wpmath0000016_5

Finite_volume_method_for_two_dimensional_diffusion_problem.html

  1. x ( Γ x ) + y ( Γ y ) + S = 0 \frac{\partial{}}{\partial{}x}\left(\Gamma{}\frac{\partial{}\varnothing{}}{% \partial{}x}\right)+\frac{\partial{}}{\partial{}y}\left(\Gamma{}\frac{\partial% {}\varnothing{}}{\partial{}y}\right)+S=0
  2. Γ \Gamma
  3. S S
  4. Δ v x ( Γ x ) d x . d y + Δ v y ( Γ y ) d x . d y + Δ v S d V = 0 \int_{\Delta{v}}\frac{\partial{}}{\partial{}x}\left(\Gamma{}\frac{\partial{}% \varnothing{}}{\partial{}x}\right)dx.dy+\int_{\Delta{v}}\frac{\partial{}}{% \partial{}y}\left(\Gamma{}\frac{\partial{}\varnothing{}}{\partial{}y}\right)dx% .dy+\int_{\Delta{v}}S_{\varnothing{}}dV=0
  5. [ Γ e A e ( x ) - Γ w A w ( x ) ] + [ Γ n A n ( x ) - Γ s A s ( x ) ] + S ¯ Δ V = 0 \left[{\Gamma{}}_{e}A_{e}\left(\frac{\partial{}\varnothing{}}{\partial{}x}% \right)-{\Gamma{}}_{w}A_{w}\left(\frac{\partial{}\varnothing{}}{\partial{}x}% \right)\right]+\left[{\Gamma{}}_{n}A_{n}\left(\frac{\partial{}\varnothing{}}{% \partial{}x}\right)-{\Gamma{}}_{s}A_{s}\left(\frac{\partial{}\varnothing{}}{% \partial{}x}\right)\right]+\bar{S}\Delta{}V=0
  6. Γ w A w ( x ) w = Γ w A w ( p - w ) δ x W P {{\Gamma{}}_{w}A_{w}\left(\frac{\partial{}\varnothing{}}{\partial{x}}\right)}_% {w}={\Gamma{}}_{w}A_{w}\frac{({\varnothing{}}_{p}-{\varnothing{}}_{w})}{{% \delta{}x}_{WP}}
  7. Γ e A e ( x ) e = Γ e A e ( e - p ) δ x P E {{\Gamma{}}_{e}A_{e}\left(\frac{\partial{}\varnothing{}}{\partial{x}}\right)}_% {e}={\Gamma{}}_{e}A_{e}\frac{({\varnothing{}}_{e}-{\varnothing{}}_{p})}{{% \delta{}x}_{PE}}
  8. Γ s A s ( x ) s = Γ s A s ( p - s ) δ x S P {{\Gamma{}}_{s}A_{s}\left(\frac{\partial{}\varnothing{}}{\partial{x}}\right)}_% {s}={\Gamma{}}_{s}A_{s}\frac{({\varnothing{}}_{p}-{\varnothing{}}_{s})}{{% \delta{}x}_{SP}}
  9. Γ n A n ( x ) n = Γ n A n ( n - p ) δ x P N {{\Gamma{}}_{n}A_{n}\left(\frac{\partial{}\varnothing{}}{\partial{x}}\right)}_% {n}={\Gamma{}}_{n}A_{n}\frac{({\varnothing{}}_{n}-{\varnothing{}}_{p})}{{% \delta{}x}_{PN}}
  10. Γ e A e ( e - p ) δ x P E - Γ w A w ( p - w ) δ x W P + Γ n A n ( n - p ) δ x P N - Γ s A s ( p - s ) δ x S P + S ¯ Δ V = 0 {\Gamma{}}_{e}A_{e}\frac{({\varnothing{}}_{e}-{\varnothing{}}_{p})}{{\delta{}x% }_{PE}}-{\Gamma{}}_{w}A_{w}\frac{({\varnothing{}}_{p}-{\varnothing{}}_{w})}{{% \delta{}x}_{WP}}+{\Gamma{}}_{n}A_{n}\frac{({\varnothing{}}_{n}-{\varnothing{}}% _{p})}{{\delta{}x}_{PN}}-{\Gamma{}}_{s}A_{s}\frac{({\varnothing{}}_{p}-{% \varnothing{}}_{s})}{{\delta{}x}_{SP}}+\bar{S}\Delta{}V=0
  11. S ¯ Δ V = S u + S p * S φ \bar{S}\Delta{}V=S_{u}+S_{p}*S_{\varphi}
  12. [ Γ w A w δ x W P + Γ w A w δ x W P + Γ w A w δ x W P + Γ w A w δ x W P + Γ w A w δ x W P + S p ] \left[\frac{{\Gamma{}}_{w}A_{w}}{{\delta{}x}_{WP}}+\frac{{\Gamma{}}_{w}A_{w}}{% {\delta{}x}_{WP}}+\frac{{\Gamma{}}_{w}A_{w}}{{\delta{}x}_{WP}}+\frac{{\Gamma{}% }_{w}A_{w}}{{\delta{}x}_{WP}}+\frac{{\Gamma{}}_{w}A_{w}}{{\delta{}x}_{WP}}+S_{% p}\right]\varnothing{}
  13. Γ w A w δ x W P + Γ w A w δ x W P + e Γ w A w δ x W P + w Γ w A w δ x W P + n Γ w A w δ x W P + s S u \frac{{\Gamma{}}_{w}A_{w}}{{\delta{}x}_{WP}}+\frac{{\Gamma{}}_{w}A_{w}}{{% \delta{}x}_{WP}}\varnothing{}_{e}+\frac{{\Gamma{}}_{w}A_{w}}{{\delta{}x}_{WP}}% \varnothing{}_{w}+\frac{{\Gamma{}}_{w}A_{w}}{{\delta{}x}_{WP}}\varnothing{}_{n% }+\frac{{\Gamma{}}_{w}A_{w}}{{\delta{}x}_{WP}}\varnothing{}_{s}+S_{u}
  14. a p = p a w + w a e + e a s + s a n + n S u a_{p}\varnothing{}_{p}=a_{w}\varnothing{}_{w}+a_{e}\varnothing{}_{e}+a_{s}% \varnothing{}_{s}+a_{n}\varnothing{}_{n}+S_{u}
  15. a W a_{W}
  16. a E a_{E}
  17. a s a_{s}
  18. a N a_{N}
  19. a P a_{P}
  20. Γ w A w δ x W P \frac{{\Gamma{}}_{w}A_{w}}{{\delta{}x}_{WP}}
  21. Γ e A e δ x P E \frac{{\Gamma{}}_{e}A_{e}}{{\delta{}x}_{PE}}
  22. Γ s A s δ x S P \frac{{\Gamma{}}_{s}A_{s}}{{\delta{}x}_{SP}}
  23. Γ n A n δ x P N \frac{{\Gamma{}}_{n}A_{n}}{{\delta{}x}_{PN}}
  24. a w + a E + a s + a N - S p a_{w}+a_{E}+a_{s}+a_{N}-S_{p}
  25. A w = A e = Δ a A_{w}=A_{e}=\Delta{}a
  26. A n = A s = Δ x A_{n}=A_{s}=\Delta{}x
  27. φ \varphi{}
  28. s u s_{u}
  29. S p S_{p}
  30. φ \varphi{}

Finite_volume_method_for_unsteady_flow.html

  1. ρ ϕ t + div ( ρ ϕ υ ) = div ( Γ grad ϕ ) + S ϕ \frac{\partial\rho\phi}{\partial t}+\operatorname{div}(\rho\phi\upsilon)=% \operatorname{div}(\Gamma\operatorname{grad}\phi)+S_{\phi}
  2. ρ \rho
  3. ϕ \phi
  4. Γ \Gamma
  5. S S
  6. div ( ρ ϕ υ ) \operatorname{div}(\rho\phi\upsilon)
  7. ϕ \phi
  8. div ( Γ grad ϕ ) \operatorname{div}(\Gamma\operatorname{grad}\phi)
  9. ϕ \phi
  10. S ϕ S_{\phi}
  11. ϕ \phi
  12. ρ ϕ t \frac{\partial\rho\phi}{\partial t}
  13. ϕ \phi
  14. c v t t + Δ t ( ρ ϕ t d t ) d V + t t + Δ t A ( n . ρ ϕ u d A ) d t = t t + Δ t A ( n . ( Γ grad ϕ ) d A ) d t + t t + Δ t c v S ϕ d V d t \int\limits_{cv}\!\!\!\int_{t}^{t+\Delta t}(\frac{\partial\rho\phi}{\partial t% }\,dt)\,dV+\int_{t}^{t+\Delta t}\!\!\!\int\limits_{A}(n.{\rho\phi u}\,dA)\,dt=% \int_{t}^{t+\Delta t}\!\!\!\int\limits_{A}(n.(\Gamma\operatorname{grad}\phi)\,% dA)\,dt+\int_{t}^{t+\Delta t}\!\!\!\int\limits_{cv}S_{\phi}\,dV\,dt
  15. ρ c T t = k T x x + S \rho c\frac{\partial T}{\partial t}=\frac{\partial\frac{k\partial T}{\partial x% }}{\partial x}+S
  16. t t + Δ t c v ρ c T t d V d t = t t + Δ t c v k T x x d V d t + t t + Δ t c v S d V d t \int_{t}^{t+\Delta t}\!\!\!\int\limits_{cv}\rho c\frac{\partial T}{\partial t}% \,dV\,dt=\int_{t}^{t+\Delta t}\!\!\!\int\limits_{cv}\frac{\partial\frac{k% \partial T}{\partial x}}{\partial x}\,dV\,dt+\int_{t}^{t+\Delta t}\!\!\!\int% \limits_{cv}S\,dV\,dt
  17. e w t t + Δ t ( ρ c T t d t ) d V = t t + Δ t [ ( k A T x ) e - ( k A T x ) w ] d t + t t + Δ t S ¯ Δ V d t \int_{e}^{w}\!\!\!\int_{t}^{t+\Delta t}(\rho c\frac{\partial T}{\partial t}\,% dt)\,dV=\int_{t}^{t+\Delta t}[(kA\frac{\partial T}{\partial x})_{e}-(kA\frac{% \partial T}{\partial x})_{w}]\,dt+\int_{t}^{t+\Delta t}\bar{S}\Delta V\,dt
  18. c v t t + Δ t ( ρ c T t d t ) d V = ρ c ( T P - T P O ) Δ V \int\limits_{cv}\!\!\!\int_{t}^{t+\Delta t}(\rho c\frac{\partial T}{\partial t% }\,dt)\,dV=\rho c(T_{P}-{T_{P}}^{O})\Delta V
  19. ρ c ( T P - T P 0 ) Δ V = t t + Δ t [ ( K e A T E - T P δ x P E ) - ( K w A T P - T W δ x W P ) ] d t + t t + Δ t S ¯ Δ V d t \rho c(T_{P}-{T_{P}}^{0})\Delta V=\int_{t}^{t+\Delta t}[(K_{e}A\frac{T_{E}-T_{% P}}{\delta x_{PE}})-(K_{w}A\frac{T_{P}-T_{W}}{\delta x_{WP}})]\,dt+\int_{t}^{t% +\Delta t}\bar{S}\Delta V\,dt
  20. θ \theta
  21. T P T_{P}
  22. I T = t t + Δ t T P d t = [ θ T P - ( 1 - θ ) T P 0 ] Δ t I_{T}=\int_{t}^{t+\Delta t}T_{P}\,dt=[\theta T_{P}-(1-\theta){T_{P}}^{0}]\Delta t
  23. Θ \Theta
  24. Θ \Theta
  25. Θ \Theta
  26. b = S u + S P T P 0 b=S_{u}+{S_{P}}{T_{P}}^{0}
  27. θ = 0 \theta=0
  28. a P T P = a w T w 0 + a e T e 0 + [ a P 0 - ( a w + a e - S P ) ] T P 0 + S u a_{P}T_{P}=a_{w}{T_{w}}^{0}+a_{e}{T_{e}}^{0}+[{a_{P}}^{0}-(a_{w}+a_{e}-S_{P})]% {T_{P}}^{0}+S_{u}
  29. a P = a P 0 a_{P}={a_{P}}^{0}
  30. δ x P E = δ x W P = Δ x \delta x_{PE}=\delta x_{WP}=\Delta x
  31. ρ c Δ x Δ t > 2 K Δ x \rho c\frac{\Delta x}{\Delta t}>\frac{2K}{\Delta x}
  32. Δ x \Delta x
  33. θ = 1 2 \theta=\frac{1}{2}
  34. a P T P = a E [ T E + T E 0 2 ] + a W [ T W + T W 0 2 ] + [ a P 0 - a E 2 - a W 2 ] T P 0 + b a_{P}T_{P}=a_{E}[\frac{T_{E}+{T_{E}}^{0}}{2}]+a_{W}[\frac{T_{W}+{T_{W}}^{0}}{2% }]+[{a_{P}}^{0}-\frac{a_{E}}{2}-\frac{a_{W}}{2}]{T_{P}}^{0}+b
  35. a P = a W + a E 2 + a P 0 - S P 2 a_{P}=\frac{a_{W}+a_{E}}{2}+{a_{P}}^{0}-\frac{S_{P}}{2}
  36. 1 2 < θ < 1 \frac{1}{2}<\theta<1
  37. T P 0 {T_{P}}^{0}
  38. a P 0 = [ a E + a W 2 ] {a_{P}}^{0}=[\frac{a_{E}+a_{W}}{2}]
  39. Δ t < ρ c Δ x 2 K \Delta t<\rho c\frac{\Delta x^{2}}{K}
  40. a P T P = a W T W + a E T E + a P 0 T P 0 + S u a_{P}T_{P}=a_{W}T_{W}+a_{E}T_{E}+{a_{P}}^{0}{T_{P}}^{0}+S_{u}
  41. a P = a P 0 + a W + a E - S P a_{P}={a_{P}}^{0}+a_{W}+a_{E}-S_{P}
  42. T 0 T^{0}
  43. Δ t \Delta t
  44. T T
  45. T 0 T^{0}

Finite_water-content_vadose_zone_flow_method.html

  1. Δ θ \Delta\theta
  2. z z
  3. θ \theta
  4. ( d z d t ) θ = K ( θ ) θ [ 1 - ( ψ ( θ ) z ) ] \left(\frac{dz}{dt}\right)_{\theta}=\frac{\partial K(\theta)}{\partial\theta}% \left[1-\left(\frac{\partial\psi(\theta)}{\partial z}\right)\right]
  5. K K
  6. ψ \psi
  7. z z
  8. θ \theta
  9. t t
  10. q q
  11. θ t + q z = 0. \frac{\partial\theta}{\partial t}+\frac{\partial q}{\partial z}=0.
  12. z t = q θ . \frac{\partial z}{\partial t}=\frac{\partial q}{\partial\theta}.
  13. q = - K ( θ ) ψ ( θ ) z + K ( θ ) , q=-K(\theta)\frac{\partial\psi(\theta)}{\partial z}+K(\theta),
  14. q θ = θ [ K ( θ ) ( 1 - ψ ( θ ) z ) ] = K ( θ ) θ ( 1 - ψ ( θ ) z ) - K ( θ ) 2 ψ ( θ ) z θ . \frac{\partial q}{\partial\theta}=\frac{\partial}{\partial\theta}\left[K(% \theta)\left(1-\frac{\partial\psi(\theta)}{\partial z}\right)\right]=\frac{% \partial K(\theta)}{\partial\theta}\left(1-\frac{\partial\psi(\theta)}{% \partial z}\right)-K(\theta)\frac{\partial^{2}\psi(\theta)}{\partial z\,% \partial\theta}.
  15. ψ \psi
  16. K s K_{s}
  17. ( d z d t ) θ = K ( θ ) θ [ 1 - ( ψ ( θ ) z ) ] . \left(\frac{dz}{dt}\right)_{\theta}=\frac{\partial K(\theta)}{\partial\theta}% \left[1-\left(\frac{\partial\psi(\theta)}{\partial z}\right)\right].
  18. q ( θ , t ) q(\theta,t)
  19. z ( θ , t ) z(\theta,t)
  20. q θ d θ = z t d θ \int\frac{\partial q}{\partial\theta}\,d\theta=\int\frac{\partial z}{\partial t% }\,d\theta
  21. j = 1 N [ q θ ] j Δ θ = j = 1 N [ z t ] j Δ θ \sum_{j=1}^{N}\left[\frac{\partial q}{\partial\theta}\right]_{j}\Delta\theta=% \sum_{j=1}^{N}\left[\frac{\partial z}{\partial t}\right]_{j}\Delta\theta
  22. N N
  23. [ q θ ] j = [ z t ] j . \left[\frac{\partial q}{\partial\theta}\right]_{j}=\left[\frac{\partial z}{% \partial t}\right]_{j}.
  24. θ d \theta_{d}
  25. θ i \theta_{i}
  26. K ( θ ) θ = K ( θ d ) - K ( θ i ) θ d - θ i . \frac{\partial K(\theta)}{\partial\theta}=\frac{K(\theta_{d})-K(\theta_{i})}{% \theta_{d}-\theta_{i}}.
  27. h p h_{p}
  28. ψ ( θ ) z = | ψ ( θ d ) | + h p z j , \frac{\partial\psi(\theta)}{\partial z}=\frac{|\psi(\theta_{d})|+h_{p}}{z_{j}},
  29. ( d z d t ) j = K ( θ d ) - K ( θ i ) θ d - θ i ( | ψ ( θ d ) | + h p z j + 1 ) . \left(\frac{dz}{dt}\right)_{j}=\frac{K(\theta_{d})-K(\theta_{i})}{\theta_{d}-% \theta_{i}}\left(\frac{|\psi(\theta_{d})|+h_{p}}{z_{j}}+1\right).
  30. j th Δ θ j\text{th}\ \Delta\theta
  31. ( d z d t ) j = K ( θ j ) - K ( θ j - 1 ) θ j - θ j - 1 \left(\frac{dz}{dt}\right)_{j}=\frac{K(\theta_{j})-K(\theta_{j-1})}{\theta_{j}% -\theta_{j-1}}
  32. j th j\text{th}
  33. K ( θ ) θ = K ( θ j ) - K ( θ i ) θ j - θ i , \frac{\partial K(\theta)}{\partial\theta}=\frac{K(\theta_{j})-K(\theta_{i})}{% \theta_{j}-\theta_{i}},
  34. ψ ( θ ) z = | ψ ( θ j ) | H j \frac{\partial\psi(\theta)}{\partial z}=\frac{|\psi(\theta_{j})|}{H_{j}}
  35. ( d z d t ) j = K ( θ j ) - K ( θ i ) θ j - θ i ( | ψ ( θ j ) | H j - 1 ) . \left(\frac{dz}{dt}\right)_{j}=\frac{K(\theta_{j})-K(\theta_{i})}{\theta_{j}-% \theta_{i}}\left(\frac{|\psi(\theta_{j})|}{H_{j}}-1\right).

Fink_protocol.html

  1. n ! n!
  2. n 3 / 3 n^{3}/3
  3. n - 1 n-1
  4. n n
  5. 1 / n 1/n

First-order_second-moment_method.html

  1. g ( x ) g(x)
  2. x x
  3. X X
  4. f X ( x ) f_{X}(x)
  5. X X
  6. g g
  7. g g
  8. μ g g ( μ ) \mu_{g}\approx g(\mu)
  9. g g
  10. σ g 2 i = 1 n j = 1 n g ( μ ) x i g ( μ ) x j cov ( X i , X j ) \sigma^{2}_{g}\approx\sum_{i=1}^{n}\sum_{j=1}^{n}\frac{\partial g(\mu)}{% \partial x_{i}}\frac{\partial g(\mu)}{\partial x_{j}}\operatorname{cov}(X_{i},% X_{j})
  11. n n
  12. x x
  13. g ( μ ) x i \frac{\partial g(\mu)}{\partial x_{i}}
  14. g g
  15. μ \mu
  16. x x
  17. μ \mu
  18. g ( x ) = g ( μ ) + i = 1 n g ( μ ) x i ( x i - μ i ) + 1 2 i = 1 n j = 1 n 2 g ( μ ) x i x j ( x i - μ i ) ( x j - μ j ) + g(x)=g(\mu)+\sum_{i=1}^{n}\frac{\partial g(\mu)}{\partial x_{i}}(x_{i}-\mu_{i}% )+\frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{n}\frac{\partial^{2}g(\mu)}{\partial x_% {i}\,\partial x_{j}}(x_{i}-\mu_{i})(x_{j}-\mu_{j})+\cdots
  19. g g
  20. μ g = E [ g ( x ) ] = - g ( x ) f X ( x ) d x \mu_{g}=E[g(x)]=\int_{-\infty}^{\infty}g(x)f_{X}(x)\,dx
  21. μ g - [ g ( μ ) + i = 1 n g ( μ ) x i ] f X ( x ) d x = - g ( μ ) f X ( x ) d x + - i = 1 n g ( μ ) x i ( x i - μ i ) f X ( x ) d x = g ( μ ) - f X ( x ) d x 1 + i = 1 n g ( μ ) x i - ( x i - μ i ) f X ( x ) d x 0 = g ( μ ) . \begin{aligned}\displaystyle\mu_{g}&\displaystyle\approx\int_{-\infty}^{\infty% }\left[g(\mu)+\sum_{i=1}^{n}\frac{\partial g(\mu)}{\partial x_{i}}\right]f_{X}% (x)\,dx\\ &\displaystyle=\int_{-\infty}^{\infty}g(\mu)f_{X}(x)\,dx+\int_{-\infty}^{% \infty}\sum_{i=1}^{n}\frac{\partial g(\mu)}{\partial x_{i}}(x_{i}-\mu_{i})f_{X% }(x)\,dx\\ &\displaystyle=g(\mu)\underbrace{\int_{-\infty}^{\infty}f_{X}(x)\,dx}_{1}+\sum% _{i=1}^{n}\frac{\partial g(\mu)}{\partial x_{i}}\underbrace{\int_{-\infty}^{% \infty}(x_{i}-\mu_{i})f_{X}(x)\,dx}_{0}\\ &\displaystyle=g(\mu).\end{aligned}
  22. g g
  23. σ g 2 = E ( [ g ( x ) - μ g ] 2 ) = - [ g ( x ) - μ g ] 2 f X ( x ) d x . \sigma_{g}^{2}=E([g(x)-\mu_{g}]^{2})=\int_{-\infty}^{\infty}[g(x)-\mu_{g}]^{2}% f_{X}(x)\,dx.
  24. σ g 2 = E ( [ g ( x ) - μ g ] 2 ) = E ( g ( x ) 2 ) - μ g 2 = - g ( x ) 2 f X ( x ) d x - μ g 2 \sigma_{g}^{2}=E([g(x)-\mu_{g}]^{2})=E(g(x)^{2})-\mu_{g}^{2}=\int_{-\infty}^{% \infty}g(x)^{2}f_{X}(x)\,dx-\mu_{g}^{2}
  25. σ g 2 - [ g ( μ ) + i = 1 n g ( μ ) x i ( x i - μ i ) ] 2 f X ( x ) d x - μ g 2 = - { g ( μ ) 2 + 2 g μ i = 1 n g ( μ ) x i ( x i - μ i ) + [ i = 1 n g ( μ ) x i ( x i - μ i ) ] 2 } f X ( x ) d x - μ g 2 = - g ( μ ) 2 f X ( x ) d x + - 2 g μ i = 1 n g ( μ ) x i ( x i - μ i ) f X ( x ) d x + - [ i = 1 n g ( μ ) x i ( x i - μ i ) ] 2 f X ( x ) d x - μ g 2 = g μ 2 - f X ( x ) d x 1 + 2 g μ i = 1 n g ( μ ) x i - ( x i - μ i ) f X ( x ) d x 0 + - [ i = 1 n j = 1 n g ( μ ) x i g ( μ ) x j ( x i - μ i ) ( x j - μ j ) ] f X ( x ) d x - μ g 2 = g ( μ ) 2 μ g 2 + i = 1 n j = 1 n g ( μ ) x i g ( μ ) x j - ( x i - μ i ) ( x j - μ j ) f ( x ) d x cov ( X i , X j ) - μ g 2 = i = 1 n j = 1 n g ( μ ) x i g ( μ ) x j cov ( X i , X j ) . \begin{aligned}\displaystyle\sigma_{g}^{2}&\displaystyle\approx\int_{-\infty}^% {\infty}\left[g(\mu)+\sum_{i=1}^{n}\frac{\partial g(\mu)}{\partial x_{i}}(x_{i% }-\mu_{i})\right]^{2}f_{X}(x)\,dx-\mu_{g}^{2}\\ &\displaystyle=\int_{-\infty}^{\infty}\left\{g(\mu)^{2}+2g_{\mu}\sum_{i=1}^{n}% \frac{\partial g(\mu)}{\partial x_{i}}(x_{i}-\mu_{i})+\left[\sum_{i=1}^{n}% \frac{\partial g(\mu)}{\partial x_{i}}(x_{i}-\mu_{i})\right]^{2}\right\}f_{X}(% x)\,dx-\mu_{g}^{2}\\ &\displaystyle=\int_{-\infty}^{\infty}g(\mu)^{2}f_{X}(x)\,dx+\int_{-\infty}^{% \infty}2\,g_{\mu}\sum_{i=1}^{n}\frac{\partial g(\mu)}{\partial x_{i}}(x_{i}-% \mu_{i})f_{X}(x)\,dx\\ &\displaystyle{}\quad{}+\int_{-\infty}^{\infty}\left[\sum_{i=1}^{n}\frac{% \partial g(\mu)}{\partial x_{i}}(x_{i}-\mu_{i})\right]^{2}f_{X}(x)\,dx-\mu_{g}% ^{2}\\ &\displaystyle=g_{\mu}^{2}\underbrace{\int_{-\infty}^{\infty}f_{X}(x)\,dx}_{1}% +2g_{\mu}\sum_{i=1}^{n}\frac{\partial g(\mu)}{\partial x_{i}}\underbrace{\int_% {-\infty}^{\infty}(x_{i}-\mu_{i})f_{X}(x)\,dx}_{0}\\ &\displaystyle{}\quad{}+\int_{-\infty}^{\infty}\left[\sum_{i=1}^{n}\sum_{j=1}^% {n}\frac{\partial g(\mu)}{\partial x_{i}}\frac{\partial g(\mu)}{\partial x_{j}% }(x_{i}-\mu_{i})(x_{j}-\mu_{j})\right]f_{X}(x)\,dx-\mu_{g}^{2}\\ &\displaystyle=\underbrace{g(\mu)^{2}}_{\mu_{g}^{2}}+\sum_{i=1}^{n}\sum_{j=1}^% {n}\frac{\partial g(\mu)}{\partial x_{i}}\frac{\partial g(\mu)}{\partial x_{j}% }\underbrace{\int_{-\infty}^{\infty}(x_{i}-\mu_{i})(x_{j}-\mu_{j})f(x)\,dx}_{% \operatorname{cov}(X_{i},X_{j})}-\mu_{g}^{2}\\ &\displaystyle=\sum_{i=1}^{n}\sum_{j=1}^{n}\frac{\partial g(\mu)}{\partial x_{% i}}\frac{\partial g(\mu)}{\partial x_{j}}\operatorname{cov}(X_{i},X_{j}).\end{aligned}
  26. g μ = g ( μ ) , g , i = g ( μ ) x i , g , i j = 2 g ( μ ) x i x j , μ i , j = E [ ( x i - μ i ) j ] g_{\mu}=g(\mu),\quad g_{,i}=\frac{\partial g(\mu)}{\partial x_{i}},\quad g_{,% ij}=\frac{\partial^{2}g(\mu)}{\partial x_{i}\,\partial x_{j}},\quad\mu_{i,j}=E% [(x_{i}-\mu_{i})^{j}]
  27. X X
  28. μ g g μ + 1 2 i = 1 n g , i i μ i , 2 \mu_{g}\approx g_{\mu}+\frac{1}{2}\sum_{i=1}^{n}g_{,ii}\;\mu_{i,2}
  29. σ g 2 g μ 2 + i = 1 n g , i 2 μ i , 2 + 1 4 i = 1 n g , i i 2 μ i , 4 + g μ i = 1 n g , i i μ i , 2 + i = 1 n g , i g , i i μ i , 3 + 1 2 i = 1 n j = i + 1 n g , i i g , j j μ i , 2 μ j , 2 + i = 1 n j = i + 1 n g , i j 2 μ i , 2 μ j , 2 - μ g 2 \begin{aligned}\displaystyle\sigma_{g}^{2}&\displaystyle\approx g_{\mu}^{2}+% \sum_{i=1}^{n}g_{,i}^{2}\,\mu_{i,2}+\frac{1}{4}\sum_{i=1}^{n}g_{,ii}^{2}\,\mu_% {i,4}+g_{\mu}\sum_{i=1}^{n}g_{,ii}\,\mu_{i,2}+\sum_{i=1}^{n}g_{,i}\,g_{,ii}\,% \mu_{i,3}\\ &\displaystyle{}\quad{}+\frac{1}{2}\sum_{i=1}^{n}\sum_{j=i+1}^{n}g_{,ii}\,g_{,% jj}\,\mu_{i,2}\,\mu_{j,2}+\sum_{i=1}^{n}\sum_{j=i+1}^{n}g_{,ij}^{2}\,\mu_{i,2}% \,\mu_{j,2}-\mu_{g}^{2}\end{aligned}
  30. g g
  31. μ g , 3 \mu_{g,3}
  32. μ g , 3 i = 1 n g , i 3 μ i , 3 \mu_{g,3}\approx\sum_{i=1}^{n}g_{,i}^{3}\;\mu_{i,3}
  33. 2 n + 1 2n+1

First_passage_percolation.html

  1. p p
  2. p p
  3. q = 1 - p q=1-p
  4. p p
  5. p p
  6. T T
  7. T ( r ) = i t ( e i ) , T(r)=\sum_{i}t(e_{i}),
  8. T T
  9. r r
  10. e i e_{i}
  11. t ( e i ) t(e_{i})

Fixed-point_subgroup.html

  1. G f G^{f}
  2. G f = { g G f ( g ) = g } . G^{f}=\{g\in G\mid f(g)=g\}.
  3. f ( g ) = ( g T ) - 1 f(g)=(g^{T})^{-1}
  4. G f G^{f}
  5. O ( n ) O(n)
  6. R f = { r R f ( r ) = r } . R^{f}=\{r\in R\mid f(r)=r\}.
  7. Aut ( R ) \operatorname{Aut}(R)
  8. R G R^{G}
  9. R g , g G R^{g},\,g\in G

Flajolet–Martin_algorithm.html

  1. h a s h ( x ) hash(x)
  2. x x
  3. [ 0 ; 2 L - 1 ] [0;2^{L}-1]
  4. 2 L - 1 2^{L}-1
  5. L L
  6. y y
  7. b i t ( y , k ) bit(y,k)
  8. k k
  9. y y
  10. y = k 0 bit ( y , k ) 2 k y=\sum_{k\geq 0}\,\text{bit}(y,k)2^{k}
  11. ρ ( y ) \rho(y)
  12. y y
  13. ρ ( y ) = min k 0 bit ( y , k ) 0 \rho(y)=\min_{k\geq 0}\,\text{bit}(y,k)\neq 0
  14. ρ ( 0 ) = L \rho(0)=L
  15. ρ ( 13 ) = ρ ( 1101 ) = 0 \rho(13)=\rho(1101)=0
  16. ρ ( 8 ) = ρ ( 0100 ) = 2 \rho(8)=\rho(0100)=2
  17. 2 k 2^{k}
  18. k k
  19. 2 - ( k + 1 ) 2^{-(k+1)}
  20. k k
  21. M M
  22. L L
  23. x x
  24. M M
  25. ρ ( hash ( x ) ) \rho(\,\text{hash}(x))
  26. B I T M A P [ i n d e x ] = 1 BITMAP[index]=1
  27. R R
  28. i i
  29. B I T M A P [ i ] = 0 BITMAP[i]=0
  30. M M
  31. 2 R ϕ 2^{R}\cdot\phi
  32. ϕ 0.77351 \phi\approx 0.77351
  33. n n
  34. M M
  35. B I T M A P [ 0 ] BITMAP[0]
  36. n / 2 n/2
  37. B I T M A P [ 1 ] BITMAP[1]
  38. n / 4 n/4
  39. i log 2 n i\gg\log_{2}n
  40. B I T M A P [ i ] BITMAP[i]
  41. i log 2 n i\ll\log_{2}n
  42. B I T M A P [ i ] BITMAP[i]
  43. i log 2 n i\approx\log_{2}n
  44. B I T M A P [ i ] BITMAP[i]
  45. ϕ 0.77351 \phi\approx 0.77351
  46. k k
  47. k k
  48. 2 R / ϕ 2^{R}/\phi
  49. R R
  50. k k\cdot\ell
  51. k k
  52. \ell
  53. \ell
  54. k k

Flory-Stockmayer_Theory.html

  1. α = p A p B \alpha=p_{A}p_{B}
  2. α c = 1 f - 1 \alpha_{c}=\frac{1}{f-1}
  3. ρ = N A f o f N A f o + 2 N A 2 \rho=\frac{N_{Afo}f}{N_{Afo}+2N_{A2}}
  4. α = p A p B ρ 1 - p A p B ( 1 - ρ ) \alpha=\frac{p_{A}p_{B}\rho}{1-p_{A}p_{B}(1-\rho)}
  5. r = N A o N B o r=\frac{N_{Ao}}{N_{Bo}}
  6. N A o p A = N B o p B N_{Ao}p_{A}=N_{Bo}p_{B}
  7. α = ( p B 2 ρ r ) 1 - ( p B 2 ) r ( 1 - ρ ) \alpha=\frac{(\frac{p_{B}^{2}\rho}{r})}{\frac{1-(p_{B}^{2})}{r(1-\rho)}}
  8. α c = 1 f - 1 = 1 3 - 1 = 1 2 \alpha_{c}=\frac{1}{f-1}=\frac{1}{3-1}=\frac{1}{2}
  9. α = ( p B 2 ρ r ) 1 - ( p B 2 r ( 1 - ρ ) ) = p B 2 r \alpha=\frac{(\frac{p_{B}^{2}\rho}{r})}{\frac{1-(p_{B}^{2}}{r(1-\rho)})}=\frac% {p_{B}^{2}}{r}
  10. p B 2 r > α c \frac{p_{B}^{2}}{r}>\alpha_{c}
  11. p B > r 2 p_{B}>\sqrt{\frac{r}{2}}
  12. p A > 1 2 r p_{A}>\sqrt{\frac{1}{2r}}
  13. p A > 1 2 r p_{A}>\sqrt{\frac{1}{2r}}
  14. p B > r 2 p_{B}>\sqrt{\frac{r}{2}}

Flotation_of_flexible_objects.html

  1. w w
  2. t t
  3. b b
  4. K s K_{s}
  5. α \alpha
  6. ϕ \phi
  7. h h
  8. ρ \rho
  9. ρ s \rho_{s}
  10. h h
  11. α \alpha
  12. ρ s \rho_{s}
  13. Π \Pi
  14. Π = U - V \Pi=U-V
  15. V V
  16. U U
  17. V V
  18. V = W H , i + W H , p - W g , p + W σ V=W_{H,i}+W_{H,p}-W_{g,p}+W_{\sigma}
  19. W H , i W_{H,i}
  20. W H , p W_{H,p}
  21. W g , p W_{g,p}
  22. W σ W_{\sigma}
  23. U U
  24. U = S + E s U=S+E_{s}
  25. S S
  26. E s E_{s}
  27. E s = K s ( 2 α ) 2 / 2 E_{s}=K_{s}(2\alpha)^{2}/2
  28. δ h \delta h
  29. δ α \delta\alpha
  30. δ Π = δ U - δ V \delta\Pi=\delta U-\delta V
  31. δ V = δ W H , i + δ W H , p - δ W g , p + δ W σ \delta V=\delta W_{H,i}+\delta W_{H,p}-\delta W_{g,p}+\delta W_{\sigma}
  32. δ W H , i = ρ g η ( x ) d x δ ϵ \delta W_{H,i}=\rho g\int\eta(x)dx\delta\epsilon
  33. δ W H , p = ρ g ( ( 2 b h cos α + b 2 cos α sin α ) δ h + ( b 3 3 sin α + b 2 h ) δ α ) \delta W_{H,p}=\rho g\left(\left(2bh\,\text{cos}\alpha+b^{2}\,\text{cos}\alpha% \,\text{sin}\alpha\right)\delta h+\left(\frac{b^{3}}{3}\,\text{sin}\alpha+b^{2% }h\right)\delta\alpha\right)
  34. δ W g , p = ρ s g t ( 2 b δ h + b 2 cos ( α ) δ α ) \delta W_{g,p}=\rho_{s}gt\left(2b\delta h+b^{2}\,\text{cos}\left(\alpha\right)% \delta\alpha\right)
  35. δ W σ = 2 σ ( sin ϕ δ h + b ( sin ( ϕ - α ) δ α ) ) \delta W_{\sigma}=2\sigma\left(\,\text{sin}\phi\delta h+b\left(\,\text{sin}% \left(\phi-\alpha\right)\delta\alpha\right)\right)
  36. δ U = σ δ + 4 K s α δ α \delta U=\sigma\delta\mathcal{L}+4K_{s}\alpha\delta\alpha
  37. η ( x ) \eta(x)
  38. δ ϵ \delta\epsilon
  39. σ \sigma
  40. ρ s \rho_{s}
  41. h h
  42. α \alpha
  43. δ Π δ h = 0 \frac{\delta\Pi}{\delta h}=0
  44. δ Π δ α = 0 \frac{\delta\Pi}{\delta\alpha}=0
  45. h h
  46. α \alpha
  47. ρ s \rho_{s}
  48. β = b 2 l c \beta=\frac{b}{2l_{c}}
  49. H = h l c H=\frac{h}{l_{c}}
  50. D = ρ s - ρ ρ t l c D=\frac{\rho_{s}-\rho}{\rho}\frac{t}{l_{c}}
  51. H max , D = 2 ( 2 + β ) 4 + 2 2 β + β 2 H_{\,\text{max},D}=\frac{2\left(\sqrt{2}+\beta\right)}{\sqrt{4+2\sqrt{2}\beta+% \beta^{2}}}
  52. α max , D = arcos ( 1 2 + 8 + β 2 ( - 2 + 2 β ) 16 + β 2 ) \alpha_{\,\text{max},D}=\,\text{arcos}\left(\sqrt{\frac{1}{2}+\frac{8+\beta^{2% }\left(-2+\sqrt{2}\beta\right)}{16+\beta^{2}}}\right)
  53. D max = β 4 + 2 + 1 β D_{\,\text{max}}=\frac{\beta}{4}+\sqrt{2}+\frac{1}{\beta}
  54. H H
  55. α \alpha
  56. D D
  57. β \beta
  58. β H \beta<<H
  59. β H \beta\sim H
  60. β H \beta>>H
  61. n n

Flow_chart_language.html

  1. ( 𝚒𝚗𝚒𝚝 , [ n 4 , x 1 0 , x 2 0 , t 0 ] ) \displaystyle\left(\mathtt{init},\ \left[n\mapsto 4,\ x1\mapsto 0,\ x2\mapsto 0% ,\ t\mapsto 0\right]\right)
  2. 𝚑𝚊𝚕𝚝 , v \left\langle\mathtt{halt},\ v\right\rangle
  3. v v

Flow_graph_(mathematics).html

  1. [ 1 2 0 0 1 1 5 - 1 - 1 ] [ x 1 x 2 x 3 ] = [ 5 5 0 ] \begin{bmatrix}1&2&0\\ 0&1&1\\ 5&-1&-1\end{bmatrix}\begin{bmatrix}x_{1}\\ x_{2}\\ x_{3}\end{bmatrix}=\begin{bmatrix}5\\ 5\\ 0\end{bmatrix}
  2. [ c 11 c 12 c 13 c 21 c 22 c 23 c 31 c 32 c 33 ] [ x 1 x 2 x 3 ] = [ y 1 y 2 y 3 ] \begin{bmatrix}c_{11}&c_{12}&c_{13}\\ c_{21}&c_{22}&c_{23}\\ c_{31}&c_{32}&c_{33}\end{bmatrix}\begin{bmatrix}x_{1}\\ x_{2}\\ x_{3}\end{bmatrix}=\begin{bmatrix}y_{1}\\ y_{2}\\ y_{3}\end{bmatrix}
  3. ( c 11 + 1 ) x 1 + c 12 x 2 + c 13 x 3 - y 1 = x 1 . \left(c_{11}+1\right)x_{1}+c_{12}x_{2}+c_{13}x_{3}-y_{1}=x_{1}\ .

Flow_in_a_Curb::Gutter_System.html

  1. v = k n R 2 / 3 S 1 / 2 v={k\over n}R^{2/3}S^{1/2}
  2. Q = A v Q=A\cdot v
  3. Q = k n A R 2 / 3 S 1 / 2 Q={k\over n}AR^{2/3}S^{1/2}
  4. A = 1 2 B y = 1 2 m y 2 A={1\over 2}By={1\over 2}my^{2}
  5. F r = v g A B F_{r}={v\over\sqrt{g{A\over B}}}
  6. E = v 2 2 g + y = Q 2 2 g A 2 + y E={v^{2}\over 2g}+y={Q^{2}\over 2gA^{2}}+y
  7. E = Q 2 2 g ( 1 2 m y 2 ) 2 + y = 2 Q 2 g m 2 y 4 + y E={Q^{2}\over{2g(\tfrac{1}{2}my^{2})^{2}}}+y={2Q^{2}\over gm^{2}y^{4}}+y
  8. d E d y = 0 = ( 2 Q 2 g m 2 ) - 4 y c 5 + 1 {\operatorname{d}\!E\over\operatorname{d}\!y}=0=({2Q^{2}\over gm^{2}}){-4\over y% _{c}^{5}}+1
  9. y c = ( 8 Q 2 g m 2 ) 1 / 5 y_{c}=({8Q^{2}\over gm^{2}})^{1}/5
  10. R e l a t i v e E r r o r ( % ) = x t - x 0 x 0 × 100 RelativeError(\%)={x_{t}-x_{0}\over x_{0}}\times 100

Flow_plasticity_theory.html

  1. σ = σ 0 \sigma=\sigma_{0}
  2. σ = σ y \sigma=\sigma_{y}
  3. d σ > 0 d\sigma>0
  4. d ε p > 0 d\varepsilon_{p}>0
  5. d σ < 0 d\sigma<0
  6. d ε = d ε e + d ε p d\varepsilon=d\varepsilon_{e}+d\varepsilon_{p}
  7. d σ d ε = d σ ( d ε e + d ε p ) 0 d\sigma\,d\varepsilon=d\sigma\,(d\varepsilon_{e}+d\varepsilon_{p})\geq 0
  8. s y m b o l σ = 𝖢 : s y m b o l ε symbol{\sigma}=\mathsf{C}:symbol{\varepsilon}
  9. 𝖢 \mathsf{C}
  10. f ( s y m b o l σ ) = 0 . f(symbol{\sigma})=0\,.
  11. f ( s y m b o l σ , s y m b o l ε p ) = 0 . f(symbol{\sigma},symbol{\varepsilon}_{p})=0\,.
  12. d σ > 0 d\sigma>0
  13. f 0 f\geq 0
  14. f / \partialsymbol σ \partial f/\partialsymbol{\sigma}
  15. d s y m b o l ε p : d s y m b o l σ 0 dsymbol{\varepsilon}_{p}:dsymbol{\sigma}\geq 0
  16. d s y m b o l σ : f s y m b o l σ 0 . dsymbol{\sigma}:\frac{\partial f}{\partial symbol{\sigma}}\geq 0\,.
  17. f < 0 f<0
  18. d s y m b o l σ : f s y m b o l σ < 0 . dsymbol{\sigma}:\frac{\partial f}{\partial symbol{\sigma}}<0\,.
  19. d s y m b o l ε = d s y m b o l ε e + d s y m b o l ε p . dsymbol{\varepsilon}=dsymbol{\varepsilon}_{e}+dsymbol{\varepsilon}_{p}\,.
  20. d s y m b o l σ : d s y m b o l ε 0 . dsymbol{\sigma}:dsymbol{\varepsilon}\geq 0\,.
  21. d s y m b o l ε p = d λ f s y m b o l σ dsymbol{\varepsilon}_{p}=d\lambda\,\frac{\partial f}{\partial symbol{\sigma}}
  22. d λ > 0 d\lambda>0
  23. f f
  24. d s y m b o l σ = 0 dsymbol{\sigma}=0
  25. d s y m b o l ε p > 0 dsymbol{\varepsilon}_{p}>0
  26. d s y m b o l ε e = 0 dsymbol{\varepsilon}_{e}=0
  27. d s y m b o l σ : f s y m b o l σ = 0 and d s y m b o l σ : d s y m b o l ε p = 0 . dsymbol{\sigma}:\frac{\partial f}{\partial symbol{\sigma}}=0\quad\,\text{and}% \quad dsymbol{\sigma}:dsymbol{\varepsilon}_{p}=0\,.
  28. d s y m b o l σ : d s y m b o l ε p 0 . dsymbol{\sigma}:dsymbol{\varepsilon}_{p}\geq 0\,.
  29. d λ d\lambda
  30. d f = 0 df=0
  31. f ( s y m b o l σ , s y m b o l ε p ) = 0 f(symbol{\sigma},symbol{\varepsilon}_{p})=0
  32. d f = f s y m b o l σ : d s y m b o l σ + f s y m b o l ε p : d s y m b o l ε p = 0 . df=\frac{\partial f}{\partial symbol{\sigma}}:dsymbol{\sigma}+\frac{\partial f% }{\partial symbol{\varepsilon}_{p}}:dsymbol{\varepsilon}_{p}=0\,.
  33. s y m b o l F = s y m b o l F e \cdotsymbol F p symbol{F}=symbol{F}^{e}\cdotsymbol{F}^{p}
  34. s y m b o l l = s y m b o l F ˙ \cdotsymbol F - 1 = ( s y m b o l F ˙ e \cdotsymbol F p + s y m b o l F e s y m b o l F ˙ p ) [ ( s y m b o l F p ) - 1 ( s y m b o l F e ) - 1 ] = s y m b o l F ˙ e ( s y m b o l F e ) - 1 + s y m b o l F e [ s y m b o l F ˙ p ( s y m b o l F p ) - 1 ] ( s y m b o l F e ) - 1 . \begin{aligned}\displaystyle symbol{l}&\displaystyle=\dot{symbol{F}}% \cdotsymbol{F}^{-1}=\left(\dot{symbol{F}}^{e}\cdotsymbol{F}^{p}+symbol{F}^{e}% \cdot\dot{symbol{F}}^{p}\right)\cdot\left[(symbol{F}^{p})^{-1}\cdot(symbol{F}^% {e})^{-1}\right]\\ &\displaystyle=\dot{symbol{F}}^{e}\cdot(symbol{F}^{e})^{-1}+symbol{F}^{e}\cdot% [\dot{symbol{F}}^{p}\cdot(symbol{F}^{p})^{-1}]\cdot(symbol{F}^{e})^{-1}\,.\end% {aligned}
  35. s y m b o l l = s y m b o l l e + s y m b o l F e \cdotsymbol L p ( s y m b o l F e ) - 1 . symbol{l}=symbol{l}^{e}+symbol{F}^{e}\cdotsymbol{L}^{p}\cdot(symbol{F}^{e})^{-% 1}\,.
  36. s y m b o l L p := s y m b o l F ˙ p ( s y m b o l F p ) - 1 symbol{L}^{p}:=\dot{symbol{F}}^{p}\cdot(symbol{F}^{p})^{-1}
  37. s y m b o l D p = 1 2 [ s y m b o l L p + ( s y m b o l L p ) T ] , s y m b o l W p = 1 2 [ s y m b o l L p - ( s y m b o l L p ) T ] . symbol{D}^{p}=\tfrac{1}{2}[symbol{L}^{p}+(symbol{L}^{p})^{T}]~{},~{}~{}symbol{% W}^{p}=\tfrac{1}{2}[symbol{L}^{p}-(symbol{L}^{p})^{T}]\,.
  38. s y m b o l C e := ( s y m b o l F e ) T \cdotsymbol F e . symbol{C}^{e}:=(symbol{F}^{e})^{T}\cdotsymbol{F}^{e}\,.
  39. s y m b o l E e := 1 2 \lnsymbol C e . symbol{E}^{e}:=\tfrac{1}{2}\lnsymbol{C}^{e}\,.
  40. s y m b o l M := 1 2 ( s y m b o l C e \cdotsymbol S + s y m b o l S \cdotsymbol C e ) symbol{M}:=\tfrac{1}{2}(symbol{C}^{e}\cdotsymbol{S}+symbol{S}\cdotsymbol{C}^{e})
  41. s y m b o l M = W s y m b o l E e = J d U d J + 2 μ dev ( s y m b o l E e ) symbol{M}=\frac{\partial W}{\partial symbol{E}^{e}}=J\,\frac{dU}{dJ}+2\mu\,\,% \text{dev}(symbol{E}^{e})
  42. s y m b o l D p = λ ˙ f s y m b o l M . symbol{D}^{p}=\dot{\lambda}\,\frac{\partial f}{\partial symbol{M}}\,.
  43. λ ˙ 0 , f 0 , λ ˙ f = 0 . \dot{\lambda}\geq 0~{},~{}~{}f\leq 0~{},~{}~{}\dot{\lambda}\,f=0\,.
  44. λ ˙ f ˙ = 0 . \dot{\lambda}\,\dot{f}=0\,.

Flower_pollination_algorithm.html

  1. p [ 0 , 1 ] p\in[0,1]
  2. x i t + 1 = x i t + L ( x i t - g * ) x_{i}^{t+1}=x_{i}^{t}+L(x_{i}^{t}-g_{*})
  3. x i t + 1 = x i t + ϵ ( x i t - x k t ) x_{i}^{t+1}=x_{i}^{t}+\epsilon(x_{i}^{t}-x_{k}^{t})
  4. x i t x_{i}^{t}
  5. g * g_{*}
  6. p p
  7. ϵ \epsilon
  8. L L
  9. L 1 s 1 + β , L\sim\frac{1}{s^{1+\beta}},
  10. β \beta
  11. s s
  12. u u
  13. v v
  14. s = u | v | 1 + β , s=\frac{u}{|v|^{1+\beta}},
  15. u N ( 0 , σ 2 ) , v N ( 0 , 1 ) , u\sim N(0,\sigma^{2}),\quad v\sim N(0,1),
  16. σ \sigma
  17. β \beta

Fluid_flow_through_porous_media.html

  1. Q = - κ A μ ( p b - p a ) L Q=\frac{-\kappa A}{\mu}\frac{(p_{b}-p_{a})}{L}
  2. Q Q
  3. κ \kappa
  4. μ \mu
  5. A A
  6. ( p b - p a ) (p_{b}-p_{a})
  7. L L
  8. Q = - κ A μ ( d p d x ) Q=\frac{-\kappa A}{\mu}\left(\frac{dp}{dx}\right)
  9. t t
  10. Δ t \Delta t
  11. x x
  12. Δ x \Delta x
  13. m m
  14. [ A ( x ) ρ ( x ) q ( x ) - A ( x + Δ x ) ρ ( x + Δ x ) q ( x + Δ x ) ] Δ t = m ( t + Δ t ) - m ( t ) . [A(x)\rho(x)q(x)-A(x+\Delta x)\rho(x+\Delta x)q(x+\Delta x)]\Delta t=m(t+% \Delta t)-m(t).
  15. m = ρ V p m=\rho V_{p}
  16. V p V_{p}
  17. x x
  18. x + Δ x x+\Delta x
  19. ρ \rho
  20. m = ρ V p = ρ ϕ V = ρ ϕ A Δ x . m=\rho V_{p}=\rho\phi V=\rho\phi A\Delta x.
  21. ϕ \phi
  22. A Δ x A\Delta x
  23. Δ x \Delta x
  24. 0
  25. - d ( ρ q ) d x = d ( ρ ϕ ) d t ( i ) \frac{-d(\rho q)}{dx}=\frac{d(\rho\phi)}{dt}~{}~{}~{}~{}\to(i)
  26. d ( ρ q ) d x + d ( ρ q ) d y + d ( ρ q ) d z = - d ( ρ ϕ ) d t \frac{d(\rho q)}{dx}+\frac{d(\rho q)}{dy}+\frac{d(\rho q)}{dz}=\frac{-d(\rho% \phi)}{dt}
  27. ρ q \rho q
  28. d ( ρ ϕ ) d t = ρ d ϕ d t + ϕ d ρ d t = ρ d ϕ d P d P d t + ϕ d ρ d P d P d t = ρ ϕ [ 1 ϕ d ϕ d P + 1 ρ d ρ d P ] d P d t = ρ ϕ [ c ϕ + c f ] d P d t ( i i ) \frac{d(\rho\phi)}{dt}=\rho\frac{d\phi}{dt}+\phi\frac{d\rho}{dt}=\rho\frac{d% \phi}{dP}\frac{dP}{dt}+\phi\frac{d\rho}{dP}\frac{dP}{dt}=\rho\phi\left[\frac{1% }{\phi}\frac{d\phi}{dP}+\frac{1}{\rho}\frac{d\rho}{dP}\right]\quad\frac{dP}{dt% }=\rho\phi[c_{\phi}+c_{f}]\frac{dP}{dt}~{}~{}~{}~{}\to(ii)
  29. c f c_{f}
  30. c ϕ c_{\phi}
  31. - d ( ρ q ) d x = - d d x [ - ρ k μ d P d x ] = k μ [ ρ d 2 P d x 2 + d ρ d x d P d x d P d x ] = ρ k μ [ d 2 P d x 2 + ( 1 ρ d ρ d P ) ( d P d x ) 2 ] = ρ k μ [ d 2 P d x 2 + c f ( d P d x ) 2 ] ( i i i ) \frac{-d(\rho q)}{dx}=\frac{-d}{dx}\left[\frac{-\rho k}{\mu}\frac{dP}{dx}% \right]\quad=\frac{k}{\mu}\left[\rho\frac{d^{2}P}{dx^{2}}+\frac{d\rho}{dx}% \frac{dP}{dx}\frac{dP}{dx}\right]\quad=\frac{\rho k}{\mu}\left[\frac{d^{2}P}{% dx^{2}}+\left(\frac{1}{\rho}\frac{d\rho}{dP}\right)\left(\frac{dP}{dx}\right)^% {2}\right]\quad=\frac{\rho k}{\mu}\left[\frac{d^{2}P}{dx^{2}}+c_{f}\left(\frac% {dP}{dx}\right)^{2}\right]\quad~{}~{}~{}~{}\to(iii)
  32. ( i i ) (ii)
  33. ( i i i ) (iii)
  34. d 2 P d x 2 + c f ( d P d x ) 2 = ϕ μ ( c f + c ϕ ) k d P d t \frac{d^{2}P}{dx^{2}}+c_{f}(\frac{dP}{dx})^{2}=\frac{\phi\mu(c_{f}+c_{\phi})}{% k}\frac{dP}{dt}
  35. d P d t = k ϕ μ c t d 2 P d x 2 , \frac{dP}{dt}=\frac{k}{\phi\mu c_{t}}\frac{d^{2}P}{dx^{2}},
  36. c t = c f + c ϕ c_{t}=c_{f}+c_{\phi}

Fluid_thread_breakup.html

  1. ω 2 = σ k ρ a 2 I 1 ( k a ) I 0 ( k a ) ( 1 - k 2 a 2 ) \omega^{2}=\frac{\sigma k}{\rho a^{2}}\frac{I_{1}\left(ka\right)}{I_{0}\left(% ka\right)}\left(1-k^{2}a^{2}\right)
  2. λ m a x = 9.02 a \lambda_{max}=9.02a
  3. k a < 1 ka<1
  4. μ A \mu_{A}
  5. μ B \mu_{B}
  6. ω = σ ( k 2 a 2 - 1 ) 2 a μ A 1 k 2 a 2 + 1 - k 2 a 2 I 0 2 ( k a ) / I 1 2 ( k a ) \omega=\frac{\sigma\left(k^{2}a^{2}-1\right)}{2a\mu_{A}}\frac{1}{k^{2}a^{2}+1-% k^{2}a^{2}I_{0}^{2}\left(ka\right)/I_{1}^{2}\left(ka\right)}
  7. ω = σ ( 1 - k 2 a 2 ) 2 a μ B 1 k 2 a 2 + 1 - k 2 a 2 K 0 2 ( k a ) / K 1 2 ( k a ) \omega=\frac{\sigma\left(1-k^{2}a^{2}\right)}{2a\mu_{B}}\frac{1}{k^{2}a^{2}+1-% k^{2}a^{2}K_{0}^{2}\left(ka\right)/K_{1}^{2}\left(ka\right)}
  8. ω = σ ( 1 - k 2 a 2 ) 2 a μ B Φ ( k a ) \omega=\frac{\sigma\left(1-k^{2}a^{2}\right)}{2a\mu_{B}}\Phi\left(ka\right)
  9. Φ \Phi
  10. Φ ( k a ) = N ( k a ) D ( k a ) \Phi\left(ka\right)=\frac{N\left(ka\right)}{D\left(ka\right)}
  11. N ( k a ) = I 1 ( k a ) Δ 1 - { k a I 0 ( k a ) - I 1 ( k a ) } Δ 2 N\left(ka\right)=I_{1}\left(ka\right)\Delta_{1}-\left\{kaI_{0}\left(ka\right)-% I_{1}\left(ka\right)\right\}\Delta_{2}
  12. D ( k a ) = μ A μ B { k a I 0 ( k a ) - I 1 ( k a ) } Δ 1 - μ A μ B { ( k 2 a 2 + 1 ) I 1 ( k a ) - k a I 0 ( k a ) } Δ 2 - { k a K 0 ( k a ) + K 1 ( k a ) } Δ 3 - { ( k 2 a 2 + 1 ) K 1 ( k a ) + k a K 0 ( k a ) } Δ 4 \begin{aligned}\displaystyle D\left(ka\right)=&\displaystyle\frac{\mu_{A}}{\mu% _{B}}\left\{kaI_{0}\left(ka\right)-I_{1}\left(ka\right)\right\}\Delta_{1}-% \frac{\mu_{A}}{\mu_{B}}\left\{\left(k^{2}a^{2}+1\right)I_{1}\left(ka\right)-% kaI_{0}\left(ka\right)\right\}\Delta_{2}\\ &\displaystyle-\left\{kaK_{0}\left(ka\right)+K_{1}\left(ka\right)\right\}% \Delta_{3}-\left\{\left(k^{2}a^{2}+1\right)K_{1}\left(ka\right)+kaK_{0}\left(% ka\right)\right\}\Delta_{4}\end{aligned}
  13. Δ \Delta
  14. Δ 1 = | k a I 0 ( k a ) - I 1 ( k a ) K 1 ( k a ) - k a K 0 ( k a ) - K 1 ( k a ) I 0 ( k a ) + k a I 1 ( k a ) - K 0 ( k a ) - K 0 ( k a ) + k a K 1 ( k a ) μ A μ B k a I 0 ( k a ) K 1 ( k a ) - k a K 0 ( k a ) | \Delta_{1}=\begin{vmatrix}kaI_{0}\left(ka\right)-I_{1}\left(ka\right)&K_{1}% \left(ka\right)&-kaK_{0}\left(ka\right)-K_{1}\left(ka\right)\\ I_{0}\left(ka\right)+kaI_{1}\left(ka\right)&-K_{0}\left(ka\right)&-K_{0}\left(% ka\right)+kaK_{1}\left(ka\right)\\ \frac{\mu_{A}}{\mu_{B}}kaI_{0}\left(ka\right)&K_{1}\left(ka\right)&-kaK_{0}% \left(ka\right)\end{vmatrix}
  15. Δ 2 = | I 1 ( k a ) K 1 ( k a ) - k a K 0 ( k a ) - K 1 ( k a ) I 0 ( k a ) - K 0 ( k a ) - K 0 ( k a ) + k a K 1 ( k a ) μ A μ B I 1 ( k a ) K 1 ( k a ) - k a K 0 ( k a ) | \Delta_{2}=\begin{vmatrix}I_{1}\left(ka\right)&K_{1}\left(ka\right)&-kaK_{0}% \left(ka\right)-K_{1}\left(ka\right)\\ I_{0}\left(ka\right)&-K_{0}\left(ka\right)&-K_{0}\left(ka\right)+kaK_{1}\left(% ka\right)\\ \frac{\mu_{A}}{\mu_{B}}I_{1}\left(ka\right)&K_{1}\left(ka\right)&-kaK_{0}\left% (ka\right)\end{vmatrix}
  16. Δ 3 = | I 1 ( k a ) k a I 0 ( k a ) - I 1 ( k a ) - k a K 0 ( k a ) - K 1 ( k a ) I 0 ( k a ) I 0 ( k a ) + k a I 1 ( k a ) - K 0 ( k a ) + k a K 1 ( k a ) μ A μ B I 1 ( k a ) μ A μ B k a I 0 ( k a ) - k a K 0 ( k a ) | \Delta_{3}=\begin{vmatrix}I_{1}\left(ka\right)&kaI_{0}\left(ka\right)-I_{1}% \left(ka\right)&-kaK_{0}\left(ka\right)-K_{1}\left(ka\right)\\ I_{0}\left(ka\right)&I_{0}\left(ka\right)+kaI_{1}\left(ka\right)&-K_{0}\left(% ka\right)+kaK_{1}\left(ka\right)\\ \frac{\mu_{A}}{\mu_{B}}I_{1}\left(ka\right)&\frac{\mu_{A}}{\mu_{B}}kaI_{0}% \left(ka\right)&-kaK_{0}\left(ka\right)\end{vmatrix}
  17. Δ 4 = | I 1 ( k a ) k a I 0 ( k a ) - I 1 ( k a ) K 1 ( k a ) I 0 ( k a ) I 0 ( k a ) + k a I 1 ( k a ) - K 0 ( k a ) μ A μ B I 1 ( k a ) μ a μ B k a I 0 ( k a ) K 1 ( k a ) | \Delta_{4}=\begin{vmatrix}I_{1}\left(ka\right)&kaI_{0}\left(ka\right)-I_{1}% \left(ka\right)&K_{1}\left(ka\right)\\ I_{0}\left(ka\right)&I_{0}\left(ka\right)+kaI_{1}\left(ka\right)&-K_{0}\left(% ka\right)\\ \frac{\mu_{A}}{\mu_{B}}I_{1}\left(ka\right)&\frac{\mu_{a}}{\mu_{B}}kaI_{0}% \left(ka\right)&K_{1}\left(ka\right)\end{vmatrix}
  18. μ A / μ B = 0.28 \mu_{A}/\mu_{B}=0.28
  19. λ = 10.66 a \lambda=10.66a

Foias_constant.html

  1. x n + 1 = ( 1 + 1 x n ) n for n = 1 , 2 , 3 , , x_{n+1}=\left(1+\frac{1}{x_{n}}\right)^{n}\,\text{ for }n=1,2,3,\ldots,
  2. α = 1.187452351126501 \alpha=1.187452351126501\ldots\,
  3. lim n x n log n n = 1 , \lim_{n\to\infty}x_{n}\frac{\log n}{n}=1,
  4. lim n x n π ( n ) = 1 , \lim_{n\to\infty}\frac{x_{n}}{\pi(n)}=1,
  5. π \pi

Force_spectrum_microscopy.html

  1. F = k x F=kx
  2. F F
  3. x x
  4. k k
  5. ( F ( v ) ) 2 = ( K ( v ) ) 2 * ( x ( v ) ) 2 \langle\left(F(v)\right)^{2}\rangle=\langle\left(K(v)\right)^{2}\rangle*% \langle\left(x(v)\right)^{2}\rangle
  6. ( F ( v ) ) 2 \langle\left(F(v)\right)^{2}\rangle
  7. ( K ( v ) ) 2 \langle\left(K(v)\right)^{2}\rangle
  8. ( x ( v ) ) 2 \langle\left(x(v)\right)^{2}\rangle
  9. M S D = ( X ( t ) ) 2 MSD=\langle\left(X(t)\right)^{2}\rangle

Formal_ball.html

  1. ( X , d ) (X,d)
  2. + \mathbb{R}^{+}
  3. B + ( X , d ) = X × + B^{+}(X,d)=X\times\mathbb{R}^{+}
  4. B ( X , d ) = X × B(X,d)=X\times\mathbb{R}
  5. \leq
  6. ( x , r ) ( y , s ) (x,r)\leq(y,s)
  7. d ( x , y ) r - s d(x,y)\leq r-s
  8. B ( X , d ) B(X,d)
  9. B + ( X , d ) B^{+}(X,d)
  10. X X

Formulae_of_shapes.html

  1. 3 \sqrt{3}
  2. 3 \sqrt{3}
  3. 3 \sqrt{3}
  4. 4 / 3 {4}/{3}
  5. 1 / 3 {1}/{3}

Foundational_relation.html

  1. ( S ) ( S A S ( x S ) ( S R - 1 { x } = ) ) , (\forall S)\left(S\subseteq A\wedge S\not=\emptyset\Rightarrow(\exists x\in S)% (S\cap R^{-1}\{x\}=\emptyset)\right),
  2. R - 1 { x } = { y | y R x } . R^{-1}\{x\}=\{y|yRx\}.

Four-spiral_semigroup.html

  1. a b ω l d c \begin{matrix}&&\mathcal{R}&&\\ &a&\longleftrightarrow&b&\\ \omega^{l}&\Big\uparrow&&\Big\updownarrow&\mathcal{L}\\ &d&\longleftrightarrow&c&\\ &&\mathcal{R}&&\end{matrix}
  2. ( r , x , y , s ) * ( t , z , w , u ) = { ( r , x - y + max ( y , z + 1 ) , max ( y - 1 , z ) - z + w , u ) if s = 0 , t = 1 ( r , x - y + max ( y , z ) , max ( y , z ) - z + w , u ) otherwise. (r,x,y,s)*(t,z,w,u)=\begin{cases}(r,x-y+\max(y,z+1),\max(y-1,z)-z+w,u)&\,\text% {if }s=0,t=1\\ (r,x-y+\max(y,z),\max(y,z)-z+w,u)&\,\text{otherwise.}\end{cases}

Four_in_a_Bed_(series_6).html

  1. \bigstar\bigstar\bigstar\bigstar\bigstar
  2. \bigstar\bigstar\bigstar\bigstar

Four_point_flexural_test.html

  1. E f E_{f}
  2. σ f \sigma_{f}
  3. ε f \varepsilon_{f}
  4. σ f \sigma_{f}
  5. σ f = 3 F L 4 b d 2 \sigma_{f}=\frac{3FL}{4bd^{2}}
  6. σ f \sigma_{f}
  7. F F
  8. L L
  9. b b
  10. d d

Fractal_tree_index.html

  1. B B
  2. B B
  3. O ( B ) O(B)
  4. O ( log B N ) O(\log_{B}N)
  5. B \sqrt{B}
  6. O ( log B N ) = O ( log B N ) O(\log_{\sqrt{B}}N)=O(\log_{B}N)
  7. B + 1 \sqrt{B}+1
  8. B B + 1 B \frac{B}{\sqrt{B}+1}\approx\sqrt{B}
  9. O ( 1 ) O(1)
  10. O ( 1 B ) O\left(\frac{1}{\sqrt{}}{B}\right)
  11. O ( log B N ) O(\log_{B}N)
  12. O ( 1 B ) O\left(\frac{1}{\sqrt{}}{B}\right)
  13. O ( log B N B ) O\left(\frac{\log_{B}N}{\sqrt{}}{B}\right)
  14. B ε B^{\varepsilon}
  15. O ( 1 B 1 - ε ) O\left(\frac{1}{B^{1-\varepsilon}}\right)
  16. O ( log B N B ) O\left(\frac{\log_{B}N}{\sqrt{}}{B}\right)
  17. O ( log B N ) O(\log_{B}N)
  18. O ( B ) O(\sqrt{B})
  19. B B
  20. O ( 1 B ) O\left(\frac{1}{B}\right)
  21. O ( log B N ) O(\log_{B}N)
  22. B \sqrt{B}
  23. N N
  24. O ( N B ) O\left(\frac{N}{B}\right)
  25. N N
  26. M M
  27. O ( N + M B ) O\left(\frac{N+M}{B}\right)
  28. N N
  29. O ( N B ) O\left(\frac{N}{B}\right)
  30. O ( B ) O(\sqrt{B})
  31. k k
  32. O ( k B ) O\left(\frac{k}{\sqrt{}}{B}\right)
  33. O ( log B N B ) O\left(\frac{\log_{B}N}{\sqrt{}}{B}\right)
  34. i i
  35. O ( log B B i 2 ) = O ( i ) O(\log_{B}B^{\frac{i}{2}})=O(i)
  36. i i
  37. B i 2 B^{\frac{i}{2}}
  38. O ( log B 2 N ) O(\log^{2}_{B}N)

Fractional-order_system.html

  1. H ( D α 1 , α 2 , , α m ) ( y 1 , y 2 , , y l ) = G ( D β 1 , β 2 , , β n ) ( u 1 , u 2 , , u k ) H(D^{\alpha_{1},\alpha_{2},\ldots,\alpha_{m}})(y_{1},y_{2},\ldots,y_{l})=G(D^{% \beta_{1},\beta_{2},\ldots,\beta_{n}})(u_{1},u_{2},\ldots,u_{k})
  2. H H
  3. G G
  4. α 1 , α 2 , , α m \alpha_{1},\alpha_{2},\ldots,\alpha_{m}
  5. β 1 , β 2 , , β n \beta_{1},\beta_{2},\ldots,\beta_{n}
  6. y i y_{i}
  7. u j u_{j}
  8. ( k = 0 m a k D α k ) y ( t ) = ( k = 0 n b k D β k ) u ( t ) \left(\sum_{k=0}^{m}a_{k}D^{\alpha_{k}}\right)y(t)=\left(\sum_{k=0}^{n}b_{k}D^% {\beta_{k}}\right)u(t)
  9. α k \alpha_{k}
  10. β k \beta_{k}
  11. α k , β k = k δ , δ R + \alpha_{k},\beta_{k}=k\delta,\quad\delta\in R^{+}
  12. α k , β k = k δ , δ = 1 q , q Z + \alpha_{k},\beta_{k}=k\delta,\quad\delta=\frac{1}{q},q\in Z^{+}
  13. q = 1 q=1
  14. G ( s ) = Y ( s ) U ( s ) = k = 0 n b k s β k k = 0 m a k s α k G(s)=\frac{Y(s)}{U(s)}=\frac{\sum_{k=0}^{n}b_{k}s^{\beta_{k}}}{\sum_{k=0}^{m}a% _{k}s^{\alpha_{k}}}
  15. α k \alpha_{k}
  16. β k \beta_{k}
  17. σ ( t ) = E D t α ε ( t ) , 0 < α < 1. \sigma(t)=E{D_{t}^{\alpha}}\varepsilon(t),\quad 0<\alpha<1.
  18. D t α 0 C x ( t ) = f ( t , x ( t ) ) , t [ 0 , T ] , x ( 0 ) = x 0 , 0 < α < 1. {{}_{0}^{C}D_{t}^{\alpha}}x(t)=f(t,x(t)),\quad t\in[0,T],\quad x(0)=x_{0},% \quad 0<\alpha<1.
  19. x ( t ) = x 0 + D t - α 0 C f ( t , x ( t ) ) = x 0 + 1 Γ ( α ) 0 t f ( s , x ( s ) ) d s ( t - s ) 1 - α , x(t)=x_{0}+{{}_{0}^{C}D_{t}^{-\alpha}}f(t,x(t))=x_{0}+\frac{1}{\Gamma(\alpha)}% \int_{0}^{t}\frac{f(s,x(s))\,ds}{(t-s)^{1-\alpha}},

Francois-Joseph_Servois.html

  1. ϕ ( x + y + ) = ϕ ( x ) + ϕ ( y ) + \phi(x+y+...)=\phi(x)+\phi(y)+...
  2. ϕ \phi
  3. f z fz
  4. f f
  5. z z
  6. f f

Free_cash_flow_to_equity.html

  1. F C F E = F C F F + N e t B o r r o w i n g - I n t e r e s t * ( 1 - t ) FCFE=FCFF+NetBorrowing-Interest*(1-t)
  2. F C F E = N I + D A - C a p e x - Δ W C + N e t B o r r o w i n g FCFE=NI+DA-Capex-\Delta WC+NetBorrowing

Free_presentation.html

  1. i I R 𝑓 j J R 𝑔 M 0 , \bigoplus_{i\in I}R\overset{f}{\to}\bigoplus_{j\in J}R\overset{g}{\to}M\to 0,
  2. F 𝑔 M 0 F\overset{g}{\to}M\to 0
  3. F 𝑓 ker g 0 F^{\prime}\overset{f}{\to}\ker g\to 0
  4. i I N f 1 j J N M R N 0. \bigoplus_{i\in I}N\overset{f\otimes 1}{\to}\bigoplus_{j\in J}N\to M\otimes_{R% }N\to 0.
  5. M R N M\otimes_{R}N
  6. f 1 f\otimes 1
  7. M R N M\otimes_{R}N
  8. R n R m M R^{\oplus n}\to R^{\oplus m}\to M
  9. 0 F ( M ) F ( R m ) F ( R n ) 0\to F(M)\to F(R^{\oplus m})\to F(R^{\oplus n})
  10. \square

Frenkel_line.html

  1. τ 0 = a 2 6 D \tau_{0}=\frac{a^{2}}{6D}
  2. a a
  3. D D
  4. τ * \tau^{*}
  5. τ 0 τ * \tau_{0}\approx\tau^{*}
  6. 3 k B 3k_{B}
  7. k B k_{B}
  8. 1 k B 1k_{B}
  9. c V = 2 k B c_{V}=2k_{B}

Frequency_selective_surface.html

  1. E = - j k η [ A + 1 k 2 ( A ) ] ( 1.1 ) E~{}=~{}-jk\eta\left[A+\frac{1}{k^{2}}\nabla(\nabla\bullet A)\right]~{}~{}~{}~% {}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(1.1)
  2. 2 A + k 2 A = - J ( 1.2 ) \nabla^{2}A+k^{2}A~{}=~{}-J~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}% ~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(1.2)
  3. k 2 = ω 2 ( μ ϵ ) = 2 π / λ ( 1.3 ) k^{2}~{}=~{}\omega^{2}(\mu\epsilon)=2\pi/\lambda~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}% ~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(1.3)
  4. J ( x , y , z ) = m n J ( α m , β n ) e j ( α m x + β n y ± γ m n z ) ( 2.1 a ) J(x,y,z)~{}=~{}\sum_{mn}~{}J(\alpha_{m},\beta_{n})~{}e^{j(\alpha_{m}x+\beta_{n% }y\pm\gamma_{mn}z)}~{}~{}~{}~{}~{}~{}(2.1a)
  5. E ( x , y , z ) = m n E ( α m , β n ) e j ( α m x + β n y ± γ m n z ) ( 2.1 b ) E(x,y,z)~{}=~{}\sum_{mn}~{}E(\alpha_{m},\beta_{n})~{}e^{j(\alpha_{m}x+\beta_{n% }y\pm\gamma_{mn}z)}~{}~{}~{}~{}~{}(2.1b)
  6. A ( x , y , z ) = m n A ( α m , β n ) e j ( α m x + β n y ± γ m n z ) ( 2.1 c ) A(x,y,z)~{}=~{}\sum_{mn}~{}A(\alpha_{m},\beta_{n})~{}e^{j(\alpha_{m}x+\beta_{n% }y\pm\gamma_{mn}z)}~{}~{}~{}~{}(2.1c)
  7. α m = k sin θ 0 cos ϕ 0 + 2 m π l x ( 2.2 a ) \alpha_{m}~{}=~{}k~{}\sin\theta_{0}~{}\cos\phi_{0}~{}+~{}\frac{2m\pi}{l_{x}}~{% }~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(2.2a)
  8. β n = k sin θ 0 sin ϕ 0 + 2 n π l y ( 2.2 b ) \beta_{n}~{}=~{}k~{}\sin\theta_{0}~{}\sin\phi_{0}~{}+~{}\frac{2n\pi}{l_{y}}~{}% ~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(2.2b)
  9. γ m n = k 2 - α m 2 - β n 2 ( 2.2 c ) \gamma_{mn}~{}=~{}\sqrt{~{}k^{2}~{}-~{}\alpha_{m}^{2}~{}-~{}\beta_{n}^{2}}~{}~% {}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(2.2c)
  10. k = 2 π / λ ( 2.3 ) k~{}=~{}2\pi/\lambda~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~% {}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(2.3)
  11. E ( α m , β n ) = j k η k 2 - α m 2 - β n 2 G m n J ( α m , β n ) ( 3.1 ) E(\alpha_{m},\beta_{n})~{}=~{}\frac{jk\eta}{\sqrt{k^{2}-\alpha_{m}^{2}-\beta_{% n}^{2}}}~{}G_{mn}~{}J(\alpha_{m},\beta_{n})~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{% }~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(3.1)
  12. G m n = ( 1 - α m 2 k 2 - α m β n k 2 - α m β n k 2 1 - β n 2 k 2 ) ( 3.2 ) G_{mn}~{}=~{}\left(\begin{matrix}1-\frac{\alpha_{m}^{2}}{k^{2}}&-\frac{\alpha_% {m}\beta_{n}}{k^{2}}\\ -\frac{\alpha_{m}\beta_{n}}{k^{2}}&1-\frac{\beta_{n}^{2}}{k^{2}}\end{matrix}% \right)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{% }~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{% }~{}~{}~{}~{}~{}(3.2)
  13. m n 1 k 2 - α m 2 - β n 2 G m n p J ( α m , β n ) e j ( α m x + β n y ) = - E i n c ( x , y ) ( 3.3 ) ~{}\sum_{mn}~{}\frac{1}{\sqrt{}}{k^{2}-\alpha_{m}^{2}-\beta_{n}^{2}}~{}G_{mnp}% ~{}J(\alpha_{m},\beta_{n})~{}e^{j(\alpha_{m}x+\beta_{n}y)}~{}=~{}-E^{inc}(x,y)% ~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(3.3)
  14. E i n c ( x , y ) = E i n c ( α 0 , β 0 ) e j ( α 0 x + β 0 y ) ( 3.4 ) ~{}E^{inc}(x,y)=~{}E^{inc}(\alpha_{0},\beta_{0})~{}e^{j(\alpha_{0}x+\beta_{0}y% )}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~% {}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(3.4)
  15. J ( x , y , z ) = j J j J j ( x , y , z ) ( 4.1 ) ~{}J(x,y,z)~{}=~{}\sum_{j}~{}J_{j}~{}J_{j}(x,y,z)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{% }~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{% }~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{% }~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{% }~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(4.1)
  16. j J j [ m n J i ( - α m , - β n ) G m n J j ( α m , β n ) k 2 - α m 2 - β n 2 ] = - J i ( - α 0 , - β 0 ) E i n c ( α 0 , β 0 ) ( 4.2 ) ~{}\sum_{j}~{}J_{j}~{}\left[~{}\sum_{mn}~{}\frac{J_{i}(-\alpha_{m},-\beta_{n})% ~{}G_{mn}~{}J_{j}(\alpha_{m},\beta_{n})}{\sqrt{}}{k^{2}-\alpha_{m}^{2}-\beta_{% n}^{2}}\right]~{}=~{}-J_{i}(-\alpha_{0},-\beta_{0})~{}\bullet~{}E^{inc}(\alpha% _{0},\beta_{0})~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(4.2)
  17. e ± j k z ~{}e^{\pm jkz}
  18. Z 0 = μ 0 ϵ 0 ~{}Z_{0}~{}=~{}\sqrt{\frac{\mu_{0}}{\epsilon_{0}}}
  19. k 0 = ω μ 0 ϵ 0 ~{}k_{0}~{}=~{}\omega\sqrt{\mu_{0}\epsilon_{0}}
  20. Z 0 = μ 0 ϵ 0 / cos θ ~{}Z_{0}~{}=~{}\sqrt{\frac{\mu_{0}}{\epsilon_{0}}}~{}/~{}\cos\theta
  21. k 0 = cos θ ω μ 0 ϵ 0 ~{}k_{0}~{}=~{}\cos\theta~{}\omega\sqrt{\mu_{0}\epsilon_{0}}
  22. Z 0 = μ 0 ϵ 0 cos θ ~{}Z_{0}~{}=~{}\sqrt{\frac{\mu_{0}}{\epsilon_{0}}}~{}\cos\theta
  23. k 0 = cos θ ω μ 0 ϵ 0 ~{}k_{0}~{}=~{}\cos\theta~{}\omega\sqrt{\mu_{0}\epsilon_{0}}
  24. Z shunt = j ω L + 1 j ω C Z\text{shunt}~{}=~{}j\omega L~{}+~{}\frac{1}{j\omega C}~{}~{}~{}~{}~{}~{}~{}~{% }~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}
  25. Z shunt = 1 - ω 2 L C j ω C Z\text{shunt}~{}=~{}\frac{1-\omega^{2}LC}{j\omega C}~{}~{}~{}~{}~{}~{}~{}~{}~{% }~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}
  26. ω 0 2 = 1 L C \omega_{0}^{2}~{}=~{}\frac{1}{LC}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}% ~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}
  27. | R | = ω C Z 0 ω 2 C 2 Z 0 2 + 4 ( 1 - ω 2 L C ) 2 |R|=\frac{\omega CZ_{0}}{\sqrt{\omega^{2}C^{2}Z_{0}^{2}~{}+~{}4~{}(1~{}-~{}% \omega^{2}LC)^{2}}}
  28. ω 1 , 2 2 = - b ± b 2 - 4 a c 2 a \omega^{2}_{1,2}~{}=~{}\frac{-b\pm\sqrt{b^{2}~{}-~{}4ac}}{2a}
  29. a = ( L C ) 2 a~{}=~{}(LC)^{2}
  30. b = - ( C 2 Z 0 2 / 4 + 2 L C ) b~{}=~{}-(C^{2}Z_{0}^{2}~{}/~{}4~{}+~{}2LC)
  31. c = 1 c~{}=~{}1
  32. Y shunt = 1 - ω 2 L C j ω L Y\text{shunt}~{}=~{}\frac{1-\omega^{2}LC}{j\omega L}~{}~{}~{}~{}~{}~{}~{}~{}~{% }~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}
  33. ω 0 2 = 1 L C \omega_{0}^{2}~{}=~{}\frac{1}{LC}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}% ~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}
  34. | T | = 4 ω L / Z 0 4 ω 2 L 2 / Z 0 2 + 4 ( 1 - ω 2 L C ) 2 |T|=\frac{4\omega L/Z_{0}}{\sqrt{4\omega^{2}L^{2}/Z_{0}^{2}~{}+~{}4~{}(1~{}-~{% }\omega^{2}LC)^{2}}}
  35. L 2 = C 1 Z 0 2 4 L_{2}~{}=~{}C_{1}~{}\frac{Z_{0}^{2}}{4}
  36. C 2 = L 1 4 Z 0 2 C_{2}~{}=~{}L_{1}~{}\frac{4}{Z_{0}^{2}}

Frequent_subtree_mining.html

  1. 𝒞 \mathcal{C}
  2. P T P\preceq T
  3. P , T 𝒞 P,T\in\mathcal{C}
  4. 𝒟 𝒞 \mathcal{D}\subseteq\mathcal{C}
  5. 𝒫 𝒞 \mathcal{P}\subset\mathcal{C}
  6. 𝒫 \mathcal{P}
  7. P 𝒫 : freq ( P , 𝒟 ) = T 𝒟 d ( P , T ) minfreq , \forall P\in\mathcal{P}:\quad\mathrm{freq}(P,\mathcal{D})=\sum\nolimits_{T\in% \mathcal{D}}d(P,T)\geq\mathrm{minfreq},
  8. d d
  9. P P P^{\prime}\preceq P
  10. T 𝒞 : d ( P , T ) d ( P , T ) . \forall T\in\mathcal{C}:\quad d(P^{\prime},T)\geq d(P,T).

Friedelin_synthase.html

  1. \rightleftharpoons

Fujita–Storm_equation.html

  1. u t = a ( u - 2 u x ) x u_{t}=a(u^{-2}u_{x})_{x}

Fulkerson–Chen–Anstee_theorem.html

  1. ( ( a 1 , b 1 ) , , ( a n , b n ) ) ((a_{1},b_{1}),\ldots,(a_{n},b_{n}))
  2. a 1 a n a_{1}\geq\cdots\geq a_{n}
  3. ( ( a 1 , b 1 ) , , ( a n , b n ) ) ((a_{1},b_{1}),\ldots,(a_{n},b_{n}))
  4. a 1 a n a_{1}\geq\cdots\geq a_{n}
  5. i = 1 n a i = i = 1 n b i \sum_{i=1}^{n}a_{i}=\sum_{i=1}^{n}b_{i}
  6. 1 k n 1\leq k\leq n
  7. i = 1 k a i i = 1 k min ( b i , k - 1 ) + i = k + 1 n min ( b i , k ) \sum^{k}_{i=1}a_{i}\leq\sum^{k}_{i=1}\min(b_{i},k-1)+\sum^{n}_{i=k+1}\min(b_{i% },k)
  8. k k
  9. 1 k < n 1\leq k<n
  10. a k > a k + 1 a_{k}>a_{k+1}
  11. k = n k=n
  12. ( a 1 , , a n ) (a_{1},\ldots,a_{n})
  13. ( b 1 , , b n ) (b_{1},\ldots,b_{n})
  14. ( a 1 * , , a n * ) (a^{*}_{1},\ldots,a^{*}_{n})
  15. a k * := | { b i | i > k , b i k } | + | { b i | i k , b i k - 1 } | a^{*}_{k}:=|\{b_{i}|i>k,b_{i}\geq k\}|+|\{b_{i}|i\leq k,b_{i}\geq k-1\}|
  16. ( a 1 * , , a n * ) (a^{*}_{1},\ldots,a^{*}_{n})
  17. ( a 1 , , a n ) (a_{1},\ldots,a_{n})
  18. ( b 1 , , b n ) (b_{1},\ldots,b_{n})
  19. ( a 1 * , , a n * ) (a^{*}_{1},\ldots,a^{*}_{n})
  20. n n
  21. a a
  22. b b
  23. a * a^{*}
  24. i = 1 k a i * = i = 1 k min ( b i , k - 1 ) + i = k + 1 n min ( b i , k ) \sum_{i=1}^{k}a^{*}_{i}=\sum^{k}_{i=1}\min(b_{i},k-1)+\sum^{n}_{i=k+1}\min(b_{% i},k)
  25. ( a , b ) (a,b)
  26. a a
  27. a * a^{*}
  28. a a
  29. ( ( a 1 , b 1 ) , , ( a n , b n ) ) ((a_{1},b_{1}),\ldots,(a_{n},b_{n}))
  30. a 1 a n a_{1}\geq\cdots\geq a_{n}
  31. i = 1 n a i = i = 1 n b i \sum_{i=1}^{n}a_{i}=\sum_{i=1}^{n}b_{i}
  32. c c
  33. ( c , b ) (c,b)
  34. c c
  35. a a

Function_of_several_real_variables.html

  1. n n
  2. n n
  3. X X
  4. n n
  5. f : X f:X\rightarrow\mathbb{R}
  6. X X
  7. X X
  8. n n
  9. n n
  10. 𝐱 \mathbf{x}
  11. [ u u n d e r l i n e , u x ] [u^{\prime}underline^{\prime},u^{\prime}x^{\prime}]
  12. [ u o v e r s e t , u 192 , u x ] [u^{\prime}overset^{\prime},u^{\prime}\u{2}192^{\prime},u^{\prime}x^{\prime}]
  13. V : X V:X\rightarrow\mathbb{R}
  14. X = { ( A , h ) 2 : 0 A , h < } X=\{(A,h)\in\mathbb{R}^{2}\,:\,0\leq A,h<\infty\}
  15. V ( A , h ) = 1 3 A h V(A,h)=\frac{1}{3}Ah
  16. V V
  17. A A
  18. h h
  19. z : 2 z:\mathbb{R}^{2}\rightarrow\mathbb{R}
  20. z ( x , y ) = a x + b y z(x,y)=ax+by
  21. a a
  22. b b
  23. a a
  24. b b
  25. ( x , y ) (x,y)
  26. z : p z:\mathbb{R}^{p}\rightarrow\mathbb{R}
  27. z ( x 1 , x 2 , , x p ) = a 1 x 1 + a 2 x 2 + + a p x p z(x_{1},x_{2},\ldots,x_{p})=a_{1}x_{1}+a_{2}x_{2}+\cdots+a_{p}x_{p}
  28. p p
  29. p p
  30. f ( s y m b o l x ) = s y m b o l x = x 1 2 + + x n 2 f(symbol{x})=\|symbol{x}\|=\sqrt{x_{1}^{2}+\cdots+x_{n}^{2}}
  31. g ( s y m b o l x ) = 1 f ( s y m b o l x ) g(symbol{x})=\frac{1}{f(symbol{x})}
  32. 𝐱 ( 0 , 0 , , 0 ) \mathbf{x}≠(0,0,...,0)
  33. z : X z:X\rightarrow\mathbb{R}
  34. X = { ( x , y ) 2 : x 2 + y 2 8 , x 0 , y 0 } X=\{(x,y)\in\mathbb{R}^{2}\,:\,x^{2}+y^{2}\leq 8\,,\,x\neq 0\,,\,y\neq 0\}
  35. z ( x , y ) = 1 2 x y x 2 + y 2 z(x,y)=\frac{1}{2xy}\sqrt{x^{2}+y^{2}}
  36. X X
  37. 8 \sqrt{8}
  38. ( x , y ) = ( 0 , 0 ) (x,y)=(0,0)
  39. ( x , y ) = ( 0 , 0 ) (x,y)=(0,0)
  40. f f
  41. ( x , y ) = ( 2 , 3 ) (x,y)=(2,\sqrt{3})
  42. X X
  43. z ( 2 , 3 ) = 1 2 2 3 ( 2 ) 2 + ( 3 ) 2 = 1 4 3 7 , z(2,\sqrt{3})=\frac{1}{2\cdot 2\cdot\sqrt{3}}\sqrt{(2)^{2}+(\sqrt{3})^{2}}=% \frac{1}{4\sqrt{3}}\sqrt{7}\,,
  44. ( x , y ) = ( 65 , 10 ) x 2 + y 2 = ( 65 ) 2 + ( 10 ) 2 > 8 (x,y)=(65,\sqrt{10})\,\Rightarrow\,x^{2}+y^{2}=(65)^{2}+(\sqrt{10})^{2}>8
  45. x x
  46. y y
  47. f f
  48. n n
  49. f f
  50. y y
  51. X X
  52. f f
  53. Y X Y⊂X
  54. f f
  55. Y Y
  56. f f
  57. f f
  58. f ( 𝟎 ) = 0 f(\mathbf{0})=0
  59. B B
  60. 𝟎 = ( 0 , , 0 ) \mathbf{0}=(0,...,0)
  61. g ( s y m b o l x ) = 1 1 + f ( s y m b o l x ) g(symbol{x})=\frac{1}{1+f(symbol{x})}
  62. h ( s y m b o l x ) = 1 + f ( s y m b o l x ) h(symbol{x})=\sqrt{1+f(symbol{x})}
  63. C C
  64. 𝟎 \mathbf{0}
  65. C C
  66. f f
  67. f f
  68. f f
  69. C C
  70. r r
  71. ( x 1 , , x n ) r (x_{1},\ldots,x_{n})\mapsto r
  72. r r
  73. f f
  74. r f : ( x 1 , , x n ) r f ( x 1 , , x n ) rf:(x_{1},\ldots,x_{n})\mapsto rf(x_{1},\ldots,x_{n})
  75. f f
  76. r = 0 r=0
  77. f f
  78. g g
  79. X X
  80. Y Y
  81. X Y X∩Y
  82. f g : ( x 1 , , x n ) f ( x 1 , , x n ) g ( x 1 , , x n ) f\,g:(x_{1},\ldots,x_{n})\mapsto f(x_{1},\ldots,x_{n})\,g(x_{1},\ldots,x_{n})
  83. f g : ( x 1 , , x n ) f ( x 1 , , x n ) g ( x 1 , , x n ) f\,g:(x_{1},\ldots,x_{n})\mapsto f(x_{1},\ldots,x_{n})\,g(x_{1},\ldots,x_{n})
  84. X Y X∩Y
  85. n n
  86. n n
  87. 1 / f : ( x 1 , , x n ) 1 / f ( x 1 , , x n ) , 1/f:(x_{1},\ldots,x_{n})\mapsto 1/f(x_{1},\ldots,x_{n}),
  88. f f
  89. f f
  90. x f ( x , a 2 , , a n ) , x\mapsto f(x,a_{2},\ldots,a_{n}),
  91. f f
  92. i = 2 , , n i=2,...,n
  93. f f
  94. x f ( a 1 + c 1 x , a 2 + c 2 x , , a n + c n x ) , x\mapsto f(a_{1}+c_{1}x,a_{2}+c_{2}x,\ldots,a_{n}+c_{n}x),
  95. 2 n 2n
  96. d ( s y m b o l x , s y m b o l y ) = d ( x 1 , , x n , y 1 , , y n ) = ( x 1 - y 1 ) 2 + + ( x n - y n ) 2 d(symbol{x},symbol{y})=d(x_{1},\ldots,x_{n},y_{1},\ldots,y_{n})=\sqrt{(x_{1}-y% _{1})^{2}+\cdots+(x_{n}-y_{n})^{2}}
  97. f f
  98. ε ε
  99. φ φ
  100. x x
  101. φ φ
  102. f f
  103. φ φ
  104. 𝐚 \mathbf{a}
  105. 2 ε
  106. f ( 𝐚 ) f(\mathbf{a})
  107. f ( 𝐚 ) f(\mathbf{a})
  108. f ( 𝐚 ) f(\mathbf{a})
  109. f ( 𝐚 ) f(\mathbf{a})
  110. f f
  111. f ( 0 , 0 ) = 0 f(0,0)=0
  112. f ( x , y ) = x 2 y x 4 + y 2 . f(x,y)=\frac{x^{2}y}{x^{4}+y^{2}}.
  113. x f ( x , 0 ) x↦f(x,0)
  114. y f ( 0 , y ) y↦f(0,y)
  115. f f
  116. ( 0 , 0 ) (0,0)
  117. f ( x , y ) = 1 / 2 f(x,y)=1/2
  118. | x | |x|
  119. ( 0 , 0 ) (0,0)
  120. f ( x , λ x ) = λ x x 2 + λ 2 f(x,\lambda x)=\frac{\lambda x}{x^{2}+\lambda^{2}}
  121. λ 0 λ≠0
  122. X X
  123. f f
  124. f f
  125. L L
  126. 𝐱 \mathbf{x}
  127. 𝐚 \mathbf{a}
  128. L = lim s y m b o l x s y m b o l a f ( s y m b o l x ) , L=\lim_{symbol{x}\rightarrow symbol{a}}f(symbol{x}),
  129. ε > 0 ε>0
  130. δ > 0 δ>0
  131. | f ( s y m b o l x ) - L | < ε |f(symbol{x})-L|<\varepsilon
  132. 𝐱 \mathbf{x}
  133. d ( s y m b o l x , s y m b o l a ) < δ . d(symbol{x},symbol{a})<\delta.
  134. 𝐚 \mathbf{a}
  135. 𝐚 \mathbf{a}
  136. f ( s y m b o l a ) = lim s y m b o l x s y m b o l a f ( s y m b o l x ) . f(symbol{a})=\lim_{symbol{x}\rightarrow symbol{a}}f(symbol{x}).
  137. 𝐚 \mathbf{a}
  138. f f
  139. f f
  140. 𝐚 \mathbf{a}
  141. f f
  142. 𝐚 \mathbf{a}
  143. f f
  144. f ( , x i , , x j , ) = f ( , x j , , x i , ) f(\ldots,x_{i},\ldots,x_{j},\ldots)=f(\ldots,x_{j},\ldots,x_{i},\ldots)
  145. i i
  146. j j
  147. 1 , 2 , , n 1,2,...,n
  148. f ( x , y , z , t ) = t 2 - x 2 - y 2 - z 2 f(x,y,z,t)=t^{2}-x^{2}-y^{2}-z^{2}
  149. x , y , z x,y,z
  150. x , y , z x,y,z
  151. f f
  152. x , y , z , t x,y,z,t
  153. t t
  154. x x
  155. y y
  156. z z
  157. ξ 1 = ξ 1 ( x 1 , x 2 , , x n ) , ξ 2 = ξ 2 ( x 1 , x 2 , , x n ) , ξ m = ξ m ( x 1 , x 2 , , x n ) , \xi_{1}=\xi_{1}(x_{1},x_{2},\ldots,x_{n}),\quad\xi_{2}=\xi_{2}(x_{1},x_{2},% \ldots,x_{n}),\ldots\xi_{m}=\xi_{m}(x_{1},x_{2},\ldots,x_{n}),
  158. ξ = ξ ( 𝐱 ) \mathbf{ξ}=\mathbf{ξ}(\mathbf{x})
  159. X X
  160. n n
  161. X X
  162. m m
  163. Ξ Ξ
  164. s y m b o l ξ : X Ξ symbol{\xi}:X\rightarrow\Xi
  165. ζ ζ
  166. ξ ( 𝐱 ) \mathbf{ξ}(\mathbf{x})
  167. Ξ Ξ
  168. ζ : Ξ \zeta:\Xi\rightarrow\mathbb{R}
  169. ζ = ζ ( ξ 1 , ξ 2 , , ξ m ) \zeta=\zeta(\xi_{1},\xi_{2},\ldots,\xi_{m})
  170. X X
  171. ζ : X \zeta:X\rightarrow\mathbb{R}
  172. ζ = ζ ( ξ 1 , ξ 2 , , ξ m ) = f ( x 1 , x 2 , , x n ) , \zeta=\zeta(\xi_{1},\xi_{2},\ldots,\xi_{m})=f(x_{1},x_{2},\ldots,x_{n}),
  173. m m
  174. n n
  175. f ( x , y ) = e x y [ sin 3 ( x - y ) - cos 2 ( x + y ) ] f(x,y)=e^{xy}[\sin 3(x-y)-\cos 2(x+y)]
  176. ( α , β , γ ) = ( α ( x , y ) , β ( x , y ) , γ ( x , y ) ) = ( x y , x - y , x + y ) (\alpha,\beta,\gamma)=(\alpha(x,y),\beta(x,y),\gamma(x,y))=(xy,x-y,x+y)
  177. f ( x , y ) = ζ ( α ( x , y ) , β ( x , y ) , γ ( x , y ) ) = ζ ( α , β , γ ) = e α [ sin ( 3 β ) - cos ( 2 γ ) ] . f(x,y)=\zeta(\alpha(x,y),\beta(x,y),\gamma(x,y))=\zeta(\alpha,\beta,\gamma)=e^% {\alpha}[\sin(3\beta)-\cos(2\gamma)]\,.
  178. x 1 f ( x 1 , x 2 , , x n ) , x 2 f ( x 1 , x 2 , x n ) , , x n f ( x 1 , x 2 , , x n ) . \frac{\partial}{\partial x_{1}}f(x_{1},x_{2},\ldots,x_{n})\,,\quad\frac{% \partial}{\partial x_{2}}f(x_{1},x_{2},\ldots x_{n})\,,\ldots,\frac{\partial}{% \partial x_{n}}f(x_{1},x_{2},\ldots,x_{n}).
  179. f f
  180. y = f ( x ) y=f(x)
  181. d y / d x dy/dx
  182. y = f ( x ) y=f(x)
  183. 2 x 1 2 f ( x 1 , x 2 , , x n ) , 2 x 1 x 2 f ( x 1 , x 2 , x n ) , , 2 x n 2 f ( x 1 , x 2 , , x n ) . \frac{\partial^{2}}{\partial x^{2}_{1}}f(x_{1},x_{2},\ldots,x_{n})\,,\quad% \frac{\partial^{2}}{\partial x_{1}x_{2}}f(x_{1},x_{2},\ldots x_{n})\,,\ldots,% \frac{\partial^{2}}{\partial x^{2}_{n}}f(x_{1},x_{2},\ldots,x_{n}).
  184. p p
  185. p x 1 p 1 x 2 p 2 x n p n f ( x 1 , x 2 , , x n ) p 1 x 1 p 1 p 2 x 2 p 2 p n x n p n f ( x 1 , x 2 , , x n ) \frac{\partial^{p}}{\partial x_{1}^{p_{1}}\partial x_{2}^{p_{2}}\ldots\partial x% _{n}^{p_{n}}}f(x_{1},x_{2},\ldots,x_{n})\equiv\frac{\partial^{p_{1}}}{\partial x% _{1}^{p_{1}}}\frac{\partial^{p_{2}}}{\partial x_{2}^{p_{2}}}\cdots\frac{% \partial^{p_{n}}}{\partial x_{n}^{p_{n}}}f(x_{1},x_{2},\ldots,x_{n})
  186. 0
  187. p p
  188. 0 x 1 0 f ( x 1 , x 2 , , x n ) = f ( x 1 , x 2 , , x n ) , 0 x n 0 f ( x 1 , x 2 , , x n ) = f ( x 1 , x 2 , , x n ) . \frac{\partial^{0}}{\partial x_{1}^{0}}f(x_{1},x_{2},\ldots,x_{n})=f(x_{1},x_{% 2},\ldots,x_{n})\,,\quad\ldots\frac{\partial^{0}}{\partial x_{n}^{0}}f(x_{1},x% _{2},\ldots,x_{n})=f(x_{1},x_{2},\ldots,x_{n})\,.
  189. p p
  190. p p
  191. f ( 𝐱 ) f(\mathbf{x})
  192. 𝐚 \mathbf{a}
  193. n n
  194. 𝐚 \mathbf{a}
  195. f ( s y m b o l x ) = f ( s y m b o l a ) + s y m b o l A ( s y m b o l a ) ( s y m b o l x - s y m b o l a ) + α ( s y m b o l x ) | s y m b o l x - s y m b o l a | f(symbol{x})=f(symbol{a})+symbol{A}(symbol{a})\cdot(symbol{x}-symbol{a})+% \alpha(symbolx)|symbol{x}-symbol{a}|
  196. α 0 α→0
  197. | 𝐱 𝐚 | 0 |\mathbf{x}−\mathbf{a}|→0
  198. f f
  199. 𝐚 \mathbf{a}
  200. f f
  201. 𝐱 = 𝐚 \mathbf{x}=\mathbf{a}
  202. f f
  203. 𝐚 \mathbf{a}
  204. 𝐚 \mathbf{a}
  205. f ( s y m b o l x ) x i | s y m b o l x = s y m b o l a = A i ( s y m b o l a ) \left.\frac{\partial f(symbol{x})}{\partial x_{i}}\right|_{symbol{x}=symbol{a}% }=A_{i}(symbol{a})
  206. i = 1 , 2 , , n i=1,2,...,n
  207. f f
  208. n n
  209. f ( s y m b o l x ) = ( x 1 , x 2 , , x n ) f ( s y m b o l x ) \nabla f(symbol{x})=\left(\frac{\partial}{\partial x_{1}},\frac{\partial}{% \partial x_{2}},\ldots,\frac{\partial}{\partial x_{n}}\right)f(symbol{x})
  210. f ∇f
  211. 𝐱 = 𝐚 ) \mathbf{x}=\mathbf{a})
  212. f ( s y m b o l x ) - f ( s y m b o l a ) = f ( s y m b o l a ) ( s y m b o l x - s y m b o l a ) + α | s y m b o l x - s y m b o l a | f(symbol{x})-f(symbol{a})=\nabla f(symbol{a})\cdot(symbol{x}-symbol{a})+\alpha% |symbol{x}-symbol{a}|
  213. · ·
  214. f f
  215. 𝐱 \mathbf{x}
  216. 𝐚 \mathbf{a}
  217. f f
  218. 𝐱 \mathbf{x}
  219. 𝐱 𝐚 \mathbf{x}→\mathbf{a}
  220. d f = f ( s y m b o l x ) x 1 | s y m b o l x = s y m b o l a d x 1 + f ( s y m b o l x ) x 2 | s y m b o l x = s y m b o l a d x 2 + f ( s y m b o l x ) x n | s y m b o l x = s y m b o l a d x n = f ( s y m b o l a ) d s y m b o l x df=\left.\frac{\partial f(symbol{x})}{\partial x_{1}}\right|_{symbol{x}=symbol% {a}}dx_{1}+\left.\frac{\partial f(symbol{x})}{\partial x_{2}}\right|_{symbol{x% }=symbol{a}}dx_{2}+\cdots\left.\frac{\partial f(symbol{x})}{\partial x_{n}}% \right|_{symbol{x}=symbol{a}}dx_{n}=\nabla f(symbol{a})\cdot dsymbol{x}
  221. f f
  222. 𝐚 \mathbf{a}
  223. f f
  224. f f
  225. d f df
  226. f f
  227. f ∇f
  228. f f
  229. f ( 𝐱 ) = c f(\mathbf{x})=c
  230. c c
  231. ( n 1 ) (n−1)
  232. d f = ( f ) d s y m b o l x = 0 df=(\nabla f)\cdot dsymbol{x}=0
  233. d 𝐱 d\mathbf{x}
  234. 𝐱 \mathbf{x}
  235. f ( 𝐱 ) = c f(\mathbf{x})=c
  236. f ∇f
  237. d 𝐱 d\mathbf{x}
  238. f ∇f
  239. d 𝐱 d\mathbf{x}
  240. n n
  241. 𝐚 \mathbf{a}
  242. x 1 f ( s y m b o l x ) | s y m b o l x = s y m b o l a , x 2 f ( s y m b o l x ) | s y m b o l x = s y m b o l a , , x n f ( s y m b o l x ) | s y m b o l x = s y m b o l a \left.\frac{\partial}{\partial x_{1}}f(symbol{x})\right|_{symbol{x}=symbol{a}}% \,,\quad\left.\frac{\partial}{\partial x_{2}}f(symbol{x})\right|_{symbol{x}=% symbol{a}}\,,\ldots,\left.\frac{\partial}{\partial x_{n}}f(symbol{x})\right|_{% symbol{x}=symbol{a}}
  243. 𝐚 \mathbf{a}
  244. f f
  245. p p
  246. 𝐚 \mathbf{a}
  247. p x 1 p 1 x 2 p 2 x n p n f ( s y m b o l x ) | s y m b o l x = s y m b o l a \left.\frac{\partial^{p}}{\partial x_{1}^{p_{1}}\partial x_{2}^{p_{2}}\ldots% \partial x_{n}^{p_{n}}}f(symbol{x})\right|_{symbol{x}=symbol{a}}
  248. p p
  249. 𝐚 \mathbf{a}
  250. f f
  251. p p
  252. f f
  253. f f
  254. f f
  255. R n R 2 R 1 f ( x 1 , x 2 , , x n ) d x 1 d x 2 d x n R f ( s y m b o l x ) d n s y m b o l x \int_{R_{n}}\cdots\int_{R_{2}}\int_{R_{1}}f(x_{1},x_{2},\ldots,x_{n})\,dx_{1}% dx_{2}\cdots dx_{n}\equiv\int_{R}f(symbol{x})\,d^{n}symbol{x}
  256. R 1 , R 2 , , R n , R_{1}\subseteq\mathbb{R}\,,\quad R_{2}\subseteq\mathbb{R}\,,\ldots,R_{n}% \subseteq\mathbb{R},
  257. R = R 1 × R 2 × R n , R n , R=R_{1}\times R_{2}\times\ldots R_{n}\,,\quad R\subseteq\mathbb{R}^{n}\,,
  258. n n
  259. R R
  260. y = f ( x ) y=f(x)
  261. x x
  262. y = f ( x ) y=f(x)
  263. x x
  264. n n
  265. n n
  266. f ( 𝐱 ) f(\mathbf{x})
  267. f f
  268. 𝐱 \mathbf{x}
  269. R R
  270. R R
  271. y 1 = f 1 ( x 1 , x 2 , , x n ) , y 2 = f 2 ( x 1 , x 2 , , x n ) , , y m = f m ( x 1 , x 2 , x n ) y_{1}=f_{1}(x_{1},x_{2},\ldots,x_{n})\,,\quad y_{2}=f_{2}(x_{1},x_{2},\ldots,x% _{n})\,,\ldots,y_{m}=f_{m}(x_{1},x_{2},\cdots x_{n})
  272. m m
  273. ( y 1 , y 2 , , y m ) [ f 1 ( x 1 , x 2 , , x n ) f 2 ( x 1 , x 2 , x n ) f m ( x 1 , x 2 , , x n ) ] [ f 1 ( x 1 , x 2 , , x n ) f 2 ( x 1 , x 2 , , x n ) f m ( x 1 , x 2 , , x n ) ] (y_{1},y_{2},\ldots,y_{m})\leftrightarrow\begin{bmatrix}f_{1}(x_{1},x_{2},% \ldots,x_{n})\\ f_{2}(x_{1},x_{2},\cdots x_{n})\\ \vdots\\ f_{m}(x_{1},x_{2},\ldots,x_{n})\end{bmatrix}\leftrightarrow\begin{bmatrix}f_{1% }(x_{1},x_{2},\ldots,x_{n})&f_{2}(x_{1},x_{2},\ldots,x_{n})&\cdots&f_{m}(x_{1}% ,x_{2},\ldots,x_{n})\end{bmatrix}
  274. m m
  275. 𝐲 = 𝐟 ( 𝐱 ) \mathbf{y}=\mathbf{f}(\mathbf{x})
  276. y = f ( ) y=f(...)
  277. ϕ : n + 1 { 0 } \phi:\mathbb{R}^{n+1}\rightarrow\{0\}
  278. ϕ ( x 1 , x 2 , , x n , y ) = 0 \phi(x_{1},x_{2},\ldots,x_{n},y)=0
  279. y = f ( x 1 , x 2 , , x n ) y=f(x_{1},x_{2},\ldots,x_{n})
  280. ϕ ( x 1 , x 2 , , x n , y ) = y - f ( x 1 , x 2 , , x n ) = 0 \phi(x_{1},x_{2},\ldots,x_{n},y)=y-f(x_{1},x_{2},\ldots,x_{n})=0
  281. ϕ : X { 0 } \phi:X\rightarrow\{0\}
  282. ϕ ( x , y , z ) = ( x a ) 2 + ( y b ) 2 + ( z c ) 2 - 1 = 0 \phi(x,y,z)=\left(\frac{x}{a}\right)^{2}+\left(\frac{y}{b}\right)^{2}+\left(% \frac{z}{c}\right)^{2}-1=0
  283. X = [ - a , a ] × [ - b , b ] × [ - c , c ] = { ( x , y , z ) 3 : - a x a , - b y b , - c z c } . X=[-a,a]\times[-b,b]\times[-c,c]=\{(x,y,z)\in\mathbb{R}^{3}\,:\,-a\leq x\leq a% ,-b\leq y\leq b,-c\leq z\leq c\}\,.
  284. ( x , y , z ) = ( 0 , 0 , 0 ) (x,y,z)=(0,0,0)
  285. a , b , c a,b,c
  286. a = b = c = r a=b=c=r
  287. r r
  288. x x
  289. y y
  290. z z
  291. ϕ : 4 { 0 } \phi:\mathbb{R}^{4}\rightarrow\{0\}
  292. ϕ ( t , x , y , z ) = C t z e t x - y z + A sin ( 3 ω t ) ( x 2 z - B y 6 ) = 0 \phi(t,x,y,z)=Ctze^{tx-yz}+A\sin(3\omega t)\left(x^{2}z-By^{6}\right)=0
  293. A , B , C , ω A,B,C,ω
  294. ( t , x , y , z ) (t,x,y,z)
  295. t = t=
  296. x = x=
  297. ϕ ( s y m b o l a , b ) = 0 ; \phi(symbol{a},b)=0;
  298. ϕ ϕ
  299. y y
  300. ( 𝐚 , b ) (\mathbf{a},b)
  301. ϕ ( s y m b o l x , y ) y | ( s y m b o l x , y ) = ( s y m b o l a , b ) 0. \left.\frac{\partial\phi(symbol{x},y)}{\partial y}\right|_{(symbol{x},y)=(% symbol{a},b)}\neq 0.
  302. b b
  303. R R
  304. ( 𝐚 , b ) (\mathbf{a},b)
  305. 𝐱 \mathbf{x}
  306. R R
  307. y y
  308. ϕ ( 𝐱 , y ) = 0 ϕ(\mathbf{x},y)=0
  309. y y
  310. 𝐱 \mathbf{x}
  311. ϕ ( 𝐱 , y ( 𝐱 ) ) = 0 ϕ(\mathbf{x},y(\mathbf{x}))=0
  312. d y = y x 1 d x 1 + y x 2 d x 2 + + y x n d x n ; dy=\frac{\partial y}{\partial x_{1}}dx_{1}+\frac{\partial y}{\partial x_{2}}dx% _{2}+\ldots+\frac{\partial y}{\partial x_{n}}dx_{n};
  313. d ϕ = ϕ x 1 d x 1 + ϕ x 2 d x 2 + + ϕ x n d x n + ϕ y d y . d\phi=\frac{\partial\phi}{\partial x_{1}}dx_{1}+\frac{\partial\phi}{\partial x% _{2}}dx_{2}+\ldots+\frac{\partial\phi}{\partial x_{n}}dx_{n}+\frac{\partial% \phi}{\partial y}dy.
  314. d y dy
  315. y y
  316. ϕ x i + ϕ y y x i = 0 \frac{\partial\phi}{\partial x_{i}}+\frac{\partial\phi}{\partial y}\frac{% \partial y}{\partial x_{i}}=0
  317. i = 1 , 2 , , n i=1,2,...,n
  318. f ( x 1 , , x n ) = g ( x 1 , , x n ) + i h ( x 1 , , x n ) , f(x_{1},\ldots,x_{n})=g(x_{1},\ldots,x_{n})+ih(x_{1},\ldots,x_{n}),
  319. g g
  320. h h
  321. z ( x , y , α , a , q ) = q 2 π [ ln ( x + i y - a e i α ) - ln ( x + i y + a e - i α ) ] z(x,y,\alpha,a,q)=\frac{q}{2\pi}\left[\ln(x+iy-ae^{i\alpha})-\ln(x+iy+ae^{-i% \alpha})\right]
  322. ρ ρ
  323. 𝐫 = ( x , y , z ) \mathbf{r}=(x,y,z)
  324. t t
  325. ρ = ρ ( 𝐫 , t ) = ρ ( x , y , z , t ) \rho=\rho(\mathbf{r},t)=\rho(x,y,z,t)
  326. 𝐯 ( 𝐫 , t ) = 𝐯 ( x , y , z , t ) = [ v x ( x , y , z , t ) , v y ( x , y , z , t ) , v z ( x , y , z , t ) ] \mathbf{v}(\mathbf{r},t)=\mathbf{v}(x,y,z,t)=[v_{x}(x,y,z,t),v_{y}(x,y,z,t),v_% {z}(x,y,z,t)]
  327. P P
  328. T T
  329. V V
  330. f ( P , V , T ) = 0 f(P,V,T)=0
  331. f ( P , V , T ) = P V - n R T = 0 f(P,V,T)=PV-nRT=0
  332. n n
  333. R R
  334. F ( x , y , ) = φ ( x , y , ) + i ψ ( x , y , ) F(x,y,\ldots)=\varphi(x,y,\ldots)+i\psi(x,y,\ldots)
  335. x x
  336. y y
  337. Y m = Y m ( θ , ϕ ) Y^{m}_{\ell}=Y^{m}_{\ell}(\theta,\phi)
  338. t t
  339. Ψ = Ψ ( 𝐫 , t ) = Ψ ( x , y , z , t ) , Φ = Φ ( 𝐩 , t ) = Φ ( p x , p y , p z , t ) \Psi=\Psi(\mathbf{r},t)=\Psi(x,y,z,t)\,,\quad\Phi=\Phi(\mathbf{p},t)=\Phi(p_{x% },p_{y},p_{z},t)

Functional_notation.html

  1. f ( x ) f(x)
  2. f - 1 ( x ) f^{-1}(x)

Functional_principal_component_analysis.html

  1. μ ( t ) = E ( X ( t ) ) \mu(t)=\,\text{E}(X(t))
  2. G ( s , t ) = Cov ( X ( s ) , X ( t ) ) = k = 1 λ k φ k ( s ) φ k ( t ) , G(s,t)=\,\text{Cov}(X(s),X(t))=\sum_{k=1}^{\infty}\lambda_{k}\varphi_{k}(s)% \varphi_{k}(t),
  3. G : L 2 ( 𝒯 ) L 2 ( 𝒯 ) , G ( f ) = 𝒯 G ( s , t ) f ( s ) d s . G:L^{2}(\mathcal{T})\rightarrow L^{2}(\mathcal{T}),\,G(f)=\int_{\mathcal{T}}G(% s,t)f(s)ds.
  4. X ( t ) - μ ( t ) = k = 1 ξ k φ k ( t ) , X(t)-\mu(t)=\sum_{k=1}^{\infty}\xi_{k}\varphi_{k}(t),
  5. ξ k = 𝒯 ( X ( t ) - μ ( t ) ) φ k ( t ) d t \xi_{k}=\int_{\mathcal{T}}(X(t)-\mu(t))\varphi_{k}(t)dt
  6. E ( ξ k ) = 0 , Var ( ξ k ) = λ k and E ( ξ k ξ l ) = 0 for k l . \,\text{E}(\xi_{k})=0,\,\text{Var}(\xi_{k})=\lambda_{k}\,\text{ and }\,\text{E% }(\xi_{k}\xi_{l})=0\,\text{ for }k\neq l.
  7. X ( t ) X m ( t ) = μ ( t ) + k = 1 m ξ k φ k ( t ) , X(t)\approx X_{m}(t)=\mu(t)+\sum_{k=1}^{m}\xi_{k}\varphi_{k}(t),
  8. E ( 𝒯 ( X ( t ) - X m ( t ) ) 2 d t ) = j > m λ j 0 as m . \mathrm{E}\left(\int_{\mathcal{T}}\left(X(t)-X_{m}(t)\right)^{2}dt\right)=\sum% _{j>m}\lambda_{j}\rightarrow 0\,\text{ as }m\rightarrow\infty.
  9. φ 1 = arg max φ = 1 { Var ( 𝒯 ( X ( t ) - μ ( t ) ) φ ( t ) d t ) } , \varphi_{1}=\underset{\|\mathbf{\varphi}\|=1}{\operatorname{arg\,max}}\left\{% \operatorname{Var}(\int_{\mathcal{T}}(X(t)-\mu(t))\varphi(t)dt)\right\},
  10. φ = ( 𝒯 φ ( t ) 2 d t ) 1 2 . \|\mathbf{\varphi}\|=\left(\int_{\mathcal{T}}\varphi(t)^{2}dt\right)^{\frac{1}% {2}}.
  11. φ k = arg max φ = 1 , φ , φ j = 0 for j = 1 , , k - 1 { Var ( 𝒯 ( X ( t ) - μ ( t ) ) φ ( t ) d t ) } , \varphi_{k}=\underset{\|\mathbf{\varphi}\|=1,\langle\varphi,\varphi_{j}\rangle% =0\,\text{ for }j=1,\dots,k-1}{\operatorname{arg\,max}}\left\{\operatorname{% Var}(\int_{\mathcal{T}}(X(t)-\mu(t))\varphi(t)dt)\right\},
  12. φ , φ j = 𝒯 φ ( t ) φ j ( t ) d t , for j = 1 , , k - 1. \langle\varphi,\varphi_{j}\rangle=\int_{\mathcal{T}}\varphi(t)\varphi_{j}(t)dt% ,\,\text{ for }j=1,\dots,k-1.
  13. μ ^ ( t i j ) = 1 n i = 1 n Y i j . \hat{\mu}(t_{ij})=\frac{1}{n}\sum_{i=1}^{n}Y_{ij}.
  14. G ^ ( s , t ) \hat{G}(s,t)
  15. G i ( t i j , t i l ) = ( Y i j - μ ^ ( t i j ) ) ( Y i l - μ ^ ( t i l ) ) , j l , i = 1 , , n . G_{i}(t_{ij},t_{il})=(Y_{ij}-\hat{\mu}(t_{ij}))(Y_{il}-\hat{\mu}(t_{il})),j% \neq l,i=1,\dots,n.
  16. G ^ ( s , t ) \hat{G}(s,t)
  17. φ ^ k \hat{\varphi}_{k}
  18. v k ^ . \hat{v_{k}}.
  19. G ~ ( s , t ) = λ k > 0 λ ^ k φ ^ k ( s ) φ ^ k ( t ) . \tilde{G}(s,t)=\sum_{\lambda_{k}>0}\hat{\lambda}_{k}\hat{\varphi}_{k}(s)\hat{% \varphi}_{k}(t).
  20. V ^ ( t ) \hat{V}(t)
  21. V ^ ( t ) \hat{V}(t)
  22. σ ^ 2 = 2 | 𝒯 | 𝒯 ( V ^ ( t ) - G ~ ( t , t ) ) d t , \hat{\sigma}^{2}=\frac{2}{|\mathcal{T}|}\int_{\mathcal{T}}(\hat{V}(t)-\tilde{G% }(t,t))dt,
  23. σ ^ 2 > 0 ; \hat{\sigma}^{2}>0;
  24. σ ^ 2 = 0. \hat{\sigma}^{2}=0.
  25. ξ ^ k = X - μ ^ , φ ^ k . \hat{\xi}_{k}=\langle X-\hat{\mu},\hat{\varphi}_{k}\rangle.
  26. ξ ^ k = λ ^ k φ ^ k T Σ ^ Y i - 1 ( Y i - μ ^ ) , \hat{\xi}_{k}=\hat{\lambda}_{k}\hat{\varphi}_{k}^{T}\hat{\Sigma}_{Y_{i}}^{-1}(% Y_{i}-\hat{\mu}),
  27. Σ ^ Y i = G ~ + σ ^ 2 𝐈 m i \hat{\Sigma}_{Y_{i}}=\tilde{G}+\hat{\sigma}^{2}\mathbf{I}_{m_{i}}
  28. G ~ \tilde{G}
  29. X p X\in\mathbb{R}^{p}
  30. X L 2 ( 𝒯 ) X\in L^{2}(\mathcal{T})
  31. p < p<\infty
  32. \infty
  33. μ = E ( X ) \mu=\,\text{E}(X)
  34. μ ( t ) = E ( X ( t ) ) \mu(t)=\,\text{E}(X(t))
  35. Cov ( X ) = Σ p × p \,\text{Cov}(X)=\Sigma_{p\times p}
  36. Cov ( X ( s ) , X ( t ) ) = G ( s , t ) \,\text{Cov}(X(s),X(t))=G(s,t)
  37. λ 1 , λ 2 , , λ p \lambda_{1},\lambda_{2},\dots,\lambda_{p}
  38. λ 1 , λ 2 , \lambda_{1},\lambda_{2},\dots
  39. 𝐯 1 , 𝐯 2 , , 𝐯 p \mathbf{v}_{1},\mathbf{v}_{2},\dots,\mathbf{v}_{p}
  40. φ 1 ( t ) , φ 2 ( t ) , \varphi_{1}(t),\varphi_{2}(t),\dots
  41. 𝐗 , 𝐘 = k = 1 p X k Y k \langle\mathbf{X},\mathbf{Y}\rangle=\sum_{k=1}^{p}X_{k}Y_{k}
  42. X , Y = 𝒯 X ( t ) Y ( t ) d t \langle X,Y\rangle=\int_{\mathcal{T}}X(t)Y(t)dt
  43. z k = X - μ , 𝐯 𝐤 , k = 1 , 2 , , p z_{k}=\langle X-\mu,\mathbf{v_{k}}\rangle,k=1,2,\dots,p
  44. ξ k = X - μ , φ k , k = 1 , 2 , \xi_{k}=\langle X-\mu,\varphi_{k}\rangle,k=1,2,\dots

Fundamental_theorem_of_algebraic_K-theory.html

  1. R [ t ] R[t]
  2. R [ t , t - 1 ] R[t,t^{-1}]
  3. K 0 , K 1 K_{0},K_{1}
  4. G i ( R ) G_{i}(R)
  5. G i ( R ) = π i ( B + f-gen-Mod R ) G_{i}(R)=\pi_{i}(B^{+}\,\text{f-gen-Mod}_{R})
  6. B + = Ω B Q B^{+}=\Omega BQ
  7. G i ( R ) = K i ( R ) , G_{i}(R)=K_{i}(R),
  8. G i ( R [ t ] ) = G i ( R ) , i 0 G_{i}(R[t])=G_{i}(R),\,i\geq 0
  9. G i ( R [ t , t - 1 ] ) = G i ( R ) G i - 1 ( R ) , i 0 , G - 1 ( R ) = 0 G_{i}(R[t,t^{-1}])=G_{i}(R)\oplus G_{i-1}(R),\,i\geq 0,\,G_{-1}(R)=0
  10. K i K_{i}
  11. K i ( R ) = π i ( B + proj-Mod R ) , i 0 K_{i}(R)=\pi_{i}(B^{+}\,\text{proj-Mod}_{R}),\,i\geq 0

Funnelsort.html

  1. N N
  2. Z Z
  3. L L
  4. O ( N L log Z N ) O(\frac{N}{L}\log_{Z}N)
  5. Z = Ω ( L 2 ) Z=\Omega(L^{2})
  6. Θ ( N log N ) \Theta(N\log N)
  7. N N
  8. N 1 / 3 N^{1/3}
  9. N 2 / 3 N^{2/3}
  10. N 1 / 3 N^{1/3}
  11. N 1 / 3 N^{1/3}
  12. k k
  13. k 3 k^{3}
  14. N 1 / 3 N^{1/3}
  15. N 1 / 3 N^{1/3}
  16. N 2 / 3 N^{2/3}
  17. k \sqrt{k}
  18. k \sqrt{k}
  19. k \sqrt{k}
  20. I 1 , I 2 , , I k I_{1},I_{2},\ldots,I_{\sqrt{k}}
  21. k \sqrt{k}
  22. O O
  23. k \sqrt{k}
  24. k \sqrt{k}
  25. 2 k 3 / 2 2k^{3/2}
  26. k \sqrt{k}
  27. O O
  28. O O
  29. k 3 / 2 k^{3/2}
  30. k 3 / 2 k^{3/2}
  31. k 3 k^{3}
  32. k 3 / 2 k^{3/2}
  33. I i I_{i}
  34. k 3 / 2 k^{3/2}
  35. k 3 / 2 k^{3/2}
  36. k 3 k^{3}
  37. O ( k 2 ) O(k^{2})
  38. S ( k ) S(k)
  39. k 1 / 2 k^{1/2}
  40. 2 k 3 / 2 2k^{3/2}
  41. O ( k 2 ) O(k^{2})
  42. k + 1 \sqrt{k}+1
  43. ( k + 1 ) S ( k ) (\sqrt{k}+1)S(\sqrt{k})
  44. S ( k ) = ( k + 1 ) S ( k ) + O ( k 2 ) . S(k)=(\sqrt{k}+1)S(\sqrt{k})+O(k^{2}).
  45. S ( k ) = O ( k 2 ) . S(k)=O(k^{2}).
  46. α \alpha
  47. α Z \alpha\sqrt{Z}
  48. Q M ( k ) Q_{M}(k)
  49. Q M ( k ) = O ( ( k 3 log Z k ) / L ) . Q_{M}(k)=O((k^{3}\log_{Z}k)/L).
  50. k α Z k\leq\alpha\sqrt{Z}
  51. k \sqrt{k}
  52. k 3 / 2 k^{3/2}
  53. k 3 / 2 + 2 k k^{3/2}+2\sqrt{k}
  54. 2 k 3 / 2 + 2 k 2k^{3/2}+2\sqrt{k}
  55. k \sqrt{k}
  56. k 3 / 2 k^{3/2}
  57. k 2 k^{2}
  58. Q M ( k ) ( 2 k 3 / 2 + 2 k ) Q M ( k ) + k 2 Q_{M}(k)\leq(2k^{3/2}+2\sqrt{k})Q_{M}(\sqrt{k})+k^{2}
  59. Q ( N ) Q(N)
  60. Q ( N ) = N 1 / 3 Q ( N 2 / 3 ) + Q M ( N 1 / 3 ) . Q(N)=N^{1/3}Q(N^{2/3})+Q_{M}(N^{1/3}).
  61. Q ( N ) = O ( ( N / L ) log Z N ) . Q(N)=O((N/L)\log_{Z}N).

Furstenberg_boundary.html

  1. D = { z ; | z | < 1 } D=\{z;|z|<1\}
  2. f ( z ) = 1 2 π 0 2 π f ^ ( e i θ ) P ( z , e i θ ) d θ f(z)=\frac{1}{2\pi}\int_{0}^{2\pi}\hat{f}(e^{i\theta})P(z,e^{i\theta})\,d\theta
  3. F ( g ) = f ( g ( 0 ) ) F(g)=f(g(0))
  4. F ( g ) = | z | = 1 f ^ ( g z ) d m ( z ) . F(g)=\int_{|z|=1}\hat{f}(gz)\,dm(z).
  5. f ( g ) = G f ( g g ) d μ ( g ) f(g)=\int_{G}f(gg^{\prime})\,d\mu(g^{\prime})
  6. f ( g ) = Π f ^ ( g p ) d ν ( p ) f(g)=\int_{\Pi}\hat{f}(gp)\,d\nu(p)
  7. f ^ \hat{f}

Fusicocca-2,10(14)-diene_synthase.html

  1. \rightleftharpoons

Fuss–Catalan_number.html

  1. A m ( p , r ) r m p + r ( m p + r m ) = r m ! i = 1 m - 1 ( m p + r - i ) = r Γ ( m p + r ) Γ ( 1 + m ) Γ ( m ( p - 1 ) + r + 1 ) . A_{m}(p,r)\equiv\frac{r}{mp+r}{\left({{mp+r}\atop{m}}\right)}=\frac{r}{m!}% \prod_{i=1}^{m-1}(mp+r-i)=r\frac{\Gamma(mp+r)}{\Gamma(1+m)\Gamma(m(p-1)+r+1)}.
  2. r r
  3. ( 2 m m ) {\textstyle\left({{2m}\atop{m}}\right)}
  4. ( 6 3 ) {\textstyle\left({{6}\atop{3}}\right)}
  5. 1 2 m + 1 ( 2 m m ) \tfrac{1}{2m+1}{\textstyle\left({{2m}\atop{m}}\right)}
  6. 1 4 ( 6 3 ) \tfrac{1}{4}{\textstyle\left({{6}\atop{3}}\right)}
  7. m m + 1 ( 2 m m ) \tfrac{m}{m+1}{\textstyle\left({{2m}\atop{m}}\right)}
  8. 3 4 ( 6 3 ) \tfrac{3}{4}{\textstyle\left({{6}\atop{3}}\right)}
  9. ( 2 m - 2 m - 1 ) {\textstyle\left({{2m-2}\atop{m-1}}\right)}
  10. ( 4 2 ) {\textstyle\left({{4}\atop{2}}\right)}
  11. A 0 ( p , r ) = 1 A_{0}(p,r)=1
  12. A 1 ( p , r ) = r A_{1}(p,r)=r
  13. A 2 ( p , 1 ) = p A_{2}(p,1)=p
  14. p = 0 p=0
  15. A m ( 0 , r ) < m t p l ( r m ) A_{m}(0,r)<mtpl>{{=}}{\left({{r}\atop{m}}\right)}
  16. A m ( 0 , 1 ) = 1 , 1 A_{m}(0,1)=1,1
  17. A m ( 0 , 2 ) = 1 , 2 , 1 A_{m}(0,2)=1,2,1
  18. A m ( 0 , 3 ) = 1 , 3 , 3 , 1 A_{m}(0,3)=1,3,3,1
  19. A m ( 0 , 4 ) = 1 , 4 , 6 , 4 , 1 A_{m}(0,4)=1,4,6,4,1
  20. p = 1 p=1
  21. A m ( 1 , 1 ) = 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , A_{m}(1,1)=1,1,1,1,1,1,1,1,1,1,\ldots
  22. A m ( 1 , 2 ) = 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , A_{m}(1,2)=1,2,3,4,5,6,7,8,9,10,\ldots
  23. A m ( 1 , 3 ) = 1 , 3 , 6 , 10 , 15 , 21 , 28 , 35 , 45 , 55 , A_{m}(1,3)=1,3,6,10,15,21,28,35,45,55,\ldots
  24. A m ( 1 , 4 ) = 1 , 4 , 10 , 20 , 35 , 56 , 84 , 120 , 165 , 220 , A_{m}(1,4)=1,4,10,20,35,56,84,120,165,220,\ldots
  25. A m ( 1 , 5 ) = 1 , 5 , 15 , 35 , 70 , 126 , 210 , 330 , 495 , 715 , A_{m}(1,5)=1,5,15,35,70,126,210,330,495,715,\ldots
  26. A m ( 1 , 6 ) = 1 , 6 , 21 , 56 , 126 , 252 , 462 , 792 , 1287 , 2002 , A_{m}(1,6)=1,6,21,56,126,252,462,792,1287,2002,\ldots
  27. m 0 m\geq 0
  28. p = 2 p=2
  29. A m ( 2 , 1 ) = 1 , 1 , 2 , 5 , 14 , 42 , 132 , 429 , 1430 , 4862 , A_{m}(2,1)=1,1,2,5,14,42,132,429,1430,4862,\ldots
  30. A m ( 2 , 2 ) = 1 , 2 , 5 , 14 , 42 , 132 , 429 , 1430 , 4862 , 16796 , = A m + 1 ( 2 , 1 ) A_{m}(2,2)=1,2,5,14,42,132,429,1430,4862,16796,\ldots=A_{m+1}(2,1)
  31. A m ( 2 , 3 ) = 1 , 3 , 9 , 28 , 90 , 297 , 1001 , 3432 , 11934 , 41990 , A_{m}(2,3)=1,3,9,28,90,297,1001,3432,11934,41990,\ldots
  32. A m ( 2 , 4 ) = 1 , 4 , 14 , 48 , 165 , 572 , 2002 , 7072 , 25194 , 90440 , A_{m}(2,4)=1,4,14,48,165,572,2002,7072,25194,90440,\ldots
  33. p = 3 p=3
  34. A m ( 3 , 1 ) = 1 , 1 , 3 , 12 , 55 , 273 , 1428 , 7752 , 43263 , 246675 , A_{m}(3,1)=1,1,3,12,55,273,1428,7752,43263,246675,\ldots
  35. A m ( 3 , 2 ) = 1 , 2 , 7 , 30 , 143 , 728 , 3876 , 21318 , 120175 , 690690 , A_{m}(3,2)=1,2,7,30,143,728,3876,21318,120175,690690,\ldots
  36. A m ( 3 , 3 ) = 1 , 3 , 12 , 55 , 273 , 1428 , 7752 , 43263 , 246675 , 1430715 , = A m + 1 ( 3 , 1 ) A_{m}(3,3)=1,3,12,55,273,1428,7752,43263,246675,1430715,\ldots=A_{m+1}(3,1)
  37. A m ( 3 , 4 ) = 1 , 4 , 18 , 88 , 455 , 2448 , 13566 , 76912 , 444015 , 2601300 , A_{m}(3,4)=1,4,18,88,455,2448,13566,76912,444015,2601300,\ldots
  38. p = 4 p=4
  39. A m ( 4 , 1 ) = 1 , 1 , 4 , 22 , 140 , 969 , 7084 , 53820 , 420732 , 3362260 , A_{m}(4,1)=1,1,4,22,140,969,7084,53820,420732,3362260,\ldots
  40. A m ( 4 , 2 ) = 1 , 2 , 9 , 52 , 340 , 2394 , 17710 , 135720 , 1068012 , 8579560 , A_{m}(4,2)=1,2,9,52,340,2394,17710,135720,1068012,8579560,\ldots
  41. A m ( 4 , 3 ) = 1 , 3 , 15 , 91 , 612 , 4389 , 32890 , 254475 , 2017356 , 16301164 , A_{m}(4,3)=1,3,15,91,612,4389,32890,254475,2017356,16301164,\ldots
  42. A m ( 4 , 4 ) = 1 , 4 , 22 , 140 , 969 , 7084 , 53820 , 420732 , 3362260 , 27343888 , = A m + 1 ( 4 , 1 ) A_{m}(4,4)=1,4,22,140,969,7084,53820,420732,3362260,27343888,\ldots=A_{m+1}(4,1)
  43. A m ( p , r ) = A m ( p , r - 1 ) + A m - 1 ( p , p + r - 1 ) A_{m}(p,r)=A_{m}(p,r-1)+A_{m-1}(p,p+r-1)
  44. A m ( p , 0 ) = 0 A_{m}(p,0)=0
  45. A 0 ( p , r ) = 1 A_{0}(p,r)=1
  46. p p
  47. A m ( p , s + r ) = k = 0 m A k ( p , r ) A m - k ( p , s ) A_{m}(p,s+r)=\sum_{k=0}^{m}A_{k}(p,r)A_{m-k}(p,s)
  48. m m
  49. B p , r ( z ) := m = 0 A m ( p , r ) z m B_{p,r}(z):=\sum_{m=0}^{\infty}A_{m}(p,r)z^{m}
  50. r r
  51. A m ( p , p ) = A m + 1 ( p , 1 ) A_{m}(p,p)=A_{m+1}(p,1)
  52. B p , 1 ( z ) = 1 + z B p , p ( z ) B_{p,1}(z)=1+zB_{p,p}(z)
  53. B p , 1 ( z ) r = B p , r ( z ) B_{p,1}(z)^{r}=B_{p,r}(z)
  54. B p , r ( z ) = [ 1 + z B p , r ( z ) p / r ] r B_{p,r}(z)=[1+zB_{p,r}(z)^{p/r}]^{r}
  55. A m ( p , r ) r m p + r ( m p + r m ) = r m ( p - 1 ) + r ( m p + r - 1 m ) = r m ( m p + r - 1 m - 1 ) A_{m}(p,r)\equiv\frac{r}{mp+r}{\left({{mp+r}\atop{m}}\right)}=\frac{r}{m(p-1)+% r}{\left({{mp+r-1}\atop{m}}\right)}=\frac{r}{m}{\left({{mp+r-1}\atop{m-1}}% \right)}

Fuzzy_cold_dark_matter.html

  1. 10 - 22 \approx 10^{-22}

G-domain.html

  1. D D
  2. D D
  3. D D
  4. u u
  5. I I
  6. u n I u^{n}\in I
  7. n n
  8. I R I\subset R
  9. R / I R/I
  10. T R T\supset R
  11. T T
  12. R R
  13. T T
  14. R R

G-measure.html

  1. μ \mu
  2. G = ( G n ) n = 1 G=\left(G_{n}\right)_{n=1}^{\infty}
  3. G n ( t ) = k = 1 n ( 1 + r cos ( 2 π m k t ) ) G_{n}(t)=\prod_{k=1}^{n}\left(1+r\cos(2\pi m^{k}t)\right)
  4. - 1 < r < 1 , m -1<r<1,m\in\mathbb{N}
  5. 𝕋 \mathbb{T}
  6. f C ( 𝕋 ) f\in C(\mathbb{T})
  7. f d μ = lim n f ( t ) k = 1 n ( 1 + r cos ( 2 π m k t ) ) d t = lim n f ( t ) G n ( t ) d t \int f\,d\mu=\lim_{n\to\infty}\int f(t)\prod_{k=1}^{n}\left(1+r\cos(2\pi m^{k}% t)\right)\,dt=\lim_{n\to\infty}\int f(t)G_{n}(t)\,dt
  8. d t dt
  9. S ( x ) = m x mod 1 S(x)=mx\,\bmod\,1
  10. k = 1 ( 1 + r k cos ( 2 π m 1 m 2 m k t ) ) \prod_{k=1}^{\infty}\left(1+r_{k}\cos(2\pi m_{1}m_{2}\cdots m_{k}t)\right)
  11. - 1 < r k < 1 , m k , m k 3 -1<r_{k}<1,m_{k}\in\mathbb{N},m_{k}\geq 3

G-prior.html

  1. ( x 1 , y 1 ) , , ( x n , y n ) (x_{1},y_{1}),\ldots,(x_{n},y_{n})
  2. x i x_{i}
  3. y i y_{i}
  4. y i = x i β + ε i . y_{i}=x_{i}^{\top}\beta+\varepsilon_{i}.
  5. ε i \varepsilon_{i}
  6. β \beta
  7. β \beta
  8. ε i \varepsilon_{i}
  9. ψ - 1 \psi^{-1}
  10. X X
  11. i i
  12. x i x_{i}^{\top}
  13. β \beta
  14. β 0 \beta_{0}
  15. ψ ( X X ) - 1 \psi(X^{\top}X)^{-1}
  16. β MVN [ β 0 , g ψ ( X X ) - 1 ] . \beta\sim\,\text{MVN}[\beta_{0},g\psi(X^{\top}X)^{-1}].

G-spectrum.html

  1. X h G X^{hG}
  2. X G X h G , X^{G}\to X^{hG},
  3. X h G X^{hG}
  4. F ( B G + , X ) G F(BG_{+},X)^{G}
  5. / 2 \mathbb{Z}/2
  6. K U h / 2 = K O KU^{h\mathbb{Z}/2}=KO
  7. X h G X h G X_{hG}\to X^{hG}
  8. A B A g G A A\otimes_{B}A\to\prod_{g\in G}A
  9. x y ( g ( x ) y ) x\otimes y\mapsto(g(x)y)

Gabor_wavelet.html

  1. f ( x ) f(x)
  2. ( Δ x ) 2 = - ( x - μ ) 2 f ( x ) f * ( x ) d x - f ( x ) f * ( x ) d x (\Delta x)^{2}=\frac{\int_{-\infty}^{\infty}(x-\mu)^{2}f(x)f^{*}(x)\,dx}{\int_% {-\infty}^{\infty}f(x)f^{*}(x)\,dx}
  3. f * ( x ) f^{*}(x)
  4. f ( x ) f(x)
  5. μ \mu
  6. μ = - x f ( x ) f * ( x ) d x - f ( x ) f * ( x ) d x \mu=\frac{\int_{-\infty}^{\infty}xf(x)f^{*}(x)\,dx}{\int_{-\infty}^{\infty}f(x% )f^{*}(x)\,dx}
  7. ( Δ k ) 2 = - ( k - k 0 ) 2 F ( k ) F * ( k ) d k - F ( k ) F * ( k ) d k (\Delta k)^{2}=\frac{\int_{-\infty}^{\infty}(k-k_{0})^{2}F(k)F^{*}(k)\,dk}{% \int_{-\infty}^{\infty}F(k)F^{*}(k)\,dk}
  8. k 0 k_{0}
  9. f ( x ) f(x)
  10. F ( x ) F(x)
  11. k 0 = - k F ( k ) F * ( k ) d k - F ( k ) F * ( k ) d k k_{0}=\frac{\int_{-\infty}^{\infty}kF(k)F^{*}(k)\,dk}{\int_{-\infty}^{\infty}F% (k)F^{*}(k)\,dk}
  12. ( Δ x ) ( Δ k ) (\Delta x)(\Delta k)
  13. 1 2 \frac{1}{2}
  14. ( Δ x ) (\Delta x)
  15. ( Δ k ) \hbar(\Delta k)
  16. f ( x ) f(x)
  17. f ( x ) = e - ( x - x 0 ) 2 / a 2 e - i k 0 ( x - x 0 ) f(x)=e^{-(x-x_{0})^{2}/a^{2}}e^{-ik_{0}(x-x_{0})}
  18. sin \sin
  19. cos \cos
  20. x 0 x_{0}
  21. a a
  22. k 0 k_{0}
  23. F ( k ) = e - ( k - k 0 ) 2 a 2 e - i x 0 ( k - k 0 ) F(k)=e^{-(k-k_{0})^{2}a^{2}}e^{-ix_{0}(k-k_{0})}

Gaelic_football_rankings.html

  1. P E X = C R O , D * C w t * S c w t PEX=CR_{O,D}*C_{wt}*Sc_{wt}
  2. C R O = 1 - Δ R T 15 ; C R D = - Δ R T 15 CR_{O}=1-\frac{\Delta RT}{15};CR_{D}=-\frac{\Delta RT}{15}
  3. Δ R T = P r e v i o u s R T A - P r e v i o u s R T B \Delta RT=PreviousRT_{A}-PreviousRT_{B}
  4. P E X PEX
  5. C R O CR_{O}
  6. C R D CR_{D}
  7. C w t C_{wt}
  8. S c w t Sc_{wt}
  9. Δ R T \Delta RT
  10. P r e v i o u s R T A PreviousRT_{A}
  11. P r e v i o u s R T B PreviousRT_{B}
  12. P E X = ( 1 - Δ R T 15 ) * C w t * S c w t = ( 1 - ( 73 + 3 ) - 70 15 ) * 1.5 * 1 = 0.6 * 1.5 * 1 = 0.9 PEX=(1-\frac{\Delta RT}{15})*C_{wt}*Sc_{wt}=(1-\frac{(73+3)-70}{15})*1.5*1=0.6% *1.5*1=0.9
  13. P E X = ( 1 - Δ R T 15 ) * C w t * S c w t = ( 1 - 84 - 96 15 ) * 3 * 1 = 1.8 * 3 * 1 = 4.8 PEX=(1-\frac{\Delta RT}{15})*C_{wt}*Sc_{wt}=(1-\frac{84-96}{15})*3*1=1.8*3*1=4.8
  14. P E X = - Δ R T 15 * C w t * S c w t = - 76 - ( 64 + 3 ) 15 * 3 * 1 = - 0.6 * 3 * 1 ) = - 1.8 PEX=-\frac{\Delta RT}{15}*C_{wt}*Sc_{wt}=-\frac{76-(64+3)}{15}*3*1=-0.6*3*1)=-% 1.8

Galactose_oxidase.html

  1. \rightleftharpoons

Gale–Ryser_theorem.html

  1. ( a 1 , , a n ) (a_{1},\ldots,a_{n})
  2. ( b 1 , , b n ) (b_{1},\ldots,b_{n})
  3. a 1 a n a_{1}\geq\cdots\geq a_{n}
  4. i = 1 n a i = i = 1 n b i \sum_{i=1}^{n}a_{i}=\sum_{i=1}^{n}b_{i}
  5. 1 k n 1\leq k\leq n
  6. i = 1 k a i i = 1 n min ( b i , k ) . \sum^{k}_{i=1}a_{i}\leq\sum^{n}_{i=1}\min(b_{i},k).
  7. b 1 b n b_{1}\geq\cdots\geq b_{n}
  8. ( a 1 , , a n ) (a_{1},\ldots,a_{n})
  9. ( b 1 , , b n ) (b_{1},\ldots,b_{n})
  10. m = i = 1 n a i m=\sum_{i=1}^{n}a_{i}
  11. ( a 1 * , , a n * ) (a^{*}_{1},\ldots,a^{*}_{n})
  12. a k * := | { b i | b i k } | a^{*}_{k}:=|\{b_{i}|b_{i}\geq k\}|
  13. ( b 1 , , b n ) (b_{1},\ldots,b_{n})
  14. ( a 1 , , a n ) (a_{1},\ldots,a_{n})
  15. ( b 1 , , b n ) (b_{1},\ldots,b_{n})
  16. ( a 1 * , , a n * ) (a^{*}_{1},\ldots,a^{*}_{n})
  17. n n
  18. a a
  19. b b
  20. a * a^{*}
  21. i = 1 k a i * = i = 1 n min ( b i , k ) \sum_{i=1}^{k}a^{*}_{i}=\sum^{n}_{i=1}\min(b_{i},k)
  22. a * a^{*}
  23. b b
  24. a a
  25. ( ( a 1 , b 1 ) , , ( a n , b n ) ) ((a_{1},b_{1}),...,(a_{n},b_{n}))
  26. ( b 1 , , b n ) (b_{1},\ldots,b_{n})
  27. ( a 1 , , a n ) (a_{1},\ldots,a_{n})
  28. a * a^{*}
  29. a * a^{*}
  30. a a
  31. b b
  32. a a
  33. n n
  34. k k
  35. 1 k < n 1\leq k<n
  36. a k > a k + 1 a_{k}>a_{k+1}
  37. k = n k=n
  38. a a
  39. b b
  40. a a
  41. i = 1 n a i = i = 1 n b i \sum_{i=1}^{n}a_{i}=\sum_{i=1}^{n}b_{i}
  42. c c
  43. c , b c,b
  44. c c
  45. a a

Gamma-curcumene_synthase.html

  1. \rightleftharpoons

Gamma-humulene_synthase.html

  1. \rightleftharpoons

Gamma-L-glutamyl-butirosin_B_gamma-glutamyl_cyclotransferase.html

  1. \rightleftharpoons

Gamma-muurolene_synthase.html

  1. \rightleftharpoons

Gamma-Re_Transition_Model.html

  1. γ - Re θ t \gamma-\mathrm{Re}_{\theta_{t}}
  2. e n e^{n}
  3. γ \gamma
  4. ( ρ γ ) t + ( ρ U j γ ) x j = P γ - E γ + x j [ ( μ + μ f σ f ) γ x j ] \frac{\partial(\rho\gamma)}{\partial t}+\frac{\partial(\rho U_{j}\gamma)}{% \partial x_{j}}=P_{\gamma}-E_{\gamma}+\frac{\partial}{\partial x_{j}}\left[% \left(\mu+\frac{\mu_{f}}{\sigma_{f}}\right)\frac{\partial\gamma}{\partial x_{j% }}\right]
  5. R e θ Re_{\theta}
  6. ( ρ R e θ t ¯ ) t + ( ρ U j R e θ t ¯ ) x j = P θ t + x j [ σ θ t ( μ + μ t ) R e θ t ¯ x j ] \frac{\partial(\rho\overline{Re_{\theta t}})}{\partial t}+\frac{\partial(\rho U% _{j}\overline{Re_{\theta t}})}{\partial x_{j}}=P_{\theta t}+\frac{\partial}{% \partial x_{j}}\left[\sigma_{\theta t}(\mu+\mu_{t})\frac{\partial\overline{Re_% {\theta t}}}{\partial x_{j}}\right]

Gamma-terpinene_synthase.html

  1. \rightleftharpoons

Gap_reduction.html

  1. O P T P ( x ) k OPT_{P}(x)\leq k
  2. O P T P ( x ) > c k OPT_{P}(x)>c\cdot k

Gardner_equation.html

  1. u t + ( 2 * a * u - 3 * b * u 2 ) * u x + 3 u x 3 = 0 \frac{\partial u}{\partial t}+(2*a*u-3*b*u^{2})*\frac{\partial u}{\partial x}+% \frac{\partial^{3}u}{\partial x^{3}}=0

Gauge_vector–tensor_gravity.html

  1. B μ , B ~ μ B_{\mu},\tilde{B}_{\mu}
  2. g μ ν g_{\mu\nu}
  3. d s = - g μ ν d x μ d x ν + ( B μ + B ~ μ ) d x μ ds=\sqrt{-g_{\mu\nu}dx^{\mu}dx^{\nu}}+(B_{\mu}+\tilde{B}_{\mu})dx^{\mu}
  4. m m
  5. S = m d τ ( 1 2 x ˙ μ x ˙ ν g μ ν + ( B μ + B ~ μ ) x ˙ μ ) . S=m\int d\tau(\frac{1}{2}\dot{x}^{\mu}\dot{x}^{\nu}g_{\mu\nu}+(B_{\mu}+\tilde{% B}_{\mu})\dot{x}^{\mu})\,.
  6. S Grav = 1 16 π G d 4 x - g R S_{\,\text{Grav}}=\frac{1}{16\pi G}\int d^{4}x\,\sqrt{-g}R
  7. R R
  8. S B = - 1 16 π G κ l 2 d 4 x - g L ( l 2 4 B μ ν B μ ν ) S_{B}=-\frac{1}{16\pi G\kappa l^{2}}\int d^{4}x\sqrt{-g}\,{L}(\frac{l^{2}}{4}B% _{\mu\nu}B^{\mu\nu})
  9. S B ~ = - 1 16 π G κ ~ l ~ 2 d 4 x - g L ( l ~ 2 4 B ~ μ ν B ~ μ ν ) S_{\tilde{B}}=-\frac{1}{16\pi G\tilde{\kappa}\tilde{l}^{2}}\int d^{4}x\sqrt{-g% }\,{L}(\frac{\tilde{l}^{2}}{4}\tilde{B}_{\mu\nu}\tilde{B}^{\mu\nu})
  10. L ( x ) = { x , for x 1 2 3 | x | 3 2 , for x 1 , {L}(x)=\left\{\begin{array}[]{ccc}x&,&\,\text{for}~{}x\gg 1\\ \frac{2}{3}|x|^{\frac{3}{2}}&,&\,\text{for}~{}x\leq 1\end{array}\right.~{},
  11. κ , κ ~ \kappa,\tilde{\kappa}
  12. l , l ~ l,\tilde{l}
  13. l < l ~ . l<\tilde{l}\,.
  14. J μ = ρ u μ J^{\mu}=\rho u^{\mu}
  15. ρ \rho
  16. u μ u^{\mu}
  17. L ( l 2 4 B μ ν B μ ν ) = l 2 4 B μ ν B μ ν , L ( l ~ 2 4 B ~ μ ν B ~ μ ν ) = l ~ 2 4 B ~ μ ν B ~ μ ν . \begin{array}[]{ccc}{L}(\frac{l^{2}}{4}B_{\mu\nu}B^{\mu\nu})&=&\frac{l^{2}}{4}% B_{\mu\nu}B^{\mu\nu}\,,\\ {L}(\frac{\tilde{l}^{2}}{4}\tilde{B}_{\mu\nu}\tilde{B}^{\mu\nu})&=&\frac{% \tilde{l}^{2}}{4}\tilde{B}_{\mu\nu}\tilde{B}^{\mu\nu}\,.\end{array}
  18. κ + κ ~ = 0 \kappa+\tilde{\kappa}=0
  19. B μ + B ~ μ = 0 . B_{\mu}+\tilde{B}_{\mu}=0\,.
  20. L ( l 2 4 B μ ν B μ ν ) = | l 2 4 B μ ν B μ ν | 3 2 , L ( l ~ 2 4 B ~ μ ν B ~ μ ν ) = l ~ 2 4 B ~ μ ν B ~ μ ν . \begin{array}[]{ccc}{L}(\frac{l^{2}}{4}B_{\mu\nu}B^{\mu\nu})&=&\left|\frac{l^{% 2}}{4}B_{\mu\nu}B^{\mu\nu}\right|^{\frac{3}{2}}\,,\\ {L}(\frac{\tilde{l}^{2}}{4}\tilde{B}_{\mu\nu}\tilde{B}^{\mu\nu})&=&\frac{% \tilde{l}^{2}}{4}\tilde{B}_{\mu\nu}\tilde{B}^{\mu\nu}\,.\end{array}
  21. B μ B_{\mu}
  22. a 0 = 4 2 κ c 2 l a_{0}=\frac{4\sqrt{2}\kappa c^{2}}{l}
  23. κ \kappa
  24. B μ , B ~ μ B_{\mu},\tilde{B}_{\mu}

Gauss'_Method.html

  1. 𝐑 𝐧 = [ R e 1 - ( 2 f - f 2 ) sin 2 ϕ n + H n ] cos ϕ n ( cos θ n 𝐈 ^ + sin θ n 𝐉 ^ ) + [ R e ( 1 - f ) 2 1 - ( 2 f - f 2 ) sin 2 ϕ n + H n ] sin ϕ n 𝐊 ^ \mathbf{R_{n}}=\left[{R_{e}\over\sqrt{1-(2f-f^{2})\sin^{2}\phi_{n}}}+H_{n}% \right]\cos\phi_{n}(\cos\theta_{n}\mathbf{\hat{I}}+\sin\theta_{n}\mathbf{\hat{% J}})+\left[{R_{e}(1-f)^{2}\over\sqrt{1-(2f-f^{2})\sin^{2}\phi_{n}}}+H_{n}% \right]\sin\phi_{n}\mathbf{\hat{K}}
  2. 𝐑 𝐧 = R e cos ϕ n cos θ n 𝐈 ^ + R e cos ϕ n sin θ n 𝐉 ^ + R e sin ϕ n 𝐊 ^ \mathbf{R_{n}}=R_{e}\cos\phi^{\prime}_{n}\cos\theta_{n}\mathbf{\hat{I}}+R_{e}% \cos\phi^{\prime}_{n}\sin\theta_{n}\mathbf{\hat{J}}+R_{e}\sin\phi^{\prime}_{n}% \mathbf{\hat{K}}
  3. ρ ^ 𝐧 = cos δ n cos α n 𝐈 ^ + cos δ n sin α n 𝐉 ^ + sin δ n 𝐊 ^ \mathbf{\hat{\rho}_{n}}=\cos\delta_{n}\cos\alpha_{n}\mathbf{\hat{I}}+\cos% \delta_{n}\sin\alpha_{n}\mathbf{\hat{J}}+\sin\delta_{n}\mathbf{\hat{K}}
  4. ρ ^ \mathbf{\hat{\rho}}
  5. τ 1 = t 1 - t 2 \tau_{1}=t_{1}-t_{2}
  6. τ 2 = t 3 - t 2 \tau_{2}=t_{3}-t_{2}
  7. τ = t 3 - t 1 \tau=t_{3}-t_{1}
  8. 𝐩 𝟏 = ρ ^ 𝟐 × ρ ^ 𝟑 \mathbf{p_{1}}=\mathbf{\hat{\rho}_{2}}\times\mathbf{\hat{\rho}_{3}}
  9. 𝐩 𝟐 = ρ ^ 𝟏 × ρ ^ 𝟑 \mathbf{p_{2}}=\mathbf{\hat{\rho}_{1}}\times\mathbf{\hat{\rho}_{3}}
  10. 𝐩 𝟑 = ρ ^ 𝟏 × ρ ^ 𝟐 \mathbf{p_{3}}=\mathbf{\hat{\rho}_{1}}\times\mathbf{\hat{\rho}_{2}}
  11. ρ ^ \mathbf{\hat{\rho}}
  12. D 0 = ρ ^ 𝟏 𝐩 𝟏 = ρ ^ 𝟏 ( ρ ^ 𝟐 × ρ ^ 𝟑 ) D_{0}=\mathbf{\hat{\rho}_{1}}\cdot\mathbf{p_{1}}=\mathbf{\hat{\rho}_{1}}\cdot(% \mathbf{\hat{\rho}_{2}}\times\mathbf{\hat{\rho}_{3}})
  13. ρ ^ \mathbf{\hat{\rho}}
  14. D 11 = 𝐑 𝟏 𝐩 𝟏 D 12 = 𝐑 𝟏 𝐩 𝟐 D 13 = 𝐑 𝟏 𝐩 𝟑 D_{11}=\mathbf{R_{1}}\cdot\mathbf{p_{1}}\qquad D_{12}=\mathbf{R_{1}}\cdot% \mathbf{p_{2}}\qquad D_{13}=\mathbf{R_{1}}\cdot\mathbf{p_{3}}
  15. D 21 = 𝐑 𝟐 𝐩 𝟏 D 22 = 𝐑 𝟐 𝐩 𝟐 D 23 = 𝐑 𝟐 𝐩 𝟑 D_{21}=\mathbf{R_{2}}\cdot\mathbf{p_{1}}\qquad D_{22}=\mathbf{R_{2}}\cdot% \mathbf{p_{2}}\qquad D_{23}=\mathbf{R_{2}}\cdot\mathbf{p_{3}}
  16. D 31 = 𝐑 𝟑 𝐩 𝟏 D 32 = 𝐑 𝟑 𝐩 𝟐 D 33 = 𝐑 𝟑 𝐩 𝟑 D_{31}=\mathbf{R_{3}}\cdot\mathbf{p_{1}}\qquad D_{32}=\mathbf{R_{3}}\cdot% \mathbf{p_{2}}\qquad D_{33}=\mathbf{R_{3}}\cdot\mathbf{p_{3}}
  17. A = 1 D 0 ( - D 12 τ 3 τ + D 22 + D 32 τ 1 τ ) A=\frac{1}{D_{0}}\left(-D_{12}\frac{\tau_{3}}{\tau}+D_{22}+D_{32}\frac{\tau_{1% }}{\tau}\right)
  18. B = 1 6 D 0 [ D 12 ( τ 3 2 - τ 2 ) τ 3 τ + D 32 ( τ 2 - τ 1 2 ) τ 1 τ ] B=\frac{1}{6D_{0}}\left[D_{12}\left(\tau_{3}^{2}-\tau^{2}\right)\frac{\tau_{3}% }{\tau}+D_{32}\left(\tau^{2}-\tau_{1}^{2}\right)\frac{\tau_{1}}{\tau}\right]
  19. E = 𝐑 𝟐 ρ ^ 𝟐 E=\mathbf{R_{2}}\cdot\mathbf{\hat{\rho}_{2}}
  20. ρ ^ \mathbf{\hat{\rho}}
  21. R 2 2 = 𝐑 𝟐 𝐑 𝟐 {R_{2}}^{2}=\mathbf{R_{2}}\cdot\mathbf{R_{2}}
  22. a = - ( A 2 + 2 A E + R 2 2 ) a=-\left(A^{2}+2AE+{R_{2}}^{2}\right)
  23. b = - 2 μ B ( A + E ) b=-2\mu B(A+E)
  24. c = - μ 2 B 2 c=-\mu^{2}B^{2}
  25. 0 = r 2 8 + a r 2 6 + b r 2 3 + c 0={r_{2}}^{8}+a{r_{2}}^{6}+b{r_{2}}^{3}+c
  26. ρ 1 = 1 D 0 [ 6 ( D 31 τ 1 τ 3 + D 21 τ τ 3 ) r 2 3 + μ D 31 ( τ 2 - τ 1 2 ) τ 1 τ 3 6 r 2 3 + μ ( τ 2 - τ 3 2 ) - D 11 ] \rho_{1}=\frac{1}{D_{0}}\left[\frac{6\left(D_{31}\dfrac{\tau_{1}}{\tau_{3}}+D_% {21}\dfrac{\tau}{\tau_{3}}\right){r_{2}}^{3}+\mu D_{31}\left(\tau^{2}-{\tau_{1% }}^{2}\right)\dfrac{\tau_{1}}{\tau_{3}}}{6{r_{2}}^{3}+\mu\left(\tau^{2}-{\tau_% {3}}^{2}\right)}-D_{11}\right]
  27. ρ 2 = A + μ B r 2 3 \rho_{2}=A+\frac{\mu B}{{r_{2}}^{3}}
  28. ρ 3 = 1 D 0 [ 6 ( D 13 τ 3 τ 1 + D 23 τ τ 1 ) r 2 3 + μ D 13 ( τ 2 - τ 3 2 ) τ 3 τ 1 6 r 2 3 + μ ( τ 2 - τ 1 2 ) - D 33 ] \rho_{3}=\frac{1}{D_{0}}\left[\frac{6\left(D_{13}\dfrac{\tau_{3}}{\tau_{1}}+D_% {23}\dfrac{\tau}{\tau_{1}}\right){r_{2}}^{3}+\mu D_{13}\left(\tau^{2}-{\tau_{3% }}^{2}\right)\dfrac{\tau_{3}}{\tau_{1}}}{6{r_{2}}^{3}+\mu\left(\tau^{2}-{\tau_% {1}}^{2}\right)}-D_{33}\right]
  29. 𝐫 𝟏 = 𝐑 𝟏 + ρ 1 ρ ^ 𝟏 \mathbf{r_{1}}=\mathbf{R_{1}}+\rho_{1}\mathbf{\hat{\rho}_{1}}
  30. 𝐫 𝟐 = 𝐑 𝟐 + ρ 2 ρ ^ 𝟐 \mathbf{r_{2}}=\mathbf{R_{2}}+\rho_{2}\mathbf{\hat{\rho}_{2}}
  31. 𝐫 𝟑 = 𝐑 𝟑 + ρ 3 ρ ^ 𝟑 \mathbf{r_{3}}=\mathbf{R_{3}}+\rho_{3}\mathbf{\hat{\rho}_{3}}
  32. ρ ^ \mathbf{\hat{\rho}}
  33. f 1 1 - 1 2 μ r 2 3 τ 1 2 f_{1}\approx 1-\frac{1}{2}\frac{\mu}{{r_{2}}^{3}}{\tau_{1}}^{2}
  34. f 3 1 - 1 2 μ r 2 3 τ 3 2 f_{3}\approx 1-\frac{1}{2}\frac{\mu}{{r_{2}}^{3}}{\tau_{3}}^{2}
  35. g 1 τ 1 - 1 6 μ r 2 3 τ 1 2 g_{1}\approx\tau_{1}-\frac{1}{6}\frac{\mu}{{r_{2}}^{3}}{\tau_{1}}^{2}
  36. g 3 τ 3 - 1 6 μ r 2 3 τ 3 2 g_{3}\approx\tau_{3}-\frac{1}{6}\frac{\mu}{{r_{2}}^{3}}{\tau_{3}}^{2}
  37. 𝐯 𝟐 = 1 f 1 g 3 - f 3 g 1 ( - f 3 𝐫 𝟏 + f 1 𝐫 𝟑 ) \mathbf{v_{2}}=\frac{1}{f_{1}g_{3}-f_{3}g_{1}}\left(-f_{3}\mathbf{r_{1}}+f_{1}% \mathbf{r_{3}}\right)

Gausson_(physics).html

  1. ( = 1 ) (\hbar=1)
  2. i ψ t = - 1 2 2 ψ x 2 - a ln | ψ | 2 ψ i{\partial\psi\over\partial t}=-\frac{1}{2}\frac{\partial^{2}\psi}{\partial x^% {2}}-a\ln|\psi|^{2}\psi
  3. ψ ( x , t ) = e - i E t ψ ( x - k t ) \frac{}{}\psi(x,t)=e^{-iEt}\psi(x-kt)
  4. y = x - k t \frac{}{}y=x-kt
  5. - 1 2 ( ψ y + i k ) 2 - a ln | ψ | 2 ψ = ( E + k 2 2 ) ψ -\frac{1}{2}\left({\frac{\partial\psi}{\partial y}+ik}\right)^{2}-a\ln|\psi|^{% 2}\psi=\left(E+\frac{k^{2}}{2}\right)\psi
  6. Ψ ( y ) = e - i k y ψ ( y ) \frac{}{}\Psi(y)=e^{-iky}\psi(y)
  7. Ψ ( y ) = N e - ω y 2 / 2 \Psi(y)=Ne^{-\omega y^{2}/2}
  8. - 1 2 2 Ψ y 2 + a ω y 2 Ψ = ( E + k 2 2 + N 2 a ) Ψ -\frac{1}{2}{\frac{\partial^{2}\Psi}{\partial y^{2}}}+a\omega y^{2}\Psi=\left(% E+\frac{k^{2}}{2}+N^{2}a\right)\Psi
  9. ( a > 0 ) (a>0)
  10. a ω = ω 2 / 2 \frac{}{}a\omega=\omega^{2}/2
  11. ω = 2 a \frac{}{}\omega=2a
  12. ψ ( x , t ) = e - i E t e i k ( x - k t ) e - a ( x - k t ) 2 \frac{}{}\psi(x,t)=e^{-iEt}e^{ik{(x-kt)}}e^{-a({x-kt})^{2}}
  13. E = a ( 1 - N 2 ) - k 2 / 2 \frac{}{}E=a(1-N^{2})-k^{2}/2

General_linear_methods.html

  1. y = f ( t , y ) , y ( t 0 ) = y 0 . y^{\prime}=f(t,y),\quad y(t_{0})=y_{0}.
  2. y ( t ) y(t)
  3. t i t_{i}
  4. y i y ( t i ) where t i = t 0 + i h , y_{i}\approx y(t_{i})\quad\,\text{where}\quad t_{i}=t_{0}+ih,
  5. Δ t \Delta t
  6. r r
  7. s s
  8. r = 1 r=1
  9. s = 1 s=1
  10. Y i Y_{i}
  11. F i , i = 1 , 2 , s F_{i},i=1,2,\dots s
  12. y i [ n - 1 ] , i = 1 , , r y_{i}^{[n-1]},i=1,\dots,r
  13. n n
  14. y [ n - 1 ] = [ y 1 [ n - 1 ] y 2 [ n - 1 ] y r [ n - 1 ] ] , y [ n ] = [ y 1 [ n ] y 2 [ n ] y r [ n ] ] , Y = [ Y 1 Y 2 Y s ] , F = [ F 1 F 2 F s ] . y^{[n-1]}=\left[\begin{matrix}y_{1}^{[n-1]}\\ y_{2}^{[n-1]}\\ \vdots\\ y_{r}^{[n-1]}\\ \end{matrix}\right],\quad y^{[n]}=\left[\begin{matrix}y_{1}^{[n]}\\ y_{2}^{[n]}\\ \vdots\\ y_{r}^{[n]}\\ \end{matrix}\right],\quad Y=\left[\begin{matrix}Y_{1}\\ Y_{2}\\ \vdots\\ Y_{s}\end{matrix}\right],\quad F=\left[\begin{matrix}F_{1}\\ F_{2}\\ \vdots\\ F_{s}\end{matrix}\right].
  15. A = [ a i j ] A=[a_{ij}]
  16. U = [ u i j ] U=[u_{ij}]
  17. Y i = j = 1 s a i j h F j + j = 1 r u i j y j [ n - 1 ] , i = 1 , 2 , , s , Y_{i}=\sum_{j=1}^{s}a_{ij}hF_{j}+\sum_{j=1}^{r}u_{ij}y_{j}^{[n-1]},\qquad i=1,% 2,\dots,s,
  18. t n t^{n}
  19. B = [ b i j ] B=[b_{ij}]
  20. V = [ v i j ] V=[v_{ij}]
  21. y i [ n ] = j = 1 s b i j h F j + j = 1 r v i j y j [ n - 1 ] , i = 1 , 2 , , r . y_{i}^{[n]}=\sum_{j=1}^{s}b_{ij}hF_{j}+\sum_{j=1}^{r}v_{ij}y_{j}^{[n-1]},% \qquad i=1,2,\dots,r.
  22. A , U , B and V A,U,B\,\text{and}V
  23. [ Y y [ n ] ] = [ A I U I B I V I ] [ F y [ n - 1 ] ] , \left[\begin{matrix}Y\\ y^{[n]}\end{matrix}\right]=\left[\begin{matrix}A\otimes I&U\otimes I\\ B\otimes I&V\otimes I\end{matrix}\right]\left[\begin{matrix}F\\ y^{[n-1]}\end{matrix}\right],
  24. \otimes
  25. F = f ( Y ) F=f(Y)
  26. y n - 1 / 2 * = y n - 2 + h ( 9 8 f ( y n - 1 ) + 3 8 f ( y n - 2 ) ) . y^{*}_{n-1/2}=y_{n-2}+h\left(\frac{9}{8}f(y_{n-1})+\frac{3}{8}f(y_{n-2})\right).
  27. y n * y^{*}_{n}
  28. y n - 1 / 2 * y^{*}_{n-1/2}
  29. y n * = 28 5 y n - 1 - 23 5 y n - 2 + h ( 32 15 f ( y n - 1 / 2 * ) - 4 f ( y n - 1 ) - 26 15 f ( y n - 2 ) ) , y^{*}_{n}=\frac{28}{5}y_{n-1}-\frac{23}{5}y_{n-2}+h\left(\frac{32}{15}f(y^{*}_% {n-1/2})-4f(y_{n-1})-\frac{26}{15}f(y_{n-2})\right),
  30. y n = 32 31 y n - 1 - 1 31 y n - 2 + h ( 5 31 f ( y * ) + 64 93 f ( y n - 1 / 2 * ) + 4 31 f ( y n - 1 ) - 1 93 f ( y n - 2 ) ) . y_{n}=\frac{32}{31}y_{n-1}-\frac{1}{31}y_{n-2}+h\left(\frac{5}{31}f(y^{*})+% \frac{64}{93}f(y^{*}_{n-1/2})+\frac{4}{31}f(y_{n-1})-\frac{1}{93}f(y_{n-2})% \right).
  31. [ 0 0 0 0 1 9 8 3 8 32 15 0 0 28 5 - 23 5 - 4 - 26 15 64 93 5 31 0 32 31 - 1 31 4 31 - 1 93 64 93 5 31 0 32 31 - 1 31 4 31 - 1 93 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 ] . \left[\begin{array}[]{ccc|cccc}0&0&0&0&1&\frac{9}{8}&\frac{3}{8}\\ \frac{32}{15}&0&0&\frac{28}{5}&-\frac{23}{5}&-4&-\frac{26}{15}\\ \frac{64}{93}&\frac{5}{31}&0&\frac{32}{31}&-\frac{1}{31}&\frac{4}{31}&-\frac{1% }{93}\\ \hline\frac{64}{93}&\frac{5}{31}&0&\frac{32}{31}&-\frac{1}{31}&\frac{4}{31}&-% \frac{1}{93}\\ 0&0&0&1&0&0&0\\ 0&0&1&0&0&0&0\\ 0&0&0&0&0&1&0\\ \end{array}\right].

General_rating_of_city_appeal.html

  1. K i K_{i}
  2. Q i Qi
  3. Q m a x Q^{max}
  4. Q m i n Q^{min}
  5. K i = f ( Q m i n , , Q i , , Q m a x ) Ki=f{(Q^{min},\ldots,Q_{i},\ldots,Q^{max})}
  6. K i = Q i - Q m i n Q m a x - Q m i n K_{i}=\frac{Q_{i}-Q^{min}}{Q^{max}-Q^{min}}
  7. G i G_{i}
  8. 0 < G i < 1 0<G_{i}<1
  9. i = 1 N G i = 1 \sum_{i=1}^{N}G_{i}=1
  10. K k K_{k}
  11. K k = K e f i = 1 N K i G i K_{k}=K_{e}f\sum_{i=1}^{N}K_{i}G_{i}
  12. K e f = 1 K_{e}f=1
  13. G ( K 1 , K 2 , , K n ) = K 1 K 2 K n n = ( i = 1 n K i ) 1 / n G(K_{1},K_{2},\ldots,K_{n})=\sqrt[n]{K_{1}K_{2}\cdots K_{n}}=\left(\prod_{i=1}% ^{n}K_{i}\right)^{1/n}

Generalized_filtering.html

  1. ( Ω , U , X , S , p , q ) (\Omega,U,X,S,p,q)
  2. Ω \Omega
  3. ω Ω \omega\in\Omega
  4. U U\in\mathbb{R}
  5. X : X × U × Ω X:X\times U\times\Omega\to\mathbb{R}
  6. S : X × U × Ω S:X\times U\times\Omega\to\mathbb{R}
  7. p ( s ~ , x ~ , u ~ m ) p(\tilde{s},\tilde{x},\tilde{u}\mid m)
  8. q ( x ~ , u ~ m ) q(\tilde{x},\tilde{u}\mid m)
  9. μ ~ \tilde{\mu}\in\mathbb{R}
  10. u ~ = [ u , u , u ′′ , ] T \tilde{u}=[u,u^{\prime},u^{\prime\prime},\ldots]^{T}
  11. p ( s ~ ( t ) | m ) p(\tilde{s}(t)|m)
  12. μ ~ ( t ) = arg min μ ~ { F ( < ~ m t p l > s ( t ) , μ ~ ) } \tilde{\mu}(t)=\underset{\tilde{\mu}}{\operatorname{arg\,min}}\{F(\tilde{<}% mtpl>{{s}}(t),\tilde{{\mu}})\}
  13. G ( s ~ , x ~ , u ~ ) = - ln p ( s ~ , x ~ , u ~ | m ) G(\tilde{s},\tilde{x},\tilde{u})=-\ln p(\tilde{s},\tilde{x},\tilde{u}|m)
  14. q q
  15. H [ q ] = E q [ - log ( q ) ] H[q]=E_{q}[-\log(q)]
  16. F ( s ~ , μ ~ ) = E q [ G ( s ~ , , ) ] - H [ q ( , | μ ~ ) ] = - ln p ( s ~ | m ) + D K L [ q ( , | μ ~ ) | | p ( , | s ~ , m ) ] F(\tilde{s},\tilde{\mu})=E_{q}[G(\tilde{s},\cdot,\cdot)]-H[q(\cdot,\cdot|% \tilde{\mu})]=-\ln p(\tilde{s}|m)+D_{KL}[q(\cdot,\cdot|\tilde{\mu})||p(\cdot,% \cdot|\tilde{s},m)]
  17. q ( x ~ , < ~ m t p l > u | μ ~ ) = 𝒩 ( μ ~ , C ) q(\tilde{x},\tilde{<}mtpl>{{u}}|\tilde{\mu})=\mathcal{N}(\tilde{\mu},C)
  18. C - 1 = Π = μ ~ μ ~ G ( μ ~ ) C^{-1}=\Pi=\partial_{\tilde{\mu}\tilde{\mu}}G(\tilde{\mu})
  19. F = G ( μ ~ ) + 1 2 ln | μ ~ μ ~ G ( μ ~ ) | F=G(\tilde{\mu})+\textstyle{1\over 2}\ln|\partial_{\tilde{\mu}\tilde{\mu}}G(% \tilde{\mu})|
  20. μ ~ ˙ = D μ ~ - μ ~ F ( s ~ , μ ~ ) \dot{\tilde{\mu}}=D\tilde{\mu}-\partial_{\tilde{\mu}}F(\tilde{s},\tilde{\mu})
  21. D D
  22. D u ~ = [ u , u ′′ , ] T D\tilde{u}=[u^{\prime},u^{\prime\prime},\ldots]^{T}
  23. μ ~ ˙ = D μ ~ - μ ~ F ( s , μ ~ ) \dot{\tilde{\mu}}=D\tilde{\mu}-\partial_{\tilde{\mu}}F(s,\tilde{\mu})
  24. S = d t F ( s ~ ( t ) , μ ~ ( t ) ) S=\int dt\,F(\tilde{s}(t),\tilde{\mu}(t))
  25. μ ~ ˙ = D μ ~ μ ~ F ( s ~ , μ ~ ) = 0 δ μ ~ S = 0 \dot{\tilde{\mu}}=D\tilde{\mu}\Leftrightarrow\partial_{\tilde{\mu}}F(\tilde{s}% ,\tilde{\mu})=0\Leftrightarrow\delta_{\tilde{\mu}}S=0
  26. < ˙ m t p l > μ ~ - D μ ~ = - μ ~ F ( s , μ ~ ) \dot{<}mtpl>{{\tilde{{\mu}}}}-D\tilde{{\mu}}=-\partial_{\tilde{\mu}}F(s,\tilde% {\mu})
  27. Δ t \Delta t
  28. Δ μ ~ \displaystyle\Delta\tilde{\mu}
  29. s \displaystyle s
  30. s ~ = g ~ ( x ~ , u ~ ) + ω ~ s s = g ( x , u ) + ω s s = x g x + u g u + ω x s ′′ = x g x ′′ + u g u ′′ + ω x ′′ x ~ ˙ = f ~ ( x ~ , u ~ ) + ω ~ x x ˙ = f ( x , u ) + ω x x ˙ = x f x + u f u + ω x x ˙ ′′ = x f x ′′ + u f u ′′ + ω x ′′ \begin{aligned}\displaystyle\tilde{s}&\displaystyle=\tilde{g}(\tilde{x},\tilde% {u})+\tilde{\omega}_{s}\\ \\ \displaystyle s&\displaystyle=g(x,u)+\omega_{s}\\ \displaystyle s^{\prime}&\displaystyle=\partial_{x}g\cdot x^{\prime}+\partial_% {u}g\cdot u^{\prime}+\omega^{\prime}_{x}\\ \displaystyle s^{\prime\prime}&\displaystyle=\partial_{x}g\cdot x^{\prime% \prime}+\partial_{u}g\cdot u^{\prime\prime}+\omega^{\prime\prime}_{x}\\ &\displaystyle\vdots\\ \end{aligned}\qquad\begin{aligned}\displaystyle\dot{\tilde{x}}&\displaystyle=% \tilde{f}(\tilde{x},\tilde{u})+\tilde{\omega}_{x}\\ \\ \displaystyle\dot{x}&\displaystyle=f(x,u)+\omega_{x}\\ \displaystyle\dot{x}^{\prime}&\displaystyle=\partial_{x}f\cdot x^{\prime}+% \partial_{u}f\cdot u^{\prime}+\omega^{\prime}_{x}\\ \displaystyle\dot{x}^{\prime\prime}&\displaystyle=\partial_{x}f\cdot x^{\prime% \prime}+\partial_{u}f\cdot u^{\prime\prime}+\omega^{\prime\prime}_{x}\\ &\displaystyle\vdots\end{aligned}
  31. ω \omega
  32. p ( s ~ , x ~ , u ~ | m ) \displaystyle p\left(\tilde{s},\tilde{x},\tilde{u}|m\right)
  33. < ~ m t p l > Σ = V Σ \tilde{<}mtpl>{{\Sigma}}=V\otimes\Sigma
  34. V V
  35. V = [ 1 0 ρ ¨ ( 0 ) 0 - ρ ¨ ( 0 ) 0 ρ ¨ ( 0 ) 0 ρ ¨ ¨ ( 0 ) ] V=\begin{bmatrix}1&0&\ddot{\rho}(0)&\cdots\\ 0&-\ddot{\rho}(0)&0&\\ \ddot{\rho}(0)&0&\ddot{\ddot{\rho}}(0)&\\ \vdots&&&\ddots\\ \end{bmatrix}
  36. < ¨ m t p l > ρ ( 0 ) \ddot{<}mtpl>{{\rho}}(0)
  37. N N
  38. [ s 1 , , s N ] T = ( E I ) s ~ ( t ) : E i j = ( i - t ) ( j - 1 ) ( j - 1 ) ! [s_{1},\dots,s_{N}]^{T}=(E\otimes I)\cdot\tilde{s}(t):\qquad E_{ij}=\frac{(i-t% )^{(j-1)}}{(j-1)!}
  39. f ( x , u , θ ) f(x,u,\theta)
  40. < ~ m t p l > Π ( x , u , θ ) \tilde{<}mtpl>{{\Pi}}(x,u,\theta)
  41. μ \mu
  42. μ ˙ \displaystyle\dot{\mu}
  43. < ˙ m t p l > μ ~ = 0 \dot{<}mtpl>{{\tilde{{\mu}}}}=0
  44. < ˙ m t p l > μ = μ ˙ = 0 μ F = 0 δ μ S = 0 \dot{<}mtpl>{{\mu}}={\dot{{\mu}}}^{\prime}=0\Rightarrow\partial_{\mu}F=0% \Rightarrow\delta_{\mu}S=0
  45. < ~ m t p l > μ = ( μ , μ ) \tilde{<}mtpl>{{\mu}}=(\mu,{\mu}^{\prime})
  46. μ ˙ \displaystyle\dot{\mu}
  47. q ( x ~ , u ~ , θ | μ ~ , μ ) = q ( x ~ , u ~ | μ ~ ) q ( θ | μ ) q(\tilde{x},\tilde{u},\theta\dots|\tilde{\mu},\mu)=q(\tilde{x},\tilde{u}|% \tilde{\mu})q(\theta|\mu)\dots
  48. s ~ \displaystyle\tilde{s}
  49. μ ~ ˙ u ( i ) \displaystyle\dot{\tilde{\mu}}_{u}^{(i)}
  50. Π ( i ) \Pi^{(i)}

Generalized_functional_linear_model.html

  1. X X
  2. β \beta
  3. β \beta
  4. X X
  5. β \beta
  6. β \beta
  7. X ( t ) , t T \textstyle X(t),t\in T
  8. T T
  9. β ( t ) , t T \beta(t),t\in T
  10. T T
  11. d w dw
  12. T T
  13. η = α + X c ( t ) β ( t ) d w ( t ) \eta=\alpha+\int X^{c}(t)\beta(t)\,dw(t)
  14. X c ( t ) = X ( t ) - E ( X ( t ) ) X^{c}(t)=X(t)-\,\text{E}(X(t))
  15. α \alpha
  16. Y Y
  17. Y Y
  18. Var ( Y X ) = σ 2 ( μ ) \rm{Var}(Y\mid X)=\sigma^{2}(\mu)
  19. E ( Y X ) = μ \rm{E}(Y\mid X)=\mu
  20. g g
  21. E ( Y X ) = μ \rm{E}(Y\mid X)=\mu
  22. η = α + X c ( t ) β ( t ) d w ( t ) \eta=\alpha+\int X^{c}(t)\beta(t)\,dw(t)
  23. μ = g ( η ) \mu=g(\eta)
  24. X c ( t ) X^{c}(t)
  25. β ( t ) \beta(t)
  26. X c ( t ) = j = 1 ξ j ρ j ( t ) and β ( t ) = j = 1 β j ρ j ( t ) , X^{c}(t)=\sum_{j=1}^{\infty}\xi_{j}\rho_{j}(t)\,\text{ and }\beta(t)=\sum_{j=1% }^{\infty}\beta_{j}\rho_{j}(t),
  27. ρ j , j = 1 , 2 , \rho_{j},j=1,2,\ldots
  28. L 2 ( d w ) , L^{2}(dw),
  29. T ρ j ( t ) ρ k ( t ) d w ( t ) = δ j k \int_{T}\rho_{j}(t)\rho_{k}(t)\,dw(t)=\delta_{jk}
  30. δ j k = 1 \delta_{jk}=1
  31. j = k j=k
  32. 0
  33. ξ j \xi_{j}
  34. ξ j = X c ( t ) ρ j ( t ) d w ( t ) \xi_{j}=\int X^{c}(t)\rho_{j}(t)\,dw(t)
  35. β j \beta_{j}
  36. β j = β ( t ) ρ j ( t ) d w ( t ) \beta_{j}=\int\beta(t)\rho_{j}(t)\,dw(t)
  37. j = 1 , 2 , j=1,2,\ldots
  38. E ( ξ j ) = 0 \,\text{E}(\xi_{j})=0
  39. j = 1 β j 2 < \sum_{j=1}^{\infty}\beta_{j}^{2}<\infty
  40. σ j 2 = Var ( ξ j ) = E ( ξ j 2 ) \sigma_{j}^{2}=\,\text{Var}(\xi_{j})=\,\text{E}(\xi_{j}^{2})
  41. j = 1 σ j 2 = E ( X c ( t ) ) 2 d w ( t ) < \sum_{j=1}^{\infty}\sigma_{j}^{2}=\int\,\text{E}(X^{c}(t))^{2}\,dw(t)<\infty
  42. ρ j \rho_{j}
  43. X c ( t ) β ( t ) d w ( t ) = j = 1 β j ξ j \int X^{c}(t)\beta(t)\,dw(t)=\sum_{j=1}^{\infty}\beta_{j}\xi_{j}
  44. η = α + X c ( t ) β ( t ) d w ( t ) = α + j = 1 β j ξ j \eta=\alpha+\int X^{c}(t)\beta(t)\,dw(t)=\alpha+\sum_{j=1}^{\infty}\beta_{j}% \xi_{j}
  45. η α + j = 1 p β j ξ j \eta\approx\alpha+\sum_{j=1}^{p}\beta_{j}\xi_{j}
  46. p p
  47. p p
  48. p p
  49. Var ( j = p + 1 β j ξ j ) = E ( ( j = p + 1 β j ξ j ) 2 ) = j = p + 1 β j σ j 2 \,\text{Var}(\sum_{j=p+1}^{\infty}\beta_{j}\xi_{j})=\,\text{E}\left(\left(\sum% _{j=p+1}^{\infty}\beta_{j}\xi_{j}\right)^{2}\right)=\sum_{j=p+1}^{\infty}\beta% _{j}\sigma_{j}^{2}
  50. E ( ( j = p + 1 β j ξ j ) 2 ) = j = p + 1 β j σ j 2 j = p + 1 β j 2 j = p + 1 σ j 2 \,\text{E}\left(\left(\sum_{j=p+1}^{\infty}\beta_{j}\xi_{j}\right)^{2}\right)=% \sum_{j=p+1}^{\infty}\beta_{j}\sigma_{j}^{2}\leq\sum_{j=p+1}^{\infty}\beta_{j}% ^{2}\ \sum_{j=p+1}^{\infty}\sigma_{j}^{2}
  51. p p\rightarrow\infty
  52. j = 1 β j 2 \sum_{j=1}^{\infty}\beta_{j}^{2}
  53. j = 1 σ j 2 \sum_{j=1}^{\infty}\sigma_{j}^{2}
  54. σ j 2 , j = 1 , 2 , \sigma_{j}^{2},j=1,2,\ldots
  55. G ( s , t ) = Cov ( X ( s ) , X ( t ) ) , s , t T G(s,t)=\,\text{Cov}(X(s),X(t)),\ s,t\in T
  56. n n
  57. ξ j 0 = 1 \xi_{j}^{0}=1
  58. β 0 = α \beta_{0}=\alpha
  59. ξ j i = X i ( t ) ρ j ( t ) d w ( t ) \xi_{j}^{i}=\int X_{i}(t)\rho_{j}(t)\,dw(t)
  60. η i = j = 0 p β j ξ j i , i = 1 , 2 , , n \eta_{i}=\sum_{j=0}^{p}\beta_{j}\xi_{j}^{i},i=1,2,\ldots,n
  61. μ i = g ( η i ) \mu_{i}=g(\eta_{i})
  62. β \beta
  63. p p
  64. p p
  65. s y m b o l β T = ( β 0 , β 1 , , β p ) symbol\beta^{T}=(\beta_{0},\beta_{1},\ldots,\beta_{p})
  66. U ( β ) = 0. U(\beta)=0.
  67. U ( β ) = i = 1 n ( Y i - μ i ) g ( η i ) ξ i / σ 2 ( μ i ) U(\beta)=\sum_{i=1}^{n}(Y_{i}-\mu_{i})g^{\prime}(\eta_{i})\xi_{i}/\sigma^{2}(% \mu_{i})
  68. s y m b o l β symbol\beta
  69. μ \mu
  70. η \eta
  71. U ( β ) = 0 U(\beta)=0
  72. s y m b o l β ^ symbol\hat{\beta}
  73. β ^ ( t ) = β ^ o + j = 1 p β ^ j ρ j ( t ) \hat{\beta}(t)=\hat{\beta}_{o}+\sum_{j=1}^{p}\hat{\beta}_{j}\rho_{j}(t)
  74. p p
  75. p p\rightarrow\infty
  76. Y i Y_{i}
  77. X i L 2 ( T ) X_{i}\in L^{2}(T)
  78. f ( y i X i ) = exp ( y i θ i - b ( θ i ) ϕ + c ( y i , ϕ ) ) f(y_{i}\mid X_{i})=\exp\left(\frac{y_{i}\theta_{i}-b(\theta_{i})}{\phi}+c(y_{i% },\phi)\right)
  79. b b
  80. c c
  81. θ i \theta_{i}
  82. ϕ \phi
  83. η i = α + X i c ( t ) β ( t ) d w ( t ) = θ i \eta_{i}=\alpha+\int X_{i}^{c}(t)\beta(t)\,dw(t)=\theta_{i}
  84. μ i = b ( θ i ) , and so μ i = b ( η i ) . \mu_{i}=b^{\prime}(\theta_{i}),\,\text{ and so }\mu_{i}=b^{\prime}(\eta_{i}).
  85. b b^{\prime}
  86. Var ( y i ) = ϕ b ′′ ( θ i ) = ϕ b ′′ ( η i ) = ϕ g ( η i ) = ϕ g ( g - 1 ( μ i ) ) ) \,\text{Var}(y_{i})=\phi b^{\prime\prime}(\theta_{i})=\phi b^{\prime\prime}(% \eta_{i})=\phi g^{\prime}(\eta_{i})=\phi g^{\prime}(g^{-1}(\mu_{i})))
  87. ϕ \phi
  88. μ = E ( Y X ) = η = α + X c ( t ) β ( t ) d w ( t ) \mu=\operatorname{E}(Y\mid X)=\eta=\alpha+\int X^{c}(t)\beta(t)\,dw(t)
  89. μ i = P ( Y i = 1 X i ) \mu_{i}=P(Y_{i}=1\mid X_{i})
  90. Var ( Y i ) = ϕ μ i ( 1 - μ i ) \operatorname{Var}(Y_{i})=\phi\mu_{i}(1-\mu_{i})
  91. ϕ \phi
  92. μ i \mu_{i}
  93. η i \eta_{i}
  94. Var ( Y i ) = ϕ μ i \operatorname{Var}(Y_{i})=\phi\mu_{i}
  95. ϕ \phi
  96. g - 1 ( E ( Y X ) ) = α + j = 1 p f j ( ξ j ) , g^{-1}(\operatorname{E}(Y\mid X))=\alpha+\sum_{j=1}^{p}f_{j}(\xi_{j}),
  97. ξ j \xi_{j}
  98. X X
  99. f j f_{j}
  100. E ( f j ( ξ j ) ) = 0. \,\text{E}(f_{j}(\xi_{j}))=0.
  101. X X
  102. f j f_{j}

Generalized_relative_entropy.html

  1. ϵ \epsilon
  2. ϵ \epsilon
  3. ϵ \epsilon
  4. ϵ \epsilon
  5. D ϵ ( ρ | | σ ) D^{\epsilon}(\rho||\sigma)
  6. ρ \rho
  7. σ \sigma
  8. Q Q
  9. I - Q I-Q
  10. ρ \rho
  11. Tr ( ρ Q ) \operatorname{Tr}(\rho Q)
  12. Tr ( σ Q ) \operatorname{Tr}(\sigma Q)
  13. ϵ \epsilon
  14. σ \sigma
  15. ρ \rho
  16. ϵ \epsilon
  17. ϵ ( 0 , 1 ) \epsilon\in(0,1)
  18. ϵ \epsilon
  19. ρ \rho
  20. σ \sigma
  21. D ϵ ( ρ | | σ ) = - log 1 ϵ min { Q , σ | 0 Q I and Q , ρ ϵ } . D^{\epsilon}(\rho||\sigma)=-\log\frac{1}{\epsilon}\min\{\langle Q,\sigma% \rangle|0\leq Q\leq I\,\text{ and }\langle Q,\rho\rangle\geq\epsilon\}~{}.
  22. D ϵ ( ρ | | σ ) 0 D^{\epsilon}(\rho||\sigma)\geq 0
  23. ρ = σ \rho=\sigma
  24. ρ \rho
  25. σ \sigma
  26. || ρ - σ || 1 = δ . ||\rho-\sigma||_{1}=\delta~{}.
  27. 0 < ϵ < 1 0<\epsilon<1
  28. log ϵ ϵ - ( 1 - ϵ ) δ D ϵ ( ρ | | σ ) log ϵ ϵ - δ . \log\frac{\epsilon}{\epsilon-(1-\epsilon)\delta}\quad\leq\quad D^{\epsilon}(% \rho||\sigma)\quad\leq\quad\log\frac{\epsilon}{\epsilon-\delta}~{}.
  29. 1 - ϵ ϵ | | ρ - σ | | 1 D ϵ ( ρ | | σ ) . \frac{1-\epsilon}{\epsilon}||\rho-\sigma||_{1}\quad\leq\quad D^{\epsilon}(\rho% ||\sigma)~{}.
  30. ϵ ( 0 , 1 ) \epsilon\in(0,1)
  31. D ϵ ( ρ | | σ ) = 0 D^{\epsilon}(\rho||\sigma)=0
  32. ρ = σ \rho=\sigma
  33. || ρ - σ || 1 = max 0 Q 1 Tr ( Q ( ρ - σ ) ) . ||\rho-\sigma||_{1}=\max_{0\leq Q\leq 1}\operatorname{Tr}(Q(\rho-\sigma))~{}.
  34. Q Q
  35. ρ - σ \rho-\sigma
  36. Q Q
  37. Tr ( Q ( ρ - σ ) ) δ \operatorname{Tr}(Q(\rho-\sigma))\leq\delta
  38. Tr ( Q ρ ) ϵ \operatorname{Tr}(Q\rho)\geq\epsilon
  39. Tr ( Q σ ) Tr ( Q ρ ) - δ ϵ - δ . \operatorname{Tr}(Q\sigma)~{}\geq~{}\operatorname{Tr}(Q\rho)-\delta~{}\geq~{}% \epsilon-\delta~{}.
  40. ϵ \epsilon
  41. 2 - D ϵ ( ρ | | σ ) ϵ - δ ϵ . 2^{-D^{\epsilon}(\rho||\sigma)}\geq\frac{\epsilon-\delta}{\epsilon}~{}.
  42. Q Q
  43. ρ - σ \rho-\sigma
  44. Q ¯ \bar{Q}
  45. I I
  46. Q Q
  47. Q ¯ = ( ϵ - μ ) I + ( 1 - ϵ + μ ) Q \bar{Q}=(\epsilon-\mu)I+(1-\epsilon+\mu)Q
  48. μ = ( 1 - ϵ ) Tr ( Q ρ ) 1 - Tr ( Q ρ ) . \mu=\frac{(1-\epsilon)\operatorname{Tr}(Q\rho)}{1-\operatorname{Tr}(Q\rho)}~{}.
  49. μ = ( 1 - ϵ + μ ) Tr ( Q ρ ) \mu=(1-\epsilon+\mu)\operatorname{Tr}(Q\rho)
  50. Tr ( Q ¯ ρ ) = ( ϵ - μ ) + ( 1 - ϵ + μ ) Tr ( Q ρ ) = ϵ . \operatorname{Tr}(\bar{Q}\rho)~{}=~{}(\epsilon-\mu)+(1-\epsilon+\mu)% \operatorname{Tr}(Q\rho)~{}=~{}\epsilon~{}.
  51. Tr ( Q ¯ σ ) = ϵ - μ + ( 1 - ϵ + μ ) Tr ( Q σ ) . \operatorname{Tr}(\bar{Q}\sigma)~{}=~{}\epsilon-\mu+(1-\epsilon+\mu)% \operatorname{Tr}(Q\sigma)~{}.
  52. μ = ( 1 - ϵ + μ ) Tr ( Q ρ ) \mu=(1-\epsilon+\mu)\operatorname{Tr}(Q\rho)
  53. Q Q
  54. μ \mu
  55. Tr ( Q ¯ σ ) = ϵ - ( 1 - ϵ + μ ) Tr ( Q ρ ) + ( 1 - ϵ + μ ) Tr ( Q σ ) \operatorname{Tr}(\bar{Q}\sigma)~{}=~{}\epsilon-(1-\epsilon+\mu)\operatorname{% Tr}(Q\rho)+(1-\epsilon+\mu)\operatorname{Tr}(Q\sigma)
  56. = ϵ - ( 1 - ϵ ) δ 1 - Tr ( Q ρ ) ϵ - ( 1 - ϵ ) δ . ~{}=~{}\epsilon-\frac{(1-\epsilon)\delta}{1-\operatorname{Tr}(Q\rho)}~{}\leq~{% }\epsilon-(1-\epsilon)\delta~{}.
  57. D ϵ ( ρ | | σ ) log ϵ ϵ - ( 1 - ϵ ) δ . D^{\epsilon}(\rho||\sigma)\geq\log\frac{\epsilon}{\epsilon-(1-\epsilon)\delta}% ~{}.
  58. log ϵ ϵ - ( 1 - ϵ ) δ = - log ( 1 - ( 1 - ϵ ) δ ϵ ) δ 1 - ϵ ϵ . \log\frac{\epsilon}{\epsilon-(1-\epsilon)\delta}~{}=~{}-\log\left(1-\frac{(1-% \epsilon)\delta}{\epsilon}\right)~{}\geq~{}\delta\frac{1-\epsilon}{\epsilon}~{}.
  59. S ( σ ) S(\sigma)
  60. σ \sigma
  61. ρ A B C \rho_{ABC}
  62. A B C \mathcal{H}_{A}\otimes\mathcal{H}_{B}\otimes\mathcal{H}_{C}
  63. S ( ρ A B C ) + S ( ρ B ) S ( ρ A B ) + S ( ρ B C ) S(\rho_{ABC})+S(\rho_{B})\leq S(\rho_{AB})+S(\rho_{BC})
  64. ρ A B , ρ B C , ρ B \rho_{AB},\rho_{BC},\rho_{B}
  65. S ( ρ | | σ ) - S ( ( ρ ) | | ( σ ) ) 0 S(\rho||\sigma)-S(\mathcal{E}(\rho)||\mathcal{E}(\sigma))\geq 0
  66. \mathcal{E}
  67. S ( ω | | τ ) S(\omega||\tau)
  68. ω , τ \omega,\tau
  69. ϵ \epsilon
  70. D ϵ ( ρ | | σ ) D ϵ ( ( ρ ) | | ( σ ) ) D^{\epsilon}(\rho||\sigma)\geq D^{\epsilon}(\mathcal{E}(\rho)||\mathcal{E}(% \sigma))
  71. \mathcal{E}
  72. ( R , I - R ) (R,I-R)
  73. ( ρ ) \mathcal{E}(\rho)
  74. ( σ ) \mathcal{E}(\sigma)
  75. R , ( ρ ) = ( R ) , ρ ϵ \langle R,\mathcal{E}(\rho)\rangle=\langle\mathcal{E}^{\dagger}(R),\rho\rangle\geq\epsilon
  76. ( ( R ) , I - ( R ) ) (\mathcal{E}^{\dagger}(R),I-\mathcal{E}^{\dagger}(R))
  77. ρ \rho
  78. σ \sigma
  79. R , ( σ ) = ( R ) , σ Q , σ \langle R,\mathcal{E}(\sigma)\rangle=\langle\mathcal{E}^{\dagger}(R),\sigma% \rangle\geq\langle Q,\sigma\rangle
  80. ( Q , I - Q ) (Q,I-Q)
  81. D ϵ ( ρ | | σ ) D^{\epsilon}(\rho||\sigma)
  82. lim n 1 n D ϵ ( ρ n | | σ n ) = lim n - 1 n log min 1 ϵ Tr ( σ n Q ) \lim_{n\rightarrow\infty}\frac{1}{n}D^{\epsilon}(\rho^{\otimes n}||\sigma^{% \otimes n})=\lim_{n\rightarrow\infty}\frac{-1}{n}\log\min\frac{1}{\epsilon}% \operatorname{Tr}(\sigma^{\otimes n}Q)
  83. = D ( ρ | | σ ) - lim n 1 n ( log 1 ϵ ) =D(\rho||\sigma)-\lim_{n\rightarrow\infty}\frac{1}{n}\left(\log\frac{1}{% \epsilon}\right)
  84. = D ( ρ | | σ ) , =D(\rho||\sigma)~{},
  85. 0 Q 1 0\leq Q\leq 1
  86. Tr ( Q ρ n ) ϵ . \operatorname{Tr}(Q\rho^{\otimes n})\geq\epsilon~{}.
  87. ρ n \rho^{\otimes n}
  88. σ n \sigma^{\otimes n}
  89. n \mathcal{E}^{\otimes n}
  90. D ϵ ( ρ n | | σ n ) D ϵ ( ( ρ ) n | | ( σ ) n ) . D^{\epsilon}(\rho^{\otimes n}||\sigma^{\otimes n})~{}\geq~{}D^{\epsilon}(% \mathcal{E}(\rho)^{\otimes n}||\mathcal{E}(\sigma)^{\otimes n})~{}.
  91. n n
  92. n n\rightarrow\infty

Generalized_spectrogram.html

  1. f f
  2. w ( t ) w(t)
  3. S P x , w ( t , f ) = G x , w ( t , f ) G x , w * ( t , f ) = | G x , w ( t , f ) | 2 S{P_{x,w}}(t,f)={G_{x,w}}(t,f)G_{{}_{x,w}}^{*}(t,f)=|{G_{x,w}}(t,f)|^{2}
  4. G x , w 1 {G_{x,{w_{1}}}}
  5. x ( t ) x(t)
  6. S P x , w 1 , w 2 ( t , f ) = G x , w 1 ( t , f ) G x , w 2 * ( t , f ) S{P_{x,{w_{1}},{w_{2}}}}(t,f)={G_{x,{w_{1}}}}(t,f)G_{{}_{x,{w_{2}}}}^{*}(t,f)
  7. G x , w 1 ( t , f ) = - w 1 ( t - τ ) x ( τ ) e - j 2 π f τ d τ {G_{x,{w_{1}}}}\left({t,f}\right)=\int_{-\infty}^{\infty}{{w_{1}}\left({t-\tau% }\right)x\left(\tau\right)\,{e^{-j2\pi\,f\,\tau}}d\tau}
  8. G x , w 2 ( t , f ) = - w 2 ( t - τ ) x ( τ ) e - j 2 π f τ d τ {G_{x,{w_{2}}}}\left({t,f}\right)=\int_{-\infty}^{\infty}{{w_{2}}\left({t-\tau% }\right)x\left(\tau\right)\,{e^{-j2\pi\,f\,\tau}}d\tau}
  9. w 1 ( t ) = w 2 ( t ) = w ( t ) w_{1}(t)=w_{2}(t)=w(t)
  10. S P x , w ( t , f ) = G x , w ( t , f ) G x , w * ( t , f ) = | G x , w ( t , f ) | 2 S{P_{x,w}}(t,f)={G_{x,w}}(t,f)G_{{}_{x,w}}^{*}(t,f)=|{G_{x,w}}(t,f)|^{2}
  11. w 1 ( t ) w_{1}(t)
  12. w 2 ( t ) w_{2}(t)
  13. w 1 ( t ) w_{1}(t)
  14. w 1 ( t ) w_{1}(t)
  15. 𝒮 𝒫 w 1 , w 2 ( t , f ) ( x , w ) = W i g ( w 1 , w 2 ) * W i g ( t , f ) ( x , w ) , \mathcal{SP}_{w_{1},w_{2}}(t,f)(x,w)=Wig(w_{1}^{\prime},w_{2}^{\prime})*Wig(t,% f)(x,w),
  16. w 1 ( s ) := w 1 ( - s ) , w 2 ( s ) := w 2 ( - s ) w_{1}^{\prime}(s):=w_{1}(-s),w_{2}^{\prime}(s):=w_{2}(-s)
  17. 𝒮 𝒫 w 1 , w 2 ( t , f ) ( x , w ) \mathcal{SP}_{w_{1},w_{2}}(t,f)(x,w)
  18. w 1 w 2 = δ w_{1}w_{2}^{\prime}=\delta
  19. δ \delta
  20. 𝒮 𝒫 w 1 , w 2 ( t , f ) ( x , w ) \mathcal{SP}_{w_{1},w_{2}}(t,f)(x,w)
  21. w 1 w 2 = δ w_{1}w_{2}^{\prime}=\delta
  22. δ \delta
  23. 𝒮 𝒫 w 1 , w 2 ( t , f ) ( x , w ) \mathcal{SP}_{w_{1},w_{2}}(t,f)(x,w)
  24. ( w 1 , w 2 ) = 1 (w_{1},w_{2})=1
  25. 𝒮 𝒫 w 1 , w 2 ( t , f ) ( x , w ) \mathcal{SP}_{w_{1},w_{2}}(t,f)(x,w)
  26. w 1 = C w 2 w_{1}=Cw_{2}
  27. C \R C\in\R

Generalized_Wiener_filter.html

  1. d d
  2. d = s + n d=s+n
  3. s T s = S \langle s^{T}s\rangle=S
  4. n T n = N \langle n^{T}n\rangle=N
  5. G G
  6. e = ( G d - s ) T ( G d - s ) e=\langle(Gd-s)^{T}(Gd-s)\rangle
  7. G G
  8. G = S ( S + N ) - 1 G=S(S+N)^{-1}
  9. s ^ = S ( S + N ) - 1 d \hat{s}=S(S+N)^{-1}d
  10. G = ( S - 1 + N - 1 ) - 1 N - 1 G=(S^{-1}+N^{-1})^{-1}N^{-1}
  11. A - 1 + B - 1 = A - 1 ( A + B ) B - 1 A^{-1}+B^{-1}=A^{-1}(A+B)B^{-1}
  12. s ^ = S ( S + N ) - 1 d \hat{s}=S(S+N)^{-1}d
  13. S + N S+N
  14. ( S + N ) S - 1 s ^ = d (S+N)S^{-1}\hat{s}=d

Generator_(circuit_theory).html

  1. [ V 1 I 2 ] = [ h 11 h 12 h 21 h 22 ] [ I 1 V 2 ] \begin{bmatrix}V_{1}\\ I_{2}\end{bmatrix}=\begin{bmatrix}h_{11}&h_{12}\\ h_{21}&h_{22}\end{bmatrix}\begin{bmatrix}I_{1}\\ V_{2}\end{bmatrix}

Generic_matrix_ring.html

  1. X 1 , X m X_{1},\dots X_{m}
  2. F n F_{n}
  3. A 1 , , A m A_{1},\dots,A_{m}
  4. X i A i X_{i}\mapsto A_{i}
  5. F n M n ( R ) F_{n}\to M_{n}(R)
  6. F n F_{n}
  7. M n ( k [ ( X l ) i j | 1 l m , 1 i , j n ] ) M_{n}(k[(X_{l})_{ij}|1\leq l\leq m,1\leq i,j\leq n])
  8. X 1 , , X m X_{1},\dots,X_{m}
  9. ( X l ) i j (X_{l})_{ij}
  10. F 1 F_{1}
  11. F n F_{n}
  12. k [ ( X l ) i j ] GL n ( k ) k[(X_{l})_{ij}]^{\operatorname{GL}_{n}(k)}
  13. F n F_{n}
  14. k t 1 , , t m k\langle t_{1},\dots,t_{m}\rangle
  15. t i X i t_{i}\mapsto X_{i}
  16. k t 1 , , t m k\langle t_{1},\dots,t_{m}\rangle
  17. F n F_{n}
  18. k [ t , , t m ] k[t,\dots,t_{m}]
  19. k m k^{m}
  20. k m k^{m}
  21. k [ t , , t m ] k k[t,\dots,t_{m}]\to k
  22. k t 1 , , t m k\langle t_{1},\dots,t_{m}\rangle
  23. Spec n ( A ) \operatorname{Spec}_{n}(A)
  24. 𝔪 \mathfrak{m}
  25. A / 𝔪 M n ( k ) A/\mathfrak{m}\approx M_{n}(k)
  26. Spec 1 ( A ) \operatorname{Spec}_{1}(A)
  27. Spec n ( A ) \operatorname{Spec}_{n}(A)
  28. n > 1 n>1

Genetic_purging.html

  1. W t = W e - δ F t W_{t}=We^{-\delta F_{t}}
  2. F t F_{t}
  3. W t = W e - δ g t W_{t}=We^{-\delta g_{t}}
  4. g t g_{t}
  5. g t = [ ( 1 - 1 2 N ) g t - 1 + 1 2 N ] [ 1 - 2 d F t - 1 ] g_{t}=\left[\left(1-\frac{1}{2N}\right)g_{t-1}+\frac{1}{2N}\right]\left[1-2dF_% {t-1}\right]
  6. < v a r > F <var>F

Geometric_measure_of_entanglement.html

  1. N N
  2. \mathcal{H}
  3. = 1 2 N \mathcal{H}=\mathcal{H}_{1}\otimes\mathcal{H}_{2}\ldots\otimes\mathcal{H}_{N}
  4. ψ \psi\in\mathcal{H}
  5. ψ = ψ 1 ψ 2 ψ N \psi=\psi_{1}\otimes\psi_{2}\ldots\otimes\psi_{N}
  6. ψ i i \psi_{i}\in\mathcal{H}_{i}
  7. ψ \psi
  8. ψ | ψ = 1 \langle\psi|\psi\rangle=1
  9. ψ - ϕ \|\psi-\phi\|
  10. ϕ = i = 1 N ϕ i , \phi=\prod_{i=1}^{N}\otimes\phi_{i},
  11. ϕ i | ϕ i = 1 \langle\phi_{i}|\phi_{i}\rangle=1
  12. ψ \psi
  13. ϕ \phi
  14. ϕ \phi
  15. ψ \psi

Geometric_mechanics.html

  1. 𝐉 : P 𝔤 * \mathbf{J}:P\to\mathfrak{g}^{*}
  2. μ 𝔤 * \mu\in\mathfrak{g}^{*}
  3. P μ = 𝐉 - 1 ( μ ) / G μ P_{\mu}=\mathbf{J}^{-1}(\mu)/G_{\mu}
  4. μ \mu

Geometric_Poisson_distribution.html

  1. 𝒫 𝒢 ( λ , θ ) \mathcal{PG}(\lambda,\theta)
  2. f N ( n ) = Pr ( N = n ) = { k = 1 n e - λ λ k k ! ( 1 - θ ) n - k θ k ( n - 1 k - 1 ) , n > 0 e - λ , n = 0 f_{N}(n)=\mathrm{Pr}(N=n)=\begin{cases}\sum_{k=1}^{n}e^{-\lambda}\frac{\lambda% ^{k}}{k!}(1-\theta)^{n-k}\theta^{k}{\left({{n-1}\atop{k-1}}\right)},&n>0\\ e^{-\lambda},&n=0\end{cases}

Geometric_quotient.html

  1. π : X Y \pi:X\to Y
  2. π - 1 ( y ) \pi^{-1}(y)
  3. U Y U\subset Y
  4. π - 1 ( U ) \pi^{-1}(U)
  5. U Y U\subset Y
  6. π # : k [ U ] k [ π - 1 ( U ) ] G \pi^{\#}:k[U]\to k[\pi^{-1}(U)]^{G}
  7. 𝒪 Y ( π * 𝒪 X ) G \mathcal{O}_{Y}\simeq(\pi_{*}\mathcal{O}_{X})^{G}
  8. k ( Y ) = k ( X ) G k(Y)=k(X)^{G}
  9. G / H G/H
  10. 𝔸 n + 1 0 n \mathbb{A}^{n+1}\setminus 0\to\mathbb{P}^{n}
  11. X ( 0 ) s X^{s}_{(0)}
  12. X ( 0 ) s X ( 0 ) s / G X^{s}_{(0)}\to X^{s}_{(0)}/G

Geometric_separator.html

  1. width ( R 0 ) / 2 n \operatorname{width}(R_{0})/2\sqrt{n}
  2. 12 n 12\sqrt{n}
  3. width ( R 0 ) / 2 n \operatorname{width}(R_{0})/2\sqrt{n}
  4. | t 1 - t 2 | 1 / n |t_{1}-t_{2}|\geq 1/\sqrt{n}
  5. | width ( R t 1 ) - width ( R t 2 ) | width ( R 0 ) / n |\operatorname{width}(R_{t_{1}})-\operatorname{width}(R_{t_{2}})|\geq% \operatorname{width}(R_{0})/\sqrt{n}
  6. width ( R 0 ) / 2 n \operatorname{width}(R_{0})/2\sqrt{n}
  7. j = 0 n - 1 | intersect ( j / n ) | n \sum_{j=0}^{\sqrt{n}-1}{|\operatorname{intersect}(j/\sqrt{n})|}\leq n
  8. | intersect ( j 0 / n ) | n |\operatorname{intersect}(j_{0}/\sqrt{n})|\leq\sqrt{n}
  9. t = j 0 / n t=j_{0}/\sqrt{n}
  10. ( N + 1 - k ) / ( 2 d ) \lfloor(N+1-k)/(2d)\rfloor
  11. 1.3 a d n 1.3{a\over d}\sqrt{n}
  12. 1.3 a d n 1.3{a\over d}\sqrt{n}

Geometry_index.html

  1. τ τ
  2. τ τ
  3. τ 5 = β - α 60 - 0.01667 α + 0.01667 β \tau_{5}=\frac{\beta-\alpha}{60^{\circ}}\approx-0.01667\alpha+0.01667\beta
  4. β > α β>α
  5. τ 4 = 360 - ( α + β ) 360 - 2 θ - 0.00709 α - 0.00709 β + 2.55 \tau_{4}=\frac{360^{\circ}-(\alpha+\beta)}{360^{\circ}-2\theta}\approx-0.00709% \alpha-0.00709\beta+2.55
  6. α α
  7. β β
  8. α α
  9. β β
  10. τ 4 = β - α 360 - θ + 180 - β 180 - θ - 0.00399 α - 0.01019 β + 2.55 \tau_{4}^{\prime}=\frac{\beta-\alpha}{360^{\circ}-\theta}+\frac{180^{\circ}-% \beta}{180^{\circ}-\theta}\approx-0.00399\alpha-0.01019\beta+2.55
  11. β > α β>α
  12. θ = c o s < s u p > 1 ( 1 / 3 ) 109.5 ° θ=cos<sup>−1(−{1}/{3})≈109.5°

Geraniol_isomerase.html

  1. \rightleftharpoons

Germacradienol_synthase.html

  1. \rightleftharpoons

Germacrene_C_synthase.html

  1. \rightleftharpoons

Germanicol_synthase.html

  1. \rightleftharpoons

GERmanium_Detector_Array.html

  1. T 0 ν β β > 2.1 10 25 y r T_{0\nu\beta\beta}>2.1\cdot 10^{25}yr

Gibbons–Tsarev_equation.html

  1. u t u x t - u x u t t + u x x + 1 = 0 ( 1 ) u_{t}u_{xt}-u_{x}u_{tt}+u_{xx}+1=0\qquad(1)
  2. N N
  3. N N
  4. A n A^{n}
  5. p i p_{i}
  6. λ i \lambda_{i}
  7. A 0 A^{0}
  8. p i λ j = - A 0 λ j p i - p j , ( 2 a ) \frac{\partial p_{i}}{\partial\lambda_{j}}=-\frac{\frac{\partial A^{0}}{% \partial\lambda_{j}}}{p_{i}-p_{j}},\qquad(2a)
  9. A 0 λ i λ j = 2 A 0 λ i A 0 λ j ( p i - p j ) 2 . ( 2 b ) \frac{\partial A^{0}}{\partial\lambda_{i}\partial\lambda_{j}}=2\frac{\frac{% \partial A^{0}}{\partial\lambda_{i}}\frac{\partial A^{0}}{\lambda_{j}}}{(p_{i}% -p_{j})^{2}}.\qquad(2b)
  10. i j i\neq j
  11. p p
  12. λ \lambda
  13. D D
  14. λ i \lambda_{i}
  15. λ i \lambda_{i}
  16. p i p_{i}
  17. p λ i = - A 0 λ j p - p i . ( 3 ) \frac{\partial p}{\partial\lambda_{i}}=-\frac{\frac{\partial A^{0}}{\partial% \lambda_{j}}}{p-p_{i}}.\qquad(3)
  18. u ( x , t ) = - 1 2 x 2 + 1 C x + 2 C u(x,t)=-\frac{1}{2}x^{2}+{{}_{C}1}x+{{}_{C}2}
  19. u ( x , t ) = exp ( [ 2 ] c x ) 1 C / [ 2 ] c + x / [ 2 ] c + 2 C + 1 2 [ 2 ] c t 2 + 3 C t + 4 C u(x,t)=\exp({{}_{c}[2]}x)\cdot{{}_{C}1}/{{}_{c}[2]}+x/{{}_{c}[2]}+{{}_{C}2}+% \frac{1}{2}\cdot{{}_{c}[2]}t^{2}+{{}_{C}3}t+{{}_{C}}4
  20. u ( x , t ) = - 1 2 ( 3 C + 1 C x + 2 C t ) 2 / 1 2 C + ( 3 C + 1 C x + 2 C t ) 2 C + 3 C u(x,t)=-\frac{1}{2}({{}_{C}}3+{{}_{C}}1x+{{}_{C}}2t)^{2}/{{}_{C}}1^{2}+({{}_{C% }}3+{{}_{C}}1x+{{}_{C}}2t){{}_{C}}2+{{}_{C}}3
  21. λ N + 1 = i = 0 N ( p - q i ) , \lambda^{N+1}=\prod_{i=0}^{N}(p-q_{i}),
  22. q i q_{i}
  23. i = 0 N q i = 0. \sum_{i=0}^{N}q_{i}=0.
  24. p = p i p=p_{i}
  25. λ = λ i \lambda=\lambda_{i}
  26. A 0 = 1 N + 1 i > j q i q j , A^{0}=\frac{1}{N+1}\sum\sum_{i>j}q_{i}q_{j},
  27. p i p_{i}
  28. A 0 A^{0}

Gilbert–Shannon–Reeds_model.html

  1. ( n k ) / 2 n {\textstyle\left({{n}\atop{k}}\right)}/2^{n}
  2. x 2 x ( mod 1 ) x\mapsto 2x\;\;(\mathop{{\rm mod}}1)
  3. 3 2 log 2 n + θ \tfrac{3}{2}\log_{2}n+\theta
  4. 3 2 log 2 n \tfrac{3}{2}\log_{2}n
  5. log 2 n \log_{2}n
  6. 3 2 log 2 n \tfrac{3}{2}\log_{2}n

Gillies'_conjecture.html

  1. If \,\text{If }
  2. A < B < M p , as B / A and M p , the number of prime divisors of M A<B<\sqrt{M_{p}}\,\text{, as }B/A\,\text{ and }M_{p}\rightarrow\infty\,\text{,% the number of prime divisors of }M
  3. in the interval [ A , B ] is Poisson-distributed with \,\text{ in the interval }[A,B]\,\text{ is Poisson-distributed with}
  4. mean { log ( log B / log A ) if A 2 p log ( log B / log 2 p ) if A < 2 p \,\text{mean }\sim\begin{cases}\log(\log B/\log A)&\,\text{ if }A\geq 2p\\ \log(\log B/\log 2p)&\,\text{ if }A<2p\end{cases}
  5. x x
  6. 2 log 2 log log x ~{}\frac{2}{\log 2}\log\log x
  7. M p M_{p}
  8. x p 2 x x\leq p\leq 2x
  9. 2 \sim 2
  10. M p M_{p}
  11. 2 log 2 p p log 2 ~{}\frac{2\log 2p}{p\log 2}

Giraud_subcategory.html

  1. 𝒜 \mathcal{A}
  2. \mathcal{B}
  3. i : 𝒜 i\colon\mathcal{B}\rightarrow\mathcal{A}
  4. i i
  5. \mathcal{B}
  6. \mathcal{B}
  7. 𝒜 \mathcal{A}
  8. i : 𝒜 i\colon\mathcal{B}\rightarrow\mathcal{A}
  9. \mathcal{B}
  10. X X
  11. \mathcal{B}
  12. i ( X ) i(X)
  13. 𝒜 \mathcal{A}
  14. a : 𝒜 a\colon\mathcal{A}\rightarrow\mathcal{B}
  15. i i
  16. 𝒞 \mathcal{C}
  17. 𝒜 \mathcal{A}
  18. 𝒜 / 𝒞 \mathcal{A}/\mathcal{C}
  19. S : 𝒜 / 𝒞 𝒜 S\colon\mathcal{A}/\mathcal{C}\rightarrow\mathcal{A}
  20. 𝒜 / 𝒞 \mathcal{A}/\mathcal{C}
  21. \mathcal{B}
  22. 𝒞 \mathcal{C}
  23. 𝒜 \mathcal{A}

Gisbert_Hasenjaeger.html

  1. Δ 2 0 \Delta^{0}_{2}
  2. Π 1 0 \Pi^{0}_{1}

GIT_quotient.html

  1. Spec A \operatorname{Spec}A
  2. Spec ( A G ) \operatorname{Spec}(A^{G})
  3. X / / G X/\!/G
  4. Spec \operatorname{Spec}
  5. G / H = G / / H = Spec k [ G ] H G/H=G/\!/H=\operatorname{Spec}k[G]^{H}

Glaeser's_continuity_theorem.html

  1. C 2 C^{2}
  2. f : U \R + f\ :\ U\rightarrow\R^{+}
  3. C 2 C^{2}
  4. \R n \R^{n}
  5. f \sqrt{f}
  6. C 1 C^{1}

Global_cascades_model.html

  1. G 0 ( x ) = k ρ k p k x k , G_{0}(x)=\sum_{k}\rho_{k}p_{k}x^{k},
  2. ρ k = { 1 k = 0 0 1 / k f ( χ ) d χ k > 0 \rho_{k}=\begin{cases}1&k=0\\ \int_{0}^{1/k}f(\chi)\,d\chi&k>0\\ \end{cases}
  3. n = G 0 ( 1 ) + G 0 ( 1 ) 2 z - G 0 ′′ ( 1 ) \langle n\rangle=G_{0}(1)+\frac{G_{0}^{\prime}(1)^{2}}{z-G_{0}^{\prime\prime}(% 1)}
  4. G 0 ′′ ( 1 ) = k k ( k - 1 ) ρ k p k = z G_{0}^{\prime\prime}(1)=\sum_{k}k(k-1)\rho_{k}p_{k}=z
  5. G 0 ′′ ( 1 ) < z G_{0}^{\prime\prime}(1)<z
  6. G 0 ′′ ( 1 ) > z G_{0}^{\prime\prime}(1)>z

Global_mode.html

  1. y ( x , t ) y(x,t)
  2. x x
  3. t t
  4. t t
  5. y ( x , t ) = y ^ ( x ) e i ω t y(x,t)=\hat{y}(x)e^{i\omega t}
  6. ω \omega
  7. ω \omega
  8. x x

Glossary_of_Principia_Mathematica.html

  1. x ^ \hat{x}
  2. R \overrightarrow{R}
  3. x R z x\overrightarrow{R}z
  4. y R z y\overrightarrow{R}z
  5. R \overleftarrow{R}
  6. R \overrightarrow{R}
  7. R \overrightarrow{R}
  8. R \overleftarrow{R}
  9. | |
  10. \upharpoonleft
  11. α R \alpha\upharpoonleft R
  12. \upharpoonright
  13. R α R\upharpoonright\alpha
  14. \uparrow
  15. α P α \alpha\upharpoonleft P\upharpoonright\alpha
  16. | | ||
  17. R | | S R||S
  18. \downarrow
  19. s m ¯ \overline{sm}
  20. R \overleftrightarrow{R}

Glossary_of_set_theory.html

  1. \gimel

Gluon_field.html

  1. s y m b o l 𝒜 n ( 𝐫 , t ) = [ 𝒜 0 n ( 𝐫 , t ) timelike , 𝒜 1 n ( 𝐫 , t ) , 𝒜 2 n ( 𝐫 , t ) , 𝒜 3 n ( 𝐫 , t ) spacelike ] = [ ϕ n ( 𝐫 , t ) , 𝐀 n ( 𝐫 , t ) ] symbol{\mathcal{A}}^{n}(\mathbf{r},t)=[\underbrace{\mathcal{A}^{n}_{0}(\mathbf% {r},t)}_{\,\text{timelike}},\underbrace{\mathcal{A}^{n}_{1}(\mathbf{r},t),% \mathcal{A}^{n}_{2}(\mathbf{r},t),\mathcal{A}^{n}_{3}(\mathbf{r},t)}_{\,\text{% spacelike}}]=[\phi^{n}(\mathbf{r},t),\mathbf{A}^{n}(\mathbf{r},t)]
  2. n = 1 , 2 , 8 n=1,2,...8
  3. 𝐫 \mathbf{r}
  4. 𝒜 α a \mathcal{A}^{a}_{\alpha}
  5. t a = λ a 2 , t_{a}=\frac{\lambda_{a}}{2}\,,
  6. 𝒜 α = t a 𝒜 α a t 1 𝒜 α 1 + t 2 𝒜 α 2 + t 8 𝒜 α 8 \mathcal{A}_{\alpha}=t_{a}\mathcal{A}^{a}_{\alpha}\equiv t_{1}\mathcal{A}^{1}_% {\alpha}+t_{2}\mathcal{A}^{2}_{\alpha}+\cdots t_{8}\mathcal{A}^{8}_{\alpha}
  7. s y m b o l 𝒜 ( 𝐫 , t ) = [ 𝒜 0 ( 𝐫 , t ) , 𝒜 1 ( 𝐫 , t ) , 𝒜 2 ( 𝐫 , t ) , 𝒜 3 ( 𝐫 , t ) ] symbol{\mathcal{A}}(\mathbf{r},t)=[\mathcal{A}_{0}(\mathbf{r},t),\mathcal{A}_{% 1}(\mathbf{r},t),\mathcal{A}_{2}(\mathbf{r},t),\mathcal{A}_{3}(\mathbf{r},t)]
  8. s y m b o l 𝒜 = t a s y m b o l 𝒜 a . symbol{\mathcal{A}}=t_{a}symbol{\mathcal{A}}^{a}\,.
  9. D μ = μ ± i g s t a 𝒜 μ a , D_{\mu}=\partial_{\mu}\pm ig_{s}t_{a}\mathcal{A}^{a}_{\mu}\,,
  10. i i
  11. g s = 4 π α s g_{s}=\sqrt{4\pi\alpha_{s}}
  12. ψ = ( ψ 1 ψ 2 ψ 3 ) , ψ ¯ = ( ψ ¯ 1 * ψ ¯ 2 * ψ ¯ 3 * ) \psi=\begin{pmatrix}\psi_{1}\\ \psi_{2}\\ \psi_{3}\end{pmatrix},\overline{\psi}=\begin{pmatrix}\overline{\psi}^{*}_{1}\\ \overline{\psi}^{*}_{2}\\ \overline{\psi}^{*}_{3}\end{pmatrix}
  13. ψ ψ
  14. ψ ¯ \overline{ψ}
  15. * *
  16. 𝒜 α n \mathcal{A}^{n}_{\alpha}
  17. 𝒜 α n e i θ ¯ ( 𝐫 , t ) ( 𝒜 α n + i g s α ) e - i θ ¯ ( 𝐫 , t ) \mathcal{A}^{n}_{\alpha}\rightarrow e^{i\bar{\theta}(\mathbf{r},t)}\left(% \mathcal{A}^{n}_{\alpha}+\frac{i}{g_{s}}\partial_{\alpha}\right)e^{-i\bar{% \theta}(\mathbf{r},t)}
  18. θ ¯ ( 𝐫 , t ) = t n θ n ( 𝐫 , t ) , \bar{\theta}(\mathbf{r},t)=t_{n}\theta^{n}(\mathbf{r},t)\,,
  19. 𝐫 \mathbf{r}
  20. χ ( 𝐫 , t ) χ(\mathbf{r},t)
  21. A A
  22. A α ( 𝐫 , t ) = A α ( 𝐫 , t ) - α χ ( 𝐫 , t ) A^{\prime}_{\alpha}(\mathbf{r},t)=A_{\alpha}(\mathbf{r},t)-\partial_{\alpha}% \chi(\mathbf{r},t)\,
  23. F F
  24. ψ ( 𝐫 , t ) e i g θ ¯ ( 𝐫 , t ) ψ ( 𝐫 , t ) \psi(\mathbf{r},t)\rightarrow e^{ig\bar{\theta}(\mathbf{r},t)}\psi(\mathbf{r},t)

Glutamate_2,3-aminomutase.html

  1. \rightleftharpoons

Glutinol_synthase.html

  1. \rightleftharpoons

Goldschmidt_alternator.html

  1. f = P U f=PU\,
  2. f = N P U f=NPU\,

Gompertz_constant.html

  1. G G
  2. G = 1 2 - 1 4 - 4 6 - 9 8 - 16 10 - 25 12 - 36 14 - 49 16 - , G=\frac{1}{2-\frac{1}{4-\frac{4}{6-\frac{9}{8-\frac{16}{10-\frac{25}{12-\frac{% 36}{14-\frac{49}{16-\dots}}}}}}}},
  3. G = 1 1 + 1 1 + 1 1 + 2 1 + 2 1 + 3 1 + 3 1 + 4 1 1 + . G=\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{2}{1+\frac{2}{1+\frac{3}{1+\frac{3}{1% +4\frac{1}{1+\dots}}}}}}}}.
  4. G G
  5. G = 0 ln ( 1 + x ) e - x d x = 0 e - x 1 + x d x = 0 1 1 1 - log ( x ) d x . G=\int_{0}^{\infty}\ln(1+x)e^{-x}dx=\int_{0}^{\infty}\frac{e^{-x}}{1+x}dx=\int% _{0}^{1}\frac{1}{1-\log(x)}dx.
  6. G G
  7. G = 0.596347362323194074341078499369279376074 G=0.596347362323194074341078499369279376074\dots
  8. G G
  9. G G
  10. G G
  11. G = - e Ei ( - 1 ) . G=-e\textrm{Ei}(-1).
  12. $\textrm{Ei}$
  13. G = - e ( γ + n = 1 ( - 1 ) n n n ! ) . G=-e\left(\gamma+\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n\cdot n!}\right).
  14. G = n = 0 ln ( n + 1 ) n ! - n = 0 C n + 1 { e n ! } - 1 2 . G=\sum_{n=0}^{\infty}\frac{\ln(n+1)}{n!}-\sum_{n=0}^{\infty}C_{n+1}\{e\cdot n!% \}-\frac{1}{2}.

Goodwin–Staton_integral.html

  1. G ( z ) = 0 e - t 2 t + z d t G(z)=\int_{0}^{\infty}\frac{e^{-t^{2}}}{t+z}\,dt
  2. 4 w ( z ) + 8 z d d z w ( z ) + ( 2 + 2 z 2 ) d 2 d z 2 w ( z ) + z d 3 d z 3 w ( z ) = 0 4w(z)+8\,z\frac{d}{dz}w(z)+(2+2\,z^{2})\frac{d^{2}}{dz^{2}}w(z)+z\frac{d^{3}}{% dz^{3}}w\left(z\right)=0
  3. G ( - z ) = - G ( z ) G(-z)=-G(z)
  4. G ( z ) = ( 1 - γ - ln ( z 2 ) - i csgn ( i z 2 ) π + 2 i π z + ( - 2 + γ + ln ( z 2 ) + i csgn ( i z 2 ) π ) z 2 + - 4 / 3 i π z 3 + ( 5 4 - 1 / 2 γ - 1 / 2 ln ( z 2 ) - 1 / 2 i csgn ( i z 2 ) π ) z 4 + O ( z 5 ) ) G(z)=\left(1-\gamma-\ln(z^{2})-i\operatorname{csgn}(iz^{2})\pi+\frac{2\,i}{% \sqrt{\pi}}z+(-2+\gamma+\ln(z^{2})+i\operatorname{csgn}(iz^{2})\pi\right)z^{2}% +\frac{-4/3\,i}{\sqrt{\pi}}z^{3}+\left(\frac{5}{4}-1/2\,\gamma-1/2\,\ln(z^{2})% -1/2\,i\operatorname{csgn}(iz^{2})\pi\right)z^{4}+O(z^{5}))

Gopakumar–Vafa_invariant.html

  1. g 0 , n 1 , β H 2 ( M , ) G W ( g , β ) q - β λ 2 g - 2 = k > 0 , r 0 , β H 2 ( M , ) B P S ( r , β ) 1 k ( 2 sin ( k λ 2 ) 2 r - 2 q k β ) \sum_{g\geq 0,n\geq 1,\beta\in H^{2}(M,\mathbb{Z})}GW(g,\beta)q^{-\beta}% \lambda^{2g-2}=\sum_{k>0,r\geq 0,\beta\in H^{2}(M,\mathbb{Z})}BPS(r,\beta)% \frac{1}{k}\left(2\sin\left(\frac{k\lambda}{2}\right)^{2r-2}q^{k\beta}\right)
  2. G W ( g , β ) GW(g,\beta)
  3. β \beta
  4. B P S ( r , β ) BPS(r,\beta)
  5. Z t o p = exp [ k > 0 , r 0 , β H 2 ( M , ) B P S ( r , β ) 1 k ( 2 sin ( k λ 2 ) 2 r - 2 q k β t ) ] . Z_{top}=\exp\left[\sum_{\begin{smallmatrix}k>0,\ r\geq 0,\\ \beta\in H^{2}(M,\mathbb{Z})\end{smallmatrix}}BPS(r,\beta)\frac{1}{k}\left(2% \sin\left(\frac{k\lambda}{2}\right)^{2r-2}q^{k\beta\cdot t}\right)\right]\ .

Gödel_operation.html

  1. 𝔉 1 ( X , Y ) = { X , Y } \mathfrak{F}_{1}(X,Y)=\{X,Y\}
  2. 𝔉 2 ( X , Y ) = E X = { ( a , b ) \isin X a \isin b } \mathfrak{F}_{2}(X,Y)=E\cdot X=\{(a,b)\isin X\mid a\isin b\}
  3. 𝔉 3 ( X , Y ) = X - Y \mathfrak{F}_{3}(X,Y)=X-Y
  4. 𝔉 4 ( X , Y ) = X Y = X ( V × Y ) = { ( a , b ) \isin X b \isin Y } \mathfrak{F}_{4}(X,Y)=X\upharpoonright Y=X\cdot(V\times Y)=\{(a,b)\isin X\mid b% \isin Y\}
  5. 𝔉 5 ( X , Y ) = X 𝔇 ( Y ) = { b \isin X a ( a , b ) \isin Y } \mathfrak{F}_{5}(X,Y)=X\cdot\mathfrak{D}(Y)=\{b\isin X\mid\exists a(a,b)\isin Y\}
  6. 𝔉 6 ( X , Y ) = X Y - 1 = { ( a , b ) \isin X ( b , a ) \isin Y } \mathfrak{F}_{6}(X,Y)=X\cdot Y^{-1}=\{(a,b)\isin X\mid(b,a)\isin Y\}
  7. 𝔉 7 ( X , Y ) = X 𝔫 𝔳 2 ( Y ) = { ( a , b , c ) \isin X ( a , c , b ) \isin Y } \mathfrak{F}_{7}(X,Y)=X\cdot\mathfrak{Cnv}_{2}(Y)=\{(a,b,c)\isin X\mid(a,c,b)% \isin Y\}
  8. 𝔉 8 ( X , Y ) = X 𝔫 𝔳 3 ( Y ) = { ( a , b , c ) \isin X ( c , a , b ) \isin Y } \mathfrak{F}_{8}(X,Y)=X\cdot\mathfrak{Cnv}_{3}(Y)=\{(a,b,c)\isin X\mid(c,a,b)% \isin Y\}
  9. G 1 ( X , Y ) = { X , Y } G_{1}(X,Y)=\{X,Y\}
  10. G 2 ( X , Y ) = X × Y G_{2}(X,Y)=X\times Y
  11. G 3 ( X , Y ) = { ( x , y ) x \isin X , y \isin Y , x \isin y } G_{3}(X,Y)=\{(x,y)\mid x\isin X,y\isin Y,x\isin y\}
  12. G 4 ( X , Y ) = X - Y G_{4}(X,Y)=X-Y
  13. G 5 ( X , Y ) = X Y G_{5}(X,Y)=X\cap Y
  14. G 6 ( X ) = X G_{6}(X)=\cup X
  15. G 7 ( X ) = dom ( X ) G_{7}(X)=\,\text{dom}(X)
  16. G 8 ( X ) = { ( x , y ) ( y , x ) \isin X } G_{8}(X)=\{(x,y)\mid(y,x)\isin X\}
  17. G 9 ( X ) = { ( x , y , z ) ( x , z , y ) \isin X } G_{9}(X)=\{(x,y,z)\mid(x,z,y)\isin X\}
  18. G 10 ( X ) = { ( x , y , z ) ( y , z , x ) \isin X } G_{10}(X)=\{(x,y,z)\mid(y,z,x)\isin X\}

Grace–Walsh–Szegő_theorem.html

  1. ζ 1 , , ζ n A \zeta_{1},\ldots,\zeta_{n}\in A
  2. ζ A \zeta\in A
  3. f ( ζ 1 , , ζ n ) = f ( ζ , , ζ ) . f(\zeta_{1},\ldots,\zeta_{n})=f(\zeta,\ldots,\zeta).\,

Graded_(mathematics).html

  1. X X
  2. I I
  3. I I
  4. X = i I X i X=\oplus_{i\in I}X_{i}
  5. X i X_{i}
  6. \mathbb{N}
  7. \mathbb{Z}
  8. X X
  9. 2 \mathbb{Z}_{2}
  10. / 2 \mathbb{Z}/2\mathbb{Z}
  11. \mathbb{Z}
  12. \mathbb{N}
  13. X 0 = X , X i = 0 X_{0}=X,X_{i}=0
  14. i 0 i\neq 0
  15. 0
  16. I I
  17. V = i I V i V=\oplus_{i\in I}V_{i}
  18. R i R_{i}
  19. R i R j R i + j R_{i}R_{j}\subseteq R_{i+j}
  20. i i
  21. \mathbb{N}
  22. \mathbb{Z}
  23. R R
  24. I I
  25. gr I R = n I n / I n + 1 \operatorname{gr}_{I}R=\oplus_{n\in\mathbb{N}}I^{n}/I^{n+1}
  26. M M
  27. i I M i \oplus_{i\in I}M_{i}
  28. R i M j M i + j R_{i}M_{j}\subseteq M_{i+j}
  29. R R
  30. M M
  31. I I
  32. gr I M = n I n M / I n + 1 M \operatorname{gr}_{I}M=\oplus_{n\in\mathbb{N}}I^{n}M/I^{n+1}M
  33. \mathbb{Z}
  34. M M
  35. d : M M : M i M i + 1 d\colon M\to M\colon M_{i}\to M_{i+1}
  36. M M
  37. d d = 0 d\circ d=0
  38. A A
  39. R R
  40. R R
  41. A i R j A i + j R i A j A_{i}R_{j}\subseteq A_{i+j}\supseteq R_{i}A_{j}
  42. d : A A d\colon A\to A
  43. A A
  44. d ( a b ) = ( d a ) b + ( - 1 ) | a | a ( d b ) d(a\cdot b)=(da)\cdot b+(-1)^{|a|}a\cdot(db)
  45. D ( a b ) = D ( a ) b + ε | a | | D | a D ( b ) , ε = ± 1 D(ab)=D(a)b+\varepsilon^{|a||D|}aD(b),\varepsilon=\pm 1
  46. ε \varepsilon
  47. 2 \mathbb{Z}_{2}
  48. y x = ( - 1 ) | x | | y | x y . yx=(-1)^{|x||y|}xy.\,
  49. | a | |a|
  50. a a
  51. 2 \mathbb{Z}_{2}
  52. [ , ] : L i L j L i + j [,]:L_{i}\otimes L_{j}\to L_{i+j}
  53. d : L i L i - 1 d:L_{i}\to L_{i-1}
  54. [ x , y ] = ( - 1 ) | x | | y | + 1 [ y , x ] , [x,y]=(-1)^{|x||y|+1}[y,x],
  55. B W ( F ) BW(F)
  56. 𝒜 \mathcal{A}
  57. 𝒜 \mathcal{A}
  58. 𝒞 \mathcal{C}
  59. F : 𝒞 𝒜 F:\mathcal{C}\rightarrow\mathcal{A}
  60. \mathbb{Z}
  61. P P
  62. ρ : P \rho\colon P\to\mathbb{N}
  63. ρ ( x ) < ρ ( x ) x < y \rho(x)<\rho(x)\implies x<y
  64. y y
  65. x ρ ( y ) = ρ ( x ) + 1 x\implies\rho(y)=\rho(x)+1

GRADELA.html

  1. σ i j = ( λ ε k k δ i j + 2 μ ε i j ) - l s 2 Δ ( λ ε k k δ i j + 2 μ ε i j ) , \sigma_{ij}=\Bigl(\lambda\varepsilon_{kk}\delta_{ij}+2\mu\varepsilon_{ij}\Bigr% )-l^{2}_{s}\,\Delta\,\Bigl(\lambda\varepsilon_{kk}\delta_{ij}+2\mu\varepsilon_% {ij}\Bigr),
  2. l s l_{s}

Graph_amalgamation.html

  1. G G
  2. H H
  3. G G
  4. H H
  5. H H
  6. G G
  7. ϕ : E ( G ) E ( H ) \phi:E(G)\to E(H)
  8. ψ : V ( G ) V ( H ) \psi:V(G)\to V(H)
  9. x x
  10. y y
  11. G G
  12. ψ ( x ) ψ ( y ) \psi(x)\neq\psi(y)
  13. x x
  14. y y
  15. e e
  16. G G
  17. ψ ( x ) \psi(x)
  18. ψ ( y ) \psi(y)
  19. ϕ ( e ) \phi(e)
  20. H H
  21. e e
  22. x V ( G ) x\in V(G)
  23. ϕ ( e ) \phi(e)
  24. ψ ( x ) H \psi(x)\in H
  25. e e
  26. x , y V ( G ) x,y\in V(G)
  27. x y x\neq y
  28. ψ ( x ) = ψ ( y ) \psi(x)=\psi(y)
  29. ψ ( e ) \psi(e)
  30. ψ ( x ) \psi(x)
  31. G G
  32. H H
  33. G G
  34. K 2 n + 1 K_{2n+1}
  35. H H
  36. K 5 K_{5}
  37. ϕ \phi
  38. ψ \psi
  39. v V ( G ) v\in V(G)
  40. ϕ ( v ) \phi(v)
  41. v 1 v_{1}
  42. u 2 u_{2}
  43. v 2 v_{2}
  44. u 2 u_{2}
  45. v 3 v_{3}
  46. u 1 u_{1}
  47. v 4 v_{4}
  48. u 3 u_{3}
  49. v 5 v_{5}
  50. u 2 u_{2}
  51. n n
  52. K 2 n + 1 K_{2n+1}

Graph_C*-algebra.html

  1. E = ( E 0 , E 1 , r , s ) E=(E^{0},E^{1},r,s)
  2. E 0 E^{0}
  3. E 1 E^{1}
  4. r , s : E 1 E 0 r,s:E^{1}\rightarrow E^{0}
  5. E E
  6. C * ( E ) C^{*}(E)
  7. { p v : v E 0 } \{p_{v}:v\in E^{0}\}
  8. { s e : e E 1 } \{s_{e}:e\in E^{1}\}
  9. s e * s e = p r ( e ) s_{e}^{*}s_{e}=p_{r(e)}
  10. e E 1 e\in E^{1}
  11. p v = s ( e ) = v s e s e * p_{v}=\sum_{s(e)=v}s_{e}s_{e}^{*}
  12. 0 < | s - 1 ( v ) | < 0<|s^{-1}(v)|<\infty
  13. s e s e * p s ( e ) s_{e}s_{e}^{*}\leq p_{s(e)}
  14. e E 1 e\in E^{1}
  15. \mathbb{C}
  16. C ( S 1 ) C(S^{1})
  17. M n ( ) M_{n}(\mathbb{C})
  18. \mathbb{C}
  19. M n ( C ( S 1 ) ) M_{n}(C(S^{1}))
  20. C ( S 1 ) C(S^{1})
  21. 𝒦 \mathcal{K}
  22. 𝒯 \mathcal{T}
  23. 𝒪 n \mathcal{O}_{n}

Graph_coloring_game.html

  1. G G
  2. χ g ( G ) \chi_{g}(G)
  3. G G
  4. G G
  5. χ ( G ) χ g ( G ) Δ ( G ) + 1 \chi(G)\leq\chi_{g}(G)\leq\Delta(G)+1
  6. χ ( G ) \chi(G)
  7. G G
  8. Δ ( G ) \Delta(G)
  9. G G
  10. k k
  11. χ g ( G ) k ( k + 1 ) \chi_{g}(G)\leq k(k+1)
  12. G G
  13. χ g ( G ) c o l g ( G ) \chi_{g}(G)\leq col_{g}(G)
  14. c o l g ( G ) col_{g}(G)
  15. G G
  16. G G
  17. c c
  18. χ g ( G ) 4 + c \chi_{g}(G)\leq 4+c
  19. 𝒞 {\mathcal{C}}
  20. χ g ( 𝒞 ) \chi_{g}({\mathcal{C}})
  21. k k
  22. G G
  23. 𝒞 {\mathcal{C}}
  24. χ g ( G ) k \chi_{g}(G)\leq k
  25. χ g ( 𝒞 ) \chi_{g}({\mathcal{C}})
  26. χ g ( ) = 4 \chi_{g}({\mathcal{F}})=4
  27. χ g ( 𝒞 ) = 5 \chi_{g}({\mathcal{C}})=5
  28. 6 χ g ( 𝒪 ) 7 6\leq\chi_{g}({\mathcal{O}})\leq 7
  29. 7 χ g ( 𝒫 ) 17 7\leq\chi_{g}({\mathcal{P}})\leq 17
  30. χ g ( 𝒫 4 ) 13 \chi_{g}({\mathcal{P}}_{4})\leq 13
  31. χ g ( 𝒫 5 ) 8 \chi_{g}({\mathcal{P}}_{5})\leq 8
  32. χ g ( 𝒫 6 ) 6 \chi_{g}({\mathcal{P}}_{6})\leq 6
  33. χ g ( 𝒫 8 ) 5 \chi_{g}({\mathcal{P}}_{8})\leq 5
  34. χ g ( 𝒯 G ) = 5 \chi_{g}({\mathcal{T}G})=5
  35. χ g ( 𝒯 k ) 3 k + 2 \chi_{g}({\mathcal{T}}_{k})\leq 3k+2
  36. 2 ω χ g ( ) 3 ω - 2 2\omega\leq\chi_{g}({\mathcal{I}})\leq 3\omega-2
  37. ω \omega
  38. G H G\square H
  39. χ g ( G ) \chi_{g}(G)
  40. χ g ( H ) \chi_{g}(H)
  41. K n , n K_{n,n}
  42. χ g ( K n , n K m , m ) \chi_{g}(K_{n,n}\square K_{m,m})
  43. n , m n,m
  44. χ g ( K 2 P 1 ) = 2 \chi_{g}(K_{2}\square P_{1})=2
  45. χ g ( K 2 P k ) = 3 \chi_{g}(K_{2}\square P_{k})=3
  46. k { 2 , 3 } k\in\{2,3\}
  47. χ g ( K 2 P k ) = 4 \chi_{g}(K_{2}\square P_{k})=4
  48. k 4 k\geq 4
  49. k 3 k\geq 3
  50. χ g ( K 2 C k ) = 4 \chi_{g}(K_{2}\square C_{k})=4
  51. k k
  52. χ g ( K 2 K k ) = k + 1 \chi_{g}(K_{2}\square K_{k})=k+1
  53. χ g ( T 1 T 2 ) 12 \chi_{g}(T_{1}\square T_{2})\leq 12
  54. χ g ( S m P 1 ) = 2 \chi_{g}(S_{m}\square P_{1})=2
  55. χ g ( S m P 2 ) = 3 \chi_{g}(S_{m}\square P_{2})=3
  56. χ g ( S m P n ) = χ g ( S m C n ) = 4 \chi_{g}(S_{m}\square P_{n})=\chi_{g}(S_{m}\square C_{n})=4
  57. n 3 n\geq 3
  58. χ g ( P 2 W n ) = 5 \chi_{g}(P_{2}\square W_{n})=5
  59. n 9 n\geq 9
  60. χ g ( P 2 K m , n ) = 5 \chi_{g}(P_{2}\square K_{m,n})=5
  61. m , n 5 m,n\geq 5
  62. G G
  63. χ g ( G ) \chi^{\prime}_{g}(G)
  64. G G
  65. χ ( G ) χ g ( G ) 2 Δ ( G ) - 1 \chi^{\prime}(G)\leq\chi^{\prime}_{g}(G)\leq 2\Delta(G)-1
  66. χ g ( G ) > 1.008 Δ ( G ) \chi^{\prime}_{g}(G)>1.008\Delta(G)
  67. Δ ( G ) \Delta(G)
  68. ϵ > 0 \epsilon>0
  69. G G
  70. χ g ( G ) ( 2 - ϵ ) Δ ( G ) \chi^{\prime}_{g}(G)\leq(2-\epsilon)\Delta(G)
  71. Δ ( G ) \Delta(G)
  72. G G
  73. a ( G ) a(G)
  74. G G
  75. G G
  76. Δ ( G ) \Delta(G)
  77. χ g ( G ) Δ ( G ) + 3 a ( G ) - 1 \chi^{\prime}_{g}(G)\leq\Delta(G)+3a(G)-1
  78. 𝒞 {\mathcal{C}}
  79. χ g ( 𝒞 ) \chi^{\prime}_{g}({\mathcal{C}})
  80. k k
  81. G G
  82. 𝒞 {\mathcal{C}}
  83. χ g ( G ) k \chi^{\prime}_{g}(G)\leq k
  84. χ g ( 𝒞 ) \chi^{\prime}_{g}({\mathcal{C}})
  85. χ g ( W 3 ) = 5 \chi^{\prime}_{g}(W_{3})=5
  86. χ g ( W n ) = n + 1 \chi^{\prime}_{g}(W_{n})=n+1
  87. n 4 n\geq 4
  88. χ g ( Δ ) Δ + 1 \chi^{\prime}_{g}({\mathcal{F}}_{\Delta})\leq\Delta+1
  89. Δ 4 \Delta\neq 4
  90. 5 χ g ( 4 ) 6 5\leq\chi^{\prime}_{g}({\mathcal{F}}_{4})\leq 6
  91. F F
  92. 4 {\mathcal{F}}_{4}
  93. χ g ( F ) 5 \chi^{\prime}_{g}(F)\leq 5
  94. c 2 c\geq 2
  95. χ g ( G ) Δ ( G ) + c \chi^{\prime}_{g}(G)\leq\Delta(G)+c
  96. G G
  97. c = 2 c=2
  98. ϵ > 0 \epsilon>0
  99. d 0 d_{0}
  100. G G
  101. δ ( G ) d 0 \delta(G)\geq d_{0}
  102. χ g ( G ) ( 1 + ϵ ) δ ( G ) \chi^{\prime}_{g}(G)\geq(1+\epsilon)\delta(G)
  103. G G
  104. i g ( G ) i_{g}(G)
  105. G G
  106. G G
  107. Δ \Delta
  108. 3 Δ - 1 2 < i g ( G ) < 3 Δ - 1 \frac{3\Delta-1}{2}<i_{g}(G)<3\Delta-1
  109. G G
  110. a a
  111. d d
  112. i g ( G ) 3 Δ ( G ) - a 2 + 8 a + 3 d - 1 i_{g}(G)\leq\left\lfloor\frac{3\Delta(G)-a}{2}\right\rfloor+8a+3d-1
  113. Δ ( G ) 5 a + 6 d \Delta(G)\geq 5a+6d
  114. i g ( G ) 3 Δ ( G ) - a 2 + 8 a + d - 1 i_{g}(G)\leq\left\lfloor\frac{3\Delta(G)-a}{2}\right\rfloor+8a+d-1
  115. G G
  116. Δ ( G ) \Delta(G)
  117. i g ( G ) 2 Δ ( G ) + 4 k - 2 i_{g}(G)\leq 2\Delta(G)+4k-2
  118. i g ( G ) 2 Δ ( G ) + 3 k - 1 i_{g}(G)\leq 2\Delta(G)+3k-1
  119. Δ ( G ) 5 k - 1 \Delta(G)\geq 5k-1
  120. i g ( G ) Δ ( G ) + 8 k - 2 i_{g}(G)\leq\Delta(G)+8k-2
  121. Δ ( G ) 5 k - 1 \Delta(G)\leq 5k-1
  122. 𝒞 {\mathcal{C}}
  123. i g ( 𝒞 ) i_{g}({\mathcal{C}})
  124. k k
  125. G G
  126. 𝒞 {\mathcal{C}}
  127. i g ( G ) k i_{g}(G)\leq k
  128. k 13 k\geq 13
  129. i g ( P k ) = 5 i_{g}(P_{k})=5
  130. k 3 k\geq 3
  131. i g ( C k ) = 5 i_{g}(C_{k})=5
  132. k 1 k\geq 1
  133. i g ( S 2 k ) = 3 k i_{g}(S_{2k})=3k
  134. k 6 k\geq 6
  135. i g ( W 2 k + 1 ) = 3 k + 2 i_{g}(W_{2k+1})=3k+2
  136. k 7 k\geq 7
  137. i g ( W 2 k ) = 3 k i_{g}(W_{2k})=3k
  138. k 13 k\geq 13
  139. G G
  140. W k W_{k}
  141. S k S_{k}
  142. i g ( G ) = 3 k 2 i_{g}(G)=\left\lceil\frac{3k}{2}\right\rceil
  143. i g ( G ) < 3 Δ ( G ) - 1 i_{g}(G)<3\Delta(G)-1
  144. Δ ( G ) \Delta(G)

Graph_realization_problem.html

  1. ( d 1 , , d n ) (d_{1},\dots,d_{n})
  2. ( d 1 , , d n ) (d_{1},\dots,d_{n})
  3. n n
  4. ( d 1 , , d n ) (d_{1},\ldots,d_{n})

Graphene_applications_as_optical_lenses.html

  1. h h
  2. " e 2 " / " h " "e^{2}"/"h"
  3. σ uni = ( π ) e 2 2 h \sigma_{\,\text{uni}}=\frac{(\pi)e^{2}}{2h}
  4. R s = t σ R_{\,\text{s}}=\frac{t}{\sigma}
  5. σ = d . c . c o n d u c t i v i t y \sigma={d.c.conductivity}

Graphical_lasso.html

  1. X 1 , X 2 , , X n X_{1},X_{2},...,X_{n}
  2. X N ( 0 , Σ ) X\sim N(0,\Sigma)
  3. Θ = Σ - 1 \Theta=\Sigma^{-1}
  4. Θ ^ \hat{\Theta}
  5. Θ ^ = argmin Θ 0 ( t r ( S Θ ) - log det ( Θ ) + λ j k | Θ j k | ) \hat{\Theta}=\underset{\Theta\geq 0}{\operatorname{argmin}}(tr(S\Theta)-\log% \det(\Theta)+\lambda\sum_{j\neq k}{|\Theta_{jk}|})
  6. S S
  7. λ \lambda

Graphoid.html

  1. I ( X , Z , Y ) I ( Y , Z , X ) I(X,Z,Y)\Leftrightarrow I(Y,Z,X)
  2. I ( X , Z , Y W ) I ( X , Z , Y ) & I ( X , Z , W ) I(X,Z,Y\cup W)\Rightarrow I(X,Z,Y)~{}\&~{}I(X,Z,W)
  3. I ( X , Z , Y W ) I ( X , Z W , Y ) I(X,Z,Y\cup W)\Rightarrow I(X,Z\cup W,Y)
  4. I ( X , Z , Y ) & I ( X , Z Y , W ) I ( X , Z , Y W ) I(X,Z,Y)~{}\&~{}I(X,Z\cup Y,W)\Rightarrow I(X,Z,Y\cup W)
  5. I ( X , Z W , Y ) & I ( X , Z Y , W ) I ( X , Z , Y W ) I(X,Z\cup W,Y)~{}\&~{}I(X,Z\cup Y,W)\Rightarrow I(X,Z,Y\cup W)
  6. I ( X , Z , Y ) iff P ( X , Y Z ) = P ( X Z ) I(X,Z,Y)\,\text{ iff }P(X,Y\mid Z)=P(X\mid Z)
  7. I c ( X , Y , Z ) ρ x y . z = 0 for every x X and y Y I_{c}(X,Y,Z)\Leftrightarrow\rho_{xy.z}=0\,\text{ for every }x\in X\,\text{ and% }y\in Y
  8. ρ x y . z \rho_{xy.z}
  9. P ( X , Z ) > 0 & P ( Y , Z ) > 0 P ( X , Y , Z ) > 0. P(X,Z)>0~{}\&~{}P(Y,Z)>0\implies P(X,Y,Z)>0.
  10. I ( X , Z , Y ) X , Z , Y G , I(X,Z,Y)\Leftrightarrow\langle X,Z,Y\rangle_{G},
  11. I ( X , Z , Y ) I ( X , Z W , Y ) I(X,Z,Y)\implies I(X,Z\cup W,Y)
  12. I ( X , Z , Y ) I ( X , Z , γ ) or I ( γ , Z , Y ) γ X Y Z . I(X,Z,Y)\implies I(X,Z,\gamma)\,\text{ or }I(\gamma,Z,Y)~{}~{}\forall~{}~{}% \gamma\notin X\cup Y\cup Z.
  13. I ( X , Z , Y ) X , Z , Y D I(X,Z,Y)\Leftrightarrow\langle X,Z,Y\rangle_{D}
  14. X , Z , Y D \langle X,Z,Y\rangle_{D}

Greek_and_Roman_artillery.html

  1. 1.1 ( 100 W ) 1 / 3 , 1.1(100W)^{1/3},

Griewank_function.html

  1. 1 + 1 4000 i = 1 n x i 2 - i = 1 n cos ( x i i ) 1+\frac{1}{4000}\sum_{i=1}^{n}x_{i}^{2}-\prod_{i=1}^{n}\cos\left(\frac{x_{i}}{% \sqrt{i}}\right)
  2. g := 1 + ( 1 / 4000 ) x [ 1 ] 2 - cos ( x [ 1 ] ) g:=1+(1/4000)\cdot x[1]^{2}-\cos(x[1])
  3. 1 2000 x [ 1 ] + sin ( x [ 1 ] ) = 0 \frac{1}{2000}\cdot x[1]+\sin(x[1])=0
  4. 1 + 1 4000 x 1 2 + 1 4000 x 2 2 - cos ( x 1 ) cos ( 1 2 x 2 2 ) 1+\frac{1}{4000}x_{1}^{2}+\frac{1}{4000}x_{2}^{2}-\cos(x_{1})\cos\left(\frac{1% }{2}x_{2}\sqrt{2}\right)
  5. { 1 + 1 4000 x 1 2 + 1 4000 x 2 2 + 1 4000 x < m t p l > 3 2 - cos ( x 1 ) cos ( 1 2 x 2 2 ) cos ( 1 3 x 3 3 ) } \left\{1+\frac{1}{4000}\,x_{1}^{2}+\frac{1}{4000}\,x_{2}^{2}+\frac{1}{4000}\,{% x_{<}mtpl>{{3}}}^{2}-\cos(x_{1})\cos\left(\frac{1}{2}x_{2}\sqrt{2}\right)\cos% \left(\frac{1}{3}x_{3}\sqrt{3}\right)\right\}

Griffiths_group.html

  1. Griff k ( X ) := Z k ( X ) hom / Z k ( X ) alg \operatorname{Griff}^{k}(X):=Z^{k}(X)_{\mathrm{hom}}/Z^{k}(X)_{\mathrm{alg}}
  2. Z k ( X ) Z^{k}(X)
  3. 𝐏 4 \mathbf{P}^{4}
  4. Griff 2 ( X ) \operatorname{Griff}^{2}(X)

Gromov_boundary.html

  1. O O
  2. X X
  3. γ : [ 0 , ) X \gamma:[0,\infty)\rightarrow X
  4. γ ( [ 0 , t ] ) \gamma([0,t])
  5. O O
  6. γ ( t ) \gamma(t)
  7. γ 1 , γ 2 \gamma_{1},\gamma_{2}
  8. K K
  9. d ( γ 1 ( t ) , γ 2 ( t ) ) K d(\gamma_{1}(t),\gamma_{2}(t))\leq K
  10. t t
  11. γ \gamma
  12. [ γ ] [\gamma]
  13. X X
  14. X = { [ γ ] | γ \partial X=\{[\gamma]|\gamma
  15. X } X\}
  16. x , y , z x,y,z
  17. ( x , y ) z = 1 / 2 ( d ( x , z ) + d ( y , z ) - d ( x , y ) ) (x,y)_{z}=1/2(d(x,z)+d(y,z)-d(x,y))
  18. z z
  19. x x
  20. y y
  21. z z
  22. x x
  23. y y
  24. p p
  25. V ( p , r ) = { q X | V(p,r)=\{q\in\partial X|
  26. γ 1 , γ 2 \gamma_{1},\gamma_{2}
  27. [ γ 1 ] = p , [ γ 2 ] = q [\gamma_{1}]=p,[\gamma_{2}]=q
  28. lim inf s , t ( γ 1 ( s ) , γ 2 ( t ) ) O r } \lim\inf_{s,t\rightarrow\infty}(\gamma_{1}(s),\gamma_{2}(t))_{O}\geq r\}
  29. r r
  30. G G
  31. G G
  32. G G
  33. X X

Gross_substitutes.html

  1. X X
  2. Y Y
  3. Δ demand ( X ) Δ price ( Y ) > 0 \frac{\Delta\,\text{demand}(X)}{\Delta\,\text{price}(Y)}>0
  4. Δ demand ( X ) Δ price ( Y ) 0 \frac{\Delta\,\text{demand}(X)}{\Delta\,\text{price}(Y)}\geq 0

Group-envy-free.html

  1. A < i B A<_{i}B
  2. { A i } i X \{A_{i}\}_{i\in X}
  3. { B i } i X \{B_{i}\}_{i\in X}
  4. A i < i B i A_{i}<_{i}B_{i}
  5. i Y A i \cup_{i\in Y}{A_{i}}

Grouped_Dirichlet_distribution.html

  1. 𝒯 n = { ( x 1 , x n ) | x i 0 , i = 1 , , n , i = 1 n x n = 1 } \mathcal{T}_{n}=\left\{\left(x_{1},\ldots x_{n}\right)\left|x_{i}\geq 0,i=1,% \cdots,n,\sum_{i=1}^{n}x_{n}=1\right.\right\}
  2. 𝐱 𝒯 n \mathbf{x}\in\mathcal{T}_{n}
  3. 𝐱 - n = ( x 1 , , x n - 1 ) \mathbf{x}_{-n}=\left(x_{1},\ldots,x_{n-1}\right)
  4. n - 1 n-1
  5. 𝒯 n \mathcal{T}_{n}
  6. 𝐱 \mathbf{x}
  7. GD n , 2 , s ( 𝐱 - n | 𝐚 , 𝐛 ) = ( i = 1 n x i a i - 1 ) ( i = 1 s x i ) b 1 ( i = s + 1 n x i ) b 2 B ( a 1 , , a s ) B ( a s + 1 , , a n ) B ( b 1 + i = 1 s a i , b 2 + i = s + 1 n a i ) \operatorname{GD}_{n,2,s}\left(\left.\mathbf{x}_{-n}\right|\mathbf{a},\mathbf{% b}\right)=\frac{\left(\prod_{i=1}^{n}x_{i}^{a_{i}-1}\right)\cdot\left(\prod_{i% =1}^{s}x_{i}\right)^{b_{1}}\cdot\left(\prod_{i=s+1}^{n}x_{i}\right)^{b_{2}}}{B% \left(a_{1},\ldots,a_{s}\right)\cdot B\left(a_{s+1},\ldots,a_{n}\right)\cdot B% \left(b_{1}+\sum_{i=1}^{s}a_{i},b_{2}+\sum_{i=s+1}^{n}a_{i}\right)}
  8. B ( 𝐚 ) B\left(\mathbf{a}\right)
  9. 𝐱 - n \mathbf{x}_{-n}
  10. GD n , m , 𝐬 ( 𝐱 - n | 𝐚 , 𝐛 ) = c m - 1 ( i = 1 n x i a i - 1 ) j = 1 m ( k = s j - 1 + 1 s j x k ) b j \operatorname{GD}_{n,m,\mathbf{s}}\left(\left.\mathbf{x}_{-n}\right|\mathbf{a}% ,\mathbf{b}\right)=c_{m}^{-1}\cdot\left(\prod_{i=1}^{n}x_{i}^{a_{i}-1}\right)% \cdot\prod_{j=1}^{m}\left(\sum_{k=s_{j-1}+1}^{s_{j}}x_{k}\right)^{b_{j}}
  11. 𝐬 = ( s 1 , , s m ) \mathbf{s}=\left(s_{1},\ldots,s_{m}\right)
  12. 0 = s 0 < s 1 s m = n 0=s_{0}<s_{1}\leqslant\cdots\leqslant s_{m}=n
  13. c m = { j = 1 m B ( a s j - 1 + 1 , , a s j ) } B ( b 1 + k = 1 s 1 , , b m + k = s m - 1 + 1 s m a k ) c_{m}=\left\{\prod_{j=1}^{m}B\left(a_{s_{j-1}+1},\ldots,a_{s_{j}}\right)\right% \}\cdot B\left(b_{1}+\sum_{k=1}^{s_{1}},\ldots,b_{m}+\sum_{k=s_{m-1}+1}^{s_{m}% }a_{k}\right)

Groupoid_scheme.html

  1. R , U R,U
  2. s , t : R U , e : U R , m : R × U , s , t R R , i : R R s,t:R\to U,e:U\to R,m:R\times_{U,s,t}R\to R,i:R\to R
  3. s e , t e s\circ e,t\circ e
  4. s m = s p 1 , t m = t p 2 s\circ m=s\circ p_{1},t\circ m=t\circ p_{2}
  5. R U R\rightrightarrows U
  6. R = U × G R=U\times G
  7. [ R U ] [R\rightrightarrows U]
  8. ( R U ) (R\rightrightarrows U)

Growth_curve_(statistics).html

  1. X = A B C + Σ 1 / 2 E X=ABC+\Sigma^{1/2}E

Gyratory_equipment.html

  1. E f f i c i e n c y F a c t o r = m o M o C ( 1 - m o ) Efficiency\;Factor=\frac{m_{o}\;M_{o}}{C\;(1-m_{o})}
  2. m o m_{\,\text{o}}
  3. M o M_{\,\text{o}}
  4. F u ( θ ) = [ c o s [ ( θ ( t ) + ϕ ) d D u ] m u r u ( d N π 30 D u ) 2 η x s i n [ ( θ ( t ) + ϕ ) d D u ] m u r u ( d N π 30 D u ) 2 η y - m u g η z ] F_{u}(\theta)=\begin{bmatrix}{cos[(\theta(t)+\phi)\frac{d}{D_{u}}]}\;m_{u}\;r_% {u}\;(\frac{dN\pi}{30D_{u}})^{2}\;\eta_{x}\\ {sin[(\theta(t)+\phi)\frac{d}{D_{u}}]}\;m_{u}\;r_{u}\;(\frac{dN\pi}{30D_{u}})^% {2}\;\eta_{y}\\ -m_{u}\;g\;\eta_{z}\end{bmatrix}
  5. F n ( θ ) = [ c o s [ - ( θ ( t ) - ϕ ) d D n ] m n r n ( d N π 30 D n ) 2 η x s i n [ - ( θ ( t ) - ϕ ) d D n ] m n r n ( d N π 30 D n ) 2 η y - m n g η z ] F_{n}(\theta)=\begin{bmatrix}{cos[-(\theta(t)-\phi)\frac{d}{D_{n}}]}\;m_{n}\;r% _{n}\;(\frac{dN\pi}{30D_{n}})^{2}\;\eta_{x}\\ {sin[-(\theta(t)-\phi)\frac{d}{D_{n}}]}\;m_{n}\;r_{n}\;(\frac{dN\pi}{30D_{n}})% ^{2}\;\eta_{y}\\ -m_{n}\;g\;\eta_{z}\end{bmatrix}
  6. θ \theta
  7. ϕ \phi
  8. m m
  9. N N
  10. η \eta
  11. v c = ( D a - d p 2 ) g 2 D a v_{c}=(D_{a}-\frac{d_{p}}{2})\sqrt{\frac{g}{2D_{a}}}
  12. D a D_{\,\text{a}}
  13. d p d_{\,\text{p}}

Half-band_filter.html

  1. f s / 4 , \scriptstyle f_{s}/4,
  2. f s \scriptstyle f_{s}

Half-exponential_function.html

  1. f ( f ( x ) ) = a b x . f(f(x))=ab^{x}.\,
  2. A A
  3. ( 0 , 1 ) (0,1)
  4. [ 0 , A ] [0,A]
  5. [ A , 1 ] [A,1]
  6. f f
  7. f ( f ( x ) ) = exp x f(f(x))=\exp x
  8. f ( x ) = { g ( x ) if x [ 0 , A ] , exp ( g - 1 ( x ) ) if x ( A , 1 ] , exp ( f ( ln ( x ) ) ) if x ( 1 , ) , ln ( f ( exp ( x ) ) ) if x ( - , 0 ) . f(x)=\begin{cases}g(x)&\mbox{if }~{}x\in[0,A],\\ \exp(g^{-1}(x))&\mbox{if }~{}x\in(A,1],\\ \exp(f(\ln(x)))&\mbox{if }~{}x\in(1,\infty),\\ \ln(f(\exp(x)))&\mbox{if }~{}x\in(-\infty,0).\\ \end{cases}

Halimadienyl-diphosphate_synthase.html

  1. \rightleftharpoons

Hall–Petresco_identity.html

  1. x m y m = ( x y ) m c 2 ( m 2 ) c 3 ( m 3 ) c m - 1 ( m m - 1 ) c m x^{m}y^{m}=(xy)^{m}c_{2}^{{\left({{m}\atop{2}}\right)}}c_{3}^{{\left({{m}\atop% {3}}\right)}}\cdots c_{m-1}^{{\left({{m}\atop{m-1}}\right)}}c_{m}

Hamiltonian_field_theory.html

  1. φ ( 𝐱 , t ) φ(\mathbf{x},t)
  2. ( ϕ , π , 𝐱 , t ) = ϕ ˙ π - ( ϕ , ϕ , ϕ / t , 𝐱 , t ) . \mathcal{H}(\phi,\pi,\mathbf{x},t)=\dot{\phi}\pi-\mathcal{L}(\phi,\nabla\phi,% \partial\phi/\partial t,\mathbf{x},t)\,.
  3. 𝐱 \mathbf{x}
  4. t t
  5. φ ( 𝐱 , t ) φ(\mathbf{x},t)
  6. π ( 𝐱 , t ) π(\mathbf{x},t)
  7. π = ϕ ˙ , ϕ ˙ ϕ t , \pi=\frac{\partial\mathcal{L}}{\partial\dot{\phi}}\,,\quad\dot{\phi}\equiv% \frac{\partial\phi}{\partial t}\,,
  8. / t ∂/∂t
  9. d / d t d/dt
  10. ( ϕ 1 , ϕ 2 , , π 1 , π 2 , , 𝐱 , t ) = i ϕ i ˙ π i - ( ϕ 1 , ϕ 2 , ϕ 1 , ϕ 2 , , ϕ 1 / t , ϕ 2 / t , , 𝐱 , t ) . \mathcal{H}(\phi_{1},\phi_{2},\ldots,\pi_{1},\pi_{2},\ldots,\mathbf{x},t)=\sum% _{i}\dot{\phi_{i}}\pi_{i}-\mathcal{L}(\phi_{1},\phi_{2},\ldots\nabla\phi_{1},% \nabla\phi_{2},\ldots,\partial\phi_{1}/\partial t,\partial\phi_{2}/\partial t,% \ldots,\mathbf{x},t)\,.
  11. π i ( 𝐱 , t ) = ϕ ˙ i . \pi_{i}(\mathbf{x},t)=\frac{\partial\mathcal{L}}{\partial\dot{\phi}_{i}}\,.
  12. H = d 3 x . H=\int d^{3}x\mathcal{H}\,.
  13. δ δ ϕ i = ϕ i - ( ϕ i ) - t ( ϕ i / t ) , \frac{\delta}{\delta\phi_{i}}=\frac{\partial}{\partial\phi_{i}}-\nabla\cdot% \frac{\partial}{\partial(\nabla\phi_{i})}-\frac{\partial}{\partial t}\frac{% \partial}{\partial(\partial\phi_{i}/\partial t)}\,,
  14. δ δ ϕ i = ϕ i - μ ( μ ϕ i ) \frac{\delta}{\delta\phi_{i}}=\frac{\partial}{\partial\phi_{i}}-\partial_{\mu}% \frac{\partial}{\partial(\partial_{\mu}\phi_{i})}
  15. A = d 3 x 𝒜 ( ϕ 1 , ϕ 2 , , π 1 , π 2 , , ϕ 1 , ϕ 2 , , π 1 , π 2 , , ϕ 1 t , ϕ 2 t , , π 1 t , π 2 t , , 𝐱 , t ) , A=\int d^{3}x\mathcal{A}\left(\phi_{1},\phi_{2},\ldots,\pi_{1},\pi_{2},\ldots,% \nabla\phi_{1},\nabla\phi_{2},\ldots,\nabla\pi_{1},\nabla\pi_{2},\ldots,\frac{% \partial\phi_{1}}{\partial t},\frac{\partial\phi_{2}}{\partial t},\ldots,\frac% {\partial\pi_{1}}{\partial t},\frac{\partial\pi_{2}}{\partial t},\ldots,% \mathbf{x},t\right)\,,
  16. B = d 3 x ( ϕ 1 , ϕ 2 , , π 1 , π 2 , , ϕ 1 , ϕ 2 , , π 1 , π 2 , , ϕ 1 t , ϕ 2 t , , π 1 t , π 2 t , , 𝐱 , t ) , B=\int d^{3}x\mathcal{B}\left(\phi_{1},\phi_{2},\ldots,\pi_{1},\pi_{2},\ldots,% \nabla\phi_{1},\nabla\phi_{2},\ldots,\nabla\pi_{1},\nabla\pi_{2},\ldots,\frac{% \partial\phi_{1}}{\partial t},\frac{\partial\phi_{2}}{\partial t},\ldots,\frac% {\partial\pi_{1}}{\partial t},\frac{\partial\pi_{2}}{\partial t},\ldots,% \mathbf{x},t\right)\,,
  17. [ A , B ] ϕ , π = d 3 x i ( 𝒜 ϕ i π i - ϕ i 𝒜 π i ) . [A,B]_{\phi,\pi}=\int d^{3}x\sum_{i}\left(\frac{\partial\mathcal{A}}{\phi_{i}}% \frac{\partial\mathcal{B}}{\pi_{i}}-\frac{\partial\mathcal{B}}{\phi_{i}}\frac{% \partial\mathcal{A}}{\pi_{i}}\right)\,.
  18. A A
  19. B B
  20. d A d t = [ A , H ] + A t \frac{dA}{dt}=[A,H]+\frac{\partial A}{\partial t}
  21. A A
  22. = 𝒯 + 𝒱 . \mathcal{H}=\mathcal{T}+\mathcal{V}\,.
  23. t + 𝐒 = 0 \frac{\partial\mathcal{H}}{\partial t}+\nabla\cdot\mathbf{S}=0
  24. 𝐒 = ( ϕ ) ϕ t \mathbf{S}=\frac{\partial\mathcal{L}}{\partial(\nabla\phi)}\frac{\partial\phi}% {\partial t}
  25. ( ϕ , μ ϕ , x μ ) \mathcal{L}(\phi,\partial_{\mu}\phi,x_{\mu})
  26. μ μ
  27. ( ϕ , ϕ x , ϕ y , ϕ z , ϕ t , x , y , z , t ) \mathcal{L}\left(\phi,\frac{\partial\phi}{\partial x},\frac{\partial\phi}{% \partial y},\frac{\partial\phi}{\partial z},\frac{\partial\phi}{\partial t},x,% y,z,t\right)

Hardgrove_Grindability_Index.html

  1. H = 13 + 6.93 D H=13+6{.}93\cdot D

Harmonic_morphism.html

  1. ϕ : ( M m , g ) ( N n , h ) \phi:(M^{m},g)\to(N^{n},h)
  2. x x
  3. M M
  4. y y
  5. N N
  6. ϕ \phi
  7. τ ( ϕ ) = i , j = 1 m g i j ( 2 ϕ γ x i x j - k = 1 m Γ ^ i j k ϕ γ x k + α , β = 1 n Γ α β γ ϕ ϕ α x i ϕ β x j ) = 0 , \tau(\phi)=\sum_{i,j=1}^{m}g^{ij}\left(\frac{\partial^{2}\phi^{\gamma}}{% \partial x_{i}\partial x_{j}}-\sum_{k=1}^{m}\hat{\Gamma}^{k}_{ij}\frac{% \partial\phi^{\gamma}}{\partial x_{k}}+\sum_{\alpha,\beta=1}^{n}\Gamma^{\gamma% }_{\alpha\beta}\circ\phi\frac{\partial\phi^{\alpha}}{\partial x_{i}}\frac{% \partial\phi^{\beta}}{\partial x_{j}}\right)=0,
  8. ϕ α = y α ϕ \phi^{\alpha}=y_{\alpha}\circ\phi
  9. Γ ^ , Γ \hat{\Gamma},\Gamma
  10. M M
  11. N N
  12. i , j = 1 m g i j ( x ) ϕ α x i ( x ) ϕ β x j ( x ) = λ 2 ( x ) h α β ( ϕ ( x ) ) , \sum_{i,j=1}^{m}g^{ij}(x)\frac{\partial\phi^{\alpha}}{\partial x_{i}}(x)\frac{% \partial\phi^{\beta}}{\partial x_{j}}(x)=\lambda^{2}(x)h^{\alpha\beta}(\phi(x)),
  13. λ : M 0 + \lambda:M\to\mathbb{R}_{0}^{+}
  14. ϕ : ( M , g ) ( N 2 , h ) \phi:(M,g)\to(N^{2},h)
  15. h h
  16. ϕ = u + i v : ( M , g ) \phi=u+iv:(M,g)\to\mathbb{C}
  17. Δ M ( ϕ ) = Δ M ( u ) + i Δ M ( v ) = 0 \Delta_{M}(\phi)=\Delta_{M}(u)+i\Delta_{M}(v)=0
  18. g ( ϕ , ϕ ) = u 2 - v 2 + 2 i g ( u , v ) = 0. g(\nabla\phi,\nabla\phi)=\|\nabla u\|^{2}-\|\nabla v\|^{2}+2ig(\nabla u,\nabla v% )=0.
  19. u , v : ( M , g ) u,v:(M,g)\to\mathbb{R}
  20. u , v \nabla u,\nabla v
  21. ϕ : ( M , g ) \phi:(M,g)\to\mathbb{C}
  22. f : ( M , g , J ) f:(M,g,J)\to\mathbb{C}
  23. ( M , g ) (M,g)
  24. ϕ - 1 ( { z 0 } ) \phi^{-1}(\{z_{0}\})
  25. ϕ : ( M , g ) ( N 2 , h ) \phi:(M,g)\to(N^{2},h)
  26. ( M 4 , g ) (M^{4},g)
  27. n \mathbb{C}^{n}
  28. ϕ : S 3 S 2 \phi:S^{3}\to S^{2}
  29. ϕ : S 7 S 4 \phi:S^{7}\to S^{4}
  30. ϕ : S 15 S 8 \phi:S^{15}\to S^{8}
  31. K H G K\subset H\subset G
  32. ϕ : G / H G / K \phi:G/H\to G/K

Hasse_derivative.html

  1. D ( r ) X n = ( n r ) X n - r , D^{(r)}X^{n}={\left({{n}\atop{r}}\right)}X^{n-r},
  2. D ( r ) = 1 r ! ( d d X ) r . D^{(r)}=\frac{1}{r!}\left(\frac{\mathrm{d}}{\mathrm{d}X}\right)^{r}\ .
  3. f = r D ( r ) ( f ) t r . f=\sum_{r}D^{(r)}(f)\cdot t^{r}\ .

Hattendorf's_theorem.html

  1. K ( x ) K(x)
  2. T x T_{x}
  3. K ( x ) = T x K(x)=\lfloor T_{x}\rfloor
  4. p j k {}_{k}p_{j}
  5. Pr ( j T x < j + k ) \mathrm{Pr}(j\leq T_{x}<j+k)
  6. π j \pi_{j}
  7. b j b_{j}
  8. L h L_{h}
  9. C h C_{h}
  10. v v
  11. Λ h \Lambda_{h}
  12. V h V_{h}
  13. 𝔼 [ L h | K ( x ) h ] \mathbb{E}[L_{h}|K(x)\geq h]
  14. L h L_{h}
  15. Λ k \Lambda_{k}
  16. π 0 , π 1 π K ( x ) \pi_{0},\pi_{1}\dots\pi_{K(x)}
  17. K ( x ) + 1 K(x)+1
  18. b K ( x ) + 1 b_{K(x)+1}
  19. K ( x ) < h K(x)<h
  20. C h = 0 C_{h}=0
  21. K ( x ) = h K(x)=h
  22. v b h + 1 vb_{h+1}
  23. C h = v b h + 1 - π h . C_{h}=vb_{h+1}-\pi_{h}.
  24. K ( x ) > h K(x)>h
  25. C h = { 0 if K ( x ) = 0 , 1 h - 1 v b h + 1 - π h if K ( x ) = h - π h if K ( x ) = h + 1 , h + 2 C_{h}=\begin{cases}0&\mbox{if }~{}K(x)=0,1\dots h-1\\ vb_{h+1}-\pi_{h}&\mbox{if }~{}K(x)=h\\ -\pi_{h}&\mbox{if }~{}K(x)=h+1,h+2\dots\\ \end{cases}
  26. L h = { 0 if K ( x ) = 0 , 1 h - 1 v b K ( x ) + 1 - π K ( x ) if K ( x ) = h v K ( x ) - h + 1 b K ( x ) + 1 - k = h K ( x ) v k - h π k if K ( x ) = h + 1 , h + 2 L_{h}=\begin{cases}0&\mbox{if }~{}K(x)=0,1\dots h-1\\ vb_{K(x)+1}-\pi_{K(x)}&\mbox{if }~{}K(x)=h\\ v^{K(x)-h+1}b_{K(x)+1}-\sum_{k=h}^{K(x)}v^{k-h}\pi_{k}&\mbox{if }~{}K(x)=h+1,h% +2\dots\end{cases}
  27. L h L_{h}
  28. L h = k = h v k - h C h . L_{h}=\sum_{k=h}^{\infty}v^{k-h}C_{h}.
  29. C h C_{h}
  30. K ( x ) > h K(x)>h
  31. L h = k = h K ( x ) - 1 v k - h C h + v K ( x ) - h C K ( x ) + k = K ( x ) + 1 v k - h C h = - k = h K ( x ) - 1 v k - h π k + v K ( x ) - h ( v b K ( x ) + 1 - π h ) = v K ( x ) - h + 1 b K ( x ) + 1 - k = h K ( x ) v k - h π k . \begin{aligned}\displaystyle L_{h}&\displaystyle=\sum_{k=h}^{K(x)-1}v^{k-h}C_{% h}+v^{K(x)-h}C_{K(x)}+\sum_{k=K(x)+1}^{\infty}v^{k-h}C_{h}\\ &\displaystyle=-\sum_{k=h}^{K(x)-1}v^{k-h}\pi_{k}+v^{K(x)-h}\left(vb_{K(x)+1}-% \pi_{h}\right)\\ &\displaystyle=v^{K(x)-h+1}b_{K(x)+1}-\sum_{k=h}^{K(x)}v^{k-h}\pi_{k}.\end{aligned}
  32. K ( x ) = h K(x)=h
  33. L h = C K ( x ) L_{h}=C_{K(x)}
  34. K ( x ) < h K(x)<h
  35. V h V_{h}
  36. V h = 𝔼 [ L h | K ( x ) h ] V_{h}=\mathbb{E}[L_{h}|K(x)\geq h]
  37. V h = k = 0 ( v k + 1 b k + h + 1 - j = 0 k v j π j + h ) p x + h k q x + h + k V_{h}=\sum_{k=0}^{\infty}\left(v^{k+1}b_{k+h+1}-\sum_{j=0}^{k}v^{j}\pi_{j+h}% \right){{}_{k}p_{x+h}}q_{x+h+k}
  38. V h V_{h}
  39. V h = 𝔼 [ L h | K ( x ) h ] = 𝔼 [ v K ( x ) - h + 1 b K ( x ) + 1 - k = h K ( x ) v k - h π k | K ( x ) h ] . \begin{aligned}\displaystyle V_{h}&\displaystyle=\mathbb{E}[L_{h}|K(x)\geq h]% \\ &\displaystyle=\mathbb{E}\left[v^{K(x)-h+1}b_{K(x)+1}-\sum_{k=h}^{K(x)}v^{k-h}% \pi_{k}|K(x)\geq h\right].\end{aligned}
  40. K ( x ) - h K(x)-h
  41. x + h x+h
  42. K ( x + h ) K(x+h)
  43. 𝔼 [ f ( K ( x ) - h ) | K ( x ) h ] = 𝔼 [ f ( K ( x + h ) ) ] \mathbb{E}[f(K(x)-h)|K(x)\geq h]=\mathbb{E}[f(K(x+h))]
  44. V h = 𝔼 [ v ( K ( x ) - h ) + 1 b ( K ( x ) - h ) + h + 1 - j = 0 K ( x ) - h v j π j + h | K ( x ) h ] = 𝔼 [ v K ( x + h ) + 1 b K ( x + h ) + h + 1 - j = 0 K ( x + h ) v j π j + h ] = k = 0 ( v k + 1 b k + h + 1 - j = 0 k v j π j + h ) Pr ( K ( x + h ) = k ) = k = 0 ( v k + 1 b k + h + 1 - j = 0 k v j π j + h ) p x + h k q x + h + k \begin{aligned}\displaystyle V_{h}&\displaystyle=\mathbb{E}\left[v^{(K(x)-h)+1% }b_{(K(x)-h)+h+1}-\sum_{j=0}^{K(x)-h}v^{j}\pi_{j+h}|K(x)\geq h\right]\\ &\displaystyle=\mathbb{E}\left[v^{K(x+h)+1}b_{K(x+h)+h+1}-\sum_{j=0}^{K(x+h)}v% ^{j}\pi_{j+h}\right]\\ &\displaystyle=\sum_{k=0}^{\infty}\left(v^{k+1}b_{k+h+1}-\sum_{j=0}^{k}v^{j}% \pi_{j+h}\right)\mathrm{Pr}(K(x+h)=k)\\ &\displaystyle={\sum_{k=0}^{\infty}\left(v^{k+1}b_{k+h+1}-\sum_{j=0}^{k}v^{j}% \pi_{j+h}\right)}{{}_{k}p_{x+h}}q_{x+h+k}\end{aligned}
  45. Λ h \Lambda_{h}
  46. C h C_{h}
  47. P V ( Δ V h ) PV(\Delta V_{h})
  48. Λ h = C h + v Δ L i a b i l i t i e s \Lambda_{h}=C_{h}+v\Delta Liabilities
  49. K ( x ) > h K(x)>h
  50. Λ h = - π h + ( v V h + 1 - V h ) \Lambda_{h}=-\pi_{h}+(vV_{h+1}-V_{h})
  51. K ( x ) = h K(x)=h
  52. Λ h = ( v b h + 1 - π h ) - V h \Lambda_{h}=(vb_{h+1}-\pi_{h})-V_{h}
  53. K ( x ) < h K(x)<h
  54. Λ h = 0 \Lambda_{h}=0
  55. Λ h = { 0 if K ( x ) = 0 , 1 h - 1 ( v b h + 1 - π h ) - V h if K ( x ) = h ( v V h + 1 - V h ) - π h if K ( x ) = h + 1 , h + 2 . \Lambda_{h}=\begin{cases}0&\mbox{if }~{}K(x)=0,1\dots h-1\\ (vb_{h+1}-\pi_{h})-V_{h}&\mbox{if }~{}K(x)=h\\ (vV_{h+1}-V_{h})-\pi_{h}&\mbox{if }~{}K(x)=h+1,h+2\dots\end{cases}.
  56. k = h v k - h Λ k = k = h v k - h [ C k + v Δ Liabilities in year ( k , k + 1 ) ] = k = h v k - h C k + k = h v k - h + 1 Δ Liabilities in year ( k , k + 1 ) = L h + k = h v k - h + 1 ( v V k + 1 - V k ) = L h + k = h v ( k + 1 ) - h + 1 V k + 1 - k = h v k - h + 1 V k \begin{aligned}\displaystyle\sum_{k=h}^{\infty}v^{k-h}\Lambda_{k}&% \displaystyle=\sum_{k=h}^{\infty}v^{k-h}[C_{k}+v\Delta\,\text{Liabilities in % year }(k,k+1)]\\ &\displaystyle=\sum_{k=h}^{\infty}v^{k-h}C_{k}+\sum_{k=h}^{\infty}v^{k-h+1}% \Delta\,\text{Liabilities in year }(k,k+1)\\ &\displaystyle=L_{h}+\sum_{k=h}^{\infty}v^{k-h+1}(vV_{k+1}-V_{k})\\ &\displaystyle=L_{h}+\sum_{k=h}^{\infty}v^{(k+1)-h+1}V_{k+1}-\sum_{k=h}^{% \infty}v^{k-h+1}V_{k}\\ \end{aligned}
  57. L h = k = h v k - h Λ k + V h . L_{h}=\sum_{k=h}^{\infty}v^{k-h}\Lambda_{k}+V_{h}.
  58. Cov ( Λ h Λ j | K ( x ) k ) = 0 \mathrm{Cov}(\Lambda_{h}\Lambda_{j}|K(x)\geq k)=0
  59. k h < j k\leq h<j
  60. Var [ L h | K ( x ) h ] = Var [ k = h v k - h Λ k + V h | K ( x ) h ] = k = h v 2 ( k - h ) Var [ Λ k | K ( x ) h ] \mathrm{Var}[L_{h}|K(x)\geq h]=\mathrm{Var}\left[\sum_{k=h}^{\infty}v^{k-h}% \Lambda_{k}+V_{h}|K(x)\geq h\right]=\sum_{k=h}^{\infty}v^{2(k-h)}\mathrm{Var}[% \Lambda_{k}|K(x)\geq h]