wpmath0000008_4

File:Bjt_forward_active_bands.svg.html

  1. E c E_{c}
  2. E f E_{f}
  3. E i E_{i}
  4. E v E_{v}

File:Canada_Olympic_Park_mid-July_2006_from_north_side_of_TransCanada_Highway.jpg.html

  1. c k l @ c k l ckl@ckl

File:Cesaro-0.3.png.html

  1. a = 0.3 + i 0.3 a=0.3+i0.3

File:Cesaro-0.5.png.html

  1. a = 0.5 + i 0.5 a=0.5+i0.5

File:Chebyshev-big.svg.html

  1. ψ ( x ) - x + log ( π ) \psi(x)-x+\log(\pi)
  2. ± 1 2 x \pm\frac{1}{2}\sqrt{x}
  3. ψ ( x ) - x < - K x \psi(x)-x<-K\sqrt{x}
  4. ψ ( x ) - x > K x \psi(x)-x>K\sqrt{x}

File:Chebyshev-smooth.svg.html

  1. ψ 1 ( x ) - x 2 2 = 0 x ψ ( t ) d t - x 2 2 \psi_{1}(x)-\frac{x^{2}}{2}=\int_{0}^{x}\psi(t)\,dt-\frac{x^{2}}{2}
  2. x = 0 x=0
  3. x = 10 6 x=10^{6}

File:Chebyshev.svg.html

  1. ψ ( x ) - x + log ( π ) \psi(x)-x+\log(\pi)
  2. ± 1 2 x \pm\frac{1}{2}\sqrt{x}
  3. ψ ( x ) - x < - K x \psi(x)-x<-K\sqrt{x}
  4. ψ ( x ) - x > K x \psi(x)-x>K\sqrt{x}

File:ChebyshevPsi.png.html

  1. ψ \psi
  2. x - log 2 π x-\log 2\pi

File:Curve_-0.10_-0.80_-0.30_-0.60.png.html

  1. α = 0.5 , β = 1.0 , δ = - 0.1 , ϵ = - 0.6 , ζ = - 0.3 , η = - 0.6 \alpha=0.5,\beta=1.0,\delta=-0.1,\epsilon=-0.6,\zeta=-0.3,\eta=-0.6

File:Curve_-0.35_0.00_-0.35_0.00.png.html

  1. α = 0.5 , β = 1.0 , δ = - 0.35 , ϵ = 0 , ζ = - 0.35 , η = 0 \alpha=0.5,\beta=1.0,\delta=-0.35,\epsilon=0,\zeta=-0.35,\eta=0

File:Curve_0.00_0.60_0.18_0.60.png.html

  1. α = 0.5 , β = 1.0 , δ = 0 , ϵ = 0.6 , ζ = 0.18 , η = 0.6 \alpha=0.5,\beta=1.0,\delta=0,\epsilon=0.6,\zeta=0.18,\eta=0.6

File:Curve_0.33_-0.38_-0.18_-0.42.png.html

  1. α = 0.5 , β = 1.0 , δ = 0.33 , ϵ = - 0.38 , ζ = - 0.18 , η = - 0.42 \alpha=0.5,\beta=1.0,\delta=0.33,\epsilon=-0.38,\zeta=-0.18,\eta=-0.42

File:Divisor-distribution.jpeg.html

  1. Δ ( x ) = D ( x ) - x log x - x ( 2 γ - 1 ) \Delta(x)=D(x)-x\log x-x(2\gamma-1)
  2. D ( x ) D(x)
  3. D ( x ) = n x d ( n ) D(x)=\sum_{n\leq x}d(n)
  4. d ( n ) d(n)
  5. γ = 0.577 \gamma=0.577\ldots
  6. x = 0 x=0
  7. x = 10 7 x=10^{7}
  8. 10 7 10^{7}
  9. Δ ( x ) \Delta(x)
  10. 2 x 7 / 22 2x^{7/22}
  11. y = 0 y=0
  12. ± x 7 / 22 \pm x^{7/22}
  13. x = 0 x=0
  14. x = 10 7 x=10^{7}

File:Divisor-summatory-big.svg.html

  1. D ( x ) - x log x - x ( 2 γ - 1 ) D(x)-x\log x-x(2\gamma-1)
  2. D ( x ) D(x)
  3. D ( x ) = n x d ( n ) D(x)=\sum_{n\leq x}d(n)
  4. d ( n ) d(n)
  5. γ = 0.577 \gamma=0.577\ldots
  6. x = 10 7 x=10^{7}
  7. ± 2 x 7 / 22 \pm 2x^{7/22}

File:Divisor-summatory.svg.html

  1. D ( x ) - x log x - x ( 2 γ - 1 ) D(x)-x\log x-x(2\gamma-1)
  2. D ( x ) D(x)
  3. D ( x ) = n x d ( n ) D(x)=\sum_{n\leq x}d(n)
  4. d ( n ) d(n)
  5. γ = 0.577 \gamma=0.577\ldots
  6. x = 10 4 x=10^{4}
  7. ± 2 x 7 / 22 \pm 2x^{7/22}

File:Divisor_cube.svg.html

  1. σ 3 ( n ) = d | n d 3 \sigma_{3}(n)=\sum_{d|n}d^{3}
  2. 1 n 250 1\leq n\leq 250

File:Gaps.jpg.html

  1. h N ( g ) h_{N}(g)
  2. N = 2 20 , 2 22 , , 2 44 N=2^{20},2^{22},\ldots,2^{44}
  3. h N ( g ) h_{N}(g)
  4. g = 30 ( = 2 3 5 ) , 60 , g=30(=2\cdot 3\cdot 5),60,\dots
  5. g = 210 = 2 3 5 7 g=210=2\cdot 3\cdot 5\cdot 7
  6. N = 2 40 , N = 2 42 , N = 2 44 N=2^{40},~{}N=2^{42},~{}N=2^{44}

File:Koch-Peano-0.37.png.html

  1. a = 0.6 + i 0.37 a=0.6+i0.37

File:Koch-Peano-0.45.png.html

  1. a = 0.6 + i 0.45 a=0.6+i0.45

File:LangevinScrew_BornChart.png.html

  1. p 3 = 1 - ω 2 r 2 1 r ϕ + ω r 1 - ω 2 r 2 t \vec{p}_{3}=\sqrt{1-\omega^{2}\,r^{2}}\,\frac{1}{r}\,\partial_{\phi}+\frac{% \omega\,r}{\sqrt{1-\omega^{2}\,r^{2}}}\,\partial_{t}
  2. ϕ = 0 , t = 0 \phi=0,\,t=0
  3. p 2 = r \vec{p}_{2}=\partial_{r}
  4. p 2 , p 3 \vec{p}_{2},\;\vec{p}_{3}

File:Liouville-harmonic.svg.html

  1. M ( n ) = k = 1 n λ ( k ) k M(n)=\sum_{k=1}^{n}\frac{\lambda(k)}{k}
  2. 1 n 10 3 1\leq n\leq 10^{3}

File:MH_Yield_Surface_3D.png.html

  1. max ( | σ 1 - σ 2 | 2 - c + K σ 1 + σ 2 2 , | σ 2 - σ 3 | 2 - c + K σ 2 + σ 3 2 , | σ 3 - σ 1 | 2 - c + K σ 3 + σ 1 2 ) = 0 \max\left(\cfrac{|\sigma_{1}-\sigma_{2}|}{2}\ -c+K\cfrac{\sigma_{1}+\sigma_{2}% }{2}\ ,\ \cfrac{|\sigma_{2}-\sigma_{3}|}{2}\ -c+K\cfrac{\sigma_{2}+\sigma_{3}}% {2}\ ,\ \cfrac{|\sigma_{3}-\sigma_{1}|}{2}\ -c+K\cfrac{\sigma_{3}+\sigma_{1}}{% 2}\right)=0

File:UKPolls.png.html

  1. x ¯ 0 = 1 5 i = - 2 2 x i \bar{x}_{0}=\frac{1}{5}\sum_{i=-2}^{2}x_{i}

File:X3+xy4+y2.gif.html

  1. a 2 x 3 + a x y 4 + a 2 y 2 = 0 a^{2}x^{3}+axy^{4}+a^{2}y^{2}=0

File:Xswyield.png.html

  1. R R
  2. Y p Y_{p}
  3. P H = H . r P_{H}=H.r

Film_grain.html

  1. G = σ 2 a G=\sigma\sqrt{2a}

Fin_(extended_surface).html

  1. Q ˙ ( x + d x ) = Q ˙ ( x ) + d Q ˙ c o n v . \dot{Q}(x+dx)=\dot{Q}(x)+d\dot{Q}_{conv}.
  2. Q ˙ ( x ) = - k A c ( d T d x ) , \dot{Q}(x)=-kA_{c}\left(\frac{dT}{dx}\right),
  3. A c A_{c}
  4. q ′′ = h ( T - T ) , q^{\prime\prime}=h\left(T-T_{\infty}\right),
  5. T T_{\infty}
  6. d Q ˙ c o n v = P h ( T - T ) d x . d\dot{Q}_{conv}=Ph\left(T-T_{\infty}\right)dx.
  7. - k A c ( d T d x ) | x + d x = - k A c ( d T d x ) | x + P h ( T - T ) d x . -kA_{c}\left.\left(\frac{dT}{dx}\right)\right|_{x+dx}=-kA_{c}\left.\left(\frac% {dT}{dx}\right)\right|_{x}+Ph\left(T-T_{\infty}\right)dx.
  8. k d d x ( A c d T d x ) + P h ( T - T ) = 0 k\frac{d}{dx}\left(A_{c}\frac{dT}{dx}\right)+Ph\left(T-T_{\infty}\right)=0
  9. k A c d 2 T d x 2 + k d A c d x d T d x + P h ( T - T ) = 0. kA_{c}\frac{d^{2}T}{dx^{2}}+k\frac{dA_{c}}{dx}\frac{dT}{dx}+Ph\left(T-T_{% \infty}\right)=0.
  10. d 2 T d x 2 = h P k A c ( T - T ) . \frac{d^{2}T}{dx^{2}}=\frac{hP}{kA_{c}}\left(T-T_{\infty}\right).
  11. θ ( x ) = C 1 e m x + C 2 e - m x , \theta(x)=C_{1}e^{mx}+C_{2}e^{-mx},
  12. m 2 = h P k A c m^{2}=\frac{hP}{kA_{c}}
  13. θ ( x ) = T ( x ) - T \theta(x)=T(x)-T_{\infty}
  14. C 1 C_{1}
  15. C 2 C_{2}
  16. θ ( x = 0 ) = T b - T \theta(x=0)=T_{b}-T_{\infty}
  17. x = L x=L
  18. h A c ( T ( L ) - T ) = - k A c ( d T d x ) | x = L , hA_{c}\left(T(L)-T_{\infty}\right)=-kA_{c}\left.\left(\frac{dT}{dx}\right)% \right|_{x=L},
  19. h θ ( L ) = - k d θ d x | x = L . h\theta(L)=-k\left.\frac{d\theta}{dx}\right|_{x=L}.
  20. h ( C 1 e m L + C 2 e - m L ) = k m ( C 2 e - m L - C 1 e m L ) . h\left(C_{1}e^{mL}+C_{2}e^{-mL}\right)=km\left(C_{2}e^{-mL}-C_{1}e^{mL}\right).
  21. C 1 C_{1}
  22. C 2 C_{2}
  23. d θ d x | x = L = 0. \left.\frac{d\theta}{dx}\right|_{x=L}=0.
  24. θ ( L ) = θ L \theta(L)=\theta_{L}
  25. lim L θ L = 0 \lim_{L\rightarrow\infty}\theta_{L}=0\,
  26. Q ˙ total = h P k A c ( C 2 - C 1 ) . \dot{Q}\text{total}=\sqrt{hPkA_{c}}(C_{2}-C_{1}).
  27. θ θ b = cosh m ( L - x ) + ( h m k ) sinh m ( L - x ) cosh m L + ( h m k ) sinh m L \frac{\theta}{\theta_{b}}=\frac{\cosh{m(L-x)}+\left(\frac{h}{mk}\right)\sinh{m% (L-x)}}{\cosh{mL}+\left(\frac{h}{mk}\right)\sinh{mL}}
  28. h P k A c θ b sinh m L + ( h / m k ) cosh m L cosh m L + ( h / m k ) sinh m L \sqrt{hPkA_{c}}\theta_{b}\frac{\sinh{mL}+(h/mk)\cosh{mL}}{\cosh{mL}+(h/mk)% \sinh{mL}}
  29. θ θ b = cosh m ( L - x ) cosh m L \frac{\theta}{\theta_{b}}=\frac{\cosh{m(L-x)}}{\cosh{mL}}
  30. h P k A c θ b tanh m L \sqrt{hPkA_{c}}\theta_{b}\tanh{mL}
  31. θ θ b = θ L θ b sinh m x + sinh m ( L - x ) sinh m L \frac{\theta}{\theta_{b}}=\frac{\frac{\theta_{L}}{\theta_{b}}\sinh{mx}+\sinh{m% (L-x)}}{\sinh{mL}}
  32. h P k A c θ b cosh m L - θ L θ b sinh m L \sqrt{hPkA_{c}}\theta_{b}\frac{\cosh{mL}-\frac{\theta_{L}}{\theta_{b}}}{\sinh{% mL}}
  33. θ θ b = e - m x \frac{\theta}{\theta_{b}}=e^{-mx}
  34. h P k A c θ b \sqrt{hPkA_{c}}\theta_{b}
  35. Q ˙ f \dot{Q}_{f}
  36. ϵ f = Q ˙ f h A c , b θ b , \epsilon_{f}=\frac{\dot{Q}_{f}}{hA_{c,b}\theta_{b}},
  37. A c , b A_{c,b}
  38. η f = Q ˙ f h A f θ b . \eta_{f}=\frac{\dot{Q}_{f}}{hA_{f}\theta_{b}}.
  39. A f A_{f}
  40. η o = Q ˙ t h A t θ b , \eta_{o}=\frac{\dot{Q}_{t}}{hA_{t}\theta_{b}},
  41. A t A_{t}
  42. Q ˙ t \dot{Q}_{t}

Fine_topology_(potential_theory).html

  1. Δ u 0 , \Delta u\geq 0,
  2. Δ \Delta
  3. \R n \R^{n}
  4. \R n \R^{n}
  5. n 2 n\geq 2
  6. U U
  7. ζ \zeta
  8. v v
  9. ζ \zeta
  10. v ( ζ ) > lim sup z ζ , z U v ( z ) . v(\zeta)>\limsup_{z\to\zeta,z\in U}v(z).
  11. U U
  12. ζ \zeta
  13. U U
  14. ζ \zeta
  15. n 2 n\geq 2
  16. F F
  17. \R n \R^{n}
  18. F F
  19. \R n \R^{n}
  20. \R n \R^{n}
  21. \R n \R^{n}
  22. \R n \R^{n}

Finite-difference_frequency-domain_method.html

  1. A x = b Ax=b
  2. 𝐉 ( 𝐱 ) e i ω t \mathbf{J}(\mathbf{x})e^{i\omega t}
  3. A x = b Ax=b
  4. A x = λ x Ax=\lambda x

Finite-dimensional_distribution.html

  1. ( X , , μ ) (X,\mathcal{F},\mu)
  2. μ \mu
  3. f * ( μ ) f_{*}(\mu)
  4. f : X k f:X\to\mathbb{R}^{k}
  5. k k\in\mathbb{N}
  6. ( Ω , , ) (\Omega,\mathcal{F},\mathbb{P})
  7. X : I × Ω 𝕏 X:I\times\Omega\to\mathbb{X}
  8. X X
  9. i 1 i k X \mathbb{P}_{i_{1}\dots i_{k}}^{X}
  10. 𝕏 k \mathbb{X}^{k}
  11. k k\in\mathbb{N}
  12. i 1 i k X ( S ) := { ω Ω | ( X i 1 ( ω ) , , X i k ( ω ) ) S } . \mathbb{P}_{i_{1}\dots i_{k}}^{X}(S):=\mathbb{P}\left\{\omega\in\Omega\left|% \left(X_{i_{1}}(\omega),\dots,X_{i_{k}}(\omega)\right)\in S\right.\right\}.
  13. i 1 i k X ( A 1 × × A k ) := { ω Ω | X i j ( ω ) A j for 1 j k } . \mathbb{P}_{i_{1}\dots i_{k}}^{X}(A_{1}\times\cdots\times A_{k}):=\mathbb{P}% \left\{\omega\in\Omega\left|X_{i_{j}}(\omega)\in A_{j}\mathrm{\,for\,}1\leq j% \leq k\right.\right\}.
  14. X X
  15. μ \mu
  16. X \mathcal{L}_{X}
  17. X X
  18. 𝕏 I \mathbb{X}^{I}
  19. I I
  20. 𝕏 \mathbb{X}
  21. X X
  22. f * ( X ) f_{*}\left(\mathcal{L}_{X}\right)
  23. 𝕏 k \mathbb{X}^{k}
  24. f : 𝕏 I 𝕏 k : σ ( σ ( t 1 ) , , σ ( t k ) ) f:\mathbb{X}^{I}\to\mathbb{X}^{k}:\sigma\mapsto\left(\sigma(t_{1}),\dots,% \sigma(t_{k})\right)
  25. t 1 , , t k t_{1},\dots,t_{k}
  26. ( μ n ) n = 1 (\mu_{n})_{n=1}^{\infty}
  27. μ n \mu_{n}
  28. μ \mu
  29. μ n \mu_{n}
  30. μ \mu

Finite_difference_method.html

  1. f ( x 0 + h ) = f ( x 0 ) + f ( x 0 ) 1 ! h + f ( 2 ) ( x 0 ) 2 ! h 2 + + f ( n ) ( x 0 ) n ! h n + R n ( x ) , f(x_{0}+h)=f(x_{0})+\frac{f^{\prime}(x_{0})}{1!}h+\frac{f^{(2)}(x_{0})}{2!}h^{% 2}+\cdots+\frac{f^{(n)}(x_{0})}{n!}h^{n}+R_{n}(x),
  2. f ( x 0 + h ) = f ( x 0 ) + f ( x 0 ) h + R 1 ( x ) , f(x_{0}+h)=f(x_{0})+f^{\prime}(x_{0})h+R_{1}(x),
  3. f ( a + h ) = f ( a ) + f ( a ) h + R 1 ( x ) , f(a+h)=f(a)+f^{\prime}(a)h+R_{1}(x),
  4. f ( a + h ) h = f ( a ) h + f ( a ) + R 1 ( x ) h {f(a+h)\over h}={f(a)\over h}+f^{\prime}(a)+{R_{1}(x)\over h}
  5. f ( a ) = f ( a + h ) - f ( a ) h - R 1 ( x ) h f^{\prime}(a)={f(a+h)-f(a)\over h}-{R_{1}(x)\over h}
  6. R 1 ( x ) R_{1}(x)
  7. f ( a ) f ( a + h ) - f ( a ) h . f^{\prime}(a)\approx{f(a+h)-f(a)\over h}.
  8. f ( x i ) - f i f^{\prime}(x_{i})-f^{\prime}_{i}
  9. f ( x i ) f^{\prime}(x_{i})
  10. f i f^{\prime}_{i}
  11. f ( x 0 + h ) f(x_{0}+h)
  12. R n ( x 0 + h ) = f ( n + 1 ) ( ξ ) ( n + 1 ) ! ( h ) n + 1 R_{n}(x_{0}+h)=\frac{f^{(n+1)}(\xi)}{(n+1)!}(h)^{n+1}
  13. x 0 < ξ < x 0 + h x_{0}<\xi<x_{0}+h
  14. f ( x i ) = f ( x 0 + i h ) f(x_{i})=f(x_{0}+ih)
  15. f ( x 0 + i h ) = f ( x 0 ) + f ( x 0 ) i h + f ′′ ( ξ ) 2 ! ( i h ) 2 , f(x_{0}+ih)=f(x_{0})+f^{\prime}(x_{0})ih+\frac{f^{\prime\prime}(\xi)}{2!}(ih)^% {2},
  16. f ( x 0 + i h ) - f ( x 0 ) i h = f ( x 0 ) + f ′′ ( ξ ) 2 ! i h , \frac{f(x_{0}+ih)-f(x_{0})}{ih}=f^{\prime}(x_{0})+\frac{f^{\prime\prime}(\xi)}% {2!}ih,
  17. f ( x 0 + i h ) - f ( x 0 ) i h = f ( x 0 ) + O ( h ) . \frac{f(x_{0}+ih)-f(x_{0})}{ih}=f^{\prime}(x_{0})+O(h).
  18. u ( x ) = 3 u ( x ) + 2. u^{\prime}(x)=3u(x)+2.\,
  19. u ( x + h ) - u ( x ) h u ( x ) \frac{u(x+h)-u(x)}{h}\approx u^{\prime}(x)
  20. u ( x + h ) = u ( x ) + h ( 3 u ( x ) + 2 ) . u(x+h)=u(x)+h(3u(x)+2).\,
  21. U t = U x x U_{t}=U_{xx}\,
  22. U ( 0 , t ) = U ( 1 , t ) = 0 U(0,t)=U(1,t)=0\,
  23. U ( x , 0 ) = U 0 ( x ) U(x,0)=U_{0}(x)\,
  24. x 0 , , x J x_{0},...,x_{J}
  25. t 0 , . , t N t_{0},....,t_{N}
  26. u ( x j , t n ) = u j n u(x_{j},t_{n})=u_{j}^{n}
  27. u ( x j , t n ) . u(x_{j},t_{n}).
  28. t n t_{n}
  29. x j x_{j}
  30. u j n + 1 - u j n k = u j + 1 n - 2 u j n + u j - 1 n h 2 . \frac{u_{j}^{n+1}-u_{j}^{n}}{k}=\frac{u_{j+1}^{n}-2u_{j}^{n}+u_{j-1}^{n}}{h^{2% }}.\,
  31. u j n + 1 u_{j}^{n+1}
  32. u j n + 1 = ( 1 - 2 r ) u j n + r u j - 1 n + r u j + 1 n u_{j}^{n+1}=(1-2r)u_{j}^{n}+ru_{j-1}^{n}+ru_{j+1}^{n}
  33. r = k / h 2 . r=k/h^{2}.
  34. u 0 n u_{0}^{n}
  35. u J n u_{J}^{n}
  36. r 1 / 2 r\leq 1/2
  37. Δ u = O ( k ) + O ( h 2 ) \Delta u=O(k)+O(h^{2})\,
  38. t n + 1 t_{n+1}
  39. x j x_{j}
  40. u j n + 1 - u j n k = u j + 1 n + 1 - 2 u j n + 1 + u j - 1 n + 1 h 2 . \frac{u_{j}^{n+1}-u_{j}^{n}}{k}=\frac{u_{j+1}^{n+1}-2u_{j}^{n+1}+u_{j-1}^{n+1}% }{h^{2}}.\,
  41. u j n + 1 u_{j}^{n+1}
  42. ( 1 + 2 r ) u j n + 1 - r u j - 1 n + 1 - r u j + 1 n + 1 = u j n (1+2r)u_{j}^{n+1}-ru_{j-1}^{n+1}-ru_{j+1}^{n+1}=u_{j}^{n}
  43. Δ u = O ( k ) + O ( h 2 ) . \Delta u=O(k)+O(h^{2}).\,
  44. t n + 1 / 2 t_{n+1/2}
  45. x j x_{j}
  46. u j n + 1 - u j n k = 1 2 ( u j + 1 n + 1 - 2 u j n + 1 + u j - 1 n + 1 h 2 + u j + 1 n - 2 u j n + u j - 1 n h 2 ) . \frac{u_{j}^{n+1}-u_{j}^{n}}{k}=\frac{1}{2}\left(\frac{u_{j+1}^{n+1}-2u_{j}^{n% +1}+u_{j-1}^{n+1}}{h^{2}}+\frac{u_{j+1}^{n}-2u_{j}^{n}+u_{j-1}^{n}}{h^{2}}% \right).\,
  47. u j n + 1 u_{j}^{n+1}
  48. ( 2 + 2 r ) u j n + 1 - r u j - 1 n + 1 - r u j + 1 n + 1 = ( 2 - 2 r ) u j n + r u j - 1 n + r u j + 1 n (2+2r)u_{j}^{n+1}-ru_{j-1}^{n+1}-ru_{j+1}^{n+1}=(2-2r)u_{j}^{n}+ru_{j-1}^{n}+% ru_{j+1}^{n}
  49. Δ u = O ( k 2 ) + O ( h 2 ) . \Delta u=O(k^{2})+O(h^{2}).\,

First-order_hold.html

  1. x s ( t ) x_{s}(t)\,
  2. = x ( t ) T n = - δ ( t - n T ) =x(t)\ T\sum_{n=-\infty}^{\infty}\delta(t-nT)
  3. = T n = - x ( n T ) δ ( t - n T ) =T\sum_{n=-\infty}^{\infty}x(nT)\delta(t-nT)
  4. x FOH ( t ) = n = - x ( n T ) tri ( t - n T T ) x_{\mathrm{FOH}}(t)\,=\sum_{n=-\infty}^{\infty}x(nT)\mathrm{tri}\left(\frac{t-% nT}{T}\right)
  5. h FOH ( t ) = 1 T tri ( t T ) = { 1 T ( 1 - | t | T ) if | t | < T 0 otherwise h_{\mathrm{FOH}}(t)\,=\frac{1}{T}\mathrm{tri}\left(\frac{t}{T}\right)=\begin{% cases}\frac{1}{T}\left(1-\frac{|t|}{T}\right)&\mbox{if }~{}|t|<T\\ 0&\mbox{otherwise}\end{cases}
  6. tri ( x ) \mathrm{tri}(x)
  7. H FOH ( f ) H_{\mathrm{FOH}}(f)\,
  8. = { h FOH ( t ) } =\mathcal{F}\{h_{\mathrm{FOH}}(t)\}
  9. = ( e i π f T - e - i π f T i 2 π f T ) 2 =\left(\frac{e^{i\pi fT}-e^{-i\pi fT}}{i2\pi fT}\right)^{2}
  10. = sinc 2 ( f T ) =\mathrm{sinc}^{2}(fT)
  11. sinc ( x ) \mathrm{sinc}(x)
  12. H FOH ( s ) H_{\mathrm{FOH}}(s)\,
  13. = { h FOH ( t ) } =\mathcal{L}\{h_{\mathrm{FOH}}(t)\}
  14. = ( e s T / 2 - e - s T / 2 s T ) 2 =\left(\frac{e^{sT/2}-e^{-sT/2}}{sT}\right)^{2}
  15. x FOH ( t ) = n = - x ( n T ) tri ( t - T - n T T ) x_{\mathrm{FOH}}(t)\,=\sum_{n=-\infty}^{\infty}x(nT)\mathrm{tri}\left(\frac{t-% T-nT}{T}\right)
  16. h FOH ( t ) = 1 T tri ( t - T T ) = { 1 T ( 1 - | t - T | T ) if | t - T | < T 0 otherwise h_{\mathrm{FOH}}(t)\,=\frac{1}{T}\mathrm{tri}\left(\frac{t-T}{T}\right)=\begin% {cases}\frac{1}{T}\left(1-\frac{|t-T|}{T}\right)&\mbox{if }~{}|t-T|<T\\ 0&\mbox{otherwise}\end{cases}
  17. tri ( x ) \mathrm{tri}(x)
  18. H FOH ( f ) H_{\mathrm{FOH}}(f)\,
  19. = { h FOH ( t ) } =\mathcal{F}\{h_{\mathrm{FOH}}(t)\}
  20. = ( 1 - e - i 2 π f T i 2 π f T ) 2 =\left(\frac{1-e^{-i2\pi fT}}{i2\pi fT}\right)^{2}
  21. = e - i 2 π f T sinc 2 ( f T ) =e^{-i2\pi fT}\mathrm{sinc}^{2}(fT)
  22. sinc ( x ) \mathrm{sinc}(x)
  23. H FOH ( s ) H_{\mathrm{FOH}}(s)\,
  24. = { h FOH ( t ) } =\mathcal{L}\{h_{\mathrm{FOH}}(t)\}
  25. = ( 1 - e - s T s T ) 2 =\left(\frac{1-e^{-sT}}{sT}\right)^{2}
  26. x s ( t ) x_{s}(t)\,
  27. = x ( t ) T n = - δ ( t - n T ) =x(t)\ T\sum_{n=-\infty}^{\infty}\delta(t-nT)
  28. = T n = - x ( n T ) δ ( t - n T ) =T\sum_{n=-\infty}^{\infty}x(nT)\delta(t-nT)
  29. x FOH ( t ) x_{\mathrm{FOH}}(t)\,
  30. = n = - ( x ( n T ) + ( x ( n T ) - x ( ( n - 1 ) T ) ) t - n T T ) rect ( t - n T T - 1 2 ) =\sum_{n=-\infty}^{\infty}\left(x(nT)+\left(x(nT)-x((n-1)T)\right)\frac{t-nT}{% T}\right)\mathrm{rect}\left(\frac{t-nT}{T}-\frac{1}{2}\right)
  31. = n = - x ( n T ) ( rect ( t - n T T - 1 2 ) - rect ( t - n T T - 3 2 ) + tri ( t - n T T - 1 ) ) =\sum_{n=-\infty}^{\infty}x(nT)\left(\mathrm{rect}\left(\frac{t-nT}{T}-\frac{1% }{2}\right)-\mathrm{rect}\left(\frac{t-nT}{T}-\frac{3}{2}\right)+\mathrm{tri}% \left(\frac{t-nT}{T}-1\right)\right)
  32. h FOH ( t ) h_{\mathrm{FOH}}(t)\,
  33. = 1 T ( rect ( t T - 1 2 ) - rect ( t T - 3 2 ) + tri ( t T - 1 ) ) =\frac{1}{T}\left(\mathrm{rect}\left(\frac{t}{T}-\frac{1}{2}\right)-\mathrm{% rect}\left(\frac{t}{T}-\frac{3}{2}\right)+\mathrm{tri}\left(\frac{t}{T}-1% \right)\right)
  34. = { 1 T ( 1 + t T ) if 0 t < T 1 T ( 1 - t T ) if T t < 2 T 0 otherwise =\begin{cases}\frac{1}{T}\left(1+\frac{t}{T}\right)&\mbox{if }~{}0\leq t<T\\ \frac{1}{T}\left(1-\frac{t}{T}\right)&\mbox{if }~{}T\leq t<2T\\ 0&\mbox{otherwise}\end{cases}
  35. rect ( x ) \mathrm{rect}(x)
  36. tri ( x ) \mathrm{tri}(x)
  37. H FOH ( f ) H_{\mathrm{FOH}}(f)\,
  38. = { h FOH ( t ) } =\mathcal{F}\{h_{\mathrm{FOH}}(t)\}
  39. = ( 1 + i 2 π f T ) ( 1 - e - i 2 π f T i 2 π f T ) 2 =(1+i2\pi fT)\left(\frac{1-e^{-i2\pi fT}}{i2\pi fT}\right)^{2}
  40. = ( 1 + i 2 π f T ) e - i 2 π f T sinc 2 ( f T ) ) =(1+i2\pi fT)e^{-i2\pi fT}\mathrm{sinc}^{2}(fT))
  41. sinc ( x ) \mathrm{sinc}(x)
  42. H FOH ( s ) H_{\mathrm{FOH}}(s)\,
  43. = { h FOH ( t ) } =\mathcal{L}\{h_{\mathrm{FOH}}(t)\}
  44. = ( 1 + s T ) ( 1 - e - s T s T ) 2 =(1+sT)\left(\frac{1-e^{-sT}}{sT}\right)^{2}

First-order_reduction.html

  1. FO L \mbox{FO}~{}\subsetneq\mbox{L}~{}

Fisher's_method.html

  1. X 2 k 2 - 2 i = 1 k ln ( p i ) , X^{2}_{2k}\sim-2\sum_{i=1}^{k}\ln(p_{i}),
  2. α ( k + 1 ) / ( 2 k ) \alpha(k+1)/(2k)
  3. α \alpha
  4. k k
  5. Z i = 1 k Z i k , Z\sim\frac{\sum_{i=1}^{k}Z_{i}}{\sqrt{k}},
  6. Z i = 1 k w i Z i i = 1 k w i 2 , Z\sim\frac{\sum_{i=1}^{k}w_{i}Z_{i}}{\sqrt{\sum_{i=1}^{k}w_{i}^{2}}},

Fisher_kernel.html

  1. U X = θ log P ( X | θ ) U_{X}=\nabla_{\theta}\log P(X|\theta)
  2. K ( X i , X j ) = U X i T I - 1 U X j K(X_{i},X_{j})=U_{X_{i}}^{T}I^{-1}U_{X_{j}}

Fixed-asset_turnover.html

  1. F i x e d A s s e t T u r n o v e r = N e t s a l e s A v e r a g e n e t f i x e d a s s e t s Fixed\ Asset\ Turnover=\frac{Net\ sales}{Average\ net\ fixed\ assets}

Fixed-point_iteration.html

  1. f f
  2. x 0 x_{0}
  3. f f
  4. x n + 1 = f ( x n ) , n = 0 , 1 , 2 , x_{n+1}=f(x_{n}),\,n=0,1,2,\dots
  5. x 0 , x 1 , x 2 , x_{0},x_{1},x_{2},\dots
  6. x x
  7. f f
  8. x x
  9. f f
  10. f ( x ) = x f(x)=x
  11. f f
  12. f ( x ) = 1 2 ( a x + x ) f(x)=\frac{1}{2}\left(\frac{a}{x}+x\right)
  13. x = a x=\sqrt{a}
  14. x 0 0 x_{0}\gg 0
  15. x n + 1 = cos x n x_{n+1}=\cos x_{n}\,
  16. f ( x ) = cos x f(x)=\cos x\,
  17. x 0 . x_{0}.
  18. | x n - x 0 | q n 1 - q | x 1 - x 0 | = C q n |x_{n}-x_{0}|\leq{q^{n}\over 1-q}|x_{1}-x_{0}|=Cq^{n}
  19. q = 0.85 q=0.85
  20. x 0 = 1 x_{0}=1
  21. q n q^{n}
  22. x n + 1 = 2 x n x_{n+1}=2x_{n}\,
  23. x 0 = 0 x_{0}=0
  24. f ( x ) = 2 x f(x)=2x\,
  25. x n + 1 = { x n 2 , x n 0 1 , x n = 0 x_{n+1}=\begin{cases}\frac{x_{n}}{2},&x_{n}\neq 0\\ 1,&x_{n}=0\end{cases}
  26. x 0 x_{0}
  27. f ( x ) = { x 2 , x 0 1 , x = 0 f(x)=\begin{cases}\frac{x}{2},&x\neq 0\\ 1,&x=0\end{cases}
  28. x = 0 x=0
  29. f ( x ) f(x)
  30. x n + 1 = x n - f ( x n ) f ( x n ) x_{n+1}=x_{n}-\frac{f(x_{n})}{f^{\prime}(x_{n})}
  31. g ( x ) = x - f ( x ) f ( x ) g(x)=x-\frac{f(x)}{f^{\prime}(x)}
  32. x n + 1 = g ( x n ) x_{n+1}=g(x_{n})
  33. x x
  34. g g
  35. x = g ( x ) = x - f ( x ) f ( x ) x=g(x)=x-\frac{f(x)}{f^{\prime}(x)}
  36. f ( x ) / f ( x ) = 0 f(x)/f^{\prime}(x)=0
  37. f ( x ) = 0 f(x)=0
  38. x x
  39. f f
  40. | x n - x | < C q 2 n |x_{n}-x|<Cq^{2^{n}}
  41. | x n - x | < C q 3 n |x_{n}-x|<Cq^{3^{n}}
  42. C q k n Cq^{k^{n}}
  43. k k\in\mathbb{N}
  44. k k
  45. y = a y y^{\prime}=ay
  46. y = 0 y=0
  47. f f
  48. L < 1 L<1
  49. x 0 . x_{0}.
  50. f f
  51. L < 1 L<1
  52. { x n , n = 0 , 1 , 2... } \{x_{n},n=0,1,2...\}
  53. | x 2 - x 1 | = | f ( x 1 ) - f ( x 0 ) | L | x 1 - x 0 | |x_{2}-x_{1}|=|f(x_{1})-f(x_{0})|\leq L|x_{1}-x_{0}|
  54. | x 3 - x 2 | = | f ( x 2 ) - f ( x 1 ) | L | x 2 - x 1 | |x_{3}-x_{2}|=|f(x_{2})-f(x_{1})|\leq L|x_{2}-x_{1}|
  55. \cdots
  56. | x n - x n - 1 | = | f ( x n - 1 ) - f ( x n - 2 ) | L | x n - 1 - x n - 2 | |x_{n}-x_{n-1}|=|f(x_{n-1})-f(x_{n-2})|\leq L|x_{n-1}-x_{n-2}|
  57. | x n - x n - 1 | L n - 1 | x 1 - x 0 | |x_{n}-x_{n-1}|\leq L^{n-1}|x_{1}-x_{0}|
  58. L < 1 L<1
  59. L n - 1 0 L^{n-1}\rightarrow 0
  60. n n\rightarrow\infty
  61. { x n } \{x_{n}\}
  62. x * x^{*}
  63. x n = f ( x n - 1 ) x_{n}=f(x_{n-1})
  64. n n
  65. x * = f ( x * ) x^{*}=f(x^{*})
  66. x * x^{*}
  67. f f

Fleiss'_kappa.html

  1. κ \kappa\,
  2. κ = P ¯ - P e ¯ 1 - P e ¯ \kappa=\frac{\bar{P}-\bar{P_{e}}}{1-\bar{P_{e}}}
  3. 1 - P e ¯ 1-\bar{P_{e}}
  4. P ¯ - P e ¯ \bar{P}-\bar{P_{e}}
  5. κ = 1 \kappa=1~{}
  6. κ 0 \kappa\leq 0
  7. p j = 1 N n i = 1 N n i j , 1 = 1 n j = 1 k n i j p_{j}=\frac{1}{Nn}\sum_{i=1}^{N}n_{ij},\quad\quad 1=\frac{1}{n}\sum_{j=1}^{k}n% _{ij}
  8. P i P_{i}\,
  9. P i = 1 n ( n - 1 ) j = 1 k n i j ( n i j - 1 ) P_{i}=\frac{1}{n(n-1)}\sum_{j=1}^{k}n_{ij}(n_{ij}-1)
  10. = 1 n ( n - 1 ) j = 1 k ( n i j 2 - n i j ) =\frac{1}{n(n-1)}\sum_{j=1}^{k}(n_{ij}^{2}-n_{ij})
  11. = 1 n ( n - 1 ) [ ( j = 1 k n i j 2 ) - ( n ) ] =\frac{1}{n(n-1)}[(\sum_{j=1}^{k}n_{ij}^{2})-(n)]
  12. P ¯ \bar{P}
  13. P i P_{i}\,
  14. P e ¯ \bar{P_{e}}
  15. κ \kappa\,
  16. P ¯ = 1 N i = 1 N P i \bar{P}=\frac{1}{N}\sum_{i=1}^{N}P_{i}
  17. = 1 N n ( n - 1 ) ( i = 1 N j = 1 k n i j 2 - N n ) =\frac{1}{Nn(n-1)}(\sum_{i=1}^{N}\sum_{j=1}^{k}n_{ij}^{2}-Nn)
  18. P e ¯ = j = 1 k p j 2 \bar{P_{e}}=\sum_{j=1}^{k}p_{j}^{2}
  19. P i P_{i}\,
  20. p j p_{j}\,
  21. n n
  22. N N
  23. k k
  24. N N
  25. n n
  26. k k
  27. P i P_{i}\,
  28. p 1 = 0 + 0 + 0 + 0 + 2 + 7 + 3 + 2 + 6 + 0 140 = 0.143 p_{1}=\frac{0+0+0+0+2+7+3+2+6+0}{140}=0.143
  29. P 2 = 1 14 ( 14 - 1 ) ( 0 2 + 2 2 + 6 2 + 4 2 + 2 2 - 14 ) = 0.253 P_{2}=\frac{1}{14(14-1)}\left(0^{2}+2^{2}+6^{2}+4^{2}+2^{2}-14\right)=0.253
  30. P ¯ \bar{P}
  31. P i P_{i}
  32. i = 1 N P i = 1.000 + 0.253 + + 0.286 + 0.286 = 3.780 \sum_{i=1}^{N}P_{i}=1.000+0.253+\cdots+0.286+0.286=3.780
  33. P ¯ = 1 ( 10 ) ( 3.780 ) = 0.378 \bar{P}=\frac{1}{(10)}(3.780)=0.378
  34. P ¯ e = 0.143 2 + 0.200 2 + 0.279 2 + 0.150 2 + 0.229 2 = 0.213 \bar{P}_{e}=0.143^{2}+0.200^{2}+0.279^{2}+0.150^{2}+0.229^{2}=0.213
  35. κ = 0.378 - 0.213 1 - 0.213 = 0.210 \kappa=\frac{0.378-0.213}{1-0.213}=0.210
  36. κ \kappa
  37. κ \kappa

Flicker_noise.html

  1. K C ox W L f \frac{K}{C_{\mathrm{ox}}\cdot WLf}
  2. C ox C_{\mathrm{ox}}

Floating-gate_MOSFET.html

  1. g m i = C i C T g m for i = [ 1 , N ] g_{mi}=\frac{C_{i}}{C_{T}}g_{m}\quad\mbox{for}~{}\quad i=[1,N]
  2. g d s F = g d s + C G D C T g m g_{dsF}=g_{ds}+\frac{C_{GD}}{C_{T}}g_{m}
  3. g m b F = g m b + C G B C T g m g_{mbF}=g_{mb}+\frac{C_{GB}}{C_{T}}g_{m}
  4. C T C_{T}

Flow_coefficient.html

  1. C v = F S G Δ P C_{v}=F\sqrt{\dfrac{SG}{\Delta P}}

Floyd_Williams.html

  1. L 2 ( Γ \ G ) L^{2}(\Gamma\backslash G)

Flue-gas_stack.html

  1. Δ P = C a h ( 1 T o - 1 T i ) \Delta P=\;C\,a\;h\;\bigg(\frac{1}{T_{o}}-\frac{1}{T_{i}}\bigg)
  2. Q = C A 2 g H T i - T o T i Q=C\;A\;\sqrt{2\;g\;H\;\frac{T_{i}-T_{o}}{T_{i}}}

Fluid_conductance.html

  1. C b = K A b C_{b}=K\frac{A}{b}
  2. C b C_{b}
  3. K K
  4. A A
  5. b b
  6. Q b = C b ( h b - h ) Q_{b}=C_{b}(h_{b}-h)\,
  7. Q b Q_{b}
  8. h b h_{b}
  9. h h
  10. Q = ( P 1 - P 2 ) C . Q=(P_{1}-P_{2})C.
  11. Q Q
  12. P 1 P_{1}
  13. P 2 P_{2}
  14. C C
  15. C = 15 d 2 C=15d^{2}
  16. d d

Fluorescence_anisotropy.html

  1. τ \tau
  2. ϕ \phi
  3. τ \tau
  4. r = r 0 1 + τ / ϕ r=\frac{r_{0}}{1+\tau/\phi}
  5. τ \tau
  6. ϕ \phi

Fluorescence_interference_contrast_microscopy.html

  1. I F L I C I_{FLIC}
  2. P e x P_{ex}
  3. P e m P_{em}
  4. n 1 n_{1}
  5. n 0 n_{0}
  6. λ e x \lambda_{ex}
  7. λ e m \lambda_{em}
  8. e e x ′′ ′′ {}^{\prime\prime}e_{ex}^{\prime\prime}
  9. P e x P_{ex}
  10. F i n F_{in}
  11. P e x F i n e e x 2 P_{ex}\propto\mid F_{in}\cdot e_{ex}\mid^{2}
  12. F i n F_{in}
  13. Φ i n \Phi_{in}
  14. Φ i n = 4 π n 1 d cos θ 1 i n λ e x \Phi_{in}=\frac{4\pi n_{1}d\cos\theta^{in}_{1}}{\lambda_{ex}}
  15. θ 1 i n \theta^{in}_{1}
  16. F i n F_{in}
  17. θ i \theta_{i}
  18. θ j \theta_{j}
  19. θ i \theta_{i}
  20. θ j \theta_{j}
  21. r i j T E = n i cos θ i - n j cos θ j n i cos θ i + n j cos θ j r i j T M = n j cos θ i - n i cos θ j n j cos θ i + n i cos θ j r^{TE}_{ij}=\frac{n_{i}\cos\theta_{i}-n_{j}\cos\theta_{j}}{n_{i}\cos\theta_{i}% +n_{j}\cos\theta_{j}}\quad r^{TM}_{ij}=\frac{n_{j}\cos\theta_{i}-n_{i}\cos% \theta_{j}}{n_{j}\cos\theta_{i}+n_{i}\cos\theta_{j}}
  22. F i n = sin γ i n [ 0 1 + r 10 T E 𝑒 i Φ i n 0 ] + cos γ i n [ cos θ 1 i n ( 1 - r 10 T M 𝑒 i Φ i n ) 0 sin θ 1 i n ( 1 + r 10 T M 𝑒 i Φ i n ) ] F_{in}=\sin\gamma_{in}\left[\begin{array}[]{c}0\\ 1+r^{TE}_{10}\,\textit{e}^{i\Phi_{in}}\\ 0\end{array}\right]+\cos\gamma_{in}\left[\begin{array}[]{c}\cos\theta^{in}_{1}% (1-r^{TM}_{10}\,\textit{e}^{i\Phi_{in}})\\ 0\\ \sin\theta^{in}_{1}(1+r^{TM}_{10}\,\textit{e}^{i\Phi_{in}})\end{array}\right]
  23. γ i n \gamma_{in}
  24. θ e x \theta_{ex}
  25. ϕ e x \phi_{ex}
  26. 𝑒 e x = [ cos ϕ e x sin θ e x sin ϕ e x sin θ e x cos θ e x ] \,\textit{e}_{ex}=\left[\begin{array}[]{c}\cos\phi_{ex}\sin\theta_{ex}\\ \sin\phi_{ex}\sin\theta_{ex}\\ \cos\theta_{ex}\end{array}\right]
  27. F i n F_{in}
  28. 𝑒 e x \,\textit{e}_{ex}
  29. P e x P_{ex}
  30. θ 1 i n \theta^{in}_{1}
  31. γ i n \gamma_{in}
  32. θ e x \theta_{ex}
  33. λ e x \lambda_{ex}
  34. F i n e e x 2 \mid F_{in}\cdot e_{ex}\mid^{2}
  35. P e x P_{ex}
  36. < F i n e e x 2 > sin θ 1 i n d θ 1 i n A i n ( θ 1 i n ) × sin θ e x d θ e x O ( θ e x ) U e x ( λ i n , θ 1 i n . θ e x ) <\mid F_{in}\cdot e_{ex}\mid^{2}>\propto\int\sin\theta_{1}^{in}d\theta_{1}^{in% }A_{in}(\theta_{1}^{in})\times\int\sin\theta_{ex}d\theta_{ex}O(\theta_{ex})U_{% ex}(\lambda_{in},\theta_{1}^{in}.\theta_{ex})
  37. U e x = sin 2 θ e x 1 + r 10 T E 𝑒 i Φ i n 2 + sin 2 θ e x cos 2 θ 1 i n 1 - r 10 T M 𝑒 i Φ i n 2 + 2 cos 2 θ e x sin 2 θ 1 i n 1 + r 10 T M 𝑒 i Φ i n 2 U_{ex}=\sin^{2}\theta_{ex}\mid 1+r^{TE}_{10}\,\textit{e}^{i\Phi_{in}}\mid^{2}+% \sin^{2}\theta_{ex}\cos^{2}\theta^{in}_{1}\mid 1-r^{TM}_{10}\,\textit{e}^{i% \Phi_{in}}\mid^{2}+2\cos^{2}\theta_{ex}\sin^{2}\theta^{in}_{1}\mid 1+r^{TM}_{1% 0}\,\textit{e}^{i\Phi_{in}}\mid^{2}
  38. O ( θ e x ) O(\theta_{ex})
  39. ϕ e x \phi_{ex}
  40. γ i n \gamma_{in}
  41. I ( λ e x ) I(\lambda_{ex})
  42. ϵ ( λ e x ) \epsilon(\lambda_{ex})
  43. P e x d λ e x I ( λ e x ) ϵ ( λ e x ) < F i n e e x 2 Align g t ; P_{ex}\propto\int d\lambda_{ex}I(\lambda_{ex})\epsilon(\lambda_{ex})<\mid F_{% in}\cdot e_{ex}\mid^{2}&gt;
  44. P e m P_{em}
  45. P e x P_{ex}
  46. P e m d λ e m Φ d e t ( λ e m ) 𝑓 ( λ e m ) < F i n e e x 2 Align g t ; P_{em}\propto\int d\lambda_{em}\Phi_{det}(\lambda_{em})\,\textit{f}(\lambda_{% em})<\mid F_{in}\cdot e_{ex}\mid^{2}&gt;
  47. I F L I C P e x P e m I_{FLIC}\propto P_{ex}P_{em}
  48. I F L I C I_{FLIC}
  49. d 𝑓 , d_{\,\textit{f}},
  50. I t h e o r y ( d 1 ) I t h e o r y ( d 0 ) = I e x p ( d 1 + d 𝑓 ) I e x p ( d 0 + d 𝑓 ) \frac{I_{theory}(d_{1})}{I_{theory}(d_{0})}=\frac{I_{exp}(d_{1}+d_{\,\textit{f% }})}{I_{exp}(d_{0}+d_{\,\textit{f}})}
  51. d 𝑓 d_{\,\textit{f}}
  52. d 𝑓 d_{\,\textit{f}}
  53. d 𝑓 d_{\,\textit{f}}
  54. d 𝑓 d_{\,\textit{f}}
  55. d 𝑓 d_{\,\textit{f}}
  56. d 𝑓 d_{\,\textit{f}}
  57. d 𝑓 d_{\,\textit{f}}
  58. 1 n m \sim 1nm

Flux_limiter.html

  1. d u i d t + 1 Δ x i [ F ( u i + 1 2 ) - F ( u i - 1 2 ) ] = 0 , \frac{du_{i}}{dt}+\frac{1}{\Delta x_{i}}\left[F\left(u_{i+\frac{1}{2}}\right)-% F\left(u_{i-\frac{1}{2}}\right)\right]=0,
  2. F ( u i + 1 2 ) F\left(u_{i+\frac{1}{2}}\right)
  3. F ( u i - 1 2 ) F\left(u_{i-\frac{1}{2}}\right)
  4. F ( u i + 1 2 ) = f i + 1 2 l o w - ϕ ( r i ) ( f i + 1 2 l o w - f i + 1 2 h i g h ) F\left(u_{i+\frac{1}{2}}\right)=f^{low}_{i+\frac{1}{2}}-\phi\left(r_{i}\right)% \left(f^{low}_{i+\frac{1}{2}}-f^{high}_{i+\frac{1}{2}}\right)
  5. F ( u i - 1 2 ) = f i - 1 2 l o w - ϕ ( r i - 1 ) ( f i - 1 2 l o w - f i - 1 2 h i g h ) F\left(u_{i-\frac{1}{2}}\right)=f^{low}_{i-\frac{1}{2}}-\phi\left(r_{i-1}% \right)\left(f^{low}_{i-\frac{1}{2}}-f^{high}_{i-\frac{1}{2}}\right)
  6. f l o w = f^{low}=
  7. f h i g h = f^{high}=
  8. ϕ ( r ) = \phi\ (r)=
  9. r r
  10. r i = u i - u i - 1 u i + 1 - u i r_{i}=\frac{u_{i}-u_{i-1}}{u_{i+1}-u_{i}}
  11. ϕ ( r ) 0 \phi\ (r)\geq 0
  12. ϕ ( r ) \phi\ (r)
  13. ϕ c m ( r ) = { r ( 3 r + 1 ) ( r + 1 ) 2 , r > 0 , lim r ϕ c m ( r ) = 3 0 , r 0 \phi_{cm}(r)=\left\{\begin{array}[]{ll}\frac{r\left(3r+1\right)}{\left(r+1% \right)^{2}},\quad r>0,\quad\lim_{r\rightarrow\infty}\phi_{cm}(r)=3\\ 0\quad\quad\,,\quad r\leq 0\end{array}\right.
  14. ϕ h c ( r ) = 1.5 ( r + | r | ) ( r + 2 ) ; lim r ϕ h c ( r ) = 3 \phi_{hc}(r)=\frac{1.5\left(r+\left|r\right|\right)}{\left(r+2\right)};\quad% \lim_{r\rightarrow\infty}\phi_{hc}(r)=3
  15. ϕ h q ( r ) = 2 ( r + | r | ) ( r + 3 ) ; lim r ϕ h q ( r ) = 4 \phi_{hq}(r)=\frac{2\left(r+\left|r\right|\right)}{\left(r+3\right)};\quad\lim% _{r\rightarrow\infty}\phi_{hq}(r)=4
  16. ϕ k n ( r ) = max [ 0 , min ( 2 r , ( 2 + r ) / 3 , 2 ) ] ; lim r ϕ k n ( r ) = 2 \phi_{kn}(r)=\max\left[0,\min\left(2r,\left(2+r\right)/3,2\right)\right];\quad% \lim_{r\rightarrow\infty}\phi_{kn}(r)=2
  17. ϕ m m ( r ) = max [ 0 , min ( 1 , r ) ] ; lim r ϕ m m ( r ) = 1 \phi_{mm}(r)=\max\left[0,\min\left(1,r\right)\right];\quad\lim_{r\rightarrow% \infty}\phi_{mm}(r)=1
  18. ϕ m c ( r ) = max [ 0 , min ( 2 r , 0.5 ( 1 + r ) , 2 ) ] ; lim r ϕ m c ( r ) = 2 \phi_{mc}(r)=\max\left[0,\min\left(2r,0.5(1+r),2\right)\right];\quad\lim_{r% \rightarrow\infty}\phi_{mc}(r)=2
  19. ϕ o s ( r ) = max [ 0 , min ( r , β ) ] , ( 1 β 2 ) ; lim r ϕ o s ( r ) = β \phi_{os}(r)=\max\left[0,\min\left(r,\beta\right)\right],\quad\left(1\leq\beta% \leq 2\right);\quad\lim_{r\rightarrow\infty}\phi_{os}(r)=\beta
  20. ϕ o p ( r ) = 1.5 ( r 2 + r ) ( r 2 + r + 1 ) ; lim r ϕ o p ( r ) = 1.5 \phi_{op}(r)=\frac{1.5\left(r^{2}+r\right)}{\left(r^{2}+r+1\right)};\quad\lim_% {r\rightarrow\infty}\phi_{op}(r)=1.5
  21. ϕ s m ( r ) = max [ 0 , min ( 2 r , ( 0.25 + 0.75 r ) , 4 ) ] ; lim r ϕ s m ( r ) = 4 \phi_{sm}(r)=\max\left[0,\min\left(2r,\left(0.25+0.75r\right),4\right)\right];% \quad\lim_{r\rightarrow\infty}\phi_{sm}(r)=4
  22. ϕ s b ( r ) = max [ 0 , min ( 2 r , 1 ) , min ( r , 2 ) ] ; lim r ϕ s b ( r ) = 2 \phi_{sb}(r)=\max\left[0,\min\left(2r,1\right),\min\left(r,2\right)\right];% \quad\lim_{r\rightarrow\infty}\phi_{sb}(r)=2
  23. ϕ s w ( r ) = max [ 0 , min ( β r , 1 ) , min ( r , β ) ] , ( 1 β 2 ) ; lim r ϕ s w ( r ) = β \phi_{sw}(r)=\max\left[0,\min\left(\beta r,1\right),\min\left(r,\beta\right)% \right],\quad\left(1\leq\beta\leq 2\right);\quad\lim_{r\rightarrow\infty}\phi_% {sw}(r)=\beta
  24. ϕ u m ( r ) = max [ 0 , min ( 2 r , ( 0.25 + 0.75 r ) , ( 0.75 + 0.25 r ) , 2 ) ] ; lim r ϕ u m ( r ) = 2 \phi_{um}(r)=\max\left[0,\min\left(2r,\left(0.25+0.75r\right),\left(0.75+0.25r% \right),2\right)\right];\quad\lim_{r\rightarrow\infty}\phi_{um}(r)=2
  25. ϕ v a 1 ( r ) = r 2 + r r 2 + 1 ; lim r ϕ v a 1 ( r ) = 1 \phi_{va1}(r)=\frac{r^{2}+r}{r^{2}+1};\quad\lim_{r\rightarrow\infty}\phi_{va1}% (r)=1
  26. ϕ v a 2 ( r ) = 2 r r 2 + 1 ; lim r ϕ v a 2 ( r ) = 0 \phi_{va2}(r)=\frac{2r}{r^{2}+1};\quad\lim_{r\rightarrow\infty}\phi_{va2}(r)=0
  27. ϕ v l ( r ) = r + | r | 1 + | r | ; lim r ϕ v l ( r ) = 2 \phi_{vl}(r)=\frac{r+\left|r\right|}{1+\left|r\right|};\quad\lim_{r\rightarrow% \infty}\phi_{vl}(r)=2
  28. ϕ ( r ) r = ϕ ( 1 r ) \frac{\phi\left(r\right)}{r}=\phi\left(\frac{1}{r}\right)
  29. r ϕ ( r ) 2 r , ( 0 r 1 ) r\leq\phi(r)\leq 2r,\left(0\leq r\leq 1\right)
  30. 1 ϕ ( r ) r , ( 1 r 2 ) 1\leq\phi(r)\leq r,\left(1\leq r\leq 2\right)
  31. 1 ϕ ( r ) 2 , ( r > 2 ) 1\leq\phi(r)\leq 2,\left(r>2\right)
  32. ϕ ( 1 ) = 1 \phi(1)=1
  33. β = 1.5 \beta=1.5
  34. ϕ m g ( u , θ ) = max ( 0 , min ( θ r , 1 + r 2 , θ ) ) , θ [ 1 , 2 ] . \phi_{mg}(u,\theta)=\max\left(0,\min\left(\theta r,\frac{1+r}{2},\theta\right)% \right),\quad\theta\in\left[1,2\right].
  35. ϕ m g \phi_{mg}
  36. θ = 1 , \theta=1,
  37. ϕ m m , \phi_{mm},
  38. θ = 2 \theta=2

Fluxional_molecule.html

  1. k = π Δ ν 2 1 / 2 2 Δ ν k=\frac{\pi\Delta\nu_{\circ}}{2^{1/2}}\sim 2\Delta\nu_{\circ}
  2. k 2 ( 500 ) = 1000 s - 1 k\sim 2(500)=1000\mathrm{s}^{-1}
  3. k Δ ν 2 ( 10 c m - 1 ) ( 300 10 8 cm / s ) 6 × 10 11 s - 1 k\sim\Delta\nu_{\circ}\sim 2(10\mathrm{cm}^{-1})(300\cdot 10^{8}\mathrm{cm/s})% \sim 6\times 10^{11}\mathrm{s}^{-1}\cdot

Flyby_anomaly.html

  1. d V V = 2 ω e R e ( cos φ i - cos φ o ) c \frac{dV}{V}=\frac{2\omega_{e}R_{e}(\cos\varphi_{i}-\cos\varphi_{o})}{c}

FO_(complexity).html

  1. E ( x , y ) E(x,y)
  2. x x
  3. y y
  4. P ( n ) P(n)
  5. n n
  6. x x
  7. n n
  8. ( n - 1 ) (n-1)
  9. y y
  10. y < x y<x
  11. b i t ( x , k ) bit(x,k)
  12. k k
  13. x x
  14. p l u s ( x , y , z ) plus(x,y,z)
  15. x + y = z x+y=z
  16. t i m e s ( x , y , z ) times(x,y,z)
  17. x * y = z x*y=z
  18. \wedge
  19. ¬ \neg
  20. \forall
  21. \exists
  22. \vee
  23. P a ( x ) P_{a}(x)
  24. a a
  25. x x
  26. a a
  27. ¬ A \neg A
  28. A A
  29. A B A\wedge B
  30. A A
  31. B B
  32. x P ( x ) \forall xP(x)
  33. P ( v ) P(v)
  34. v v
  35. x x
  36. P ( x 1 , , x c ) P(x_{1},\dots,x_{c})
  37. x i x_{i}
  38. n i n_{i}
  39. P ( n 1 , , n c ) P(n_{1},\dots,n_{c})
  40. , \forall,\exists
  41. \land
  42. \lor
  43. n n
  44. x , y x,y
  45. k k
  46. P P
  47. k k
  48. ϕ \phi
  49. x x
  50. P P
  51. ( P i ) i N (P_{i})_{i\in N}
  52. P 0 ( x ) = f a l s e P_{0}(x)=false
  53. P i ( x ) = ϕ ( P i - 1 , x ) P_{i}(x)=\phi(P_{i-1},x)
  54. ϕ \phi
  55. P i - 1 P_{i-1}
  56. P P
  57. ( P i ) (P_{i})
  58. ( ϕ P , x ) ( y ) (\phi_{P,x})(y)
  59. ( P i ) (P_{i})
  60. y y
  61. P P
  62. k k
  63. 2 n k 2^{n^{k}}
  64. P i P_{i}
  65. 2 n O ( 1 ) 2^{n^{O(1)}}
  66. P P
  67. P i ( x ) P_{i}(x)
  68. P i + 1 ( x ) P_{i+1}(x)
  69. ϕ \phi
  70. P P
  71. ϕ P , x \phi_{P,x}
  72. ψ P , x \psi_{P,x}
  73. ψ ( P , x ) = ϕ ( P , x ) P ( x ) \psi(P,x)=\phi(P,x)\vee P(x)
  74. P P
  75. n k n^{k}
  76. n k n^{k}
  77. n O ( 1 ) n^{O(1)}
  78. k k
  79. u , v , x , y u,v,x,y
  80. k k
  81. ( ϕ u , v ) ( x , y ) (\phi_{u,v})(x,y)
  82. n n
  83. ( z i ) (z_{i})
  84. z 1 = x , z n = y z_{1}=x,z_{n}=y
  85. i < n i<n
  86. ϕ ( z i , z i + 1 ) \phi(z_{i},z_{i+1})
  87. ϕ \phi
  88. ϕ ( x , y ) \phi(x,y)
  89. u u
  90. v v
  91. x x
  92. y y
  93. ϕ u , v \phi_{u,v}
  94. u u
  95. v v
  96. ϕ ( u , v ) \phi(u,v)
  97. ϕ u , v \phi_{u,v}
  98. ψ u , v \psi_{u,v}
  99. ψ ( u , v ) = ϕ ( u , v ) x ( x = v ¬ ϕ ( u , x ) ) \psi(u,v)=\phi(u,v)\wedge\forall x(x=v\vee\neg\phi(u,x))
  100. 0 , ( n - 1 ) 0,(n-1)
  101. t ( n ) t(n)
  102. t ( n ) t(n)
  103. t ( n ) t(n)
  104. t ( n ) t(n)
  105. ( x P ) Q (\forall xP)Q
  106. ( x ( P Q ) ) (\forall x(P\Rightarrow Q))
  107. ( x P ) Q (\exists xP)Q
  108. ( x ( P Q ) ) (\exists x(P\vee Q))
  109. ( Q 1 x 1 , ϕ 1 ) ( Q k x k , ϕ k ) (Q_{1}x_{1},\phi_{1})...(Q_{k}x_{k},\phi_{k})
  110. ϕ i \phi_{i}
  111. Q i Q_{i}
  112. \forall
  113. \exists
  114. Q Q
  115. [ Q ] t ( n ) [Q]^{t(n)}
  116. Q Q
  117. t ( n ) t(n)
  118. k * t ( n ) k*t(n)
  119. k k
  120. t ( n ) t(n)
  121. t ( n ) t(n)
  122. t ( n ) t(n)
  123. ( log n ) i (\log n)^{i}
  124. t ( n ) t(n)
  125. t ( n ) t(n)
  126. ( log n ) O ( 1 ) (\log n)^{O(1)}
  127. n O ( 1 ) n^{O(1)}
  128. 2 n O ( 1 ) 2^{n^{O(1)}}
  129. succ ( x , y ) \rm{succ}(x,y)
  130. x + 1 = y x+1=y
  131. plus ( a , b , c ) = ( DTC v , x , y , z succ ( v , y ) succ ( z , x ) ) ( a , b , c , 0 ) \rm{plus}(a,b,c)=(\rm{DTC}_{v,x,y,z}\rm{succ}(v,y)\land\rm{succ}(z,x))(a,b,c,0)
  132. bit ( a , b ) = ( DTC a , b , a , b ψ ) ( a , b , 1 , 0 ) \rm{bit}(a,b)=(\rm{DTC}_{a,b,a^{\prime},b^{\prime}}\psi)(a,b,1,0)
  133. ψ = if b = 0 then ( if m ( a = m + m + 1 ) then ( a = 1 b = 0 ) else ) else ( succ ( b , b ) ( a + a = a a + a + 1 = a ) \psi=\,\text{if }b=0\,\text{ then }(\,\text{if }\exists m(a=m+m+1)\,\text{ % then }(a^{\prime}=1\land b^{\prime}=0)\,\text{ else }\bot)\,\text{ else }(\rm{% succ}(b^{\prime},b)\land(a+a=a^{\prime}\lor a+a+1=a^{\prime})
  134. a a
  135. b > 0 b>0
  136. a a
  137. b - 1 b-1

Fold_equity.html

  1. Fold Equity = likelihood that opponent folds * gain in equity if opponent(s) fold \,\text{Fold Equity}\,=\,\text{likelihood that opponent folds }*\,\text{ gain % in equity if opponent(s) fold}
  2. 70 % * 68.5 % = 48.2925 % 70\%*68.5\%=48.2925\%
  3. 31.5 % + 48.2925 % = 79.7925 % 31.5\%+48.2925\%=79.7925\%

Folded_normal_distribution.html

  1. 1 2 [ erf ( x + μ σ 2 ) + erf ( x - μ σ 2 ) ] \frac{1}{2}\left[\mbox{erf}~{}\left(\frac{x+\mu}{\sigma\sqrt{2}}\right)+\mbox{% erf}~{}\left(\frac{x-\mu}{\sigma\sqrt{2}}\right)\right]
  2. μ Y = σ 2 π e ( - μ 2 / 2 σ 2 ) + μ ( 1 - 2 Φ ( - μ σ ) ) \mu_{Y}=\sigma\sqrt{\tfrac{2}{\pi}}\,e^{(-\mu^{2}/2\sigma^{2})}+\mu\left(1-2\,% \Phi(\tfrac{-\mu}{\sigma})\right)
  3. σ Y 2 = μ 2 + σ 2 - μ Y 2 \sigma_{Y}^{2}=\mu^{2}+\sigma^{2}-\mu_{Y}^{2}
  4. f Y ( x ; μ , σ ) = 1 σ 2 π e - ( x - μ ) 2 2 σ 2 + 1 σ 2 π e - ( x + μ ) 2 2 σ 2 f_{Y}(x;\mu,\sigma)=\frac{1}{\sigma\sqrt{2\pi}}\,e^{-\frac{(x-\mu)^{2}}{2% \sigma^{2}}}+\frac{1}{\sigma\sqrt{2\pi}}\,e^{-\frac{(x+\mu)^{2}}{2\sigma^{2}}}
  5. F Y ( x ; μ , σ ) = 1 2 [ erf ( x + μ σ 2 ) + erf ( x - μ σ 2 ) ] F_{Y}(x;\mu,\sigma)=\frac{1}{2}\left[\mbox{erf}~{}\left(\frac{x+\mu}{\sigma% \sqrt{2}}\right)+\mbox{erf}~{}\left(\frac{x-\mu}{\sigma\sqrt{2}}\right)\right]
  6. μ Y = σ 2 π exp ( - μ 2 2 σ 2 ) - μ erf ( - μ 2 σ ) , \mu_{Y}=\sigma\sqrt{\frac{2}{\pi}}\,\,\exp\left(\frac{-\mu^{2}}{2\sigma^{2}}% \right)-\mu\,\mbox{erf}~{}\left(\frac{-\mu}{\sqrt{2}\sigma}\right),
  7. σ Y 2 = μ 2 + σ 2 - μ Y 2 . \sigma_{Y}^{2}=\mu^{2}+\sigma^{2}-\mu_{Y}^{2}.
  8. { σ 4 f ′′ ( x ) + 2 σ 2 x f ( x ) + ( - μ 2 + σ 2 + x 2 ) f ( x ) = 0 f ( 0 ) = 2 / π 1 σ e - μ 2 2 σ 2 f ( 0 ) = 0 \begin{cases}\sigma^{4}f^{\prime\prime}(x)+2\sigma^{2}xf^{\prime}(x)+\left(-% \mu^{2}+\sigma^{2}+x^{2}\right)f(x)=0\\ f(0)=\sqrt{2/\pi}\,\frac{1}{\sigma}\,e^{-\frac{\mu^{2}}{2\sigma^{2}}}\\ f^{\prime}(0)=0\end{cases}
  9. μ = 0 μ=0
  10. Y Y
  11. ( Y / σ ) < s u p > 2 (Y/σ)<sup>2

Folded_unipole_antenna.html

  1. P = I 2 R P=I^{2}R

Force_concentration.html

  1. ( P o w e r 2 P o w e r 1 ) 2 \left(\frac{Power_{2}}{Power_{1}}\right)^{2}
  2. d d t N 1 = - c 2 N 2 \frac{d}{dt}N_{1}=-c_{2}N_{2}
  3. d d t N 2 = - c 1 N 1 \frac{d}{dt}N_{2}=-c_{1}N_{1}
  4. N 1 N_{1}
  5. d d t N 2 \frac{d}{dt}N_{2}
  6. c 1 c_{1}
  7. d 2 d t 2 N 1 = c 2 c 1 N 1 \frac{d^{2}}{dt^{2}}N_{1}=c_{2}c_{1}N_{1}
  8. d 2 d t 2 N 2 = c 2 c 1 N 2 \frac{d^{2}}{dt^{2}}N_{2}=c_{2}c_{1}N_{2}
  9. N 1 N_{1}
  10. N 2 N_{2}

Form_factor_(electronics).html

  1. X rms = 1 T t 0 t 0 + T [ f ( t ) ] 2 d t X_{\mathrm{rms}}=\sqrt{{1\over{T}}{\int_{t_{0}}^{t_{0}+T}{[f(t)]}^{2}\,dt}}
  2. X arv = 1 T t 0 t 0 + T | x ( t ) | d t X_{\mathrm{arv}}={1\over{T}}{\int_{t_{0}}^{t_{0}+T}{|x(t)|\,dt}}
  3. k f k_{\mathrm{f}}
  4. k k
  5. k f = R M S ARV = 1 T t 0 t 0 + T [ f ( t ) ] 2 d t 1 T t 0 t 0 + T | x ( t ) | d t = T t 0 t 0 + T [ f ( t ) ] 2 d t t 0 t 0 + T | x ( t ) | d t k_{\mathrm{f}}=\frac{\mathrm{}}{RMS}\mathrm{ARV}=\frac{\sqrt{{1\over{T}}{\int_% {t_{0}}^{t_{0}+T}{[f(t)]}^{2}\,dt}}}{{1\over{T}}{\int_{t_{0}}^{t_{0}+T}{|x(t)|% \,dt}}}=\frac{\sqrt{T\int_{t_{0}}^{t_{0}+T}{{[f(t)]}^{2}\,dt}}}{\int_{t_{0}}^{% t_{0}+T}{|x(t)|\,dt}}
  6. X rms X_{\mathrm{rms}}
  7. X arv X_{\mathrm{arv}}
  8. RMS total = RMS 1 2 + RMS 2 2 + + RMS n 2 \mathrm{RMS}_{\mathrm{total}}=\sqrt{{{\mathrm{RMS}_{1}}^{2}}+{{\mathrm{RMS}_{2% }}^{2}}+...+{{\mathrm{RMS}_{n}}^{2}}}
  9. RMS total = RMS 1 + RMS 2 + + RMS n \mathrm{RMS}_{\mathrm{total}}=\mathrm{RMS}_{1}+\mathrm{RMS}_{2}+...+\mathrm{% RMS}_{n}
  10. ARV total = ARV 1 + ARV 2 + + ARV n \mathrm{ARV}_{\mathrm{total}}=\mathrm{ARV}_{1}+\mathrm{ARV}_{2}+...+\mathrm{% ARV}_{n}
  11. k f tot = RMS tot ARV tot = RMS 1 + + RMS n ARV 1 + + ARV n k_{\mathrm{f}_{\mathrm{tot}}}=\frac{\mathrm{RMS}_{\mathrm{tot}}}{\mathrm{ARV}_% {\mathrm{tot}}}=\frac{\mathrm{RMS}_{1}+...+\mathrm{RMS}_{n}}{\mathrm{ARV}_{1}+% ...+\mathrm{ARV}_{n}}
  12. a a
  13. k f k_{\mathrm{f}}
  14. k a = X max X rms k_{\mathrm{a}}=\frac{X_{\mathrm{max}}}{X_{\mathrm{rms}}}
  15. k av = X max X arv k_{\mathrm{av}}=\frac{X_{\mathrm{max}}}{X_{\mathrm{arv}}}
  16. k av k a k f k_{\mathrm{av}}\geq k_{\mathrm{a}}\geq k_{\mathrm{f}}
  17. k av = k a k f k_{\mathrm{av}}=k_{\mathrm{a}}k_{\mathrm{f}}
  18. k f = k av k a k_{\mathrm{f}}=\frac{k_{\mathrm{av}}}{k_{\mathrm{a}}}
  19. a a
  20. 8 sin ( t ) 8\sin(t)
  21. f ( t ) = a sin ( t ) , a = 8 f(t)=a\sin(t),\ a=8
  22. D = τ T D=\frac{\tau}{T}
  23. τ \tau
  24. T T
  25. τ = T , D = 1 \tau=T,D=1
  26. D D = 1 D = T τ \frac{\sqrt{D}}{D}=\frac{1}{\sqrt{D}}=\sqrt{\frac{T}{\tau}}
  27. D = 1 2 D=\frac{1}{2}
  28. k f = k f frs 2 k_{\mathrm{f}}=k_{\mathrm{f}_{\mathrm{frs}}}\sqrt{2}
  29. a 2 \frac{a}{\sqrt{2}}
  30. a 2 π a\frac{2}{\pi}
  31. π 2 2 1.11072073 \frac{\pi}{2\sqrt{2}}\approx 1.11072073
  32. a 2 \frac{a}{2}
  33. a π \frac{a}{\pi}
  34. π 2 1.5707963 \frac{\pi}{2}\approx 1.5707963
  35. a 2 \frac{a}{\sqrt{2}}
  36. a 2 π a\frac{2}{\pi}
  37. π 2 2 \frac{\pi}{2\sqrt{2}}
  38. a a
  39. a a
  40. a a = 1 \frac{a}{a}=1
  41. a D a\sqrt{D}
  42. a D aD
  43. 1 D = T τ \frac{1}{\sqrt{D}}=\sqrt{\frac{T}{\tau}}
  44. a 3 \frac{a}{\sqrt{3}}
  45. a 2 \frac{a}{2}
  46. 2 3 1.15470054 \frac{2}{\sqrt{3}}\approx 1.15470054
  47. a 3 \frac{a}{\sqrt{3}}
  48. a 2 \frac{a}{2}
  49. 2 3 \frac{2}{\sqrt{3}}
  50. 1 3 \frac{1}{\sqrt{3}}
  51. 1 2 \frac{1}{2}
  52. 2 3 \frac{2}{\sqrt{3}}

Form_factor_(quantum_field_theory).html

  1. ε μ N ¯ ( α ( q 2 ) γ μ + β ( q 2 ) q μ + κ ( q 2 ) σ μ ν q ν ) N \varepsilon_{\mu}\bar{N}\left(\alpha(q^{2})\gamma^{\mu}+\beta(q^{2})q^{\mu}+% \kappa(q^{2})\sigma^{\mu\nu}q_{\nu}\right)N\,
  2. q μ q^{\mu}
  3. α , β , κ \alpha,\beta,\kappa

Formation_and_evolution_of_the_Solar_System.html

  1. × 10 2 1 \times 10^{2}1

Formation_matrix.html

  1. L ( θ ) L(\theta)
  2. L ( θ ) L(\theta)
  3. L ( θ ) L(\theta)
  4. L ( θ ) L(\theta)
  5. j i j j^{ij}
  6. g i j g^{ij}
  7. g i j g_{ij}
  8. g i k g k j = δ i j g_{ik}g^{kj}=\delta_{i}^{j}

Fourier–Bessel_series.html

  1. f : [ 0 , b ] f:[0,b]\rightarrow\mathbb{R}
  2. ( J α ) n ( x ) := J α ( u α , n b x ) (J_{\alpha})_{n}(x):=J_{\alpha}\left(\frac{u_{\alpha,n}}{b}x\right)
  3. f ( x ) n = 1 c n J α ( u α , n b x ) f(x)\sim\sum_{n=1}^{\infty}c_{n}J_{\alpha}\left(\frac{u_{\alpha,n}}{b}x\right)
  4. f , g = 0 b x f ( x ) g ( x ) d x \langle f,g\rangle=\int_{0}^{b}xf(x)g(x)\mathrm{d}x
  5. 0 1 x J α ( x u α , n ) J α ( x u α , m ) d x = δ m n 2 [ J α + 1 ( u α , n ) ] 2 \int_{0}^{1}xJ_{\alpha}(xu_{\alpha,n})\,J_{\alpha}(xu_{\alpha,m})\,dx=\frac{% \delta_{mn}}{2}[J_{\alpha+1}(u_{\alpha,n})]^{2}
  6. c n = f , ( J α ) n ( J α ) n , ( J α ) n = 0 b x f ( x ) ( J α ) n ( x ) d x 1 2 ( b ( J α ± 1 ) n ( b ) ) 2 c_{n}=\frac{\langle f,(J_{\alpha})_{n}\rangle}{\langle(J_{\alpha})_{n},(J_{% \alpha})_{n}\rangle}=\frac{\int_{0}^{b}xf(x)(J_{\alpha})_{n}(x)\mathrm{d}x}{% \frac{1}{2}(b(J_{\alpha\pm 1})_{n}(b))^{2}}
  7. b f ( b ) + c f ( b ) = 0 bf^{\prime}(b)+cf(b)=0
  8. c c
  9. f ( x ) n = 1 b n J α ( γ n x / b ) f(x)\sim\sum_{n=1}^{\infty}b_{n}J_{\alpha}(\gamma_{n}x/b)
  10. γ n \gamma_{n}
  11. x J α ( x ) + c J α ( x ) xJ^{\prime}_{\alpha}(x)+cJ_{\alpha}(x)
  12. b n b_{n}
  13. b n = 2 γ n 2 b 2 ( c 2 + γ n 2 - α 2 ) J α 2 ( γ n ) 0 b J α ( γ n x / b ) f ( x ) x d x b_{n}=\frac{2\gamma_{n}^{2}}{b^{2}(c^{2}+\gamma_{n}^{2}-\alpha^{2})J_{\alpha}^% {2}(\gamma_{n})}\int_{0}^{b}J_{\alpha}(\gamma_{n}x/b)\,f(x)\,x\,dx

Fractional_cascading.html

  1. | L i | + 1 2 | L i + 1 | + 1 4 | L i + 2 | + + 1 2 j | L i + j | + , |L_{i}|+\frac{1}{2}|L_{i+1}|+\frac{1}{4}|L_{i+2}|+\cdots+\frac{1}{2^{j}}|L_{i+% j}|+\cdots,
  2. | M i | = | L i | ( 1 + 1 2 + 1 4 + ) 2 n = O ( n ) , \sum|M_{i}|=\sum|L_{i}|(1+\frac{1}{2}+\frac{1}{4}+\cdots)\leq 2n=O(n),

Fractional_factorial_design.html

  1. - -
  2. + +
  3. - - --
  4. + - +-
  5. - + -+
  6. + + ++

Fractional_sodium_excretion.html

  1. F E N a < m t p l 100 × sodium urinary × creatinine plasma sodium plasma × creatinine urinary FE_{Na}<mtpl>{{=}}100\times\frac{\rm sodium_{urinary}\times creatinine_{plasma% }}{\rm sodium_{plasma}\times creatinine_{urinary}}

Frank–Tamm_formula.html

  1. d E dE
  2. d ω d\omega
  3. d E d x d ω = q 2 4 π μ ( ω ) ω ( 1 - c 2 v 2 n 2 ( ω ) ) \frac{dE}{dx\,d\omega}=\frac{q^{2}}{4\pi}\mu(\omega)\omega{\left(1-\frac{c^{2}% }{v^{2}n^{2}(\omega)}\right)}
  4. β = v c > 1 n ( ω ) \beta=\frac{v}{c}>\frac{1}{n(\omega)}
  5. μ ( ω ) \mu(\omega)
  6. n ( ω ) n(\omega)
  7. q q
  8. v v
  9. c c
  10. d E d x = q 2 4 π v > c n ( ω ) μ ( ω ) ω ( 1 - c 2 v 2 n 2 ( ω ) ) d ω \frac{dE}{dx}=\frac{q^{2}}{4\pi}\int_{v>\frac{c}{n(\omega)}}\mu(\omega)\omega{% \left(1-\frac{c^{2}}{v^{2}n^{2}(\omega)}\right)}d\omega
  11. ω \omega
  12. v v
  13. c n ( ω ) \frac{c}{n(\omega)}

Franz_Hofmeister.html

  1. F - SO 4 2 - > HPO 4 2 - > acetate > Cl - > NO 3 - > Br - > ClO 3 - > I - > ClO 4 - > SCN - \mathrm{F^{-}\approx SO_{4}^{2-}>HPO_{4}^{2-}>acetate>Cl^{-}>NO_{3}^{-}>Br^{-}% >ClO_{3}^{-}>I^{-}>ClO_{4}^{-}>SCN^{-}}
  2. NH 4 + > K + > Na + > Li + > Mg 2 + > Ca 2 + > guanidinium \mathrm{NH_{4}^{+}>K^{+}>Na^{+}>Li^{+}>Mg^{2+}>Ca^{2+}>guanidinium}

Frattini's_argument.html

  1. \square
  2. \square

Frequency_divider.html

  1. f i n f_{in}
  2. f o u t = f i n n f_{out}=\frac{f_{in}}{n}
  3. n n
  4. f i n / 2 f_{in}/2
  5. f i n / 2 f_{in}/2
  6. 3 f i n / 2 3f_{in}/2
  7. f i n / 2 f_{in}/2

Frequency_separation.html

  1. ± \pm

Freundlich_equation.html

  1. x m = K p 1 / n \frac{x}{m}=Kp^{1/n}
  2. log x m = log K + 1 n log p \log\frac{x}{m}=\log K+\frac{1}{n}\log p
  3. x m = K c 1 / n \frac{x}{m}=Kc^{1/n}
  4. log x m = log K + 1 n log c \log\frac{x}{m}=\log K+\frac{1}{n}\log c

Frey_curve.html

  1. y 2 = x ( x - a ) ( x + b ) y^{2}=x(x-a^{\ell})(x+b^{\ell})
  2. a + b = c . a^{\ell}+b^{\ell}=c^{\ell}.
  3. a + b = c , a^{\ell}+b^{\ell}=c^{\ell},
  4. y 2 = x ( x - a ) ( x + b ) y^{2}=x(x-a^{\ell})(x+b^{\ell})
  5. y 2 = x ( x - a ) ( x - c ) . y^{2}=x(x-a^{\ell})(x-c^{\ell}).

Fréchet_surface.html

  1. f : M X f:M\to X
  2. ρ ( f , g ) = inf σ max x M d ( f ( x ) , g ( σ ( x ) ) ) , \rho(f,g)=\inf_{\sigma}\max_{x\in M}d\left(f(x),g(\sigma(x))\right),
  3. ρ ( f , g ) = 0. \rho(f,g)=0.

Friction_loss.html

  1. h l = f D ( L D ) ( V 2 2 g ) h_{l}=f_{D}\left(\frac{L}{D}\right)\left(\frac{V^{2}}{2g}\right)

Friedrichs'_inequality.html

  1. W 0 k , p ( Ω ) W_{0}^{k,p}(\Omega)
  2. u L p ( Ω ) d k ( | α | = k D α u L p ( Ω ) p ) 1 / p . \|u\|_{L^{p}(\Omega)}\leq d^{k}\left(\sum_{|\alpha|=k}\|\mathrm{D}^{\alpha}u\|% _{L^{p}(\Omega)}^{p}\right)^{1/p}.
  3. L p ( Ω ) \|\cdot\|_{L^{p}(\Omega)}
  4. D α u = | α | u x 1 α 1 x n α n . \mathrm{D}^{\alpha}u=\frac{\partial^{|\alpha|}u}{\partial_{x_{1}}^{\alpha_{1}}% \cdots\partial_{x_{n}}^{\alpha_{n}}}.

Fritz_Carlson.html

  1. ( n = 1 | a n | ) 4 π 2 n = 1 | a n | 2 n = 1 n 2 | a n | 2 . \left(\sum_{n=1}^{\infty}|a_{n}|\right)^{4}\leq\pi^{2}\sum_{n=1}^{\infty}|a_{n% }|^{2}\,\sum_{n=1}^{\infty}n^{2}|a_{n}|^{2}~{}.

Full_state_feedback.html

  1. x ¯ ˙ = 𝐀 x ¯ + 𝐁 u ¯ ; \dot{\underline{x}}=\mathbf{A}\underline{x}+\mathbf{B}\underline{u};
  2. y ¯ = 𝐂 x ¯ + 𝐃 u ¯ \underline{y}=\mathbf{C}\underline{x}+\mathbf{D}\underline{u}
  3. | s 𝐈 - 𝐀 | = 0. \left|s\,\textbf{I}-\,\textbf{A}\right|=0.
  4. u ¯ \underline{u}
  5. u ¯ = - 𝐊 x ¯ \underline{u}=-\mathbf{K}\underline{x}
  6. x ¯ ˙ = ( 𝐀 - 𝐁𝐊 ) x ¯ ; \dot{\underline{x}}=(\mathbf{A}-\mathbf{B}\mathbf{K})\underline{x};
  7. y ¯ = ( 𝐂 - 𝐃𝐊 ) x ¯ . \underline{y}=(\mathbf{C}-\mathbf{D}\mathbf{K})\underline{x}.
  8. det [ s 𝐈 - ( 𝐀 - 𝐁 𝐊 ) ] \det\left[s\,\textbf{I}-\left(\,\textbf{A}-\,\textbf{B}\,\textbf{K}\right)\right]
  9. 𝐊 \,\textbf{K}
  10. x ¯ ˙ = [ 0 1 - 2 - 3 ] x ¯ + [ 0 1 ] u ¯ \dot{\underline{x}}=\begin{bmatrix}0&1\\ -2&-3\end{bmatrix}\underline{x}+\begin{bmatrix}0\\ 1\end{bmatrix}\underline{u}
  11. s = - 1 s=-1
  12. s = - 2 s=-2
  13. s = - 1 s=-1
  14. s = - 5 s=-5
  15. s 2 + 6 s + 5 = 0 s^{2}+6s+5=0
  16. 𝐊 = [ k 1 k 2 ] \mathbf{K}=\begin{bmatrix}k_{1}&k_{2}\end{bmatrix}
  17. | s 𝐈 - ( 𝐀 - 𝐁𝐊 ) | = det [ s - 1 2 + k 1 s + 3 + k 2 ] = s 2 + ( 3 + k 2 ) s + ( 2 + k 1 ) \left|s\mathbf{I}-\left(\mathbf{A}-\mathbf{B}\mathbf{K}\right)\right|=\det% \begin{bmatrix}s&-1\\ 2+k_{1}&s+3+k_{2}\end{bmatrix}=s^{2}+(3+k_{2})s+(2+k_{1})
  18. 𝐊 = [ 3 3 ] \mathbf{K}=\begin{bmatrix}3&3\end{bmatrix}
  19. u ¯ = - 𝐊 x ¯ \underline{u}=-\mathbf{K}\underline{x}

Fulton–Hansen_connectedness_theorem.html

  1. Z Z
  2. f : Z P n × P n f:Z\to P^{n}\times P^{n}
  3. dim f ( Z ) > n \dim f(Z)>n
  4. f - 1 Δ f^{-1}\Delta
  5. Δ \Delta
  6. P n × P n P^{n}\times P^{n}
  7. Z = V × W Z=V\times W
  8. f f

Function_representation.html

  1. f ( x 1 , x 2 , , x n ) f(x_{1},x_{2},...,x_{n})
  2. f ( x 1 , x 2 , , x n ) 0 f(x_{1},x_{2},...,x_{n})\geq 0
  3. f ( x 1 , x 2 , , x n ) < 0 f(x_{1},x_{2},...,x_{n})<0
  4. f ( x 1 , x 2 , , x n ) = 0 f(x_{1},x_{2},...,x_{n})=0
  5. C k C^{k}

Functional_completeness.html

  1. \land
  2. \lor
  3. ¬ \neg
  4. \to
  5. \leftrightarrow
  6. A B \displaystyle A\to B
  7. { ¬ , , } \{\neg,\land,\lor\}
  8. \lor
  9. A B := ¬ ( ¬ A ¬ B ) . A\lor B:=\neg(\neg A\land\neg B).
  10. \land
  11. \lor
  12. \vee
  13. \rightarrow
  14. A B := ( A B ) B . \ A\vee B:=(A\rightarrow B)\rightarrow B.
  15. ¬ \neg
  16. { , , } \{\land,\lor,\rightarrow\}
  17. { ¬ , , , , } \{\neg,\land,\lor,\to,\leftrightarrow\}
  18. \vee
  19. \wedge
  20. \top
  21. \bot
  22. ¬ \neg
  23. \top
  24. \bot
  25. \leftrightarrow
  26. ↮ \not\leftrightarrow
  27. ¬ \neg
  28. \vee
  29. \wedge
  30. \top
  31. \rightarrow
  32. \leftrightarrow
  33. \vee
  34. \wedge
  35. \bot
  36. ↛ \not\rightarrow
  37. ↮ \not\leftrightarrow
  38. \vee
  39. \wedge
  40. \bot
  41. \bot
  42. ↮ \not\leftrightarrow
  43. ↮ \not\leftrightarrow
  44. ↛ \not\to
  45. ↚ \not\leftarrow
  46. ↛ \not\to
  47. ↚ \not\leftarrow
  48. ↛ \not\to
  49. ↚ \not\leftarrow
  50. ↛ \not\to
  51. \top
  52. ↚ \not\leftarrow
  53. \top
  54. ↛ \not\to
  55. \leftrightarrow
  56. ↚ \not\leftarrow
  57. \leftrightarrow
  58. \lor
  59. \leftrightarrow
  60. \bot
  61. \lor
  62. \leftrightarrow
  63. ↮ \not\leftrightarrow
  64. \lor
  65. ↮ \not\leftrightarrow
  66. \top
  67. \land
  68. \leftrightarrow
  69. \bot
  70. \land
  71. \leftrightarrow
  72. ↮ \not\leftrightarrow
  73. \land
  74. ↮ \not\leftrightarrow
  75. \top
  76. \vee
  77. ↚ \not\leftarrow
  78. ↛ \not\rightarrow

Functional_response.html

  1. f ( R ) = a R 1 + a h R \begin{aligned}\displaystyle f(R)&\displaystyle=\frac{aR}{1+ahR}\end{aligned}

Fundamental_plane_(elliptical_galaxies).html

  1. R e I e - 0.83 ± 0.08 R_{e}\propto\langle I\rangle_{e}^{-0.83\pm 0.08}
  2. R e R_{e}
  3. I e \langle I\rangle_{e}
  4. R e R_{e}
  5. L e = π I e R e 2 L_{e}=\pi\langle I\rangle_{e}R_{e}^{2}
  6. L e I e I e - 1.66 L_{e}\propto\langle I\rangle_{e}\langle I\rangle_{e}^{-1.66}
  7. I e L - 3 / 2 \langle I\rangle_{e}\sim L^{-3/2}
  8. L e σ o 4 L_{e}\sim\sigma_{o}^{4}
  9. ( log R e , I e , log σ ) \left(\log R_{e},\langle I\rangle_{e},\log\sigma\right)
  10. log R e \log\,R_{e}
  11. 0.26 ( I e / μ B ) + log σ o 0.26\,(\langle I\rangle_{e}/\mu_{B})+\log\sigma_{o}
  12. log R e = 0.36 ( I e / μ B ) + 1.4 log σ o \log R_{e}=0.36\,(\langle I\rangle_{e}/\mu_{B})+1.4\,\log\sigma_{o}
  13. D n - σ o D_{n}-\sigma_{o}
  14. D n kpc = 2.05 ( σ 100 km / s ) 1.33 \frac{D_{n}}{\,\text{kpc}}=2.05\,\left(\frac{\sigma}{100\,\,\text{km}/\,\text{% s}}\right)^{1.33}
  15. D n D_{n}
  16. 20.75 μ B 20.75\mu_{B}
  17. M V = - 23.04 M_{V}=-23.04
  18. M M^{\prime}

Fundamental_thermodynamic_relation.html

  1. d U = T d S - P d V \mathrm{d}U=T\,\mathrm{d}S-P\,\mathrm{d}V\,
  2. d F = - S d T - P d V \mathrm{d}F=-S\,\mathrm{d}T-P\,\mathrm{d}V\,
  3. d H = V d P + T d S . dH=VdP+TdS\;.
  4. d U = δ Q - δ W \mathrm{d}U=\delta Q-\delta W\,
  5. δ Q \delta Q
  6. δ W \delta W
  7. d S = δ Q T \mathrm{d}S=\frac{\delta Q}{T}\,
  8. δ Q = T d S \delta Q=T\,\mathrm{d}S\,
  9. d U = T d S - δ W \mathrm{d}U=T\,\mathrm{d}S-\delta W\,
  10. d U = T d S - P d V \mathrm{d}U=T\,\mathrm{d}S-P\,\mathrm{d}V\,
  11. U U
  12. S S
  13. V V
  14. d U = T d S - i X i d x i + j μ j d N j \mathrm{d}U=T\,\mathrm{d}S-\sum_{i}X_{i}\,\mathrm{d}x_{i}+\sum_{j}\mu_{j}\,% \mathrm{d}N_{j}\,
  15. X i X_{i}
  16. x i x_{i}
  17. μ j \mu_{j}
  18. j j
  19. E E
  20. S = k log [ Ω ( E ) ] S=k\log\left[\Omega\left(E\right)\right]\,
  21. Ω ( E ) \Omega\left(E\right)
  22. E E
  23. E + δ E E+\delta E
  24. δ E \delta E
  25. δ E \delta E
  26. δ E \delta E
  27. δ E \delta E
  28. d S = δ Q T dS=\frac{\delta Q}{T}
  29. Ω ( E ) \Omega\left(E\right)
  30. 1 k T β d log [ Ω ( E ) ] d E \frac{1}{kT}\equiv\beta\equiv\frac{d\log\left[\Omega\left(E\right)\right]}{dE}\,
  31. X d x Xdx
  32. E r E_{r}
  33. X = - d E r d x X=-\frac{dE_{r}}{dx}
  34. δ E \delta E
  35. X = - d E r d x X=-\left\langle\frac{dE_{r}}{dx}\right\rangle\,
  36. Ω ( E ) \Omega\left(E\right)
  37. d E r d x \frac{dE_{r}}{dx}
  38. Y Y
  39. Y + δ Y Y+\delta Y
  40. Ω Y ( E ) \Omega_{Y}\left(E\right)
  41. Ω ( E ) = Y Ω Y ( E ) \Omega\left(E\right)=\sum_{Y}\Omega_{Y}\left(E\right)\,
  42. X = - 1 Ω ( E ) Y Y Ω Y ( E ) X=-\frac{1}{\Omega\left(E\right)}\sum_{Y}Y\Omega_{Y}\left(E\right)\,
  43. Ω ( E ) \Omega\left(E\right)
  44. E E
  45. E + δ E E+\delta E
  46. d E r d x \frac{dE_{r}}{dx}
  47. Y Y
  48. Y + δ Y Y+\delta Y
  49. N Y ( E ) = Ω Y ( E ) δ E Y d x N_{Y}\left(E\right)=\frac{\Omega_{Y}\left(E\right)}{\delta E}Ydx\,
  50. Y d x δ E Ydx\leq\delta E
  51. E E
  52. E + δ E E+\delta E
  53. Ω \Omega
  54. E + δ E E+\delta E
  55. E + δ E E+\delta E
  56. N Y ( E + δ E ) N_{Y}\left(E+\delta E\right)
  57. N Y ( E ) - N Y ( E + δ E ) N_{Y}\left(E\right)-N_{Y}\left(E+\delta E\right)\,
  58. Ω \Omega
  59. δ E \delta E
  60. E E
  61. E + δ E E+\delta E
  62. N Y ( E ) N_{Y}\left(E\right)
  63. N Y ( E + δ E ) N_{Y}\left(E+\delta E\right)
  64. ( Ω x ) E = - Y Y ( Ω Y E ) x = ( ( Ω X ) E ) x \left(\frac{\partial\Omega}{\partial x}\right)_{E}=-\sum_{Y}Y\left(\frac{% \partial\Omega_{Y}}{\partial E}\right)_{x}=\left(\frac{\partial\left(\Omega X% \right)}{\partial E}\right)_{x}\,
  65. Ω \Omega
  66. ( log ( Ω ) x ) E = β X + ( X E ) x \left(\frac{\partial\log\left(\Omega\right)}{\partial x}\right)_{E}=\beta X+% \left(\frac{\partial X}{\partial E}\right)_{x}\,
  67. ( S x ) E = X T \left(\frac{\partial S}{\partial x}\right)_{E}=\frac{X}{T}\,
  68. ( S E ) x = 1 T \left(\frac{\partial S}{\partial E}\right)_{x}=\frac{1}{T}\,
  69. d S = ( S E ) x d E + ( S x ) E d x = d E T + X T d x dS=\left(\frac{\partial S}{\partial E}\right)_{x}dE+\left(\frac{\partial S}{% \partial x}\right)_{E}dx=\frac{dE}{T}+\frac{X}{T}dx\,
  70. d E = T d S - X d x dE=TdS-Xdx\,

Fuzzy_subalgebra.html

  1. x 1 , , x n ( S ( x 1 ) and . . and S ( x n ) S ( h ( x 1 , , x n ) ) \forall x_{1},...,\forall x_{n}(S(x_{1})\and.....\and S(x_{n})\rightarrow S(h(% x_{1},...,x_{n}))
  2. \odot
  3. s ( d 1 ) s ( d n ) s ( h ( d 1 , , d n ) ) s(d_{1})\odot...\odot s(d_{n})\leq s({h}(d_{1},...,d_{n}))
  4. \odot
  5. s ( u ) = 1 s({u})=1
  6. s ( x ) s ( y ) s ( x y ) s(x)\odot s(y)\leq s(x\cdot y)

G-factor_(physics).html

  1. s y m b o l μ = g e 2 m s y m b o l S , symbol\mu=g{e\over 2m}symbolS,
  2. s y m b o l μ = g μ N s y m b o l I = g e 2 m p s y m b o l I , symbol{\mu}=g{\mu\text{N}\over\hbar}symbol{I}=g{e\over 2m\text{p}}symbol{I},
  3. s y m b o l μ s = g e μ B s y m b o l S symbol{\mu}_{\,\text{s}}=g\text{e}{\mu\text{B}\over\hbar}symbol{S}
  4. g s = | g e | = - g e . g\text{s}=|g\text{e}|=-g\text{e}.
  5. μ z = - g s μ B m s \mu\text{z}=-g\text{s}\mu\text{B}m\text{s}
  6. s y m b o l μ L = - g L μ B s y m b o l L , symbol{\mu}\text{L}=-g\text{L}{\mu_{\mathrm{B}}\over\hbar}symbol{L},
  7. μ z = g L μ B m l \mu\text{z}=g\text{L}\mu\text{B}m\text{l}
  8. s y m b o l μ = - g J μ B s y m b o l J symbol{\mu}=-g\text{J}{\mu\text{B}\over\hbar}symbol{J}
  9. s y m b o l μ = g e 2 m μ s y m b o l S , symbol\mu=g{e\over 2m_{\mu}}symbol{S},

G_(disambiguation).html

  1. G p , q m , n G_{p,q}^{m,n}

Gabor_atom.html

  1. g , n ( x ) = g ( x - a ) e 2 π i b n x , - < , n < , g_{\ell,n}(x)=g(x-a\ell)e^{2\pi ibnx},\quad-\infty<\ell,n<\infty,
  2. a a
  3. b b
  4. g g

Gallery_of_named_graphs.html

  1. n n
  2. n n
  3. K n K_{n}
  4. K 1 K_{1}
  5. K 2 K_{2}
  6. K 3 K_{3}
  7. K 4 K_{4}
  8. K 5 K_{5}
  9. K 6 K_{6}
  10. K 7 K_{7}
  11. K 8 K_{8}
  12. K n , m K_{n,m}
  13. n = 1 n=1
  14. K 2 , 2 K_{2,2}
  15. C 4 C_{4}
  16. K 2 , 3 K_{2,3}
  17. K 3 , 3 K_{3,3}
  18. K 2 , 4 K_{2,4}
  19. K 3 , 4 K_{3,4}
  20. n n
  21. C n C_{n}
  22. C 3 C_{3}
  23. C 4 C_{4}
  24. C 5 C_{5}
  25. C 6 C_{6}
  26. C 3 C_{3}
  27. C 4 C_{4}
  28. C 5 C_{5}
  29. C 6 C_{6}
  30. n = 8 n=8
  31. m = 12 m=12
  32. n = 6 n=6
  33. m = 12 m=12
  34. n = 20 n=20
  35. m = 30 m=30
  36. n = 12 n=12
  37. m = 30 m=30
  38. W 4 W_{4}
  39. W 9 W_{9}

Gallium_trichloride.html

  1. \overrightarrow{\leftarrow}

Galois::Counter_Mode.html

  1. x 128 + x 7 + x 2 + x + 1 x^{128}+x^{7}+x^{2}+x+1
  2. GHASH ( H , A , C ) = X m + n + 1 \,\text{GHASH}(H,A,C)=X_{m+n+1}
  3. X i = { 0 for i = 0 ( X i - 1 A i ) H for i = 1 , , m - 1 ( X m - 1 ( A m * 0 128 - v ) ) H for i = m ( X i - 1 C i - m ) H for i = m + 1 , , m + n - 1 ( X m + n - 1 ( C n * 0 128 - u ) ) H for i = m + n ( X m + n ( len ( A ) len ( C ) ) ) H for i = m + n + 1 X_{i}=\begin{cases}0&\,\text{for }i=0\\ (X_{i-1}\oplus A_{i})\cdot H&\,\text{for }i=1,\ldots,m-1\\ (X_{m-1}\oplus(A^{*}_{m}\lVert 0^{128-v}))\cdot H&\,\text{for }i=m\\ (X_{i-1}\oplus C_{i-m})\cdot H&\,\text{for }i=m+1,\ldots,m+n-1\\ (X_{m+n-1}\oplus(C^{*}_{n}\lVert 0^{128-u}))\cdot H&\,\text{for }i=m+n\\ (X_{m+n}\oplus(\operatorname{len}(A)\lVert\operatorname{len}(C)))\cdot H&\,% \text{for }i=m+n+1\\ \end{cases}
  4. \lVert
  5. 128 {}^{128}

Galvanostat.html

  1. R = U I {R}={U\over I}
  2. U c = R v × I o U_{c}={R_{v}\times I_{o}}
  3. I o I_{o}
  4. U c U_{c}
  5. R v R_{v}
  6. U U
  7. R x R_{x}
  8. R l o a d R_{load}
  9. I I
  10. I = U R x + R l o a d I=\frac{U}{R_{x}+R_{load}}
  11. R x R_{x}
  12. R l o a d R_{load}
  13. I I
  14. R x R_{x}
  15. I U R x I\cong\frac{U}{R_{x}}

Gambling_and_information_theory.html

  1. I ( X ; Y ) \displaystyle I(X;Y)
  2. W ( b , p ) = 𝔼 [ log 2 S ( X ) ] = i = 1 m p i log 2 b i o i W(b,p)=\mathbb{E}[\log_{2}S(X)]=\sum_{i=1}^{m}p_{i}\log_{2}b_{i}o_{i}
  3. m m
  4. i i
  5. p i p_{i}
  6. b i b_{i}
  7. o i o_{i}
  8. o i = 2 o_{i}=2
  9. i i
  10. b = p b=p\,
  11. max b W ( b , p ) = i p i log 2 o i - H ( p ) \max_{b}W(b,p)=\sum_{i}p_{i}\log_{2}o_{i}-H(p)\,
  12. H ( p ) H(p)
  13. 𝔼 log K t = log K 0 + i = 1 t H i \mathbb{E}\log K_{t}=\log K_{0}+\sum_{i=1}^{t}H_{i}
  14. K 0 K_{0}
  15. K t K_{t}
  16. H i H_{i}

Gambling_mathematics.html

  1. n p q \sqrt{npq}
  2. n n
  3. p p
  4. q q
  5. 2 b n p q 2b\sqrt{npq}
  6. b b
  7. n n
  8. p = 18 / 38 p=18/38
  9. q = 20 / 38 q=20/38
  10. 2 $ 1 100 18 / 38 20 / 38 $ 9.99 2\cdot\$1\cdot\sqrt{100\cdot 18/38\cdot 20/38}\approx\$9.99
  11. 100 $ 1 2 / 38 $ 5.26 100\cdot\$1\cdot 2/38\approx\$5.26
  12. - $ 5.26 - 3 $ 9.99 -\$5.26-3\cdot\$9.99
  13. - $ 5.26 + 3 $ 9.99 -\$5.26+3\cdot\$9.99
  14. 18 / 38 20 / 38 0.499 \sqrt{18/38\cdot 20/38}\approx 0.499
  15. v v

Ganea_conjecture.html

  1. cat ( X × S n ) = cat ( X ) + 1 , n > 0 \,\text{cat}(X\times S^{n})=\,\text{cat}(X)+1,n>0\,\!
  2. cat ( X × Y ) cat ( X ) + cat ( Y ) \,\text{cat}(X\times Y)\leq\,\text{cat}(X)+\,\text{cat}(Y)
  3. cat ( M - p ) = cat ( M ) - 1 , \,\text{cat}(M-{p})=\,\text{cat}(M)-1,

Gas_thermometer.html

  1. V T V\propto T\,
  2. V T = k \frac{V}{T}=k

Gaugino_condensation.html

  1. λ α a λ β b δ a b ϵ α β Λ 3 \langle\lambda^{a}_{\alpha}\lambda^{b}_{\beta}\rangle\sim\delta^{ab}\epsilon_{% \alpha\beta}\Lambda^{3}
  2. λ \lambda
  3. Λ \Lambda
  4. W α D ¯ 2 D α V W_{\alpha}\equiv\overline{D}^{2}D_{\alpha}V
  5. W α a W β b = λ α a λ β b δ a b ϵ α β Λ 3 \langle W^{a}_{\alpha}W^{b}_{\beta}\rangle=\langle\lambda^{a}_{\alpha}\lambda^% {b}_{\beta}\rangle\sim\delta^{ab}\epsilon_{\alpha\beta}\Lambda^{3}
  6. W α W β W_{\alpha}W_{\beta}
  7. λ α a λ β b \lambda^{a}_{\alpha}\lambda^{b}_{\beta}

Gauss's_principle_of_least_constraint.html

  1. N N
  2. Z = def k = 1 N m k | d 2 𝐫 k d t 2 - 𝐅 k m k | 2 Z\ \stackrel{\mathrm{def}}{=}\ \sum_{k=1}^{N}m_{k}\left|\frac{d^{2}\mathbf{r}_% {k}}{dt^{2}}-\frac{\mathbf{F}_{k}}{m_{k}}\right|^{2}
  3. m k m_{k}
  4. 𝐫 k \mathbf{r}_{k}
  5. 𝐅 k \mathbf{F}_{k}
  6. k th \mathrm{k^{th}}
  7. Z = k = 1 N | d 2 𝐫 k d t 2 | 2 Z=\sum_{k=1}^{N}\left|\frac{d^{2}\mathbf{r}_{k}}{dt^{2}}\right|^{2}
  8. T T
  9. T = def 1 2 k = 1 N | d 𝐫 k d t | 2 T\ \stackrel{\mathrm{def}}{=}\ \frac{1}{2}\sum_{k=1}^{N}\left|\frac{d\mathbf{r% }_{k}}{dt}\right|^{2}
  10. d s 2 ds^{2}
  11. 3 N 3N
  12. d s 2 = def k = 1 N | d 𝐫 k | 2 ds^{2}\ \stackrel{\mathrm{def}}{=}\ \sum_{k=1}^{N}\left|d\mathbf{r}_{k}\right|% ^{2}
  13. ( d s d t ) 2 = 2 T \left(\frac{ds}{dt}\right)^{2}=2T
  14. Z Z
  15. 2 T 2T
  16. K = def k = 1 N | d 2 𝐫 k d s 2 | 2 K\ \stackrel{\mathrm{def}}{=}\ \sum_{k=1}^{N}\left|\frac{d^{2}\mathbf{r}_{k}}{% ds^{2}}\right|^{2}
  17. K \sqrt{K}
  18. 3 N 3N
  19. K K

Gauss_Moutinho_Cordeiro.html

  1. 1 / n 1/n

Gauss_sum.html

  1. G ( χ ) := G ( χ , ψ ) = χ ( r ) ψ ( r ) G(\chi):=G(\chi,\psi)=\sum\chi(r)\cdot\psi(r)
  2. χ ¯ \overline{χ}
  3. G ( χ ) / | G ( χ ) | , G(\chi)\ /\ |G(\chi)|,
  4. χ ¯ \overline{χ}
  5. e 2 π i r 2 p \sum e^{\frac{2\pi ir^{2}}{p}}
  6. G ( χ ) = a = 1 N χ ( a ) e 2 π i a / N . G(\chi)=\sum_{a=1}^{N}\chi(a)e^{2\pi ia/N}.
  7. | G ( χ ) | = N , |G(\chi)|=\sqrt{N},
  8. G ( χ ) = μ ( N / N 0 ) χ 0 ( N / N 0 ) G ( χ 0 ) G(\chi)=\mu(N/N_{0})\chi_{0}(N/N_{0})G(\chi_{0})~{}
  9. G ( χ ¯ ) = χ ( - 1 ) G ( χ ) ¯ , G(\overline{\chi})=\chi(-1)\overline{G(\chi)},
  10. χ ¯ \overline{χ}
  11. G ( χ χ ) = χ ( N ) χ ( N ) G ( χ ) G ( χ ) . G(\chi\chi^{\prime})=\chi(N^{\prime})\chi^{\prime}(N)G(\chi)G(\chi^{\prime}).
  12. G ( χ χ ) = G ( χ ) G ( χ ) J ( χ , χ ) . G(\chi\chi^{\prime})=\frac{G(\chi)G(\chi^{\prime})}{J(\chi,\chi^{\prime})}.

Gaussian_filter.html

  1. g ( x ) = a π e - a x 2 g(x)=\sqrt{\frac{a}{\pi}}\cdot e^{-a\cdot x^{2}}
  2. g ^ ( f ) = e - π 2 f 2 a \hat{g}(f)=e^{-\frac{\pi^{2}f^{2}}{a}}
  3. f f
  4. g ( x ) = 1 2 π σ e - x 2 2 σ 2 g(x)=\frac{1}{\sqrt{2\pi}\cdot\sigma}\cdot e^{-\frac{x^{2}}{2\sigma^{2}}}
  5. g ^ ( f ) = e - f 2 2 σ f 2 \hat{g}(f)=e^{-\frac{f^{2}}{2\sigma_{f}^{2}}}
  6. a a
  7. σ \sigma
  8. g ( x ) g(x)
  9. σ f \sigma_{f}
  10. g ^ ( f ) \hat{g}(f)
  11. σ σ f = 1 2 π \sigma\cdot\sigma_{f}=\frac{1}{2\pi}
  12. g ( x , y ) = 1 2 π σ 2 e - x 2 + y 2 2 σ 2 g(x,y)=\frac{1}{2\pi\sigma^{2}}\cdot e^{-\frac{x^{2}+y^{2}}{2\sigma^{2}}}
  13. x ( - , ) x\in(-\infty,\infty)
  14. σ σ f = N 2 π \sigma\cdot\sigma_{f}=\frac{N}{2\pi}
  15. f c = σ f = 1 2 π σ f_{c}=\sigma_{f}=\frac{1}{2\pi\sigma}
  16. σ \sigma
  17. f c = F s 2 π σ f_{c}=\frac{F_{s}}{2\pi\sigma}
  18. F s F_{s}
  19. 2 \sqrt{2}
  20. f c = 2 ln ( c ) σ f f_{c}=\sqrt{2\ln(c)}\cdot\sigma_{f}
  21. 2 \sqrt{2}
  22. n n
  23. ( n 2 - 1 ) / 12 \sqrt{({n}^{2}-1)/12}
  24. m m
  25. n 1 , , n m {n}_{1},\dots,{n}_{m}
  26. σ = n 1 2 + + n m 2 - m 12 \sigma=\sqrt{\frac{{n}_{1}^{2}+\cdots+{n}_{m}^{2}-m}{12}}
  27. 6 σ - 1 6{\sigma}-1
  28. σ {\sigma}
  29. 2 {\sqrt{2}}
  30. σ {\sigma}

Gaussian_measure.html

  1. N \sqrt{N}
  2. γ n ( A ) = 1 2 π n A exp ( - 1 2 x n 2 ) d λ n ( x ) \gamma^{n}(A)=\frac{1}{\sqrt{2\pi}^{n}}\int_{A}\exp\left(-\frac{1}{2}\|x\|_{% \mathbb{R}^{n}}^{2}\right)\,\mathrm{d}\lambda^{n}(x)
  3. d γ n d λ n ( x ) = 1 2 π n exp ( - 1 2 x n 2 ) . \frac{\mathrm{d}\gamma^{n}}{\mathrm{d}\lambda^{n}}(x)=\frac{1}{\sqrt{2\pi}^{n}% }\exp\left(-\frac{1}{2}\|x\|_{\mathbb{R}^{n}}^{2}\right).
  4. γ μ , σ 2 n ( A ) := 1 2 π σ 2 n A exp ( - 1 2 σ 2 x - μ n 2 ) d λ n ( x ) . \gamma_{\mu,\sigma^{2}}^{n}(A):=\frac{1}{\sqrt{2\pi\sigma^{2}}^{n}}\int_{A}% \exp\left(-\frac{1}{2\sigma^{2}}\|x-\mu\|_{\mathbb{R}^{n}}^{2}\right)\,\mathrm% {d}\lambda^{n}(x).
  5. γ μ , σ 2 n \gamma_{\mu,\sigma^{2}}^{n}
  6. λ n γ n λ n \lambda^{n}\ll\gamma^{n}\ll\lambda^{n}
  7. \ll
  8. γ n ( A ) = sup { γ n ( K ) | K A , K is compact } , \gamma^{n}(A)=\sup\{\gamma^{n}(K)|K\subseteq A,K\mbox{ is compact}~{}\},
  9. d ( T h ) * ( γ n ) d γ n ( x ) = exp ( h , x n - 1 2 h n 2 ) , \frac{\mathrm{d}(T_{h})_{*}(\gamma^{n})}{\mathrm{d}\gamma^{n}}(x)=\exp\left(% \langle h,x\rangle_{\mathbb{R}^{n}}-\frac{1}{2}\|h\|_{\mathbb{R}^{n}}^{2}% \right),
  10. Z Normal ( μ , σ 2 ) ( Z A ) = γ μ , σ 2 n ( A ) . Z\sim\mathrm{Normal}(\mu,\sigma^{2})\implies\mathbb{P}(Z\in A)=\gamma_{\mu,% \sigma^{2}}^{n}(A).

Gårding's_inequality.html

  1. ( L u ) ( x ) = 0 | α | , | β | k ( - 1 ) | α | D α ( A α β ( x ) D β u ( x ) ) , (Lu)(x)=\sum_{0\leq|\alpha|,|\beta|\leq k}(-1)^{|\alpha|}\mathrm{D}^{\alpha}% \left(A_{\alpha\beta}(x)\mathrm{D}^{\beta}u(x)\right),
  2. | α | , | β | = k ξ α A α β ( x ) ξ β > θ | ξ | 2 k for all x Ω , ξ n { 0 } . \sum_{|\alpha|,|\beta|=k}\xi^{\alpha}A_{\alpha\beta}(x)\xi^{\beta}>\theta|\xi|% ^{2k}\mbox{ for all }~{}x\in\Omega,\xi\in\mathbb{R}^{n}\setminus\{0\}.
  3. A α β L ( Ω ) for all | α | , | β | k . A_{\alpha\beta}\in L^{\infty}(\Omega)\mbox{ for all }~{}|\alpha|,|\beta|\leq k.
  4. B [ u , u ] + G u L 2 ( Ω ) 2 C u H k ( Ω ) 2 for all u H 0 k ( Ω ) , B[u,u]+G\|u\|_{L^{2}(\Omega)}^{2}\geq C\|u\|_{H^{k}(\Omega)}^{2}\mbox{ for all% }~{}u\in H_{0}^{k}(\Omega),
  5. B [ v , u ] = 0 | α | , | β | k Ω A α β ( x ) D α u ( x ) D β v ( x ) d x B[v,u]=\sum_{0\leq|\alpha|,|\beta|\leq k}\int_{\Omega}A_{\alpha\beta}(x)% \mathrm{D}^{\alpha}u(x)\mathrm{D}^{\beta}v(x)\,\mathrm{d}x
  6. { - Δ u ( x ) = f ( x ) , x Ω ; u ( x ) = 0 , x Ω ; \begin{cases}-\Delta u(x)=f(x),&x\in\Omega;\\ u(x)=0,&x\in\partial\Omega;\end{cases}
  7. B [ u , v ] = f , v for all v H 0 1 ( Ω ) , B[u,v]=\langle f,v\rangle\mbox{ for all }~{}v\in H_{0}^{1}(\Omega),
  8. B [ u , v ] = Ω u ( x ) v ( x ) d x , B[u,v]=\int_{\Omega}\nabla u(x)\cdot\nabla v(x)\,\mathrm{d}x,
  9. f , v = Ω f ( x ) v ( x ) d x . \langle f,v\rangle=\int_{\Omega}f(x)v(x)\,\mathrm{d}x.
  10. B [ u , u ] C u H 1 ( Ω ) 2 - G u L 2 ( Ω ) 2 for all u H 0 1 ( Ω ) . B[u,u]\geq C\|u\|_{H^{1}(\Omega)}^{2}-G\|u\|_{L^{2}(\Omega)}^{2}\mbox{ for all% }~{}u\in H_{0}^{1}(\Omega).
  11. B [ u , u ] K u H 1 ( Ω ) 2 for all u H 0 1 ( Ω ) , B[u,u]\geq K\|u\|_{H^{1}(\Omega)}^{2}\mbox{ for all }~{}u\in H_{0}^{1}(\Omega),

General_frame.html

  1. 𝐅 = F , R , V \mathbf{F}=\langle F,R,V\rangle
  2. F , R \langle F,R\rangle
  3. \Box
  4. A = { x F ; y F ( x R y y A ) } \Box A=\{x\in F;\,\forall y\in F\,(x\,R\,y\to y\in A)\}
  5. F , R , \langle F,R,\Vdash\rangle
  6. F , R \langle F,R\rangle
  7. { x F ; x p } V \{x\in F;\,x\Vdash p\}\in V
  8. { x F ; x A } \{x\in F;\,x\Vdash A\}
  9. x A x\Vdash A
  10. \Vdash
  11. x F x\in F
  12. F , R \langle F,R\rangle
  13. F , R , 𝒫 ( F ) \langle F,R,\mathcal{P}(F)\rangle
  14. 𝒫 ( F ) \mathcal{P}(F)
  15. 𝐅 = F , R , V \mathbf{F}=\langle F,R,V\rangle
  16. A V ( x A y A ) \forall A\in V\,(x\in A\Leftrightarrow y\in A)
  17. x = y x=y
  18. A V ( x A y A ) \forall A\in V\,(x\in\Box A\Rightarrow y\in A)
  19. x R y x\,R\,y
  20. F , R , \langle F,R,{\Vdash}\rangle
  21. F , R , V \langle F,R,V\rangle
  22. V = { { x F ; x A } ; A is a formula } . V=\big\{\{x\in F;\,x\Vdash A\};\,A\hbox{ is a formula}\big\}.
  23. 𝐆 = G , S , W \mathbf{G}=\langle G,S,W\rangle
  24. 𝐅 = F , R , V \mathbf{F}=\langle F,R,V\rangle
  25. G , S \langle G,S\rangle
  26. F , R \langle F,R\rangle
  27. W = { A G ; A V } . W=\{A\cap G;\,A\in V\}.
  28. f : 𝐅 𝐆 f\colon\mathbf{F}\to\mathbf{G}
  29. F , R \langle F,R\rangle
  30. G , S \langle G,S\rangle
  31. f - 1 [ A ] V f^{-1}[A]\in V
  32. A W A\in W
  33. 𝐅 i = F i , R i , V i \mathbf{F}_{i}=\langle F_{i},R_{i},V_{i}\rangle
  34. i I i\in I
  35. 𝐅 = F , R , V \mathbf{F}=\langle F,R,V\rangle
  36. { F i ; i I } \{F_{i};\,i\in I\}
  37. { R i ; i I } \{R_{i};\,i\in I\}
  38. V = { A F ; i I ( A F i V i ) } . V=\{A\subseteq F;\,\forall i\in I\,(A\cap F_{i}\in V_{i})\}.
  39. 𝐅 = F , R , V \mathbf{F}=\langle F,R,V\rangle
  40. 𝐆 = G , S , W \mathbf{G}=\langle G,S,W\rangle
  41. x y A V ( x A y A ) , x\sim y\iff\forall A\in V\,(x\in A\Leftrightarrow y\in A),
  42. G = F / G=F/{\sim}
  43. \sim
  44. x / , y / S A V ( x A y A ) , \langle x/{\sim},y/{\sim}\rangle\in S\iff\forall A\in V\,(x\in\Box A% \Rightarrow y\in A),
  45. A / W A V . A/{\sim}\in W\iff A\in V.
  46. F , R , \langle F,R,{\Vdash}\rangle
  47. F , R , \langle F,R,{\Vdash}\rangle
  48. 𝐅 = F , R , V \mathbf{F}=\langle F,R,V\rangle
  49. 𝒫 ( F ) , , , - \langle\mathcal{P}(F),\cap,\cup,-\rangle
  50. \Box
  51. V , , , - , \langle V,\cap,\cup,-,\Box\rangle
  52. 𝐅 + \mathbf{F}^{+}
  53. 𝐀 + = F , R , V \mathbf{A}_{+}=\langle F,R,V\rangle
  54. 𝐀 = A , , , - , \mathbf{A}=\langle A,\wedge,\vee,-,\Box\rangle
  55. A , , , - \langle A,\wedge,\vee,-\rangle
  56. 𝐀 + \mathbf{A}_{+}
  57. x R y a A ( a x a y ) x\,R\,y\iff\forall a\in A\,(\Box a\in x\Rightarrow a\in y)
  58. ( 𝐀 + ) + (\mathbf{A}_{+})^{+}
  59. 𝐀 \mathbf{A}
  60. 𝐅 \mathbf{F}
  61. ( 𝐅 + ) + (\mathbf{F}^{+})_{+}
  62. ( ) + (\cdot)^{+}
  63. ( ) + (\cdot)_{+}
  64. F , , V \langle F,\leq,V\rangle
  65. \leq
  66. A B = ( - A B ) A\to B=\Box(-A\cup B)
  67. 𝐅 = F , , V \mathbf{F}=\langle F,\leq,V\rangle
  68. A V ( x A y A ) \forall A\in V\,(x\in A\Leftrightarrow y\in A)
  69. x y x\leq y
  70. V { F - A ; A V } V\cup\{F-A;\,A\in V\}
  71. 𝐅 = F , , V \mathbf{F}=\langle F,\leq,V\rangle
  72. 𝐅 + = V , , , , \mathbf{F}^{+}=\langle V,\cap,\cup,\to,\emptyset\rangle
  73. 𝐀 = A , , , , 0 \mathbf{A}=\langle A,\wedge,\vee,\to,0\rangle
  74. 𝐀 + = F , , V \mathbf{A}_{+}=\langle F,\leq,V\rangle
  75. \leq
  76. { x F ; a x } , \{x\in F;\,a\in x\},
  77. a A a\in A
  78. ( ) + (\cdot)^{+}
  79. ( ) + (\cdot)_{+}

Generalizability_theory.html

  1. X = T + e X=T+e

Generalizations_of_Fibonacci_numbers.html

  1. φ \varphi
  2. F n = φ n - ( - φ ) - n 5 . F_{n}=\frac{\varphi^{n}-(-\varphi)^{-n}}{\sqrt{5}}.
  3. F e ( x ) = φ x - φ - x 5 Fe(x)=\frac{\varphi^{x}-\varphi^{-x}}{\sqrt{5}}
  4. F o ( x ) = φ x + φ - x 5 Fo(x)=\frac{\varphi^{x}+\varphi^{-x}}{\sqrt{5}}
  5. F i b ( x ) = φ x - cos ( x π ) φ - x 5 Fib(x)=\frac{\varphi^{x}-\cos(x\pi)\varphi^{-x}}{\sqrt{5}}
  6. F i b ( 3 + 4 i ) - 5248.5 - 14195.9 i Fib(3+4i)\approx-5248.5-14195.9i
  7. g ( 1 ) φ n - ( - φ ) - n 5 + g ( 0 ) φ n - 1 - ( - φ ) 1 - n 5 , g(1){{\varphi^{n}-(-\varphi)^{-n}}\over{\sqrt{5}}}+g(0){{\varphi^{n-1}-(-% \varphi)^{1-n}}\over{\sqrt{5}}}\,,
  8. φ \varphi
  9. a φ n + b ( - φ ) - n a\varphi^{n}+b(-\varphi)^{-n}
  10. L n = φ n + ( - φ ) - n L_{n}=\varphi^{n}+(-\varphi)^{-n}
  11. φ n = ( 1 2 ( 1 + 5 ) ) n = 1 2 ( L ( n ) + F ( n ) 5 ) . \varphi^{n}=\left(\frac{1}{2}\left(1+\sqrt{5}\right)\right)^{n}=\frac{1}{2}% \left(L(n)+F(n)\sqrt{5}\right).
  12. L ( n ) = F ( n - 1 ) + F ( n + 1 ) . L\left(n\right)=F\left(n-1\right)+F\left(n+1\right).\,
  13. 1 + 5 2 \frac{1+\sqrt{5}}{2}
  14. 1 + 2 1+\sqrt{2}
  15. n + n 2 + 4 2 \frac{n+\sqrt{n^{2}+4}}{2}
  16. 1 + 19 + 3 33 3 + 19 - 3 33 3 3 \tfrac{1+\sqrt[3]{19+3\sqrt{33}}+\sqrt[3]{19-3\sqrt{33}}}{3}
  17. T ( n ) = 3 b ( 1 3 ( a + + a - + 1 ) ) n b 2 - 2 b + 4 T(n)=\left\lfloor 3\,b\frac{\left(\frac{1}{3}\left(a_{+}+a_{-}+1\right)\right)% ^{n}}{b^{2}-2b+4}\right\rceil
  18. \lfloor\cdot\rceil
  19. a ± = ( 19 ± 3 33 ) 1 / 3 a_{\pm}=\left(19\pm 3\sqrt{33}\right)^{1/3}
  20. b = ( 586 + 102 33 ) 1 / 3 b=\left(586+102\sqrt{33}\right)^{1/3}
  21. x + x - n = 2 x+x^{-n}=2\,
  22. r = k = 0 n - 1 r - k r=\sum_{k=0}^{n-1}r^{-k}
  23. ϕ = 1 + 1 ϕ \phi=1+\frac{1}{\phi}
  24. F k ( n ) = [ r k - 1 ( r - 1 ) ( n + 1 ) r - 2 n ] F_{k}^{(n)}=\left[\frac{r^{k-1}(r-1)}{(n+1)r-2n}\right]
  25. x + x - n = 2 x+x^{-n}=2
  26. F m + 2 ( n ) 2 m \frac{F_{m+2}^{(n)}}{2^{m}}
  27. F n := F ( n ) := { b if n = 0 ; a if n = 1 ; F ( n - 1 ) + F ( n - 2 ) if n > 1. F_{n}:=F(n):=\begin{cases}b&\mbox{if }~{}n=0;\\ a&\mbox{if }~{}n=1;\\ F(n-1)+F(n-2)&\mbox{if }~{}n>1.\\ \end{cases}
  28. F n ( 0 ) = F n F_{n}^{(0)}=F_{n}
  29. F n ( r + 1 ) = i = 0 n F i F n - i ( r ) F_{n}^{(r+1)}=\sum_{i=0}^{n}F_{i}F_{n-i}^{(r)}
  30. F n + 1 ( r + 1 ) = F n ( r + 1 ) + F n - 1 ( r + 1 ) + F n ( r ) F_{n+1}^{(r+1)}=F_{n}^{(r+1)}+F_{n-1}^{(r+1)}+F_{n}^{(r)}
  31. s ( r ) ( x ) = k = 0 F n ( r ) x n = ( x 1 - x - x 2 ) r . s^{(r)}(x)=\sum_{k=0}^{\infty}F^{(r)}_{n}x^{n}=\left(\frac{x}{1-x-x^{2}}\right% )^{r}.
  32. F n ( r ) = r ! F n ( r ) ( 1 ) F_{n}^{(r)}=r!F_{n}^{(r)}(1)
  33. F n ( 1 ) = ( n L n - F n ) / 5 F_{n}^{(1)}=(nL_{n}-F_{n})/5
  34. F n + 1 ( 1 ) = 2 F n ( 1 ) + F n - 1 ( 1 ) - 2 F n - 2 ( 1 ) - F n - 3 ( 1 ) . F_{n+1}^{(1)}=2F_{n}^{(1)}+F_{n-1}^{(1)}-2F_{n-2}^{(1)}-F_{n-3}^{(1)}.
  35. L ( n ) = 2 F ( n - 1 ) + F ( n ) L(n)=2F(n-1)+F(n)

Generalizations_of_Pauli_matrices.html

  1. j k jk
  2. k > j k>j
  3. k = d k=d
  4. h d d = 2 d ( d - 1 ) ( h 1 d - 1 ( 1 - d ) ) . ~{}~{}~{}h_{d}^{d}=\sqrt{\tfrac{2}{d(d-1)}}\left(h_{1}^{d-1}\oplus(1-d)\right)% ~{}.
  5. d d
  6. d × d d×d
  7. 𝔤 𝔩 \mathfrak{gl}
  8. d d
  9. 𝔰 𝔲 \mathfrak{su}
  10. d d
  11. d d
  12. σ 1 \sigma_{1}
  13. σ 3 \sigma_{3}
  14. σ 1 2 = σ 3 2 = I , σ 1 σ 3 = - σ 3 σ 1 = e π i σ 3 σ 1 . \sigma_{1}^{2}=\sigma_{3}^{2}=I,\;\sigma_{1}\sigma_{3}=-\sigma_{3}\sigma_{1}=e% ^{\pi i}\sigma_{3}\sigma_{1}.
  15. W = 1 2 [ 1 1 1 - 1 ] . W=\tfrac{1}{\sqrt{2}}\begin{bmatrix}1&1\\ 1&-1\end{bmatrix}.
  16. σ 1 , σ 3 \sigma_{1},\;\sigma_{3}
  17. σ 1 = W σ 3 W * . \;\sigma_{1}=W\sigma_{3}W^{*}.
  18. d d
  19. ω = e x p ( 2 π i / d ) ω=exp(2πi/d)
  20. ω 1 ω≠1
  21. 1 + ω + + ω d - 1 = 0. 1+\omega+\cdots+\omega^{d-1}=0.
  22. d d
  23. Σ 1 = [ 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 ] \Sigma_{1}=\begin{bmatrix}0&0&0&\cdots&0&1\\ 1&0&0&\cdots&0&0\\ 0&1&0&\cdots&0&0\\ 0&0&1&\cdots&0&0\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&0&\cdots&1&0\\ \end{bmatrix}
  24. Σ 3 = [ 1 0 0 0 0 ω 0 0 0 0 ω 2 0 0 0 0 ω d - 1 ] . \Sigma_{3}=\begin{bmatrix}1&0&0&\cdots&0\\ 0&\omega&0&\cdots&0\\ 0&0&\omega^{2}&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\cdots&\omega^{d-1}\end{bmatrix}.
  25. Σ 1 d = Σ 3 d = I \Sigma_{1}^{d}=\Sigma_{3}^{d}=I
  26. Σ 3 Σ 1 = ω Σ 1 Σ 3 = e 2 π i / d Σ 1 Σ 3 , \;\Sigma_{3}\Sigma_{1}=\omega\Sigma_{1}\Sigma_{3}=e^{2\pi i/d}\Sigma_{1}\Sigma% _{3},
  27. Σ 3 Σ 1 Σ 3 d - 1 Σ 1 d - 1 = ω . \;\Sigma_{3}\Sigma_{1}\Sigma_{3}^{d-1}\Sigma_{1}^{d-1}=\omega~{}.
  28. W = 1 2 [ 1 1 1 ω 2 - 1 ] = 1 2 [ 1 1 1 ω d - 1 ] . W=\tfrac{1}{\sqrt{2}}\begin{bmatrix}1&1\\ 1&\omega^{2-1}\end{bmatrix}=\tfrac{1}{\sqrt{2}}\begin{bmatrix}1&1\\ 1&\omega^{d-1}\end{bmatrix}.
  29. W = 1 d [ 1 1 1 1 1 ω d - 1 ω 2 ( d - 1 ) ω ( d - 1 ) 2 1 ω d - 2 ω 2 ( d - 2 ) ω ( d - 1 ) ( d - 2 ) 1 ω ω 2 ω d - 1 ] . W=\frac{1}{\sqrt{d}}\begin{bmatrix}1&1&1&\cdots&1\\ 1&\omega^{d-1}&\omega^{2(d-1)}&\cdots&\omega^{(d-1)^{2}}\\ 1&\omega^{d-2}&\omega^{2(d-2)}&\cdots&\omega^{(d-1)(d-2)}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 1&\omega&\omega^{2}&\cdots&\omega^{d-1}\end{bmatrix}~{}.
  30. Σ 1 = W Σ 3 W * , \;\Sigma_{1}=W\Sigma_{3}W^{*}~{},
  31. W W
  32. 𝔤 𝔩 \mathfrak{gl}
  33. 𝔤 𝔩 \mathfrak{gl}
  34. 𝔤 𝔩 \mathfrak{gl}
  35. 𝔤 𝔩 \mathfrak{gl}
  36. d d→∞

Generalized_complex_structure.html

  1. \oplus
  2. X + ξ , Y + η = 1 2 ( ξ ( Y ) + η ( X ) ) . \langle X+\xi,Y+\eta\rangle=\frac{1}{2}(\xi(Y)+\eta(X)).
  3. 𝒥 : 𝐓 𝐓 * 𝐓 𝐓 * {\mathcal{J}}:\mathbf{T}\oplus\mathbf{T}^{*}\rightarrow\mathbf{T}\oplus\mathbf% {T}^{*}
  4. 𝒥 2 = - Id , and 𝒥 ( X + ξ ) , 𝒥 ( Y + η ) = X + ξ , Y + η . {\mathcal{J}}^{2}=-{\rm Id},\ \ \mbox{ and }~{}\ \ \langle{\mathcal{J}}(X+\xi)% ,{\mathcal{J}}(Y+\eta)\rangle=\langle X+\xi,Y+\eta\rangle.
  5. - 1 \sqrt{-1}
  6. L L
  7. ( 𝐓 𝐓 * ) (\mathbf{T}\oplus\mathbf{T}^{*})\otimes\mathbb{C}
  8. L = { X + ξ ( 𝐓 𝐓 * ) : 𝒥 ( X + ξ ) = - 1 ( X + ξ ) } L=\{X+\xi\in(\mathbf{T}\oplus\mathbf{T}^{*})\otimes\mathbb{C}\ :\ {\mathcal{J}% }(X+\xi)=\sqrt{-1}(X+\xi)\}
  9. L L ¯ = 0 L\cap\overline{L}=0
  10. , = 0 \langle\ell,\ell^{\prime}\rangle=0
  11. , L . \ell,\ell^{\prime}\in L.
  12. - 1 \sqrt{-1}
  13. [ X + ξ , Y + η ] = [ X , Y ] + X η - Y ξ - 1 2 d ( i ( X ) η - i ( Y ) ξ ) [X+\xi,Y+\eta]=[X,Y]+\mathcal{L}_{X}\eta-\mathcal{L}_{Y}\xi-\frac{1}{2}d(i(X)% \eta-i(Y)\xi)
  14. X \mathcal{L}_{X}
  15. \oplus
  16. \oplus
  17. \oplus
  18. X + ξ , Y + η = 1 2 ( ξ ( Y ) + η ( X ) ) = 1 2 ( ϵ ( Y , X ) + ϵ ( X , Y ) ) = 0 \langle X+\xi,Y+\eta\rangle=\frac{1}{2}(\xi(Y)+\eta(X))=\frac{1}{2}(\epsilon(Y% ,X)+\epsilon(X,Y))=0
  19. \oplus
  20. \otimes
  21. \oplus
  22. X + ξ X + ξ + i X B X+\xi\longrightarrow X+\xi+i_{X}B
  23. \oplus
  24. \otimes
  25. \otimes
  26. \oplus
  27. \otimes
  28. \oplus
  29. \otimes
  30. \oplus
  31. \otimes
  32. \oplus
  33. \otimes
  34. Φ = e B + i ω Ω \Phi=e^{B+i\omega}\Omega
  35. \otimes
  36. \oplus
  37. \otimes
  38. \otimes
  39. \oplus
  40. \otimes
  41. E E ¯ = Δ 𝐂 E\cap\overline{E}=\Delta\otimes\mathbf{C}
  42. \otimes
  43. \oplus
  44. \otimes
  45. \oplus
  46. \otimes
  47. \partial
  48. ϕ = e i ω \phi=e^{i\omega}
  49. ϕ \phi
  50. \oplus
  51. \otimes
  52. O ( 2 n , 2 n ) U ( n , n ) . \frac{O(2n,2n)}{U(n,n)}.
  53. \oplus
  54. \otimes
  55. × \times
  56. × \times

Generalized_dihedral_group.html

  1. Dih ( H ) = H ϕ Z 2 \mathrm{Dih}(H)=H\rtimes_{\phi}Z_{2}
  2. \infty\infty
  3. \infty

Generalized_inverse.html

  1. A n × m A\in\mathbb{R}^{n\times m}
  2. A g m × n A^{\mathrm{g}}\in\mathbb{R}^{m\times n}
  3. A g A^{\mathrm{g}}
  4. A A
  5. A A g A = A AA^{\mathrm{g}}A=A
  6. A n × m A\in\mathbb{R}^{n\times m}
  7. A g m × n , A^{\mathrm{g}}\in\mathbb{R}^{m\times n},
  8. A A g A = A AA^{\mathrm{g}}A=A
  9. A g A A g = A g A^{\mathrm{g}}AA^{\mathrm{g}}=A^{\mathrm{g}}
  10. ( A A g ) T = A A g (AA^{\mathrm{g}})^{\mathrm{T}}=AA^{\mathrm{g}}
  11. ( A g A ) T = A g A (A^{\mathrm{g}}A)^{\mathrm{T}}=A^{\mathrm{g}}A
  12. A g A^{\mathrm{g}}
  13. A A
  14. A A
  15. A A
  16. n × m n\times m
  17. n > m n>m
  18. n < m n<m
  19. A left - 1 = ( A T A ) - 1 A T A_{\mathrm{left}}^{-1}=\left(A^{\mathrm{T}}A\right)^{-1}A^{\mathrm{T}}
  20. A left - 1 A = I m A_{\mathrm{left}}^{-1}A=I_{m}
  21. I m I_{m}
  22. m × m m\times m
  23. A right - 1 = A T ( A A T ) - 1 A_{\mathrm{right}}^{-1}=A^{\mathrm{T}}\left(AA^{\mathrm{T}}\right)^{-1}
  24. A A right - 1 = I n AA_{\mathrm{right}}^{-1}=I_{n}
  25. I n I_{n}
  26. n × n n\times n
  27. A x = b Ax=b
  28. x x
  29. x = A g b + [ I - A g A ] w x=A^{\mathrm{g}}b+[I-A^{\mathrm{g}}A]w
  30. A g A^{\mathrm{g}}
  31. A . A.
  32. A g b A^{\mathrm{g}}b
  33. A A g b = b . AA^{\mathrm{g}}b=b.

Generalized_linear_array_model.html

  1. 𝐘 \mathbf{Y}
  2. d d
  3. n 1 × n 2 × × n d n_{1}\times n_{2}\times\ldots\times n_{d}
  4. 𝐲 = 𝐯𝐞𝐜 ( 𝐘 ) \mathbf{y}=\,\textbf{vec}(\mathbf{Y})
  5. n 1 n 2 n 3 n d n_{1}n_{2}n_{3}\cdots n_{d}
  6. 𝐗 = 𝐗 d 𝐗 d - 1 𝐗 1 . \mathbf{X}=\mathbf{X}_{d}\otimes\mathbf{X}_{d-1}\otimes\ldots\otimes\mathbf{X}% _{1}.
  7. 𝐲 \mathbf{y}
  8. 𝐗 \mathbf{X}
  9. 𝐗 𝐖 ~ δ 𝐗 s y m b o l θ ^ = 𝐗 𝐖 ~ δ 𝐳 ~ , \mathbf{X}^{\prime}\tilde{\mathbf{W}}_{\delta}\mathbf{X}\hat{symbol\theta}=% \mathbf{X}^{\prime}\tilde{\mathbf{W}}_{\delta}\tilde{\mathbf{z}},
  10. s y m b o l θ ~ \tilde{symbol\theta}
  11. s y m b o l θ symbol\theta
  12. s y m b o l θ ^ \hat{symbol\theta}
  13. 𝐖 δ \mathbf{W}_{\delta}
  14. w i i - 1 = ( η i μ i ) 2 var ( y i ) , w_{ii}^{-1}=\left(\frac{\partial\eta_{i}}{\partial\mu_{i}}\right)^{2}\,\text{% var}(y_{i}),
  15. 𝐳 = s y m b o l η + 𝐖 δ - 1 ( 𝐲 - s y m b o l μ ) \mathbf{z}=symbol\eta+\mathbf{W}_{\delta}^{-1}(\mathbf{y}-symbol\mu)
  16. s y m b o l η = 𝐗 s y m b o l θ symbol\eta=\mathbf{X}symbol\theta
  17. 𝐗 𝐖 ~ δ 𝐗 \mathbf{X}^{\prime}\tilde{\mathbf{W}}_{\delta}\mathbf{X}
  18. 𝐗 . \mathbf{X}.
  19. 𝐗 = 𝐗 2 𝐗 1 , \mathbf{X}=\mathbf{X}_{2}\otimes\mathbf{X}_{1},
  20. 𝐗 1 s y m b o l Θ 𝐗 2 \mathbf{X}_{1}symbol\Theta\mathbf{X}_{2}^{\prime}
  21. s y m b o l Θ symbol\Theta
  22. G ( 𝐗 1 ) 𝐖 G ( 𝐗 2 ) G(\mathbf{X}_{1})^{\prime}\mathbf{W}G(\mathbf{X}_{2})
  23. 𝐖 \mathbf{W}
  24. G ( 𝐌 ) G(\mathbf{M})
  25. r × c r\times c
  26. 𝐌 \mathbf{M}
  27. G ( 𝐌 ) = ( 𝐌 𝟏 ) * ( 𝟏 𝐌 ) G(\mathbf{M})=(\mathbf{M}\otimes\mathbf{1}^{\prime})*(\mathbf{1}^{\prime}% \otimes\mathbf{M})
  28. * *
  29. 𝟏 \mathbf{1}
  30. c c
  31. d d
  32. d d
  33. d d

Generalized_Maxwell_model.html

  1. N + 1 N+1
  2. E i E_{i}
  3. η i \eta_{i}
  4. τ i = η i E i \tau_{i}=\frac{\eta_{i}}{E_{i}}
  5. = =
  6. E 0 ϵ + E_{0}\epsilon+
  7. n = 1 N ( i 1 = 1 N - n + 1 ( i a = i a - 1 + 1 N - ( n - a ) + 1 ( i n = i n - 1 + 1 N ( ( E 0 + j { i 1 , , i n } E j ) ( k { i 1 , , i n } τ k ) ) ) ) ) n ϵ t n \sum^{N}_{n=1}{\left({\sum^{N-n+1}_{i_{1}=1}{...\left({\sum^{N-\left({n-a}% \right)+1}_{i_{a}=i_{a-1}+1}{...\left({\sum^{N}_{i_{n}=i_{n-1}+1}{\left({\left% ({E_{0}+\sum_{j\in\left\{{i_{1},...,i_{n}}\right\}}{E_{j}}}\right)\left({\prod% _{k\in\left\{{i_{1},...,i_{n}}\right\}}{\tau_{k}}}\right)}\right)}}\right)...}% }\right)...}}\right)\frac{\partial^{n}{\epsilon}}{\partial{t}^{n}}}
  8. ( i = 1 N - 1 ( j = i + 1 N τ i τ j ) ) 2 σ t 2 {\left({\sum^{N-1}_{i=1}{\left({\sum^{N}_{j=i+1}{\tau_{i}\tau_{j}}}\right)}}% \right)}\frac{\partial^{2}{\sigma}}{\partial{t}^{2}}
  9. + + +...+
  10. ( i 1 = 1 N - n + 1 ( i a = i a - 1 + 1 N - ( n - a ) + 1 ( i n = i n - 1 + 1 N ( j { i 1 , , i n } τ j ) ) ) ) n σ t n \left({\sum^{N-n+1}_{i_{1}=1}{...\left({\sum^{N-\left({n-a}\right)+1}_{i_{a}=i% _{a-1}+1}{...\left({\sum^{N}_{i_{n}=i_{n-1}+1}{\left({\prod_{j\in\left\{{i_{1}% ,...,i_{n}}\right\}}{\tau_{j}}}\right)}}\right)...}}\right)...}}\right)\frac{% \partial^{n}{\sigma}}{\partial{t}^{n}}
  11. + + +...+
  12. ( i = 1 N τ i ) N σ t N \left({\prod^{N}_{i=1}{\tau_{i}}}\right)\frac{\partial^{N}{\sigma}}{\partial{t% }^{N}}
  13. = =
  14. E 0 ϵ + E_{0}\epsilon+
  15. ( i = 1 N ( E 0 + E i ) τ i ) ϵ t + {\left({\sum^{N}_{i=1}{\left({E_{0}+E_{i}}\right)\tau_{i}}}\right)}\frac{% \partial{\epsilon}}{\partial{t}}+
  16. ( i = 1 N - 1 ( j = i + 1 N ( E 0 + E i + E j ) τ i τ j ) ) 2 ϵ t 2 {\left({\sum^{N-1}_{i=1}{\left({\sum^{N}_{j=i+1}{\left({E_{0}+E_{i}+E_{j}}% \right)\tau_{i}\tau_{j}}}\right)}}\right)}\frac{\partial^{2}{\epsilon}}{% \partial{t}^{2}}
  17. + + +...+
  18. ( i 1 = 1 N - n + 1 ( i a = i a - 1 + 1 N - ( n - a ) + 1 ( i n = i n - 1 + 1 N ( ( E 0 + j { i 1 , , i n } E j ) ( k { i 1 , , i n } τ k ) ) ) ) ) n ϵ t n \left({\sum^{N-n+1}_{i_{1}=1}{...\left({\sum^{N-\left({n-a}\right)+1}_{i_{a}=i% _{a-1}+1}{...\left({\sum^{N}_{i_{n}=i_{n-1}+1}{\left({\left({E_{0}+\sum_{j\in% \left\{{i_{1},...,i_{n}}\right\}}{E_{j}}}\right)\left({\prod_{k\in\left\{{i_{1% },...,i_{n}}\right\}}{\tau_{k}}}\right)}\right)}}\right)...}}\right)...}}% \right)\frac{\partial^{n}{\epsilon}}{\partial{t}^{n}}
  19. + + +...+
  20. ( E 0 + j = 1 N E j ) ( i = 1 N τ i ) N ϵ t N \left({E_{0}+\sum_{j=1}^{N}E_{j}}\right)\left({\prod^{N}_{i=1}{\tau_{i}}}% \right)\frac{\partial^{N}{\epsilon}}{\partial{t}^{N}}
  21. N + 1 = 2 N+1=2
  22. N + 1 N+1
  23. E i E_{i}
  24. η i \eta_{i}
  25. τ i = η i E i \tau_{i}=\frac{\eta_{i}}{E_{i}}
  26. = =
  27. n = 1 N ( η 0 + i 1 = 1 N - n + 1 ( i a = i a - 1 + 1 N - ( n - a ) + 1 ( i n = i n - 1 + 1 N ( ( j { i 1 , , i n } E j ) ( k { i 1 , , i n } τ k ) ) ) ) ) n ϵ t n \sum^{N}_{n=1}{\left({\eta_{0}+\sum^{N-n+1}_{i_{1}=1}{...\left({\sum^{N-\left(% {n-a}\right)+1}_{i_{a}=i_{a-1}+1}{...\left({\sum^{N}_{i_{n}=i_{n-1}+1}{\left({% \left({\sum_{j\in\left\{{i_{1},...,i_{n}}\right\}}{E_{j}}}\right)\left({\prod_% {k\in\left\{{i_{1},...,i_{n}}\right\}}{\tau_{k}}}\right)}\right)}}\right)...}}% \right)...}}\right)\frac{\partial^{n}{\epsilon}}{\partial{t}^{n}}}
  28. ( i = 1 N - 1 ( j = i + 1 N τ i τ j ) ) 2 σ t 2 {\left({\sum^{N-1}_{i=1}{\left({\sum^{N}_{j=i+1}{\tau_{i}\tau_{j}}}\right)}}% \right)}\frac{\partial^{2}{\sigma}}{\partial{t}^{2}}
  29. + + +...+
  30. ( i 1 = 1 N - n + 1 ( i a = i a - 1 + 1 N - ( n - a ) + 1 ( i n = i n - 1 + 1 N ( j { i 1 , , i n } τ j ) ) ) ) n σ t n \left({\sum^{N-n+1}_{i_{1}=1}{...\left({\sum^{N-\left({n-a}\right)+1}_{i_{a}=i% _{a-1}+1}{...\left({\sum^{N}_{i_{n}=i_{n-1}+1}{\left({\prod_{j\in\left\{{i_{1}% ,...,i_{n}}\right\}}{\tau_{j}}}\right)}}\right)...}}\right)...}}\right)\frac{% \partial^{n}{\sigma}}{\partial{t}^{n}}
  31. + + +...+
  32. ( i = 1 N τ i ) N σ t N \left({\prod^{N}_{i=1}{\tau_{i}}}\right)\frac{\partial^{N}{\sigma}}{\partial{t% }^{N}}
  33. = =
  34. ( η 0 + i = 1 N E i τ i ) ϵ t + {\left({\eta_{0}+\sum^{N}_{i=1}{E_{i}\tau_{i}}}\right)}\frac{\partial{\epsilon% }}{\partial{t}}+
  35. ( η 0 + i = 1 N - 1 ( j = i + 1 N ( E i + E j ) τ i τ j ) ) 2 ϵ t 2 {\left({\eta_{0}+\sum^{N-1}_{i=1}{\left({\sum^{N}_{j=i+1}{\left({E_{i}+E_{j}}% \right)\tau_{i}\tau_{j}}}\right)}}\right)}\frac{\partial^{2}{\epsilon}}{% \partial{t}^{2}}
  36. + + +...+
  37. ( η 0 + i 1 = 1 N - n + 1 ( i a = i a - 1 + 1 N - ( n - a ) + 1 ( i n = i n - 1 + 1 N ( ( j { i 1 , , i n } E j ) ( k { i 1 , , i n } τ k ) ) ) ) ) n ϵ t n \left({\eta_{0}+\sum^{N-n+1}_{i_{1}=1}{...\left({\sum^{N-\left({n-a}\right)+1}% _{i_{a}=i_{a-1}+1}{...\left({\sum^{N}_{i_{n}=i_{n-1}+1}{\left({\left({\sum_{j% \in\left\{{i_{1},...,i_{n}}\right\}}{E_{j}}}\right)\left({\prod_{k\in\left\{{i% _{1},...,i_{n}}\right\}}{\tau_{k}}}\right)}\right)}}\right)...}}\right)...}}% \right)\frac{\partial^{n}{\epsilon}}{\partial{t}^{n}}
  38. + + +...+
  39. ( η 0 + ( j = 1 N E j ) ( i = 1 N τ i ) ) N ϵ t N \left({\eta_{0}+\left({\sum_{j=1}^{N}E_{j}}\right)\left({\prod^{N}_{i=1}{\tau_% {i}}}\right)}\right)\frac{\partial^{N}{\epsilon}}{\partial{t}^{N}}

Generalized_quantifier.html

  1. { x | x is a boy } { x | x sleeps } \{x\,|\,x\mbox{ is a boy}~{}\}\subseteq\{x\,|\,x\mbox{ sleeps}~{}\}
  2. a , b \langle a,b\rangle
  3. e , t ; t , t ; e , t , t ; e , e , t ; e , t , e , t , t ; \langle e,t\rangle;\qquad\langle t,t\rangle;\qquad\langle\langle e,t\rangle,t% \rangle;\qquad\langle e,\langle e,t\rangle\rangle;\qquad\langle\langle e,t% \rangle,\langle\langle e,t\rangle,t\rangle\rangle;\qquad\ldots
  4. D e D_{e}
  5. { 0 , 1 } \{0,1\}
  6. e , t \langle e,t\rangle
  7. D t D e D_{t}^{D_{e}}
  8. a , b \langle a,b\rangle
  9. a a
  10. b b
  11. D b D a D_{b}^{D_{a}}
  12. e , t \langle e,t\rangle
  13. e , t \langle e,t\rangle
  14. e , t , e , t , t \langle\langle e,t\rangle,\langle\langle e,t\rangle,t\rangle\rangle
  15. A B A\subseteq B
  16. λ x . s l e e p ( x ) \lambda x.sleep^{\prime}(x)
  17. D e D_{e}
  18. λ x . x \lambda x.x
  19. e , t \langle e,t\rangle
  20. λ X . λ Y . X Y \lambda X.\lambda Y.X\subseteq Y
  21. ( λ X . λ Y . X Y ) ( B ) ( S ) (\lambda X.\lambda Y.X\subseteq Y)(B)(S)
  22. ( λ Y . B Y ) ( S ) (\lambda Y.B\subseteq Y)(S)
  23. B S B\subseteq S
  24. e , t , t \langle\langle e,t\rangle,t\rangle
  25. X Y X\subseteq Y
  26. X Y X\subseteq Y
  27. λ X . λ Y . X Y = \lambda X.\lambda Y.X\cap Y=\emptyset
  28. λ X . λ Y . | X Y | = 3 \lambda X.\lambda Y.|X\cap Y|=3
  29. D ( A ) ( B ) D ( A ) ( A B ) D(A)(B)\leftrightarrow D(A)(A\cap B)

Generating_function_(physics).html

  1. F = F 1 ( q , Q , t ) F=F_{1}(q,Q,t)\,\!
  2. p = F 1 q p=~{}~{}\frac{\partial F_{1}}{\partial q}\,\!
  3. P = - F 1 Q P=-\frac{\partial F_{1}}{\partial Q}\,\!
  4. F = F 2 ( q , P , t ) = F 1 - Q P F=F_{2}(q,P,t)=F_{1}-QP\,\!
  5. p = F 2 q p=~{}~{}\frac{\partial F_{2}}{\partial q}\,\!
  6. Q = F 2 P Q=~{}~{}\frac{\partial F_{2}}{\partial P}\,\!
  7. F = F 3 ( p , Q , t ) = F 1 + q p F=F_{3}(p,Q,t)=F_{1}+qp\,\!
  8. q = - F 3 p q=-\frac{\partial F_{3}}{\partial p}\,\!
  9. P = - F 3 Q P=-\frac{\partial F_{3}}{\partial Q}\,\!
  10. F = F 4 ( p , P , t ) = F 1 + q p - Q P F=F_{4}(p,P,t)=F_{1}+qp-QP\,\!
  11. q = - F 4 p q=-\frac{\partial F_{4}}{\partial p}\,\!
  12. Q = F 4 P Q=~{}~{}\frac{\partial F_{4}}{\partial P}\,\!
  13. H = a P 2 + b Q 2 . H=aP^{2}+bQ^{2}.
  14. H = 1 2 q 2 + p 2 q 4 2 , H=\frac{1}{2q^{2}}+\frac{p^{2}q^{4}}{2},
  15. P = p q 2 and Q = - 1 q . P=pq^{2}\,\text{ and }Q=\frac{-1}{q}.\,
  16. H = Q 2 2 + P 2 2 , H=\frac{Q^{2}}{2}+\frac{P^{2}}{2},
  17. F = F 3 ( p , Q ) . F=F_{3}(p,Q).
  18. P = - F 3 Q , P=-\frac{\partial F_{3}}{\partial Q},
  19. p Q 2 = - F 3 Q \frac{p}{Q^{2}}=-\frac{\partial F_{3}}{\partial Q}
  20. F 3 ( p , Q ) = p Q F_{3}(p,Q)=\frac{p}{Q}
  21. q = - F 3 p = - 1 Q q=-\frac{\partial F_{3}}{\partial p}=\frac{-1}{Q}

Generator_matrix.html

  1. [ n , k , d ] q [n,k,d]_{q}
  2. k × n k\times n
  3. G = [ I k | P ] G=\begin{bmatrix}I_{k}|P\end{bmatrix}
  4. I k I_{k}
  5. G = [ I k | P ] G=\begin{bmatrix}I_{k}|P\end{bmatrix}
  6. H = [ - P | I n - k ] H=\begin{bmatrix}-P^{\top}|I_{n-k}\end{bmatrix}
  7. P P^{\top}
  8. P P
  9. C C
  10. C C^{\perp}
  11. U G = [ I k | P ] UG=\begin{bmatrix}I_{k}|P\end{bmatrix}
  12. [ I k | P ] \begin{bmatrix}I_{k}|P\end{bmatrix}

Genus_of_a_multiplicative_sequence.html

  1. 1 + p 1 z + p 2 z 2 + = ( 1 + q 1 z + q 2 z 2 + ) ( 1 + r 1 z + r 2 z 2 + ) 1+p_{1}z+p_{2}z^{2}+\dots=(1+q_{1}z+q_{2}z^{2}+\cdots)(1+r_{1}z+r_{2}z^{2}+\cdots)
  2. j K j ( p 1 , p 2 , ) z j = j K j ( q 1 , q 2 , ) z j k K k ( r 1 , r 2 , ) z k \sum_{j}K_{j}(p_{1},p_{2},\cdots)z^{j}=\sum_{j}K_{j}(q_{1},q_{2},\cdots)z^{j}% \sum_{k}K_{k}(r_{1},r_{2},\cdots)z^{k}
  3. K = 1 + K 1 + K 2 + K=1+K_{1}+K_{2}+\cdots
  4. K ( p 1 , p 2 , p 3 , ) = Q ( z 1 ) Q ( z 2 ) Q ( z 3 ) K(p_{1},p_{2},p_{3},\cdots)=Q(z_{1})Q(z_{2})Q(z_{3})\cdots
  5. Φ ( X ) = K ( p 1 , p 2 , p 3 , ) \Phi(X)=K(p_{1},p_{2},p_{3},\cdots)
  6. z tanh ( z ) = k 0 2 2 k B 2 k z k ( 2 k ) ! = 1 + z 3 - z 2 45 + {\sqrt{z}\over\tanh(\sqrt{z})}=\sum_{k\geq 0}{2^{2k}B_{2k}z^{k}\over(2k)!}=1+{% z\over 3}-{z^{2}\over 45}+\cdots
  7. B 2 k B_{2k}
  8. L 0 = 1 L_{0}=1
  9. L 1 = 1 3 p 1 L_{1}=\tfrac{1}{3}p_{1}
  10. L 2 = 1 45 ( 7 p 2 - p 1 2 ) L_{2}=\tfrac{1}{45}(7p_{2}-p_{1}^{2})
  11. L 3 = 1 945 ( 62 p 3 - 13 p 1 p 2 + 2 p 1 3 ) L_{3}=\tfrac{1}{945}(62p_{3}-13p_{1}p_{2}+2p_{1}^{3})
  12. L 4 = 1 14175 ( 381 p 4 - 71 p 1 p 3 - 19 p 2 2 + 22 p 1 2 p 2 - 3 p 1 4 ) L_{4}=\tfrac{1}{14175}(381p_{4}-71p_{1}p_{3}-19p_{2}^{2}+22p_{1}^{2}p_{2}-3p_{% 1}^{4})
  13. p i = p i ( M ) p_{i}=p_{i}(M)
  14. [ M ] [M]
  15. σ ( M ) \sigma(M)
  16. σ ( M ) = L n ( p 1 ( M ) , , p n ( M ) ) , [ M ] . \sigma(M)=\langle L_{n}(p_{1}(M),\dots,p_{n}(M)),[M]\rangle.
  17. z 1 - exp ( - z ) = 1 + 1 2 z + i = 1 ( - 1 ) i + 1 B 2 i ( 2 i ) ! z 2 i \frac{z}{1-\exp(-z)}=1+\frac{1}{2}z+\sum_{i=1}^{\infty}(-1)^{i+1}\frac{B_{2i}}% {(2i)!}z^{2i}
  18. B 2 k B_{2k}
  19. T d 0 = 1 Td_{0}=1
  20. T d 1 = 1 2 c 1 Td_{1}=\tfrac{1}{2}c_{1}
  21. T d 2 = 1 12 ( c 2 + c 1 2 ) Td_{2}=\tfrac{1}{12}(c_{2}+c_{1}^{2})
  22. T d 3 = 1 24 c 1 c 2 Td_{3}=\tfrac{1}{24}c_{1}c_{2}
  23. T d 4 = 1 720 ( - c 1 4 + 4 c 2 c 1 2 + 3 c 2 2 + c 3 c 1 - c 4 ) Td_{4}=\tfrac{1}{720}(-c_{1}^{4}+4c_{2}c_{1}^{2}+3c_{2}^{2}+c_{3}c_{1}-c_{4})
  24. Td n ( n ) = 1 \mathrm{Td}_{n}(\mathbb{CP}^{n})=1
  25. Q ( z ) = z / 2 sinh ( z / 2 ) = 1 - z / 24 + 7 z 2 / 5760 - . Q(z)={\sqrt{z}/2\over\sinh(\sqrt{z}/2)}=1-z/24+7z^{2}/5760-\cdots.
  26. A ^ 0 = 1 \hat{A}_{0}=1
  27. A ^ 1 = - 1 24 p 1 \hat{A}_{1}=-\tfrac{1}{24}p_{1}
  28. A ^ 2 = 1 5760 ( - 4 p 2 + 7 p 1 2 ) \hat{A}_{2}=\tfrac{1}{5760}(-4p_{2}+7p_{1}^{2})
  29. A ^ 3 = 1 967680 ( - 16 p 3 + 44 p 2 p 1 - 31 p 1 3 ) \hat{A}_{3}=\tfrac{1}{967680}(-16p_{3}+44p_{2}p_{1}-31p_{1}^{3})
  30. A ^ 4 = 1 464486400 ( - 192 p 4 + 512 p 3 p 1 + 208 p 2 2 - 904 p 2 p 1 2 + 381 p 1 4 ) \hat{A}_{4}=\tfrac{1}{464486400}(-192p_{4}+512p_{3}p_{1}+208p_{2}^{2}-904p_{2}% p_{1}^{2}+381p_{1}^{4})
  31. 2 {\mathbb{Z}}_{2}
  32. 2 {\mathbb{Z}}_{2}
  33. f 2 = 1 - 2 δ f 2 + ϵ f 4 {f^{\prime}}^{2}=1-2\delta f^{2}+\epsilon f^{4}
  34. δ = ϵ = 1 , f ( z ) = tanh ( z ) \delta=\epsilon=1,f(z)=\tanh(z)
  35. δ = - 1 / 8 , ϵ = 0 , f ( z ) = 2 sinh ( z / 2 ) \delta=-1/8,\epsilon=0,f(z)=2\sinh(z/2)
  36. 1 3 δ p 1 \tfrac{1}{3}\delta p_{1}
  37. 1 90 [ ( - 4 δ 2 + 18 ϵ ) p 2 + ( 7 δ 2 - 9 ϵ ) p 1 2 ] \tfrac{1}{90}\big[(-4\delta^{2}+18\epsilon)p_{2}+(7\delta^{2}-9\epsilon)p_{1}^% {2}\big]
  38. 1 1890 [ ( 16 δ 3 + 108 δ ϵ ) p 3 + ( - 44 δ 3 + 18 δ ϵ ) p 2 p 1 + ( 31 δ 3 - 27 δ ϵ ) p 1 3 ] \tfrac{1}{1890}\big[(16\delta^{3}+108\delta\epsilon)p_{3}+(-44\delta^{3}+18% \delta\epsilon)p_{2}p_{1}+(31\delta^{3}-27\delta\epsilon)p_{1}^{3}\big]
  39. 1 113400 [ ( - 192 δ 4 + 1728 δ 2 ϵ + 1512 ϵ 2 ) p 4 + ( 512 δ 4 + 432 δ 2 ϵ - 1512 ϵ 2 ) p 3 p 1 + ( 208 δ 4 - 1872 δ 2 ϵ + 1512 ϵ 2 ) p 2 2 + ( - 904 δ 4 + 1836 δ 2 ϵ - 756 ϵ 2 ) p 2 p 1 2 + ( 381 δ 4 - 594 δ 2 ϵ + 189 ϵ 2 ) p 1 4 ] \tfrac{1}{113400}\big[(-192\delta^{4}+1728\delta^{2}\epsilon+1512\epsilon^{2})% p_{4}+(512\delta^{4}+432\delta^{2}\epsilon-1512\epsilon^{2})p_{3}p_{1}+(208% \delta^{4}-1872\delta^{2}\epsilon+1512\epsilon^{2})p_{2}^{2}+(-904\delta^{4}+1% 836\delta^{2}\epsilon-756\epsilon^{2})p_{2}p_{1}^{2}+(381\delta^{4}-594\delta^% {2}\epsilon+189\epsilon^{2})p_{1}^{4}\big]
  40. Q ( z ) = z / σ L ( z ) = exp ( k 2 2 G 2 k ( τ ) z 2 k ( 2 k ) ! ) Q(z)=z/\sigma_{L}(z)=\exp\left(\sum_{k\geq 2}{2G_{2k}(\tau)z^{2k}\over(2k)!}\right)

Geodesic_grid.html

  1. sin γ 1 2 = a 1 2 r u . \sin\frac{\gamma_{1}}{2}=\frac{a_{1}}{2r_{u}}.
  2. s s
  3. γ s γ 1 / s \gamma_{s}\equiv\gamma_{1}/s
  4. sin γ s 2 = a s 2 r u , \sin\frac{\gamma_{s}}{2}=\frac{a_{s}}{2r_{u}},
  5. a s = 2 r u 1 - cos ( γ 1 / s ) 2 . a_{s}=2r_{u}\sqrt{\frac{1-\cos(\gamma_{1}/s)}{2}}.
  6. s > 1 s>1
  7. V s = 20 t = 1 , , s 2 r i , s , t A s , t 3 . V_{s}=20\sum_{t=1,\ldots,s^{2}}\frac{r_{i,s,t}A_{s,t}}{3}.
  8. f s V s 4 π r u 3 / 3 f_{s}\equiv\frac{V_{s}}{4\pi r_{u}^{3}/3}
  9. s s
  10. ( s + 1 ) ( s + 2 ) / 2 (s+1)(s+2)/2
  11. a < s u b > s / r u a<sub>s/r_{u}

Geodesic_manifold.html

  1. \mathbb{R}
  2. n \mathbb{R}^{n}
  3. 𝕊 n \mathbb{S}^{n}
  4. 𝕋 n \mathbb{T}^{n}
  5. M := 2 { 0 } M:=\mathbb{R}^{2}\setminus\{0\}

Geodesics_as_Hamiltonian_flows.html

  1. H = m v 2 / 2 = p 2 / 2 m H=mv^{2}/2=p^{2}/2m
  2. γ : I M \gamma:I\to M
  3. E ( γ ) = 1 2 I g ( γ ˙ ( t ) , γ ˙ ( t ) ) d t , E(\gamma)=\frac{1}{2}\int_{I}g(\dot{\gamma}(t),\dot{\gamma}(t))\,dt,
  4. γ ˙ ( t ) \dot{\gamma}(t)
  5. γ \gamma
  6. t I t\in I
  7. g ( , ) g(\cdot,\cdot)
  8. d 2 x a d t 2 + Γ a d x b d t b c d x c d t = 0 \frac{d^{2}x^{a}}{dt^{2}}+\Gamma^{a}{}_{bc}\frac{dx^{b}}{dt}\frac{dx^{c}}{dt}=0
  9. Γ a b c \Gamma^{a}{}_{bc}
  10. T * M | U U × n T^{*}M|_{U}\simeq U\times\mathbb{R}^{n}
  11. H ( x , p ) = 1 2 g a b ( x ) p a p b . H(x,p)=\frac{1}{2}g^{ab}(x)p_{a}p_{b}.
  12. δ a c {\delta^{a}}_{c}
  13. x ˙ a = H p a = g a b ( x ) p b \dot{x}^{a}=\frac{\partial H}{\partial p_{a}}=g^{ab}(x)p_{b}
  14. p ˙ a = - H x a = - 1 2 g b c ( x ) x a p b p c . \dot{p}_{a}=-\frac{\partial H}{\partial x^{a}}=-\frac{1}{2}\frac{\partial g^{% bc}(x)}{\partial x^{a}}p_{b}p_{c}.
  15. d H d t = H x a x ˙ a + H p a p ˙ a = - p ˙ a x ˙ a + x ˙ a p ˙ a = 0. \frac{dH}{dt}=\frac{\partial H}{\partial x^{a}}\dot{x}^{a}+\frac{\partial H}{% \partial p_{a}}\dot{p}_{a}=-\dot{p}_{a}\dot{x}^{a}+\dot{x}^{a}\dot{p}_{a}=0.
  16. M E = { ( x , p ) T * M : H ( x , p ) = E } M_{E}=\{(x,p)\in T^{*}M:H(x,p)=E\}
  17. T * M = E 0 M E T^{*}M=\bigcup_{E\geq 0}M_{E}

Geometric_function_theory.html

  1. f : U V f:U\rightarrow V\qquad
  2. U , V n U,V\subset\mathbb{C}^{n}
  3. u 0 u_{0}
  4. u 0 u_{0}
  5. G G
  6. Ω \Omega
  7. f : G Ω f:G\to\Omega
  8. f ( G ) = Ω f(G)=\Omega
  9. f f
  10. f f
  11. f f
  12. f - 1 f^{-1}
  13. z 0 z_{0}
  14. D 1 ( D 1 ) D_{1}(D_{1}\neq\mathbb{C})
  15. D 1 D_{1}
  16. w = f ( z ) w=f(z)
  17. D 1 D_{1}
  18. | w | < 1 |w|<1
  19. f ( z 0 ) = 0 f(z_{0})=0
  20. f ( z 0 ) > 0 f^{\prime}(z_{0})>0
  21. D 1 D_{1}
  22. D 2 D_{2}
  23. \mathbb{C}
  24. w = f ( z ) w=f(z)
  25. D 1 D_{1}
  26. w = g ( z ) w=g(z)
  27. D 2 D_{2}
  28. g - 1 f g^{-1}f
  29. D 1 D_{1}
  30. D 2 D_{2}
  31. g - 1 g^{-1}
  32. D 1 D_{1}
  33. D 2 D_{2}
  34. \mathbb{C}
  35. 2 - 2 g 2-2g\,
  36. π : S S \pi:S^{\prime}\to S\,
  37. χ ( S ) = N χ ( S ) . \chi(S^{\prime})=N\cdot\chi(S).\,
  38. χ ( S ) = N χ ( S ) - P S ( e P - 1 ) \chi(S^{\prime})=N\cdot\chi(S)-\sum_{P\in S^{\prime}}(e_{P}-1)

Geostrophic_current.html

  1. f v = 1 ρ p x fv=\frac{1}{\rho}\frac{\partial p}{\partial x}
  2. f u = - 1 ρ p y fu=-\frac{1}{\rho}\frac{\partial p}{\partial y}
  3. f f
  4. u x + v y = 0 \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0
  5. u t - f v = - 1 ρ p x \frac{\partial u}{\partial t}-fv=-\frac{1}{\rho}\frac{\partial p}{\partial x}
  6. v t + f u = - 1 ρ p y \frac{\partial v}{\partial t}+fu=-\frac{1}{\rho}\frac{\partial p}{\partial y}
  7. u t = v t = 0 \frac{\partial u}{\partial t}=\frac{\partial v}{\partial t}=0
  8. u v e i ω t u\propto v\propto e^{i\omega t}
  9. ω = 0 \omega=0

Geothermal_heat_pump.html

  1. G H G S a v i n g s = H L ( F I A F U E × 1000 k g t o n - E I C O P × 3600 s e c h r ) GHG\ Savings=HL\left(\frac{FI}{AFUE\times 1000\frac{kg}{ton}}-\frac{EI}{COP% \times 3600\frac{sec}{hr}}\right)

GHK_flux_equation.html

  1. Φ S = P S z S 2 V m F 2 R T [ S ] i - [ S ] o exp ( - z S V m F / R T ) 1 - exp ( - z S V m F / R T ) \Phi_{S}=P_{S}z_{S}^{2}\frac{V_{m}F^{2}}{RT}\frac{[\mbox{S}~{}]_{i}-[\mbox{S}~% {}]_{o}\exp(-z_{S}V_{m}F/RT)}{1-\exp(-z_{S}V_{m}F/RT)}
  2. Φ \Phi
  3. lim Φ S 0 V m 0 \lim_{\Phi_{S}\rightarrow 0}V_{m}\neq 0
  4. lim V m 0 Φ S 0 \lim_{V_{m}\rightarrow 0}\Phi_{S}\neq 0
  5. V m = 0 V_{m}=0
  6. Φ S = 0 0 \Phi_{S}=\frac{0}{0}
  7. 0 0 \frac{0}{0}
  8. lim V m 0 Φ S = P S z S 2 F 2 R T [ V m ( [ S ] i - [ S ] o exp ( - z S V m F / R T ) ) ] [ 1 - exp ( - z S V m F / R T ) ] \lim_{V_{m}\rightarrow 0}\Phi_{S}=P_{S}\frac{z_{S}^{2}F^{2}}{RT}\frac{[V_{m}([% \mbox{S}~{}]_{i}-[\mbox{S}~{}]_{o}\exp(-z_{S}V_{m}F/RT))]^{\prime}}{[1-\exp(-z% _{S}V_{m}F/RT)]^{\prime}}
  9. [ f ] [f]^{\prime}
  10. lim V m 0 Φ S = P S z S F ( [ S ] i - [ S ] o ) \lim_{V_{m}\rightarrow 0}\Phi_{S}=P_{S}z_{S}F([\mbox{S}~{}]_{i}-[\mbox{S}~{}]_% {o})
  11. V m = 0 V_{m}=0
  12. Φ S 0 \Phi_{S}\neq 0
  13. ( [ S ] i - [ S ] o ) 0 ([\mbox{S}~{}]_{i}-[\mbox{S}~{}]_{o})\neq 0
  14. lim Φ S 0 V m 0 \lim_{\Phi_{S}\rightarrow 0}V_{m}\neq 0
  15. Φ S = 0 \Phi_{S}=0
  16. Φ S = 0 = P S z S 2 F 2 R T V m ( [ S ] i - [ S ] o exp ( - z S V m F / R T ) ) 1 - exp ( - z S V m F / R T ) \Phi_{S}=0=P_{S}\frac{z_{S}^{2}F^{2}}{RT}\frac{V_{m}([\mbox{S}~{}]_{i}-[\mbox{% S}~{}]_{o}\exp(-z_{S}V_{m}F/RT))}{1-\exp(-z_{S}V_{m}F/RT)}
  17. [ S ] i - [ S ] o exp ( - z S V m F / R T ) = 0 [\mbox{S}~{}]_{i}-[\mbox{S}~{}]_{o}\exp(-z_{S}V_{m}F/RT)=0
  18. V m = - R T z S F ln ( [ S ] i [ S ] o ) V_{m}=-\frac{RT}{z_{S}F}\ln\left(\frac{[\mbox{S}~{}]_{i}}{[\mbox{S}~{}]_{o}}\right)
  19. Φ S | i o = P S z S 2 V m F 2 R T [ S ] i for V m 0 \Phi_{S|i\to o}=P_{S}z_{S}^{2}\frac{V_{m}F^{2}}{RT}[\mbox{S}~{}]_{i}\ \mbox{% for}~{}\ V_{m}\gg\;0
  20. Φ S | o i = P S z S 2 V m F 2 R T [ S ] o for V m 0 \Phi_{S|o\to i}=P_{S}z_{S}^{2}\frac{V_{m}F^{2}}{RT}[\mbox{S}~{}]_{o}\ \mbox{% for}~{}\ V_{m}\ll\;0
  21. Φ \Phi

Ghost_Leg.html

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

Gibbs_isotherm.html

  1. - d γ = Γ 1 d μ 1 + Γ 2 d μ 2 , -\mathrm{d}\gamma\ =\Gamma_{1}\mathrm{d}\mu_{1}\,+\Gamma_{2}\mathrm{d}\mu_{2}\,,
  2. γ \gamma\,\!
  3. Γ \Gamma\,\!
  4. μ \mu\,\!
  5. α α
  6. β β
  7. α α
  8. β β
  9. α α
  10. β β
  11. α α
  12. β β
  13. α α
  14. β β
  15. i i
  16. i i
  17. i i
  18. α α
  19. β β
  20. Γ i = n i TOTAL - n i α - n i β A , \Gamma_{i}=\frac{{n_{i}}^{\,\text{TOTAL}}-{n_{i}}^{\alpha}\,-{n_{i}}^{\beta}\,% }{A}\,,
  21. i i
  22. n n
  23. α α
  24. β β
  25. A A
  26. Γ Γ
  27. i i
  28. Γ i 1 = Γ i - Γ 1 ( C i α - C i β C 1 α - C 1 β ) . \Gamma_{i}^{1}=\Gamma_{i}-\Gamma_{1}\,\left(\frac{{C_{i}}^{\alpha}\,-{C_{i}}^{% \beta}\,}{{C_{1}}^{\alpha}\,-{C_{1}}^{\beta}\,}\right)\,.
  29. α α
  30. β β
  31. G = G α + G β + G S , G=G^{\alpha}\,+G^{\beta}\,+G^{\mathrm{S}}\,,
  32. G G
  33. α α
  34. β β
  35. G = U + p V - T S + i = 1 k μ i n i , G=U+pV-TS+\sum_{i=1}^{k}\mu_{i}\,\mathrm{n}_{i}\,,
  36. U U
  37. p p
  38. V V
  39. T T
  40. S S
  41. i i
  42. α α
  43. β β
  44. d G = α , β , S ( d U + p d V + V d p - T d S - S d T + i = 1 k μ i d n i + i = 1 k n i d μ i ) + A d γ + γ d A , \mathrm{d}G=\sum_{\alpha,\beta,S}\,\left(\mathrm{d}U+p\mathrm{d}V\,+V\mathrm{d% }p\,-T\mathrm{d}S\,-S\mathrm{d}T\,+\sum_{i=1}^{k}\mu_{i}\,\mathrm{d}n_{i}\,+% \sum_{i=1}^{k}\mathrm{n}_{i}\,\mathrm{d}\mu_{i}\,\right)+A\mathrm{d}\gamma\,+% \gamma\mathrm{d}A\,,
  45. A A
  46. γ γ
  47. d U = δ q + δ w , \mathrm{d}U=\delta\,q+\delta\,w\,,
  48. q q
  49. w w
  50. δ q + δ w = α , β , S ( T d S - p d V - δ w non-pV ) . \delta\,q+\delta\,w=\sum_{\alpha,\beta,S}\,\left(T\mathrm{d}S\,-p\mathrm{d}V\,% -\delta\,w_{\,\text{non-pV}}\right)\,.
  51. γ d A γdA
  52. d G = α , β , S ( V d p - S d T + i = 1 k μ i d n i + i = 1 k n i d μ i ) + A d γ , \mathrm{d}G=\sum_{\alpha,\beta,S}\,\left(V\mathrm{d}p\,-S\mathrm{d}T\,+\sum_{i% =1}^{k}\mu_{i}\,\mathrm{d}n_{i}\,+\sum_{i=1}^{k}\mathrm{n}_{i}\,\mathrm{d}\mu_% {i}\,\right)+A\mathrm{d}\gamma\,,
  53. d G = V d p - S d T + i = 1 k μ i d n i . \mathrm{d}G=V\mathrm{d}p\,-S\mathrm{d}T\,+\sum_{i=1}^{k}\mu_{i}\,\mathrm{d}n_{% i}\,.
  54. α α
  55. β β
  56. i = 1 k n i α d μ i + i = 1 k n i β d μ i + i = 1 k n i S d μ i + A d γ = 0 . \sum_{i=1}^{k}\mathrm{n_{i}}^{\alpha}\,\mathrm{d}\mu_{i}\,+\sum_{i=1}^{k}% \mathrm{n_{i}}^{\beta}\,\mathrm{d}\mu_{i}\,+\sum_{i=1}^{k}\mathrm{n_{i}}^{% \mathrm{S}}\,\mathrm{d}\mu_{i}\,+A\mathrm{d}\gamma\,=0\,.
  57. α α
  58. β β
  59. i = 1 k n i α d μ i = i = 1 k n i β d μ i = 0 . \sum_{i=1}^{k}\mathrm{n_{i}}^{\alpha}\,\mathrm{d}\mu_{i}\,=\sum_{i=1}^{k}% \mathrm{n_{i}}^{\beta}\,\mathrm{d}\mu_{i}\,=0\,.
  60. i = 1 k n i S d μ i + A d γ = 0 . \sum_{i=1}^{k}\mathrm{n_{i}}^{\mathrm{S}}\,\mathrm{d}\mu_{i}\,+A\mathrm{d}% \gamma\,=0\,.
  61. - d γ = Γ 1 d μ 1 + Γ 2 d μ 2 . -\mathrm{d}\gamma\ =\Gamma_{1}\mathrm{d}\mu_{1}\,+\Gamma_{2}\mathrm{d}\mu_{2}\,.
  62. i i
  63. μ i = μ i o + R T ln a i , \mu_{i}={\mu_{i}}^{o}+RT\ln a_{i}\,,
  64. i i
  65. i i
  66. R R
  67. T T
  68. i i
  69. d μ i = R T d a i a i = R T d ln f C i , \mathrm{d}\mu_{i}=RT\frac{\mathrm{d}a_{i}}{a_{i}}=RT\mathrm{d}\ln fC_{i}\,,
  70. f f
  71. i i
  72. C C
  73. i i
  74. α α
  75. β β
  76. i i
  77. i i
  78. Γ i = - 1 R T ( γ ln C i ) T , p . \Gamma_{i}=-\frac{1}{RT}\left(\frac{\partial\gamma}{\partial\ln C_{i}}\right)_% {T,p}\,.
  79. R Na z = R z- + z Na + . \mathrm{R}\,\mathrm{Na_{\,\text{z}}}=\mathrm{R}^{\,\text{z-}}+\mathrm{z}\,% \mathrm{Na^{\,\text{+}}}\,.
  80. - d γ = Γ RNaz W d μ RNaz + Γ NaCl W d μ NaCl . -\mathrm{d}\gamma=\Gamma_{\,\text{RNaz}}^{W}\,\mathrm{d}\mu_{\,\text{RNaz}}\,+% \Gamma_{\,\text{NaCl}}^{W}\,\mathrm{d}\mu_{\,\text{NaCl}}\,.
  81. - d γ = m R T Γ i W d ln C i , -\mathrm{d}\gamma=mRT\Gamma_{i}^{W}\,\mathrm{d}\ln C_{i}\,,
  82. m m
  83. m m
  84. Γ S = - 1 2 R T ( γ ln C ) T , p , \Gamma_{S}=-\frac{1}{2RT}\left(\frac{\partial\gamma}{\partial\ln C}\right)_{T,% p}\,,
  85. Γ S \Gamma_{S}
  86. Γ S = - 1 R T ( γ ln C ) T , p . \Gamma_{S}=-\frac{1}{RT}\left(\frac{\partial\gamma}{\partial\ln C}\right)_{T,p% }\,.

Gingerbreadman_map.html

  1. { x n + 1 = 1 - y n + | x n | y n + 1 = x n \begin{cases}x_{n+1}=1-y_{n}+|x_{n}|\\ y_{n+1}=x_{n}\end{cases}

Glare_(vision).html

  1. L b L_{b}
  2. UGR = 8 log 0.25 L b n ( L n 2 ω n p n 2 ) , \mathrm{UGR}=8\log\frac{0.25}{L_{b}}\sum_{n}\left(L_{n}^{2}\frac{\omega_{n}}{p% _{n}^{2}}\right),
  3. log \log
  4. L n L_{n}
  5. n n
  6. ω n \omega_{n}
  7. p n p_{n}

Glauber–Sudarshan_P_representation.html

  1. P ( α ) P(\alpha)
  2. ρ ^ \hat{\rho}
  3. { | α } \{|\alpha\rangle\}
  4. ρ ^ = P ( α ) | α α | d 2 α , d 2 α d Re ( α ) d Im ( α ) . \hat{\rho}=\int P(\alpha)|{\alpha}\rangle\langle{\alpha}|\,d^{2}\alpha,\qquad d% ^{2}\alpha\equiv d\,{\rm Re}(\alpha)\,d\,{\rm Im}(\alpha).
  5. ρ ^ A = j , k c j , k a ^ j a ^ k . \hat{\rho}_{A}=\sum_{j,k}c_{j,k}\cdot\hat{a}^{j}\hat{a}^{\dagger k}.
  6. I ^ = 1 π | α α | d 2 α , \hat{I}=\frac{1}{\pi}\int|{\alpha}\rangle\langle{\alpha}|\,d^{2}\alpha,
  7. ρ A ( a ^ , a ^ ) = 1 π j , k c j , k a ^ j | α α | a ^ k d 2 α = 1 π j , k c j , k α j | α α | α * k d 2 α = 1 π j , k c j , k α j α * k | α α | d 2 α = 1 π ρ A ( α , α * ) | α α | d 2 α , \begin{aligned}\displaystyle\rho_{A}(\hat{a},\hat{a}^{\dagger})&\displaystyle=% \frac{1}{\pi}\sum_{j,k}\int c_{j,k}\cdot\hat{a}^{j}|{\alpha}\rangle\langle{% \alpha}|\hat{a}^{\dagger k}\,d^{2}\alpha\\ &\displaystyle=\frac{1}{\pi}\sum_{j,k}\int c_{j,k}\cdot\alpha^{j}|{\alpha}% \rangle\langle{\alpha}|\alpha^{*k}\,d^{2}\alpha\\ &\displaystyle=\frac{1}{\pi}\int\sum_{j,k}c_{j,k}\cdot\alpha^{j}\alpha^{*k}|{% \alpha}\rangle\langle{\alpha}|\,d^{2}\alpha\\ &\displaystyle=\frac{1}{\pi}\int\rho_{A}(\alpha,\alpha^{*})|{\alpha}\rangle% \langle{\alpha}|\,d^{2}\alpha,\end{aligned}
  8. P ( α ) = 1 π ρ A ( α , α * ) . P(\alpha)=\frac{1}{\pi}\rho_{A}(\alpha,\alpha^{*}).
  9. χ N ( β ) = tr ( ρ ^ e i β a ^ e i β * a ^ ) \chi_{N}(\beta)=\operatorname{tr}(\hat{\rho}\cdot e^{i\beta\cdot\hat{a}^{% \dagger}}e^{i\beta^{*}\cdot\hat{a}})
  10. P ( α ) = 1 π 2 χ N ( β ) e - β α * + β * α d 2 β . P(\alpha)=\frac{1}{\pi^{2}}\int\chi_{N}(\beta)e^{-\beta\alpha^{*}+\beta^{*}% \alpha}\,d^{2}\beta.
  11. P ( α ) = e | α | 2 π - β | ρ ^ | β e | β | 2 - β α * + β * α d 2 β . P(\alpha)=\frac{e^{|\alpha|^{2}}}{\pi}\int\langle-\beta|\hat{\rho}|\beta% \rangle e^{|\beta|^{2}-\beta\alpha^{*}+\beta^{*}\alpha}\,d^{2}\beta.
  12. ρ ^ \hat{\rho}
  13. { | n } \{|n\rangle\}
  14. P ( α ) = n k n | ρ ^ | k n ! k ! 2 π r ( n + k ) ! e r 2 - i ( n - k ) θ [ ( - r ) n + k δ ( r ) ] , P(\alpha)=\sum_{n}\sum_{k}\langle n|\hat{\rho}|k\rangle\frac{\sqrt{n!k!}}{2\pi r% (n+k)!}e^{r^{2}-i(n-k)\theta}\left[\left(-\frac{\partial}{\partial r}\right)^{% n+k}\delta(r)\right],
  15. P ( α ) \scriptstyle P(\alpha)\,
  16. n ^ 𝐤 , s = 1 e ω / k B T - 1 . \langle\hat{n}_{\mathbf{k},s}\rangle=\frac{1}{e^{\hbar\omega/k_{B}T}-1}.
  17. P ( { α 𝐤 , s } ) = 𝐤 , s 1 π n ^ 𝐤 , s e - | α | 2 / n ^ 𝐤 , s . P(\{\alpha_{\mathbf{k},s}\})=\prod_{\mathbf{k},s}\frac{1}{\pi\langle\hat{n}_{% \mathbf{k},s}\rangle}e^{-|\alpha|^{2}/\langle\hat{n}_{\mathbf{k},s}\rangle}.
  18. | ψ = c 0 | α 0 + c 1 | α 1 |\psi\rangle=c_{0}|\alpha_{0}\rangle+c_{1}|\alpha_{1}\rangle
  19. 1 = | c 0 | 2 + | c 1 | 2 + 2 e - ( | α 0 | 2 + | α 1 | 2 ) / 2 Re ( c 0 * c 1 e α 0 * α 1 ) . 1=|c_{0}|^{2}+|c_{1}|^{2}+2e^{-(|\alpha_{0}|^{2}+|\alpha_{1}|^{2})/2}% \operatorname{Re}\left(c_{0}^{*}c_{1}e^{\alpha_{0}^{*}\alpha_{1}}\right).
  20. | α 0 |\alpha_{0}\rangle
  21. | α 1 |\alpha_{1}\rangle
  22. - α | ρ ^ | α = - α | ψ ψ | α \langle-\alpha|\hat{\rho}|\alpha\rangle=\langle-\alpha|\psi\rangle\langle\psi|\alpha\rangle
  23. P ( α ) \displaystyle P(\alpha)
  24. ψ | n ^ | ψ \displaystyle\langle\psi|\hat{n}|\psi\rangle

Gleason's_theorem.html

  1. α 1 , α 2 , α 3 , , α n \alpha_{1},\alpha_{2},\alpha_{3},...,\alpha_{n}
  2. x i x_{i}
  3. α i \alpha_{i}
  4. x i x_{i}
  5. P ( 0 ) = 0 P(0)=0
  6. P ( y ) 0 P(y)\geq 0
  7. y L y\in L
  8. j = 1 n P ( x j ) = 1 \sum_{j=1}^{n}P(x_{j})=1
  9. x 1 , x 2 , x 3 , , x n x_{1},x_{2},x_{3},...,x_{n}
  10. P ( y ) = Σ { P ( x j ) | x j y } P(y)=\Sigma\{P(x_{j})|x_{j}\leq y\}
  11. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  12. 3 \geq 3
  13. P ( x ) = 𝐱 , W 𝐱 P(x)=\langle\mathbf{x},W\mathbf{x}\rangle
  14. x L x\in L
  15. , \langle\,,\,\rangle
  16. 𝐱 \mathbf{x}
  17. x x
  18. x 0 x_{0}
  19. P ( x 0 ) = 1 P(x_{0})=1
  20. x 0 x_{0}
  21. P ( x ) = | 𝐱 𝟎 , 𝐱 | 2 P(x)=\left|\langle\mathbf{x_{0}},\mathbf{x}\rangle\right|^{2}
  22. x L x\in L
  23. u W u , u u\rightarrow\langle Wu,u\rangle

Glicko_rating_system.html

  1. R D RD
  2. R D 0 RD_{0}
  3. R D = min ( R D 0 2 + c 2 t , 350 ) RD=\min\left(\sqrt{{RD_{0}}^{2}+c^{2}t},350\right)
  4. t t
  5. c c
  6. 350 = 50 2 + 100 c 2 350=\sqrt{50^{2}+100c^{2}}
  7. c c
  8. ( 350 2 - 50 2 ) / 100 = c \sqrt{(350^{2}-50^{2})/100}=c
  9. r = r 0 + q 1 R D 2 + 1 d 2 i = 1 m g ( R D i ) ( s i - E ( s | r , r i , R D i ) ) r=r_{0}+\frac{q}{\frac{1}{RD^{2}}+\frac{1}{d^{2}}}\sum_{i=1}^{m}{g(RD_{i})(s_{% i}-E(s|r,r_{i},RD_{i}))}
  10. g ( R D i ) = 1 1 + 3 q 2 ( R D i 2 ) π 2 g(RD_{i})=\frac{1}{\sqrt{1+\frac{3q^{2}(RD_{i}^{2})}{\pi^{2}}}}
  11. E ( s | r , r i , R D i ) = 1 1 + 10 ( g ( R D i ) ( r - r i ) - 400 ) E(s|r,r_{i},RD_{i})=\frac{1}{1+10^{\left(\frac{g(RD_{i})(r-r_{i})}{-400}\right% )}}
  12. q = l n ( 10 ) 400 = 0.00575646273 q=\frac{ln(10)}{400}=0.00575646273
  13. d 2 = 1 q 2 i = 1 m ( g ( R D i ) ) 2 E ( s | r , r i , R D i ) ( 1 - E ( s | r , r i , R D i ) ) d^{2}=\frac{1}{q^{2}\sum_{i=1}^{m}{(g(RD_{i}))^{2}E(s|r,r_{i},RD_{i})(1-E(s|r,% r_{i},RD_{i}))}}
  14. r i r_{i}
  15. s i s_{i}
  16. 1 2 \frac{1}{2}
  17. R D = ( 1 R D 2 + 1 d 2 ) - 1 RD^{\prime}=\sqrt{\left(\frac{1}{RD^{2}}+\frac{1}{d^{2}}\right)^{-1}}

Globally_hyperbolic_manifold.html

  1. J + ( p ) J^{+}(p)
  2. J - ( p ) J^{-}(p)
  3. J - ( p ) J + ( q ) J^{-}(p)\cap J^{+}(q)
  4. \mathbb{R}
  5. C 0 C^{0}
  6. \mathbb{R}

Glossary_of_semisimple_groups.html

  1. A l t k n Alt^{k}\ {\mathbb{C}}^{n}

Glycosylphosphatidylinositol_phospholipase_D.html

  1. \rightleftharpoons

Godunov's_scheme.html

  1. Q i n = 1 Δ x x i - 1 / 2 x i + 1 / 2 q ( t n , x ) d x Q^{n}_{i}=\frac{1}{\Delta x}\int_{x_{i-1/2}}^{x_{i+1/2}}q(t^{n},x)\,dx
  2. x i = x low + ( i - 1 / 2 ) Δ x x_{i}=x_{\,\text{low}}+\left(i-1/2\right)\Delta x
  3. t n = n Δ t t^{n}=n\Delta t
  4. q t + ( f ( q ) ) x = 0. q_{t}+(f(q))_{x}=0.
  5. [ x i - 1 / 2 , x i + 1 / 2 ] , [x_{i-1/2},x_{i+1/2}],
  6. t Q i ( t ) = - 1 Δ x ( f ( q ( t , x i + 1 / 2 ) ) - f ( q ( t , x i - 1 / 2 ) ) ) , \frac{\partial}{\partial t}Q_{i}(t)=-\frac{1}{\Delta x}\left(f(q(t,x_{i+1/2}))% -f(q(t,x_{i-1/2}))\right),
  7. t = t n t=t^{n}
  8. t = t n + 1 t=t^{n+1}
  9. Q i n + 1 = Q i n - 1 Δ x t n t n + 1 ( f ( q ( t , x i + 1 / 2 ) ) - f ( q ( t , x i - 1 / 2 ) ) ) d t . Q^{n+1}_{i}=Q^{n}_{i}-\frac{1}{\Delta x}\int_{t^{n}}^{t^{n+1}}\left(f(q(t,x_{i% +1/2}))-f(q(t,x_{i-1/2}))\right)\,dt.
  10. t n t n + 1 f ( q ( t , x i - 1 / 2 ) ) d t \int_{t^{n}}^{t^{n+1}}f(q(t,x_{i-1/2}))\,dt
  11. Q i n Q^{n}_{i}
  12. t n t n + 1 f ( q ( t , x i - 1 / 2 ) ) d t Δ t f ( Q i - 1 n , Q i n ) , \int_{t^{n}}^{t^{n+1}}f(q(t,x_{i-1/2}))\,dt\approx\Delta tf^{\downarrow}\left(% Q^{n}_{i-1},Q^{n}_{i}\right),
  13. f ( q l , q r ) f^{\downarrow}\left(q_{l},q_{r}\right)
  14. f ( q l , q r ) = f ( q l ) if q l = q r , f^{\downarrow}(q_{l},q_{r})=f(q_{l})\quad\,\text{ if }\quad q_{l}=q_{r},
  15. f f^{\downarrow}
  16. f ( q ) > 0 f^{\prime}(q)>0
  17. f ( q l , q r ) = f ( q l ) f^{\downarrow}(q_{l},q_{r})=f(q_{l})
  18. Q i n + 1 = Q i n - λ ( f ^ i + 1 / 2 n - f ^ i - 1 / 2 n ) , λ = Δ t Δ x , f ^ i - 1 / 2 n = f ( Q i - 1 n , Q i n ) Q^{n+1}_{i}=Q^{n}_{i}-\lambda\left(\hat{f}^{n}_{i+1/2}-\hat{f}^{n}_{i-1/2}% \right),\quad\lambda=\frac{\Delta t}{\Delta x},\quad\hat{f}^{n}_{i-1/2}=f^{% \downarrow}\left(Q^{n}_{i-1},Q^{n}_{i}\right)
  19. f ( q ) = a q f(q)=aq
  20. a > 0 a>0
  21. Q i n + 1 = Q i n - ν ( Q i n - Q i - 1 n ) , ν = a Δ t Δ x , Q^{n+1}_{i}=Q^{n}_{i}-\nu\left(Q^{n}_{i}-Q^{n}_{i-1}\right),\quad\nu=a\frac{% \Delta t}{\Delta x},
  22. ν = | a Δ t Δ x | 1 \nu=\left|a\frac{\Delta t}{\Delta x}\right|\leq 1
  23. t = ( n + 1 ) Δ t t=(n+1)\Delta t\,
  24. t = n Δ t {t=n\Delta t}\,
  25. t = ( n + 1 ) Δ t {t=(n+1)\Delta t}\,
  26. Δ x {\Delta x}\,
  27. Δ x {\Delta x}\,
  28. Δ t {\Delta t}\,
  29. Δ t {\Delta t}\,
  30. Δ t {\Delta t}\,
  31. | a max | Δ t < Δ x / 2 |a_{\max}|\Delta t<\Delta x/2\,
  32. | a max | |a_{\max}|\,

Godunov's_theorem.html

  1. x j = j Δ x x_{j}=j\,\Delta x
  2. t n = n Δ t t^{n}=n\,\Delta t
  3. m = 1 M β m φ j + m n + 1 = m = 1 M α m φ j + m n . ( 1 ) \sum\limits_{m=1}^{M}{\beta_{m}}\varphi_{j+m}^{n+1}=\sum\limits_{m=1}^{M}{% \alpha_{m}\varphi_{j+m}^{n}}.\quad\quad(1)
  4. φ j n + 1 \varphi_{j}^{n+1}
  5. n + 1 n+1
  6. j j
  7. n n
  8. β m \beta_{m}
  9. φ j n + 1 \varphi_{j}^{n+1}
  10. φ j n \varphi_{j}^{n}
  11. φ j n + 1 \varphi_{j}^{n+1}
  12. φ j n + 1 = m M γ m φ j + m n . ( 2 ) \varphi_{j}^{n+1}=\sum\limits_{m}^{M}{\gamma_{m}\varphi_{j+m}^{n}}.\quad\quad(2)
  13. γ m 0 , m . ( 3 ) \gamma_{m}\geq 0,\quad\forall m.\quad\quad(3)
  14. φ j n \varphi_{j}^{n}
  15. j j
  16. φ j n φ j + 1 n φ j + m n \varphi_{j}^{n}\leq\varphi_{j+1}^{n}\leq\cdots\leq\varphi_{j+m}^{n}
  17. φ j n + 1 φ j + 1 n + 1 φ j + m n + 1 \varphi_{j}^{n+1}\leq\varphi_{j+1}^{n+1}\leq\cdots\leq\varphi_{j+m}^{n+1}
  18. φ j n + 1 - φ j - 1 n + 1 = m M γ m ( φ j + m n - φ j + m - 1 n ) 0. ( 4 ) \varphi_{j}^{n+1}-\varphi_{j-1}^{n+1}=\sum\limits_{m}^{M}{\gamma_{m}\left({% \varphi_{j+m}^{n}-\varphi_{j+m-1}^{n}}\right)}\geq 0.\quad\quad(4)
  19. γ p < 0 \gamma_{p}<0
  20. p p
  21. φ j n \varphi_{j}^{n}\quad
  22. φ i n = 0 , i < k ; φ i n = 1 , i k . ( 5 ) \varphi_{i}^{n}=0,\quad i<k;\quad\varphi_{i}^{n}=1,\quad i\geq k.\quad\quad(5)
  23. φ j n + 1 - φ j - 1 n + 1 = m M γ m ( φ j + m n - φ j + m - 1 n ) = { 0 , [ j + m k ] γ m , [ j + m = k ] ( 6 ) \varphi_{j}^{n+1}-\varphi_{j-1}^{n+1}=\sum\limits_{m}^{M}{\gamma_{m}}\left({% \varphi_{j+m}^{n}-\varphi_{j+m-1}^{n}}\right)=\left\{{\begin{array}[]{*{20}c}{% 0,}&{\left[{j+m\neq k}\right]}\\ {\gamma_{m},}&{\left[{j+m=k}\right]}\\ \end{array}}\right.\quad\quad(6)
  24. j = k - p j=k-p
  25. φ k - p n + 1 - φ k - p - 1 n + 1 = γ p ( φ k n - φ k - 1 n ) < 0 , ( 7 ) \varphi_{k-p}^{n+1}-\varphi_{k-p-1}^{n+1}={\gamma_{p}\left({\varphi_{k}^{n}-% \varphi_{k-1}^{n}}\right)}<0,\quad\quad(7)
  26. φ j n + 1 \varphi_{j}^{n+1}
  27. γ p < 0 \gamma_{p}<0
  28. φ t + c φ x = 0 , t > 0 , x ( 10 ) {{\partial\varphi}\over{\partial t}}+c{{\partial\varphi}\over{\partial x}}=0,% \quad t>0,\quad x\in\mathbb{R}\quad\quad(10)
  29. σ = | c | Δ t Δ x , ( 11 ) \sigma=\left|c\right|{{\Delta t}\over{\Delta x}}\in\mathbb{N},\quad\quad(11)
  30. σ \sigma
  31. φ ( 0 , x ) = ( x Δ x - 1 2 ) 2 - 1 4 , φ j 0 = ( j - 1 2 ) 2 - 1 4 . ( 12 ) \varphi\left({0,x}\right)=\left({{x\over{\Delta x}}-{1\over 2}}\right)^{2}-{1% \over 4},\quad\varphi_{j}^{0}=\left({j-{1\over 2}}\right)^{2}-{1\over 4}.\quad% \quad(12)
  32. φ ( t , x ) = ( x - c t Δ x - 1 2 ) 2 - 1 4 . ( 13 ) \varphi\left({t,x}\right)=\left({{{x-ct}\over{\Delta x}}-{1\over 2}}\right)^{2% }-{1\over 4}.\quad\quad(13)
  33. φ j 1 = ( j - σ - 1 2 ) 2 - 1 4 , φ j 0 = ( j - 1 2 ) 2 - 1 4 . ( 14 ) \varphi_{j}^{1}=\left({j-\sigma-{1\over 2}}\right)^{2}-{1\over 4},\quad\varphi% _{j}^{0}=\left({j-{1\over 2}}\right)^{2}-{1\over 4}.\quad\quad(14)
  34. ( j - σ - 1 2 ) 2 - 1 4 = m M γ m { ( j + m - 1 2 ) 2 - 1 4 } . ( 15 ) \left({j-\sigma-{1\over 2}}\right)^{2}-{1\over 4}=\sum\limits_{m}^{M}{\gamma_{% m}\left\{{\left({j+m-{1\over 2}}\right)^{2}-{1\over 4}}\right\}}.\quad\quad(15)
  35. γ m 0 \gamma_{m}\geq 0
  36. ( j - σ - 1 2 ) 2 - 1 4 0 , j . ( 16 ) \left({j-\sigma-{1\over 2}}\right)^{2}-{1\over 4}\geq 0,\quad\forall j.\quad% \quad(16)
  37. σ > 0 , σ \sigma>0,\quad\sigma\notin\mathbb{N}
  38. j j
  39. j > σ > ( j - 1 ) j>\sigma>\left(j-1\right)
  40. ( j - σ ) > 0 \left({j-\sigma}\right)>0
  41. ( j - σ - 1 ) < 0 \left({j-\sigma-1}\right)<0
  42. ( j - σ - 1 2 ) 2 - 1 4 = ( j - σ ) ( j - σ - 1 ) < 0 , ( 17 ) \left({j-\sigma-{1\over 2}}\right)^{2}-{1\over 4}=\left(j-\sigma\right)\left(j% -\sigma-1\right)<0,\quad\quad(17)
  43. σ = | c | Δ t Δ x \sigma=\left|c\right|{{\Delta t}\over{\Delta x}}\in\mathbb{N}

Golden–Thompson_inequality.html

  1. tr e A + B tr ( e A e B ) \operatorname{tr}\,e^{A+B}\leq\operatorname{tr}\left(e^{A}e^{B}\right)

Goodman–Nguyen–van_Fraassen_algebra.html

  1. Ω ^ \hat{\Omega}
  2. Ω ^ \hat{\Omega}
  3. A ^ \hat{A}
  4. F ^ \hat{F}
  5. Ω ^ \hat{\Omega}
  6. F ^ \hat{F}
  7. P ^ \hat{P}
  8. P ^ \hat{P}
  9. P ^ \hat{P}
  10. P ^ \hat{P}
  11. A ^ \hat{A}
  12. P ^ \hat{P}
  13. P ^ \hat{P}
  14. P ^ \hat{P}
  15. P ^ \hat{P}

Good–Turing_frequency_estimation.html

  1. R red R\text{red}
  2. R black R\text{black}
  3. R green R\text{green}
  4. R ¯ \bar{R}
  5. R x R_{x}
  6. ( N r ) r = 0 , 1 , (N_{r})_{r=0,1,\ldots}
  7. R x R_{x}
  8. N r = | { x R x = r } | N_{r}=|\{x\mid R_{x}=r\}|
  9. N 1 N_{1}
  10. N = r = 1 r N r . N=\sum_{r=1}^{\infty}rN_{r}.
  11. p 0 = N 1 N . p_{0}=\frac{N_{1}}{N}.
  12. p r = ( r + 1 ) S ( N r + 1 ) N S ( N r ) . p_{r}=\frac{(r+1)S(N_{r+1})}{NS(N_{r})}.
  13. ( r + 1 ) S ( N r + 1 ) N . \frac{(r+1)S(N_{r+1})}{N}.
  14. S ( ) S()
  15. log N r \log N_{r}
  16. log r \log r
  17. N r N_{r}
  18. log Z r \log Z_{r}
  19. log r \log r
  20. Z r = N r 0.5 ( t - q ) , Z_{r}=\frac{N_{r}}{0.5(t-q)},
  21. N q , N r , N t N_{q},N_{r},N_{t}
  22. S ( N r ) = N r S(N_{r})=N_{r}
  23. S ( N r ) S(N_{r})

Goursat's_lemma.html

  1. G G
  2. G G^{\prime}
  3. H H
  4. G × G G\times G^{\prime}
  5. p 1 : H G p_{1}:H\rightarrow G
  6. p 2 : H G p_{2}:H\rightarrow G^{\prime}
  7. H H
  8. G G
  9. G G^{\prime}
  10. N N
  11. p 2 p_{2}
  12. N N^{\prime}
  13. p 1 p_{1}
  14. N N
  15. G G
  16. N N^{\prime}
  17. G G^{\prime}
  18. H H
  19. G / N × G / N G/N\times G^{\prime}/N^{\prime}
  20. G / N G / N G/N\approx G^{\prime}/N^{\prime}
  21. N N
  22. N N^{\prime}
  23. G × { e } G\times\{e^{\prime}\}
  24. { e } × G \{e\}\times G^{\prime}
  25. N N
  26. N N^{\prime}
  27. p 2 p_{2}
  28. g G g\in G
  29. h = ( g , g ) H h=(g,g^{\prime})\in H
  30. p 1 p_{1}
  31. p 1 ( N ) p_{1}(N)
  32. g p 1 ( N ) = p 1 ( h ) p 1 ( N ) = p 1 ( h N ) = p 1 ( N h ) = p 1 ( N ) g gp_{1}(N)=p_{1}(h)p_{1}(N)=p_{1}(hN)=p_{1}(Nh)=p_{1}(N)g
  33. N N
  34. G × { e } G\times\{e^{\prime}\}
  35. ( g , e ) N = ( g , e ) ( p 1 ( N ) × { e } ) = g p 1 ( N ) × { e } = p 1 ( N ) g × { e } = ( p 1 ( N ) × { e } ) ( g , e ) = N ( g , e ) (g,e^{\prime})N=(g,e^{\prime})(p_{1}(N)\times\{e^{\prime}\})=gp_{1}(N)\times\{% e^{\prime}\}=p_{1}(N)g\times\{e^{\prime}\}=(p_{1}(N)\times\{e^{\prime}\})(g,e^% {\prime})=N(g,e^{\prime})
  36. N N^{\prime}
  37. { e } × G \{e\}\times G^{\prime}
  38. G G
  39. G × { e } G\times\{e^{\prime}\}
  40. G / N G/N
  41. g N gN
  42. ( G × { e } ) / N (G\times\{e^{\prime}\})/N
  43. ( g , e ) N (g,e^{\prime})N
  44. g G g\in G
  45. G / N G^{\prime}/N^{\prime}
  46. g N g^{\prime}N^{\prime}
  47. g G g^{\prime}\in G^{\prime}
  48. H G / N × G / N H\rightarrow G/N\times G^{\prime}/N^{\prime}
  49. ( g , g ) ( g N , g N ) (g,g^{\prime})\mapsto(gN,g^{\prime}N^{\prime})
  50. H H
  51. { ( g N , g N ) | ( g , g ) H } \{(gN,g^{\prime}N^{\prime})|(g,g^{\prime})\in H\}
  52. G / N G / N G/N\rightarrow G^{\prime}/N^{\prime}
  53. g N = N g N = N gN=N\Rightarrow g^{\prime}N^{\prime}=N^{\prime}
  54. g N = N gN=N
  55. ( g , e ) N = N (g,e^{\prime})N=N
  56. ( g , e ) N H (g,e^{\prime})\in N\subset H
  57. ( e , g ) = ( g , g ) ( g - 1 , e ) H (e,g^{\prime})=(g,g^{\prime})(g^{-1},e^{\prime})\in H
  58. ( e , g ) N (e,g^{\prime})\in N^{\prime}
  59. g N = N g^{\prime}N^{\prime}=N^{\prime}
  60. g N = N g N = N g^{\prime}N^{\prime}=N^{\prime}\Rightarrow gN=N

Graded_Lie_algebra.html

  1. 𝔤 {\mathfrak{g}}
  2. 𝔤 = i 𝔤 i {\mathfrak{g}}=\bigoplus_{i\in{\mathbb{Z}}}{\mathfrak{g}}_{i}
  3. [ 𝔤 i , 𝔤 j ] 𝔤 i + j . [{\mathfrak{g}}_{i},{\mathfrak{g}}_{j}]\subseteq{\mathfrak{g}}_{i+j}.
  4. X = ( 0 1 0 0 ) , Y = ( 0 0 1 0 ) , X=\left(\begin{matrix}0&1\\ 0&0\end{matrix}\right),\quad Y=\left(\begin{matrix}0&0\\ 1&0\end{matrix}\right),
  5. H = ( 1 0 0 - 1 ) . H=\left(\begin{matrix}1&0\\ 0&-1\end{matrix}\right).
  6. [ - , - ] : E k E E [-,-]:E\otimes_{k}E\rightarrow E
  7. [ E i , E j ] E i + j [E_{i},E_{j}]\subseteq E_{i+j}
  8. [ x , y ] = - ( - 1 ) i j [ y , x ] [x,y]=-(-1)^{ij}\,[y,x]
  9. ( - 1 ) i k [ x , [ y , z ] ] + ( - 1 ) i j [ y , [ z , x ] ] + ( - 1 ) j k [ z , [ x , y ] ] = 0 (-1)^{ik}[x,[y,z]]+(-1)^{ij}[y,[z,x]]+(-1)^{jk}[z,[x,y]]=0
  10. [ x , [ x , x ] ] = 0 [x,[x,x]]=0
  11. A = i A i A=\bigoplus_{i\in{\mathbb{Z}}}A_{i}
  12. Der ( A ) = l Der l ( A ) \hbox{Der}(A)=\bigoplus_{l}\hbox{Der}_{l}(A)
  13. [ x , y ] = ( - 1 ) ϵ ( deg x ) ϵ ( deg y ) [ y , x ] [x,y]=(-1)^{\epsilon(\hbox{deg}\ x)\epsilon(\hbox{deg}\ y)}[y,x]
  14. ( - 1 ) ϵ ( deg x ) ϵ ( deg z ) [ x , [ y , z ] ] + ( - 1 ) ϵ ( deg y ) ϵ ( deg x ) [ y , [ z , x ] ] + ( - 1 ) ϵ ( deg z ) ϵ ( deg y ) [ z , [ x , y ] ] = 0. (-1)^{\epsilon(\hbox{deg}\ x)\epsilon(\hbox{deg}\ z)}[x,[y,z]]+(-1)^{\epsilon(% \hbox{deg}\ y)\epsilon(\hbox{deg}\ x)}[y,[z,x]]+(-1)^{\epsilon(\hbox{deg}\ z)% \epsilon(\hbox{deg}\ y)}[z,[x,y]]=0.

Gradient_theorem.html

  1. φ : U n \varphi:U\subseteq\mathbb{R}^{n}\to\mathbb{R}
  2. φ ( 𝐪 ) - φ ( 𝐩 ) = γ [ 𝐩 , 𝐪 ] φ ( 𝐫 ) d 𝐫 . \varphi\left(\mathbf{q}\right)-\varphi\left(\mathbf{p}\right)=\int_{\gamma[% \mathbf{p},\,\mathbf{q}]}\nabla\varphi(\mathbf{r})\cdot d\mathbf{r}.
  3. d d t ( φ 𝐫 ) ( t ) = φ ( 𝐫 ( t ) ) 𝐫 ( t ) \frac{d}{dt}(\varphi\circ\mathbf{r})(t)=\nabla\varphi(\mathbf{r}(t))\cdot% \mathbf{r}^{\prime}(t)
  4. γ φ ( 𝐮 ) d 𝐮 = a b φ ( 𝐫 ( t ) ) 𝐫 ( t ) d t = a b d d t φ ( 𝐫 ( t ) ) d t = φ ( 𝐫 ( b ) ) - φ ( 𝐫 ( a ) ) = φ ( 𝐪 ) - φ ( 𝐩 ) \begin{aligned}\displaystyle\int_{\gamma}\nabla\varphi(\mathbf{u})\cdot d% \mathbf{u}&\displaystyle=\int_{a}^{b}\nabla\varphi(\mathbf{r}(t))\cdot\mathbf{% r}^{\prime}(t)dt\\ &\displaystyle=\int_{a}^{b}\frac{d}{dt}\varphi(\mathbf{r}(t))dt=\varphi(% \mathbf{r}(b))-\varphi(\mathbf{r}(a))=\varphi\left(\mathbf{q}\right)-\varphi% \left(\mathbf{p}\right)\end{aligned}
  5. γ y d x + x d y = 0 π - tan - 1 ( 3 4 ) ( 5 sin t ( - 5 sin t ) + 5 cos t ( 5 cos t ) ) d t = 0 π - tan - 1 ( 3 4 ) 25 ( - sin 2 t + cos 2 t ) d t = 0 π - tan - 1 ( 3 4 ) 25 cos ( 2 t ) d t = 25 2 sin ( 2 t ) | 0 π - tan - 1 ( 3 4 ) = 25 2 sin ( 2 π - 2 tan - 1 ( 3 4 ) ) = - 25 2 sin ( 2 tan - 1 ( 3 4 ) ) = - 25 ( 3 4 ) ( 3 4 ) 2 + 1 = - 12. \begin{aligned}\displaystyle\int_{\gamma}ydx+xdy&\displaystyle=\int_{0}^{\pi-{% \tan}^{-1}(\frac{3}{4})}(5\sin t(-5\sin t)+5\cos t(5\cos t))dt\\ &\displaystyle=\int_{0}^{\pi-{\tan}^{-1}(\frac{3}{4})}25(-{\sin}^{2}t+{\cos}^{% 2}t)dt\\ &\displaystyle=\int_{0}^{\pi-{\tan}^{-1}(\frac{3}{4})}25\cos(2t)dt\\ &\displaystyle=\left.\tfrac{25}{2}\sin(2t)\right|_{0}^{\pi-{\tan}^{-1}(\tfrac{% 3}{4})}\\ &\displaystyle=\tfrac{25}{2}\sin(2\pi-2{\tan}^{-1}(\tfrac{3}{4}))\\ &\displaystyle=-\tfrac{25}{2}\sin(2{\tan}^{-1}(\tfrac{3}{4}))\\ &\displaystyle=-\frac{25(\tfrac{3}{4})}{{(\tfrac{3}{4})}^{2}+1}=-12.\end{aligned}
  6. γ y d x + x d y = γ ( x y ) ( d x , d y ) = x y | ( 5 , 0 ) ( - 4 , 3 ) = - 4 3 - 5 0 = - 12 \int_{\gamma}ydx+xdy=\int_{\gamma}\nabla(xy)\cdot(dx,dy)=xy|_{(5,0)}^{(-4,3)}=% -4\cdot 3-5\cdot 0=-12
  7. γ | 𝐱 | α - 1 𝐱 d 𝐱 = 1 α + 1 γ ( α + 1 ) | 𝐱 | ( α + 1 ) - 2 𝐱 d 𝐱 = 1 α + 1 γ ( | 𝐱 | α + 1 ) d 𝐱 = | 𝐪 | α + 1 - | 𝐩 | α + 1 α + 1 \begin{aligned}\displaystyle\int_{\gamma}|\mathbf{x}|^{\alpha-1}\mathbf{x}% \cdot d\mathbf{x}&\displaystyle=\frac{1}{\alpha+1}\int_{\gamma}(\alpha+1)|% \mathbf{x}|^{(\alpha+1)-2}\mathbf{x}\cdot d\mathbf{x}\\ &\displaystyle=\frac{1}{\alpha+1}\int_{\gamma}\nabla(|\mathbf{x}|^{\alpha+1})% \cdot d\mathbf{x}=\frac{|\mathbf{q}|^{\alpha+1}-|\mathbf{p}|^{\alpha+1}}{% \alpha+1}\end{aligned}
  8. 𝐅 ( 𝐫 ) = k q i = 1 n Q i ( 𝐫 - 𝐩 i ) | 𝐫 - 𝐩 i | 3 \mathbf{F}(\mathbf{r})=kq\sum_{i=1}^{n}\frac{Q_{i}(\mathbf{r}-\mathbf{p}_{i})}% {|\mathbf{r}-\mathbf{p}_{i}|^{3}}
  9. W = γ 𝐅 ( 𝐫 ) d 𝐫 = γ ( k q i = 1 n Q i ( 𝐫 - 𝐩 i ) | 𝐫 - 𝐩 i | 3 ) d 𝐫 = k q i = 1 n ( Q i γ ( 𝐫 - 𝐩 i ) | 𝐫 - 𝐩 i | 3 d 𝐫 ) W=\int_{\gamma}\mathbf{F}(\mathbf{r})\cdot d\mathbf{r}=\int_{\gamma}\bigg(kq% \sum_{i=1}^{n}\frac{Q_{i}(\mathbf{r}-\mathbf{p}_{i})}{|\mathbf{r}-\mathbf{p}_{% i}|^{3}}\bigg)\cdot d\mathbf{r}=kq\sum_{i=1}^{n}\bigg(Q_{i}\int_{\gamma}\frac{% (\mathbf{r}-\mathbf{p}_{i})}{|\mathbf{r}-\mathbf{p}_{i}|^{3}}\cdot d\mathbf{r}\bigg)
  10. ( 𝐫 - 𝐩 i ) | 𝐫 - 𝐩 i | 3 = - ( 1 | 𝐫 - 𝐩 i | ) . \frac{(\mathbf{r}-\mathbf{p}_{i})}{|\mathbf{r}-\mathbf{p}_{i}|^{3}}=-\nabla% \left(\frac{1}{|\mathbf{r}-\mathbf{p}_{i}|}\right).
  11. W = - k q i = 1 n ( Q i γ ( 1 | 𝐫 - 𝐩 i | ) d 𝐫 ) = k q i = 1 n Q i ( 1 | 𝐚 - 𝐩 i | - 1 | 𝐛 - 𝐩 i | ) W=-kq\sum_{i=1}^{n}\bigg(Q_{i}\int_{\gamma}\nabla\left(\frac{1}{|\mathbf{r}-% \mathbf{p}_{i}|}\right)\cdot d\mathbf{r}\bigg)=kq\sum_{i=1}^{n}Q_{i}\left(% \frac{1}{|\mathbf{a}-\mathbf{p}_{i}|}-\frac{1}{|\mathbf{b}-\mathbf{p}_{i}|}\right)
  12. f ( 𝐱 ) := γ [ 𝐚 , 𝐱 ] 𝐅 ( 𝐮 ) d 𝐮 f(\mathbf{x}):=\int_{\gamma[\mathbf{a},\mathbf{x}]}\mathbf{F}(\mathbf{u})\cdot d% \mathbf{u}
  13. f 𝐯 ( 𝐱 ) = lim t 0 f ( 𝐱 + t 𝐯 ) - f ( 𝐱 ) t = lim t 0 γ [ 𝐚 , 𝐱 + t 𝐯 ] 𝐅 ( 𝐮 ) d 𝐮 - γ [ 𝐚 , 𝐱 ] 𝐅 ( 𝐮 ) d 𝐮 t = lim t 0 1 t γ [ 𝐱 , 𝐱 + t 𝐯 ] 𝐅 ( 𝐮 ) d 𝐮 \begin{aligned}\displaystyle\frac{\partial f}{\partial\mathbf{v}}(\mathbf{x})&% \displaystyle=\lim_{t\to 0}\frac{f(\mathbf{x}+t\mathbf{v})-f(\mathbf{x})}{t}\\ &\displaystyle=\lim_{t\to 0}\frac{\int_{\gamma[\mathbf{a},\mathbf{x}+t\mathbf{% v}]}\mathbf{F}(\mathbf{u})\cdot d\mathbf{u}-\int_{\gamma[\mathbf{a},\mathbf{x}% ]}\mathbf{F}(\mathbf{u})\cdot d\mathbf{u}}{t}\\ &\displaystyle=\lim_{t\to 0}\frac{1}{t}\int_{\gamma[\mathbf{x},\mathbf{x}+t% \mathbf{v}]}\mathbf{F}(\mathbf{u})\cdot d\mathbf{u}\end{aligned}
  14. f ( 𝐱 ) = ( f ( 𝐱 ) x 1 , f ( 𝐱 ) x 2 , , f ( 𝐱 ) x n ) = ( 𝐅 ( 𝐱 ) 𝐞 1 , 𝐅 ( 𝐱 ) 𝐞 2 , , 𝐅 ( 𝐱 ) 𝐞 n ) = 𝐅 ( 𝐱 ) \nabla f(\mathbf{x})=\bigg(\frac{\partial f(\mathbf{x})}{\partial x_{1}},\frac% {\partial f(\mathbf{x})}{\partial x_{2}},...,\frac{\partial f(\mathbf{x})}{% \partial x_{n}}\bigg)=(\mathbf{F}(\mathbf{x})\cdot\mathbf{e}_{1},\mathbf{F}(% \mathbf{x})\cdot\mathbf{e}_{2},...,\mathbf{F}(\mathbf{x})\cdot\mathbf{e}_{n})=% \mathbf{F}(\mathbf{x})
  15. U e ( 𝐫 ) := - γ [ 𝐚 , 𝐫 ] 𝐅 e ( 𝐮 ) d 𝐮 U_{e}(\mathbf{r}):=-\int_{\gamma[\mathbf{a},\mathbf{r}]}\mathbf{F}_{e}(\mathbf% {u})\cdot d\mathbf{u}
  16. γ ϕ = γ d ϕ \int_{\partial\gamma}\phi=\int_{\gamma}d\phi
  17. Ω ω = Ω d ω \int_{\partial\Omega}\omega=\int_{\Omega}d\omega

Graduate_Texts_in_Mathematics.html

  1. C * C^{*}

Graeffe's_method.html

  1. p ( x ) = ( x - x 1 ) ( x - x 2 ) ( x - x n ) p(x)=(x-x_{1})(x-x_{2})\cdots(x-x_{n})\,
  2. p ( - x ) = ( - 1 ) n ( x + x 1 ) ( x + x 2 ) ( x + x n ) . p(-x)=(-1)^{n}\,(x+x_{1})(x+x_{2})\cdots(x+x_{n}).\,
  3. q ( x ) q(x)
  4. x 1 2 , x 2 2 , , x n 2 x_{1}^{2},\,x_{2}^{2},\dots,\,x_{n}^{2}
  5. q ( x ) = ( x - x 1 2 ) ( x - x 2 2 ) ( x - x n 2 ) . q(x)=(x-x_{1}^{2})(x-x_{2}^{2})\cdots(x-x_{n}^{2}).\,
  6. x 2 - x k 2 = ( x - x k ) ( x + x k ) x^{2}-x_{k}^{2}=(x-x_{k})(x+x_{k})
  7. q ( x 2 ) = ( x 2 - x 1 2 ) ( x 2 - x 2 2 ) ( x 2 - x n 2 ) = ( - 1 ) n p ( x ) p ( - x ) . q(x^{2})=(x^{2}-x_{1}^{2})(x^{2}-x_{2}^{2})\cdots(x^{2}-x_{n}^{2})=(-1)^{n}\,p% (x)p(-x).\,
  8. p ( x ) = x n + a 1 x n - 1 + + a n - 1 x + a n p(x)=x^{n}+a_{1}x^{n-1}+\cdots+a_{n-1}x+a_{n}\,
  9. q ( x ) = x n + b 1 x n - 1 + + b n - 1 x + b n , q(x)=x^{n}+b_{1}x^{n-1}+\cdots+b_{n-1}x+b_{n},\,
  10. b k = ( - 1 ) k a k 2 + 2 j = 0 k - 1 ( - 1 ) j a j a 2 k - j , with a 0 = b 0 = 1. b_{k}=(-1)^{k}\,a_{k}^{2}+2\sum_{j=0}^{k-1}(-1)^{j}\,a_{j}a_{2k-j},\,\text{ % with }a_{0}=b_{0}=1.\,
  11. p ( x ) = p e ( x 2 ) + x p o ( x 2 ) q ( x ) = ( - 1 ) n ( p e ( x ) 2 - x p o ( x ) 2 ) . p(x)=p_{e}(x^{2})+x\,p_{o}(x^{2})\;\implies\;q(x)=(-1)^{n}(p_{e}(x)^{2}-x\,p_{% o}(x)^{2}).\,
  12. q k ( y ) = y n + a k 1 y n - 1 + + a k n - 1 y + a k n q^{k}(y)=y^{n}+{a^{k}}_{1}\,y^{n-1}+\cdots+{a^{k}}_{n-1}\,y+{a^{k}}_{n}\,
  13. y 1 = x 1 2 k , y 2 = x 2 2 k , , y n = x n 2 k y_{1}=x_{1}^{2^{k}},\,y_{2}=x_{2}^{2^{k}},\,\dots,\,y_{n}=x_{n}^{2^{k}}
  14. ρ > 1 \rho>1
  15. | x k | ρ | x k + 1 | |x_{k}|\geq\rho\,|x_{k+1}|
  16. ρ 2 k 1 + 2 k ( ρ - 1 ) \rho^{2^{k}}\geq 1+2^{k}(\rho-1)
  17. a 1 k = - ( y 1 + y 2 + + y n ) a 2 k = y 1 y 2 + y 1 y 3 + + y n - 1 y n a n k = ( - 1 ) n ( y 1 y 2 y n ) . \begin{aligned}\displaystyle a^{k}_{\;1}&\displaystyle=-(y_{1}+y_{2}+\cdots+y_% {n})\\ \displaystyle a^{k}_{\;2}&\displaystyle=y_{1}y_{2}+y_{1}y_{3}+\cdots+y_{n-1}y_% {n}\\ &\displaystyle\;\vdots\\ \displaystyle a^{k}_{\;n}&\displaystyle=(-1)^{n}(y_{1}y_{2}\cdots y_{n}).\end{aligned}
  18. x 1 , , x n x_{1},\dots,x_{n}
  19. ρ > 1 \rho>1
  20. | x m | ρ | x m + 1 | |x_{m}|\geq\rho|x_{m+1}|
  21. y 1 , y 2 , , y n y_{1},y_{2},...,y_{n}
  22. ρ 2 k \rho^{2^{k}}
  23. a 1 k - y 1 a^{k}_{\;1}\approx-y_{1}
  24. a 2 k y 1 y 2 a^{k}_{\;2}\approx y_{1}y_{2}
  25. y 1 - a 1 k , y 2 - a 2 k / a 1 k , y n - a n k / a n - 1 k . y_{1}\approx-a^{k}_{\;1},\;y_{2}\approx-a^{k}_{\;2}/a^{k}_{\;1},\;\dots\;y_{n}% \approx-a^{k}_{\;n}/a^{k}_{\;n-1}.
  26. q m ( y ) q^{m}(y)
  27. q m - 1 ( x ) q^{m-1}(x)
  28. q m - 2 ( x ) q^{m-2}(x)
  29. q m - 1 ( x ) q^{m-1}(x)
  30. x + 1 = x + 2 = = x + d x_{\ell+1}=x_{\ell+2}=\dots=x_{\ell+d}
  31. | ( a + i m - 1 ) 2 a + i m | \left|\frac{(a^{m-1}_{\;\ell+i})^{2}}{a^{m}_{\;\ell+i}}\right|
  32. ( d i ) {\left({{d}\atop{i}}\right)}
  33. i = 0 , 1 , , d i=0,1,\dots,d
  34. ε \varepsilon
  35. ε 2 = 0 \varepsilon^{2}=0
  36. p ( x + ε ) = p ( x ) + ε p ( x ) p(x+\varepsilon)=p(x)+\varepsilon\,p^{\prime}(x)
  37. x m - ε x_{m}-\varepsilon
  38. ( x m - ε ) 2 k = x m 2 k - ε 2 k x m 2 k - 1 = y m + ε y ˙ m . (x_{m}-\varepsilon)^{2^{k}}=x_{m}^{2^{k}}-\varepsilon\,{2^{k}}\,x_{m}^{2^{k}-1% }=y_{m}+\varepsilon\,\dot{y}_{m}.
  39. x m x_{m}
  40. x m = - 2 k y m y ˙ m . x_{m}=-\tfrac{2^{k}\,y_{m}}{\dot{y}_{m}}.
  41. n M 2 nM^{2}
  42. n 2 k - 1 M 2 k n^{2^{k}-1}M^{2^{k}}
  43. c = α e - 2 k r , c=\alpha\,e^{-2^{k}\,r},
  44. α = c | c | \alpha=\frac{c}{|c|}
  45. r = - 2 - k log | c | r=-2^{-k}\log|c|
  46. 2 k 2^{k}
  47. c 3 = c 1 + c 2 = | c 1 | ( α 1 + α 2 | c 2 | | c 1 | ) c_{3}=c_{1}+c_{2}=|c_{1}|\cdot\left(\alpha_{1}+\alpha_{2}\tfrac{|c_{2}|}{|c_{1% }|}\right)
  48. c 1 c_{1}
  49. r 1 < r 2 r_{1}<r_{2}
  50. α 3 = s | s | \alpha_{3}=\tfrac{s}{|s|}
  51. r 3 = r 1 + 2 - k log | s | r_{3}=r_{1}+2^{-k}\,\log{|s|}
  52. s = α 1 + α 2 e 2 k ( r 1 - r 2 ) . s=\alpha_{1}+\alpha_{2}\,e^{2^{k}(r_{1}-r_{2})}.
  53. a 0 , a 1 , , a n a_{0},a_{1},\dots,a_{n}
  54. ( α m , r m ) (\alpha_{m},r_{m})
  55. m = 0 , , n m=0,\dots,n
  56. { ( m , r m ) : m = 0 , , n } \{(m,r_{m}):\;m=0,\dots,n\}

Grand_potential.html

  1. Φ G = def U - T S - μ N \Phi_{G}\ \stackrel{\mathrm{def}}{=}\ U-TS-\mu N
  2. d Φ G = d U - T d S - S d T - μ d N - N d μ = - P d V - S d T - N d μ \begin{aligned}\displaystyle d\Phi_{G}&\displaystyle=dU-TdS-SdT-\mu dN-Nd\mu\\ &\displaystyle=-PdV-SdT-Nd\mu\end{aligned}
  3. d U = T d S - P d V + μ d N dU=TdS-PdV+\mu dN
  4. Ω = def F - μ N = U - T S - μ N \Omega\ \stackrel{\mathrm{def}}{=}\ F-\mu N=U-TS-\mu N
  5. Ω = - P V \Omega=-PV\,\;
  6. λ V \lambda V
  7. λ \lambda
  8. V V
  9. ( P / V ) μ , T = 0 (\partial\langle P\rangle/\partial V)_{\mu,T}=0
  10. ( N / V ) μ , T = N / V (\partial\langle N\rangle/\partial V)_{\mu,T}=N/V
  11. Φ G = - P V \Phi_{G}=-\langle P\rangle V
  12. G = N μ G=\langle N\rangle\mu
  13. Φ G \Phi_{G}
  14. Φ G = - P V \Phi_{G}=-\langle P\rangle V
  15. Φ G - P V \Phi_{G}\neq-\langle P\rangle V
  16. Φ G = - k B T ln ( Ξ ) = - k B T Z 1 e β μ \Phi_{G}=-k_{B}T\ln(\Xi)=-k_{B}TZ_{1}e^{\beta\mu}

Grandi's_series.html

  1. n = 0 ( - 1 ) n \sum_{n=0}^{\infty}(-1)^{n}
  2. 1 + 1 + 1 + 1 + 1 - 1 - 1 + 1 + 1 - 1 - 1 + 1 + 1 - 1 - 1 + 1 + 1 1+1+1+1+1-1-1+1+1-1-1+1+1-1-1+1+1\cdots

Graph_embedding.html

  1. G G
  2. G G
  3. e e
  4. e e
  5. 3 \mathbb{R}^{3}
  6. 2 \mathbb{R}^{2}
  7. G G
  8. G G

Graphical_timeline_from_Big_Bang_to_Heat_Death.html

  1. s = { log 10 log 10 y e a r if y e a r > 10 , corresponding to y e a r = 10 10 s 0 if 0.1 y e a r 10 - log 10 ( - log 10 y e a r ) if y e a r < 0.1 , corresponding to y e a r = 10 - 10 - s s=\begin{cases}\log_{10}\log_{10}year&\mbox{if }~{}year>10\mbox{ , % corresponding to }~{}year=10^{10^{s}}\\ 0&\mbox{if }~{}0.1\leq year\leq 10\\ -\log_{10}(-\log_{10}year)&\mbox{if }~{}year<0.1\mbox{ , corresponding to }~{}% year=10^{-10^{-s}}\end{cases}
  2. s e c o n d / 31557600 second/31557600

Graphical_timeline_of_the_Stelliferous_Era.html

  1. 10 × log 10 year 10\times\log_{10}\mathrm{year}
  2. 10 × log 10 1000000 = 10 × 6 = 60 10\times\log_{10}1000000=10\times 6=60

Grassmann's_law_(optics).html

  1. ( R 1 , G 1 , B 1 ) (R_{1},G_{1},B_{1})
  2. ( R 2 , G 2 , B 2 ) (R_{2},G_{2},B_{2})
  3. ( R , G , B ) (R,G,B)
  4. R = R 1 + R 2 R=R_{1}+R_{2}\,
  5. G = G 1 + G 2 G=G_{1}+G_{2}\,
  6. B = B 1 + B 2 B=B_{1}+B_{2}\,
  7. I ( λ ) I(\lambda)
  8. R = 0 I ( λ ) r ¯ ( λ ) d λ R=\int_{0}^{\infty}I(\lambda)\,\bar{r}(\lambda)\,d\lambda
  9. G = 0 I ( λ ) g ¯ ( λ ) d λ G=\int_{0}^{\infty}I(\lambda)\,\bar{g}(\lambda)\,d\lambda
  10. B = 0 I ( λ ) b ¯ ( λ ) d λ B=\int_{0}^{\infty}I(\lambda)\,\bar{b}(\lambda)\,d\lambda
  11. I I
  12. r ¯ ( λ ) , g ¯ ( λ ) , b ¯ ( λ ) \bar{r}(\lambda),\bar{g}(\lambda),\bar{b}(\lambda)

Gravitational_wave.html

  1. h h
  2. h = 0.5 h=0.5
  3. h 10 - 20 h\approx 10^{-20}
  4. d h d t \frac{\mathrm{d}h}{\mathrm{d}t}
  5. λ \lambda
  6. c c
  7. h + h_{\,+}
  8. h × h_{\,\times}
  9. x x
  10. y y
  11. m 1 m_{1}
  12. m 2 m_{2}
  13. r r
  14. P = d E d t = - 32 5 G 4 c 5 ( m 1 m 2 ) 2 ( m 1 + m 2 ) r 5 P=\frac{\mathrm{d}E}{\mathrm{d}t}=-\frac{32}{5}\,\frac{G^{4}}{c^{5}}\,\frac{(m% _{1}m_{2})^{2}(m_{1}+m_{2})}{r^{5}}
  15. r r
  16. × 10 1 1 \times 10^{1}1
  17. m 1 m_{1}
  18. m 2 m_{2}
  19. × 10 3 0 \times 10^{3}0
  20. × 10 2 4 \times 10^{2}4
  21. × 10 2 6 \times 10^{2}6
  22. × 10 3 6 \times 10^{3}6
  23. × 10 - 15 \times 10^{-}15
  24. × 10 1 3 \times 10^{1}3
  25. × 10 8 \times 10^{8}
  26. × 10 2 8 \times 10^{2}8
  27. d r d t = - 64 5 G 3 c 5 ( m 1 m 2 ) ( m 1 + m 2 ) r 3 \frac{\mathrm{d}r}{\mathrm{d}t}=-\frac{64}{5}\,\frac{G^{3}}{c^{5}}\,\frac{(m_{% 1}m_{2})(m_{1}+m_{2})}{r^{3}}
  28. 8 = 2.828 \sqrt{8}=2.828
  29. × 10 - 20 \times 10^{-}20
  30. × 10 - 13 \times 10^{-}13
  31. × 10 - 9 \times 10^{-}9
  32. × 10 1 0 \times 10^{1}0
  33. × 10 3 6 \times 10^{3}6
  34. × 10 9 \times 10^{9}
  35. × 10 8 \times 10^{8}
  36. × 10 - 6 \times 10^{-}6
  37. × 10 6 \times 10^{6}
  38. t = 5 256 c 5 G 3 r 4 ( m 1 m 2 ) ( m 1 + m 2 ) t=\frac{5}{256}\,\frac{c^{5}}{G^{3}}\,\frac{r^{4}}{(m_{1}m_{2})(m_{1}+m_{2})}
  39. × 10 3 0 \times 10^{3}0
  40. × 10 2 3 \times 10^{2}3
  41. × 10 1 0 \times 10^{1}0
  42. × 10 8 \times 10^{8}
  43. × 10 8 \times 10^{8}
  44. × 10 1 3 \times 10^{1}3
  45. × 10 6 \times 10^{6}
  46. θ \theta
  47. R R
  48. h + = - 1 R G 2 c 4 2 m 1 m 2 r ( 1 + cos 2 θ ) cos [ 2 ω ( t - R ) ] , h_{+}=-\frac{1}{R}\,\frac{G^{2}}{c^{4}}\,\frac{2m_{1}m_{2}}{r}(1+\cos^{2}% \theta)\cos\left[2\omega(t-R)\right],
  49. h × = - 1 R G 2 c 4 4 m 1 m 2 r ( cos θ ) sin [ 2 ω ( t - R ) ] . h_{\times}=-\frac{1}{R}\,\frac{G^{2}}{c^{4}}\,\frac{4m_{1}m_{2}}{r}\,(\cos{% \theta})\sin\left[2\omega(t-R)\right].
  50. ω = G ( m 1 + m 2 ) / r 3 . \omega=\sqrt{G(m_{1}+m_{2})/r^{3}}.
  51. x x
  52. y y
  53. θ = π / 2 \theta=\pi/2
  54. cos ( θ ) = 0 \cos(\theta)=0
  55. h × h_{\times}
  56. h + = - 1 R G 2 c 4 4 m 1 m 2 r = - 1 R 1.7 × 10 - 10 m . h_{+}=-\frac{1}{R}\,\frac{G^{2}}{c^{4}}\,\frac{4m_{1}m_{2}}{r}=-\frac{1}{R}\,1% .7\times 10^{-10}\,\mathrm{m}.
  57. 10 - 16 Hz < f < 10 4 Hz 10^{-16}\,\mathrm{Hz}<f<10^{4}\,\mathrm{Hz}
  58. 1 / R 1/R
  59. h h
  60. h 5 × 10 - 20 h\sim 5\times 10^{-20}
  61. h 2 × 10 - 13 / Hz h\sim{2\times 10^{-13}/\sqrt{\mathrm{Hz}}}
  62. h 2 × 10 - 17 / Hz h\sim{2\times 10^{-17}/\sqrt{\mathrm{Hz}}}
  63. h 2 × 10 - 20 / Hz h\sim{2\times 10^{-20}/\sqrt{\mathrm{Hz}}}
  64. g μ ν g_{\mu\nu}\,
  65. \nabla\,
  66. G μ ν G_{\mu\nu}
  67. T μ ν T_{\mu\nu}
  68. G μ ν = 8 π G N c 4 T μ ν , G_{\mu\nu}=\frac{8\pi G_{N}}{c^{4}}T_{\mu\nu},
  69. G N G_{N}
  70. c c
  71. G N = 1 = c G_{N}=1=c
  72. ( t , r , θ , ϕ ) (t,r,\theta,\phi)\,
  73. η μ ν \eta_{\mu\nu}\,
  74. η μ ν = [ - 1 0 0 0 0 1 0 0 0 0 r 2 0 0 0 0 r 2 sin 2 θ ] . \eta_{\mu\nu}=\begin{bmatrix}-1&0&0&0\\ 0&1&0&0\\ 0&0&r^{2}&0\\ 0&0&0&r^{2}\sin^{2}\theta\end{bmatrix}.
  75. g μ ν g_{\mu\nu}
  76. det g \det g
  77. h ¯ α β η α β - | det g | g α β \bar{h}^{\alpha\beta}\equiv\eta^{\alpha\beta}-\sqrt{|\det g|}g^{\alpha\beta}\,
  78. β h ¯ α β = 0 , \nabla_{\beta}\,\bar{h}^{\alpha\beta}=0,
  79. \nabla
  80. h ¯ α β = - 16 π τ α β \Box\bar{h}^{\alpha\beta}=-16\pi\tau^{\alpha\beta}\,
  81. = - t 2 + Δ \Box=-\partial_{t}^{2}+\Delta\,
  82. τ α β \tau^{\alpha\beta}\,
  83. h ¯ α β \bar{h}^{\alpha\beta}\,
  84. η α β \eta^{\alpha\beta}\,
  85. h ¯ α β \bar{h}^{\alpha\beta}\,
  86. τ α β \tau^{\alpha\beta}\,
  87. T α β T^{\alpha\beta}\,
  88. h ¯ α β = - 16 π T α β \Box\bar{h}^{\alpha\beta}=-16\pi T^{\alpha\beta}\,
  89. h ¯ α β = 0 \Box\bar{h}^{\alpha\beta}=0\,
  90. h ¯ α β \bar{h}^{\alpha\beta}\,
  91. 1 / r 1/r\,
  92. A ( t - r , θ , ϕ ) / r A(t-r,\theta,\phi)/r\,
  93. A A\,
  94. r = 0 r=0
  95. h ¯ α β = 1 r [ 0 0 0 0 0 0 0 0 0 0 A + ( t - r , θ , ϕ ) A × ( t - r , θ , ϕ ) 0 0 A × ( t - r , θ , ϕ ) - A + ( t - r , θ , ϕ ) ] [ 0 0 0 0 0 0 0 0 0 0 h + ( t - r , r , θ , ϕ ) h × ( t - r , r , θ , ϕ ) 0 0 h × ( t - r , r , θ , ϕ ) - h + ( t - r , r , θ , ϕ ) ] \begin{array}[]{lcl}\bar{h}^{\alpha\beta}&=&\frac{1}{r}\,\begin{bmatrix}0&0&0&% 0\\ 0&0&0&0\\ 0&0&A_{+}(t-r,\theta,\phi)&A_{\times}(t-r,\theta,\phi)\\ 0&0&A_{\times}(t-r,\theta,\phi)&-A_{+}(t-r,\theta,\phi)\end{bmatrix}\\ \\ &\equiv&\begin{bmatrix}0&0&0&0\\ 0&0&0&0\\ 0&0&h_{+}(t-r,r,\theta,\phi)&h_{\times}(t-r,r,\theta,\phi)\\ 0&0&h_{\times}(t-r,r,\theta,\phi)&-h_{+}(t-r,r,\theta,\phi)\end{bmatrix}\end{% array}\,
  96. h ¯ α β = - 16 π τ α β \Box\bar{h}^{\alpha\beta}=-16\pi\tau^{\alpha\beta}\,
  97. h ¯ α β ( t , x ) = - 16 π G γ δ α β ( t , x ; t , x ) τ γ δ ( t , x ) d t d 3 x \bar{h}^{\alpha\beta}(t,\vec{x})=-16\pi\int\,G^{\alpha\beta}_{\gamma\delta}(t,% \vec{x};t^{\prime},\vec{x}^{\prime})\,\tau^{\gamma\delta}(t^{\prime},\vec{x}^{% \prime})\,\mathrm{d}t^{\prime}\,\mathrm{d}^{3}x^{\prime}
  98. G γ δ α β ( t , x ; t , x ) = 1 4 π δ γ α δ δ β δ ( t ± | x - x | - t ) | x - x | G^{\alpha\beta}_{\gamma\delta}(t,\vec{x};t^{\prime},\vec{x}^{\prime})=\frac{1}% {4\pi}\delta_{\gamma}^{\alpha}\,\delta_{\delta}^{\beta}\,\frac{\delta(t\pm|% \vec{x}-\vec{x}^{\prime}|-t^{\prime})}{|\vec{x}-\vec{x}^{\prime}|}
  99. h ¯ α β ( t , x ) = - 4 τ α β ( t - | x - x | , x ) | x - x | d 3 x \bar{h}^{\alpha\beta}(t,\vec{x})=-4\int\,\frac{\tau^{\alpha\beta}(t-|\vec{x}-% \vec{x}^{\prime}|,\vec{x}^{\prime})}{|\vec{x}-\vec{x}^{\prime}|}\,\mathrm{d}^{% 3}x^{\prime}
  100. h ¯ α β ( t , x ) - 4 r τ α β ( t - r , x ) d 3 x \bar{h}^{\alpha\beta}(t,\vec{x})\approx-\frac{4}{r}\,\int\,\tau^{\alpha\beta}(% t-r,\vec{x}^{\prime})\,\mathrm{d}^{3}x^{\prime}
  101. r = | x | r=|\vec{x}|
  102. τ i j ( t - r , x ) d 3 x = x i x j k l τ k l ( t - r , x ) d 3 x \int\,\tau^{ij}(t-r,\vec{x}^{\prime})\,\mathrm{d}^{3}x^{\prime}=\int\,x^{% \prime i}x^{\prime j}\nabla_{k}\nabla_{l}\tau^{kl}(t-r,\vec{x}^{\prime})\,% \mathrm{d}^{3}x^{\prime}
  103. h ¯ i j ( t , x ) - 4 r x i x j k l τ k l ( t - r , x ) d 3 x \bar{h}^{ij}(t,\vec{x})\approx-\frac{4}{r}\,\int\,x^{\prime i}x^{\prime j}% \nabla_{k}\nabla_{l}\tau^{kl}(t-r,\vec{x}^{\prime})\,\mathrm{d}^{3}x^{\prime}
  104. β h ¯ α β = 0 \nabla_{\beta}\,\bar{h}^{\alpha\beta}=0
  105. β τ α β = 0 \nabla_{\beta}\,\tau^{\alpha\beta}=0
  106. 0 0 τ 00 = j k τ j k \nabla_{0}\nabla_{0}\tau^{00}=\nabla_{j}\nabla_{k}\tau^{jk}
  107. h ¯ i j ( t , x ) - 4 r d 2 d t 2 x i x j τ 00 ( t - r , x ) d 3 x \bar{h}^{ij}(t,\vec{x})\approx-\frac{4}{r}\,\frac{\mathrm{d}^{2}}{\mathrm{d}t^% {2}}\,\int\,x^{\prime i}x^{\prime j}\tau^{00}(t-r,\vec{x}^{\prime})\,\mathrm{d% }^{3}x^{\prime}
  108. τ 00 = ρ \tau^{00}=\rho
  109. M 1 M_{1}
  110. M 2 M_{2}
  111. x 1 \vec{x}_{1}
  112. x 2 \vec{x}_{2}
  113. ρ ( t - r , x ) = M 1 δ 3 ( x - x 1 ( t - r ) ) + M 2 δ 3 ( x - x 2 ( t - r ) ) \rho(t-r,\vec{x}^{\prime})=M_{1}\delta^{3}(\vec{x}^{\prime}-\vec{x}_{1}(t-r))+% M_{2}\delta^{3}(\vec{x}^{\prime}-\vec{x}_{2}(t-r))
  114. h ¯ i j ( t , x ) - 4 r d 2 d t 2 { M 1 x 1 i ( t - r ) x 1 j ( t - r ) + M 2 x 2 i ( t - r ) x 2 j ( t - r ) } \bar{h}^{ij}(t,\vec{x})\approx-\frac{4}{r}\,\frac{\mathrm{d}^{2}}{\mathrm{d}t^% {2}}\,\left\{M_{1}x_{1}^{i}(t-r)x_{1}^{j}(t-r)+M_{2}x_{2}^{i}(t-r)x_{2}^{j}(t-% r)\right\}
  115. h ¯ i j ( t , x ) - 4 r M 1 M 2 R n i ( t - r ) n j ( t - r ) \bar{h}^{ij}(t,\vec{x})\approx-\frac{4}{r}\,\frac{M_{1}M_{2}}{R}\,n^{i}(t-r)n^% {j}(t-r)
  116. n = x 1 / | x 1 | \vec{n}=\vec{x}_{1}/|\vec{x}_{1}|
  117. x 1 ( t - r ) \vec{x}_{1}(t-r)

Greedy_algorithm_for_Egyptian_fractions.html

  1. x y = 1 y / x + ( - y ) mod x y y / x \frac{x}{y}=\frac{1}{\lceil y/x\rceil}+\frac{(-y)\bmod x}{y\lceil y/x\rceil}
  2. 7 15 = 1 3 + 2 15 = 1 3 + 1 8 + 1 120 . \frac{7}{15}=\frac{1}{3}+\frac{2}{15}=\frac{1}{3}+\frac{1}{8}+\frac{1}{120}.
  3. 5 121 = 1 25 + 1 757 + 1 763309 + 1 873960180913 + 1 1527612795642093418846225 , \frac{5}{121}=\frac{1}{25}+\frac{1}{757}+\frac{1}{763309}+\frac{1}{87396018091% 3}+\frac{1}{1527612795642093418846225},
  4. 5 121 = 1 33 + 1 121 + 1 363 . \frac{5}{121}=\frac{1}{33}+\frac{1}{121}+\frac{1}{363}.
  5. y / x + 1 \lfloor y/x\rfloor+1
  6. y / x \lceil y/x\rceil
  7. 1 2 + 1 3 + 1 7 + 1 43 = 1805 1806 \frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1}{43}=\frac{1805}{1806}
  8. ( - y ) mod x y y / x = 2 y ( y + 2 ) / 3 \frac{(-y)\bmod x}{y\lceil y/x\rceil}=\frac{2}{y(y+2)/3}
  9. 1 , 2 3 , 3 7 , 4 17 , 5 31 , 6 109 , 7 253 , 8 97 , 9 271 , 1,\frac{2}{3},\frac{3}{7},\frac{4}{17},\frac{5}{31},\frac{6}{109},\frac{7}{253% },\frac{8}{97},\frac{9}{271},\dots
  10. φ = 1 1 + 1 2 + 1 9 + 1 145 + 1 37986 + \varphi=\frac{1}{1}+\frac{1}{2}+\frac{1}{9}+\frac{1}{145}+\frac{1}{37986}+\cdots
  11. x y = 1 d + x d - y y d , \frac{x}{y}=\frac{1}{d}+\frac{xd-y}{yd},

Green's_function_(many-body_theory).html

  1. ψ ( 𝐱 ) \psi(\mathbf{x})
  2. ψ ( 𝐱 , t ) = e i K t ψ ( 𝐱 ) e - i K t , \psi(\mathbf{x},t)=\mathrm{e}^{\mathrm{i}Kt}\psi(\mathbf{x})\mathrm{e}^{-% \mathrm{i}Kt},
  3. ψ ¯ ( 𝐱 , t ) = [ ψ ( 𝐱 , t ) ] \bar{\psi}(\mathbf{x},t)=[\psi(\mathbf{x},t)]^{\dagger}
  4. K = H - μ N K=H-\mu N
  5. ψ ( 𝐱 , τ ) = e K τ ψ ( 𝐱 ) e - K τ \psi(\mathbf{x},\tau)=\mathrm{e}^{K\tau}\psi(\mathbf{x})\mathrm{e}^{-K\tau}
  6. ψ ¯ ( 𝐱 , τ ) = e K τ ψ ( 𝐱 ) e - K τ . \bar{\psi}(\mathbf{x},\tau)=\mathrm{e}^{K\tau}\psi^{\dagger}(\mathbf{x})% \mathrm{e}^{-K\tau}.
  7. ψ ¯ ( 𝐱 , τ ) \bar{\psi}(\mathbf{x},\tau)
  8. ψ ( 𝐱 , τ ) \psi(\mathbf{x},\tau)
  9. 2 n 2n
  10. G ( n ) ( 1 n | 1 n ) = i n T ψ ( 1 ) ψ ( n ) ψ ¯ ( n ) ψ ¯ ( 1 ) , G^{(n)}(1\ldots n|1^{\prime}\ldots n^{\prime})=\mathrm{i}^{n}\langle T\psi(1)% \ldots\psi(n)\bar{\psi}(n^{\prime})\ldots\bar{\psi}(1^{\prime})\rangle,
  11. j j
  12. 𝐱 j , t j \mathbf{x}_{j},t_{j}
  13. j j^{\prime}
  14. 𝐱 j , t j \mathbf{x}_{j}^{\prime},t_{j}^{\prime}
  15. T T
  16. 𝒢 ( n ) ( 1 n | 1 n ) = T ψ ( 1 ) ψ ( n ) ψ ¯ ( n ) ψ ¯ ( 1 ) , \mathcal{G}^{(n)}(1\ldots n|1^{\prime}\ldots n^{\prime})=\langle T\psi(1)% \ldots\psi(n)\bar{\psi}(n^{\prime})\ldots\bar{\psi}(1^{\prime})\rangle,
  17. j j
  18. 𝐱 j , τ j \mathbf{x}_{j},\tau_{j}
  19. τ j \tau_{j}
  20. 0
  21. β = 1 k B T \beta=\frac{1}{k_{B}T}
  22. n = 1 n=1
  23. 𝒢 ( 𝐤 , ω n ) = 1 - i ω n + ξ 𝐤 , \mathcal{G}(\mathbf{k},\omega_{n})=\frac{1}{-\mathrm{i}\omega_{n}+\xi_{\mathbf% {k}}},
  24. G R ( 𝐤 , ω ) = 1 - ( ω + i η ) + ξ 𝐤 , G^{\mathrm{R}}(\mathbf{k},\omega)=\frac{1}{-(\omega+\mathrm{i}\eta)+\xi_{% \mathbf{k}}},
  25. ω n = [ 2 n + θ ( - ζ ) ] π / β \omega_{n}={[2n+\theta(-\zeta)]\pi}/{\beta}
  26. ζ \zeta
  27. + 1 +1
  28. - 1 -1
  29. [ , ] = [ , ] - ζ [\ldots,\ldots]=[\ldots,\ldots]_{-\zeta}
  30. n = 1 n=1
  31. 𝒢 ( 𝐱 τ | 𝐱 τ ) = 𝐤 d 𝐤 1 β ω n 𝒢 ( 𝐤 , ω n ) e i 𝐤 ( 𝐱 - 𝐱 ) - i ω n ( τ - τ ) , \mathcal{G}(\mathbf{x}\tau|\mathbf{x}^{\prime}\tau^{\prime})=\int_{\mathbf{k}}% d\mathbf{k}\frac{1}{\beta}\sum_{\omega_{n}}\mathcal{G}(\mathbf{k},\omega_{n})% \mathrm{e}^{\mathrm{i}\mathbf{k}\cdot(\mathbf{x}-\mathbf{x}^{\prime})-\mathrm{% i}\omega_{n}(\tau-\tau^{\prime})},
  32. ( L / 2 π ) d (L/2\pi)^{d}
  33. G T ( 𝐱 t | 𝐱 t ) = 𝐤 d 𝐤 d ω 2 π G T ( 𝐤 , ω ) e i 𝐤 ( 𝐱 - 𝐱 ) - i ω ( t - t ) . G^{\mathrm{T}}(\mathbf{x}t|\mathbf{x}^{\prime}t^{\prime})=\int_{\mathbf{k}}d% \mathbf{k}\int\frac{\mathrm{d}\omega}{2\pi}G^{\mathrm{T}}(\mathbf{k},\omega)% \mathrm{e}^{\mathrm{i}\mathbf{k}\cdot(\mathbf{x}-\mathbf{x}^{\prime})-\mathrm{% i}\omega(t-t^{\prime})}.
  34. G R ( 𝐱 t | 𝐱 t ) = - i [ ψ ( 𝐱 , t ) , ψ ¯ ( 𝐱 , t ) ] Θ ( t - t ) G^{\mathrm{R}}(\mathbf{x}t|\mathbf{x}^{\prime}t^{\prime})=-\mathrm{i}\langle[% \psi(\mathbf{x},t),\bar{\psi}(\mathbf{x}^{\prime},t^{\prime})]\rangle\Theta(t-% t^{\prime})
  35. G A ( 𝐱 t | 𝐱 t ) = i [ ψ ( 𝐱 , t ) , ψ ¯ ( 𝐱 , t ) ] Θ ( t - t ) , G^{\mathrm{A}}(\mathbf{x}t|\mathbf{x}^{\prime}t^{\prime})=\mathrm{i}\langle[% \psi(\mathbf{x},t),\bar{\psi}(\mathbf{x}^{\prime},t^{\prime})]\rangle\Theta(t^% {\prime}-t),
  36. G T ( 𝐤 , ω ) = [ 1 + ζ n ( ω ) ] G R ( 𝐤 , ω ) - ζ n ( ω ) G A ( 𝐤 , ω ) , G^{\mathrm{T}}(\mathbf{k},\omega)=[1+\zeta n(\omega)]G^{\mathrm{R}}(\mathbf{k}% ,\omega)-\zeta n(\omega)G^{\mathrm{A}}(\mathbf{k},\omega),
  37. n ( ω ) = 1 e β ω - ζ n(\omega)=\frac{1}{\mathrm{e}^{\beta\omega}-\zeta}
  38. β \beta
  39. 0
  40. β \beta
  41. 𝒢 ( τ , τ ) = 𝒢 ( τ - τ ) . \mathcal{G}(\tau,\tau^{\prime})=\mathcal{G}(\tau-\tau^{\prime}).
  42. τ - τ \tau-\tau^{\prime}
  43. - β -\beta
  44. β \beta
  45. 𝒢 ( τ ) \mathcal{G}(\tau)
  46. β \beta
  47. 𝒢 ( τ - β ) = ζ 𝒢 ( τ ) , \mathcal{G}(\tau-\beta)=\zeta\mathcal{G}(\tau),
  48. 0 < τ < β 0<\tau<\beta
  49. 𝒢 ( ω n ) = 0 β d τ 𝒢 ( τ ) e i ω n τ . \mathcal{G}(\omega_{n})=\int_{0}^{\beta}\mathrm{d}\tau\,\mathcal{G}(\tau)\,% \mathrm{e}^{\mathrm{i}\omega_{n}\tau}.
  50. 𝒢 ( τ ) \mathcal{G}(\tau)
  51. τ = 0 \tau=0
  52. 𝒢 ( ω n ) 1 / | ω n | \mathcal{G}(\omega_{n})\sim 1/|\omega_{n}|
  53. ρ ( 𝐤 , ω ) = 1 𝒵 α , α 2 π δ ( E α - E α - ω ) | α | ψ 𝐤 | α | 2 ( e - β E α - ζ e - β E α ) , \rho(\mathbf{k},\omega)=\frac{1}{\mathcal{Z}}\sum_{\alpha,\alpha^{\prime}}2\pi% \delta(E_{\alpha}-E_{\alpha^{\prime}}-\omega)\;|\langle\alpha|\psi_{\mathbf{k}% }^{\dagger}|\alpha^{\prime}\rangle|^{2}\left(\mathrm{e}^{-\beta E_{\alpha^{% \prime}}}-\zeta\mathrm{e}^{-\beta E_{\alpha}}\right),
  54. | α |\alpha\rangle
  55. H - μ N H-\mu N
  56. E α E_{\alpha}
  57. 𝒢 ( 𝐤 , ω n ) = - d ω 2 π ρ ( 𝐤 , ω ) - i ω n + ω . \mathcal{G}(\mathbf{k},\omega_{n})=\int_{-\infty}^{\infty}\frac{\mathrm{d}% \omega^{\prime}}{2\pi}\frac{\rho(\mathbf{k},\omega^{\prime})}{-\mathrm{i}% \omega_{n}+\omega^{\prime}}.
  58. G R ( 𝐤 , ω ) = - d ω 2 π ρ ( 𝐤 , ω ) - ( ω + i η ) + ω , G^{\mathrm{R}}(\mathbf{k},\omega)=\int_{-\infty}^{\infty}\frac{\mathrm{d}% \omega^{\prime}}{2\pi}\frac{\rho(\mathbf{k},\omega^{\prime})}{-(\omega+\mathrm% {i}\eta)+\omega^{\prime}},
  59. η 0 + \eta\rightarrow 0^{+}
  60. - i η -\mathrm{i}\eta
  61. G R G^{\mathrm{R}}
  62. G A G^{\mathrm{A}}
  63. G R ( ω ) G^{\mathrm{R}}(\omega)
  64. G A ( ω ) G^{\mathrm{A}}(\omega)
  65. 𝒢 ( ω n ) \mathcal{G}(\omega_{n})
  66. ω n \omega_{n}
  67. G R G^{\mathrm{R}}
  68. lim η 0 + 1 x ± i η = P 1 x i π δ ( x ) , \lim_{\eta\rightarrow 0^{+}}\frac{1}{x\pm\mathrm{i}\eta}={P}\frac{1}{x}\mp i% \pi\delta(x),
  69. P P
  70. ρ ( 𝐤 , ω ) = 2 I m G R ( 𝐤 , ω ) . \rho(\mathbf{k},\omega)=2\mathrm{Im}\,G^{\mathrm{R}}(\mathbf{k},\omega).
  71. G R ( 𝐤 , ω ) G^{\mathrm{R}}(\mathbf{k},\omega)
  72. Re G R ( 𝐤 , ω ) = - 2 P - d ω 2 π Im G R ( 𝐤 , ω ) ω - ω , \mathrm{Re}\,G^{\mathrm{R}}(\mathbf{k},\omega)=-2P\int_{-\infty}^{\infty}\frac% {\mathrm{d}\omega^{\prime}}{2\pi}\frac{\mathrm{Im}\,G^{\mathrm{R}}(\mathbf{k},% \omega^{\prime})}{\omega-\omega^{\prime}},
  73. P P
  74. - d ω 2 π ρ ( 𝐤 , ω ) = 1 , \int_{-\infty}^{\infty}\frac{\mathrm{d}\omega}{2\pi}\rho(\mathbf{k},\omega)=1,
  75. G R ( ω ) 1 | ω | G^{\mathrm{R}}(\omega)\sim\frac{1}{|\omega|}
  76. | ω | |\omega|\rightarrow\infty
  77. G ( 𝐤 , z ) = - d x 2 π ρ ( 𝐤 , x ) - z + x , G(\mathbf{k},z)=\int_{-\infty}^{\infty}\frac{\mathrm{d}x}{2\pi}\frac{\rho(% \mathbf{k},x)}{-z+x},
  78. 𝒢 \mathcal{G}
  79. G R G^{\mathrm{R}}
  80. 𝒢 ( 𝐤 , ω n ) = G ( 𝐤 , i ω n ) \mathcal{G}(\mathbf{k},\omega_{n})=G(\mathbf{k},\mathrm{i}\omega_{n})
  81. G R ( 𝐤 , ω ) = G ( 𝐤 , ω + i η ) . G^{\mathrm{R}}(\mathbf{k},\omega)=G(\mathbf{k},\omega+\mathrm{i}\eta).
  82. G A G^{\mathrm{A}}
  83. G ( 𝐤 , z ) G(\mathbf{k},z)
  84. ρ ( 𝐤 , x ) \rho(\mathbf{k},x)
  85. 𝒢 ( 𝐱 , τ | 𝐱 , τ ) = T ψ ( 𝐱 , τ ) ψ ¯ ( 𝐱 , τ ) . \mathcal{G}(\mathbf{x},\tau|\mathbf{x}^{\prime},\tau^{\prime})=\langle T\psi(% \mathbf{x},\tau)\bar{\psi}(\mathbf{x}^{\prime},\tau^{\prime})\rangle.
  86. 𝒢 ( 𝐱 , τ | 𝟎 , 0 ) \mathcal{G}(\mathbf{x},\tau|\mathbf{0},0)
  87. τ > 0 \tau>0
  88. 𝒢 ( 𝐱 , τ | 𝟎 , 0 ) = 1 𝒵 α e - β E α α | ψ ( 𝐱 , τ ) ψ ¯ ( 𝟎 , 0 ) | α . \mathcal{G}(\mathbf{x},\tau|\mathbf{0},0)=\frac{1}{\mathcal{Z}}\sum_{\alpha^{% \prime}}\mathrm{e}^{-\beta E_{\alpha^{\prime}}}\langle\alpha^{\prime}|\psi(% \mathbf{x},\tau)\bar{\psi}(\mathbf{0},0)|\alpha^{\prime}\rangle.
  89. 𝒢 ( 𝐱 , τ | 𝟎 , 0 ) = 1 𝒵 α , α e - β E α α | ψ ( 𝐱 , τ ) | α α | ψ ¯ ( 𝟎 , 0 ) | α . \mathcal{G}(\mathbf{x},\tau|\mathbf{0},0)=\frac{1}{\mathcal{Z}}\sum_{\alpha,% \alpha^{\prime}}\mathrm{e}^{-\beta E_{\alpha^{\prime}}}\langle\alpha^{\prime}|% \psi(\mathbf{x},\tau)|\alpha\rangle\langle\alpha|\bar{\psi}(\mathbf{0},0)|% \alpha^{\prime}\rangle.
  90. | α |\alpha\rangle
  91. | α |\alpha^{\prime}\rangle
  92. H - μ N H-\mu N
  93. 𝒢 ( 𝐱 , τ | 𝟎 , 0 ) = 1 𝒵 α , α e - β E α e τ ( E α - E α ) α | ψ ( 𝐱 ) | α α | ψ ( 𝟎 ) | α . \mathcal{G}(\mathbf{x},\tau|\mathbf{0},0)=\frac{1}{\mathcal{Z}}\sum_{\alpha,% \alpha^{\prime}}\mathrm{e}^{-\beta E_{\alpha^{\prime}}}\mathrm{e}^{\tau(E_{% \alpha^{\prime}}-E_{\alpha})}\langle\alpha^{\prime}|\psi(\mathbf{x})|\alpha% \rangle\langle\alpha|\psi^{\dagger}(\mathbf{0})|\alpha^{\prime}\rangle.
  94. 𝒢 ( 𝐤 , ω n ) = 1 𝒵 α , α e - β E α 1 - ζ e β ( E α - E α ) - i ω n + E α - E α 𝐤 d 𝐤 α | ψ ( 𝐤 ) | α α | ψ ( 𝐤 ) | α . \mathcal{G}(\mathbf{k},\omega_{n})=\frac{1}{\mathcal{Z}}\sum_{\alpha,\alpha^{% \prime}}\mathrm{e}^{-\beta E_{\alpha^{\prime}}}\frac{1-\zeta\mathrm{e}^{\beta(% E_{\alpha^{\prime}}-E_{\alpha})}}{-\mathrm{i}\omega_{n}+E_{\alpha}-E_{\alpha^{% \prime}}}\int_{\mathbf{k}^{\prime}}d\mathbf{k}^{\prime}\langle\alpha|\psi(% \mathbf{k})|\alpha^{\prime}\rangle\langle\alpha^{\prime}|\psi^{\dagger}(% \mathbf{k}^{\prime})|\alpha\rangle.
  95. | α | ψ ( 𝐤 ) | α | 2 , |\langle\alpha^{\prime}|\psi^{\dagger}(\mathbf{k})|\alpha\rangle|^{2},
  96. 1 = 1 𝒵 α α | e - β ( H - μ N ) [ ψ 𝐤 , ψ 𝐤 ] - ζ | α , 1=\frac{1}{\mathcal{Z}}\sum_{\alpha}\langle\alpha|\mathrm{e}^{-\beta(H-\mu N)}% [\psi_{\mathbf{k}},\psi_{\mathbf{k}}^{\dagger}]_{-\zeta}|\alpha\rangle,
  97. 1 = 1 𝒵 α , α e - β E α ( α | ψ 𝐤 | α α | ψ 𝐤 | α - ζ α | ψ 𝐤 | α α | ψ 𝐤 | α ) . 1=\frac{1}{\mathcal{Z}}\sum_{\alpha,\alpha^{\prime}}\mathrm{e}^{-\beta E_{% \alpha}}\left(\langle\alpha|\psi_{\mathbf{k}}|\alpha^{\prime}\rangle\langle% \alpha^{\prime}|\psi_{\mathbf{k}}^{\dagger}|\alpha\rangle-\zeta\langle\alpha|% \psi_{\mathbf{k}}^{\dagger}|\alpha^{\prime}\rangle\langle\alpha^{\prime}|\psi_% {\mathbf{k}}|\alpha\rangle\right).
  98. 1 = 1 𝒵 α , α ( e - β E α - ζ e - β E α ) | α | ψ 𝐤 | α | 2 , 1=\frac{1}{\mathcal{Z}}\sum_{\alpha,\alpha^{\prime}}\left(\mathrm{e}^{-\beta E% _{\alpha^{\prime}}}-\zeta\mathrm{e}^{-\beta E_{\alpha}}\right)|\langle\alpha|% \psi_{\mathbf{k}}^{\dagger}|\alpha^{\prime}\rangle|^{2},
  99. ρ \rho
  100. ψ 𝐤 | α \psi_{\mathbf{k}}^{\dagger}|\alpha^{\prime}\rangle
  101. E α + ξ 𝐤 E_{\alpha^{\prime}}+\xi_{\mathbf{k}}
  102. ξ 𝐤 = ϵ 𝐤 - μ \xi_{\mathbf{k}}=\epsilon_{\mathbf{k}}-\mu
  103. ρ 0 ( 𝐤 , ω ) = 1 𝒵 2 π δ ( ξ 𝐤 - ω ) α α | ψ 𝐤 ψ 𝐤 | α ( 1 - ζ e - β ξ 𝐤 ) e - β E α . \rho_{0}(\mathbf{k},\omega)=\frac{1}{\mathcal{Z}}\,2\pi\delta(\xi_{\mathbf{k}}% -\omega)\sum_{\alpha^{\prime}}\langle\alpha^{\prime}|\psi_{\mathbf{k}}\psi_{% \mathbf{k}}^{\dagger}|\alpha^{\prime}\rangle(1-\zeta\mathrm{e}^{-\beta\xi_{% \mathbf{k}}})\mathrm{e}^{-\beta E_{\alpha^{\prime}}}.
  104. α | ψ 𝐤 ψ 𝐤 | α = α | ( 1 + ζ ψ 𝐤 ψ 𝐤 ) | α , \langle\alpha^{\prime}|\psi_{\mathbf{k}}\psi_{\mathbf{k}}^{\dagger}|\alpha^{% \prime}\rangle=\langle\alpha^{\prime}|(1+\zeta\psi_{\mathbf{k}}^{\dagger}\psi_% {\mathbf{k}})|\alpha^{\prime}\rangle,
  105. [ 1 + ζ n ( ξ 𝐤 ) ] 𝒵 [1+\zeta n(\xi_{\mathbf{k}})]\mathcal{Z}
  106. ρ 0 ( 𝐤 , ω ) = 2 π δ ( ξ 𝐤 - ω ) . \rho_{0}(\mathbf{k},\omega)=2\pi\delta(\xi_{\mathbf{k}}-\omega).
  107. 𝒢 0 ( 𝐤 , ω ) = 1 - i ω n + ξ 𝐤 \mathcal{G}_{0}(\mathbf{k},\omega)=\frac{1}{-\mathrm{i}\omega_{n}+\xi_{\mathbf% {k}}}
  108. G 0 R ( 𝐤 , ω ) = 1 - ( ω + i η ) + ξ 𝐤 . G_{0}^{\mathrm{R}}(\mathbf{k},\omega)=\frac{1}{-(\omega+\mathrm{i}\eta)+\xi_{% \mathbf{k}}}.
  109. β \beta\rightarrow\infty
  110. ρ ( 𝐤 , ω ) = 2 π α [ δ ( E α - E 0 - ω ) | α | ψ 𝐤 | 0 | 2 - ζ δ ( E 0 - E α - ω ) | 0 | ψ 𝐤 | α | 2 ] \rho(\mathbf{k},\omega)=2\pi\sum_{\alpha}\left[\delta(E_{\alpha}-E_{0}-\omega)% |\langle\alpha|\psi_{\mathbf{k}}^{\dagger}|0\rangle|^{2}-\zeta\delta(E_{0}-E_{% \alpha}-\omega)|\langle 0|\psi_{\mathbf{k}}^{\dagger}|\alpha\rangle|^{2}\right]
  111. α = 0 \alpha=0
  112. ω \omega
  113. ψ ( 𝐱 , τ ) = φ α ( 𝐱 ) ψ α ( τ ) , \psi(\mathbf{x},\tau)=\varphi_{\alpha}(\mathbf{x})\psi_{\alpha}(\tau),
  114. ψ α \psi_{\alpha}
  115. α \alpha
  116. φ α ( 𝐱 ) \varphi_{\alpha}(\mathbf{x})
  117. 𝒢 α 1 α n | β 1 β n ( n ) ( τ 1 τ n | τ 1 τ n ) = T ψ α 1 ( τ 1 ) ψ α n ( τ n ) ψ ¯ β n ( τ n ) ψ ¯ β 1 ( τ 1 ) \mathcal{G}^{(n)}_{\alpha_{1}\ldots\alpha_{n}|\beta_{1}\ldots\beta_{n}}(\tau_{% 1}\ldots\tau_{n}|\tau_{1}^{\prime}\ldots\tau_{n}^{\prime})=\langle T\psi_{% \alpha_{1}}(\tau_{1})\ldots\psi_{\alpha_{n}}(\tau_{n})\bar{\psi}_{\beta_{n}}(% \tau_{n}^{\prime})\ldots\bar{\psi}_{\beta_{1}}(\tau_{1}^{\prime})\rangle
  118. G ( n ) G^{(n)}
  119. 𝒢 α β ( τ | τ ) = 1 β ω n 𝒢 α β ( ω n ) e - i ω n ( τ - τ ) \mathcal{G}_{\alpha\beta}(\tau|\tau^{\prime})=\frac{1}{\beta}\sum_{\omega_{n}}% \mathcal{G}_{\alpha\beta}(\omega_{n})\,\mathrm{e}^{-\mathrm{i}\omega_{n}(\tau-% \tau^{\prime})}
  120. G α β ( t | t ) = - d ω 2 π G α β ( ω ) e - i ω ( t - t ) . G_{\alpha\beta}(t|t^{\prime})=\int_{-\infty}^{\infty}\frac{\mathrm{d}\omega}{2% \pi}\,G_{\alpha\beta}(\omega)\,\mathrm{e}^{-\mathrm{i}\omega(t-t^{\prime})}.
  121. 𝒢 α β \mathcal{G}_{\alpha\beta}
  122. 𝒢 α β ( τ | τ ) = 𝒢 α β ( τ - τ ) \mathcal{G}_{\alpha\beta}(\tau|\tau^{\prime})=\mathcal{G}_{\alpha\beta}(\tau-% \tau^{\prime})
  123. 𝒢 α β ( τ ) = 𝒢 α β ( τ + β ) , \mathcal{G}_{\alpha\beta}(\tau)=\mathcal{G}_{\alpha\beta}(\tau+\beta),
  124. τ < 0 \tau<0
  125. ρ α β ( ω ) = 1 𝒵 m , n 2 π δ ( E n - E m - ω ) m | ψ α | n n | ψ β | m ( e - β E m - ζ e - β E n ) , \rho_{\alpha\beta}(\omega)=\frac{1}{\mathcal{Z}}\sum_{m,n}2\pi\delta(E_{n}-E_{% m}-\omega)\;\langle m|\psi_{\alpha}|n\rangle\langle n|\psi_{\beta}^{\dagger}|m% \rangle\left(\mathrm{e}^{-\beta E_{m}}-\zeta\mathrm{e}^{-\beta E_{n}}\right),
  126. m m
  127. n n
  128. 𝒢 α β ( ω n ) = - d ω 2 π ρ α β ( ω ) - i ω n + ω \mathcal{G}_{\alpha\beta}(\omega_{n})=\int_{-\infty}^{\infty}\frac{\mathrm{d}% \omega^{\prime}}{2\pi}\frac{\rho_{\alpha\beta}(\omega^{\prime})}{-\mathrm{i}% \omega_{n}+\omega^{\prime}}
  129. G α β R ( ω ) = - d ω 2 π ρ α β ( ω ) - ( ω + i η ) + ω . G^{\mathrm{R}}_{\alpha\beta}(\omega)=\int_{-\infty}^{\infty}\frac{\mathrm{d}% \omega^{\prime}}{2\pi}\frac{\rho_{\alpha\beta}(\omega^{\prime})}{-(\omega+% \mathrm{i}\eta)+\omega^{\prime}}.
  130. [ H - μ N , ψ α ] = ξ α ψ α , [H-\mu N,\psi_{\alpha}^{\dagger}]=\xi_{\alpha}\psi_{\alpha}^{\dagger},
  131. | n |n\rangle
  132. ( H - μ N ) | n = E n | n , (H-\mu N)|n\rangle=E_{n}|n\rangle,
  133. ψ α | n \psi_{\alpha}|n\rangle
  134. ( H - μ N ) ψ α | n = ( E n - ξ α ) ψ α | n , (H-\mu N)\psi_{\alpha}|n\rangle=(E_{n}-\xi_{\alpha})\psi_{\alpha}|n\rangle,
  135. ψ α | n \psi_{\alpha}^{\dagger}|n\rangle
  136. ( H - μ N ) ψ α | n = ( E n + ξ α ) ψ α | n . (H-\mu N)\psi_{\alpha}^{\dagger}|n\rangle=(E_{n}+\xi_{\alpha})\psi_{\alpha}^{% \dagger}|n\rangle.
  137. m | ψ α | n n | ψ β | m = δ ξ α , ξ β δ E n , E m + ξ α m | ψ α | n n | ψ β | m . \langle m|\psi_{\alpha}|n\rangle\langle n|\psi_{\beta}^{\dagger}|m\rangle=% \delta_{\xi_{\alpha},\xi_{\beta}}\delta_{E_{n},E_{m}+\xi_{\alpha}}\langle m|% \psi_{\alpha}|n\rangle\langle n|\psi_{\beta}^{\dagger}|m\rangle.
  138. ρ α β ( ω ) = 1 𝒵 m , n 2 π δ ( ξ α - ω ) δ ξ α , ξ β m | ψ α | n n | ψ β | m e - β E m ( 1 - ζ e - β ξ α ) , \rho_{\alpha\beta}(\omega)=\frac{1}{\mathcal{Z}}\sum_{m,n}2\pi\delta(\xi_{% \alpha}-\omega)\delta_{\xi_{\alpha},\xi_{\beta}}\langle m|\psi_{\alpha}|n% \rangle\langle n|\psi_{\beta}^{\dagger}|m\rangle\mathrm{e}^{-\beta E_{m}}\left% (1-\zeta\mathrm{e}^{-\beta\xi_{\alpha}}\right),
  139. ρ α β ( ω ) = 1 𝒵 m 2 π δ ( ξ α - ω ) δ ξ α , ξ β m | ψ α ψ β e - β ( H - μ N ) | m ( 1 - ζ e - β ξ α ) , \rho_{\alpha\beta}(\omega)=\frac{1}{\mathcal{Z}}\sum_{m}2\pi\delta(\xi_{\alpha% }-\omega)\delta_{\xi_{\alpha},\xi_{\beta}}\langle m|\psi_{\alpha}\psi_{\beta}^% {\dagger}\mathrm{e}^{-\beta(H-\mu N)}|m\rangle\left(1-\zeta\mathrm{e}^{-\beta% \xi_{\alpha}}\right),
  140. m | ψ α ψ β | m = δ α , β m | ζ ψ α ψ α + 1 | m \langle m|\psi_{\alpha}\psi_{\beta}^{\dagger}|m\rangle=\delta_{\alpha,\beta}% \langle m|\zeta\psi_{\alpha}^{\dagger}\psi_{\alpha}+1|m\rangle
  141. ρ α β = 2 π δ ( ξ α - ω ) δ α β , \rho_{\alpha\beta}=2\pi\delta(\xi_{\alpha}-\omega)\delta_{\alpha\beta},
  142. 𝒢 α β ( ω n ) = δ α β - i ω n + ξ β \mathcal{G}_{\alpha\beta}(\omega_{n})=\frac{\delta_{\alpha\beta}}{-\mathrm{i}% \omega_{n}+\xi_{\beta}}
  143. G α β ( ω ) = δ α β - ( ω + i η ) + ξ β . G_{\alpha\beta}(\omega)=\frac{\delta_{\alpha\beta}}{-(\omega+\mathrm{i}\eta)+% \xi_{\beta}}.

Green's_function_for_the_three-variable_Laplace_equation.html

  1. 2 u ( 𝐱 ) = f ( 𝐱 ) \nabla^{2}u(\mathbf{x})=f(\mathbf{x})
  2. 2 \nabla^{2}
  3. 3 \mathbb{R}^{3}
  4. f ( 𝐱 ) f(\mathbf{x})
  5. u ( 𝐱 ) u(\mathbf{x})
  6. 2 \nabla^{2}
  7. u ( 𝐱 ) u(\mathbf{x})
  8. f ( 𝐱 ) f(\mathbf{x})
  9. u ( 𝐱 ) = 𝐱 G ( 𝐱 , 𝐱 ) f ( 𝐱 ) d 𝐱 u(\mathbf{x})=\int_{\mathbf{x}^{\prime}}G(\mathbf{x},\mathbf{x^{\prime}})f(% \mathbf{x^{\prime}})d\mathbf{x}^{\prime}
  10. G ( 𝐱 , 𝐱 ) G(\mathbf{x},\mathbf{x^{\prime}})
  11. 𝐱 \mathbf{x}
  12. 𝐱 \mathbf{x^{\prime}}
  13. 2 G ( 𝐱 , 𝐱 ) = δ ( 𝐱 - 𝐱 ) \nabla^{2}G(\mathbf{x},\mathbf{x^{\prime}})=\delta(\mathbf{x}-\mathbf{x^{% \prime}})
  14. δ ( 𝐱 - 𝐱 ) \delta(\mathbf{x}-\mathbf{x^{\prime}})
  15. 𝐄 = - ϕ ( 𝐱 ) \mathbf{E}=-\mathbf{\nabla}\phi(\mathbf{x})
  16. 𝐄 = ρ ( 𝐱 ) ε 0 \mathbf{\nabla}\cdot\mathbf{E}=\frac{\rho(\mathbf{x})}{\varepsilon_{0}}
  17. - 2 ϕ ( 𝐱 ) = ρ ( 𝐱 ) ε 0 -\mathbf{\nabla}^{2}\phi(\mathbf{x})=\frac{\rho(\mathbf{x})}{\varepsilon_{0}}
  18. ϕ ( 𝐱 ) \phi(\mathbf{x})
  19. q q
  20. 𝐱 \mathbf{x^{\prime}}
  21. ρ ( 𝐱 ) = q δ ( 𝐱 - 𝐱 ) \rho(\mathbf{x})=q\delta(\mathbf{x}-\mathbf{x^{\prime}})
  22. - ε 0 q 2 ϕ ( 𝐱 ) = δ ( 𝐱 - 𝐱 ) -\frac{\varepsilon_{0}}{q}\mathbf{\nabla}^{2}\phi(\mathbf{x})=\delta(\mathbf{x% }-\mathbf{x^{\prime}})
  23. G ( 𝐱 , 𝐱 ) G(\mathbf{x},\mathbf{x^{\prime}})
  24. - ε 0 q 2 -\frac{\varepsilon_{0}}{q}\nabla^{2}
  25. q q
  26. ϕ ( 𝐱 ) \phi(\mathbf{x})
  27. ϕ ( 𝐱 ) = 𝐱 G ( 𝐱 , 𝐱 ) ρ ( 𝐱 ) d 𝐱 \phi(\mathbf{x})=\int_{\mathbf{x}^{\prime}}G(\mathbf{x},\mathbf{x^{\prime}})% \rho(\mathbf{x^{\prime}})d\mathbf{x}^{\prime}
  28. 2 G ( 𝐱 , 𝐱 ) = δ ( 𝐱 - 𝐱 ) \nabla^{2}G(\mathbf{x},\mathbf{x^{\prime}})=\delta(\mathbf{x}-\mathbf{x^{% \prime}})
  29. G ( 𝐱 , 𝐱 ) = - 1 4 π 1 | 𝐱 - 𝐱 | , G(\mathbf{x},\mathbf{x^{\prime}})=-\frac{1}{4\pi}\cdot\frac{1}{|\mathbf{x}-% \mathbf{x^{\prime}}|},
  30. 𝐱 = ( x , y , z ) \mathbf{x}=(x,y,z)
  31. δ \,\!\delta
  32. - 1 / ( 4 π ) \,\!-1/(4\pi)
  33. 1 | 𝐱 - 𝐱 | = [ ( x - x ) 2 + ( y - y ) 2 + ( z - z ) 2 ] - 1 2 . \frac{1}{|\mathbf{x}-\mathbf{x^{\prime}}|}=[(x-x^{\prime})^{2}+(y-y^{\prime})^% {2}+(z-z^{\prime})^{2}]^{-\frac{1}{2}}.
  34. 1 | 𝐱 - 𝐱 | = l = 0 r < l r > l + 1 P l ( cos γ ) , \frac{1}{|\mathbf{x}-\mathbf{x^{\prime}}|}=\sum_{l=0}^{\infty}\frac{r_{<}^{l}}% {r_{>}^{l+1}}P_{l}(\cos\gamma),
  35. ( r , θ , φ ) \,\!(r,\theta,\varphi)
  36. γ \,\!\gamma
  37. ( 𝐱 , 𝐱 ) (\mathbf{x},\mathbf{x^{\prime}})
  38. cos γ = cos θ cos θ + sin θ sin θ cos ( φ - φ ) . \cos\gamma=\cos\theta\cos\theta^{\prime}+\sin\theta\sin\theta^{\prime}\cos(% \varphi-\varphi^{\prime}).
  39. 1 | 𝐱 - 𝐱 | = 1 π R R m = - e i m ( φ - φ ) Q m - 1 2 ( χ ) \frac{1}{|\mathbf{x}-\mathbf{x^{\prime}}|}=\frac{1}{\pi\sqrt{RR^{\prime}}}\sum% _{m=-\infty}^{\infty}e^{im(\varphi-\varphi^{\prime})}Q_{m-\frac{1}{2}}(\chi)
  40. χ = R 2 + R 2 + ( z - z ) 2 2 R R \chi=\frac{R^{2}+{R^{\prime}}^{2}+(z-z^{\prime})^{2}}{2RR^{\prime}}
  41. Q m - 1 2 ( χ ) \,\!Q_{m-\frac{1}{2}}(\chi)
  42. ( R , φ , z ) \,\!(R,\varphi,z)
  43. 1 | 𝐱 - 𝐱 | = π 2 R R ( χ 2 - 1 ) 1 / 2 m = - ( - 1 ) m Γ ( m + 1 / 2 ) P - 1 2 m ( χ χ 2 - 1 ) e i m ( φ - φ ) \frac{1}{|\mathbf{x}-\mathbf{x^{\prime}}|}=\sqrt{\frac{\pi}{2RR^{\prime}(\chi^% {2}-1)^{1/2}}}\sum_{m=-\infty}^{\infty}\frac{(-1)^{m}}{\Gamma(m+1/2)}P_{-\frac% {1}{2}}^{m}\biggl(\frac{\chi}{\sqrt{\chi^{2}-1}}\biggr)e^{im(\varphi-\varphi^{% \prime})}
  44. 1 | 𝐱 - 𝐱 | = 0 J 0 ( k R 2 + R 2 - 2 R R cos ( φ - φ ) ) e - k ( z > - z < ) d k , \frac{1}{|\mathbf{x}-\mathbf{x^{\prime}}|}=\int_{0}^{\infty}J_{0}\biggl(k\sqrt% {R^{2}+{R^{\prime}}^{2}-2RR^{\prime}\cos(\varphi-\varphi^{\prime})}\biggr)e^{-% k(z_{>}-z_{<})}\,dk,
  45. z > ( z < ) \,\!z_{>}(z_{<})
  46. z \,\!z
  47. z \,\!z^{\prime}
  48. 1 | 𝐱 - 𝐱 | = 2 π 0 K 0 ( k R 2 + R 2 - 2 R R cos ( φ - φ ) ) cos k ( z - z ) d k . \frac{1}{|\mathbf{x}-\mathbf{x^{\prime}}|}=\frac{2}{\pi}\int_{0}^{\infty}K_{0}% \biggl(k\sqrt{R^{2}+{R^{\prime}}^{2}-2RR^{\prime}\cos(\varphi-\varphi^{\prime}% )}\biggr)\cos{k(z-z^{\prime})}\,dk.