wpmath0000005_7

Large_Plasma_Device.html

  1. × \times

Large_sieve.html

  1. S { 1 , , N } S\subset\{1,\cdots,N\}
  2. f p ^ ( a ) \widehat{f_{p}}(a)
  3. f ^ ( a / p ) \widehat{f}(a/p)
  4. f ^ \widehat{f}
  5. f ^ ( a / p ) = f p ^ ( a ) \widehat{f}(a/p)=\widehat{f_{p}}(a)
  6. f ^ ( x ) \widehat{f}(x)
  7. | f | 2 = | S | |f|_{2}=\sqrt{|S|}
  8. | f ^ | 2 = | f | 2 |\widehat{f}|_{2}=|f|_{2}
  9. sup v | A v | W / | v | V \sup_{v}|Av|_{W}/|v|_{V}
  10. sup w | A * w | V * / | w | W * \sup_{w}|A^{*}w|^{*}_{V}/|w|^{*}_{W}

Laser_rangefinder.html

  1. D = c t 2 D=\frac{ct}{2}
  2. t = φ ω t=\frac{\varphi}{\omega}
  3. D = 1 2 c t = 1 2 c φ ω = c 4 π f ( N π + Δ φ ) = λ 4 ( N + Δ N ) D=\frac{1}{2}ct=\frac{1}{2}\frac{c\varphi}{\omega}=\frac{c}{4\pi f}(N\pi+% \Delta\varphi)=\frac{\lambda}{4}(N+\Delta N)

Latin_letters_used_in_mathematics.html

  1. 𝔸 \mathbb{A}
  2. \mathcal{B}
  3. 𝔹 \mathbb{B}
  4. \mathcal{B}
  5. \mathbb{C}
  6. s y m b o l E symbol{E}
  7. \mathcal{E}
  8. \mathbb{H}
  9. \mathcal{H}
  10. - 1 2 + 1 2 i 3 -\frac{1}{2}+\frac{1}{2}i\sqrt{3}
  11. \mathcal{L}
  12. \mathbb{N}
  13. ( 0 , 0 , , 0 ) (0,0,\ldots,0)
  14. \mathbb{P}
  15. \mathbb{Q}
  16. \mathbb{R}
  17. n \mathbb{R}^{n}
  18. 𝒮 \mathcal{S}
  19. \mathbb{Z}

Law_of_total_cumulance.html

  1. κ ( X 1 , , X n ) = π κ ( κ ( X i : i B Y ) : B π ) , \kappa(X_{1},\dots,X_{n})=\sum_{\pi}\kappa(\kappa(X_{i}:i\in B\mid Y):B\in\pi),
  2. π \pi
  3. μ 3 ( X ) = E ( μ 3 ( X Y ) ) + μ 3 ( E ( X Y ) ) + 3 cov ( E ( X Y ) , var ( X Y ) ) . \mu_{3}(X)=E(\mu_{3}(X\mid Y))+\mu_{3}(E(X\mid Y))+3\,\operatorname{cov}(E(X% \mid Y),\operatorname{var}(X\mid Y)).\,
  4. κ ( X 1 , X 2 , X 3 , X 4 ) \kappa(X_{1},X_{2},X_{3},X_{4})\,
  5. = κ ( κ ( X 1 , X 2 , X 3 , X 4 Y ) ) =\kappa(\kappa(X_{1},X_{2},X_{3},X_{4}\mid Y))\,
  6. + κ ( κ ( X 1 , X 2 , X 3 Y ) , κ ( X 4 Y ) ) + κ ( κ ( X 1 , X 2 , X 4 Y ) , κ ( X 3 Y ) ) + κ ( κ ( X 1 , X 3 , X 4 Y ) , κ ( X 2 Y ) ) + κ ( κ ( X 2 , X 3 , X 4 Y ) , κ ( X 1 Y ) ) } ( partitions of the 3 + 1 form ) \left.\begin{matrix}&{}+\kappa(\kappa(X_{1},X_{2},X_{3}\mid Y),\kappa(X_{4}% \mid Y))\\ \\ &{}+\kappa(\kappa(X_{1},X_{2},X_{4}\mid Y),\kappa(X_{3}\mid Y))\\ \\ &{}+\kappa(\kappa(X_{1},X_{3},X_{4}\mid Y),\kappa(X_{2}\mid Y))\\ \\ &{}+\kappa(\kappa(X_{2},X_{3},X_{4}\mid Y),\kappa(X_{1}\mid Y))\end{matrix}% \right\}(\mathrm{partitions}\ \mathrm{of}\ \mathrm{the}\ 3+1\ \mathrm{form})
  7. + κ ( κ ( X 1 , X 2 Y ) , κ ( X 3 , X 4 Y ) ) + κ ( κ ( X 1 , X 3 Y ) , κ ( X 2 , X 4 Y ) ) + κ ( κ ( X 1 , X 4 Y ) , κ ( X 2 , X 3 Y ) ) } ( partitions of the 2 + 2 form ) \left.\begin{matrix}&{}+\kappa(\kappa(X_{1},X_{2}\mid Y),\kappa(X_{3},X_{4}% \mid Y))\\ \\ &{}+\kappa(\kappa(X_{1},X_{3}\mid Y),\kappa(X_{2},X_{4}\mid Y))\\ \\ &{}+\kappa(\kappa(X_{1},X_{4}\mid Y),\kappa(X_{2},X_{3}\mid Y))\end{matrix}% \right\}(\mathrm{partitions}\ \mathrm{of}\ \mathrm{the}\ 2+2\ \mathrm{form})
  8. + κ ( κ ( X 1 , X 2 Y ) , κ ( X 3 Y ) , κ ( X 4 Y ) ) + κ ( κ ( X 1 , X 3 Y ) , κ ( X 2 Y ) , κ ( X 4 Y ) ) + κ ( κ ( X 1 , X 4 Y ) , κ ( X 2 Y ) , κ ( X 3 Y ) ) + κ ( κ ( X 2 , X 3 Y ) , κ ( X 1 Y ) , κ ( X 4 Y ) ) + κ ( κ ( X 2 , X 4 Y ) , κ ( X 1 Y ) , κ ( X 3 Y ) ) + κ ( κ ( X 3 , X 4 Y ) , κ ( X 1 Y ) , κ ( X 2 Y ) ) } ( partitions of the 2 + 1 + 1 form ) \left.\begin{matrix}&{}+\kappa(\kappa(X_{1},X_{2}\mid Y),\kappa(X_{3}\mid Y),% \kappa(X_{4}\mid Y))\\ \\ &{}+\kappa(\kappa(X_{1},X_{3}\mid Y),\kappa(X_{2}\mid Y),\kappa(X_{4}\mid Y))% \\ \\ &{}+\kappa(\kappa(X_{1},X_{4}\mid Y),\kappa(X_{2}\mid Y),\kappa(X_{3}\mid Y))% \\ \\ &{}+\kappa(\kappa(X_{2},X_{3}\mid Y),\kappa(X_{1}\mid Y),\kappa(X_{4}\mid Y))% \\ \\ &{}+\kappa(\kappa(X_{2},X_{4}\mid Y),\kappa(X_{1}\mid Y),\kappa(X_{3}\mid Y))% \\ \\ &{}+\kappa(\kappa(X_{3},X_{4}\mid Y),\kappa(X_{1}\mid Y),\kappa(X_{2}\mid Y))% \end{matrix}\right\}(\mathrm{partitions}\ \mathrm{of}\ \mathrm{the}\ 2+1+1\ % \mathrm{form})
  9. + κ ( κ ( X 1 Y ) , κ ( X 2 Y ) , κ ( X 3 Y ) , κ ( X 4 Y ) ) . {}+\kappa(\kappa(X_{1}\mid Y),\kappa(X_{2}\mid Y),\kappa(X_{3}\mid Y),\kappa(X% _{4}\mid Y)).\,
  10. X = y = 1 Y W y . X=\sum_{y=1}^{Y}W_{y}.\,
  11. κ n ( W 1 + + W m ) = κ n ( W 1 ) + + κ n ( W m ) . \kappa_{n}(W_{1}+\cdots+W_{m})=\kappa_{n}(W_{1})+\cdots+\kappa_{n}(W_{m}).\,
  12. κ 4 ( X ) = κ ( X , X , X , X ) \kappa_{4}(X)=\kappa(X,X,X,X)\,
  13. = κ 1 ( κ 4 ( X Y ) ) + 4 κ ( κ 3 ( X Y ) , κ 1 ( X Y ) ) + 3 κ 2 ( κ 2 ( X Y ) ) =\kappa_{1}(\kappa_{4}(X\mid Y))+4\kappa(\kappa_{3}(X\mid Y),\kappa_{1}(X\mid Y% ))+3\kappa_{2}(\kappa_{2}(X\mid Y))\,
  14. + 6 κ ( κ 2 ( X Y ) , κ 1 ( X Y ) , κ 1 ( X Y ) ) + κ 4 ( κ 1 ( X Y ) ) {}+6\kappa(\kappa_{2}(X\mid Y),\kappa_{1}(X\mid Y),\kappa_{1}(X\mid Y))+\kappa% _{4}(\kappa_{1}(X\mid Y))\,
  15. = κ 1 ( Y κ 4 ( W ) ) + 4 κ ( Y κ 3 ( W ) , Y κ 1 ( W ) ) + 3 κ 2 ( Y κ 2 ( W ) ) =\kappa_{1}(Y\kappa_{4}(W))+4\kappa(Y\kappa_{3}(W),Y\kappa_{1}(W))+3\kappa_{2}% (Y\kappa_{2}(W))\,
  16. + 6 κ ( Y κ 2 ( W ) , Y κ 1 ( W ) , Y κ 1 ( W ) ) + κ 4 ( Y κ 1 ( W ) ) {}+6\kappa(Y\kappa_{2}(W),Y\kappa_{1}(W),Y\kappa_{1}(W))+\kappa_{4}(Y\kappa_{1% }(W))\,
  17. = κ 4 ( W ) κ 1 ( Y ) + 4 κ 3 ( W ) κ 1 ( W ) κ 2 ( Y ) + 3 κ 2 ( W ) 2 κ 2 ( Y ) =\kappa_{4}(W)\kappa_{1}(Y)+4\kappa_{3}(W)\kappa_{1}(W)\kappa_{2}(Y)+3\kappa_{% 2}(W)^{2}\kappa_{2}(Y)\,
  18. + 6 κ 2 ( W ) κ 1 ( W ) 2 κ 3 ( Y ) + κ 1 ( W ) 4 κ 4 ( Y ) {}+6\kappa_{2}(W)\kappa_{1}(W)^{2}\kappa_{3}(Y)+\kappa_{1}(W)^{4}\kappa_{4}(Y)\,
  19. = κ 4 ( W ) + 4 κ 3 ( W ) κ 1 ( W ) + 3 κ 2 ( W ) 2 + 6 κ 2 ( W ) κ 1 ( W ) 2 + κ 1 ( W ) 4 . =\kappa_{4}(W)+4\kappa_{3}(W)\kappa_{1}(W)+3\kappa_{2}(W)^{2}+6\kappa_{2}(W)% \kappa_{1}(W)^{2}+\kappa_{1}(W)^{4}.\,
  20. = E ( W 4 ) =E(W^{4})\,
  21. κ n ( X ) = p κ n ( F ) + q κ n ( G ) + π < 1 ^ κ | π | ( Y ) B π ( κ | B | ( F ) - κ | B | ( G ) ) \kappa_{n}(X)=p\kappa_{n}(F)+q\kappa_{n}(G)+\sum_{\pi<\widehat{1}}\kappa_{% \left|\pi\right|}(Y)\prod_{B\in\pi}(\kappa_{\left|B\right|}(F)-\kappa_{\left|B% \right|}(G))
  22. π < 1 ^ \pi<\widehat{1}
  23. κ 3 ( X ) = p κ 3 ( F ) + q κ 3 ( G ) + 3 p q ( κ 2 ( F ) - κ 2 ( G ) ) ( κ 1 ( F ) - κ 1 ( G ) ) + p q ( q - p ) ( κ 1 ( F ) - κ 1 ( G ) ) 3 . \kappa_{3}(X)=p\kappa_{3}(F)+q\kappa_{3}(G)+3pq(\kappa_{2}(F)-\kappa_{2}(G))(% \kappa_{1}(F)-\kappa_{1}(G))+pq(q-p)(\kappa_{1}(F)-\kappa_{1}(G))^{3}.\,

Lead_time.html

  1. O L T V = j Q u a n t i t y j O L T j T o t a l Q u a n t i t y D e l i v e r OLTV=\frac{\sum_{j}{Quantity_{j}\cdot OLT_{j}}}{{}_{TotalQuantityDeliver}}\,

Leakage_inductance.html

  1. L o c p r i = L P = L P σ + L M L_{oc}^{pri}=L_{P}=L_{P}^{\sigma}+L_{M}
  2. L P σ = L P ( 1 - k ) L_{P}^{\sigma}=L_{P}\cdot{(1-k)}
  3. L M = L P k L_{M}=L_{P}\cdot{k}
  4. L o c p r i L_{oc}^{pri}
  5. L P L_{P}
  6. L P σ L_{P}^{\sigma}
  7. L M L_{M}
  8. L o c s e c = L S = L S σ + L M L_{oc}^{sec}=L_{S}=L_{S}^{\sigma}+L_{M}
  9. L S σ = L S ( 1 - k ) L_{S}^{\sigma}=L_{S}\cdot{(1-k)}
  10. L M = L S k L_{M}=L_{S}\cdot{k}
  11. L o c s e c L_{oc}^{sec}
  12. L S L_{S}
  13. L S σ L_{S}^{\sigma}
  14. L M a 2 \frac{L_{M}}{a^{2}}
  15. a a
  16. k = M / L P L S k=M/\sqrt{L_{P}L_{S}}
  17. a = N P / N S = v P / v S = i S / i P = L P / L S a=N_{P}/N_{S}=v_{P}/v_{S}=i_{S}/i_{P}=\sqrt{L_{P}/L_{S}}
  18. v P = R P i P + d Ψ P d t v_{P}=R_{P}i_{P}+\frac{d\Psi{{}_{P}}}{dt}
  19. v S = - R S i S - d Ψ S d t v_{S}=-R_{S}i_{S}-\frac{d\Psi{{}_{S}}}{dt}
  20. Ψ P = L P i P - M i S \Psi_{P}=L_{P}i_{P}-Mi_{S}
  21. Ψ S = L S i S - M i P \Psi_{S}=L_{S}i_{S}-Mi_{P}
  22. σ = 1 - M 2 L P L S = 1 - k 2 = L s c L o c = L s c s e c L P = L s c p r i L S = i o c i s c \sigma=1-\frac{M^{2}}{L_{P}L_{S}}=1-k^{2}=\frac{L_{sc}}{L_{oc}}=\frac{L_{sc}^{% sec}}{L_{P}}=\frac{L_{sc}^{pri}}{L_{S}}=\frac{i_{oc}}{i_{sc}}
  23. L M = a M L_{M}=a{M}
  24. L P σ = L P - a M L_{P}^{\sigma}=L_{P}-a{M}
  25. L S σ = L S - a M L_{S}^{\sigma}=L_{S}-a{M}
  26. L S σ = a 2 L S - a M L_{S}^{\sigma\prime}=a^{2}L_{S}-aM
  27. R S = a 2 R S R_{S}^{\prime}=a^{2}R_{S}
  28. V S = a V S V_{S}^{\prime}=aV_{S}
  29. I S = I S / a I_{S}^{\prime}=I_{S}/a
  30. k = M / L P L S k=M/\sqrt{L_{P}L_{S}}
  31. a = L P / L S a=\sqrt{L_{P}/L_{S}}
  32. a M = L P / L S * k * L P L S = k L P aM=\sqrt{L_{P}/L_{S}}*k*\sqrt{L_{P}L_{S}}=kL_{P}
  33. L P σ = L S σ = L P * ( 1 - k ) L_{P}^{\sigma}=L_{S}^{\sigma\prime}=L_{P}*(1-k)
  34. L M = k L P L_{M}=kL_{P}
  35. σ = 1 - M 2 L P L S = 1 - a 2 M 2 L P a 2 L S = 1 - L M 2 L P L S = 1 - 1 L P L M . L S L M = 1 - 1 ( 1 + σ P ) ( 1 + σ S ) \sigma=1-\frac{M^{2}}{L_{P}L_{S}}=1-\frac{a^{2}M^{2}}{L_{P}a^{2}L_{S}}=1-\frac% {L_{M}^{2}}{L_{P}L_{S}^{\prime}}=1-\frac{1}{\frac{L_{P}}{L_{M}}.\frac{L_{S}^{% \prime}}{L_{M}}}=1-\frac{1}{(1+\sigma_{P})(1+\sigma_{S})}

Least_mean_squares_filter.html

  1. 𝐗 \scriptstyle\mathbf{X}
  2. 𝐲 \scriptstyle\mathbf{y}
  3. s y m b o l β ^ = ( 𝐗 𝐓 𝐗 ) - 1 𝐗 𝐓 s y m b o l y . symbol{\hat{\beta}}=(\mathbf{X}^{\mathbf{T}}\mathbf{X})^{-1}\mathbf{X}^{% \mathbf{T}}symboly.
  4. 𝐡 ( n ) \mathbf{h}(n)
  5. 𝐡 ^ ( n ) \hat{\mathbf{h}}(n)
  6. 𝐡 ( n ) \mathbf{h}(n)
  7. x ( n ) x(n)
  8. d ( n ) d(n)
  9. e ( n ) e(n)
  10. y ( n ) y(n)
  11. v ( n ) v(n)
  12. h ( n ) h(n)
  13. n n
  14. p p
  15. { } H \{\cdot\}^{H}
  16. 𝐱 ( n ) = [ x ( n ) , x ( n - 1 ) , , x ( n - p + 1 ) ] T \mathbf{x}(n)=\left[x(n),x(n-1),\dots,x(n-p+1)\right]^{T}
  17. 𝐡 ( n ) = [ h 0 ( n ) , h 1 ( n ) , , h p - 1 ( n ) ] T , 𝐡 ( n ) p \mathbf{h}(n)=\left[h_{0}(n),h_{1}(n),\dots,h_{p-1}(n)\right]^{T},\quad\mathbf% {h}(n)\in\mathbb{C}^{p}
  18. y ( n ) = 𝐡 H ( n ) 𝐱 ( n ) y(n)=\mathbf{h}^{H}(n)\cdot\mathbf{x}(n)
  19. d ( n ) = y ( n ) + ν ( n ) d(n)=y(n)+\nu(n)
  20. 𝐡 ^ ( n ) \hat{\mathbf{h}}(n)
  21. n n
  22. e ( n ) = d ( n ) - y ^ ( n ) = d ( n ) - 𝐡 ^ H ( n ) 𝐱 ( n ) e(n)=d(n)-\hat{y}(n)=d(n)-\hat{\mathbf{h}}^{H}(n)\cdot\mathbf{x}(n)
  23. ( R - 1 P ) (R^{-1}P)
  24. W n + 1 = W n - μ ε [ n ] W_{n+1}=W_{n}-\mu\nabla\varepsilon[n]
  25. ε \varepsilon
  26. 𝐡 ( n ) \mathbf{h}(n)
  27. C ( n ) = E { | e ( n ) | 2 } C(n)=E\left\{|e(n)|^{2}\right\}
  28. e ( n ) e(n)
  29. E { } E\{\cdot\}
  30. C ( n ) C(n)
  31. 𝐡 ^ H C ( n ) = 𝐡 ^ H E { e ( n ) e * ( n ) } = 2 E { 𝐡 ^ H ( e ( n ) ) e * ( n ) } \nabla_{\hat{\mathbf{h}}^{H}}C(n)=\nabla_{\hat{\mathbf{h}}^{H}}E\left\{e(n)\,e% ^{*}(n)\right\}=2E\left\{\nabla_{\hat{\mathbf{h}}^{H}}(e(n))\,e^{*}(n)\right\}
  32. \nabla
  33. 𝐡 ^ H ( e ( n ) ) = 𝐡 ^ H ( d ( n ) - 𝐡 ^ H 𝐱 ( n ) ) = - 𝐱 ( n ) \nabla_{\hat{\mathbf{h}}^{H}}(e(n))=\nabla_{\hat{\mathbf{h}}^{H}}\left(d(n)-% \hat{\mathbf{h}}^{H}\cdot\mathbf{x}(n)\right)=-\mathbf{x}(n)
  34. C ( n ) = - 2 E { 𝐱 ( n ) e * ( n ) } \nabla C(n)=-2E\left\{\mathbf{x}(n)\,e^{*}(n)\right\}
  35. C ( n ) \nabla C(n)
  36. C ( n ) \nabla C(n)
  37. 𝐡 ^ ( n + 1 ) = 𝐡 ^ ( n ) - μ 2 C ( n ) = 𝐡 ^ ( n ) + μ E { 𝐱 ( n ) e * ( n ) } \hat{\mathbf{h}}(n+1)=\hat{\mathbf{h}}(n)-\frac{\mu}{2}\nabla C(n)=\hat{% \mathbf{h}}(n)+\mu\,E\left\{\mathbf{x}(n)\,e^{*}(n)\right\}
  38. μ 2 \frac{\mu}{2}
  39. E { 𝐱 ( n ) e * ( n ) } E\left\{\mathbf{x}(n)\,e^{*}(n)\right\}
  40. E { 𝐱 ( n ) e * ( n ) } {E}\left\{\mathbf{x}(n)\,e^{*}(n)\right\}
  41. E ^ { 𝐱 ( n ) e * ( n ) } = 1 N i = 0 N - 1 𝐱 ( n - i ) e * ( n - i ) \hat{E}\left\{\mathbf{x}(n)\,e^{*}(n)\right\}=\frac{1}{N}\sum_{i=0}^{N-1}% \mathbf{x}(n-i)\,e^{*}(n-i)
  42. N N
  43. N = 1 N=1
  44. E ^ { 𝐱 ( n ) e * ( n ) } = 𝐱 ( n ) e * ( n ) \hat{E}\left\{\mathbf{x}(n)\,e^{*}(n)\right\}=\mathbf{x}(n)\,e^{*}(n)
  45. 𝐡 ^ ( n + 1 ) = 𝐡 ^ ( n ) + μ 𝐱 ( n ) e * ( n ) \hat{\mathbf{h}}(n+1)=\hat{\mathbf{h}}(n)+\mu\mathbf{x}(n)\,e^{*}(n)
  46. p p
  47. p = p=
  48. μ = \mu=
  49. 𝐡 ^ ( 0 ) = zeros ( p ) \hat{\mathbf{h}}(0)=\operatorname{zeros}(p)
  50. n = 0 , 1 , 2 , n=0,1,2,...
  51. 𝐱 ( n ) = [ x ( n ) , x ( n - 1 ) , , x ( n - p + 1 ) ] T \mathbf{x}(n)=\left[x(n),x(n-1),\dots,x(n-p+1)\right]^{T}
  52. e ( n ) = d ( n ) - 𝐡 ^ H ( n ) 𝐱 ( n ) e(n)=d(n)-\hat{\mathbf{h}}^{H}(n)\mathbf{x}(n)
  53. 𝐡 ^ ( n + 1 ) = 𝐡 ^ ( n ) + μ e * ( n ) 𝐱 ( n ) \hat{\mathbf{h}}(n+1)=\hat{\mathbf{h}}(n)+\mu\,e^{*}(n)\mathbf{x}(n)
  54. μ \mu
  55. μ \mu
  56. μ \mu
  57. μ \mu
  58. 0 < μ < 2 λ max 0<\mu<\frac{2}{\lambda_{\mathrm{max}}}
  59. λ max \lambda_{\max}
  60. 𝐑 = E { 𝐱 ( n ) 𝐱 H ( n ) } {\mathbf{R}}=E\{{\mathbf{x}}(n){\mathbf{x}^{H}}(n)\}
  61. h ^ ( n ) \hat{h}(n)
  62. μ = 2 λ max + λ min , \mu=\frac{2}{\lambda_{\mathrm{max}}+\lambda_{\mathrm{min}}},
  63. λ min \lambda_{\min}
  64. μ \mu
  65. λ min \lambda_{\min}
  66. λ max \lambda_{\max}
  67. λ min \lambda_{\min}
  68. 𝐑 {\mathbf{R}}
  69. 𝐑 = σ 2 𝐈 {\mathbf{R}}=\sigma^{2}{\mathbf{I}}
  70. σ 2 \sigma^{2}
  71. μ \mu
  72. h ^ ( n ) \hat{h}(n)
  73. 0 < μ < 2 tr [ 𝐑 ] , 0<\mu<\frac{2}{\mathrm{tr}\left[{\mathbf{R}}\right]},
  74. tr [ 𝐑 ] \mathrm{tr}[{\mathbf{R}}]
  75. 𝐑 {\mathbf{R}}
  76. h ^ ( n ) \hat{h}(n)
  77. μ \mu
  78. x ( n ) x(n)
  79. μ \mu
  80. p = p=
  81. μ = \mu=
  82. 𝐡 ^ ( 0 ) = zeros ( p ) \hat{\mathbf{h}}(0)=\operatorname{zeros}(p)
  83. n = 0 , 1 , 2 , n=0,1,2,...
  84. 𝐱 ( n ) = [ x ( n ) , x ( n - 1 ) , , x ( n - p + 1 ) ] T \mathbf{x}(n)=\left[x(n),x(n-1),\dots,x(n-p+1)\right]^{T}
  85. e ( n ) = d ( n ) - 𝐡 ^ H ( n ) 𝐱 ( n ) e(n)=d(n)-\hat{\mathbf{h}}^{H}(n)\mathbf{x}(n)
  86. 𝐡 ^ ( n + 1 ) = 𝐡 ^ ( n ) + μ e * ( n ) 𝐱 ( n ) 𝐱 H ( n ) 𝐱 ( n ) \hat{\mathbf{h}}(n+1)=\hat{\mathbf{h}}(n)+\frac{\mu\,e^{*}(n)\mathbf{x}(n)}{% \mathbf{x}^{H}(n)\mathbf{x}(n)}
  87. v ( n ) = 0 v(n)=0
  88. μ o p t = 1 \mu_{opt}=1
  89. x ( n ) x(n)
  90. 𝐡 ( n ) \mathbf{h}(n)
  91. v ( n ) 0 v(n)\neq 0
  92. μ o p t = E [ | y ( n ) - y ^ ( n ) | 2 ] E [ | e ( n ) | 2 ] \mu_{opt}=\frac{E\left[\left|y(n)-\hat{y}(n)\right|^{2}\right]}{E\left[|e(n)|^% {2}\right]}
  93. v ( n ) v(n)
  94. x ( n ) x(n)
  95. Λ ( n ) = | 𝐡 ( n ) - 𝐡 ^ ( n ) | 2 \Lambda(n)=\left|\mathbf{h}(n)-\hat{\mathbf{h}}(n)\right|^{2}
  96. E [ Λ ( n + 1 ) ] = E [ | 𝐡 ^ ( n ) + μ e * ( n ) 𝐱 ( n ) 𝐱 H ( n ) 𝐱 ( n ) - 𝐡 ( n ) | 2 ] E\left[\Lambda(n+1)\right]=E\left[\left|\hat{\mathbf{h}}(n)+\frac{\mu\,e^{*}(n% )\mathbf{x}(n)}{\mathbf{x}^{H}(n)\mathbf{x}(n)}-\mathbf{h}(n)\right|^{2}\right]
  97. E [ Λ ( n + 1 ) ] = E [ | 𝐡 ^ ( n ) + μ ( v * ( n ) + y * ( n ) - y ^ * ( n ) ) 𝐱 ( n ) 𝐱 H ( n ) 𝐱 ( n ) - 𝐡 ( n ) | 2 ] E\left[\Lambda(n+1)\right]=E\left[\left|\hat{\mathbf{h}}(n)+\frac{\mu\,\left(v% ^{*}(n)+y^{*}(n)-\hat{y}^{*}(n)\right)\mathbf{x}(n)}{\mathbf{x}^{H}(n)\mathbf{% x}(n)}-\mathbf{h}(n)\right|^{2}\right]
  98. δ = 𝐡 ^ ( n ) - 𝐡 ( n ) \mathbf{\delta}=\hat{\mathbf{h}}(n)-\mathbf{h}(n)
  99. r ( n ) = y ^ ( n ) - y ( n ) r(n)=\hat{y}(n)-y(n)
  100. E [ Λ ( n + 1 ) ] = E [ | δ ( n ) - μ ( v ( n ) + r ( n ) ) 𝐱 ( n ) 𝐱 H ( n ) 𝐱 ( n ) | 2 ] E\left[\Lambda(n+1)\right]=E\left[\left|\mathbf{\delta}(n)-\frac{\mu\,\left(v(% n)+r(n)\right)\mathbf{x}(n)}{\mathbf{x}^{H}(n)\mathbf{x}(n)}\right|^{2}\right]
  101. E [ Λ ( n + 1 ) ] = E [ ( δ ( n ) - μ ( v ( n ) + r ( n ) ) 𝐱 ( n ) 𝐱 H ( n ) 𝐱 ( n ) ) H ( δ ( n ) - μ ( v ( n ) + r ( n ) ) 𝐱 ( n ) 𝐱 H ( n ) 𝐱 ( n ) ) ] E\left[\Lambda(n+1)\right]=E\left[\left(\mathbf{\delta}(n)-\frac{\mu\,\left(v(% n)+r(n)\right)\mathbf{x}(n)}{\mathbf{x}^{H}(n)\mathbf{x}(n)}\right)^{H}\left(% \mathbf{\delta}(n)-\frac{\mu\,\left(v(n)+r(n)\right)\mathbf{x}(n)}{\mathbf{x}^% {H}(n)\mathbf{x}(n)}\right)\right]
  102. E [ Λ ( n + 1 ) ] = Λ ( n ) + E [ ( μ ( v ( n ) - r ( n ) ) 𝐱 ( n ) 𝐱 H ( n ) 𝐱 ( n ) ) H ( μ ( v ( n ) - r ( n ) ) 𝐱 ( n ) 𝐱 H ( n ) 𝐱 ( n ) ) ] - 2 E [ μ | r ( n ) | 2 𝐱 H ( n ) 𝐱 ( n ) ] E\left[\Lambda(n+1)\right]=\Lambda(n)+E\left[\left(\frac{\mu\,\left(v(n)-r(n)% \right)\mathbf{x}(n)}{\mathbf{x}^{H}(n)\mathbf{x}(n)}\right)^{H}\left(\frac{% \mu\,\left(v(n)-r(n)\right)\mathbf{x}(n)}{\mathbf{x}^{H}(n)\mathbf{x}(n)}% \right)\right]-2E\left[\frac{\mu|r(n)|^{2}}{\mathbf{x}^{H}(n)\mathbf{x}(n)}\right]
  103. E [ Λ ( n + 1 ) ] = Λ ( n ) + μ 2 E [ | e ( n ) | 2 ] 𝐱 H ( n ) 𝐱 ( n ) - 2 μ E [ | r ( n ) | 2 ] 𝐱 H ( n ) 𝐱 ( n ) E\left[\Lambda(n+1)\right]=\Lambda(n)+\frac{\mu^{2}E\left[|e(n)|^{2}\right]}{% \mathbf{x}^{H}(n)\mathbf{x}(n)}-\frac{2\mu E\left[|r(n)|^{2}\right]}{\mathbf{x% }^{H}(n)\mathbf{x}(n)}
  104. d E [ Λ ( n + 1 ) ] d μ = 0 \frac{dE\left[\Lambda(n+1)\right]}{d\mu}=0
  105. 2 μ E [ | e ( n ) | 2 ] - 2 E [ | r ( n ) | 2 ] = 0 2\mu E\left[|e(n)|^{2}\right]-2E\left[|r(n)|^{2}\right]=0
  106. μ = E [ | r ( n ) | 2 ] E [ | e ( n ) | 2 ] \mu=\frac{E\left[|r(n)|^{2}\right]}{E\left[|e(n)|^{2}\right]}

Lebesgue_point.html

  1. f f
  2. k \mathbb{R}^{k}
  3. x x
  4. f f
  5. lim r 0 + 1 | B ( x , r ) | B ( x , r ) | f ( y ) - f ( x ) | d y = 0. \lim_{r\rightarrow 0^{+}}\frac{1}{|B(x,r)|}\int_{B(x,r)}\!|f(y)-f(x)|\,\mathrm% {d}y=0.
  6. B ( x , r ) B(x,r)
  7. x x
  8. r > 0 r>0
  9. | B ( x , r ) | |B(x,r)|
  10. f f
  11. f f
  12. f L 1 ( k ) f\in L^{1}(\mathbb{R}^{k})
  13. x x

Left_quotient.html

  1. L 1 L_{1}
  2. L 2 L_{2}
  3. L 1 L_{1}
  4. L 2 L_{2}
  5. L 2 L_{2}
  6. L 1 L_{1}
  7. L 1 \ L 2 = { w | x ( ( x L 1 ) ( x w L 2 ) ) } L_{1}\backslash L_{2}=\{w\ |\ \exists x((x\in L_{1})\land(xw\in L_{2}))\}
  8. L 2 L_{2}
  9. L 1 L_{1}

Left_recursion.html

  1. 1 + 2 + 3 1+2+3
  2. 1 + 2 1+2
  3. + 3 {}+3
  4. A A
  5. A + A α A\Rightarrow^{+}A\alpha
  6. + \Rightarrow^{+}
  7. α \alpha
  8. A A α A\to A\alpha
  9. α \alpha
  10. 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 + 𝑇𝑒𝑟𝑚 \mathit{Expression}\to\mathit{Expression}+\mathit{Term}
  11. A 0 β 0 A 1 α 0 A_{0}\to\beta_{0}A_{1}\alpha_{0}
  12. A 1 β 1 A 2 α 1 A_{1}\to\beta_{1}A_{2}\alpha_{1}
  13. \cdots
  14. A n β n A 0 α n A_{n}\to\beta_{n}A_{0}\alpha_{n}
  15. β 0 , β 1 , , β n \beta_{0},\beta_{1},\ldots,\beta_{n}
  16. α 0 , α 1 , , α n \alpha_{0},\alpha_{1},\ldots,\alpha_{n}
  17. A 0 β 0 A 1 α 0 + A 1 α 0 β 1 A 2 α 1 α 0 + + A 0 α n α 1 α 0 A_{0}\Rightarrow\beta_{0}A_{1}\alpha_{0}\Rightarrow^{+}A_{1}\alpha_{0}% \Rightarrow\beta_{1}A_{2}\alpha_{1}\alpha_{0}\Rightarrow^{+}\cdots\Rightarrow^% {+}A_{0}\alpha_{n}\dots\alpha_{1}\alpha_{0}
  18. A 0 A_{0}
  19. A A
  20. A A A\rightarrow A
  21. A A α 1 A α n β 1 β m A\rightarrow A\alpha_{1}\mid\ldots\mid A\alpha_{n}\mid\beta_{1}\mid\ldots\mid% \beta_{m}
  22. α \alpha
  23. β \beta
  24. A A
  25. A A
  26. A β 1 A β m A A\rightarrow\beta_{1}A^{\prime}\mid\ldots\mid\beta_{m}A^{\prime}
  27. A A^{\prime}
  28. A α 1 A α n A ϵ A^{\prime}\rightarrow\alpha_{1}A^{\prime}\mid\ldots\mid\alpha_{n}A^{\prime}\mid\epsilon
  29. 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 + 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝐼𝑛𝑡𝑒𝑔𝑒𝑟 𝑆𝑡𝑟𝑖𝑛𝑔 \mathit{Expression}\rightarrow\mathit{Expression}+\mathit{Expression}\mid% \mathit{Integer}\mid\mathit{String}
  30. 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝐼𝑛𝑡𝑒𝑔𝑒𝑟 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝑆𝑡𝑟𝑖𝑛𝑔 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 \mathit{Expression}\rightarrow\mathit{Integer}\,\mathit{Expression}^{\prime}% \mid\mathit{String}\,\mathit{Expression}^{\prime}
  31. 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 + 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 ϵ \mathit{Expression}^{\prime}\rightarrow{}+\mathit{Expression}\,\mathit{% Expression}^{\prime}\mid\epsilon
  32. A 1 , , A n A_{1},\ldots,A_{n}
  33. A i A_{i}
  34. A i α i A_{i}\rightarrow\alpha_{i}
  35. α i \alpha_{i}
  36. α i \alpha_{i}
  37. A j A_{j}
  38. j < i j<i
  39. β i \beta_{i}
  40. α i \alpha_{i}
  41. A j A_{j}
  42. A i α i A_{i}\rightarrow\alpha_{i}
  43. A j α j A_{j}\rightarrow\alpha_{j}
  44. A i α j β i A_{i}\rightarrow\alpha_{j}\beta_{i}
  45. A i A_{i}
  46. 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 - 𝑇𝑒𝑟𝑚 𝑇𝑒𝑟𝑚 \mathit{Expression}\rightarrow\mathit{Expression}\,-\,\mathit{Term}\mid\mathit% {Term}
  47. 𝑇𝑒𝑟𝑚 𝑇𝑒𝑟𝑚 * 𝐹𝑎𝑐𝑡𝑜𝑟 𝐹𝑎𝑐𝑡𝑜𝑟 \mathit{Term}\rightarrow\mathit{Term}\,*\,\mathit{Factor}\mid\mathit{Factor}
  48. 𝐹𝑎𝑐𝑡𝑜𝑟 ( 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 ) 𝐼𝑛𝑡𝑒𝑔𝑒𝑟 \mathit{Factor}\rightarrow(\mathit{Expression})\mid\mathit{Integer}
  49. 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝑇𝑒𝑟𝑚 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 \mathit{Expression}\rightarrow\mathit{Term}\ \mathit{Expression}^{\prime}
  50. 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 - 𝑇𝑒𝑟𝑚 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 ϵ \mathit{Expression}^{\prime}\rightarrow{}-\mathit{Term}\ \mathit{Expression}^{% \prime}\mid\epsilon
  51. 𝑇𝑒𝑟𝑚 𝐹𝑎𝑐𝑡𝑜𝑟 𝑇𝑒𝑟𝑚 \mathit{Term}\rightarrow\mathit{Factor}\ \mathit{Term}^{\prime}
  52. 𝑇𝑒𝑟𝑚 * 𝐹𝑎𝑐𝑡𝑜𝑟 𝑇𝑒𝑟𝑚 ϵ \mathit{Term}^{\prime}\rightarrow{}*\mathit{Factor}\ \mathit{Term}^{\prime}\mid\epsilon
  53. 𝐹𝑎𝑐𝑡𝑜𝑟 ( 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 ) 𝐼𝑛𝑡𝑒𝑔𝑒𝑟 \mathit{Factor}\rightarrow(\mathit{Expression})\mid\mathit{Integer}

Left–right_symmetry.html

  1. [ S U ( 2 ) W × U ( 1 ) Y ] 2 {[SU(2)_{W}\times U(1)_{Y}]\over\mathbb{Z}_{2}}
  2. S U ( 2 ) L × S U ( 2 ) R × U ( 1 ) B - L 2 . {SU(2)_{L}\times SU(2)_{R}\times U(1)_{B-L}\over\mathbb{Z}_{2}}.
  3. Q = I 3 L + I 3 R + B - L 2 Q=I_{3L}+I_{3R}+\frac{B-L}{2}
  4. I 3 L , R \!I_{3L,R}
  5. S U ( 3 ) C × S U ( 2 ) L × S U ( 2 ) R × U ( 1 ) B - L 6 {SU(3)_{C}\times SU(2)_{L}\times SU(2)_{R}\times U(1)_{B-L}\over\mathbb{Z}_{6}}
  6. S U ( 3 ) C × S U ( 2 ) L × S U ( 2 ) R × U ( 1 ) B - L 6 2 . {SU(3)_{C}\times SU(2)_{L}\times SU(2)_{R}\times U(1)_{B-L}\over\mathbb{Z}_{6}% }\rtimes\mathbb{Z}_{2}.
  7. ( 3 ¯ , 1 , 2 ) - 1 3 (\bar{3},1,2)_{-{1\over 3}}
  8. ( 3 , 2 , 1 ) 1 3 ( 3 ¯ , 1 , 2 ) - 1 3 . (3,2,1)_{1\over 3}\oplus(\bar{3},1,2)_{-{1\over 3}}.
  9. ( 1 , 2 , 1 ) - 1 ( 1 , 1 , 2 ) 1 . (1,2,1)_{-1}\oplus(1,1,2)_{1}.
  10. ( 1 , 3 , 1 ) 2 ( 1 , 1 , 3 ) 2 . (1,3,1)_{2}\oplus(1,1,3)_{2}.

Legendre_chi_function.html

  1. χ ν ( z ) = k = 0 z 2 k + 1 ( 2 k + 1 ) ν . \chi_{\nu}(z)=\sum_{k=0}^{\infty}\frac{z^{2k+1}}{(2k+1)^{\nu}}.
  2. χ ν ( z ) = 1 2 [ Li ν ( z ) - Li ν ( - z ) ] . \chi_{\nu}(z)=\frac{1}{2}\left[\operatorname{Li}_{\nu}(z)-\operatorname{Li}_{% \nu}(-z)\right].
  3. χ ν ( z ) = 2 - ν z Φ ( z 2 , ν , 1 / 2 ) . \chi_{\nu}(z)=2^{-\nu}z\,\Phi(z^{2},\nu,1/2).\,
  4. χ 2 ( x ) + χ 2 ( 1 / x ) = π 2 4 - i π 2 | ln x | ( x > 0 ) . \chi_{2}(x)+\chi_{2}(1/x)=\frac{\pi^{2}}{4}-\frac{i\pi}{2}|\ln x|\qquad(x>0).
  5. d d x χ 2 ( x ) = arctanh x x . \frac{d}{dx}\chi_{2}(x)=\frac{{\rm arctanh\,}x}{x}.
  6. 0 π / 2 arcsin ( r sin θ ) d θ = χ 2 ( r ) \int_{0}^{\pi/2}\arcsin(r\sin\theta)d\theta=\chi_{2}\left(r\right)
  7. 0 π / 2 arctan ( r sin θ ) d θ = - 1 2 0 π r θ cos θ 1 + r 2 sin 2 θ d θ = 2 χ 2 ( 1 + r 2 - 1 r ) \int_{0}^{\pi/2}\arctan(r\sin\theta)d\theta=-\frac{1}{2}\int_{0}^{\pi}\frac{r% \theta\cos\theta}{1+r^{2}\sin^{2}\theta}d\theta=2\chi_{2}\left(\frac{\sqrt{1+r% ^{2}}-1}{r}\right)
  8. 0 π / 2 arctan ( p sin θ ) arctan ( q sin θ ) d θ = π χ 2 ( 1 + p 2 - 1 p 1 + q 2 - 1 q ) \int_{0}^{\pi/2}\arctan(p\sin\theta)\arctan(q\sin\theta)d\theta=\pi\chi_{2}% \left(\frac{\sqrt{1+p^{2}}-1}{p}\cdot\frac{\sqrt{1+q^{2}}-1}{q}\right)
  9. 0 α 0 β d x d y 1 - x 2 y 2 = χ 2 ( α β ) if | α β | 1 \int_{0}^{\alpha}\int_{0}^{\beta}\frac{dxdy}{1-x^{2}y^{2}}=\chi_{2}(\alpha% \beta)\qquad{\rm if}~{}~{}|\alpha\beta|\leq 1

Legendre_sieve.html

  1. S ( A , P ) = a A d a ; d P μ ( d ) = d P μ ( d ) | A d | , S(A,P)=\sum_{a\in A}\sum_{d\mid a;\,d\mid P}\mu(d)=\sum_{d\mid P}\mu(d)|A_{d}|,
  2. μ \mu
  3. A d A_{d}
  4. S ( A , P ) = | { n : n A , ( n , P ) = 1 } | S(A,P)=|\{n:n\in A,(n,P)=1\}|
  5. S ( A , P ) \displaystyle S(A,P)
  6. X \lfloor X\rfloor
  7. 2 π ( z ) 2^{\pi(z)}
  8. π ( z ) \pi(z)
  9. π ( X ) \pi(X)
  10. S ( A , P ) π ( X ) - π ( z ) + 1 , S(A,P)\geq\pi(X)-\pi(z)+1,\,

Leibniz_formula_for_π.html

  1. π \pi
  2. 1 - 1 3 + 1 5 - 1 7 + 1 9 - = π 4 . 1\,-\,\frac{1}{3}\,+\,\frac{1}{5}\,-\,\frac{1}{7}\,+\,\frac{1}{9}\,-\,\cdots\;% =\;\frac{\pi}{4}.\!
  3. n = 0 ( - 1 ) n 2 n + 1 = π 4 . \sum_{n=0}^{\infty}\,\frac{(-1)^{n}}{2n+1}\;=\;\frac{\pi}{4}.\!
  4. arctan x = x - x 3 3 + x 5 5 - x 7 7 + \arctan x=x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\frac{x^{7}}{7}+\cdots
  5. π \pi
  6. s = 1 s=1
  7. β ( 1 ) β(1)
  8. π 4 = arctan ( 1 ) = 0 1 1 1 + x 2 d x = 0 1 ( k = 0 n ( - 1 ) k x 2 k + ( - 1 ) n + 1 x 2 n + 2 1 + x 2 ) d x = k = 0 n ( - 1 ) k 2 k + 1 + ( - 1 ) n + 1 0 1 x 2 n + 2 1 + x 2 d x . {\begin{aligned}\displaystyle\frac{\pi}{4}&\displaystyle=\arctan(1)\;=\;\int_{% 0}^{1}\frac{1}{1+x^{2}}\,dx\\ &\displaystyle=\int_{0}^{1}\left(\sum_{k=0}^{n}(-1)^{k}x^{2k}+\frac{(-1)^{n+1}% \,x^{2n+2}}{1+x^{2}}\right)\,dx\\ &\displaystyle=\sum_{k=0}^{n}\frac{(-1)^{k}}{2k+1}+(-1)^{n+1}\int_{0}^{1}\frac% {x^{2n+2}}{1+x^{2}}\,dx.\end{aligned}}
  9. 0 < 0 1 x 2 n + 2 1 + x 2 d x < 0 1 x 2 n + 2 d x = 1 2 n + 3 0 as n . 0<\int_{0}^{1}\frac{x^{2n+2}}{1+x^{2}}\,dx\;<\;\int_{0}^{1}x^{2n+2}\,dx\;=\;% \frac{1}{2n+3}\;\rightarrow\;0\,\text{ as }n\rightarrow\infty.\!
  10. n n→∞
  11. π 4 = k = 0 ( - 1 ) k 2 k + 1 . \frac{\pi}{4}\;=\;\sum_{k=0}^{\infty}\frac{(-1)^{k}}{2k+1}.
  12. π \pi
  13. 1 2 k + 1 < 10 - 10 \scriptstyle\frac{1}{2k+1}<10^{-10}
  14. k > 10 10 - 1 2 \scriptstyle k>\frac{10^{10}-1}{2}
  15. π \pi
  16. π 4 = n = 0 ( 1 4 n + 1 - 1 4 n + 3 ) = n = 0 2 ( 4 n + 1 ) ( 4 n + 3 ) \frac{\pi}{4}=\sum_{n=0}^{\infty}\bigg(\frac{1}{4n+1}-\frac{1}{4n+3}\bigg)=% \sum_{n=0}^{\infty}\frac{2}{(4n+1)(4n+3)}
  17. π \pi
  18. π 2 - 2 k = 1 N / 2 ( - 1 ) k - 1 2 k - 1 m = 0 E 2 m N 2 m + 1 \frac{\pi}{2}-2\sum_{k=1}^{N/2}\frac{(-1)^{k-1}}{2k-1}\sim\sum_{m=0}^{\infty}% \frac{E_{2m}}{N^{2m+1}}\!
  19. N N
  20. N N
  21. π \pi
  22. π / 4 = ( p 1 ( mod 4 ) p p - 1 ) ( p 3 ( mod 4 ) p p + 1 ) = 3 4 5 4 7 8 11 12 13 12 17 16 \pi/4=\left(\prod_{p\equiv 1\;\;(\mathop{{\rm mod}}4)}\frac{p}{p-1}\right)% \cdot\left(\prod_{p\equiv 3\;\;(\mathop{{\rm mod}}4)}\frac{p}{p+1}\right)=% \frac{3}{4}\cdot\frac{5}{4}\cdot\frac{7}{8}\cdot\frac{11}{12}\cdot\frac{13}{12% }\cdot\frac{17}{16}\cdots

Lenstra–Lenstra–Lovász_lattice_basis_reduction_algorithm.html

  1. 𝐁 = { 𝐛 1 , 𝐛 2 , , 𝐛 d } \mathbf{B}=\{\mathbf{b}_{1},\mathbf{b}_{2},\dots,\mathbf{b}_{d}\}
  2. d n \ d\leq n
  3. O ( d 5 n log 3 B ) O(d^{5}n\log^{3}B)\,
  4. b i b_{i}
  5. 𝐁 = { 𝐛 0 , 𝐛 1 , , 𝐛 n } , \mathbf{B}=\{\mathbf{b}_{0},\mathbf{b}_{1},\dots,\mathbf{b}_{n}\},
  6. 𝐁 * = { 𝐛 0 * , 𝐛 1 * , , 𝐛 n * } , \mathbf{B}^{*}=\{\mathbf{b}^{*}_{0},\mathbf{b}^{*}_{1},\dots,\mathbf{b}^{*}_{n% }\},
  7. μ i , j = 𝐛 i , 𝐛 j * 𝐛 j * , 𝐛 j * \mu_{i,j}=\frac{\langle\mathbf{b}_{i},\mathbf{b}^{*}_{j}\rangle}{\langle% \mathbf{b}^{*}_{j},\mathbf{b}^{*}_{j}\rangle}
  8. 1 j < i n 1\leq j<i\leq n
  9. B B
  10. δ \delta
  11. 1 j < i n : | μ i , j | 0.5 1\leq j<i\leq n\colon\left|\mu_{i,j}\right|\leq 0.5
  12. : δ 𝐛 k - 1 * 2 𝐛 k * 2 + μ k , k - 1 2 𝐛 k - 1 * 2 \colon\delta\|\mathbf{b}^{*}_{k-1}\|^{2}\leq\|\mathbf{b}^{*}_{k}\|^{2}+\mu_{k,% k-1}^{2}\|\mathbf{b}^{*}_{k-1}\|^{2}
  13. δ \delta
  14. δ \delta
  15. δ = 3 4 \delta=\frac{3}{4}
  16. δ = 1 \delta=1
  17. δ \delta
  18. c i > 1 c_{i}>1
  19. c 1 c_{1}
  20. c 2 c_{2}
  21. \triangleright
  22. b 0 , b 1 , , b n Z m b_{0},b_{1},\dots,b_{n}\in Z^{m}
  23. \triangleright
  24. δ \delta
  25. 1 4 < δ < 1 \frac{1}{4}<\delta<1
  26. δ = 3 4 \delta=\frac{3}{4}
  27. o r t h o := g r a m S c h m i d t ( { b 0 , , b n } ) = { b 0 * , , b n * } ortho:=gramSchmidt(\{b_{0},\dots,b_{n}\})=\{b_{0}^{*},\dots,b_{n}^{*}\}
  28. μ i , j := b i , b j * b j * , b j * \mu_{i,j}:=\frac{\langle b_{i},b_{j}^{*}\rangle}{\langle b_{j}^{*},b_{j}^{*}\rangle}
  29. b i , b j * b_{i},b_{j}^{*}
  30. k = 1 k=1
  31. k n k\leq n
  32. j j
  33. k - 1 k-1
  34. 0
  35. | μ k , j | > 1 2 |\mu_{k,j}|>\frac{1}{2}
  36. b k = b k - μ k , j b j b_{k}=b_{k}-\lfloor\mu_{k,j}\rceil b_{j}
  37. o r t h o ortho
  38. μ i , j \mu_{i,j}
  39. o r t h o := g r a m S c h m i d t ( { b 0 , , b n } ) = { b 0 * , , b n * } ortho:=gramSchmidt(\{b_{0},\dots,b_{n}\})=\{b_{0}^{*},\dots,b_{n}^{*}\}
  40. b i b_{i}
  41. b k * , b k * ( δ - ( μ k , k - 1 ) 2 ) b k - 1 * , b k - 1 * \langle b_{k}^{*},b_{k}^{*}\rangle\geq(\delta-(\mu_{k,k-1})^{2})\langle b_{k-1% }^{*},b_{k-1}^{*}\rangle
  42. k = k + 1 k=k+1
  43. b k b_{k}
  44. b k - 1 b_{k-1}
  45. o r t h o ortho
  46. μ i , j \mu_{i,j}
  47. k = max ( k - 1 , 1 ) k=\max(k-1,1)
  48. b 0 , b 1 , , b n b_{0},b_{1},\dots,b_{n}
  49. 𝐛 1 , 𝐛 2 , 𝐛 3 Z 3 \mathbf{b}_{1},\mathbf{b}_{2},\mathbf{b}_{3}\in Z^{3}
  50. [ 1 - 1 3 1 0 5 1 2 6 ] \begin{bmatrix}1&-1&3\\ 1&0&5\\ 1&2&6\end{bmatrix}
  51. b 1 * = b 1 = [ 1 1 1 ] , B 1 = b 1 * , b 1 * = [ 1 1 1 ] [ 1 1 1 ] = 3 b_{1}^{*}=b_{1}=\begin{bmatrix}1\\ 1\\ 1\end{bmatrix},B_{1}=\langle b_{1}^{*},b_{1}^{*}\rangle=\begin{bmatrix}1\\ 1\\ 1\end{bmatrix}\begin{bmatrix}1\\ 1\\ 1\end{bmatrix}=3
  52. i = 2 i=2
  53. j = 1 j=1
  54. μ 2 , 1 = b 2 , b 1 * B 1 = [ - 1 0 2 ] [ 1 1 1 ] 3 = 1 3 ( < 1 2 ) \mu_{2,1}=\frac{\langle b_{2},b_{1}^{*}\rangle}{B_{1}}=\frac{\begin{bmatrix}-1% \\ 0\\ 2\end{bmatrix}\begin{bmatrix}1\\ 1\\ 1\end{bmatrix}}{3}=\frac{1}{3}(<\frac{1}{2})
  55. b 2 * = b 2 - μ 2 , 1 b 1 * = [ - 1 0 2 ] - 1 3 [ 1 1 1 ] = [ - 4 3 - 1 3 5 3 ] . b_{2}^{*}=b_{2}-\mu_{2,1}b_{1}^{*}=\begin{bmatrix}-1\\ 0\\ 2\end{bmatrix}-\frac{1}{3}\begin{bmatrix}1\\ 1\\ 1\end{bmatrix}=\begin{bmatrix}\frac{-4}{3}\\ \frac{-1}{3}\\ \frac{5}{3}\end{bmatrix}.
  56. B 2 = b 2 * , b 2 * = [ - 4 3 - 1 3 5 3 ] [ - 4 3 - 1 3 5 3 ] = 14 3 . B_{2}=\langle b_{2}^{*},b_{2}^{*}\rangle=\begin{bmatrix}\frac{-4}{3}\\ \frac{-1}{3}\\ \frac{5}{3}\end{bmatrix}\begin{bmatrix}\frac{-4}{3}\\ \frac{-1}{3}\\ \frac{5}{3}\end{bmatrix}=\frac{14}{3}.
  57. 𝐤 := 2 \mathbf{k}:=2
  58. μ 2 , 1 \mu_{2,1}
  59. i = 3 i=3
  60. j = 1 , 2 j=1,2
  61. μ i , j \mu_{i,j}
  62. B i B_{i}
  63. μ 3 , 1 = b 3 , b 1 * B 1 = [ 3 5 6 ] [ 1 1 1 ] 3 = 14 3 ( > 1 2 ) \mu_{3,1}=\frac{\langle b_{3},b_{1}^{*}\rangle}{B_{1}}=\frac{\begin{bmatrix}3% \\ 5\\ 6\end{bmatrix}\begin{bmatrix}1\\ 1\\ 1\end{bmatrix}}{3}=\frac{14}{3}(>\frac{1}{2})
  64. b 3 * = b 3 - μ 3 , 1 b 1 * = [ 3 5 6 ] - 14 3 [ 1 1 1 ] = [ - 5 3 1 3 4 3 ] b_{3}^{*}=b_{3}-\mu_{3,1}b_{1}^{*}=\begin{bmatrix}3\\ 5\\ 6\end{bmatrix}-\frac{14}{3}\begin{bmatrix}1\\ 1\\ 1\end{bmatrix}=\begin{bmatrix}\frac{-5}{3}\\ \frac{1}{3}\\ \frac{4}{3}\end{bmatrix}
  65. μ 3 , 2 = b 3 , b 2 * B 2 = [ 3 5 6 ] [ - 4 3 - 1 3 5 3 ] 14 3 = 13 14 ( > 1 2 ) \mu_{3,2}=\frac{\langle b_{3},b_{2}^{*}\rangle}{B_{2}}=\frac{\begin{bmatrix}3% \\ 5\\ 6\end{bmatrix}\begin{bmatrix}\frac{-4}{3}\\ \frac{-1}{3}\\ \frac{5}{3}\end{bmatrix}}{\frac{14}{3}}=\frac{13}{14}(>\frac{1}{2})
  66. b 3 * = b 3 * - μ 3 , 2 b 2 * = [ - 5 3 1 3 4 3 ] - 13 14 [ - 4 3 - 1 3 5 3 ] = [ - 18 42 27 42 - 9 42 ] = [ - 6 14 9 14 - 3 14 ] b_{3}^{*}=b_{3}^{*}-\mu_{3,2}b_{2}^{*}=\begin{bmatrix}\frac{-5}{3}\\ \frac{1}{3}\\ \frac{4}{3}\end{bmatrix}-\frac{13}{14}\begin{bmatrix}\frac{-4}{3}\\ \frac{-1}{3}\\ \frac{5}{3}\end{bmatrix}=\begin{bmatrix}\frac{-18}{42}\\ \frac{27}{42}\\ \frac{-9}{42}\end{bmatrix}=\begin{bmatrix}\frac{-6}{14}\\ \frac{9}{14}\\ \frac{-3}{14}\end{bmatrix}
  67. B 3 = b 3 * , b 3 * = [ - 6 14 9 14 - 3 14 ] [ - 6 14 9 14 - 3 14 ] = 126 196 = 9 14 B_{3}=\langle b_{3}^{*},b_{3}^{*}\rangle=\begin{bmatrix}\frac{-6}{14}\\ \frac{9}{14}\\ \frac{-3}{14}\end{bmatrix}\begin{bmatrix}\frac{-6}{14}\\ \frac{9}{14}\\ \frac{-3}{14}\end{bmatrix}=\frac{126}{196}=\frac{9}{14}
  68. k 3 k\leq 3
  69. b 3 b_{3}
  70. μ 3 , 1 \mu_{3,1}
  71. μ 3 , 2 \mu_{3,2}
  72. μ 3 , 1 > 1 2 \mid\mu_{3,1}\mid>\frac{1}{2}
  73. r = 0.5 + μ 3 , l = 5 r=\lfloor 0.5+\mu_{3,l}\rfloor=5
  74. b 3 = b 3 - 5 b 1 = [ 3 5 6 ] - [ 5 5 5 ] = [ - 2 0 1 ] b_{3}=b_{3}-5b_{1}=\begin{bmatrix}3\\ 5\\ 6\end{bmatrix}-\begin{bmatrix}5\\ 5\\ 5\end{bmatrix}=\begin{bmatrix}-2\\ 0\\ 1\end{bmatrix}
  75. μ 3 , 1 = μ 3 , l - r μ 1 , 1 = - 1 3 ( < 1 2 ) \mu_{3,1}=\mu_{3,l}-r\mu_{1,1}=\frac{-1}{3}(<\frac{1}{2})
  76. μ 3 , 1 = μ 3 , 1 - r = 14 3 - 5 = - 1 3 \mu_{3,1}=\mu_{3,1}-r=\frac{14}{3}-5=\frac{-1}{3}
  77. μ 3 , 2 > 1 2 \mid\mu_{3,2}\mid>\frac{1}{2}
  78. r = 0.5 + μ 3 , 2 = 1 r=\lfloor 0.5+\mu_{3,2}\rfloor=1
  79. b 3 = b 3 - b 2 = [ 3 5 6 ] - [ - 1 0 2 ] = [ 4 5 4 ] b_{3}=b_{3}-b_{2}=\begin{bmatrix}3\\ 5\\ 6\end{bmatrix}-\begin{bmatrix}-1\\ 0\\ 2\end{bmatrix}=\begin{bmatrix}4\\ 5\\ 4\end{bmatrix}
  80. μ 3 , 2 = μ 3 , 2 - r μ 2 , 2 = 13 14 - 1 = - 1 14 \mu_{3,2}=\mu_{3,2}-r\mu_{2,2}=\frac{13}{14}-1=\frac{-1}{14}
  81. μ 3 , 2 = μ 3 , 2 - 1 = - 1 14 ( < 1 2 ) \mu_{3,2}=\mu_{3,2}-1=\frac{-1}{14}(<\frac{1}{2})
  82. B 3 < ( 3 4 - μ 3 , 2 2 ) B 2 B_{3}<(\frac{3}{4}-\mu_{3,2}^{2})B_{2}
  83. b 3 b_{3}
  84. b 2 b_{2}
  85. k k
  86. [ 1 4 - 1 1 5 0 1 4 2 ] \begin{bmatrix}1&4&-1\\ 1&5&0\\ 1&4&2\end{bmatrix}
  87. b 1 * = b 1 = [ 1 1 1 ] , B 1 = 3 b_{1}^{*}=b_{1}=\begin{bmatrix}1\\ 1\\ 1\end{bmatrix},B_{1}=3
  88. μ 2 , 1 = b 2 , b 1 * B 1 = [ 4 5 4 ] [ 1 1 1 ] 3 = 13 3 ( > 1 2 ) \mu_{2,1}=\frac{\langle b_{2},b_{1}^{*}\rangle}{B_{1}}=\frac{\begin{bmatrix}4% \\ 5\\ 4\end{bmatrix}\begin{bmatrix}1\\ 1\\ 1\end{bmatrix}}{3}=\frac{13}{3}(>\frac{1}{2})
  89. b 2 * = b 2 - μ 2 , 1 b 1 * = [ 4 5 4 ] - 13 3 [ 1 1 1 ] = [ - 1 3 2 3 - 1 3 ] b_{2}^{*}=b_{2}-\mu_{2,1}b_{1}^{*}=\begin{bmatrix}4\\ 5\\ 4\end{bmatrix}-\frac{13}{3}\begin{bmatrix}1\\ 1\\ 1\end{bmatrix}=\begin{bmatrix}\frac{-1}{3}\\ \frac{2}{3}\\ \frac{-1}{3}\end{bmatrix}
  90. B 2 = b 2 * , b 2 * = 2 3 B_{2}=\langle b_{2}^{*},b_{2}^{*}\rangle=\frac{2}{3}
  91. μ 2 , 1 > 1 2 \mid\mu_{2,1}\mid>\frac{1}{2}
  92. r = 0.5 + μ 2 , l = 4 r=\lfloor 0.5+\mu_{2,l}\rfloor=4
  93. b 2 = b 2 - 4 b 1 = [ 4 5 4 ] - [ 4 4 4 ] = [ 0 1 0 ] b_{2}=b_{2}-4b_{1}=\begin{bmatrix}4\\ 5\\ 4\end{bmatrix}-\begin{bmatrix}4\\ 4\\ 4\end{bmatrix}=\begin{bmatrix}0\\ 1\\ 0\end{bmatrix}
  94. μ 2 , 1 = μ 2 , 1 - 4 μ 1 , 1 = 13 3 - 4 = 1 3 ( < 1 2 ) \mu_{2,1}=\mu_{2,1}-4\mu_{1,1}=\frac{13}{3}-4=\frac{1}{3}(<\frac{1}{2})
  95. B 2 < ( 3 4 - μ 2 , 1 2 ) B 1 B_{2}<(\frac{3}{4}-\mu_{2,1}^{2})B_{1}
  96. b 2 b_{2}
  97. b 1 b_{1}
  98. [ 0 1 - 1 1 0 0 0 1 2 ] \begin{bmatrix}0&1&-1\\ 1&0&0\\ 0&1&2\end{bmatrix}
  99. R 4 R^{4}
  100. [ 1 , 0 , 0 , 10000 r 2 ] , [ 0 , 1 , 0 , 10000 r ] , [1,0,0,10000r^{2}],[0,1,0,10000r],
  101. [ 0 , 0 , 1 , 10000 ] [0,0,1,10000]
  102. [ a , b , c , 10000 ( a r 2 + b r + c ) ] [a,b,c,10000(ar^{2}+br+c)]
  103. a r 2 + b r + c ar^{2}+br+c
  104. x 2 - x - 1 x^{2}-x-1

Leonard_Eugene_Dickson.html

  1. k 7 k\geq 7
  2. ( 3 k + 1 ) / ( 2 k - 1 ) [ 1.5 k ] + 1 (3^{k}+1)/(2^{k}-1)\leq[1.5^{k}]+1
  3. k 6 k\geq 6

Lepton_number.html

  1. L = n - n ¯ L=n_{\ell}-n_{\overline{\ell}}
  2. L < s u b > e L<sub>e

Lerch_zeta_function.html

  1. L ( λ , α , s ) = n = 0 exp ( 2 π i λ n ) ( n + α ) s . L(\lambda,\alpha,s)=\sum_{n=0}^{\infty}\frac{\exp(2\pi i\lambda n)}{(n+\alpha)% ^{s}}.
  2. Φ ( z , s , α ) = n = 0 z n ( n + α ) s . \Phi(z,s,\alpha)=\sum_{n=0}^{\infty}\frac{z^{n}}{(n+\alpha)^{s}}.
  3. Φ ( exp ( 2 π i λ ) , s , α ) = L ( λ , α , s ) . \,\Phi(\exp(2\pi i\lambda),s,\alpha)=L(\lambda,\alpha,s).
  4. Φ ( z , s , a ) = 1 Γ ( s ) 0 t s - 1 e - a t 1 - z e - t d t \Phi(z,s,a)=\frac{1}{\Gamma(s)}\int_{0}^{\infty}\frac{t^{s-1}e^{-at}}{1-ze^{-t% }}\,dt
  5. ( a ) > 0 ( s ) > 0 z < 1 ( a ) > 0 ( s ) > 1 z = 1. \Re(a)>0\wedge\Re(s)>0\wedge z<1\vee\Re(a)>0\wedge\Re(s)>1\wedge z=1.
  6. Φ ( z , s , a ) = - Γ ( 1 - s ) 2 π i 0 ( + ) ( - t ) s - 1 e - a t 1 - z e - t d t \Phi(z,s,a)=-\frac{\Gamma(1-s)}{2\pi i}\int_{0}^{(+\infty)}\frac{(-t)^{s-1}e^{% -at}}{1-ze^{-t}}\,dt
  7. ( a ) > 0 ( s ) < 0 z < 1 \Re(a)>0\wedge\Re(s)<0\wedge z<1
  8. t = log ( z ) + 2 k π i , k Z . t=\log(z)+2k\pi i,k\in Z.
  9. Φ ( z , s , a ) = 1 2 a s + 0 z t ( a + t ) s d t + 2 a s - 1 0 sin ( s arctan ( t ) - t a log ( z ) ) ( 1 + t 2 ) s / 2 ( e 2 π a t - 1 ) d t \Phi(z,s,a)=\frac{1}{2a^{s}}+\int_{0}^{\infty}\frac{z^{t}}{(a+t)^{s}}\,dt+% \frac{2}{a^{s-1}}\int_{0}^{\infty}\frac{\sin(s\arctan(t)-ta\log(z))}{(1+t^{2})% ^{s/2}(e^{2\pi at}-1)}\,dt
  10. ( a ) > 0 | z | < 1 \Re(a)>0\wedge|z|<1
  11. Φ ( z , s , a ) = 1 2 a s + log s - 1 ( 1 / z ) z a Γ ( 1 - s , a log ( 1 / z ) ) + 2 a s - 1 0 sin ( s arctan ( t ) - t a log ( z ) ) ( 1 + t 2 ) s / 2 ( e 2 π a t - 1 ) d t \Phi(z,s,a)=\frac{1}{2a^{s}}+\frac{\log^{s-1}(1/z)}{z^{a}}\Gamma(1-s,a\log(1/z% ))+\frac{2}{a^{s-1}}\int_{0}^{\infty}\frac{\sin(s\arctan(t)-ta\log(z))}{(1+t^{% 2})^{s/2}(e^{2\pi at}-1)}\,dt
  12. ( a ) > 0. \Re(a)>0.
  13. ζ ( s , α ) = L ( 0 , α , s ) = Φ ( 1 , s , α ) . \,\zeta(s,\alpha)=L(0,\alpha,s)=\Phi(1,s,\alpha).
  14. Li s ( z ) = z Φ ( z , s , 1 ) . \,\textrm{Li}_{s}(z)=z\Phi(z,s,1).
  15. χ n ( z ) = 2 - n z Φ ( z 2 , n , 1 / 2 ) . \,\chi_{n}(z)=2^{-n}z\Phi(z^{2},n,1/2).
  16. ζ ( s ) = Φ ( 1 , s , 1 ) . \,\zeta(s)=\Phi(1,s,1).
  17. η ( s ) = Φ ( - 1 , s , 1 ) . \,\eta(s)=\Phi(-1,s,1).
  18. L ( λ , α , s ) L(\lambda,\alpha,s)
  19. Φ ( z , s , a ) = z n Φ ( z , s , a + n ) + k = 0 n - 1 z k ( k + a ) s \Phi(z,s,a)=z^{n}\Phi(z,s,a+n)+\sum_{k=0}^{n-1}\frac{z^{k}}{(k+a)^{s}}
  20. Φ ( z , s - 1 , a ) = ( a + z z ) Φ ( z , s , a ) \Phi(z,s-1,a)=\left(a+z\frac{\partial}{\partial z}\right)\Phi(z,s,a)
  21. Φ ( z , s + 1 , a ) = - 1 s a Φ ( z , s , a ) . \Phi(z,s+1,a)=-\,\frac{1}{s}\frac{\partial}{\partial a}\Phi(z,s,a).
  22. Φ ( z , s , q ) = 1 1 - z n = 0 ( - z 1 - z ) n k = 0 n ( - 1 ) k ( n k ) ( q + k ) - s . \Phi(z,s,q)=\frac{1}{1-z}\sum_{n=0}^{\infty}\left(\frac{-z}{1-z}\right)^{n}% \sum_{k=0}^{n}(-1)^{k}{\left({{n}\atop{k}}\right)}(q+k)^{-s}.
  23. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  24. | log ( z ) | < 2 π ; s 1 , 2 , 3 , ; a 0 , - 1 , - 2 , |\log(z)|<2\pi;s\neq 1,2,3,\dots;a\neq 0,-1,-2,\dots
  25. Φ ( z , s , a ) = z - a [ Γ ( 1 - s ) ( - log ( z ) ) s - 1 + k = 0 ζ ( s - k , a ) log k ( z ) k ! ] \Phi(z,s,a)=z^{-a}\left[\Gamma(1-s)\left(-\log(z)\right)^{s-1}+\sum_{k=0}^{% \infty}\zeta(s-k,a)\frac{\log^{k}(z)}{k!}\right]
  26. Φ ( z , n , a ) = z - a { k = 0 k n - 1 ζ ( n - k , a ) log k ( z ) k ! + [ ψ ( n ) - ψ ( a ) - log ( - log ( z ) ) ] log n - 1 ( z ) ( n - 1 ) ! } , \Phi(z,n,a)=z^{-a}\left\{\sum_{{k=0}\atop k\neq n-1}^{\infty}\zeta(n-k,a)\frac% {\log^{k}(z)}{k!}+\left[\psi(n)-\psi(a)-\log(-\log(z))\right]\frac{\log^{n-1}(% z)}{(n-1)!}\right\},
  27. ψ ( n ) \psi(n)
  28. Φ ( z , s , a + x ) = k = 0 Φ ( z , s + k , a ) ( s ) k ( - x ) k k ! ; | x | < ( a ) , \Phi(z,s,a+x)=\sum_{k=0}^{\infty}\Phi(z,s+k,a)(s)_{k}\frac{(-x)^{k}}{k!};|x|<% \Re(a),
  29. ( s ) k (s)_{k}
  30. Φ ( z , s , a ) = k = 0 n z k ( a + k ) s + z n m = 0 ( 1 - m - s ) m Li s + m ( z ) ( a + n ) m m ! ; a - n \Phi(z,s,a)=\sum_{k=0}^{n}\frac{z^{k}}{(a+k)^{s}}+z^{n}\sum_{m=0}^{\infty}(1-m% -s)_{m}\operatorname{Li}_{s+m}(z)\frac{(a+n)^{m}}{m!};\ a\rightarrow-n
  31. Φ ( z , s , a ) = 1 a s + m = 0 ( 1 - m - s ) m Li s + m ( z ) a m m ! ; | a | < 1 , \Phi(z,s,a)=\frac{1}{a^{s}}+\sum_{m=0}^{\infty}(1-m-s)_{m}\operatorname{Li}_{s% +m}(z)\frac{a^{m}}{m!};|a|<1,
  32. Li s ( z ) \operatorname{Li}_{s}(z)
  33. s - s\rightarrow-\infty
  34. Φ ( z , s , a ) = z - a Γ ( 1 - s ) k = - [ 2 k π i - log ( z ) ] s - 1 e 2 k π a i \Phi(z,s,a)=z^{-a}\Gamma(1-s)\sum_{k=-\infty}^{\infty}[2k\pi i-\log(z)]^{s-1}e% ^{2k\pi ai}
  35. | a | < 1 ; ( s ) < 0 ; z ( - , 0 ) |a|<1;\Re(s)<0;z\notin(-\infty,0)
  36. Φ ( - z , s , a ) = z - a Γ ( 1 - s ) k = - [ ( 2 k + 1 ) π i - log ( z ) ] s - 1 e ( 2 k + 1 ) π a i \Phi(-z,s,a)=z^{-a}\Gamma(1-s)\sum_{k=-\infty}^{\infty}[(2k+1)\pi i-\log(z)]^{% s-1}e^{(2k+1)\pi ai}
  37. | a | < 1 ; ( s ) < 0 ; z ( 0 , ) . |a|<1;\Re(s)<0;z\notin(0,\infty).
  38. Φ ( z , s , a ) = 1 2 a s + 1 z a k = 1 e - 2 π i ( k - 1 ) a Γ ( 1 - s , a ( - 2 π i ( k - 1 ) - log ( z ) ) ) ( - 2 π i ( k - 1 ) - log ( z ) ) 1 - s + e 2 π i k a Γ ( 1 - s , a ( 2 π i k - log ( z ) ) ) ( 2 π i k - log ( z ) ) 1 - s \Phi(z,s,a)=\frac{1}{2a^{s}}+\frac{1}{z^{a}}\sum_{k=1}^{\infty}\frac{e^{-2\pi i% (k-1)a}\Gamma(1-s,a(-2\pi i(k-1)-\log(z)))}{(-2\pi i(k-1)-\log(z))^{1-s}}+% \frac{e^{2\pi ika}\Gamma(1-s,a(2\pi ik-\log(z)))}{(2\pi ik-\log(z))^{1-s}}
  39. | a | < 1 ; ( s ) < 0. |a|<1;\Re(s)<0.

Lerner_index.html

  1. L = P - M C P L=\frac{P-MC}{P}
  2. E d E_{d}
  3. L = - 1 E d L=\frac{-1}{E_{d}}
  4. E d E_{d}

Leslie_matrix.html

  1. n x n_{x}
  2. s x s_{x}
  3. f x f_{x}
  4. n 0 n_{0}
  5. b x + 1 b_{x+1}
  6. f x = s x b x + 1 . f_{x}=s_{x}b_{x+1}.
  7. n 0 n_{0}
  8. s x s_{x}
  9. n x + 1 = s x n x n_{x+1}=s_{x}n_{x}
  10. [ n 0 n 1 n ω - 1 ] t + 1 = [ f 0 f 1 f 2 f ω - 2 f ω - 1 s 0 0 0 0 0 0 s 1 0 0 0 0 0 s 2 0 0 0 0 0 s ω - 2 0 ] [ n 0 n 1 n ω - 1 ] t \begin{bmatrix}n_{0}\\ n_{1}\\ \vdots\\ n_{\omega-1}\\ \end{bmatrix}_{t+1}=\begin{bmatrix}f_{0}&f_{1}&f_{2}&\ldots&f_{\omega-2}&f_{% \omega-1}\\ s_{0}&0&0&\ldots&0&0\\ 0&s_{1}&0&\ldots&0&0\\ 0&0&s_{2}&\ldots&0&0\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&0&\ldots&s_{\omega-2}&0\end{bmatrix}\begin{bmatrix}n_{0}\\ n_{1}\\ \vdots\\ n_{\omega-1}\end{bmatrix}_{t}
  11. ω \omega
  12. 𝐧 t + 1 = 𝐋𝐧 t \mathbf{n}_{t+1}=\mathbf{L}\mathbf{n}_{t}
  13. 𝐧 t = 𝐋 t 𝐧 0 \mathbf{n}_{t}=\mathbf{L}^{t}\mathbf{n}_{0}
  14. 𝐧 t \mathbf{n}_{t}
  15. 𝐋 \mathbf{L}
  16. 𝐋 \mathbf{L}
  17. λ \lambda
  18. λ \lambda
  19. f x + s x = 1 f_{x}+s_{x}=1
  20. x x
  21. N 0 N_{0}

Level_set_method.html

  1. φ \varphi
  2. φ \varphi
  3. φ \varphi
  4. Γ \Gamma
  5. φ \varphi
  6. Γ \Gamma
  7. φ \varphi
  8. Γ = { ( x , y ) | φ ( x , y ) = 0 } , \Gamma=\{(x,y)|\,\varphi(x,y)=0\},\,
  9. Γ \Gamma
  10. φ \varphi
  11. φ \varphi
  12. Γ \Gamma
  13. Γ \Gamma
  14. v v
  15. φ \varphi
  16. φ t = v | φ | . \frac{\partial\varphi}{\partial t}=v|\nabla\varphi|.
  17. | | |\cdot|
  18. t t
  19. 𝐑 2 \mathbf{R}^{2}

Leverage_(finance).html

  1. DOL = EBIT + Fixed Costs EBIT \mathrm{DOL}=\frac{\mathrm{EBIT\;+\;Fixed\;Costs}}{\mathrm{EBIT}}
  2. DFL = EBIT EBIT - Total Interest Expense \mathrm{DFL}=\frac{\mathrm{EBIT}}{\mathrm{EBIT\;-\;Total\;Interest\;Expense}}
  3. DCL = DOL * DFL = EBIT + Fixed Costs EBIT - Total Interest Expense \mathrm{DCL}=\mathrm{DOL*DFL}=\frac{\mathrm{EBIT\;+\;Fixed\;Costs}}{\mathrm{% EBIT\;-\;Total\;Interest\;Expense}}
  4. Operating leverage = Revenue - Variable Cost Revenue - Variable Cost - Fixed Cost = Revenue - Variable Cost Operating Income \mathrm{Operating\;leverage}=\frac{\mathrm{Revenue}-\mathrm{Variable\;Cost}}{% \mathrm{Revenue}-\mathrm{Variable\;Cost}-\mathrm{Fixed\;Cost}}=\frac{\mathrm{% Revenue}-\mathrm{Variable\;Cost}}{\mathrm{Operating\;Income}}
  5. Financial leverage = Total Debt Shareholders Equity \mathrm{Financial\;leverage}=\frac{\mathrm{Total\;Debt}}{\mathrm{Shareholders^% {\prime}\;Equity}}

Lévy's_continuity_theorem.html

  1. φ \varphi
  2. φ n ( t ) φ ( t ) t , \varphi_{n}(t)\to\varphi(t)\quad\forall t\in\mathbb{R},

Lévy_distribution.html

  1. erfc ( c 2 ( x - μ ) ) \textrm{erfc}\left(\sqrt{\frac{c}{2(x-\mu)}}\right)
  2. \infty
  3. c / 2 ( erfc - 1 ( 1 / 2 ) ) 2 c/2(\textrm{erfc}^{-1}(1/2))^{2}\,
  4. μ = 0 \mu=0
  5. c 3 \frac{c}{3}
  6. μ = 0 \mu=0
  7. \infty
  8. 1 + 3 γ + ln ( 16 π c 2 ) 2 \frac{1+3\gamma+\ln(16\pi c^{2})}{2}
  9. γ \gamma
  10. e i μ t - - 2 i c t e^{i\mu t-\sqrt{-2ict}}
  11. x μ x\geq\mu
  12. f ( x ; μ , c ) = c 2 π e - c 2 ( x - μ ) ( x - μ ) 3 / 2 f(x;\mu,c)=\sqrt{\frac{c}{2\pi}}~{}~{}\frac{e^{-\frac{c}{2(x-\mu)}}}{(x-\mu)^{% 3/2}}
  13. μ \mu
  14. c c
  15. F ( x ; μ , c ) = erfc ( c 2 ( x - μ ) ) F(x;\mu,c)=\textrm{erfc}\left(\sqrt{\frac{c}{2(x-\mu)}}\right)
  16. erfc ( z ) \textrm{erfc}(z)
  17. μ \mu
  18. μ \mu
  19. μ \mu
  20. \infty
  21. f ( x ; μ , c ) d x = f ( y ; 0 , 1 ) d y f(x;\mu,c)dx=f(y;0,1)dy\,
  22. y = x - μ c y=\frac{x-\mu}{c}\,
  23. φ ( t ; μ , c ) = e i μ t - - 2 i c t . \varphi(t;\mu,c)=e^{i\mu t-\sqrt{-2ict}}.
  24. α = 1 / 2 \alpha=1/2
  25. β = 1 \beta=1
  26. φ ( t ; μ , c ) = e i μ t - | c t | 1 / 2 ( 1 - i sign ( t ) ) . \varphi(t;\mu,c)=e^{i\mu t-|ct|^{1/2}~{}(1-i~{}\textrm{sign}(t))}.
  27. μ = 0 \mu=0
  28. m n = def c 2 π 0 e - c / 2 x x n x 3 / 2 d x m_{n}\ \stackrel{\mathrm{def}}{=}\ \sqrt{\frac{c}{2\pi}}\int_{0}^{\infty}\frac% {e^{-c/2x}\,x^{n}}{x^{3/2}}\,dx
  29. M ( t ; c ) = def c 2 π 0 e - c / 2 x + t x x 3 / 2 d x M(t;c)\ \stackrel{\mathrm{def}}{=}\ \sqrt{\frac{c}{2\pi}}\int_{0}^{\infty}% \frac{e^{-c/2x+tx}}{x^{3/2}}\,dx
  30. t > 0 t>0
  31. lim x f ( x ; μ , c ) = c 2 π 1 x 3 / 2 . \lim_{x\rightarrow\infty}f(x;\mu,c)=\sqrt{\frac{c}{2\pi}}~{}\frac{1}{x^{3/2}}.
  32. μ = 0 \mu=0
  33. { f ( x ) ( - c - 3 μ + 3 x ) + 2 ( x - μ ) 2 f ( x ) = 0 , f ( 0 ) = e c 2 μ ( - c μ ) 3 / 2 2 π c } \left\{f(x)(-c-3\mu+3x)+2(x-\mu)^{2}f^{\prime}(x)=0,f(0)=\frac{e^{\frac{c}{2% \mu}}\left(-\frac{c}{\mu}\right)^{3/2}}{\sqrt{2\pi}c}\right\}
  34. X Levy ( μ , c ) X\sim\textrm{Levy}(\mu,c)\,
  35. k X + b Levy ( k μ + b , k c ) kX+b\sim\textrm{Levy}(k\mu+b,kc)\,
  36. X Levy ( 0 , c ) X\,\sim\,\textrm{Levy}(0,c)
  37. X Inv-Gamma ( 1 2 , c 2 ) X\,\sim\,\textrm{Inv-Gamma}(\tfrac{1}{2},\tfrac{c}{2})
  38. Y Normal ( μ , σ 2 ) Y\,\sim\,\textrm{Normal}(\mu,\sigma^{2})
  39. ( Y - μ ) - 2 Levy ( 0 , 1 / σ 2 ) {(Y-\mu)}^{-2}\sim\,\textrm{Levy}(0,1/\sigma^{2})
  40. X Normal ( μ , 1 σ ) X\sim\textrm{Normal}(\mu,\tfrac{1}{\sqrt{\sigma}})\,
  41. ( X - μ ) - 2 Levy ( 0 , σ ) {(X-\mu)}^{-2}\sim\textrm{Levy}(0,\sigma)\,
  42. X Levy ( μ , c ) X\,\sim\,\textrm{Levy}(\mu,c)
  43. X Stable ( 1 / 2 , 1 , c , μ ) X\,\sim\,\textrm{Stable}(1/2,1,c,\mu)\,
  44. X Levy ( 0 , c ) X\,\sim\,\textrm{Levy}(0,c)
  45. X Scale-inv- χ 2 ( 1 , c ) X\,\sim\,\textrm{Scale-inv-}\chi^{2}(1,c)
  46. X Levy ( μ , c ) X\,\sim\,\textrm{Levy}(\mu,c)
  47. ( X - μ ) - 1 2 FoldedNormal ( 0 , 1 / c ) {(X-\mu)}^{-\tfrac{1}{2}}\sim\,\textrm{FoldedNormal}(0,1/\sqrt{c})
  48. X = F - 1 ( U ) = c ( Φ - 1 ( 1 - U / 2 ) ) 2 + μ X=F^{-1}(U)=\frac{c}{(\Phi^{-1}(1-U/2))^{2}}+\mu
  49. μ \mu
  50. c c
  51. Φ ( x ) \Phi(x)
  52. α \alpha
  53. c = α 2 c=\alpha^{2}

Liang–Barsky_algorithm.html

  1. x = x 0 + u ( x 1 - x 0 ) = x 0 + u Δ x x=x_{0}+u(x_{1}-x_{0})=x_{0}+u\Delta x\,\!
  2. y = y 0 + u ( y 1 - y 0 ) = y 0 + u Δ y y=y_{0}+u(y_{1}-y_{0})=y_{0}+u\Delta y\,\!
  3. x min x 0 + u Δ x x max x_{\,\text{min}}\leq x_{0}+u\Delta x\leq x_{\,\text{max}}\,\!
  4. y min y 0 + u Δ y y max y_{\,\text{min}}\leq y_{0}+u\Delta y\leq y_{\,\text{max}}\,\!
  5. u p k q k , k = 1 , 2 , 3 , 4 up_{k}\leq q_{k},\quad k=1,2,3,4\,\!
  6. p 1 = - Δ x , q 1 = x 0 - x min p_{1}=-\Delta x,q_{1}=x_{0}-x_{\,\text{min}}\,\!
  7. p 2 = Δ x , q 2 = x max - x 0 p_{2}=\Delta x,q_{2}=x_{\,\text{max}}-x_{0}\,\!
  8. p 3 = - Δ y , q 3 = y 0 - y min p_{3}=-\Delta y,q_{3}=y_{0}-y\text{min}\,\!
  9. p 4 = Δ y , q 4 = y max - y 0 p_{4}=\Delta y,q_{4}=y\text{max}-y_{0}\,\!
  10. p k = 0 p_{k}=0
  11. k k
  12. q k < 0 q_{k}<0
  13. p k < 0 p_{k}<0
  14. p k > 0 p_{k}>0
  15. p k p_{k}
  16. u = q k p k u=\frac{q_{k}}{p_{k}}
  17. u 1 u_{1}
  18. u 2 u_{2}
  19. u 1 u_{1}
  20. p k < 0 p_{k}<0
  21. u 1 u_{1}
  22. { 0 , q k p k } \left\{0,\frac{q_{k}}{p_{k}}\right\}
  23. u 2 u_{2}
  24. p k > 0 p_{k}>0
  25. u 2 u_{2}
  26. { 1 , q k p k } \left\{1,\frac{q_{k}}{p_{k}}\right\}
  27. u 1 > u 2 u_{1}>u_{2}

Liberal_paradox.html

  1. A \succ_{A}
  2. B \succ_{B}
  3. A \succ_{A}
  4. A \succ_{A}
  5. A \succ_{A}
  6. B \succ_{B}
  7. B \succ_{B}
  8. B \succ_{B}
  9. F : R e l ( X ) N 𝒫 ( X ) F:{Rel}(X)^{N}\rightarrow\mathcal{P}(X)

Lidstone_series.html

  1. f ( z ) = n = 0 [ A n ( 1 - z ) f ( 2 n ) ( 0 ) + A n ( z ) f ( 2 n ) ( 1 ) ] + k = 1 N C k sin ( k π z ) . f(z)=\sum_{n=0}^{\infty}\left[A_{n}(1-z)f^{(2n)}(0)+A_{n}(z)f^{(2n)}(1)\right]% +\sum_{k=1}^{N}C_{k}\sin(k\pi z).
  2. h ( θ ; f ) = lim sup r 1 r log | f ( r e i θ ) | h(\theta;f)=\underset{r\to\infty}{\limsup}\,\frac{1}{r}\log|f(re^{i\theta})|\,
  3. t = sup θ [ 0 , 2 π ) h ( θ ; f ) t=\sup_{\theta\in[0,2\pi)}h(\theta;f)\,
  4. N π t < ( N + 1 ) π . N\pi\leq t<(N+1)\pi.\,

Lie_ring.html

  1. [ x , y ] = x y - y x [x,y]=xy-yx
  2. L L
  3. [ , ] [\cdot,\cdot]
  4. [ x + y , z ] = [ x , z ] + [ y , z ] , [ z , x + y ] = [ z , x ] + [ z , y ] [x+y,z]=[x,z]+[y,z],\quad[z,x+y]=[z,x]+[z,y]
  5. [ x , [ y , z ] ] + [ y , [ z , x ] ] + [ z , [ x , y ] ] = 0 [x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0\quad
  6. [ x , x ] = 0 [x,x]=0\quad
  7. [ x , y ] = x y - y x [x,y]=xy-yx
  8. G G
  9. ( x , y ) = x - 1 y - 1 x y (x,y)=x^{-1}y^{-1}xy
  10. G = G 0 G 1 G 2 G n G=G_{0}\supseteq G_{1}\supseteq G_{2}\supseteq\cdots\supseteq G_{n}\supseteq\cdots
  11. G G
  12. ( G i , G j ) (G_{i},G_{j})
  13. G i + j G_{i+j}
  14. i , j i,j
  15. L = G i / G i + 1 L=\bigoplus G_{i}/G_{i+1}
  16. [ x G i , y G j ] = ( x , y ) G i + j [xG_{i},yG_{j}]=(x,y)G_{i+j}
  17. ( x , y ) (x,y)

Lie_theory.html

  1. { exp ( j t ) = cosh ( t ) + j sinh ( t ) : t R } , \{\exp(jt)=\cosh(t)+j\sinh(t):t\in R\},
  2. { exp ( ϵ t ) = 1 + ϵ t : t R } . \{\exp(\epsilon t)=1+\epsilon t:t\in R\}.

Life_table.html

  1. q x \,q_{x}
  2. x \,x
  3. ( x + 1 ) \,(x+1)
  4. p x \,p_{x}
  5. x \,x
  6. ( x + 1 ) \,(x+1)
  7. p x = 1 - q x \,p_{x}=1-q_{x}
  8. x \,\ell_{x}
  9. x \,x
  10. 0 \,\ell_{0}
  11. x + 1 = x ( 1 - q x ) = x p x \,\ell_{x+1}=\ell_{x}\cdot(1-q_{x})=\ell_{x}\cdot p_{x}
  12. x + 1 x = p x \,{\ell_{x+1}\over\ell_{x}}=p_{x}
  13. d x \,d_{x}
  14. x \,x
  15. d x = x - x + 1 = x ( 1 - p x ) = x q x \,d_{x}=\ell_{x}-\ell_{x+1}=\ell_{x}\cdot(1-p_{x})=\ell_{x}\cdot q_{x}
  16. p x t \,{}_{t}p_{x}
  17. x \,x
  18. t \,t
  19. x + t \,x+t
  20. p x t = x + t x \,{}_{t}p_{x}={\ell_{x+t}\over\ell_{x}}
  21. q x t k \,{}_{t\mid k}q_{x}
  22. x \,x
  23. t \,t
  24. k \,k
  25. q x t k = p x t q x + t k = x + t - x + t + k x \,{}_{t\mid k}q_{x}={}_{t}p_{x}\cdot{}_{k}q_{x+t}={\ell_{x+t}-\ell_{x+t+k}% \over\ell_{x}}
  26. m x \,m_{x}

Light_dressed_state.html

  1. H = i 1 2 m i [ 𝐩 i - z i c 𝐀 ( 𝐫 𝐢 , 𝐭 ) ] 2 + V ( { 𝐫 i } ) , ( 1 ) H=\sum_{i}\frac{1}{2m_{i}}\left[\mathbf{p}_{i}-\frac{z_{i}}{c}\mathbf{A(% \mathbf{r}_{i},t)}\right]^{2}+V(\{\mathbf{r}_{i}\}),\ \ \ \ \ \ \ \ \ \ \ (1)
  2. 𝐀 \mathbf{A}
  3. 𝐀 \mathbf{A}
  4. 𝐀 ( t + T ) = 𝐀 ( t ) \mathbf{A}(t+T)=\mathbf{A}(t)
  5. i i\,
  6. 𝐫 i \mathbf{r}_{i}\,
  7. 𝐩 i \mathbf{p}_{i}\,
  8. m i m_{i}\,
  9. z i z_{i}\,
  10. c c\,
  11. H ( t + T ) = H ( t ) . H(t+T)=H(t)\,.
  12. ψ ( 𝐫 , t ) \psi(\mathbf{r},t)
  13. i t ψ ( { 𝐫 i } , t ) = H ( t ) ψ ( { 𝐫 i } , t ) i\hbar\frac{\partial}{\partial t}\psi(\{\mathbf{r}_{i}\},t)=H(t)\psi(\{\mathbf% {r}_{i}\},t)
  14. ψ ( { 𝐫 i } , t ) = exp [ - i E t / ] ϕ ( { 𝐫 i } , t ) \psi(\{\mathbf{r}_{i}\},t)=\exp[-iEt/\hbar]\phi(\{\mathbf{r}_{i}\},t)
  15. ϕ \phi\,
  16. ϕ ( { 𝐫 i } , t + T ) = ϕ ( { 𝐫 i } , t ) . \phi(\{\mathbf{r}_{i}\},t+T)=\phi(\{\mathbf{r}_{i}\},t).
  17. ψ ( { 𝐫 i } , t ) = exp [ - i E t / ] n = - exp [ i n ω t ] ϕ n ( { 𝐫 i } ) ( 2 ) \psi(\{\mathbf{r}_{i}\},t)=\exp[-iEt/\hbar]\sum_{n=-\infty}^{\infty}\exp[in% \omega t]\phi_{n}(\{\mathbf{r}_{i}\})\ \ \ \ \ \ \ \ \ \ \ (2)
  18. ω ( = 2 π / T ) \omega(=2\pi/T)\,
  19. E E\,
  20. n n\,
  21. n n\,
  22. n n\,
  23. n N n\ll N\,
  24. N N\,

Limb_darkening.html

  1. I ( ψ ) I ( 0 ) = k = 0 N a k cos k ( ψ ) \frac{I(\psi)}{I(0)}=\sum_{k=0}^{N}a_{k}\,\textrm{cos}^{k}(\psi)
  2. k = 0 N a k = 1 \sum_{k=0}^{N}a_{k}=1
  3. a 0 = 1 - a 1 - a 2 a_{0}=1-a_{1}-a_{2}\,
  4. a 1 = 0.93 a_{1}=0.93\,
  5. a 2 = - 0.23 a_{2}=-0.23\,
  6. I ( ψ ) I ( 0 ) = 1 + k = 1 N A k ( 1 - cos ( ψ ) ) k \frac{I(\psi)}{I(0)}=1+\sum_{k=1}^{N}A_{k}\,(1-\cos(\psi))^{k}
  7. cos ( ψ ) = cos 2 ( θ ) - cos 2 ( Ω ) sin ( Ω ) \cos(\psi)=\frac{\sqrt{\cos^{2}(\theta)-\cos^{2}(\Omega)}}{\sin(\Omega)}
  8. I m = I ( ψ ) d ω d ω I_{m}=\frac{\int I(\psi)d\omega}{\int d\omega}
  9. I m I ( 0 ) = 2 k = 0 N a k k + 2 \frac{I_{m}}{I(0)}=2\sum_{k=0}^{N}\frac{a_{k}}{k+2}

Limiting_reagent.html

  1. 2 C 6 H 6 ( l ) + 15 O 2 ( g ) 12 C O 2 ( g ) + 6 H 2 O ( l ) 2\,\text{C}_{6}\,\text{H}_{6}(l)+15\,\text{O}_{2}(g)\rightarrow 12\,\text{C}\,% \text{O}_{2}(g)+6\,\text{H}_{2}\,\text{O}(l)
  2. 1.5 mol C 6 H 6 × 15 mol O 2 2 mol C 6 H 6 = 11.25 mol O 2 1.5\ \mbox{mol}~{}\,\,\text{C}_{6}\,\text{H}_{6}\times\frac{15\ \mbox{mol}~{}% \,\,\text{O}_{2}}{2\ \mbox{mol}~{}\,\,\text{C}_{6}\,\text{H}_{6}}=11.25\ \mbox% {mol}~{}\,\,\text{O}_{2}
  3. required: mol O 2 mol C 6 H 6 = 15 mol O 2 2 mol C 6 H 6 = 7.5 mol O 2 \,\text{required:}\;\frac{\mbox{mol}~{}\,\,\text{O}_{2}}{\mbox{mol}~{}\,\,% \text{C}_{6}\,\text{H}_{6}}=\frac{15\ \mbox{mol}~{}\,\,\text{O}_{2}}{2\ \mbox{% mol}~{}\,\,\text{C}_{6}\,\text{H}_{6}}=7.5\ \mbox{mol}~{}\,\,\text{O}_{2}
  4. actual: mol O 2 mol C 6 H 6 = 18 mol O 2 1.5 mol C 6 H 6 = 12 mol O 2 \,\text{actual:}\;\frac{\mbox{mol}~{}\,\,\text{O}_{2}}{\mbox{mol}~{}\,\,\text{% C}_{6}\,\text{H}_{6}}=\frac{18\ \mbox{mol}~{}\,\,\text{O}_{2}}{1.5\ \mbox{mol}% ~{}\,\,\text{C}_{6}\,\text{H}_{6}}=12\ \mbox{mol}~{}\,\,\text{O}_{2}
  5. Fe 2 O 3 ( s ) + 2 Al ( s ) 2 Fe ( l ) + Al 2 O 3 ( s ) \,\text{Fe}_{2}\,\text{O}_{3}(s)+2\,\text{Al}(s)\rightarrow 2\,\text{Fe}(l)+\,% \text{Al}_{2}\,\text{O}_{3}(s)\,
  6. mol Fe 2 O 3 = grams Fe 2 O 3 g/mol Fe 2 O 3 \mbox{mol}~{}\,\,\,\text{Fe}_{2}\,\text{O}_{3}=\frac{\,\text{grams}\,\,\,\text% {Fe}_{2}\,\text{O}_{3}}{\,\text{g/mol}\,\,\,\text{Fe}_{2}\,\text{O}_{3}}\,
  7. mol Fe 2 O 3 = 20.0 g 159.7 g/mol = 0.125 mol \mbox{mol}~{}\,\,\,\text{Fe}_{2}\,\text{O}_{3}=\frac{20.0\,\,\,\text{g}}{159.7% \,\,\,\text{g/mol}}=0.125\,\,\mbox{mol}~{}
  8. mol Fe = 0.125 mol Fe 2 O 3 × 2 mol Fe 1 mol Fe 2 O 3 = 0.250 mol Fe \mbox{mol}~{}\,\,\,\text{Fe}=0.125\ \mbox{mol}~{}\,\,\text{Fe}_{2}\,\text{O}_{% 3}\times\frac{2\ \mbox{mol}~{}\,\,\text{Fe}}{1\ \mbox{mol}~{}\,\,\text{Fe}_{2}% \,\text{O}_{3}}=0.250\ \mbox{mol}~{}\,\,\text{Fe}
  9. mol Al = grams Al g/mol Al \mbox{mol}~{}\,\,\,\text{Al}=\frac{\,\text{grams Al}}{\,\text{g/mol Al}}\,
  10. mol Al = 8.00 g 26.98 g/mol = 0.297 mol \mbox{mol}~{}\,\,\,\text{Al}=\frac{8.00\,\,\,\text{g}}{26.98\,\,\,\text{g/mol}% }=0.297\,\,\mbox{mol}~{}\,
  11. mol Fe = 0.297 mol Al × 2 mol Fe 2 mol Al = 0.297 mol Fe \mbox{mol}~{}\,\,\,\text{Fe}=0.297\ \mbox{mol}~{}\,\,\text{Al}\times\frac{2\ % \mbox{mol}~{}\,\,\text{Fe}}{2\ \mbox{mol}~{}\,\,\text{Al}}=0.297\ \mbox{mol}~{% }\,\,\text{Fe}
  12. Moles of Reagent X Coefficient of Reagent X \frac{\mbox{Moles of Reagent X }~{}}{\mbox{Coefficient of Reagent X}~{}}

Lindelöf_hypothesis.html

  1. ζ ( 1 2 + i t ) is 𝒪 ( t ε ) , \zeta\left(\frac{1}{2}+it\right)\mbox{ is }~{}\mathcal{O}(t^{\varepsilon}),
  2. ζ ( 1 2 + i t ) is o ( t ε ) . \zeta\left(\frac{1}{2}+it\right)\mbox{ is }~{}o(t^{\varepsilon}).
  3. 1 T 0 T | ζ ( 1 / 2 + i t ) | 2 k d t = O ( T ε ) \frac{1}{T}\int_{0}^{T}|\zeta(1/2+it)|^{2k}\,dt=O(T^{\varepsilon})
  4. 0 T | ζ ( 1 / 2 + i t ) | 2 k d t = T j = 0 k 2 c k , j log ( T ) k 2 - j + o ( T ) \int_{0}^{T}|\zeta(1/2+it)|^{2k}\,dt=T\sum_{j=0}^{k^{2}}c_{k,j}\log(T)^{k^{2}-% j}+o(T)
  5. ( 42 / 9 ! ) p ( ( 1 - p - 1 ) 4 ( 1 + 4 p - 1 + p - 2 ) ) (42/9!)\prod_{p}\left((1-p^{-1})^{4}(1+4p^{-1}+p^{-2})\right)
  6. p n + 1 - p n p n 1 / 2 + ε p_{n+1}-p_{n}\ll p_{n}^{1/2+\varepsilon}\,

Line_search.html

  1. 𝐱 * \mathbf{x}^{*}
  2. f : n f:\mathbb{R}^{n}\to\mathbb{R}
  3. f f
  4. 𝐱 \mathbf{x}
  5. k = 0 \displaystyle k=0
  6. 𝐱 0 \mathbf{x}_{0}
  7. 𝐩 k \mathbf{p}_{k}
  8. α k \displaystyle\alpha_{k}
  9. h ( α ) = f ( 𝐱 k + α 𝐩 k ) h(\alpha)=f(\mathbf{x}_{k}+\alpha\mathbf{p}_{k})
  10. α + \alpha\in\mathbb{R}_{+}
  11. 𝐱 k + 1 = 𝐱 k + α k 𝐩 k \mathbf{x}_{k+1}=\mathbf{x}_{k}+\alpha_{k}\mathbf{p}_{k}
  12. k = k + 1 \displaystyle k=k+1
  13. f ( 𝐱 k ) \|\nabla f(\mathbf{x}_{k})\|
  14. h ( α k ) = 0 h^{\prime}(\alpha_{k})=0
  15. f ( x ) f(x)
  16. 1 ϕ ( x 2 - x 1 ) = x 4 - x 1 = x 2 - x 3 = ϕ ( x 2 - x 4 ) = ϕ ( x 3 - x 1 ) = ϕ 2 ( x 4 - x 3 ) \frac{1}{\phi}(x_{2}-x_{1})=x_{4}-x_{1}=x_{2}-x_{3}=\phi(x_{2}-x_{4})=\phi(x_{% 3}-x_{1})=\phi^{2}(x_{4}-x_{3})
  17. ϕ = 1 2 ( 1 + 5 ) 1.618 \phi=\frac{1}{2}(1+\sqrt{5})\approx 1.618

Linear_code.html

  1. 𝔽 q n \mathbb{F}_{q}^{n}
  2. 𝔽 q \mathbb{F}_{q}
  3. 𝔽 q n \mathbb{F}_{q}^{n}
  4. 𝔽 q n \mathbb{F}_{q}^{n}
  5. G = ( I k | A ) G=(I_{k}|A)
  6. I k I_{k}
  7. k × k k\times k
  8. k × ( n - k ) k\times(n-k)
  9. ϕ : 𝔽 q n 𝔽 q n - k \phi:\mathbb{F}_{q}^{n}\to\mathbb{F}_{q}^{n-k}
  10. min c C , c c 0 d ( c , c 0 ) = min c C , c c 0 d ( c - c 0 , 0 ) = min c C , c 0 d ( c , 0 ) = d . \min_{c\in C,\ c\neq c_{0}}d(c,c_{0})=\min_{c\in C,c\neq c_{0}}d(c-c_{0},0)=% \min_{c\in C,c\neq 0}d(c,0)=d.
  11. s y m b o l H s y m b o l c T = s y m b o l 0 symbol{H}\cdot symbol{c}^{T}=symbol{0}
  12. i = 1 n ( c i s y m b o l H i ) = s y m b o l 0 \sum_{i=1}^{n}(c_{i}\cdot symbol{H_{i}})=symbol{0}
  13. s y m b o l H i symbol{H_{i}}
  14. i t h i^{th}
  15. s y m b o l H symbol{H}
  16. c i = 0 c_{i}=0
  17. s y m b o l H i symbol{H_{i}}
  18. c i 0 c_{i}\neq 0
  19. d d
  20. { s y m b o l H j | j S } \{symbol{H_{j}}|j\in S\}
  21. S S
  22. i = 1 n ( c i s y m b o l H i ) = j S ( c j s y m b o l H j ) + j S ( c j s y m b o l H j ) = s y m b o l 0 \sum_{i=1}^{n}(c_{i}\cdot symbol{H_{i}})=\sum_{j\in S}(c_{j}\cdot symbol{H_{j}% })+\sum_{j\notin S}(c_{j}\cdot symbol{H_{j}})=symbol{0}
  23. s y m b o l c symbol{c^{\prime}}
  24. c j = 0 c_{j}^{{}^{\prime}}=0
  25. j S j\notin S
  26. s y m b o l c C symbol{c^{\prime}}\in C
  27. s y m b o l H s y m b o l c T = s y m b o l 0 symbol{H}\cdot symbol{c^{\prime}}^{T}=symbol{0}
  28. d w t ( s y m b o l c ) d\leq wt(symbol{c^{\prime}})
  29. s y m b o l H symbol{H}
  30. r 2 r\geq 2
  31. [ 2 r - 1 , 2 r - r - 1 , 3 ] 2 [2^{r}-1,2^{r}-r-1,3]_{2}
  32. d = 3 d=3
  33. [ 7 , 4 , 3 ] 2 [7,4,3]_{2}
  34. s y m b o l G = ( 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 1 1 1 0 0 0 1 1 0 1 ) , symbol{G}=\begin{pmatrix}1\ 0\ 0\ 0\ 1\ 1\ 0\\ 0\ 1\ 0\ 0\ 0\ 1\ 1\\ 0\ 0\ 1\ 0\ 1\ 1\ 1\\ 0\ 0\ 0\ 1\ 1\ 0\ 1\end{pmatrix},
  35. s y m b o l H = ( 1 0 1 1 1 0 0 1 1 1 0 0 1 0 0 1 1 1 0 0 1 ) symbol{H}=\begin{pmatrix}1\ 0\ 1\ 1\ 1\ 0\ 0\\ 1\ 1\ 1\ 0\ 0\ 1\ 0\\ 0\ 1\ 1\ 1\ 0\ 0\ 1\end{pmatrix}
  36. [ 2 r , r , 2 r - 1 ] 2 [2^{r},r,2^{r-1}]_{2}
  37. i t h i^{th}
  38. i i
  39. 2 r - 1 2^{r-1}
  40. 2 r - 2 - 1 2^{r-2}-1
  41. [ 8 , 3 , 4 ] 2 [8,3,4]_{2}
  42. s y m b o l G H a d = ( 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 ) symbol{G}_{Had}=\begin{pmatrix}0\ 0\ 0\ 0\ 1\ 1\ 1\ 1\\ 0\ 0\ 1\ 1\ 0\ 0\ 1\ 1\\ 0\ 1\ 0\ 1\ 0\ 1\ 0\ 1\end{pmatrix}
  43. s y m b o l G H a d symbol{G}_{Had}
  44. [ 7 , 3 , 4 ] 2 [7,3,4]_{2}
  45. 𝔽 q n \mathbb{F}_{q}^{n}
  46. 𝔽 q n \mathbb{F}_{q}^{n}
  47. k + d n + 1 k+d\leq n+1
  48. n × n n\times n
  49. M : 𝔽 q n 𝔽 q n M\colon\mathbb{F}_{q}^{n}\to\mathbb{F}_{q}^{n}
  50. 2 2 m \mathbb{Z}_{2}^{2m}
  51. 4 m \mathbb{Z}_{4}^{m}
  52. 2 2 m \mathbb{Z}_{2}^{2m}
  53. 4 m \mathbb{Z}_{4}^{m}

Linear_complementarity_problem.html

  1. w = M z + q {w}={Mz}+{q}\,
  2. w 0 , z 0 {w}\geq 0,{z}\geq 0\,
  3. w i z i = 0 {w}_{i}{z}_{i}=0\,
  4. w {w}\,
  5. z {z}\,
  6. M z + q 0 {Mz}+{q}\geq{0}\,
  7. z 0 {z}\geq{0}\,
  8. z T ( M z + q ) = 0 {z}^{\mathrm{T}}({Mz}+{q})=0\,
  9. f ( z ) = z T ( M z + q ) f({z})={z}^{\mathrm{T}}({Mz}+{q})\,
  10. M z + q 0 {Mz}+{q}\geq{0}\,
  11. z 0 {z}\geq{0}\,
  12. f ( x ) = c T x + 1 2 x T Q x f({x})={c}^{T}{x}+\frac{1}{2}{x}^{T}{Qx}\,
  13. A x b {Ax}\geq{b}\,
  14. x 0 {x}\geq{0}\,
  15. q = [ c - b ] {q}=\left[\begin{array}[]{c}{c}\\ -{b}\end{array}\right]\,
  16. M = [ Q - A T A 0 ] {M}=\left[\begin{array}[]{cc}{Q}&-{A}^{T}\\ {A}&0\end{array}\right]\,
  17. v = Q x - A T λ + c {v}={Q}{x}-{A}^{T}{\lambda}+{c}\,
  18. s = A x - b {s}={A}{x}-{b}\,
  19. x , λ , v , s 0 {x},{\lambda},{v},{s}\geq{0}\,
  20. x T v + λ T s = 0 {x}^{T}{v}+{\lambda}^{T}{s}={0}\,
  21. v {v}\,
  22. λ {\lambda}\,
  23. s {s}\,
  24. x , s {x},{s}\,
  25. v , λ {v},{\lambda}\,
  26. z = [ x λ ] {z}=\left[\begin{array}[]{c}{x}\\ {\lambda}\end{array}\right]\,
  27. w = [ v s ] {w}=\left[\begin{array}[]{c}{v}\\ {s}\end{array}\right]\,
  28. x {x}\,
  29. Q {Q}\,
  30. v {v}\,
  31. Q x = A T λ - c {Q}{x}={A}^{T}{\lambda}-{c}\,
  32. x = Q - 1 ( A T λ - c ) {x}={Q}^{-1}({A}^{T}{\lambda}-{c})\,
  33. A {A}\,
  34. b {b}\,
  35. A x - b = A Q - 1 ( A T λ - c ) - b {A}{x}-{b}={A}{Q}^{-1}({A}^{T}{\lambda}-{c})-{b}\,
  36. s {s}\,
  37. s = ( A Q - 1 A T ) λ + ( - A Q - 1 c - b ) {s}=({A}{Q}^{-1}{A}^{T}){\lambda}+(-{A}{Q}^{-1}{c}-{b})\,
  38. M = def ( A Q - 1 A T ) {M}\,\overset{\underset{\mathrm{def}}{}}{=}\,({A}{Q}^{-1}{A}^{T})\,
  39. q = def ( - A Q - 1 c - b ) {q}\,\overset{\underset{\mathrm{def}}{}}{=}\,(-{A}{Q}^{-1}{c}-{b})\,
  40. s {s}\,
  41. λ {\lambda}\,
  42. x {x}\,
  43. λ {\lambda}\,
  44. A e q x = b e q {A}_{eq}{x}={b}_{eq}\,
  45. μ {\mu}\,
  46. b e q {b}_{eq}\,
  47. M {M}\,
  48. q {q}\,
  49. s = M λ + q {s}={M}{\lambda}+{q}\,
  50. M = def ( [ A 0 ] [ Q A e q T - A e q 0 ] - 1 [ A T 0 ] ) {M}~{}\overset{\underset{\mathrm{def}}{}}{=}~{}\left(\left[\begin{array}[]{cc}% {A}&{0}\end{array}\right]\left[\begin{array}[]{cc}{Q}&{A}_{eq}^{T}\\ -{A}_{eq}&{0}\end{array}\right]^{-1}\left[\begin{array}[]{cc}{A}^{T}\\ {0}\end{array}\right]\right)\,
  51. q = def ( - [ A 0 ] [ Q A e q T - A e q 0 ] - 1 [ c b e q ] - b ) {q}~{}\overset{\underset{\mathrm{def}}{}}{=}~{}\left(-\left[\begin{array}[]{cc% }{A}&{0}\end{array}\right]\left[\begin{array}[]{cc}{Q}&{A}_{eq}^{T}\\ -{A}_{eq}&{0}\end{array}\right]^{-1}\left[\begin{array}[]{c}{c}\\ {b}_{eq}\end{array}\right]-{b}\right)\,
  52. λ {\lambda}\,
  53. x {x}\,
  54. μ {\mu}\,
  55. [ x μ ] = [ Q A e q T - A e q 0 ] - 1 [ A T λ - c - b e q ] \left[\begin{array}[]{c}{x}\\ {\mu}\end{array}\right]=\left[\begin{array}[]{cc}{Q}&{A}_{eq}^{T}\\ -{A}_{eq}&{0}\end{array}\right]^{-1}\left[\begin{array}[]{c}{A}^{T}{\lambda}-{% c}\\ -{b}_{eq}\end{array}\right]\,

Linear_complex_structure.html

  1. J : V V J:V\rightarrow V
  2. J 2 = - Id V . J^{2}=-\rm{Id}_{V}.
  3. J J
  4. V V
  5. J J
  6. 1 −1
  7. i i
  8. V V
  9. ( x + i y ) v = x v + y J ( v ) (x+iy)v=xv+yJ(v)
  10. x , y x,y
  11. v v
  12. V V
  13. V V
  14. W W
  15. J w = i w Jw=iw
  16. w W w∈W
  17. 𝐂 \mathbf{C}
  18. 𝐂 = 𝐑 [ x ] / ( x 2 + 1 ) , \mathbf{C}=\mathbf{R}[x]/(x^{2}+1),
  19. 𝐂 \mathbf{C}
  20. V V
  21. 𝐂 \mathbf{C}
  22. V V
  23. 𝐂 E n d ( V ) \mathbf{C}→End(V)
  24. i i
  25. i i
  26. i i
  27. E n d ( V ) End(V)
  28. J J
  29. n n
  30. V V
  31. 2 n 2n
  32. V V
  33. J J
  34. e , f e,f
  35. J e = f Je=f
  36. J f = e Jf=−e
  37. V V
  38. V V
  39. A : V V A:V→V
  40. A A
  41. J J
  42. A J = J A . AJ=JA.
  43. U U
  44. V V
  45. J J
  46. U U
  47. J U = U . JU=U.
  48. i ( λ v ) = ( i λ ) v = ( λ i ) v = λ ( i v ) \qquad i(\lambda v)=(i\lambda)v=(\lambda i)v=\lambda(iv)\qquad
  49. { e 1 , e 2 , , e n } \left\{e_{1},e_{2},\dots,e_{n}\right\}
  50. { i e 1 , i e 2 , , i e n } , \left\{ie_{1},ie_{2},\dots,ie_{n}\right\},
  51. 𝐂 n = 𝐑 n 𝐑 𝐂 \mathbf{C}^{n}=\mathbf{R}^{n}\otimes_{\mathbf{R}}\mathbf{C}
  52. 𝐂 n = 𝐂 𝐑 𝐑 n . \mathbf{C}^{n}=\mathbf{C}\otimes_{\mathbf{R}}\mathbf{R}^{n}.
  53. { e 1 , i e 1 , e 2 , i e 2 , , e n , i e n } , \left\{e_{1},ie_{1},e_{2},ie_{2},\dots,e_{n},ie_{n}\right\},
  54. J 2 n = [ 0 - 1 1 0 0 - 1 1 0 0 - 1 1 0 ] = [ J 2 J 2 J 2 ] . J_{2n}=\begin{bmatrix}0&-1\\ 1&0\\ &&0&-1\\ &&1&0\\ &&&&\ddots\\ &&&&&\ddots\\ &&&&&&0&-1\\ &&&&&&1&0\end{bmatrix}=\begin{bmatrix}J_{2}\\ &J_{2}\\ &&\ddots\\ &&&J_{2}\end{bmatrix}.
  55. 𝐂 m 𝐂 n \mathbf{C}^{m}\oplus\mathbf{C}^{n}
  56. 𝐂 m + n . \mathbf{C}^{m+n}.
  57. { e 1 , e 2 , , e n , i e 1 , i e 2 , , i e n } , \left\{e_{1},e_{2},\dots,e_{n},ie_{1},ie_{2},\dots,ie_{n}\right\},
  58. J 2 n = [ 0 - I n I n 0 ] . J_{2n}=\begin{bmatrix}0&-I_{n}\\ I_{n}&0\end{bmatrix}.
  59. { A G L ( 2 n , 𝐑 ) A J = J A } . \left\{A\in GL(2n,\mathbf{R})\mid AJ=JA\right\}.
  60. [ J , A ] = 0 ; [J,A]=0;
  61. [ J , - ] . [J,-].
  62. A J - J A = 0 , AJ-JA=0,
  63. [ A , J ] = 0 , [A,J]=0,
  64. J ( v , w ) = ( - w , v ) . J(v,w)=(-w,v).\,
  65. J = [ 0 - I V I V 0 ] J=\begin{bmatrix}0&-I_{V}\\ I_{V}&0\end{bmatrix}
  66. I V I_{V}
  67. 𝐂 𝐑 V . \mathbf{C}\otimes_{\mathbf{R}}V.
  68. B B
  69. V V
  70. J J
  71. B B
  72. B ( J u , J v ) = B ( u , v ) B(Ju,Jv)=B(u,v)
  73. u , v V u,v∈V
  74. J J
  75. B B
  76. B ( J u , v ) = - B ( u , J v ) B(Ju,v)=-B(u,Jv)
  77. g g
  78. V V
  79. J J
  80. g g
  81. J J
  82. J J
  83. ω ω
  84. J J
  85. ω ( J u , J v ) = ω ( u , v ) ) ω(Ju,Jv)=ω(u,v))
  86. ω ω
  87. J J
  88. ω ω
  89. ω ( u , J u ) > 0 \omega(u,Ju)>0
  90. u u
  91. V V
  92. J J
  93. ω ω
  94. ω ω
  95. J J
  96. g J ( u , v ) = ω ( u , J v ) g_{J}(u,v)=\omega(u,Jv)
  97. J J
  98. ω ω
  99. J J
  100. V = V . V^{\mathbb{C}}=V\otimes_{\mathbb{R}}\mathbb{C}.
  101. v z ¯ = v z ¯ \overline{v\otimes z}=v\otimes\bar{z}
  102. J ( v z ) = J ( v ) z . J(v\otimes z)=J(v)\otimes z.
  103. V = V + V - V^{\mathbb{C}}=V^{+}\oplus V^{-}
  104. 𝒫 ± = 1 2 ( 1 i J ) . \mathcal{P}^{\pm}={1\over 2}(1\mp iJ).
  105. V ± = { v 1 J v i : v V } . V^{\pm}=\{v\otimes 1\mp Jv\otimes i:v\in V\}.
  106. W W W ¯ . W^{\mathbb{C}}\cong W\oplus\overline{W}.
  107. ( V * ) = ( V * ) + ( V * ) - (V^{*})^{\mathbb{C}}=(V^{*})^{+}\oplus(V^{*})^{-}
  108. Λ r U = p + q = r ( Λ p S ) ( Λ q T ) . \Lambda^{r}U=\bigoplus_{p+q=r}(\Lambda^{p}S)\otimes(\Lambda^{q}T).
  109. Λ r V = p + q = r Λ p , q V J \Lambda^{r}\,V^{\mathbb{C}}=\bigoplus_{p+q=r}\Lambda^{p,q}\,V_{J}
  110. Λ p , q V J = def ( Λ p V + ) ( Λ q V - ) . \Lambda^{p,q}\,V_{J}\;\stackrel{\mathrm{def}}{=}\,(\Lambda^{p}\,V^{+})\otimes(% \Lambda^{q}\,V^{-}).
  111. dim Λ r V = ( 2 n r ) dim Λ p , q V J = ( n p ) ( n q ) . \dim_{\mathbb{C}}\Lambda^{r}\,V^{\mathbb{C}}={2n\choose r}\qquad\dim_{\mathbb{% C}}\Lambda^{p,q}\,V_{J}={n\choose p}{n\choose q}.

Linear_discriminant_analysis.html

  1. x {\vec{x}}
  2. x \vec{x}
  3. p ( x | y = 0 ) p(\vec{x}|y=0)
  4. p ( x | y = 1 ) p(\vec{x}|y=1)
  5. ( μ 0 , Σ 0 ) \left(\vec{\mu}_{0},\Sigma_{0}\right)
  6. ( μ 1 , Σ 1 ) \left(\vec{\mu}_{1},\Sigma_{1}\right)
  7. ( x - μ 0 ) T Σ 0 - 1 ( x - μ 0 ) + ln | Σ 0 | - ( x - μ 1 ) T Σ 1 - 1 ( x - μ 1 ) - ln | Σ 1 | < T (\vec{x}-\vec{\mu}_{0})^{T}\Sigma_{0}^{-1}(\vec{x}-\vec{\mu}_{0})+\ln|\Sigma_{% 0}|-(\vec{x}-\vec{\mu}_{1})^{T}\Sigma_{1}^{-1}(\vec{x}-\vec{\mu}_{1})-\ln|% \Sigma_{1}|\ <\ T
  8. Σ 0 = Σ 1 = Σ \Sigma_{0}=\Sigma_{1}=\Sigma
  9. x T Σ 0 - 1 x = x T Σ 1 - 1 x {\vec{x}}^{T}\Sigma_{0}^{-1}\vec{x}={\vec{x}}^{T}\Sigma_{1}^{-1}\vec{x}
  10. x T Σ i - 1 μ i = μ i T Σ i - 1 x {\vec{x}}^{T}{\Sigma_{i}}^{-1}\vec{\mu_{i}}={\vec{\mu_{i}}}^{T}{\Sigma_{i}}^{-% 1}\vec{x}
  11. Σ i \Sigma_{i}
  12. w x > c \vec{w}\cdot\vec{x}>c
  13. w = Σ - 1 ( μ 1 - μ 0 ) \vec{w}=\Sigma^{-1}(\vec{\mu}_{1}-\vec{\mu}_{0})
  14. c = 1 2 ( T - μ 0 T Σ 0 - 1 μ 0 + μ 1 T Σ 1 - 1 μ 1 ) c=\frac{1}{2}(T-{\vec{\mu_{0}}}^{T}\Sigma_{0}^{-1}{\vec{\mu_{0}}}+{\vec{\mu_{1% }}}^{T}\Sigma_{1}^{-1}{\vec{\mu_{1}}})
  15. x \vec{x}
  16. x \vec{x}
  17. x \vec{x}
  18. w \vec{w}
  19. x \vec{x}
  20. w \vec{w}
  21. μ 0 , μ 1 \vec{\mu}_{0},\vec{\mu}_{1}
  22. Σ 0 , Σ 1 \Sigma_{0},\Sigma_{1}
  23. w x \vec{w}\cdot\vec{x}
  24. w μ i \vec{w}\cdot\vec{\mu}_{i}
  25. w T Σ i w \vec{w}^{T}\Sigma_{i}\vec{w}
  26. i = 0 , 1 i=0,1
  27. S = σ between 2 σ within 2 = ( w μ 1 - w μ 0 ) 2 w T Σ 1 w + w T Σ 0 w = ( w ( μ 1 - μ 0 ) ) 2 w T ( Σ 0 + Σ 1 ) w S=\frac{\sigma_{\,\text{between}}^{2}}{\sigma_{\,\text{within}}^{2}}=\frac{(% \vec{w}\cdot\vec{\mu}_{1}-\vec{w}\cdot\vec{\mu}_{0})^{2}}{\vec{w}^{T}\Sigma_{1% }\vec{w}+\vec{w}^{T}\Sigma_{0}\vec{w}}=\frac{(\vec{w}\cdot(\vec{\mu}_{1}-\vec{% \mu}_{0}))^{2}}{\vec{w}^{T}(\Sigma_{0}+\Sigma_{1})\vec{w}}
  28. w ( Σ 0 + Σ 1 ) - 1 ( μ 1 - μ 0 ) \vec{w}\propto(\Sigma_{0}+\Sigma_{1})^{-1}(\vec{\mu}_{1}-\vec{\mu}_{0})
  29. w \vec{w}
  30. w \vec{w}
  31. w \vec{w}
  32. w μ 0 \vec{w}\cdot\vec{\mu}_{0}
  33. w μ 1 \vec{w}\cdot\vec{\mu}_{1}
  34. w x > c \vec{w}\cdot\vec{x}>c
  35. c = w 1 2 ( μ 0 + μ 1 ) = 1 2 μ 1 t Σ - 1 μ 1 - 1 2 μ 0 t Σ - 1 μ 0 c=\vec{w}\cdot\frac{1}{2}(\vec{\mu}_{0}+\vec{\mu}_{1})=\frac{1}{2}\vec{\mu}_{1% }^{t}\Sigma^{-1}\vec{\mu}_{1}-\frac{1}{2}\vec{\mu}_{0}^{t}\Sigma^{-1}\vec{\mu}% _{0}
  36. μ i \mu_{i}
  37. Σ \Sigma
  38. Σ b = 1 C i = 1 C ( μ i - μ ) ( μ i - μ ) T \Sigma_{b}=\frac{1}{C}\sum_{i=1}^{C}(\mu_{i}-\mu)(\mu_{i}-\mu)^{T}
  39. μ \mu
  40. w \vec{w}
  41. S = w T Σ b w w T Σ w S=\frac{\vec{w}^{T}\Sigma_{b}\vec{w}}{\vec{w}^{T}\Sigma\vec{w}}
  42. w \vec{w}
  43. Σ - 1 Σ b \Sigma^{-1}\Sigma_{b}
  44. Σ - 1 Σ b \Sigma^{-1}\Sigma_{b}
  45. Σ b \Sigma_{b}
  46. Σ b \Sigma_{b}
  47. Σ = ( 1 - λ ) Σ + λ I \Sigma=(1-\lambda)\Sigma+\lambda I\,
  48. I I
  49. λ \lambda
  50. p ( x | c = i ) p(\vec{x}|c=i)
  51. P ( c | x ) P(c|\vec{x})

Linear_grammar.html

  1. { a i b i | i 0 } \{a^{i}b^{i}\;|\;i\geq 0\}
  2. L M L\cap M

Linear_phase.html

  1. , sin ( ω t + θ ) , ,\ \sin(\omega t+\theta),
  2. τ , \tau,
  3. A ( ω ) sin ( ω ( t - τ ) + θ ) = A ( ω ) sin ( ω t + θ - ω τ ) , A(\omega)\cdot\sin(\omega(t-\tau)+\theta)=A(\omega)\cdot\sin(\omega t+\theta-% \omega\tau),
  4. A ( ω ) A(\omega)
  5. ω τ \omega\tau
  6. - τ -\tau
  7. ω τ , \omega\tau,
  8. n = - h [ n ] sin ( ω ( n - α ) + β ) = 0 \sum_{n=-\infty}^{\infty}h[n]\cdot\sin(\omega\cdot(n-\alpha)+\beta)=0
  9. α , β \alpha,\beta

Linear_probability_model.html

  1. Y Y
  2. X X
  3. Pr ( Y = 1 | X = x ) = x β . \Pr(Y=1|X=x)=x^{\prime}\beta.
  4. E [ Y | X ] = Pr ( Y = 1 | X ) = x β , E[Y|X]=\Pr(Y=1|X)=x^{\prime}\beta,
  5. Var ( Y | X = x ) \operatorname{Var}(Y|X=x)
  6. β \beta
  7. [ 0 , 1 ] [0,1]

Linear_probing.html

  1. H ( x , i ) = ( H ( x ) + i ) ( mod n ) . H(x,i)=(H(x)+i)\;\;(\mathop{{\rm mod}}n).\,

Link_budget.html

  1. P R X = P T X + G T X - L T X - L F S - L M + G R X - L R X P_{RX}=P_{TX}+G_{TX}-L_{TX}-L_{FS}-L_{M}+G_{RX}-L_{RX}\,
  2. P R X P_{RX}
  3. P T X P_{TX}
  4. G T X G_{TX}
  5. L T X L_{TX}
  6. L F S L_{FS}
  7. L M L_{M}
  8. G R X G_{RX}
  9. L R X L_{RX}
  10. L F S L_{FS}
  11. L F S L_{FS}
  12. L F S L_{FS}
  13. L F S L_{FS}

Linkage_(mechanical).html

  1. M = 6 n - i = 1 j ( 6 - f i ) = 6 ( N - 1 - j ) + i = 1 j f i , M=6n-\sum_{i=1}^{j}(6-f_{i})=6(N-1-j)+\sum_{i=1}^{j}\ f_{i},
  2. M = i = 1 j f i . M=\sum_{i=1}^{j}\ f_{i}.
  3. M = i = 1 j f i - 6. M=\sum_{i=1}^{j}\ f_{i}-6.
  4. M = 3 ( N - 1 - j ) + i = 1 j f i , M=3(N-1-j)+\sum_{i=1}^{j}\ f_{i},
  5. M = i = 1 j f i , M=\sum_{i=1}^{j}\ f_{i},
  6. M = i = 1 j f i - 3. M=\sum_{i=1}^{j}\ f_{i}-3.
  7. M = 3 ( N - 1 - j ) + j = 1 , M=3(N-1-j)+j=1,\!
  8. j = ( 3 / 2 ) N - 2. j=(3/2)N-2.\!

Linkage_principle.html

  1. 0 , ω 0,ω
  2. i i
  3. i i
  4. v i ( 𝐗 ) = u ( X i , 𝐗 - i ) , v_{i}(\mathbf{X})=u(X_{i},\mathbf{X}_{-i}),
  5. u u
  6. N 1 N−1
  7. Y 1 , , Y N - 1 Y_{1},\cdots,Y_{N-1}
  8. X 2 , , X N X_{2},\cdots,X_{N}
  9. G ( x ) G(\cdot\mid x)
  10. Y 1 Y_{1}
  11. X 1 = x X_{1}=x
  12. G ( z x ) Pr ( Y 1 < z X 1 = x ) G(z\mid x)\equiv\Pr\left(Y_{1}<z\mid X_{1}=x\right)
  13. g ( x ) g(\cdot\mid x)
  14. v ( x , y ) = E [ V 1 X 1 = x , Y 1 = y ] v(x,y)=E\left[V_{1}\mid X_{1}=x,Y_{1}=y\right]
  15. x x
  16. y y
  17. v v
  18. y y
  19. x x
  20. v ( 0 , 0 ) = 0 v(0,0)=0
  21. A A
  22. W A ( z , x ) W^{A}(z,x)
  23. x x
  24. z z
  25. W 1 A ( x , z ) W_{1}^{A}(x,z)
  26. W 2 A ( x , z ) W_{2}^{A}(x,z)
  27. ( x , z ) (x,z)
  28. I I
  29. W I ( z , x ) = β I ( z ) , W^{I}(z,x)=\beta^{I}(z),
  30. I I II
  31. W I I ( z , x ) = E [ β I I ( Y 1 ) X 1 = x , Y 1 < z ] . W^{II}(z,x)=E\left[\beta^{II}(Y_{1})\mid X_{1}=x,Y_{1}<z\right].
  32. A A
  33. B B
  34. x x
  35. W 2 A ( x , x ) W 2 B ( x , x ) ; W_{2}^{A}(x,x)\geq W_{2}^{B}(x,x);
  36. A A
  37. B B
  38. x x
  39. - z v ( x , y ) g ( y x ) d y - G ( z x ) W A ( z , x ) \int_{-\infty}^{z}v(x,y)g(y\mid x)dy-G(z\mid x)W^{A}(z,x)
  40. z = x z=x
  41. v ( x , x ) g ( x x ) - g ( x x ) W A ( x , x ) + G ( x x ) W 1 A ( x , x ) = 0 , v(x,x)g(x\mid x)-g(x\mid x)W^{A}(x,x)+G(x\mid x)W_{1}^{A}(x,x)=0,
  42. W 1 A ( x , x ) = g ( x x ) G ( x x ) v ( x , x ) - g ( x x ) G ( x x ) W A ( x , x ) . W_{1}^{A}(x,x)=\frac{g(x\mid x)}{G(x\mid x)}v(x,x)-\frac{g(x\mid x)}{G(x\mid x% )}W^{A}(x,x).
  43. Δ ( x ) = W A ( x , x ) - W B ( x , x ) , \Delta(x)=W^{A}(x,x)-W^{B}(x,x),
  44. Δ ( x ) = - g ( x x ) G ( x x ) Δ ( x ) + ( W 2 A ( x , x ) - W 2 B ( x , x ) ) \Delta^{\prime}(x)=-\frac{g(x\mid x)}{G(x\mid x)}\Delta(x)+\left(W_{2}^{A}(x,x% )-W_{2}^{B}(x,x)\right)
  45. Δ ( 0 ) = 0 Δ(0)=0
  46. Δ ( x ) Δ(x)
  47. Δ ( x ) Δ′(x)
  48. x x
  49. Δ ( x ) 0 Δ(x)≥0
  50. G ( z ) G(z\mid\cdot)
  51. W 2 I I ( x , x ) 0 W_{2}^{II}(x,x)\geq 0
  52. W 2 I ( x , x ) = 0 W_{2}^{I}(x,x)=0
  53. W 2 I I ( x , x ) W 2 I ( x , x ) W_{2}^{II}(x,x)\geq W_{2}^{I}(x,x)
  54. S S
  55. β ^ ( S , X 1 ) \widehat{\beta}(S,X_{1})
  56. W ^ I ( z , x ) = E [ β ^ ( S , z ) | X 1 = x ] \widehat{W}^{I}(z,x)=E\left[\widehat{\beta}(S,z)\big|X_{1}=x\right]
  57. x x
  58. S S
  59. W ^ 2 I ( z , x ) 0 , \widehat{W}_{2}^{I}(z,x)\geq 0,
  60. W ^ 2 I ( x , x ) W 2 I ( x , x ) \widehat{W}_{2}^{I}(x,x)\geq W_{2}^{I}(x,x)
  61. X < s u b > 1 X<sub>1

List_of_finite_simple_groups.html

  1. n ! 2 \frac{n!}{2}
  2. q 1 2 n ( n + 1 ) ( n + 1 , q - 1 ) i = 1 n ( q i + 1 - 1 ) \frac{q^{\frac{1}{2}n(n+1)}}{(n+1,q-1)}\prod_{i=1}^{n}(q^{i+1}-1)
  3. q n 2 ( 2 , q - 1 ) i = 1 n ( q 2 i - 1 ) \frac{q^{n^{2}}}{(2,q-1)}\prod_{i=1}^{n}(q^{2i}-1)
  4. q n 2 ( 2 , q - 1 ) i = 1 n ( q 2 i - 1 ) \frac{q^{n^{2}}}{(2,q-1)}\prod_{i=1}^{n}(q^{2i}-1)
  5. q n ( n - 1 ) ( q n - 1 ) ( 4 , q n - 1 ) i = 1 n - 1 ( q 2 i - 1 ) \frac{q^{n(n-1)}(q^{n}-1)}{(4,q^{n}-1)}\prod_{i=1}^{n-1}(q^{2i}-1)
  6. q 36 ( 3 , q - 1 ) i { 2 , 5 , 6 , 8 , 9 , 12 } ( q i - 1 ) \frac{q^{36}}{(3,q-1)}\prod_{i\in\{2,5,6,8,9,12\}}(q^{i}-1)
  7. q 63 ( 2 , q - 1 ) i { 2 , 6 , 8 , 10 , 12 , 14 , 18 } ( q i - 1 ) \frac{q^{63}}{(2,q-1)}\prod_{i\in\{2,6,8,10,12,14,18\}}(q^{i}-1)
  8. q 120 i { 2 , 8 , 12 , 14 , 18 , 20 , 24 , 30 } ( q i - 1 ) q^{120}\prod_{i\in\{2,8,12,14,18,20,24,30\}}(q^{i}-1)
  9. q 24 i { 2 , 6 , 8 , 12 } ( q i - 1 ) q^{24}\prod_{i\in\{2,6,8,12\}}(q^{i}-1)
  10. q 6 i { 2 , 6 } ( q i - 1 ) q^{6}\prod_{i\in\{2,6\}}(q^{i}-1)
  11. q n ( n + 1 ) / 2 ( n + 1 , q + 1 ) i = 1 n ( q i + 1 + ( - 1 ) i ) \frac{q^{n(n+1)/2}}{(n+1,q+1)}\prod_{i=1}^{n}(q^{i+1}+(-1)^{i})
  12. q n ( n - 1 ) ( 4 , q n + 1 ) ( q n + 1 ) i = 1 n - 1 ( q 2 i - 1 ) \frac{q^{n(n-1)}}{(4,q^{n}+1)}(q^{n}+1)\prod_{i=1}^{n-1}(q^{2i}-1)
  13. q 36 ( 3 , q + 1 ) i { 2 , 5 , 6 , 8 , 9 , 12 } ( ( - q ) i - 1 ) \frac{q^{36}}{(3,q+1)}\prod_{i\in\{2,5,6,8,9,12\}}((-q)^{i}-1)
  14. q 12 ( q 8 + q 4 + 1 ) ( q 6 - 1 ) ( q 2 - 1 ) q^{12}(q^{8}+q^{4}+1)(q^{6}-1)(q^{2}-1)
  15. q 2 ( q 2 + 1 ) ( q - 1 ) q^{2}(q^{2}+1)(q-1)
  16. q 12 ( q 6 + 1 ) ( q 4 - 1 ) ( q 3 + 1 ) ( q - 1 ) q^{12}(q^{6}+1)(q^{4}-1)(q^{3}+1)(q-1)
  17. q 3 ( q 3 + 1 ) ( q - 1 ) q^{3}(q^{3}+1)(q-1)
  18. × 10 3 3 \times 10^{3}3
  19. × 10 5 3 \times 10^{5}3
  20. q 1 2 n ( n + 1 ) ( n + 1 , q - 1 ) i = 1 n ( q i + 1 - 1 ) \frac{q^{\frac{1}{2}n(n+1)}}{(n+1,q-1)}\prod_{i=1}^{n}(q^{i+1}-1)
  21. q n 2 ( 2 , q - 1 ) i = 1 n ( q 2 i - 1 ) \frac{q^{n^{2}}}{(2,q-1)}\prod_{i=1}^{n}(q^{2i}-1)
  22. q n 2 ( 2 , q - 1 ) i = 1 n ( q 2 i - 1 ) \frac{q^{n^{2}}}{(2,q-1)}\prod_{i=1}^{n}(q^{2i}-1)
  23. q n ( n - 1 ) ( q n - 1 ) ( 4 , q n - 1 ) i = 1 n - 1 ( q 2 i - 1 ) \frac{q^{n(n-1)}(q^{n}-1)}{(4,q^{n}-1)}\prod_{i=1}^{n-1}(q^{2i}-1)
  24. 1 ( n + 1 , q + 1 ) q n ( n + 1 ) / 2 i = 1 n ( q i + 1 - ( - 1 ) i + 1 ) {1\over(n+1,q+1)}q^{n(n+1)/2}\prod_{i=1}^{n}(q^{i+1}-(-1)^{i+1})
  25. 1 ( 4 , q n + 1 ) q n ( n - 1 ) ( q n + 1 ) i = 1 n - 1 ( q 2 i - 1 ) {1\over(4,q^{n}+1)}q^{n(n-1)}(q^{n}+1)\prod_{i=1}^{n-1}(q^{2i}-1)

List_of_knot_theory_topics.html

  1. 3 \mathbb{R}^{3}

List_of_NP-complete_problems.html

  1. A , B , C 0 \textstyle A,B,C\geq 0
  2. x , y x,y
  3. A x 2 + B y - C = 0 Ax^{2}+By-C=0

List_of_stochastic_processes_topics.html

  1. S S

Little_Wing.html

  1. 2 4 \tfrac{2}{4}
  2. 4 4 \tfrac{4}{4}

Littlewood–Offord_problem.html

  1. v 1 , v 2 , , v n d v_{1},v_{2},\cdots,v_{n}\in\mathbb{R}^{d}
  2. A d A\subset\mathbb{R}^{d}
  3. ( c log n / n ) 2 n \Big(c\,\log n/\sqrt{n}\Big)\,2^{n}
  4. ( n n / 2 ) 2 n 1 n {n\choose\lfloor{n/2}\rfloor}\approx 2^{n}\,\frac{1}{\sqrt{n}}

Little–Parks_effect.html

  1. 𝐁 = 0 = × 𝐀 \mathbf{B}=0=\nabla\times\mathbf{A}
  2. φ \varphi
  3. φ = q P 𝐀 d 𝐱 , \varphi=\frac{q}{\hbar}\int_{P}\mathbf{A}\cdot d\mathbf{x},
  4. × 𝐀 = 𝐁 \nabla\times\mathbf{A}=\mathbf{B}
  5. Δ φ = q Φ B . \Delta\varphi=\frac{q\Phi_{B}}{\hbar}.
  6. Δ Φ B = 2 π / 2 e = h / 2 e . \Delta\Phi_{B}=2\pi\hbar/2e=h/2e.

Local_boundedness.html

  1. | f ( x ) | M |f(x)|\leq M
  2. d ( f ( x ) , a ) M d\left(f(x),a\right)\leq M
  3. f ( x ) = 1 x 2 + 1 f(x)=\frac{1}{x^{2}+1}\,
  4. f ( x ) = 2 x + 3 f(x)=2x+3\,
  5. f ( x ) = 1 x f(x)=\frac{1}{x}\,
  6. | f ( x ) | M |f(x)|\leq M
  7. f n ( x ) = x n f_{n}(x)=\frac{x}{n}
  8. | f n ( x ) | M |f_{n}(x)|\leq M
  9. f n ( x ) = 1 x 2 + n 2 f_{n}(x)=\frac{1}{x^{2}+n^{2}}
  10. | f n ( x ) | M |f_{n}(x)|\leq M
  11. f n ( x ) = x + n f_{n}(x)=x+n

Local_diffeomorphism.html

  1. f : X Y f:X\to Y\,
  2. f ( U ) f(U)\,
  3. f | U : U f ( U ) f|_{U}:U\to f(U)\,

Locally_compact_group.html

  1. L p L^{p}

Locally_compact_quantum_group.html

  1. A A
  2. A 0 A_{\geq 0}
  3. A A
  4. A A
  5. ϕ : A 0 [ 0 , ] \phi:A_{\geq 0}\to[0,\infty]
  6. ϕ ( a 1 + a 2 ) = ϕ ( a 1 ) + ϕ ( a 2 ) \phi(a_{1}+a_{2})=\phi(a_{1})+\phi(a_{2})
  7. a 1 , a 2 A 0 a_{1},a_{2}\in A_{\geq 0}
  8. ϕ ( r a ) = r ϕ ( a ) \phi(r\cdot a)=r\cdot\phi(a)
  9. r [ 0 , ) r\in[0,\infty)
  10. a A 0 a\in A_{\geq 0}
  11. ϕ \phi
  12. A A
  13. ϕ + := { a A 0 ϕ ( a ) < } \mathcal{M}_{\phi}^{+}:=\{a\in A_{\geq 0}\mid\phi(a)<\infty\}
  14. ϕ \phi
  15. A A
  16. 𝒩 ϕ := { a A ϕ ( a * a ) < } \mathcal{N}_{\phi}:=\{a\in A\mid\phi(a^{*}a)<\infty\}
  17. ϕ \phi
  18. A A
  19. ϕ := Span ϕ + = 𝒩 ϕ * 𝒩 ϕ \mathcal{M}_{\phi}:=\,\text{Span}~{}\mathcal{M}_{\phi}^{+}=\mathcal{N}_{\phi}^% {*}\mathcal{N}_{\phi}
  20. ϕ \phi
  21. A A
  22. ϕ \phi
  23. A A
  24. ϕ \phi
  25. ϕ ( a ) 0 \phi(a)\neq 0
  26. a A 0 a\in A_{\geq 0}
  27. ϕ \phi
  28. { a A 0 ϕ ( a ) λ } \{a\in A_{\geq 0}\mid\phi(a)\leq\lambda\}
  29. A A
  30. λ [ 0 , ] \lambda\in[0,\infty]
  31. ϕ \phi
  32. ϕ + \mathcal{M}_{\phi}^{+}
  33. A 0 A_{\geq 0}
  34. 𝒩 ϕ \mathcal{N}_{\phi}
  35. ϕ \mathcal{M}_{\phi}
  36. A A
  37. ϕ \phi
  38. A A
  39. A A
  40. α = ( α t ) t \alpha=(\alpha_{t})_{t\in\mathbb{R}}
  41. A A
  42. α s α t = α s + t \alpha_{s}\circ\alpha_{t}=\alpha_{s+t}
  43. s , t s,t\in\mathbb{R}
  44. α \alpha
  45. a A a\in A
  46. A \mathbb{R}\to A
  47. t α t ( a ) t\mapsto{\alpha_{t}}(a)
  48. α \alpha
  49. A A
  50. α \alpha
  51. z z\in\mathbb{C}
  52. I ( z ) := { y | ( y ) | | ( z ) | } I(z):=\{y\in\mathbb{C}\mid|\Im(y)|\leq|\Im(z)|\}
  53. f : I ( z ) A f:I(z)\to A
  54. I ( z ) I(z)
  55. y 0 y_{0}
  56. I ( z ) I(z)
  57. lim y y 0 f ( y ) - f ( y 0 ) y - y 0 \displaystyle\lim_{y\to y_{0}}\frac{f(y)-f(y_{0})}{y-y_{0}}
  58. A A
  59. I ( z ) I(z)
  60. I ( z ) I(z)
  61. z z\in\mathbb{C}\setminus\mathbb{R}
  62. D z := { a A There exists a norm-regular f : I ( z ) A such that f ( t ) = α t ( a ) for all t } . D_{z}:=\{a\in A\mid\,\text{There exists a norm-regular}~{}f:I(z)\to A~{}\,% \text{such that}~{}f(t)={\alpha_{t}}(a)~{}\,\text{for all}~{}t\in\mathbb{R}\}.
  63. α z : D z A \alpha_{z}:D_{z}\to A
  64. α z ( a ) := f ( z ) {\alpha_{z}}(a):=f(z)
  65. f f
  66. α z \alpha_{z}
  67. ( α z ) z (\alpha_{z})_{z\in\mathbb{C}}
  68. α \alpha
  69. z D z \cap_{z\in\mathbb{C}}D_{z}
  70. A A
  71. A A
  72. A A
  73. ϕ : A 0 [ 0 , ] \phi:A_{\geq 0}\to[0,\infty]
  74. A A
  75. ϕ \phi
  76. A A
  77. ϕ \phi
  78. A A
  79. ( σ t ) t (\sigma_{t})_{t\in\mathbb{R}}
  80. A A
  81. ϕ \phi
  82. σ \sigma
  83. ϕ σ t = ϕ \phi\circ\sigma_{t}=\phi
  84. t t\in\mathbb{R}
  85. a Dom ( σ i / 2 ) a\in\,\text{Dom}(\sigma_{i/2})
  86. ϕ ( a * a ) = ϕ ( σ i / 2 ( a ) [ σ i / 2 ( a ) ] * ) \phi(a^{*}a)=\phi(\sigma_{i/2}(a)[\sigma_{i/2}(a)]^{*})
  87. A A
  88. B B
  89. π : A M ( B ) \pi:A\to M(B)
  90. π [ A ] B \pi[A]B
  91. B B
  92. π \pi
  93. π ¯ : M ( A ) M ( B ) \overline{\pi}:M(A)\to M(B)
  94. ω : A \omega:A\to\mathbb{C}
  95. 1 1
  96. A A
  97. ω \omega
  98. ω ¯ : M ( A ) \overline{\omega}:M(A)\to\mathbb{C}
  99. M ( A ) M(A)
  100. 𝒢 = ( A , Δ ) \mathcal{G}=(A,\Delta)
  101. A A
  102. Δ : A M ( A A ) \Delta:A\to M(A\otimes A)
  103. Δ ι ¯ Δ = ι Δ ¯ Δ \overline{\Delta\otimes\iota}\circ\Delta=\overline{\iota\otimes\Delta}\circ\Delta
  104. { ω id ¯ ( Δ ( a ) ) | ω A * , a A } \left\{\overline{\omega\otimes\,\text{id}}(\Delta(a))~{}\big|~{}\omega\in A^{*% },~{}a\in A\right\}
  105. { id ω ¯ ( Δ ( a ) ) | ω A * , a A } \left\{\overline{\,\text{id}\otimes\omega}(\Delta(a))~{}\big|~{}\omega\in A^{*% },~{}a\in A\right\}
  106. A A
  107. ϕ \phi
  108. A A
  109. ϕ ( ω id ¯ ( Δ ( a ) ) ) = ω ¯ ( 1 M ( A ) ) ϕ ( a ) \phi\!\left(\overline{\omega\otimes\,\text{id}}(\Delta(a))\right)=\overline{% \omega}(1_{M(A)})\cdot\phi(a)
  110. ω A * \omega\in A^{*}
  111. a ϕ + a\in\mathcal{M}_{\phi}^{+}
  112. ψ \psi
  113. A A
  114. ψ ( id ω ¯ ( Δ ( a ) ) ) = ω ¯ ( 1 M ( A ) ) ψ ( a ) \psi\!\left(\overline{\,\text{id}\otimes\omega}(\Delta(a))\right)=\overline{% \omega}(1_{M(A)})\cdot\psi(a)
  115. ω A * \omega\in A^{*}
  116. a ϕ + a\in\mathcal{M}_{\phi}^{+}
  117. ψ \psi
  118. ψ \psi

Location_model.html

  1. a a\,
  2. b b\,
  3. o o\,
  4. c c\,
  5. P P\,
  6. c c\,
  7. P P\,
  8. a a\,
  9. b b\,
  10. P + c = P 1 P+c=P1\,
  11. P 1 P1\,
  12. c c\,
  13. P 1 P1\,
  14. o o\,
  15. a a\,
  16. b b\,
  17. c c\,
  18. d d\,
  19. o o\,
  20. c c\,
  21. o o\,
  22. u u\,
  23. d d\,
  24. U ( d , d 1 ) = u - r | d - d 1 | U(d,d_{1})=u-r|d-d_{1}|\,
  25. u u\,
  26. r r\,
  27. d d\,
  28. d 1 d_{1}\,
  29. | d - d 1 | |d-d_{1}|\,
  30. C S CS\,
  31. U ( d , d 1 ) - P = C S U(d,d_{1})-P=CS\,
  32. d d\,
  33. P P\,
  34. u * u^{*}\,
  35. U ( d , d 1 ) - P u * U(d,d_{1})-P\geq u^{*}\,
  36. u - u * - r | d - d 1 | - P 0 u-u^{*}-r|d-d_{1}|-P\geq 0\,

Lock-in_amplifier.html

  1. U in ( t ) U_{\mathrm{in}}(t)
  2. U out ( t ) U_{\mathrm{out}}(t)
  3. U out ( t ) = 1 T t - T t sin [ 2 π f ref s + φ ] U in ( s ) d s U_{\mathrm{out}}(t)=\frac{1}{T}\int_{t-T}^{t}{\sin\left[2\pi f_{\mathrm{ref}}% \cdot s+\varphi\right]U_{\mathrm{in}}(s)}\;\mathrm{d}s
  4. U out = V s i g cos θ U_{\mathrm{out}}=V_{sig}\cos\theta
  5. V s i g V_{sig}
  6. θ \theta
  7. X = V s i g cos θ X=V_{sig}\cos\theta
  8. Y = V s i g sin θ Y=V_{sig}\sin\theta
  9. R = X 2 + Y 2 = V s i g R=\sqrt{X^{2}+Y^{2}}=V_{sig}
  10. tan θ = Y / X \tan\theta=Y/X
  11. 2 \sqrt{2}

Lockhart–Martinelli_parameter.html

  1. χ \chi
  2. χ = m m g ρ g ρ , \chi=\frac{m_{\ell}}{m_{g}}\sqrt{\frac{\rho_{g}}{\rho_{\ell}}},
  3. m m_{\ell}
  4. m g m_{g}
  5. ρ g \rho_{g}
  6. ρ \rho_{\ell}

LOFAR.html

  1. z z
  2. 6 < z < 10 6<z<10
  3. 1.5 < z < 7 1.5<z<7
  4. 10 15 - 10 20.5 10^{15}-10^{20.5}
  5. 10 6 10^{6}

Logarithmic_units.html

  1. log b a = log c a log c b , \log_{b}a=\frac{\log_{c}a}{\log_{c}b},
  2. Log ( a ) = ( log b a ) [ log b ] = ( log c a ) [ log c ] . \mathrm{Log}(a)=(\log_{b}a)[\log b]=(\log_{c}a)[\log c].\,
  3. ln W = log e W = Log ( W ) log e . \ln W=\log_{e}W=\frac{\mathrm{Log}(W)}{\log e}.

Logarithmically_concave_function.html

  1. f ( θ x + ( 1 - θ ) y ) f ( x ) θ f ( y ) 1 - θ f(\theta x+(1-\theta)y)\geq f(x)^{\theta}f(y)^{1-\theta}
  2. x , y d o m f x,y∈domf
  3. l o g f log∘f
  4. log f ( θ x + ( 1 - θ ) y ) θ log f ( x ) + ( 1 - θ ) log f ( y ) \log f(\theta x+(1-\theta)y)\geq\theta\log f(x)+(1-\theta)\log f(y)
  5. x , y d o m f x,y∈domf
  6. f ( θ x + ( 1 - θ ) y ) f ( x ) θ f ( y ) 1 - θ f(\theta x+(1-\theta)y)\leq f(x)^{\theta}f(y)^{1-\theta}
  7. x , y d o m f x,y∈domf
  8. x x
  9. f ( x ) > 0 f(x) > 0
  10. f ( x ) 2 f ( x ) f ( x ) f ( x ) T f(x)\nabla^{2}f(x)\preceq\nabla f(x)\nabla f(x)^{T}
  11. f ( x ) 2 f ( x ) - f ( x ) f ( x ) T f(x)\nabla^{2}f(x)-\nabla f(x)\nabla f(x)^{T}
  12. f ( x ) f ′′ ( x ) ( f ( x ) ) 2 f(x)f^{\prime\prime}(x)\leq(f^{\prime}(x))^{2}
  13. f f
  14. g g
  15. l o g f logf
  16. l o g g logg
  17. log f ( x ) + log g ( x ) = log ( f ( x ) g ( x ) ) \log\,f(x)+\log\,g(x)=\log(f(x)g(x))
  18. f g fg
  19. f ( x , y ) f(x,y)
  20. g ( x ) = f ( x , y ) d y g(x)=\int f(x,y)dy
  21. h ( x , y ) h(x,y)
  22. f ( x - y ) g ( y ) f(x-y)g(y)
  23. f f
  24. g g
  25. ( f * g ) ( x ) = f ( x - y ) g ( y ) d y = h ( x , y ) d y (f*g)(x)=\int f(x-y)g(y)dy=\int h(x,y)dy

Logarithmically_convex_function.html

  1. log f {\log}\circ f
  2. f f
  3. f f
  4. exp \exp
  5. log f \log\circ f
  6. g : x x 2 g:x\mapsto x^{2}
  7. log g : x log x 2 = 2 log | x | {\log}\circ g:x\mapsto\log x^{2}=2\log|x|
  8. g g
  9. x e x 2 x\mapsto e^{x^{2}}
  10. x log e x 2 = x 2 x\mapsto\log e^{x^{2}}=x^{2}

Log–log_plot.html

  1. y = a x k y=ax^{k}
  2. y = a x k , y=ax^{k},
  3. log y = k log x + log a . \log y=k\log x+\log a.
  4. X = log x X=\log x
  5. Y = log y , Y=\log y,
  6. Y = m X + b Y=mX+b
  7. log 10 F ( x ) = m log 10 x + b , \log_{10}F(x)=m\log_{10}x+b,
  8. F ( x ) = x m 10 b , F(x)=x^{m}\cdot 10^{b},
  9. log [ F ( x 1 ) ] = m log ( x 1 ) + b , \log[F(x_{1})]=m\log(x_{1})+b,\,
  10. log [ F ( x 2 ) ] = m log ( x 2 ) + b . \mathrm{log}[F(x_{2})]=m\log(x_{2})+b.\,
  11. m = log ( F 2 ) - log ( F 1 ) log ( x 2 ) - log ( x 1 ) = log ( F 2 / F 1 ) log ( x 2 / x 1 ) , m=\frac{\mathrm{log}(F_{2})-\mathrm{log}(F_{1})}{\log(x_{2})-\log(x_{1})}=% \frac{\log(F_{2}/F_{1})}{\log(x_{2}/x_{1})},\,
  12. log ( x 1 / x 2 ) = - log ( x 2 / x 1 ) . \log(x_{1}/x_{2})=-\log(x_{2}/x_{1}).\,
  13. m = log ( F 1 / F 0 ) log ( x 1 / x 0 ) m=\frac{\log(F_{1}/F_{0})}{\log(x_{1}/x_{0})}
  14. log ( F 1 / F 0 ) = m log ( x 1 / x 0 ) = log [ ( x 1 / x 0 ) m ] . \log(F_{1}/F_{0})=m\log(x_{1}/x_{0})=\log[(x_{1}/x_{0})^{m}].\,
  15. F 1 F 0 = ( x 1 x 0 ) m \frac{F_{1}}{F_{0}}=\left(\frac{x_{1}}{x_{0}}\right)^{m}
  16. F 1 = F 0 x 0 m x m , F_{1}=\frac{F_{0}}{x_{0}^{m}}\,\,x^{m},\,
  17. F ( x ) = constant x m . F(x)=\mathrm{constant}\cdot x^{m}.
  18. F ( x ) = F 0 ( x x 0 ) log ( F 1 / F 0 ) log ( x 1 / x 0 ) , F(x)={F_{0}}\left(\frac{x}{x_{0}}\right)^{\frac{\log(F_{1}/F_{0})}{\log(x_{1}/% x_{0})}},
  19. F ( x ) = constant x m F(x)=\mathrm{constant}\cdot x^{m}
  20. F ( x ) = constant x m . F(x)=\mathrm{constant}\cdot x^{m}.
  21. A ( x ) = x 0 x 1 F ( x ) = constant m + 1 x m + 1 : [ x 0 , x 1 ] A(x)=\int_{x_{0}}^{x_{1}}F(x)=\frac{\mathrm{constant}}{m+1}\cdot x^{m+1}:[x_{0% },x_{1}]
  22. constant = F 0 x 0 m \mathrm{constant}=\frac{F_{0}}{x_{0}^{m}}
  23. A = F 0 / x 0 m m + 1 ( x 1 m + 1 - x 0 m + 1 ) A=\frac{F_{0}/x_{0}^{m}}{m+1}\cdot(x_{1}^{m+1}-x_{0}^{m+1})
  24. log A = log [ F 0 / x 0 m m + 1 ( x 1 m + 1 - x 0 m + 1 ) ] = log F 0 m + 1 - log 1 x 0 m + log ( x 1 m + 1 - x 0 m + 1 ) \log A=\log\left[\frac{F_{0}/x_{0}^{m}}{m+1}\cdot(x_{1}^{m+1}-x_{0}^{m+1})% \right]=\log\frac{F_{0}}{m+1}-\log\frac{1}{x_{0}^{m}}+\log(x_{1}^{m+1}-x_{0}^{% m+1})
  25. log A = log F 0 m + 1 + log ( x 1 m + 1 - x 0 m + 1 x 0 m ) = log F 0 m + 1 + log ( x 1 m x 0 m x 1 - x 0 m + 1 x 0 m ) \log A=\log\frac{F_{0}}{m+1}+\log\left(\frac{x_{1}^{m+1}-x_{0}^{m+1}}{x_{0}^{m% }}\right)=\log\frac{F_{0}}{m+1}+\log\left(\frac{x_{1}^{m}}{x_{0}^{m}}\cdot x_{% 1}-\frac{x_{0}^{m+1}}{x_{0}^{m}}\right)
  26. A = F 0 m + 1 [ x 1 ( x 1 x 0 ) m - x 0 ] A=\frac{F_{0}}{m+1}\cdot\left[x_{1}\cdot\left(\frac{x_{1}}{x_{0}}\right)^{m}-x% _{0}\right]
  27. A ( m = - 1 ) = x 0 x 1 F ( x ) = x 0 x 1 constant x = F 0 x 0 - 1 x 0 x 1 1 x = F 0 x 0 ln x : [ x 0 , x 1 ] A_{(m=-1)}=\int_{x_{0}}^{x_{1}}F(x)=\int_{x_{0}}^{x_{1}}\frac{\mathrm{constant% }}{x}=\frac{F_{0}}{x_{0}^{-1}}\int_{x_{0}}^{x_{1}}\frac{1}{x}=F_{0}\cdot x_{0}% \cdot\ln x:[x_{0},x_{1}]
  28. A ( m = - 1 ) = F 0 x 0 ln x 1 x 0 A_{(m=-1)}=F_{0}\cdot x_{0}\cdot\ln\frac{x_{1}}{x_{0}}
  29. M t = A R t b Y t c U t , M_{t}=AR_{t}^{b}Y_{t}^{c}U_{t},
  30. m t = a + b r t + c y t + u t , m_{t}=a+br_{t}+cy_{t}+u_{t},
  31. Q t = A N t α K t β U t , Q_{t}=AN_{t}^{\alpha}K_{t}^{\beta}U_{t},
  32. α \alpha
  33. β \beta
  34. q t = a + α n t + β k t + u t q_{t}=a+\alpha n_{t}+\beta k_{t}+u_{t}

Long-tail_traffic.html

  1. ρ ( k ) k - α \rho(k)\sim k^{-\alpha}
  2. P ( a ) = ( μ a a ! ) e - μ , P(a)=\left(\frac{\mu^{a}}{a!}\right)e^{-\mu},
  3. μ \mu
  4. P ( d ) = ( λ d d ! ) e - λ , P(d)=\left(\frac{\lambda^{d}}{d!}\right)e^{-\lambda},
  5. λ \lambda
  6. P [ T t ] = e - t h , P[T\geq\ t]=e^{\frac{-t}{h}},
  7. P [ X > x ] x - α , as x , 0 < α < 2 P[X>x]\sim x^{-\alpha},\ \,\text{as}\ x\to\infty,0<\alpha<2
  8. p ( x ) = α k α x - α - 1 , α , k > 0 , x k p(x)=\alpha k^{\alpha}x^{-\alpha-1},\ \alpha,k>0,\ x\geq k
  9. F ( x ) = P [ X x ] = 1 - ( k x ) α F(x)=P[X\leq\ x]=1-\left(\frac{k}{x}\right)^{\alpha}
  10. 1 \mathcal{1}
  11. α \alpha

Loop_algebra.html

  1. 𝐠 \mathbf{g}
  2. 𝐠 \mathbf{g}
  3. 𝔤 C ( S 1 ) \mathfrak{g}\otimes C^{\infty}(S^{1})
  4. [ g 1 f 1 , g 2 f 2 ] = [ g 1 , g 2 ] f 1 f 2 [g_{1}\otimes f_{1},g_{2}\otimes f_{2}]=[g_{1},g_{2}]\otimes f_{1}f_{2}
  5. 𝐠 \mathbf{g}
  6. 𝐠 \mathbf{g}
  7. 𝐠 \mathbf{g}
  8. 𝐠 \mathbf{g}
  9. G G
  10. g t n g\otimes t^{n}
  11. g e - i n σ g\otimes e^{-in\sigma}
  12. S < s u p > 1 S<sup>1

Loop_group.html

  1. M M
  2. G G
  3. L G LG
  4. L G = { γ : S 1 G | γ C ( S 1 , G ) } , LG=\{\gamma:S^{1}\to G|\gamma\in C(S^{1},G)\},
  5. L G LG
  6. G G
  7. L G LG
  8. θ θ
  9. γ : θ S 1 γ ( θ ) G , \gamma:\theta\in S^{1}\mapsto\gamma(\theta)\in G,
  10. L G LG
  11. ( γ 1 γ 2 ) ( θ ) γ 1 ( θ ) γ 2 ( θ ) . (\gamma_{1}\gamma_{2})(\theta)\equiv\gamma_{1}(\theta)\gamma_{2}(\theta).
  12. G G
  13. γ - 1 : γ - 1 ( θ ) γ ( θ ) - 1 , \gamma^{-1}:\gamma^{-1}(\theta)\equiv\gamma(\theta)^{-1},
  14. e : θ e G . e:\theta\mapsto e\in G.
  15. L G LG
  16. G G
  17. L G LG
  18. Ω G \Omega G\,
  19. G G
  20. e 1 : L G G e_{1}:LG\to G
  21. L G LG
  22. 1 1
  23. G G
  24. L G LG
  25. 1 Ω G L G G 1 1\to\Omega G\to LG\to G\to 1
  26. L G LG
  27. L G = Ω G G LG=\Omega G\rtimes G
  28. Ω G ΩG
  29. G G
  30. Ω G ΩG
  31. Ω G ΩG
  32. Ω G ΩG

Loop_optimization.html

  1. [ 0 1 1 0 ] \left[\begin{array}[]{cc}0&1\\ 1&0\end{array}\right]

Lorentz_scalar.html

  1. x μ = ( c t , 𝐱 ) x^{\mu}=(ct,\mathbf{x})
  2. 𝐱 = 𝐯 t \mathbf{x}=\mathbf{v}t
  3. 𝐯 \mathbf{v}
  4. c c
  5. x μ x μ = η μ ν x μ x ν = ( c t ) 2 - 𝐱 𝐱 = def ( c τ ) 2 x_{\mu}x^{\mu}=\eta_{\mu\nu}x^{\mu}x^{\nu}=(ct)^{2}-\mathbf{x}\cdot\mathbf{x}% \ \stackrel{\mathrm{def}}{=}\ (c\tau)^{2}
  6. τ \tau
  7. η μ ν = η μ ν = ( 1 0 0 0 0 - 1 0 0 0 0 - 1 0 0 0 0 - 1 ) \eta^{\mu\nu}=\eta_{\mu\nu}=\begin{pmatrix}1&0&0&0\\ 0&-1&0&0\\ 0&0&-1&0\\ 0&0&0&-1\end{pmatrix}
  8. η μ ν = η μ ν = ( - 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) \eta^{\mu\nu}=\eta_{\mu\nu}=\begin{pmatrix}-1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{pmatrix}
  9. s s
  10. x μ x μ = η μ ν x μ x ν = 𝐱 𝐱 - ( c t ) 2 = def s 2 x_{\mu}x^{\mu}=\eta_{\mu\nu}x^{\mu}x^{\nu}=\mathbf{x}\cdot\mathbf{x}-(ct)^{2}% \ \stackrel{\mathrm{def}}{=}\ s^{2}
  11. v μ = def d x μ d τ = ( c d t d τ , d t d τ d 𝐱 d t ) = ( γ c , γ 𝐯 ) = γ ( c , 𝐯 ) v^{\mu}\ \stackrel{\mathrm{def}}{=}\ {dx^{\mu}\over d\tau}=\left(c{dt\over d% \tau},{dt\over d\tau}{d\mathbf{x}\over dt}\right)=\left(\gamma c,\gamma{% \mathbf{v}}\right)=\gamma\left(c,{\mathbf{v}}\right)
  12. γ = def 1 1 - 𝐯 𝐯 c 2 \gamma\ \stackrel{\mathrm{def}}{=}\ {1\over{\sqrt{1-{{\mathbf{v}\cdot\mathbf{v% }}\over c^{2}}}}}
  13. v μ v μ = - c 2 v_{\mu}v^{\mu}=-c^{2}\,
  14. a μ = def d v μ d τ a^{\mu}\ \stackrel{\mathrm{def}}{=}\ {dv^{\mu}\over d\tau}
  15. 0 = 1 2 d d τ ( v μ v μ ) = d v μ d τ v μ = a μ v μ 0={1\over 2}{d\over d\tau}\left(v_{\mu}v^{\mu}\right)={dv_{\mu}\over d\tau}v^{% \mu}=a_{\mu}v^{\mu}
  16. d E d τ = 𝐅 𝐯 {dE\over d\tau}=\mathbf{F}\cdot{\mathbf{v}}
  17. E E
  18. 𝐅 \mathbf{F}
  19. p μ = m v μ = ( γ m c , γ m 𝐯 ) = ( γ m c , 𝐩 ) = ( E c , 𝐩 ) p^{\mu}=mv^{\mu}=\left(\gamma mc,\gamma{m\mathbf{v}}\right)=\left(\gamma mc,{% \mathbf{p}}\right)=\left({E\over c},{\mathbf{p}}\right)
  20. m m
  21. 𝐩 \mathbf{p}
  22. E = γ m c 2 E=\gamma mc^{2}\,
  23. u u
  24. 𝐮 2 \mathbf{u}_{2}
  25. u u
  26. p p
  27. p μ u μ = - E 1 p_{\mu}u^{\mu}=-{E_{1}}
  28. E 1 E_{1}
  29. E 1 = γ 1 γ 2 m 1 c 2 - γ 2 𝐩 1 𝐮 2 {E_{1}}=\gamma_{1}\gamma_{2}m_{1}c^{2}-\gamma_{2}\mathbf{p}_{1}\cdot\mathbf{u}% _{2}
  30. E 1 E_{1}
  31. p μ p μ = - ( m c ) 2 p_{\mu}p^{\mu}=-(mc)^{2}\,
  32. m 0 m_{0}
  33. γ m 0 \gamma m_{0}
  34. ( p μ u μ / c ) 2 + p μ p μ = E 1 2 c 2 - ( m c ) 2 = ( γ 1 2 - 1 ) ( m c ) 2 = γ 1 2 𝐯 1 𝐯 1 m 2 = 𝐩 1 𝐩 1 \left(p_{\mu}u^{\mu}/c\right)^{2}+p_{\mu}p^{\mu}={E_{1}^{2}\over c^{2}}-(mc)^{% 2}=\left(\gamma_{1}^{2}-1\right)(mc)^{2}=\gamma_{1}^{2}{{\mathbf{v}_{1}\cdot% \mathbf{v}_{1}}}m^{2}=\mathbf{p}_{1}\cdot\mathbf{p}_{1}
  35. v 1 2 = 𝐯 1 𝐯 1 = 𝐩 1 𝐩 1 c 4 E 1 2 v_{1}^{2}=\mathbf{v}_{1}\cdot\mathbf{v}_{1}={{\mathbf{p}_{1}\cdot\mathbf{p}_{1% }c^{4}}\over{E_{1}^{2}}}

Lorenz_gauge_condition.html

  1. μ A μ = 0 \partial_{\mu}A^{\mu}=0
  2. A μ A μ + μ f A^{\mu}\to A^{\mu}+\partial^{\mu}f
  3. f f
  4. μ μ f = 0 \partial^{\mu}\partial_{\mu}f=0
  5. μ A μ A μ = , μ 0 \partial_{\mu}A^{\mu}\equiv A^{\mu}{}_{,\mu}=0\!
  6. A μ A^{\mu}
  7. A + 1 c 2 φ t = 0. \nabla\cdot{\vec{A}}+\frac{1}{c^{2}}\frac{\partial\varphi}{\partial t}=0.
  8. A \vec{A}
  9. φ \,\varphi
  10. A + 1 c φ t = 0. \nabla\cdot{\vec{A}}+\frac{1}{c}\frac{\partial\varphi}{\partial t}=0.
  11. × E = - B t = × - A t \nabla\times\vec{E}=-\frac{\partial\vec{B}}{\partial t}=\nabla\times-\frac{% \partial\vec{A}}{\partial t}
  12. × ( E + A t ) = 0 \nabla\times(\vec{E}+\frac{\partial\vec{A}}{\partial t})=0
  13. φ \,\varphi
  14. - φ = E + A t -\nabla\,\varphi=\vec{E}+\frac{\partial\vec{A}}{\partial t}
  15. E = - φ - A t \vec{E}=-\nabla\,\varphi-\frac{\partial\vec{A}}{\partial t}
  16. × B = × ( × A ) = ( A ) - 2 A = μ 0 J + 1 c 2 E t = μ 0 J - 1 c 2 φ t - 1 c 2 2 A t 2 {\nabla\times\vec{B}}={\nabla\times(\nabla\times\vec{A})}={\nabla(\nabla\cdot% \vec{A})-\nabla^{2}\vec{A}}={\mu_{0}\vec{J}+\frac{1}{c^{2}}\frac{\partial\vec{% E}}{\partial t}}={\mu_{0}\vec{J}-\frac{1}{c^{2}}\nabla\frac{\partial\varphi}{% \partial t}-\frac{1}{c^{2}}\frac{\partial^{2}\vec{A}}{\partial t^{2}}}
  17. ( A + 1 c 2 φ t ) = μ 0 J - 1 c 2 2 A t 2 + 2 A \nabla(\nabla\cdot\vec{A}+\frac{1}{c^{2}}\frac{\partial\varphi}{\partial t})=% \mu_{0}\vec{J}-\frac{1}{c^{2}}\frac{\partial^{2}\vec{A}}{\partial t^{2}}+% \nabla^{2}\vec{A}
  18. A = [ 1 c 2 2 t 2 - 2 ] A = μ 0 J \Box\vec{A}=\left[\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}-\nabla^{2% }\right]\vec{A}=\mu_{0}\vec{J}
  19. φ = [ 1 c 2 2 t 2 - 2 ] φ = 1 ϵ 0 ρ . \Box\varphi=\left[\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}-\nabla^{2% }\right]\varphi=\frac{1}{\epsilon_{0}}\rho\,.
  20. c = 1 ϵ 0 μ 0 c=\frac{1}{\sqrt{\epsilon_{0}\mu_{0}}}
  21. \Box
  22. ρ \rho
  23. J \vec{J}
  24. E \vec{E}
  25. B \vec{B}\,
  26. φ \,\varphi
  27. A \vec{A}
  28. E = - φ - A t \vec{E}=-\nabla\varphi-\frac{\partial\vec{A}}{\partial t}
  29. B = × A . \vec{B}=\nabla\times\vec{A}\,.
  30. φ \,\varphi
  31. A \vec{A}

Lotka's_law.html

  1. 1 / n a 1/n^{a}
  2. X n Y = C X^{n}Y=C
  3. Y = C / X n , Y=C/X^{n},\,
  4. C C
  5. n 2 n\approx 2

Low_(complexity).html

  1. P {\oplus}\hbox{P}
  2. P {\oplus}\hbox{P}

Lower_hybrid_oscillation.html

  1. ω = [ ( Ω i Ω e ) - 1 + ω p i - 2 ] - 1 / 2 \omega=[(\Omega_{i}\Omega_{e})^{-1}+\omega_{pi}^{-2}]^{-1/2}

LSZ_reduction_formula.html

  1. | { p } in |\{p\}\ \mathrm{in}\rangle
  2. | { p } out |\{p\}\ \mathrm{out}\rangle
  3. S f i = { q } out | { p } in S_{fi}=\langle\{q\}\ \mathrm{out}|\{p\}\ \mathrm{in}\rangle
  4. = 1 2 μ φ μ φ - 1 2 m 0 2 φ 2 + int \mathcal{L}=\frac{1}{2}\partial_{\mu}\varphi\partial^{\mu}\varphi-\frac{1}{2}m% _{0}^{2}\varphi^{2}+\mathcal{L}_{\mathrm{int}}
  5. int \mathcal{L}_{\mathrm{int}}
  6. g φ ψ ¯ ψ g\ \varphi\bar{\psi}\psi
  7. ( 2 + m 0 2 ) φ ( x ) = j 0 ( x ) \left(\partial^{2}+m_{0}^{2}\right)\varphi(x)=j_{0}(x)
  8. int \mathcal{L}_{\mathrm{int}}
  9. j 0 = int φ j_{0}=\frac{\partial\mathcal{L}_{\mathrm{int}}}{\partial\varphi}
  10. m m
  11. φ φ
  12. ( 2 + m 2 ) φ ( x ) = j 0 ( x ) + ( m 2 - m 0 2 ) φ ( x ) = j ( x ) \left(\partial^{2}+m^{2}\right)\varphi(x)=j_{0}(x)+\left(m^{2}-m_{0}^{2}\right% )\varphi(x)=j(x)
  13. 2 + m 2 \partial^{2}+m^{2}
  14. Δ ret ( x ) = i θ ( x 0 ) d 3 k ( 2 π ) 3 2 ω k ( e - i k x - e i k x ) k 0 = ω k ω k = 𝐤 2 + m 2 \Delta_{\mathrm{ret}}(x)=i\theta\left(x^{0}\right)\int\frac{\mathrm{d}^{3}k}{(% 2\pi)^{3}2\omega_{k}}\left(e^{-ik\cdot x}-e^{ik\cdot x}\right)_{k^{0}=\omega_{% k}}\qquad\omega_{k}=\sqrt{\mathbf{k}^{2}+m^{2}}
  15. φ ( x ) = Z φ in ( x ) + d 4 y Δ ret ( x - y ) j ( y ) \varphi(x)=\sqrt{Z}\varphi_{\mathrm{in}}(x)+\int\mathrm{d}^{4}y\Delta_{\mathrm% {ret}}(x-y)j(y)
  16. Z \sqrt{Z}
  17. ( 2 + m 2 ) φ in ( x ) = 0 , \left(\partial^{2}+m^{2}\right)\varphi_{\mathrm{in}}(x)=0,
  18. φ in ( x ) = d 3 k { f k ( x ) a in ( 𝐤 ) + f k * ( x ) a in ( 𝐤 ) } \varphi_{\mathrm{in}}(x)=\int\mathrm{d}^{3}k\left\{f_{k}(x)a_{\mathrm{in}}(% \mathbf{k})+f^{*}_{k}(x)a^{\dagger}_{\mathrm{in}}(\mathbf{k})\right\}
  19. f k ( x ) = e - i k x ( 2 π ) 3 2 ( 2 ω k ) 1 2 | k 0 = ω k f_{k}(x)=\left.\frac{e^{-ik\cdot x}}{(2\pi)^{\frac{3}{2}}(2\omega_{k})^{\frac{% 1}{2}}}\right|_{k^{0}=\omega_{k}}
  20. a in ( 𝐤 ) = i d 3 x f k * ( x ) 0 φ in ( x ) a_{\mathrm{in}}(\mathbf{k})=i\int\mathrm{d}^{3}xf^{*}_{k}(x)% \overleftrightarrow{\partial}_{0}\varphi_{\mathrm{in}}(x)
  21. g 0 f = g 0 f - f 0 g . {\mathrm{g}}{\overleftrightarrow{\partial}}_{0}f=\mathrm{g}\partial_{0}f-f% \partial_{0}\mathrm{g}.
  22. [ a in ( 𝐩 ) , a in ( 𝐪 ) ] = 0 ; [ a in ( 𝐩 ) , a in ( 𝐪 ) ] = δ 3 ( 𝐩 - 𝐪 ) ; [a_{\mathrm{in}}(\mathbf{p}),a_{\mathrm{in}}(\mathbf{q})]=0;\quad[a_{\mathrm{% in}}(\mathbf{p}),a^{\dagger}_{\mathrm{in}}(\mathbf{q})]=\delta^{3}(\mathbf{p}-% \mathbf{q});
  23. | k 1 , , k n in = 2 ω k 1 a in ( 𝐤 1 ) 2 ω k n a in ( 𝐤 n ) | 0 \left|k_{1},\ldots,k_{n}\ \mathrm{in}\right\rangle=\sqrt{2\omega_{k_{1}}}a_{% \mathrm{in}}^{\dagger}(\mathbf{k}_{1})\ldots\sqrt{2\omega_{k_{n}}}a_{\mathrm{% in}}^{\dagger}(\mathbf{k}_{n})|0\rangle
  24. φ ( x ) Z φ in ( x ) as x 0 - \varphi(x)\sim\sqrt{Z}\varphi_{\mathrm{in}}(x)\qquad\mathrm{as}\quad x^{0}\to-\infty
  25. j ( x ) j(x)
  26. m m
  27. | α |\alpha\rangle
  28. | β |\beta\rangle
  29. f ( x ) f(x)
  30. ( 2 + m 2 ) f ( x ) = 0 (\partial^{2}+m^{2})f(x)=0
  31. lim x 0 - d 3 x α | f ( x ) 0 φ ( x ) | β = Z d 3 x α | f ( x ) 0 φ in ( x ) | β \lim_{x^{0}\to-\infty}\int\mathrm{d}^{3}x\langle\alpha|f(x)\overleftrightarrow% {\partial}_{0}\varphi(x)|\beta\rangle=\sqrt{Z}\int\mathrm{d}^{3}x\langle\alpha% |f(x)\overleftrightarrow{\partial}_{0}\varphi_{\mathrm{in}}(x)|\beta\rangle
  32. f f
  33. φ ( x ) = Z φ out ( x ) + d 4 y Δ adv ( x - y ) j ( y ) \varphi(x)=\sqrt{Z}\varphi_{\mathrm{out}}(x)+\int\mathrm{d}^{4}y\Delta_{% \mathrm{adv}}(x-y)j(y)
  34. lim x 0 d 3 x α | f ( x ) 0 φ ( x ) | β = Z d 3 x α | f ( x ) 0 φ out ( x ) | β \lim_{x^{0}\to\infty}\int\mathrm{d}^{3}x\langle\alpha|f(x)\overleftrightarrow{% \partial}_{0}\varphi(x)|\beta\rangle=\sqrt{Z}\int\mathrm{d}^{3}x\langle\alpha|% f(x)\overleftrightarrow{\partial}_{0}\varphi_{\mathrm{out}}(x)|\beta\rangle
  35. = β out | T φ ( y 1 ) φ ( y n ) | α p in \mathcal{M}=\langle\beta\ \mathrm{out}|\mathrm{T}\varphi(y_{1})\ldots\varphi(y% _{n})|\alpha\ p\ \mathrm{in}\rangle
  36. \mathcal{M}
  37. φ ( y 1 ) φ ( y n ) \varphi(y_{1})\cdots\varphi(y_{n})
  38. β β
  39. p p
  40. α α
  41. \mathcal{M}
  42. p p
  43. = 2 ω p β out | T [ φ ( y 1 ) φ ( y n ) ] a in ( 𝐩 ) | α in \mathcal{M}=\sqrt{2\omega_{p}}\ \left\langle\beta\ \mathrm{out}\bigg|\mathrm{T% }\left[\varphi(y_{1})\ldots\varphi(y_{n})\right]a_{\mathrm{in}}^{\dagger}(% \mathbf{p})\bigg|\alpha\ \mathrm{in}\right\rangle
  44. p p
  45. = 2 ω p β out | T [ φ ( y 1 ) φ ( y n ) ] a in ( 𝐩 ) - a out ( 𝐩 ) T [ φ ( y 1 ) φ ( y n ) ] | α in \mathcal{M}=\sqrt{2\omega_{p}}\ \left\langle\beta\ \mathrm{out}\bigg|\mathrm{T% }\left[\varphi(y_{1})\ldots\varphi(y_{n})\right]a_{\mathrm{in}}^{\dagger}(% \mathbf{p})-a_{\mathrm{out}}^{\dagger}(\mathbf{p})\mathrm{T}\left[\varphi(y_{1% })\ldots\varphi(y_{n})\right]\bigg|\alpha\ \mathrm{in}\right\rangle
  46. a out a_{\mathrm{out}}^{\dagger}
  47. = - i 2 ω p d 3 x f p ( x ) 0 β out | T [ φ ( y 1 ) φ ( y n ) ] φ in ( x ) - φ out ( x ) T [ φ ( y 1 ) φ ( y n ) ] | α in \mathcal{M}=-i\sqrt{2\omega_{p}}\ \int\mathrm{d}^{3}xf_{p}(x)% \overleftrightarrow{\partial_{0}}\left\langle\beta\ \mathrm{out}\bigg|\mathrm{% T}\left[\varphi(y_{1})\ldots\varphi(y_{n})\right]\varphi_{\mathrm{in}}(x)-% \varphi_{\mathrm{out}}(x)\mathrm{T}\left[\varphi(y_{1})\ldots\varphi(y_{n})% \right]\bigg|\alpha\ \mathrm{in}\right\rangle
  48. = - i 2 ω p Z { lim x 0 - d 3 x f p ( x ) 0 β out | T [ φ ( y 1 ) φ ( y n ) ] φ ( x ) | α in - lim x 0 d 3 x f p ( x ) 0 β out | φ ( x ) T [ φ ( y 1 ) φ ( y n ) ] | α in } \mathcal{M}=-i\sqrt{\frac{2\omega_{p}}{Z}}\left\{\lim_{x^{0}\to-\infty}\int% \mathrm{d}^{3}xf_{p}(x)\overleftrightarrow{\partial_{0}}\langle\beta\ \mathrm{% out}|\mathrm{T}\left[\varphi(y_{1})\ldots\varphi(y_{n})\right]\varphi(x)|% \alpha\ \mathrm{in}\rangle-\lim_{x^{0}\to\infty}\int\mathrm{d}^{3}xf_{p}(x)% \overleftrightarrow{\partial_{0}}\langle\beta\ \mathrm{out}|\varphi(x)\mathrm{% T}\left[\varphi(y_{1})\ldots\varphi(y_{n})\right]|\alpha\ \mathrm{in}\rangle\right\}
  49. φ ( x ) φ(x)
  50. = - i 2 ω p Z ( lim x 0 - - lim x 0 ) d 3 x f p ( x ) 0 β out | T [ φ ( x ) φ ( y 1 ) φ ( y n ) ] | α in \mathcal{M}=-i\sqrt{\frac{2\omega_{p}}{Z}}\left(\lim_{x^{0}\to-\infty}-\lim_{x% ^{0}\to\infty}\right)\int\mathrm{d}^{3}xf_{p}(x)\overleftrightarrow{\partial_{% 0}}\langle\beta\ \mathrm{out}|\mathrm{T}\left[\varphi(x)\varphi(y_{1})\ldots% \varphi(y_{n})\right]|\alpha\ \mathrm{in}\rangle
  51. x x
  52. β out | T [ φ ( x ) φ ( y 1 ) φ ( y n ) ] | α in = η ( x ) \langle\beta\ \mathrm{out}|\mathrm{T}\left[\varphi(x)\varphi(y_{1})\ldots% \varphi(y_{n})\right]|\alpha\ \mathrm{in}\rangle=\eta(x)
  53. = i 2 ω p Z d ( x 0 ) 0 d 3 x f p ( x ) 0 η ( x ) \mathcal{M}=i\sqrt{\frac{2\omega_{p}}{Z}}\int\mathrm{d}(x^{0})\partial_{0}\int% \mathrm{d}^{3}xf_{p}(x)\overleftrightarrow{\partial_{0}}\eta(x)
  54. = i 2 ω p Z d 4 x { f p ( x ) 0 2 η ( x ) - η ( x ) 0 2 f p ( x ) } \mathcal{M}=i\sqrt{\frac{2\omega_{p}}{Z}}\int\mathrm{d}^{4}x\left\{f_{p}(x)% \partial_{0}^{2}\eta(x)-\eta(x)\partial_{0}^{2}f_{p}(x)\right\}
  55. 0 2 f p ( x ) = ( Δ - m 2 ) f p ( x ) \partial_{0}^{2}f_{p}(x)=\left(\Delta-m^{2}\right)f_{p}(x)
  56. \mathcal{M}
  57. = i 2 ω p Z d 4 x f p ( x ) ( 0 2 - Δ + m 2 ) η ( x ) \mathcal{M}=i\sqrt{\frac{2\omega_{p}}{Z}}\int\mathrm{d}^{4}xf_{p}(x)\left(% \partial_{0}^{2}-\Delta+m^{2}\right)\eta(x)
  58. = i ( 2 π ) 3 2 Z 1 2 d 4 x e - i p x ( + m 2 ) β out | T [ φ ( x ) φ ( y 1 ) φ ( y n ) ] | α in \mathcal{M}=\frac{i}{(2\pi)^{\frac{3}{2}}Z^{\frac{1}{2}}}\int\mathrm{d}^{4}xe^% {-ip\cdot x}\left(\Box+m^{2}\right)\langle\beta\ \mathrm{out}|\mathrm{T}\left[% \varphi(x)\varphi(y_{1})\ldots\varphi(y_{n})\right]|\alpha\ \mathrm{in}\rangle
  59. p 1 , , p n out | q 1 , , q m in = i = 1 m { d 4 x i i e - i q i x i ( x i + m 2 ) ( 2 π ) 3 2 Z 1 2 } j = 1 n { d 4 y j i e i p j y j ( y j + m 2 ) ( 2 π ) 3 2 Z 1 2 } 0 | T φ ( x 1 ) φ ( x m ) φ ( y 1 ) φ ( y n ) | 0 \langle p_{1},\ldots,p_{n}\ \mathrm{out}|q_{1},\ldots,q_{m}\ \mathrm{in}% \rangle=\int\prod_{i=1}^{m}\left\{\mathrm{d}^{4}x_{i}\frac{ie^{-iq_{i}\cdot x_% {i}}\left(\Box_{x_{i}}+m^{2}\right)}{(2\pi)^{\frac{3}{2}}Z^{\frac{1}{2}}}% \right\}\prod_{j=1}^{n}\left\{\mathrm{d}^{4}y_{j}\frac{ie^{ip_{j}\cdot y_{j}}% \left(\Box_{y_{j}}+m^{2}\right)}{(2\pi)^{\frac{3}{2}}Z^{\frac{1}{2}}}\right\}% \langle 0|\mathrm{T}\varphi(x_{1})\ldots\varphi(x_{m})\varphi(y_{1})\ldots% \varphi(y_{n})|0\rangle
  60. Γ ( p 1 , , p n ) = i = 1 n { d 4 x i e i p i x i } 0 | T φ ( x 1 ) φ ( x n ) | 0 \Gamma\left(p_{1},\ldots,p_{n}\right)=\int\prod_{i=1}^{n}\left\{\mathrm{d}^{4}% x_{i}e^{ip_{i}\cdot x_{i}}\right\}\langle 0|\mathrm{T}\ \varphi(x_{1})\ldots% \varphi(x_{n})|0\rangle
  61. p 1 , , p n out | q 1 , , q m in = i = 1 m { - i ( p i 2 - m 2 ) ( 2 π ) 3 2 Z 1 2 } j = 1 n { - i ( q j 2 - m 2 ) ( 2 π ) 3 2 Z 1 2 } Γ ( p 1 , , p n ; - q 1 , , - q m ) \langle p_{1},\ldots,p_{n}\ \mathrm{out}|q_{1},\ldots,q_{m}\ \mathrm{in}% \rangle=\prod_{i=1}^{m}\left\{-\frac{i\left(p_{i}^{2}-m^{2}\right)}{(2\pi)^{% \frac{3}{2}}Z^{\frac{1}{2}}}\right\}\prod_{j=1}^{n}\left\{-\frac{i\left(q_{j}^% {2}-m^{2}\right)}{(2\pi)^{\frac{3}{2}}Z^{\frac{1}{2}}}\right\}\Gamma\left(p_{1% },\ldots,p_{n};-q_{1},\ldots,-q_{m}\right)
  62. Ψ ( x ) = s = ± d p ~ ( b s 𝐩 u s 𝐩 e i p x + d s 𝐩 v s 𝐩 e - i p x ) , \Psi(x)=\sum_{s=\pm}\int\!\mathrm{d}\tilde{p}\big(b^{s}\textbf{p}u^{s}\textbf{% p}\mathrm{e}^{ip\cdot x}+d^{\dagger s}\textbf{p}v^{s}\textbf{p}\mathrm{e}^{-ip% \cdot x}\big),
  63. b s 𝐩 b^{s}\textbf{p}
  64. 𝐩 \,\textbf{p}
  65. s = ± s=\pm
  66. d s 𝐩 d^{\dagger s}\textbf{p}
  67. s s
  68. u s 𝐩 u^{s}\textbf{p}
  69. v s 𝐩 v^{s}\textbf{p}
  70. ( p / + m ) u s 𝐩 = 0 (p\!\!\!/+m)u^{s}\textbf{p}=0
  71. ( p / - m ) v s 𝐩 = 0 (p\!\!\!/-m)v^{s}\textbf{p}=0
  72. d p ~ := d 3 p / ( 2 π ) 3 2 ω 𝐩 \mathrm{d}\tilde{p}:=\mathrm{d}^{3}p/(2\pi)^{3}2\omega\textbf{p}
  73. ω 𝐩 = 𝐩 2 + m 2 \omega\textbf{p}=\sqrt{\,\textbf{p}^{2}+m^{2}}
  74. | α in |\alpha\ \mathrm{in}\rangle
  75. | β out |\beta\ \mathrm{out}\rangle
  76. = β out | α in , \mathcal{M}=\langle\beta\ \mathrm{out}|\alpha\ \mathrm{in}\rangle,
  77. n n
  78. n n^{\prime}
  79. n n
  80. { 𝐩 1 , , 𝐩 n } \{\,\textbf{p}_{1},...,\,\textbf{p}_{n}\}
  81. { s 1 , , s n } \{s_{1},...,s_{n}\}
  82. { 𝐤 1 , , 𝐤 n } \{\,\textbf{k}_{1},...,\,\textbf{k}_{n^{\prime}}\}
  83. { σ 1 , , σ n } \{\sigma_{1},...,\sigma_{n^{\prime}}\}
  84. | α in = | 𝐩 1 s 1 , , 𝐩 n s n and | β out = | 𝐤 1 σ 1 , , 𝐤 n σ n . |\alpha\ \mathrm{in}\rangle=|\,\textbf{p}_{1}^{s_{1}},...,\,\textbf{p}_{n}^{s_% {n}}\rangle\quad\,\text{and}\quad|\beta\ \mathrm{out}\rangle=|\,\textbf{k}_{1}% ^{\sigma_{1}},...,\,\textbf{k}_{n^{\prime}}^{\sigma_{n^{\prime}}}\rangle.
  85. | α in |\alpha\ \mathrm{in}\rangle
  86. b 𝐩 1 , in s 1 b^{\dagger s_{1}}_{\,\textbf{p}_{1},\mathrm{in}}
  87. = β out | b 𝐩 1 , in s 1 - b 𝐩 1 , out s 1 | α in , \mathcal{M}=\langle\beta\ \mathrm{out}|b^{\dagger s_{1}}_{\,\textbf{p}_{1},% \mathrm{in}}-b^{\dagger s_{1}}_{\,\textbf{p}_{1},\mathrm{out}}|\alpha^{\prime}% \ \mathrm{in}\rangle,
  88. α \alpha
  89. b s 𝐩 = d 3 x e i p x Ψ ¯ ( x ) γ 0 u s 𝐩 , b^{\dagger s}\textbf{p}=\int\!\mathrm{d}^{3}x\;\mathrm{e}^{ip\cdot x}\bar{\Psi% }(x)\gamma^{0}u^{s}\textbf{p},
  90. Ψ ¯ ( x ) = Ψ ( x ) β \bar{\Psi}(x)=\Psi^{\dagger}(x)\beta
  91. Ψ in \Psi\text{in}
  92. Ψ out \Psi\text{out}
  93. = d 3 x 1 e i p 1 x 1 β out | Ψ ¯ in ( x 1 ) γ 0 u 𝐩 1 s 1 - Ψ ¯ out ( x 1 ) γ 0 u 𝐩 1 s 1 | α in . \mathcal{M}=\int\!\mathrm{d}^{3}x_{1}\;\mathrm{e}^{ip_{1}\cdot x_{1}}\langle% \beta\ \mathrm{out}|\bar{\Psi}\text{in}(x_{1})\gamma^{0}u^{s_{1}}_{\,\textbf{p% }_{1}}-\bar{\Psi}\text{out}(x_{1})\gamma^{0}u^{s_{1}}_{\,\textbf{p}_{1}}|% \alpha^{\prime}\ \mathrm{in}\rangle.
  94. lim x 0 - d 3 x β | e i p x Ψ ¯ ( x ) γ 0 u 𝐩 s | α = Z d 3 x β | e i p x Ψ ¯ in ( x ) γ 0 u 𝐩 s | α , \lim_{x^{0}\rightarrow-\infty}\int\!\mathrm{d}^{3}x\langle\beta|\mathrm{e}^{ip% \cdot x}\bar{\Psi}(x)\gamma^{0}u^{s}_{\,\textbf{p}}|\alpha\rangle=\sqrt{Z}\int% \!\mathrm{d}^{3}x\langle\beta|\mathrm{e}^{ip\cdot x}\bar{\Psi}\text{in}(x)% \gamma^{0}u^{s}_{\,\textbf{p}}|\alpha\rangle,
  95. = 1 Z ( lim x 1 0 - - lim x 1 0 + ) d 3 x 1 e i p 1 x 1 β out | Ψ ¯ ( x 1 ) γ 0 u 𝐩 1 s 1 | α in , \mathcal{M}=\frac{1}{\sqrt{Z}}\Big(\lim_{x_{1}^{0}\rightarrow-\infty}-\lim_{x^% {0}_{1}\rightarrow+\infty}\Big)\int\!\mathrm{d}^{3}x_{1}\;\mathrm{e}^{ip_{1}% \cdot x_{1}}\langle\beta\ \mathrm{out}|\bar{\Psi}(x_{1})\gamma^{0}u^{s_{1}}_{% \,\textbf{p}_{1}}|\alpha^{\prime}\ \mathrm{in}\rangle,
  96. = - 1 Z d 4 x 1 0 ( e i p 1 x 1 β out | Ψ ¯ ( x 1 ) γ 0 u 𝐩 1 s 1 | α in ) \mathcal{M}=-\frac{1}{\sqrt{Z}}\int\!\mathrm{d}^{4}x_{1}\partial_{0}\big(% \mathrm{e}^{ip_{1}\cdot x_{1}}\langle\beta\ \mathrm{out}|\bar{\Psi}(x_{1})% \gamma^{0}u^{s_{1}}_{\,\textbf{p}_{1}}|\alpha^{\prime}\ \mathrm{in}\rangle\big)
  97. = - 1 Z d 4 x 1 ( 0 e i p 1 x 1 η ( x 1 ) + e i p 1 x 1 0 η ( x 1 ) ) γ 0 u 𝐩 1 s 1 , =-\frac{1}{\sqrt{Z}}\int\!\mathrm{d}^{4}x_{1}(\partial_{0}\mathrm{e}^{ip_{1}% \cdot x_{1}}\eta(x_{1})+\mathrm{e}^{ip_{1}\cdot x_{1}}\partial_{0}\eta(x_{1})% \big)\gamma^{0}u^{s_{1}}_{\,\textbf{p}_{1}},
  98. η ( x 1 ) := β out | Ψ ¯ ( x 1 ) | α in \eta(x_{1}):=\langle\beta\ \mathrm{out}|\bar{\Psi}(x_{1})|\alpha^{\prime}\ % \mathrm{in}\rangle
  99. e i p x u s 𝐩 \mathrm{e}^{ip\cdot x}u^{s}\textbf{p}
  100. ( - i / + m ) e i p x u s 𝐩 = 0. (-i\partial\!\!\!/+m)\mathrm{e}^{ip\cdot x}u^{s}\textbf{p}=0.
  101. γ 0 0 e i p x u s 𝐩 \gamma^{0}\partial_{0}\mathrm{e}^{ip\cdot x}u^{s}\textbf{p}
  102. = i Z d 4 x 1 e i p 1 x 1 ( i μ η ( x 1 ) γ μ + η ( x 1 ) m ) u 𝐩 1 s 1 . \mathcal{M}=\frac{i}{\sqrt{Z}}\int\!\mathrm{d}^{4}x_{1}\mathrm{e}^{ip_{1}\cdot x% _{1}}\big(i\partial_{\mu}\eta(x_{1})\gamma^{\mu}+\eta(x_{1})m\big)u^{s_{1}}_{% \,\textbf{p}_{1}}.
  103. = i Z d 4 x 1 e i p 1 x 1 [ ( i / x 1 + m ) u 𝐩 1 s 1 ] α 1 β out | Ψ ¯ α 1 ( x 1 ) | α in . \mathcal{M}=\frac{i}{\sqrt{Z}}\int\!\mathrm{d}^{4}x_{1}\mathrm{e}^{ip_{1}\cdot x% _{1}}[(i{\partial\!\!\!/}_{x_{1}}+m)u^{s_{1}}_{\,\textbf{p}_{1}}]_{\alpha_{1}}% \langle\beta\ \mathrm{out}|\bar{\Psi}_{\alpha_{1}}(x_{1})|\alpha^{\prime}\ % \mathrm{in}\rangle.
  104. β out | Ψ ¯ α 1 ( x 1 ) | α in = β out | b 𝐤 1 , out σ 1 Ψ ¯ α 1 ( x 1 ) - Ψ ¯ α 1 ( x 1 ) b 𝐤 1 , in σ 1 | α in . \langle\beta\ \mathrm{out}|\bar{\Psi}_{\alpha_{1}}(x_{1})|\alpha^{\prime}\ % \mathrm{in}\rangle=\langle\beta^{\prime}\ \mathrm{out}|b^{\sigma_{1}}_{\,% \textbf{k}_{1},\mathrm{out}}\bar{\Psi}_{\alpha_{1}}(x_{1})-\bar{\Psi}_{\alpha_% {1}}(x_{1})b^{\sigma_{1}}_{\,\textbf{k}_{1},\mathrm{in}}|\alpha^{\prime}\ % \mathrm{in}\rangle.
  105. ( Ψ ¯ γ 0 u s 𝐩 ) = u ¯ s 𝐩 γ 0 Ψ (\bar{\Psi}\gamma^{0}u^{s}\textbf{p})^{\dagger}=\bar{u}^{s}\textbf{p}\gamma^{0}\Psi
  106. u ¯ s 𝐩 := u s 𝐩 β \bar{u}^{s}\textbf{p}:=u^{\dagger s}\textbf{p}\beta
  107. β out | Ψ ¯ α 1 ( x 1 ) | α in = 1 Z ( lim y 1 0 - lim y 1 0 - ) d 3 y 1 e - i k 1 y 1 [ u ¯ 𝐤 1 σ 1 γ 0 ] β 1 β out | T [ Ψ β 1 ( y 1 ) Ψ ¯ α 1 ( x 1 ) ] | α in . \langle\beta\ \mathrm{out}|\bar{\Psi}_{\alpha_{1}}(x_{1})|\alpha^{\prime}\ % \mathrm{in}\rangle=\frac{1}{\sqrt{Z}}\Big(\lim_{y^{0}_{1}\rightarrow\infty}-% \lim_{y^{0}_{1}\rightarrow-\infty}\Big)\int\!\mathrm{d}^{3}y_{1}\mathrm{e}^{-% ik_{1}\cdot y_{1}}[\bar{u}^{\sigma_{1}}_{\,\textbf{k}_{1}}\gamma^{0}]_{\beta_{% 1}}\langle\beta^{\prime}\ \mathrm{out}|\mathrm{T}[\Psi_{\beta_{1}}(y_{1})\bar{% \Psi}_{\alpha_{1}}(x_{1})]|\alpha^{\prime}\ \mathrm{in}\rangle.
  108. Ψ β 1 ( y 1 ) \Psi_{\beta_{1}}(y_{1})
  109. β out | Ψ ¯ α 1 ( x 1 ) | α in = i Z d 4 y 1 e - i k 1 y 1 [ u ¯ 𝐤 1 σ 1 ( - i / y 1 + m ) ] β 1 β out | T [ Ψ β 1 ( y 1 ) Ψ ¯ α 1 ( x 1 ) ] | α in . \langle\beta\ \mathrm{out}|\bar{\Psi}_{\alpha_{1}}(x_{1})|\alpha^{\prime}\ % \mathrm{in}\rangle=\frac{i}{\sqrt{Z}}\int\!\mathrm{d}^{4}y_{1}\mathrm{e}^{-ik_% {1}\cdot y_{1}}[\bar{u}^{\sigma_{1}}_{\,\textbf{k}_{1}}(-i\partial\!\!\!/_{y_{% 1}}+m)]_{\beta_{1}}\langle\beta^{\prime}\ \mathrm{out}|\mathrm{T}[\Psi_{\beta_% {1}}(y_{1})\bar{\Psi}_{\alpha_{1}}(x_{1})]|\alpha^{\prime}\ \mathrm{in}\rangle.
  110. β out | α in = j = 1 n d 4 x j i e i p j x j Z [ ( i / x j + m ) u 𝐩 j s j ] α j l = 1 n d 4 y l i e - i k l y l Z [ u ¯ 𝐤 l σ l ( - i / y l + m ) ] β l 0 | T [ Ψ β 1 ( y 1 ) Ψ β n ( y n ) Ψ ¯ α 1 ( x 1 ) Ψ ¯ α n ( x n ) ] | 0 . \langle\beta\ \mathrm{out}|\alpha\ \mathrm{in}\rangle=\int\!\prod_{j=1}^{n}% \mathrm{d}^{4}x_{j}\frac{i\mathrm{e}^{ip_{j}x_{j}}}{\sqrt{Z}}[(i{\partial\!\!% \!/}_{x_{j}}+m)u^{s_{j}}_{\,\textbf{p}_{j}}]_{\alpha_{j}}\prod_{l=1}^{n^{% \prime}}\mathrm{d}^{4}y_{l}\frac{i\mathrm{e}^{-ik_{l}y_{l}}}{\sqrt{Z}}[\bar{u}% ^{\sigma_{l}}_{\,\textbf{k}_{l}}(-i{\partial\!\!\!/}_{y_{l}}+m)]_{\beta_{l}}% \langle 0|\mathrm{T}[\Psi_{\beta_{1}}(y_{1})...\Psi_{\beta_{n^{\prime}}}(y_{n^% {\prime}})\bar{\Psi}_{\alpha_{1}}(x_{1})...\bar{\Psi}_{\alpha_{n}}(x_{n})]|0\rangle.
  111. u s 𝐩 u^{s}\textbf{p}
  112. v s 𝐩 v^{s}\textbf{p}
  113. Ψ \Psi
  114. Ψ ¯ \bar{\Psi}
  115. Z Z
  116. | p |p\rangle
  117. 0 | φ ( x ) | p = Z 0 | φ in ( x ) | p + d 4 y Δ ret ( x - y ) 0 | j ( y ) | p \langle 0|\varphi(x)|p\rangle=\sqrt{Z}\langle 0|\varphi_{\mathrm{in}}(x)|p% \rangle+\int\mathrm{d}^{4}y\Delta_{\mathrm{ret}}(x-y)\langle 0|j(y)|p\rangle
  118. φ φ
  119. φ ( x ) = e i P x φ ( 0 ) e - i P x \varphi(x)=e^{iP\cdot x}\varphi(0)e^{-iP\cdot x}
  120. e - i p x 0 | φ ( 0 ) | p = Z e - i p x 0 | φ in ( 0 ) | p + d 4 y Δ ret ( x - y ) 0 | j ( y ) | p e^{-ip\cdot x}\langle 0|\varphi(0)|p\rangle=\sqrt{Z}e^{-ip\cdot x}\langle 0|% \varphi_{\mathrm{in}}(0)|p\rangle+\int\mathrm{d}^{4}y\Delta_{\mathrm{ret}}(x-y% )\langle 0|j(y)|p\rangle
  121. p p
  122. 0 = 0 + d 4 y δ 4 ( x - y ) 0 | j ( y ) | p ; 0 | j ( x ) | p = 0 0=0+\int\mathrm{d}^{4}y\delta^{4}(x-y)\langle 0|j(y)|p\rangle;\quad% \Leftrightarrow\quad\langle 0|j(x)|p\rangle=0
  123. 0 | φ ( x ) | p = Z 0 | φ in ( x ) | p \langle 0|\varphi(x)|p\rangle=\sqrt{Z}\langle 0|\varphi_{\mathrm{in}}(x)|p\rangle
  124. Z Z
  125. 0 | φ in ( x ) | p = d 3 q ( 2 π ) 3 2 ( 2 ω q ) 1 2 e - i q x 0 | a in ( 𝐪 ) | p = d 3 q ( 2 π ) 3 2 e - i q x 0 | a in ( 𝐪 ) a in ( 𝐩 ) | 0 \langle 0|\varphi_{\mathrm{in}}(x)|p\rangle=\int\frac{\mathrm{d}^{3}q}{(2\pi)^% {\frac{3}{2}}(2\omega_{q})^{\frac{1}{2}}}e^{-iq\cdot x}\langle 0|a_{\mathrm{in% }}(\mathbf{q})|p\rangle=\int\frac{\mathrm{d}^{3}q}{(2\pi)^{\frac{3}{2}}}e^{-iq% \cdot x}\langle 0|a_{\mathrm{in}}(\mathbf{q})a^{\dagger}_{\mathrm{in}}(\mathbf% {p})|0\rangle
  126. a in a^{\dagger}_{\mathrm{in}}
  127. 0 | φ in ( x ) | p = e - i p x ( 2 π ) 3 2 \langle 0|\varphi_{\mathrm{in}}(x)|p\rangle=\frac{e^{-ip\cdot x}}{(2\pi)^{% \frac{3}{2}}}
  128. 0 | φ ( 0 ) | p = Z ( 2 π ) 3 \langle 0|\varphi(0)|p\rangle=\sqrt{\frac{Z}{(2\pi)^{3}}}
  129. Z Z
  130. 0 | φ ( 0 ) | p \langle 0|\varphi(0)|p\rangle

LTI_system_theory.html

  1. x 1 ( t ) x_{1}(t)\,
  2. y 1 ( t ) , y_{1}(t),\,
  3. x 2 ( t ) x_{2}(t)\,
  4. y 2 ( t ) , y_{2}(t),\,
  5. a 1 x 1 ( t ) + a 2 x 2 ( t ) a_{1}x_{1}(t)+a_{2}x_{2}(t)\,
  6. a 1 y 1 ( t ) + a 2 y 2 ( t ) a_{1}y_{1}(t)+a_{2}y_{2}(t)\,
  7. a 1 a_{1}
  8. a 2 a_{2}
  9. c 1 , c 2 , , c k c_{1},c_{2},\ldots,c_{k}
  10. k c k x k ( t ) \sum_{k}c_{k}\,x_{k}(t)
  11. k c k y k ( t ) . \sum_{k}c_{k}\,y_{k}(t).\,
  12. c ω c_{\omega}
  13. x ω x_{\omega}
  14. ω \omega
  15. x ( t ) x(t)
  16. y ( t ) y(t)
  17. x ( t - T ) x(t-T)
  18. y ( t - T ) y(t-T)
  19. A e s t Ae^{st}
  20. A A
  21. s s
  22. B e s t Be^{st}
  23. B B
  24. B / A B/A
  25. s s
  26. y ( t ) = x ( t ) * h ( t ) y(t)=x(t)*h(t)\,
  27. = def - x ( t - τ ) h ( τ ) d τ {}\quad\stackrel{\mathrm{def}}{=}\ \int_{-\infty}^{\infty}x(t-\tau)\cdot h(% \tau)\,\operatorname{d}\tau
  28. = - x ( τ ) h ( t - τ ) d τ , {}\quad=\int_{-\infty}^{\infty}x(\tau)\cdot h(t-\tau)\,\operatorname{d}\tau,
  29. h ( t ) \textstyle h(t)
  30. x ( τ ) = δ ( τ ) . \textstyle x(\tau)=\delta(\tau).
  31. y ( t ) \textstyle y(t)
  32. x ( τ ) . \textstyle x(\tau).
  33. h ( - τ ) , \textstyle h(-\tau),
  34. t . \textstyle t.
  35. t \textstyle t
  36. h ( τ ) \textstyle h(\tau)
  37. τ , \textstyle\tau,
  38. y ( t ) \textstyle y(t)
  39. x \textstyle x
  40. t , \textstyle t,
  41. { x ( u - τ ) ; u } \textstyle\{x(u-\tau);\ u\}
  42. x ( u - τ ) \textstyle x(u-\tau)
  43. u \textstyle u
  44. τ . \textstyle\tau.
  45. { x } \textstyle\{x\}\,
  46. { x ( u ) ; u } . \textstyle\{x(u);\ u\}.
  47. { x } , \textstyle\{x\},
  48. { y } . \textstyle\{y\}.
  49. y ( t ) = def O t { x } , y(t)\ \stackrel{\,\text{def}}{=}\ O_{t}\{x\},
  50. O t \textstyle O_{t}
  51. t . \textstyle t.
  52. y ( t ) \textstyle y(t)
  53. x \textstyle x
  54. t . \textstyle t.
  55. t , \textstyle t,
  56. O \textstyle O
  57. O t { - c τ x τ ( u ) d τ ; u } = - c τ y τ ( t ) O t { x τ } d τ . O_{t}\left\{\int_{-\infty}^{\infty}c_{\tau}\ x_{\tau}(u)\,\operatorname{d}\tau% ;\ u\right\}=\int_{-\infty}^{\infty}c_{\tau}\ \underbrace{y_{\tau}(t)}_{O_{t}% \{x_{\tau}\}}\,\operatorname{d}\tau.\,
  58. O t { x ( u - τ ) ; u } = y ( t - τ ) = def O t - τ { x } . \begin{aligned}\displaystyle O_{t}\{x(u-\tau);\ u\}&\displaystyle\stackrel{% \quad}{=}\ y(t-\tau)\\ &\displaystyle\stackrel{\,\text{def}}{=}\ O_{t-\tau}\{x\}.\end{aligned}
  59. h ( t ) = def O t { δ ( u ) ; u } . \textstyle h(t)\ \stackrel{\,\text{def}}{=}\ O_{t}\{\delta(u);\ u\}.
  60. h ( t - τ ) h(t-\tau)\,
  61. = def O t - τ { δ ( u ) ; u } {}\stackrel{\,\text{def}}{=}\ O_{t-\tau}\{\delta(u);\ u\}
  62. = O t { δ ( u - τ ) ; u } . {}=O_{t}\{\delta(u-\tau);\ u\}.\,
  63. x ( t ) * h ( t ) = - x ( τ ) h ( t - τ ) d τ = - x ( τ ) O t { δ ( u - τ ) ; u } d τ , \begin{aligned}\displaystyle x(t)*h(t)&\displaystyle=\int_{-\infty}^{\infty}x(% \tau)\cdot h(t-\tau)\,\operatorname{d}\tau\\ &\displaystyle=\int_{-\infty}^{\infty}x(\tau)\cdot O_{t}\{\delta(u-\tau);\ u\}% \,\operatorname{d}\tau,\end{aligned}
  64. c τ = x ( τ ) \textstyle c_{\tau}=x(\tau)
  65. x τ ( u ) = δ ( u - τ ) . \textstyle x_{\tau}(u)=\delta(u-\tau).
  66. x ( t ) * h ( t ) = O t { - x ( τ ) δ ( u - τ ) d τ ; u } = O t { x ( u ) ; u } = def y ( t ) . \begin{aligned}\displaystyle x(t)*h(t)&\displaystyle=O_{t}\left\{\int_{-\infty% }^{\infty}x(\tau)\cdot\delta(u-\tau)\,\operatorname{d}\tau;\ u\right\}\\ &\displaystyle=O_{t}\left\{x(u);\ u\right\}\\ &\displaystyle\ \stackrel{\,\text{def}}{=}\ y(t).\end{aligned}
  67. { x } , \textstyle\{x\},
  68. f = λ f \mathcal{H}f=\lambda f
  69. λ \lambda
  70. A e s t Ae^{st}
  71. A , s A,s\in\mathbb{C}
  72. x ( t ) = A e s t x(t)=Ae^{st}
  73. h ( t ) h(t)
  74. - h ( t - τ ) A e s τ d τ \int_{-\infty}^{\infty}h(t-\tau)Ae^{s\tau}\,\operatorname{d}\tau
  75. - h ( τ ) A e s ( t - τ ) d τ f = - h ( τ ) A e s t e - s τ d τ = A e s t - h ( τ ) e - s τ d τ = A e s t Input f H ( s ) Scalar λ , \begin{aligned}\displaystyle\overbrace{\int_{-\infty}^{\infty}h(\tau)\,Ae^{s(t% -\tau)}\,\operatorname{d}\tau}^{\mathcal{H}f}&\displaystyle=\int_{-\infty}^{% \infty}h(\tau)\,Ae^{st}e^{-s\tau}\,\operatorname{d}\tau&\displaystyle=Ae^{st}% \int_{-\infty}^{\infty}h(\tau)\,e^{-s\tau}\,\operatorname{d}\tau\\ &\displaystyle=\overbrace{\underbrace{Ae^{st}}_{\,\text{Input}}}^{f}\overbrace% {\underbrace{H(s)}_{\,\text{Scalar}}}^{\lambda},\end{aligned}
  76. H ( s ) = def - h ( t ) e - s t d t H(s)\ \stackrel{\,\text{def}}{=}\ \int_{-\infty}^{\infty}h(t)e^{-st}\,% \operatorname{d}t
  77. A , s A,s\in\mathbb{C}
  78. A e s t Ae^{st}
  79. H ( s ) H(s)
  80. A e s t Ae^{st}
  81. H ( s ) H(s)
  82. v ( t ) = e i ω t v(t)=e^{i\omega t}
  83. v a ( t ) = e i ω ( t + a ) v_{a}(t)=e^{i\omega(t+a)}
  84. H [ v a ] ( t ) = e i ω a H [ v ] ( t ) H[v_{a}](t)=e^{i\omega a}H[v](t)
  85. e i ω a e^{i\omega a}
  86. H [ v a ] ( t ) = H [ v ] ( t + a ) H[v_{a}](t)=H[v](t+a)
  87. H H
  88. H [ v ] ( t + a ) = e i ω a H [ v ] ( t ) H[v](t+a)=e^{i\omega a}H[v](t)
  89. t = 0 t=0
  90. H [ v ] ( τ ) = e i ω τ H [ v ] ( 0 ) H[v](\tau)=e^{i\omega\tau}H[v](0)
  91. e i ω τ e^{i\omega\tau}
  92. H ( s ) = def { h ( t ) } = def - h ( t ) e - s t d t H(s)\ \stackrel{\,\text{def}}{=}\ \mathcal{L}\{h(t)\}\ \stackrel{\,\text{def}}% {=}\ \int_{-\infty}^{\infty}h(t)e^{-st}\,\operatorname{d}t
  93. e j ω t e^{j\omega t}
  94. ω \omega\in\mathbb{R}
  95. j = def - 1 j\ \stackrel{\,\text{def}}{=}\ \sqrt{-1}
  96. H ( j ω ) = { h ( t ) } H(j\omega)=\mathcal{F}\{h(t)\}
  97. H ( s ) H(s)
  98. H ( j ω ) H(j\omega)
  99. y ( t ) = ( h * x ) ( t ) = def - h ( t - τ ) x ( τ ) d τ = def - 1 { H ( s ) X ( s ) } . y(t)=(h*x)(t)\ \stackrel{\,\text{def}}{=}\ \int_{-\infty}^{\infty}h(t-\tau)x(% \tau)\,\operatorname{d}\tau\ \stackrel{\,\text{def}}{=}\ \mathcal{L}^{-1}\{H(s% )X(s)\}.
  100. exp ( s t ) \exp({st})
  101. d d t ( c 1 x 1 ( t ) + c 2 x 2 ( t ) ) = c 1 x 1 ( t ) + c 2 x 2 ( t ) \frac{\operatorname{d}}{\operatorname{d}t}\left(c_{1}x_{1}(t)+c_{2}x_{2}(t)% \right)=c_{1}x^{\prime}_{1}(t)+c_{2}x^{\prime}_{2}(t)
  102. d d t x ( t - τ ) = x ( t - τ ) \frac{\operatorname{d}}{\operatorname{d}t}x(t-\tau)=x^{\prime}(t-\tau)
  103. { d d t x ( t ) } = s X ( s ) \mathcal{L}\left\{\frac{\operatorname{d}}{\operatorname{d}t}x(t)\right\}=sX(s)
  104. 𝒜 { x ( t ) } = def t - a t + a x ( λ ) d λ . \mathcal{A}\left\{x(t)\right\}\ \stackrel{\,\text{def}}{=}\ \int_{t-a}^{t+a}x(% \lambda)\,\operatorname{d}\lambda.
  105. 𝒜 { c 1 x 1 ( t ) + c 2 x 2 ( t ) } = t - a t + a ( c 1 x 1 ( λ ) + c 2 x 2 ( λ ) ) d λ = c 1 t - a t + a x 1 ( λ ) d λ + c 2 t - a t + a x 2 ( λ ) d λ = c 1 𝒜 { x 1 ( t ) } + c 2 𝒜 { x 2 ( t ) } , \begin{aligned}\displaystyle\mathcal{A}\left\{c_{1}x_{1}(t)+c_{2}x_{2}(t)% \right\}&\displaystyle=\int_{t-a}^{t+a}\left(c_{1}x_{1}(\lambda)+c_{2}x_{2}(% \lambda)\right)\,\operatorname{d}\lambda\\ &\displaystyle=c_{1}\int_{t-a}^{t+a}x_{1}(\lambda)\,\operatorname{d}\lambda+c_% {2}\int_{t-a}^{t+a}x_{2}(\lambda)\,\operatorname{d}\lambda\\ &\displaystyle=c_{1}\mathcal{A}\left\{x_{1}(t)\right\}+c_{2}\mathcal{A}\left\{% x_{2}(t)\right\},\end{aligned}
  106. 𝒜 { x ( t - τ ) } = t - a t + a x ( λ - τ ) d λ = ( t - τ ) - a ( t - τ ) + a x ( ξ ) d ξ = 𝒜 { x } ( t - τ ) , \begin{aligned}\displaystyle\mathcal{A}\left\{x(t-\tau)\right\}&\displaystyle=% \int_{t-a}^{t+a}x(\lambda-\tau)\,\operatorname{d}\lambda\\ &\displaystyle=\int_{(t-\tau)-a}^{(t-\tau)+a}x(\xi)\,\operatorname{d}\xi\\ &\displaystyle=\mathcal{A}\{x\}(t-\tau),\end{aligned}
  107. 𝒜 \mathcal{A}
  108. Π ( t ) \Pi(t)
  109. 𝒜 { x ( t ) } = - Π ( λ - t 2 a ) x ( λ ) d λ , \mathcal{A}\left\{x(t)\right\}=\int_{-\infty}^{\infty}\Pi\left(\frac{\lambda-t% }{2a}\right)x(\lambda)\,\operatorname{d}\lambda,
  110. Π ( t ) = def { 1 if | t | < 1 2 , 0 if | t | > 1 2 . \Pi(t)\ \stackrel{\,\text{def}}{=}\ \begin{cases}1&\,\text{if }|t|<\frac{1}{2}% ,\\ 0&\,\text{if }|t|>\frac{1}{2}.\end{cases}
  111. h ( t ) = 0 t < 0 , h(t)=0\quad\forall t<0,
  112. h ( t ) h(t)
  113. x ( t ) < \ \|x(t)\|_{\infty}<\infty
  114. y ( t ) < \ \|y(t)\|_{\infty}<\infty
  115. x ( t ) x(t)
  116. y ( t ) y(t)
  117. h ( t ) h(t)
  118. h ( t ) 1 = - | h ( t ) | d t < . \ \|h(t)\|_{1}=\int_{-\infty}^{\infty}|h(t)|\,\operatorname{d}t<\infty.
  119. s = j ω s=j\omega
  120. t < 0 t<0\,
  121. t > 0 t>0\,
  122. x ( t ) x(t)
  123. x [ n ] = def x ( n T ) n , x[n]\ \stackrel{\,\text{def}}{=}\ x(nT)\qquad\forall\,n\in\mathbb{Z},
  124. 1 / ( 2 T ) 1/(2T)
  125. { x [ m - k ] ; m } \{x[m-k];\ m\}
  126. { x [ m - k ] ; for all integer values of m } \{x[m-k];\ \mbox{for all integer values of m}~{}\}
  127. { x } \{x\}\,
  128. { x [ m ] ; m } . \{x[m];\ m\}.
  129. { x } \{x\}
  130. { y } . \{y\}.
  131. O O
  132. y [ n ] = def O n { x } . y[n]\ \stackrel{\,\text{def}}{=}\ O_{n}\{x\}.
  133. x [ m ] = δ [ m ] , x[m]=\delta[m],
  134. h [ n ] = def O n { δ [ m ] ; m } . h[n]\ \stackrel{\,\text{def}}{=}\ O_{n}\{\delta[m];\ m\}.\,
  135. O O
  136. O n { k = - c k x k [ m ] ; m } = k = - c k O n { x k } . O_{n}\left\{\sum_{k=-\infty}^{\infty}c_{k}\cdot x_{k}[m];\ m\right\}=\sum_{k=-% \infty}^{\infty}c_{k}\cdot O_{n}\{x_{k}\}.\,
  137. O n { x [ m - k ] ; m } \displaystyle O_{n}\{x[m-k];\ m\}
  138. { h } , \{h\},\,
  139. x [ m ] k = - x [ k ] δ [ m - k ] , x[m]\equiv\sum_{k=-\infty}^{\infty}x[k]\cdot\delta[m-k],\,
  140. { x } \{x\}\,
  141. y [ n ] = O n { x } \displaystyle y[n]=O_{n}\{x\}
  142. c k = x [ k ] c_{k}=x[k]\,
  143. x k [ m ] = δ [ m - k ] . x_{k}[m]=\delta[m-k].\,
  144. O n { δ [ m - k ] ; m } \displaystyle O_{n}\{\delta[m-k];\ m\}
  145. y [ n ] y[n]\,
  146. = k = - x [ k ] h [ n - k ] =\sum_{k=-\infty}^{\infty}x[k]\cdot h[n-k]\,
  147. = k = - x [ n - k ] h [ k ] , =\sum_{k=-\infty}^{\infty}x[n-k]\cdot h[k],
  148. O n O_{n}\,
  149. f = λ f \mathcal{H}f=\lambda f
  150. λ \lambda
  151. z n = e s T n z^{n}=e^{sTn}
  152. n n\in\mathbb{Z}
  153. T T\in\mathbb{R}
  154. z = e s T , z , s z=e^{sT},\ z,s\in\mathbb{C}
  155. x [ n ] = z n x[n]=\,\!z^{n}
  156. h [ n ] h[n]
  157. m = - h [ n - m ] z m \sum_{m=-\infty}^{\infty}h[n-m]\,z^{m}
  158. m = - h [ m ] z ( n - m ) = z n m = - h [ m ] z - m = z n H ( z ) \sum_{m=-\infty}^{\infty}h[m]\,z^{(n-m)}=z^{n}\sum_{m=-\infty}^{\infty}h[m]\,z% ^{-m}=z^{n}H(z)
  159. H ( z ) = def m = - h [ m ] z - m H(z)\ \stackrel{\,\text{def}}{=}\ \sum_{m=-\infty}^{\infty}h[m]z^{-m}
  160. z n z^{n}
  161. H ( z ) H(z)
  162. H ( z ) = 𝒵 { h [ n ] } = n = - h [ n ] z - n H(z)=\mathcal{Z}\{h[n]\}=\sum_{n=-\infty}^{\infty}h[n]z^{-n}
  163. e j ω n e^{j\omega n}
  164. ω \omega\in\mathbb{R}
  165. z n z^{n}
  166. z = e j ω z=e^{j\omega}
  167. H ( e j ω ) = { h [ n ] } H(e^{j\omega})=\mathcal{F}\{h[n]\}
  168. H ( z ) H(z)
  169. H ( e j ω ) H(e^{j\omega})
  170. y [ n ] = ( h * x ) [ n ] = m = - h [ n - m ] x [ m ] = 𝒵 - 1 { H ( z ) X ( z ) } . y[n]=(h*x)[n]=\sum_{m=-\infty}^{\infty}h[n-m]x[m]=\mathcal{Z}^{-1}\{H(z)X(z)\}.
  171. z n z^{n}
  172. D { x [ n ] } = def x [ n - 1 ] D\{x[n]\}\ \stackrel{\,\text{def}}{=}\ x[n-1]
  173. D ( c 1 x 1 [ n ] + c 2 x 2 [ n ] ) = c 1 x 1 [ n - 1 ] + c 2 x 2 [ n - 1 ] = c 1 D x 1 [ n ] + c 2 D x 2 [ n ] D\left(c_{1}x_{1}[n]+c_{2}x_{2}[n]\right)=c_{1}x_{1}[n-1]+c_{2}x_{2}[n-1]=c_{1% }Dx_{1}[n]+c_{2}Dx_{2}[n]
  174. D { x [ n - m ] } = x [ n - m - 1 ] = x [ ( n - 1 ) - m ] = D { x } [ n - m ] D\{x[n-m]\}=x[n-m-1]=x[(n-1)-m]=D\{x\}[n-m]\,
  175. 𝒵 { D x [ n ] } = z - 1 X ( z ) . \mathcal{Z}\left\{Dx[n]\right\}=z^{-1}X(z).
  176. 𝒜 { x [ n ] } = def k = n - a n + a x [ k ] \mathcal{A}\left\{x[n]\right\}\ \stackrel{\,\text{def}}{=}\ \sum_{k=n-a}^{n+a}% x[k]
  177. 𝒜 { c 1 x 1 [ n ] + c 2 x 2 [ n ] } = k = n - a n + a ( c 1 x 1 [ k ] + c 2 x 2 [ k ] ) = c 1 k = n - a n + a x 1 [ k ] + c 2 k = n - a n + a x 2 [ k ] = c 1 𝒜 { x 1 [ n ] } + c 2 𝒜 { x 2 [ n ] } , \begin{aligned}\displaystyle\mathcal{A}\left\{c_{1}x_{1}[n]+c_{2}x_{2}[n]% \right\}&\displaystyle=\sum_{k=n-a}^{n+a}\left(c_{1}x_{1}[k]+c_{2}x_{2}[k]% \right)\\ &\displaystyle=c_{1}\sum_{k=n-a}^{n+a}x_{1}[k]+c_{2}\sum_{k=n-a}^{n+a}x_{2}[k]% \\ &\displaystyle=c_{1}\mathcal{A}\left\{x_{1}[n]\right\}+c_{2}\mathcal{A}\left\{% x_{2}[n]\right\},\end{aligned}
  178. 𝒜 { x [ n - m ] } = k = n - a n + a x [ k - m ] = k = ( n - m ) - a ( n - m ) + a x [ k ] = 𝒜 { x } [ n - m ] , \begin{aligned}\displaystyle\mathcal{A}\left\{x[n-m]\right\}&\displaystyle=% \sum_{k=n-a}^{n+a}x[k-m]\\ &\displaystyle=\sum_{k^{\prime}=(n-m)-a}^{(n-m)+a}x[k^{\prime}]\\ &\displaystyle=\mathcal{A}\left\{x\right\}[n-m],\end{aligned}
  179. h [ n ] h[n]
  180. h [ n ] = 0 n < 0 , h[n]=0\ \forall n<0,
  181. h [ n ] h[n]
  182. x [ n ] < \ \|x[n]\|_{\infty}<\infty
  183. y [ n ] < \ \|y[n]\|_{\infty}<\infty
  184. x [ n ] x[n]
  185. y [ n ] y[n]
  186. h [ n ] h[n]
  187. h [ n ] 1 = def n = - | h [ n ] | < . \|h[n]\|_{1}\ \stackrel{\,\text{def}}{=}\ \sum_{n=-\infty}^{\infty}|h[n]|<\infty.
  188. | z | = 1 |z|=1

Ludwig_Schläfli.html

  1. n n
  2. n n
  3. n n
  4. n n
  5. n n
  6. n = 2 , 3 n=2,3
  7. n n
  8. x , y , x,y,\ldots
  9. n n
  10. 1 , 2 , 3 , 1,2,3,\ldots
  11. n - 1 n-1
  12. n - 2 n-2
  13. n - 3 n-3
  14. x , y , x,y,\ldots
  15. x , y , x^{\prime},y^{\prime},\ldots
  16. ( x - x ) 2 + ( y - y ) 2 + \sqrt{(x^{\prime}-x)^{2}+(y^{\prime}-y)^{2}+\cdots}
  17. n n
  18. n = 2 , 3 n=2,3
  19. n n
  20. x , y , x,y,\ldots
  21. n n
  22. 1 , 2 , 3 , 1,2,3,\ldots
  23. n - 1 n-1
  24. n - 2 n-2
  25. n - 3 n-3
  26. x , y , x,y,\ldots
  27. x , y , x^{\prime},y^{\prime},\ldots
  28. ( x - x ) 2 + ( y - y ) 2 + \sqrt{(x^{\prime}-x)^{2}+(y^{\prime}-y)^{2}+\cdots}
  29. n n
  30. 𝐑 n \mathbf{R}^{n}
  31. n n
  32. n n

Luminous_efficacy.html

  1. 0 y λ J λ d λ 0 J λ d λ , \frac{\int^{\infty}_{0}y_{\lambda}J_{\lambda}d\lambda}{\int^{\infty}_{0}J_{% \lambda}d\lambda},
  2. 0 y λ J λ d λ \int^{\infty}_{0}y_{\lambda}J_{\lambda}d\lambda
  3. y λ y_{\lambda}
  4. J λ J_{\lambda}
  5. 0 J λ d λ \int^{\infty}_{0}J_{\lambda}d\lambda
  6. J λ J_{\lambda}

Luneburg_lens.html

  1. ϵ r \epsilon_{r}
  2. n n
  3. 2 \sqrt{2}
  4. n = ϵ r = 2 - ( r R ) 2 n=\sqrt{\epsilon_{r}}=\sqrt{2-\left(\frac{r}{R}\right)^{2}}
  5. R R
  6. n = ϵ r = n 0 1 + ( r R ) 2 n=\sqrt{\epsilon_{r}}=\frac{n_{0}}{1+\left(\frac{r}{R}\right)^{2}}
  7. ( r , θ ) \scriptstyle{(r,\theta)}
  8. ( r 1 , θ 1 ) \scriptstyle{(r_{1},\theta_{1})}
  9. ( r 2 , θ 2 ) \scriptstyle{(r_{2},\theta_{2})}
  10. T = ( r 1 , θ 1 ) ( r 2 , θ 2 ) n ( r ) c ( r d θ ) 2 + d r 2 = 1 c θ 1 θ 2 n ( r ) r 2 + ( d r d θ ) 2 d θ T=\int_{(r_{1},\theta_{1})}^{(r_{2},\theta_{2})}\frac{n(r)}{c}\sqrt{(rd\theta)% ^{2}+dr^{2}}=\frac{1}{c}\int_{\theta_{1}}^{\theta_{2}}n(r)\sqrt{r^{2}+\left(% \frac{dr}{d\theta}\right)^{2}}d\theta
  11. c \scriptstyle{c}
  12. T \scriptstyle{T}
  13. r \scriptstyle{r}
  14. θ \scriptstyle{\theta}
  15. L ( r , r ) = n ( r ) r 2 + r 2 \scriptstyle{L(r,r^{\prime})=n(r)\sqrt{{r^{\prime}}^{2}+r^{2}}}
  16. r \scriptstyle{r^{\prime}}
  17. d r d θ \scriptstyle{\frac{dr}{d\theta}}
  18. n ( r ) r 2 + r 2 - n ( r ) r 2 r 2 + r 2 = h n(r)\sqrt{{r^{\prime}}^{2}+r^{2}}-n(r)\frac{{r^{\prime}}^{2}}{\sqrt{{r^{\prime% }}^{2}+r^{2}}}=h
  19. h \scriptstyle{h}
  20. r \scriptstyle{r}
  21. θ \scriptstyle{\theta}
  22. d θ = h r ( n ( r ) ) 2 r 2 - h 2 d r d\theta=\frac{h}{r\sqrt{(n(r))^{2}r^{2}-h^{2}}}dr
  23. h \scriptstyle{h}
  24. θ \scriptstyle{\theta}
  25. r \scriptstyle{r}
  26. θ \scriptstyle{\theta}
  27. r \scriptstyle{r}

Luria–Delbrück_experiment.html

  1. r m - ln ( m ) - 1.24 = 0. \frac{r}{m}-\ln(m)-1.24=0.
  2. μ = β log ( ρ ) 1 - exp ( - β t ) \mu=\frac{\beta\log(\rho)}{1-\exp(-\beta t)}
  3. ρ = N s N \rho=\frac{N_{s}}{N}
  4. G ( z , μ , ϕ ) = exp ( μ ϕ ( 1 z - 1 ) log ( 1 - ϕ z ) ) G(z,\mu,\phi)=\exp\left(\frac{\mu}{\phi}\left(\frac{1}{z}-1\right)\log\left(1-% \phi z\right)\right)

Machine_that_always_halts.html

  1. M 1 , M 2 , M_{1},M_{2},\ldots
  2. T 1 , T 2 , T_{1},\ldots T_{2},\ldots
  3. T T
  4. T i ( i ) + 1 T_{i}(i)+1\,
  5. T j ( j ) = T j ( j ) + 1 T_{j}(j)=T_{j}(j)+1\,
  6. Π 2 0 \Pi^{0}_{2}

Mackey_space.html

  1. X X
  2. X X^{\prime}
  3. σ ( X , X ) \sigma(X^{\prime},X)
  4. X X^{\prime}
  5. σ \sigma

Mackey_topology.html

  1. ( X , X ) (X,X^{\prime})
  2. X X
  3. X X^{\prime}
  4. τ ( X , X ) \tau(X,X^{\prime})
  5. X X
  6. X X^{\prime}
  7. ( X , τ ) (X,\tau)
  8. X X^{\prime}
  9. τ = τ ( X , X ) \tau=\tau(X,X^{\prime})
  10. ( X , τ ) (X,\tau)
  11. τ = τ ( X , X ) = β ( X , X ) \tau=\tau(X,X^{\prime})=\beta(X,X^{\prime})

Magic_hexagon.html

  1. ( 3 n 2 - 3 n + 1 ) (3n^{2}-3n+1)
  2. s = 1 2 ( 3 n 2 - 3 n + 1 ) ( 3 n 2 - 3 n + 2 ) = 9 n 4 - 18 n 3 + 18 n 2 - 9 n + 2 2 s={1\over{2}}(3n^{2}-3n+1)(3n^{2}-3n+2)={9n^{4}-18n^{3}+18n^{2}-9n+2\over{2}}
  3. M = s r = 9 n 4 - 18 n 3 + 18 n 2 - 9 n + 2 2 ( 2 n - 1 ) M={s\over{r}}={9n^{4}-18n^{3}+18n^{2}-9n+2\over{2(2n-1)}}
  4. M = ( 9 n 3 4 - 27 n 2 8 + 45 n 16 - 27 32 ) + 5 32 ( 2 n - 1 ) M=\left(\frac{9n^{3}}{4}-\frac{27n^{2}}{8}+\frac{45n}{16}-\frac{27}{32}\right)% +\frac{5}{32\left(2n-1\right)}
  5. 32 M = 72 n 3 - 108 n 2 + 90 n - 27 + 5 2 n - 1 32M=72n^{3}-108n^{2}+90n-27+{5\over 2n-1}
  6. 5 2 n - 1 \frac{5}{2n-1}
  7. n 1 n\geq 1
  8. n = 1 n=1
  9. n = 3 n=3
  10. 6 n 2 6n^{2}
  11. S = 3 n 2 ( 6 n 2 + 1 ) {S}={3n^{2}(6n^{2}+1)}
  12. M = S R = 3 n 2 ( 6 n 2 + 1 ) 2 n M={S\over R}={3n^{2}(6n^{2}+1)\over 2n}
  13. 17 + 20 + 22 + 21 + 2 + 6 + 10 + 14 + 3 + 16 + 12 + 7 {17+20+22+21+2+6+10+14+3+16+12+7}
  14. = 5 + 11 + 19 + 9 + 8 + 13 + 4 + 1 + 24 + 15 + 23 + 18 {=5+11+19+9+8+13+4+1+24+15+23+18}
  15. = 150 {=150}

Magic_series.html

  1. 1 π 3 e ( e n ) n n 3 - 3 5 n 2 + 2 7 n \frac{1}{\pi}\cdot\sqrt{\frac{3}{e}}\cdot\frac{(en)^{n}}{n^{3}-\frac{3}{5}n^{2% }+\frac{2}{7}n}
  2. 1 n 3 ( 1 + 3 5 n + 31 420 n 2 + ) \tfrac{1}{n^{3}}\left(1+\tfrac{3}{5n}+\tfrac{31}{420n^{2}}+\cdots\right)
  3. n 3 - 3 5 n 2 + ( 2 7 + 1 2100 ) n + . n^{3}-\tfrac{3}{5}n^{2}+\left(\tfrac{2}{7}+\tfrac{1}{2100}\right)n+\cdots.

Magnetic_core.html

  1. L = n 2 A L L=n^{2}A_{L}

Magnetoception.html

  1. V i n d = - d ϕ d t V_{ind}=-\frac{d\phi}{dt}

Magnetohydrodynamic_generator.html

  1. 𝐅 = Q ( 𝐯 × 𝐁 ) \mathbf{F}=Q\cdot(\mathbf{v}\times\mathbf{B})

Magnetosonic_wave.html

  1. ω 2 k 2 = c 2 v s 2 + v A 2 c 2 + v A 2 \frac{\omega^{2}}{k^{2}}=c^{2}\,\frac{v_{s}^{2}+v_{A}^{2}}{c^{2}+v_{A}^{2}}

Magnetostatics.html

  1. J \scriptstyle\vec{J}
  2. B = 0 \vec{\nabla}\cdot\vec{B}=0
  3. S B d S = 0 \oint_{S}\vec{B}\cdot\mathrm{d}\vec{S}=0
  4. × H = J \vec{\nabla}\times\vec{H}=\vec{J}
  5. C H d l = I enc \oint_{C}\vec{H}\cdot\mathrm{d}\vec{l}=I_{\mathrm{enc}}
  6. S S
  7. d S \scriptstyle d\vec{S}
  8. 𝐇 \mathbf{H}
  9. C C
  10. l \scriptstyle\vec{l}
  11. I enc \scriptstyle I\text{enc}
  12. J \scriptstyle\vec{J}
  13. D / t \scriptstyle\partial\vec{D}/\partial t
  14. J \scriptstyle\vec{J}
  15. B / t \scriptstyle\partial\vec{B}/\partial t
  16. E \scriptstyle\vec{E}
  17. J \scriptstyle\vec{J}
  18. B = μ 0 4 π I d l × r ^ r 2 \vec{B}=\frac{\mu_{0}}{4\pi}I\int{\frac{\mathrm{d}\vec{l}\times\hat{r}}{r^{2}}}
  19. B \scriptstyle\vec{B}
  20. B = × A , \vec{B}=\nabla\times\vec{A},
  21. A = μ 0 4 π J r d V \vec{A}=\frac{\mu_{0}}{4\pi}\int{\frac{\vec{J}}{r}dV}
  22. J \scriptstyle\vec{J}
  23. B = μ 0 ( M + H ) . \vec{B}=\mu_{0}(\vec{M}+\vec{H}).
  24. × H = 0. \nabla\times\vec{H}=0.
  25. H = - U , \vec{H}=-\nabla U,
  26. U U
  27. 2 U = M . \nabla^{2}U=\nabla\cdot\vec{M}.
  28. M , \scriptstyle\nabla\cdot\vec{M},
  29. ρ M \rho_{M}
  30. J M = × M . \vec{J_{M}}=\nabla\times\vec{M}.

Malfatti_circles.html

  1. n n
  2. n n
  3. n 3 n≤3
  4. a a
  5. b b
  6. c c
  7. x x
  8. y y
  9. z z
  10. x x
  11. a a
  12. y y
  13. b b
  14. z z
  15. c c
  16. a b y x abyx
  17. a c z x aczx
  18. b c z y bczy
  19. x x
  20. y y
  21. z z
  22. a a
  23. b b
  24. c c
  25. r r
  26. s = ( a + b + c ) / 2 s=(a+b+c)/2
  27. d d
  28. e e
  29. f f
  30. a a
  31. b b
  32. c c
  33. r 1 = r 2 ( s - a ) ( s + d - r - e - f ) , r_{1}=\frac{r}{2(s-a)}(s+d-r-e-f),
  34. r 2 = r 2 ( s - b ) ( s + e - r - d - f ) , r_{2}=\frac{r}{2(s-b)}(s+e-r-d-f),
  35. r 3 = r 2 ( s - c ) ( s + f - r - d - e ) . r_{3}=\frac{r}{2(s-c)}(s+f-r-d-e).

Mandelstam_variables.html

  1. diag ( 1 , - 1 , - 1 , - 1 ) \mathrm{diag}(1,-1,-1,-1)
  2. s , t , u s,t,u
  3. s = ( p 1 + p 2 ) 2 = ( p 3 + p 4 ) 2 s=(p_{1}+p_{2})^{2}=(p_{3}+p_{4})^{2}\,
  4. t = ( p 1 - p 3 ) 2 = ( p 2 - p 4 ) 2 t=(p_{1}-p_{3})^{2}=(p_{2}-p_{4})^{2}\,
  5. u = ( p 1 - p 4 ) 2 = ( p 2 - p 3 ) 2 u=(p_{1}-p_{4})^{2}=(p_{2}-p_{3})^{2}\,
  6. s , t , u s,t,u
  7. s , t , u s,t,u
  8. s = ( p 1 + p 2 ) 2 = p 1 2 + p 2 2 + 2 p 1 p 2 2 p 1 p 2 s=(p_{1}+p_{2})^{2}=p_{1}^{2}+p_{2}^{2}+2p_{1}\cdot p_{2}\approx 2p_{1}\cdot p% _{2}\,
  9. p 1 2 = m 1 2 p_{1}^{2}=m_{1}^{2}
  10. p 2 2 = m 2 2 . p_{2}^{2}=m_{2}^{2}.
  11. E 2 = 𝐩 𝐩 + m 0 2 E^{2}=\mathbf{p}\cdot\mathbf{p}+{m_{0}}^{2}
  12. E 2 𝐩 𝐩 E^{2}\approx\mathbf{p}\cdot\mathbf{p}
  13. s s\approx\,
  14. 2 p 1 p 2 2p_{1}\cdot p_{2}\approx\,
  15. 2 p 3 p 4 2p_{3}\cdot p_{4}\,
  16. t t\approx\,
  17. - 2 p 1 p 3 -2p_{1}\cdot p_{3}\approx\,
  18. - 2 p 2 p 4 -2p_{2}\cdot p_{4}\,
  19. u u\approx\,
  20. - 2 p 1 p 4 -2p_{1}\cdot p_{4}\approx\,
  21. - 2 p 3 p 2 -2p_{3}\cdot p_{2}\,
  22. s + t + u = m 1 2 + m 2 2 + m 3 2 + m 4 2 s+t+u=m_{1}^{2}+m_{2}^{2}+m_{3}^{2}+m_{4}^{2}\,
  23. m i m_{i}
  24. i i
  25. p i 2 = m i 2 ( 1 ) p_{i}^{2}=m_{i}^{2}\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(1)\,
  26. p 1 + p 2 = p 3 + p 4 p_{1}+p_{2}=p_{3}+p_{4}\,
  27. p 1 = - p 2 + p 3 + p 4 ( 2 ) p_{1}=-p_{2}+p_{3}+p_{4}\quad\quad\quad\quad\quad\quad\quad(2)\,
  28. s = ( p 1 + p 2 ) 2 = p 1 2 + p 2 2 + 2 p 1 p 2 s=(p_{1}+p_{2})^{2}=p_{1}^{2}+p_{2}^{2}+2p_{1}\cdot p_{2}\,
  29. t = ( p 1 - p 3 ) 2 = p 1 2 + p 3 2 - 2 p 1 p 3 t=(p_{1}-p_{3})^{2}=p_{1}^{2}+p_{3}^{2}-2p_{1}\cdot p_{3}\,
  30. u = ( p 1 - p 4 ) 2 = p 1 2 + p 4 2 - 2 p 1 p 4 u=(p_{1}-p_{4})^{2}=p_{1}^{2}+p_{4}^{2}-2p_{1}\cdot p_{4}\,
  31. s + t + u = m 1 2 + m 2 2 + m 3 2 + m 4 2 + 2 p 1 2 + 2 p 1 p 2 - 2 p 1 p 3 - 2 p 1 p 4 s+t+u=m_{1}^{2}+m_{2}^{2}+m_{3}^{2}+m_{4}^{2}+2p_{1}^{2}+2p_{1}\cdot p_{2}-2p_% {1}\cdot p_{3}-2p_{1}\cdot p_{4}\,
  32. 2 p 1 2 + 2 p 1 p 2 - 2 p 1 p 3 - 2 p 1 p 4 = 2 p 1 ( p 1 + p 2 - p 3 - p 4 ) = 0 2p_{1}^{2}+2p_{1}\cdot p_{2}-2p_{1}\cdot p_{3}-2p_{1}\cdot p_{4}=2p_{1}\cdot(p% _{1}+p_{2}-p_{3}-p_{4})=0\,
  33. s + t + u = m 1 2 + m 2 2 + m 3 2 + m 4 2 s+t+u=m_{1}^{2}+m_{2}^{2}+m_{3}^{2}+m_{4}^{2}\,

Mantel_test.html

  1. n ( n - 1 ) 2 {\frac{n(n-1)}{2}}
  2. n - 1 n-1
  3. n ( n - 1 ) / 2 n(n-1)/2

Mapping_cone_(topology).html

  1. C f C_{f}
  2. C f Cf
  3. f : X Y f\colon X\to Y
  4. C f C_{f}
  5. ( X × I ) Y (X\times I)\sqcup Y
  6. ( x , 0 ) ( x , 0 ) (x,0)\sim(x^{\prime},0)\,
  7. ( x , 1 ) f ( x ) (x,1)\sim f(x)\,
  8. I I
  9. X × I X\times I
  10. f : ( X , x 0 ) ( Y , y 0 ) , f\colon(X,x_{0})\to(Y,y_{0}),
  11. f : x 0 y 0 f\colon x_{0}\mapsto y_{0}
  12. x 0 × I {x_{0}}\times I
  13. ( x 0 , t ) ( x 0 , t ) . (x_{0},t)\sim(x_{0},t^{\prime})\,.
  14. y 0 . y_{0}.
  15. X X
  16. α : S 1 X \alpha\colon S^{1}\to X
  17. i : A X i\colon A\to X
  18. E * ( X , A ) = E * ( X / A , * ) = E ~ * ( X / A ) E_{*}(X,A)=E_{*}(X/A,*)=\tilde{E}_{*}(X/A)
  19. f : X Y f\colon X\rightarrow Y
  20. H * \mathbb{}H_{*}
  21. f : X Y f:X\rightarrow Y
  22. H * \mathbb{}H_{*}
  23. { } C f \{\cdot\}\hookrightarrow C_{f}
  24. H * \mathbb{}H_{*}
  25. H * ( C f , p t ) = 0 \mathbb{}H_{*}(C_{f},pt)=0

Mapping_cylinder.html

  1. f f
  2. X X
  3. Y Y
  4. M f = ( ( [ 0 , 1 ] × X ) Y ) / M_{f}=(([0,1]\times X)\amalg Y)\,/\,\sim
  5. ( 0 , x ) f ( x ) for each x X . (0,x)\sim f(x)\quad\,\text{for each }x\in X.
  6. M f M_{f}
  7. X × [ 0 , 1 ] X\times[0,1]
  8. Y Y
  9. f f
  10. { 1 } × X \{1\}\times X
  11. X X
  12. f ( X ) Y f(X)\subset Y
  13. M f M_{f}
  14. M f Y M_{f}\to Y
  15. Y y y Y M f Y\ni y\mapsto y\in Y\subset M_{f}
  16. R R
  17. R : M f × I M f R:M_{f}\times I\rightarrow M_{f}
  18. ( [ t , x ] , s ) [ s t , x ] ([t,x],s)\mapsto[s\cdot t,x]
  19. Y Y
  20. [ 0 , x ] = [ s 0 , x ] [0,x]=[s\cdot 0,x]
  21. s s
  22. f : X Y f:X\to Y
  23. { 1 } × X \{1\}\times X
  24. M f M_{f}
  25. f : X Y f\colon X\to Y
  26. M f M_{f}
  27. f ~ : X M f \tilde{f}\colon X\to M_{f}
  28. M f Y M_{f}\to Y
  29. M f M_{f}
  30. X M f Y X\to M_{f}\to Y
  31. M f M_{f}
  32. f ~ \tilde{f}
  33. f : X Y f\colon X\to Y
  34. f ~ : X M f \tilde{f}\colon X\to M_{f}
  35. f : X Y f\colon X\rightarrow Y
  36. X Y X\subset Y
  37. f f
  38. X X
  39. Y , Y,
  40. X X
  41. Y , Y,
  42. f : X Y f\colon X\to Y
  43. id X : X X \,\text{id}_{X}\colon X\to X
  44. X 1 f 1 X 2 f 2 X 3 X_{1}\to_{f_{1}}X_{2}\to_{f_{2}}X_{3}\to\cdots
  45. O ( n ) O ( n + 1 ) O(n)\subset O(n+1)
  46. ( i [ 0 , 1 ] × X i ) / ( ( 0 , x i ) ( 1 , f ( x i ) ) ) \Bigl(\coprod_{i}[0,1]\times X_{i}\Bigr)/((0,x_{i})\sim(1,f(x_{i})))

Marginal_revenue.html

  1. R ( q ) = P ( q ) q R(q)=P(q)\cdot q
  2. R ( q ) = P ( q ) + P ( q ) q R^{\prime}(q)=P(q)+P^{\prime}(q)\cdot q
  3. P ( q ) = 0 P^{\prime}(q)=0
  4. P ( q ) < 0 P^{\prime}(q)<0
  5. q q
  6. R = d R d Q R^{\prime}=\frac{dR}{dQ}
  7. R = R + ( d R d Q ) Q R^{\prime}=\langle R\rangle+\left(\frac{d\langle R\rangle}{dQ}\right)Q
  8. R = P \langle R\rangle=P
  9. R = R [ 1 + d R d Q Q R ] R^{\prime}=\langle R\rangle\left[1+\frac{d\langle R\rangle}{dQ}\frac{Q}{% \langle R\rangle}\right]
  10. R = R ( 1 + 1 / e R ) R^{\prime}=\langle R\rangle(1+1/e_{\langle R\rangle})

Marginal_revenue_productivity_theory_of_wages.html

  1. M R P = M P P × AR MRP=MPP\times\,\text{AR}\,\!
  2. M R P = M P P × Price MRP=MPP\times\,\text{Price}\,\!

Mark_and_recapture.html

  1. N = K n k , N=\frac{Kn}{k},
  2. k / K k/K
  3. n / N n/N
  4. k K = n N . \frac{k}{K}=\frac{n}{N}.
  5. N = K n k , N=\frac{Kn}{k},
  6. N = K n k = 10 × 15 5 = 30 N=\frac{Kn}{k}=\frac{10\times 15}{5}=30
  7. N ( K + 1 ) ( n + 1 ) k + 1 - 1 N\approx\frac{(K+1)(n+1)}{k+1}-1
  8. N ( K + 1 ) ( n + 1 ) k + 1 - 1 = 11 × 16 6 - 1 = 28.3 N\approx\frac{(K+1)(n+1)}{k+1}-1=\frac{11\times 16}{6}-1=28.3
  9. var ( N ) = ( K + 1 ) ( n + 1 ) ( K - k ) ( n - k ) ( k + 1 ) ( k + 1 ) ( k + 2 ) . \operatorname{var}(N)=\frac{(K+1)(n+1)(K-k)(n-k)}{(k+1)(k+1)(k+2)}.
  10. N ( K - 1 ) ( n - 1 ) k - 2 ± ( K - 1 ) ( n - 1 ) ( K - k + 1 ) ( n - k + 1 ) ( k - 2 ) ( k - 2 ) ( k - 3 ) N\approx\frac{(K-1)(n-1)}{k-2}\pm\sqrt{\frac{(K-1)(n-1)(K-k+1)(n-k+1)}{(k-2)(k% -2)(k-3)}}

Market_value_added.html

  1. M V A = V - K MVA=V-K

Markov_random_field.html

  1. X u X v X V { u , v } if { u , v } E X_{u}\perp\!\!\!\perp X_{v}\mid X_{V\setminus\{u,v\}}\quad\,\text{if }\{u,v\}\notin E
  2. X v X V cl ( v ) X ne ( v ) X_{v}\perp\!\!\!\perp X_{V\setminus\operatorname{cl}(v)}\mid X_{\operatorname{% ne}(v)}
  3. X A X B X S X_{A}\perp\!\!\!\perp X_{B}\mid X_{S}
  4. P ( X = x ) = C cl ( G ) ϕ C ( x C ) P(X=x)=\prod_{C\in\operatorname{cl}(G)}\phi_{C}(x_{C})
  5. f k f_{k}
  6. P ( X = x ) = 1 Z exp ( k w k f k ( x { k } ) ) P(X=x)=\frac{1}{Z}\exp\left(\sum_{k}w_{k}^{\top}f_{k}(x_{\{k\}})\right)
  7. w k f k ( x { k } ) = i = 1 N k w k , i f k , i ( x { k } ) w_{k}^{\top}f_{k}(x_{\{k\}})=\sum_{i=1}^{N_{k}}w_{k,i}\cdot f_{k,i}(x_{\{k\}})
  8. Z = x 𝒳 exp ( k w k f k ( x { k } ) ) . Z=\sum_{x\in\mathcal{X}}\exp\left(\sum_{k}w_{k}^{\top}f_{k}(x_{\{k\}})\right).
  9. 𝒳 \mathcal{X}
  10. f k , i f_{k,i}
  11. f k , i ( x { k } ) = 1 f_{k,i}(x_{\{k\}})=1
  12. x { k } x_{\{k\}}
  13. N k = | dom ( C k ) | N_{k}=|\operatorname{dom}(C_{k})|
  14. f k , i f_{k,i}
  15. w k , i = log ϕ ( c k , i ) w_{k,i}=\log\phi(c_{k,i})
  16. c k , i c_{k,i}
  17. C k C_{k}
  18. 𝒳 \mathcal{X}
  19. Z [ J ] = x 𝒳 exp ( k w k f k ( x { k } ) + v J v x v ) Z[J]=\sum_{x\in\mathcal{X}}\exp\left(\sum_{k}w_{k}^{\top}f_{k}(x_{\{k\}})+\sum% _{v}J_{v}x_{v}\right)
  20. E [ X v ] = 1 Z Z [ J ] J v | J v = 0 . E[X_{v}]=\frac{1}{Z}\left.\frac{\partial Z[J]}{\partial J_{v}}\right|_{J_{v}=0}.
  21. C [ X u , X v ] = 1 Z 2 Z [ J ] J u J v | J u = 0 , J v = 0 . C[X_{u},X_{v}]=\frac{1}{Z}\left.\frac{\partial^{2}Z[J]}{\partial J_{u}\partial J% _{v}}\right|_{J_{u}=0,J_{v}=0}.
  22. X = ( X v ) v V 𝒩 ( s y m b o l μ , Σ ) X=(X_{v})_{v\in V}\sim\mathcal{N}(symbol\mu,\Sigma)
  23. ( Σ - 1 ) u v = 0 if { u , v } E . (\Sigma^{-1})_{uv}=0\quad\,\text{if}\quad\{u,v\}\notin E.
  24. V = { v 1 , , v i } V^{\prime}=\{v_{1},\ldots,v_{i}\}
  25. W = { w 1 , , w j } W^{\prime}=\{w_{1},\ldots,w_{j}\}
  26. u V , W u\notin V^{\prime},W^{\prime}
  27. o o
  28. ϕ k \phi_{k}
  29. o o

Marshall–Lerner_condition.html

  1. e e
  2. N x = X - Q e N_{x}=X-Qe
  3. N x e = X e - e Q e - Q \frac{\partial N_{x}}{\partial e}=\frac{\partial X}{\partial e}-e\frac{% \partial Q}{\partial e}-Q
  4. N x e 1 X = X e 1 X - e X Q e - Q X \frac{\partial N_{x}}{\partial e}\frac{1}{X}=\frac{\partial X}{\partial e}% \frac{1}{X}-\frac{e}{X}\frac{\partial Q}{\partial e}-\frac{Q}{X}
  5. X = e Q X=eQ
  6. N x e 1 X = X e 1 X - 1 Q Q e - 1 e \frac{\partial N_{x}}{\partial e}\frac{1}{X}=\frac{\partial X}{\partial e}% \frac{1}{X}-\frac{1}{Q}\frac{\partial Q}{\partial e}-\frac{1}{e}
  7. N x e e X = X e e X - Q e e Q - 1 \frac{\partial N_{x}}{\partial e}\frac{e}{X}=\frac{\partial X}{\partial e}% \frac{e}{X}-\frac{\partial Q}{\partial e}\frac{e}{Q}-1
  8. N x e e X = η X e - η Q e - 1 \frac{\partial N_{x}}{\partial e}\frac{e}{X}=\eta_{Xe}-\eta_{Qe}-1
  9. η X e \eta_{Xe}
  10. η Q e \eta_{Qe}
  11. N x N_{x}
  12. η X e - η Q e - 1 > 0 η X e - η Q e > 1 \eta_{Xe}-\eta_{Qe}-1>0\Rightarrow\eta_{Xe}-\eta_{Qe}>1
  13. η X e + | η Q e | > 1 \eta_{Xe}+\left|\eta_{Qe}\right|>1

Martin's_axiom.html

  1. 𝔠 \mathfrak{c}
  2. 0 \aleph_{0}
  3. 𝔠 \mathfrak{c}
  4. 0 \aleph_{0}
  5. 2 0 2^{\aleph_{0}}
  6. 2 0 = 𝔠 2^{\aleph_{0}}=\mathfrak{c}
  7. 1 \aleph_{1}

Mass_deficit.html

  1. M def = 4 π 0 R c [ ρ i ( r ) - ρ ( r ) ] r 2 d r M_{\mathrm{def}}=4\pi\int_{0}^{R_{c}}\left[\rho_{i}(r)-\rho(r)\right]r^{2}dr

Mass_flow_meter.html

  1. Q m = K u - I u ω 2 2 K d 2 τ Q_{m}=\frac{K_{u}-I_{u}\omega^{2}}{2Kd^{2}}\tau

Mathematics_of_general_relativity.html

  1. p \scriptstyle\,p
  2. p \scriptstyle\,p
  3. ( r , s ) \scriptstyle\,(r,\,s)
  4. r \scriptstyle\,r
  5. s \scriptstyle\,s
  6. ( r , s ) \scriptstyle\,(r,\,s)
  7. p \scriptstyle\,p
  8. ( T p ) r M s \scriptstyle\,(T_{p})^{r}{}_{s}M
  9. dim ( T p ) r M s = n r + s \scriptstyle\dim(T_{p})^{r}{}_{s}M\;=\;n^{r+s}
  10. T = T a 1 a r b 1 b s x a 1 x a r d x b 1 d x b s T\;\!=\;\!{T^{a_{1}\ldots a_{r}}}_{{b_{1}}\ldots{b_{s}}}\frac{\partial}{% \partial x^{a_{1}}}\otimes\ldots\otimes\frac{\partial}{\partial x^{a_{r}}}% \otimes dx^{b_{1}}\otimes\ldots\otimes dx^{b_{s}}
  11. x a i \scriptstyle\frac{\partial}{\partial x^{a_{i}}}
  12. d x b j \scriptstyle dx^{b_{j}}
  13. T \scriptstyle T
  14. T a b = T b a \scriptstyle T_{ab}\;=\;T_{ba}
  15. P \scriptstyle P
  16. P a b = - P b a \scriptstyle P_{ab}\;=\;-P_{ba}
  17. d s 2 = g a b d x a d x b ds^{2}=g_{ab}\,dx^{a}\,dx^{b}
  18. g = g a b d x a d x b g=g_{ab}\,dx^{a}\otimes dx^{b}
  19. R = R a b g a b \scriptstyle R\;=\;R^{ab}g_{ab}
  20. K = R a b c d R a b c d \scriptstyle K\;=\;R^{abcd}R_{abcd}
  21. p \scriptstyle p
  22. p \scriptstyle p
  23. U a = x ˙ a \scriptstyle U^{a}\;=\;\dot{x}^{a}
  24. A a = x ¨ a \scriptstyle A^{a}\;=\;\ddot{x}^{a}
  25. J a \scriptstyle\,J^{a}
  26. T a b \scriptstyle\,T^{ab}
  27. F a b \scriptstyle\,F^{ab}
  28. Γ ( T M ) × Γ ( T M ) Γ ( T M ) \scriptstyle\Gamma(TM)\times\Gamma(TM)\;\rightarrow\;\Gamma(TM)
  29. Γ ( T M ) \scriptstyle\,\Gamma(TM)
  30. e i e j = Γ j i k e k \nabla_{e_{i}}e_{j}=\Gamma^{k}_{ji}e_{k}
  31. Γ j i k = Γ i j k \scriptstyle\Gamma^{k}_{ji}\;=\;\Gamma^{k}_{ij}
  32. γ \scriptstyle\gamma
  33. A = γ ( 0 ) \scriptstyle A\;=\;\gamma(0)
  34. B = γ ( t ) \scriptstyle B\;=\;\gamma(t)
  35. X ( t ) = Π 0 , t , γ X ( 0 ) X(t)\,=\Pi_{0,t,\gamma}X(0)
  36. X ( t ) \scriptstyle\,X(t)
  37. d d t X i ( t ) = C ( t ) X i ( t ) = Γ j k i X j ( t ) C k ( t ) \frac{d}{dt}X^{i}(t)=\nabla_{C(t)}X^{i}(t)=\Gamma^{i}_{jk}X^{j}(t)C^{k}(t)
  38. C j ( t ) \scriptstyle\,C^{j}(t)
  39. γ ( t ) \scriptstyle\gamma(t)
  40. X \scriptstyle X
  41. A \scriptstyle\vec{A}
  42. X \scriptstyle X
  43. B \scriptstyle\vec{B}
  44. B \scriptstyle\vec{B}
  45. X \scriptstyle X
  46. A \scriptstyle\vec{A}
  47. γ ( t ) \scriptstyle\gamma\,(t)
  48. X = γ ( 0 ) \scriptstyle X\;=\;\gamma(0)
  49. A = d d t γ ( 0 ) \scriptstyle\vec{A}\;=\;{d\over dt}\gamma(0)
  50. A B ( X ) = lim ϵ 0 1 ϵ [ Π ( ϵ , 0 , γ ) B ( γ [ ϵ ] ) - B ( X ) ] \nabla_{\vec{A}}\vec{B}(X)=\lim_{\epsilon\rightarrow 0}\frac{1}{\epsilon}\left% [\Pi_{(\epsilon,0,\gamma)}\vec{B}(\gamma[\epsilon])-\vec{B}(X)\right]
  51. B \scriptstyle\vec{B}
  52. A \scriptstyle\vec{A}
  53. Π \scriptstyle\,\Pi
  54. Y X = X a Y b ; b x a = ( X a + , b Γ b c a X c ) Y b x a \nabla_{\vec{Y}}\vec{X}=X^{a}{}_{;b}Y^{b}\frac{\partial}{\partial x^{a}}=(X^{a% }{}_{,b}+\Gamma^{a}_{bc}X^{c})Y^{b}\frac{\partial}{\partial x^{a}}
  55. X \scriptstyle X
  56. X \scriptstyle\nabla\vec{X}
  57. X = X a x a ; b d x b = ( X a + , b Γ b c a X c ) x a d x b \nabla\vec{X}=X^{a}{}_{;b}\frac{\partial}{\partial x^{a}}\otimes dx^{b}=(X^{a}% {}_{,b}+\Gamma^{a}_{bc}X^{c})\frac{\partial}{\partial x^{a}}\otimes dx^{b}
  58. D a T d e b c = a T d e b c = T d e ; a b c D_{a}T^{b\dots c}_{d\dots e}=\nabla_{a}T^{b\dots c}_{d\dots e}=T^{b\dots c}_{d% \dots e;a}
  59. a ( X b + Y b ) \displaystyle\nabla_{a}(X^{b}+Y^{b})
  60. a T b = a ( T c g b c ) = g b c a T c \nabla_{a}T^{b}=\nabla_{a}(T_{c}g^{bc})=g^{bc}\nabla_{a}T_{c}
  61. X \scriptstyle\mathcal{L}_{X}
  62. X \scriptstyle X
  63. X ϕ = X a a ϕ = X a ϕ x a \mathcal{L}_{X}\phi=X^{a}\nabla_{a}\phi=X^{a}\frac{\partial\phi}{\partial x^{a}}
  64. X T a b = X c c T a b + ( a X c ) T c b + ( b X c ) T a c = X c T a b , c + X , a c T c b + X , b c T a c \mathcal{L}_{X}T_{ab}=X^{c}\nabla_{c}T_{ab}+(\nabla_{a}X^{c})T_{cb}+(\nabla_{b% }X^{c})T_{ac}=X^{c}T_{ab,c}+X^{c}_{,a}T_{cb}+X^{c}_{,b}T_{ac}
  65. X T a 1 a r = b 1 b s X c ( c T a 1 a r ) b 1 b s - ( c X a 1 ) T c a r - b 1 b s - ( c X a r ) T a 1 a r - 1 c + b 1 b s ( b 1 X c ) T a 1 a r + c b s + ( b s X c ) T a 1 a r b 1 b s - 1 c \begin{aligned}&\displaystyle\mathcal{L}_{X}T^{a_{1}\ldots a_{r}}{}_{b_{1}% \ldots b_{s}}=X^{c}(\nabla_{c}T^{a_{1}\ldots a_{r}}{}_{b_{1}\ldots b_{s}})-\\ &\displaystyle\quad(\nabla_{c}X^{a_{1}})T^{c\ldots a_{r}}{}_{b_{1}\ldots b_{s}% }-\ldots-(\nabla_{c}X^{a_{r}})T^{a_{1}\ldots a_{r-1}c}{}_{b_{1}\ldots b_{s}}+% \\ &\displaystyle\quad(\nabla_{b_{1}}X^{c})T^{a_{1}\ldots a_{r}}{}_{c\ldots b_{s}% }+\ldots+(\nabla_{b_{s}}X^{c})T^{a_{1}\ldots a_{r}}{}_{b_{1}\ldots b_{s-1}c}% \end{aligned}
  66. a \scriptstyle\nabla_{a}
  67. ~ a \scriptstyle\tilde{\nabla}_{a}
  68. a \scriptstyle\partial_{a}
  69. X g a b \displaystyle\mathcal{L}_{X}g_{ab}
  70. μ V ν = μ V ν - Γ ρ V ρ μ ν \nabla_{\mu}V_{\nu}=\partial_{\mu}V_{\nu}-\Gamma^{\rho}{}_{\mu\nu}V_{\rho}
  71. m [ V μ ν ] = m V μ ν - Γ ρ V ρ m ν - Γ ρ V ρ m μ \nabla_{m}[V_{\mu\nu}]=\partial_{m}V_{\mu\nu}-\Gamma^{\rho}{}_{m\nu}V_{\rho}-% \Gamma^{\rho}{}_{m\mu}V_{\rho}
  72. σ , μ V ν = σ [ μ V ν ] = σ [ μ V ν - Γ ρ V ρ μ ν ] \nabla_{\sigma,\mu}V_{\nu}=\nabla_{\sigma}[\nabla_{\mu}V_{\nu}]=\nabla_{\sigma% }[\partial_{\mu}V_{\nu}-\Gamma^{\rho}{}_{\mu\nu}V_{\rho}]
  73. σ [ μ V ν ] - σ [ Γ ρ V ρ μ ν ] \nabla_{\sigma}[\partial_{\mu}V_{\nu}]-\nabla_{\sigma}[\Gamma^{\rho}{}_{\mu\nu% }V_{\rho}]
  74. = [ σ [ μ V ν ] - Γ ρ σ μ ν V ρ - Γ ρ μ σ ν V ρ - Γ ρ ρ σ μ V ν ] - [ σ [ Γ ρ V ρ μ ν ] - Γ α Γ ρ σ ν V ρ α μ - Γ α Γ ρ σ μ V ρ α ν ] =[\partial_{\sigma}[\partial_{\mu}V_{\nu}]-\Gamma^{\rho}{}_{\mu\nu}\partial_{% \sigma}V_{\rho}-\Gamma^{\rho}{}_{\sigma\nu}\partial_{\mu}V_{\rho}-\Gamma^{\rho% }{}_{\sigma\mu}\partial_{\rho}V_{\nu}]-[\partial_{\sigma}[\Gamma^{\rho}{}_{\mu% \nu}V_{\rho}]-\Gamma^{\alpha}{}_{\sigma\nu}\Gamma^{\rho}{}_{\alpha\mu}V_{\rho}% -\Gamma^{\alpha}{}_{\sigma\mu}\Gamma^{\rho}{}_{\alpha\nu}V_{\rho}]
  75. σ , μ V ν \nabla_{\sigma,\mu}V_{\nu}
  76. σ \sigma
  77. μ \mu
  78. μ , σ V ν \nabla_{\mu,\sigma}V_{\nu}
  79. [ ν [ σ V ν ] - Γ ρ σ σ ν V ρ - Γ ρ μ μ ν V ρ - Γ ρ ρ μ σ V ν ] - [ μ [ Γ ρ V ρ σ ν ] - Γ α Γ ρ μ ν V ρ α σ - Γ α Γ ρ μ σ V ρ α ν ] [\partial_{\nu}[\partial_{\sigma}V_{\nu}]-\Gamma^{\rho}{}_{\sigma\nu}\partial_% {\sigma}V_{\rho}-\Gamma^{\rho}{}_{\mu\nu}\partial_{\mu}V_{\rho}-\Gamma^{\rho}{% }_{\mu\sigma}\partial_{\rho}V_{\nu}]-[\partial_{\mu}[\Gamma^{\rho}{}_{\sigma% \nu}V_{\rho}]-\Gamma^{\alpha}{}_{\mu\nu}\Gamma^{\rho}{}_{\alpha\sigma}V_{\rho}% -\Gamma^{\alpha}{}_{\mu\sigma}\Gamma^{\rho}{}_{\alpha\nu}V_{\rho}]
  80. σ [ μ V ν ] - Γ ρ σ μ ν V ρ - Γ ρ μ σ ν V ρ - Γ ρ ρ σ μ V ν - σ [ Γ ρ V ρ μ ν ] + Γ α Γ ρ σ ν V ρ α μ + Γ α Γ ρ σ μ V ρ α ν \partial_{\sigma}[\partial_{\mu}V_{\nu}]-\Gamma^{\rho}{}_{\mu\nu}\partial_{% \sigma}V_{\rho}-\Gamma^{\rho}{}_{\sigma\nu}\partial_{\mu}V_{\rho}-\Gamma^{\rho% }{}_{\sigma\mu}\partial_{\rho}V_{\nu}-\partial_{\sigma}[\Gamma^{\rho}{}_{\mu% \nu}V_{\rho}]+\Gamma^{\alpha}{}_{\sigma\nu}\Gamma^{\rho}{}_{\alpha\mu}V_{\rho}% +\Gamma^{\alpha}{}_{\sigma\mu}\Gamma^{\rho}{}_{\alpha\nu}V_{\rho}
  81. - μ [ σ V ν ] + Γ ρ μ σ ν V ρ + Γ ρ σ μ ν V ρ + Γ ρ ρ μ σ V ν + μ [ Γ ρ V ρ σ ν ] - Γ α Γ ρ μ ν V ρ α σ - Γ α Γ ρ μ σ V ρ α ν μ Γ ρ V ρ σ ν - σ Γ ρ V ρ μ ν + Γ α Γ ρ σ ν V ρ α μ - Γ α Γ ρ μ ν V ρ α σ \frac{-\partial_{\mu}[\partial_{\sigma}V_{\nu}]+\Gamma^{\rho}{}_{\sigma\nu}% \partial_{\mu}V_{\rho}+\Gamma^{\rho}{}_{\mu\nu}\partial_{\sigma}V_{\rho}+% \Gamma^{\rho}{}_{\mu\sigma}\partial_{\rho}V_{\nu}+\partial_{\mu}[\Gamma^{\rho}% {}_{\sigma\nu}V_{\rho}]-\Gamma^{\alpha}{}_{\mu\nu}\Gamma^{\rho}{}_{\alpha% \sigma}V_{\rho}-\Gamma^{\alpha}{}_{\mu\sigma}\Gamma^{\rho}{}_{\alpha\nu}V_{% \rho}}{\partial_{\mu}\Gamma^{\rho}{}_{\sigma\nu}V_{\rho}-\partial_{\sigma}% \Gamma^{\rho}{}_{\mu\nu}V_{\rho}+\Gamma^{\alpha}{}_{\sigma\nu}\Gamma^{\rho}{}_% {\alpha\mu}V_{\rho}-\Gamma^{\alpha}{}_{\mu\nu}\Gamma^{\rho}{}_{\alpha\sigma}V_% {\rho}}
  82. V ρ V_{\rho}
  83. ( μ Γ ρ - σ ν σ Γ ρ + μ ν Γ α Γ ρ σ ν - α μ Γ α Γ ρ μ ν ) α σ V ρ (\partial_{\mu}\Gamma^{\rho}{}_{\sigma\nu}-\partial_{\sigma}\Gamma^{\rho}{}_{% \mu\nu}+\Gamma^{\alpha}{}_{\sigma\nu}\Gamma^{\rho}{}_{\alpha\mu}-\Gamma^{% \alpha}{}_{\mu\nu}\Gamma^{\rho}{}_{\alpha\sigma})V_{\rho}
  84. σ μ V ν - μ σ V ν = ( μ Γ ρ - σ ν σ Γ ρ + μ ν Γ α Γ ρ σ ν - α μ Γ α Γ ρ μ ν ) α σ V ρ \nabla_{\sigma}\nabla_{\mu}V_{\nu}-\nabla_{\mu}\nabla_{\sigma}V_{\nu}=(% \partial_{\mu}\Gamma^{\rho}{}_{\sigma\nu}-\partial_{\sigma}\Gamma^{\rho}{}_{% \mu\nu}+\Gamma^{\alpha}{}_{\sigma\nu}\Gamma^{\rho}{}_{\alpha\mu}-\Gamma^{% \alpha}{}_{\mu\nu}\Gamma^{\rho}{}_{\alpha\sigma})V_{\rho}
  85. R σ μ ν ρ V ρ = ( μ Γ ρ - σ ν σ Γ ρ + μ ν Γ α Γ ρ σ ν - α μ Γ α Γ ρ μ ν ) α σ V ρ R^{\rho}_{\sigma\mu\nu}V_{\rho}=(\partial_{\mu}\Gamma^{\rho}{}_{\sigma\nu}-% \partial_{\sigma}\Gamma^{\rho}{}_{\mu\nu}+\Gamma^{\alpha}{}_{\sigma\nu}\Gamma^% {\rho}{}_{\alpha\mu}-\Gamma^{\alpha}{}_{\mu\nu}\Gamma^{\rho}{}_{\alpha\sigma})% V_{\rho}
  86. R σ μ ν ρ = μ Γ ρ - σ ν σ Γ ρ + μ ν Γ α Γ ρ σ ν - α μ Γ α Γ ρ μ ν α σ R^{\rho}_{\sigma\mu\nu}=\partial_{\mu}\Gamma^{\rho}{}_{\sigma\nu}-\partial_{% \sigma}\Gamma^{\rho}{}_{\mu\nu}+\Gamma^{\alpha}{}_{\sigma\nu}\Gamma^{\rho}{}_{% \alpha\mu}-\Gamma^{\alpha}{}_{\mu\nu}\Gamma^{\rho}{}_{\alpha\sigma}
  87. g ρ λ R σ μ ν λ = R ρ σ μ ν g_{\rho\lambda}R^{\lambda}_{\sigma\mu\nu}=R_{\rho\sigma\mu\nu}
  88. g ρ μ R ρ σ μ ν = R σ ν g^{\rho\mu}R_{\rho\sigma\mu\nu}=R_{\sigma\nu}
  89. ρ \rho
  90. μ \mu
  91. g σ ν R σ ν = R g^{\sigma\nu}R_{\sigma\nu}=R
  92. R σ μ ν ρ R^{\rho}_{\sigma\mu\nu}
  93. R ρ σ μ ν R_{\rho\sigma\mu\nu}
  94. R σ ν R_{\sigma\nu}
  95. R R
  96. T a b = ; b 0 T^{ab}{}_{;b}\,=0
  97. T a b = , b 0 T^{ab}{}_{,b}\,=0
  98. G a b + Λ g a b = 8 π G c 4 T a b G_{ab}+\Lambda g_{ab}={8\pi G\over c^{4}}T_{ab}
  99. G a b \scriptstyle G_{ab}
  100. Λ \scriptstyle\Lambda
  101. c \scriptstyle c
  102. G \scriptstyle G
  103. U \scriptstyle\vec{U}
  104. U U = 0 \scriptstyle\nabla_{\vec{U}}\vec{U}\;=\;0
  105. x a \scriptstyle x^{a}
  106. U a = d x a d τ \scriptstyle U^{a}=\frac{dx^{a}}{d\tau}
  107. x ¨ a + Γ a b c x ˙ b x ˙ c = 0 \ddot{x}^{a}+{\Gamma^{a}}_{bc}\,\dot{x}^{b}\,\dot{x}^{c}=0
  108. ˙ \scriptstyle\dot{}
  109. d / d τ \scriptstyle d/d\tau

Mathieu_function.html

  1. d 2 y d x 2 + [ a - 2 q cos ( 2 x ) ] y = 0. \frac{d^{2}y}{dx^{2}}+[a-2q\cos(2x)]y=0.
  2. d 2 y d u 2 - [ a - 2 q cosh ( 2 u ) ] y = 0 \frac{d^{2}y}{du^{2}}-[a-2q\cosh(2u)]y=0
  3. u = i x u=ix
  4. t = cos ( x ) t=\cos(x)
  5. ( 1 - t 2 ) d 2 y d t 2 - t d y d t + ( a + 2 q ( 1 - 2 t 2 ) ) y = 0. (1-t^{2})\frac{d^{2}y}{dt^{2}}-t\,\frac{dy}{dt}+(a+2q(1-2t^{2}))\,y=0.
  6. t = - 1 , 1 t=-1,1
  7. F ( a , q , x ) = exp ( i μ x ) P ( a , q , x ) F(a,q,x)=\exp(i\mu\,x)\,P(a,q,x)
  8. μ \mu
  9. x x
  10. π \pi
  11. a = 1 , q = 1 5 , μ 1 + 0.0995 i a=1,\,q=\frac{1}{5},\,\mu\approx 1+0.0995i
  12. C ( a , q , x ) C(a,q,x)
  13. x x
  14. C ( a , q , 0 ) = 1 C(a,q,0)=1
  15. C ( a , q , 0 ) = 0 C^{\prime}(a,q,0)=0
  16. S ( a , q , x ) S(a,q,x)
  17. S ( a , q , 0 ) = 1 S^{\prime}(a,q,0)=1
  18. S ( a , q , 0 ) = 0 S(a,q,0)=0
  19. C ( a , q , x ) = F ( a , q , x ) + F ( a , q , - x ) 2 F ( a , q , 0 ) C(a,q,x)=\frac{F(a,q,x)+F(a,q,-x)}{2F(a,q,0)}
  20. S ( a , q , x ) = F ( a , q , x ) - F ( a , q , - x ) 2 F ( a , q , 0 ) . S(a,q,x)=\frac{F(a,q,x)-F(a,q,-x)}{2F^{\prime}(a,q,0)}.
  21. C ( a , 0 , x ) = cos ( a x ) , S ( a , 0 , x ) = sin ( a x ) a , C(a,0,x)=\cos(\sqrt{a}x),\;S(a,0,x)=\frac{\sin(\sqrt{a}x)}{\sqrt{a}},
  22. C ( a , q , x ) cos ( a x ) , S ( a , q , x ) sin ( a x ) a . C(a,q,x)\approx\cos(\sqrt{a}x),\;\;S(a,q,x)\approx\frac{\sin(\sqrt{a}x)}{\sqrt% {a}}.
  23. q q
  24. a a
  25. 2 π 2\pi
  26. a n ( q ) , b n ( q ) a_{n}(q),\,b_{n}(q)
  27. C E ( n , q , x ) , S E ( n , q , x ) CE(n,q,x),\,SE(n,q,x)
  28. π \pi
  29. C ( a n ( q ) , q , x ) = C E ( n , q , x ) C E ( n , q , 0 ) C\left(a_{n}(q),q,x\right)=\frac{CE(n,q,x)}{CE(n,q,0)}
  30. S ( b n ( q ) , q , x ) = S E ( n , q , x ) S E ( n , q , 0 ) . S\left(b_{n}(q),q,x\right)=\frac{SE(n,q,x)}{SE^{\prime}(n,q,0)}.
  31. C E ( 1 , 1 , x ) CE(1,1,x)

Matrix_calculus.html

  1. f ( x 1 , x 2 , x 3 ) f(x_{1},x_{2},x_{3})
  2. f = f x 1 x 1 ^ + f x 2 x 2 ^ + f x 3 x 3 ^ \nabla f=\frac{\partial f}{\partial x_{1}}\hat{x_{1}}+\frac{\partial f}{% \partial x_{2}}\hat{x_{2}}+\frac{\partial f}{\partial x_{3}}\hat{x_{3}}
  3. x i ^ \hat{x_{i}}
  4. x i x_{i}
  5. 1 i 3 1\leq i\leq 3
  6. 𝐱 \mathbf{x}
  7. f = f 𝐱 = [ f x 1 f x 2 f x 3 ] . \nabla f=\frac{\partial f}{\partial\mathbf{x}}=\begin{bmatrix}\frac{\partial f% }{\partial x_{1}}&\frac{\partial f}{\partial x_{2}}&\frac{\partial f}{\partial x% _{3}}\\ \end{bmatrix}.
  8. y x \frac{\partial y}{\partial x}
  9. 𝐲 x \frac{\partial\mathbf{y}}{\partial x}
  10. 𝐘 x \frac{\partial\mathbf{Y}}{\partial x}
  11. y 𝐱 \frac{\partial y}{\partial\mathbf{x}}
  12. 𝐲 𝐱 \frac{\partial\mathbf{y}}{\partial\mathbf{x}}
  13. y 𝐗 \frac{\partial y}{\partial\mathbf{X}}
  14. 𝐲 = [ y 1 y 2 y m ] \mathbf{y}=\begin{bmatrix}y_{1}\\ y_{2}\\ \vdots\\ y_{m}\\ \end{bmatrix}
  15. 𝐲 x = [ y 1 x y 2 x y m x ] . \frac{\partial\mathbf{y}}{\partial x}=\begin{bmatrix}\frac{\partial y_{1}}{% \partial x}\\ \frac{\partial y_{2}}{\partial x}\\ \vdots\\ \frac{\partial y_{m}}{\partial x}\\ \end{bmatrix}.
  16. 𝐲 x \frac{\partial\mathbf{y}}{\partial x}
  17. \rightarrow
  18. 𝐱 = [ x 1 x 2 x n ] \mathbf{x}=\begin{bmatrix}x_{1}\\ x_{2}\\ \vdots\\ x_{n}\\ \end{bmatrix}
  19. y 𝐱 = [ y x 1 y x 2 y x n ] . \frac{\partial y}{\partial\mathbf{x}}=\left[\frac{\partial y}{\partial x_{1}}% \ \ \frac{\partial y}{\partial x_{2}}\ \ \cdots\ \ \frac{\partial y}{\partial x% _{n}}\right].
  20. u f ( x ) = f ( x ) u \nabla_{{u}}{f}({x})=\nabla f({x})\cdot{u}
  21. 𝐮 f = f 𝐱 𝐮 . \nabla_{\mathbf{u}}f=\frac{\partial f}{\partial\mathbf{x}}\mathbf{u}.
  22. 𝐲 = [ y 1 y 2 y m ] \mathbf{y}=\begin{bmatrix}y_{1}\\ y_{2}\\ \vdots\\ y_{m}\\ \end{bmatrix}
  23. 𝐱 = [ x 1 x 2 x n ] \mathbf{x}=\begin{bmatrix}x_{1}\\ x_{2}\\ \vdots\\ x_{n}\\ \end{bmatrix}
  24. 𝐲 𝐱 = [ y 1 x 1 y 1 x 2 y 1 x n y 2 x 1 y 2 x 2 y 2 x n y m x 1 y m x 2 y m x n ] . \frac{\partial\mathbf{y}}{\partial\mathbf{x}}=\begin{bmatrix}\frac{\partial y_% {1}}{\partial x_{1}}&\frac{\partial y_{1}}{\partial x_{2}}&\cdots&\frac{% \partial y_{1}}{\partial x_{n}}\\ \frac{\partial y_{2}}{\partial x_{1}}&\frac{\partial y_{2}}{\partial x_{2}}&% \cdots&\frac{\partial y_{2}}{\partial x_{n}}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{\partial y_{m}}{\partial x_{1}}&\frac{\partial y_{m}}{\partial x_{2}}&% \cdots&\frac{\partial y_{m}}{\partial x_{n}}\\ \end{bmatrix}.
  25. d 𝐟 ( 𝐯 ) = 𝐟 𝐱 𝐯 . d\,\mathbf{f}(\mathbf{v})=\frac{\partial\mathbf{f}}{\partial\mathbf{x}}\mathbf% {v}.
  26. 𝐘 x = [ y 11 x y 12 x y 1 n x y 21 x y 22 x y 2 n x y m 1 x y m 2 x y m n x ] . \frac{\partial\mathbf{Y}}{\partial x}=\begin{bmatrix}\frac{\partial y_{11}}{% \partial x}&\frac{\partial y_{12}}{\partial x}&\cdots&\frac{\partial y_{1n}}{% \partial x}\\ \frac{\partial y_{21}}{\partial x}&\frac{\partial y_{22}}{\partial x}&\cdots&% \frac{\partial y_{2n}}{\partial x}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{\partial y_{m1}}{\partial x}&\frac{\partial y_{m2}}{\partial x}&\cdots&% \frac{\partial y_{mn}}{\partial x}\\ \end{bmatrix}.
  27. y 𝐗 = [ y x 11 y x 21 y x p 1 y x 12 y x 22 y x p 2 y x 1 q y x 2 q y x p q ] . \frac{\partial y}{\partial\mathbf{X}}=\begin{bmatrix}\frac{\partial y}{% \partial x_{11}}&\frac{\partial y}{\partial x_{21}}&\cdots&\frac{\partial y}{% \partial x_{p1}}\\ \frac{\partial y}{\partial x_{12}}&\frac{\partial y}{\partial x_{22}}&\cdots&% \frac{\partial y}{\partial x_{p2}}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{\partial y}{\partial x_{1q}}&\frac{\partial y}{\partial x_{2q}}&\cdots&% \frac{\partial y}{\partial x_{pq}}\\ \end{bmatrix}.
  28. 𝐗 y ( 𝐗 ) = y ( 𝐗 ) 𝐗 \nabla_{\mathbf{X}}y(\mathbf{X})=\frac{\partial y(\mathbf{X})}{\partial\mathbf% {X}}
  29. 𝐘 f = tr ( f 𝐗 𝐘 ) . \nabla_{\mathbf{Y}}f=\operatorname{tr}\left(\frac{\partial f}{\partial\mathbf{% X}}\mathbf{Y}\right).
  30. \to
  31. 𝐅 𝐗 = [ 𝐅 X 1 , 1 𝐅 X n , 1 𝐅 X 1 , m 𝐅 X n , m ] , \frac{\partial\mathbf{F}}{\partial\mathbf{X}}=\begin{bmatrix}\frac{\partial% \mathbf{F}}{\partial X_{1,1}}&\cdots&\frac{\partial\mathbf{F}}{\partial X_{n,1% }}\\ \vdots&\ddots&\vdots\\ \frac{\partial\mathbf{F}}{\partial X_{1,m}}&\cdots&\frac{\partial\mathbf{F}}{% \partial X_{n,m}}\\ \end{bmatrix},
  32. 𝐅 𝐗 i j \frac{\partial\mathbf{F}}{\partial\mathbf{X}_{ij}}
  33. d 𝐅 ( 𝐘 ) = tr ( 𝐅 𝐗 𝐘 ) , d\mathbf{F}(\mathbf{Y})=\operatorname{tr}\left(\frac{\partial\mathbf{F}}{% \partial\mathbf{X}}\mathbf{Y}\right),
  34. ϕ \phi
  35. n × m n\times m
  36. 𝐗 = ( x i , j ) \mathbf{X}=(x_{i,j})
  37. ϕ ( 𝐗 ) 𝐗 = [ ϕ x 1 , 1 ϕ x 1 , q ϕ x n , 1 ϕ x n , q ] \frac{\partial\mathbf{\phi}(\mathbf{X})}{\partial\mathbf{X}}=\begin{bmatrix}% \frac{\partial\mathbf{\phi}}{\partial x_{1,1}}&\cdots&\frac{\partial\mathbf{% \phi}}{\partial x_{1,q}}\\ \vdots&\ddots&\vdots\\ \frac{\partial\mathbf{\phi}}{\partial x_{n,1}}&\cdots&\frac{\partial\mathbf{% \phi}}{\partial x_{n,q}}\\ \end{bmatrix}
  38. 𝐅 = ( f s , t ) \mathbf{F}=(f_{s,t})
  39. m × n m\times n
  40. n × m n\times m
  41. 𝐗 \mathbf{X}
  42. 𝐅 ( 𝐗 ) 𝐗 = [ f 1 , 1 𝐗 f 1 , p 𝐗 f m , 1 𝐗 f m , p 𝐗 ] \frac{\partial\mathbf{F}(\mathbf{X})}{\partial\mathbf{X}}=\begin{bmatrix}\frac% {\partial f_{1,1}}{\partial\mathbf{X}}&\cdots&\frac{\partial f_{1,p}}{\partial% \mathbf{X}}\\ \vdots&\ddots&\vdots\\ \frac{\partial f_{m,1}}{\partial\mathbf{X}}&\cdots&\frac{\partial f_{m,p}}{% \partial\mathbf{X}}\\ \end{bmatrix}
  43. D 𝐅 ( 𝐗 ) = vec 𝐅 ( 𝐗 ) ( vec 𝐗 ) . \mathrm{D}\,\mathbf{F}\left(\mathbf{X}\right)=\frac{\partial\,\mathrm{vec}\ % \mathbf{F}\left(\mathbf{X}\right)}{\partial\left(\mathrm{vec}\ \mathbf{X}% \right)^{\prime}}.
  44. 𝐲 𝐱 \frac{\partial\mathbf{y}}{\partial\mathbf{x}}
  45. y 𝐱 , \frac{\partial y}{\partial\mathbf{x}},
  46. 𝐲 𝐱 , \frac{\partial\mathbf{y}}{\partial\mathbf{x}^{\prime}},
  47. y 𝐱 \frac{\partial y}{\partial\mathbf{x}}
  48. 𝐲 x , \frac{\partial\mathbf{y}}{\partial x},
  49. 𝐲 𝐱 , \frac{\partial\mathbf{y}}{\partial\mathbf{x}},
  50. y 𝐱 \frac{\partial y}{\partial\mathbf{x}}
  51. 𝐲 x \frac{\partial\mathbf{y}}{\partial x}
  52. 𝐲 𝐱 , \frac{\partial\mathbf{y}}{\partial\mathbf{x}},
  53. y 𝐱 \frac{\partial y}{\partial\mathbf{x}}
  54. 𝐲 x \frac{\partial\mathbf{y}}{\partial x}
  55. y 𝐱 \frac{\partial y}{\partial\mathbf{x}^{\prime}}
  56. 𝐲 x , \frac{\partial\mathbf{y}}{\partial x},
  57. 𝐲 𝐱 . \frac{\partial\mathbf{y}}{\partial\mathbf{x}}.
  58. y 𝐗 \frac{\partial y}{\partial\mathbf{X}}
  59. 𝐘 x , \frac{\partial\mathbf{Y}}{\partial x},
  60. 𝐘 x , \frac{\partial\mathbf{Y}}{\partial x},
  61. 𝐘 x \frac{\partial\mathbf{Y}}{\partial x}
  62. y 𝐗 \frac{\partial y}{\partial\mathbf{X}}
  63. 𝐘 x \frac{\partial\mathbf{Y}}{\partial x}
  64. y 𝐗 \frac{\partial y}{\partial\mathbf{X}}
  65. y 𝐗 , \frac{\partial y}{\partial\mathbf{X}^{\prime}},
  66. y 𝐱 , 𝐲 x , 𝐲 𝐱 , y 𝐗 \frac{\partial y}{\partial\mathbf{x}},\frac{\partial\mathbf{y}}{\partial x},% \frac{\partial\mathbf{y}}{\partial\mathbf{x}},\frac{\partial y}{\partial% \mathbf{X}}
  67. 𝐘 x \frac{\partial\mathbf{Y}}{\partial x}
  68. y x \frac{\partial y}{\partial x}
  69. 𝐲 x \frac{\partial\mathbf{y}}{\partial x}
  70. 𝐘 x \frac{\partial\mathbf{Y}}{\partial x}
  71. y 𝐱 \frac{\partial y}{\partial\mathbf{x}}
  72. 𝐲 𝐱 \frac{\partial\mathbf{y}}{\partial\mathbf{x}}
  73. 𝐘 𝐱 \frac{\partial\mathbf{Y}}{\partial\mathbf{x}}
  74. y 𝐗 \frac{\partial y}{\partial\mathbf{X}}
  75. 𝐲 𝐗 \frac{\partial\mathbf{y}}{\partial\mathbf{X}}
  76. 𝐘 𝐗 \frac{\partial\mathbf{Y}}{\partial\mathbf{X}}
  77. y 𝐱 = [ y x 1 y x 2 y x n ] . \frac{\partial y}{\partial\mathbf{x}}=\left[\frac{\partial y}{\partial x_{1}}% \frac{\partial y}{\partial x_{2}}\cdots\frac{\partial y}{\partial x_{n}}\right].
  78. 𝐲 x = [ y 1 x y 2 x y m x ] . \frac{\partial\mathbf{y}}{\partial x}=\begin{bmatrix}\frac{\partial y_{1}}{% \partial x}\\ \frac{\partial y_{2}}{\partial x}\\ \vdots\\ \frac{\partial y_{m}}{\partial x}\\ \end{bmatrix}.
  79. 𝐲 𝐱 = [ y 1 x 1 y 1 x 2 y 1 x n y 2 x 1 y 2 x 2 y 2 x n y m x 1 y m x 2 y m x n ] . \frac{\partial\mathbf{y}}{\partial\mathbf{x}}=\begin{bmatrix}\frac{\partial y_% {1}}{\partial x_{1}}&\frac{\partial y_{1}}{\partial x_{2}}&\cdots&\frac{% \partial y_{1}}{\partial x_{n}}\\ \frac{\partial y_{2}}{\partial x_{1}}&\frac{\partial y_{2}}{\partial x_{2}}&% \cdots&\frac{\partial y_{2}}{\partial x_{n}}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{\partial y_{m}}{\partial x_{1}}&\frac{\partial y_{m}}{\partial x_{2}}&% \cdots&\frac{\partial y_{m}}{\partial x_{n}}\\ \end{bmatrix}.
  80. y 𝐗 = [ y x 11 y x 21 y x p 1 y x 12 y x 22 y x p 2 y x 1 q y x 2 q y x p q ] . \frac{\partial y}{\partial\mathbf{X}}=\begin{bmatrix}\frac{\partial y}{% \partial x_{11}}&\frac{\partial y}{\partial x_{21}}&\cdots&\frac{\partial y}{% \partial x_{p1}}\\ \frac{\partial y}{\partial x_{12}}&\frac{\partial y}{\partial x_{22}}&\cdots&% \frac{\partial y}{\partial x_{p2}}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{\partial y}{\partial x_{1q}}&\frac{\partial y}{\partial x_{2q}}&\cdots&% \frac{\partial y}{\partial x_{pq}}\\ \end{bmatrix}.
  81. 𝐘 x = [ y 11 x y 12 x y 1 n x y 21 x y 22 x y 2 n x y m 1 x y m 2 x y m n x ] . \frac{\partial\mathbf{Y}}{\partial x}=\begin{bmatrix}\frac{\partial y_{11}}{% \partial x}&\frac{\partial y_{12}}{\partial x}&\cdots&\frac{\partial y_{1n}}{% \partial x}\\ \frac{\partial y_{21}}{\partial x}&\frac{\partial y_{22}}{\partial x}&\cdots&% \frac{\partial y_{2n}}{\partial x}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{\partial y_{m1}}{\partial x}&\frac{\partial y_{m2}}{\partial x}&\cdots&% \frac{\partial y_{mn}}{\partial x}\\ \end{bmatrix}.
  82. d 𝐗 = [ d x 11 d x 12 d x 1 n d x 21 d x 22 d x 2 n d x m 1 d x m 2 d x m n ] . d\mathbf{X}=\begin{bmatrix}dx_{11}&dx_{12}&\cdots&dx_{1n}\\ dx_{21}&dx_{22}&\cdots&dx_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ dx_{m1}&dx_{m2}&\cdots&dx_{mn}\\ \end{bmatrix}.
  83. y 𝐱 = [ y x 1 y x 2 y x n ] . \frac{\partial y}{\partial\mathbf{x}}=\begin{bmatrix}\frac{\partial y}{% \partial x_{1}}\\ \frac{\partial y}{\partial x_{2}}\\ \vdots\\ \frac{\partial y}{\partial x_{n}}\\ \end{bmatrix}.
  84. 𝐲 x = [ y 1 x y 2 x y m x ] . \frac{\partial\mathbf{y}}{\partial x}=\left[\frac{\partial y_{1}}{\partial x}% \frac{\partial y_{2}}{\partial x}\cdots\frac{\partial y_{m}}{\partial x}\right].
  85. 𝐲 𝐱 = [ y 1 x 1 y 2 x 1 y m x 1 y 1 x 2 y 2 x 2 y m x 2 y 1 x n y 2 x n y m x n ] . \frac{\partial\mathbf{y}}{\partial\mathbf{x}}=\begin{bmatrix}\frac{\partial y_% {1}}{\partial x_{1}}&\frac{\partial y_{2}}{\partial x_{1}}&\cdots&\frac{% \partial y_{m}}{\partial x_{1}}\\ \frac{\partial y_{1}}{\partial x_{2}}&\frac{\partial y_{2}}{\partial x_{2}}&% \cdots&\frac{\partial y_{m}}{\partial x_{2}}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{\partial y_{1}}{\partial x_{n}}&\frac{\partial y_{2}}{\partial x_{n}}&% \cdots&\frac{\partial y_{m}}{\partial x_{n}}\\ \end{bmatrix}.
  86. y 𝐗 = [ y x 11 y x 12 y x 1 q y x 21 y x 22 y x 2 q y x p 1 y x p 2 y x p q ] . \frac{\partial y}{\partial\mathbf{X}}=\begin{bmatrix}\frac{\partial y}{% \partial x_{11}}&\frac{\partial y}{\partial x_{12}}&\cdots&\frac{\partial y}{% \partial x_{1q}}\\ \frac{\partial y}{\partial x_{21}}&\frac{\partial y}{\partial x_{22}}&\cdots&% \frac{\partial y}{\partial x_{2q}}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{\partial y}{\partial x_{p1}}&\frac{\partial y}{\partial x_{p2}}&\cdots&% \frac{\partial y}{\partial x_{pq}}\\ \end{bmatrix}.
  87. 𝐲 𝐱 \frac{\partial\mathbf{y}}{\partial\mathbf{x}}
  88. 𝐚 𝐱 = \frac{\partial\mathbf{a}}{\partial\mathbf{x}}=
  89. 𝟎 \mathbf{0}
  90. 𝐱 𝐱 = \frac{\partial\mathbf{x}}{\partial\mathbf{x}}=
  91. 𝐈 \mathbf{I}
  92. 𝐀𝐱 𝐱 = \frac{\partial\mathbf{A}\mathbf{x}}{\partial\mathbf{x}}=
  93. 𝐀 \mathbf{A}
  94. 𝐀 \mathbf{A}^{\top}
  95. 𝐱 𝐀 𝐱 = \frac{\partial\mathbf{x}^{\top}\mathbf{A}}{\partial\mathbf{x}}=
  96. 𝐀 \mathbf{A}^{\top}
  97. 𝐀 \mathbf{A}
  98. a 𝐮 𝐱 = \frac{\partial a\mathbf{u}}{\partial\,\mathbf{x}}=
  99. a 𝐮 𝐱 a\frac{\partial\mathbf{u}}{\partial\mathbf{x}}
  100. a 𝐮 𝐱 = \frac{\partial a\mathbf{u}}{\partial\mathbf{x}}=
  101. a 𝐮 𝐱 + 𝐮 a 𝐱 a\frac{\partial\mathbf{u}}{\partial\mathbf{x}}+\mathbf{u}\frac{\partial a}{% \partial\mathbf{x}}
  102. a 𝐮 𝐱 + a 𝐱 𝐮 a\frac{\partial\mathbf{u}}{\partial\mathbf{x}}+\frac{\partial a}{\partial% \mathbf{x}}\mathbf{u}^{\top}
  103. 𝐀𝐮 𝐱 = \frac{\partial\mathbf{A}\mathbf{u}}{\partial\mathbf{x}}=
  104. 𝐀 𝐮 𝐱 \mathbf{A}\frac{\partial\mathbf{u}}{\partial\mathbf{x}}
  105. 𝐮 𝐱 𝐀 \frac{\partial\mathbf{u}}{\partial\mathbf{x}}\mathbf{A}^{\top}
  106. ( 𝐮 + 𝐯 ) 𝐱 = \frac{\partial(\mathbf{u}+\mathbf{v})}{\partial\mathbf{x}}=
  107. 𝐮 𝐱 + 𝐯 𝐱 \frac{\partial\mathbf{u}}{\partial\mathbf{x}}+\frac{\partial\mathbf{v}}{% \partial\mathbf{x}}
  108. 𝐠 ( 𝐮 ) 𝐱 = \frac{\partial\mathbf{g(u)}}{\partial\mathbf{x}}=
  109. 𝐠 ( 𝐮 ) 𝐮 𝐮 𝐱 \frac{\partial\mathbf{g(u)}}{\partial\mathbf{u}}\frac{\partial\mathbf{u}}{% \partial\mathbf{x}}
  110. 𝐮 𝐱 𝐠 ( 𝐮 ) 𝐮 \frac{\partial\mathbf{u}}{\partial\mathbf{x}}\frac{\partial\mathbf{g(u)}}{% \partial\mathbf{u}}
  111. 𝐟 ( 𝐠 ( 𝐮 ) ) 𝐱 = \frac{\partial\mathbf{f(g(u))}}{\partial\mathbf{x}}=
  112. 𝐟 ( 𝐠 ) 𝐠 𝐠 ( 𝐮 ) 𝐮 𝐮 𝐱 \frac{\partial\mathbf{f(g)}}{\partial\mathbf{g}}\frac{\partial\mathbf{g(u)}}{% \partial\mathbf{u}}\frac{\partial\mathbf{u}}{\partial\mathbf{x}}
  113. 𝐮 𝐱 𝐠 ( 𝐮 ) 𝐮 𝐟 ( 𝐠 ) 𝐠 \frac{\partial\mathbf{u}}{\partial\mathbf{x}}\frac{\partial\mathbf{g(u)}}{% \partial\mathbf{u}}\frac{\partial\mathbf{f(g)}}{\partial\mathbf{g}}
  114. y 𝐱 = 𝐱 y \frac{\partial y}{\partial\mathbf{x}}=\nabla_{\mathbf{x}}y
  115. a 𝐱 = \frac{\partial a}{\partial\mathbf{x}}=
  116. 𝟎 \mathbf{0}^{\top}
  117. 𝟎 \mathbf{0}
  118. a u 𝐱 = \frac{\partial au}{\partial\mathbf{x}}=
  119. a u 𝐱 a\frac{\partial u}{\partial\mathbf{x}}
  120. ( u + v ) 𝐱 = \frac{\partial(u+v)}{\partial\mathbf{x}}=
  121. u 𝐱 + v 𝐱 \frac{\partial u}{\partial\mathbf{x}}+\frac{\partial v}{\partial\mathbf{x}}
  122. u v 𝐱 = \frac{\partial uv}{\partial\mathbf{x}}=
  123. u v 𝐱 + v u 𝐱 u\frac{\partial v}{\partial\mathbf{x}}+v\frac{\partial u}{\partial\mathbf{x}}
  124. g ( u ) 𝐱 = \frac{\partial g(u)}{\partial\mathbf{x}}=
  125. g ( u ) u u 𝐱 \frac{\partial g(u)}{\partial u}\frac{\partial u}{\partial\mathbf{x}}
  126. f ( g ( u ) ) 𝐱 = \frac{\partial f(g(u))}{\partial\mathbf{x}}=
  127. f ( g ) g g ( u ) u u 𝐱 \frac{\partial f(g)}{\partial g}\frac{\partial g(u)}{\partial u}\frac{\partial u% }{\partial\mathbf{x}}
  128. ( 𝐮 𝐯 ) 𝐱 = 𝐮 𝐯 𝐱 = \frac{\partial(\mathbf{u}\cdot\mathbf{v})}{\partial\mathbf{x}}=\frac{\partial% \mathbf{u}^{\top}\mathbf{v}}{\partial\mathbf{x}}=
  129. 𝐮 𝐯 𝐱 + 𝐯 𝐮 𝐱 \mathbf{u}^{\top}\frac{\partial\mathbf{v}}{\partial\mathbf{x}}+\mathbf{v}^{% \top}\frac{\partial\mathbf{u}}{\partial\mathbf{x}}
  130. 𝐮 𝐱 , 𝐯 𝐱 \frac{\partial\mathbf{u}}{\partial\mathbf{x}},\frac{\partial\mathbf{v}}{% \partial\mathbf{x}}
  131. 𝐮 𝐱 𝐯 + 𝐯 𝐱 𝐮 \frac{\partial\mathbf{u}}{\partial\mathbf{x}}\mathbf{v}+\frac{\partial\mathbf{% v}}{\partial\mathbf{x}}\mathbf{u}
  132. 𝐮 𝐱 , 𝐯 𝐱 \frac{\partial\mathbf{u}}{\partial\mathbf{x}},\frac{\partial\mathbf{v}}{% \partial\mathbf{x}}
  133. ( 𝐮 𝐀𝐯 ) 𝐱 = 𝐮 𝐀𝐯 𝐱 = \frac{\partial(\mathbf{u}\cdot\mathbf{A}\mathbf{v})}{\partial\mathbf{x}}=\frac% {\partial\mathbf{u}^{\top}\mathbf{A}\mathbf{v}}{\partial\mathbf{x}}=
  134. 𝐮 𝐀 𝐯 𝐱 + 𝐯 𝐀 𝐮 𝐱 \mathbf{u}^{\top}\mathbf{A}\frac{\partial\mathbf{v}}{\partial\mathbf{x}}+% \mathbf{v}^{\top}\mathbf{A}^{\top}\frac{\partial\mathbf{u}}{\partial\mathbf{x}}
  135. 𝐮 𝐱 , 𝐯 𝐱 \frac{\partial\mathbf{u}}{\partial\mathbf{x}},\frac{\partial\mathbf{v}}{% \partial\mathbf{x}}
  136. 𝐮 𝐱 𝐀𝐯 + 𝐯 𝐱 𝐀 𝐮 \frac{\partial\mathbf{u}}{\partial\mathbf{x}}\mathbf{A}\mathbf{v}+\frac{% \partial\mathbf{v}}{\partial\mathbf{x}}\mathbf{A}^{\top}\mathbf{u}
  137. 𝐮 𝐱 , 𝐯 𝐱 \frac{\partial\mathbf{u}}{\partial\mathbf{x}},\frac{\partial\mathbf{v}}{% \partial\mathbf{x}}
  138. ( 𝐚 𝐱 ) 𝐱 = ( 𝐱 𝐚 ) 𝐱 = \frac{\partial(\mathbf{a}\cdot\mathbf{x})}{\partial\mathbf{x}}=\frac{\partial(% \mathbf{x}\cdot\mathbf{a})}{\partial\mathbf{x}}=
  139. 𝐚 𝐱 𝐱 = 𝐱 𝐚 𝐱 = \frac{\partial\mathbf{a}^{\top}\mathbf{x}}{\partial\mathbf{x}}=\frac{\partial% \mathbf{x}^{\top}\mathbf{a}}{\partial\mathbf{x}}=
  140. 𝐚 \mathbf{a}^{\top}
  141. 𝐚 \mathbf{a}
  142. 𝐛 𝐀𝐱 𝐱 = \frac{\partial\mathbf{b}^{\top}\mathbf{A}\mathbf{x}}{\partial\mathbf{x}}=
  143. 𝐛 𝐀 \mathbf{b}^{\top}\mathbf{A}
  144. 𝐀 𝐛 \mathbf{A}^{\top}\mathbf{b}
  145. 𝐱 𝐀𝐱 𝐱 = \frac{\partial\mathbf{x}^{\top}\mathbf{A}\mathbf{x}}{\partial\mathbf{x}}=
  146. 𝐱 ( 𝐀 + 𝐀 ) \mathbf{x}^{\top}(\mathbf{A}+\mathbf{A}^{\top})
  147. ( 𝐀 + 𝐀 ) 𝐱 (\mathbf{A}+\mathbf{A}^{\top})\mathbf{x}
  148. 𝐱 𝐀𝐱 𝐱 = \frac{\partial\mathbf{x}^{\top}\mathbf{A}\mathbf{x}}{\partial\mathbf{x}}=
  149. 2 𝐱 𝐀 2\mathbf{x}^{\top}\mathbf{A}
  150. 2 𝐀𝐱 2\mathbf{A}\mathbf{x}
  151. 2 𝐱 𝐀𝐱 𝐱 2 = \frac{\partial^{2}\mathbf{x}^{\top}\mathbf{A}\mathbf{x}}{\partial\mathbf{x}^{2% }}=
  152. 𝐀 + 𝐀 \mathbf{A}+\mathbf{A}^{\top}
  153. 2 𝐱 𝐀𝐱 𝐱 2 = \frac{\partial^{2}\mathbf{x}^{\top}\mathbf{A}\mathbf{x}}{\partial\mathbf{x}^{2% }}=
  154. 2 𝐀 2\mathbf{A}
  155. ( 𝐱 𝐱 ) 𝐱 = 𝐱 𝐱 𝐱 = \frac{\partial(\mathbf{x}\cdot\mathbf{x})}{\partial\mathbf{x}}=\frac{\partial% \mathbf{x}^{\top}\mathbf{x}}{\partial\mathbf{x}}=
  156. 2 𝐱 2\mathbf{x}^{\top}
  157. 2 𝐱 2\mathbf{x}
  158. ( 𝐚 𝐮 ) 𝐱 = 𝐚 𝐮 𝐱 = \frac{\partial(\mathbf{a}\cdot\mathbf{u})}{\partial\mathbf{x}}=\frac{\partial% \mathbf{a}^{\top}\mathbf{u}}{\partial\mathbf{x}}=
  159. 𝐚 𝐮 𝐱 \mathbf{a}^{\top}\frac{\partial\mathbf{u}}{\partial\mathbf{x}}
  160. 𝐮 𝐱 \frac{\partial\mathbf{u}}{\partial\mathbf{x}}
  161. 𝐮 𝐱 𝐚 \frac{\partial\mathbf{u}}{\partial\mathbf{x}}\mathbf{a}
  162. 𝐮 𝐱 \frac{\partial\mathbf{u}}{\partial\mathbf{x}}
  163. 𝐚 𝐱 𝐱 𝐛 𝐱 = \frac{\partial\;\,\textbf{a}^{\top}\,\textbf{x}\,\textbf{x}^{\top}\,\textbf{b}% }{\partial\;\,\textbf{x}}=
  164. 𝐱 ( 𝐚 𝐛 + 𝐛 𝐚 ) \,\textbf{x}^{\top}(\,\textbf{a}\,\textbf{b}^{\top}+\,\textbf{b}\,\textbf{a}^{% \top})
  165. ( 𝐚 𝐛 + 𝐛 𝐚 ) 𝐱 (\,\textbf{a}\,\textbf{b}^{\top}+\,\textbf{b}\,\textbf{a}^{\top})\,\textbf{x}
  166. ( 𝐀 𝐱 + 𝐛 ) 𝐂 ( 𝐃 𝐱 + 𝐞 ) 𝐱 = \frac{\partial\;(\,\textbf{A}\,\textbf{x}+\,\textbf{b})^{\top}\,\textbf{C}(\,% \textbf{D}\,\textbf{x}+\,\textbf{e})}{\partial\;\,\textbf{x}}=
  167. ( 𝐃 𝐱 + 𝐞 ) 𝐂 𝐀 + ( 𝐀 𝐱 + 𝐛 ) 𝐂 𝐃 (\,\textbf{D}\,\textbf{x}+\,\textbf{e})^{\top}\,\textbf{C}^{\top}\,\textbf{A}+% (\,\textbf{A}\,\textbf{x}+\,\textbf{b})^{\top}\,\textbf{C}\,\textbf{D}
  168. 𝐃 𝐂 ( 𝐀 𝐱 + 𝐛 ) + 𝐀 𝐂 ( 𝐃 𝐱 + 𝐞 ) \,\textbf{D}^{\top}\,\textbf{C}^{\top}(\,\textbf{A}\,\textbf{x}+\,\textbf{b})+% \,\textbf{A}^{\top}\,\textbf{C}(\,\textbf{D}\,\textbf{x}+\,\textbf{e})
  169. 𝐱 - 𝐚 𝐱 = \frac{\partial\;\|\mathbf{x}-\mathbf{a}\|}{\partial\;\mathbf{x}}=
  170. ( 𝐱 - 𝐚 ) 𝐱 - 𝐚 \frac{(\mathbf{x}-\mathbf{a})^{\top}}{\|\mathbf{x}-\mathbf{a}\|}
  171. 𝐱 - 𝐚 𝐱 - 𝐚 \frac{\mathbf{x}-\mathbf{a}}{\|\mathbf{x}-\mathbf{a}\|}
  172. 𝐲 x \frac{\partial\mathbf{y}}{\partial x}
  173. 𝐚 x = \frac{\partial\mathbf{a}}{\partial x}=
  174. 𝟎 \mathbf{0}
  175. a 𝐮 x = \frac{\partial a\mathbf{u}}{\partial x}=
  176. a 𝐮 x a\frac{\partial\mathbf{u}}{\partial x}
  177. 𝐀𝐮 x = \frac{\partial\mathbf{A}\mathbf{u}}{\partial x}=
  178. 𝐀 𝐮 x \mathbf{A}\frac{\partial\mathbf{u}}{\partial x}
  179. 𝐮 x 𝐀 \frac{\partial\mathbf{u}}{\partial x}\mathbf{A}^{\top}
  180. 𝐮 x = \frac{\partial\mathbf{u}^{\top}}{\partial x}=
  181. ( 𝐮 x ) \left(\frac{\partial\mathbf{u}}{\partial x}\right)^{\top}
  182. ( 𝐮 + 𝐯 ) x = \frac{\partial(\mathbf{u}+\mathbf{v})}{\partial x}=
  183. 𝐮 x + 𝐯 x \frac{\partial\mathbf{u}}{\partial x}+\frac{\partial\mathbf{v}}{\partial x}
  184. ( 𝐮 × 𝐯 ) x = \frac{\partial(\mathbf{u}\times\mathbf{v})}{\partial x}=
  185. 𝐮 × 𝐯 x + 𝐮 x × 𝐯 \mathbf{u}\times\frac{\partial\mathbf{v}}{\partial x}+\frac{\partial\mathbf{u}% }{\partial x}\times\mathbf{v}
  186. 𝐠 ( 𝐮 ) x = \frac{\partial\mathbf{g(u)}}{\partial x}=
  187. 𝐠 ( 𝐮 ) 𝐮 𝐮 x \frac{\partial\mathbf{g(u)}}{\partial\mathbf{u}}\frac{\partial\mathbf{u}}{% \partial x}
  188. 𝐮 x 𝐠 ( 𝐮 ) 𝐮 \frac{\partial\mathbf{u}}{\partial x}\frac{\partial\mathbf{g(u)}}{\partial% \mathbf{u}}
  189. 𝐟 ( 𝐠 ( 𝐮 ) ) x = \frac{\partial\mathbf{f(g(u))}}{\partial x}=
  190. 𝐟 ( 𝐠 ) 𝐠 𝐠 ( 𝐮 ) 𝐮 𝐮 x \frac{\partial\mathbf{f(g)}}{\partial\mathbf{g}}\frac{\partial\mathbf{g(u)}}{% \partial\mathbf{u}}\frac{\partial\mathbf{u}}{\partial x}
  191. 𝐮 x 𝐠 ( 𝐮 ) 𝐮 𝐟 ( 𝐠 ) 𝐠 \frac{\partial\mathbf{u}}{\partial x}\frac{\partial\mathbf{g(u)}}{\partial% \mathbf{u}}\frac{\partial\mathbf{f(g)}}{\partial\mathbf{g}}
  192. 𝐠 ( 𝐮 ) 𝐮 \frac{\partial\mathbf{g(u)}}{\partial\mathbf{u}}
  193. 𝐟 ( 𝐠 ) 𝐠 \frac{\partial\mathbf{f(g)}}{\partial\mathbf{g}}
  194. tr ( 𝐀 ) = tr ( 𝐀 ) {\rm tr}(\mathbf{A})={\rm tr}(\mathbf{A^{\top}})
  195. tr ( 𝐀𝐁𝐂𝐃 ) = tr ( 𝐁𝐂𝐃𝐀 ) = tr ( 𝐂𝐃𝐀𝐁 ) = tr ( 𝐃𝐀𝐁𝐂 ) {\rm tr}(\mathbf{ABCD})={\rm tr}(\mathbf{BCDA})={\rm tr}(\mathbf{CDAB})={\rm tr% }(\mathbf{DABC})
  196. tr ( 𝐀𝐗𝐁𝐗 𝐂 ) 𝐗 : \frac{\partial{\rm tr}(\mathbf{AXBX^{\top}C})}{\partial\mathbf{X}}:
  197. d tr ( 𝐀𝐗𝐁𝐗 𝐂 ) = d tr ( 𝐂𝐀𝐗𝐁𝐗 ) = tr ( d ( 𝐂𝐀𝐗𝐁𝐗 ) ) = tr ( 𝐂𝐀𝐗 d ( 𝐁𝐗 ) + d ( 𝐂𝐀𝐗 ) 𝐁𝐗 ) = tr ( 𝐂𝐀𝐗 d ( 𝐁𝐗 ) ) + tr ( d ( 𝐂𝐀𝐗 ) 𝐁𝐗 ) = tr ( 𝐂𝐀𝐗𝐁 d ( 𝐗 ) ) + tr ( 𝐂𝐀 ( d 𝐗 ) 𝐁𝐗 ) = tr ( 𝐂𝐀𝐗𝐁 ( d 𝐗 ) ) + tr ( 𝐂𝐀 ( d 𝐗 ) 𝐁𝐗 ) = tr ( ( 𝐂𝐀𝐗𝐁 ( d 𝐗 ) ) ) + tr ( 𝐂𝐀 ( d 𝐗 ) 𝐁𝐗 ) = tr ( ( d 𝐗 ) 𝐁 𝐗 𝐀 𝐂 ) + tr ( 𝐂𝐀 ( d 𝐗 ) 𝐁𝐗 ) = tr ( 𝐁 𝐗 𝐀 𝐂 ( d 𝐗 ) ) + tr ( 𝐁𝐗 𝐂𝐀 ( d 𝐗 ) ) = tr ( ( 𝐁 𝐗 𝐀 𝐂 + 𝐁𝐗 𝐂𝐀 ) d 𝐗 ) \begin{aligned}\displaystyle d\,{\rm tr}(\mathbf{AXBX^{\top}C})&\displaystyle=% d\,{\rm tr}(\mathbf{CAXBX^{\top}})={\rm tr}(d(\mathbf{CAXBX^{\top}}))\\ &\displaystyle={\rm tr}(\mathbf{CAX}d(\mathbf{BX^{\top}})+d(\mathbf{CAX})% \mathbf{BX^{\top}})\\ &\displaystyle={\rm tr}(\mathbf{CAX}d(\mathbf{BX^{\top}}))+{\rm tr}(d(\mathbf{% CAX})\mathbf{BX^{\top}})\\ &\displaystyle={\rm tr}(\mathbf{CAXB}d(\mathbf{X^{\top}}))+{\rm tr}(\mathbf{CA% }(d\mathbf{X})\mathbf{BX^{\top}})\\ &\displaystyle={\rm tr}(\mathbf{CAXB}(d\mathbf{X})^{\top})+{\rm tr}(\mathbf{CA% }(d\mathbf{X})\mathbf{BX^{\top}})\\ &\displaystyle={\rm tr}\left((\mathbf{CAXB}(d\mathbf{X})^{\top})^{\top}\right)% +{\rm tr}(\mathbf{CA}(d\mathbf{X})\mathbf{BX^{\top}})\\ &\displaystyle={\rm tr}((d\mathbf{X})\mathbf{B^{\top}X^{\top}A^{\top}C^{\top}}% )+{\rm tr}(\mathbf{CA}(d\mathbf{X})\mathbf{BX^{\top}})\\ &\displaystyle={\rm tr}(\mathbf{B^{\top}X^{\top}A^{\top}C^{\top}}(d\mathbf{X})% )+{\rm tr}(\mathbf{BX^{\top}}\mathbf{CA}(d\mathbf{X}))\\ &\displaystyle={\rm tr}\left((\mathbf{B^{\top}X^{\top}A^{\top}C^{\top}}+% \mathbf{BX^{\top}}\mathbf{CA})d\mathbf{X}\right)\end{aligned}
  198. tr ( 𝐀𝐗𝐁𝐗 𝐂 ) 𝐗 = 𝐁 𝐗 𝐀 𝐂 + 𝐁𝐗 𝐂𝐀 . \frac{\partial{\rm tr}(\mathbf{AXBX^{\top}C})}{\partial\mathbf{X}}=\mathbf{B^{% \top}X^{\top}A^{\top}C^{\top}}+\mathbf{BX^{\top}}\mathbf{CA}.
  199. y 𝐗 \frac{\partial y}{\partial\mathbf{X}}
  200. a 𝐗 = \frac{\partial a}{\partial\mathbf{X}}=
  201. 𝟎 \mathbf{0}^{\top}
  202. 𝟎 \mathbf{0}
  203. a u 𝐗 = \frac{\partial au}{\partial\mathbf{X}}=
  204. a u 𝐗 a\frac{\partial u}{\partial\mathbf{X}}
  205. ( u + v ) 𝐗 = \frac{\partial(u+v)}{\partial\mathbf{X}}=
  206. u 𝐗 + v 𝐗 \frac{\partial u}{\partial\mathbf{X}}+\frac{\partial v}{\partial\mathbf{X}}
  207. u v 𝐗 = \frac{\partial uv}{\partial\mathbf{X}}=
  208. u v 𝐗 + v u 𝐗 u\frac{\partial v}{\partial\mathbf{X}}+v\frac{\partial u}{\partial\mathbf{X}}
  209. g ( u ) 𝐗 = \frac{\partial g(u)}{\partial\mathbf{X}}=
  210. g ( u ) u u 𝐗 \frac{\partial g(u)}{\partial u}\frac{\partial u}{\partial\mathbf{X}}
  211. f ( g ( u ) ) 𝐗 = \frac{\partial f(g(u))}{\partial\mathbf{X}}=
  212. f ( g ) g g ( u ) u u 𝐗 \frac{\partial f(g)}{\partial g}\frac{\partial g(u)}{\partial u}\frac{\partial u% }{\partial\mathbf{X}}
  213. g ( 𝐔 ) X i j = \frac{\partial g(\mathbf{U})}{\partial X_{ij}}=
  214. tr ( g ( 𝐔 ) 𝐔 𝐔 X i j ) {\rm tr}\left(\frac{\partial g(\mathbf{U})}{\partial\mathbf{U}}\frac{\partial% \mathbf{U}}{\partial X_{ij}}\right)
  215. tr ( ( g ( 𝐔 ) 𝐔 ) 𝐔 X i j ) {\rm tr}\left(\left(\frac{\partial g(\mathbf{U})}{\partial\mathbf{U}}\right)^{% \top}\frac{\partial\mathbf{U}}{\partial X_{ij}}\right)
  216. 𝐔 X i j , \frac{\partial\mathbf{U}}{\partial X_{ij}},
  217. tr ( 𝐗 ) 𝐗 = \frac{\partial{\rm tr}(\mathbf{X})}{\partial\mathbf{X}}=
  218. 𝐈 \mathbf{I}
  219. tr ( 𝐔 + 𝐕 ) 𝐗 = \frac{\partial{\rm tr}(\mathbf{U}+\mathbf{V})}{\partial\mathbf{X}}=
  220. tr ( 𝐔 ) 𝐗 + tr ( 𝐕 ) 𝐗 \frac{\partial{\rm tr}(\mathbf{U})}{\partial\mathbf{X}}+\frac{\partial{\rm tr}% (\mathbf{V})}{\partial\mathbf{X}}
  221. tr ( a 𝐔 ) 𝐗 = \frac{\partial{\rm tr}(a\mathbf{U})}{\partial\mathbf{X}}=
  222. a tr ( 𝐔 ) 𝐗 a\frac{\partial{\rm tr}(\mathbf{U})}{\partial\mathbf{X}}
  223. tr ( 𝐠 ( 𝐗 ) ) 𝐗 = \frac{\partial{\rm tr}(\mathbf{g(X)})}{\partial\mathbf{X}}=
  224. 𝐠 ( 𝐗 ) \mathbf{g}^{\prime}(\mathbf{X})
  225. ( 𝐠 ( 𝐗 ) ) (\mathbf{g}^{\prime}(\mathbf{X}))^{\top}
  226. ( 𝐀𝐗 ) 𝐗 = tr ( 𝐗𝐀 ) 𝐗 = \frac{\partial\top(\mathbf{AX})}{\partial\mathbf{X}}=\frac{\partial{\rm tr}(% \mathbf{XA})}{\partial\mathbf{X}}=
  227. 𝐀 \mathbf{A}
  228. 𝐀 \mathbf{A}^{\top}
  229. tr ( 𝐀𝐗 ) 𝐗 = tr ( 𝐗 𝐀 ) 𝐗 = \frac{\partial{\rm tr}(\mathbf{AX^{\top}})}{\partial\mathbf{X}}=\frac{\partial% {\rm tr}(\mathbf{X^{\top}A})}{\partial\mathbf{X}}=
  230. 𝐀 \mathbf{A}^{\top}
  231. 𝐀 \mathbf{A}
  232. tr ( 𝐗 𝐀𝐗 ) 𝐗 = \frac{\partial{\rm tr}(\mathbf{X^{\top}AX})}{\partial\mathbf{X}}=
  233. 𝐗 ( 𝐀 + 𝐀 ) \mathbf{X}^{\top}(\mathbf{A}+\mathbf{A}^{\top})
  234. ( 𝐀 + 𝐀 ) 𝐗 (\mathbf{A}+\mathbf{A}^{\top})\mathbf{X}
  235. tr ( 𝐗 - 𝟏 𝐀 ) 𝐗 = \frac{\partial{\rm tr}(\mathbf{X^{-1}A})}{\partial\mathbf{X}}=
  236. - ( 𝐗 - 1 ) 𝐀 ( 𝐗 - 1 ) -(\mathbf{X}^{-1})^{\top}\mathbf{A}(\mathbf{X}^{-1})^{\top}
  237. - 𝐗 - 1 𝐀 𝐗 - 1 -\mathbf{X}^{-1}\mathbf{A}^{\top}\mathbf{X}^{-1}
  238. tr ( 𝐀𝐗𝐁 ) 𝐗 = tr ( 𝐁𝐀𝐗 ) 𝐗 = \frac{\partial{\rm tr}(\mathbf{AXB})}{\partial\mathbf{X}}=\frac{\partial{\rm tr% }(\mathbf{BAX})}{\partial\mathbf{X}}=
  239. 𝐁𝐀 \mathbf{BA}
  240. 𝐀 𝐁 \mathbf{A^{\top}B^{\top}}
  241. tr ( 𝐀𝐗𝐁𝐗 𝐂 ) 𝐗 = \frac{\partial{\rm tr}(\mathbf{AXBX^{\top}C})}{\partial\mathbf{X}}=
  242. 𝐁𝐗 𝐂𝐀 + 𝐁 𝐗 𝐀 𝐂 \mathbf{BX^{\top}CA}+\mathbf{B^{\top}X^{\top}A^{\top}C^{\top}}
  243. 𝐀 𝐂 𝐗𝐁 + 𝐂𝐀𝐗𝐁 \mathbf{A^{\top}C^{\top}XB^{\top}}+\mathbf{CAXB}
  244. tr ( 𝐗 n ) 𝐗 = \frac{\partial{\rm tr}(\mathbf{X}^{n})}{\partial\mathbf{X}}=
  245. n 𝐗 n - 1 n\mathbf{X}^{n-1}
  246. n ( 𝐗 n - 1 ) n(\mathbf{X}^{n-1})^{\top}
  247. tr ( 𝐀𝐗 n ) 𝐗 = \frac{\partial{\rm tr}(\mathbf{A}\mathbf{X}^{n})}{\partial\mathbf{X}}=
  248. i = 0 n - 1 𝐗 i 𝐀𝐗 n - i - 1 \sum_{i=0}^{n-1}\mathbf{X}^{i}\mathbf{A}\mathbf{X}^{n-i-1}
  249. i = 0 n - 1 ( 𝐗 i 𝐀𝐗 n - i - 1 ) \sum_{i=0}^{n-1}(\mathbf{X}^{i}\mathbf{A}\mathbf{X}^{n-i-1})^{\top}
  250. tr ( e 𝐗 ) 𝐗 = \frac{\partial{\rm tr}(e^{\mathbf{X}})}{\partial\mathbf{X}}=
  251. e 𝐗 e^{\mathbf{X}}
  252. ( e 𝐗 ) (e^{\mathbf{X}})^{\top}
  253. tr ( sin ( 𝐗 ) ) 𝐗 = \frac{\partial{\rm tr}(\sin(\mathbf{X}))}{\partial\mathbf{X}}=
  254. cos ( 𝐗 ) \cos(\mathbf{X})
  255. ( cos ( 𝐗 ) ) (\cos(\mathbf{X}))^{\top}
  256. | 𝐗 | 𝐗 = \frac{\partial|\mathbf{X}|}{\partial\mathbf{X}}=
  257. cofactor ( X ) = | 𝐗 | 𝐗 - 1 \operatorname{cofactor}(X)^{\top}=|\mathbf{X}|\mathbf{X}^{-1}
  258. cofactor ( X ) = | 𝐗 | ( 𝐗 - 1 ) \operatorname{cofactor}(X)=|\mathbf{X}|(\mathbf{X}^{-1})^{\top}
  259. ln | a 𝐗 | 𝐗 = \frac{\partial\ln|a\mathbf{X}|}{\partial\mathbf{X}}=
  260. d ln a u d x = 1 a u d ( a u ) d x = 1 a u a d u d x = 1 u d u d x = d ln u d x . \frac{d\,\ln au}{dx}=\frac{1}{au}\frac{d(au)}{dx}=\frac{1}{au}a\frac{du}{dx}=% \frac{1}{u}\frac{du}{dx}=\frac{d\,\ln u}{dx}.
  261. | 𝐀𝐗𝐁 | 𝐗 = \frac{\partial|\mathbf{AXB}|}{\partial\mathbf{X}}=
  262. | 𝐀𝐗𝐁 | 𝐗 - 1 |\mathbf{AXB}|\mathbf{X}^{-1}
  263. | 𝐀𝐗𝐁 | ( 𝐗 - 1 ) |\mathbf{AXB}|(\mathbf{X}^{-1})^{\top}
  264. | 𝐗 n | 𝐗 = \frac{\partial|\mathbf{X}^{n}|}{\partial\mathbf{X}}=
  265. n | 𝐗 n | 𝐗 - 1 n|\mathbf{X}^{n}|\mathbf{X}^{-1}
  266. n | 𝐗 n | ( 𝐗 - 1 ) n|\mathbf{X}^{n}|(\mathbf{X}^{-1})^{\top}
  267. ln | 𝐗 𝐗 | 𝐗 = \frac{\partial\ln|\mathbf{X}^{\top}\mathbf{X}|}{\partial\mathbf{X}}=
  268. 2 𝐗 + 2\mathbf{X}^{+}
  269. 2 ( 𝐗 + ) 2(\mathbf{X}^{+})^{\top}
  270. ln | 𝐗 𝐗 | 𝐗 + = \frac{\partial\ln|\mathbf{X}^{\top}\mathbf{X}|}{\partial\mathbf{X}^{+}}=
  271. - 2 𝐗 -2\mathbf{X}
  272. - 2 𝐗 -2\mathbf{X}^{\top}
  273. | 𝐗 𝐀𝐗 | 𝐗 = \frac{\partial|\mathbf{X^{\top}}\mathbf{A}\mathbf{X}|}{\partial\mathbf{X}}=
  274. 2 | 𝐗 𝐀𝐗 | 𝐗 - 1 2|\mathbf{X^{\top}}\mathbf{A}\mathbf{X}|\mathbf{X}^{-1}
  275. 2 | 𝐗 𝐀𝐗 | ( 𝐗 - 1 ) 2|\mathbf{X^{\top}}\mathbf{A}\mathbf{X}|(\mathbf{X}^{-1})^{\top}
  276. | 𝐗 𝐀𝐗 | 𝐗 = \frac{\partial|\mathbf{X^{\top}}\mathbf{A}\mathbf{X}|}{\partial\mathbf{X}}=
  277. 2 | 𝐗 𝐀𝐗 | ( 𝐗 𝐀 𝐗 ) - 1 𝐗 𝐀 2|\mathbf{X^{\top}}\mathbf{A}\mathbf{X}|(\mathbf{X^{\top}A^{\top}X})^{-1}% \mathbf{X^{\top}A^{\top}}
  278. 2 | 𝐗 𝐀𝐗 | 𝐀𝐗 ( 𝐗 𝐀𝐗 ) - 1 2|\mathbf{X^{\top}}\mathbf{A}\mathbf{X}|\mathbf{AX}(\mathbf{X^{\top}AX})^{-1}
  279. | 𝐗 𝐀𝐗 | 𝐗 = \frac{\partial|\mathbf{X^{\top}}\mathbf{A}\mathbf{X}|}{\partial\mathbf{X}}=
  280. | 𝐗 𝐀𝐗 | ( ( 𝐗 𝐀𝐗 ) - 1 𝐗 𝐀 |\mathbf{X^{\top}}\mathbf{A}\mathbf{X}|((\mathbf{X^{\top}AX})^{-1}\mathbf{X^{% \top}A}
  281. + ( 𝐗 𝐀 𝐗 ) - 1 𝐗 𝐀 ) +(\mathbf{X^{\top}A^{\top}X})^{-1}\mathbf{X^{\top}A^{\top}})
  282. | 𝐗 𝐀𝐗 | ( 𝐀𝐗 ( 𝐗 𝐀𝐗 ) - 1 |\mathbf{X^{\top}}\mathbf{A}\mathbf{X}|(\mathbf{AX}(\mathbf{X^{\top}AX})^{-1}
  283. + 𝐀 𝐗 ( 𝐗 𝐀 𝐗 ) - 1 ) +\mathbf{A^{\top}X}(\mathbf{X^{\top}A^{\top}X})^{-1})
  284. 𝐘 x \frac{\partial\mathbf{Y}}{\partial x}
  285. a 𝐔 x = \frac{\partial a\mathbf{U}}{\partial x}=
  286. a 𝐔 x a\frac{\partial\mathbf{U}}{\partial x}
  287. 𝐀𝐔𝐁 x = \frac{\partial\mathbf{AUB}}{\partial x}=
  288. 𝐀 𝐔 x 𝐁 \mathbf{A}\frac{\partial\mathbf{U}}{\partial x}\mathbf{B}
  289. ( 𝐔 + 𝐕 ) x = \frac{\partial(\mathbf{U}+\mathbf{V})}{\partial x}=
  290. 𝐔 x + 𝐕 x \frac{\partial\mathbf{U}}{\partial x}+\frac{\partial\mathbf{V}}{\partial x}
  291. ( 𝐔𝐕 ) x = \frac{\partial(\mathbf{U}\mathbf{V})}{\partial x}=
  292. 𝐔 𝐕 x + 𝐔 x 𝐕 \mathbf{U}\frac{\partial\mathbf{V}}{\partial x}+\frac{\partial\mathbf{U}}{% \partial x}\mathbf{V}
  293. ( 𝐔 𝐕 ) x = \frac{\partial(\mathbf{U}\otimes\mathbf{V})}{\partial x}=
  294. 𝐔 𝐕 x + 𝐔 x 𝐕 \mathbf{U}\otimes\frac{\partial\mathbf{V}}{\partial x}+\frac{\partial\mathbf{U% }}{\partial x}\otimes\mathbf{V}
  295. ( 𝐔 𝐕 ) x = \frac{\partial(\mathbf{U}\circ\mathbf{V})}{\partial x}=
  296. 𝐔 𝐕 x + 𝐔 x 𝐕 \mathbf{U}\circ\frac{\partial\mathbf{V}}{\partial x}+\frac{\partial\mathbf{U}}% {\partial x}\circ\mathbf{V}
  297. 𝐔 - 1 x = \frac{\partial\mathbf{U}^{-1}}{\partial x}=
  298. - 𝐔 - 1 𝐔 x 𝐔 - 1 -\mathbf{U}^{-1}\frac{\partial\mathbf{U}}{\partial x}\mathbf{U}^{-1}
  299. 2 𝐔 - 1 x y = \frac{\partial^{2}\mathbf{U}^{-1}}{\partial x\partial y}=
  300. 𝐔 - 1 ( 𝐔 x 𝐔 - 1 𝐔 y - 2 𝐔 x y + 𝐔 y 𝐔 - 1 𝐔 x ) 𝐔 - 1 \mathbf{U}^{-1}\left(\frac{\partial\mathbf{U}}{\partial x}\mathbf{U}^{-1}\frac% {\partial\mathbf{U}}{\partial y}-\frac{\partial^{2}\mathbf{U}}{\partial x% \partial y}+\frac{\partial\mathbf{U}}{\partial y}\mathbf{U}^{-1}\frac{\partial% \mathbf{U}}{\partial x}\right)\mathbf{U}^{-1}
  301. 𝐠 ( x 𝐀 ) x = \frac{\partial\,\mathbf{g}(x\mathbf{A})}{\partial x}=
  302. 𝐀𝐠 ( x 𝐀 ) = 𝐠 ( x 𝐀 ) 𝐀 \mathbf{A}\mathbf{g}^{\prime}(x\mathbf{A})=\mathbf{g}^{\prime}(x\mathbf{A})% \mathbf{A}
  303. e x 𝐀 x = \frac{\partial e^{x\mathbf{A}}}{\partial x}=
  304. 𝐀 e x 𝐀 = e x 𝐀 𝐀 \mathbf{A}e^{x\mathbf{A}}=e^{x\mathbf{A}}\mathbf{A}
  305. g ( 𝐮 ) x = \frac{\partial g(\mathbf{u})}{\partial x}=
  306. g ( 𝐮 ) 𝐮 𝐮 x \frac{\partial g(\mathbf{u})}{\partial\mathbf{u}}\cdot\frac{\partial\mathbf{u}% }{\partial x}
  307. ( 𝐮 𝐯 ) x = \frac{\partial(\mathbf{u}\cdot\mathbf{v})}{\partial x}=
  308. 𝐮 𝐯 x + 𝐮 x 𝐯 \mathbf{u}\cdot\frac{\partial\mathbf{v}}{\partial x}+\frac{\partial\mathbf{u}}% {\partial x}\cdot\mathbf{v}
  309. | 𝐔 | x = \frac{\partial|\mathbf{U}|}{\partial x}=
  310. | 𝐔 | tr ( 𝐔 - 1 𝐔 x ) |\mathbf{U}|{\rm tr}\left(\mathbf{U}^{-1}\frac{\partial\mathbf{U}}{\partial x}\right)
  311. ln | 𝐔 | x = \frac{\partial\ln|\mathbf{U}|}{\partial x}=
  312. tr ( 𝐔 - 1 𝐔 x ) {\rm tr}\left(\mathbf{U}^{-1}\frac{\partial\mathbf{U}}{\partial x}\right)
  313. 2 | 𝐔 | x 2 = \frac{\partial^{2}|\mathbf{U}|}{\partial x^{2}}=
  314. | 𝐔 | [ tr ( 𝐔 - 1 2 𝐔 x 2 ) + ( tr ( 𝐔 - 1 𝐔 x ) ) 2 - tr ( ( 𝐔 - 1 𝐔 x ) ( 𝐔 - 1 𝐔 x ) ) ] |\mathbf{U}|\left[{\rm tr}\left(\mathbf{U}^{-1}\frac{\partial^{2}\mathbf{U}}{% \partial x^{2}}\right)+\left({\rm tr}\left(\mathbf{U}^{-1}\frac{\partial% \mathbf{U}}{\partial x}\right)\right)^{2}-{\rm tr}\left(\left(\mathbf{U}^{-1}% \frac{\partial\mathbf{U}}{\partial x}\right)\left(\mathbf{U}^{-1}\frac{% \partial\mathbf{U}}{\partial x}\right)\right)\right]
  315. g ( 𝐔 ) x = \frac{\partial g(\mathbf{U})}{\partial x}=
  316. tr ( g ( 𝐔 ) 𝐔 𝐔 x ) {\rm tr}\left(\frac{\partial g(\mathbf{U})}{\partial\mathbf{U}}\frac{\partial% \mathbf{U}}{\partial x}\right)
  317. tr ( ( g ( 𝐔 ) 𝐔 ) 𝐔 x ) {\rm tr}\left(\left(\frac{\partial g(\mathbf{U})}{\partial\mathbf{U}}\right)^{% \top}\frac{\partial\mathbf{U}}{\partial x}\right)
  318. tr ( 𝐠 ( x 𝐀 ) ) x = \frac{\partial\,{\rm tr}(\mathbf{g}(x\mathbf{A}))}{\partial x}=
  319. tr ( 𝐀𝐠 ( x 𝐀 ) ) {\rm tr}(\mathbf{A}\mathbf{g}^{\prime}(x\mathbf{A}))
  320. tr ( e x 𝐀 ) x = \frac{\partial\,{\rm tr}(e^{x\mathbf{A}})}{\partial x}=
  321. tr ( 𝐀 e x 𝐀 ) {\rm tr}(\mathbf{A}e^{x\mathbf{A}})
  322. d ( tr ( 𝐗 ) ) = d({\rm tr}(\mathbf{X}))=
  323. tr ( d 𝐗 ) {\rm tr}(d\mathbf{X})
  324. d ( | 𝐗 | ) = d(|\mathbf{X}|)=
  325. | 𝐗 | tr ( 𝐗 - 1 d 𝐗 ) |\mathbf{X}|{\rm tr}(\mathbf{X}^{-1}d\mathbf{X})
  326. d ( ln | 𝐗 | ) = d(\ln|\mathbf{X}|)=
  327. tr ( 𝐗 - 1 d 𝐗 ) {\rm tr}(\mathbf{X}^{-1}d\mathbf{X})
  328. d ( 𝐀 ) = d(\mathbf{A})=
  329. 0
  330. d ( a 𝐗 ) = d(a\mathbf{X})=
  331. a d 𝐗 a\,d\mathbf{X}
  332. d ( 𝐗 + 𝐘 ) = d(\mathbf{X}+\mathbf{Y})=
  333. d 𝐗 + d 𝐘 d\mathbf{X}+d\mathbf{Y}
  334. d ( 𝐗𝐘 ) = d(\mathbf{X}\mathbf{Y})=
  335. ( d 𝐗 ) 𝐘 + 𝐗 ( d 𝐘 ) (d\mathbf{X})\mathbf{Y}+\mathbf{X}(d\mathbf{Y})
  336. d ( 𝐗 𝐘 ) = d(\mathbf{X}\otimes\mathbf{Y})=
  337. ( d 𝐗 ) 𝐘 + 𝐗 ( d 𝐘 ) (d\mathbf{X})\otimes\mathbf{Y}+\mathbf{X}\otimes(d\mathbf{Y})
  338. d ( 𝐗 𝐘 ) = d(\mathbf{X}\circ\mathbf{Y})=
  339. ( d 𝐗 ) 𝐘 + 𝐗 ( d 𝐘 ) (d\mathbf{X})\circ\mathbf{Y}+\mathbf{X}\circ(d\mathbf{Y})
  340. d ( 𝐗 ) = d(\mathbf{X}^{\top})=
  341. ( d 𝐗 ) (d\mathbf{X})^{\top}
  342. d ( 𝐗 H ) = d(\mathbf{X}^{\rm H})=
  343. ( d 𝐗 ) H (d\mathbf{X})^{\rm H}
  344. d y = a d x dy=a\,dx
  345. d y d x = a \frac{dy}{dx}=a
  346. d y = 𝐚 d 𝐱 dy=\mathbf{a}\,d\mathbf{x}
  347. d y d 𝐱 = 𝐚 \frac{dy}{d\mathbf{x}}=\mathbf{a}
  348. d y = tr ( 𝐀 d 𝐗 ) dy={\rm tr}(\mathbf{A}\,d\mathbf{X})
  349. d y d 𝐗 = 𝐀 \frac{dy}{d\mathbf{X}}=\mathbf{A}
  350. d 𝐲 = 𝐚 d x d\mathbf{y}=\mathbf{a}\,dx
  351. d 𝐲 d x = 𝐚 \frac{d\mathbf{y}}{dx}=\mathbf{a}
  352. d 𝐲 = 𝐀 d 𝐱 d\mathbf{y}=\mathbf{A}\,d\mathbf{x}
  353. d 𝐲 d 𝐱 = 𝐀 \frac{d\mathbf{y}}{d\mathbf{x}}=\mathbf{A}
  354. d 𝐘 = 𝐀 d x d\mathbf{Y}=\mathbf{A}\,dx
  355. d 𝐘 d x = 𝐀 \frac{d\mathbf{Y}}{dx}=\mathbf{A}
  356. 𝟎 \mathbf{0}
  357. 𝟎 \mathbf{0}
  358. 𝟎 \mathbf{0}
  359. 𝐘 x , \frac{\partial\mathbf{Y}}{\partial x},
  360. y 𝐗 . \frac{\partial y}{\partial\mathbf{X}}.