wpmath0000003_2

Clairaut's_equation.html

  1. y ( x ) = x d y d x + f ( d y d x ) . y(x)=x\frac{dy}{dx}+f\left(\frac{dy}{dx}\right).
  2. d y d x = d y d x + x d 2 y d x 2 + f ( d y d x ) d 2 y d x 2 , \frac{dy}{dx}=\frac{dy}{dx}+x\frac{d^{2}y}{dx^{2}}+f^{\prime}\left(\frac{dy}{% dx}\right)\frac{d^{2}y}{dx^{2}},
  3. 0 = ( x + f ( d y d x ) ) d 2 y d x 2 . 0=\left(x+f^{\prime}\left(\frac{dy}{dx}\right)\right)\frac{d^{2}y}{dx^{2}}.
  4. 0 = d 2 y d x 2 0=\frac{d^{2}y}{dx^{2}}
  5. 0 = x + f ( d y d x ) . 0=x+f^{\prime}\left(\frac{dy}{dx}\right).
  6. y ( x ) = C x + f ( C ) , y(x)=Cx+f(C),\,
  7. 0 = x + f ( d y d x ) , 0=x+f^{\prime}\left(\frac{dy}{dx}\right),
  8. u = x u x + y u y + f ( u x , u y ) . \displaystyle u=xu_{x}+yu_{y}+f(u_{x},u_{y}).
  9. f ( p ) = p 2 f(p)=p^{2}
  10. f ( p ) = p 3 f(p)=p^{3}

Clarence_Irving_Lewis.html

  1. \Diamond
  2. \Diamond
  3. ¬ \neg\Diamond
  4. and ¬ \and\neg
  5. \Diamond
  6. \square
  7. \Diamond
  8. \square

Class_number_problem.html

  1. d - d\to-\infty
  2. h ( d ) as d - . h(d)\to\infty\,\text{ as }d\to-\infty.
  3. d = - 3 , - 4 , - 7 , - 8 , - 11 , - 19 , - 43 , - 67 , - 163. d=-3,-4,-7,-8,-11,-19,-43,-67,-163.
  4. d = - 12 , - 16 , - 27 , - 28. d=-12,-16,-27,-28.
  5. d = - 4 , - 8 , - 12 , - 16 , - 28. d=-4,-8,-12,-16,-28.
  6. 𝐐 ( k ) \mathbf{Q}(\sqrt{k})
  7. - 1 , - 2 , - 3 , - 7 , - 11 , - 19 , - 43 , - 67 , - 163. -1,-2,-3,-7,-11,-19,-43,-67,-163.

Classical_electromagnetism.html

  1. 𝐅 = q 𝐄 + q 𝐯 × 𝐁 \mathbf{F}=q\mathbf{E}+q\mathbf{v}\times\mathbf{B}
  2. 𝐅 = q 0 𝐄 \mathbf{F}=q_{0}\mathbf{E}
  3. 𝐄 ( 𝐫 ) = 1 4 π ε 0 i = 1 n q i ( 𝐫 - 𝐫 i ) | 𝐫 - 𝐫 i | 3 \mathbf{E(r)}=\frac{1}{4\pi\varepsilon_{0}}\sum_{i=1}^{n}\frac{q_{i}\left(% \mathbf{r}-\mathbf{r}_{i}\right)}{\left|\mathbf{r}-\mathbf{r}_{i}\right|^{3}}
  4. 𝐄 ( 𝐫 ) = 1 4 π ε 0 ρ ( 𝐫 ) ( 𝐫 - 𝐫 ) | 𝐫 - 𝐫 | 3 d 3 𝐫 \mathbf{E(r)}=\frac{1}{4\pi\varepsilon_{0}}\int\frac{\rho(\mathbf{r^{\prime}})% \left(\mathbf{r}-\mathbf{r^{\prime}}\right)}{\left|\mathbf{r}-\mathbf{r^{% \prime}}\right|^{3}}\mathrm{d^{3}}\mathbf{r^{\prime}}
  5. ρ ( 𝐫 ) \rho(\mathbf{r^{\prime}})
  6. 𝐫 - 𝐫 \mathbf{r}-\mathbf{r^{\prime}}
  7. d 3 𝐫 \mathrm{d^{3}}\mathbf{r^{\prime}}
  8. φ ( 𝐫 ) = - C 𝐄 d 𝐥 \varphi\mathbf{(r)}=-\int_{C}\mathbf{E}\cdot\mathrm{d}\mathbf{l}
  9. φ ( 𝐫 ) = 1 4 π ε 0 i = 1 n q i | 𝐫 - 𝐫 i | \varphi\mathbf{(r)}=\frac{1}{4\pi\varepsilon_{0}}\sum_{i=1}^{n}\frac{q_{i}}{% \left|\mathbf{r}-\mathbf{r}_{i}\right|}
  10. φ ( 𝐫 ) = 1 4 π ε 0 ρ ( 𝐫 ) | 𝐫 - 𝐫 | d 3 𝐫 \varphi\mathbf{(r)}=\frac{1}{4\pi\varepsilon_{0}}\int\frac{\rho(\mathbf{r^{% \prime}})}{|\mathbf{r}-\mathbf{r^{\prime}}|}\,\mathrm{d^{3}}\mathbf{r^{\prime}}
  11. ρ ( 𝐫 ) \rho(\mathbf{r^{\prime}})
  12. 𝐫 - 𝐫 \mathbf{r}-\mathbf{r^{\prime}}
  13. d 3 𝐫 \mathrm{d^{3}}\mathbf{r^{\prime}}
  14. 𝐄 ( 𝐫 ) = - φ ( 𝐫 ) . \mathbf{E(r)}=-\nabla\varphi\mathbf{(r)}.
  15. φ = 1 4 π ε 0 q | 𝐫 - 𝐫 q ( t r e t ) | - 𝐯 q ( t r e t ) c ( 𝐫 - 𝐫 q ( t r e t ) ) \varphi=\frac{1}{4\pi\varepsilon_{0}}\frac{q}{\left|\mathbf{r}-\mathbf{r}_{q}(% t_{ret})\right|-\frac{\mathbf{v}_{q}(t_{ret})}{c}\cdot(\mathbf{r}-\mathbf{r}_{% q}(t_{ret}))}
  16. 𝐀 = μ 0 4 π q 𝐯 q ( t r e t ) | 𝐫 - 𝐫 q ( t r e t ) | - 𝐯 q ( t r e t ) c ( 𝐫 - 𝐫 q ( t r e t ) ) . \mathbf{A}=\frac{\mu_{0}}{4\pi}\frac{q\mathbf{v}_{q}(t_{ret})}{\left|\mathbf{r% }-\mathbf{r}_{q}(t_{ret})\right|-\frac{\mathbf{v}_{q}(t_{ret})}{c}\cdot(% \mathbf{r}-\mathbf{r}_{q}(t_{ret}))}.

Classical_general_equilibrium_model.html

  1. p p
  2. w w
  3. ω \omega
  4. π \pi
  5. L D L^{D}
  6. Y S Y^{S}
  7. Y S ( L D ) Y^{S}(L^{D})
  8. p π = p Y S - w L D p\cdot\pi=p\cdot Y^{S}-w\cdot L^{D}
  9. π = Y S - w p L D = Y S - ω L D \pi=Y^{S}-\frac{w}{p}\cdot L^{D}=Y^{S}-\omega\cdot L^{D}
  10. d Y S ( L D ) d L D = ω \frac{dY^{S}(L^{D})}{dL^{D}}=\omega
  11. Y D = π + ω L S Y^{D}=\pi+\omega\cdot L^{S}
  12. U = Y D - D ( L S ) U=Y^{D}-D(L^{S})
  13. U = π + ω L S - D ( L S ) U=\pi+\omega\cdot L^{S}-D(L^{S})
  14. d D ( L S ) d L S = ω \frac{dD(L^{S})}{dL^{S}}=\omega

Classical_orthogonal_polynomials.html

  1. Q , L : \R \R Q,L:\R\to\R
  2. n 𝒩 0 \forall\,n\in\mathcal{N}_{0}
  3. f n : \R \R f_{n}:\R\to\R
  4. Q ( x ) f n ′′ + L ( x ) f n + λ n f n = 0 Q(x)\,f_{n}^{\prime\prime}+L(x)\,f_{n}^{\prime}+\lambda_{n}f_{n}=0
  5. λ n \R \lambda_{n}\in\R
  6. P n P_{n}
  7. W : + W:\mathbb{R}\rightarrow\mathbb{R}^{+}
  8. deg P n = n , n = 0 , 1 , 2 , P m ( x ) P n ( x ) W ( x ) d x = 0 , m n . \begin{aligned}&\displaystyle\deg P_{n}=n~{},\quad n=0,1,2,\ldots\\ &\displaystyle\int P_{m}(x)\,P_{n}(x)\,W(x)\,dx=0~{},\quad m\neq n~{}.\end{aligned}
  9. P n P_{n}
  10. P n ( x ) 2 W ( x ) d x = 1 . \int P_{n}(x)^{2}W(x)\,dx=1~{}.
  11. (Jacobi) \displaystyle\,\text{(Jacobi)}
  12. α , β > - 1 \alpha,\,\beta>-1
  13. P n ( α , β ) ( z ) = ( - 1 ) n 2 n n ! ( 1 - z ) - α ( 1 + z ) - β d n d z n { ( 1 - z ) α ( 1 + z ) β ( 1 - z 2 ) n } . P_{n}^{(\alpha,\beta)}(z)=\frac{(-1)^{n}}{2^{n}n!}(1-z)^{-\alpha}(1+z)^{-\beta% }\frac{d^{n}}{dz^{n}}\left\{(1-z)^{\alpha}(1+z)^{\beta}(1-z^{2})^{n}\right\}~{}.
  14. P n ( α , β ) ( 1 ) = ( n + α n ) , P_{n}^{(\alpha,\beta)}(1)={n+\alpha\choose n},
  15. - 1 1 ( 1 - x ) α ( 1 + x ) β P m ( α , β ) ( x ) P n ( α , β ) ( x ) d x = 2 α + β + 1 2 n + α + β + 1 Γ ( n + α + 1 ) Γ ( n + β + 1 ) Γ ( n + α + β + 1 ) n ! δ n m . \begin{aligned}&\displaystyle\int_{-1}^{1}(1-x)^{\alpha}(1+x)^{\beta}P_{m}^{(% \alpha,\beta)}(x)P_{n}^{(\alpha,\beta)}(x)\;dx\\ &\displaystyle\quad=\frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1}\frac{\Gamma(n% +\alpha+1)\Gamma(n+\beta+1)}{\Gamma(n+\alpha+\beta+1)n!}\delta_{nm}.\end{aligned}
  16. ( 1 - x 2 ) y ′′ + ( β - α - ( α + β + 2 ) x ) y + n ( n + α + β + 1 ) y = 0 . (1-x^{2})y^{\prime\prime}+(\beta-\alpha-(\alpha+\beta+2)x)y^{\prime}+n(n+% \alpha+\beta+1)y=0~{}.
  17. α = β \alpha=\beta
  18. γ = α + 1 / 2 \gamma=\alpha+1/2
  19. α = β = 0 \alpha=\beta=0
  20. P 0 ( x ) = 1 , P 1 ( x ) = x , P 2 ( x ) = 3 x 2 - 1 2 , P 3 ( x ) = 5 x 3 - 3 x 2 , P_{0}(x)=1,\,P_{1}(x)=x,\,P_{2}(x)=\frac{3x^{2}-1}{2},\,P_{3}(x)=\frac{5x^{3}-% 3x}{2},\ldots
  21. α = β = ± 1 / 2 \alpha=\beta=\pm 1/2
  22. H n ( x ) = ( - 1 ) n e x 2 d n d x n e - x 2 = e x 2 / 2 ( x - d d x ) n e - x 2 / 2 H_{n}(x)=(-1)^{n}e^{x^{2}}\frac{d^{n}}{dx^{n}}e^{-x^{2}}=e^{x^{2}/2}\bigg(x-% \frac{d}{dx}\bigg)^{n}e^{-x^{2}/2}\,\!
  23. - H n ( x ) H m ( x ) e - x 2 d x = π 2 n n ! δ m n , \int_{-\infty}^{\infty}H_{n}(x)H_{m}(x)e^{-x^{2}}dx=\sqrt{\pi}2^{n}n!\delta_{% mn}~{},
  24. y ′′ - 2 x y + 2 n y = 0 . y^{\prime\prime}-2xy^{\prime}+2n\,y=0~{}.
  25. L n ( α ) ( x ) = x - α e x n ! d n d x n ( e - x x n + α ) L_{n}^{(\alpha)}(x)={x^{-\alpha}e^{x}\over n!}{d^{n}\over dx^{n}}\left(e^{-x}x% ^{n+\alpha}\right)
  26. α = 0 \alpha=0
  27. 0 x α e - x L n ( α ) ( x ) L m ( α ) ( x ) d x = Γ ( n + α + 1 ) n ! δ n , m , \int_{0}^{\infty}x^{\alpha}e^{-x}L_{n}^{(\alpha)}(x)L_{m}^{(\alpha)}(x)dx=% \frac{\Gamma(n+\alpha+1)}{n!}\delta_{n,m}~{},
  28. x y ′′ + ( α + 1 - x ) y + n y = 0 . x\,y^{\prime\prime}+(\alpha+1-x)\,y^{\prime}+n\,y=0~{}.
  29. Q ( x ) f ′′ + L ( x ) f + λ f = 0 Q(x)\,f^{\prime\prime}+L(x)\,f^{\prime}+\lambda f=0\,
  30. D ( f ) = Q f ′′ + L f D(f)=Qf^{\prime\prime}+Lf^{\prime}\,
  31. R ( x ) = e L ( x ) Q ( x ) d x R(x)=e^{\int\frac{L(x)}{Q(x)}\,dx}\,
  32. W ( x ) = R ( x ) Q ( x ) W(x)=\frac{R(x)}{Q(x)}\,
  33. 1 W ( x ) d n d x n ( W ( x ) [ Q ( x ) ] n ) . \frac{1}{W(x)}\ \frac{d^{n}}{dx^{n}}\left(W(x)[Q(x)]^{n}\right).
  34. P n ( x ) = 1 e n W ( x ) d n d x n ( W ( x ) [ Q ( x ) ] n ) P_{n}(x)=\frac{1}{{e_{n}}W(x)}\ \frac{d^{n}}{dx^{n}}\left(W(x)[Q(x)]^{n}\right)
  35. λ n = - n ( n - 1 2 Q ′′ + L ) . \lambda_{n}=-n\left(\frac{n-1}{2}Q^{\prime\prime}+L^{\prime}\right).
  36. Q ′′ Q^{\prime\prime}
  37. L L^{\prime}
  38. R ( x ) = e L ( x ) Q ( x ) d x R(x)=e^{\int\frac{L(x)}{Q(x)}\,dx}\,
  39. ( R y ) = R y ′′ + R y = R y ′′ + R L Q y . (Ry^{\prime})^{\prime}=R\,y^{\prime\prime}+R^{\prime}\,y^{\prime}=R\,y^{\prime% \prime}+\frac{R\,L}{Q}\,y^{\prime}.
  40. Q y ′′ + L y + λ y = 0 Q\,y^{\prime\prime}+L\,y^{\prime}+\lambda y=0\,
  41. R y ′′ + R L Q y + R λ Q y = 0 R\,y^{\prime\prime}+\frac{R\,L}{Q}\,y^{\prime}+\frac{R\,\lambda}{Q}\,y=0\,
  42. ( R y ) + R λ Q y = 0. (Ry^{\prime})^{\prime}+\frac{R\,\lambda}{Q}\,y=0.\,
  43. S ( x ) = R ( x ) = e L ( x ) 2 Q ( x ) d x . S(x)=\sqrt{R(x)}=e^{\int\frac{L(x)}{2\,Q(x)}\,dx}.\,
  44. S = S L 2 Q . S^{\prime}=\frac{S\,L}{2\,Q}.
  45. Q y ′′ + L y + λ y = 0 Q\,y^{\prime\prime}+{L}\,y^{\prime}+\lambda y=0\,
  46. S y ′′ + S L Q y + S λ Q y = 0 S\,y^{\prime\prime}+\frac{S\,L}{Q}\,y^{\prime}+\frac{S\,\lambda}{Q}\,y=0\,
  47. S y ′′ + 2 S y + S λ Q y = 0 S\,y^{\prime\prime}+2\,S^{\prime}\,y^{\prime}+\frac{S\,\lambda}{Q}\,y=0\,
  48. ( S y ) ′′ = S y ′′ + 2 S y + S ′′ y (S\,y)^{\prime\prime}=S\,y^{\prime\prime}+2\,S^{\prime}\,y^{\prime}+S^{\prime% \prime}\,y
  49. ( S y ) ′′ + ( S λ Q - S ′′ ) y = 0 , (S\,y)^{\prime\prime}+\left(\frac{S\,\lambda}{Q}-S^{\prime\prime}\right)\,y=0,\,
  50. u ′′ + ( λ Q - S ′′ S ) u = 0. u^{\prime\prime}+\left(\frac{\lambda}{Q}-\frac{S^{\prime\prime}}{S}\right)\,u=% 0.\,
  51. W Q r WQ^{r}\,
  52. 1 W ( x ) [ Q ( x ) ] r d n - r d x n - r ( W ( x ) [ Q ( x ) ] n ) \frac{1}{W(x)[Q(x)]^{r}}\ \frac{d^{n-r}}{dx^{n-r}}\left(W(x)[Q(x)]^{n}\right)
  53. Q y ′′ + ( r Q + L ) y + [ λ n - λ r ] y = 0 {Q}\,y^{\prime\prime}+(rQ^{\prime}+L)\,y^{\prime}+[{\lambda}_{n}-{\lambda}_{r}% ]\,y=0\,
  54. λ r = - r ( r - 1 2 Q ′′ + L ) {\lambda}_{r}=-r\left(\frac{r-1}{2}Q^{\prime\prime}+L^{\prime}\right)
  55. ( R Q r y ) + [ λ n - λ r ] R Q r - 1 y = 0 (RQ^{r}y^{\prime})^{\prime}+[{\lambda}_{n}-{\lambda}_{r}]RQ^{r-1}\,y=0\,
  56. P n [ r ] = a P n + 1 [ r + 1 ] + b P n [ r + 1 ] + c P n - 1 [ r + 1 ] P_{n}^{[r]}=aP_{n+1}^{[r+1]}+bP_{n}^{[r+1]}+cP_{n-1}^{[r+1]}
  57. P n [ r ] = ( a x + b ) P n [ r + 1 ] + c P n - 1 [ r + 1 ] P_{n}^{[r]}=(ax+b)P_{n}^{[r+1]}+cP_{n-1}^{[r+1]}
  58. Q P n [ r + 1 ] = ( a x + b ) P n [ r ] + c P n - 1 [ r ] QP_{n}^{[r+1]}=(ax+b)P_{n}^{[r]}+cP_{n-1}^{[r]}
  59. 2 T m ( x ) T n ( x ) = T m + n ( x ) + T m - n ( x ) 2\,T_{m}(x)\,T_{n}(x)=T_{m+n}(x)+T_{m-n}(x)\,
  60. H 2 n ( x ) = ( - 4 ) n n ! L n ( - 1 / 2 ) ( x 2 ) H_{2n}(x)=(-4)^{n}\,n!\,L_{n}^{(-1/2)}(x^{2})
  61. H 2 n + 1 ( x ) = 2 ( - 4 ) n n ! x L n ( 1 / 2 ) ( x 2 ) H_{2n+1}(x)=2(-4)^{n}\,n!\,x\,L_{n}^{(1/2)}(x^{2})
  62. Q f ¨ n + L f ˙ n + λ n f n = 0 Q\ddot{f}_{n}+L\dot{f}_{n}+\lambda_{n}f_{n}=0
  63. ( R / Q ) f m (R/Q)f_{m}
  64. R f m f ¨ n + R Q L f m f ˙ n + R Q λ n f m f n = 0 Rf_{m}\ddot{f}_{n}+\frac{R}{Q}Lf_{m}\dot{f}_{n}+\frac{R}{Q}\lambda_{n}f_{m}f_{% n}=0
  65. R f n f ¨ m + R Q L f n f ˙ m + R Q λ m f n f m = 0 Rf_{n}\ddot{f}_{m}+\frac{R}{Q}Lf_{n}\dot{f}_{m}+\frac{R}{Q}\lambda_{m}f_{n}f_{% m}=0
  66. a b [ R ( f m f ¨ n - f n f ¨ m ) + R Q L ( f m f ˙ n - f n f ˙ m ) ] d x + ( λ n - λ m ) a b R Q f m f n d x = 0 \int_{a}^{b}\left[R(f_{m}\ddot{f}_{n}-f_{n}\ddot{f}_{m})+\frac{R}{Q}L(f_{m}% \dot{f}_{n}-f_{n}\dot{f}_{m})\right]\,dx+(\lambda_{n}-\lambda_{m})\int_{a}^{b}% \frac{R}{Q}f_{m}f_{n}\,dx=0
  67. d d x [ R ( f m f ˙ n - f n f ˙ m ) ] = R ( f m f ¨ n - f n f ¨ m ) + R L Q ( f m f ˙ n - f n f ˙ m ) \frac{d}{dx}\left[R(f_{m}\dot{f}_{n}-f_{n}\dot{f}_{m})\right]=R(f_{m}\ddot{f}_% {n}-f_{n}\ddot{f}_{m})\,\,+\,\,R\frac{L}{Q}(f_{m}\dot{f}_{n}-f_{n}\dot{f}_{m})
  68. [ R ( f m f ˙ n - f n f ˙ m ) ] a b + ( λ n - λ m ) a b R Q f m f n d x = 0 \left[R(f_{m}\dot{f}_{n}-f_{n}\dot{f}_{m})\right]_{a}^{b}\,\,+\,\,(\lambda_{n}% -\lambda_{m})\int_{a}^{b}\frac{R}{Q}f_{m}f_{n}\,dx=0
  69. λ m λ n \lambda_{m}\neq\lambda_{n}
  70. m n m\neq n
  71. a b R Q f m f n d x = 0 \int_{a}^{b}\frac{R}{Q}f_{m}f_{n}\,dx=0
  72. m n m\neq n
  73. P n ( α , β ) P_{n}^{(\alpha,\beta)}
  74. [ 0 , ) [0,\infty)
  75. L n ( α ) L_{n}^{(\alpha)}
  76. L n \ L_{n}
  77. ( - , ) (-\infty,\infty)
  78. H n H_{n}\,
  79. α \alpha
  80. β \beta
  81. P n ( α , β ) P_{n}^{(\alpha,\beta)}
  82. Q ( x ) = 1 - x 2 Q(x)=1-x^{2}\,
  83. L ( x ) = β - α - ( α + β + 2 ) x L(x)=\beta-\alpha-(\alpha+\beta+2)\,x
  84. α \alpha
  85. β \beta
  86. α \alpha
  87. β \beta
  88. ( 1 - x 2 ) y ′′ + ( β - α - [ α + β + 2 ] x ) y + λ y = 0 with λ = n ( n + 1 + α + β ) (1-x^{2})\,y^{\prime\prime}+(\beta-\alpha-[\alpha+\beta+2]\,x)\,y^{\prime}+{% \lambda}\,y=0\qquad\mathrm{with}\qquad\lambda=n(n+1+\alpha+\beta)\,
  89. α \alpha
  90. β \beta
  91. C n ( α ) C_{n}^{(\alpha)}
  92. C n ( α ) ( x ) = Γ ( 2 α + n ) Γ ( α + 1 / 2 ) Γ ( 2 α ) Γ ( α + n + 1 / 2 ) P n ( α - 1 / 2 , α - 1 / 2 ) ( x ) . C_{n}^{(\alpha)}(x)=\frac{\Gamma(2\alpha\!+\!n)\,\Gamma(\alpha\!+\!1/2)}{% \Gamma(2\alpha)\,\Gamma(\alpha\!+\!n\!+\!1/2)}\!\ P_{n}^{(\alpha-1/2,\alpha-1/% 2)}(x).
  93. Q ( x ) = 1 - x 2 Q(x)=1-x^{2}\,
  94. L ( x ) = - ( 2 α + 1 ) x L(x)=-(2\alpha+1)\,x
  95. α \alpha
  96. C n ( 0 ) ( 1 ) = 2 n C_{n}^{(0)}(1)=\frac{2}{n}
  97. α \alpha
  98. C n ( α ) C_{n}^{(\alpha)}
  99. C n ( α + 1 ) ( x ) = 1 2 α d d x C n + 1 ( α ) ( x ) C_{n}^{(\alpha+1)}(x)=\frac{1}{2\alpha}\!\ \frac{d}{dx}C_{n+1}^{(\alpha)}(x)
  100. C n ( α + m ) ( x ) = Γ ( α ) 2 m Γ ( α + m ) C n + m ( α ) [ m ] ( x ) . C_{n}^{(\alpha+m)}(x)=\frac{\Gamma(\alpha)}{2^{m}\Gamma(\alpha+m)}\!\ C_{n+m}^% {(\alpha)[m]}(x).
  101. α \alpha
  102. ( 1 - x 2 ) y ′′ - 2 x y + λ y = 0 with λ = n ( n + 1 ) . (1-x^{2})\,y^{\prime\prime}-2x\,y^{\prime}+{\lambda}\,y=0\qquad\mathrm{with}% \qquad\lambda=n(n+1).\,
  103. d d x [ ( 1 - x 2 ) y ] + λ y = 0. \frac{d}{dx}[(1-x^{2})\,y^{\prime}]+\lambda\,y=0.\,
  104. ( n + 1 ) P n + 1 ( x ) = ( 2 n + 1 ) x P n ( x ) - n P n - 1 ( x ) . (n+1)\,P_{n+1}(x)=(2n+1)x\,P_{n}(x)-n\,P_{n-1}(x).\,
  105. P n + 1 [ r + 1 ] ( x ) = P n - 1 [ r + 1 ] ( x ) + ( 2 n + 1 ) P n [ r ] ( x ) . P_{n+1}^{[r+1]}(x)=P_{n-1}^{[r+1]}(x)+(2n+1)\,P_{n}^{[r]}(x).\,
  106. P n ( x ) = 1 2 n n ! d n d x n ( [ x 2 - 1 ] n ) . P_{n}(x)=\,\frac{1}{2^{n}n!}\ \frac{d^{n}}{dx^{n}}\left([x^{2}-1]^{n}\right).
  107. P ( m ) ( x ) P_{\ell}^{(m)}(x)
  108. \ell
  109. m m
  110. 0 m 0\leqslant m\leqslant\ell
  111. P ( m ) ( x ) = ( - 1 ) m ( 1 - x 2 ) m / 2 P [ m ] ( x ) . P_{\ell}^{(m)}(x)=(-1)^{m}\,(1-x^{2})^{m/2}\ P_{\ell}^{[m]}(x).\,
  112. ( + 1 - m ) P + 1 ( m ) ( x ) = ( 2 + 1 ) x P ( m ) ( x ) - ( + m ) P - 1 ( m ) ( x ) . (\ell+1-m)\,P_{\ell+1}^{(m)}(x)=(2\ell+1)x\,P_{\ell}^{(m)}(x)-(\ell+m)\,P_{% \ell-1}^{(m)}(x).\,
  113. P m ( m ) , P m + 1 ( m ) , P m + 2 ( m ) , P_{m}^{(m)},P_{m+1}^{(m)},P_{m+2}^{(m)},\dots
  114. P ( m ) ( x ) P_{\ell}^{(m)}(x)
  115. ( 1 - x 2 ) y ′′ - 2 x y + [ λ - m 2 1 - x 2 ] y = 0 with λ = ( + 1 ) . (1-x^{2})\,y^{\prime\prime}-2xy^{\prime}+\left[\lambda-\frac{m^{2}}{1-x^{2}}% \right]\,y=0\qquad\mathrm{with}\qquad\lambda=\ell(\ell+1).\,
  116. ( 1 - x 2 ) y ′′ - x y + λ y = 0 with λ = n 2 . (1-x^{2})\,y^{\prime\prime}-x\,y^{\prime}+{\lambda}\,y=0\qquad\mathrm{with}% \qquad\lambda=n^{2}.\,
  117. T n + 1 ( x ) = 2 x T n ( x ) - T n - 1 ( x ) . T_{n+1}(x)=2x\,T_{n}(x)-T_{n-1}(x).\,
  118. T n ( x ) = Γ ( 1 / 2 ) 1 - x 2 ( - 2 ) n Γ ( n + 1 / 2 ) d n d x n ( [ 1 - x 2 ] n - 1 / 2 ) . T_{n}(x)=\frac{\Gamma(1/2)\sqrt{1-x^{2}}}{(-2)^{n}\,\Gamma(n+1/2)}\ \frac{d^{n% }}{dx^{n}}\left([1-x^{2}]^{n-1/2}\right).
  119. T n ( x ) = cos ( n arccos ( x ) ) . T_{n}(x)=\cos(n\,\arccos(x)).
  120. U n U_{n}\,
  121. U n = 1 n + 1 T n + 1 . U_{n}=\frac{1}{n+1}\,T_{n+1}^{\prime}.\,
  122. L n ( α ) L_{n}^{(\alpha)}
  123. α \alpha
  124. α = 0 \alpha=0
  125. L n ( x ) = L n ( 0 ) ( x ) . L_{n}(x)=L_{n}^{(0)}(x).\,
  126. x y ′′ + ( α + 1 - x ) y + λ y = 0 with λ = n . x\,y^{\prime\prime}+(\alpha+1-x)\,y^{\prime}+{\lambda}\,y=0\,\text{ with }% \lambda=n.\,
  127. ( x α + 1 e - x y ) + λ x α e - x y = 0. (x^{\alpha+1}\,e^{-x}\,y^{\prime})^{\prime}+{\lambda}\,x^{\alpha}\,e^{-x}\,y=0.\,
  128. ( n + 1 ) L n + 1 ( α ) ( x ) = ( 2 n + 1 + α - x ) L n ( α ) ( x ) - ( n + α ) L n - 1 ( α ) ( x ) . (n+1)\,L_{n+1}^{(\alpha)}(x)=(2n+1+\alpha-x)\,L_{n}^{(\alpha)}(x)-(n+\alpha)\,% L_{n-1}^{(\alpha)}(x).\,
  129. L n ( α ) ( x ) = x - α e x n ! d n d x n ( x n + α e - x ) . L_{n}^{(\alpha)}(x)=\frac{x^{-\alpha}e^{x}}{n!}\ \frac{d^{n}}{dx^{n}}\left(x^{% n+\alpha}\,e^{-x}\right).
  130. α \alpha
  131. L n ( α ) L_{n}^{(\alpha)}
  132. L n ( α + 1 ) ( x ) = - d d x L n + 1 ( α ) ( x ) L_{n}^{(\alpha+1)}(x)=-\frac{d}{dx}L_{n+1}^{(\alpha)}(x)
  133. L n ( α + m ) ( x ) = ( - 1 ) m L n + m ( α ) [ m ] ( x ) . L_{n}^{(\alpha+m)}(x)=(-1)^{m}L_{n+m}^{(\alpha)[m]}(x).
  134. u = x α - 1 2 e - x / 2 L n ( α ) ( x ) u=x^{\frac{\alpha-1}{2}}e^{-x/2}L_{n}^{(\alpha)}(x)
  135. u ′′ + 2 x u + [ λ x - 1 4 - α 2 - 1 4 x 2 ] u = 0 with λ = n + α + 1 2 . u^{\prime\prime}+\frac{2}{x}\,u^{\prime}+\left[\frac{\lambda}{x}-\frac{1}{4}-% \frac{\alpha^{2}-1}{4x^{2}}\right]\,u=0\,\text{ with }\lambda=n+\frac{\alpha+1% }{2}.\,
  136. = α - 1 2 \ell=\frac{\alpha-1}{2}
  137. n + 1 n\geq\ell+1
  138. u = x e - x / 2 L n - - 1 ( 2 + 1 ) ( x ) u=x^{\ell}e^{-x/2}L_{n-\ell-1}^{(2\ell+1)}(x)
  139. u ′′ + 2 x u + [ λ x - 1 4 - ( + 1 ) x 2 ] u = 0 with λ = n . u^{\prime\prime}+\frac{2}{x}\,u^{\prime}+\left[\frac{\lambda}{x}-\frac{1}{4}-% \frac{\ell(\ell+1)}{x^{2}}\right]\,u=0\,\text{ with }\lambda=n.\,
  140. u = x e - x / 2 L n + [ 2 + 1 ] ( x ) . u=x^{\ell}e^{-x/2}L_{n+\ell}^{[2\ell+1]}(x).
  141. ( n ! ) (n!)
  142. y ′′ - 2 x y + λ y = 0 , with λ = 2 n . y^{\prime\prime}-2xy^{\prime}+{\lambda}\,y=0,\qquad\mathrm{with}\qquad\lambda=% 2n.\,
  143. ( e - x 2 y ) + e - x 2 λ y = 0. (e^{-x^{2}}\,y^{\prime})^{\prime}+e^{-x^{2}}\,\lambda\,y=0.\,
  144. ( e - x 2 / 2 y ) ′′ + ( λ + 1 - x 2 ) ( e - x 2 / 2 y ) = 0. (e^{-x^{2}/2}\,y)^{\prime\prime}+({\lambda}+1-x^{2})(e^{-x^{2}/2}\,y)=0.\,
  145. H n + 1 ( x ) = 2 x H n ( x ) - 2 n H n - 1 ( x ) . H_{n+1}(x)=2x\,H_{n}(x)-2n\,H_{n-1}(x).\,
  146. H n ( x ) = ( - 1 ) n e x 2 d n d x n ( e - x 2 ) . H_{n}(x)=(-1)^{n}\,e^{x^{2}}\ \frac{d^{n}}{dx^{n}}\left(e^{-x^{2}}\right).
  147. H 0 ( x ) = 1 H_{0}(x)=1\,
  148. H 1 ( x ) = 2 x H_{1}(x)=2x\,
  149. H 2 ( x ) = 4 x 2 - 2 H_{2}(x)=4x^{2}-2\,
  150. H 3 ( x ) = 8 x 3 - 12 x H_{3}(x)=8x^{3}-12x\,
  151. H 4 ( x ) = 16 x 4 - 48 x 2 + 12 H_{4}(x)=16x^{4}-48x^{2}+12\,
  152. ψ n ( x ) = ( h n ) - 1 / 2 e - x 2 / 2 H n ( x ) . {\psi}_{n}(x)=(h_{n})^{-1/2}\,e^{-x^{2}/2}H_{n}(x).\,
  153. ( - , ) (-\infty,\infty)
  154. ψ ′′ + ( λ + 1 - x 2 ) ψ = 0. \psi^{\prime\prime}+({\lambda}+1-x^{2})\psi=0.\,
  155. e - x 2 / 2 e^{-x^{2}/2}
  156. e - x 2 e^{-x^{2}}
  157. H e n ( x ) = 2 - n / 2 H n ( x 2 ) . He_{n}(x)=2^{-n/2}\,H_{n}\left(\frac{x}{\sqrt{2}}\right).
  158. T n \ T_{n}
  159. U n \ U_{n}
  160. P n \ P_{n}
  161. H n \ H_{n}
  162. - 1 , 1 -1,1\,
  163. - 1 , 1 -1,1\,
  164. - 1 , 1 -1,1\,
  165. - , -\infty,\infty
  166. W ( x ) W(x)\,
  167. ( 1 - x 2 ) - 1 / 2 (1-x^{2})^{-1/2}\,
  168. ( 1 - x 2 ) 1 / 2 (1-x^{2})^{1/2}\,
  169. 1 1\,
  170. e - x 2 e^{-x^{2}}
  171. T n ( 1 ) = 1 T_{n}(1)=1\,
  172. U n ( 1 ) = n + 1 U_{n}(1)=n+1\,
  173. P n ( 1 ) = 1 P_{n}(1)=1\,
  174. 2 n 2^{n}\,
  175. { π : n = 0 π / 2 : n 0 \left\{\begin{matrix}\pi&:~{}n=0\\ \pi/2&:~{}n\neq 0\end{matrix}\right.
  176. π / 2 \pi/2\,
  177. 2 2 n + 1 \frac{2}{2n+1}
  178. 2 n n ! π 2^{n}\,n!\,\sqrt{\pi}
  179. 2 n - 1 2^{n-1}\,
  180. 2 n 2^{n}\,
  181. ( 2 n ) ! 2 n ( n ! ) 2 \frac{(2n)!}{2^{n}\,(n!)^{2}}\,
  182. 2 n 2^{n}\,
  183. k n k^{\prime}_{n}\,
  184. 0 0\,
  185. 0 0\,
  186. 0 0\,
  187. 0 0\,
  188. Q Q\,
  189. 1 - x 2 1-x^{2}\,
  190. 1 - x 2 1-x^{2}\,
  191. 1 - x 2 1-x^{2}\,
  192. 1 1\,
  193. L L\,
  194. - x -x\,
  195. - 3 x -3x\,
  196. - 2 x -2x\,
  197. - 2 x -2x\,
  198. R ( x ) = e L ( x ) Q ( x ) d x R(x)=e^{\int\frac{L(x)}{Q(x)}\,dx}
  199. ( 1 - x 2 ) 1 / 2 (1-x^{2})^{1/2}\,
  200. ( 1 - x 2 ) 3 / 2 (1-x^{2})^{3/2}\,
  201. 1 - x 2 1-x^{2}\,
  202. e - x 2 e^{-x^{2}}\,
  203. λ n {\lambda}_{n}\,
  204. n 2 n^{2}\,
  205. n ( n + 2 ) n(n+2)\,
  206. n ( n + 1 ) n(n+1)\,
  207. 2 n 2n\,
  208. e n e_{n}\,
  209. ( - 2 ) n Γ ( n + 1 / 2 ) π (-2)^{n}\,\frac{\Gamma(n+1/2)}{\sqrt{\pi}}\,
  210. 2 ( - 2 ) n Γ ( n + 3 / 2 ) ( n + 1 ) π 2(-2)^{n}\,\frac{\Gamma(n+3/2)}{(n+1)\,\sqrt{\pi}}\,
  211. ( - 2 ) n n ! (-2)^{n}\,n!\,
  212. ( - 1 ) n (-1)^{n}\,
  213. a n a_{n}\,
  214. 2 2\,
  215. 2 2\,
  216. 2 n + 1 n + 1 \frac{2n+1}{n+1}\,
  217. 2 2\,
  218. b n b_{n}\,
  219. 0 0\,
  220. 0 0\,
  221. 0 0\,
  222. 0 0\,
  223. c n c_{n}\,
  224. 1 1\,
  225. 1 1\,
  226. n n + 1 \frac{n}{n+1}\,
  227. 2 n 2n\,
  228. L n ( α ) L_{n}^{(\alpha)}
  229. L n \ L_{n}
  230. 0 , 0,\infty\,
  231. 0 , 0,\infty\,
  232. W ( x ) W(x)\,
  233. x α e - x x^{\alpha}e^{-x}\,
  234. e - x e^{-x}\,
  235. ( - 1 ) n n ! \frac{(-1)^{n}}{n!}\,
  236. ( - 1 ) n n ! \frac{(-1)^{n}}{n!}\,
  237. h n h_{n}\,
  238. Γ ( n + α + 1 ) n ! \frac{\Gamma(n+\alpha+1)}{n!}\,
  239. 1 1\,
  240. k n k_{n}\,
  241. ( - 1 ) n n ! \frac{(-1)^{n}}{n!}\,
  242. ( - 1 ) n n ! \frac{(-1)^{n}}{n!}\,
  243. k n k^{\prime}_{n}\,
  244. ( - 1 ) n + 1 ( n + α ) ( n - 1 ) ! \frac{(-1)^{n+1}(n+\alpha)}{(n-1)!}\,
  245. ( - 1 ) n + 1 n ( n - 1 ) ! \frac{(-1)^{n+1}n}{(n-1)!}\,
  246. Q Q\,
  247. x x\,
  248. x x\,
  249. L L\,
  250. α + 1 - x \alpha+1-x\,
  251. 1 - x 1-x\,
  252. R ( x ) = e L ( x ) Q ( x ) d x R(x)=e^{\int\frac{L(x)}{Q(x)}\,dx}
  253. x α + 1 e - x x^{\alpha+1}\,e^{-x}\,
  254. x e - x x\,e^{-x}\,
  255. λ n {\lambda}_{n}\,
  256. n n\,
  257. n n\,
  258. e n e_{n}\,
  259. n ! n!\,
  260. n ! n!\,
  261. a n a_{n}\,
  262. - 1 n + 1 \frac{-1}{n+1}\,
  263. - 1 n + 1 \frac{-1}{n+1}\,
  264. b n b_{n}\,
  265. 2 n + 1 + α n + 1 \frac{2n+1+\alpha}{n+1}\,
  266. 2 n + 1 n + 1 \frac{2n+1}{n+1}\,
  267. c n c_{n}\,
  268. n + α n + 1 \frac{n+\alpha}{n+1}\,
  269. n n + 1 \frac{n}{n+1}\,
  270. C n ( α ) C_{n}^{(\alpha)}
  271. P n ( α , β ) P_{n}^{(\alpha,\beta)}
  272. - 1 , 1 -1,1\,
  273. - 1 , 1 -1,1\,
  274. W ( x ) W(x)\,
  275. ( 1 - x 2 ) α - 1 / 2 (1-x^{2})^{\alpha-1/2}\,
  276. ( 1 - x ) α ( 1 + x ) β (1-x)^{\alpha}(1+x)^{\beta}\,
  277. C n ( α ) ( 1 ) = Γ ( n + 2 α ) n ! Γ ( 2 α ) C_{n}^{(\alpha)}(1)=\frac{\Gamma(n+2\alpha)}{n!\,\Gamma(2\alpha)}\,
  278. α 0 \alpha\neq 0
  279. P n ( α , β ) ( 1 ) = Γ ( n + 1 + α ) n ! Γ ( 1 + α ) P_{n}^{(\alpha,\beta)}(1)=\frac{\Gamma(n+1+\alpha)}{n!\,\Gamma(1+\alpha)}\,
  280. h n h_{n}\,
  281. π 2 1 - 2 α Γ ( n + 2 α ) n ! ( n + α ) ( Γ ( α ) ) 2 \frac{\pi\,2^{1-2\alpha}\Gamma(n+2\alpha)}{n!(n+\alpha)(\Gamma(\alpha))^{2}}
  282. 2 α + β + 1 Γ ( n + α + 1 ) Γ ( n + β + 1 ) n ! ( 2 n + α + β + 1 ) Γ ( n + α + β + 1 ) \frac{2^{\alpha+\beta+1}\,\Gamma(n\!+\!\alpha\!+\!1)\,\Gamma(n\!+\!\beta\!+\!1% )}{n!(2n\!+\!\alpha\!+\!\beta\!+\!1)\Gamma(n\!+\!\alpha\!+\!\beta\!+\!1)}
  283. k n k_{n}\,
  284. Γ ( 2 n + 2 α ) Γ ( 1 / 2 + α ) n ! 2 n Γ ( 2 α ) Γ ( n + 1 / 2 + α ) \frac{\Gamma(2n+2\alpha)\Gamma(1/2+\alpha)}{n!\,2^{n}\,\Gamma(2\alpha)\Gamma(n% +1/2+\alpha)}\,
  285. Γ ( 2 n + 1 + α + β ) n ! 2 n Γ ( n + 1 + α + β ) \frac{\Gamma(2n+1+\alpha+\beta)}{n!\,2^{n}\,\Gamma(n+1+\alpha+\beta)}\,
  286. k n k^{\prime}_{n}\,
  287. 0 0\,
  288. ( α - β ) Γ ( 2 n + α + β ) ( n - 1 ) ! 2 n Γ ( n + 1 + α + β ) \frac{(\alpha-\beta)\,\Gamma(2n+\alpha+\beta)}{(n-1)!\,2^{n}\,\Gamma(n+1+% \alpha+\beta)}\,
  289. Q Q\,
  290. 1 - x 2 1-x^{2}\,
  291. 1 - x 2 1-x^{2}\,
  292. L L\,
  293. - ( 2 α + 1 ) x -(2\alpha+1)\,x\,
  294. β - α - ( α + β + 2 ) x \beta-\alpha-(\alpha+\beta+2)\,x\,
  295. R ( x ) = e L ( x ) Q ( x ) d x R(x)=e^{\int\frac{L(x)}{Q(x)}\,dx}
  296. ( 1 - x 2 ) α + 1 / 2 (1-x^{2})^{\alpha+1/2}\,
  297. ( 1 - x ) α + 1 ( 1 + x ) β + 1 (1-x)^{\alpha+1}(1+x)^{\beta+1}\,
  298. λ n {\lambda}_{n}\,
  299. n ( n + 2 α ) n(n+2\alpha)\,
  300. n ( n + 1 + α + β ) n(n+1+\alpha+\beta)\,
  301. e n e_{n}\,
  302. ( - 2 ) n n ! Γ ( 2 α ) Γ ( n + 1 / 2 + α ) Γ ( n + 2 α ) Γ ( α + 1 / 2 ) \frac{(-2)^{n}\,n!\,\Gamma(2\alpha)\,\Gamma(n\!+\!1/2\!+\!\alpha)}{\Gamma(n\!+% \!2\alpha)\Gamma(\alpha\!+\!1/2)}
  303. ( - 2 ) n n ! (-2)^{n}\,n!\,
  304. a n a_{n}\,
  305. 2 ( n + α ) n + 1 \frac{2(n+\alpha)}{n+1}\,
  306. ( 2 n + 1 + α + β ) ( 2 n + 2 + α + β ) 2 ( n + 1 ) ( n + 1 + α + β ) \frac{(2n+1+\alpha+\beta)(2n+2+\alpha+\beta)}{2(n+1)(n+1+\alpha+\beta)}
  307. b n b_{n}\,
  308. 0 0\,
  309. ( α 2 - β 2 ) ( 2 n + 1 + α + β ) 2 ( n + 1 ) ( 2 n + α + β ) ( n + 1 + α + β ) \frac{({\alpha}^{2}-{\beta}^{2})(2n+1+\alpha+\beta)}{2(n+1)(2n+\alpha+\beta)(n% +1+\alpha+\beta)}
  310. c n c_{n}\,
  311. n + 2 α - 1 n + 1 \frac{n+2{\alpha}-1}{n+1}\,
  312. ( n + α ) ( n + β ) ( 2 n + 2 + α + β ) ( n + 1 ) ( n + 1 + α + β ) ( 2 n + α + β ) \frac{(n+\alpha)(n+\beta)(2n+2+\alpha+\beta)}{(n+1)(n+1+\alpha+\beta)(2n+% \alpha+\beta)}
  313. h n = P n 2 ( x ) W ( x ) d x h_{n}=\int P_{n}^{2}(x)W(x)dx
  314. P n ( x ) = k n x n + k n x n - 1 + + k ( n ) P_{n}(x)=k_{n}x^{n}+k^{\prime}_{n}x^{n-1}+\cdots+k^{(n)}

Classical_test_theory.html

  1. X X
  2. T T
  3. E E
  4. X X
  5. ρ X T 2 {\rho^{2}_{XT}}
  6. σ T 2 {\sigma^{2}_{T}}
  7. σ X 2 {\sigma^{2}_{X}}
  8. ρ X T 2 = < m t p l > σ T 2 σ X 2 {\rho^{2}_{XT}}=\frac{<}{m}tpl>{{\sigma^{2}_{T}}}{{\sigma^{2}_{X}}}
  9. ρ X T 2 = < m t p l > σ T 2 σ X 2 = σ T 2 σ T 2 + σ E 2 {\rho^{2}_{XT}}=\frac{<}{m}tpl>{{\sigma^{2}_{T}}}{{\sigma^{2}_{X}}}=\frac{{% \sigma^{2}_{T}}}{{\sigma^{2}_{T}}+{\sigma^{2}_{E}}}
  10. ε ( X i ) = ε ( X i ) {\varepsilon}(X_{i})={\varepsilon}(X^{\prime}_{i})
  11. σ E i 2 = σ E i 2 {\sigma}^{2}_{E_{i}}={\sigma}^{2}_{E^{\prime}_{i}}
  12. ρ X X = σ X X σ X σ X = σ T 2 σ X 2 = ρ X T 2 {\rho}_{XX^{\prime}}=\frac{{\sigma}_{XX^{\prime}}}{{\sigma}_{X}{\sigma}_{X^{% \prime}}}=\frac{{\sigma}_{T}^{2}}{{\sigma}_{X}^{2}}={\rho}_{XT}^{2}
  13. α {\alpha}
  14. k k
  15. u j u_{j}
  16. j = 1 , , k j=1,\ldots,k
  17. i i
  18. X i = j = 1 k U i j X_{i}=\sum_{j=1}^{k}{U_{ij}}
  19. α = k k - 1 ( 1 - j = 1 k σ U j 2 σ X 2 ) \alpha=\frac{k}{k-1}\left(1-\frac{\sum_{j=1}^{k}{\sigma^{2}_{U_{j}}}}{\sigma^{% 2}_{X}}\right)
  20. α {\alpha}
  21. α {\alpha}
  22. α {\alpha}
  23. α {\alpha}
  24. α {\alpha}
  25. α {\alpha}

Clausen_function.html

  1. Cl 2 ( φ ) = - 0 φ log | 2 sin x 2 | d x : \operatorname{Cl}_{2}(\varphi)=-\int_{0}^{\varphi}\log\Bigg|2\sin\frac{x}{2}% \Bigg|\,dx:
  2. 0 < φ < 2 π 0<\varphi<2\pi\,
  3. Cl 2 ( φ ) = k = 1 sin k φ k 2 = sin φ + sin 2 φ 2 2 + sin 3 φ 3 2 + sin 4 φ 4 2 + \operatorname{Cl}_{2}(\varphi)=\sum_{k=1}^{\infty}\frac{\sin k\varphi}{k^{2}}=% \sin\varphi+\frac{\sin 2\varphi}{2^{2}}+\frac{\sin 3\varphi}{3^{2}}+\frac{\sin 4% \varphi}{4^{2}}+\,\cdots
  4. π , \pi,\,
  5. k k\in\mathbb{Z}\,
  6. sin k π = 0 \sin k\pi=0
  7. Cl 2 ( m π ) = 0 , m = 0 , ± 1 , ± 2 , ± 3 , \,\text{Cl}_{2}(m\pi)=0,\quad m=0,\,\pm 1,\,\pm 2,\,\pm 3,\,\cdots
  8. θ = π 3 + 2 m π [ m ] \theta=\frac{\pi}{3}+2m\pi\quad[m\in\mathbb{Z}]
  9. Cl 2 ( π 3 + 2 m π ) = 1.01494160 \,\text{Cl}_{2}\left(\frac{\pi}{3}+2m\pi\right)=1.01494160\cdots
  10. θ = - π 3 + 2 m π [ m ] \theta=-\frac{\pi}{3}+2m\pi\quad[m\in\mathbb{Z}]
  11. Cl 2 ( - π 3 + 2 m π ) = - 1.01494160 \,\text{Cl}_{2}\left(-\frac{\pi}{3}+2m\pi\right)=-1.01494160\cdots
  12. Cl 2 ( θ + 2 m π ) = Cl 2 ( θ ) \,\text{Cl}_{2}(\theta+2m\pi)=\,\text{Cl}_{2}(\theta)
  13. Cl 2 ( - θ ) = - Cl 2 ( θ ) \,\text{Cl}_{2}(-\theta)=-\,\text{Cl}_{2}(\theta)
  14. S z ( θ ) = k = 1 sin k θ k z \operatorname{S}_{z}(\theta)=\sum_{k=1}^{\infty}\frac{\sin k\theta}{k^{z}}
  15. C z ( θ ) = k = 1 cos k θ k z \operatorname{C}_{z}(\theta)=\sum_{k=1}^{\infty}\frac{\cos k\theta}{k^{z}}
  16. Cl 2 m + 2 ( θ ) = k = 1 sin k θ k 2 m + 2 \operatorname{Cl}_{2m+2}(\theta)=\sum_{k=1}^{\infty}\frac{\sin k\theta}{k^{2m+% 2}}
  17. Cl 2 m + 1 ( θ ) = k = 1 cos k θ k 2 m + 1 \operatorname{Cl}_{2m+1}(\theta)=\sum_{k=1}^{\infty}\frac{\cos k\theta}{k^{2m+% 1}}
  18. Sl 2 m + 2 ( θ ) = k = 1 cos k θ k 2 m + 2 \operatorname{Sl}_{2m+2}(\theta)=\sum_{k=1}^{\infty}\frac{\cos k\theta}{k^{2m+% 2}}
  19. Sl 2 m + 1 ( θ ) = k = 1 sin k θ k 2 m + 1 \operatorname{Sl}_{2m+1}(\theta)=\sum_{k=1}^{\infty}\frac{\sin k\theta}{k^{2m+% 1}}
  20. Gl m ( θ ) \operatorname{Gl}_{m}(\theta)\,
  21. θ \,\theta\,
  22. B 2 n - 1 ( x ) = 2 ( - 1 ) n ( 2 n - 1 ) ! ( 2 π ) 2 n - 1 k = 1 sin 2 π k x k 2 n - 1 B_{2n-1}(x)=\frac{2(-1)^{n}(2n-1)!}{(2\pi)^{2n-1}}\,\sum_{k=1}^{\infty}\frac{% \sin 2\pi kx}{k^{2n-1}}
  23. B 2 n ( x ) = 2 ( - 1 ) n - 1 ( 2 n ) ! ( 2 π ) 2 n k = 1 cos 2 π k x k 2 n B_{2n}(x)=\frac{2(-1)^{n-1}(2n)!}{(2\pi)^{2n}}\,\sum_{k=1}^{\infty}\frac{\cos 2% \pi kx}{k^{2n}}
  24. x = θ / 2 π \,x=\theta/2\pi\,
  25. Sl 2 m ( θ ) = ( - 1 ) m - 1 ( 2 π ) 2 m 2 ( 2 m ) ! B 2 m ( θ 2 π ) \,\text{Sl}_{2m}(\theta)=\frac{(-1)^{m-1}(2\pi)^{2m}}{2(2m)!}B_{2m}\left(\frac% {\theta}{2\pi}\right)
  26. Sl 2 m - 1 ( θ ) = ( - 1 ) m ( 2 π ) 2 m - 1 2 ( 2 m - 1 ) ! B 2 m - 1 ( θ 2 π ) \,\text{Sl}_{2m-1}(\theta)=\frac{(-1)^{m}(2\pi)^{2m-1}}{2(2m-1)!}B_{2m-1}\left% (\frac{\theta}{2\pi}\right)
  27. B n ( x ) \,B_{n}(x)\,
  28. B n B n ( 0 ) \,B_{n}\equiv B_{n}(0)\,
  29. B n ( x ) = j = 0 n ( n j ) B j x n - j B_{n}(x)=\sum_{j=0}^{n}{\left({{n}\atop{j}}\right)}B_{j}x^{n-j}
  30. Sl 1 ( θ ) = π 2 - θ 2 \,\text{Sl}_{1}(\theta)=\frac{\pi}{2}-\frac{\theta}{2}
  31. Sl 2 ( θ ) = π 2 6 - π θ 2 + θ 2 4 \,\text{Sl}_{2}(\theta)=\frac{\pi^{2}}{6}-\frac{\pi\theta}{2}+\frac{\theta^{2}% }{4}
  32. Sl 3 ( θ ) = π 2 θ 6 - π θ 2 4 + θ 3 12 \,\text{Sl}_{3}(\theta)=\frac{\pi^{2}\theta}{6}-\frac{\pi\theta^{2}}{4}+\frac{% \theta^{3}}{12}
  33. Sl 4 ( θ ) = π 4 90 - π 2 θ 2 12 + π θ 3 12 - θ 4 48 \,\text{Sl}_{4}(\theta)=\frac{\pi^{4}}{90}-\frac{\pi^{2}\theta^{2}}{12}+\frac{% \pi\theta^{3}}{12}-\frac{\theta^{4}}{48}
  34. 0 < θ < π 0<\theta<\pi
  35. Cl 2 ( 2 θ ) = 2 Cl 2 ( θ ) - 2 Cl 2 ( π - θ ) \operatorname{Cl}_{2}(2\theta)=2\operatorname{Cl}_{2}(\theta)-2\operatorname{% Cl}_{2}(\pi-\theta)
  36. Cl 2 ( π 2 ) = G \operatorname{Cl}_{2}\left(\frac{\pi}{2}\right)=G
  37. Cl 2 ( π 4 ) - Cl 2 ( 3 π 4 ) = G 2 \operatorname{Cl}_{2}\left(\frac{\pi}{4}\right)-\operatorname{Cl}_{2}\left(% \frac{3\pi}{4}\right)=\frac{G}{2}
  38. 2 Cl 2 ( π 3 ) = 3 Cl 2 ( 2 π 3 ) 2\operatorname{Cl}_{2}\left(\frac{\pi}{3}\right)=3\operatorname{Cl}_{2}\left(% \frac{2\pi}{3}\right)
  39. θ \,\theta\,
  40. x \,x\,
  41. [ 0 , θ ] . \,[0,\theta].\,
  42. Cl 3 ( 2 θ ) = 4 Cl 3 ( θ ) + 4 Cl 3 ( π - θ ) \operatorname{Cl}_{3}(2\theta)=4\operatorname{Cl}_{3}(\theta)+4\operatorname{% Cl}_{3}(\pi-\theta)
  43. Cl 4 ( 2 θ ) = 8 Cl 4 ( θ ) - 8 Cl 4 ( π - θ ) \operatorname{Cl}_{4}(2\theta)=8\operatorname{Cl}_{4}(\theta)-8\operatorname{% Cl}_{4}(\pi-\theta)
  44. Cl 5 ( 2 θ ) = 16 Cl 5 ( θ ) + 16 Cl 5 ( π - θ ) \operatorname{Cl}_{5}(2\theta)=16\operatorname{Cl}_{5}(\theta)+16\operatorname% {Cl}_{5}(\pi-\theta)
  45. Cl 6 ( 2 θ ) = 32 Cl 6 ( θ ) - 32 Cl 6 ( π - θ ) \operatorname{Cl}_{6}(2\theta)=32\operatorname{Cl}_{6}(\theta)-32\operatorname% {Cl}_{6}(\pi-\theta)
  46. m , m 1 \,m,\,\,m\geq 1
  47. Cl m + 1 ( 2 θ ) = 2 m [ Cl m + 1 ( θ ) + ( - 1 ) m Cl m + 1 ( π - θ ) ] \operatorname{Cl}_{m+1}(2\theta)=2^{m}\Bigg[\operatorname{Cl}_{m+1}(\theta)+(-% 1)^{m}\operatorname{Cl}_{m+1}(\pi-\theta)\Bigg]
  48. m 1 \,m\in\mathbb{Z}\geq 1\,
  49. Cl 2 m ( π 2 ) = 2 2 m - 1 [ Cl 2 m ( π 4 ) - Cl 2 m ( 3 π 4 ) ] = β ( 2 m ) \,\text{Cl}_{2m}\left(\frac{\pi}{2}\right)=2^{2m-1}\left[\,\text{Cl}_{2m}\left% (\frac{\pi}{4}\right)-\,\text{Cl}_{2m}\left(\frac{3\pi}{4}\right)\right]=\beta% (2m)
  50. β ( x ) \,\beta(x)\,
  51. Cl 2 ( 2 θ ) = - 0 2 θ log | 2 sin x 2 | d x \operatorname{Cl}_{2}(2\theta)=-\int_{0}^{2\theta}\log\Bigg|2\sin\frac{x}{2}% \Bigg|\,dx
  52. sin 2 x = 2 sin x 2 cos x 2 \sin 2x=2\sin\frac{x}{2}\cos\frac{x}{2}
  53. - 0 2 θ log | ( 2 sin x 4 ) ( 2 cos x 4 ) | d x = -\int_{0}^{2\theta}\log\Bigg|\left(2\sin\frac{x}{4}\right)\left(2\cos\frac{x}{% 4}\right)\Bigg|\,dx=
  54. - 0 2 θ log | 2 sin x 4 | d x - 0 2 θ log | 2 cos x 4 | d x = -\int_{0}^{2\theta}\log\Bigg|2\sin\frac{x}{4}\Bigg|\,dx-\int_{0}^{2\theta}\log% \Bigg|2\cos\frac{x}{4}\Bigg|\,dx=
  55. x = 2 y , d x = 2 d y x=2y,dx=2\,dy
  56. \Rightarrow
  57. - 2 0 θ log | 2 sin x 2 | d x - 2 0 θ log | 2 cos x 2 | d x = -2\int_{0}^{\theta}\log\Bigg|2\sin\frac{x}{2}\Bigg|\,dx-2\int_{0}^{\theta}\log% \Bigg|2\cos\frac{x}{2}\Bigg|\,dx=
  58. 2 Cl 2 ( θ ) - 2 0 θ log | 2 cos x 2 | d x 2\,\operatorname{Cl}_{2}(\theta)-2\int_{0}^{\theta}\log\Bigg|2\cos\frac{x}{2}% \Bigg|\,dx
  59. y = π - x , x = π - y , d x = - d y y=\pi-x,\,x=\pi-y,\,dx=-dy
  60. cos ( x - y ) = cos x cos y - sin x sin y \cos(x-y)=\cos x\cos y-\sin x\sin y
  61. cos ( π - y 2 ) = sin y 2 \cos\left(\frac{\pi-y}{2}\right)=\sin\frac{y}{2}\Rightarrow
  62. Cl 2 ( 2 θ ) = 2 Cl 2 ( θ ) - 2 0 θ log | 2 cos x 2 | d x = \operatorname{Cl}_{2}(2\theta)=2\,\operatorname{Cl}_{2}(\theta)-2\int_{0}^{% \theta}\log\Bigg|2\cos\frac{x}{2}\Bigg|\,dx=
  63. 2 Cl 2 ( θ ) + 2 π π - θ log | 2 sin y 2 | d y = 2\,\operatorname{Cl}_{2}(\theta)+2\int_{\pi}^{\pi-\theta}\log\Bigg|2\sin\frac{% y}{2}\Bigg|\,dy=
  64. Cl 2 ( θ ) - 2 Cl 2 ( π - θ ) + 2 Cl 2 ( π ) \,\operatorname{Cl}_{2}(\theta)-2\,\operatorname{Cl}_{2}(\pi-\theta)+2\,% \operatorname{Cl}_{2}(\pi)
  65. Cl 2 ( π ) = 0 \operatorname{Cl}_{2}(\pi)=0
  66. Cl 2 ( 2 θ ) = 2 Cl 2 ( θ ) - 2 Cl 2 ( π - θ ) . \operatorname{Cl}_{2}(2\theta)=2\,\operatorname{Cl}_{2}(\theta)-2\,% \operatorname{Cl}_{2}(\pi-\theta)\,.\,\Box
  67. d d θ Cl 2 m + 2 ( θ ) = d d θ k = 1 sin k θ k 2 m + 2 = k = 1 cos k θ k 2 m + 1 = Cl 2 m + 1 ( θ ) \frac{d}{d\theta}\operatorname{Cl}_{2m+2}(\theta)=\frac{d}{d\theta}\sum_{k=1}^% {\infty}\frac{\sin k\theta}{k^{2m+2}}=\sum_{k=1}^{\infty}\frac{\cos k\theta}{k% ^{2m+1}}=\operatorname{Cl}_{2m+1}(\theta)
  68. d d θ Cl 2 m + 1 ( θ ) = d d θ k = 1 cos k θ k 2 m + 1 = - k = 1 sin k θ k 2 m = - Cl 2 m ( θ ) \frac{d}{d\theta}\operatorname{Cl}_{2m+1}(\theta)=\frac{d}{d\theta}\sum_{k=1}^% {\infty}\frac{\cos k\theta}{k^{2m+1}}=-\sum_{k=1}^{\infty}\frac{\sin k\theta}{% k^{2m}}=-\operatorname{Cl}_{2m}(\theta)
  69. d d θ Sl 2 m + 2 ( θ ) = d d θ k = 1 cos k θ k 2 m + 2 = - k = 1 sin k θ k 2 m + 1 = - Sl 2 m + 1 ( θ ) \frac{d}{d\theta}\operatorname{Sl}_{2m+2}(\theta)=\frac{d}{d\theta}\sum_{k=1}^% {\infty}\frac{\cos k\theta}{k^{2m+2}}=-\sum_{k=1}^{\infty}\frac{\sin k\theta}{% k^{2m+1}}=-\operatorname{Sl}_{2m+1}(\theta)
  70. d d θ Sl 2 m + 1 ( θ ) = d d θ k = 1 sin k θ k 2 m + 1 = k = 1 cos k θ k 2 m = Sl 2 m ( θ ) \frac{d}{d\theta}\operatorname{Sl}_{2m+1}(\theta)=\frac{d}{d\theta}\sum_{k=1}^% {\infty}\frac{\sin k\theta}{k^{2m+1}}=\sum_{k=1}^{\infty}\frac{\cos k\theta}{k% ^{2m}}=\operatorname{Sl}_{2m}(\theta)
  71. d d θ Cl 2 ( θ ) = d d θ [ - 0 θ log | 2 sin x 2 | d x ] = - log | 2 sin θ 2 | = Cl 1 ( θ ) \frac{d}{d\theta}\operatorname{Cl}_{2}(\theta)=\frac{d}{d\theta}\left[-\int_{0% }^{\theta}\log\Bigg|2\sin\frac{x}{2}\Bigg|\,dx\,\right]=-\log\Bigg|2\sin\frac{% \theta}{2}\Bigg|=\operatorname{Cl}_{1}(\theta)
  72. 0 < z < 1 0<z<1
  73. Ti 2 ( z ) = 0 z tan - 1 x x d x = k = 0 ( - 1 ) k z 2 k + 1 ( 2 k + 1 ) 2 \operatorname{Ti}_{2}(z)=\int_{0}^{z}\frac{\tan^{-1}x}{x}\,dx=\sum_{k=0}^{% \infty}(-1)^{k}\frac{z^{2k+1}}{(2k+1)^{2}}
  74. Ti 2 ( tan θ ) = θ log ( tan θ ) + 1 2 Cl 2 ( 2 θ ) + 1 2 Cl 2 ( π - 2 θ ) \operatorname{Ti}_{2}(\tan\theta)=\theta\log(\tan\theta)+\frac{1}{2}% \operatorname{Cl}_{2}(2\theta)+\frac{1}{2}\operatorname{Cl}_{2}(\pi-2\theta)
  75. Ti 2 ( tan θ ) = 0 tan θ tan - 1 x x d x \operatorname{Ti}_{2}(\tan\theta)=\int_{0}^{\tan\theta}\frac{\tan^{-1}x}{x}\,dx
  76. 0 tan θ tan - 1 x x d x = tan - 1 x log x | 0 tan θ - 0 tan θ log x 1 + x 2 d x = \int_{0}^{\tan\theta}\frac{\tan^{-1}x}{x}\,dx=\tan^{-1}x\log x\,\Bigg|_{0}^{% \tan\theta}-\int_{0}^{\tan\theta}\frac{\log x}{1+x^{2}}\,dx=
  77. θ log tan θ - 0 tan θ log x 1 + x 2 d x \theta\log{\tan\theta}-\int_{0}^{\tan\theta}\frac{\log x}{1+x^{2}}\,dx
  78. x = tan y , y = tan - 1 x , d y = d x 1 + x 2 x=\tan y,\,y=\tan^{-1}x,\,dy=\frac{dx}{1+x^{2}}\,
  79. θ log tan θ - 0 θ log ( tan y ) d y \theta\log{\tan\theta}-\int_{0}^{\theta}\log(\tan y)\,dy
  80. y = x / 2 , d y = d x / 2 y=x/2,\,dy=dx/2\,
  81. θ log tan θ - 1 2 0 2 θ log ( tan x 2 ) d x = \theta\log{\tan\theta}-\frac{1}{2}\int_{0}^{2\theta}\log\left(\tan\frac{x}{2}% \right)\,dx=
  82. θ log tan θ - 1 2 0 2 θ log ( sin ( x / 2 ) cos ( x / 2 ) ) d x = \theta\log{\tan\theta}-\frac{1}{2}\int_{0}^{2\theta}\log\left(\frac{\sin(x/2)}% {\cos(x/2)}\right)\,dx=
  83. θ log tan θ - 1 2 0 2 θ log ( 2 sin ( x / 2 ) 2 cos ( x / 2 ) ) d x = \theta\log{\tan\theta}-\frac{1}{2}\int_{0}^{2\theta}\log\left(\frac{2\sin(x/2)% }{2\cos(x/2)}\right)\,dx=
  84. θ log tan θ - 1 2 0 2 θ log ( 2 sin x 2 ) d x + 1 2 0 2 θ log ( 2 cos x 2 ) d x = \theta\log{\tan\theta}-\frac{1}{2}\int_{0}^{2\theta}\log\left(2\sin\frac{x}{2}% \right)\,dx+\frac{1}{2}\int_{0}^{2\theta}\log\left(2\cos\frac{x}{2}\right)\,dx=
  85. θ log tan θ + 1 2 Cl 2 ( 2 θ ) + 1 2 0 2 θ log ( 2 cos x 2 ) d x \theta\log{\tan\theta}+\frac{1}{2}\operatorname{Cl}_{2}(2\theta)+\frac{1}{2}% \int_{0}^{2\theta}\log\left(2\cos\frac{x}{2}\right)\,dx
  86. x = ( π - y ) x=(\pi-y)\,
  87. 0 2 θ log ( 2 cos x 2 ) d x = Cl 2 ( π - 2 θ ) - Cl 2 ( π ) = Cl 2 ( π - 2 θ ) \int_{0}^{2\theta}\log\left(2\cos\frac{x}{2}\right)\,dx=\operatorname{Cl}_{2}(% \pi-2\theta)-\operatorname{Cl}_{2}(\pi)=\operatorname{Cl}_{2}(\pi-2\theta)
  88. Ti 2 ( tan θ ) = θ log tan θ + 1 2 Cl 2 ( 2 θ ) + 1 2 Cl 2 ( π - 2 θ ) . \operatorname{Ti}_{2}(\tan\theta)=\theta\log{\tan\theta}+\frac{1}{2}% \operatorname{Cl}_{2}(2\theta)+\frac{1}{2}\operatorname{Cl}_{2}(\pi-2\theta)\,% .\,\Box
  89. 0 < z < 1 0<z<1
  90. Cl 2 ( 2 π z ) = 2 π log ( G ( 1 - z ) G ( 1 + z ) ) - 2 π log ( sin π z π ) \operatorname{Cl}_{2}(2\pi z)=2\pi\log\left(\frac{G(1-z)}{G(1+z)}\right)-2\pi% \log\left(\frac{\sin\pi z}{\pi}\right)
  91. Cl 2 ( 2 π z ) = 2 π log ( G ( 1 - z ) G ( z ) ) - 2 π log Γ ( z ) - 2 π log ( sin π z π ) \operatorname{Cl}_{2}(2\pi z)=2\pi\log\left(\frac{G(1-z)}{G(z)}\right)-2\pi% \log\Gamma(z)-2\pi\log\left(\frac{\sin\pi z}{\pi}\right)
  92. Cl 2 m ( θ ) = ( Li 2 m ( e i θ ) ) , m 1 \operatorname{Cl}_{2m}(\theta)=\Im(\operatorname{Li}_{2m}(e^{i\theta})),\quad m% \in\mathbb{Z}\geq 1
  93. Cl 2 m + 1 ( θ ) = ( Li 2 m + 1 ( e i θ ) ) , m 0 \operatorname{Cl}_{2m+1}(\theta)=\Re(\operatorname{Li}_{2m+1}(e^{i\theta})),% \quad m\in\mathbb{Z}\geq 0
  94. Li n ( z ) = k = 1 z k k n Li n ( e i θ ) = k = 1 ( e i θ ) k k n = k = 1 e i k θ k n \,\text{Li}_{n}(z)=\sum_{k=1}^{\infty}\frac{z^{k}}{k^{n}}\quad\Rightarrow\,% \text{Li}_{n}\left(e^{i\theta}\right)=\sum_{k=1}^{\infty}\frac{\left(e^{i% \theta}\right)^{k}}{k^{n}}=\sum_{k=1}^{\infty}\frac{e^{ik\theta}}{k^{n}}
  95. e i θ = cos θ + i sin θ e^{i\theta}=\cos\theta+i\sin\theta
  96. ( cos θ + i sin θ ) k = cos k θ + i sin k θ Li n ( e i θ ) = k = 1 cos k θ k n + i k = 1 sin k θ k n (\cos\theta+i\sin\theta)^{k}=\cos k\theta+i\sin k\theta\quad\Rightarrow\,\text% {Li}_{n}\left(e^{i\theta}\right)=\sum_{k=1}^{\infty}\frac{\cos k\theta}{k^{n}}% +i\,\sum_{k=1}^{\infty}\frac{\sin k\theta}{k^{n}}
  97. Li 2 m ( e i θ ) = k = 1 cos k θ k 2 m + i k = 1 sin k θ k 2 m = Sl 2 m ( θ ) + i Cl 2 m ( θ ) \,\text{Li}_{2m}\left(e^{i\theta}\right)=\sum_{k=1}^{\infty}\frac{\cos k\theta% }{k^{2m}}+i\,\sum_{k=1}^{\infty}\frac{\sin k\theta}{k^{2m}}=\,\text{Sl}_{2m}(% \theta)+i\,\text{Cl}_{2m}(\theta)
  98. Li 2 m + 1 ( e i θ ) = k = 1 cos k θ k 2 m + 1 + i k = 1 sin k θ k 2 m + 1 = Cl 2 m + 1 ( θ ) + i Sl 2 m + 1 ( θ ) \,\text{Li}_{2m+1}\left(e^{i\theta}\right)=\sum_{k=1}^{\infty}\frac{\cos k% \theta}{k^{2m+1}}+i\,\sum_{k=1}^{\infty}\frac{\sin k\theta}{k^{2m+1}}=\,\text{% Cl}_{2m+1}(\theta)+i\,\text{Sl}_{2m+1}(\theta)
  99. Cl 2 m ( q π p ) = 1 ( 2 p ) 2 m ( 2 m - 1 ) ! j = 1 p sin ( q j π p ) [ ψ 2 m - 1 ( j 2 p ) + ( - 1 ) q ψ 2 m - 1 ( j + p 2 p ) ] \,\text{Cl}_{2m}\left(\frac{q\pi}{p}\right)=\frac{1}{(2p)^{2m}(2m-1)!}\,\sum_{% j=1}^{p}\sin\left(\tfrac{qj\pi}{p}\right)\,\left[\psi_{2m-1}\left(\tfrac{j}{2p% }\right)+(-1)^{q}\psi_{2m-1}\left(\tfrac{j+p}{2p}\right)\right]
  100. p \,p\,
  101. q \,q\,
  102. q / p \,q/p\,
  103. 0 < q / p < 1 \,0<q/p<1\,
  104. Cl 2 m ( q π p ) = k = 1 sin ( k q π / p ) k 2 m \,\text{Cl}_{2m}\left(\frac{q\pi}{p}\right)=\sum_{k=1}^{\infty}\frac{\sin(kq% \pi/p)}{k^{2m}}
  105. k p + 1 , \,kp+1,\,
  106. k p + 2 , \,kp+2,\,
  107. k p + p \,kp+p\,
  108. Cl 2 m ( q π p ) = \,\text{Cl}_{2m}\left(\frac{q\pi}{p}\right)=
  109. k = 0 sin [ ( k p + 1 ) q π p ] ( k p + 1 ) 2 m + k = 0 sin [ ( k p + 2 ) q π p ] ( k p + 2 ) 2 m + k = 0 sin [ ( k p + 3 ) q π p ] ( k p + 3 ) 2 m + \sum_{k=0}^{\infty}\frac{\sin\left[(kp+1)\frac{q\pi}{p}\right]}{(kp+1)^{2m}}+% \sum_{k=0}^{\infty}\frac{\sin\left[(kp+2)\frac{q\pi}{p}\right]}{(kp+2)^{2m}}+% \sum_{k=0}^{\infty}\frac{\sin\left[(kp+3)\frac{q\pi}{p}\right]}{(kp+3)^{2m}}+% \,\cdots\,
  110. + k = 0 sin [ ( k p + p - 2 ) q π p ] ( k p + p - 2 ) 2 m + k = 0 sin [ ( k p + p - 1 ) q π p ] ( k p + p - 1 ) 2 m + k = 0 sin [ ( k p + p ) q π p ] ( k p + p ) 2 m +\sum_{k=0}^{\infty}\frac{\sin\left[(kp+p-2)\frac{q\pi}{p}\right]}{(kp+p-2)^{2% m}}+\sum_{k=0}^{\infty}\frac{\sin\left[(kp+p-1)\frac{q\pi}{p}\right]}{(kp+p-1)% ^{2m}}+\sum_{k=0}^{\infty}\frac{\sin\left[(kp+p)\frac{q\pi}{p}\right]}{(kp+p)^% {2m}}
  111. Cl 2 m ( q π p ) = j = 1 p { k = 0 sin [ ( k p + j ) q π p ] ( k p + j ) 2 m } = \,\text{Cl}_{2m}\left(\frac{q\pi}{p}\right)=\sum_{j=1}^{p}\Bigg\{\sum_{k=0}^{% \infty}\frac{\sin\left[(kp+j)\frac{q\pi}{p}\right]}{(kp+j)^{2m}}\Bigg\}=
  112. j = 1 p 1 p 2 m { k = 0 sin [ ( k p + j ) q π p ] ( k + ( j / p ) ) 2 m } \sum_{j=1}^{p}\frac{1}{p^{2m}}\Bigg\{\sum_{k=0}^{\infty}\frac{\sin\left[(kp+j)% \frac{q\pi}{p}\right]}{(k+(j/p))^{2m}}\Bigg\}
  113. sin ( x + y ) = sin x cos y + cos x sin y , \,\sin(x+y)=\sin x\cos y+\cos x\sin y,\,
  114. sin [ ( k p + j ) q π p ] = sin ( k q π + q j π p ) = sin k q π cos q j π p + cos k q π sin q j π p \sin\left[(kp+j)\frac{q\pi}{p}\right]=\sin\left(kq\pi+\frac{qj\pi}{p}\right)=% \sin kq\pi\cos\frac{qj\pi}{p}+\cos kq\pi\sin\frac{qj\pi}{p}
  115. sin m π 0 , cos m π ( - 1 ) m m = 0 , ± 1 , ± 2 , ± 3 , \sin m\pi\equiv 0,\quad\,\cos m\pi\equiv(-1)^{m}\quad\Leftrightarrow m=0,\,\pm 1% ,\,\pm 2,\,\pm 3,\,\cdots
  116. sin [ ( k p + j ) q π p ] = ( - 1 ) k q sin q j π p \sin\left[(kp+j)\frac{q\pi}{p}\right]=(-1)^{kq}\sin\frac{qj\pi}{p}
  117. Cl 2 m ( q π p ) = j = 1 p 1 p 2 m sin ( q j π p ) { k = 0 ( - 1 ) k q ( k + ( j / p ) ) 2 m } \,\text{Cl}_{2m}\left(\frac{q\pi}{p}\right)=\sum_{j=1}^{p}\frac{1}{p^{2m}}\sin% \left(\frac{qj\pi}{p}\right)\,\Bigg\{\sum_{k=0}^{\infty}\frac{(-1)^{kq}}{(k+(j% /p))^{2m}}\Bigg\}
  118. k = 0 ( - 1 ) k q ( k + ( j / p ) ) 2 m = k = 0 ( - 1 ) ( 2 k ) q ( ( 2 k ) + ( j / p ) ) 2 m + k = 0 ( - 1 ) ( 2 k + 1 ) q ( ( 2 k + 1 ) + ( j / p ) ) 2 m = \sum_{k=0}^{\infty}\frac{(-1)^{kq}}{(k+(j/p))^{2m}}=\sum_{k=0}^{\infty}\frac{(% -1)^{(2k)q}}{((2k)+(j/p))^{2m}}+\sum_{k=0}^{\infty}\frac{(-1)^{(2k+1)q}}{((2k+% 1)+(j/p))^{2m}}=
  119. k = 0 1 ( 2 k + ( j / p ) ) 2 m + ( - 1 ) q k = 0 1 ( 2 k + 1 + ( j / p ) ) 2 m = \sum_{k=0}^{\infty}\frac{1}{(2k+(j/p))^{2m}}+(-1)^{q}\,\sum_{k=0}^{\infty}% \frac{1}{(2k+1+(j/p))^{2m}}=
  120. 1 2 p [ k = 0 1 ( k + ( j / 2 p ) ) 2 m + ( - 1 ) q k = 0 1 ( k + ( j + p 2 p ) ) 2 m ] \frac{1}{2^{p}}\left[\sum_{k=0}^{\infty}\frac{1}{(k+(j/2p))^{2m}}+(-1)^{q}\,% \sum_{k=0}^{\infty}\frac{1}{(k+\left(\frac{j+p}{2p}\right))^{2m}}\right]
  121. m 1 \,m\in\mathbb{Z}\geq 1\,
  122. ψ m ( z ) = ( - 1 ) m + 1 m ! k = 0 1 ( k + z ) m + 1 \psi_{m}(z)=(-1)^{m+1}m!\,\sum_{k=0}^{\infty}\frac{1}{(k+z)^{m+1}}
  123. 1 2 2 m ( 2 m - 1 ) ! [ ψ 2 m - 1 ( j 2 p ) + ( - 1 ) q ψ 2 m - 1 ( j + p 2 p ) ] \frac{1}{2^{2m}(2m-1)!}\left[\psi_{2m-1}\left(\tfrac{j}{2p}\right)+(-1)^{q}% \psi_{2m-1}\left(\tfrac{j+p}{2p}\right)\right]
  124. Cl 2 m ( q π p ) = 1 ( 2 p ) 2 m ( 2 m - 1 ) ! j = 1 p sin ( q j π p ) [ ψ 2 m - 1 ( j 2 p ) + ( - 1 ) q ψ 2 m - 1 ( j + p 2 p ) ] \,\text{Cl}_{2m}\left(\frac{q\pi}{p}\right)=\frac{1}{(2p)^{2m}(2m-1)!}\,\sum_{% j=1}^{p}\sin\left(\tfrac{qj\pi}{p}\right)\,\left[\psi_{2m-1}\left(\tfrac{j}{2p% }\right)+(-1)^{q}\psi_{2m-1}\left(\tfrac{j+p}{2p}\right)\right]
  125. s n m ( θ ) = - 0 θ x m log n - m - 1 | 2 sin x 2 | d x \mathcal{L}s_{n}^{m}(\theta)=-\int_{0}^{\theta}x^{m}\log^{n-m-1}\Bigg|2\sin% \frac{x}{2}\Bigg|\,dx
  126. Cl 2 ( θ ) = s 2 0 ( θ ) \,\text{Cl}_{2}(\theta)=\mathcal{L}s_{2}^{0}(\theta)
  127. Li 2 ( e i θ ) = ζ ( 2 ) - θ ( 2 π - θ ) / 4 + i Cl 2 ( θ ) \operatorname{Li}_{2}(e^{i\theta})=\zeta(2)-\theta(2\pi-\theta)/4+i% \operatorname{Cl}_{2}(\theta)
  128. 0 θ 2 π 0\leq\theta\leq 2\pi
  129. Λ ( θ ) = - 0 θ log | 2 sin ( t ) | d t = Cl 2 ( 2 θ ) / 2 \Lambda(\theta)=-\int_{0}^{\theta}\log|2\sin(t)|\,dt=\operatorname{Cl}_{2}(2% \theta)/2
  130. 0 θ log | sec ( t ) | d t = Λ ( θ + π / 2 ) + θ log 2. \int_{0}^{\theta}\log|\sec(t)|\,dt=\Lambda(\theta+\pi/2)+\theta\log 2.
  131. θ / π \theta/\pi
  132. θ / π = p / q \theta/\pi=p/q
  133. sin ( n θ ) \sin(n\theta)
  134. Cl s ( θ ) \operatorname{Cl}_{s}(\theta)
  135. Cl 2 ( θ ) θ = 1 - log | θ | + n = 1 ζ ( 2 n ) n ( 2 n + 1 ) ( θ 2 π ) 2 n \frac{\operatorname{Cl}_{2}(\theta)}{\theta}=1-\log|\theta|+\sum_{n=1}^{\infty% }\frac{\zeta(2n)}{n(2n+1)}\left(\frac{\theta}{2\pi}\right)^{2n}
  136. | θ | < 2 π |\theta|<2\pi
  137. ζ ( s ) \zeta(s)
  138. Cl 2 ( θ ) θ = 3 - log [ | θ | ( 1 - θ 2 4 π 2 ) ] - 2 π θ log ( 2 π + θ 2 π - θ ) + n = 1 ζ ( 2 n ) - 1 n ( 2 n + 1 ) ( θ 2 π ) n . \frac{\operatorname{Cl}_{2}(\theta)}{\theta}=3-\log\left[|\theta|\left(1-\frac% {\theta^{2}}{4\pi^{2}}\right)\right]-\frac{2\pi}{\theta}\log\left(\frac{2\pi+% \theta}{2\pi-\theta}\right)+\sum_{n=1}^{\infty}\frac{\zeta(2n)-1}{n(2n+1)}% \left(\frac{\theta}{2\pi}\right)^{n}.
  139. ζ ( n ) - 1 \zeta(n)-1
  140. Cl 2 ( π 2 ) = G \operatorname{Cl}_{2}\left(\frac{\pi}{2}\right)=G
  141. Cl 2 ( π 3 ) = 3 π log ( G ( 2 3 ) G ( 1 3 ) ) - 3 π log Γ ( 1 3 ) + π log ( 2 π 3 ) \operatorname{Cl}_{2}\left(\frac{\pi}{3}\right)=3\pi\log\left(\frac{G\left(% \frac{2}{3}\right)}{G\left(\frac{1}{3}\right)}\right)-3\pi\log\Gamma\left(% \frac{1}{3}\right)+\pi\log\left(\frac{2\pi}{\sqrt{3}}\right)
  142. Cl 2 ( 2 π 3 ) = 2 π log ( G ( 2 3 ) G ( 1 3 ) ) - 2 π log Γ ( 1 3 ) + 2 π 3 log ( 2 π 3 ) \operatorname{Cl}_{2}\left(\frac{2\pi}{3}\right)=2\pi\log\left(\frac{G\left(% \frac{2}{3}\right)}{G\left(\frac{1}{3}\right)}\right)-2\pi\log\Gamma\left(% \frac{1}{3}\right)+\frac{2\pi}{3}\log\left(\frac{2\pi}{\sqrt{3}}\right)
  143. Cl 2 ( π 4 ) = 2 π log ( G ( 7 8 ) G ( 1 8 ) ) - 2 π log Γ ( 1 8 ) + π 4 log ( 2 π 2 - 2 ) \operatorname{Cl}_{2}\left(\frac{\pi}{4}\right)=2\pi\log\left(\frac{G\left(% \frac{7}{8}\right)}{G\left(\frac{1}{8}\right)}\right)-2\pi\log\Gamma\left(% \frac{1}{8}\right)+\frac{\pi}{4}\log\left(\frac{2\pi}{\sqrt{2-\sqrt{2}}}\right)
  144. Cl 2 ( 3 π 4 ) = 2 π log ( G ( 5 8 ) G ( 3 8 ) ) - 2 π log Γ ( 3 8 ) + 3 π 4 log ( 2 π 2 + 2 ) \operatorname{Cl}_{2}\left(\frac{3\pi}{4}\right)=2\pi\log\left(\frac{G\left(% \frac{5}{8}\right)}{G\left(\frac{3}{8}\right)}\right)-2\pi\log\Gamma\left(% \frac{3}{8}\right)+\frac{3\pi}{4}\log\left(\frac{2\pi}{\sqrt{2+\sqrt{2}}}\right)
  145. Cl 2 ( π 6 ) = 2 π log ( G ( 11 12 ) G ( 1 12 ) ) - 2 π log Γ ( 1 12 ) + π 6 log ( 2 π 2 3 - 1 ) \operatorname{Cl}_{2}\left(\frac{\pi}{6}\right)=2\pi\log\left(\frac{G\left(% \frac{11}{12}\right)}{G\left(\frac{1}{12}\right)}\right)-2\pi\log\Gamma\left(% \frac{1}{12}\right)+\frac{\pi}{6}\log\left(\frac{2\pi\sqrt{2}}{\sqrt{3}-1}\right)
  146. Cl 2 ( 5 π 6 ) = 2 π log ( G ( 7 12 ) G ( 5 12 ) ) - 2 π log Γ ( 5 12 ) + 5 π 6 log ( 2 π 2 3 + 1 ) \operatorname{Cl}_{2}\left(\frac{5\pi}{6}\right)=2\pi\log\left(\frac{G\left(% \frac{7}{12}\right)}{G\left(\frac{5}{12}\right)}\right)-2\pi\log\Gamma\left(% \frac{5}{12}\right)+\frac{5\pi}{6}\log\left(\frac{2\pi\sqrt{2}}{\sqrt{3}+1}\right)
  147. Cl 2 m ( 0 ) = Cl 2 m ( π ) = Cl 2 m ( 2 π ) = 0 \operatorname{Cl}_{2m}\left(0\right)=\operatorname{Cl}_{2m}\left(\pi\right)=% \operatorname{Cl}_{2m}\left(2\pi\right)=0
  148. Cl 2 m ( π 2 ) = β ( 2 m ) \operatorname{Cl}_{2m}\left(\frac{\pi}{2}\right)=\beta(2m)
  149. Cl 2 m + 1 ( 0 ) = Cl 2 m + 1 ( 2 π ) = ζ ( 2 m + 1 ) \operatorname{Cl}_{2m+1}\left(0\right)=\operatorname{Cl}_{2m+1}\left(2\pi% \right)=\zeta(2m+1)
  150. Cl 2 m + 1 ( π ) = - η ( 2 m + 1 ) = - ( 2 2 m - 1 2 2 m ) ζ ( 2 m + 1 ) \operatorname{Cl}_{2m+1}\left(\pi\right)=-\eta(2m+1)=-\left(\frac{2^{2m}-1}{2^% {2m}}\right)\zeta(2m+1)
  151. Cl 2 m + 1 ( π 2 ) = - 1 2 2 m + 1 η ( 2 m + 1 ) = - ( 2 2 m - 1 2 4 m + 1 ) ζ ( 2 m + 1 ) \operatorname{Cl}_{2m+1}\left(\frac{\pi}{2}\right)=-\frac{1}{2^{2m+1}}\eta(2m+% 1)=-\left(\frac{2^{2m}-1}{2^{4m+1}}\right)\zeta(2m+1)
  152. G = β ( 2 ) G=\beta(2)
  153. β ( x ) \beta(x)
  154. η ( x ) \eta(x)
  155. ζ ( x ) \zeta(x)
  156. β ( x ) = k = 0 ( - 1 ) k ( 2 k + 1 ) x \beta(x)=\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(2k+1)^{x}}
  157. 0 θ Cl 2 m ( x ) d x = ζ ( 2 m + 1 ) - Cl 2 m + 1 ( θ ) \int_{0}^{\theta}\operatorname{Cl}_{2m}(x)\,dx=\zeta(2m+1)-\operatorname{Cl}_{% 2m+1}(\theta)
  158. 0 θ Cl 2 m + 1 ( x ) d x = Cl 2 m + 2 ( θ ) \int_{0}^{\theta}\operatorname{Cl}_{2m+1}(x)\,dx=\operatorname{Cl}_{2m+2}(\theta)
  159. 0 θ Sl 2 m ( x ) d x = Sl 2 m + 1 ( θ ) \int_{0}^{\theta}\operatorname{Sl}_{2m}(x)\,dx=\operatorname{Sl}_{2m+1}(\theta)
  160. 0 θ Sl 2 m + 1 ( x ) d x = ζ ( 2 m + 2 ) - Cl 2 m + 2 ( θ ) \int_{0}^{\theta}\operatorname{Sl}_{2m+1}(x)\,dx=\zeta(2m+2)-\operatorname{Cl}% _{2m+2}(\theta)
  161. G \,G\,
  162. log 2 \,\log 2\,
  163. ζ ( 2 ) \,\zeta(2)\,
  164. ζ ( 3 ) \,\zeta(3)\,
  165. 0 θ log ( sin x ) d x = - 1 2 Cl 2 ( 2 θ ) - θ log 2 \int_{0}^{\theta}\log(\sin x)\,dx=-\tfrac{1}{2}\,\text{Cl}_{2}(2\theta)-\theta\log 2
  166. 0 θ log ( cos x ) d x = 1 2 Cl 2 ( π - 2 θ ) - θ log 2 \int_{0}^{\theta}\log(\cos x)\,dx=\tfrac{1}{2}\,\text{Cl}_{2}(\pi-2\theta)-% \theta\log 2
  167. 0 θ log ( tan x ) d x = - 1 2 Cl 2 ( 2 θ ) - 1 2 Cl 2 ( π - 2 θ ) \int_{0}^{\theta}\log(\tan x)\,dx=-\tfrac{1}{2}\,\text{Cl}_{2}(2\theta)-\tfrac% {1}{2}\,\text{Cl}_{2}(\pi-2\theta)
  168. 0 θ log ( 1 + cos x ) d x = 2 Cl 2 ( π - θ ) - θ log 2 \int_{0}^{\theta}\log(1+\cos x)\,dx=2\,\text{Cl}_{2}(\pi-\theta)-\theta\log 2
  169. 0 θ log ( 1 - cos x ) d x = - 2 Cl 2 ( θ ) - θ log 2 \int_{0}^{\theta}\log(1-\cos x)\,dx=-2\,\text{Cl}_{2}(\theta)-\theta\log 2
  170. 0 θ log ( 1 + sin x ) d x = 2 G - 2 Cl 2 ( π 2 + θ ) - θ log 2 \int_{0}^{\theta}\log(1+\sin x)\,dx=2G-2\,\text{Cl}_{2}\left(\frac{\pi}{2}+% \theta\right)-\theta\log 2
  171. 0 θ log ( 1 - sin x ) d x = - 2 G + 2 Cl 2 ( π 2 - θ ) - θ log 2 \int_{0}^{\theta}\log(1-\sin x)\,dx=-2G+2\,\text{Cl}_{2}\left(\frac{\pi}{2}-% \theta\right)-\theta\log 2

Clique_(graph_theory).html

  1. n n
  2. n 2 n 2 \scriptstyle\lfloor\frac{n}{2}\rfloor\cdot\lceil\frac{n}{2}\rceil

Closed-form_expression.html

  1. a x 2 + b x + c = 0 , ax^{2}+bx+c=0,\,
  2. x = - b ± b 2 - 4 a c 2 a x={-b\pm\sqrt{b^{2}-4ac}\over 2a}
  3. n n
  4. f ( x ) = i = 0 x 2 i f(x)=\sum_{i=0}^{\infty}{x\over 2^{i}}
  5. f ( x ) = 2 x f(x)=2x
  6. e - x 2 e^{-x^{2}}
  7. erf ( x ) = 2 π 0 x e - t 2 d t . \operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-t^{2}}\,\mathrm{d}t.

Closed_and_exact_differential_forms.html

  1. 𝐑 2 { 0 } , \mathbf{R}^{2}\setminus\{0\},
  2. S 1 d θ = 2 π , \oint_{S^{1}}d\theta=2\pi,
  3. d θ = 1 x 2 + y 2 ( - y d x + x d y ) , d\theta=\frac{1}{x^{2}+y^{2}}\left(-y\,dx+x\,dy\right),
  4. H d R 1 ( 𝐑 2 { 0 } ) 𝐑 , H^{1}_{dR}(\mathbf{R}^{2}\setminus\{0\})\cong\mathbf{R},
  5. ω \omega
  6. d f df
  7. d θ : d\theta:
  8. ω = d f + k d θ , \omega=df+k\cdot d\theta,
  9. k = 1 2 π S 1 ω \textstyle{k=\frac{1}{2\pi}\oint_{S^{1}}\omega}
  10. α = f ( x , y ) d x + g ( x , y ) d y \alpha=f(x,y)\,dx+g(x,y)\,dy
  11. d α = ( g x - f y ) d x d y d\alpha=(g_{x}-f_{y})\,dx\wedge dy\,
  12. α \alpha
  13. f y = g x . f_{y}=g_{x}.\,
  14. d h = h x d x + h y d y . dh=h_{x}\,dx+h_{y}\,dy.\,
  15. K d + d K = j 1 * - j 0 * . Kd+dK=j_{1}^{*}-j_{0}^{*}.
  16. a ( x , t ) d x p + 1 0 , a ( x , t ) d t d x p ( 0 1 a ( x , t ) d t ) d x p , a(x,t)\,dx^{p+1}\mapsto 0,\;a(x,t)\,dt\,dx^{p}\mapsto\left(\int_{0}^{1}a(x,t)% \,dt\right)dx^{p},
  17. F j 1 = id , F j 0 = Q . F\circ j_{1}=\mathrm{id},\;F\circ j_{0}=Q.
  18. j 1 * F * = id , j 0 * F * = 0. j_{1}^{*}\circ F^{*}=\mathrm{id},\;j_{0}^{*}\circ F^{*}=0.
  19. ζ - η = d β \zeta-\eta=d\beta\,
  20. B ( 𝐫 ) \vec{B}(\mathbf{r})
  21. A ( 𝐫 ) \vec{A}(\mathbf{r})
  22. 3 . \mathbb{R}^{3}\,.
  23. j . \vec{j}\,.
  24. 𝐈 := j 1 ( x , y , z ) d x 2 d x 3 + j 2 ( x , y , z ) d x 3 d x 1 + j 3 ( x , y , z ) d x 1 d x 2 . \mathbf{I}:=j_{1}(x,y,z)\,{\rm d}x_{2}\wedge{\rm d}x_{3}+j_{2}(x,y,z)\,{\rm d}% x_{3}\wedge{\rm d}x_{1}+j_{3}(x,y,z)\,{\rm d}x_{1}\wedge{\rm d}x_{2}.
  25. B \vec{B}
  26. Φ B := B 1 d x 2 d x 3 + , \Phi_{B}:=B_{1}{\rm d}x_{2}\wedge{\rm d}x_{3}+\cdots,
  27. A \vec{A}
  28. 𝐀 \mathbf{A}
  29. B = curl A = { A 3 x 2 - A 2 x 3 , A 1 x 3 - A 3 x 1 , A 2 x 1 - A 1 x 2 } , or Φ B = d 𝐀 . \vec{B}={\rm curl\,\,}\vec{A}=\left\{\frac{\partial A_{3}}{\partial x_{2}}-% \frac{\partial A_{2}}{\partial x_{3}},\frac{\partial A_{1}}{\partial x_{3}}-% \frac{\partial A_{3}}{\partial x_{1}},\frac{\partial A_{2}}{\partial x_{1}}-% \frac{\partial A_{1}}{\partial x_{2}}\right\},\,\text{ or }\Phi_{B}={\rm d}% \mathbf{A}.
  30. A \vec{A}
  31. 𝐀 := A 1 d x 1 + A 2 d x 2 + A 3 d x 3 . \mathbf{A}:=A_{1}\,{\rm d}x_{1}+A_{2}\,{\rm d}x_{2}+A_{3}\,{\rm d}x_{3}.
  32. div B 0 , {\rm div\,\,}\vec{B}\equiv 0,
  33. div A = ! 0 {\rm div\,\,}\vec{A}\stackrel{!}{=}0
  34. A i ( r ) = μ 0 j i ( r ) d x 1 d x 2 d x 3 4 π | r - r | . A_{i}(\vec{r})=\int\frac{\mu_{0}j_{i}(\vec{r}^{\,{}^{\prime}})\,\,dx_{1}^{% \prime}dx_{2}^{\prime}dx_{3}^{\prime}}{4\pi|\vec{r}-\vec{r}^{\,{}^{\prime}}|}\,.
  35. μ 0 \mu_{0}
  36. E \vec{E}
  37. ϕ ( x 1 , x 2 , x 3 ) \,\phi(x_{1},x_{2},x_{3})
  38. ρ ( x 1 , x 2 , x 3 ) \rho(x_{1},x_{2},x_{3})
  39. E \vec{E}
  40. B , \vec{B},
  41. ρ \rho
  42. j , \vec{j},
  43. ϕ \,\phi
  44. A \vec{A}
  45. A i , A_{i}\,,
  46. j i , j_{i}^{\prime}\,,
  47. t := t - | r - r | c , t^{\prime}:=t-\frac{|\vec{r}-\vec{r}^{\,{}^{\prime}}|}{c}\,,

Closed_form.html

  1. α \alpha
  2. d α = 0 d\alpha=0

Closed_graph_theorem.html

  1. { ( x , y ) X × Y T x = y } . \{(x,y)\in X\times Y\mid Tx=y\}.
  2. X × Y X\times Y
  3. f : X Y f:X\to Y
  4. u : X Y u:X\to Y
  5. X × Y X\times Y
  6. K σ δ K_{\sigma\delta}
  7. K σ δ K_{\sigma\delta}
  8. K σ δ K_{\sigma\delta}
  9. u : X Y u:X\to Y
  10. X × Y X\times Y

Closed_system.html

  1. j = 1 m a i j N j = b i \sum_{j=1}^{m}a_{ij}N_{j}=b_{i}
  2. N j N_{j}
  3. a i j a_{ij}

Closure_operator.html

  1. cl : 𝒫 ( S ) 𝒫 ( S ) \operatorname{cl}:\mathcal{P}(S)\rightarrow\mathcal{P}(S)
  2. X , Y S X,Y\subseteq S
  3. X cl ( X ) X\subseteq\operatorname{cl}(X)
  4. X Y cl ( X ) cl ( Y ) X\subseteq Y\Rightarrow\operatorname{cl}(X)\subseteq\operatorname{cl}(Y)
  5. cl ( cl ( X ) ) = cl ( X ) \operatorname{cl}(\operatorname{cl}(X))=\operatorname{cl}(X)
  6. cl ( X 1 X n ) = cl ( X 1 ) cl ( X n ) \operatorname{cl}(X_{1}\cup\dots\cup X_{n})=\operatorname{cl}(X_{1})\cup\dots% \cup\operatorname{cl}(X_{n})
  7. n 𝒩 n\in\mathcal{N}
  8. n = 0 n=0
  9. cl ( ) = \operatorname{cl}(\varnothing)=\varnothing
  10. cl ( X ) = { cl ( Y ) : Y X and Y finite } . \operatorname{cl}(X)=\bigcup\left\{\operatorname{cl}(Y):Y\subseteq X\,\text{ % and }Y\,\text{ finite}\right\}.
  11. \cup

Cobb–Douglas_production_function.html

  1. β = 1 - α \beta=1-\alpha
  2. Y = A L β K α Y=AL^{\beta}K^{\alpha}
  3. α α
  4. β β
  5. α = 0.45 α=0.45
  6. 1 % 1\%
  7. 0.45 % 0.45\%
  8. α + β = 1 α+β=1
  9. α + β > 1 α+β>1
  10. α + β = 1 α+β=1
  11. α α
  12. β β
  13. α = β = 1 α=β=1
  14. u ( x 1 , x 2 ) = x 1 α x 2 β ; u(x_{1},x_{2})=x_{1}^{\alpha}x_{2}^{\beta};
  15. u ~ ( x ) = i = 1 L x i λ i , x = ( x 1 , , x L ) . \tilde{u}(x)=\prod_{i=1}^{L}x_{i}^{\lambda_{i}},\qquad x=(x_{1},\cdots,x_{L}).
  16. x x 1 λ x\mapsto x^{\frac{1}{\lambda}}
  17. x > 0 x>0
  18. u ( x ) = u ~ ( x ) 1 λ u(x)=\tilde{u}(x)^{\frac{1}{\lambda}}
  19. u ( x ) = i = 1 L x i α i , i = 1 L α i = 1. u(x)=\prod_{i=1}^{L}x_{i}^{\alpha_{i}},\qquad\sum_{i=1}^{L}\alpha_{i}=1.
  20. ln u ( x ) = i = 1 L α i ln x i \ln u(x)=\sum_{i=1}^{L}{\alpha_{i}}\ln x_{i}
  21. max x i = 1 L α i ln x i s.t. i = 1 L p i x i = w \max_{x}\sum_{i=1}^{L}{\alpha_{i}}\ln x_{i}\quad\,\text{ s.t. }\quad\sum_{i=1}% ^{L}p_{i}x_{i}=w
  22. j : x j = w α j p j . \forall j:\qquad x_{j}^{\star}=\frac{w\alpha_{j}}{p_{j}}.
  23. j j
  24. ln ( Y ) = a 0 + i a i ln ( I i ) \ln(Y)=a_{0}+\sum_{i}a_{i}\ln(I_{i})
  25. Y = Output Y=\,\text{Output}
  26. I i = Inputs I_{i}=\,\text{Inputs}
  27. a i = Model coefficients a_{i}=\,\text{Model coefficients}
  28. Y = ( I 1 ) a 1 * ( I 2 ) a 2 Y=(I_{1})^{a_{1}}*(I_{2})^{a_{2}}\cdots
  29. Y = K α L β Y=K^{\alpha}L^{\beta}
  30. ln ( Y ) \displaystyle\ln(Y)
  31. Y = A ( α K γ + ( 1 - α ) L γ ) 1 γ , Y=A\left(\alpha K^{\gamma}+(1-\alpha)L^{\gamma}\right)^{\frac{1}{\gamma}},
  32. γ = 0 γ=0
  33. Y = A K α L 1 - α , Y=AK^{\alpha}L^{1-\alpha},
  34. ln ( Y ) = ln ( A ) + 1 γ ln ( α K γ + ( 1 - α ) L γ ) \ln(Y)=\ln(A)+\frac{1}{\gamma}\ln\left(\alpha K^{\gamma}+(1-\alpha)L^{\gamma}\right)
  35. lim γ 0 ln ( Y ) = ln ( A ) + α ln ( K ) + ( 1 - α ) ln ( L ) . \lim_{\gamma\to 0}\ln(Y)=\ln(A)+\alpha\ln(K)+(1-\alpha)\ln(L).
  36. Y = A K α L 1 - α Y=AK^{\alpha}L^{1-\alpha}

Cobordism.html

  1. W = M N \partial W=M\sqcup N
  2. { ( x 1 , x 2 , , x n ) 𝐑 n x n 0 } . \{(x_{1},x_{2},\ldots,x_{n})\in\mathbf{R}^{n}\mid x_{n}\geq 0\}.
  3. W = i ( M ) j ( N ) . \partial W=i(M)\sqcup j(N)~{}.
  4. M M M\sqcup M^{\prime}
  5. M M M\sqcup M^{\prime}
  6. M M M\sqcup M^{\prime}
  7. Ω * G \Omega^{G}_{*}
  8. Ω * G \Omega^{G}_{*}
  9. Ω * P L ( X ) , Ω * T O P ( X ) \Omega_{*}^{PL}(X),\Omega_{*}^{TOP}(X)
  10. N := ( M - int im φ ) φ | 𝐒 p × 𝐒 q - 1 ( 𝐃 p + 1 × 𝐒 q - 1 ) N:=(M-\operatorname{int~{}im}\varphi)\cup_{\varphi|_{\mathbf{S}^{p}\times% \mathbf{S}^{q-1}}}(\mathbf{D}^{p+1}\times\mathbf{S}^{q-1})
  11. W := ( M × I ) 𝐒 p × 𝐃 q × { 1 } ( 𝐃 p + 1 × 𝐃 q ) W:=(M\times I)\cup_{\mathbf{S}^{p}\times\mathbf{D}^{q}\times\{1\}}(\mathbf{D}^% {p+1}\times\mathbf{D}^{q})
  12. 𝔑 n \mathfrak{N}_{n}
  13. Ω n O \Omega_{n}^{\,\text{O}}
  14. [ M ] + [ N ] = [ M N ] [M]+[N]=[M\sqcup N]
  15. 𝔑 n \mathfrak{N}_{n}
  16. [ ] [\emptyset]
  17. [ M ] + [ M ] = [ ] [M]+[M]=[\emptyset]
  18. M M = ( M × [ 0 , 1 ] ) M\sqcup M=\partial(M\times[0,1])
  19. 𝔑 n \mathfrak{N}_{n}
  20. [ M ] [ N ] = [ M × N ] [M][N]=[M\times N]
  21. 𝔑 * = n 0 𝔑 n \mathfrak{N}_{*}=\sum\limits_{n\geq 0}\mathfrak{N}_{n}
  22. [ M ] 𝔑 n [M]\in\mathfrak{N}_{n}
  23. [ M ] = 0 𝔑 n [M]=0\in\mathfrak{N}_{n}
  24. 𝔑 * = 𝐙 2 [ x i i 1 , i 2 j - 1 ] \mathfrak{N}_{*}=\mathbf{Z}_{2}[x_{i}\mid i\geq 1,i\neq 2^{j}-1]
  25. [ M ] = [ N ] 𝔑 n [M]=[N]\in\mathfrak{N}_{n}
  26. ( i 1 , i 2 , , i k ) (i_{1},i_{2},\ldots,i_{k})
  27. i 1 , i 2 j - 1 i\geq 1,i\neq 2^{j}-1
  28. i 1 + i 2 + + i k = n i_{1}+i_{2}+\dots+i_{k}=n
  29. w i 1 ( M ) w i 2 ( M ) w i k ( M ) , [ M ] = w i 1 ( N ) w i 2 ( N ) w i k ( N ) , [ N ] 𝐙 2 \langle w_{i_{1}}(M)w_{i_{2}}(M)\cdots w_{i_{k}}(M),[M]\rangle=\langle w_{i_{1% }}(N)w_{i_{2}}(N)\cdots w_{i_{k}}(N),[N]\rangle\in\mathbf{Z}_{2}
  30. w i ( M ) H i ( M ; 𝐙 2 ) w_{i}(M)\in H^{i}(M;\mathbf{Z}_{2})
  31. [ M ] H n ( M ; 𝐙 2 ) [M]\in H_{n}(M;\mathbf{Z}_{2})
  32. 𝐙 2 \mathbf{Z}_{2}
  33. 𝔑 0 = 𝐙 2 , 𝔑 1 = 0 , 𝔑 2 = 𝐙 2 , 𝔑 3 = 0 , 𝔑 4 = 𝐙 2 𝐙 2 , 𝔑 5 = 𝐙 2 . \begin{aligned}\displaystyle\mathfrak{N}_{0}&\displaystyle=\mathbf{Z}_{2},\\ \displaystyle\mathfrak{N}_{1}&\displaystyle=0,\\ \displaystyle\mathfrak{N}_{2}&\displaystyle=\mathbf{Z}_{2},\\ \displaystyle\mathfrak{N}_{3}&\displaystyle=0,\\ \displaystyle\mathfrak{N}_{4}&\displaystyle=\mathbf{Z}_{2}\oplus\mathbf{Z}_{2}% ,\\ \displaystyle\mathfrak{N}_{5}&\displaystyle=\mathbf{Z}_{2}.\end{aligned}
  34. χ ( M ) 𝐙 2 \chi(M)\in\mathbf{Z}_{2}
  35. χ ( 𝐏 2 i 1 ( 𝐑 ) × × 𝐏 2 i k ( 𝐑 ) ) = 1. \chi\left(\mathbf{P}^{2i_{1}}(\mathbf{R})\times\dots\times\mathbf{P}^{2i_{k}}(% \mathbf{R})\right)=1.
  36. χ : 𝔑 2 i 𝐙 2 \chi:\mathfrak{N}_{2i}\to\mathbf{Z}_{2}
  37. ν ~ : M X k \tilde{\nu}:M\to X_{k}
  38. Ω * G \Omega^{G}_{*}
  39. M ( - N ) M\sqcup(-N)
  40. M ( - M ) M\sqcup(-M)
  41. Ω * SO . \Omega_{*}^{\,\text{SO}}.
  42. Ω * SO 𝐐 = 𝐐 [ y 4 i | i 1 ] , \Omega_{*}^{\,\text{SO}}\otimes\mathbf{Q}=\mathbf{Q}[y_{4i}|i\geq 1],
  43. y 4 i = [ 𝐏 2 i ( 𝐂 ) ] Ω 4 i SO y_{4i}=\left[\mathbf{P}^{2i}(\mathbf{C})\right]\in\Omega_{4i}^{\,\text{SO}}
  44. Ω * SO \Omega_{*}^{\,\text{SO}}
  45. Ω 0 SO \displaystyle\Omega_{0}^{\,\text{SO}}
  46. σ ( 𝐏 2 i 1 ( 𝐂 ) × × 𝐏 2 i k ( 𝐂 ) ) = 1. \sigma\left(\mathbf{P}^{2i_{1}}(\mathbf{C})\times\dots\times\mathbf{P}^{2i_{k}% }(\mathbf{C})\right)=1.
  47. σ : Ω 4 i SO 𝐙 \sigma:\Omega_{4i}^{\,\text{SO}}\to\mathbf{Z}
  48. Ω n G ( X ) \Omega^{G}_{n}(X)
  49. Ω G n ( X ) \Omega^{n}_{G}(X)
  50. Ω * G ( X ) \Omega_{*}^{G}(X)
  51. Ω G * ( X ) \Omega^{*}_{G}(X)
  52. Ω n G = Ω n G ( pt ) \Omega_{n}^{G}=\Omega_{n}^{G}(\,\text{pt})
  53. Ω n G ( X ) \Omega^{G}_{n}(X)
  54. { Ω n G ( X ) H n ( X ) ( M , f ) f * [ M ] \begin{cases}\Omega^{G}_{n}(X)\to H_{n}(X)\\ (M,f)\mapsto f_{*}[M]\end{cases}
  55. Ω G n ( X ) \Omega^{n}_{G}(X)
  56. Ω n G ( X ) = p + q = n H p ( X ; Ω q G ( pt ) ) . \Omega^{G}_{n}(X)=\sum\limits_{p+q=n}H_{p}(X;\Omega^{G}_{q}(\,\text{pt})).

Cochran's_theorem.html

  1. i = 1 n U i 2 = Q 1 + + Q k \sum_{i=1}^{n}U_{i}^{2}=Q_{1}+\cdots+Q_{k}
  2. r 1 + + r k = n r_{1}+\cdots+r_{k}=n
  3. Q i = j = 1 n k = 1 n U j B j , k ( i ) U k . Q_{i}=\sum_{j=1}^{n}\sum_{k=1}^{n}U_{j}B_{j,k}^{(i)}U_{k}.
  4. C ( i ) j i B ( j ) C^{(i)}\equiv\sum_{j\neq i}B^{(j)}
  5. j i r j = N - r i \sum_{j\neq i}r_{j}=N-r_{i}
  6. B ( i ) + C ( i ) = I N × N B^{(i)}+C^{(i)}=I_{N\times N}
  7. [ λ 1 0 0 0 λ 2 0 0 0 0 0 0 λ r i 0 0 0 0... 0 0 0 0 0 0... 0 ] . \begin{bmatrix}\lambda_{1}&0&...&...&...&0\\ 0&\lambda_{2}&0&...&...&0\\ 0&...&...&...&...&0\\ 0&...&0&\lambda_{r_{i}}&0&...\\ 0&...&&0&0...&0\\ 0&...&&0&...&...\\ 0&...&&0&0...&0\end{bmatrix}.
  8. ( N - r i ) (N-r_{i})
  9. C ( i ) = I - B ( i ) C^{(i)}=I-B^{(i)}
  10. ( N - r i ) × ( N - r i ) (N-r_{i})\times(N-r_{i})
  11. C ( 1 ) = B ( 2 ) + j > 2 B ( j ) C^{(1)}=B^{(2)}+\sum_{j>2}B^{(j)}
  12. C ( 1 ) C^{(1)}
  13. ( N - r i ) × ( N - r i ) (N-r_{i})\times(N-r_{i})
  14. j > 2 B ( j ) \sum_{j>2}B^{(j)}
  15. S T B ( i ) S S^{\mathrm{T}}B^{(i)}S
  16. r 1 + + r i - 1 + 1 r_{1}+...+r_{i-1}+1
  17. r 1 + + r i r_{1}+...+r_{i}
  18. U i U_{i}^{\prime}
  19. U i U_{i}
  20. φ i ( t ) = \displaystyle\varphi_{i}(t)=
  21. φ ( t 1 , t 2 t k ) = \displaystyle\varphi(t_{1},t_{2}...t_{k})=
  22. U i = X i - μ σ U_{i}=\frac{X_{i}-\mu}{\sigma}
  23. i = 1 n U i 2 = i = 1 n ( X i - X ¯ σ ) 2 + n ( X ¯ - μ σ ) 2 \sum_{i=1}^{n}U_{i}^{2}=\sum_{i=1}^{n}\left(\frac{X_{i}-\overline{X}}{\sigma}% \right)^{2}+n\left(\frac{\overline{X}-\mu}{\sigma}\right)^{2}
  24. X ¯ \overline{X}
  25. σ 2 \sigma^{2}
  26. ( X i - μ ) 2 = ( X i - X ¯ + X ¯ - μ ) 2 \sum(X_{i}-\mu)^{2}=\sum(X_{i}-\overline{X}+\overline{X}-\mu)^{2}
  27. ( X i - μ ) 2 = ( X i - X ¯ ) 2 + ( X ¯ - μ ) 2 + 2 ( X i - X ¯ ) ( X ¯ - μ ) . \sum(X_{i}-\mu)^{2}=\sum(X_{i}-\overline{X})^{2}+\sum(\overline{X}-\mu)^{2}+2% \sum(X_{i}-\overline{X})(\overline{X}-\mu).
  28. ( X ¯ - X i ) = 0 , \sum(\overline{X}-X_{i})=0,
  29. ( X i - μ ) 2 = ( X i - X ¯ ) 2 + n ( X ¯ - μ ) 2 , \sum(X_{i}-\mu)^{2}=\sum(X_{i}-\overline{X})^{2}+n(\overline{X}-\mu)^{2},
  30. ( X i - μ σ ) 2 = ( X i - X ¯ σ ) 2 + n ( X ¯ - μ σ ) 2 = Q 1 + Q 2 . \sum\left(\frac{X_{i}-\mu}{\sigma}\right)^{2}=\sum\left(\frac{X_{i}-\overline{% X}}{\sigma}\right)^{2}+n\left(\frac{\overline{X}-\mu}{\sigma}\right)^{2}=Q_{1}% +Q_{2}.
  31. ( X i - X ¯ ) 2 σ 2 χ n - 1 2 . \sum\left(X_{i}-\overline{X}\right)^{2}\sim\sigma^{2}\chi^{2}_{n-1}.
  32. n ( X ¯ - μ ) 2 σ 2 χ 1 2 , n(\overline{X}-\mu)^{2}\sim\sigma^{2}\chi^{2}_{1},
  33. n ( X ¯ - μ ) 2 1 n - 1 ( X i - X ¯ ) 2 χ 1 2 1 n - 1 χ n - 1 2 F 1 , n - 1 \frac{n\left(\overline{X}-\mu\right)^{2}}{\frac{1}{n-1}\sum\left(X_{i}-% \overline{X}\right)^{2}}\sim\frac{\chi^{2}_{1}}{\frac{1}{n-1}\chi^{2}_{n-1}}% \sim F_{1,n-1}
  34. σ ^ 2 = 1 n ( X i - X ¯ ) 2 . \widehat{\sigma}^{2}=\frac{1}{n}\sum\left(X_{i}-\overline{X}\right)^{2}.
  35. n σ ^ 2 σ 2 χ n - 1 2 \frac{n\widehat{\sigma}^{2}}{\sigma^{2}}\sim\chi^{2}_{n-1}
  36. σ ^ 2 \widehat{\sigma}^{2}
  37. Y N n ( 0 , σ 2 I n ) Y\sim N_{n}(0,\sigma^{2}I_{n})
  38. I n I_{n}
  39. A 1 , , A k A_{1},\ldots,A_{k}
  40. i = 1 k A i = I n \sum_{i=1}^{k}A_{i}=I_{n}
  41. r i = R a n k ( A i ) r_{i}=Rank(A_{i})
  42. i = 1 k r i = n , \sum_{i=1}^{k}r_{i}=n,
  43. Y T A i Y σ 2 χ r i 2 Y^{T}A_{i}Y\sim\sigma^{2}\chi^{2}_{r_{i}}
  44. A i A_{i}
  45. Y T A i Y Y^{T}A_{i}Y
  46. Y T A j Y Y^{T}A_{j}Y
  47. i j . i\neq j.

Codex_Bezae.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}
  4. 𝔓 \mathfrak{P}
  5. 𝔓 \mathfrak{P}
  6. 𝔓 \mathfrak{P}
  7. 𝔓 \mathfrak{P}
  8. 𝔓 \mathfrak{P}
  9. 𝔓 \mathfrak{P}

Codimension.html

  1. codim ( W ) = dim ( V ) - dim ( W ) . \operatorname{codim}(W)=\dim(V)-\dim(W).
  2. dim ( W ) + codim ( W ) = dim ( V ) . \dim(W)+\operatorname{codim}(W)=\dim(V).
  3. codim ( N ) = dim ( M ) - dim ( N ) . \operatorname{codim}(N)=\dim(M)-\dim(N).
  4. codim ( W ) = dim ( V / W ) = dim coker ( W V ) = dim ( V ) - dim ( W ) , \operatorname{codim}(W)=\dim(V/W)=\dim\operatorname{coker}(W\to V)=\dim(V)-% \dim(W),

Coding_theory.html

  1. X : Ω 𝒳 X:\Omega\rightarrow\mathcal{X}
  2. x 𝒳 x\in\mathcal{X}
  3. [ X = x ] \mathbb{P}[X=x]
  4. Σ \Sigma
  5. C : 𝒳 Σ * C:\mathcal{X}\rightarrow\Sigma^{*}
  6. Σ + \Sigma^{+}
  7. C ( x ) C(x)
  8. x x
  9. l ( C ( x ) ) l(C(x))
  10. l ( C ) = x 𝒳 l ( C ( x ) ) [ X = x ] l(C)=\sum_{x\in\mathcal{X}}l(C(x))\mathbb{P}[X=x]
  11. C ( x 1 , , x k ) = C ( x 1 ) C ( x 2 ) C ( x k ) C(x_{1},...,x_{k})=C(x_{1})C(x_{2})...C(x_{k})
  12. C ( ϵ ) = ϵ C(\epsilon)=\epsilon
  13. C : 𝒳 Σ * C:\mathcal{X}\rightarrow\Sigma^{*}
  14. C : 𝒳 * Σ * C:\mathcal{X}^{*}\rightarrow\Sigma^{*}
  15. C : 𝒳 Σ * C:\mathcal{X}\rightarrow\Sigma^{*}
  16. C ( x 1 ) C(x_{1})
  17. C ( x 2 ) C(x_{2})

Coefficient_of_performance.html

  1. C O P = Q W COP=\frac{Q}{W}
  2. Q Q
  3. W W
  4. C O P h e a t i n g = | Q H | W = | Q C | + W W COP_{heating}=\frac{|Q_{H}|}{W}=\frac{|Q_{C}|+W}{W}
  5. C O P c o o l i n g = | Q C | W COP_{cooling}=\frac{|Q_{C}|}{W}
  6. Q C Q_{C}
  7. Q H Q_{H}
  8. Q h o t = Q c o l d + W Q_{hot}=Q_{cold}+W
  9. W = Q h o t - Q c o l d W=Q_{hot}-Q_{cold}
  10. Q h o t Q_{hot}
  11. Q c o l d Q_{cold}
  12. C O P h e a t i n g = Q h o t Q h o t - Q c o l d COP_{heating}=\frac{Q_{hot}}{Q_{hot}-Q_{cold}}
  13. Q h o t T h o t = Q c o l d T c o l d \frac{Q_{hot}}{T_{hot}}=\frac{Q_{cold}}{T_{cold}}
  14. Q c o l d = Q h o t T c o l d T h o t Q_{cold}=\frac{Q_{hot}T_{cold}}{T_{hot}}
  15. T h o t T_{hot}
  16. T c o l d T_{cold}
  17. C O P h e a t i n g = T h o t T h o t - T c o l d COP_{heating}=\frac{T_{hot}}{T_{hot}-T_{cold}}
  18. C O P c o o l i n g = Q c o l d Q h o t - Q c o l d = T c o l d T h o t - T c o l d COP_{cooling}=\frac{Q_{cold}}{Q_{hot}-Q_{cold}}=\frac{T_{cold}}{T_{hot}-T_{% cold}}
  19. C O P h e a t i n g COP_{heating}
  20. C O P c o o l i n g COP_{cooling}
  21. T h o t {T_{hot}}
  22. T c o l d {T_{cold}}
  23. T h o t T_{hot}
  24. T c o l d T_{cold}
  25. C O P h e a t i n g COP_{heating}
  26. C O P h e a t i n g COP_{heating}
  27. C O P c o o l i n g COP_{cooling}

Coefficients_of_potential.html

  1. ϕ 1 = p 11 Q 1 + + p 1 n Q n ϕ 2 = p 21 Q 1 + + p 2 n Q n ϕ n = p n 1 Q 1 + + p n n Q n . \begin{matrix}\phi_{1}=p_{11}Q_{1}+\cdots+p_{1n}Q_{n}\\ \phi_{2}=p_{21}Q_{1}+\cdots+p_{2n}Q_{n}\\ \vdots\\ \phi_{n}=p_{n1}Q_{1}+\cdots+p_{nn}Q_{n}\end{matrix}.
  2. p 21 p_{21}
  3. p i j = ϕ i Q j = ( ϕ i Q j ) Q 1 , , Q j - 1 , Q j + 1 , , Q n , p_{ij}={\partial\phi_{i}\over\partial Q_{j}}=\left({\partial\phi_{i}\over% \partial Q_{j}}\right)_{Q_{1},...,Q_{j-1},Q_{j+1},...,Q_{n}},
  4. p i j = 1 4 π ϵ 0 S j S j f j d a j R j i . p_{ij}=\frac{1}{4\pi\epsilon_{0}S_{j}}\int_{S_{j}}\frac{f_{j}da_{j}}{R_{ji}}.
  5. ϕ P = j = 1 n 1 4 π ϵ 0 S j σ j d a j R j \phi_{P}=\sum_{j=1}^{n}\frac{1}{4\pi\epsilon_{0}}\int_{S_{j}}\frac{\sigma_{j}% da_{j}}{R_{j}}
  6. ϕ i = j = 1 n 1 4 π ϵ 0 S j σ j d a j R j i (i=1, 2…, n) , \phi_{i}=\sum_{j=1}^{n}\frac{1}{4\pi\epsilon_{0}}\int_{S_{j}}\frac{\sigma_{j}% da_{j}}{R_{ji}}\mbox{ (i=1, 2..., n)}~{},
  7. σ j σ j = f j , \frac{\sigma_{j}}{\langle\sigma_{j}\rangle}=f_{j},
  8. σ j = σ j f j = Q j S j f j . \sigma_{j}=\langle\sigma_{j}\rangle f_{j}=\frac{Q_{j}}{S_{j}}f_{j}.
  9. ϕ i = j = 1 n Q j 4 π ϵ 0 S j S j f j d a j R j i \phi_{i}=\sum_{j=1}^{n}\frac{Q_{j}}{4\pi\epsilon_{0}S_{j}}\int_{S_{j}}\frac{f_% {j}da_{j}}{R_{ji}}
  10. ϕ i = j = 1 n p i j Q j (i = 1, 2, …, n) , \phi_{i}=\sum_{j=1}^{n}p_{ij}Q_{j}\mbox{ (i = 1, 2, ..., n)}~{},
  11. p i j = 1 4 π ϵ 0 S j S j f j d a j R j i . p_{ij}=\frac{1}{4\pi\epsilon_{0}S_{j}}\int_{S_{j}}\frac{f_{j}da_{j}}{R_{ji}}.
  12. ϕ 1 = p 11 Q 1 + p 12 Q 2 ϕ 2 = p 21 Q 1 + p 22 Q 2 . \begin{matrix}\phi_{1}=p_{11}Q_{1}+p_{12}Q_{2}\\ \phi_{2}=p_{21}Q_{1}+p_{22}Q_{2}\end{matrix}.
  13. ϕ 1 = ( p 11 - p 12 ) Q ϕ 2 = ( p 21 - p 22 ) Q , \begin{matrix}\phi_{1}=(p_{11}-p_{12})Q\\ \phi_{2}=(p_{21}-p_{22})Q\end{matrix},
  14. Δ ϕ = ϕ 1 - ϕ 2 = ( p 11 + p 22 - p 12 - p 21 ) Q . \Delta\phi=\phi_{1}-\phi_{2}=(p_{11}+p_{22}-p_{12}-p_{21})Q.
  15. C = 1 p 11 + p 22 - 2 p 12 . C=\frac{1}{p_{11}+p_{22}-2p_{12}}.
  16. ϕ i = j = 1 n p i j Q j (i = 1,2,…n) \phi_{i}=\sum_{j=1}^{n}p_{ij}Q_{j}\mbox{ (i = 1,2,...n)}~{}
  17. Q i = j = 1 n c i j ϕ j (i = 1,2,…n) Q_{i}=\sum_{j=1}^{n}c_{ij}\phi_{j}\mbox{ (i = 1,2,...n)}~{}
  18. C = c 11 c 22 - c 12 2 c 11 + c 22 + 2 c 12 C=\frac{c_{11}c_{22}-c_{12}^{2}}{c_{11}+c_{22}+2c_{12}}

Coequalizer.html

  1. ~{}\sim
  2. x X x\in X
  3. f ( x ) g ( x ) f(x)\sim g(x)
  4. S = { f ( x ) g ( x ) - 1 | x X } S=\{f(x)g(x)^{-1}\ |\ x\in X\}
  5. S 1 S^{1}
  6. f , g : X Y f,g:X\to Y
  7. F : C D F:C\to D

Cofiniteness.html

  1. 𝒯 = { A X A = or X A is finite } \mathcal{T}=\{A\subseteq X\mid A=\varnothing\mbox{ or }~{}X\setminus A\mbox{ % is finite}~{}\}
  2. U A := x A G x U_{A}:=\bigcap_{x\in A}G_{x}
  3. X i \prod X_{i}
  4. U i \prod U_{i}
  5. U i X i U_{i}\subset X_{i}
  6. U i = X i U_{i}=X_{i}
  7. M i \bigoplus M_{i}
  8. α i M i \alpha_{i}\in M_{i}
  9. α i = 0 \alpha_{i}=0

Cohen–Macaulay_ring.html

  1. R G R^{G}
  2. R R
  3. K [ [ x , y ] ] / ( x 2 , x y ) K[[x,y]]/(x^{2},xy)
  4. K [ [ x , y , z ] ] / ( x y , x z ) K[[x,y,z]]/(xy,xz)
  5. x - z x-z
  6. K [ [ w , x , y , z ] ] / ( w y , w z , x y , x z ) K[[w,x,y,z]]/(wy,wz,xy,xz)
  7. w - x w-x

Coherence_(philosophical_gambling_strategy).html

  1. Price ( Red Sox ) + Price ( Yankees ) Price ( Red Sox or Yankees ) \,\text{Price}(\,\text{Red Sox})+\,\text{Price}(\,\text{Yankees})\neq\,\text{% Price}(\,\text{Red Sox or Yankees})\,
  2. Price ( complete game ) × Price ( Red Sox win complete game ) Price ( Red Sox win ) \,\text{Price}(\,\text{complete game})\times\,\text{Price}(\,\text{Red Sox win% }\mid\,\text{complete game})\neq\,\text{Price}(\,\text{Red Sox win})
  3. Price ( complete game ) × Price ( Red Sox win complete game ) - Price ( Red Sox win ) . \,\text{Price}(\,\text{complete game})\times\,\text{Price}(\,\text{Red Sox win% }\mid\,\text{complete game})-\,\text{Price}(\,\text{Red Sox win}).

Coherent_sheaf.html

  1. ( X , 𝒪 X ) (X,\mathcal{O}_{X})
  2. \mathcal{F}
  3. 𝒪 X \mathcal{O}_{X}
  4. \mathcal{F}
  5. 𝒪 X \mathcal{O}_{X}
  6. x X x\in X
  7. U X U\subset X
  8. | U \mathcal{F}|_{U}
  9. \mathcal{F}
  10. U U
  11. 𝒪 X n | U | U \mathcal{O}_{X}^{n}|_{U}\to\mathcal{F}|_{U}
  12. n n\in\mathbb{N}
  13. U X U\subset X
  14. n n\in\mathbb{N}
  15. φ : 𝒪 X n | U | U \varphi\colon\mathcal{O}_{X}^{n}|_{U}\to\mathcal{F}|_{U}
  16. 𝒪 X \mathcal{O}_{X}
  17. φ \varphi
  18. 𝒪 X \mathcal{O}_{X}
  19. x X x\in X
  20. U U
  21. | U \mathcal{F}|_{U}
  22. \mathcal{F}
  23. U U
  24. 𝒪 X n | U 𝒪 X m | U \mathcal{O}_{X}^{n}|_{U}\to\mathcal{O}_{X}^{m}|_{U}
  25. n n
  26. m m
  27. 𝒪 X \mathcal{O}_{X}
  28. 𝒪 X \mathcal{O}_{X}
  29. \mathcal{F}
  30. 𝒪 X \mathcal{O}_{X}
  31. U i U_{i}
  32. X X
  33. 𝒪 ( I i ) | U i 𝒪 ( J i ) | U i | U i 0 \mathcal{O}^{(I_{i})}|_{U_{i}}\to\mathcal{O}^{(J_{i})}|_{U_{i}}\to\mathcal{F}|% _{U_{i}}\to 0
  34. 𝒪 X \mathcal{O}_{X}
  35. U i = Spec A i U_{i}=\operatorname{Spec}A_{i}
  36. F | U i = M ~ i F|_{U_{i}}=\widetilde{M}_{i}
  37. O U i O_{U_{i}}
  38. M ~ i \widetilde{M}_{i}
  39. M i M_{i}
  40. M i M_{i}
  41. 𝒪 X \mathcal{O}_{X}
  42. 𝒪 X \mathcal{O}_{X}
  43. \mathcal{F}
  44. X X
  45. p X p\in X
  46. U U
  47. p p
  48. | U \mathcal{F}|_{U}
  49. 𝒪 X | U \mathcal{O}_{X}|_{U}
  50. p \mathcal{F}_{p}
  51. \mathcal{F}
  52. p p
  53. ( 𝒪 X ) p (\mathcal{O}_{X})_{p}
  54. p p
  55. \mathcal{F}
  56. p \mathcal{F}_{p}
  57. n n
  58. p X p\in X
  59. \mathcal{F}
  60. n . n.
  61. X = Spec ( R ) X=\operatorname{Spec}(R)
  62. 𝒪 X \mathcal{O}_{X}
  63. ( X , 𝒪 X ) (X,\mathcal{O}_{X})
  64. ( X , 𝒪 X ) (X,\mathcal{O}_{X})
  65. Γ ( X , 𝒪 X ) \Gamma(X,\mathcal{O}_{X})
  66. 𝒪 X \mathcal{O}_{X}

Cointerpretability.html

  1. Σ 1 \Sigma_{1}

Collaborative_filtering.html

  1. r u , i = aggr u U r u , i r_{u,i}=\operatorname{aggr}_{u^{\prime}\in U}r_{u^{\prime},i}
  2. r u , i = 1 N u U r u , i r_{u,i}=\frac{1}{N}\sum\limits_{u^{\prime}\in U}r_{u^{\prime},i}
  3. r u , i = k u U simil ( u , u ) r u , i r_{u,i}=k\sum\limits_{u^{\prime}\in U}\operatorname{simil}(u,u^{\prime})r_{u^{% \prime},i}
  4. r u , i = r u ¯ + k u U simil ( u , u ) ( r u , i - r u ¯ ) r_{u,i}=\bar{r_{u}}+k\sum\limits_{u^{\prime}\in U}\operatorname{simil}(u,u^{% \prime})(r_{u^{\prime},i}-\bar{r_{u^{\prime}}})
  5. k = 1 / u U | simil ( u , u ) | k=1/\sum_{u^{\prime}\in U}|\operatorname{simil}(u,u^{\prime})|
  6. r u ¯ \bar{r_{u}}
  7. simil ( x , y ) = i I x y ( r x , i - r x ¯ ) ( r y , i - r y ¯ ) i I x y ( r x , i - r x ¯ ) 2 i I x y ( r y , i - r y ¯ ) 2 \operatorname{simil}(x,y)=\frac{\sum\limits_{i\in I_{xy}}(r_{x,i}-\bar{r_{x}})% (r_{y,i}-\bar{r_{y}})}{\sqrt{\sum\limits_{i\in I_{xy}}(r_{x,i}-\bar{r_{x}})^{2% }\sum\limits_{i\in I_{xy}}(r_{y,i}-\bar{r_{y}})^{2}}}
  8. simil ( x , y ) = cos ( x , y ) = x y || x || × || y || = i I x y r x , i r y , i i I x r x , i 2 i I y r y , i 2 \operatorname{simil}(x,y)=\cos(\vec{x},\vec{y})=\frac{\vec{x}\cdot\vec{y}}{||% \vec{x}||\times||\vec{y}||}=\frac{\sum\limits_{i\in I_{xy}}r_{x,i}r_{y,i}}{% \sqrt{\sum\limits_{i\in I_{x}}r_{x,i}^{2}}\sqrt{\sum\limits_{i\in I_{y}}r_{y,i% }^{2}}}
  9. O ( M ) O(M)
  10. O ( N ) O(N)
  11. n n

Colligative_properties.html

  1. p = p A x A + p B x B + p=p^{\star}_{\rm A}x_{\rm A}+p^{\star}_{\rm B}x_{\rm B}+\cdots
  2. p i p^{\star}_{\rm i}
  3. x i x_{\rm i}
  4. p B = 0 p^{\star}_{\rm B}=0
  5. p = p A x A p=p^{\star}_{\rm A}x_{\rm A}
  6. Δ p = p A - p = p A ( 1 - x A ) = p A x B \Delta p=p^{\star}_{\rm A}-p=p^{\star}_{\rm A}(1-x_{\rm A})=p^{\star}_{\rm A}x% _{\rm B}
  7. i i
  8. Δ p = 3 p A x B \Delta p=3p^{\star}_{\rm A}x_{\rm B}
  9. T b T_{\rm b}
  10. Δ T b = T b ( s o l u t i o n ) - T b ( s o l v e n t ) = i K b m \Delta T_{\rm b}=T_{\rm b}(solution)-T_{\rm b}(solvent)=i\cdot K_{b}\cdot m
  11. μ A ( T b ) = μ A ( T b ) + R T ln x A = μ A ( g , 1 a t m ) \mu_{A}(T_{b})=\mu_{A}^{\star}(T_{b})+RT\ln x_{A}\ =\mu_{A}^{\star}(g,1atm)
  12. K b = R M T b 2 / Δ H vap K_{b}=RMT_{b}^{2}/\Delta H_{\mathrm{vap}}
  13. T f T_{\rm f}
  14. Δ T f = T f ( s o l u t i o n ) - T f ( s o l v e n t ) = - i K f m \Delta T_{\rm f}=T_{\rm f}(solution)-T_{\rm f}(solvent)=-i\cdot K_{f}\cdot m
  15. K f = R M T f 2 / Δ H fus K_{f}=RMT_{f}^{2}/\Delta H_{\mathrm{fus}}
  16. p V = n R T pV=nRT
  17. Π V = n R T i \Pi V=nRTi
  18. Π \Pi
  19. c = n / V c=n/V
  20. Π = n R T i V = c R T i \Pi=\frac{nRTi}{V}=cRTi

Colombeau_algebra.html

  1. 𝒟 ( ) \mathcal{D}^{\prime}(\mathbb{R})
  2. \mathbb{R}
  3. ( A ( ) , , + ) (A(\mathbb{R}),\circ,+)
  4. 𝒟 ( ) \mathcal{D}^{\prime}(\mathbb{R})
  5. A ( ) A(\mathbb{R})
  6. 1 1
  7. A ( ) A(\mathbb{R})
  8. \partial
  9. A ( ) A(\mathbb{R})
  10. \partial
  11. 𝒟 ( ) \mathcal{D}^{\prime}(\mathbb{R})
  12. \circ
  13. C ( ) × C ( ) C(\mathbb{R})\times C(\mathbb{R})
  14. C ( ) C(\mathbb{R})
  15. C k ( ) C^{k}(\mathbb{R})
  16. k k
  17. C ( ) × C ( ) C(\mathbb{R})\times C(\mathbb{R})
  18. C ( ) × C ( ) C^{\infty}(\mathbb{R})\times C^{\infty}(\mathbb{R})
  19. C M ( n ) / C N ( n ) . C^{\infty}_{M}(\mathbb{R}^{n})/C^{\infty}_{N}(\mathbb{R}^{n}).
  20. n \mathbb{R}^{n}
  21. C M ( n ) C^{\infty}_{M}(\mathbb{R}^{n})
  22. n \mathbb{R}^{n}
  23. f : + C ( n ) {f:}\mathbb{R}_{+}\to C^{\infty}(\mathbb{R}^{n})
  24. n \mathbb{R}^{n}
  25. sup x K | | α | ( x 1 ) α 1 ( x n ) α n f ε ( x ) | = O ( ε - N ) ( ε 0 ) . \sup_{x\in K}\left|\frac{\partial^{|\alpha|}}{(\partial x_{1})^{\alpha_{1}}% \cdots(\partial x_{n})^{\alpha_{n}}}f_{\varepsilon}(x)\right|=O(\varepsilon^{-% N})\qquad(\varepsilon\to 0).
  26. C N ( n ) C^{\infty}_{N}(\mathbb{R}^{n})
  27. ϕ ε δ \phi_{\varepsilon}\to\delta

Color_histogram.html

  1. f L 1 ( n ) f\in L^{1}(\mathbb{R}^{n})
  2. H H
  3. H ( f ) ( y ) = μ { x : f ( x ) y } H(f)(y)=\mu\{x:f(x)\leq y\}
  4. μ \mu
  5. H ( f ) H(f)
  6. h ( f ) = H ( f ) h(f)=H(f)^{\prime}

Color_theory.html

  1. Color harmony = f ( Col 1 , 2 , 3 , , n ) ( I D + C E + C X + P + T ) \,\text{Color harmony}=f(\,\text{Col}1,2,3,\dots,n)\cdot(ID+CE+CX+P+T)

Commitment_scheme.html

  1. 2 k 2^{k}
  2. C o m m i t k Commit_{k}
  3. x , x x,x^{\prime}
  4. o p e n , o p e n open,open^{\prime}
  5. x x x\neq x^{\prime}
  6. C o m m i t k ( x , o p e n ) = C o m m i t k ( x , o p e n ) Commit_{k}(x,open)=Commit_{k}(x^{\prime},open^{\prime})
  7. ( t , ϵ ) (t,\epsilon)
  8. x , x , o p e n , o p e n x,x^{\prime},open,open^{\prime}
  9. x x x\neq x^{\prime}
  10. C o m m i t ( x , o p e n ) = C o m m i t ( x , o p e n ) Commit(x,open)=Commit(x^{\prime},open^{\prime})
  11. ϵ \epsilon
  12. U k U_{k}
  13. 2 k 2^{k}
  14. x x x\neq x^{\prime}
  15. { C o m m i t k ( x , U k ) } k 𝒩 \{Commit_{k}(x,U_{k})\}_{k\in\mathcal{N}}
  16. { C o m m i t k ( x , U k ) } k 𝒩 \{Commit_{k}(x^{\prime},U_{k})\}_{k\in\mathcal{N}}
  17. ( h , f ( x ) , b h ( x ) ) (h,f(x),b\oplus h(x))
  18. \oplus
  19. G ( Y ) R G(Y)\oplus R
  20. G ( Y ) = G ( Y ) R G(Y^{\prime})=G(Y)\oplus R
  21. c = g x h r c=g^{x}h^{r}

Common_collector.html

  1. A v = v out v in 1 {A_{\mathrm{v}}}={v_{\mathrm{out}}\over v_{\mathrm{in}}}\approx 1
  2. r in β 0 R E r_{\mathrm{in}}\approx\beta_{0}R_{\mathrm{E}}
  3. r out R E R source β 0 r_{\mathrm{out}}\approx{R_{\mathrm{E}}}\|{R_{\mathrm{source}}\over\beta_{0}}
  4. r out R source β 0 r_{\mathrm{out}}\approx{R_{\mathrm{source}}\over\beta_{0}}
  5. β = g m r π \beta=g_{m}r_{\pi}
  6. A i = i out i in {A_{\mathrm{i}}}={i_{\mathrm{out}}\over i_{\mathrm{in}}}
  7. β 0 + 1 \beta_{0}+1
  8. β 0 \approx\beta_{0}
  9. β 0 1 \beta_{0}\gg 1
  10. A v = v out v in {A_{\mathrm{v}}}={v_{\mathrm{out}}\over v_{\mathrm{in}}}
  11. g m R E g m R E + 1 {g_{m}R_{\mathrm{E}}\over g_{m}R_{\mathrm{E}}+1}
  12. 1 \approx 1
  13. g m R E 1 g_{m}R_{\mathrm{E}}\gg 1
  14. r in = v in i in r_{\mathrm{in}}=\frac{v_{\mathrm{in}}}{i_{\mathrm{in}}}
  15. r π + ( β 0 + 1 ) R E r_{\pi}+(\beta_{0}+1)R_{\mathrm{E}}
  16. β 0 R E \approx\beta_{0}R_{\mathrm{E}}
  17. ( g m R E 1 ) ( β 0 1 ) (g_{m}R_{\mathrm{E}}\gg 1)\wedge(\beta_{0}\gg 1)
  18. r out = v out i out r_{\mathrm{out}}=\frac{v_{\mathrm{out}}}{i_{\mathrm{out}}}
  19. R E ( r π + R source β 0 + 1 ) R_{\mathrm{E}}\parallel\left({r_{\pi}+R_{\mathrm{source}}\over\beta_{0}+1}\right)
  20. 1 g m + R source β 0 \approx{1\over g_{m}}+{R_{\mathrm{source}}\over\beta_{0}}
  21. ( β 0 1 ) ( r in R source ) (\beta_{0}\gg 1)\wedge(r_{\mathrm{in}}\gg R_{\mathrm{source}})
  22. R source R_{\mathrm{source}}
  23. R E = R L r O R_{\mathrm{E}}=R_{\mathrm{L}}\parallel r_{\mathrm{O}}
  24. ( β + 1 ) v in - v out R S + r π = v out ( 1 R L + 1 r O ) . (\beta+1)\frac{v_{\mathrm{in}}-v_{\mathrm{out}}}{R_{\mathrm{S}}+r_{\pi}}=v_{% \mathrm{out}}\left(\frac{1}{R_{\mathrm{L}}}+\frac{1}{r_{\mathrm{O}}}\right)\ .
  25. 1 R E = 1 R L + 1 r O \frac{1}{R_{\mathrm{E}}}=\frac{1}{R_{\mathrm{L}}}+\frac{1}{r_{\mathrm{O}}}
  26. R = R S + r π β + 1 . R=\frac{R_{\mathrm{S}}+r_{\pi}}{\beta+1}\ .
  27. A v = v out v in = 1 1 + R R E . A_{\mathrm{v}}=\frac{v_{\mathrm{out}}}{v_{\mathrm{in}}}=\frac{1}{1+\frac{R}{R_% {\mathrm{E}}}}\ .
  28. R E R_{\mathrm{E}}
  29. R in = v in i b = R S + r π 1 - A v R_{\mathrm{in}}=\frac{v_{\mathrm{in}}}{i_{\mathrm{b}}}=\frac{R_{\mathrm{S}}+r_% {\pi}}{1-A_{\mathrm{v}}}
  30. = ( R S + r π ) ( 1 + R E R ) =\left(R_{\mathrm{S}}+r_{\pi}\right)\left(1+\frac{R_{\mathrm{E}}}{R}\right)
  31. = R S + r π + ( β + 1 ) R E . =R_{\mathrm{S}}+r_{\pi}+(\beta+1)R_{\mathrm{E}}\ .
  32. r O r_{\mathrm{O}}
  33. R L R_{\mathrm{L}}
  34. R L R_{\mathrm{L}}
  35. R E R_{\mathrm{E}}
  36. R L R_{\mathrm{L}}
  37. β \beta
  38. R L R_{\mathrm{L}}
  39. R S R_{\mathrm{S}}
  40. R out = v x i x . R_{\mathrm{out}}=\frac{v_{\mathrm{x}}}{i_{\mathrm{x}}}\ .
  41. ( β + 1 ) i b = i x - v x R E , (\beta+1)i_{\mathrm{b}}=i_{\mathrm{x}}-\frac{v_{\mathrm{x}}}{R_{\mathrm{E}}}\ ,
  42. R E R_{\mathrm{E}}
  43. v x v_{\mathrm{x}}
  44. v x = i b ( R S + r π ) . v_{\mathrm{x}}=i_{\mathrm{b}}\left(R_{\mathrm{S}}+r_{\pi}\right)\ .
  45. R out = v x i x = R R E , R_{\mathrm{out}}=\frac{v_{\mathrm{x}}}{i_{\mathrm{x}}}=R\parallel R_{\mathrm{E% }}\ ,
  46. R R
  47. R R
  48. β \beta
  49. R R

Common_emitter.html

  1. R E R_{\,\text{E}}
  2. G m = g m G_{m}=g_{m}
  3. g m R E + 1 g_{m}R_{\,\text{E}}+1
  4. A v v out v in = - g m R C g m R E + 1 - R C R E ( where g m R E 1 ) . A_{\,\text{v}}\triangleq\frac{v_{\,\text{out}}}{v_{\,\text{in}}}=\frac{-g_{m}R% _{\,\text{C}}}{g_{m}R_{\,\text{E}}+1}\approx-\frac{R_{\,\text{C}}}{R_{\,\text{% E}}}\qquad(\,\text{where}\quad g_{m}R_{\,\text{E}}\gg 1).\,
  5. R C / R E R_{\,\text{C}}/R_{\,\text{E}}
  6. A i i out i in A_{\,\text{i}}\triangleq\frac{i_{\,\text{out}}}{i_{\,\text{in}}}\,
  7. β \beta\,
  8. β \beta
  9. A v v out v in A_{\,\text{v}}\triangleq\frac{v_{\,\text{out}}}{v_{\,\text{in}}}\,
  10. - β R C r π + ( β + 1 ) R E \begin{matrix}-\frac{\beta R_{\,\text{C}}}{r_{\pi}+(\beta+1)R_{\,\text{E}}}% \end{matrix}\,
  11. - g m R C \approx-g_{m}R_{\,\text{C}}
  12. r in v in i in r_{\,\text{in}}\triangleq\frac{v_{\,\text{in}}}{i_{\,\text{in}}}\,
  13. r π + ( β + 1 ) R E r_{\pi}+(\beta+1)R_{\,\text{E}}\,
  14. r π r_{\pi}
  15. r out v out i out r_{\,\text{out}}\triangleq\frac{v_{\,\text{out}}}{i_{\,\text{out}}}\,
  16. R C R_{\,\text{C}}\,
  17. R C R_{\,\text{C}}
  18. R E = 0 Ω R_{\,\text{E}}=0\,\Omega
  19. R E R_{\,\text{E}}\,
  20. A v A_{\,\text{v}}\,
  21. C CB C_{\,\text{CB}}\,
  22. C CB ( 1 - A v ) C_{\,\text{CB}}(1-A_{\,\text{v}})\,
  23. A v A_{\,\text{v}}\,
  24. r s ( 1 - A V ) C CB r_{\,\text{s}}(1-A_{\,\text{V}})C_{\,\text{CB}}\,
  25. r s r_{\,\text{s}}\,
  26. | A v | \left|A_{\,\text{v}}\right|\,
  27. r s r_{\,\text{s}}\,

Compact_operator.html

  1. U X U\subset X
  2. V Y V\subset Y
  3. T ( U ) V T(U)\subset V
  4. ( x n ) n (x_{n})_{n\in\mathbb{N}}
  5. ( T x n ) n (Tx_{n})_{n\in\mathbb{N}}
  6. i d X id_{X}
  7. B ( Y , Z ) K ( X , Y ) B ( W , X ) K ( W , Z ) . B(Y,Z)\circ K(X,Y)\circ B(W,X)\subseteq K(W,Z).
  8. i d X id_{X}
  9. i d X - T id_{X}-T
  10. im ( i d X - T ) \operatorname{im}\,(id_{X}-T)
  11. ( λ K + I ) u = f (\lambda K+I)u=f\,
  12. T T
  13. \mathcal{H}
  14. T : T:\mathcal{H}\to\mathcal{H}
  15. T = n = 1 λ n f n , g n , T=\sum_{n=1}^{\infty}\lambda_{n}\langle f_{n},\cdot\rangle g_{n}\,,
  16. f 1 , f 2 , f_{1},f_{2},\ldots
  17. g 1 , g 2 , g_{1},g_{2},\ldots
  18. λ 1 , λ 2 , \lambda_{1},\lambda_{2},\ldots
  19. λ N + k = 0 \lambda_{N+k}=0
  20. N 𝒩 , N\in\mathcal{N},
  21. k = 1 , 2 , k=1,2,\dots
  22. T = n = 1 N λ n f n , g n . T=\sum_{n=1}^{N}\lambda_{n}\langle f_{n},\cdot\rangle g_{n}\,.
  23. , \langle\cdot,\cdot\rangle
  24. ( x n ) (x_{n})
  25. ( T x n ) (Tx_{n})
  26. p \ell^{p}
  27. ( T f ) ( x ) = 0 x f ( t ) g ( t ) d t . (Tf)(x)=\int_{0}^{x}f(t)g(t)\,\mathrm{d}t.
  28. ( T f ) ( x ) = Ω k ( x , y ) f ( y ) d y (Tf)(x)=\int_{\Omega}k(x,y)f(y)\,\mathrm{d}y

Compactly_generated_group.html

  1. K = n ( K K - 1 ) n = G . \langle K\rangle=\bigcup_{n\in\mathbb{N}}(K\cup K^{-1})^{n}=G.
  2. G = n K n . G=\bigcup_{n\in\mathbb{N}}K^{n}.

Companion_matrix.html

  1. p ( t ) = c 0 + c 1 t + + c n - 1 t n - 1 + t n , p(t)=c_{0}+c_{1}t+\cdots+c_{n-1}t^{n-1}+t^{n}~{},
  2. C ( p ) = [ 0 0 0 - c 0 1 0 0 - c 1 0 1 0 - c 2 0 0 1 - c n - 1 ] . C(p)=\begin{bmatrix}0&0&\dots&0&-c_{0}\\ 1&0&\dots&0&-c_{1}\\ 0&1&\dots&0&-c_{2}\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&\dots&1&-c_{n-1}\end{bmatrix}.
  3. C v i = C i v 1 = v i + 1 Cv_{i}=C^{i}v_{1}=v_{i+1}
  4. V V
  5. K C C KCC
  6. C C
  7. C ( p ) C(p)
  8. p p
  9. C ( p ) C(p)
  10. p p
  11. A A
  12. K K
  13. A A
  14. K K
  15. A A
  16. A A
  17. n n
  18. 𝐯 \mathbf{v}
  19. V = K n V=K^{n}
  20. A A
  21. K [ A ] K[A]
  22. V = K [ A ] / ( p ( A ) ) V=K[A]/(p(A))
  23. A A
  24. A A
  25. A A
  26. p ( t ) p(t)
  27. V C ( p ) V - 1 = diag ( λ 1 , , λ n ) VC(p)V^{-1}=\operatorname{diag}(\lambda_{1},\dots,\lambda_{n})
  28. V V
  29. λ λ
  30. C C
  31. Tr C m = i = 1 n λ i m . \mathrm{Tr}C^{m}=\sum_{i=1}^{n}\lambda_{i}^{m}~{}.
  32. p ( t ) = c 0 + c 1 t + + c n - 1 t n - 1 + t n p(t)=c_{0}+c_{1}t+\cdots+c_{n-1}t^{n-1}+t^{n}\,
  33. C T ( p ) = [ 0 1 0 0 0 0 1 0 0 0 0 1 - c 0 - c 1 - c 2 - c n - 1 ] C^{T}(p)=\begin{bmatrix}0&1&0&\cdots&0\\ 0&0&1&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\cdots&1\\ -c_{0}&-c_{1}&-c_{2}&\cdots&-c_{n-1}\end{bmatrix}
  34. C T [ a k a k + 1 a k + n - 1 ] = [ a k + 1 a k + 2 a k + n ] . C^{T}\begin{bmatrix}a_{k}\\ a_{k+1}\\ \vdots\\ a_{k+n-1}\end{bmatrix}=\begin{bmatrix}a_{k+1}\\ a_{k+2}\\ \vdots\\ a_{k+n}\end{bmatrix}.
  35. ( 1 , t , t < s u p > 2 , , t n - 1 ) (1,t,t<sup>2,...,t^{n-1})

Comparison_of_topologies.html

  1. τ 1 τ 2 \tau_{1}\subseteq\tau_{2}
  2. τ 1 τ 2 \tau_{1}\neq\tau_{2}

Competitive_inhibition.html

  1. V max V_{\max}
  2. K d K_{d}
  3. K m K_{m}
  4. K d K_{d}
  5. K app m K\text{app}_{m}
  6. V 0 V_{0}
  7. V 0 = V max [ S ] K app m + [ S ] V_{0}=\frac{V_{\max}\,[S]}{K\text{app}_{m}+[S]}
  8. K app m = K m ( 1 + [ I ] / K i ) K\text{app}_{m}=K_{m}(1+[I]/K_{i})
  9. K i K_{i}
  10. [ I ] [I]
  11. V max V_{\max}
  12. K app m K\text{app}_{m}
  13. V max / 2 V_{\max}/2
  14. V max V_{\max}
  15. V max V_{\max}
  16. E + S k 1 k - 1 E S k 2 E + P E+S\,\overset{k_{1}}{\underset{}{}}{k_{-1}}\rightleftharpoons\,ES\,\overset{k_% {2}}{\longrightarrow}\,E+P
  17. E I + S k - 3 k 3 E + S + I k 1 k - 1 E S + I k 2 E + P + I EI+S\,\overset{k_{-3}}{\underset{}{}}{k_{3}}\rightleftharpoons\,E+S+I\,% \overset{k_{1}}{\underset{}{}}{k_{-1}}\rightleftharpoons\,ES+I\,\overset{k_{2}% }{\longrightarrow}\,E+P+I
  18. d [ E ] d t = d [ E S ] d t = d [ E I ] d t = 0. \frac{d[E]}{dt}=\frac{d[ES]}{dt}=\frac{d[EI]}{dt}=0.
  19. [ E ] 0 = [ E ] + [ E S ] + [ E I ] [E]_{0}=[E]+[ES]+[EI]
  20. d [ E ] d t = 0 = - k 1 [ E ] [ S ] + k - 1 [ E S ] + k 2 [ E S ] - k 3 [ E ] [ I ] + k - 3 [ E I ] \frac{d[E]}{dt}=0=-k_{1}[E][S]+k_{-1}[ES]+k_{2}[ES]-k_{3}[E][I]+k_{-3}[EI]
  21. d [ E S ] d t = 0 = k 1 [ E ] [ S ] - k - 1 [ E S ] - k 2 [ E S ] \frac{d[ES]}{dt}=0=k_{1}[E][S]-k_{-1}[ES]-k_{2}[ES]
  22. d [ E I ] d t = 0 = k 3 [ E ] [ I ] - k - 3 [ E I ] \frac{d[EI]}{dt}=0=k_{3}[E][I]-k_{-3}[EI]
  23. [ S ] [S]
  24. [ I ] [I]
  25. [ E ] 0 [E]_{0}
  26. V 0 = d [ P ] / d t = k 2 [ E S ] V_{0}=d[P]/dt=k_{2}[ES]
  27. [ E S ] [ES]
  28. [ S ] [S]
  29. [ I ] [I]
  30. [ E ] 0 [E]_{0}
  31. k 1 [ E ] [ S ] = ( k - 1 + k 2 ) [ E S ] k_{1}[E][S]=(k_{-1}+k_{2})[ES]\,\!
  32. k 1 [ S ] k_{1}[S]
  33. [ E ] = ( k - 1 + k 2 ) [ E S ] k 1 [ S ] [E]=\frac{(k_{-1}+k_{2})[ES]}{k_{1}[S]}
  34. ( k - 1 + k 2 ) / k 1 (k_{-1}+k_{2})/k_{1}
  35. K m K_{m}
  36. [ E ] = K m [ E S ] [ S ] [E]=\frac{K_{m}[ES]}{[S]}
  37. 0 = k 3 [ I ] K m [ E S ] [ S ] - k - 3 [ E I ] 0=\frac{k_{3}[I]K_{m}[ES]}{[S]}-k_{-3}[EI]
  38. [ E I ] = K m k 3 [ I ] [ E S ] k - 3 [ S ] [EI]=\frac{K_{m}k_{3}[I][ES]}{k_{-3}[S]}
  39. K i = k - 3 / k 3 K_{i}=k_{-3}/k_{3}
  40. [ E I ] = K m [ I ] [ E S ] K i [ S ] [EI]=\frac{K_{m}[I][ES]}{K_{i}[S]}
  41. [ E ] 0 = K m [ E S ] [ S ] + [ E S ] + K m [ I ] [ E S ] K i [ S ] [E]_{0}=\frac{K_{m}[ES]}{[S]}+[ES]+\frac{K_{m}[I][ES]}{K_{i}[S]}
  42. [ E ] 0 = [ E S ] ( K m [ S ] + 1 + K m [ I ] K i [ S ] ) = [ E S ] K m K i + K i [ S ] + K m [ I ] K i [ S ] [E]_{0}=[ES]\left(\frac{K_{m}}{[S]}+1+\frac{K_{m}[I]}{K_{i}[S]}\right)=[ES]% \frac{K_{m}K_{i}+K_{i}[S]+K_{m}[I]}{K_{i}[S]}
  43. [ E S ] = K i [ S ] [ E ] 0 K m K i + K i [ S ] + K m [ I ] [ES]=\frac{K_{i}[S][E]_{0}}{K_{m}K_{i}+K_{i}[S]+K_{m}[I]}
  44. V 0 V_{0}
  45. V 0 = k 2 [ E S ] = k 2 K i [ S ] [ E ] 0 K m K i + K i [ S ] + K m [ I ] V_{0}=k_{2}[ES]=\frac{k_{2}K_{i}[S][E]_{0}}{K_{m}K_{i}+K_{i}[S]+K_{m}[I]}
  46. V 0 = k 2 [ E ] 0 [ S ] K m + [ S ] + K m [ I ] K i V_{0}=\frac{k_{2}[E]_{0}[S]}{K_{m}+[S]+K_{m}\frac{[I]}{K_{i}}}
  47. V max = k 2 [ E ] 0 V_{\max}=k_{2}[E]_{0}
  48. V 0 = V max [ S ] K m ( 1 + [ I ] K i ) + [ S ] V_{0}=\frac{V_{\max}[S]}{K_{m}(1+\frac{[I]}{K_{i}})+[S]}

Complete_Fermi–Dirac_integral.html

  1. F j ( x ) = 1 Γ ( j + 1 ) 0 t j e t - x + 1 d t . F_{j}(x)=\frac{1}{\Gamma(j+1)}\int_{0}^{\infty}\frac{t^{j}}{e^{t-x}+1}\,dt.
  2. - Li j + 1 ( - e x ) , -\operatorname{Li}_{j+1}(-e^{x}),
  3. Li s ( z ) \operatorname{Li}_{s}(z)
  4. F 0 ( x ) = ln ( 1 + exp ( x ) ) . F_{0}(x)=\ln(1+\exp(x)).\,
  5. s = 1 s=1
  6. Li 1 ( z ) = - ln ( 1 - z ) . \operatorname{Li}_{1}(z)=-\ln(1-z).

Complete_partial_order.html

  1. f ( D ) Q f(D)\subseteq Q
  2. D P D\subseteq P
  3. f ( sup D ) = sup f ( D ) f(\sup D)=\sup f(D)
  4. D P D\subseteq P

Completeness_(order_theory).html

  1. \vee
  2. \wedge
  3. \vee
  4. \wedge
  5. \wedge
  6. \vee
  7. \vee
  8. \wedge
  9. \vee
  10. \wedge
  11. \vee
  12. \wedge
  13. \wedge

Complex_manifold.html

  1. { z 𝐂 n : z < 1 } . \left\{z\in\mathbf{C}^{n}\ :\ \|z\|<1\right\}.
  2. { z = ( z 1 , z 2 , , z n ) 𝐂 n : | z i | < 1 , for all i = 1 , , n } . \left\{z=(z_{1},z_{2},\dots,z_{n})\in\mathbf{C}^{n}\ :\ |z_{i}|<1,\mbox{ for % all }~{}i=1,\dots,n\right\}.
  3. N J ( X , Y ) = [ X , Y ] + J [ J X , Y ] + J [ X , J Y ] - [ J X , J Y ] . N_{J}(X,Y)=[X,Y]+J[JX,Y]+J[X,JY]-[JX,JY]\ .

Complex_multiplication.html

  1. K = ( - d ) d > 0 K=\mathbb{Q}(\sqrt{-d})\ d>0
  2. ω 1 , ω 2 \omega_{1},\omega_{2}
  3. f f
  4. f ( z ) f(z)
  5. f ( λ z ) f(\lambda z)
  6. λ \lambda
  7. K K
  8. K K
  9. / [ i ] θ \mathbb{C}/\mathbb{Z}[i]\ \theta\ \;
  10. Y 2 = 4 X 3 - a X Y^{2}=4X^{3}-aX\ \,
  11. Y - i Y , X - X Y\rightarrow-iY,\ \ \ \ X\rightarrow-X
  12. ω 1 , ω 2 \omega_{1},\omega_{2}
  13. z z
  14. \mathbb{C}
  15. ( z ; L ) = ( z ; ω 1 , ω 2 ) = 1 z 2 + n 2 + m 2 0 { 1 ( z + m ω 1 + n ω 2 ) 2 - 1 ( m ω 1 + n ω 2 ) 2 } \wp(z;L)=\wp(z;\omega_{1},\omega_{2})=\frac{1}{z^{2}}+\sum_{n^{2}+m^{2}\neq 0}% \left\{\frac{1}{(z+m\omega_{1}+n\omega_{2})^{2}}-\frac{1}{\left(m\omega_{1}+n% \omega_{2}\right)^{2}}\right\}
  16. g 2 = 60 ( m , n ) ( 0 , 0 ) ( m ω 1 + n ω 2 ) - 4 g_{2}=60\sum_{(m,n)\neq(0,0)}(m\omega_{1}+n\omega_{2})^{-4}
  17. g 3 = 140 ( m , n ) ( 0 , 0 ) ( m ω 1 + n ω 2 ) - 6 . g_{3}=140\sum_{(m,n)\neq(0,0)}(m\omega_{1}+n\omega_{2})^{-6}.
  18. \wp^{\prime}
  19. \wp
  20. z ( 1 , ( z ) , ( z ) ) 3 z\mapsto(1,\wp(z),\wp^{\prime}(z))\in\mathbb{C}^{3}
  21. / L \mathbb{C}/L
  22. E = { x 3 | x 0 x 2 2 = 4 x 1 3 - g 2 x 0 2 x 1 - g 3 x 0 3 } E=\left\{x\in\mathbb{C}^{3}\right.\left|\;x_{0}x_{2}^{2}=4x_{1}^{3}-g_{2}x_{0}% ^{2}x_{1}-g_{3}x_{0}^{3}\;\right\}
  23. / L \mathbb{C}/L
  24. E E
  25. K K
  26. K K\subset\mathbb{C}
  27. L L
  28. K K
  29. / L \mathbb{C}/L
  30. 𝔬 \mathfrak{o}
  31. τ = ω 1 / ω 2 Im τ > 0 \tau=\omega_{1}/\omega_{2}\ \,\text{Im}\tau>0
  32. Δ ( L ) = g 2 ( L ) 3 - 27 g 3 ( L ) 3 \Delta(L)=g_{2}(L)^{3}-27g_{3}(L)^{3}
  33. J ( τ ) = J ( E ) = J ( L ) = 2 6 3 3 g 2 ( L ) 3 / Δ ( L ) . J(\tau)=J(E)=J(L)=2^{6}3^{3}g_{2}(L)^{3}/\Delta(L)\ .
  34. E E
  35. K K
  36. E E
  37. e π 163 = 262537412640768743.99999999999925007 e^{\pi\sqrt{163}}=262537412640768743.99999999999925007\dots\,
  38. e π 163 = 640320 3 + 743.99999999999925007 e^{\pi\sqrt{163}}=640320^{3}+743.99999999999925007\dots\,
  39. 𝐙 [ 1 + - 163 2 ] \mathbf{Z}\left[\frac{1+\sqrt{-163}}{2}\right]
  40. ( 1 + - 163 ) / 2 (1+\sqrt{-163})/2
  41. e π 163 = 12 3 ( 231 2 - 1 ) 3 + 743.99999999999925007 e^{\pi\sqrt{163}}=12^{3}(231^{2}-1)^{3}+743.99999999999925007\dots\,

Complexity_class.html

  1. DTIME ( f ( n ) ) DTIME ( f ( n ) \sdot log 2 ( f ( n ) ) ) \operatorname{DTIME}\big(f(n)\big)\subsetneq\operatorname{DTIME}\big(f(n)\sdot% \log^{2}(f(n))\big)
  2. DSPACE ( f ( n ) ) DSPACE ( f ( n ) \sdot log ( f ( n ) ) ) \operatorname{DSPACE}\big(f(n)\big)\subsetneq\operatorname{DSPACE}\big(f(n)% \sdot\log(f(n))\big)

Composition_(combinatorics).html

  1. ( 1 1 1 1 n ) \big(\,\overbrace{1\,\square\,1\,\square\,\ldots\,\square\,1\,\square\,1}^{n}% \,\big)
  2. ( n - 1 k - 1 ) {n-1\choose k-1}
  3. k = 1 n ( n - 1 k - 1 ) = 2 n - 1 . \sum_{k=1}^{n}{n-1\choose k-1}=2^{n-1}.
  4. ( n + k - 1 k - 1 ) {n+k-1\choose k-1}
  5. ( k n ) ( 1 ) a A = [ x n ] ( a A x a ) k {\left({{k}\atop{n}}\right)}_{(1)_{a\in A}}=[x^{n}](\sum_{a\in A}x^{a})^{k}

Compositional_data.html

  1. κ \scriptstyle\kappa
  2. 𝒞 [ ] \scriptstyle\mathcal{C}[\cdot]
  3. 𝒞 [ x 1 , x 2 , , x D ] = [ x 1 i = 1 D x i , x 2 i = 1 D x i , , x D i = 1 D x i ] , \mathcal{C}[x_{1},x_{2},\dots,x_{D}]=\left[\frac{x_{1}}{\sum_{i=1}^{D}x_{i}},% \frac{x_{2}}{\sum_{i=1}^{D}x_{i}},\dots,\frac{x_{D}}{\sum_{i=1}^{D}x_{i}}% \right],
  4. [ ] [\cdot]
  5. 𝒮 D = { 𝐱 = [ x 1 , x 2 , , x D ] D | x i > 0 , i = 1 , 2 , , D ; i = 1 D x i = κ } . \mathcal{S}^{D}=\left\{\mathbf{x}=[x_{1},x_{2},\dots,x_{D}]\in\mathbb{R}^{D}% \left|x_{i}>0,i=1,2,\dots,D;\sum_{i=1}^{D}x_{i}=\kappa\right.\right\}.
  6. 𝒮 D \scriptstyle\mathcal{S}^{D}
  7. κ \scriptstyle\kappa
  8. κ \scriptstyle\kappa
  9. 𝒮 D \scriptstyle\mathcal{S}^{D}

Compound_interest.html

  1. S = P ( 1 + j n ) n t S=P\left(1+\frac{j}{n}\right)^{nt}
  2. S = 1500 ( 1 + 0.043 4 ) 4 × 6 = 1938.84 S=1500\left(1+\frac{0.043}{4}\right)^{4\times 6}=1938.84
  3. A ( t ) = A 0 ( 1 + r n ) n t A(t)=A_{0}\left(1+\frac{r}{n}\right)^{\lfloor nt\rfloor}
  4. t t
  5. n n
  6. n t n\cdot t
  7. r r
  8. n t \lfloor nt\rfloor
  9. a ( t ) = 1 + t r a(t)=1+tr\,
  10. a ( t ) = ( 1 + r n ) n t a(t)=\left(1+\frac{r}{n}\right)^{nt}
  11. A ( t ) = A 0 e r t . A(t)=A_{0}e^{rt}.
  12. r 0 r_{0}
  13. r 0 = n ln ( 1 + r ) r_{0}=n\,\ln(1+r)
  14. δ t = a ( t ) a ( t ) \delta_{t}=\frac{a^{\prime}(t)}{a(t)}\,
  15. a ( n ) = e 0 n δ t d t , a(n)=e^{\int_{0}^{n}\delta_{t}\,dt}\ ,
  16. a ( 0 ) = 1 a(0)=1
  17. d a ( t ) = δ t a ( t ) d t da(t)=\delta_{t}a(t)\,dt\,
  18. δ = ln ( 1 + r ) \delta=\ln(1+r)\,
  19. a ( t ) = e t δ a(t)=e^{t\delta}\,
  20. δ t = p + s 1 + r s e s t \delta_{t}=p+{s\over{1+rse^{st}}}
  21. r 2 = [ ( 1 + r 1 n 1 ) n 1 n 2 - 1 ] n 2 , r_{2}=\left[\left(1+\frac{r_{1}}{n_{1}}\right)^{\frac{n_{1}}{n_{2}}}-1\right]{% n_{2}},
  22. R = n ln ( 1 + r / n ) , R=n\ln{\left(1+r/n\right)},
  23. P = L i 1 - 1 ( 1 + i ) n P=\frac{Li}{1-\frac{1}{(1+i)^{n}}}
  24. P = L i 1 - e - n ln ( 1 + i ) P=\frac{Li}{1-e^{-n\ln(1+i)}}
  25. L 1 = ( 1 + i ) L - P L_{1}=(1+i)L-P
  26. L 1 = 0 L_{1}=0
  27. L = P 1 + i L=\frac{P}{1+i}
  28. L 2 = ( 1 + i ) L 1 - P L_{2}=(1+i)L_{1}-P
  29. L 2 = ( 1 + i ) ( ( 1 + i ) L - P ) - P L_{2}=(1+i)((1+i)L-P)-P
  30. L 2 = 0 L_{2}=0
  31. L = P 1 + i + P ( 1 + i ) 2 L=\frac{P}{1+i}+\frac{P}{(1+i)^{2}}
  32. L = P j = 1 n 1 ( 1 + i ) j L=P\sum_{j=1}^{n}\frac{1}{(1+i)^{j}}
  33. L = P i ( 1 - 1 ( 1 + i ) n ) L=\frac{P}{i}\left(1-\frac{1}{(1+i)^{n}}\right)
  34. P = L i 1 - 1 ( 1 + i ) n = L i 1 - e - n ln ( 1 + i ) P=\frac{Li}{1-\frac{1}{(1+i)^{n}}}=\frac{Li}{1-e^{-n\ln(1+i)}}
  35. = =
  36. = =
  37. = =
  38. I < 8 % I<8\%
  39. i 1 i<<1
  40. ln ( 1 + i ) i \ln(1+i)\approx i
  41. P L i 1 - e - n i = L n n i 1 - e - n i P\approx\frac{Li}{1-e^{-ni}}=\frac{L}{n}\frac{ni}{1-e^{-ni}}
  42. Y n i = T I Y\equiv ni=TI
  43. P 0 L n P_{0}\equiv\frac{L}{n}
  44. P 0 P_{0}
  45. n n
  46. P P 0 Y 1 - e - Y P\approx P_{0}\frac{Y}{1-e^{-Y}}
  47. f ( Y ) Y 1 - e - Y - Y 2 f(Y)\equiv\frac{Y}{1-e^{-Y}}-\frac{Y}{2}
  48. f ( Y ) = f ( - Y ) f(Y)=f(-Y)
  49. Y Y
  50. Y 1 - e - Y \frac{Y}{1-e^{-Y}}
  51. Y Y
  52. Y / 2 Y/2
  53. X = 1 2 Y = 1 2 I T X=\frac{1}{2}Y=\frac{1}{2}IT
  54. P P 0 2 X 1 - e - 2 X P\approx P_{0}\frac{2X}{1-e^{-2X}}
  55. P P 0 ( 1 + X + X 2 3 - 1 45 X 4 + ) P\approx P_{0}\left(1+X+\frac{X^{2}}{3}-\frac{1}{45}X^{4}+...\right)
  56. X X
  57. P P 0 ( 1 + X + X 2 3 ) P\approx P_{0}\left(1+X+\frac{X^{2}}{3}\right)
  58. X 1 X\leq 1
  59. T = 3 T=3
  60. I = .178 I=.178
  61. X = 1 2 I T = .675 X=\frac{1}{2}IT=.675
  62. P P 0 ( 1 + X + 1 3 X 2 ) = $ 333.33 ( 1 + .675 + .675 2 / 3 ) = $ 608.96 P\approx P_{0}\left(1+X+\frac{1}{3}X^{2}\right)=\$333.33(1+.675+.675^{2}/3)=\$% 608.96
  63. P = $ 608.02 P=\$608.02

Compressed_air_energy_storage.html

  1. p V = n R T = constant pV=nRT=\operatorname{constant}
  2. T = T A = T B T=T_{A}=T_{B}
  3. W A B = V A V B p d V = V A V B n R T V d V = n R T V A V B 1 V d V = n R T ( ln V B - ln V A ) = n R T ln V B V A = n R T ln p A p B = p A V A ln p A p B \begin{aligned}\displaystyle W_{A\to B}&\displaystyle=\int_{V_{A}}^{V_{B}}pdV=% \int_{V_{A}}^{V_{B}}\frac{nRT}{V}dV=nRT\int_{V_{A}}^{V_{B}}\frac{1}{V}dV\\ &\displaystyle=nRT(\ln{V_{B}}-\ln{V_{A}})=nRT\ln{\frac{V_{B}}{V_{A}}}=nRT\ln{% \frac{p_{A}}{p_{B}}}=p_{A}V_{A}\ln{\frac{p_{A}}{p_{B}}}\\ \end{aligned}
  4. p V = p A V A = p B V B pV=p_{A}V_{A}=p_{B}V_{B}
  5. V B V A = p A p B \frac{V_{B}}{V_{A}}=\frac{p_{A}}{p_{B}}
  6. p p
  7. V V
  8. n n
  9. R R
  10. W = p B v B ln p A p B W=p_{B}v_{B}\ln\frac{p_{A}}{p_{B}}

Compression_(physics).html

  1. x x
  2. x x
  3. x x
  4. x x
  5. x x
  6. x x

Compressive_strength.html

  1. σ = E ϵ \sigma=E\epsilon
  2. σ = F A \sigma=\frac{F}{A}
  3. σ e = F A 0 \sigma_{e}=\frac{F}{A_{0}}
  4. ϵ e = l - l 0 l 0 \epsilon_{e}=\frac{l-l_{0}}{l_{0}}
  5. ( ϵ e * , σ e * ) (\epsilon_{e}^{*},\sigma_{e}^{*})
  6. σ e * = F * A 0 \sigma_{e}^{*}=\frac{F^{*}}{A_{0}}
  7. ϵ e * = l * - l 0 l 0 \epsilon_{e}^{*}=\frac{l^{*}-l_{0}}{l_{0}}

Computability.html

  1. λ \lambda
  2. λ \lambda
  3. f ( x ) f(x)
  4. g ( x ) g(x)
  5. h ( x , y ) h(x,y)
  6. f ( x ) = h ( x , g ( x ) ) f(x)=h(x,g(x))
  7. ( n + 1 ) (n+1)
  8. ( n + 1 ) (n+1)
  9. ( n + 1 ) (n+1)
  10. d > 0 d>0
  11. ( n + d + 1 ) (n+d+1)
  12. ( n + 1 ) (n+1)
  13. ( n + d + 1 ) (n+d+1)
  14. ( n + 1 ) (n+1)
  15. ( n + d + 1 ) (n+d+1)
  16. n + 1 n+1
  17. M M^{\prime}
  18. M M^{\prime}
  19. M M^{\prime}
  20. 1 = n = 1 1 2 n = 1 2 + 1 4 + 1 8 + 1 16 + 1=\sum_{n=1}^{\infty}\frac{1}{2^{n}}=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac% {1}{16}+\cdots

Concyclic_points.html

  1. R = a 2 b 2 c 2 ( a + b + c ) ( - a + b + c ) ( a - b + c ) ( a + b - c ) . R=\sqrt{\frac{a^{2}b^{2}c^{2}}{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}}.
  2. C A D = C B D \angle CAD=\angle CBD
  3. R = 1 4 ( a b + c d ) ( a c + b d ) ( a d + b c ) ( s - a ) ( s - b ) ( s - c ) ( s - d ) , R=\frac{1}{4}\sqrt{\frac{(ab+cd)(ac+bd)(ad+bc)}{(s-a)(s-b)(s-c)(s-d)}},
  4. A C B D = A B C D + B C A D . AC\cdot BD=AB\cdot CD+BC\cdot AD.
  5. A X X C = B X X D . \displaystyle AX\cdot XC=BX\cdot XD.
  6. T T
  7. T T

Conditional_(computer_programming).html

  1. x < 0 x<0
  2. x = 0 x=0
  3. x > 0 x>0

Conditional_expectation.html

  1. E ( X H ) = ω H X ( ω ) | H | \operatorname{E}(X\mid H)=\frac{\sum_{\omega\in H}X(\omega)}{|H|}
  2. X ( ω ) \scriptstyle X(\omega)
  3. 𝒳 \scriptstyle\mathcal{X}
  4. E ( X H ) = x 𝒳 x | { ω H X ( ω ) = x } | | H | \operatorname{E}(X\mid H)=\sum_{x\in\mathcal{X}}x\,\frac{|\{\omega\in H\mid X(% \omega)=x\}|}{|H|}
  5. ( Ω , , P ) \scriptstyle(\Omega,\mathcal{F},P)
  6. H \scriptstyle H\in\mathcal{F}
  7. P ( H ) > 0 \scriptstyle P(H)>0
  8. E ( X H ) = E ( 1 H X ) P ( H ) = x 𝒳 x P ( d x H ) , \operatorname{E}(X\mid H)=\frac{\operatorname{E}(1_{H}X)}{\operatorname{P}(H)}% =\int_{x\in\mathcal{X}}x\operatorname{P}(dx\mid H),
  9. 𝒳 \scriptstyle\mathcal{X}
  10. P ( A H ) = P ( A H ) P ( H ) \operatorname{P}(A\mid H)=\frac{\operatorname{P}(A\cap H)}{\operatorname{P}(H)}
  11. 𝒴 \scriptstyle\mathcal{Y}
  12. 𝒴 \scriptstyle\mathcal{Y}
  13. g : y E ( X Y = y ) . g:y\mapsto\operatorname{E}(X\mid Y=y).
  14. g Y \scriptstyle g\circ Y
  15. E ( X Y ) : ω E ( X Y = Y ( ω ) ) . \operatorname{E}(X\mid Y):\omega\mapsto\operatorname{E}(X\mid Y=Y(\omega)).
  16. E ( X Y ) \scriptstyle\operatorname{E}(X\mid Y)
  17. H y ϵ = { ω Y ( ω ) - y < ϵ } \scriptstyle H_{y}^{\epsilon}=\{\omega\mid\|Y(\omega)-y\|<\epsilon\}
  18. ϵ 0 \scriptstyle\epsilon\to 0
  19. P ( H y ϵ ) > 0 \scriptstyle P(H_{y}^{\epsilon})>0
  20. ϵ > 0 \scriptstyle\epsilon>0
  21. g : y lim ϵ 0 E ( X H y ε ) g:y\mapsto\lim_{\epsilon\to 0}\operatorname{E}(X\mid H_{y}^{\varepsilon})
  22. ( Ω , , P ) \scriptstyle(\Omega,\mathcal{F},\operatorname{P})
  23. 𝒜 = \scriptstyle\mathcal{A}=\mathcal{F}
  24. \scriptstyle\mathcal{B}
  25. 𝒞 \scriptstyle\mathcal{C}
  26. ( Ω , , P ) \scriptstyle(\Omega,\mathcal{F},\operatorname{P})
  27. X : Ω n \scriptstyle X:\Omega\to\mathbb{R}^{n}
  28. \scriptstyle\mathcal{H}\subseteq\mathcal{F}
  29. \scriptstyle\mathcal{F}
  30. \scriptstyle\mathcal{H}
  31. E ( X ) \scriptstyle\operatorname{E}(X\mid\mathcal{H})
  32. \scriptstyle\mathcal{H}
  33. Ω n \scriptstyle\Omega\to\mathbb{R}^{n}
  34. H E ( X ) d P | = H X d P for each H \int_{H}\operatorname{E}(X\mid\mathcal{H})\;dP_{|\mathcal{H}}=\int_{H}X\;dP% \qquad\,\text{for each}\quad H\in\mathcal{H}
  35. E ( X ) \scriptstyle\operatorname{E}(X\mid\mathcal{H})
  36. μ X : F F X \scriptstyle\mu^{X}:F\mapsto\int_{F}X
  37. F \scriptstyle F\in\mathcal{F}
  38. ( Ω , ) \scriptstyle(\Omega,\mathcal{F})
  39. P \scriptstyle P
  40. h \scriptstyle h
  41. \scriptstyle\mathcal{H}
  42. \scriptstyle\mathcal{F}
  43. μ X h = μ | X \scriptstyle\mu^{X}\circ h=\mu^{X}_{|\mathcal{H}}
  44. μ X \scriptstyle\mu^{X}
  45. \scriptstyle\mathcal{H}
  46. P h = P | \scriptstyle P\circ h=P_{|\mathcal{H}}
  47. P \scriptstyle P
  48. \scriptstyle\mathcal{H}
  49. μ X h \scriptstyle\mu^{X}\circ h
  50. P h \scriptstyle P\circ h
  51. P h ( H ) = 0 P ( h ( H ) ) = 0 μ X ( h ( H ) ) = 0 μ X h ( H ) = 0 \scriptstyle P\circ h(H)=0\Leftrightarrow P(h(H))=0\Rightarrow\mu^{X}(h(H))=0% \Leftrightarrow\mu^{X}\circ h(H)=0
  52. E ( X ) = d μ | X d P | = d ( μ X h ) d ( P h ) \operatorname{E}(X\mid\mathcal{H})=\frac{d\mu^{X}_{|\mathcal{H}}}{dP_{|% \mathcal{H}}}=\frac{d(\mu^{X}\circ h)}{d(P\circ h)}
  53. ( U , Σ ) \scriptstyle(U,\Sigma)
  54. Y : Ω U \scriptstyle Y:\Omega\to U
  55. Σ \scriptstyle\Sigma
  56. g : U n \scriptstyle g:U\to\mathbb{R}^{n}
  57. g ( Y ) f ( Y ) = X f ( Y ) for every Σ -measurable function f : U n \int g(Y)f(Y)=\int Xf(Y)\qquad\,\text{for every }\Sigma\,\text{-measurable % function}\quad f:U\to\mathbb{R}^{n}
  58. g ( Y ) \scriptstyle g(Y)
  59. E ( X Y ) \scriptstyle\operatorname{E}(X\mid Y)
  60. Y \scriptstyle Y
  61. = σ ( Y ) := Y - 1 ( Σ ) := { Y - 1 ( B ) : B Σ } \mathcal{H}=\sigma(Y):=Y^{-1}\left(\Sigma\right):=\{Y^{-1}(B):B\in\Sigma\}
  62. E ( X Y ) = E ( X ) = d ( μ X Y - 1 ) d ( P Y - 1 ) Y \operatorname{E}(X\mid Y)=\operatorname{E}(X\mid\mathcal{H})=\frac{d(\mu^{X}% \circ Y^{-1})}{d(P\circ Y^{-1})}\circ Y
  63. E ( X ) \scriptstyle\operatorname{E}(X\mid\mathcal{H})
  64. E ( X H ) \scriptstyle\operatorname{E}(X\mid H)
  65. \scriptstyle\mathcal{H}
  66. Ω n \scriptstyle\Omega\to\mathbb{R}^{n}
  67. n \scriptstyle\mathbb{R}^{n}
  68. H \scriptstyle H
  69. n \scriptstyle\mathcal{F}\to\mathbb{R}^{n}
  70. H E ( X H ) \scriptstyle H\mapsto\operatorname{E}(X\mid H)
  71. \scriptstyle\mathcal{H}
  72. E ( X ) \scriptstyle{E}(X\mid\mathcal{H})
  73. \scriptstyle\mathcal{H}
  74. Y Y
  75. ( U , Σ ) (U,\Sigma)
  76. Σ 𝒫 ( U ) \Sigma\subset\mathcal{P}(U)
  77. U U
  78. U U
  79. Y : Ω U Y\colon\Omega\to U
  80. Y - 1 ( B ) Y^{-1}(B)\in\mathcal{F}
  81. B Σ B\in\Sigma
  82. Y Y
  83. Y : Σ \mathbb{P}_{Y}:\Sigma\to\mathbb{R}
  84. Y ( B ) = ( Y - 1 ( B ) ) \mathbb{P}_{Y}(B)=\mathbb{P}(Y^{-1}(B))
  85. X : Ω X:\Omega\to\mathbb{R}
  86. Y \mathbb{P}_{Y}
  87. E ( X Y ) : U \operatorname{E}(X\mid Y):U\to\mathbb{R}
  88. Y - 1 ( B ) X d = B E ( X Y ) d Y , \int_{Y^{-1}(B)}X\;d\operatorname{\mathbb{P}}=\int_{B}\operatorname{E}(X\mid Y% )\;d\operatorname{\mathbb{P}_{Y}},
  89. B Σ B\in\Sigma
  90. μ : Σ \mu:\Sigma\to\mathbb{R}
  91. μ ( B ) = Y - 1 ( B ) X d \mu(B)=\int_{Y^{-1}(B)}X\;d\operatorname{\mathbb{P}}
  92. μ \mu
  93. Y \mathbb{P}_{Y}
  94. Y ( B ) = 0 \mathbb{P}_{Y}(B)=0
  95. ( Y - 1 ( B ) ) = 0 \mathbb{P}(Y^{-1}(B))=0
  96. μ \mu
  97. Y \mathbb{P}_{Y}
  98. E ( X Y ) \operatorname{E}(X\mid Y)
  99. \square
  100. E ( X Y ) Y = E ( X Y - 1 ( Σ ) ) . \operatorname{E}(X\mid Y)\circ Y=\operatorname{E}\left(X\mid Y^{-1}\left(% \Sigma\right)\right).
  101. Y - 1 ( B ) X d = Y - 1 ( B ) ( E ( X Y ) Y ) d . \int_{Y^{-1}(B)}X\ d\operatorname{\mathbb{P}}=\int_{Y^{-1}(B)}(\operatorname{E% }(X\mid Y)\circ Y)\ d\operatorname{\mathbb{P}}.
  102. X X
  103. E ( X Y ) Y \operatorname{E}(X\mid Y)\circ Y
  104. Y - 1 ( B ) Y^{-1}(B)
  105. B Σ B\in\Sigma
  106. E ( X Y = y ) = x 𝒳 x P ( X = x Y = y ) = x 𝒳 x P ( X = x , Y = y ) P ( Y = y ) , \operatorname{E}(X\mid Y=y)=\sum_{x\in\mathcal{X}}x\ \operatorname{P}(X=x\mid Y% =y)=\sum_{x\in\mathcal{X}}x\ \frac{\operatorname{P}(X=x,Y=y)}{\operatorname{P}% (Y=y)},
  107. 𝒳 \mathcal{X}
  108. E ( X Y = y ) = 𝒳 x f X ( x Y = y ) d x \operatorname{E}(X\mid Y=y)=\int_{\mathcal{X}}xf_{X}(x\mid Y=y)\,dx
  109. f X ( x Y = y ) = f X , Y ( x , y ) P ( Y = y ) f_{X}(x\mid Y=y)=\frac{f_{X,Y}(x,y)}{\operatorname{P}(Y=y)}
  110. E ( X Y = y ) = 𝒳 x f X Y ( x y ) d x \operatorname{E}(X\mid Y=y)=\int_{\mathcal{X}}xf_{X\mid Y}(x\mid y)\,dx
  111. f X Y ( x y ) = f X , Y ( x , y ) f Y ( y ) f_{X\mid Y}(x\mid y)=\frac{f_{X,Y}(x,y)}{f_{Y}(y)}
  112. \scriptstyle\mathcal{H}
  113. Z \scriptstyle Z
  114. X X
  115. \mathcal{H}
  116. E ( X ) = E ( X ) E(X\mid\mathcal{H})=E(X)
  117. B B\in\mathcal{H}
  118. X X
  119. 1 B 1_{B}
  120. B X d P = E ( X 1 B ) = E ( X ) E ( 1 B ) = E ( X ) P ( B ) = B E ( X ) d P \int_{B}XdP=E(X1_{B})=E(X)E(1_{B})=E(X)P(B)=\int_{B}E(X)dP
  121. E ( X ) E(X)
  122. X X
  123. σ ( Y , ) \sigma(Y,\mathcal{H})
  124. E ( X Y ) = E ( X ) E ( Y ) E(XY\mid\mathcal{H})=E(X)\,E(Y\mid\mathcal{H})
  125. X X
  126. \mathcal{H}
  127. Y Y
  128. X , Y X,Y
  129. 𝒢 , \mathcal{G},\mathcal{H}
  130. X X
  131. \mathcal{H}
  132. Y Y
  133. 𝒢 \mathcal{G}
  134. E ( E ( X Y 𝒢 ) ) = E ( X ) E ( Y ) = E ( E ( X Y ) 𝒢 ) . E(E(XY\mid\mathcal{G})\mid\mathcal{H})=E(X)E(Y)=E(E(XY\mid\mathcal{H})\mid% \mathcal{G}).
  135. X X
  136. \mathcal{H}
  137. E ( X ) = X E(X\mid\mathcal{H})=X
  138. E ( f ( Z ) Z ) = f ( Z ) \operatorname{E}(f(Z)\mid Z)=f(Z)
  139. E ( Z Z ) = Z \operatorname{E}(Z\mid Z)=Z
  140. X X
  141. \mathcal{H}
  142. E ( X Y ) = X E ( Y ) E(XY\mid\mathcal{H})=X\,E(Y\mid\mathcal{H})
  143. E ( f ( Z ) Y Z ) = f ( Z ) E ( Y Z ) \operatorname{E}(f(Z)Y\mid Z)=f(Z)\operatorname{E}(Y\mid Z)
  144. E ( E ( X ) ) = E ( X ) E(E(X\mid\mathcal{H}))=E(X)
  145. 1 2 \mathcal{H}_{1}\subset\mathcal{H}_{2}\subset\mathcal{F}
  146. E ( E ( X 2 ) 1 ) = E ( X 1 ) = E ( E ( X | 1 ) 2 ) E(E(X\mid\mathcal{H}_{2})\mid\mathcal{H}_{1})=E(X\mid\mathcal{H}_{1})=E(E(X|% \mathcal{H}_{1})\mid\mathcal{H}_{2})
  147. \mathcal{H}
  148. σ ( Z ) \sigma(Z)\subset\mathcal{H}
  149. E ( E ( X ) Z ) = E ( X Z ) E(E(X\mid\mathcal{H})\mid Z)=E(X\mid Z)
  150. Z = E ( X ) Z=E(X\mid\mathcal{H})
  151. \mathcal{H}
  152. E ( Z Z ) = Z \operatorname{E}(Z\mid Z)=Z
  153. E ( X E ( X ) ) = E ( X ) E(X\mid E(X\mid\mathcal{H}))=E(X\mid\mathcal{H})
  154. E ( X 1 + X 2 ) = E ( X 1 ) + E ( X 2 ) E(X_{1}+X_{2}\mid\mathcal{H})=E(X_{1}\mid\mathcal{H})+E(X_{2}\mid\mathcal{H})
  155. E ( a X ) = a E ( X ) E(aX\mid\mathcal{H})=a\,E(X\mid\mathcal{H})
  156. a \R a\in\R
  157. X 0 X\geq 0
  158. E ( X ) 0 E(X\mid\mathcal{H})\geq 0
  159. X 1 X 2 X_{1}\leq X_{2}
  160. E ( X 1 ) E ( X 2 ) E(X_{1}\mid\mathcal{H})\leq E(X_{2}\mid\mathcal{H})
  161. 0 X n X 0\leq X_{n}\uparrow X
  162. E ( X n ) E ( X ) E(X_{n}\mid\mathcal{H})\uparrow E(X\mid\mathcal{H})
  163. X n X X_{n}\to X
  164. | X n | Y |X_{n}|\leq Y
  165. Y L 1 Y\in L^{1}
  166. E ( X n ) E ( X ) E(X_{n}\mid\mathcal{H})\to E(X\mid\mathcal{H})
  167. E ( inf n X n ) > - \textstyle E(\inf_{n}X_{n}\mid\mathcal{H})>-\infty
  168. E ( lim inf n X n ) lim inf n E ( X n ) \textstyle E(\liminf_{n\to\infty}X_{n}\mid\mathcal{H})\leq\liminf_{n\to\infty}% E(X_{n}\mid\mathcal{H})
  169. f : f\colon\mathbb{R}\rightarrow\mathbb{R}
  170. f ( E ( X ) ) E ( f ( X ) ) f(E(X\mid\mathcal{H}))\leq E(f(X)\mid\mathcal{H})
  171. Var ( X ) = E ( ( X - E ( X ) ) 2 ) \operatorname{Var}(X\mid\mathcal{H})=\operatorname{E}\bigl((X-\operatorname{E}% (X\mid\mathcal{H}))^{2}\mid\mathcal{H}\bigr)
  172. Var ( X ) = E ( X 2 ) - ( E ( X ) ) 2 \operatorname{Var}(X\mid\mathcal{H})=\operatorname{E}(X^{2}\mid\mathcal{H})-% \bigl(\operatorname{E}(X\mid\mathcal{H})\bigr)^{2}
  173. Var ( X ) = E ( Var ( X ) ) + Var ( E ( X ) ) \operatorname{Var}(X)=\operatorname{E}(\operatorname{Var}(X\mid\mathcal{H}))+% \operatorname{Var}(\operatorname{E}(X\mid\mathcal{H}))
  174. X X
  175. E ( X n ) E ( X ) E(X\mid\mathcal{H}_{n})\to E(X\mid\mathcal{H})
  176. 1 2 \mathcal{H}_{1}\subset\mathcal{H}_{2}\subset\cdots
  177. = σ ( n = 1 n ) \textstyle\mathcal{H}=\sigma(\bigcup_{n=1}^{\infty}\mathcal{H}_{n})
  178. 1 2 \mathcal{H}_{1}\supset\mathcal{H}_{2}\supset\cdots
  179. = n = 1 n \textstyle\mathcal{H}=\bigcap_{n=1}^{\infty}\mathcal{H}_{n}
  180. L 2 L^{2}
  181. X , Y X,Y
  182. \mathcal{H}
  183. Y Y
  184. E ( Y ( X - E ( X ) ) ) = 0 E(Y(X-E(X\mid\mathcal{H})))=0
  185. E ( X ) E(X\mid\mathcal{H})
  186. X X
  187. \mathcal{H}
  188. X E ( X ) X\mapsto\operatorname{E}(X\mid\mathcal{H})
  189. E ( X E ( Y ) ) = E ( E ( X ) E ( Y ) ) = E ( E ( X ) Y ) \operatorname{E}(X\operatorname{E}(Y\mid\mathcal{H}))=\operatorname{E}\left(% \operatorname{E}(X\mid\mathcal{H})\operatorname{E}(Y\mid\mathcal{H})\right)=% \operatorname{E}(\operatorname{E}(X\mid\mathcal{H})Y)
  190. L P s ( Ω ; ) L P s ( Ω ; ) L^{s}_{P}(\Omega;\mathcal{F})\rightarrow L^{s}_{P}(\Omega;\mathcal{H})
  191. E | E ( X ) | s E | X | s \operatorname{E}|\operatorname{E}(X\mid\mathcal{H})|^{s}\leq\operatorname{E}|X% |^{s}

Conditional_probability_distribution.html

  1. p Y ( y X = x ) = P ( Y = y X = x ) = P ( X = x Y = y ) P ( X = x ) . p_{Y}(y\mid X=x)=P(Y=y\mid X=x)=\frac{P(X=x\ \cap Y=y)}{P(X=x)}.
  2. P ( X = x ) P(X=x)
  3. P ( X = x ) . P(X=x).
  4. P ( Y = y X = x ) P ( X = x ) = P ( X = x Y = y ) = P ( X = x Y = y ) P ( Y = y ) . P(Y=y\mid X=x)P(X=x)=P(X=x\ \cap Y=y)=P(X=x\mid Y=y)P(Y=y).
  5. f Y ( y X = x ) = f X , Y ( x , y ) f X ( x ) , f_{Y}(y\mid X=x)=\frac{f_{X,Y}(x,y)}{f_{X}(x)},
  6. f X ( x ) > 0 f_{X}(x)>0
  7. f Y ( y X = x ) f X ( x ) = f X , Y ( x , y ) = f X ( x Y = y ) f Y ( y ) . f_{Y}(y\mid X=x)f_{X}(x)=f_{X,Y}(x,y)=f_{X}(x\mid Y=y)f_{Y}(y).
  8. ( Ω , , P ) (\Omega,\mathcal{F},P)
  9. 𝒢 \mathcal{G}\subseteq\mathcal{F}
  10. σ \sigma
  11. \mathcal{F}
  12. X : Ω X:\Omega\to\mathbb{R}
  13. σ \sigma
  14. 1 \mathcal{R}^{1}
  15. \mathbb{R}
  16. μ : 1 × Ω \mu:\mathcal{R}^{1}\times\Omega\to\mathbb{R}
  17. μ ( , ω ) \mu(\cdot,\omega)
  18. 1 \mathcal{R}^{1}
  19. ω Ω \omega\in\Omega
  20. μ ( H , ) = P ( X H | 𝒢 ) \mu(H,\cdot)=P(X\in H|\mathcal{G})
  21. H 1 H\in\mathcal{R}^{1}
  22. ω Ω \omega\in\Omega
  23. μ ( , ω ) : 1 \mu(\cdot,\omega):\mathcal{R}^{1}\to\mathbb{R}
  24. X X
  25. 𝒢 \mathcal{G}
  26. E [ X | 𝒢 ] = - x μ ( d x , ) E[X|\mathcal{G}]=\int_{-\infty}^{\infty}x\,\mu(dx,\cdot)
  27. A 𝒜 A\in\mathcal{A}\supseteq\mathcal{B}
  28. 𝟏 A ( ω ) = { 1 if ω A , 0 if ω A , \mathbf{1}_{A}(\omega)=\begin{cases}1&\,\text{if }\omega\in A,\\ 0&\,\text{if }\omega\notin A,\end{cases}
  29. E ( 𝟏 A ) = P ( A ) . \operatorname{E}(\mathbf{1}_{A})=\operatorname{P}(A).\;
  30. \scriptstyle\mathcal{B}
  31. P ( | ) : 𝒜 × Ω ( 0 , 1 ) \scriptstyle\operatorname{P}(\cdot|\mathcal{B}):\mathcal{A}\times\Omega\to(0,1)
  32. P ( A | ) \scriptstyle\operatorname{P}(A|\mathcal{B})
  33. P ( A | ) = E ( 𝟏 A | ) \operatorname{P}(A|\mathcal{B})=\operatorname{E}(\mathbf{1}_{A}|\mathcal{B})\;
  34. P ( A | ) \scriptstyle\operatorname{P}(A|\mathcal{B})
  35. \scriptstyle\mathcal{B}
  36. B P ( A | ) ( ω ) d P ( ω ) = P ( A B ) for all A 𝒜 , B . \int_{B}\operatorname{P}(A|\mathcal{B})(\omega)\,\operatorname{d}\operatorname% {P}(\omega)=\operatorname{P}(A\cap B)\qquad\,\text{for all}\quad A\in\mathcal{% A},B\in\mathcal{B}.
  37. P ( | ) ( ω ) \scriptstyle\operatorname{P}(\cdot|\mathcal{B})(\omega)
  38. = { , Ω } \mathcal{B}=\{\emptyset,\Omega\}
  39. P ( A | { , Ω } ) P ( A ) . \operatorname{P}\!\left(A|\{\emptyset,\Omega\}\right)\equiv\operatorname{P}(A).
  40. A A\in\mathcal{B}
  41. P ( A | ) = 1 A . \operatorname{P}(A|\mathcal{B})=1_{A}.

Conditional_sentence.html

  1. p q \scriptstyle p\Rightarrow q

Configuration_space.html

  1. 3 × SO ( 3 ) \mathbb{R}^{3}\times\mathrm{SO}(3)
  2. 3 \mathbb{R}^{3}
  3. SO ( 3 ) \mathrm{SO}(3)
  4. 3 \mathbb{R}^{3}
  5. SO ( 3 ) \mathrm{SO}(3)
  6. C n X C_{n}X
  7. X X
  8. C n X = F n X / Σ n C_{n}X=F_{n}X/\Sigma_{n}
  9. F n X = { ( x 1 , , x n ) X n : x i x j i j } F_{n}X=\{(x_{1},\cdots,x_{n})\in X^{n}:x_{i}\neq x_{j}\forall\ i\neq j\}
  10. Σ n \Sigma_{n}
  11. F n X F_{n}X
  12. C n X C_{n}X
  13. X X
  14. F n X F_{n}X
  15. X X
  16. X n . X^{n}.
  17. C n 2 C_{n}\mathbb{R}^{2}
  18. π : E n C n \pi\colon E_{n}\to C_{n}
  19. C n × X C n C_{n}\times X\to C_{n}
  20. p C n p\in C_{n}
  21. X X
  22. F n m F_{n}\mathbb{R}^{m}
  23. m m
  24. F n F_{n}\mathbb{R}
  25. F n 2 F_{n}\mathbb{R}^{2}
  26. K ( π , 1 ) K(\pi,1)
  27. F n m F_{n}\mathbb{R}^{m}
  28. m 3 m\geq 3

Conformal_field_theory.html

  1. n n
  2. L n = z = 0 d z 2 π i z n + 1 T z z , L_{n}=\oint_{z=0}\frac{dz}{2\pi i}z^{n+1}T_{zz}~{},
  3. T z z = 1 2 ( z ϕ ) 2 . T_{zz}=\tfrac{1}{2}(\partial_{z}\phi)^{2}~{}.
  4. c c
  5. L n W ( z ) = - z n + 1 z W ( z ) - ( n + 1 ) Δ z n W ( z ) L_{n}W(z)=-z^{n+1}\frac{\partial}{\partial z}W(z)-(n+1)\Delta z^{n}W(z)
  6. L ¯ n W ( z ) = 0 . \bar{L}_{n}W(z)=0~{}.
  7. Δ Δ
  8. W W
  9. i > 1 i>1
  10. S O ( d + 1 , 1 ) SO(d+1,1)
  11. S O ( d , 2 ) SO(d,2)
  12. [ P μ , P ν ] = 0 , [P_{\mu},P_{\nu}]=0,
  13. [ D , K μ ] = - K μ , [D,K_{\mu}]=-K_{\mu},
  14. [ D , P μ ] = P μ , [D,P_{\mu}]=P_{\mu},
  15. [ K μ , K ν ] = 0 , [K_{\mu},K_{\nu}]=0,
  16. [ K μ , P ν ] = η μ ν D - i M μ ν , [K_{\mu},P_{\nu}]=\eta_{\mu\nu}D-iM_{\mu\nu},
  17. P P
  18. D D
  19. K μ K_{\mu}
  20. μ ξ ν + ν ξ μ = ξ η μ ν , \partial_{\mu}\xi_{\nu}+\partial_{\nu}\xi_{\mu}=\partial\cdot\xi\eta_{\mu\nu},% ~{}
  21. z ¯ ξ ( z ) = 0 = z ξ ( z ¯ ) \partial_{\bar{z}}\xi(z)=0=\partial_{z}\xi(\bar{z})
  22. z < s u p > n z z<sup>n∂_{z}

Conformal_group.html

  1. Q ( T x ) = λ 2 Q ( x ) Q(Tx)=\lambda^{2}Q(x)

Congruence_of_squares.html

  1. x 2 - y 2 = n x^{2}-y^{2}=n\,\!
  2. x 2 y 2 ( mod n ) , x ± y ( mod n ) x^{2}\equiv y^{2}\;\;(\mathop{{\rm mod}}n)\hbox{ , }x\not\equiv\pm y\;\;(% \mathop{{\rm mod}}n)
  3. x 2 - y 2 0 ( mod n ) , ( x + y ) ( x - y ) 0 ( mod n ) x^{2}-y^{2}\equiv 0\;\;(\mathop{{\rm mod}}n)\hbox{ , }(x+y)(x-y)\equiv 0\;\;(% \mathop{{\rm mod}}n)
  4. x 2 y 2 ( mod n ) \textstyle x^{2}\equiv y^{2}\;\;(\mathop{{\rm mod}}n)
  5. x 2 y ( mod n ) \textstyle x^{2}\equiv y\;\;(\mathop{{\rm mod}}n)
  6. 6 2 = 36 1 1 2 ( mod n ) \textstyle 6^{2}=36\equiv 1\equiv 1^{2}\;\;(\mathop{{\rm mod}}n)
  7. ( gcd [ 6 - 1 , 35 ] ) ( gcd [ 6 + 1 , 35 ] ) = ( 5 ) ( 7 ) = 35. (\gcd[6-1,35])\cdot(\gcd[6+1,35])=(5)\cdot(7)=35.
  8. 41 2 32 : 42 2 115 : 43 2 200 ( mod 1649 ) , 41^{2}\equiv 32:42^{2}\equiv 115:43^{2}\equiv 200\;\;(\mathop{{\rm mod}}1649),
  9. 32 = 2 5 : 200 = ( 2 3 ) ( 5 2 ) , 32=2^{5}:200=(2^{3})\cdot(5^{2}),
  10. ( 32 ) ( 200 ) = ( 2 5 + 3 ) ( 5 2 ) = ( ( 2 4 ) ( 5 ) ) 2 = 80 2 (32)\cdot(200)=(2^{5+3})\cdot(5^{2})=((2^{4})\cdot(5))^{2}=80^{2}
  11. ( 32 ) ( 200 ) = 80 2 ( 41 2 ) ( 43 2 ) 114 2 ( mod 1649 ) (32)\cdot(200)=80^{2}\equiv(41^{2})\cdot(43^{2})\equiv 114^{2}\;\;(\mathop{{% \rm mod}}1649)
  12. ( gcd [ 114 - 80 , 1649 ] ) ( gcd [ 114 + 80 , 1649 ] ) = ( 17 ) ( 97 ) = 1649. (\gcd[114-80,1649])\cdot(\gcd[114+80,1649])=(17)\cdot(97)=1649.

Congruence_subgroup.html

  1. Γ ( r ) := { [ a b c d ] Γ : a d ± 1 , b c 0 mod r } . \Gamma(r):=\left\{\begin{bmatrix}a&b\\ c&d\end{bmatrix}\in\Gamma:a\equiv d\equiv\pm 1,~{}b\equiv c\equiv 0\mod r% \right\}.
  2. Γ 1 ( r ) := { [ a b c d ] Γ : a d 1 , c 0 mod r } . \Gamma_{1}(r):=\left\{\begin{bmatrix}a&b\\ c&d\end{bmatrix}\in\Gamma:a\equiv d\equiv 1,~{}c\equiv 0\mod r\right\}.
  3. Γ 0 ( r ) := { [ a b c d ] Γ : c 0 mod r } . \Gamma_{0}(r):=\left\{\begin{bmatrix}a&b\\ c&d\end{bmatrix}\in\Gamma:c\equiv 0\mod r\right\}.
  4. R Γ k = 0 p - 1 S T k ( R Γ ) R_{\Gamma}\cup\bigcup_{k=0}^{p-1}ST^{k}(R_{\Gamma})
  5. w ( t ) = a t + b c t + d w(t)=\frac{at+b}{ct+d}
  6. Sp ( n , 𝐙 ) = { S SL ( 2 n , 𝐙 ) : S [ 0 I n - I n 0 ] S = [ 0 I n - I n 0 ] } \mathrm{Sp}(n,\mathrm{\mathbf{Z}})=\left\{S\in\mathrm{SL}(2n,\mathrm{\mathbf{Z% }}):S\begin{bmatrix}0&I_{n}\\ -I_{n}&0\end{bmatrix}S^{\top}=\begin{bmatrix}0&I_{n}\\ -I_{n}&0\end{bmatrix}\right\}
  7. \,{}^{\top}
  8. Γ ϑ ( n ) \Gamma_{\vartheta}^{(n)}
  9. [ C A ] B D [^{A}_{C}\,{}^{B}{}_{D}]
  10. A B AB^{\top}
  11. C D CD^{\top}

Conjugate_element_(field_theory).html

  1. 1 3 = { 1 - 1 2 + 3 2 i - 1 2 - 3 2 i \sqrt[3]{1}=\begin{cases}\ \ 1\\ -\frac{1}{2}+\frac{\sqrt{3}}{2}i\\ -\frac{1}{2}-\frac{\sqrt{3}}{2}i\end{cases}
  2. ( x + 1 2 ) 2 + 3 4 = x 2 + x + 1. \left(x+\frac{1}{2}\right)^{2}+\frac{3}{4}=x^{2}+x+1.

Connected_sum.html

  1. # \#
  2. A # B A\#B
  3. A A
  4. B B
  5. S m S^{m}
  6. M # S m M\#S^{m}
  7. M M
  8. g g
  9. k k
  10. M 1 M_{1}
  11. M 2 M_{2}
  12. V V
  13. M 1 M_{1}
  14. M 2 M_{2}
  15. ψ : N M 1 V N M 2 V \psi:N_{M_{1}}V\to N_{M_{2}}V
  16. ψ \psi
  17. N 1 V N M 1 V V N M 2 V V N 2 V , N_{1}\setminus V\cong N_{M_{1}}V\setminus V\to N_{M_{2}}V\setminus V\cong N_{2% }\setminus V,
  18. N M i V N_{M_{i}}V
  19. N i N_{i}
  20. V V
  21. M i M_{i}
  22. N M 2 V V N M 2 V V N_{M_{2}}V\setminus V\to N_{M_{2}}V\setminus V
  23. v v / | v | 2 v\mapsto v/|v|^{2}
  24. M 1 M_{1}
  25. M 2 M_{2}
  26. V V
  27. ( M 1 V ) N 1 V = N 2 V ( M 2 V ) (M_{1}\setminus V)\bigcup_{N_{1}\setminus V=N_{2}\setminus V}(M_{2}\setminus V)
  28. ( M 1 , V ) # ( M 2 , V ) . (M_{1},V)\#(M_{2},V).
  29. V V
  30. ψ \psi
  31. V V
  32. V V
  33. V V
  34. V V
  35. V V
  36. V V
  37. M i M_{i}
  38. ψ \psi
  39. e ( N M 1 V ) = - e ( N M 2 V ) . e(N_{M_{1}}V)=-e(N_{M_{2}}V).
  40. S O ( 2 ) SO(2)
  41. V V
  42. H 1 ( V ) H^{1}(V)
  43. ψ \psi
  44. H 1 ( V ) H^{1}(V)
  45. V V
  46. V V
  47. M M
  48. V V
  49. M M
  50. n n

Connection_(principal_bundle).html

  1. 𝔤 \mathfrak{g}
  2. Ω 1 ( P , 𝔤 ) C ( P , T * P 𝔤 ) \Omega^{1}(P,\mathfrak{g})\cong C^{\infty}(P,T^{*}P\otimes\mathfrak{g})
  3. ad g ( R g * ω ) = ω \hbox{ad}_{g}(R_{g}^{*}\omega)=\omega
  4. ad g \operatorname{ad}_{g}
  5. 𝔤 \mathfrak{g}
  6. ad g X = d d t g exp ( t X ) g - 1 | t = 0 \operatorname{ad}_{g}X=\frac{d}{dt}g\exp(tX)g^{-1}\bigl|_{t=0}
  7. ξ 𝔤 \xi\in\mathfrak{g}
  8. P × 𝔤 P\times\mathfrak{g}
  9. d π : T P T M {\mathrm{d}}\pi\colon TP\to TM
  10. H p g = d ( R g ) p ( H p ) H_{pg}=\mathrm{d}(R_{g})_{p}(H_{p})
  11. 𝔤 \mathfrak{g}
  12. 𝔤 \mathfrak{g}
  13. H = q - 1 Γ ( T M ) T P . H=q^{-1}\Gamma(TM)\subset TP.
  14. 𝔤 \mathfrak{g}
  15. 𝔤 P := P × G 𝔤 . \mathfrak{g}_{P}:=P\times^{G}\mathfrak{g}.
  16. P × G W P\times^{G}W
  17. P × G W P\times^{G}W
  18. P × G W P\times^{G}W
  19. P × G W P\times^{G}W
  20. P × G W P\times^{G}W
  21. P × G W P\times^{G}W
  22. 𝔤 \mathfrak{g}
  23. Ω = d ω + 1 2 [ ω ω ] . \Omega=d\omega+\tfrac{1}{2}[\omega\wedge\omega].
  24. 𝔤 P \mathfrak{g}_{P}
  25. Θ = d θ + ω θ . \Theta=\mathrm{d}\theta+\omega\wedge\theta.

Connection_(vector_bundle).html

  1. : Γ ( E ) Γ ( E T * M ) \nabla:\Gamma(E)\to\Gamma(E\otimes T^{*}M)
  2. ( σ f ) = ( σ ) f + σ d f \nabla(\sigma f)=(\nabla\sigma)f+\sigma\otimes df
  3. X : Γ ( E ) Γ ( E ) \nabla_{X}:\Gamma(E)\to\Gamma(E)
  4. X ( σ 1 + σ 2 ) = X σ 1 + X σ 2 \displaystyle\nabla_{X}(\sigma_{1}+\sigma_{2})=\nabla_{X}\sigma_{1}+\nabla_{X}% \sigma_{2}
  5. Ω r ( E ) = Γ ( E r T * M ) . \Omega^{r}(E)=\Gamma(E\otimes\textstyle\bigwedge^{r}T^{*}M).
  6. Ω 0 ( E ) = Γ ( E ) . \Omega^{0}(E)=\Gamma(E).\,
  7. : Ω 0 ( E ) Ω 1 ( E ) . \nabla:\Omega^{0}(E)\to\Omega^{1}(E).
  8. d : Ω r ( E ) Ω r + 1 ( E ) . d^{\nabla}:\Omega^{r}(E)\to\Omega^{r+1}(E).
  9. ( 1 - 2 ) ( f σ ) = f ( 1 σ - 2 σ ) (\nabla_{1}-\nabla_{2})(f\sigma)=f(\nabla_{1}\sigma-\nabla_{2}\sigma)
  10. ( 1 - 2 ) Ω 1 ( M ; End E ) . (\nabla_{1}-\nabla_{2})\in\Omega^{1}(M;\mathrm{End}\,E).
  11. ψ ( X σ ) = X H ( ψ ( σ ) ) \psi(\nabla_{X}\sigma)=X^{H}(\psi(\sigma))
  12. σ = σ α e α \sigma=\sigma^{\alpha}e_{\alpha}
  13. σ = ( d σ α + ω α σ β β ) e α \nabla\sigma=(\mathrm{d}\sigma^{\alpha}+\omega^{\alpha}\!{}_{\beta}\sigma^{% \beta})e_{\alpha}
  14. ω α e α β = e β . \omega^{\alpha}\!{}_{\beta}\,e_{\alpha}=\nabla e_{\beta}.
  15. ω α = β ω i α d β x i . \omega^{\alpha}\!{}_{\beta}={\omega_{i}}^{\alpha}\!{}_{\beta}\,\mathrm{d}x^{i}.
  16. ϖ = t - 1 ω t + t - 1 d t . \varpi=t^{-1}\omega t+t^{-1}\mathrm{d}t.
  17. X σ = X i ( i σ α + ω i α σ β β ) e α . \nabla_{X}\sigma=X^{i}(\partial_{i}\,\sigma^{\alpha}+{\omega_{i}}^{\alpha}\!{}% _{\beta}\sigma^{\beta})e_{\alpha}.
  18. γ ˙ ( t ) σ = 0 \nabla_{\dot{\gamma}(t)}\sigma=0
  19. τ γ : E x E y \tau_{\gamma}:E_{x}\to E_{y}\,
  20. Hol x = { τ γ : γ is a loop based at x } . \mathrm{Hol}_{x}=\{\tau_{\gamma}:\gamma\,\text{ is a loop based at }x\}.\,
  21. F Ω 2 ( End E ) = Γ ( End E Λ 2 T * M ) . F^{\nabla}\in\Omega^{2}(\mathrm{End}\,E)=\Gamma(\mathrm{End}\,E\otimes\Lambda^% {2}T^{*}M).
  22. F ( X , Y ) ( s ) = X Y s - Y X s - [ X , Y ] s F^{\nabla}(X,Y)(s)=\nabla_{X}\nabla_{Y}s-\nabla_{Y}\nabla_{X}s-\nabla_{[X,Y]}s
  23. ( d ) 2 σ = F σ . (d^{\nabla})^{2}\sigma=F^{\nabla}\wedge\sigma.

Conservative_vector_field.html

  1. 𝐯 \mathbf{v}
  2. φ \varphi
  3. 𝐯 = φ . \mathbf{v}=\nabla\varphi.
  4. φ \nabla\varphi
  5. φ \varphi
  6. φ \varphi
  7. 𝐯 \mathbf{v}
  8. S 3 S\subseteq\mathbb{R}^{3}
  9. P P
  10. S S
  11. A A
  12. B B
  13. 𝐯 = φ \mathbf{v}=\nabla\varphi
  14. P 𝐯 d 𝐫 = φ ( B ) - φ ( A ) . \int_{P}\mathbf{v}\cdot d\mathbf{r}=\varphi(B)-\varphi(A).
  15. 𝐯 d 𝐫 = 0 \oint\mathbf{v}\cdot d\mathbf{r}=0
  16. 𝐯 \mathbf{v}
  17. × 𝐯 = 0. \nabla\times\mathbf{v}=\mathbf{0}.
  18. φ \varphi
  19. × φ = 0. \nabla\times\nabla\varphi=\mathbf{0}.
  20. S S
  21. S S
  22. S S
  23. z z
  24. S = 3 { ( 0 , 0 , z ) | z } S=\mathbb{R}^{3}\setminus\{(0,0,z)~{}|~{}z\in\mathbb{R}\}
  25. 𝐯 = ( - y x 2 + y 2 , x x 2 + y 2 , 0 ) . \mathbf{v}=\left(\frac{-y}{x^{2}+y^{2}},\frac{x}{x^{2}+y^{2}},0\right).
  26. 𝐯 \mathbf{v}
  27. S S
  28. 𝐯 \mathbf{v}
  29. 𝐯 \mathbf{v}
  30. x , y x,y
  31. 2 π 2\pi
  32. 𝐯 = 𝐞 ϕ / r \mathbf{v}=\mathbf{e_{\phi}}/r
  33. 𝐯𝐞 ϕ d ϕ = 2 π \int\mathbf{v}\mathbf{e_{\phi}}\mathrm{d}\phi=2\pi
  34. 𝐯 \mathbf{v}
  35. arg ( x + i y ) \arg(x+iy)
  36. ϕ \phi
  37. H dR 1 H_{\mathrm{dR}}^{1}
  38. 𝐯 \mathbf{v}
  39. s y m b o l ω symbol{\omega}
  40. s y m b o l ω = × 𝐯 . symbol{\omega}=\nabla\times\mathbf{v}.
  41. ζ \zeta\;
  42. 𝐯 \mathbf{v}
  43. × 𝐯 = 𝟎 \nabla\times\mathbf{v}=\mathbf{0}
  44. 𝐅 \mathbf{F}
  45. 𝐅 G \mathbf{F}_{G}
  46. m m
  47. M M
  48. r r
  49. 𝐅 G = - G m M 𝐫 ^ r 2 , \mathbf{F}_{G}=-\frac{GmM\hat{\mathbf{r}}}{r^{2}},
  50. G G
  51. 𝐫 ^ \hat{\mathbf{r}}
  52. M M
  53. m m
  54. 𝐅 G = - Φ G \mathbf{F}_{G}=-\nabla\Phi_{G}
  55. Φ G = - G m M r \Phi_{G}=-\frac{GmM}{r}
  56. A A
  57. B B
  58. W = 𝐅 d 𝐫 = 0. W=\oint\mathbf{F}\cdot d\mathbf{r}=0.

Consistency_(statistics).html

  1. 𝐛 \mathbf{b}
  2. supp ( 𝐛 ) = { i : 𝐛 i 0 } \operatorname{supp}(\mathbf{b})=\{i:\mathbf{b}_{i}\neq 0\}
  3. 𝐛 i \mathbf{b}_{i}
  4. i i
  5. 𝐛 \mathbf{b}
  6. 𝐛 ^ \hat{\mathbf{b}}
  7. 𝐛 \mathbf{b}
  8. P ( supp ( 𝐛 ^ ) = supp ( 𝐛 ) ) 1 P(\operatorname{supp}(\hat{\mathbf{b}})=\operatorname{supp}(\mathbf{b}))\rightarrow 1
  9. n n\rightarrow\infty

Constant_function.html

  1. y ( x ) = 4 y(x)=4
  2. y ( x ) y(x)
  3. x x
  4. y ( x ) = c y(x)=c
  5. y = c y=c
  6. y ( x ) = 2 y(x)=2
  7. y = 2 y=2
  8. c = 2 c=2
  9. y = c y=c
  10. ( 0 , c ) (0,c)
  11. f ( x ) = c , c 0 f(x)=c\,,\,\,c\neq 0
  12. f ( x ) = 0 f(x)=0
  13. ( c ) = 0 (c)^{\prime}=0
  14. y ( x ) = - 2 y(x)=-\sqrt{2}
  15. y ( x ) = ( - 2 ) = 0 y^{\prime}(x)=(-\sqrt{2})^{\prime}=0
  16. x ~ : Y X \tilde{x}:Y\rightarrow X
  17. x ~ ( y ) = x \tilde{x}(y)=x
  18. y Y y\in Y
  19. f : X Y f:X\rightarrow Y
  20. f ( y ) = f ( y ) f(y)=f(y^{\prime})
  21. y , y Y y,y^{\prime}\in Y
  22. f f
  23. X X
  24. X 1 X^{1}
  25. h o m ( 1 , X ) hom(1,X)
  26. h o m ( X × Y , Z ) h o m ( X ( h o m ( Y , Z ) ) hom(X\times Y,Z)\cong hom(X(hom(Y,Z))
  27. λ : 1 × X X X × 1 : ρ \lambda:1\times X\cong X\cong X\times 1:\rho
  28. p 1 p_{1}
  29. p 2 p_{2}
  30. ( * , x ) (*,x)
  31. ( x , * ) (x,*)
  32. x x
  33. * *

Constructible_polygon.html

  1. 2 ( 2 n ) + 1. 2^{(2^{n})}+1.

Constructible_universe.html

  1. Def ( X ) := { { y y X and ( X , ) Φ ( y , z 1 , , z n ) } | Φ is a first-order formula and z 1 , , z n X } . \operatorname{Def}(X):=\Bigl\{\{y\mid y\in X\,\text{ and }(X,\in)\models\Phi(y% ,z_{1},\ldots,z_{n})\}~{}\Big|~{}\Phi\,\text{ is a first-order formula and }z_% {1},\ldots,z_{n}\in X\Bigr\}.
  2. L 0 := . L_{0}:=\varnothing.
  3. L α + 1 := Def ( L α ) . L_{\alpha+1}:=\operatorname{Def}(L_{\alpha}).
  4. λ \lambda
  5. L λ := α < λ L α . L_{\lambda}:=\bigcup_{\alpha<\lambda}L_{\alpha}.
  6. L := α 𝐎𝐫𝐝 L α . L:=\bigcup_{\alpha\in\mathbf{Ord}}L_{\alpha}.
  7. L α = β < α Def ( L β ) L_{\alpha}=\bigcup_{\beta<\alpha}\operatorname{Def}(L_{\beta})\!
  8. L α { L α } L_{\alpha}\cup\{L_{\alpha}\}
  9. L ω 1 CK L_{\omega_{1}^{\mathrm{CK}}}
  10. ω 1 CK \omega_{1}^{\mathrm{CK}}
  11. L ω 1 CK L_{\omega_{1}^{\mathrm{CK}}}
  12. S L α S\in L_{\alpha}
  13. T L β + 1 T\in L_{\beta+1}
  14. T = { x L β : x S Φ ( x , z i ) } = { x S : Φ ( x , z i ) } T=\{x\in L_{\beta}:x\in S\wedge\Phi(x,z_{i})\}=\{x\in S:\Phi(x,z_{i})\}
  15. z i z_{i}
  16. L β L_{\beta}
  17. L α L_{\alpha}
  18. w i w_{i}
  19. L β L_{\beta}
  20. w i w_{i}
  21. z i z_{i}
  22. L α L_{\alpha}
  23. V = L V=L
  24. L β L_{\beta}
  25. K = L γ K=L_{\gamma}
  26. T = { x L β : x S Φ ( x , z i ) } = { x L γ : x S Φ ( x , w i ) } T=\{x\in L_{\beta}:x\in S\wedge\Phi(x,z_{i})\}=\{x\in L_{\gamma}:x\in S\wedge% \Phi(x,w_{i})\}
  27. L β L_{\beta}
  28. L γ L_{\gamma}
  29. L γ + 1 L_{\gamma+1}
  30. L 𝒫 ( S ) L α + 1 L\cap\mathcal{P}(S)\subseteq L_{\alpha+1}
  31. y ( y s ( y L ω + 1 and ( a ( a y a L 5 and O r d ( a ) ) b ( b y b L ω and O r d ( b ) ) ) ) ) \forall y(y\in s\iff(y\in L_{\omega+1}\and(\forall a(a\in y\iff a\in L_{5}\and Ord% (a))\forall b(b\in y\iff b\in L_{\omega}\and Ord(b)))))
  32. O r d ( a ) Ord(a)
  33. c a ( d c ( d a and e d ( e c ) ) ) . \forall c\in a(\forall d\in c(d\in a\and\forall e\in d(e\in c))).
  34. L λ ( A ) = α < λ L α ( A ) L_{\lambda}(A)=\bigcup_{\alpha<\lambda}L_{\alpha}(A)\!
  35. L ( A ) = α L α ( A ) L(A)=\bigcup_{\alpha}L_{\alpha}(A)\!

Constructive_proof.html

  1. a a
  2. b b
  3. a b a^{b}
  4. 2 2 \sqrt{2}^{\sqrt{2}}
  5. ( 2 2 ) 2 = 2 (\sqrt{2}^{\sqrt{2}})^{\sqrt{2}}=2
  6. 2 \sqrt{2}
  7. q = 2 2 q=\sqrt{2}^{\sqrt{2}}
  8. q q
  9. a a
  10. b b
  11. 2 \sqrt{2}
  12. q q
  13. a a
  14. 2 2 \sqrt{2}^{\sqrt{2}}
  15. b b
  16. 2 \sqrt{2}
  17. ( 2 2 ) 2 = 2 ( 2 2 ) = 2 2 = 2. \left(\sqrt{2}^{\sqrt{2}}\right)^{\sqrt{2}}=\sqrt{2}^{(\sqrt{2}\cdot\sqrt{2})}% =\sqrt{2}^{2}=2.
  18. 2 2 \sqrt{2}^{\sqrt{2}}
  19. a = 2 , b = log 2 9 , a b = 3 . a=\sqrt{2}\,,\quad b=\log_{2}9\,,\quad a^{b}=3\,.
  20. log 2 9 \log_{2}9
  21. m n m\over n
  22. f ( x ) = { 0 if Goldbach’s conjecture is false 1 if Goldbach’s conjecture is true f(x)=\begin{cases}0&\mbox{if Goldbach's conjecture is false}\\ 1&\mbox{if Goldbach's conjecture is true}\end{cases}

Contact_geometry.html

  1. PT * M = T * M / where, for ω i T p * M , ω 1 ω 2 λ 0 : ω 1 = λ ω 2 . \,\text{PT}^{*}M=\,\text{T}^{*}M/\!\sim\ \,\text{ where, for }\omega_{i}\in\,% \text{T}^{*}_{p}M,\ \ \omega_{1}\sim\omega_{2}\ \iff\ \exists\ \lambda\neq 0\ % :\ \omega_{1}=\lambda\omega_{2}.
  2. α ( d α ) k 0 where ( d α ) k = d α d α k - times . \alpha\wedge(\,\text{d}\alpha)^{k}\neq 0\ \,\text{where}\ (\,\text{d}\alpha)^{% k}=\underbrace{\,\text{d}\alpha\wedge\ldots\wedge\,\text{d}\alpha}_{k-\,\text{% times}}.
  3. L = { ( q , p ) T * N | H ( q , p ) = E } L=\{(q,p)\in T^{*}N|H(q,p)=E\}
  4. i Y d λ i_{Y}d\lambda

Continuation.html

  1. c r , fell ( r , c ) \forall c\exists r,\mbox{fell}~{}(r,c)
  2. r c , fell ( r , c ) \exists r\forall c,\mbox{fell}~{}(r,c)
  3. x y , saw ( x , y ) \exists x\forall y,\mbox{saw}~{}(x,y)
  4. y x , saw ( x , y ) \forall y\exists x,\mbox{saw}~{}(x,y)

Continuity_correction.html

  1. P ( X x ) = P ( X < x + 1 ) P(X\leq x)=P(X<x+1)
  2. P ( Y x + 1 / 2 ) P(Y\leq x+1/2)
  3. P ( X x ) = P ( X < x + 1 ) P ( Y x + 1 / 2 ) P(X\leq x)=P(X<x+1)\approx P(Y\leq x+1/2)

Continuity_equation.html

  1. 𝐣 = ρ 𝐮 \mathbf{j}=\rho\mathbf{u}
  2. S d 𝐒 \int\!\!\!\!\int_{S}d\mathbf{S}
  3. q q
  4. Σ \Sigma
  5. ρ t + 𝐣 = 0 \frac{\partial\rho}{\partial t}+\nabla\cdot\mathbf{j}=0\,
  6. 𝐉 = - ρ t \nabla\cdot\mathbf{J}=-{\partial\rho\over\partial t}
  7. × 𝐇 = 𝐉 + 𝐃 t . \nabla\times\mathbf{H}=\mathbf{J}+\frac{\partial\mathbf{D}}{\partial t}.
  8. ( × 𝐇 ) = 𝐉 + ( 𝐃 ) t , \nabla\cdot(\nabla\times\mathbf{H})=\nabla\cdot\mathbf{J}+\frac{\partial(% \nabla\cdot\mathbf{D})}{\partial t},
  9. 𝐉 + ( 𝐃 ) t = 0. \nabla\cdot\mathbf{J}+\frac{\partial(\nabla\cdot\mathbf{D})}{\partial t}=0.
  10. 𝐃 = ρ , \nabla\cdot\mathbf{D}=\rho,\,
  11. 𝐉 + ρ t = 0. \nabla\cdot\mathbf{J}+{\partial\rho\over\partial t}=0.\,
  12. ρ t + ( ρ 𝐮 ) = 0 {\partial\rho\over\partial t}+\nabla\cdot(\rho\mathbf{u})=0
  13. 𝐮 = 0 , \nabla\cdot\mathbf{u}=0,
  14. u t + 𝐪 = 0 \frac{\partial u}{\partial t}+\nabla\cdot\mathbf{q}=0
  15. Ψ \Psi
  16. Ψ = Ψ ( 𝐫 , t ) \Psi=\Psi(\mathbf{r},t)
  17. ρ ( 𝐫 , t ) = Ψ * ( 𝐫 , t ) Ψ ( 𝐫 , t ) = | Ψ ( 𝐫 , t ) | 2 \rho(\mathbf{r},t)=\Psi^{*}(\mathbf{r},t)\Psi(\mathbf{r},t)=|\Psi(\mathbf{r},t% )|^{2}\,\!
  18. P = P 𝐫 V ( t ) = V Ψ * Ψ d V = V | Ψ | 2 d V P=P_{\mathbf{r}\in V}(t)=\int_{V}\Psi^{*}\Psi dV=\int_{V}|\Psi|^{2}dV\,
  19. 𝐣 ( 𝐫 , t ) = 2 m i [ Ψ * ( Ψ ) - Ψ ( Ψ * ) ] . \mathbf{j}(\mathbf{r},t)=\frac{\hbar}{2mi}\left[\Psi^{*}\left(\nabla\Psi\right% )-\Psi\left(\nabla\Psi^{*}\right)\right].
  20. 𝐣 + ρ t = 0 𝐣 + | Ψ | 2 t = 0. \nabla\cdot\mathbf{j}+\frac{\partial\rho}{\partial t}=0\rightleftharpoons% \nabla\cdot\mathbf{j}+\frac{\partial|\Psi|^{2}}{\partial t}=0.
  21. - 2 2 m 2 Ψ + U Ψ = i Ψ t , \displaystyle-\frac{\hbar^{2}}{2m}\nabla^{2}\Psi+U\Psi=i\hbar\frac{\partial% \Psi}{\partial t},
  22. ρ t = | Ψ | 2 t = t ( Ψ * Ψ ) = Ψ * Ψ t + Ψ Ψ * t . \frac{\partial\rho}{\partial t}=\frac{\partial|\Psi|^{2}}{\partial t}=\frac{% \partial}{\partial t}\left(\Psi^{*}\Psi\right)=\Psi^{*}\frac{\partial\Psi}{% \partial t}+\Psi\frac{\partial\Psi^{*}}{\partial t}.
  23. Ψ * Ψ / t \scriptstyle\Psi^{*}\partial\Psi/\partial t\,\!
  24. Ψ Ψ * / t \scriptstyle\Psi\partial\Psi^{*}/\partial t\,\!
  25. Ψ * Ψ t = 1 i [ - 2 Ψ * 2 m 2 Ψ + U Ψ * Ψ ] , Ψ Ψ * t = - 1 i [ - 2 Ψ 2 m 2 Ψ * + U Ψ Ψ * ] , \begin{aligned}&\displaystyle\Psi^{*}\frac{\partial\Psi}{\partial t}=\frac{1}{% i\hbar}\left[-\frac{\hbar^{2}\Psi^{*}}{2m}\nabla^{2}\Psi+U\Psi^{*}\Psi\right],% \\ &\displaystyle\Psi\frac{\partial\Psi^{*}}{\partial t}=-\frac{1}{i\hbar}\left[-% \frac{\hbar^{2}\Psi}{2m}\nabla^{2}\Psi^{*}+U\Psi\Psi^{*}\right],\\ \end{aligned}
  26. ρ t \displaystyle\frac{\partial\rho}{\partial t}
  27. 𝐣 \displaystyle\nabla\cdot\mathbf{j}
  28. ρ t = - 𝐣 \displaystyle\frac{\partial\rho}{\partial t}=-\nabla\cdot\mathbf{j}
  29. J = ( c ρ , j x , j y , j z ) J=\left(c\rho,j_{x},j_{y},j_{z}\right)
  30. μ J μ = ρ t + 𝐣 \partial_{\mu}J^{\mu}=\frac{\partial\rho}{\partial t}+\nabla\cdot\mathbf{j}
  31. μ J μ = 0 \partial_{\mu}J^{\mu}=0
  32. μ J μ = 0 \partial_{\mu}J^{\mu}=0
  33. ν T μ ν = 0 \partial_{\nu}T^{\mu\nu}=0
  34. T ν ; μ μ = 0. T^{\mu}_{\;\;\nu;\mu}=0.
  35. μ T μ ν = - Γ μ λ μ T λ ν - Γ μ λ ν T μ λ , \partial_{\mu}T^{\mu\nu}=-\Gamma^{\mu}_{\mu\lambda}T^{\lambda\nu}-\Gamma^{\nu}% _{\mu\lambda}T^{\mu\lambda},

Continuous_linear_extension.html

  1. X X
  2. 𝖳 \mathsf{T}
  3. X X
  4. 𝖳 \mathsf{T}
  5. 𝖳 \mathsf{T}
  6. X X
  7. Y Y
  8. 𝖳 ~ \tilde{\mathsf{T}}
  9. X X
  10. Y Y
  11. 𝖳 \mathsf{T}
  12. c c
  13. 𝖳 ~ \tilde{\mathsf{T}}
  14. c c
  15. [ a , b ] [a,b]
  16. f r 1 1 [ a , x 1 ) + r 2 1 [ x 1 , x 2 ) + + r n 1 [ x n - 1 , b ] f\equiv r_{1}\mathit{1}_{[a,x_{1})}+r_{2}\mathit{1}_{[x_{1},x_{2})}+\cdots+r_{% n}\mathit{1}_{[x_{n-1},b]}
  17. r 1 , , r n r_{1},\ldots,r_{n}
  18. a = x 0 < x 1 < < x n - 1 < x n = b a=x_{0}<x_{1}<\ldots<x_{n-1}<x_{n}=b
  19. 1 S \mathit{1}_{S}
  20. S S
  21. [ a , b ] [a,b]
  22. L L^{\infty}
  23. 𝒮 \mathcal{S}
  24. 𝖨 ( i = 1 n r i 1 [ x i - 1 , x i ) ) = i = 1 n r i ( x i - x i - 1 ) \mathsf{I}\left(\sum_{i=1}^{n}r_{i}\mathit{1}_{[x_{i-1},x_{i})}\right)=\sum_{i% =1}^{n}r_{i}(x_{i}-x_{i-1})
  25. 𝖨 \mathsf{I}
  26. 𝒮 \mathcal{S}
  27. \mathbb{R}
  28. 𝒫 𝒞 \mathcal{PC}
  29. [ a , b ] [a,b]
  30. L L^{\infty}
  31. 𝒮 \mathcal{S}
  32. 𝒫 𝒞 \mathcal{PC}
  33. 𝖨 \mathsf{I}
  34. 𝖨 ~ \tilde{\mathsf{I}}
  35. 𝒫 𝒞 \mathcal{PC}
  36. \mathbb{R}
  37. 𝒫 𝒞 \mathcal{PC}
  38. f 𝒫 𝒞 f\in\mathcal{PC}
  39. a b f ( x ) d x = 𝖨 ~ ( f ) \int_{a}^{b}f(x)dx=\tilde{\mathsf{I}}(f)
  40. T : S Y T:S\rightarrow Y
  41. S ¯ = X \bar{S}=X
  42. Y Y
  43. S S
  44. X X
  45. S S
  46. X X
  47. \mathbb{R}
  48. \mathbb{R}

Contract_curve.html

  1. x 2 1 x_{2}^{1}
  2. u 1 u^{1}
  3. u 2 u^{2}
  4. u 0 2 u_{0}^{2}
  5. ω 1 t o t \omega_{1}^{tot}
  6. ω 2 t o t \omega_{2}^{tot}
  7. max x 1 1 , x 2 1 , x 1 2 , x 2 2 u 1 ( x 1 1 , x 2 1 ) \max_{x_{1}^{1},x_{2}^{1},x_{1}^{2},x_{2}^{2}}u^{1}(x_{1}^{1},x_{2}^{1})
  8. x 1 1 + x 1 2 ω 1 t o t x_{1}^{1}+x_{1}^{2}\leq\omega_{1}^{tot}
  9. x 2 1 + x 2 2 ω 2 t o t x_{2}^{1}+x_{2}^{2}\leq\omega_{2}^{tot}
  10. u 2 ( x 1 2 , x 2 2 ) u 0 2 u^{2}(x_{1}^{2},x_{2}^{2})\geq u_{0}^{2}
  11. max x 1 1 , x 2 1 , x 1 2 , x 2 2 b u 1 ( x 1 1 , x 2 1 ) + ( 1 - b ) u 2 ( x 1 2 , x 2 2 ) \max_{x_{1}^{1},x_{2}^{1},x_{1}^{2},x_{2}^{2}}b\cdot u^{1}(x_{1}^{1},x_{2}^{1}% )+(1-b)\cdot u^{2}(x_{1}^{2},x_{2}^{2})
  12. x 1 1 + x 1 2 ω 1 t o t x_{1}^{1}+x_{1}^{2}\leq\omega_{1}^{tot}
  13. x 2 1 + x 2 2 ω 2 t o t x_{2}^{1}+x_{2}^{2}\leq\omega_{2}^{tot}
  14. u 1 ( x 1 1 , x 2 1 ) u 0 1 u^{1}(x_{1}^{1},x_{2}^{1})\geq u_{0}^{1}
  15. u 2 ( x 1 2 , x 2 2 ) u 0 2 u^{2}(x_{1}^{2},x_{2}^{2})\geq u_{0}^{2}
  16. u 1 u^{1}

Control_chart.html

  1. x ¯ \bar{x}
  2. x ¯ \bar{x}
  3. < v a r > n > 1000 <var>n>1000

Convertible_bond.html

  1. Δ = δ C δ S δ C = Δ × δ S . \Delta=\frac{\delta C}{\delta S}\Rightarrow\delta C=\Delta\times\delta S.
  2. 0 < Δ < 1 0<\Delta<1
  3. δ C < δ S \delta C<\delta S

Conway_group.html

  1. T 2 A ( τ ) T_{2A}(\tau)
  2. T 4 A ( τ ) T_{4A}(\tau)
  3. j 4 A ( τ ) = T 4 A ( τ ) + 24 = ( η 2 ( 2 τ ) η ( τ ) η ( 4 τ ) ) 24 = ( ( η ( τ ) η ( 4 τ ) ) 4 + 4 2 ( η ( 4 τ ) η ( τ ) ) 4 ) 2 = 1 q + 24 + 276 q + 2048 q 2 + 11202 q 3 + 49152 q 4 + \begin{aligned}\displaystyle j_{4A}(\tau)&\displaystyle=T_{4A}(\tau)+24\\ &\displaystyle=\Big(\tfrac{\eta^{2}(2\tau)}{\eta(\tau)\,\eta(4\tau)}\Big)^{24}% \\ &\displaystyle=\Big(\big(\tfrac{\eta(\tau)}{\eta(4\tau)}\big)^{4}+4^{2}\big(% \tfrac{\eta(4\tau)}{\eta(\tau)}\big)^{4}\Big)^{2}\\ &\displaystyle=\frac{1}{q}+24+276q+2048q^{2}+11202q^{3}+49152q^{4}+\dots\end{aligned}

Cooley–Tukey_FFT_algorithm.html

  1. X k = n = 0 N - 1 x n e - 2 π i N n k , X_{k}=\sum_{n=0}^{N-1}x_{n}e^{-\frac{2\pi i}{N}nk},
  2. k k
  3. 0
  4. N - 1 N-1
  5. ( x 2 m = x 0 , x 2 , , x N - 2 ) (x_{2m}=x_{0},x_{2},\ldots,x_{N-2})
  6. ( x 2 m + 1 = x 1 , x 3 , , x N - 1 ) (x_{2m+1}=x_{1},x_{3},\ldots,x_{N-1})
  7. x n x_{n}
  8. n = 2 m n={2m}
  9. n = 2 m + 1 n={2m+1}
  10. X k = m = 0 N / 2 - 1 x 2 m e - 2 π i N ( 2 m ) k + m = 0 N / 2 - 1 x 2 m + 1 e - 2 π i N ( 2 m + 1 ) k \begin{matrix}X_{k}&=&\sum\limits_{m=0}^{N/2-1}x_{2m}e^{-\frac{2\pi i}{N}(2m)k% }+\sum\limits_{m=0}^{N/2-1}x_{2m+1}e^{-\frac{2\pi i}{N}(2m+1)k}\end{matrix}
  11. e - 2 π i N k e^{-\frac{2\pi i}{N}k}
  12. x 2 m x_{2m}
  13. x 2 m + 1 x_{2m+1}
  14. x n x_{n}
  15. x 2 m x_{2m}
  16. E k E_{k}
  17. x 2 m + 1 x_{2m+1}
  18. O k O_{k}
  19. X k = m = 0 N / 2 - 1 x 2 m e - 2 π i N / 2 m k DFT of even - indexed part of x m + e - 2 π i N k m = 0 N / 2 - 1 x 2 m + 1 e - 2 π i N / 2 m k DFT of odd - indexed part of x m = E k + e - 2 π i N k O k . \begin{matrix}X_{k}=\underbrace{\sum\limits_{m=0}^{N/2-1}x_{2m}e^{-\frac{2\pi i% }{N/2}mk}}_{\mathrm{DFT\;of\;even-indexed\;part\;of\;}x_{m}}{}+e^{-\frac{2\pi i% }{N}k}\underbrace{\sum\limits_{m=0}^{N/2-1}x_{2m+1}e^{-\frac{2\pi i}{N/2}mk}}_% {\mathrm{DFT\;of\;odd-indexed\;part\;of\;}x_{m}}=E_{k}+e^{-\frac{2\pi i}{N}k}O% _{k}.\end{matrix}
  20. E k + N 2 = E k E_{{k}+\frac{N}{2}}=E_{k}
  21. O k + N 2 = O k O_{{k}+\frac{N}{2}}=O_{k}
  22. X k = { E k + e - 2 π i N k O k for 0 k < N / 2 E k - N / 2 + e - 2 π i N k O k - N / 2 for N / 2 k < N . \begin{matrix}X_{k}&=&\left\{\begin{matrix}E_{k}+e^{-\frac{2\pi i}{N}k}O_{k}&% \mbox{for }~{}0\leq k<N/2\\ \\ E_{k-N/2}+e^{-\frac{2\pi i}{N}k}O_{k-N/2}&\mbox{for }~{}N/2\leq k<N.\\ \end{matrix}\right.\end{matrix}
  23. e - 2 π i k / N e^{-2\pi ik/N}
  24. e - 2 π i N ( k + N / 2 ) = e - 2 π i k N - π i = e - π i e - 2 π i k N = - e - 2 π i k N \begin{matrix}e^{\frac{-2\pi i}{N}(k+N/2)}&=&e^{\frac{-2\pi ik}{N}-{\pi i}}\\ &=&e^{-\pi i}e^{\frac{-2\pi ik}{N}}\\ &=&-e^{\frac{-2\pi ik}{N}}\end{matrix}
  25. 0 k < N 2 0\leq k<\frac{N}{2}
  26. X k = E k + e - 2 π i N k O k X k + N 2 = E k - e - 2 π i N k O k \begin{matrix}X_{k}&=&E_{k}+e^{-\frac{2\pi i}{N}k}O_{k}\\ X_{k+\frac{N}{2}}&=&E_{k}-e^{-\frac{2\pi i}{N}k}O_{k}\end{matrix}
  27. E k E_{k}
  28. O k exp ( - 2 π i k / N ) O_{k}\exp(-2\pi ik/N)
  29. exp [ - 2 π i k / N ] \exp[-2\pi ik/N]
  30. k = N 2 k 1 + k 2 k=N_{2}k_{1}+k_{2}
  31. n = N 1 n 2 + n 1 n=N_{1}n_{2}+n_{1}
  32. N 1 n 2 N 2 k 1 N_{1}n_{2}N_{2}k_{1}
  33. X N 2 k 1 + k 2 = n 1 = 0 N 1 - 1 n 2 = 0 N 2 - 1 x N 1 n 2 + n 1 e - 2 π i N 1 N 2 ( N 1 n 2 + n 1 ) ( N 2 k 1 + k 2 ) X_{N_{2}k_{1}+k_{2}}=\sum_{n_{1}=0}^{N_{1}-1}\sum_{n_{2}=0}^{N_{2}-1}x_{N_{1}n% _{2}+n_{1}}e^{-\frac{2\pi i}{N_{1}N_{2}}\cdot(N_{1}n_{2}+n_{1})\cdot(N_{2}k_{1% }+k_{2})}
  34. = n 1 = 0 N 1 - 1 [ e - 2 π i N n 1 k 2 ] ( n 2 = 0 N 2 - 1 x N 1 n 2 + n 1 e - 2 π i N 2 n 2 k 2 ) e - 2 π i N 1 n 1 k 1 =\sum_{n_{1}=0}^{N_{1}-1}\left[e^{-\frac{2\pi i}{N}n_{1}k_{2}}\right]\left(% \sum_{n_{2}=0}^{N_{2}-1}x_{N_{1}n_{2}+n_{1}}e^{-\frac{2\pi i}{N_{2}}n_{2}k_{2}% }\right)e^{-\frac{2\pi i}{N_{1}}n_{1}k_{1}}
  35. E k E_{k}
  36. O k O_{k}
  37. E k E_{k}
  38. O k O_{k}
  39. N log 2 N 2 \frac{N\log_{2}{N}}{2}
  40. log 2 N + N - 3 \log_{2}{N}+N-3

Coordinate_rotations_and_reflections.html

  1. Rot ( θ ) = [ cos θ - sin θ sin θ cos θ ] , \mathrm{Rot}(\theta)=\begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{bmatrix},
  2. Ref ( θ ) = [ cos 2 θ sin 2 θ sin 2 θ - cos 2 θ ] . \mathrm{Ref}(\theta)=\begin{bmatrix}\cos 2\theta&\sin 2\theta\\ \sin 2\theta&-\cos 2\theta\end{bmatrix}.
  3. Ref ( θ ) Ref ( ϕ ) = Rot ( 2 ( θ - ϕ ) ) , \mathrm{Ref}(\theta)\,\mathrm{Ref}(\phi)=\mathrm{Rot}(2(\theta-\phi)),
  4. Rot ( θ ) Rot ( ϕ ) = Rot ( θ + ϕ ) , \mathrm{Rot}(\theta)\,\mathrm{Rot}(\phi)=\mathrm{Rot}(\theta+\phi),
  5. Rot ( θ ) Ref ( ϕ ) = Ref ( ϕ + θ / 2 ) , \mathrm{Rot}(\theta)\,\mathrm{Ref}(\phi)=\mathrm{Ref}(\phi+\theta/2),
  6. Ref ( ϕ ) Rot ( θ ) = Ref ( ϕ - θ / 2 ) . \mathrm{Ref}(\phi)\,\mathrm{Rot}(\theta)=\mathrm{Ref}(\phi-\theta/2).

Copper(I)_oxide.html

  1. P n 3 ¯ m \scriptstyle Pn\bar{3}m

Corporate_Average_Fuel_Economy.html

  1. n A + n B + n C + n D n A f A + n B f B + n C f C + n D f D \frac{n_{A}+n_{B}+n_{C}+n_{D}}{\frac{n_{A}}{f_{A}}+\frac{n_{B}}{f_{B}}+\frac{n% _{C}}{f_{C}}+\frac{n_{D}}{f_{D}}}
  2. 4 1 15 + 1 13 + 1 17 + 1 100 = 18.83 \frac{4}{\frac{1}{15}+\frac{1}{13}+\frac{1}{17}+\frac{1}{100}}=18.83
  3. 15 + 13 + 17 + 100 4 = 36.25 \frac{15+13+17+100}{4}=36.25

Correspondence_(mathematics).html

  1. C C
  2. D D
  3. C o p × D 𝐒𝐞𝐭 C^{op}\times D\to\mathbf{Set}

Cosmic_distance_ladder.html

  1. 5 log 10 D = m - M - 10 , 5\cdot\log_{10}D=m-M-10,
  2. M - m = - 2.5 log 10 ( F 1 / F 2 ) . \ M-m=-2.5\log_{10}(F_{1}/F_{2})\,.
  3. 5 log 10 d = V + ( 3.34 ) log 10 P - ( 2.45 ) ( V - I ) + 7.52 . 5\log_{10}{d}=V+(3.34)\log_{10}{P}-(2.45)(V-I)+7.52\,.
  4. 5 log 10 d = V + ( 3.37 ) log 10 P - ( 2.55 ) ( V - I ) + 7.48 . 5\log_{10}{d}=V+(3.37)\log_{10}{P}-(2.55)(V-I)+7.48\,.
  5. ω = Δ θ Δ t , \omega=\frac{\Delta\theta}{\Delta t}\,,
  6. d = V e j ω , \ d=\frac{V_{ej}}{\omega}\,,
  7. 1.4 M 1.4M_{\odot}
  8. M B M V - 19.3 ± 0.3 . \ M_{B}\approx M_{V}\approx-19.3\pm 0.3\,.
  9. M V max = - 9.96 - 2.31 log 10 x ˙ . \ M^{\max}_{V}=-9.96-2.31\log_{10}\dot{x}\,.
  10. x ˙ \dot{x}
  11. Φ ( m ) = A e ( m - m 0 ) 2 / 2 σ 2 \ \Phi(m)=Ae^{(m-m_{0})^{2}/2\sigma^{2}}\,
  12. N ( M ) e 0.307 M ( 1 - e 3 ( M * - M ) ) . \ N(M)\propto e^{0.307M}(1-e^{3(M^{*}-M)})\,.
  13. log 10 ( D ) = 1.333 log ( σ ) + C . \log_{10}(D)=1.333\log(\sigma)+C\,.

Cost-of-living_index.html

  1. P K ( p 0 , p 1 , u ) = C ( u , p 1 ) C ( u , p 0 ) P_{K}(p^{0},p^{1},u)=\frac{C(u,p^{1})}{C(u,p^{0})}
  2. P K ( p 0 , p 1 , q ) = C ( f ( q ) , p 1 ) C ( f ( q ) , p 0 ) P_{K}(p^{0},p^{1},q)=\frac{C(f(q),p^{1})}{C(f(q),p^{0})}

Costate_equations.html

  1. λ ˙ ( t ) = - H x \dot{\lambda}(t)=-\frac{\partial H}{\partial x}
  2. λ ( t ) \lambda(t)

Coulomb_barrier.html

  1. U c o u l = k q 1 q 2 r = 1 4 π ϵ 0 q 1 q 2 r U_{coul}=k{{q_{1}\,q_{2}}\over r}={1\over{4\pi\epsilon_{0}}}{{q_{1}\,q_{2}}% \over r}
  2. U c o u l = k Z 1 Z 2 e 2 r U_{coul}={{k\,Z_{1}\,Z_{2}\,e^{2}}\over r}

Coupled_cluster.html

  1. H | Ψ = E | Ψ H|{\Psi}\rangle=E|{\Psi}\rangle
  2. H H
  3. | Ψ |{\Psi}\rangle
  4. | Ψ = e T | Φ 0 |{\Psi}\rangle=e^{T}|{\Phi_{0}}\rangle
  5. | Φ 0 |{\Phi_{0}}\rangle
  6. T T
  7. | Φ 0 |{\Phi_{0}}\rangle
  8. T = T 1 + T 2 + T 3 + T=T_{1}+T_{2}+T_{3}+\cdots
  9. T 1 T_{1}
  10. T 2 T_{2}
  11. T 1 = i a t a i a ^ a a ^ i , T_{1}=\sum_{i}\sum_{a}t_{a}^{i}\hat{a}^{a}\hat{a}_{i},
  12. T 2 = 1 4 i , j a , b t a b i j a ^ a a ^ b a ^ j a ^ i , T_{2}=\frac{1}{4}\sum_{i,j}\sum_{a,b}t_{ab}^{ij}\hat{a}^{a}\hat{a}^{b}\hat{a}_% {j}\hat{a}_{i},
  13. T n = 1 ( n ! ) 2 i 1 , i 2 , , i n a 1 , a 2 , , a n t a 1 , a 2 , , a n i 1 , i 2 , , i n a ^ a 1 a ^ a 2 a ^ a n a ^ i n a ^ i 2 a ^ i 1 . T_{n}=\frac{1}{(n!)^{2}}\sum_{i_{1},i_{2},\ldots,i_{n}}\sum_{a_{1},a_{2},% \ldots,a_{n}}t_{a_{1},a_{2},\ldots,a_{n}}^{i_{1},i_{2},\ldots,i_{n}}\hat{a}^{a% _{1}}\hat{a}^{a_{2}}\ldots\hat{a}^{a_{n}}\hat{a}_{i_{n}}\ldots\hat{a}_{i_{2}}% \hat{a}_{i_{1}}.
  14. ( a ^ a = ) a ^ a (\hat{a}^{\dagger}_{a}=)\hat{a}^{a}
  15. a ^ i \hat{a}_{i}
  16. | Φ 0 |{\Phi_{0}}\rangle
  17. T 1 T_{1}
  18. T 2 T_{2}
  19. | Φ 0 |{\Phi_{0}}\rangle
  20. T 1 T_{1}
  21. T 2 T_{2}
  22. t a i t_{a}^{i}
  23. t a b i j t_{ab}^{ij}
  24. | Ψ |{\Psi}\rangle
  25. e T e^{T}
  26. T 1 T_{1}
  27. T 2 T_{2}
  28. T T
  29. e T = 1 + T + 1 2 ! T 2 + = 1 + T 1 + T 2 + 1 2 T 1 2 + T 1 T 2 + 1 2 T 2 2 + e^{T}=1+T+\frac{1}{2!}T^{2}+\cdots=1+T_{1}+T_{2}+\frac{1}{2}T_{1}^{2}+T_{1}T_{% 2}+\frac{1}{2}T_{2}^{2}+\cdots
  30. T 1 T_{1}
  31. T 2 T_{2}
  32. T 5 T_{5}
  33. T 6 T_{6}
  34. T T
  35. T T
  36. T = T 1 + + T n T=T_{1}+...+T_{n}
  37. T n T_{n}
  38. H | Ψ 0 = H e T | Φ 0 = E e T | Φ 0 H|{\Psi_{0}}\rangle=He^{T}|{\Phi_{0}}\rangle=Ee^{T}|{\Phi_{0}}\rangle
  39. e - T e^{-T}
  40. T T
  41. | Φ 0 |{\Phi_{0}}\rangle
  42. | Φ * |{\Phi^{*}}\rangle
  43. | Φ i a |{\Phi_{i}^{a}}\rangle
  44. | Φ i j a b |{\Phi_{ij}^{ab}}\rangle
  45. Φ 0 | e - T H e T | Φ 0 = E Φ 0 | Φ 0 = E \langle{\Phi_{0}}|e^{-T}He^{T}|{\Phi_{0}}\rangle=E\langle{\Phi_{0}}|{\Phi_{0}}% \rangle=E
  46. Φ * | e - T H e T | Φ 0 = E Φ * | Φ 0 = 0 \langle{\Phi^{*}}|e^{-T}He^{T}|{\Phi_{0}}\rangle=E\langle{\Phi^{*}}|{\Phi_{0}}% \rangle=0
  47. e - T e T = 1 e^{-T}e^{T}=1
  48. Φ 0 | e - ( T 1 + T 2 ) H e ( T 1 + T 2 ) | Φ 0 = E \langle{\Phi_{0}}|e^{-(T_{1}+T_{2})}He^{(T_{1}+T_{2})}|{\Phi_{0}}\rangle=E
  49. Φ i a | e - ( T 1 + T 2 ) H e ( T 1 + T 2 ) | Φ 0 = 0 \langle{\Phi_{i}^{a}}|e^{-(T_{1}+T_{2})}He^{(T_{1}+T_{2})}|{\Phi_{0}}\rangle=0
  50. Φ i j a b | e - ( T 1 + T 2 ) H e ( T 1 + T 2 ) | Φ 0 = 0 \langle{\Phi_{ij}^{ab}}|e^{-(T_{1}+T_{2})}He^{(T_{1}+T_{2})}|{\Phi_{0}}\rangle=0
  51. H ¯ \bar{H}
  52. H ¯ = e - T H e T = H + [ H , T ] + ( 1 / 2 ) [ [ H , T ] , T ] + = ( H e T ) C \bar{H}=e^{-T}He^{T}=H+[H,T]+(1/2)[[H,T],T]+...=(He^{T})_{C}
  53. T T
  54. T T
  55. T = T 1 + T 2 + T 3 . T=T_{1}+T_{2}+T_{3}.
  56. | Φ 0 |{\Phi_{0}}\rangle
  57. | Ψ 0 |{\Psi_{0}}\rangle
  58. | Ψ 0 = ( 1 + C ) | Φ 0 |{\Psi_{0}}\rangle=(1+C)|{\Phi_{0}}\rangle
  59. C = j = 1 N C j C=\sum_{j=1}^{N}C_{j}
  60. T T
  61. T N T_{N}
  62. C 1 = T 1 C_{1}=T_{1}
  63. C 2 = T 2 + 1 2 ( T 1 ) 2 C_{2}=T_{2}+\frac{1}{2}(T_{1})^{2}
  64. C 3 = T 3 + T 1 T 2 + 1 6 ( T 1 ) 3 C_{3}=T_{3}+T_{1}T_{2}+\frac{1}{6}(T_{1})^{3}
  65. C 4 = T 4 + 1 2 ( T 2 ) 2 + T 1 T 3 + 1 2 ( T 1 ) 2 T 2 + 1 24 ( T 1 ) 4 C_{4}=T_{4}+\frac{1}{2}(T_{2})^{2}+T_{1}T_{3}+\frac{1}{2}(T_{1})^{2}T_{2}+% \frac{1}{24}(T_{1})^{4}
  66. S = I S I S=\sum_{I}S_{I}
  67. Φ | ( H - E 0 ) e S | Φ = 0 \langle\Phi|(H-E_{0})e^{S}|\Phi\rangle=0
  68. Φ i 1 i n a 1 a n | ( H - E 0 ) e S | Φ = 0 \langle\Phi_{i_{1}\ldots i_{n}}^{a_{1}\ldots a_{n}}|(H-E_{0})e^{S}|\Phi\rangle=0
  69. i 1 < < i n i_{1}<\cdots<i_{n}
  70. a 1 < < a n a_{1}<\cdots<a_{n}
  71. n = 1 , , M s n=1,\dots,M_{s}
  72. | Φ i 1 i n a 1 a n |\Phi_{i_{1}\ldots i_{n}}^{a_{1}\ldots a_{n}}\rangle
  73. | Φ |\Phi\rangle
  74. M s M_{s}
  75. e S e^{S}
  76. H e S He^{S}
  77. 1 2 S 2 2 \frac{1}{2}S_{2}^{2}

Cousin_prime.html

  1. B 4 = ( 1 7 + 1 11 ) + ( 1 13 + 1 17 ) + ( 1 19 + 1 23 ) + . B_{4}=\left(\frac{1}{7}+\frac{1}{11}\right)+\left(\frac{1}{13}+\frac{1}{17}% \right)+\left(\frac{1}{19}+\frac{1}{23}\right)+\cdots.

Cousin_problems.html

  1. H 0 ( M , 𝐊 ) 𝜑 H 0 ( M , 𝐊 / 𝐎 ) . H^{0}(M,\mathbf{K})\xrightarrow{\varphi}H^{0}(M,\mathbf{K}/\mathbf{O}).
  2. H 0 ( M , 𝐊 ) 𝜑 H 0 ( M , 𝐊 / 𝐎 ) H 1 ( M , 𝐎 ) H^{0}(M,\mathbf{K})\xrightarrow{\varphi}H^{0}(M,\mathbf{K}/\mathbf{O})\to H^{1% }(M,\mathbf{O})
  3. H 0 ( M , 𝐊 * ) ϕ H 0 ( M , 𝐊 * / 𝐎 * ) . H^{0}(M,\mathbf{K}^{*})\xrightarrow{\phi}H^{0}(M,\mathbf{K}^{*}/\mathbf{O}^{*}).
  4. H 0 ( M , 𝐊 * ) ϕ H 0 ( M , 𝐊 * / 𝐎 * ) H 1 ( M , 𝐎 * ) H^{0}(M,\mathbf{K}^{*})\xrightarrow{\phi}H^{0}(M,\mathbf{K}^{*}/\mathbf{O}^{*}% )\to H^{1}(M,\mathbf{O}^{*})
  5. 0 2 π i 𝐎 exp 𝐎 * 0 0\to 2\pi i\mathbb{Z}\to\mathbf{O}\xrightarrow{\exp}\mathbf{O}^{*}\to 0
  6. 2 π i \scriptstyle{2\pi i\mathbb{Z}}
  7. H 2 ( M , ) \scriptstyle{H^{2}(M,\mathbb{Z})}
  8. H 1 ( M , 𝐎 ) H 1 ( M , 𝐎 * ) 2 π i H 2 ( M , ) H 2 ( M , 𝐎 ) . H^{1}(M,\mathbf{O})\to H^{1}(M,\mathbf{O}^{*})\to 2\pi iH^{2}(M,\mathbb{Z})\to H% ^{2}(M,\mathbf{O}).
  9. q > 0 q>0
  10. H 2 ( M , ) = 0 \scriptstyle{H^{2}(M,\mathbb{Z})=0}

Covariant_derivative.html

  1. u v \nabla_{u}{v}
  2. u v ( P ) \nabla_{u}{v}(P)
  3. u v \nabla_{u}{v}
  4. ( 𝐞 r , 𝐞 θ ) ({\mathbf{e}}_{r},{\mathbf{e}}_{\theta})
  5. 𝐞 r {\mathbf{e}}_{r}
  6. 𝐞 θ {\mathbf{e}}_{\theta}
  7. 𝐯 𝐮 \nabla_{\mathbf{v}}{\mathbf{u}}
  8. M M
  9. ( \R n , ; ) (\R^{n},\langle\cdot;\cdot\rangle)
  10. Ψ : \R d U \R n \vec{\Psi}:\R^{d}\supset U\rightarrow\R^{n}
  11. Ψ ( p ) M \vec{\Psi}(p)\in M
  12. { Ψ x i | p : i { 1 , , d } } \left\{\left.\frac{\partial\vec{\Psi}}{\partial x^{i}}\right|_{p}:i\in\{1,% \dots,d\}\right\}
  13. \R n \R^{n}
  14. g i j = Ψ x i ; Ψ x j g_{ij}=\left\langle\frac{\partial\vec{\Psi}}{\partial x^{i}};\frac{\partial% \vec{\Psi}}{\partial x^{j}}\right\rangle
  15. V = v j Ψ x j \vec{V}=v^{j}\frac{\partial\vec{\Psi}}{\partial x^{j}}\quad
  16. V x i = v j x i Ψ x j + v j 2 Ψ x i x j \quad\frac{\partial\vec{V}}{\partial x^{i}}=\frac{\partial v^{j}}{\partial x^{% i}}\frac{\partial\vec{\Psi}}{\partial x^{j}}+v^{j}\frac{\partial^{2}\vec{\Psi}% }{\partial x^{i}\,\partial x^{j}}
  17. 2 Ψ x i x j = Γ k Ψ x k i j + n \frac{\partial^{2}\vec{\Psi}}{\partial x^{i}\,\partial x^{j}}=\Gamma^{k}{}_{ij% }\frac{\partial\vec{\Psi}}{\partial x^{k}}+\vec{n}
  18. 𝐞 i V \nabla_{{\mathbf{e}}_{i}}\vec{V}
  19. i V \nabla_{i}\vec{V}
  20. 𝐞 i V := V x i - n = ( v k x i + v j Γ k ) i j Ψ x k . \nabla_{{\mathbf{e}}_{i}}\vec{V}:=\frac{\partial\vec{V}}{\partial x^{i}}-\vec{% n}=\left(\frac{\partial v^{k}}{\partial x^{i}}+v^{j}\Gamma^{k}{}_{ij}\right)% \frac{\partial\vec{\Psi}}{\partial x^{k}}.
  21. n \vec{n}
  22. 2 Ψ x i x j ; Ψ x l = Γ k Ψ x k ; Ψ x l i j = Γ k g k l i j \left\langle\frac{\partial^{2}\vec{\Psi}}{\partial x^{i}\,\partial x^{j}};% \frac{\partial\vec{\Psi}}{\partial x^{l}}\right\rangle=\Gamma^{k}{}_{ij}\left% \langle\frac{\partial\vec{\Psi}}{\partial x^{k}};\frac{\partial\vec{\Psi}}{% \partial x^{l}}\right\rangle=\Gamma^{k}{}_{ij}\,g_{kl}
  23. g a b x c = 2 Ψ x c x a ; Ψ x b + Ψ x a ; 2 Ψ x c x b \frac{\partial g_{ab}}{\partial x^{c}}=\left\langle\frac{\partial^{2}\vec{\Psi% }}{\partial x^{c}\,\partial x^{a}};\frac{\partial\vec{\Psi}}{\partial x^{b}}% \right\rangle+\left\langle\frac{\partial\vec{\Psi}}{\partial x^{a}};\frac{% \partial^{2}\vec{\Psi}}{\partial x^{c}\,\partial x^{b}}\right\rangle
  24. g j k x i + g k i x j - g i j x k = 2 2 Ψ x i x j ; Ψ x k \frac{\partial g_{jk}}{\partial x^{i}}+\frac{\partial g_{ki}}{\partial x^{j}}-% \frac{\partial g_{ij}}{\partial x^{k}}=2\left\langle\frac{\partial^{2}\vec{% \Psi}}{\partial x^{i}\,\partial x^{j}};\frac{\partial\vec{\Psi}}{\partial x^{k% }}\right\rangle
  25. g k l Γ k = i j 1 2 ( g j l x i + g l i x j - g i j x l ) . g_{kl}\Gamma^{k}{}_{ij}=\frac{1}{2}\left(\frac{\partial g_{jl}}{\partial x^{i}% }+\frac{\partial g_{li}}{\partial x^{j}}-\frac{\partial g_{ij}}{\partial x^{l}% }\right).
  26. ( 𝐯 f ) p (\nabla_{\mathbf{v}}f)_{p}
  27. ϕ : [ - 1 , 1 ] M \phi:[-1,1]\to M
  28. ϕ ( 0 ) = p \phi(0)=p
  29. ϕ ( 0 ) = 𝐯 \phi^{\prime}(0)=\mathbf{v}
  30. ( 𝐯 f ) p = ( f ϕ ) ( 0 ) = lim t 0 t - 1 ( f ( ϕ ( t ) ) - f ( p ) ) . (\nabla_{\mathbf{v}}f)_{p}=(f\circ\phi)^{\prime}(0)=\lim_{t\to 0}t^{-1}(f(\phi% (t))-f(p)).
  31. 𝐯 f \nabla_{\mathbf{v}}f
  32. ( 𝐯 f ) p (\nabla_{\mathbf{v}}f)_{p}
  33. \nabla
  34. ( 𝐯 𝐮 ) p (\nabla_{\mathbf{v}}{\mathbf{u}})_{p}
  35. ( 𝐮 , 𝐯 ) ({\mathbf{u}},{\mathbf{v}})
  36. ( 𝐯 𝐮 ) p (\nabla_{\mathbf{v}}{\mathbf{u}})_{p}
  37. 𝐯 {\mathbf{v}}
  38. ( g 𝐱 + h 𝐲 𝐮 ) p = ( 𝐱 𝐮 ) p g + ( 𝐲 𝐮 ) p h (\nabla_{g{\mathbf{x}}+h{\mathbf{y}}}{\mathbf{u}})_{p}=(\nabla_{\mathbf{x}}{% \mathbf{u}})_{p}g+(\nabla_{\mathbf{y}}{\mathbf{u}})_{p}h
  39. ( 𝐯 𝐮 ) p (\nabla_{\mathbf{v}}{\mathbf{u}})_{p}
  40. 𝐮 {\mathbf{u}}
  41. ( 𝐯 ( 𝐮 + 𝐰 ) ) p = ( 𝐯 𝐮 ) p + ( 𝐯 𝐰 ) p (\nabla_{\mathbf{v}}({\mathbf{u}}+{\mathbf{w}}))_{p}=(\nabla_{\mathbf{v}}{% \mathbf{u}})_{p}+(\nabla_{\mathbf{v}}{\mathbf{w}})_{p}
  42. ( 𝐯 𝐮 ) p (\nabla_{\mathbf{v}}{\mathbf{u}})_{p}
  43. ( 𝐯 ( f 𝐮 ) ) p = f ( p ) ( 𝐯 𝐮 ) p + ( 𝐯 f ) p 𝐮 p (\nabla_{\mathbf{v}}(f{\mathbf{u}}))_{p}=f(p)(\nabla_{\mathbf{v}}{\mathbf{u}})% _{p}+(\nabla_{\mathbf{v}}f)_{p}{\mathbf{u}}_{p}
  44. 𝐯 f \nabla_{\mathbf{v}}f
  45. 𝐯 𝐮 \nabla_{\mathbf{v}}\mathbf{u}
  46. ( 𝐯 𝐮 ) p (\nabla_{\mathbf{v}}\mathbf{u})_{p}
  47. ( 𝐯 𝐮 ) p (\nabla_{\mathbf{v}}{\mathbf{u}})_{p}
  48. α \alpha
  49. ( 𝐯 α ) p (\nabla_{\mathbf{v}}\alpha)_{p}
  50. ( 𝐯 α ) p (\nabla_{\mathbf{v}}\alpha)_{p}
  51. ( 𝐯 α ) p ( 𝐮 p ) = 𝐯 ( α ( 𝐮 ) ) p - α p ( ( 𝐯 𝐮 ) p ) . (\nabla_{\mathbf{v}}\alpha)_{p}({\mathbf{u}}_{p})=\nabla_{\mathbf{v}}(\alpha({% \mathbf{u}}))_{p}-\alpha_{p}((\nabla_{\mathbf{v}}{\mathbf{u}})_{p}).
  52. φ \varphi
  53. ψ \psi\,
  54. 𝐯 ( φ ψ ) p = ( 𝐯 φ ) p ψ ( p ) + φ ( p ) ( 𝐯 ψ ) p , \nabla_{\mathbf{v}}(\varphi\otimes\psi)_{p}=(\nabla_{\mathbf{v}}\varphi)_{p}% \otimes\psi(p)+\varphi(p)\otimes(\nabla_{\mathbf{v}}\psi)_{p},
  55. φ \varphi
  56. ψ \psi
  57. 𝐯 ( φ + ψ ) p = ( 𝐯 φ ) p + ( 𝐯 ψ ) p . \nabla_{\mathbf{v}}(\varphi+\psi)_{p}=(\nabla_{\mathbf{v}}\varphi)_{p}+(\nabla% _{\mathbf{v}}\psi)_{p}.
  58. ( Y T ) ( α 1 , α 2 , , X 1 , X 2 , ) = Y ( T ( α 1 , α 2 , , X 1 , X 2 , ) ) (\nabla_{Y}T)(\alpha_{1},\alpha_{2},\ldots,X_{1},X_{2},\ldots)=Y(T(\alpha_{1},% \alpha_{2},\ldots,X_{1},X_{2},\ldots))
  59. - T ( Y α 1 , α 2 , , X 1 , X 2 , ) - T ( α 1 , Y α 2 , , X 1 , X 2 , ) - -T(\nabla_{Y}\alpha_{1},\alpha_{2},\ldots,X_{1},X_{2},\ldots)-T(\alpha_{1},% \nabla_{Y}\alpha_{2},\ldots,X_{1},X_{2},\ldots)-\ldots
  60. - T ( α 1 , α 2 , , Y X 1 , X 2 , ) - T ( α 1 , α 2 , , X 1 , Y X 2 , ) - -T(\alpha_{1},\alpha_{2},\ldots,\nabla_{Y}X_{1},X_{2},\ldots)-T(\alpha_{1},% \alpha_{2},\ldots,X_{1},\nabla_{Y}X_{2},\ldots)-\ldots
  61. x i , i = 0 , 1 , 2 , x^{i},\ i=0,1,2,\dots
  62. 𝐞 i = x i \mathbf{e}_{i}={\partial\over\partial x^{i}}
  63. Γ k 𝐞 k \Gamma^{k}{\mathbf{e}}_{k}\,
  64. 𝐞 j {\mathbf{e}}_{j}\,
  65. 𝐞 i {\mathbf{e}}_{i}\,
  66. 𝐞 i 𝐞 j = Γ k 𝐞 k i j , \nabla_{{\mathbf{e}}_{i}}{\mathbf{e}}_{j}=\Gamma^{k}{}_{ij}{\mathbf{e}}_{k},
  67. Γ i j k \Gamma^{k}_{\ ij}
  68. 𝐯 = v i e i {\mathbf{v}}=v^{i}e_{i}
  69. 𝐮 = u j e j {\mathbf{u}}=u^{j}e_{j}
  70. 𝐯 𝐮 = v i 𝐞 i u j 𝐞 j = v i 𝐞 i u j 𝐞 j = v i u j 𝐞 i 𝐞 j + v i 𝐞 j 𝐞 i u j = v i u j Γ k 𝐞 k i j + v i u j x i 𝐞 j \nabla_{\mathbf{v}}{\mathbf{u}}=\nabla_{v^{i}{\mathbf{e}}_{i}}u^{j}{\mathbf{e}% }_{j}=v^{i}\nabla_{{\mathbf{e}}_{i}}u^{j}{\mathbf{e}}_{j}=v^{i}u^{j}\nabla_{{% \mathbf{e}}_{i}}{\mathbf{e}}_{j}+v^{i}{\mathbf{e}}_{j}\nabla_{{\mathbf{e}}_{i}% }u^{j}=v^{i}u^{j}\Gamma^{k}{}_{ij}{\mathbf{e}}_{k}+v^{i}{\partial u^{j}\over% \partial x^{i}}{\mathbf{e}}_{j}
  71. 𝐯 𝐮 = ( v i u j Γ k + i j v i u k x i ) 𝐞 k \nabla_{\mathbf{v}}{\mathbf{u}}=\left(v^{i}u^{j}\Gamma^{k}{}_{ij}+v^{i}{% \partial u^{k}\over\partial x^{i}}\right){\mathbf{e}}_{k}
  72. 𝐞 j 𝐮 = j 𝐮 = ( u i x j + u k Γ i ) j k 𝐞 i \nabla_{{\mathbf{e}}_{j}}{\mathbf{u}}=\nabla_{j}{\mathbf{u}}=\left(\frac{% \partial u^{i}}{\partial x^{j}}+u^{k}\Gamma^{i}{}_{jk}\right){\mathbf{e}}_{i}
  73. 𝐞 j θ = ( θ i x j - θ k Γ k ) i j 𝐞 * i \nabla_{{\mathbf{e}}_{j}}{\mathbf{\theta}}=\left(\frac{\partial\theta_{i}}{% \partial x^{j}}-\theta_{k}\Gamma^{k}{}_{ij}\right){\mathbf{e}^{*}}^{i}
  74. 𝐞 * i ( 𝐞 j ) = δ i j {\mathbf{e}^{*}}^{i}({\mathbf{e}}_{j})={\delta^{i}}_{j}
  75. e c e_{c}
  76. ( e c T ) a 1 a r = b 1 b s x c T a 1 a r + b 1 b s Γ a 1 T d a 2 a r d c + b 1 b s + Γ a r T a 1 a r - 1 d d c b 1 b s (\nabla{e_{c}}T)^{a_{1}\ldots a_{r}}{}_{b_{1}\ldots b_{s}}=\frac{\partial}{% \partial x^{c}}T^{a_{1}\ldots a_{r}}{}_{b_{1}\ldots b_{s}}+\,\Gamma^{a_{1}}{}_% {dc}T^{da_{2}\ldots a_{r}}{}_{b_{1}\ldots b_{s}}+\cdots+\Gamma^{a_{r}}{}_{dc}T% ^{a_{1}\ldots a_{r-1}d}{}_{b_{1}\ldots b_{s}}
  77. - Γ d T a 1 a r b 1 c - d b 2 b s - Γ d T a 1 a r b s c . b 1 b s - 1 d -\,\Gamma^{d}{}_{b_{1}c}T^{a_{1}\ldots a_{r}}{}_{db_{2}\ldots b_{s}}-\cdots-% \Gamma^{d}{}_{b_{s}c}T^{a_{1}\ldots a_{r}}{}_{b_{1}\ldots b_{s-1}d}.
  78. + Γ a i d c +\Gamma^{a_{i}}{}_{dc}
  79. a i a_{i}
  80. - Γ d b i c -\Gamma^{d}{}_{b_{i}c}
  81. b i b_{i}
  82. - Γ d T a 1 a r d c . b 1 b s -\Gamma^{d}{}_{dc}T^{a_{1}\ldots a_{r}}{}_{b_{1}\ldots b_{s}}.
  83. - g \sqrt{-g}
  84. ( - g ) ; c = ( - g ) , c - - g Γ d d c (\sqrt{-g})_{;c}=(\sqrt{-g})_{,c}-\sqrt{-g}\,\Gamma^{d}{}_{dc}
  85. ϕ \displaystyle\phi\,
  86. ϕ ; a a ϕ \displaystyle\phi_{;a}\equiv\partial_{a}\phi
  87. λ a \lambda^{a}\,
  88. λ a ; b b λ a + Γ a λ c b c \lambda^{a}{}_{;b}\equiv\partial_{b}\lambda^{a}+\Gamma^{a}{}_{bc}\lambda^{c}
  89. λ a \lambda_{a}\,
  90. λ a ; c c λ a - Γ b λ b c a \lambda_{a;c}\equiv\partial_{c}\lambda_{a}-\Gamma^{b}{}_{ca}\lambda_{b}
  91. τ a b \tau^{ab}\,
  92. τ a b ; c c τ a b + Γ a τ d b c d + Γ b τ a d c d \tau^{ab}{}_{;c}\equiv\partial_{c}\tau^{ab}+\Gamma^{a}{}_{cd}\tau^{db}+\Gamma^% {b}{}_{cd}\tau^{ad}
  93. τ a b \tau_{ab}\,
  94. τ a b ; c c τ a b - Γ d τ d b c a - Γ d τ a d c b \tau_{ab;c}\equiv\partial_{c}\tau_{ab}-\Gamma^{d}{}_{ca}\tau_{db}-\Gamma^{d}{}% _{cb}\tau_{ad}
  95. τ a b \tau^{a}{}_{b}\,
  96. τ a b ; c c τ a + b Γ a τ d c d - b Γ d τ a c b d \tau^{a}{}_{b;c}\equiv\partial_{c}\tau^{a}{}_{b}+\Gamma^{a}{}_{cd}\tau^{d}{}_{% b}-\Gamma^{d}{}_{cb}\tau^{a}{}_{d}
  97. τ a b ; c ( 𝐞 c τ ) a b \tau^{ab}{}_{;c}\equiv(\nabla_{{\mathbf{e}}_{c}}\tau)^{ab}
  98. λ a ; b c λ a ; c b \lambda_{a;bc}\neq\lambda_{a;cb}\,
  99. λ a ; b c - λ a ; c b = R d λ d a b c \lambda_{a;bc}-\lambda_{a;cb}=R^{d}{}_{abc}\lambda_{d}
  100. R d a b c R^{d}{}_{abc}\,
  101. λ a - ; b c λ a = ; c b - R a λ d d b c \lambda^{a}{}_{;bc}-\lambda^{a}{}_{;cb}=-R^{a}{}_{dbc}\lambda^{d}
  102. τ a b - ; c d τ a b = ; d c - R a τ e b e c d - R b τ a e e c d \tau^{ab}{}_{;cd}-\tau^{ab}{}_{;dc}=-R^{a}{}_{ecd}\tau^{eb}-R^{b}{}_{ecd}\tau^% {ae}
  103. τ a b = λ a μ b \tau^{ab}=\lambda^{a}\mu^{b}\,
  104. e j 𝐯 = def v s e s ; j v i = ; j v i + , j v k Γ i k j \nabla_{e_{j}}{\mathbf{v}}\ \stackrel{\mathrm{def}}{=}\ v^{s}{}_{;j}e_{s}\;\;% \;\;\;\;v^{i}{}_{;j}=v^{i}{}_{,j}+v^{k}\Gamma^{i}{}_{kj}
  105. v i , j v^{i}{}_{,j}
  106. v k Γ i k j v^{k}\Gamma^{i}{}_{kj}
  107. e j 𝐯 = def v i | | j \nabla_{e_{j}}{\mathbf{v}}\ \stackrel{\mathrm{def}}{=}\ v^{i}{}_{||j}
  108. X T \nabla_{X}T
  109. T T
  110. p p
  111. X X
  112. p p
  113. γ ( t ) \gamma(t)
  114. D t T = γ ˙ ( t ) T . D_{t}T=\nabla_{\dot{\gamma}(t)}T.
  115. T T
  116. γ ( t ) \gamma(t)
  117. γ ˙ ( t ) \dot{\gamma}(t)
  118. γ \gamma
  119. γ ˙ ( t ) γ ˙ ( t ) \nabla_{\dot{\gamma}(t)}\dot{\gamma}(t)
  120. \partial

Covariant_transformation.html

  1. 𝐯 = i v i 𝐞 i = j v j 𝐞 j . \mathbf{v}=\sum_{i}v^{i}{\mathbf{e}}_{i}=\sum_{j}{v^{\prime}\,}^{j}\mathbf{e}^% {\prime}_{j}.
  2. x i , i = 0 , 1 , x^{i},\;i=0,1,\dots
  3. x j , j = 0 , 1 , {x^{\prime}}^{j},j=0,1,\dots
  4. x i {x}^{i}
  5. x i ( x j ) , j = 0 , 1 , {x}^{i}({x^{\prime}}^{j}),j=0,1,\dots
  6. f x i = f x j x j x i \frac{\partial f}{\partial{x^{\prime}}^{i}}=\frac{\partial f}{\partial{x}^{j}}% \;\frac{\partial{x}^{j}}{\partial{x^{\prime}}^{i}}
  7. f , i = def f x i f_{,i}\ \stackrel{\mathrm{def}}{=}\ \frac{\partial f}{\partial x^{i}}
  8. x i x^{i}
  9. i = 0 , 1 , i=0,1,\dots
  10. f ( x 0 , x 1 , ) f\;(x^{0},x^{1},\dots)
  11. d c / d λ dc/d\lambda
  12. 𝐯 [ f ] = def d f d λ = d d λ f ( c ( λ ) ) {\mathbf{v}}[f]\ \stackrel{\mathrm{def}}{=}\ \frac{df}{d\lambda}=\frac{d\;\;}{% d\lambda}f(c(\lambda))
  13. 𝐯 [ f ] = d x i d λ f x i {\mathbf{v}}[f]=\frac{dx^{i}}{d\lambda}\frac{\partial f}{\partial x^{i}}
  14. / x i \partial/\partial x^{i}
  15. 𝐯 = d x i d λ x i = d x i d λ 𝐞 i {\mathbf{v}}=\frac{dx^{i}}{d\lambda}\frac{\partial\;\;}{\partial x^{i}}=\frac{% dx^{i}}{d\lambda}{\mathbf{e}}_{i}
  16. 𝐞 i = / x i {\mathbf{e}}_{i}=\partial/\partial x^{i}
  17. x i , i = 0 , 1 , {x^{\prime}}^{i},\;i=0,1,\dots
  18. x i {x^{i}}
  19. x i ( x j ) , j = 0 , 1 , x^{i}({x^{\prime}}^{j}),j=0,1,\dots
  20. 𝐞 i = / x i {\mathbf{e}}^{\prime}_{i}={\partial}/{\partial{x^{\prime}}^{i}}
  21. 𝐞 i {\mathbf{e}}_{i}
  22. 𝐞 i = x i = x j x i x j = x j x i 𝐞 j {\mathbf{e}}^{\prime}_{i}=\frac{\partial}{\partial{x^{\prime}}^{i}}=\frac{% \partial x^{j}}{\partial{x^{\prime}}^{i}}\frac{\partial}{\partial x^{j}}=\frac% {\partial x^{j}}{\partial{x^{\prime}}^{i}}{\mathbf{e}}_{j}
  23. v i v^{i}
  24. 𝐞 i {\mathbf{e}}_{i}
  25. 𝐞 i {\mathbf{e}}^{\prime}_{i}
  26. v i {v^{\prime}}^{i}
  27. 𝐯 = v i 𝐞 i = v i 𝐞 i {\mathbf{v}}=v^{i}{\mathbf{e}}_{i}={v^{\prime}}^{i}{\mathbf{e}}^{\prime}_{i}
  28. v i = d x i d λ and v i = d x i d λ v^{i}=\frac{dx^{i}}{d\lambda}\;\mbox{ and }~{}\;{v^{\prime}}^{i}=\frac{d{x^{% \prime}}^{i}}{d\lambda}
  29. v i = d x i d λ = x i x j d x j d λ = x i x j v j {v^{\prime}}^{i}=\frac{d{x^{\prime}}^{i}}{d\lambda\;\;}=\frac{\partial{x^{% \prime}}^{i}}{\partial x^{j}}\frac{dx^{j}}{d\lambda}=\frac{\partial{x^{\prime}% }^{i}}{\partial x^{j}}{v}^{j}
  30. x i x^{i}
  31. d x i dx^{i}
  32. d x i = x i x j d x j d{x^{\prime}}^{i}=\frac{\partial{x^{\prime}}^{i}}{\partial{x}^{j}}{dx}^{j}
  33. f ( 𝐯 + 𝐰 ) = f ( 𝐯 ) + f ( 𝐰 ) f({\mathbf{v}}+{\mathbf{w}})=f({\mathbf{v}})+f({\mathbf{w}})
  34. f ( α 𝐯 ) = α f ( 𝐯 ) f(\alpha{\mathbf{v}})=\alpha f({\mathbf{v}})
  35. 𝐞 i {\mathbf{e}}_{i}
  36. 𝐞 i {\mathbf{e}}_{i}
  37. ω 0 {\omega}^{0}
  38. 𝐞 0 {\mathbf{e}}_{0}
  39. ω 0 {\omega}^{0}
  40. 𝐯 = v i 𝐞 i {\mathbf{v}}=v^{i}{\mathbf{e}}_{i}
  41. ω 0 ( 𝐯 ) = ω 0 ( v i 𝐞 i ) = v i ω 0 ( 𝐞 i ) = v 0 \omega^{0}({\mathbf{v}})=\omega^{0}(v^{i}{\mathbf{e}}_{i})=v^{i}\omega^{0}({% \mathbf{e}}_{i})=v^{0}
  42. ω i \omega^{i}
  43. 𝐞 i {\mathbf{e}}_{i}
  44. σ [ 𝐮 ] := σ , 𝐮 \sigma[{\mathbf{u}}]:=\langle\sigma,{\mathbf{u}}\rangle
  45. σ , 𝐮 \langle\sigma,{\mathbf{u}}\rangle
  46. T ( σ , , ρ , 𝐮 , , 𝐯 ) or as T σ ρ 𝐮 𝐯 T(\sigma,\ldots,\rho,{\mathbf{u}},\ldots,{\mathbf{v}})\;\,\text{ or as }\;{T^{% \sigma\ldots\rho}}_{{\mathbf{u}}\ldots{\mathbf{v}}}
  47. 𝐮 , 𝐯 {\mathbf{u}},{\mathbf{v}}
  48. ω i ω j \omega^{i}\ldots\omega^{j}
  49. 𝐞 k 𝐞 l {\mathbf{e}}_{k}\ldots{\mathbf{e}}_{l}
  50. T ( ω i , , ω j , 𝐞 k 𝐞 l ) = T i j k l T(\omega^{i},\ldots,\omega^{j},{\mathbf{e}}_{k}\ldots{\mathbf{e}}_{l})={T^{i% \ldots j}}_{k\ldots l}
  51. T i j k l {T^{i\ldots j}}_{k\ldots l}
  52. A i j = x l x i x m x j A l m {A^{\prime}}_{ij}=\frac{\partial x^{l}}{\partial{x^{\prime}}^{i}}\frac{% \partial x^{m}}{\partial{x^{\prime}}^{j}}A_{lm}
  53. A i j = x i x l x j x m A l m {A^{\prime}\,}^{ij}=\frac{\partial{x^{\prime}}^{i}}{\partial x^{l}}\frac{% \partial{x^{\prime}}^{j}}{\partial x^{m}}A^{lm}
  54. A i = j x i x l x m x j A l m {A^{\prime}\,}^{i}{}_{j}=\frac{\partial{x^{\prime}}^{i}}{\partial x^{l}}\frac{% \partial x^{m}}{\partial{x^{\prime}}^{j}}A^{l}{}_{m}

Cover_(topology).html

  1. X X
  2. X X
  3. C = { U α : α A } C=\{U_{\alpha}:\alpha\in A\}
  4. U α U_{\alpha}
  5. C C
  6. X X
  7. X α A U α . X\subseteq\bigcup_{\alpha\in A}U_{\alpha}.
  8. Y α A U α Y\subseteq\bigcup_{\alpha\in A}U_{\alpha}
  9. { α A : U α N ( x ) } \left\{\alpha\in A:U_{\alpha}\cap N(x)\neq\varnothing\right\}
  10. D = V β B D=V_{\beta\in B}
  11. U α A when β α V β U α U_{\alpha\in A}\qquad\mbox{when}~{}\qquad\forall\beta\ \exists\alpha\ V_{\beta% }\subseteq U_{\alpha}
  12. ϕ : B A \phi:B\rightarrow A
  13. V β U ϕ ( β ) V_{\beta}\subseteq U_{\phi(\beta)}
  14. β B \beta\in B
  15. a 0 < a 1 < < a n a_{0}<a_{1}<...<a_{n}
  16. a 0 < b 0 < a 1 < a 2 < < a n < b 1 a_{0}<b_{0}<a_{1}<a_{2}<...<a_{n}<b_{1}

Cramér–Rao_bound.html

  1. θ \theta
  2. x x
  3. f ( x ; θ ) f(x;\theta)
  4. θ ^ \hat{\theta}
  5. θ \theta
  6. I ( θ ) I(\theta)
  7. var ( θ ^ ) 1 I ( θ ) \mathrm{var}(\hat{\theta})\geq\frac{1}{I(\theta)}
  8. I ( θ ) I(\theta)
  9. I ( θ ) = E [ ( ( x ; θ ) θ ) 2 ] = - E [ 2 ( x ; θ ) θ 2 ] I(\theta)=\mathrm{E}\left[\left(\frac{\partial\ell(x;\theta)}{\partial\theta}% \right)^{2}\right]=-\mathrm{E}\left[\frac{\partial^{2}\ell(x;\theta)}{\partial% \theta^{2}}\right]
  10. ( x ; θ ) = log ( f ( x ; θ ) ) \ell(x;\theta)=\log(f(x;\theta))
  11. E \mathrm{E}
  12. x x
  13. θ ^ \hat{\theta}
  14. e ( θ ^ ) = I ( θ ) - 1 var ( θ ^ ) e(\hat{\theta})=\frac{I(\theta)^{-1}}{{\rm var}(\hat{\theta})}
  15. e ( θ ^ ) 1. e(\hat{\theta})\leq 1.
  16. T ( X ) T(X)
  17. θ \theta
  18. E { T ( X ) } = ψ ( θ ) E\{T(X)\}=\psi(\theta)
  19. var ( T ) [ ψ ( θ ) ] 2 I ( θ ) \mathrm{var}(T)\geq\frac{[\psi^{\prime}(\theta)]^{2}}{I(\theta)}
  20. ψ ( θ ) \psi^{\prime}(\theta)
  21. ψ ( θ ) \psi(\theta)
  22. θ \theta
  23. I ( θ ) I(\theta)
  24. θ ^ \hat{\theta}
  25. b ( θ ) = E { θ ^ } - θ b(\theta)=E\{\hat{\theta}\}-\theta
  26. ψ ( θ ) = b ( θ ) + θ \psi(\theta)=b(\theta)+\theta
  27. ψ ( θ ) \psi(\theta)
  28. ( ψ ( θ ) ) 2 / I ( θ ) (\psi^{\prime}(\theta))^{2}/I(\theta)
  29. θ ^ \hat{\theta}
  30. b ( θ ) b(\theta)
  31. var ( θ ^ ) [ 1 + b ( θ ) ] 2 I ( θ ) . \mathrm{var}\left(\hat{\theta}\right)\geq\frac{[1+b^{\prime}(\theta)]^{2}}{I(% \theta)}.
  32. b ( θ ) = 0 b(\theta)=0
  33. E ( ( θ ^ - θ ) 2 ) [ 1 + b ( θ ) ] 2 I ( θ ) + b ( θ ) 2 , \mathrm{E}\left((\hat{\theta}-\theta)^{2}\right)\geq\frac{[1+b^{\prime}(\theta% )]^{2}}{I(\theta)}+b(\theta)^{2},
  34. s y m b o l θ = [ θ 1 , θ 2 , , θ d ] T d symbol{\theta}=\left[\theta_{1},\theta_{2},\dots,\theta_{d}\right]^{T}\in% \mathbb{R}^{d}
  35. f ( x ; s y m b o l θ ) f(x;symbol{\theta})
  36. d × d d\times d
  37. I m , k I_{m,k}
  38. I m , k = E [ θ m log f ( x ; s y m b o l θ ) θ k log f ( x ; s y m b o l θ ) ] = - E [ 2 θ m θ k log f ( x ; s y m b o l θ ) ] . I_{m,k}=\mathrm{E}\left[\frac{\partial}{\partial\theta_{m}}\log f\left(x;% symbol{\theta}\right)\frac{\partial}{\partial\theta_{k}}\log f\left(x;symbol{% \theta}\right)\right]=-\mathrm{E}\left[\frac{\partial^{2}}{\partial\theta_{m}% \partial\theta_{k}}\log f\left(x;symbol{\theta}\right)\right].
  39. s y m b o l T ( X ) symbol{T}(X)
  40. s y m b o l T ( X ) = ( T 1 ( X ) , , T d ( X ) ) T symbol{T}(X)=(T_{1}(X),\ldots,T_{d}(X))^{T}
  41. E [ s y m b o l T ( X ) ] \mathrm{E}[symbol{T}(X)]
  42. s y m b o l ψ ( s y m b o l θ ) symbol{\psi}(symbol{\theta})
  43. s y m b o l T ( X ) symbol{T}(X)
  44. cov s y m b o l θ ( s y m b o l T ( X ) ) s y m b o l ψ ( s y m b o l θ ) s y m b o l θ [ I ( s y m b o l θ ) ] - 1 ( s y m b o l ψ ( s y m b o l θ ) s y m b o l θ ) T \mathrm{cov}_{symbol{\theta}}\left(symbol{T}(X)\right)\geq\frac{\partial symbol% {\psi}\left(symbol{\theta}\right)}{\partial symbol{\theta}}[I\left(symbol{% \theta}\right)]^{-1}\left(\frac{\partial symbol{\psi}\left(symbol{\theta}% \right)}{\partial symbol{\theta}}\right)^{T}
  45. A B A\geq B
  46. A - B A-B
  47. s y m b o l ψ ( s y m b o l θ ) / s y m b o l θ \partial symbol{\psi}(symbol{\theta})/\partial symbol{\theta}
  48. i j ij
  49. ψ i ( s y m b o l θ ) / θ j \partial\psi_{i}(symbol{\theta})/\partial\theta_{j}
  50. s y m b o l T ( X ) symbol{T}(X)
  51. s y m b o l θ symbol{\theta}
  52. s y m b o l ψ ( s y m b o l θ ) = s y m b o l θ symbol{\psi}\left(symbol{\theta}\right)=symbol{\theta}
  53. cov s y m b o l θ ( s y m b o l T ( X ) ) I ( s y m b o l θ ) - 1 . \mathrm{cov}_{symbol{\theta}}\left(symbol{T}(X)\right)\geq I\left(symbol{% \theta}\right)^{-1}.
  54. var s y m b o l θ ( T m ( X ) ) = [ cov s y m b o l θ ( s y m b o l T ( X ) ) ] m m [ I ( s y m b o l θ ) - 1 ] m m ( [ I ( s y m b o l θ ) ] m m ) - 1 . \mathrm{var}_{symbol{\theta}}\left(T_{m}(X)\right)=\left[\mathrm{cov}_{symbol{% \theta}}\left(symbol{T}(X)\right)\right]_{mm}\geq\left[I\left(symbol{\theta}% \right)^{-1}\right]_{mm}\geq\left(\left[I\left(symbol{\theta}\right)\right]_{% mm}\right)^{-1}.
  55. f ( x ; θ ) f(x;\theta)
  56. T ( X ) T(X)
  57. x x
  58. f ( x ; θ ) > 0 f(x;\theta)>0
  59. θ log f ( x ; θ ) \frac{\partial}{\partial\theta}\log f(x;\theta)
  60. x x
  61. θ \theta
  62. T T
  63. θ [ T ( x ) f ( x ; θ ) d x ] = T ( x ) [ θ f ( x ; θ ) ] d x \frac{\partial}{\partial\theta}\left[\int T(x)f(x;\theta)\,dx\right]=\int T(x)% \left[\frac{\partial}{\partial\theta}f(x;\theta)\right]\,dx
  64. f ( x ; θ ) f(x;\theta)
  65. x x
  66. θ \theta
  67. f ( x ; θ ) f(x;\theta)
  68. θ \theta
  69. f ( x ; θ ) f(x;\theta)
  70. 2 θ 2 [ T ( x ) f ( x ; θ ) d x ] = T ( x ) [ 2 θ 2 f ( x ; θ ) ] d x . \frac{\partial^{2}}{\partial\theta^{2}}\left[\int T(x)f(x;\theta)\,dx\right]=% \int T(x)\left[\frac{\partial^{2}}{\partial\theta^{2}}f(x;\theta)\right]\,dx.
  71. I ( θ ) = - E [ 2 θ 2 log f ( X ; θ ) ] . I(\theta)=-\mathrm{E}\left[\frac{\partial^{2}}{\partial\theta^{2}}\log f(X;% \theta)\right].
  72. var ( θ ^ ) 1 I ( θ ) = 1 - E [ 2 θ 2 log f ( X ; θ ) ] . \mathrm{var}\left(\widehat{\theta}\right)\geq\frac{1}{I(\theta)}=\frac{1}{-% \mathrm{E}\left[\frac{\partial^{2}}{\partial\theta^{2}}\log f(X;\theta)\right]}.
  73. T = t ( X ) T=t(X)
  74. ψ ( θ ) \psi(\theta)
  75. X X
  76. E ( T ) = ψ ( θ ) {\rm E}(T)=\psi(\theta)
  77. θ \theta
  78. var ( t ( X ) ) [ ψ ( θ ) ] 2 I ( θ ) . {\rm var}(t(X))\geq\frac{[\psi^{\prime}(\theta)]^{2}}{I(\theta)}.
  79. X X
  80. f ( x ; θ ) f(x;\theta)
  81. T = t ( X ) T=t(X)
  82. ψ ( θ ) \psi(\theta)
  83. V V
  84. V = θ ln f ( X ; θ ) = 1 f ( X ; θ ) θ f ( X ; θ ) V=\frac{\partial}{\partial\theta}\ln f(X;\theta)=\frac{1}{f(X;\theta)}\frac{% \partial}{\partial\theta}f(X;\theta)
  85. V V
  86. E ( V ) {\rm E}(V)
  87. E ( V ) = x f ( x ; θ ) [ 1 f ( x ; θ ) θ f ( x ; θ ) ] d x = θ x f ( x ; θ ) d x = 0 {\rm E}\left(V\right)=\int_{x}f(x;\theta)\left[\frac{1}{f(x;\theta)}\frac{% \partial}{\partial\theta}f(x;\theta)\right]dx=\frac{\partial}{\partial\theta}% \int_{x}f(x;\theta)dx=0
  88. cov ( V , T ) {\rm cov}(V,T)
  89. V V
  90. T T
  91. cov ( V , T ) = E ( V T ) {\rm cov}(V,T)={\rm E}(VT)
  92. E ( V ) = 0 {\rm E}(V)=0
  93. cov ( V , T ) = E ( T [ 1 f ( X ; θ ) θ f ( X ; θ ) ] ) = x t ( x ) [ θ f ( x ; θ ) ] d x = θ [ x t ( x ) f ( x ; θ ) d x ] = ψ ( θ ) {\rm cov}(V,T)={\rm E}\left(T\cdot\left[\frac{1}{f(X;\theta)}\frac{\partial}{% \partial\theta}f(X;\theta)\right]\right)=\int_{x}t(x)\left[\frac{\partial}{% \partial\theta}f(x;\theta)\right]\,dx=\frac{\partial}{\partial\theta}\left[% \int_{x}t(x)f(x;\theta)\,dx\right]=\psi^{\prime}(\theta)
  94. var ( T ) var ( V ) | cov ( V , T ) | = | ψ ( θ ) | \sqrt{{\rm var}(T){\rm var}(V)}\geq\left|{\rm cov}(V,T)\right|=\left|\psi^{% \prime}(\theta)\right|
  95. var ( T ) [ ψ ( θ ) ] 2 var ( V ) = [ ψ ( θ ) ] 2 I ( θ ) {\rm var}(T)\geq\frac{[\psi^{\prime}(\theta)]^{2}}{{\rm var}(V)}=\frac{[\psi^{% \prime}(\theta)]^{2}}{I(\theta)}
  96. s y m b o l x N d ( s y m b o l μ ( s y m b o l θ ) , s y m b o l C ( s y m b o l θ ) ) symbol{x}\sim N_{d}\left(symbol{\mu}\left(symbol{\theta}\right),{symbolC}\left% (symbol{\theta}\right)\right)
  97. I m , k = s y m b o l μ T θ m s y m b o l C - 1 s y m b o l μ θ k + 1 2 tr ( s y m b o l C - 1 s y m b o l C θ m s y m b o l C - 1 s y m b o l C θ k ) I_{m,k}=\frac{\partial symbol{\mu}^{T}}{\partial\theta_{m}}{symbolC}^{-1}\frac% {\partial symbol{\mu}}{\partial\theta_{k}}+\frac{1}{2}\mathrm{tr}\left({% symbolC}^{-1}\frac{\partial{symbolC}}{\partial\theta_{m}}{symbolC}^{-1}\frac{% \partial{symbolC}}{\partial\theta_{k}}\right)
  98. w [ n ] w[n]
  99. N N
  100. θ \theta
  101. σ 2 \sigma^{2}
  102. w [ n ] N ( θ s y m b o l 1 , σ 2 s y m b o l I ) . w[n]\sim\mathbb{N}_{N}\left(\theta{symbol1},\sigma^{2}{symbolI}\right).
  103. I ( θ ) = ( \partialsymbol μ ( θ ) θ ) T s y m b o l C - 1 ( \partialsymbol μ ( θ ) θ ) = i = 1 N 1 σ 2 = N σ 2 , I(\theta)=\left(\frac{\partialsymbol{\mu}(\theta)}{\partial\theta}\right)^{T}{% symbolC}^{-1}\left(\frac{\partialsymbol{\mu}(\theta)}{\partial\theta}\right)=% \sum^{N}_{i=1}\frac{1}{\sigma^{2}}=\frac{N}{\sigma^{2}},
  104. var ( θ ^ ) σ 2 N . \mathrm{var}\left(\hat{\theta}\right)\geq\frac{\sigma^{2}}{N}.
  105. μ \mu
  106. σ 2 \sigma^{2}
  107. T = i = 1 n ( X i - μ ) 2 n . T=\frac{\sum_{i=1}^{n}\left(X_{i}-\mu\right)^{2}}{n}.
  108. σ 2 \sigma^{2}
  109. E ( T ) = σ 2 E(T)=\sigma^{2}
  110. var ( T ) = var ( X - μ ) 2 n = 1 n [ E { ( X - μ ) 4 } - ( E { ( X - μ ) 2 } ) 2 ] \mathrm{var}(T)=\frac{\mathrm{var}(X-\mu)^{2}}{n}=\frac{1}{n}\left[E\left\{(X-% \mu)^{4}\right\}-\left(E\left\{(X-\mu)^{2}\right\}\right)^{2}\right]
  111. 3 ( σ 2 ) 2 3(\sigma^{2})^{2}
  112. ( σ 2 ) 2 (\sigma^{2})^{2}
  113. var ( T ) = 2 ( σ 2 ) 2 n . \mathrm{var}(T)=\frac{2(\sigma^{2})^{2}}{n}.
  114. V = σ 2 log L ( σ 2 , X ) V=\frac{\partial}{\partial\sigma^{2}}\log L(\sigma^{2},X)
  115. L L
  116. V = σ 2 log [ 1 2 π σ 2 e - ( X - μ ) 2 / 2 σ 2 ] = ( X - μ ) 2 2 ( σ 2 ) 2 - 1 2 σ 2 V=\frac{\partial}{\partial\sigma^{2}}\log\left[\frac{1}{\sqrt{2\pi\sigma^{2}}}% e^{-(X-\mu)^{2}/{2\sigma^{2}}}\right]=\frac{(X-\mu)^{2}}{2(\sigma^{2})^{2}}-% \frac{1}{2\sigma^{2}}
  117. I = - E ( V σ 2 ) = - E ( - ( X - μ ) 2 ( σ 2 ) 3 + 1 2 ( σ 2 ) 2 ) = σ 2 ( σ 2 ) 3 - 1 2 ( σ 2 ) 2 = 1 2 ( σ 2 ) 2 . I=-E\left(\frac{\partial V}{\partial\sigma^{2}}\right)=-E\left(-\frac{(X-\mu)^% {2}}{(\sigma^{2})^{3}}+\frac{1}{2(\sigma^{2})^{2}}\right)=\frac{\sigma^{2}}{(% \sigma^{2})^{3}}-\frac{1}{2(\sigma^{2})^{2}}=\frac{1}{2(\sigma^{2})^{2}}.
  118. n n
  119. n n
  120. n 2 ( σ 2 ) 2 . \frac{n}{2(\sigma^{2})^{2}}.
  121. var ( T ) 1 I . \mathrm{var}(T)\geq\frac{1}{I}.
  122. T = i = 1 n ( X i - μ ) 2 n + 2 . T=\frac{\sum_{i=1}^{n}\left(X_{i}-\mu\right)^{2}}{n+2}.
  123. var ( T ) = 2 n ( σ 2 ) 2 ( n + 2 ) 2 . \mathrm{var}(T)=\frac{2n(\sigma^{2})^{2}}{(n+2)^{2}}.
  124. ( 1 - n n + 2 ) σ 2 = 2 σ 2 n + 2 \left(1-\frac{n}{n+2}\right)\sigma^{2}=\frac{2\sigma^{2}}{n+2}
  125. MSE ( T ) = ( 2 n ( n + 2 ) 2 + 4 ( n + 2 ) 2 ) ( σ 2 ) 2 = 2 ( σ 2 ) 2 n + 2 \mathrm{MSE}(T)=\left(\frac{2n}{(n+2)^{2}}+\frac{4}{(n+2)^{2}}\right)(\sigma^{% 2})^{2}=\frac{2(\sigma^{2})^{2}}{n+2}