130 3 - 23184 3 = 57316 3 - 37030 3 = 97290 3 - 92184 3 = 218316 3 - 217350 3 \begin{matrix}\mathrm{Cabtaxi}(8)&=&137513849003496&=&22944^{3}+50058^{3}\\ &&&=&36547^{3}+44597^{3}\\ &&&=&36984^{3}+44298^{3}\\ &&&=&52164^{3}-16422^{3}\\ &&&=&53130^{3}-23184^{3}\\ &&&=&57316^{3}-37030^{3}\\ &&&=&97290^{3}-92184^{3}\\ &&&=&218316^{3}-217350^{3}\end{matrix}
  9. Cabtaxi ( 9 ) = 424910390480793000 = 645210 3 + 538680 3 = 649565 3 + 532315 3 = 752409 3 - 101409 3 = 759780 3 - 239190 3 = 773850 3 - 337680 3 = 834820 3 - 539350 3 = 1417050 3 - 1342680 3 = 3179820 3 - 3165750 3 = 5960010 3 - 5956020 3 \begin{matrix}\mathrm{Cabtaxi}(9)&=&424910390480793000&=&645210^{3}+538680^{3}% \\ &&&=&649565^{3}+532315^{3}\\ &&&=&752409^{3}-101409^{3}\\ &&&=&759780^{3}-239190^{3}\\ &&&=&773850^{3}-337680^{3}\\ &&&=&834820^{3}-539350^{3}\\ &&&=&1417050^{3}-1342680^{3}\\ &&&=&3179820^{3}-3165750^{3}\\ &&&=&5960010^{3}-5956020^{3}\end{matrix}
  10. Cabtaxi ( 10 ) = 933528127886302221000 = 77480130 3 - 77428260 3 = 41337660 3 - 41154750 3 = 18421650 3 - 17454840 3 = 10852660 3 - 7011550 3 = 10060050 3 - 4389840 3 = 9877140 3 - 3109470 3 = 9781317 3 - 1318317 3 = 9773330 3 - 84560 3 = 8444345 3 + 6920095 3 = 8387730 3 + 7002840 3 \begin{matrix}\mathrm{Cabtaxi}(10)&=&933528127886302221000&=&77480130^{3}-7742% 8260^{3}\\ &&&=&41337660^{3}-41154750^{3}\\ &&&=&18421650^{3}-17454840^{3}\\ &&&=&10852660^{3}-7011550^{3}\\ &&&=&10060050^{3}-4389840^{3}\\ &&&=&9877140^{3}-3109470^{3}\\ &&&=&9781317^{3}-1318317^{3}\\ &&&=&9773330^{3}-84560^{3}\\ &&&=&8444345^{3}+6920095^{3}\\ &&&=&8387730^{3}+7002840^{3}\end{matrix}

Cache_algorithms.html

  1. T = m * T m + T h + E T=m*T_{m}+T_{h}+E
  2. T T
  3. m m
  4. T m T_{m}
  5. T h T_{h}
  6. E E
  7. Q < s u b > 0 Q<sub>0

Calculus_of_negligence.html

  1. P L > B PL>B

Calibrated_airspeed.html

  1. C A S = a 0 5 [ ( q c P 0 + 1 ) 2 7 - 1 ] CAS=a_{0}\sqrt{5\left[\left(\frac{q_{c}}{P_{0}}+1\right)^{\frac{2}{7}}-1\right]}
  2. q c q_{c}
  3. P 0 P_{0}
  4. a 0 {a_{0}}
  5. C A S = a 0 [ ( q c P 0 + 1 ) × ( 7 ( C A S a 0 ) 2 - 1 ) 2.5 / ( 6 2.5 × 1.2 3.5 ) ] ( 1 / 7 ) CAS=a_{0}\left[\left(\frac{q_{c}}{P_{0}}+1\right)\times\left(7\left(\frac{CAS}% {a_{0}}\right)^{2}-1\right)^{2.5}/\left(6^{2.5}\times 1.2^{3.5}\right)\right]^% {(1/7)}
  6. C A S CAS
  7. a 0 a_{0}
  8. P 0 P_{0}
  9. a 0 a_{0}
  10. P 0 P_{0}
  11. a 0 a_{0}
  12. q c q_{c}
  13. P 0 P_{0}
  14. H 2 0 H_{2}0