wpmath0000012_9

Normal_moveout.html

  1. t 2 = t 0 2 + x 2 v 2 t^{2}=t_{0}^{2}+\frac{x^{2}}{v^{2}}
  2. t 0 t_{0}

Normal_order_of_an_arithmetic_function.html

  1. ( 1 - ε ) g ( n ) f ( n ) ( 1 + ε ) g ( n ) (1-\varepsilon)g(n)\leq f(n)\leq(1+\varepsilon)g(n)\,

Normalized_frequency_(fiber_optics).html

  1. V = 2 π a λ n 1 2 - n 2 2 = 2 π a λ NA , V={2\pi a\over\lambda}\sqrt{{n_{1}}^{2}-{n_{2}}^{2}}\quad={2\pi a\over\lambda}% \mathrm{NA},
  2. V 2 2 ( g g + 2 ) , {V^{2}\over 2}\left({g\over g+2}\right)\quad,

Normalized_frequency_(unit).html

  1. f \scriptstyle f
  2. f / f s , \scriptstyle f/f_{s},
  3. f s = 1 / T \scriptstyle f_{s}=1/T
  4. f s / 2 \scriptstyle f_{s}/2
  5. f s \scriptstyle f_{s}
  6. T \scriptstyle T
  7. f s \scriptstyle f_{s}
  8. T \scriptstyle T
  9. f s \scriptstyle f_{s}
  10. f s \scriptstyle f_{s}
  11. T \scriptstyle T
  12. f , \scriptstyle f,
  13. f / f s \scriptstyle f/f_{s}
  14. f T . \scriptstyle f\cdot T.
  15. f s / 2 \scriptstyle f_{s}/2
  16. ω . \scriptstyle\omega.
  17. ω \scriptstyle\omega
  18. f s \scriptstyle f_{\mathrm{s}}

NPDGamma_experiment.html

  1. n + p d + γ \vec{n}+p\to d+\gamma
  2. A γ A_{\gamma}
  3. h π h_{\pi}
  4. f π f_{\pi}
  5. A γ = O ( 10 - 8 ) A_{\gamma}=O\left(10^{-8}\right)

Nullity_theorem.html

  1. [ A B C D ] - 1 = [ E F G H ] . \begin{bmatrix}A&B\\ C&D\end{bmatrix}^{-1}=\begin{bmatrix}E&F\\ G&H\end{bmatrix}.
  2. nullity A \displaystyle\operatorname{nullity}\,A

Numerical_sign_problem.html

  1. μ \mu
  2. Z Z
  3. exp ( - S ) \exp(-S)
  4. S S
  5. σ \sigma
  6. Z = D σ ρ [ σ ] Z=\int D\sigma\;\rho[\sigma]
  7. D σ D\sigma
  8. σ ( x ) \sigma(x)
  9. ρ [ σ ] = det ( M ( μ , σ ) ) exp ( - S [ σ ] ) \rho[\sigma]=\det(M(\mu,\sigma))\exp(-S[\sigma])
  10. S S
  11. M ( μ , σ ) M(\mu,\sigma)
  12. A [ σ ] A[\sigma]
  13. ρ [ σ ] \rho[\sigma]
  14. A ρ = D σ A [ σ ] ρ [ σ ] D σ ρ [ σ ] . \langle A\rangle_{\rho}=\frac{\int D\sigma\;A[\sigma]\;\rho[\sigma]}{\int D% \sigma\;\rho[\sigma]}.
  15. ρ [ σ ] \rho[\sigma]
  16. A ρ \langle A\rangle_{\rho}
  17. ρ [ σ ] \rho[\sigma]
  18. μ \mu
  19. μ 0 \mu\neq 0
  20. det ( M ( μ , σ ) ) \det(M(\mu,\sigma))
  21. ρ ( σ ) \rho(\sigma)
  22. ρ [ σ ] = p [ σ ] exp ( i θ [ σ ] ) \rho[\sigma]=p[\sigma]\,\exp(i\theta[\sigma])
  23. p [ σ ] p[\sigma]
  24. A ρ = D σ A [ σ ] exp ( i θ [ σ ] ) p [ σ ] D σ exp ( i θ [ σ ] ) p [ σ ] = A [ σ ] exp ( i θ [ σ ] ) p exp ( i θ [ σ ] ) p \langle A\rangle_{\rho}=\frac{\int D\sigma A[\sigma]\exp(i\theta[\sigma])\;p[% \sigma]}{\int D\sigma\exp(i\theta[\sigma])\;p[\sigma]}=\frac{\langle A[\sigma]% \exp(i\theta[\sigma])\rangle_{p}}{\langle\exp(i\theta[\sigma])\rangle_{p}}
  25. p [ σ ] p[\sigma]
  26. exp ( i θ [ σ ] ) \exp(i\theta[\sigma])
  27. exp ( i θ [ σ ] ) p \langle\exp(i\theta[\sigma])\rangle_{p}
  28. exp ( i θ [ σ ] ) p exp ( - f V / T ) \langle\exp(i\theta[\sigma])\rangle_{p}\propto\exp(-fV/T)
  29. V V
  30. T T
  31. f f
  32. ρ [ σ ] = p [ σ ] ρ [ σ ] p [ σ ] \rho[\sigma]=p[\sigma]\frac{\rho[\sigma]}{p[\sigma]}
  33. p [ σ ] p[\sigma]
  34. μ = 0 \mu=0
  35. ρ [ σ ] p [ σ ] p exp ( - f V / T ) \left\langle\frac{\rho[\sigma]}{p[\sigma]}\right\rangle_{p}\propto\exp(-fV/T)
  36. μ \mu
  37. μ \mu
  38. μ \mu

Numerov's_method.html

  1. ( d 2 d x 2 + a ( x ) ) y ( x ) = 0 \left(\frac{d^{2}}{dx^{2}}+a(x)\right)y(x)=0
  2. a ( x ) a(x)
  3. x n x_{n}
  4. x n - 1 x_{n-1}
  5. x n x_{n}
  6. y n + 1 = ( 2 - 5 h 2 6 a n ) y n - ( 1 + h 2 12 a n - 1 ) y n - 1 1 + h 2 12 a n + 1 y_{n+1}=\frac{\left(2-\frac{5h^{2}}{6}a_{n}\right)y_{n}-\left(1+\frac{h^{2}}{1% 2}a_{n-1}\right)y_{n-1}}{1+\frac{h^{2}}{12}a_{n+1}}
  7. a n = a ( x n ) a_{n}=a(x_{n})
  8. y n = y ( x n ) y_{n}=y(x_{n})
  9. x n x_{n}
  10. h = x n - x n - 1 h=x_{n}-x_{n-1}
  11. d 2 d x 2 y = f ( x , y ) \frac{d^{2}}{dx^{2}}y=f(x,y)
  12. y n + 1 = 2 y n - y n - 1 + 1 12 h 2 ( f n + 1 + 10 f n + f n - 1 ) . y_{n+1}=2y_{n}-y_{n-1}+\tfrac{1}{12}h^{2}(f_{n+1}+10f_{n}+f_{n-1}).
  13. f ( x , y ) = - a ( x ) y ( x ) f(x,y)=-a(x)y(x)
  14. [ - 2 2 μ ( 1 r 2 r 2 r - l ( l + 1 ) r 2 ) + V ( r ) ] R ( r ) = E R ( r ) \left[-{\hbar^{2}\over 2\mu}\left(\frac{1}{r}{\partial^{2}\over\partial r^{2}}% r-{l(l+1)\over r^{2}}\right)+V(r)\right]R(r)=ER(r)
  15. [ 2 r 2 - l ( l + 1 ) r 2 + 2 μ 2 ( E - V ( r ) ) ] u ( r ) = 0 \left[{\partial^{2}\over\partial r^{2}}-{l(l+1)\over r^{2}}+{2\mu\over\hbar^{2% }}\left(E-V(r)\right)\right]u(r)=0
  16. u ( r ) = r R ( r ) u(r)=rR(r)
  17. a ( x ) = 2 μ 2 ( E - V ( x ) ) - l ( l + 1 ) x 2 a(x)=\frac{2\mu}{\hbar^{2}}\left(E-V(x)\right)-\frac{l(l+1)}{x^{2}}
  18. y ( x ) y(x)
  19. x 0 x_{0}
  20. y ( x ) = y ( x 0 ) + ( x - x 0 ) y ( x 0 ) + ( x - x 0 ) 2 2 ! y ′′ ( x 0 ) + ( x - x 0 ) 3 3 ! y ′′′ ( x 0 ) + ( x - x 0 ) 4 4 ! y ′′′′ ( x 0 ) + ( x - x 0 ) 5 5 ! y ′′′′′ ( x 0 ) + 𝒪 ( h 6 ) y(x)=y(x_{0})+(x-x_{0})y^{\prime}(x_{0})+\frac{(x-x_{0})^{2}}{2!}y^{\prime% \prime}(x_{0})+\frac{(x-x_{0})^{3}}{3!}y^{\prime\prime\prime}(x_{0})+\frac{(x-% x_{0})^{4}}{4!}y^{\prime\prime\prime\prime}(x_{0})+\frac{(x-x_{0})^{5}}{5!}y^{% \prime\prime\prime\prime\prime}(x_{0})+\mathcal{O}(h^{6})
  21. x x
  22. x 0 x_{0}
  23. h = x - x 0 h=x-x_{0}
  24. x = x 0 + h x=x_{0}+h
  25. y ( x 0 + h ) = y ( x 0 ) + h y ( x 0 ) + h 2 2 ! y ′′ ( x 0 ) + h 3 3 ! y ′′′ ( x 0 ) + h 4 4 ! y ′′′′ ( x 0 ) + h 5 5 ! y ′′′′′ ( x 0 ) + 𝒪 ( h 6 ) y(x_{0}+h)=y(x_{0})+hy^{\prime}(x_{0})+\frac{h^{2}}{2!}y^{\prime\prime}(x_{0})% +\frac{h^{3}}{3!}y^{\prime\prime\prime}(x_{0})+\frac{h^{4}}{4!}y^{\prime\prime% \prime\prime}(x_{0})+\frac{h^{5}}{5!}y^{\prime\prime\prime\prime\prime}(x_{0})% +\mathcal{O}(h^{6})
  26. y ( x 0 - h ) y(x_{0}-h)
  27. y ( x 0 - h ) = y ( x 0 ) - h y ( x 0 ) + h 2 2 ! y ′′ ( x 0 ) - h 3 3 ! y ′′′ ( x 0 ) + h 4 4 ! y ′′′′ ( x 0 ) - h 5 5 ! y ′′′′′ ( x 0 ) + 𝒪 ( h 6 ) y(x_{0}-h)=y(x_{0})-hy^{\prime}(x_{0})+\frac{h^{2}}{2!}y^{\prime\prime}(x_{0})% -\frac{h^{3}}{3!}y^{\prime\prime\prime}(x_{0})+\frac{h^{4}}{4!}y^{\prime\prime% \prime\prime}(x_{0})-\frac{h^{5}}{5!}y^{\prime\prime\prime\prime\prime}(x_{0})% +\mathcal{O}(h^{6})
  28. x n x_{n}
  29. h h
  30. ( x n - 1 , y n - 1 ) (x_{n-1},y_{n-1})
  31. ( x n + 1 , y n + 1 ) (x_{n+1},y_{n+1})
  32. y n - 1 y_{n-1}
  33. y n + 1 y_{n+1}
  34. y n + 1 = y ( x n + h ) = y ( x n ) + h y ( x n ) + h 2 2 ! y ′′ ( x n ) + h 3 3 ! y ′′′ ( x n ) + h 4 4 ! y ′′′′ ( x n ) + h 5 5 ! y ′′′′′ ( x n ) + 𝒪 ( h 6 ) y_{n+1}=y(x_{n}+h)=y(x_{n})+hy^{\prime}(x_{n})+\frac{h^{2}}{2!}y^{\prime\prime% }(x_{n})+\frac{h^{3}}{3!}y^{\prime\prime\prime}(x_{n})+\frac{h^{4}}{4!}y^{% \prime\prime\prime\prime}(x_{n})+\frac{h^{5}}{5!}y^{\prime\prime\prime\prime% \prime}(x_{n})+\mathcal{O}(h^{6})
  35. y n - 1 = y ( x n - h ) = y ( x n ) - h y ( x n ) + h 2 2 ! y ′′ ( x n ) - h 3 3 ! y ′′′ ( x n ) + h 4 4 ! y ′′′′ ( x n ) - h 5 5 ! y ′′′′′ ( x n ) + 𝒪 ( h 6 ) y_{n-1}=y(x_{n}-h)=y(x_{n})-hy^{\prime}(x_{n})+\frac{h^{2}}{2!}y^{\prime\prime% }(x_{n})-\frac{h^{3}}{3!}y^{\prime\prime\prime}(x_{n})+\frac{h^{4}}{4!}y^{% \prime\prime\prime\prime}(x_{n})-\frac{h^{5}}{5!}y^{\prime\prime\prime\prime% \prime}(x_{n})+\mathcal{O}(h^{6})
  36. y n - 1 + y n + 1 = 2 y n + h 2 y n ′′ + h 4 12 y n ′′′′ + 𝒪 ( h 6 ) y_{n-1}+y_{n+1}=2y_{n}+{h^{2}}y^{\prime\prime}_{n}+\frac{h^{4}}{12}y^{\prime% \prime\prime\prime}_{n}+\mathcal{O}(h^{6})
  37. y n ′′ y^{\prime\prime}_{n}
  38. y n ′′ = - a n y n y^{\prime\prime}_{n}=-a_{n}y_{n}
  39. h 2 a n y n = 2 y n - y n - 1 - y n + 1 + h 4 12 y n ′′′′ + 𝒪 ( h 6 ) h^{2}a_{n}y_{n}=2y_{n}-y_{n-1}-y_{n+1}+\frac{h^{4}}{12}y^{\prime\prime\prime% \prime}_{n}+\mathcal{O}(h^{6})
  40. y ′′′′ ( x ) = - d 2 d x 2 [ a ( x ) y ( x ) ] y^{\prime\prime\prime\prime}(x)=-\frac{d^{2}}{dx^{2}}\left[a(x)y(x)\right]
  41. d 2 d x 2 \frac{d^{2}}{dx^{2}}
  42. a n y n a_{n}y_{n}
  43. h 2 a n y n = 2 y n - y n - 1 - y n + 1 - h 4 12 a n - 1 y n - 1 - 2 a n y n + a n + 1 y n + 1 h 2 + 𝒪 ( h 6 ) h^{2}a_{n}y_{n}=2y_{n}-y_{n-1}-y_{n+1}-\frac{h^{4}}{12}\frac{a_{n-1}y_{n-1}-2a% _{n}y_{n}+a_{n+1}y_{n+1}}{h^{2}}+\mathcal{O}(h^{6})
  44. y n + 1 y_{n+1}
  45. y n + 1 = ( 2 - 5 h 2 6 a n ) y n - ( 1 + h 2 12 a n - 1 ) y n - 1 1 + h 2 12 a n + 1 + 𝒪 ( h 6 ) . y_{n+1}=\frac{\left(2-\frac{5h^{2}}{6}a_{n}\right)y_{n}-\left(1+\frac{h^{2}}{1% 2}a_{n-1}\right)y_{n-1}}{1+\frac{h^{2}}{12}a_{n+1}}+\mathcal{O}(h^{6}).
  46. h 6 h^{6}

Observer_effect_(physics).html

  1. F ^ \hat{F}

Obstacle_problem.html

  1. J = D | u | 2 d x J=\int_{D}|\nabla u|^{2}\mathrm{d}x
  2. D D
  3. u u
  4. u u
  5. ϕ \phi
  6. ( x ) (x)
  7. ϕ \phi
  8. ( x ) (x)
  9. J ( u ) = D 1 + | u | 2 d x . J(u)=\int_{D}\sqrt{1+|\nabla u|^{2}}\,\mathrm{d}x.
  10. J ( u ) = D | u | 2 d x . J(u)=\int_{D}|\nabla u|^{2}\mathrm{d}x.
  11. ϕ \phi
  12. ( x ) (x)
  13. u ( x ) u(x)
  14. x x
  15. D D
  16. f ( x ) f(x)
  17. D D
  18. D D
  19. φ \varphi
  20. ( x ) (x)
  21. D D
  22. φ | D \scriptstyle\varphi|_{\partial D}
  23. φ \varphi
  24. ( x ) (x)
  25. D D
  26. f f
  27. K = { u ( x ) H 1 ( D ) : u | D = f ( x ) and u φ } , K=\left\{u(x)\in H^{1}(D):u|_{\partial D}=f(x)\,\text{ and }u\geq\varphi\right\},
  28. J ( u ) = D | u | 2 d x J(u)=\int_{D}|\nabla u|^{2}\mathrm{d}x
  29. u ( x ) u(x)
  30. K K
  31. K K
  32. u K u\in K
  33. D u , ( v - u ) d x 0 v K , \int_{D}\langle{\nabla u},{\nabla(v-u)}\rangle\mathrm{d}x\geq 0\qquad\forall v% \in K,
  34. u u
  35. K K
  36. a ( u , v - u ) f ( v - u ) v K . a(u,v-u)\geq f(v-u)\qquad\forall v\in K.\,
  37. a ( u , v ) a(u,v)
  38. f ( v ) f(v)
  39. C 1 , 1 C^{1,1}
  40. C 1 , 1 \scriptstyle C^{1,1}
  41. ϕ ( x ) \scriptstyle\phi(x)
  42. σ ( r ) \scriptstyle\sigma(r)
  43. | ϕ ( x ) - ϕ ( y ) | σ ( | x - y | ) \scriptstyle|\phi(x)-\phi(y)|\leq\sigma(|x-y|)
  44. u ( x ) \scriptstyle u(x)
  45. C σ ( 2 r ) \scriptstyle C\sigma(2r)
  46. σ ( r ) \scriptstyle\sigma(r)
  47. C r σ ( 2 r ) \scriptstyle Cr\sigma(2r)
  48. { x : u ( x ) - ϕ ( x ) = t } \scriptstyle\{x:u(x)-\phi(x)=t\}
  49. t > 0 \scriptstyle t>0
  50. C 1 , α \scriptstyle C^{1,\alpha}
  51. C 1 , α \scriptstyle C^{1,\alpha}
  52. C 1 \scriptstyle C^{1}

Ocean_dynamics.html

  1. D u D t = - 1 ρ p x + f v + 1 ρ τ x z \frac{Du}{Dt}=-\frac{1}{\rho}\frac{\partial p}{\partial x}+fv+\frac{1}{\rho}% \frac{\partial\tau_{x}}{\partial z}
  2. D v D t = - 1 ρ p y - f u + 1 ρ τ y z \frac{Dv}{Dt}=-\frac{1}{\rho}\frac{\partial p}{\partial y}-fu+\frac{1}{\rho}% \frac{\partial\tau_{y}}{\partial z}
  3. p z = - ρ g \frac{\partial p}{\partial z}=-\rho g
  4. u x + v y + w z = 0 \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{% \partial z}=0
  5. T t + u T x + v T y + w T z = Q \frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T% }{\partial y}+w\frac{\partial T}{\partial z}=Q
  6. S t + u S x + v S y + w S z = ( E - P ) S ( z = 0 ) \frac{\partial S}{\partial t}+u\frac{\partial S}{\partial x}+v\frac{\partial S% }{\partial y}+w\frac{\partial S}{\partial z}=(E-P)S(z=0)

Octal_game.html

  1. 0. 3 ˙ 0.\dot{3}
  2. 0.0 7 ˙ 0.0\dot{7}
  3. 0.1 0.1

Odds_algorithm.html

  1. I 1 , I 2 , , I n I_{1},\,I_{2},\,\dots,\,I_{n}
  2. I k = 1 \,I_{k}=1
  3. I k = 0 \,I_{k}=0
  4. p k = P ( I k = 1 ) \,p_{k}=P(\,I_{k}\,=1)
  5. q k = 1 - p k \,q_{k}=\,1-p_{k}
  6. r k = p k / q k \,r_{k}=p_{k}/q_{k}
  7. r k \,r_{k}
  8. r n + r n - 1 + r n - 2 + , r_{n}+r_{n-1}+r_{n-2}\,+\cdots,\,
  9. R s = r n + r n - 1 + r n - 2 + + r s . R_{s}=\,r_{n}+r_{n-1}+r_{n-2}+\cdots+r_{s}.\,
  10. Q s = q n q n - 1 q s . Q_{s}=q_{n}q_{n-1}\cdots q_{s}.\,
  11. s \,s
  12. w = Q s R s \,w=Q_{s}R_{s}
  13. w = Q s R s \,w=Q_{s}R_{s}
  14. R s 1 \,R_{s}\geq\,1
  15. w \,w
  16. 1 / e = 0.368 \,1/e=0.368\dots

On_the_Sphere_and_Cylinder.html

  1. A C = 2 π r 2 + 2 π r h = 2 π r ( r + h ) . A_{C}=2\pi r^{2}+2\pi rh=2\pi r(r+h).\,
  2. V C = π r 2 h . V_{C}=\pi r^{2}h.\,
  3. A S = 4 π r 2 . A_{S}=4\pi r^{2}.\,
  4. V S = 4 3 π r 3 . V_{S}=\frac{4}{3}\pi r^{3}.

One-class_classification.html

  1. P P
  2. U U

Online_machine_learning.html

  1. f : X Y f:X\to Y
  2. X X
  3. Y Y
  4. p ( x , y ) p(x,y)
  5. X × Y X\times Y
  6. V : Y × Y V:Y\times Y\to\mathbb{R}
  7. V ( f ( x ) , y ) V(f(x),y)
  8. f ( x ) f(x)
  9. y y
  10. f f\in\mathcal{H}
  11. \mathcal{H}
  12. I [ f ] = 𝔼 [ V ( f ( x ) , y ) ] = V ( f ( x ) , y ) d p ( x , y ) . I[f]=\mathbb{E}[V(f(x),y)]=\int V(f(x),y)\,dp(x,y)\ .
  13. p ( x , y ) p(x,y)
  14. ( x 1 , y 1 ) , , ( x n , y n ) (x_{1},y_{1}),\ldots,(x_{n},y_{n})
  15. p ( x , y ) p(x,y)
  16. f ^ \hat{f}
  17. f 1 , f 2 , f_{1},f_{2},\ldots
  18. f t + 1 f_{t+1}
  19. f t f_{t}
  20. ( x t , y t ) (x_{t},y_{t})
  21. f t f_{t}
  22. ( x t , y t ) (x_{t},y_{t})
  23. f t + 1 f_{t+1}
  24. f t f_{t}
  25. ( x 1 , y 1 ) , , ( x t , y t ) (x_{1},y_{1}),\ldots,(x_{t},y_{t})
  26. ( x 1 , y 1 ) , ( x 2 , y 2 ) , (x_{1},y_{1}),(x_{2},y_{2}),\ldots
  27. ( x 1 , y 1 ) , , ( x n , y n ) (x_{1},y_{1}),\ldots,(x_{n},y_{n})
  28. X = d X=\mathbb{R}^{d}
  29. Y Y\subseteq\mathbb{R}
  30. = { w , : w d } \mathcal{H}=\{\langle w,\cdot\rangle:w\in\mathbb{R}^{d}\}
  31. X X
  32. \mathbb{R}
  33. f f\in\mathcal{H}
  34. w d w\in\mathbb{R}^{d}
  35. V ( , ) V(\cdot,\cdot)
  36. w t + 1 w t - γ t V ( w t , x t , y t ) , w_{t+1}\leftarrow w_{t}-\gamma_{t}\nabla V(\langle w_{t},x_{t}\rangle,y_{t})\ ,
  37. w 1 0 w_{1}\leftarrow 0
  38. V ( w t , x t , y t ) \nabla V(\langle w_{t},x_{t}\rangle,y_{t})
  39. ( x t , y t ) (x_{t},y_{t})
  40. w t w_{t}
  41. γ t > 0 \gamma_{t}>0
  42. w 1 , w 2 , w_{1},w_{2},\ldots
  43. I [ w ] I[w]
  44. ( x 1 , y 1 ) , ( x 2 , y 2 ) , (x_{1},y_{1}),(x_{2},y_{2}),\ldots
  45. p ( x , y ) p(x,y)
  46. V ( , ) V(\cdot,\cdot)
  47. I [ w ] I[w]
  48. I [ w t ] - I [ w ] I[w_{t}]-I[w^{\ast}]
  49. w w^{\ast}
  50. I [ w ] I[w]
  51. I n [ w ] = 1 n i = 1 n V ( w , x i , y i ) . I_{n}[w]=\frac{1}{n}\sum_{i=1}^{n}V(\langle w,x_{i}\rangle,y_{i})\ .
  52. V ( , ) V(\cdot,\cdot)
  53. I n [ w ] I_{n}[w]
  54. I n [ w t ] - I n [ w n ] I_{n}[w_{t}]-I_{n}[w^{\ast}_{n}]
  55. w n w^{\ast}_{n}
  56. I n [ w ] I_{n}[w]
  57. x t x_{t}
  58. w t w_{t}
  59. y t y_{t}
  60. w t w_{t}
  61. w t + 1 w_{t+1}
  62. w 1 , w 2 , w_{1},w_{2},\ldots
  63. w w^{\ast}
  64. R T ( w ) = t = 1 T V ( w t , x t , y t ) - t = 1 T V ( w , x t , y t ) . R_{T}(w^{\ast})=\sum_{t=1}^{T}V(\langle w_{t},x_{t}\rangle,y_{t})-\sum_{t=1}^{% T}V(\langle w^{\ast},x_{t}\rangle,y_{t})\ .
  65. O ( T ) O(\sqrt{T})
  66. { γ t } \{\gamma_{t}\}
  67. X X
  68. O ( d ) O(d)
  69. V ( w , x i , y i ) = ( x i T w - y i ) 2 V(\langle w,x_{i}\rangle,y_{i})=(x_{i}^{T}w-y_{i})^{2}
  70. x i d x_{i}\in\mathbb{R}^{d}
  71. w i d w_{i}\in\mathbb{R}^{d}
  72. y i y_{i}\in\mathbb{R}
  73. { x i , y i } \{x_{i},y_{i}\}
  74. w * = ( X T X ) - 1 X T Y w^{*}=(X^{T}X)^{-1}X^{T}Y
  75. X X
  76. Y Y
  77. i i
  78. X X
  79. i i
  80. d d
  81. Y Y
  82. i i
  83. 1 1
  84. O ( i d 2 ) O(id^{2})
  85. n n
  86. i = 1 , , n i=1,\ldots,n
  87. O ( n 2 d 2 ) O(n^{2}d^{2})
  88. X T X X^{T}X
  89. w 0 d w_{0}\in\mathbb{R}^{d}
  90. Γ 0 d x d \Gamma_{0}\in\mathbb{R}^{dxd}
  91. Γ i = Γ i - 1 - Γ i - 1 x i x i T Γ i - 1 1 + x i T Γ i - 1 x i \Gamma_{i}=\Gamma_{i-1}-\frac{\Gamma_{i-1}x_{i}x_{i}^{T}\Gamma_{i-1}}{1+x_{i}^% {T}\Gamma_{i-1}x_{i}}
  92. w i = w i - 1 - Γ i x i ( x i T w i - 1 - y i ) w_{i}=w_{i-1}-\Gamma_{i}x_{i}(x_{i}^{T}w_{i-1}-y_{i})
  93. n n
  94. O ( n d 2 ) O(nd^{2})
  95. i i
  96. O ( d 2 ) O(d^{2})
  97. Γ i \Gamma_{i}
  98. w i = w i - 1 - Γ i x n ( x i T w i - 1 - y i ) w_{i}=w_{i-1}-\Gamma_{i}x_{n}(x_{i}^{T}w_{i-1}-y_{i})
  99. w i = w i - 1 - γ i x i ( x i T w i - 1 - y i ) w_{i}=w_{i-1}-\gamma_{i}x_{i}(x_{i}^{T}w_{i-1}-y_{i})
  100. Γ i d × d \Gamma_{i}\in\mathbb{R}^{d\times d}
  101. γ i \gamma_{i}\in\mathbb{R}
  102. n n
  103. O ( n d ) O(nd)
  104. i i
  105. O ( d ) O(d)
  106. γ i \gamma_{i}

Open_mapping_theorem_(functional_analysis).html

  1. U = B 1 X ( 0 ) , V = B 1 Y ( 0 ) U=B_{1}^{X}(0),V=B_{1}^{Y}(0)
  2. X = k k U X=\bigcup_{k\in\mathbb{N}}kU
  3. Y = A ( X ) = A ( k k U ) = k A ( k U ) . Y=A(X)=A\left(\bigcup_{k\in\mathbb{N}}kU\right)=\bigcup_{k\in\mathbb{N}}A(kU).
  4. k : ( A ( k U ) ¯ ) . \exists k\in\mathbb{N}:\qquad\left(\overline{A(kU)}\right)^{\circ}\neq\varnothing.
  5. B r ( c ) ( A ( k U ) ¯ ) A ( k U ) ¯ . B_{r}(c)\subseteq\left(\overline{A(kU)}\right)^{\circ}\subseteq\overline{A(kU)}.
  6. c , c + r v B r ( c ) A ( k U ) ¯ . c,c+rv\in B_{r}(c)\subseteq\overline{A(kU)}.
  7. r v A ( k U ) ¯ + A ( k U ) ¯ A ( k U ) + A ( k U ) ¯ A ( 2 k U ) ¯ , rv\in\overline{A(kU)}+\overline{A(kU)}\subseteq\overline{A(kU)+A(kU)}\subseteq% \overline{A(2kU)},
  8. V A ( L U ) ¯ . V\subseteq\overline{A\left(LU\right)}.
  9. y Y , ε > 0 , x X : x X L y Y and y - A x X < ε . ( 1 ) \forall y\in Y,\forall\varepsilon>0,\exists x\in X:\qquad\|x\|_{X}\leq L\|y\|_% {Y}\quad\,\text{and}\quad\|y-Ax\|_{X}<\varepsilon.\qquad(1)
  10. V A ( 2 L U ) V⊆A(2LU)
  11. y V y∈V
  12. x n < L 2 n - 1 and y - A ( x 1 + x 2 + + x n ) < 1 2 n . ( 2 ) \|x_{n}\|<\frac{L}{2^{n-1}}\quad\,\text{and}\quad\left\|y-A(x_{1}+x_{2}+\cdots% +x_{n})\right\|<\frac{1}{2^{n}}.\qquad(2)
  13. x n + 1 < L 2 n and y - A ( x 1 + x 2 + + x n ) - A ( x n + 1 ) < 1 2 n + 1 , \|x_{n+1}\|<\frac{L}{2^{n}}\quad\,\text{and}\quad\left\|y-A(x_{1}+x_{2}+\cdots% +x_{n})-A(x_{n+1})\right\|<\frac{1}{2^{n+1}},
  14. s n = x 1 + x 2 + + x n . s_{n}=x_{1}+x_{2}+\cdots+x_{n}.
  15. x X x∈X
  16. A x = y Ax=y
  17. x = lim n s n n = 1 x n < 2 L . \|x\|=\lim_{n\to\infty}\|s_{n}\|\leq\sum_{n=1}^{\infty}\|x_{n}\|<2L.
  18. y y
  19. A ( 2 L U ) A(2LU)
  20. V A ( 2 L U ) V⊆A(2LU)
  21. V / 2 L U V/2LU
  22. A : X Y A:X→Y
  23. X X / N 𝛼 Y X\to X/N\overset{\alpha}{\to}Y
  24. X / N X/N
  25. X X / N X→X/N
  26. u : X Y u:X→Y
  27. u ^ : X / ker ( u ) Y \hat{u}:X/\ker(u)\to Y

Optical_format.html

  1. O F = p w 2 + h 2 16000 OF=\frac{p\sqrt{w^{2}+h^{2}}}{16000}

Optical_heterodyne_detection.html

  1. I ( E sig cos ( ω sig t + φ ) + E LO cos ( ω LO t ) ) 2 = E sig 2 + E LO 2 + 2 E LO E sig cos ( ω sig t + φ ) cos ( ω LO t ) I\propto\left(E_{\mathrm{sig}}\cos(\omega_{\mathrm{sig}}t+\varphi)+E_{\mathrm{% LO}}\cos(\omega_{\mathrm{LO}}t)\right)^{2}=E_{\mathrm{sig}}^{2}+E_{\mathrm{LO}% }^{2}+2E_{\mathrm{LO}}E_{\mathrm{sig}}\cos(\omega_{\mathrm{sig}}t+\varphi)\cos% (\omega_{\mathrm{LO}}t)
  2. E LO E sig E_{\mathrm{LO}}E_{\mathrm{sig}}
  3. E sig 2 E_{\mathrm{sig}}^{2}
  4. E sig 2 E_{\mathrm{sig}}^{2}

Optical_phase_space.html

  1. 𝐮 ( 𝐱 , t ) = 𝐮 𝟎 e i ( 𝐤 𝐱 - w t ) \mathbf{u}(\mathbf{x},t)=\mathbf{u_{0}}e^{i(\mathbf{k}\cdot\mathbf{x}-wt)}
  2. \cdot
  3. a ^ \widehat{a}^{\dagger}
  4. a ^ \widehat{a}
  5. E i E_{i}
  6. E ^ i \widehat{E}_{i}
  7. E i E_{i}
  8. E ^ i = u i * ( 𝐱 , t ) a ^ + u i ( 𝐱 , t ) a ^ \widehat{E}_{i}=u_{i}^{*}(\mathbf{x},t)\widehat{a}^{\dagger}+u_{i}(\mathbf{x},% t)\widehat{a}
  9. H ^ = ω ( a ^ a ^ + 1 / 2 ) \widehat{H}=\hbar\omega(\widehat{a}^{\dagger}\widehat{a}+1/2)
  10. ω \omega
  11. [ a ^ , a ^ ] = 1 [\widehat{a},\widehat{a}^{\dagger}]=1
  12. a ^ | α = α | α \widehat{a}|\alpha\rangle=\alpha|\alpha\rangle
  13. α \alpha
  14. N ^ = a ^ a ^ , \widehat{N}=\widehat{a}^{\dagger}\widehat{a},
  15. q ^ = 1 2 ( a ^ + a ^ ) \widehat{q}=\tfrac{1}{\sqrt{2}}(\widehat{a}^{\dagger}+\widehat{a})
  16. p ^ = i 2 ( a ^ - a ^ ) \widehat{p}=\tfrac{i}{\sqrt{2}}(\widehat{a}^{\dagger}-\widehat{a})
  17. a ^ \widehat{a}
  18. [ q ^ , p ^ ] \displaystyle\left[\widehat{q},\widehat{p}\right]
  19. q ^ \widehat{q}
  20. p ^ \widehat{p}
  21. q ^ | q = q | q \widehat{q}|q\rangle=q|q\rangle
  22. p ^ | p = p | p \widehat{p}|p\rangle=p|p\rangle
  23. q | q = δ ( q - q ) \langle q|q^{\prime}\rangle=\delta(q-q^{\prime})
  24. p | p = δ ( p - p ) \langle p|p^{\prime}\rangle=\delta(p-p^{\prime})
  25. - | q q | d q = 1 \int_{-\infty}^{\infty}|q\rangle\langle q|dq=1
  26. - | p p | d p = 1 \int_{-\infty}^{\infty}|p\rangle\langle p|dp=1
  27. α \alpha
  28. α | q ^ | α = 2 - 1 / 2 ( α | a ^ | α + α | a ^ | α ) = 2 - 1 / 2 ( α * α | α + α α | α ) \langle\alpha|\widehat{q}|\alpha\rangle=2^{-1/2}(\langle\alpha|\widehat{a}^{% \dagger}|\alpha\rangle+\langle\alpha|\widehat{a}|\alpha\rangle)=2^{-1/2}(% \alpha^{*}\langle\alpha|\alpha\rangle+\alpha\langle\alpha|\alpha\rangle)
  29. α | α = e ( - 1 / 2 ) ( | α | 2 + | α | 2 ) + α * α \langle\alpha^{\prime}|\alpha\rangle=e^{(-1/2)(|\alpha^{\prime}|^{2}+|\alpha|^% {2})+\alpha^{\prime*}\alpha}
  30. α | q ^ | α = 2 - 1 / 2 ( α * + α ) = q α \langle\alpha|\widehat{q}|\alpha\rangle=2^{-1/2}(\alpha^{*}+\alpha)=q_{\alpha}
  31. α | p ^ | α = i 2 - 1 / 2 ( α * - α ) = p α \langle\alpha|\widehat{p}|\alpha\rangle=i2^{-1/2}(\alpha^{*}-\alpha)=p_{\alpha}
  32. α = 2 - 1 / 2 ( α | q ^ | α + i α | p ^ | α ) = 2 - 1 / 2 ( q α + i p α ) \alpha=2^{-1/2}(\langle\alpha|\widehat{q}|\alpha\rangle+i\langle\alpha|% \widehat{p}|\alpha\rangle)=2^{-1/2}(q_{\alpha}+ip_{\alpha})
  33. α \alpha
  34. q α = α | q ^ | α q_{\alpha}=\langle\alpha|\widehat{q}|\alpha\rangle
  35. p α = α | p ^ | α p_{\alpha}=\langle\alpha|\widehat{p}|\alpha\rangle
  36. q ^ \widehat{q}
  37. p ^ \widehat{p}
  38. | α |\alpha\rangle
  39. Δ q Δ p 1 / 2 \Delta q\Delta p\geq 1/2
  40. Δ q \Delta q
  41. Δ p \Delta p
  42. α \alpha
  43. θ \theta
  44. U ^ ( θ ) = e - i θ N ^ \widehat{U}(\theta)=e^{-i\theta\widehat{N}}
  45. U ^ ( θ ) a ^ U ^ ( θ ) = a ^ e - i θ \widehat{U}(\theta)^{\dagger}\widehat{a}\widehat{U}(\theta)=\widehat{a}e^{-i\theta}
  46. d / d θ ( U ^ a ^ U ^ ) = i N ^ U ^ a ^ U ^ - i U ^ a ^ U ^ N ^ = U ^ i [ N ^ , a ^ ] U ^ d/d\theta(\widehat{U}^{\dagger}\widehat{a}\widehat{U})=i\widehat{N}\widehat{U}% ^{\dagger}\widehat{a}\widehat{U}-i\widehat{U}^{\dagger}\widehat{a}\widehat{U}% \widehat{N}=\widehat{U}^{\dagger}i[\widehat{N},\widehat{a}]\widehat{U}
  47. = U ^ i ( a ^ a ^ a ^ - a ^ a ^ a ^ ) U ^ = U ^ i [ a ^ , a ^ ] a ^ U ^ = - i U ^ a ^ U ^ =\widehat{U}^{\dagger}i(\widehat{a}^{\dagger}\widehat{a}\widehat{a}-\widehat{a% }\widehat{a}^{\dagger}\widehat{a})\widehat{U}=\widehat{U}^{\dagger}i[\widehat{% a}^{\dagger},\widehat{a}]\widehat{a}\widehat{U}=-i\widehat{U}^{\dagger}% \widehat{a}\widehat{U}
  48. U ^ ( θ ) | α = | α e - i θ \widehat{U}(\theta)|\alpha\rangle=|\alpha e^{-i\theta}\rangle
  49. a ^ ( U ^ | α ) = U ^ a ^ e - i θ | α \widehat{a}(\widehat{U}|\alpha\rangle)=\widehat{U}\widehat{a}e^{-i\theta}|\alpha\rangle
  50. a ^ ( U ^ | α ) = U ^ α e - i θ | α = α e - i θ ( U ^ | α ) \widehat{a}(\widehat{U}|\alpha\rangle)=\widehat{U}\alpha e^{-i\theta}|\alpha% \rangle=\alpha e^{-i\theta}(\widehat{U}|\alpha\rangle)
  51. ( α e - i θ , U ^ | α ) (\alpha e^{-i\theta},\widehat{U}|\alpha\rangle)
  52. a ^ U ^ | α \widehat{a}\widehat{U}|\alpha\rangle
  53. ( α e - i θ = 2 - 1 / 2 [ q α c o s ( θ ) + p α s i n ( θ ) ] + i 2 - 1 / 2 [ - q α c s i n ( θ ) + p α c o s ( θ ) ] , U ^ | α = | α e - i θ ) (\alpha e^{-i\theta}=2^{-1/2}[q_{\alpha}cos(\theta)+p_{\alpha}sin(\theta)]+i2^% {-1/2}[-q_{\alpha}csin(\theta)+p_{\alpha}cos(\theta)],\widehat{U}|\alpha% \rangle=|\alpha e^{-i\theta}\rangle)
  54. D ^ ( α ) = e α a ^ - α * a ^ \widehat{D}(\alpha)=e^{\alpha\widehat{a}^{\dagger}-\alpha^{*}\widehat{a}}
  55. D ^ a ^ D ^ = a ^ + α \widehat{D}^{\dagger}\widehat{a}\widehat{D}=\widehat{a}+\alpha
  56. δ α \delta\alpha
  57. D ^ \widehat{D}
  58. D ^ \widehat{D}^{\dagger}
  59. e X = k = 0 X k k ! e^{X}=\sum_{k=0}^{\infty}\frac{X^{k}}{k!}
  60. δ α \delta\alpha
  61. D ^ ( δ α ) a ^ D ^ ( δ α ) = i , j ( δ α * a ^ - δ α a ^ ) i a ^ ( δ α a ^ - δ α * a ^ ) j / i ! j ! \widehat{D}^{\dagger}(\delta\alpha)\widehat{a}\widehat{D}(\delta\alpha)=\sum_{% i,j}(\delta\alpha^{*}\widehat{a}-\delta\alpha\widehat{a}^{\dagger})^{i}% \widehat{a}(\delta\alpha\widehat{a}^{\dagger}-\delta\alpha^{*}\widehat{a})^{j}% /i!j!
  62. = a ^ + ( δ α * a ^ - δ α a ^ ) a ^ + a ^ ( δ α a ^ - δ α * a ^ ) + O ( δ α 2 , ( δ α * ) 2 ) =\widehat{a}+(\delta\alpha^{*}\widehat{a}-\delta\alpha\widehat{a}^{\dagger})% \widehat{a}+\widehat{a}(\delta\alpha\widehat{a}^{\dagger}-\delta\alpha^{*}% \widehat{a})+O(\delta\alpha^{2},(\delta\alpha^{*})^{2})
  63. = a ^ + a ^ ( δ α a ^ - δ α * a ^ ) - ( δ α a ^ - δ α * a ^ ) a ^ =\widehat{a}+\widehat{a}(\delta\alpha\widehat{a}^{\dagger}-\delta\alpha^{*}% \widehat{a})-(\delta\alpha\widehat{a}^{\dagger}-\delta\alpha^{*}\widehat{a})% \widehat{a}
  64. = a ^ + [ a ^ , δ α a ^ - δ α * a ^ ] = a ^ + δ α [ a ^ , a ^ ] - δ α * [ a ^ , a ^ ] =\widehat{a}+[\widehat{a},\delta\alpha\widehat{a}^{\dagger}-\delta\alpha^{*}% \widehat{a}]=\widehat{a}+\delta\alpha[\widehat{a},\widehat{a}^{\dagger}]-% \delta\alpha^{*}[\widehat{a},\widehat{a}]
  65. D ^ ( δ α ) a ^ D ^ ( δ α ) = a ^ + δ α \widehat{D}^{\dagger}(\delta\alpha)\widehat{a}\widehat{D}(\delta\alpha)=% \widehat{a}+\delta\alpha
  66. ( D ^ ( δ α ) ) k a ^ ( D ^ ( δ α ) ) k = a ^ + k δ α (\widehat{D}^{\dagger}(\delta\alpha))^{k}\widehat{a}(\widehat{D}(\delta\alpha)% )^{k}=\widehat{a}+k\delta\alpha
  67. D ^ ( δ α ) a ^ D ^ ( δ α ) = a ^ + δ α \widehat{D}^{\dagger}(\delta\alpha)\widehat{a}\widehat{D}(\delta\alpha)=% \widehat{a}+\delta\alpha
  68. a ^ D ^ ( - α ) | α = D ^ ( - α ) ( a ^ - α ) | α \widehat{a}\widehat{D}(-\alpha)|\alpha\rangle=\widehat{D}(-\alpha)(\widehat{a}% -\alpha)|\alpha\rangle
  69. = D ^ ( - α ) ( a ^ | α - α | α ) =\widehat{D}(-\alpha)(\widehat{a}|\alpha\rangle-\alpha|\alpha\rangle)
  70. = D ^ ( - α ) ( α | α - α | α ) =\widehat{D}(-\alpha)(\alpha|\alpha\rangle-\alpha|\alpha\rangle)
  71. a ^ ( D ^ ( - α ) | α ) = 0 \widehat{a}(\widehat{D}(-\alpha)|\alpha\rangle)=0
  72. D ^ ( - α ) | α = | 0 \widehat{D}(-\alpha)|\alpha\rangle=|0\rangle
  73. | α = D ^ ( α ) | 0 |\alpha\rangle=\widehat{D}(\alpha)|0\rangle

Optical_properties_of_carbon_nanotubes.html

  1. d = a π ( n 2 + n m + m 2 ) . d=\frac{a}{\pi}\sqrt{(n^{2}+nm+m^{2})}.
  2. = =

Optimal_decision.html

  1. d d
  2. D D
  3. o = f ( d ) o=f(d)
  4. O O
  5. U O ( o ) U_{O}(o)
  6. d d
  7. U D ( d ) = U O ( f ( d ) ) . U_{D}(d)\ =\ U_{O}(f(d)).\,
  8. d opt d_{\mathrm{opt}}
  9. U D ( d ) U_{D}(d)
  10. d opt = arg max d D U D ( d ) . d_{\mathrm{opt}}=\arg\max\limits_{d\in D}U_{D}(d).\,
  11. o o
  12. d ; d;
  13. U O ( o ) U_{O}(o)
  14. o ; o;
  15. d d
  16. U D ( d ) . U_{D}(d).
  17. d d
  18. p ( o | d ) p(o|d)
  19. U D ( d ) U_{D}(d)
  20. d d
  21. d d
  22. E U D ( d ) = p ( o | d ) U ( o ) d o \,\text{E}U_{D}(d)=\int{p(o|d)U(o)do}\,
  23. O O
  24. d opt d_{\mathrm{opt}}
  25. E U D ( d ) \,\text{E}U_{D}(d)
  26. d opt = arg max d D E U D ( d ) . d_{\mathrm{opt}}=\arg\max\limits_{d\in D}\,\text{E}U_{D}(d).\,

Optimal_projection_equations.html

  1. 𝐱 ^ r ( t ) \hat{\mathbf{x}}_{r}(t)
  2. n r = d i m ( 𝐱 ^ r ( t ) ) n_{r}=dim(\hat{\mathbf{x}}_{r}(t))
  3. n = d i m ( 𝐱 ( t ) ) n=dim({\mathbf{x}}(t))
  4. 𝐱 ^ ˙ r ( t ) = A r ( t ) 𝐱 ^ r ( t ) + B r ( t ) 𝐮 ( t ) + K r ( t ) ( 𝐲 ( t ) - C r ( t ) 𝐱 ^ r ( t ) ) , 𝐱 ^ r ( 0 ) = 𝐱 r ( 0 ) , \dot{\hat{\mathbf{x}}}_{r}(t)=A_{r}(t)\hat{\mathbf{x}}_{r}(t)+B_{r}(t){\mathbf% {u}}(t)+K_{r}(t)\left({\mathbf{y}}(t)-C_{r}(t)\hat{\mathbf{x}}_{r}(t)\right),% \hat{\mathbf{x}}_{r}(0)={\mathbf{x}}_{r}(0),
  5. 𝐮 ( t ) = - L r ( t ) 𝐱 ^ r ( t ) . {\mathbf{u}}(t)=-L_{r}(t)\hat{\mathbf{x}}_{r}(t).
  6. 𝐱 ^ ˙ r ( t ) = F r ( t ) 𝐱 ^ r ( t ) + K r ( t ) 𝐲 ( t ) , 𝐱 ^ r ( 0 ) = 𝐱 r ( 0 ) , \dot{\hat{\mathbf{x}}}_{r}(t)=F_{r}(t)\hat{\mathbf{x}}_{r}(t)+K_{r}(t){\mathbf% {y}}(t),\hat{\mathbf{x}}_{r}(0)={\mathbf{x}}_{r}(0),
  7. 𝐮 ( t ) = - L r ( t ) 𝐱 ^ r ( t ) , {\mathbf{u}}(t)=-L_{r}(t)\hat{\mathbf{x}}_{r}(t),
  8. F r ( t ) = A r ( t ) - B r ( t ) L r ( t ) - K r ( t ) C r ( t ) . {\mathbf{}}F_{r}(t)=A_{r}(t)-B_{r}(t)L_{r}(t)-K_{r}(t)C_{r}(t).
  9. F r ( t ) , K r ( t ) , L r ( t ) {\mathbf{}}F_{r}(t),K_{r}(t),L_{r}(t)
  10. 𝐱 r ( 0 ) {\mathbf{x}}_{r}(0)
  11. τ ( t ) {\mathbf{}}\tau(t)
  12. n {\mathbf{}}n
  13. n r . {\mathbf{}}n_{r}.
  14. τ 2 ( t ) = τ ( t ) . {\mathbf{}}\tau^{2}(t)=\tau(t).
  15. τ ( t ) {\mathbf{}}\tau_{\perp}(t)
  16. I n - τ ( t ) {\mathbf{}}I_{n}-\tau(t)
  17. I n {\mathbf{}}I_{n}
  18. n {\mathbf{}}n
  19. P ˙ ( t ) = A ( t ) P ( t ) + P ( t ) A ( t ) - P ( t ) C ( t ) W - 1 ( t ) C ( t ) P ( t ) + V ( t ) \dot{P}(t)=A(t)P(t)+P(t)A^{\prime}(t)-P(t)C^{\prime}(t)W^{-1}(t)C(t)P(t)+V(t)
  20. + τ ( t ) P ( t ) C ( t ) W - 1 ( t ) C ( t ) P ( t ) τ ( t ) , +\tau_{\perp}(t)P(t)C^{\prime}(t)W^{-1}(t)C(t)P(t)\tau^{\prime}_{\perp}(t),
  21. P ( 0 ) = E ( 𝐱 ( 0 ) 𝐱 ( 0 ) ) , P(0)=E\left({\mathbf{x}}(0){\mathbf{x}}^{\prime}(0)\right),
  22. - S ˙ ( t ) = A ( t ) S ( t ) + S ( t ) A ( t ) - S ( t ) B ( t ) R - 1 ( t ) B ( t ) S ( t ) + Q ( t ) -\dot{S}(t)=A^{\prime}(t)S(t)+S(t)A(t)-S(t)B(t)R^{-1}(t)B^{\prime}(t)S(t)+Q(t)
  23. + τ ( t ) S ( t ) B ( t ) R - 1 ( t ) B ( t ) S ( t ) τ ( t ) , +\tau^{\prime}_{\perp}(t)S(t)B(t)R^{-1}(t)B^{\prime}(t)S(t)\tau_{\perp}(t),
  24. S ( T ) = F . {\mathbf{}}S(T)=F.
  25. n = n r {\mathbf{}}n=n_{r}
  26. τ ( t ) = I n , τ ( t ) = 0 \tau(t)=I_{n},\tau_{\perp}(t)=0
  27. n r < n {\mathbf{}}n_{r}<n
  28. τ ( t ) . {\mathbf{}}\tau(t).
  29. τ ( t ) {\mathbf{}}\tau(t)
  30. Ψ 1 ( t ) = ( A ( t ) - B ( t ) R - 1 ( t ) B ( t ) S ( t ) ) P ^ ( t ) + P ^ ( t ) ( A ( t ) - B ( t ) R - 1 ( t ) B ( t ) S ( t ) ) \Psi_{1}(t)=(A(t)-B(t)R^{-1}(t)B^{\prime}(t)S(t))\hat{P}(t)+\hat{P}(t)(A(t)-B(% t)R^{-1}(t)B^{\prime}(t)S(t))^{\prime}
  31. + P ( t ) C ( t ) W - 1 ( t ) C ( t ) P ( t ) , {\mathbf{}}+P(t)C^{\prime}(t)W^{-1}(t)C(t)P(t),
  32. Ψ 2 ( t ) = ( A ( t ) - P ( t ) C ( t ) W - 1 ( t ) C ( t ) ) S ^ ( t ) + S ^ ( t ) ( A ( t ) - P ( t ) C ( t ) W - 1 ( t ) C ( t ) ) \Psi_{2}(t)=(A(t)-P(t)C^{\prime}(t)W^{-1}(t)C(t))^{\prime}\hat{S}(t)+\hat{S}(t% )(A(t)-P(t)C^{\prime}(t)W^{-1}(t)C(t))
  33. + S ( t ) B ( t ) R - 1 ( t ) B ( t ) S ( t ) . {\mathbf{}}+S(t)B(t)R^{-1}(t)B^{\prime}(t)S(t).
  34. P ^ ˙ ( t ) = 1 / 2 ( τ ( t ) Ψ 1 ( t ) + Ψ 1 ( t ) τ ( t ) ) , P ^ ( 0 ) = E ( 𝐱 ( 0 ) ) E ( 𝐱 ( 0 ) ) , r a n k ( P ^ ( t ) ) = n r \dot{\hat{P}}(t)=1/2\left(\tau(t)\Psi_{1}(t)+\Psi_{1}(t)\tau^{\prime}(t)\right% ),\hat{P}(0)=E({\mathbf{x}}(0))E({\mathbf{x}}(0))^{\prime},rank(\hat{P}(t))=n_% {r}
  35. - S ^ ˙ ( t ) = 1 / 2 ( τ ( t ) Ψ 2 ( t ) + Ψ 2 ( t ) τ ( t ) ) , S ^ ( T ) = 0 , r a n k ( S ^ ( t ) ) = n r -\dot{\hat{S}}(t)=1/2\left(\tau^{\prime}(t)\Psi_{2}(t)+\Psi_{2}(t)\tau(t)% \right),\hat{S}(T)=0,rank(\hat{S}(t))=n_{r}
  36. τ ( t ) = P ^ ( t ) S ^ ( t ) ( P ^ ( t ) S ^ ( t ) ) * . {\mathbf{}}\tau(t)=\hat{P}(t)\hat{S}(t)\left(\hat{P}(t)\hat{S}(t)\right)^{*}.
  37. A * = A ( A 3 ) + A . {\mathbf{}}A^{*}=A(A^{3})^{+}A.
  38. P ( t ) , S ( t ) , P ^ ( t ) , S ^ ( t ) {\mathbf{}}P(t),S(t),\hat{P}(t),\hat{S}(t)
  39. F r ( t ) , K r ( t ) , L r ( t ) {\mathbf{}}F_{r}(t),K_{r}(t),L_{r}(t)
  40. 𝐱 r ( 0 ) {\mathbf{x}}_{r}(0)
  41. F r ( t ) = H ( t ) ( A ( t ) - P ( t ) C ( t ) W - 1 ( t ) C ( t ) - B ( t ) R - 1 ( t ) B ( t ) S ( t ) ) G ( t ) + H ˙ ( t ) G ( t ) , {\mathbf{}}F_{r}(t)=H(t)\left(A(t)-P(t)C^{\prime}(t)W^{-1}(t)C(t)-B(t)R^{-1}(t% )B^{\prime}(t)S(t)\right)G(t)+\dot{H}(t)G^{\prime}(t),
  42. K r ( t ) = H ( t ) P ( t ) C ( t ) W - 1 ( t ) , {\mathbf{}}K_{r}(t)=H(t)P(t)C^{\prime}(t)W^{-1}(t),
  43. L r ( t ) = R - 1 ( t ) B ( t ) S ( t ) G ( t ) , {\mathbf{}}L_{r}(t)=R^{-1}(t)B^{\prime}(t)S(t)G^{\prime}(t),
  44. 𝐱 r ( 0 ) = H ( 0 ) E ( 𝐱 ( 0 ) ) . {\mathbf{x}}_{r}(0)=H(0)E({\mathbf{x}}(0)).
  45. G ( t ) , H ( t ) {\mathbf{}}G(t),H(t)
  46. G ( t ) H ( t ) = τ ( t ) , G ( t ) H ( t ) = I n r {\mathbf{}}G^{\prime}(t)H(t)=\tau(t),G(t)H^{\prime}(t)=I_{n_{r}}
  47. P ^ ( t ) S ^ ( t ) {\mathbf{}}\hat{P}(t)\hat{S}(t)
  48. τ ( t ) P ^ ( t ) = P ^ ( t ) τ ( t ) = P ^ ( t ) , τ ( t ) S ^ ( t ) = S ^ ( t ) τ ( t ) = S ^ ( t ) {\mathbf{}}\tau(t)\hat{P}(t)=\hat{P}(t)\tau^{\prime}(t)=\hat{P}(t),\tau^{% \prime}(t)\hat{S}(t)=\hat{S}(t)\tau(t)=\hat{S}(t)
  49. P ˙ ( t ) = A ( t ) P ( t ) + P ( t ) A ( t ) - P ( t ) C ( t ) W - 1 ( t ) C ( t ) P ( t ) + V ( t ) + τ ( t ) Ψ 1 ( t ) τ ( t ) , \dot{P}(t)=A(t)P(t)+P(t)A^{\prime}(t)-P(t)C^{\prime}(t)W^{-1}(t)C(t)P(t)+V(t)+% \tau_{\perp}(t)\Psi_{1}(t)\tau^{\prime}_{\perp}(t),
  50. P ( 0 ) = E ( 𝐱 ( 0 ) 𝐱 ( 0 ) ) , P(0)=E\left({\mathbf{x}}(0){\mathbf{x}}^{\prime}(0)\right),
  51. - S ˙ ( t ) = A ( t ) S ( t ) + S ( t ) A ( t ) - S ( t ) B ( t ) R - 1 ( t ) B ( t ) S ( t ) + Q ( t ) + τ Ψ 2 ( t ) τ ( t ) , -\dot{S}(t)=A^{\prime}(t)S(t)+S(t)A(t)-S(t)B(t)R^{-1}(t)B^{\prime}(t)S(t)+Q(t)% +\tau^{\prime}_{\perp}\Psi_{2}(t)\tau_{\perp}(t),
  52. S ( T ) = F . {\mathbf{}}S(T)=F.
  53. T {\mathbf{}}T
  54. n r < n {\mathbf{}}n_{r}<n
  55. Ψ i 1 = ( A i - B i ( B i S i + 1 B i + R i ) - 1 B i S i + 1 A i ) ) P ^ i ( A i - B i ( B i S i + 1 B i + R i ) - 1 B i S i + 1 A i ) ) {\mathbf{}}\Psi^{1}_{i}=\left(A_{i}-B_{i}(B^{\prime}_{i}S_{i+1}B_{i}+R_{i})^{-% 1}B^{\prime}_{i}S_{i+1}A_{i})\right)\hat{P}_{i}\left(A_{i}-B_{i}(B^{\prime}_{i% }S_{i+1}B_{i}+R_{i})^{-1}B^{\prime}_{i}S_{i+1}A_{i})\right)^{\prime}
  56. + A i P i C i ( C i P i C i + W i ) - 1 C i P i A i {\mathbf{}}+A_{i}P_{i}C^{\prime}_{i}(C_{i}P_{i}C^{\prime}_{i}+W_{i})^{-1}C_{i}% P_{i}A^{\prime}_{i}
  57. Ψ i + 1 2 = ( A i - A i P i C i ( C i P i C i + W i ) - 1 C i ) S ^ i + 1 ( A i - A i P i C i ( C i P i C i + W i ) - 1 C i ) {\mathbf{}}\Psi^{2}_{i+1}=\left(A_{i}-A_{i}P_{i}C^{\prime}_{i}(C_{i}P_{i}C^{% \prime}_{i}+W_{i})^{-1}C_{i}\right)^{\prime}\hat{S}_{i+1}\left(A_{i}-A_{i}P_{i% }C^{\prime}_{i}(C_{i}P_{i}C^{\prime}_{i}+W_{i})^{-1}C_{i}\right)
  58. + A i S i + 1 B i ( B i S i + 1 B i + R i ) - 1 B i S i + 1 A i {\mathbf{}}+A^{\prime}_{i}S_{i+1}B_{i}(B^{\prime}_{i}S_{i+1}B_{i}+R_{i})^{-1}B% ^{\prime}_{i}S_{i+1}A_{i}
  59. P i + 1 = A i ( P i - P i C i ( C i P i C i + W i ) - 1 C i P i ) A i + V i + τ i + 1 Ψ i 1 τ i + 1 , P 0 = E ( 𝐱 0 𝐱 0 ) {\mathbf{}}P_{i+1}=A_{i}\left(P_{i}-P_{i}C^{\prime}_{i}\left(C_{i}P_{i}C^{% \prime}_{i}+W_{i}\right)^{-1}C_{i}P_{i}\right)A^{\prime}_{i}+V_{i}+\tau_{\perp i% +1}\Psi^{1}_{i}\tau^{\prime}_{\perp i+1},P_{0}=E\left({\mathbf{x}}_{0}{\mathbf% {x^{\prime}}}_{0}\right)
  60. S i = A i ( S i + 1 - S i + 1 B i ( B i S i + 1 B i + R i ) - 1 B i S i + 1 ) A i + Q i + τ i Ψ i + 1 2 τ i , S N = F {\mathbf{}}S_{i}=A^{\prime}_{i}\left(S_{i+1}-S_{i+1}B_{i}\left(B^{\prime}_{i}S% _{i+1}B_{i}+R_{i}\right)^{-1}B^{\prime}_{i}S_{i+1}\right)A_{i}+Q_{i}+\tau^{% \prime}_{\perp i}\Psi^{2}_{i+1}\tau_{\perp i},S_{N}=F
  61. P ^ i + 1 = 1 / 2 ( τ i + 1 Ψ i 1 + Ψ i 1 τ i + 1 ) , P ^ 0 = E ( 𝐱 ( 0 ) ) E ( 𝐱 ( 0 ) ) , r a n k ( P ^ i ) = n r {\mathbf{}}\hat{P}_{i+1}=1/2(\tau_{i+1}\Psi_{i}^{1}+\Psi_{i}^{1}\tau^{\prime}_% {i+1}),\hat{P}_{0}=E({\mathbf{x}}(0))E({\mathbf{x}}(0))^{\prime},rank(\hat{P}_% {i})=n_{r}
  62. S ^ i = 1 / 2 ( τ i Ψ i + 1 2 + Ψ i + 1 2 τ i ) , S ^ N = 0 , r a n k ( S ^ i ) = n r {\mathbf{}}\hat{S}_{i}=1/2(\tau^{\prime}_{i}\Psi_{i+1}^{2}+\Psi_{i+1}^{2}\tau_% {i}),\hat{S}_{N}=0,rank(\hat{S}_{i})=n_{r}
  63. τ i = P ^ i S ^ i ( P ^ i S ^ i ) * . {\mathbf{}}\tau_{i}=\hat{P}_{i}\hat{S}_{i}\left(\hat{P}_{i}\hat{S}_{i}\right)^% {*}.
  64. P i , S i , P ^ i , S ^ i {\mathbf{}}P_{i},S_{i},\hat{P}_{i},\hat{S}_{i}
  65. F i r , K i r , L i r {\mathbf{}}F_{i}^{r},K_{i}^{r},L_{i}^{r}
  66. 𝐱 0 r {\mathbf{x}}_{0}^{r}
  67. F i r = H i + 1 ( A i - P i C i ( C i P i C i + W i ) - 1 C i - B i ( B i S i + 1 B i + R i ) - 1 B i S i + 1 ) G i , {\mathbf{}}F_{i}^{r}=H_{i+1}\left(A_{i}-P_{i}C^{\prime}_{i}\left(C_{i}P_{i}C^{% \prime}_{i}+W_{i}\right)^{-1}C_{i}-B_{i}\left(B^{\prime}_{i}S_{i+1}B_{i}+R_{i}% \right)^{-1}B^{\prime}_{i}S_{i+1}\right)G^{\prime}_{i},
  68. K i r = H i + 1 P i C i ( C i P i C i + W i ) - 1 , {\mathbf{}}K_{i}^{r}=H_{i+1}P_{i}C^{\prime}_{i}\left(C_{i}P_{i}C^{\prime}_{i}+% W_{i}\right)^{-1},
  69. L i r = ( B i S i + 1 B i + R i ) - 1 B i S i + 1 G i , {\mathbf{}}L_{i}^{r}=\left(B^{\prime}_{i}S_{i+1}B_{i}+R_{i}\right)^{-1}B^{% \prime}_{i}S_{i+1}G^{\prime}_{i},
  70. 𝐱 0 r = H 0 E ( 𝐱 0 ) . {\mathbf{x}}_{0}^{r}=H_{0}E({\mathbf{x}}_{0}).
  71. G i , H i {\mathbf{}}G_{i},H_{i}
  72. G i H i = τ i , G i H i = I n r {\mathbf{}}G^{\prime}_{i}H_{i}=\tau_{i},G_{i}H^{\prime}_{i}=I_{n_{r}}
  73. P ^ i S ^ i {\mathbf{}}\hat{P}_{i}\hat{S}_{i}
  74. τ i P ^ i = P ^ i τ i = P ^ i , τ i S ^ i = S ^ i τ i = S ^ i {\mathbf{}}\tau_{i}\hat{P}_{i}=\hat{P}_{i}\tau^{\prime}_{i}=\hat{P}_{i},\tau^{% \prime}_{i}\hat{S}_{i}=\hat{S}_{i}\tau_{i}=\hat{S}_{i}
  75. N {\mathbf{}}N

Optional_stopping_theorem.html

  1. τ τ
  2. τ τ
  3. c c∈ℕ
  4. τ c τ≤c
  5. τ τ
  6. 𝔼 [ τ ] < \mathbb{E}[\tau]<\infty
  7. c c
  8. 𝔼 [ | X t + 1 - X t | | t ] c \mathbb{E}\bigl[|X_{t+1}-X_{t}|\,\big|\,{\mathcal{F}}_{t}\bigr]\leq c
  9. c c
  10. 𝔼 [ X τ ] = 𝔼 [ X 0 ] . \mathbb{E}[X_{\tau}]=\mathbb{E}[X_{0}].
  11. X X
  12. 𝔼 [ X τ ] 𝔼 [ X 0 ] , \mathbb{E}[X_{\tau}]\geq\mathbb{E}[X_{0}],
  13. 𝔼 [ X τ ] 𝔼 [ X 0 ] , \mathbb{E}[X_{\tau}]\leq\mathbb{E}[X_{0}],
  14. τ = τ=∞
  15. X X
  16. τ τ
  17. a 0 a≥0
  18. 0
  19. m a m≥a
  20. a a
  21. a = p m + ( 1 p ) 0 a=pm+(1–p)0
  22. p p
  23. m m
  24. 0
  25. p = a / m p=a/m
  26. X X
  27. 0
  28. m –m
  29. + m +m
  30. τ τ
  31. X X
  32. ± m ±m
  33. + m +m
  34. m −m
  35. X X
  36. m m
  37. m m
  38. M := c M:=c
  39. X t τ = X 0 + s = 0 τ and t - 1 ( X s + 1 - X s ) , t 0 , X_{t}^{\tau}=X_{0}+\sum_{s=0}^{\tau\and t-1}(X_{s+1}-X_{s}),\quad t\in{\mathbb% {N}}_{0},
  40. M := | X 0 | + s = 0 τ - 1 | X s + 1 - X s | = | X 0 | + s = 0 | X s + 1 - X s | 𝟏 { τ > s } M:=|X_{0}|+\sum_{s=0}^{\tau-1}|X_{s+1}-X_{s}|=|X_{0}|+\sum_{s=0}^{\infty}|X_{s% +1}-X_{s}|\cdot\mathbf{1}_{\{\tau>s\}}
  41. 𝔼 [ M ] = 𝔼 [ | X 0 | ] + s = 0 𝔼 [ | X s + 1 - X s | 𝟏 { τ > s } ] \mathbb{E}[M]=\mathbb{E}[|X_{0}|]+\sum_{s=0}^{\infty}\mathbb{E}\bigl[|X_{s+1}-% X_{s}|\cdot\mathbf{1}_{\{\tau>s\}}\bigr]
  42. M M
  43. s s
  44. τ τ
  45. 𝔼 [ M ] = 𝔼 [ | X 0 | ] + s = 0 𝔼 [ 𝔼 [ | X s + 1 - X s | | s ] 𝟏 { τ > s } c 1 { τ > s } a.s. by (b) ] 𝔼 [ | X 0 | ] + c s = 0 ( τ > s ) = 𝔼 [ | X 0 | ] + c 𝔼 [ τ ] < , \begin{aligned}\displaystyle\mathbb{E}[M]&\displaystyle=\mathbb{E}[|X_{0}|]+% \sum_{s=0}^{\infty}\mathbb{E}\bigl[\underbrace{\mathbb{E}\bigl[|X_{s+1}-X_{s}|% \big|{\mathcal{F}}_{s}\bigr]\cdot\mathbf{1}_{\{\tau>s\}}}_{\leq\,c\,\mathbf{1}% _{\{\tau>s\}}\,\text{ a.s. by (b)}}\bigr]\\ &\displaystyle\leq\mathbb{E}[|X_{0}|]+c\sum_{s=0}^{\infty}\mathbb{P}(\tau>s)\\ &\displaystyle=\mathbb{E}[|X_{0}|]+c\,\mathbb{E}[\tau]<\infty,\\ \end{aligned}
  46. M M
  47. 𝔼 [ X τ ] = lim t 𝔼 [ X t τ ] . \mathbb{E}[X_{\tau}]=\lim_{t\to\infty}\mathbb{E}[X_{t}^{\tau}].
  48. 𝔼 [ X t τ ] = 𝔼 [ X 0 ] , t 0 , \mathbb{E}[X_{t}^{\tau}]=\mathbb{E}[X_{0}],\quad t\in{\mathbb{N}}_{0},
  49. 𝔼 [ X τ ] = 𝔼 [ X 0 ] . \mathbb{E}[X_{\tau}]=\mathbb{E}[X_{0}].
  50. X X

Order_of_integration_(calculus).html

  1. D f ( x , y ) d x d y , \iint_{D}\ f(x,y)\ dx\,dy,
  2. a z a x h ( y ) d y d x \int_{a}^{z}\,\int_{a}^{x}\,h(y)\,dy\,dx
  3. a z d x a x h ( y ) d y \int_{a}^{z}dx\,\int_{a}^{x}\,h(y)\,dy
  4. a z d x a x h ( y ) d y = a z h ( y ) d y y z d x = a z ( z - y ) h ( y ) d y . \int_{a}^{z}\ dx\ \int_{a}^{x}\ h(y)\ dy\ =\int_{a}^{z}\ h(y)\ dy\ \ \int_{y}^% {z}\ dx=\int_{a}^{z}\ \left(z-y\right)h(y)\,dy\ .
  5. a z f ( x ) g ( x ) d x = [ f ( x ) g ( x ) ] a z - a z f ( x ) g ( x ) d x \int_{a}^{z}f(x)g^{\prime}(x)\,dx=\left[f(x)g(x)\right]_{a}^{z}-\int_{a}^{z}f^% {\prime}(x)g(x)\,dx\!
  6. f ( x ) = a x h ( y ) d y and g ( x ) = 1 . f(x)=\int_{a}^{x}\ h(y)\,dy\,\text{ and }g^{\prime}(x)=1\ .
  7. 1 ( 2 π i ) 2 L * d τ 1 τ 1 - t L * g ( τ ) d τ τ - τ 1 = 1 4 g ( t ) , \frac{1}{(2\pi i)^{2}}\int_{L}^{*}\frac{d{\tau}_{1}}{{\tau}_{1}-t}\ \int_{L}^{% *}\ g(\tau)\frac{d\tau}{\tau-\tau_{1}}=\frac{1}{4}g(t)\ ,
  8. 1 ( 2 π i ) 2 L * g ( τ ) d τ ( L * d τ 1 ( τ 1 - t ) ( τ - τ 1 ) ) = 0 . \frac{1}{(2\pi i)^{2}}\int_{L}^{*}g(\tau)\ d\tau\left(\int_{L}^{*}\frac{d\tau_% {1}}{\left(\tau_{1}-t\right)\left(\tau-\tau_{1}\right)}\right)=0\ .
  9. L * d τ 1 τ 1 - t = L * d τ 1 τ 1 - t = π i . \int_{L}^{*}\frac{d\tau_{1}}{\tau_{1}-t}=\int_{L}^{*}\frac{d\tau_{1}}{\tau_{1}% -t}=\pi\ i\ .
  10. L * \int_{L}^{*}
  11. 1 x 2 - y 2 ( x 2 + y 2 ) 2 d y = [ y x 2 + y 2 ] 1 = - 1 1 + x 2 [ x 1 ] . \int_{1}^{\infty}\frac{x^{2}-y^{2}}{\left(x^{2}+y^{2}\right)^{2}}\ dy=\left[% \frac{y}{x^{2}+y^{2}}\right]_{1}^{\infty}=-\frac{1}{1+x^{2}}\ \left[x\geq 1% \right]\ .
  12. 1 ( 1 x 2 - y 2 ( x 2 + y 2 ) 2 d y ) d x = - π 4 . \int_{1}^{\infty}\left(\int_{1}^{\infty}\frac{x^{2}-y^{2}}{\left(x^{2}+y^{2}% \right)^{2}}\ dy\right)\ dx=-\frac{\pi}{4}\ .
  13. 1 ( 1 x 2 - y 2 ( x 2 + y 2 ) 2 d x ) d y = π 4 . \int_{1}^{\infty}\left(\int_{1}^{\infty}\frac{x^{2}-y^{2}}{\left(x^{2}+y^{2}% \right)^{2}}\ dx\right)\ dy=\frac{\pi}{4}\ .
  14. a d x ( c d y f ( x , y ) ) \int_{a}^{\infty}dx\ \left(\int_{c}^{\infty}dy\ f(x,\ y)\right)

Ordinal_analysis.html

  1. T T
  2. α \alpha
  3. o o
  4. T T
  5. o o
  6. α \alpha
  7. R R
  8. ω \omega
  9. α \alpha
  10. T T
  11. R R
  12. Σ 1 1 \Sigma^{1}_{1}
  13. Σ 1 0 \Sigma^{0}_{1}
  14. ω 1 CK \omega_{1}^{\mathrm{CK}}
  15. n \mathcal{E}^{n}
  16. Π 1 1 - 𝖢𝖠 0 \Pi^{1}_{1}\mbox{-}~{}\mathsf{CA}_{0}
  17. I D < ω ID_{<\omega}
  18. Σ 2 1 - 𝖠𝖢 + 𝖡𝖨 \Sigma^{1}_{2}\mbox{-}~{}\mathsf{AC}+\mathsf{BI}

Ordinal_collapsing_function.html

  1. ψ \psi
  2. Ω \Omega
  3. ω 1 \omega_{1}
  4. ε \varepsilon
  5. ω 1 \omega_{1}
  6. ψ \psi
  7. α \alpha
  8. ψ ( α ) \psi(\alpha)
  9. α \alpha
  10. ψ ( β ) \psi(\beta)
  11. β < α \beta<\alpha
  12. ψ ( α ) \psi(\alpha)
  13. C ( α ) C(\alpha)
  14. 0
  15. 1 1
  16. ω \omega
  17. Ω \Omega
  18. ψ α \psi\upharpoonright_{\alpha}
  19. ψ \psi
  20. β < α \beta<\alpha
  21. C ( α ) 0 = { 0 , 1 , ω , Ω } C(\alpha)_{0}=\{0,1,\omega,\Omega\}
  22. C ( α ) n + 1 = C ( α ) n { β 1 + β 2 , β 1 β 2 , β 1 β 2 : β 1 , β 2 C ( α ) n } { ψ ( β ) : β C ( α ) n β < α } C(\alpha)_{n+1}=C(\alpha)_{n}\cup\{\beta_{1}+\beta_{2},\beta_{1}\beta_{2},{% \beta_{1}}^{\beta_{2}}:\beta_{1},\beta_{2}\in C(\alpha)_{n}\}\cup\{\psi(\beta)% :\beta\in C(\alpha)_{n}\land\beta<\alpha\}
  23. n n
  24. C ( α ) C(\alpha)
  25. C ( α ) n C(\alpha)_{n}
  26. n n
  27. ψ ( α ) \psi(\alpha)
  28. C ( α ) C(\alpha)
  29. ψ ( α ) \psi(\alpha)
  30. 0
  31. 1 1
  32. ω \omega
  33. Ω \Omega
  34. ψ \psi
  35. α \alpha
  36. ψ \psi
  37. Ω \Omega
  38. ψ \psi
  39. ψ \psi
  40. ψ \psi
  41. C ( 0 ) C(0)
  42. 0
  43. 1 1
  44. 2 2
  45. 3 3
  46. ω \omega
  47. ω + 1 \omega+1
  48. ω + 2 \omega+2
  49. ω 2 \omega 2
  50. ω 3 \omega 3
  51. ω 2 \omega^{2}
  52. ω 3 \omega^{3}
  53. ω ω \omega^{\omega}
  54. ω ω ω \omega^{\omega^{\omega}}
  55. Ω \Omega
  56. Ω + 1 \Omega+1
  57. Ω ω \Omega\omega
  58. Ω Ω \Omega^{\Omega}
  59. ε 0 \varepsilon_{0}
  60. ω \omega
  61. ω ω \omega^{\omega}
  62. ω ω ω \omega^{\omega^{\omega}}
  63. Ω \Omega
  64. ε Ω + 1 \varepsilon_{\Omega+1}
  65. Ω \Omega
  66. Ω Ω \Omega^{\Omega}
  67. Ω Ω Ω \Omega^{\Omega^{\Omega}}
  68. ψ ( 0 ) = ε 0 \psi(0)=\varepsilon_{0}
  69. C ( 1 ) C(1)
  70. 0
  71. 1 1
  72. ω \omega
  73. Ω \Omega
  74. ε 0 \varepsilon_{0}
  75. ε 1 \varepsilon_{1}
  76. ψ ( 1 ) = ε 1 \psi(1)=\varepsilon_{1}
  77. ψ ( α ) = ε α \psi(\alpha)=\varepsilon_{\alpha}
  78. α \alpha
  79. α < ε α \alpha<\varepsilon_{\alpha}
  80. ψ ( α ) = ε α = ϕ 1 ( α ) \psi(\alpha)=\varepsilon_{\alpha}=\phi_{1}(\alpha)
  81. α ζ 0 \alpha\leq\zeta_{0}
  82. ζ 0 = ϕ 2 ( 0 ) \zeta_{0}=\phi_{2}(0)
  83. α ε α \alpha\mapsto\varepsilon_{\alpha}
  84. ϕ \phi
  85. ϕ 1 ( α ) = ε α \phi_{1}(\alpha)=\varepsilon_{\alpha}
  86. ψ ( ζ 0 ) = ζ 0 \psi(\zeta_{0})=\zeta_{0}
  87. ψ ( ζ 0 + 1 ) \psi(\zeta_{0}+1)
  88. ζ 0 \zeta_{0}
  89. ϕ 1 : α ε α \phi_{1}\colon\alpha\mapsto\varepsilon_{\alpha}
  90. C ( α ) C(\alpha)
  91. α Ω \alpha\leq\Omega
  92. ψ \psi
  93. ζ 0 \zeta_{0}
  94. ψ ( α ) = ζ 0 \psi(\alpha)=\zeta_{0}
  95. ζ 0 α Ω \zeta_{0}\leq\alpha\leq\Omega
  96. ψ ( Ω ) = ζ 0 \psi(\Omega)=\zeta_{0}
  97. ψ ( Ω + 1 ) \psi(\Omega+1)
  98. Ω \Omega
  99. C ( α ) C(\alpha)
  100. ψ ( Ω ) = ζ 0 \psi(\Omega)=\zeta_{0}
  101. C ( Ω + 1 ) C(\Omega+1)
  102. 0
  103. 1 1
  104. ω \omega
  105. Ω \Omega
  106. ϕ 1 : α ε α \phi_{1}\colon\alpha\mapsto\varepsilon_{\alpha}
  107. ζ 0 \zeta_{0}
  108. ζ 0 \zeta_{0}
  109. C ( Ω + 1 ) C(\Omega+1)
  110. ε ζ 0 + 1 \varepsilon_{\zeta_{0}+1}
  111. ε \varepsilon
  112. ζ 0 \zeta_{0}
  113. ψ ( Ω ) = ζ 0 \psi(\Omega)=\zeta_{0}
  114. ψ \psi
  115. ψ ( Ω + 1 ) = ε ζ 0 + 1 \psi(\Omega+1)=\varepsilon_{\zeta_{0}+1}
  116. Ω \Omega
  117. ζ 0 \zeta_{0}
  118. ψ \psi
  119. ψ ( Ω + α ) = ε ζ 0 + α \psi(\Omega+\alpha)=\varepsilon_{\zeta_{0}+\alpha}
  120. α ζ 1 = ϕ 2 ( 1 ) \alpha\leq\zeta_{1}=\phi_{2}(1)
  121. ψ ( Ω + ζ 0 ) = ε ζ 0 2 \psi(\Omega+\zeta_{0})=\varepsilon_{\zeta_{0}2}
  122. ζ 0 \zeta_{0}
  123. ζ 1 = ϕ 2 ( 1 ) \zeta_{1}=\phi_{2}(1)
  124. α ε α \alpha\mapsto\varepsilon_{\alpha}
  125. ζ 0 \zeta_{0}
  126. ζ 1 \zeta_{1}
  127. ζ 0 \zeta_{0}
  128. ε \varepsilon
  129. ψ ( Ω 2 ) = ζ 1 \psi(\Omega 2)=\zeta_{1}
  130. ψ ( Ω ( 1 + α ) ) = ϕ 2 ( α ) \psi(\Omega(1+\alpha))=\phi_{2}(\alpha)
  131. α ϕ 3 ( 0 ) \alpha\leq\phi_{3}(0)
  132. ϕ 2 \phi_{2}
  133. ϕ 1 : α ε α \phi_{1}\colon\alpha\mapsto\varepsilon_{\alpha}
  134. ϕ 3 ( 0 ) \phi_{3}(0)
  135. ϕ 2 \phi_{2}
  136. ψ ( Ω 2 ) = ϕ 3 ( 0 ) \psi(\Omega^{2})=\phi_{3}(0)
  137. ψ ( Ω α ) = ϕ 1 + α ( 0 ) \psi(\Omega^{\alpha})=\phi_{1+\alpha}(0)
  138. Γ 0 \Gamma_{0}
  139. α ϕ α ( 0 ) \alpha\mapsto\phi_{\alpha}(0)
  140. ψ ( Ω Ω ) = Γ 0 \psi(\Omega^{\Omega})=\Gamma_{0}
  141. ψ ( Ω Ω + Ω α ) = ϕ Γ 0 + α ( 0 ) \psi(\Omega^{\Omega}+\Omega^{\alpha})=\phi_{\Gamma_{0}+\alpha}(0)
  142. α Γ 1 \alpha\leq\Gamma_{1}
  143. Γ 1 \Gamma_{1}
  144. α ϕ α ( 0 ) \alpha\mapsto\phi_{\alpha}(0)
  145. α Γ α \alpha\mapsto\Gamma_{\alpha}
  146. ϕ ( 1 , 0 , α ) \phi(1,0,\alpha)
  147. ψ ( Ω Ω ( 1 + α ) ) = Γ α \psi(\Omega^{\Omega}(1+\alpha))=\Gamma_{\alpha}
  148. ϕ ( 1 , 1 , 0 ) \phi(1,1,0)
  149. α Γ α \alpha\mapsto\Gamma_{\alpha}
  150. ψ ( Ω Ω + 1 ) \psi(\Omega^{\Omega+1})
  151. ϕ ( 2 , 0 , 0 ) \phi(2,0,0)
  152. α ϕ ( 1 , α , 0 ) \alpha\mapsto\phi(1,\alpha,0)
  153. ψ ( Ω Ω 2 ) \psi(\Omega^{\Omega 2})
  154. ψ ( Ω Ω 2 ) \psi(\Omega^{\Omega^{2}})
  155. ϕ ( α , β , γ ) \phi(\alpha,\beta,\gamma)
  156. ψ ( Ω Ω ω ) \psi(\Omega^{\Omega^{\omega}})
  157. ϕ ( ) \phi(\ldots)
  158. ψ ( Ω Ω Ω ) \psi(\Omega^{\Omega^{\Omega}})
  159. ϕ ( ) \phi(\ldots)
  160. ψ ( ε Ω + 1 ) \psi(\varepsilon_{\Omega+1})
  161. ψ ( Ω ) \psi(\Omega)
  162. ψ ( Ω Ω ) \psi(\Omega^{\Omega})
  163. ψ ( Ω Ω Ω ) \psi(\Omega^{\Omega^{\Omega}})
  164. ψ \psi
  165. ψ \psi
  166. δ \delta
  167. ω \omega
  168. ω \omega
  169. ε 0 \varepsilon_{0}
  170. Ω \Omega
  171. α \alpha
  172. δ β 1 γ 1 + + δ β k γ k \delta^{\beta_{1}}\gamma_{1}+\ldots+\delta^{\beta_{k}}\gamma_{k}
  173. k k
  174. γ 1 , , γ k \gamma_{1},\ldots,\gamma_{k}
  175. δ \delta
  176. β 1 > β 2 > > β k \beta_{1}>\beta_{2}>\cdots>\beta_{k}
  177. β k = 0 \beta_{k}=0
  178. δ \delta
  179. δ = ω \delta=\omega
  180. α = δ α \alpha=\delta^{\alpha}
  181. β i \beta_{i}
  182. α \alpha
  183. α < δ \alpha<\delta
  184. k 1 k\leq 1
  185. γ 1 = α \gamma_{1}=\alpha
  186. α \alpha
  187. ε Ω + 1 \varepsilon_{\Omega+1}
  188. Ω \Omega
  189. γ i < Ω \gamma_{i}<\Omega
  190. β i < α \beta_{i}<\alpha
  191. α < ε Ω + 1 \alpha<\varepsilon_{\Omega+1}
  192. Ω \Omega
  193. Ω \Omega
  194. α \alpha
  195. < Ω <\Omega
  196. Ω \Omega
  197. α \alpha
  198. ψ \psi
  199. ψ \psi
  200. ψ ( α ) = ψ ( β ) \psi(\alpha)=\psi(\beta)
  201. β < α \beta<\alpha
  202. C ( α ) = C ( β ) C(\alpha)=C(\beta)
  203. β \beta^{\prime}
  204. β β < α \beta\leq\beta^{\prime}<\alpha
  205. C ( α ) C(\alpha)
  206. ψ \psi
  207. ψ ( α ) \psi(\alpha)
  208. C ( α ) C(\alpha)
  209. C ( β ) C(\beta)
  210. C ( α ) C(\alpha)
  211. γ = ψ ( α ) \gamma=\psi(\alpha)
  212. ψ \psi
  213. ε \varepsilon
  214. β ω β \beta\mapsto\omega^{\beta}
  215. C ( α ) C(\alpha)
  216. C ( α ) C(\alpha)
  217. δ \delta
  218. ε \varepsilon
  219. α \alpha
  220. ψ ( β ) < δ \psi(\beta)<\delta
  221. β < α \beta<\alpha
  222. Ω \Omega
  223. C ( α ) C(\alpha)
  224. δ \delta
  225. C C^{\prime}
  226. Ω \Omega
  227. δ \delta
  228. C C^{\prime}
  229. δ \delta
  230. ε \varepsilon
  231. C C^{\prime}
  232. ψ ( β ) \psi(\beta)
  233. β < α \beta<\alpha
  234. 0
  235. 1 1
  236. ω \omega
  237. Ω \Omega
  238. C C ( α ) C^{\prime}\supseteq C(\alpha)
  239. ψ ( α ) δ \psi(\alpha)\leq\delta
  240. δ C ( α ) \delta\not\in C(\alpha)
  241. ε \varepsilon
  242. ψ \psi
  243. ψ \psi
  244. ψ \psi
  245. ε \varepsilon
  246. δ \delta
  247. ε \varepsilon
  248. ψ \psi
  249. α \alpha
  250. β \beta
  251. ψ ( β ) < δ \psi(\beta)<\delta
  252. ψ ( α ) δ \psi(\alpha)\leq\delta
  253. ψ ( α ) < δ \psi(\alpha)<\delta
  254. α \alpha
  255. ψ ( α ) = δ \psi(\alpha)=\delta
  256. ψ ( α ) = δ \psi(\alpha)=\delta
  257. C ( α ) C(\alpha)
  258. γ \gamma
  259. ε Ω + 1 \varepsilon_{\Omega+1}
  260. Ω \Omega
  261. δ \delta
  262. δ \delta
  263. ε Ω + 1 \varepsilon_{\Omega+1}
  264. Ω \Omega
  265. δ \delta
  266. C ( α ) C(\alpha)
  267. ψ ( β ) < δ \psi(\beta)<\delta
  268. β < α \beta<\alpha
  269. α \alpha
  270. ψ ( α ) = δ \psi(\alpha)=\delta
  271. ψ ( α ) = ψ ( β ) \psi(\alpha)=\psi(\beta)
  272. β < α \beta<\alpha
  273. C ( α ) = C ( β ) C(\alpha)=C(\beta)
  274. α \alpha
  275. ψ ( α ) = δ \psi(\alpha)=\delta
  276. γ \gamma
  277. γ \gamma
  278. γ \gamma
  279. ε 0 \varepsilon_{0}
  280. γ \gamma
  281. ε \varepsilon
  282. δ \delta
  283. γ \gamma
  284. ε \varepsilon
  285. δ < γ \delta<\gamma
  286. δ \delta
  287. δ \delta
  288. γ \gamma
  289. γ \gamma
  290. γ = δ \gamma=\delta
  291. ε \varepsilon
  292. δ = ψ ( α ) \delta=\psi(\alpha)
  293. α < ε Ω + 1 \alpha<\varepsilon_{\Omega+1}
  294. α \alpha
  295. ψ \psi
  296. Ω \Omega
  297. α \alpha
  298. δ \delta
  299. C ( α ) C(\alpha)
  300. α \alpha
  301. C ( α + 1 ) = C ( α ) C(\alpha+1)=C(\alpha)
  302. ψ \psi
  303. α \alpha
  304. ψ ( α + 1 ) = ψ ( α ) = δ \psi(\alpha+1)=\psi(\alpha)=\delta
  305. α \alpha
  306. Ω \Omega
  307. Ω \Omega
  308. ψ \psi
  309. ε 0 = ψ ( 0 ) \varepsilon_{0}=\psi(0)
  310. ε 1 = ψ ( 1 ) \varepsilon_{1}=\psi(1)
  311. ε 0 \varepsilon_{0}
  312. ω ω ε 0 + ω \omega^{\omega^{\varepsilon_{0}+\omega}}
  313. ε 0 ω ω {\varepsilon_{0}}^{\omega^{\omega}}
  314. ψ ( 0 ) ω ω \psi(0)^{\omega^{\omega}}
  315. ε 2 = ψ ( 2 ) \varepsilon_{2}=\psi(2)
  316. ε 1 \varepsilon_{1}
  317. ε 0 \varepsilon_{0}
  318. ω ω ε 1 + ε 0 + 1 \omega^{\omega^{\varepsilon_{1}+\varepsilon_{0}+1}}
  319. ε 1 ε 0 ω {\varepsilon_{1}}^{\varepsilon_{0}\omega}
  320. ψ ( 1 ) ψ ( 0 ) ω \psi(1)^{\psi(0)\,\omega}
  321. ζ 0 = ψ ( Ω ) \zeta_{0}=\psi(\Omega)
  322. ε \varepsilon
  323. Ω \Omega
  324. Ω \Omega
  325. ε \varepsilon
  326. ψ ( ε Ω + 1 ) \psi(\varepsilon_{\Omega+1})
  327. ψ \psi
  328. ψ \psi
  329. Ω \Omega
  330. α \alpha
  331. α \alpha
  332. ψ ( α ) = δ \psi(\alpha)=\delta
  333. ε \varepsilon
  334. δ \delta
  335. δ \delta
  336. ψ ( ψ ( Ω ) + 1 ) \psi(\psi(\Omega)+1)
  337. ψ ( Ω ) = ζ 0 \psi(\Omega)=\zeta_{0}
  338. ψ \psi
  339. ζ 0 \zeta_{0}
  340. Ω \Omega
  341. ε 0 \varepsilon_{0}
  342. δ \delta
  343. δ = ψ ( α ) \delta=\psi(\alpha)
  344. α \alpha
  345. Ω \Omega
  346. δ \delta
  347. Ω \Omega
  348. ψ \psi
  349. ψ \psi
  350. ψ ( Ω ω + 1 ψ ( Ω ) + ψ ( Ω ω ) ψ ( Ω 2 ) 42 ) ψ ( 1729 ) ω \psi(\Omega^{\omega+1}\,\psi(\Omega)+\psi(\Omega^{\omega})^{\psi(\Omega^{2})}4% 2)^{\psi(1729)\,\omega}
  351. ϕ 1 ( ϕ ω + 1 ( ϕ 2 ( 0 ) ) + ϕ ω ( 0 ) ϕ 3 ( 0 ) 42 ) ϕ 1 ( 1729 ) ω \phi_{1}(\phi_{\omega+1}(\phi_{2}(0))+\phi_{\omega}(0)^{\phi_{3}(0)}42)^{\phi_% {1}(1729)\,\omega}
  352. ψ ( Ω Ω ) \psi(\Omega^{\Omega})
  353. ψ ( Ω ψ ( Ω ) ) = ϕ ϕ 2 ( 0 ) ( 0 ) \psi(\Omega^{\psi(\Omega)})=\phi_{\phi_{2}(0)}(0)
  354. Ω \Omega
  355. ψ \psi
  356. ψ ( Ω ψ ( Ω ) ) = ϕ ϕ 2 ( 0 ) ( 0 ) \psi(\Omega^{\psi(\Omega)})=\phi_{\phi_{2}(0)}(0)
  357. ψ ( Ω ) ψ ( Ω ) = ϕ 2 ( 0 ) ϕ 2 ( 0 ) \psi(\Omega)^{\psi(\Omega)}=\phi_{2}(0)^{\phi_{2}(0)}
  358. Ω ψ ( Ω ) \Omega^{\psi(\Omega)}
  359. Ω \Omega
  360. ψ ( Ω ) \psi(\Omega)
  361. ψ ( Ω Ω ) \psi(\Omega^{\Omega})
  362. ψ ( Ω ) ψ ( Ω ) \psi(\Omega)^{\psi(\Omega)}
  363. ψ ( Ω + 1 ) \psi(\Omega+1)
  364. Ω \Omega
  365. δ \delta
  366. α = δ β 1 γ 1 + + δ β k γ k \alpha=\delta^{\beta_{1}}\gamma_{1}+\cdots+\delta^{\beta_{k}}\gamma_{k}
  367. δ \delta
  368. ω \omega
  369. ψ ( ) \psi(\cdots)
  370. Ω \Omega
  371. k k
  372. α = 0 \alpha=0
  373. β k \beta_{k}
  374. γ k \gamma_{k}
  375. α \alpha
  376. γ k \gamma_{k}
  377. γ k \gamma_{k}
  378. γ k \gamma_{k}
  379. γ k \gamma_{k}
  380. β k \beta_{k}
  381. δ β k γ k \delta^{\beta_{k}}\gamma_{k}
  382. δ β k ( γ k - 1 ) + δ β k \delta^{\beta_{k}}(\gamma_{k}-1)+\delta^{\beta_{k}}
  383. β k \beta_{k}
  384. γ k \gamma_{k}
  385. β k \beta_{k}
  386. δ β k γ k \delta^{\beta_{k}}\gamma_{k}
  387. δ β k ( γ k - 1 ) + δ β k - 1 δ \delta^{\beta_{k}}(\gamma_{k}-1)+\delta^{\beta_{k}-1}\delta
  388. δ \delta
  389. δ \delta
  390. ω \omega
  391. 0
  392. 1 1
  393. 2 2
  394. 3 3
  395. δ \delta
  396. δ = ψ ( 0 ) \delta=\psi(0)
  397. δ \delta
  398. ω \omega
  399. ω ω \omega^{\omega}
  400. ω ω ω \omega^{\omega^{\omega}}
  401. δ = ψ ( α + 1 ) \delta=\psi(\alpha+1)
  402. δ \delta
  403. ψ ( α ) \psi(\alpha)
  404. ψ ( α ) ψ ( α ) \psi(\alpha)^{\psi(\alpha)}
  405. ψ ( α ) ψ ( α ) ψ ( α ) \psi(\alpha)^{\psi(\alpha)^{\psi(\alpha)}}
  406. δ = ψ ( α ) \delta=\psi(\alpha)
  407. α \alpha
  408. δ \delta
  409. ψ \psi
  410. α \alpha
  411. ψ \psi
  412. δ = ψ ( α ) \delta=\psi(\alpha)
  413. α \alpha
  414. Ω \Omega
  415. α \alpha
  416. ρ < α \rho<\alpha
  417. ψ \psi
  418. ρ \rho
  419. α \alpha
  420. ρ \rho
  421. ξ h ( ψ ( ξ ) ) \xi\mapsto h(\psi(\xi))
  422. Ω \Omega
  423. α \alpha
  424. α \alpha
  425. Ω \Omega
  426. Ω \Omega
  427. α = h ( Ω ) \alpha=h(\Omega)
  428. ψ ( ξ ) \psi(\xi)
  429. ξ \xi
  430. 0
  431. ξ h ( ψ ( ξ ) ) \xi\mapsto h(\psi(\xi))
  432. 0
  433. h ( ψ ( 0 ) ) h(\psi(0))
  434. h ( ψ ( h ( ψ ( 0 ) ) ) ) h(\psi(h(\psi(0))))
  435. ρ \rho
  436. ψ ( α ) = ψ ( ρ ) \psi(\alpha)=\psi(\rho)
  437. ψ ( 0 ) \psi(0)
  438. ψ ( h ( ψ ( 0 ) ) ) \psi(h(\psi(0)))
  439. ψ ( h ( ψ ( h ( ψ ( 0 ) ) ) ) ) \psi(h(\psi(h(\psi(0)))))
  440. ψ ( Ω ) \psi(\Omega)
  441. ψ ( 0 ) \psi(0)
  442. ψ ( ψ ( 0 ) ) \psi(\psi(0))
  443. ψ ( ψ ( ψ ( 0 ) ) ) \psi(\psi(\psi(0)))
  444. ρ = ψ ( Ω ) = ζ 0 \rho=\psi(\Omega)=\zeta_{0}
  445. ψ \psi
  446. Ω \Omega
  447. ψ ( Ω 2 ) \psi(\Omega 2)
  448. ψ ( 0 ) \psi(0)
  449. ψ ( Ω + ψ ( 0 ) ) \psi(\Omega+\psi(0))
  450. ψ ( Ω + ψ ( Ω + ψ ( 0 ) ) ) \psi(\Omega+\psi(\Omega+\psi(0)))
  451. ψ \psi
  452. ρ = Ω + ψ ( Ω 2 ) = Ω + ζ 1 \rho=\Omega+\psi(\Omega 2)=\Omega+\zeta_{1}
  453. ψ \psi
  454. Ω 2 \Omega 2
  455. ψ ( Ω 2 ) \psi(\Omega^{2})
  456. ψ ( 0 ) \psi(0)
  457. ψ ( Ω ψ ( 0 ) ) \psi(\Omega\psi(0))
  458. ψ ( Ω ψ ( Ω ψ ( 0 ) ) ) \psi(\Omega\psi(\Omega\psi(0)))
  459. ψ \psi
  460. ρ = Ω ψ ( Ω 2 ) \rho=\Omega\psi(\Omega^{2})
  461. ψ ( Ω 2 3 + Ω ) \psi(\Omega^{2}3+\Omega)
  462. ψ ( 0 ) \psi(0)
  463. ψ ( Ω 2 3 + ψ ( 0 ) ) \psi(\Omega^{2}3+\psi(0))
  464. ψ ( Ω 2 3 + ψ ( Ω 2 3 + ψ ( 0 ) ) ) \psi(\Omega^{2}3+\psi(\Omega^{2}3+\psi(0)))
  465. ψ \psi
  466. ρ = Ω 2 3 + ψ ( Ω 2 3 + Ω ) \rho=\Omega^{2}3+\psi(\Omega^{2}3+\Omega)
  467. ψ ( Ω Ω ) \psi(\Omega^{\Omega})
  468. ψ ( 0 ) \psi(0)
  469. ψ ( Ω ψ ( 0 ) ) \psi(\Omega^{\psi(0)})
  470. ψ ( Ω ψ ( Ω ψ ( 0 ) ) ) \psi(\Omega^{\psi(\Omega^{\psi(0)})})
  471. ψ \psi
  472. ρ = Ω ψ ( Ω Ω ) \rho=\Omega^{\psi(\Omega^{\Omega})}
  473. ψ ( Ω Ω 3 ) \psi(\Omega^{\Omega}3)
  474. ψ ( 0 ) \psi(0)
  475. ψ ( Ω Ω 2 + Ω ψ ( 0 ) ) \psi(\Omega^{\Omega}2+\Omega^{\psi(0)})
  476. ψ ( Ω Ω 2 + Ω ψ ( Ω Ω 2 + Ω ψ ( 0 ) ) ) \psi(\Omega^{\Omega}2+\Omega^{\psi(\Omega^{\Omega}2+\Omega^{\psi(0)})})
  477. ψ \psi
  478. ρ = Ω Ω 2 + Ω ψ ( Ω Ω 3 ) \rho=\Omega^{\Omega}2+\Omega^{\psi(\Omega^{\Omega}3)}
  479. ψ ( Ω Ω + 1 ) \psi(\Omega^{\Omega+1})
  480. ψ ( 0 ) \psi(0)
  481. ψ ( Ω Ω ψ ( 0 ) ) \psi(\Omega^{\Omega}\psi(0))
  482. ψ ( Ω Ω ψ ( Ω Ω ψ ( 0 ) ) ) \psi(\Omega^{\Omega}\psi(\Omega^{\Omega}\psi(0)))
  483. ψ \psi
  484. ρ = Ω Ω ψ ( Ω Ω + 1 ) \rho=\Omega^{\Omega}\psi(\Omega^{\Omega+1})
  485. ψ ( Ω Ω 2 + Ω 3 ) \psi(\Omega^{\Omega^{2}+\Omega 3})
  486. ψ ( 0 ) \psi(0)
  487. ψ ( Ω Ω 2 + Ω 2 + ψ ( 0 ) ) \psi(\Omega^{\Omega^{2}+\Omega 2+\psi(0)})
  488. ψ ( Ω Ω 2 + Ω 2 + ψ ( Ω Ω 2 + Ω 2 + ψ ( 0 ) ) ) \psi(\Omega^{\Omega^{2}+\Omega 2+\psi(\Omega^{\Omega^{2}+\Omega 2+\psi(0)})})
  489. ω 2 \omega^{2}
  490. 0
  491. ω \omega
  492. ω 2 \omega 2
  493. ω 3 \omega 3
  494. ψ ( ω ω ) \psi(\omega^{\omega})
  495. ψ ( 1 ) \psi(1)
  496. ψ ( ω ) \psi(\omega)
  497. ψ ( ω 2 ) \psi(\omega^{2})
  498. ψ ( ω 3 ) \psi(\omega^{3})
  499. ψ ( Ω ) ω \psi(\Omega)^{\omega}
  500. 1 1
  501. ψ ( Ω ) \psi(\Omega)
  502. ψ ( Ω ) 2 \psi(\Omega)^{2}
  503. ψ ( Ω ) 3 \psi(\Omega)^{3}
  504. ψ ( Ω + 1 ) \psi(\Omega+1)
  505. ψ ( Ω ) \psi(\Omega)
  506. ψ ( Ω ) ψ ( Ω ) \psi(\Omega)^{\psi(\Omega)}
  507. ψ ( Ω ) ψ ( Ω ) ψ ( Ω ) \psi(\Omega)^{\psi(\Omega)^{\psi(\Omega)}}
  508. ψ ( Ω + ω ) \psi(\Omega+\omega)
  509. ψ ( Ω ) \psi(\Omega)
  510. ψ ( Ω + 1 ) \psi(\Omega+1)
  511. ψ ( Ω + 2 ) \psi(\Omega+2)
  512. ψ ( Ω + 3 ) \psi(\Omega+3)
  513. ψ ( Ω ω ) \psi(\Omega\omega)
  514. ψ ( 0 ) \psi(0)
  515. ψ ( Ω ) \psi(\Omega)
  516. ψ ( Ω 2 ) \psi(\Omega 2)
  517. ψ ( Ω 3 ) \psi(\Omega 3)
  518. ψ ( Ω ω ) \psi(\Omega^{\omega})
  519. ψ ( 1 ) \psi(1)
  520. ψ ( Ω ) \psi(\Omega)
  521. ψ ( Ω 2 ) \psi(\Omega^{2})
  522. ψ ( Ω 3 ) \psi(\Omega^{3})
  523. ψ ( Ω ψ ( 0 ) ) \psi(\Omega^{\psi(0)})
  524. ψ ( Ω ω ) \psi(\Omega^{\omega})
  525. ψ ( Ω ω ω ) \psi(\Omega^{\omega^{\omega}})
  526. ψ ( Ω ω ω ω ) \psi(\Omega^{\omega^{\omega^{\omega}}})
  527. ψ ( 0 ) \psi(0)
  528. ψ ( Ω ψ ( Ω ) ) \psi(\Omega^{\psi(\Omega)})
  529. ψ ( Ω ψ ( 0 ) ) \psi(\Omega^{\psi(0)})
  530. ψ ( Ω ψ ( ψ ( 0 ) ) ) \psi(\Omega^{\psi(\psi(0))})
  531. ψ ( Ω ψ ( ψ ( ψ ( 0 ) ) ) ) \psi(\Omega^{\psi(\psi(\psi(0)))})
  532. ψ ( Ω ) \psi(\Omega)
  533. ψ ( ε Ω + 1 ) \psi(\varepsilon_{\Omega+1})
  534. ψ ( Ω ) \psi(\Omega)
  535. ψ ( Ω Ω ) \psi(\Omega^{\Omega})
  536. ψ ( Ω Ω Ω ) \psi(\Omega^{\Omega^{\Omega}})
  537. ψ ( Ω Ω ω ) \psi(\Omega^{\Omega^{\omega}})
  538. ψ ( Ω Ω 3 ) \psi(\Omega^{\Omega^{3}})
  539. ψ ( Ω Ω 2 ψ ( 0 ) ) \psi(\Omega^{\Omega^{2}\psi(0)})
  540. ψ ( Ω Ω 2 ω ω ) \psi(\Omega^{\Omega^{2}\omega^{\omega}})
  541. ψ ( Ω Ω 2 ω 3 ) \psi(\Omega^{\Omega^{2}\omega^{3}})
  542. ψ ( Ω Ω 2 ω 2 7 ) \psi(\Omega^{\Omega^{2}\omega^{2}7})
  543. ψ ( Ω Ω 2 ( ω 2 6 + ω ) ) \psi(\Omega^{\Omega^{2}(\omega^{2}6+\omega)})
  544. ψ ( Ω Ω 2 ( ω 2 6 + 1 ) ) \psi(\Omega^{\Omega^{2}(\omega^{2}6+1)})
  545. ψ ( Ω Ω 2 ω 2 6 + Ω ψ ( Ω Ω 2 ω 2 6 + Ω ψ ( 0 ) ) ) \psi(\Omega^{\Omega^{2}\omega^{2}6+\Omega\psi(\Omega^{\Omega^{2}\omega^{2}6+% \Omega\psi(0)})})
  546. α \alpha
  547. ψ ( ε Ω + 1 ) \psi(\varepsilon_{\Omega+1})
  548. f α ( n ) f_{\alpha}(n)
  549. n n
  550. f α f_{\alpha}
  551. f ω ω ( n ) f_{\omega^{\omega}}(n)
  552. n n n^{n}
  553. f ψ ( Ω ω ) ( n ) f_{\psi(\Omega^{\omega})}(n)
  554. A ( n , n ) A(n,n)
  555. f ψ ( ε Ω + 1 ) ( n ) f_{\psi(\varepsilon_{\Omega+1})}(n)
  556. α \alpha
  557. ε 0 \varepsilon_{0}
  558. ψ \psi
  559. ψ \psi
  560. C ( α ) C(\alpha)
  561. ψ ( 0 ) = ω ω \psi(0)=\omega^{\omega}
  562. 0
  563. 1 1
  564. ω \omega
  565. ψ ( 1 ) = ω ω 2 \psi(1)=\omega^{\omega^{2}}
  566. ψ ( ω ) = ω ω ω \psi(\omega)=\omega^{\omega^{\omega}}
  567. ψ ( ψ ( 0 ) ) = ω ω ω ω \psi(\psi(0))=\omega^{\omega^{\omega^{\omega}}}
  568. ψ ( Ω ) = ε 0 \psi(\Omega)=\varepsilon_{0}
  569. ψ ( Ω + 1 ) = ε 0 ω \psi(\Omega+1)={\varepsilon_{0}}^{\omega}
  570. ψ ( Ω 2 ) = ε 1 \psi(\Omega 2)=\varepsilon_{1}
  571. Ω \Omega
  572. ψ ( Ω 2 ) = ϕ 2 ( 0 ) \psi(\Omega^{2})=\phi_{2}(0)
  573. ψ ( Ω 3 ) = ϕ 3 ( 0 ) \psi(\Omega^{3})=\phi_{3}(0)
  574. Ω ω \Omega^{\omega}
  575. ψ ( Ω ω ) = ϕ ω ( 0 ) \psi(\Omega^{\omega})=\phi_{\omega}(0)
  576. ψ ( Ω ω ) \psi(\Omega^{\omega})
  577. ψ \psi
  578. ψ ( 0 ) = ω 2 \psi(0)=\omega^{2}
  579. ψ ( 1 ) = ω 3 \psi(1)=\omega^{3}
  580. ψ ( ψ ( 0 ) ) = ω ω 2 \psi(\psi(0))=\omega^{\omega^{2}}
  581. ψ ( Ω ) = ε 0 \psi(\Omega)=\varepsilon_{0}
  582. ψ ( Ω + 1 ) = ε 0 ω \psi(\Omega+1)=\varepsilon_{0}\omega
  583. ψ ( Ω 2 ) = ε 1 \psi(\Omega 2)=\varepsilon_{1}
  584. ψ ( Ω 3 ) = ε 2 \psi(\Omega 3)=\varepsilon_{2}
  585. Ω \Omega
  586. ψ ( Ω ω ) = ε ω = ϕ 1 ( ω ) \psi(\Omega\omega)=\varepsilon_{\omega}=\phi_{1}(\omega)
  587. ψ \psi
  588. ψ ( ε Ω + 1 ) \psi(\varepsilon_{\Omega+1})
  589. ψ ( ε Ω + 1 + 1 ) \psi(\varepsilon_{\Omega+1}+1)
  590. ε Ω + 1 \varepsilon_{\Omega+1}
  591. C ( α ) C(\alpha)
  592. α \alpha
  593. ε \varepsilon
  594. ψ \psi
  595. ψ ( Ω + 1 ) \psi(\Omega+1)
  596. ε ϕ 2 ( 0 ) + 1 \varepsilon_{\phi_{2}(0)+1}
  597. ε Ω + 1 \varepsilon_{\Omega+1}
  598. ψ 1 ( α ) \psi_{1}(\alpha)
  599. Ω \Omega
  600. Ω 2 \Omega_{2}
  601. ψ 1 \psi_{1}
  602. α \alpha
  603. Ω 2 \Omega_{2}
  604. ψ 1 \psi_{1}
  605. Ω = ω 1 \Omega=\omega_{1}
  606. Ω 2 = ω 2 \Omega_{2}=\omega_{2}
  607. ψ 1 ( 0 ) = ε Ω + 1 \psi_{1}(0)=\varepsilon_{\Omega+1}
  608. ψ 1 ( α ) = ε Ω + 1 + α \psi_{1}(\alpha)=\varepsilon_{\Omega+1+\alpha}
  609. ψ 1 ( Ω ) = ε Ω 2 \psi_{1}(\Omega)=\varepsilon_{\Omega 2}
  610. ψ 1 ( ψ 1 ( 0 ) ) = ε Ω + ε Ω + 1 \psi_{1}(\psi_{1}(0))=\varepsilon_{\Omega+\varepsilon_{\Omega+1}}
  611. ζ Ω + 1 \zeta_{\Omega+1}
  612. Ω \Omega
  613. ξ ε ξ \xi\mapsto\varepsilon_{\xi}
  614. ψ 1 ( 0 ) \psi_{1}(0)
  615. ψ 1 ( ψ 1 ( 0 ) ) \psi_{1}(\psi_{1}(0))
  616. ψ 1 ( α ) = ζ Ω + 1 \psi_{1}(\alpha)=\zeta_{\Omega+1}
  617. Ω 2 \Omega_{2}
  618. ψ ( Ω ) \psi(\Omega)
  619. ψ 1 ( Ω 2 ) = ζ Ω + 1 \psi_{1}(\Omega_{2})=\zeta_{\Omega+1}
  620. ψ 1 ( Ω 2 + 1 ) = ε ζ Ω + 1 + 1 \psi_{1}(\Omega_{2}+1)=\varepsilon_{\zeta_{\Omega+1}+1}
  621. ψ 1 \psi_{1}
  622. ψ 1 ( ε Ω 2 + 1 ) \psi_{1}(\varepsilon_{\Omega_{2}+1})
  623. ψ 1 ( Ω 2 ) \psi_{1}(\Omega_{2})
  624. ψ 1 ( Ω 2 Ω 2 ) \psi_{1}({\Omega_{2}}^{\Omega_{2}})
  625. ψ \psi
  626. ψ ( α ) \psi(\alpha)
  627. 0
  628. 1 1
  629. ω \omega
  630. Ω \Omega
  631. Ω 2 \Omega_{2}
  632. ψ 1 \psi_{1}
  633. ψ \psi
  634. α \alpha
  635. ψ \psi
  636. ψ ( ψ 1 ( 0 ) ) \psi(\psi_{1}(0))
  637. ψ ( ψ 1 ( 0 ) + 1 ) \psi(\psi_{1}(0)+1)
  638. ε ψ ( ψ 1 ( 0 ) ) + 1 \varepsilon_{\psi(\psi_{1}(0))+1}
  639. ε \varepsilon
  640. Ω \Omega
  641. Ω 2 \Omega_{2}
  642. Ω 2 \Omega_{2}
  643. ψ ( α ) \psi(\alpha)
  644. 0
  645. 1 1
  646. ω \omega
  647. Ω \Omega
  648. Ω 2 \Omega_{2}
  649. ψ 1 \psi_{1}
  650. ψ \psi
  651. α \alpha
  652. ψ 1 \psi_{1}
  653. α \alpha
  654. ψ ( Ω 2 ) \psi(\Omega_{2})
  655. ψ ( ψ 1 ( Ω 2 ) ) \psi(\psi_{1}(\Omega_{2}))
  656. ψ ( ψ 1 ( Ω 2 ) ) = ψ ( ζ Ω + 1 ) \psi(\psi_{1}(\Omega_{2}))=\psi(\zeta_{\Omega+1})
  657. Ω 2 \Omega_{2}
  658. ψ 1 \psi_{1}
  659. Ω 2 \Omega_{2}
  660. ψ \psi
  661. Ω 2 \Omega_{2}
  662. ψ 1 \psi_{1}
  663. ψ \psi
  664. ψ 1 \psi_{1}
  665. ω + 1 \omega+1
  666. Ω 1 , Ω 2 , , Ω ω \Omega_{1},\Omega_{2},\ldots,\Omega_{\omega}
  667. ω + 1 \omega+1
  668. 1 1
  669. ω \omega
  670. ψ \psi
  671. θ \theta
  672. ψ 0 ( ε Ω ω + 1 ) \psi_{0}(\varepsilon_{\Omega_{\omega}+1})
  673. Π 1 1 \Pi^{1}_{1}
  674. α \alpha
  675. ψ \psi
  676. C ( α , β ) C(\alpha,\beta)
  677. 0
  678. 1 1
  679. ω \omega
  680. Ω \Omega
  681. β \beta
  682. ψ α \psi\upharpoonright_{\alpha}
  683. ψ ( α ) \psi(\alpha)
  684. ρ \rho
  685. C ( α , ρ ) Ω = ρ C(\alpha,\rho)\cap\Omega=\rho
  686. σ \sigma
  687. C ( α , 0 ) C(\alpha,0)
  688. ψ ( α ) \psi(\alpha)
  689. C ( α , 0 ) = C ( α , σ ) C(\alpha,0)=C(\alpha,\sigma)
  690. ψ \psi
  691. σ \sigma
  692. Ω \Omega
  693. C ( α , σ ) C(\alpha,\sigma)
  694. C ~ ( α , β ) \tilde{C}(\alpha,\beta)
  695. 0
  696. 1 1
  697. ω \omega
  698. Ω \Omega
  699. β \beta
  700. ψ ~ α \tilde{\psi}\upharpoonright_{\alpha}
  701. ψ ~ ( α ) \tilde{\psi}(\alpha)
  702. ρ \rho
  703. C ~ ( α , ρ ) Ω = ρ \tilde{C}(\alpha,\rho)\cap\Omega=\rho
  704. α C ~ ( α , ρ ) \alpha\in\tilde{C}(\alpha,\rho)
  705. ψ ~ \tilde{\psi}
  706. ψ \psi
  707. α < ζ 0 \alpha<\zeta_{0}
  708. ζ 0 = φ 2 ( 0 ) \zeta_{0}=\varphi_{2}(0)
  709. ψ ~ ( α ) = ψ ( α ) \tilde{\psi}(\alpha)=\psi(\alpha)
  710. α C ~ ( α , ρ ) \alpha\in\tilde{C}(\alpha,\rho)
  711. ψ \psi
  712. ζ 0 \zeta_{0}
  713. ζ 0 α Ω \zeta_{0}\leq\alpha\leq\Omega
  714. ψ ~ \tilde{\psi}
  715. ψ ~ ( ζ 0 ) = ε ζ 0 + 1 \tilde{\psi}(\zeta_{0})=\varepsilon_{\zeta_{0}+1}
  716. α C ~ ( α , ρ ) \alpha\in\tilde{C}(\alpha,\rho)
  717. ψ ~ ( ζ 0 ) > ζ 0 \tilde{\psi}(\zeta_{0})>\zeta_{0}
  718. ψ ~ ( Ω ) = ζ 0 \tilde{\psi}(\Omega)=\zeta_{0}
  719. Ω C ( α , ρ ) \Omega\in C(\alpha,\rho)
  720. ρ \rho
  721. ψ ~ \tilde{\psi}
  722. ψ \psi
  723. ψ ~ \tilde{\psi}
  724. ψ ( Ω + 1 + α ) = ψ ~ ( ψ ~ ( Ω ) + α ) \psi(\Omega+1+\alpha)=\tilde{\psi}(\tilde{\psi}(\Omega)+\alpha)
  725. α \alpha
  726. ψ ( Ω 2 ) = ψ ~ ( Ω + 1 ) \psi(\Omega 2)=\tilde{\psi}(\Omega+1)
  727. Δ 2 1 \Delta^{1}_{2}
  728. α Ω α \alpha\mapsto\Omega_{\alpha}
  729. C ( ) C(\cdot)
  730. Π 3 \Pi_{3}
  731. Ξ ( α ) \Xi(\alpha)
  732. α \alpha
  733. α Ξ ( α ) \alpha\mapsto\Xi(\alpha)
  734. Π 2 1 \Pi^{1}_{2}
  735. Σ 1 \Sigma_{1}

Organic_nonlinear_optical_materials.html

  1. π - π \pi-\pi

Ornstein–Uhlenbeck_operator.html

  1. Δ = div \Delta=\mathrm{div}\circ\nabla
  2. f H 1 2 = ( 1 - Δ ) 1 / 2 f L 2 2 . \big\|f\big\|_{H^{1}}^{2}=\big\|(1-\Delta)^{1/2}f\big\|_{L^{2}}^{2}.
  3. γ n ( A ) := A ( 2 π ) - n / 2 exp ( - | x | 2 / 2 ) d x . \gamma^{n}(A):=\int_{A}(2\pi)^{-n/2}\exp(-|x|^{2}/2)\,\mathrm{d}x.
  4. 𝔼 [ f v ] = 𝔼 [ f δ v ] . \mathbb{E}\big[\nabla f\cdot v\big]=\mathbb{E}\big[f\delta v\big].
  5. δ v ( x ) = i = 1 n ( x i v i ( x ) - v i x i ( x ) ) . \delta v(x)=\sum_{i=1}^{n}\left(x_{i}v^{i}(x)-\frac{\partial v^{i}}{\partial x% _{i}}(x)\right).
  6. L := - δ , L:=-\delta\circ\nabla,
  7. δ ( f g ) = - f g - f L g . \delta(f\nabla g)=-\nabla f\cdot\nabla g-fLg.
  8. L f ( x ) = Δ f ( x ) - x f ( x ) . Lf(x)=\Delta f(x)-x\cdot\nabla f(x).
  9. 𝔻 1 , 2 \mathbb{D}^{1,2}
  10. 𝔼 [ D F , v H ] = 𝔼 [ F δ v ] \mathbb{E}\big[\langle\mathrm{D}F,v\rangle_{H}\big]=\mathbb{E}\big[F\delta v\big]
  11. 𝔻 1 , 2 \mathbb{D}^{1,2}
  12. L := - δ D . L:=-\delta\circ\mathrm{D}.

Orthogonality_principle.html

  1. x ^ = H y + c \hat{x}=Hy+c
  2. x ^ \hat{x}
  3. E { ( x ^ - x ) y T } = 0 , E\{(\hat{x}-x)y^{T}\}=0,
  4. E { x ^ - x } = 0. E\{\hat{x}-x\}=0.
  5. σ x 2 . \sigma_{x}^{2}.
  6. y = x + w , y=x+w,
  7. σ w 2 . \sigma_{w}^{2}.
  8. x ^ = h y + c \hat{x}=hy+c
  9. x ^ = h y + c \hat{x}=hy+c
  10. 0 = E { ( x ^ - x ) y } 0=E\{(\hat{x}-x)y\}
  11. 0 = E { ( h x + h w + c - x ) ( x + w ) } 0=E\{(hx+hw+c-x)(x+w)\}
  12. 0 = h ( σ x 2 + σ w 2 ) + c m - σ x 2 0=h(\sigma_{x}^{2}+\sigma_{w}^{2})+cm-\sigma_{x}^{2}
  13. 0 = E { x ^ - x } 0=E\{\hat{x}-x\}
  14. 0 = E { h x + h w + c - x } 0=E\{hx+hw+c-x\}
  15. 0 = ( h - 1 ) m + c . 0=(h-1)m+c.
  16. h = σ x 2 - m 2 ( σ x 2 - m 2 ) + σ w 2 , c = σ w 2 ( σ x 2 - m 2 ) + σ w 2 m , h=\frac{\sigma_{x}^{2}-m^{2}}{(\sigma_{x}^{2}-m^{2})+\sigma_{w}^{2}},\quad c=% \frac{\sigma_{w}^{2}}{(\sigma_{x}^{2}-m^{2})+\sigma_{w}^{2}}m,
  17. x ^ = σ x 2 - m 2 ( σ x 2 - m 2 ) + σ w 2 y + σ w 2 ( σ x 2 - m 2 ) + σ w 2 m . \hat{x}=\frac{\sigma_{x}^{2}-m^{2}}{(\sigma_{x}^{2}-m^{2})+\sigma_{w}^{2}}y+% \frac{\sigma_{w}^{2}}{(\sigma_{x}^{2}-m^{2})+\sigma_{w}^{2}}m.
  18. σ w 2 \sigma_{w}^{2}
  19. σ x 2 - m 2 \sigma_{x}^{2}-m^{2}
  20. V V
  21. x , y = E { x H y } \langle x,y\rangle=E\{x^{H}y\}
  22. W W
  23. V V
  24. x ^ W \hat{x}\in W
  25. x V x\in V
  26. E x - x ^ 2 E\|x-\hat{x}\|^{2}
  27. x ^ \hat{x}
  28. x x
  29. V V
  30. x x
  31. y y
  32. W W
  33. y y
  34. W W
  35. x x
  36. x ^ \hat{x}
  37. W W
  38. e e
  39. W W
  40. W W
  41. V V
  42. x x
  43. V V
  44. x ^ W \hat{x}\in W
  45. W W
  46. E { ( x - x ^ ) y T } = 0 E\{(x-\hat{x})y^{T}\}=0
  47. y W . y\in W.
  48. x x
  49. x = x ^ + e x=\hat{x}+e\,
  50. x ^ = i c i p i \hat{x}=\sum_{i}c_{i}p_{i}
  51. x x
  52. W W
  53. p 1 , p 2 , . p_{1},p_{2},\ldots.
  54. c i c_{i}
  55. e 2 \left\|e\right\|^{2}
  56. x - i c i p i , p j = 0. \left\langle x-\sum_{i}c_{i}p_{i},p_{j}\right\rangle=0.
  57. x , p j = i c i p i , p j = i c i p i , p j . \left\langle x,p_{j}\right\rangle=\left\langle\sum_{i}c_{i}p_{i},p_{j}\right% \rangle=\sum_{i}c_{i}\left\langle p_{i},p_{j}\right\rangle.
  58. n n
  59. p i p_{i}
  60. [ x , p 1 x , p 2 x , p n ] = [ p 1 , p 1 p 2 , p 1 p n , p 1 p 1 , p 2 p 2 , p 2 p n , p 2 p 1 , p n p 2 , p n p n , p n ] [ c 1 c 2 c n ] . \begin{bmatrix}\left\langle x,p_{1}\right\rangle\\ \left\langle x,p_{2}\right\rangle\\ \vdots\\ \left\langle x,p_{n}\right\rangle\end{bmatrix}=\begin{bmatrix}\left\langle p_{% 1},p_{1}\right\rangle&\left\langle p_{2},p_{1}\right\rangle&\cdots&\left% \langle p_{n},p_{1}\right\rangle\\ \left\langle p_{1},p_{2}\right\rangle&\left\langle p_{2},p_{2}\right\rangle&% \cdots&\left\langle p_{n},p_{2}\right\rangle\\ \vdots&\vdots&\ddots&\vdots\\ \left\langle p_{1},p_{n}\right\rangle&\left\langle p_{2},p_{n}\right\rangle&% \cdots&\left\langle p_{n},p_{n}\right\rangle\end{bmatrix}\begin{bmatrix}c_{1}% \\ c_{2}\\ \vdots\\ c_{n}\end{bmatrix}.
  61. p i p_{i}
  62. [ c 1 c 2 c n ] = [ p 1 , p 1 p 2 , p 1 p n , p 1 p 1 , p 2 p 2 , p 2 p n , p 2 p 1 , p n p 2 , p n p n , p n ] - 1 [ x , p 1 x , p 2 x , p n ] , \begin{bmatrix}c_{1}\\ c_{2}\\ \vdots\\ c_{n}\end{bmatrix}=\begin{bmatrix}\left\langle p_{1},p_{1}\right\rangle&\left% \langle p_{2},p_{1}\right\rangle&\cdots&\left\langle p_{n},p_{1}\right\rangle% \\ \left\langle p_{1},p_{2}\right\rangle&\left\langle p_{2},p_{2}\right\rangle&% \cdots&\left\langle p_{n},p_{2}\right\rangle\\ \vdots&\vdots&\ddots&\vdots\\ \left\langle p_{1},p_{n}\right\rangle&\left\langle p_{2},p_{n}\right\rangle&% \cdots&\left\langle p_{n},p_{n}\right\rangle\end{bmatrix}^{-1}\begin{bmatrix}% \left\langle x,p_{1}\right\rangle\\ \left\langle x,p_{2}\right\rangle\\ \vdots\\ \left\langle x,p_{n}\right\rangle\end{bmatrix},
  63. c i c_{i}

OSA-UCS.html

  1. Δ U C S = ( 2 L 2 - 2 L 1 ) 2 + ( j 2 - j 1 ) 2 + ( g 2 - g 1 ) 2 = 2 Δ L 2 + Δ j 2 + Δ g 2 \Delta UCS=\sqrt{(\sqrt{2}L_{2}-\sqrt{2}L_{1})^{2}+(j_{2}-j_{1})^{2}+(g_{2}-g_% {1})^{2}}=\sqrt{2\Delta L^{2}+\Delta j^{2}+\Delta g^{2}}
  2. K = 4.4934 x 2 + 4.3034 y 2 - 4.276 x y - 1.3744 x - 2.5643 y + 1.8103 K=4.4934x^{2}+4.3034y^{2}-4.276xy-1.3744x-2.5643y+1.8103
  3. Y 0 = K * Y Y_{0}=K*Y
  4. L = 1 2 ( 5.9 ( Y 0 1 / 3 - 2 3 + 0.042 ( Y 0 - 30 ) 1 / 3 ) - 14.3993 ) L=\frac{1}{\sqrt{2}}(5.9(Y_{0}^{1/3}-\frac{2}{3}+0.042(Y_{0}-30)^{1/3})-14.3993)
  5. C = 1 + 0.042 ( Y 0 - 30 ) 1 / 3 Y 0 1 / 3 - 2 3 C=1+\frac{0.042(Y_{0}-30)^{1/3}}{Y_{0}^{1/3}-\frac{2}{3}}
  6. [ R G B ] = [ 0.7990 0.4194 - 0.1648 - 0.4493 1.3265 0.0927 - 0.1149 0.3394 0.7170 ] [ X Y Z ] \begin{bmatrix}R\\ G\\ B\end{bmatrix}=\begin{bmatrix}0.7990&0.4194&-0.1648\\ -0.4493&1.3265&0.0927\\ -0.1149&0.3394&0.7170\end{bmatrix}\begin{bmatrix}X\\ Y\\ Z\end{bmatrix}
  7. a = - 13.7 R 1 / 3 + 17.7 G 1 / 3 - 4 B 1 / 3 a=-13.7R^{1/3}+17.7G^{1/3}-4B^{1/3}
  8. b = 1.7 R 1 / 3 + 8 G 1 / 3 - 9.7 B 1 / 3 b=1.7R^{1/3}+8G^{1/3}-9.7B^{1/3}
  9. g = C * a g=C*a
  10. j = C * b j=C*b

Oscar_Lanford.html

  1. g ( x ) = T ( g ) ( x ) = - λ g ( g ( - λ x ) ) , g ( - 2 ) = 1 , g ′′ ( 0 ) < 0 , λ = - g ( 1 ) > 0 g(x)=T(g)(x)=-\lambda g(g(-\lambda x)),g(-2)=1,g^{\prime\prime}(0)<0,\lambda=-% g(1)>0
  2. f ( x ) = c x ( 1 - x ) f(x)=cx(1-x)
  3. b ( n ) = a ( n + 1 ) - a ( n ) b(n)=a(n+1)-a(n)
  4. f ( x ) = c sin ( π x ) f(x)=c\sin(\pi x)
  5. lim n b ( n ) / b ( n + 1 ) \lim_{n\to\infty}b(n)/b(n+1)
  6. d = 4.6692016091029... d=4.6692016091029...

Oseen_equations.html

  1. - ρ 𝐔 𝐮 = - p + μ 2 𝐮 , 𝐮 = 0 , \begin{aligned}\displaystyle-\rho\mathbf{U}\cdot\nabla\mathbf{u}&\displaystyle% =-\nabla p\,+\,\mu\nabla^{2}\mathbf{u},\\ \displaystyle\nabla\cdot\mathbf{u}&\displaystyle=0,\end{aligned}
  2. 𝐮 \displaystyle\mathbf{u}

Osipkov–Merritt_model.html

  1. f = f ( E ) f=f(E)
  2. f = f ( Q ) = f ( E + J 2 / 2 r a 2 ) f=f(Q)=f(E+J^{2}/2r_{a}^{2})
  3. 2 Q = v r 2 + ( 1 + r 2 / r a 2 ) v t 2 + 2 Φ ( r ) 2Q=v_{r}^{2}+(1+r^{2}/r_{a}^{2})v_{t}^{2}+2\Phi(r)
  4. ρ ( r ) = 2 π f ( E , J ) v t d v t d v r \rho(r)=2\pi\int\int f(E,J)v_{t}dv_{t}dv_{r}
  5. ρ ( r ) = 2 π r 2 Φ 0 d Q f ( Q ) 0 2 r 2 ( Q - Φ ) / ( 1 + r 2 / r a 2 ) d J 2 [ 2 ( Q - Φ ) - ( J 2 / r 2 ) ( 1 + r 2 / r a 2 ) ] - 1 / 2 \rho(r)={2\pi\over r^{2}}\int_{\Phi}^{0}dQf(Q)\int_{0}^{2r^{2}(Q-\Phi)/(1+r^{2% }/r_{a}^{2})}dJ^{2}\left[2(Q-\Phi)-(J^{2}/r^{2})(1+r^{2}/r_{a}^{2})\right]^{-1% /2}
  6. ρ ( r ) = 4 π 1 + r 2 / r a 2 Φ 0 d Q 2 ( Q - Φ ) f ( Q ) . \rho(r)={4\pi\over 1+r^{2}/r_{a}^{2}}\int_{\Phi}^{0}dQ\sqrt{2(Q-\Phi)}f(Q).
  7. f ( Q ) = 2 4 π 2 d d Q Q 0 d Φ Φ - Q d ρ d Φ , ρ ( Φ ) = [ 1 + r ( Φ ) 2 / r a 2 ] ρ [ r ( Φ ) ] . f(Q)={\sqrt{2}\over 4\pi^{2}}{d\over dQ}\int_{Q}^{0}{d\Phi\over\sqrt{\Phi-Q}}{% d\rho^{^{\prime}}\over d\Phi},\ \ \ \ \ \rho^{^{\prime}}(\Phi)=\left[1+r(\Phi)% ^{2}/r_{a}^{2}\right]\rho\left[r(\Phi)\right].
  8. σ r 2 σ t 2 = 1 + r 2 r a 2 . {\sigma_{r}^{2}\over\sigma_{t}^{2}}=1+{r^{2}\over r_{a}^{2}}.
  9. σ r σ t \sigma_{r}\gg\sigma_{t}
  10. r r a r\gg r_{a}
  11. σ r σ t \sigma_{r}\approx\sigma_{t}
  12. r r a r\ll r_{a}
  13. f = f ( E - J 2 / 2 r a 2 ) f=f(E-J^{2}/2r_{a}^{2})

Osmotic_coefficient.html

  1. φ = μ A * - μ A R T M A i b i \varphi=\frac{\mu_{A}^{*}-\mu_{A}}{RTM_{A}\sum_{i}b_{i}}\,
  2. φ = - μ A * - μ A R T ln x A \varphi=-\frac{\mu_{A}^{*}-\mu_{A}}{RT\ln x_{A}}\,
  3. μ A * \mu_{A}^{*}
  4. μ A \mu_{A}
  5. ln x A = - ln ( 1 + M A i b i ) - M A i b i , \ln x_{A}=-\ln(1+M_{A}\sum_{i}b_{i})\approx-M_{A}\sum_{i}b_{i},
  6. G e x G_{ex}
  7. R T b ( 1 - φ ) = G e x - b d G e x d b RTb(1-\varphi)=G_{ex}-b\frac{dG_{ex}}{db}
  8. R T ln γ = d G e x d b RT\ln\gamma=\frac{dG_{ex}}{db}
  9. d ( ( φ - 1 ) b ) = b d ln γ d((\varphi-1)b)=bd\ln\gamma
  10. ( φ - 1 ) i b i (\varphi-1)\sum_{i}b_{i}
  11. - 2 3 A I 3 / 2 -\frac{2}{3}AI^{3/2}

Oswald_efficiency_number.html

  1. C D = C D 0 + ( C L ) 2 π e 0 A R C_{D}=C_{D_{0}}+\frac{(C_{L})^{2}}{\pi e_{0}AR}
  2. C D C_{D}\;
  3. C D 0 C_{D_{0}}\;
  4. C L C_{L}\;
  5. π \pi\;
  6. e 0 e_{0}\;
  7. A R AR
  8. C L C_{L}
  9. C L C_{L}
  10. C D = c d 0 + c d 2 ( C L ) 2 + ( C L ) 2 π e A R C_{D}=c_{d_{0}}+c_{d_{2}}(C_{L})^{2}+\frac{(C_{L})^{2}}{\pi eAR}
  11. c d 0 c_{d_{0}}\;
  12. c d 2 c_{d_{2}}\;
  13. e e\;
  14. C D C_{D}
  15. C D 0 = c d 0 C_{D_{0}}=c_{d_{0}}
  16. 1 e 0 = 1 e + π A R c d 2 \frac{1}{e_{0}}=\frac{1}{e}+\pi ARc_{d_{2}}
  17. c d 2 > 0 c_{d_{2}}>0
  18. e 0 < e e_{0}<e

Overlap–save_method.html

  1. x [ n ] x[n]
  2. h [ n ] h[n]
  3. y [ n ] = x [ n ] * h [ n ] = def m = - h [ m ] x [ n - m ] = m = 1 M h [ m ] x [ n - m ] , y[n]=x[n]*h[n]\ \stackrel{\mathrm{def}}{=}\ \sum_{m=-\infty}^{\infty}h[m]\cdot x% [n-m]=\sum_{m=1}^{M}h[m]\cdot x[n-m],\,
  4. x k [ n ] = def { x [ n + k L ] 1 n L + M - 1 0 otherwise . x_{k}[n]\ \stackrel{\mathrm{def}}{=}\begin{cases}x[n+kL]&1\leq n\leq L+M-1\\ 0&\textrm{otherwise}.\end{cases}
  5. y k [ n ] = def x k [ n ] * h [ n ] y_{k}[n]\ \stackrel{\mathrm{def}}{=}\ x_{k}[n]*h[n]\,
  6. y [ n ] = m = 1 M h [ m ] x k [ n - k L - m ] = x k [ n - k L ] * h [ n ] = def y k [ n - k L ] . \begin{aligned}\displaystyle y[n]=\sum_{m=1}^{M}h[m]\cdot x_{k}[n-kL-m]&% \displaystyle=x_{k}[n-kL]*h[n]\\ &\displaystyle\stackrel{\mathrm{def}}{=}\ y_{k}[n-kL].\end{aligned}
  7. x k , N [ n ] = def = - x k [ n - N ] , x_{k,N}[n]\ \stackrel{\mathrm{def}}{=}\ \sum_{\ell=-\infty}^{\infty}x_{k}[n-% \ell N],
  8. ( x k , N ) * h (x_{k,N})*h\,
  9. x k * h x_{k}*h\,
  10. x k [ n ] x_{k}[n]\,
  11. h [ n ] h[n]\,
  12. y k [ n ] = DFT - 1 ( DFT ( x k [ n ] ) DFT ( h [ n ] ) ) , y_{k}[n]=\scriptstyle\,\text{DFT}^{-1}\displaystyle(\ \scriptstyle\,\text{DFT}% \displaystyle(x_{k}[n])\cdot\scriptstyle\,\text{DFT}\displaystyle(h[n])\ ),
  13. N log 2 ( N ) + N N - M + 1 . \frac{N\log_{2}(N)+N}{N-M+1}.\,

Ōno's_lexical_law.html

  1. X 0 , x , X 1 X_{0},x,X_{1}
  2. Y 0 , y , Y 1 Y_{0},y,Y_{1}
  3. ( X 0 , Y 0 ) , ( x , y ) , ( X 1 , Y 1 ) (X_{0},Y_{0}),(x,y),(X_{1},Y_{1})
  4. y - Y 0 Y 1 - Y 0 x - X 0 X 1 - X 0 \frac{y-Y_{0}}{Y_{1}-Y_{0}}\approx\frac{x-X_{0}}{X_{1}-X_{0}}

P-adic_order.html

  1. p p
  2. p p
  3. p p
  4. n n
  5. p p
  6. n n
  7. p p
  8. 0
  9. \infty
  10. p p
  11. n n
  12. n / d n/d
  13. n n
  14. d d
  15. p p
  16. n / d n/d
  17. p p
  18. n n
  19. p p
  20. n n
  21. p p
  22. d d
  23. p p
  24. d d
  25. ν p : 𝐙 𝐍 \nu_{p}:\,\textbf{Z}\to\,\textbf{N}
  26. ν p ( n ) = { max { v : p v n } if n 0 if n = 0 \nu_{p}(n)=\begin{cases}\mathrm{max}\{v\in\mathbb{N}:p^{v}\mid n\}&\,\text{if % }n\neq 0\\ \infty&\,\text{if }n=0\end{cases}
  27. ν p : 𝐐 𝐙 \nu_{p}:\,\textbf{Q}\to\,\textbf{Z}
  28. ν p ( a b ) = ν p ( a ) - ν p ( b ) . \nu_{p}\left(\frac{a}{b}\right)=\nu_{p}(a)-\nu_{p}(b).
  29. ν p ( m n ) = ν p ( m ) + ν p ( n ) . \nu_{p}(m\cdot n)=\nu_{p}(m)+\nu_{p}(n)~{}.
  30. ν p ( m + n ) inf { ν p ( m ) , ν p ( n ) } . \nu_{p}(m+n)\geq\inf\{\nu_{p}(m),\nu_{p}(n)\}.
  31. ν p ( m ) ν p ( n ) \nu_{p}(m)\neq\nu_{p}(n)
  32. ν p ( m + n ) = inf { ν p ( m ) , ν p ( n ) } . \nu_{p}(m+n)=\inf\{\nu_{p}(m),\nu_{p}(n)\}.
  33. inf \inf
  34. | | p : 𝐐 𝐑 |\,\cdot\,|_{p}:\,\textbf{Q}\to\,\textbf{R}
  35. | x | p = { p - ν p ( x ) if x 0 0 if x = 0 |x|_{p}=\begin{cases}p^{-\nu_{p}(x)}&\,\text{if }x\neq 0\\ 0&\,\text{if }x=0\end{cases}
  36. | a | p 0 \displaystyle|a|_{p}\geq 0
  37. d : 𝐐 × 𝐐 𝐑 d:\,\textbf{Q}\times\,\textbf{Q}\to\,\textbf{R}
  38. d ( x , y ) = | x - y | p . d(x,y)=|x-y|_{p}.

P-adically_closed_field.html

  1. v ( p ) = 1 v(p)=1
  2. ( F , w ) (F,w)
  3. w ( x ) = v ( x ) w(x)=v(x)
  4. { x F : w ( x ) 0 } \{x\in F:w(x)\geq 0\}
  5. { x F : w ( x ) > 0 } \{x\in F:w(x)>0\}
  6. w ( 2 + i ) = 1 w(2+i)=1
  7. w ( 2 - i ) = 0 w(2-i)=0
  8. X 2 + 1 X^{2}+1
  9. w ( 3 ) = 1 w(3)=1
  10. ( F , w ) (F,w)
  11. v ( π ) = 1 v(\pi)=1
  12. v ( p ) v(p)
  13. v ( π ) = 1 v(\pi)=1
  14. 𝔭 \mathfrak{p}
  15. 𝔭 \mathfrak{p}
  16. γ ( z ) = 1 π z q - z ( z q - z ) 2 - 1 \gamma(z)=\frac{1}{\pi}\,\frac{z^{q}-z}{(z^{q}-z)^{2}-1}
  17. z q - z ± 1 z^{q}-z\neq\pm 1
  18. γ ( z ) \gamma(z)
  19. 1 π \frac{1}{\pi}
  20. - 1 -1
  21. p \mathbb{Q}_{p}

P-Laplacian.html

  1. p p
  2. 1 < p < 1<p<\infty
  3. ( | u | p - 2 u ) . \nabla\cdot(|\nabla u|^{p-2}\nabla u).
  4. | | p - 2 |\nabla\cdot|^{p-2}
  5. | u | p - 2 = [ ( u x 1 ) 2 + + ( u x n ) 2 ] p - 2 2 \quad|\nabla u|^{p-2}=\left[\textstyle\left(\frac{\partial u}{\partial x_{1}}% \right)^{2}+\cdots+\left(\frac{\partial u}{\partial x_{n}}\right)^{2}\right]^{% \frac{p-2}{2}}
  6. p = 2 p=2
  7. ( | u | p - 2 u ) = 0 \nabla\cdot(|\nabla u|^{p-2}\nabla u)=0
  8. Ω \Omega
  9. J ( u ) = | u | p d x J(u)=\int|\nabla u|^{p}\,dx
  10. W 1 , p ( Ω ) W^{1,p}(\Omega)

P108.html

  1. 𝔓 \mathfrak{P}

P111.html

  1. 𝔓 \mathfrak{P}

Palierne_equation.html

  1. G * = G m * ( 1 + 5 ϕ H * ) G^{*}=G^{*}_{m}(1+5\phi H^{*})
  2. G * G^{*}
  3. G m * G^{*}_{m}
  4. ϕ \phi
  5. H * H^{*}
  6. H * = ( G d * - G m * ) ( 19 G d * + 16 G m * ) + ( 4 σ / R ) ( 5 G d * + 2 G m * ) ( 2 G d * + 3 G m * ) ( 19 G d * + 16 G m * ) + ( 40 σ / R ) ( G d * + G m * ) H^{*}=\frac{(G^{*}_{d}-G^{*}_{m})(19G^{*}_{d}+16G^{*}_{m})+(4\sigma/R)(5G^{*}_% {d}+2G^{*}_{m})}{(2G^{*}_{d}+3G^{*}_{m})(19G^{*}_{d}+16G^{*}_{m})+(40\sigma/R)% (G^{*}_{d}+G^{*}_{m})}
  7. G d * G^{*}_{d}
  8. σ \sigma
  9. R R
  10. H * H^{*}
  11. H * = G d * - G m * 2 G d * + 3 G m * H^{*}=\frac{G^{*}_{d}-G^{*}_{m}}{2G^{*}_{d}+3G^{*}_{m}}
  12. ϕ \phi
  13. G * = G m * 1 + 3 ϕ H * 1 - 2 ϕ H * G^{*}=G^{*}_{m}\frac{1+3\phi H^{*}}{1-2\phi H^{*}}

Pappus_chain.html

  1. 𝐏 n 𝐔 ¯ + 𝐏 n 𝐕 ¯ = ( r U + r n ) + ( r V - r n ) = r U + r V \overline{\mathbf{P}_{n}\mathbf{U}}+\overline{\mathbf{P}_{n}\mathbf{V}}=\left(% r_{U}+r_{n}\right)+\left(r_{V}-r_{n}\right)=r_{U}+r_{V}
  2. ( x n , y n ) = ( r ( 1 + r ) 2 [ n 2 ( 1 - r ) 2 + r ] , n r ( 1 - r ) n 2 ( 1 - r ) 2 + r ) \left(x_{n},y_{n}\right)=\left(\frac{r(1+r)}{2[n^{2}(1-r)^{2}+r]}~{},~{}\frac{% nr(1-r)}{n^{2}(1-r)^{2}+r}\right)
  3. r n = ( 1 - r ) r 2 [ n 2 ( 1 - r ) 2 + r ] r_{n}=\frac{(1-r)r}{2[n^{2}(1-r)^{2}+r]}

Papyrus_10.html

  1. 𝔓 \mathfrak{P}

Papyrus_100.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}
  4. 𝔓 \mathfrak{P}
  5. 𝔓 \mathfrak{P}

Papyrus_101.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}

Papyrus_102.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}

Papyrus_103.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}

Papyrus_105.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}

Papyrus_106.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}
  4. 𝔓 \mathfrak{P}
  5. 𝔓 \mathfrak{P}
  6. 𝔓 \mathfrak{P}
  7. 𝔓 \mathfrak{P}
  8. 𝔐 \mathfrak{M}
  9. 𝔓 \mathfrak{P}
  10. 𝔓 \mathfrak{P}
  11. 𝔓 \mathfrak{P}
  12. 𝔓 \mathfrak{P}
  13. 𝔓 \mathfrak{P}
  14. 𝔐 \mathfrak{M}
  15. 𝔓 \mathfrak{P}
  16. 𝔓 \mathfrak{P}
  17. 𝔓 \mathfrak{P}
  18. 𝔓 \mathfrak{P}
  19. 𝔓 \mathfrak{P}
  20. 𝔓 \mathfrak{P}
  21. 𝔓 \mathfrak{P}
  22. 𝔓 \mathfrak{P}
  23. 𝔓 \mathfrak{P}
  24. 𝔓 \mathfrak{P}
  25. 𝔓 \mathfrak{P}
  26. 𝔓 \mathfrak{P}
  27. 𝔐 \mathfrak{M}
  28. 𝔓 \mathfrak{P}
  29. 𝔓 \mathfrak{P}
  30. 𝔓 \mathfrak{P}
  31. 𝔓 \mathfrak{P}
  32. 𝔐 \mathfrak{M}
  33. 𝔓 \mathfrak{P}
  34. 𝔓 \mathfrak{P}
  35. υ ς ¯ \overline{υς}
  36. 𝔓 \mathfrak{P}
  37. 𝔓 \mathfrak{P}
  38. 𝔓 \mathfrak{P}
  39. 𝔓 \mathfrak{P}
  40. 𝔓 \mathfrak{P}
  41. 𝔐 \mathfrak{M}
  42. 𝔓 \mathfrak{P}
  43. 𝔓 \mathfrak{P}
  44. 𝔓 \mathfrak{P}
  45. 𝔓 \mathfrak{P}
  46. 𝔐 \mathfrak{M}
  47. 𝔓 \mathfrak{P}
  48. 𝔓 \mathfrak{P}
  49. 𝔓 \mathfrak{P}
  50. 𝔐 \mathfrak{M}
  51. 𝔓 \mathfrak{P}
  52. 𝔓 \mathfrak{P}
  53. 𝔓 \mathfrak{P}
  54. 𝔓 \mathfrak{P}
  55. 𝔓 \mathfrak{P}
  56. 𝔓 \mathfrak{P}
  57. 𝔐 \mathfrak{M}
  58. 𝔓 \mathfrak{P}
  59. 𝔓 \mathfrak{P}
  60. 𝔓 \mathfrak{P}
  61. 𝔓 \mathfrak{P}
  62. 𝔐 \mathfrak{M}
  63. 𝔓 \mathfrak{P}
  64. 𝔓 \mathfrak{P}
  65. 𝔓 \mathfrak{P}
  66. 𝔐 \mathfrak{M}
  67. 𝔓 \mathfrak{P}
  68. 𝔓 \mathfrak{P}
  69. 𝔓 \mathfrak{P}
  70. 𝔓 \mathfrak{P}
  71. 𝔐 \mathfrak{M}
  72. 𝔓 \mathfrak{P}
  73. 𝔓 \mathfrak{P}
  74. 𝔓 \mathfrak{P}
  75. 𝔐 \mathfrak{M}
  76. 𝔓 \mathfrak{P}
  77. 𝔓 \mathfrak{P}
  78. 𝔓 \mathfrak{P}
  79. 𝔓 \mathfrak{P}
  80. 𝔐 \mathfrak{M}
  81. 𝔓 \mathfrak{P}
  82. 𝔓 \mathfrak{P}
  83. 𝔓 \mathfrak{P}
  84. 𝔐 \mathfrak{M}
  85. 𝔓 \mathfrak{P}
  86. 𝔓 \mathfrak{P}
  87. 𝔓 \mathfrak{P}
  88. 𝔓 \mathfrak{P}
  89. 𝔐 \mathfrak{M}
  90. 𝔓 \mathfrak{P}
  91. 𝔓 \mathfrak{P}
  92. 𝔓 \mathfrak{P}
  93. 𝔓 \mathfrak{P}
  94. 𝔓 \mathfrak{P}
  95. 𝔐 \mathfrak{M}
  96. 𝔓 \mathfrak{P}
  97. 𝔓 \mathfrak{P}
  98. 𝔓 \mathfrak{P}
  99. 𝔐 \mathfrak{M}
  100. 𝔓 \mathfrak{P}
  101. 𝔓 \mathfrak{P}
  102. 𝔓 \mathfrak{P}
  103. 𝔐 \mathfrak{M}
  104. 𝔓 \mathfrak{P}
  105. 𝔓 \mathfrak{P}
  106. 𝔓 \mathfrak{P}
  107. 𝔓 \mathfrak{P}
  108. 𝔓 \mathfrak{P}
  109. 𝔐 \mathfrak{M}
  110. 𝔓 \mathfrak{P}
  111. 𝔓 \mathfrak{P}
  112. 𝔓 \mathfrak{P}
  113. 𝔐 \mathfrak{M}
  114. 𝔓 \mathfrak{P}
  115. 𝔓 \mathfrak{P}
  116. 𝔓 \mathfrak{P}
  117. 𝔐 \mathfrak{M}
  118. 𝔓 \mathfrak{P}
  119. 𝔓 \mathfrak{P}
  120. 𝔓 \mathfrak{P}
  121. 𝔐 \mathfrak{M}
  122. 𝔓 \mathfrak{P}
  123. 𝔓 \mathfrak{P}
  124. 𝔓 \mathfrak{P}
  125. 𝔓 \mathfrak{P}
  126. 𝔐 \mathfrak{M}
  127. 𝔓 \mathfrak{P}
  128. 𝔓 \mathfrak{P}
  129. 𝔓 \mathfrak{P}
  130. 𝔐 \mathfrak{M}
  131. 𝔓 \mathfrak{P}
  132. 𝔓 \mathfrak{P}
  133. 𝔓 \mathfrak{P}
  134. 𝔐 \mathfrak{M}
  135. 𝔓 \mathfrak{P}
  136. 𝔓 \mathfrak{P}

Papyrus_107.html

  1. 𝔓 \mathfrak{P}
  2. υ ς ¯ \overline{υς}
  3. 𝔓 \mathfrak{P}
  4. υ ς ¯ \overline{υς}
  5. 𝔐 \mathfrak{M}
  6. υ ς ¯ \overline{υς}
  7. υ ς ¯ \overline{υς}
  8. 𝔓 \mathfrak{P}
  9. 𝔐 \mathfrak{M}
  10. 𝔓 \mathfrak{P}
  11. 𝔐 \mathfrak{M}
  12. 𝔓 \mathfrak{P}
  13. 𝔓 \mathfrak{P}
  14. 𝔐 \mathfrak{M}
  15. 𝔓 \mathfrak{P}
  16. 𝔓 \mathfrak{P}
  17. 𝔓 \mathfrak{P}
  18. 𝔐 \mathfrak{M}
  19. 𝔓 \mathfrak{P}
  20. 𝔓 \mathfrak{P}
  21. 𝔓 \mathfrak{P}
  22. 𝔐 \mathfrak{M}
  23. 𝔓 \mathfrak{P}
  24. 𝔓 \mathfrak{P}
  25. 𝔓 \mathfrak{P}

Papyrus_108.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}
  4. 𝔓 \mathfrak{P}
  5. 𝔓 \mathfrak{P}
  6. 𝔓 \mathfrak{P}
  7. 𝔓 \mathfrak{P}
  8. 𝔐 \mathfrak{M}
  9. 𝔓 \mathfrak{P}
  10. 𝔓 \mathfrak{P}
  11. 𝔓 \mathfrak{P}
  12. 𝔓 \mathfrak{P}
  13. 𝔓 \mathfrak{P}
  14. 𝔐 \mathfrak{M}
  15. 𝔓 \mathfrak{P}
  16. 𝔓 \mathfrak{P}
  17. 𝔐 \mathfrak{M}
  18. 𝔓 \mathfrak{P}
  19. 𝔓 \mathfrak{P}
  20. 𝔐 \mathfrak{M}
  21. 𝔓 \mathfrak{P}
  22. 𝔓 \mathfrak{P}
  23. 𝔐 \mathfrak{M}
  24. 𝔓 \mathfrak{P}
  25. 𝔐 \mathfrak{M}
  26. 𝔓 \mathfrak{P}
  27. 𝔐 \mathfrak{M}
  28. 𝔓 \mathfrak{P}
  29. 𝔓 \mathfrak{P}
  30. 𝔓 \mathfrak{P}

Papyrus_109.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}
  4. 𝔐 \mathfrak{M}
  5. 𝔓 \mathfrak{P}
  6. 𝔓 \mathfrak{P}
  7. 𝔐 \mathfrak{M}
  8. 𝔓 \mathfrak{P}
  9. 𝔐 \mathfrak{M}
  10. 𝔓 \mathfrak{P}
  11. 𝔐 \mathfrak{M}
  12. 𝔓 \mathfrak{P}
  13. 𝔐 \mathfrak{M}
  14. 𝔓 \mathfrak{P}
  15. 𝔓 \mathfrak{P}

Papyrus_11.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}

Papyrus_110.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}
  4. 𝔓 \mathfrak{P}
  5. 𝔓 \mathfrak{P}
  6. 𝔓 \mathfrak{P}
  7. 𝔓 \mathfrak{P}
  8. 𝔐 \mathfrak{M}
  9. 𝔓 \mathfrak{P}
  10. 𝔓 \mathfrak{P}
  11. 𝔓 \mathfrak{P}
  12. 𝔓 \mathfrak{P}
  13. 𝔐 \mathfrak{M}
  14. 𝔓 \mathfrak{P}
  15. 𝔐 \mathfrak{M}
  16. 𝔓 \mathfrak{P}
  17. 𝔓 \mathfrak{P}
  18. 𝔐 \mathfrak{M}
  19. 𝔓 \mathfrak{P}
  20. 𝔓 \mathfrak{P}
  21. 𝔓 \mathfrak{P}

Papyrus_111.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. ι η υ ¯ \overline{ιηυ}
  4. 𝔓 \mathfrak{P}
  5. 𝔓 \mathfrak{P}
  6. 𝔐 \mathfrak{M}
  7. 𝔓 \mathfrak{P}
  8. 𝔓 \mathfrak{P}
  9. 𝔐 \mathfrak{M}
  10. 𝔓 \mathfrak{P}
  11. 𝔓 \mathfrak{P}
  12. 𝔐 \mathfrak{M}
  13. 𝔓 \mathfrak{P}
  14. 𝔓 \mathfrak{P}

Papyrus_112.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}

Papyrus_113.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}

Papyrus_114.html

  1. 𝔓 \mathfrak{P}
  2. Θ Σ ¯ \overline{ΘΣ}
  3. Θ Σ ¯ \overline{ΘΣ}
  4. 𝔓 \mathfrak{P}
  5. Θ Σ ¯ \overline{ΘΣ}
  6. Θ Σ ¯ \overline{ΘΣ}
  7. 𝔓 \mathfrak{P}
  8. 𝔓 \mathfrak{P}
  9. 𝔓 \mathfrak{P}
  10. 𝔐 \mathfrak{M}
  11. 𝔓 \mathfrak{P}

Papyrus_116.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}

Papyrus_117.html

  1. 𝔓 \mathfrak{P}

Papyrus_118.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}

Papyrus_119.html

  1. 𝔓 \mathfrak{P}

Papyrus_12.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}

Papyrus_120.html

  1. 𝔓 \mathfrak{P}

Papyrus_121.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}

Papyrus_122.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}

Papyrus_123.html

  1. 𝔓 \mathfrak{P}

Papyrus_124.html

  1. 𝔓 \mathfrak{P}

Papyrus_14.html

  1. 𝔓 \mathfrak{P}

Papyrus_15.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}

Papyrus_16.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}

Papyrus_17.html

  1. 𝔓 \mathfrak{P}

Papyrus_18.html

  1. 𝔓 \mathfrak{P}
  2. Ι Η ¯ \overline{ΙΗ}
  3. Χ Ρ ¯ \overline{ΧΡ}
  4. θ υ ¯ \overline{θυ}
  5. θ ω ¯ \overline{θω}

Papyrus_19.html

  1. 𝔓 \mathfrak{P}

Papyrus_20.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}
  4. 𝔓 \mathfrak{P}
  5. 𝔓 \mathfrak{P}

Papyrus_21.html

  1. 𝔓 \mathfrak{P}

Papyrus_22.html

  1. 𝔓 \mathfrak{P}
  2. Π Σ ¯ \overline{ΠΣ}
  3. Π Ν Α ¯ \overline{ΠΝΑ}
  4. Π Ρ Σ ¯ \overline{ΠΡΣ}
  5. Π Ρ Α ¯ \overline{ΠΡΑ}
  6. Ι Η Σ ¯ \overline{ΙΗΣ}
  7. Α Ν Ο Σ ¯ \overline{ΑΝΟΣ}

Papyrus_23.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}

Papyrus_25.html

  1. 𝔓 \mathfrak{P}

Papyrus_26.html

  1. 𝔓 \mathfrak{P}

Papyrus_29.html

  1. 𝔓 \mathfrak{P}

Papyrus_30.html

  1. 𝔓 \mathfrak{P}

Papyrus_31.html

  1. 𝔓 \mathfrak{P}

Papyrus_32.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}
  4. 𝔓 \mathfrak{P}

Papyrus_33.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}

Papyrus_34.html

  1. 𝔓 \mathfrak{P}

Papyrus_35.html

  1. 𝔓 \mathfrak{P}

Papyrus_36.html

  1. 𝔓 \mathfrak{P}

Papyrus_38.html

  1. 𝔓 \mathfrak{P}

Papyrus_39.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}
  4. 𝔓 \mathfrak{P}
  5. 𝔓 \mathfrak{P}

Papyrus_40.html

  1. 𝔓 \mathfrak{P}

Papyrus_41.html

  1. 𝔓 \mathfrak{P}

Papyrus_43.html

  1. 𝔓 \mathfrak{P}

Papyrus_44.html

  1. 𝔓 \mathfrak{P}

Papyrus_47.html

  1. 𝔓 \mathfrak{P}

Papyrus_48.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}
  4. 𝔓 \mathfrak{P}

Papyrus_49.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}
  4. 𝔓 \mathfrak{P}
  5. 𝔓 \mathfrak{P}
  6. 𝔓 \mathfrak{P}
  7. 𝔓 46 \mathfrak{P}^{46}
  8. 𝔓 46 \mathfrak{P}^{46}
  9. 𝔓 46 \mathfrak{P}^{46}
  10. 𝔓 46 \mathfrak{P}^{46}
  11. 𝔓 \mathfrak{P}
  12. 𝔓 \mathfrak{P}

Papyrus_50.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}

Papyrus_51.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}
  4. 𝔓 \mathfrak{P}

Papyrus_53.html

  1. 𝔓 \mathfrak{P}

Papyrus_54.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}
  4. 𝔓 \mathfrak{P}
  5. 𝔓 \mathfrak{P}

Papyrus_55.html

  1. 𝔓 \mathfrak{P}

Papyrus_56.html

  1. 𝔓 \mathfrak{P}

Papyrus_57.html

  1. 𝔓 \mathfrak{P}

Papyrus_59.html

  1. 𝔓 \mathfrak{P}

Papyrus_6.html

  1. 𝔓 \mathfrak{P}

Papyrus_60.html

  1. 𝔓 \mathfrak{P}

Papyrus_61.html

  1. 𝔓 \mathfrak{P}

Papyrus_62.html

  1. 𝔓 \mathfrak{P}

Papyrus_63.html

  1. 𝔓 \mathfrak{P}

Papyrus_65.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}

Papyrus_68.html

  1. 𝔓 \mathfrak{P}

Papyrus_69.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}

Papyrus_7.html

  1. 𝔓 \mathfrak{P}

Papyrus_70.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}
  4. 𝔓 \mathfrak{P}

Papyrus_71.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}

Papyrus_72.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. Δ Υ Μ Ι ¯ \overline{ΔΥΜΙ}
  4. ρ ρ α ¯ \overline{ρρα}
  5. ρ α α ¯ \overline{ραα}
  6. Ν ω ε ¯ \overline{Νωε}
  7. Μ ι χ α η ς ¯ \overline{Μιχαης}
  8. ν ω ¯ \overline{νω}
  9. 𝔓 \mathfrak{P}
  10. 𝔓 \mathfrak{P}
  11. 𝔓 \mathfrak{P}

Papyrus_73.html

  1. 𝔓 \mathfrak{P}

Papyrus_74.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}

Papyrus_76.html

  1. 𝔓 \mathfrak{P}

Papyrus_77.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}

Papyrus_78.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}
  4. 𝔓 \mathfrak{P}

Papyrus_79.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}

Papyrus_8.html

  1. 𝔓 \mathfrak{P}

Papyrus_80.html

  1. 𝔓 \mathfrak{P}

Papyrus_81.html

  1. 𝔓 \mathfrak{P}

Papyrus_82.html

  1. 𝔓 \mathfrak{P}

Papyrus_83.html

  1. 𝔓 \mathfrak{P}

Papyrus_84.html

  1. 𝔓 \mathfrak{P}

Papyrus_85.html

  1. 𝔓 \mathfrak{P}

Papyrus_86.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}

Papyrus_87.html

  1. 𝔓 \mathfrak{P}

Papyrus_88.html

  1. 𝔓 \mathfrak{P}

Papyrus_89.html

  1. 𝔓 \mathfrak{P}

Papyrus_9.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}

Papyrus_90.html

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

Papyrus_91.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}

Papyrus_92.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}
  4. 𝔓 \mathfrak{P}

Papyrus_93.html

  1. 𝔓 \mathfrak{P}

Papyrus_94.html

  1. 𝔓 \mathfrak{P}

Papyrus_95.html

  1. 𝔓 \mathfrak{P}

Papyrus_96.html

  1. 𝔓 \mathfrak{P}

Papyrus_97.html

  1. 𝔓 \mathfrak{P}

Papyrus_98.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}

Papyrus_99.html

  1. 𝔓 \mathfrak{P}

Papyrus_Oxyrhynchus_208_+_1781.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}
  4. 𝔓 \mathfrak{P}
  5. 𝔓 \mathfrak{P}

Parallel-TEBD.html

  1. | Ψ |\Psi\rangle
  2. | Ψ = i 1 , . . , i N = 1 M α 1 , . . , α N - 1 = 0 χ Γ α 1 [ 1 ] i 1 λ α 1 [ 1 ] Γ α 1 α 2 [ 2 ] i 2 λ α 2 [ 2 ] Γ α 2 α 3 [ 3 ] i 3 λ α 3 [ 3 ] . . Γ α N - 2 α N - 1 [ N - 1 ] i N - 1 λ α N - 1 [ N - 1 ] Γ α N - 1 [ N ] i N | i 1 , i 2 , . . , i N - 1 , i N |\Psi\rangle=\sum\limits_{i_{1},..,i_{N}=1}^{M}\sum\limits_{\alpha_{1},..,% \alpha_{N-1}=0}^{\chi}\Gamma^{[1]i_{1}}_{\alpha_{1}}\lambda^{[1]}_{\alpha_{1}}% \Gamma^{[2]i_{2}}_{\alpha_{1}\alpha_{2}}\lambda^{[2]}_{\alpha_{2}}\Gamma^{[3]i% _{3}}_{\alpha_{2}\alpha_{3}}\lambda^{[3]}_{\alpha_{3}}\cdot..\cdot\Gamma^{[{N-% 1}]i_{N-1}}_{\alpha_{N-2}\alpha_{N-1}}\lambda^{[N-1]}_{\alpha_{N-1}}\Gamma^{[N% ]i_{N}}_{\alpha_{N-1}}|{i_{1},i_{2},..,i_{N-1},i_{N}}\rangle
  3. λ α l [ l ] \lambda^{[l]}_{\alpha_{l}}
  4. Γ s {\Gamma}^{\prime}s
  5. Γ α 0 [ 0 ] i 0 \Gamma^{[0]i_{0}}_{\alpha_{0}}
  6. Γ α l - 1 α l [ l ] i l , l = 1 , 2 k - 1 \Gamma^{[l]i_{l}}_{\alpha_{l-1}\alpha_{l}},l=1,2k-1
  7. λ α l [ l ] , l = 0 , 2 k - 2 \lambda^{[l]}_{\alpha_{l}},l=0,2k-2
  8. e - i δ H k , k + 1 . e^{\frac{-i\delta}{\hbar}H_{k,k+1}}.
  9. | Ψ ( t + δ ) = e - i δ F 2 e - i δ G e - i δ F 2 | Ψ ( t ) , |\Psi(t+\delta)\rangle=e^{{-i\delta}\frac{F}{2}}e^{{-i\delta}G}e^{{-i\delta}% \frac{F}{2}}|\Psi(t)\rangle,
  10. F k = 0 N 2 - 1 ( H 2 k , 2 k + 1 ) = k = 0 N 2 - 1 ( F 2 k ) , F\equiv\sum_{k=0}^{\frac{N}{2}-1}(H_{2k,2k+1})=\sum_{k=0}^{\frac{N}{2}-1}(F_{2% k}),
  11. G k = 0 N 2 - 2 ( H 2 k + 1 , 2 k + 2 ) = k = 0 N 2 - 2 ( G 2 k + 1 ) . G\equiv\sum_{k=0}^{\frac{N}{2}-2}(H_{2k+1,2k+2})=\sum_{k=0}^{\frac{N}{2}-2}(G_% {2k+1}).
  12. e - i δ 2 F 2 k , e - i δ G 2 k + 1 . e^{{-i}\frac{\delta}{2}F_{2k}},e^{{-i\delta}{G_{2k+1}}}.
  13. e - i δ 2 F m * 2 k , e - i δ 2 F m * 2 k + 2 , , e - i δ 2 F ( m + 1 ) * 2 k - 2 e^{{-i}\frac{\delta}{2}F_{m*2k}},e^{{-i}\frac{\delta}{2}F_{m*2k+2}},...,e^{{-i% }\frac{\delta}{2}F_{(m+1)*2k-2}}

Parametric_family.html

  1. f X f_{X}
  2. θ \theta
  3. f X ( ; θ ) f_{X}(\cdot\,;\theta)
  4. θ \theta
  5. θ \theta
  6. { f X ( ; θ ) θ Θ } \{f_{X}(\cdot\,;\theta)\mid\theta\in\Theta\}
  7. Θ \Theta
  8. θ \theta

Parametrix.html

  1. P ( D ) u ( x ) = δ ( x ) , P(D){u(x)}=\delta(x),\,
  2. P ( x , D ) u ( x ) = δ ( x ) + ω ( x ) , P(x,D){u(x)}=\delta(x)+\omega(x),\,
  3. L L + - I , L + L - I L\circ L^{+}-I,\quad L^{+}\circ L-I
  4. L * P = 1 + R L*P=1+R
  5. L * P * ( 1 - R + R * R - R * R * R + ) = 1 L*P*(1-R+R*R-R*R*R+\cdots)=1
  6. P - P * R + P * R * R - P * R * R * R + P-P*R+P*R*R-P*R*R*R+\cdots

Paramylodon.html

  1. 5 / 4 5/4

Pareto_priority_index.html

  1. PPI = savings × probability of success cost × time of completion \,\text{PPI}=\frac{\,\text{savings}\times\,\text{probability of success}}{\,% \text{cost}\times\,\text{time of completion}}

Parity_problem_(sieve_theory).html

  1. \mathbb{N}
  2. 1 , 2 , 3 , 1,2,3,\dots
  3. n n\in\mathbb{N}
  4. n > 1 n>1
  5. n n
  6. 1 1
  7. \mathbb{P}
  8. = { 2 , 3 , 5 , 7 , 11 , } \mathbb{P}=\{2,3,5,7,11,\dots\}\subset\mathbb{N}
  9. n n\in\mathbb{N}
  10. n > 1 n>1
  11. n = p 1 p 2 p k , n=p_{1}p_{2}\dots p_{k},
  12. p 1 , p 2 , , p k p_{1}\in\mathbb{P},\ p_{2}\in\mathbb{P},\ \dots,\ p_{k}\in\mathbb{P}
  13. 6 m + 1 6m+1
  14. m = 1 , 2 , m=1,2,\dots
  15. k m + l km+l
  16. 1 l < k 1\leq l<k
  17. ( l , k ) = 1 (l,k)=1
  18. m = 0 , 1 , 2 , m=0,1,2,\dots
  19. 0 \mathbb{N}_{0}
  20. 1 \mathbb{N}_{1}
  21. \mathbb{N}
  22. n 0 n\in\mathbb{N}_{0}
  23. n n
  24. n 1 n\in\mathbb{N}_{1}
  25. n n
  26. 0 \mathbb{N}_{0}
  27. 1 \mathbb{N}_{1}
  28. x 1 x\geq 1
  29. n 0 ( x ) n_{0}(x)
  30. n 1 ( x ) n_{1}(x)
  31. n 0 ( x ) n_{0}(x)
  32. n n
  33. 0 \mathbb{N}_{0}
  34. n x n\leq x
  35. n 1 ( x ) n_{1}(x)
  36. n n
  37. 1 \mathbb{N}_{1}
  38. n x n\leq x
  39. n 0 ( x ) n_{0}(x)
  40. n 1 ( x ) n_{1}(x)
  41. n 0 ( x ) = 1 2 x + O ( x e - c ln x ) , n 1 ( x ) = 1 2 x + O ( x e - c ln x ) ; c > 0. n_{0}(x)=\frac{1}{2}x+O\left(xe^{-c\sqrt{\ln x}}\right),n_{1}(x)=\frac{1}{2}x+% O\left(xe^{-c\sqrt{\ln x}}\right);c>0.
  42. n 0 ( x ) n 1 ( x ) 1 2 x , n_{0}(x)\sim n_{1}(x)\sim\frac{1}{2}x,
  43. n 0 ( x ) n_{0}(x)
  44. n 1 ( x ) n_{1}(x)
  45. n 1 ( x ) - n 0 ( x ) = O ( x e - c ln x ) , n_{1}(x)-n_{0}(x)=O\left(xe^{-c\sqrt{\ln x}}\right),
  46. k 2 k\geq 2
  47. l 1 , l 2 , l r l_{1},l_{2},\dots l_{r}
  48. 1 r < φ ( k ) 1\leq r<\varphi(k)
  49. 1 l j < k 1\leq l_{j}<k
  50. ( l j , k ) = 1 (l_{j},k)=1
  51. l j l_{j}
  52. k k
  53. j = 1 , 2 , r . j=1,2,\dots r.
  54. 𝔸 \mathbb{A}
  55. k n + l j kn+l_{j}
  56. j r j\leq r
  57. 𝔸 \mathbb{A}
  58. k k
  59. * \mathbb{N}^{*}
  60. 𝔸 \mathbb{A}
  61. 0 * \mathbb{N}^{*}_{0}
  62. * \mathbb{N}^{*}
  63. 1 * \mathbb{N}^{*}_{1}
  64. * \mathbb{N}^{*}
  65. n * ( x ) = n x n * 1 ; n 0 * ( x ) = n x n 0 * 1 ; n 1 * ( x ) = n x n 1 * 1. n^{*}(x)=\displaystyle\sum_{\begin{array}[]{c}n\leq x\\ n\in\mathbb{N}^{*}\end{array}}1;n^{*}_{0}(x)=\displaystyle\sum_{\begin{array}[% ]{c}n\leq x\\ n\in\mathbb{N}^{*}_{0}\end{array}}1;n^{*}_{1}(x)=\displaystyle\sum_{\begin{% array}[]{c}n\leq x\\ n\in\mathbb{N}^{*}_{1}\end{array}}1.
  66. x + x\to+\infty
  67. n 1 * ( x ) - n 0 * ( x ) C n * ( x ) ( ln x ) 2 ( r φ ( k ) - 1 ) , n^{*}_{1}(x)-n^{*}_{0}(x)\sim Cn^{*}(x)(\ln x)^{2\left(\frac{r}{\varphi(k)}-1% \right)},
  68. C C
  69. 𝔸 \mathbb{A}
  70. 𝐀 \mathbf{A}
  71. 6 m + 1 6m+1
  72. m = 1 , 2 , m=1,2,\dots
  73. n 1 * ( x ) - n 0 * ( x ) π 8 3 n * ( x ) ln x , x + . n^{*}_{1}(x)-n^{*}_{0}(x)\sim\frac{\pi}{8\sqrt{3}}\frac{n^{*}(x)}{\ln x},x\to+\infty.

Parser_combinator.html

  1. e m p t y ( j ) = { j } empty(j)=\{j\}
  2. t e r m ( x , j ) = { { } , j # i n p u t { j + 1 } , j t h element of i n p u t = x { } , otherwise term(x,j)=\begin{cases}\left\{\right\},&j\geq\#input\\ \left\{j+1\right\},&j^{th}\mbox{ element of }~{}input=x\\ \left\{\right\},&\mbox{otherwise}\end{cases}
  3. ( p q ) ( j ) = p ( j ) q ( j ) (p\oplus q)(j)=p(j)\cup q(j)
  4. ( p q ) ( j ) = { q ( k ) : k p ( j ) } (p\circledast q)(j)=\bigcup\{q(k):k\in p(j)\}

Parseval–Gutzmer_formula.html

  1. f ( z ) = k = 0 a k z k , f(z)=\sum^{\infty}_{k=0}a_{k}z^{k},
  2. 0 2 π | f ( r e i ϑ ) | 2 d ϑ = 2 π k = 0 | a k | 2 r 2 k . \int^{2\pi}_{0}|f(re^{i\vartheta})|^{2}\,\mathrm{d}\vartheta=2\pi\sum^{\infty}% _{k=0}|a_{k}|^{2}r^{2k}.
  3. a n = 1 2 π i γ f ( z ) z n + 1 d z a_{n}=\frac{1}{2\pi i}\int_{\gamma}\frac{f(z)}{z^{n+1}}\,\mathrm{d}\ z
  4. x ¯ x = | x | 2 \overline{x}{x}=|x|^{2}
  5. 0 2 π | f ( r e i ϑ ) | 2 d ϑ = 0 2 π f ( r e i ϑ ) f ( r e i ϑ ) ¯ d ϑ \int^{2\pi}_{0}|f(re^{i\vartheta})|^{2}\,\mathrm{d}\vartheta=\int^{2\pi}_{0}{f% (re^{i\vartheta})}\overline{f(re^{i\vartheta})}\,\mathrm{d}\vartheta
  6. = 0 2 π f ( r e i ϑ ) k = 0 a k ( r e i ϑ ) k ¯ d ϑ =\int^{2\pi}_{0}{f(re^{i\vartheta})}{\sum^{\infty}_{k=0}\overline{a_{k}(re^{i% \vartheta})^{k}}}\,\mathrm{d}\vartheta
  7. = k = 0 0 2 π f ( r e i ϑ ) a k ¯ ( r k ) ( e i ϑ ) k , d ϑ =\sum^{\infty}_{k=0}\int^{2\pi}_{0}\frac{{f(re^{i\vartheta})}\overline{a_{k}}(% r^{k})}{(e^{i\vartheta})^{k}},\mathrm{d}\vartheta
  8. = k = 0 ( 2 π a k ¯ r 2 k ) ( 1 2 π i 0 2 π f ( r e i ϑ ) ( r e i ϑ ) k + 1 r i e i ϑ ) d ϑ =\sum^{\infty}_{k=0}({2\pi}{\overline{a_{k}}r^{2k}})(\frac{1}{2{\pi}i}\int^{2% \pi}_{0}\frac{{f(re^{i\vartheta})}}{(re^{i\vartheta})^{k+1}}{rie^{i\vartheta}}% )\mathrm{d}\vartheta
  9. = k = 0 ( 2 π a k ¯ r 2 k ) a k = 2 π k = 0 | a k | 2 r 2 k =\sum^{\infty}_{k=0}({2\pi}{\overline{a_{k}}r^{2k}}){a_{k}}={2\pi}\sum^{\infty% }_{k=0}{|a_{k}|^{2}r^{2k}}
  10. k = 0 | a k | 2 r 2 k M r 2 \sum^{\infty}_{k=0}|a_{k}|^{2}r^{2k}\leq{M_{r}}^{2}
  11. M r = sup { | f ( z ) | : | z | = r } M_{r}=\sup\{|f(z)|:|z|=r\}
  12. 0 2 π | f ( r e i ϑ ) | 2 d ϑ 2 π | m a x ϑ [ 0 , 2 π ) ( f ( r e i ϑ ) ) | 2 = 2 π | m a x | z | = r ( f ( z ) ) | 2 = 2 π ( M r ) 2 \int^{2\pi}_{0}|f(re^{i\vartheta})|^{2}\,\mathrm{d}\vartheta\leq 2\pi|max_{% \vartheta\in[0,2\pi)}(f(re^{i\vartheta}))|^{2}=2\pi|max_{|z|=r}(f(z))|^{2}=2% \pi(M_{r})^{2}

Parthasarathy's_theorem.html

  1. X X
  2. Y Y
  3. [ 0 , 1 ] [0,1]
  4. X \mathcal{M}_{X}
  5. X X
  6. Y \mathcal{M}_{Y}
  7. A X A_{X}
  8. X X
  9. A Y A_{Y}
  10. k ( x , y ) k(x,y)
  11. 0 x , y 1 0\leq x,y\leq 1
  12. k ( x , y ) k(x,y)
  13. y = ϕ k ( x ) y=\phi_{k}(x)
  14. k = 1 , 2 , , n k=1,2,\ldots,n
  15. ϕ k ( x ) \phi_{k}(x)
  16. k ( μ , λ ) = y = 0 1 x = 0 1 k ( y , x ) d μ ( x ) d λ ( y ) = x = 0 1 y = 0 1 k ( x , y ) d λ ( y ) d μ ( x ) . k(\mu,\lambda)=\int_{y=0}^{1}\int_{x=0}^{1}k(y,x)\,d\mu(x)\,d\lambda(y)=\int_{% x=0}^{1}\int_{y=0}^{1}k(x,y)\,d\lambda(y)\,d\mu(x).
  17. max μ X inf λ A Y k ( μ , λ ) = inf λ A Y max μ X k ( μ , λ ) . \max_{\mu\in{\mathcal{M}}_{X}}\,\inf_{\lambda\in A_{Y}}k(\mu,\lambda)=\inf_{% \lambda\in A_{Y}}\,\max_{\mu\in{\mathcal{M}}_{X}}k(\mu,\lambda).
  18. k ( , ) k(\cdot,\cdot)
  19. X X
  20. max μ X inf λ Y k ( μ , λ ) inf λ Y max μ X k ( μ , λ ) \max_{\mu\in{\mathcal{M}}_{X}}\,\inf_{\lambda\in{\mathcal{M}}_{Y}}k(\mu,% \lambda)\neq\inf_{\lambda\in{\mathcal{M}}_{Y}}\,\max_{\mu\in{\mathcal{M}}_{X}}% k(\mu,\lambda)

Partial_autocorrelation_function.html

  1. z t z_{t}
  2. α ( k ) \alpha(k)
  3. z t z_{t}
  4. z t + k z_{t+k}
  5. z t z_{t}
  6. z t + 1 z_{t+1}
  7. z t + k - 1 z_{t+k-1}
  8. z t z_{t}
  9. z t + k z_{t+k}
  10. α ( 1 ) = Cor ( z t + 1 , z t ) , \alpha(1)=\operatorname{Cor}(z_{t+1},z_{t}),
  11. α ( k ) = Cor ( z t + k - P t , k ( z t + k ) , z t - P t , k ( z t ) ) , for k 2 , \alpha(k)=\operatorname{Cor}(z_{t+k}-P_{t,k}(z_{t+k}),\,z_{t}-P_{t,k}(z_{t})),% \,\text{ for }k\geq 2,
  12. P t , k ( x ) P_{t,k}(x)
  13. x x
  14. x t + 1 , , x t + k - 1 x_{t+1},\dots,x_{t+k-1}
  15. ± 1.96 / n \pm 1.96/\sqrt{n}

Partial_element_equivalent_circuit.html

  1. E i ( r , t ) = J ( r , t ) σ + A ( r , t ) t + ϕ ( r , t ) \vec{E}^{i}(\vec{r},t)=\frac{\vec{J}(\vec{r},t)}{\sigma}+\frac{\partial\vec{A}% (\vec{r},t)}{\partial t}+\nabla\phi(\vec{r},t)
  2. E i \vec{E}^{i}
  3. J \vec{J}
  4. A \vec{A}
  5. ϕ \phi
  6. σ \sigma
  7. r \vec{r}
  8. L p α β = μ 4 π 1 a α a β v α v β 1 | r α - r β | d v α d v β L_{p_{\alpha\beta}}=\frac{\mu}{4\pi}\frac{1}{a_{\alpha}a_{\beta}}\int_{v_{% \alpha}}\int_{v_{\beta}}\frac{1}{|\vec{r}_{\alpha}-\vec{r}_{\beta}|}dv_{\alpha% }dv_{\beta}
  9. α \alpha
  10. β \beta
  11. P i j = 1 S i S j 1 4 π ϵ 0 S i S j 1 | r i - r j | d S j d S i P_{ij}=\frac{1}{S_{i}S_{j}}\frac{1}{4\pi\epsilon_{0}}\int_{S_{i}}\int_{S_{j}}% \frac{1}{|\vec{r}_{i}-\vec{r}_{j}|}\;dS_{j}\;dS_{i}
  12. R γ = l γ a γ σ γ . R_{\gamma}=\frac{l_{\gamma}}{a_{\gamma}\sigma_{\gamma}}.

Partial_leverage.html

  1. ( PL j ) i = ( X j [ j ] ) i 2 k = 1 n ( X j [ j ] ) k 2 \left(\mathrm{PL}_{j}\right)_{i}=\frac{\left(X_{j\bullet[j]}\right)_{i}^{2}}{% \sum_{k=1}^{n}\left(X_{j\bullet[j]}\right)_{k}^{2}}
  2. p \mathbb{R}^{p}
  3. H = X ( X X ) - 1 X . H=X(X^{\prime}X)^{-1}X^{\prime}.\,

Partial_molar_property.html

  1. Z Z
  2. P P
  3. T T
  4. Z = Z ( T , P , n 1 , n 2 , ) . Z=Z(T,P,n_{1},n_{2},\cdots).
  5. Z = Z ( n 1 , n 2 , ) Z=Z(n_{1},n_{2},\cdots)
  6. Z Z
  7. λ \lambda
  8. Z ( λ n 1 , λ n 2 , ) = λ Z ( n 1 , n 2 , ) . Z(\lambda n_{1},\lambda n_{2},\cdots)=\lambda Z(n_{1},n_{2},\cdots).
  9. Z = i = 1 m n i Z i ¯ , Z=\sum_{i=1}^{m}n_{i}\bar{Z_{i}},
  10. Z i ¯ \bar{Z_{i}}
  11. Z Z
  12. i i
  13. Z i ¯ = ( Z n i ) T , P , n j i . \bar{Z_{i}}=\left(\frac{\partial Z}{\partial n_{i}}\right)_{T,P,n_{j\neq i}}.
  14. Z i ¯ \bar{Z_{i}}
  15. λ \lambda
  16. Z i ¯ ( λ n 1 , λ n 2 , ) = Z i ¯ ( n 1 , n 2 , ) . \bar{Z_{i}}(\lambda n_{1},\lambda n_{2},\cdots)=\bar{Z_{i}}(n_{1},n_{2},\cdots).
  17. λ = 1 / n T \lambda=1/n_{T}
  18. n T = n 1 + n 2 + n_{T}=n_{1}+n_{2}+\cdots
  19. Z i ¯ ( x 1 , x 2 , ) = Z i ¯ ( n 1 , n 2 , ) , \bar{Z_{i}}(x_{1},x_{2},\cdots)=\bar{Z_{i}}(n_{1},n_{2},\cdots),
  20. x i = n i n T x_{i}=\frac{n_{i}}{n_{T}}
  21. i i
  22. i = 1 m x i = 1 , \sum_{i=1}^{m}x_{i}=1,
  23. m - 1 m-1
  24. Z i ¯ = Z i ¯ ( x 1 , x 2 , , x m - 1 ) . \bar{Z_{i}}=\bar{Z_{i}}(x_{1},x_{2},\cdots,x_{m-1}).
  25. Δ z M = z - i x i z i * . \Delta z^{M}=z-\sum_{i}x_{i}z^{*}_{i}.
  26. * *
  27. M M
  28. z z
  29. z = i x i Z i ¯ , z=\sum_{i}x_{i}\bar{Z_{i}},
  30. Δ z M = i x i ( Z i ¯ - z i * ) . \Delta z^{M}=\sum_{i}x_{i}(\bar{Z_{i}}-z_{i}^{*}).
  31. H i ¯ = U i ¯ + P V i ¯ , \bar{H_{i}}=\bar{U_{i}}+P\bar{V_{i}},
  32. A i ¯ = U i ¯ - T S i ¯ , \bar{A_{i}}=\bar{U_{i}}-T\bar{S_{i}},
  33. G i ¯ = H i ¯ - T S i ¯ , \bar{G_{i}}=\bar{H_{i}}-T\bar{S_{i}},
  34. P P
  35. V V
  36. T T
  37. S S
  38. d U = T d S - P d V + i μ i d n i , dU=TdS-PdV+\sum_{i}\mu_{i}dn_{i},\,
  39. d H = T d S + V d P + i μ i d n i , dH=TdS+VdP+\sum_{i}\mu_{i}dn_{i},\,
  40. d A = - S d T - P d V + i μ i d n i , dA=-SdT-PdV+\sum_{i}\mu_{i}dn_{i},\,
  41. d G = - S d T + V d P + i μ i d n i , dG=-SdT+VdP+\sum_{i}\mu_{i}dn_{i},\,
  42. μ i \mu_{i}
  43. μ i = ( U n i ) S , V = ( H n i ) S , P = ( A n i ) T , V = ( G n i ) T , P . \mu_{i}=\left(\frac{\partial U}{\partial n_{i}}\right)_{S,V}=\left(\frac{% \partial H}{\partial n_{i}}\right)_{S,P}=\left(\frac{\partial A}{\partial n_{i% }}\right)_{T,V}=\left(\frac{\partial G}{\partial n_{i}}\right)_{T,P}.
  44. G i ¯ \bar{G_{i}}
  45. μ i ( x 1 , x 2 , , x m ) \mu_{i}(x_{1},x_{2},\cdots,x_{m})
  46. Z 1 ¯ \bar{Z_{1}}
  47. 2 2
  48. 1 1
  49. Z Z
  50. Z ( n 1 ) Z(n_{1})
  51. n 1 n_{1}
  52. Z 1 ¯ \bar{Z_{1}}
  53. Z 2 ¯ \bar{Z_{2}}
  54. Z = Z 1 ¯ n 1 + Z 2 ¯ n 2 . Z=\bar{Z_{1}}n_{1}+\bar{Z_{2}}n_{2}.
  55. V 1 ¯ = V ~ 1 ϕ + b V ~ 1 ϕ b . \bar{V_{1}}={}^{\phi}\tilde{V}_{1}+b\frac{\partial{}^{\phi}\tilde{V}_{1}}{% \partial b}.

Partial_regression_plot.html

  1. Y [ i ] versus X i [ i ] Y_{\bullet[i]}\mathrm{\ versus\ }X_{i\bullet[i]}
  2. β i \beta_{i}

Partial_residual_plot.html

  1. Residuals + β ^ i X i versus X i \mathrm{Residuals}+\hat{\beta}_{i}X_{i}\mathrm{\ versus\ }X_{i}
  2. β ^ i \hat{\beta}_{i}
  3. β ^ i X i versus X i . \hat{\beta}_{i}X_{i}\mathrm{\ versus\ }X_{i}.

Partial_word.html

  1. u : { 0 , , n - 1 } A u:\{0,\ldots,n-1\}\rightarrow A
  2. A A
  3. k { 0 , , n - 1 } k\in\{0,\ldots,n-1\}

Particle_accelerator.html

  1. B B

Partition_function_(mathematics).html

  1. X i X_{i}
  2. x i x_{i}
  3. H ( x 1 , x 2 , ) H(x_{1},x_{2},\dots)
  4. Z ( β ) = x i exp ( - β H ( x 1 , x 2 , ) ) Z(\beta)=\sum_{x_{i}}\exp\left(-\beta H(x_{1},x_{2},\dots)\right)
  5. { X 1 , X 2 , } \{X_{1},X_{2},\cdots\}
  6. β \beta
  7. x i x_{i}
  8. X i X_{i}
  9. X i X_{i}
  10. Z ( β ) = exp ( - β H ( x 1 , x 2 , ) ) d x 1 d x 2 Z(\beta)=\int\exp\left(-\beta H(x_{1},x_{2},\dots)\right)dx_{1}dx_{2}\cdots
  11. X i X_{i}
  12. Z ( β ) = tr ( exp ( - β H ) ) Z(\beta)=\mbox{tr}~{}\left(\exp\left(-\beta H\right)\right)
  13. X i X_{i}
  14. Z = 𝒟 ϕ exp ( - β H [ ϕ ] ) Z=\int\mathcal{D}\phi\exp\left(-\beta H[\phi]\right)
  15. β \beta
  16. H H
  17. X X
  18. β \beta
  19. H H
  20. { H k ( x 1 , ) } \{H_{k}(x_{1},\cdots)\}
  21. X i X_{i}
  22. β \beta
  23. Z ( β ) = x i exp ( - k β k H k ( x i ) ) Z(\beta)=\sum_{x_{i}}\exp\left(-\sum_{k}\beta_{k}H_{k}(x_{i})\right)
  24. β = ( β 1 , β 2 , ) \beta=(\beta_{1},\beta_{2},\cdots)
  25. H k H_{k}
  26. Z ( β ) = tr [ exp ( - k β k H k ) ] Z(\beta)=\mbox{tr}~{}\left[\,\exp\left(-\sum_{k}\beta_{k}H_{k}\right)\right]
  27. H k H_{k}
  28. β k ( - log Z ) = H k = E [ H k ] \frac{\partial}{\partial\beta_{k}}\left(-\log Z\right)=\langle H_{k}\rangle=% \mathrm{E}\left[H_{k}\right]
  29. H k \langle H_{k}\rangle
  30. H k H_{k}
  31. E [ ] \mathrm{E}[\;]
  32. β \beta
  33. β \beta
  34. H ( x 1 , x 2 , ) = s V ( s ) H(x_{1},x_{2},\dots)=\sum_{s}V(s)\,
  35. X = { x 1 , x 2 , } X=\{x_{1},x_{2},\dots\}
  36. exp ( - β H ( x 1 , x 2 , ) ) \exp\left(-\beta H(x_{1},x_{2},\dots)\right)
  37. ( x 1 , x 2 , ) (x_{1},x_{2},\dots)
  38. ( x 1 , x 2 , ) (x_{1},x_{2},\dots)
  39. P ( x 1 , x 2 , ) = 1 Z ( β ) exp ( - β H ( x 1 , x 2 , ) ) P(x_{1},x_{2},\dots)=\frac{1}{Z(\beta)}\exp\left(-\beta H(x_{1},x_{2},\dots)\right)
  40. ( x 1 , x 2 , ) (x_{1},x_{2},\dots)
  41. 0 P ( x 1 , x 2 , ) 1 0\leq P(x_{1},x_{2},\dots)\leq 1
  42. ( x 1 , x 2 , ) (x_{1},x_{2},\dots)
  43. β \beta
  44. i t H itH
  45. β H \beta H
  46. β \beta
  47. log ( Z ( β ) ) \log(Z(\beta))
  48. β \beta
  49. E [ H ] = H = - log ( Z ( β ) ) β {E}[H]=\langle H\rangle=-\frac{\partial\log(Z(\beta))}{\partial\beta}
  50. f = x i f ( x 1 , x 2 , ) P ( x 1 , x 2 , ) = 1 Z ( β ) x i f ( x 1 , x 2 , ) exp ( - β H ( x 1 , x 2 , ) ) \begin{aligned}\displaystyle\langle f\rangle&\displaystyle=\sum_{x_{i}}f(x_{1}% ,x_{2},\dots)P(x_{1},x_{2},\dots)\\ &\displaystyle=\frac{1}{Z(\beta)}\sum_{x_{i}}f(x_{1},x_{2},\dots)\exp\left(-% \beta H(x_{1},x_{2},\dots)\right)\end{aligned}
  51. S = - k B ln P = - k B x i P ( x 1 , x 2 , ) ln P ( x 1 , x 2 , ) = k B ( β H + log Z ( β ) ) \begin{aligned}\displaystyle S&\displaystyle=-k_{B}\langle\ln P\rangle\\ &\displaystyle=-k_{B}\sum_{x_{i}}P(x_{1},x_{2},\dots)\ln P(x_{1},x_{2},\dots)% \\ &\displaystyle=k_{B}(\beta\langle H\rangle+\log Z(\beta))\end{aligned}
  52. β \beta
  53. g i j ( β ) = 2 β i β j ( - log Z ( β ) ) = ( H i - H i ) ( H j - H j ) g_{ij}(\beta)=\frac{\partial^{2}}{\partial\beta^{i}\partial\beta^{j}}\left(-% \log Z(\beta)\right)=\langle\left(H_{i}-\langle H_{i}\rangle\right)\left(H_{j}% -\langle H_{j}\rangle\right)\rangle
  54. β \beta
  55. g i j ( β ) \displaystyle g_{ij}(\beta)
  56. P ( x ) P(x)
  57. P ( x 1 , x 2 , ) P(x_{1},x_{2},\dots)
  58. X k X_{k}
  59. β \beta
  60. J k J_{k}
  61. Z ( β , J ) = Z ( β , J 1 , J 2 , ) = x i exp ( - β H ( x 1 , x 2 , ) + n J n x n ) \begin{aligned}\displaystyle Z(\beta,J)&\displaystyle=Z(\beta,J_{1},J_{2},% \dots)\\ &\displaystyle=\sum_{x_{i}}\exp\left(-\beta H(x_{1},x_{2},\dots)+\sum_{n}J_{n}% x_{n}\right)\end{aligned}
  62. E [ x k ] = x k = J k log Z ( β , J ) | J = 0 {E}[x_{k}]=\langle x_{k}\rangle=\left.\frac{\partial}{\partial J_{k}}\log Z(% \beta,J)\right|_{J=0}
  63. x k x_{k}
  64. C ( x j , x k ) C(x_{j},x_{k})
  65. x j x_{j}
  66. x k x_{k}
  67. C ( x j , x k ) = J j J k log Z ( β , J ) | J = 0 C(x_{j},x_{k})=\left.\frac{\partial}{\partial J_{j}}\frac{\partial}{\partial J% _{k}}\log Z(\beta,J)\right|_{J=0}
  68. H = 1 2 n x n D x n H=\frac{1}{2}\sum_{n}x_{n}Dx_{n}
  69. C ( x j , x k ) C(x_{j},x_{k})

Password_Authenticated_Key_Exchange_by_Juggling.html

  1. G G
  2. g g
  3. q q
  4. s s
  5. s 0 s\neq 0
  6. x 1 R [ 0 , q - 1 ] x_{1}\in_{R}[0,q-1]
  7. x 2 R ( 0 , q - 1 ] x_{2}\in_{R}(0,q-1]
  8. g x 1 g^{x_{1}}
  9. g x 2 g^{x_{2}}
  10. x 1 x_{1}
  11. x 2 x_{2}
  12. x 3 R [ 0 , q - 1 ] x_{3}\in_{R}[0,q-1]
  13. x 4 R ( 0 , q - 1 ] x_{4}\in_{R}(0,q-1]
  14. g x 3 g^{x_{3}}
  15. g x 4 g^{x_{4}}
  16. x 3 x_{3}
  17. x 4 x_{4}
  18. g x 2 , g x 4 1 g^{x_{2}},g^{x_{4}}\neq 1
  19. A = g ( x 1 + x 3 + x 4 ) x 2 s A=g^{(x_{1}+x_{3}+x_{4})x_{2}s}
  20. x 2 s x_{2}s
  21. g x 1 + x 3 + x 4 g^{x_{1}+x_{3}+x_{4}}
  22. B = g ( x 1 + x 2 + x 3 ) x 4 s B=g^{(x_{1}+x_{2}+x_{3})x_{4}s}
  23. x 4 s x_{4}s
  24. K = ( B / g x 2 x 4 s ) x 2 = g ( x 1 + x 3 ) x 2 x 4 s K=(B/g^{x_{2}x_{4}s})^{x_{2}}=g^{(x_{1}+x_{3})x_{2}x_{4}s}
  25. K = ( A / g x 2 x 4 s ) x 4 = g ( x 1 + x 3 ) x 2 x 4 s K=(A/g^{x_{2}x_{4}s})^{x_{4}}=g^{(x_{1}+x_{3})x_{2}x_{4}s}
  26. K K
  27. κ = H ( K ) \kappa=H(K)
  28. g x 1 , g x 2 g^{x_{1}},g^{x_{2}}
  29. g x 3 , g x 4 , B = g ( x 1 + x 2 + x 3 ) x 4 s g^{x_{3}},g^{x_{4}},B=g^{(x_{1}+x_{2}+x_{3})x_{4}s}
  30. A = g ( x 1 + x 3 + x 4 ) x 2 s A=g^{(x_{1}+x_{3}+x_{4})x_{2}s}
  31. H ( H ( κ ) ) H(H(\kappa))
  32. H ( κ ) H(\kappa)

Paul_Gerber.html

  1. V = μ r ( 1 - 1 c d r d t ) 2 V=\frac{\mu}{r\left(1-\frac{1}{c}\frac{dr}{dt}\right)^{2}}
  2. V = μ r [ 1 + 2 c d r d t + 3 c 2 ( d r d t ) 2 ] V=\frac{\mu}{r}\left[1+\frac{2}{c}\frac{dr}{dt}+\frac{3}{c^{2}}\left(\frac{dr}% {dt}\right)^{2}\right]
  3. c 2 = 6 π μ a ( 1 - ϵ 2 ) Ψ c^{2}=\frac{6\pi\mu}{a(1-\epsilon^{2})\Psi}
  4. μ = 4 π 2 a 3 τ 2 \mu=\frac{4\pi^{2}a^{3}}{\tau^{2}}
  5. Ψ = 24 π 3 a 2 τ 2 c 2 ( 1 - ϵ 2 ) \Psi=24\pi^{3}\frac{a^{2}}{\tau^{2}c^{2}(1-\epsilon^{2})}
  6. ϵ = 24 π 3 a 2 T 2 c 2 ( 1 - e 2 ) \epsilon=24\pi^{3}\frac{a^{2}}{T^{2}c^{2}(1-e^{2})}

Peakon.html

  1. e - | x | e^{-|x|}
  2. u t - u x x t + ( b + 1 ) u u x = b u x u x x + u u x x x , u_{t}-u_{xxt}+(b+1)uu_{x}=bu_{x}u_{xx}+uu_{xxx},\,
  3. u ( x , t ) u(x,t)
  4. m ( x , t ) m(x,t)
  5. m = u - u x x m=u-u_{xx}
  6. m t + m x u + b m u x = 0. m_{t}+m_{x}u+bmu_{x}=0.\,
  7. u ( x , t ) = c e - | x - c t | u(x,t)=c\,e^{-|x-ct|}
  8. m = u - u x x = c δ ( x - c t ) m=u-u_{xx}=c\,\delta(x-ct)
  9. m u x mu_{x}
  10. m = ( G / 2 ) * u m=(G/2)*u
  11. G ( x ) = exp ( - | x | ) G(x)=\exp(-|x|)
  12. t u + x [ u 2 2 + G 2 * ( b u 2 2 + ( 3 - b ) u x 2 2 ) ] = 0. \partial_{t}u+\partial_{x}\left[\frac{u^{2}}{2}+\frac{G}{2}*\left(\frac{bu^{2}% }{2}+\frac{(3-b)u_{x}^{2}}{2}\right)\right]=0.
  13. u = m 1 e - | x - x 1 | + m 2 e - | x - x 2 | u=m_{1}\,e^{-|x-x_{1}|}+m_{2}\,e^{-|x-x_{2}|}
  14. u ( x , t ) = i = 1 n m i ( t ) e - | x - x i ( t ) | , u(x,t)=\sum_{i=1}^{n}m_{i}(t)\,e^{-|x-x_{i}(t)|},
  15. x i ( t ) x_{i}(t)
  16. m i ( t ) m_{i}(t)
  17. x ˙ k = i = 1 n m i e - | x k - x i | , m ˙ k = ( b - 1 ) i = 1 n m k m i sgn ( x k - x i ) e - | x k - x i | ( k = 1 , , n ) \dot{x}_{k}=\sum_{i=1}^{n}m_{i}e^{-|x_{k}-x_{i}|},\qquad\dot{m}_{k}=(b-1)\sum_% {i=1}^{n}m_{k}m_{i}\operatorname{sgn}(x_{k}-x_{i})e^{-|x_{k}-x_{i}|}\qquad(k=1% ,\dots,n)
  18. x k x_{k}
  19. x = x k x=x_{k}
  20. m k m_{k}
  21. u x u_{x}
  22. exp ( - | x | ) \exp(-|x|)
  23. x ˙ k = u ( x k ) , m ˙ k = - ( b - 1 ) m k u x ( x k ) ( k = 1 , , n ) . \dot{x}_{k}=u(x_{k}),\qquad\dot{m}_{k}=-(b-1)m_{k}u_{x}(x_{k})\qquad(k=1,\dots% ,n).
  24. x 1 ( t ) = log ( λ 1 - λ 2 ) 2 ( λ 1 - λ 3 ) 2 ( λ 2 - λ 3 ) 2 a 1 a 2 a 3 j < k λ j 2 λ k 2 ( λ j - λ k ) 2 a j a k x 2 ( t ) = log j < k ( λ j - λ k ) 2 a j a k λ 1 2 a 1 + λ 2 2 a 2 + λ 3 2 a 3 x 3 ( t ) = log ( a 1 + a 2 + a 3 ) m 1 ( t ) = j < k λ j 2 λ k 2 ( λ j - λ k ) 2 a j a k λ 1 λ 2 λ 3 j < k λ j λ k ( λ j - λ k ) 2 a j a k m 2 ( t ) = ( λ 1 2 a 1 + λ 2 2 a 2 + λ 3 2 a 3 ) j < k ( λ j - λ k ) 2 a j a k ( λ 1 a 1 + λ 2 a 2 + λ 3 a 3 ) j < k λ j λ k ( λ j - λ k ) 2 a j a k m 3 ( t ) = a 1 + a 2 + a 3 λ 1 a 1 + λ 2 a 2 + λ 3 a 3 \begin{aligned}\displaystyle x_{1}(t)&\displaystyle=\log\frac{(\lambda_{1}-% \lambda_{2})^{2}(\lambda_{1}-\lambda_{3})^{2}(\lambda_{2}-\lambda_{3})^{2}a_{1% }a_{2}a_{3}}{\sum_{j<k}\lambda_{j}^{2}\lambda_{k}^{2}(\lambda_{j}-\lambda_{k})% ^{2}a_{j}a_{k}}\\ \displaystyle x_{2}(t)&\displaystyle=\log\frac{\sum_{j<k}(\lambda_{j}-\lambda_% {k})^{2}a_{j}a_{k}}{\lambda_{1}^{2}a_{1}+\lambda_{2}^{2}a_{2}+\lambda_{3}^{2}a% _{3}}\\ \displaystyle x_{3}(t)&\displaystyle=\log(a_{1}+a_{2}+a_{3})\\ \displaystyle m_{1}(t)&\displaystyle=\frac{\sum_{j<k}\lambda_{j}^{2}\lambda_{k% }^{2}(\lambda_{j}-\lambda_{k})^{2}a_{j}a_{k}}{\lambda_{1}\lambda_{2}\lambda_{3% }\sum_{j<k}\lambda_{j}\lambda_{k}(\lambda_{j}-\lambda_{k})^{2}a_{j}a_{k}}\\ \displaystyle m_{2}(t)&\displaystyle=\frac{\left(\lambda_{1}^{2}a_{1}+\lambda_% {2}^{2}a_{2}+\lambda_{3}^{2}a_{3}\right)\sum_{j<k}(\lambda_{j}-\lambda_{k})^{2% }a_{j}a_{k}}{\left(\lambda_{1}a_{1}+\lambda_{2}a_{2}+\lambda_{3}a_{3}\right)% \sum_{j<k}\lambda_{j}\lambda_{k}(\lambda_{j}-\lambda_{k})^{2}a_{j}a_{k}}\\ \displaystyle m_{3}(t)&\displaystyle=\frac{a_{1}+a_{2}+a_{3}}{\lambda_{1}a_{1}% +\lambda_{2}a_{2}+\lambda_{3}a_{3}}\end{aligned}
  25. a k ( t ) = a k ( 0 ) e t / λ k a_{k}(t)=a_{k}(0)e^{t/\lambda_{k}}
  26. a k ( 0 ) a_{k}(0)
  27. λ k \lambda_{k}
  28. a k a_{k}
  29. λ k \lambda_{k}

Pebble_automaton.html

  1. { 1 , 2 , , n } \{1,2,\dots,n\}
  2. D P A n DPA_{n}
  3. P A n PA_{n}
  4. D P A = n D P A n DPA=\bigcup_{n}DPA_{n}
  5. P A = n P A n PA=\bigcup_{n}PA_{n}
  6. T W A D P A TWA\subsetneq DPA
  7. P A R E G PA\subsetneq REG
  8. D P A = P A DPA=PA
  9. P A n P A n + 1 PA_{n}\subsetneq PA_{n+1}
  10. D P A n D P A n + 1 DPA_{n}\subsetneq DPA_{n+1}
  11. F O + T C FO+TC
  12. F O + pos T C FO+\,\text{pos}\,TC
  13. P A F O + T C PA\subseteq FO+TC
  14. P A = F O + pos T C PA=FO+\,\text{pos}\,TC

Peierls_transition.html

  1. T T i = M i m e \frac{T}{T_{i}}=\sqrt{\frac{M_{i}}{m_{e}}}

Pentagon.html

  1. H e i g h t = S i d e 5 - 5 + 5 + 5 8 Height=Side\cdot\frac{\sqrt{5-\sqrt{5}}+\sqrt{5+\sqrt{5}}}{\sqrt{8}}
  2. W i d t h = S i d e 1 + 5 2 . Width=Side\cdot\frac{1+\sqrt{5}}{2}.
  3. A = t 2 25 + 10 5 4 = 5 t 2 tan ( 54 ) 4 1.720477401 t 2 . A=\frac{{t^{2}\sqrt{25+10\sqrt{5}}}}{4}=\frac{5t^{2}\tan(54^{\circ})}{4}% \approx 1.720477401t^{2}.
  4. t = R 5 - 5 2 = 2 R sin 36 = 2 R sin π 5 1.17557050458 R . t=R\ {\sqrt{\frac{5-\sqrt{5}}{2}}}=2R\sin 36^{\circ}=2R\sin\frac{\pi}{5}% \approx 1.17557050458R.
  5. A = 1 2 P a A=\frac{1}{2}Pa
  6. A = 1 2 × 5 t 1 × t tan ( 54 ) 2 A=\frac{1}{2}\times\frac{5t}{1}\times\frac{t\tan(54^{\circ})}{2}
  7. A = 1 2 × 5 t 2 tan ( 54 ) 2 A=\frac{1}{2}\times\frac{5t^{2}\tan(54^{\circ})}{2}
  8. A = 5 t 2 tan ( 54 ) 4 . A=\frac{5t^{2}\tan(54^{\circ})}{4}.
  9. d t = φ = 1 + 5 2 , \frac{d}{t}=\varphi=\frac{1+\sqrt{5}}{2}\ ,
  10. d = t × φ = R 5 + 5 2 = 2 R cos 18 = 2 R cos π 10 1.90211303259 R . d=t\times\varphi\ =R\ {\sqrt{\frac{5+\sqrt{5}}{2}}}=2R\cos 18^{\circ}=2R\cos% \frac{\pi}{10}\approx 1.90211303259R.
  11. 10 ( 5 - 1 ) r = t 50 + 10 5 . 10(\sqrt{5}-1)r=t\sqrt{50+10\sqrt{5}}.
  12. 5 / 2 \scriptstyle\sqrt{5}/2
  13. tan ( ϕ / 2 ) = 1 - cos ( ϕ ) sin ( ϕ ) , \tan(\phi/2)=\frac{1-\cos(\phi)}{\sin(\phi)}\ ,
  14. h = 5 - 1 4 . h=\frac{\sqrt{5}-1}{4}\ .
  15. a 2 = 1 - h 2 ; a = 1 2 5 + 5 2 . a^{2}=1-h^{2}\ ;\ a=\frac{1}{2}\sqrt{\frac{5+\sqrt{5}}{2}}\ .
  16. s 2 = ( 1 - h ) 2 + a 2 = ( 1 - h ) 2 + 1 - h 2 = 1 - 2 h + h 2 + 1 - h 2 = 2 - 2 h = 2 - 2 ( 5 - 1 4 ) s^{2}=(1-h)^{2}+a^{2}=(1-h)^{2}+1-h^{2}=1-2h+h^{2}+1-h^{2}=2-2h=2-2\left(\frac% {\sqrt{5}-1}{4}\right)
  17. = 5 - 5 2 . =\frac{5-\sqrt{5}}{2}\ .
  18. s = 5 - 5 2 , s=\sqrt{\frac{5-\sqrt{5}}{2}}\ ,
  19. 1 + 5 4 \tfrac{1+\sqrt{5}}{4}
  20. 5 \sqrt{5}
  21. 5 \sqrt{5}
  22. 5 \sqrt{5}
  23. 1 + 5 4 \tfrac{1+\sqrt{5}}{4}
  24. 1 + 5 4 \tfrac{1+\sqrt{5}}{4}
  25. O P ¯ \overline{OP}
  26. 1 + 5 4 \tfrac{1+\sqrt{5}}{4}
  27. 0 = cos 90 0=\cos 90
  28. = cos ( 72 + 18 ) =\cos(72+18)
  29. = cos 72 cos 18 - sin 72 sin 18 =\cos 72\cos 18-\sin 72\sin 18
  30. = ( 2 cos 2 36 - 1 ) 1 + cos 36 2 - 2 sin 36 cos 36 1 - cos 36 2 =(2\cos^{2}36-1)\sqrt{\tfrac{1+\cos 36}{2}}-2\sin 36\cos 36\sqrt{\tfrac{1-\cos 3% 6}{2}}
  31. 0 = ( 2 u 2 - 1 ) 1 + u 2 - 2 1 - u 2 u 1 - u 2 2 1 - u 2 u 1 - u 2 = ( 2 u 2 - 1 ) 1 + u 2 2 1 + u 1 - u u 1 - u = ( 2 u 2 - 1 ) 1 + u 2 u ( 1 - u ) = 2 u 2 - 1 2 u - 2 u 2 = 2 u 2 - 1 0 = 4 u 2 - 2 u - 1 u = 2 + ( - 2 ) 2 - 4 ( 4 ) ( - 1 ) 2 ( 4 ) u = 2 + 20 8 u = 1 + 5 4 \begin{aligned}\displaystyle 0&\displaystyle{}=(2u^{2}-1)\sqrt{\tfrac{1+u}{2}}% -2\sqrt{1-u^{2}}\cdot u\sqrt{\tfrac{1-u}{2}}\\ \displaystyle 2\sqrt{1-u^{2}}\cdot u\sqrt{\tfrac{1-u}{2}}&\displaystyle{}=(2u^% {2}-1)\sqrt{\tfrac{1+u}{2}}\\ \displaystyle 2\sqrt{1+u}\sqrt{1-u}\cdot u\sqrt{1-u}&\displaystyle{}=(2u^{2}-1% )\sqrt{1+u}\\ \displaystyle 2u(1-u)&\displaystyle{}=2u^{2}-1\\ \displaystyle 2u-2u^{2}&\displaystyle{}=2u^{2}-1\\ \displaystyle 0&\displaystyle{}=4u^{2}-2u-1\\ \displaystyle u&\displaystyle{}=\tfrac{2+\sqrt{(-2)^{2}-4(4)(-1)}}{2(4)}\\ \displaystyle u&\displaystyle{}=\tfrac{2+\sqrt{20}}{8}\\ \displaystyle u&\displaystyle{}=\tfrac{1+\sqrt{5}}{4}\end{aligned}

Perceived_visual_angle.html

  1. tan θ = S / D \tan\theta=S/D\,
  2. R / n = tan θ R/n=\tan\theta\,
  3. S / D = tan θ S^{\prime}/D^{\prime}=\tan\theta^{\prime}\,
  4. S / D = tan θ S^{\prime}/D^{\prime}=\tan\theta\,

Percentage-of-completion_method.html

  1. I n c k = [ i = 1 m C o s t s i i = 1 N E ( C o s t s i ) [ i = 1 N R e v e n u e s i - i = 1 N E ( C o s t s i ) ] ] - i = 1 m - 1 I n c i . Inc_{k}=[\frac{\sum_{i=1}^{m}Costs_{i}}{\sum_{i=1}^{N}E(Costs_{i})}\cdot[\sum_% {i=1}^{N}Revenues_{i}-\sum_{i=1}^{N}E(Costs_{i})]]-\sum_{i=1}^{m-1}Inc_{i}.\,
  2. m = m=\,
  3. N = N=\,
  4. k = k=\,

Perfect_measure.html

  1. A 1 A A 2 and μ ( f - 1 ( A 2 A 1 ) ) = 0. A_{1}\subseteq A\subseteq A_{2}\mbox{ and }~{}\mu\big(f^{-1}(A_{2}\setminus A_% {1})\big)=0.

Performic_acid.html

  1. HCOOH + H 2 O 2 HCO 2 OH + H 2 O \rm HCOOH+H_{2}O_{2}\leftrightharpoons HCO_{2}OH+H_{2}O

Periodic_annual_increment.html

  1. P A I = Y 2 - Y 1 T 2 - T 1 PAI=\frac{Y_{2}-Y_{1}}{T_{2}-T_{1}}
  2. 34 - 14 10 - 5 = 4 f e e t / y e a r \frac{34-14}{10-5}=4feet/year

Periodic_table_(crystal_structure).html

  1. 2 2 3 \scriptstyle 2\sqrt{\frac{2}{3}}
  2. c a = \scriptstyle\frac{c}{a}=
  3. 4 2 3 \scriptstyle 4\sqrt{\frac{2}{3}}
  4. c 2 a \scriptstyle\frac{c}{2a}

Permutation_graph.html

  1. G ¯ \overline{G}

Perovskite.html

  1. t = R A + R O 2 ( R B + R O ) t=\frac{R_{A}+R_{O}}{\sqrt{2}(R_{B}+R_{O})}

Pervaded_volume.html

  1. V R 3 V\approx R^{3}

Perveance.html

  1. P P
  2. I I
  3. U a U_{a}
  4. I = P U a 3 2 {I}={P}\cdot U_{a}^{\frac{3}{2}}
  5. U a U_{a}
  6. I U a 3 2 \frac{I}{U_{a}^{\frac{3}{2}}}
  7. K K
  8. K = < m t p l > I I 0 2 β 3 γ 3 ( 1 - γ 2 f e ) {K}=\frac{<}{m}tpl>{{I}}{{I_{0}}}\cdot\frac{{2}}{{\beta}^{3}{\gamma}^{3}}\cdot% (1-\gamma^{2}f_{e})
  9. I 0 = 4 π ε 0 m c 3 e 17 k A {I_{0}}=4\pi\varepsilon_{0}\cdot\frac{mc^{3}}{e}\approx 17\mathrm{kA}
  10. β \mathbf{\beta}
  11. γ \mathbf{\gamma}
  12. f e f_{e}

Petersson_trace_formula.html

  1. \mathcal{F}
  2. S k ( Γ ( 1 ) ) S_{k}(\Gamma(1))
  3. k > 2 k>2
  4. S L 2 ( ) SL_{2}(\mathbb{Z})
  5. m , n m,n
  6. Γ ( k - 1 ) ( 4 π m n ) k - 1 f f ¯ ( m ) f ( n ) = δ m n + 2 π i - k c > 0 S ( m , n ; c ) c J k - 1 ( 4 π m n c ) , \frac{\Gamma(k-1)}{(4\pi\sqrt{mn})^{k-1}}\sum_{f\in\mathcal{F}}\bar{f}(m)f(n)=% \delta_{mn}+2\pi i^{-k}\sum_{c>0}\frac{S(m,n;c)}{c}J_{k-1}\left(\frac{4\pi% \sqrt{mn}}{c}\right),
  7. δ \delta
  8. S S
  9. J J

Pettis_integral.html

  1. f : X V f\colon X\to V
  2. ( X , Σ , μ ) (X,\Sigma,\mu)
  3. V V
  4. V V
  5. V * V^{*}
  6. V V
  7. φ , x = φ [ x ] \langle\varphi,x\rangle=\varphi[x]
  8. E Σ E\in\Sigma
  9. f f
  10. E E
  11. e V e\in V
  12. φ , e = E φ , f ( x ) d μ ( x ) for all functionals φ V * . \langle\varphi,e\rangle=\int_{E}\langle\varphi,f(x)\rangle\,d\mu(x)\,\text{ % for all functionals }\varphi\in V^{*}.
  13. e e
  14. f f
  15. E E
  16. e e
  17. E f μ \int_{E}f\mu
  18. E f ( t ) d μ ( t ) \int_{E}f(t)\,d\mu(t)
  19. μ [ f 1 E ] \mu[f1_{E}]
  20. X X
  21. φ f \varphi\circ f
  22. φ X * \varphi\in X^{*}
  23. ( Ω , , ) (\Omega,\mathcal{F},\mathbb{P})
  24. V V
  25. v n : Ω V v_{n}:\Omega\to V
  26. 𝔼 [ v n ] \mathbb{E}[v_{n}]
  27. v n v_{n}
  28. X X
  29. 𝔼 [ v n ] \mathbb{E}[v_{n}]
  30. V V
  31. v ¯ N := 1 N n = 1 N v n \bar{v}_{N}:=\frac{1}{N}\sum_{n=1}^{N}v_{n}
  32. v ¯ N \bar{v}_{N}
  33. 𝔼 [ v ¯ N ] = 1 N n = 1 N 𝔼 [ v n ] \mathbb{E}[\bar{v}_{N}]=\frac{1}{N}\sum_{n=1}^{N}\mathbb{E}[v_{n}]
  34. V V
  35. 1 N n = 1 N 𝔼 [ v ¯ n ] \frac{1}{N}\sum_{n=1}^{N}\mathbb{E}[\bar{v}_{n}]
  36. V V
  37. λ V \lambda\in V
  38. φ , 𝔼 [ v ¯ N ] - λ 0 \langle\varphi,\mathbb{E}[\bar{v}_{N}]-\lambda\rangle\to 0
  39. φ V * \varphi\in V^{*}
  40. 𝔼 [ v ¯ N ] λ \mathbb{E}[\bar{v}_{N}]\to\lambda
  41. X X
  42. 𝔼 [ v ¯ N ] \mathbb{E}[\bar{v}_{N}]
  43. λ \lambda