wpmath0000011_3

Quantifier_shift.html

  1. x , y R x y y x R x y \forall x\,,\exists y\,Rxy\vdash\exists y\,\forall x\,Rxy
  2. \exist y x R x y x \exist y R x y \exist y\,\forall x\,Rxy\vdash\forall x\,\exist y\,Rxy

Quantile_regression.html

  1. Y Y
  2. F Y ( y ) = P ( Y y ) F_{Y}(y)=P(Y\leq y)
  3. τ \tau
  4. Q Y ( τ ) = F Y - 1 ( τ ) = inf { y : F Y ( y ) τ } Q_{Y}(\tau)=F_{Y}^{-1}(\tau)=\inf\left\{y:F_{Y}(y)\geq\tau\right\}
  5. τ [ 0 , 1 ] . \tau\in[0,1].
  6. ρ τ ( y ) = | y ( τ - 𝕀 ( y < 0 ) ) | \rho_{\tau}(y)=|y(\tau-\mathbb{I}_{(y<0)})|
  7. 𝕀 \mathbb{I}
  8. Y - u Y-u
  9. u u
  10. min 𝑢 E ( ρ τ ( Y - u ) ) = min 𝑢 ( τ - 1 ) - u ( y - u ) d F Y ( y ) + τ u ( y - u ) d F Y ( y ) . \underset{u}{\min}E(\rho_{\tau}(Y-u))=\underset{u}{\min}(\tau-1)\int_{-\infty}% ^{u}(y-u)dF_{Y}(y)+\tau\int_{u}^{\infty}(y-u)dF_{Y}(y).
  11. q τ q_{\tau}
  12. 0 = ( 1 - τ ) - q τ d F Y ( y ) - τ q τ d F Y ( y ) . 0=(1-\tau)\int_{-\infty}^{q_{\tau}}dF_{Y}(y)-\tau\int_{q_{\tau}}^{\infty}dF_{Y% }(y).
  13. 0 = F Y ( q τ ) - τ , 0=F_{Y}(q_{\tau})-\tau,
  14. F Y ( q τ ) = τ . F_{Y}(q_{\tau})=\tau.
  15. q τ q_{\tau}
  16. τ \tau
  17. Y Y
  18. τ = 0.5 \tau=0.5
  19. L ( u ) = ( τ - 1 ) 9 y i < u ( y i - u ) + τ 9 y i u ( y i - u ) = 0.5 9 ( - y i < u ( y i - u ) + y i u ( y i - u ) ) . L(u)=\frac{(\tau-1)}{9}\sum_{y_{i}<u}(y_{i}-u)+\frac{\tau}{9}\sum_{y_{i}\geq u% }(y_{i}-u)=\frac{0.5}{9}\left(-\sum_{y_{i}<u}(y_{i}-u)+\sum_{y_{i}\geq u}(y_{i% }-u)\right).
  20. 0.5 / 9 {0.5/9}
  21. τ = 0.5 \tau=0.5
  22. L ( 3 ) i = 1 2 - ( i - 3 ) + i = 3 9 ( i - 3 ) = [ ( 2 + 1 ) + ( 0 + 1 + 2 + + 6 ) ] = 24. L(3)\propto\sum_{i=1}^{2}-(i-3)+\sum_{i=3}^{9}(i-3)=[(2+1)+(0+1+2+...+6)]=24.
  23. ( 3 ) - ( 6 ) = - 3 (3)-(6)=-3
  24. L ( 5 ) i = 1 4 i + i = 0 4 i = 20 , L(5)\propto\sum_{i=1}^{4}i+\sum_{i=0}^{4}i=20,
  25. 0.5 / 9 {0.5/9}
  26. τ = 0.5 \tau=0.5
  27. q τ q_{\tau}
  28. - 0.5 - q ( y - q ) d F Y ( y ) + 0.5 q ( y - q ) d F Y ( y ) . -0.5\int_{-\infty}^{q}(y-q)dF_{Y}(y)+0.5\int_{q}^{\infty}(y-q)dF_{Y}(y).
  29. - q 1 d F Y ( y ) - q 1 d F Y ( y ) . \int_{-\infty}^{q}1dF_{Y}(y)-\int_{q}^{\infty}1dF_{Y}(y).
  30. F Y ( q ) F_{Y}(q)
  31. 1 - F Y ( q ) 1-F_{Y}(q)
  32. F Y ( q ) < 0.5 F_{Y}(q)<0.5
  33. τ \tau
  34. τ \tau
  35. q ^ τ = arg min q R i = 1 n ρ τ ( y i - q ) , \hat{q}_{\tau}=\underset{q\in R}{\mbox{arg min}~{}}\sum_{i=1}^{n}\rho_{\tau}(y% _{i}-q),
  36. = arg min q R [ ( τ - 1 ) y i < q ( y i - q ) + τ y i q ( y i - q ) ] =\underset{q\in R}{\mbox{arg min}~{}}\left[(\tau-1)\sum_{y_{i}<q}(y_{i}-q)+% \tau\sum_{y_{i}\geq q}(y_{i}-q)\right]
  37. τ \tau
  38. Q Y | X ( τ ) = X β τ Q_{Y|X}(\tau)=X\beta_{\tau}
  39. Y Y
  40. β τ \beta_{\tau}
  41. β τ = arg min β R k E ( ρ τ ( Y - X β ) ) . \beta_{\tau}=\underset{\beta\in R^{k}}{\mbox{arg min}~{}}E(\rho_{\tau}(Y-X% \beta)).
  42. β \beta
  43. β τ ^ = arg min β R k i = 1 n ( ρ τ ( Y i - X β ) ) . \hat{\beta_{\tau}}=\underset{\beta\in R^{k}}{\mbox{arg min}~{}}\sum_{i=1}^{n}(% \rho_{\tau}(Y_{i}-X\beta)).
  44. min β + , β - , u + , u - R 2 k × R + 2 n { τ 1 n u + + ( 1 - τ ) 1 n u - | X ( β + - β - ) + u + - u - = Y } , \underset{\beta^{+},\beta^{-},u^{+},u^{-}\in R^{2k}\times R_{+}^{2n}}{\min}% \left\{\tau 1_{n}^{{}^{\prime}}u^{+}+(1-\tau)1_{n}^{{}^{\prime}}u^{-}|X(\beta^% {+}-\beta^{-})+u^{+}-u^{-}=Y\right\},
  45. β j + = max ( β j , 0 ) \beta_{j}^{+}=\max(\beta_{j},0)
  46. β j - = - min ( β j , 0 ) \beta_{j}^{-}=-\min(\beta_{j},0)
  47. u j + = max ( u j , 0 ) u_{j}^{+}=\max(u_{j},0)
  48. u j - = - min ( u j , 0 ) . u_{j}^{-}=-\min(u_{j},0).
  49. τ ( 0 , 1 ) \tau\in(0,1)
  50. β ^ τ \hat{\beta}_{\tau}
  51. n ( β ^ τ - β τ ) 𝑑 N ( 0 , τ ( 1 - τ ) D - 1 Ω x D - 1 ) , \sqrt{n}(\hat{\beta}_{\tau}-\beta_{\tau})\overset{d}{\rightarrow}N(0,\tau(1-% \tau)D^{-1}\Omega_{x}D^{-1}),
  52. D = E ( f Y ( X β ) X X ) D=E(f_{Y}(X\beta)XX^{\prime})
  53. Ω x = E ( X X ) . \Omega_{x}=E(X^{\prime}X).
  54. a > 0 a>0
  55. τ [ 0 , 1 ] \tau\in[0,1]
  56. β ^ ( τ ; a Y , X ) = a β ^ ( τ ; Y , X ) , \hat{\beta}(\tau;aY,X)=a\hat{\beta}(\tau;Y,X),
  57. β ^ ( τ ; - a Y , X ) = - a β ^ ( 1 - τ ; Y , X ) . \hat{\beta}(\tau;-aY,X)=-a\hat{\beta}(1-\tau;Y,X).
  58. γ R k \gamma\in R^{k}
  59. τ [ 0 , 1 ] \tau\in[0,1]
  60. β ^ ( τ ; Y + X γ , X ) = β ^ ( τ ; Y , X ) + γ . \hat{\beta}(\tau;Y+X\gamma,X)=\hat{\beta}(\tau;Y,X)+\gamma.
  61. A A
  62. p × p p\times p
  63. τ [ 0 , 1 ] \tau\in[0,1]
  64. β ^ ( τ ; Y , X A ) = A - 1 β ^ ( τ ; Y , X ) . \hat{\beta}(\tau;Y,XA)=A^{-1}\hat{\beta}(\tau;Y,X).
  65. h h
  66. h ( Q Y | X ( τ ) ) Q h ( Y ) | X ( τ ) . h(Q_{Y|X}(\tau))\equiv Q_{h(Y)|X}(\tau).
  67. Y = ln ( W ) Y=\ln(W)
  68. Q Y | X ( τ ) = X β τ Q_{Y|X}(\tau)=X\beta_{\tau}
  69. Q W | X ( τ ) = exp ( X β τ ) Q_{W|X}(\tau)=\exp(X\beta_{\tau})
  70. E ( ln ( Y ) ) ln ( E ( Y ) ) . \operatorname{E}(\ln(Y))\neq\ln(\operatorname{E}(Y)).
  71. Y c = max ( 0 , Y ) Y^{c}=\max(0,Y)
  72. Q Y | X = X β τ Q_{Y|X}=X\beta_{\tau}
  73. Q Y c | X ( τ ) = max ( 0 , X β τ ) Q_{Y^{c}|X}(\tau)=\max(0,X\beta_{\tau})

Quasi-derivative.html

  1. lim t 0 + f ( g ( t ) ) - f ( x 0 ) t = u ( g ( 0 ) ) . \lim_{t\to 0^{+}}\frac{f(g(t))-f(x_{0})}{t}=u(g^{\prime}(0)).

Quaternary-amine-transporting_ATPase.html

  1. \rightleftharpoons

Quercitrinase.html

  1. \rightleftharpoons

Queuine_tRNA-ribosyltransferase.html

  1. \rightleftharpoons

Quinate_O-hydroxycinnamoyltransferase.html

  1. \rightleftharpoons

R-linalool_synthase.html

  1. \rightleftharpoons

Rabinowitsch_trick.html

  1. g 0 , g 1 , , g m K [ x 0 , x 1 , , x n ] g_{0},g_{1},\dots,g_{m}\in K[x_{0},x_{1},\dots,x_{n}]
  2. 1 = g 0 ( x 0 , x 1 , , x n ) ( 1 - x 0 f ( x 1 , , x n ) ) + i = 1 m g i ( x 0 , x 1 , , x n ) f i ( x 1 , , x n ) 1=g_{0}(x_{0},x_{1},\dots,x_{n})(1-x_{0}f(x_{1},\dots,x_{n}))+\sum_{i=1}^{m}g_% {i}(x_{0},x_{1},\dots,x_{n})f_{i}(x_{1},\dots,x_{n})
  3. K [ x 0 , x 1 , , x n ] K[x_{0},x_{1},\dots,x_{n}]
  4. x 0 , x 1 , , x n x_{0},x_{1},\dots,x_{n}
  5. x 0 = 1 / f ( x 1 , , x n ) x_{0}=1/f(x_{1},\dots,x_{n})
  6. 1 = i = 1 m g i ( 1 / f ( x 1 , , x n ) , x 1 , , x n ) f i ( x 1 , , x n ) 1=\sum_{i=1}^{m}g_{i}(1/f(x_{1},\dots,x_{n}),x_{1},\dots,x_{n})f_{i}(x_{1},% \dots,x_{n})
  7. K ( x 1 , , x n ) K(x_{1},\dots,x_{n})
  8. K [ x 1 , , x n ] K[x_{1},\dots,x_{n}]
  9. 1 = i = 1 m h i ( x 1 , , x n ) f i ( x 1 , , x n ) f ( x 1 , , x n ) r 1=\frac{\sum_{i=1}^{m}h_{i}(x_{1},\dots,x_{n})f_{i}(x_{1},\dots,x_{n})}{f(x_{1% },\dots,x_{n})^{r}}
  10. h 1 , , h m K [ x 1 , , x n ] h_{1},\dots,h_{m}\in K[x_{1},\dots,x_{n}]
  11. f ( x 1 , , x n ) r = i = 1 m h i ( x 1 , , x n ) f i ( x 1 , , x n ) f(x_{1},\dots,x_{n})^{r}=\sum_{i=1}^{m}h_{i}(x_{1},\dots,x_{n})f_{i}(x_{1},% \dots,x_{n})
  12. f r f^{r}

Rademacher_complexity.html

  1. S = ( z 1 , z 2 , , z m ) Z m S=(z_{1},z_{2},\dots,z_{m})\in Z^{m}
  2. \mathcal{H}
  3. Z Z
  4. \mathcal{H}
  5. ^ S ( ) = 2 m 𝔼 [ sup h | i = 1 m σ i h ( z i ) | | S ] \widehat{\mathcal{R}}_{S}(\mathcal{H})=\frac{2}{m}\mathbb{E}\left[\sup_{h\in% \mathcal{H}}\left|\sum_{i=1}^{m}\sigma_{i}h(z_{i})\right|\ \bigg|\ S\right]
  6. σ 1 , σ 2 , , σ m \sigma_{1},\sigma_{2},\dots,\sigma_{m}
  7. Pr ( σ i = + 1 ) = Pr ( σ i = - 1 ) = 1 / 2 \Pr(\sigma_{i}=+1)=\Pr(\sigma_{i}=-1)=1/2
  8. i = 1 , 2 , , m i=1,2,\dots,m
  9. P P
  10. Z Z
  11. \mathcal{H}
  12. P P
  13. m m
  14. m ( ) = 𝔼 [ ^ S ( ) ] \mathcal{R}_{m}(\mathcal{H})=\mathbb{E}\left[\widehat{\mathcal{R}}_{S}(% \mathcal{H})\right]
  15. S = ( z 1 , z 2 , , z m ) S=(z_{1},z_{2},\dots,z_{m})
  16. P P
  17. C C
  18. { 0 , 1 } \{0,1\}
  19. d d
  20. C d m C\sqrt{\frac{d}{m}}
  21. g i g_{i}
  22. σ i \sigma_{i}
  23. g i g_{i}
  24. g i 𝒩 ( 0 , 1 ) g_{i}\sim\mathcal{N}\left(0,1\right)

Radiative_transfer_equation_and_diffusion_theory_for_photon_transport_in_biological_tissue.html

  1. L ( r , s ^ , t ) ( W m 2 s r ) L(\vec{r},\hat{s},t)(\frac{W}{m^{2}sr})
  2. r \vec{r}
  3. s ^ \hat{s}
  4. t t
  5. Φ ( r , t ) = 4 π L ( r , s ^ , t ) d Ω ( W m 2 ) \Phi(\vec{r},t)=\int_{4\pi}L(\vec{r},\hat{s},t)d\Omega(\frac{W}{m^{2}})
  6. F ( r ) = - + Φ ( r , t ) d t ( J m 2 ) F(\vec{r})=\int_{-\infty}^{+\infty}\Phi(\vec{r},t)dt(\frac{J}{m^{2}})
  7. J ( r , t ) = 4 π s ^ L ( r , s ^ , t ) d Ω ( W m 2 ) \vec{J}(\vec{r},t)=\int_{4\pi}\hat{s}L(\vec{r},\hat{s},t)d\Omega(\frac{W}{m^{2% }})
  8. L ( r , s ^ , t ) L(\vec{r},\hat{s},t)
  9. n n
  10. g g
  11. L ( r , s ^ , t ) / c t = - s ^ L ( r , s ^ , t ) - μ t L ( r , s ^ , t ) + μ s 4 π L ( r , s ^ , t ) P ( s ^ s ^ ) d Ω + S ( r , s ^ , t ) \frac{\partial L(\vec{r},\hat{s},t)/c}{\partial t}=-\hat{s}\cdot\nabla L(\vec{% r},\hat{s},t)-\mu_{t}L(\vec{r},\hat{s},t)+\mu_{s}\int_{4\pi}L(\vec{r},\hat{s}^% {\prime},t)P(\hat{s}^{\prime}\cdot\hat{s})d\Omega^{\prime}+S(\vec{r},\hat{s},t)
  12. c c
  13. = =
  14. P ( s ^ , s ^ ) P(\hat{s}^{\prime},\hat{s})
  15. s ^ \hat{s}^{\prime}
  16. d Ω d\Omega
  17. s ^ \hat{s}
  18. s ^ \hat{s}^{\prime}
  19. s ^ \hat{s}
  20. P ( s ^ , s ^ ) = P ( s ^ s ^ ) P(\hat{s}^{\prime},\hat{s})=P(\hat{s}^{\prime}\cdot\hat{s})
  21. g = 4 π ( s ^ s ^ ) P ( s ^ s ^ ) d Ω g=\int_{4\pi}(\hat{s}^{\prime}\cdot\hat{s})P(\hat{s}^{\prime}\cdot\hat{s})d\Omega
  22. S ( r , s ^ , t ) S(\vec{r},\hat{s},t)
  23. x x
  24. y y
  25. z z
  26. r \vec{r}
  27. θ \theta
  28. ϕ \phi
  29. s ^ \hat{s}
  30. t t
  31. Y Y
  32. L ( r , s ^ , t ) n = 0 1 m = - n n L n , m ( r , t ) Y n , m ( s ^ ) L(\vec{r},\hat{s},t)\approx\ \sum_{n=0}^{1}\sum_{m=-n}^{n}L_{n,m}(\vec{r},t)Y_% {n,m}(\hat{s})
  33. L L
  34. Φ ( r , t ) \Phi(\vec{r},t)
  35. J ( r , t ) \vec{J}(\vec{r},t)
  36. L 0 , 0 ( r , t ) Y 0 , 0 ( s ^ ) = Φ ( r , t ) 4 π L_{0,0}(\vec{r},t)Y_{0,0}(\hat{s})=\frac{\Phi(\vec{r},t)}{4\pi}
  37. m = - 1 1 L 1 , m ( r , t ) Y 1 , m ( s ^ ) = 3 4 π J ( r , t ) s ^ \sum_{m=-1}^{1}L_{1,m}(\vec{r},t)Y_{1,m}(\hat{s})=\frac{3}{4\pi}\vec{J}(\vec{r% },t)\cdot\hat{s}
  38. L ( r , s ^ , t ) = 1 4 π Φ ( r , t ) + 3 4 π J ( r , t ) s ^ L(\vec{r},\hat{s},t)=\frac{1}{4\pi}\Phi(\vec{r},t)+\frac{3}{4\pi}\vec{J}(\vec{% r},t)\cdot\hat{s}
  39. 4 π 4\pi
  40. s ^ \hat{s}
  41. Φ ( r , t ) c t + μ a Φ ( r , t ) + J ( r , t ) = S ( r , t ) \frac{\partial\Phi(\vec{r},t)}{c\partial t}+\mu_{a}\Phi(\vec{r},t)+\nabla\cdot% \vec{J}(\vec{r},t)=S(\vec{r},t)
  42. J ( r , t ) c t + ( μ a + μ s ) J ( r , t ) + 1 3 Φ ( r , t ) = 0 \frac{\partial\vec{J}(\vec{r},t)}{c\partial t}+(\mu_{a}+\mu_{s}^{\prime})\vec{% J}(\vec{r},t)+\frac{1}{3}\nabla\Phi(\vec{r},t)=0
  43. J ( r , t ) \vec{J}(\vec{r},t)
  44. J ( r , t ) = - Φ ( r , t ) 3 ( μ a + μ s ) \vec{J}(\vec{r},t)=\frac{-\nabla\Phi(\vec{r},t)}{3(\mu_{a}+\mu_{s}^{\prime})}
  45. 1 c Φ ( r , t ) t + μ a Φ ( r , t ) - [ D Φ ( r , t ) ] = S ( r , t ) \frac{1}{c}\frac{\partial\Phi(\vec{r},t)}{\partial t}+\mu_{a}\Phi(\vec{r},t)-% \nabla\cdot[D\nabla\Phi(\vec{r},t)]=S(\vec{r},t)
  46. D = 1 3 ( μ a + μ s ) D=\frac{1}{3(\mu_{a}+\mu_{s}^{\prime})}
  47. = ( 1 - g ) =(1-g)
  48. D D
  49. S ( r , t ) S(\vec{r},t)
  50. S ( r , t , r , t ) = δ ( r - r ) δ ( t - t ) S(\vec{r},t,\vec{r^{\prime}},t^{\prime})=\delta(\vec{r}-\vec{r^{\prime}})% \delta(t-t^{\prime})
  51. r \vec{r}
  52. r \vec{r^{\prime}}
  53. t t^{\prime}
  54. Φ ( r , t ; r , t ) = c [ 4 π D c ( t - t ) ] 3 / 2 exp [ - r - r 2 4 D c ( t - t ) ] exp [ - μ a c ( t - t ) ] \Phi(\vec{r},t;\vec{r^{\prime}},t)=\frac{c}{[4\pi Dc(t-t^{\prime})]^{3/2}}\exp% \left[-\frac{\mid\vec{r}-\vec{r^{\prime}}\mid^{2}}{4Dc(t-t^{\prime})}\right]% \exp[-\mu_{a}c(t-t^{\prime})]
  55. exp [ - μ a c ( t - t ) ] \exp\left[-\mu_{a}c(t-t^{\prime})\right]
  56. S ( r ) = δ ( r ) S(\vec{r})=\delta(\vec{r})
  57. Φ ( r ) = 1 4 π D r exp ( - μ eff r ) \Phi(\vec{r})=\frac{1}{4\pi Dr}\exp(-\mu_{\mathrm{eff}}r)
  58. μ eff = μ a D \mu_{\mathrm{eff}}=\sqrt{\frac{\mu_{a}}{D}}
  59. R F R_{F}
  60. s ^ n ^ < 0 L ( r , s ^ , t ) s ^ n ^ d Ω = s ^ n ^ > 0 R F ( s ^ n ^ ) L ( r , s ^ , t ) s ^ n ^ d Ω \int_{\hat{s}\cdot\hat{n}<0}L(\vec{r},\hat{s},t)\hat{s}\cdot\hat{n}d\Omega=% \int_{\hat{s}\cdot\hat{n}>0}R_{F}(\hat{s}\cdot\hat{n})L(\vec{r},\hat{s},t)\hat% {s}\cdot\hat{n}d\Omega
  61. n ^ \hat{n}
  62. L L
  63. Φ \Phi
  64. J \vec{J}
  65. Φ ( r , t ) 4 + J ( r , t ) n ^ 2 = R Φ Φ ( r , t ) 4 - R J J ( r , t ) n ^ 2 \frac{\Phi(\vec{r},t)}{4}+\vec{J}(\vec{r},t)\cdot\frac{\hat{n}}{2}=R_{\Phi}% \frac{\Phi(\vec{r},t)}{4}-R_{J}\vec{J}(\vec{r},t)\cdot\frac{\hat{n}}{2}
  66. R Φ = 0 π / 2 2 sin θ cos θ R F ( cos θ ) d θ R_{\Phi}=\int_{0}^{\pi/2}2\sin\theta\cos\theta R_{F}(\cos\theta)d\theta
  67. R J = 0 π / 2 3 sin θ ( cos θ ) 2 R F ( cos θ ) d θ R_{J}=\int_{0}^{\pi/2}3\sin\theta(\cos\theta)^{2}R_{F}(\cos\theta)d\theta
  68. J ( r , t ) = - D Φ ( r , t ) \vec{J}(\vec{r},t)=-D\nabla\Phi(\vec{r},t)
  69. Φ ( r , t ) = A z Φ ( r , t ) z \Phi(\vec{r},t)=A_{z}\frac{\partial\Phi(\vec{r},t)}{\partial z}
  70. A z = 2 D 1 + R eff 1 - R eff A_{z}=2D\frac{1+R_{\mathrm{eff}}}{1-R_{\mathrm{eff}}}
  71. R eff = R Φ + R J 2 - R Φ + R J R_{\mathrm{eff}}=\frac{R_{\Phi}+R_{J}}{2-R_{\Phi}+R_{J}}
  72. Φ ( z = 0 , t ) \Phi(z=0,t)
  73. z z
  74. Φ ( z = - A z , t ) Φ ( z = 0 , t ) - A z Φ ( r , t ) z | z = 0 \left.\Phi(z=-A_{z},t)\approx\Phi(z=0,t)-A_{z}\frac{\partial\Phi(\vec{r},t)}{% \partial z}\right|_{z=0}
  75. Φ ( r , t ) = A z Φ ( r , t ) z \Phi(\vec{r},t)=A_{z}\frac{\partial\Phi(\vec{r},t)}{\partial z}
  76. z z
  77. - A -A
  78. R R
  79. z z
  80. = - 2 D =-2D
  81. g g
  82. = 0 =0
  83. ( 1 - g (1-g
  84. ) )
  85. g g
  86. l l
  87. a a
  88. l l
  89. + 2 z +2z
  90. Φ ( r , θ , z ; r , θ , z ) = 1 4 π D ρ exp ( - μ eff ρ ) \Phi_{\infty}(r,\theta,z;r^{\prime},\theta^{\prime},z^{\prime})=\frac{1}{4\pi D% \rho}\exp(-\mu_{\mathrm{eff}}\rho)
  91. ρ \rho
  92. ( r , θ , z ) (r,\theta,z)
  93. ( r , θ , z ) (r^{\prime},\theta^{\prime},z^{\prime})
  94. Φ ( r , θ , z ; r , θ , z ) = a Φ ( r , θ , z ; r , θ , z ) - a Φ ( r , θ , z ; r , θ , - z - 2 z b ) \Phi(r,\theta,z;r^{\prime},\theta^{\prime},z^{\prime})=a^{\prime}\Phi_{\infty}% (r,\theta,z;r^{\prime},\theta^{\prime},z^{\prime})-a^{\prime}\Phi_{\infty}(r,% \theta,z;r^{\prime},\theta^{\prime},-z^{\prime}-2z_{b})
  95. R R
  96. ( r ) (r)
  97. R d ( r ) = D Φ z | z = 0 = a z ( 1 + μ eff ρ 1 ) exp ( - μ eff ρ 1 ) 4 π ρ 1 3 + a ( z + 4 D ) ( 1 + μ eff ρ 2 ) exp ( - μ eff ρ 2 ) 4 π ρ 2 3 \left.R_{d}(r)=D\frac{\partial\Phi}{\partial z}\right|_{z=0}=\frac{a^{\prime}z% ^{\prime}(1+\mu_{\mathrm{eff}}\rho_{1})\exp(-\mu_{\mathrm{eff}}\rho_{1})}{4\pi% \rho_{1}^{3}}+\frac{a^{\prime}(z^{\prime}+4D)(1+\mu_{\mathrm{eff}}\rho_{2})% \exp(-\mu_{\mathrm{eff}}\rho_{2})}{4\pi\rho_{2}^{3}}
  98. ρ 1 \rho_{1}
  99. ( r , 0 , 0 ) (r,0,0)
  100. ( 0 , 0 , z ) (0,0,z^{\prime})
  101. ρ 2 \rho_{2}
  102. ( 0 , 0 , - z - 2 z (0,0,-z^{\prime}-2z
  103. ) )

Raffinose—raffinose_alpha-galactosyltransferase.html

  1. \rightleftharpoons

Randomized_block_design.html

  1. \vdots
  2. \vdots
  3. \vdots
  4. \cdots
  5. \vdots
  6. Y i j = μ + T i + B j + random error Y_{ij}=\mu+T_{i}+B_{j}+\mathrm{random\ error}
  7. Y ¯ \overline{Y}
  8. Y ¯ i - Y ¯ \overline{Y}_{i\cdot}-\overline{Y}
  9. Y ¯ i \overline{Y}_{i\cdot}
  10. Y ¯ j - Y ¯ \overline{Y}_{\cdot j}-\overline{Y}
  11. Y ¯ j \overline{Y}_{\cdot j}

Range_tree.html

  1. O ( log d n + k ) O(\log^{d}n+k)
  2. O ( n log d - 1 n ) O(n\log^{d-1}n)
  3. O ( log d - 1 n + k ) O(\log^{d-1}n+k)
  4. O ( n ( log n log log n ) d - 1 ) O\left(n\left(\frac{\log n}{\log\log n}\right)^{d-1}\right)

Rasta_filtering.html

  1. T ( z ) = ( k * ( n - ( N - 1 ) / 2 ) * z - n ) / ( 1 - ρ / x ) T(z)=(k*\sum(n-(N-1)/2)*z^{-n})/(1-\rho/x)\,\!

Raucaffricine_beta-glucosidase.html

  1. \rightleftharpoons

Rec._709.html

  1. [ 0..1 ] [0..1]
  2. V = { 4.500 L L < 0.018 1.099 L 0.45 - 0.099 L 0.018 V=\begin{cases}4.500L&L<0.018\\ 1.099L^{0.45}-0.099&L\geq 0.018\end{cases}
  3. L = { V 4.5 V < 0.081 ( V + 0.099 1.099 ) 1 0.45 V > 0.081 L=\begin{cases}\dfrac{V}{4.5}&V<0.081\\ \left(\dfrac{V+0.099}{1.099}\right)^{\frac{1}{0.45}}&V>0.081\end{cases}

Receivables_turnover_ratio.html

  1. Receivable Turnover Ratio = Net receivable sales Average net receivables \mathrm{Receivable\ Turnover\ Ratio}={\mathrm{Net\ receivable\ sales}\over% \mathrm{Average\ net\ receivables}}

Receptor_protein_serine::threonine_kinase.html

  1. \rightleftharpoons

Red_blood_cell_indices.html

  1. M C V = H c t R B C MCV=\frac{Hct}{RBC}
  2. M C H = H b R B C MCH=\frac{Hb}{RBC}
  3. M C H C = H b H c t MCHC=\frac{Hb}{Hct}

Regular_modal_logic.html

  1. A ¬ ¬ A \Diamond A\equiv\lnot\Box\lnot A
  2. ( A B ) C ( A B ) C . (A\land B)\to C\vdash(\Box A\land\Box B)\to\Box C.

Relative_biological_effectiveness.html

  1. R B E = D X D R RBE=\frac{D_{X}}{D_{R}}

Reptation.html

  1. τ τ
  2. M M
  3. L L
  4. τ τ
  5. v v
  6. f f
  7. M e M_{e}
  8. n n
  9. l l
  10. n e n_{e}
  11. n e n_{e}
  12. d = l n e d=l\sqrt{n_{e}}
  13. n n
  14. A A
  15. A = n n e A=\dfrac{n}{n_{e}}
  16. L L
  17. L = A d = n l n e n e = n l n e L=Ad=\dfrac{nl\sqrt{n_{e}}}{n_{e}}=\dfrac{nl}{\sqrt{n_{e}}}
  18. t t
  19. t = l 2 n 3 μ n e k T t=\dfrac{l^{2}n^{3}\mu}{n_{e}kT}
  20. μ \mu
  21. k k
  22. T T
  23. M M
  24. M e M_{e}
  25. M < 10 M e M<10M_{e}
  26. 10 M e 10M_{e}
  27. M - 2 M^{-2}
  28. M 3 M^{3}

Retinol_O-fatty-acyltransferase.html

  1. \rightleftharpoons

Retinyl-palmitate_esterase.html

  1. \rightleftharpoons

Return_ratio.html

  1. T = - i r i t . T=-\frac{i_{r}}{i_{t}}\ .
  2. i r = g m v π . i_{r}=g_{m}v_{\pi}\ .
  3. i f = R D / / r O R D / / r O + R F + r π / / R S i t . i_{f}=\frac{R_{D}//r_{O}}{R_{D}//r_{O}+R_{F}+r_{\pi}//R_{S}}\ i_{t}\ .
  4. v π = - i f ( r π / / R S ) . v_{\pi}=-i_{f}\ (r_{\pi}//R_{S})\ .
  5. T = g m ( r π / / R S ) R D / / r O R D / / r O + R F + r π / / R S . T=g_{m}(r_{\pi}//R_{S})\ \frac{R_{D}//r_{O}}{R_{D}//r_{O}+R_{F}+r_{\pi}//R_{S}% }\ .
  6. G = v o u t i i n = ( 1 - g m R F ) R 1 R 2 R F + R 1 + R 2 + g m R 1 R 2 , G=\frac{v_{out}}{i_{in}}=\frac{(1-g_{m}R_{F})R_{1}R_{2}}{R_{F}+R_{1}+R_{2}+g_{% m}R_{1}R_{2}}\ ,
  7. G = G T 1 + T + G 0 1 1 + T , G=\ G_{\infty}\frac{T}{1+T}+G_{0}\frac{1}{1+T}\ \ ,
  8. G = - R F , G_{\infty}=-R_{F}\ ,
  9. G 0 = R 1 R 2 R F + R 1 + R 2 . G_{0}=\frac{R_{1}R_{2}}{R_{F}+R_{1}+R_{2}}\ .

Reverberation_mapping.html

  1. G M = f R BLR ( Δ V ) 2 . GM_{\bullet}=fR_{\mathrm{BLR}}(\Delta V)^{2}.

Rhamnulokinase.html

  1. \rightleftharpoons

Rhamnulose-1-phosphate_aldolase.html

  1. \rightleftharpoons

Rhenium-osmium_dating.html

  1. ( Os 187 Os 188 ) present = ( Os 187 Os 188 ) initial + ( Re 187 Os 188 ) ( e λ t - 1 ) , \left(\frac{{}^{187}\mathrm{Os}}{{}^{188}\mathrm{Os}}\right)_{\mathrm{present}% }=\left(\frac{{}^{187}\mathrm{Os}}{{}^{188}\mathrm{Os}}\right)_{\mathrm{% initial}}+\left(\frac{{}^{187}\mathrm{Re}}{{}^{188}\mathrm{Os}}\right)\cdot(e^% {\lambda t}-1),

Rhind_Mathematical_Papyrus_2::n_table.html

  1. 2 3 n = 1 2 n + 1 6 n \frac{2}{3n}=\frac{1}{2n}+\frac{1}{6n}
  2. 2 n = 1 2 1 n + 3 2 1 n \frac{2}{n}=\frac{1}{2}\frac{1}{n}+\frac{3}{2}\frac{1}{n}
  3. 2 n = 1 3 1 n + 5 3 1 n \frac{2}{n}=\frac{1}{3}\frac{1}{n}+\frac{5}{3}\frac{1}{n}
  4. 2 m n = 1 m 1 k + 1 n 1 k \frac{2}{mn}=\frac{1}{m}\frac{1}{k}+\frac{1}{n}\frac{1}{k}
  5. 2 n = 1 n + 1 2 n + 1 3 n + 1 6 n \frac{2}{n}=\frac{1}{n}+\frac{1}{2n}+\frac{1}{3n}+\frac{1}{6n}
  6. 1 2 k \frac{1}{2^{k}}
  7. 1 / 2 + 1 / 4 + 1 / 8 + 1 / 16 + 1 / 32 + 1 / 64 + ( 5 r o ) 1/2+1/4+1/8+1/16+1/32+1/64+(5\ ro)

Riboflavin_kinase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Riboflavin_phosphotransferase.html

  1. \rightleftharpoons

Riboflavinase.html

  1. \rightleftharpoons

Ribokinase.html

  1. \rightleftharpoons

Ribose-5-phosphate_adenylyltransferase.html

  1. \rightleftharpoons

Ribose-5-phosphate—ammonia_ligase.html

  1. \rightleftharpoons

Ribose_1,5-bisphosphate_phosphokinase.html

  1. \rightleftharpoons

Ribosomal-protein-alanine_N-acetyltransferase.html

  1. \rightleftharpoons

Ribosylnicotinamide_kinase.html

  1. \rightleftharpoons

Ribosylpyrimidine_nucleosidase.html

  1. \rightleftharpoons

Ribulokinase.html

  1. \rightleftharpoons

Rice–Shapiro_theorem.html

  1. { n φ n A } \{n\mid\varphi_{n}\in A\}
  2. φ n \varphi_{n}
  3. n n
  4. ψ \psi
  5. ψ A \psi\in A\Leftrightarrow\exists
  6. θ ψ \theta\subseteq\psi
  7. θ A . \theta\in A.
  8. x 1 , x 2 , , x n x_{1},x_{2},...,x_{n}
  9. θ ψ \theta\subseteq\psi
  10. x { x 1 , x 2 , , x n } x\in\{x_{1},x_{2},...,x_{n}\}
  11. ψ ( x ) \psi(x)
  12. θ ( x ) \theta(x)
  13. { n : φ n A } \{n:\varphi_{n}\in A\}
  14. { θ : θ = φ n n A } \{\theta:\theta=\varphi_{n}\forall n\in A\}
  15. n A n\in A
  16. \exists
  17. θ \theta
  18. φ n \varphi_{n}
  19. θ c ( θ ) A \theta\wedge c(\theta)\in A
  20. c ( θ ) c(\theta)
  21. θ \theta

Ricinine_nitrilase.html

  1. \rightleftharpoons

Riemannian_Penrose_inequality.html

  1. m A 16 π . m\geq\sqrt{\frac{A}{16\pi}}.
  2. m = A 16 π , m=\sqrt{\frac{A}{16\pi}},

RNA-3'-phosphate_cyclase.html

  1. \rightleftharpoons

RNA_ligase_(ATP).html

  1. \rightleftharpoons

RNA_uridylyltransferase.html

  1. \rightleftharpoons

Robust_associations_of_massive_baryonic_objects.html

  1. M M_{\odot}

Rosmarinate_synthase.html

  1. \rightleftharpoons

Rotating_wheel_space_station.html

  1. a = ω 2 r a=\omega^{2}r
  2. ω \omega
  3. r r
  4. a a
  5. 9.81 m / s 2 9.81m/s^{2}

Rubber_cis-polyprenylcistransferase.html

  1. \rightleftharpoons

Ruin_theory.html

  1. X t = x + c t - i = 1 N t ξ i for t 0. X_{t}=x+ct-\sum_{i=1}^{N_{t}}\xi_{i}\quad\,\text{ for t}\geq 0.
  2. ψ ( x ) = x { τ < } \psi(x)=\mathbb{P}^{x}\{\tau<\infty\}
  3. τ = inf { t > 0 : X ( t ) < 0 } \scriptstyle\tau=\inf\{t>0\,:\,X(t)<0\}
  4. inf = \scriptstyle\inf\varnothing=\infty
  5. ψ ( x ) = ( 1 - λ μ c ) n = 0 ( λ μ c ) n ( 1 - F l n ( x ) ) \psi(x)=\left(1-\frac{\lambda\mu}{c}\right)\sum_{n=0}^{\infty}\left(\frac{% \lambda\mu}{c}\right)^{n}(1-F^{\ast n}_{l}(x))
  6. F l n ( x ) \scriptstyle F^{\ast n}_{l}(x)
  7. F l ( x ) = 1 μ 0 x ( 1 - F ( u ) ) d u . F_{l}(x)=\frac{1}{\mu}\int_{0}^{x}\left(1-F(u)\right)\,\text{d}u.
  8. ψ ( x ) = λ μ c e - ( 1 μ - λ β ) x . \psi(x)=\frac{\lambda\mu}{c}e^{-\left(\frac{1}{\mu}-\frac{\lambda}{\beta}% \right)x}.
  9. X t = x + c t - i = 1 N t ξ i for t 0 , X_{t}=x+ct-\sum_{i=1}^{N_{t}}\xi_{i}\quad\,\text{ for }t\geq 0,
  10. ( N t ) t 0 (N_{t})_{t\geq 0}
  11. ( ξ i ) i (\xi_{i})_{i\in\mathbb{N}}
  12. ξ i > 0 \xi_{i}>0
  13. ( N t ) t 0 (N_{t})_{t\geq 0}
  14. ( ξ i ) i (\xi_{i})_{i\in\mathbb{N}}
  15. m ( x ) = 𝔼 x [ e - δ τ K τ ] m(x)=\mathbb{E}^{x}[e^{-\delta\tau}K_{\tau}]
  16. δ \delta
  17. K τ K_{\tau}
  18. 𝔼 x \mathbb{E}^{x}
  19. x \mathbb{P}^{x}
  20. m ( x ) = 𝔼 x [ e - δ τ w ( X τ - , X τ ) 𝕀 ( τ < ) ] m(x)=\mathbb{E}^{x}[e^{-\delta\tau}w(X_{\tau-},X_{\tau})\mathbb{I}(\tau<\infty)]
  21. δ \delta
  22. w ( X τ - , X τ ) w(X_{\tau-},X_{\tau})
  23. X τ - X_{\tau-}
  24. X τ X_{\tau}
  25. 𝔼 x \mathbb{E}^{x}
  26. x \mathbb{P}^{x}
  27. 𝕀 ( τ < ) \mathbb{I}(\tau<\infty)
  28. τ \tau
  29. e - δ τ e^{-\delta\tau}
  30. τ \tau
  31. x { τ < } \mathbb{P}^{x}\{\tau<\infty\}
  32. δ = 0 , w ( x 1 , x 2 ) = 1 \delta=0,w(x_{1},x_{2})=1
  33. x { X τ - < x , X τ < y } \mathbb{P}^{x}\{X_{\tau-}<x,X_{\tau}<y\}
  34. δ = 0 , w ( x 1 , x 2 ) = 𝕀 ( x 1 < x , x 2 < y ) \delta=0,w(x_{1},x_{2})=\mathbb{I}(x_{1}<x,x_{2}<y)
  35. x { X τ - - X τ < z } \mathbb{P}^{x}\{X_{\tau-}-X_{\tau}<z\}
  36. δ = 0 , w ( x 1 , x 2 ) = 𝕀 ( x 1 + x 2 < z ) \delta=0,w(x_{1},x_{2})=\mathbb{I}(x_{1}+x_{2}<z)
  37. 𝔼 x [ e - δ τ - s X τ - - z X τ ] \mathbb{E}^{x}[e^{-\delta\tau-sX_{\tau-}-zX_{\tau}}]
  38. w ( x 1 , x 2 ) = e - s x 1 - z x 2 w(x_{1},x_{2})=e^{-sx_{1}-zx_{2}}
  39. 𝔼 x [ X τ - j X τ k ] \mathbb{E}^{x}[X_{\tau-}^{j}X_{\tau}^{k}]
  40. δ = 0 , w ( x 1 , x 2 ) = x 1 j x 2 k \delta=0,w(x_{1},x_{2})=x_{1}^{j}x_{2}^{k}

Rule_of_Sarrus.html

  1. M = ( a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ) , M=\begin{pmatrix}a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\end{pmatrix},
  2. det ( M ) = | a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 | = a 11 a 22 a 33 + a 12 a 23 a 31 + a 13 a 21 a 32 - a 31 a 22 a 13 - a 32 a 23 a 11 - a 33 a 21 a 12 . \det(M)=\begin{vmatrix}a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\end{vmatrix}=a_{11}a_{22}a_{33}+a_{12}a_{23}a_{31}+a_{13}% a_{21}a_{32}-a_{31}a_{22}a_{13}-a_{32}a_{23}a_{11}-a_{33}a_{21}a_{12}.
  3. det ( M ) = | a 11 a 12 a 21 a 22 | = a 11 a 22 - a 21 a 12 . \det(M)=\begin{vmatrix}a_{11}&a_{12}\\ a_{21}&a_{22}\end{vmatrix}=a_{11}a_{22}-a_{21}a_{12}.

Rules_of_passage_(logic).html

  1. Q x [ ¬ α ( x ) ] ¬ Q x [ α ( x ) ] . Qx[\lnot\alpha(x)]\leftrightarrow\lnot Q^{\prime}x[\alpha(x)].
  2. Q x [ β α ( x ) ] ( β Q x α ( x ) ) . \ Qx[\beta\alpha(x)]\leftrightarrow(\beta Qx\alpha(x)).
  3. \exist x [ α ( x ) γ ( x ) ] ( \exist x α ( x ) \exist x γ ( x ) ) . \exist x[\alpha(x)\gamma(x)]\leftrightarrow(\exist x\alpha(x)\exist x\gamma(x)).
  4. Q x [ β and α ( x ) ] ( β and Q x α ( x ) ) . \ Qx[\beta\and\alpha(x)]\leftrightarrow(\beta\and Qx\alpha(x)).
  5. x [ α ( x ) γ ( x ) ] ( x α ( x ) x γ ( x ) ) . \forall x\,[\alpha(x)\land\gamma(x)]\leftrightarrow(\forall x\,\alpha(x)\land% \forall x\,\gamma(x)).
  6. \exist x [ α ( x ) and γ ( x ) ] ( \exist x α ( x ) and \exist x γ ( x ) ) . \exist x[\alpha(x)\and\gamma(x)]\rightarrow(\exist x\alpha(x)\and\exist x% \gamma(x)).
  7. ( x α ( x ) x γ ( x ) ) x [ α ( x ) γ ( x ) ] . (\forall x\,\alpha(x)\forall x\,\gamma(x))\rightarrow\forall x\,[\alpha(x)% \gamma(x)].
  8. ( x α ( x ) and x γ ( x ) ) x [ α ( x ) and γ ( x ) ] . (\exists x\,\alpha(x)\and\forall x\,\gamma(x))\rightarrow\exists x\,[\alpha(x)% \and\gamma(x)].

Runs_produced.html

  1. R P = R + R B I - H R RP=R+RBI-HR
  2. R P = ( R + R B I ) / 2 RP=(R+RBI)/2

S-adenosylhomocysteine_deaminase.html

  1. \rightleftharpoons

S-formylglutathione_hydrolase.html

  1. \rightleftharpoons

S-linalool_synthase.html

  1. \rightleftharpoons

S-methyl-5'-thioadenosine_phosphorylase.html

  1. \rightleftharpoons

S-methyl-5-thioribose_kinase.html

  1. \rightleftharpoons

S-succinylglutathione_hydrolase.html

  1. \rightleftharpoons

Sabinene-hydrate_synthase.html

  1. \rightleftharpoons

Salicyl-alcohol_beta-D-glucosyltransferase.html

  1. \rightleftharpoons

Salutaridinol_7-O-acetyltransferase.html

  1. \rightleftharpoons

Sampling_design.html

  1. P ( S ) P(S)
  2. S . S.
  3. P ( S ) P(S)
  4. P ( S ) = q N sample ( S ) × ( 1 - q ) ( N pop - N sample ( S ) ) P(S)=q^{N\text{sample}(S)}\times(1-q)^{(N\text{pop}-N\text{sample}(S))}
  5. q q
  6. N sample ( S ) N\text{sample}(S)
  7. S S
  8. N pop N\text{pop}

Saturated-surface-dry.html

  1. A = M s s d - M d r y M d r y A=\frac{M_{ssd}-M_{dry}}{M_{dry}}

Schanuel's_lemma.html

  1. X = { ( p , q ) P P : ϕ ( p ) = ϕ ( q ) } . X=\{(p,q)\in P\oplus P^{\prime}:\phi(p)=\phi^{\prime}(q)\}.
  2. \in
  3. \in
  4. \in
  5. ker π \displaystyle\ker\pi
  6. 0 K X P 0. 0\rightarrow K^{\prime}\rightarrow X\rightarrow P\rightarrow 0.
  7. 0 K X P 0 , 0\rightarrow K\rightarrow X\rightarrow P^{\prime}\rightarrow 0,

Scheffé's_method.html

  1. C = i = 1 r c i μ i C=\sum_{i=1}^{r}c_{i}\mu_{i}
  2. i = 1 r c i = 0. \sum_{i=1}^{r}c_{i}=0.
  3. C ^ = i = 1 r c i Y ¯ i \hat{C}=\sum_{i=1}^{r}c_{i}\bar{Y}_{i}
  4. s C ^ 2 = σ ^ e 2 i = 1 r c i 2 n i , s_{\hat{C}}^{2}=\hat{\sigma}_{e}^{2}\sum_{i=1}^{r}\frac{c_{i}^{2}}{n_{i}},
  5. σ ^ e 2 \hat{\sigma}_{e}^{2}
  6. C ^ ± s C ^ ( r - 1 ) F α ; r - 1 ; N - r \hat{C}\pm\,s_{\hat{C}}\sqrt{\left(r-1\right)F_{\alpha;r-1;N-r}}

Schottky_anomaly.html

  1. S = 0 T ( C v T ) d T S=\int_{0}^{T}\!\left(\frac{C_{v}}{T}\right)dT\,
  2. C Schottky = R ( Δ T ) 2 e Δ / T [ 1 + e Δ / T ] 2 C_{\rm Schottky}=R\left(\frac{\Delta}{T}\right)^{2}\frac{e^{\Delta/T}}{[1+e^{% \Delta/T}]^{2}}\,

Schur–Weyl_duality.html

  1. n n n \mathbb{C}^{n}\otimes\mathbb{C}^{n}\otimes\cdots\otimes\mathbb{C}^{n}
  2. σ ( v 1 v 2 v k ) = v σ - 1 ( 1 ) v σ - 1 ( 2 ) v σ - 1 ( k ) . \sigma(v_{1}\otimes v_{2}\otimes\cdots\otimes v_{k})=v_{\sigma^{-1}(1)}\otimes v% _{\sigma^{-1}(2)}\otimes\cdots\otimes v_{\sigma^{-1}(k)}.
  3. g ( v 1 v 2 v k ) = g v 1 g v 2 g v k , g G L n . g(v_{1}\otimes v_{2}\otimes\cdots\otimes v_{k})=gv_{1}\otimes gv_{2}\otimes% \cdots\otimes gv_{k},\quad g\in GL_{n}.
  4. n n n = D π k D ρ n D . \mathbb{C}^{n}\otimes\mathbb{C}^{n}\otimes\cdots\otimes\mathbb{C}^{n}=\sum_{D}% \pi_{k}^{D}\otimes\rho_{n}^{D}.
  5. π k D \pi_{k}^{D}
  6. ρ n D \rho_{n}^{D}
  7. End ( n n n ) . \mathrm{End}_{\mathbb{C}}(\mathbb{C}^{n}\otimes\mathbb{C}^{n}\otimes\cdots% \otimes\mathbb{C}^{n}).
  8. n n = S 2 n Λ 2 n . \mathbb{C}^{n}\otimes\mathbb{C}^{n}=S^{2}\mathbb{C}^{n}\oplus\Lambda^{2}% \mathbb{C}^{n}.

Schwarzschild_criterion.html

  1. - d T d Z < g C p -\frac{dT}{dZ}<\frac{g}{C_{p}}
  2. g g
  3. C p C_{p}

Scopoletin_glucosyltransferase.html

  1. \rightleftharpoons

Scyllo-inosamine_4-kinase.html

  1. \rightleftharpoons

Scytalone_dehydratase.html

  1. \rightleftharpoons

Secondary_flow.html

  1. c 1 c_{1}
  2. w 1 = d c 1 d z w_{1}=\frac{dc_{1}}{dz}
  3. w s = - 2 e ( d c 1 d z ) w_{s}=-2e\left(\frac{dc_{1}}{dz}\right)

Sedoheptulokinase.html

  1. \rightleftharpoons

Segmentation-based_object_categorization.html

  1. w i j w_{ij}
  2. ( i , j ) E (i,j)\in E
  3. i i
  4. j j
  5. V 1 , , V k V_{1},\cdots,V_{k}
  6. V V
  7. V i V_{i}
  8. V i , V j V_{i},V_{j}
  9. A A
  10. B B
  11. w ( A , B ) = i A , j B w i j w(A,B)=\sum\limits_{i\in A,j\in B}w_{ij}
  12. ncut ( A , B ) = w ( A , B ) w ( A , V ) + w ( A , B ) w ( B , V ) \operatorname{ncut}(A,B)=\frac{w(A,B)}{w(A,V)}+\frac{w(A,B)}{w(B,V)}
  13. nassoc ( A , B ) = w ( A , A ) w ( A , V ) + w ( B , B ) w ( B , V ) \operatorname{nassoc}(A,B)=\frac{w(A,A)}{w(A,V)}+\frac{w(B,B)}{w(B,V)}
  14. ( S , S ¯ ) (S,\overline{S})
  15. G G
  16. ncut ( S , S ¯ ) \operatorname{ncut}(S,\overline{S})
  17. nassoc ( S , S ¯ ) \operatorname{nassoc}(S,\overline{S})
  18. ncut ( S , S ¯ ) = 2 - nassoc ( S , S ¯ ) \operatorname{ncut}(S,\overline{S})=2-\operatorname{nassoc}(S,\overline{S})
  19. ( S * , S ¯ * ) (S^{*},{\overline{S}}^{*})
  20. ncut ( S , S ¯ ) \operatorname{ncut}(S,\overline{S})
  21. nassoc ( S , S ¯ ) \operatorname{nassoc}(S,\overline{S})
  22. ( S * , S ¯ * ) (S^{*},{\overline{S}}^{*})
  23. ncut ( S , S ¯ ) \operatorname{ncut}(S,\overline{S})
  24. ( S , S ¯ ) (S,\overline{S})
  25. ncut ( S , S ¯ ) \operatorname{ncut}(S,\overline{S})
  26. d ( i ) = j w i j d(i)=\sum\limits_{j}w_{ij}
  27. n × n n\times n
  28. d d
  29. W W
  30. n × n n\times n
  31. W i j = w i j W_{ij}=w_{ij}
  32. min ( S , S ¯ ) ncut ( S , S ¯ ) = min y y T ( D - W ) y y T D y \min\limits_{(S,\overline{S})}\operatorname{ncut}(S,\overline{S})=\min\limits_% {y}\frac{y^{T}(D-W)y}{y^{T}Dy}
  33. y i { 1 , - b } y_{i}\in\{1,-b\}
  34. - b -b
  35. y t D 1 = 0 y^{t}D1=0
  36. y T ( D - W ) y y T D y \frac{y^{T}(D-W)y}{y^{T}Dy}
  37. y y
  38. ( D - W ) y = λ D y (D-W)y=\lambda Dy
  39. G = ( V , E ) G=(V,E)
  40. D D
  41. W W
  42. ( D - W ) y = λ D y (D-W)y=\lambda Dy
  43. O ( n 3 ) O(n^{3})
  44. n n
  45. O ( n ) O(n)
  46. O ( n ) O(n)
  47. O ( n ) O(n)
  48. n n
  49. Θ \Theta
  50. Θ \Theta
  51. E ( m , Θ ) E(m,\Theta)
  52. E ( m , Θ ) = ϕ x ( D | m x ) + ϕ x ( m x | Θ ) + Ψ x y ( m x , m y ) + ϕ ( D | m x , m y ) E(m,\Theta)=\sum\phi_{x}(D|m_{x})+\phi_{x}(m_{x}|\Theta)+\sum\Psi_{xy}(m_{x},m% _{y})+\phi(D|m_{x},m_{y})
  53. ϕ x ( D | m x ) + ϕ x ( m x | Θ ) \phi_{x}(D|m_{x})+\phi_{x}(m_{x}|\Theta)
  54. Ψ x y ( m x , m y ) + ϕ ( D | m x , m y ) \Psi_{xy}(m_{x},m_{y})+\phi(D|m_{x},m_{y})
  55. ϕ x ( D | m x ) \phi_{x}(D|m_{x})
  56. ϕ x ( m x | Θ ) \phi_{x}(m_{x}|\Theta)
  57. Θ \Theta
  58. Ψ x y ( m x , m y ) \Psi_{xy}(m_{x},m_{y})
  59. ϕ ( D | m x , m y ) \phi(D|m_{x},m_{y})
  60. m * m^{*}
  61. i w i E ( m , Θ i ) \sum\limits_{i}w_{i}E(m,\Theta_{i})
  62. w i w_{i}
  63. Θ i \Theta_{i}
  64. m * = arg min m i w i E ( m , Θ i ) m^{*}=\arg\min\limits_{m}\sum\limits_{i}w_{i}E(m,\Theta_{i})
  65. Θ 1 , , Θ s \Theta_{1},\cdots,\Theta_{s}
  66. E ( m , Θ i ) E(m,\Theta_{i})
  67. w i = g ( Θ i | Z ) w_{i}=g(\Theta_{i}|Z)

Segregation_in_materials.html

  1. G = p e + P E - k T [ ln ( n ! N ! ) - ln ( n - p ) ! p ! ( N - P ) ! P ! ] G=pe+PE-kT[\ln(n!N!)-\ln(n-p)!p!(N-P)!P!]
  2. X b X_{b}
  3. X b X b 0 - X b = X c 1 - X c exp \frac{X_{b}}{X_{b}^{0}-X_{b}}=\frac{X_{c}}{1-X_{c}}\exp
  4. ( - Δ G R T ) \left(\frac{-\Delta\,G}{RT}\right)
  5. X b 0 X_{b}^{0}
  6. X b X_{b}
  7. X c X_{c}
  8. Δ G \Delta\,G
  9. Δ G \Delta\,G
  10. E e l E_{el}
  11. E e l = 24 π K μ 0 r 0 ( r 1 - r 0 ) 2 3 K + 4 μ 0 E_{el}=\frac{24\pi\,K\,\mu\,_{0}r_{0}(r_{1}-r_{0})^{2}}{3K\,+4\mu\,_{0}}
  12. K K\,
  13. μ 0 , \mu\,_{0},
  14. r 0 , r_{0},
  15. r 1 , r_{1},
  16. X b X b 0 - X b = X c X c 0 exp \frac{X_{b}}{X_{b}^{0}-X_{b}}=\frac{X_{c}}{X_{c}^{0}}\exp
  17. ( - Δ G R T ) \left(\frac{-\Delta\ G^{\prime}}{RT}\right)
  18. Δ G = Δ G + Δ G s o l \Delta\,G=\Delta\,G^{\prime}+\Delta\,G_{sol}
  19. X c 0 X_{c}^{0}
  20. X c 0 = e x p ( Δ G s o l R T ) X_{c}^{0}=exp\left(\frac{\Delta\,G_{sol}}{RT}\right)
  21. X c X c 0 X_{c}\leq X_{c}^{0}
  22. X c 0 X_{c}^{0}
  23. X b X b 0 - X b = e x p ( - Δ G R T ) \frac{X_{b}}{X_{b}^{0}-X_{b}}=exp\left(\frac{-\Delta\,G^{\prime}}{RT}\right)
  24. ω \omega\,
  25. ω \omega\,
  26. ω \omega\,
  27. X b X b 0 - X b = X c 1 - X c e x p [ - Δ G - Z 1 ω X b X b 0 R T ] \frac{X_{b}}{X_{b}^{0}-X_{b}}=\frac{X_{c}}{1-X_{c}}exp\left[\frac{-\Delta\,G-Z% _{1}\omega\,\frac{X_{b}}{X_{b}^{0}}}{RT}\right]
  28. ω \omega\,
  29. ω \omega\,
  30. X b X_{b}
  31. X s X_{s}
  32. Δ G s = Δ H s - T Δ S \Delta\,G_{s}=\Delta\,H_{s}-T\Delta\,S
  33. - Δ H s = γ 0 s - γ 1 s - 2 H m Z X c ( 1 - X c ) -\Delta\,H_{s}=\gamma\,_{0}^{s}-\gamma\,_{1}^{s}-\frac{2H_{m}}{ZX_{c}(1-X_{c})}
  34. [ Z 1 ( X c - X s ) + Z v ( X c - 1 2 ) ] + 24 π K μ 0 r 0 ( r 1 - r 0 ) 2 3 K + 4 μ 0 \left[Z_{1}(X_{c}-X_{s})+Z_{v}\left(X_{c}-\frac{1}{2}\right)\right]+\frac{24% \pi\,K\,\mu\,_{0}r_{0}(r_{1}-r_{0})^{2}}{3K\,+4\mu\,_{0}}
  35. γ 0 \gamma\,_{0}
  36. γ 1 \gamma\,_{1}
  37. H 1 H_{1}
  38. Z 1 Z_{1}
  39. Z v Z_{v}
  40. E ( e l ) E_{(}el)
  41. Δ G c h e m \Delta\,G_{chem}
  42. Δ G c h e m = Δ G s + ( E B - E A ) Θ \Delta\,G_{chem}=\Delta\,G_{s}+(E_{B}-E_{A})\Theta\,
  43. E A E_{A}
  44. E B E_{B}
  45. X b 0 X_{b}^{0}
  46. X b X c = exp ( - Δ G R T ) X c 0 \frac{X_{b}}{X_{c}}=\frac{\exp\left(\frac{-\Delta\,G^{\prime}}{RT}\right)}{X_{% c}^{0}}
  47. X b ( t ) - X b ( 0 ) X b ( ) - X b ( 0 ) = 1 - exp ( F D t β 2 f 2 ) \frac{X_{b}(t)-X_{b}(0)}{X_{b}(\infty\,)-X_{b}(0)}=1-\exp\left(\frac{FDt}{% \beta\,^{2}f^{2}}\right)
  48. e r f c ( F D t β 2 f 2 ) 1 2 erfc\left(\frac{FDt}{\beta\,^{2}f^{2}}\right)^{\frac{1}{2}}
  49. X b ( t ) X_{b}(t)
  50. f = a 3 b - 2 f=a^{3}b^{-2}
  51. X b ( t ) - X b ( 0 ) X b ( ) - X b ( 0 ) = 2 β f F D t π = 2 β b 2 a 3 F D t π \frac{X_{b}(t)-X_{b}(0)}{X_{b}(\infty\,)-X_{b}(0)}=\frac{2}{\beta\,f}\sqrt{% \frac{FDt}{\pi\,}}=\frac{2}{\beta\,}\frac{b^{2}}{a^{3}}\sqrt{\frac{FDt}{\pi\,}}

Seiberg–Witten_invariant.html

  1. D A ϕ = 0 D^{A}\phi=0
  2. F A + = σ ( ϕ ) + i ω F^{+}_{A}=\sigma(\phi)+i\omega
  3. ω \omega
  4. ϕ = 0 \phi=0
  5. ω \omega
  6. ω / 2 π \omega/2\pi
  7. ( c 1 ( s ) 2 - 2 χ ( M ) - 3 s i g n ( M ) ) / 4. (c_{1}(s)^{2}-2\chi(M)-3sign(M))/4.

Selenide,_water_dikinase.html

  1. \rightleftharpoons

Self-concordant_function.html

  1. f : f:\mathbb{R}\rightarrow\mathbb{R}
  2. | f ′′′ ( x ) | 2 f ′′ ( x ) 3 / 2 . |f^{\prime\prime\prime}(x)|\leq 2f^{\prime\prime}(x)^{3/2}.
  3. g ( x ) : n g(x):\mathbb{R}^{n}\rightarrow\mathbb{R}

Semi-s-cobordism.html

  1. M W M\hookrightarrow W
  2. M - W M^{-}\hookrightarrow W
  3. K = ker ( π 1 ( M - ) π 1 ( W ) ) K=\ker(\pi_{1}(M^{-})\twoheadrightarrow\pi_{1}(W))
  4. π 1 ( M - ) \pi_{1}(M^{-})
  5. 1 K π 1 ( M - ) π 1 ( M ) 1 1\rightarrow K\rightarrow\pi_{1}(M^{-})\rightarrow\pi_{1}(M)\rightarrow 1
  6. π 1 ( M ) \pi_{1}(M)

Semisimple_algebraic_group.html

  1. k k
  2. S L n ( k ) SL_{n}(k)

Sepiapterin_deaminase.html

  1. \rightleftharpoons

Serine-ethanolaminephosphate_phosphodiesterase.html

  1. \rightleftharpoons

Serine-phosphoethanolamine_synthase.html

  1. \rightleftharpoons

Serine_C-palmitoyltransferase.html

  1. \rightleftharpoons

Serine_O-acetyltransferase.html

  1. \rightleftharpoons

Serine—glyoxylate_transaminase.html

  1. \rightleftharpoons

Serine—pyruvate_transaminase.html

  1. \rightleftharpoons

Serine—tRNA_ligase.html

  1. \rightleftharpoons

Shape_context.html

  1. h i ( k ) = # { q p i : ( q - p i ) bin ( k ) } h_{i}(k)=\#\{q\neq p_{i}:(q-p_{i})\in\mbox{bin}~{}(k)\}
  2. p i p_{i}
  3. α \alpha
  4. C S = 1 2 k = 1 K [ g ( k ) - h ( k ) ] 2 g ( k ) + h ( k ) C_{S}=\frac{1}{2}\sum_{k=1}^{K}\frac{[g(k)-h(k)]^{2}}{g(k)+h(k)}
  5. C A = 1 2 ( cos ( θ 1 ) sin ( θ 1 ) ) - ( cos ( θ 2 ) sin ( θ 2 ) ) C_{A}=\frac{1}{2}\begin{Vmatrix}{\displaystyle\left({{\cos(\theta_{1})}\atop{% \sin(\theta_{1})}}\right)}-{\displaystyle\left({{\cos(\theta_{2})}\atop{\sin(% \theta_{2})}}\right)}\end{Vmatrix}
  6. θ 1 \theta_{1}
  7. θ 2 \theta_{2}
  8. C = ( 1 - β ) C S + β C A C=(1-\beta)C_{S}+\beta C_{A}\!\,
  9. H ( π ) = i C ( p i , q π ( i ) ) H(\pi)=\sum_{i}C\left(p_{i},q_{\pi(i)}\right)
  10. O ( N 3 ) O(N^{3})
  11. T : 2 2 T:\mathbb{R}^{2}\to\mathbb{R}^{2}
  12. T ( p ) = A p + o T(p)=Ap+o\!
  13. A A
  14. o = 1 n i = 1 n ( p i - q π ( i ) ) , A = ( Q + P ) t o=\frac{1}{n}\sum_{i=1}^{n}\left(p_{i}-q_{\pi(i)}\right),A=(Q^{+}P)^{t}
  15. P = ( 1 p 11 p 12 1 p n 1 p n 2 ) P=\begin{pmatrix}1&p_{11}&p_{12}\\ \vdots&\vdots&\vdots\\ 1&p_{n1}&p_{n2}\end{pmatrix}
  16. Q Q\!
  17. Q + Q^{+}\!
  18. Q Q\!
  19. T ( x , y ) = ( f x ( x , y ) , f y ( x , y ) ) T(x,y)=\left(f_{x}(x,y),f_{y}(x,y)\right)
  20. f ( x , y ) = a 1 + a x x + a y y + i = 1 n ω i U ( ( x i , y i ) - ( x , y ) ) , f(x,y)=a_{1}+a_{x}x+a_{y}y+\sum_{i=1}^{n}\omega_{i}U\left(\begin{Vmatrix}(x_{i% },y_{i})-(x,y)\end{Vmatrix}\right),
  21. U ( r ) U(r)\!
  22. U ( r ) = r 2 log r 2 U(r)=r^{2}\log r^{2}\!
  23. v i v_{i}
  24. p i = ( x i , y i ) p_{i}=(x_{i},y_{i})
  25. f x f_{x}
  26. v i v_{i}
  27. x x^{\prime}
  28. p i p_{i}
  29. f y f_{y}
  30. y y^{\prime}
  31. H [ f ] = i = 1 n ( v i - f ( x i , y i ) ) 2 + λ I f H[f]=\sum_{i=1}^{n}(v_{i}-f(x_{i},y_{i}))^{2}+\lambda I_{f}
  32. I f I_{f}\!
  33. λ \lambda\!
  34. ( x i , y i ) and ( x i , y i ) (x_{i},y_{i})\mbox{ and }~{}(x^{\prime}_{i},y^{\prime}_{i})
  35. P P\!
  36. Q Q\!
  37. D s c ( P , Q ) = 1 n p P arg min q Q C ( p , T ( q ) ) + 1 m q Q arg min p P C ( p , T ( q ) ) D_{sc}(P,Q)=\frac{1}{n}\sum_{p\in P}\arg\underset{q\in Q}{\min}C(p,T(q))+\frac% {1}{m}\sum_{q\in Q}\arg\underset{p\in P}{\min}C(p,T(q))
  38. D a c ( P , Q ) = 1 n i = 1 n Δ Z 2 G ( Δ ) [ I P ( p i + Δ ) - I Q ( T ( q π ( i ) ) + Δ ) ] 2 D_{ac}(P,Q)=\frac{1}{n}\sum_{i=1}^{n}\sum_{\Delta\in Z^{2}}G(\Delta)\left[I_{P% }(p_{i}+\Delta)-I_{Q}(T(q_{\pi(i)})+\Delta)\right]^{2}
  39. I P I_{P}\!
  40. I Q I_{Q}\!
  41. I Q I_{Q}\!
  42. G G\!
  43. D b e ( P , Q ) D_{be}(P,Q)\!\,

Shift_space.html

  1. 𝐱 = ( x n ) n M \mathbf{x}=(x_{n})_{n\in M}
  2. M = M=\mathbb{N}
  3. M = M=\mathbb{Z}
  4. x n x_{n}
  5. n M n\in M
  6. σ \sigma
  7. ( σ ( 𝐱 ) ) ( n ) = x n + 1 (\sigma(\mathbf{x}))(n)=x_{n+1}
  8. M = M=\mathbb{N}
  9. A A^{\mathbb{N}}
  10. S A S\subseteq A^{\mathbb{N}}
  11. ( 𝐱 k ) k 0 (\mathbf{x}_{k})_{k\geq 0}
  12. lim k 𝐱 k \lim_{k\to\infty}\mathbf{x}_{k}
  13. σ ( S ) = S \sigma(S)=S
  14. ( S , σ ) (S,\sigma)
  15. A A^{\mathbb{N}}
  16. A A^{\mathbb{N}}
  17. A = { a , b } A=\{a,b\}

Shikimate_O-hydroxycinnamoyltransferase.html

  1. \rightleftharpoons

Shock_capturing_method.html

  1. U t + F x + G y + H z = 0 \frac{\partial{U}}{\partial t}+\frac{\partial{F}}{\partial x}+\frac{\partial{G% }}{\partial y}+\frac{\partial{H}}{\partial z}=0
  2. U = [ ρ ρ u ρ v ρ w ρ e t ] F = [ ρ u ρ u 2 + p ρ u v ρ u w ( ρ e t + p ) u ] G = [ ρ v ρ v u ρ v 2 + p ρ v w ( ρ e t + p ) v ] H = [ ρ w ρ w u ρ w v ρ w 2 + p ( ρ e t + p ) w ] {U}=\left[\begin{array}[]{c}\rho\\ \rho u\\ \rho v\\ \rho w\\ \rho e_{t}\\ \end{array}\right]\quad{F}=\left[\begin{array}[]{c}\rho u\\ \rho u^{2}+p\\ \rho uv\\ \rho uw\\ (\rho e_{t}+p)u\\ \end{array}\right]\quad{G}=\left[\begin{array}[]{c}\rho v\\ \rho vu\\ \rho v^{2}+p\\ \rho vw\\ (\rho e_{t}+p)v\\ \end{array}\right]\quad{H}=\left[\begin{array}[]{c}\rho w\\ \rho wu\\ \rho wv\\ \rho w^{2}+p\\ (\rho e_{t}+p)w\\ \end{array}\right]\qquad
  3. e t e_{t}
  4. e t = e + u 2 + v 2 + w 2 2 + g z e_{t}=e+\frac{u^{2}+v^{2}+w^{2}}{2}+gz

Short_division.html

  1. 125 4 ) 500 ¯ \begin{matrix}\quad 125\\ 4\overline{)500}\\ \end{matrix}
  2. 4 ) 500 ¯ 125 \begin{matrix}4\underline{)500}\\ \ \ 125\\ \end{matrix}
  3. 4 ) 9 1 5 3 0. 2 0 ¯ 2 3 7. 5 \begin{matrix}4\underline{)9^{1}5^{3}0.^{2}0}\\ \ \ 2\ 3\ 7.\ 5\\ \end{matrix}
  4. 2 ) 950 ¯ 5 ) 475 ¯ 5 ) 95 ¯ 19 \begin{matrix}2\underline{)950}\\ 5\underline{)475}\\ 5\underline{)\ 95}\\ \ \ \ 19\\ \end{matrix}
  5. 7 ) 16 2 7 6 6 3 2 4 1 6 0 4 9 0 \begin{matrix}7)16^{2}7^{6}6^{3}2^{4}1^{6}0^{4}9^{0}\end{matrix}

Sialate_O-acetylesterase.html

  1. \rightleftharpoons

Sieve_of_Sundaram.html

  1. i , j , 1 i j i,j\in\mathbb{N},\ 1\leq i\leq j
  2. i + j + 2 i j n i+j+2ij\leq n
  3. 1 j k / 2 1\leq j\leq\lfloor k/2\rfloor

Signal-flow_graph.html

  1. x 1 \displaystyle x_{\mathrm{1}}
  2. x 2 = f 21 ( x 1 ) + f 23 ( x 3 ) x_{2}=f_{21}(x_{1})+f_{23}(x_{3})
  3. x 3 = f 31 ( x 1 ) + f 32 ( x 2 ) + f 33 ( x 3 ) , x_{3}=f_{31}(x_{1})+f_{32}(x_{2})+f_{33}(x_{3})\ ,
  4. x j \displaystyle x_{\mathrm{j}}
  5. t j k t_{jk}
  6. x k x_{k}
  7. x j x_{j}
  8. x x
  9. m m
  10. m m
  11. s s
  12. ω \omega
  13. z z
  14. V i n V_{in}
  15. V i n V_{in}
  16. m m
  17. I o u t I_{out}
  18. m m
  19. A × m A\times m
  20. Z = a X + b Y Z=aX+bY
  21. N \displaystyle N
  22. N \displaystyle N
  23. N \displaystyle N
  24. k = 1 N c jk x k = y j \begin{aligned}\displaystyle\sum_{\mathrm{k}=1}^{\mathrm{N}}c_{\mathrm{jk}}x_{% \mathrm{k}}&\displaystyle=y_{\mathrm{j}}\end{aligned}
  25. k = 1 N c jk x k - y j = 0 \begin{aligned}\displaystyle\sum_{\mathrm{k}=1}^{\mathrm{N}}c_{\mathrm{jk}}x_{% \mathrm{k}}-y_{\mathrm{j}}&\displaystyle=0\end{aligned}
  26. k = 1 N c jk x k + x j - y j = x j \begin{aligned}\displaystyle\sum_{\mathrm{k=1}}^{\mathrm{N}}c_{\mathrm{jk}}x_{% \mathrm{k}}+x_{\mathrm{j}}-y_{\mathrm{j}}&\displaystyle=x_{\mathrm{j}}\end{aligned}
  27. k = 1 N ( c jk + δ jk ) x k - y j = x j \begin{aligned}\displaystyle\sum_{\mathrm{k=1}}^{\mathrm{N}}(c_{\mathrm{jk}}+% \delta_{\mathrm{jk}})x_{\mathrm{k}}-y_{\mathrm{j}}&\displaystyle=x_{\mathrm{j}% }\end{aligned}
  28. x 1 = ( 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}\ ,
  29. x k = k = 1 M ( G kj ) y j \begin{aligned}\displaystyle x_{\mathrm{k}}&\displaystyle=\sum_{\mathrm{k}=1}^% {\mathrm{M}}(G_{\mathrm{kj}})y_{\mathrm{j}}\end{aligned}
  30. V 2 = a 12 V 1 . V_{2}=a_{12}V_{1}\,.
  31. G = y 2 x 1 G=\frac{y_{2}}{x_{1}}
  32. = G ( T T + 1 ) + G 0 ( 1 T + 1 ) . =G_{\infty}\left(\frac{T}{T+1}\right)+G_{0}\left(\frac{1}{T+1}\right)\ .
  33. G = lim T G ; G 0 = lim T 0 G . G_{\infty}=\lim_{T\to\infty}G\ ;\ G_{0}=\lim_{T\to 0}G\ .
  34. G = y 2 x 1 G=\frac{y_{2}}{x_{1}}
  35. = G 0 + A 1 - β A . =G_{0}+\frac{A}{1-\beta A}\ .
  36. T = - β A , T=-\beta A\ ,
  37. G = lim T G = G 0 - 1 β . G_{\infty}=\lim_{T\to\infty}G=G_{0}-\frac{1}{\beta}\ .
  38. G = G 0 + 1 β - T 1 + T G=G_{0}+\frac{1}{\beta}\frac{-T}{1+T}
  39. = G 0 + ( G 0 - G ) - T 1 + T =G_{0}+(G_{0}-G_{\infty})\frac{-T}{1+T}
  40. = G T 1 + T + G 0 1 1 + T , =G_{\infty}\frac{T}{1+T}+G_{0}\frac{1}{1+T}\ ,
  41. 1 s L M \frac{1}{s\mathrm{L}_{\mathrm{M}}}\,
  42. x j \displaystyle x_{\mathrm{j}}

Similarities_between_Wiener_and_LMS.html

  1. s [ n ] s[n]
  2. x [ n ] x[n]
  3. x [ n ] = k = 0 N - 1 h k s [ n - k ] + w [ n ] x[n]=\sum_{k=0}^{N-1}h_{k}s[n-k]+w[n]
  4. h k h_{k}
  5. w [ n ] w[n]
  6. x ^ [ n ] \hat{x}[n]
  7. x ^ [ n ] = k = 0 N - 1 h ^ k s [ n - k ] \hat{x}[n]=\sum_{k=0}^{N-1}\hat{h}_{k}s[n-k]
  8. h ^ k \hat{h}_{k}
  9. e [ n ] = x [ n ] - x ^ [ n ] e[n]=x[n]-\hat{x}[n]
  10. E E
  11. E = n = - e [ n ] 2 E=\sum_{n=-\infty}^{\infty}e[n]^{2}
  12. E = n = - ( x [ n ] - x ^ [ n ] ) 2 E=\sum_{n=-\infty}^{\infty}(x[n]-\hat{x}[n])^{2}
  13. E = n = - ( x [ n ] 2 - 2 x [ n ] x ^ [ n ] + x ^ [ n ] 2 ) E=\sum_{n=-\infty}^{\infty}(x[n]^{2}-2x[n]\hat{x}[n]+\hat{x}[n]^{2})
  14. n n
  15. E = 0 \nabla E=0
  16. E h ^ i = 0 \frac{\partial E}{\partial\hat{h}_{i}}=0
  17. i = 0 , 1 , 2 , , N - 1 i=0,1,2,...,N-1
  18. E h ^ i = h ^ i n = - [ x [ n ] 2 - 2 x [ n ] x ^ [ n ] + x ^ [ n ] 2 ] \frac{\partial E}{\partial\hat{h}_{i}}=\frac{\partial}{\partial\hat{h}_{i}}% \sum_{n=-\infty}^{\infty}[x[n]^{2}-2x[n]\hat{x}[n]+\hat{x}[n]^{2}]
  19. x ^ [ n ] \hat{x}[n]
  20. E h ^ i = h ^ i n = - [ x [ n ] 2 - 2 x [ n ] k = 0 N - 1 h ^ k s [ n - k ] + ( k = 0 N - 1 h ^ k s [ n - k ] ) 2 ] \frac{\partial E}{\partial\hat{h}_{i}}=\frac{\partial}{\partial\hat{h}_{i}}% \sum_{n=-\infty}^{\infty}[x[n]^{2}-2x[n]\sum_{k=0}^{N-1}\hat{h}_{k}s[n-k]+(% \sum_{k=0}^{N-1}\hat{h}_{k}s[n-k])^{2}]
  21. E h ^ i = n = - [ - 2 x [ n ] s [ n - i ] + 2 ( k = 0 N - 1 h ^ k s [ n - k ] ) s [ n - i ] ] \frac{\partial E}{\partial\hat{h}_{i}}=\sum_{n=-\infty}^{\infty}[-2x[n]s[n-i]+% 2(\sum_{k=0}^{N-1}\hat{h}_{k}s[n-k])s[n-i]]
  22. R x y ( i ) = n = - x [ n ] y [ n - i ] R_{xy}(i)=\sum_{n=-\infty}^{\infty}x[n]y[n-i]
  23. E h ^ i = - 2 R x s [ i ] + 2 k = 0 N - 1 h ^ k R s s [ i - k ] = 0 \frac{\partial E}{\partial\hat{h}_{i}}=-2R_{xs}[i]+2\sum_{k=0}^{N-1}\hat{h}_{k% }R_{ss}[i-k]=0
  24. R x s [ i ] = k = 0 N - 1 h ^ k R s s [ i - k ] R_{xs}[i]=\sum_{k=0}^{N-1}\hat{h}_{k}R_{ss}[i-k]
  25. i = 0 , 1 , 2 , , N - 1 i=0,1,2,...,N-1
  26. n n
  27. E = ( d [ n ] - y [ n ] ) 2 E=(d[n]-y[n])^{2}
  28. E w = w ( d [ n ] - y [ n ] ) 2 \frac{\partial E}{\partial w}=\frac{\partial}{\partial w}(d[n]-y[n])^{2}
  29. E w = 2 ( d [ n ] - y [ n ] ) w ( d [ n ] - k = 0 N - 1 w ^ k x [ n - k ] ) \frac{\partial E}{\partial w}=2(d[n]-y[n])\frac{\partial}{\partial w}(d[n]-% \sum_{k=0}^{N-1}\hat{w}_{k}x[n-k])
  30. E w = - 2 ( e [ n ] ) ( x [ n - i ] ) \frac{\partial E}{\partial w}=-2(e[n])(x[n-i])
  31. μ \mu
  32. w [ n + 1 ] = w [ n ] - μ E w w[n+1]=w[n]-\mu\frac{\partial E}{\partial w}
  33. w [ n + 1 ] = w [ n ] + 2 μ ( e [ n ] ) ( x [ n - i ] ) w[n+1]=w[n]+2\mu(e[n])(x[n-i])

Sinapate_1-glucosyltransferase.html

  1. \rightleftharpoons

Sinapine_esterase.html

  1. \rightleftharpoons

Sinapoylglucose—choline_O-sinapoyltransferase.html

  1. \rightleftharpoons

Sinapoylglucose—malate_O-sinapoyltransferase.html

  1. \rightleftharpoons

Sinapoylglucose—sinapoylglucose_O-sinapoyltransferase.html

  1. \rightleftharpoons

Singular_spectrum_analysis.html

  1. { X ( t ) : t = 1 , , N } \{X(t):t=1,\ldots,N\}
  2. M M
  3. M × M M\times M
  4. 𝐂 X {\textbf{C}}_{X}
  5. X ( t ) X(t)
  6. 𝐂 X {\textbf{C}}_{X}
  7. c i j c_{ij}
  8. | i - j | |i-j|
  9. c i j = 1 N - | i - j | t = 1 N - | i - j | X ( t ) X ( t + | i - j | ) . c_{ij}=\frac{1}{N-|i-j|}\sum_{t=1}^{N-|i-j|}X(t)X(t+|i-j|).
  10. 𝐂 X {\textbf{C}}_{X}
  11. N × M N^{\prime}\times M
  12. ${\textbf D}$
  13. M M
  14. X ( t ) {\it X(t)}
  15. N = N - M + 1 N^{\prime}=N-M+1
  16. 𝐂 X = 1 N 𝐃 t 𝐃 . {\textbf{C}}_{X}=\frac{1}{N^{\prime}}{\textbf{D}}^{\rm t}{\textbf{D}}.
  17. M M
  18. 𝐄 k {\textbf{E}}_{k}
  19. 𝐂 X {\textbf{C}}_{X}
  20. λ k \lambda_{k}
  21. 𝐂 X {\textbf{C}}_{X}
  22. 𝐄 k {\textbf{E}}_{k}
  23. 𝐂 X {\textbf{C}}_{X}
  24. X ( t ) X(t)
  25. λ k 1 / 2 \lambda^{1/2}_{k}
  26. 𝐂 X . {\textbf{C}}_{X}.
  27. 𝐀 k {\textbf{A}}_{k}
  28. A k ( t ) = j = 1 M X ( t + j - 1 ) E k ( j ) . A_{k}(t)=\sum_{j=1}^{M}X(t+j-1)E_{k}(j).
  29. M M
  30. λ k \lambda_{k}
  31. λ k 1 / 2 \lambda^{1/2}_{k}
  32. k k
  33. k * = S k^{*}=S
  34. D D
  35. 𝐑 K {\textbf{R}}_{K}
  36. R K ( t ) = 1 M t k 𝐾 j = L t U t A k ( t - j + 1 ) E k ( j ) ; R_{K}(t)=\frac{1}{M_{t}}\sum_{k\in{\textit{K}}}\sum_{j={L_{t}}}^{U_{t}}A_{k}(t% -j+1)E_{k}(j);
  37. K K
  38. M t M_{t}
  39. L t L_{t}
  40. U t U_{t}
  41. L L
  42. N N
  43. { X l ( t ) : l = 1 , , L ; t = 1 , , N } \{X_{l}(t):l=1,\dots,L;t=1,\dots,N\}
  44. L L
  45. M M
  46. L M L\leq M
  47. M M
  48. ${\textbf X}$
  49. 𝐂 X {\textbf{C}}_{X}
  50. 𝐄 k {\textbf{E}}_{k}
  51. M M
  52. 𝕏 = ( x 1 , , x N ) \mathbb{X}=(x_{1},\ldots,x_{N})
  53. N N
  54. L L
  55. ( 1 < L < N ) \ (1<L<N)
  56. K = N - L + 1 K=N-L+1
  57. 𝕏 \mathbb{X}
  58. L × K L\!\times\!K
  59. 𝐗 = [ X 1 : : X K ] = ( x i j ) i , j = 1 L , K = [ x 1 x 2 x 3 x K x 2 x 3 x 4 x K + 1 x 3 x 4 x 5 x K + 2 x L x L + 1 x L + 2 x N ] \mathbf{X}=[X_{1}:\ldots:X_{K}]=(x_{ij})_{i,j=1}^{L,K}=\begin{bmatrix}x_{1}&x_% {2}&x_{3}&\ldots&x_{K}\\ x_{2}&x_{3}&x_{4}&\ldots&x_{K+1}\\ x_{3}&x_{4}&x_{5}&\ldots&x_{K+2}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ x_{L}&x_{L+1}&x_{L+2}&\ldots&x_{N}\\ \end{bmatrix}
  60. X i = ( x i , , x i + L - 1 ) T ( 1 i K ) X_{i}=(x_{i},\ldots,x_{i+L-1})^{\mathrm{T}}\;\quad(1\leq i\leq K)
  61. L L
  62. 𝐗 \mathbf{X}
  63. 𝐗 \mathbf{X}
  64. x i j x_{ij}
  65. i + j = const i+j=\,{\rm const}
  66. 𝐗 \mathbf{X}
  67. 𝐒 = 𝐗𝐗 T \mathbf{S}=\mathbf{X}\mathbf{X}^{\mathrm{T}}
  68. λ 1 , , λ L \lambda_{1},\ldots,\lambda_{L}
  69. 𝐒 \mathbf{S}
  70. λ 1 λ L 0 \lambda_{1}\geq\ldots\geq\lambda_{L}\geq 0
  71. U 1 , , U L U_{1},\ldots,U_{L}
  72. 𝐒 \mathbf{S}
  73. d = rank 𝐗 = max { i , such that λ i > 0 } d=\mathop{\mathrm{rank}}\mathbf{X}=\max\{i,\ \mbox{such that}~{}\ \lambda_{i}>0\}
  74. d = L d=L
  75. V i = 𝐗 T U i / λ i V_{i}=\mathbf{X}^{\mathrm{T}}U_{i}/\sqrt{\lambda_{i}}
  76. ( i = 1 , , d ) (i=1,\ldots,d)
  77. 𝐗 \mathbf{X}
  78. 𝐗 = 𝐗 1 + + 𝐗 d , \mathbf{X}=\mathbf{X}_{1}+\ldots+\mathbf{X}_{d},
  79. 𝐗 i = λ i U i V i T \mathbf{X}_{i}=\sqrt{\lambda_{i}}U_{i}V_{i}^{\mathrm{T}}
  80. ( λ i , U i , V i ) (\sqrt{\lambda_{i}},U_{i},V_{i})
  81. i i
  82. U i U_{i}
  83. 𝐗 \mathbf{X}
  84. λ i \sqrt{\lambda_{i}}
  85. 𝐗 \mathbf{X}
  86. λ i V i = 𝐗 T U i \sqrt{\lambda_{i}}V_{i}=\mathbf{X}^{\mathrm{T}}U_{i}
  87. { 1 , , d } \{1,\ldots,d\}
  88. m m
  89. I 1 , , I m I_{1},\ldots,I_{m}
  90. I = { i 1 , , i p } I=\{i_{1},\ldots,i_{p}\}
  91. 𝐗 I \mathbf{X}_{I}
  92. I I
  93. 𝐗 I = 𝐗 i 1 + + 𝐗 i p \mathbf{X}_{I}=\mathbf{X}_{i_{1}}+\ldots+\mathbf{X}_{i_{p}}
  94. I = I 1 , , I m I=I_{1},\ldots,I_{m}
  95. 𝐗 \mathbf{X}
  96. 𝐗 = 𝐗 I 1 + + 𝐗 I m . \mathbf{X}=\mathbf{X}_{I_{1}}+\ldots+\mathbf{X}_{I_{m}}.
  97. 𝐗 I j \mathbf{X}_{I_{j}}
  98. N N
  99. 𝐗 I k \mathbf{X}_{I_{k}}
  100. 𝕏 ~ ( k ) = ( x ~ 1 ( k ) , , x ~ N ( k ) ) \widetilde{\mathbb{X}}^{(k)}=(\widetilde{x}^{(k)}_{1},\ldots,\widetilde{x}^{(k% )}_{N})
  101. x 1 , , x N x_{1},\ldots,x_{N}
  102. m m
  103. x n = k = 1 m x ~ n ( k ) ( n = 1 , 2 , , N ) . x_{n}=\sum\limits_{k=1}^{m}\widetilde{x}^{(k)}_{n}\ \ (n=1,2,\ldots,N).
  104. L L
  105. N N\rightarrow\infty
  106. N N
  107. L L
  108. L L
  109. L L
  110. π / 2 \pi/2
  111. 𝕏 \mathbb{X}
  112. d < L d<L
  113. d d
  114. d d
  115. x n = k = 1 d b k x n - k x_{n}=\sum_{k=1}^{d}b_{k}x_{n-k}
  116. L > d L>d
  117. U 1 , , U d U_{1},\ldots,U_{d}
  118. L L
  119. x n = k = 1 L - 1 a k x n - k x_{n}=\sum_{k=1}^{L-1}a_{k}x_{n-k}
  120. ( a L - 1 , , a 1 ) T (a_{L-1},\ldots,a_{1})^{\mathrm{T}}
  121. U 1 , , U d U_{1},\ldots,U_{d}
  122. L x × L y L_{x}\times L_{y}
  123. x n = s n + e n x_{n}=s_{n}+e_{n}
  124. s n = k = 1 r a k s n - k s_{n}=\sum_{k=1}^{r}a_{k}s_{n-k}
  125. e n e_{n}
  126. x n = k = 1 r a k x n - k + e n x_{n}=\sum_{k=1}^{r}a_{k}x_{n-k}+e_{n}
  127. k / N k/N
  128. s n = k = 1 r a k s n - k s_{n}=\sum_{k=1}^{r}a_{k}s_{n-k}
  129. s n = k C k ρ k n e i 2 π ω k n s_{n}=\sum_{k}C_{k}\rho_{k}^{n}e^{i2\pi\omega_{k}n}
  130. ω k \omega_{k}
  131. ρ k \rho_{k}
  132. C k C_{k}
  133. C k C_{k}
  134. n n
  135. r r
  136. span ( U 1 , , U r ) \mathop{\mathrm{span}}(U_{1},\ldots,U_{r})
  137. x n = s n + e n x_{n}=s_{n}+e_{n}
  138. s n = k = 1 r a k s n - k s_{n}=\sum_{k=1}^{r}a_{k}s_{n-k}
  139. e n e_{n}
  140. L L
  141. U i = ( u 1 , , u L ) T U_{i}=(u_{1},\ldots,u_{L})^{\mathrm{T}}
  142. 2 L - 1 2L-1
  143. x ~ s \widetilde{x}_{s}
  144. L s K L\leq s\leq K

Six_exponentials_theorem.html

  1. exp ( x i y j ) , ( 1 i d , 1 j l ) . \exp(x_{i}y_{j}),\quad(1\leq i\leq d,1\leq j\leq l).
  2. = { λ : e λ ¯ } . \mathcal{L}=\{\lambda\in\mathbb{C}\,:\,e^{\lambda}\in\overline{\mathbb{Q}}\}.
  3. M = ( λ 11 λ 12 λ 13 λ 21 λ 22 λ 23 ) M=\begin{pmatrix}\lambda_{11}&\lambda_{12}&\lambda_{13}\\ \lambda_{21}&\lambda_{22}&\lambda_{23}\end{pmatrix}
  4. e x 1 y 1 , e x 1 y 2 , e x 2 y 1 , e x 2 y 2 , e γ x 2 / x 1 . e^{x_{1}y_{1}},e^{x_{1}y_{2}},e^{x_{2}y_{1}},e^{x_{2}y_{2}},e^{\gamma x_{2}/x_% {1}}.
  5. e x 1 y 1 - β 11 , e x 1 y 2 - β 12 , e x 2 y 1 - β 21 , e x 2 y 2 - β 22 , e x 3 y 1 - β 31 , e x 3 y 2 - β 32 . e^{x_{1}y_{1}-\beta_{11}},e^{x_{1}y_{2}-\beta_{12}},e^{x_{2}y_{1}-\beta_{21}},% e^{x_{2}y_{2}-\beta_{22}},e^{x_{3}y_{1}-\beta_{31}},e^{x_{3}y_{2}-\beta_{32}}.
  6. e x 1 y 1 - β 11 , e x 1 y 2 - β 12 , e x 2 y 1 - β 21 , e x 2 y 2 - β 22 , e ( γ x 2 / x 1 ) - α . e^{x_{1}y_{1}-\beta_{11}},e^{x_{1}y_{2}-\beta_{12}},e^{x_{2}y_{1}-\beta_{21}},% e^{x_{2}y_{2}-\beta_{22}},e^{(\gamma x_{2}/x_{1})-\alpha}.
  7. β 0 + i = 1 n β i log α i , \beta_{0}+\sum_{i=1}^{n}\beta_{i}\log\alpha_{i},
  8. x 1 y 1 , x 1 y 2 , x 2 y 1 , x 2 y 2 , x 1 / x 2 . x_{1}y_{1},\,x_{1}y_{2},\,x_{2}y_{1},\,x_{2}y_{2},\,x_{1}/x_{2}.
  9. e z e^{z}
  10. G m G_{m}
  11. \C \C
  12. G = G m 2 G=G_{m}^{2}
  13. u : \C u:\C
  14. G ( \C ) G(\C)
  15. L L
  16. l l
  17. \C \C
  18. u ( l ) u(l)
  19. G G
  20. L L
  21. \Q \Q
  22. x 1 , x 2 , x 3 , x_{1},x_{2},x_{3},...
  23. u ( \C ) u(\C)
  24. G ( \C ) G(\C)
  25. G G
  26. G = G m 2 G=G_{m}^{2}
  27. G = G m 3 G=G_{m}^{3}
  28. G = G m G=G_{m}
  29. E E
  30. G = E G=E
  31. E E^{\prime}
  32. E , E E,E^{\prime}
  33. e z e^{z}
  34. \wp
  35. , \wp,\wp^{\prime}
  36. g 2 , g 3 , g 2 , g 3 g_{2},g_{3},g_{2}^{\prime},g_{3}^{\prime}
  37. e y 1 z , e y 2 z e^{y_{1}z},e^{y_{2}z}
  38. x 1 , x 2 , x 3 x_{1},x_{2},x_{3}
  39. L L
  40. L = \Q x 1 + \Q x 2 + \Q x 3 L=\Q x_{1}+\Q x_{2}+\Q x_{3}
  41. G = G m G=G_{m}
  42. E E
  43. E E
  44. u ( \C ) u(\C)
  45. G ( \C ) G(\C)
  46. L L
  47. x 1 , x 2 x_{1},x_{2}
  48. \Q \Q
  49. L L
  50. \Q \Q
  51. x 1 , x 2 , x 3 x_{1},x_{2},x_{3}
  52. \R c \R c
  53. c c
  54. G = E G=E
  55. E E^{\prime}

Sl2-triple.html

  1. [ h , e ] = 2 e , [ h , f ] = - 2 f , [ e , f ] = h . [h,e]=2e,\quad[h,f]=-2f,\quad[e,f]=h.
  2. h = [ 1 0 0 - 1 ] , e = [ 0 1 0 0 ] , f = [ 0 0 1 0 ] h=\begin{bmatrix}1&0\\ 0&-1\end{bmatrix},\quad e=\begin{bmatrix}0&1\\ 0&0\end{bmatrix},\quad f=\begin{bmatrix}0&0\\ 1&0\end{bmatrix}
  3. g = j g j , [ h , a ] = j a for a g j . g=\bigoplus_{j\in\mathbb{Z}}g_{j},\quad[h,a]=ja{\ \ }\textrm{ for }{\ \ }a\in g% _{j}.

Smallest-circle_problem.html

  1. O ( N ) O(N)
  2. O ( 1 / i ) O(1/i)
  3. O ( N ) O(N)

Sn-glycerol-3-phosphate_1-galactosyltransferase.html

  1. \rightleftharpoons

Sn-glycerol-3-phosphate_2-alpha-galactosyltransferase.html

  1. \rightleftharpoons

Social_value_orientations.html

  1. 0 0^{\circ}
  2. 45 45^{\circ}
  3. - 45 -45^{\circ}
  4. U ( π s , π o ) = a * π s + b * π o U_{(\pi_{s},\pi_{o})}=a*\pi_{s}+b*\pi_{o}
  5. π s \pi_{s}
  6. π o \pi_{o}
  7. a a
  8. b b
  9. a a
  10. U ( π s , π o ) = a * π s + b * π o - c * | π s - π o | U_{(\pi_{s},\pi_{o})}=a*\pi_{s}+b*\pi_{o}-c*|\pi_{s}-\pi_{o}|

Sokolov–Ternov_effect.html

  1. ξ ( t ) = A ( 1 - e - t τ ) \xi(t)=A\Bigl(1-e^{-\frac{t}{\tau}}\Bigr)
  2. A = 8 3 / 15 0.924 \scriptstyle A\;=\;8\sqrt{3}/15\;\approx\;0.924
  3. τ \scriptstyle\tau
  4. τ = A 2 m c e 2 ( m c 2 E ) 2 ( H 0 H ) 3 \tau=A{{{\hbar}^{2}}\over{mc{e}^{2}}}\Bigl({{mc^{2}}\over{E}}\Bigr)^{2}\Bigl({% H_{0}\over H}\Bigr)^{3}
  5. A \scriptstyle A
  6. m \scriptstyle m
  7. e \scriptstyle e
  8. c \scriptstyle c
  9. H 0 4.414 × 10 13 gauss \scriptstyle H_{0}\;\approx\;4.414\,\times\,10^{13}\,\,\text{gauss}
  10. H \scriptstyle H
  11. E \scriptstyle E
  12. A A

Solar_car_racing.html

  1. η { η b E + P x v } = { W C r r 1 + N C r r 2 v + 1 2 ρ C d A v 2 } x + W h + N a W v 2 2 g \eta\left\{\eta_{b}E+\frac{Px}{v}\right\}=\left\{WC_{rr1}+NC_{rr2}v+\frac{1}{2% }\rho C_{d}Av^{2}\right\}x+Wh+\frac{N_{a}Wv^{2}}{2g}
  2. η { η b E v / x + P } = { W C r r 1 v + 1 2 ρ C d A v 3 } \eta\left\{\eta_{b}Ev/x+P\right\}=\left\{WC_{rr1}v+\frac{1}{2}\rho C_{d}Av^{3}\right\}

Sorbitol-6-phosphatase.html

  1. \rightleftharpoons

Souček_space.html

  1. lim k u k = u in L 1 ( Ω ; 𝐑 m ) \lim_{k\to\infty}u_{k}=u\mbox{ in }~{}L^{1}(\Omega;\mathbf{R}^{m})
  2. lim k u k = v \lim_{k\to\infty}\nabla u_{k}=v
  3. ( u , v ) := u L 1 + v M , \|(u,v)\|:=\|u\|_{L^{1}}+\|v\|_{M},

Spectral_flux_density.html

  1. F ( x , t ; ν ) = Ω I ( x , t ; n ^ , ν ) n ^ d ω ( n ^ ) {F}({x},t;\nu)=\oint_{\Omega}\ I({x},t;{\hat{n}},\nu)\,{\hat{n}}\,d\omega({% \hat{n}})
  2. I ( x , t ; n ^ , ν ) I({x},t;{\hat{n}},\nu)
  3. x {x}
  4. t t\!
  5. ν \nu\!
  6. n ^ {\hat{n}}
  7. x {x}
  8. d ω ( n ^ ) d\omega({\hat{n}})
  9. n ^ {\hat{n}}
  10. Ω \Omega\!
  11. F ( x , t ; ν ) = Ω + I ( x , t ; n ^ , ν ) cos ( θ ( n ^ ) ) d ω ( n ^ ) F({x},t;\nu)=\int_{\Omega^{{}^{+}}}I({x},t;{\hat{n}},\nu)\,\cos\,(\theta({\hat% {n}}))\,d\omega({\hat{n}})
  12. Ω + \Omega^{{}^{+}}
  13. θ ( n ^ ) \theta({\hat{n}})
  14. n ^ {\hat{n}}
  15. cos ( θ ( n ^ ) ) \cos\,(\theta({\hat{n}}))
  16. F ( x , t ; ν ) F({x},t;\nu)
  17. I ( 𝐱 , t ; 𝐫 < s u b > 1 , < v a r > ν < / v a r > ) I(\mathbf{x},t;\mathbf{r}<sub>1,<var>ν</var>)

Sphinganine-1-phosphate_aldolase.html

  1. \rightleftharpoons

Sphingomyelin_synthase.html

  1. \rightleftharpoons

Sphingosine_beta-galactosyltransferase.html

  1. \rightleftharpoons

Sphingosine_cholinephosphotransferase.html

  1. \rightleftharpoons

Sphingosine_N-acyltransferase.html

  1. \rightleftharpoons

Starch_synthase.html

  1. \rightleftharpoons

Stars_and_bars_(combinatorics).html

  1. ( n - 1 k - 1 ) \textstyle{n-1\choose k-1}
  2. ( 3 - 1 2 - 1 ) = 2 {\textstyle\left({{3-1}\atop{2-1}}\right)}=2
  3. ( ( n + 1 k - 1 ) ) \left(\!{\textstyle\left({{n+1}\atop{k-1}}\right)}\!\right)
  4. ( n + k - 1 k - 1 ) = ( n + k - 1 n ) {\textstyle\left({{n+k-1}\atop{k-1}}\right)}={\textstyle\left({{n+k-1}\atop{n}% }\right)}
  5. ( ( k n ) ) \left(\!{\textstyle\left({{k}\atop{n}}\right)}\!\right)
  6. ( ( 3 + 1 2 - 1 ) ) = 4 \left(\!{\textstyle\left({{3+1}\atop{2-1}}\right)}\!\right)=4
  7. ( n - 1 k - 1 ) {\textstyle\left({{n-1}\atop{k-1}}\right)}
  8. ( ( n + 1 k - 1 ) ) \left(\!{\textstyle\left({{n+1}\atop{k-1}}\right)}\!\right)
  9. ( n + k - 1 k - 1 ) {\textstyle\left({{n+k-1}\atop{k-1}}\right)}
  10. ( 7 - 1 3 - 1 ) = 15 \textstyle{7-1\choose 3-1}=15

Steinberg_group_(K-theory).html

  1. St ( A ) \operatorname{St}(A)
  2. A A
  3. A A
  4. K K
  5. K 2 K_{2}
  6. K 3 K_{3}
  7. A A
  8. St ( A ) \operatorname{St}(A)
  9. e p q ( λ ) := 𝟏 + a p q ( λ ) {e_{pq}}(\lambda):=\mathbf{1}+{a_{pq}}(\lambda)
  10. 𝟏 \mathbf{1}
  11. a p q ( λ ) {a_{pq}}(\lambda)
  12. λ \lambda
  13. ( p , q ) (p,q)
  14. p q p\neq q
  15. e i j ( λ ) e i j ( μ ) \displaystyle e_{ij}(\lambda)e_{ij}(\mu)
  16. r r
  17. A A
  18. St r ( A ) {\operatorname{St}_{r}}(A)
  19. x i j ( λ ) {x_{ij}}(\lambda)
  20. 1 i j r 1\leq i\neq j\leq r
  21. λ A \lambda\in A
  22. St ( A ) \operatorname{St}(A)
  23. St r ( A ) St r + 1 ( A ) {\operatorname{St}_{r}}(A)\to{\operatorname{St}_{r+1}}(A)
  24. x i j ( λ ) e i j ( λ ) {x_{ij}}(\lambda)\mapsto{e_{ij}}(\lambda)
  25. φ : St ( A ) GL ( A ) \varphi:\operatorname{St}(A)\to{\operatorname{GL}_{\infty}}(A)
  26. K K
  27. K 1 K_{1}
  28. K 1 ( A ) {K_{1}}(A)
  29. φ : St ( A ) GL ( A ) \varphi:\operatorname{St}(A)\to{\operatorname{GL}_{\infty}}(A)
  30. K 1 K_{1}
  31. GL ( A ) {\operatorname{GL}_{\infty}}(A)
  32. φ \varphi
  33. K 2 K_{2}
  34. K 2 ( A ) {K_{2}}(A)
  35. K K
  36. φ : St ( A ) GL ( A ) \varphi:\operatorname{St}(A)\to{\operatorname{GL}_{\infty}}(A)
  37. 1 K 2 ( A ) St ( A ) GL ( A ) K 1 ( A ) 1. 1\to{K_{2}}(A)\to\operatorname{St}(A)\to{\operatorname{GL}_{\infty}}(A)\to{K_{% 1}}(A)\to 1.
  38. K 2 ( A ) = H 2 ( E ( A ) ; ) {K_{2}}(A)={H_{2}}(E(A);\mathbb{Z})
  39. K 3 K_{3}
  40. K 3 ( A ) = H 3 ( St ( A ) ; ) {K_{3}}(A)={H_{3}}(\operatorname{St}(A);\mathbb{Z})

Stellar-wind_bubble.html

  1. n > 0.1 cm - 3 n>0.1\mbox{ cm}~{}^{-3}

Steroid-lactonase.html

  1. \rightleftharpoons

Steroid_N-acetylglucosaminyltransferase.html

  1. \rightleftharpoons

Sterol_3beta-glucosyltransferase.html

  1. \rightleftharpoons

Sterol_esterase.html

  1. \rightleftharpoons

Sterol_O-acyltransferase.html

  1. \rightleftharpoons

Steryl-beta-glucosidase.html

  1. \rightleftharpoons

Stipitatonate_decarboxylase.html

  1. \rightleftharpoons

Stochastic_ordering.html

  1. A A
  2. B B
  3. A A
  4. B B
  5. Pr ( A > x ) Pr ( B > x ) for all x ( - , ) , \Pr(A>x)\leq\Pr(B>x)\,\text{ for all }x\in(-\infty,\infty),
  6. Pr ( ) \Pr(\cdot)
  7. A B A\preceq B
  8. A s t B A\leq_{st}B
  9. Pr ( A > x ) < Pr ( B > x ) \Pr(A>x)<\Pr(B>x)
  10. x x
  11. A A
  12. B B
  13. A B A\prec B
  14. A B A\preceq B
  15. u u
  16. E [ u ( A ) ] E [ u ( B ) ] {\rm E}[u(A)]\leq{\rm E}[u(B)]
  17. u u
  18. A B A\preceq B
  19. u ( A ) u ( B ) u(A)\preceq u(B)
  20. u : n u:\mathbb{R}^{n}\to\mathbb{R}
  21. A i A_{i}
  22. B i B_{i}
  23. A i B i A_{i}\preceq B_{i}
  24. i i
  25. u ( A 1 , , A n ) u ( B 1 , , B n ) u(A_{1},\dots,A_{n})\preceq u(B_{1},\dots,B_{n})
  26. i = 1 n A i i = 1 n B i \sum_{i=1}^{n}A_{i}\preceq\sum_{i=1}^{n}B_{i}
  27. i i
  28. A ( i ) B ( i ) A_{(i)}\preceq B_{(i)}
  29. A i A_{i}
  30. B i B_{i}
  31. A i B i A_{i}\preceq B_{i}
  32. i i
  33. A B A\preceq B
  34. A A
  35. B B
  36. C C
  37. c Pr ( C = c ) = 1 \sum_{c}\Pr(C=c)=1
  38. Pr ( A > u | C = c ) Pr ( B > u | C = c ) \Pr(A>u|C=c)\leq\Pr(B>u|C=c)
  39. u u
  40. c c
  41. Pr ( C = c ) > 0 \Pr(C=c)>0
  42. A B A\preceq B
  43. A B A\preceq B
  44. E [ A ] = E [ B ] {\rm E}[A]={\rm E}[B]
  45. A = 𝑑 B A\overset{d}{=}B
  46. A ( 0 ) B A\preceq_{(0)}B
  47. A B A\leq B
  48. d \mathbb{R}^{d}
  49. A A
  50. d \mathbb{R}^{d}
  51. B B
  52. E [ f ( A ) ] E [ f ( B ) ] for all bounded, increasing functions f : d {\rm E}[f(A)]\leq{\rm E}[f(B)]\,\text{ for all bounded, increasing functions }% f:\mathbb{R}^{d}\longrightarrow\mathbb{R}
  53. A A
  54. B B
  55. Pr ( A > 𝐱 ) Pr ( B > 𝐱 ) for all 𝐱 d \Pr(A>\mathbf{x})\leq\Pr(B>\mathbf{x})\,\text{ for all }\mathbf{x}\in\mathbb{R% }^{d}
  56. A A
  57. B B
  58. Pr ( A 𝐱 ) Pr ( B 𝐱 ) for all 𝐱 d \Pr(A\leq\mathbf{x})\geq\Pr(B\leq\mathbf{x})\,\text{ for all }\mathbf{x}\in% \mathbb{R}^{d}
  59. A A
  60. B B
  61. E [ f ( A ) ] E [ f ( B ) ] {\rm E}[f(A)]\leq{\rm E}[f(B)]
  62. f : d f:\mathbb{R}^{d}\longrightarrow\mathbb{R}
  63. 𝒢 \mathcal{G}
  64. 𝒢 \mathcal{G}
  65. X X
  66. F F
  67. f f
  68. r ( t ) = d d t ( - log ( 1 - F ( t ) ) ) = f ( t ) 1 - F ( t ) . r(t)=\frac{d}{dt}(-\log(1-F(t)))=\frac{f(t)}{1-F(t)}.
  69. X X
  70. Y Y
  71. F F
  72. G G
  73. r r
  74. q q
  75. X X
  76. Y Y
  77. X h r Y X\leq_{hr}Y
  78. r ( t ) q ( t ) r(t)\geq q(t)
  79. t 0 t\geq 0
  80. 1 - F ( t ) 1 - G ( t ) \frac{1-F(t)}{1-G(t)}
  81. t t
  82. X X
  83. Y Y
  84. f ( t ) f\left(t\right)
  85. g ( t ) g\left(t\right)
  86. g ( t ) f ( t ) \frac{g\left(t\right)}{f\left(t\right)}
  87. t t
  88. X X
  89. Y Y
  90. X X
  91. Y Y
  92. X l r Y X\leq_{lr}Y
  93. A A
  94. B B
  95. u u
  96. E [ u ( A ) ] E [ u ( B ) ] {\rm E}[u(A)]\leq{\rm E}[u(B)]
  97. u ( x ) = - exp ( - α x ) u(x)=-\exp(-\alpha x)
  98. α \alpha
  99. ( P α ) α F ({P}_{\alpha})_{\alpha\in F}
  100. ( E , ) (E,\preceq)
  101. α F \alpha\in F
  102. ( F , ) (F,\preceq)
  103. ( X α ) α (X_{\alpha})_{\alpha}
  104. X α X_{\alpha}
  105. P α {P}_{\alpha}
  106. X α X β X_{\alpha}\preceq X_{\beta}
  107. α β \alpha\preceq\beta

Stokes_drift.html

  1. s y m b o l ξ ˙ = s y m b o l ξ t = s y m b o l u ( s y m b o l ξ , t ) , \dot{symbol{\xi}}\,=\,\frac{\partial symbol{\xi}}{\partial t}\,=\,symbol{u}(% symbol{\xi},t),
  2. s y m b o l ξ ( s y m b o l α , t 0 ) = s y m b o l α . symbol{\xi}(symbol{\alpha},t_{0})\,=\,symbol{\alpha}.
  3. s y m b o l u ¯ E \displaystyle\overline{symbol{u}}_{E}
  4. s y m b o l u ¯ S = s y m b o l u ¯ L - s y m b o l u ¯ E . \overline{symbol{u}}_{S}\,=\,\overline{symbol{u}}_{L}\,-\,\overline{symbol{u}}% _{E}.
  5. u = u ^ sin ( k x - ω t ) , u=\hat{u}\sin\left(kx-\omega t\right),
  6. k u ^ / ω k\hat{u}/\omega
  7. x = ξ ( ξ 0 , t ) : x=\xi(\xi_{0},t):
  8. < ˙ m t p l > ξ = u ( ξ , t ) = u ^ sin ( k ξ - ω t ) , \dot{<}mtpl>{{\xi}}=\,{u}({\xi},t)=\hat{u}\sin\,\left(k\xi-\omega t\right),
  9. ξ ( ξ 0 , t ) ξ 0 + u ^ ω cos ( k ξ 0 - ω t ) + k u ^ 2 2 ω 2 sin 2 ( k ξ 0 - ω t ) + k u ^ 2 2 ω t . \xi(\xi_{0},t)\approx\xi_{0}+\frac{\hat{u}}{\omega}\cos(k\xi_{0}-\omega t)+% \frac{k\hat{u}^{2}}{2\omega^{2}}\sin 2(k\xi_{0}-\omega t)+\frac{k\hat{u}^{2}}{% 2\omega}t.
  10. 1 2 k u ^ 2 / ω . \tfrac{1}{2}k\hat{u}^{2}/\omega.
  11. η = a cos ( k x - ω t ) , \eta\,=\,a\,\cos\,\left(kx-\omega t\right),
  12. φ = ω k a e k z sin ( k x - ω t ) . \varphi\,=\,\frac{\omega}{k}\,a\;\,\text{e}^{kz}\,\sin\,\left(kx-\omega t% \right).
  13. ω 2 = g k . \omega^{2}\,=\,g\,k.
  14. ξ x = x + φ x d t = x - a e k z sin ( k x - ω t ) , ξ z = z + φ z d t = z + a e k z cos ( k x - ω t ) . \begin{aligned}\displaystyle\xi_{x}&\displaystyle=\,x\,+\,\int\,\frac{\partial% \varphi}{\partial x}\;\,\text{d}t\,=\,x\,-\,a\,\,\text{e}^{kz}\,\sin\,\left(kx% -\omega t\right),\\ \displaystyle\xi_{z}&\displaystyle=\,z\,+\,\int\,\frac{\partial\varphi}{% \partial z}\;\,\text{d}t\,=\,z\,+\,a\,\,\text{e}^{kz}\,\cos\,\left(kx-\omega t% \right).\end{aligned}
  15. u ¯ S = u x ( s y m b o l ξ , t ) ¯ - u x ( s y m b o l x , t ) ¯ = [ u x ( s y m b o l x , t ) + ( ξ x - x ) u x ( s y m b o l x , t ) x + ( ξ z - z ) u x ( s y m b o l x , t ) z + ] ¯ - u x ( s y m b o l x , t ) ¯ ( ξ x - x ) 2 ξ x x t ¯ + ( ξ z - z ) 2 ξ x z t ¯ = [ - a e k z sin ( k x - ω t ) ] [ - ω k a e k z sin ( k x - ω t ) ] ¯ + [ a e k z cos ( k x - ω t ) ] [ ω k a e k z cos ( k x - ω t ) ] ¯ = ω k a 2 e 2 k z [ sin 2 ( k x - ω t ) + cos 2 ( k x - ω t ) ] ¯ = ω k a 2 e 2 k z . \begin{aligned}\displaystyle\overline{u}_{S}&\displaystyle=\,\overline{u_{x}(% symbol{\xi},t)}\,-\,\overline{u_{x}(symbol{x},t)}\\ &\displaystyle=\,\overline{\left[u_{x}(symbol{x},t)\,+\,\left(\xi_{x}-x\right)% \,\frac{\partial u_{x}(symbol{x},t)}{\partial x}\,+\,\left(\xi_{z}-z\right)\,% \frac{\partial u_{x}(symbol{x},t)}{\partial z}\,+\,\cdots\right]}-\,\overline{% u_{x}(symbol{x},t)}\\ &\displaystyle\approx\,\overline{\left(\xi_{x}-x\right)\,\frac{\partial^{2}\xi% _{x}}{\partial x\,\partial t}}\,+\,\overline{\left(\xi_{z}-z\right)\,\frac{% \partial^{2}\xi_{x}}{\partial z\,\partial t}}\\ &\displaystyle=\,\overline{\bigg[-a\,\,\text{e}^{kz}\,\sin\,\left(kx-\omega t% \right)\bigg]\,\bigg[-\omega\,k\,a\,\,\text{e}^{kz}\,\sin\,\left(kx-\omega t% \right)\bigg]}\\ &\displaystyle+\,\overline{\bigg[a\,\,\text{e}^{kz}\,\cos\,\left(kx-\omega t% \right)\bigg]\,\bigg[\omega\,k\,a\,\,\text{e}^{kz}\,\cos\,\left(kx-\omega t% \right)\bigg]}\\ &\displaystyle=\,\overline{\omega\,k\,a^{2}\,\,\text{e}^{2kz}\,\bigg[\sin^{2}% \,\left(kx-\omega t\right)+\cos^{2}\,\left(kx-\omega t\right)\bigg]}\\ &\displaystyle=\,\omega\,k\,a^{2}\,\,\text{e}^{2kz}.\end{aligned}

Stokes_wave.html

  1. η ( x , t ) = a { cos θ + 1 2 ( k a ) cos 2 θ + 3 8 ( k a ) 2 cos 3 θ } + 𝒪 ( ( k a ) 4 ) , Φ ( x , z , t ) = a ω k e k z sin θ + 𝒪 ( ( k a ) 4 ) , c = ω k = ( 1 + 1 2 ( k a ) 2 ) g k + 𝒪 ( ( k a ) 4 ) , and θ ( x , t ) = k x - ω t , \begin{aligned}\displaystyle\eta(x,t)=&\displaystyle a\left\{\cos\theta+\tfrac% {1}{2}(ka)\,\cos 2\theta+\tfrac{3}{8}(ka)^{2}\,\cos 3\theta\right\}\\ &\displaystyle+\mathcal{O}\left((ka)^{4}\right),\\ \displaystyle\Phi(x,z,t)=&\displaystyle a\frac{\omega}{k}\,\,\text{e}^{kz}\,% \sin\theta+\mathcal{O}\left((ka)^{4}\right),\\ \displaystyle c=&\displaystyle\frac{\omega}{k}=\left(1+\tfrac{1}{2}(ka)^{2}% \right)\,\sqrt{\frac{g}{k}}+\mathcal{O}\left((ka)^{4}\right),\,\text{ and}\\ \displaystyle\theta(x,t)=&\displaystyle kx-\omega t,\end{aligned}
  2. H = 2 a ( 1 + 3 8 k 2 a 2 ) . H=2a\,\left(1+\tfrac{3}{8}\,k^{2}a^{2}\right).
  3. η ( x , t ) = a { cos θ + k a 3 - σ 2 4 σ 3 cos 2 θ } + 𝒪 ( ( k a ) 3 ) , Φ ( x , z , t ) = a ω k cosh k ( z + h ) sinh k h × { sin θ + k a 3 cosh 2 k ( z + h ) 8 sinh 3 k h sin 2 θ } - ( k a ) 2 1 2 sinh 2 k h g t k + 𝒪 ( ( k a ) 3 ) , c = ω k = g k σ + 𝒪 ( ( k a ) 2 ) , σ = tanh k h and θ ( x , t ) = k x - ω t . \begin{aligned}\displaystyle\eta(x,t)=&\displaystyle a\,\left\{\cos\,\theta+ka% \,\frac{3-\sigma^{2}}{4\,\sigma^{3}}\,\cos\,2\theta\right\}\\ &\displaystyle+\mathcal{O}\left((ka)^{3}\right),\\ \displaystyle\Phi(x,z,t)=&\displaystyle a\,\frac{\omega}{k}\,\frac{\cosh\,k(z+% h)}{\sinh\,kh}\\ &\displaystyle\times\left\{\sin\,\theta+ka\,\frac{3\cosh\,2k(z+h)}{8\,\sinh^{3% }\,kh}\,\sin\,2\theta\right\}\\ &\displaystyle-(ka)^{2}\,\frac{1}{2\,\sinh\,2kh}\,\frac{g\,t}{k}+\mathcal{O}% \left((ka)^{3}\right),\\ \displaystyle c=&\displaystyle\frac{\omega}{k}=\sqrt{\frac{g}{k}\,\sigma}+% \mathcal{O}\left((ka)^{2}\right),\\ \displaystyle\sigma=&\displaystyle\tanh\,kh\quad\,\text{and}\quad\theta(x,t)=% kx-\omega t.\end{aligned}
  4. 𝒮 \mathcal{S}
  5. 𝒮 = k a 3 - tanh 2 k h 4 tanh 3 k h . \mathcal{S}=ka\,\frac{3-\tanh^{2}\,kh}{4\,\tanh^{3}\,kh}.
  6. 𝒮 \mathcal{S}
  7. lim k h 𝒮 = 1 2 k a . \lim_{kh\to\infty}\mathcal{S}=\frac{1}{2}\,ka.
  8. 𝒮 \mathcal{S}
  9. lim k h 0 𝒮 = 3 4 k a ( k h ) 3 , \lim_{kh\downarrow 0}\mathcal{S}=\frac{3}{4}\,\frac{ka}{(kh)^{3}},
  10. lim k h 0 𝒮 = 3 32 π 2 H λ 2 h 3 = 3 32 π 2 𝒰 , \lim_{kh\downarrow 0}\mathcal{S}=\frac{3}{32\,\pi^{2}}\,\frac{H\,\lambda^{2}}{% h^{3}}=\frac{3}{32\,\pi^{2}}\,\mathcal{U},
  11. 𝒰 H λ 2 h 3 . \mathcal{U}\equiv\frac{H\,\lambda^{2}}{h^{3}}.
  12. 𝒰 \mathcal{U}
  13. ω 2 = ( g k tanh k h ) { 1 + 9 - 10 σ 2 + 9 σ 4 8 σ 4 ( k a ) 2 } + 𝒪 ( ( k a ) 4 ) , with σ = tanh k h . \begin{aligned}\displaystyle\omega^{2}=&\displaystyle\left(gk\,\tanh\,kh\right% )\;\left\{1+\frac{9-10\,\sigma^{2}+9\,\sigma^{4}}{8\,\sigma^{4}}\,(ka)^{2}% \right\}\\ &\displaystyle+\mathcal{O}\left((ka)^{4}\right),\qquad\,\text{with}\\ \displaystyle\sigma=&\displaystyle\tanh\,kh.\end{aligned}
  14. lim k h ω 2 = g k { 1 + ( k a ) 2 } + 𝒪 ( ( k a ) 4 ) , \lim_{kh\to\infty}\omega^{2}=gk\,\left\{1+\left(ka\right)^{2}\right\}+\mathcal% {O}\left((ka)^{4}\right),
  15. lim k h 0 ω 2 = k 2 g h { 1 + 9 8 ( k a ) 2 ( k h ) 4 } + 𝒪 ( ( k a ) 4 ) . \lim_{kh\downarrow 0}\omega^{2}=k^{2}\,gh\,\left\{1+\frac{9}{8}\,\frac{\left(% ka\right)^{2}}{\left(kh\right)^{4}}\right\}+\mathcal{O}\left((ka)^{4}\right).
  16. 1 7 \frac{1}{7}
  17. η λ = A [ cosh ( x - c t λ ) - 1 ] , \frac{\eta}{\lambda}=A\,\left[\cosh\,\left(\frac{x-ct}{\lambda}\right)-1\right],
  18. A = 1 3 sinh ( 1 2 ) 1.108 , A=\frac{1}{\sqrt{3}\,\sinh\left(\frac{1}{2}\right)}\approx 1.108,
  19. - 1 2 λ ( x - c t ) 1 2 λ , -\tfrac{1}{2}\,\lambda\leq(x-ct)\leq\tfrac{1}{2}\,\lambda,
  20. \color G r a y ( t + 𝐮 s y m b o l ) ( Φ t + 1 2 | 𝐮 | 2 + g η ) = 0 {\color{Gray}{\Bigl(\frac{\partial}{\partial t}+\mathbf{u}\cdot symbol{\nabla}% \Bigr)\,\left(\frac{\partial\Phi}{\partial t}+\tfrac{1}{2}\,|\mathbf{u}|^{2}+g% \,\eta\right)=0}}
  21. \color G r a y 2 Φ t 2 + g Φ z + 𝐮 s y m b o l Φ t + 1 2 t ( | 𝐮 | 2 ) + 1 2 𝐮 s y m b o l ( | 𝐮 | 2 ) = 0 {\color{Gray}{\Rightarrow\quad\frac{\partial^{2}\Phi}{\partial t^{2}}+g\,\frac% {\partial\Phi}{\partial z}+\mathbf{u}\cdot symbol{\nabla}\frac{\partial\Phi}{% \partial t}+\tfrac{1}{2}\,\frac{\partial}{\partial t}\left(|\mathbf{u}|^{2}% \right)+\tfrac{1}{2}\,\mathbf{u}\cdot symbol{\nabla}\left(|\mathbf{u}|^{2}% \right)=0}}
  22. Φ z = 0 at z = - h . \frac{\partial\Phi}{\partial z}=0\qquad\,\text{ at }z=-h.
  23. f ( x , y , η , t ) = [ f ] 0 + η [ f z ] 0 + 1 2 η 2 [ 2 f z 2 ] 0 + f(x,y,\eta,t)=\left[f\right]_{0}+\eta\,\left[\frac{\partial f}{\partial z}% \right]_{0}+\frac{1}{2}\,\eta^{2}\,\left[\frac{\partial^{2}f}{\partial z^{2}}% \right]_{0}+\cdots
  24. 𝒪 \mathcal{O}
  25. 𝒪 \mathcal{O}
  26. η \displaystyle\eta
  27. ε { 2 Φ 1 t 2 + g Φ 1 z } + ε 2 { 2 Φ 2 t 2 + g Φ 2 z + η 1 z ( 2 Φ 1 t 2 + g Φ 1 z ) + t ( | 𝐮 1 | 2 ) } + ε 3 { 2 Φ 3 t 2 + g Φ 3 z + η 1 z ( 2 Φ 2 t 2 + g Φ 2 z ) + η 2 z ( 2 Φ 1 t 2 + g Φ 1 z ) + 2 t ( 𝐮 1 𝐮 2 ) + 1 2 η 1 2 2 z 2 ( 2 Φ 1 t 2 + g Φ 1 z ) + η 1 2 t z ( | 𝐮 1 | 2 ) + 1 2 𝐮 1 s y m b o l ( | 𝐮 1 | 2 ) } + 𝒪 ( ε 4 ) = 0 , at z = 0. \begin{aligned}&\displaystyle\varepsilon\,\left\{\frac{\partial^{2}\Phi_{1}}{% \partial t^{2}}+g\,\frac{\partial\Phi_{1}}{\partial z}\right\}\\ &\displaystyle+\varepsilon^{2}\,\left\{\frac{\partial^{2}\Phi_{2}}{\partial t^% {2}}+g\,\frac{\partial\Phi_{2}}{\partial z}+\eta_{1}\,\frac{\partial}{\partial z% }\left(\frac{\partial^{2}\Phi_{1}}{\partial t^{2}}+g\,\frac{\partial\Phi_{1}}{% \partial z}\right)+\frac{\partial}{\partial t}\left(|\mathbf{u}_{1}|^{2}\right% )\right\}\\ &\displaystyle+\varepsilon^{3}\,\left\{\frac{\partial^{2}\Phi_{3}}{\partial t^% {2}}+g\,\frac{\partial\Phi_{3}}{\partial z}+\eta_{1}\,\frac{\partial}{\partial z% }\left(\frac{\partial^{2}\Phi_{2}}{\partial t^{2}}+g\,\frac{\partial\Phi_{2}}{% \partial z}\right)\right.\\ &\displaystyle\qquad\quad\left.+\eta_{2}\,\frac{\partial}{\partial z}\left(% \frac{\partial^{2}\Phi_{1}}{\partial t^{2}}+g\,\frac{\partial\Phi_{1}}{% \partial z}\right)+2\,\frac{\partial}{\partial t}\left(\mathbf{u}_{1}\cdot% \mathbf{u}_{2}\right)\right.\\ &\displaystyle\qquad\quad\left.+\tfrac{1}{2}\,\eta_{1}^{2}\,\frac{\partial^{2}% }{\partial z^{2}}\left(\frac{\partial^{2}\Phi_{1}}{\partial t^{2}}+g\,\frac{% \partial\Phi_{1}}{\partial z}\right)+\eta_{1}\,\frac{\partial^{2}}{\partial t% \,\partial z}\left(|\mathbf{u}_{1}|^{2}\right)+\tfrac{1}{2}\,\mathbf{u}_{1}% \cdot symbol{\nabla}\left(|\mathbf{u}_{1}|^{2}\right)\right\}\\ &\displaystyle+\mathcal{O}\left(\varepsilon^{4}\right)=0,\qquad\,\text{at }z=0% .\end{aligned}
  28. η ( x , t ) = η ( x - c t ) and 𝐮 ( x , z , t ) = 𝐮 ( x - c t , z ) . \eta(x,t)=\eta(x-ct)\quad\,\text{and}\quad\mathbf{u}(x,z,t)=\mathbf{u}(x-ct,z).
  29. Φ ( x , z , t ) = β x - γ t + φ ( x - c t , z ) , \Phi(x,z,t)=\beta x-\gamma t+\varphi(x-ct,z),
  30. θ = k x - ω t = k ( x - c t ) , \theta=kx-\omega t=k\left(x-ct\right),
  31. η = n = 1 A n cos ( n θ ) . \eta=\sum_{n=1}^{\infty}A_{n}\,\cos\,(n\theta).
  32. Φ = β x - γ t + n = 1 B n [ cosh ( n k ( z + h ) ) ] sin ( n θ ) , \Phi=\beta x-\gamma t+\sum_{n=1}^{\infty}B_{n}\,\biggl[\cosh\,\left(nk\,(z+h)% \right)\biggr]\,\sin\,(n\theta),
  33. f x = + k f θ and f t = - ω f θ . \frac{\partial f}{\partial x}=+k\,\frac{\partial f}{\partial\theta}\qquad\,% \text{and}\qquad\frac{\partial f}{\partial t}=-\omega\,\frac{\partial f}{% \partial\theta}.
  34. ω = ω 0 + ε ω 1 + ε 2 ω 2 + . \omega=\omega_{0}+\varepsilon\,\omega_{1}+\varepsilon^{2}\,\omega_{2}+\cdots.

Stone–Geary_utility_function.html

  1. U = i ( q i - γ i ) β i U=\prod_{i}(q_{i}-\gamma_{i})^{\beta_{i}}
  2. U U
  3. q i q_{i}
  4. i i
  5. β \beta
  6. γ \gamma
  7. γ i = 0 \gamma_{i}=0
  8. q i = γ i + β i p i ( y - j γ j p j ) q_{i}=\gamma_{i}+\frac{\beta_{i}}{p_{i}}(y-\sum_{j}\gamma_{j}p_{j})
  9. y y
  10. p i p_{i}
  11. i i

Stranski–Krastanov_growth.html

  1. μ ( n ) = μ + [ φ a - φ a ( n ) + ε d ( n ) + ε e ( n ) ] \mu(n)=\mu_{\infty}+[\varphi_{a}-\varphi_{a}^{\prime}(n)+\varepsilon_{d}(n)+% \varepsilon_{e}(n)]
  2. μ \mu_{\infty}
  3. φ a \varphi_{a}
  4. φ a ( n ) \varphi_{a}^{\prime}(n)
  5. ε d ( n ) \varepsilon_{d}(n)
  6. ε e ( n ) \varepsilon_{e}(n)
  7. φ a \varphi_{a}
  8. φ a ( n ) \varphi_{a}^{\prime}(n)
  9. ε d ( n ) \varepsilon_{d}(n)
  10. ε e ( n ) \varepsilon_{e}(n)
  11. ε d , e ( n ) μ \varepsilon_{d,e}(n)\ll\mu_{\infty}
  12. d μ d n \frac{d\mu}{dn}
  13. d μ d n < 0 \frac{d\mu}{dn}<0
  14. d μ d n > 0 \frac{d\mu}{dn}>0
  15. a f - a s a s \frac{a_{f}-a_{s}}{a_{s}}
  16. a f a_{f}
  17. a s a_{s}
  18. h C h_{C}
  19. h C h_{C}
  20. h C h_{C}

Strengthening_mechanisms_of_materials.html

  1. Δ σ y = G b ρ \Delta\sigma_{y}={Gb\sqrt{\rho_{\perp}}}
  2. G G
  3. b b
  4. ρ \rho_{\perp}
  5. Δ τ = G b c ϵ 3 / 2 \Delta\tau=Gb\sqrt{c}\epsilon^{3/2}
  6. c c
  7. ϵ \epsilon
  8. Δ τ = G b L - 2 r \Delta\tau={Gb\over L-2r}
  9. Δ τ = γ π r b L \Delta\tau={\gamma\pi r\over bL}
  10. σ y = σ y , 0 + k d x \sigma_{y}=\sigma_{y,0}+{k\over{d^{x}}}
  11. k k
  12. d d
  13. σ y , 0 \sigma_{y,0}
  14. σ y = m o d i f i e d = σ y , 0 + σ c o m p r e s s i v e \sigma_{y=modified}=\sigma_{y,0}+\sigma_{compressive}

Streptomycin-6-phosphatase.html

  1. \rightleftharpoons

Streptomycin_6-kinase.html

  1. \rightleftharpoons

Stress_functions.html

  1. σ i j , i = 0 \sigma_{ij,i}=0\,
  2. σ \sigma
  3. σ i j , k k + 1 1 + ν σ k k , i j = 0 \sigma_{ij,kk}+\frac{1}{1+\nu}\sigma_{kk,ij}=0
  4. σ = × × Φ \sigma=\nabla\times\nabla\times\Phi
  5. σ i j = ε i k m ε j l n Φ k l , m n \sigma_{ij}=\varepsilon_{ikm}\varepsilon_{jln}\Phi_{kl,mn}
  6. σ x = 2 Φ y y z z + 2 Φ z z y y - 2 2 Φ y z y z \sigma_{x}=\frac{\partial^{2}\Phi_{yy}}{\partial z\partial z}+\frac{\partial^{% 2}\Phi_{zz}}{\partial y\partial y}-2\frac{\partial^{2}\Phi_{yz}}{\partial y% \partial z}
  7. σ x y = - 2 Φ x y z z - 2 Φ z z x y + 2 Φ y z x z + 2 Φ z x y z \sigma_{xy}=-\frac{\partial^{2}\Phi_{xy}}{\partial z\partial z}-\frac{\partial% ^{2}\Phi_{zz}}{\partial x\partial y}+\frac{\partial^{2}\Phi_{yz}}{\partial x% \partial z}+\frac{\partial^{2}\Phi_{zx}}{\partial y\partial z}
  8. σ y = 2 Φ x x z z + 2 Φ z z x x - 2 2 Φ z x z x \sigma_{y}=\frac{\partial^{2}\Phi_{xx}}{\partial z\partial z}+\frac{\partial^{% 2}\Phi_{zz}}{\partial x\partial x}-2\frac{\partial^{2}\Phi_{zx}}{\partial z% \partial x}
  9. σ y z = - 2 Φ y z x x - 2 Φ x x y z + 2 Φ z x y x + 2 Φ x y z x \sigma_{yz}=-\frac{\partial^{2}\Phi_{yz}}{\partial x\partial x}-\frac{\partial% ^{2}\Phi_{xx}}{\partial y\partial z}+\frac{\partial^{2}\Phi_{zx}}{\partial y% \partial x}+\frac{\partial^{2}\Phi_{xy}}{\partial z\partial x}
  10. σ z = 2 Φ y y x x + 2 Φ x x y y - 2 2 Φ x y x y \sigma_{z}=\frac{\partial^{2}\Phi_{yy}}{\partial x\partial x}+\frac{\partial^{% 2}\Phi_{xx}}{\partial y\partial y}-2\frac{\partial^{2}\Phi_{xy}}{\partial x% \partial y}
  11. σ z x = - 2 Φ z x y y - 2 Φ y y z x + 2 Φ x y z y + 2 Φ y z x y \sigma_{zx}=-\frac{\partial^{2}\Phi_{zx}}{\partial y\partial y}-\frac{\partial% ^{2}\Phi_{yy}}{\partial z\partial x}+\frac{\partial^{2}\Phi_{xy}}{\partial z% \partial y}+\frac{\partial^{2}\Phi_{yz}}{\partial x\partial y}
  12. Φ m n \Phi_{mn}
  13. ε \varepsilon
  14. \nabla
  15. Φ m n \Phi_{mn}
  16. Φ i j = [ A 0 0 0 B 0 0 0 C ] \Phi_{ij}=\begin{bmatrix}A&0&0\\ 0&B&0\\ 0&0&C\end{bmatrix}
  17. σ x = 2 B z 2 + 2 C y 2 \sigma_{x}=\frac{\partial^{2}B}{\partial z^{2}}+\frac{\partial^{2}C}{\partial y% ^{2}}
  18. σ y z = - 2 A y z \sigma_{yz}=-\frac{\partial^{2}A}{\partial y\partial z}
  19. σ y = 2 C x 2 + 2 A z 2 \sigma_{y}=\frac{\partial^{2}C}{\partial x^{2}}+\frac{\partial^{2}A}{\partial z% ^{2}}
  20. σ z x = - 2 B z x \sigma_{zx}=-\frac{\partial^{2}B}{\partial z\partial x}
  21. σ z = 2 A y 2 + 2 B x 2 \sigma_{z}=\frac{\partial^{2}A}{\partial y^{2}}+\frac{\partial^{2}B}{\partial x% ^{2}}
  22. σ x y = - 2 C x y \sigma_{xy}=-\frac{\partial^{2}C}{\partial x\partial y}
  23. 4 A + 4 B + 4 C = 3 ( 2 A x 2 + 2 B y 2 + 2 C z 2 ) / ( 2 - ν ) , \nabla^{4}A+\nabla^{4}B+\nabla^{4}C=3\left(\frac{\partial^{2}A}{\partial x^{2}% }+\frac{\partial^{2}B}{\partial y^{2}}+\frac{\partial^{2}C}{\partial z^{2}}% \right)/(2-\nu),
  24. C C
  25. φ \varphi
  26. σ x = 2 φ y 2 ; σ y = 2 φ x 2 ; σ x y = - 2 φ x y - ( f x y + f y x ) \sigma_{x}=\frac{\partial^{2}\varphi}{\partial y^{2}}~{};~{}~{}\sigma_{y}=% \frac{\partial^{2}\varphi}{\partial x^{2}}~{};~{}~{}\sigma_{xy}=-\frac{% \partial^{2}\varphi}{\partial x\partial y}-(f_{x}y+f_{y}x)
  27. f x f_{x}
  28. f y f_{y}
  29. σ r r = 1 r φ r + 1 r 2 2 φ θ 2 ; σ θ θ = 2 φ r 2 ; σ r θ = σ θ r = - r ( 1 r φ θ ) \sigma_{rr}=\frac{1}{r}\frac{\partial\varphi}{\partial r}+\frac{1}{r^{2}}\frac% {\partial^{2}\varphi}{\partial\theta^{2}}~{};~{}~{}\sigma_{\theta\theta}=\frac% {\partial^{2}\varphi}{\partial r^{2}}~{};~{}~{}\sigma_{r\theta}=\sigma_{\theta r% }=-\frac{\partial}{\partial r}\left(\frac{1}{r}\frac{\partial\varphi}{\partial% \theta}\right)
  30. Φ m n \Phi_{mn}
  31. Φ i j = [ 0 C B C 0 A B A 0 ] \Phi_{ij}=\begin{bmatrix}0&C&B\\ C&0&A\\ B&A&0\end{bmatrix}
  32. σ x = - 2 2 A y z \sigma_{x}=-2\frac{\partial^{2}A}{\partial y\partial z}
  33. σ y z = - 2 A x 2 + 2 B y x + 2 C z x \sigma_{yz}=-\frac{\partial^{2}A}{\partial x^{2}}+\frac{\partial^{2}B}{% \partial y\partial x}+\frac{\partial^{2}C}{\partial z\partial x}
  34. σ y = - 2 2 B z x \sigma_{y}=-2\frac{\partial^{2}B}{\partial z\partial x}
  35. σ z x = - 2 B y 2 + 2 C z y + 2 A x y \sigma_{zx}=-\frac{\partial^{2}B}{\partial y^{2}}+\frac{\partial^{2}C}{% \partial z\partial y}+\frac{\partial^{2}A}{\partial x\partial y}
  36. σ z = - 2 2 C x y \sigma_{z}=-2\frac{\partial^{2}C}{\partial x\partial y}
  37. σ x y = - 2 C z 2 + 2 A x z + 2 B y z \sigma_{xy}=-\frac{\partial^{2}C}{\partial z^{2}}+\frac{\partial^{2}A}{% \partial x\partial z}+\frac{\partial^{2}B}{\partial y\partial z}

Succinate—CoA_ligase_(ADP-forming).html

  1. \rightleftharpoons

Succinate—CoA_ligase_(GDP-forming).html

  1. \rightleftharpoons

Succinyl-CoA_hydrolase.html

  1. \rightleftharpoons

Succinyl-diaminopimelate_desuccinylase.html

  1. \rightleftharpoons

Succinyldiaminopimelate_transaminase.html

  1. \rightleftharpoons

Succinylglutamate_desuccinylase.html

  1. \rightleftharpoons

Succinylornithine_transaminase.html

  1. \rightleftharpoons

Sucrose-phosphatase.html

  1. \rightleftharpoons

Sucrose-phosphate_synthase.html

  1. \rightleftharpoons

Sucrose:sucrose_fructosyltransferase.html

  1. \rightleftharpoons

Sucrose_6F-alpha-galactosyltransferase.html

  1. \rightleftharpoons

Sucrose_synthase.html

  1. \rightleftharpoons

Sucrose—1,6-alpha-glucan_3(6)-alpha-glucosyltransferase.html

  1. \rightleftharpoons

Sugar-phosphatase.html

  1. \rightleftharpoons

Sugar-terminal-phosphatase.html

  1. \rightleftharpoons

SUHA_(computer_science).html

  1. P ( h ( a ) = h ( b ) ) = 1 m . P(h(a)=h(b))=\frac{1}{m}.
  2. α \alpha
  3. α = n m \alpha=\tfrac{n}{m}
  4. Θ ( α + 1 ) \Theta(\alpha+1)\,
  5. Θ ( α + 1 ) \Theta(\alpha+1)\,
  6. α = n m \alpha=\tfrac{n}{m}
  7. α = 30 10 \alpha=\tfrac{30}{10}
  8. α = 3 \alpha=3\,

Sulfate-transporting_ATPase.html

  1. \rightleftharpoons

Sulfate_adenylyltransferase.html

  1. \rightleftharpoons

Sulfate_adenylyltransferase_(ADP).html

  1. \rightleftharpoons

Sulfinoalanine_decarboxylase.html

  1. \rightleftharpoons

Sulfoacetaldehyde_acetyltransferase.html

  1. \rightleftharpoons

Sulfopyruvate_decarboxylase.html

  1. \rightleftharpoons

Super-logarithm.html

  1. x y {{}^{y}x}
  2. slog b ( z ) \,\mathrm{slog}_{b}(z)
  3. slog b ( b z ) = slog b ( z ) + 1 \,\mathrm{slog}_{b}(b^{z})=\mathrm{slog}_{b}(z)+1
  4. slog b ( 1 ) = 0. \,\mathrm{slog}_{b}(1)=0.
  5. b , b b , b b b b,b^{b},b^{b^{b}}
  6. slog b ( z ) { slog b ( b z ) - 1 if z 0 - 1 + z if 0 < z 1 slog b ( log b ( z ) ) + 1 if 1 < z \mathrm{slog}_{b}(z)\approx\begin{cases}\mathrm{slog}_{b}(b^{z})-1&\,\text{if % }z\leq 0\\ -1+z&\,\text{if }0<z\leq 1\\ \mathrm{slog}_{b}(\log_{b}(z))+1&\,\text{if }1<z\\ \end{cases}
  7. C 0 C^{0}
  8. C 0 C^{0}
  9. slog b ( z ) { slog b ( b z ) - 1 if z 0 - 1 + 2 log ( b ) 1 + log ( b ) z + 1 - log ( b ) 1 + log ( b ) z 2 if 0 < z 1 slog b ( log b ( z ) ) + 1 if 1 < z \mathrm{slog}_{b}(z)\approx\begin{cases}\mathrm{slog}_{b}(b^{z})-1&\,\text{if % }z\leq 0\\ -1+\frac{2\log(b)}{1+\log(b)}z+\frac{1-\log(b)}{1+\log(b)}z^{2}&\,\text{if }0<% z\leq 1\\ \mathrm{slog}_{b}(\log_{b}(z))+1&\,\text{if }1<z\end{cases}
  10. C 1 C^{1}
  11. A f ( f ( x ) ) = A f ( x ) + 1 \,A_{f}(f(x))=A_{f}(x)+1
  12. A f ( x ) A_{f}(x)
  13. A f ( x ) = A f ( x ) + c A^{\prime}_{f}(x)=A_{f}(x)+c
  14. slog b ( 1 ) = 0 \mathrm{slog}_{b}(1)=0
  15. slog b ( z ) = slog b ( log b ( z ) ) + 1 \,\mathrm{slog}_{b}(z)=\mathrm{slog}_{b}(\log_{b}(z))+1
  16. slog b ( z ) > - 2 \,\mathrm{slog}_{b}(z)>-2
  17. i > j . i>j.
  18. Θ ( N 2 ) \,\Theta(N^{2})
  19. k = 1 \,k=1
  20. Θ ( N log N ) \,\Theta(N\log N)
  21. k = 2 \,k=2
  22. Θ ( N log log N ) \,\Theta(N\log\log N)
  23. k = 3 \,k=3
  24. Θ ( N slog N ) \,\Theta(N\,\mathrm{slog}\,N)
  25. k = 4 \,k=4
  26. k = 5 \,k=5
  27. k > 5 \,k>5
  28. sexp b ( z ) ~{}{\rm sexp}_{b}(z)~{}
  29. b ~{}b~{}
  30. s l o g b = s e x p b - 1 slogb=sexp_{b}^{-1}
  31. slog b ( z ) ~{}{\rm slog}_{b}(z)~{}
  32. z ~{}z~{}
  33. b = e ~{}b=e~{}
  34. L 0.318 + 1.337 i L\approx 0.318+1.337{\!~{}\rm i}
  35. L * 0.318 - 1.337 i L^{*}\approx 0.318-1.337{\!~{}\rm i}
  36. L = ln ( L ) L=\ln(L)
  37. z = L ~{}z=L~{}
  38. z = L * ~{}z=L^{*}
  39. - 2 -2

Super-recursive_algorithm.html

  1. Δ 2 0 \Delta^{0}_{2}

Superconducting_radio_frequency.html

  1. Q o = ω U P d Q_{o}=\frac{\omega U}{P_{d}}
  2. U = μ 0 2 | H | 2 d V U=\frac{\mu_{0}}{2}\int{|\overrightarrow{H}|^{2}dV}
  3. P d = R s 2 | H | 2 d S P_{d}=\frac{R_{s}}{2}\int{|\overrightarrow{H}|^{2}dS}
  4. G = ω μ 0 | H | 2 d V | H | 2 d S G=\frac{\omega\mu_{0}\int{|\overrightarrow{H}|^{2}dV}}{\int{|\overrightarrow{H% }|^{2}dS}}
  5. Q o = G R s Q_{o}=\frac{G}{R_{s}}\cdot
  6. R s n o r m a l = ω μ 0 2 σ R_{s\ normal}=\sqrt{\frac{\omega\mu_{0}}{2\sigma}}
  7. R s = R B C S + R r e s R_{s}=R_{BCS}+R_{res}
  8. R B C S 2 × 10 - 4 ( f 1.5 × 10 9 ) 2 e - 17.67 / T T R_{BCS}\simeq 2\times 10^{-4}\left(\frac{f}{1.5\times 10^{9}}\right)^{2}\frac{% e^{-17.67/T}}{T}
  9. R H = H e x t 2 H c 2 R n 9.49 × 10 - 12 H e x t f R_{H}=\frac{H_{ext}}{2H_{c2}}R_{n}\approx 9.49\times 10^{-12}H_{ext}\sqrt{f}
  10. V w a k e = q ω o R 2 Q o e j ω o t e - ω t 2 Q L = k q e j ω o t e - ω t 2 Q L V_{wake}=\frac{q\omega_{o}R}{2Q_{o}}\ e^{j\omega_{o}t}\ e^{-\frac{\omega t}{2Q% _{L}}}=kq\ e^{j\omega_{o}t}\ e^{-\frac{\omega t}{2Q_{L}}}
  11. R = ( E d l ) 2 P d = V 2 P d R=\frac{\left(\int{\overrightarrow{E}\cdot dl}\right)^{2}}{P_{d}}=\frac{V^{2}}% {P_{d}}
  12. k = ω o R 2 Q o k=\frac{\omega_{o}R}{2Q_{o}}
  13. R Q o = V 2 ω U = 2 ( E d l ) 2 ω μ o | H | 2 d V = 2 k ω o \frac{R}{Q_{o}}=\frac{V^{2}}{\omega U}=\frac{2\left(\int{\overrightarrow{E}% \cdot dl}\right)^{2}}{\omega\mu_{o}\int{|\overrightarrow{H}|^{2}dV}}=\frac{2k}% {\omega_{o}}
  14. V s s w a k e = V w a k e ( 1 1 - e - τ e j δ - 1 2 ) V_{ss\ wake}=V_{wake}\left(\frac{1}{1-e^{-\tau}e^{j\delta}}-\frac{1}{2}\right)
  15. τ = ω T b 2 Q L \tau=\frac{\omega T_{b}}{2Q_{L}}
  16. η C = { T c o l d T w a r m - T c o l d , if T c o l d < T w a r m - T c o l d 1 , otherwise \eta_{C}=\begin{cases}\frac{T_{cold}}{T_{warm}-T_{cold}},&\mbox{if }~{}T_{cold% }<T_{warm}-T_{cold}\\ 1,&\mbox{otherwise}\end{cases}
  17. P w a r m = P c o l d η C η p P_{warm}=\frac{P_{cold}}{\eta_{C}\ \eta_{p}}

Supermodule.html

  1. E = E 0 E 1 E=E_{0}\oplus E_{1}
  2. E i A j E i + j E_{i}A_{j}\subseteq E_{i+j}
  3. a x = ( - 1 ) | a | | x | x a a\cdot x=(-1)^{|a||x|}x\cdot a
  4. ϕ : E F \phi:E\to F\,
  5. ϕ ( x + y ) = ϕ ( x ) + ϕ ( y ) \phi(x+y)=\phi(x)+\phi(y)\,
  6. ϕ ( x a ) = ϕ ( x ) a \phi(x\cdot a)=\phi(x)\cdot a\,
  7. ϕ ( E i ) F i \phi(E_{i})\subseteq F_{i}\,
  8. ϕ ( E i ) F i \phi(E_{i})\subseteq F_{i}
  9. ϕ ( E i ) F 1 - i . \phi(E_{i})\subseteq F_{1-i}.
  10. ϕ ( x a ) = ϕ ( x ) a ϕ ( a x ) = ( - 1 ) | a | | ϕ | a ϕ ( x ) . \phi(x\cdot a)=\phi(x)\cdot a\qquad\phi(a\cdot x)=(-1)^{|a||\phi|}a\cdot\phi(x).
  11. ( a ϕ ) ( x ) = a ϕ ( x ) ( ϕ a ) ( x ) = ϕ ( a x ) . \begin{aligned}\displaystyle(a\cdot\phi)(x)&\displaystyle=a\cdot\phi(x)\\ \displaystyle(\phi\cdot a)(x)&\displaystyle=\phi(a\cdot x).\end{aligned}
  12. 𝐇𝐨𝐦 0 ( E , F ) = Hom ( E , F ) . \mathbf{Hom}_{0}(E,F)=\mathrm{Hom}(E,F).

Superstatistics.html

  1. E E
  2. exp ( - β E ) \exp(-\beta E)
  3. β \beta
  4. β \beta
  5. f ( β ) f(\beta)
  6. B ( E ) = 0 d β f ( β ) exp ( - β E ) . B(E)=\int_{0}^{\infty}d\beta f(\beta)\exp(-\beta E).
  7. Z = i = 1 W B ( E i ) , Z=\sum_{i=1}^{W}B(E_{i}),
  8. { E i } i = 1 W \{E_{i}\}_{i=1}^{W}
  9. E i E_{i}
  10. p i = 1 Z B ( E i ) . p_{i}=\frac{1}{Z}B(E_{i}).
  11. β \beta

Surface_diffusion.html

  1. Γ = ν e - E diff / k B T (eq. 1) \Gamma=\nu e^{-E_{\mathrm{diff}}/k_{B}T}\qquad\,\text{(eq. 1)}
  2. Δ r 2 = a Γ t \scriptstyle\sqrt{\langle\Delta r^{2}\rangle}=a\sqrt{\Gamma t}\,\!

Surface_force.html

  1. f s = p A f_{s}=p\cdot A
  2. f o r c e a r e a = N m 2 \frac{force}{area}=\mathrm{\frac{N}{m^{2}}}
  3. ( l e n g t h ) ( w i d t h ) = m m = m 2 (length)\cdot(width)=\mathrm{m\cdot m}=\mathrm{m^{2}}
  4. 5 N m 2 = 5 P a 5\mathrm{\frac{N}{m^{2}}}=5\mathrm{Pa}
  5. 20 m 2 20\mathrm{m^{2}}
  6. 5 P a 20 m 2 = 100 N 5\mathrm{Pa}\cdot 20\mathrm{m^{2}}=100\mathrm{N}

Surface_phonon.html

  1. m i m_{i}
  2. - j , β ϕ i α , j β u j , β -\sum_{j,\beta}\phi_{i\alpha,j\beta}u_{j,\beta}
  3. ϕ i α , j β \phi_{i\alpha,j\beta}
  4. u l , m , κ , α = ( m κ ) v l , κ , α ( ω , q ) e i [ ω t - q x ( l , m ) ] u_{l,m,\kappa,\alpha}=\sqrt{(m_{\kappa})}v_{l,\kappa,\alpha}(\omega,q)e^{i[% \omega t-qx(l,m)]}
  5. E = 2 k 2 2 m E=\frac{\hbar^{2}k^{2}}{2m}
  6. Δ E = | E i - E s | = ω \Delta E=|E_{i}-E_{s}|=\hbar\omega
  7. Δ K = k i ( s i n θ i ) - k s ( s i n θ s ) = \Delta K=k_{i}(sin\theta_{i})-k_{s}(sin\theta_{s})=

Sym-norspermidine_synthase.html

  1. \rightleftharpoons

Synaptic_weight.html

  1. 𝐱 \,\textbf{x}
  2. 𝐲 \,\textbf{y}
  3. w w
  4. y j = i w i j x i or 𝐲 = w 𝐱 y_{j}=\sum_{i}w_{ij}x_{i}~{}~{}\textrm{or}~{}~{}\,\textbf{y}=w\,\textbf{x}

Synephrine_dehydratase.html

  1. \rightleftharpoons

Szász–Mirakjan–Kantorovich_operator.html

  1. [ 𝒯 n ( f ) ] ( x ) = n e - n x k = 0 ( n x ) k k ! k / n ( k + 1 ) / n f ( t ) d t [\mathcal{T}_{n}(f)](x)=ne^{-nx}\sum_{k=0}^{\infty}{\frac{(nx)^{k}}{k!}\int_{k% /n}^{(k+1)/n}f(t)\,dt}
  2. x [ 0 , ) x\in[0,\infty)\subset\mathbb{R}
  3. n n\in\mathbb{N}

Szász–Mirakyan_operator.html

  1. [ 𝒮 n ( f ) ] ( x ) \left[\mathcal{S}_{n}(f)\right](x)
  2. e - n x k = 0 ( n x ) k k ! f ( k n ) e^{-nx}\sum_{k=0}^{\infty}{\frac{(nx)^{k}}{k!}f\left(\frac{k}{n}\right)}
  3. x [ 0 , ) x\in[0,\infty)\subset\mathbb{R}
  4. n n\in\mathbb{N}
  5. f f
  6. ( 𝒮 n ( f ) ) n (\mathcal{S}_{n}(f))_{n\in\mathbb{N}}
  7. n n
  8. 𝒮 n ( f ) f \mathcal{S}_{n}(f)\geq f
  9. f f
  10. m \leq m
  11. 𝒮 n ( f ) \mathcal{S}_{n}(f)
  12. n n
  13. f f
  14. ( 0 , ) (0,\infty)
  15. 𝒮 n ( f ) \mathcal{S}_{n}(f)
  16. f f
  17. n n\rightarrow\infty

Szegő_polynomial.html

  1. - π π f ( e i θ ) g ( e i θ ) ¯ d μ \int_{-\pi}^{\pi}f(e^{i\theta})\overline{g(e^{i\theta})}\,d\mu

T2-induced_deoxynucleotide_kinase.html

  1. \rightleftharpoons

Taft_equation.html

  1. log ( k s k CH3 ) = ρ * σ * + δ E s \log\left(\frac{k_{s}}{k_{\,\text{CH3}}}\right)=\rho^{*}\sigma^{*}+\delta E_{s}
  2. σ * = ( 1 2.48 ρ * ) [ log ( k s k CH3 ) B - log ( k s k CH3 ) A ] \sigma^{*}=\left(\frac{1}{2.48\rho^{*}}\right)\Bigg[\log\left(\frac{k_{s}}{k_{% \,\text{CH3}}}\right)_{B}-\log\left(\frac{k_{s}}{k_{\,\text{CH3}}}\right)_{A}\Bigg]
  3. E s = 1 δ log ( k s k CH3 ) E_{s}=\frac{1}{\delta}\log\left(\frac{k_{s}}{k_{\,\text{CH3}}}\right)
  4. log ( k s k CH3 ) = ρ * σ * \log\left(\frac{k_{s}}{k_{\,\text{CH3}}}\right)=\rho^{*}\sigma^{*}
  5. log ( k s k CH3 ) = δ E s \log\left(\frac{k_{s}}{k_{\,\text{CH3}}}\right)=\delta E_{s}

Tagatose-6-phosphate_kinase.html

  1. \rightleftharpoons

Tagatose-bisphosphate_aldolase.html

  1. \rightleftharpoons

Tagatose_kinase.html

  1. \rightleftharpoons

Tail_value_at_risk.html

  1. VaR α ( X ) \operatorname{VaR}_{\alpha}(X)
  2. α \alpha
  3. X X
  4. 0 < α < 1 0<\alpha<1
  5. TVaR α ( X ) = E [ - X | X - VaR α ( X ) ] = E [ - X | X x α ] , \operatorname{TVaR}_{\alpha}(X)=\operatorname{E}[-X|X\leq-\operatorname{VaR}_{% \alpha}(X)]=\operatorname{E}[-X|X\leq x^{\alpha}],
  6. x α x^{\alpha}
  7. α \alpha
  8. x α = inf { x : Pr ( X x ) > α } x^{\alpha}=\inf\{x\in\mathbb{R}:\Pr(X\leq x)>\alpha\}
  9. X X
  10. p 1 p\geq 1

Tannase.html

  1. \rightleftharpoons

Target_income_sales.html

  1. Target Income Sales (in Units) \displaystyle\,\text{Target Income Sales (in Units)}

Tartrate_decarboxylase.html

  1. \rightleftharpoons

Tartronate-semialdehyde_synthase.html

  1. \rightleftharpoons

Tartronate_O-hydroxycinnamoyltransferase.html

  1. \rightleftharpoons

Tate's_algorithm.html

  1. \mathbb{Q}
  2. c p = [ E ( p ) : E 0 ( p ) ] , c_{p}=[E(\mathbb{Q}_{p}):E^{0}(\mathbb{Q}_{p})],
  3. E 0 ( p ) E^{0}(\mathbb{Q}_{p})
  4. p \mathbb{Q}_{p}
  5. y 2 + a 1 x y + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 . y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}.
  6. a i , m = a i / π m a_{i,m}=a_{i}/\pi^{m}
  7. b 2 = a 1 2 + 4 a 2 b_{2}=a_{1}^{2}+4a_{2}
  8. b 4 = a 1 a 3 + 2 a 4 b_{4}=a_{1}a_{3}+2a_{4}
  9. b 6 = a 3 2 + 4 a 6 b_{6}=a_{3}^{2}+4a_{6}
  10. b 8 = a 1 2 a 6 - a 1 a 3 a 4 + 4 a 2 a 6 + a 2 a 3 2 - a 4 2 b_{8}=a_{1}^{2}a_{6}-a_{1}a_{3}a_{4}+4a_{2}a_{6}+a_{2}a_{3}^{2}-a_{4}^{2}
  11. c 4 = b 2 2 - 24 b 4 c_{4}=b_{2}^{2}-24b_{4}
  12. c 6 = - b 2 3 + 36 b 2 b 4 - 216 b 6 c_{6}=-b_{2}^{3}+36b_{2}b_{4}-216b_{6}
  13. Δ = - b 2 2 b 8 - 8 b 4 3 - 27 b 6 2 + 9 b 2 b 4 b 6 \Delta=-b_{2}^{2}b_{8}-8b_{4}^{3}-27b_{6}^{2}+9b_{2}b_{4}b_{6}
  14. j = c 4 3 / Δ . j=c_{4}^{3}/\Delta.
  15. P ( T ) = T 3 + a 2 , 1 T 2 + a 4 , 2 T + a 6 , 3 . P(T)=T^{3}+a_{2,1}T^{2}+a_{4,2}T+a_{6,3}.
  16. Y 2 + a 3 , 2 Y - a 6 , 4 Y^{2}+a_{3,2}Y-a_{6,4}

Tau-protein_kinase.html

  1. \rightleftharpoons

Taurine-transporting_ATPase.html

  1. \rightleftharpoons

Taurine—2-oxoglutarate_transaminase.html

  1. \rightleftharpoons

Taurine—pyruvate_aminotransferase.html

  1. \rightleftharpoons

Taurocyamine_kinase.html

  1. \rightleftharpoons

Taxadien-5alpha-ol_O-acetyltransferase.html

  1. \rightleftharpoons

Taxadiene_synthase.html

  1. \rightleftharpoons

Taylor–Green_vortex.html

  1. 𝐯 = ( u , v , w ) \mathbf{v}=(u,v,w)
  2. t = 0 t=0
  3. u = A cos a x sin b y sin c z , u=A\cos ax\sin by\sin cz,
  4. v = B sin a x cos b y sin c z , v=B\sin ax\cos by\sin cz,
  5. w = C sin a x sin b y cos c z . w=C\sin ax\sin by\cos cz.
  6. 𝐯 = 0 \nabla\cdot\mathbf{v}=0
  7. A a + B b + C c = 0 Aa+Bb+Cc=0
  8. u x + v y = 0 , \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0,
  9. u t + u u x + v u y = - 1 ρ p x + ν ( 2 u x 2 + 2 u y 2 ) , \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u% }{\partial y}=-\frac{1}{\rho}\frac{\partial p}{\partial x}+\nu\left(\frac{% \partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}\right),
  10. v t + u v x + v v y = - 1 ρ p y + ν ( 2 v x 2 + 2 v y 2 ) . \frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v% }{\partial y}=-\frac{1}{\rho}\frac{\partial p}{\partial y}+\nu\left(\frac{% \partial^{2}v}{\partial x^{2}}+\frac{\partial^{2}v}{\partial y^{2}}\right).
  11. 0 x , y 2 π 0\leq x,y\leq 2\pi
  12. u = sin x cos y F ( t ) , v = - cos x sin y F ( t ) , u=\sin x\cos y\,F(t),\qquad\qquad v=-\cos x\sin y\,F(t),
  13. F ( t ) = e - 2 ν t F(t)=e^{-2\nu t}
  14. ν \nu
  15. A = a = b = 1 A=a=b=1
  16. F ( t ) = 1 - 2 ν t + O ( t 2 ) F(t)=1-2\nu t+O(t^{2})
  17. p p
  18. p = ρ 4 ( cos 2 x + cos 2 y ) F 2 ( t ) . p=\frac{\rho}{4}\left(\cos 2x+\cos 2y\right)F^{2}(t).
  19. 𝐯 = × s y m b o l ψ \mathbf{v}=\nabla\times symbol{\psi}
  20. 𝐯 \mathbf{v}
  21. s y m b o l ψ = sin x sin y F ( t ) 𝐳 ^ . symbol{\psi}=\sin x\sin yF(t)\,\hat{\mathbf{z}}.
  22. s y m b o l ω = × 𝐯 symbol{\mathbf{\omega}}=\nabla\times\mathbf{v}
  23. s y m b o l ω = 2 sin x sin y F ( t ) 𝐳 ^ . symbol{\mathbf{\omega}}=2\sin x\sin y\,F(t)\hat{\mathbf{z}}.

Technology_of_television.html

  1. c d / m 2 cd/m^{2}

Teichoic-acid-transporting_ATPase.html

  1. \rightleftharpoons

Template:False.html

  1. \bot

Template:Ident.html

  1. \equiv

Template:Models.html

  1. \models

Template:True.html

  1. \top

Temporal_finitism.html

  1. \infty

Term_Discrimination.html

  1. A A
  2. A k A_{k}
  3. k k
  4. Q ( A ) Q(A)
  5. A A
  6. k k
  7. D V k = Q ( A ) - Q ( A k ) DV_{k}=Q(A)-Q(A_{k})
  8. A A
  9. k k
  10. C C
  11. D i D_{i}
  12. C C
  13. D i D_{i}
  14. C C
  15. k k
  16. K K

Terrace_ledge_kink_model.html

  1. Δ G = ϵ k i n k - ϵ a d a t o m ( 1 ) \Delta G=\epsilon_{kink}-\epsilon_{adatom}\qquad(1)
  2. ϕ \phi
  3. Δ G = ϵ k i n k - ϵ a d a t o m = 3 ϕ - ϕ = 2 ϕ ( 2 ) \Delta G=\epsilon_{kink}-\epsilon_{adatom}=3\phi-\phi=2\phi\qquad(2)
  4. Δ G = ϵ s u r f a c e a t o m - ϵ a d a t o m ( 3 ) \Delta G=\epsilon_{surfaceatom}-\epsilon_{adatom}\qquad(3)
  5. n a d a t o m = n 0 e - Δ G a d a t o m k B T ( 4 ) n_{adatom}=n_{0}e^{\frac{-\Delta G_{adatom}}{k_{B}T}}\qquad(4)

Tetraacyldisaccharide_4'-kinase.html

  1. \rightleftharpoons

Tetrahydrodipicolinate_N-acetyltransferase.html

  1. \rightleftharpoons

Tetrahydrofolate_synthase.html

  1. \rightleftharpoons

The_Sea_Island_Mathematical_Manual.html

  1. C D x D F F H - D G \tfrac{CDxDF}{FH-DG}
  2. D G x D F F H - D G \tfrac{DGxDF}{FH-DG}

Theanine_hydrolase.html

  1. \rightleftharpoons

Thermal_time_scale.html

  1. τ t h = total kinetic energy rate of energy loss = G M 2 R L \tau_{th}=\frac{\mbox{total kinetic energy}~{}}{\mbox{rate of energy loss}~{}}% =\cfrac{GM^{2}}{RL}

Thermodynamic_integration.html

  1. U A U_{A}
  2. U B U_{B}
  3. U A U_{A}
  4. U B U_{B}
  5. U ( λ ) = U A + λ ( U B - U A ) U(\lambda)=U_{A}+\lambda(U_{B}-U_{A})
  6. λ \lambda
  7. λ \lambda
  8. λ = 0 \lambda=0
  9. λ = 1 \lambda=1
  10. Q ( N , V , T , λ ) = s exp [ - U s ( λ ) / k B T ] Q(N,V,T,\lambda)=\sum_{s}\exp[-U_{s}(\lambda)/k_{B}T]
  11. U s ( λ ) U_{s}(\lambda)
  12. s s
  13. U ( λ ) U(\lambda)
  14. F ( N , V , T , λ ) = - k B T ln Q ( N , V , T , λ ) F(N,V,T,\lambda)=-k_{B}T\ln Q(N,V,T,\lambda)
  15. Δ F ( A B ) = 0 1 d λ F ( λ ) λ = - 0 1 d λ k B T Q Q λ = 0 1 d λ k B T Q s 1 k B T exp [ - U s ( λ ) / k B T ] U s ( λ ) λ = 0 1 d λ U ( λ ) λ λ \Delta F(A\rightarrow B)=\int_{0}^{1}d\lambda\frac{\partial F(\lambda)}{% \partial\lambda}=-\int_{0}^{1}d\lambda\frac{k_{B}T}{Q}\frac{\partial Q}{% \partial\lambda}=\int_{0}^{1}d\lambda\frac{k_{B}T}{Q}\sum_{s}\frac{1}{k_{B}T}% \exp[-U_{s}(\lambda)/k_{B}T]\frac{\partial U_{s}(\lambda)}{\partial\lambda}=% \int_{0}^{1}d\lambda\left\langle\frac{\partial U(\lambda)}{\partial\lambda}% \right\rangle_{\lambda}
  16. λ \lambda
  17. U ( λ ) U(\lambda)
  18. λ \lambda
  19. U ( λ ) U(\lambda)
  20. λ \lambda
  21. λ \lambda

Theta_solvent.html

  1. 1 / 2 1/2
  2. Π \Pi
  3. v s v_{s}
  4. Δ μ 1 = - v s Π \Delta\mu_{1}=-v_{s}\Pi
  5. Π R T = c M + B c 2 + B 3 c 3 \frac{\Pi}{RT}=\frac{c}{M}+Bc^{2}+B_{3}c^{3}...
  6. Δ μ 1 = Δ μ 1 i d e a l + Δ μ 1 e x c e s s \Delta\mu_{1}=\Delta\mu_{1}^{ideal}+\Delta\mu_{1}^{excess}
  7. B = - Δ μ 1 e x c e s s v s c 2 B=\frac{-\Delta\mu_{1}^{excess}}{{v_{s}}{c^{2}}}

Thiamine-diphosphate_kinase.html

  1. \rightleftharpoons

Thiamine-phosphate_diphosphorylase.html

  1. \rightleftharpoons

Thiamine-phosphate_kinase.html

  1. \rightleftharpoons

Thiamine-triphosphatase.html

  1. \rightleftharpoons

Thiamine_diphosphokinase.html

  1. \rightleftharpoons

Thiamine_kinase.html

  1. \rightleftharpoons

Thiocyanate_hydrolase.html

  1. \rightleftharpoons

Threonine-phosphate_decarboxylase.html

  1. \rightleftharpoons

Threonine_aldolase.html

  1. \rightleftharpoons

Threonine_synthase.html

  1. \rightleftharpoons

Threonine—tRNA_ligase.html

  1. \rightleftharpoons

Thymidine-triphosphatase.html

  1. \rightleftharpoons

Thymidine_phosphorylase.html

  1. \rightleftharpoons

Thymidylate_5'-phosphatase.html

  1. \rightleftharpoons

Thyroid-hormone_transaminase.html

  1. \rightleftharpoons

Times_interest_earned.html

  1. Times-Interest-Earned = EBIT or EBITDA Interest Charges \mbox{Times-Interest-Earned}~{}=\frac{\mbox{EBIT or EBITDA}~{}}{\mbox{Interest% Charges}~{}}

Time–temperature_superposition.html

  1. E ( t , T ) = E ( a T t , T 0 ) . E(t,T)=E(a_{\rm T}\,t,T_{0})\,.
  2. T > T 0 a T > 1 \displaystyle T>T_{0}\quad\implies\quad a_{\rm T}>1
  3. M ( ω , T ) \displaystyle M^{\prime}(\omega,T)
  4. a T = η T η T0 a_{\rm T}=\frac{\eta_{\rm T}}{\eta_{\rm{T0}}}
  5. log a T = - C 1 ( T - T 0 ) C 2 + ( T - T 0 ) \log a_{\rm T}=-\frac{C_{1}(T-T_{0})}{C_{2}+(T-T_{0})}
  6. log a T = - C 1 g ( T - T g ) C 2 g + ( T - T g ) = log ( η T η T g ) \log a_{\rm T}=-\frac{C^{g}_{1}(T-T_{g})}{C^{g}_{2}+(T-T_{g})}=\log\left(\frac% {\eta_{\rm T}}{\eta_{T_{g}}}\right)
  7. C 1 = C 1 C 2 C 2 + ( T 0 - T 0 ) and C 2 = C 2 + ( T 0 - T 0 ) . C^{\prime}_{1}=\frac{C_{1}\,C_{2}}{C_{2}+(T^{\prime}_{0}-T_{0})}\qquad{\rm and% }\qquad C^{\prime}_{2}=C_{2}+(T^{\prime}_{0}-T_{0})\,.
  8. C 1 0 = C 1 g C 2 g C 2 g + ( T 0 - T g ) and C 2 0 = C 2 g + ( T 0 - T g ) . C^{0}_{1}=\frac{C^{g}_{1}\,C^{g}_{2}}{C^{g}_{2}+(T_{0}-T_{g})}\qquad{\rm and}% \qquad C^{0}_{2}=C^{g}_{2}+(T_{0}-T_{g})\,.
  9. log ( a T ) = E a 2.303 R ( 1 T - 1 T 0 ) \log(a_{\rm T})=\frac{E_{a}}{2.303R}\left(\frac{1}{T}-\frac{1}{T_{0}}\right)

Toeplitz_algebra.html

  1. T f + K T_{f}+K\;

Toeplitz_operator.html

  1. T g = P M g | H 2 , T_{g}=PM_{g}|_{H^{2}},

Topological_game.html

  1. P 1 P_{1}
  2. P 2 P_{2}
  3. G ( X , Φ ) G(X,\Phi)
  4. X X
  5. | X | |X|
  6. Φ : X 𝒫 ( X ) \Phi:X\mapsto\mathcal{P}(X)
  7. x X x\in X
  8. Φ ( x ) 𝒫 = { Y | Y X } \Phi(x)\in\mathcal{P}=\{Y\;|\;Y\subset X\}
  9. P 1 P_{1}
  10. x x
  11. Φ ( x ) = \Phi(x)=\emptyset
  12. [ X ] κ = { Y X | | Y | = κ } [X]^{\kappa}=\{Y\subset X\;|\;|Y|=\kappa\}
  13. [ X ] < κ = { Y X | | Y | < κ } [X]^{<\kappa}=\{Y\subset X\;|\;|Y|<\kappa\}
  14. [ X ] > κ = { Y X | | Y | > κ } [X]^{>\kappa}=\{Y\subset X\;|\;|Y|>\kappa\}
  15. κ \kappa
  16. E ¯ \bar{E}
  17. E E
  18. X X
  19. ( X ) \mathcal{F}(X)
  20. X X
  21. ω \omega
  22. ω = { 0 , 1 , 2 , } \omega=\{0,1,2,\cdots\}
  23. 2 ω = { 0 , 1 } × { 0 , 1 } × 2^{\omega}=\{0,1\}\times\{0,1\}\times\cdots
  24. ω ω = ω × ω × \omega^{\omega}=\omega\times\omega\times\cdots
  25. P i P_{i}
  26. P - i P_{-i}
  27. P i G P_{i}\uparrow G
  28. P i P_{i}
  29. G G
  30. G G
  31. P 1 G P_{1}\uparrow G
  32. P 2 G P_{2}\uparrow G
  33. G 1 G_{1}
  34. G 2 G_{2}
  35. ( P 1 G 1 P 1 G 2 ) ( P 2 G 1 P 2 G 2 ) (P_{1}\uparrow G_{1}\Leftrightarrow P_{1}\uparrow G_{2})\land(P_{2}\uparrow G_% {1}\Leftrightarrow P_{2}\uparrow G_{2})
  36. P i P_{i}
  37. P - i P_{-i}
  38. P - i P_{-i}
  39. R R
  40. J J
  41. 𝔈 \mathfrak{E}
  42. Y Y
  43. ( E 𝔈 ) ( E E Y ) ( E 𝔈 ) (E\in\mathfrak{E})\land(E\subset E^{\prime}\subset Y)\Rightarrow(E^{\prime}\in% \mathfrak{E})
  44. ( n < ω E n 𝔈 ) ( m < ω | E m 𝔈 ) (\cup_{n<\omega}E_{n}\in\mathfrak{E})\Rightarrow(\exists\;m<\omega\;|\;E_{m}% \in\mathfrak{E})
  45. ( A n | n < ω ) (A_{n}\;|\;n<\omega)
  46. Y Y
  47. H ( A n ) H(A_{n})
  48. Y Y
  49. ( A H ( A n ) ) ( A A Y ) ( A H ( A n ) ) (A\in H(A_{n}))\land(A\subset A^{\prime}\subset Y)\Rightarrow(A^{\prime}\in H(% A_{n}))
  50. ( B 0 , B 1 , H ( A n ) ) ( n < ω B n H ( A n ) ) (B_{0},B_{1},\cdots\in H(A_{n}))\Rightarrow(\cap_{n<\omega}B_{n}\in H(A_{n}))
  51. X X
  52. Y Y
  53. S ( X , Y , 𝔈 , H ) S(X,Y,\mathfrak{E},H)
  54. E 0 , E 1 , 𝔈 E_{0},E_{1},\cdots\in\mathfrak{E}
  55. X E 0 E 1 X\supset E_{0}\supset E_{1}\supset\cdots
  56. P 2 P_{2}
  57. X H ( E n | n < ω ) X\in H(E_{n}\;|\;n<\omega)
  58. P 2 S ( X , Y , 𝔈 , H ) P_{2}\uparrow S(X,Y,\mathfrak{E},H)
  59. X X
  60. S ( X , Y ) S(X,Y)
  61. Y Y
  62. 𝔈 = [ X ] > ω \mathfrak{E}=[X]^{>\omega}
  63. H ( A n ) = { A X | A ( n < ω E ¯ n ) } H(A_{n})=\{A\subset X\;|\;A\supset(\cap_{n<\omega}\bar{E}_{n})\}
  64. P 1 S ( X , J ) P_{1}\uparrow S(X,J)
  65. J X J\setminus X
  66. P 2 S ( X , J ) P_{2}\uparrow S(X,J)
  67. X X
  68. X X
  69. J J
  70. P 2 S ( X , J ) P_{2}\uparrow S(X,J)
  71. J X J\setminus X
  72. X X
  73. P 1 S ( X , J ) P_{1}\uparrow S(X,J)
  74. X X
  75. J J
  76. P 1 S ( X , J ) P_{1}\uparrow S(X,J)
  77. { X J | P 2 S ( X , J ) } \{X\subset J\;|\;P_{2}\uparrow S(X,J)\}
  78. P 1 P_{1}
  79. P 2 P_{2}

Topological_indistinguishability.html

  1. { f α : X Y α } \{f_{\alpha}:X\to Y_{\alpha}\}
  2. f α f_{\alpha}
  3. f α ( x ) = f α ( y ) f_{\alpha}(x)=f_{\alpha}(y)
  4. α \alpha
  5. \darr x = { y X : y x } = cl { x } \mathop{\darr}x=\{y\in X:y\leq x\}=\textrm{cl}\{x\}
  6. \uarr x = { y X : x y } = 𝒩 x . \mathop{\uarr}x=\{y\in X:x\leq y\}=\bigcap\mathcal{N}_{x}.
  7. [ x ] = \darr x \uarr x . [x]={\mathop{\darr}x}\cap{\mathop{\uarr}x}.
  8. [ x ] = cl { x } = 𝒩 x . [x]=\textrm{cl}\{x\}=\bigcap\mathcal{N}_{x}.
  9. f α : X Y α f_{\alpha}:X\to Y_{\alpha}

Toroidal_inductors_and_transformers.html

  1. b f A = 0 bf{A}=0
  2. ρ = 0 \rho=0\,
  3. 1 c 2 ϕ t \frac{1}{c^{2}}\frac{\partial\phi}{\partial t}
  4. × 𝐀 = 𝐁 \nabla\times\mathbf{A}=\mathbf{B}
  5. × 𝐁 = μ 0 𝐣 \nabla\times\mathbf{B}=\mu_{0}\mathbf{j}
  6. E t 0 \frac{\partial E}{\partial t}\rightarrow 0
  7. ϕ \phi\,
  8. 𝐁 = × 𝐀 . \mathbf{B}=\nabla\times\mathbf{A}.
  9. 𝐄 = - ϕ - 𝐀 t \mathbf{E}=-\nabla\phi-\frac{\partial\mathbf{A}}{\partial t}
  10. 𝐀 t \frac{\partial\mathbf{A}}{\partial t}
  11. ϕ \nabla\phi\,
  12. ϕ \nabla\phi\,
  13. 𝐄𝐌𝐅 = p a t h 𝐄 d l = - p a t h 𝐀 t d l = - t p a t h 𝐀 d l = - t s u r f a c e 𝐁 d s \mathbf{EMF}=\oint_{path}\mathbf{E}\cdot{\rm d}l=-\oint_{path}\frac{\partial% \mathbf{A}}{\partial t}\cdot{\rm d}l=-\frac{\partial}{\partial t}\oint_{path}% \mathbf{A}\cdot{\rm d}l=-\frac{\partial}{\partial t}\int_{surface}\mathbf{B}% \cdot{\rm d}s

Trace_diagram.html

  1. 𝒟 = ( V 1 V 2 V n , E ) \mathcal{D}=(V_{1}\sqcup V_{2}\sqcup V_{n},E)
  2. ( 𝐮 × 𝐯 ) 𝐰 = 𝐮 ( 𝐯 × 𝐰 ) = ( 𝐰 × 𝐮 ) 𝐯 = det ( 𝐮𝐯𝐰 ) . (\mathbf{u}\times\mathbf{v})\cdot\mathbf{w}=\mathbf{u}\cdot(\mathbf{v}\times% \mathbf{w})=(\mathbf{w}\times\mathbf{u})\cdot\mathbf{v}=\det(\mathbf{u}\mathbf% {v}\mathbf{w}).
  3. ( 𝐱 × 𝐮 ) ( 𝐯 × 𝐰 ) = ( 𝐱 𝐯 ) ( 𝐮 𝐰 ) - ( 𝐱 𝐰 ) ( 𝐮 𝐯 ) , (\mathbf{x}\times\mathbf{u})\cdot(\mathbf{v}\times\mathbf{w})=(\mathbf{x}\cdot% \mathbf{v})(\mathbf{u}\cdot\mathbf{w})-(\mathbf{x}\cdot\mathbf{w})(\mathbf{u}% \cdot\mathbf{v}),
  4. \propto
  5. G k G^{k}
  6. ( g 1 , , g k ) (g_{1},\ldots,g_{k})
  7. ( a g 1 a - 1 , , a g k a - 1 ) (ag_{1}a^{-1},\ldots,ag_{k}a^{-1})
  8. a G a\in G

Trace_identity.html

  1. tr ( A n ) - tr ( A ) tr ( A n - 1 ) + + ( - 1 ) n det ( A ) = 0. {\rm tr}(A^{n})-{\rm tr}(A){\rm tr}(A^{n-1})+\cdots+(-1)^{n}\det(A)=0.\,
  2. tr ( A ) = tr ( A T ) . {\rm tr}(A)={\rm tr}(A\text{T}).\,

Trans-feruloyl-CoA_hydratase.html

  1. \rightleftharpoons

Trans-feruloyl-CoA_synthase.html

  1. \rightleftharpoons

Trans-hexaprenyltranstransferase.html

  1. \rightleftharpoons

Trans-L-3-hydroxyproline_dehydratase.html

  1. \rightleftharpoons

Trans-octaprenyltranstransferase.html

  1. \rightleftharpoons

Trans-pentaprenyltranstransferase.html

  1. \rightleftharpoons

Trans-zeatin_O-beta-D-glucosyltransferase.html

  1. \rightleftharpoons

Trehalose-phosphatase.html

  1. \rightleftharpoons

Trehalose_6-phosphate_phosphorylase.html

  1. \rightleftharpoons

Trehalose_O-mycolyltransferase.html

  1. \rightleftharpoons

Triacetate-lactonase.html

  1. \rightleftharpoons

Triacylglycerol—sterol_O-acyltransferase.html

  1. \rightleftharpoons

Triadic_closure.html

  1. G = ( V , E ) G=(V,E)
  2. N = | V | N=|V|
  3. M = | E | M=|E|
  4. d i d_{i}
  5. i i
  6. j j
  7. k k
  8. i i
  9. δ ( i ) \delta(i)
  10. δ ( G ) = 1 3 i V δ ( i ) \delta(G)=\frac{1}{3}\sum_{i\in V}\ \delta(i)
  11. τ ( i ) = ( d i 2 ) \tau(i)={\left({{d_{i}}\atop{2}}\right)}
  12. d i 2 d_{i}\geq 2
  13. τ ( G ) = 1 3 i V τ ( i ) \tau(G)=\frac{1}{3}\sum_{i\in V}\ \tau(i)
  14. i i
  15. d i 2 d_{i}\geq 2
  16. c ( i ) c(i)
  17. i i
  18. i i
  19. δ ( i ) τ ( i ) \frac{\delta(i)}{\tau(i)}
  20. C ( G ) C(G)
  21. G G
  22. C ( G ) = 1 N 2 i V , d i 2 c ( i ) C(G)=\frac{1}{N_{2}}\sum_{i\in V,d_{i}\geq 2}c(i)
  23. N 2 N_{2}
  24. T ( G ) = 3 δ ( G ) τ ( G ) T(G)=\frac{3\delta(G)}{\tau(G)}

Trichodiene_synthase.html

  1. \rightleftharpoons

Tricubic_interpolation.html

  1. f ( x , y , z ) = i = 0 3 j = 0 3 k = 0 3 a i j k x i y j z k . f(x,y,z)=\sum_{i=0}^{3}\sum_{j=0}^{3}\sum_{k=0}^{3}a_{ijk}x^{i}y^{j}z^{k}.
  2. a i j k a_{ijk}
  3. CINT x ( a - 1 , a 0 , a 1 , a 2 ) \mathrm{CINT}_{x}(a_{-1},a_{0},a_{1},a_{2})
  4. a - 1 a_{-1}
  5. a 0 a_{0}
  6. a 1 a_{1}
  7. a 2 a_{2}
  8. x x
  9. CINT x ( u - 1 , u 0 , u 1 , u 2 ) = 𝐯 x ( u - 1 , u 0 , u 1 , u 2 ) \mathrm{CINT}_{x}(u_{-1},u_{0},u_{1},u_{2})=\mathbf{v}_{x}\cdot\left(u_{-1},u_% {0},u_{1},u_{2}\right)
  10. 𝐯 x \mathbf{v}_{x}
  11. x x
  12. s ( i , j , k ) \displaystyle s(i,j,k)
  13. i , j , k { - 1 , 0 , 1 , 2 } i,j,k\in\{-1,0,1,2\}
  14. x , y , z [ 0 , 1 ] x,y,z\in[0,1]
  15. CINT \mathrm{CINT}
  16. 64 × 64 {64\times 64}
  17. 64 × 64 {64\times 64}
  18. CINT x \mathrm{CINT}_{x}

Trihydroxystilbene_synthase.html

  1. \rightleftharpoons

Trimetaphosphatase.html

  1. \rightleftharpoons

Trimethylamine-oxide_aldolase.html

  1. \rightleftharpoons

Triokinase.html

  1. \rightleftharpoons

Triphosphatase.html

  1. \rightleftharpoons

Triphosphate—protein_phosphotransferase.html

  1. \rightleftharpoons

Triphosphoribosyl-dephospho-CoA_synthase.html

  1. \rightleftharpoons

TRNA-queuosine_beta-mannosyltransferase.html

  1. \rightleftharpoons

TRNA-uridine_aminocarboxypropyltransferase.html

  1. \rightleftharpoons

TRNA_isopentenyltransferase.html

  1. \rightleftharpoons

TRNA_nucleotidyltransferase.html

  1. \rightleftharpoons

Tropinesterase.html

  1. \rightleftharpoons

Tropomyosin_kinase.html

  1. \rightleftharpoons

Trypanothione_synthase.html

  1. \rightleftharpoons

Tryptophan_dimethylallyltransferase.html

  1. \rightleftharpoons

Tryptophan_transaminase.html

  1. \rightleftharpoons

Tryptophanamidase.html

  1. \rightleftharpoons

Tryptophanase.html

  1. \rightleftharpoons

Tryptophan—phenylpyruvate_transaminase.html

  1. \rightleftharpoons

Tryptophan—tRNA_ligase.html

  1. \rightleftharpoons

Tubulin—tyrosine_ligase.html

  1. \rightleftharpoons

Tukey's_range_test.html

  1. μ i - μ j \mu_{i}-\mu_{j}\,
  2. q s = Y A - Y B S E , q_{s}=\frac{Y_{A}-Y_{B}}{SE},
  3. y ¯ \bar{y}
  4. y ¯ \bar{y}
  5. q = ( y ¯ m a x - y ¯ m i n ) S 2 / n q=\frac{(\overline{y}_{max}-\overline{y}_{min})}{S\sqrt{2/n}}
  6. 2 \sqrt{2}
  7. y ¯ i - y ¯ j ± q α ; k ; N - k 2 σ ^ ε 2 n i , j = 1 , , k i j . \bar{y}_{i\bullet}-\bar{y}_{j\bullet}\pm\frac{q_{\alpha;k;N-k}}{\sqrt{2}}% \widehat{\sigma}_{\varepsilon}\sqrt{\frac{2}{n}}\qquad i,j=1,\ldots,k\quad i% \neq j.
  8. σ ^ ε \widehat{\sigma}_{\varepsilon}
  9. y ¯ i - y ¯ j ± q α ; k ; N - k 2 σ ^ ε 1 n i + 1 n j \bar{y}_{i\bullet}-\bar{y}_{j\bullet}\pm\frac{q_{\alpha;k;N-k}}{\sqrt{2}}% \widehat{\sigma}_{\varepsilon}\sqrt{\frac{1}{n}_{i}+\frac{1}{n}_{j}}\qquad

Turán's_inequalities.html

  1. P n ( x ) 2 > P n - 1 ( x ) P n + 1 ( x ) for - 1 < x < 1. \,\!P_{n}(x)^{2}>P_{n-1}(x)P_{n+1}(x)\,\text{ for }-1<x<1.
  2. H n ( x ) 2 - H n - 1 ( x ) H n + 1 ( x ) = ( n - 1 ) ! i = 0 n - 1 2 n - i i ! H i ( x ) 2 > 0 H_{n}(x)^{2}-H_{n-1}(x)H_{n+1}(x)=(n-1)!\cdot\sum_{i=0}^{n-1}\frac{2^{n-i}}{i!% }H_{i}(x)^{2}>0
  3. T n ( x ) 2 - T n - 1 ( x ) T n + 1 ( x ) = 1 - x 2 > 0 for - 1 < x < 1. \!T_{n}(x)^{2}-T_{n-1}(x)T_{n+1}(x)=1-x^{2}>0\,\text{ for }-1<x<1.

Turán_number.html

  1. T ( n , k , r ) ( n r ) ( k r ) - 1 . T(n,k,r)\geq{\left({{n}\atop{r}}\right)}{{\left({{k}\atop{r}}\right)}}^{-1}.

Tweedie_distribution.html

  1. var ( Y ) = a [ E ( Y ) ] p , \,\text{var}\,(Y)=a[\,\text{E}\,(Y)]^{p},
  2. P λ , θ ( Z A ) = A exp [ θ z - λ κ ( θ ) ] ν λ ( d z ) , P_{\lambda,\theta}(Z\in A)=\int_{A}\exp[\theta\cdot z-\lambda\kappa(\theta)]% \cdot\nu_{\lambda}\,(dz),
  3. κ ( θ ) = λ - 1 log e θ z ν λ ( d z ) ; \kappa(\theta)=\lambda^{-1}\log\int e^{\theta z}\cdot\nu_{\lambda}\,(dz);
  4. Z + = Z 1 + + Z n , Z_{+}=Z_{1}+\cdots+Z_{n},
  5. Z + E D * ( θ , λ 1 + + λ n ) . Z_{+}\sim ED^{*}(\theta,\lambda_{1}+\cdots+\lambda_{n}).
  6. Y = Z / λ E D ( μ , σ 2 ) , Y=Z/\lambda\sim ED(\mu,\sigma^{2}),
  7. w = i = 1 n w i , w=\sum_{i=1}^{n}w_{i},
  8. w - 1 i = 1 n w i Y i E D ( μ , σ 2 / w ) . w^{-1}\sum_{i=1}^{n}w_{i}Y_{i}\sim ED(\mu,\sigma^{2}/w).
  9. Y Z = Y / σ 2 . Y\mapsto Z=Y/\sigma^{2}.
  10. c E D ( μ , σ 2 ) = E D ( c μ , c 2 - p σ 2 ) , cED(\mu,\sigma^{2})=ED(c\mu,c^{2-p}\sigma^{2}),
  11. τ ( θ ) = κ ( θ ) = μ . \tau(\theta)=\kappa^{\prime}(\theta)=\mu.
  12. V ( μ ) = τ [ τ - 1 ( μ ) ] . V(\mu)=\tau^{\prime}[\tau^{-1}(\mu)].
  13. τ - 1 ( μ ) μ = 1 V ( μ ) . \frac{\partial\tau^{-1}(\mu)}{\partial\mu}=\frac{1}{V(\mu)}.
  14. κ ( θ ) θ = τ ( θ ) . \frac{\partial\kappa(\theta)}{\partial\theta}=\tau(\theta).
  15. K * ( s ) = log [ E ( e s Z ) ] = λ [ κ ( θ + s ) - κ ( θ ) ] , K^{*}(s)=\log[\,\text{E}(e^{sZ})]=\lambda[\kappa(\theta+s)-\kappa(\theta)],
  16. K ( s ) = log [ E ( e s Y ) ] = λ [ κ ( θ + s / λ ) - κ ( θ ) ] , K(s)=\log[\,\text{E}(e^{sY})]=\lambda[\kappa(\theta+s/\lambda)-\kappa(\theta)],
  17. κ p ( θ ) = { α - 1 α ( θ α - 1 ) α p 1 , 2 , - log ( - θ ) p = 2 , e θ p = 1 , \kappa_{p}(\theta)=\begin{cases}\dfrac{\alpha-1}{\alpha}\left(\dfrac{\theta}{% \alpha-1}\right)^{\alpha}&\quad p\neq 1,2,\\ -\log(-\theta)&\quad p=2,\\ e^{\theta}&\quad p=1,\end{cases}
  18. α = p - 2 p - 1 . \alpha=\dfrac{p-2}{p-1}.
  19. K p * ( s ; θ , λ ) = { λ κ p ( θ ) [ ( 1 + s / θ ) α - 1 ] p 1 , 2 , - λ log ( 1 + s / θ ) p = 2 , λ e θ ( e s - 1 ) p = 1 , K^{*}_{p}(s;\theta,\lambda)=\begin{cases}\lambda\kappa_{p}(\theta)[(1+s/\theta% )^{\alpha}-1]&\quad p\neq 1,2,\\ -\lambda\log(1+s/\theta)&\quad p=2,\\ \lambda e^{\theta}(e^{s}-1)&\quad p=1,\end{cases}
  20. K p ( s ; θ , λ ) = { λ κ p ( θ ) { [ 1 + s / ( θ λ ) ] α - 1 } p 1 , 2 , - λ log [ 1 + s / ( θ λ ) ] p = 2 , λ e θ ( e s / λ - 1 ) p = 1. K_{p}(s;\theta,\lambda)=\begin{cases}\lambda\kappa_{p}(\theta)\left\{[1+s/(% \theta\lambda)]^{\alpha}-1\right\}&\quad p\neq 1,2,\\ -\lambda\log[1+s/(\theta\lambda)]&\quad p=2,\\ \lambda e^{\theta}(e^{s/\lambda}-1)&\quad p=1.\end{cases}
  21. var ( Z ) E ( Z ) p . \mathrm{var}(Z)\propto\mathrm{E}(Z)^{p}.
  22. p ( 0 , 1 ) , p\notin(0,1),
  23. μ > 0 \mu>0
  24. σ 2 > 0 \sigma^{2}>0
  25. c - 1 E D ( c μ , σ 2 c 2 - p ) T w p ( μ , c 0 σ 2 ) c^{-1}ED(c\mu,\sigma^{2}c^{2-p})\rightarrow Tw_{p}(\mu,c_{0}\sigma^{2})
  26. c 0 c\downarrow 0
  27. c c\rightarrow\infty
  28. var ( Y ) = a μ p , \,\text{var}\,(Y)=a\mu^{p},
  29. S ( f ) 1 / f γ , S(f)\propto 1/f^{\gamma},
  30. Y = ( Y i : i = 0 , 1 , 2 , , N ) Y=(Y_{i}:i=0,1,2,\ldots,N)
  31. μ ^ = E ( Y i ) , \hat{\mu}=\,\text{E}(Y_{i}),
  32. y i = Y i - μ ^ , y_{i}=Y_{i}-\hat{\mu},
  33. σ ^ 2 = E ( y i 2 ) , \hat{\sigma}^{2}=\,\text{E}(y_{i}^{2}),
  34. r ( k ) = E ( y i , y i + k ) / E ( y i 2 ) r(k)=\,\text{E}(y_{i},y_{i+k})/\,\text{E}(y_{i}^{2})
  35. r ( k ) k - d L ( k ) r(k)\sim k^{-d}L(k)
  36. Y i ( m ) = ( Y i m - m + 1 + + Y i m ) / m . Y_{i}^{(m)}=(Y_{im-m+1}+\cdots+Y_{im})/m.
  37. var [ Y ( m ) ] = σ ^ 2 m - d \,\text{var}[Y^{(m)}]=\hat{\sigma}^{2}m^{-d}
  38. lim k r ( k ) / k - d = ( 2 - d ) ( 1 - d ) / 2. \lim_{k\to\infty}r(k)/k^{-d}=(2-d)(1-d)/2.
  39. Z i ( m ) = m Y i ( m ) , Z_{i}^{(m)}=mY_{i}^{(m)},
  40. Z i ( m ) = ( Y i m - m + 1 + + Y i m ) . Z_{i}^{(m)}=(Y_{im-m+1}+\cdots+Y_{im}).
  41. var [ Z i ( m ) ] = m 2 var [ Y ( m ) ] = ( σ ^ 2 / μ ^ 2 - d ) E [ Z i ( m ) ] 2 - d \,\text{var}[Z_{i}^{(m)}]=m^{2}\,\text{var}[Y^{(m)}]=(\hat{\sigma}^{2}/\hat{% \mu}^{2-d})\,\text{E}[Z_{i}^{(m)}]^{2-d}
  42. μ ^ \hat{\mu}
  43. σ ^ 2 \hat{\sigma}^{2}
  44. D = 2 - H = 2 - p / 2. D=2-H=2-p/2.
  45. R D ( m ) = R D ( m ref ) ( m m ref ) 1 - D s RD(m)=RD(m\text{ref})\left(\frac{m}{m\text{ref}}\right)^{1-D_{s}}
  46. G ( s ) = exp [ λ α - 1 α ( θ α - 1 ) α { ( 1 - 1 θ + s θ ) α - 1 } ] G(s)=\exp\left[\lambda\frac{\alpha-1}{\alpha}\left(\frac{\theta}{\alpha-1}% \right)^{\alpha}\left\{\left(1-\frac{1}{\theta}+\frac{s}{\theta}\right)^{% \alpha}-1\right\}\right]
  47. var ( Y ) = a E ( Y ) b + E ( Y ) , \,\text{var}\,(Y)=a\,\text{E}(Y)^{b}+\,\text{E}(Y),
  48. ρ ¯ ( E ) = { 2 N - E 2 / π | E | < 2 N 0 | E | > 2 N \bar{\rho}(E)=\begin{cases}\sqrt{2N-E^{2}}/\pi&\quad\left|E\right|<\sqrt{2N}\\ 0&\quad\left|E\right|>\sqrt{2N}\end{cases}
  49. η ¯ ( E ) = 1 2 π [ E 2 N - E 2 + 2 N arcsin ( E 2 N ) + π N ] . \bar{\eta}(E)=\frac{1}{2\pi}\left[E\sqrt{2N-E^{2}}+2N\arcsin\left(\frac{E}{% \sqrt{2N}}\right)+\pi N\right].
  50. e n = η ¯ ( E ) = - E n d E ρ ¯ ( E ) . e_{n}=\bar{\eta}(E)=\int\limits_{-\infty}^{E_{n}}dE^{\prime}\bar{\rho}(E^{% \prime}).
  51. | D ¯ n | = | n - η ¯ ( E n ) | \left|\bar{D}_{n}\right|=\left|n-\bar{\eta}(E_{n})\right|
  52. ψ ( x ) = p ^ k x log p ^ = n x Λ ( n ) \psi(x)=\sum_{\hat{p}^{k}\leq x}\log\hat{p}=\sum_{n\leq x}\Lambda(n)
  53. p ^ k \hat{p}^{k}
  54. Λ ( n ) \Lambda(n)
  55. ψ 0 ( x ) = x - ρ x ρ ρ - ln 2 π - 1 2 ln ( 1 - x - 2 ) \psi_{0}(x)=x-\sum_{\rho}\frac{x^{\rho}}{\rho}-\ln 2\pi-\frac{1}{2}\ln(1-x^{-2})
  56. ψ 0 ( x ) = lim ε 0 ψ ( x - ε ) + ψ ( x + ε ) 2 . \psi_{0}(x)=\lim_{\varepsilon\rightarrow 0}\frac{\psi(x-\varepsilon)+\psi(x+% \varepsilon)}{2}.
  57. Δ ( x ) = | ψ ( x ) - x | < x log 2 ( x ) / ( 8 π ) \Delta(x)=\left|\psi(x)-x\right|<\sqrt{x}\log^{2}(x)/(8\pi)
  58. x > 73.2 x>73.2

Two-dimensional_graph.html

  1. y = f ( x ) y=f(x)
  2. f ( x ) = x 3 - 9 x f(x)={{x^{3}}-9x}\!
  3. y = f ( x ) y=f(x)
  4. ( x - a ) 2 + ( y - b ) 2 = 1. (x-a)^{2}+(y-b)^{2}=1.

Two-dimensional_space.html

  1. A = π r 2 A=\pi r^{2}
  2. r r
  3. 𝐀 𝐁 = A 1 B 1 + A 2 B 2 \mathbf{A}\cdot\mathbf{B}=A_{1}B_{1}+A_{2}B_{2}
  4. 𝐀 \|\mathbf{A}\|
  5. 𝐀 𝐁 = 𝐀 𝐁 cos θ , \mathbf{A}\cdot\mathbf{B}=\|\mathbf{A}\|\,\|\mathbf{B}\|\cos\theta,
  6. 𝐀 𝐀 = 𝐀 2 , \mathbf{A}\cdot\mathbf{A}=\|\mathbf{A}\|^{2},
  7. 𝐀 = 𝐀 𝐀 , \|\mathbf{A}\|=\sqrt{\mathbf{A}\cdot\mathbf{A}},
  8. f = f x 𝐢 + f y 𝐣 \nabla f=\frac{\partial f}{\partial x}\mathbf{i}+\frac{\partial f}{\partial y}% \mathbf{j}
  9. C f d s = a b f ( 𝐫 ( t ) ) | 𝐫 ( t ) | d t . \int\limits_{C}f\,ds=\int_{a}^{b}f(\mathbf{r}(t))|\mathbf{r}^{\prime}(t)|\,dt.
  10. a < b a<b
  11. C 𝐅 ( 𝐫 ) d 𝐫 = a b 𝐅 ( 𝐫 ( t ) ) 𝐫 ( t ) d t . \int\limits_{C}\mathbf{F}(\mathbf{r})\cdot\,d\mathbf{r}=\int_{a}^{b}\mathbf{F}% (\mathbf{r}(t))\cdot\mathbf{r}^{\prime}(t)\,dt.
  12. f ( x , y ) , f(x,y),
  13. D f ( x , y ) d x d y . \iint\limits_{D}f(x,y)\,dx\,dy.
  14. φ : U 2 \varphi:U\subseteq\mathbb{R}^{2}\to\mathbb{R}
  15. φ ( 𝐪 ) - φ ( 𝐩 ) = γ [ 𝐩 , 𝐪 ] φ ( 𝐫 ) d 𝐫 . \varphi\left(\mathbf{q}\right)-\varphi\left(\mathbf{p}\right)=\int_{\gamma[% \mathbf{p},\,\mathbf{q}]}\nabla\varphi(\mathbf{r})\cdot d\mathbf{r}.
  16. C ( L d x + M d y ) = D ( M x - L y ) d x d y \oint_{C}(L\,dx+M\,dy)=\iint_{D}\left(\frac{\partial M}{\partial x}-\frac{% \partial L}{\partial y}\right)\,dx\,dy

Type_inhabitation.html

  1. τ \tau
  2. Γ \Gamma
  3. λ \lambda
  4. Γ M : τ \Gamma\vdash M:\tau
  5. τ \tau

Tyramine_N-feruloyltransferase.html

  1. \rightleftharpoons

Tyrosine_decarboxylase.html

  1. \rightleftharpoons

Tyrosine_phenol-lyase.html

  1. \rightleftharpoons

Tyrosine—arginine_ligase.html

  1. \rightleftharpoons

Tyrosine—tRNA_ligase.html

  1. \rightleftharpoons

Ubiquitin—calmodulin_ligase.html

  1. \rightleftharpoons

UDP-2-acetamido-4-amino-2,4,6-trideoxyglucose_transaminase.html

  1. \rightleftharpoons

UDP-galactose—UDP-N-acetylglucosamine_galactose_phosphotransferase.html

  1. \rightleftharpoons

UDP-galacturonate_decarboxylase.html

  1. \rightleftharpoons

UDP-glucose_4,6-dehydratase.html

  1. \rightleftharpoons

UDP-glucose—glycoprotein_glucose_phosphotransferase.html

  1. \rightleftharpoons

UDP-glucose—hexose-1-phosphate_uridylyltransferase.html

  1. \rightleftharpoons

UDP-glucuronate_decarboxylase.html

  1. \rightleftharpoons

UDP-N-acetylglucosamine_1-carboxyvinyltransferase.html

  1. \rightleftharpoons

UDP-N-acetylglucosamine_diphosphorylase.html

  1. \rightleftharpoons

UDP-N-acetylglucosamine—dolichyl-phosphate_N-acetylglucosaminephosphotransferase.html

  1. \rightleftharpoons

UDP-N-acetylglucosamine—lysosomal-enzyme_N-acetylglucosaminephosphotransferase.html

  1. \rightleftharpoons

UDP-N-acetylmuramate—L-alanine_ligase.html

  1. \rightleftharpoons

UDP-N-acetylmuramoyl-L-alanine—D-glutamate_ligase.html

  1. \rightleftharpoons

UDP-N-acetylmuramoyl-L-alanyl-D-glutamate—L-lysine_ligase.html

  1. \rightleftharpoons

UDP-N-acetylmuramoyl-tripeptide—D-alanyl-D-alanine_ligase.html

  1. \rightleftharpoons

UDP-N-acetylmuramoylpentapeptide-lysine_N6-alanyltransferase.html

  1. \rightleftharpoons

UDP-sugar_diphosphatase.html

  1. \rightleftharpoons

Ultrahyperbolic_equation.html

  1. 2 u x 1 2 + + 2 u x n 2 - 2 u y 1 2 - - 2 u y n 2 = 0. ( 1 ) \frac{\partial^{2}u}{\partial x_{1}^{2}}+\cdots+\frac{\partial^{2}u}{\partial x% _{n}^{2}}-\frac{\partial^{2}u}{\partial y_{1}^{2}}-\cdots-\frac{\partial^{2}u}% {\partial y_{n}^{2}}=0.\qquad\qquad(1)
  2. a i j u x i x j a_{ij}u_{x_{i}x_{j}}

UMP_kinase.html

  1. \rightleftharpoons

Unbiased_rendering.html

  1. n n
  2. n \sqrt{n}
  3. n n

Undecaprenol_kinase.html

  1. \rightleftharpoons

Undecaprenyl-diphosphatase.html

  1. \rightleftharpoons

Undecaprenyl-phosphate_galactose_phosphotransferase.html

  1. \rightleftharpoons

Undecaprenyl-phosphate_mannosyltransferase.html

  1. \rightleftharpoons

Undecaprenyldiphospho-muramoylpentapeptide_beta-N-acetylglucosaminyltransferase.html

  1. \rightleftharpoons

Unger_model.html

  1. | H N E X T ( f ) | 2 |H_{NEXT}(f)|^{2}
  2. 10 log ( | H N E X T ( f ) | 2 ) = { - 66 + 6 log ( f ) d B f < 20 K H z - 50.5 + 15 log ( f ) d B f 20 K H z 10\log(|H_{NEXT}(f)|^{2})=\begin{cases}-66+6\log(f)dB&f<20KHz\\ -50.5+15\log(f)dB&f>=20KHz\end{cases}
  3. 10 log ( | H N E X T ( f ) | 2 ) = { - 59.2 + 4 log ( f ) d B f < 20 K H z - 42.2 + 14 log ( f ) d B f 20 K H z 10\log(|H_{NEXT}(f)|^{2})=\begin{cases}-59.2+4\log(f)dB&f<20KHz\\ -42.2+14\log(f)dB&f>=20KHz\end{cases}

Uniform_5-polytope.html

  1. A ~ 4 {\tilde{A}}_{4}
  2. C ~ 4 {\tilde{C}}_{4}
  3. B ~ 4 {\tilde{B}}_{4}
  4. D ~ 4 {\tilde{D}}_{4}
  5. F ~ 4 {\tilde{F}}_{4}
  6. A ~ 4 {\tilde{A}}_{4}
  7. D ~ 4 {\tilde{D}}_{4}
  8. C ~ 3 {\tilde{C}}_{3}
  9. I ~ 1 {\tilde{I}}_{1}
  10. B ~ 3 {\tilde{B}}_{3}
  11. I ~ 1 {\tilde{I}}_{1}
  12. A ~ 3 {\tilde{A}}_{3}
  13. I ~ 1 {\tilde{I}}_{1}
  14. C ~ 2 {\tilde{C}}_{2}
  15. I ~ 1 {\tilde{I}}_{1}
  16. I ~ 1 {\tilde{I}}_{1}
  17. H ~ 2 {\tilde{H}}_{2}
  18. I ~ 1 {\tilde{I}}_{1}
  19. I ~ 1 {\tilde{I}}_{1}
  20. A ~ 2 {\tilde{A}}_{2}
  21. I ~ 1 {\tilde{I}}_{1}
  22. I ~ 1 {\tilde{I}}_{1}
  23. I ~ 1 {\tilde{I}}_{1}
  24. I ~ 1 {\tilde{I}}_{1}
  25. I ~ 1 {\tilde{I}}_{1}
  26. I ~ 1 {\tilde{I}}_{1}
  27. A ~ 2 {\tilde{A}}_{2}
  28. A ~ 2 {\tilde{A}}_{2}
  29. A ~ 2 {\tilde{A}}_{2}
  30. B ~ 2 {\tilde{B}}_{2}
  31. A ~ 2 {\tilde{A}}_{2}
  32. G ~ 2 {\tilde{G}}_{2}
  33. B ~ 2 {\tilde{B}}_{2}
  34. B ~ 2 {\tilde{B}}_{2}
  35. B ~ 2 {\tilde{B}}_{2}
  36. G ~ 2 {\tilde{G}}_{2}
  37. G ~ 2 {\tilde{G}}_{2}
  38. G ~ 2 {\tilde{G}}_{2}
  39. A F ^ 4 {\widehat{AF}}_{4}
  40. D H ¯ 4 {\bar{DH}}_{4}
  41. H ¯ 4 {\bar{H}}_{4}
  42. B H ¯ 4 {\bar{BH}}_{4}
  43. K ¯ 4 {\bar{K}}_{4}
  44. P ¯ 4 {\bar{P}}_{4}
  45. B P ¯ 4 {\bar{BP}}_{4}
  46. F R ¯ 4 {\bar{FR}}_{4}
  47. D P ¯ 4 {\bar{DP}}_{4}
  48. N ¯ 4 {\bar{N}}_{4}
  49. O ¯ 4 {\bar{O}}_{4}
  50. S ¯ 4 {\bar{S}}_{4}
  51. M ¯ 4 {\bar{M}}_{4}
  52. R ¯ 4 {\bar{R}}_{4}

Uniformly_most_powerful_test.html

  1. X X
  2. f θ ( x ) f_{\theta}(x)
  3. θ Θ \theta\in\Theta
  4. Θ \Theta
  5. Θ 0 \Theta_{0}
  6. Θ 1 \Theta_{1}
  7. H 0 H_{0}
  8. θ Θ 0 \theta\in\Theta_{0}
  9. H 1 H_{1}
  10. θ Θ 1 \theta\in\Theta_{1}
  11. ϕ ( x ) \phi(x)
  12. ϕ ( x ) = { 1 if x R 0 if x A \phi(x)=\begin{cases}1&\,\text{if }x\in R\\ 0&\,\text{if }x\in A\end{cases}
  13. H 1 H_{1}
  14. X R X\in R
  15. H 0 H_{0}
  16. X A X\in A
  17. A R A\cup R
  18. ϕ ( x ) \phi(x)
  19. α \alpha
  20. ϕ ( x ) \phi^{\prime}(x)
  21. sup θ Θ 0 E θ ϕ ( X ) = α α = sup θ Θ 0 E θ ϕ ( X ) \sup_{\theta\in\Theta_{0}}\;\operatorname{E}_{\theta}\phi^{\prime}(X)=\alpha^{% \prime}\leq\alpha=\sup_{\theta\in\Theta_{0}}\;\operatorname{E}_{\theta}\phi(X)\,
  22. E θ ϕ ( X ) = 1 - β 1 - β = E θ ϕ ( X ) θ Θ 1 . \operatorname{E}_{\theta}\phi^{\prime}(X)=1-\beta^{\prime}\leq 1-\beta=% \operatorname{E}_{\theta}\phi(X)\quad\forall\theta\in\Theta_{1}.
  23. l ( x ) = f θ 1 ( x ) / f θ 0 ( x ) l(x)=f_{\theta_{1}}(x)/f_{\theta_{0}}(x)
  24. l ( x ) l(x)
  25. x x
  26. θ 1 θ 0 \theta_{1}\geq\theta_{0}
  27. x x
  28. H 1 H_{1}
  29. ϕ ( x ) = { 1 if x > x 0 0 if x < x 0 \phi(x)=\begin{cases}1&\,\text{if }x>x_{0}\\ 0&\,\text{if }x<x_{0}\end{cases}
  30. x 0 x_{0}
  31. E θ 0 ϕ ( X ) = α \operatorname{E}_{\theta_{0}}\phi(X)=\alpha
  32. H 0 : θ θ 0 vs. H 1 : θ > θ 0 . H_{0}:\theta\leq\theta_{0}\,\text{ vs. }H_{1}:\theta>\theta_{0}.
  33. H 0 : θ = θ 0 vs. H 1 : θ > θ 0 . H_{0}:\theta=\theta_{0}\,\text{ vs. }H_{1}:\theta>\theta_{0}.
  34. f θ ( x ) = c ( θ ) h ( x ) exp ( π ( θ ) T ( x ) ) f_{\theta}(x)=c(\theta)h(x)\exp(\pi(\theta)T(x))
  35. π ( θ ) \pi(\theta)
  36. X = ( X 0 , X 1 , , X M - 1 ) X=(X_{0},X_{1},\dots,X_{M-1})
  37. N N
  38. θ m \theta m
  39. R R
  40. f θ ( X ) = ( 2 π ) - M N / 2 | R | - M / 2 exp { - 1 2 n = 0 M - 1 ( X n - θ m ) T R - 1 ( X n - θ m ) } = f_{\theta}(X)=(2\pi)^{-MN/2}|R|^{-M/2}\exp\left\{-\frac{1}{2}\sum_{n=0}^{M-1}(% X_{n}-\theta m)^{T}R^{-1}(X_{n}-\theta m)\right\}=
  41. = ( 2 π ) - M N / 2 | R | - M / 2 exp { - 1 2 n = 0 M - 1 ( θ 2 m T R - 1 m ) } =(2\pi)^{-MN/2}|R|^{-M/2}\exp\left\{-\frac{1}{2}\sum_{n=0}^{M-1}(\theta^{2}m^{% T}R^{-1}m)\right\}
  42. exp { - 1 2 n = 0 M - 1 X n T R - 1 X n } exp { θ m T R - 1 n = 0 M - 1 X n } \exp\left\{-\frac{1}{2}\sum_{n=0}^{M-1}X_{n}^{T}R^{-1}X_{n}\right\}\exp\left\{% \theta m^{T}R^{-1}\sum_{n=0}^{M-1}X_{n}\right\}
  43. T ( X ) = m T R - 1 n = 0 M - 1 X n . T(X)=m^{T}R^{-1}\sum_{n=0}^{M-1}X_{n}.
  44. ϕ ( T ) = { 1 if T > t 0 0 if T < t 0 \phi(T)=\begin{cases}1&\,\text{if }T>t_{0}\\ 0&\,\text{if }T<t_{0}\end{cases}
  45. E θ 0 ϕ ( T ) = α \operatorname{E}_{\theta_{0}}\phi(T)=\alpha
  46. α \alpha
  47. H 0 : θ θ 0 H_{0}:\theta\leq\theta_{0}
  48. H 1 : θ > θ 0 H_{1}:\theta>\theta_{0}
  49. θ 1 \theta_{1}
  50. θ 1 > θ 0 \theta_{1}>\theta_{0}
  51. θ 2 \theta_{2}
  52. θ 2 < θ 0 \theta_{2}<\theta_{0}

Universal_composability.html

  1. P 1 P_{1}
  2. P 2 P_{2}
  3. P 1 P_{1}
  4. P 2 P_{2}
  5. P 1 P_{1}
  6. 𝒜 \mathcal{A}
  7. 𝖠𝗎𝗍𝗁 \mathcal{F}_{\mathsf{Auth}}
  8. m m
  9. P P
  10. P P
  11. 𝖠𝗎𝗍𝗁 \mathcal{F}_{\mathsf{Auth}}
  12. 𝒜 \mathcal{A}
  13. m , P m,P
  14. 𝖲𝖾𝖼 \mathcal{F}_{\mathsf{Sec}}
  15. 𝖲𝖾𝖼 \mathcal{F}_{\mathsf{Sec}}
  16. 𝖠𝗎𝗍𝗁 \mathcal{F}_{\mathsf{Auth}}
  17. 𝖠𝗇𝗈𝗇 \mathcal{F}_{\mathsf{Anon}}
  18. m m
  19. P P
  20. P P
  21. 𝖯𝗌𝖾𝗎 \mathcal{F}_{\mathsf{Pseu}}
  22. 𝖯𝗌𝖾𝗎 \mathcal{F}_{\mathsf{Pseu}}
  23. m m
  24. n y m nym
  25. m , n y m m,nym

Universal_Soil_Loss_Equation.html

  1. A = R K L S C P A=RKLSCP
  2. K = A / R K=A/R

Upwind_scheme.html

  1. u t + a u x = 0 \qquad\frac{\partial u}{\partial t}+a\frac{\partial u}{\partial x}=0
  2. x x
  3. a a
  4. i i
  5. i i
  6. a a
  7. i i
  8. a a
  9. u / x \partial u/\partial x
  10. ( 1 ) u i n + 1 - u i n Δ t + a u i n - u i - 1 n Δ x = 0 for a > 0 \quad(1)\qquad\frac{u_{i}^{n+1}-u_{i}^{n}}{\Delta t}+a\frac{u_{i}^{n}-u_{i-1}^% {n}}{\Delta x}=0\quad\,\text{for}\quad a>0
  11. ( 2 ) u i n + 1 - u i n Δ t + a u i + 1 n - u i n Δ x = 0 for a < 0 \quad(2)\qquad\frac{u_{i}^{n+1}-u_{i}^{n}}{\Delta t}+a\frac{u_{i+1}^{n}-u_{i}^% {n}}{\Delta x}=0\quad\,\text{for}\quad a<0
  12. a + = max ( a , 0 ) , a - = min ( a , 0 ) \qquad\qquad a^{+}=\,\text{max}(a,0)\,,\qquad a^{-}=\,\text{min}(a,0)
  13. u x - = u i n - u i - 1 n Δ x , u x + = u i + 1 n - u i n Δ x \qquad\qquad u_{x}^{-}=\frac{u_{i}^{n}-u_{i-1}^{n}}{\Delta x}\,,\qquad u_{x}^{% +}=\frac{u_{i+1}^{n}-u_{i}^{n}}{\Delta x}
  14. ( 3 ) u i n + 1 = u i n - Δ t [ a + u x - + a - u x + ] \quad(3)\qquad u_{i}^{n+1}=u_{i}^{n}-\Delta t\left[a^{+}u_{x}^{-}+a^{-}u_{x}^{% +}\right]
  15. c = | a Δ t Δ x | 1. \qquad\qquad c=\left|\frac{a\Delta t}{\Delta x}\right|\leq 1.
  16. u x - u_{x}^{-}
  17. u x - = 3 u i n - 4 u i - 1 n + u i - 2 n 2 Δ x \qquad\qquad u_{x}^{-}=\frac{3u_{i}^{n}-4u_{i-1}^{n}+u_{i-2}^{n}}{2\Delta x}
  18. u x + u_{x}^{+}
  19. u x + = - u i + 2 n + 4 u i + 1 n - 3 u i n 2 Δ x \qquad\qquad u_{x}^{+}=\frac{-u_{i+2}^{n}+4u_{i+1}^{n}-3u_{i}^{n}}{2\Delta x}
  20. u x - u_{x}^{-}
  21. u x - = 2 u i + 1 + 3 u i - 6 u i - 1 + u i - 2 6 Δ x \qquad\qquad u_{x}^{-}=\frac{2u_{i+1}+3u_{i}-6u_{i-1}+u_{i-2}}{6\Delta x}
  22. u x + u_{x}^{+}
  23. u x + = - u i + 2 + 6 u i + 1 - 3 u i - 2 u i - 1 6 Δ x \qquad\qquad u_{x}^{+}=\frac{-u_{i+2}+6u_{i+1}-3u_{i}-2u_{i-1}}{6\Delta x}

Uracil-5-carboxylate_decarboxylase.html

  1. \rightleftharpoons

Uracilylalanine_synthase.html

  1. \rightleftharpoons

Urate-ribonucleotide_phosphorylase.html

  1. \rightleftharpoons

Urea_carboxylase.html

  1. \rightleftharpoons

Ureidoglycolate_hydrolase.html

  1. \rightleftharpoons

Ureidosuccinase.html

  1. \rightleftharpoons

Urethanase.html

  1. \rightleftharpoons

Uridine_kinase.html

  1. \rightleftharpoons

Uridine_nucleosidase.html

  1. \rightleftharpoons

Uridine_phosphorylase.html

  1. \rightleftharpoons

Uronolactonase.html

  1. \rightleftharpoons

UTP-monosaccharide-1-phosphate_uridylyltransferase.html

  1. \rightleftharpoons

UTP—hexose-1-phosphate_uridylyltransferase.html

  1. \rightleftharpoons

UTP—xylose-1-phosphate_uridylyltransferase.html

  1. \rightleftharpoons

UVW_mapping.html

  1. 2 \mathbb{R}^{2}
  2. 3 \mathbb{R}^{3}
  3. 2 \mathbb{R}^{2}
  4. 2 \mathbb{R}^{2}

Vague_topology.html

  1. I μ ( f ) := X f d μ . I_{\mu}(f):=\int_{X}f\,d\mu.

Valentinus_Otho.html

  1. π 355 113 \pi\approx\tfrac{355}{113}

Valine_decarboxylase.html

  1. \rightleftharpoons

Valine—3-methyl-2-oxovalerate_transaminase.html

  1. \rightleftharpoons

Valine—pyruvate_transaminase.html

  1. \rightleftharpoons

Valine—tRNA_ligase.html

  1. \rightleftharpoons

Vanillin_synthase.html

  1. \rightleftharpoons

Variational_vector_field.html

  1. d Fl X t : T M T M d\mathrm{Fl}_{X}^{t}:TM\to TM

Varimax_rotation.html

  1. R VARIMAX = arg max R ( 1 p j = 1 k i = 1 p ( Λ R ) i j 4 - j = 1 k ( 1 p i = 1 p ( Λ R ) i j 2 ) 2 ) . R_{\mathrm{VARIMAX}}=\operatorname{arg}\max_{R}\left(\frac{1}{p}\sum_{j=1}^{k}% \sum_{i=1}^{p}(\Lambda R)^{4}_{ij}-\sum_{j=1}^{k}\left(\frac{1}{p}\sum_{i=1}^{% p}(\Lambda R)^{2}_{ij}\right)^{2}\right).

Vaughan's_identity.html

  1. f ( n ) Λ ( n ) \sum f(n)\Lambda(n)

Vedic_square.html

  1. ( ( / 9 ) × , { 1 , } ) ((\mathbb{Z}/9\mathbb{Z})^{\times},\{1,\circ\})
  2. / 9 \mathbb{Z}/9\mathbb{Z}
  3. \circ
  4. a , b a,b
  5. ( ( / 9 ) × , { 1 , } ) ((\mathbb{Z}/9\mathbb{Z})^{\times},\{1,\circ\})
  6. a b a\circ b
  7. ( a × b ) mod 9 (a\times b)\mod{9}
  8. 6 3 = 9 6\circ 3=9
  9. a { 1 , , 9 } a\in\{1,\cdots,9\}
  10. 9 a = 6. 9\circ a=6.
  11. { 1 , 2 , 4 , 5 , 7 , 8 } \{1,2,4,5,7,8\}
  12. / 9 \mathbb{Z}/9\mathbb{Z}
  13. \circ
  14. { 1 , 2 , 4 , 5 , 7 , 8 } \{1,2,4,5,7,8\}
  15. \circ

Venkatesan_Guruswami.html

  1. G F ( 2 m ) GF(2^{m})

Vesicle-fusing_ATPase.html

  1. \rightleftharpoons

Vetispiradiene_synthase.html

  1. \rightleftharpoons

Vicianin_beta-glucosidase.html

  1. \rightleftharpoons

View_factor.html

  1. F A B F_{A\rightarrow B}
  2. A A
  3. B B
  4. S i S_{i}
  5. j = 1 n F S i S j = 1 \sum_{j=1}^{n}{F_{S_{i}\rightarrow S_{j}}}=1
  6. F A A = 0 F_{A\rightarrow A}=0
  7. F A A > 0 F_{A\rightarrow A}>0
  8. F 1 ( 2 , 3 ) = F 1 2 + F 1 3 F_{1\rightarrow(2,3)}=F_{1\rightarrow 2}+F_{1\rightarrow 3}
  9. F B A F_{B\rightarrow A}
  10. F A B F_{A\rightarrow B}
  11. A A A_{A}
  12. A B A_{B}
  13. A A F A B = A B F B A A_{A}F_{A\rightarrow B}=A_{B}F_{B\rightarrow A}
  14. d A 1 \hbox{d}A_{1}
  15. d A 2 \hbox{d}A_{2}
  16. F 1 2 = cos θ 1 cos θ 2 π s 2 d A 2 F_{1\rightarrow 2}=\frac{\cos\theta_{1}\cos\theta_{2}}{\pi s^{2}}\hbox{d}A_{2}
  17. θ 1 \theta_{1}
  18. θ 2 \theta_{2}
  19. A 1 A_{1}
  20. A 2 A_{2}
  21. F 1 2 = 1 A 1 A 1 A 2 cos θ 1 cos θ 2 π s 2 d A 2 d A 1 F_{1\rightarrow 2}=\frac{1}{A_{1}}\int_{A_{1}}\int_{A_{2}}\frac{\cos\theta_{1}% \cos\theta_{2}}{\pi s^{2}}\,\hbox{d}A_{2}\,\hbox{d}A_{1}

Vinorine_synthase.html

  1. \rightleftharpoons

Viola–Jones_object_detection_framework.html

  1. M = 162 , 336 M=162,336
  2. h ( 𝐱 ) = sign ( j = 1 M α j h j ( 𝐱 ) ) h(\mathbf{x})=\,\text{sign}\left(\sum_{j=1}^{M}\alpha_{j}h_{j}(\mathbf{x})\right)
  3. f j f_{j}
  4. h j ( 𝐱 ) = { - s j if f j < θ j s j otherwise h_{j}(\mathbf{x})=\begin{cases}-s_{j}&\,\text{if }f_{j}<\theta_{j}\\ s_{j}&\,\text{otherwise}\end{cases}
  5. θ j \theta_{j}
  6. s j ± 1 s_{j}\in\pm 1
  7. α j \alpha_{j}
  8. N N
  9. ( 𝐱 i , y i ) {(\mathbf{x}^{i},y^{i})}
  10. i i
  11. y i = 1 y^{i}=1
  12. y i = - 1 y^{i}=-1
  13. w 1 i = 1 N w^{i}_{1}=\frac{1}{N}
  14. i i
  15. f j f_{j}
  16. j = 1 , , M j=1,...,M
  17. θ j , s j \theta_{j},s_{j}
  18. θ j , s j = arg min θ , s i = 1 N w j i ε j i \theta_{j},s_{j}=\arg\min_{\theta,s}\;\sum_{i=1}^{N}w^{i}_{j}\varepsilon^{i}_{j}
  19. ε j i = { 0 if y i = h j ( 𝐱 i , θ j , s j ) 1 otherwise \varepsilon^{i}_{j}=\begin{cases}0&\,\text{if }y^{i}=h_{j}(\mathbf{x}^{i},% \theta_{j},s_{j})\\ 1&\,\text{otherwise}\end{cases}
  20. α j \alpha_{j}
  21. h j h_{j}
  22. w j + 1 i w_{j+1}^{i}
  23. i i
  24. h ( 𝐱 ) = sign ( j = 1 M α j h j ( 𝐱 ) ) h(\mathbf{x})=\,\text{sign}\left(\sum_{j=1}^{M}\alpha_{j}h_{j}(\mathbf{x})\right)
  25. F = i = 1 K f i . F=\prod_{i=1}^{K}f_{i}.
  26. D = i = 1 K d i . D=\prod_{i=1}^{K}d_{i}.
  27. 10 - 6 10^{-6}

Viomycin_kinase.html

  1. \rightleftharpoons

Vitamin_B12-transporting_ATPase.html

  1. \rightleftharpoons

Vitexin_beta-glucosyltransferase.html

  1. \rightleftharpoons

Volta_potential.html

  1. Q Q
  2. C C
  3. Q = C Δ ψ Q=C\Delta\psi
  4. Δ ψ \Delta\psi

Volume_contraction.html

  1. n b = O s m b × T B W b n_{b}=Osm_{b}\times TBW_{b}
  2. n a = n b - n l o s t N a + - n l o s t K + n_{a}=n_{b}-n_{lostNa^{+}}-n_{lostK^{+}}
  3. O s m a = n a T B W b - V l o s t Osm_{a}=\frac{n_{a}}{TBW_{b}-V_{lost}}
  4. V I C F a = n I C F a O s m a = V I C F b × O s m b - n l o s t K + O s m a V_{ICFa}=\frac{n_{ICFa}}{Osm_{a}}=\frac{V_{ICFb}\times Osm_{b}-n_{lostK^{+}}}{% Osm_{a}}
  5. V E C F a = n E C F a O s m a = V E C F b × O s m b - n l o s t N a + O s m a V_{ECFa}=\frac{n_{ECFa}}{Osm_{a}}=\frac{V_{ECFb}\times Osm_{b}-n_{lostNa^{+}}}% {Osm_{a}}
  6. V l o s t I C F = V I C F b - V I C F a V_{lostICF}=V_{ICFb}-V_{ICFa}
  7. V l o s t E C F = V E C F b - V E C F a V_{lostECF}=V_{ECFb}-V_{ECFa}

Vomilenine_glucosyltransferase.html

  1. \rightleftharpoons

Wait-for_graph.html

  1. P i P_{i}
  2. P j P_{j}
  3. P j P_{j}
  4. P i P_{i}
  5. P i P_{i}
  6. P j P_{j}

Wallman_compactification.html

  1. \mathcal{F}
  2. \mathcal{F}

Wax-ester_hydrolase.html

  1. \rightleftharpoons

Weakly_measurable_function.html

  1. g f : X 𝐊 : x g ( f ( x ) ) g\circ f\colon X\to\mathbf{K}\colon x\mapsto g(f(x))

Weibel_instability.html

  1. γ \gamma
  2. n b 0 n_{b0}
  3. v 0 𝐳 v_{0}\mathbf{z}
  4. n p 0 = n b 0 n_{p0}=n_{b0}
  5. - v 0 𝐳 -v_{0}\mathbf{z}
  6. 𝐁 𝟎 = 𝐄 𝟎 = 0 \mathbf{B_{0}}=\mathbf{E_{0}}=0
  7. 𝐱 ^ \mathbf{\hat{x}}
  8. 𝐤 = k 𝐱 ^ \mathbf{k}=k\mathbf{\hat{x}}
  9. 𝐄 𝟏 = A e i ( k x - ω t ) 𝐳 \mathbf{E_{1}}=Ae^{i(kx-\omega t)}\mathbf{z}
  10. t - i ω \frac{\partial}{\partial t}\rightarrow-i\omega
  11. i k 𝐱 ^ \nabla\rightarrow ik\mathbf{\hat{x}}
  12. × 𝐄 𝟏 = - 𝐁 𝟏 t i 𝐤 × 𝐄 𝟏 = i ω 𝐁 𝟏 𝐁 𝟏 = 𝐲 ^ k ω E 1 \nabla\times\mathbf{E_{1}}=-\frac{\partial\mathbf{B_{1}}}{\partial t}% \Rightarrow i\mathbf{k}\times\mathbf{E_{1}}=i\omega\mathbf{B_{1}}\Rightarrow% \mathbf{B_{1}}=\mathbf{\hat{y}}\frac{k}{\omega}E_{1}
  13. 𝐯 𝐛 = 𝐯 𝐛𝟎 + 𝐯 𝐛𝟏 \mathbf{v_{b}}=\mathbf{v_{b0}}+\mathbf{v_{b1}}
  14. n b = n b 0 + n b 1 n_{b}=n_{b0}+n_{b1}
  15. 𝐉 𝐛𝟏 = - e n b 𝐯 𝐛 = - e n b 0 𝐯 𝐛𝟏 + - e n b 1 𝐯 𝐛𝟎 \mathbf{J_{b1}}=-en_{b}\mathbf{v_{b}}=-en_{b0}\mathbf{v_{b1}}+-en_{b1}\mathbf{% v_{b0}}
  16. m ( 𝐯 𝐛 t + ( 𝐯 𝐛 ) 𝐯 𝐛 ) = - e 𝐄 - e 𝐯 𝐛 × 𝐁 m(\frac{\partial\mathbf{v_{b}}}{\partial t}+(\mathbf{v_{b}}\cdot\nabla)\mathbf% {v_{b}})=-e\mathbf{E}-e\mathbf{v_{b}}\times\mathbf{B}
  17. 𝐯 𝐛𝟎 t = 𝐯 𝐛𝟎 = 0 \frac{\partial\mathbf{v_{b0}}}{\partial t}=\nabla\cdot\mathbf{v_{b0}}=0
  18. - i ω m 𝐯 𝐛𝟏 = - e 𝐄 𝟏 - e 𝐯 𝐛𝟎 × 𝐁 𝟏 -i\omega m\mathbf{v_{b1}}=-e\mathbf{E_{1}}-e\mathbf{v_{b0}}\times\mathbf{B_{1}}
  19. v b 1 z = e E 1 m i ω v_{b1z}=\frac{eE_{1}}{mi\omega}
  20. v b 1 x = e E 1 m i ω k v b 0 ω v_{b1x}=\frac{eE_{1}}{mi\omega}\frac{kv_{b0}}{\omega}
  21. n b 1 n_{b1}
  22. n b t + ( n b 𝐯 𝐛 ) = 0 \frac{\partial n_{b}}{\partial t}+\nabla\cdot(n_{b}\mathbf{v_{b}})=0
  23. n b 0 t = n b 0 = 0 \frac{\partial n_{b0}}{\partial t}=\nabla n_{b0}=0
  24. n b 1 = n b 0 k ω v b 1 x n_{b1}=n_{b0}\frac{k}{\omega}v_{b1x}
  25. J b 1 x = - n b 0 e 2 E 1 k v b 0 i m ω 2 J_{b1x}=-n_{b0}e^{2}E_{1}\frac{kv_{b0}}{im\omega^{2}}
  26. J b 1 z = - n b 0 e 2 E 1 1 i m ω ( 1 + k 2 v b 0 2 ω 2 ) J_{b1z}=-n_{b0}e^{2}E_{1}\frac{1}{im\omega}(1+\frac{k^{2}v_{b0}^{2}}{\omega^{2% }})
  27. v 0 v_{0}
  28. v 0 2 v_{0}^{2}
  29. 𝐉 𝟏 = - 2 n b 0 e 2 E 1 1 i m ω ( 1 + k 2 v b 0 2 ω 2 ) 𝐳 ^ \mathbf{J_{1}}=-2n_{b0}e^{2}E_{1}\frac{1}{im\omega}(1+\frac{k^{2}v_{b0}^{2}}{% \omega^{2}})\mathbf{\hat{z}}
  30. × 𝐄 𝟏 = i ω 𝐁 𝟏 \nabla\times\mathbf{E_{1}}=i\omega\mathbf{B_{1}}
  31. × 𝐁 𝟏 = μ 0 𝐉 𝟏 - i ω ϵ 0 μ 0 𝐄 𝟏 \nabla\times\mathbf{B_{1}}=\mu_{0}\mathbf{J_{1}}-i\omega\epsilon_{0}\mu_{0}% \mathbf{E_{1}}
  32. × 𝐄 𝟏 = - 2 𝐄 𝟏 + ( 𝐄 𝟏 ) = k 2 𝐄 𝟏 + i 𝐤 ( i 𝐤 𝐄 𝟏 ) = k 2 𝐄 𝟏 = i ω × 𝐁 𝟏 = i ω c 2 ϵ 0 𝐉 𝟏 + ω 2 c 2 𝐄 𝟏 \Rightarrow\nabla\times\nabla\mathbf{E_{1}}=-\nabla^{2}\mathbf{E_{1}}+\nabla(% \nabla\cdot\mathbf{E_{1}})=k^{2}\mathbf{E_{1}}+i\mathbf{k}(i\mathbf{k}\cdot% \mathbf{E_{1}})=k^{2}\mathbf{E_{1}}=i\omega\nabla\times\mathbf{B_{1}}=\frac{i% \omega}{c^{2}\epsilon_{0}}\mathbf{J_{1}}+\frac{\omega^{2}}{c^{2}}\mathbf{E_{1}}
  33. c = 1 ϵ 0 μ 0 c=\frac{1}{\epsilon_{0}\mu_{0}}
  34. ω p = 2 n b 0 e 2 ϵ 0 m \omega_{p}=\frac{2n_{b0}e^{2}}{\epsilon_{0}m}
  35. k 2 - ω 2 c 2 = - ω p 2 c 2 ( 1 + k 2 v 0 2 ω 2 ) ω 4 - ω 2 ( ω p 2 + k 2 c 2 ) - ω p 2 k 2 v 0 2 = 0 k^{2}-\frac{\omega^{2}}{c^{2}}=-\frac{\omega_{p}^{2}}{c^{2}}(1+\frac{k^{2}v_{0% }^{2}}{\omega^{2}})\Rightarrow\omega^{4}-\omega^{2}(\omega_{p}^{2}+k^{2}c^{2})% -\omega_{p}^{2}k^{2}v_{0}^{2}=0
  36. ω 2 = 1 2 ( ω p 2 + k 2 c 2 ± ( ω p 2 + k 2 c 2 ) 2 + 4 ω p 2 k 2 v 0 2 ) \omega^{2}=\frac{1}{2}(\omega_{p}^{2}+k^{2}c^{2}\pm\sqrt{(\omega_{p}^{2}+k^{2}% c^{2})^{2}+4\omega_{p}^{2}k^{2}v_{0}^{2}})
  37. I m ( ω ) 0 Im(\omega)\neq 0
  38. k k
  39. v 0 c v_{0}<<c
  40. ( ω p 2 + k 2 c 2 ) 2 + 4 ω p 2 k 2 v 0 2 = ( ω p 2 + k 2 c 2 ) ( 1 + 4 ω p 2 k 2 v 0 2 ( ω p 2 + k 2 c 2 ) 2 ) 1 / 2 ( ω p 2 + k 2 c 2 ) ( 1 + 2 ω p 2 k 2 v 0 2 ( ω p 2 + k 2 c 2 ) 2 ) \sqrt{(\omega_{p}^{2}+k^{2}c^{2})^{2}+4\omega_{p}^{2}k^{2}v_{0}^{2}}=(\omega_{% p}^{2}+k^{2}c^{2})(1+\frac{4\omega_{p}^{2}k^{2}v_{0}^{2}}{(\omega_{p}^{2}+k^{2% }c^{2})^{2}})^{1/2}\approx(\omega_{p}^{2}+k^{2}c^{2})(1+\frac{2\omega_{p}^{2}k% ^{2}v_{0}^{2}}{(\omega_{p}^{2}+k^{2}c^{2})^{2}})
  41. ω 2 = - ω p 2 k 2 v 0 2 ω p 2 + k 2 c 2 < 0 \omega^{2}=\frac{-\omega_{p}^{2}k^{2}v_{0}^{2}}{\omega_{p}^{2}+k^{2}c^{2}}<0
  42. ω \omega
  43. ω = i γ \omega=i\gamma
  44. γ = ω p k v 0 ( ω p 2 + k 2 c 2 ) 1 / 2 = ω p v 0 c 1 ( 1 + ω p 2 k 2 c 2 ) 1 / 2 \gamma=\frac{\omega_{p}kv_{0}}{(\omega_{p}^{2}+k^{2}c^{2})^{1/2}}=\omega_{p}% \frac{v_{0}}{c}\frac{1}{(1+\frac{\omega_{p}^{2}}{k^{2}c^{2}})^{1/2}}
  45. I m ( ω ) > 0 Im(\omega)>0
  46. 𝐄 𝟏 = A 𝐳 ^ e γ t e i k x \mathbf{E_{1}}=A\mathbf{\hat{z}}e^{\gamma t}e^{ikx}
  47. 𝐁 𝟏 = 𝐲 ^ k ω E 1 = 𝐲 ^ k i γ A e γ t e i k x \mathbf{B_{1}}=\mathbf{\hat{y}}\frac{k}{\omega}E_{1}=\mathbf{\hat{y}}\frac{k}{% i\gamma}Ae^{\gamma t}e^{ikx}
  48. 90 o 90^{o}
  49. | B 1 | | E 1 | = k γ c v 0 1 \frac{|B_{1}|}{|E_{1}|}=\frac{k}{\gamma}\propto\frac{c}{v_{0}}>>1
  50. γ \gamma
  51. γ ω p v 0 c ω c B m e ω p v 0 c \gamma\sim\omega_{p}\frac{v_{0}}{c}\sim\omega_{c}\Rightarrow B\sim\frac{m}{e}% \omega_{p}\frac{v_{0}}{c}

Weighted-average_life.html

  1. WAL = i = 1 n P i P t i , \,\text{WAL}=\sum_{i=1}^{n}\frac{P_{i}}{P}t_{i},
  2. P P
  3. P i P_{i}
  4. i i
  5. P i P \frac{P_{i}}{P}
  6. i i
  7. t i t_{i}
  8. i i
  9. t i t_{i}
  10. 1 12 ( i + α - 1 ) \frac{1}{12}(i+\alpha-1)
  11. α \alpha
  12. 15 + 1 / 24 15.04 15+1/24\approx 15.04
  13. WAL × r × P , \,\text{WAL}\times r\times P,
  14. WAL × r \,\text{WAL}\times r
  15. t i = i / 12 t_{i}=i/12
  16. WAL \displaystyle\,\text{WAL}
  17. i = 1 n Q i r 12 = r 12 i = 1 n Q i , \sum_{i=1}^{n}Q_{i}\frac{r}{12}=\frac{r}{12}\sum_{i=1}^{n}Q_{i},
  18. Q i Q_{i}
  19. i = 1 n i P i = i = 1 n Q i \sum_{i=1}^{n}iP_{i}=\sum_{i=1}^{n}Q_{i}
  20. i P i iP_{i}
  21. Q i Q_{i}
  22. Q n = P n , Q n - 1 = P n + P n - 1 Q_{n}=P_{n},Q_{n-1}=P_{n}+P_{n-1}
  23. 20 + 2 30 + 3 50 = 230 20+2\cdot 30+3\cdot 50=230
  24. 100 + 80 + 50 = 230 100+80+50=230
  25. Q i Q_{i}
  26. i P i iP_{i}
  27. A n An
  28. A n - P An-P
  29. WAL = A n - P P r \,\text{WAL}=\frac{An-P}{Pr}
  30. WAL × r \,\text{WAL}\times r
  31. WAL × r = A n - P P \,\text{WAL}\times r=\frac{An-P}{P}

Weighted_space.html

  1. U U\subset\mathbb{R}
  2. \mathbb{R}
  3. U \|\cdot\|_{U}
  4. f U = sup x U | f ( x ) | \|f\|_{U}=\sup_{x\in U}{|f(x)|}
  5. f = sup x U | f ( x ) 1 1 + x 2 | \|f\|=\sup_{x\in U}{\left|f(x)\tfrac{1}{1+x^{2}}\right|}
  6. f = sup x U | f ( x ) x 4 | \|f\|=\sup_{x\in U}{\left|f(x)x^{4}\right|}
  7. 1 1 + x m \tfrac{1}{1+x^{m}}

Wilberforce_pendulum.html

  1. f a l t = f R - f T = 0.1 Hz f_{alt}=f_{R}-f_{T}=0.1\;\mathrm{Hz}
  2. T a l t = 1 / f a l t = 10 s T_{alt}=1/f_{alt}=10\;\mathrm{s}

Wind.html

  1. E = 1 2 ρ A v 3 t E=\frac{1}{2}\rho Av^{3}t
  2. P = d E / d t = 1 2 ρ A v 3 P=dE/dt=\frac{1}{2}\rho Av^{3}

Wind_stress.html

  1. τ \tau
  2. U h U_{h}
  3. τ wind = ρ air C D U h 2 , \tau\text{wind}=\rho\text{air}C_{D}U_{h}^{2},
  4. ρ air \rho\text{air}
  5. C D C_{D}
  6. C D C_{D}
  7. U h U_{h}
  8. U h U_{h}
  9. C D C_{D}

Window_operator.html

  1. \triangle
  2. M , w ϕ u , M , u ϕ R w u M,w\models\triangle\phi\iff\forall u,M,u\models\phi\Rightarrow Rwu
  3. M = ( W , R , f ) M=(W,R,f)
  4. w , u W w,u\in W
  5. \square
  6. \Diamond

X-Machine_Testing.html

  1. \otimes
  2. \otimes
  3. \otimes
  4. \cup
  5. \otimes
  6. \otimes
  7. \cup
  8. \otimes
  9. \otimes
  10. \otimes
  11. \cup
  12. \otimes
  13. \otimes
  14. \cup
  15. \cup
  16. \otimes

Xanthine_phosphoribosyltransferase.html

  1. \rightleftharpoons

Xenobiotic-transporting_ATPase.html

  1. \rightleftharpoons

Xylitol_kinase.html

  1. \rightleftharpoons

Xyloglucan-specific_endo-beta-1,4-glucanase.html

  1. \rightleftharpoons

Xyloglucan-specific_exo-beta-1,4-glucanase.html

  1. \rightleftharpoons

Xylonate_dehydratase.html

  1. \rightleftharpoons

Xylono-1,4-lactonase.html

  1. \rightleftharpoons

Xylosylprotein_4-beta-galactosyltransferase.html

  1. \rightleftharpoons

Xylulokinase.html

  1. \rightleftharpoons

Yangian.html

  1. t i j ( p ) t_{ij}^{(p)}
  2. [ t i j ( p + 1 ) , t k l ( q ) ] - [ t i j ( p ) , t k l ( q + 1 ) ] = - ( t k j ( p ) t i l ( q ) - t k j ( q ) t i l ( p ) ) . [t_{ij}^{(p+1)},t_{kl}^{(q)}]-[t_{ij}^{(p)},t_{kl}^{(q+1)}]=-(t_{kj}^{(p)}t_{% il}^{(q)}-t_{kj}^{(q)}t_{il}^{(p)}).
  3. t i j ( - 1 ) = δ i j t_{ij}^{(-1)}=\delta_{ij}
  4. T ( z ) = p - 1 t i j ( p ) z - p + 1 T(z)=\sum_{p\geq-1}t_{ij}^{(p)}z^{-p+1}
  5. \otimes
  6. R 12 ( z - w ) T 1 ( z ) T 2 ( w ) = T 2 ( w ) T 1 ( z ) R 12 ( z - w ) . \displaystyle{R_{12}(z-w)T_{1}(z)T_{2}(w)=T_{2}(w)T_{1}(z)R_{12}(z-w).}
  7. ( Δ id ) T ( z ) = T 12 ( z ) T 13 ( z ) , ( ε id ) T ( z ) = I , ( s id ) T ( z ) = T ( z ) - 1 . (\Delta\otimes\mathrm{id})T(z)=T_{12}(z)T_{13}(z),\,\,(\varepsilon\otimes% \mathrm{id})T(z)=I,\,\,(s\otimes\mathrm{id})T(z)=T(z)^{-1}.
  8. ( z - w ) (z-w)
  9. T ( z ) T(z)
  10. S ( z ) = T ( z ) σ T ( - z ) , \displaystyle{S(z)=T(z)\sigma T(-z),}
  11. σ ( E i j ) = ( - 1 ) i + j E 2 N - j + 1 , 2 N - i + 1 . \displaystyle{\sigma(E_{ij})=(-1)^{i+j}E_{2N-j+1,2N-i+1}.}

Yates_analysis.html

  1. - - - ---
  2. + - - +--
  3. - + - -+-
  4. + + - ++-
  5. - - + --+
  6. + - + +-+
  7. - + + -++
  8. + + + +++
  9. t = e s e t=\frac{e}{s_{e}}
  10. response = constant + 0.5 X i \textrm{response}=\textrm{constant}+0.5X_{i}
  11. response = constant + 0.5 ( all effect estimates down to and including the effect of interest ) \textrm{response}=\textrm{constant}+0.5\mathrm{(all\ effect\ estimates\ down\ % to\ and\ including\ the\ effect\ of\ interest)}
  12. response = constant \textrm{response}=\textrm{constant}
  13. response = constant + 0.5 ( all factor and interaction estimates ) \textrm{response}=\textrm{constant}+0.5\mathrm{(all\ factor\ and\ interaction% \ estimates)}

Yield_gap.html

  1. Yield Gap = Yield Ratio of Equity Yield Ratio of Bond \mbox{Yield Gap}~{}=\frac{\mbox{Yield Ratio of Equity}~{}}{\mbox{Yield Ratio % of Bond}~{}}

Young's_lattice.html

  1. μ ( q , p ) = { ( - 1 ) | p | - | q | if the skew diagram p / q is a disconnected union of squares (no common edges); 0 otherwise . \mu(q,p)=\begin{cases}(-1)^{|p|-|q|}&\,\text{if the skew diagram }p/q\,\text{ % is a disconnected union of squares}\\ &\,\text{(no common edges);}\\ 0&\,\text{otherwise}.\end{cases}
  2. n + + 3 + 2 + 1 n+\cdots+3+2+1
  3. 1 + + 1 n terms \displaystyle\underbrace{1+\cdots\cdots\cdots+1}_{n\,\text{ terms}}

Yttria-stabilized_zirconia.html

  1. \leftrightarrow
  2. \leftrightarrow
  3. \leftrightarrow

Yurii_Shirokov.html

  1. ~{}\hbar~{}
  2. 0 ~{}\hbar\rightarrow 0~{}

Z-farnesyl_diphosphate_synthase.html

  1. \rightleftharpoons

Z-group.html

  1. G ( m , n , r ) = a , b | a n = b m = 1 , a b = a r G(m,n,r)=\langle a,b|a^{n}=b^{m}=1,a^{b}=a^{r}\rangle

Zeatin_9-aminocarboxyethyltransferase.html

  1. \rightleftharpoons

Zeatin_O-beta-D-xylosyltransferase.html

  1. \rightleftharpoons

Zn2+-exporting_ATPase.html

  1. \rightleftharpoons

Vivification.html

  1. C 1 C 2 C n C_{1}\sqcup C_{2}\ldots\sqcup C_{n}
  2. C 1 , C 2 , C n C_{1},C_{2},\ldots C_{n}
  3. PIANIST(Jill) ORGANIST(Jill) \textrm{PIANIST(Jill)}\vee\textrm{ORGANIST(Jill)}
  4. $\textrm{KEYBOARD-PLAYER(Jill)}$
  5. $\textrm{ORGANIST}$