wpmath0000008_9

Photon_polarization.html

  1. α x , α y \alpha_{x},\alpha_{y}
  2. α x = α y = def α . \alpha_{x}=\alpha_{y}\ \stackrel{\mathrm{def}}{=}\ \alpha.
  3. α \alpha
  4. θ \theta
  5. | ψ = ( cos θ sin θ ) exp ( i α ) . |\psi\rangle=\begin{pmatrix}\cos\theta\\ \sin\theta\end{pmatrix}\exp\left(i\alpha\right).
  6. | x = def ( 1 0 ) |x\rangle\ \stackrel{\mathrm{def}}{=}\ \begin{pmatrix}1\\ 0\end{pmatrix}
  7. | y = def ( 0 1 ) |y\rangle\ \stackrel{\mathrm{def}}{=}\ \begin{pmatrix}0\\ 1\end{pmatrix}
  8. | ψ = cos θ exp ( i α ) | x + sin θ exp ( i α ) | y = ψ x | x + ψ y | y . |\psi\rangle=\cos\theta\exp\left(i\alpha\right)|x\rangle+\sin\theta\exp\left(i% \alpha\right)|y\rangle=\psi_{x}|x\rangle+\psi_{y}|y\rangle.
  9. α x \alpha_{x}
  10. α y \alpha_{y}
  11. π / 2 \pi/2
  12. | ψ = 1 2 ( 1 ± i ) exp ( i α x ) |\psi\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\ \pm i\end{pmatrix}\exp\left(i\alpha_{x}\right)
  13. | R = def 1 2 ( 1 i ) |R\rangle\ \stackrel{\mathrm{def}}{=}\ {1\over\sqrt{2}}\begin{pmatrix}1\\ i\end{pmatrix}
  14. | L = def 1 2 ( 1 - i ) |L\rangle\ \stackrel{\mathrm{def}}{=}\ {1\over\sqrt{2}}\begin{pmatrix}1\\ -i\end{pmatrix}
  15. | ψ = ψ R | R + ψ L | L |\psi\rangle=\psi_{R}|R\rangle+\psi_{L}|L\rangle
  16. ψ R = R | ψ = 1 2 ( cos θ exp ( i α x ) - i sin θ exp ( i α y ) ) \psi_{R}=\langle R|\psi\rangle=\frac{1}{\sqrt{2}}(\cos\theta\exp(i\alpha_{x})-% i\sin\theta\exp(i\alpha_{y}))
  17. ψ L = L | ψ = 1 2 ( cos θ exp ( i α x ) + i sin θ exp ( i α y ) ) . \psi_{L}=\langle L|\psi\rangle=\frac{1}{\sqrt{2}}(\cos\theta\exp(i\alpha_{x})+% i\sin\theta\exp(i\alpha_{y})).
  18. 1 = | ψ R | 2 + | ψ L | 2 . 1=|\psi_{R}|^{2}+|\psi_{L}|^{2}.
  19. | ψ = def ( ψ x ψ y ) = ( cos θ exp ( i α x ) sin θ exp ( i α y ) ) . |\psi\rangle\ \stackrel{\mathrm{def}}{=}\ \begin{pmatrix}\psi_{x}\\ \psi_{y}\end{pmatrix}=\begin{pmatrix}\cos\theta\exp\left(i\alpha_{x}\right)\\ \sin\theta\exp\left(i\alpha_{y}\right)\end{pmatrix}.
  20. e i ω t e^{i\omega t}
  21. ( x ( t ) y ( t ) ) = ( ( e i ω t ψ x ) ( e i ω t ψ y ) ) = [ e i ω t ( ψ x ψ y ) ] = ( e i ω t | ψ ) . \begin{pmatrix}x(t)\\ y(t)\end{pmatrix}=\begin{pmatrix}\Re(e^{i\omega t}\psi_{x})\\ \Re(e^{i\omega t}\psi_{y})\end{pmatrix}=\Re\left[e^{i\omega t}\begin{pmatrix}% \psi_{x}\\ \psi_{y}\end{pmatrix}\right]=\Re\left(e^{i\omega t}|\psi\rangle\right).
  22. M ( | ψ ) = { ( x ( t ) , y ( t ) ) | t } M(|\psi\rangle)=\left.\left\{\Big(x(t),\,y(t)\Big)\,\right|\,\forall\,t\right\}
  23. M ( e i α | ψ ) = M ( | ψ ) , α M(e^{i\alpha}|\psi\rangle)=M(|\psi\rangle),\ \alpha\in\mathbb{R}
  24. | ψ |\psi\rangle
  25. e i α | ψ e^{i\alpha}|\psi\rangle
  26. 1 / 2 1/\sqrt{2}
  27. c = 1 8 π [ 𝐄 2 ( 𝐫 , t ) + 𝐁 2 ( 𝐫 , t ) ] . \mathcal{E}_{c}=\frac{1}{8\pi}\left[\mathbf{E}^{2}(\mathbf{r},t)+\mathbf{B}^{2% }(\mathbf{r},t)\right].
  28. c = 𝐄 2 8 π \mathcal{E}_{c}=\frac{\mid\mathbf{E}\mid^{2}}{8\pi}
  29. f x = 𝐄 2 cos 2 θ 𝐄 2 = ψ x * ψ x = cos 2 θ f_{x}=\frac{\mid\mathbf{E}\mid^{2}\cos^{2}\theta}{\mid\mathbf{E}\mid^{2}}=\psi% _{x}^{*}\psi_{x}=\cos^{2}\theta
  30. f y = sin 2 θ f_{y}=\sin^{2}\theta
  31. ψ x * ψ x + ψ y * ψ y = ψ | ψ = 1. \psi_{x}^{*}\psi_{x}+\psi_{y}^{*}\psi_{y}=\langle\psi|\psi\rangle=1.
  32. s y m b o l 𝒫 = 1 4 π c 𝐄 ( 𝐫 , t ) × 𝐁 ( 𝐫 , t ) . symbol{\mathcal{P}}={1\over 4\pi c}\mathbf{E}(\mathbf{r},t)\times\mathbf{B}(% \mathbf{r},t).
  33. 𝒫 z c = c . \mathcal{P}_{z}c=\mathcal{E}_{c}.
  34. s y m b o l = 𝐫 × s y m b o l 𝒫 = 1 4 π c 𝐫 × [ 𝐄 ( 𝐫 , t ) × 𝐁 ( 𝐫 , t ) ] . symbol{\mathcal{L}}=\mathbf{r}\times symbol{\mathcal{P}}={1\over 4\pi c}% \mathbf{r}\times\left[\mathbf{E}(\mathbf{r},t)\times\mathbf{B}(\mathbf{r},t)% \right].
  35. z z
  36. z z
  37. = 𝐄 2 8 π ω ( R | ψ 2 - L | ψ 2 ) = 1 ω c ( ψ R 2 - ψ L 2 ) \mathcal{L}={{\mid\mathbf{E}\mid^{2}}\over{8\pi\omega}}\left(\mid\langle R|% \psi\rangle\mid^{2}-\mid\langle L|\psi\rangle\mid^{2}\right)={1\over\omega}% \mathcal{E}_{c}\left(\mid\psi_{R}\mid^{2}-\mid\psi_{L}\mid^{2}\right)
  38. f x = ψ x * ψ x = cos 2 θ . f_{x}=\psi_{x}^{*}\psi_{x}=\cos^{2}\theta.\,
  39. θ \theta
  40. | ψ = ( cos θ sin θ ) |\psi\rangle=\begin{pmatrix}\cos\theta\\ \sin\theta\end{pmatrix}
  41. | ψ = ( cos θ exp ( i α x ) sin θ exp ( i α y ) ) = ( exp ( i α x ) 0 0 exp ( i α y ) ) ( cos θ sin θ ) = def U ^ | ψ . |\psi^{\prime}\rangle=\begin{pmatrix}\cos\theta\exp\left(i\alpha_{x}\right)\\ \sin\theta\exp\left(i\alpha_{y}\right)\end{pmatrix}=\begin{pmatrix}\exp\left(i% \alpha_{x}\right)&0\\ 0&\exp\left(i\alpha_{y}\right)\end{pmatrix}\begin{pmatrix}\cos\theta\\ \sin\theta\end{pmatrix}\ \stackrel{\mathrm{def}}{=}\ \hat{U}|\psi\rangle.
  42. ψ | = ψ | U ^ \langle\psi^{\prime}|=\langle\psi|\hat{U}^{\dagger}
  43. U U^{\dagger}
  44. ψ | ψ = ψ | U ^ U ^ | ψ = ψ | ψ = 1. \langle\psi^{\prime}|\psi^{\prime}\rangle=\langle\psi|\hat{U}^{\dagger}\hat{U}% |\psi\rangle=\langle\psi|\psi\rangle=1.
  45. U ^ U ^ = I , \hat{U}^{\dagger}\hat{U}=I,
  46. U ^ I + i H ^ \hat{U}\approx I+i\hat{H}
  47. U ^ I - i H ^ . \hat{U}^{\dagger}\approx I-i\hat{H}^{\dagger}.
  48. I = U ^ U ^ ( I - i H ^ ) ( I + i H ^ ) I - i H ^ + i H ^ . I=\hat{U}^{\dagger}\hat{U}\approx\left(I-i\hat{H}^{\dagger}\right)\left(I+i% \hat{H}\right)\approx I-i\hat{H}^{\dagger}+i\hat{H}.
  49. H ^ = H ^ . \hat{H}=\hat{H}^{\dagger}.
  50. | ψ - | ψ = i H ^ | ψ . |\psi^{\prime}\rangle-|\psi\rangle=i\hat{H}|\psi\rangle.
  51. ϵ = ω \epsilon=\hbar\omega
  52. \hbar
  53. N N
  54. V V
  55. N ω N\hbar\omega
  56. N ω V {N\hbar\omega\over V}
  57. N N
  58. N ω V = c = 𝐄 2 8 π . {N\hbar\omega\over V}=\mathcal{E}_{c}=\frac{\mid\mathbf{E}\mid^{2}}{8\pi}.
  59. N = V 8 π ω 𝐄 2 . N=\frac{V}{8\pi\hbar\omega}\mid\mathbf{E}\mid^{2}.
  60. 𝒫 z = N ω c V = N k z V \mathcal{P}_{z}={N\hbar\omega\over cV}={N\hbar k_{z}\over V}
  61. p z = k z . p_{z}=\hbar k_{z}.\,
  62. = 1 ω c ( ψ R 2 - ψ L 2 ) = N V ( ψ R 2 - ψ L 2 ) \mathcal{L}={1\over\omega}\mathcal{E}_{c}\left(\mid\psi_{R}\mid^{2}-\mid\psi_{% L}\mid^{2}\right)={N\hbar\over V}\left(\mid\psi_{R}\mid^{2}-\mid\psi_{L}\mid^{% 2}\right)
  63. l z = ( ψ R 2 - ψ L 2 ) . l_{z}=\hbar\left(\mid\psi_{R}\mid^{2}-\mid\psi_{L}\mid^{2}\right).
  64. ψ R 2 \mid\psi_{R}\mid^{2}
  65. \hbar
  66. ψ L 2 \mid\psi_{L}\mid^{2}
  67. - -\hbar
  68. ± \pm\hbar
  69. \hbar
  70. | R |R\rangle
  71. | L |L\rangle
  72. S ^ = def | R R | - | L L | = ( 0 - i i 0 ) . \hat{S}\ \stackrel{\mathrm{def}}{=}\ |R\rangle\langle R|-|L\rangle\langle L|=% \begin{pmatrix}0&-i\\ i&0\end{pmatrix}.
  73. | R |R\rangle
  74. | L |L\rangle
  75. ψ | S ^ | ψ = ψ R 2 - ψ L 2 . \langle\psi|\hat{S}|\psi\rangle=\mid\psi_{R}\mid^{2}-\mid\psi_{L}\mid^{2}.
  76. | s |s\rangle
  77. | R |R\rangle
  78. | L |L\rangle
  79. | ψ = s = - 1 , 1 a s exp ( i α x - i s θ ) | s |\psi\rangle=\sum_{s=-1,1}a_{s}\exp\left(i\alpha_{x}-is\theta\right)|s\rangle
  80. s = - 1 , 1 a s 2 = 1. \sum_{s=-1,1}\mid a_{s}\mid^{2}=1.
  81. S ^ d i θ \hat{S}_{d}\rightarrow i{\partial\over\partial\theta}
  82. S ^ d - i θ . \hat{S}_{d}^{\dagger}\rightarrow-i{\partial\over\partial\theta}.
  83. exp ( i α x - i s θ ) | s . \exp\left(i\alpha_{x}-is\theta\right)|s\rangle.
  84. S ^ d exp ( i α x - i s θ ) | s i θ exp ( i α x - i s θ ) | s = s [ exp ( i α x - i s θ ) | s ] . \hat{S}_{d}\exp\left(i\alpha_{x}-is\theta\right)|s\rangle\rightarrow i{% \partial\over\partial\theta}\exp\left(i\alpha_{x}-is\theta\right)|s\rangle=s% \left[\exp\left(i\alpha_{x}-is\theta\right)|s\rangle\right].
  85. l ^ z = S ^ d . \hat{l}_{z}=\hbar\hat{S}_{d}.
  86. R | x y | R , \langle R|x\rangle\langle y|R\rangle,
  87. y | R R | x , \langle y|R\rangle\langle R|x\rangle,
  88. y | L L | x \langle y|L\rangle\langle L|x\rangle\dots
  89. 1 4 | ( A ^ B ^ - B ^ A ^ ) x | x | 2 A ^ x 2 B ^ x 2 . \frac{1}{4}|\langle(\hat{A}\hat{B}-\hat{B}\hat{A})x|x\rangle|^{2}\leq\|\hat{A}% x\|^{2}\|\hat{B}x\|^{2}.
  90. Δ ψ A ^ Δ ψ B ^ 1 2 | [ A ^ , B ^ ] ψ | \Delta_{\psi}\hat{A}\,\Delta_{\psi}\hat{B}\geq\frac{1}{2}\left|\left\langle% \left[{\hat{A}},{\hat{B}}\right]\right\rangle_{\psi}\right|
  91. X ^ ψ = ψ | X ^ ψ \left\langle\hat{X}\right\rangle_{\psi}=\left\langle\psi|\hat{X}\psi\right\rangle
  92. Δ ψ X ^ = X ^ 2 ψ - X ^ ψ 2 . \Delta_{\psi}\hat{X}=\sqrt{\langle{\hat{X}}^{2}\rangle_{\psi}-\langle{\hat{X}}% \rangle_{\psi}^{2}}.
  93. [ A ^ , B ^ ] = def A ^ B ^ - B ^ A ^ \left[{\hat{A}},{\hat{B}}\right]\ \stackrel{\mathrm{def}}{=}\ \hat{A}\hat{B}-% \hat{B}\hat{A}
  94. Δ ψ l ^ z Δ ψ θ 2 , \Delta_{\psi}\hat{l}_{z}\,\Delta_{\psi}{\theta}\geq\frac{\hbar}{2},

Pi_helix.html

  1. 3 cos Ω = 1 - 4 cos 2 [ ( ϕ + ψ ) / 2 ] 3\cos\Omega=1-4\cos^{2}\left[\left(\phi+\psi\right)/2\right]

Pi_Josephson_junction.html

  1. π \pi
  2. π \pi
  3. 2 π 2\pi
  4. ϕ \phi
  5. ϕ + 2 π n \phi+2\pi n
  6. 0 ϕ < 2 π 0\leq\phi<2\pi
  7. ϕ = π \phi=\pi
  8. ϕ = π \phi=\pi
  9. ϕ = π \phi=\pi
  10. ϕ = π \phi=\pi
  11. π \pi
  12. π \pi
  13. π \pi
  14. π \pi
  15. π \pi
  16. π \pi
  17. π \pi
  18. π \pi
  19. π \pi
  20. π \pi

Picture_language.html

  1. L = { a n , n | n > 0 } L=\left\{a^{n,n}|n>0\right\}
  2. a a
  3. L L
  4. L \in L

Pierce_oscillator.html

  1. C L = [ C 1 + C i ] × [ C 2 + C o ] [ C 1 + C i + C 2 + C o ] + C S C_{L}={[C_{1}+C_{i}]\times[C_{2}+C_{o}]\over[C_{1}+C_{i}+C_{2}+C_{o}]}+C_{% \mathrm{S}}

Pinch_(plasma_physics).html

  1. × B = μ 0 J \nabla\times\vec{B}=\mu_{0}\vec{J}
  2. B = B z ( r ) z ^ \vec{B}=B_{z}(r)\hat{z}
  3. μ 0 J = 1 r d d θ B z r ^ - d d r B z θ ^ \mu_{0}\vec{J}=\frac{1}{r}\frac{d}{d\theta}B_{z}\hat{r}-\frac{d}{dr}B_{z}\hat{\theta}
  4. μ 0 J = - d d r B z θ ^ \mu_{0}\vec{J}=-\frac{d}{dr}B_{z}\hat{\theta}
  5. d d r ( p + B z 2 2 μ 0 ) = 0 \frac{d}{dr}\left(p+\frac{B_{z}^{2}}{2\mu_{0}}\right)=0
  6. × B = μ 0 J \nabla\times\vec{B}=\mu_{0}\vec{J}
  7. B = B θ ( r ) θ ^ \vec{B}=B_{\theta}(r)\hat{\theta}
  8. μ 0 J = 1 r d d r ( r B θ ) z ^ - d d z B θ r ^ \mu_{0}\vec{J}=\frac{1}{r}\frac{d}{dr}(rB_{\theta})\hat{z}-\frac{d}{dz}B_{% \theta}\hat{r}
  9. μ 0 J = 1 r d d r ( r B θ ) z ^ \mu_{0}\vec{J}=\frac{1}{r}\frac{d}{dr}(rB_{\theta})\hat{z}
  10. d d r ( p + B θ 2 2 μ 0 ) + B θ 2 μ 0 r = 0 \frac{d}{dr}\left(p+\frac{B_{\theta}^{2}}{2\mu_{0}}\right)+\frac{B_{\theta}^{2% }}{\mu_{0}r}=0
  11. × B = μ 0 J \nabla\times\vec{B}=\mu_{0}\vec{J}
  12. B = B θ ( r ) θ ^ + B z ( r ) z ^ \vec{B}=B_{\theta}(r)\hat{\theta}+B_{z}(r)\hat{z}
  13. μ 0 J = 1 r d d r ( r B θ ) z ^ - d d r B z θ ^ \mu_{0}\vec{J}=\frac{1}{r}\frac{d}{dr}(rB_{\theta})\hat{z}-\frac{d}{dr}B_{z}% \hat{\theta}
  14. d d r ( p + B z 2 + B θ 2 2 μ 0 ) + B θ 2 μ 0 r = 0 \frac{d}{dr}\left(p+\frac{B_{z}^{2}+B_{\theta}^{2}}{2\mu_{0}}\right)+\frac{B_{% \theta}^{2}}{\mu_{0}r}=0
  15. ( p + B 2 2 μ 0 ) - B 2 μ 0 κ = 0 \nabla_{\perp}\left(p+\frac{B^{2}}{2\mu_{0}}\right)-\frac{B^{2}}{\mu_{0}}\vec{% \kappa}=0
  16. κ = ( b ) b \vec{\kappa}=(\vec{b}\cdot\nabla)\vec{b}
  17. 2 N k ( T e + T i ) = < m t p l > μ 0 2Nk(T_{e}+T_{i})=\frac{<}{m}tpl>{{\mu_{0}}}
  18. 1 4 2 J 0 t 2 = W kin + Δ W E z + Δ W B z + Δ W k - < m t p l > μ 0 8 π I 2 ( a ) - 1 2 G m ¯ 2 N 2 ( a ) + 1 2 π a 2 ϵ 0 ( E r 2 ( a ) - E ϕ 2 ( a ) ) \begin{aligned}\displaystyle\frac{1}{4}\frac{\partial^{2}J_{0}}{\partial t^{2}% }&\displaystyle=W_{\perp\,\text{kin}}+\Delta W_{E_{z}}+\Delta W_{B_{z}}+\Delta W% _{k}-\frac{<}{m}tpl>{{\mu_{0}}}{8\pi}I^{2}(a)\\ &\displaystyle{}-\frac{1}{2}G\overline{m}^{2}N^{2}(a)+\frac{1}{2}\pi a^{2}% \epsilon_{0}\left(E_{r}^{2}(a)-E_{\phi}^{2}(a)\right)\\ \end{aligned}
  19. < m t p l > μ 0 8 π I 2 ( a ) + 1 2 G m ¯ 2 N 2 ( a ) = Δ W B z + Δ W k \frac{<}{m}tpl>{{\mu_{0}}}{8\pi}I^{2}(a)+\frac{1}{2}G\overline{m}^{2}N^{2}(a)=% \Delta W_{B_{z}}+\Delta W_{k}

Pipe_network_analysis.html

  1. Δ Q = head loss c - head loss c c n ( head loss c Q c + head loss c c Q c c ) , \Delta Q=\frac{\sum{\scriptstyle\,\text{head loss}_{c}}-\sum{\scriptstyle\,% \text{head loss}_{cc}}}{n\cdot(\sum\frac{\,\text{head loss}_{c}}{Q_{c}}+\sum% \frac{\,\text{head loss}_{cc}}{Q_{cc}})},

Pisarenko_harmonic_decomposition.html

  1. x ( n ) x(n)
  2. p p
  3. p + 1 p+1
  4. M × M M\times M
  5. ( p + 1 ) × ( p + 1 ) (p+1)\times(p+1)
  6. V m i n ( z ) = k = 0 p v m i n ( k ) z - k V_{min}(z)=\sum_{k=0}^{p}v_{min}(k)z^{-k}
  7. P ^ P H D ( e j ω ) = 1 | 𝐞 H 𝐯 m i n | 2 \hat{P}_{PHD}(e^{j\omega})=\frac{1}{|\mathbf{e}^{H}\mathbf{v}_{min}|^{2}}
  8. 𝐯 m i n \mathbf{v}_{min}
  9. e = [ 1 e j ω e j 2 ω e j ( M - 1 ) ω ] T e=\begin{bmatrix}1&e^{j\omega}&e^{j2\omega}&\cdots&e^{j(M-1)\omega}\end{% bmatrix}^{T}

Piston_motion_equations.html

  1. ω = 2 π RPM 60 \omega=\frac{2\pi\cdot\mathrm{RPM}}{60}
  2. l 2 = r 2 + x 2 - 2 r x cos A l^{2}=r^{2}+x^{2}-2\cdot r\cdot x\cdot\cos A
  3. l 2 - r 2 = x 2 - 2 r x cos A l^{2}-r^{2}=x^{2}-2\cdot r\cdot x\cdot\cos A
  4. l 2 - r 2 = x 2 - 2 r x cos A + r 2 [ ( cos 2 A + sin 2 A ) - 1 ] l^{2}-r^{2}=x^{2}-2\cdot r\cdot x\cdot\cos A+r^{2}[(\cos^{2}A+\sin^{2}A)-1]
  5. l 2 - r 2 + r 2 - r 2 sin 2 A = x 2 - 2 r x cos A + r 2 cos 2 A l^{2}-r^{2}+r^{2}-r^{2}\sin^{2}A=x^{2}-2\cdot r\cdot x\cdot\cos A+r^{2}\cos^{2}A
  6. l 2 - r 2 sin 2 A = ( x - r cos A ) 2 l^{2}-r^{2}\sin^{2}A=(x-r\cdot\cos A)^{2}
  7. x - r cos A = l 2 - r 2 sin 2 A x-r\cdot\cos A=\sqrt{l^{2}-r^{2}\sin^{2}A}
  8. x = r cos A + l 2 - r 2 sin 2 A x=r\cos A+\sqrt{l^{2}-r^{2}\sin^{2}A}
  9. x = d x d A = - r sin A + ( 1 2 ) . ( - 2 ) . r 2 sin A cos A l 2 - r 2 sin 2 A = - r sin A - r 2 sin A cos A l 2 - r 2 sin 2 A \begin{array}[]{lcl}x^{\prime}&=&\frac{dx}{dA}\\ &=&-r\sin A+\frac{(\frac{1}{2}).(-2).r^{2}\sin A\cos A}{\sqrt{l^{2}-r^{2}\sin^% {2}A}}\\ &=&-r\sin A-\frac{r^{2}\sin A\cos A}{\sqrt{l^{2}-r^{2}\sin^{2}A}}\end{array}
  10. x ′′ = d 2 x d A 2 = - r cos A - r 2 cos 2 A l 2 - r 2 sin 2 A - - r 2 sin 2 A l 2 - r 2 sin 2 A - r 2 sin A cos A . ( - 1 2 ) ( - 2 ) . r 2 sin A cos A ( l 2 - r 2 sin 2 A ) 3 = - r cos A - r 2 ( cos 2 A - sin 2 A ) l 2 - r 2 sin 2 A - r 4 sin 2 A cos 2 A ( l 2 - r 2 sin 2 A ) 3 \begin{array}[]{lcl}x^{\prime\prime}&=&\frac{d^{2}x}{dA^{2}}\\ &=&-r\cos A-\frac{r^{2}\cos^{2}A}{\sqrt{l^{2}-r^{2}\sin^{2}A}}-\frac{-r^{2}% \sin^{2}A}{\sqrt{l^{2}-r^{2}\sin^{2}A}}-\frac{r^{2}\sin A\cos A.(-\frac{1}{2})% \cdot(-2).r^{2}\sin A\cos A}{\left(\sqrt{l^{2}-r^{2}\sin^{2}A}\right)^{3}}\\ &=&-r\cos A-\frac{r^{2}(\cos^{2}A-\sin^{2}A)}{\sqrt{l^{2}-r^{2}\sin^{2}A}}-% \frac{r^{4}\sin^{2}A\cos^{2}A}{\left(\sqrt{l^{2}-r^{2}\sin^{2}A}\right)^{3}}% \end{array}
  11. A = ω t A=\omega t\,
  12. d A d t = ω \frac{dA}{dt}=\omega
  13. d 2 A d t 2 = 0 \frac{d^{2}A}{dt^{2}}=0
  14. x x\,
  15. v = d x d t = d x d A d A d t = d x d A ω = x ω \begin{array}[]{lcl}v&=&\frac{dx}{dt}\\ &=&\frac{dx}{dA}\cdot\frac{dA}{dt}\\ &=&\frac{dx}{dA}\cdot\ \omega\\ &=&x^{\prime}\cdot\omega\\ \end{array}
  16. a = d 2 x d t 2 = d d t d x d t = d d t ( d x d A d A d t ) = d d t ( d x d A ) d A d t + d x d A d d t ( d A d t ) = d d A ( d x d A ) ( d A d t ) 2 + d x d A d 2 A d t 2 = d 2 x d A 2 ( d A d t ) 2 + d x d A d 2 A d t 2 = d 2 x d A 2 ω 2 + d x d A 0 = x ′′ ω 2 \begin{array}[]{lcl}a&=&\frac{d^{2}x}{dt^{2}}\\ &=&\frac{d}{dt}\frac{dx}{dt}\\ &=&\frac{d}{dt}(\frac{dx}{dA}\cdot\frac{dA}{dt})\\ &=&\frac{d}{dt}(\frac{dx}{dA})\cdot\frac{dA}{dt}+\frac{dx}{dA}\cdot\frac{d}{dt% }(\frac{dA}{dt})\\ &=&\frac{d}{dA}(\frac{dx}{dA})\cdot(\frac{dA}{dt})^{2}+\frac{dx}{dA}\cdot\frac% {d^{2}A}{dt^{2}}\\ &=&\frac{d^{2}x}{dA^{2}}\cdot(\frac{dA}{dt})^{2}+\frac{dx}{dA}\cdot\frac{d^{2}% A}{dt^{2}}\\ &=&\frac{d^{2}x}{dA^{2}}\cdot\omega^{2}+\frac{dx}{dA}\cdot 0\\ &=&x^{\prime\prime}\cdot\omega^{2}\\ \end{array}

Pitometer_log.html

  1. p T o t a l = p S t a t i c + p D y n a m i c p_{Total}=p_{Static}+p_{Dynamic}\,\!
  2. p T o t a l = p S t a t i c + ρ v W a t e r 2 2 p_{Total}=p_{Static}+\rho\cdot\frac{v_{Water}^{2}}{2}\,\!
  3. v W a t e r = 2 ( p T o t a l - p S t a t i c ) ρ v_{Water}=\sqrt{\frac{2\cdot\left(p_{Total}-p_{Static}\right)}{\rho}}\,\!

Pivotal_quantity.html

  1. X = ( X 1 , X 2 , , X n ) X=(X_{1},X_{2},\ldots,X_{n})
  2. θ \theta
  3. g ( X , θ ) g(X,\theta)
  4. θ \theta
  5. g g
  6. μ \mu
  7. σ 2 \sigma^{2}
  8. z = x - μ σ , z=\frac{x-\mu}{\sigma},
  9. N ( 0 , 1 ) N(0,1)
  10. N ( μ , σ 2 / n ) , N(\mu,\sigma^{2}/n),
  11. z = X ¯ - μ σ / n z=\frac{\overline{X}-\mu}{\sigma/\sqrt{n}}
  12. N ( 0 , 1 ) . N(0,1).
  13. n n
  14. X = ( X 1 , X 2 , , X n ) X=(X_{1},X_{2},\ldots,X_{n})
  15. μ \mu
  16. σ 2 \sigma^{2}
  17. g ( x , X ) = n x - X ¯ s g(x,X)=\sqrt{n}\frac{x-\overline{X}}{s}
  18. X ¯ = 1 n i = 1 n X i \overline{X}=\frac{1}{n}\sum_{i=1}^{n}{X_{i}}
  19. s 2 = 1 n - 1 i = 1 n ( X i - X ¯ ) 2 s^{2}=\frac{1}{n-1}\sum_{i=1}^{n}{(X_{i}-\overline{X})^{2}}
  20. μ \mu
  21. σ 2 \sigma^{2}
  22. g ( x , X ) g(x,X)
  23. x x
  24. X X
  25. x = μ x=\mu
  26. g ( μ , X ) g(\mu,X)
  27. ν = n - 1 \nu=n-1
  28. μ \mu
  29. g g
  30. g ( μ , X ) g(\mu,X)
  31. μ \mu
  32. σ \sigma
  33. X 1 , , X n X_{1},\ldots,X_{n}
  34. X n + 1 ; X_{n+1};
  35. n n
  36. ( X i , Y i ) (X_{i},Y_{i})^{\prime}
  37. ρ \rho
  38. ρ \rho
  39. r = 1 n - 1 i = 1 n ( X i - X ¯ ) ( Y i - Y ¯ ) s X s Y r=\frac{\frac{1}{n-1}\sum_{i=1}^{n}(X_{i}-\overline{X})(Y_{i}-\overline{Y})}{s% _{X}s_{Y}}
  40. s X 2 , s Y 2 s_{X}^{2},s_{Y}^{2}
  41. X X
  42. Y Y
  43. r r
  44. n r - ρ 1 - ρ 2 N ( 0 , 1 ) \sqrt{n}\frac{r-\rho}{1-\rho^{2}}\Rightarrow N(0,1)
  45. z = tanh - 1 r = 1 2 ln 1 + r 1 - r z=\rm{tanh}^{-1}r=\frac{1}{2}\ln\frac{1+r}{1-r}
  46. z z
  47. n ( z - ζ ) N ( 0 , 1 ) \sqrt{n}(z-\zeta)\Rightarrow N(0,1)
  48. ζ = tanh - 1 ρ \zeta={\rm tanh}^{-1}\rho
  49. n n
  50. z z
  51. r r
  52. Var ( z ) 1 n - 3 . \operatorname{Var}(z)\approx\frac{1}{n-3}.

Pixel_aspect_ratio.html

  1. 123 / 11 12{3}/{11}
  2. 143 / 4 14{3}/{4}
  3. 131 / 2 13{1}/{2}
  4. 12 3 11 ÷ 13 1 2 = 10 11 12\tfrac{3}{11}\div 13\tfrac{1}{2}=\tfrac{10}{11}
  5. 14 3 4 ÷ 13 1 2 = 59 54 14\tfrac{3}{4}\div 13\tfrac{1}{2}=\tfrac{59}{54}
  6. 4320 4739 \frac{4320}{4739}
  7. 5760 4739 \frac{5760}{4739}
  8. 30 33 \frac{30}{33}
  9. 40 33 \frac{40}{33}
  10. 9 10 \frac{9}{10}
  11. 6 5 \frac{6}{5}
  12. 24 27 \frac{24}{27}
  13. 32 27 \frac{32}{27}
  14. x 480 × 10 11 = 4 3 x = 480 × 11 × 4 10 × 3 = 704 \frac{x}{480}\times\frac{10}{11}=\frac{4}{3}\Rightarrow x=\frac{480\times 11% \times 4}{10\times 3}=704
  15. x 576 × 59 54 = 4 3 x = 576 × 54 × 4 59 × 3 702.915254 \frac{x}{576}\times\frac{59}{54}=\frac{4}{3}\Rightarrow x=\frac{576\times 54% \times 4}{59\times 3}\approx 702.915254
  16. 4 3 ÷ 704 576 = \color b l u e 12 11 \frac{4}{3}\div\frac{704}{576}=\color{blue}{\frac{12}{11}}
  17. 16 9 ÷ 704 576 = \color b l u e 16 11 \frac{16}{9}\div\frac{704}{576}=\color{blue}{\frac{16}{11}}
  18. 925 ¯ \overline{925}
  19. 09 ¯ \overline{09}
  20. 456790123 ¯ \overline{456790123}
  21. 45 ¯ \overline{45}
  22. 90 ¯ \overline{90}
  23. 21 ¯ \overline{21}
  24. 3 ¯ \overline{3}

Plane_stress.html

  1. σ = [ σ 11 0 0 0 σ 22 0 0 0 0 ] [ σ x 0 0 0 σ y 0 0 0 0 ] \sigma=\begin{bmatrix}\sigma_{11}&0&0\\ 0&\sigma_{22}&0\\ 0&0&0\end{bmatrix}\equiv\begin{bmatrix}\sigma_{x}&0&0\\ 0&\sigma_{y}&0\\ 0&0&0\end{bmatrix}
  2. x x
  3. y y
  4. z z
  5. x x
  6. y y
  7. σ = [ 500 P a 0 0 0 - 4000 P a 0 0 0 0 ] \sigma=\begin{bmatrix}500\mathrm{Pa}&0&0\\ 0&-4000\mathrm{Pa}&0\\ 0&0&0\end{bmatrix}
  8. σ = [ σ 11 σ 12 0 σ 21 σ 22 0 0 0 0 ] [ σ x τ x y 0 τ y x σ y 0 0 0 0 ] \sigma=\begin{bmatrix}\sigma_{11}&\sigma_{12}&0\\ \sigma_{21}&\sigma_{22}&0\\ 0&0&0\end{bmatrix}\equiv\begin{bmatrix}\sigma_{x}&\tau_{xy}&0\\ \tau_{yx}&\sigma_{y}&0\\ 0&0&0\end{bmatrix}
  9. σ i j = [ σ 11 σ 12 σ 21 σ 22 ] [ σ x τ x y τ y x σ y ] \sigma_{ij}=\begin{bmatrix}\sigma_{11}&\sigma_{12}\\ \sigma_{21}&\sigma_{22}\end{bmatrix}\equiv\begin{bmatrix}\sigma_{x}&\tau_{xy}% \\ \tau_{yx}&\sigma_{y}\end{bmatrix}
  10. ε i j = [ ε 11 ε 12 0 ε 21 ε 22 0 0 0 ε 33 ] \varepsilon_{ij}=\begin{bmatrix}\varepsilon_{11}&\varepsilon_{12}&0\\ \varepsilon_{21}&\varepsilon_{22}&0\\ 0&0&\varepsilon_{33}\end{bmatrix}\,\!
  11. ε 33 \varepsilon_{33}\,\!
  12. P P\,\!
  13. ( σ x , σ y , τ x y ) (\sigma_{x},\sigma_{y},\tau_{xy})\,\!
  14. P P\,\!
  15. σ n \sigma_{\mathrm{n}}\,\!
  16. τ n \tau_{\mathrm{n}}\,\!
  17. x x\,\!
  18. y y\,\!
  19. P P\,\!
  20. 𝐧 \mathbf{n}\,\!
  21. θ \theta\,\!
  22. cos θ \cos\theta\,\!
  23. x x\,\!
  24. σ n = 1 2 ( σ x + σ y ) + 1 2 ( σ x - σ y ) cos 2 θ + τ x y sin 2 θ \sigma_{\mathrm{n}}=\frac{1}{2}(\sigma_{x}+\sigma_{y})+\frac{1}{2}(\sigma_{x}-% \sigma_{y})\cos 2\theta+\tau_{xy}\sin 2\theta\,\!
  25. τ n = - 1 2 ( σ x - σ y ) sin 2 θ + τ x y cos 2 θ \tau_{\mathrm{n}}=-\frac{1}{2}(\sigma_{x}-\sigma_{y})\sin 2\theta+\tau_{xy}% \cos 2\theta\,\!
  26. θ \theta\,\!
  27. ( σ x , σ y , τ x y ) (\sigma_{x},\sigma_{y},\tau_{xy})\,\!
  28. y y\,\!
  29. z z\,\!
  30. τ n \tau_{\mathrm{n}}\,\!
  31. τ n = - 1 2 ( σ x - σ y ) sin 2 θ + τ x y cos 2 θ = 0 \tau_{\mathrm{n}}=-\frac{1}{2}(\sigma_{x}-\sigma_{y})\sin 2\theta+\tau_{xy}% \cos 2\theta=0\,\!
  32. tan 2 θ p = 2 τ x y σ x - σ y \tan 2\theta_{\mathrm{p}}=\frac{2\tau_{xy}}{\sigma_{x}-\sigma_{y}}\,\!
  33. θ p \theta_{\mathrm{p}}\,\!
  34. 90 90^{\circ}\,\!
  35. θ \theta\,\!
  36. σ n \sigma_{\mathrm{n}}\,\!
  37. d σ n d θ = 0 \frac{d\sigma_{\mathrm{n}}}{d\theta}=0\,\!
  38. σ 1 \sigma_{1}\,\!
  39. σ 2 \sigma_{2}\,\!
  40. σ max \sigma_{\mathrm{max}}\,\!
  41. σ min \sigma_{\mathrm{min}}\,\!
  42. θ p \theta_{\mathrm{p}}\,\!
  43. σ n \sigma_{\mathrm{n}}\,\!
  44. σ n \sigma_{\mathrm{n}}\,\!
  45. τ n \tau_{\mathrm{n}}\,\!
  46. [ σ n - 1 2 ( σ x + σ y ) ] 2 + τ n 2 = [ 1 2 ( σ x - σ y ) ] 2 + τ x y 2 \left[\sigma_{\mathrm{n}}-\tfrac{1}{2}(\sigma_{x}+\sigma_{y})\right]^{2}+\tau_% {\mathrm{n}}^{2}=\left[\tfrac{1}{2}(\sigma_{x}-\sigma_{y})\right]^{2}+\tau_{xy% }^{2}\,\!
  47. ( σ n - σ avg ) 2 + τ n 2 = R 2 (\sigma_{\mathrm{n}}-\sigma_{\mathrm{avg}})^{2}+\tau_{\mathrm{n}}^{2}=R^{2}\,\!
  48. R = [ 1 2 ( σ x - σ y ) ] 2 + τ x y 2 and σ avg = 1 2 ( σ x + σ y ) R=\sqrt{\left[\tfrac{1}{2}(\sigma_{x}-\sigma_{y})\right]^{2}+\tau_{xy}^{2}}% \quad\,\text{and}\quad\sigma_{\mathrm{avg}}=\tfrac{1}{2}(\sigma_{x}+\sigma_{y}% )\,\!
  49. R R\,\!
  50. [ σ avg , 0 ] [\sigma_{\mathrm{avg}},0]\,\!
  51. τ n = 0 \tau_{\mathrm{n}}=0\,\!
  52. σ 1 = σ max = 1 2 ( σ x + σ y ) + [ 1 2 ( σ x - σ y ) ] 2 + τ x y 2 \sigma_{1}=\sigma_{\mathrm{max}}=\tfrac{1}{2}(\sigma_{x}+\sigma_{y})+\sqrt{% \left[\tfrac{1}{2}(\sigma_{x}-\sigma_{y})\right]^{2}+\tau_{xy}^{2}}\,\!
  53. σ 2 = σ min = 1 2 ( σ x + σ y ) - [ 1 2 ( σ x - σ y ) ] 2 + τ x y 2 \sigma_{2}=\sigma_{\mathrm{min}}=\tfrac{1}{2}(\sigma_{x}+\sigma_{y})-\sqrt{% \left[\tfrac{1}{2}(\sigma_{x}-\sigma_{y})\right]^{2}+\tau_{xy}^{2}}\,\!
  54. τ x y = 0 \tau_{xy}=0\,\!
  55. σ x = σ 1 \sigma_{x}=\sigma_{1}\,\!
  56. σ y = σ 2 \sigma_{y}=\sigma_{2}\,\!
  57. σ n \sigma_{\mathrm{n}}\,\!
  58. τ n \tau_{\mathrm{n}}\,\!
  59. τ x y = 0 \tau_{xy}=0\,\!
  60. σ n = 1 2 ( σ 1 + σ 2 ) + 1 2 ( σ 1 - σ 2 ) cos 2 θ \sigma_{\mathrm{n}}=\frac{1}{2}(\sigma_{1}+\sigma_{2})+\frac{1}{2}(\sigma_{1}-% \sigma_{2})\cos 2\theta\,\!
  61. τ n = - 1 2 ( σ 1 - σ 2 ) sin 2 θ \tau_{\mathrm{n}}=-\frac{1}{2}(\sigma_{1}-\sigma_{2})\sin 2\theta\,\!
  62. τ max \tau_{\mathrm{max}}\,\!
  63. sin 2 θ = 1 \sin 2\theta=1\,\!
  64. θ = 45 \theta=45^{\circ}\,\!
  65. τ max = 1 2 ( σ 1 - σ 2 ) \tau_{\mathrm{max}}=\frac{1}{2}(\sigma_{1}-\sigma_{2})\,\!
  66. τ min \tau_{\mathrm{min}}\,\!
  67. sin 2 θ = - 1 \sin 2\theta=-1\,\!
  68. θ = 135 \theta=135^{\circ}\,\!
  69. τ min = - 1 2 ( σ 1 - σ 2 ) \tau_{\mathrm{min}}=-\frac{1}{2}(\sigma_{1}-\sigma_{2})\,\!

Planum_Boreum.html

  1. d = 78.5 - 54.7 360 × 2 π × 3400 km = 1412 km d=\frac{78.5-54.7}{360}\times 2\pi\ \times 3400\mbox{ km}~{}=1412\mbox{ km}~{}

Plasma_modeling.html

  1. f ( x , v , t ) f(\vec{x},\vec{v},t)
  2. x \vec{x}
  3. v \vec{v}

Plasma_parameter.html

  1. Λ = 4 π n λ D 3 \Lambda=4\pi n\lambda_{D}^{3}
  2. N D N_{D}
  3. N D = 4 π 3 n λ D 3 N_{D}=\frac{4\pi}{3}n\lambda_{D}^{3}
  4. 4 π / 3 4\pi/3
  5. λ D = ϵ 0 k T e n e q e 2 \lambda_{D}=\sqrt{\frac{\epsilon_{0}kT_{e}}{n_{e}q_{e}^{2}}}
  6. N D = ( ϵ 0 k T e ) 3 / 2 q e 3 n e 1 / 2 N_{D}=\frac{(\epsilon_{0}kT_{e})^{3/2}}{q_{e}^{3}n_{e}^{1/2}}
  7. ϵ p = Λ - 1 \epsilon_{p}=\Lambda^{-1}
  8. Γ \Gamma
  9. Γ = E C k T e \Gamma=\frac{E_{\mathrm{C}}}{kT_{e}}
  10. E C = q e 2 4 π ϵ 0 r E_{\mathrm{C}}=\frac{q_{e}^{2}}{4\pi\epsilon_{0}\langle r\rangle}
  11. r \langle r\rangle
  12. Γ = q e 2 4 π ϵ 0 k T e 4 π n e 3 3 \Gamma=\frac{q_{e}^{2}}{4\pi\epsilon_{0}kT_{e}}\sqrt[3]{\frac{4\pi n_{e}}{3}}
  13. Γ Λ - 2 / 3 \Gamma\sim\Lambda^{-2/3}
  14. Γ s = q s 2 4 π ϵ 0 k T s 4 π n s 3 3 \Gamma_{s}=\frac{q_{s}^{2}}{4\pi\epsilon_{0}kT_{s}}\sqrt[3]{\frac{4\pi n_{s}}{% 3}}
  15. Λ 1 ( Γ 1 ) \Lambda\ll 1~{}(\Gamma\gg 1)
  16. Λ 1 ( Γ 1 ) \Lambda\gg 1~{}(\Gamma\ll 1)

Plateau–Rayleigh_instability.html

  1. R ( z ) = R 0 + A k cos ( k z ) \scriptstyle R\left(z\right)\;=\;R_{0}\,+\,A_{k}\cos\left(kz\right)
  2. R 0 \scriptstyle R_{0}
  3. A k \scriptstyle A_{k}
  4. z \scriptstyle z
  5. k \scriptstyle k
  6. k R 0 < 1 \scriptstyle kR_{0}\;<\;1
  7. k R 0 0.697 kR_{0}\;\simeq\;0.697

Plutonium.html

  1. U 92 238 + 0 1 n 92 239 U β - 23.5 min 93 239 Np β - 2.3565 d 94 239 Pu \mathrm{{}^{238}_{\ 92}U\ +\ ^{1}_{0}n\ \longrightarrow\ ^{239}_{\ 92}U\ % \xrightarrow[23.5\ min]{\beta^{-}}\ ^{239}_{\ 93}Np\ \xrightarrow[2.3565\ d]{% \beta^{-}}\ ^{239}_{\ 94}Pu}
  2. U 92 238 + 1 2 D 93 238 Np + 2 0 1 n ; 93 238 Np β - 2.117 d 94 238 Pu \mathrm{{}^{238}_{\ 92}U\ +\ ^{2}_{1}D\ \longrightarrow\ ^{238}_{\ 93}Np\ +\ 2% \ ^{1}_{0}n\quad;\quad^{238}_{\ 93}Np\ \xrightarrow[2.117\ d]{\beta^{-}}\ ^{23% 8}_{\ 94}Pu}

Poinsot's_ellipsoid.html

  1. s y m b o l ω symbol\omega
  2. s y m b o l ω symbol\omega
  3. s y m b o l ω symbol\omega
  4. T T
  5. d T d t = 0 \frac{dT}{dt}=0
  6. 𝐈 \mathbf{I}
  7. s y m b o l ω symbol\omega
  8. T = 1 2 s y m b o l ω 𝐈 s y m b o l ω = 1 2 I 1 ω 1 2 + 1 2 I 2 ω 2 2 + 1 2 I 3 ω 3 2 T=\frac{1}{2}symbol\omega\cdot\mathbf{I}\cdot symbol\omega=\frac{1}{2}I_{1}% \omega_{1}^{2}+\frac{1}{2}I_{2}\omega_{2}^{2}+\frac{1}{2}I_{3}\omega_{3}^{2}
  9. ω k \omega_{k}
  10. s y m b o l ω symbol\omega
  11. I k I_{k}
  12. s y m b o l ω symbol\omega
  13. s y m b o l ω symbol\omega
  14. 𝐋 \mathbf{L}
  15. d 𝐋 d t = 0 \frac{d\mathbf{L}}{dt}=0
  16. 𝐋 \mathbf{L}
  17. 𝐈 \mathbf{I}
  18. s y m b o l ω symbol\omega
  19. 𝐋 = 𝐈 s y m b o l ω \mathbf{L}=\mathbf{I}\cdot symbol\omega
  20. T = 1 2 s y m b o l ω 𝐋 . T=\frac{1}{2}symbol\omega\cdot\mathbf{L}.
  21. s y m b o l ω symbol\omega
  22. 𝐋 \mathbf{L}
  23. 𝐋 \mathbf{L}
  24. s y m b o l ω symbol\omega
  25. 𝐋 \mathbf{L}
  26. s y m b o l ω symbol\omega
  27. 𝐋 \mathbf{L}
  28. 𝐋 \mathbf{L}
  29. s y m b o l ω symbol\omega
  30. s y m b o l ω symbol\omega
  31. 𝐋 \mathbf{L}
  32. d T d s y m b o l ω = 𝐈 s y m b o l ω = 𝐋 . \frac{dT}{dsymbol\omega}=\mathbf{I}\cdot symbol\omega=\mathbf{L}.
  33. s y m b o l ω symbol\omega
  34. 𝐋 \mathbf{L}
  35. s y m b o l ω symbol\omega
  36. L L
  37. T T
  38. L 2 = L 1 2 + L 2 2 + L 3 2 L^{2}=L_{1}^{2}+L_{2}^{2}+L_{3}^{2}
  39. T = L 1 2 2 I 1 + L 2 2 2 I 2 + L 3 2 2 I 3 T=\frac{L_{1}^{2}}{2I_{1}}+\frac{L_{2}^{2}}{2I_{2}}+\frac{L_{3}^{2}}{2I_{3}}
  40. L k L_{k}
  41. I k I_{k}
  42. 𝐋 \mathbf{L}
  43. 𝐋 \mathbf{L}
  44. 𝐋 \mathbf{L}
  45. 𝐋 \mathbf{L}
  46. 𝐋 \mathbf{L}
  47. s y m b o l ω symbol\omega

Point-to-point_Lee_model.html

  1. L = L 0 + γ g log d - 10 ( log F A - 2 log ( H E T 30 ) ) L=L_{0}\;+\;\gamma g\log d\;-10(\log{F_{A}}-2\log(\frac{H_{ET}}{30}))
  2. γ \gamma\;
  3. L 0 = G B + G M + 20 ( log λ - log d ) - 22 L_{0}\;=\;G_{B}\;+\;G_{M}\;+\;20\;(\log\lambda\;-\log d\;)\;-\;22
  4. λ \lambda
  5. F A = F B H F B G F M H F M G F F F_{A}\;=\;F_{BH}\;F_{BG}\;F_{MH}\;F_{MG}\;F_{F}
  6. F 1 = ( h B 30.48 ) 2 F_{1}\;=\;\big(\frac{h_{B}}{30.48}\big)^{2}
  7. F 2 = G B 4 F_{2}\;=\;\frac{G_{B}}{4}
  8. F 3 = { h M 3 if, h m > 3 ( h M 3 ) 2 if, h M 3 F_{3}\;=\;\begin{cases}\frac{h_{M}}{3}\mbox{ if,}~{}h_{m}>3\\ (\frac{h_{M}}{3})^{2}\mbox{ if,}~{}h_{M}\leq 3\end{cases}
  9. F 4 = G M F_{4}\;=\;G_{M}
  10. F 5 = ( f 900 ) - n for 2 < n < 3 F_{5}\;=\;(\frac{f}{900})^{-n}\mbox{ for }~{}2<n<3

Point_source.html

  1. θ \theta
  2. θ λ / D \theta<<\lambda/D
  3. λ \lambda
  4. D D

Pointwise_mutual_information.html

  1. pmi ( x ; y ) log p ( x , y ) p ( x ) p ( y ) = log p ( x | y ) p ( x ) = log p ( y | x ) p ( y ) . \operatorname{pmi}(x;y)\equiv\log\frac{p(x,y)}{p(x)p(y)}=\log\frac{p(x|y)}{p(x% )}=\log\frac{p(y|x)}{p(y)}.
  2. p ( x , y ) p(x,y)
  3. pmi ( x ; y ) = pmi ( y ; x ) \operatorname{pmi}(x;y)=\operatorname{pmi}(y;x)
  4. p ( x | y ) p(x|y)
  5. p ( y | x ) = 1 p(y|x)=1
  6. - pmi ( x ; y ) min [ - log p ( x ) , - log p ( y ) ] . -\infty\leq\operatorname{pmi}(x;y)\leq\min\left[-\log p(x),-\log p(y)\right].
  7. pmi ( x ; y ) \operatorname{pmi}(x;y)
  8. p ( x | y ) p(x|y)
  9. p ( x ) p(x)
  10. p m i ( x ; y ) pmi(x;y)
  11. I ( X ; Y ) \operatorname{I}(X;Y)
  12. pmi ( x ; y ) = h ( x ) + h ( y ) - h ( x , y ) = h ( x ) - h ( x | y ) = h ( y ) - h ( y | x ) \begin{aligned}\displaystyle\operatorname{pmi}(x;y)&\displaystyle=&% \displaystyle h(x)+h(y)-h(x,y)\\ &\displaystyle=&\displaystyle h(x)-h(x|y)\\ &\displaystyle=&\displaystyle h(y)-h(y|x)\end{aligned}
  13. h ( x ) h(x)
  14. - log 2 p ( X = x ) -\log_{2}p(X=x)
  15. npmi ( x ; y ) = pmi ( x ; y ) - log [ p ( x , y ) ] \operatorname{npmi}(x;y)=\frac{\operatorname{pmi}(x;y)}{-\log\left[p(x,y)% \right]}
  16. pmi ( x ; y z ) = pmi ( x ; y ) + pmi ( x ; z | y ) \operatorname{pmi}(x;yz)=\operatorname{pmi}(x;y)+\operatorname{pmi}(x;z|y)
  17. pmi ( x ; y ) + pmi ( x ; z | y ) \displaystyle\operatorname{pmi}(x;y)+\operatorname{pmi}(x;z|y)

Poisson_random_measure.html

  1. ( E , 𝒜 , μ ) (E,\mathcal{A},\mu)
  2. σ \sigma
  3. μ \mu
  4. μ \mu
  5. { N A } A 𝒜 \{N_{A}\}_{A\in\mathcal{A}}
  6. ( Ω , , P ) (\Omega,\mathcal{F},\mathrm{P})
  7. A 𝒜 , N A \forall A\in\mathcal{A},\quad N_{A}
  8. μ ( A ) \mu(A)
  9. A 1 , A 2 , , A n 𝒜 A_{1},A_{2},\ldots,A_{n}\in\mathcal{A}
  10. ω Ω N ( ω ) \forall\omega\in\Omega\;N_{\bullet}(\omega)
  11. ( E , 𝒜 ) (E,\mathcal{A})
  12. μ 0 \mu\equiv 0
  13. N 0 N\equiv 0
  14. μ \mu
  15. Z Z
  16. μ ( E ) \mu(E)
  17. X 1 , X 2 , X_{1},X_{2},\ldots
  18. μ μ ( E ) \frac{\mu}{\mu(E)}
  19. N ( ω ) = i = 1 Z ( ω ) δ X i ( ω ) ( ) N_{\cdot}(\omega)=\sum\limits_{i=1}^{Z(\omega)}\delta_{X_{i}(\omega)}(\cdot)
  20. δ c ( A ) \delta_{c}(A)
  21. c c
  22. N N
  23. μ \mu
  24. N N
  25. E E
  26. μ \mu

Poisson–Boltzmann_equation.html

  1. 2 ψ = 2 ψ x 2 + 2 ψ y 2 + 2 ψ z 2 = - ρ e ϵ ϵ 0 \nabla^{2}\psi=\frac{\partial^{2}\psi}{\partial x^{2}}+\frac{\partial^{2}\psi}% {\partial y^{2}}+\frac{\partial^{2}\psi}{\partial z^{2}}=-\frac{\rho_{e}}{% \epsilon\epsilon_{0}}
  2. ρ e \rho_{e}
  3. ϵ \epsilon
  4. ϵ 0 \epsilon_{0}
  5. c i = c i 0 e - W i k B T c_{i}=c^{0}_{i}\cdot e^{\frac{-W_{i}}{k_{B}T}}
  6. c i 0 c^{0}_{i}
  7. W i W_{i}
  8. k B k_{B}
  9. T T
  10. ψ \psi
  11. W + = e ψ W^{+}=e\psi
  12. W - = - e ψ W^{-}=-e\psi
  13. c - = c 0 e e ψ ( x , y , z ) k B T c^{-}=c_{0}\cdot e^{\frac{e\psi(x,y,z)}{k_{B}T}}
  14. c + = c 0 e - e ψ ( x , y , z ) k B T c^{+}=c_{0}\cdot e^{\frac{-e\psi(x,y,z)}{k_{B}T}}
  15. ρ e = e ( c + - c - ) = c 0 e [ e - e ψ ( x , y , z ) k B T - e e ψ ( x , y , z ) k B T ] \rho_{e}=e{(c^{+}-c^{-})}=c_{0}e\cdot[e^{\frac{-e\psi(x,y,z)}{k_{B}T}}-e^{% \frac{e\psi(x,y,z)}{k_{B}T}}]
  16. d 2 ψ d x 2 \frac{d^{2}\psi}{dx^{2}}
  17. c 0 e ϵ ϵ 0 [ e e ψ ( x ) k B T - e - e ψ ( x ) k B T ] \frac{c_{0}e}{\epsilon\epsilon_{0}}\cdot[e^{\frac{e\psi(x)}{k_{B}T}}-e^{\frac{% -e\psi(x)}{k_{B}T}}]
  18. d 2 ψ d r 2 + L r d ψ d r = e ψ - δ e - ψ \frac{d^{2}\psi}{dr^{2}}+\frac{L}{r}\frac{d\psi}{dr}=e^{\psi}-\delta e^{-\psi}
  19. ψ = e Φ k T \psi=\frac{e\Phi}{kT}
  20. R e D = k T 4 π e 2 n e 0 R_{eD}=\sqrt{\frac{kT}{4\pi e^{2}n_{e0}}}
  21. n e 0 n_{e0}
  22. ψ = ψ 0 e - K x \psi=\psi_{0}e^{-Kx}
  23. e | ψ | k B T e\left|\psi\right|\ll k_{B}T
  24. ψ 25 m V \psi\leq 25mV
  25. λ D \lambda_{D}
  26. K = 2 c 0 e 2 ϵ ϵ 0 k B T K=\sqrt{\frac{2c_{0}e^{2}}{\epsilon\epsilon_{0}k_{B}T}}
  27. λ D = K - 1 \lambda_{D}=K^{-1}
  28. 25 C 25^{\circ}C
  29. λ D = 3.04 A c 0 L m o l \lambda_{D}=\frac{3.04A}{\sqrt{c_{0}\frac{L}{mol}}}
  30. K = e 2 ϵ ϵ 0 k B T Σ c i Z i 2 K=\sqrt{\frac{e^{2}}{\epsilon\epsilon_{0}k_{B}T}\Sigma c_{i}{Z_{i}}^{2}}
  31. y e ψ k B T y\equiv\frac{e\psi}{k_{B}T}
  32. e - K x = ( e y / 2 - 1 ) ( e y 0 / 2 + 1 ) ( e y / 2 + 1 ) ( e y 0 / 2 - 1 ) e^{-Kx}=\frac{(e^{y/2}-1)(e^{y_{0}/2}+1)}{(e^{y/2}+1)(e^{y_{0}/2}-1)}
  33. e y / 2 e^{y/2}
  34. e y / 2 = e y 0 / 2 + 1 + ( e y 0 / 2 - 1 ) * e - K x e y 0 / 2 + 1 - ( e y 0 / 2 - 1 ) * e - K x e^{y/2}=\frac{e^{y_{0}/2}+1+(e^{y_{0}/2}-1)*e^{-Kx}}{e^{y_{0}/2}+1-(e^{y_{0}/2% }-1)*e^{-Kx}}
  35. y = 2 ln e y 0 / 2 + 1 + ( e y 0 / 2 - 1 ) * e - K x e y 0 / 2 + 1 - ( e y 0 / 2 - 1 ) * e - K x y=2\ln\frac{e^{y_{0}/2}+1+(e^{y_{0}/2}-1)*e^{-Kx}}{e^{y_{0}/2}+1-(e^{y_{0}/2}-% 1)*e^{-Kx}}
  36. y e ψ k B T y\equiv\frac{e\psi}{k_{B}T}
  37. ψ \psi
  38. ψ = 2 k B T e * ln e y 0 / 2 + 1 + ( e y 0 / 2 - 1 ) * e - K x e y 0 / 2 + 1 - ( e y 0 / 2 - 1 ) * e - K x \psi=\frac{2k_{B}T}{e}*\ln\frac{e^{y_{0}/2}+1+(e^{y_{0}/2}-1)*e^{-Kx}}{e^{y_{0% }/2}+1-(e^{y_{0}/2}-1)*e^{-Kx}}
  39. y 0 = e ψ 0 k B T y_{0}=\frac{e\psi_{0}}{k_{B}T}
  40. G e l \vartriangle G^{el}
  41. τ q U ( τ ) d τ \int\limits^{\tau}qU(\tau^{\prime})\,d\tau^{\prime}
  42. τ q \tau q
  43. Δ G e l \Delta G^{el}
  44. V ( k T i c i [ 1 - exp ( - z i q U k T ) ] + p f U - - ϵ ( U ) 2 8 π ) d V \int\limits_{V}(kT\sum_{i}c_{i}^{\infty}[1-\exp(\frac{-z_{i}qU}{kT})]+p^{f}U-% \frac{-\epsilon(\vec{\nabla}U)^{2}}{8\pi})dV
  45. Δ G e l \Delta G^{el}
  46. Δ G e f + Δ G e m + Δ G m o b + Δ G s o l v \Delta G^{ef}+\Delta G^{em}+\Delta G^{mob}+\Delta G^{solv}
  47. Δ G e f \Delta G^{ef}
  48. V p f U 2 d V \int\limits_{V}\frac{p^{f}U}{2}dV
  49. Δ G e m \Delta G^{em}
  50. V i c i z i q U 2 d V \int\limits_{V}\frac{\sum_{i}c_{i}z_{i}qU}{2}dV
  51. Δ G m o b \Delta G^{mob}
  52. k T V c i ln c i c i d V kT\int\limits_{V}c_{i}\ln\frac{c_{i}}{c_{i}^{\infty}}dV
  53. Δ G s o l v \Delta G^{solv}
  54. k T V i c i [ 1 - exp ( - z i q U k T ) ] d V kT\int\limits_{V}\sum_{i}c_{i}^{\infty}[1-\exp(\frac{-z_{i}qU}{kT})]dV
  55. Δ G o u t \Delta G^{out}
  56. Δ G e m + Δ G m o b + Δ G s o l v \Delta G^{em}+\Delta G^{mob}+\Delta G^{solv}
  57. f 1 f_{1}
  58. f 0 - f 0 + e E z τ 0 m f 0 v z ( 1 - e - τ τ 0 - 0 t e m e t - τ τ 0 ρ [ r - v ( t - t ) ] × f 0 v d t f^{0}-f_{0}+\frac{eE_{z}\tau_{0}}{m}\frac{\partial f_{0}}{\partial v_{z}}(1-e^% {\frac{-\tau}{\tau_{0}}}-\int\limits_{0}^{t}\frac{e}{m}e{{}^{\frac{t-\tau^{% \prime}}{\tau_{0}}}}\bigtriangledown\rho[r-v(t-t^{\prime})]\times\frac{% \partial f_{0}}{\partial v}dt^{\prime}
  59. ρ \rho
  60. ρ 1 + ρ 2 \rho_{1}+\rho_{2}
  61. ρ 1 \rho_{1}
  62. \approx
  63. a E z 2 λ D 1 e - λ D 1 z \frac{aE_{z}}{2\lambda_{D1}}e^{-\lambda_{D1}z}
  64. ρ 2 \rho_{2}
  65. \approx
  66. n e π G ( i λ D 1 ) e - t τ 0 - λ D 1 z 3 3 ϵ 0 ϵ r λ D 1 ( 1 - e 1 - 2 n e 2 t 2 m ϵ 0 ϵ r ) \frac{ne\sqrt{\pi}G(i\lambda_{D1})e^{\frac{-t}{\tau_{0}}-\lambda_{D1}z}}{3% \sqrt{3}\epsilon_{0}\epsilon_{r}\lambda_{D1}}(1-e^{1-\sqrt{\frac{2ne^{2}t^{2}}% {m\epsilon_{0}\epsilon_{r}}}})
  67. λ \lambda

Poker_probability_(Omaha).html

  1. ( 52 4 ) = 270 , 725 {52\choose 4}=270,725
  2. ( 13 1 ) \begin{matrix}{13\choose 1}\end{matrix}
  3. ( 4 4 ) \begin{matrix}{4\choose 4}\end{matrix}
  4. ( 13 1 ) ( 12 1 ) \begin{matrix}{13\choose 1}{12\choose 1}\end{matrix}
  5. ( 4 3 ) ( 3 1 ) \begin{matrix}{4\choose 3}{3\choose 1}\end{matrix}
  6. ( 13 1 ) ( 12 1 ) \begin{matrix}{13\choose 1}{12\choose 1}\end{matrix}
  7. ( 4 3 ) \begin{matrix}{4\choose 3}\end{matrix}
  8. ( 13 2 ) \begin{matrix}{13\choose 2}\end{matrix}
  9. ( 4 2 ) \begin{matrix}{4\choose 2}\end{matrix}
  10. ( 13 2 ) \begin{matrix}{13\choose 2}\end{matrix}
  11. ( 4 2 ) ( 2 1 ) 2 \begin{matrix}{4\choose 2}{2\choose 1}^{2}\end{matrix}
  12. ( 13 2 ) \begin{matrix}{13\choose 2}\end{matrix}
  13. ( 4 2 ) \begin{matrix}{4\choose 2}\end{matrix}
  14. ( 13 1 ) ( 12 2 ) \begin{matrix}{13\choose 1}{12\choose 2}\end{matrix}
  15. ( 4 2 ) ( 2 1 ) \begin{matrix}{4\choose 2}{2\choose 1}\end{matrix}
  16. ( 13 1 ) ( 12 2 ) \begin{matrix}{13\choose 1}{12\choose 2}\end{matrix}
  17. ( 4 2 ) ( 2 1 ) \begin{matrix}{4\choose 2}{2\choose 1}\end{matrix}
  18. ( 13 1 ) ( 12 2 ) ( 2 1 ) \begin{matrix}{13\choose 1}{12\choose 2}{2\choose 1}\end{matrix}
  19. ( 4 2 ) ( 2 1 ) 2 \begin{matrix}{4\choose 2}{2\choose 1}^{2}\end{matrix}
  20. ( 13 1 ) ( 12 2 ) \begin{matrix}{13\choose 1}{12\choose 2}\end{matrix}
  21. ( 4 2 ) ( 2 1 ) \begin{matrix}{4\choose 2}{2\choose 1}\end{matrix}
  22. ( 13 1 ) ( 12 2 ) \begin{matrix}{13\choose 1}{12\choose 2}\end{matrix}
  23. ( 4 2 ) × 2 ! \begin{matrix}{4\choose 2}\times 2!\end{matrix}
  24. ( 13 4 ) \begin{matrix}{13\choose 4}\end{matrix}
  25. ( 4 1 ) \begin{matrix}{4\choose 1}\end{matrix}
  26. ( 13 4 ) ( 4 3 ) \begin{matrix}{13\choose 4}{4\choose 3}\end{matrix}
  27. ( 4 1 ) ( 3 1 ) \begin{matrix}{4\choose 1}{3\choose 1}\end{matrix}
  28. ( 13 4 ) ( 4 2 ) ÷ 2 ! \begin{matrix}{13\choose 4}{4\choose 2}\div 2!\end{matrix}
  29. ( 4 2 ) ( 2 1 ) \begin{matrix}{4\choose 2}{2\choose 1}\end{matrix}
  30. ( 13 4 ) ( 4 2 ) \begin{matrix}{13\choose 4}{4\choose 2}\end{matrix}
  31. ( 4 1 ) ( 3 2 ) × 2 ! \begin{matrix}{4\choose 1}{3\choose 2}\times 2!\end{matrix}
  32. ( 13 4 ) \begin{matrix}{13\choose 4}\end{matrix}
  33. 4 ! \begin{matrix}4!\end{matrix}
  34. r r
  35. ( 13 - r ) × 4 = 52 - 4 r \begin{matrix}(13-r)\times 4=52-4r\end{matrix}
  36. ( r 0 ) \begin{matrix}{r\choose 0}\end{matrix}
  37. ( 52 - 4 r 4 ) \begin{matrix}{52-4r\choose 4}\end{matrix}
  38. ( r 1 ) \begin{matrix}{r\choose 1}\end{matrix}
  39. ( 4 1 ) ( 52 - 4 r 3 ) \begin{matrix}{4\choose 1}{52-4r\choose 3}\end{matrix}
  40. ( r 1 ) \begin{matrix}{r\choose 1}\end{matrix}
  41. ( 4 2 ) ( 52 - 4 r 2 ) \begin{matrix}{4\choose 2}{52-4r\choose 2}\end{matrix}
  42. ( r 1 ) \begin{matrix}{r\choose 1}\end{matrix}
  43. ( 4 3 ) ( 52 - 4 r 1 ) \begin{matrix}{4\choose 3}{52-4r\choose 1}\end{matrix}
  44. ( r 1 ) \begin{matrix}{r\choose 1}\end{matrix}
  45. ( 4 4 ) \begin{matrix}{4\choose 4}\end{matrix}
  46. ( r 2 ) \begin{matrix}{r\choose 2}\end{matrix}
  47. ( 4 1 ) 2 ( 52 - 4 r 2 ) \begin{matrix}{4\choose 1}^{2}{52-4r\choose 2}\end{matrix}
  48. ( r 1 ) ( r - 1 1 ) \begin{matrix}{r\choose 1}{r-1\choose 1}\end{matrix}
  49. ( 4 2 ) ( 4 1 ) ( 52 - 4 r 1 ) \begin{matrix}{4\choose 2}{4\choose 1}{52-4r\choose 1}\end{matrix}
  50. ( r 1 ) ( r - 1 1 ) \begin{matrix}{r\choose 1}{r-1\choose 1}\end{matrix}
  51. ( 4 3 ) ( 4 1 ) \begin{matrix}{4\choose 3}{4\choose 1}\end{matrix}
  52. ( r 2 ) \begin{matrix}{r\choose 2}\end{matrix}
  53. ( 4 2 ) 2 \begin{matrix}{4\choose 2}^{2}\end{matrix}
  54. ( r 3 ) \begin{matrix}{r\choose 3}\end{matrix}
  55. ( 4 1 ) 3 ( 52 - 4 r 1 ) \begin{matrix}{4\choose 1}^{3}{52-4r\choose 1}\end{matrix}
  56. ( r 1 ) ( r - 1 2 ) \begin{matrix}{r\choose 1}{r-1\choose 2}\end{matrix}
  57. ( 4 2 ) ( 4 1 ) 2 \begin{matrix}{4\choose 2}{4\choose 1}^{2}\end{matrix}
  58. ( r 4 ) \begin{matrix}{r\choose 4}\end{matrix}
  59. ( 4 1 ) 4 \begin{matrix}{4\choose 1}^{4}\end{matrix}
  60. ( 52 3 ) = 22 , 100 {52\choose 3}=22,100
  61. ( 52 4 ) = 270 , 725 {52\choose 4}=270,725
  62. ( 52 5 ) = 2 , 598 , 960 {52\choose 5}=2,598,960
  63. ( 48 3 ) = 17 , 296 {48\choose 3}=17,296
  64. ( 48 4 ) = 194 , 580 {48\choose 4}=194,580
  65. ( 48 5 ) = 1 , 712 , 304 {48\choose 5}=1,712,304
  66. ( 3 3 ) ( 4 2 ) = 6 {3\choose 3}{4\choose 2}=6
  67. ( 4 3 ) ( 4 2 ) = 24 {4\choose 3}{4\choose 2}=24
  68. ( 5 3 ) ( 4 2 ) = 60 {5\choose 3}{4\choose 2}=60
  69. ( 5 5 ) = 1 , \begin{matrix}{5\choose 5}=1\end{matrix},
  70. ( 6 5 ) = 6 \begin{matrix}{6\choose 5}=6\end{matrix}
  71. ( 7 5 ) = 21 \begin{matrix}{7\choose 5}=21\end{matrix}
  72. r r
  73. P f P_{f}
  74. P f = ( r 3 ) ( 4 1 ) 3 < m t p l > ( 52 3 ) . P_{f}=\frac{{r\choose 3}{4\choose 1}^{3}}{<}mtpl>{{52\choose 3}}.
  75. ( 8 3 ) = 56 \begin{matrix}{8\choose 3}=56\end{matrix}
  76. ( 9 3 ) = 84 \begin{matrix}{9\choose 3}=84\end{matrix}
  77. ( 52 3 ) = 22 , 100 \begin{matrix}{52\choose 3}=22,100\end{matrix}
  78. ( r 3 ) ( 4 1 ) 3 \begin{matrix}{r\choose 3}{4\choose 1}^{3}\end{matrix}
  79. P t P_{t}
  80. ( 52 4 ) = 270 , 725 \begin{matrix}{52\choose 4}=270,725\end{matrix}
  81. ( r 4 ) ( 4 1 ) 4 \begin{matrix}{r\choose 4}{4\choose 1}^{4}\end{matrix}
  82. ( r 1 ) ( 4 2 ) ( r - 1 2 ) ( 4 1 ) 2 \begin{matrix}{r\choose 1}{4\choose 2}{r-1\choose 2}{4\choose 1}^{2}\end{matrix}
  83. ( r 3 ) ( 4 1 ) 3 ( 52 - 4 r 1 ) \begin{matrix}{r\choose 3}{4\choose 1}^{3}{52-4r\choose 1}\end{matrix}
  84. P t P_{t}
  85. ( 52 5 ) = 2 , 598 , 960 \begin{matrix}{52\choose 5}=2,598,960\end{matrix}
  86. P r P_{r}
  87. ( r 5 ) ( 4 1 ) 5 \begin{matrix}{r\choose 5}{4\choose 1}^{5}\end{matrix}
  88. ( r 1 ) ( 4 2 ) ( r - 1 3 ) ( 4 1 ) 3 \begin{matrix}{r\choose 1}{4\choose 2}{r-1\choose 3}{4\choose 1}^{3}\end{matrix}
  89. ( r 4 ) ( 4 1 ) 4 ( 52 - 4 r 1 ) \begin{matrix}{r\choose 4}{4\choose 1}^{4}{52-4r\choose 1}\end{matrix}
  90. ( r 1 ) ( 4 3 ) ( r - 1 2 ) ( 4 1 ) 2 \begin{matrix}{r\choose 1}{4\choose 3}{r-1\choose 2}{4\choose 1}^{2}\end{matrix}
  91. ( r 2 ) ( 4 2 ) 2 ( r - 2 1 ) ( 4 1 ) \begin{matrix}{r\choose 2}{4\choose 2}^{2}{r-2\choose 1}{4\choose 1}\end{matrix}
  92. ( r 1 ) ( 4 2 ) ( r - 1 2 ) ( 4 1 ) 2 ( 52 - 4 r 1 ) \begin{matrix}{r\choose 1}{4\choose 2}{r-1\choose 2}{4\choose 1}^{2}{52-4r% \choose 1}\end{matrix}
  93. ( r 3 ) ( 4 1 ) 3 ( 52 - 4 r 2 ) \begin{matrix}{r\choose 3}{4\choose 1}^{3}{52-4r\choose 2}\end{matrix}
  94. P r P_{r}
  95. ( 49 2 ) = 1 , 176 \begin{matrix}{49\choose 2}=1,176\end{matrix}
  96. ( 52 3 ) ( 3 3 ) ( 49 2 ) = 25 , 989 , 600 {52\choose 3}{3\choose 3}{49\choose 2}=25,989,600
  97. ( 4 3 ) = 4 \begin{matrix}{4\choose 3}=4\end{matrix}
  98. ( 48 2 ) = 1 , 128 \begin{matrix}{48\choose 2}=1,128\end{matrix}
  99. ( 52 4 ) ( 4 3 ) ( 48 2 ) = 1 , 221 , 511 , 200 {52\choose 4}{4\choose 3}{48\choose 2}=1,221,511,200
  100. ( 5 3 ) = 10 \begin{matrix}{5\choose 3}=10\end{matrix}
  101. ( 47 2 ) = 1 , 081 \begin{matrix}{47\choose 2}=1,081\end{matrix}
  102. ( 52 5 ) ( 5 3 ) ( 47 2 ) = 28 , 094 , 757 , 600 {52\choose 5}{5\choose 3}{47\choose 2}=28,094,757,600
  103. 10 × ( 4 2 ) + ( 4 3 ) = 64 \begin{matrix}10\times{4\choose 2}+{4\choose 3}=64\end{matrix}
  104. 64 × ( 4 1 ) = 256 64\times{4\choose 1}=256
  105. 432 × ( 4 1 ) + 64 × ( 4 1 ) ( 39 1 ) = 11 , 712 432\times{4\choose 1}+64\times{4\choose 1}{39\choose 1}=11,712
  106. 1 , 208 × ( 4 1 ) + 432 × ( 4 1 ) ( 39 1 ) + 64 × ( 4 1 ) ( 39 2 ) = 261 , 920 1,208\times{4\choose 1}+432\times{4\choose 1}{39\choose 1}+64\times{4\choose 1% }{39\choose 2}=261,920
  107. ( 13 1 ) ( 4 3 ) + ( 13 1 ) ( 4 2 ) ( 12 1 ) ( 4 1 ) = 3 , 796 {13\choose 1}{4\choose 3}+{13\choose 1}{4\choose 2}{12\choose 1}{4\choose 1}=3% ,796
  108. 64 × ( 4 1 ) ( 3 1 ) ( 3 1 ) = 2 , 304 \begin{matrix}64\times{4\choose 1}{3\choose 1}{3\choose 1}=2,304\end{matrix}
  109. ( 13 1 ) ( 4 3 ) ( 12 1 ) ( 4 1 ) + ( 13 2 ) ( 4 2 ) 2 + [ ( 13 2 ) ( 4 2 ) ( 12 2 ) ( 4 1 ) 2 - 2 , 304 ] = 85 , 368 {13\choose 1}{4\choose 3}{12\choose 1}{4\choose 1}+{13\choose 2}{4\choose 2}^{% 2}+\left[{13\choose 2}{4\choose 2}{12\choose 2}{4\choose 1}^{2}-2,304\right]=8% 5,368
  110. 64 × ( 4 1 ) ( 3 1 ) ( 3 2 ) = 2 , 304 \begin{matrix}64\times{4\choose 1}{3\choose 1}{3\choose 2}=2,304\end{matrix}
  111. 64 × ( 4 1 ) ( 3 2 ) ( 3 1 ) 2 = 6 , 912 \begin{matrix}64\times{4\choose 1}{3\choose 2}{3\choose 1}^{2}=6,912\end{matrix}
  112. 432 × ( 4 1 ) ( 3 1 ) = 20 , 736 \begin{matrix}432\times{4\choose 1}{3\choose 1}=20,736\end{matrix}
  113. 64 × ( 4 1 ) ( 3 1 ) ( 3 1 ) ( 10 1 ) ( 3 1 ) = 69 , 120 \begin{matrix}64\times{4\choose 1}{3\choose 1}{3\choose 1}{10\choose 1}{3% \choose 1}=69,120\end{matrix}
  114. 64 × ( 4 1 ) ( 10 1 ) ( 3 2 ) = 7 , 680 \begin{matrix}64\times{4\choose 1}{10\choose 1}{3\choose 2}=7,680\end{matrix}
  115. ( 13 1 ) ( 4 3 ) ( 12 1 ) ( 4 2 ) {13\choose 1}{4\choose 3}{12\choose 1}{4\choose 2}
  116. 3 , 744 3,744\,
  117. + ( 13 1 ) ( 4 3 ) ( 12 2 ) ( 4 1 ) 2 - 2 , 304 +{13\choose 1}{4\choose 3}{12\choose 2}{4\choose 1}^{2}-2,304
  118. 52 , 608 52,608\,
  119. + ( 13 2 ) ( 4 2 ) 2 ( 11 2 ) ( 4 1 ) - 6 , 912 +{13\choose 2}{4\choose 2}^{2}{11\choose 2}{4\choose 1}-6,912
  120. 116 , 640 116,640\,
  121. + ( 13 1 ) ( 4 2 ) ( 12 3 ) ( 4 1 ) 3 - 97 , 536 +{13\choose 1}{4\choose 2}{12\choose 3}{4\choose 1}^{3}-97,536
  122. 1 , 000 , 704 1,000,704\,
  123. 1 , 173 , 696 1,173,696\,
  124. ( 13 1 ) = 13 {13\choose 1}=13
  125. ( 13 1 ) ( 48 1 ) = 624 {13\choose 1}{48\choose 1}=624
  126. ( 13 3 ) - 64 = 222 \begin{matrix}{13\choose 3}-64=222\end{matrix}
  127. ( 13 4 ) - 432 = 283 \begin{matrix}{13\choose 4}-432=283\end{matrix}
  128. ( 13 5 ) - 1 , 208 = 79 \begin{matrix}{13\choose 5}-1,208=79\end{matrix}
  129. 222 × ( 4 1 ) = 888 222\times{4\choose 1}=888
  130. 283 × ( 4 1 ) + 222 × ( 4 1 ) ( 10 1 ) ( 3 1 ) = 27 , 772 283\times{4\choose 1}+222\times{4\choose 1}{10\choose 1}{3\choose 1}=27,772
  131. 79 × ( 4 1 ) + 283 × ( 4 1 ) ( 9 1 ) ( 3 1 ) + 222 × ( 4 1 ) ( 10 2 ) ( 3 1 ) 2 = 390 , 520 79\times{4\choose 1}+283\times{4\choose 1}{9\choose 1}{3\choose 1}+222\times{4% \choose 1}{10\choose 2}{3\choose 1}^{2}=390,520
  132. n n
  133. 4 n 4^{n}
  134. ( n n - 1 ) ( 4 1 ) ( 3 1 ) = 12 n \begin{matrix}{n\choose n-1}{4\choose 1}{3\choose 1}=12n\end{matrix}
  135. n - 1 n-1
  136. ( n n - x ) ( 4 1 ) ( 3 1 ) x \begin{matrix}{n\choose n-x}{4\choose 1}{3\choose 1}^{x}\end{matrix}
  137. n - x n-x
  138. 4 3 - 4 = 60 4^{3}-4=60
  139. 64 × 60 = 3 , 840 64\times 60=3,840\,
  140. 4 + ( 4 3 ) ( 4 1 ) ( 3 1 ) = 52 \begin{matrix}4+{4\choose 3}{4\choose 1}{3\choose 1}=52\end{matrix}
  141. 4 4 - 52 = 204 4^{4}-52=204
  142. 4 + ( 5 4 ) ( 4 1 ) ( 3 1 ) + ( 5 3 ) ( 4 1 ) ( 3 1 ) 2 = 424 \begin{matrix}4+{5\choose 4}{4\choose 1}{3\choose 1}+{5\choose 3}{4\choose 1}{% 3\choose 1}^{2}=424\end{matrix}
  143. 4 5 - 424 = 600 4^{5}-424=600
  144. 432 × 204 = 88 , 128 432\times 204=88,128\,
  145. 1 , 208 × 600 = 724 , 800 1,208\times 600=724,800\,
  146. 222 × 60 = 13 , 320 222\times 60=13,320\,
  147. 283 × 204 = 57 , 732 283\times 204=57,732\,
  148. 79 × 600 = 47 , 400 79\times 600=47,400\,
  149. ( 5 2 ) \begin{matrix}{5\choose 2}\end{matrix}
  150. ( 4 1 ) 2 ( 32 2 ) \begin{matrix}{4\choose 1}^{2}{32\choose 2}\end{matrix}
  151. ( 5 2 ) ( 2 1 ) \begin{matrix}{5\choose 2}{2\choose 1}\end{matrix}
  152. ( 4 2 ) ( 4 1 ) ( 32 1 ) \begin{matrix}{4\choose 2}{4\choose 1}{32\choose 1}\end{matrix}
  153. ( 5 2 ) ( 2 1 ) \begin{matrix}{5\choose 2}{2\choose 1}\end{matrix}
  154. ( 4 3 ) ( 4 1 ) \begin{matrix}{4\choose 3}{4\choose 1}\end{matrix}
  155. ( 5 2 ) \begin{matrix}{5\choose 2}\end{matrix}
  156. ( 4 2 ) 2 \begin{matrix}{4\choose 2}^{2}\end{matrix}
  157. ( 5 3 ) \begin{matrix}{5\choose 3}\end{matrix}
  158. ( 4 1 ) 3 ( 32 1 ) \begin{matrix}{4\choose 1}^{3}{32\choose 1}\end{matrix}
  159. ( 5 3 ) ( 3 1 ) \begin{matrix}{5\choose 3}{3\choose 1}\end{matrix}
  160. ( 4 2 ) ( 4 1 ) 2 \begin{matrix}{4\choose 2}{4\choose 1}^{2}\end{matrix}
  161. ( 5 4 ) \begin{matrix}{5\choose 4}\end{matrix}
  162. ( 4 1 ) 4 \begin{matrix}{4\choose 1}^{4}\end{matrix}
  163. s s
  164. P f P_{f}
  165. P f = < m t p l > ( 13 - s 3 ) ( 48 3 ) P_{f}=\frac{<}{m}tpl>{{13-s\choose 3}}{{48\choose 3}}
  166. 39 - ( 4 - s ) = 35 + s 39-(4-s)=35+s
  167. P t = ( 13 - s 4 ) + ( 13 - s 3 ) ( 35 + s 1 ) < m t p l > ( 48 4 ) P_{t}=\frac{{13-s\choose 4}+{13-s\choose 3}{35+s\choose 1}}{<}mtpl>{{48\choose 4}}
  168. P r = ( 13 - s 5 ) + ( 13 - s 4 ) ( 35 + s 1 ) + ( 13 - s 3 ) ( 35 + s 2 ) < m t p l > ( 48 5 ) P_{r}=\frac{{13-s\choose 5}+{13-s\choose 4}{35+s\choose 1}+{13-s\choose 3}{35+% s\choose 2}}{<}mtpl>{{48\choose 5}}
  169. 2 P f 2P_{f}
  170. 2 P t 2P_{t}
  171. 2 P r 2P_{r}
  172. s s
  173. F f F_{f}
  174. F f = ( s 1 ) F_{f}={s\choose 1}
  175. s s
  176. n 42 n_{42}
  177. F t F_{t}
  178. F t = ( s 1 ) ( 45 1 ) - n 42 F_{t}={s\choose 1}{45\choose 1}-n_{42}
  179. s s
  180. n 52 n_{52}
  181. n 53 n_{53}
  182. F r F_{r}
  183. F r = ( s 1 ) ( 45 2 ) - n 52 - 2 n 53 F_{r}={s\choose 1}{45\choose 2}-n_{52}-2n_{53}
  184. ( 4 + 2 - 1 2 ) = 10 \begin{matrix}{4+2-1\choose 2}\end{matrix}=10
  185. P 1 P_{1}
  186. P 2 P_{2}
  187. P P
  188. P = P 1 + P 2 P=P_{1}+P_{2}
  189. ( 4 1 ) 3 \begin{matrix}{4\choose 1}^{3}\end{matrix}
  190. ( 4 1 ) 2 ( 3 1 ) \begin{matrix}{4\choose 1}^{2}{3\choose 1}\end{matrix}
  191. ( 4 1 ) 2 ( 2 1 ) \begin{matrix}{4\choose 1}^{2}{2\choose 1}\end{matrix}
  192. ( 4 1 ) ( 3 1 ) 2 \begin{matrix}{4\choose 1}{3\choose 1}^{2}\end{matrix}
  193. r 444 r_{444}
  194. r 443 r_{443}
  195. r 442 r_{442}
  196. r 433 r_{433}
  197. C = 64 r 444 + 48 r 443 + 32 r 442 + 36 r 433 . C=64r_{444}+48r_{443}+32r_{442}+36r_{433}.
  198. P f = C < m t p l > ( 48 3 ) = C 17 , 296 . P_{f}=\frac{C}{<}mtpl>{{48\choose 3}}=\frac{C}{17,296}.
  199. 1 6 \begin{matrix}\frac{1}{6}\end{matrix}

Polar_curve.html

  1. Δ Q = a x + b y + c z . \Delta_{Q}=a{\partial\over\partial x}+b{\partial\over\partial y}+c{\partial% \over\partial z}.
  2. x f x ( p , q , r ) + y f y ( p , q , r ) + z f z ( p , q , r ) = 0. x{\partial f\over\partial x}(p,q,r)+y{\partial f\over\partial y}(p,q,r)+z{% \partial f\over\partial z}(p,q,r)=0.
  3. μ n f ( p , q , r ) + λ μ n - 1 Δ Q f ( p , q , r ) + 1 2 λ 2 μ n - 2 Δ Q 2 f ( p , q , r ) + \mu^{n}f(p,q,r)+\lambda\mu^{n-1}\Delta_{Q}f(p,q,r)+\frac{1}{2}\lambda^{2}\mu^{% n-2}\Delta_{Q}^{2}f(p,q,r)+\dots
  4. λ n f ( a , b , c ) + μ λ n - 1 Δ P f ( a , b , c ) + 1 2 μ 2 λ n - 2 Δ P 2 f ( a , b , c ) + . \lambda^{n}f(a,b,c)+\mu\lambda^{n-1}\Delta_{P}f(a,b,c)+\frac{1}{2}\mu^{2}% \lambda^{n-2}\Delta_{P}^{2}f(a,b,c)+\dots.
  5. 1 p ! Δ Q p f ( p , q , r ) = 1 ( n - p ) ! Δ P n - p f ( a , b , c ) . \frac{1}{p!}\Delta_{Q}^{p}f(p,q,r)=\frac{1}{(n-p)!}\Delta_{P}^{n-p}f(a,b,c).
  6. Δ ( x , y , z ) 2 f ( a , b , c ) = x 2 2 f x 2 ( a , b , c ) + 2 x y 2 f x y ( a , b , c ) + = 0. \Delta_{(x,y,z)}^{2}f(a,b,c)=x^{2}{\partial^{2}f\over\partial x^{2}}(a,b,c)+2% xy{\partial^{2}f\over\partial x\partial y}(a,b,c)+\dots=0.
  7. H ( f ) = [ 2 f x 2 2 f x y 2 f x z 2 f y x 2 f y 2 2 f y z 2 f z x 2 f z y 2 f z 2 ] , H(f)=\begin{bmatrix}\frac{\partial^{2}f}{\partial x^{2}}&\frac{\partial^{2}f}{% \partial x\,\partial y}&\frac{\partial^{2}f}{\partial x\,\partial z}\\ \\ \frac{\partial^{2}f}{\partial y\,\partial x}&\frac{\partial^{2}f}{\partial y^{% 2}}&\frac{\partial^{2}f}{\partial y\,\partial z}\\ \\ \frac{\partial^{2}f}{\partial z\,\partial x}&\frac{\partial^{2}f}{\partial z\,% \partial y}&\frac{\partial^{2}f}{\partial z^{2}}\end{bmatrix},

Polar_modulation.html

  1. P ( n ) = I 2 ( n ) + Q 2 ( n ) P(n)=\sqrt{I^{2}(n)+Q^{2}(n)}

Polar_set_(potential_theory).html

  1. Z Z
  2. \R n \R^{n}
  3. n 2 n\geq 2
  4. u u
  5. \R n \R^{n}
  6. Z { x : u ( x ) = - } . Z\subseteq\{x:u(x)=-\infty\}.
  7. - -\infty
  8. \infty
  9. \R n \R^{n}
  10. \R n \R^{n}
  11. \R n . \R^{n}.

Polarization_of_an_algebraic_form.html

  1. F ( u ( 1 ) , , u ( d ) ) = 1 d ! λ 1 λ d f ( λ 1 u ( 1 ) + + λ d u ( d ) ) | λ = 0 . F({u}^{(1)},\dots,{u}^{(d)})=\frac{1}{d!}\frac{\partial}{\partial\lambda_{1}}% \dots\frac{\partial}{\partial\lambda_{d}}f(\lambda_{1}{u}^{(1)}+\dots+\lambda_% {d}{u}^{(d)})|_{\lambda=0}.
  2. f ( x ) = x 2 + 3 x y + 2 y 2 . f({x})=x^{2}+3xy+2y^{2}.
  3. F ( x ( 1 ) , x ( 2 ) ) = x ( 1 ) x ( 2 ) + 3 2 x ( 2 ) y ( 1 ) + 3 2 x ( 1 ) y ( 2 ) + 2 y ( 1 ) y ( 2 ) . F({x}^{(1)},{x}^{(2)})=x^{(1)}x^{(2)}+\frac{3}{2}x^{(2)}y^{(1)}+\frac{3}{2}x^{% (1)}y^{(2)}+2y^{(1)}y^{(2)}.
  4. F ( x ( 1 ) , y ( 1 ) , x ( 2 ) , y ( 2 ) , x ( 3 ) , y ( 3 ) ) = x ( 1 ) x ( 2 ) x ( 3 ) + 2 3 x ( 1 ) y ( 2 ) y ( 3 ) + 2 3 x ( 3 ) y ( 1 ) y ( 2 ) + 2 3 x ( 2 ) y ( 3 ) y ( 1 ) . F(x^{(1)},y^{(1)},x^{(2)},y^{(2)},x^{(3)},y^{(3)})=x^{(1)}x^{(2)}x^{(3)}+\frac% {2}{3}x^{(1)}y^{(2)}y^{(3)}+\frac{2}{3}x^{(3)}y^{(1)}y^{(2)}+\frac{2}{3}x^{(2)% }y^{(3)}y^{(1)}.
  5. A = d A d . A=\bigoplus_{d}A_{d}.
  6. A d S y m d k n A_{d}\cong Sym^{d}k^{n}
  7. A d S y m d V * . A_{d}\cong Sym^{d}V^{*}.
  8. A S y m V * A\cong Sym^{\cdot}V^{*}

Polyharmonic_spline.html

  1. y ( 𝐱 ) = i = 1 N w i ϕ ( || 𝐱 - 𝐜 i || ) + 𝐯 T [ 1 𝐱 ] y(\mathbf{x})\,=\,\sum_{i=1}^{N}w_{i}\,\phi(||\mathbf{x}-\mathbf{c}_{i}||)+% \mathbf{v}^{T}\,\begin{bmatrix}1\\ \mathbf{x}\end{bmatrix}
  2. 𝐱 = [ x 1 , x 2 , , x n x ] T \mathbf{x}=[x_{1},x_{2},\cdots,x_{nx}]^{T}
  3. 𝐜 i = [ c 1 , i , c 2 , i , , c n x , i ] T \mathbf{c}_{i}=[c_{1,i},c_{2,i},\cdots,c_{nx,i}]^{T}
  4. 𝐱 \mathbf{x}
  5. 𝐰 = [ w 1 , w 2 , , w N ] T \mathbf{w}=[w_{1},w_{2},\cdots,w_{N}]^{T}
  6. 𝐯 = [ v 1 , v 2 , , v n x + 1 ] T \mathbf{v}=[v_{1},v_{2},\cdots,v_{nx+1}]^{T}
  7. 𝐯 \mathbf{v}
  8. 𝐜 i \mathbf{c}_{i}
  9. ϕ ( r ) = { r k with k = 1 , 3 , 5 , , r k ln ( r ) with k = 2 , 4 , 6 , r = || 𝐱 - 𝐜 i || 2 = ( 𝐱 - 𝐜 i ) T ( 𝐱 - 𝐜 i ) \begin{matrix}\phi(r)=\begin{cases}r^{k}&\mbox{with }~{}k=1,3,5,\dots,\\ r^{k}\ln(r)&\mbox{with }~{}k=2,4,6,\dots\end{cases}\\ r=||\mathbf{x}-\mathbf{c}_{i}||_{2}=\sqrt{(\mathbf{x}-\mathbf{c}_{i})^{T}\,(% \mathbf{x}-\mathbf{c}_{i})}\end{matrix}
  10. ϕ ( r ) = r 2 \phi(r)=r^{2}
  11. ϕ ( r ) = { r k - 1 ln ( r r ) for r < 1 r k ln ( r ) for r 1 \phi(r)=\begin{cases}r^{k-1}\ln(r^{r})&\mbox{for }~{}r<1\\ r^{k}\ln(r)&\mbox{for }~{}r\geq 1\end{cases}
  12. w i w_{i}
  13. v j v_{j}
  14. N N
  15. ( 𝐜 i , y i ) (\mathbf{c}_{i},y_{i})
  16. n x + 1 nx+1
  17. 0 = i = 1 N w i , 0 = i = 1 N w i c j , i ( j = 1 , 2 , , n x ) 0=\sum_{i=1}^{N}w_{i},\;\;0=\sum_{i=1}^{N}w_{i}\,c_{j,i}\;\;\;(j=1,2,...,nx)
  18. [ 𝐀 𝐕 T 𝐕 𝟎 ] [ 𝐰 𝐯 ] = [ 𝐲 𝟎 ] \begin{bmatrix}\mathbf{A}&\mathbf{V}^{T}\\ \mathbf{V}&\mathbf{0}\end{bmatrix}\;\begin{bmatrix}\mathbf{w}\\ \mathbf{v}\end{bmatrix}\;=\;\begin{bmatrix}\mathbf{y}\\ \mathbf{0}\end{bmatrix}\;\;\;\;
  19. A i , j = ϕ ( || 𝐜 i - 𝐜 j || ) , 𝐕 = [ 1 1 1 𝐜 1 𝐜 2 𝐜 N ] , 𝐲 = [ y 1 , y 2 , , y N ] T A_{i,j}=\phi(||\mathbf{c}_{i}-\mathbf{c}_{j}||),\;\;\;\mathbf{V}=\begin{% bmatrix}1&1&\cdots&1\\ \mathbf{c}_{1}&\mathbf{c}_{2}&\cdots&\mathbf{c}_{N}\end{bmatrix},\;\;\;\mathbf% {y}=[y_{1},y_{2},\cdots,y_{N}]^{T}
  20. 𝐱 \mathbf{x}

Polynomial_function_theorems_for_zeros.html

  1. p ( x ) = a n x n + a n - 1 x n - 1 + + a 2 x 2 + a 1 x + a 0 p(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{2}x^{2}+a_{1}x+a_{0}
  2. a i ( i = 0 , 1 , 2 , , n ) a_{i}\,(i=0,1,2,\ldots,n)
  3. a n 0 a_{n}\neq 0
  4. p ( z ) = a n z n + a n - 1 z n - 1 + + a 2 z 2 + a 1 z + a 0 = 0 p(z)=a_{n}z^{n}+a_{n-1}z^{n-1}+\cdots+a_{2}z^{2}+a_{1}z+a_{0}=0
  5. z z
  6. p ( x ) p(x)
  7. z z
  8. z z
  9. p ( x ) p(x)
  10. z z
  11. z z
  12. p ( x ) p(x)
  13. n 1 n\geq 1
  14. n 1 n\geq 1
  15. n n
  16. p ( x ) = a 1 x + a 0 p(x)=a_{1}x+a_{0}
  17. - a 0 a 1 \frac{-a_{0}}{a_{1}}
  18. p ( x ) = a 2 x 2 + a 1 x + a 0 p(x)=a_{2}x^{2}+a_{1}x+a_{0}
  19. - a 1 + a 1 2 - 4 a 2 a 0 2 a 2 \frac{-a_{1}+\sqrt{{a_{1}}^{2}-4a_{2}a_{0}}}{2a_{2}}
  20. - a 1 - a 1 2 - 4 a 2 a 0 2 a 2 \frac{-a_{1}-\sqrt{{a_{1}}^{2}-4a_{2}a_{0}}}{2a_{2}}
  21. p ( x ) p(x)
  22. x - c x-c
  23. p ( c ) p(c)
  24. p ( x ) = x 3 + 2 x - 3 p(x)=x^{3}+2x-3
  25. x - 2 x-2
  26. p ( 2 ) = 2 3 + 2 ( 2 ) - 3 = 9 p(2)=2^{3}+2(2)-3=9
  27. p ( x ) p(x)
  28. x + 1 x+1
  29. p ( - 1 ) = ( - 1 ) 3 + 2 ( - 1 ) - 3 = - 6 p(-1)=(-1)^{3}+2(-1)-3=-6
  30. p ( x ) p(x)
  31. p ( x ) p(x)
  32. x - 1 x-1
  33. p ( 1 ) = ( 1 ) 3 + 2 ( 1 ) - 3 = 0 p(1)=(1)^{3}+2(1)-3=0
  34. p ( x ) p(x)

Polynomial_lemniscate.html

  1. | p ( z ) | = c . |p(z)|=c.
  2. p ( z ) p ¯ ( z ¯ ) p(z)\bar{p}(\bar{z})
  3. ( x 2 + y 2 ) 2 = 2 ( x 2 - y 2 ) (x^{2}+y^{2})^{2}=2(x^{2}-y^{2})\,

Polynomial_matrix.html

  1. P = n = 0 p A ( n ) x n = A ( 0 ) + A ( 1 ) x + A ( 2 ) x 2 + + A ( p ) x p P=\sum_{n=0}^{p}A(n)x^{n}=A(0)+A(1)x+A(2)x^{2}+\cdots+A(p)x^{p}
  2. A ( i ) A(i)
  3. A ( p ) A(p)
  4. P = ( 1 x 2 x 0 2 x 2 3 x + 2 x 2 - 1 0 ) = ( 1 0 0 0 0 2 2 - 1 0 ) + ( 0 0 1 0 2 0 3 0 0 ) x + ( 0 1 0 0 0 0 0 1 0 ) x 2 . P=\begin{pmatrix}1&x^{2}&x\\ 0&2x&2\\ 3x+2&x^{2}-1&0\end{pmatrix}=\begin{pmatrix}1&0&0\\ 0&0&2\\ 2&-1&0\end{pmatrix}+\begin{pmatrix}0&0&1\\ 0&2&0\\ 3&0&0\end{pmatrix}x+\begin{pmatrix}0&1&0\\ 0&0&0\\ 0&1&0\end{pmatrix}x^{2}.
  5. M n ( R [ X ] ) M_{n}(R[X])
  6. ( M n ( R ) ) [ X ] (M_{n}(R))[X]

Polyphase_matrix.html

  1. h , g \scriptstyle h,\,g
  2. a 0 \scriptstyle a_{0}
  3. a 1 , d 1 \scriptstyle a_{1},\,d_{1}
  4. a 1 \displaystyle a_{1}
  5. \scriptstyle\downarrow
  6. h e = h 2 a 0 , e = a 0 2 h o = ( h 1 ) 2 a 0 , o = ( a 0 1 ) 2 \begin{array}[]{rclcrcl}h_{\mbox{e}~{}}&=&h\downarrow 2&&a_{0,\mbox{e}~{}}&=&a% _{0}\downarrow 2\\ h_{\mbox{o}~{}}&=&(h\leftarrow 1)\downarrow 2&&a_{0,\mbox{o}~{}}&=&(a_{0}% \leftarrow 1)\downarrow 2\end{array}
  7. \scriptstyle\leftarrow
  8. \scriptstyle\rightarrow
  9. δ = ( , 0 , 0 , 1 0 - th position , 0 , 0 , ) \delta=(\dots,0,0,\underset{0-\mbox{th position}~{}}{1},0,0,\dots)
  10. a 1 \displaystyle a_{1}
  11. P = ( h e h o 1 g e g o 1 ) ( a 1 d 1 ) = P ( a 0 , e a 0 , o ) \begin{aligned}\displaystyle P&\displaystyle=\begin{pmatrix}h_{\mbox{e}~{}}&h_% {\mbox{o}~{}}\rightarrow 1\\ g_{\mbox{e}~{}}&g_{\mbox{o}~{}}\rightarrow 1\end{pmatrix}\\ \displaystyle\begin{pmatrix}a_{1}\\ d_{1}\end{pmatrix}&\displaystyle=P\cdot\begin{pmatrix}a_{0,\mbox{e}~{}}\\ a_{0,\mbox{o}~{}}\end{pmatrix}\end{aligned}
  12. P \scriptstyle P
  13. 2 × 2 \scriptstyle 2\,\times\,2
  14. det P = h e g o - h o g e A A P = I c k det P = c δ k \begin{aligned}\displaystyle\det P&\displaystyle=h_{\mbox{e}~{}}\cdot g_{\mbox% {o}~{}}-h_{\mbox{o}~{}}\cdot g_{\mbox{e}~{}}\\ \displaystyle\exists A\ A\cdot P&\displaystyle=I\iff\exists c\ \exists k\ \det P% =c\cdot\delta\rightarrow k\end{aligned}
  15. P \scriptstyle P
  16. P - 1 det P = ( g o 1 - h o 1 - g e h e ) P^{-1}\cdot\det P=\begin{pmatrix}g_{\mbox{o}~{}}\rightarrow 1&-h_{\mbox{o}~{}}% \rightarrow 1\\ -g_{\mbox{e}~{}}&h_{\mbox{e}~{}}\end{pmatrix}
  17. P * \scriptstyle P^{*}
  18. P \scriptstyle P
  19. P * = ( h e * g e * h o * 1 g o * 1 ) P^{*}=\begin{pmatrix}h_{\mbox{e}~{}}^{*}&g_{\mbox{e}~{}}^{*}\\ h_{\mbox{o}~{}}^{*}\leftarrow 1&g_{\mbox{o}~{}}^{*}\leftarrow 1\end{pmatrix}
  20. || a 1 || 2 2 + || d 1 || 2 2 = || a 0 || 2 2 ||a_{1}||_{2}^{2}+||d_{1}||_{2}^{2}=||a_{0}||_{2}^{2}
  21. P P * = I P\cdot P^{*}=I
  22. h e * h e + h o * h o = δ g e * g e + g o * g o = δ h e * g e + h o * g o = 0 \begin{aligned}\displaystyle h_{\mbox{e}~{}}^{*}\cdot h_{\mbox{e}~{}}+h_{\mbox% {o}~{}}^{*}\cdot h_{\mbox{o}~{}}&\displaystyle=\delta\\ \displaystyle g_{\mbox{e}~{}}^{*}\cdot g_{\mbox{e}~{}}+g_{\mbox{o}~{}}^{*}% \cdot g_{\mbox{o}~{}}&\displaystyle=\delta\\ \displaystyle h_{\mbox{e}~{}}^{*}\cdot g_{\mbox{e}~{}}+h_{\mbox{o}~{}}^{*}% \cdot g_{\mbox{o}~{}}&\displaystyle=0\end{aligned}
  23. x P x 2 [ P - 1 2 - 1 x 2 , P 2 x 2 ] \forall x\ \|P\cdot x\|_{2}\in\left[\|P^{-1}\|_{2}^{-1}\cdot\|x\|_{2},\|P\|_{2% }\cdot\|x\|_{2}\right]
  24. 2 × 2 \scriptstyle 2\,\times\,2
  25. F \scriptstyle\|\cdot\|_{F}
  26. Z \scriptstyle Z
  27. p ( z ) = 1 2 Z P ( z ) F 2 q ( z ) = | det [ Z P ( z ) ] | 2 P 2 = max { p ( z ) + p ( z ) 2 - q ( z ) : z | z | = 1 } P - 1 2 - 1 = min { p ( z ) - p ( z ) 2 - q ( z ) : z | z | = 1 } \begin{aligned}\displaystyle p(z)&\displaystyle=\frac{1}{2}\cdot\|ZP(z)\|_{F}^% {2}\\ \displaystyle q(z)&\displaystyle=\left|\det[ZP(z)]\right|^{2}\\ \displaystyle\|P\|_{2}&\displaystyle=\max\left\{\sqrt{p(z)+\sqrt{p(z)^{2}-q(z)% }}:z\in\mathbb{C}\ \land\ |z|=1\right\}\\ \displaystyle\|P^{-1}\|_{2}^{-1}&\displaystyle=\min\left\{\sqrt{p(z)-\sqrt{p(z% )^{2}-q(z)}}:z\in\mathbb{C}\ \land\ |z|=1\right\}\end{aligned}
  28. n × n n\times n
  29. P 2 = max { λ max [ Z P * ( z ) Z P ( z ) ] : z | z | = 1 } = max { Z P ( z ) 2 : z | z | = 1 } P - 1 2 - 1 = min { λ min [ Z P * ( z ) Z P ( z ) ] : z | z | = 1 } \begin{aligned}\displaystyle\|P\|_{2}&\displaystyle=\sqrt{\max\left\{\lambda_{% \mbox{max}~{}}\left[ZP^{*}(z)\cdot ZP(z)\right]:z\in\mathbb{C}\ \land\ |z|=1% \right\}}\\ &\displaystyle=\max\left\{\|ZP(z)\|_{2}:z\in\mathbb{C}\ \land\ |z|=1\right\}\\ \displaystyle\|P^{-1}\|_{2}^{-1}&\displaystyle=\sqrt{\min\left\{\lambda_{\mbox% {min}~{}}\left[ZP^{*}(z)\cdot ZP(z)\right]:z\in\mathbb{C}\ \land\ |z|=1\right% \}}\end{aligned}

Polyproline_helix.html

  1. 3 cos Ω = 1 - 4 cos 2 [ ( ϕ + ψ ) / 2 ] 3\cos\Omega=1-4\cos^{2}\left[\left(\phi+\psi\right)/2\right]

Polytope_model.html

  1. ( p , t ) = ( i , 2 j + i ) (p,\,t)=(i,\,2j+i)

Polywell.html

  1. β = p p m a g = n k B T ( B 2 / 2 μ 0 ) \beta=\frac{p}{p_{mag}}=\frac{nk_{B}T}{(B^{2}/2\mu_{0})}
  2. P fusion = n A n B σ v A , B E fusion P\text{fusion}=n_{A}n_{B}\langle\sigma v_{A,B}\rangle E\text{fusion}
  3. P fusion P\text{fusion}
  4. σ v A , B \langle\sigma v_{A,B}\rangle

Pontecorvo–Maki–Nakagawa–Sakata_matrix.html

  1. [ ν e ν μ ν τ ] = [ U e 1 U e 2 U e 3 U μ 1 U μ 2 U μ 3 U τ 1 U τ 2 U τ 3 ] [ ν 1 ν 2 ν 3 ] . \begin{bmatrix}{\nu_{e}}\\ {\nu_{\mu}}\\ {\nu_{\tau}}\end{bmatrix}=\begin{bmatrix}U_{e1}&U_{e2}&U_{e3}\\ U_{\mu 1}&U_{\mu 2}&U_{\mu 3}\\ U_{\tau 1}&U_{\tau 2}&U_{\tau 3}\end{bmatrix}\begin{bmatrix}\nu_{1}\\ \nu_{2}\\ \nu_{3}\end{bmatrix}.
  2. [ 1 0 0 0 c 23 s 23 0 - s 23 c 23 ] [ c 13 0 s 13 e - i δ C P 0 1 0 - s 13 e i δ C P 0 c 13 ] [ c 12 s 12 0 - s 12 c 12 0 0 0 1 ] \displaystyle\begin{bmatrix}1&0&0\\ 0&c_{23}&s_{23}\\ 0&-s_{23}&c_{23}\end{bmatrix}\begin{bmatrix}c_{13}&0&s_{13}e^{-i\delta_{CP}}\\ 0&1&0\\ -s_{13}e^{i\delta_{CP}}&0&c_{13}\end{bmatrix}\begin{bmatrix}c_{12}&s_{12}&0\\ -s_{12}&c_{12}&0\\ 0&0&1\end{bmatrix}
  3. ν = ν c \nu=\nu^{c}
  4. sin 2 2 θ 12 = 0.857 ± 0.024 sin 2 2 θ 23 > 0.95 sin 2 2 θ 13 = 0.095 ± 0.010 \begin{aligned}\displaystyle\sin^{2}2\theta_{12}&\displaystyle=0.857\pm 0.024% \\ \displaystyle\sin^{2}2\theta_{23}&\displaystyle>0.95\\ \displaystyle\sin^{2}2\theta_{13}&\displaystyle=0.095\pm 0.010\\ \end{aligned}
  5. θ 12 [ ] = 33.36 - 0.78 + 0.81 θ 23 [ ] = 40.0 - 1.5 + 2.1 or 50.4 - 1.3 + 1.3 θ 13 [ ] = 8.66 - 0.46 + 0.44 δ CP [ ] = 300 - 138 + 66 \begin{aligned}\displaystyle\theta_{12}[^{\circ}]&\displaystyle=33.36^{+0.81}_% {-0.78}\\ \displaystyle\theta_{23}[^{\circ}]&\displaystyle=40.0^{+2.1}_{-1.5}~{}\textrm{% or}~{}50.4^{+1.3}_{-1.3}\\ \displaystyle\theta_{13}[^{\circ}]&\displaystyle=8.66^{+0.44}_{-0.46}\\ \displaystyle\delta_{\textrm{CP}}[^{\circ}]&\displaystyle=300^{+66}_{-138}\\ \end{aligned}

Portal:India::Quiz::Archive5.html

  1. \infty

Poset_topology.html

  1. σ V \sigma\subseteq V
  2. ρ , σ . ρ σ Δ ρ Δ \forall\rho,\sigma.\;\ \rho\subseteq\sigma\in\Delta\Rightarrow\rho\in\Delta
  3. Γ Δ \Gamma\subseteq\Delta
  4. ρ , σ . ρ σ Γ ρ Γ \forall\rho,\sigma.\;\ \rho\subseteq\sigma\in\Gamma\Rightarrow\rho\in\Gamma

Post's_inversion_formula.html

  1. sup t > 0 f ( t ) e b t < \sup_{t>0}\frac{f(t)}{e^{bt}}<\infty
  2. f ( t ) = - 1 { F ( s ) } = lim k ( - 1 ) k k ! ( k t ) k + 1 F ( k ) ( k t ) f(t)=\mathcal{L}^{-1}\{F(s)\}=\lim_{k\to\infty}\frac{(-1)^{k}}{k!}\left(\frac{% k}{t}\right)^{k+1}F^{(k)}\left(\frac{k}{t}\right)

Potassium_persulfate.html

  1. \overrightarrow{\leftarrow}

Potential_density.html

  1. P P
  2. P 0 P_{0}
  3. ρ θ \rho_{\theta}
  4. P 0 P_{0}
  5. σ θ = ρ θ - 1000 \sigma_{\theta}=\rho_{\theta}-1000
  6. σ 4 \sigma_{4}
  7. γ n \gamma^{n}
  8. ρ = ρ ( P , T , S 1 , S 2 , ) \rho=\rho(P,T,S_{1},S_{2},...)
  9. T T
  10. P P
  11. S n S_{n}
  12. ρ θ = ρ ( P 0 , θ , S 1 , S 2 , ) \rho_{\theta}=\rho(P_{0},\theta,S_{1},S_{2},...)
  13. θ \theta
  14. P 0 P_{0}

Potential_Determining_Ion.html

  1. NaCl ( s ) Na ( a q ) + + Cl ( a q ) - \mathrm{NaCl}_{(s)}\leftrightarrow\mathrm{Na}_{(aq)}^{+}+\mathrm{Cl}_{(aq)}^{-}
  2. a Na + a Cl - = K s p a_{\mathrm{Na}^{+}}\cdot a_{\mathrm{Cl}^{-}}=K_{sp}

Potential_method.html

  1. T amortized ( o ) = T actual ( o ) + C ( Φ ( S after ) - Φ ( S before ) ) , T_{\mathrm{amortized}}(o)=T_{\mathrm{actual}}(o)+C\cdot(\Phi(S_{\mathrm{after}% })-\Phi(S_{\mathrm{before}})),
  2. O = o 1 , o 2 , O=o_{1},o_{2},\dots
  3. T amortized ( O ) = i T amortized ( o i ) T_{\mathrm{amortized}}(O)=\sum_{i}T_{\mathrm{amortized}}(o_{i})
  4. T actual ( O ) = i T actual ( o i ) T_{\mathrm{actual}}(O)=\sum_{i}T_{\mathrm{actual}}(o_{i})
  5. T amortized ( O ) = i ( T actual ( o i ) + C ( Φ ( S i + 1 ) - Φ ( S i ) ) ) = T actual ( O ) + C ( Φ ( S final ) - Φ ( S initial ) ) T_{\mathrm{amortized}}(O)=\sum_{i}\left(T_{\mathrm{actual}}(o_{i})+C\cdot(\Phi% (S_{i+1})-\Phi(S_{i}))\right)=T_{\mathrm{actual}}(O)+C\cdot(\Phi(S_{\mathrm{% final}})-\Phi(S_{\mathrm{initial}}))
  6. T actual ( O ) = T amortized ( O ) + C ( Φ ( S initial ) - Φ ( S final ) ) T_{\mathrm{actual}}(O)=T_{\mathrm{amortized}}(O)+C\cdot(\Phi(S_{\mathrm{% initial}})-\Phi(S_{\mathrm{final}}))
  7. Φ ( S final ) 0 \Phi(S_{\mathrm{final}})\geq 0
  8. Φ ( S initial ) = 0 \Phi(S_{\mathrm{initial}})=0
  9. T actual ( O ) T amortized ( O ) T_{\mathrm{actual}}(O)\leq T_{\mathrm{amortized}}(O)

Potential_vorticity.html

  1. 10 - 6 K m 2 k g s 1 P V U {10^{-6}\cdot K\cdot m^{2}\over kg\cdot s}\equiv 1PVU
  2. PV = 1 ρ ζ a θ {\rm PV}=\frac{1}{\rho}\,\zeta\,^{a}\cdot\,\nabla\theta
  3. ρ \rho
  4. ζ a \zeta^{a}
  5. θ \nabla\theta

Poussin_proof.html

  1. k = 1 η d ( k ) η ln η + 2 γ - 1 , \frac{\sum_{k=1}^{\eta}d(k)}{\eta}\approx\ln\eta+2\gamma-1,
  2. p η { η p } π ( η ) 1 - γ , \frac{\sum_{p\leq\eta}\left\{\frac{\eta}{p}\right\}}{\pi(\eta)}\approx 1-\gamma,

Power_automorphism.html

  1. P o t ( G ) Pot(G)
  2. G G

Power_dividers_and_directional_couplers.html

  1. P a , b P_{\mathrm{a,b}}
  2. C 3 , 1 = 10 log ( P 3 P 1 ) dB C_{3,1}=10\log{\left(\frac{P_{3}}{P_{1}}\right)}\quad\rm{dB}
  3. L i 2 , 1 = - 10 log ( P 2 P 1 ) dB L_{i2,1}=-10\log{\left(\frac{P_{2}}{P_{1}}\right)}\quad\rm{dB}
  4. L c 2 , 1 = - 10 log ( 1 - P 3 P 1 ) dB L_{c2,1}=-10\log{\left(1-\frac{P_{3}}{P_{1}}\right)}\quad\rm{dB}
  5. I 4 , 1 = - 10 log ( P 4 P 1 ) dB I_{4,1}=-10\log{\left(\frac{P_{4}}{P_{1}}\right)}\quad\rm{dB}
  6. I 3 , 2 = - 10 log ( P 3 P 2 ) dB I_{3,2}=-10\log{\left(\frac{P_{3}}{P_{2}}\right)}\quad\rm{dB}
  7. D 3 , 4 = - 10 log ( P 4 P 3 ) = - 10 log ( P 4 P 1 ) + 10 log ( P 3 P 1 ) dB D_{3,4}=-10\log{\left(\frac{P_{4}}{P_{3}}\right)}=-10\log{\left(\frac{P_{4}}{P% _{1}}\right)}+10\log{\left(\frac{P_{3}}{P_{1}}\right)}\quad\rm{dB}
  8. D 3 , 4 = I 4 , 1 - C 3 , 1 dB D_{3,4}=I_{4,1}-C_{3,1}\quad\rm{dB}
  9. 𝐒 = [ 0 τ κ 0 τ 0 0 κ κ 0 0 τ 0 κ τ 0 ] \mathbf{S}=\begin{bmatrix}0&\tau&\kappa&0\\ \tau&0&0&\kappa\\ \kappa&0&0&\tau\\ 0&\kappa&\tau&0\end{bmatrix}
  10. τ \tau
  11. κ \kappa
  12. τ \tau
  13. κ \kappa
  14. τ τ ¯ + κ κ ¯ = 1 \tau\overline{\tau}+\kappa\overline{\kappa}=1
  15. τ \tau
  16. L ( dB ) = - 20 log | τ | L(\mathrm{dB})=-20\log|\tau|
  17. κ \kappa
  18. C ( dB ) = 20 log | κ | C(\mathrm{dB})=20\log|\kappa|
  19. τ \tau
  20. κ \kappa
  21. κ = i κ I \kappa=i\kappa_{\mathrm{I}}
  22. τ 2 + κ I 2 = 1 \tau^{2}+{\kappa_{\mathrm{I}}}^{2}=1
  23. 2 \scriptstyle\sqrt{2}
  24. 2 \scriptstyle\sqrt{2}
  25. 𝐒 = 1 2 [ 0 - i - 1 0 - i 0 0 - 1 - 1 0 0 - i 0 - 1 - i 0 ] \mathbf{S}=\frac{1}{\sqrt{2}}\begin{bmatrix}0&-i&-1&0\\ -i&0&0&-1\\ -1&0&0&-i\\ 0&-1&-i&0\end{bmatrix}
  26. 2 \scriptstyle\sqrt{2}
  27. 𝐒 = 1 2 [ 0 - i - i 0 - i 0 0 i - i 0 0 - i 0 i - i 0 ] \mathbf{S}=\frac{1}{\sqrt{2}}\begin{bmatrix}0&-i&-i&0\\ -i&0&0&i\\ -i&0&0&-i\\ 0&i&-i&0\end{bmatrix}
  28. 2 λ / 4 \scriptstyle\sqrt{2}\lambda/4
  29. C 3 , 1 = 20 log n C_{3,1}=20\log n
  30. n = 2 \scriptstyle n=\sqrt{2}

Power_iteration.html

  1. b k + 1 = A b k A b k . b_{k+1}=\frac{Ab_{k}}{\|Ab_{k}\|}.
  2. b 0 b_{0}
  3. ( b k ) \left(b_{k}\right)
  4. ( b k ) \left(b_{k}\right)
  5. b k = e i ϕ k v 1 + r k b_{k}=e^{i\phi_{k}}v_{1}+r_{k}
  6. v 1 v_{1}
  7. r k 0 \|r_{k}\|\rightarrow 0
  8. e i ϕ k e^{i\phi_{k}}
  9. ( b k ) \left(b_{k}\right)
  10. e i ϕ k = 1 e^{i\phi_{k}}=1
  11. ( μ k ) \left(\mu_{k}\right)
  12. μ k = b k * A b k b k * b k \mu_{k}=\frac{b_{k}^{*}Ab_{k}}{b_{k}^{*}b_{k}}

Prandtl–Meyer_expansion_fan.html

  1. w 2 > w 1 w_{2}>w_{1}
  2. v 2 = v 1 v_{2}=v_{1}
  3. Δ s = s 2 - s 1 \Delta s=s_{2}-s_{1}
  4. Δ s R = l n [ ( p 2 p 1 ) 1 / ( γ - 1 ) ( ρ 2 ρ 1 ) - γ / ( γ - 1 ) ] γ + 1 12 γ 2 ( p 2 - p 1 p 1 ) 3 γ + 1 12 γ 2 [ ρ 1 w 1 2 p 1 ( 1 - w 2 w 1 ) ] 3 \begin{aligned}\displaystyle\frac{\Delta s}{R}&\displaystyle=ln\bigg[\bigg(% \frac{p_{2}}{p_{1}}\bigg)^{1/(\gamma-1)}\bigg(\frac{\rho_{2}}{\rho_{1}}\bigg)^% {-\gamma/(\gamma-1)}\bigg]\\ &\displaystyle\approx\frac{\gamma+1}{12\gamma^{2}}\bigg(\frac{p_{2}-p_{1}}{p_{% 1}}\bigg)^{3}\\ &\displaystyle\approx\frac{\gamma+1}{12\gamma^{2}}\bigg[\frac{\rho_{1}w_{1}^{2% }}{p_{1}}\bigg(1-\frac{w_{2}}{w_{1}}\bigg)\bigg]^{3}\end{aligned}
  5. R R
  6. γ \gamma
  7. ρ \rho
  8. p p
  9. s s
  10. w w
  11. w 2 > w 1 w_{2}>w_{1}
  12. Δ s < 0 \Delta s<0
  13. Δ s 0 \Delta s\rightarrow 0
  14. M 1 M\geq 1
  15. μ = arcsin ( c u ) = arcsin ( 1 M ) \textstyle\mu=\arcsin\left(\frac{c}{u}\right)=\arcsin\left(\frac{1}{M}\right)
  16. u > c u>c
  17. u c u\geq c
  18. arcsin ( ) \arcsin()
  19. μ 1 = arcsin ( 1 M 1 ) \mu_{1}=\arcsin\left(\frac{1}{M_{1}}\right)
  20. μ 2 = arcsin ( 1 M 2 ) \mu_{2}=\arcsin\left(\frac{1}{M_{2}}\right)
  21. p 0 p_{0}
  22. T 0 T_{0}
  23. ρ 0 \rho_{0}
  24. M 2 M_{2}
  25. T 2 T 1 = ( 1 + γ - 1 2 M 1 2 1 + γ - 1 2 M 2 2 ) p 2 p 1 = ( 1 + γ - 1 2 M 1 2 1 + γ - 1 2 M 2 2 ) γ / ( γ - 1 ) ρ 2 ρ 1 = ( 1 + γ - 1 2 M 1 2 1 + γ - 1 2 M 2 2 ) 1 / ( γ - 1 ) . \begin{array}[]{lcl}\frac{T_{2}}{T_{1}}&=&\bigg(\frac{1+\frac{\gamma-1}{2}M_{1% }^{2}}{1+\frac{\gamma-1}{2}M_{2}^{2}}\bigg)\\ \frac{p_{2}}{p_{1}}&=&\bigg(\frac{1+\frac{\gamma-1}{2}M_{1}^{2}}{1+\frac{% \gamma-1}{2}M_{2}^{2}}\bigg)^{\gamma/(\gamma-1)}\\ \frac{\rho_{2}}{\rho_{1}}&=&\bigg(\frac{1+\frac{\gamma-1}{2}M_{1}^{2}}{1+\frac% {\gamma-1}{2}M_{2}^{2}}\bigg)^{1/(\gamma-1)}.\end{array}
  26. M 2 M_{2}
  27. M 1 M_{1}
  28. θ \theta
  29. θ = ν ( M 2 ) - ν ( M 1 ) \theta=\nu(M_{2})-\nu(M_{1})\,
  30. ν ( M ) \nu(M)\,
  31. ν ( M ) \displaystyle\nu(M)
  32. ν ( 1 ) = 0. \nu(1)=0.\,
  33. M 1 M_{1}
  34. ν ( M 1 ) \nu(M_{1})\,
  35. ν ( M 2 ) \nu(M_{2})\,
  36. ν ( M 2 ) \nu(M_{2})\,
  37. M 2 M_{2}
  38. θ m a x \theta_{max}
  39. \infty
  40. ν \nu\,
  41. ν max \nu\text{max}\,
  42. ν max = π 2 ( γ + 1 γ - 1 - 1 ) . \nu\text{max}=\frac{\pi}{2}\bigg(\sqrt{\frac{\gamma+1}{\gamma-1}}-1\bigg).
  43. θ max = ν max - ν ( M 1 ) . \theta\text{max}=\nu\text{max}-\nu(M_{1}).\,
  44. M 1 M_{1}

Pre-charge.html

  1. I = C ( d V / d T ) I=C(dV/dT)

Preclosure_operator.html

  1. X X
  2. [ ] p [\quad]_{p}
  3. [ ] p : 𝒫 ( X ) 𝒫 ( X ) [\quad]_{p}:\mathcal{P}(X)\to\mathcal{P}(X)
  4. 𝒫 ( X ) \mathcal{P}(X)
  5. X X
  6. [ ] p = [\varnothing]_{p}=\varnothing\!
  7. A [ A ] p A\subseteq[A]_{p}
  8. [ A B ] p = [ A ] p [ B ] p [A\cup B]_{p}=[A]_{p}\cup[B]_{p}
  9. A B A\subseteq B
  10. [ A ] p [ B ] p [A]_{p}\subseteq[B]_{p}
  11. A A
  12. [ A ] p = A [A]_{p}=A
  13. U X U\subset X
  14. A = X U A=X\setminus U
  15. [ A ] p cl ( A ) [A]_{p}\subseteq\operatorname{cl}(A)
  16. A X A\subset X
  17. d d
  18. X X
  19. [ A ] p = { x X : d ( x , A ) = 0 } [A]_{p}=\{x\in X:d(x,A)=0\}
  20. X X
  21. [ ] seq [\quad]_{\mbox{seq}}~{}
  22. 𝒯 \mathcal{T}
  23. ( X , 𝒯 ) (X,\mathcal{T})
  24. 𝒯 seq \mathcal{T}_{\mbox{seq}}~{}
  25. [ ] seq [\quad]_{\mbox{seq}}~{}
  26. 𝒯 \mathcal{T}
  27. 𝒯 seq = 𝒯 \mathcal{T}_{\mbox{seq}}~{}=\mathcal{T}

Preconditioner.html

  1. P P
  2. A A
  3. P - 1 A P^{-1}A
  4. A A
  5. T = P - 1 T=P^{-1}
  6. P P
  7. P P
  8. T = P - 1 T=P^{-1}
  9. T = P - 1 T=P^{-1}
  10. P P
  11. T = P - 1 T=P^{-1}
  12. A A
  13. A x = b Ax=b
  14. x x
  15. A A
  16. A P - 1 P x = b AP^{-1}Px=b
  17. A P - 1 y = b AP^{-1}y=b
  18. y y
  19. P x = y Px=y
  20. x x
  21. P - 1 ( A x - b ) = 0 P^{-1}(Ax-b)=0
  22. P P
  23. P - 1 A P^{-1}A
  24. A P - 1 , AP^{-1},
  25. P - 1 A P^{-1}A
  26. A P - 1 AP^{-1}
  27. P - 1 P^{-1}
  28. P P
  29. P - 1 P^{-1}
  30. P - 1 P^{-1}
  31. P = I P=I
  32. P - 1 = I . P^{-1}=I.
  33. P = A P=A
  34. P - 1 A = A P - 1 = I , P^{-1}A=AP^{-1}=I,
  35. P - 1 = A - 1 , P^{-1}=A^{-1},
  36. P P
  37. P - 1 P^{-1}
  38. A x - b = 0 Ax-b=0
  39. P - 1 ( A x - b ) = 0. P^{-1}(Ax-b)=0.
  40. A x - b = 0 Ax-b=0
  41. 𝐱 n + 1 = 𝐱 n - γ n ( A 𝐱 n - 𝐛 ) , n 0. \mathbf{x}_{n+1}=\mathbf{x}_{n}-\gamma_{n}(A\mathbf{x}_{n}-\mathbf{b}),\ n\geq 0.
  42. P - 1 ( A x - b ) = 0 , P^{-1}(Ax-b)=0,
  43. 𝐱 n + 1 = 𝐱 n - γ n P - 1 ( A 𝐱 n - 𝐛 ) , n 0. \mathbf{x}_{n+1}=\mathbf{x}_{n}-\gamma_{n}P^{-1}(A\mathbf{x}_{n}-\mathbf{b}),% \ n\geq 0.
  44. P - 1 ( A x - b ) = 0 P^{-1}(Ax-b)=0
  45. A x - b = 0. Ax-b=0.
  46. A A
  47. P P
  48. P - 1 A P^{-1}A
  49. P P
  50. P - 1 A P^{-1}A
  51. P P
  52. T = P - 1 T=P^{-1}
  53. r r
  54. T T
  55. T r . Tr.
  56. T T
  57. T ( r ) T(r)
  58. r r
  59. r r
  60. r r
  61. P - 1 A P^{-1}A
  62. P - 1 P^{-1}
  63. A A
  64. P = diag ( A ) . P=\mathrm{diag}(A).
  65. A i i 0 , i A_{ii}\neq 0,\forall i
  66. P i j - 1 = δ i j A i j . P^{-1}_{ij}=\frac{\delta_{ij}}{A_{ij}}.
  67. A A
  68. A T - I F , \|AT-I\|_{F},
  69. F \|\cdot\|_{F}
  70. T = P - 1 T=P^{-1}
  71. T T
  72. A A
  73. A x = λ x Ax=\lambda x
  74. A A
  75. P - 1 A P^{-1}A
  76. P P
  77. A A
  78. P - 1 A P^{-1}A
  79. α \alpha
  80. A x = λ x Ax=\lambda x
  81. ( A - α I ) - 1 x = μ x (A-\alpha I)^{-1}x=\mu x
  82. α \alpha
  83. λ \lambda_{\star}
  84. ( A - λ I ) x = 0 (A-\lambda_{\star}I)x=0
  85. T ( A - λ I ) x = 0 T(A-\lambda_{\star}I)x=0
  86. T T
  87. 𝐱 n + 1 = 𝐱 n - γ n T ( A - λ I ) ) 𝐱 n , n 0. \mathbf{x}_{n+1}=\mathbf{x}_{n}-\gamma_{n}T(A-\lambda_{\star}I))\mathbf{x}_{n}% ,\ n\geq 0.
  88. T = ( A - λ I ) + T=(A-\lambda_{\star}I)^{+}
  89. γ n = 1 \gamma_{n}=1
  90. I - ( A - λ I ) + ( A - λ I ) I-(A-\lambda_{\star}I)^{+}(A-\lambda_{\star}I)
  91. P P_{\star}
  92. λ \lambda_{\star}
  93. T = ( A - λ I ) + T=(A-\lambda_{\star}I)^{+}
  94. λ \lambda_{\star}
  95. λ ~ \tilde{\lambda}_{\star}
  96. T = ( I - P ~ ) ( A - λ ~ I ) - 1 ( I - P ~ ) T=(I-\tilde{P}_{\star})(A-\tilde{\lambda}_{\star}I)^{-1}(I-\tilde{P}_{\star})
  97. P ~ \tilde{P}_{\star}
  98. P P_{\star}
  99. ( A - λ ~ I ) (A-\tilde{\lambda}_{\star}I)
  100. λ \lambda_{\star}
  101. λ n \lambda_{n}
  102. 𝐱 n + 1 = 𝐱 n - γ n T ( A - λ n I ) 𝐱 n , n 0. \mathbf{x}_{n+1}=\mathbf{x}_{n}-\gamma_{n}T(A-\lambda_{n}I)\mathbf{x}_{n},\ n% \geq 0.
  103. λ n = ρ ( x n ) \lambda_{n}=\rho(x_{n})
  104. ρ ( ) \rho(\cdot)
  105. T = ( d i a g ( A ) ) - 1 T=(diag(A))^{-1}
  106. T = ( d i a g ( A - λ n I ) ) - 1 . T=(diag(A-\lambda_{n}I))^{-1}.
  107. T A - 1 T\approx A^{-1}
  108. T A - 1 T\approx A^{-1}
  109. λ n \lambda_{n}
  110. F ( 𝐱 ) F(\mathbf{x})
  111. - F ( 𝐚 ) -\nabla F(\mathbf{a})
  112. 𝐱 n + 1 = 𝐱 n - γ n F ( 𝐱 n ) , n 0. \mathbf{x}_{n+1}=\mathbf{x}_{n}-\gamma_{n}\nabla F(\mathbf{x}_{n}),\ n\geq 0.
  113. 𝐱 n + 1 = 𝐱 n - γ n P - 1 F ( 𝐱 n ) , n 0. \mathbf{x}_{n+1}=\mathbf{x}_{n}-\gamma_{n}P^{-1}\nabla F(\mathbf{x}_{n}),\ n% \geq 0.
  114. F ( 𝐱 ) = 1 2 𝐱 T A 𝐱 - 𝐱 T 𝐛 F(\mathbf{x})=\frac{1}{2}\mathbf{x}^{T}A\mathbf{x}-\mathbf{x}^{T}\mathbf{b}
  115. 𝐱 \mathbf{x}
  116. 𝐛 \mathbf{b}
  117. A A
  118. A 𝐱 = 𝐛 A\mathbf{x}=\mathbf{b}
  119. F ( 𝐱 ) = A 𝐱 - 𝐛 \nabla F(\mathbf{x})=A\mathbf{x}-\mathbf{b}
  120. F ( 𝐱 ) F(\mathbf{x})
  121. 𝐱 n + 1 = 𝐱 n - γ n P - 1 ( A 𝐱 n - 𝐛 ) , n 0. \mathbf{x}_{n+1}=\mathbf{x}_{n}-\gamma_{n}P^{-1}(A\mathbf{x}_{n}-\mathbf{b}),% \ n\geq 0.
  122. ρ ( 𝐱 ) = 𝐱 T A 𝐱 𝐱 T 𝐱 , \rho(\mathbf{x})=\frac{\mathbf{x}^{T}A\mathbf{x}}{\mathbf{x}^{T}\mathbf{x}},
  123. 𝐱 \mathbf{x}
  124. A A
  125. A A
  126. ρ ( 𝐱 ) \nabla\rho(\mathbf{x})
  127. A 𝐱 - ρ ( 𝐱 ) 𝐱 A\mathbf{x}-\rho(\mathbf{x})\mathbf{x}
  128. ρ ( 𝐱 ) \rho(\mathbf{x})
  129. 𝐱 n + 1 = 𝐱 n - γ n P - 1 ( A 𝐱 n - ρ ( 𝐱 𝐧 ) 𝐱 𝐧 ) , n 0. \mathbf{x}_{n+1}=\mathbf{x}_{n}-\gamma_{n}P^{-1}(A\mathbf{x}_{n}-\rho(\mathbf{% x_{n}})\mathbf{x_{n}}),\ n\geq 0.
  130. 𝐱 n + 1 = 𝐱 n - γ n P n - 1 F ( 𝐱 n ) , n 0. \mathbf{x}_{n+1}=\mathbf{x}_{n}-\gamma_{n}P_{n}^{-1}\nabla F(\mathbf{x}_{n}),% \ n\geq 0.

Preferential_attachment.html

  1. P ( k ) = B ( k + a , γ ) B ( k 0 + a , γ - 1 ) , P(k)={\mathrm{B}(k+a,\gamma)\over\mathrm{B}(k_{0}+a,\gamma-1)},
  2. B ( x , y ) = Γ ( x ) Γ ( y ) Γ ( x + y ) , \mathrm{B}(x,y)={\Gamma(x)\Gamma(y)\over\Gamma(x+y)},
  3. γ = 2 + k 0 + a m . \gamma=2+{k_{0}+a\over m}.
  4. P ( k ) k - γ . P(k)\propto k^{-\gamma}.

Preferential_concentration.html

  1. S t k τ p τ f = ρ p d 2 ϵ 1 / 2 18 ρ f ν 3 / 2 Stk\equiv\frac{\tau_{p}}{\tau_{f}}=\frac{\rho_{p}d^{2}\epsilon^{1/2}}{18\rho_{% f}\nu^{3/2}}
  2. τ p \tau_{p}
  3. τ f \tau_{f}
  4. ρ p \rho_{p}
  5. ρ f \rho_{f}
  6. ν \nu
  7. ϵ \epsilon
  8. S t k 1 Stk\sim 1
  9. S t k 1 Stk\ll 1
  10. S t k 1 Stk\gg 1

Presentation_of_a_monoid.html

  1. R = { u 1 = v 1 , , u n = v n } R=\{u_{1}=v_{1},\cdots,u_{n}=v_{n}\}
  2. p , q | p q = 1 \langle p,q\,|\;pq=1\rangle
  3. a , b | a b a = b a a , b b a = b a b \langle a,b\,|\;aba=baa,bba=bab\rangle
  4. a i b j ( b a ) k a^{i}b^{j}(ba)^{k}
  5. ( X ; T ) (X;T)
  6. ( X X - 1 ) * (X\cup X^{-1})^{*}
  7. X X
  8. T ( X X - 1 ) * × ( X X - 1 ) * T\subseteq(X\cup X^{-1})^{*}\times(X\cup X^{-1})^{*}
  9. T e T^{\mathrm{e}}
  10. T c T^{\mathrm{c}}
  11. Inv 1 X | T . \mathrm{Inv}^{1}\langle X|T\rangle.
  12. ρ X \rho_{X}
  13. X X
  14. Inv 1 X | T \mathrm{Inv}^{1}\langle X|T\rangle
  15. ( X ; T ) (X;T)
  16. Inv 1 X | T = ( X X - 1 ) * / ( T ρ X ) c . \mathrm{Inv}^{1}\langle X|T\rangle=(X\cup X^{-1})^{*}/(T\cup\rho_{X})^{\mathrm% {c}}.
  17. ( X X - 1 ) * ({X\cup X^{-1}})^{*}
  18. ( X X - 1 ) + ({X\cup X^{-1}})^{+}
  19. ( X ; T ) (X;T)
  20. Inv X | T \mathrm{Inv}\langle X|T\rangle
  21. ( X ; T ) (X;T)
  22. X X
  23. FIM ( X ) \mathrm{FIM}(X)
  24. FIS ( X ) \mathrm{FIS}(X)
  25. FIM ( X ) = Inv 1 X | = ( X X - 1 ) * / ρ X , \mathrm{FIM}(X)=\mathrm{Inv}^{1}\langle X|\varnothing\rangle=({X\cup X^{-1}})^% {*}/\rho_{X},
  26. FIS ( X ) = Inv X | = ( X X - 1 ) + / ρ X . \mathrm{FIS}(X)=\mathrm{Inv}\langle X|\varnothing\rangle=({X\cup X^{-1}})^{+}/% \rho_{X}.

Pressure_head.html

  1. ψ = p γ = p ρ g \psi=\frac{p}{\gamma}=\frac{p}{\rho\,g}
  2. ψ \psi
  3. p p
  4. γ \gamma
  5. ρ \rho
  6. g g
  7. h v + z elevation + ψ = C h_{v}+z\text{elevation}+\psi=C\,
  8. h v h_{v}
  9. z elevation z\text{elevation}
  10. ψ \psi
  11. C C
  12. ψ \psi\,
  13. g g
  14. p > 0 p>0
  15. p > 0 p>0
  16. p < 0 p<0
  17. g > 0 g>0
  18. ψ > 0 \psi>0
  19. ψ < 0 \psi<0
  20. p < 0 p<0
  21. g < 0 g<0

Primary_pseudoperfect_number.html

  1. p | N 1 p + 1 N = 1 , \sum_{p|N}\frac{1}{p}+\frac{1}{N}=1,
  2. p | N N p + 1 = N . \sum_{p|N}\frac{N}{p}+1=N.

Prime_gap.html

  1. g n = p n + 1 - p n . g_{n}=p_{n+1}-p_{n}.
  2. p n + 1 = 2 + i = 1 n g i p_{n+1}=2+\sum_{i=1}^{n}g_{i}
  3. P # + 2 , P # + 3 , , P # + ( Q - 1 ) P\#+2,P\#+3,\ldots,P\#+(Q-1)
  4. lim sup n g n log p n = . \limsup_{n\to\infty}\frac{g_{n}}{\log p_{n}}=\infty.
  5. lim n g n p n = 0 \lim_{n\to\infty}\frac{g_{n}}{p_{n}}=0
  6. g n < p n θ , g_{n}<p_{n}^{\theta},\,
  7. ζ ( 1 / 2 + i t ) = O ( t c ) \zeta(1/2+it)=O(t^{c})\,
  8. π ( x + x θ ) - π ( x ) x θ log ( x ) \pi(x+x^{\theta})-\pi(x)\sim\frac{x^{\theta}}{\log(x)}
  9. lim inf n g n log p n = 0 \liminf_{n\to\infty}\frac{g_{n}}{\log p_{n}}=0
  10. lim inf n g n log p n ( log log p n ) 2 < . \liminf_{n\to\infty}\frac{g_{n}}{\sqrt{\log p_{n}}(\log\log p_{n})^{2}}<\infty.
  11. lim inf n g n < 7 10 7 \liminf_{n\to\infty}g_{n}<7\cdot 10^{7}
  12. g n > c log n log log n log log log log n ( log log log n ) 2 g_{n}>\frac{c\log n\log\log n\log\log\log\log n}{(\log\log\log n)^{2}}
  13. g n log n log log n log log log log n log log log n g_{n}\gg\frac{\log n\log\log n\log\log\log\log n}{\log\log\log n}
  14. g n = O ( p n ln p n ) , g_{n}=O(\sqrt{p_{n}}\ln p_{n}),
  15. g n = O ( ( ln p n ) 2 ) . g_{n}=O\left((\ln p_{n})^{2}\right).
  16. p n 1 / n p_{n}^{1/n}\,
  17. p n p_{n}\,
  18. p n + 1 1 / ( n + 1 ) < p n 1 / n for all n 1. p_{n+1}^{1/(n+1)}<p_{n}^{1/n}\,\text{ for all }n\geq 1.
  19. g n = p n + 1 - p n g_{n}=p_{n+1}-p_{n}
  20. g n < ( log p n ) 2 - log p n for all n > 4. g_{n}<(\log p_{n})^{2}-\log p_{n}\,\text{ for all }n>4.
  21. × 10 1 8 \times 10^{1}8
  22. g n < p n g_{n}<\sqrt{p_{n}}\,
  23. g n < 2 p n + 1. g_{n}<2\sqrt{p_{n}}+1.\,

Primitive_recursive_arithmetic.html

  1. S ( x ) 0 S(x)\neq 0
  2. S ( x ) = S ( y ) x = y , S(x)=S(y)~{}\to~{}x=y,
  3. f ( 0 , y 1 , , y n ) = g ( y 1 , , y n ) f(0,y_{1},\ldots,y_{n})=g(y_{1},\ldots,y_{n})
  4. f ( S ( x ) , y 1 , , y n ) = h ( x , f ( x , y 1 , , y n ) , y 1 , , y n ) f(S(x),y_{1},\ldots,y_{n})=h(x,f(x,y_{1},\ldots,y_{n}),y_{1},\ldots,y_{n})
  5. x + 0 = x x+0=x
  6. x + S ( y ) = S ( x + y ) x+S(y)=S(x+y)
  7. x 0 = 0 x\cdot 0=0
  8. x S ( y ) = x y + x x\cdot S(y)=x\cdot y+x
  9. φ ( 0 ) \varphi(0)
  10. φ ( x ) φ ( S ( x ) ) \varphi(x)\to\varphi(S(x))
  11. φ ( y ) \varphi(y)
  12. φ . \varphi.
  13. F ( 0 ) = G ( 0 ) F ( S ( x ) ) = H ( x , F ( x ) ) G ( S ( x ) ) = H ( x , G ( x ) ) F ( x ) = G ( x ) . {F(0)=G(0)\quad F(S(x))=H(x,F(x))\quad G(S(x))=H(x,G(x))\over F(x)=G(x)}.
  14. F ( x ) = G ( x ) F ( A ) = G ( A ) A = B F ( A ) = F ( B ) A = B A = C B = C . {F(x)=G(x)\over F(A)=G(A)}\qquad{A=B\over F(A)=F(B)}\qquad{A=B\quad A=C\over B% =C}.
  15. P ( 0 ) = 0 \displaystyle P(0)=0
  16. | x - y | + | u - v | = 0 |x-y|+|u-v|=0\!
  17. | x - y | | u - v | = 0 |x-y|\cdot|u-v|=0\!
  18. 1 - ˙ | x - y | = 0 1\dot{-}|x-y|=0

Primon_gas.html

  1. | p |p\rangle
  2. H | p = E p | p H|p\rangle=E_{p}|p\rangle
  3. E p = E 0 log p E_{p}=E_{0}\log p\,
  4. k p k_{p}
  5. p p
  6. | n = | k 2 , k 3 , k 5 , k 7 , k 11 , , k p , |n\rangle=|k_{2},k_{3},k_{5},k_{7},k_{11},\ldots,k_{p},\ldots\rangle
  7. n n
  8. n = 2 k 2 3 k 3 5 k 5 7 k 7 11 k 11 p k p n=2^{k_{2}}\cdot 3^{k_{3}}\cdot 5^{k_{5}}\cdot 7^{k_{7}}\cdot 11^{k_{11}}% \cdots p^{k_{p}}\cdots
  9. E ( n ) = p k p E p = E 0 p k p log p = E 0 log n E(n)=\sum_{p}k_{p}E_{p}=E_{0}\cdot\sum_{p}k_{p}\log p=E_{0}\log n
  10. Z ( T ) := n = 1 exp ( - E ( n ) k B T ) = n = 1 exp ( - E 0 log n k B T ) = n = 1 1 n s = ζ ( s ) Z(T):=\sum_{n=1}^{\infty}\exp\left(\frac{-E(n)}{k_{B}T}\right)=\sum_{n=1}^{% \infty}\exp\left(\frac{-E_{0}\log n}{k_{B}T}\right)=\sum_{n=1}^{\infty}\frac{1% }{n^{s}}=\zeta(s)
  11. μ ( n ) \mu(n)

Principal_ideal_ring.html

  1. R = i = 1 n R i R=\prod_{i=1}^{n}R_{i}
  2. A = i = 1 n A i A=\prod_{i=1}^{n}A_{i}
  3. A i A_{i}
  4. ( x 1 , , x n ) R = A (x_{1},\ldots,x_{n})R=A
  5. / n \mathbb{Z}/n\mathbb{Z}
  6. R 1 , , R n R_{1},\ldots,R_{n}
  7. R = i = 1 n R i R=\prod_{i=1}^{n}R_{i}
  8. I = i = 1 n P i a i I=\prod_{i=1}^{n}P_{i}^{a_{i}}
  9. R / I i = 1 n R / P i a i R/I\cong\prod_{i=1}^{n}R/P_{i}^{a_{i}}
  10. R / P i a i R/P_{i}^{a_{i}}
  11. R / P i a i R/P_{i}^{a_{i}}
  12. R P i / P i a i R P i R_{P_{i}}/P_{i}^{a_{i}}R_{P_{i}}
  13. R P i R_{P_{i}}
  14. A = k [ x , y ] A=k[x,y]
  15. 𝔪 = x , y \mathfrak{m}=\langle x,y\rangle
  16. R = A / 𝔪 2 R=A/\mathfrak{m}^{2}
  17. ( 𝒫 ( X ) , Δ , ) (\mathcal{P}(X),\Delta,\cap)
  18. Δ \Delta
  19. 𝒫 ( X ) \mathcal{P}(X)
  20. I = ( I ) I=(\bigcup I)
  21. i = 1 n R i \prod_{i=1}^{n}R_{i}
  22. i = 1 n R i \prod_{i=1}^{n}R_{i}
  23. σ \sigma
  24. D [ x , σ ] D[x,\sigma]

Principle_of_covariance.html

  1. m d v d t = F , m\frac{d\vec{v}}{dt}=\vec{F},
  2. m m
  3. v \vec{v}
  4. F \vec{F}
  5. t t
  6. m d u a d s = q F a b u b , m\frac{du^{a}}{ds}=qF^{ab}u_{b},
  7. m m
  8. q q
  9. d s ds
  10. u a u^{a}
  11. F a b F^{ab}

Principle_of_maximum_work.html

  1. - d U -dU\,
  2. d S dS\,
  3. δ W \delta W\,
  4. d S w dS_{w}\,
  5. δ Q \delta Q\,
  6. d S h dS_{h}\,
  7. T T\,
  8. - d U = δ Q + δ W -dU=\delta Q+\delta W\,
  9. d S + d S h + d S w 0 dS+dS_{h}+dS_{w}\geq 0\,
  10. d S w = 0 dS_{w}=0\,
  11. δ Q = T d S h \delta Q=TdS_{h}\,
  12. d S w dS_{w}
  13. δ Q \delta Q
  14. d S h dS_{h}
  15. δ W - ( d U - T d S ) \delta W\leq-(dU-TdS)

Prior_knowledge_for_pattern_recognition.html

  1. ( s y m b o l x i , y i ) (symbol{x}_{i},y_{i})
  2. T θ : s y m b o l x T θ s y m b o l x T_{\theta}:symbol{x}\mapsto T_{\theta}symbol{x}
  3. θ \theta
  4. f ( s y m b o l x ) f(symbol{x})
  5. s y m b o l x symbol{x}
  6. f ( s y m b o l x ) = f ( T θ s y m b o l x ) , s y m b o l x , θ . f(symbol{x})=f(T_{\theta}symbol{x}),\quad\forall symbol{x},\theta.
  7. θ = 0 \theta=0
  8. T 0 s y m b o l x = s y m b o l x T_{0}symbol{x}=symbol{x}
  9. θ | θ = 0 f ( T θ s y m b o l x ) = 0. \left.\frac{\partial}{\partial\theta}\right|_{\theta=0}f(T_{\theta}symbol{x})=0.
  10. f f
  11. f f
  12. f ( s y m b o l x ) = y 𝒫 , s y m b o l x 𝒫 , f(symbol{x})=y_{\mathcal{P}},\ \forall symbol{x}\in\mathcal{P},
  13. y 𝒫 y_{\mathcal{P}}
  14. 𝒫 \mathcal{P}

Prism_compressor.html

  1. n ϕ 2 , m = H 2 , m + ( M - 1 ) ( H 1 , m ± n ϕ 2 , ( m - 1 ) ) \nabla_{n}\phi_{2,m}=H_{2,m}+(M^{-1})\bigg(H_{1,m}\pm\nabla_{n}\phi_{2,(m-1)}\bigg)
  2. n 2 ϕ 2 , m = n H 2 , m + ( n M - 1 ) ( H 1 , m ± n ϕ 2 , ( m - 1 ) ) + ( M - 1 ) ( n H 1 , m ± n 2 ϕ 2 , ( m - 1 ) ) \nabla_{n}^{2}\phi_{2,m}=\nabla_{n}H_{2,m}+(\nabla_{n}M^{-1})\bigg(H_{1,m}\pm% \nabla_{n}\phi_{2,(m-1)}\bigg)+(M^{-1})\bigg(\nabla_{n}H_{1,m}\pm\nabla_{n}^{2% }\phi_{2,(m-1)}\bigg)
  3. n 3 ϕ 2 , m = n 2 H 2 , m + ( n 2 M - 1 ) ( H 1 , m ± n ϕ 2 , ( m - 1 ) ) + 2 ( n M - 1 ) ( n H 1 , m ± n 2 ϕ 2 , ( m - 1 ) ) + ( M - 1 ) ( n 2 H 1 , m ± n 3 ϕ 2 , ( m - 1 ) ) \nabla_{n}^{3}\phi_{2,m}=\nabla_{n}^{2}H_{2,m}+(\nabla_{n}^{2}M^{-1})\bigg(H_{% 1,m}\pm\nabla_{n}\phi_{2,(m-1)}\bigg)+2(\nabla_{n}M^{-1})\bigg(\nabla_{n}H_{1,% m}\pm\nabla_{n}^{2}\phi_{2,(m-1)}\bigg)+(M^{-1})\bigg(\nabla_{n}^{2}H_{1,m}\pm% \nabla_{n}^{3}\phi_{2,(m-1)}\bigg)
  4. n = / n \nabla_{n}=\partial/\partial n
  5. M = k 1 , m k 2 , m \,M=k_{1,m}k_{2,m}
  6. k 1 , m = c o s ψ 1 , m / c o s ϕ 1 , m \,k_{1,m}=cos\psi_{1,m}/cos\phi_{1,m}
  7. k 2 , m = c o s ϕ 2 , m / c o s ψ 2 , m \,k_{2,m}=cos\phi_{2,m}/cos\psi_{2,m}
  8. H 1 , m = ( t a n ϕ 1 , m ) / n m \,H_{1,m}=(tan\phi_{1,m})/n_{m}
  9. H 2 , m = ( t a n ϕ 2 , m ) / n m \,H_{2,m}=(tan\phi_{2,m})/n_{m}

PRO_(category_theory).html

  1. \mathbb{N}
  2. Δ \Delta
  3. P P
  4. C C
  5. P P
  6. C C
  7. P P
  8. C C
  9. Alg P C \mathrm{Alg}_{P}^{C}
  10. P P
  11. C C
  12. \mathbb{N}
  13. C C
  14. C C
  15. Δ \Delta
  16. C C
  17. Δ \Delta
  18. C C
  19. C C
  20. P P
  21. C C
  22. C C

Probabilistic_CTL.html

  1. ϕ : := p | ¬ p | ϕ ϕ | ϕ ϕ | 𝒫 λ ( ϕ 𝒰 ϕ ) | 𝒫 λ ( ϕ ) \phi::=p|\neg p|\phi\lor\phi|\phi\land\phi|\mathcal{P}_{\sim\lambda}(\phi% \mathcal{U}\phi)|\mathcal{P}_{\sim\lambda}(\square\phi)
  2. { < , , , > } \sim\in\{<,\leq,\geq,>\}
  3. λ \lambda
  4. K = S , s i , 𝒯 , L K=\langle S,s^{i},\mathcal{T},L\rangle
  5. S S
  6. s i S s^{i}\in S
  7. 𝒯 \mathcal{T}
  8. 𝒯 : S × S [ 0 , 1 ] \mathcal{T}:S\times S\to[0,1]
  9. s S s\in S
  10. s S 𝒯 ( s , s ) = 1 \sum_{s^{\prime}\in S}\mathcal{T}(s,s^{\prime})=1
  11. L L
  12. L : S 2 A L:S\to 2^{A}
  13. σ \sigma
  14. s 0 s_{0}
  15. s 0 s 1 s n s_{0}\to s_{1}\to\dots\to s_{n}\to\dots
  16. σ [ n ] \sigma[n]
  17. σ \sigma
  18. n n
  19. σ n \sigma\uparrow n
  20. μ m \mu_{m}
  21. n n
  22. μ m ( { σ X : σ n = s 0 s n } ) = 𝒯 ( s 0 , s 1 ) × × 𝒯 ( s n - 1 , s n ) \mu_{m}(\{\sigma\in X:\sigma\uparrow n=s_{0}\to\dots\to s_{n}\})=\mathcal{T}(s% _{0},s_{1})\times\dots\times\mathcal{T}(s_{n-1},s_{n})
  23. n = 0 n=0
  24. μ m ( { σ X : σ 0 = s 0 } ) = 1 \mu_{m}(\{\sigma\in X:\sigma\uparrow 0=s_{0}\})=1
  25. s K f s\models_{K}f
  26. σ K f \sigma\models_{K}f
  27. s K a s\models_{K}a
  28. a L ( s ) a\in L(s)
  29. s K ¬ f s\models_{K}\neg f
  30. s K f s\models_{K}f
  31. s K f 1 f 2 s\models_{K}f_{1}\lor f_{2}
  32. s K f 1 s\models_{K}f_{1}
  33. s K f 2 s\models_{K}f_{2}
  34. s K f 1 f 2 s\models_{K}f_{1}\land f_{2}
  35. s K f 1 s\models_{K}f_{1}
  36. s K f 2 s\models_{K}f_{2}
  37. s K 𝒫 λ ( f 1 𝒰 f 2 ) s\models_{K}\mathcal{P}_{\sim\lambda}(f_{1}\mathcal{U}f_{2})
  38. μ m ( { σ : σ [ 0 ] = s ( i ) σ [ i ] K f 2 ( 0 j < i ) σ [ j ] K f 1 } ) λ \mu_{m}(\{\sigma:\sigma[0]=s\land(\exists i)\sigma[i]\models_{K}f_{2}\land(% \forall 0\leq j<i)\sigma[j]\models_{K}f_{1}\})\sim\lambda
  39. s K 𝒫 λ ( f ) s\models_{K}\mathcal{P}_{\sim\lambda}(\square f)
  40. μ m ( { σ : σ [ 0 ] = s ( i 0 ) σ [ i ] K f } ) λ \mu_{m}(\{\sigma:\sigma[0]=s\land(\forall i\geq 0)\sigma[i]\models_{K}f\})\sim\lambda

ProbCons.html

  1. x i x_{i}
  2. y i y_{i}
  3. a * a^{*}
  4. P ( x i y i | x , y ) = d e f P r [ x i y i in some a | x , y ] = alignment a with x i - y i P r [ a | x , y ] = alignment a 𝟏 { x i - y i a } P r [ a | x , y ] \begin{aligned}\displaystyle P(x_{i}\sim y_{i}|x,y)&\displaystyle\stackrel{def% }{=}Pr[x_{i}\sim y_{i}\,\text{ in some a }|x,y]\\ &\displaystyle=\sum_{\,\text{alignment a with }x_{i}-y_{i}}Pr[a|x,y]\\ &\displaystyle=\sum_{\,\text{alignment a}}\mathbf{1}\{x_{i}-y_{i}\in a\}Pr[a|x% ,y]\end{aligned}
  5. 𝟏 { x i y i a } \mathbf{1}\{x_{i}\sim y_{i}\in a\}
  6. x i x_{i}
  7. y i y_{i}
  8. a * a^{*}
  9. a a
  10. E P r [ a | x , y ] ( a c c ( a * , a ) ) \displaystyle E_{Pr[a|x,y]}(acc(a^{*},a))
  11. E ( x , y ) = arg max a * E P r [ a | x , y ] ( a c c ( a * , a ) ) E(x,y)=\arg\max_{a^{*}}\;E_{Pr[a|x,y]}(acc(a^{*},a))
  12. 𝒮 \mathcal{S}
  13. P ( x i - y i | x , y ) = 1 | 𝒮 | z 1 k | z | P ( x i z i | x , z ) P ( z i y i | z , y ) P^{\prime}(x_{i}-y_{i}|x,y)=\frac{1}{|\mathcal{S}|}\sum_{z}\sum_{1\leq k\leq|z% |}P(x_{i}\sim z_{i}|x,z)\cdot P(z_{i}\sim y_{i}|z,y)

Prod.html

  1. \prod

Product_forecasting.html

  1. f ( t ) 1 - F ( t ) = p + q m N ( t ) \frac{f(t)}{1-F(t)}=p+\frac{q}{m}N(t)
  2. V = ( H H T R T U ) + ( H H T R M R R R R U ) V=(HH\cdot TR\cdot TU)+(HH\cdot TR\cdot MR\cdot RR\cdot RU)

Product_metric.html

  1. d p ( 𝐱 1 , , 𝐱 n ) = ( d 1 ( 𝐱 1 ) , , d n ( 𝐱 n ) ) p d_{p}(\mathbf{x}_{1},\dots,\mathbf{x}_{n})=\|(d_{1}(\mathbf{x}_{1}),\dots,d_{n% }(\mathbf{x}_{n}))\|_{p}
  2. ( X , d X ) (X,d_{X})
  3. ( Y , d Y ) (Y,d_{Y})
  4. 1 p + 1\leq p\leq+\infty
  5. p p
  6. d p d_{p}
  7. X × Y X\times Y
  8. d p ( ( x 1 , y 1 ) , ( x 2 , y 2 ) ) := ( d X ( x 1 , x 2 ) p + d Y ( y 1 , y 2 ) p ) 1 / p d_{p}\left((x_{1},y_{1}),(x_{2},y_{2})\right):=\left(d_{X}(x_{1},x_{2})^{p}+d_% {Y}(y_{1},y_{2})^{p}\right)^{1/p}
  9. 1 p < ; 1\leq p<\infty;
  10. d ( ( x 1 , y 1 ) , ( x 2 , y 2 ) ) := max { d X ( x 1 , x 2 ) , d Y ( y 1 , y 2 ) } . d_{\infty}\left((x_{1},y_{1}),(x_{2},y_{2})\right):=\max\left\{d_{X}(x_{1},x_{% 2}),d_{Y}(y_{1},y_{2})\right\}.
  11. x 1 , x 2 X x_{1},x_{2}\in X
  12. y 1 , y 2 Y y_{1},y_{2}\in Y

Progressively_measurable_process.html

  1. ( Ω , , ) (\Omega,\mathcal{F},\mathbb{P})
  2. ( 𝕏 , 𝒜 ) (\mathbb{X},\mathcal{A})
  3. { t t 0 } \{\mathcal{F}_{t}\mid t\geq 0\}
  4. \mathcal{F}
  5. X : [ 0 , ) × Ω 𝕏 X:[0,\infty)\times\Omega\to\mathbb{X}
  6. [ 0 , T ] [0,T]
  7. 0 \mathbb{N}_{0}
  8. [ 0 , ) [0,\infty)
  9. X X
  10. t t
  11. [ 0 , t ] × Ω 𝕏 [0,t]\times\Omega\to\mathbb{X}
  12. ( s , ω ) X s ( ω ) (s,\omega)\mapsto X_{s}(\omega)
  13. Borel ( [ 0 , t ] ) t \mathrm{Borel}([0,t])\otimes\mathcal{F}_{t}
  14. X X
  15. t \mathcal{F}_{t}
  16. P [ 0 , ) × Ω P\subseteq[0,\infty)\times\Omega
  17. X s ( ω ) := χ P ( s , ω ) X_{s}(\omega):=\chi_{P}(s,\omega)
  18. χ P \chi_{P}
  19. P P
  20. P P
  21. [ 0 , ) × Ω [0,\infty)\times\Omega
  22. Prog \mathrm{Prog}
  23. X X
  24. Prog \mathrm{Prog}
  25. L 2 ( B ) L^{2}(B)
  26. X : [ 0 , T ] × Ω n X:[0,T]\times\Omega\to\mathbb{R}^{n}
  27. 0 T X t d B t \int_{0}^{T}X_{t}\,\mathrm{d}B_{t}
  28. B B
  29. Prog \mathrm{Prog}
  30. L 2 ( [ 0 , T ] × Ω ; n ) L^{2}([0,T]\times\Omega;\mathbb{R}^{n})\,

Projected_dynamical_system.html

  1. d x ( t ) d t = Π K ( x ( t ) , - F ( x ( t ) ) ) \frac{dx(t)}{dt}=\Pi_{K}(x(t),-F(x(t)))
  2. N K ( x ) = { p V | p , x - x * 0 , x * K } . N_{K}(x)=\{p\in V|\langle p,x-x^{*}\rangle\geq 0,\forall x^{*}\in K\}.
  3. T K ( x ) = h > 0 1 h ( K - x ) ¯ . T_{K}(x)=\overline{\bigcup_{h>0}\frac{1}{h}(K-x)}.
  4. P K ( x ) P_{K}(x)
  5. x - P K ( x ) x - y \|x-P_{K}(x)\|\leq\|x-y\|
  6. Π K ( x , v ) = lim δ 0 + P K ( x + δ v ) - x δ . \Pi_{K}(x,v)=\lim_{\delta\to 0^{+}}\frac{P_{K}(x+\delta v)-x}{\delta}.
  7. d x ( t ) d t = Π K ( x ( t ) , - F ( x ( t ) ) ) . \frac{dx(t)}{dt}=\Pi_{K}(x(t),-F(x(t))).
  8. [ 0 , ) [0,\infty)
  9. d x ( t ) d t = P T K ( x ( t ) ) ( - F ( x ( t ) ) ) \frac{dx(t)}{dt}=P_{T_{K}(x(t))}(-F(x(t)))
  10. d x ( t ) d t = - F ( x ( t ) ) - P N K ( x ( t ) ) ( - F ( x ( t ) ) ) . \frac{dx(t)}{dt}=-F(x(t))-P_{N_{K}(x(t))}(-F(x(t))).

Projective_connection.html

  1. 𝔤 {\mathfrak{g}}
  2. 𝔥 {\mathfrak{h}}
  3. 𝔤 = 𝔰 𝔩 ( n + 1 , ) {\mathfrak{g}}={\mathfrak{s}}{\mathfrak{l}}(n+1,{\mathbb{R}})
  4. 𝔤 {\mathfrak{g}}
  5. ( λ v i w j a j i ) , ( v i ) 1 × n , ( w j ) n × 1 , ( a j i ) n × n , λ = - i a i i \left(\begin{matrix}\lambda&v^{i}\\ w_{j}&a_{j}^{i}\end{matrix}\right),\quad(v^{i})\in{\mathbb{R}}^{1\times n},(w_% {j})\in{\mathbb{R}}^{n\times 1},(a_{j}^{i})\in{\mathbb{R}}^{n\times n},\lambda% =-\sum_{i}a_{i}^{i}
  6. 𝔥 {\mathfrak{h}}

Prokhorov's_theorem.html

  1. ( S , ρ ) (S,\rho)
  2. 𝒫 ( S ) \mathcal{P}(S)
  3. S S
  4. K 𝒫 ( S ) K\subset\mathcal{P}(S)
  5. K K
  6. 𝒫 ( S ) \mathcal{P}(S)
  7. 𝒫 ( S ) \mathcal{P}(S)
  8. ( S , ρ ) (S,\rho)
  9. ( S , ρ ) (S,\rho)
  10. d 0 d_{0}
  11. 𝒫 ( S ) \mathcal{P}(S)
  12. K 𝒫 ( S ) K\subset\mathcal{P}(S)
  13. K K
  14. ( 𝒫 ( S ) , d 0 ) (\mathcal{P}(S),d_{0})
  15. ( μ n ) (\mu_{n})
  16. 𝒫 ( k ) \mathcal{P}(\mathbb{R}^{k})
  17. k k
  18. ( μ n k ) (\mu_{n_{k}})
  19. μ 𝒫 ( k ) \mu\in\mathcal{P}(\mathbb{R}^{k})
  20. μ n k \mu_{n_{k}}
  21. μ \mu
  22. ( μ n ) (\mu_{n})
  23. 𝒫 ( k ) \mathcal{P}(\mathbb{R}^{k})
  24. ( μ n k ) (\mu_{n_{k}})
  25. μ 𝒫 ( k ) \mu\in\mathcal{P}(\mathbb{R}^{k})
  26. ( μ n ) (\mu_{n})
  27. μ \mu
  28. ( S , ρ ) (S,\rho)
  29. Π \Pi
  30. S S
  31. Π \Pi
  32. { μ n } Π \{\mu_{n}\}\subset\Pi
  33. Π \Pi

Prony's_method.html

  1. f ( t ) f(t)
  2. N N
  3. f ^ ( t ) = i = 1 N A i e σ i t cos ( 2 π f i t + ϕ i ) \hat{f}(t)=\sum_{i=1}^{N}A_{i}e^{\sigma_{i}t}\cos(2\pi f_{i}t+\phi_{i})
  4. f ( t ) f(t)
  5. f ^ ( t ) = i = 1 N A i e σ i t cos ( 2 π f i t + ϕ i ) = i = 1 N 1 2 A i e ± j ϕ i e λ i t \begin{aligned}\displaystyle\hat{f}(t)&\displaystyle=\sum_{i=1}^{N}A_{i}e^{% \sigma_{i}t}\cos(2\pi f_{i}t+\phi_{i})\\ &\displaystyle=\sum_{i=1}^{N}\frac{1}{2}A_{i}e^{\pm j\phi_{i}}e^{\lambda_{i}t}% \end{aligned}
  6. λ i = σ i ± j ω i \lambda_{i}=\sigma_{i}\pm j\omega_{i}
  7. σ i \sigma_{i}
  8. ϕ i \phi_{i}
  9. f i f_{i}
  10. A i A_{i}
  11. j j
  12. j 2 = - 1 j^{2}=-1
  13. M M
  14. f ^ ( t ) \hat{f}(t)
  15. n n
  16. N N
  17. F n = f ^ ( Δ t n ) = m = 1 M B m e λ m Δ t n . F_{n}=\hat{f}(\Delta_{t}n)=\sum_{m=1}^{M}B_{m}e^{\lambda_{m}\Delta_{t}n}.
  18. f ^ ( t ) \hat{f}(t)
  19. B a = 1 2 A i e ϕ i j , B b = 1 2 A i e - ϕ i j , λ a = σ i + j ω i , λ b = σ i - j ω i , \begin{aligned}\displaystyle B_{a}&\displaystyle=\frac{1}{2}A_{i}e^{\phi_{i}j}% ,\\ \displaystyle B_{b}&\displaystyle=\frac{1}{2}A_{i}e^{-\phi_{i}j},\\ \displaystyle\lambda_{a}&\displaystyle=\sigma_{i}+j\omega_{i},\\ \displaystyle\lambda_{b}&\displaystyle=\sigma_{i}-j\omega_{i},\end{aligned}
  20. B a e λ a t + B b e λ b t = 1 2 A i e ϕ i j e ( σ i + j ω i ) t + 1 2 A i e - ϕ i j e ( σ i - j ω i ) t = A i e σ i t cos ( ω i t + ϕ i ) . \begin{aligned}\displaystyle B_{a}e^{\lambda_{a}t}+B_{b}e^{\lambda_{b}t}&% \displaystyle=\frac{1}{2}A_{i}e^{\phi_{i}j}e^{(\sigma_{i}+j\omega_{i})t}+\frac% {1}{2}A_{i}e^{-\phi_{i}j}e^{(\sigma_{i}-j\omega_{i})t}\\ &\displaystyle=A_{i}e^{\sigma_{i}t}\cos(\omega_{i}t+\phi_{i}).\end{aligned}
  21. f ^ ( Δ t n ) = - m = 1 M f ^ [ Δ t ( n - m ) ] P m . \hat{f}(\Delta_{t}n)=-\sum_{m=1}^{M}\hat{f}[\Delta_{t}(n-m)]P_{m}.
  22. m = 1 M + 1 P m x m - 1 = m = 1 M ( x - e λ m ) . \sum_{m=1}^{M+1}P_{m}x^{m-1}=\prod_{m=1}^{M}\left(x-e^{\lambda_{m}}\right).
  23. P m P_{m}
  24. [ F M F N - 1 ] = - [ F M - 1 F 0 F N - 2 F N - M - 1 ] [ P 1 P M ] . \begin{bmatrix}F_{M}\\ \vdots\\ F_{N-1}\end{bmatrix}=-\begin{bmatrix}F_{M-1}&\dots&F_{0}\\ \vdots&\ddots&\vdots\\ F_{N-2}&\dots&F_{N-M-1}\end{bmatrix}\begin{bmatrix}P_{1}\\ \vdots\\ P_{M}\end{bmatrix}.
  25. N 2 M N\neq 2M
  26. P m P_{m}
  27. P m P_{m}
  28. x M + m = 1 M P m x m - 1 . x^{M}+\sum_{m=1}^{M}P_{m}x^{m-1}.
  29. m m
  30. e λ m e^{\lambda_{m}}
  31. e λ m e^{\lambda_{m}}
  32. F n F_{n}
  33. B m B_{m}
  34. [ F k 1 F k M ] = [ ( e λ 1 ) k 1 ( e λ M ) k 1 ( e λ 1 ) k M ( e λ M ) k M ] [ B 1 B M ] , \begin{bmatrix}F_{k_{1}}\\ \vdots\\ F_{k_{M}}\end{bmatrix}=\begin{bmatrix}(e^{\lambda_{1}})^{k_{1}}&\dots&(e^{% \lambda_{M}})^{k_{1}}\\ \vdots&\ddots&\vdots\\ (e^{\lambda_{1}})^{k_{M}}&\dots&(e^{\lambda_{M}})^{k_{M}}\end{bmatrix}\begin{% bmatrix}B_{1}\\ \vdots\\ B_{M}\end{bmatrix},
  35. M M
  36. k i k_{i}
  37. M M
  38. λ m \lambda_{m}
  39. e λ m e^{\lambda_{m}}
  40. e λ m = e λ m + q 2 π j e^{\lambda_{m}}=e^{\lambda_{m}\,+\,q2\pi j}
  41. q q
  42. | I m ( λ m ) | = | ω m | < 1 2 Δ t . \left|Im(\lambda_{m})\right|=\left|\omega_{m}\right|<\frac{1}{2\Delta_{t}}.

Proof_by_example.html

  1. φ ( β / α ) ¯ \underline{\varphi(\beta/\alpha)}\,\!
  2. α φ \exists\alpha\,\varphi\,\!

Proof_of_knowledge.html

  1. x x
  2. L L
  3. W ( x ) W(x)
  4. R = { ( x , w ) : x L , w W ( x ) } R=\{(x,w):x\in L,w\in W(x)\}
  5. R R
  6. κ \kappa
  7. P P
  8. V V
  9. ( x , w ) R (x,w)\in R
  10. w w
  11. x x
  12. V V
  13. P r ( P ( x , w ) V ( x ) 1 ) = 1 Pr(P(x,w)\leftrightarrow V(x)\rightarrow 1)=1
  14. E E
  15. P ~ \tilde{P}
  16. P ~ \tilde{P}
  17. R R
  18. W ( x ) W(x)
  19. x x
  20. κ \kappa
  21. κ \kappa
  22. E E
  23. P ~ \tilde{P}
  24. P ~ \tilde{P}
  25. E P ~ ( x ) ( x ) W ( x ) { } E^{\tilde{P}(x)}(x)\in W(x)\cup\{\bot\}
  26. Pr ( E P ~ ( x ) ( x ) W ( x ) ) Pr ( P ~ ( x ) V ( x ) 1 ) - κ ( x ) . \Pr(E^{\tilde{P}(x)}(x)\in W(x))\geq\Pr(\tilde{P}(x)\leftrightarrow V(x)% \rightarrow 1)-\kappa(x).
  27. \bot
  28. E E
  29. κ ( x ) \kappa(x)
  30. V V
  31. x x
  32. w w
  33. E E
  34. E E
  35. w w
  36. P ~ \tilde{P}
  37. P ~ \tilde{P}
  38. w w
  39. κ ( x ) \kappa(x)
  40. 2 - 80 2^{-80}
  41. 1 / poly ( | x | ) 1/\mathrm{poly}(|x|)
  42. g \langle g\rangle
  43. g g
  44. L = { x | g w = x } L=\{x|g^{w}=x\}
  45. x x
  46. g \langle g\rangle
  47. w w
  48. g w = x g^{w}=x
  49. G q G_{q}
  50. q q
  51. g g
  52. x = log g y x=\log_{g}y
  53. r r
  54. t = g r t=g^{r}
  55. c c
  56. c c
  57. s = r + c x s=r+cx
  58. g s = t y c g^{s}=ty^{c}
  59. Σ \Sigma
  60. y 1 y_{1}
  61. y 2 y_{2}
  62. g 1 g_{1}
  63. g 2 g_{2}
  64. Z q Z_{q}
  65. x 1 x_{1}
  66. x 2 x_{2}
  67. y 1 = g 1 x 1 y 2 = g 2 x 2 y_{1}=g_{1}^{x_{1}}\land y_{2}=g_{2}^{x_{2}}
  68. x 2 = a x 1 + b x_{2}=ax_{1}+b
  69. x 2 x_{2}
  70. x 1 x_{1}
  71. y 1 = g 1 x y 2 = ( g 2 a ) x g 2 b y_{1}=g_{1}^{x}\land y_{2}={(g_{2}^{a})}^{x}g_{2}^{b}
  72. P K { ( x ) : y 1 = g 1 x y 2 = ( g 2 a ) x g 2 b } , PK\{(x):y_{1}=g_{1}^{x}\land y_{2}={(g_{2}^{a})}^{x}g_{2}^{b}\},

Proof_of_Stein's_example.html

  1. d ( 𝐱 ) = 𝐱 d(\mathbf{x})=\mathbf{x}
  2. R ( θ , d ) = 𝔼 θ [ | θ - 𝐗 | 2 ] R(\theta,d)=\mathbb{E}_{\theta}[|\mathbf{\theta-X}|^{2}]
  3. = ( θ - 𝐱 ) T ( θ - 𝐱 ) ( 1 2 π ) n / 2 e ( - 1 / 2 ) ( θ - 𝐱 ) T ( θ - 𝐱 ) m ( d x ) =\int(\mathbf{\theta-x})^{T}(\mathbf{\theta-x})\left(\frac{1}{2\pi}\right)^{n/% 2}e^{(-1/2)(\mathbf{\theta-x})^{T}(\mathbf{\theta-x})}m(dx)
  4. = n . =n.\,
  5. d ( 𝐱 ) = 𝐱 - α | 𝐱 | 2 𝐱 d^{\prime}(\mathbf{x})=\mathbf{x}-\frac{\alpha}{|\mathbf{x}|^{2}}\mathbf{x}
  6. α = n - 2 \alpha=n-2
  7. d d^{\prime}
  8. d d
  9. R ( θ , d ) = 𝔼 θ [ | θ - 𝐗 + α | 𝐗 | 2 𝐗 | 2 ] R(\theta,d^{\prime})=\mathbb{E}_{\theta}\left[\left|\mathbf{\theta-X}+\frac{% \alpha}{|\mathbf{X}|^{2}}\mathbf{X}\right|^{2}\right]
  10. = 𝔼 θ [ | θ - 𝐗 | 2 + 2 ( θ - 𝐗 ) T α | 𝐗 | 2 𝐗 + α 2 | 𝐗 | 4 | 𝐗 | 2 ] =\mathbb{E}_{\theta}\left[|\mathbf{\theta-X}|^{2}+2(\mathbf{\theta-X})^{T}% \frac{\alpha}{|\mathbf{X}|^{2}}\mathbf{X}+\frac{\alpha^{2}}{|\mathbf{X}|^{4}}|% \mathbf{X}|^{2}\right]
  11. = 𝔼 θ [ | θ - 𝐗 | 2 ] + 2 α 𝔼 θ [ ( θ - 𝐗 ) 𝐓 𝐗 | 𝐗 | 2 ] + α 2 𝔼 θ [ 1 | 𝐗 | 2 ] =\mathbb{E}_{\theta}\left[|\mathbf{\theta-X}|^{2}\right]+2\alpha\mathbb{E}_{% \theta}\left[\frac{\mathbf{(\theta-X)^{T}X}}{|\mathbf{X}|^{2}}\right]+\alpha^{% 2}\mathbb{E}_{\theta}\left[\frac{1}{|\mathbf{X}|^{2}}\right]
  12. α \alpha
  13. h : 𝐱 h ( 𝐱 ) h:\mathbf{x}\mapsto h(\mathbf{x})\in\mathbb{R}
  14. 1 i n 1\leq i\leq n
  15. h h
  16. x i x_{i}
  17. 𝔼 θ [ ( θ i - X i ) h ( 𝐗 ) | X j = x j ( j i ) ] = ( θ i - x i ) h ( 𝐱 ) ( 1 2 π ) n / 2 e - ( 1 / 2 ) ( 𝐱 - θ ) T ( 𝐱 - θ ) m ( d x i ) \mathbb{E}_{\theta}[(\theta_{i}-X_{i})h(\mathbf{X})|X_{j}=x_{j}(j\neq i)]=\int% (\theta_{i}-x_{i})h(\mathbf{x})\left(\frac{1}{2\pi}\right)^{n/2}e^{-(1/2)% \mathbf{(x-\theta)}^{T}\mathbf{(x-\theta)}}m(dx_{i})
  18. = [ h ( 𝐱 ) ( 1 2 π ) n / 2 e - ( 1 / 2 ) ( 𝐱 - θ ) T ( 𝐱 - θ ) ] x i = - - h x i ( 𝐱 ) ( 1 2 π ) n / 2 e - ( 1 / 2 ) ( 𝐱 - θ ) T ( 𝐱 - θ ) m ( d x i ) =\left[h(\mathbf{x})\left(\frac{1}{2\pi}\right)^{n/2}e^{-(1/2)\mathbf{(x-% \theta)}^{T}\mathbf{(x-\theta)}}\right]^{\infty}_{x_{i}=-\infty}-\int\frac{% \partial h}{\partial x_{i}}(\mathbf{x})\left(\frac{1}{2\pi}\right)^{n/2}e^{-(1% /2)\mathbf{(x-\theta)}^{T}\mathbf{(x-\theta)}}m(dx_{i})
  19. = - 𝔼 θ [ h x i ( 𝐗 ) | X j = x j ( j i ) ] . =-\mathbb{E}_{\theta}\left[\frac{\partial h}{\partial x_{i}}(\mathbf{X})|X_{j}% =x_{j}(j\neq i)\right].
  20. 𝔼 θ [ ( θ i - X i ) h ( 𝐗 ) ] = - 𝔼 θ [ h x i ( 𝐗 ) ] . \mathbb{E}_{\theta}[(\theta_{i}-X_{i})h(\mathbf{X})]=-\mathbb{E}_{\theta}\left% [\frac{\partial h}{\partial x_{i}}(\mathbf{X})\right].
  21. h ( 𝐱 ) = x i | 𝐱 | 2 . h(\mathbf{x})=\frac{x_{i}}{|\mathbf{x}|^{2}}.
  22. h h
  23. h x i = 1 | 𝐱 | 2 - 2 x i 2 | 𝐱 | 4 \frac{\partial h}{\partial x_{i}}=\frac{1}{|\mathbf{x}|^{2}}-\frac{2x_{i}^{2}}% {|\mathbf{x}|^{4}}
  24. 𝔼 θ [ ( θ - 𝐗 ) 𝐓 𝐗 | 𝐗 | 2 ] = i = 1 n 𝔼 θ [ ( θ i - X i ) X i | 𝐗 | 2 ] \mathbb{E}_{\theta}\left[\frac{\mathbf{(\theta-X)^{T}X}}{|\mathbf{X}|^{2}}% \right]=\sum_{i=1}^{n}\mathbb{E}_{\theta}\left[(\theta_{i}-X_{i})\frac{X_{i}}{% |\mathbf{X}|^{2}}\right]
  25. = - i = 1 n 𝔼 θ [ 1 | 𝐗 | 2 - 2 X i 2 | 𝐗 | 4 ] =-\sum_{i=1}^{n}\mathbb{E}_{\theta}\left[\frac{1}{|\mathbf{X}|^{2}}-\frac{2X_{% i}^{2}}{|\mathbf{X}|^{4}}\right]
  26. = - ( n - 2 ) 𝔼 θ [ 1 | 𝐗 | 2 ] . =-(n-2)\mathbb{E}_{\theta}\left[\frac{1}{|\mathbf{X}|^{2}}\right].
  27. d d^{\prime}
  28. R ( θ , d ) = n - 2 α ( n - 2 ) 𝔼 θ [ 1 | 𝐗 | 2 ] + α 2 𝔼 θ [ 1 | 𝐗 | 2 ] . R(\theta,d^{\prime})=n-2\alpha(n-2)\mathbb{E}_{\theta}\left[\frac{1}{|\mathbf{% X}|^{2}}\right]+\alpha^{2}\mathbb{E}_{\theta}\left[\frac{1}{|\mathbf{X}|^{2}}% \right].
  29. α \alpha
  30. α = n - 2 , \alpha=n-2,\,
  31. R ( θ , d ) = R ( θ , d ) - ( n - 2 ) 2 𝔼 θ [ 1 | 𝐗 | 2 ] R(\theta,d^{\prime})=R(\theta,d)-(n-2)^{2}\mathbb{E}_{\theta}\left[\frac{1}{|% \mathbf{X}|^{2}}\right]
  32. R ( θ , d ) < R ( θ , d ) . R(\theta,d^{\prime})<R(\theta,d).
  33. d d
  34. h ( 𝐗 ) = 𝐗 | 𝐗 | 2 . h(\mathbf{X})=\frac{\mathbf{X}}{|\mathbf{X}|^{2}}.
  35. 𝐱 = 0 \mathbf{x}=0
  36. h ( 𝐗 ) = 𝐗 ϵ + | 𝐗 | 2 h(\mathbf{X})=\frac{\mathbf{X}}{\epsilon+|\mathbf{X}|^{2}}
  37. ϵ 0 \epsilon\to 0

Proof_of_the_Euler_product_formula_for_the_Riemann_zeta_function.html

  1. n = 1 1 n s = p prime 1 1 - p - s \sum_{n=1}^{\infty}\frac{1}{n^{s}}=\prod_{p\,\text{ prime}}\frac{1}{1-p^{-s}}
  2. ζ ( s ) = n = 1 1 n s = 1 + 1 2 s + 1 3 s + 1 4 s + 1 5 s + \zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}=1+\frac{1}{2^{s}}+\frac{1}{3^{s}}+% \frac{1}{4^{s}}+\frac{1}{5^{s}}+\ldots
  3. p prime 1 1 - p - s = 1 1 - 2 - s 1 1 - 3 - s 1 1 - 5 - s 1 1 - 7 - s 1 1 - p - s \prod_{p\,\text{ prime}}\frac{1}{1-p^{-s}}=\frac{1}{1-2^{-s}}\cdot\frac{1}{1-3% ^{-s}}\cdot\frac{1}{1-5^{-s}}\cdot\frac{1}{1-7^{-s}}\cdots\frac{1}{1-p^{-s}}\cdots
  4. ζ ( s ) = 1 + 1 2 s + 1 3 s + 1 4 s + 1 5 s + \zeta(s)=1+\frac{1}{2^{s}}+\frac{1}{3^{s}}+\frac{1}{4^{s}}+\frac{1}{5^{s}}+\ldots
  5. 1 2 s ζ ( s ) = 1 2 s + 1 4 s + 1 6 s + 1 8 s + 1 10 s + \frac{1}{2^{s}}\zeta(s)=\frac{1}{2^{s}}+\frac{1}{4^{s}}+\frac{1}{6^{s}}+\frac{% 1}{8^{s}}+\frac{1}{10^{s}}+\ldots
  6. ( 1 - 1 2 s ) ζ ( s ) = 1 + 1 3 s + 1 5 s + 1 7 s + 1 9 s + 1 11 s + 1 13 s + \left(1-\frac{1}{2^{s}}\right)\zeta(s)=1+\frac{1}{3^{s}}+\frac{1}{5^{s}}+\frac% {1}{7^{s}}+\frac{1}{9^{s}}+\frac{1}{11^{s}}+\frac{1}{13^{s}}+\ldots
  7. 1 3 s ( 1 - 1 2 s ) ζ ( s ) = 1 3 s + 1 9 s + 1 15 s + 1 21 s + 1 27 s + 1 33 s + \frac{1}{3^{s}}\left(1-\frac{1}{2^{s}}\right)\zeta(s)=\frac{1}{3^{s}}+\frac{1}% {9^{s}}+\frac{1}{15^{s}}+\frac{1}{21^{s}}+\frac{1}{27^{s}}+\frac{1}{33^{s}}+\ldots
  8. ( 1 - 1 3 s ) ( 1 - 1 2 s ) ζ ( s ) = 1 + 1 5 s + 1 7 s + 1 11 s + 1 13 s + 1 17 s + \left(1-\frac{1}{3^{s}}\right)\left(1-\frac{1}{2^{s}}\right)\zeta(s)=1+\frac{1% }{5^{s}}+\frac{1}{7^{s}}+\frac{1}{11^{s}}+\frac{1}{13^{s}}+\frac{1}{17^{s}}+\ldots
  9. ( 1 - 1 11 s ) ( 1 - 1 7 s ) ( 1 - 1 5 s ) ( 1 - 1 3 s ) ( 1 - 1 2 s ) ζ ( s ) = 1 \ldots\left(1-\frac{1}{11^{s}}\right)\left(1-\frac{1}{7^{s}}\right)\left(1-% \frac{1}{5^{s}}\right)\left(1-\frac{1}{3^{s}}\right)\left(1-\frac{1}{2^{s}}% \right)\zeta(s)=1
  10. ζ ( s ) = 1 ( 1 - 1 2 s ) ( 1 - 1 3 s ) ( 1 - 1 5 s ) ( 1 - 1 7 s ) ( 1 - 1 11 s ) \zeta(s)=\frac{1}{\left(1-\frac{1}{2^{s}}\right)\left(1-\frac{1}{3^{s}}\right)% \left(1-\frac{1}{5^{s}}\right)\left(1-\frac{1}{7^{s}}\right)\left(1-\frac{1}{1% 1^{s}}\right)\ldots}
  11. ζ ( s ) = p prime 1 1 - p - s \zeta(s)=\prod_{p\,\text{ prime}}\frac{1}{1-p^{-s}}
  12. ( s ) > 1 \Re(s)>1
  13. s = 1 s=1
  14. ( 1 - 1 11 ) ( 1 - 1 7 ) ( 1 - 1 5 ) ( 1 - 1 3 ) ( 1 - 1 2 ) ζ ( 1 ) = 1 \ldots\left(1-\frac{1}{11}\right)\left(1-\frac{1}{7}\right)\left(1-\frac{1}{5}% \right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{2}\right)\zeta(1)=1
  15. ( 10 11 ) ( 6 7 ) ( 4 5 ) ( 2 3 ) ( 1 2 ) ζ ( 1 ) = 1 \ldots\left(\frac{10}{11}\right)\left(\frac{6}{7}\right)\left(\frac{4}{5}% \right)\left(\frac{2}{3}\right)\left(\frac{1}{2}\right)\zeta(1)=1
  16. ( 10 6 4 2 1 11 7 5 3 2 ) ζ ( 1 ) = 1 \left(\frac{\ldots\cdot 10\cdot 6\cdot 4\cdot 2\cdot 1}{\ldots\cdot 11\cdot 7% \cdot 5\cdot 3\cdot 2}\right)\zeta(1)=1
  17. ζ ( 1 ) = 1 + 1 2 + 1 3 + 1 4 + 1 5 + \zeta(1)=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\ldots
  18. 1 + 1 2 + 1 3 + 1 4 + 1 5 + = 2 3 5 7 11 1 2 4 6 10 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\ldots=\frac{2\cdot 3\cdot 5% \cdot 7\cdot 11\cdot\ldots}{1\cdot 2\cdot 4\cdot 6\cdot 10\cdot\ldots}
  19. 1 1 - p - s = 1 + 1 p s + 1 p 2 s + 1 p 3 s + + 1 p k s + \frac{1}{1-p^{-s}}=1+\frac{1}{p^{s}}+\frac{1}{p^{2s}}+\frac{1}{p^{3s}}+\ldots+% \frac{1}{p^{ks}}+\ldots
  20. ( s ) > 1 \Re(s)>1
  21. | ζ ( s ) - p q ( 1 1 - p - s ) | < n = q + 1 1 n σ \left|\zeta(s)-\prod_{p\leq q}\left(\frac{1}{1-p^{-s}}\right)\right|<\sum_{n=q% +1}^{\infty}\frac{1}{n^{\sigma}}

Proofs_of_trigonometric_identities.html

  1. sin θ = opposite hypotenuse = a h \sin\theta=\frac{\mathrm{opposite}}{\mathrm{hypotenuse}}=\frac{a}{h}
  2. cos θ = adjacent hypotenuse = b h \cos\theta=\frac{\mathrm{adjacent}}{\mathrm{hypotenuse}}=\frac{b}{h}
  3. tan θ = opposite adjacent = a b \tan\theta=\frac{\mathrm{opposite}}{\mathrm{adjacent}}=\frac{a}{b}
  4. cot θ = adjacent opposite = b a \cot\theta=\frac{\mathrm{adjacent}}{\mathrm{opposite}}=\frac{b}{a}
  5. sec θ = hypotenuse adjacent = h b \sec\theta=\frac{\mathrm{hypotenuse}}{\mathrm{adjacent}}=\frac{h}{b}
  6. csc θ = hypotenuse opposite = h a \csc\theta=\frac{\mathrm{hypotenuse}}{\mathrm{opposite}}=\frac{h}{a}
  7. a b = ( a h ) ( b h ) \frac{a}{b}=\frac{\left(\frac{a}{h}\right)}{\left(\frac{b}{h}\right)}
  8. tan θ = opposite adjacent = ( opposite hypotenuse ) ( adjacent hypotenuse ) = sin θ cos θ \tan\theta=\frac{\mathrm{opposite}}{\mathrm{adjacent}}=\frac{\left(\frac{% \mathrm{opposite}}{\mathrm{hypotenuse}}\right)}{\left(\frac{\mathrm{adjacent}}% {\mathrm{hypotenuse}}\right)}=\frac{\sin\theta}{\cos\theta}
  9. cot θ = adjacent opposite = ( adjacent adjacent ) ( opposite adjacent ) = 1 tan θ = cos θ sin θ \cot\theta=\frac{\mathrm{adjacent}}{\mathrm{opposite}}=\frac{\left(\frac{% \mathrm{adjacent}}{\mathrm{adjacent}}\right)}{\left(\frac{\mathrm{opposite}}{% \mathrm{adjacent}}\right)}=\frac{1}{\tan\theta}=\frac{\cos\theta}{\sin\theta}
  10. sec θ = 1 cos θ = hypotenuse adjacent \sec\theta=\frac{1}{\cos\theta}=\frac{\mathrm{hypotenuse}}{\mathrm{adjacent}}
  11. csc θ = 1 sin θ = hypotenuse opposite \csc\theta=\frac{1}{\sin\theta}=\frac{\mathrm{hypotenuse}}{\mathrm{opposite}}
  12. tan θ = opposite adjacent = ( opposite × hypotenuse opposite × adjacent ) ( adjacent × hypotenuse opposite × adjacent ) = ( hypotenuse adjacent ) ( hypotenuse opposite ) = sec θ csc θ \tan\theta=\frac{\mathrm{opposite}}{\mathrm{adjacent}}=\frac{\left(\frac{% \mathrm{opposite}\times\mathrm{hypotenuse}}{\mathrm{opposite}\times\mathrm{% adjacent}}\right)}{\left(\frac{\mathrm{adjacent}\times\mathrm{hypotenuse}}{% \mathrm{opposite}\times\mathrm{adjacent}}\right)}=\frac{\left(\frac{\mathrm{% hypotenuse}}{\mathrm{adjacent}}\right)}{\left(\frac{\mathrm{hypotenuse}}{% \mathrm{opposite}}\right)}=\frac{\sec\theta}{\csc\theta}
  13. tan θ = sin θ cos θ = ( 1 csc θ ) ( 1 sec θ ) = ( csc θ sec θ csc θ ) ( csc θ sec θ sec θ ) = sec θ csc θ \tan\theta=\frac{\sin\theta}{\cos\theta}=\frac{\left(\frac{1}{\csc\theta}% \right)}{\left(\frac{1}{\sec\theta}\right)}=\frac{\left(\frac{\csc\theta\sec% \theta}{\csc\theta}\right)}{\left(\frac{\csc\theta\sec\theta}{\sec\theta}% \right)}=\frac{\sec\theta}{\csc\theta}
  14. cot θ = csc θ sec θ \cot\theta=\frac{\csc\theta}{\sec\theta}
  15. sin ( π / 2 - θ ) = cos θ \sin\left(\pi/2-\theta\right)=\cos\theta
  16. cos ( π / 2 - θ ) = sin θ \cos\left(\pi/2-\theta\right)=\sin\theta
  17. tan ( π / 2 - θ ) = cot θ \tan\left(\pi/2-\theta\right)=\cot\theta
  18. cot ( π / 2 - θ ) = tan θ \cot\left(\pi/2-\theta\right)=\tan\theta
  19. sec ( π / 2 - θ ) = csc θ \sec\left(\pi/2-\theta\right)=\csc\theta
  20. csc ( π / 2 - θ ) = sec θ \csc\left(\pi/2-\theta\right)=\sec\theta
  21. sin 2 ( x ) + cos 2 ( x ) = 1 \sin^{2}(x)+\cos^{2}(x)=1\,
  22. sin 2 ( x ) + cos 2 ( x ) = 1 \sin^{2}(x)+\cos^{2}(x)=1
  23. cos 2 ( x ) \cos^{2}(x)
  24. sin 2 ( x ) \sin^{2}(x)
  25. tan 2 ( x ) + 1 = sec 2 ( x ) \tan^{2}(x)+1\ =\sec^{2}(x)
  26. 1 + cot 2 ( x ) = csc 2 ( x ) 1\ +\cot^{2}(x)=\csc^{2}(x)
  27. 1 + cot 2 ( x ) = csc 2 ( x ) 1\ +\cot^{2}(x)=\csc^{2}(x)
  28. csc 2 ( x ) - cot 2 ( x ) = 1 \csc^{2}(x)-\cot^{2}(x)=1
  29. csc 2 ( x ) + sec 2 ( x ) - cot 2 ( x ) = 2 + tan 2 ( x ) \csc^{2}(x)+\sec^{2}(x)-\cot^{2}(x)=2\ +\tan^{2}(x)
  30. a 2 + b 2 = h 2 a^{2}+b^{2}=h^{2}
  31. csc 2 ( x ) + sec 2 ( x ) = h 2 a 2 + h 2 b 2 = a 2 + b 2 a 2 + a 2 + b 2 b 2 = 2 + b 2 a 2 + a 2 b 2 \csc^{2}(x)+\sec^{2}(x)=\frac{h^{2}}{a^{2}}+\frac{h^{2}}{b^{2}}=\frac{a^{2}+b^% {2}}{a^{2}}+\frac{a^{2}+b^{2}}{b^{2}}=2\ +\frac{b^{2}}{a^{2}}+\frac{a^{2}}{b^{% 2}}
  32. 2 + b 2 a 2 + a 2 b 2 = 2 + tan 2 ( x ) + cot 2 ( x ) 2\ +\frac{b^{2}}{a^{2}}+\frac{a^{2}}{b^{2}}=2\ +\tan^{2}(x)+\cot^{2}(x)
  33. csc 2 ( x ) + sec 2 ( x ) - cot 2 ( x ) = 2 + tan 2 ( x ) \csc^{2}(x)+\sec^{2}(x)-\cot^{2}(x)=2\ +\tan^{2}(x)
  34. α \alpha
  35. β \beta
  36. α + β \alpha+\beta
  37. α + β \alpha+\beta
  38. α \alpha
  39. \therefore
  40. \therefore
  41. R P Q = α RPQ=\alpha
  42. O Q A = 90 - α OQA=90-\alpha
  43. R Q O = α , R Q P = 90 - α RQO=\alpha,RQP=90-\alpha
  44. R P Q = α RPQ=\alpha
  45. R P Q = π 2 - R Q P = π 2 - ( π 2 - R Q O ) = R Q O = α RPQ=\tfrac{\pi}{2}-RQP=\tfrac{\pi}{2}-(\tfrac{\pi}{2}-RQO)=RQO=\alpha
  46. O P = 1 OP=1
  47. P Q = sin β PQ=\sin\beta
  48. O Q = cos β OQ=\cos\beta
  49. A Q O Q = sin α \frac{AQ}{OQ}=\sin\alpha\,
  50. A Q = sin α cos β AQ=\sin\alpha\cos\beta
  51. P R P Q = cos α \frac{PR}{PQ}=\cos\alpha\,
  52. P R = cos α sin β PR=\cos\alpha\sin\beta
  53. sin ( α + β ) = P B = R B + P R = A Q + P R = sin α cos β + cos α sin β \sin(\alpha+\beta)=PB=RB+PR=AQ+PR=\sin\alpha\cos\beta+\cos\alpha\sin\beta
  54. - β -\beta
  55. β \beta
  56. sin ( α - β ) = sin α cos - β + cos α sin - β \sin(\alpha-\beta)=\sin\alpha\cos-\beta+\cos\alpha\sin-\beta
  57. sin ( α - β ) = sin α cos β - cos α sin β \sin(\alpha-\beta)=\sin\alpha\cos\beta-\cos\alpha\sin\beta
  58. e i φ = cos φ + i sin φ e^{i\varphi}=\cos\varphi+i\sin\varphi
  59. α \alpha
  60. β \beta
  61. e i ( α + β ) = cos ( α + β ) + i sin ( α + β ) e^{i(\alpha+\beta)}=\cos(\alpha+\beta)+i\sin(\alpha+\beta)
  62. e i ( α + β ) = e i α e i β = ( cos α + i sin α ) ( cos β + i sin β ) e^{i(\alpha+\beta)}=e^{i\alpha}e^{i\beta}=(\cos\alpha+i\sin\alpha)(\cos\beta+i% \sin\beta)
  63. e i ( α + β ) = ( cos α cos β - sin α sin β ) + i ( sin α cos β + sin β cos α ) e^{i(\alpha+\beta)}=(\cos\alpha\cos\beta-\sin\alpha\sin\beta)+i(\sin\alpha\cos% \beta+\sin\beta\cos\alpha)
  64. cos ( α + β ) = cos α cos β - sin α sin β \cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta
  65. sin ( α + β ) = sin α cos β + sin β cos α \sin(\alpha+\beta)=\sin\alpha\cos\beta+\sin\beta\cos\alpha
  66. O P = 1 OP=1\,
  67. P Q = sin β PQ=\sin\beta\,
  68. O Q = cos β OQ=\cos\beta\,
  69. O A O Q = cos α \frac{OA}{OQ}=\cos\alpha\,
  70. O A = cos α cos β OA=\cos\alpha\cos\beta\,
  71. R Q P Q = sin α \frac{RQ}{PQ}=\sin\alpha\,
  72. R Q = sin α sin β RQ=\sin\alpha\sin\beta\,
  73. cos ( α + β ) = O B = O A - B A = O A - R Q = cos α cos β - sin α sin β \cos(\alpha+\beta)=OB=OA-BA=OA-RQ=\cos\alpha\cos\beta\ -\sin\alpha\sin\beta\,
  74. - β -\beta
  75. β \beta
  76. cos ( α - β ) = cos α cos - β - sin α sin - β \cos(\alpha-\beta)=\cos\alpha\cos-\beta\ -\sin\alpha\sin-\beta\,
  77. cos ( α - β ) = cos α cos β + sin α sin β \cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta\,
  78. cos ( α + β ) = sin ( π / 2 - ( α + β ) ) = sin ( ( π / 2 - α ) - β ) = sin ( π / 2 - α ) cos β - cos ( π / 2 - α ) sin β = cos α cos β - sin α sin β \begin{aligned}\displaystyle\cos(\alpha+\beta)&\displaystyle=\sin\left(\pi/2-(% \alpha+\beta)\right)\\ &\displaystyle=\sin\left((\pi/2-\alpha)-\beta\right)\\ &\displaystyle=\sin\left(\pi/2-\alpha\right)\cos\beta-\cos\left(\pi/2-\alpha% \right)\sin\beta\\ &\displaystyle=\cos\alpha\cos\beta-\sin\alpha\sin\beta\\ \end{aligned}
  79. tan ( α + β ) = sin ( α + β ) cos ( α + β ) = sin α cos β + cos α sin β cos α cos β - sin α sin β \tan(\alpha+\beta)=\frac{\sin(\alpha+\beta)}{\cos(\alpha+\beta)}=\frac{\sin% \alpha\cos\beta+\cos\alpha\sin\beta}{\cos\alpha\cos\beta-\sin\alpha\sin\beta}
  80. cos α cos β \cos\alpha\cos\beta
  81. tan ( α + β ) = tan α + tan β 1 - tan α tan β \tan(\alpha+\beta)=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}
  82. β \beta
  83. α \alpha
  84. tan ( - β ) = - tan β \tan(-\beta)=-\tan\beta
  85. tan ( α - β ) = tan α + tan ( - β ) 1 - tan α tan ( - β ) = tan α - tan β 1 + tan α tan β \tan(\alpha-\beta)=\frac{\tan\alpha+\tan(-\beta)}{1-\tan\alpha\tan(-\beta)}=% \frac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}
  86. cot ( α + β ) = cos ( α + β ) sin ( α + β ) = cos α cos β - sin α sin β sin α cos β + cos α sin β \cot(\alpha+\beta)=\frac{\cos(\alpha+\beta)}{\sin(\alpha+\beta)}=\frac{\cos% \alpha\cos\beta-\sin\alpha\sin\beta}{\sin\alpha\cos\beta+\cos\alpha\sin\beta}
  87. sin α sin β \sin\alpha\sin\beta
  88. cot ( α + β ) = cot α cot β - 1 cot α + cot β \cot(\alpha+\beta)=\frac{\cot\alpha\cot\beta-1}{\cot\alpha+\cot\beta}
  89. cot θ = 1 tan θ \cot\theta=\frac{1}{\tan\theta}
  90. cot ( α + β ) = 1 - tan α tan β tan α + tan β = 1 tan α tan β - 1 1 tan α + 1 tan β = cot α cot β - 1 cot α + cot β \cot(\alpha+\beta)=\frac{1-\tan\alpha\tan\beta}{\tan\alpha+\tan\beta}=\frac{% \frac{1}{\tan\alpha\tan\beta}-1}{\frac{1}{\tan\alpha}+\frac{1}{\tan\beta}}=% \frac{\cot\alpha\cot\beta-1}{\cot\alpha+\cot\beta}
  91. cot ( - β ) = - cot β \cot(-\beta)=-\cot\beta
  92. cot ( α - β ) = cot α cot ( - β ) - 1 cot α + cot ( - β ) = cot α cot β + 1 cot β - cot α \cot(\alpha-\beta)=\frac{\cot\alpha\cot(-\beta)-1}{\cot\alpha+\cot(-\beta)}=% \frac{\cot\alpha\cot\beta+1}{\cot\beta-\cot\alpha}
  93. sin ( 2 θ ) = 2 sin θ cos θ \sin(2\theta)=2\sin\theta\cos\theta\,
  94. cos ( 2 θ ) = cos 2 θ - sin 2 θ \cos(2\theta)=\cos^{2}\theta-\sin^{2}\theta\,
  95. cos ( 2 θ ) = 2 cos 2 θ - 1 \cos(2\theta)=2\cos^{2}\theta-1\,
  96. cos ( 2 θ ) = 1 - 2 sin 2 θ \cos(2\theta)=1-2\sin^{2}\theta\,
  97. tan ( 2 θ ) = 2 tan θ 1 - tan 2 θ = 2 cot θ - tan θ \tan(2\theta)=\frac{2\tan\theta}{1-\tan^{2}\theta}=\frac{2}{\cot\theta-\tan% \theta}\,
  98. cot ( 2 θ ) = cot 2 θ - 1 2 cot θ = cot θ - tan θ 2 \cot(2\theta)=\frac{\cot^{2}\theta-1}{2\cot\theta}=\frac{\cot\theta-\tan\theta% }{2}\,
  99. e i φ = cos φ + i sin φ e^{i\varphi}=\cos\varphi+i\sin\varphi
  100. e i 2 φ = ( cos φ + i sin φ ) 2 e^{i2\varphi}=(\cos\varphi+i\sin\varphi)^{2}
  101. e i 2 φ = cos 2 φ + i sin 2 φ e^{i2\varphi}=\cos 2\varphi+i\sin 2\varphi
  102. ( cos φ + i sin φ ) 2 = cos 2 φ + i sin 2 φ (\cos\varphi+i\sin\varphi)^{2}=\cos 2\varphi+i\sin 2\varphi
  103. i ( 2 sin φ cos φ ) + cos 2 φ - sin 2 φ = cos 2 φ + i sin 2 φ i(2\sin\varphi\cos\varphi)+\cos^{2}\varphi-\sin^{2}\varphi\ =\cos 2\varphi+i% \sin 2\varphi
  104. cos 2 φ - sin 2 φ = cos 2 φ \cos^{2}\varphi-\sin^{2}\varphi\ =\cos 2\varphi
  105. 2 sin φ cos φ = sin 2 φ 2\sin\varphi\cos\varphi=\sin 2\varphi
  106. cos θ 2 = ± 1 + cos θ 2 , \cos\frac{\theta}{2}=\pm\,\sqrt{\frac{1+\cos\theta}{2}},\,
  107. sin θ 2 = ± 1 - cos θ 2 . \sin\frac{\theta}{2}=\pm\,\sqrt{\frac{1-\cos\theta}{2}}.\,
  108. tan θ 2 = ± 1 - cos θ 1 + cos θ . \tan\frac{\theta}{2}=\pm\,\sqrt{\frac{1-\cos\theta}{1+\cos\theta}}.\,
  109. tan θ 2 = sin θ 1 + cos θ . \tan\frac{\theta}{2}=\frac{\sin\theta}{1+\cos\theta}.\,
  110. tan θ 2 = 1 - cos θ sin θ . \tan\frac{\theta}{2}=\frac{1-\cos\theta}{\sin\theta}.\,
  111. tan θ 2 = csc θ - cot θ . \tan\frac{\theta}{2}=\csc\theta-\cot\theta.\,
  112. cot θ 2 = ± 1 + cos θ 1 - cos θ = 1 + cos θ sin θ = sin θ 1 - cos θ = csc θ + cot θ . \cot\frac{\theta}{2}=\pm\,\sqrt{\frac{1+\cos\theta}{1-\cos\theta}}=\frac{1+% \cos\theta}{\sin\theta}=\frac{\sin\theta}{1-\cos\theta}=\csc\theta+\cot\theta.\,
  113. ψ + θ + ϕ = π = \psi+\theta+\phi=\pi=
  114. ψ \psi
  115. θ \theta
  116. ϕ \phi
  117. tan ( ψ ) + tan ( θ ) + tan ( ϕ ) = tan ( ψ ) tan ( θ ) tan ( ϕ ) . \tan(\psi)+\tan(\theta)+\tan(\phi)=\tan(\psi)\tan(\theta)\tan(\phi).
  118. ψ = π - θ - ϕ tan ( ψ ) = tan ( π - θ - ϕ ) = - tan ( θ + ϕ ) = - tan θ - tan ϕ 1 - tan θ tan ϕ = tan θ + tan ϕ tan θ tan ϕ - 1 ( tan θ tan ϕ - 1 ) tan ψ = tan θ + tan ϕ tan ψ tan θ tan ϕ - tan ψ = tan θ + tan ϕ tan ψ tan θ tan ϕ = tan ψ + tan θ + tan ϕ \begin{aligned}\displaystyle\psi&\displaystyle=\pi-\theta-\phi\\ \displaystyle\tan(\psi)&\displaystyle=\tan(\pi-\theta-\phi)\\ &\displaystyle=-\tan(\theta+\phi)\\ &\displaystyle=\frac{-\tan\theta-\tan\phi}{1-\tan\theta\tan\phi}\\ &\displaystyle=\frac{\tan\theta+\tan\phi}{\tan\theta\tan\phi-1}\\ \displaystyle(\tan\theta\tan\phi-1)\tan\psi&\displaystyle=\tan\theta+\tan\phi% \\ \displaystyle\tan\psi\tan\theta\tan\phi-\tan\psi&\displaystyle=\tan\theta+\tan% \phi\\ \displaystyle\tan\psi\tan\theta\tan\phi&\displaystyle=\tan\psi+\tan\theta+\tan% \phi\\ \end{aligned}
  119. ψ + θ + ϕ = π 2 = \psi+\theta+\phi=\tfrac{\pi}{2}=
  120. cot ( ψ ) + cot ( θ ) + cot ( ϕ ) = cot ( ψ ) cot ( θ ) cot ( ϕ ) \cot(\psi)+\cot(\theta)+\cot(\phi)=\cot(\psi)\cot(\theta)\cot(\phi)
  121. ψ \psi
  122. θ \theta
  123. ϕ \phi
  124. ψ + θ + ϕ = π 2 \psi+\theta+\phi=\tfrac{\pi}{2}\,
  125. ( π 2 - ψ ) + ( π 2 - θ ) + ( π 2 - ϕ ) = 3 π 2 - ( ψ + θ + ϕ ) = 3 π 2 - π 2 = π \therefore(\tfrac{\pi}{2}-\psi)+(\tfrac{\pi}{2}-\theta)+(\tfrac{\pi}{2}-\phi)=% \tfrac{3\pi}{2}-(\psi+\theta+\phi)=\tfrac{3\pi}{2}-\tfrac{\pi}{2}=\pi
  126. sin θ ± sin ϕ = 2 sin ( θ ± ϕ 2 ) cos ( θ ϕ 2 ) \sin\theta\pm\sin\phi=2\sin\left(\frac{\theta\pm\phi}{2}\right)\cos\left(\frac% {\theta\mp\phi}{2}\right)
  127. cos θ + cos ϕ = 2 cos ( θ + ϕ 2 ) cos ( θ - ϕ 2 ) \cos\theta+\cos\phi=2\cos\left(\frac{\theta+\phi}{2}\right)\cos\left(\frac{% \theta-\phi}{2}\right)
  128. cos θ - cos ϕ = - 2 sin ( θ + ϕ 2 ) sin ( θ - ϕ 2 ) \cos\theta-\cos\phi=-2\sin\left(\frac{\theta+\phi}{2}\right)\sin\left(\frac{% \theta-\phi}{2}\right)
  129. sin ( α + β ) = sin α cos β + cos α sin β \sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta
  130. sin ( α - β ) = sin α cos β - cos α sin β \sin(\alpha-\beta)=\sin\alpha\cos\beta-\cos\alpha\sin\beta
  131. sin ( α + β ) + sin ( α - β ) = sin α cos β + cos α sin β + sin α cos β - cos α sin β = 2 sin α cos β \sin(\alpha+\beta)+\sin(\alpha-\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta+% \sin\alpha\cos\beta-\cos\alpha\sin\beta=2\sin\alpha\cos\beta
  132. sin ( α + β ) - sin ( α - β ) = sin α cos β + cos α sin β - sin α cos β + cos α sin β = 2 cos α sin β \sin(\alpha+\beta)-\sin(\alpha-\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta-% \sin\alpha\cos\beta+\cos\alpha\sin\beta=2\cos\alpha\sin\beta
  133. α + β = θ \alpha+\beta=\theta
  134. α - β = ϕ \alpha-\beta=\phi
  135. α = θ + ϕ 2 \therefore\alpha=\frac{\theta+\phi}{2}
  136. β = θ - ϕ 2 \beta=\frac{\theta-\phi}{2}
  137. θ \theta
  138. ϕ \phi
  139. sin θ + sin ϕ = 2 sin ( θ + ϕ 2 ) cos ( θ - ϕ 2 ) \sin\theta+\sin\phi=2\sin\left(\frac{\theta+\phi}{2}\right)\cos\left(\frac{% \theta-\phi}{2}\right)
  140. sin θ - sin ϕ = 2 cos ( θ + ϕ 2 ) sin ( θ - ϕ 2 ) = 2 sin ( θ - ϕ 2 ) cos ( θ + ϕ 2 ) \sin\theta-\sin\phi=2\cos\left(\frac{\theta+\phi}{2}\right)\sin\left(\frac{% \theta-\phi}{2}\right)=2\sin\left(\frac{\theta-\phi}{2}\right)\cos\left(\frac{% \theta+\phi}{2}\right)
  141. sin θ ± sin ϕ = 2 sin ( θ ± ϕ 2 ) cos ( θ ϕ 2 ) \sin\theta\pm\sin\phi=2\sin\left(\frac{\theta\pm\phi}{2}\right)\cos\left(\frac% {\theta\mp\phi}{2}\right)
  142. cos ( α + β ) = cos α cos β - sin α sin β \cos(\alpha+\beta)=\cos\alpha\cos\beta\ -\sin\alpha\sin\beta
  143. cos ( α - β ) = cos α cos β + sin α sin β \cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta
  144. cos ( α + β ) + cos ( α - β ) = cos α cos β - sin α sin β + cos α cos β + sin α sin β = 2 cos α cos β \cos(\alpha+\beta)+\cos(\alpha-\beta)=\cos\alpha\cos\beta\ -\sin\alpha\sin% \beta+\cos\alpha\cos\beta+\sin\alpha\sin\beta=2\cos\alpha\cos\beta
  145. cos ( α + β ) - cos ( α - β ) = cos α cos β - sin α sin β - cos α cos β - sin α sin β = - 2 sin α sin β \cos(\alpha+\beta)-\cos(\alpha-\beta)=\cos\alpha\cos\beta\ -\sin\alpha\sin% \beta-\cos\alpha\cos\beta-\sin\alpha\sin\beta=-2\sin\alpha\sin\beta
  146. θ \theta
  147. ϕ \phi
  148. cos θ + cos ϕ = 2 cos ( θ + ϕ 2 ) cos ( θ - ϕ 2 ) \cos\theta+\cos\phi=2\cos\left(\frac{\theta+\phi}{2}\right)\cos\left(\frac{% \theta-\phi}{2}\right)
  149. cos θ - cos ϕ = - 2 sin ( θ + ϕ 2 ) sin ( θ - ϕ 2 ) \cos\theta-\cos\phi=-2\sin\left(\frac{\theta+\phi}{2}\right)\sin\left(\frac{% \theta-\phi}{2}\right)
  150. O A = O D = 1 OA=OD=1\,
  151. A B = sin θ AB=\sin\theta\,
  152. C D = tan θ CD=\tan\theta\,
  153. sin θ < θ < tan θ \sin\theta<\theta<\tan\theta\,
  154. sin θ θ < 1 if 0 < θ \frac{\sin\theta}{\theta}<1\ \ \ \mathrm{if}\ \ \ 0<\theta\,
  155. sin θ θ = sin ( - θ ) - θ < 1 \frac{\sin\theta}{\theta}=\frac{\sin(-\theta)}{-\theta}<1\,
  156. sin θ θ < 1 if θ 0 \frac{\sin\theta}{\theta}<1\ \ \ \mathrm{if}\ \ \ \theta\neq 0\,
  157. tan θ θ > 1 if 0 < θ < π 2 \frac{\tan\theta}{\theta}>1\ \ \ \mathrm{if}\ \ \ 0<\theta<\frac{\pi}{2}\,
  158. lim θ 0 sin θ = 0 \lim_{\theta\to 0}{\sin\theta}=0\,
  159. lim θ 0 cos θ = 1 \lim_{\theta\to 0}{\cos\theta}=1\,
  160. lim θ 0 sin θ θ = 1 \lim_{\theta\to 0}{\frac{\sin\theta}{\theta}}=1
  161. sin θ < θ < tan θ \sin\theta<\theta<\tan\theta\,
  162. sin θ θ < 1 < tan θ θ \frac{\sin\theta}{\theta}<1<\frac{\tan\theta}{\theta}\,
  163. tan θ = sin θ cos θ \tan\theta=\frac{\sin\theta}{\cos\theta}
  164. 1 < sin θ θ cos θ \therefore 1<\frac{\sin\theta}{\theta\cos\theta}
  165. cos θ \cos\theta
  166. cos θ < sin θ θ \cos\theta<\frac{\sin\theta}{\theta}
  167. cos θ < sin θ θ < 1 \cos\theta<\frac{\sin\theta}{\theta}<1
  168. cos θ \cos\theta
  169. θ 0 \theta\to 0
  170. lim θ 0 cos θ = 1 \lim_{\theta\to 0}{\cos\theta}=1\,
  171. lim θ 0 sin θ θ = 1 \lim_{\theta\to 0}{\frac{\sin\theta}{\theta}}=1
  172. lim θ 0 1 - cos θ θ = 0 \lim_{\theta\to 0}\frac{1-\cos\theta}{\theta}=0
  173. 1 - cos θ θ \displaystyle\frac{1-\cos\theta}{\theta}
  174. lim θ 0 1 - cos θ θ 2 = 1 2 \lim_{\theta\to 0}\frac{1-\cos\theta}{\theta^{2}}=\frac{1}{2}
  175. 1 - cos θ θ 2 = sin θ θ × sin θ θ × 1 1 + cos θ . \frac{1-\cos\theta}{\theta^{2}}=\frac{\sin\theta}{\theta}\times\frac{\sin% \theta}{\theta}\times\frac{1}{1+\cos\theta}.\,
  176. sin [ arctan ( x ) ] = x 1 + x 2 \sin[\arctan(x)]=\frac{x}{\sqrt{1+x^{2}}}
  177. sin 2 θ + cos 2 θ = 1 \sin^{2}\theta+\cos^{2}\theta=1
  178. cos 2 θ \cos^{2}\theta
  179. cos 2 θ = 1 tan 2 θ + 1 \cos^{2}\theta=\frac{1}{\tan^{2}\theta+1}
  180. θ = arctan ( x ) \theta=\arctan(x)
  181. 1 - sin 2 [ arctan ( x ) ] = 1 tan 2 [ arctan ( x ) ] + 1 1-\sin^{2}[\arctan(x)]=\frac{1}{\tan^{2}[\arctan(x)]+1}
  182. tan [ arctan ( x ) ] x \tan[\arctan(x)]\equiv x
  183. sin [ arctan ( x ) ] = x x 2 + 1 \sin[\arctan(x)]=\frac{x}{\sqrt{x^{2}+1}}

Propeller_(aeronautics).html

  1. η = propulsive power out shaft power in = thrust axial speed resistance torque rotational speed . \eta=\frac{\hbox{propulsive power out}}{\hbox{shaft power in}}=\frac{\hbox{% thrust}\cdot\hbox{axial speed}}{\hbox{resistance torque}\cdot\hbox{rotational % speed}}.

Proper_forcing_axiom.html

  1. λ \lambda
  2. [ λ ] ω [\lambda]^{\omega}
  3. \subseteq
  4. 1 \aleph_{1}
  5. 2 0 = 2 2^{\aleph_{0}}=\aleph_{2}
  6. 1 \aleph_{1}
  7. P ( ω ) P(\omega)
  8. κ \kappa
  9. κ \kappa
  10. P α : α κ \langle P_{\alpha}\,\colon\alpha\leq\kappa\rangle
  11. Q α : α < κ \langle Q_{\alpha}\,\colon\alpha<\kappa\rangle
  12. N N
  13. H λ H_{\lambda}
  14. λ \lambda
  15. P κ N P_{\kappa}\in N
  16. α κ N \alpha\in\kappa\cap N
  17. p p
  18. ( N , P α ) (N,P_{\alpha})
  19. p p
  20. q P κ / G P α N [ G P α ] q\in P_{\kappa}/G_{P_{\alpha}}\cap N[G_{P_{\alpha}}]
  21. r P κ r\in P_{\kappa}
  22. r r
  23. N N
  24. r r
  25. P α P_{\alpha}
  26. p p
  27. p p
  28. r r
  29. [ α , κ ) [\alpha,\kappa)
  30. q q
  31. q q
  32. N N
  33. κ \kappa
  34. α = 0 \alpha=0

Proportional_hazards_model.html

  1. λ 0 ( t ) \lambda_{0}(t)
  2. t t
  3. λ 0 ( t ) \lambda_{0}(t)
  4. x x
  5. x x
  6. λ ( t | X ) = λ 0 ( t ) exp ( β 1 X 1 + + β p X p ) = λ 0 ( t ) exp ( X β ) . \lambda(t|X)=\lambda_{0}(t)\exp(\beta_{1}X_{1}+\cdots+\beta_{p}X_{p})=\lambda_% {0}(t)\exp(X\beta^{\prime}).
  7. L ( β ) = i : C i = 1 θ i j : Y j Y i θ j , L(\beta)=\prod_{i:C_{i}=1}\frac{\theta_{i}}{\sum_{j:Y_{j}\geq Y_{i}}\theta_{j}},
  8. ( β ) = i : C i = 1 ( X i β - log j : Y j Y i θ j ) . \ell(\beta)=\sum_{i:C_{i}=1}\left(X_{i}\beta^{\prime}-\log\sum_{j:Y_{j}\geq Y_% {i}}\theta_{j}\right).
  9. ( β ) = i : C i = 1 ( X i - j : Y j Y i θ j X j j : Y j Y i θ j ) , \ell^{\prime}(\beta)=\sum_{i:C_{i}=1}\left(X_{i}-\frac{\sum_{j:Y_{j}\geq Y_{i}% }\theta_{j}X_{j}}{\sum_{j:Y_{j}\geq Y_{i}}\theta_{j}}\right),
  10. ′′ ( β ) = - i : C i = 1 ( j : Y j Y i θ j X j X j j : Y j Y i θ j - j : Y j Y i θ j X j × j : Y j Y i θ j X j [ j : Y j Y i θ j ] 2 ) . \ell^{\prime\prime}(\beta)=-\sum_{i:C_{i}=1}\left(\frac{\sum_{j:Y_{j}\geq Y_{i% }}\theta_{j}X_{j}X_{j}^{\prime}}{\sum_{j:Y_{j}\geq Y_{i}}\theta_{j}}-\frac{% \sum_{j:Y_{j}\geq Y_{i}}\theta_{j}X_{j}\times\sum_{j:Y_{j}\geq Y_{i}}\theta_{j% }X_{j}^{\prime}}{[\sum_{j:Y_{j}\geq Y_{i}}\theta_{j}]^{2}}\right).
  11. L ( β ) = j i H j θ i = 0 m - 1 [ i : Y i t j θ i - m i H j θ i ] . L(\beta)=\prod_{j}\frac{\prod_{i\in H_{j}}\theta_{i}}{\prod_{\ell=0}^{m-1}[% \sum_{i:Y_{i}\geq t_{j}}\theta_{i}-\frac{\ell}{m}\sum_{i\in H_{j}}\theta_{i}]}.
  12. ( β ) = j ( i H j X i β - = 0 m - 1 log ( i : Y i t j θ i - m i H j θ i ) ) , \ell(\beta)=\sum_{j}\left(\sum_{i\in H_{j}}X_{i}\beta^{\prime}-\sum_{\ell=0}^{% m-1}\log\left(\sum_{i:Y_{i}\geq t_{j}}\theta_{i}-\frac{\ell}{m}\sum_{i\in H_{j% }}\theta_{i}\right)\right),
  13. ( β ) = j ( i H j X i - = 0 m - 1 i : Y i t j θ i X i - m i H j θ i X i i : Y i t j θ i - m i H j θ i ) , \ell^{\prime}(\beta)=\sum_{j}\left(\sum_{i\in H_{j}}X_{i}-\sum_{\ell=0}^{m-1}% \frac{\sum_{i:Y_{i}\geq t_{j}}\theta_{i}X_{i}-\frac{\ell}{m}\sum_{i\in H_{j}}% \theta_{i}X_{i}}{\sum_{i:Y_{i}\geq t_{j}}\theta_{i}-\frac{\ell}{m}\sum_{i\in H% _{j}}\theta_{i}}\right),
  14. ′′ ( β ) = - j = 0 m - 1 ( i : Y i t j θ i X i X i - m i H j θ i X i X i ϕ j , , m - Z j , , m × Z j , , m ϕ j , , m 2 ) , \ell^{\prime\prime}(\beta)=-\sum_{j}\sum_{\ell=0}^{m-1}\left(\frac{\sum_{i:Y_{% i}\geq t_{j}}\theta_{i}X_{i}X_{i}^{\prime}-\frac{\ell}{m}\sum_{i\in H_{j}}% \theta_{i}X_{i}X_{i}^{\prime}}{\phi_{j,\ell,m}}-\frac{Z_{j,\ell,m}\times Z_{j,% \ell,m}^{\prime}}{\phi_{j,\ell,m}^{2}}\right),
  15. ϕ j , , m = i : Y i t j θ i - m i H j θ i \phi_{j,\ell,m}=\sum_{i:Y_{i}\geq t_{j}}\theta_{i}-\frac{\ell}{m}\sum_{i\in H_% {j}}\theta_{i}
  16. Z j , , m = i : Y i t j θ i X i - m i H j θ i X i . Z_{j,\ell,m}=\sum_{i:Y_{i}\geq t_{j}}\theta_{i}X_{i}-\frac{\ell}{m}\sum_{i\in H% _{j}}\theta_{i}X_{i}.
  17. λ ( t | X ) = λ 0 ( t ) + β 1 X 1 + + β p X p = λ 0 ( t ) + X β . \lambda(t|X)=\lambda_{0}(t)+\beta_{1}X_{1}+\cdots+\beta_{p}X_{p}=\lambda_{0}(t% )+X\beta^{\prime}.
  18. λ ( t | X ) \lambda(t|X)
  19. λ 0 ( t ) \lambda_{0}(t)
  20. ( β ) = j ( i H j X i β - = 0 m - 1 log ( i : Y i t j θ i - m i H j θ i ) ) + λ β 1 , \ell(\beta)=\sum_{j}\left(\sum_{i\in H_{j}}X_{i}\beta^{\prime}-\sum_{\ell=0}^{% m-1}\log\left(\sum_{i:Y_{i}\geq t_{j}}\theta_{i}-\frac{\ell}{m}\sum_{i\in H_{j% }}\theta_{i}\right)\right)+\lambda\|\beta\|_{1},

Protein_precipitation.html

  1. Δ G = Δ H - T Δ S . \Delta G=\Delta H-T\Delta S.
  2. PO 4 3 - > SO 4 2 - > COO - > Cl - \mathrm{PO_{4}^{3-}>SO_{4}^{2-}>COO^{-}>Cl^{-}}
  3. NH 4 + > K + > Na + \mathrm{NH_{4}^{+}>K^{+}>Na^{+}}
  4. log S = B - K I \log S=B-KI\,
  5. I = 1 2 i = 1 n c i z i 2 I=\begin{matrix}\frac{1}{2}\end{matrix}\sum_{i=1}^{n}c_{i}z_{i}^{2}
  6. log S = k / e 2 + log S 0 \log S=k/e^{2}+\log S^{0}\,
  7. ln ( S ) + p S = X - a C \ln(S)+pS=X-aC\,
  8. x = ( μ i - μ i 0 ) R T x=(\mu_{i}-\mu_{i}^{0})RT

Proto-Indo-European_nominals.html

  1. root + suffix stem + ending word \underbrace{\underbrace{\mathrm{root+suffix}}_{\mathrm{stem}}+\mathrm{ending}}% _{\mathrm{word}}

Pruning_(decision_trees).html

  1. T T
  2. T T
  3. T T
  4. i i
  5. i - 1 i-1
  6. T T
  7. S S
  8. e r r ( T , S ) err(T,S)
  9. e r r ( p r u n e ( T , t ) , S ) - e r r ( T , S ) | l e a v e s ( T ) | - | l e a v e s ( p r u n e ( T , t ) ) | \frac{err(prune(T,t),S)-err(T,S)}{|leaves(T)|-|leaves(prune(T,t))|}
  10. p r u n e ( T , t ) prune(T,t)
  11. t t
  12. T T

Pseudo-Zernike_polynomials.html

  1. V n m ( x , y ) = R n m ( x , y ) e j m arctan ( y x ) V_{nm}(x,y)=R_{nm}(x,y)e^{jm\arctan(\frac{y}{x})}
  2. x 2 + y 2 1 , n 0 , | m | n x^{2}+y^{2}\leq 1,n\geq 0,|m|\leq n
  3. 0 2 π 0 1 r [ V n l ( r cos θ , r sin θ ) ] * × V m k ( r cos θ , r sin θ ) d r d θ = π n + 1 δ m n δ k l , \int_{0}^{2\pi}\int_{0}^{1}r[V_{nl}(r\cos\theta,r\sin\theta)]^{*}\times V_{mk}% (r\cos\theta,r\sin\theta)drd\theta=\frac{\pi}{n+1}\delta_{mn}\delta_{kl},
  4. r 2 = x 2 + y 2 r^{2}=x^{2}+y^{2}
  5. x = r cos θ x=r\cos\theta
  6. y = r sin θ y=r\sin\theta
  7. R n m R_{nm}
  8. R n m ( x , y ) = s = 0 n - | m | D n , | m | , s ( x 2 + y 2 ) ( n - s ) / 2 R_{nm}(x,y)=\sum_{s=0}^{n-|m|}D_{n,|m|,s}(x^{2}+y^{2})^{(n-s)/2}
  9. D n , m , s = ( - 1 ) s ( 2 n + 1 - s ) ! s ! ( n - m - s ) ! ( n + m - s + 1 ) ! . D_{n,m,s}=(-1)^{s}\frac{(2n+1-s)!}{s!(n-m-s)!(n+m-s+1)!}.
  10. R 0 , 0 = 1 R_{0,0}=1
  11. R 1 , 0 = - 2 + 3 r R_{1,0}=-2+3r
  12. R 1 , 1 = r R_{1,1}=r
  13. R 2 , 0 = 3 + 10 r 2 - 12 r R_{2,0}=3+10r^{2}-12r
  14. R 2 , 1 = 5 r 2 - 4 r R_{2,1}=5r^{2}-4r
  15. R 2 , 2 = r 2 R_{2,2}=r^{2}
  16. R 3 , 0 = - 4 + 35 r 3 - 60 r 2 + 30 r R_{3,0}=-4+35r^{3}-60r^{2}+30r
  17. R 3 , 1 = 21 r 3 - 30 r 2 + 10 r R_{3,1}=21r^{3}-30r^{2}+10r
  18. R 3 , 2 = 7 r 3 - 6 r 2 R_{3,2}=7r^{3}-6r^{2}
  19. R 3 , 3 = r 3 R_{3,3}=r^{3}
  20. R 4 , 0 = 5 + 126 r 4 - 280 r 3 + 210 r 2 - 60 r R_{4,0}=5+126r^{4}-280r^{3}+210r^{2}-60r
  21. R 4 , 1 = 84 r 4 - 168 r 3 + 105 r 2 - 20 r R_{4,1}=84r^{4}-168r^{3}+105r^{2}-20r
  22. R 4 , 2 = 36 r 4 - 56 r 3 + 21 r 2 R_{4,2}=36r^{4}-56r^{3}+21r^{2}
  23. R 4 , 3 = 9 r 4 - 8 r 3 R_{4,3}=9r^{4}-8r^{3}
  24. R 4 , 4 = r 4 R_{4,4}=r^{4}
  25. R 5 , 0 = - 6 + 462 r 5 - 1260 r 4 + 1260 r 3 - 560 r 2 + 105 r R_{5,0}=-6+462r^{5}-1260r^{4}+1260r^{3}-560r^{2}+105r
  26. R 5 , 1 = 330 r 5 - 840 r 4 + 756 r 3 - 280 r 2 + 35 r R_{5,1}=330r^{5}-840r^{4}+756r^{3}-280r^{2}+35r
  27. R 5 , 2 = 165 r 5 - 360 r 4 + 252 r 3 - 56 r 2 R_{5,2}=165r^{5}-360r^{4}+252r^{3}-56r^{2}
  28. R 5 , 3 = 55 r 5 - 90 r 4 + 36 r 3 R_{5,3}=55r^{5}-90r^{4}+36r^{3}
  29. R 5 , 4 = 11 r 5 - 10 r 4 R_{5,4}=11r^{5}-10r^{4}
  30. R 5 , 5 = r 5 R_{5,5}=r^{5}
  31. n n
  32. l l
  33. A n l = n + 1 π 0 2 π 0 1 [ V n l ( r cos θ , r sin θ ) ] * f ( r cos θ , r sin θ ) r d r d θ A_{nl}=\frac{n+1}{\pi}\int_{0}^{2\pi}\int_{0}^{1}[V_{nl}(r\cos\theta,r\sin% \theta)]^{*}f(r\cos\theta,r\sin\theta)rdrd\theta
  34. n = 0 , n=0,\ldots\infty
  35. l l
  36. | l | n |l|\leq n
  37. f ( x , y ) = n = 0 l = - n + n A n l V n l ( x , y ) . f(x,y)=\sum_{n=0}^{\infty}\sum_{l=-n}^{+n}A_{nl}V_{nl}(x,y).

Pullback_attractor.html

  1. φ \varphi
  2. ( X , d ) (X,d)
  3. ( Ω , , ) (\Omega,\mathcal{F},\mathbb{P})
  4. ϑ : × Ω Ω \vartheta:\mathbb{R}\times\Omega\to\Omega
  5. 𝒜 \mathcal{A}
  6. x 0 X x_{0}\in X
  7. φ ( t , ω ) x 0 𝒜 \varphi(t,\omega)x_{0}\to\mathcal{A}
  8. t + t\to+\infty
  9. a X a\in X
  10. 𝒜 \mathcal{A}
  11. x 0 X x_{0}\in X
  12. t n + t_{n}\to+\infty
  13. d ( φ ( t n , ω ) x 0 , a ) 0 d\left(\varphi(t_{n},\omega)x_{0},a\right)\to 0
  14. n n\to\infty
  15. ω \omega
  16. t t
  17. t + t\to+\infty
  18. t t
  19. t t
  20. lim t + φ ( t , ϑ - t ω ) \lim_{t\to+\infty}\varphi(t,\vartheta_{-t}\omega)
  21. B ( ω ) X B(\omega)\subseteq X
  22. Ω B ( ω ) := { x X | t n + , b n B ( ϑ - t n ω ) s . t . φ ( t n , ϑ - t n ω ) b n x as n } . \Omega_{B}(\omega):=\left\{x\in X\left|\exists t_{n}\to+\infty,\exists b_{n}% \in B(\vartheta_{-t_{n}}\omega)\mathrm{\,s.t.\,}\varphi(t_{n},\vartheta_{-t_{n% }}\omega)b_{n}\to x\mathrm{\,as\,}n\to\infty\right.\right\}.
  23. Ω B ( ω ) = t 0 s t φ ( s , ϑ - s ω ) B ( ϑ - s ω ) ¯ . \Omega_{B}(\omega)=\bigcap_{t\geq 0}\overline{\bigcup_{s\geq t}\varphi(s,% \vartheta_{-s}\omega)B(\vartheta_{-s}\omega)}.
  24. 𝒜 ( ω ) \mathcal{A}(\omega)
  25. \mathbb{P}
  26. 𝒜 ( ω ) \mathcal{A}(\omega)
  27. 𝒜 ( ω ) X \mathcal{A}(\omega)\subseteq X
  28. ω dist ( x , 𝒜 ( ω ) ) \omega\mapsto\mathrm{dist}(x,\mathcal{A}(\omega))
  29. ( , ( X ) ) (\mathcal{F},\mathcal{B}(X))
  30. x X x\in X
  31. 𝒜 ( ω ) \mathcal{A}(\omega)
  32. φ ( t , ω ) ( 𝒜 ( ω ) ) = 𝒜 ( ϑ t ω ) \varphi(t,\omega)(\mathcal{A}(\omega))=\mathcal{A}(\vartheta_{t}\omega)
  33. 𝒜 ( ω ) \mathcal{A}(\omega)
  34. B X B\subseteq X
  35. lim t + dist ( φ ( t , ϑ - t ω ) ( B ) , 𝒜 ( ω ) ) = 0 \lim_{t\to+\infty}\mathrm{dist}\left(\varphi(t,\vartheta_{-t}\omega)(B),% \mathcal{A}(\omega)\right)=0
  36. dist ( x , A ) := inf a A d ( x , a ) , \mathrm{dist}(x,A):=\inf_{a\in A}d(x,a),
  37. dist ( B , A ) := sup b B inf a A d ( b , a ) . \mathrm{dist}(B,A):=\sup_{b\in B}\inf_{a\in A}d(b,a).
  38. K K
  39. 𝒜 ( ω ) = B Ω B ( ω ) ¯ , \mathcal{A}(\omega)=\overline{\bigcup_{B}\Omega_{B}(\omega)},
  40. B X B\subseteq X
  41. ϑ \vartheta
  42. D X D\subseteq X
  43. ( 𝒜 ( ) D ) > 0 , \mathbb{P}\left(\mathcal{A}(\cdot)\subseteq D\right)>0,
  44. 𝒜 ( ω ) = Ω D ( ω ) \mathcal{A}(\omega)=\Omega_{D}(\omega)
  45. \mathbb{P}

Pure_spinor.html

  1. v 2 = Q ( v ) . v^{2}=Q(v).\,
  2. ψ Γ μ ψ = 0. \psi\Gamma^{\mu}\psi=0.\,
  3. ( 2 n n - 4 ) {2n\choose n-4}

Pure_subgroup.html

  1. S S
  2. G G
  3. S S
  4. n t h n^{th}
  5. G G
  6. n t h n^{th}
  7. S S

Purification_of_quantum_state.html

  1. H A H_{A}
  2. H B H_{B}
  3. | ψ H A H B |\psi\rangle\in H_{A}\otimes H_{B}
  4. | ψ ψ | |\psi\rangle\langle\psi|
  5. H B H_{B}
  6. tr B ( | ψ ψ | ) = ρ . \operatorname{tr_{B}}\left(|\psi\rangle\langle\psi|\right)=\rho.
  7. | ψ |\psi\rangle
  8. ρ \rho
  9. ρ = i = 1 n p i | i i | \rho=\sum_{i=1}^{n}p_{i}|i\rangle\langle i|
  10. { | i } \{|i\rangle\}
  11. H B H_{B}
  12. { | i } \{|i^{\prime}\rangle\}
  13. | ψ H A H B |\psi\rangle\in H_{A}\otimes H_{B}
  14. | ψ = i p i | i | i . |\psi\rangle=\sum_{i}\sqrt{p_{i}}|i\rangle\otimes|i^{\prime}\rangle.
  15. tr B ( | ψ ψ | ) = tr B [ ( i p i | i | i ) ( j p j j | j | ) ] \operatorname{tr_{B}}\left(|\psi\rangle\langle\psi|\right)=\operatorname{tr_{B% }}\left[\left(\sum_{i}\sqrt{p_{i}}|i\rangle\otimes|i^{\prime}\rangle\right)% \left(\sum_{j}\sqrt{p_{j}}\langle j|\otimes\langle j^{\prime}|\right)\right]
  16. = tr B ( i , j p i p j | i j | | i j | ) = i , j δ i j p i p j | i j | = ρ . =\operatorname{tr_{B}}\left(\sum_{i,j}\sqrt{p_{i}p_{j}}|i\rangle\langle j|% \otimes|i^{\prime}\rangle\langle j^{\prime}|\right)=\sum_{i,j}\delta_{ij}\sqrt% {p_{i}p_{j}}|i\rangle\langle j|=\rho.
  17. | ψ |\psi\rangle

Putnam_model.html

  1. B 1 / 3 Size Productivity = Effort 1 / 3 Time 4 / 3 \frac{B^{1/3}\cdot\textrm{Size}}{\textrm{Productivity}}=\textrm{Effort}^{1/3}% \cdot\textrm{Time}^{4/3}
  2. Effort = [ Size Productivity Time 4 / 3 ] 3 B \textrm{Effort}=\left[\frac{\textrm{Size}}{\textrm{Productivity}\cdot\textrm{% Time}^{4/3}}\right]^{3}\cdot B
  3. Process Productivity = Size [ Effort B ] 1 / 3 Time 4 / 3 \textrm{Process\ Productivity}=\frac{\textrm{Size}}{\left[\frac{\textrm{Effort% }}{B}\right]^{1/3}\cdot\textrm{Time}^{4/3}}

Pyrolant.html

  1. B a C r O 4 BaCrO_{\mathrm{4}}

Pythagorean_quadruple.html

  1. a 2 + b 2 + c 2 = d 2 a^{2}+b^{2}+c^{2}=d^{2}
  2. ( a , b , c , d ) (a,b,c,d)
  3. ( a , b , c , d ) (a,b,c,d)
  4. a = m 2 + n 2 - p 2 - q 2 , a=m^{2}+n^{2}-p^{2}-q^{2},\,
  5. b = 2 ( m q + n p ) , b=2(mq+np),\,
  6. c = 2 ( n q - m p ) , c=2(nq-mp),\,
  7. d = m 2 + n 2 + p 2 + q 2 , d=m^{2}+n^{2}+p^{2}+q^{2},\,
  8. ( m 2 + n 2 + p 2 + q 2 ) 2 = ( 2 m q + 2 n p ) 2 + ( 2 n q - 2 m p ) 2 + ( m 2 + n 2 - p 2 - q 2 ) 2 . (m^{2}+n^{2}+p^{2}+q^{2})^{2}=(2mq+2np)^{2}+(2nq-2mp)^{2}+(m^{2}+n^{2}-p^{2}-q% ^{2})^{2}.
  9. a a
  10. b b
  11. a 2 + b 2 a^{2}+b^{2}
  12. p 2 < a 2 + b 2 p^{2}<a^{2}+b^{2}
  13. c = ( a 2 + b 2 - p 2 ) / ( 2 p ) c=(a^{2}+b^{2}-p^{2})/(2p)
  14. d = ( a 2 + b 2 + p 2 ) / ( 2 p ) d=(a^{2}+b^{2}+p^{2})/(2p)
  15. p = d - c p={d-c}
  16. a , b a,b
  17. l 2 + m 2 l^{2}+m^{2}
  18. n 2 < l 2 + m 2 . n^{2}<l^{2}+m^{2}.
  19. c = l 2 + m 2 - n 2 n c=\frac{l^{2}+m^{2}-n^{2}}{n}
  20. d = l 2 + m 2 + n 2 n . d=\frac{l^{2}+m^{2}+n^{2}}{n}.
  21. ( a , b , c , d ) (a,b,c,d)
  22. ( m , n , p , q ) (m,n,p,q)
  23. E ( α ) E(\alpha)
  24. α ( ) α ¯ \alpha(\cdot)\overline{\alpha}
  25. α = m + n i + p j + q k \alpha=m+ni+pj+qk
  26. \mathbb{H}
  27. i , j , k i,j,k
  28. E ( α ) = ( m 2 + n 2 - p 2 - q 2 2 n p - 2 m q 2 m p + 2 n q 2 m q + 2 n p m 2 - n 2 + p 2 - q 2 2 p q - 2 m n 2 n q - 2 m p 2 m n + 2 p q m 2 - n 2 - p 2 + q 2 ) , E(\alpha)=\begin{pmatrix}m^{2}+n^{2}-p^{2}-q^{2}&2np-2mq&2mp+2nq\\ 2mq+2np&m^{2}-n^{2}+p^{2}-q^{2}&2pq-2mn\\ 2nq-2mp&2mn+2pq&m^{2}-n^{2}-p^{2}+q^{2}\\ \end{pmatrix},
  29. 1 d E ( α ) \frac{1}{d}E(\alpha)
  30. SO ( 3 , ) \in\,\text{SO}(3,\mathbb{Q})
  31. x 2 + y 2 + z 2 = m 2 x^{2}+y^{2}+z^{2}=m^{2}

Q-gamma_function.html

  1. Γ q ( x ) = ( 1 - q ) 1 - x n = 0 1 - q n + 1 1 - q n + x = ( 1 - q ) 1 - x ( q ; q ) ( q x ; q ) \Gamma_{q}(x)=(1-q)^{1-x}\prod_{n=0}^{\infty}\frac{1-q^{n+1}}{1-q^{n+x}}=(1-q)% ^{1-x}\,\frac{(q;q)_{\infty}}{(q^{x};q)_{\infty}}
  2. Γ q ( x + 1 ) = 1 - q x 1 - q Γ q ( x ) = [ x ] q Γ q ( x ) \Gamma_{q}(x+1)=\frac{1-q^{x}}{1-q}\Gamma_{q}(x)=[x]_{q}\Gamma_{q}(x)
  3. Γ q ( n ) = [ n - 1 ] q ! \Gamma_{q}(n)=[n-1]_{q}!
  4. lim q 1 ± Γ q ( x ) = Γ ( x ) . \lim_{q\to 1\pm}\Gamma_{q}(x)=\Gamma(x).
  5. 0 1 log Γ q ( x ) d x = ζ ( 2 ) log q + log q - 1 q 6 + log ( q - 1 ; q - 1 ) ( q > 1 ) . \int_{0}^{1}\log\Gamma_{q}(x)dx=\frac{\zeta(2)}{\log q}+\log\sqrt{\frac{q-1}{% \sqrt[6]{q}}}+\log(q^{-1};q^{-1})_{\infty}\quad(q>1).

Q-Vandermonde_identity.html

  1. ( m + n k ) q = j ( m k - j ) q ( n j ) q q j ( m - k + j ) . {\left({{m+n}\atop{k}}\right)}_{\!\!q}=\sum_{j}{\left({{m}\atop{k-j}}\right)}_% {\!\!q}{\left({{n}\atop{j}}\right)}_{\!\!q}q^{j(m-k+j)}.
  2. m a x ( 0 , k m ) j m i n ( n , k ) . max(0,k−m)≤j≤min(n,k).
  3. B q ( n , k ) B_{q}(n,k)
  4. B q ( n , k ) = q - k ( n - k ) ( n k ) q 2 . B_{q}(n,k)=q^{-k(n-k)}{\left({{n}\atop{k}}\right)}_{\!\!q^{2}}.
  5. q - 1 q^{-1}
  6. B q ( m + n , k ) = q n k j q - ( m + n ) j B q ( m , k - j ) B q ( n , j ) . B_{q}(m+n,k)=q^{nk}\sum_{j}q^{-(m+n)j}B_{q}(m,k-j)B_{q}(n,j).
  7. ( 1 + x ) m ( 1 + x ) n (1+x)^{m}(1+x)^{n}
  8. ( 1 + x ) ( 1 + q x ) ( 1 + q m + n - 1 x ) (1+x)(1+qx)\cdots\left(1+q^{m+n-1}x\right)
  9. ( 1 + x ) ( 1 + q x ) ( 1 + q m + n - 1 x ) = k q k ( k - 1 ) 2 ( m + n k ) q x k . (1+x)(1+qx)\cdots\left(1+q^{m+n-1}x\right)=\sum_{k}q^{\frac{k(k-1)}{2}}{\left(% {{m+n}\atop{k}}\right)}_{\!\!q}x^{k}.
  10. ( 1 + x ) ( 1 + q x ) ( 1 + q m + n - 1 x ) = ( ( 1 + x ) ( 1 + q m - 1 x ) ) ( ( 1 + ( q m x ) ) ( 1 + q ( q m x ) ) ( 1 + q n - 1 ( q m x ) ) ) (1+x)(1+qx)\cdots\left(1+q^{m+n-1}x\right)=\left((1+x)\cdots(1+q^{m-1}x)\right% )\left(\left(1+(q^{m}x)\right)\left(1+q(q^{m}x)\right)\cdots\left(1+q^{n-1}(q^% {m}x)\right)\right)
  11. ( 1 + x ) ( 1 + q x ) ( 1 + q m + n - 1 x ) = ( i q i ( i - 1 ) 2 ( m i ) q x i ) ( i q m i + i ( i - 1 ) 2 ( n i ) q x i ) . (1+x)(1+qx)\cdots\left(1+q^{m+n-1}x\right)=\left(\sum_{i}q^{\frac{i(i-1)}{2}}{% \left({{m}\atop{i}}\right)}_{\!\!q}x^{i}\right)\cdot\left(\sum_{i}q^{mi+\frac{% i(i-1)}{2}}{\left({{n}\atop{i}}\right)}_{\!\!q}x^{i}\right).
  12. k j ( q j ( m - k + j ) + k ( k - 1 ) 2 ( m k - j ) q ( n j ) q ) x k . \sum_{k}\sum_{j}\left(q^{j(m-k+j)+\frac{k(k-1)}{2}}{\left({{m}\atop{k-j}}% \right)}_{\!\!q}{\left({{n}\atop{j}}\right)}_{\!\!q}\right)x^{k}.
  13. x x
  14. ( A + B ) m ( A + B ) n (A+B)^{m}(A+B)^{n}

Qualitative_economics.html

  1. + +\,\!
  2. G D P = f ( M ) GDP=f(M)\,\!
  3. o r or\,\!
  4. d f ( M ) d M > 0. \frac{df(M)}{dM}>0.
  5. - -\,\!
  6. G D P = f ( T ) GDP=f(T)\,\!
  7. o r or\,\!
  8. d f ( T ) d T < 0. \frac{df(T)}{dT}<0.
  9. + +\,\!
  10. - -\,\!
  11. G D P = f ( M , T ) GDP=f(M,T)\,\!
  12. o r or\,\!
  13. f ( M , T ) M > 0 , \frac{\partial f(M,T)}{\partial M}>0,
  14. f ( M , T ) T < 0. \frac{\partial f(M,T)}{\partial T}<0.

Quantal_response_equilibrium.html

  1. P i j = exp ( λ E U i j ( P - i ) ) k exp ( λ E U i k ( P - i ) ) P_{ij}=\frac{\exp(\lambda EU_{ij}(P_{-i}))}{\sum_{k}{\exp(\lambda EU_{ik}(P_{-% i}))}}
  2. P i j P_{ij}
  3. E U i j ( P - i ) ) EU_{ij}(P_{-i}))
  4. P - i P_{-i}

Quantities_of_information.html

  1. p log p p\log p\,
  2. lim p 0 + p log p = 0 \lim_{p\rightarrow 0+}p\log p=0
  3. I ( m ) = log ( 1 p ( m ) ) = - log ( p ( m ) ) I(m)=\log\left(\frac{1}{p(m)}\right)=-\log(p(m))\,
  4. p ( m ) = Pr ( M = m ) p(m)=\mathrm{Pr}(M=m)
  5. M M
  6. M M
  7. m m
  8. H ( M ) = 𝔼 { I ( M ) } = m M p ( m ) I ( m ) = - m M p ( m ) log p ( m ) . H(M)=\mathbb{E}\{I(M)\}=\sum_{m\in M}p(m)I(m)=-\sum_{m\in M}p(m)\log p(m).
  9. 𝔼 { } \mathbb{E}\{\}
  10. p ( m ) = 1 / M p(m)=1/M
  11. H ( M ) = log | M | . H(M)=\log|M|.
  12. H ( p 1 , p 2 , , p k ) = - i = 1 k p i log p i , H(p_{1},p_{2},\ldots,p_{k})=-\sum_{i=1}^{k}p_{i}\log p_{i},
  13. p i 0 p_{i}\geq 0
  14. i = 1 k p i = 1. \sum_{i=1}^{k}p_{i}=1.
  15. H b ( p ) = H ( p , 1 - p ) = - p log p - ( 1 - p ) log ( 1 - p ) . H_{\mbox{b}}~{}(p)=H(p,1-p)=-p\log p-(1-p)\log(1-p).\,
  16. X X
  17. Y Y
  18. X X
  19. Y Y
  20. H ( X , Y ) = 𝔼 X , Y [ - log p ( x , y ) ] = - x , y p ( x , y ) log p ( x , y ) H(X,Y)=\mathbb{E}_{X,Y}[-\log p(x,y)]=-\sum_{x,y}p(x,y)\log p(x,y)\,
  21. X X
  22. Y Y
  23. Y Y
  24. X X
  25. Y = y Y=y
  26. H ( X | y ) = 𝔼 < m t p l > X | Y [ - log p ( x | y ) ] = - x X p ( x | y ) log p ( x | y ) H(X|y)=\mathbb{E}_{<}mtpl>{{X|Y}}[-\log p(x|y)]=-\sum_{x\in X}p(x|y)\log p(x|y)
  27. p ( x | y ) = p ( x , y ) p ( y ) p(x|y)=\frac{p(x,y)}{p(y)}
  28. x x
  29. y y
  30. X X
  31. Y Y
  32. X X
  33. Y Y
  34. H ( X | Y ) = 𝔼 Y { H ( X | y ) } = - y Y p ( y ) x X p ( x | y ) log p ( x | y ) = x , y p ( x , y ) log p ( y ) p ( x , y ) . H(X|Y)=\mathbb{E}_{Y}\{H(X|y)\}=-\sum_{y\in Y}p(y)\sum_{x\in X}p(x|y)\log p(x|% y)=\sum_{x,y}p(x,y)\log\frac{p(y)}{p(x,y)}.
  35. H ( X | Y ) = H ( X , Y ) - H ( Y ) . H(X|Y)=H(X,Y)-H(Y).\,

Quantum_annealing.html

  1. exp ( - Δ / k B T ) \exp{(-\Delta/k_{B}T)}
  2. T T
  3. k B k_{B}
  4. Δ \Delta
  5. Δ \Delta
  6. w w
  7. exp ( - Δ 1 / 2 w / Γ ) \exp{(-\Delta^{1/2}w/\Gamma)}
  8. Γ \Gamma
  9. ( w Δ - 1 / 2 ) (w\ll\Delta^{-1/2})
  10. N N
  11. Δ \Delta
  12. N N
  13. τ \tau
  14. exp ( N 1 / 2 ) \exp{(N^{1/2})}
  15. τ \tau
  16. exp ( N ) \exp{(N)}
  17. O ( N 1 / 2 ) O(N^{1/2})
  18. Δ \Delta
  19. N N

Quantum_cloning.html

  1. U | ψ A | e B = | ψ A | ψ B U|\psi\rangle_{A}|e\rangle_{B}=|\psi\rangle_{A}|\psi\rangle_{B}
  2. U U
  3. | ψ A |\psi\rangle_{A}
  4. | e B |e\rangle_{B}
  5. | ψ A |\psi\rangle_{A}

Quantum_cohomology.html

  1. H 2 ( X ) = H 2 ( X , 𝐙 ) / torsion H_{2}(X)=H_{2}(X,\mathbf{Z})/\mathrm{torsion}
  2. λ = A H 2 ( X ) λ A e A , \lambda=\sum_{A\in H_{2}(X)}\lambda_{A}e^{A},
  3. λ A \lambda_{A}
  4. e A e^{A}
  5. e A e B = e A + B e^{A}e^{B}=e^{A+B}
  6. λ A \lambda_{A}
  7. e A e^{A}
  8. 2 c 1 ( A ) 2c_{1}(A)
  9. c 1 c_{1}
  10. H * ( X ) = H * ( X , 𝐙 ) / torsion H^{*}(X)=H^{*}(X,\mathbf{Z})/\mathrm{torsion}
  11. Q H * ( X , Λ ) = H * ( X ) 𝐙 Λ . QH^{*}(X,\Lambda)=H^{*}(X)\otimes_{\mathbf{Z}}\Lambda.
  12. i a i λ i . \sum_{i}a_{i}\otimes\lambda_{i}.
  13. deg ( a i λ i ) = deg ( a i ) + deg ( λ i ) . \deg(a_{i}\otimes\lambda_{i})=\deg(a_{i})+\deg(\lambda_{i}).
  14. a a 1 a\mapsto a\otimes 1
  15. H 2 ( X ) H_{2}(X)
  16. X ( a * b ) A c = G W 0 , 3 X , A ( a , b , c ) . \int_{X}(a*b)_{A}\smile c=GW_{0,3}^{X,A}(a,b,c).
  17. a * b := A H 2 ( X ) ( a * b ) A e A . a*b:=\sum_{A\in H_{2}(X)}(a*b)_{A}\otimes e^{A}.
  18. Q H * ( X , Λ ) Q H * ( X , Λ ) Q H * ( X , Λ ) QH^{*}(X,\Lambda)\otimes QH^{*}(X,\Lambda)\to QH^{*}(X,\Lambda)
  19. G W 0 , 3 X , 0 ( a , b , c ) = X a b c ; GW_{0,3}^{X,0}(a,b,c)=\int_{X}a\smile b\smile c;
  20. ( a * b ) 0 = a b . (a*b)_{0}=a\smile b.
  21. H 2 ( X ) \ell\in H^{2}(X)
  22. H * ( X ) 𝐙 [ ] / 3 . H^{*}(X)\cong\mathbf{Z}[\ell]/\ell^{3}.
  23. X ( i * j ) 0 k = G W 0 , 3 X , 0 ( i , j , k ) = δ ( i + j + k , 4 ) \int_{X}(\ell^{i}*\ell^{j})_{0}\smile\ell^{k}=GW_{0,3}^{X,0}(\ell^{i},\ell^{j}% ,\ell^{k})=\delta(i+j+k,4)
  24. X ( i * j ) L k = G W 0 , 3 X , L ( i , j , k ) = δ ( i + j + k , 5 ) , \int_{X}(\ell^{i}*\ell^{j})_{L}\smile\ell^{k}=GW_{0,3}^{X,L}(\ell^{i},\ell^{j}% ,\ell^{k})=\delta(i+j+k,5),
  25. * = 2 e 0 + 0 e L = 2 , \ell*\ell=\ell^{2}e^{0}+0e^{L}=\ell^{2},
  26. * 2 = 0 e 0 + 1 e L = e L . \ell*\ell^{2}=0e^{0}+1e^{L}=e^{L}.
  27. e L e^{L}
  28. 6 = 2 c 1 ( L ) 6=2c_{1}(L)
  29. Q H * ( X , 𝐙 [ q ] ) 𝐙 [ , q ] / ( 3 = q ) . QH^{*}(X,\mathbf{Z}[q])\cong\mathbf{Z}[\ell,q]/(\ell^{3}=q).
  30. deg ( a * b ) = deg ( a ) + deg ( b ) \deg(a*b)=\deg(a)+\deg(b)
  31. b * a = ( - 1 ) deg ( a ) deg ( b ) a * b . b*a=(-1)^{\deg(a)\deg(b)}a*b.
  32. 1 H 0 ( X ) 1\in H^{0}(X)
  33. Q H * ( X , Λ ) Q H * ( X , Λ ) R QH^{*}(X,\Lambda)\otimes QH^{*}(X,\Lambda)\to R
  34. i a i λ i , j b j μ j = i , j ( λ i ) 0 ( μ j ) 0 X a i b j . \left\langle\sum_{i}a_{i}\otimes\lambda_{i},\sum_{j}b_{j}\otimes\mu_{j}\right% \rangle=\sum_{i,j}(\lambda_{i})_{0}(\mu_{j})_{0}\int_{X}a_{i}\smile b_{j}.
  35. a * b , c = a , b * c . \langle a*b,c\rangle=\langle a,b*c\rangle.
  36. , \langle,\rangle
  37. , \langle,\rangle
  38. * a : H H H *_{a}:H\otimes H\to H
  39. x * a y , z := n A 1 n ! G W 0 , n + 3 X , A ( x , y , z , a , , a ) . \langle x*_{a}y,z\rangle:=\sum_{n}\sum_{A}\frac{1}{n!}GW_{0,n+3}^{X,A}(x,y,z,a% ,\ldots,a).

Quantum_critical_point.html

  1. k B T k_{B}T

Quantum_finite_automata.html

  1. σ = ( σ 0 , σ 1 , , σ k ) \sigma=(\sigma_{0},\sigma_{1},\cdots,\sigma_{k})
  2. σ i \sigma_{i}
  3. Σ \Sigma
  4. Pr ( σ ) \operatorname{Pr}(\sigma)
  5. Σ = { α } \Sigma=\{\alpha\}
  6. α Σ \alpha\in\Sigma
  7. U α U_{\alpha}
  8. { U α | α Σ } \{U_{\alpha}|\alpha\in\Sigma\}
  9. U α U_{\alpha}
  10. q 0 Q q_{0}\in Q
  11. α β γ \alpha\beta\gamma\cdots
  12. q = U γ U β U α q 0 . q=\cdots U_{\gamma}U_{\beta}U_{\alpha}q_{0}.
  13. U α U_{\alpha}
  14. { U α } \{U_{\alpha}\}
  15. { U α } \{U_{\alpha}\}
  16. { U α } \{U_{\alpha}\}
  17. { U α } \{U_{\alpha}\}
  18. P N \mathbb{C}P^{N}
  19. P N \mathbb{C}P^{N}
  20. P N \mathbb{C}P^{N}
  21. N N
  22. N N
  23. | ψ |\psi\rangle
  24. N N
  25. | ψ P N |\psi\rangle\in\mathbb{C}P^{N}
  26. N N
  27. \|\cdot\|
  28. N × N N\times N
  29. U α U_{\alpha}
  30. α Σ \alpha\in\Sigma
  31. α \alpha
  32. | ψ |\psi\rangle
  33. | ψ |\psi^{\prime}\rangle
  34. | ψ = U α | ψ |\psi^{\prime}\rangle=U_{\alpha}|\psi\rangle
  35. ( P N , Σ , { U α | α Σ } ) (\mathbb{C}P^{N},\Sigma,\{U_{\alpha}|\alpha\in\Sigma\})
  36. N × N N\times N
  37. P P
  38. N N
  39. | ψ |\psi\rangle
  40. | ψ |\psi\rangle
  41. ψ | P | ψ = P | ψ 2 \langle\psi|P|\psi\rangle=\|P|\psi\rangle\|^{2}
  42. σ = ( σ 0 , σ 1 , , σ k ) \sigma=(\sigma_{0},\sigma_{1},\cdots,\sigma_{k})
  43. Pr ( σ ) = P U σ k U σ 1 U σ 0 | ψ 2 \operatorname{Pr}(\sigma)=\|PU_{\sigma_{k}}\cdots U_{\sigma_{1}}U_{\sigma_{0}}% |\psi\rangle\|^{2}
  44. | ψ |\psi\rangle
  45. \varnothing
  46. Pr ( ) = P | ψ 2 \operatorname{Pr}(\varnothing)=\|P|\psi\rangle\|^{2}
  47. U α U_{\alpha}
  48. | ψ |\psi\rangle
  49. σ \sigma
  50. p p
  51. σ \sigma
  52. | ψ |\psi\rangle
  53. p < Pr ( σ ) p<\operatorname{Pr}(\sigma)
  54. | ψ = a 1 | S 1 + a 2 | S 2 = [ a 1 a 2 ] |\psi\rangle=a_{1}|S_{1}\rangle+a_{2}|S_{2}\rangle=\begin{bmatrix}a_{1}\\ a_{2}\end{bmatrix}
  55. a 1 , a 2 a_{1},a_{2}
  56. [ a 1 * a 2 * ] [ a 1 a 2 ] = a 1 * a 1 + a 2 * a 2 = 1 \begin{bmatrix}a^{*}_{1}\;\;a^{*}_{2}\end{bmatrix}\begin{bmatrix}a_{1}\\ a_{2}\end{bmatrix}=a_{1}^{*}a_{1}+a_{2}^{*}a_{2}=1
  57. U 0 = [ 0 1 1 0 ] U_{0}=\begin{bmatrix}0&1\\ 1&0\end{bmatrix}
  58. U 1 = [ 1 0 0 1 ] U_{1}=\begin{bmatrix}1&0\\ 0&1\end{bmatrix}
  59. S 1 S_{1}
  60. P = [ 1 0 0 0 ] P=\begin{bmatrix}1&0\\ 0&0\end{bmatrix}
  61. | S 1 |S_{1}\rangle
  62. | S 2 |S_{2}\rangle
  63. ( 1 * ( 01 * 0 ) * ) * (1^{*}(01^{*}0)^{*})^{*}\,\!
  64. a 1 a_{1}
  65. a 2 a_{2}
  66. U 0 U_{0}
  67. U 1 U_{1}
  68. Q \mathcal{H}_{Q}
  69. Q = accept reject non-halting \mathcal{H}_{Q}=\mathcal{H}_{\mbox{accept}}~{}\oplus\mathcal{H}_{\mbox{reject}% }~{}\oplus\mathcal{H}_{\mbox{non-halting}}~{}
  70. Q Q
  71. Q \mathcal{H}_{Q}
  72. Q acc Q Q_{\mbox{acc}}~{}\subset Q
  73. Q rej Q Q_{\mbox{rej}}~{}\subset Q
  74. accept = span { | q : | q Q acc } \mathcal{H}_{\mbox{accept}}~{}=\operatorname{span}\{|q\rangle:|q\rangle\in Q_{% \mbox{acc}}~{}\}
  75. P acc P_{\mbox{acc}}~{}
  76. P rej P_{\mbox{rej}}~{}
  77. P non P_{\mbox{non}}~{}
  78. P acc : Q accept P\mbox{acc}~{}:\mathcal{H}_{Q}\to\mathcal{H}_{\mbox{accept}}~{}
  79. | ψ |\psi\rangle
  80. α \alpha
  81. | ψ = U α | ψ |\psi^{\prime}\rangle=U_{\alpha}|\psi\rangle
  82. | ψ |\psi^{\prime}\rangle
  83. P P
  84. accept \mathcal{H}_{\mbox{accept}}~{}
  85. reject \mathcal{H}_{\mbox{reject}}~{}
  86. non-halting \mathcal{H}_{\mbox{non-halting}}~{}
  87. Pr acc ( σ ) = P acc | ψ 2 \operatorname{Pr}_{\mbox{acc}}~{}(\sigma)=\|P_{\mbox{acc}}~{}|\psi^{\prime}% \rangle\|^{2}
  88. P non P_{\mbox{non}}~{}
  89. κ \kappa
  90. ( Q ; Σ ; δ ; q 0 ; Q acc ; Q rej ) (Q;\Sigma;\delta;q_{0};Q_{\mbox{acc}}~{};Q_{\mbox{rej}}~{})
  91. Q Q
  92. Σ \Sigma
  93. Q acc Q\mbox{acc}~{}
  94. Q rej Q\mbox{rej}~{}
  95. | ψ = | q 0 |\psi\rangle=|q_{0}\rangle
  96. δ \delta
  97. δ : Q × Σ × Q \delta:Q\times\Sigma\times Q\to\mathbb{C}
  98. U α | q 1 = q 2 Q δ ( q 1 , α , q 2 ) | q 2 U_{\alpha}|q_{1}\rangle=\sum_{q_{2}\in Q}\delta(q_{1},\alpha,q_{2})|q_{2}\rangle
  99. P N \mathbb{C}P^{N}
  100. P r = | P | ψ | 2 {Pr}=|\langle P|\psi\rangle|^{2}
  101. | P |P\rangle
  102. | ψ |\psi\rangle
  103. P N \mathbb{C}P^{N}
  104. P N \mathbb{C}P^{N}

Quantum_Markov_chain.html

  1. ( E , ρ ) (E,\rho)
  2. ρ \rho
  3. E E
  4. E : E:\mathcal{B}\otimes\mathcal{B}\to\mathcal{B}
  5. \mathcal{B}
  6. Tr ρ ( b 1 b 2 ) = Tr ρ E ( b 1 , b 2 ) \operatorname{Tr}\rho(b_{1}\otimes b_{2})=\operatorname{Tr}\rho E(b_{1},b_{2})
  7. b 1 , b 2 b_{1},b_{2}\in\mathcal{B}

Quantum_mutual_information.html

  1. p ( x ) = y p ( x , y ) , p ( y ) = x p ( x , y ) . p(x)=\sum_{y}p(x,y)\;,\;p(y)=\sum_{x}p(x,y).
  2. I ( X , Y ) = S ( p ( x ) ) + S ( p ( y ) ) - S ( p ( x , y ) ) \;I(X,Y)=S(p(x))+S(p(y))-S(p(x,y))
  3. S ( p ( x ) ) + S ( p ( y ) ) \;S(p(x))+S(p(y))
  4. = - ( x p x log p ( x ) + y p y log p ( y ) ) \;=-(\sum_{x}p_{x}\log p(x)+\sum_{y}p_{y}\log p(y))
  5. = - ( x ( y p ( x , y ) log y p ( x , y ) ) + y ( x p ( x , y ) log x p ( x , y ) ) ) \;=-(\sum_{x}\;(\sum_{y^{\prime}}p(x,y^{\prime})\log\sum_{y^{\prime}}p(x,y^{% \prime}))+\sum_{y}(\sum_{x^{\prime}}p(x^{\prime},y)\log\sum_{x^{\prime}}p(x^{% \prime},y)))
  6. = - ( x , y p ( x , y ) ( log y p ( x , y ) + log x p ( x , y ) ) ) \;=-(\sum_{x,y}p(x,y)(\log\sum_{y^{\prime}}p(x,y^{\prime})+\log\sum_{x^{\prime% }}p(x^{\prime},y)))
  7. = - x , y p ( x , y ) log p ( x ) p ( y ) . \;=-\sum_{x,y}p(x,y)\log p(x)p(y).
  8. I ( X , Y ) = x , y p ( x , y ) log p ( x , y ) p ( x ) p ( y ) . I(X,Y)=\sum_{x,y}p(x,y)\log\frac{p(x,y)}{p(x)p(y)}.
  9. H = H A H B . H=H_{A}\otimes H_{B}.
  10. S ( ρ A B ) = - Tr ρ A B log ρ A B . S(\rho^{AB})=-\operatorname{Tr}\rho^{AB}\log\rho^{AB}.
  11. ρ A = Tr B ρ A B \rho^{A}=\operatorname{Tr}_{B}\;\rho^{AB}
  12. S ( ρ A ) . \;S(\rho^{A}).
  13. I ( ρ A B ) = S ( ρ A ) + S ( ρ B ) - S ( ρ A B ) . \;I(\rho^{AB})=S(\rho^{A})+S(\rho^{B})-S(\rho^{AB}).
  14. I ( ρ A B ) = S ( ρ A B ρ A ρ B ) I(\rho^{AB})=S(\rho^{AB}\|\rho^{A}\otimes\rho^{B})
  15. S ( ) S(\cdot\|\cdot)

Quantum_potential.html

  1. Q = - 2 2 m 2 R R \quad Q=-\frac{\hbar^{2}}{2m}\frac{\nabla^{2}R}{R}
  2. i ψ t = ( - 2 2 m 2 + V ) ψ i\hbar\frac{\partial\psi}{\partial t}=\left(-\frac{\hbar^{2}}{2m}\nabla^{2}+V% \right)\psi\quad
  3. ψ = R exp i S / \quad\psi=R\exp iS/\hbar
  4. R R
  5. S S
  6. R R
  7. ψ \psi
  8. S / S/\hbar
  9. R t = - 1 2 m [ R 2 S + 2 R S ] , \frac{\partial R}{\partial t}=-\frac{1}{2m}\left[R\nabla^{2}S+2\nabla R\cdot% \nabla S\right]\;,
  10. ρ = R 2 \rho=R^{2}
  11. ρ / t + ( ρ v ) = 0 \partial\rho/\partial t+\nabla\cdot(\rho v)=0
  12. ρ \rho
  13. S t = - [ ( S ) 2 2 m + V + Q ] , \frac{\partial S}{\partial t}=-\left[\frac{\left(\nabla S\right)^{2}}{2m}+V+Q% \right]\;,
  14. Q Q
  15. \hbar
  16. S S
  17. S S
  18. R R
  19. Q Q
  20. ψ \psi
  21. ψ ( 𝐫 𝟏 , 𝐫 𝟐 , t ) \psi(\mathbf{r_{1}},\mathbf{r_{2}},\,t)
  22. m m
  23. Q ( 𝐫 𝟏 , 𝐫 𝟐 , t ) = - 2 2 m ( 1 2 + 2 2 ) R ( 𝐫 𝟏 , 𝐫 𝟐 , t ) R ( 𝐫 𝟏 , 𝐫 𝟐 , t ) \quad Q(\mathbf{r_{1}},\mathbf{r_{2}},\,t)=-\frac{\hbar^{2}}{2m}\frac{(\nabla_% {1}^{2}+\nabla_{2}^{2})R(\mathbf{r_{1}},\mathbf{r_{2}},\,t)}{R(\mathbf{r_{1}},% \mathbf{r_{2}},\,t)}
  24. 1 2 \nabla_{1}^{2}
  25. 2 2 \nabla_{2}^{2}
  26. n n
  27. Q ( 𝐫 𝟏 , , 𝐫 𝐧 , t ) = - 2 2 i = 1 n i 2 m i \quad Q(\mathbf{r_{1}},...,\mathbf{r_{n}},\,t)=-\frac{\hbar^{2}}{2}\sum_{i=1}^% {n}\frac{\nabla_{i}^{2}}{m_{i}}
  28. ψ \psi
  29. ψ ( 𝐫 𝟏 , 𝐫 𝟐 , t ) = ψ A ( 𝐫 𝟏 , t ) ψ B ( 𝐫 𝟐 , t ) \psi(\mathbf{r_{1}},\mathbf{r_{2}},\,t)=\psi_{A}(\mathbf{r_{1}},\,t)\psi_{B}(% \mathbf{r_{2}},\,t)
  30. R R
  31. Q ( 𝐫 𝟏 , 𝐫 𝟐 , t ) = - 2 2 m ( 1 2 R A ( 𝐫 𝟏 , t ) R A ( 𝐫 𝟏 , t ) + 2 2 R B ( 𝐫 𝟐 , t ) R B ( 𝐫 𝟐 , t ) ) = Q A ( 𝐫 𝟏 , t ) + Q B ( 𝐫 𝟐 , t ) Q(\mathbf{r_{1}},\mathbf{r_{2}},\,t)=-\frac{\hbar^{2}}{2m}(\frac{\nabla_{1}^{2% }R_{A}(\mathbf{r_{1}},\,t)}{R_{A}(\mathbf{r_{1}},\,t)}+\frac{\nabla_{2}^{2}R_{% B}(\mathbf{r_{2}},\,t)}{R_{B}(\mathbf{r_{2}},\,t)})=Q_{A}(\mathbf{r_{1}},\,t)+% Q_{B}(\mathbf{r_{2}},\,t)
  32. ψ \psi
  33. ψ ( 𝐫 𝟏 , 𝐫 𝟐 , t ) = ψ A ( 𝐫 𝟏 , t ) ψ B ( 𝐫 𝟐 , t ) \psi(\mathbf{r_{1}},\mathbf{r_{2}},\,t)=\psi_{A}(\mathbf{r_{1}},\,t)\psi_{B}(% \mathbf{r_{2}},\,t)
  34. n n
  35. n n
  36. n n
  37. R R
  38. ρ = R 2 \rho=R^{2}\quad
  39. 2 ρ = ρ 1 / 2 = ( 1 2 ρ - 1 / 2 ρ ) = 1 2 ( ρ - 1 / 2 ρ ) = 1 2 [ ( ρ - 1 / 2 ) ρ + ρ - 1 / 2 2 ρ ] \nabla^{2}\sqrt{\rho}=\nabla\nabla\rho^{1/2}=\nabla(\frac{1}{2}\rho^{-1/2}% \nabla\rho)=\frac{1}{2}\nabla(\rho^{-1/2}\nabla\rho)=\frac{1}{2}\left[(\nabla% \rho^{-1/2})\nabla\rho+\rho^{-1/2}\nabla^{2}\rho\right]
  40. Q = - 2 2 m 2 ρ ρ = - 2 4 m [ 2 ρ ρ - 1 2 ( ρ ) 2 ρ 2 ] Q=-\frac{\hbar^{2}}{2m}\frac{\nabla^{2}\sqrt{\rho}}{\sqrt{\rho}}=-\frac{\hbar^% {2}}{4m}\left[\frac{\nabla^{2}\rho}{\rho}-\frac{1}{2}\frac{(\nabla\rho)^{2}}{% \rho^{2}}\right]
  41. F Q = - Q F_{Q}=-\nabla Q
  42. F Q = 2 4 m [ 3 ρ ρ - ( ρ ρ ) 2 ρ 2 - ( 2 ρ ρ - ρ ρ ρ 2 ) ρ ρ ] F_{Q}=\frac{\hbar^{2}}{4m}\left[\frac{\nabla^{3}\rho}{\rho}-\frac{\nabla(% \nabla\rho\cdot\nabla\rho)}{2\rho^{2}}-\left(\frac{\nabla^{2}\rho}{\rho}-\frac% {\nabla\rho\cdot\nabla\rho}{\rho^{2}}\right)\frac{\nabla\rho}{\rho}\right]
  43. x x
  44. p p
  45. x x
  46. x x
  47. x x
  48. p p
  49. Q = - 2 x 2 ρ / ( 2 m ρ ) Q=-\hbar^{2}\,\partial^{2}_{x}{\sqrt{\rho}}/(2m{\sqrt{\rho}})
  50. = ρ ( x ln ρ ) 2 d x = - ρ x 2 ln ρ d x \mathcal{I}=\int\rho\,(\partial_{x}\ln\rho)^{2}dx=-\int\rho\,\partial^{2}_{x}% \ln\rho\,dx
  51. ρ Q d x = 2 8 m \int\rho\,Q\,dx=\frac{\hbar^{2}}{8m}\mathcal{I}
  52. E W [ ρ ] = d r ρ 2 ( ln ρ ) 2 / 8 m = ( 2 / 8 m ) d r ( ρ ) 2 / ρ = d r ρ Q E_{W}[\rho]=\int dr\rho\hbar^{2}(\nabla\ln\rho)^{2}/8m=(\hbar^{2}/8m)\int dr(% \nabla\rho)^{2}/\rho=\int dr\rho\,Q
  53. 𝐕 2 = ( ρ and 𝐬 ) 2 ( m ρ ) 2 = ( ρ ) 2 𝐬 2 - ( ρ 𝐬 ) ( m ρ ) 2 \mathbf{V}^{2}=\frac{(\nabla\rho\and\mathbf{s})^{2}}{(m\rho)^{2}}=\frac{(% \nabla\rho)^{2}\mathbf{s}^{2}-(\nabla\rho\cdot\mathbf{s})}{(m\rho)^{2}}
  54. | 𝐬 | = / 2 |\mathbf{s}|=\hbar/2
  55. | 𝐕 | = 2 ρ m ρ |\mathbf{V}|=\frac{\hbar}{2}\frac{\nabla\rho}{m\rho}
  56. \hbar
  57. Q = - 1 2 m 𝐯 S 2 - 1 2 𝐯 S Q=-\frac{1}{2}m\mathbf{v}_{S}^{2}-\frac{1}{2}\nabla\cdot\mathbf{v}_{S}
  58. 𝐯 = 𝐯 B + 𝐯 S × 𝐬 \mathbf{v}=\mathbf{v}_{B}+\mathbf{v}_{S}\times\mathbf{s}
  59. 𝐯 B = S m \mathbf{v}_{B}=\frac{\nabla S}{m}
  60. 𝐯 S × 𝐬 \mathbf{v}_{S}\times\mathbf{s}
  61. 𝐯 S = R 2 2 m R 2 \mathbf{v}_{S}=\frac{\nabla R^{2}}{2mR^{2}}
  62. 𝐬 \mathbf{s}
  63. ψ = ( 𝐄 - i 𝐁 ) / 2 \psi=(\mathbf{E}-i\mathbf{B})/\sqrt{2}
  64. ψ * ψ = ( 𝐄 2 + 𝐁 2 ) / 2 \psi^{*}\cdot\psi=(\mathbf{E}^{2}+\mathbf{B}^{2})/2
  65. Q ( q ) = 2 4 m { S ; q } Q(q)=\frac{\hbar^{2}}{4m}\{S;q\}
  66. { ; } \{\,;\,\}
  67. { S ; q } = ( S ′′′ / S ) - ( 3 / 2 ) ( S ′′ / S ) 2 \{S;q\}=(S\,^{\prime\prime\prime}/S\,^{\prime})-(3/2)(S\,^{\prime\prime}/S\,^{% \prime})^{2}
  68. Q ( q ) = - 2 2 m Δ R R Q(q)=-\frac{\hbar^{2}}{2m}\frac{\Delta R}{R}
  69. R e x p ( i S / ) R\,exp(iS/\hbar)
  70. \hbar
  71. C i , j C\ell_{i,j}
  72. Φ L ( 𝐫 , t ) \Phi_{L}(\mathbf{r},t)
  73. Φ R ( 𝐫 , t ) = Φ ~ L ( 𝐫 , t ) \Phi_{R}(\mathbf{r},t)=\tilde{\Phi}_{L}(\mathbf{r},t)
  74. ρ c ( 𝐫 , t ) = Φ L ( 𝐫 , t ) Φ ~ L ( 𝐫 , t ) \rho_{c}(\mathbf{r},t)=\Phi_{L}(\mathbf{r},t)\tilde{\Phi}_{L}(\mathbf{r},t)
  75. B B
  76. Tr B ρ c {\rm Tr}B\rho_{c}
  77. \scriptstyle\Box
  78. = 1 \hbar=1
  79. Q = - 1 2 m ρ ρ Q=-\frac{1}{2m}\frac{\quad\Box\sqrt{\rho}}{\sqrt{\rho}}
  80. R / R \Box R/R
  81. ψ ( 𝐱 , t ) \psi(\mathbf{x},t)
  82. d 3 x d^{3}x
  83. t t
  84. Q = - ( 1 / 2 m ) R / R Q=-(1/2m)\,\Box R/R
  85. | ψ | 2 |\psi|^{2}
  86. Ψ [ ψ ( 𝐱 , t ) ] = R [ ψ ( 𝐱 , t ) ] e S [ ψ ( 𝐱 , t ) ] \Psi\left[\psi(\mathbf{x},t)\right]=R\left[\psi(\mathbf{x},t)\right]e^{S\left[% \psi(\mathbf{x},t)\right]}
  87. R [ ψ ( 𝐱 , t ) ] , S [ ψ ( 𝐱 , t ) ] R\left[\psi(\mathbf{x},t)\right],S\left[\psi(\mathbf{x},t)\right]
  88. Q [ ψ ( 𝐱 , t ) ] = - ( 1 / 2 R ) d 3 x δ 2 R / δ ψ 2 Q\left[\psi(\mathbf{x},t)\right]=-(1/2R)\int d^{3}x\,\delta^{2}R/\delta\psi^{2}
  89. V V
  90. Q Q

Quantum_relative_entropy.html

  1. - log q j . \;-\log q_{j}.
  2. - j p j log q j . \;-\sum_{j}p_{j}\log q_{j}.
  3. - j p j log p j , \;-\sum_{j}p_{j}\log p_{j},
  4. - j p j log q j - ( - j p j log p j ) = j p j log p j - j p j log q j \;-\sum_{j}p_{j}\log q_{j}-\left(-\sum_{j}p_{j}\log p_{j}\right)=\sum_{j}p_{j}% \log p_{j}-\sum_{j}p_{j}\log q_{j}
  5. D KL ( P Q ) = j p j log p j q j . D_{\mathrm{KL}}(P\|Q)=\sum_{j}p_{j}\log\frac{p_{j}}{q_{j}}\!.
  6. S ( ρ ) = - Tr ρ log ρ . S(\rho)=-\operatorname{Tr}\rho\log\rho.
  7. S ( ρ σ ) = - Tr ρ log σ - S ( ρ ) = Tr ρ log ρ - Tr ρ log σ = Tr ρ ( log ρ - log σ ) . S(\rho\|\sigma)=-\operatorname{Tr}\rho\log\sigma-S(\rho)=\operatorname{Tr}\rho% \log\rho-\operatorname{Tr}\rho\log\sigma=\operatorname{Tr}\rho(\log\rho-\log% \sigma).
  8. S ( ρ σ ) = S(\rho\|\sigma)=\infty
  9. supp ( ρ ) ker ( σ ) 0. \,\text{supp}(\rho)\cap\,\text{ker}(\sigma)\neq 0.
  10. D KL ( P Q ) = j p j log p j q j 0 , D_{\mathrm{KL}}(P\|Q)=\sum_{j}p_{j}\log\frac{p_{j}}{q_{j}}\geq 0,
  11. D KL ( P Q ) = j p j log p j q j = j ( - log q j p j ) ( p j ) . D_{\mathrm{KL}}(P\|Q)=\sum_{j}p_{j}\log\frac{p_{j}}{q_{j}}=\sum_{j}(-\log\frac% {q_{j}}{p_{j}})(p_{j}).
  12. D KL ( P Q ) = j ( - log q j p j ) ( p j ) - log ( j q j p j p j ) = 0. D_{\mathrm{KL}}(P\|Q)=\sum_{j}(-\log\frac{q_{j}}{p_{j}})(p_{j})\geq-\log(\sum_% {j}\frac{q_{j}}{p_{j}}p_{j})=0.
  13. S ( ρ σ ) = Tr ρ ( log ρ - log σ ) . S(\rho\|\sigma)=\operatorname{Tr}\rho(\log\rho-\log\sigma).
  14. ρ = i p i v i v i * , σ = i q i w i w i * . \rho=\sum_{i}p_{i}v_{i}v_{i}^{*}\;,\;\sigma=\sum_{i}q_{i}w_{i}w_{i}^{*}.
  15. log ρ = i ( log p i ) v i v i * , log σ = i ( log q i ) w i w i * . \log\rho=\sum_{i}(\log p_{i})v_{i}v_{i}^{*}\;,\;\log\sigma=\sum_{i}(\log q_{i}% )w_{i}w_{i}^{*}.
  16. S ( ρ σ ) S(\rho\|\sigma)
  17. = k p k log p k - i , j ( p i log q j ) | v i * w j | 2 =\sum_{k}p_{k}\log p_{k}-\sum_{i,j}(p_{i}\log q_{j})|v_{i}^{*}w_{j}|^{2}
  18. = i p i ( log p i - j log q j | v i * w j | 2 ) =\sum_{i}p_{i}(\log p_{i}-\sum_{j}\log q_{j}|v_{i}^{*}w_{j}|^{2})
  19. = i p i ( log p i - j ( log q j ) P i j ) , \;=\sum_{i}p_{i}(\log p_{i}-\sum_{j}(\log q_{j})P_{ij}),
  20. i p i ( log p i - log ( j q j P i j ) ) \geq\sum_{i}p_{i}(\log p_{i}-\log(\sum_{j}q_{j}P_{ij}))
  21. = i p i ( log p i - log ( j q j P i j ) ) . \;=\sum_{i}p_{i}(\log p_{i}-\log(\sum_{j}q_{j}P_{ij})).
  22. S ( ρ σ ) i p i log p i r i 0. S(\rho\|\sigma)\geq\sum_{i}p_{i}\log\frac{p_{i}}{r_{i}}\geq 0.
  23. i p i ( log p i - j ( log q j ) P i j ) i p i ( log p i - log ( j q j P i j ) ) \sum_{i}p_{i}(\log p_{i}-\sum_{j}(\log q_{j})P_{ij})\geq\sum_{i}p_{i}(\log p_{% i}-\log(\sum_{j}q_{j}P_{ij}))
  24. H = k H k H=\otimes_{k}H_{k}
  25. D REE ( ρ ) = min σ S ( ρ σ ) \;D_{\mathrm{REE}}(\rho)=\min_{\sigma}S(\rho\|\sigma)
  26. D REE ( ρ ) = 0 \;D_{\mathrm{REE}}(\rho)=0
  27. S ( ρ A | | I A / n A ) = log ( n A ) - S ( ρ A ) , S(\rho_{A}||I_{A}/n_{A})=\mathrm{log}(n_{A})-S(\rho_{A}),\;
  28. S ( ρ A B | | ρ A ρ B ) = S ( ρ A ) + S ( ρ B ) - S ( ρ A B ) = I ( A : B ) , S(\rho_{AB}||\rho_{A}\otimes\rho_{B})=S(\rho_{A})+S(\rho_{B})-S(\rho_{AB})=I(A% :B),
  29. S ( ρ A B | | ρ A I B / n B ) = log ( n B ) + S ( ρ A ) - S ( ρ A B ) = log ( n B ) - S ( B | A ) , S(\rho_{AB}||\rho_{A}\otimes I_{B}/n_{B})=\mathrm{log}(n_{B})+S(\rho_{A})-S(% \rho_{AB})=\mathrm{log}(n_{B})-S(B|A),

Quantum_vortex.html

  1. C 𝐯 d 𝐥 = m C ϕ d 𝐥 = m Δ ϕ , \oint_{C}\mathbf{v}\cdot\,d\mathbf{l}=\frac{\hbar}{m}\oint_{C}\nabla\phi\cdot% \,d\mathbf{l}=\frac{\hbar}{m}\Delta\phi,
  2. \hbar
  3. 2 π 2\pi
  4. Δ ϕ \Delta\phi
  5. Δ ϕ = 2 π n \Delta\phi=2\pi n
  6. C 𝐯 d 𝐥 = 2 π m n . \oint_{C}\mathbf{v}\cdot\,d\mathbf{l}=\frac{2\pi\hbar}{m}n.
  7. Φ = S 𝐁 𝐧 ^ d 2 x = S 𝐀 d 𝐥 . \Phi=\oint_{S}\mathbf{B}\cdot\mathbf{\hat{n}}\,d^{2}x=\oint_{\partial S}% \mathbf{A}\cdot d\mathbf{l}.
  8. 𝐣 s = - n s e s 2 m 𝐀 + n s e s m ϕ \mathbf{j}_{s}=-\frac{n_{s}e_{s}^{2}}{m}\mathbf{A}+\frac{n_{s}e_{s}\hbar}{m}% \mathbf{\nabla}\phi
  9. Φ = - m n s e 2 S 𝐣 s d 𝐥 + e s S ϕ d 𝐥 \Phi=-\frac{m}{n_{s}e^{2}}\oint_{\partial S}\mathbf{j}_{s}\cdot d\mathbf{l}+% \frac{\hbar}{e_{s}}\oint_{\partial S}\mathbf{\nabla}\phi\cdot d\mathbf{l}
  10. 𝐣 s = 0 \mathbf{j}_{s}=0
  11. S \partial S
  12. Φ = e s S ϕ d 𝐥 = e s Δ ϕ = 2 π e s n . \Phi=\frac{\hbar}{e_{s}}\oint_{\partial S}\mathbf{\nabla}\phi\cdot d\mathbf{l}% =\frac{\hbar}{e_{s}}\Delta\phi=\frac{2\pi\hbar}{e_{s}}n.

Quasi-Frobenius_Lie_algebra.html

  1. ( 𝔤 , [ , ] , β ) (\mathfrak{g},[\,\,\,,\,\,\,],\beta)
  2. k k
  3. ( 𝔤 , [ , ] ) (\mathfrak{g},[\,\,\,,\,\,\,])
  4. β : 𝔤 × 𝔤 k \beta:\mathfrak{g}\times\mathfrak{g}\to k
  5. 𝔤 \mathfrak{g}
  6. k k
  7. β ( [ X , Y ] , Z ) + β ( [ Z , X ] , Y ) + β ( [ Y , Z ] , X ) = 0 \beta\left(\left[X,Y\right],Z\right)+\beta\left(\left[Z,X\right],Y\right)+% \beta\left(\left[Y,Z\right],X\right)=0
  8. X X
  9. Y Y
  10. Z Z
  11. 𝔤 \mathfrak{g}
  12. β \beta
  13. f : 𝔤 k f:\mathfrak{g}\to k
  14. β ( X , Y ) = f ( [ X , Y ] ) , \beta(X,Y)=f(\left[X,Y\right]),
  15. ( 𝔤 , [ , ] , β ) (\mathfrak{g},[\,\,\,,\,\,\,],\beta)
  16. ( 𝔤 , [ , ] , β ) (\mathfrak{g},[\,\,\,,\,\,\,],\beta)
  17. 𝔤 \mathfrak{g}
  18. \triangleleft
  19. β ( [ X , Y ] , Z ) = β ( Z Y , X ) \beta\left(\left[X,Y\right],Z\right)=\beta\left(Z\triangleleft Y,X\right)
  20. [ X , Y ] = X Y - Y X \left[X,Y\right]=X\triangleleft Y-Y\triangleleft X
  21. ( 𝔤 , ) (\mathfrak{g},\triangleleft)

Quasi-Newton_method.html

  1. g g
  2. x n + 1 = x n - [ J g ( x n ) ] - 1 g ( x n ) x_{n+1}=x_{n}-[J_{g}(x_{n})]^{-1}g(x_{n})\,\!
  3. [ J g ( x n ) ] - 1 [J_{g}(x_{n})]^{-1}
  4. J g ( x n ) J_{g}(x_{n})
  5. g g
  6. x n x_{n}
  7. J g ( x n ) J_{g}(x_{n})
  8. J g ( x n ) J_{g}(x_{n})
  9. J g ( x o ) J_{g}(x_{o})
  10. g g
  11. f f
  12. g g
  13. f f
  14. g g
  15. f f
  16. B B
  17. B - 1 B^{-1}
  18. f ( x ) f(x)
  19. f ( x ) f(x)
  20. f ( x k + Δ x ) f ( x k ) + f ( x k ) T Δ x + 1 2 Δ x T B Δ x , f(x_{k}+\Delta x)\approx f(x_{k})+\nabla f(x_{k})^{T}\Delta x+\frac{1}{2}% \Delta x^{T}{B}\,\Delta x,
  21. f \nabla f
  22. B B
  23. Δ x \Delta x
  24. f ( x k + Δ x ) f ( x k ) + B Δ x \nabla f(x_{k}+\Delta x)\approx\nabla f(x_{k})+B\,\Delta x
  25. Δ x = - B - 1 f ( x k ) , \Delta x=-B^{-1}\nabla f(x_{k}),\,
  26. B B
  27. f ( x k + Δ x ) = f ( x k ) + B Δ x , \nabla f(x_{k}+\Delta x)=\nabla f(x_{k})+B\,\Delta x,
  28. B B
  29. B B
  30. B T = B B^{T}=B
  31. B k + 1 B_{k+1}
  32. B k B_{k}
  33. B k + 1 = argmin B B - B k V B_{k+1}=\textrm{argmin}_{B}\|B-B_{k}\|_{V}
  34. V V
  35. B 0 = I * x B_{0}=I*x
  36. B 0 B_{0}
  37. x k x_{k}
  38. B k B_{k}
  39. Δ x k = - α k B k - 1 f ( x k ) \Delta x_{k}=-\alpha_{k}B_{k}^{-1}\nabla f(x_{k})
  40. α \alpha
  41. x k + 1 = x k + Δ x k x_{k+1}=x_{k}+\Delta x_{k}
  42. f ( x k + 1 ) \nabla f(x_{k+1})
  43. y k = f ( x k + 1 ) - f ( x k ) , y_{k}=\nabla f(x_{k+1})-\nabla f(x_{k}),
  44. B k + 1 \displaystyle B_{k+1}
  45. H k + 1 = B k + 1 - 1 \displaystyle H_{k+1}=B_{k+1}^{-1}
  46. B k B_{k}
  47. α k \alpha_{k}
  48. B k + 1 \displaystyle B_{k+1}
  49. B k + 1 = \displaystyle B_{k+1}=
  50. H k + 1 = B k + 1 - 1 = H_{k+1}=B_{k+1}^{-1}=
  51. ( I - y k Δ x k T y k T Δ x k ) B k ( I - Δ x k y k T y k T Δ x k ) + y k y k T y k T Δ x k \left(I-\frac{y_{k}\,\Delta x_{k}^{T}}{y_{k}^{T}\,\Delta x_{k}}\right)B_{k}% \left(I-\frac{\Delta x_{k}y_{k}^{T}}{y_{k}^{T}\,\Delta x_{k}}\right)+\frac{y_{% k}y_{k}^{T}}{y_{k}^{T}\,\Delta x_{k}}
  52. H k + Δ x k Δ x k T y k T Δ x k - H k y k y k T H k T y k T H k y k H_{k}+\frac{\Delta x_{k}\Delta x_{k}^{T}}{y_{k}^{T}\,\Delta x_{k}}-\frac{H_{k}% y_{k}y_{k}^{T}H_{k}^{T}}{y_{k}^{T}H_{k}y_{k}}
  53. B k + y k y k T y k T Δ x k - B k Δ x k ( B k Δ x k ) T Δ x k T B k Δ x k B_{k}+\frac{y_{k}y_{k}^{T}}{y_{k}^{T}\Delta x_{k}}-\frac{B_{k}\Delta x_{k}(B_{% k}\Delta x_{k})^{T}}{\Delta x_{k}^{T}B_{k}\,\Delta x_{k}}
  54. ( I - Δ x k y k T y k T Δ x k ) T H k ( I - y k Δ x k T y k T Δ x k ) + Δ x k Δ x k T y k T Δ x k \left(I-\frac{\Delta x_{k}y_{k}^{T}}{y_{k}^{T}\Delta x_{k}}\right)^{T}H_{k}% \left(I-\frac{y_{k}\Delta x_{k}^{T}}{y_{k}^{T}\Delta x_{k}}\right)+\frac{% \Delta x_{k}\Delta x_{k}^{T}}{y_{k}^{T}\,\Delta x_{k}}
  55. B k + y k - B k Δ x k Δ x k T Δ x k Δ x k T B_{k}+\frac{y_{k}-B_{k}\Delta x_{k}}{\Delta x_{k}^{T}\,\Delta x_{k}}\,\Delta x% _{k}^{T}
  56. H k + ( Δ x k - H k y k ) Δ x k T H k Δ x k T H k y k H_{k}+\frac{(\Delta x_{k}-H_{k}y_{k})\Delta x_{k}^{T}H_{k}}{\Delta x_{k}^{T}H_% {k}\,y_{k}}
  57. ( 1 - φ k ) B k + 1 B F G S + φ k B k + 1 D F P , φ [ 0 , 1 ] (1-\varphi_{k})B_{k+1}^{BFGS}+\varphi_{k}B_{k+1}^{DFP},\qquad\varphi\in[0,1]
  58. B k + ( y k - B k Δ x k ) ( y k - B k Δ x k ) T ( y k - B k Δ x k ) T Δ x k B_{k}+\frac{(y_{k}-B_{k}\,\Delta x_{k})(y_{k}-B_{k}\,\Delta x_{k})^{T}}{(y_{k}% -B_{k}\,\Delta x_{k})^{T}\,\Delta x_{k}}
  59. H k + ( Δ x k - H k y k ) ( Δ x k - H k y k ) T ( Δ x k - H k y k ) T y k H_{k}+\frac{(\Delta x_{k}-H_{k}y_{k})(\Delta x_{k}-H_{k}y_{k})^{T}}{(\Delta x_% {k}-H_{k}y_{k})^{T}y_{k}}