wpmath0000006_9

Optical_molasses.html

  1. T d = Γ / 2 k b T_{d}=\hbar\Gamma/{2k_{b}}
  2. Γ \Gamma
  3. \hbar
  4. k b k_{b}

Optical_parametric_oscillator.html

  1. ω p \omega_{p}
  2. ω s , ω i \omega_{s},\omega_{i}
  3. ω s + ω i = ω p \omega_{s}+\omega_{i}=\omega_{p}
  4. ω s = ω i = ω p / 2 \omega_{s}=\omega_{i}=\omega_{p}/2

Optical_theorem.html

  1. σ tot = 4 π k Im f ( 0 ) , \sigma_{\mathrm{tot}}=\frac{4\pi}{k}~{}\mathrm{Im}\,f(0),
  2. σ tot \sigma_{\mathrm{tot}}
  3. Im f ( k ^ , k ^ ) = k 4 π f ( k ^ , k ^ ′′ ) f ( k ^ ′′ , k ^ ) d k ^ ′′ . \mathrm{Im}~{}f({\hat{k}}^{\prime},{\hat{k}})=\frac{k}{4\pi}\int f({\hat{k}}^{% \prime},{\hat{k}}^{\prime\prime})f({\hat{k}}^{\prime\prime},{\hat{k}})~{}d{% \hat{k}}^{\prime\prime}.
  4. n = 1 + 2 π N f ( 0 ) k 2 , n=1+2\pi\frac{Nf(0)}{k^{2}},
  5. ψ ( r ) e i k z + f ( θ ) e i k r r . \psi({r})\approx e^{ikz}+f(\theta)\frac{e^{ikr}}{r}.
  6. 1 / r 2 1/r^{2}
  7. z z
  8. r = x 2 + y 2 + z 2 z + x 2 + y 2 2 z . r=\sqrt{x^{2}+y^{2}+z^{2}}\approx z+\frac{x^{2}+y^{2}}{2z}.
  9. ψ \psi
  10. 1 / r 1/r
  11. 1 / z 1/z
  12. | ψ | 2 | e i k z + f ( θ ) z e i k z e i k ( x 2 + y 2 ) / 2 z | 2 = 1 + f ( θ ) z e i k ( x 2 + y 2 ) / 2 z + f * ( θ ) z e - i k ( x 2 + y 2 ) / 2 z + | f ( θ ) | 2 z 2 . \begin{aligned}\displaystyle|\psi|^{2}&\displaystyle\approx\left|e^{ikz}+\frac% {f(\theta)}{z}e^{ikz}e^{ik(x^{2}+y^{2})/2z}\right|^{2}\\ &\displaystyle=1+\frac{f(\theta)}{z}e^{ik(x^{2}+y^{2})/2z}+\frac{f^{*}(\theta)% }{z}e^{-ik(x^{2}+y^{2})/2z}+\frac{|f(\theta)|^{2}}{z^{2}}.\end{aligned}
  13. 1 / z 2 1/z^{2}
  14. c + c * = 2 Re c c+c^{*}=2\operatorname{Re}{c}
  15. | ψ | 2 1 + 2 Re [ f ( θ ) z e i k ( x 2 + y 2 ) / 2 z ] . |\psi|^{2}\approx 1+2\operatorname{Re}{\left[\frac{f(\theta)}{z}e^{ik(x^{2}+y^% {2})/2z}\right]}.
  16. - -\infty
  17. \infty
  18. f ( θ ) = f ( 0 ) f(\theta)=f(0)
  19. | ψ | 2 d x d y A + 2 Re [ f ( 0 ) z - e i k x 2 / 2 z d x - e i k y 2 / 2 z d y ] , \int|\psi|^{2}\;dx\,dy\approx A+2\operatorname{Re}\left[\frac{f(0)}{z}\int_{-% \infty}^{\infty}e^{ikx^{2}/2z}dx\int_{-\infty}^{\infty}e^{iky^{2}/2z}dy\right],
  20. | ψ | 2 d a = A + 2 Re [ f ( 0 ) z 2 π i z k ] = A - 4 π k Im [ f ( 0 ) ] . \begin{aligned}\displaystyle\int|\psi|^{2}\;da&\displaystyle=A+2\operatorname{% Re}\left[\frac{f(0)}{z}\,\frac{2\pi iz}{k}\right]\\ &\displaystyle=A-\frac{4\pi}{k}\,\operatorname{Im}[f(0)].\end{aligned}
  21. ( 4 π / k ) Im [ f ( 0 ) ] (4\pi/k)\operatorname{Im}[f(0)]

Optical_vortex.html

  1. π \pi
  2. ψ e i m ϕ e - r 2 , \psi\propto e^{im\phi}e^{-r^{2}},\!
  3. π \pi
  4. ± 2 q \pm 2q

Optimum_programming.html

  1. n / 2 n/2

Orbifold_notation.html

  1. S 2 S^{2}
  2. E 2 E^{2}
  3. H 2 H^{2}
  4. 1 , 2 , 3 , 1,2,3,\dots
  5. \infty
  6. × \times
  7. × \times
  8. \infty
  9. n - 1 n \frac{n-1}{n}
  10. n - 1 2 n \frac{n-1}{2n}
  11. × \times
  12. 4 ¯ \overline{4}
  13. n ¯ \overline{n}
  14. n ¯ \overline{n}
  15. n ¯ \overline{n}
  16. n ¯ \overline{n}
  17. n ¯ \overline{n}
  18. n ¯ \overline{n}
  19. 2 ¯ \overline{2}
  20. 2 ¯ \overline{2}
  21. 2 ¯ \overline{2}
  22. 2 ¯ \overline{2}
  23. 2 ¯ \overline{2}
  24. 2 ¯ \overline{2}
  25. 1 ¯ \overline{1}
  26. 1 ¯ \overline{1}
  27. 1 ¯ \overline{1}
  28. 1 ¯ \overline{1}
  29. 1 ¯ \overline{1}
  30. 1 ¯ \overline{1}

Order_dimension.html

  1. = ( < 1 , , < t ) \mathcal{R}=(<_{1},\dots,<_{t})
  2. P = = i = 1 t < i . P=\bigcap\mathcal{R}=\bigcap_{i=1}^{t}<_{i}.
  3. x y x\leq y
  4. x i y i x_{i}\leq y_{i}
  5. = ( < 1 , , < t ) \mathcal{R}=(<_{1},\dots,<_{t})
  6. < P = <_{P}=\bigcap\mathcal{R}
  7. Θ ( log log n ) \Theta(\log\log n)
  8. O ( log log n ) O(\log\log n)
  9. dim k \textrm{dim}_{k}

Ordinal_arithmetic.html

  1. \cup
  2. α < β γ + α < γ + β \alpha<\beta\Rightarrow\gamma+\alpha<\gamma+\beta
  3. α < β α + γ β + γ \alpha<\beta\Rightarrow\alpha+\gamma\leq\beta+\gamma
  4. 3 + ω = 0 + ω = ω 3+\omega=0+\omega=\omega
  5. 3 0 3\neq 0
  6. \Rightarrow
  7. \Rightarrow
  8. ω n \omega^{n}
  9. n < ω ω n \bigcup_{n<\omega}\omega^{n}
  10. ω ω \omega^{\omega}
  11. ω n 1 c 1 + ω n 2 c 2 + + ω n k c k \omega^{n_{1}}c_{1}+\omega^{n_{2}}c_{2}+\cdots+\omega^{n_{k}}c_{k}
  12. ω ω \omega^{\omega}
  13. 2 0 2^{\aleph_{0}}
  14. 0 \aleph_{0}
  15. 0 \aleph_{0}
  16. ω β 1 c 1 + ω β 2 c 2 + + ω β k c k \omega^{\beta_{1}}c_{1}+\omega^{\beta_{2}}c_{2}+\cdots+\omega^{\beta_{k}}c_{k}
  17. c 1 , c 2 , , c k c_{1},c_{2},\ldots,c_{k}
  18. β 1 > β 2 > > β k 0 \beta_{1}>\beta_{2}>\ldots>\beta_{k}\geq 0
  19. β 1 \beta_{1}
  20. α \alpha
  21. β 1 α \beta_{1}\leq\alpha
  22. β 1 = α \beta_{1}=\alpha
  23. α = ω α \alpha=\omega^{\alpha}
  24. ω β 1 + ω β 2 + + ω β k \omega^{\beta_{1}}+\omega^{\beta_{2}}+\cdots+\omega^{\beta_{k}}
  25. β 1 β 2 β k 0 \beta_{1}\geq\beta_{2}\geq\ldots\geq\beta_{k}\geq 0
  26. ω \omega
  27. β 1 < α \beta_{1}<\alpha
  28. β i \beta_{i}
  29. β i \beta_{i}
  30. ω ( ω ( ω 7 6 + ω + 42 ) 1729 + ω 9 + 88 ) 3 + ω ( ω ω ) 5 + 65537 \omega^{\left(\omega^{\left(\omega^{7}\cdot 6+\omega+42\right)}\cdot 1729+% \omega^{9}+88\right)}\cdot 3+\omega^{\left(\omega^{\omega}\right)}\cdot 5+65537
  31. 0 , 1 = ω 0 , ω = ω 1 , ω ω , ω ω ω , . 0,\,1=\omega^{0},\,\omega=\omega^{1},\,\omega^{\omega},\,\omega^{\omega^{% \omega}},\,\ldots\,.
  32. ω β c + ω β c = ω β c , \omega^{\beta}c+\omega^{\beta^{\prime}}c^{\prime}=\omega^{\beta^{\prime}}c^{% \prime}\,,
  33. β > β \beta^{\prime}>\beta
  34. β = β \beta^{\prime}=\beta
  35. ω β ( c + c ) \omega^{\beta}(c+c^{\prime})
  36. β < β \beta^{\prime}<\beta
  37. 0 < α = ω β 1 c 1 + + ω β k c k 0<\alpha=\omega^{\beta_{1}}c_{1}+\cdots+\omega^{\beta_{k}}c_{k}
  38. 0 < β 0<\beta^{\prime}
  39. α ω β = ω β 1 + β \alpha\omega^{\beta^{\prime}}=\omega^{\beta_{1}+\beta^{\prime}}\,
  40. α n = ω β 1 c 1 n + ω β 2 c 2 + + ω β k c k , \alpha n=\omega^{\beta_{1}}c_{1}n+\omega^{\beta_{2}}c_{2}+\cdots+\omega^{\beta% _{k}}c_{k}\,,
  41. β 1 \beta_{1}
  42. c 1 c_{1}
  43. β 2 \beta_{2}
  44. c 2 c_{2}
  45. ω α 1 n 1 + + ω α k n k \omega^{\alpha_{1}}n_{1}+\cdots+\omega^{\alpha_{k}}n_{k}
  46. α 1 > > α k \alpha_{1}>\cdots>\alpha_{k}
  47. ω ω β 1 ω ω β m n k ( ω α k - 1 - α k + 1 ) n k - 1 n 2 ( ω α 1 - α 2 + 1 ) n 1 \omega^{\omega^{\beta_{1}}}\cdots\omega^{\omega^{\beta_{m}}}n_{k}(\omega^{% \alpha_{k-1}-\alpha_{k}}+1)n_{k-1}\cdots n_{2}(\omega^{\alpha_{1}-\alpha_{2}}+% 1)n_{1}
  48. α k = ω β 1 + + ω β m \alpha_{k}=\omega^{\beta_{1}}+\cdots+\omega^{\beta_{m}}
  49. β 1 β m \beta_{1}\geq\cdots\geq\beta_{m}
  50. ϵ 0 \epsilon_{0}
  51. ω \omega
  52. S S
  53. S ( S ( S ( S ( 0 ) ) ) ) S(S(S(S(0))))
  54. ϵ 0 \epsilon_{0}
  55. ϵ 0 \epsilon_{0}
  56. ϵ 0 \epsilon_{0}
  57. ω 1 \omega_{1}
  58. S T S\cup T
  59. S T S\cup T
  60. α = ω γ 1 k 1 + + ω γ n k n \alpha=\omega^{\gamma_{1}}\cdot k_{1}+\cdots+\omega^{\gamma_{n}}\cdot k_{n}
  61. β = ω γ 1 j 1 + + ω γ n j n \beta=\omega^{\gamma_{1}}\cdot j_{1}+\cdots+\omega^{\gamma_{n}}\cdot j_{n}
  62. α # β = ω γ 1 ( k 1 + j 1 ) + + ω γ n ( k n + j n ) . \alpha\#\beta=\omega^{\gamma_{1}}\cdot(k_{1}+j_{1})+\cdots+\omega^{\gamma_{n}}% \cdot(k_{n}+j_{n}).
  63. x 5 + x 4 + x 3 + x 2 + x + 1 = ( x + 1 ) ( x 4 + x 2 + 1 ) = ( x 2 + x + 1 ) ( x 3 + 1 ) x^{5}+x^{4}+x^{3}+x^{2}+x+1=(x+1)(x^{4}+x^{2}+1)=(x^{2}+x+1)(x^{3}+1)

Ore_condition.html

  1. G = x , y G=\langle x,y\rangle\,
  2. F [ G ] F[G]\,

Ore_extension.html

  1. δ ( r 1 r 2 ) = σ ( r 1 ) δ ( r 2 ) + δ ( r 1 ) r 2 . \delta(r_{1}r_{2})=\sigma(r_{1})\delta(r_{2})+\delta(r_{1})r_{2}.
  2. x r = σ ( r ) x + δ ( r ) . xr=\sigma(r)x+\delta(r).

Orientation_(computer_vision).html

  1. f ( 𝐱 ) = g ( 𝐱 𝐧 ^ ) f(\mathbf{x})=g(\mathbf{x}\cdot\hat{\mathbf{n}})
  2. f f
  3. 𝐱 \mathbf{x}
  4. g g
  5. 𝐧 ^ \hat{\mathbf{n}}
  6. f f
  7. 𝐧 ^ \hat{\mathbf{n}}
  8. 𝐧 ^ \hat{\mathbf{n}}
  9. f f
  10. 𝐧 ^ \hat{\mathbf{n}}
  11. 𝐧 ~ = - 𝐧 ^ \tilde{\mathbf{n}}=-\hat{\mathbf{n}}
  12. g ~ ( x ) = g ( - x ) \tilde{g}(x)=g(-x)
  13. f f
  14. f ( 𝐱 ) = g ~ ( 𝐱 𝐧 ~ ) f(\mathbf{x})=\tilde{g}(\mathbf{x}\cdot\tilde{\mathbf{n}})
  15. 𝐧 ~ = - 𝐧 ^ \tilde{\mathbf{n}}=-\hat{\mathbf{n}}
  16. f f
  17. 𝐧 ^ \hat{\mathbf{n}}
  18. 𝐧 ^ \hat{\mathbf{n}}
  19. 𝐧 ^ \hat{\mathbf{n}}

Ornstein–Uhlenbeck_process.html

  1. d x t = θ ( μ - x t ) d t + σ d W t dx_{t}=\theta(\mu-x_{t})\,dt+\sigma\,dW_{t}
  2. θ > 0 \theta>0
  3. μ \mu
  4. σ > 0 \sigma>0
  5. W t W_{t}
  6. f t = θ x [ ( x - μ ) f ] + σ 2 2 2 f x 2 \frac{\partial f}{\partial t}=\theta\frac{\partial}{\partial x}[(x-\mu)f]+% \frac{\sigma^{2}}{2}\frac{\partial^{2}f}{\partial x^{2}}
  7. μ = 0 \mu=0
  8. D = σ 2 / 2 D=\sigma^{2}/2
  9. y y
  10. f ( x , t ) = θ 2 π D ( 1 - e - 2 θ t ) exp { - θ 2 D [ ( x - y e - θ t ) 2 1 - e - 2 θ t ] } f(x,t)=\sqrt{\frac{\theta}{2\pi D(1-e^{-2\theta t})}}\exp\left\{\frac{-\theta}% {2D}\left[\frac{(x-ye^{-\theta t})^{2}}{1-e^{-2\theta t}}\right]\right\}
  11. μ \mu
  12. σ 2 / ( 2 θ ) \sigma^{2}/(2\theta)
  13. f s ( x ) = θ π σ 2 e - θ ( x - μ ) 2 / σ 2 . f_{s}(x)=\sqrt{\frac{\theta}{\pi\sigma^{2}}}\,e^{-\theta(x-\mu)^{2}/\sigma^{2}}.
  14. k k
  15. γ \gamma
  16. T T
  17. x ( t ) x(t)
  18. x 0 x_{0}
  19. θ \displaystyle\theta
  20. σ \sigma
  21. D = σ 2 / 2 = k B T / γ D=\sigma^{2}/2=k_{B}T/\gamma
  22. γ x ˙ ( t ) = - k ( x ( t ) - x 0 ) + ξ ( t ) \gamma\dot{x}(t)=-k(x(t)-x_{0})+\xi(t)
  23. ξ ( t ) \xi(t)
  24. ξ ( t 1 ) ξ ( t 2 ) = 2 k B T γ δ ( t 1 - t 2 ) . \langle\xi(t_{1})\xi(t_{2})\rangle=2k_{B}T\,\gamma\,\delta(t_{1}-t_{2}).
  25. E = k ( x - x 0 ) 2 / 2 = k B T / 2 \langle E\rangle=k\langle(x-x_{0})^{2}\rangle/2=k_{B}T/2
  26. μ \mu
  27. σ \sigma
  28. θ \theta
  29. var ( x t ) = σ 2 2 θ . \operatorname{var}(x_{t})={\sigma^{2}\over 2\theta}.\,
  30. f ( x t , t ) = x t e θ t f(x_{t},t)=x_{t}e^{\theta t}\,
  31. d f ( x t , t ) \displaystyle df(x_{t},t)
  32. x t e θ t = x 0 + 0 t e θ s θ μ d s + 0 t σ e θ s d W s x_{t}e^{\theta t}=x_{0}+\int_{0}^{t}e^{\theta s}\theta\mu\,ds+\int_{0}^{t}% \sigma e^{\theta s}\,dW_{s}\,
  33. x t = x 0 e - θ t + μ ( 1 - e - θ t ) + e - θ t 0 t σ e θ s d W s . x_{t}=x_{0}e^{-\theta t}+\mu(1-e^{-\theta t})+e^{-\theta t}\int_{0}^{t}\sigma e% ^{\theta s}\,dW_{s}.\,
  34. E ( x t ) = x 0 e - θ t + μ ( 1 - e - θ t ) E(x_{t})=x_{0}e^{-\theta t}+\mu(1-e^{-\theta t})\!
  35. cov ( x s , x t ) \displaystyle\operatorname{cov}(x_{s},x_{t})
  36. t t\rightarrow\infty
  37. x t = μ + σ 2 θ e - θ t W e 2 θ t x_{t}=\mu+{\sigma\over\sqrt{2\theta}}e^{-\theta t}W_{e^{2\theta t}}
  38. x t = x 0 e - θ t + μ ( 1 - e - θ t ) + σ 2 θ e - θ t W e 2 θ t - 1 . x_{t}=x_{0}e^{-\theta t}+\mu(1-e^{-\theta t})+{\sigma\over\sqrt{2\theta}}e^{-% \theta t}W_{e^{2\theta t}-1}.
  39. n n
  40. X n X_{n}
  41. n n
  42. X [ n t ] - n / 2 n \frac{X_{[nt]}-n/2}{\sqrt{n}}
  43. n n
  44. X X
  45. σ x γ d W t \sigma\,x^{\gamma}\,dW_{t}
  46. γ = 1 / 2 \gamma=1/2
  47. γ = 0 \gamma=0

Orthogonal_trajectory.html

  1. g ( x , y ) = C g(x,y)=C
  2. C C
  3. f ( x , y ) f(x,y)
  4. f g = 0 \nabla f\cdot\nabla g=0
  5. f ( x , y ) f(x,y)
  6. f f
  7. g g
  8. r ( t ) \vec{r}(t)
  9. g ( x , y ) g(x,y)
  10. d d t ( r ( t ) ) = g \frac{d}{dt}\left(\vec{r}(t)\right)=\nabla g
  11. g g
  12. r ( t ) \vec{r}(t)
  13. x 2 + y 2 = c x^{2}+y^{2}=c
  14. { d y d x = f ( x , y ) d k d x = - 1 f ( x , y ) \left\{\begin{matrix}\frac{\mathrm{d}y}{\mathrm{d}x}=f(x,y)\\ \frac{\mathrm{d}k}{\mathrm{d}x}=-\frac{1}{f(x,y)}\end{matrix}\right.
  15. x 2 + y 2 = c 2 x + 2 y d y d x = 0 x^{2}+y^{2}=c\Rightarrow 2x+2y\frac{\mathrm{d}y}{\mathrm{d}x}=0
  16. d y d x = - x y \frac{\mathrm{d}y}{\mathrm{d}x}=\frac{-x}{y}
  17. d y d x = y x \frac{\mathrm{d}y}{\mathrm{d}x}=\frac{y}{x}
  18. y = m x y=mx
  19. r - R = 0 r-R=0
  20. R R
  21. f f
  22. f ( r - R ) = 0 \nabla f\cdot\nabla(r-R)=0
  23. ( f r , 1 r f θ ) ( r ( r - R ) , 1 r θ ( r - R ) ) = 0 \left(\frac{\partial f}{\partial r},\frac{1}{r}\frac{\partial f}{\partial% \theta}\right)\cdot\left(\frac{\partial}{\partial r}(r-R),\frac{1}{r}\frac{% \partial}{\partial\theta}(r-R)\right)=0
  24. f r = 0 f = f ( θ ) \frac{\partial f}{\partial r}=0\quad\Rightarrow\quad f=f(\theta)
  25. f ( θ ) f(\theta)
  26. f ( θ ) f(\theta)
  27. f ( θ ) = 0 f(\theta)=0
  28. f f
  29. θ = c o n s t a n t \theta=constant

Oseledets_theorem.html

  1. C ( x , 0 ) = I n for all x X C(x,0)=I_{n}{\rm~{}for~{}all~{}}x\in X
  2. C ( x , t + s ) = C ( x ( t ) , s ) C ( x , t ) for all x X and t , s T C(x,t+s)=C(x(t),s)\,C(x,t){\rm~{}for~{}all~{}}x\in X{\rm~{}and~{}}t,s\in T
  3. x log C ( x , t ) x\rightarrow\log\|C(x,t)\|
  4. x log C ( x , t ) - 1 x\rightarrow\log\|C(x,t)^{-1}\|
  5. λ = lim t 1 t log C ( x , t ) u u \lambda=\lim_{t\to\infty}{1\over t}\log{\|C(x,t)u\|\over\|u\|}
  6. g / x \partial g/\partial x
  7. lim t 1 t 0 t f ( x ( s ) ) d s = 1 μ ( X ) X f ( x ) μ ( d x ) \lim_{t\to\infty}{1\over t}\int_{0}^{t}f(x(s))\,ds={1\over\mu(X)}\int_{X}f(x)% \,\mu(dx)

Osmotic_concentration.html

  1. osmol / L = i φ i n i C i \mathrm{osmol/L}=\sum_{i}\varphi_{i}\,n_{i}C_{i}
  2. o s m o l a l i t y = i φ i n i m i osmolality=\sum_{i}\varphi_{i}\,n_{i}m_{i}

Ostwald–Freundlich_equation.html

  1. R R
  2. p p e q = exp ( R c r i t i c a l R ) \frac{p}{p_{eq}}=\exp{\left(\frac{R_{critical}}{R}\right)}
  3. R c r i t i c a l = 2 γ V A t o m k B T R_{critical}=\frac{2\cdot\gamma\cdot V_{Atom}}{k_{B}\cdot T}
  4. V A t o m V_{Atom}
  5. k B k_{B}
  6. γ \gamma
  7. \cdot
  8. p e q p_{eq}
  9. p p
  10. T T
  11. p ( r 1 , r 2 ) = P - γ ρ v a p o r ( ρ l i q u i d - ρ v a p o r ) ( 1 r 1 + 1 r 2 ) p(r_{1},r_{2})=P-\frac{\gamma\,\rho\,_{vapor}}{(\rho\,_{liquid}-\rho\,_{vapor}% )}\left(\frac{1}{r_{1}}+\frac{1}{r_{2}}\right)
  12. p ( r ) p(r)
  13. r r
  14. P P
  15. r = r=\infty
  16. p e q p_{eq}
  17. γ \gamma
  18. ρ v a p o r \rho\,_{vapor}
  19. ρ l i q u i d \rho\,_{liquid}
  20. r 1 r_{1}
  21. r 2 r_{2}
  22. p ( r 1 , r 2 ) = P - γ ρ v a p o r ( ρ l i q u i d - ρ v a p o r ) ( 1 r 1 + 1 r 2 ) p(r_{1},r_{2})=P-\frac{\gamma\,\rho\,_{vapor}}{(\rho\,_{liquid}-\rho\,_{vapor}% )}\left(\frac{1}{r_{1}}+\frac{1}{r_{2}}\right)
  23. r = r 1 = r 2 r=r_{1}=r_{2}
  24. p ( r ) = P - 2 γ ρ v a p o r ( ρ l i q u i d - ρ v a p o r ) r p(r)=P-\frac{2\gamma\,\rho\,_{vapor}}{(\rho\,_{liquid}-\rho\,_{vapor})r}
  25. γ \gamma
  26. γ \gamma
  27. γ \gamma
  28. ρ l i q u i d ρ v a p o r \rho\,_{liquid}\gg\rho\,_{vapor}
  29. ρ l i q u i d - ρ v a p o r ρ l i q u i d \rho\,_{liquid}-\rho\,_{vapor}\approx\rho\,_{liquid}
  30. p ( r ) P + 2 γ ρ v a p o r ρ l i q u i d r p(r)\approx P+\frac{2\gamma\,\rho\,_{vapor}}{\rho\,_{liquid}\cdot r}
  31. ρ v a p o r = m v a p o r V = M W n V = M W P R T = M W P N 0 k B T \rho\,_{vapor}=\frac{m_{vapor}}{V}=\frac{MW\cdot n}{V}=\frac{MW\cdot P}{RT}=% \frac{MW\cdot P}{N_{0}k_{B}T}
  32. m v a p o r m_{vapor}
  33. V V
  34. M W MW
  35. n n
  36. V V
  37. R R
  38. N 0 k B N_{0}k_{B}
  39. N 0 N_{0}
  40. k B k_{B}
  41. T T
  42. M W N 0 = \frac{MW}{N_{0}}=
  43. ( M W N 0 ) ρ l i q u i d = \frac{\left(\frac{MW}{N_{0}}\right)}{\rho\,_{liquid}}=
  44. = V m o l e c u l e =V_{molecule}
  45. p ( r ) P + 2 γ V m o l e c u l e P k B T r = P + R c r i t i c a l P r p(r)\approx P+\frac{2\gamma V_{molecule}P}{k_{B}Tr}=P+\frac{R_{critical}P}{r}
  46. R c r i t i c a l = 2 γ V m o l e c u l e k B T R_{critical}=\frac{2\gamma V_{molecule}}{k_{B}T}
  47. p ( r ) - P P R c r i t i c a l r \frac{p(r)-P}{P}\approx\frac{R_{critical}}{r}
  48. p ( r ) P = 1 - P - p ( r ) P \frac{p(r)}{P}=1-\frac{P-p(r)}{P}
  49. log ( p ( r ) P ) = log ( 1 - P - p ( r ) P ) \log\left(\frac{p(r)}{P}\right)=\log\left(1-\frac{P-p(r)}{P}\right)
  50. p ( r ) P p(r)\approx P
  51. P - p ( r ) P 1 \frac{P-p(r)}{P}\ll 1
  52. x 1 x\ll 1
  53. log ( 1 - x ) - x \log\left(1-x\right)\approx-x
  54. log ( p ( r ) P ) p ( r ) - P P \log\left(\frac{p(r)}{P}\right)\approx\frac{p(r)-P}{P}
  55. log ( p ( r ) P ) R c r i t i c a l r \log\left(\frac{p(r)}{P}\right)\approx\frac{R_{critical}}{r}

P-chart.html

  1. p ¯ ± 3 p ¯ ( 1 - p ¯ ) n \bar{p}\pm 3\sqrt{\frac{\bar{p}(1-\bar{p})}{n}}
  2. p ¯ i = j = 1 n { 1 if x i j defective 0 otherwise n \bar{p}_{i}=\frac{\sum_{j=1}^{n}\begin{cases}1&\mbox{if }~{}x_{ij}\mbox{ % defective}\\ 0&\mbox{otherwise}\end{cases}}{n}
  3. p ¯ ± 3 p ¯ ( 1 - p ¯ ) n \bar{p}\pm 3\sqrt{\frac{\bar{p}(1-\bar{p})}{n}}
  4. p ¯ \bar{p}
  5. n ( 3 δ ) 2 p ¯ ( 1 - p ¯ ) n\geq\left(\frac{3}{\delta}\right)^{2}\bar{p}(1-\bar{p})
  6. n > 3 2 ( 1 - p ¯ ) p ¯ n>\frac{3^{2}(1-\bar{p})}{\bar{p}}
  7. p ¯ ± 3 p ¯ ( 1 - p ¯ ) n i \bar{p}\pm 3\sqrt{\frac{\bar{p}(1-\bar{p})}{n_{i}}}
  8. p ¯ ± 3 p ¯ ( 1 - p ¯ ) n ¯ \bar{p}\pm 3\sqrt{\frac{\bar{p}(1-\bar{p})}{\bar{n}}}
  9. n ¯ \bar{n}
  10. i = 1 m n i m \frac{\sum_{i=1}^{m}n_{i}}{m}
  11. p ^ i \hat{p}_{i}
  12. Z i = p ^ i - p ¯ p ¯ ( 1 - p ¯ ) n i Z_{i}=\frac{\hat{p}_{i}-\bar{p}}{\sqrt{\frac{\bar{p}(1-\bar{p})}{n_{i}}}}

P-form_electrodynamics.html

  1. 𝐀 𝐀 + d α \mathbf{A}\rightarrow\mathbf{A}+d\alpha
  2. d * 𝐉 = 0 d*\mathbf{J}=0
  3. 𝐅 = d 𝐀 \mathbf{F}=d\mathbf{A}
  4. d * 𝐅 = * 𝐉 d*\mathbf{F}=*\mathbf{J}
  5. S = M [ 1 2 𝐅 * 𝐅 - 𝐀 * 𝐉 ] S=\int_{M}\left[\frac{1}{2}\mathbf{F}\wedge*\mathbf{F}-\mathbf{A}\wedge*% \mathbf{J}\right]
  6. 𝐁 𝐁 + d α \mathbf{B}\rightarrow\mathbf{B}+d\mathbf{\alpha}
  7. d * 𝐉 = 0 d*\mathbf{J}=0
  8. 𝐂 = d 𝐁 \mathbf{C}=d\mathbf{B}
  9. d * 𝐂 = * 𝐉 d*\mathbf{C}=*\mathbf{J}
  10. S = M [ 1 2 𝐂 * 𝐂 + ( - 1 ) p 𝐁 * 𝐉 ] S=\int_{M}\left[\frac{1}{2}\mathbf{C}\wedge*\mathbf{C}+(-1)^{p}\mathbf{B}% \wedge*\mathbf{J}\right]

P::poly.html

  1. Σ 2 P \Sigma_{2}^{\rm P}
  2. Σ 2 P Π 2 P \Sigma_{2}^{\rm P}\cap\Pi_{2}^{\rm P}
  3. Σ 2 P Π 2 P \Sigma_{2}^{\rm P}\cap\Pi_{2}^{\rm P}
  4. Σ 2 P Π 2 P \Sigma_{2}^{\rm P}\cap\Pi_{2}^{\rm P}
  5. x Prob [ R Bad ( x ) ] R 1 e n . \forall x\,\mbox{Prob}~{}_{R}[R\in\mbox{Bad}~{}(x)]\leq\frac{1}{e^{n}}.
  6. Prob [ x R Bad ( x ) ] R 2 n e n < 1. \mbox{Prob}~{}_{R}[\exists x\,R\in\mbox{Bad}~{}(x)]\leq\frac{2^{n}}{e^{n}}<1.

Padé_approximant.html

  1. R ( x ) = j = 0 m a j x j 1 + k = 1 n b k x k = a 0 + a 1 x + a 2 x 2 + + a m x m 1 + b 1 x + b 2 x 2 + + b n x n R(x)=\frac{\sum_{j=0}^{m}a_{j}x^{j}}{1+\sum_{k=1}^{n}b_{k}x^{k}}=\frac{a_{0}+a% _{1}x+a_{2}x^{2}+\cdots+a_{m}x^{m}}{1+b_{1}x+b_{2}x^{2}+\cdots+b_{n}x^{n}}
  2. f ( 0 ) = R ( 0 ) f ( 0 ) = R ( 0 ) f ′′ ( 0 ) = R ′′ ( 0 ) f ( m + n ) ( 0 ) = R ( m + n ) ( 0 ) \begin{array}[]{rcl}f(0)&=&R(0)\\ f^{\prime}(0)&=&R^{\prime}(0)\\ f^{\prime\prime}(0)&=&R^{\prime\prime}(0)\\ &\vdots&\\ f^{(m+n)}(0)&=&R^{(m+n)}(0)\end{array}
  3. f ( x ) - R ( x ) = c m + n + 1 x m + n + 1 + c m + n + 2 x m + n + 2 + f(x)-R(x)=c_{m+n+1}x^{m+n+1}+c_{m+n+2}x^{m+n+2}+\cdots
  4. a 0 , a 1 , , a m , b 1 , , b n a_{0},a_{1},\dots,a_{m},b_{1},\dots,b_{n}
  5. [ m / n ] f ( x ) . [m/n]_{f}(x).\,
  6. T N ( x ) = c 0 + c 1 x + c 2 x 2 + + c N x N T_{N}(x)=c_{0}+c_{1}x+c_{2}x^{2}+\cdots+c_{N}x^{N}
  7. c k = f ( k ) ( 0 ) k ! . c_{k}=\frac{f^{(k)}(0)}{k!}.
  8. R ( x ) = P ( x ) / Q ( x ) = T m + n ( x ) mod x m + n + 1 R(x)=P(x)/Q(x)=T_{m+n}(x)\,\text{ mod }x^{m+n+1}
  9. P ( x ) = Q ( x ) T m + n ( x ) + K ( x ) x m + n + 1 P(x)=Q(x)T_{m+n}(x)+K(x)x^{m+n+1}
  10. T m + n ( x ) T_{m+n}(x)
  11. x m + n + 1 x^{m+n+1}
  12. r 0 = p , r 1 = q , r k - 1 = q k r k + r k + 1 r_{0}=p,\;r_{1}=q,\quad r_{k-1}=q_{k}r_{k}+r_{k+1}
  13. deg r k + 1 < deg r k \deg r_{k+1}<\deg r_{k}\,
  14. r k + 1 = 0 r_{k+1}=0
  15. u 0 = 1 , v 0 = 0 , u 1 = 0 , v 1 = 1 , u k + 1 = u k - 1 - q k u k , v k + 1 = v k - 1 - q k v k u_{0}=1,\;v_{0}=0,\quad u_{1}=0,\;v_{1}=1,\quad u_{k+1}=u_{k-1}-q_{k}u_{k},\;v% _{k+1}=v_{k-1}-q_{k}v_{k}
  16. r k ( x ) = u k ( x ) p ( x ) + v k ( x ) q ( x ) r_{k}(x)=u_{k}(x)p(x)+v_{k}(x)q(x)
  17. r 0 = x m + n + 1 , r 1 = T m + n ( x ) r_{0}=x^{m+n+1},\;r_{1}=T_{m+n}(x)
  18. v k v_{k}
  19. P = r k , Q = v k P=r_{k},\;Q=v_{k}
  20. z = 1 f ( z ) , \sum_{z=1}^{\infty}f(z),
  21. ζ R ( s ) = z = 1 R ( z ) z s , \zeta_{R}(s)=\sum_{z=1}^{\infty}\frac{R(z)}{z^{s}},
  22. R ( x ) = [ m / n ] f ( x ) R(x)=[m/n]_{f}(x)\,
  23. j = 0 n a j ζ R ( s - j ) = j = 0 m b j ζ 0 ( s - j ) , \sum_{j=0}^{n}a_{j}\zeta_{R}(s-j)=\sum_{j=0}^{m}b_{j}\zeta_{0}(s-j),
  24. f ( x ) | x - r | p f(x)\sim\left|x-r\right|^{p}
  25. [ n / n + 1 ] g ( x ) \left[n/n+1\right]_{g}\left(x\right)
  26. g = f f g=\frac{f^{\prime}}{f}
  27. sin ( x ) ( 12671 / 4363920 ) x 5 - ( 2363 / 18183 ) x 3 + x 1 + ( 445 / 12122 ) x 2 + ( 601 / 872784 ) x 4 + ( 121 / 16662240 ) x 6 \sin(x)\approx\frac{(12671/4363920)x^{5}-(2363/18183)x^{3}+x}{1+(445/12122)x^{% 2}+(601/872784)x^{4}+(121/16662240)x^{6}}
  28. exp ( x ) 1 + ( 1 / 2 ) x + ( 1 / 9 ) x 2 + ( 1 / 72 ) x 3 + ( 1 / 1008 ) x 4 + ( 1 / 30240 ) x 5 1 - ( 1 / 2 ) x + ( 1 / 9 ) x 2 - ( 1 / 72 ) x 3 + ( 1 / 1008 ) x 4 - ( 1 / 30240 ) x 5 \exp(x)\approx\frac{1+(1/2)x+(1/9)x^{2}+(1/72)x^{3}+(1/1008)x^{4}+(1/30240)x^{% 5}}{1-(1/2)x+(1/9)x^{2}-(1/72)x^{3}+(1/1008)x^{4}-(1/30240)x^{5}}
  29. sn ( z | 3 ) - ( 9853969 / 39583665 ) z 5 - ( 1493060 / 2638911 ) z 3 + z 1 + ( 968375 / 879637 ) z 2 - ( 1167506 / 7916733 ) z 4 + ( 867043 / 2159109 ) z 6 \mathrm{sn}(z|3)\approx\frac{-(9853969/39583665)z^{5}-(1493060/2638911)z^{3}+z% }{1+(968375/879637)z^{2}-(1167506/7916733)z^{4}+(867043/2159109)z^{6}}
  30. J 5 ( x ) - ( 107 / 28416000 ) x 7 + ( 1 / 3840 ) x 5 1 + ( 151 / 5550 ) x 2 + ( 1453 / 3729600 ) x 4 + ( 1339 / 358041600 ) x 6 + ( 2767 / 120301977600 ) x 8 J_{5}(x)\approx\frac{-(107/28416000)x^{7}+(1/3840)x^{5}}{1+(151/5550)x^{2}+(14% 53/3729600)x^{4}+(1339/358041600)x^{6}+(2767/120301977600)x^{8}}
  31. erf ( x ) ( 2 / 15 ) ( 49140 x + 3570 x 3 + 739 x 5 ) π ( 165 x 4 + 1330 x 2 + 3276 ) \mathrm{erf}(x)\approx\frac{(2/15)\cdot(49140x+3570x^{3}+739x^{5})}{\sqrt{\pi}% \cdot(165x^{4}+1330x^{2}+3276)}
  32. C ( x ) ( 1 / 135 ) ( 990791 x 9 π 4 - 147189744 x 5 π 2 + 8714684160 x ) ( 1749 π 4 x 8 + 523536 π 2 x 4 + 64553216 ) C(x)\approx\frac{(1/135)\cdot(990791x^{9}\pi^{4}-147189744x^{5}\pi^{2}+8714684% 160x)}{(1749\pi^{4}x^{8}+523536\pi^{2}x^{4}+64553216)}
  33. w 3 ( z ) 1 - z 2 / 108 + z 4 / 45360 1 + ( 17 / 108 ) z 2 + z 4 / 1008 w_{3}(z)\approx\frac{1-z^{2}/108+z^{4}/45360}{1+(17/108)z^{2}+z^{4}/1008}

Padovan_polynomials.html

  1. P n ( x ) = { 1 , if n = 1 0 , if n = 2 x , if n = 3 x P n - 2 ( x ) + P n - 3 ( x ) , if n 4. P_{n}(x)=\left\{\begin{matrix}1,&\mbox{if }~{}n=1\\ 0,&\mbox{if }~{}n=2\\ x,&\mbox{if }~{}n=3\\ xP_{n-2}(x)+P_{n-3}(x),&\mbox{if }~{}n\geq 4.\end{matrix}\right.
  2. P 1 ( x ) = 1 P_{1}(x)=1\,
  3. P 2 ( x ) = 0 P_{2}(x)=0\,
  4. P 3 ( x ) = x P_{3}(x)=x\,
  5. P 4 ( x ) = 1 P_{4}(x)=1\,
  6. P 5 ( x ) = x 2 P_{5}(x)=x^{2}\,
  7. P 6 ( x ) = 2 x P_{6}(x)=2x\,
  8. P 7 ( x ) = x 3 + 1 P_{7}(x)=x^{3}+1\,
  9. P 8 ( x ) = 3 x 2 P_{8}(x)=3x^{2}\,
  10. P 9 ( x ) = x 4 + 3 x P_{9}(x)=x^{4}+3x\,
  11. P 10 ( x ) = 4 x 3 + 1 P_{10}(x)=4x^{3}+1\,
  12. P 11 ( x ) = x 5 + 6 x 2 . P_{11}(x)=x^{5}+6x^{2}.\,
  13. n = 1 P n ( x ) t n = t 1 - x t 2 - t 3 . \sum_{n=1}^{\infty}P_{n}(x)t^{n}=\frac{t}{1-xt^{2}-t^{3}}.

Padovan_sequence.html

  1. P ( 1 ) = P ( 2 ) = P ( 3 ) = 1 , P(1)=P(2)=P(3)=1,
  2. P ( n ) = P ( n - 2 ) + P ( n - 3 ) . P(n)=P(n-2)+P(n-3).
  3. P ( n ) = P ( n - 1 ) + P ( n - 5 ) P(n)=P(n-1)+P(n-5)
  4. P ( m ) P(m)
  5. P ( m - 2 ) + P ( m - 3 ) P(m-2)+P(m-3)
  6. Perrin ( n ) = P ( n + 1 ) + P ( n - 10 ) . \mathrm{Perrin}(n)=P(n+1)+P(n-10).\,
  7. m = 0 n P ( m ) = P ( n + 5 ) - 2. \sum_{m=0}^{n}P(m)=P(n+5)-2.
  8. m = 0 n P ( 2 m ) = P ( 2 n + 3 ) - 1 \sum_{m=0}^{n}P(2m)=P(2n+3)-1
  9. m = 0 n P ( 2 m + 1 ) = P ( 2 n + 4 ) - 1 \sum_{m=0}^{n}P(2m+1)=P(2n+4)-1
  10. m = 0 n P ( 3 m ) = P ( 3 n + 2 ) \sum_{m=0}^{n}P(3m)=P(3n+2)
  11. m = 0 n P ( 3 m + 1 ) = P ( 3 n + 3 ) - 1 \sum_{m=0}^{n}P(3m+1)=P(3n+3)-1
  12. m = 0 n P ( 3 m + 2 ) = P ( 3 n + 4 ) - 1 \sum_{m=0}^{n}P(3m+2)=P(3n+4)-1
  13. m = 0 n P ( 5 m ) = P ( 5 n + 1 ) . \sum_{m=0}^{n}P(5m)=P(5n+1).
  14. m = 0 n P ( m ) 2 = P ( n + 2 ) 2 - P ( n - 1 ) 2 - P ( n - 3 ) 2 \sum_{m=0}^{n}P(m)^{2}=P(n+2)^{2}-P(n-1)^{2}-P(n-3)^{2}
  15. m = 0 n P ( m ) 2 P ( m + 1 ) = P ( n ) P ( n + 1 ) P ( n + 2 ) \sum_{m=0}^{n}P(m)^{2}P(m+1)=P(n)P(n+1)P(n+2)
  16. m = 0 n P ( m ) P ( m + 2 ) = P ( n + 2 ) P ( n + 3 ) - 1. \sum_{m=0}^{n}P(m)P(m+2)=P(n+2)P(n+3)-1.
  17. P ( n ) 2 - P ( n + 1 ) P ( n - 1 ) = P ( - n - 7 ) . P(n)^{2}-P(n+1)P(n-1)=P(-n-7).\,
  18. 2 m + n = k ( m n ) = P ( k - 2 ) . \sum_{2m+n=k}{m\choose n}=P(k-2).
  19. ( 6 0 ) + ( 5 2 ) + ( 4 4 ) = 1 + 10 + 1 = 12 = P ( 10 ) . {6\choose 0}+{5\choose 2}+{4\choose 4}=1+10+1=12=P(10).\,
  20. x 3 - x - 1 = 0. x^{3}-x-1=0.\,
  21. P ( n ) = a p n + b q n + c r n P\left(n\right)=ap^{n}+bq^{n}+cr^{n}
  22. P ( n ) - a p n P\left(n\right)-a{p^{n}}
  23. n 0 n\geq 0
  24. p n - 1 s \frac{p^{n-1}}{s}
  25. G ( P ( n ) ; x ) = 1 + x 1 - x 2 - x 3 . G(P(n);x)=\frac{1+x}{1-x^{2}-x^{3}}.
  26. n = 0 P ( n ) 2 n = 12 5 . \sum_{n=0}^{\infty}\frac{P(n)}{2^{n}}=\frac{12}{5}.
  27. n = 0 P ( n ) α n = α 2 ( α + 1 ) α 3 - α - 1 . \sum_{n=0}^{\infty}\frac{P(n)}{\alpha^{n}}=\frac{\alpha^{2}(\alpha+1)}{\alpha^% {3}-\alpha-1}.

Paleothermometer.html

  1. 155.8 * 10 - 6 155.8*10^{-6}
  2. 2005 * 10 - 6 2005*10^{-6}
  3. O = 1000 * ( ( 18 O / 16 O ) / ( 18 O / 16 O ) S M O W - 1 ) O=1000*((^{18}O/^{16}O)/(^{18}O/^{16}O)_{SMOW}-1)
  4. σ [ LMA ] = c P m a r g i n ( 1 - P m a r g i n ) r \sigma[\,\text{LMA}]=c\sqrt{\frac{P_{margin}(1-P_{margin})}{r}}

Pandigital_number.html

  1. b b - 1 + d = 2 b - 1 d b b - 1 - d b^{b-1}+\sum_{d=2}^{b-1}db^{b-1-d}
  2. 2 n - 1 2^{n}-1
  3. b x b^{x}

Panel_analysis.html

  1. y i t = a + b x i t + ϵ i t y_{it}=a+bx_{it}+\epsilon_{it}
  2. ϵ i t \epsilon_{it}
  3. ϵ i t \epsilon_{it}
  4. i i
  5. t t
  6. ϵ i t \epsilon_{it}
  7. i i
  8. t t

Panel_data.html

  1. person year income age sex 1 2001 1300 27 1 1 2002 1600 28 1 1 2003 2000 29 1 2 2001 2000 38 2 2 2002 2300 39 2 2 2003 2400 40 2 \begin{matrix}\mathrm{person}&\mathrm{year}&\mathrm{income}&\mathrm{age}&% \mathrm{sex}\\ 1&2001&1300&27&1\\ 1&2002&1600&28&1\\ 1&2003&2000&29&1\\ 2&2001&2000&38&2\\ 2&2002&2300&39&2\\ 2&2003&2400&40&2\end{matrix}
  2. person year income age sex 1 2001 1600 23 1 1 2002 1500 24 1 2 2001 1900 41 2 2 2002 2000 42 2 2 2003 2100 43 2 3 2002 3300 34 1 \begin{matrix}\mathrm{person}&\mathrm{year}&\mathrm{income}&\mathrm{age}&% \mathrm{sex}\\ 1&2001&1600&23&1\\ 1&2002&1500&24&1\\ 2&2001&1900&41&2\\ 2&2002&2000&42&2\\ 2&2003&2100&43&2\\ 3&2002&3300&34&1\end{matrix}
  3. X i t , i = 1 , , N t = 1 , , T , X_{it},\;i=1,\dots,N\;t=1,\dots,T,
  4. i i
  5. t t
  6. y i t = α + β X i t + u i t . y_{it}=\alpha+\beta^{\prime}X_{it}+u_{it}.
  7. y i t = α + β X i t + u i t , y_{it}=\alpha+\beta^{\prime}X_{it}+u_{it},
  8. u i t = μ i + ν i t . u_{it}=\mu_{i}+\nu_{it}.
  9. μ i \mu_{i}
  10. μ i i.i.d. N ( 0 , σ μ 2 ) \mu_{i}\sim\,\text{i.i.d.}N(0,\sigma^{2}_{\mu})
  11. ν i t i.i.d. N ( 0 , σ ν 2 ) , \nu_{it}\sim\,\text{i.i.d.}N(0,\sigma^{2}_{\nu}),

Paper_bag_problem.html

  1. V = w 3 ( h / ( π w ) - 0.142 ( 1 - 10 ( - h / w ) ) ) , V=w^{3}\left(h/\left(\pi w\right)-0.142\left(1-10^{\left(}-h/w\right)\right)% \right),
  2. V = w 3 ( h / ( π w ) - 0.071 ( 1 - 10 ( - 2 h / w ) ) ) V=w^{3}\left(h/\left(\pi w\right)-0.071\left(1-10^{\left(}-2h/w\right)\right)\right)
  3. V = 1 π - 0.142 0.9 V=\frac{1}{\pi}-0.142\cdot 0.9

Paraformer.html

  1. v ( t ) = L d i ( t ) d t v(t)=L\frac{di(t)}{dt}
  2. v ( t ) = I d l ( t ) d t v(t)=I\frac{dl(t)}{dt}

Parallel_projection.html

  1. 2 3 81.65 % \sqrt{\tfrac{2}{3}}\approx 81.65\%

Parallelizable_manifold.html

  1. M \scriptstyle M
  2. { V 1 , , V n } \{V_{1},\dots,V_{n}\}
  3. p \scriptstyle p
  4. M \scriptstyle M
  5. { V 1 ( p ) , , V n ( p ) } \{V_{1}(p),\dots,V_{n}(p)\}
  6. p \scriptstyle p
  7. M . \scriptstyle M.
  8. M \scriptstyle M
  9. M \scriptstyle M

Parameter_space.html

  1. Θ 1 Θ 2 . \Theta_{1}\leq\Theta_{2}.
  2. x n + 1 = r x n ( 1 - x n ) \qquad x_{n+1}=rx_{n}(1-x_{n})
  3. y ( t ) = A sin ( ω t + ϕ ) , y(t)=A\cdot\sin(\omega t+\phi),
  4. R + × R + × S 1 . R^{+}\times R^{+}\times S^{1}.

Parameterized_post-Newtonian_formalism.html

  1. γ \gamma
  2. g i j g_{ij}
  3. β \beta
  4. g 00 g_{00}
  5. β 1 \beta_{1}
  6. 1 2 ρ 0 v 2 \textstyle\frac{1}{2}\rho_{0}v^{2}
  7. β 2 \beta_{2}
  8. ρ 0 / U \rho_{0}/U
  9. β 3 \beta_{3}
  10. ρ 0 Π \rho_{0}\Pi
  11. β 4 \beta_{4}
  12. p p
  13. ζ \zeta
  14. η \eta
  15. Δ 1 \Delta_{1}
  16. g 0 j g_{0j}
  17. ρ 0 v \rho_{0}v
  18. Δ 2 \Delta_{2}
  19. g μ ν g_{\mu\nu}
  20. i i
  21. j j
  22. γ = β = β 1 = β 2 = β 3 = β 4 = Δ 1 = Δ 2 = 1 \gamma=\beta=\beta_{1}=\beta_{2}=\beta_{3}=\beta_{4}=\Delta_{1}=\Delta_{2}=1
  23. ζ = η = 0 \zeta=\eta=0
  24. γ = γ \gamma=\gamma
  25. β = β \beta=\beta
  26. α 1 = 7 Δ 1 + Δ 2 - 4 γ - 4 \alpha_{1}=7\Delta_{1}+\Delta_{2}-4\gamma-4
  27. α 2 = Δ 2 + ζ - 1 \alpha_{2}=\Delta_{2}+\zeta-1
  28. α 3 = 4 β 1 - 2 γ - 2 - ζ \alpha_{3}=4\beta_{1}-2\gamma-2-\zeta
  29. ζ 1 = ζ \zeta_{1}=\zeta
  30. ζ 2 = 2 β + 2 β 2 - 3 γ - 1 \zeta_{2}=2\beta+2\beta_{2}-3\gamma-1
  31. ζ 3 = β 3 - 1 \zeta_{3}=\beta_{3}-1
  32. ζ 4 = β 4 - γ \zeta_{4}=\beta_{4}-\gamma
  33. ξ \xi
  34. 3 η = 12 β - 3 γ - 9 + 10 ξ - 3 α 1 + 2 α 2 - 2 ζ 1 - ζ 2 3\eta=12\beta-3\gamma-9+10\xi-3\alpha_{1}+2\alpha_{2}-2\zeta_{1}-\zeta_{2}
  35. α 1 \alpha_{1}
  36. α 2 \alpha_{2}
  37. α 3 \alpha_{3}
  38. ζ 1 \zeta_{1}
  39. ζ 2 \zeta_{2}
  40. ζ 3 \zeta_{3}
  41. ζ 4 \zeta_{4}
  42. α 3 \alpha_{3}
  43. γ = β = 1 \gamma=\beta=1
  44. α 1 = α 2 = α 3 = ζ 1 = ζ 2 = ζ 3 = ζ 4 = ξ = 0 \alpha_{1}=\alpha_{2}=\alpha_{3}=\zeta_{1}=\zeta_{2}=\zeta_{3}=\zeta_{4}=\xi=0
  45. g 00 = - 1 + 2 U - 2 β U 2 - 2 ξ Φ W + ( 2 γ + 2 + α 3 + ζ 1 - 2 ξ ) Φ 1 + 2 ( 3 γ - 2 β + 1 + ζ 2 + ξ ) Φ 2 + 2 ( 1 + ζ 3 ) Φ 3 + 2 ( 3 γ + 3 ζ 4 - 2 ξ ) Φ 4 - ( ζ 1 - 2 ξ ) A - ( α 1 - α 2 - α 3 ) w 2 U - α 2 w i w j U i j + ( 2 α 3 - α 1 ) w i V i + O ( ϵ 3 ) \begin{matrix}g_{00}=-1+2U-2\beta U^{2}-2\xi\Phi_{W}+(2\gamma+2+\alpha_{3}+% \zeta_{1}-2\xi)\Phi_{1}+2(3\gamma-2\beta+1+\zeta_{2}+\xi)\Phi_{2}\\ \ +2(1+\zeta_{3})\Phi_{3}+2(3\gamma+3\zeta_{4}-2\xi)\Phi_{4}-(\zeta_{1}-2\xi)A% -(\alpha_{1}-\alpha_{2}-\alpha_{3})w^{2}U\\ \ -\alpha_{2}w^{i}w^{j}U_{ij}+(2\alpha_{3}-\alpha_{1})w^{i}V_{i}+O(\epsilon^{3% })\end{matrix}
  46. g 0 i = - 1 2 ( 4 γ + 3 + α 1 - α 2 + ζ 1 - 2 ξ ) V i - 1 2 ( 1 + α 2 - ζ 1 + 2 ξ ) W i - 1 2 ( α 1 - 2 α 2 ) w i U - α 2 w j U i j + O ( ϵ 5 2 ) g_{0i}=-\textstyle\frac{1}{2}(4\gamma+3+\alpha_{1}-\alpha_{2}+\zeta_{1}-2\xi)V% _{i}-\textstyle\frac{1}{2}(1+\alpha_{2}-\zeta_{1}+2\xi)W_{i}-\textstyle\frac{1% }{2}(\alpha_{1}-2\alpha_{2})w^{i}U-\alpha_{2}w^{j}U_{ij}+O(\epsilon^{\frac{5}{% 2}})\;
  47. g i j = ( 1 + 2 γ U ) δ i j + O ( ϵ 2 ) g_{ij}=(1+2\gamma U)\delta_{ij}+O(\epsilon^{2})\;
  48. ϵ \epsilon
  49. U U
  50. w i w^{i}
  51. w 2 = δ i j w i w j w^{2}=\delta_{ij}w^{i}w^{j}
  52. δ i j = 1 \delta_{ij}=1
  53. i = j i=j
  54. 0
  55. U U
  56. U i j U_{ij}
  57. Φ W \Phi_{W}
  58. A A
  59. Φ 1 \Phi_{1}
  60. Φ 2 \Phi_{2}
  61. Φ 3 \Phi_{3}
  62. Φ 4 \Phi_{4}
  63. V i V_{i}
  64. W i W_{i}
  65. U ( 𝐱 , t ) = ρ ( 𝐱 , t ) | 𝐱 - 𝐱 | d 3 x U(\mathbf{x},t)=\int{\rho(\mathbf{x}^{\prime},t)\over|\mathbf{x}-\mathbf{x}^{% \prime}|}d^{3}x^{\prime}
  66. U i j = ρ ( 𝐱 , t ) ( x - x ) i ( x - x ) j | 𝐱 - 𝐱 | 3 d 3 x U_{ij}=\int{\rho(\mathbf{x}^{\prime},t)(x-x^{\prime})_{i}(x-x^{\prime})_{j}% \over|\mathbf{x}-\mathbf{x}^{\prime}|^{3}}d^{3}x^{\prime}
  67. Φ W = ρ ( 𝐱 , t ) ρ ( 𝐱 ′′ , t ) ( x - x ) i | 𝐱 - 𝐱 | 3 ( ( x - x ′′ ) i | 𝐱 - 𝐱 | - ( x - x ′′ ) i | 𝐱 - 𝐱 ′′ | ) d 3 x d 3 x ′′ \Phi_{W}=\int{\rho(\mathbf{x}^{\prime},t)\rho(\mathbf{x}^{\prime\prime},t)(x-x% ^{\prime})_{i}\over|\mathbf{x}-\mathbf{x}^{\prime}|^{3}}\left({(x^{\prime}-x^{% \prime\prime})^{i}\over|\mathbf{x}-\mathbf{x}^{\prime}|}-{(x-x^{\prime\prime})% ^{i}\over|\mathbf{x}^{\prime}-\mathbf{x}^{\prime\prime}|}\right)d^{3}x^{\prime% }d^{3}x^{\prime\prime}
  68. A = ρ ( 𝐱 , t ) ( 𝐯 ( 𝐱 , t ) ( 𝐱 - 𝐱 ) ) 2 | 𝐱 - 𝐱 | 3 d 3 x A=\int{\rho(\mathbf{x}^{\prime},t)\left(\mathbf{v}(\mathbf{x}^{\prime},t)\cdot% (\mathbf{x}-\mathbf{x}^{\prime})\right)^{2}\over|\mathbf{x}-\mathbf{x}^{\prime% }|^{3}}d^{3}x^{\prime}
  69. Φ 1 = ρ ( 𝐱 , t ) 𝐯 ( 𝐱 , t ) 2 | 𝐱 - 𝐱 | d 3 x \Phi_{1}=\int{\rho(\mathbf{x}^{\prime},t)\mathbf{v}(\mathbf{x}^{\prime},t)^{2}% \over|\mathbf{x}-\mathbf{x}^{\prime}|}d^{3}x^{\prime}
  70. Φ 2 = ρ ( 𝐱 , t ) U ( 𝐱 , t ) | 𝐱 - 𝐱 | d 3 x \Phi_{2}=\int{\rho(\mathbf{x}^{\prime},t)U(\mathbf{x}^{\prime},t)\over|\mathbf% {x}-\mathbf{x}^{\prime}|}d^{3}x^{\prime}
  71. Φ 3 = ρ ( 𝐱 , t ) Π ( 𝐱 , t ) | 𝐱 - 𝐱 | d 3 x \Phi_{3}=\int{\rho(\mathbf{x}^{\prime},t)\Pi(\mathbf{x}^{\prime},t)\over|% \mathbf{x}-\mathbf{x}^{\prime}|}d^{3}x^{\prime}
  72. Φ 4 = p ( 𝐱 , t ) | 𝐱 - 𝐱 | d 3 x \Phi_{4}=\int{p(\mathbf{x}^{\prime},t)\over|\mathbf{x}-\mathbf{x}^{\prime}|}d^% {3}x^{\prime}
  73. V i = ρ ( 𝐱 , t ) v ( 𝐱 , t ) i | 𝐱 - 𝐱 | d 3 x V_{i}=\int{\rho(\mathbf{x}^{\prime},t)v(\mathbf{x}^{\prime},t)_{i}\over|% \mathbf{x}-\mathbf{x}^{\prime}|}d^{3}x^{\prime}
  74. W i = ρ ( 𝐱 , t ) ( 𝐯 ( 𝐱 , t ) ( 𝐱 - 𝐱 ) ) ( x - x ) i | 𝐱 - 𝐱 | 3 d 3 x W_{i}=\int{\rho(\mathbf{x}^{\prime},t)\left(\mathbf{v}(\mathbf{x}^{\prime},t)% \cdot(\mathbf{x}-\mathbf{x}^{\prime})\right)(x-x^{\prime})_{i}\over|\mathbf{x}% -\mathbf{x}^{\prime}|^{3}}d^{3}x^{\prime}
  75. ρ \rho
  76. Π \Pi
  77. p p
  78. 𝐯 \mathbf{v}
  79. T 00 = ρ ( 1 + Π + 𝐯 2 + 2 U ) T^{00}=\rho(1+\Pi+\mathbf{v}^{2}+2U)
  80. T 0 i = ρ ( 1 + Π + 𝐯 2 + 2 U + p / ρ ) v i T^{0i}=\rho(1+\Pi+\mathbf{v}^{2}+2U+p/\rho)v^{i}
  81. T i j = ρ ( 1 + Π + 𝐯 2 + 2 U + p / ρ ) v i v j + p δ i j ( 1 - 2 γ U ) T^{ij}=\rho(1+\Pi+\mathbf{v}^{2}+2U+p/\rho)v^{i}v^{j}+p\delta^{ij}(1-2\gamma U)
  82. g μ ν g_{\mu\nu}\,
  83. ϕ \phi\,
  84. K μ K_{\mu}\,
  85. B μ ν B_{\mu\nu}\,
  86. η μ ν \eta_{\mu\nu}\,
  87. t t\,
  88. g μ ν ( 0 ) = diag ( - c 0 , c 1 , c 1 , c 1 ) g^{(0)}_{\mu\nu}=\mbox{diag}~{}(-c_{0},c_{1},c_{1},c_{1})\,
  89. ϕ 0 \phi_{0}\,
  90. K μ ( 0 ) K^{(0)}_{\mu}\,
  91. B μ ν ( 0 ) B^{(0)}_{\mu\nu}\,
  92. h μ ν = g μ ν - g μ ν ( 0 ) h_{\mu\nu}=g_{\mu\nu}-g^{(0)}_{\mu\nu}\,
  93. ϕ - ϕ 0 \phi-\phi_{0}\,
  94. K μ - K μ ( 0 ) K_{\mu}-K^{(0)}_{\mu}\,
  95. B μ ν - B μ ν ( 0 ) B_{\mu\nu}-B^{(0)}_{\mu\nu}\,
  96. h μ ν h_{\mu\nu}\,
  97. h 00 h_{00}\,
  98. O ( 2 ) O(2)\,
  99. h 00 = 2 α U h_{00}=2\alpha U\,
  100. U U\,
  101. α \alpha\,
  102. G G\,
  103. g 00 = - c 0 + 2 α U g_{00}=-c_{0}+2\alpha U\,
  104. g 0 j = 0 g_{0j}=0\,
  105. g i j = δ i j c 1 g_{ij}=\delta_{ij}c_{1}\,
  106. G today = α / c 0 c 1 = 1 G_{\mbox{today}~{}}=\alpha/c_{0}c_{1}=1\,
  107. h i j h_{ij}\,
  108. O ( 2 ) O(2)\,
  109. h 0 j h_{0j}\,
  110. O ( 3 ) O(3)\,
  111. h 00 h_{00}\,
  112. O ( 4 ) O(4)\,
  113. g μ ν g_{\mu\nu}\,
  114. 𝐠 = f s y m b o l η \mathbf{g}=fsymbol{\eta}\,
  115. γ = - 1 \gamma=-1\,
  116. 𝐠 = f 1 𝐝 t 𝐝 t + f 2 s y m b o l η \mathbf{g}=f_{1}\mathbf{d}t\otimes\mathbf{d}t+f_{2}symbol{\eta}\,
  117. α 1 = - 4 ( γ + 1 ) \alpha_{1}=-4(\gamma+1)\,
  118. ξ = β \xi=\beta\,
  119. ξ \xi
  120. α 2 \alpha_{2}
  121. α 2 \alpha_{2}\,
  122. α 2 < 4 × 10 - 7 \alpha_{2}<4\times 10^{-7}\,
  123. γ = 1 + ω 2 + ω \gamma=\textstyle\frac{1+\omega}{2+\omega}\,
  124. γ - 1 < 2.3 × 10 - 5 \gamma-1<2.3\times 10^{-5}\,
  125. ω \omega\,
  126. α 2 \alpha_{2}\,
  127. α 2 < 4 × 10 - 7 \alpha_{2}<4\times 10^{-7}\,
  128. γ - 1 \gamma-1
  129. 2.3 2.3
  130. 10 - 5 10^{-5}
  131. β - 1 \beta-1
  132. 3 3
  133. 10 - 3 10^{-3}
  134. β - 1 \beta-1
  135. 2.3 2.3
  136. 10 - 4 10^{-4}
  137. η N = 4 β - γ - 3 \eta_{N}=4\beta-\gamma-3
  138. ξ \xi
  139. 0.001 0.001
  140. α 1 \alpha_{1}
  141. 10 - 4 10^{-4}
  142. α 2 \alpha_{2}
  143. 4 4
  144. 10 - 7 10^{-7}
  145. α 3 \alpha_{3}
  146. 4 4
  147. 10 - 20 10^{-20}
  148. η N \eta_{N}
  149. 9 9
  150. 10 - 4 10^{-4}
  151. ζ 1 \zeta_{1}
  152. 0.02 0.02
  153. ζ 2 \zeta_{2}
  154. 4 4
  155. 10 - 5 10^{-5}
  156. ζ 3 \zeta_{3}
  157. 10 - 8 10^{-8}
  158. ζ 4 \zeta_{4}
  159. 0.006 0.006
  160. 6 ζ 4 = 3 α 3 + 2 ζ 1 - 3 ζ 3 6\zeta_{4}=3\alpha_{3}+2\zeta_{1}-3\zeta_{3}
  161. | ζ 4 | < 0.4 |\zeta_{4}|<0.4

Parametric_model.html

  1. 𝒫 = { P θ | θ Θ } . \mathcal{P}=\big\{P_{\theta}\ \big|\ \theta\in\Theta\big\}.
  2. 𝒫 = { f θ | θ Θ } . \mathcal{P}=\big\{f_{\theta}\ \big|\ \theta\in\Theta\big\}.
  3. 𝒫 = { p λ ( j ) = λ j j ! e - λ , j = 0 , 1 , 2 , 3 , | λ > 0 } , \mathcal{P}=\Big\{\ p_{\lambda}(j)=\tfrac{\lambda^{j}}{j!}e^{-\lambda},\ j=0,1% ,2,3,\dots\ \Big|\ \lambda>0\ \Big\},
  4. 𝒫 = { f θ ( x ) = 1 2 π σ e - 1 2 σ 2 ( x - μ ) 2 | μ , σ > 0 } . \mathcal{P}=\Big\{\ f_{\theta}(x)=\tfrac{1}{\sqrt{2\pi}\sigma}e^{-\frac{1}{2% \sigma^{2}}(x-\mu)^{2}}\ \Big|\ \mu\in\mathbb{R},\sigma>0\ \Big\}.
  5. 𝒫 = { f θ ( x ) = β λ ( x - μ λ ) β - 1 exp ( - ( x - μ λ ) β ) 1 { x > μ } | λ > 0 , β > 0 , μ } . \mathcal{P}=\Big\{\ f_{\theta}(x)=\tfrac{\beta}{\lambda}\left(\tfrac{x-\mu}{% \lambda}\right)^{\beta-1}\!\exp\!\big(\!-\!\big(\tfrac{x-\mu}{\lambda}\big)^{% \beta}\big)\,\mathbf{1}_{\{x>\mu\}}\ \Big|\ \lambda>0,\,\beta>0,\,\mu\in% \mathbb{R}\ \Big\}.
  6. μ \scriptstyle\mathcal{M}_{\mu}
  7. 𝒫 = { P θ | θ Θ } μ \mathcal{P}\!=\!\{P_{\theta}|\,\theta\in\Theta\}\subseteq\mathcal{M}_{\mu}
  8. θ s ( θ ) = d P θ / d μ \theta\mapsto s(\theta)=\sqrt{dP_{\theta}/d\mu}
  9. s ˙ ( θ ) = ( s ˙ 1 ( θ ) , , s ˙ k ( θ ) ) \dot{s}(\theta)=(\dot{s}_{1}(\theta),\,\ldots,\,\dot{s}_{k}(\theta))
  10. s ( θ + h ) - s ( θ ) - s ˙ ( θ ) h = o ( | h | ) as h 0 , \lVert s(\theta+h)-s(\theta)-\dot{s}(\theta)^{\prime}h\rVert=o(|h|)\ \ \,\text% {as }h\to 0,
  11. θ s ˙ ( θ ) \theta\mapsto\dot{s}(\theta)
  12. I ( θ ) = 4 s ˙ ( θ ) s ˙ ( θ ) d μ I(\theta)=4\int\dot{s}(\theta)\dot{s}(\theta)^{\prime}d\mu
  13. z θ = f θ f θ 𝟏 { f θ > 0 } z_{\theta}=\frac{\nabla f_{\theta}}{f_{\theta}}\cdot\mathbf{1}_{\{f_{\theta}>0\}}
  14. I θ = z θ z θ d P θ I_{\theta}=\int\!z_{\theta}z_{\theta}^{\prime}\,dP_{\theta}
  15. n \scriptstyle\sqrt{n}

Paravector.html

  1. 𝐯𝐯 = 𝐯 𝐯 \mathbf{v}\mathbf{v}=\mathbf{v}\cdot\mathbf{v}
  2. 𝐯 = 𝐮 + 𝐰 , \mathbf{v}=\mathbf{u}+\mathbf{w},
  3. ( 𝐮 + 𝐰 ) 2 = 𝐮𝐮 + 𝐮𝐰 + 𝐰𝐮 + 𝐰𝐰 , (\mathbf{u}+\mathbf{w})^{2}=\mathbf{u}\mathbf{u}+\mathbf{u}\mathbf{w}+\mathbf{% w}\mathbf{u}+\mathbf{w}\mathbf{w},
  4. 𝐮 𝐮 + 2 𝐮 𝐰 + 𝐰 𝐰 = 𝐮 𝐮 + 𝐮𝐰 + 𝐰𝐮 + 𝐰 𝐰 , \mathbf{u}\cdot\mathbf{u}+2\mathbf{u}\cdot\mathbf{w}+\mathbf{w}\cdot\mathbf{w}% =\mathbf{u}\cdot\mathbf{u}+\mathbf{u}\mathbf{w}+\mathbf{w}\mathbf{u}+\mathbf{w% }\cdot\mathbf{w},
  5. 𝐮 𝐰 = 1 2 ( 𝐮𝐰 + 𝐰𝐮 ) . \mathbf{u}\cdot\mathbf{w}=\frac{1}{2}\left(\mathbf{u}\mathbf{w}+\mathbf{w}% \mathbf{u}\right).
  6. 𝐮𝐰 + 𝐰𝐮 = 0 \mathbf{u}\mathbf{w}+\mathbf{w}\mathbf{u}=0
  7. C 3 C\ell_{3}
  8. { 1 , { 𝐞 1 , 𝐞 2 , 𝐞 3 } , { 𝐞 23 , 𝐞 31 , 𝐞 12 } , 𝐞 123 } , \{1,\{\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3}\},\{\mathbf{e}_{23},\mathbf% {e}_{31},\mathbf{e}_{12}\},\mathbf{e}_{123}\},
  9. 𝐞 23 = 𝐞 2 𝐞 3 . \mathbf{e}_{23}=\mathbf{e}_{2}\mathbf{e}_{3}.
  10. 1 1
  11. { 𝐞 1 , 𝐞 2 , 𝐞 3 } \{\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3}\}
  12. { 𝐞 23 , 𝐞 31 , 𝐞 12 } \{\mathbf{e}_{23},\mathbf{e}_{31},\mathbf{e}_{12}\}
  13. 𝐞 123 \mathbf{e}_{123}
  14. 𝐞 i 𝐞 j + 𝐞 j 𝐞 i = 2 δ i j \mathbf{e}_{i}\mathbf{e}_{j}+\mathbf{e}_{j}\mathbf{e}_{i}=2\delta_{ij}
  15. 𝐞 i 𝐞 j = - 𝐞 j 𝐞 i ; i j \mathbf{e}_{i}\mathbf{e}_{j}=-\mathbf{e}_{j}\mathbf{e}_{i}\,\,;i\neq j
  16. 𝐞 123 \mathbf{e}_{123}
  17. - 1 -1
  18. 𝐞 123 2 = 𝐞 1 𝐞 2 𝐞 3 𝐞 1 𝐞 2 𝐞 3 = 𝐞 2 𝐞 3 𝐞 2 𝐞 3 = - 𝐞 3 𝐞 3 = - 1. \mathbf{e}_{123}^{2}=\mathbf{e}_{1}\mathbf{e}_{2}\mathbf{e}_{3}\mathbf{e}_{1}% \mathbf{e}_{2}\mathbf{e}_{3}=\mathbf{e}_{2}\mathbf{e}_{3}\mathbf{e}_{2}\mathbf% {e}_{3}=-\mathbf{e}_{3}\mathbf{e}_{3}=-1.
  19. 𝐞 123 \mathbf{e}_{123}
  20. C ( 3 ) C\ell(3)
  21. i i
  22. 𝐞 123 \mathbf{e}_{123}
  23. 1 1
  24. { 𝐞 1 , 𝐞 2 , 𝐞 3 } \{\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3}\}
  25. { i 𝐞 1 , i 𝐞 2 , i 𝐞 3 } \{i\mathbf{e}_{1},i\mathbf{e}_{2},i\mathbf{e}_{3}\}
  26. i i
  27. { 1 , 𝐞 1 , 𝐞 2 , 𝐞 3 } \{1,\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3}\}
  28. C 3 C\ell_{3}
  29. 1 = 𝐞 0 1=\mathbf{e}_{0}
  30. { 𝐞 μ } , \{\mathbf{e}_{\mu}\},
  31. μ \mu
  32. 0
  33. 3 3
  34. \dagger
  35. ( A B ) = B A (AB)^{\dagger}=B^{\dagger}A^{\dagger}
  36. 𝐚 = 𝐚 \mathbf{a}^{\dagger}=\mathbf{a}
  37. 1 = 1 1^{\dagger}=1
  38. 1 1
  39. 𝐞 1 \mathbf{e}_{1}
  40. 𝐞 2 \mathbf{e}_{2}
  41. 𝐞 3 \mathbf{e}_{3}
  42. 𝐞 12 \mathbf{e}_{12}
  43. 𝐞 23 \mathbf{e}_{23}
  44. 𝐞 31 \mathbf{e}_{31}
  45. 𝐞 123 \mathbf{e}_{123}
  46. ¯ \bar{}
  47. 𝐚 ¯ = - 𝐚 \bar{\mathbf{a}}=-\mathbf{a}
  48. 1 ¯ = 1 \bar{1}=1
  49. A B ¯ = B ¯ A ¯ \overline{AB}=\overline{B}\,\,\overline{A}
  50. 1 1
  51. 𝐞 1 \mathbf{e}_{1}
  52. 𝐞 2 \mathbf{e}_{2}
  53. 𝐞 3 \mathbf{e}_{3}
  54. 𝐞 12 \mathbf{e}_{12}
  55. 𝐞 23 \mathbf{e}_{23}
  56. 𝐞 31 \mathbf{e}_{31}
  57. 𝐞 123 \mathbf{e}_{123}
  58. A B ¯ = A ¯ B ¯ \overline{AB}^{\dagger}=\overline{A}^{\dagger}\overline{B}^{\dagger}
  59. 1 1
  60. 𝐞 1 \mathbf{e}_{1}
  61. 𝐞 2 \mathbf{e}_{2}
  62. 𝐞 3 \mathbf{e}_{3}
  63. 𝐞 12 \mathbf{e}_{12}
  64. 𝐞 23 \mathbf{e}_{23}
  65. 𝐞 31 \mathbf{e}_{31}
  66. 𝐞 123 \mathbf{e}_{123}
  67. C 3 C\ell_{3}
  68. p p
  69. p p
  70. p S = 1 2 ( p + p ¯ ) , \langle p\rangle_{S}=\frac{1}{2}(p+\overline{p}),
  71. p V = 1 2 ( p - p ¯ ) \langle p\rangle_{V}=\frac{1}{2}(p-\overline{p})
  72. p p
  73. p R = 1 2 ( p + p ) , \langle p\rangle_{R}=\frac{1}{2}(p+p^{\dagger}),
  74. p I = 1 2 ( p - p ) \langle p\rangle_{I}=\frac{1}{2}(p-p^{\dagger})
  75. p R S = p S R p R S \langle p\rangle_{RS}=\langle p\rangle_{SR}\equiv\langle\langle p\rangle_{R}% \rangle_{S}
  76. p R V = p V R p R V \langle p\rangle_{RV}=\langle p\rangle_{VR}\equiv\langle\langle p\rangle_{R}% \rangle_{V}
  77. p I V = p V I p I V \langle p\rangle_{IV}=\langle p\rangle_{VI}\equiv\langle\langle p\rangle_{I}% \rangle_{V}
  78. p I S = p S I p I S \langle p\rangle_{IS}=\langle p\rangle_{SI}\equiv\langle\langle p\rangle_{I}% \rangle_{S}
  79. C 3 C\ell_{3}
  80. 𝐞 123 = i \mathbf{e}_{123}=i
  81. - 𝐞 23 = i -\mathbf{e}_{23}=i
  82. - 𝐞 31 = j -\mathbf{e}_{31}=j
  83. - 𝐞 12 = k -\mathbf{e}_{12}=k
  84. u u
  85. v v
  86. u v ¯ S . \langle u\bar{v}\rangle_{S}.
  87. u u
  88. u u ¯ S , \langle u\bar{u}\rangle_{S},
  89. η μ ν = 𝐞 μ 𝐞 ¯ ν S \eta_{\mu\nu}=\langle\mathbf{e}_{\mu}\bar{\mathbf{e}}_{\nu}\rangle_{S}
  90. η 00 = 𝐞 0 𝐞 ¯ 0 = 1 ( 1 ) S = 1 , \eta_{00}=\langle\mathbf{e}_{0}\bar{\mathbf{e}}_{0}\rangle=\langle 1(1)\rangle% _{S}=1,
  91. η 11 = 𝐞 1 𝐞 ¯ 1 = 𝐞 1 ( - 𝐞 1 ) S = - 1 , \eta_{11}=\langle\mathbf{e}_{1}\bar{\mathbf{e}}_{1}\rangle=\langle\mathbf{e}_{% 1}(-\mathbf{e}_{1})\rangle_{S}=-1,
  92. η 01 = 𝐞 0 𝐞 ¯ 1 = 1 ( - 𝐞 1 ) S = 0. \eta_{01}=\langle\mathbf{e}_{0}\bar{\mathbf{e}}_{1}\rangle=\langle 1(-\mathbf{% e}_{1})\rangle_{S}=0.
  93. u u
  94. v v
  95. B = u v ¯ V B=\langle u\bar{v}\rangle_{V}
  96. { 𝐞 μ 𝐞 ¯ ν V } , \{\langle\mathbf{e}_{\mu}\bar{\mathbf{e}}_{\nu}\rangle_{V}\},
  97. 𝐞 0 𝐞 ¯ k V = - 𝐞 k , \langle\mathbf{e}_{0}\bar{\mathbf{e}}_{k}\rangle_{V}=-\mathbf{e}_{k},
  98. 𝐞 j 𝐞 ¯ k V = - 𝐞 j k \langle\mathbf{e}_{j}\bar{\mathbf{e}}_{k}\rangle_{V}=-\mathbf{e}_{jk}
  99. j , k j,k
  100. F = 𝐄 + i 𝐁 , F=\mathbf{E}+i\mathbf{B},
  101. 𝐄 = 𝐄 \mathbf{E}^{\dagger}=\mathbf{E}
  102. 𝐁 = 𝐁 \mathbf{B}^{\dagger}=\mathbf{B}
  103. i i
  104. W = i θ j 𝐞 j + η j 𝐞 j , W=i\theta^{j}\mathbf{e}_{j}+\eta^{j}\mathbf{e}_{j},
  105. θ j \theta^{j}
  106. η j \eta^{j}
  107. u u
  108. v v
  109. w w
  110. T = u v ¯ w I T=\langle u\bar{v}w\rangle_{I}
  111. { 𝐞 μ 𝐞 ¯ ν 𝐞 λ I } , \{\langle\mathbf{e}_{\mu}\bar{\mathbf{e}}_{\nu}\mathbf{e}_{\lambda}\rangle_{I}\},
  112. { i 𝐞 ρ } \{i\mathbf{e}_{\rho}\}
  113. { 𝐞 μ 𝐞 ¯ ν 𝐞 λ 𝐞 ¯ ρ I S } , \{\langle\mathbf{e}_{\mu}\bar{\mathbf{e}}_{\nu}\mathbf{e}_{\lambda}\bar{% \mathbf{e}}_{\rho}\rangle_{IS}\},
  114. i = 𝐞 1 𝐞 2 𝐞 3 i=\mathbf{e}_{1}\mathbf{e}_{2}\mathbf{e}_{3}
  115. = 𝐞 0 0 - 𝐞 1 1 - 𝐞 2 2 - 𝐞 3 3 , \partial=\mathbf{e}_{0}\partial_{0}-\mathbf{e}_{1}\partial_{1}-\mathbf{e}_{2}% \partial_{2}-\mathbf{e}_{3}\partial_{3},
  116. = ¯ S = ¯ S \square=\langle\bar{\partial}\partial\rangle_{S}=\langle\partial\bar{\partial}% \rangle_{S}
  117. = 𝐞 1 1 + 𝐞 2 2 + 𝐞 3 3 , \nabla=\mathbf{e}_{1}\partial_{1}+\mathbf{e}_{2}\partial_{2}+\mathbf{e}_{3}% \partial_{3},
  118. = 0 - , \partial=\partial_{0}-\nabla,
  119. 𝐞 0 = 1 \mathbf{e}_{0}=1
  120. e f ( x ) 𝐞 3 = ( f ( x ) ) e f ( x ) 𝐞 3 𝐞 3 , \partial e^{f(x)\mathbf{e}_{3}}=(\partial f(x))e^{f(x)\mathbf{e}_{3}}\mathbf{e% }_{3},
  121. f ( x ) f(x)
  122. ( L ) = 𝐞 0 0 L + ( 1 L ) 𝐞 1 + ( 2 L ) 𝐞 2 + ( 3 L ) 𝐞 3 (L\partial)=\mathbf{e}_{0}\partial_{0}L+(\partial_{1}L)\mathbf{e}_{1}+(% \partial_{2}L)\mathbf{e}_{2}+(\partial_{3}L)\mathbf{e}_{3}
  123. p p
  124. p p ¯ = 0. p\bar{p}=0.
  125. P 𝐤 = 1 2 ( 1 + 𝐤 ^ ) , P_{\mathbf{k}}=\frac{1}{2}(1+\hat{\mathbf{k}}),
  126. 𝐤 ^ \hat{\mathbf{k}}
  127. P 𝐤 P_{\mathbf{k}}
  128. P ¯ 𝐤 \bar{P}_{\mathbf{k}}
  129. P ¯ 𝐤 = 1 2 ( 1 - 𝐤 ^ ) , \bar{P}_{\mathbf{k}}=\frac{1}{2}(1-\hat{\mathbf{k}}),
  130. P 𝐤 + P ¯ 𝐤 = 1 P_{\mathbf{k}}+\bar{P}_{\mathbf{k}}=1
  131. P 𝐤 = P 𝐤 P 𝐤 = P 𝐤 P 𝐤 P 𝐤 = P_{\mathbf{k}}=P_{\mathbf{k}}P_{\mathbf{k}}=P_{\mathbf{k}}P_{\mathbf{k}}P_{% \mathbf{k}}=...
  132. P 𝐤 P ¯ 𝐤 = 0. P_{\mathbf{k}}\bar{P}_{\mathbf{k}}=0.
  133. 𝐤 ^ = P 𝐤 - P ¯ 𝐤 , \hat{\mathbf{k}}=P_{\mathbf{\mathbf{k}}}-\bar{P}_{\mathbf{k}},
  134. 𝐤 ^ \hat{\mathbf{k}}
  135. P 𝐤 P_{\mathbf{\mathbf{k}}}
  136. P ¯ 𝐤 \bar{P}_{\mathbf{\mathbf{k}}}
  137. 1 1
  138. - 1 -1
  139. f ( 𝐤 ^ ) f(\hat{\mathbf{k}})
  140. f ( 𝐤 ^ ) = f ( 1 ) P 𝐤 + f ( - 1 ) P ¯ 𝐤 . f(\hat{\mathbf{k}})=f(1)P_{\mathbf{k}}+f(-1)\bar{P}_{\mathbf{k}}.
  141. f ( 𝐤 ^ ) P 𝐤 = f ( 1 ) P 𝐤 , f(\hat{\mathbf{k}})P_{\mathbf{k}}=f(1)P_{\mathbf{k}},
  142. f ( 𝐤 ^ ) P ¯ 𝐤 = f ( - 1 ) P ¯ 𝐤 . f(\hat{\mathbf{k}})\bar{P}_{\mathbf{k}}=f(-1)\bar{P}_{\mathbf{k}}.
  143. C 3 C\ell_{3}
  144. { P ¯ 3 , P 3 𝐞 1 , P 3 , 𝐞 1 P 3 } \{\bar{P}_{3},P_{3}\mathbf{e}_{1},P_{3},\mathbf{e}_{1}P_{3}\}
  145. p = p 0 𝐞 0 + p 1 𝐞 1 + p 2 𝐞 2 + p 3 𝐞 3 p=p^{0}\mathbf{e}_{0}+p^{1}\mathbf{e}_{1}+p^{2}\mathbf{e}_{2}+p^{3}\mathbf{e}_% {3}
  146. p = ( p 0 + p 3 ) P 3 + ( p 0 - p 3 ) P ¯ 3 + ( p 1 + i p 2 ) 𝐞 1 P 3 + ( p 1 - i p 2 ) P 3 𝐞 1 p=(p^{0}+p^{3})P_{3}+(p^{0}-p^{3})\bar{P}_{3}+(p^{1}+ip^{2})\mathbf{e}_{1}P_{3% }+(p^{1}-ip^{2})P_{3}\mathbf{e}_{1}
  147. P 3 P_{3}
  148. P ¯ 3 \bar{P}_{3}
  149. p p
  150. { u , v } \{u,v\}
  151. w w
  152. p = u P ¯ 3 + v P 3 + w 𝐞 1 P 3 + w P 3 𝐞 1 p=u\bar{P}_{3}+vP_{3}+w\mathbf{e}_{1}P_{3}+w^{\dagger}P_{3}\mathbf{e}_{1}
  153. = 2 P 3 u + 2 P ¯ 3 v - 2 𝐞 1 P 3 w - 2 P 3 𝐞 1 w \partial=2P_{3}\partial_{u}+2\bar{P}_{3}\partial_{v}-2\mathbf{e}_{1}P_{3}% \partial_{w^{\dagger}}-2P_{3}\mathbf{e}_{1}\partial_{w}
  154. ( n 2 ) \begin{pmatrix}n\\ 2\end{pmatrix}
  155. ( n m ) \begin{pmatrix}n\\ m\end{pmatrix}
  156. C ( n ) C\ell(n)
  157. 2 n 2^{n}
  158. \dagger
  159. 0
  160. 1 1
  161. 2 2
  162. 3 3
  163. 4 4
  164. 5 5
  165. 6 6
  166. 7 7
  167. \vdots
  168. C ( 3 ) C\ell(3)
  169. 𝐞 0 \mathbf{e}_{0}
  170. σ 0 \sigma_{0}
  171. ( 1 0 0 1 ) \begin{pmatrix}1&&0\\ 0&&1\end{pmatrix}
  172. 𝐞 1 \mathbf{e}_{1}
  173. σ 1 \sigma_{1}
  174. ( 0 1 1 0 ) \begin{pmatrix}0&&1\\ 1&&0\end{pmatrix}
  175. 𝐞 2 \mathbf{e}_{2}
  176. σ 2 \sigma_{2}
  177. ( 0 - i i 0 ) \begin{pmatrix}0&&-i\\ i&&0\end{pmatrix}
  178. 𝐞 3 \mathbf{e}_{3}
  179. σ 3 \sigma_{3}
  180. ( 1 0 0 - 1 ) \begin{pmatrix}1&&0\\ 0&&-1\end{pmatrix}
  181. P 3 = ( 1 0 0 0 ) ; P ¯ 3 = ( 0 0 0 1 ) ; P 3 𝐞 1 = ( 0 1 0 0 ) ; 𝐞 1 P 3 = ( 0 0 1 0 ) . {P_{3}}=\begin{pmatrix}1&0\\ 0&0\end{pmatrix}\,;\bar{P}_{3}=\begin{pmatrix}0&0\\ 0&1\end{pmatrix}\,;{P_{3}}\mathbf{e}_{1}=\begin{pmatrix}0&1\\ 0&0\end{pmatrix}\,;\mathbf{e}_{1}{P}_{3}=\begin{pmatrix}0&0\\ 1&0\end{pmatrix}.
  182. Ψ = ψ 11 P 3 - ψ 12 P 3 𝐞 1 + ψ 21 𝐞 1 P 3 + ψ 22 P ¯ 3 , \Psi=\psi_{11}P_{3}-\psi_{12}P_{3}\mathbf{e}_{1}+\psi_{21}\mathbf{e}_{1}P_{3}+% \psi_{22}\bar{P}_{3},
  183. ψ j k \psi_{jk}
  184. Ψ ( ψ 11 ψ 12 ψ 21 ψ 22 ) \Psi\rightarrow\begin{pmatrix}\psi_{11}&\psi_{12}\\ \psi_{21}&\psi_{22}\end{pmatrix}
  185. Ψ ¯ ( ψ 22 - ψ 12 - ψ 21 ψ 11 ) , \bar{\Psi}\rightarrow\begin{pmatrix}\psi_{22}&-\psi_{12}\\ -\psi_{21}&\psi_{11}\end{pmatrix},
  186. Ψ S ψ 11 + ψ 22 2 ( 1 0 0 1 ) = T r [ ψ ] 2 𝟏 2 × 2 \langle\Psi\rangle_{S}\rightarrow\frac{\psi_{11}+\psi_{22}}{2}\begin{pmatrix}1% &0\\ 0&1\end{pmatrix}=\frac{Tr[\psi]}{2}\mathbf{1}_{2\times 2}
  187. Ψ V ( 0 ψ 12 ψ 21 0 ) \langle\Psi\rangle_{V}\rightarrow\begin{pmatrix}0&\psi_{12}\\ \psi_{21}&0\end{pmatrix}
  188. Ψ R 1 2 ( ψ 11 + ψ 11 * ψ 12 + ψ 21 * ψ 21 + ψ 12 * ψ 22 + ψ 22 * ) \langle\Psi\rangle_{R}\rightarrow\frac{1}{2}\begin{pmatrix}\psi_{11}+\psi_{11}% ^{*}&\psi_{12}+\psi_{21}^{*}\\ \psi_{21}+\psi_{12}^{*}&\psi_{22}+\psi_{22}^{*}\end{pmatrix}
  189. Ψ I 1 2 ( ψ 11 - ψ 11 * ψ 12 - ψ 21 * ψ 21 - ψ 12 * ψ 22 - ψ 22 * ) \langle\Psi\rangle_{I}\rightarrow\frac{1}{2}\begin{pmatrix}\psi_{11}-\psi_{11}% ^{*}&\psi_{12}-\psi_{21}^{*}\\ \psi_{21}-\psi_{12}^{*}&\psi_{22}-\psi_{22}^{*}\end{pmatrix}
  190. 2 n 2^{n}
  191. 𝐞 1 \mathbf{e}_{1}
  192. σ 3 σ 1 \sigma_{3}\otimes\sigma_{1}
  193. 𝐞 2 \mathbf{e}_{2}
  194. σ 3 σ 2 \sigma_{3}\otimes\sigma_{2}
  195. 𝐞 3 \mathbf{e}_{3}
  196. σ 3 σ 3 \sigma_{3}\otimes\sigma_{3}
  197. 𝐞 4 \mathbf{e}_{4}
  198. σ 2 σ 0 \sigma_{2}\otimes\sigma_{0}
  199. 𝐞 1 \mathbf{e}_{1}
  200. σ 0 σ 3 σ 1 \sigma_{0}\otimes\sigma_{3}\otimes\sigma_{1}
  201. 𝐞 2 \mathbf{e}_{2}
  202. σ 0 σ 3 σ 2 \sigma_{0}\otimes\sigma_{3}\otimes\sigma_{2}
  203. 𝐞 3 \mathbf{e}_{3}
  204. σ 0 σ 3 σ 3 \sigma_{0}\otimes\sigma_{3}\otimes\sigma_{3}
  205. 𝐞 4 \mathbf{e}_{4}
  206. σ 0 σ 2 σ 0 \sigma_{0}\otimes\sigma_{2}\otimes\sigma_{0}
  207. 𝐞 5 \mathbf{e}_{5}
  208. σ 3 σ 1 σ 0 \sigma_{3}\otimes\sigma_{1}\otimes\sigma_{0}
  209. 𝐞 6 \mathbf{e}_{6}
  210. σ 1 σ 1 σ 0 \sigma_{1}\otimes\sigma_{1}\otimes\sigma_{0}
  211. 𝐞 7 \mathbf{e}_{7}
  212. σ 2 σ 1 σ 0 \sigma_{2}\otimes\sigma_{1}\otimes\sigma_{0}
  213. s p i n ( n ) spin(n)
  214. s p i n ( 3 ) spin(3)
  215. s u ( 2 ) su(2)
  216. s p i n ( 3 ) spin(3)
  217. S L ( 2 , C ) SL(2,C)
  218. S O ( 3 , 1 ) SO(3,1)
  219. S L ( 2 , C ) SL(2,C)
  220. s u ( N ) su(N)
  221. s p i n ( 6 ) spin(6)
  222. s u ( 4 ) su(4)
  223. s p i n ( 5 ) spin(5)
  224. s p ( 4 ) sp(4)
  225. s p ( 4 ) sp(4)
  226. U S p ( 4 ) USp(4)
  227. S U ( 4 ) SU(4)

Paraxial_approximation.html

  1. sin θ θ , tan θ θ and cos θ 1. \sin\theta\approx\theta,\quad\tan\theta\approx\theta\quad\,\text{and}\quad\cos% \theta\approx 1.
  2. cos θ 1 - θ 2 2 . \cos\theta\approx 1-{\theta^{2}\over 2}\ .

Partial_order_reduction.html

  1. a m p l e ( s ) = e n a b l e d ( s ) = {ample(s)=}\iff{enabled(s)=}
  2. α \alpha
  3. e n a b l e d ( s ) a m p l e ( s ) enabled(s)\neq ample(s)
  4. α \alpha
  5. T ( s ) T(s)
  6. a T ( s ) b 1 , , b n T ( s ) \forall a\in T(s)\forall b_{1},...,b_{n}\notin T(s)
  7. b 1 , , b n , a b_{1},...,b_{n},a
  8. s s^{\prime}
  9. a , b 1 , , b n a,b_{1},...,b_{n}
  10. s s^{\prime}
  11. s s
  12. a T ( s ) \exists a\in T(s)
  13. b 1 , , b n T ( s ) \forall b_{1},...,b_{n}\notin T(s)
  14. b 1 , , b n , a b_{1},...,b_{n},a

Partially_observable_Markov_decision_process.html

  1. ( S , A , T , R , Ω , O , γ ) (S,A,T,R,\Omega,O,\gamma)
  2. S S
  3. A A
  4. T T
  5. R : S × A R:S\times A\to\mathbb{R}
  6. Ω \Omega
  7. O O
  8. γ [ 0 , 1 ] \gamma\in[0,1]
  9. s S s\in S
  10. a A a\in A
  11. s s^{\prime}
  12. T ( s s , a ) T(s^{\prime}\mid s,a)
  13. o Ω o\in\Omega
  14. O ( o s , a ) O(o\mid s^{\prime},a)
  15. R ( s , a ) R(s,a)
  16. E [ t = 0 γ t r t ] E\left[\sum_{t=0}^{\infty}\gamma^{t}r_{t}\right]
  17. γ \gamma
  18. γ = 0 \gamma=0
  19. γ = 1 \gamma=1
  20. a a
  21. o o
  22. b = τ ( b , a , o ) b^{\prime}=\tau(b,a,o)
  23. s s^{\prime}
  24. o Ω o\in\Omega
  25. O ( o s , a ) O(o\mid s^{\prime},a)
  26. b b
  27. S S
  28. b ( s ) b(s)
  29. s s
  30. b ( s ) b(s)
  31. a a
  32. o o
  33. b ( s ) = η O ( o s , a ) s S T ( s s , a ) b ( s ) b^{\prime}(s^{\prime})=\eta O(o\mid s^{\prime},a)\sum_{s\in S}T(s^{\prime}\mid s% ,a)b(s)
  34. η = 1 / Pr ( o b , a ) \eta=1/\Pr(o\mid b,a)
  35. Pr ( o b , a ) = s S O ( o s , a ) s S T ( s s , a ) b ( s ) \Pr(o\mid b,a)=\sum_{s^{\prime}\in S}O(o\mid s^{\prime},a)\sum_{s\in S}T(s^{% \prime}\mid s,a)b(s)
  36. B B
  37. S S
  38. ( B , A , τ , r , γ ) (B,A,\tau,r,\gamma)
  39. B B
  40. A A
  41. τ \tau
  42. r : B × A r:B\times A\to\mathbb{R}
  43. γ \gamma
  44. γ \gamma
  45. τ \tau
  46. r r
  47. τ \tau
  48. τ ( b , a , b ) = o Ω P r ( b | b , a , o ) Pr ( o | a , b ) , \tau(b,a,b^{\prime})=\sum_{o\in\Omega}Pr(b^{\prime}|b,a,o)\Pr(o|a,b),
  49. Pr ( o | a , b ) \Pr(o|a,b)
  50. P r ( b | b , a , o ) = { 1 if the belief update with arguments b , a , o returns b 0 otherwise . Pr(b^{\prime}|b,a,o)=\begin{cases}1&\,\text{if the belief update with % arguments }b,a,o\,\text{ returns }b^{\prime}\\ 0&\,\text{otherwise }\end{cases}.
  51. r r
  52. r ( b , a ) = s S b ( s ) R ( s , a ) r(b,a)=\sum_{s\in S}b(s)R(s,a)
  53. π \pi
  54. a = π ( b ) a=\pi(b)
  55. b b
  56. R R
  57. π \pi
  58. b 0 b_{0}
  59. V π ( b 0 ) = t = 0 γ t r ( b t , a t ) = t = 0 γ t E [ R ( s t , a t ) b 0 , π ] V^{\pi}(b_{0})=\sum_{t=0}^{\infty}\gamma^{t}r(b_{t},a_{t})=\sum_{t=0}^{\infty}% \gamma^{t}E\Bigl[R(s_{t},a_{t})\mid b_{0},\pi\Bigr]
  60. γ < 1 \gamma<1
  61. π * \pi^{*}
  62. π * = argmax 𝜋 V π ( b 0 ) \pi^{*}=\underset{\pi}{\mbox{argmax}~{}}\ V^{\pi}(b_{0})
  63. b 0 b_{0}
  64. π * \pi^{*}
  65. V * V^{*}
  66. V * ( b ) = max a A [ r ( b , a ) + γ o Ω O ( o b , a ) V * ( τ ( b , a , o ) ) ] V^{*}(b)=\max_{a\in A}\Bigl[r(b,a)+\gamma\sum_{o\in\Omega}O(o\mid b,a)V^{*}(% \tau(b,a,o))\Bigr]
  67. V * V^{*}
  68. ϵ \epsilon

Particle-beam_weapon.html

  1. 10 9 10^{9}

Particle_shower.html

  1. X 0 X_{0}
  2. X 0 X_{0}
  3. X = X 0 ln ( E 0 / E c ) ln 2 , X=X_{0}\frac{\ln(E_{0}/E_{\mathrm{c}})}{\ln 2},
  4. X 0 X_{0}
  5. E c E_{\mathrm{c}}
  6. E c = 800 Mev / ( Z + 1.2 ) E_{\mathrm{c}}=800\,\mathrm{Mev}/(Z+1.2)
  7. 2 R M 2R_{\mathrm{M}}
  8. d E d t = E 0 b ( b t ) a - 1 e - b t Γ ( a ) \frac{dE}{dt}=E_{0}b\frac{(bt)^{a-1}e^{-bt}}{\Gamma(a)}
  9. t = X / X 0 t=X/X_{0}
  10. E 0 E_{0}
  11. a a
  12. b b
  13. λ = A N A σ abs \lambda=\frac{A}{N_{A}\sigma_{\mathrm{abs}}}
  14. d Π ( E , x ) d x = 2 E Γ ( u , x ) γ ( u , E ) d u + E Π ( u , x ) π ( u , u - E ) d u - 0 E Π ( E , x ) π ( E , E - u ) d u d Γ ( E , x ) d x = E Π ( u , x ) π ( u , E ) d u - 0 E Γ ( E , x ) γ ( E , u ) d u . \begin{aligned}\displaystyle\frac{d\Pi(E,x)}{dx}&\displaystyle=2\int_{E}^{% \infty}\Gamma(u,x)\gamma(u,E)du+\int_{E}^{\infty}\Pi(u,x)\pi(u,u-E)du-\int_{0}% ^{E}\Pi(E,x)\pi(E,E-u)du\\ \displaystyle\frac{d\Gamma(E,x)}{dx}&\displaystyle=\int_{E}^{\infty}\Pi(u,x)% \pi(u,E)du-\int_{0}^{E}\Gamma(E,x)\gamma(E,u)du.\end{aligned}

Particular_values_of_Riemann_zeta_function.html

  1. ζ ( 0 ) = - B 1 = - 1 2 . \zeta(0)=-B_{1}=-\tfrac{1}{2}.\!
  2. lim ϵ 0 ± ζ ( 1 + ϵ ) = ± \lim_{\epsilon\to 0^{\pm}}\zeta(1+\epsilon)=\pm\infty
  3. ζ ( 2 n ) = ( - 1 ) n + 1 B 2 n ( 2 π ) 2 n 2 ( 2 n ) ! \zeta(2n)=(-1)^{n+1}\frac{B_{2n}(2\pi)^{2n}}{2(2n)!}\!
  4. ζ ( 2 ) = 1 + 1 2 2 + 1 3 2 + = π 2 6 = 1.6449 \zeta(2)=1+\frac{1}{2^{2}}+\frac{1}{3^{2}}+\cdots=\frac{\pi^{2}}{6}=1.6449\dots\!
  5. ζ ( 4 ) = 1 + 1 2 4 + 1 3 4 + = π 4 90 = 1.0823 \zeta(4)=1+\frac{1}{2^{4}}+\frac{1}{3^{4}}+\cdots=\frac{\pi^{4}}{90}=1.0823\dots\!
  6. ζ ( 6 ) = 1 + 1 2 6 + 1 3 6 + = π 6 945 = 1.0173... \zeta(6)=1+\frac{1}{2^{6}}+\frac{1}{3^{6}}+\cdots=\frac{\pi^{6}}{945}=1.0173..% .\dots\!
  7. ζ ( 8 ) = 1 + 1 2 8 + 1 3 8 + = π 8 9450 = 1.00407... \zeta(8)=1+\frac{1}{2^{8}}+\frac{1}{3^{8}}+\cdots=\frac{\pi^{8}}{9450}=1.00407% ...\dots\!
  8. ζ ( 10 ) = 1 + 1 2 10 + 1 3 10 + = π 10 93555 = 1.000994... \zeta(10)=1+\frac{1}{2^{10}}+\frac{1}{3^{10}}+\cdots=\frac{\pi^{10}}{93555}=1.% 000994...\dots\!
  9. ζ ( 12 ) = 1 + 1 2 12 + 1 3 12 + = 691 π 12 638512875 = 1.000246 \zeta(12)=1+\frac{1}{2^{12}}+\frac{1}{3^{12}}+\cdots=\frac{691\pi^{12}}{638512% 875}=1.000246\dots\!
  10. ζ ( 14 ) = 1 + 1 2 14 + 1 3 14 + = 2 π 14 18243225 = 1.0000612 \zeta(14)=1+\frac{1}{2^{14}}+\frac{1}{3^{14}}+\cdots=\frac{2\pi^{14}}{18243225% }=1.0000612\dots\!
  11. A n ζ ( n ) = B n π n A_{n}\zeta(n)=B_{n}\pi^{n}\,\!
  12. ζ ( 2 n ) = = 1 1 2 n = η n π 2 n , \zeta(2n)=\sum_{\ell=1}^{\infty}\frac{1}{\ell^{2n}}=\eta_{n}\pi^{2n},
  13. η 1 = 1 / 6 ; η n = = 1 n - 1 ( - 1 ) - 1 η n - ( 2 + 1 ) ! + ( - 1 ) n + 1 n ( 2 n + 1 ) ! . \begin{aligned}\displaystyle\eta_{1}&\displaystyle=1/6;\\ \displaystyle\eta_{n}&\displaystyle=\sum_{\ell=1}^{n-1}(-1)^{\ell-1}\frac{\eta% _{n-\ell}}{(2\ell+1)!}+(-1)^{n+1}\frac{n}{(2n+1)!}.\end{aligned}
  14. ζ ( 2 n ) = 1 n + 1 2 k = 1 n - 1 ζ ( 2 k ) ζ ( 2 n - 2 k ) , n > 1 \zeta(2n)=\frac{1}{n+\frac{1}{2}}\sum_{k=1}^{n-1}\zeta(2k)\zeta(2n-2k),n>1
  15. d d x cot ( x ) = - 1 - cot 2 ( x ) \frac{d}{dx}\cot(x)=-1-\cot^{2}(x)
  16. n = 0 ζ ( 2 n ) x 2 n = - π x 2 cot ( π x ) = - 1 2 + π 2 6 x 2 + π 4 90 x 4 + π 6 945 x 6 + \sum_{n=0}^{\infty}\zeta(2n)x^{2n}=-\frac{\pi x}{2}\cot(\pi x)=-\frac{1}{2}+% \frac{\pi^{2}}{6}x^{2}+\frac{\pi^{4}}{90}x^{4}+\frac{\pi^{6}}{945}x^{6}+\cdots
  17. lim n ζ ( 2 n ) = 1 , \lim_{n\rightarrow\infty}\zeta(2n)=1,
  18. n , n n\in\mathbb{N},n\rightarrow\infty
  19. | B 2 n | 2 ( 2 n ) ! ( 2 π ) 2 n \left|B_{2n}\right|\sim\frac{2(2n)!}{(2\pi)^{2n}}
  20. ζ ( 1 ) = 1 + 1 2 + 1 3 + = \zeta(1)=1+\frac{1}{2}+\frac{1}{3}+\cdots=\infty\!
  21. ζ ( 3 ) = 1 + 1 2 3 + 1 3 3 + = 1.20205 \zeta(3)=1+\frac{1}{2^{3}}+\frac{1}{3^{3}}+\cdots=1.20205\dots\!
  22. ζ ( 5 ) = 1 + 1 2 5 + 1 3 5 + = 1.03692 \zeta(5)=1+\frac{1}{2^{5}}+\frac{1}{3^{5}}+\cdots=1.03692\dots\!
  23. ζ ( 7 ) = 1 + 1 2 7 + 1 3 7 + = 1.00834 \zeta(7)=1+\frac{1}{2^{7}}+\frac{1}{3^{7}}+\cdots=1.00834\dots\!
  24. ζ ( 9 ) = 1 + 1 2 9 + 1 3 9 + = 1.002008 \zeta(9)=1+\frac{1}{2^{9}}+\frac{1}{3^{9}}+\cdots=1.002008\dots\!
  25. ζ ( 5 ) = 1 294 π 5 - 72 35 n = 1 1 n 5 ( e 2 π n - 1 ) - 2 35 n = 1 1 n 5 ( e 2 π n + 1 ) ζ ( 5 ) = 12 n = 1 1 n 5 sinh ( π n ) - 39 20 n = 1 1 n 5 ( e 2 π n - 1 ) - 1 20 n = 1 1 n 5 ( e 2 π n + 1 ) \begin{aligned}\displaystyle\zeta(5)&\displaystyle=\frac{1}{294}\pi^{5}-\frac{% 72}{35}\sum_{n=1}^{\infty}\frac{1}{n^{5}(e^{2\pi n}-1)}-\frac{2}{35}\sum_{n=1}% ^{\infty}\frac{1}{n^{5}(e^{2\pi n}+1)}\\ \displaystyle\zeta(5)&\displaystyle=12\sum_{n=1}^{\infty}\frac{1}{n^{5}\sinh(% \pi n)}-\frac{39}{20}\sum_{n=1}^{\infty}\frac{1}{n^{5}(e^{2\pi n}-1)}-\frac{1}% {20}\sum_{n=1}^{\infty}\frac{1}{n^{5}(e^{2\pi n}+1)}\end{aligned}
  26. ζ ( 7 ) = 19 56700 π 7 - 2 n = 1 1 n 7 ( e 2 π n - 1 ) \zeta(7)=\frac{19}{56700}\pi^{7}-2\sum_{n=1}^{\infty}\frac{1}{n^{7}(e^{2\pi n}% -1)}\!
  27. S ± ( s ) = n = 1 1 n s ( e 2 π n ± 1 ) S_{\pm}(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}(e^{2\pi n}\pm 1)}
  28. 0 = A n ζ ( n ) - B n π n + C n S - ( n ) + D n S + ( n ) 0=A_{n}\zeta(n)-B_{n}\pi^{n}+C_{n}S_{-}(n)+D_{n}S_{+}(n)\,
  29. ζ ( - n ) = - B n + 1 n + 1 . \zeta(-n)=-\frac{B_{n+1}}{n+1}.
  30. ζ ( - 2 n ) = 0. \zeta(-2n)=0.\,
  31. ζ ( - 1 ) = - 1 12 \zeta(-1)=-\frac{1}{12}
  32. ζ ( - 3 ) = 1 120 \zeta(-3)=\frac{1}{120}
  33. ζ ( - 5 ) = - 1 252 \zeta(-5)=-\frac{1}{252}
  34. ζ ( - 7 ) = 1 240 . \zeta(-7)=\frac{1}{240}.
  35. ζ ( - 2 n ) = ( - 1 ) n ( 2 n ) ! 2 ( 2 π ) 2 n ζ ( 2 n + 1 ) . \zeta^{\prime}(-2n)=(-1)^{n}\frac{(2n)!}{2(2\pi)^{2n}}\zeta(2n+1).
  36. ζ ( - 2 ) = - ζ ( 3 ) 4 π 2 \zeta^{\prime}(-2)=-\frac{\zeta(3)}{4\pi^{2}}
  37. ζ ( - 4 ) = 3 4 π 4 ζ ( 5 ) \zeta^{\prime}(-4)=\frac{3}{4\pi^{4}}\zeta(5)
  38. ζ ( - 6 ) = - 45 8 π 6 ζ ( 7 ) \zeta^{\prime}(-6)=-\frac{45}{8\pi^{6}}\zeta(7)
  39. ζ ( - 8 ) = 315 4 π 8 ζ ( 9 ) . \zeta^{\prime}(-8)=\frac{315}{4\pi^{8}}\zeta(9).
  40. ζ ( 0 ) = - 1 2 ln ( 2 π ) - 0.918938533 \zeta^{\prime}(0)=-\frac{1}{2}\ln(2\pi)\approx-0.918938533\ldots
  41. ζ ( - 1 ) = 1 12 - ln A - 0.1654211437 \zeta^{\prime}(-1)=\frac{1}{12}-\ln A\approx-0.1654211437\ldots
  42. k = 2 ζ ( k ) x k - 1 = - ψ 0 ( 1 - x ) - γ \sum_{k=2}^{\infty}\zeta(k)x^{k-1}=-\psi_{0}(1-x)-\gamma
  43. k = 2 ( ζ ( k ) - 1 ) = 1 \sum_{k=2}^{\infty}(\zeta(k)-1)=1
  44. k = 1 ( ζ ( 2 k ) - 1 ) = 3 4 \sum_{k=1}^{\infty}(\zeta(2k)-1)=\frac{3}{4}
  45. k = 1 ( ζ ( 2 k + 1 ) - 1 ) = 1 4 \sum_{k=1}^{\infty}(\zeta(2k+1)-1)=\frac{1}{4}
  46. k = 2 ( - 1 ) k ( ζ ( k ) - 1 ) = 1 2 . \sum_{k=2}^{\infty}(-1)^{k}(\zeta(k)-1)=\frac{1}{2}.
  47. k = 2 ( - 1 ) k ζ ( k ) k = γ \sum_{k=2}^{\infty}(-1)^{k}\frac{\zeta(k)}{k}=\gamma
  48. k = 2 ζ ( k ) - 1 k = 1 - γ \sum_{k=2}^{\infty}\frac{\zeta(k)-1}{k}=1-\gamma
  49. k = 2 ( - 1 ) k ζ ( k ) - 1 k = ln 2 + γ - 1 \sum_{k=2}^{\infty}(-1)^{k}\frac{\zeta(k)-1}{k}=\ln 2+\gamma-1
  50. ζ ( k ) = lim ε 0 ζ ( k + ε ) + ζ ( k - ε ) 2 , \zeta(k)=\lim_{\varepsilon\to 0}\frac{\zeta(k+\varepsilon)+\zeta(k-\varepsilon% )}{2},
  51. k = 1 ( - 1 ) k ζ ( k ) k = 0 \sum_{k=1}^{\infty}(-1)^{k}\frac{\zeta(k)}{k}=0
  52. k = 1 ζ ( k ) - 1 k = 0 \sum_{k=1}^{\infty}\frac{\zeta(k)-1}{k}=0
  53. k = 1 ( - 1 ) k ζ ( k ) - 1 k = ln 2 \sum_{k=1}^{\infty}(-1)^{k}\frac{\zeta(k)-1}{k}=\ln 2

Pascal's_rule.html

  1. ( n - 1 k ) + ( n - 1 k - 1 ) = ( n k ) for 1 k n {n-1\choose k}+{n-1\choose k-1}={n\choose k}\quad\,\text{for }1\leq k\leq n
  2. ( n k ) {n\choose k}
  3. ( n k ) + ( n k - 1 ) = ( n + 1 k ) for 1 k n + 1 {n\choose k}+{n\choose k-1}={n+1\choose k}\quad\,\text{for }1\leq k\leq n+1
  4. ( a b ) {a\choose b}
  5. ( n k ) {n\choose k}
  6. ( n - 1 k - 1 ) n-1\choose k-1
  7. ( n - 1 k ) n-1\choose k
  8. ( n k ) {n\choose k}
  9. ( n - 1 k - 1 ) + ( n - 1 k ) . {n-1\choose k-1}+{n-1\choose k}.
  10. ( n k ) + ( n k - 1 ) = ( n + 1 k ) . {n\choose k}+{n\choose k-1}={n+1\choose k}.
  11. ( n k ) + ( n k - 1 ) = n ! k ! ( n - k ) ! + n ! ( k - 1 ) ! ( n - k + 1 ) ! = n ! [ n + 1 - k k ! ( n + 1 - k ) ! + k k ! ( n + 1 - k ) ! ] = n ! ( n + 1 ) k ! ( n + 1 - k ) ! = ( n + 1 k ) \begin{aligned}\displaystyle{n\choose k}+{n\choose k-1}&\displaystyle=\frac{n!% }{k!(n-k)!}+\frac{n!}{(k-1)!(n-k+1)!}\\ &\displaystyle=n!\left[\frac{n+1-k}{k!(n+1-k)!}+\frac{k}{k!(n+1-k)!}\right]\\ &\displaystyle=\frac{n!(n+1)}{k!(n+1-k)!}={\left({{n+1}\atop{k}}\right)}\end{aligned}
  12. n , k 1 , k 2 , k 3 , , k p , p * n,k_{1},k_{2},k_{3},\dots,k_{p},p\in\mathbb{N}^{*}\,\!
  13. n = k 1 + k 2 + k 3 + + k p n=k_{1}+k_{2}+k_{3}+\cdots+k_{p}\,\!
  14. ( n - 1 k 1 - 1 , k 2 , k 3 , , k p ) + ( n - 1 k 1 , k 2 - 1 , k 3 , , k p ) + + ( n - 1 k 1 , k 2 , k 3 , , k p - 1 ) \displaystyle{}\quad{n-1\choose k_{1}-1,k_{2},k_{3},\dots,k_{p}}+{n-1\choose k% _{1},k_{2}-1,k_{3},\dots,k_{p}}+\cdots+{n-1\choose k_{1},k_{2},k_{3},\dots,k_{% p}-1}

Pascal's_simplex.html

  1. m \wedge^{m}
  2. n m \wedge^{m}_{n}
  3. n m - 1 \vartriangle^{m-1}_{n}
  4. n m = n m - 1 \wedge^{m}_{n}=\vartriangle^{m-1}_{n}
  5. | x | n = | k | = n ( n k ) x k ; x m , k 0 m , n 0 , m |x|^{n}=\sum_{|k|=n}{{\left({{n}\atop{k}}\right)}x^{k}};\ \ x\in\mathbb{R}^{m}% ,\ k\in\mathbb{N}^{m}_{0},\ n\in\mathbb{N}_{0},\ m\in\mathbb{N}
  6. | x | = i = 1 m x i , | k | = i = 1 m k i , x k = i = 1 m x i k i \textstyle|x|=\sum_{i=1}^{m}{x_{i}},\ |k|=\sum_{i=1}^{m}{k_{i}},\ x^{k}=\prod_% {i=1}^{m}{x_{i}^{k_{i}}}
  7. 4 \wedge^{4}
  8. 1 \wedge^{1}
  9. n 1 = n 0 \wedge^{1}_{n}=\vartriangle^{0}_{n}
  10. ( x 1 ) n = k 1 = n ( n k 1 ) x 1 k 1 ; k 1 , n 0 (x_{1})^{n}=\sum_{k_{1}=n}{n\choose k_{1}}x_{1}^{k_{1}};\ \ k_{1},n\in\mathbb{% N}_{0}
  11. n 0 \vartriangle^{0}_{n}
  12. ( n n ) \textstyle{n\choose n}
  13. 2 \wedge^{2}
  14. n 2 = n 1 \wedge^{2}_{n}=\vartriangle^{1}_{n}
  15. ( x 1 + x 2 ) n = k 1 + k 2 = n ( n k 1 , k 2 ) x 1 k 1 x 2 k 2 ; k 1 , k 2 , n 0 (x_{1}+x_{2})^{n}=\sum_{k_{1}+k_{2}=n}{n\choose k_{1},k_{2}}x_{1}^{k_{1}}x_{2}% ^{k_{2}};\ \ k_{1},k_{2},n\in\mathbb{N}_{0}
  16. n 1 \vartriangle^{1}_{n}
  17. ( n n , 0 ) , ( n n - 1 , 1 ) , , ( n 1 , n - 1 ) , ( n 0 , n ) \textstyle{n\choose n,0},{n\choose n-1,1},\cdots,{n\choose 1,n-1},{n\choose 0,n}
  18. 3 \wedge^{3}
  19. n 3 = n 2 \wedge^{3}_{n}=\vartriangle^{2}_{n}
  20. ( x 1 + x 2 + x 3 ) n = k 1 + k 2 + k 3 = n ( n k 1 , k 2 , k 3 ) x 1 k 1 x 2 k 2 x 3 k 3 ; k 1 , k 2 , k 3 , n 0 (x_{1}+x_{2}+x_{3})^{n}=\sum_{k_{1}+k_{2}+k_{3}=n}{n\choose k_{1},k_{2},k_{3}}% x_{1}^{k_{1}}x_{2}^{k_{2}}x_{3}^{k_{3}};\ \ k_{1},k_{2},k_{3},n\in\mathbb{N}_{0}
  21. n 2 \vartriangle^{2}_{n}
  22. ( n n , 0 , 0 ) , ( n n - 1 , 1 , 0 ) , , ( n 1 , n - 1 , 0 ) , ( n 0 , n , 0 ) ( n n - 1 , 0 , 1 ) , ( n n - 2 , 1 , 1 ) , , ( n 0 , n - 1 , 1 ) ( n 1 , 0 , n - 1 ) , ( n 0 , 1 , n - 1 ) ( n 0 , 0 , n ) \begin{aligned}\displaystyle\textstyle{n\choose n,0,0}&\displaystyle,% \textstyle{n\choose n-1,1,0},\cdots\cdots,{n\choose 1,n-1,0},{n\choose 0,n,0}% \\ \displaystyle\textstyle{n\choose n-1,0,1}&\displaystyle,\textstyle{n\choose n-% 2,1,1},\cdots\cdots,{n\choose 0,n-1,1}\\ &\displaystyle\vdots\\ \displaystyle\textstyle{n\choose 1,0,n-1}&\displaystyle,\textstyle{n\choose 0,% 1,n-1}\\ \displaystyle\textstyle{n\choose 0,0,n}\end{aligned}
  23. n m = n m - 1 \wedge^{m}_{n}=\vartriangle^{m-1}_{n}
  24. n m = n m + 1 \vartriangle^{m}_{n}=\wedge^{m+1}_{n}
  25. n m = n m - 1 n m = n m + 1 \wedge^{m}_{n}=\vartriangle^{m-1}_{n}\subset\ \vartriangle^{m}_{n}=\wedge^{m+1% }_{n}
  26. m \wedge^{m}
  27. m + 1 \wedge^{m+1}
  28. m m + 1 \wedge^{m}\subset\wedge^{m+1}
  29. 1 \wedge^{1}
  30. 2 \wedge^{2}
  31. 3 \wedge^{3}
  32. 4 \wedge^{4}
  33. 0 m \wedge^{m}_{0}
  34. 1 m \wedge^{m}_{1}
  35. 2 m \wedge^{m}_{2}
  36. 3 m \wedge^{m}_{3}
  37. n m + 1 = n m \wedge^{m+1}_{n}=\vartriangle^{m}_{n}
  38. n m - 1 = n m \vartriangle^{m-1}_{n}=\wedge^{m}_{n}
  39. n m + 1 = n m n m - 1 = n m \wedge^{m+1}_{n}=\vartriangle^{m}_{n}\supset\vartriangle^{m-1}_{n}=\wedge^{m}_% {n}
  40. n i + 1 = n i n m > i = n m > i + 1 \wedge^{i+1}_{n}=\vartriangle^{i}_{n}\subset\vartriangle^{m>i}_{n}=\wedge^{m>i% +1}_{n}
  41. 1 = n 1 = n 0 n m - 1 = n m 1=\wedge^{1}_{n}=\vartriangle^{0}_{n}\subset\vartriangle^{m-1}_{n}=\wedge^{m}_% {n}
  42. ( ( n - 1 ) + ( m - 1 ) ( m - 1 ) ) + ( n + ( m - 2 ) ( m - 2 ) ) = ( n + ( m - 1 ) ( m - 1 ) ) , {(n-1)+(m-1)\choose(m-1)}+{n+(m-2)\choose(m-2)}={n+(m-1)\choose(m-1)},
  43. | b d p | n = | k | = n ( n k ) b d p k ; b , d , n 0 , k , p 0 m , p : p 1 = 0 , p i = ( n + 1 ) i - 2 \left|b^{dp}\right|^{n}=\sum_{|k|=n}{{\left({{n}\atop{k}}\right)}b^{dp\cdot k}% };\ \ b,d\in\mathbb{N},\ n\in\mathbb{N}_{0},\ k,p\in\mathbb{N}_{0}^{m},\ p:\ p% _{1}=0,p_{i}=(n+1)^{i-2}
  44. b d p = ( b d p 1 , , b d p m ) m , p k = i = 1 m p i k i 0 \textstyle b^{dp}=(b^{dp_{1}},\cdots,b^{dp_{m}})\in\mathbb{N}^{m},\ p\cdot k={% \sum_{i=1}^{m}{p_{i}k_{i}}}\in\mathbb{N}_{0}

Patch_antenna.html

  1. < m t p l > δ f f r e s = Z 0 2 R r a d d W \frac{<}{m}tpl>{{\delta f}}{{f_{res}}}=\frac{{Z_{0}}}{{2R_{rad}}}\frac{d}{W}
  2. Z 0 Z_{0}
  3. R r a d R_{rad}
  4. < m t p l > δ f f r e s = 1.2 ( d W ) \frac{<}{m}tpl>{{\delta f}}{{f_{res}}}=1.2\left({\frac{d}{W}}\right)
  5. T M 10 TM_{10}
  6. T M 01 TM_{01}
  7. T M 10 TM_{10}
  8. T M 01 TM_{01}

Path_tracing.html

  1. 1 π \frac{1}{\pi}

Pathfinding.html

  1. O ( | V | + | E | ) O(|V|+|E|)
  2. O ( | V | | E | ) O(|V||E|)
  3. O ( | E | log ( | V | ) ) O(|E|\log(|V|))

Patterson_function.html

  1. P ( u , v , w ) = h k l | F h k l | 2 e - 2 π i ( h u + k v + l w ) . P(u,v,w)=\sum\limits_{hkl}\left|F_{hkl}\right|^{2}\;e^{-2\pi i(hu+kv+lw)}.
  2. P ( u ) = ρ ( r ) * ρ ( - r ) . P(\vec{u})=\rho(\vec{r})*\rho(-\vec{r}).
  3. f ( x ) = δ ( x ) + 3 δ ( x - 2 ) + δ ( x - 5 ) + 3 δ ( x - 8 ) + 5 δ ( x - 10 ) . f(x)=\delta(x)+3\delta(x-2)+\delta(x-5)+3\delta(x-8)+5\delta(x-10).\,
  4. P ( u ) = 5 δ ( u + 10 ) + 18 δ ( u + 8 ) + 9 δ ( u + 6 ) + 6 δ ( u + 5 ) + 6 δ ( u + 3 ) + 18 δ ( u + 2 ) + 45 δ ( u ) P(u)=5\delta(u+10)+18\delta(u+8)+9\delta(u+6)+6\delta(u+5)+6\delta(u+3)+18% \delta(u+2)+45\delta(u)\,
  5. + 18 δ ( u - 2 ) + 6 δ ( u - 3 ) + 6 δ ( u - 5 ) + 9 δ ( u - 6 ) + 18 δ ( u - 8 ) + 5 δ ( u - 10 ) . {}+18\delta(u-2)+6\delta(u-3)+6\delta(u-5)+9\delta(u-6)+18\delta(u-8)+5\delta(% u-10).\,

Peaucellier–Lipkin_linkage.html

  1. x = B P = P D x=\ell_{BP}=\ell_{PD}
  2. y = O B y=\ell_{OB}
  3. h = A P h=\ell_{AP}
  4. O B O D = y ( y + 2 x ) = y 2 + 2 x y \ell_{OB}\cdot\ell_{OD}=y(y+2x)=y^{2}+2xy
  5. O A 2 = ( y + x ) 2 + h 2 {\ell_{OA}}^{2}=(y+x)^{2}+h^{2}
  6. A D 2 = x 2 + h 2 {\ell_{AD}}^{2}=x^{2}+h^{2}
  7. O A 2 - A D 2 = y 2 + 2 x y = O B O D {\ell_{OA}}^{2}-{\ell_{AD}}^{2}=y^{2}+2xy=\ell_{OB}\cdot\ell_{OD}
  8. O B O D = k 2 \ell_{OB}\cdot\ell_{OD}=k^{2}

Peek's_law.html

  1. e v = m v g v r ln ( S r ) e_{v}=m_{v}g_{v}r\ln\left({S\over r}\right)
  2. δ = ρ ρ S A T P \delta={\rho\over\rho_{SATP}}
  3. g v = g 0 δ ( 1 + c δ r ) g_{v}=g_{0}\delta\left(1+{c\over\sqrt{\delta r}}\right)

Peetre's_inequality.html

  1. ( 1 + | x | 2 1 + | y | 2 ) t 2 | t | ( 1 + | x - y | 2 ) | t | . \left(\frac{1+|x|^{2}}{1+|y|^{2}}\right)^{t}\leq 2^{|t|}(1+|x-y|^{2})^{|t|}.

Peetre_theorem.html

  1. Γ ( E ) , and Γ ( F ) \Gamma^{\infty}(E),\ \hbox{and}\ \Gamma^{\infty}(F)
  2. D : Γ ( E ) Γ ( F ) D:\Gamma^{\infty}(E)\rightarrow\Gamma^{\infty}(F)
  3. D = i D j k D=i_{D}\circ j^{k}
  4. j k : Γ E J k E j^{k}:\Gamma^{\infty}E\rightarrow J^{k}E
  5. i D : J k E F i_{D}:J^{k}E\rightarrow F
  6. | α | k sup y B δ ( x 2 k ) | α s k ( y ) | 1 M k ( δ 2 ) k \sum_{|\alpha|\leq k}\ \sup_{y\in B^{\prime}_{\delta}(x_{2k})}|\nabla^{\alpha}% s_{k}(y)|\leq\frac{1}{M_{k}}\left(\frac{\delta}{2}\right)^{k}
  7. M k = | α | k sup | α ρ | . M_{k}=\sum_{|\alpha|\leq k}\sup|\nabla^{\alpha}\rho|.
  8. ρ 2 k ( y ) := ρ ( y - x 2 k δ ) \rho_{2k}(y):=\rho\left(\frac{y-x_{2k}}{\delta}\right)
  9. max | α | k sup y B δ ( x 2 k ) | α ( ρ 2 k s 2 k ) | 2 - k . \max_{|\alpha|\leq k}\ \sup_{y\in B^{\prime}_{\delta}(x_{2k})}|\nabla^{\alpha}% (\rho_{2k}s_{2k})|\leq 2^{-k}.
  10. q ( y ) = k = 1 ρ 2 k ( y ) s 2 k ( y ) , q(y)=\sum_{k=1}^{\infty}\rho_{2k}(y)s_{2k}(y),
  11. ρ \rho
  12. lim k | D q ( x 2 k ) | C \lim_{k\rightarrow\infty}|Dq(x_{2k})|\geq C
  13. lim k D q ( x 2 k + 1 ) = 0 \lim_{k\rightarrow\infty}Dq(x_{2k+1})=0
  14. Γ ( E ) \Gamma^{\infty}(E)
  15. D : Γ ( E ) Γ ( F ) D:\Gamma^{\infty}(E)\rightarrow\Gamma^{\infty}(F)
  16. π D p = p . \pi\circ D_{p}=p.
  17. D = i D j k D=i_{D}\circ j^{k}
  18. i D i_{D}

Penrose_graphical_notation.html

  1. g a b g^{ab}\,
  2. g a b g_{ab}\,
  3. ε a b n \varepsilon_{ab\ldots n}
  4. ϵ a b n \epsilon^{ab\ldots n}
  5. ε a b n ϵ a b n \varepsilon_{ab\ldots n}\,\epsilon^{ab\ldots n}
  6. = n ! =n!
  7. γ α β χ = - γ β α χ {\gamma_{\alpha\beta}}^{\chi}=-{\gamma_{\beta\alpha}}^{\chi}
  8. γ a b c {\gamma_{ab}}^{c}
  9. δ b a \delta^{a}_{b}
  10. β a ξ a \beta_{a}\,\xi^{a}
  11. g a b g b c = δ a c = g c b g b a g_{ab}\,g^{bc}=\delta_{a}^{c}=g^{cb}\,g_{ba}
  12. Q ( a b n ) Q^{(ab\ldots n)}
  13. Q a b = Q [ a b ] + Q ( a b ) {}_{Q^{ab}=Q^{[ab]}+Q^{(ab)}}
  14. E [ a b n ] E_{[ab\ldots n]}
  15. E a b = E [ a b ] + E ( a b ) {}_{E_{ab}=E_{[ab]}+E_{(ab)}}
  16. det 𝐓 = det ( T b a ) \det\mathbf{T}=\det\left(T^{a}_{\ b}\right)
  17. 𝐓 - 1 = ( T b a ) - 1 \mathbf{T}^{-1}=\left(T^{a}_{\ b}\right)^{-1}
  18. \nabla
  19. 12 a ( ξ f λ f b [ c ( d D g h ] e ) b ) 12\nabla_{a}\left(\xi^{f}\,\lambda^{(d}_{fb[c}\,D^{e)b}_{gh]}\right)
  20. = 12 ( ξ f ( a λ f b [ c ( d ) D g h ] e ) b + ( a ξ f ) λ f b [ c ( d D g h ] e ) b + ξ f λ f b [ c ( d ( a D g h ] e ) b ) ) =12\left(\xi^{f}(\nabla_{a}\lambda^{(d}_{fb[c})\,D^{e)b}_{gh]}+(\nabla_{a}\xi^% {f})\lambda^{(d}_{fb[c}\,D^{e)b}_{gh]}+\xi^{f}\lambda^{(d}_{fb[c}\,(\nabla_{a}% D^{e)b}_{gh]})\right)
  21. ε a c ϵ a c = n ! \varepsilon_{a...c}\epsilon^{a...c}=n!
  22. R a b = R a c b c R_{ab}=R_{acb}^{\ \ \ c}
  23. ( a b - b a ) ξ d (\nabla_{a}\,\nabla_{b}-\nabla_{b}\,\nabla_{a})\,\mathbf{\xi}^{d}
  24. = R a b c d ξ c =R_{abc}^{\ \ \ d}\,\mathbf{\xi}^{c}
  25. [ a R b c ] d e = 0 \nabla_{[a}R_{bc]d}^{\ \ \ e}=0

Pentadiagonal_matrix.html

  1. ( c 1 d 1 e 1 0 0 b 1 c 2 d 2 e 2 a 1 b 2 0 a 2 e n - 3 0 d n - 2 e n - 2 a n - 3 b n - 2 c n - 1 d n - 1 0 0 a n - 2 b n - 1 c n ) . \begin{pmatrix}c_{1}&d_{1}&e_{1}&0&\cdots&\cdots&0\\ b_{1}&c_{2}&d_{2}&e_{2}&\ddots&&\vdots\\ a_{1}&b_{2}&\ddots&\ddots&\ddots&\ddots&\vdots\\ 0&a_{2}&\ddots&\ddots&\ddots&e_{n-3}&0\\ \vdots&\ddots&\ddots&\ddots&\ddots&d_{n-2}&e_{n-2}\\ \vdots&&\ddots&a_{n-3}&b_{n-2}&c_{n-1}&d_{n-1}\\ 0&\cdots&\cdots&0&a_{n-2}&b_{n-1}&c_{n}\end{pmatrix}.
  2. 5 n - 6 5n-6

Pentagonal_prism.html

  1. h 3 4 5 ( 5 + 2 5 ) \frac{h^{3}}{4}\sqrt{5(5+2\sqrt{5})}

Pentation.html

  1. a [ 5 ] b a[5]b
  2. a b a\uparrow\uparrow\uparrow b
  3. a 3 b a\uparrow^{3}b
  4. a b a\uparrow b
  5. a b a^{b}
  6. x a x x\mapsto ax
  7. b b
  8. a b a\uparrow\uparrow b
  9. x a x x\mapsto a\uparrow x
  10. b b
  11. a b a\uparrow\uparrow\uparrow b
  12. x a x x\mapsto a\uparrow\uparrow x
  13. b b
  14. a b = a b 3 a\uparrow\uparrow\uparrow b=a\rightarrow b\rightarrow 3
  15. a b {{}_{b}a}
  16. A ( n , m ) A(n,m)
  17. A ( m - 1 , A ( m , n - 1 ) ) A(m-1,A(m,n-1))
  18. A ( 1 , n ) = a n A(1,n)=an
  19. A ( m , 1 ) = a A(m,1)=a
  20. a b = A ( 4 , b ) a\uparrow\uparrow\uparrow b=A(4,b)
  21. a 3 b a\uparrow^{3}b
  22. 1 3 b = 1 1\uparrow^{3}b=1
  23. a 3 1 = a a\uparrow^{3}1=a
  24. a 3 0 = 1 a\uparrow^{3}0=1
  25. a 3 - 1 = 0 a\uparrow^{3}-1=0
  26. 2 3 2 = 2 2 = 2 2 = 4 2\uparrow^{3}2={{}^{2}2}=2^{2}=4
  27. 2 3 3 = 2 2 2 = 2 4 = 2 2 2 2 = 2 2 4 = 2 16 = 65 , 536 2\uparrow^{3}3={{}^{{}^{2}2}2}={{}^{4}2}=2^{2^{2^{2}}}=2^{2^{4}}=2^{16}=65,536
  28. 2 3 4 = 2 2 2 2 = 2 65 , 536 = 2 2 2 2 (a power tower of height 65,536) exp 10 65 , 533 ( 4.29508 ) 2\uparrow^{3}4={{}^{{}^{{}^{2}2}2}2}={{}^{65,536}2}=2^{2^{2^{\cdot^{\cdot^{% \cdot^{2}}}}}}\mbox{ (a power tower of height 65,536) }~{}\approx\exp_{10}^{65% ,533}(4.29508)
  29. exp 10 ( n ) = 10 n \exp_{10}(n)=10^{n}
  30. 3 3 2 = 3 3 = 3 3 3 = 3 27 = 7 , 625 , 597 , 484 , 987 3\uparrow^{3}2={{}^{3}3}=3^{3^{3}}=3^{27}=7,625,597,484,987
  31. 3 3 3 = 3 3 3 = 3 7 , 625 , 597 , 484 , 987 = 3 3 3 3 (a power tower of height 7,625,597,484,987) exp 10 7 , 625 , 597 , 484 , 986 ( 1.09902 ) 3\uparrow^{3}3={{}^{{}^{3}3}3}={{}^{7,625,597,484,987}3}=3^{3^{3^{\cdot^{\cdot% ^{\cdot^{3}}}}}}\mbox{ (a power tower of height 7,625,597,484,987) }~{}\approx% \exp_{10}^{7,625,597,484,986}(1.09902)
  32. 4 3 2 = 4 4 = 4 4 4 4 = 4 4 256 exp 10 3 ( 2.19 ) 4\uparrow^{3}2={{}^{4}4}=4^{4^{4^{4}}}=4^{4^{256}}\approx\exp_{10}^{3}(2.19)
  33. 5 3 2 = 5 5 = 5 5 5 5 5 = 5 5 5 3125 exp 10 4 ( 3.33928 ) 5\uparrow^{3}2={{}^{5}5}=5^{5^{5^{5^{5}}}}=5^{5^{5^{3125}}}\approx\exp_{10}^{4% }(3.33928)

Peres–Horodecki_criterion.html

  1. ρ \rho
  2. A A
  3. B B
  4. ρ \rho
  5. A B \mathcal{H}_{A}\otimes\mathcal{H}_{B}
  6. ρ = i j k l p k l i j | i j | | k l | \rho=\sum_{ijkl}p^{ij}_{kl}|i\rangle\langle j|\otimes|k\rangle\langle l|
  7. ρ T B := I T ( ρ ) = i j k l p k l i j | i j | ( | k l | ) T = i j k l p k l i j | i j | | l k | \rho^{T_{B}}:=I\otimes T(\rho)=\sum_{ijkl}p^{ij}_{kl}|i\rangle\langle j|% \otimes(|k\rangle\langle l|)^{T}=\sum_{ijkl}p^{ij}_{kl}|i\rangle\langle j|% \otimes|l\rangle\langle k|
  8. I T ( ρ ) I\otimes T(\rho)
  9. ρ = ( A 11 A 12 A 1 n A 21 A 22 A n 1 A n n ) \rho=\begin{pmatrix}A_{11}&A_{12}&\dots&A_{1n}\\ A_{21}&A_{22}&&\\ \vdots&&\ddots&\\ A_{n1}&&&A_{nn}\end{pmatrix}
  10. n = dim A n=\dim\mathcal{H}_{A}
  11. m = dim B m=\dim\mathcal{H}_{B}
  12. ρ T B = ( A 11 T A 12 T A 1 n T A 21 T A 22 T A n 1 T A n n T ) \rho^{T_{B}}=\begin{pmatrix}A_{11}^{T}&A_{12}^{T}&\dots&A_{1n}^{T}\\ A_{21}^{T}&A_{22}^{T}&&\\ \vdots&&\ddots&\\ A_{n1}^{T}&&&A_{nn}^{T}\end{pmatrix}
  13. ρ \rho\;\!
  14. ρ T B \rho^{T_{B}}
  15. ρ T B \rho^{T_{B}}
  16. ρ \rho\;\!
  17. ρ T A = ( ρ T B ) T \rho^{T_{A}}=(\rho^{T_{B}})^{T}
  18. ρ = p | Ψ - Ψ - | + ( 1 - p ) I 4 \rho=p|\Psi^{-}\rangle\langle\Psi^{-}|+(1-p)\frac{I}{4}
  19. | Ψ - |\Psi^{-}\rangle
  20. ρ = 1 4 ( 1 - p 0 0 0 0 p + 1 - 2 p 0 0 - 2 p p + 1 0 0 0 0 1 - p ) \rho=\frac{1}{4}\begin{pmatrix}1-p&0&0&0\\ 0&p+1&-2p&0\\ 0&-2p&p+1&0\\ 0&0&0&1-p\end{pmatrix}
  21. ρ T B = 1 4 ( 1 - p 0 0 - 2 p 0 p + 1 0 0 0 0 p + 1 0 - 2 p 0 0 1 - p ) \rho^{T_{B}}=\frac{1}{4}\begin{pmatrix}1-p&0&0&-2p\\ 0&p+1&0&0\\ 0&0&p+1&0\\ -2p&0&0&1-p\end{pmatrix}
  22. ( 1 - 3 p ) / 4 (1-3p)/4
  23. p > 1 / 3 p>1/3
  24. ρ = p i ρ i A ρ i B \rho=\sum p_{i}\rho^{A}_{i}\otimes\rho^{B}_{i}
  25. ρ T B = I T ( ρ ) = p i ρ i A ( ρ i B ) T \rho^{T_{B}}=I\otimes T(\rho)=\sum p_{i}\rho^{A}_{i}\otimes(\rho^{B}_{i})^{T}
  26. ρ T B \rho^{T_{B}}
  27. ρ \rho\;\!
  28. ρ T B \rho^{T_{B}}
  29. I Λ ( ρ ) I\otimes\Lambda(\rho)
  30. B ( B ) B(\mathcal{H}_{B})
  31. B ( A ) B(\mathcal{H}_{A})
  32. B ( B ) B(\mathcal{H}_{B})
  33. B ( A ) B(\mathcal{H}_{A})
  34. dim ( B ) = 2 \textrm{dim}(\mathcal{H}_{B})=2
  35. dim ( A ) = 2 or 3 \textrm{dim}(\mathcal{H}_{A})=2\;\textrm{or}\;3
  36. Λ = Λ 1 + Λ 2 T , \Lambda=\Lambda_{1}+\Lambda_{2}\circ T,
  37. Λ 1 \Lambda_{1}
  38. Λ 2 \Lambda_{2}
  39. ρ T B \rho^{T_{B}}
  40. I Λ ( ρ ) I\otimes\Lambda(\rho)
  41. dim ( A B ) 6 \textrm{dim}(\mathcal{H}_{A}\otimes\mathcal{H}_{B})\leq 6
  42. 1 1 1\oplus 1
  43. 1 n 1\oplus n
  44. 2 2 2\oplus 2

Perfect_fluid.html

  1. ρ m \rho_{m}
  2. T μ ν = ( ρ m + p c 2 ) U μ U ν + p η μ ν T^{\mu\nu}=\left(\rho_{m}+\frac{p}{c^{2}}\right)\,U^{\mu}U^{\nu}+p\,\eta^{\mu% \nu}\,
  3. η μ ν = D i a g [ - 1 , 1 , 1 , 1 ] \eta_{\mu\nu}=Diag[-1,1,1,1]
  4. T μ ν = ( ρ m + p c 2 ) U μ U ν - p η μ ν T^{\mu\nu}=\left(\rho_{m}+\frac{p}{c^{2}}\right)\,U^{\mu}U^{\nu}-p\,\eta^{\mu% \nu}\,
  5. η μ ν = D i a g [ 1 , - 1 , - 1 , - 1 ] \eta_{\mu\nu}=Diag[1,-1,-1,-1]
  6. [ ρ e 0 0 0 0 p 0 0 0 0 p 0 0 0 0 p ] \left[\begin{matrix}\rho_{e}&0&0&0\\ 0&p&0&0\\ 0&0&p&0\\ 0&0&0&p\end{matrix}\right]
  7. ρ e = ρ m c 2 \rho_{e}=\rho_{m}c^{2}
  8. p p

Perfect_ruler.html

  1. n n
  2. { 0 , a 2 , , a n } { 0 , 1 , 2 , , n } \{0,a_{2},\cdots,a_{n}\}\subset\{0,1,2,\ldots,n\}
  3. m m
  4. k m k\leq m
  5. k = a i - a j k=a_{i}-a_{j}
  6. i , j i,j
  7. m m
  8. 7 7
  9. { 0 , 1 , 3 , 7 } \{0,1,3,7\}
  10. 1 , 2 , 3 , 4 1,2,3,4
  11. 1 = 1 - 0 1=1-0
  12. 2 = 3 - 1 2=3-1
  13. 3 = 3 - 0 3=3-0
  14. 4 = 7 - 3 4=7-3
  15. n n
  16. a n a_{n}

Perpetual_bond.html

  1. i y \frac{i}{y}
  2. i i
  3. y y

Personal_identity_number_(Sweden).html

  1. 811228 - 987 x 811228\mbox{-}~{}987x
  2. [ 8 2 = 16 1 + 6 7 ] + 1 + [ 1 2 = 2 ] + 2 + [ 2 2 = 4 ] + 8 + [ 9 2 = 18 1 + 8 9 ] + 8 + [ 7 2 = 14 1 + 4 5 ] = 46 \left[\begin{matrix}\underbrace{8\cdot 2=16}\\ 1+6\to 7\end{matrix}\right]+1+\left[1\cdot 2=2\right]+2+\left[2\cdot 2=4\right% ]+8+\left[\begin{matrix}\underbrace{9\cdot 2=18}\\ 1+8\to 9\end{matrix}\right]+8+\left[\begin{matrix}\underbrace{7\cdot 2=14}\\ 1+4\to 5\end{matrix}\right]=46
  3. 46 46
  4. 6 6
  5. x = 10 - 6 = 4 x=10-6=4
  6. 4 4
  7. 50 50
  8. 0
  9. 10 10
  10. x = 10 - 0 = 10 x=10-0=10
  11. 0

Perturbative_QCD.html

  1. α s \alpha_{s}

Petrov_classification.html

  1. X a b 1 2 C a b m n X m n X^{ab}\rightarrow\frac{1}{2}\,{C^{ab}}_{mn}X^{mn}
  2. λ \lambda
  3. X a b X^{ab}
  4. 1 2 C a b m n X m n = λ X a b \frac{1}{2}\,{C^{ab}}_{mn}\,X^{mn}=\lambda\,X^{ab}
  5. U a b = - l [ a m b ] U_{ab}=-l_{[a}m_{b]}
  6. V a b = n [ a m b ] V_{ab}=n_{[a}m_{b]}
  7. W a b = m [ a m ¯ b ] - n [ a l b ] . W_{ab}=m_{[a}\bar{m}_{b]}-n_{[a}l_{b]}.
  8. C a b c d = Ψ 0 U a b U c d + Ψ 1 ( U a b W c d + W a b U c d ) + Ψ 2 ( V a b U c d + U a b V c d + W a b W c d ) + Ψ 3 ( V a b W c d + W a b V c d ) + Ψ 4 V a b V c d \begin{aligned}\displaystyle C_{abcd}&\displaystyle=\Psi_{0}U_{ab}U_{cd}\\ &\displaystyle\,\,\,+\Psi_{1}(U_{ab}W_{cd}+W_{ab}U_{cd})\\ &\displaystyle\,\,\,+\Psi_{2}(V_{ab}U_{cd}+U_{ab}V_{cd}+W_{ab}W_{cd})\\ &\displaystyle\,\,\,+\Psi_{3}(V_{ab}W_{cd}+W_{ab}V_{cd})\\ &\displaystyle\,\,\,+\Psi_{4}V_{ab}V_{cd}\end{aligned}
  9. { Ψ j } \{\Psi_{j}\}
  10. Ψ 0 = 0 \Psi_{0}=0
  11. Ψ 0 = Ψ 1 = 0 \Psi_{0}=\Psi_{1}=0
  12. Ψ 0 = Ψ 1 = Ψ 3 = Ψ 4 = 0 \Psi_{0}=\Psi_{1}=\Psi_{3}=\Psi_{4}=0
  13. Ψ 0 = Ψ 1 = Ψ 2 = 0 \Psi_{0}=\Psi_{1}=\Psi_{2}=0
  14. Ψ 0 = Ψ 1 = Ψ 2 = Ψ 3 = 0 \Psi_{0}=\Psi_{1}=\Psi_{2}=\Psi_{3}=0
  15. Ψ 0 = Ψ 1 = Ψ 2 = Ψ 3 = Ψ 4 = 0 \Psi_{0}=\Psi_{1}=\Psi_{2}=\Psi_{3}=\Psi_{4}=0
  16. M M
  17. C C
  18. p M p\in M
  19. p p
  20. p p
  21. C a b c d C_{abcd}
  22. C a b c d C_{abcd}
  23. k ( p ) k(p)
  24. C a b c d k d = 0 C_{abcd}\,k^{d}=0
  25. k k
  26. C a b c d C_{abcd}
  27. C a b c d C_{abcd}
  28. k ( p ) k(p)
  29. C a b c d k b k d = 0 = C a b c d * k b k d C_{abcd}\,k^{b}k^{d}=0={{}^{*}C}_{abcd}\,k^{b}k^{d}
  30. k k
  31. C a b c d C_{abcd}
  32. k k
  33. C a b c d k b k d = α k a k c C_{abcd}\,k^{b}k^{d}=\alpha k_{a}k_{c}
  34. C a b c d * k b k d = β k a k c {}^{*}C_{abcd}\,k^{b}k^{d}=\beta k_{a}k_{c}
  35. α β 0 \alpha\beta\neq 0
  36. k k
  37. C a b c d C_{abcd}
  38. k k
  39. k k^{\prime}
  40. C a b c d k b k d = α k a k c C_{abcd}\,k^{b}k^{d}=\alpha k_{a}k_{c}
  41. C a b c d * k b k d = β k a k c {}^{*}C_{abcd}\,k^{b}k^{d}=\beta k_{a}k_{c}
  42. α β 0 \alpha\beta\neq 0
  43. C a b c d k b k d = γ k a k c C_{abcd}\,k^{\prime b}k^{\prime d}=\gamma k^{\prime}_{a}k^{\prime}_{c}
  44. C a b c d * k b k d = δ k a k c {}^{*}C_{abcd}\,k^{\prime b}k^{\prime d}=\delta k^{\prime}_{a}k^{\prime}_{c}
  45. γ δ 0 \gamma\delta\neq 0
  46. C a b c d * {{}^{*}C}_{abcd}
  47. p p
  48. O ( r - 3 ) O(r^{-3})
  49. r r
  50. O ( r - 4 ) O(r^{-4})
  51. O ( r - 2 ) O(r^{-2})
  52. O ( r - 1 ) O(r^{-1})
  53. d d
  54. l l
  55. n n
  56. d - 2 d-2
  57. l l
  58. n n
  59. l l

Peukert's_law.html

  1. C p = I k t , C_{p}=I^{k}t,
  2. C p C_{p}
  3. I I
  4. t t
  5. k k
  6. t = H ( C I H ) k t=H\left(\frac{C}{IH}\right)^{k}
  7. H H
  8. C C
  9. I I
  10. k k
  11. t t
  12. 20 ( 100 10 20 ) 1.2 20{\left(\frac{100}{10\cdot 20}\right)^{1.2}}
  13. I t = C ( C I H ) k - 1 , It=C\left(\frac{C}{IH}\right)^{k-1},
  14. I t It
  15. I I

Pharmacode.html

  1. n n
  2. 2 n 2^{n}
  3. 2 × 2 n 2\times 2^{n}

Phase_plane.html

  1. d x d t = A x + B y d y d t = C x + D y \begin{aligned}\displaystyle\frac{dx}{dt}&\displaystyle=Ax+By\\ \displaystyle\frac{dy}{dt}&\displaystyle=Cx+Dy\end{aligned}
  2. d d t ( x y ) = ( A B C D ) ( x y ) \displaystyle\frac{d}{dt}\begin{pmatrix}x\\ y\\ \end{pmatrix}=\begin{pmatrix}A&B\\ C&D\\ \end{pmatrix}\begin{pmatrix}x\\ y\\ \end{pmatrix}
  3. d y d x = C x + D y A x + B y \frac{dy}{dx}=\frac{Cx+Dy}{Ax+By}
  4. det ( 𝐀 - λ 𝐈 ) = 0 \det(\mathbf{A}-\lambda\mathbf{I})=0
  5. 𝐀𝐱 = λ 𝐱 \mathbf{A}\mathbf{x}=\lambda\mathbf{x}
  6. x = [ k 1 k 2 ] c 1 e λ 1 t + [ k 3 k 4 ] c 2 e λ 2 t . x=\begin{bmatrix}k_{1}\\ k_{2}\end{bmatrix}c_{1}e^{\lambda_{1}t}+\begin{bmatrix}k_{3}\\ k_{4}\end{bmatrix}c_{2}e^{\lambda_{2}t}.
  7. λ 2 - ( A + D ) λ + ( A D - B C ) = 0 \lambda^{2}-(A+D)\lambda+(AD-BC)=0
  8. λ 2 - p λ + q = 0 \lambda^{2}-p\lambda+q=0
  9. p = A + D = tr ( 𝐀 ) , p=A+D=\mathrm{tr}(\mathbf{A})\,,
  10. q = A D - B C = det ( 𝐀 ) . q=AD-BC=\det(\mathbf{A})\,.
  11. λ = 1 2 ( p ± Δ ) \lambda=\frac{1}{2}(p\pm\sqrt{\Delta})\,
  12. Δ = p 2 - 4 q . \Delta=p^{2}-4q\,.

Phosphor_thermometry.html

  1. τ \tau
  2. I = I o e - t τ \!\,I=I_{o}e^{\frac{-t}{\tau}}
  3. ϕ = t a n ( 2 π f τ ) \!\,\phi=tan(2{\pi}f{\tau})

Photo_CD.html

  1. 𝑅𝐺𝐵 = { 1.099 𝑅𝐺𝐵 0.45 - 0.99 , 𝑅𝐺𝐵 0.018 - 1.099 ( - 𝑅𝐺𝐵 ) 0.45 + 0.99 , 𝑅𝐺𝐵 - 0.018 4.5 𝑅𝐺𝐵 - 0.018 < 𝑅𝐺𝐵 < 0.018 \mathit{RGB^{\prime}=\begin{cases}1.099\cdot RGB^{0.45}-0.99,&RGB\geq 0.018\\ -1.099\cdot(-RGB)^{0.45}+0.99,&RGB\leq-0.018\\ 4.5\cdot RGB&-0.018<RGB<0.018\\ \end{cases}}
  2. Y = 0.299 R + 0.587 G + 0.114 B Y^{\prime}=0.299R^{\prime}+0.587G^{\prime}+0.114B^{\prime}
  3. B ′′ - Y = - 0.299 R - 0.587 G + 0.886 B B^{\prime\prime}-Y^{\prime}=-0.299\cdot R^{\prime}-0.587\cdot G^{\prime}+0.886% \cdot B^{\prime}
  4. R ′′ - Y = 0.701 R - 0.587 G - 0.114 B R^{\prime\prime}-Y^{\prime}=0.701\cdot R^{\prime}-0.587\cdot G^{\prime}-0.114% \cdot B^{\prime}
  5. Y ′′ = 255 / 1.402 Y Y^{\prime\prime}={255/1.402}\cdot Y^{\prime}
  6. C 1 = 111.40 ( B ′′ - Y ) + 156 C1=111.40\cdot(B^{\prime\prime}-Y^{\prime})+156
  7. C 2 = 135.64 ( R ′′ - Y ) + 137 C2=135.64\cdot(R^{\prime\prime}-Y^{\prime})+137

Photoacoustic_imaging.html

  1. H ( r , t ) H(\vec{r},t)
  2. p ( r , t ) p(\vec{r},t)
  3. 2 p ( r , t ) - 1 v s 2 2 t 2 p ( r , t ) = - β C p t H ( r , t ) ( 1 ) , \nabla^{2}p(\vec{r},t)-\frac{1}{v_{s}^{2}}\frac{\partial^{2}}{\partial{t^{2}}}% p(\vec{r},t)=-\frac{\beta}{C_{p}}\frac{\partial}{\partial t}H(\vec{r},t)\qquad% \qquad\quad\quad(1),
  4. v s v_{s}
  5. β \beta
  6. C p C_{p}
  7. p ( r , t ) = β 4 π C p d r | r - r | H ( r , t ) t | t = t - | r - r | / v s ( 2 ) . \left.p(\vec{r},t)=\frac{\beta}{4\pi C_{p}}\int\frac{d\vec{r^{\prime}}}{|\vec{% r}-\vec{r^{\prime}}|}\frac{\partial H(\vec{r^{\prime}},t^{\prime})}{\partial t% ^{\prime}}\right|_{t^{\prime}=t-|\vec{r}-\vec{r^{\prime}}|/v_{s}}\qquad\quad\,% \,\,\,(2).
  8. p ( r , t ) = 1 4 π v s 2 t [ 1 v s t d r p 0 ( r ) δ ( t - | r - r | v s ) ] ( 3 ) , p(\vec{r},t)=\frac{1}{4\pi v_{s}^{2}}\frac{\partial}{\partial t}\left[\frac{1}% {v_{s}t}\int d\vec{r^{\prime}}p_{0}(\vec{r^{\prime}})\delta\left(t-\frac{|\vec% {r}-\vec{r^{\prime}}|}{v_{s}}\right)\right]\qquad\,(3),
  9. p 0 p_{0}
  10. p 0 p_{0}
  11. p 0 ( r ) = Ω 0 d Ω 0 Ω 0 [ 2 p ( r 0 , v s t ) - 2 v s t p ( r 0 , v s t ) ( v s t ) ] | t = | r - r 0 | / v s , ( 4 ) , \left.p_{0}(\vec{r})=\int_{\Omega_{0}}\frac{d\Omega_{0}}{\Omega_{0}}\left[2p(% \vec{r_{0}},v_{s}t)-2v_{s}t\frac{\partial p(\vec{r_{0}},v_{s}t)}{\partial(v_{s% }t)}\right]\right|_{t=|\vec{r}-\vec{r_{0}}|/v_{s}},\qquad\quad(4),
  12. Ω 0 \Omega_{0}
  13. S 0 S_{0}
  14. r \vec{r}
  15. S 0 S_{0}
  16. d Ω 0 = d S 0 | r - r 0 | 2 n ^ 0 s . ( r - r 0 ) | r - r 0 | . d\Omega_{0}=\frac{dS_{0}}{|\vec{r}-\vec{r_{0}}|^{2}}\frac{\hat{n}_{0}^{s}.(% \vec{r}-\vec{r_{0}})}{|\vec{r}-\vec{r_{0}}|}.

Photon_diffusion.html

  1. F = - c 12 π σ U , \vec{F}=-\frac{c}{12\pi\sigma}\vec{\nabla}U,
  2. F = 0 , \vec{\nabla}\cdot\vec{F}=0,
  3. σ \sigma
  4. 2 U - 1 σ U σ = 0. \nabla^{2}U-\frac{1}{\sigma}\vec{\nabla}U\cdot\vec{\nabla}\sigma=0.

Photon_sphere.html

  1. r = 3 G M c 2 r=\frac{3GM}{c^{2}}
  2. d s 2 = ( 1 - 2 G M r c 2 ) c 2 d t 2 - ( 1 - 2 G M r c 2 ) - 1 d r 2 - r 2 ( sin 2 θ d ϕ 2 + d θ 2 ) ds^{2}=\left(1-\frac{2GM}{rc^{2}}\right)c^{2}dt^{2}-\left(1-\frac{2GM}{rc^{2}}% \right)^{-1}dr^{2}-r^{2}(\textrm{sin}^{2}\theta d\phi^{2}+d\theta^{2})
  3. ( 1 - 2 G M r c 2 ) c 2 d t 2 = r 2 sin 2 θ d ϕ 2 \left(1-\frac{2GM}{rc^{2}}\right)c^{2}dt^{2}=r^{2}\textrm{sin}^{2}\theta d\phi% ^{2}
  4. d ϕ d t = c r sin θ 1 - R s r \frac{d\phi}{dt}=\frac{c}{r\textrm{sin}\theta}\sqrt{1-\frac{R_{s}}{r}}
  5. d ϕ d t \frac{d\phi}{dt}
  6. d 2 r d τ 2 + Γ μ ν r u μ u ν = 0. \frac{d^{2}r}{d\tau^{2}}+\Gamma^{r}_{\mu\nu}u^{\mu}u^{\nu}=0.
  7. Γ \Gamma
  8. Γ t t r = B B 2 , Γ r r r , Γ θ θ r , Γ ϕ ϕ r = - B r sin 2 θ \Gamma^{r}_{tt}=\frac{BB^{\prime}}{2},\;\Gamma^{r}_{rr},\;\Gamma^{r}_{\theta% \theta},\;\Gamma^{r}_{\phi\phi}=-Br\sin^{2}\theta
  9. B = d B d r , B = 1 - R s r B^{\prime}=\frac{dB}{dr},B=1-\frac{R_{s}}{r}
  10. θ \theta
  11. d r d τ , d 2 r d τ 2 , d θ d τ = 0 \frac{dr}{d\tau},\;\frac{d^{2}r}{d\tau^{2}},\;\frac{d\theta}{d\tau}=0
  12. ( d ϕ d t ) 2 = c 2 R s 2 r 3 sin 2 θ \left(\frac{d\phi}{dt}\right)^{2}=\frac{c^{2}R_{s}}{2r^{3}\sin^{2}\theta}
  13. c R s 2 r = c 1 - R s r c\sqrt{\frac{R_{s}}{2r}}=c\sqrt{1-\frac{R_{s}}{r}}
  14. θ = π 2 \theta=\frac{\pi}{2}
  15. ϕ \phi
  16. θ = π 2 \theta=\frac{\pi}{2}
  17. r = 3 2 R s r=\frac{3}{2}R_{s}

Physical_Coding_Sublayer.html

  1. x 58 + x 39 + 1 x^{58}+x^{39}+1
  2. x 7 + x 6 + 1 x^{7}+x^{6}+1

Physical_Medium_Dependent.html

  1. x 7 + x 6 + 1 x^{7}+x^{6}+1

Picard_horn.html

  1. PSL 2 ( 𝐙 [ i ] ) \operatorname{PSL}_{2}(\mathbf{Z}[i])
  2. Υ 1 \Upsilon_{1}

Picard–Fuchs_equation.html

  1. j = g 2 3 g 2 3 - 27 g 3 2 j=\frac{g_{2}^{3}}{g_{2}^{3}-27g_{3}^{2}}
  2. g 2 g_{2}
  3. g 3 g_{3}
  4. y 2 = 4 x 3 - g 2 x - g 3 . y^{2}=4x^{3}-g_{2}x-g_{3}.\,
  5. / Γ \mathbb{H}/\Gamma
  6. { } \mathbb{C}\cup\{\infty\}
  7. \mathbb{H}
  8. Γ \Gamma
  9. d 2 y d j 2 + 1 j d y d j + 31 j - 4 144 j 2 ( 1 - j ) 2 y = 0. \frac{d^{2}y}{dj^{2}}+\frac{1}{j}\frac{dy}{dj}+\frac{31j-4}{144j^{2}(1-j)^{2}}% y=0.\,
  10. d 2 f d j 2 + 1 - 1968 j + 2654208 j 2 4 j 2 ( 1 - 1728 j ) 2 f = 0. \frac{d^{2}f}{dj^{2}}+\frac{1-1968j+2654208j^{2}}{4j^{2}(1-1728j)^{2}}f=0.\,
  11. y ( j ) = P { 0 1 1 / 6 1 / 4 0 j - 1 / 6 3 / 4 0 } y(j)=P\left\{\begin{matrix}0&1&\infty&\\ {1/6}&{1/4}&0&j\\ {-1/6\;}&{3/4}&0&\end{matrix}\right\}\,
  12. 2 S τ ( j ) = 1 - 1 4 ( 1 - j ) 2 + 1 - 1 9 j 2 + 1 - 1 4 - 1 9 j ( 1 - j ) = 3 4 ( 1 - j ) 2 + 8 9 j 2 + 23 36 j ( 1 - j ) 2S\tau(j)=\frac{1-\frac{1}{4}}{(1-j)^{2}}+\frac{1-\frac{1}{9}}{j^{2}}+\frac{1-% \frac{1}{4}-\frac{1}{9}}{j(1-j)}=\frac{3}{4(1-j)^{2}}+\frac{8}{9j^{2}}+\frac{2% 3}{36j(1-j)}

Pierpont_prime.html

  1. k 2 n + 1 divides 2 2 m + 1. k\cdot 2^{n}+1\,\text{ divides }2^{2^{m}}+1.\,

Pierre_Raymond_de_Montmort.html

  1. n a + n ( n - 1 ) 1 2 Δ a + n ( n - 1 ) ( n - 2 ) 1 2 3 Δ 2 a + , na+\frac{n(n-1)}{1\cdot 2}\Delta a+\frac{n(n-1)(n-2)}{1\cdot 2\cdot 3}\Delta^{% 2}a+\cdots,

Planar_lamina.html

  1. m m
  2. ρ ( x , y ) \rho\ (x,y)
  3. m = ρ ( x , y ) d x d y m=\int\int{}\rho\ (x,y)\,dx\,dy
  4. ( M y m , M x m ) \left(\frac{M_{y}}{m},\frac{M_{x}}{m}\right)
  5. M y M_{y}
  6. M x M_{x}
  7. M y = lim m , n i = 1 m j = 1 n x ρ i j * ( x , i j * y ) i j * Δ \Alpha = x ρ ( x , y ) d x d y M_{y}=\lim_{m,n\to\infty}\,\sum_{i=1}^{m}\,\sum_{j=1}^{n}\,x{{}_{ij}}^{*}\,% \rho\ (x{{}_{ij}}^{*},y{{}_{ij}}^{*})\,\Delta\Alpha=\iint{}x\,\rho\ (x,y)\,dx% \,dy
  8. M x = lim m , n i = 1 m j = 1 n y ρ i j * ( x , i j * y ) i j * Δ \Alpha = y ρ ( x , y ) d x d y M_{x}=\lim_{m,n\to\infty}\,\sum_{i=1}^{m}\,\sum_{j=1}^{n}\,y{{}_{ij}}^{*}\,% \rho\ (x{{}_{ij}}^{*},y{{}_{ij}}^{*})\,\Delta\Alpha=\iint{}y\,\rho\ (x,y)\,dx% \,dy
  9. x = 0 x=0
  10. x = y x=y
  11. y = 4 - x y=4-x
  12. ρ ( x , y ) = 2 x + 3 y + 2 \rho\ (x,y)\,=2x+3y+2
  13. m = 0 2 x 4 - x 2 x + 3 y + 2 d y d x m=\int_{0}^{2}{\int_{x}^{4-x}}{}\,2x+3y+2\,dy\,dx
  14. m = 0 2 ( 2 x y + 3 y 2 2 + 2 y ) | x 4 - x d x m=\int_{0}^{2}(2xy+\frac{3y^{2}}{2}+2y)|_{x}^{4-x}\,dx
  15. m = 0 2 - 4 x 2 - 8 x + 32 d x m=\int_{0}^{2}-4x^{2}-8x+32\,dx
  16. m = ( - 4 x 3 3 - 4 x 2 + 32 x ) | 0 2 m=(\frac{-4x^{3}}{3}-4x^{2}+32x)|_{0}^{2}
  17. m = 112 3 m=\frac{112}{3}
  18. M y = 0 2 x 4 - x x ( 2 x + 3 y + 2 ) d y d x M_{y}=\int_{0}^{2}{\int_{x}^{4-x}}{}{}x\,(2x+3y+2)\,dy\,dx
  19. M y = 0 2 ( 2 x 2 y + 3 x y 2 2 + 2 x y ) | x 4 - x d x M_{y}=\int_{0}^{2}(2x^{2}y+\frac{3xy^{2}}{2}+2xy)|_{x}^{4-x}\,dx
  20. M y = 0 2 - 4 x 3 - 8 x 2 + 32 x d x M_{y}=\int_{0}^{2}-4x^{3}-8x^{2}+32x\,dx
  21. M y = ( - x 4 - 8 x 3 3 + 16 x 2 ) | 0 2 M_{y}=(-x^{4}-\frac{8x^{3}}{3}+16x^{2})|_{0}^{2}
  22. M y = 80 3 M_{y}=\frac{80}{3}
  23. M x = 0 2 x 4 - x y ( 2 x + 3 y + 2 ) d y d x M_{x}=\int_{0}^{2}{\int_{x}^{4-x}}{}{}y\,(2x+3y+2)\,dy\,dx
  24. M x = 0 2 ( x y 2 + y 3 + y 2 ) | x 4 - x d x M_{x}=\int_{0}^{2}(xy^{2}+y^{3}+y^{2})|_{x}^{4-x}\,dx
  25. M x = 0 2 - 2 x 3 + 4 x 2 - 40 x + 80 d x M_{x}=\int_{0}^{2}-2x^{3}+4x^{2}-40x+80\,dx
  26. M x = ( - x 4 2 + 4 x 3 3 - 20 x 2 + 80 x ) | 0 2 M_{x}=\left.\left(\frac{-x^{4}}{2}+\frac{4x^{3}}{3}-20x^{2}+80x\right)\right|_% {0}^{2}
  27. M x = 248 3 M_{x}=\frac{248}{3}
  28. ( 80 3 112 3 , 248 3 112 3 ) = ( 5 7 , 31 14 ) \left(\frac{\frac{80}{3}}{\frac{112}{3}},\frac{\frac{248}{3}}{\frac{112}{3}}% \right)=\left(\frac{5}{7},\frac{31}{14}\right)

Planck_force.html

  1. F P = m P c t P = c 4 G = 1.21027 × 10 44 N. F\text{P}=\frac{m\text{P}c}{t\text{P}}=\frac{c^{4}}{G}=1.21027\times 10^{44}% \mbox{ N.}~{}
  2. F P = G m 2 r G 2 F\text{P}=\frac{Gm^{2}}{r\text{G}^{2}}
  3. r G = r s 2 = G m c 2 . r\text{G}=\frac{r\text{s}}{2}=\frac{Gm}{c^{2}}.
  4. F P = m c 2 G m c 2 = c 4 G . F\text{P}=\frac{mc^{2}}{\frac{Gm}{c^{2}}}=\frac{c^{4}}{G}.
  5. F = m c 2 m c = m 2 c 3 . F=\frac{mc^{2}}{\frac{\hbar}{mc}}=\frac{m^{2}c^{3}}{\hbar}.
  6. m P c = G m P c 2 \frac{\hbar}{m\text{P}c}=\frac{Gm\text{P}}{c^{2}}
  7. c = G m P 2 . c\hbar=Gm\text{P}^{2}.
  8. G μ ν = 8 π G c 4 T μ ν G_{\mu\nu}=8\pi\frac{G}{c^{4}}T_{\mu\nu}
  9. G μ ν G_{\mu\nu}
  10. T μ ν T_{\mu\nu}

Planck_postulate.html

  1. E = n h ν E=nh\nu\,
  2. n n
  3. h h
  4. ν \nu

Plane_partition.html

  1. n i , j n_{i,j}
  2. n i , j n i , j + 1 and n i , j n i + 1 , j n_{i,j}\geq n_{i,j+1}\quad\mbox{and}~{}\quad n_{i,j}\geq n_{i+1,j}\,
  3. n i , j n_{i,j}
  4. n = i , j n i , j n=\sum_{i,j}n_{i,j}\,
  5. 1 1 1 1 1 1 1 1 1 2 1 2 1 3 \begin{matrix}1&1&1\end{matrix}\qquad\begin{matrix}1&1\\ 1&\end{matrix}\qquad\begin{matrix}1\\ 1\\ 1&\end{matrix}\qquad\begin{matrix}2&1&\end{matrix}\qquad\begin{matrix}2\\ 1&\end{matrix}\qquad\begin{matrix}3\end{matrix}
  6. n n
  7. n n
  8. λ = ( 𝐲 1 , 𝐲 2 , , 𝐲 n ) \lambda=(\mathbf{y}_{1},\mathbf{y}_{2},\ldots,\mathbf{y}_{n})
  9. 𝐲 i 0 3 \mathbf{y}_{i}\in\mathbb{Z}_{\geq 0}^{3}
  10. 𝐚 = ( a 1 , a 2 , a 3 ) λ \mathbf{a}=(a_{1},a_{2},a_{3})\in\lambda
  11. 𝐲 = ( y 1 , y 2 , y 3 ) \mathbf{y}=(y_{1},y_{2},y_{3})
  12. 0 y i a i 0\leq y_{i}\leq a_{i}
  13. i = 1 , 2 , 3 i=1,2,3
  14. n i , j n_{i,j}
  15. ( i - 1 , j - 1 , * ) (i-1,j-1,*)
  16. * *
  17. n i , j n_{i,j}
  18. n i , j n_{i,j}
  19. n i , j n_{i,j}
  20. n i , j n_{i,j}
  21. ( i - 1 , j - 1 , y 3 ) (i-1,j-1,y_{3})
  22. 0 y 3 < n i , j - 1 0\leq y_{3}<n_{i,j}-1
  23. ( 0 0 0 0 0 1 0 1 0 1 0 0 1 1 0 ) 2 1 1 1 \left(\begin{smallmatrix}0\\ 0\\ 0\end{smallmatrix}\begin{smallmatrix}0\\ 0\\ 1\end{smallmatrix}\begin{smallmatrix}0\\ 1\\ 0\end{smallmatrix}\begin{smallmatrix}1\\ 0\\ 0\end{smallmatrix}\begin{smallmatrix}1\\ 1\\ 0\end{smallmatrix}\right)\qquad\Longleftrightarrow\qquad\begin{matrix}2&1\\ 1&1\end{matrix}
  24. n i , j n_{i,j}
  25. S 3 S_{3}
  26. S 3 S_{3}
  27. S 3 S_{3}
  28. 3 1 3 1 2 1 1 2 1 1 1 1 1 1 1 1 1 1 \begin{smallmatrix}3&1\end{smallmatrix}\quad\begin{smallmatrix}3\\ 1\end{smallmatrix}\quad\begin{smallmatrix}2&1&1\end{smallmatrix}\quad\begin{% smallmatrix}2\\ 1\\ 1\end{smallmatrix}\quad\begin{smallmatrix}1&1&1\\ 1\end{smallmatrix}\quad\begin{smallmatrix}1&1\\ 1\\ 1\end{smallmatrix}
  29. n = 0 PL ( n ) x n = k = 1 1 ( 1 - x k ) k = 1 + x + 3 x 2 + 6 x 3 + 13 x 4 + 24 x 5 + . \sum_{n=0}^{\infty}\mbox{PL}~{}(n)\,x^{n}=\prod_{k=1}^{\infty}\frac{1}{(1-x^{k% })^{k}}=1+x+3x^{2}+6x^{3}+13x^{4}+24x^{5}+\cdots.
  30. M ( a , b , c ) M(a,b,c)
  31. a × b × c a\times b\times c
  32. M ( a , b , 1 ) = ( a + b a ) . M(a,b,1)={\left({{a+b}\atop{a}}\right)}.
  33. M ( a , b , c ) M(a,b,c)
  34. M ( a , b , c ) = i = 1 a j = 1 b k = 1 c i + j + k - 1 i + j + k - 2 . M(a,b,c)=\prod_{i=1}^{a}\prod_{j=1}^{b}\prod_{k=1}^{c}\frac{i+j+k-1}{i+j+k-2}.
  35. n n
  36. PL ( n ) ζ ( 3 ) 7 / 36 12 π ( n 2 ) - 25 / 36 exp ( 3 ζ ( 3 ) 1 / 3 ( n 2 ) 2 / 3 + ζ ( - 1 ) ) , \mathrm{PL}(n)\sim\frac{\zeta(3)^{7/36}}{\sqrt{12\pi}}\ \left(\frac{n}{2}% \right)^{-25/36}\ \exp\left(3\ \zeta(3)^{1/3}\left(\frac{n}{2}\right)^{2/3}+% \zeta^{\prime}(-1)\right)\ ,
  37. n - 2 / 3 ln PL ( n ) 2.00945 - 0.69444 n - 2 / 3 ln n - 1.14631 n - 2 / 3 . n^{-2/3}\ln\mathrm{PL}(n)\sim 2.00945-0.69444\ n^{-2/3}\ \ln n-1.14631\ n^{-2/% 3}\ .
  38. 4 2 1 3 1 \begin{matrix}4&2&1\\ 3&1\end{matrix}
  39. 4 3 2 1 1 \begin{matrix}4&3\\ 2&1\\ 1\end{matrix}
  40. a × b × c a\times b\times c

Plastic_number.html

  1. 108 + 12 69 3 + 108 - 12 69 3 6 \frac{\sqrt[3]{108+12\sqrt{69}}+\sqrt[3]{108-12\sqrt{69}}}{6}
  2. x 3 = x + 1 . x^{3}=x+1\,.
  3. ρ = 108 + 12 69 3 + 108 - 12 69 3 6 . \rho=\frac{\sqrt[3]{108+12\sqrt{69}}+\sqrt[3]{108-12\sqrt{69}}}{6}\,.
  4. 1 + 2 1+\sqrt{2}
  5. ( - 23 , ρ ) \mathbb{Q}(\sqrt{-23},\rho)
  6. ( - 23 ) \mathbb{Q}(\sqrt{-23})
  7. ρ = 1 + 1 + 1 + 3 3 3 . \rho=\sqrt[3]{1+\sqrt[3]{1+\sqrt[3]{1+\cdots}}}\,.
  8. ( - 1 2 ± 3 2 i ) 1 2 + 1 6 23 3 3 + ( - 1 2 3 2 i ) 1 2 - 1 6 23 3 3 - 0.662359 ± 0.56228 i , \left(-\frac{1}{2}\pm\frac{\sqrt{3}}{2}i\right)\sqrt[3]{\frac{1}{2}+\frac{1}{6% }\sqrt{\frac{23}{3}}}+\left(-\frac{1}{2}\mp\frac{\sqrt{3}}{2}i\right)\sqrt[3]{% \frac{1}{2}-\frac{1}{6}\sqrt{\frac{23}{3}}}\approx-0.662359\pm 0.56228i,
  9. 1 ρ \frac{1}{\sqrt{\rho}}
  10. ρ = 1 c cosh ( 1 3 cosh - 1 ( 3 c ) ) , c = cos ( 2 π 12 ) = sin ( 2 π 6 ) = 3 2 . \rho=\tfrac{1}{c}\cosh\left(\tfrac{1}{3}\cosh^{-1}(3c)\right),\qquad c=\cos% \left(\tfrac{2\pi}{12}\right)=\sin\left(\tfrac{2\pi}{6}\right)=\tfrac{\sqrt{3}% }{2}\,.

Plating_efficiency.html

  1. PE = # cells on day 1 # cells plated on day 0 × 100 \mathrm{PE}=\frac{\#\,\,\text{cells on day 1}}{\#\,\,\text{cells plated on day% 0}}\times 100
  2. PE = # colonies counted # cells inoculated × 100 \mathrm{PE}=\frac{\#\,\,\text{colonies counted}}{\#\,\,\text{cells inoculated}% }\times 100

Pleione_(star).html

  1. 4 / 3 {4}/{3}
  2. 4 / 3 {4}/{3}
  3. m A {m_{A}}
  4. m P {m_{P}}
  5. m A m P = 2.512 ( 5.05 - 3.62 ) = 3.73 \frac{m_{A}}{m_{P}}=2.512^{(5.05-3.62)}=3.73

Plimpton_322.html

  1. a 2 + b 2 = c 2 \scriptstyle a^{2}+b^{2}=c^{2}
  2. s 2 l 2 \scriptstyle\tfrac{s^{2}}{l^{2}}
  3. d 2 l 2 \scriptstyle\tfrac{d^{2}}{l^{2}}
  4. ( p 2 - q 2 , 2 p q , p 2 + q 2 ) \scriptstyle(p^{2}-q^{2},\,2pq,\,p^{2}+q^{2})
  5. x - 1 x = c \scriptstyle x-\tfrac{1}{x}=c

Plug_flow_reactor_model.html

  1. τ \tau
  2. τ \tau
  3. F i ( x ) - F i ( x + d x ) + A t d x ν i r = 0 F_{i}(x)-F_{i}(x+dx)+A_{t}dx\nu_{i}r=0
  4. u = v ˙ A t = 4 v ˙ π D 2 u=\frac{\dot{v}}{A_{t}}=\frac{4\dot{v}}{\pi D^{2}}
  5. F i = A t u C i F_{i}=A_{t}uC_{i}\,
  6. A t u [ C i ( x ) - C i ( x + d x ) ] + A t d x ν i r = 0 A_{t}u[C_{i}(x)-C_{i}(x+dx)]+A_{t}dx\nu_{i}r=0\,
  7. u d C i d x = ν i r u\frac{dC_{i}}{dx}=\nu_{i}r
  8. τ \tau
  9. C A ( x ) = C A 0 e - k τ C_{A}(x)=C_{A0}e^{-k\tau}\,
  10. t t
  11. t + τ t+\tau
  12. τ \tau
  13. τ \tau

Pluriharmonic_function.html

  1. G n G⊆ ℂ^{n}
  2. f : G f:G→ ℂ
  3. C [ u I s u p , u 2 ] C[u^{\prime}Isup^{\prime},u^{\prime}2^{\prime}]
  4. f f
  5. { a + b z z } n \{a+bz\mid z\in{\mathbb{C}}\}\subset\mathbb{C}^{n}
  6. a , b n a,b∈ ℂ^{n}
  7. z f ( a + b z ) z\mapsto f(a+bz)
  8. { z a + b z G } \{z\in{\mathbb{C}}\mid a+bz\in G\}\subset\mathbb{C}

Pluripolar_set.html

  1. G n G\subset{\mathbb{C}}^{n}
  2. f : G { - } f\colon G\to{\mathbb{R}}\cup\{-\infty\}
  3. - -\infty
  4. 𝒫 := { z G f ( z ) = - } {\mathcal{P}}:=\{z\in G\mid f(z)=-\infty\}
  5. 2 n - 2 2n-2
  6. f f
  7. log | f | \log|f|
  8. f f

Plurisubharmonic_function.html

  1. f : G { - } , f\colon G\to{\mathbb{R}}\cup\{-\infty\},
  2. G n G\subset{\mathbb{C}}^{n}
  3. { a + b z z } n \{a+bz\mid z\in{\mathbb{C}}\}\subset{\mathbb{C}}^{n}
  4. a , b n a,b\in{\mathbb{C}}^{n}
  5. z f ( a + b z ) z\mapsto f(a+bz)
  6. { z a + b z G } . \{z\in{\mathbb{C}}\mid a+bz\in G\}.
  7. X X
  8. f : X { - } f\colon X\to{\mathbb{R}}\cup\{-\infty\}
  9. φ : Δ X \varphi\colon\Delta\to X
  10. f φ : Δ { - } f\circ\varphi\colon\Delta\to{\mathbb{R}}\cup\{-\infty\}
  11. Δ \Delta\subset{\mathbb{C}}
  12. f f
  13. C 2 C^{2}
  14. f f
  15. L f = ( λ i j ) L_{f}=(\lambda_{ij})
  16. λ i j = 2 f z i z ¯ j \lambda_{ij}=\frac{\partial^{2}f}{\partial z_{i}\partial\bar{z}_{j}}
  17. C 2 C^{2}
  18. - 1 ¯ f \sqrt{-1}\partial\bar{\partial}f
  19. n \mathbb{C}^{n}
  20. f ( z ) = | z | 2 f(z)=|z|^{2}
  21. - 1 ¯ f \sqrt{-1}\partial\overline{\partial}f
  22. n \mathbb{C}^{n}
  23. g g
  24. - 1 ¯ g = ω \sqrt{-1}\partial\overline{\partial}g=\omega
  25. ω \omega
  26. g g
  27. 1 \mathbb{C}^{1}
  28. u ( z ) = log ( z ) u(z)=\log(z)
  29. f f
  30. f ( 0 ) = - - 1 2 π C f z ¯ d z d z ¯ z f(0)=-\frac{\sqrt{-1}}{2\pi}\int_{C}\frac{\partial f}{\partial\bar{z}}\frac{% dzd\bar{z}}{z}
  31. - 1 π ¯ log | z | = d d c log | z | \frac{\sqrt{-1}}{\pi}\partial\overline{\partial}\log|z|=dd^{c}\log|z|
  32. f f
  33. c > 0 c>0
  34. c f c\cdot f
  35. f 1 f_{1}
  36. f 2 f_{2}
  37. f 1 + f 2 f_{1}+f_{2}
  38. f f
  39. ϕ : \phi:\mathbb{R}\to\mathbb{R}
  40. ϕ f \phi\circ f
  41. f 1 f_{1}
  42. f 2 f_{2}
  43. f ( x ) := max ( f 1 ( x ) , f 2 ( x ) ) f(x):=\max(f_{1}(x),f_{2}(x))
  44. f 1 , f 2 , f_{1},f_{2},\dots
  45. f ( x ) := lim n f n ( x ) f(x):=\lim_{n\to\infty}f_{n}(x)
  46. f f
  47. lim sup x x 0 f ( x ) = f ( x 0 ) \limsup_{x\to x_{0}}f(x)=f(x_{0})
  48. f f
  49. D D
  50. sup x D f ( x ) = f ( x 0 ) \sup_{x\in D}f(x)=f(x_{0})
  51. x 0 D x_{0}\in D
  52. f f
  53. f : M f:\;M\mapsto{\mathbb{R}}
  54. f - 1 ( ] - , c ] ) f^{-1}(]-\infty,c])
  55. c c\in{\mathbb{R}}
  56. - 1 ( ¯ f - ω ) \sqrt{-1}(\partial\bar{\partial}f-\omega)
  57. ω \omega
  58. C C^{\infty}

Plutonium-239.html

  1. ν ¯ e \bar{\nu}_{e}
  2. 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}

Plücker_formula.html

  1. d * = d ( d - 1 ) d^{*}=d(d-1)\,
  2. d * = d ( d - 1 ) - 2 δ - 3 κ . d^{*}=d(d-1)-2\delta-3\kappa.\,
  3. κ * = 3 d ( d - 2 ) - 6 δ - 8 κ . \kappa^{*}=3d(d-2)-6\delta-8\kappa.\,
  4. d = d * ( d * - 1 ) - 2 δ * - 3 κ * , d=d^{*}(d^{*}-1)-2\delta^{*}-3\kappa^{*},\,
  5. κ = 3 d * ( d * - 2 ) - 6 δ * - 8 κ * . \kappa=3d^{*}(d^{*}-2)-6\delta^{*}-8\kappa^{*}.\,
  6. g = 1 2 ( d - 1 ) ( d - 2 ) - δ - κ . g={1\over 2}(d-1)(d-2)-\delta-\kappa.
  7. g = 1 2 ( d * - 1 ) ( d * - 2 ) - δ * - κ * g={1\over 2}(d^{*}-1)(d^{*}-2)-\delta^{*}-\kappa^{*}
  8. d * = d ( d - 1 ) d^{*}=d(d-1)\,
  9. δ * = 1 2 d ( d - 2 ) ( d - 3 ) ( d + 3 ) \delta^{*}={1\over 2}d(d-2)(d-3)(d+3)
  10. κ * = 3 d ( d - 2 ) \kappa^{*}=3d(d-2)\,
  11. g = 1 2 ( d - 1 ) ( d - 2 ) . g={1\over 2}(d-1)(d-2).

Pohlmeyer_charge.html

  1. P T r exp i T μ d σ A σ ( μ ) ( σ ) P\,Tr\,\exp iT_{\mu}\oint d\sigma A_{\sigma}^{(\mu)}(\sigma)
  2. A σ μ A_{\sigma}^{\mu}
  3. X μ \partial X^{\mu}

Poincaré_recurrence_theorem.html

  1. ( X , Σ , μ ) (X,\Sigma,\mu)
  2. f : X X f\colon X\to X
  3. E Σ E\in\Sigma
  4. x x
  5. E E
  6. f n ( x ) E f^{n}(x)\notin E
  7. n > 0 n>0
  8. E E
  9. E E
  10. μ ( { x E : there exists N such that f n ( x ) E for all n > N } ) = 0. \mu\left(\{x\in E:\,\text{ there exists }N\,\text{ such that }f^{n}(x)\notin E% \,\text{ for all }n>N\}\right)=0.
  11. X X
  12. Σ \Sigma
  13. f f
  14. ε > 0 \varepsilon>0
  15. T 0 > 0 T_{0}>0
  16. T 0 T_{0}
  17. | | ψ ( T ) - | ψ ( 0 ) | < ε ||\psi(T)\rangle-|\psi(0)\rangle|<\varepsilon
  18. | ψ ( t ) |\psi(t)\rangle
  19. | ψ ( t ) = n = 0 c n exp ( - i E n t ) | ϕ n |\psi(t)\rangle=\sum_{n=0}^{\infty}c_{n}\exp(-iE_{n}t)|\phi_{n}\rangle
  20. E n E_{n}
  21. = 1 \hbar=1
  22. | ϕ n |\phi_{n}\rangle
  23. | | ψ ( T ) - | ψ ( 0 ) | 2 = 2 n = 0 | c n | 2 [ 1 - cos ( E n T ) ] ||\psi(T)\rangle-|\psi(0)\rangle|^{2}=2\sum_{n=0}^{\infty}|c_{n}|^{2}[1-\cos(E% _{n}T)]
  24. n = N + 1 | c n | 2 [ 1 - cos ( E n T ) ] 2 n = N + 1 | c n | 2 \sum_{n=N+1}^{\infty}|c_{n}|^{2}[1-\cos(E_{n}T)]\leq 2\sum_{n=N+1}^{\infty}|c_% {n}|^{2}
  25. n = 0 | c n | 2 \sum_{n=0}^{\infty}|c_{n}|^{2}
  26. n = 0 N | c n | 2 [ 1 - cos ( E n T ) ] \sum_{n=0}^{N}|c_{n}|^{2}[1-\cos(E_{n}T)]
  27. k n k_{n}
  28. | E n T - 2 π k n | < δ |E_{n}T-2\pi k_{n}|<\delta
  29. δ > 0 \delta>0
  30. 1 - cos ( E n T ) < δ 2 2 . 1-\cos(E_{n}T)<\frac{\delta^{2}}{2}.
  31. 2 n = 0 N | c n | 2 [ 1 - cos ( E n T ) ] < δ 2 n = 0 N | c n | 2 < δ 2 2\sum_{n=0}^{N}|c_{n}|^{2}[1-\cos(E_{n}T)]<\delta^{2}\sum_{n=0}^{N}|c_{n}|^{2}% <\delta^{2}

Point_accepted_mutation.html

  1. n n
  2. n n
  3. A A
  4. A A
  5. A A
  6. A A
  7. j j
  8. m ( j ) m(j)
  9. f ( j ) f(j)
  10. j j
  11. n ( j ) n(j)
  12. N N
  13. f ( j ) = n ( j ) N f(j)=\frac{n(j)}{N}
  14. m ( j ) = i = 1 , i j 20 A ( i , j ) n ( j ) m(j)=\frac{\sum_{i=1,i\neq j}^{20}A(i,j)}{n(j)}
  15. 1 N f ( j ) = 1 n ( j ) = m ( j ) i = 1 , i j 20 A ( i , j ) \frac{1}{Nf(j)}=\frac{1}{n(j)}=\frac{m(j)}{\sum_{i=1,i\neq j}^{20}A(i,j)}
  16. M M
  17. M ( i , j ) M(i,j)
  18. j j
  19. i i
  20. M ( i , j ) = λ A ( i , j ) m ( j ) i = 1 , i j 20 A ( i , j ) = λ A ( i , j ) N f ( j ) M(i,j)=\lambda A(i,j)\frac{m(j)}{\sum_{i=1,i\neq j}^{20}A(i,j)}=\frac{\lambda A% (i,j)}{Nf(j)}
  21. λ \lambda
  22. M M
  23. M ( j , j ) = 1 - i = 1 , i j 20 M ( i , j ) M(j,j)=1-\sum_{i=1,i\neq j}^{20}M(i,j)
  24. M ( j , j ) = 1 - λ m ( j ) M(j,j)=1-\lambda m(j)
  25. M ( j , j ) = 1 - i = 1 , i j 20 M ( i , j ) M(j,j)=1-\sum_{i=1,i\neq j}^{20}M(i,j)
  26. M ( j , j ) = 1 - i = 1 , i j 20 λ m ( j ) A ( i , j ) i = 1 , i j 20 A ( i , j ) M(j,j)=1-\sum_{i=1,i\neq j}^{20}\frac{\lambda m(j)A(i,j)}{\sum_{i=1,i\neq j}^{% 20}A(i,j)}
  27. M ( j , j ) = 1 - i = 1 , i j 20 λ m ( j ) A ( i , j ) i = 1 , i j 20 A ( i , j ) M(j,j)=1-\frac{\sum_{i=1,i\neq j}^{20}\lambda m(j)A(i,j)}{\sum_{i=1,i\neq j}^{% 20}A(i,j)}
  28. λ \lambda
  29. m ( j ) m(j)
  30. i i
  31. M ( j , j ) = 1 - λ m ( j ) i = 1 , i j 20 A ( i , j ) i = 1 , i j 20 A ( i , j ) M(j,j)=1-\frac{\lambda m(j)\sum_{i=1,i\neq j}^{20}A(i,j)}{\sum_{i=1,i\neq j}^{% 20}A(i,j)}
  32. M ( j , j ) = 1 - λ m ( j ) M(j,j)=1-\lambda m(j)
  33. f ( j ) M ( i , j ) = λ N A ( i , j ) = λ N A ( j , i ) = f ( i ) M ( j , i ) f(j)M(i,j)=\frac{\lambda}{N}A(i,j)=\frac{\lambda}{N}A(j,i)=f(i)M(j,i)
  34. f ( j ) M ( i , j ) = f ( i ) M ( j , i ) f(j)M(i,j)=f(i)M(j,i)
  35. M M
  36. M M
  37. λ \lambda
  38. n n
  39. n ( j ) M ( j , j ) n(j)M(j,j)
  40. j j
  41. j = 1 20 n ( j ) M ( j , j ) = j = 1 20 n ( j ) - λ j = 1 20 n ( j ) m ( j ) = N - N λ j = 1 20 f ( j ) m ( j ) \sum_{j=1}^{20}n(j)M(j,j)=\sum_{j=1}^{20}n(j)-\lambda\sum_{j=1}^{20}n(j)m(j)=N% -N\lambda\sum_{j=1}^{20}f(j)m(j)
  42. λ \lambda
  43. 0.99 = 1 - λ j = 1 20 f ( j ) m ( j ) 0.99=1-\lambda\sum_{j=1}^{20}f(j)m(j)
  44. λ \lambda
  45. M n M_{n}
  46. M 1 M_{1}
  47. M n = M 1 n M_{n}=M_{1}^{n}
  48. j j
  49. i i
  50. PAM n ( i , j ) = l o g f ( j ) M n ( i , j ) f ( i ) f ( j ) = l o g f ( j ) M n ( i , j ) f ( i ) f ( j ) = l o g M n ( i , j ) f ( i ) \,\text{PAM}_{n}(i,j)=log\frac{f(j)M_{n}(i,j)}{f(i)f(j)}=log\frac{f(j)M^{n}(i,% j)}{f(i)f(j)}=log\frac{M^{n}(i,j)}{f(i)}
  51. M ( i , j ) M(i,j)
  52. PAM n ( i , j ) \,\text{PAM}_{n}(i,j)
  53. i i
  54. j j
  55. n n
  56. i i
  57. j j
  58. j j
  59. j j
  60. f ( j ) f(j)
  61. i i
  62. M n ( i , j ) M_{n}(i,j)
  63. f ( j ) M n ( i , j ) f(j)M_{n}(i,j)
  64. i i
  65. j j
  66. f ( i ) f(i)
  67. f ( j ) f(j)
  68. f ( i ) f ( j ) f(i)f(j)
  69. M M
  70. f ( i ) M ( i , j ) = f ( j ) M ( j , i ) f(i)M(i,j)=f(j)M(j,i)
  71. M M
  72. f ( i ) M n ( i , j ) = f ( j ) M n ( j , i ) f(i)M^{n}(i,j)=f(j)M^{n}(j,i)
  73. M M
  74. f ( i ) M ( i , j ) = f ( j ) M ( j , i ) f(i)M(i,j)=f(j)M(j,i)
  75. k k
  76. f ( i ) M k ( i , j ) = f ( j ) M k ( j , i ) f(i)M^{k}(i,j)=f(j)M^{k}(j,i)
  77. M k + 1 = M k M M^{k+1}=M^{k}\cdot M
  78. f ( i ) M k + 1 ( i , j ) = f ( i ) n = 0 N M k ( i , n ) M ( n , j ) f(i)M^{k+1}(i,j)=f(i)\sum^{N}_{n=0}M^{k}(i,n)M(n,j)
  79. f ( i ) M k + 1 ( i , j ) = n = 0 N ( f ( i ) M k ( i , n ) ) M ( n , j ) f(i)M^{k+1}(i,j)=\sum^{N}_{n=0}(f(i)M^{k}(i,n))M(n,j)
  80. M k M^{k}
  81. f ( i ) M k + 1 ( i , j ) = n = 0 N ( f ( n ) M k ( n , i ) ) M ( n , j ) f(i)M^{k+1}(i,j)=\sum^{N}_{n=0}(f(n)M^{k}(n,i))M(n,j)
  82. f ( i ) M k + 1 ( i , j ) = n = 0 N M k ( n , i ) ( f ( n ) M ( n , j ) ) f(i)M^{k+1}(i,j)=\sum^{N}_{n=0}M^{k}(n,i)(f(n)M(n,j))
  83. M M
  84. f ( i ) M k + 1 ( i , j ) = n = 0 N M k ( n , i ) ( f ( j ) M ( j , n ) ) f(i)M^{k+1}(i,j)=\sum^{N}_{n=0}M^{k}(n,i)(f(j)M(j,n))
  85. f ( i ) M k + 1 ( i , j ) = f ( j ) n = 0 N M ( j , n ) M k ( n , i ) f(i)M^{k+1}(i,j)=f(j)\sum^{N}_{n=0}M(j,n)M^{k}(n,i)
  86. f ( i ) M k + 1 ( i , j ) = f ( j ) M k + 1 ( j , i ) f(i)M^{k+1}(i,j)=f(j)M^{k+1}(j,i)
  87. k = 1 k=1
  88. k = 2 k=2
  89. k k
  90. k k
  91. PAM n ( i , j ) = l o g f ( i ) M n ( i , j ) f ( i ) f ( j ) = l o g f ( j ) M n ( j , i ) f ( j ) f ( i ) = PAM n ( j , i ) \,\text{PAM}_{n}(i,j)=log\frac{f(i)M^{n}(i,j)}{f(i)f(j)}=log\frac{f(j)M^{n}(j,% i)}{f(j)f(i)}=\,\text{PAM}_{n}(j,i)
  92. n n
  93. m m
  94. m 100 = 1 - e - n 100 \frac{m}{100}=1-e^{-\frac{n}{100}}
  95. m m
  96. n n
  97. K K
  98. N N
  99. M M
  100. E ( M ) = N - N ( 1 - 1 N ) K E(M)=N-N(1-\frac{1}{N})^{K}
  101. E ( M ) N = 1 - ( 1 - 1 N ) K \frac{E(M)}{N}=1-(1-\frac{1}{N})^{K}
  102. n n
  103. n 100 = K N \frac{n}{100}=\frac{K}{N}
  104. m m
  105. m 100 = E ( M ) N \frac{m}{100}=\frac{E(M)}{N}
  106. m m
  107. n n
  108. m 100 = 1 - ( 1 - 1 N ) n N 100 \frac{m}{100}=1-(1-\frac{1}{N})^{\frac{nN}{100}}
  109. N N
  110. m 100 = 1 - e - n 100 \frac{m}{100}=1-e^{-\frac{n}{100}}
  111. M M
  112. j = 1 20 n ( j ) M n ( j , j ) \sum_{j=1}^{20}n(j)M^{n}(j,j)
  113. j = 1 20 n ( j ) M n ( j , j ) N = j = 1 20 f ( j ) M n ( j , j ) = 1 - m 100 \frac{\sum_{j=1}^{20}n(j)M^{n}(j,j)}{N}=\sum_{j=1}^{20}f(j)M^{n}(j,j)=1-\frac{% m}{100}
  114. n n
  115. T = K 2 r T=\frac{K}{2r}
  116. K K
  117. r r

Point_groups_in_three_dimensions.html

  1. 2 ¯ n \overline{2}{n}
  2. n ¯ \overline{n}
  3. 2 ¯ n \overline{2}{n}
  4. 2 ¯ n \overline{2}{n}
  5. n ¯ \overline{n}
  6. 2 ¯ n \overline{2}{n}
  7. 4 ¯ \overline{4}
  8. 3 ¯ \overline{3}
  9. 3 ¯ \overline{3}
  10. 2 \sqrt{2}
  11. 3 ¯ \overline{3}
  12. 3 ¯ \overline{3}
  13. 5 ¯ \overline{5}
  14. 3 ¯ \overline{3}
  15. 5 ¯ \overline{5}
  16. 3 ¯ \overline{3}
  17. \cong
  18. A n A_{n}
  19. D n D_{n}
  20. E 6 E_{6}
  21. E 7 E_{7}
  22. E 8 E_{8}

Point_groups_in_two_dimensions.html

  1. R ( θ ) = [ cos θ - sin θ sin θ cos θ ] R(\theta)=\begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\\ \end{bmatrix}
  2. S ( θ ) = [ cos θ sin θ sin θ - cos θ ] S(\theta)=\begin{bmatrix}\cos\theta&\sin\theta\\ \sin\theta&-\cos\theta\\ \end{bmatrix}
  3. \oplus
  4. \cong

Point_process.html

  1. 𝔑 \mathfrak{N}
  2. 𝒩 \mathcal{N}
  3. 𝔑 \mathfrak{N}
  4. Φ B : 𝔑 + , ϱ ϱ ( B ) \Phi_{B}:\mathfrak{N}\to\mathbb{Z}_{+},\varrho\mapsto\varrho(B)
  5. ξ : Ω 𝔑 \xi:\Omega\to\mathfrak{N}
  6. ( Ω , , P ) (\Omega,\mathcal{F},P)
  7. ( 𝔑 , 𝒩 ) (\mathfrak{N},\mathcal{N})
  8. ξ = i = 1 N δ X i , \xi=\sum_{i=1}^{N}\delta_{X_{i}},
  9. δ \delta
  10. X i X_{i}
  11. X i X_{i}
  12. ξ ( x ) 1 \xi(x)\leq 1
  13. x d x\in\mathbb{R}^{d}
  14. E ξ ( B ) := E ( ξ ( B ) ) for every B . E\xi(B):=E\bigl(\xi(B)\bigr)\quad\,\text{for every }B\in\mathcal{B}.
  15. Ψ N ( f ) \Psi_{N}(f)
  16. [ 0 , ) [0,\infty)
  17. Ψ N ( f ) = E [ exp ( - N ( f ) ) ] \Psi_{N}(f)=E[\exp(-N(f))]
  18. n n
  19. ξ n , \xi^{n},
  20. S n S^{n}
  21. ξ n ( A 1 × × A n ) = i = 1 n ξ ( A i ) \xi^{n}(A_{1}\times\cdots\times A_{n})=\prod_{i=1}^{n}\xi(A_{i})
  22. ( S n , B ( S n ) ) . (S^{n},B(S^{n})).
  23. E ξ n ( ) E\xi^{n}(\cdot)
  24. n n
  25. S = d S=\mathbb{R}^{d}
  26. ξ \xi
  27. ρ ( k ) : ( d ) k [ 0 , ) \rho^{(k)}:(\mathbb{R}^{d})^{k}\to[0,\infty)
  28. B 1 , , B k B_{1},\ldots,B_{k}
  29. E ( i ξ ( B i ) ) = B 1 × × B k ρ ( k ) ( x 1 , , x k ) d x 1 d x k . E\left(\prod_{i}\xi(B_{i})\right)=\int_{B_{1}\times\cdots\times B_{k}}\rho^{(k% )}(x_{1},\ldots,x_{k})\,dx_{1}\cdots dx_{k}.
  30. ξ d \xi\subset\mathbb{R}^{d}
  31. ξ + x := i = 1 N δ X i + x \xi+x:=\sum_{i=1}^{N}\delta_{X_{i}+x}
  32. ξ \xi
  33. x d . x\in\mathbb{R}^{d}.
  34. E ξ ( ) = λ E\xi(\cdot)=\lambda\|\cdot\|
  35. λ 0 \lambda\geq 0
  36. \|\cdot\|
  37. λ \lambda
  38. d \mathbb{R}^{d}
  39. d \mathbb{R}^{d}
  40. d . \mathbb{R}^{d}.
  41. ξ \xi
  42. ξ ( B 1 ) , , ξ ( B n ) \xi(B_{1}),\ldots,\xi(B_{n})
  43. B 1 , , B n . B_{1},\ldots,B_{n}.
  44. B B
  45. ξ ( B ) \xi(B)
  46. λ B , \lambda\|B\|,
  47. \|\cdot\|
  48. B 1 , , B n B_{1},\ldots,B_{n}
  49. k 1 , , k n k_{1},\ldots,k_{n}
  50. Pr [ ξ ( B i ) = k i , 1 i n ] = i e - λ B i ( λ B i ) k i k i ! . \Pr[\xi(B_{i})=k_{i},1\leq i\leq n]=\prod_{i}e^{-\lambda\|B_{i}\|}\frac{(% \lambda\|B_{i}\|)^{k_{i}}}{k_{i}!}.
  51. λ \lambda
  52. λ . \lambda.
  53. λ B \lambda\|B\|
  54. B λ ( x ) d x \stackrel{}{}\int_{B}\lambda(x)\,dx
  55. λ \lambda
  56. d . \mathbb{R}^{d}.
  57. λ B \lambda\|B\|
  58. Λ \Lambda
  59. Λ \Lambda
  60. ξ \xi
  61. Λ ( ) \Lambda(\cdot)
  62. ξ ( B ) \xi(B)
  63. Λ ( B ) \Lambda(B)
  64. B . B.
  65. B 1 , , B n B_{1},\ldots,B_{n}
  66. Λ ( B 1 ) , , Λ ( B n ) , \Lambda(B_{1}),\ldots,\Lambda(B_{n}),
  67. ξ ( B 1 ) , , ξ ( B n ) \xi(B_{1}),\ldots,\xi(B_{n})
  68. E ξ ( ) = E Λ ( ) E\xi(\cdot)=E\Lambda(\cdot)
  69. λ . \lambda\|\cdot\|.
  70. Λ ( ) \Lambda(\cdot)
  71. Λ ( ) \Lambda(\cdot)
  72. λ ( ) \lambda(\cdot)
  73. Λ ( B ) = a.s. B λ ( x ) d x , \Lambda(B)\stackrel{\,\text{a.s.}}{=}\int_{B}\lambda(x)\,dx,
  74. λ ( ) \lambda(\cdot)
  75. λ ( y ) = exp ( X ( y ) ) \lambda(y)=\exp(X(y))
  76. X ( . ) X(.)
  77. λ ( y ) = X Φ h ( X , y ) \lambda(y)=\sum_{X\in\Phi}h(X,y)
  78. Φ ( ) \Phi(\cdot)
  79. h ( , ) h(\cdot,\cdot)
  80. λ ( y ) = X Φ h ( X , y ) \lambda(y)=\sum_{X\in\Phi}h(X,y)
  81. Φ ( ) \Phi(\cdot)
  82. h ( . , . ) h(.,.)
  83. λ ( y ) = h ( x , y ) L ( d x ) \lambda(y)=\int h(x,y)L(dx)
  84. L ( ) L(\cdot)
  85. h ( . , . ) h(.,.)
  86. λ ( y ) = X 1 2 ( y ) + + X k 2 ( y ) \lambda(y)=X_{1}^{2}(y)+\cdots+X_{k}^{2}(y)
  87. X i ( ) X_{i}(\cdot)
  88. λ ( y ) = λ / ( 1 + exp ( - X ( y ) ) ) \lambda(y)=\lambda^{\star}/(1+\exp(-X(y)))
  89. X ( ) X(\cdot)
  90. λ > 0 \lambda^{\star}>0
  91. B B
  92. Var ( ξ ( B ) ) Var ( ξ α ( B ) ) , \operatorname{Var}(\xi(B))\geq\operatorname{Var}(\xi_{\alpha}(B)),
  93. ξ α \xi_{\alpha}
  94. α ( ) := E ξ ( ) = E Λ ( ) . \alpha(\cdot):=E\xi(\cdot)=E\Lambda(\cdot).
  95. X k = j = 1 k T j for k 1. X_{k}=\sum_{j=1}^{k}T_{j}\quad\,\text{for }k\geq 1.
  96. λ ( t H t ) = lim Δ t 0 1 Δ t P ( One event occurs in the time-interval [ t , t + Δ t ] H t ) , \lambda(t\mid H_{t})=\lim_{\Delta t\to 0}\frac{1}{\Delta t}{P}(\,\text{One % event occurs in the time-interval}\,[t,t+\Delta t]\mid H_{t}),
  97. Λ ( s , u ) = s u λ ( t | H t ) d t \Lambda(s,u)=\int_{s}^{u}\lambda(t|H_{t})\mathrm{d}t
  98. N N
  99. n n
  100. n \mathbb{R}^{n}
  101. λ p ( x ) = lim δ 0 1 | B δ ( x ) | P { One event occurs in B δ ( x ) σ [ N ( B δ ( x ) ) ] } , \lambda_{p}(x)=\lim_{\delta\to 0}\frac{1}{|B_{\delta}(x)|}{P}\{\,\text{One % event occurs in }\,B_{\delta}(x)\mid\sigma[N\setminus(B_{\delta}(x))]\},
  102. B δ ( x ) B_{\delta}(x)
  103. x x
  104. δ \delta
  105. σ [ N ( B δ ( x ) ) ] \sigma[N\setminus(B_{\delta}(x))]
  106. N N
  107. B δ ( x ) B_{\delta}(x)

Poisson_kernel.html

  1. P r ( θ ) = n = - r | n | e i n θ = 1 - r 2 1 - 2 r cos θ + r 2 = Re ( 1 + r e i θ 1 - r e i θ ) , 0 r < 1. P_{r}(\theta)=\sum_{n=-\infty}^{\infty}r^{|n|}e^{in\theta}=\frac{1-r^{2}}{1-2r% \cos\theta+r^{2}}=\operatorname{Re}\left(\frac{1+re^{i\theta}}{1-re^{i\theta}}% \right),\ \ \ 0\leq r<1.
  2. D = { z : | z | < 1 } D=\{z:|z|<1\}
  3. u ( r e i θ ) = 1 2 π - π π P r ( θ - t ) f ( e i t ) d t , 0 r < 1 u(re^{i\theta})=\frac{1}{2\pi}\int_{-\pi}^{\pi}P_{r}(\theta-t)f(e^{it})\,% \mathrm{d}t,\ \ \ 0\leq r<1
  4. A r f ( e 2 π i x ) = k 𝐙 f k r | k | e 2 π i k x . A_{r}f(e^{2\pi ix})=\sum_{k\in\mathbf{Z}}f_{k}r^{|k|}e^{2\pi ikx}.
  5. u ( x + i y ) = 1 π - P y ( x - t ) f ( t ) d t u(x+iy)=\frac{1}{\pi}\int_{-\infty}^{\infty}P_{y}(x-t)f(t)dt
  6. y > 0 y>0
  7. P y ( x ) = y x 2 + y 2 . P_{y}(x)=\frac{y}{x^{2}+y^{2}}.
  8. f L p ( ) f\in L^{p}(\mathbb{R})
  9. u H p u\in H^{p}
  10. u H p = f L p \|u\|_{H^{p}}=\|f\|_{L^{p}}
  11. L p ( ) L^{p}(\mathbb{R})
  12. B r B_{r}
  13. P ( x , ζ ) = r 2 - | x | 2 r ω n - 1 | x - ζ | n P(x,\zeta)=\frac{r^{2}-|x|^{2}}{r\omega_{n-1}|x-\zeta|^{n}}
  14. x B r x\in B_{r}
  15. ζ S \zeta\in S
  16. B r B_{r}
  17. ω n - 1 \omega_{n-1}
  18. P [ u ] ( x ) = S u ( ζ ) P ( x , ζ ) d σ ( ζ ) . P[u](x)=\int_{S}u(\zeta)P(x,\zeta)d\sigma(\zeta).\,
  19. B r B_{r}
  20. ( t , x ) = ( t , x 1 , , x n ) . (t,x)=(t,x_{1},\dots,x_{n}).
  21. H n + 1 = { ( t ; 𝐱 ) 𝐑 n + 1 t > 0 } . H^{n+1}=\{(t;\mathbf{x})\in\mathbf{R}^{n+1}\mid t>0\}.
  22. P ( t , x ) = c n t ( t 2 + | x | 2 ) ( n + 1 ) / 2 P(t,x)=c_{n}\frac{t}{(t^{2}+|x|^{2})^{(n+1)/2}}
  23. c n = Γ [ ( n + 1 ) / 2 ] π ( n + 1 ) / 2 . c_{n}=\frac{\Gamma[(n+1)/2]}{\pi^{(n+1)/2}}.
  24. K ( t , ξ ) = e - 2 π t | ξ | K(t,\xi)=e^{-2\pi t|\xi|}
  25. P ( t , x ) = ( K ( t , ) ) ( x ) = 𝐑 n e - 2 π t | ξ | e - 2 π i ξ x d ξ . P(t,x)=\mathcal{F}(K(t,\cdot))(x)=\int_{\mathbf{R}^{n}}e^{-2\pi t|\xi|}e^{-2% \pi i\xi\cdot x}\,d\xi.
  26. P [ u ] ( t , x ) = [ P ( t , ) * u ] ( x ) P[u](t,x)=[P(t,\cdot)*u](x)

Poisson_regression.html

  1. 𝐱 n \mathbf{x}\in\mathbb{R}^{n}
  2. log ( E ( Y 𝐱 ) ) = α + β 𝐱 , \log(\operatorname{E}(Y\mid\mathbf{x}))=\alpha+\mathbf{\beta}^{\prime}\mathbf{% x},
  3. α \alpha\in\mathbb{R}
  4. β n \mathbf{\beta}\in\mathbb{R}^{n}
  5. log ( E ( Y 𝐱 ) ) = s y m b o l θ 𝐱 , \log(\operatorname{E}(Y\mid\mathbf{x}))=symbol{\theta}^{\prime}\mathbf{x},\,
  6. E ( Y 𝐱 ) = e s y m b o l θ 𝐱 . \operatorname{E}(Y\mid\mathbf{x})=e^{symbol{\theta}^{\prime}\mathbf{x}}.\,
  7. E ( Y x ) = e θ x \operatorname{E}(Y\mid x)=e^{\theta^{\prime}x}\,
  8. p ( y x ; θ ) = [ E ( Y x ) ] y × e - E ( Y x ) y ! = e y θ x e - e θ x y ! p(y\mid x;\theta)=\frac{[\operatorname{E}(Y\mid x)]^{y}\times e^{-% \operatorname{E}(Y\mid x)}}{y!}=\frac{e^{y\theta^{\prime}x}e^{-e^{\theta^{% \prime}x}}}{y!}
  9. x i n + 1 , i = 1 , , m x_{i}\in\mathbb{R}^{n+1},\,i=1,\ldots,m
  10. y 1 , , y m y_{1},\ldots,y_{m}\in\mathbb{R}
  11. p ( y 1 , , y m x 1 , , x m ; θ ) = i = 1 m e y i θ x i e - e θ x i y i ! . p(y_{1},\ldots,y_{m}\mid x_{1},\ldots,x_{m};\theta)=\prod_{i=1}^{m}\frac{e^{y_% {i}\theta^{\prime}x_{i}}e^{-e^{\theta^{\prime}x_{i}}}}{y_{i}!}.
  12. L ( θ X , Y ) = i = 1 m e y i θ x i e - e θ x i y i ! L(\theta\mid X,Y)=\prod_{i=1}^{m}\frac{e^{y_{i}\theta^{\prime}x_{i}}e^{-e^{% \theta^{\prime}x_{i}}}}{y_{i}!}
  13. ( θ X , Y ) = log L ( θ X , Y ) = i = 1 m ( y i θ x i - e θ x i - log ( y i ! ) ) \ell(\theta\mid X,Y)=\log L(\theta\mid X,Y)=\sum_{i=1}^{m}\left(y_{i}\theta^{% \prime}x_{i}-e^{\theta^{\prime}x_{i}}-\log(y_{i}!)\right)
  14. ( θ X , Y ) = i = 1 m ( y i θ x i - e θ x i ) \ell(\theta\mid X,Y)=\sum_{i=1}^{m}\left(y_{i}\theta^{\prime}x_{i}-e^{\theta^{% \prime}x_{i}}\right)
  15. ( θ X , Y ) θ = 0 \frac{\partial\ell(\theta\mid X,Y)}{\partial\theta}=0
  16. - ( θ X , Y ) -\ell(\theta\mid X,Y)
  17. log ( E ( Y x ) ) = log ( exposure ) + θ x \log{(\operatorname{E}(Y\mid x))}=\log{(\,\text{exposure})}+\theta^{\prime}x
  18. log ( E ( Y x ) ) - log ( exposure ) = log ( E ( Y x ) exposure ) = θ x \log{(\operatorname{E}(Y\mid x))}-\log{(\,\text{exposure})}=\log{\left(\frac{% \operatorname{E}(Y\mid x)}{\,\text{exposure}}\right)}=\theta^{\prime}x
  19. i = 1 m log ( p ( y i ; e θ x ) ) , \sum_{i=1}^{m}\log(p(y_{i};e^{\theta^{\prime}x})),
  20. p ( y i ; e θ x ) p(y_{i};e^{\theta^{\prime}x})
  21. e θ x e^{\theta^{\prime}x}
  22. i = 1 m log ( p ( y i ; e θ x ) ) - λ θ 2 2 , \sum_{i=1}^{m}\log(p(y_{i};e^{\theta^{\prime}x}))-\lambda\left\|\theta\right\|% _{2}^{2},
  23. λ \lambda

Polarity_(international_relations).html

  1. Concentration t = i = 1 N t ( S i t ) 2 - 1 N t 1 - 1 N t \,\text{Concentration}_{t}=\sqrt{\frac{\sum_{i=1}^{N_{t}}(S_{it})^{2}-\frac{1}% {N_{t}}}{1-\frac{1}{N_{t}}}}
  2. i = 1 n ( S i t ) 2 \sum_{i=1}^{n}(S_{it})^{2}

Polarizable_vacuum.html

  1. d s 2 = - 1 κ 2 d t 2 + κ 2 ( d x 2 + d y 2 + d z 2 ) ds^{2}=-\frac{1}{\kappa^{2}}dt^{2}+\kappa^{2}\,(dx^{2}+dy^{2}+dz^{2})
  2. κ 2 = K \kappa^{2}=K
  3. κ = exp ( m / r ) = 1 + m / r + O ( 1 r 2 ) \kappa=\exp(m/r)=1+m/r+O\left(\frac{1}{r^{2}}\right)
  4. d s 2 = - exp ( 2 ϕ ) + exp ( - 2 ϕ ) ( d x 2 + d y 2 + d z 2 ds^{2}=-\exp(2\,\phi)+\exp(-2\phi)(dx^{2}+dy^{2}+dz^{2}
  5. ϕ = 0 \Box\phi=0

Polarization_identity.html

  1. x \|x\|\,
  2. x , y \langle x,\ y\rangle\,
  3. \|\cdot\|
  4. x 2 = x , x \|x\|^{2}=\langle x,\ x\rangle
  5. x V x\in V
  6. 2 𝐮 2 + 2 𝐯 2 = 𝐮 + 𝐯 2 + 𝐮 - 𝐯 2 . 2\|\,\textbf{u}\|^{2}+2\|\,\textbf{v}\|^{2}=\|\,\textbf{u}+\,\textbf{v}\|^{2}+% \|\,\textbf{u}-\,\textbf{v}\|^{2}.
  7. x , y = 1 4 ( x + y 2 - x - y 2 ) x , y V . \langle x,\ y\rangle=\frac{1}{4}\left(\|x+y\|^{2}-\|x-y\|^{2}\right)\ \forall% \ x,y\in V\ .
  8. x , y = 1 4 ( x + y 2 - x - y 2 + i x + i y 2 - i x - i y 2 ) x , y V ; \langle x,\ y\rangle=\frac{1}{4}\left(\|x+y\|^{2}-\|x-y\|^{2}+i\|x+iy\|^{2}-i% \|x-iy\|^{2}\right)\ \forall\ x,y\in V;
  9. 𝐮 𝐯 = 1 2 ( 𝐮 + 𝐯 2 - 𝐮 2 - 𝐯 2 ) , ( 1 ) 𝐮 𝐯 = 1 2 ( 𝐮 2 + 𝐯 2 - 𝐮 - 𝐯 2 ) , ( 2 ) 𝐮 𝐯 = 1 4 ( 𝐮 + 𝐯 2 - 𝐮 - 𝐯 2 ) . ( 3 ) \begin{array}[]{lr}\,\textbf{u}\cdot\,\textbf{v}=\displaystyle\frac{1}{2}\left% (\|\,\textbf{u}+\,\textbf{v}\|^{2}-\|\,\textbf{u}\|^{2}-\|\,\textbf{v}\|^{2}% \right),&(1)\\ \,\textbf{u}\cdot\,\textbf{v}=\displaystyle\frac{1}{2}\left(\|\,\textbf{u}\|^{% 2}+\|\,\textbf{v}\|^{2}-\|\,\textbf{u}-\,\textbf{v}\|^{2}\right),&(2)\\ \,\textbf{u}\cdot\,\textbf{v}=\displaystyle\frac{1}{4}\left(\|\,\textbf{u}+\,% \textbf{v}\|^{2}-\|\,\textbf{u}-\,\textbf{v}\|^{2}\right).&(3)\end{array}
  10. 𝐮 - 𝐯 2 = 𝐮 2 + 𝐯 2 - 2 ( 𝐮 𝐯 ) . \|\,\textbf{u}-\,\textbf{v}\|^{2}=\|\,\textbf{u}\|^{2}+\|\,\textbf{v}\|^{2}-2(% \,\textbf{u}\cdot\,\textbf{v}).
  11. 𝐮 𝐯 = 𝐮 𝐯 cos θ , \,\textbf{u}\cdot\,\textbf{v}=\|\,\textbf{u}\|\,\|\,\textbf{v}\|\cos\theta,
  12. 𝐯 2 = 𝐯 𝐯 . \|\,\textbf{v}\|^{2}=\,\textbf{v}\cdot\,\textbf{v}.
  13. 𝐮 + 𝐯 2 = ( 𝐮 + 𝐯 ) ( 𝐮 + 𝐯 ) = ( 𝐮 𝐮 ) + ( 𝐮 𝐯 ) + ( 𝐯 𝐮 ) + ( 𝐯 𝐯 ) = 𝐮 2 + 𝐯 2 + 2 ( 𝐮 𝐯 ) , \begin{aligned}\displaystyle\|\,\textbf{u}+\,\textbf{v}\|^{2}&\displaystyle=(% \,\textbf{u}+\,\textbf{v})\cdot(\,\textbf{u}+\,\textbf{v})\\ &\displaystyle=(\,\textbf{u}\cdot\,\textbf{u})+(\,\textbf{u}\cdot\,\textbf{v})% +(\,\textbf{v}\cdot\,\textbf{u})+(\,\textbf{v}\cdot\,\textbf{v})\\ &\displaystyle=\|\,\textbf{u}\|^{2}+\|\,\textbf{v}\|^{2}+2(\,\textbf{u}\cdot\,% \textbf{v}),\end{aligned}
  14. 𝐮 - 𝐯 2 = 𝐮 2 + 𝐯 2 - 2 ( 𝐮 𝐯 ) . \|\,\textbf{u}-\,\textbf{v}\|^{2}=\|\,\textbf{u}\|^{2}+\|\,\textbf{v}\|^{2}-2(% \,\textbf{u}\cdot\,\textbf{v}).
  15. v = v , v . \|v\|=\sqrt{\langle v,v\rangle}.
  16. u , v = u v cos θ ; ( - π < θ π ) , \langle u,\ v\rangle=\|u\|\|v\|\cos\theta\ ;\ (-\pi<\theta\leq\pi)\ ,
  17. u , v u v . \langle u,\ v\rangle\leq\|u\|\|v\|\ .
  18. u , v = 0 \langle u,\ v\rangle=0\,
  19. u , v = 1 2 ( u + v 2 - u 2 - v 2 ) , u , v = 1 2 ( u 2 + v 2 - u - v 2 ) , u , v = 1 4 ( u + v 2 - u - v 2 ) . \begin{array}[]{l}\langle u,v\rangle=\frac{1}{2}\left(\|u+v\|^{2}-\|u\|^{2}-\|% v\|^{2}\right),\\ \langle u,v\rangle=\frac{1}{2}\left(\|u\|^{2}+\|v\|^{2}-\|u-v\|^{2}\right),\\ \langle u,v\rangle=\frac{1}{4}\left(\|u+v\|^{2}-\|u-v\|^{2}\right).\end{array}
  20. Q ( v ) = B ( v , v ) , Q(v)=B(v,v),\,\!
  21. 2 B ( u , v ) = Q ( u + v ) - Q ( u ) - Q ( v ) , 2 B ( u , v ) = Q ( u ) + Q ( v ) - Q ( u - v ) , 4 B ( u , v ) = Q ( u + v ) - Q ( u - v ) . \begin{aligned}\displaystyle 2B(u,v)&\displaystyle=Q(u+v)-Q(u)-Q(v),\\ \displaystyle 2B(u,v)&\displaystyle=Q(u)+Q(v)-Q(u-v),\\ \displaystyle 4B(u,v)&\displaystyle=Q(u+v)-Q(u-v).\end{aligned}
  22. v , u \langle v,u\rangle
  23. u , v \langle u,v\rangle
  24. Re u , v = 1 2 ( u + v 2 - u 2 - v 2 ) , Re u , v = 1 2 ( u 2 + v 2 - u - v 2 ) , Re u , v = 1 4 ( u + v 2 - u - v 2 ) . \begin{array}[]{l}\,\text{Re}\langle u,v\rangle=\frac{1}{2}\left(\|u+v\|^{2}-% \|u\|^{2}-\|v\|^{2}\right),\\ \,\text{Re}\langle u,v\rangle=\frac{1}{2}\left(\|u\|^{2}+\|v\|^{2}-\|u-v\|^{2}% \right),\\ \,\text{Re}\langle u,v\rangle=\frac{1}{4}\left(\|u+v\|^{2}-\|u-v\|^{2}\right).% \end{array}
  25. Im u , v = Re u , - i v \,\text{Im}\langle u,v\rangle=\,\text{Re}\langle u,-iv\rangle
  26. Im u , v = 1 2 ( u - i v 2 - u 2 - v 2 ) , Im u , v = 1 2 ( u 2 + v 2 - u + i v 2 ) , Im u , v = 1 4 ( u - i v 2 - u + i v 2 ) . \begin{array}[]{l}\,\text{Im}\langle u,v\rangle=\frac{1}{2}\left(\|u-iv\|^{2}-% \|u\|^{2}-\|v\|^{2}\right),\\ \,\text{Im}\langle u,v\rangle=\frac{1}{2}\left(\|u\|^{2}+\|v\|^{2}-\|u+iv\|^{2% }\right),\\ \,\text{Im}\langle u,v\rangle=\frac{1}{4}\left(\|u-iv\|^{2}-\|u+iv\|^{2}\right% ).\end{array}
  27. u , v = 4 - 1 k = 0 3 i k u + i k v 2 . \langle u,v\rangle=4^{-1}\sum_{k=0}^{3}i^{k}\|u+i^{k}v\|^{2}.

Polarizer.html

  1. I = I 0 cos 2 θ i , I=I_{0}\cos^{2}\theta_{i},
  2. cos 2 θ \cos^{2}\theta
  3. I I 0 = 1 2 . \frac{I}{I_{0}}=\frac{1}{2}.
  4. I = I 0 f f 0 [ 1 + λ ( f 0 - f ) 2 c ] cos 2 θ i I=I_{0}\frac{f}{f}_{0}\left[1+\frac{\lambda(f_{0}-f)}{2c}\right]\cos^{2}\theta% _{i}
  5. f 0 f_{0}
  6. f f
  7. λ \lambda
  8. c c

Polymatroid.html

  1. f f
  2. E E
  3. E P f := { x E | e U x ( e ) f ( U ) , U E } EP_{f}:=\{x\in\mathbb{R}^{E}|\sum_{e\in U}x(e)\leq f(U),\forall U\subseteq E\}
  4. P f := E P f { x E | x 0 } P_{f}:=EP_{f}\cap\{x\in\mathbb{R}^{E}|x\geq 0\}
  5. P f P_{f}
  6. E P f EP_{f}
  7. f f
  8. P f P_{f}
  9. f 0 f\geq 0
  10. E P f EP_{f}
  11. f ( ) 0 f(\emptyset)\geq 0
  12. E P EP
  13. f f
  14. f ( ) = 0 f(\emptyset)=0
  15. E P f = E P EP_{f}=EP
  16. f ( ) = 0 f(\emptyset)=0
  17. w + E : S E , e S w ( e ) f ( S ) w\in\mathbb{R}_{+}^{E}:\forall S\subseteq E,\sum_{e\in S}w(e)\geq f(S)

Polytomous_Rasch_model.html

  1. X n i = x { 0 , 1 , , m i } X_{ni}=x\in\{0,1,\dots,m_{i}\}\,
  2. m i m_{i}
  3. X n i X_{ni}
  4. m i m_{i}
  5. X n i = x X_{ni}=x
  6. Pr { X n i = x , x > 0 } = exp k = 1 x ( β n - τ k i ) 1 + j = 1 m i exp k = 1 j ( β n - τ k i ) ; \Pr\{X_{ni}=x,x>0\}=\frac{\exp{{\sum_{k=1}^{x}(\beta_{n}}-{\tau_{ki}})}}{1+% \sum_{j=1}^{m_{i}}\exp{{\sum_{k=1}^{j}(\beta_{n}}-{\tau_{ki}})}};
  7. Pr { X n i = 0 } = 1 1 + j = 1 m i exp k = 1 j ( β n - τ k i ) \Pr\{X_{ni}=0\}=\frac{1}{1+\sum_{j=1}^{m_{i}}\exp{{\sum_{k=1}^{j}(\beta_{n}}-{% \tau_{ki}})}}
  8. τ k i \tau_{ki}
  9. β n \beta_{n}
  10. m i m_{i}
  11. Pr { X n i = x } = exp k = 0 x ( β n - τ k i ) j = 0 m i exp k = 0 j ( β n - τ k i ) \Pr\{X_{ni}=x\}=\frac{\exp{{\sum_{k=0}^{x}(\beta_{n}}-{\tau_{ki}})}}{\sum_{j=0% }^{m_{i}}\exp{{\sum_{k=0}^{j}(\beta_{n}}-{\tau_{ki}})}}
  12. τ 0 i \tau_{0i}
  13. Pr { X n i = x } = exp k = 0 x ( β n - ( δ i - τ k ) ) j = 0 m exp k = 0 j ( β n - ( δ i - τ k ) ) \Pr\{X_{ni}=x\}=\frac{\exp{{\sum_{k=0}^{x}(\beta_{n}}-({\delta_{i}-\tau_{k}}))% }}{\sum_{j=0}^{m}\exp{{\sum_{k=0}^{j}(\beta_{n}}-{(\delta_{i}-\tau_{k}}))}}
  14. δ i \delta_{i}
  15. τ k \tau_{k}
  16. τ 0 \tau_{0}
  17. τ k i \tau_{ki}
  18. τ 1 i \tau_{1i}
  19. Y n k = y { 0 , 1 } , k { 0 , 1 , , m } Y_{nk}=y\in\{0,1\},k\in\{0,1,\dots,m\}\,
  20. Ω = def { 1 , , 1 , 0 , , 0 } \Omega^{\prime}\ \stackrel{\mathrm{def}}{=}\ \{1,\dots,1,0,\dots,0\}
  21. 0 , 0 0 0,0\Leftrightarrow 0
  22. 1 , 0 1 1,0\Leftrightarrow 1
  23. 1 , 1 2 1,1\Leftrightarrow 2
  24. P n x i = exp ( β n - τ k i ) 1 + exp ( β n - τ k i ) , k = x , P_{nxi}=\frac{\exp({\beta_{n}}-{\tau_{ki}})}{1+\exp({\beta_{n}}-{\tau_{ki}})},% \ k=x,\,
  25. Y n k i = 1 Y_{nki}=1
  26. Q n x i = 1 - P n x i Q_{nxi}=1-P_{nxi}
  27. P n 1 Q n 2 Q n 1 Q n 2 + P n 1 Q n 2 + P n 1 P n 2 . \frac{P_{n1}Q_{n2}}{Q_{n1}Q_{n2}+P_{n1}Q_{n2}+P_{n1}P_{n2}}.
  28. exp k = 1 1 ( β n - τ k ) 1 + j = 1 2 exp k = 1 j ( β n - τ k ) \frac{\exp{{\sum_{k=1}^{1}(\beta_{n}}-{\tau_{k}})}}{1+\sum_{j=1}^{2}\exp{{\sum% _{k=1}^{j}(\beta_{n}}-{\tau_{k}})}}
  29. P r { X n i = 1 } Pr\{X_{ni}=1\}
  30. { 0 , 0 } , { 1 , 0 } , \{0,0\},\{1,0\},
  31. { 1 , 1 } \{1,1\}
  32. Ω \Omega^{\prime}

Poncelet's_closure_theorem.html

  1. σ \sigma
  2. σ \sigma
  3. τ \tau
  4. τ σ \tau\sigma
  5. τ σ \tau\sigma

Ponderomotive_force.html

  1. 𝐅 p = \mathbf{F}_{\,\text{p}}=
  2. - e 2 4 m ω 2 -\frac{e^{2}}{4m\omega^{2}}
  3. \nabla
  4. E 2 E^{2}
  5. x ¨ = g ( x ) cos ( ω t ) , \ddot{x}=g(x)\cos(\omega t),
  6. g ( x ) g(x)
  7. x = x 0 + x 1 x=x_{0}+x_{1}
  8. x 0 x_{0}
  9. x 1 x_{1}
  10. x 1 x 0 x_{1}\ll x_{0}
  11. x 0 x_{0}
  12. x 0 ¨ + x 1 ¨ = [ g ( x 0 ) + x 1 g ( x 0 ) ] cos ( ω t ) \ddot{x_{0}}+\ddot{x_{1}}=\left[g(x_{0})+x_{1}g^{\prime}(x_{0})\right]\cos(% \omega t)
  13. x 0 ¨ x 1 ¨ \ddot{x_{0}}\ll\ddot{x_{1}}
  14. x 1 x_{1}
  15. g ( x 0 ) x 1 g ( x 0 ) g(x_{0})\gg x_{1}g^{\prime}(x_{0})
  16. x 1 ¨ = g ( x 0 ) cos ( ω t ) \ddot{x_{1}}=g(x_{0})\cos(\omega t)
  17. x 1 x_{1}
  18. x 0 x_{0}
  19. x 1 = - g ( x 0 ) ω 2 cos ( ω t ) x_{1}=-\frac{g(x_{0})}{\omega^{2}}\cos(\omega t)
  20. 2 π / ω 2\pi/\omega
  21. x 0 ¨ = - g ( x 0 ) g ( x 0 ) 2 ω 2 \ddot{x_{0}}=-\frac{g(x_{0})g^{\prime}(x_{0})}{2\omega^{2}}
  22. x 0 ¨ = - 1 4 ω 2 d d x [ g ( x ) 2 ] | x = x 0 \Rightarrow\ddot{x_{0}}=-\frac{1}{4\omega^{2}}\left.\frac{d}{dx}\left[g(x)^{2}% \right]\right|_{x=x_{0}}
  23. n ¯ ( x ) = n 0 exp [ - e κ T Φ P ( x ) ] \bar{n}(x)=n_{0}\exp\left[-\frac{e}{\kappa T}\Phi_{\,\text{P}}(x)\right]
  24. Φ P \Phi_{\,\text{P}}
  25. Φ P ( x ) = m 4 ω 2 [ g ( x ) ] 2 \Phi_{\,\text{P}}(x)=\frac{m}{4\omega^{2}}\left[g(x)\right]^{2}
  26. x ¨ = h ( x ) + g ( x ) cos ( ω t ) \ddot{x}=h(x)+g(x)\cos(\omega t)
  27. h ( x ) = 0 h(x)=0
  28. x 0 ¨ = h ( x 0 ) - g ( x 0 ) g ( x 0 ) 2 ω 2 \ddot{x_{0}}=h(x_{0})-\frac{g(x_{0})g^{\prime}(x_{0})}{2\omega^{2}}

Porkchop_plot.html

  1. C 3 = v 2 C_{3}=v_{\infty}^{2}\,\!
  2. v v_{\infty}\,
  3. 1 2 m v 2 \frac{1}{2}mv^{2}
  4. ϵ \epsilon

Portal:Gravitation::Einstein's_concept.html

  1. G a b = 8 π T a b G_{ab}=8\pi T_{ab}
  2. G a b G_{ab}
  3. T a b T_{ab}

Portal:Gravitation::Mass.html

  1. E / c 2 E/c^{2}

Portal:Gravitation::Newton's_concept.html

  1. F = G m 1 m 2 r 2 F=G\frac{m_{1}m_{2}}{r^{2}}
  2. F F
  3. m 1 m_{1}
  4. m 2 m_{2}
  5. r r
  6. G G

Portal:Gravitation::Where_to_start.html

  1. d = g t 2 2 \ d={gt^{2}\over 2}

Positive_energy_theorem.html

  1. Q Q
  2. P P
  3. M Q 2 + P 2 , M\geq\sqrt{Q^{2}+P^{2}},

Potential_temperature.html

  1. P P
  2. P 0 P_{0}
  3. θ \theta
  4. θ = T ( P 0 P ) R / c p , \theta=T\left(\frac{P_{0}}{P}\right)^{R/c_{p}},
  5. T T
  6. R R
  7. c p c_{p}
  8. R / c p = 0.286 R/c_{p}=0.286
  9. θ z > 0 \frac{\partial\theta}{\partial z}>0
  10. θ z < 0 \frac{\partial\theta}{\partial z}<0
  11. d h = T d s + v d p , dh=T\,ds+v\,dp,
  12. d h dh
  13. T T
  14. d s ds
  15. v v
  16. p p
  17. d h = v d p . dh=v\,dp.
  18. p v = R T pv=RT
  19. d p p = c p R d T T , \frac{dp}{p}={\frac{c_{p}}{R}\frac{dT}{T}},
  20. d h = c p d T dh=c_{p}dT
  21. p v pv
  22. ( p 1 p 0 ) R / c p = T 1 T 0 , \left(\frac{p_{1}}{p_{0}}\right)^{R/c_{p}}=\frac{T_{1}}{T_{0}},
  23. T 0 T_{0}
  24. p 0 p_{0}
  25. T 0 = T 1 ( p 0 p 1 ) R / c p θ . T_{0}=T_{1}\left(\frac{p_{0}}{p_{1}}\right)^{R/c_{p}}\equiv\theta.

Potentiostat.html

  1. R = E I {R}={E\over I}
  2. I o = E c R v I_{o}={E_{c}\over R_{v}}
  3. I o I_{o}
  4. E c E_{c}
  5. R v R_{v}
  6. i i
  7. R R
  8. R m R_{\textrm{m}}
  9. E i E_{\textrm{i}}
  10. R m R_{\textrm{m}}
  11. Z 1 Z_{1}
  12. R m R_{\textrm{m}}
  13. Z 2 Z_{2}
  14. E out = A ( E + - E - ) = A ( E i - E r ) E_{\textrm{out}}=A\,(E^{+}-E^{-})=A\,(E_{\textrm{i}}-E_{\textrm{r}})
  15. A A
  16. I c = E out Z 1 + Z 2 I_{\textrm{c}}=\frac{E_{\textrm{out}}}{Z_{1}+Z_{2}}
  17. I c = E r Z 2 I_{\textrm{c}}=\frac{E_{\textrm{r}}}{Z_{2}}
  18. E r = Z 2 Z 1 + Z 2 E out = β E out , E_{\textrm{r}}=\frac{Z_{2}}{Z_{1}+Z_{2}}\,E_{\textrm{out}}=\beta\,E_{\textrm{% out}},
  19. β \beta
  20. β = Z 2 Z 1 + Z 2 \beta=\frac{Z_{2}}{Z_{1}+Z_{2}}
  21. E r E i = β A 1 + β A \frac{E_{\textrm{r}}}{E_{\textrm{i}}}=\frac{\beta\,A}{1+\beta\,A}
  22. β \beta
  23. A A
  24. E i = E r E_{\textrm{i}}=E_{\textrm{r}}

Pound–Rebka_experiment.html

  1. f r = 1 - v / c 1 + v / c f e . f_{r}=\sqrt{\frac{1-v/c}{1+v/c}}f_{e}.
  2. f r = 1 - 2 G M ( R + h ) c 2 1 - 2 G M R c 2 f e . f_{r}=\sqrt{\frac{1-\dfrac{2GM}{(R+h)c^{2}}}{1-\dfrac{2GM}{Rc^{2}}}}f_{e}.
  3. 1 - v / c 1 + v / c 1 - 2 G M ( R + h ) c 2 1 - 2 G M R c 2 = 1. \sqrt{\frac{1-v/c}{1+v/c}\cdot\frac{1-\dfrac{2GM}{(R+h)c^{2}}}{1-\dfrac{2GM}{% Rc^{2}}}}=1.
  4. h R h\ll R
  5. v g h c v\approx\frac{gh}{c}

Pourbaix_diagram.html

  1. E H = E 0 - 0.0592 n log [ C ] c [ D ] d [ A ] a [ B ] b { V } E_{H}=E^{0}-\frac{0.0592}{n}\log\frac{[C]^{c}[D]^{d}}{[A]^{a}[B]^{b}}\ \{V\}
  2. pH = - log 10 ( a H + ) = log 10 ( 1 a H + ) \mathrm{pH}=-\log_{10}(a_{\textrm{H}^{+}})=\log_{10}\left(\frac{1}{a_{\textrm{% H}^{+}}}\right)
  3. 2 H 2 O + 2 e - H 2 ( g ) + 2 O H - 2H_{2}O+2e^{-}\rightarrow H_{2}(g)+2OH^{-}
  4. 2 H 3 O + + 2 e - H 2 ( g ) + 2 H 2 O 2H_{3}O^{+}+2e^{-}\rightarrow H_{2}(g)+2H_{2}O
  5. E H = - 0.0591 * p H { V } E_{H}=-0.0591*pH\ \{V\}
  6. 6 H 2 O 4 H 3 O + + O 2 ( g ) + 4 e - 6H_{2}O\rightarrow 4H_{3}O^{+}+O_{2}(g)+4e^{-}
  7. E H = 1.229 V - 0.0591 * p H { V } E_{H}=1.229V-0.0591*pH\ \{V\}
  8. p E = E H 0.0591 V pE=\frac{E_{H}}{0.0591V}

Powder_diffraction.html

  1. | G | = q = 2 k sin ( θ ) = 4 π λ sin ( θ ) . |G|=q=2k\sin(\theta)=\frac{4\pi}{\lambda}\sin(\theta).\,
  2. B cos ( θ ) = k λ D + η sin ( θ ) , B\cdot\cos(\theta)=\frac{k\lambda}{D}+\eta\cdot\sin(\theta),
  3. B cos ( θ ) \displaystyle B\cdot\cos(\theta)
  4. sin ( θ ) \displaystyle\sin(\theta)
  5. η \displaystyle\eta
  6. k λ D \displaystyle\frac{k\lambda}{D}
  7. I ( q ) = i = 1 N j = 1 N f i ( q ) f j ( q ) sin ( q r i j ) q r i j , I(q)=\sum_{i=1}^{N}\sum_{j=1}^{N}f_{i}(q)f_{j}(q)\frac{\sin(qr_{ij})}{qr_{ij}},

Power_domains.html

  1. \vee

Power_MOSFET.html

  1. C i s s = C G S + C G D C o s s = C G D + C D S C r s s = C G D \begin{matrix}C_{iss}&=&C_{GS}+C_{GD}\\ C_{oss}&=&C_{GD}+C_{DS}\\ C_{rss}&=&C_{GD}\end{matrix}
  2. C G D = C o x D × C G D j ( V G D ) C o x D + C G D j ( V G D ) C_{GD}=\frac{C_{oxD}\times C_{GDj}\left(V_{GD}\right)}{C_{oxD}+C_{GDj}\left(V_% {GD}\right)}
  3. w G D j = 2 ϵ S i V G D q N w_{GDj}=\sqrt{\frac{2\epsilon_{Si}V_{GD}}{qN}}
  4. ϵ S i \epsilon_{Si}
  5. C G D j = A G D ϵ S i w G D j C_{GDj}=A_{GD}\frac{\epsilon_{Si}}{w_{GDj}}
  6. C G D j ( V G D ) = A G D q ϵ S i N 2 V G D C_{GDj}\left(V_{GD}\right)=A_{GD}\sqrt{\frac{q\epsilon_{Si}N}{2V_{GD}}}
  7. V i n + V G S V_{in}+V_{GS}
  8. V i n V_{in}
  9. R D S o n R_{DSon}

Power_of_a_point.html

  1. h = s 2 - r 2 , h=s^{2}-r^{2},
  2. 𝐏𝐓 ¯ 2 = 𝐏𝐌 ¯ × 𝐏𝐍 ¯ = 𝐏𝐀 ¯ × 𝐏𝐁 ¯ = ( s - r ) × ( s + r ) = s 2 - r 2 = h . \mathbf{\overline{PT}}^{2}=\mathbf{\overline{PM}}\times\mathbf{\overline{PN}}=% \mathbf{\overline{PA}}\times\mathbf{\overline{PB}}=(s-r)\times(s+r)=s^{2}-r^{2% }=h.
  3. R 2 = s 2 - r 2 = p R^{2}=s^{2}-r^{2}=p\,
  4. A P A Q AP\cdot AQ\,
  5. A P A Q = A R A S AP\cdot AQ=AR\cdot AS\,
  6. A P A Q = A R A S AP\cdot AQ=AR\cdot AS\,
  7. A P A Q = A R A S AP\cdot AQ=AR\cdot AS\,
  8. A P A P = A R A S AP\cdot AP=AR\cdot AS\,
  9. A P 2 = A R A S AP^{2}=AR\cdot AS\,
  10. ( A 1 A 2 ) 2 - r 1 2 - r 2 2 (A_{1}A_{2})^{2}-r_{1}^{2}-r_{2}^{2}\,
  11. r 1 r 2 cos φ r_{1}r_{2}\cos\varphi\,

Precision_rectifier.html

  1. R L R_{\mathrm{L}}
  2. R 2 R_{2}
  3. - R 2 / R 1 -R_{2}/R_{1}

Precision_Time_Protocol.html

  1. t t
  2. o ( t ) o(t)
  3. t t
  4. o ( t ) = s ( t ) - m ( t ) \ o(t)=s(t)-m(t)
  5. s ( t ) s(t)
  6. t t
  7. m ( t ) m(t)
  8. t t
  9. T 1 T1
  10. T 1 T1^{\prime}
  11. T 1 T1
  12. T 1 T1
  13. d d
  14. T 2 T2
  15. T 2 T2^{\prime}
  16. T 2 T2^{\prime}
  17. T 1 T1
  18. T 1 T1^{\prime}
  19. T 2 T2
  20. T 2 T2^{\prime}
  21. d d
  22. o ~ \tilde{o}
  23. T 1 - T 1 = o ~ + d \ T1^{\prime}-T1=\tilde{o}+d
  24. T 2 - T 2 = - o ~ + d \ T2^{\prime}-T2=-\tilde{o}+d
  25. o ~ = ( T 1 - T 1 - T 2 + T 2 ) / 2 \tilde{o}=(T1^{\prime}-T1-T2^{\prime}+T2)/2
  26. o ~ \tilde{o}

Preferential_entailment.html

  1. F F
  2. \leq
  3. F F
  4. \leq
  5. F pref G F\models\text{pref}G
  6. F F
  7. \leq
  8. G G

Prewellordering.html

  1. \leq
  2. x y y x x\leq y\land y\nleq x
  3. \leq
  4. X X
  5. \sim
  6. x y x y y x x\sim y\iff x\leq y\land y\leq x
  7. \sim
  8. X X
  9. \leq
  10. X / X/\sim
  11. X X
  12. X X
  13. ϕ : X O r d \phi:X\to Ord
  14. x y ϕ ( x ) ϕ ( y ) x\leq y\iff\phi(x)\leq\phi(y)
  15. ϕ : X O r d \phi:X\to Ord
  16. x X x\in X
  17. α < ϕ ( x ) \alpha<\phi(x)
  18. y X y\in X
  19. ϕ ( y ) = α \phi(y)=\alpha
  20. s y m b o l Γ symbol{\Gamma}
  21. \mathcal{F}
  22. \mathcal{F}
  23. \leq
  24. P P
  25. X X
  26. \mathcal{F}
  27. \leq
  28. s y m b o l Γ symbol{\Gamma}
  29. P P
  30. < * <^{*}\,
  31. * \leq^{*}
  32. s y m b o l Γ symbol{\Gamma}
  33. x , y X x,y\in X
  34. x < * y x P [ y P { x y y x } ] x<^{*}y\iff x\in P\land[y\notin P\lor\{x\leq y\land y\not\leq x\}]
  35. x * y x P [ y P x y ] x\leq^{*}y\iff x\in P\land[y\notin P\lor x\leq y]
  36. s y m b o l Γ symbol{\Gamma}
  37. s y m b o l Γ symbol{\Gamma}
  38. s y m b o l Γ symbol{\Gamma}
  39. s y m b o l Π 1 1 symbol{\Pi}^{1}_{1}\,
  40. s y m b o l Σ 2 1 symbol{\Sigma}^{1}_{2}
  41. n ω n\in\omega
  42. s y m b o l Π 2 n + 1 1 symbol{\Pi}^{1}_{2n+1}
  43. s y m b o l Σ 2 n + 2 1 symbol{\Sigma}^{1}_{2n+2}
  44. s y m b o l Γ symbol{\Gamma}
  45. X X\in\mathcal{F}
  46. A , B X A,B\subseteq X
  47. A A
  48. B B
  49. s y m b o l Γ symbol{\Gamma}
  50. A B A\cup B
  51. A * , B * A^{*},B^{*}\,
  52. s y m b o l Γ symbol{\Gamma}
  53. A * A A^{*}\subseteq A
  54. B * B B^{*}\subseteq B
  55. s y m b o l Γ symbol{\Gamma}
  56. s y m b o l Γ symbol{\Gamma}
  57. X X\in\mathcal{F}
  58. A , B X A,B\subseteq X
  59. A A
  60. B B
  61. s y m b o l Γ symbol{\Gamma}
  62. C X C\subseteq X
  63. C C
  64. X C X\setminus C
  65. s y m b o l Γ symbol{\Gamma}
  66. A C A\subseteq C
  67. B C = B\cap C=\emptyset
  68. s y m b o l Π 1 1 symbol{\Pi}^{1}_{1}
  69. s y m b o l Σ 1 1 symbol{\Sigma}^{1}_{1}
  70. A A
  71. B B
  72. X X
  73. C C
  74. X X
  75. C C
  76. A A
  77. B B

Primary_field.html

  1. K μ K_{\mu}
  2. x = 0 x=0
  3. [ K μ , 𝒪 ( 0 ) ] = 0 [K_{\mu},\mathcal{O}(0)]=0
  4. P μ P_{\mu}
  5. L n , L ¯ n , - < n < L_{n},\bar{L}_{n},-\infty<n<\infty
  6. L n , L ¯ n L_{n},\bar{L}_{n}
  7. L n , L ¯ n L_{n},\bar{L}_{n}
  8. L 1 , L ¯ 1 L_{1},\bar{L}_{1}
  9. D 6 D\leq 6
  10. K μ K_{\mu}

Primary_ideal.html

  1. 𝔮 \mathfrak{q}
  2. x y 𝔮 xy\in\mathfrak{q}
  3. x 𝔮 x\in\mathfrak{q}
  4. y 𝔮 y\in\mathfrak{q}
  5. x , y 𝔮 x,y\in\sqrt{\mathfrak{q}}
  6. 𝔮 \sqrt{\mathfrak{q}}
  7. 𝔮 \mathfrak{q}
  8. R = k [ x , y , z ] / ( x y - z 2 ) R=k[x,y,z]/(xy-z^{2})
  9. 𝔭 = ( x ¯ , z ¯ ) \mathfrak{p}=(\overline{x},\overline{z})
  10. 𝔮 = 𝔭 2 \mathfrak{q}=\mathfrak{p}^{2}
  11. 𝔭 \mathfrak{p}
  12. 𝔮 = 𝔭 \sqrt{\mathfrak{q}}=\mathfrak{p}
  13. x ¯ y ¯ = z ¯ 2 𝔭 2 = 𝔮 \overline{x}\overline{y}={\overline{z}}^{2}\in\mathfrak{p}^{2}=\mathfrak{q}
  14. x ¯ 𝔮 \overline{x}\not\in\mathfrak{q}
  15. y ¯ n 𝔮 {\overline{y}}^{n}\not\in\mathfrak{q}
  16. 𝔮 \mathfrak{q}
  17. 𝔮 \mathfrak{q}
  18. ( x ¯ ) ( x ¯ 2 , x ¯ z ¯ , y ¯ ) (\overline{x})\cap({\overline{x}}^{2},\overline{x}\overline{z},\overline{y})
  19. ( x ¯ ) (\overline{x})
  20. 𝔭 \mathfrak{p}
  21. ( x ¯ 2 , x ¯ z ¯ , y ¯ ) ({\overline{x}}^{2},\overline{x}\overline{z},\overline{y})
  22. ( x ¯ , y ¯ , z ¯ ) (\overline{x},\overline{y},\overline{z})
  23. A A P A\to A_{P}

Primary_School_Leaving_Examination.html

  1. T = 50 + 10 x - μ σ T=50+10{x-\mu\over\sigma}

Prime_decomposition_(3-manifold).html

  1. n n
  2. M = M # S n . M=M\#S^{n}.

Primitive_permutation_group.html

  1. S 3 S_{3}
  2. X = { 1 , 2 , 3 } X=\{1,2,3\}
  3. η = ( 1 2 3 2 3 1 ) . \eta=\begin{pmatrix}1&2&3\\ 2&3&1\end{pmatrix}.
  4. S 3 S_{3}
  5. η \eta
  6. S 4 S_{4}
  7. { 1 , 2 , 3 , 4 } \{1,2,3,4\}
  8. σ = ( 1 2 3 4 2 3 4 1 ) . \sigma=\begin{pmatrix}1&2&3&4\\ 2&3&4&1\end{pmatrix}.
  9. σ \sigma
  10. ( X 1 , X 2 ) (X_{1},X_{2})
  11. X 1 = { 1 , 3 } X_{1}=\{1,3\}
  12. X 2 = { 2 , 4 } X_{2}=\{2,4\}
  13. σ \sigma
  14. σ ( X 1 ) = X 2 \sigma(X_{1})=X_{2}
  15. σ ( X 2 ) = X 1 \sigma(X_{2})=X_{1}
  16. S n S_{n}
  17. { 1 , , n } \{1,\ldots,n\}
  18. A n A_{n}
  19. { 1 , , n } \{1,\ldots,n\}

Principal_curvature.html

  1. I I ( X , Y ) I\!I(X,Y)
  2. [ I I i j ] = [ I I ( X 1 , X 1 ) I I ( X 1 , X 2 ) I I ( X 2 , X 1 ) I I ( X 2 , X 2 ) ] . \left[I\!I_{ij}\right]=\begin{bmatrix}I\!I(X_{1},X_{1})&I\!I(X_{1},X_{2})\\ I\!I(X_{2},X_{1})&I\!I(X_{2},X_{2})\end{bmatrix}.
  3. [ I I i j ] \left[I\!I_{ij}\right]
  4. I I ( X i , X j ) I\!I(X_{i},X_{j})
  5. K ( X i , X j ) = k i k j K(X_{i},X_{j})=k_{i}k_{j}
  6. i , j i,j
  7. i j i\neq j

Principal_ideal_theorem.html

  1. I O L IO_{L}

Prism_coupler.html

  1. exp ( - α ( x ) d x + i β w x ) \exp\left(-\int\alpha(x)\,dx+i\beta_{w}x\right)
  2. β w \beta_{w}
  3. A α ( x ) exp ( - α ( x ) d x + i β w x ) A\sqrt{\alpha(x)}\exp\left(-\int\alpha(x)\,dx+i\beta_{w}x\right)
  4. f ( x ) exp ( - i β i n x ) f(x)\exp(-i\beta_{in}x)
  5. A f ( x ) α ( x ) exp ( - α ( x ) d x ) Af(x)\sqrt{\alpha(x)}\exp\left(-\int\alpha(x)\,dx\right)
  6. β m \beta_{m}
  7. θ m \theta_{m}
  8. β m = 2 π λ 0 n p cos θ m \beta_{m}=\frac{2\pi}{\lambda_{0}}n_{p}\cos\theta_{m}
  9. n p n_{p}
  10. β m = k n 1 cos θ m \beta_{m}=kn_{1}\cos\theta_{m}
  11. n 1 n_{1}
  12. β m \beta_{m}
  13. β m > k n 1 \beta_{m}>kn_{1}
  14. sin θ m > 1 \sin\theta_{m}>1

Private_language_argument.html

  1. x quus y = { x + y for x , y < 57 5 otherwise \,\text{x quus y}=\begin{cases}\,\text{x + y}&\,\text{for }x,y<57\\ 5&\,\text{otherwise}\end{cases}

Probabilistic_latent_semantic_analysis.html

  1. d d
  2. c c
  3. P ( c | d ) P(c|d)
  4. w w
  5. P ( w | c ) P(w|c)
  6. d d
  7. w w
  8. c c
  9. ( w , d ) (w,d)
  10. P ( w , d ) = c P ( c ) P ( d | c ) P ( w | c ) = P ( d ) c P ( c | d ) P ( w | c ) P(w,d)=\sum_{c}P(c)P(d|c)P(w|c)=P(d)\sum_{c}P(c|d)P(w|c)
  11. w w
  12. d d
  13. c c
  14. P ( d | c ) P(d|c)
  15. P ( w | c ) P(w|c)
  16. d d
  17. P ( c | d ) P(c|d)
  18. P ( w | c ) P(w|c)
  19. c d + w c cd+wc

Problem_of_Apollonius.html

  1. ( x s - x 1 ) 2 + ( y s - y 1 ) 2 = ( r s - s 1 r 1 ) 2 \left(x_{s}-x_{1}\right)^{2}+\left(y_{s}-y_{1}\right)^{2}=\left(r_{s}-s_{1}r_{% 1}\right)^{2}
  2. ( x s - x 2 ) 2 + ( y s - y 2 ) 2 = ( r s - s 2 r 2 ) < m t p l > 2 \left(x_{s}-x_{2}\right)^{2}+\left(y_{s}-y_{2}\right)^{2}=\left(r_{s}-s_{2}r_{% 2}\right)^{<}mtpl>{{2}}
  3. ( x s - x 3 ) 2 + ( y s - y 3 ) 2 = ( r s - s 3 r 3 ) 2 . \left(x_{s}-x_{3}\right)^{2}+\left(y_{s}-y_{3}\right)^{2}=\left(r_{s}-s_{3}r_{% 3}\right)^{2}.
  4. x s = M + N r s x_{s}=M+Nr_{s}
  5. y s = P + Q r s y_{s}=P+Qr_{s}
  6. ( X 1 | X 2 ) := v 1 w 2 + v 2 w 1 + 𝐜 1 𝐜 2 - s 1 s 2 r 1 r 2 . \left(X_{1}|X_{2}\right):=v_{1}w_{2}+v_{2}w_{1}+\mathbf{c}_{1}\cdot\mathbf{c}_% {2}-s_{1}s_{2}r_{1}r_{2}.
  7. ( X 1 - X 2 | X 1 - X 2 ) = 2 ( v 1 - v 2 ) ( w 1 - w 2 ) + ( 𝐜 1 - 𝐜 2 ) ( 𝐜 1 - 𝐜 2 ) - ( s 1 r 1 - s 2 r 2 ) 2 . \left(X_{1}-X_{2}|X_{1}-X_{2}\right)=2\left(v_{1}-v_{2}\right)\left(w_{1}-w_{2% }\right)+\left(\mathbf{c}_{1}-\mathbf{c}_{2}\right)\cdot\left(\mathbf{c}_{1}-% \mathbf{c}_{2}\right)-\left(s_{1}r_{1}-s_{2}r_{2}\right)^{2}.
  8. ( X 1 - X 2 | X 1 - X 2 ) = ( X 1 | X 1 ) - 2 ( X 1 | X 2 ) + ( X 2 | X 2 ) . \left(X_{1}-X_{2}|X_{1}-X_{2}\right)=\left(X_{1}|X_{1}\right)-2\left(X_{1}|X_{% 2}\right)+\left(X_{2}|X_{2}\right).
  9. - 2 ( X 1 | X 2 ) = | 𝐜 1 - 𝐜 2 | 2 - ( s 1 r 1 - s 2 r 2 ) 2 . -2\left(X_{1}|X_{2}\right)=\left|\mathbf{c}_{1}-\mathbf{c}_{2}\right|^{2}-% \left(s_{1}r_{1}-s_{2}r_{2}\right)^{2}.
  10. | 𝐜 1 - 𝐜 2 | 2 = ( r 1 - r 2 ) 2 . \left|\mathbf{c}_{1}-\mathbf{c}_{2}\right|^{2}=\left(r_{1}-r_{2}\right)^{2}.
  11. | 𝐜 1 - 𝐜 2 | 2 = ( r 1 + r 2 ) 2 . \left|\mathbf{c}_{1}-\mathbf{c}_{2}\right|^{2}=\left(r_{1}+r_{2}\right)^{2}.
  12. ( X sol | X sol ) = ( X sol | X 1 ) = ( X sol | X 2 ) = ( X sol | X 3 ) = 0 \left(X_{\mathrm{sol}}|X_{\mathrm{sol}}\right)=\left(X_{\mathrm{sol}}|X_{1}% \right)=\left(X_{\mathrm{sol}}|X_{2}\right)=\left(X_{\mathrm{sol}}|X_{3}\right% )=0
  13. 𝐎𝐏 ¯ 𝐎𝐏 ¯ = R 2 . \overline{\mathbf{OP}}\cdot\overline{\mathbf{OP^{\prime}}}=R^{2}.
  14. cos θ = d s 2 + d non 2 - d T 2 2 d s d non C ± . \cos\theta=\frac{d_{\mathrm{s}}^{2}+d_{\mathrm{non}}^{2}-d_{\mathrm{T}}^{2}}{2% d_{\mathrm{s}}d_{\mathrm{non}}}\equiv C_{\pm}.
  15. θ = ± 2 atan ( 1 - C 1 + C ) . \theta=\pm 2\ \mathrm{atan}\left(\sqrt{\frac{1-C}{1+C}}\right).
  16. X 3 A 1 ¯ X 3 A 2 ¯ = X 3 B 1 ¯ X 3 B 2 ¯ \overline{X_{3}A_{1}}\cdot\overline{X_{3}A_{2}}=\overline{X_{3}B_{1}}\cdot% \overline{X_{3}B_{2}}
  17. ( k 1 + k 2 + k 3 + k s ) 2 = 2 ( k 1 2 + k 2 2 + k 3 2 + k s 2 ) \left(k_{1}+k_{2}+k_{3}+k_{s}\right)^{2}=2\,\left(k_{1}^{2}+k_{2}^{2}+k_{3}^{2% }+k_{s}^{2}\right)

Procept.html

  1. 3 + 4 3+4
  2. n = 0 ( a n ) \sum_{n=0}^{\infty}(a_{n})
  3. f ( x ) = 3 x + 2 f(x)=3x+2

Product_measure.html

  1. ( X 1 , Σ 1 ) (X_{1},\Sigma_{1})
  2. ( X 2 , Σ 2 ) (X_{2},\Sigma_{2})
  3. Σ 1 \Sigma_{1}
  4. Σ 2 \Sigma_{2}
  5. X 1 X_{1}
  6. X 2 X_{2}
  7. μ 1 \mu_{1}
  8. μ 2 \mu_{2}
  9. Σ 1 Σ 2 \Sigma_{1}\otimes\Sigma_{2}
  10. X 1 × X 2 X_{1}\times X_{2}
  11. B 1 × B 2 B_{1}\times B_{2}
  12. B 1 Σ 1 B_{1}\in\Sigma_{1}
  13. B 2 Σ 2 . B_{2}\in\Sigma_{2}.
  14. μ 1 × μ 2 \mu_{1}\times\mu_{2}
  15. ( X 1 × X 2 , Σ 1 Σ 2 ) (X_{1}\times X_{2},\Sigma_{1}\otimes\Sigma_{2})
  16. ( μ 1 × μ 2 ) ( B 1 × B 2 ) = μ 1 ( B 1 ) μ 2 ( B 2 ) (\mu_{1}\times\mu_{2})(B_{1}\times B_{2})=\mu_{1}(B_{1})\mu_{2}(B_{2})
  17. B 1 Σ 1 , B 2 Σ 2 B_{1}\in\Sigma_{1},\ B_{2}\in\Sigma_{2}
  18. σ \sigma
  19. ( μ 1 × μ 2 ) ( E ) = X 2 μ 1 ( E y ) d μ 2 ( y ) = X 1 μ 2 ( E x ) d μ 1 ( x ) , (\mu_{1}\times\mu_{2})(E)=\int_{X_{2}}\mu_{1}(E^{y})\,d\mu_{2}(y)=\int_{X_{1}}% \mu_{2}(E_{x})\,d\mu_{1}(x),
  20. E x = { y X 2 | ( x , y ) E } E_{x}=\{y\in X_{2}|(x,y)\in E\}
  21. E y = { x X 1 | ( x , y ) E } E^{y}=\{x\in X_{1}|(x,y)\in E\}
  22. ( X 1 , Σ 1 , μ 1 ) (X_{1},\Sigma_{1},\mu_{1})
  23. ( X 2 , Σ 2 , μ 2 ) (X_{2},\Sigma_{2},\mu_{2})

Product_term.html

  1. A B A\wedge B
  2. A ( ¬ B ) ( ¬ C ) A\wedge(\neg B)\wedge(\neg C)
  3. ¬ A \neg A
  4. n n
  5. x 1 , , x n {x_{1},\dots,x_{n}}
  6. n n

Proizvolov's_identity.html

  1. A 1 < A 2 < < A N . A_{1}<A_{2}<\cdots<A_{N}.
  2. B 1 > B 2 > > B N . B_{1}>B_{2}>\cdots>B_{N}.
  3. | A 1 - B 1 | + | A 2 - B 2 | + + | A N - B N | |A_{1}-B_{1}|+|A_{2}-B_{2}|+\cdots+|A_{N}-B_{N}|
  4. | A 1 - B 1 | + | A 2 - B 2 | + | A 3 - B 3 | = | 2 - 6 | + | 3 - 4 | + | 5 - 1 | = 4 + 1 + 4 = 9 , |A_{1}-B_{1}|+|A_{2}-B_{2}|+|A_{3}-B_{3}|=|2-6|+|3-4|+|5-1|=4+1+4=9,

Proj_construction.html

  1. S S
  2. S = i 0 S i S=\bigoplus_{i\geq 0}S_{i}
  3. S + = i > 0 S i . S_{+}=\bigoplus_{i>0}S_{i}.
  4. V ( a ) = { p Proj S a p } , V(a)=\{p\in\operatorname{Proj}\,S\mid a\subseteq p\},
  5. ( a i ) i I (a_{i})_{i\in I}
  6. V ( a i ) = V ( Σ a i ) \bigcap V(a_{i})=V(\Sigma a_{i})
  7. V ( a i ) = V ( Π a i ) \bigcup V(a_{i})=V(\Pi a_{i})
  8. D ( a ) = { p Proj S a p } . D(a)=\{p\in\operatorname{Proj}\,S\mid a\;\not\subseteq\;p\}.
  9. S + S_{+}
  10. O X ( U ) O_{X}(U)
  11. f : U p U S ( p ) f\colon U\to\bigcup_{p\in U}S_{(p)}
  12. S ( p ) S_{(p)}
  13. S p S_{p}
  14. S ( p ) S_{(p)}
  15. O X ( U ) O_{X}(U)
  16. O X O_{X}
  17. O X O_{X}
  18. S ( p ) S_{(p)}
  19. M ~ \tilde{M}
  20. O X O_{X}
  21. M ~ \tilde{M}
  22. O X O_{X}
  23. O X ( 1 ) O_{X}(1)
  24. O X O_{X}
  25. O X O_{X}
  26. O ( n ) = i = 1 n O ( 1 ) O(n)=\bigotimes_{i=1}^{n}O(1)
  27. O X O_{X}
  28. N ( n ) = N O ( n ) N(n)=N\otimes O(n)
  29. A n = Proj A [ x 0 , , x n ] . \mathbb{P}^{n}_{A}=\operatorname{Proj}\,A[x_{0},\ldots,x_{n}].
  30. S = A [ x 0 , , x n ] S=A[x_{0},\ldots,x_{n}]
  31. x i x_{i}
  32. x i x_{i}
  33. x i x_{i}
  34. O X O_{X}
  35. O X O_{X}
  36. S = i 0 S i S=\bigoplus_{i\geq 0}S_{i}
  37. S i S_{i}
  38. O X O_{X}
  39. O X ( U ) O_{X}(U)
  40. S ( U ) = i 0 S i ( U ) S(U)=\bigoplus_{i\geq 0}S_{i}(U)
  41. S 0 = O X S_{0}=O_{X}
  42. ( 𝐏𝐫𝐨𝐣 S ) | p - 1 ( U ) = Proj ( S ( U ) ) . (\operatorname{\mathbf{Proj}}\,S)|_{p^{-1}(U)}=\operatorname{Proj}(S(U)).
  43. Y U Y_{U}
  44. Y U = Proj S ( U ) , Y_{U}=\operatorname{Proj}\,S(U),
  45. p U : Y U U p_{U}\colon Y_{U}\to U
  46. p U p_{U}
  47. O X ( U ) O_{X}(U)
  48. p U p_{U}
  49. Y U Y_{U}
  50. S 1 S_{1}
  51. S 0 S_{0}
  52. O X , x O_{X,x}
  53. O X , x O_{X,x}
  54. Y U Y_{U}
  55. \mathcal{E}
  56. X X
  57. 𝐒𝐲𝐦 O X ( ) \mathbf{Sym}_{O_{X}}(\mathcal{E})
  58. O X O_{X}
  59. ( ) \mathbb{P}(\mathcal{E})
  60. \mathcal{E}
  61. p : ( ) X p:\mathbb{P}(\mathcal{E})\to X
  62. x X x\in X
  63. x x
  64. ( ( x ) ) \mathbb{P}(\mathcal{E}(x))
  65. ( x ) := O X k ( x ) \mathcal{E}(x):=\mathcal{E}\otimes_{O_{X}}k(x)
  66. k ( x ) k(x)
  67. 𝒮 \mathcal{S}
  68. O X O_{X}
  69. 𝒮 1 \mathcal{S}_{1}
  70. 𝒮 1 \mathcal{S}_{1}
  71. 𝐏𝐫𝐨𝐣 𝒮 \mathbf{Proj}\mathcal{S}
  72. ( 𝒮 1 ) \mathbb{P}(\mathcal{S}_{1})
  73. X X
  74. ( ) \mathbb{P}(\mathcal{E})
  75. \mathcal{E}
  76. n + 1 n+1
  77. ( ) \mathbb{P}(\mathcal{E})
  78. X X
  79. n n
  80. U = Spec ( A ) U=\mathrm{Spec}(A)
  81. \mathcal{E}
  82. ( ) | p - 1 ( U ) Proj A [ x 0 , , x n ] = A n = U n , \mathbb{P}(\mathcal{E})|_{p^{-1}(U)}\simeq\operatorname{Proj}\,A[x_{0},\dots,x% _{n}]=\mathbb{P}^{n}_{A}=\mathbb{P}^{n}_{U},
  83. ( ) \mathbb{P}(\mathcal{E})